Pressure vessels
© Carl T. F. Ross, 2011
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© Carl T. F. Ross, 2011
Pressure vessels External pressure technology Second edition
Carl T. F. Ross
Oxford
Cambridge
Philadelphia
© Carl T. F. Ross, 2011
New Delhi
Published by Woodhead Publishing Limited, 80 High Street, Sawston, Cambridge CB22 3HJ, UK www.woodheadpublishing.com Woodhead Publishing, 1518 Walnut Street, Suite 1100, Philadelphia, PA 19102-3406, USA Woodhead Publishing India Private Limited, G-2, Vardaan House, 7/28 Ansari Road, Daryaganj, New Delhi – 110002, India www.woodheadpublishingindia.com First edition 2001, Horwood Publishing Limited Second edition 2011, Woodhead Publishing Limited © Carl T. F. Ross, 2011 The author has asserted his moral rights. This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials. Neither the author nor the publisher, nor anyone else associated with this publication, shall be liable for any loss, damage or liability directly or indirectly caused or alleged to be caused by this book. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming and recording, or by any information storage or retrieval system, without permission in writing from Woodhead Publishing Limited. The consent of Woodhead Publishing Limited does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from Woodhead Publishing Limited for such copying. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library. ISBN 978-0-85709-248-9 (print) ISBN 978-0-85709-249-6 (online) The publisher’s policy is to use permanent paper from mills that operate a sustainable forestry policy, and which has been manufactured from pulp which is processed using acid-free and elemental chlorine-free practices. Furthermore, the publisher ensures that the text paper and cover board used have met acceptable environmental accreditation standards. Typeset by Toppan Best-set Premedia Limited Printed by TJI Digital, Padstow, Cornwall, UK
© Carl T. F. Ross, 2011
Contents
Author contact details Preface Acknowledgements Notation 1
ix xiii xv xvii
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8
An overview of pressure vessels under external pressure Pressure vessel types The spherical pressure vessel Cylinder/cone/dome pressure hulls Other vessels that withstand external pressure Weakening effect on ring-stiffeners owing to tilt Bulkheads Materials of construction Pressure, depth and compressibility
1 1 1 4 7 8 8 9 13
2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10
Axisymmetric deformation of pressure vessels Axisymmetric yield failure Unstiffened circular cylinders and spheres Ring-stiffened circular cylinders Axisymmetric deformation of thin-walled cones and domes Thick-walled cones and domes Ring-stiffeners Plastic collapse Experimental procedure Theoretical plastic analysis Conclusions
15 15 15 16 30 52 77 83 85 95 96
3 3.1 3.2 3.3
Shell instability of pressure vessels Shell instability of thin-walled circular cylinders Instability of thin-walled conical shells Buckling of orthotropic cylinders and cones
100 100 111 117 v
© Carl T. F. Ross, 2011
vi
Contents
3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11
Buckling of thin-walled domes Boundary conditions The legs of off-shore drilling rigs Some buckling formulae for domes and cones Inelastic instability Higher order elements for conical shells Higher order elements for hemi-ellipsoidal domes Varying thickness cylinders
124 138 141 142 144 151 159 163
4 4.1 4.2
165 165
4.3
General instability of pressure vessels General instability of ring-stiffened circular cylinders Inelastic general instability of ring-stiffened circular cylinders General instability of ring-stiffened conical shells
5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8
Vibration of pressure vessel shells Free vibration of unstiffened circular cylinders and cones Free vibration of ring-stiffened cylinders and cones Free vibrations of domes Higher order elements for thin-walled cones Higher order elements for thin-walled domes Effects of pressure on vibration Effects of added virtual mass Effects of damping
192 192 201 205 214 216 217 220 220
6 6.1 6.2 6.3 6.4
221 221 229 236
6.5
Vibration of pressure vessel shells in water Free vibration of ring-stiffened cones in water Free vibration of domes in water Vibration of domes under external water pressure Vibration of unstiffened and ring-stiffened circular cylinders and cones under external hydrostatic pressure Effect of tank size
7 7.1 7.2 7.3 7.4 7.5
Novel pressure hull designs Design of dome ends Design of cylindrical body Ring-stiffened or corrugated prolate domes A submarine for the oceans of Europa Conclusions
280 280 284 290 291 292
8 8.1
Vibration and collapse of novel pressure hulls Buckling of corrugated circular cylinders under external hydrostatic pressure
293
© Carl T. F. Ross, 2011
179 184
243 275
293
Contents 8.2 8.3 8.4 8.5 8.6 9 9.1 9.2 9.3 10 10.1 10.2 10.3 10.4 11 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 11.10 11.11 11.12 11.13 11.14 12 12.1
Buckling of a corrugated carbon-fibre-reinforced plastic (CFRP) cylinder Vibration of CFRP corrugated circular cylinder under external hydrostatic pressure Vibration and instability of tube-stiffened axisymmetric shells under external hydrostatic pressure Collapse of dome cup ends under external hydrostatic pressure A redesign of the corrugated food can
vii
303 316 324 334 346
Design of submarine pressure hulls to withstand buckling under external hydrostatic pressure Introduction The designs Conclusions
355 355 356 360
Nonlinear analyses of model submarine pressure hulls using ANSYS Introduction Experimental analysis Theoretical analysis Conclusions
361 361 364 368 372
Star wars underwater: deep-diving underwater pressure vessels for missile defence systems Introduction The design Manpower and living conditions Power requirements Environmental control and life support systems External requirements Size of elliptical structure Central spherical shell Connecting walkways Material property requirements Choice of material Pressure hull designs Required wall thickness Conclusions
375 375 377 379 380 381 384 385 385 385 386 386 390 390 391
Vibration of a thin-walled shell under external water pressure using ANSYS Introduction
393 393
© Carl T. F. Ross, 2011
viii
Contents
12.2 12.3 12.4 12.5 12.6
Experimental method Theoretical basis of the finite element method Vibration analysis of a prolate dome in air Vibration analysis of the prolate dome in water Vibration analysis of the prolate dome under external pressure Conclusions
12.7
394 395 399 406 415 418
References
419
Appendix I Computer program for axisymmetric stresses in circular cylinders stiffened by equal-strength ring frames
428
Appendix II Computer program for axisymmetric stresses in circular cylinders stiffened by unequal-strength ring frames
432
Appendix III Computer programs for shell instability
444
Appendix IV Computer programs for general instability
448
Appendix V Conversion tables of imperial units to SI
460
Index
463
© Carl T. F. Ross, 2011
Author contact details
Prof. Carl T. F. Ross 6 Hurstville Drive Waterlooville Hants PO7 7NB UK E-mail:
[email protected]
ix © Carl T. F. Ross, 2011
Dedication
To my grandson Nathan
xi © Carl T. F. Ross, 2011
AOVo DYDiODEOe A Quick Guide to API 510 &eUtiÀed PUeVVuUe 9eVVeO IQVSectoU 6\OODEuV ([DPSOe QueVtioQV DQd :oUked AQVZeUV Clifford Matthews, Matthews Engineering Training Limited, UK 7Ke API IQdiYiduDO &eUtiÀcDtioQ PUoJUDPV I&PV DUe ZeOO eVtDEOiVKed ZoUOdZide iQ tKe oiO JDV DQd SetUoOeuP iQduVtUieV 7KiV Quick Guide iV uQiTue iQ SUoYidiQJ ViPSOe DcceVViEOe DQd ZeOOVtUuctuUed JuidDQce IoU DQ\oQe Vtud\iQJ tKe API 510 &eUtiÀed PUeVVuUe 9eVVeO IQVSectoU V\OODEuV E\ VuPPDUiViQJ DQd KeOSiQJ tKeP tKUouJK tKe V\OODEuV DQd SUoYidiQJ PuOtiSOe e[DPSOe TueVtioQV DQd ZoUked DQVZeUV 7ecKQicDO VtDQdDUdV DUe UeIeUeQced IUoP tKe API ¶Eod\ oI kQoZOedJe· IoU tKe e[DPiQDtioQ ie API 510 PUeVVuUe YeVVeO iQVSectioQ DOteUDtioQ UeUDtiQJ API 5 PUeVVuUe YeVVeO iQVSectioQ API 5P 51 'DPDJe PecKDQiVPV API 5P 5 :eOdiQJ A60(9III 9eVVeO deViJQ A60(9 1'( DQd A60( I; :eOdiQJ TuDOiÀcDtioQV ISBN: 978-1-84569-755-6 (print) ISBN: 978-0-85709-102-4 (online) 336 pages 170 x 112mm paperback 2010 For full contents visit www.woodheadpublishing.com/quickguide
© Carl T. F. Ross, 2011
Preface
This up-to-date second edition of the technology of external pressure vessels, covers problems that arise in submarine pressure hulls, torpedoes, aircraft fuselages, space shuttles, underwater storage vessels, intercontinental missiles, oil drilling rigs, grain storage silos, cooling towers, valves, tunnels, under-soil tubes, immersed tubes, medical equipment and tin can containers of food and everyday products. Stress analysts, designers, consultants and manufacturers need to take into account the effects of external pressures to prevent buckling of external walls of descending submarines, ascending or descending rockets, aircraft and space shuttles by devising countermeasures in their choice of structural design and materials. Computer programs (codes in QuickBASIC) are published in the Appendices. The present author, from long experience in engineering research and design, covers the problem, its remediation, and practical applications. He records experimental and theoretical work including plastic collapse, and provides many design charts. Chapter 1 introduces the subject of pressure vessels under external pressure and discusses the main modes of failure of these vessels. Chapter 2 gives theoretical and experimental solutions for the axisymmetric mode of failure, together with some of its history. Chapter 3 gives theoretical and experimental solutions for the shell instability mode of failure, also known as lobar buckling, together with some of its history. Chapter 4 provides a similar treatise on general instability, where the entire ring–shell combination buckles bodily in its flank. Chapter 5 is on the vibrations of shells, and Chapter 6 extends this work to vibrations in water; both experimental and theoretical analyses are considered. In Chapter 6, the theoretical and experimental ‘nonlinear’ effects of the vibration of shells under external water pressure are also considered, together with the possible effects of dynamic buckling. Chapter 7 introduces many novel pressure hull designs, some of which are structurally more efficient than conventional pressure hulls. Chapter 8 xiii © Carl T. F. Ross, 2011
xiv
Preface
gives some experimental and theoretical work on novel pressure ‘hull’ designs and food cans, including the use of composite materials. Chapter 9 is on the use of design charts to design pressure hulls, including the design charts of the present author and those of PD 5500 (previously BS 5500) and shows that the charts of the present author are more efficient and easier to use than PD 5500. Chapter 10 is on the detrimental effects of geometrical and material nonlinearity on these vessels; using the commercial computer package ANSYS. Chapter 11 is on a conceptual design of a ‘star wars underwater’ system and is of much strategic importance. Chapter 12 analyses the vibration of a dome shell in air and under external water pressure, using the commercial computer package ANSYS. There are also several Appendices, some of which provide ‘listings’ of computer programs, written in QuickBASIC. Carl T. F. Ross
© Carl T. F. Ross, 2011
Acknowledgements
The author would like to thank the following of his co-researchers for their valuable contributions: Drs Mike Mackney, Kevin Port, Andrew Little and Frank Abraham; Terry Johns, Grant Waterman, Bob Maguire, Emile, David Richards, Geoff Lafolley-Lane, David Sawkins, James Etheridge, Daniel Short, Astrit Spahiu, Terry Whittaker, Mohammad Al-Enezi, Chris Bull, Paul Smith, Angela Etheridge, Anouska Terry, and Philipp Köster. His thanks are extended to his sister, Helen Facey, for the considerable care and devotion she gave to the typing of this book. Without her help, the production of this book would not have been possible.
xv © Carl T. F. Ross, 2011
Notation
Unless otherwise stated the following notation is used: a Af bf [B] c [Cv] D [D] [DC] Df e E E1 E2 Ex Ey Ez f G [G] G12 h [H] I IZ K
radius of circular cylindrical shell cross-sectional area of a stiffener width of frame web in contact with shell plating a matrix relating strain to displacement, i.e. {ε} = [B]{Ui} speed of sound in water = % (K/ρF), or c = cos α a matrix of viscous damping terms Et 3/[12(1−ν 2)] a matrix of orthotropic material constants, which allow for the element’s material properties a matrix of directional cosines diameter of equivalent ring-stiffener combination distance of frame centroid from the mid-surface of the shell Young’s modulus of elasticity Young’s modulus in the direction of the fibres Young’s modulus perpendicular to the fibres Young’s modulus in the meridional direction Young’s modulus in the circumferential direction out-of-plane Young’s modulus frequency rigidity modulus Σ[g]e = a type of elemental ‘mass’ matrix for the fluid in-plane shear modulus thickness of shell plating Σ[h]e = a type of elemental ‘stiffness’ matrix for the fluid second moment of area of a ring’s cross-section about its centroid and an axis parallel to the axis of the vessel second moment of area of a ring’s cross-section about its centroid and an axis perpendicular to the axis of the vessel bulk modulus of water xvii © Carl T. F. Ross, 2011
xviii [K] [KG] [K*] L, l Lb [M] n N [N] [N ] P, p Pcr {Pi} {Po} R, r {R} Rf s se [S]
Notation
t t′ u ux {U 0} {Ui} {uo} v [ v] ve w wx,θ, etc. x z
stiffness matrix geometrical stiffness matrix dependent on external pressure overall stiffness matrix (= [K] + [KG]) length length between bulkheads mass matrix number of circumferential waves number of ring-stiffeners a matrix of shape functions normal component of the shape function (structural) pressure theoretical buckling pressure vector of nodal acoustic pressures vector of peak values of nodal acoustic pressures radius external vector of forcing functions radius of frame centroid length along a fluid–structure interface surface, or s = sin α elemental area of fluid/structure interface Σ[s]e = an elemental matrix for the fluid–structure interaction shell thickness or time shell thickness of equivalent ring–cylinder combination displacement in the x direction ∂u/∂x vector of nodal displacements (global) vector of nodal displacements (local) vector of peak values of nodal displacements displacement in the y direction v cos α volume of element displacement in the z direction ∂2w/∂x∂θ, etc. direction along the axis of the vessel radial direction of the vessel
α γ {ε} {εL} ζ θ λ
half cone angle (of conical element) shear strain vector of strains owing to small deflections vector of strains owing to large deflections x/P rotational displacement Windenburg’s thinness ratio
© Carl T. F. Ross, 2011
Notation λ′
xix
ν ν12 ρ ρF σ1 σ2 σx σϕ τ τϕx ϕ, θ χ ω
thinness ratio for general instability, corrugated and composite vessels Poisson’s ratio in-plane Poisson’s ratio material density fluid density principal longitudinal stresses principal transverse stresses in-plane meridional stress (of conical element) in-plane circumferential stress (of conical element) in-plane shear stress in-plane shear stress (of conical element) circumferential direction flexural or twisting (strain) radian frequency
CMC CON VMC
constant meridional curvature element conical element varying meridional curvature element
Superscripts e indicates an elemental property T indicates the transpose of the matrix
© Carl T. F. Ross, 2011
1 An overview of pressure vessels under external pressure
Abstract: This chapter gives an overview of pressure vessels under external pressure, highlighting their main modes of failure. These modes of failure, called shell instability (or lobar buckling), general instability and axisymmetric deformation, are described with the aid of photographs. Both circular cylinders and domes are considered, together with ring-stiffened circular cones. Material properties of suitable materials of construction, including a number of metals and composites, are presented. Key words: submarine pressure hulls, shell instability, deformation, construction materials.
1.1
Pressure vessel types
Structures designed to withstand external water pressure usually take the form of thin-walled curved shells constructed from metals because it is usually more efficient for such structures to withstand their pressure loading in a membrane manner rather than through bending. However, most underwater pressure vessels are not of this shape, because other shapes lend themselves more readily for other important purposes besides structural efficiency. For example, a submarine pressure hull of cylindrical shape would be more manoeuvrable at sea than one of spherical shape. Furthermore, the submarine pressure hull of cylindrical shape would present fewer difficulties during docking than a spherical one of the same volume and, in any case, a cylindrical pressure hull would be more efficient than a spherical one for housing large numbers of personnel, as the former can be made very long. Precise construction of a cylindrical vessel is usually more easily achieved than that of a similar spherical vessel. Nevertheless, the spherical pressure vessel is also considered in this chapter, as it is useful for the socalled miniature submarine, and also for deep-diving bathyscaphes.
1.2
The spherical pressure vessel
A spherical pressure vessel is usually constructed in the form of a thinwalled spherical shell with a pressure-tight hatch to allow access. In the case 1 © Carl T. F. Ross, 2011
2
Pressure vessels Hatch Pressure hull
Casing
Water
Water
Atmospheric pressure
1.1 The spherical pressure hull.
of mini-submarines, the pressure hull is usually covered with casing to improve hydrodynamic streamlining, as shown in Fig. 1.1. Whereas the pressure hull is under external pressure, the casing or hydrodynamic hull is only in a state of hydrostatic stress,1 so that the latter is unlikely to fail owing to static pressure. Thus, the casing can be constructed from plating that is relatively thin compared with the plating required for the pressure hull, although this casing must be strong enough to withstand hydrodynamic forces under motion and stresses owing to underwater currents and self-weight. For the pressure hull, however, the stresses caused by the external pressure can be very large. For example, a spherical pressure hull of diameter 5 m, diving to a depth of 1000 m, has to withstand a load of 786 MN (78 870 tonf). Perhaps the most famous spherically shaped pressure hulls are those developed by Auguste Picard, namely the Trieste and the FRNS-2, which operated at depths of 4.8 km (3 miles). On 22 January 1960, the Trieste dived to a depth of 11.52 km (7.16 miles), and stayed at that depth for the duration of about half an hour. Under uniform external pressure, a thin-walled spherical pressure hull can collapse either because of axisymmetric yield or by buckling in a lobar manner, as shown in Fig. 1.2. The mode of failure depends on a number of factors, including the thickness– radius ratio of the vessel and the mechanical properties of its material of construction. For example, for Picard’s Trieste, which was 15.24 cm (6 in) thick and 1.83 m (6 ft) diameter, the maximum membrane stresses at a depth of 11.52 km (7.16 miles) would have been about 347.5 MPa (50 400 lbf/in2), so that it is likely that its design was intended to eliminate lobar buckling. If Picard had used a safety factor of about 1.7, it would have been necessary to construct the vessel from a high-tensile steel with a 0.1% proof stress of about 590 MPa (38.26 tonf/in2). Another famous ‘spherical’ shell was that of Bushnell’s ‘Turtle’, Fig. 1.3. Bushnell produced this submarine in wood, in 1776 to fight the British in America’s War of Independence.
© Carl T. F. Ross, 2011
An overview of pressure vessels under external pressure
1.2 Lobar buckling of a thin-walled spherical shell.
1.3 Bushnell’s ‘Turtle’ (1776).
© Carl T. F. Ross, 2011
3
4
Pressure vessels
1.3
Cylinder/cone/dome pressure hulls
Most submarine pressure hulls take the form of a cylinder/cone/dome construction, surrounded by a casing, where the purpose of the latter is to improve the hydrodynamic streamlining, as shown in Fig. 1.4. As for the spherical pressure hull submarine, the casing is in a state of hydrostatic stress2 and it can therefore be constructed from relatively thin plating as the only loads likely to cause its failure are those caused by hydrodynamic forces, underwater currents and self-weight. However, for a pressure hull constructed from a combination of cylinders, cones and domes, these structures can fail either through axisymmetric yield or by buckling, as shown in Figs 1.5–1.8. Very often, the pressure required to cause shell instability of a thin-walled circular cylinder or cone is only a fraction of that necessary to cause axisymmetric yield resulting from a bulk stress. Thus, unstiffened thin-walled cirPressure hull Water Water Atmospheric pressure
Casing
1.4 Cylinder/cone/dome pressure hull.
1.5 Shell instability of thin-walled circular cylinders under external pressure.
© Carl T. F. Ross, 2011
An overview of pressure vessels under external pressure
5
cular cylinders and cones are structurally inefficient at withstanding external pressure, particularly if the vessels are long; one way of improving their structural efficiency is to stiffen them with suitably sized ring-stiffeners, spaced at suitable distances apart. The ring-stiffeners can be internal or external, or both, but theoretical studies3 have revealed that internal ringstiffeners are structurally more efficient than external ones, partly because
1.6 Lobar buckling of a hemi-ellipsoidal prolate or hemispherical dome under external pressure.
1.7 Axisymmetric buckling of an oblate dome under external pressure.
© Carl T. F. Ross, 2011
6
Pressure vessels
1.8 Axisymmetric buckling of a circular cylinder.
1.9 General instability of ring-stiffened circular cylinders.
of their increased curvature and partly because an internal ring-stiffener which otherwise had the same cross-sectional properties as an external ring-stiffener would weigh less. If, however, the ring-stiffeners were not strong enough to prevent structural instability, there is a possibility that the entire ring-shell combination could buckle bodily, as shown in Figs 1.9 and 1.10. This form of buckling is called general instability. Similarly, under external pressure, the buckling mode for thin-walled hemi-ellipsoidal prolate domes and thin-walled hemispherical domes tends to be of lobar form, and in the flanks of the vessels, as shown in Fig. 1.6, and the buckling mode for hemi-ellipsoidal oblate domes tends to be of axisymmetric form and in the noses of the vessels, as shown in Fig. 1.7. Thus, if the buckling resistance of hemi-ellipsoidal prolate domes and hemispherical domes is required to be improved, this can be achieved by introducing
© Carl T. F. Ross, 2011
An overview of pressure vessels under external pressure
7
1.10 General instability of ring-stiffened circular cones.
ring-stiffeners to the flanks of these vessels. Similarly, if the buckling resistance of hemi-ellipsoidal oblate domes is required to be improved, this can be achieved by introducing meridional stiffening, particularly to the noses of such vessels. It should be emphasised that owing to initial geometric imperfections the experimentally obtained buckling pressures can very often be considerably lower than predictions based on elastic theory, and therefore it is usually best to design the pressure hull so that buckling is eliminated and any likely failure will be caused by axisymmetric yield. Theoretical estimates of failure pressures based on axisymmetric yield are usually much better than those based on instability, providing the vessel is constructed properly. One question that readers who are not experts in the field may raise is why the ends of a submarine should be blocked off by doubly curved domes instead of, say, flat plates. The reason is that as flat plates have no meridional curvature, they will have to resist the effects of pressure in flexure and, because of this, in order for them to have equal strength to the circular cylindrical shell to which they are attached, their required thickness may be over ten times that of the circular cylindrical shell.
1.3.1 Thin-walled conical shells It should be pointed out that thin-walled conical shells of large apical angle can fail axisymmetrically, owing to either buckling or yield.
1.4
Other vessels that withstand external pressure
Other vessels that are required to withstand external pressure include the containment vessels of ships’ nuclear reactors, the legs of off-shore drilling
© Carl T. F. Ross, 2011
8
Pressure vessels
rigs, torpedoes, rockets, cooling towers, silos, aircraft fuselages, immersed tubes, tunnels, under-soil pipes, medical pressure vessels and food cans. If an accident takes place in a nuclear-powered ship and if the ship sinks, it is very important that the nuclear reactor is kept watertight, and this can best be achieved by surrounding the nuclear reactor with a containment vessel. Similarly, if a rocket or an aircraft ascends or descends in the atmosphere, the fuselage experiences external pressure. For off-shore drilling rigs, it is usually required that some sections of these structures withstand external water pressure, in addition to supporting the weight of the platform. The additional axial compression in the legs of the drilling rig can further weaken the legs, and much work has been done on this by Galletly et al.4
1.5
Weakening effect on ring-stiffeners owing to tilt
The weakening effect of tilt on ring-stiffeners results in localised failure if the stiffeners are not perpendicular to the axis of the cylinder. The strength of such stiffeners can be estimated by an extension of standard asymmetrical beam theory,1,2 or by Kendrick’s method.5
1.6
Bulkheads
As an additional safety precaution, it is usually desirable to introduce internal watertight bulkheads to submersibles, as shown in Fig. 1.11. These internal watertight bulkheads can be used to isolate a damaged compartment, in the event of an accident, so that personnel and perhaps even the vessel itself can be saved. Internal watertight bulkheads can appear in the form of cross-stiffened circular plates of variable thickness.
Pressure hull Bulkhead Bulkhead
Ring stiffeners
1.11 Pressure hull with internal watertight bulkheads.
© Carl T. F. Ross, 2011
An overview of pressure vessels under external pressure
9
External bulkhead
Ring stiffeners
Pressure hull
1.12 Pressure hull with an external bulkhead.
Sometimes, in addition to internal watertight bulkheads, it is desirable to introduce external bulkheads, as shown in Fig. 1.12. External bulkheads have two main purposes, namely to improve the general instability characteristics of the submersible and also as a means of attaching the casing to the pressure hull. The design of internal watertight bulkheads are not discussed here, but a computer program for designing a cross-stiffened plate is available.6
1.7
Materials of construction
The greatest depth of the ocean can be found in the Mariana’s Trench which is about 7.16 miles (11.52 km) deep. The average depth of the ocean is somewhere between 3 and 4 miles (4.83 to 6.44 km), but a large submarine can, at present, dive only to about 2000 ft (609.6 m). The reason for this is that as a submarine dives deeper into the ocean, the external pressure acting on its pressure hull increases. Thus, it is necessary to increase the wall thickness of the hull with an increase in its operating depth so that if the hull is made from a popular structural material, such as a high-strength steel, the vessel eventually has no reserve buoyancy and sinks like a stone. If the pressure vessel is not required to dive to great depths, a suitable material of construction could be mild steel, as it is relatively cheap and easy to weld. The main reason for its ease of welding is that, when it yields, it can strain to about 40 times the value of strain at first yield before it starts to strain-harden. This property obviates the necessity to carry out extensive stress-relieving of a welded pressure vessel. A more suitable alternative to mild steel is a high-tensile steel, but many high-tensile steels are difficult to weld, and all require extensive stress-relieving through heat treatment. Other materials with a better strength–weight ratio than mild
© Carl T. F. Ross, 2011
10
Pressure vessels
steel include aluminium alloy, titanium, glass-reinforced plastic (GRP), carbon fibre-reinforced plastic (CFRP), metal matrix composites (MMC), and ceramics. The materials for pressure vessels must not only be capable of withstanding the pressure at the required depth but must also have suitable characteristics to withstand the other factors that the operating environment imposes upon them. Such factors include: (a) (b)
(c) (d) (e) (f) (g)
Resistance to corrosion. High strength-to-density ratio; this is required for the structure to be able to obtain a positive buoyancy and therefore have a greater load carrying capacity. Cost of material. Fabrication properties: having chosen a specific material and a certain structural design, is it possible to manufacture it? Pressure hull design. Susceptibility to temperature and fire protection. Operating life span of material.
Unfortunately, as for most material requirements for specialised projects, there is not one material that satisfies all the particular requirements and therefore compromises have to be made in less critical areas of the materials properties, or its behaviour.
1.7.1 Possible materials and their problems Currently, the main materials used for specialised pressure hull designs are: (a) high-strength steels; (b) aluminium alloys; (c) titanium alloys; and (d) composites.
1.7.2 General corrosion In the marine environment, corrosion has been extensively studied and data generated regarding corrosion rates, making it relatively easy to predict and compensate for. The attack of submerged surfaces is governed principally by the rate of diffusion of oxygen through layers of rust and marine organisms. For steels the diffusion rate usually ranges from 3 to 6 mm per year and it is substantially independent of water temperature and tidal velocity, except where industrial pollution leads to higher rates. Certain marine organisms can also generate additional concentration cell and sulfur compound effects.6
© Carl T. F. Ross, 2011
An overview of pressure vessels under external pressure
11
1.7.3 Stress corrosion cracking Stress corrosion cracking is a form of localised failure which is more severe under the combined action of stress and corrosion than would be expected from the sum of the individual effects acting alone.7 There are many variables affecting the instigation of stress corrosion cracking and among these are alloy composition, tensile stress (internal or applied), corrosive environment, temperature and time. There are methods of relieving the internal stress and it is possible to solve the susceptibility of materials to stress corrosion cracking by using fracture mechanics. Therefore, although this is a problem, it is not one that cannot be predicted.
1.7.4 Other factors Other factors leading to failure are: (a) (b) (c)
brittle fracture; fatigue fracture; and fabrication-induced problems: i.e. stresses induced to welding and heat-affected zones around the weld area.
1.7.5 Material properties Tables 1.1 to 1.5 give material properties of various materials that are, or can be used for submersible pressure hulls. HY80 is the most commonly used steel for submarine hulls, its also commonly used for commercial applications including pressure vessels, storage tanks and merchant ships.8 Such alloys are attractive because of their availability, low cost, fabricability and high strength/density ratios. They have the disadvantage of being anodic to most other structural alloys and, therefore, vulnerable to corrosion when used in mixed structures, although these problems can be avoided by special design modifications.9 It is also difficult or impossible to obtain matching strength in weld metal and base metal and it is therefore necessary for the
Table 1.1 High tensile steels
Material
Specific density
Young’s modulus (GPa)
Compressive yield strength (MPa)
Heat treatment
HY80 HY100 HY130 HY180
7.8 7.8 7.8 7.8
207 207 207 207
550 690 890 1240
Q&T Q&T Q&T Q&T
© Carl T. F. Ross, 2011
12
Pressure vessels
Table 1.2 Aluminium alloys
Material
Specific density
Tensile strength (MPa)
Yield strength 0.2% (MPa)
Young’s modulus (GPa)
5086-H116 6061-T6 7075-T6 7075-T73 L65
2.8 2.8 2.9 2.9 2.8
290 310 572 434 –
207 276 503 400 390
70 70 70 70 70
Table 1.3 Titanium alloys Material
UTS* (MPa)
Yield strength (MPa)
Density (g cm−3)
6-4 Alloy (annealed) 6-2-1-1 Alloy 6-4 STOA Alloy C.P. Grade 2
896 869 870 345
827 724 830 276
4.5 4.5 4.5 4.5
* Ultimate tensile strength. Table 1.4 Composites
Material GRP (epoxy/S-glass UD*) GRP (epoxy/S-glass filament wound) CFRP (epoxy/HS UD) CFRP (epoxy/HS filament wound) MMC (6061 Al/SiC fibre UD) MMC (6061 Al/ alumina fibre UD)
Density (g cm−3)
Fibre volume fraction
2.1
0.67
65
1200
1
2.1
0.67
50
1000
3.2
1.7 1.7
0.67 0.67
210 170
1200 1000
3.0 5.1
2.7
0.5
140
3000
11
3.1
0.5
190
3100
15
Young’s modulus (GPa)
Compressive yield strength (MPa)
Relative cost
* Unidirectional. Table 1.5 Carbon nanotubes
Material
UTS* (GPa)
Young’s modulus (GPa)
Density (kg m−3)
Cost
Carbon nanotubes
150
940
1350
?
* Ultimate tensile strength.
© Carl T. F. Ross, 2011
An overview of pressure vessels under external pressure
13
welds to be thicker than the surrounding base metal or for welds to be located in light stress areas.10 As shown in Table 1.3, titanium alloys have high strength/density ratios and are therefore ideal materials for pressure hull designs where payload is an important characteristic. Unfortunately, they are very high cost at about 5.5 times more expensive than aluminium alloys, and this constrains their use from a commercial point of view.10 The most commonly used composite for marine structures such as ships, is GRP-based. The main reason for this is not only its very high strength/ density ratio but also its low cost compared with other composites. Metal matrix composites have many advantages over both GRPs and FRPs but they are still in the development stages, their cost is high, and there are fabrication difficulties on large structures (presently limited to components up to about 500 mm in diameter).11,12 Wenk13 has designed a submarine to dive to a depth of 4.57 km (15 000 ft); to achieve this he had to specify aluminium alloy as the metal of construction. More research is required on the suitability of composites and ceramics for such vessels, together with the use of carbon nanotubes; details of these are given in Tables 1.4 and 1.5.
1.8
Pressure, depth and compressibility
The pressure P at a depth h of a liquid is given by: P = ρ gh
[1.1]
in SI units, or: P = ρw h
[1.2]
in Imperial gravitational units where ρ is density, 1020 kg m−3 (salt water) or 1000 kg m−3 (pure water) and ρw is specific weight, 64 lbf ft−3 (salt water) or 62.5 lbf ft−3 (pure water). Thus, at a depth of 500 m in salt water, the pressure is: P = 1020 × 9.81 × 500 = 5.0 MPa or P = 50 bar At the bottom of the Mariana’s Trench, where the depth of water is about 11.52 km, the pressure is 1152.7 bar or 16 718 psi or 7.46 tonf in−2. In salt water, a pressure of 1 bar is approximately equivalent to a 10 m depth of water.
© Carl T. F. Ross, 2011
14
Pressure vessels
1.8.1 Compressibility Although, for most purposes, water can be regarded as practically incompressible, it is in fact compressible, where:
εv =
δV P = V K
where εv is volumetric strain, δV is the change in volume over a volume (V), P is pressure and K is the bulk modulus (2.05 GPa). Thus, at a depth of 500 m, in salt water, the volumetric strain is about 0.24%, that is the density of water, because of compressibility, increases by 0.24%. Similarly, at the bottom of the Mariana’s Trench the density of water owing to compressibility increases by 5.6%. The density of the water also increases with increased salinity and with frigid temperatures which are just above the freezing point of water.
© Carl T. F. Ross, 2011
2 Axisymmetric deformation of pressure vessels
Abstract: Theoretical analyses are provided of the axisymmetric deformation of ring-stiffened circular cylinders, cones, and domes. Solutions for both thin-walled and thick-walled shells are presented. For a thin-walled ring-stiffened circular cylinder, the solution is based on beam-on-elastic-foundation theory, whereas for thin- and thick-walled cones and domes the solutions are based on the finite element method. Finite element solutions for thin-walled axisymmetric shells are extended to cater for orthotropic shells. Key words: axisymmetric deformation, cylinders, cones, domes, beam-onelastic-foundation theory, finite element analysis, orthotropic shells.
2.1
Axisymmetric yield failure
Failure owing to axisymmetric yield is perhaps the most important mode of failure of vessels under external pressure, as sensible design should prevent failure owing to instability. The difficulty of predicting the loss in buckling resistance of a vessel owing to the detrimental effects of its initial geometrical imperfections, which, in a vessel under external water pressure, can cause a catastrophic fall in its buckling resistance, leads to axisymmetric yield failure. In this chapter, theoretical analyses of the axisymmetric deformation of ring-stiffened circular cylinders, cones and domes are provided. Solutions for both thin-walled and thick-walled shells are presented. For a thin-walled ring-stiffened circular cylinder, the solution is based on beam-on-elasticfoundation theory, but for thin- and thick-walled cones and domes the solutions are based on the finite element method. Finite element solutions for thin-walled axisymmetric shells are extended to cater for orthotropic shells. Firstly, however, it is necessary to consider unstiffened circular cylinders and spheres.
2.2
Unstiffened circular cylinders and spheres
For an unstiffened thin-walled circular cylinder under uniform pressure:
σ H = hoop stress = pa / t σ L = longitudinal stress = pa /(2t ) 15 © Carl T. F. Ross, 2011
16
Pressure vessels
where p is pressure (positive if internal); a is internal radius if p is positive and external radius if p is negative (or external pressure); and t is wall thickness. For a thin-walled sphere:
σ = membrane stress = pa /(2t ) The above formula starts to breakdown when t/a > 1/30 For a thick-walled circular cylinder under internal pressure:
σ H (maximum ) =
p(R12 + R22 ) ( R22 − R12 )
For a thick-walled circular cylinder under external pressure:
σ H (maximum ) =
−2 pR22 ( R22 − R12 )
where R1 is the internal radius and R2 is the external radius. In both the above instances, the maximum value of hoop stress occurs on the internal surface of the thick-walled cylinder (r = R1). For a thick-walled sphere under uniform internal pressure and r = R1
σ H (maximum) =
p (R23 + 2 R13 ) 2 ( R23 − R13 )
and the maximum radial stress is:
σ R (maximum) = − p For a thick-walled sphere under uniform external pressure and r = R1
σ H (maximum ) =
−3 pR23 2 ( R23 − R13 )
σ R (maximum) = − p
2.3
Ring-stiffened circular cylinders
Thin-walled circular cylinders under uniform external pressure can collapse through nonsymmetric bifurcation buckling (lobar buckling) at a pressure that might only be a fraction of that needed to cause axisymmetric yield. To improve the structural efficiency of these vessels, ring-stiffeners are introduced. Ring-stiffeners, however, can cause large bending stresses near the ring-stiffener or bulkhead and, because of this, more elaborate solutions have been produced.
© Carl T. F. Ross, 2011
Axisymmetric deformation of pressure vessels
17
One of the earliest solutions presented for the axisymmetric deformation of ring-stiffened circular cylinders was that of von Sanden and Gunther in 1920,14 based on the differential equation [2.1]: d 4 w 12(1 − v 2 )w 12 (1 − v 2 ) p + = dx 4 t 2 a2 Et 3
[2.1]
where w = radial deflection (positive inwards) a = mean radius of cylinder t = wall thickness E = Young’s modulus ν = Poisson’s ratio p = pressure (positive external) Unfortunately, however, in presenting their solution, the authors accidentally interchanged two of the signs and consequently published an incorrect solution. Although the error was discovered in the following year by Hovgaard,15 a number of other authors have republished the original incorrect solution. The incorrect circumferential stress formula can be in error by as much as 20% for some cylindrical shells. Another, smaller, error in the von Sanden and Gunther solution is that their differential equation did not fully take into account the loading on the shell caused by the pressure normal to it, the so-called Viterbo effect.16 The Viterbo effect, however, is only about 1%, and its inclusion in the differential equation [2.1] is shown in equation [2.2]: d 4 w 12 (1 − v 2 ) w 12 (1 − v 2 ) p (1 − v / 2 ) = + dx 4 t 2 a2 Et 3
[2.2]
Faulkner17 publishes the corrected von Sanden and Gunther formula as: Py =
σ yp ( t / a) 1−γG
where Py = the pressure to cause yield at the mid-thickness of the shell at mid-bay σyp = yield stress or 0.2% proof stress A (1 − ν / 2 ) γ = A + bf t + 2 Nt / α A = Af(a/Rf)2 ν = Poisson’s ratio α L = 1.285L / at Af = cross-sectional area of frame bf = web thickness of the stiffener
© Carl T. F. Ross, 2011
18
Pressure vessels
L = unsupported length of shell = Lf − bf Lf = spacing of ring stiffeners 2(sinh (α L / 2 ) cos (α L / 2 ) + cosh (α L / 2 ) sin (α L / 2 )) G= sinh (α L) + sin (α L) Another deficiency with both equations [2.1] and [2.2] is that they do not include the beam-column effect, which causes the deformation to be nonlinear, and this can be quite large in certain instances. Salerno and Pulos18 introduced the beam-column effect in the original differential equation, which is shown in equation [2.3]. The beam-column effect can increase the maximum, longitudinal stress by about 10% for many vessels: d 4 w 6 (1 − v 2 ) pa d 2 w 12 (1 − v 2 ) w 12 (1 − v 2 ) p (1 − v / 2 ) + = + dx 4 dx 2 Et 3 t 2 a2 Et 3
[2.3]
A further improvement to the differential equation [2.3] was made by Wilson19 when he solved the differential equation [2.4] using a Fourier cosine transformation: d 4 w ⎡ v 6 (1 − v 2 ) pa ⎤ d 2 w 12 (1 − v 2 ) w 12 (1 − v 2 ) p (1 − v / 2 ) = + + ⎥ dx 2 + Et 3 dx 4 ⎢⎣ a 2 Et 3 t 2 a2 ⎦ [2.4] Similarly to Salerno and Pulos,18 Wilson19 solved this differential equation for a shell stiffened by equal-size ring-stiffeners, where the shell deformed symmetrically about mid-span, as shown by Fig. 2.1. The boundary conditions assumed by Wilson were:
Frame t
bf Shell Deformation line
Axis of cylinder
2.1 Ring-stiffened circular cylinder, stiffened by equal-size stiffening rings.
© Carl T. F. Ross, 2011
Axisymmetric deformation of pressure vessels (a) w is symmetrical about x = 0 (mid-span); dw = 0 at x = ± L / 2; (b) dx Et 3 d 3w (c ) = G1w − H1 at x = ± L / 2 6 (1 − v 2 ) dx 3
19
[2.5]
where G1 = E(Af/a2f + bft/a2) H1 = pbf(1 − ν/2) Af = cross-sectional area of a frame af = radius of centroid of frame bf = width of shell in direct contact with the frame’s web L = frame spacing Ross20 solved Wilson’s differential equation using the method of Salerno and Pulos as follows:
α 4 = 3 (1 − v 2 ) / ( a 2 t 2 )
β2 =
pa 3 t 2v + 2Et 12 (1 − v 2 )
C0 = pa 2 (1 − v / 2 ) / ( Et )
to give the differential equation [2.6]: 4 12 (1 − v 2 ) C0 d4w 4 2 d w 4 + + = 4 α β 4 α w t 2 a2 dx 4 dx 4
[2.6]
For F1 = α
(1 − α 2 β 2 )
and
F2 = α
(1 + α 2 β 2 )
the complete solution of equation [2.6] is w = A1 cosh F1 x cos F2 x + A2 sinh F1 x sin F2 x + A3 cosh F1 x sin F2 x + A4 sinh F1 x cos F2 x + C0
[2.7]
and some of its derivatives are dw = ( A1 F1 + A2 F2 )sinh F1 x cos F2 x dx + ( A2 F1 − A1 F2 )cosh F1 x sin F2 x + ( A3 F2 + A4 F1 )cosh F1 x cos F2 x + ( A3 F1 − A4 F2 )sinh F1 x sin F2 x
© Carl T. F. Ross, 2011
[2.8]
20
Pressure vessels d 2w = [ A1 (F12 − F22 ) + 2 A2 F1 F2 ]cosh F1 x cos F2 x dx 2 + [ A2 (F12 − F22 ) − 2 A1 F1 F2 ]sinh F1 x sin F2 x + [ A3 (F12 − F22 ) − 2 A4 F1 F2 ]cosh F1 x sin F2 x + [ A4 (F12 − F22 ) + 2 A3 F1 F2 ]sinh F1 x cos F2 x
[2.9]
d 3w = [ A1 F1 (F12 − 3F22 ) + A2 F2 (3F12 − F22 ]sinh F1 x cos F2 x dx 3 + [ A1 F2 (F22 − 3F12 ) + A2 F1 (F12 − 3F22 ]cosh F1 x sin F2 x + [ A3 F2 (3F12 − F22 ) + A4 F1 (F12 − 3F22 ]cosh F1 x cos F2 x + [ A3 F1 (F12 − 3F22 ) + A4 F2 (F22 − 3F12 ]sinh F1 x cos F2 x
[2.10]
The only unknown parts of the derivations shown above are the arbitrary constants, and these can be solved by assuming that certain conditions exist at the boundary, as described earlier in equation [2.5]. For the circular cylindrical shell element, stiffened by equal-strength frames, it can be seen from the boundary conditions (a) given in equation [2.5] that the animetric (asymmetric) terms must vanish because of the symmetry of w about midspan, i.e. A3 = A4 = 0 Thus, there are only two unknowns in equation [2.7], namely A1 and A2, and these are obtained from conditions (b) and (c) in equation [2.5] as follows: A1 = N1 / D
A2 = N 2 / D
where N1 = −(G1C0 − H1 ) × (F1 cosh 0.5F1L sin 0.5F2 L + F2 sinh 0.5F1L cos 0.5F2 L) N 2 = (G1C0 − H1 ) × (F1 sinh 0.5F1L cos 0.5F2 L − F2 cosh 0.5F1L sin 0.5F2 L) and D1 = D2 = {Et 3 / 12(1 − ν 2 )}{2 F1 F2 (F12 + F22 )(cosh F1L − cos F2 L)} + 0.5G1 (F1 sin F2 L + F2 sinh F1L) The stress distributions across the bay can be obtained by substituting A1 and A2 into equations [2.7] and [2.9], and then by substituting equations [2.7] and [2.9] into equations [2.11] and [2.12]:
© Carl T. F. Ross, 2011
Axisymmetric deformation of pressure vessels Hoop stress = −
21
Ew pav Et d2w ⎞ ⎛w − ± +v 2 ⎟ 2 ⎜ 2 a 2t 2 (1 − v ) ⎝ a dx ⎠
[2.11]
pa Et ⎛ vw d 2 w ⎞ ± + ⎜ ⎟ 2t 2 (1 − v 2 ) ⎝ a 2 dx 2 ⎠
[2.12]
Longitudinal stress = −
2.3.1 Circular cylinder stiffened by unequally sized rings Unfortunately, however, all the previous solutions are based on the assumption that the shell is supported by equal-strength ring-stiffeners and, as a result of this, the shell deforms symmetrically about mid-span. In practice, however, the shell may be stiffened by unequally sized stiffeners at unequal spacings and, because of the added effect of rigid bulkheads, the shell does not deform symmetrically between many pairs of adjacent stiffeners, as shown in Fig. 2.2. In 1970, Ross21 overcame this problem by publishing a solution for a multi-bay circular cylinder, stiffened by unequally sized stiffeners, which can be internal or external or any combination of the two, as shown in Fig. 2.3. The solution is based on the differential equation [2.13]:
Large stiffener
Shell plating
Deformation line Axis of cylinder
Small stiffener
w
N th ring
(i +1) ring
i th ring x
Bay 1
ti
Li a0
R1 a1
Bay i
Clamped end
Bay 0
Bay N
Clamped end
1st ring
2nd ring
2.2 Circular cylinder stiffened by unequal-size rings.
RN
Centre line x
2.3 Ring-stiffened cylinder.
© Carl T. F. Ross, 2011
22
Pressure vessels
d 4 w ⎡ v 6 (1 − v 2 ) ai Pi ⎤ d 2 w 12 (1 − v 2 ) w 12 (1 − v 2 ) + + = ( P − 0.5vP1 ) ⎥ dx 2 + Eti3 Eti3 ti2 ai2 dx 4 ⎢⎣ ai2 ⎦ [2.13] where P and P1 are the lateral and axial pressures, respectively, and the other terms refer to the ith bay. By substituting
α i4 = 3 (1 − v 2 ) / ( ai2 ti2 ) βi2 = {( P1ai3 / 2Eti ) + [vti2 / 12 (1 − v 2 )]} C0 i = ai2 ( P − 0.5vP1 ) / ( Eti ) then equation [2.13] takes the form of equation [2.14]: 2 d4w 4 2 d w + 4 α β + 4α i4 w = [12 (1 − v 2 ) C0 i / (ti2 ai2 )] i i dx 4 dx 2
[2.14]
and the complete solution of equation [2.14] is given by equation [2.15]: w = A1 cosh(F1i x)cos( F2 i x) + A2 sinh(F1i x)sin(F2 i x) + A3 cosh(F1i x)ssin( F2 i x) + A4 sinh(F1i x)cos(F2 i x) + C0 i
[2.15]
dw = ( A3 F2 i + A4 F1i )cosh(F1i x)cos(F2 i x) dx + ( A3 F1i − A4 F2 i )sinh( F1ii x)sin(F2 i x) + ( A2 F1i − A1 F2 i )cosh(F1i x)sin( F2 i x) + ( A1 F1i + A2 F2 i )sinh(F1i x)cos(F2 i x)
[2.16]
Now,
d 2w = [ A1 (F12i − F22i ) + 2 A2 F1i F2 i ]cosh(F1i x)cos(F2 i x) dx 2 + [ A2 (F12i − F22i ) − 2 A1 F1i F2 i ]sinh( F1i x)sin(F2 i x) + [ A2 (F12i − F22i ) − 2 A4 F1i F2 i ]cosh( F1i x)sin(F2 i x) + [ A1 (F12i − F22i ) + 2 A3 F1i F2 i ]sinh( F1i x)cos(F2 i x)
[2.17]
d 3w = [ A3 F2 i (3F12i − F22i ) + A4 F1i (F12i − 3F22i )]cosh(F1i x)cos(F F2 i x) dx 3 + [ A3 F1i (F12i − 3F22i ) + A4 F2 i (F22i − 3F12i )]sinh(F1i x)sin(F2 i x) + [ A1 F2 i (F22i − 3F12i ) + A2 F1i (F12i − 3F22i )]cosh(F1i x)sin(F2 i x) + [ A1 F1i (F12i − 3F22i ) + A2 F2 i (3F12i − F22i )]sinh(F1i x)cos(F2 i x) [2.18]
© Carl T. F. Ross, 2011
Axisymmetric deformation of pressure vessels
23
Now the ith bay is between the ith ring and the (i + 1)th ring and to solve the arbitrary constants for the ith bay the slope and deflection must be equated at the boundaries, i.e. w = wi ⎫ ⎬ at x = 0 dw / dx = θ i ⎭
and
w = w(i +1) ⎫ ⎬ at x = Li dw / dx = θ(i +1) ⎭
Substituting these boundary conditions into equations [2.15] and [2.16], the arbitrary constants can be obtained in terms of the slopes and deflections at the rings, as follows: – A1 = w1 − C0 i A2 = ψ 11wi + ψ 12θ i + ψ 13w(i +1) + ψ 14θ(i +1) + ψ 15 A3 = ( − F1i / F2 i ) [ψ 6 wi + (ψ 7 − 1 / F1i ) θ i + ψ 8(i +1) + ψ 9θ(i +1) + ψ 10 ] A4 = ψ 6 wi + ψ 7θ i + ψ 8 w(i +1) + ψ 9θ(i +1) + ψ 10 where F1i = αi(1 − α i2 − βi2)0.5 F2i = αi(1 + α i2βi2)0.5 C1i = cosh(F1iLi)cos(F2iLi) C2i = sinh(F1iLi)sin(F2iLi) C3i = cosh(F1iLi)sin(F2iLi) C4i = sinh(F1iLi)cos(F2iLi) Di = Eh3i /[12(1 − ν2)] ψ1 = −[F1iC4i − C1iF2i cot(F2iLi) − C1iF1i coth(F1iLi) − F2iC3i] ψ2 = −[−F1isin(F2iLi)]/[F2isinh(F1iLi)] ψ3 = −F2icot(F2iLi) − F1icoth(F1iLi) ψ4 = (F1iC3i/F2i − C4i) × [F2i cot(F2iLi) + F1i coth(F1iLi)] − (F 1i2 /F2i + F2i)C2i ψ5 = −C0i{−F1iC4i + (C1i − 1) × [F2i cot(F2iLi) + F1i coth(F1iLi)] + F2iC3i} ψ6 = ψ1/ψ4 ψ7 = ψ2/ψ4 ψ8 = ψ3/ψ4 ψ9 = 1/ψ4 ψ10 = ψ5/ ψ4 ψ11 = −C1i/C2i + ψ6 [F1i coth(F1iLi)/F2i − cot(F2iLi)] ψ12 = −coth(F1iLi)/F2i + ψ7 [F1i coth(F1iLi)/F2i − cot(F2iLi)] ψ13 = 1/C2i + ψ8 [F1i coth(F1iLi)/F2i − cot(F2iLi)] ψ14 = ψ9 [F1i coth(F1iLi)/F2i − cot(F2iLi)] ψ15 = C0i (C1i − 1)/C2i + ψ10 [F1i coth(F1iLi)/F2i − cot(F2iLi)] The next step is to obtain the bending moments and shearing forces at the stations in terms of the slopes and deflections at these positions, and
© Carl T. F. Ross, 2011
24
Pressure vessels
this can be achieved by considering equilibrium at the ith stiffener, as shown in Fig. 2.4. In the ith bay the bending moment/unit length at any distance x from the ith ring is given by ⎛ d 2 w vw ⎞ M x = − Di ⎜ 2 + 2 ⎟ ⎝ dx ai ⎠ If x = 0: ⎛ d 2 w vw ⎞ Mi( 0 ) = − Di ⎜ 2 + 2 i ⎟ ⎝ dx ai ⎠ x = 0
= − Di [ A1 ( F12i − F22i ) + 2 A2 F1i F2 i + vwi / ai2 ]
= −γ 1wi − γ 2θ i − γ 3w(i +1) − γ 4θ(i +1) − γ 5
[2.20] In the ith bay, the shearing force/unit length at any distance x from the ith ring is given by Qx = − Di
d 3w ⎛ dw ⎞ − 0.5Pa i i⎜ ⎝ dx ⎟⎠ dx 3
Therefore Qi(0) = Qx at x = 0 ⎛ d 3w ⎞ ∴ Qx = − Di ⎜ 3 ⎟ − 0.5P1aiθ i ⎝ dx ⎠ = − Di [ A3 F2 i ( 3F12i − F22i ) + A4 F1i ( F12i − 3F22i )] − 0.5P1aiθ i = −γ 6 wi − γ 7θ i − γ 8 w(i +1) − γ 9θ(i +1) − γ 10
[2.21]
Now, ⎛ d 2 w vw(i +1) ⎞ Mi(Li ) = − Di ⎜ 2 + ⎟ ⎝ dx ai2 ⎠ x = Li
= − Di {[ A1 ( F12i − F22i ) + 2 A2 F1i F2 i ]C1i + [ A2 ( F12i − F22i ) − 2 A1 F1i F2 i ]C2 i + [ A3 ( F12i − F22i ) − 2 A4 F1i F2 i ]C3i
+ [ A4 ( F12i − F22i ) + 2 A3 F1i F2 i ]C4 i + vw(i +1) / ai2 } bi
Pi
Mi(0)
M i(Li–1)
[2.22] Q i(0)
Q i(Li–1)
i th bay (a)
i th bay (b)
2.4 Equilibrium at ith ring: (a) moments; and (b) shearing forces.
© Carl T. F. Ross, 2011
Axisymmetric deformation of pressure vessels
25
Mi ( Li ) = −φ1i wi − φ2 iθ i − φ3i w( i + 1) − φ4 iθ( i + 1) − φ5i and ⎛ d 3w ⎞ Qi (Li ) = − Di ⎜ 3 ⎟ − 0.5P1aiθ (i + 1) ⎝ dx ⎠ x = Li
= − Di {[ A1 F1i ( F12i − 3F22i ) + A2 F2 i ( 3F12i − F22i )]C4 i + [ A1 F2 i ( F22i − 3F12i ) + A2 F1i ( F12i − 3F22i )]C3i
+ [ A3 F2 i ( 3F12i − F22i ) + A4 F1i ( F12i − 3F22i )]C1i
+ [ A3 F1i ( F12i − 3F22i ) + A4 F2 i ( F22i − 3F12i )]C2 i } − 0.5P1aiθ (i + 1) Qi (Li ) = −φ6 i wi − φ7 iθ i − φ8 i w(i + 1) − φ9 iθ (i + 1) − φ10 i where γ1 = Di[(F 1i2 − F 2i2 ) + 2ψ11F1iF2i + ν/a2i ] γ2 = 2Diψ12F1iF2i γ3 = 2Diψ13F1iF2i γ4 = −2Diψ14F1iF2i γ5 = Di[2ψ15F1iF2i − C0i(F 1i2 − F 2i2 )] γ6 = −2DiF1i(F 1i2 + F 2i2 )ψ6 γ7 = Di[(3F 1i2 − F 2i2 ) − 2F1i(F 1i2 − F 2i2 )ψ7] + 0.5P1ai γ8 = −2DiF1i(F 1i2 + F 2i2 )ψ8 γ9 = −2DiF1i(F 1i2 + F 2i2 )ψ9 γ10 = −2DiF1i(F 1i2 + F 2i2 )ψ10 ⎧ φ1i = Di ⎨[ F12i − F22i + 2ψ 11 F1i F2 i ]C1i + [ψ 11 ( F12i − F22i ) − 2 F1i F2 i ]C2 i ⎩ ⎡ − F1i ( F12i − F22i ) ⎤ ⎫ + ψ6 ⎢ − 2 F1i F2 i ⎥ C3i − ψ 6 ( F12i + F22i ) C4 i ⎬ F 2i ⎣ ⎦ ⎭
φ2 i = Di {2 F1i F2 iψ 12C1i + ψ 12 ( F12i − F22i ) C2 i
⎛ − F1i ( F12i − F22i ) ⎞⎤ ⎡ ( F12i − F22i ) +⎢ +ψ7⎜ − 2 F1i F2 i ⎟ ⎥ C3i ⎝ ⎠⎦ F F 2i 2i ⎣ 2 2 + [ 2 F1i − ψ 7 ( F1i + F2 i )]C4 i }
φ3i = Di {2 F1i F2 iψ 13C1i + ψ 13 ( F12i − F22i ) C2 i
⎞⎤ ⎡ ⎛ − F1i ( F12i − F22i ) + ⎢ψ 8 ⎜ − 2 F1i F2 i ⎟ ⎥ C3i ⎝ ⎠⎦ F 2 i ⎣ 2 2 2 − ψ 8 ( F1i + F2 i ) C4 i + v / ai }
© Carl T. F. Ross, 2011
[2.23]
26
Pressure vessels
⎧ φ4 i = Di ⎨ 2 F1i F2 iψ 14C1i ( F12i − F22i ) ψ 14C2 i ⎩ ⎫ ⎡ − F1i ( F12i − F22i ) ⎤ + ψ9 ⎢ − 2 F1i F2 i ⎥ C3i − ψ 9 ( F12i + F22i ) C4 i ⎬ F 2 i ⎭ ⎣ ⎦ ⎧ φ5i = Di ⎨[ −C0 i ( F12i − F22i ) + 2 F1i F2 iψ 15 ]C1i ⎩ + [ψ 15 ( F12i − F22i ) + 2C0 i F1i F2 i ]C2 i
⎡ − F1i ( F12i − F22i ) ⎤ ⎫ + ψ 10 ⎢ − 2 F1i F2 i ⎥ C3i − ψ 10 ( F12i + F22i ) C4 i ⎬ F2 i ⎣ ⎦ ⎭ ⎧ φ6 i = Di ⎨[ Fi ( F12i − 3F22i ) + ψ 11 F2 i ( 3F12i − F22i )]C4 i ⎩ + [ F2 i ( F22i − 3F12i ) + ψ 11 F1i ( F12i − 3F22i )]C3i − 2ψ 6 F1i ( F12i + F22i ) C1i
F2 ⎡ ⎤ ⎫ + ψ 6 ⎢ F2 i ( F22i − 3F12i ) − 1i ( F12i − 3F22i )⎥ C2 i ⎬ F2 i ⎣ ⎦ ⎭ ⎧ φ7 i = Di ⎨ψ 12 F2 i ( 3F12i − F22i ) C4 i + ψ 12 F1i ( F12i − 3F22i ) C3i ⎩ + [( 3F12i − F22i ) − 2ψ 7 F1i ( F12i + F22i )]C1i
⎡ F1i ( F12i − 3F22i ) ⎫ F2 ⎛ ⎞⎤ + ψ 7 ⎜ F2 i ( F22i − 3F12i ) − 12i ( F12i − 3F22i )⎟ ⎥ C2 i ⎬ ⎢ ⎝ ⎠ F2 i F2 i ⎣ ⎦ ⎭
φ8 i = Di {ψ 13 F2 i ( 3F12i − F22i ) C4 i + ψ 13 F1i ( F12i − 3F22i ) C3i − 2ψ 8 F1i ( F12i + F22i ) C1i F2 ⎡ ⎤ ⎫ + ψ 8 ⎢ F2 i ( F22i − 3F12i ) − 1i ( F12i − 3F22i ) ⎥ C2 i ⎬ F2 i ⎣ ⎦ ⎭ ⎧ φ9 i = Di ⎨ψ 14 F2 i ( 3F12i − F22i ) C4 i + ψ 14 F1i ( F12i − 3F22i ) C3i ⎩ − 2ψ 9 F1i ( F12i + F22i ) C1i F2 ⎡ ⎤ ⎫ + ψ 9 ⎢ F2 i ( F22i − 3F12i ) − 1i ( F12i − 3F22i )⎥ C2 i ⎬ + 0.5P1ai F2 i ⎣ ⎦ ⎭ ⎧ φ10 i = Di ⎨[ −C0 i F1i ( F12i − 3F22i ) + ψ 15 F2 i ( 3F12i − F22i )]C4 i ⎩ + [ −C0 i F2 i ( F22i − 3F12i ) + ψ 15 F1i ( F12i − 3F22i )]C3i − 2ψ 10 F1i ( F12i + F22i ) C1i
F2 ⎡ ⎤ ⎫ + ψ 10 ⎢ F2 i ( F22i − 3F12i ) − 12i ( F12i − 3F22i )⎥ C2 i ⎬ F2 i ⎣ ⎦ ⎭
© Carl T. F. Ross, 2011
Axisymmetric deformation of pressure vessels
27
The equilibrium of the ith ring requires that clockwise moments − anticlockwise moments = zero; M(i − L )(Li ) a(i − L ) − Mi(0) ai + (EI i / Ri )θ i = 0
[2.24]
and outward forces − inward forces = zero; Q(i − L )(Li −1) a(i − L ) − Qi(0) ai + EAFi wi / Ri − Pbi ai + 0.5vP1bi ai = 0
[2.25]
Substitution of equations [2.20] to [2.23] into equations [2.24] and [2.25] leads to the following sets of simultaneous equations. −φ1(i −1) a(i −1)w(i −1) − φ2(i −1) a(i −1)θ(i −1) + ( −φ3(i −1) a(i −1) + γ 1ai ) wi
+ ( −φ4(i −1) a(i −1) + γ 2 ai − EI i / Ri ) θ i + γ 3 ai w(i +1) + γ 4 aiθ(i +1) = φ5(i −1) a(i −1) − γ 5 ai
[2.26]
−φ6(i − 1) a(i − 1)w(i − 1) − φ7(i − 1) a(i − 1)θ (i − 1) + ( −φ8(i − 1) a(i − 1) + γ 6 ai − EAFi / Ri ) wi
+ ( −φ9(i − 1) a(i − 1) + γ 7 ai ) θ i + γ 8 ai w(i + 1) + γ 9 aiθ (i + 1) = Pbi ai (1 − vPi / 2 P ) + φ10(i − 1) a(i − 1) − γ 10 ai
[2.27]
Solution of these simultaneous equations results in the slope and deflection of the ring-stiffeners and back-substitution of these into the appropriate equations results in the deflection and stress values. If the vessel is clamped at its end and there are N ring-stiffeners between the extremities, then, at i = 1:
φ1( 0 ) = φ2( 0 ) = φ6( 0 ) = φ7( 0 ) = 0 and at i = N:
γ3 =γ4 =γ8 =γ9 = 0 Equations for stress are given by: Hoop stress = − Longitudinal stress = −
Ew vP1a Ehi ⎛ w d2w ⎞ − ± +v 2 ⎟ 2 ⎜ 2 2 h 2 (1 − v ) ⎝ a a dx ⎠ P1a Ehi ⎛ vw d 2 w ⎞ ± + ⎜ ⎟ 2 h 2 (1 − v 2 ) ⎝ a 2 dx 2 ⎠
[2.28]
where hi = ti = shell thickness.
2.3.2 Comparisons between experiment and theory Comparisons between experiment and theory for model number 321 (Fig. 2.5 and Table 3.4) are shown in Figs 2.6–2.10, where they can be seen to be
© Carl T. F. Ross, 2011
Pressure vessels 0.75″
bF
L1
bf
L
0.7″
0.62″
t = 0.08″ 5.125″ Axis of cylinder
0.002
0.001
1st Bay
0
2nd Bay
3rd Bay
3rd Frame
0.002 2nd Frame
0.003
1st Frame
0.003
0.001 0
Closure plate
Inward radial deflection (in)
2.5 Model number 3.
2.6 Deflection of longitudinal generator at 100 lbf in−2 (Model number 3).
–6000
–6000
–4000
–4000
–2000
–2000
0
0
+6000 +8000
1st Bay
2nd Bay
3rd Bay
3rd Frame
+4000 2nd Frame
+2000
+4000 1st Frame
+2000 Closure plate
Stress (lbf in–2)
28
+6000 +8000
2.7 Longitudinal stress of the outermost fibre at 100 lbf in−2 (Model number 3).
© Carl T. F. Ross, 2011
–4000
–2000
–2000
+2000
1st Bay
2nd Bay
3rd Bay
0
3rd Frame
0
2nd Frame
–4000
1st Frame
–6000
Closure plate
Stress (lbf in–2)
–6000
+2000
2.8 Circumferential stress of the outermost fibre at 100 lbf in−2 (Model number 3). +2000
+2000
0
–6000
–8000
–8000
–10000
–10000
–12000
–12000
–14000
1st Bay
2nd Bay
3rd Bay
3rd Frame
–6000
2nd Frame
–4000
1st Frame
–4000
Closure plate
–2000
–14000
2.9 Longitudinal stress of the innermost fibre at 100 lbf in−2 (Model number 3).
–4000
–2000
–2000
0
1st Bay
2nd Bay
3rd Bay
3rd Frame
–4000
2nd Frame
–6000
1st Frame
–6000
Closure plate
Stress (lbf in–2)
Stress (lbf in–2)
0 –2000
0
2.10 Circumferential stress of the innermost fibre at 100 lbf in−2 (Model number 3).
© Carl T. F. Ross, 2011
30
Pressure vessels Table 2.1 Details of model No. 3: L1 = unsupported length of the end bays; L = unsupported length of a typical bay; bF = web width of the first and last stiffening ring; bf = web width of a typical ring-stiffener; E = Young’s modulus = 10.3 × 106 lbf in−2; ν = Poisson’s ratio = 0.32 (assumed); N = number of ring-stiffeners L1
L
bF
bf
N
2.5 in
2.25 in
0.4 in
0.325 in
5
in good agreement. Model number 3 was made from HE9/WP aluminium alloy, with the geometrical properties shown in Table 2.1 and Fig. 2.5.
2.3.3 Advantages of the beam-on-elastic-foundation theory Some submarine structural engineers might criticise the above solution, preferring one based on the finite element method. However, the author has found that for determining stresses in thin-walled circular cylinders, solutions based on the differential equation are much better than the finite element solutions, from the point of view of both precision and efficiency. For a finite element solution to be precise, it is necessary to take hundreds of elements for a typical vessel; as each elemental stiffness matrix is of order 6 × 6, computation is uneconomical when compared with the solution shown in this section, where only a few elements are required. It must be emphasised that the solution given in 2.3.1 is also a computational one, as it is based on determining slopes and deflections at the ringstiffeners (or nodes), and then through substitution obtaining the deflection and stress distributions across each bay. Computer programs for the stress analysis of thin-walled cylinders are given in Appendices I and II.
2.4
Axisymmetric deformation of thin-walled cones and domes
Solution of axisymmetric problems involving cones and domes by analytical methods is very difficult, and, in those instances, the finite element method provides a simpler and better solution. For thin-walled cones, the solution is based on the methods of Grafton and Strome22 and Zienkiewicz,23 except that the thin-walled conical element was allowed to have linear taper, as shown in Fig. 2.11. From Fig. 2.11, it can be seen that the element has two nodal circles at its ends, and that each nodal circle has three global degrees
© Carl T. F. Ross, 2011
Axisymmetric deformation of pressure vessels
31
w2 w 0,y 0
w
u2
w1 q1
u1
L
x=0 R1
q2
u,x x=1
x 0,u 0
t1 f
t2
R2
2.11 Thin-walled conical element.
of freedom, namely u0i , w0i and θi at node i. A brief derivation of the element is given below. In terms of the six degrees of freedom of the element, the u and the w displacements can be expressed as: 0 0 ξ ⎧ u ⎫ ⎡( 1 − ξ ) ⎨ ⎬=⎢ 2 3 2 3 (1 − 3ξ + 2ξ ) L (ξ − 2ξ + ξ ) 0 ⎩w ⎭ ⎣ 0 ⎧ui ⎫ ⎪w ⎪ ⎪ i⎪ ⎪θ i ⎪ × ⎨ ⎬ = [ N ] {ui } ⎪u j ⎪ ⎪w j ⎪ ⎪ ⎪ ⎩θ j ⎭
0 0 ⎤ 3 ( 3ξ − 2ξ ) L ( −ξ 2 + ξ 3 )⎥⎦ 2
[2.29]
where θ = dw/dx and ξ = x/L. According to Novozhilov,24 the four strain components are given by equation [2.30]. du / dx ⎧ ⎫ ⎧ε x ⎫ ⎪ ⎪ + w cos u sin / r φ φ ( ) ⎪ε ⎪ ⎪ ⎪⎪ ⎪ H⎪ ⎪ −d 2 w / dx 2 ⎨ ⎬=⎨ ⎬ ⎪χx ⎪ ⎪ ⎪ dw sin φ ⎪⎩ χ H ⎪⎭ ⎪ ⎪ − dx r ⎪⎩ ⎪⎭
© Carl T. F. Ross, 2011
[2.30]
32
Pressure vessels
so that the ‘strain’ matrix [B] becomes: 0 0 ⎡ −1 / L ⎢ sin φ cos φ cos φ ⎢ (1 − ξ ) L (ξ − 2ξ 2 + ξ 3 ) (1 − 3ξ 2 + 2ξ 3 ) r r r ⎢ [ B] = ⎢ 2 0 − ( −6 + 12ξ ) / L − ( −4 + 6ξ ) L ⎢ φ sin sin φ ⎢ 0 − ( −6ξ − 6ξ 2 ) − ( −1 + 4ξ − 3ξ 2 ) ⎢⎣ r (rL) 1/ L 0 0 ⎤ ⎥ sin φ cos cos φ φ ⎥ L ( −ξ 2 + ξ 3 ) ξ ( 3ξ 2 − 2ξ 3 ) r r r ⎥ [2.31] 0 − ( −6 − 12ξ ) / L2 − ( −2 + 6ξ ) L ⎥ ⎥ sin φ sin φ ⎥ − ( −6ξ + 6ξ 2 ) − ( −2 − 3ξ 2 ) 0 rL r ⎥⎦ ( ) The elemental stiffness matrix [k] is given by
[ k ] = ∫ [ B]T [ D][ B] d ( A)
[2.32]
where d(A) = 2πr dr; 0 0 ⎤ v 1 0 0 ⎥ ⎥ 0 t 2 / 12 vt 2 / 12 ⎥ ⎥ 0 vt 2 / 12 t 2 / 12 ⎦
⎡1 ⎢ Et ⎢v [ D] = 1 − v2 ⎢0 ⎢ ⎣0
[2.33]
for an isotropic material; and t = (1 − ξ ) t1 + ξt 2
[2.34]
The elemental stiffness matrix in global coordinates [k0] is given by
[k 0 ] = [Ξ]T [k ][Ξ]
[2.35]
where ⎡c ⎢− s ⎢ ⎢0 [Ξ ] = ⎢ ⎢0 ⎢0 ⎢ ⎣0
s c 0 0 0 0
0 0 0 0 1 0 0 c 0 −s 0 0
0 0 0 s c 0
0⎤ 0⎥ ⎥ 0⎥ ⎥ 0⎥ 0⎥ ⎥ 1⎦
and c = cos ϕ and s = sin ϕ.
© Carl T. F. Ross, 2011
[2.36]
Axisymmetric deformation of pressure vessels
33
Integration of equation [2.32] can be carried out using four Gauss points per element, although the author has found that by using three Gauss points no loss in precision was detected for a number of examples and, in certain instances, locking25 was avoided. The elemental load vector in global coordinates is given by: {qi } = [ Ξ]
T
∫ [N ]
T
⎣ 0 P ⎦ d ( A)
[2.37]
where P = pressure normal to surface. In the computer program, P is allowed to vary linearly between adjacent nodal circles, as follows: P = Pi (1 − ξ ) + Pj (ξ )
[2.38]
where Pi = lateral pressure at node i and Pj = lateral pressure at node j.
2.4.1 Comparisons with FEA and Woinowsky-Kreiger The comparison between the finite element solution of this section and the analytical solution of Timoshenko and Woinowsky-Kreiger26 for the thinwalled spherical shell cap of Fig. 2.12, is shown in Figs 2.13 and 2.14. To obtain the results of Figs 2.13 and 2.14, the shell was divided into 10 equal meridional length elements, where node 1 was at the nose of the shell and node 11 was at the encastré end (or wall). Figure 2.13 shows the distribution of hoop forces/unit length and Fig. 2.14 shows the distribution of meridional forces/unit length. A computer program in BASIC for analysing thin-walled axisymmetric shells, using the conical shell element, is published on the net, and other versions of this program are available directly from the author.
1 lbf in–2
Nose Wall
35° 35°
2.12 Spherical shell cap.
© Carl T. F. Ross, 2011
34
Pressure vessels 0 Internal Thin-walled External conical shell Timoshenko and Woinowsky-Kreiger (average)
lbf in–1
–10 –20 –30 –40 –50 11
10
Wall
9 Nodes
8
7
6
5
4
3
2
2.13 Hoop forces/unit length for cap.
40
lbf in–1
30 20 10 0
11
–10 Wall
10
9
8
7
6
5
4
3
2
Nodes
2.14 Meridional bending moment/unit length for cap.
2.4.2 Axisymmetric varying meridional curvature (AVMC) element The conical shell element described in Section 2.3 can also be used for domes, cylinders and cylinder/cone/dome combinations, but a more suitable element for an axisymmetric varying meridional curvature element (AVMC) was presented by Cook,27 as shown in Fig. 2.15. In this instance,
{ε }T = ⎣⎢ε s εφ χ s χφ ⎦⎥
© Carl T. F. Ross, 2011
[2.39]
Axisymmetric deformation of pressure vessels x=1
35
b2
s
w
bc b1
x = –1
R2
R1 Axis
2.15 Varying meridional curvature element.
where
ε s = ∂u / ∂s − w∂β / ∂s εφ = (1 / r ) (u sin β + w cos β )
[2.40]
χ s = −∂ 2 w / ∂s 2 − u∂ 2 β / ∂s 2 − ∂u / ∂s.∂β / ∂s
χφ = − (1 / r ) [1 / r∂ 2 w / ∂φ 2 + ( ∂w / ∂s + u∂β / ∂s ) sin β ]
Cook assumed that β was given by
β = α 0 + α 1 s + α 2 s2
[2.41]
and he determined these constants with the aid of the boundary conditions: at s = 0, β = β1, and at s = L, β = β2, where L is the arc length; and equation [2.42]:
∫
L
0
L
sin ( β − βc ) ds ≈ ∫ ( β − βc ) ds = 0
[2.42]
0
where it is assumed that the slope, which is small, is tan (β = βc) = dy/dx. Substituting the above boundary conditions into equation [2.41] results in the following values for the constants:
α 0 = β1 α 1 = (6βc − 4β1 + 2β2 ) α 2 =
( 3β1 + 3β2 − 6βc ) L2
[2.43]
If the arc length L is not known, Cook has shown that it can be approximated by equation [2.44]: L≈l+
l ⎡ 4 −1⎤ ⎧(β1 − βc ) ⎫ ⎢⎣(β1 − βc ) (β 2 − βc )⎥⎦ ⎢ ⎬ ⎥⎨ 60 ⎣ −1 4 ⎦ ⎩(β 2 − βc )⎭
[2.44]
where l = chord length. The radius r can be determined from equation [2.45]: s
r = R1 + ∫ sin βds
[2.45]
0
© Carl T. F. Ross, 2011
36
Pressure vessels
where
β ≈ β c + dy / dx
[2.46]
Equation [2.45] can be integrated explicitly for a hemi-ellipsoidal dome, and for an axisymmetric shell of arbitrary meridional shape it can be integrated numerically. In the meridional direction, it is convenient to assume that u = α3 + α4s. Cook27 shows that the rotational displacement θi is given by:
θ i = [ ∂w / ∂s + u.∂β / ∂s ]i
[2.47]
but if this is approximated by:
θ i = ∂wi / ∂s
[2.48]
then [k] can be calculated from equation [2.32] and [k0] from equation [2.35] where: ⎡ c1 s1 0 ⎤ ⎢− s c 0 ⎥ 03 ⎢ 1 1 ⎥ ⎢ 0 0 1 ⎥ [Ξ ] = ⎢ ⎥ c s 0 2 2 ⎢ ⎥ ⎢ 03 − s1 c2 0 ⎥ ⎢ ⎥ 0 0 1⎥⎦ ⎢⎣
[2.49]
and c1 = cos β1, s1 = sin β1; c2 = cos β2, s2 = sin β2 An alternative method of determining the arc length of the element, together with its associated parameters, is to introduce a mid-side node. This mid-side node need not be used for determining additional nodal displacements, but simply the shape of the element.
2.4.3 Axisymmetric constant meridional curvature (ACMC) element For the special case of the thin-walled axisymmetric shell of constant meridional curvature, ACMC28 (Figs 2.16 and 2.17), {ε} is given by ⎧ε s ⎫ ⎪ε ⎪ ⎪ φ⎪ {ε } = ⎨ ⎬ = [ B] {Ui } ⎪ χs ⎪ ⎪⎩ χφ ⎪⎭
© Carl T. F. Ross, 2011
[2.50]
Axisymmetric deformation of pressure vessels
Nodal circle 1
f
b1
r
s
w
v b
x
u
b2
Nodal circle 2
2.16 Axisymmetric shell element.
x=1 q2 2
w
b
s,x r
b2
u
x = –1 q1
b1
1 A
Ri
Rc1
r
Rc2
y
a a
x y
2.17 Longitudinal section through element.
© Carl T. F. Ross, 2011
37
38
Pressure vessels
where ⎧ ∂u + w ⎫ ⎪ ∂s R1 ⎪ ⎪ ⎪ ⎪ 1 [u sin β + w cos β ] ⎪ ⎪⎪ r ⎪⎪ {ε } = ⎨ 2 1 ∂w 1 ∂u ⎬ ⎪− ⎪ + ⎪ α 2 R12 ∂ξ 2 α R12 ∂ξ ⎪ ⎪ ⎪ ⎪ − 1 ⎛ ∂w − α u⎞ sin β ⎪ ⎜ ⎟ ⎠ ⎪⎩ α 1 Ri ⎝ ∂ξ ⎪⎭
[2.51]
The assumed displacement functions are:
(1 − ξ ) u1 + u2 2 2 (ξ 3 − 3ξ + 2 ) w + (1 + ξ ) (1 − ξ )2 α R θ w= 1 1 1 4 4 ( −ξ 3 + 3ξ + 2 ) w − (1 − ξ ) (1 + ξ )2 α R θ + 2 11 1 4 4 u=
(1 − ξ )
[2.52]
or ⎧u ⎫ ⎨ ⎬ = [ N ] {U1 } ⎩w ⎭ where [N] = a matrix of shape functions; and ⎧u1 ⎫ ⎪w ⎪ ⎪ 1⎪ ⎪θ1 ⎪ {Ui } = ⎨ ⎬ ⎪u2 ⎪ ⎪w2 ⎪ ⎪ ⎪ ⎩θ 2 ⎭ Hence, by substituting equations [2.52] into [2.51], [B] can be obtained, as shown in Table 2.2. Now,
[k ] = ∫ [ B]T [ D][ B] 2πr ds
and
[k 0 ] = [Ξ]T [k ][Ξ]
[2.53]
where [D] is obtained from equation [2.33] and [Ξ] is obtained from equation [2.36] and β=A−ψ r = Ri sin(90 − A + ψ) − Y
© Carl T. F. Ross, 2011
© Carl T. F. Ross, 2011
2rR1
2rR1 4r
− ( −1 − 2ξ + 3ξ 2 ) sin β
−3 (ξ 2 − 1) sin β
(1 − ξ ) sin β
4r − ( 3ξ − 1) 2αR1
4r
2r
(1 + ξ ) (1 − ξ )2αR1 cos β
−3ξ 2α 2R1
(ξ 3 − 3ξ + 2) cos β
(1 − ξ ) sin β
4
(1 + ξ ) (1 − ξ )2α
−1 2αR1
ξ 3 − 3ξ + 2 4R1
−1 2αR1
2rR1
(1 + ξ ) sin β
1 2αR12
2r
(1 + ξ ) sin β
1 2αR1
Table 2.2 [B] for axisymmetric constant meridional curvature (ACMC) element
4αrR1
−3 ( −ξ 2 + 1) sin β
3ξ 2α 2R12
4r
( −ξ 3 + 3ξ + 2) cos β
−ξ 3 + 3ξ + 2 4R1
4r
(1 − 2ξ − 3ξ2 ) sin β
− ( 3ξ + 1) 2αR1
2
− (1 − ξ ) (1 + ξ ) αR1 cos β 4r
2
− (1 − ξ ) (1 + ξ ) α 4
40
Pressure vessels
Y = Ri sin (90 − A + α) − Rc1 s = R1ψ ds = R1dψ but
ξ = ψ / α, therefore ds = R1α dξ. The assumed lateral pressure distribution is given by ps =
(1 − ξ ) 2
p1 +
(1 + ξ ) 2
p2
where, p1 = lateral pressure at ξ = −1 p2 = lateral pressure at ξ = +1 The vector of nodal forces is obtained from:
{q} = [Ξ]T ∫ [ N ]T ps 2πr ds
[2.54]
In a computer program for analysing axisymmetric elements of constant meridional curvature,28 four Gauss points were used in the meridian of each element. The use of three Gauss points is likely to give equally good results, with the added advantage of computational economy and the avoidance of locking25 for certain cases. Comparison of the computer solution with that of Timoshenko and Woinowsky-Kreiger26 for the spherical cap of Fig. 2.12 realised near-identical results with those of the conical element in Section 2.3 using a similar mathematical model.
2.4.4 Comparisons between the ACMC and AVMC elements To test the ACMC and the AVMC elements, experimental and theoretical investigations were carried out on two of the thin-walled hemi-ellipsoidal domes referred to in Section 3.4.2 and shown in Fig. 2.18. These domes were constructed in solid urethane plastic (SUP), in the manner described in Section 3.4.2, and their aspect ratios (AR) were 1.5 and 3, where: Aspect ratio =
Dome height Base radius
© Carl T. F. Ross, 2011
Axisymmetric deformation of pressure vessels
41
2.18 Hemi-ellipsoidal domes of aspect ratio: (left) 3; and (right) 1.5.
The domes were firmly clamped at their bases and they were subjected to external hydrostatic pressure in the tank shown in Fig. 2.19. Before carrying out the experimental investigation, the conforming properties of the two elements were examined, using these two domes as a standard. For both domes, the number of elements taken to model the vessel was varied from two to 16, in increments of two and, in Figs 2.20 and 2.21, plots are made of the variation of stress with mesh refinement. The stresses were calculated at the mid-meridian of each dome; from Figs 2.20 and 2.21, it can be seen that both elements appear to conform very rapidly. The figures also show better agreement for the hoop stresses than for the meridional stresses, but the latter were small in magnitude in any case. From these results, it can be seen that the stresses varied very little when 10 or more elements were adopted. Therefore 14 elements were used to model the domes for the comparisons between the theoretical and experimental results. For each dome, 10 strain gauges were placed at five positions on each dome, as shown in Figs 2.22 and 2.23. It was necessary to use one pair of strain gauges at each position, where the strain gauges were placed at 90° to each other and in the directions of the principal strains, because the problem was a two-dimensional one. Tables 2.3 and 2.4 show mean recorded strains at various pressures on the internal surfaces of these vessels. The slight nonlinearity that appeared to occur with these experimental results was attributed to the fact that the strain gauges used were intended for metals and not for plastics. Figures 2.24–2.27 show comparisons between the theoretical and the experimental stress distributions for these two domes when they were
© Carl T. F. Ross, 2011
Securing annulus
8 M12 securing bolts
Rubber annulus seal
Relief valve Dome aspect ratio 3.0
Pressure gauge Water
Pressure inlet
2.19 Pressure tank with test shell in position.
Internal stress (compressive) (MN m–2)
11
10
9
CMC hoop VMC hoop CMC meridional VMC meridional
8
7
6
0
2
4
8 10 12 6 Number of elements
14
16
2.20 Variation in stress with mesh refinement aspect ratio 1.5.
© Carl T. F. Ross, 2011
Axisymmetric deformation of pressure vessels
43
Internal stress (compressive) (MN m–2)
4
3
2
1 CMC hoop VMC hoop CMC meridional VMC meridional 0
2
4
10 12 6 8 Number of elements
14
16
150 111
69
35 11
2.21 Variation in stress with mesh refinement aspect ratio 3.0.
1 2 3
4 5
6A 7 8
9
B C
Circumferential gauges
D
10 E Meridional gauges
2.22 Strain gauge positions and directions on 1.5 aspect ratio dome (mm).
subjected to an external pressure of 45 lbf in−2 for the dome of 1.5 AR and to a pressure of 15 lbf in−2 for the dome of 3.0 AR. From these figures, it can be seen that both the ACMC and the AVMC elements tend to predict stresses of a higher magnitude than those obtained experimentally, although all the lines appear to follow similar patterns. The figures also show good
© Carl T. F. Ross, 2011
Pressure vessels 95 65 50 35
44
8 A 9 2 B 10 3 C 11 4 D
280
220
150
1
5
12 E
6
13 F
7
14
Circumferential gauges
G Meridional gauges
2.23 Strain gauge positions and directions on 3.0 aspect ratio dome.
Table 2.3 Experimental strains along dome meridian, aspect ratio 1.5 Mean recorded strain με at pressures of (lbf in−2) [MPa]: Gauge
10 [0.069]
20 [0.138]
30 [0.207]
40 [0.276]
45 [0.31]
1 2 3 4 5 6 7 8 9 10
−314 −534 −472 −416 −104 −96 −162 −270 −256 −140
−588 −1054 −914 −822 −214 −290 −296 −568 −548 −264
−850 −1600 −1360 −1249 −311 −603 −426 −878 −818 −368
−1105 −2199 −1838 −1677 −397 −950 −529 −1171 −1089 −457
−1198 −2465 −2067 −1910 −437 −1138 −570 −1315 −1215 −522
agreement between the predictions of the ACMC element and those of the AVMC element. It is believed that the main reason why the experimental stresses lay on lines ‘lower’ than the lines for the theoretical stresses was that the strain gauges used were meant for metals and not for plastics. The domes were constructed very precisely, as can be seen by the out-ofcircularity plots shown in Figs 2.28 and 2.29, where the meridians on which the gauges lay are also indicated.
© Carl T. F. Ross, 2011
Axisymmetric deformation of pressure vessels
45
Table 2.4 Experimental strains along dome meridian, aspect ratio 3.0 Mean recorded strain με at pressures of (lbf in−2) [MPa]: Gauge
7.5 [0.0517]
10 [0.069]
12.5 [0.0862]
15 [0.103]
1 2 3 4 5 6 7 8 9 10 11 12 13 14
−471 −489 −479 −492 −443 −345 −197 −12 −49 −77 −67 −56 −69 −71
−603 −633 −621 −653 −589 −457 −247 −10 −61 −91 −81 −65 −84 −91
−771 −803 −793 −831 −733 −573 −307 −6 −75 −107 −102 −97 −119 −129
−933 −978 −979 −1018 −876 −681 −352 −7 −96 −136 −123 −115 −140 −155
8
Stress (compressive) (MN m–2)
7
6
5
4
3
2
1
0
ACMC AVMC Experimental
100 50 Axial coordinate (mm)
150
2.24 Internal meridional stress distribution aspect ratio 1.5. Dashed curve shows another experimental possibility.
© Carl T. F. Ross, 2011
Pressure vessels
Stress (compressive) (MN m–2)
15
10
5
ACMC AVMC Experimental
0
50 100 Axial coordinate (mm)
150
2.25 Internal circumferential stress distribution aspect ratio 1.5.
3 Stress (compressive) (MN m–2)
46
2
1 ACMC AVMC Experimental 0
100 200 Axial coordinate (mm)
300
2.26 Internal meridional stress distribution aspect ratio 3.0.
© Carl T. F. Ross, 2011
Stress (compressive) (MN m–2)
Axisymmetric deformation of pressure vessels
47
5 4 3 2 ACMC AVMC Experimental
1
300
200 100 Axial coordinate (mm)
0
2.27 Internal circumferential stress distribution aspect ratio 3.0.
Aspect ratio 3.0
73 mm
Specimen 3.0 Mag Filter ×50 N Talyrond 10″Arm
Ra
nk
Tay lo
s D ice r H o bson Le
t er Location of meridional gauges
1 Division = 0.002″
2.28 Out-of-circularity plot for 3.0 aspect ratio dome.
2.4.5 Tapered cylindrical shell element Another useful element in this family of thin-walled axisymmetric elements is the tapered thin-walled circular cylindrical shell element of Fig. 2.30.29 Its usefulness is that, when used to model mathematically circular cylindrical shells, it is computationally more efficient than the elements described earlier. A brief description of the derivation of this element is given below. The cylinder is assumed to have a linear variation in thickness, as shown in Fig. 2.30, so that:
© Carl T. F. Ross, 2011
48
Pressure vessels
Aspect ratio 1.5
77 mm
Specimen 1.5 Mag Filter ×100 N Talyrond 5″Arm
Ra
nk
Tay lo
D st e ic e r H o bson Le
r
Location of meridional gauges
1 Division = 0.001″
2.29 Out-of-circularity plot for 1.5 aspect ratio dome.
z,w
w1
w2
q1
q2 a 2
1
x,u
l t1
t2
2.30 Tapered thin-walled circular cylinder.
t=
(t2 − t1 ) l
x + t1 = mx + c
[2.55]
Suitable displacement distributions for axial and radial displacements are: u = α1 + α 2 x
w = α 3 + α 4 x + α 5 x2 + α 6 x3
[2.56]
In terms of nodal displacements, these become: u = u1 + (u2 − u1 )( x / l )
© Carl T. F. Ross, 2011
[2.57]
Axisymmetric deformation of pressure vessels
49
−3w 2θ 3w θ 2w θ 2w θ w = w1 + θ1 x + ⎛⎜ 2 1 − 1 + 2 2 − 2 ⎞⎟ x 2 + ⎛⎜ 3 1 + 21 − 3 2 + 22 ⎞⎟ x 3 ⎝ l ⎝ l l ⎠ l l l ⎠ l l
[2.58]
and
For a circular cylinder of varying thickness, the bending strain energy is given by: Ub =
{
}
1 2 2 D ( χ x + χ y ) − 2 (1 − v) ( χ x χ y − χ xy ) dx dy 2 ∫∫
[2.59]
and the membrane strain energy by: Um =
1 (σ xε x + σ yε y + τ xyγ xy ) t dx dy 2 ∫∫
[2.60]
As σx and σy are principal stresses,
τ xy = γ xy = χ xy = 0 Furthermore, as deflections are small,
χy ≈ 0 so that 2
Ub =
1 ⎛ d 2w ⎞ D ⎜ 2 ⎟ dx dy ∫∫ ⎝ dx ⎠ 2
[2.61]
where, D = Et3/[12(1 − ν2)] σx = longitudinal stress εx = longitudinal strain σy = hoop stress εy = hoop strain τxy = shear stress in the plane of the cylinder’s shell γxy = shear strain in the plane of the cylinder’s shell χx = curvature in the longitudinal direction χy = curvature in the hoop direction χxy = twist in the plane of the cylinder’s shell For a two-dimensional system of stress,
σx =
Et
(1 − v 2 )
( ε x + vε y )
and
σy =
© Carl T. F. Ross, 2011
Et
(1 − v 2 )
( ε y + vε x )
50
Pressure vessels
but
εx =
du dx
and
εy =
w a
Therefore Um =
⎡⎛ du vw ⎞ 2 E ⎛ du w ⎞ ⎤ dx dy t + ⎟⎥ ⎟⎠ − 2 (1 − v ) ⎜⎝ 2 ∫∫ ⎢⎜ ⎝ 2 (1 − v ) dx a ⎠ ⎦ ⎣ dx a
[2.62]
Substituting the appropriate displacement configurations into the strain energy expression, the stiffness matrix of the element is given by
[ k ] = [ kA ] + [ k1 ] + [ k2 ] + + [ k8 ]
[2.63]
where [kA] = elemental stiffness matrix for the cylinder under axial load only. It can be seen that these matrices are of order 2 × 2, 4 × 4 and 6 × 6, and the displacement vectors corresponding to them are given by
{u1 u2 } {w1 θ1 w2 θ 2 } {u1 w1 θ1 u2 w2 θ 2 } Thus to obtain the stiffness matrix of the tapered cylinder, it will be necessary to construct a 6 × 6 matrix and add together the components of each of the following matrices in their appropriate positions:
[ kA ] =
2 πaE ⎛ ⎡ m / 2 − m / 2 ⎤ ⎡ c / l + (1 − v2 ) ⎜⎝ ⎢⎣− m / 2 m / 2 ⎥⎦ ⎢⎣−c / l
− c / l ⎤⎞ c / l ⎥⎦⎟⎠
[2.64]
sym. ⎤ ⎡ 42 / 5 2 ⎢ ⎥ πaEm ⎢ 12l / 5 4l / 5 ⎥ [ k1 ] = ⎥ 12 (1 − v 2 ) ⎢ −42 / 5 −12l / 5 42 / 5 ⎢ ⎥ 2 2 ⎣ 30l / 5 8l / 5 −30l / 5 22l / 5⎦
[2.65]
sym. ⎤ ⎡ 48 / 5l ⎢ ⎥ πaEm c ⎢ 14 / 5 16l / 15 ⎥ [ k2 ] = 2 ⎥ 4 (1 − v ) ⎢ −48 / 5l −14 / 5 48 / 5l ⎢ ⎥ ⎣ 34 / 5 26l / 15 −34 / 5 76l / 15⎦
[2.66]
3
2
© Carl T. F. Ross, 2011
Axisymmetric deformation of pressure vessels ⎡ 12 / l 2 ⎢ πaEmc 2 ⎢ 4 / l [ k3 ] = 4 (1 − v 2 ) ⎢ −12 / l 2 ⎢ ⎣ 8/l ⎡ 24 / l 3 ⎢ πaEc 3 ⎢ 12 / l 2 [ k4 ] = 12 (1 − v 2 ) ⎢ −24 / l 3 ⎢ 2 ⎣ 12 / l
2 −4 / l 12 / l 2 2 −8 / l
8l −12 / l 2 4/l
sym.⎤ ⎥ ⎥ ⎥ ⎥ 6 ⎦
24 / l 3 −12 / l 2
sym.⎤ ⎥ ⎥ ⎥ ⎥ 8/l ⎦
51
[2.67]
[2.68]
sym.⎤ ⎡ 0 ⎢ −3l 2 / 20 ⎥ 0 ⎢ ⎥ 3 0 0 ⎥ 2Emvπ ⎢ −l / 30 [ k5 ] = ⎢ ⎥ 2 3 2 3l / 20 l / 30 0 (1 − v ) l ⎢ 0 ⎥ ⎢ −7l 2 / 20 ⎥ 0 0 7l 2 / 20 0 ⎢ 3 ⎥ 3 0 0 −l / 20 0 0 ⎦ ⎣ l / 20 [2.69] sym.⎤ ⎡ 0 ⎢ −1 / 2 ⎥ 0 ⎢ ⎥ 0 ⎥ 2Ecvπ ⎢ −l / 12 0 [ k6 ] = ⎥ 2 ⎢ (1 − v ) ⎢ 0 1 / 2 l / 12 0 ⎥ ⎢ −1 / 2 ⎥ 0 0 1/ 2 0 ⎢ ⎥ 0 ⎦ −l / 12 0 0 0 ⎣ l / 12
[2.70]
sym. ⎤ ⎡ 6l 2 / 35 ⎢ l 3 / 30 ⎥ 4 l / 140 E πm ⎢ ⎥ [ k7 ] = ⎥ (1 − v2 ) a ⎢9l 2 / 70 l 3 / 30 4l 2 / 7 ⎢ 3 ⎥ 4 3 4 ⎣ −l / 35 −l / 140 −l / 14 l / 84 ⎦
[2.71]
sym. ⎤ ⎡ 26l / 35 2 3 ⎢ ⎥ 2l / 105 Ecπ ⎢ 22l / 210 ⎥ [ k8 ] = 2 2 ⎥ (1 − v ) a ⎢ 9l / 35 13l / 210 26l / 35 ⎢ ⎥ 2 3 2 3 ⎣ −13l / 210 −3l / 210 −22l / 210 2l / 105⎦ [2.72]
© Carl T. F. Ross, 2011
52
Pressure vessels
For a cylinder of uniform thickness, m = 0 and c = t = thickness. Hence, the elemental stiffness matrix is given by: ⎡ 2a / l ⎢ ⎢ −v ⎢ ⎢ −vl ⎢ πEt ⎢ 6 [k ] = (1 − v2 ) ⎢⎢ −2a / l ⎢ ⎢ −v ⎢ ⎢ vl ⎢ ⎣ 6
⎛ 24az + 26l ⎞ ⎜⎝ 3 ⎟ 35a ⎠ l ⎛ 12az 11l 2 ⎞ ⎜⎝ l 2 + 105a ⎟⎠
⎛ 8az 2l 3 ⎞ ⎜⎝ l + 105a ⎟⎠
v
vl / 6
⎛ −24az + 9l ⎞ ⎜⎝ 3 ⎟ l 35a ⎠
⎛ −12az 13l 2 ⎞ ⎜⎝ l 2 + 210a ⎟⎠
⎛ 12az 13l 2 ⎞ ⎜⎝ l 2 − 210a ⎟⎠
⎛ 4az 3l 3 ⎞ − ⎝⎜ l 210a ⎟⎠
2a / l ⎛ 24az + 26l ⎞ ⎜⎝ 3 ⎟ l 35a ⎠ ⎛ −12az 22l 2 ⎞ ⎜⎝ l 2 − 210a ⎟⎠
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎛ 8az 2l 3 ⎞ ⎥ ⎜⎝ l + 105a ⎟⎠ ⎥ ⎦ sym.
v − vl 6
[2.73]
where z = t2/12. Using a solution based on the above element, comparison is made in Fig. 2.31 with the semi-numerical solution30 for a vertical tank of varying wall thickness subjected to a linearly varying lateral pressure. The tank was assumed to be clamped at both ends. From Fig. 2.31 it can be seen that there is good agreement between the two sets of results.
2.5
Thick-walled cones and domes
The thin-walled solution described in the earlier sections in this chapter tends to break down if the wall thickness–radius ratio exceeds 1 : 30, and for these cases it is necessary to use a thick-shell theory. For both thick-walled cones and domes, the theory described here is that of Ahmad et al.31 For the thick-walled cone, the element is shown in Fig. 2.32, where it can be seen that it is described by two end nodal circles, and that its thickness is assumed to vary linearly with length. As there are two nodes, and the element is conical, the matrix shape functions can be assumed to be:
[ N ] = [ N1 N 2 ]
[2.74]
© Carl T. F. Ross, 2011
Axisymmetric deformation of pressure vessels
53
Radial deflection (in)
0.07 0.06 0.05 0.04 0.03
Numerical solution
0.02
Analytical solution Top
0.01 Bottom 20"
40"
0.8"
Shell
0.6" 120"
1"
6 p.s.i.
Lateral pressure
36 p.s.i.
0.4" 120"
2.31 Radial deflection of a longitudinal generator.
r ,w0 w0
Node 1 u10
t1 q1
Node 2
f t2
Axis
x
x0 , u0
2.32 Thick conical shell.
where, N1 =
1 (1 − ξ ) 2
N2 =
1 (1 + ξ ) 2
ξ, η = local coordinates (they can be curvilinear for some elements)
© Carl T. F. Ross, 2011
54
Pressure vessels
The radius r at ξ and η is given by r = N 1r1 + N 2 r2 + 0.5t1 N 1η cos φ + 0.5t2 N 2 η cos φ
[2.75]
Similarly, x 0 = N1 x10 + N 2 x20 + 0.5t1 N1η sin φ + 0.5t 2 N 2η sin φ
[2.76]
There are three nodal displacements at each node, making a total of six nodal displacements {U 0i } for an element, as follows: ⎧w10 ⎫ ⎪ 0⎪ ⎪u1 ⎪ ⎪θ ⎪ {Ui0 } = ⎪⎨ 10 ⎪⎬ ⎪w2 ⎪ ⎪u 0 ⎪ ⎪ 2⎪ ⎪⎩θ 2 ⎪⎭ The displacements w0 and u0 are given by: ⎧w 0 ⎫ ⎡wi0 ⎤ ti ⎧ − sin φi ⎫ = N ⎨ 0 ⎬ ∑ i ⎢ 0 ⎥ + ∑ Niηi ⎨ ⎬θi 2 ⎩ cos φi ⎭ ⎩u ⎭ ⎣ui ⎦
[2.77]
The derivatives w and u are given by: ⎡ ∂u 0 ⎢ ∂x 0 ⎢ 0 ⎢ ∂u ⎢⎣ ∂r
⎡ ∂u 0 ∂w 0 ⎤ ⎢ ∂ξ ∂x 0 ⎥ −1 ⎢ = J ⎥ [ ]⎢ 0 ∂w 0 ⎥ ∂u ⎢ ⎥ ∂r ⎦ ⎣ ∂η
∂w 0 ⎤ ∂ξ ⎥ ⎥ ∂w 0 ⎥ ⎥ ∂η ⎦
[2.78]
where the Jacobian [J] is defined as: ⎡ ∂x 0 ⎢ ∂ξ [J] = ⎢ 0 ⎢ ∂x ⎢ ⎣ ∂η
∂r ⎤ ⎡ ∂N 1 ∂N 2 ⎤ 0 ∂ξ ⎥ ⎢ ⎡ x1 r1 ⎤ ⎥ = ∂η ∂ξ ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎥ ∂r ⎣ x2 r2 ⎦ ⎢ ⎥ 0 0 ⎦ ⎥ ⎣ ∂η ⎦ ⎡0.5t η ∂N1 0.5t η ∂N 2 ⎤ 1 2 ⎡sin φ cos φ ⎤ ∂ξ ∂ξ ⎥ ⎢ +⎢ ⎢ ⎥ ⎣sin φ cos φ ⎥⎦ ⎢⎣ 0.5t1 N1 0.5t 2 N 2 ⎥⎦
or ⎡ x10 0 ⎣ x2
[ J ] = [ TT ] ⎢
r1 ⎤ ⎡sin φ cos φ ⎤ + [ TT 2 ] ⎢ ⎥ ⎥ r2 ⎦ ⎣sin φ cos φ ⎦
© Carl T. F. Ross, 2011
[2.79]
Axisymmetric deformation of pressure vessels
55
where, ⎡− 1 [ TT ] = ⎢ 2 ⎢ ⎣ 0
1⎤ 2⎥ ⎥ 0⎦
and ⎡ −t1η ⎢ [ TT2 ] = ⎢ 4 ⎢ t1 (1 − ξ ) ⎢⎣ 4
t 2η ⎤ 4 ⎥ ⎥ t 2 (1 + ξ ) ⎥ 4 ⎥⎦
and x01 r1 x02 r2 are nodal coordinates in global axes. Hence, ⎡ ∂u 0 ⎢ ∂ξ ⎢ ⎢ ∂u 0 ⎢ ⎣ ∂η
∂w 0 ⎤ ∂ξ ⎥ u0 ⎥ = [ TT ] ⎡⎢ 1 0 ∂w 0 ⎥ ⎣u2 ⎥ ∂η ⎦
w10 ⎤ ⎡ cos φθ1 + [ TT22 ] ⎢ 0⎥ w2 ⎦ ⎣cos φθ 2
− sin φθ1 ⎤ − sin φθ 2 ⎥⎦
The relationship between local and global derivatives is given by: ⎡ ∂u ⎢ ∂x ⎢ ⎢ ∂u ⎢⎣ ∂r 1
∂w ⎤ ⎡ ∂u 0 ⎢ 0 ∂x ⎥ T ∂x ⎥ = [ DC ] ⎢ 0 ∂w ⎥ ⎢ ∂u ⎢⎣ ∂r ∂r 1 ⎥⎦
∂w 0 ⎤ ∂x 0 ⎥ ⎥ [ DC ] ∂w 0 ⎥ ∂r ⎥⎦
[2.80]
−∂r ⎤ ∂ξ ⎥ ⎥ ∂x 0 ⎥ ⎥ ∂ξ ⎦
[2.81]
where,
[ DC ] =
⎡ ∂x 0 ⎢ ∂ξ 1 ⎢ 2 ∂x 0 ∂r 2 ⎢ ∂r ⎢ + ∂ξ ∂ξ ⎣ ∂ξ
= a matrix of directional cosines; u, w = local displacements ⎫ ⎬ see Fig. 2.33 x, r 1 = local axes ⎭
© Carl T. F. Ross, 2011
56
Pressure vessels w0, r
r1 , w0 x,u
Line η = constant
x0 , u0
2.33 Local and global axes.
⎡ ∂u ⎢ ∂x ⎢ ⎢ ∂u ⎢⎣ ∂r 1
∂w ⎤ ∂x ⎥ ⎥ ∂w ⎥ ∂r 1 ⎥⎦ A11 ( DC 12 u10 + DC 22 w10 ) ⎤ ⎡ A11 ( DC 11u10 + DC 21w10 ) ⎢ + A DC u0 + DC w0 + A DC u0 + DC w0 ⎥ 12 ( 11 2 21 2 ) 12 ( 12 2 22 2 ) ⎥ =⎢ 0 0 0 ⎢ A21 ( DC 11u1 + DC 21w1 ) A21 ( DC 12 u1 + DC 22 w10 ) ⎥ ⎢ 0 0 0 0 ⎥ ⎣ + A22 ( DC 11u2 + DC 21w2 ) + A22 ( DC 12 u2 + DC 22 w2 ) ⎦
C11θ 1 ( DC 12 cos φ − DC 22 sin φ ) ⎤ ⎡ C11θ 1 ( DC 11 cos φ − DC 21 sin φ ) ⎢ +C θ ( DC cos φ − DC sin φ ) +C θ ( DC cos φ − DC sin φ )⎥ 12 2 11 21 12 2 12 22 ⎥ +⎢ C21θ 1 ( DC 12 cos φ − DC 22 sin φ ) ⎥ ⎢ C21θ 1 ( DC 11 cos φ − DC 21 sin φ ) ⎢ ⎥ ⎣ +C22θ 2 ( DC 11 cos φ − DC 21 sin φ ) +C22θ 2 ( DC 12 cos φ − DC 22 sin φ )⎦
where,
[ A] = [ DC ]T [ J −1 ][ TT ] [C ] = [ DC ]T [ J −1 ][ TT 2 ] The matrix of strains {ε1} is given by: ⎧ ∂u ⎫ ⎪ ∂x ⎪ ⎪ ⎪ ⎪ ⎪ w0 0 1 {ε } = ⎨ ⎬ = [ B ′ ] {ui } r ⎪ ⎪ ⎪ ∂u ∂w ⎪ ⎪ ∂r 1 + ∂x ⎪ ⎩ ⎭ where [B′] is given in Table 2.5.
© Carl T. F. Ross, 2011
[2.82]
© Carl T. F. Ross, 2011
⎡ A11DC21 ⎢ ⎢ ⎢ N1 [B ′ ] = ⎢⎢ r ⎢ ⎢ A DC ⎢ 21 21 ⎢⎣ + A11DC22
w 10 A12DC21 N2 r
C11 (DC11 cos φ − DC21 sin φ )
−0.5t1ηN1 sin φ r
A11DC11
0
A21DC11 + A11DC12
C 21 (DC11 cos φ − DC21 sin φ ) A22DC21 +C11 (DC12 cos φ − DC22 sin φ ) + A12DC22
w 20
u10
θ1
Table 2.5 [B′] matrix for thick conical shell
A22DC11 + A12DC12
0
A12DC11
u20 C12 (DC11 cos φ − DC21 sin φ ) ⎤ ⎥ ⎥ ⎥ −0.5t 2ηN 2 sin φ ⎥ r ⎥ ⎥ C 22 (DC11 cos φ − DC21 sin φ ) ⎥⎥ +C12 (DC12 cos φ − DC22 sin φ )⎦⎥
θ2
58
Pressure vessels
The elemental stiffness matrix in global coordinates is obtained numerically from equation [2.83]:
[k 0 ] = ∫−1 ∫−1 [ B′ ]T [ D][ B′ ] 2πr (det dξ dη) 1
1
[2.83]
where, 0 ⎡1 v ⎤ E ⎢ ⎥ 0 v 1 [ D] = (1 − v2 ) ⎢⎢0 0 (1 − v) / 2k ⎥⎥ ⎣ ⎦ and k is a factor to account for the shear strain energy (taken as 1.2).31 The external load matrix for lateral pressure in global coordinates is obtained by equating work done, as follows:
{Ui0 }T { pi0 } = − ∫ {ux ,r }
T
0
= − {U
{ p( ) } d ( A) = ∫ {U } [ N ]{ p( ) } d ( A) 0 T i
x 0 ,r
x 0 ,r
} ∫ [ N ][ DC ]T { p1 } d ( A)
0 T i
or
{ p10 } = − ∫ [ N ]T [ DC ]T { p1 } d ( A)
[2.84]
where, ⎡ N1 ⎣0
[N ] = ⎢
0 N1
−0.5t1 N1η sin φ1 0.5t1 N1η cos φ1
N2 0
0 N2
−0.5t1 N1η sin φ1 ⎤ 0.5t 2 N 2η cos φ2 ⎥⎦
and p1 is the pressure perpendicular to either the outer surface or the inner surface.
2.5.1 Parabolic element A more suitable element for thick-walled cones and domes is that shown in Fig. 2.34. The element is of parabolic shape and of parabolic taper, and it is described by three nodal circles. As the element has three nodes and
© Carl T. F. Ross, 2011
Axisymmetric deformation of pressure vessels r ,w0
59
wi0
1
ui0 i=2
fi 3
x0 , u0
2.34 Three-node parabolic element.
is of parabolic shape, it is convenient to describe it with the following matrix of shape functions:
[ N ] = [ N1 N 2 N 3 ] where, 1 N 1 = − (1 − ξ ) ξ 2 N 2 = (1 − ξ 2 ) N3 =
1 (1 + ξ ) ξ 2
The radius r at ξ and η is given by: r = N1r1 + N 2 r2 + N 3r3 + 0.5t1 N1η cos φ1 + 0.5t 2 N 2η cos φ2 + 0.5t3 N 3η cos φ3 [2.85] and x 0 = N1 x10 + N 2 x20 + N 3 x30 + 0.5t1 N1η sin φ1 + 0.5t 2 N 2η sin φ2 + 0.5t3 N 3η sin φ3
© Carl T. F. Ross, 2011
[2.86]
60
Pressure vessels
There are a total of nine nodal displacements: ⎧w10 ⎫ ⎪ 0⎪ ⎪u1 ⎪ ⎪θ1 ⎪ ⎪ 0⎪ ⎪w2 ⎪ 0 {Ui } = ⎪⎨u20 ⎪⎬ ⎪θ ⎪ ⎪ 2⎪ ⎪w30 ⎪ ⎪ 0⎪ ⎪u3 ⎪ ⎪θ 3 ⎪ ⎩ ⎭ The displacements w0 and u0 are obtained from equation [2.77], and their derivatives from equation [2.78]. The Jacobian [J] is obtained from equation [2.79]: ⎡ x10 [ J ] = [ TT ] ⎢⎢ x20 ⎢⎣ x30
r1 ⎤ ⎡ sin φ1 r2 ⎥⎥ + [ TT 2 ] ⎢sin φ2 ⎢ ⎢⎣ sin φ3 r3 ⎥⎦
cos φ1 ⎤ cos φ2 ⎥ ⎥ cos φ3 ⎥⎦
[2.87]
where in this instance: ⎡ ∂N 1
[ TT ] = ⎢⎢ ∂ξ ⎢⎣ 0
∂N 2 ∂ξ 0
⎡0.5t η ∂N1 1 ∂ξ [ TT2 ] = ⎢⎢ ⎢⎣0.5t1 N1 ∂N 1 1 = − +ξ ∂ξ 2
∂N 3 ⎤ ∂ξ ⎥ ⎥ 0 ⎥⎦ ∂N 2 ∂ξ 0.5t 2 N 2 0.5t 2η
∂N 2 = −2ξ ∂ξ
∂N 3 ⎤ ∂ξ ⎥ ⎥ 0.5t3 N 3 ⎥⎦ 0.5t3η
∂N 3 1 = − +ξ ∂ξ 2
From equation [2.80], the derivatives with respect to local axes are given by:
© Carl T. F. Ross, 2011
Axisymmetric deformation of pressure vessels
61
⎡ ∂u ⎢ ∂x ⎢ ⎢ ⎢ ∂u ⎢ 1 ⎣ ∂r
∂w ⎤ ⎡u10 w10 ⎤ ⎡θ 1 cos φ1 −θ 1 sin φ1 ⎤ ∂x ⎥ ⎥ ⎢u0 w0 ⎥ DC + C ⎢θ cos φ −θ sin φ ⎥ DC = A ] [ ]⎢ 2 ] 2 [ 2 2 2 [ ⎥ [ ]⎢ 2 ⎥ ⎥ ⎢⎣u30 w30 ⎥⎦ ⎢⎣θ 3 cos φ3 −θ 3 sin φ3 ⎥⎦ ∂w ⎥ ⎥ ∂r 1 ⎦ ⎡⎛ A11 ( DC 11u10 + DC 21w10 ) ⎞ ⎛ A11 ( DC12 u10 + DC 22 w10 ) ⎞ ⎤ ⎢⎜ ⎜ ⎟⎥ 0 0 ⎟ C 12 u20 + DC 22 w20 )⎟ ⎥ ⎢⎜ + A12 ( DC 11u2 + DC 21w2 )⎟ ⎜ + A12 ( DC ⎢⎜⎝ + A DC u0 + DC w0 ⎟⎠ ⎜⎝ + A DC u0 + DC w0 ⎟⎠ ⎥ 13 ( 11 3 21 3 ) 13 ( 12 3 22 3 ) ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ 0 0 0 0 ⎢⎛ A21 ( DC11u1 + DC 21w1 ) ⎞ ⎛ A21 ( DC 12 u1 + DC 22 w1 ) ⎞ ⎥ ⎢⎜ ⎜ 0 0 ⎟ 0 0 ⎟⎥ ⎢⎜ + A22 ( DC 11u2 + DC 21w2 )⎟ ⎜ + A22 ( DC 12 u2 + DC 22 w2 )⎟ ⎥ ⎢⎜⎝ + A ( DC u0 + DC w0 )⎟⎠ ⎜⎝ + A ( DC u0 + DC w0 )⎟⎠ ⎥ 22 3 23 11 3 21 3 23 12 3 ⎣ ⎦ ⎡⎛ C11θ 1 ( DC 11 cos φ1 − DC 21 sin φ1 ) ⎞ ⎢⎜ +C θ ( DC cos φ − DC sin φ )⎟ 11 2 21 2 ⎢⎜ 12 2 ⎟ ⎝ ⎢ +C13θ 3 ( DC 11 cos φ3 − DC 21 sin φ3 ) ⎠ ⎢ +⎢ ⎢⎛ C21θ 1 ( DC 11 cos φ1 − DC 21 sin φ1 ) ⎞ ⎢⎜ ⎢⎜ +C22θ 2 ( DC 11 cos φ2 − DC 21 sin φ2 )⎟⎟ ⎢⎣⎝ +C23θ 3 ( DC 11 cos φ3 − DC 21 sin φ3 ) ⎠
⎛ C11θ 1 ( DC 12 cos φ1 − DC 22 sin φ1 ) ⎞ ⎤ ⎜ +C12θ 2 ( DC 12 cos φ2 − DC 22 sin φ2 )⎟ ⎥ ⎜ ⎟⎥ ⎝ +C13θ 3 ( DC 12 cos φ3 − DC 22 sin φ3 ) ⎠ ⎥ ⎥ ⎥ ⎛ C21θ 1 ( DC 12 cos φ1 − DC 22 sin φ1 ) ⎞ ⎥ ⎜ +C22θ 2 ( DC 12 cos φ2 − DC 22 sin φ2 )⎟ ⎥⎥ ⎜ ⎟ ⎝ +C23θ 3 ( DC12 cos φ3 − DC 22 sin φ3 ) ⎠ ⎥⎦
where [DC] is defined in equation [2.81]. The matrix of strains is defined in equation [2.82] and [B′] is given in Table 2.6. The matrix of nodal external forces owing to pressure is obtained from equation [2.84], where
(−t1 N1η sin φ1 ) / 2 N 2 0 (t1 N1η cos φ1 ) / 2 0 N 2 (−t2 N 2η sin φ2 ) / 2 N 3 0 (−t3 N 3η sin φ3 ) / 2 ⎤ (t2 N 2η cos φ2 ) / 2 0 N 3 (t3 N 3η cos φ3 ) / 2 ⎥⎦
⎡ N1 ⎣0
[N ] = ⎢
0 N1
For both instances of the thick-walled shell, numerical integration was carried out to determine [k0] and {Pi}. Four Gauss points were used in the ξ-direction and two Gauss points in the η-direction. The results obtained using computer programs for analysing thick-walled cones and domes28 are compared with the analytical solution of Timoshenko
© Carl T. F. Ross, 2011
© Carl T. F. Ross, 2011
A12DC21 N2 r
C11 (DC11 cos φ1 − DC21 sin φ1 )
−t1N1η sin φ1 2r
A11DC11
0
u30
A13DC11
0
A23DC11 + A13DC12
A13DC21
N3 r
A23DC21 + A13DC22
A21DC11 + A11DC12
C13 (DC11 cos φ3 − DC21 sin φ3 ) ⎤ ⎥ ⎥ ⎥ −t 3N 3η sin φ3 ⎥ 2r ⎥ ⎥ C 23 (DC11 cos φ3 − DC21 sin φ3 ) ⎥⎥ +C13 (DC12 cos φ3 − DC22 sin φ3 )⎦⎥
θ3
C 21 (DC11 cos φ1 − DC21 sin φ1 ) A22DC21 +C11 (DC12 cos φ1 − DC21 sin φ1 ) + A12DC22
w 20
θ1
u10
w 30
⎡ A11DC21 ⎢ ⎢ ⎢ N1 [B ′ ] = ⎢⎢ r ⎢ ⎢ A DC ⎢ 21 21 ⎢⎣ + A11DC22
w 10
−t 2N 2η sin φ2 2r
C12 (DC11 cos φ2 − DC21 sin φ2 )
θ2
A22DC11 C 22 (DC11 cos φ2 − DC21 sin φ2 ) + A12DC12 +C12 (DC12 cos φ2 − DC22 sinφ2 )
0
A21DC11
u20
Table 2.6 [B′] matrix for the thick-walled three-node parabolic element
Axisymmetric deformation of pressure vessels
63
and Woinowsky-Kreiger26 for the shell cap of Fig. 2.12 (Fig. 2.35). The results from Fig. 2.35 show good agreement between the various solutions, and this was found to be particularly encouraging as the chosen shell was thin but the theories of the present section were based on thick-walled shells. Comparison of Ahmad and co-workers’ solution31 for the thick shell of Fig. 2.36 with the results obtained from the computer program for the three-
Angle f (degrees) 0
31.5
28.0 24.5 21.0 17.5 14.0 10.5
7.0
3.5
−1 −2 −3
Thick conical element Thick (three node) parabolic element
−4 −5
Timoshenko and Woinowsky-Kreiger26
−6
Mean hoop stress (Ibf in−2)
−7 −8 −9 −10 −11 −12 −13 −14 −15 −16 −17 −18
2.35 Mean hoop stress distribution for cap of 3 in. thickness.
© Carl T. F. Ross, 2011
0
64
Pressure vessels 100 lbf in−2
Wall Nose
35º
35º
2.36 Shell cap of 9 in. thickness.
node element is given in Fig. 2.37. These can be seen to be in good agreement.
2.5.2 Four-node element The results show that the three-node element gives better predictions than the two-node element, and to test whether four- or five-node elements would make even better predictions, the following cones were developed. For the four-node element of Fig. 2.38, the assumed displacement functions for u0 and w0 are: u 0 = a + bξ + cξ 2 + dξ 3
w0 = e + fξ + gξ 2 + hξ 3
[2.88]
which can be seen to be of cubic form. To determine the shape functions for u0 and w0, consider the boundary conditions for u0, which from Fig. 2.38 are: at ξ = −1, u 0 = u10 at ξ = − 13 , u 0 = u20 at ξ = 13 , u 0 = u30 at ξ = 1, u0 = u40
∴ u10 ∴ u20 ∴ u30 ∴ u40
= a−b+c−d = a − 13 b + 91 c − 271 d = a + 13 b + 91 c + 271 d = a+b+c+d
Solving the above simultaneous equations, the following expressions are obtained for the constants:
© Carl T. F. Ross, 2011
Axisymmetric deformation of pressure vessels Angle f (degrees)
Wall 100
31.5
28.0 24.5 21.0 17.5 14.0 10.5 7.0
65 Nose
3.5
0
External Internal
0
Ahmad et al.31 (average)
Hoop stress (Ibf in−2)
−100
−200
−300
−400
−500
−600
−700
−800
2.37 Hoop stresses for cap of 9 in. thickness.
a = − 161 (u40 + u10 ) + 169 (u30 + u20 ) b = − 161 ( u40 − u10 ) + 169 ( 3u30 − 3u20 )
c= d=
9 16 9 16
(u40 + u10 − u30 − u20 ) (u40 − u10 − 3u30 + 3u20 )
Similar expressions can be derived for the constants e to h, and substituting these constants back into the displacement functions, the shape functions for u0 and w0 are given by:
© Carl T. F. Ross, 2011
66
Pressure vessels r ,w0
1
wi0
•
2
•
3
ui0
•
4
•
fi
x0 , u0
2.38 Four-node cubic element.
N1 = N2 =
1 16 9 16
( −1 + ξ + 9ξ 2 − 9ξ 3 ) (1 − 3ξ − ξ 2 + 3ξ 3 )
N3 = N4 =
9 16 1 16
(1 + 3ξ − ξ 2 − 3ξ 3 ) ( −1 − ξ + 9ξ 2 + 9ξ 3 )
In a manner similar to that adopted for the two- and three-node elements, [B′] is obtained, as in Table 2.7, and [k0] can be calculated from equation [2.83].
2.5.3 Five-node element The five-node element is shown in Fig. 2.39, and the displacement function for u0, which is a quartic function, is given in equation [2.89]. It must be emphasised that as the shape function for w0 is the same as for u0, it will only be necessary to consider the latter: u 0 = a + bξ + cξ 2 + dξ 3 + eξ 4
[2.89]
From Fig. 2.39 it can be seen that the boundary conditions are as follows: at ξ = −1, u 0 = u10 at ξ = −0.5, u 0 = u20 at ξ = 0, u 0 = u30 at ξ = 0.5, u 0 = u40 at ξ = 1, u 0 = u50
∴ u10 = a − b + c − d + e b c d e ∴ u20 = a − + − + 2 4 8 16 ∴ u30 = a b c d e ∴ u40 = a + + + + 2 4 8 16 ∴ u50 = a + b + c + d + e
© Carl T. F. Ross, 2011
© Carl T. F. Ross, 2011
u30
A13DC11
0
A23DC11 + A13DC12
A13DC21
N3 r
A23DC21 + A13DC22
N4 r
−t 3N 3η sin φ3 2r
A24DC11 C 23 (DC11 cos φ3 − DC21 sin φ3 ) +C13 (DC12 cos φ3 − DC22 sin φ3 ) + A14DC22
A14DC21
w 40
C13 (DC11 cos φ3 − DC21 sin φ3 )
θ3
C 21 (DC11 cos φ1 − DC21 sin φ1 ) A22DC21 +C11 (DC12 cos φ1 − DC22 sin φ1 ) + A12DC22
N2 r
−t1N1η sin φ1 2r
0
A21DC11 + A11DC12
A11DC21
w 20
C11 (DC11 cos φ1 − DC21 sin φ1 )
θ1
A11DC11
u10
w 30
⎡ A11DC21 ⎢ ⎢ ⎢ N1 [B ′ ] = ⎢⎢ r ⎢ ⎢ A DC ⎢ 21 21 ⎢⎣ + A11DC22
w 10
Table 2.7 [B′] matrix for the four-node element
A24DC11 + A14DC12
0
A14DC11
u40
A22DC11 + A12DC12
0
A12DC11
u20
C14 (DC11 cos φ4 − DC21 sin φ4 ) ⎤ ⎥ ⎥ ⎥ −t 4N4η sin φ4 ⎥ 2r ⎥ ⎥ C 24 (DC11 cos φ4 − DC21 sin φ4 ) ⎥⎥ +C14 (DC12 cos φ4 − DC22 sin φ4 )⎦⎥
θ4
C 22 (DC11 cos φ2 − DC21 sin φ2 ) +C12 (DC12 cos φ2 − DC22 sinφ2 )
−t 2N 2η sin φ2 2r
C12 (DC11 cos φ2 − DC21 sin φ2 )
θ2
68
Pressure vessels r ,w0
1
wi0
•
2
•
3
•
4
ui0
•
5
•
fi
x0 , u0
2.39 Five-node quartic element.
Solving the above simultaneous equations, the following expressions are obtained for the constants: a = u30 b = − 61 (u50 − u10 ) + 43 (u40 − u20 ) c = − 61 (u50 + u10 ) − 5u30 + 83 (u40 + u20 ) d=
2 3
(u50 − u10 ) − 43 (u40 − u20 )
e=
2 3
(u50 + u10 ) + 4u30 − 83 (u40 + u20 )
Substituting these constants back into the displacement function for u0, the shape functions for both u0 and w0 are obtained as follows: N1 =
1 6
[ξ − ξ 2 − 4ξ 3 + 4ξ 4 ]
N2 =
1 3
[ −4ξ + 8ξ 2 + 4ξ 3 − 8ξ 4 ]
N 3 = [1 − 5ξ 2 + 4ξ 4 ] N4 =
1 3
[4ξ + 8ξ 2 − 4ξ 3 − 8ξ 4 ]
N5 =
1 6
[ −ξ − ξ 2 + 4ξ 3 + 4ξ 4 ]
In a manner similar to that adopted for the two- and three-node elements, [B′] is obtained, as is shown in Table 2.8, and [k0] can be obtained from
© Carl T. F. Ross, 2011
© Carl T. F. Ross, 2011
u40
A14DC11
0
A24DC11 + A14DC12
A14DC21
N4 r
A24DC21 +A A14DC22
A21DC11 + A11DC12
0
A11DC11
u10
w 40
⎡ A11DC21 ⎢ ⎢ ⎢ ⎢ ⎢ N1 ⎢ r [B ′ ] = ⎢ ⎢ ⎢ A21DC21 ⎢ ⎢ + A11DC22 ⎢ ⎢ ⎢⎣
w 10
N2 r
A12DC21
w 20
C 24 (DC11 cos φ4 −DC21 sin φ4 ) +C14 (DC12 cos φ4 −DC22 sin φ4 )
−t 4N4η sin φ4 2r
C14 (DC11 cos φ4 −DC21 sin φ4 )
θ4
A25DC21 + A15DC22
N5 r
A15DC21
w 50
C21 (DC11 cos φ1 A22DC21 −DC21 sin φ1 ) + A12DC22 +C11 (DC12 cos φ1 −DC22 sin φ1 )
−t1N1η sin φ1 2r
C11 (DC11 cos φ1 −DC21 sin φ1 )
θ1
Table 2.8 [B′] matrix for the five-node element
A25DC11 + A25DC12
0
A15DC11
u50
A22DC11 + A12DC12
0
A12DC11
u20
C15 (DC11 cos φ5 ⎤ −DC21 sin φ5 ) ⎥ ⎥ ⎥ ⎥ −t 5N5η sin φ5 ⎥ ⎥ 2r ⎥ ⎥ C 25 (DC11 cos φ5 ⎥ ⎥ −DC21 sin φ5 ) ⎥ +C15 (DC12 cos φ5 ⎥ ⎥ −DC22 sin φ5 ) ⎥⎦
θ5
C22 (DC11 cos φ2 −DC21 sin φ2 ) +C12 (DC12 cos φ2 −DC22 sin φ2 )
−t 2N 2η sin φ2 2r
C12 (DC11 cos φ1 −DC21 sin φ2 )
θ2
A23DC21 + A13DC22
N3 r
A13DC21
w 30
A23DC11 + A13DC12
0
A13DC11
u30
C 23 (DC11 cos φ3 −DC21 sin φ3 ) +C13 (DC12 cos φ3 −DC22 sin φ3 )
−t 3N 3η sin φ3 2r
C13 (DC11 cos φ3 −DC21 sin φ3 )
θ3
70
Pressure vessels 35.0 31.5 28.0 24.5 21.0 17.5 14.0 10.5 0
7.0
3.5
0
−1 −2 −3 −4 −5
Timoshenko and Woinowsky-Kreiger26 Five node quartic element Four node cubic element
−7 −8 3 in
−10 35°
−11
in
−9
90
Mean hoop stress (Ibf in−2)
Three node parabolic element −6
−12 −13 −14 −15 −16
2.40 Hoop stress distributions for 3-in. shell cap.
equation [2.83]. To compare the results predicted by the three-, four- and five-node elements, once again, the 3-in.-thick shell cap of Timoshenko and Woinowsky-Kreiger26 was used, and the results are shown in Fig. 2.40.
2.5.4 Comparisons between various elements It can be seen that the four- and five-node elements give slightly better results than the three-node element. One advantage of the four- and fivenode elements over the three-node element is that they require fewer elements to model the dome.
© Carl T. F. Ross, 2011
Axisymmetric deformation of pressure vessels
71
2.5.5 Comparison with a tapered dome A further test of the four-and five-node elements was carried out with the aid of the tapered Araldite models shown in Figs 2.41 and 2.42. The models were subjected to external hydrostatic pressure in the tank shown in Fig. 2.43, and 10 strain gauges were attached to the inner surface of each model in the positions shown in Figs 2.44 and 2.45. It was necessary to use two strain gauges at each position and to place the gauges perpendicular to each other, and in the directions of the principal strains, as the problem was of a plane stress type on the surfaces of the domes.
12.8
90.25
17.2
17.2 198.8
2.41 Experimental shell section: dome 1 (dimensions are in mm).
21.46
88.83
9.95
9.95 198.5
2.42 Experimental shell section: dome 2 (dimensions are in mm).
© Carl T. F. Ross, 2011
72
Pressure vessels Clamping flange Rubber seal Pressure tank
Araldite dome
Water Pressure gauge
Water pump connection
2.43 Method of pressurising dome.
Experimentally obtained strains are given in Tables 2.9 and 2.10. The observed strains show a little nonlinearity, which is attributed to the fact that the gauges were meant for metals and not for Araldite. The following material properties were assumed for Araldite: E = 2758 MPa and v = 0.345. Comparisons between experiment and the theoretical predictions from the five-node element are shown in Figs 2.46–2.49. Comparison is also made, in Figs 2.50–2.53, of the theoretical stresses in these domes as predicted by the three- and four-node elements; it can be seen that agreement between the two sets of results is very good.
2.5.6 Orthotropic element It is a relatively simple matter to extend the previous finite element theories for isotropic thin-walled axisymmetric shells to the orthotropic cases. For the finite element given in Section 2.4, the following matrix of material constants applies:
© Carl T. F. Ross, 2011
Axisymmetric deformation of pressure vessels
φ
1
6
7
2
8 3 9
10
4
f1−6
=
11°
f2−7
=
35°
f3−8
=
55°
f4−9
=
77°
f5−10
=
90°
5
2.44 Strain gauge positions: dome 1.
φ 1
7
2
3
4
5
6
8
9
10
f1−6
=
23°
f2−7
=
37°
f3−8
=
50°
f4−9
=
67°
f5−10
=
90°
2.45 Strain gauge positions: dome 2.
© Carl T. F. Ross, 2011
73
74
Pressure vessels
Table 2.9 Experimental strain results for dome 1
Gauge
345 kPa (50 lbf in−2)
517 kPa (75 lbf in−2)
690 kPa (100 lbf in−2)
862 kPa (125 lbf in−2)
Hoop strains, με
1 2 3 4 5
−234 −404 −347 −310.5 −305
−361 −582.5 −511.5 −470 −466
−477.3 −762 −670.7 −625.3 −618
−58 −950 −840 −778 −767
Meridional strains, με
6 7 8 9 10
448 −173.5 −357.5 −350.5 −344
483 −281.5 −529.5 −517 −502
602 −378 −687.3 −674 −647.3
688 −477 −851 −833 −790
D11 =
Ex t (1 − v x v y )
D33 =
Ex t 3 12 (1 − vx vy )
D22 =
Ey t (1 − v x v y )
D44 =
Ey t 3 12 (1 − vx vy )
D12 =
vy E x t (1 − v x v y )
D34
vy E x t 3 12 (1 − vx vy )
D21 =
vx E y t (1 − v x v y )
D43
vx E y t 3 12 (1 − vx vy )
All other Dij = 0 where, Ex = Young’s modulus in meridional direction Ey = Young’s modulus in hoop direction vx = Poisson’s ratio due to a direct stress in the x-direction vy = Poisson’s ratio due to a direct stress in the y-direction vxEy = vyEx t = shell thickness. Similarly, for the thick-walled axisymmetric shells in Section 2.5, D11 = Ex / (1 − vx vy )
D22 = Ey / (1 − vx vy )
D12 = D21 = vy Ex / (1 − vx vy ) = vx Ey / (1 − vx vy ) D33 = G / k All other Dij = 0
© Carl T. F. Ross, 2011
© Carl T. F. Ross, 2011
Meridional strains, με
Hoop strains, με
−239.5 −226.5 −215
−103 −93.5
10
−236
−99.5
7 −98.5
−178.5
−74.5
6
8
−232
−106.5
5
9
−218 −223.5
−99.5 −99.5
3
−189
−87
2
4
−190.5
345 kPa (50 lbf in−2)
−93.5
172.5 kPa (25 lbf in−2)
1
Gauge
Table 2.10 Experimental strain results for dome 2
−327
−341.5
−369
−364
−284
−352.5
−338
−332
−290.5
−284
517 kPa (75 lbf in−2)
−440
−461.5
−497
−485
−384
−472
−456.5
−441
−386.5
−348
690 kPa (100 lbf in−2)
−553
−581
−633
−633
−501
−595
−572
−553
−489
−390
862 kPa (125 lbf in−2)
Angle f (degrees)
85 75
0
65
55
45
35
25
15
5
0
Meridional stress (MN m−2)
−1
−2
−3
External stress
−4
Internal stress Five node element program Experimental results (internal)
−5
2.46 Meridional stress distribution for dome 1 at 690 kPa (100 lbf in−2). Angle f (degrees)
85 0
75
65
55
45
35
25
15
5 0
−0.5 External stress Hoop stress (MN m−2)
−1.0
Internal stress Five node element program Experimental results (internal)
−1.5
−2.0
−2.5
−3.0
2.47 Hoop stress distribution for dome 1 at 690 kPa (100 lbf in−2).
© Carl T. F. Ross, 2011
Axisymmetric deformation of pressure vessels Angle f (degrees)
80 0
77
70
60
50
40
30
20
10
0
−1
Meridional stress (MN m−2)
−2
−3
−4 External stress −5
Internal stress Five node element program
−6
Experimental results (internal)
−7
2.48 Meridional stress distribution in dome 2 at 690 kPa (100 lbf in−2).
2.6
Ring-stiffeners
The ring element consists of one nodal circle and shares a common node with the two adjacent axisymmetric shell elements on either side of it. The stiffness components of the ring element are partly flexural and partly inplane, and the flexural component of stiffness is now derived. Owing to the moment Mi on the ring-stiffener at node i, the stiffener rotates out of its plane (turned inside-out) by an angle θi, as shown in Fig. 2.54. According to Roark and Young,32
θi =
Mi Rf EI z
[2.90]
© Carl T. F. Ross, 2011
78
Pressure vessels Angle f (degrees) 80 0
70
60
50
40
30
20
10
0
−0.2 −0.4 −0.6
Hoop stress (MN m−2)
−0.8 −1.0
External stress Internal stress Five node element program Experimental results (internal)
−1.2 −1.4 −1.6 −1.8 −2.0 −2.2 −2.4 −2.6
2.49 Hoop stress distribution for dome 2 at 690 kPa (100 lbf in−2).
where, Rf = radius of centroid of frame, together with the width of plate in direct contact with the frame’s web; Iz = second moment of area about the z–z plane of the frame, together with the width of plate in direct contact with the frame’s web; E = Young’s modulus of elasticity of the frame, or its equivalent if the shell has a different E. Therefore, Mi =
EI zθ i Rf
© Carl T. F. Ross, 2011
85 0
Angle f (degrees) 75
65
55
45
35
25
15
5 0
Meridional stress (MN m−2)
−1
−2
−3
−4
External stress Internal stress Three node element program Four node element program
−5
2.50 Meridional stress distribution for dome 1 at 690 kPa (100 lbf in−2). Angle f (degrees)
85 0
−0.5
75
65
55
45
35
25
15
5 0
External stress
Hoop stress (MN m−2)
Internal stress Three node element program −1.0
Four node element program
−1.5
−2.0
−2.5
−3.0
2.51 Hoop stress distribution for dome 1 at 690 kPa (100 lbf in−2).
© Carl T. F. Ross, 2011
80
Pressure vessels Angle f (degrees) 80 0
70
60
50
40
30
20
10
0
−1
Meridional stress (MN m−2)
−2
−3
−4
−5 External stress Internal stress −6
Four node element program Three node element program
−7
2.52 Meridional stress distribution in dome 2 at 690 kPa (100 lbf in−2).
and the bending strain energy is: Ub =
πEI z2θ i2 Mi2 2πRf = 2E Rf
[2.91]
As previously reported26 U b = 12 kbθ i2 =
πEI zθ i2 Rf
[2.92]
Therefore, kb, the bending component of stiffness about the z–z plane is: kb =
2πEI z Rf
[2.93]
© Carl T. F. Ross, 2011
Axisymmetric deformation of pressure vessels
81
Angle f (degrees) 80 0
70
60
50
40
30
20
10
0
−0.2 −0.4 −0.6
External stress Internal stress
−1.0
Four node element program
Hoop stress (MN m−2)
−0.8
Three node element program
−1.2 −1.4 −1.6 −1.8 −2.0 −2.2 −2.4 −2.6
2.53 Hoop stress distribution in dome 2 at 690 kPA (100 lbf in−2).
The in-plane (circumferential) stiffness is derived as follows. The in-plane strain energy in the circumferential direction UH is given by: UH =
σ H2 × volume 2E
[2.94]
where σH, the hoop stress in the ring is EεH. However, the hoop strain in the ring is given by
εH =
wi Rf
σH =
Ewi Rf
[2.95]
so that
© Carl T. F. Ross, 2011
82
Pressure vessels z
Mi
qi Axis
Mi
z
2.54 Out-of-plane bending of a ring. Mi = couple at node i.
Therefore, UH =
E 2 wi2 1 πEAf wi2 1 × × R × A = = kH wi2 2 π f f Rf2 Rf 2E 2
[2.96]
where the in-plane (circumferential component of stiffness in the ring is given by: kH =
2πEAf Rf
[2.97]
i.e. the elemental stiffness matrix of the ring, in terms of wi and θi is: wi θ i 2 πE ⎡ Af 0 ⎤ wi [k ] = Rf ⎢⎣ 0 I z ⎥⎦ θ i
[2.98]
There is, however, another component of stiffness that is relatively small but which can be included for greater precision. This component results
© Carl T. F. Ross, 2011
Axisymmetric deformation of pressure vessels
83
from the change of curvature with circumferential direction owing to the deflection w, i.e. the curvature in the circumferential direction is given by:
χH =
wi Rf2
[2.99]
and the bending strain energy in the circumferential direction is given by: U bH = (EIχ H2 / 2)(2 πRf ) = 0.5kbH wi2
[2.100]
so that the component of flexural stiffness due to bending of the stiffener about its x–x axis is given by kbH =
2 πEI Rf3
[2.101]
where I is the second moment of area of stiffener about its x–x axis, which is an axis through the centroid of the ring-shell combination, parallel to the axis of the cylinder/cone/dome. The improved [k] for the ring is now given by: wi
[k ] = 2.7
2 πE ⎡( Af + I / R Rf ⎢⎣ 0
2 f
)
θi 0 ⎤ wi I z ⎥⎦ θ i
[2.102]
Plastic collapse
It is likely that an axisymmetric shell can withstand a pressure somewhat higher than that based on first yield.33 The theories provided in this chapter are based on failure at first yield, but the total plastic collapse load is likely to be higher than this, particularly for thick shells. For thin shells, which have negligible bending resistance, the plastic collapse pressure is likely to be only fractionally higher than the stress to cause yield in the circumferential direction. This is because, for thin shells, the maximum circumferential stress is a bulk stress, whereas the maximum meridional stress is only a local stress and will cease to resist bending once it becomes plastic (i.e. the shell will simply rotate at the points where it has become plastic owing to meridional bending). The designer must remember that, for thin shells, large bulk stresses, such as hoop stresses, are inherently more dangerous than large meridional bending stresses, which are local. In this chapter, a theoretical axisymmetric plastic buckling analysis is carried out based on an element similar to that of Grafton and Strome,22 and on the nonlinear finite element method of Turner et al.34 Also presented are experimental tests carried out on a number of thin-walled circular cylinders and cones,35 which were tested to failure under uniform external
© Carl T. F. Ross, 2011
84
Pressure vessels
pressure. Comparisons were made between experiment and theory, and good agreement was found.
2.7.1 Experimental apparatus This section describes tests carried out on two thin-walled circular conical shells, namely cone C and cone 9, and on three thin-walled circular cylindrical shells, namely cylinders 4, 5 and 6. All of the vessels were machined carefully from solid billets of EN1A steel, and the details of these vessels are now given.
2.7.2 Cone C The geometrical details of this vessel are given in Fig. 2.55 and Table 2.11, from which it can be seen that cone C was a thin-walled vessel that was stiffened by two quite heavy ring stiffeners. Cone C was from a series of three vessels shown in Fig. 2.56. The out-of-roundness of cone C was found to be 0.005 mm. This measurement was taken around the outer circumference of the vessel, in the region of the expected failure zone in the central bay. The out-of-roundness was defined as the difference between the maximum inward and outward radial deviations from the mean mid-length circumference, where the latter was obtained from a least-square fit.
2.7.3 Cone 9 The geometrical details of cone 9 are presented in Fig. 2.57. Cone 9 was one of a series of three thin-walled cones as shown in Fig. 2.58. The out-ofroundness for cone 9 was found to be 0.007 mm. This out-of-roundness was measured around the external circumference of the vessel at its mid-length in the central bay.
2.7.4 Cylinders 4, 5 and 6 The geometrical measurements for the circular cylinders 4, 5 and 6 are given in Fig. 2.59 and Table 2.12. A photograph illustrating the three circular cylinders is shown in Fig. 2.60. The out-of-roundness of these vessels was 0.0227, 0.0108 and 0.0205 mm for cylinders 4, 5 and 6, respectively. The out-of-roundness values were measured at the mid-lengths of the vessels on their external surfaces, in the central bays. The test tank used for all the models is shown in Fig. 2.61. The material properties of the vessels were found to be as follows: • Cones A, B and C: yield stress σyp is 288 MPa and Young’s modulus E is 190 GPa.
© Carl T. F. Ross, 2011
Axisymmetric deformation of pressure vessels
85
8 9
7 6
13 5 14
4 3 2
15 1 11 10 12
17
18
16
2.55 Geometrical measurements of cone C.
• Cone 9: σyp is 231 MPa and E is 193 GPa. • Cylinders 4, 5 and 6: σyp is 244 MPa and E is 200 GPa. • For all vessels: Poisson’s ratio v is 0.3 (assumed).
2.8
Experimental procedure
All five models were tested to destruction, under uniform external pressure, in the test tank shown in Fig. 2.61. Water was used as the pressure-raising liquid, and this was pumped into the test tank with the aid of a hand-
© Carl T. F. Ross, 2011
86
Pressure vessels Table 2.11 Geometrical details of cone C (mm) (average thickness = 1.082 mm; cone angle = 19.15°) Dimension
Measurement
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
151.867 19.057 7.631 31.755 7.590 25.427 15.874 76.230 38.112 139.693 101.650 40.358 49.343 74.423 97.320 103.733 75.949 103.283
2.56 Cones A, B and C (from right to left).
© Carl T. F. Ross, 2011
Axisymmetric deformation of pressure vessels
87
211.98
15.88 7.61
15.853
10 38.096
88.79 19.07
7.63
dia. 110.518
57.164 dia. 76.443
2.57 Geometrical details of cone 9 (mm).
2.58 Cones 7, 8 and 9 (from left to right).
© Carl T. F. Ross, 2011
dia. 76.2
dia. 101.66
dia. 139.67
0.79
88
Pressure vessels b Area Bay 0
Bay 1
Bay 2 d
t
r1
r2
r3
r4
Cylinder centre line
2.59 Geometrical details of cylinders 4, 5 and 6.
Table 2.12 Geometrical dimensions of cylinders 4, 5 and 6 (mm) Cylinder 4
Cylinder 5
Cylinder 6
Unsupported length Bay 0 Bay 1 Bay 2
9.254 25.416 9.154
9.296 18.966 9.389
9.621 12.661 9.348
Stiffener 1 Breadth Depth Area Radius
6.032 6.011 36.258 55.101
6.133 5.997 36.780 55.117
5.919 6.003 35.532 55.092
Stiffener 2 Breadth Depth Area Radius
5.956 6.012 35.807 55.101
6.083 5.991 36.443 55.114
6.024 6.000 36.144 55.090
Mean shell radius Shell thickness Internal radius External radius
51.472 1.245 50.850 52.095
51.484 1.267 50.851 52.118
51.467 1.246 50.844 52.090
© Carl T. F. Ross, 2011
Axisymmetric deformation of pressure vessels
89
2.60 Circular cylinders 4, 5 and 6 (from right to left).
Bolt
Hose from pump
Model
Closure plate
2.61 Test tank with model.
operated hydraulic pump. The hose connecting the hydraulic pump to the test tank was only about 2 m long; hence line losses were negligible.
2.8.1 Cone C All three cones in this series of vessels collapsed initially in their first bays through plastic axisymmetric buckling, as shown in Figs 2.62 and 2.63. The
© Carl T. F. Ross, 2011
90
Pressure vessels
vessels also collapsed through plastic lobar buckling in the second bay, but this was not of interest in this study. After cones A, B and C collapsed axisymmetrically in their first bays, the uniform pressure fell. On increasing this pressure, an experimentally obtained ring stiffener formed in the first bay of each vessel. This ring stiffener had the effect of increasing the strength of each vessel in its first bay,
2.62 The collapsed models: A, B and C (from left to right).
2.63 Bottom view of the collapsed cone C.
© Carl T. F. Ross, 2011
Axisymmetric deformation of pressure vessels
91
Table 2.13 Collapse pressures of cones A, B and C
Cone
Axisymmetric collapse pressure (MPa)
Lobar buckling pressure (MPa)
A B C
5.97 6.03 6.21
6.48 6.76 6.62
so that it was possible to increase the pressure to a value greater than that which caused plastic axisymmetric buckling. On achieving these higher pressures, each vessel failed by lobar buckling (or shell instability, in its central bay, as shown in Fig. 2.12. The two sets of collapse pressures are shown for each vessel in Table 2.13. From Table 2.13, it can be seen that the axisymmetric collapse pressure for cone B was slightly higher than for cone A, and that the axisymmetric collapse pressure for cone C was slightly higher than that for cone B. The reason for this was that the length of the mid-bay of cone C was shorter than that of cone B, and that the length of cone B was shorter than that of cone A. Thus, if the mid-bay is shorter, it gives more support to the first bay. The maximum values of measured hoop and meridional strains in the first bay were between −3000 and −4000 microstrain, thereby indicating that the vessels failed plastically.
2.8.2 Cone 9 Cone 9 collapsed through axisymmetric buckling at a uniform external pressure of 52.41 bar. This mode of collapse was completely unexpected, as the length of the mid-bay of cone 9 was about 4.66 times that of the first bay in which the first collapse took place. The reason the collapse took place in the first bay, rather than in the central bay, may have been because the mean diameter of the first bay was about 28.3% larger than the mean diameter of the central bay. After cone 9 had collapsed axisymmetrically in the first bay, the pressure fell to 31.03 bar. On increasing this pressure an experimentally obtained ring stiffener formed in the first bay, which strengthened the first bay, so that the pressure could be increased further. When the pressure reached 48.28 bar, the central bay collapsed through shell instability, as shown in Fig. 2.64. It was assumed, however, that the central bay collapsed at a pressure of 52.41 bar, because the vessel had previously withstood this pressure.
© Carl T. F. Ross, 2011
92
Pressure vessels
2.64 Collapsed modes for cones 7, 8 and 9 (from left to right).
Table 2.14 Experimental collapse pressures
Cylinder
Pressure (bar)
4 5 6
97.24 111.72 131.72
Figure 2.64 shows the collapsed modes of cones 7, 8 and 9, where it can be seen that cone 9 collapsed axisymmetrically in its first bay. Cone 9 also collapsed through lobar buckling in its second bay, as indeed did cones 7 and 8, but this mode of failure was not of interest in this study.
2.8.3 Circular cylinders 4, 5 and 6 All three cylinders collapsed in their central bays through plastic axisymmetric buckling. It was expected that these vessels would fail in their central bays and, for this reason, these bays were heavily strain gauged. The uniform external pressures that the vessels collapsed under are given in Table 2.14.
© Carl T. F. Ross, 2011
Axisymmetric deformation of pressure vessels
93
Ten linear strain gauges were attached to the inside surfaces of each of their central bays. Plots of the variation of hoop strain versus pressure are given for each of these vessels in Figs 2.65–2.67. From these figures it can be seen that the vessels collapsed through plastic axisymmetric deformation, the measured hoop strain reaching nearly 12 000 microstrain for cylinder 4 and 6000 microstrain for cylinders 5 and 6.
0
9.66
0
Pressure (MPa) 5.52
Hoop strain/microstrain
−2000 −4000 −6000
−8000
−10000 −12000
2.65 Pressure–hoop strain relationship for cylinder 4.
0
0
Pressure (MPa) 5.52
Hoop strain/microstrain
−1000 −2000 −3000 −4000 −5000 −6000
2.66 Pressure–hoop strain relationship for cylinder 5.
© Carl T. F. Ross, 2011
11.03
94
Pressure vessels
0
13.79
0
Pressure (MPa) 6.9 −1000
Hoop strain/microstrain
−2000 −3000 −4000 −5000 −6000 −7000 −8000 −9000 −10000
2.67 Pressure–hoop strain relationship for cylinder 6.
2.68 Collapsed modes of cylinders 4, 5 and 6 (from left to right).
The experimental observations for these vessels showed that they all suffered initial plastic axisymmetric deformation as a consequence of large hoop stresses but that, as the radial deflection increased, the axial loads became more and more significant until plastic axisymmetric buckling took place in the form of a three-hinge mechanism. The middle hinge was a sagging hinge and the two outer hinges were hogging hinges, as shown in Fig. 2.68.
© Carl T. F. Ross, 2011
Axisymmetric deformation of pressure vessels
2.9
95
Theoretical plastic analysis
The finite element adopted was based on the axisymmetric conical shell element of Grafton and Strome.22 It was of truncated form and had two end ring nodes, with six degrees of freedom per element. The computational analysis, which used the author’s computer program ‘PLASCONE’, allowed for both geometrical and material nonlinearity. It is based on the incremental method of analysis and is now described with the aid of Table 2.15. In the theoretical analysis, step 1 was to load the structure with a relatively small load that was well below the point where either geometrical or material nonlinearity would commence. After calculating the stresses and deflections at the end of step 1, the geometrical stiffness matrix was calculated as was the new geometry of the structure formed as a consequence of the resulting deflections. Similarly, in step 2, the structure was subjected to another small incremental load, and the stresses and deflections resulting from this load were calculated. They were added to the stresses and deflections at the end of step 1. If the resulting stress in any element exceeded its yield stress, based on the Hencky–von Mises theory of elastic failure,1,2 the Young’s modulus in that element was made equal to 1/50th of the elastic Young’s modulus. Furthermore, if the yield stress in any element exceeded its yield stress by a factor of 1.1, based on the Hencky–von Mises stress, the Young’s modulus in that element was made equal to 1/100th of the elastic Young’s modulus. The new stiffness and geometrical stiffness matrices were calculated, and the geometry of the structure updated. The process was repeated until the structure failed through plastic axisymmetric buckling, with the formation of three circumferential plastic hinges. Each vessel was assumed to be simply supported at one end, the larger end for the cones, and clamped at the other end. By the condition clamped, it was assumed that all the deflections, except for the axial displacement, were zero at that end. Plots of pressure versus axial displacement at the
Table 2.15 Incremental non-linear method ([K0] = stiffness matrix; [K G0 ] = geometrical stiffness matrix; {δq0} = a vector of nodal incremental loads; {δu0} = a vector of incremental nodal displacements) Step
{δq0} Stiffness matrix
{δu0} Displacements
1 2 3 ¯ n Σ
{δq 01}[K 00 (0)] + [K G0 (0)] {δq 02}[K 00 (u10)] + [K G0 (u01)] {δq03}[K 00 (u02)] + [K G0 (u02)] 0 0 {δq n0}[K 00 (un−1 )] + [K G0 (un−1 )] 0 {q n}
{δu 01}{u 10} = {δu 01} {δu 02}{u 02} = {u 01} + {δu 02} {δu 03}{u 03} = {u 02} + {δu 03} 0 {δu n0}{u n0} = {u n−1 } + {δu n0} 0 {un}
© Carl T. F. Ross, 2011
96
Pressure vessels
clamped end of the vessel are given for all the vessels in Figs 2.69–2.73. From these figures, it can be seen that there was good agreement between experiment and theory. The experimental tests showed that the collapse mechanism for the five vessels described in detail was through plastic axisymmetric buckling where, after initial yielding in the circumferential direction at mid-bay, the effects of axial load became more important with increasing radial deflection so that the axial load eventually caused a three-hinge plastic buckling mechanism to occur. Previous experimental work on the plastic axisymmetric collapse of thin-walled ring-stiffened cylinders was carried out by Lunchick,36 but it was not possible to analyse Lunchick’s models by the present analysis as that author omitted to give full details of his vessels.
2.10
Conclusions
The experimental tests showed that all five vessels collapsed through plastic axisymmetric buckling. The experimental observations also showed that these vessels initially suffered plastic axisymmetric deformation at their mid-bays, and that, as the inward radial deflections increased, the effect of
7
6
Pressure (MPa)
5
4
3
2
1
0
2
4
6 8 10 12 14 16 Axial deflection (mm x 10−3)
18
20
2.69 Theoretical plot of pressure versus axial deflection for cone C.
© Carl T. F. Ross, 2011
Axisymmetric deformation of pressure vessels
97
5
4
Pressure (MPa)
3
2
1
0
8
16
24
32
36
Axial deflection (mm x 10−3)
2.70 Theoretical plot of pressure versus axial deflection for cone 9.
11
Pressure (MPa)
9
7
5
3
1 0
1
2
3
4
5
6
7
8
9
Axial deflection (mm x 10−2)
2.71 Theoretical plot of pressure versus axial deflection for cylinder 4.
© Carl T. F. Ross, 2011
15
13
11
9
Pressure (MPa)
7
5
3
1 0
1
2
3
4
5
6
7
8
9
Axial deflection (mm x 10−2)
2.72 Theoretical plot of pressure versus axial deflection for cylinder 5.
13
11
9
Pressure (MPa)
7
5
3
1 0
1
2
3 4 5 6 7 Axial deflection (mm x 10−2)
8
9
2.73 Theoretical plot of pressure versus axial deflection for cylinder 6.
© Carl T. F. Ross, 2011
Axisymmetric deformation of pressure vessels
99
axial pressure became more and more significant, resulting in a three-hinge buckling mechanism for each vessel. Comparisons between experiment and the nonlinear axisymmetric finite element solution showed good agreement for plastic axisymmetric buckling. Snap-thru buckling of shallow domes is described in Section 3.8.1.
© Carl T. F. Ross, 2011
3 Shell instability of pressure vessels
Abstract: This chapter deals with the shell instability (or lobar buckling) of unstiffened circular cylinders and cones under external hydrostatic pressure, together with domes. Solutions for the instability of thin-walled circular cylinders, cones and domes under uniform external pressure are described. Some of the theories are based on an analytical approach and some on a numerical one, namely the finite element method. Experimental results are provided and compared with those obtained by theory. The effects of initial out-of-circularity and inelastic buckling are also considered and two design charts are provided, which are easier to use than PD 5500. Key words: shell instability, thin-walled cylinders, out-of-circularity, buckling, design charts.
3.1
Shell instability of thin-walled circular cylinders
Under uniform external pressure, a thin-walled circular cylinder may buckle in the manner shown in Fig. 1.5, usually at a fraction of that pressure required to cause axisymmetric yield. If the circular cylinder is very long, its buckling resistance is very small, the vessel suffering failure in a flattening mode (i.e. ovalling). According to Bryan,37 the elastic instability pressure for an infinitely long circular cylinder under uniform lateral pressure is given by equation [3.1]: Pcr =
E ⎛ t⎞ 4 (1 − v 2 ) ⎝ a ⎠
3
[3.1]
where Pcr = buckling pressure t = wall thickness of circular cylinder a = mean radius of circular cylindrical shell E = Young’s modulus ν = Poisson’s ratio In their famous paper, Windenburg and Trilling38 state that equation [3.1] applies to long, thin, accurately made tubes, under uniform lateral pressure, when l > 4.9a (a/t)0.5 and a/t > 10, where l = length of tube. 100 © Carl T. F. Ross, 2011
Shell instability of pressure vessels
101
To demonstrate the strength of a long thin-walled tube, under lateral pressure, consider a tube with the following properties: E = 2 × 1011 N m2; ν = 0.3; yield stress = σyp = 300 MPa; a = 2 m; t = 2 × 10−2 m; l = 500 m. Now, a 2 = 100 > 10 = t 2 × 10 −2 and 4.9a ( a / t )
1/ 2
= 4.9 × 2 (100 )
1/ 2
= 98
l = 500 > 98 i.e. the Bryan formula applies. From equation [3.1], the lateral pressure to cause buckling, Pcr is given by: 2 × 1011 ⎛ 1 ⎞ ⎜ ⎟ = 54 945 Pa = 0.0549 MPa 4 × 0.91 ⎝ 100 ⎠ 3
Pcr =
Based on axisymmetric yield, the resistance of the vessel to withstand uniform lateral pressure is given by the well-known expression:1 P=
σ yp × t 1 = 300 MPa × = 3 MPa 100 a
i.e. the elastic instability pressure is only about 1/55th of the pressure to cause axisymmetric yield.
3.1.1 Von Mises formula However, for shorter tubes firmly supported at their ends, the buckling resistance can be considerably increased. Von Mises39 gives equation [3.2], which is the elastic instability pressure for a thin-walled circular cylindrical shell, simply supported at its ends and subjected to the combined action of uniform lateral and axial pressure: Pcr =
E ( t / a) ⎡⎣ n2 − 1 + 0.5 ( πa / l )2 ⎤⎦ 2 2 ⎧⎪ ⎡ 2 t2 πa ⎞ ⎤ ⎫⎪ 1 ⎛ ×⎨ n − 1+ + 2 ⎢ 2 2 ⎝ l ⎠ ⎥⎦ ⎬ 12a2 (1 − v) ⎣ ⎪⎩ ⎡⎣ n2 ( l / πa) + 1⎤⎦ ⎪⎭
[3.2]
where l = length of shell between adjacent supports and n = number of circumferential waves or lobes into which the vessel buckles, as shown in Fig. 3.1. The normal method of using equation [3.2] is to calculate Pcr for various values of n, and to select the lowest or minimum value of Pcr. The
© Carl T. F. Ross, 2011
102
Pressure vessels
n=2
n=3
n=4
3.1 Circumferential wave patterns for buckling modes.
value of n corresponding to the minimum value of Pcr is said to be the buckling eigenmode. Although, with the aid of a computer, it is simple enough to use equation [3.2], Windenburg and Trilling38 have provided a simpler version that predicts similar buckling pressures to that of von Mises: Pcr =
(1 − v )
2 0.75
2.42 E ( t / 2a)
5/ 2
⎡⎣( l / 2a) − 0.447 ( t / 2a)1 / 2 ⎤⎦
[3.3]
If ν = 0.3, then 2.6 E ( t / d ) 1/ 2 l / d − 0.45 ( t / d ) 5/ 2
Pcr =
[3.4]
where d = 2a = mean shell diameter. Equation [3.3] is also known as the David Taylor Model Basin (DTMB) formula. If, for example, it is required to increase the buckling resistance of the vessel of Section 3.1, which is extremely low, it will be convenient to stiffen the tube with suitably sized rings at intervals of (say) 2 m. Hence, according to equation [3.4], the inter-bay elastic instability buckling pressure (shell instability) is given by: Pcr =
2.6 × 2 × 1011 × ( 2 × 10 −2 / 4)
(2 / 4 ) − 0.45 ( 2 × 10 / 4) −2
2.5
0.5
=
919 239 = 1.963 MPa 0.4682
From this calculation, it can be seen that the elastic instability resistance of the vessel has been increased by a factor of about 36, by introducing stiffening rings to the very long tube of Section 3.1. It must, however, be pointed out that if the ring-stiffeners are not strong enough, the entire ring-shell combination can buckle bodily, and this mode of failure is known as general instability (Fig. 1.9).
3.1.2 Sturm’s models Sturm40 produced a simply supported and a fixed-edges solution, based on elastic theory, together with some experimental results for eight carefully © Carl T. F. Ross, 2011
Shell instability of pressure vessels
103
1'' 1/2'' Rubber tube seal
Model under external pressure
Aluminium alloy end disc cap
5/16'' 1/8''
3.2 End connection for Sturm’s models.
machined models which had initial out-of-roundness values less than 0.033t. The models were constructed in aluminium alloy and were sealed, as shown in Fig. 3.2. In order to determine the effects of end-conditions, Sturm,40 Nash41 and Ross42 produced fixed-edges solutions, and, in Table 3.1, comparison is made between these solutions and the simply supported solutions of Sturm, Reynolds, Ross and von Mises, together with the experimental results of Sturm. The solutions of Ross were based on the Kendrick strain energy43 expressions (see Section 4.1.1). In Table 3.1, comparison is also made with Windenburg’s thinness ratio λ, a parameter that can be used for detecting whether or not elastic instability solutions apply (see Section 3.1.3). From Table 3.1, it can be seen that the experimental results of Sturm agreed best with the simply supported solutions, but this was not altogether surprising, because the experimentally obtained boundary conditions were probably nearest the simply supported edges case. Table 3.1 also shows that the theoretical solutions, based on fixed ends, tended to agree with each other and predicted buckling pressures considerably higher than the simply supported edges solutions.
3.1.3 Windenburg’s models Experiments carried out earlier by Windenburg and Trilling38 on models soldered firmly at their ends (Table 3.2) do not, however, agree with these © Carl T. F. Ross, 2011
© Carl T. F. Ross, 2011
λ
4.897 0
3.720 5
3.064 3
2.129 1
1.827 3
1.457 7
1.457 7
3.671 7
No.
23
24
25
26
27
28
29
30
51.7 (4)
505 (6,7)
505 (6,7)
321 (5)
226 (4,5)
95 (3)
71.5 (3)
35.8 (2)
Sturm40
Fixed edges
50.4 (4)
463 (6)
463 (6)
294 (5)
230 (4)
106 (3)
74 (3)
38.5 (2)
Nash41
50 (4)
459 (6)
459 (6)
291 (5)
228 (4)
105 (3)
73.4 (3)
38 (2)
Ross42
34.2 (3)
358 (6)
358 (6)
216 (4)
152 (3)
62 (3)
45.2 (2)
26.3 (2)
Sturm40
36.4 (3)
346 (5)
346 (5)
217 (4)
159 (4)
76.2 (3)
51.6 (2)
29.7 (2)
Reynolds44
Simply supported edges
34.6 (3)
333 (5)
333 (5)
210 (4)
157 (4)
80 (3)
49.5 (2)
33.8 (2)
von Mises39
33.2 (3)
328 (5)
328 (5)
205 (4)
149 (4)
72.8 (3)
46.9 (2)
26.5 (2)
Ross42
34 (3)
339 (6)
330 (6)
240 (4,5)
155 (3,4)
81 (3)
53 (2,3)
29 (2)
Experimental
Table 3.1 Comparison of theoretical solutions with experimental observations from Sturm’s models40 (buckling pressures are in lbf in−2; the number of lobes is given in parentheses)
Shell instability of pressure vessels
105
Table 3.2 Comparison of theoretical solutions with experimental observations from Windenburg and Trilling’s models38 (buckling pressures are in lbf in−2; the number of lobes is given in parentheses)
No.
Out-ofroundness
Thinness ratio λ
von Mises39
Nash41
Experimental
33 46 47 50 56 58 60 61 62 67 69 70 71
0.11t 0.13t 0.16t 0.16t 0.16t 0.16t 0.11t 0.14t 0.12t 0.15t 0.16t 0.15t 0.14t
1.229 1.334 2.427 1.080 2.368 1.632 1.123 3.205 2.219 1.646 1.569 2.147 1.182
193 (12) 209 (13) 56.6 (8) 301 (17) 28.2 (11) 165 (9) 351 (12) 42 (5) 84.6 (6) 205 (6) 313 (5) 153 (4) 579 (8)
266 (14) 288 (14) 80.6 (9) 463 (19) 40 (13) 229 (10) 489 (13) 60.8 (6) 122 (8) 290 (7) 440 (7) 224 (5) 795 (9)
139 (13,14) 163 (13) 65 (9) 195 (19) 31 (11) 159 (–) 199 (14) 48 (5, 6) 89 (6) 209 (6) 288 (5, 6) 149 (4) 327 (–)
3 9 2 4 3 9 9 8 4 3 8 4 9
theories. In general, these theories tend to overestimate the buckling pressures for many typical vessels. Windenburg and Trilling concluded that this was because many of the cylinders buckled inelastically owing to the initial out-of-circularity of the models, i.e. because of initial geometrical imperfections, various parts of the shell became plastic, triggering off inelastic instability at a pressure less than that required to cause elastic instability. For this reason, the models of Windenburg and Trilling are among the most important of their time, as fabricated pressure vessels, even those made today, may have appreciable out-of-circularity, and as a result are likely to suffer inelastic instability. Windenburg and Trilling38 introduced their thinness ratio λ, which can be used to determine experimentally the plastic reduction factor (PKD), where:
λ=
4
{(l / d )
2
(t / d )3 } × (σ yp / E )
[3.5]
Other experimental results on machined stiffened models were carried out by Ross3 and Reynolds44 (Table 3.3). Details of Ross’s models 1, 2 and 3, together with the models which failed by general instability (see 4.1.1), are shown on Table 3.4. These models were machined from a thick-walled tube, made from HE9-WP aluminium alloy, with the following properties: E = 10.3 × 106 lbf in−2; 0.1% proof stress = 21 700 lbf in−2;
© Carl T. F. Ross, 2011
106
Pressure vessels Table 3.3 Buckling pressures (lbf in−2) for models of Ross3 and Reynolds44 (the number of lobes is given in parentheses) No.
von Mises39
Nash41
Experimental
Thinness ratio, λ
1 2 3 U12 U22
363 (8) 443 (8) 709 (10) 1764 (13) 959 (14)
500 (9) 617 (9) 1041 (11) 3040 (16) 1569 (17)
335 350 402 975 735
1.184 1.081 0.882 0.853 1.034
(8) (9) (12) (–) (–)
2 0 6 7 9
Table 3.4 Dimensions of models 1 to 7 (inches) (internal diameter = 10.25 in; h = 0.08 in (models 1–6), 0.081 in (model 7); N = number of ring stiffeners) No.
L1
L
Lb
bF
bf
d
N
1 2 3 4 5 6 7
3.5 3.75 2.5 1.6 1.6 1.6 –
4.5 3.75 2.25 2.0 2.0 2.0 10.0
26.625 24.275 15.775 19.92 19.92 19.92 10.0
0.325 0.40 0.40 0.08 0.08 0.08 –
0.325 0.325 0.325 0.08 0.08 0.08 –
0.62 0.62 0.62 0.08 0.12 0.16 –
5 5 5 9 9 9 –
0.2% proof stress = 22 600 lbf in−2; nominal peak stress = 26 900 lbf in−2; ν = 0.32 (assumed). Table 3.4 also shows details of the machined model No. 7, which had no stiffeners and was made from mild steel with the following properties: E = 28 × 106 lbf in−2; σyp = 40 700 lbf in−2; UTS = 65 200 lbf in−2; ν = 0.32 (assumed). Measurements of the initial out-of-roundness of model Nos. 1–6 revealed a maximum value of 0.003 in, and similar measurements on model No. 7 revealed a maximum initial out-of-roundness of 0.004 in. The models were tested to destruction in the tank shown in Fig. 3.3.
3.1.4 Design charts To obtain the design charts for the inelastic instability of circular cylinders under uniform external pressure, Pcr was calculated by the von Mises
© Carl T. F. Ross, 2011
Shell instability of pressure vessels Scale (ins) 5
3
Outlet to pressure gauge
1 0
107
24 HT bolts 5/8" BSF
'O' Rings As below
Model No.1 Flexible hose from pump
1/4" BSF cap head screws every 20°
Centre spindle
Closure plate
'O' Rings
Pressure tight cable gland
3.3 Test tank with attachments.
formula.39 The formula assumed that the cylinder was supported at its ends. This assumption was probably quite reasonable for the practical case, as neither the web of a typical ring frame, nor the thin-walled shell could resist much bending; in practice there may have been some degree of rotational restraint at the ends of the shell, but this was not very much. A design chart for the shell instability of accurately machined circular cylinders under hydrostatic pressure is given in Fig. 3.4. This chart uses the experimental observations of Sturm,40 Reynolds,44 Seleim and Roorda45 and Ross,3 Ross and Kimber,46 and Ross et al.47
© Carl T. F. Ross, 2011
Pressure vessels 1\Lambda (thinness ratio)
108
3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.0
2.0
4.0
6.0
8.0
10.0
PKD (Pcr/Pexp)
3.4 Design chart for the shell instability of machined circular cylinders.
From Fig. 3.4, it can be seen that the vessels whose values of 1/λ were less than 0.9, probably failed by elastic instability as PKD was approximately one. The three vessels that had the largest values of 1/λ actually failed by axisymmetric deformation, but Ross and Johns35 have shown that there is a link between plastic axisymmetric deformation and inelastic shell instability. Recent tests carried out at the University of Portsmouth found that in some instances of inelastic shell instability, the cylinder initially suffers plastic axisymmetric deformation before failing by inelastic shell instability. The ability to withstand pressure is reduced if the cylinder is manufactured with an initial out-of-circularity. Additionally, by plotting 1/λ against (Pcr /Pexp), the possible asymptotic behaviour of the graph was avoided in the region of axisymmetric failure. Figure 3.5 provides a design chart for some soldered and welded vessels, which had moderate values of initial out-of-circularity and which failed through shell instability. The soldered models of Windenburg and Trilling38 had initial out-of-circularity values which did not exceed 0.16t, but the initial out-of-circularity values of the models of Reynolds44 were not given. Here again, it can be seen that vessels which had values of 1/λ < 0.6, had a PKD not very far from unity and failed elastically, where Pexp = experimentally obtained buckling pressure and PKD = Pcr /Pexp. The design process therefore calculates Pcr and either λ or λ′ for the appropriate vessel and obtains PKD from the design chart, so that the predicted buckling pressure Ppred is: Ppred =Pcr /PKD Previously, Ross48 and Ross et al.47 presented design charts that were obtained by plotting λ or λ′ against Pcr /Pexp. However, for shorter and thicker vessels, the developed design curves were practically asymptotic to the horizontal axis, thus making it very difficult to measure PKD.
© Carl T. F. Ross, 2011
Shell instability of pressure vessels
109
1.6 1.4 1.2
1\Lambda
1.0 0.8 0.6
Safe side
0.4 0.2 0 0
0.5
1.0
1.5
2.0
2.5
3.0
PKD
3.5 Design chart for shell instability of soldered and welded circular cylinders.
By plotting 1/λ or 1/λ′ against Pcr /Pexp, the asymptotic behaviour of the design charts is avoided, so that more satisfactory design curves are obtained. Additionally, the design charts produced in the present report, use more experimental data than the design charts of Ross48 and Ross et al.47 and this makes them more reliable for design purposes. In order to demonstrate the use of Fig. 3.5, the vessel of Section 3.1 is assumed to be stiffened by suitably sized, equally spaced rings, 1 m apart. From equation [3.4]: Pcr = 4.213 MPa. From equation [3.5]:
λ=
4
{(1 / 4) / (2 × 10 2
−2
/ 4)
3
}
(300 × 106 / 2 × 1011 ) = 1.03
From the line of Fig. 3.5, the plastic reduction factor PKD is: PKD = Pcr /Pexp = 2.2 so that buckling pressure = 4.213/2.2 = 1.92 MPa For this vessel, therefore, the buckling pressure as calculated by equation [3.4] has to be reduced by the plastic reduction factor, which in this instance is 2.2. From Fig. 3.5, it can be seen that some models fail at an even lower pressure than that predicted by the von Mises formula and the plastic reduction
© Carl T. F. Ross, 2011
110
Pressure vessels
factor and, because of this, it is advisable to use a larger safety factor for guarding against instability than for guarding against axisymmetric yield. A computer program for calculating the shell instability buckling pressure of a thin-walled cylinder is provided in Appendix III. Failure against instability should be avoided at all costs, because of the difficulty of predicting buckling pressures. It is important to note that the plastic reduction factor must always be greater than one.
3.1.5 The ‘Holland’ The design chart of Fig. 3.5 can also be applied to the submarine ‘Holland’ (Fig. 3.6) and its 3/8th scale model; the latter was tested to destruction by the US Navy.49 The details are given in Table 3.5. The calculations for Ppred are given in Table 3.6, where the following assumptions are made: E = 30 × 106 lbf in−2; σyp = 34 160 lbf in−2; ν = 0.3.
3.6 The submarine ‘Holland’.
Table 3.5 Details of the submarine ‘Holland’ and its 3/8 scale model (inches)
t Size of frames Lf bf a Number of frames
Holland
Model
0.3 2 1/2 × 2 1/2 × 5/16 angles 18 2.5 48 –
0.109 1 × 1 × 1/8 angles 6.75 1.0 18 17
© Carl T. F. Ross, 2011
Shell instability of pressure vessels
111
Table 3.6 Predicted collapse pressures (psi) for the submarine ‘Holland’ and its 3/8 scale model (psi) (figures in parentheses = n)
λ Pcr PKD Ppred Pexp
Holland
Model
1.026 311.6 (15) 2.22 140.4 –
1.045 290.7 (15) 2.12 137.1 150
Axis
3.7 Axisymmetric buckling of a cone of large apical angle.
From Table 3.6, it can be seen that Ppred was similar for the submarine ‘Holland’ and its model as required by the US Navy; this was a remarkable achievement for the year 1899. It can also be seen from Table 3.6, that the collapse depth for the ‘Holland’ was about 316 ft; thus, its maximum safety factor in its just-built condition, was about 6.32.
3.2
Instability of thin-walled conical shells
To determine by analytical methods the uniform external pressure required to cause the elastic instability of a thin-walled conical shell is very difficult, and for these cases it is more convenient to use the finite element method (FEM). If the cone is of small apical angle, then buckling will take place in a lobar manner, as shown in Fig. 1.5, but if the apical angle of the cone is large, the vessel can buckle axisymmetrically, as shown in Fig. 3.7. In this instance, both axisymmetric yield and axisymmetric buckling must be considered. The solution presented in this section is based on the elastic instability of thin-walled cones of small apical angle, and it must be emphasised that
© Carl T. F. Ross, 2011
112
Pressure vessels u,x
f a w
u0
0
Rj r
Ri
w
w
v
f
j
i
l
3.8 Truncated conical shell element.
as these vessels are sensitive to initial geometrical imperfections, they too can suffer inelastic instability at buckling pressures considerably less than that causing elastic instability. Thus, with the aid of a sufficient number of experimentally obtained results, the plastic reduction factor for a particular thin-walled cone, of a certain apical angle, can be determined in a manner similar to that adopted for circular cylinders. This plastic reduction factor must then be divided into the theoretical elastic buckling pressure for the conical shell in question, to obtain the (reduced) predicted inelastic buckling pressure. A brief description of the solution, based on small deflection elastic theory, is now presented. The element, which was first developed in 1974,50 is a truncated cone with two nodal circles at its ends, as shown by Fig. 3.8. Each node has four degrees of freedom (u0, v, w0 and θ), making a total of eight degrees of freedom per element. To obtain the stiffness matrix, a modified form of Novozhilov’s vector of strains24 was used, as follows: ⎧ux ⎫ ⎪1 ⎪ 1 ⎪ os α ) ⎧ε x ⎫ ⎪ r ν φ + r (u sin α − w co ⎪ ⎪ε ⎪ ⎪ ⎪ ⎪ φ ⎪ ⎪ 1 u + ν − 1 ν sin x ⎪ x ⎪⎪γ xφ ⎪⎪ ⎪⎪ r φ ⎪ r {ε } = ⎨ ⎬ = ⎨ ⎬ w χ ⎪ x ⎪ ⎪ xx ⎪ ⎪ χφ ⎪ ⎪ 1 ⎪ sin α 1 wx ⎪ ⎪ ⎪ 2 wφφ + 2 ν φ cos α + ⎪ r r r ⎪⎩ χ xφ ⎪⎭ ⎪ ⎪ 1 1 1 ⎪ ⎪ ⎛1 ⎞ ⎪⎩2 ⎜⎝ r wxφ − r 2 wφ sin α + r ν x cos α − r 2 ν sin α cos α ⎟⎠ ⎪⎭
© Carl T. F. Ross, 2011
[3.6]
Shell instability of pressure vessels
113
The assumed displacement functions were: u = [ui (1 − ξ ) + u j ξ ] cos nφ
ν = [ν i (1 − ξ ) + ν j ξ ] sin nφ
w = [(1 − 3ξ 2 + 2ξ 3 ) wi + l (ξ − 2ξ 2 + ξ 3 )θ i
[3.7]
+ ( 3ξ 2 − 2ξ 3 ) w j + l ( −ξ 2 + ξ 3 )θ j ] cos nφ
where
ξ = x/l or
{U } = [ N ]{U i }
[3.8]
and
[ N ] = a matrix of shape functions; ⎧ui0 ⎫ ⎪ν ⎪ ⎪ i ⎪ placements = ⎨ 0 ⎬ {Ui } = a vector of nodal disp ⎪wi ⎪ ⎪⎩θ i ⎪⎭
[3.9]
The assumed displacement functions for {U} included a sinusoidal distribution in the circumferential direction to cater for the lobes, and also to simplify computation. The stiffness matrix was given by T [k ] = lπ [ DC ]T ∫0 r [ B1 ] [ D][ B1 ] dξ [ DC ] 1
[3.10]
where r = Ri + (Rj − Ri)ξ; and [B] = [B1]*(either cos nϕ or sin nϕ) (Table 3.7). If c = cos α and s = sin α, then: ⎡c ⎢0 ⎢ ⎢s ⎢ 0 [ DC ] = ⎢⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣
0 −s 1 0 0 c 0 0
04
0 0 0 1 c 0 s 0
0 1 0 0
⎤ ⎥ ⎥ 04 ⎥ ⎥ ⎥ −s 0⎥ ⎥ 0 0⎥ c 0⎥ ⎥ 0 1⎦⎥
© Carl T. F. Ross, 2011
[3.11]
© Carl T. F. Ross, 2011
0 0 −1 / l ⎧ 2 ⎪(1 − ξ ) sinα / r n (1 − ξ ) α / r − (1 − 3ξ + 2ξ 3 ) cos α / r ⎪ 0 ⎪ −n (1 − ξ ) / r [ −1 / l − (1 − ξ ) sinα / r ] ⎪ 0 0 6 12 ξ) / l 2 − + ( ⎪ 2 ⎪ 1 ⎡ n 2 3 ⎪ ⎢ − r 2 (1 − 3ξ + 2ξ ) + rl ⎪ ⎣ [B ′ ] = ⎨ 0 n (1 − ξ ) cos α / r 2 ⎤ ⎪ × ( −6ξ + 6ξ 2 ) sinα ⎥ ⎪ ⎦ ⎪ n (1 − ξ ) ⎡ n ⎪ ⎡ 2 2 ⎢ − cos α / rl − 2 ⎢ − ( −6ξ + 6ξ ) / rl + 2 ⎪ r2 r ⎣ r ⎣ ⎪ 0 ⎤ ⎤ ⎪ × (1 − 3ξ 2 + 2ξ 3 ) sinα ⎥ × cos α sin nα ⎥ ⎪ ⎦ ⎩ ⎦ 0 0 ⎫ − (3ξ 2 − 2ξ 3 ) cos α / r −l ( −ξ 2 + ξ 3 ) cos α / r ⎪ ⎪ 0 0 ⎪ ⎪ (6 − 12ξ ) / l 2 (−2 + 6ξ ) / l ⎪ 2 2 1 ⎪ 1 ⎡ n ⎡ nl 2 3 2 3 + − − − − + ξ + 3 ξ 2 ξ ξ ) ) ⎪ ⎢ r2 ( ⎢ r2 ( rl r ⎪ ⎣ ⎣ ⎬ ⎤ ⎤ × (6ξ − 6ξ 2 ) sinα ⎥ × ( −2ξ + 3ξ 2 ) sinα ⎥ ⎪⎪ ⎦ ⎦ ⎪ n nl ⎪ ⎡ n ⎡ n 2 2 2 ⎢ − (6ξ − 6ξ ) + 2 2 ⎢ − ( −2ξ + 3ξ ) + 2 ⎪ r r ⎣ rl ⎣ r ⎪ ⎤ ⎤ ⎪ 2 3 2 3 × (3ξ − 2ξ ) sinα ⎥ × ( −ξ + ξ ) sinα ⎥ ⎪ ⎦ ⎦ ⎭
Table 3.7 ‘Strain’ matrix for conical shell element
⎤ × (ξ − 2ξ 2 + ξ 3 ) sinα ⎥ ⎦
⎤ × (1 − 4ξ + 3ξ 2 ) sinα ⎥ ⎦ nl ⎡ n 2 2 ⎢ − (1 − 4ξ + 3ξ ) + 2 r ⎣ r
1 ⎡ n2 2 3 ⎢ − r 2 l (ξ − 2ξ + ξ ) + r ⎣
0 −l (ξ − 2ξ + ξ 3 ) cos α / r 0 (−4 + 6ξ ) / l 2
0
0
1/ l ξ sinα / r −nξ / r 0
2 (cos α / rl − ξ cos α sinα / r 2 )
nξ cos/ r
0 nξ / r (1 / l − ξ sinα / r ) 0
Shell instability of pressure vessels
115
The relationship between local and global displacements was:
{U i } = [ DC ] {U 10 } = [u1v1w1θ 1 u2 v2w2θ 2 ] and ⎡1 ⎢v ⎢ Et ⎢0 [ D] = (1 − v2 ) ⎢⎢0 ⎢0 ⎢ ⎣0
0 0 0 0 ⎤ ⎥ 1 0 0 0 0 ⎥ 0 (1 − v) / 2 0 0 0 ⎥ ⎥ 2 2 t / 12 vt / 12 0 0 0 ⎥ ⎥ vt 2 / 12 t 2 / 12 0 0 0 ⎥ 2 0 0 0 0 (1 − v) t / 24 ⎦ v
[3.12]
where [DC] = a matrix of directional cosines; [D] = a matrix of material constants for the isotropic case; and α = cone angle. To obtain the geometrical stiffness matrix, it must be remembered that the prebuckling in-plane meridional stress is:
σx ≈
Pr 2t cos α
and the prebuckling in-plane hoop stress is:
σφ ≈
Pr cos α t
However, as σx and σϕ are principal stresses, then the shear stress in the x–ϕ plane
τ xφ = φ Thus, the additional strains owing to large displacements {εL} are a modified version of Stricklin et al.51 as follows: ⎡ wx 1⎢ {ε L } = 2⎢ 0 ⎣
⎤ ⎧ wx ⎫ ⎪ ⎪ (v + wφ ) ⎥⎥ ⎨ (v + wφ ) ⎬ ⎪ ⎪ R ⎦⎩ R ⎭ 0
[3.13]
where ¯v = v cos α. Using the same notation as Zienkiewicz:23 ⎧ wx ⎫ ⎪ ⎪ ⎨ v + wφ ⎬ = [G ] {U } ⎪⎩ R ⎪⎭
[3.14]
© Carl T. F. Ross, 2011
116
Pressure vessels
Therefore the geometrical stiffness matrix [k1] was obtained from:
[k1 ] = [ DC ]T ∫vol [G]T [σ ][G] d ( vol ) [ DC ] = [ DC ]
∫ [G] [σ ][G] Rdφdxdz [ DC ] = [ DC ] πRlt ∫ [G ] [σ ][G ] dξ [ DC ] T
T
vol
T
1
1 T
1
0
1 T ⎡σ x T = [ DC ] Rlt ∫ [G1 ] ⎢ 0 ⎣0
[3.15]
0⎤ 1 [G ] dξ [ DC ] σ φ ⎥⎦
where [G] = [G1] *(either cos nϕ or sin nϕ). 1 ⎡0 0 (−6ξ + 6ξ 2 ) (1 − 4ξ + 3ξ 2 ) 0 ⎢ l 1 [G ] = ⎢ (1 − ξ ) c n nl ⎢0 − (1 − 3ξ 2 + 2ξ 3 ) − (ξ − 2ξ 2 + ξ 3 ) 0 ⎢⎣ R R R 1 0 (6ξ − 6ξ 2 ) (−2ξ + 3ξ 2 ) ⎤⎥ l ⎥ ξc n nl − ( 3ξ 2 − 2ξ 3 ) − ( −ξ 2 + ξ 3 )⎥ ⎥⎦ R R R
[3.16]
where c = cos α. The elastic instability pressures for various values of n were obtained from the eigenvalue equation:
[ K ] − [ KG ] = 0
[3.17]
[ K ] = ∑ [k ]
[3.18]
[ KG ] = ∑ [k1 ]
[3.19]
where
A computer program for determining the buckling pressures of thinwalled cones under external pressure for various values of n is presented in reference 28. This program was used to analyse a thin-walled truncated conical shell with an apical angle of 10°. The FEM assumed that the cone was fixed at its left end and simply supported at its right end, and these assumptions yielded a buckling pressure of 340 lbf in−2, with four lobes. This result was similar to Volmir’s prediction52 of 350 lbf in−2, where both ends were assumed to be simply supported.
3.2.1 Niordson’s method Niordson53 provides a formula for the elastic instability of perfect conical shells of small taper ratio, where he applies the von Mises formula
© Carl T. F. Ross, 2011
Shell instability of pressure vessels
117
y ,w y0 , w0 L/2
x ,u R
r j node
i node
x 0 , u0
R1 t
R2
3.9 Equivalent ‘cylinder’.
to an equivalent cylinder, as shown in Fig. 3.9, where a = R and l = L = generator length.
3.2.2 Truncated conical shells Short truncated conical shells can also fail by inelastic instability. To obtain the design chart for perfect vessels,50 Pcr was calculated using the FEM of Ross54 for the truncated circular conical shells. The left ends of the vessels were assumed to be fixed and their right ends were assumed to be clamped. In this instance, the calculation for λ assumed that the truncated conical shell was an equivalent circular cylindrical shell32 where the dimensions for R and L were as shown in Fig. 3.9. The design chart for this case is shown in Fig. 3.10, where the experimental observations are from the machined vessels of Ross48 and Ross and Kimber.46 There are few other published data on thin-walled truncated conical shells and most studies have vital information missing.
3.3
Buckling of orthotropic cylinders and cones
If a circular cylinder or cone is made from glass-reinforced plastic (GRP) or carbon fibre reinforced plastic (CFRP), it may be convenient for it to have orthotropic material properties, where the axes of orthotropy of the material lie in the axial and circumferential directions. This arrangement is particularly suitable for orthotropic circular cylinders and cones under uniform external pressure, as their buckling resistance depends to a large extent on the material properties of the vessel in its circumferential direction.
© Carl T. F. Ross, 2011
118
Pressure vessels 2.5 2.4 2.3 2.2 2.1 2.0 1.9 1.8 1.7 1.6 1.5 1.4 1/l
1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 Pcr/Pexp
3.10 Design chart for shell instability of machined truncated conical shells.
3.3.1 Orthotropic element The elemental stiffness and geometrical stiffness matrices55 can be obtained explicitly for an orthotropic circular cylinder under uniform external pressure, as follows.
© Carl T. F. Ross, 2011
Shell instability of pressure vessels
119
The vector of strains in this instance was a modified form of Novozhilov,24 as shown by equation [3.20]:
ε x = ux εφ =
γ xφ =
1 w νφ − R R
1 uφ + ν x R
χ x = wxx wφφ ν φ + R2 R2 1 1 = 2 ⎛⎜ wxφ + ν x ⎞⎟ ⎝R R ⎠
χφ = χ xφ
[3.20]
where u, v, w = displacements in the x, ϕ and z directions, respectively (Fig. 3.11); R = mean shell radius; εx, εϕ = direct strains in the x and ϕ directions, respectively; γxϕ = shear strain in the x–ϕ plane; χx, χϕ = flexural strains in the x and ϕ directions, respectively; χxϕ = twist in the x–ϕ plane. The assumed displacement functions incorporated a sinusoidal variation in the circumferential direction to simplify computation, as follows: u = [ui (1 − ξ ) + u j ξ ] cos nφ
ν = [ν i (1 − ξ ) + ν j ξ ] sin nφ
w = [(1 − 3ξ 2 + 2ξ 3 ) wi + l (ξ 2 − 2ξ 2 + ξ 3 )θ i
+ ( 3ξ 2 − 2ξ 3 ) w j + l ( −ξ 2 + ξ 3 )θ j ] cos nφ
[3.21]
where ξ = x/l; θ = rotational displacement; and n = number of circumferential waves or lobes. The matrix of elastic constants allowed for orthotropic properties was as follows: ⎡E F 1 ⎢ F E1 ⎢ ⎢0 0 [ D] = ⎢ ⎢0 0 ⎢0 0 ⎢ ⎣0 0
0 0
0 0 0
A 0 B C1 0 C B1 0 0 0 f
2R
i
0⎤ 0 ⎥⎥ 0⎥ ⎥ 0⎥ 0⎥ ⎥ D⎦
0 0 0
u
v
w x
[3.22]
j
3.11 Circular cylindrical element.
© Carl T. F. Ross, 2011
f
w
120
Pressure vessels
where E = Ext/(1 − νxνϕ) F1 = νxE F = νϕE1 = F1 (assumed) A = Gt B = Ext3/[12(1 − νxνϕ)] C = νxB = C1 (assumed) B1 = Eϕt3/[12(1 − νxνϕ)] D = Gt3/12 G = rigidity modulus Ex = elastic modulus in x direction Eϕ = elastic modulus in ϕ direction νx = Poisson’s ratio owing to stress in x direction νϕ = Poisson’s ratio owing to stress in ϕ direction t = shell thickness Substituting equation [3.21] into equation [3.20] and combining with equation [3.22], the elemental stiffness matrix for an orthotropic cylinder is given by ⎡ K11 ⎢K 21 [K ] = ⎢ ⎢ ↓ ⎢ ⎣ K81
K12 K18 ⎤ K 22 K 28 ⎥ ⎥ ↓ ↓ ⎥ ⎥ K82 K88 ⎦
where K11 = E/l2 + An2/(3R2) K12 = −F1n/(2Rl) + An/(2Rl) K13 = F1/(2Rl) K14 = F1/(12R) K15 = −E/l2 + An2/(6R2) K16 = F1n/(2Rl) − An/(2Rl) K17 = F1/(2Rl) K18 = −F1/(12R) K22 = E1n2/(3R2) + A/l2 + 4D/(R2l2) K23 = −7E1n/(20R2) − 4Dn/(R2l2) K24 = −E1nl/(20R2) K25 = Fn/(2Rl) + An/(2Rl) K26 = E1n2/(6R2) − A/l2 − 4D/(R2l2) K27 = −3E1n/(20R2) + 4Dn/(R2l2) K28 = Enl/(30R2) K33 = 13E1/(35R2) + 12B/l4 − 6(C1+ C)ζ/(5R2l2) + 13B1ζ2/(35R4) + 24 Dn2/(5R2l2)
© Carl T. F. Ross, 2011
[3.23]
Shell instability of pressure vessels
121
K34 = 11E1l/(210R2) + 6B/l3 − C1ζ/(10R2l) − 11Cζ/(10R2l) + 11B1ζ 2l/(210R4) + 2Dn2/(5R2l) K35 = −F/(2Rl) K36 = 3E1n/(20R2) + 4Dn/(R2l 2) K37 = 9E1/(70R2) − 12B/l4 + 6C1ζ/(5R2l 2) + 6Cζ/(5R2l 2) + 9B1ζ 2l/(70R4) − 24Dn2/(5R2l 2) K38 = −13E1/(420R2) + 6B/l 3 − C1ζ/(10R2l) − Cζ/(10R2l) − 13B1ζ 2l/(420R4) + 2Dn2/(5R2l) K44 = E1l 2/(105R2) + 4B/l 2 −2(C1+ C)ζ/(15R2) + B1l 2ζ 2/(105R4) + 8Dn2/(15R2) K45 = −F/(12R) K46 = −E1nl/(30R2) K47 = −13E1l/(420R2) − 6B/l 3 + C1ζ/(10R2l) + Cζ/(10R2l) + 13B1ζ 2l/(420R4) − 2Dn2/(5R2l) K48 = −E1l 2/(140R2) + 2B/l 2 + C1ζ/(30R2) + Cζ/(30R2) − B1l 2ζ 2l/(140R4) − 12Dn2/(15R2) K55 = E1/l 2 + An2/3R2 K56 = F1n/(2Rl) − An/(2Rl) K57 = −F1/(2Rl) K58 = F1/(12R) K66 = E1n2/(3R2) + 4D/(R2l 2) + A/l 2 K67 = −7E1n/(20R2) − 4Dn/(R2l 2) K68 = E1nl/(20R2) K77 = 13E1/(35R2) + 12B/l4 − 6(C1+ C)ζ/(5R2l 2) + 13B1ζ 2/(35R4) + 24Dn2/(5R2l 2) K78 = −11E1l/(210R2) − 6B/l3 + C1ζ/(10R2l) + 11Cζ/(10R2l) − 11B1ζ 2l/(210R4) − 4Dn2/(10R2l) K88 = E1l 2/(105R2) + 4B/l 2 − 2(C1 + C)ζ/(15R2) + B1ζ 2l 2/(105R4) + 8Dn2/(15R2) ζ = 1 − n2 To allow for geometrical nonlinearity, the additional strains resulting from large displacements were assumed to be as follows: in the x direction:
δε x =
wx2 2
in the circumferential direction:
δε φ =
1 ⎡ ν wφ ⎤ + 2 ⎢⎣ R R ⎥⎦
2
[3.24]
in the x–ϕ plane:
δγ xφ =
2wx wφ R
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The matrix of stresses, just before buckling, was assumed to be: ⎡σ x ⎢0 ⎢ ⎣⎢ 0
0 σφ 0
0 ⎤ 0 ⎥ ⎥ τ xφ ⎥⎦
[3.25]
where σx = pR/(2t); σϕ = pR/t; τxϕ = 0; and p = uniform external pressure. From equations [3.21], [3.24] and [3.25], the geometrical stiffness matrix for the cylindrical shell element was found to be: ⎡ KG11 ⎢K G 21 [ KG ] = ⎢ ⎢ ↓ ⎢ ⎣ KG 81
KG12 KG18 ⎤ KG 22 KG 28 ⎥ ⎥ ↓ ↓ ⎥ ⎥ KG 82 KG 88 ⎦
[3.26]
where KG22 = σϕ / (3R2) KG23 = −7σϕ n / (20R2) KG24 = σϕ / (6R2) KG27 = σϕ nl / (30R2) KG28 = σϕ nl / (30R2) KG33 = 6σx / (5l 2) + 13 σϕ n2 / (35R2) KG34 = σx / (10l) + 11 σϕ n2l / (210R2) KG36 = −3σϕ n / (20R2) KG37 = −6σx / (5l 2) + 9σϕn2 / (70R2) KG38 = σx / (10l) – 13 σϕ n2l / (420R2) KG44 = 2σx / 15 + σϕ n2l 2 / (105 R2) KG46 = −σϕ nl / (30R2) KG47 = −σx / (10l) + 13 σϕ n2l / (420R2) KG48 = −σx / 30 – σϕ n2l 2 / (140R2) KG66 = σϕ / (3R2) KG67 = −7σϕ n / (20R2) KG68 = σϕ nl / (20R2) KG77 = 6σx / (5l 2) + 13 σϕ n2 / (35R2) KG78 = −σx / (10l) −11 σϕ n2l / (210R2) KG88 = 2σx / 15 + σϕ n2l 2 / (105R2) To test the theory used here, comparison is made in Table 3.8 of the experimental buckling pressure of the isotropic model No. 7 of Table 3.4 with the FEM solution and the David Taylor Model Basin (DTMB) formula of equation [3.4]. The experimental strain measurements of model No. 7 revealed that, at 95.1% of the collapse pressure, the maximum observed
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Table 3.8 Buckling pressures (psi) and number of lobes (in parentheses) for model No. 7
λ
DTMB formula (simply supported)
FEM, top fixed and bottom clamped
FEM, top and bottom fixed
Experimental
1.424
427
533 (6)
596 (6)
568 (5)
Table 3.9 Buckling pressures (Pa) for orthotropic conical shells (Pcr / Ex) × 106 Model number
R2
l
Eϕ × 109
νϕx
Singer56
FEM
1 2 3 4 5
0.15 0.15 0.2 0.25 0.5
0.1 0.1 0.2 0.3 0.8
5 3.86 2.59 3.86 2.59
0.022 0.09 0.234 0.09 0.234
0.0409 0.2041 0.3048 0.0419 0.0307
0.0415 (16) 0.2426 (14) 0.466 (10) 0.0463 (12) 0.0467 (10)
circumferential stress was only about two-thirds of the yield stress, thereby indicating that the model failed elastically. Furthermore, as the model was firmly clamped around its edges, it is not surprising that the clamped- and fixed-edges solutions agreed best with the experimental buckling pressure.
3.3.2 Buckling of orthotropic conical shells For the buckling of thin-walled orthtropic conical shells under uniform external pressure, the element described in Section 3.2 can be adapted if the matrix of material constants given in equation [3.22] is used instead of the matrix of material constants for the isotropic case given in equation [3.12]. Comparison between the experiments of Singer56 and the FEM of this section is made in Table 3.9. Singer’s model No. 1 was constructed from plywood and his models Nos. 2–5 were made from fiberglass-reinforced epoxy. As Singer presented his results with the aid of a dimensionless parameter, it was necessary to make certain numerical assumptions for the FEM solution. In all instances the radius of the smaller end, R1 (Fig. 3.12), was assumed to be 0.1 m and Ex was assumed to be 1 × 1011 Pa. The wall thickness for
© Carl T. F. Ross, 2011
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Pressure vessels
l x 30°
2R1
2R2
3.12 Truncated conical shell (Singer56).
model No. 5 was assumed to be 3.48 × 10−4 m and the wall thickness for all the other models was assumed to be 3.47 × 10−4 m. Where applicable, the units in Table 3.9 are in metres and Newtons, and the figures in parentheses represent the predicted number of circumferential waves or lobes (n) into which the vessels buckle. From Table 3.9, it can be seen that comparison of the theory of Singer and the FEM solution is reasonable for three of the five models. The slightly higher buckling pressures of the FEM solution, compared with those predicted by Singer, were probably because Singer assumed simply supported ends, but the FEM solution assumed that the smaller diameter was fixed and the larger diameter was simply supported.
3.4
Buckling of thin-walled domes
The dome ends of a submarine can take various forms, from oblate hemiellipsoids to prolate hemi-ellipsoids, and from hemispheres to torispherical caps. In this section, the elastic instability of hemi-ellipsoidal and hemispherical shells is considered.
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3.4.1 Varying meridional curvature element If either a prolate hemi-ellipsoidal shell or a hemispherical shell is subjected to uniform external pressure, it can buckle in a lobar manner in its flank, as shown by Fig. 1.6. Similarly, if an oblate hemi-ellipsoidal shell is subjected to uniform external pressure, it can buckle axisymmetrically, in its nose, as shown by Fig. 1.7. The following theories,57 which are based on the FEM, are for a constant meridional curvature element (CMC) and also for a varying curvature elements (VMC). Novozhilov24 gives the vector of strains for a doubly curved axisymmetric element, as follows: ⎧ε s ⎫ ⎪ε ⎪ ⎪ φ ⎪ ⎪ ε sφ ⎪ {ε } = ⎨ ⎬ ⎪K s ⎪ ⎪Kφ ⎪ ⎪ ⎪ ⎩ K sφ ⎭
[3.27]
where 1 ∂u w + R1α ∂ξ R1
εs =
1 ⎛ ∂v ⎞ ε φ = ⎜ + u sin β + w cos β ⎟ ⎝ ⎠ r ∂φ 1 ⎛ r ∂v ∂u ⎞ − v sin β + ⎟ ε sφ = ⎜ r ⎝ R1α ∂ξ ∂φ ⎠ Ks =
1 ∂u −1 ∂ 2 w ∂2 β − u + 2 2 2 2 2 R1 α ∂ξ ∂s R1 α ∂ξ
[3.28]
The term u(∂2β/∂ξ 2) is zero for the constant curvature case: 1 ⎡ ∂ 2 w cos β ∂v 1 ⎛ 1 ∂w ⎤ ⎞ Kφ = ⎢ − + − u⎟ sin β ⎥ ⎠ r ⎣ r∂ φ 2 r ∂φ R1 ⎜⎝ α ∂ξ ⎦ 1 ∂u ⎞ 2 ⎛ −1 ∂ 2 w sin β ∂w cos β ∂v sin β cos β + + − v+ K sφ = ⎜ r ∂φ R1α ∂ξ r R1 ∂φ ⎟⎠ r ⎝ R1α ∂ξ ∂φ The assumed displacement functions were: u=
(1 − ξ ) 2
ui cos nφ +
(1 + ξ ) 2
u j cos nφ
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(1 + ξ ) vi sin nφ + v j sin nφ 2 2 (ξ 3 − 3ξ + 2) w cos nφ + (1 + ξ ) (1 − ξ )2 R αθ cos nφ w= i i 1 4 4 ( −ξ 3 + 3ξ + 2) w cos nφ − (1 − ξ ) (1 + ξ )2 R αθ cos nφ + j 1 j 4 4 v=
(1 − ξ )
[3.29]
where
ξ=
s Rα
or, in matrix form, ⎧u ⎫ ⎪ ⎪ ⎨ v ⎬ = [ N ]{U i } ⎪w ⎪ ⎩ ⎭ where {Ui} = a matrix of nodal displacements [N] = a matrix of ‘shape functions’ = [N′] * (cos nϕ) or (sin nϕ) These displacement functions, which are with respect to the meridian of the shell, assume a linear variation for u and v, and a parabolic variation for w. The assumptions for the displacement functions in the circumferential direction allow for a variation in {U} to cater for the lobes, and also to simplify computation. Substitution of the displacement functions and their derivatives into the strain relationships yields the following expressions for strain: ⎡ −1 (ξ 3 − 3ξ + 2) (1 + ξ ) (1 − ξ )2 α 1 0 εs = ⎢ 4 R1 4 2 R1α ⎣ 2 R1α 2 3 ( −ξ + 3ξ + 2) − (1 − ξ ) (1 + ξ ) α ⎤ cos nφ U { i} ⎥ 4 R1 4 ⎦
0
(1 + ξ ) (1 − ξ 2) R1α 1 ⎡1− ξ n (1 − ξ ) (ξ 3 − 3ξ + 2) εφ = ⎢ cos β sin β cos β 2 2 r⎣ 2 4 n (1 + ξ ) ( −ξ 3 + 3ξ + 2) (1 + ξ ) sin β cos β 2 2 4 2 ⎤ − (1 − ξ ) (1 + ξ ) R1α cos β ⎥ cos nφ {U i } 4 ⎦
© Carl T. F. Ross, 2011
Shell instability of pressure vessels 1 ⎡ − n (1 − ξ ) ⎛ − (1 − ξ ) sin β r ⎞ ε sφ = ⎢ − ⎜⎝ ⎟ 0 0 r⎣ 2 2 2 R1α ⎠ r ⎞ ⎛ − (1 + ξ ) sin β ⎤ + ⎜⎝ ⎟⎠ 0 0 ⎥ sin nφ {U i } 2 2 R1α ⎦ ⎡ −1 Ks = ⎢ 2 ⎣ 2 R1 α
0
−3ξ 2 R12α 2
− ( −1 + 3ξ ) 0 2 R1α
1 2 R12α
127
− n (1 + ξ ) 2
3ξ 2 R12α 2
− (1 + 3ξ ) ⎤ cos nφ {U i } 2 R1α ⎥⎦
1 ⎡ − (1 − ξ ) sin β − n (1 − ξ ) cos β Kφ = − ⎢ r⎣ 2 R1 2r 2 3 2 ⎛ − n (ξ − 3ξ + 2) ( 3ξ − 3) sin β ⎞ + ⎜⎝ ⎟⎠ 4r 4R α 1
⎛ − n (1 + ξ ) (1 − ξ ) R1α (1 − 2ξ + 3ξ 2 ) sin β ⎞ + ⎜⎝ ⎟⎠ 4 4r 2
− (1 + ξ ) sin β 2 R1
2
n (1 + ξ ) cos β 2r
⎛ − n2 ( −ξ 3 + 3ξ + 2) 3 ( −ξ 2 + 1) sin β ⎞ + ⎜⎝ ⎟⎠ 4r 4 R1α ⎛ n 2 ( 1 − ξ ) (1 + ξ )2 (1 − 2ξ − 3ξ 2 ) sin β ⎞ ⎤ cos nφ U R − α { i} 1 ⎜⎝ ⎟⎠ ⎥ 4r 4 ⎦ [3.30] 1 ⎡ − n (1 − ξ ) ⎛ sin β cos β (1 − ξ ) cos β ⎞ K sφ = ⎢ − ⎜⎝ − ⎟ r ⎣ R1 r R1α ⎠ ⎛ − n sin β (ξ 3 − 3ξ + 2) n ( 3ξ 2 − 3) ⎞ + ⎜⎝ r 2 2 R α ⎟⎠ 1
⎛ − n sin β (1 + ξ ) (1 − ξ ) R1α n ( −1 − 2ξ + 3ξ 2 ) ⎞ + ⎜⎝ ⎟⎠ 2 2r 2
− n (1 + ξ ) ⎛ − sin β cos β (1 + ξ ) cos β ⎞ + ⎜⎝ ⎟ R1 r R1α ⎠
⎛ − n sin β ( −ξ 3 + 3ξ + 2) 3n ( −ξ 2 + 1) ⎞ + ⎜⎝ 2r 2 R1ξ ⎟⎠
⎛ n sin β (1 − ξ ) (1 + ξ )2 R1α n (1 − 2ξ − 3ξ 2 ) ⎞ ⎤ − ⎜⎝ ⎟⎠ ⎥ sin nφ {U i } 2r 2 ⎦ In matrix form, these appear as: T {ξ} = [ξs ξφ ξsφ Ks Kφ Ksφ ] = [ B] {U i }
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Pressure vessels
The stiffness matrix is given by: 1
2n
[k ] = ∫∫ [ B]T [ D][ B] dx dy = ∫−1 ∫0 [ B]T [ D][ B] r dφ R1α dξ = πR1α ∫ r [ B1 ] [ D][ B1 ] dξ 1
T
−1
[3.31]
where [B1] = [B]/(either cos nϕ or sin nϕ); and [D] = matrix of elastic constants (equation [3.12]) In global coordinates,
[K 0 ] = [ DC ]T [k ][ DC ] where ⎡ζ 1 ⎣04
[ DC ] = ⎢
04 ⎤ ζ 2 ⎥⎦
⎡ c1 ⎢ 0 [ζ 1 ] = ⎢ ⎢ − s1 ⎢ ⎣ 0
0 s1 1 0 0 c1 0 0
0⎤ 0⎥ ⎥ 0⎥ ⎥ 1⎦
[3.33]
⎡ c2 ⎢ 0 [ζ 2 ] = ⎢ ⎢ − s2 ⎢ ⎣ 0
0 s2 1 0 0 c2 0 0
0⎤ 0⎥ ⎥ 0⎥ ⎥ 1⎦
[3.34]
[3.32]
and c1 = cos β1; c2 = cos β 2; s1 = sin β1; s2 = sin β 2. The wall thickness was assumed to vary linearly along the meridian of the shell, as follows: t=
(1 − ξ ) 2
t1 +
(1 + ξ ) 2
t2
where t1 = wall thickness at node 1; and t2 = wall thickness at node 2. To obtain the geometrical stiffness for this element, it was necessary to consider the additional strains resulting from large deflections,40 as follows:
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Shell instability of pressure vessels
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⎧ 1 ⎛ ∂w u ⎞ 2 ⎫ − ⎟ ⎪ ⎜⎝ ⎪ 2 ∂s R1 ⎠ ⎪ ⎪ ⎧δε s ⎫ ⎪ 2 ⎪⎪ ⎞⎤ ⎪ ⎪ 1 ⎡ 1 ⎛ ∂w ⎪ {ε L } = ⎨δε φ ⎬ = ⎨ ⎢ ⎜ − ν cos β ⎟ ⎥ ⎬ ⎠⎦ ⎪δε ⎪ ⎪ 2 ⎣ r ⎝ ∂φ ⎪ ⎩ sφ ⎭ ⎪ ⎞ ⎤⎪ ⎛ ∂w u ⎞ ⎡ 1 ⎛ ∂w − ⎟⎢ ⎜ − ν cos β ⎟ ⎥ ⎪ ⎪⎜ ⎠ ⎦ ⎪⎭ ⎪⎩⎝ ∂s R1 ⎠ ⎣ r ⎝ ∂φ
[3.35]
Now the additional strain energy resulting from large deflections is given by: 1 2
{U i }T [G]T [G]{U i }
so that the geometrical stiffness matrix was given by 1
T
2π
[k1 ] = ∫ t [G] [σ ][G] R1α dξrdφ = R1α ∫−1 ∫0 t [G]T [σ ][G] rdξdφ
[3.36]
= πR1α ∫ t [G1 ] [σ ][G1 ] rdξ 1
T
−1
where [G ] = [G]/(either cos nϕ or sin nϕ as given in Table 3.10) 1
⎡σ s ⎣0
[σ ] = ⎢
0⎤ σ φ ⎥⎦
[3.37]
For hemi-ellipsoidal thin-walled domes, the prebuckling membrane stresses58 were approximated by:
σs = σφ =
{
− pa2
t 2 ( a2 cos2 β + b2 sin 2 β )
1/ 2
}
− pa2 {b2 − ( a2 − b2 ) cos2 β }
{
t 2b2 ( a2 cos2 β + b2 sin 2 β )
1/ 2
}
[3.38]
Table 3.10 [G1] for a doubly curved axisymmetric element
(3ξ2 − 3)
− (1 − ξ ) 2R1
0
0
− (1 − ξ ) cos β 2r
− (1 + ξ ) 2R1
0
0
− (1 + ξ ) cos β 2r
4R1α
−n (ξ − 3ξ + 2) 4r (−3ξ2 + 3) 4R1α 2
−n ( −ξ 3 + 3ξ + 2) 4r
© Carl T. F. Ross, 2011
(1 − ξ )2 − 2 (1 + ξ ) (1 − ξ ) 4 −n (1 + ξ ) (1 − ξ ) R1α 4r (1 + ξ )2 − 2 (1 − ξ ) (1 + ξ ) 4 2
−n (1 − ξ ) (1 + ξ ) R1α 4r 2
130
Pressure vessels
where p = uniform pressure (external positive); a = radius of hemiellipsoidal dome at base; and b = height of hemi-ellipsoidal dome. Integration in the circumferential direction was carried out explicitly, and in the meridional direction integration was carried out numerically, using four Gauss points per element. A computer program for determining the elastic instability pressures for thin-walled domes under uniform external pressure is available.9
3.4.2 Thin-walled domes To compare theory with experiment, 45 thin-walled domes were tested to destruction under uniform external pressure. The domes (Fig. 3.13) consisted of ten different profiles, varying from oblate hemi-ellipsoids of aspect ratio (AR) 0.25 to prolate hemi-ellipsoids of AR 4.0, where AR = dome height/base radius. The domes were constructed in solid urethane plastic (SUP), and their base diameters and wall thicknesses were 0.2 m and 2 mm, respectively. Each series of domes was constructed very precisely by thermosetting the liquid plastic between two carefully machined aluminium alloy male and female moulds, the out-of-roundness of some of the prolate domes being only about ±0.04 mm. By experiment, the material properties of SUP were found to be: E = 2.9 GPa; ρ = 1200 kg/m3; and ν = 0.3 (assumed). Table 3.9 shows a comparison between experiment and the FEM of this section, where the figures in parentheses represent the predicted number of circumferential waves or lobes into which the vessels buckle.
3.13 Ten hemi-ellipsoidal oblate/prolate domes.
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Table 3.11 Buckling pressures for SUP domes Experimental buckling pressure Pcr (MPa) Aspect ratio
SUP 1
SUP 2
SUP 3
SUP 4
SUP 5
Mean
CMC
4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.7 0.444 0.25
0.086 0.099 0.145 0.179 0.358 0.517 1.137 – – –
0.103 0.103 0.151 0.199 0.386 0.475 1.365 1.075 0.282 0.062
0.089 0.103 0.158 0.220 0.358 0.461 1.337 1.062 0.282 0.062
0.103 0.103 0.145 0.206 0.358 0.461 1.324 – 0.275 0.055
0.103 0.089 0.151 0.199 0.358 0.537 1.248 – 0.262 0.062
0.097 0.100 0.150 0.201 0.364 0.490 1.280 1.070 0.276 0.060
0.112 0.136 0.167 0.225 0.323 0.552 1.412 0.851 0.317 0.108
9 5 2 4 1 5
4 4 7 9 0 6
7 1
6 4 5 6 4 9
7 1
4 4 5 8 4 9
8 2
4 6 7 9 4 7
0 1
(5) (5) (6) (7) (8) (9) (10) (0) (0) (0)
CMC, constant meridional curvature.
From Table 3.11, it can be seen that comparison between experiment and theory is reasonable for these domes, and also that the hemispherical dome had the largest resistance to buckling. The very oblate and prolate domes had abysmal resistance to buckling, their buckling pressures being less than one-tenth of the buckling pressures of the hemispherical dome. The experimentally obtained buckling pressures were, in general, lower than their theoretical predictions, and this was attributed to geometrical imperfections in the vessels, the hemispherical and prolate hemi-ellipsoids being sensitive to out-of-circularity in their flanks, and the oblate hemiellipsoids being sensitive to the loss of meridional curvature in their noses. In addition to the tests carried out on the domes of Table 3.11, two more domes from this series were tested to destruction, in much more carefully observed experiments. These domes have already been described in Section 2.4 and their out-of-circularity plots are shown in Figs 2.28 and 2.29. The domes were expected to buckle in a lobar manner, and for this reason 20 strain gauges were attached to the inner surfaces of these vessels in the circumferential direction, in their flanks, at the position where it was expected that the maximum buckling deflection would take place. The strain gauge readings were taken at several pressures, and at small increments of pressure especially near the anticipated buckling pressures. Figures 3.14 and 3.15 show circumferential strain plots superimposed over the out-of-circularity plots, at pressures just below the experimentally obtained buckling pressures. These figures appear to show that the vessels buckled elastically, and the buckling pressures together with the number of
© Carl T. F. Ross, 2011
132
Pressure vessels 10 9
11 12
8
13
7
14
6 Specimen 1·5 Mag Filter x100 N Talyrond
15
5
5''arm 16
4 Ra D r nk ste Tayl or Hobson Leice 2000 3000
17
18
3
2 19 20
1
Gauge no. 1 Division=0.001''
3.14 Microstrain readings recorded at a pressure of 0.483 MPa for the 1.5 aspect ratio dome.
lobes (n) are shown in Table 3.12. For further details of the VMC element, see Section 3.4.3. Each model buckled with the appearance of a solitary lobe in its flank, and details of these lobes are shown in Fig. 3.16. On removal of the pressure, the models appeared to regain their original shapes.
3.4.3 VMC element For the VMC (Fig. 3.17), the angle β varies along the meridian, so that in equation [3.28]:
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Shell instability of pressure vessels
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7 8
6
9
5
10
4
11
3
Specimen 3·0 Cone Filter Mag x50 N Talyrond
12
2
10''Arm 13
1 Ra
nk
Tay lor
r D ste H o b s o n L ei c e
14
20
1000 2000 15
3000 16
19 18
17 Gauge no. 1 Division=0·002''
3.15 Microstrain readings recorded at a pressure of 0.165 MPa for the 3.0 AR dome.
Table 3.12 Buckling pressures (MPa) and number of lobes n AR
Experimental
CMC
VMC
1.5 3.0
0.496 (10) 0.172 (6, 7)
0.552 (9) 0.167 (6)
0.548 (9) 0.167 (6)
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Pressure vessels
(a)
(b)
3.16 Position and size of the buckled regions: (a) shell AR 1.5, meridional length 107 mm, circumferential length 83 mm, distance of top of lobe from base 27 mm; and (b) shell AR 3.0, meridional length 115 mm, circumferential length 75 mm, distance of top of lobe from base 50 mm.
A w
x=1
u
q
j
b
bj
v w
x = −1 i bi
r
Rs
Axis of symmetry
f
Rj
Part section on A–A
A
3.17 VMC element.
∂ 2 β / ∂s 2 ≠ 0 Thus, it is a simple matter to extend the solution of Section 3.4.1, by assuming that
β = a0 + a1 s + a2 s 2
[3.39]
Cook27 shows that the three constants in equation [3.39], namely a0, a1, a2, can be determined from the following three conditions:
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Shell instability of pressure vessels
135
β = β1 at s = 0 β = β2
at s = L
and
∫
L
0
L
sin ( β − βc ) ds ∫ ( β − βc ) ds = 0 0
so that a0 = β1
a1 = ( 6βc − 4β1 − 2β 2 ) a2 =
( 3β1 + 3β 2 − 6βc ) L2
[3.40]
where 1/R1 = −∂β/∂s; L = arc length of element; and βc = chord angle. Cook shows that: l ⎧(β1 − βc ) ⎫ ⎡ 4 −1⎤ ⎧(β1 − βc ) ⎫ ⎨ ⎨ ⎬ ⎬ 60 ⎩(β 2 − βc )⎭ ⎢⎣ −1 4 ⎥⎦ ⎩(β 2 − βc )⎭ T
Ll+
[3.41]
and s
r = R1 + ∫ sin β ds
[3.42]
β β c + dy / dx
[3.43]
⎧( β1 − βc ) ⎫ y = [ x (1 − x / l ) ( x 2 / l ) ( −1 + x / l )] ⎨ ⎬ ⎩( β 2 − βc )⎭
[3.44]
0
and
where
and l = chord length. For hemi-ellipsoids, it is a simple matter to carry out the necessary integrations, and for axisymmetric elements of more complex shape, integration can be carried out numerically with the aid of an additional mid-side node. The mid-side node need only be used to carry out the integrations and need not be used to determine the stiffness matrices. However, for a more comprehensive element, it may be advisable to include a mid-side node to determine the elemental stiffness matrix. Comparison is made between theory (VMC) and experiment in Table 3.13 for two SUP models, whose base diameters and wall thicknesses were 0.5 m and 4 mm, respectively. These models were constructed in a similar manner to those described in Section 3.4.2. The out-of-roundness of model L1 was 1.42 mm and that of model L2 was 1.51 mm. From Table 3.13, it can be seen that comparison between experiment and theory was reasonable, but that model L1 buckled at a pressure a little higher than model L2. This may be partly because of the larger
© Carl T. F. Ross, 2011
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Pressure vessels Table 3.13 Buckling pressures and number of lobes n (in parentheses) of models L1 and L2 (MPa)
Model L1 Model L2
FEM
Experimental
0.202 (8) 0.202 (8)
0.214 (8) 0.203 (–)
3.18 Model L2, with rupture.
out-of-roundness of model L2 and partly because model L1 had 20 equally spaced resistance electrical strain gauges fitted in a circumferential direction around its flank to its inner surface, whereas model L2 had no strain gauges attached to it at all, i.e. the strain gauges on model L1 may have strengthened it a little in the circumferential direction. The important point about both models L1 and L2 is that they appeared to buckle elastically, with the appearance of a solitary lobe in their flanks, and that on removal of the pressure the models appeared to regain their original shapes. Further increases in pressure caused the vessels to buckle in the same positions, but at progressively lower pressures; the models eventually ruptured, as shown in Fig. 3.18.
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3.4.4 Near hemispherical domes The investigation described in Section 3.4.2 shows that a hemispherical dome buckles in a lobar manner, in its flank, as shown in Fig. 3.19. However, Galletly et al.59 state that if there is flatness in the nose of a hemispherical dome, it can buckle axisymmetrically; in any case, most theories assume that a hemispherical dome buckles axisymmetrically. From Table 3.11, it can be seen that the hemispherical dome buckles asymmetrically (n > 1), and that the dome of AR = 0.7 buckles axisymmetrically (n = 0). To determine the transition point between axisymmetric buckling and lobar buckling, a theoretical investigation is now carried out on oblate hemi-ellipsoidal domes, whose AR varies between 0.7 and 1.0. These domes are assumed to be the same series as those referred to in Table 3.11 , and their theoretical buckling pressures Pcr, together with their associated values of the number of circumferential waves n, as predicted by the VMC element, are shown in Table 3.14. From Table 3.14, it can be seen that the transition point between lobar buckling (n > 1) and axisymmetric buckling (n = 0) is for a dome with an AR somewhere between 0.85 and 0.87. Table 3.14 also shows that when the AR is reduced from 1.0 to 0.95, the value of n decreases from 11 to 3.
3.19 Lobar buckling of a hemispherical dome.
Table 3.14 Buckling pressures (MPa) and number of lobes n (in parentheses) for oblate domes Aspect ratio
Pcr (n)
0.8
0.85
0.87
0.9
0.95
1.0
1.021 (0)
1.147 (0)
1.194 (2)
1.265 (2)
1.378 (3)
1.427 (11)
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Pressure vessels
Inspection of the buckling eigenmodes for AR 0.87–0.95 showed that these were of the complex forms shown in Fig. 5.20. These findings appear to agree with the observations of Galletly et al.59 (see their Figure 12).
3.5
Boundary conditions
Boundary conditions play an important role in the experimentally obtained buckling pressures for many vessels, fixed boundaries usually resulting in a higher buckling pressure than simply supported edges. However, the effect of rotational restraint at a boundary for a thin shell is small, as thin shells have little resistance to bending. In this context, axial restraint at the boundary plays a much more significant role in increasing buckling pressures than does rotational restraint, i.e. a clamped boundary only yields a slightly higher buckling pressure than a simply supported one. Boundary conditions can be classified as follows: Simply supported: w = 0; u ≠ 0; v ≠ 0; θ ≠ 0 Clamped: u ≠ 0; v = w = θ = 0 Fixed: u = v = w = θ = 0 To determine the effects of stiffener size on shell instability, Ross and Johns60 tested to destruction three machine-stiffened circular cylinders under uniform external pressure. The three vessels had similar geometrical properties, the main difference being in the size of the ring-stiffeners, as shown in Table 3.15. The mechanical properties of TVR1 and TVR2 were found to be similar, but TVR3 proved to be slightly different. For TVR1 and TVR2: Young’s modulus = 29.3 × 106 lbf in−2; yield stress = 38 300 lbf in−2; and nominal peak stress = 79 300 lbf in−2. For TVR3: Young’s modulus = 29.3 × 106 lbf in−2; yield stress = 43 820 lbf in−2; and nominal peak stress = 84 690 lbf in−2. For all models, it is assumed v = 0.3. The models were very precisely made, as shown by the out-of-circularity plots for TVR2 in Fig. 3.20. They were tested in the tank shown in Fig. 3.21 and all three models buckled in a lobar manner, as shown in Fig. 1.5. Comparison is made in Table 3.16 of the experimentally obtained buckling pressures, the analytical solutions of von Mises and Ross, and the FEM solution described in Section 3.2. The von Mises solution assumes simply supported edges and the Ross solution,42 which was based on the elastic theory of Kendrick,43 assumed fixed edges. One FEM solution assumed that the left end was fixed and the right end was clamped, whereas the other one assumed that both edges of the shell were fixed at the ring-stiffeners. Sixteen equal length elements were used to model the shell for both FEM solutions. From Table 3.16 it can be seen that the buckling pressures of these vessels were larger than the simply supported case, and also that the experimental
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Table 3.15 Dimensions of the models (inches)
4.000
0.75
0.675
L1
b1
d1
h1
b2
L2
L3 d2
h2
h3
Model Number
L1
L2
L3
TVR1
1.5
4.0
1.5
0.725 0.725
0.750
0.750 0.025 0.0251
0.025
TVR2
1.5
4.0
1.5
0.725 0.725
0.375
0.375 0.025 0.0250
0.025
TVR3
1.5
4.0
1.5
0.725 0.725 0.0625 0.0625 0.025 0.0250
0.025
d1
d2
b1
b2
h1
h2
h3
buckling pressures of the models increased in proportion to the sizes of the stiffening rings. It was also interesting to note from Table 3.16 that the relatively simple analytical fixed-edges solution compared favorably with the much more complex FEM solution.
3.5.1 End closures To test the effect of various types of end closures on thin-walled cylinders, Galletly and Aylward61 carried out a thorough experimental and theoretical investigation on several vessels. They sealed the ends of their machined circular cylinders with the following types of end closure: (a) (b) (c) (d) (e) (f)
conical (α = 75°); conical (α = 45°); torispherical; oblate ellipsoidal; toriconical (α = 45°), and hemispherical,
where α = cone angle. Galletly and Aylward found the experimental buckling pressures of these vessels to vary from 1.275 MPa for (a) to 3.86 MPa for (b). They also found that comparisons between the experimental buckling pressures and the theoretical predictions of BOSOR69 were good.
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Pressure vessels
Specimen TVR 2 Filter Mag 4 Taylor-Hobson Talyrond
il Fa
Tra c
e e of inter nal surfac
1 Division = 0·0001''
Specimen TVR 2 Mag Filter 4 Taylor-Hobson Talyrond
il Fa
Tra c
e e of enter nal surfac
3.20 Out-of-circularity plots for TVR2 at mid-length.
© Carl T. F. Ross, 2011
Shell instability of pressure vessels 4 3 2 ins
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16 HT bolts 5/8" BSF
1 0 Outlet to pressure gauge
'O' rings As below Model No. 3
Flexible hose from pump
1/4" BSF
'O' ring
bolts
Closure plate
3.21 Test tank with attachments.
Table 3.16 Comparison of theoretical and experimental buckling pressures (lbf in−2) (the number of lobes into which the vessels buckle are given in parentheses)
Model
λ
von Mises (simply Ross supported) (fixed)
TVR1 TVR2 TVR3
1.629 9 1.629 9 1.747 9
245 (6) 245 (6) 244 (6)
3.6
The legs of off-shore drilling rigs
345 (7) 345 (7) 343 (7)
FEM FEM (clamped/fixed) (fixed)
Experimental
305 (6) 305 (6) 303 (6)
320 (6) 316 (6) 304 (6)
343 (7) 343 (7) 340 (7)
It is possible for the legs of off-shore drilling rigs to be subjected to a uniform lateral pressure together with axial pressure of a different magnitude to the lateral pressure, because of the effects of the weight of the rig. Much work on this topic has been carried out by Galletly et al.4 and by Walker and co-workers63–65 and it is a simple matter to extend the theory given in Section 3.2 to cater for this effect. Care must, however, be taken
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Pressure vessels
to ensure that the rig’s legs are not prone to suffer from other modes of failure, including buckling.
3.7
Some buckling formulae for domes and cones
3.7.1 Von Kármán and Tsien’s formula For the buckling of a perfect thin-walled sphere under uniform external pressure, von Kármán and Tsien66 give the following theoretical expression: Pcr =
2 Et 2 a2
[3.45]
[ 3 (1 − v 2 )]
However, experiments67 on the thin-walled spherical shells show that equation [3.45] overpredicts buckling pressures, and Roark and Young32 state that the ‘probable’ actual minimum buckling pressure is given by: Pcr = 0.365E ( t / a)
2
[3.46]
3.7.2 Spherical shell cap For a spherical shell cap (Fig. 3.22) with a half-centred angle between 20° and 60°, and where: 400 ≤ a / t ≤ 2000
t
R
q
q
0
3.22 Spherical shell cap.
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Kloppel and Jungbluth67 give the following empirical formula: t (θ ° − 20°) ⎤ ⎡ (0.07a / t ) ⎤ ⎡ 1− Pcr = ⎢1 − 0.175 (0.3E ) ⎛⎜⎝ ⎞⎟⎠ 20° ⎥⎦ ⎢⎣ 400 ⎥⎦ a ⎣
2
[3.47]
3.7.3 Truncated conical shell For a perfect thin-walled truncated conical shell with closed ends and under uniform external pressure, Roark states that the von Mises formula for circular cylinders can be used, if the slant length of the cone is substituted for the length of the cylinder, and a is taken as: a = ( RA + RB ) / ( 2 cos a)
[3.48]
where the notation is shown in Fig. 3.23. Thus, if this approach is applied to the Windenburg and Trilling formula given in equation [3.4], the elastic buckling pressure for a perfect thinwalled conical shell is given by: 2.6 E ( t / dm ) ( cos α ) (l / dm ) − 0.45 (t / dm )1 / 2 (cos α )−0.5 5/ 2
Pcr =
3/ 2
where dm = RA + RB = mean diameter of the cone. The above formula can be used to analyse the thin-walled cone described in reference 31, whose details are: RA = 25 in, RB = 50 in, t = 1 in, l = 141.78 in, and α = 10°, so that dm = 75 in and: 2.6 × 10 × 106 (1 / 75) ( 0.9848 ) = 521.6 / 1.838 (141.78 / 75) − 0.45 (1 / 75)0.55 (0.9848 )−0.5 2.5
Pcr =
1.5
i.e. Pcr = 284 lbf in−2 which compares with Volmir’s prediction of 350 lbf in−2 and the FEM prediction of 340 lbf in−2.
u,x a RA
u0
θ w
0
w
RB
w φ
l
3.23 Truncated conical shell.
© Carl T. F. Ross, 2011
v
144
Pressure vessels
3.8
Inelastic instability
It must be emphasised that buckling formulae based on small-deflection elastic theory tend to predict higher buckling pressure than those obtained experimentally. The main reason for this is that, in practice, most pressure vessels have geometrical imperfections, and these geometrical imperfections can cause serious loss of buckling resistance, or ‘plastic knockdown’. This effect is worsened for thicker and shorter vessels and, conversely, is of less importance for long vessels of small thickness-to-radius ratios. However, sensible structural design should eliminate possible failure owing to instability because of the difficulty of predicting the effects of geometrical imperfections, if this is at all possible; but if it is not possible to eliminate failure caused by instability, the so-called safety factor against buckling should be made very large. One method of determining inelastic buckling pressures of imperfect vessels is to determine their plastic reduction factors from experimental results, in a manner similar to that described in Section 3.1.3. For the inelastic buckling of thin-walled domes under external pressure, such a method was produced by Galletly et al.,68 who used a factor ¯λ , where:
λ = 1.285
{(a / t ) (σ yp / E )}
[3.49]
Galletly et al., who compared experimental results with the famous shell buckling program BOSOR5,69 stated that if a hemispherical dome tended to be flat at its apex, then it was more likely to fail axisymmetrically, with its nose denting inwards, than through lobar buckling. Other studies on domes, including torispherical ones, were made by Galletly et al.59 and by Newland.70
3.8.1 Snap-thru buckling The investigation carried out in Section 3.4 on the collapse of hemiellipsoidal domes under uniform external pressure, considered nonsymmetric and symmetric bifurcation buckling. That investigation gave poor results for the oblate domes which may have failed through snap-thru buckling, where the curvature of the nose of the dome is reversed. In this instance, the compressive stresses increase nonlinearly with increase in external pressure, until the vessel fails in its nose through snap-thru buckling.54 For slightly less oblate domes, Galletly et al.68 and Blachut71 have shown that they can buckle in a lobar manner, where the lobe forms nonsymmetrically about the nose, but not necessarily in the flank. In this section, a theoretical and an experimental investigation is reported on seven hemi-ellipsoidal oblate domes, which were tested to destruction
© Carl T. F. Ross, 2011
Shell instability of pressure vessels
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under external hydrostatic pressure. Four of these domes were constructed in GRP and three in SUP. The domes were of AR 0.25, 0.444 and 0.7, where: AR =
dome height base radius
The nominal base radius of all the domes was 100 mm. The GRP domes were made from glass chopped strand mat, where the moulds used were some accurately made SUP domes. The SUP domes were made by injecting liquid SUP between carefully machined male and female aluminium alloy domes and then thermosetting the SUP domes. Thus the SUP domes were accurately made. The glass fibre domes were manufactured by wet laying chopped strand glass fibre mat around the external surfaces of the relevant SUP moulds. The base diameter of the SUP moulds was 0.2 m and the wall thicknesses of the SUP domes was 2 mm. The temperature of the workshop in which the GRP domes were manufactured, was kept between 15 and 20 °C. The materials were brought into the workshop at least one day before they were required, to allow the temperatures of the materials to stabilise. Two of the GRP domes, namely AR 0.25 and AR 0.444, are shown in Figs 3.24 and 3.25. The geometrical details of the GRP domes, including their values of initial out-of-roundness e were measured on a Mitutoyo co-ordinate measuring machine with a touch-trigger probe. The geometrical details of these
3.24 GRP dome (AR 0.25 and 0.444), top.
3.25 GRP dome (AR 0.25 and 0.444), bottom.
© Carl T. F. Ross, 2011
146
Pressure vessels
Table 3.17 Geometrical details of GRP domes
Model
Initial out-ofroundness e (mm)
Base diameter (mm)
Dome height (mm)
Wall thickness (mm)
AR AR AR AR
0.3144 0.2892 0.4805 0.2395
200.8 200.8 200.8 200.8
20.68 40.55 73.05 72.56
5.30 4.00 2.75 3.34
0.25 0.444 0.7A 0.7B
Securing bolts Securing annulus Rubber annulus seal Relief valve Dome Pressure inlet
3.26 Test tank with dome.
GRP domes are given in Table 3.17. Strain gauges were attached to the concave faces of the domes, as these were free of water pressure. The domes were tested under external water pressure in the test tank shown schematically in Fig. 3.26. From Fig. 3.26, it can be seen that the test rig allowed visual inspection of the concave surfaces of the domes and that as these surfaces were free of water, the strain gauges did not require water proofing. It must be emphasised that, initially, the GRP domes were porous to the effects of water, so that before testing them they needed water proofing. This was achieved by the application of several coats of marine varnish to the convex faces of the domes. Before applying any water pressure to the domes, a relief valve was left open to the atmosphere to allow any trapped air in the system to be pumped out. When the bulk of the trapped air was pumped out, the relief valve was made watertight and the experiment commenced. Each dome had six Cu/Ni linear strain gauges attached to its inner surface; three in the meridional direction and three in the circumferential direction. One pair of strain gauges was attached to the nose of each dome and one pair attached near the dome base in the flank of the vessel. The third pair of strain gauges was attached in between these two pairs. Each vessel in turn was subjected to a gradually increasing external water pressure. The pressure was applied by a hand-driven hydraulic pump. Strain gauge readings were taken at several values of external water pressure. Each
© Carl T. F. Ross, 2011
Shell instability of pressure vessels
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3.27 Snap-thru failure of AR 0.25.
vessel was tested to destruction. The vessels AR 0.25 and AR 0.444 clearly failed by axisymmetric snap-thru buckling, as shown in Figs 3.27 and 3.28. The vessel AR 0.7A, failed non-symmetrically in a lobar manner, as shown in Fig. 3.29, whereas the vessel AR 0.7B failed in the flange, as shown in Fig. 3.30. This may have been because the flange was machined flat to achieve water tightness and this may have weakened the flange in AR 0.7B. Flat tensile specimens were manufactured in a similar manner to the dome shells. These revealed the following material properties:GRP: Tensile modulus = E = 5.44 GPa; Tensile fracture stress = 141.9 MPa; and Poisson’s ratio = v = 0.3 (assumed). The volume fraction was found to be 6.37%. SUP: Tensile modulus = E = 2.9 GPa; Tensile fracture stress = 70 MPa; and Poisson’s ratio = v = 0.3 (assumed).
© Carl T. F. Ross, 2011
148
Pressure vessels
3.28 Snap-thru failure of AR 0.44.
3.29 Lobar failure of AR 0.7A.
© Carl T. F. Ross, 2011
Shell instability of pressure vessels
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3.30 Failure in flange of AR 0.7B.
Table 3.18 Buckling pressures of the hemi-ellipsoidal domes Dome
Material
E (GPa)
t (mm)
Pexp (bar)
AR AR AR AR AR AR AR
SUP SUP SUP GRP GRP GRP GRP
2.9 2.9 2.9 5.44 5.44 5.44 5.44
2.0 2.0 2.0 5.3 4.0 2.75 3.34
0.6 2.76 10.70 8.28 12.07 31.38 40.00
0.25 0.444 0.7 0.25 0.444 0.7A 0.7B
The experimentally obtained buckling pressures of the four GRP domes, together with the three SUP domes are shown in Table 3.18, where: t = wall thickness (mm) E = tensile modulus (GPa) Pexp = experimentally obtained buckling pressures (bar). The theoretical analysis was via FEM, which allowed for both material and geometrical nonlinearity, as described in Chapter 2. The finite element used for this analysis was the thin-walled truncated conical shell element described in Ross.50 Plots of pressure against the axial deflection at the nose of the vessel are given for all the vessels in Figs 3.31 to 3.37. From Figs 3.31 to 3.37, it can be seen there was some non-convergence, particularly for the AR 0.7 series. This may have been because this series of vessels did not fail axisymmetrically, but by a lobar mode, where the lobe was unsymmetrical about the nose of the vessel, as shown in Fig. 3.29. For
© Carl T. F. Ross, 2011
150
Pressure vessels 0.6
Pressure (bar)
0.5 0.4 0.3 0.2 0.1 0 0
1
2
3
4
5
6
Deflection (mm)
3.31 Theoretical pressure–axial deflection plot for AR 0.25 (SUP).
4 3.5
Pressure (bar)
3 2.5 2 1.5 1 0.5 0 0
2
4
6
8
Deflection (mm)
3.32 Theoretical pressure–axial deflection plot for AR 0.444 (SUP).
AR 0.7B, it failed in the flange. Table 3.19 gives the theoretical buckling pressures (Pcr), as obtained from Figs 3.31 to 3.37. These buckling pressures are compared with the experimentally obtained buckling pressures (Pexp); convergence was not good for the AR 0.7 series. The flatter oblate hemiellipsoidal domes appeared to fail though a snap-thru axisymmetric buckling mode, in the noses of the vessels. The AR 0.7 series appeared to fail by a lobar mode, where the lobe was non-symmetrical about the nose of the
© Carl T. F. Ross, 2011
Shell instability of pressure vessels
151
14
Pressure (bar)
12 10 8 6 4 2 0 0
2
4
6
8
10
Deflection (mm)
3.33 Theoretical pressure–axial deflection plot for AR 0.7 (SUP).
16 14 Pressure (bar)
12 10 8 6 4 2 0 0
2
4
6
8
10
Deflection (mm)
3.34 Theoretical pressure–axial deflection plot for AR 0.25 (GRP).
domes. The AR 0.7B model failed in the flange, near its flank. Comparisons between experiment and the nonlinear finite element theory were good.
3.9
Higher order elements for conical shells
A deficiency with the elements described in Sections 3.2 and 3.4 is that they assume a linear variation for the displacements in the meridional and
© Carl T. F. Ross, 2011
152
Pressure vessels 16 14
Pressure (bar)
12 10 8 6 4 2 0 0
2
4
6
8
10
Deflection (mm)
3.35 Theoretical pressure–axial deflection plot for AR 0.444 (GRP).
50 45
Pressure (bar)
40 35 30 25 20 15 10 5 0 0
1
2
3
4
5
Deflection (mm)
3.36 Theoretical pressure–axial deflection plot for AR 0.7A (GRP).
azimuthal directions. Rajagopalan72 suggested that the elements could be improved by using higher order polynomials in these two directions. Additionally, by using ‘mid-side’ nodes, together with Guyan reduction,73 he showed that the size of the system matrices need not be increased. In 1995, Ross74 presented two conical shell elements using Rajagopalan’s idea. One element used cubic relationships for all the u, v and w displacements and the other used quadratic relationships for u and v, and cubic relationship for w.
© Carl T. F. Ross, 2011
Shell instability of pressure vessels
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50 45
Pressure (bar)
40 35 30 25 20 15 10 5 0 0
1
3
2
4
5
Deflection (mm)
3.37 Theoretical pressure–axial deflection plot for AR 0.7B (GRP).
Table 3.19 Buckling pressures for the hemi-ellipsoidal domes (bar) Model
Material
Pexp
Pcr
AR AR AR AR AR AR AR
SUP SUP SUP GRP GRP GRP GRP
0.6 2.76 10.70 8.28 12.07 31.38 40.00
0.55 2.5 12.0 10.3 13.0 40.0 45.0
0.25 0.444 0.7 0.25 0.444 0.7A 0.7B
3.9.1 The element ALLCUBE For this element, the displacement functions for u, v and w were all assumed to be of cubic form. Now the displacement function for w is of the Hermitian form shown by equation [3.7], but the displacement functions for u and v are obtained as follows. If the nodal displacements for u are u1, u2, u3 and u4 and the nodal displacements for v are v1, v2, v3, and v4, whose positions are as shown in Fig. 3.38, then: u = α1 + α 2 x + α 3 x2 + α 4 x3 where αi = unknown constants, to be determined.
© Carl T. F. Ross, 2011
[3.50]
154
Pressure vessels l l/3
l/3 1
l/3 4
3
x, u 2
Axis
3.38 Nodal displacement positions for u and v of cubic form for conical shells.
The boundary conditions are: @ x = 0, @ x = l, @x = l/3 @ x = 2l / 3,
u = u1 u = u2 u = u3 u = u4
[3.51]
Substituting each of the above boundary conditions into equation [3.50], four simultaneous equations are obtained that, on solving, result in the following values for αi.
α 1 = u1 3 α 2 = ( −11u1 / 6 + u2 / 3 + 3u3 − 3u4 / 2 ) l 9 α 3 = 2 ( 2u1 − u2 − 5u3 + 4u4 ) 2l 9 α 4 = 3 ( −u1 + u2 + 3u3 − 3u4 ) 2l
[3.52]
Substituting equations [3.52] into equation [3.50] and rearranging, the following is obtained for u: u = u1 (1 − 11ξ / 2 + 9ξ 2 − 9ξ 3 / 2 ) + u2 (ξ − 9ξ 2 / 2 + 9ξ 3 / 2 )
+ u3 (9ξ − 45ξ 2 / 2 + 27ξ 3 / 2 )
+ u4 ( −9ξ / 2 + 18ξ 2 − 27ξ 3 / 2 ) u = u1 F1 + u2 F2 + u3 F3 + u4 F4 where
ξ=x l F1 = 1 − 11ξ 2 + 9ξ 2 − 9ξ 3 2
© Carl T. F. Ross, 2011
[3.53] [3.54]
Shell instability of pressure vessels
155
F2 = ξ − 9ξ 2 2 + 9ξ 3 2 F3 = 9ξ − 45ξ 2 2 + 27ξ 3 2 F4 = −9ξ 2 + 18ξ 2 − 27ξ 3 2 Similarly,
ν = ν 1F1 + ν 2 F2 + ν 3 F3 + ν 4 F4
[3.55]
Hence, it is convenient to assume the following displacement functions for u, v and w: u = (u1F1 + u2 F2 + u3 F3 + u4 F4 ) cos nφ
ν = (ν 1F1 + ν 2 F2 + ν 3 F3 + ν 4 F4 ) sin nφ
w = [(1 − 3ξ 2 + 2ξ 3 ) w1 + l (ξ − 2ξ 2 + ξ 3 )θ 1
+ ( 3ξ 2 − 2ξ 2 ) w2 + l ( −ξ 2 + ξ 3 )θ 2 ] cos nφ
[3.56]
or
{U } = [ N ]{U i } The displacement functions for u and v are very different from those assumed by Rajagopalan72 in his Chapter 6. The stiffness matrix was obtained from the strain-displacement relationships of equation [3.10]. The mass matrix was obtained from equation [5.2]. It is evident that all these matrices are of order 12 × 12, but to simplify computations, Guyan reduction73 is used to eliminate the displacements u3, v3, u4, v4, in a manner similar to that of Rajagopalan.72 Thus, these matrices will now be of order 8 × 8, so that the original programs can be utilised with minimal changes.
3.9.2 The element QUQUCUBE For the QUQUCUBE element, the displacement functions for u and v were assumed to be of quadratic form in the meridional direction, whereas the displacement function for w was of the cubic form assumed in equation [3.7]. In this instance, the positions of the nodal displacements for u and v are shown in Fig. 3.39. Let: u = α1 + α 2 x + α 3 x2
[3.57]
© Carl T. F. Ross, 2011
156
Pressure vessels l/2
l/2
x, u 3
1
2
Axis
3.39 Nodal displacement positions for quadratic form.
having the boundary conditions: x = 0, u = u1 @ x = l, u = u2 @ x = l / 2 u = u3
[3.58]
By substituting equations [3.58] into equation [3.57], three simultaneous equations are obtained, the solution of which results in the following expressions for αi:
α 1 = u1 1 α 2 = ( −3u1 − u2 + 4u3 ) l 2 α 3 = 2 (u1 + u2 − 2u3 ) l
[3.59]
Substituting equations [3.59] into equation [3.57] and rearranging, the following is obtained: u = F1u1 + F2u2 + F3u3
[3.60]
where F1 = 1 − 3ξ + 2ξ 2 F2 = −ξ + 2ξ 2 F3 = 4ξ − 4ξ 2
[3.61]
Similarly,
ν = F1v1 + F2 v2 + F3v3
[3.62]
Hence, the displacement functions for u, v and w are: u = ( F1u1 + F2u2 + F3u3 ) cos nφ
ν = ( F1ν 1 + F2ν 2 + F3ν 3 ) sin nφ
w = [(1 − 3ξ 2 + 2ξ 3 ) w1 + l (ξ − 2ξ 2 + ξ 3 )θ 1
+ ( 3ξ 2 − 2ξ 3 ) w2 + l ( −ξ 2 + ξ 3 )θ 2 ] cos nφ
© Carl T. F. Ross, 2011
[3.63]
Shell instability of pressure vessels
157
Substitution of the above displacement functions and their derivatives into equations [3.10] and [3.15] leads to the stiffness, geometrical and mass matrices for the QUQUCUBE element. Once again, Guyan reduction can be used to eliminate the u3 and v3 displacements, so that these matrices can be reduced from order 10 × 10 to order 8 × 8. This process enables the computer program for ALLCUBE to be modified quite easily to incorporate the QUQUCUBE element.
3.9.3 Computer analysis In this section, comparisons are made between various analyses for instability. The three elements of Ross, namely LILICUBE (linear–linear– cubic50), ALLCUBE and QUQUCUBE, are compared with Rajagopalan’s LILICUB (linear–linear–cubic) and REDCUBE (reduced-all-cubic), using Kendrick’s75 example in Table 3.20. Kendrick’s example is shown in Fig. 3.40, where four equal length elements were used to model this cylinder. From the results of Table 3.20, it can be seen that the simple LILICUBE element of Ross gives results very similar to the more sophisticated analyses, and better than the LILICUB element of Rajagopalan; this is despite the fact that only four equal length elements were used.
Table 3.20 Buckling pressures (lbf in−2) for Kendrick’s example
n
Kendrick75
Rajagopalan72 Ross von Mises39 LILICUB REDCUBE LILICUBE QUQUCUBE ALLCUBE
10 11 12 13
841 774 749 753
843 776 751 755
929 836 805 805
805 – 743 –
791 758 751 764
787.7 753.9 745.3 757.7
786.5 753.0 744.6 757.1
1''
106.5'' I
Axis 40''
5
3.40 Kendrick’s example: u1 = v1 = w1 = 0; E = 30 × 106 lbf in−2; and v = 0.3.
© Carl T. F. Ross, 2011
158
Pressure vessels
10°
Simply supported edge 50'' (u1 = v1 = w1 = 0)
Simply supported 100'' edge (w19 = 0)
141.78''
3.41 Simply supported cone.
Table 3.21 Buckling pressures (lbf in−2) for cone Ross n
LILICUBE
QUQUCUBE
ALLCUBE
3 4 5
425.0 339.3 403.5
421.8 337.2 400.9
422.7 337.4 401.0
Comparisons are also made between Ross’ solutions for the thin-walled circular cone shown in Fig. 3.41. For this cone, 18 elements were used with the mesh previously described.21 The assumed boundary conditions were simply supported at the ends and the results are shown in Table 3.21. The predicted buckling pressure by Volmir’s formula52 was 350 lbf in−2; this compares favourably with all three solutions and shows that the much simpler solution of Ross, namely LILICUBE is satisfactory. The buckling pressures predicted by QUQUCUBE are almost identical to those predicted by the more complicated ALLCUBE element.
3.9.4 Conclusions The results show that the simpler LILICUBE element of Ross is quite suitable for predicting the elastic instability buckling pressures of perfect thinwalled circular cylinders and cones of small apical angle. It must be emphasised that these theories do not take into account the effects of initial imperfections in the shells, which can cause a catastrophic decrease in the buckling pressures of these vessels. However, the theories can be used for such cases, providing a plastic reduction factor is divided into the theoretical elastic instability buckling pressures.
© Carl T. F. Ross, 2011
Shell instability of pressure vessels
3.10
159
Higher order elements for hemi-ellipsoidal domes
Using the same method as for the cones, two higher order elements were developed by Ross76 for domes of varying individual curvature.
3.10.1 The all-cubic (CCC) element For this element, the displacement functions for u, v and w were all assumed to be of cubic form. Now the displacement function of w is of the Hermitian form shown by equation [3.42], but the displacement functions for u and v were obtained as follows. For the node element of Fig. 3.42, the assumed displacement functions for u and v are given by the following cubic forms: u = α + bζ + cζ 2 + dζ 3 v = e + fζ + gζ 2 + hζ 3 To determine the shape functions for u and v, consider the boundary conditions for u, as follows: at ζ = − 1, u = u1 at ζ = − 13 , u = u3 at ζ = 13 , u = u4 at ζ = 1, u = u2
∴u1 ∴u3 ∴u4 ∴u2
= = = =
− b + c − d; − b/3 + c/9 − d/27; + b/3 + c/9 + d/27; +b+c+d
a a a a
Solving these four simultaneous equations, the following expressions are obtained for the constants:
ς=1 ς = 1/3 2
ς = –1/3 4 3
ς = –1
1
3.42 Nodal displacement positions for u and v of cubic form for hemiellipsoidal domes.
© Carl T. F. Ross, 2011
160
Pressure vessels 1 9 (u2 + u1 ) + (u3 + u4 ) 16 16 1 9 b = − (u2 − u1 ) + ( 3u4 − 3u3 ) 16 16 9 c = (u1 + u2 − u3 − u4 ) 16 9 d = (u2 − u1 + 3u3 − 3u4 ) 16 a=−
[3.64]
Similar expressions can be obtained for the constants from e to h, and substituting these constants into the displacement functions, the shape functions for u and v are given by: 1 (−1 + ζ + 9ζ 2 − 9ζ 3 ) 16 1 N 2 = ( −1 − ζ + 9ζ 2 + 9ζ 3 ) 16 9 N 3 = (1 − 3ζ − ζ 2 + 3ζ 3 ) 16 9 N 4 = (1 + 3ζ − ζ 2 − 3ζ 3 ) 16 N1 =
[3.65]
i.e. u = ( N 1u1 + N 2u2 + N 3u3 + N 4u4 ) cos nφ
[3.66]
ν = ( N 1u1 + N 2u2 + N 3u3 + N 4u4 ) sin nφ
[3.67]
and These displacement functions are very different to those used by Rajagopalan72 in his chapter 6. The [B] matrix was obtained from ∂N 1 ∂ζ ∂N 2 DN 2 = ∂ζ ∂N 3 DN 3 = ∂ζ ∂N 4 DN 4 = ∂ζ DN 1 =
[3.68]
α = L / ( 2 R1 )
[3.69]
The geometrical stiffness matrix [k1] was obtained from equation [3.35], and the mass matrix [m] was obtained from equation [5.2].
© Carl T. F. Ross, 2011
Shell instability of pressure vessels
161
It is evident that all these matrices are of order 12 × 12, but to simplify computation, Guyan reduction73 was used to eliminate the u3, v3, u4 and v4 displacements in a manner similar to that of Rajagopalan. Thus, these matrices were of order 8 × 8, so that the original programs for the VMC element could be used with minimal changes.
3.10.2 The quadratic-quadratic-cubic (QQC) element For this element, the displacement functions for u and v were assumed to be in quadratic form along the meridian of the element, as shown in Fig. 3.43. The positions of the nodal displacements for u and v are shown in Fig. 3.43. The assumed boundary conditions were: at ζ = −1, u = u1 and ν = ν 1 at ζ = 1, u = u2 and ν = ν 2 at ζ = 0, u = u3 and ν = ν 3
[3.70]
This resulted in the following displacement functions for u and v: u = N 1.u1 + N 2.u2 + N 3.u3
[3.71]
v = N 1.v1 + N 2.v2 + N 3.v3
[3.72]
where N 1 = − 21 (1 − ζ )ζ N 2 = 21 (1 + ζ )ζ N 3 = (1 − ζ 2 )
[3.73]
ς=1 ς=0
ς = –1
2
3
1
3.43 Three-node varying meridional curvature element.
© Carl T. F. Ross, 2011
162
Pressure vessels
and their derivatives with respect to ζ are ∂N 1 ∂ζ ∂N 2 DN 2 = ∂ζ ∂N 3 DN 3 = ∂ζ DN 1 =
[3.74]
Substitution of the above displacement functions, together with their derivatives into equations [3.10], [3.15] and [5.2], will lead to the stiffness, geometrical and mass matrices for the QQC element. Here, again, Guyan reduction was used to eliminate the u3 and v3 displacements, so that the 10 × 10 matrices became of order 8 × 8. Hence, it was a simple matter to modify the computer program for the CCC element to incorporate the QQC element.
3.10.3 Computational analysis In this section, comparison is made for buckling ten hemi-ellipsoidal domes (Fig. 3.13), with the theoretical predictions of three varying meridional curvature elements. The models were made in SUP, where the SUP liquid was poured between machined male and female aluminium alloy moulds and then thermoset. The models had an internal base diameter of 0.2 m and a wall thickness of 0.002 m. SUP was found to have the following properties: Young’s modulus = 2.89 × 109 N m−2; Poisson’s ratio = 0.3 assumed; and density = 1230 kg m−3. The three elements of Ross, the VMC element (linear-linear-cubic), the CCC element and the QQC element, are compared with the experimentally obtained buckling pressures for the ten hemi-ellipsoidal domes of Ross and Mackney57 in Table 3.22, where: AR =
dome height base radius
From Table 3.22, it can be seen that the CCC and the QQC elements give better results than the VMC element for all hemi-ellipsoidal domes, except for the oblate dome of AR 0.7. However, it must be emphasised that this dome is prone to even the slightest variations in meridional curvature. A slight increase in meridional curvature results in a much higher buckling pressure. The flatter oblate domes probably collapsed through snap-thru
© Carl T. F. Ross, 2011
Shell instability of pressure vessels
163
Table 3.22 Buckling pressures (MPa) for hemi-ellipsoidal domes AR
Experiment
VMC
4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.7 0.44 0.25
0.097 0.100 0.150 0.201 0.364 0.490 1.280 1.070 0.276 0.060
0.111 0.135 0.168 0.224 0.323 0.549 1.426 0.944 0.380 0.132
CCC (5) (5) (6) (7) (8) (9) (11) (0) (0) (0)
0.110 0.134 0.166 0.222 0.321 0.548 1.398 0.865 0.361 0.124
QQC (5) (5) (6) (7) (7) (9) (11) (0) (0) (0)
0.110 0.134 0.166 0.222 0.321 0.548 1.398 0.866 0.376 0.124
(5) (5) (6) (7) (7) (9) (11) (0) (0) (0)
buckling, and not through axisymmetric bifurcation buckling as assumed in the theory of this section.
3.11
Varying thickness cylinders
Thin-walled circular cylinders with a stepped variation in wall thickness, appear in various branches of engineering, including ocean engineering, agricultural engineering and aerospace engineering. In ocean engineering, such structures appear as the legs of off-shore structures and as underwater storage containers, whereas in aeronautical engineering, such structures appear as the fuselages of spacecraft. Under external hydrostatic pressure, such structures can fail through non-symmetric bifurcation buckling or shell instability at a pressure that might only be a small fraction of that needed to cause axisymmetric yield. If the unsupported meridional length of the vessel is small, then the cylinder can fail through axisymmetric collapse, as shown in Figs 1.7 and 1.8. For a varying thickness circular cylinder, the vessel can fail through either shell instability or axisymmetric collapse or, by a combination of these two modes of failure. Earlier work by Esslinger and Geier77 and also by Malik et al.78 were only concerned with the elastic instability of geometrically perfect vessels. From the theory of Esslinger and Geier and according to Rajagopalan,72 the former authors quoted the following semi-empirical formula to predict the elastic instability of varying thickness circular cylinders under hydrostatic pressure: L / Pcr = ∑ ( Li / Pi ) from i = 1 to N
© Carl T. F. Ross, 2011
164
Pressure vessels
where L = the entire length of the vessel between supports. Pcr = theoretical buckling pressure based on the von Mises theory. Pi = theoretical buckling pressure of the ith bay of the vessel, assuming that it is of length L and thickness ti. Li = length of the element i. The problem with this theory is that it does not take into account the distribution of wall thickness, nor does it allow for inelastic buckling. In 2000, Ross et al.79 presented a paper which successfully took these into account. They presented both theoretical and experimental results.
© Carl T. F. Ross, 2011
4 General instability of pressure vessels
Abstract: Mathematical theories for general instability, the process where the entire ring-shell combination of a circular cylinder or cone buckles bodily in its flank (between adjacent bulkheads) are described. Some theories are based on an analytical approach and some on a numerical one, namely the finite element method. Predictions of failure using theories are compared with experimental observations. The detrimental effects of initial out-of-circularity, together with inelastic buckling are also considered. Two design charts are provided; one for ring-stiffened circular cylinders and the other for ring-stiffened cones. Key words: general instability, buckling, initial out-of-circularity, circular cylinders, cones, design charts.
4.1
General instability of ring-stiffened circular cylinders
Tokugawa80 first identified the general instability mode of failure in 1929. His method of analysis was based on the elastic instability of an isolated ring, with no allowances for any support from bulkheads. He assumed that an effective width of shell plating buckled with the ring stiffener. One simple formula that can be used as a first stab on the design estimate for the ring-stiffener is that of Levy:81 p′ = 3EI / R3
[4.1]
where p′ = load per unit length; I = second moment of area of the ring’s cross-section about its centroidal axis, which is parallel to the axis of the cylinder; and R = radius of curvature of the ring centroid. This mode of failure, which is based on elastic instability, assumes that the ring buckles in a flattening mode (i.e. n = 2). It is a simple matter to extend Levy’s formula to the form shown by equation [4.2]: p′ = 3EI f / Rf3
[4.2]
where If = second moment of area of the ring-shell combination about its centroidal axis x–x, shown in Fig. 4.1; Rf = radius of curvature of the centroid of the ring-shell combination; Leff = effective width of the shell plating that 165 © Carl T. F. Ross, 2011
166
Pressure vessels Ring Shell x
x
Leff
Rf
Axis
4.1 Ring-shell combination.
buckles with the ring-stiffener. Several authors provide an estimate for the effective width of shell plating (Leff), including Sechler.82,83 The two main problems with equation [4.2] are that it allows neither for the increase in buckling resistance resulting from the support given by adjacent bulkheads nor for the decrease in buckling resistance caused by inelastic buckling and initial out-of-circularity. An important approximate formula for the general instability of ring-stiffened circular cylinders was presented by Bryant,84 giving a buckling pressure: Pcr = Pf + Ps where Pf = ( n2 − 1) EI c / ( R3 Lf ) ; ( Et / R ) λ 4 ; Ps = ( n 2 − 1 + λ 2 / 2 ) ( n 2 + λ 2 )2 n = number of circumferential waves or lobes; λ = πR/Lb; Ic = second moment of area of the cross-section of a ring-stiffener and the effective width of shell in contact with it; Lf = stiffener spacing; Lb = length between adjacent and bulkheads. It can be seen from the above formula for Pcr, that it has a frame component, namely Pf and a shell component, namely Ps; this is not unlike Tokugawa’s formula produced some 25 years earlier! Rajagopalan72 has shown that the Bryant formula can break down for certain geometries. In any case, since the advent of the digital computer, there is no need to use approximate formulae; even for design.
© Carl T. F. Ross, 2011
General instability of pressure vessels
167
4.1.1 Kendrick Part I A major breakthrough on the elastic instability of ring-stiffened circular cylinders was achieved by Kendrick.43 His solution, which assumed simply supported edges, was based on the Rayleigh–Ritz method. From first principles, Kendrick derived the following expressions for the bending strain energy of the shell: U b = {Eh3 / 24a (1 − v2 )} ×∫
2π
0
∫ {a w Lb
2
0
+ (wθθ + w ) / a2 + 2vwxx (wθθ + w ) 2
2 xx
}
+ 2 (1 − v) (wxθ + ν x / 2 − uθ / 2a) dθ dx 2
[4.3]
and for the extensional strain energy of the shell: U e = {Eah / 2 (1 − v2 )} ×∫
2π
×∫
2π
×∫
2π
0
∫ {u
+ (ν θ − w ) / a2 + 2vux (ν θ − w ) / a
∫ (ν
2 x
+ wx2 ) dθ dx + ( N oy / 2a)
2 θ
+ wθ2 − 2ν θw) dθ dx
Lb
2
2 x
0
+ (1 − ν ) (ν x + uθ / a2 ) 2} dθ dx + (aN ox / 2 ) 0
0
Lb
0
∫ (u Lb
0
[4.4]
where E = Young’s modulus; v = Poisson’s ratio; Nox = −0.5 pa; Noy = −pahLf/(Af + hLf); u, v, w are defined in Fig. 4.2; p = applied pressure; h = shell thickness, t; a = mean radius of shell; Af = cross-sectional area of a frame; Lf = frame spacing. The bending strain energy of the ring-stiffeners is: r=N
2π
Vb = ( EI / 2a3 ) ∑ ∫ (wθθ + w ) dθ 2
[4.5]
0
r =1
and the extensional strain energy of the ring-stiffeners is: r=N
Ve = ( EAf / 2a) ∑ r =1
+ ( N of / 2a) ∫
{∫
2π
0
2π
0
[(wθθ + w)(e / a) − (νθ − w)]2 dθ
(uθ2 + wθ2 − 2νθw) dθ
}
© Carl T. F. Ross, 2011
[4.6]
168
Pressure vessels U V
W φ
X
W
R R
L
4.2 Notation for the deflections of a circular cylinder.
Shell e Ring centroid
a Axis of cylinder
x
4.3 Ring-stiffener.
where N = number of ring stiffeners; Nof = −paLfAf/(Af + hLf); e = eccentricity of the ring centroid from the mid-surface of the shell (positive inwards) (Fig. 4.3); I = second moment of area of the ring’s cross-section about its centroid and parallel to the axis of the cylinder. The potential owing to radial pressure is: Wr = − ( pa / 2 ) ∫
2π
0
∫ ( 2wu Lb
x
0
+ 2wν θ / a − ν θ ux − w 2 / a) dθ dx
[4.7]
and the potential owing to axial pressure is: Wa = ( pa / 2 ) ∫
2π
0
{u( x =0) − u( x =L ) } dθ
[4.8]
b
Kendrick assumed the buckling configuration of equation [4.9], which corresponds to simply supported edges: w = A sin
πx cos nθ Lb
ν = B sin
πx sin nθ Lb
u = C cos
© Carl T. F. Ross, 2011
πx cos nθ Lb
[4.9]
General instability of pressure vessels
169
where n = number of circumferential lobes or waves into which the vessels buckle; and A, B and C are constants. Substitution of equation [4.9] and its various derivatives into equations [4.3]–[4.8] results in the following expression for the total strain energy and potential, Ut: U t = U b + U e + Vb + Vc + Wr + Wa = {Eh3 / 24a (1 − ν 2 )}
× {π 5 a2 W14 A2 / L3b + π ( n2 − 1) LbW11 A2 / a2 + 2 π 3 n2 νW12 A2 / Lb 2
− 2 π 3νW12 A2 / Lb + 2 π 3 n2 (1 − ν ) W12 A2 / Lb + π 3 (1 − ν ) W12 B2 / 2 Lb + πn2 (1 − ν ) LbC 2 / 4a2 − 2 π 3 (1 − ν ) nW12 AB / Lb − 2 π 2 n2 (1 − ν ) W13 AC / a + π 2 n (1 − ν ) W13 BC / a} + {Eah / 2 (1 − ν 2 )}
× {0.5π 3 C 2 / Lb + πLbW11 ( nB − A) / a2 − 2 π 2 nνW10 BC / a 2
+ 2 π 2 νW10 AC / a + 0.5π 3 (1 − ν ) W12 B2 / Lb
+ 0.25πn2 Lb (1 − ν ) C 2 / a2 − π 2 n (1 − ν ) W13 BC / a} − 0.25π 3pa2 W12 B2 / Lb − 0.25π 3pa2 W12 A2 / Lb + 0.25πpn2Lb K0C 2 + 0.5πpn 2 Lb K0W11 A2 − πpnLb K0 W11 AB + ( 0.5EI / a3 ) π ( n2 − 1) W0 A2 + (0.5EAf / a) 2
× {( 2 πn3 e / a) AB − ( 2 πne / a) AB − ( 2 πn2 e / a) + ( 2 πe / a) A2
}
+ ⎡⎣ π ( n2 − 1) e 2 / a2 ⎤⎦ A2 + π ( nB − A) W0 2
2
+ 0.25πpn ( N − 1) K1C + 0.5πpnK1W0 ( nA2 − 2 AB) 2
2
+ π 2 paW10 AC − πpnLbW11 AB − 0.5π 2 panW10 BC + 0.5πpLbW11 A2
[4.10]
Now, according to the Rayleigh–Ritz theory, ∂U t ∂U t ∂U t = = =0 ∂A ∂B ∂C which, on application to equation [4.10], results in the following three simultaneous equations:
( a10 + pa11 ) A2 + ( a12 + pa13 ) B2 + ( a14 + pa15 ) C 2 = 0 ( a12 + pa13 ) A2 + ( a18 + pa19 ) B2 + ( a16 + pa17 ) C 2 = 0 ( a14 + pa15 ) A2 + ( a16 + pa17 ) B2 + ( a20 + pa21 ) C 2 = 0
[4.11]
As the solution A = B = C = zero is not of interest in this problem, the following determinant is obtained:
© Carl T. F. Ross, 2011
170
Pressure vessels ⎡ a10 + pa11 ⎢ a + pa 13 ⎢ 12 ⎢⎣ a14 + pa15
a12 + pa13 a18 + pa19 a16 + pa17
a14 + pa15 ⎤ a16 + pa17 ⎥ = 0 ⎥ a20 + pa21 ⎥⎦
[4.12]
The determinant expands to a32 p3 + a33 p2 + a34 p + a35 = 0
[4.13]
where the root of interest is the lowest positive one that can be calculated by the Newton–Raphson iterative process: W0 = W10 = W11 = W12 = W13 = W14 = 0.5 2 2 a10 = ( πEW0 / a ) ⎡⎣( n2 − 1) I / a 2 + Af ( n2 − 1) e 2 / a 2 + 1 + 2e / a − 2 n2 e / a ⎤⎦
{
{
}
+ {Eh3 / 12a (1 − ν 2 )} π 5 a 2W14 / L3b + ( 2 π 3W12 / Lb ) × [ν ( n − 1) + n (1 − ν )] 2
2
2 + πLbW11 ⎡⎣12 / h2 + ( n2 − 1) / a 2 ⎤⎦
}
a11 = πLbW11 (1 + n2 K0 ) − 0.5π 3 a 2W12 / Lb + πn2 K1W0 a12 = πnEAf W0 ( n2 e / a − e / a 2 − 1 / a )
− {πnEh / a (1 − ν 2 )} {π 2 h2 (1 − ν ) W12 / 12Lb + LbW11 }
a13 a14 a15 a16 a17 a18 a19 a20 a21 a23 a24 a25 a26 a27 a28 a29 a30 a31 a32 a33
= − πn {K1W0 + LbW11 (1 + K0 )} = {π2 Eh/(1 − ν 2)}{−h2n2(1 − ν)W13/12a2 + νW10} = π2 aW10 = {π2 Eh/(1 − ν 2)}{0.5n(1 − ν)W13(h2/12a2) − 1) − νnW10} = −0.5π2 anW10 = πn2 EAfW0/a + {Eh/(1 − ν 2)} × {π2a(1 − ν)W12(h2/24a2 + 0.5)/Lb + n2LbW11/a} = −0.5π3a2W12/Lb = {πEh/(1 − ν 2)}{0.25n2(1 − ν)Lb(h2/12a2 + 1)/a + 0.5π2 a/Lb} = 0.5πn2{K1(N − 1) + LbK0} = −a216 + a18a20 = −2a16a17 + a18a21 + a19a20 = −a217 + a19a21 = −a12a20 + a14a16 = −a12a21 + a13a20 + a14a17 + a15a16 = −a13a21 + a15a17 = a12a16 − a14a18 = a12a17 + a13a16 + a14a19 − a15a18 = a13a17 − a15a19 = a11a25 + a13a28 + a15a31 = a10a25 + a11a24 + a12a28 + a13a27 + a14a31 + a15a30
© Carl T. F. Ross, 2011
General instability of pressure vessels
171
a34 = a10a24 + a11a23 + a12a27 + a13a26 + a14a30 + a15a29 a35 = a10a23 + a12a26 + a14a29 A computer program for determining the general instability pressure of a ring-stiffened cylinder is provided in Appendix IV.
4.1.2 Kendrick Part III In addition to this solution, Kendrick85,86 produced a number of other general instability solutions, some of which are believed to be more realistic than the solution presented in Section 4.1.1. For example, Kendrick assumed the buckling configuration given in equation [4.14], which also corresponds to simply supported edges: u = A1 cos nθ cos ( πx / Lb ) ν = B1 sin nθ sin ( πx / Lb ) + B2 sin nθ [1 − cos ( 2 πx / Lf )] w = C1 cos nθ sin ( πx / Lb ) + C2 cos nθ [1 − cos ( 2 πx / Lf )]
[4.14]
Equation [4.14] is shown diagrammatically in Fig. 4.4. This configuration assumes that the shell plating between adjacent ring-stiffeners suffers additional displacements, unlike his Part I buckling configuration, given in equation [4.9] and shown diagrammatically in Fig. 4.5. Neither Kaminsky87 nor Nash88,89 accepted Kendrick’s assumed buckling configuration, and produced clamped and fixed-edges solutions, where the former was based on Kendrick’s strain energy expressions.43
Axis
4.4 Kendrick Part III buckling configuration.
Axis
4.5 Kendrick Part I buckling configuration.
© Carl T. F. Ross, 2011
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Pressure vessels
Kaminsky assumed the following buckling configuration: w = A [1 − cos ( 2 πx / Lb )] cos nθ
ν = B [1 − cos ( 2 πx / Lb )] sin nθ u = C cos ( πx / Lb ) cos nθ
[4.15]
and Nash,88,89 who used his own strain energy expressions, assumed the following buckling configuration: w = A [1 − cos ( 2 πx / Lb )] cos nθ
ν = B [1 − cos ( 2 πx / Lb )] sin nθ u = C sin ( 2 πx / Lb ) cos nθ
[4.16]
Nash’s energy expressions considered numerous extra terms, including twisting of the rings. The buckling configurations given by equations [4.9], [4.15] and [4.16] yield the following boundary conditions: for simply supported edges (equation [4.9]): w = 0 at x = 0 and x = Lb d 2 w / dx 2 = 0 at x = 0 and x = Lb dw / dx ≠ 0 at x = 0 and x = Lb u ≠ 0 at x = 0 and x = Lb for clamped edges (equation [4.15]):
ν = w = 0 at x = 0 and x = Lb d 2 w / dx 2 ≠ 0 at x = 0 and x = Lb dw / dx = 0 at x = 0 and x = Lb u ≠ 0 at x = 0 and x = Lb for fixed edges (equation [4.16]): u = ν = w = 0 at x = 0 and x = Lb d 2 w / dx 2 ≠ 0 at x = 0 and x = Lb dw / dx = 0 at x = 0 and x = Lb u = 0 at x = 0 and x = Lb
4.1.3 Galletly’s results To test the above solutions, Galletly et al.90 compared these solutions with their experimental results for six machine-stiffened models, as shown in Table 4.1, where the values given are in terms of the non-dimensional parameter Pcr/Pexp, where Pcr = theoretical buckling pressure and Pexp = experimental buckling pressure. From Table 4.1, it can be seen that comparison between experiment and Kendrick’s solutions is good, and that both Kaminsky87 and Nash88,89
© Carl T. F. Ross, 2011
General instability of pressure vessels
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Table 4.1 General instability results for the models of Galletly et al.90 Model no.a
Kendrick Part I43
Kendrick Part III85
Kaminsky87
Nash88,89
1 2 3 4 5 6
1.012 0.989 0.994 0.960 1.200 1.160
0.976 0.937 0.920 0.842 0.997 0.967
1.637 1.607 1.714 1.534 1.589 1.542
1.595 1.570 1.705 1.355 1.513 1.474
(4) (4) (3) (3) (3) (3)
a
The figures in parentheses represent the number of lobes (n) into which the vessels buckle.
overestimate the collapse pressures. It may be, however, that the main reason why the Kendrick solutions agreed best with experiment was that these models were not firmly clamped at their ends but simply secured by studs, where pressure tightness was achieved because the hydrostatic pressure caused the joints to be in compression. It appears that 12 studs were used on a diameter of about 8 in (i.e. the studs were spaced at over eight diameters apart). Kendrick argues that for ring-stiffened circular cylinders rotational restraint at the ends is small and localised because thin-walled shells have a relatively small resistance to bending, and thus solutions based on buckling configurations similar to those adopted by Kaminsky and Nash will overestimate buckling pressures. In this context, the present author agrees with Kendrick. However, later on in this chapter, it is shown that axial restraint at the edges increases the experimentally obtained buckling pressures because shells have a relatively high resistance to in-plane deformation. It must be emphasised that initial imperfections can cause experimental buckling pressures to be even lower than those predicted by Kendrick, particularly if the ring-stiffened cylinder is short and thick. Such vessels are said to suffer plastic knockdown and fail through inelastic instability, as discussed in chapter 3. The correlation between Kaminsky87 and Nash88,89 was poor, but this was probably caused by the different buckling configurations assumed for u, because when the present author applied Nash’s boundary conditions to the Kendrick strain energy expressions, resulting in solution (2b) of reference 42, he obtained good correlation between Nash and solution (2b). This finding appears to indicate that, whereas the strain energy expressions of Kendrick and Nash were different, they yielded similar results for the same boundary conditions.
© Carl T. F. Ross, 2011
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Pressure vessels
4.1.4 Varying stiffener sizes One of the problems with Kendrick’s solutions is that they are for circular cylinders, stiffened by uniform-size stiffeners, at uniform spacing, and all the stiffeners are attached to either the internal surface or the external surface. In practice such symmetrical vessels are seldom encountered and, because of this, the present author91, 92 developed a finite element solution for the general instability of ring-reinforced circular cylinders under uniform external pressure. This solution is easily achieved by introducing a stiffness matrix for a ring-stiffener, which is used in conjunction with the stiffness matrix of the circular cylindrical shell element described in Section 3.3.1. The various components of the ring-stiffness matrix are simply added to the appropriate positions in the stiffness matrix of the circular cylindrical shell, in an eigenvalue problem similar to equation [3.17]. The ring-stiffener is shown in Fig. 4.6, and its matrix of shape functions [N] is given by:91 0 0 ⎤ ⎡cos nφ ⎢ sin nφ 0 ⎥ [N ] = ⎢ 0 ⎥ ⎢⎣ 0 0 cos nφ ⎥⎦
[4.17]
The strain matrix, which was based on the work of Kendrick43 is given by 2 ⎧ε φ ⎫ ⎧(νφ − w ) / R − (wφφ + w ) e / R ⎫ ⎪ ⎪ ⎪ ⎪ {ε } = ⎨ χφ ⎬ = ⎨ (wφφ + w ) / R2 ⎬ ⎪χ ⎪ ⎪ ⎪ θi / R ⎩ x⎭ ⎩ ⎭
[4.18]
x,u v φ
z
w
4.6 Buckling configuration for ring (n = 4).
© Carl T. F. Ross, 2011
General instability of pressure vessels
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or
{ε } = [ B] {u}. Therefore,
νi
wi
⎡ n cos φ / R 0 [ B] = ⎢⎢ ⎢⎣ 0
( −1 − e / R + n e / R ) cos nφ / R (1 − n2 ) cos nφ / R2 2
0
θi 0 ⎤ 0 ⎥⎥ 1 / R ⎥⎦
[4.19]
where θi = the rotation of the ith ring about the z axis; and [B] = a matrix relating strains and displacements. The relationship between the stress {σ} and strain {ε} matrices was taken as
{σ } = [ D] {ε } and a matrix of material constants: ⎡ Af ⎢0 D = E [ ] ⎢ ⎢⎣ 0
0 0⎤ I 0⎥ ⎥ 0 I z ⎥⎦
[4.20]
where Af = cross-sectional area of ring; I = second moment of area of ring cross-section about the x axis; and Iz = second moment of area about the z axis. The stiffness matrix for a ring of uniform section was determined from: 2π
[ k ] = ∫0 [ B]T [ D][ B] Rdφ
[4.21]
This was found to be: n2 Af / R 2 ⎡ ⎢ ⎡ nAf ( −1 − e / R + n2 e / R ) ⎤ [ k ] = πER ⎢⎢ ⎢ ⎥ R2 ⎣ ⎦ ⎢ 0 ⎢⎣
nAf ( −1 − e / R + n2 e / R ) / R 2
0 ⎤ ⎥ 1 A ⎡ f −1 − e / R + n 2 e / R 2 + 2 2⎤ ⎥ − 1 0 n ( ) ( ) ⎢⎣ R 2 ⎥⎦ R4 ⎥ 0 2 I z / R 2 ⎦⎥ The parameters used in the computation include the following: p = pressure (external positive); n = number of lobes in circumferential direction;
© Carl T. F. Ross, 2011
[4.22]
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Pressure vessels
e x0
z0
4.7 Frame with shell.
t = shell thickness; R = radius of centroid of frame section; Af = cross-sectional area of frame; I = second moment of area of frame about the x0 axis (Fig. 4.7); Iz = second moment of area of frame about the z0 axis (Fig. 4.7); L = length between stiffeners; e = distance of centroid of frame section from mid-surface of shell, as shown in Fig. 4.7 (negative if frame is on outside surface of shell); u,v,w = displacement in x, ϕ and z directions; x,z = rectangular coordinates; ϕ = angular coordinate; E = elastic modulus; σ = stress. To obtain the geometrical stiffness matrix of the ring [kG], the additional strain owing to large displacements92 was considered: ⎧1 ν w 2 ⎫ {δε L } = ⎨ ⎛⎜ + φ ⎞⎟ ⎬ ⎩2 ⎝ R
R ⎠ ⎪⎭
[4.23]
Using the same notation as Zienkiewicz:23
{
}
ν wφ = [G ] {u} + R R
[4.24]
that is,
[G ] = (1/ R ) [sin nφ − n sin nφ ]
[4.25]
The geometrical stiffness matrix was obtained from:
[ kG ] = ∫vol [GT ][σ ][G ] d ( vol )
[4.26]
which for a ring of uniform cross-section was:
[ kG ] = Af ∫0 [GT ][σ ][G ] R dφ 2π
© Carl T. F. Ross, 2011
[4.27]
General instability of pressure vessels
177
The following approximation was made for the hoop stress in a ring:
σ = pRL / ( Af + Lt )
[4.28]
that is,
[ kG ] =
pAf L 2π [sin nφ − n sin nφ ][sin nφ − n sin nφ ] dφ ( Af + Lt ) ∫0
⎡ ν i wi ⎤ πpAf L ⎢ 1 − n⎥ [ kG ] = ⎥ ( Af + Lt ) ⎢ ⎢⎣ − n n2 ⎥⎦
[4.29]
[4.30]
When no ring appeared at a particular nodal point, Af was set to zero. A computer program for the general instability of an orthotropic ringstiffened cylinder is given in Appendix IV.
4.1.5 End conditions To determine the effects of end conditions, comparison is made in Table 4.2 of the experimentally obtained collapse pressure of the ring-reinforced circular cylinder of reference 93 with various analytical and numerical solutions, where solution (2b) from reference 42 was a fixed-edges solution based on Kendrick’s method.43 The vessel was firmly clamped at its ends by 1 /4-inch bolts, spaced at about 0.547 in apart (i.e. 2.19 bolt diameters). From Table 4.2, it can be seen that the fixed-edges solutions agree best with experiment, but there is some doubt if these edges are realistic for the practical case. Comparison is also made in Table 4.3 of models 4, 5 and 6 from reference 3 with numerically and analytically determined buckling pressures. These models were secured at their ends by 1/4 inch bolts, spaced at about 1.93 in (i.e. a bolt spacing of 7.72 bolt diameters). Details of the buckling pressures of these models, which were machined from a thick-walled tube of HE9-WP aluminium alloy are given in Table 4.4, from which it can be seen that by having a slightly larger bolt spacing
Table 4.2 General instability collapse pressures of circular cylinder P3 (lbf in−2) Analytical
Finite element solution
Simply supported
Solution (2b) (fixed)
Simply supported
Clamped
Fixed
Experimental
218.1
338.4
229.6 (4)a
237.7 (4)
330.8 (4)
322
a
The figures in parentheses represent the number of lobes.
© Carl T. F. Ross, 2011
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Pressure vessels
Table 4.3 General instability collapse pressures of circular cylinders (lbf in−2) Analytical
Finite element solutions
Kendrick Solution Model (simply Kaminsky (2b) Simply no. supported)a (clamped) (fixed) supported Clamped Fixed 4 5 6 a
94.6 (4) 126 (4) 170 (3)
156 (5) 189 (4) 241 (4)
147 (5) 182 (4) 233 (4)
95.6 (4) 121 (4) 160 (4)
99.7 (4) 125 (4) 164 (4)
Experimental
141 (5) 120 (4) 173 (4) 148 (4) 212 (4) 188 (4)
The number of nodes is indicated in parentheses. Table 4.4 General instability buckling pressures (lbf in−2) for DTMB machined models
Model no.
Kendrick (simply supported)a
Kaminsky (clamped)
Solution (2b) (fixed)
Solution (2a) (partially fixed)
Experimental
DD-6 DD-2A DD-3A DD-4A
558 331 232 197
910 576 536 485
969 524 488 362
636 467 322 260
700 409 378 289
(3) (2) (2) (2)
(3) (3) (3) (2)
(4) (3) (3) (2)
(3) (2) (2) (2)
(3) (3) (2) (2)
a
The figures in parentheses represent n, the number of waves or lobes into which the vessels buckle.
on the edge clamping rings than that for model P3,93 the fixed-edges solutions overestimated the experimental buckling pressures. It can also be seen that the finite element clamped solution predicted buckling pressures only fractionally higher than the simply-supported edges solutions, thereby indicating that axial restraint at the boundaries has a more significant effect on buckling resistance than does rotational restraint at these positions. Another conclusion that can be arrived from the observations of Table 4.3 is that the models must have been partially fixed at their boundaries, particularly in the axial direction. Similar conclusions are shown in Table 4.4, where comparison is made of the experimental results obtained from the DD8-2 series of machined models94 with various analytical solutions. Nevertheless, for vessels that buckle inelastically, the above arguments are purely academic, as such vessels often fail at pressures less than those predicted by even the simply supported edges solutions. Table 4.5 shows a comparison between theoretical and experimental buckling pressures for the machined models Pl, P2, P1/A and P2/A, all of
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General instability of pressure vessels
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Table 4.5 Theoretical and experimental buckling pressures (lbf in−2) Model no.
Simply supported
Fixed
Experimental
P1 P2 P1/A P2/A
1401 1776 1488 1740
2284 2951 2421 2875
865 940 1005 1100
which failed inelastically. The models were firmly clamped at their edges by 1 /4-inch bolts, for P1 and P2 the spacing of the bolts was the same as that used for P3, but for P1/A and P2/A only four bolts were used at each end. From Table 4.5 it can be seen that the experimentally obtained buckling pressures are considerably lower than the theoretical ones, and also that edge conditions play a less significant role when vessels buckle plastically. Further geometrical details of models Pl, P2, P1/A, P2/A and P3 are given in Table 4.6. Careful measurements of the models found P1, P2 and P3 to have a maximum initial out-of-circularity of about 0.001 in, and P1/A and P2/A to have a maximum initial out-of-circularity of 0.002 in. The mechanical properties of these models were as follows: P1 and P2: E = 29 × 106 lbf in−2; σyp = 23 500 lbf in−2; and nominal peak stress = 48 700 lbf in−2. P1/A: E = 30 × 106 lbf in−2; σyp = 26 700 lbf in−2; and nominal peak stress = 51 000 lbf in−2. P2/A: E = 29 × 106 lbf in−2; σyp = 27 300 lbf in−2; and nominal peak stress = 51 000 lbf in−2. P3: E = 29.3 × 106 lbf in−2; σyp = 57 500 lbf in−2; and nominal peak stress = 99 200 lbf in−2. Despite the fact that the models were machined very precisely, as shown by a typical set of out-of circularity plots for model P1 in Fig. 4.8, the plastic knockdown was as high as 89% for one of these models when compared with a simply supported edges solution.
4.2
Inelastic general instability of ring-stiffened circular cylinders
In a manner similar to that adopted in Section 3.1.3, an attempt is now made to produce a chart which can be used to determine the plastic reduction factor for the general instability of a perfect ring-stiffened circular cylinder. In this instance, instead of using λ, a similar parameter, namely λ′, another thinness ratio, will be used:
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Pressure vessels
Table 4.6 Dimensions of the P-series models (inches) xi
bi di h i=1
P1
i 1 2 3 4 5 6 7 8 9
i 1 2 3 4 5 6 7 8 9
i 1 2 3 4 5 6 7 8 9
i=2
P2
xi
di
bi
xi
di
bi
0.5010 1.2027 2.1451 3.0878 4.0296 4.9706 5.9117 6.8535 7.5555
0.0803 0.0603 0.0603 0.0604 0.0604 0.0604 0.0603 0.0603 0.0805
0.0620 0.0613 0.0635 0.0620 0.0615 0.0605 0.0617 0.0620 0.0620
0.5004 1.2008 2.1408 3.0813 4.0221 4.9628 5.9036 6.8447 7.5462
0.1203 0.1003 0.0801 0.0804 0.0804 0.0803 0.0800 0.1000 0.1200
0.0607 0.0601 0.0600 0.0609 0.0607 0.0608 0.0608 0.0613 0.0617
P1/A
P2/A
xi
di
bi
xi
di
bi
0.5017 1.2002 2.1416 3.0809 4.0209 4.9599 5.8994 6.8411 7.5384
0.0805 0.0606 0.0611 0.0611 0.0606 0.0604 0.0602 0.0601 0.0802
0.0622 0.0162 0.0609 0.0613 0.0609 0.0600 0.0615 0.0608 0.0605
0.5000 1.1996 2.1379 3.0780 4.0174 4.9573 5.8982 6.8381 7.5379
0.1201 0.1008 0.0803 0.0802 0.0802 0.0802 0.0800 0.0997 0.1194
0.0617 0.0609 0.0612 0.0604 0.0584 0.0610 0.0616 0.0611 0.0616
P3 xi
di
bi
0.6152 1.5253 2.4355 3.3428 4.2560 5.1645 6.0745 6.9850 7.8950
0.0304 0.0303 0.0304 0.0304 0.0301 0.0302 0.0301 0.0302 0.0302
0.0300 0.0325 0.0310 0.0320 0.0310 0.0310 0.0300 0.0310 0.0300
For P1 h = 0.0615 in, Lb = 8.0565 in internal diameter = 4.0003 in For P2 h = 0.0606 in, Lb = 8.0470 in internal diameter = 3.9990 in For P1/A h = 0.0608 in, Lb = 8.0390 in internal diameter = 3.9980 in For P2/A h = 0.05987 in, Lb = 8.0360 in internal diameter = 4.0006 in For P3 h = 0.0303 in, Lb = 8.5100 in internal diameter = 3.9997 in
© Carl T. F. Ross, 2011
and
and
and
and
and
General instability of pressure vessels
SG no.7
l Fai
SG .6 no
Specimen P1 Bay no.7 Mag. Filter Taylor-Hobson Talyrond
Tra c
e of i n t e r n al s u rf a
ce
SG = strain gauge
SG no.7
l Fai
SG .6 no Specimen P1 Bay no.7 Filter Mag. Taylor-Hobson Talyrond
Tra
c e of e rf a c xtern al s u
e
4.8 Out-of-circularity plots for model P1 (one division = 0.0001 in).
© Carl T. F. Ross, 2011
181
182
Pressure vessels 0·06'' Ring
0·06''
0·06''
x
Shell x
0·94''
4.9 Ring-shell combination for P1 and P1/A.
λ′ =
4
{(L
b
/ Df ) / (t ′ / Df ) 2
3
}×
(σ yp / E )
[4.31]
where Df = diameter of the centroid of a typical ring-shell combination; and t′ = equivalent shell thickness. To demonstrate how λ′, Df and t′ can be determined, the models P1, P2 P1/A and P2/A, all of which failed inelastically, are considered.
4.2.1 P1 and P1/A A typical ring-shell combination for these vessels is shown in Fig. 4.9. From elementary theory: A = area of section = 0.06 in2; y¯ = 0.036 in; Ixx = 9.974 × 10−5 in4; I = Ixx − y¯2 × A = 3.02 × 10−5 in4; where: y¯ = distance of centroid of the ring-shell combination from x–x; Df = (4 + y¯) × 2 = 4.067 in; I = (0.94 × t′3)/12 = 3.02 × 10−5; ∴t′ = 0.073 in. Therefore, for P1, λ′ = 0.816 and Pcr / Pexp = 1.62; and for P1/A, λ′ = 0.856 and Pcr / Pexp = 1.48.
4.2.2 P2 and P2/A Similarly, for P2 and P2/A, A = 0.0612 in2; y¯ = 0.0355 in; Ixx = 1.182 × 10−4 in4 I = 4.11 × 10−5 in4 ∴t′ = 0.0807 in;
© Carl T. F. Ross, 2011
General instability of pressure vessels
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Df = 4.071 in; Lb = 8.04. Therefore, for P2, λ′ = 0.757 and Pcr / Pexp = 1.89; and for P2/A, λ′ = 0.816 and Pcr / Pexp = 1.58. A plot of 1/λ′ against Pcr /Pexp for these four models, together with the results of other machined models95–97 is shown in Fig. 4.10, which can be seen to be of similar form to Fig. 3.4. It should be noted that the plastic reduction factor Pcr /Pexp, must always be larger than unity, where: Pcr = theoretical general instability pressure, based on simply supported edges, calculated using ANSYS; and Pexp = experimentally obtained general instability pressure. It must be emphasised that for Fig. 4.10 only very precisely made machinestiffened models were used, with little out-of-circularity, and that, in practice, the out-of-circularity will be larger, resulting in a larger plastic reduction factor. That is, if λ′ were plotted against (Pcr/Pexp) for vessels with a larger out-of-circularity, the resulting graph would lie on a line somewhat higher
3.6 3.4 3.2 3.0 2.8 Unsafe side
2.6 2.4
1/λ'
2.2 2.0 1.8 Safe side
1.6 1.4 1.2 1.0
Ross cylinders P4, P5 and P6 Ross cylinders 4, 5 and 6 Ross cylinders P1, P2, P1/A and P2/A Ross cylinders PHR1, PHR2 and PHR3 Reynolds cylinders DD-6, DD-2A, DD-3A and DD-4A Seleim cylinders 1, 3, 5, 8 and 10
0.8 0.6 0.4 0.2 0.0 0
1
2
3
4
5
6
7
8
9
10
Pcr / Pexp
4.10 Design chart for the general instability of machined ring-stiffened circular cylinders using ANSYS.
© Carl T. F. Ross, 2011
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Pressure vessels
3.6 3.4 3.2 3.0 2.8 2.6 2.4 2.2 1/λ'
2.0 1.8 1.6 1.4 1.2 1.0
Ross cylinders P4, P5 and P6 Ross cylinders 4, 5 and 6 Ross cylinders P1, P2, P1/A and P2/A Ross cylinders PHR1, PHR2 and PHR3 Reynolds cylinders DD-6, DD-2A, DD-3A and DD-4A Seleim cylinders 1, 3, 5, 8 and 10
0.8 0.6 0.4 0.2 0.0 0
1
2
3
4
5 6 Pcr / Pexp
7
8
9
10
4.11 Design chart for the general instability of ring-stiffened cylinders using Kendrick Part I.
than that indicated in Fig. 4.10. It must also be emphasised that more experimental data are required to give Fig. 4.10 any validity, and also that the vessels used for such an investigation should include different geometries from those of this figure. In addition, two other charts are presented in Figs 4.11 and 4.12, where Pcr was calculated using Kendrick Part I and Kendrick Part III, respectively.
4.3
General instability of ring-stiffened conical shells
Theoretical analysis of these vessels by analytical methods is very difficult, and preferred solutions rely on the finite element method, which can be obtained by combining the stiffness matrices of the ring-stiffeners described in Section 4.1.3 with the stiffness matrices of the conical elements described in Section 3.2. Using such an approach, the present author analysed the ring-stiffened truncated conical shells of Singer,56 Nos. 6, 7 and 8 in Table 4.7. It should be brought to the attention of the reader that in Singer’s
© Carl T. F. Ross, 2011
General instability of pressure vessels
185
3.6 3.4 3.2 3.0 2.8 2.6 2.4 2.2 1/λ'
2.0 1.8 1.6 1.4 1.2 1.0
Ross cylinders P4, P5 and P6 Ross cylinders 4, 5 and 6 Ross cylinders P1, P2, P1/A and P2/A Ross cylinders PHR1, PHR2 and PHR3 Reynolds cylinders DD-6, DD-2A, DD-3A and DD-4A Seleim cylinders 1, 3, 5, 8 and 10
0.8 0.6 0.4 0.2 0.0 0
1
2
3
4
5
6
7
8
9
Pcr / Pexp
4.12 Design charts for the general instability of ring-stiffened cylinders, using Kendrick Part III.
Table 4.7 Buckling pressures (P/E × 106) for ring-stiffened cones Singer56 Model no. 6 7 8
Angle (deg) 30 30 20
Stiffened shell 1.263 0.0557 3.54
Finite element solution Equivalent cylinder 1.234 0.045 3.04
Simply supported a
1.104 (10) 0.0462 (9) 3.149 (5)
Clamped
Fixed
1.243 (11) 0.0484 (9) 3.415 (5)
1.423 (12) 0.0649 (11) 4.36 (6)
a The figures in parentheses are the numbers of circumferential lobes or waves, n. P = pressure; E = Young’s modulus.
© Carl T. F. Ross, 2011
186
Pressure vessels
original study,56 a small error appeared in the third column of Table 1b, where the dimensionless parameter for cone No. 8 should have been 155 and not 98.4. In addition, as Singer used a dimensionless parameter, and because the finite element solution was a numerical one, it was necessary to make some numerical assumptions. For the numerical solution, it was assumed that the slant length of the vessels was 100 in, and they were stiffened by ten equally spaced ring-stiffeners. The cone angle for model Nos. 6 and 7 was 30°, and for model No. 8 it was 20°. From Table 4.7 it can be seen that there is reasonable agreement between the simply-supported and clamped solutions, based on the finite element method, and Singer’s two solutions. The main reasons for discrepancy may have been partly that Singer presented his results with reference to a dimensionless parameter and partly because of Singer’s assumptions that his ring-stiffened cones were equivalent to simply supported orthotropic conical shells. It is interesting to note that Singer’s simpler equivalent cylindrical shell solution gave relatively good results. Nevertheless, the above arguments are academic for ring-stiffened cones that buckle inelastically, as shorter and thicker vessels are likely to suffer plastic knockdown. Details of experimental results for vessels that fall into this category are shown in Table 4.8, and details of the buckling pressures are shown in Table 4.9. As can be seen from the results of Table 4.9, despite the fact that the models were machined very precisely (Figs 4.13–4.15), the plastic knockdown was quite large for cone 3. It must be brought to the reader’s attention that the theoretical solution was based on the finite element method for Fig. 4.16 and this theory assumed that the larger ends of the cones were fixed whereas their smaller ends were clamped. For similar charts of the inelastic instability of ring-stiffened cones, Kendrick Parts I and III were used, respectively, for two more design charts, as shown in Figs 4.17 and 4.18, respectively. Before buckling the cones, vibration tests were carried out on these vessels and, for this reason, Figs 4.13–4.15 show out-of-circularity plots before and after vibration. Furthermore, according to the experimental observations of references 98 and 99, cone 1 appeared to buckle elastically, whereas cones 2 and 3 appeared to buckle inelastically. In order to determine the plastic reduction factor, it is suggested that a thinness ratio similar to λ′ as described in Section 4.2 is used in the same manner as that described in Section 3.1.3. In this instance, the finite element was used to calculate Pcr, where the circular truncated conical shell and the ring stiffener elements of Ross91 were used; the computer program was called RCONEBUR. These elements are described in much detail in this book. Once again, the boundary conditions for these vessels were assumed to be fixed at one end and clamped at the other end. The calculation for λ′
© Carl T. F. Ross, 2011
General instability of pressure vessels
187
Table 4.8 Geometrical details of ring-stiffened cones Model no.
N
b (mm)
d (mm)
e (mm)
1 2 3
6 6 7
1.016 1.016 1.016
1.016 1.524 2.032
−0.826 −1.080 −1.333
101.6 mm
38.1 mm
211.0 mm
0.635 mm Cone no.
A
B
1
1.0
1.0
2
1.0
1.5
3
1.0
2.0
B A
d = depth of ring-stiffener or ‘rectangular’ cross-section. e = ring eccentricity (negative as the rings were external). R1 = radius of shell at small end = 1.905 cm. R2 = radius of shell at large end = 5.08 cm. h = shell thickness = 0.635 mm.
Table 4.9 Buckling pressures for the three cones Cone number
Experimental (MPa)
Elastic theory (MPa)
1 2 3
2.98 (4)a 3.93 (4) 4.10 (3, 4)
3.55 (4) 5.48 (4) 6.65 (3)
a
The number of lobes is given in parentheses.
© Carl T. F. Ross, 2011
188
Pressure vessels
Major buckle
0.0001''
0.0001''
(a)
(b)
4.13 Initial out-of-circularity plots for cone 1: (a) before vibration; and (b) after vibration.
Buckle
0.00005''
0.00005''
(a)
(b)
4.14 Initial out-of-circularity plots for cone 2: (a) before vibration; and (b) after vibration.
© Carl T. F. Ross, 2011
General instability of pressure vessels
189
0.0001''
0.00005''
Buckle
(a)
(b)
4.15 Initial out-of-circularity plots for cone 3: (a) before vibration; and (b) after vibration.
2.0 1.8 1.6 1.4
1 / λ'
1.2 Safe side
1.0 0.8 0.6
Cones Cones Cones Cones
0.4 0.2
0.0
1.0
2.0
3.0
4.0
5.0
1, 2, 3 4, 5, 6 7, 8, 9 10, 11, 12
6.0
Pcr / Pexp
4.16 Design chart for the general instability of machined ring-stiffened circular conical shells using RCONEBUR.
© Carl T. F. Ross, 2011
190
Pressure vessels 2.0
1.8
1.6
1.4
1.2 1 / λ'
Safe side 1.0
0.8
0.6 Cones 1, 2, 3 Cones 4, 5, 6
0.4
Cones 7, 8, 9 Cones 10, 11, 12
0.2
0.0 0.0
1.0
2.0
3.0
4.0
5.0
PKD1 (Pcr1 / Pexp )
4.17 Design chart for the general instability of ring-stiffened cones using Kendrick Part I, buckling pressure, namely Pcr1.
was based on an equivalent ring-stiffened circular cylinder and it is described in detail by Ross et al.96,100–102 and Fig. 3.9. The design chart, shown in Fig. 4.16 is somewhat limited owing to the lack of available experimental data, but it appears that machined vessels which had a value of 1/λ′ <0.55, appeared to fail through elastic instability. Ross et al.103 also presented a design chart for the inelastic instability of ring-stiffened cones, adapting Kendrick’s method85 (Fig. 4.15). Instructions on how to use this chart are given in Chapter 9. This design chart is for near
© Carl T. F. Ross, 2011
General instability of pressure vessels
191
2 1.8 1.6 1.4
1 / λ'
1.2 1 0.8 0.6 0.4 0.2 0 0
0.5
1
1.5 2 2.5 PKD3 (Pcr3 / Pexp )
3
3.5
4.18 Design chart for the general instability of ring-stiffened cones, using Kendrick Part III, buckling pressure, namely Pcr3. Symbols: 䉱, cones 1, 2 and 3; 䉭, cones 4, 5 and 6; 䊐, cones 7, 8 and 9; 䊏, cones 10, 11 and 12; 䉬, cones 13, 14 and 15.
geometrically perfect vessels and, for geometrically imperfect vessels, the graph will be to the right of that for perfect vessels, so that the value of PKD will be larger for vessels with larger values of initial out-of-circularity. The problem with this theory is that it does not take into account the distribution of wall thickness, nor inelastic buckling. In 2000, Ross et al.79 presented a method which successfully takes these into account; both theoretical and experimental results were given.
© Carl T. F. Ross, 2011
5 Vibration of pressure vessel shells
Abstract: Both theoretical and experimental investigations are presented on the free vibration of unstiffened and ring-stiffened thin-walled circular cylinders, cones and domes. The theoretical investigations are carried out by the finite element method, and derivations of various mass matrices are presented. Experimental tests are described in detail and experimental results given. The vessels considered are unstiffened and ring-stiffened circular cylinders and cones, together with thin-walled domes. Comparisons between experiment and theory are shown to be good. Key words: ring-stiffening, circular cylinders, cones, domes, resonance, vibration, finite elements.
The dynamical equation for free vibrations without damping is given by
[K ] − ω 2 [M ] = 0
[5.1]
where: ω = circular frequency (rad s−1); [K] = system stiffness matrix = Σ [k]; [M] = system mass matrix = Σ [m]; [k] = elemental stiffness matrix; [m] = elemental mass matrix.
5.1
Free vibration of unstiffened circular cylinders and cones
Thin-walled circular cylinders and cones vibrate in numerous modes, but the fundamental modes tend to be of lobar form, as shown in Figs 5.1 and 5.2, where the vessel vibrates with n circumferential waves or lobes. These modes of vibration tend to be of similar form to the buckling modes shown in Fig. 1.5, although, when n = 1, a cantilever mode is obtained, which is, in general, not related to the buckling mode. The elemental mass matrix92 for the circular cylindrical element of Fig. 3.11 is given by:
[ m] = ∫ [ N ]T ρ [ N ] d ( vol )
[5.2]
where [N] = matrix of shape functions, which is obtained from equation [3.21]; and ρ = density. As a sinusoidal variation has been assumed in the 192 © Carl T. F. Ross, 2011
Vibration of pressure vessel shells
193
5.1 Lobar eigenmode of vibration.
n=0
n=1
n=2
n=3
5.2 Circumferential wave pattern for cones and cylinders.
circumferential direction; equation [5.2] can be simplified to the form shown in equation [5.3]: T [ m] = π ∫ [ N 1 ] ρ [ N 1 ] RtLdξ 1
where: R = mean radius of shell; t = wall thickness; L = length of cylinder;
© Carl T. F. Ross, 2011
[5.3]
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Pressure vessels
ξ = x/L; n = number of lobes or circumferential waves into which the vessel vibrates; and 0 ⎡ (1 − ξ ) ⎢ [ N ] = ⎢ 0 (1 − ξ ) ⎢⎣ 0 0
0 0
1
ξ 0 0 ξ 0 0
(1 − 3ξ 2 + 2ξ 3 )
0 0 L (ξ − 2ξ 2 + ξ 3 )
⎤ ⎥ ⎥ ( 3ξ 2 − 2ξ 3 ) L ( −ξ 2 + ξ 3 )⎥⎦ = [ N ] (either cos nφ or sin nφ ) 0 0
0 0
[5.4]
Hence, by substituting and integrating, the elemental mass matrix is: ⎡ m11 ⎢m 21 [ m] = πRLρt ⎢ ⎢ ↓ ⎢ ⎣ m81
→ m18 ⎤ ⎥ ⎥ ⎥ ⎥ → m88 ⎦
m12
m82
[5.5]
which is symmetrical and where m11 m15 m22 m26 m33
= = = = =
1/3 1/6 1/3 1/6 13/35
m34 m37 m38 m44 m47
= = = = =
11L/210 9/70 −13L/420 L2/105 13L/420
m48 m55 m77 m78 m88
= = = = =
−L2/140 1/3 13/35 −11L/210 L2/105
5.1.1 Forsberg’s model To test the element, comparison is made in Table 5.1 of the free vibration characteristics of the thin-walled circular cylinder described in reference 104. This cylinder, which had fixed edges, had the following dimensions and properties: R = 1.0 in t = 0.002 in L = 10 in E = 30 × 106 lbf in−2 ν = 0.35 ρ = 7.35 × 10−4 lbf s2 in−4 The mode shapes in the u and w directions are shown in Fig. 5.3. From Table 5.1, it can be seen that the finite element solutions agree very favourably with the other two, and this was very encouraging as the finite element solutions used only 32 or 40 equal-length elements to represent © Carl T. F. Ross, 2011
Vibration of pressure vessel shells
195
Table 5.1 Results for circular cylinder, corresponding to the fundamental frequency n = 4
Frequencya (Hz) ûN vˆ N w ˆ N
VFD solution,c 128 grid intervals
Finite element solution
‘Exact’ solutionb
32 elements
40 elements
508.3 ±0.017 99 +0.250 7 +1.0
513.0 ±0.017 97 +0.250 7 +1.0
520.0 ±0.018 04 +0.251 0 +1.0
515.6 ±0.018 03 +0.250 9 +1.0
ûN = maximum normalised u displacement for this frequency; vˆN = maximum normalised v displacement for this frequency; w ˆ N = maximum normalised w displacement for this frequency. b The ‘exact’ solution refers to the predictions of Forsberg’s104 solution. c The VFD solution refers to the predictions of the variational finite difference solution of Abdulla and Galletly.105 a
Mid-bay
u
0.02 0.01 0.00 0.0
0.1
0.2
0.3
0.4
0.1
0.2
0.3
0.4
Mid-bay
w
1.0 0.5 0.0 0.0
ε
0.5
ε
0.5
5.3 Mode shape of cylinder with fixed edges.
the whole cylinder. In fact, because of symmetry about the mid-bay of the cylinder, only half the cylinder was considered, with either 16 or 20 elements, respectively.
5.1.2 Warburton’s model A second cylinder described by Abdulla and Galletly,105 which had simply supported ends, was also analysed by the finite element solution and compared with the solutions of Warburton,106 Webster107 and Percy et al.108 (Table 5.2). Once again the properties of symmetry were used about midspan and, because of this, the finite element solution could only yield even © Carl T. F. Ross, 2011
196
Pressure vessels
Table 5.2 Resonant frequencies (Hz) for a simply supported circular cylinder (R = 2 in, L = 8 in, t = 0.1 in, E = 30 × 106 lbf in−2, ν = 0.3, ρ = 7.35 × 10−4 lbf s2 in−4)
n
m
Warburton (exact)106
Webster (single element)107
Percy108 and many others (multi-element)
Present solution
2
1 2 3 4 5 6
2046.8 5637.6 8935.3 11405 13245 14775
2046.8 5637.6 8955.9 11486 15716 19798
2078.5 5806.3 9281.4 11893 13822 15413
2047.2 – 8956.1 – 13273 –
3
1 2 3 4 5 6
2199.3 4041.9 6620.0 9124.0 11357 13384
2199.3 4042.3 6643.6 9241.5 14398 19156
2213.5 4180.1 6987.4 9734.0 12161 14315
2199.0 – 6635.8 – 11394 –
meridional modes (i.e. m = 1, 3, 5, etc.). For the finite element solution, onehalf of the cylinder was divided into 16 equal-length elements. The single element solution of Webster,107 which had 20 degrees of freedom, gave good results for the lower modes, but small errors appeared to occur for some of the higher meridional modes. The solution of Percy et al.108 employed 40 degrees of freedom and also appeared to produce small errors for the higher meridional modes. However, comparison between the present solution and the exact solution of Warburton was excellent.
5.1.3 Elemental mass matrix For the free vibration of thin-walled cones (Fig. 3.8), the elemental mass matrix is given by equation [5.6]: T [ m] = [Ξ]T ρtlπ ∫0 r [ N 1 ] [ N 1 ] dξ [Ξ] 1
where: r = Ri (1 − ξ) + Rjξ; Ri = radius at nodal circle i; Rj = radius at nodal circle j; [Ξ] = matrix of directional cosines (see Section 3.2); [N1] is as defined in equation [5.4].
© Carl T. F. Ross, 2011
[5.6]
Vibration of pressure vessel shells
197
5.1.4 Abdulla and Galletly’s model In addition to being applicable to thin-walled cylinders and cones, this element can also be used for complex meridional shapes, such as cooling towers, domes and hour-glass figures. The element will now be compared with various experimental and theoretical results. The first such comparison is made in Table 5.3 with the cone–cylinder combination of Abdulla and Galletly,105 which is shown in Fig. 5.4. From Table 5.3, it can be seen that there is very little difference between the two finite element solutions, despite the fact that one solution used over twice as many elements as the other. Comparison between the finite element solutions and the experimental results is good for n = 3, but somewhat disappointing for n = 2.
Table 5.3 Resonant frequencies (Hz) for cone–cylinder combination
n
1 1
2 3
FEM solution
Experimental
VFD solution of Abdulla and Galletly105
12 elements
25 elements
856 886
974 900
978.8 902.8
974.8 899.9
2.7
94
in
Meridional mode no. m
60°
4.242 in
tcone = 0·104 in tcylinder = 0·252 in E = 6·5 x 106 lbf in–2 n = 0·35 r = 1·59 x 10–4 lbf s2 in–4
4.063 in
5.4 Cone–cylinder combination.
© Carl T. F. Ross, 2011
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Pressure vessels
Frequency (Hz)
1000
Experimental Theoretical m=2
750 500
m=1
250 0 3
4
5
n
6
7
8
5.5 Frequencies for a 14.2 ° conical shell.
Experimental Theoretical
Frequency (Hz)
1000
m=2
750 500
m=1
250 0 8
9
n
10
11
12
5.6 Frequencies for a 30.2 ° conical shell.
5.1.5 Lindholm and Hu’s models Comparison is also made in Figs 5.5–5.8 of the finite element solution with the experimental results of Lindholm and Hu109 for four conical shells, simply supported at their ends. The finite element solution assumed that the vessels were hinged at their smaller ends and simply supported at their larger ones. From these figures it can be seen that comparison between experiment and theory is good, particularly for the frequencies corresponding to first meridional mode (i.e. m = 1).
© Carl T. F. Ross, 2011
Vibration of pressure vessel shells m=2
Frequency (Hz)
1000
199
Experimental Theoretical
750 m=1
500 250 0 8
9
n
10
11
12
5.7 Frequencies for a 45.1 ° conical shell.
Frequency (Hz)
1000
Experimental Theoretical m=2
750 500
m=1
250 0 8
9
n
10
11
12
5.8 Frequencies for a 60.5 ° conical shell.
5.1.6 Ferrybridge cooling towers Figures 5.9 and 5.10 show the resonant frequencies110 of the Ferrybridge cooling towers, which were constructed in concrete, and also for their 1 : 576 scale nickel model. The theoretical results were obtained using the conical element, as described in this chapter, employing 25 elements. For both cases the mesh was finer at the larger end and also at the throat. For the concrete cooling towers, the following material properties were assumed to apply: E = 4 × 106 lbf in−2; ν = 0.17; and ρ = 2.248 × 10−4 lbf s2 in−2. The experimental observations on Ferrybridge tower 2B showed the two lowest resonant frequencies to be 0.59 Hz (at n = 4) and 0.66 Hz (at n = 5); these values compare favourably with the theoretical predictions of Fig. 5.9. From Fig. 5.10, it can be seen that the experimental and theoretical results compare favourably for the nickel model.
© Carl T. F. Ross, 2011
Pressure vessels 1.8 1.6 1.4
Frequency (Hz)
1.2 1.0 0.8 0.6 0.4 0.2 0 0
1 2 3 4 5 Circumferential wave number n
6
5.9 Theoretical resonant frequencies of the Ferrybridge cooling towers.
Experimental Theoretical
1400 1200
Frequency (Hz)
200
1000 800 600 400 200 0 0
1 2 3 4 5 Circumferential wave number n
6
5.10 Resonant frequencies of the nickel model of the Ferrybridge cooling towers.
© Carl T. F. Ross, 2011
Vibration of pressure vessel shells
5.2
201
Free vibration of ring-stiffened cylinders and cones
The vibration modes for ring-stiffened cylinders and cones are similar to those described in Section 5.1. For these vessels, it will be necessary to obtain the elemental mass matrix of a ring-stiffener: 2π
[ m] = ∫ [ N ]T ρ [ N ] d ( vol ) = ρ Af R ∫0 [ N ]T [ N ] dφ
0 0 ⎤ ⎡cos nφ 0 0 ⎤ ⎡cos nφ 2π = ρ Af R ∫ ⎢ 0 sin nφ 0 ⎥⎢ 0 sin nφ 0 ⎥ dφ 0 ⎢ ⎥⎢ ⎥ ⎢⎣ 0 0 cos nφ ⎥⎦ ⎢⎣ 0 0 cos nφ ⎥⎦
ui vi wi ⎡1 0 0 ⎤ ⎢0 1 0 ⎥ π m = ρ A R [ ] f ⎢ ⎥ ⎢⎣0 0 1 ⎥⎦
[5.7]
where ρ = density of ring material; R = radius of ring centroid; and Af = cross-sectional area of ring. To obtain the mass matrix of the ring–shell combination, the process is to add the various components of the mass matrix of the ring-stiffener to the appropriate positions in the mass matrix for the circular cylindrical or conical shell element, in a manner similar to that described for the stiffness matrix of the ring–shell combination described in Section 4.1.4.
5.2.1 Forsberg’s solution To test the element, comparison is made in Table 5.4 of various resonant frequencies of a ring-stiffened cylinder, with the predictions of this solution, and those predicted by the exact solution of Forsberg111 and the finite
Table 5.4 Resonant frequencies (Hz) for a ring-stiffened circular cylinder n
m
Exact111
Finite difference112
Finite element solution
2
1 2 3
887 2327 3972
889 2326 3968
848 – 3735
3
1 2 3
1485 1924 2881
1489 1932 2897
1465 – 2771
4
1 2 3
2340 2421 2716
2334 2421 2737
2298 – 2707
© Carl T. F. Ross, 2011
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Pressure vessels
difference solution of Bushnell.112 The cylinder had a mean radius of 3.045 in, a wall thickness of 0.06 in and it was stiffened by three equally spaced ‘heavy’ ring-stiffeners. The vessel was assumed to be simply supported at its ends, and its overall length was assumed to be 16.2 in. From Table 5.4 it can be seen that the finite difference solution agrees very favourably with the exact solution, and also that the finite element solution agrees well, but less favourably, with the exact solution of Forsberg. However, as the finite element solution used only 16 equal-length elements to model half the vessel whereas the finite difference solution required 100 stations to model the whole vessel, the findings were particularly encouraging for the precision of the finite element solution. Furthermore, as the property of symmetry about mid-span was used for the finite element method, it was only possible to obtain the even meridional modes (i.e. m = 1, 3, 5, etc.). A typical meridional mode for m = 1 for Bushnell’s cylinder is shown in Fig. 5.11, which clearly demonstrates the effects of the rings.
5.2.2 Weingarten’s cylindrical shells
Ring
Ring
Ring
A comparison is also made in Figs 5.12 and 5.13 for the predictions of the finite element solution with the experimental observations of Weingarten113 for his two ring-stiffened circular cylinders. Weingarten’s models were machined from a solid bar of aluminium alloy. The wall thickness of the models was 0.06 in and, for the results of Fig. 5.13, the model’s rib heights were 0.095 in. After carrying out the first series of vibration experiments, the model’s rib heights were machined down to 0.045 in, and the vibration experiments repeated to produce the results of Fig. 5.12. Weingarten’s solution,113 which assumed simply supported ends and adopted Donnel-type equations, tended to overestimate the resonant frequencies. Figures 5.12 and 5.13 show good agreement between the finite element solution and experiment. It must be emphasised that the finite
5.11 Vibration configuration of a longitudinal generator of Bushnell’s cylinder.
© Carl T. F. Ross, 2011
Vibration of pressure vessel shells 14000 12000
203
Theoretical Experimental (m = 3) Experimental (m = 2) Experimental (m = 1)
Frequency (Hz)
10000 8000 6000 4000 2000
2 3 4 5 6 7 8 9 10 11 12 Number of circumferential waves n
5.12 Comparison of theory and experiment for a ring-stiffened circular cylinder of rib height 0.045 in.
22000 20000
Theoretical Experimental (m = 3) Experimental (m = 2) Experimental (m = 1)
Frequency (Hz)
18000 16000 14000 12000 10000 8000 6000 4000
m=3 m=2 m=1
2000 2 3 4 5 6 7 8 9 10 11 12 Number of circumferential waves n
5.13 Comparison of theory and experiment for a ring-stiffened circular cylinder of rib height 0.095 in.
© Carl T. F. Ross, 2011
204
Pressure vessels
element solution has the added advantage over Weingarten’s solution in that it can cater for boundary conditions other than simple supports. For the finite element solution, the following assumptions were made for Young’s modulus and density: E = 10.3 × 106 lbf/in2 and ρ = 2.4 × 10−4 lbf s2 in−4.
5.2.3 Weingarten’s conical shells For ring-stiffened conical shells, comparison is made in Figs 5.14 and 5.15 of the finite element solution with the experimental observations of Weingarten.113 Once again Weingarten only used one ring-stiffened cone of cone angle 20 °. The model had a wall thickness of 0.06 in and, for his first set of experimental results, the rib height was 0.095 in. To obtain the second set of experimental results, the rib height of the models was machined down to 0.045 in. From Figs 5.14 and 5.15 it can be seen that the finite element solution compared more favourably with the experimental results for the cone with the smaller rib height than for the cone with the larger rib height. The reason for this may be attributed to the fact that the ribs of Weingarten’s model lay perpendicular to the meridian of the cone, but the finite element solution assumed that the ribs were perpendicular to the axis of the cone, and this discrepancy was obviously worse for the cone with the larger rib height.
24000 22000
Theoretical Experimental (m = 3) Experimental (m = 2) Experimental (m = 1)
20000
Frequency (Hz)
18000 16000 14000 12000 10000 8000 6000 4000 2000 2 3 4 5 6 7 8 9 10 11 12 Number of circumferential waves n
5.14 Comparison of theory and experiment for a ring-stiffened conical shell of rib height 0.045 in.
© Carl T. F. Ross, 2011
Vibration of pressure vessel shells
28000 26000
205
Theoretical Experimental (m = 3) Experimental (m = 2) Experimental (m = 1)
24000 22000
Frequency (Hz)
20000 18000 16000 14000 12000 10000 8000 6000 4000 2000 2 3 4 5 6 7 8 9 10 11 12 Number of circumferential waves n
5.15 Comparison of theory and experiment for a ring-stiffened conical shell of rib height 0.095 in.
5.2.4 Vibration experiments Other vibration experiments98,99 were carried out for the three ringstiffened cones of Fig. 5.16, whose details are given in Table 4.8. From Fig. 5.16 it can be seen that the stiffeners on these cones were perpendicular to the axes of the cones. The cones were vibrated by exciting their flanks through two elastic straps attached to electromagnetic vibrators, as shown in Fig. 5.17. In order to obtain even values of n, the electromagnetic vibrators were vibrated in-phase; for odd values of n the electromagnetic shakers were vibrated 180 ° out of phase. Control of the vibration was carried out with the aid of a frequency response analyser (FRA) and a transducer (microphone), as shown in Fig. 5.18. (The results are given in Figs 6.4–6.6).
5.3
Free vibrations of domes
Thin-walled hemi-ellipsoidal prolate domes vibrate in many modes,114,115 as shown by Figs 5.19 and 5.20, but the modes of interest in this context are those shown in Fig. 5.20, as these modes have the lowest resonant frequencies of vibration. Similarly, for oblate hemi-ellipsoids, the modes of vibration corresponding to the lowest resonant frequencies are those shown in Fig. 5.21.
© Carl T. F. Ross, 2011
5.16 Ring-stiffened cones.
Elastic strap movement
Elastic strap movement
Cone
5.17 Method of excitation of ring-stiffened cones.
Frequency response analyser Output
Input Charge amplifier
Amplifier
Vibrators
Oscilloscope
Transducer
5.18 Block diagram showing the excitation and detection instrumentation.
© Carl T. F. Ross, 2011
Vibration of pressure vessel shells
207
5.19 Nodal pattern for complex eigenmode.
Plans (dotted lines show displacement form)
Elevations (dotted lines show nodal outline) (a)
(b)
(c)
5.20 Lobar eigenmodes for prolate domes: (a) n = 1, m = 1; (b) n = 2, m = 2; and (c) n = 4, m = 2.
Elevations (dotted lines show displacement form)
Plans (dotted lines show nodal outline)
5.21 Axisymmetric eigenmodes for oblate domes.
© Carl T. F. Ross, 2011
208
Pressure vessels
Another reason why the modes of Figs 5.20 and 5.21 are of interest is that, as they are of similar form to the static buckling modes shown in Figs 1.6 and 1.7, there is a distinct possibility that a form of dynamic buckling occurs when a vessel under external pressure is resonated. The free vibration of domes can be analysed by the thin-walled truncated conical shell element described in Section 3.2, the constant meridional curvature element (CMC) described in Section 3.4.1, or the varying meridional curvature element (VMC) described in Section 3.4.3.
5.3.1 Method of vibrating domes in air The ten hemi-ellipsoidal domes described in Section 3.4.2 were vibrated in air. The fundamental resonant frequencies for the oblate domes tended to be of axisymmetric form and in the noses of the domes and, therefore, they were excited in the manner shown in Fig. 5.22. The fundamental resonant frequencies for the hemispherical dome and the prolate hemi-ellipsoidal domes tended to be of lobar form and in the flanks of the vessels and, for this reason, these domes were resonated in the manner shown in Fig. 5.23. Comparisons are made in Tables 5.5 and 5.6 of the experimental results for these domes and the theoretical results, using the three finite elements. For the hemispherical dome and the prolate hemi-ellipsoidal domes, comparison is also made in Figs 5.24–5.30 of the
5.22 Method of excitation for oblate domes (n = 0).
(a)
(b)
5.23 Method of excitation for prolate domes (a) Fundamental frequency (n = 1). (b) Lobar vibration (n = 2, 3, etc.).
© Carl T. F. Ross, 2011
© Carl T. F. Ross, 2011
n = 0, m = 3
n = 0, m = 4
n = 0, m = 5
n = 0, m = 6
1585 1103 696
– 1905 1909 1871 – 1302 1439 1450 1401 – 960 1114 1129 1076 –
2282 2287 2203 – 1998 2009 1921 – 1786 1797 1739 –
2817 2821 2693 – 2794 2806 2690 – 2671 2693 2644 –
3524 3526 3371 3756 3773 3654 3731 3786 3718
n=2
n=3
n=5
1
370 452 524 627 881 1113 1470
369 437 528 653 828 1073 1409
363 434 527 654 829 1074 1409
351 425 521 650 828 1073 1406
511 608 797 943 1362 1720 2390
487 590 730 926 1211 1640 2217
488 591 731 927 1212 1641 2217
492 595 735 931 1217 1645 2219
431 545 661 880 1243 1650 2435
415 502 626 811 1102 1587 2349
415 502 626 812 1102 1588 2349
420 508 633 819 1110 1596 2356
453 522 641 840 1182 1685 2475
434 501 605 770 1046 1543 2422
434 501 605 770 1047 1543 2422
439 508 613 780 1058 1556 2435
568 624 711 862 1185 – –
523 574 657 798 1051 1537 2490
523 574 657 798 1051 1538 2490
528 581 666 809 1065 1555 2510
Aspect ratio, 2 experimental, 3 varying meridional curvature element, 4 constant meridional curvature element, 5 conical element.
4.0 3.5 3.0 2.5 2.0 1.5 1.0
AR1 Exp.2 VMC3 CMC4 CON5 Exp. VMC CMC CON Exp. VMC CMC CON Exp. VMC CMC CON Exp. VMC CMC CON
n=1
n=4
Aspect ratio, 2 experimental, 3 varying meridional curvature element, 4 constant meridional curvature element, 5 conical element.
1601 1122 722
Table 5.6 Resonant frequencies (Hz) in air for prolate domes
1
0.7 1571 1598 0.44 1104 1115 0.25 836 711
AR1 Exp.2 VMC3 CMC4 CON5 Exp. VMC CMC CON Exp. VMC CMC CON Exp. VMC CMC CON Exp. VMC CMC CON
n = 0, m = 2
Table 5.5 Resonant frequencies (Hz) in air for oblate domes
2600 2500 2400 2300 2200 2100
Frequency (Hz)
2000 1900 1800 1700 1600 1500 Experimental result FEM result
1400 1300 1200 1100 1000 0
1
2
3
4
5
Number of circumferential lobes n
5.24 Variation of resonant frequencies with the number of lobes (n) for a dome of aspect ratio 1.0. 1800 1700
Frequency (Hz)
1600 1500 1400 1300 1200 Experimental result FEM result
1100 1000 0
1
2
3
4
5
Number of circumferential lobes n
5.25 Variation of resonant frequencies with the number of lobes (n) for a dome of aspect ratio 1.5.
© Carl T. F. Ross, 2011
1500 1400
Frequency (Hz)
1300 1200 1100 1000 900 Experimental result FEM result 800 700 0
1
2
3
4
5
Number of circumferential lobes n
5.26 Variation of resonant frequencies with the number of lobes (n) for a dome of aspect ratio 2.0. 1200 1100
Frequency (Hz)
1000 900 800 700 Experimental result FEM result
600 500 400 0
1
2
3
4
5
Number of circumferential lobes n
5.27 Variation of resonant frequencies with the number of lobes (n) for a dome of aspect ratio 2.5.
© Carl T. F. Ross, 2011
Pressure vessels 850
Frequency (Hz)
800 750 700 650 600 Experimental result FEM result
550 500 0
1
2
3
4
5
Number of circumferential lobes n
5.28 Variation of resonant frequencies with the number of lobes (n) for a dome of aspect ratio 3.0.
700 650 600 Frequency (Hz)
212
550 500 450 Experimental result FEM result
400 350 300 0
1
2
3
4
5
6
Number of circumferential lobes n
5.29 Variation of resonant frequencies with the number of lobes (n) for a dome of aspect ratio 3.5.
© Carl T. F. Ross, 2011
Vibration of pressure vessel shells
213
650 600
Frequency (Hz)
550 500 450 400 Experimental result FEM result
350 300 0
1
2
3
4
5
Number of circumferential lobes n
5.30 Variation of resonant frequencies with the number of lobes (n) for a dome of aspect ratio 4.0.
experimental results with those predicted using the VMC element. From these figures, it can be seen that, in all cases, the finite element solution predicts resonant frequencies that are lower than the experimentally obtained values. It can also be seen that, in all cases except for the hemispherical dome, a minimum resonant frequency is found for a finite value of n. Conversely, for the hemispherical dome, it appears that a maximum resonant frequency is found when n approaches infinity. The experimental measurements taken around a circumference for a typical dome show the circumferential eigenmodes to be reasonably pure, particularly for the lower values of n, as shown in Fig. 5.31.
5.3.2 Vibration of a large dome Comparison is also made in Fig. 5.32 between the experimental and theoretical results (VMC) for the large prolate hemi-ellipsoidal dome described in Section 3.4.3. From Fig. 5.32, it can be seen that, as for the smaller domes, the finite element graph lies a little below the experimental one, and that both predict the fundamental resonant eigenmode to have five lobes.
© Carl T. F. Ross, 2011
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Pressure vessels
+ –
–
+ + –
+ –
– +
(a) AR = 3, n = 1
(d) AR = 3, n = 4
+
–
+ –
– +
+
– –
+
– +
–
+ (e) AR = 3, n = 5
(b) AR = 3, n = 2
+
–
+
+ In-phase lobe – Out-of-phase lobe
– + –
(c) AR = 3, n = 3
5.31 Nodal lobar patterns for a dome, taken around a circumference.
5.4
Higher order elements for thin-walled cones
In a manner similar to that used in chapter 4, Ross74 has developed higher order elements for the vibration of thin-walled cones. The elements adopted here were as follows:
© Carl T. F. Ross, 2011
Vibration of pressure vessel shells
215
550
Experiment
500
Frequency (Hz)
450
FEM
400
350
300 Aspect ratio = 1·9759 250 0
1
2
3
4
5
6
7
8
Number of circumferential lobes n
5.32 Variation of resonant frequency with n for the large SUP dome of aspect ratio 2 : 1.
(a) LILICUBE, where the ‘u’ and ‘v’ displacements were of linear form and the ‘w’ displacement was of cubic form;91 (b) QUQUCUBE, where the ‘u’ and ‘v’ displacements were of quartic form and the ‘w’ displacement was of cubic form; and (c) ALLCUBE where all three displacements were of cubic form. Comparisons are made between these solutions for the circular cylinder of Warburton106 which had fixed ends. The mesh adopted the property of symmetry and 16 elements were chosen to model half the length of the vessels as described by Zienkiewicz23 (pp. 110 and 111). The results are shown in Table 5.7. The exact lowest natural frequency for this vessel, as predicted by Warburton,106 was 508.3 Hz; this compares favourably with the predictions of all three elements for n = 4. The predicted resonant frequency by the element LILICUBE appears to be a little less for n = 2 than the predictions of the more sophisticated elements. Comparisons are also made with the predictions of these three elements and the results of Abdulla and Galletly105 and their model, shown in Fig. 5.4. The results are shown in Table 5.8, which also gives the predictions by Galletly’s two solutions. Galletly’s two solutions105 were based on the variational finite differences (VFD) method and in his second solution Galletly
© Carl T. F. Ross, 2011
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Pressure vessels
Table 5.7 Resonant frequencies (Hz) for Warburton’s circular cylinder n
LILICUBE
QUQUCUBE
ALLCUBE
2 3 4 5
1414.6 740.8 523.0 551.1
1487.3 749.6 520.2 549.7
1511.2 755.8 522.4 551.2
Table 5.8 Resonant frequencies (Hz) for Galletly’s model Galletly n
LILICUBE
QUQUCUBE
ALLCUBE
1
2
Experiment
2 3
978.8 902.8
990.9 906.6
995.4 908.4
974 900
863 897
856 886
Table 5.9 Resonant frequencies (Hz) in air for oblate domes n=0 AR
Exp
VMC
CCC
QQC
0.7 0.44 0.25
1571 1104 836
1598 1115 711
1900 1271 783
1877 1267 782
removed the effects of axial restraint. The results in Table 5.8 show good agreement between Ross’s three solutions and Galletly’s first solution, but not very good agreement with Galletly’s second solution. The reason for this may be because Galletly assumed no axial restraint for his second solution.
5.5
Higher order elements for thin-walled domes
For thin-walled domes,76 comparisons are made with the CCC, QQC and VMC elements described in chapter 4. The comparisons are shown in Tables 5.8 and 5.9 with the experimentally obtained results of Ross and Johns,114 and the three solutions of Ross, for the ten hemi-ellipsoidal domes described in Section 5.3.
© Carl T. F. Ross, 2011
Vibration of pressure vessel shells
217
From Table 5.9, it can be seen that the CCC and QQC elements give worse results than the simpler VMC element for the domes of aspect ratio 0.7 and 0.44, but better results for the dome of aspect ratio 0.25. This may be attributed to the fact that all three elements adopted four degrees of freedom, namely, u, v, w and θ, but that for these oblate domes, v was zero. From Table 5.9, it can be seen that the CCC and the QQC elements gave better results than the VMC elements for all modes, except for n = 1. It may be that, for the n = 1 mode, v was of a much simpler form than assumed. The argument that can be held, therefore, is that for n = 0 and n = 1, where v was either zero or of simple form, the assumption that v varied either quadratically or cubically was less justified than assuming that v varied linearly.
5.5.1 Conclusions The results show that by assuming quadratic and cubic variations with respect to the meridian of hemi-ellipsoidal dome shells, for the meridional and circumferential displacements, good results are obtained for the majority of the vessels. In addition, the more sophisticated elements, the CCC element and the QQC element, give worse results than the simpler VMC element, when the circumferential displacement is zero, as in the instances when n = 0 and n = 1.
5.6
Effects of pressure on vibration
To determine the effects of pressure on vibration, the free vibration equation [5.1] takes the modified form of equation [5.8]
([ K ] + [ KG ]) − ω 2 [ M ] = 0
[5.8]
where the geometrical stiffness matrix of the system is: [KG] = Σ [kG]; and the elemental geometrical stiffness matrix is [kG]. The method of solving equation [5.8] is to apply a pressure to the vessel and, from static analysis, to calculate the stresses in the vessel and, hence, [KG]. Once [KG] is determined, the resonant frequencies can be calculated for that pressure. Other resonant frequencies can be calculated for other values of pressure by the same process.
5.6.1 Effect of pressure on vibrations To demonstrate the use of equation [5.8], analysis is made of model P3 (Table 4.6), where the pressure is varied from an internal one of −120 lbf in−2 to an external one of +240 lbf in−2, the model being assumed to have clamped
© Carl T. F. Ross, 2011
218
Pressure vessels
2000
2
bf +8 in –2 0l bf in –
bf
in –2
0l
0l
lbf
1000
+2
00
lbf
in –2
+1 60
Frequency (Hz)
–4
–1 20
lbf
in –2 in –2
1500
+2
40
lbf
in –2
500
2
3
4
5
6
Number of circumferential waves n
5.33 Effect of pressure on the vibration of a ring-stiffened circular cylinder.
ends, with an elastic buckling pressure of 237.7 lbf in−2. The results are shown in Fig. 5.33, where it can be seen that increased external pressure decreases the magnitude of the resonant frequencies, and that increased internal pressure increases the resonant frequencies. Furthermore, it can be seen that as the experimental buckling pressure is approached the resonant frequencies approach zero, and also that the resonant circumferential vibration eigenmode becomes similar to the circumferential static buckling eigenmode (i.e. n = 4).
© Carl T. F. Ross, 2011
Vibration of pressure vessel shells –160 kN m–2
800
0 kN m–2
700
600
Frequency (Hz)
219
80 kN m–2
500
400
160 kN m–2
300 197 kN m–2 200
100
0
2 4 6 8 10 Number of circumferential lobes n
5.34 Effect of pressure on the vibration of a hemi-ellipsoidal prolate dome of aspect ratio 2 : 1.
5.6.2 Effect of pressure on a large dome To demonstrate the effects of uniform pressure on the large SUP domes described in Section 3.4.3, comparison is made in Fig. 5.34 of the variation of its fundamental resonant frequencies with pressure, where it can be seen that, as the external pressure (positive) is increased, the resonant frequencies decrease, and also that, as the static buckling pressure of 0.202 MPa is approached, the fundamental circumferential eigenmodes of vibration become similar to the circumferential static buckling eigenmode (i.e. n = 8). This observation shows that as a submersible sinks deeper into the water there is a distinct possibility that the vessel can buckle at a pressure somewhat less than the static buckling pressure, as a result of the combined effects of external pressure and an exciting force triggering off the fundamental resonant frequency.
© Carl T. F. Ross, 2011
220
Pressure vessels
5.7
Effects of added virtual mass
In this chapter, the effects of the water vibrating with the vessels have not been considered, but these are discussed in chapter 6. It should be emphasised that the effect of water on a shell is to further decrease its resonant frequencies.
5.8
Effects of damping
The free vibration characteristics examined in this chapter neglected damping, but both experimental and theoretical investigations have found damping to have little effect on the magnitudes of the resonant frequencies, even if the structure is vibrated in water. Some of the computer programs described in the present chapter appear in reference 28, and the others can be obtained directly from the author.
© Carl T. F. Ross, 2011
6 Vibration of pressure vessel shells in water
Abstract: Theoretical solutions and experimental observations on the free vibration of ring-stiffened cones and domes in water are presented. For domes, the nonlinear effects of pressure on vibrating vessels are also considered. The theoretical analysis, which allows for a compressible fluid but neglects damping, is based on the finite element method. Key words: vibrations, pressure vessels, cones, domes, finite element analysis.
6.1
Free vibration of ring-stiffened cones in water
The modes of vibration for ring-stiffened cones under water are similar to that in air or in vacuo, as described in chapter 5. The main difference, however, is that the vibration of water with the structures causes the magnitudes of the resonant frequencies to decrease, compared with their magnitudes in air or in vacuo. Experimental observations have shown that damping owing to water has a negligible effect on the magnitudes of its resonant frequencies. Zienkiewicz and Newton116 showed, through the use of variational calculus, that the wave equation of a compressible fluid is given by equation [6.1]. Similarly, they showed that the vibration equation of a structure in a fluid, neglecting damping, is given by equation [6.2].
[ H ] { p} + [G ] ∂ 2 { p} / ∂t 2 − ω 2 [ S ] {ui } = 0
[6.1]
([ K ] − ω 2 [ M ]) {ui } − (1 / ρF ) [ S ]T { p} = 0
[6.2]
where
[ H ] = Σ [ he ] [G ] = Σ [ g e ] [ S ] = Σ [ se ] ⎛∂ N T ∂ N ∂ N T ∂ N ∂ N T ∂ N ⎞ [ he ] = ∫vol ⎜⎝ [∂x] [∂x ] + [∂y] [∂y ] + [∂z] [∂z ]⎟⎠ d ( vol ) 1
[ g e ] = c 2 ∫vol [ N ]T [ N ] d ( vol ) [ se ] = ρF ∫vol [ N ]T ⎡⎣ N ⎤⎦ d (a) 221 © Carl T. F. Ross, 2011
222
Pressure vessels
and {p} = vector of nodal pressures; d(a) = elemental area at fluid-structure interface; ¯ ] = matrix of shape functions defining interface motion; [N ρF = fluid density; c = speed of sound in fluid. From equations [6.1] and [6.2] it can be seen that both equations incorporate displacement degrees of freedom, namely {ui}, and pressure degrees of freedom, namely {p}, where, in equation [6.2], the term [S]T{p}/ρF represents the effect of the fluid motion on the fluid–structure interface. Thus, it is evident that equations [6.1] and [6.2] can be coupled together in the matrix equation [6.3] ⎡[ K ] − [ S ]T / ρF ⎤ ⎧ui ⎫ ⎡[ M ] 2 ⎢ ⎥ ⎨ p⎬−ω × ⎢ S [H ] ⎦ ⎩ ⎭ ⎣[ ] ⎣ 0
0 ⎤ ⎧ui ⎫ ⎨ ⎬ = {0} [G ]⎥⎦ ⎩ p ⎭
[6.3]
The solution of equation [6.3] is not very convenient from a computational point of view because the equation is unsymmetrical, but by rearranging the equations Irons117 provided the equivalent symmetrical form shown by equation [6.4]: ⎡ ρF [ K ] ⎢ 0 ⎣
0 ⎤ ⎧ui ⎫ 2 ⎨ ⎬−ω [G ]⎥⎦ ⎩ p ⎭
⎡ ρF [ M ] + [ S ]T [ H ]−1 [ S ] ×⎢ ⎢⎣ [G ][ H ]−1 [ S ]
[ S ]T [ H ]−1 [G ]⎤ ⎧ui ⎫ ⎥⎨ ⎬ = 0 [G ][ H ]−1 [G ] ⎥⎦ ⎩ p ⎭
[6.4]
For the present problem, equation [6.4] is of banded form, and because of this it is convenient to determine the eigenvalues and eigenmodes through the Jennings simultaneous iteration process.118 For an incompressible fluid, the speed of sound in the fluid approaches infinity, therefore equation [6.4] takes the simplified form of equation [6.5]:
ρF [ K ] − ω 2 ( ρF [ M ] + [ S ]T [ H ]−1 [ S ]) = 0
[6.5]
where the added virtual mass is:
(1 / ρF ) [ S ]T [ H ]−1 [ S ]
[6.6]
6.1.1 Fluid element The shell element and its stiffness and mass matrices are the same as those described in Sections 3.2 and 5.1.3 whereas, to represent the fluid motion, the fluid element used was a ‘solid’ annular one,119,120 which had an isoparametric cross-section, as shown in Fig. 6.1. Thus the shell element, which had
© Carl T. F. Ross, 2011
Vibration of pressure vessel shells in water z
6
223
5
7 4
f
8
1
2
3
r
6.1 Cross-section of annular fluid element.
two ring nodes at its ends, had a total of eight displacement degrees of freedom whereas the fluid element had a total of eight pressure degrees of freedom. The pressure variation in the fluid owing to vibration is assumed to be:
{ p} = [ N ] { pi } where [N] = [N′]cos nϕ and [N′] = [N1, N2, N3, ... , N8] and N1 = 41 (1 − ξ ) (1 − η ) ( −ξ − η − 1) N2 =
1 2
(1 − ξ 2 ) (1 − η )
N 3 = 41 (1 + ξ ) (1 − η ) (ξ − η − 1) N 4 = 12 (1 + ξ ) (1 − η 2 ) N 5 = 41 (1 + ξ ) (1 + η ) (ξ + η − 1) N6 =
1 2
(1 − ξ 2 ) (1 + η )
N 7 = 41 (1 − ξ ) (1 + η ) ( −ξ + η − 1) N8 = 12 (1 − ξ ) (1 − η 2 ) where ξ and η are non-dimensional coordinate systems. To represent the fluid motion, the elemental matrices [he], [ge] and [se] were derived as follows: ⎧∂ N T ∂ N ∂ N T ∂ N ∂ N T ∂ N ⎫ [ he ] = ∫vol ⎨ [∂r ] [∂r ] + [∂z] [∂z ] + [∂y] [∂y ] ⎬ d ( vol ) ⎩
⎭
where d ( vol ) = dr dz dy = dr dzr dφ = det [ J ] dξ dη r dφ
© Carl T. F. Ross, 2011
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Pressure vessels
where ⎡ ∂N 1 ⎢ ∂ξ [ J ] = the Jacobian matrix = ⎢ ⎢ ∂N 1 ⎢⎣ ∂η
∂N 8 ⎤ ⎡ r1 ∂ξ ⎥ ⎢ ⎥ ∂N 8 ⎥ ⎢ … ⎢⎣r8 ∂η ⎥⎦
…
z1 ⎤ ⎥ ⎥ z8 ⎥⎦
and r and z are as defined in Fig. 6.1. Hence, T T 2 π +1 +1 ⎧ ∂ N ∂ N ∂ N ∂ N [ he ] = ∫0 ∫−1 ∫−1 ⎨ [∂r ] [∂r ] + [∂z] [∂z ] ⎩
1 ∂[N ] ∂[N ]⎫ ⎬ r det [ J ] dξ dη dφ ∂φ ⎭ r 2 ∂φ T
+
Explicit integration can be carried out in the ϕ direction to give ⎧ ∂ [ N ′ ]T ∂ [ N ′ ] ∂ [ N ′ ]T ∂ [ N ′ ] + ⎨ −1 ∫−1 ∂r ∂z ∂z ⎩ ∂r 2 n ⎫ T + 2 [ N ′ ] [ N ′ ]⎬ r det [ J ] dξ dη r ⎭
[ he ] = π ∫
+1
+1
Integration with respect to ξ and η was carried out within the computer program using four Gauss points in both directions. Similarly, [g] was obtained from: 1
2π
+1
+1
[ g e ] = c 2 ∫0 ∫−1 ∫−1 cos2 nφ [ N ′ ]T [ N ′ ] r det [ J ] dξ dη dφ which on integrating with respect to ϕ gives: π
+1
+1
[ g e ] = c 2 ∫−1 ∫−1 [ N ′ ]T [ N ′ ] r det [ J ] dξ dη The fluid-structure interaction matrix [se] is given by:
[ se ] = ρF ∫s [ N ]T [ N ] dse e
e
where s is the contact area between the fluid and the shell, [N′] is the fluid ¯ ] is the w component of the shell shape function [N]. shape function, and [N e The integral ds may be represented by ds dy, where ds is taken along the meridian of the shell and dy is assumed to be tangential to the circumference (i.e. dy = r dϕ). For the situation illustrated in Fig. 6.1, the fluid element is above the shell element, with the contact area being nodal circles 1, 2 and 3 of the fluid element. The normal component of the shell shape function w is acting vertically upwards. In order to integrate over the contact surface area of dse (ds r dϕ), the nondimensional coordinate η was set to −1. Therefore the other coordinate ξ varied between −1 and + 1 over the
© Carl T. F. Ross, 2011
Vibration of pressure vessel shells in water
225
contact area, where ds = ldξ, l = one half of the meridional length of the shell element. The elemental matrix [se] may now be written as: 2π
+1
[ se ] = ρF ∫0 ∫−1 [ N ]T [ N ] r dφ l dξ
or
+1
[ se ] = πρF ∫−1 [ N ′ ]T [ N ′ ] rl dξ
where
[ N ] = [0
0 N1
N2
N4 ]
0 0 N3
when the interface is 1, 2, 3 and
[ N ] = [0
0 − N1
N2
N4 ]
0 0 − N3
when the interface is 5, 6, 7. N1 = N2
(ξ 3 − 3ξ + 2 )
N3 =
4
(1 − ξ − ξ 2 + ξ 3 ) l = 4
N4
( 2 + 3ξ − ξ 3 ) 4
(1 + ξ − ξ 2 − ξ 3 ) l =− 4
6.1.2 Out-of-roundness Before carrying out any vibration experiments, and immediately after the vessels were machined, initial out-of-circularity plots were made for all three vessels, as shown in Figs 4.13–4.15. After vibrating the vessels in air and then in water, further out-of-circularity plots were carried out (Figs 4.13– 4.15) and these revealed that the vessels had changed very little, if at all. The vessels were vibrated in a large tank as shown in Fig. 6.2, where it can be seen that they were excited in their flanks in a manner similar to that shown in Fig. 5.17. The electrical equipment used for the tests was the same as that described in Section 5.2.4 except that a hydrophone was used as the transducer. After vibrating the vessels in air, the vessels were vibrated while fully submerged in water, then with water on the external surface only, and, finally, with water on the inside only. Apart from in vacuo vibration, the theoretical investigations were for the vessels fully submerged and for the vessels with water on the outside only, and typical meshes are shown in Fig. 6.3. In Fig. 6.3 only half the tank is shown, as the structures were axisymmetric. In order to invert [H], it was necessary to set to zero the additional pressure p, owing to vibration, at certain points in the mathematical models, and this was done on the free surfaces at the tops of the fluid meshes. The tank boundary was assumed to be a natural boundary (i.e. ∂p/∂n = 0; where n refers to a line normal to the tank boundary). A comparison between experiment and theory, for the case of free flood and the case of water on the external surface only, is shown in
© Carl T. F. Ross, 2011
226
Pressure vessels
6.2 Underwater vibration tests.
(a)
(b)
(c)
6.3 Meshes adopted for fluid/structure: (a) mesh for cones 1 and 2 (water external); (b) mesh for cone 3 (water external); and (c) mesh for cones 1 and 2 (free flood).
© Carl T. F. Ross, 2011
Vibration of pressure vessel shells in water
227
3000
Frequency (Hz)
2500
2000
1500
1000
500
1
2
3
4
5
n
6.4 Variation of frequency with n (cone 1). Experiment —; theory - - -; in air ×; external 䊉; free flood ⵧ. 4000
Frequency (Hz)
3000
2000
1000
1
2
3
4
5
n
6.5 Variation of frequency with n (cone 2). (See Fig. 6.4 for key).
Figs 6.4–6.6, where the theoretical results in air/in vacuo are also shown. It can be seen that the effect of water was to decrease considerably the magnitudes of the resonant frequencies, although there was little difference between the results for the free flood conditions and the water-external cases.
© Carl T. F. Ross, 2011
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Pressure vessels
Frequency (Hz)
4000
3000
2000
1000
1
2
3
4
5
n
6.6 Variation of frequency with n (cone 3). (See Fig. 6.4 for key). 3000
Frequency (Hz)
2500
2000
1500
1000
500
1
2
3
4
5
n
6.7 Variation of frequency with n (experimental only; water internal and air external). Model no. 1 䊉; model no. 2 ⵧ; model no. 3 ×.
The figures also show that the experimental and the theoretical results are similar; in general, the experimental results lie a little below the theoretical ones. It is expected that if a more refined mesh were taken, especially for the fluid, a better comparison would have been obtained. Indeed this has been achieved by Wunderlich.121 For interest only, vibration tests were also carried out for the cones with water on the inside only, and these results are shown in Fig. 6.7.
© Carl T. F. Ross, 2011
Vibration of pressure vessel shells in water
6.2
229
Free vibration of domes in water
As for cones, the modes of vibration of thin-walled hemi-ellipsoidal shells in water are of similar form to those in air or in vacuo. The small SUP hemi-ellipsoidal domes described in Section 3.4 were vibrated in the free flood condition, and also with water on the external surface only. In addition to these vibration tests, the large SUP dome referred to in Section 3.4 was vibrated in the free flood condition. For both investigations, the fluid element adopted was the same as that described in Section 6.1.1.
6.2.1 Vibration of domes in water The small SUP hemi-ellipsoidal domes were modelled with a thin-walled conical element (CON), a constant meridional curvature element (CMC) and a varying meridional curvature element (VMC). In order to invert [H], it was necessary to set to zero the additional pressure p owing to vibrations at certain points, and this was done on the nodes on the free surfaces. The tank boundary was taken as a natural boundary (i.e. ∂p/∂n = 0; in this instance, n is a line normal to the tank boundary). Typical meshes are shown in Fig. 6.8, and the theoretical and experimental results are shown in Tables 6.1–6.4, from which it can be seen that there is good agreement among all three elements and, in general, with the experimental results as well. Ironically, the VMC element appears to break down for the model of aspect ratio 4 when n = 1. Similarly, the CON element appears to break down for larger values of n, and all three elements produce poor predictions for n = 1. This latter effect may be attributed to the difficulty of modelling the fluid motion for the cantilever mode, i.e. for n = 1. Two other comparisons are also shown diagrammatically in Figs 6.9 and 6.10 for the domes of aspect ratio 1 and 4, where the VMC element was used for the theoretical analyses From Figs 6.9 and 6.10, it can be seen that the effect of water on the magnitudes of the resonant frequencies was enormous, but that there was little difference between the results for the free flood case and the waterexternal case. The good agreement for the resonant frequencies in air at n = 1, especially for the dome of aspect ratio 4, appears to indicate that it was difficult to model the fluid in the cantilever mode. Figure 6.10 also shows that there is excellent agreement between experiment and theory, in water, for the dome of aspect ratio 4, for values of n greater than 1. Comparison between experiment and theory for the oblate domes was found to be somewhat patchy, and not as good as for the prolate domes.
© Carl T. F. Ross, 2011
230
(a)
(a)
Pressure vessels
AR = 4
AR = 4
(b)
(b)
AR = 1
AR = 1
(c)
(c)
AR = 0.25
AR = 0.25
6.8 Meshes for dome/fluid: (a) AR = 4; (b) AR = 1; (c) AR = 0.25.
© Carl T. F. Ross, 2011
© Carl T. F. Ross, 2011
n = 0, m = 4
n = 0, m = 5
n = 0, m = 6
362 300 191
362 283 174
484 424 404
485 394 361
487 396 365
474 392 360
822 587 503
740 575 448
745 583 451
705 539 433
– – –
971 910 789
975 929 824
923 858 796
– – –
n=4
n=5
113 138 171 215 278 369 511
113 138 170 215 278 370 509
137 175 217 266 366 513 767
135 164 204 262 349 490 720
136 165 205 263 350 491 722
140 169 210 268 355 496 727
129 158 204 260 364 503 887
127 154 193 252 346 513 821
128 155 194 252 347 514 822
134 162 201 261 358 526 836
149 170 214 269 389 586 964
146 169 204 261 358 540 905
146 169 204 261 359 541 906
153 178 215 274 375 560 929
190 202 250 296 412 – 1005
1100 1102 1103
1075 1100 1102
189 208 239 291 387 577 986
189 208 239 292 387 577 987
197 219 252 308 409 604 1020
VMC CMC CON
Aspect ratio, 2 experimental, 3 varying meridional curvature element, 4 constant meridional curvature element, 5 conical element.
132 146 173 216 277 369 509
86 91 124 196 226 303 414
n=3
4.0 3.5 3.0 2.5 2.0 1.5 1.0
n=2
Exp.2 VMC3 CMC4 CON5 Exp. VMC CMC CON Exp. VMC CMC CON Exp. VMC CMC CON Exp.
n=1
AR1
1
1100 1102 1103
Aspect ratio, 2 experimental, 3 varying meridional curvature element, 4 constant meridional curvature element, 5 conical element.
362 298 189
Table 6.2 Resonant frequencies (Hz) for prolate domes (water external)
1
n = 0, m = 3
Exp.2 VMC3 CMC4 CON5 Exp. VMC CMC CON Exp. VMC CMC CON Exp. VMC CMC CON Exp. VMC CMC CON
0.7 359 0.44 325 0.25 246
AR1
n = 0, m = 2
Table 6.1 Resonant frequencies (Hz) for oblate domes (water external)
© Carl T. F. Ross, 2011
n = 0, m = 4
n = 0, m = 5
n = 0, m = 6
291 208 133
285 194 122
344 280 259
378 377 288
378 377 293
378 373 279
454 358 354
495 393 377
498 399 378
483 378 377
– – –
659 645 559
662 661 589
621 616 569
– – –
945 1052 1059
1
91 99 117 145 185 244 332
78 95 116 145 185 244 333
77 94 115 144 185 243 331
96 116 144 184 243 337 487
97 117 145 184 244 338 488
99 120 148 188 247 342 492
93 116 143 191 270 364 592
92 110 139 180 247 362 570
92 111 139 181 247 363 571
97 116 145 187 255 372 580
107 127 158 204 283 393 602
106 123 148 189 259 388 639
106 123 148 189 259 388 641
112 129 156 199 271 402 658
137 155 187 218 317 414 664
139 153 175 213 282 419 707
Aspect ratio, 2 experimental, 3 varying meridional curvature element, 4 constant meridional curvature element, 5 conical element.
77 60 97 115 139 202 266
139 153 175 213 283 419 708
145 160 185 226 298 440 733
97 117 151 178 259 336 505
n=5
4.0 3.5 3.0 2.5 2.0 1.5 1.0
n=3
831 972 1055
Exp. VMC CMC CON Exp. VMC CMC CON Exp. VMC CMC CON Exp. VMC CMC CON
n=2
947 1068 1101
AR1 Exp.2 VMC3 CMC4 CON5
n=1
n=4
Aspect ratio, 2 experimental, 3 varying meridional curvature element, 4 constant meridional curvature element, 5 conical element.
290 206 132
Table 6.4 Resonant frequencies (Hz) for prolate domes (free flood)
1
n = 0, m = 3
Exp.2 VMC3 CMC4 CON5 Exp. VMC CMC CON Exp. VMC CMC CON Exp. VMC CMC CON Exp. VMC CMC CON
0.7 269 0.44 185 0.25 123
AR1
n = 0, m = 2
Table 6.3 Resonant frequencies (Hz) for oblate domes (free flood)
2500 2250 Theoretical Experimental In air Water external Free flood
Resonant frequency (Hz)
2000 1750 Experiment Theory
1500 1250 1000 750
Theory 500
Experiment Theory Experiment
250 1
2 3 4 5 Number of lobes (n)
6.9 Plot of frequency against n for AR = 1.
675
Theoretical Experimental In air Water external Free flood
600
Resonant frequency (Hz)
525 450 Experiment Theory
375 300 225 150
Theory Theory
75
Experiment Experiment 1
2 3 4 Number of lobes (n)
5
6.10 Plot of frequency against n for AR = 4.
© Carl T. F. Ross, 2011
234
Pressure vessels
6.2.2 Vibration of domes in free flood Vibration experiments were also carried out on the large SUP dome,119 and a comparison is made between experiment and theory for the free flood case in Fig. 6.11, where the VMC element was used to model the dome. The mesh for this example is shown in Fig. 6.12, where the additional pressures, owing to vibration, were set to zero on the free surface. The tank boundary was assumed to be a natural boundary (i.e. ∂p/∂n = 0; where n refers to a line normal to the tank boundary). From Fig. 6.11, it can be seen that comparison between experiment and theory is good, but that the theoretical results lie on a line below the experimental ones. The figure also shows that the effect of water on the magnitude of the resonant frequencies is quite dramatic.
160 150
Aspect ratio = 1.9759
140 Experiment Frequency (Hz)
130
120
FEM
110
100
90 80 0
6 1 2 3 4 5 Number of circumferential lobes (n)
7
8
6.11 Variation of resonant frequency with n for the large SUP dome, in free flood.
© Carl T. F. Ross, 2011
Vibration of pressure vessel shells in water
235
0.75 m radius
Axisymmetric fluid elements
Boundary
1075 m
Axisymmetric shell elements
Free surace
e ctiv
fle Re
6.12 Mesh for large SUP dome. It is symmetric about the centreline.
© Carl T. F. Ross, 2011
236
Pressure vessels
6.3
Vibration of domes under external water pressure
So far the investigations have been purely linear and do not take into account the nonlinear effects of external pressure.122–125 As stated in chapter 5, when a submersible sinks deeper into the oceans, the external water pressure increases, and so do the compressive stresses in the structure. The effect of the latter is to cause the stiffness of the structure to decrease, with a subsequent decrease in the magnitude of the resonant frequencies until, when the static buckling pressure is approached, the resonant frequencies approach zero. Furthermore, investigations have shown that, when the static buckling pressure is approached, the fundamental circumferential eigenmode of vibration becomes of similar form to the circumferential static buckling mode, thus producing the possibility of a form of dynamic buckling at a pressure less than that required to cause static buckling. For nonlinear vibrations under water, equation [6.2] takes the form of equation [6.7];
([ K ] + [ KG ] − ω 2 [ M ]) {ui } − [S ]T { p} / ρF = 0
[6.7]
where [KG] = the system geometrical stiffness matrix, which is dependent on the magnitude of the external pressure. Equation [6.7] is coupled with equation [6.1] in a manner similar to that shown in Section 6.1, except that: [K] = [K] + [KG]
[6.8]
in equations [6.3] and [6.4]. The eigenvalues and eigenmodes for the modified equation must be calculated for each value of pressure, because [KG] is dependent on the pressure. In this instance, the prebuckling stresses were calculated from equations [3.38], and once these were known it was a simple matter to determine [KG]. A comparison can be made with some of the small SUP domes, and also with the large SUP dome L2.
6.3.1 Vibration of domes under external water pressure It was not possible to vibrate all the small SUP domes under external water pressure because of the limits of the experimental apparatus available. This apparatus was similar to that used in Section 5.3.1 except for the tank, which is shown in Figs 6.13 and 6.14. To excite the prolate domes, a bell crank mechanism was adopted, where the pivot position was changed, depending on whether an even or odd number of lobes was required, as shown in Fig. 6.14. The oblate domes were excited by applying an up-and-down motion to the noses of these vessels via a rubber strap and an electromagnetic shaker, as shown in Fig. 6.13.
© Carl T. F. Ross, 2011
Excitation force
Vibrator connecting rod Accelerometer Clamping plate Rubber seal Oblate dome Pressure tank
Water
Water pump connection
6.13 Method of exciting oblate domes under pressure.
Excitation force Vibrator connecting rod Bell crank support
Rotating microphone holder
Clamping plate Rubber seal Rubber straps
Pivot Microphone
Pressure tank Prolate dome
Water
Water pump connection
6.14 Method of exciting the small prolate domes under pressure.
© Carl T. F. Ross, 2011
238
Pressure vessels P=0 Axis
Axis
P=0
6.15 Mesh for dome/fluid (small closed tank). AR = 1.
6.16 Mesh for dome/fluid (small closed tank). AR = 0.44.
Typical meshes are shown in Figs 6.15–6.20, where, because the tank was closed, and it was necessary to invert [H], P was assumed to be zero at the positions shown. These positions were at the top of the fluid, and also where the shell displacements were zero. Otherwise the tank boundary was assumed to be a natural one (i.e. ∂p/∂n = 0, where n was a line normal to the tank boundary). Comparison is made in Figs 6.21–6.27 between experimental and theoretical frequencies for different values of pressure ratio, where P = applied pressure; and Pcr = experimentally obtained static buckling pressure (Table 3.11). From Figs 6.21–6.23 it can be seen that, for the oblate domes, as the pressure ratio is increased, the resonant frequencies tend to decrease, especially for the dome of aspect ratio 0.25, although this effect does not appear to be particularly significant for these vessels. The same argument does not, however, apply to the prolate domes, as can be seen from Figs 6.24–6.27, where an increase in the pressure ratio considerably decreases the resonant frequencies, and also causes the fundamental circumferential eigenmodes of vibration to become of similar form to the static buckling circumferential eigenmodes (Table 3.11). The fundamental resonant frequency of vibration at n = 1 (i.e. for the cantilever mode) did not appear to have been affected by increasing the pressure ratio.
© Carl T. F. Ross, 2011
P=0
Axis
Axis
P=0
6.18 Mesh for dome/fluid (small closed tank). AR = 3.5. 6.17 Mesh for dome/fluid (small closed tank). AR = 4. P=0
Axis
Axis
P=0
6.19 Mesh for dome/fluid (small closed tank). AR = 3.
6.20 Mesh for dome/fluid (small closed tank). AR = 2.5.
© Carl T. F. Ross, 2011
240
Pressure vessels 360 340 320 300 280 260
Frequency (Hz)
240 220 200 180 160 140 120 100 80
n=0 m=2
60 40 20
Experiment Theory
0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Pressure ratio (p/Pcr)
6.21 Pressure effects on resonant frequencies of small hemiellipsoidal dome of aspect ratio 0.25.
The experimental results for the oblate domes were not as good as those for the prolate domes, but this may be because of the experimental setup, where there was a possibility of dynamic pressure being produced by the up-and-down motion of the shell. That is, because of the up-and-down motion of the shell, there may have been dynamic suction when the shell moved upwards, and excess pressure when the shell moved downwards. This effect was less likely to occur for the prolate domes as their eigenmodes were of a sinusoidal nature, so that if part of the flank of a dome moved outwards another part of its flank moved inwards by a similar amount, and at the same time. Unfortunately, it was not possible to excite the hemispherical dome experimentally when it was in the tank, because of the limitations of the apparatus that was available. However, a theoretical investigation of the hemispherical dome is shown in Fig. 6.28, from which it can be seen that when the pressure ratio was zero, a maximum value of resonant frequency
© Carl T. F. Ross, 2011
Vibration of pressure vessel shells in water
241
360 340 320 300 280 260
Frequency (Hz)
240 220 200 180 160 140 120
n=0 m=2
100 80 60 40 20
Experiment Theory
0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Pressure ratio (p/Pcr)
6.22 Pressure effects on resonant frequencies of small hemi-ellipsoidal dome of aspect ratio 0.44.
appeared to occur when n approached infinity, but when p = 0.95Pcr a saddle point was found at n = 3, and a minimum value of resonant frequency was found at n = 10 (i.e. the static buckling eigenmode was approached).
6.3.2 Vibration of a large dome under external water pressure Another investigation on the effects of external pressure on the magnitudes of resonant frequencies involved the large SUP dome, namely L2, described in Section 3.4.3. This shell was vibrated with two electromagnetic shakers, as shown in Fig. 6.29, with an electrical rig similar to that described in Section 5.2.4, except that a microphone was used as the transducer. The eigenmodes and frequencies were measured at various values of externally applied pressure, and comparison was made with the theoretical predictions obtained from the mathematical model of Fig. 6.30. In this mathematical
© Carl T. F. Ross, 2011
242
Pressure vessels 380 360 340 320 300 280
Frequency (Hz)
260 240 220 200 180
n=0 m=2
160 140 120 100 80 60 40
Theory Experiment
20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Pressure ratio (p/Pcr)
6.23 Pressure effects on resonant frequencies of small hemi-ellipsoidal dome of aspect ratio 0.7.
model, as the vessel was closed, and it was required to invert [H], it was necessary to assume that p was zero at the point shown in Fig. 6.30. As for the smaller SUP domes, this position was at the top of the water, where the shell displacements were zero. Otherwise the tank boundary was assumed to be a natural one (i.e. ∂p/∂n = 0, where in this case n is a line normal to the tank boundary). Comparison is made between experiment and theory for dome L2, for which the resonant frequencies are plotted for different values of pressure ratio in Fig. 6.31, where p = applied pressure; and the experimental static buckling pressure for dome L2 Pcr = 0.2027 MPa. Comparisons between the experimental resonant frequencies for dome L2 in the two different size tanks are shown in Fig. 6.32. From these results, it can be seen that, as the external pressure increases, the resonant frequencies decrease and the circumferential eigenmodes of
© Carl T. F. Ross, 2011
Vibration of pressure vessel shells in water
360 340 320
Theory Experiment Pressure ratio (p/Pcr)
0 0.25 0.5 0.75 0.95 p=0
300 280
p=0
260
p = 0.25 Pcr p = 0.25 Pcr
240 Frequency (Hz)
243
p = 0.5 Pcr
220
p = 0.5 Pcr
200 180
p = 0.75 Pcr
160
p = 0.75 Pcr
140
p = 0.95 Pcr
120
p = 0.95 Pcr
100 80 60 40 20 0
1
2 3 4 5 6 Number of circumferential waves (n)
6.24 Pressure effects on resonant frequencies of small hemi-ellipsoidal dome of aspect ratio 2.5.
vibration become of similar form to the circumferential static buckling eigenmodes, so that there is a possibility of a form of dynamic buckling occurring at a pressure less than that required to cause static buckling.
6.4
Vibration of unstiffened and ring-stiffened circular cylinders and cones under external hydrostatic pressure
In this section, reports are presented of experimental and theoretical investigations into the following three types of vessel:126–131 (a) Vibration of circular cylindrical shells under external water pressure; (b) Vibration of ring-stiffened circular cylinders under external water pressure; (c) Vibration of ring-stiffened cones under external water pressure.
© Carl T. F. Ross, 2011
244
Pressure vessels
360 p=0 p=0
340 320
p = 0.25 Pcr
300
p = 0.25 Pcr
Frequency (Hz)
280 260
p = 0.5 Pcr
240
p = 0.5 Pcr
220
p = 0.75 Pcr
200
p = 0.75 Pcr
180
p = 0.95 Pcr
160 140
p = 0.95 Pcr
120 100 80 60 40
Theory Experiment Pressure ratio (p/Pcr)
0 0.25 0.5 0.75 0.95
20 0
1
2 3 4 5 6 Number of circumferential waves (n)
6.25 Pressure effects on resonant frequencies of small hemi-ellipsoidal dome of aspect ratio 3.0.
6.4.1 Vibration of circular cylindrical shells under external water pressure Experimental method Experiments were carried out on three machined thin-walled circular cylinders made in steel, with the geometrical properties shown in Fig. 6.33 and the following mechanical properties: Young’s modulus: 2.05 × 1011 N m−2; Poisson’s ratio: 0.30 (assumed); and density: 7827.0 kg m−3. For shell number 1, 2 and 3, the wall thickness was 1.0, 1.0 and 0.98 mm, respectively, and the length was 300, 225 and 150 mm, respectively. The top and bottom clamping flanges had thickness 0.005 and 0.009 m, respectively, outside diameters 0.2185 and 0.2095 m, respectively, and inside diameters 0.1995 and 0.1995 m, respectively. A photograph of the three models is shown in Fig. 6.34. The assumed properties of the water were: sonic speed c = 1490 m s−1 and density ρF = 1000 kg m−3.
© Carl T. F. Ross, 2011
Vibration of pressure vessel shells in water
245
360 340 320
Theory Experiment Pressure ratio (p/Pcr)
300
Frequency (Hz)
280
0 0.25 0.5 0.75 0.95
260
p=0
240
p=0 p = 0.25 Pcr
220
p = 0.25 Pcr p = 0.5 Pcr p = 0.5 Pcr p = 0.75 Pcr p = 0.75 Pcr p = 0.95 Pcr
200 180 160 140
p = 0.95 Pcr
120 100 80 60 40 20 0
1
2
3
4
5
6
Number of circumferential waves (n)
6.26 Pressure effects on resonant frequencies of small hemi-ellipsoidal dome of aspect ratio 3.5.
The vessels were tested in an enclosed tank, as shown in Figs 6.35 and 6.36, where it can be seen that access to the internal surfaces of the cylinders was possible. The cylinders were pressure proofed at their tops with the aid of a rubber gasket and at their bases with the aid of a silicone sealant. It should be emphasized that, as the vessels were under external pressure, it was a relatively simple matter to pressure proof the cylinders. The water pressure was exerted with the aid of a hand-operated hydraulic pump, after it was ensured that the bulk of the trapped air was expelled to the atmosphere, via the pressure release valve, shown on the top left of the tank in Fig. 6.35.
© Carl T. F. Ross, 2011
246
Pressure vessels 360 340 320 300 280
Theory Experiment Pressure ratio (p/Pcr)
0 0.25 0.5 0.75 0.95
260
Frequency (Hz)
240
p=0
220
p = 0.25 Pcr
200
p = 0.5 Pcr
180 160
p = 0.75 Pcr
140 120
p = 0.95 Pcr
100 80 60 40 20 0
1
2 3 4 5 6 Number of circumferential waves (n)
6.27 Pressure effects on resonant frequencies of small hemi-ellipsoidal dome of aspect ratio 4.0.
From Fig. 6.36, it can be seen that the inner surfaces of the cylinders were open to the atmosphere and readily accessible. It was from this opening that excitation of the cylinders was carried out via an electromagnetic shaker, whose vertical motion was transmitted through a rod on to a horizontal rubber strap. The horizontal strap was attached across the diameter of each cylinder, at the vessel’s mid-length, with the aid of an adhesive, as shown in Figs 6.36–6.38. When the distance a in Fig. 6.37 was made equal to b, in-phase readings were obtained (that is even values of n were found), and when the distances a and b were unequal, out-of-phase readings were obtained (that is odd values of n were found). Other equipment used included an accelerometer, an amplifier, a charge amplifier, a frequency response analyser and an oscilloscope, as shown by the circuit diagram of Fig. 5.18. The frequency response analyser was used to control the vibrator, via an amplifier, and at the same time the frequency response analyser was used to collect the output signals from the transducer,
© Carl T. F. Ross, 2011
Vibration of pressure vessel shells in water
247
1900 1800
p=0
1700 1600
p = 0.25 Pcr
1500 1400 p = 0.5 Pcr
Frequency (Hz)
1300 1200 1100
p = 0.75 Pcr
1000 900 800
p = 0.95 Pcr
700 600 500 400 300 200 100 0
1
2
3 4 5 6 7 8 9 10 11 12 13 Number of circumferential lobes (n)
6.28 Vibration of small hemispherical dome under pressure theory, AR = 1.0.
Rotating microphone holder
Vibrator support Clamping plate Vent valve
Microphone
Rubber seal Rubber strap
Vibrator
Tank
Water
Water pump connection
6.29 Test tank for vibrating the large dome under pressure.
© Carl T. F. Ross, 2011
Axis
p=0
6.30 Mesh for fluid and dome L2 in a large closed tank.
Pe = 0.0
300
Pe = 0.25 Pt = 0.0
Experiment Theory
280 260
Pe = 0.5 Pt = 0.25
240
Frequency (Hz)
220
Pt = 0.5
200
Pe = 0.75
180
Pe = 0.9
160 Pt = 0.75
140 120
Pt = 0.9
100 80 60 40 20 0 1
2
3
4
5
6
7
8
9
10
n
6.31 Variation of resonant frequency with n for different pressure ratios, p/Pcr (model L2).
© Carl T. F. Ross, 2011
Vibration of pressure vessel shells in water
249
380 360 340 320 300 280 260
Frequency (Hz)
240
Resonant frequencies for the dome vibrated in the big tank
220 200 180 160 140 120 100
Resonant frequencies for the dome vibrated in the small tank
80 60 40 20 0
1
2
3 4 5 6 7 8 9 10 11 12 Number of circumferential waves (n)
6.32 Comparison of resonant frequencies obtained experimentally from the big tank and the small tank for the dome with AR = 4.0.
via a charge amplifier. The frequency response analyser was also connected to an oscilloscope, which resulted in the appearance of a Lissajous figure. The accelerometer consisted of an enclosed slice of piezoelectric material, which when subjected to a vibrating force generated a small electric current proportional to the magnitude of the vibrating force. The accelerometer was stuck to the cylinder wall by a thin layer of plasticine. The accelerometer’s main purpose was to measure the amplitudes and phase angles at various positions on the inner surface of each cylinder, especially around the vessel’s circumference at its mid-length. The resonant frequencies were determined by finding the maximum amplitude at the anti-nodes, combined with the best distribution of nodes and anti-nodes for that particular response.
© Carl T. F. Ross, 2011
Stud bolt
9 mm Top flange 5 mm Rubber gasket 1 mm (approx.) Tank top
Water
Air
Axis
Adhesive
Rubber strap Tank well Cylinder wall
99.75 mm
Bottom flange 9 mm Silicone sealant
5 mm
Closure plate
104.75 mm
6.33 Details of cylinders and their attachments.
6.34 Photograph of the three model cylinders. © Carl T. F. Ross, 2011
Vibration of pressure vessel shells in water
251
6.35 Photograph of test rig and pressure tank used.
When a resonant frequency was found, the accelerometer was rotated around the circumference of the cylinder to determine the phase angles, nodes and anti-nodes so that the circumferential eigenmode corresponding to that resonant frequency could be observed. The accelerometer was also moved longitudinally to ensure that the number of half-waves in the longitudinal direction corresponded to one (that is m = 1).
© Carl T. F. Ross, 2011
Pressure vessels Excitation force
Rod connected to vibrator Clamping plate Rubber seal
Pressure tank
Rubber strap
Cylinder
Water
Water pump connector
6.36 Method of exciting the vessels under pressure.
Rod from vibrator Motion
252
a Cylinder wall
b Motion
Rubber
6.37 Position of the vibrating rod on the rubber strap, where the latter is shown connected to the cylinder wall.
© Carl T. F. Ross, 2011
Vibration of pressure vessel shells in water
253
6.38 Photograph showing the method of vibrating the cylinders.
Comparison between experiment and theory The finite element meshes, used for the computational analyses, are shown in Figs 6.39–6.41. In order to invert [H], it was necessary to specify at least one pressure degree of freedom, and as the circular cylinders were enclosed, it was assumed that p = 0 at the positions shown in Figs 6.39–6.41. It was believed that the chosen position for p = 0 was probably the best for these cases, as this point was the uppermost point of the water and also because the displacement degree of freedom was zero here. The theoretical buckling pressures for these vessels were calculated by the von Mises theory39 for circular cylinders with simply supported ends, and these values are shown below. The figures in brackets represent the number of circumferential lobes n. Cylinder 1: Pcr = 0.62 MPa (n = 5) = 89.9 lb in−2; Cylinder 2: Pcr = 0.847 MPa (n = 6) = 122.8 lb in−2; and Cylinder 3: Pcr = 1.218 MPa (n = 7) = 176.6 lb in−2. Plots of the resonant frequencies against n are shown in Figs 6.42–6.44, where it can be seen that, in general, there is good agreement between experiment and theory, and also that, as the buckling pressures were approached, the circumferential eigenmodes of vibration became similar to the buckling eigenmodes. In order to prevent dynamic instability occurring, the maximum external pressures that the vessels were subjected to were not allowed to exceed about 0.75 of the von Mises buckling pressures for each respective case. It is likely that these vessels had some degree of partial
© Carl T. F. Ross, 2011
254
Pressure vessels P=0
420 mm
420 mm
Axis
Axis
P=0
147 mm 147 mm
6.40 Fluid mesh for cylinder 2. 6.39 Fluid mesh for cylinder 1.
fixity at their ends and that their true theoretical elastic instability pressures were a little higher than the predictions by the von Mises formula. Conclusions The results show that the finite element method appears to adequately describe the motion of circular cylinders under external water pressure. More importantly, as the external pressure is increased the compressive membrane stresses in the vessels also increase, and this causes their resonant frequencies to decrease because of the resulting decrease in the bending stiffness of these vessels. In Figs 6.42–6.44, the value of n, corresponding to the lowest resonant frequency, increases with an increase in external water pressure, and tends towards the values obtained by the von Mises formula. Although compressibility of the water was taken into account, it is likely that, if the water were
© Carl T. F. Ross, 2011
Vibration of pressure vessel shells in water
255
420 mm
Axis
P=0
147 mm
6.41 Fluid mesh for cylinder 3.
assumed to be incompressible, there would not be a significant change in the theoretically determined resonant frequencies of their eigenmodes. As the cylinders were of different lengths, the depth of water below their bases was different, and although this was allowed for in the various mathematical models, it is likely that this effect made little impression on the investigation, as the eigenmodes were predominantly of a radial nature, rather than of a longitudinal nature.
6.4.2 Ring-stiffened circular cylinders Experimental apparatus The three ring-reinforced circular cylinders were carefully machined from a solid billet of low carbon free cutting steel 220 M07, to the dimensions shown in Fig. 6.45. It can be seen that the main difference between the
© Carl T. F. Ross, 2011
256
Pressure vessels 1750 Theoretical P=0 P = 0.28 MPa Experimental P = 0.47 MPa
1500
P=0 P = 0.28 MPa P = 0.47 MPa
Frequency (Hz)
1250
1000
750
500
250
0
2
4
6
8
10
12
n
6.42 Results for cylinder 1.
models was the depth of their ring stiffeners, which increased with the model number. The material properties of the vessels were: Young’s modulus E = 1.9 × 105 MPa; Poisson’s ratio v = 0.3 (assumed); and density ρ = 7800 kg m−3 (assumed). The excitor rod consisted of a plastic T-piece, which was attached to the inside of each cylinder with the aid of rubber sucker pads. The top of the excitor rod was attached to an electromagnetic vibrator, whose up and down reciprocating motion was transmitted to the inside surface of the cylinder via the sucker pads. Thus, the cylinders were quite easily excited in the modes shown in Figs 5.1and 5.2. No problem occurred when either even or odd modes were excited. It is believed that the odd modes were easily excited because of the flexibility of the excitor rod in its transverse direction. Monitoring of the resonant frequencies was carried out with the aid of an electromagnetic transducer. The transducer was of a variable inductance type, and its operation was based on displacement measurement. Each vessel was tested, in turn, in the small enclosed pressure tank shown in Fig. 6.46. The models were hung from the tank top by bolting the flange
© Carl T. F. Ross, 2011
Vibration of pressure vessel shells in water
257
1750 Theoretical P=0 P = 0.38 MPa Experimental P = 0.64 MPa
1500
P=0 P = 0.38 MPa P = 0.64 MPa
Frequency (Hz)
1250
1000
750
500
250
0
2
4
6
8
10
12
n
6.43 Results for cylinder 2.
of each cylinder to the underside of the tank top. Sealing against leakage was achieved with the aid of ‘O’ rings in the tank top. Leakage, however, was not a problem, as all the components were machined and also because the vessel’s joints were in compression. The arrangement allowed easy access to the inside of each cylinder, which could then be vibrated through the excitor rod. The transducer consisted of an E-shaped permanent magnet and a coil of wire wrapped around it. Its principle was based on Faraday’s law; i.e. when the gap between the transducer and the wall of the cylinder was varied, a current flowed through the coil wrapped around the permanent magnet and the magnitude of the current was related to the size of the gap between the transducer and the cylinder wall. This induced current in the transducer was magnified by a charge amplifier. The circuit diagram is shown in Fig. 6.47. Figure 6.47 shows that a frequency response analyser (FRA) was used to control the experiment, where the electromagnetic shaker was controlled through the output channel of the FRA, via a voltage amplifier, and the
© Carl T. F. Ross, 2011
258
Pressure vessels 1750 P=0
Theoretical P=0 P = 0.55 MPa Experimental P = 0.92 MPa
1500
P = 0.55 MPa P = 0.92 MPa
Frequency (Hz)
1250
1000
750
500
250
0
2
4
6
8
10
12
n
6.44 Results for cylinder 3.
induced current emitted from the transducer was connected to the input arm of the FRA via a charge amplifier. This resulted in the appearance of a Lissajous figure on the FRA, and observation of this figure on the cathode ray tube enabled the vibration study to take place. The modes of vibration were observed for various values of external water pressure, where the latter was applied to the cylinder via a hand-operated hydraulic pump, through an inlet in the side of the pressure tank, as shown in Fig. 6.46. Experimental procedure The excess air trapped in the tank was expelled through a bleed valve in the tank top, and then the bleed valve was sealed. The modes of vibration of each cylinder were then observed at zero pressure. This was done by adjusting the forcing frequency of the electromagnetic shaker until a resonant frequency occurred. The circumferential mode of vibration was then observed by rotating the transducer around the inner circumference of each vessel and measuring the amplitude. By varying the forcing frequency, it
© Carl T. F. Ross, 2011
Vibration of pressure vessel shells in water 15.2
259
22.9
101.6
t d b Cylinder
b
d
t
N
c4
1.6
1.5
1.125
9
c5
1.6
2
1.125
9
c6
1.6
2.5
1.125
9
6.45 Geometrical details of cylinders 4, 5 and 6 (N = number of ring stiffeners; all dimensions are in mm). 1
7
2
8
3 9 4 10 5
11 6 12
1. Cast iron sleeve 2. Tank bolts 3. Tank top 4. Tank
5. Transducer 6. Sucker pads 7. Excitor rod 8. Measuring cylinder
6.46 Experimental apparatus.
© Carl T. F. Ross, 2011
9. Cylinder extension 10. Cylinder 11. End cap bolts 12. End cap
260
Pressure vessels Electromagnetic vibrator
Voltage amplifier
Transducer Cylinder wall
Charge amplifier
Frequency response analyser O/P I/P
6.47 Circuit diagram of electrical and electronic equipment.
was possible to obtain a number of resonant frequencies for five different values of external water pressure, namely 0, 10, 20, 30 and 40 bar. The theoretical elastic instability pressures were obtained by assuming that the tops of the vessels were fixed and the bottoms of the vessels were clamped. By the term ‘fixed’, it was assumed that all the displacements were zero, and by the term ‘clamped’, it was assumed that all the displacements were zero except for the axial displacement at the base, which was free to move upwards. The theoretical elastic instability pressures Pcr for cylinders 4, 5 and 6 were 72.8, 91.7 and 120.0 bar, respectively, and the number of circumferential waves n that form when the vessels buckle was 3. Finite elements The element used to represent the shell was the thin-walled truncated conical shell,50 which had four degrees of freedom, namely u°, v°, w° and θ. The element91 used to represent the ring stiffeners had three degrees of freedom, namely w°, v° and θ. The element used to represent the fluid motion was the solid annular element shown in Fig. 6.1, which had a crosssection of an eight-node isoparametric form and one degree of freedom, namely p.
© Carl T. F. Ross, 2011
Vibration of pressure vessel shells in water
261
Comparison between theory and experiment The structural and fluid meshes are shown in Fig. 6.48. Previous work126–131 on optimizing the finite element model found that meshes such as those shown in Fig. 6.48 were adequate. A slightly more refined mesh or a slightly cruder mesh would have made very little difference. In order to invert the matrix, it was necessary to set the acoustic pressure to zero, at least at one point in the fluid mesh. This posed a problem, as the fluid had no free surface, so it was decided to make this point at the position marked P in Fig. 6.48. The reason for choosing this point was that this was also the point where the vector of nodal displacements was zero. Previous work119 where good agreement was found between experiment and theory showed that this assumption was satisfactory. The chosen structural mesh included the end closure plate at the bottom of the cylinder, which was modelled with one conical element of large apical angle.
Fixed
P Fluid mesh 2 22 elements
Structure mesh 11 elements
6.48 Finite element meshes for the fluid and structure.
© Carl T. F. Ross, 2011
262
Pressure vessels
Comparison between the experimental results and the theoretical results is shown for all three cylinders in Tables 6.5–6.7. These tables show the variation of the resonant frequencies with the number of circumferential waves in which the vessels vibrate, for different values of externally applied water pressure. The values for externally applied water pressure are in units of bar. Table 6.5 Theoretical (T) and experimental (E) resonant frequencies (Hz) for cylinder 4 n=2
n=3
n=4
n=5
Pressure (bar)
T
E
T
E
T
E
T
E
0 10 20 30 40
1161 1149 1137 1125 1113
1177 1165 1144 1127 1117
1029 968 903 833 756
1068 1026 961 889 812
1526 1426 1319 1203 1073
1533 1446 1334 1217 1118
2478 2366 2247 2122 1988
2441 2327 2202 2073 1937
Table 6.6 Theoretical (T) and experimental (E) resonant frequencies (Hz) for cylinder 5 n=2
n=3
n=4
n=5
Pressure (bar)
T
E
T
E
T
E
T
E
0 10 20 30 40
1166 1153 1141 1130 1118
1199 1178 1164 1155 1144
1125 1071 1013 952 887
1181 1133 1079 1018 958
1814 1733 1647 1558 1462
1789 1716 1639 1548 1456
3004 2912 2819 2722 2621
2878 2784 2692 2593 2494
Table 6.7 Theoretical (T) and experimental (E) resonant frequencies (Hz) for cylinder 6 n=2
n=3
n=4
n=5
Pressure (bar)
T
E
T
E
T
E
T
E
0 10 20 30 40
1173 1162 1150 1139 1127
1178 1175 1167 1159 1147
1259 1211 1162 1110 1055
1283 1268 1266 1156 1146
2179 2113 2045 1975 1901
2061 2015 1984 1884 1789
3648 3575 3501 3425 3347
3583 3506 3425 3341 3253
© Carl T. F. Ross, 2011
Vibration of pressure vessel shells in water
263
Tables 6.5–6.7 also show that as the depths of the ring stiffeners increase, the magnitudes of the resonant frequencies also increase. For example, if comparison is made with cylinder 4 for n = 3, the experimentally obtained frequencies at zero pressure increase by about 10% for cylinder 5 and by about 20% for cylinder 6. Similarly, when the external water pressure is 40 bar, the magnitudes of the experimentally obtained resonant frequencies increase by about 18 and 41% for cylinders 5 and 6, respectively, at n = 3. From Tables 6.5–6.7, it can be seen that there is good agreement between the experimental and theoretical values of resonant frequency. Additionally, it can be seen that, as the externally applied water pressure is increased, the resonant frequencies decrease and also that, as the static buckling pressures are approached, the lowest circumferential eigenmodes of resonant vibration, namely the values of n, become of similar form to the static buckling eigenmodes. The resonance can be triggered by out-of-balance rotating or reciprocating machinery, or by the motion of fluid itself. Conclusions The investigation has shown that the finite element method is quite suitable for analysing the vibration characteristics of ring-stiffened circular cylinders under external water pressure. The investigation has also shown that, as the externally applied water pressure is increased, the resonant frequencies of vibration decrease, owing to the fact that the circumferential bending stiffness of the vessels decrease, and that the lowest excitation frequency to cause resonance does so with a circumferential mode shape where n = 3. This type of deformation is very similar to the deformed shape that results from static buckling owing to external pressure and suggests the possibility of a dynamic instability occurring at pressures below the static buckling pressure. The theoretical and experimental results have also shown that when compared with cylinder 4 at n = 3, the resonant frequencies increase in magnitude by about 10 and 20% for cylinders 5 and 6, respectively, at zero pressure. However, when the external water pressure is raised to 40 bar, these resonant frequencies decrease by about 18 and 41% for cylinders 5 and 6, respectively.
6.4.3 Ring-stiffened circular cones Experimental equipment The experimental investigation was carried out on the three ring-stiffened circular conical shells shown in Fig. 6.49, with the nominal dimensions shown in Fig. 6.50. The ring stiffeners were equally spaced, the only difference between the vessels being the height of the ring stiffeners. All three
© Carl T. F. Ross, 2011
Pressure vessels
6.49 Ring-stiffened cones.
211.0
2
3
4
5
6 38.1
1
1.016
101.6
264
B
Cone number
A
B
4 5 6
1.016 1.016 1.016
1.016 1.5 2.0
A
6.50 Geometrical details of ring-stiffened cones (all dimensions in mm).
© Carl T. F. Ross, 2011
Vibration of pressure vessel shells in water
265
cones were machined from the same solid billet of steel with the following material properties: type: mild steel EN1A; Young’s modulus: E = 1.88 × 1011 N m−2; Poisson’s ratio: v = 0.3 (assumed); density: ρ = 7800 kg m−3. In general, the measurement of the vibration of cones and domes under external pressure is difficult, because access to the external surfaces of these vessels is not normally possible. Previously (see reference 125), excitation was achieved by the use of rubber strips stuck to the inner walls of these vessels, these strips in turn being connected to an electromagnetic vibrator. The amplitudes of the vibrations were normally measured using a piezoelectric accelerometer, which was placed manually around an internal circumference. Owing to the small size of the test tank and the cones under investigation, manual access was not possible, and therefore an alternative approach was required. An investigation into various measuring methods was carried out, including the use of an accelerometer, via a machine operated from above the tank, and the use of variable capacitance transducers. It was considered that, for the present study, variable inductance transducers were the most suitable; these are generally used for displacement measurement. This method was most suitable partly because the response time of these devices was good and partly because they were suitable for ferrous materials. Additionally, such transducers are relatively inexpensive. Figure 6.51 shows a schematic representation of the type used for the present investigation.
Cone wall (ferromagnetic)
Flux path
To charge amplifier
E-piece (permanent magnet)
Air gap = 0.5 mm (approx.)
6.51 Schematic of transducer arrangement.
© Carl T. F. Ross, 2011
266
Pressure vessels
6.52 Measuring device, cone and tank top.
The scientific principles on which these transducers are based are now described. When used for measuring vibrations, no external current is connected to the coil. As the cone vibrates, the magnetic field around the coil, created by the E-piece and cone wall, is altered, resulting in an induced current in the coil, this current being proportional to the rate of change of magnetic flux (Faraday’s law). The induced current forms the output signal. It may be possible to use these transducers to excite as well as measure. However, it was decided that, in this study, the cones would be vibrated mechanically. In order to carry out the tests in the small space available, a measuring device had to be made, as shown in Figs 6.53 and 6.54. This measuring device was snugly fitted into the hole in the tank top, which was open to the atmosphere. Figure 6.54 illustrates the various components and their functions. The cast iron sleeve (3) was manufactured to fit the existing hole in the pressure tank top; it was held in place by three set screws. There was a zero stamped on the edge of the sleeve corresponding to another stamped on the tank top, permitting accurate repositioning. This sleeve acted as a bearing for the mild steel cylinder. The mild steel cylinder (1) was free to rotate and was hollow, thus leaving access for the vibrating device. The top lip was graduated in steps of 5°, enabling accurate positioning of the transducer (4). Its position could be fixed by using the simple clamping device (2), although in practice
© Carl T. F. Ross, 2011
Vibration of pressure vessel shells in water
6.53 Component parts of measuring device.
1 2
3
4
To pump 5
6.54 Cross-section of pressure tank layout.
© Carl T. F. Ross, 2011
267
268
Pressure vessels
6.55 Excitor rod attached to cone.
this is not found to be necessary. The transducer had a bearing fitted in the end of the brass cylinder and held in place by Loctite. The position of the brass cylinder could be varied to permit adjustment of the air gap between the cone and the transducer. It was fixed in place by the use of a clamping set screw (5). The wires from the transducer ran up through the centre of the brass cylinder and then up through the inside of the steel cylinder. Allocation was made to fit two transducers, 180° apart, but only one was used. The excitor rod, shown in Fig. 6.55, was constructed from a plastic rod, one end of which was attached to the electromagnetic excitor and the other to a T-piece. Running at right angles to the rod through the T-piece was a curved flexible plastic pipe; at each end of this pipe was a rubber sucker pad. These pads were attached to the inner wall of the cone; thus the vibrations were transmitted through the rod/pipe/sucker pad to the cone. This method was found to work reasonably well, creating both even and odd circumferential modes of vibration. A gantry was designed and manufactured with brackets to hold the electromagnetic excitor perpendicular to the tank top. Allowance was made for clamping the gantry to the table, thus providing a rigid support. The arrangement of the sections was chosen in order to make the structure as stiff as possible, thus avoiding excessive transmissions of vibrations. Figure 6.47 shows the circuit diagram for the electrical and electronic equipment used. Experimental method and results Before carrying out any experimental work, the elastic buckling pressure of each cone was calculated, using a linear finite element analysis. Each cone
© Carl T. F. Ross, 2011
Vibration of pressure vessel shells in water
269
was then set up in the tank, with water on the outside, and a search was made for the resonant frequencies by the following technique. The transducer was positioned above one of the sucker pads, near an anti-node, and the cone excited, the frequency being increased in steps of 5 Hz until the output from the transducer was seen to peak. A finer search was then made in this region, by both altering the transducer position and frequency, in steps of 1 Hz, until a maximum output was reached. The frequency was held at this value, and the amplitude and phase angle were recorded at 5° intervals around the circumference of the cone. Using these data, a plot of amplitude versus angular position of the transducer was made, the number of peaks and phase angles establishing which circumferential mode (n), to which this frequency corresponded. This technique was repeated until as many of the modes from n = 2 to n = 6 were found. Once these resonant frequencies were established, the cone was vibrated at a particular natural mode and the transducer fixed in a position such that a peak was recorded. The external pressure was then increased, the output peak being kept by adjustment of the frequency until about one-half of the estimated buckling pressure was reached. The change in resonant frequency as a function of pressure was recorded at regular intervals. This was repeated for all resonant frequencies for the circumferential eigenmodes from n = 2 to n = 6. Initially, cone 4 was vibrated in air, so that the resonant frequencies could be compared with previous results. The two sets of results are shown in Fig. 6.56. For both studies the large flange was firmly clamped. The differences between the two sets of results
6000 This study Previous study11
Frequency (Hz)
4875
3750
2625
1500 1
2
3
4 Mode n
5
6
6.56 Frequency versus mode n for cone 4 in air.
© Carl T. F. Ross, 2011
7
270
Pressure vessels
were attributed to the fact that for this study the smaller flange was blocked off by a steel closure plate, this having not been present for the previous study. The estimated linear elastic buckling pressures are shown in Table 6.8. It was necessary to determine these before vibration of the cones under external water pressure, in order to avoid structure failure. For the theoretical buckling of all three cones, the longitudinal mode was found to be one half-wave and the circumferential modes were found to consist of the waves (that is m = 1 and n = 3; n = number of circumferential waves). Figures 6.57–6.59 show the variation of experimentally determined resonant frequencies under increasing external water pressure. Computational investigations were carried out using several different combinations of structural and fluid meshes. Earlier investigations with the Table 6.8 Theoretical buckling pressures
Cone number
Buckling pressure (MN m−2)
Buckling pressure (lb in−2)
4 5 6
8.5 10.3 12.8
1233 1494 1856
4500 p=0 p = 150 lb in–2 p = 300 lb in–2 p = 450 lb in–2 Frequency (Hz)
3300
2100
900 1
2
3
4 Mode n
5
6
7
6.57 Experimental variation of frequency with external pressure (cone 4).
© Carl T. F. Ross, 2011
Vibration of pressure vessel shells in water 5000 p=0 p = 200 lb in–2 p = 400 lb in–2 p = 600 lb in–2 Frequency (Hz)
3633
2267
900 1
2
3
4 Mode n
5
6
7
6.58 Experimental variation of frequency with external pressure (cone 5).
3000 p=0 p = 200 lb in–2 p = 400 lb in–2 p = 600 lb in–2 Frequency (Hz)
2300
1600
900 1
2
3 Mode n
4
5
6.59 Experimental variation of frequency with external pressure (cone 6).
© Carl T. F. Ross, 2011
271
272
Pressure vessels Q
P Structure mesh C, 8 elements
Fluid mesh 3
Fluid mesh 4
6.60 Structure and fluid meshes.
structural mesh terminating at the small flange gave poor results. After several attempts the structural mesh C shown in Fig. 6.60 was adopted; here the closure plate at the small end was modelled with the aid of the conical element described earlier, the small flange being treated as an extra ring stiffener. It was mainly the modelling of the structure that led to good results. Node P remained fixed for all combinations. The use of different end fixing conditions for node Q was also investigated. It was found that this had a small effect on the resonant frequency for n = 2 and a negligible effect on the higher modes. To compare theoretical and experimental results for the different meshes, cone 4 with no external pressure was used. The results for the two fluid meshes shown in Fig. 6.60 used in conjunction with structural mesh C are given in Fig. 6.61 (a) and (b). It can be seen that both combinations gave good results and, as expected, a refining of the fluid mesh gave increased accuracy. Similar results were obtained for cones 5 and 6 as shown in Fig. 6.61 (c) and (d). Using the combination of structural mesh C and fluid mesh 4, results were obtained for various external pressures. Comparison between theory and experiment Comparisons between theory and experiment are shown in Figs 6.61– 6.64 where it can be seen that agreement is good. Figure 6.62 shows the
© Carl T. F. Ross, 2011
Vibration of pressure vessel shells in water 5000 Experiment Theory
3625
Frequency (Hz)
Frequency (Hz)
4500
2750 1875 1000
1
2
3
4 5 Mode n (a)
6
2750 1875
1
3633
2267
2
3
4 5 Mode n (c)
6
7
6
7
5500 Experiment Theory
Frequency (Hz)
Frequency (Hz)
3625
Experiment Theory
900 1
7
4500
1000
273
2
3
4 5 Mode n (b)
6
7
Experiment Theory 3967
2433 900 1
2
3
4 5 Mode n (d)
6.61 Comparison of theory and experiment (zero pressure): (a) cone 4, mesh 3; (b) cone 4, mesh 4; (c) cone 5, mesh 4; (d) cone 6, mesh 4.
experimental results for pressures of 75, 300 and 450 lb in−2, whereas Figs 6.63 and 6.64 show the experimental results for 200 and 600 lb in−2. Therefore, it can be concluded that the finite element method is suitable for analysing the vibration of ring-stiffened conical shells under external water pressure. The results also show that as the external water pressure was increased, the compressive membrane stresses in the vessel increased, causing their resonant frequencies to decrease, because of the resulting decrease in the bending stiffness of these vessels. The amounts by which the frequencies change for a given increase in external pressure is governed by the relative effect of the [KG] matrix on the [K*] matrix for that particular eigenmode; that is, the greatest change was for m = 1 and n = 3, because this was the static buckling eigenmode for all three cases. Additionally, the figures show that, as the static buckling pressure of a vessel was approached, the eigenmode corresponding to the lowest frequency became similar to the static buckling eigenmode. This highlights the danger of any movement that may trigger a form of dynamic buckling as the static instability pressure is approached.
© Carl T. F. Ross, 2011
274
Pressure vessels 4500
p = 75 lb /in–2 p = 300 lb /in–2 p = 450 lb /in–2 p = 700 lb /in–2 Experiment Theory Frequency (Hz)
3167
1833
500
1
2
3
4 Mode n
5
6
7
6.62 Comparison of theory and experiment (cone 4).
Conclusions The results of the investigation show that as the external water pressure surrounding a ring-stiffened conical shell is increased, its circumferential bending stiffness decreases and so too does its resonant frequencies. In addition, as the external water pressure is increased, the circumferential modes of vibration take forms similar to the static buckling modes, indicating that, at pressures approaching static buckling pressure, there is a possibility that such vessels can buckle dynamically, owing to fluid motion or the motion of unbalanced rotating machinery. This can be seen clearly in Figs 6.62–6.64, where the frequency fall was largest when m = 1 and n = 3, the static buckling eigenmode. The investigation has shown that a relatively simple finite element model provides good results, and that using variable inductance transducers for experiments involving the vibrations of walled axisymmetric shells is successful.
© Carl T. F. Ross, 2011
Vibration of pressure vessel shells in water
275
4800 p = 200 lb in–2 p = 600 lb in–2 p = 1200 lb in–2 Experiment Theory
Frequency (Hz)
3300
1800
300 1
2
3
4 Mode n
5
6
7
6.63 Comparison of theory and experiment (cone 5).
6.5
Effect of tank size
To determine the effects of the size of the tank on vibration, comparisons are made in Figs 6.65–6.67 of the experimentally obtained resonant frequencies of the small prolate domes vibrating in the small closed tank, with the experimentally obtained values for the same domes vibrating in the larger open tank, as shown in Fig. 6.68. From Figs 6.65–6.67 it can be seen that the experimental results for the two tanks differ more for lower values of n than for the higher values. This may be because the amplitudes of vibration at the lower values of resonant frequency are larger than those at the higher values of the resonant frequency (i.e. if the amplitude of vibration is large the size of the tank, and whether or nor it is closed, are important features, Fig. 6.68). Computer programs for these problems can be obtained directly from the author.
© Carl T. F. Ross, 2011
Pressure vessels 5500
p = 200 lb in–2 p = 600 lb in–2 p = 1200 lb in–2 Experiment Theory
4250
Frequency (Hz)
276
3000
1750
300 1
2
3
4 Mode n
5
6
7
6.64 Comparison of theory and experiment (cone 6).
© Carl T. F. Ross, 2011
Vibration of pressure vessel shells in water 400 380 360 340 320 300 280 Frequency (Hz)
260 240
Resonant frequencies for the dome vibrated in the big tank
220 200 180 160 140
Resonant frequencies for the dome vibrated in the small tank
120 100 80 60 40 20 0
1
2 3 4 5 6 7 8 9 10 11 12 Number of circumferential waves (n)
6.65 Comparison of resonant frequencies obtained experimentally from the big tank and small tank for the dome with AR = 3.5.
© Carl T. F. Ross, 2011
277
Pressure vessels
380 360 340 Resonant frequencies for the dome vibrated in the big tank
320 300 280 260 Frequency (Hz)
278
240 220 200 180 160
Resonant frequencies for the dome vibrated in the small tank
140 120 100 80 60 40 20 0
1
2 3 4 5 6 7 8 9 10 11 12 Number of circumferential waves (n)
6.66 Comparison of resonant frequencies obtained experimentally from the big tank and small tank for the dome with AR = 3.0.
© Carl T. F. Ross, 2011
Vibration of pressure vessel shells in water
380 360
Resonant frequencies for the dome vibrated in the small tank
340 320 300 280 260 Frequency (Hz)
240 Resonant frequencies for the dome vibrated in the big tank
220 200 180 160 140 120 100 80 60 40 20 0
1
2 3 4 5 6 7 8 9 10 11 12 Number of circumferential waves (n)
6.67 Comparison of resonant frequencies obtained experimentally from the big tank and small tank for the dome with AR = 2.5.
Vibrator Bell crank mechanism
1.5 m
1.075 m
Water Small dome
6.68 Testing a small dome in the large open tank.
© Carl T. F. Ross, 2011
279
7 Novel pressure hull designs
Abstract: Several variations to traditional pressure hull designs are described. Two of these designs, one of which involves the cylinder/cone body and the other the dome ends, were previously published by the present author. A third design, which involves stiffening the pressure hull with circumferential tubes, was developed by the present author in 1977. The author’s invention of a corrugated submarine pressure hull is shown to be structurally more efficient than the traditional ring-stiffened one. Key words: submarines, pressure hull, domes, tube-stiffening, ringstiffening, drilling rig.
7.1
Design of dome ends
As already stated in chapter 1, a dome end, which is convex to the effects of pressure, can suffer instability at a pressure which may only be a fraction of that required to cause axisymmetric yield. If the dome is geometrically imperfect and it suffers inelastic instability, its buckling resistance can be even further decreased. The proposal here is to invert the dome ends, as shown in Fig. 7.1, so that, because buckling is virtually eliminated, the domes do not have to be constructed as precisely as would have been necessary had they been convex to the effects of external pressure. It should be pointed out here that initial imperfections have a serious and detrimental effect on the buckling resistance of domes when they are convex to external pressure, and that precise construction of domes is very difficult because of their double curvature. It will now be shown that by inverting the dome ends so that they are concave to the effects of external pressure their structural efficiency will be greatly improved.132–133
7.1.1 A novel dome design To determine whether or not the dome design of Fig. 7.1 was satisfactory the cylinder–dome combination of Fig. 7.2 was considered. As the maximum membrane stress in a dome is half the magnitude of the maximum membrane stress in a cylinder of the same diameter and wall thickness, it was 280 © Carl T. F. Ross, 2011
Novel pressure hull designs Pressure hull
Hydrodynamic hull Dome Water pressure
Axis
281
end
Atmospheric pressure
Dome end
7.1 Submarine pressure hull with inverted dome ends.
Locally thickened region
Atmospheric pressure
m 10 Axis
Water pressure
Locally thickened region
10 m
2m
7.2 Dome–cylinder under external pressure.
decided to let the diameter of the dome be twice the diameter of the cylinder. The wall thickness of both the dome and cylinder was set at 3 cm, except at the joint between the dome and cylinder, where it was gradually increased to a maximum value of 8 cm. The vessel was subjected to an external pressure of 1 MPa and 21 elements were used to model it. The calculations for the stresses were carried out by using the computer program THINCONE,28 and some typical stress distributions, together with a screen dump of the deflected form of the vessel, are shown in Figs 7.3–7.7. To compare the proposed design with the traditional one, it will be necessary to make use of equation [7.1],32 which calculates the elastic instability pressure Pcr of domes convex to the effects of pressure, as shown in Fig. 7.8:
© Carl T. F. Ross, 2011
282
Pressure vessels
Axis
7.3 Screen dump of the deflected form of the vessel.
200
100
0 Node 11
Node 22
Stress (MPa)
−100
−200 −300 −400 −500
7.4 External meridional stress in cylinder.
θ − 20 ⎤ ⎛ 0.07R ⎞ ⎡ ⎛t⎞ Pcr = ⎢1 − 0.175 ⎜⎝ 1 − ⎟⎠ ( 0.3E ) ⎜⎝ ⎟⎠ ⎥ 20 ⎦ 400t R ⎣
2
[7.1]
Substituting t = 3 cm, R = 10 m and E = 2 × 1011 Pa into equation [7.1]: Pcr = 4.64 × 10 5 N m −2 which is considerably less than the pressure the dome can withstand if it is concave to the effects of pressure.
© Carl T. F. Ross, 2011
Novel pressure hull designs
283
0 Node 11 −100
Node 1
Stress (MPa)
−200
−300
−400
−500
7.5 External meridional stress in dome.
Stress (MPa)
0 Node 22 Node 11 −100
−200
−300
7.6 Internal hoop stress in cylinder.
To obtain a value of 1 MPa for Pcr, t would have to be about 4.5 cm (i.e. it has to be 50% larger than that of the proposed design). Even with an increase in its wall thickness of 50% it is unlikely that the vessel of Fig. 7.8 has the strength of the dome of Fig. 7.2, as initial imperfections cause a catastrophic decrease in the buckling resistance of the vessel of Fig. 7.8.
© Carl T. F. Ross, 2011
284
Pressure vessels
Stress (MPa)
200
100
0 Node 11
Node 1
−100
−200
7.7 Internal hoop stress in dome.
t
R θ θ 0
7.8 Dome (convex to pressure).
7.2
Design of cylindrical body
Here again, when a cylinder is subjected to uniform external pressure, its buckling resistance is very low, as demonstrated in chapter 2. One method of improving this buckling resistance is to introduce stiffening rings, although the vessel can still fail through general instability, as described in chapter 4. In this section, two proposals are presented that provide alternatives to stiffening rings, although there is no reason why rings cannot be incorporated into these designs. In addition to these, a proposal is made for improving interframe shell instability characteristics.
© Carl T. F. Ross, 2011
Novel pressure hull designs
285
7.2.1 Cylinder/cone design The first proposal here is to eliminate the ring-stiffeners, by making the body of the pressure hull of a ‘swedged’ form, as shown in Fig. 7.9. This process considerably increases the buckling resistance of cylinders under uniform external pressure. To determine the buckling resistance of swedged shaped cylinders, the vessels of Figs 7.10 and 7.11 were considered, and these buckling pressures were compared with those of equivalent ringstiffened cylinders. The buckling pressures of the swedged vessels were obtained from the computer program CONEBUCKLE28 and the buckling End plate
Shell
End plate
Axis of symmetry
7.9 Swedged shaped pressure hull.
2.5 m
2m
1m
Axis 11 m
7.10 Example 1: swedged vessel 1 (t = 3 cm).
5m
4.5 m
1m
11 m
7.11 Example 2: swedged vessel 2 (t = 6 cm).
© Carl T. F. Ross, 2011
Axis
286
Pressure vessels
Table 7.1 Buckling pressures of swedged vessels and their ring-stiffened equivalents Example 1
Example 2
Swedged vessel 1 (MPa)
Ring-stiffened 1 (MPa)
Swedged vessel 2 (MPa)
Ring-stiffened 2 (MPa)
3.607 (2)a
2.533 (4)
4.319 (3)
2.458 (6)
a
The number of nodes is given in parentheses.
pressures of the ring-stiffened equivalents were obtained from the program described in reference 55. Comparisons can be made between the results for swedged vessels and those for their ring-stiffened equivalents as shown in Table 7.1. It can be seen that swedged vessel 1 has a buckling resistance about 42% larger than its ring-stiffened equivalent, and that swedged vessel 2 has a buckling resistance about 76% larger than its ring-stiffened equivalent. In all instances, the vessels were assumed to be fixed at their left ends and clamped at their right ends. The figures in brackets in Table 7.1 represent the number of lobes into which the vessels will buckle. It is true that the ring-stiffeners, which are structurally more efficient that those chosen in the present section, could produce higher buckling pressures than those given in Table 7.1, but it is equally true that the swedged vessels could have been made more structurally efficient. For example, some of the weight from the conical elements could have been redistributed to the cylindrical elements, thereby producing a ‘flange’ effect, and the chosen cone angles were not necessarily the best from a structural point of view.
7.2.2 Tube-stiffened submarine The second proposal in this section is based on an idea by Harris,134 who suggested that the pressure hull should be stiffened by tubes placed longitudinally along it; in reply the present author suggested that it may be more desirable to have circumferential tubes, as shown in Fig. 7.12 (A similar design was developed independently by Sub Sea Oil Services.135). Furthermore, if the tubes are subjected to internal pressure so that the hull is in a state of initial tension, then the buckling resistance of the hull can be even further increased. A further possible development of this type of hull is that it could be made to be ‘intelligent’; that is, when the external pressure on the cylinder body is increased, the internal pressure in the tube-stiffeners could be increased at the same time, so that the vessel could dive to even greater depths.
© Carl T. F. Ross, 2011
Novel pressure hull designs
Tubes under internal pressure
287
Axis
Hull
7.12 Tube-stiffened pressure hull.
Ring stiffeners Axis
7.13 Swedged vessel with ring-stiffeners.
Tubes under internal pressure Axis
7.14 Swedged vessel with tube-stiffeners.
Variations on the themes described in Sections 7.2.1 and 7.2.2 can take the forms shown in Figs 7.13 and 7.14.
7.2.3 Shell instability of a corrugated cylinder The examples in Section 7.2.1 were based on guarding against general instability, but a similar approach can be adopted to improve resistance against interframe shell instability, as shown in Figs 7.15 and 7.16, where two conical elements are used. Comparison is made in Table 7.2 of the buckling resistances of these vessels with the buckling resistances of their circular cylindrical shell equivalents. All vessels were assumed to be fixed at their left ends and clamped at their right ends, and they were modelled with ten equal-length elements. The material properties of these vessels were assumed to be: E = 2 × 1011 N m−2 and ν = 0.3. The circular cylinders were assumed to have the following geometries: cylinder 3: R = 4.95 m
© Carl T. F. Ross, 2011
288
Pressure vessels 1m
1m Shell
5m
5m
4.9 m
Axis
7.15 Example 3: vessel 3 with improved shell instability resistance (t = 2 cm).
0.5 m
0.5 m Shell
2m 2m 1.9 m Axis
7.16 Example 4: vessel 4 with improved shell instability resistance (t = 2 cm).
Table 7.2 Buckling pressures of cones and their circular cylindrical shell equivalents Example 3
Example 4
Vessel 3 (MPa)
Cylinder 3 (MPa)
Vessel 4 (MPa)
Cylinder 4 (MPa)
1.556 (20)
0.675 (17)
12.745 (14)
6.058 (12)
and t = 2.01 cm; and cylinder 4: R = 1.95 m and t = 2.04 cm. Table 7.2, shows that cylinders 3 and 4 have a resistance to shell instability some 2.31 and 2.10-fold greater than their respective circular cylindrical shell equivalents. Readers are advised to consult reference 17, which gives a good summary on the strength and design of submarine structures.
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7.2.4 Method of manufacturing a corrugated submarine pressure hull Although full-scale manufacture of these vessels in metals may at first appear to be difficult, it can be achieved if the vessels are constructed from truncated ring-stiffened conical shells, as shown in Figs 7.17 and 7.18; this can be seen to be similar to the method of construction described by Smith.11 However, the true importance of the corrugated pressure hull is likely to be for vessels made from composite materials11 such as glass, boron and carbon fibres, and metal matrix composites, where construction of corrugated pressure hulls is likely to prove easier than ring-stiffened vessels. It should be emphasized that the strength of some of these man-made materials is considerably greater than that of high-tensile steel, as shown in Table 1.1–1.5. If a pressure hull is made in this form from a metal, it may be possible to machine the conical shell elements, so that the initial out-ofcircularity is reduced.
7.2.5 Design of an underwater drilling rig Ross and La Folley-Lane136 presented a conceptual design of an underwater drilling rig, which is intended for use in deep waters (Fig. 7.19). It can be seen that the main pressure hull for this vessel is a thick-walled toroidal shell. Ring stiffeners Truncated conical shells
7.17 Ring-stiffened corrugated pressure hull.
Axis
7.18 A truncated conical shell element.
© Carl T. F. Ross, 2011
Bolts
290
Pressure vessels
7.19 Underwater drilling rig.
7.3
Ring-stiffened or corrugated prolate domes
Many submarine designers prefer blocking off the forward end of a submarine pressure hull by an oblate pressure hull dome, rather than by a prolate pressure hull dome, as the former is usually much stronger than the latter.
© Carl T. F. Ross, 2011
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7.20 Ring-stiffened prolate dome.
The problem with the oblate dome is that it is not hydrodynamically efficient and that it requires a free-flood prolate hydrodynamic casing to surround it. Much of the space in-between these two domes is undesirably wasted. To avoid the occurrence of this wasted space, it should be feasible to replace the oblate pressure hull dome with a prolate pressure hull dome if the latter is suitably ring-stiffened or corrugated in its flank,57 as shown in Fig. 7.20. Similarly, for greater strength, an oblate dome can be crossstiffened in the form of a grid, which is symmetrical about the dome’s nose.
7.4
A submarine for the oceans of Europa
Europa is a moon of Jupiter, which has oceans some 60 miles (100 km) deep. As Europa has an elliptical orbit around Jupiter, it is believed that because of the resulting stretching and compressing of Europa owing to this orbit, it has liquid water under its icy crust, which is some 6 miles (10 km) thick. Moreover, because of this stretching and compressing of Europa by Jupiter, it is believed that Europa has hydrothermal vents in its oceans’ bottoms. Thus, it is likely that Europa may have extraterrestrial life; supported by chemosynthesis and powered by the output of its hydrothermal vents. The present author designed a robotic submarine for the oceans of Europa137 (Fig. 7.21). As the hydrostatic pressure is so large at the bottom of Europa’s oceans, it was suggested that the vessel be made from a material with a high strength–weight ratio, such as a material matrix composite, or possibly even from carbon nanotubes. As the submarine has to drill or melt its way through 6 miles (10 km) of ice, it needs a lot of power and could perhaps be powered by a small nuclear reactor. As the nuclear reactor has to be made watertight, it was suggested that the shape of the submarine was as shown in Fig. 7.21, so that it could house the nuclear reactor. As Europa is
© Carl T. F. Ross, 2011
292
Pressure vessels
7.21 A submarine for Europa.
about 500 million miles away from Earth, a manned flight is not advisable. The submarine is a hover submarine; capable of moving in any direction.
7.5
Conclusions
The proposal of inverting the dome ends so that they avoid buckling appears to make a weight-saving of over 50% with the example chosen. In fact, the weight-saving should be considerably larger than this, as initial imperfections further decrease the buckling resistance of the traditional design. Furthermore, as initial imperfections of the domes are of less importance with the proposed design, the cost of manufacturing the domes based on the proposed design is considerably less. Calculations for the swedged vessels, and for the vessels of Figs 7.15 and 7.16 show these vessels to be considerably stronger than their traditional equivalents. The pipe-stiffened cylinder needs experimental back-up.
© Carl T. F. Ross, 2011
8 Vibration and collapse of novel pressure hulls
Abstract: The buckling and vibration of corrugated pressure vessels and tube-stiffened vessels and the collapse of inverted dome cups are discussed. Experimental results are compared with those from theoretical models using finite element analysis for cylinders in air and water. Dome cup ends were structurally more efficient than dome cap ends. A redesign of the corrugated food can is presented and shown to result in significant savings in manufacturing costs. Key words: circular cylinders, buckling, vibration, food cans, corrugation, pressure hulls.
8.1
Buckling of corrugated circular cylinders under external hydrostatic pressure
8.1.1 Previous research on corrugated vessels In 1987, Ross133 presented an alternative design to the ring-stiffened vessel as shown in Fig. 8.1. Ross showed that the corrugated vessel was structurally more efficient than a ring-stiffened equivalent. In 1991, Yuan et al.138 showed that the corrugated cylinder could be made more structurally efficient by increasing the cone angles to certain optimum values. It should be noted that if the cone angles are too large the vessel fails axisymmetrically, as shown by Liang et al.139 These reported studies were of a theoretical nature and experimental work to verify the results was carried out by Ross et al.140–143 In 1999, Ross and Waterman144 carried out more experimental work and used this and existing results to produce a design chart. Experimental investigations have shown that in a similar way to unstiffened and ring-stiffened circular cylinders, the vessels were prone to suffer plastic knockdown owing to initial out-of-circularity. To allow for this, Ross and Palmer140 presented a thinness ratio, which was based on the thinness ratio of Windenburg and Trilling;38 the formula for which is given by the following equation: 2 3 λ = ⎡⎣( Lb / D1 ) / ( t 1 D1 ) ⎤⎦
0.25
×
(
σ yp / E )
[8.1] 293
© Carl T. F. Ross, 2011
294
Pressure vessels Water Water Water
Atmospheric pressure
Casing
8.1 Corrugated pressure hull.
where: Lb = length between bulkheads; D1 = equivalent diameter; t1 = equivalent thickness; σyp = yield stress; E = Young’s modulus of elasticity.
8.1.2 Theoretical analysis To determine by analytical methods the uniform external pressure required to cause the elastic instability of thin walled conical shells is very difficult and for these cases it is more convenient to use the finite element method. If the cone is of small apical angle then buckling will take place in a lobar manner as shown in Fig. 8.2, but if the angle of the cone is large, the vessel can buckle axisymmetrically. It must be emphasised that as these vessels are sensitive to initial geometric imperfections, they can suffer inelastic instability at buckling pressures considerably less than those causing elastic instability. This is where the importance of experimentally obtained results is crucial, to enable a plastic reduction factor to be obtained for a particular vessel. This plastic reduction factor (PKD) must be divided into the theoretical elastic buckling pressure for the vessel in question, to obtain the reduced predicted buckling pressure: Ppred = Pcr / PKD where PKD = Pcr/Pexp; Pcr is the theoretical buckling pressure, based on the finite element method and Pexp = experimentally obtained buckling pressure. The theoretical solution based on the small-deflection elastic theory uses the element first developed by Ross50 in 1974 of a truncated cone with two nodal circles at its ends.
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8.2 Buckled corrugated circular cylinders.
8.1.3 Experimental procedure Nine steel corrugated cylindrical models were tested to destruction.144,145 These models were AST1, AST2, AST3, AST4, AST5, AST6, AST7, AST8 and AST9, as shown in Fig. 8.3. Each vessel was carefully measured to determine its initial out-of-circularity, which was defined as the difference between the maximum inward and outward deviations of the external surface at the midpoint of the cylinder. This was achieved with the use of a Mitutoyo BN706 co-ordinate measuring machine, together with a touch-trigger probe. The longitudinal profiles were of a sinusoidal nature, but for the theoretical analysis, it was convenient to represent each corrugation with two truncated finite element conical shell elements; previous work by Ross and co-workers140,143 had found this to be a satisfactory procedure. The geometrical details are listed in Table 8.1; a sketch of the cylinder is shown in Fig. 8.4. Figure 8.5 shows details of a typical corrugation. N = number of corrugations; t = wall thickness = 0.25 mm; Ls = length of corrugation = 3.57 mm; Ro = RE = external radius of cylinder; Ri = RI = internal radius of cylinder; D1 = RE + RI = 99 mm. Young’s modulus and Poisson’s ratio were not measured, because the cylinders were made from typical carbon steels, so that the assumed values for these constants were reasonable. To determine the value for yield stress, three specimens of length 150 mm and width 15 mm were cut from the vessel. Material and tensile tests were performed on the Lloyd’s materials testing machine. Figure 8.6 shows the graph of load against extension.
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Pressure vessels
8.3 The AST series of steel corrugated cylindrical models.
Table 8.1 Geometrical details of AST series (mm)
Model
Number of corrugations (N)
Out-of-circularity
Length between bulkheads (Lb)
AST1 AST2 AST3 AST4 AST5 AST6 AST7 AST8 AST9
1 2 3 5 6 8 9 11 12
0.15 0.16 0.07 0.08 0.18 0.15 0.08 0.15 0.16
3.57 7.14 10.71 17.85 21.42 28.56 32.13 39.27 42.84
Closure plate
Closure plate
Ls Ri
Ro
Lb
8.4 Sketch of cylinder.
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3.57
0.55 73°
8.5 Details of a typical corrugation (mm).
2500 000
Load (N)
2000 000 1500 000
1000 000
0 000
0000 0092 0188 0279 0371 0400 0556 0651 0744 0837 0930 1024 1117 1210 1304 1397 1490 1584 1677 1770 1884 1967 2060 2144 2237 2330 2424 2517 2610 2704 2797
500 000
Extension (mm)
8.6 Graph of tensile test.
Material properties were found to be: Young’s modulus = E = 200 GPa (assumed); Poisson’s ratio = ν = 0.3 (assumed); yield stress = σyp = 405 MPa (measured). The cylinders were blocked off at their ends by end closure plates. These closure plates were simply push-fitted onto the ends of the cylinders and sealed with silicone. The silicone was initially in a semi-liquid form, which hardened after a few hours of exposure to the atmosphere, creating a watertight seal.
8.1.4 Pressure system The models were tested in the test tank shown in Fig. 8.7. Water was used as the pressure raising fluid. A Tangye Hydra hand-pump was used to apply a pressure to the tank.
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Pressure vessels
8.1.5 Experimental procedure The pressure tank was partially filled with water before placing a vessel into it. The vessel, which was previously made watertight as described earlier, was placed in the pressure tank as shown in Fig. 8.7. Before raising the water pressure, the bleed screw situated in the lid of the test tank was left open so that any trapped air could be pumped out. When the air had been pumped out and water flowed from the bleed hole, the screw to the bleed hole was tightened, so that the bleed hole was sealed and the experiment was ready to commence. Figure 8.8 shows the vessels in their collapsed state. AST 4–9 collapsed in an inelastic lobar manner whereas AST 1, 2 and 3, the vessels of smallest overall length, collapsed axisymmetrically. The experimentally obtained results are shown in Table 8.2. It can be seen from Table 8.2 that the general trend is that as Lb increases the experimentally obtained buckling pressure decreases. However, AST6 and AST8 do not follow this pattern; the Water release valve
Bolts
Gasket seal
Water under pressure End cap
Atmospheric pressure
8.7 Test tank and model.
8.8 The collapsed models.
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Table 8.2 Experimental buckling pressures (Pexp) Model
Out-of-circularity
Lb (mm)
Pressure at collapse (bar)
AST1 AST2 AST3 AST4 AST5 AST6 AST7 AST8 AST9
0.15 0.16 0.07 0.08 0.18 0.15 0.08 0.15 0.16
3.57 7.14 10.71 17.85 21.42 28.56 32.13 39.27 42.84
13.1 12.8 12.7 10.8 8.8 6.9 8.0 5.7 7.1
buckling pressure for AST8 should be higher than that of AST9 and the buckling pressure for AST6 should be higher than AST7. This is typical of vessels of this type, where occasionally low experimental buckling pressures, which cannot be described as rogue results, are observed. The results also cannot be explained by the out-of-circularity measurement as this is larger for AST9, which has a higher buckling pressure than both the shorter vessels AST8 and AST6. It is because of results like these that the present study was conducted, and why a traditional nonlinear finite element is unlikely to prove successful.
8.1.6 Determination of the design chart To calculate the theoretical buckling of the vessels a computer program called CONEBUCK,28 based on finite element analysis, was used. The element adopted in the program was the thin-walled truncated conical element of Ross.50 This element has been described earlier in Section 3.2. The theoretical buckling pressures Pcr obtained by this program are shown in Table 8.3. It should be noted that the buckling pressures were based on elastic theory for perfect vessels. For imperfectly shaped vessels, which buckle inelastically, the experimental buckling pressure may be a small fraction of the theoretical values.
8.1.7 Calculation of the thinness ratios The design procedure is to first calculate theoretical pressure Pcr and thinness ratio λ. The theoretical pressure is then used to calculate the actual collapse pressure Ppred by dividing Pcr by the plastic knockdown factor PKD, (Pcr/Pexp). This is obtained from the design curve, where 1/λ is plotted against Pcr/Pexp. To obtain the design pressure it is necessary to divide the actual
© Carl T. F. Ross, 2011
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Pressure vessels
Table 8.3 Theoretical buckling pressures (Pcr) Model
Lb (mm)
Pcr (bar)
AST1 AST2 AST3 AST4 AST5 AST6 AST7 AST8 AST9
3.57 7.14 10.71 17.85 21.42 28.56 32.13 39.27 42.84
315.8 68.9 40.6 25.2 19.5 15.0 13.5 11.2 10.0
Ds/2
t NA
Ds/2 α
tw
8.9 Swedge leg (assumed to be a parallelogram).
collapse pressure by a large safety factor. The thinness ratio λ is calculated from equation [8.1]. The equation uses an equivalent wall thickness t1 to enable the corrugated cylinder to be represented as an unstiffened circular cylinder. The thickness t1 is such that the flexural stiffness of the equivalent unstiffened vessel is the same as the corrugated one; this is calculated from: ⎡Lb (t 1 )3 ⎤ / 12 = I NA N + Σ [( Lf t 3 )] / 12 × 2 ⎣ ⎦ where INA = tw × (D3s/12) × 2; t1 = equivalent wall thickness; tw = width of a leg corrugation (Fig. 8.9); tw = t/sin α; Ds = depth of a leg of the corrugation; NA = neutral axis of swedge centroid; Lf = length of flat part of vessel.
© Carl T. F. Ross, 2011
(8.2)
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The model AST9 is now used as an example of how to calculate the thinness ratio λ and the plastic knockdown factor.
8.1.8 A typical calculation for λ and PKD (for AST9) The following parameters were defined: Ds = 0.55 mm; N = 12; Ls = 3.57; Lb = 42.84; α = tan−1 (2 × Ds)/Ls = tan−1 [(2 × 0.55)/3.57]; α = 17.125°; tw = t/sin α = 0.25/sin 17.125; tw = 0.849 mm; INA = tw × (D3s/12) × 2 = 0.849 × (0.553/12) × 2; INA = 0.235. As the AST series has no flat lengths on the vessel from equation [8.2] [Lb (t 1 )3 ]/ 12 = I NA × N t 1 = [( I NA × N × 12)/ Lb ]1 / 3 = [(0.235 × 12 × 12)/ 42.84]1 / 3 t 1 = 0.429 Using equation [8.1], the thinness ratio can now be calculated:
λ ′ = [(Lb / D1 )2 /(t 1 / D1 )3 ]0.25 × (σ yp / E ) λ ′ = [(42.84 / 99)2 /(0.429 / 99)3 ]0.25 ×
(
410 × 10 6 / 200 × 10 9
)
l ′ = 1.752 PKD = Pcr / Pexp PKD = 10 / 7.1 PKD = 1.408 The results for the calculations for the remaining vessels are given in Table 8.4. Table 8.5 shows thinness ratios and PKD for other vessels,140–144 where e = measured out-of-circularity. Models excluded from Tables 8.4 and 8.5 were partially corrugated vessels with values of e/t1 > 2.91. Models AST1, AST2 and AST3 collapsed axisymmetrically and were, therefore, analysed
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Pressure vessels
Table 8.4 Calculations for thinness ratio and PKD Model
e/t1
λ′
1/λ′
PKD
AST1 AST2 AST3 AST4 AST5 AST6 AST7 AST8 AST9
0.349 0.373 0.163 0.186 0.419 0.349 0.186 0.349 0.373
0.506 0.716 0.876 1.315 1.239 1.431 1.518 1.678 1.752
1.976 1.397 1.141 0.884 0.807 0.699 0.659 0.596 0.570
24.10 5.39 3.20 2.16 2.21 2.17 1.67 1.95 1.41
Table 8.5 Thinness ratios for other series Model
e/t1
λ′
1/λ′
PKD
GAW1 GAW2 GAW3 GAW4 GAW5 GAW6 DF MBS MBL CA
1.02 2.91 1.53 0.42 2.75 2.09 0.185 0.185 0.365 0.46
1.483 1.716 2.015 2.275 2.508 2.722 3.57 2.67 3.53 2.73
0.674 0.583 0.496 0.44 0.40 0.367 0.28 0.37 0.28 0.37
2.77 2.23 2.39 1.48 1.36 1.35 1.15 1.61 1.69 1.45
by the non-linear finite element computer program described in Section 2.7. This gave poor results. To obtain the design chart, the results for this series has been combined with the provisional design chart of Ross and Waterman,144 by plotting l/λ′ against PKD in Fig. 8.10 to create a more comprehensive chart; other results140–143 were also used.
8.1.9 Conclusions The apparent rogue results of AST7 and AST8 from this study and from the GAW and MBL series from previous research, appear to indicate that traditional nonlinear finite element solutions are not suitable for such vessels. The theoretical method presented in this section seems to be preferable for the design of these vessels; it allows for the effects of plastic knockdown owing to initial imperfections. To obtain the design pressure, it is
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2.5
1/thinness ratio
2
1.5
AST series GAW series Other vessels Design curve
1
0.5
0 0
5
10
15 PKD
20
25
30
8.10 Design chart.
advisable to divide the predicted collapse pressure obtained with the aid of the design charts, by a large safety factor. The results have shown that although the larger vessels fail by general instability, as the length and number of corrugations decrease, the vessels tend to fail in an axisymmetric manner. Vessels whose values of 1/λ′, were less than 1 failed by general instability. The three vessels, which had the largest values of 1/λ′ failed by plastic axisymmetric deformation. Ross and Johns35 have shown that there is a link between these two modes of failure for circular cylinders. Consequently, there is probably a similar link for corrugated circular cylinders. As a result more experimental work is required for shorter vessels with greater values of 1/λ′, so that the design chart can be used with a greater degree of reliability. Experimental tests carried out by Ross et al.,146 found that vessels which were corrugated in a direction perpendicular to the above series, were not as structurally efficient.
8.2
Buckling of a corrugated carbon-fibre-reinforced plastic (CFRP) cylinder
A corrugated circular cylindrical pressure vessel was produced from carbon fibre and its performance under external pressure investigated. It was a 0/90/0 lay-up (0 degrees being parallel to the axis of the cylinder), with an overall thickness of 0.5 mm. Figure 8.11 shows the geometry of the corrugations.
© Carl T. F. Ross, 2011
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Pressure vessels Lamination details, 0/90/0 (measured from axis of cylinder)
3.5
103.00
Enlargement of base of vessel
72.2 Dia
Base
Top
15.0
10.50 100.50 Extent of FE model
Base plate
0.5
10.0
Enlargement of top of vessel
105.5 Ref
φ71.50 φ74.00
8.11 Details of CFRP vessel geometry (mm).
8.2.1 Finite element analysis The theoretical analysis of the buckling of pressure vessels has been carried out using a finite element program (CONEBUCK) written by Ross.28 The program uses a truncated conical, axisymmetric element with two nodal circles at each end; each node has four degrees of freedom. A reduction process devised by Irons147 allows the size of the stiffness matrix to be reduced by a factor of about four thus providing a very efficient program in terms of computer memory and speed. The program has recently been modified to enable the materials matrix to be determined for laminated composites. Hence, the new program (BCLAM)148 enables a laminate to be defined accounting for the number of layers, the material properties of each layer (E1, E2, G12 and ν12), and the direction of the fibres relative to the axis of the vessel, where E1 = tensile modulus in direction 1 (or x); E2 = tensile modulus in direction 2 (or y); G12 = rigidity modulus in plane 1–2; ν12 = Poisson’s ratio in direction 1, owing to a stress in direction 2.
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8.2.2 Extension of empirical buckling formulae to a CFRP vessel Ross and Palmer140 showed that for a steel corrugated vessel, an equivalent thickness could be obtained, which enabled the empirical formulae to be used for these vessels. The procedure was to determine the second moment of area of a section parallel to the centreline, about the element’s neutral axis. Increased structural stiffness to resist buckling is provided by the larger second moment of area produced by the elemental corrugations, as shown in Fig. 8.12. The ring stiffeners of conventional submarine structure increase the second moment of area of the section about the neutral axis in the same way, as shown in Fig. 8.13. Using the same method as described by Ross and Palmer,140 the equivalent thickness for the 0/90/0 CFRP vessel was derived here by treating the wall of the vessel as a combination of rectangular and trapezoidal beams as shown in Fig. 8.14. Figure 8.15 shows the direction of the fibres, and the
Top
Base
8.12 I
(VESSEL)
increased by corrugations.
8.13 I
(VESSEL)
increased by ring-stiffening.
A
Top
B
C
0° 90° 0°
8.14 Rectangular and trapezoidal beams.
© Carl T. F. Ross, 2011
D
306
Pressure vessels E = 10 GPa
0° t
E = 130 GPa
90° 0°
E = 10 GPa
5
8.15 0/90/0 Lay-up of compound beam.
A
B
C
D
5
5
5
5
65
65
65
65
8.16 Equivalent widths with E = 10 GPa.
values of the moduli along the beam (i.e. into the paper). There was a factor of 13 between the E values of the material along and perpendicular to the fibres. The overall wall thickness t was built up by the three layers of unidirectional material. The ‘width’ of the centre portion of each beam section (A, B, C and D in Figs 8.14 and 8.16), was increased in proportion to the modulus. The resultant equivalent cylinder was then treated as if it were made of one material, which had a Young’s modulus equal to that of the less stiff material. The second moment of area for a trapezoidal section about its neutral axis (NA) as shown in Fig. 8.17, is quoted by Roark and Young32 as: I=
bd 2 (d + a2 ) 12
[8.3]
This provided the dimensions of an equivalent cylinder (i.e. one with the same second moment of area). The standard equation for the second moment of area of a rectangular beam was then used to provide the equivalent thickness, where the width b of the beam is the length of the cylinder (vessel). A brief outline of the calculations made is now provided.
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NA
d
a
b
8.17 Second moment of area of a trapezoidal section.
The location of the neutral axis of the vessel was established, using equation [8.4]: N
y = ∑ Ai yi AT
[8.4]
I
where y¯ = distance to the neutral axis of entire beam; yi = distance to the ith section of the beam; Ai = area of the ith section of the beam; AT = total area of the beam. The second moment of area of the vessel about the neutral axis of the vessel was then determined, using equation [8.5]: I NA VESSEL = 4 ( I A + I B + I C + I D ) + I EXTRAS
[8.5]
where the ‘extras’ include the incomplete corrugations, and the part of the vessel that fits into the end cap. Thus, I NA VESSEL = 65.38 × 10 −12 m 4
[8.6]
For an equivalent vessel: I NA VESSEL =
b (t 1 ) 12
3
[8.7]
and t1 =
3
12 × 65.38 × 10 −12 0.1005
[8.8]
Hence, t 1 = 0.001984 m
[8.9]
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Pressure vessels
Table 8.6 Predicted buckling pressures for CFRP vessel Critical pressure, Pcr (MPa) Finite element program (BCLAM)
CFRP
Von Mises
DTMB
Simply supported
Clamped
2.47
2.60
0.813
0.852
Using this value of t1, Pcr was calculated, where Pcr was obtained by the von Mises formula and also by the David Taylor Model Basin (DTMB) formula. The theoretical formulae, together with the finite element program, were used to provide the buckling pressures for the CFRP vessel; these are listed in Table 8.6.
8.2.3 Extension of the thinness ratio to a CFRP vessel Cylindrical cylinders are prone to plastic knockdown owing to initial outof-roundness, and a thinness ratio λ was defined by Windenburg and Trilling38 to determine experimentally the plastic knockdown factor (PKD). Ross and Palmer140 extended this to cover corrugated pressure vessels, by introducing an equivalent thickness t1 and a corresponding λ′.
λ ′ = 4 ( L / D)2 (t 1 / D) × σ yp / E 3
[8.10]
where L = length of vessel; D = diameter of vessel; t1 = equivalent thickness of cylinder; σyp = yield stress; E = Young’s modulus. The phenomenon of yield does not exist for composite materials. Indeed the prediction of failure becomes a complex subject. There are many criteria, to predict failure, and the one used in this work was chosen as the least complicated. The Tsai–Hill (deviatoric strain energy) failure criteria for composite materials states that failure will occur when: 2 σ 12 σ 22 τ 12 σσ + 2 + 2 − 1 22 = 1 2 F1 F2 F12 F1
© Carl T. F. Ross, 2011
[8.11]
Vibration and collapse of novel pressure hulls
309
where σ1 = principal longitudinal stresses, along the fibres; σ2 = principal transverse stresses, normal to the fibres; τ12 = in-plane shear stress; F1 = uniaxial longitudinal stress to failure, along the fibres; F2 = uniaxial transverse stress to failure, normal to the fibres. The material used for the construction of this vessel was SE84HT/ HSC/200/400/37%, a unidirectional carbon fibre, used in pre-preg form. A series of tests were performed on unidirectional tension test specimens, in accordance with the CRAG procedures as described by Curtis149 and Curtis et al.150 The test pieces were made from the same batch of material used to produce the vessel, and provided the following material properties: E1 = 130 × 109 Pa E2 = 10 × 109 Pa G12 = 7 × 109 Pa v12 = 0.28 F1 = 814 × 106 Pa F2 = 135 × 106 Pa It was assumed that failure would occur first at the largest diameter portion of the corrugation, and that the hoop stress would be twice the longitudinal stress (as for thin-walled pressure vessel theory). It was further assumed that the in-plane shear stress at this point would be zero. The Tsai–Hill formula then reduces to:
σ 12 (σ 12 / 4 ) σ 1 (σ 1 / 2 ) − =1 + F12 F22 F12
[8.12]
Substituting the values above gives:
σ 1 = 262.9 × 106 Pa When σ1 reaches this value, ‘first ply failure’ occurs. This can, therefore, be considered as equivalent to the yield stress for the composite material, and can be used in the thinness ratio formula:
λ ′ = 4 ( L / D)2 / (t 1 / D) × (σ EQ / E ) 3
where L = 0.1005 m; D = 0.06968 m; t1 = 0.001984 m; E = 10 × 109 Pa; σEQ = 262.9 × 106 Pa.
© Carl T. F. Ross, 2011
[8.13]
310
Pressure vessels
The Young’s modulus used in the DTMB formula (chapter 3) was also required for the compound beam calculation. 2 3 0.1005 ⎞ ⎛ 0.07318 ⎞ 262.9 × 10 6 ×⎜ × λ ′ = 4 ⎛⎜ ⎟ ⎟ ⎝ 0.07318 ⎠ ⎝ 0.001984 ⎠ 10 × 10 9
[8.14]
Hence
λ ′ = 2.84 and 1 / λ ′ = 0.352 A design chart proposed by Ross et al.97 is shown in Fig. 3.5. The predicted pressure can be approximated for given values of λ′. From Fig. 3.5, PKD = 1.2, therefore the theoretical buckling pressures for von Mises and the DTMB formulae, are little changed. If Fig. 8.10 is used, then PKD = 2.21, which gives a better result, but still not good enough. The reason for this may be that Fig. 8.10 is for steel vessels and different design charts are required for composites. For comparison, five hypothetical circular cylinders of similar size were also investigated, HV1, HV2, HV3, HV4 and HV5 and their details are given in Table 8.7. Using the appropriate formulae, the critical pressures were calculated, as shown in Table 8.8. Also listed, are the results from the finite element program (BCLAM). The end conditions assumed were that the top of the vessels were fixed, and two cases for the end conditions of the base were computed, namely simply supported and clamped. From Fig. 3.4, the data used to derive the plastic knockdown factor for the hypothetical vessels is provided in Table 8.9, and the equivalent values for the CFRP vessel in Table 8.10. The pressures at which the vessels would be expected to buckle Pexp were calculated, and these are displayed in Table 8.11.
Table 8.7 Geometric details of vessels
HV1 HV2 HV3 HV4 HV5 CFRP
Diameter (mm)
Length (mm)
Wall thickness (mm)
Modulus (GPa)
75 75 70 70 75 73.18*
100.5 100.5 100.5 100.5 100.5 100.5
1.5 2.0 1.5 1.5 0.3 1.984*
200 200 200 65 200 10*
*Equivalent values.
© Carl T. F. Ross, 2011
Vibration and collapse of novel pressure hulls
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Table 8.8 Predicted buckling pressures (Pcr) for the CFRP and the hypothetical vessels Critical pressure Pcr (MPa) Finite element program (BCLAM)
HV1 HV2 HV3 HV4 HV5 CFRP
Von Mises39
Windenburg and Trilling38
Simply supported
Clamped
23.20 49.70 26.90 7.55 0.408 2.47
24.70 51.20 27.40 8.04 0.431 2.60
9.70 20.00 10.70 3.15 0.171 0.813
10.60 21.70 11.50 3.44 0.178 0.852
Table 8.9 Thinness ratio data for the hypothetical vessels σyp
λ
1/λ
Pcr/Pexp
Vessel
(MPa)
(–)
(–)
(–)
HV1 HV2 HV3 HV4 HV5
300 300 300 180 300
0.843 0.679 0.829 1.160 2.30
1.186 1.47 1.21 0.860 0.435
2.1 3.0 2.2 1.2 1.2
Table 8.10 Thinness ratio data for the CFRP vessel σyp
λ′
1/λ′
Pcr/Pexp
Vessel
(MPa)
(–)
(–)
(–)
CFRP
262.9
2.84
0.352
2.21
8.2.4 Experimental method The out-of-roundness was measured on a Mitutoyo co-ordinate measuring machine, using a proximity probe. The difference between the maximum inward and outward deviations of an external circumference was measured in a number of places along the length of the vessel, and the results are listed in Table 8.12.
© Carl T. F. Ross, 2011
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Pressure vessels
Table 8.11 Predicted experimental buckling pressure for the vessels, using the plastic knockdown factor (PKD) Predicted experimental buckling pressure, Ppred (MPa) Finite element program (BCLAM)
HV1 HV2 HV3 HV4 HV5 CFRP 1
Von Mises39
Windenburg and Trilling38
Simply supported
Clamped
(1)1
(2)
(3)
(4)
(1) / (4)
11.10 16.60 12.20 6.29 0.340 1.12
11.8 17.1 12.5 6.70 0.359 1.18
4.62 6.68 4.86 2.63 0.143 0.678
5.05 7.23 5.23 2.87 0.148 0.710
2.19 2.29 2.34 2.19 2.30 1.66
Column number
Table 8.12 Measurements of out-of-roundness
1 2 3
Height from base (mm)
Out-of-roundness (mm)
33 53 73
0.178 0.135 0.112
Eight strain gauges were attached to the inside of the vessel. They were equally spaced, and located at approximately mid-height, to provide values of hoop strain during the test. An end closure plate (Fig. 8.18 and Table 8.13, item 6) was fitted to the base of the vessel, and the top of the vessel was attached to the top of the tank by means of the sleeve (Fig. 8.18 and Table 8.13, item 7). A silicone RTV sealant was used at each of these joints to prevent water ingress. The whole vessel was then protected by a thin layer of polyurethane varnish, to ensure that water was not absorbed by the carbon fibre. The vessel was supported from the top of the tank, which averted any bending stresses owing to the effects of self-weight and/or buoyancy. The pressure measurements were made at the top of the pump, which was approximately at the same height as the top of the vessel. The pressure was increased in steps of 10 psi (0.6895 bar) and readings taken of each of the strain gauges. A reading of 100 psi (6.895 bar) was achieved, but failure occurred before the last three gauges could be read,
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Table 8.13 Parts list for assembly drawing Item No.
Description
Quantity
1 2 3 4 5 6 7
Tank Top plate Index tube Locking bolt Carbon fibre vessel Base plate (closure plate) Sleeve
1 1 1 1 1 1 1
2 1
3
Signal from frequency response analyser
4 To amplifier
To pump
7
5 6
8.18 Assembly drawing of test tank.
and so this was deemed the experimental buckling pressure. There was a 5 psi (0.3447 bar) drop in pressure as the vessel failed. The actual buckling pressure was extremely close to the predicted finite element values; the actual was between the ‘simply-supported’ (0.678 MPa) and the ‘clamped’ (0.710 MPa), but did not agree with the modified von Mises and DTMB formulae.
© Carl T. F. Ross, 2011
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Pressure vessels
8.2.5 Conclusions The FE program BCLAM was found to accurately predict the four lobe buckling mode of a corrugated carbon fibre vessel under external pressure. This failure mode was found experimentally, and strain gauge data shows this in Fig. 8.19. The predicted buckling pressure was higher than the experimental values, by approximately 20%. The actual buckling pressure was expected to be lower than the FE results, because the latter assumes perfect circular geometry, i.e. no out-of-roundness. Using the Tsai–Hill failure criteria and the composite beam analogy enabled the thinness ratio to be applied to the CFRP vessel. The Ross design charts, Fig. 3.4, were then used to compensate for the out-of-roundness using the plastic knockdown factor. The buckling pressure predicted by the finite element program then agreed with the experimental result to within a few percent. The treatment of the CFRP vessel as a compound beam enabled the von Mises and the Windenburg and Trilling formulae to predict buckling pressures. The two empirical formulae were consistent; similar values were predicted for each application of the formulae, and this included the CFRP vessel. The buckling pressures calculated for the latter vessel, however, were almost a factor of three higher than those generated by the finite element calculations; and these were close to the actual buckling pressure of 0.689 MPa as shown in Table 8.14.
Vessel 103 – buckling Compressive strain (in microstrain) 0 315
800
1.03 bar (15 psi)
45
600
2.07 bar (30 psi)
400 3.45 bar (50 psi)
200 90
270
4.83 bar (70 psi) 6.21 bar (90 psi) 6.90 bar (100 psi)
135
225 180
8.19 Graph of strain gauge readings for increasing pressure; the formation of four lobes can be seen.
© Carl T. F. Ross, 2011
© Carl T. F. Ross, 2011
Pressure (bar)
1.03
2.06
3.44
4.82
6.20
6.89
Pressure (MPa)
0.103
0.206
0.344
0.482
0.620
0.689
Gauge number:
100
90
70
50
30
15
Pressure (psi)
92
29
11
20
19
11
(μstrain)
0
θ (degrees)
1
226
255
189
162
97
54
(μstrain)
45
2
158
130
117
62
36
18
(μstrain)
90
3
Table 8.14 Strain gauge readings for increasing external pressure
790
521
385
251
153
85
(μstrain)
135
4
490
492
360
281
166
86
(μstrain)
180
5
0
913
612
406
228
117
(μstrain)
225
6
0
192
168
106
66
38
(μstrain)
270
7
0
648
485
323
192
103
(μstrain)
315
8
316
Pressure vessels
It can be seen from Table 8.11 that the von Mises and DTMB formulae disagreed with the respective finite element predictions for the hypothetical vessels (as well as for the CFRP vessel); the buckling formulae gave a factor of (just over) two for the finite element predictions for the hypothetical vessels. The increase from a factor of two, to a factor of nearly three (for the CFRP vessel) implies that inaccuracies were introduced by the calculation of the equivalent thickness. The results were encouraging for the finite element solution, but further testing is needed to confirm the trends identified in this study and to provide a measure of the likely scatter from a larger sample of experimental models.
8.3
Vibration of CFRP corrugated circular cylinder under external hydrostatic pressure
8.3.1 Experimental method A corrugated cylindrical pressure vessel was produced in carbon fibre;151 the vessel is shown in Fig. 8.20. It was a 0/90/0 lay-up, where 0° is defined to be parallel to the axis of the cylinder. Figure 8.11 shows the geometry of the vessel, including the details of the corrugations. The material used for the construction of this vessel was SE84HT/ HSC/200/400/37%, a unidirectional carbon fibre, used in pre-preg form. A series of tests were performed on unidirectional tension test specimens, in accordance with the CRAG procedures as previously described.149,150 The test pieces were made from the same batch of material used to produce the vessel, and provided the following material properties: E1 = tensile modulus in direction 1 (or x) = 130 × 109 Pa; E2 = tensile modulus in direction 2 (or y) = 10 × 109 Pa; G12 = rigidity modulus in the 1–2 plane = 7 × 109 Pa; ν12 = Poisson’s ratio in direction 1 owing to stress in direction 2 = 0.28. Excitation of the vessel was achieved using a magnetic transducer as an actuator. The input signal was supplied by a frequency response analyser (FRA), and the steel strip (item 11 in Fig. 8.21) bonded to the CFRP vessel responded to the changes in the magnetic field. The eight steel strips of dimension 0.0508 mm (0.002″) thick, 4 mm high shims, were bonded around the inner circumference to coincide with the transducer, as shown in Fig. 6.51. There was a small gap between each one, to minimise any local stiffening effect around the circumference). The response of the vessel to the input signal was measured by a Technimeasure 309A accelerometer fixed to the inner wall, and this was fed back to the FRA via a charge amplifier. The FRA displayed readings of the amplitude and phase angle characterising the vibration of the vessel. A useful graphical display of the response was
© Carl T. F. Ross, 2011
Vibration and collapse of novel pressure hulls
317
8.20 Corrugated carbon fibre cylinder.
provided by a cathode ray oscilloscope, which was connected across the output from the accelerometer. A diagram showing the connection of the electrical equipment is given in Fig. 6.47. Vibration testing was carried out in air first, and then under external water pressure. The procedure was similar for both tests. The index tube (Fig. 8.21, item 3) was rotated until it was in line with the accelerometer (Fig. 8.21, item 7). A frequency sweep was performed to identify the resonant frequencies. A resonance dwell test was then undertaken at each of these resonant conditions. This involved rotating the index tube through a small angle, and recording the amplitude and phase angle of vibration. These were then plotted to identify the mode of vibration. A typical phase angle plot is given in Fig. 8.22, and a typical displacement plot in Fig. 8.23. The phase angle plot exhibits three lobes, indicating that three lobes are in phase with the forcing frequency, hence n = 3. The amplitude plot, however, shows a peak value when at an anti-nodal point on the circumference,
© Carl T. F. Ross, 2011
318
Pressure vessels
2
3
To frequency response analyser
4
5
6
To amplifier
1
7
12
11 8
10 9 1. 2. 3. 4. 5. 6.
Pressure tank Top plate Index tube Locking bolt O-ring (small) nitrile O-ring (large) nitrile
7. Accelerometer, Technimeasure 309A 8. Carbon fibre vessel 9. Base plate (closure plate) 10. E-M transducer (large) 11. Steel strip 0.0508 mm (0.002″) thick 12. Sleeve
8.21 CFRP vessel in test tank.
whether fully in-phase or fully out-of-phase. Figure 8.23 displays six maxima, confirming that n = 3 here also.
8.3.2 Theoretical analysis The theoretical analysis was via the finite element method, where a pair of coupled matrix equations were merged together and then solved, as described in Section 6.1.
8.3.3 Finite element programs The theoretical analysis of the vibration of pressure vessels in air was carried out using a finite element program (VIBCONE).28 The program used a truncated conical, axisymmetric element with two nodal circles at each end; each node had four degrees of freedom. A reduction process devised by Irons147 enabled the size of the stiffness matrix to be reduced by a factor of about four, thus providing a very efficient program in terms of computer memory and speed. VIBCONE was modified to enable the materials matrix to be determined for laminated composites. The new program
© Carl T. F. Ross, 2011
Vibration and collapse of novel pressure hulls
319
Phase angle of vibration Freq. = 646.2 Hz, in water, P=0 350 0 10 340 20 30 330 320
40
300
50
310 200
300
60 70
290 100 280
80
270
90 100
260
110
250
120
240
130
230 140
220 210
150 200
190 180
170
160
Phase angle, in degrees
8.22 A typical n = 3 phase angle plot.
(VCLAM) enabled a vessel of up to 20 layers to be defined accounting for the material properties of each layer (E1, E2, G12 and ν12) and the direction of the fibres relative to the axis of the vessel. The element used to represent the fluid motion was the solid annular element shown in Fig. 6.1, which had a cross-section of an eight-node isoparametric form and one degree of freedom, namely the acoustic pressure. The program for isotropic materials, SUBPRESS, which uses axisymmetric structural elements and isoparametric fluid elements, was similarly modified to extend its use to orthotropic materials. This new program, SUBPRESC, enabled the vibration characteristics of carbon fibre vessels to be predicted for various constant external water pressures. The structure–fluid mesh used was of the form shown in Fig. 8.24.
8.3.4 Results The finite element predictions (via VCLAM) in air were determined for two cases of end conditions imposed by the closure plate. These were the
© Carl T. F. Ross, 2011
320
Pressure vessels Amplitude of vibration Freq. = 646.2 Hz, in water, P=0 350 0 10 340 20 330 30 320 40 300 310
50 200
300
60 70
290 100 280
80
270
90
260
100 110
250
120
240
130
230 140
220 210
150 200
190 180 Gap 1, B&K=1
170
160
Gap 2, B&K=3.16
8.23 A typical n = 3 amplitude plot.
Table 8.15 Comparison of the theoretical and experimental results in air Finite element program (VCLAM) Experimental results
Simply supported
Clamped
n
Frequency (Hz)
Frequency (Hz)
Frequency (Hz)
2 3 4 5 6
2409 1826 3377 5470 –
3130 2233 3174 4810 6897
3357 2308 3247 4876 6911
simply-supported and clamped ends, respectively. The actual end conditions for vessels with closure plates fitted and sealed with a silicone RTV sealant were believed to be somewhere between simply-supported and clamped. The results are listed in Table 8.15, and compared graphically with the experimental values in Fig. 8.25. Although there was some variation between the theoretical and experimental frequencies, Fig. 8.25 shows similar graphs © Carl T. F. Ross, 2011
Vibration and collapse of novel pressure hulls Structure
321
Fluid
R36.0 1
1
56
1
2
55 1
2
3
3
4
4
5
5
6
6
7
7
8
8
9
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
9
10
10
11
11
12
12
13
13
14
14
15
15
16
16
17
17
18
18
19
R37.25
54 2 53 3 52 4 51 5 50 6 49 7 48 8 47 9 46 10 45 11 44 12 43 13 42 14 41 15 40 16 39 17 38 18 R60.0
57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93
112 111 36 110 35 109 34 108 33 107 32 106 31 105 30 104 29 103 28 102 27 101 26 100 25 99 24 98 23 97 22 96 21 95 20 94 19
113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149
R83.5
8.24 Structure–fluid mesh.
for both. Furthermore, the graphs agree in that the lowest frequency occurs when three lobes are formed (n = 3). The results of the testing in water are listed in Table 8.16, and presented graphically in Fig. 8.26. The experimental and the theoretical results are in reasonable agreement. It can also be seen that as the external water pressure is increased, the values of the resonant frequencies decreased, except for n = 2, where the frequency rises slightly, to a peak at 1.72 bar (25 psi), before dropping off. The finite element prediction, however, shows the graph to be a continual decrease, albeit a very gentle one. All the finite element derived graphs occur at slightly higher frequencies than the corresponding experimental curves, indicating that the model is slightly stiffer than the actual structure. © Carl T. F. Ross, 2011
322
Pressure vessels Number of lobes formed at resonant frequencies 7000
Frequency (Hz)
6000 Finite element, simply supported
5000 4000
Finite element, clamped
3000
Experimental
2000 1000 0 2
3
5 4 Number of lobes, n
6
8.25 Graph showing the theoretical and experimental results in air.
Variation of frequency with pressure CFRP vessel 0/90/0 lay-up 1800
N = 5, FE
1600
N = 5, EXP
Frequency (Hz)
1400
N = 4, FE
1200
N = 4, EXP
1000 800
N = 2, FE
600
N = 2, EXP
400
N = 3, FE
200
N = 3, EXP
0 0
0.5
1 1.5 Pressure (bar)
2
2.5
8.26 Graph showing the theoretical and experimental results in water.
8.3.5 Conclusions The predictions of the vibration characteristics in air, of corrugated carbon fibre pressure vessels using the finite element program VCLAM agreed reasonably well with experimental results. The resonant frequencies were more pronounced when the vessels were immersed in water, making them easier to find. The experimental results showed that the resonant frequencies of corrugated carbon fibre vessels decreased dramatically when immersed in water. Furthermore, the frequencies of the various modes
© Carl T. F. Ross, 2011
© Carl T. F. Ross, 2011
645.5
1012
1595
3
4
5
1550
999
637
767
0.345
896.0
772.7
1073
1693
2
3
4
5
1673
1058
764.0
893.9
Finite element predictions from SUBPRESC
763.5
0
2
Experimental results
Number of lobes, n
Pressure (bar)
1653
1042
755.2
891.8
1532
987
636
774
0.689
1633
1026
746.3
889.7
1539
972
632
771
1.03
1612
1010
737.3
887.5
1514
955.5
625
780.5
1.38
1590
993.7
728.2
885.4
1498
940.5
617
783.5
1.72
1568
977.1
719.0
883.3
1482
923
607.5
782.5
2.07
1545
960.1
709.7
881.1
1471
906
598
787
2.41
Table 8.16 Comparison of the theoretical and experimental results in water, showing the changes in resonant frequency for increasing external pressure
1522
942.8
700.2
879.0
1455
884
587
779
2.76
324
Pressure vessels
continued to decrease as the external pressure was increased. This behaviour was predicted by the program SUBPRESC and, although the program gave resonant frequencies slightly (but consistently) higher than the experimental results, the overall correlation was good.
8.4
Vibration and instability of tube-stiffened axisymmetric shells under external hydrostatic pressure
The tube-stiffened circular cylinder54 has been discussed in Section 7.2.2. In 1990, Ross54 suggested that if the internal tubes were themselves subjected to internal pressurisation, that the subsequent reaction in the circumferential direction could place the pressure hull in an initial state of tension and, thus, in a state best able to resist the compressive forces, associated with external hydrostatic pressure. By subsequently raising the pressure within the internal tubes as the external water pressure rises, the hull can be thought of as proactively intelligent, being able to predict forces and parallel them with increasing diving depth. In this section, the buckling and vibration of the tube-stiffened dome of Fig. 8.27 is considered. The modes of vibration of thin-walled prolate domes tend to be of a lobar form and appear in the vessels’ flank, where the total number of observed waves in the circumferential are n and the total number of half waves in the meridional directions are m. These modes appear to be
8.27 Dome with stiffeners applied to internal surface.
© Carl T. F. Ross, 2011
Vibration and collapse of novel pressure hulls
325
of a similar pattern to that which can be observed when the vessel fails through static buckling. It was also observed that the eigenmodes for buckling and vibration became similar as the external pressure approached the static buckling pressure. This demonstrated that such vessels can suffer from dynamic instability, and that this instability can occur at a pressure much less than that required to cause static buckling. This appeared to verify the link between buckling and vibration, and this is why both have been presented in this work. Vibrations emitted from engines and other machinery can produce resonances that can induce failure of the vessel by way of dynamic buckling. As such modes are the forerunners to vessel failure, it is important for engineers to allow themselves an opportunity to observe the vessel’s responses at varying frequencies and pressures. It is also important to remember that vibrations in a military world can be detected and identified as a signature. With this in mind, research concerning the vessel’s responses to frequencies is of much importance. When considering the tube stiffened vessel, the possibilities of utilising the tubes on the internal hull to provide the submarine with a fuel cell and hence, replace the conventional engine with a low-noise system, is very innovative. Using a tube-stiffened vessel together with a fuel cell produces little traceable signature, and we have thus created truly the first real monster of the deep, a stealth submarine.
8.4.1 Construction of internally tube-stiffened domes The dome was pre-made and constructed in solid urethane plastic (SUP), with an aspect ratio 2, where the aspect ratio is described as the ratio between the dome height divided by dome base radius. The material properties of the dome were obtained and demonstrated by Ross and Popken152 on the buckling of tube-stiffened prolate domes under external water pressure. For the dome material: Young’s modulus, E = 2.9 GPa; density, ρ = 1200 kg m−3; Poisson’s ratio, ν = 0.3. The tubing was made from nylon and had the following properties and dimensions: internal diameter = 5.5 mm; external diameter = 9.5 mm; wall thickness = 2.0 mm; Young’s modulus, E = 70 MPa; 0.1% proof stress = 6.37 MPa; and nominal peak stress = 8.79 MPa. © Carl T. F. Ross, 2011
326
Pressure vessels
As can be seen from a comparison of the Young’s modulus, the tubing material was considerably less stiff than that of the domes. Therefore, the tubing could be considered as not completely satisfactorily stiffening the dome. It was decided that to bond the tubing to the internal surface of the dome; Evostick impact glue was to be used. The tubing would bond to the dome, to a depth of 285 mm, where this depth was measured perpendicularly from the dome base. As the tubing had previously been used for another experiment, the tubing had to be adequately cleaned to ensure that the correct surface was tubing and not old glue. Firstly, the glue was removed physically by a stripping action; this was found not only to be tiring, but impractical. To overcome this, the glue was removed by the application of acetone with a clean rag, resulting in break down and removal of the glue. To be sure of the complete removal of the old glue, a further application of acetone was made to the tubing just before the Evostick was applied; this may have been time consuming, but the benefits of getting it right first time more than justified the time spent. Following the method of Ross and Popken,152 a spare dome of equal dimensions was placed on its base and above the test vessel. The test dome was inverted and placed into a support in order to maintain its position during the application of the tubing. The tubing was then wrapped around the spare dome in order that the spare dome would act as a reel, and thus allow better control of the tubing (Fig. 8.28). The tubing was experimentally wrapped around the internal surface until one revolution had been completed. The beginning and the end of the revolution were marked with a pen to ensure that only a defined section had Evostick applied, correspondingly; the dome was also marked to provide a marked section for the impact glue to be applied. Upon the application of the glue to the dome and the tubing, both sections were left for 8 min to allow the impact glue to settle. The tubing was then applied to the dome and firmly held in place for 1 min. The bond was then allowed to set further for a period of 5 min. After which time the next section of tubing was measured out.
8.28 Dome and reel.
© Carl T. F. Ross, 2011
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It should be noted that, after the first section, all subsequent sections have to bond to the tubing directly above them. The first complete revolution of tubing was left for 24 h; this was to ensure that the bond with the dome was as secure as it could be, and hence, provide a good platform from which to build. At a distance of 114 mm from the dome lip, a gap in the tubing coil of about 38 mm was left; this was to allow a space to enable the vessel to be vibrated and the responses to be measured by an accelerometer. It was also later used for the application of strain gauges for the buckling of the vessel. The vessel was constructed over a week in an attempt to ensure a good solid construction (Fig. 8.28).
8.4.2 Experimental method Dome out-of-roundness The importance of initial geometrical imperfections within the dome can be seen when considering how imperfections can lead to the vessel failing plastically. When comparing this dome to others used before, the results show that this vessel shows only positive imperfections. The out-of-circularity measurements were made at 30° intervals, with point 0° being the datum point, around the circumference where it was expected that the vessel would fail. Figure 8.29 shows the plot obtained around the vessel’s circumference using a dial gauge and a 3 jaw chuck.
1 1.2 11
2
1 0.8 0.6
10
3
0.4 0.2 0 4
9
5
8
7
6
8.29 Out-of-roundness plot (mm).
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Buckling experiment The internal tubing was subjected to an internal water pressure of 13.8 bar and the strain output from the ten circumferential strain gauges was recorded. Maintaining this internal tubing pressure, the external water pressure around the dome was increased until the dome buckled. Many strain recordings were taken. An experimental plot of the variation of strain around the circumference for various values of external water pressure is shown in Fig. 8.30, which also shows that buckling was of a lobar nature. This lobar pattern was exaggerated with an increase in external water pressure. When the tubing was first pressurised, the vessel was placed into a state of tension; this is noted by the positive response on the axis. The graph also describes that the vessel buckled with n = 5, where this was calculated from the angle of a peak to an adjacent peak. This observed angle was divided into 360°, and gave the total number of waves in the circumference n. From Fig. 8.30, the largest strain observed was at gauge six. When compared with the buckling pressure of the vessel at 3 bar, the failure of the vessel was clearly near gauge 6. From video evidence, the vessel did clearly fail near gauge 6; it can also be seen that, from Fig. 8.30, the deformation in the vessel was most clear near gauge 6 throughout all the readings taken. Vibration tests The tube-stiffened dome was placed into the tank where both the vibration and buckling of the vessel were to be undertaken. The dome was secured 500 0 0
100
200
300
–500
400
Internal tube pressure 13.8 bar 1
Microstrain
–1000
bar
1.5 bar
–1500
2
–2000
2.2 bar
bar
2.4 bar
–2500
2.6 bar
–3000
2.7 bar
–3500
2.9 bar
–4000 Gauge position in degrees
8.30 Experimental results for the buckling of the tube-stiffened dome.
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in the vessel by way of 24 studs and nuts, and one rubber gasket. To mechanically vibrate the dome, the two mechanical vibrators were to be set 180° apart. This positioning was achieved by the holder design described above, and it was believed that this design would not significantly dampen the frequencies applied to the dome (Fig. 8.31). The design of the holder utilised the studs on the pressure vessel for locating, and providing stability for the vibrating modules. The modules were pre-attached to a mount, which required the holder to provide locating holes for the modules to be securely attached. The modules had a 4 mm threaded section to allow for inserts. It was these inserts that when threaded into the modules, were then applied to the dome surface to vibrate the dome. The inserts selected were metal knitting needles, which were cut down to total length of 110 mm, with a 10 mm long by 4 mm diameter thread section at one end. To each end, a single sucker (of the sort of sucker readily obtained at pet stores and used in fish tanks) was fixed. The equipment used to vibrate the dome and measure the responses, shown in Fig. 6.47 were as follows: (a)
Frequency analyser (FRA): used to input the signal at the correct frequency, and show the output of the vessel pertaining to the input. (b) Frequency amplifier: used to amplify the signal from the FRA and input it into the vibrating modules via the switch box. (c) Switch box: attached by way of wiring to the vibrating modules and the amplifier; two settings are available, odd and even. These were used in the corresponding search for odd and even values of the circumferential waves.
8.31 Dome set-up with holder.
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(d) Accelerometer charge amplifier: the accelerometer was wired to its charge amplifier which was in turn wired to the FRA, as the accelerometer measures the responses of the dome. (e) The test tank (Fig. 8.32). Vibration of the dome in air was not very successful, but vibration of the dome in water was highly successful. The dome was vibrated in water under the following conditions: (a)
With water on the outside surface of the dome and with the inside containing water at zero internal pressure. (b) With the domes subjected to various values of external water pressure from 0 to 2 bar, and the internal tubing subjected to an internal water pressure of 10 bar. Observations of vibrating eigenmodes Typical plots of some of the experimentally observed circumferential eigenmodes in vibration showed that they were of a circumferential nature.
4
5
6
7
8
3 9
2
10 11
12 1 13
8.32 Large tank arrangement with vessel secured for vibration testing: 1, water pressure; 2, pressure tank; 3, rubber gasket; 4, nut; 5, bolt; 6, vibration modules; 7, vibration module holder; 8, top ring; 9, tap for bleed valve; 10, bleed valve; 11, vibration applicators; 12, tube-stiffened dome; 13, pump valve.
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8.4.3 Theory The theoretical analysis was based on the finite element, where the truncated varying meridional curvatures (VMC) axisymmetric element, as shown in Fig. 3.8, was used to model and represent the dome used. Vibration theory The element used to model the fluid was a solid annular one, whose crosssection was of eight-node isoparametric form. Buckling theory The geometrical stiffness matrix [KG] and the method used was the same as that previously described.50
8.4.4 Comparison between theory and experiment Vibration Using bending theory for compound beams,1,2 an equivalent thickness for the tube-stiffened dome was obtained. The equivalent thickness of the dome was found to be 5.15 mm, an increase of 1.15 mm over the original unstiffened SUP dome. The equivalent density of the tube-stiffened dome was found to be 2039 kg m−3. Comparison between theory (T) and experiment (E) is shown in Table 8.17, for various values of external hydrostatic pressure. It can be seen from Table 8.17, that there is good agreement between experiment and theory and also that as the external water pressure is increased the resonant frequencies decrease. Figure 8.33 shows experimental results for frequencies against ‘n’ for various values of external water pressure. Table 8.17 Theoretical (T) and experimental (E) resonant frequencies (Hz) for tube-stiffened dome n=2 Pressure T (bar) 0 1 1.5 2
n=3 E
T
n=4 E
T
113.3 105.0 124.4 117.0 136.8 113.2 124.0 110.0 136.0 113.1 123.8 110.0 135.6 113.1 123.6 107.0 135.2
n=5
n=6
E
T
E
T
E
137.0 127.0 124.0 120.0
153.8 152.3 151.6 150.9
– 154.0 146.0 141.0
177.8 175.7 174.6 173.5
209.0 189.0 184.0 178.0
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Pressure vessels n = 4, internal pressure 10 bar, external pressure 0 bar, 135 Hz
34
35
36 150
2
3 4
33
5 100
32
6 7
31 30
8
50
29
9
28
10
27
11 12
26
13
25 14
24 15
23 16
22 21
20
19
18
17
8.33 Comparison of (n) value under external pressures, and internal tube pressure.
Buckling Comparison between experimental results for various domes is shown in Table 8.18. The theoretical buckling pressure for JKE1 was 3.23 bar and that for L1 and L2 was 2.02 bar (8), where the figures in parentheses represent the number of lobes n. It can be seen from Table 8.18, that the effect of the tubes was to increase the experimental buckling pressure by 35.5%, and the additional stiffening effect of internal tube pressure, resulted in a further increase of experimental buckling pressure of about 9.1%. Conclusions The vibration of the stiffened tube showed how the resonant frequencies decreased with increase of external water pressure. It also showed that, as the static buckling pressure was approached, there was a possibility that the vessel could fail through dynamic buckling, at a lower pressure than that which would have caused static buckling. The results show that an internal tube pressure of 10 bar, did not affect the resonant frequency distribution to any great extent. It must be
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Table 8.18 Buckling pressures of domes (bar) Model
Stiffening
Internal tube pressure
Experimental buckling pressure
L1 L2 500/2/9 500/2/15 JKE1
None None Tubing Tubing Tubing
– – 0 8 13.8
2.14(8)a 2.03 2.75 2.9 3.0
a
The number of lobes is given in parentheses.
remembered that the Young’s modulus of the tubing was only 0.024 of that of the dome. The results do suggest that tubing with a Young’s modulus that approaches that of the domes, would have better served this experiment. As the results shown in Fig. 8.33 showed a slight change when an internal pressure was applied, it suggests that the idea of tube stiffening in the circumferential direction is worth exploring further. Before the vessel was buckled, an estimation based on previous investigations as to where the vessel would fail, suggested that the vessel would buckle in its flank, at a perpendicular distance from the base of between 101.6 and 127 mm. It was at this position that the gap in the tubing was located. It was assumed that the risk to an early rupture of the dome could be avoided by limiting the vibration gap to 38.1 mm. The theoretical analysis predicted that the best estimation for the buckling pressure that could be obtained was 3.23 bar with a lobar pattern of n = 7. The experiment showed that the vessel ruptured at 3 bar. This was an improvement on previous year’s work by 0.1 bar. This was not a large increase until it was realised that the Young’s modulus of the tubing is only 0.024 of the SUP domes; rather like attempting to structurally stiffen steel with wooden stiffeners. Once the rupture had occurred, it was clearly visible that the tubing was still intact, and thus had not been flexed off the vessel during the experimentation. The experimentation was recorded on video and provided the authors with a second and closer examination of the buckling behaviour. The recording clearly showed that the rupture occurred in the flank of the vessel, followed by rapid failure around the lip. This demonstrated that the pressure was seeking the weakest point in the structure, which was located within the 38.1 mm gap. When referring to the plotting of the buckling results, it can be clearly seen that the highest strain recorded was around gauge six, and was around the 215 degree mark. The vessel had its internal tubes pressurised to a pressure of 13.8 bar before buckling; the effects of this can be viewed on the plotting of results
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in Fig. 8.33. To this end it can be observed from the results that this high pressure resulted in only a small dome deflection; this again showed how ineffective the tubing was at translating its internal pressure to an effective resistance to the external hydrostatic pressure. This backs up the results and conclusions for the vibration work. The results generated by the out-of-roundness testing, showed that because the vessel was geometrically imperfect, these imperfections grew as the vessel was subjected to compression. This caused the vessel to begin to fail plastically. Thus, the vessel failed by plastic instability, and this could have been another reason why the vessel failed at a pressure lower than that indicated by the theoretical analysis. The vibration experiments were identified as a success as they consistently followed the theoretical analysis. It must be noted that the theory was conducted after all vibration results were completed and compiled, and left no room for the results to be manoeuvred furthermore. The results showed that to successfully gain any credible results the dome must be placed in water, as this had the effect of lowering the respective frequencies, whilst distributing them over a wider frequency band. This was also backed up by the theory, as it indicated that the results in air are within a much shorter frequency band and cancel each other out. In water, the bandwidth is sufficient to obtain credible results as shown by the results obtained experimentally.
8.5
Collapse of dome cup ends under external hydrostatic pressure
In 1987, Ross132 (Section 7.1), presented a radical design for submarine dome ends. He suggested that the dome ends should be inverted, so that they are concave to the effects of pressure, as shown in Fig. 8.34; he named these ends, dome cups. It can be seen from Fig. 8.34 that the hydrodynamic efficiency of the submarine is not decreased. Additionally, it can be seen from Fig. 8.34 that the dome cups are largely in tension and unlikely to buckle. Ross’ mathematical models of equivalent dome caps and cups showed, when the latter were adopted, that there was a weight saving of over 30%. Additionally, Ross argued that as the dome cups were unlikely to buckle, they can be made less precisely, thus making a further saving in manufacturing cost. It should be emphasised that geometrically imperfect axisymmetric shells become more perfect with increasing internal pressure, which is effectively the loading condition of the dome cup. This is in direct contrast to the geometrically imperfect dome cap, which becomes more geometrically imperfect with increasing external hydrostatic pressure, thus resulting in lower values of experimentally obtained buckling pressures.
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Water pressure
335
Cup end
Water pressure Atmospheric pressure
Axis
Cup end Pressure hull
8.34 Dome cup ends.
In 1995, Khan and Uddin,153 correctly criticised Ross’ design by stating that high stresses would occur at the dome–cylinder joint. However, Ross had stated as much in his 1987 paper and had recommended that the wall thickness of the vessel should be increased at the dome–cylinder joint. All the above studies were of a theoretical nature. To rectify this deficiency, Ross and Rotherham154 conducted an experimental study on nine hemi-ellipsoidal domes, which varied in shape from flat oblate domes (Fig. 3.13), with an aspect ratio (AR) of 0.25, to very tall domes with an AR of 4, where AR =
dome height base radius
The domes were subjected to uniform hydrostatic pressure on their concave faces and the experimental results revealed that the strongest dome cups were of extreme aspect ratio, the hemispherical dome cup having poor resistance to ‘internal’ pressure. As the adoption of prolate dome cups results in a loss of internal space in the submarine pressure hull and as the flat oblate dome was just as strong as the tall prolate dome, it was decided to study a flat oblate dome of AR = 0.25. Four dome cups and one dome cap were made from fibreglass.155
8.5.1 Manufacturing glass-reinforced plastic (GRP) dome cups The mould was made by joining together a machined wooden annulus and an accurately made plastic dome,156 the two being joined together with plasticine; the mould is shown in Fig. 8.35. The plasticine was smoothed to form a gentle contour at the dome–cylinder joint. This latter process was carried out to reduce the stress concentration at this joint. The mould was waxed with beeswax and left for about 10 min before being ‘buffed up’ with
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Pressure vessels
8.35 Mould for the dome cups.
a clean dry cloth. Two more layers of beeswax were applied in a similar manner and then the release agent was applied, followed by a gel coat. Before applying the gel coat, 5% of a pigment was added to it, together with a 2% of a catalyst; the whole mixture was mixed thoroughly and left to stand for about 5 min. The resin that was applied to the glass fibre mat also required 2% of a catalyst mixed into it. A layer of resin was placed on top of the dry gel coat and then a layer of mat was placed on top, where the latter had to be prewetted with a resin. The mat was a random strand mat. Care was taken to remove any trapped air bubbles. Further pre-wetted layers of mat were applied to lower layers of wet mat. Excess matting around the edges was trimmed and the glass fibre dome was allowed to dry fully over a period of 24 h or more. The dome cup was then removed from the mould, as shown in Fig. 8.36. The flanges of the dome cups were machined smooth and flat so that they could be made pressure-tight when connected to the top of the testing tank. The domes were then varnished with a marine varnish to make them watertight. It must be emphasised that glass fibre matter was porous and, because of this, the domes had to be varnished. Details of the domes are shown in Table 8.19, and in Figs 8.37–8.40, where it can be seen that AGC1 and AGC3 had one additional layer of glass fibre on the internal face of the dome– cylinder joint, whereas AGC2 and AGC4 had no additional layers of matting at this joint. The major difference between AGC4 and the other models was that it was the only model that had a section of nylon tubing incorporated into the
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8.36 A dome cup.
Table 8.19 Details of the dome cups Model
Number of layers
Condition of dome–cylinder joint
AGC1 AGC2 AGC3 AGC4
3 2 2 2
Reinforced with an extra layer of matting Not reinforced Reinforced with an extra layer of matting Tube stiffened, but not reinforced
201.6
8.5
25.94
5.5
4.5
15∅
8.37 The dimensions of AGC1 (mm).
vessel. It can be seen from Fig. 8.40 that the tubing was located between two layers of fibreglass at the junction between the cylinder and the dome. The internal diameter of the nylon tubing was 4.63 mm and its external diameter was 8 mm.
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Pressure vessels 4.45
3.09
25.94
3.35
15∅
179∅
8.38 The dimensions of AGC2 (mm).
6.72
3.41
24.9
4.02
15∅
176.8∅ 201.4∅
8.39 The dimensions of AGC3 (mm).
10.5
2.89
27.6
2.83
15∅
179.7∅ 201.3∅
8.40 The dimensions of AGC4 (mm).
8.5.2 Experimental process Tensile tests were carried out on several fibreglass specimens of varying geometric properties. The first three specimens, a, b and c, were made with two layers of random strand mat and a final layer of gel coat. The second three specimens, d, e and f, were made from three layers of random strand mat without a top layer of gel coat. The final three specimens, g, h and j were made using four layers of random strand mat and a top layer of gel coat. The thickness (T) of each of the specimens was measured at several positions (Tables 8.20–8.22). Specimen j tapered a little and therefore a fourth
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Table 8.20 Two-layer specimens Thickness T (mm) a b c
2.39 2.40 2.47
2.61 2.42 2.47
Average T (mm) 2.42 2.85 2.47
2.47 2.47 2.47 2.47
Table 8.21 Three-layer specimens Thickness T (mm) d e f
2.97 3.23 3.15
3.08 3.17 3.09
Average T (mm) 3.26 3.17 3.18
3.10 3.19 3.14 3.14
Table 8.22 Four-layer specimens Thickness T (mm) g h j
3.80 3.83 3.65
3.81 3.96 3.46
Average T (mm) 3.45 3.61
3.26
2.85
3.68 3.80 3.31 3.60
measurement was taken. Each of the specimens had a gauge length of 200 mm and a width of 20 mm. The data obtained from the tensile tests were plotted and an example for each of the different numbers of layers is shown in Figs 8.41–8.43. The experimentally obtained tensile moduli and fracture stresses are shown in Table 8.23, and the average values are given in Table 8.24. On inspection of these results, it can be seen that both the fracture tensile stress and the tensile modulus increase as the number of layers increase. It should also be noted that the difference in values between the three-layer sample and the four-layer sample is quite considerable. This is particularly true when it is compared with the difference in values between the three- and the twolayer samples. This significant difference may occur since the three-layer samples did not have an extra top layer of gel coat applied to them, as the other samples did.
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Pressure vessels 6000 Load (N)
5000 4000 3000 2000 1000 0 0
2
1
3 4 Extension (mm)
5
6
7
Load (N)
8.41 Plot to show load versus extension for specimen ‘a’ of two layers. 8000 7000 6000 5000 4000 3000 2000 1000 0 0
2
1
3 4 Extension (mm)
5
6
7
8.42 Plot to show load versus extension for specimen ‘e’ of three layers.
Load (N)
10 000 9000 8000 7000 6000 5000 4000 3000 2000 1000 0 0
1
2
3 4 5 Extension (mm)
6
7
8
8.43 Plot to show load versus extension for specimen ‘g’ of four layers.
For each of the tensile specimens the volume fraction was calculated by using the equation: ∑t Vf = T where t represents the average thickness of one layer of random strand mat fibreglass and T represents the average thickness of the sample. This
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Table 8.23 Material properties
Number of layers
Specimen
Tensile modulus, E (GN m−2)
Fracture tensile stress, σy (MN m−2)
2 2 2 3 3 3 4 4 4
a b c d e f g h j
4.18 3.90 3.90 4.03 4.65 3.66 4.52 4.26 4.84
108 83.7 94.4 91.7 113 106 122 172 105
Table 8.24 Average material properties Number of layers
Average E (GN m−2)
Average σy (MN m−2)
2 3 4
3.99 4.11 4.54
95.4 104 133
Table 8.25 Volume fractions Number of layers
Average volume fraction Vf
Average resin used (%)
2 3 4
0.61 0.72 0.84
39 28 16
calculation establishes the fraction of fibreglass matting used in the making of each sample compared with the amount of resin. Table 8.25 shows the calculated average volume fraction for the various layered samples. From this table, it appears that the highest percentage of resin was used when the two-layer models were made. However, it must be noted that, because of the random orientation of the matting, the random strand mat is thicker when measured dry. Once this random mat is wetted with resin, the resin is able to penetrate these gaps. It is this extra resin that is not accounted for. Similarly, the first layer of mat always has more resin used in wetting, as it is being laid on a dry surface of gel coat, as explained earlier.
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8.5.3 Experimental tests In order to make the joint between the dome flange and the tank top watertight, it was necessary to machine flat the flanges of the dome cups. This had the effect of making the flange somewhat thinner than the dome wall, and thus, a point of weakness. Figure 8.44 shows a schematic drawing of a typical dome cup attached to the test tank. The tank and the dome cup AGC3 are shown in Fig. 8.45. Before testing could begin, the air in the tank had to be pumped out. This was done by opening the bleed valve and pumping water into the tank until the water could be seen flowing out of the bleed valve. Once the air had been evacuated from the tank, the valve was closed.
8.5.4 Experimental results After testing, the vessels were inspected for the location and the types of failure; Fig. 8.46 shows the vessel AGC1 after failure. The inspection was made easier by the use of a camcorder. Vessel AGC1 failed around the flange; this meant that the cup–cylinder part of the vessel could have withstood more pressure. The vessel AGC2 failed at the dome–cylinder stress concentration, and Fig. 8.47 shows this vessel after testing. The vessel AGC3
8.44 Test tank and dome cup.
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8.45 The test tank with the vessel ACG3 secured for testing.
8.46 Vessel AGC1 after testing with rupture around the flange.
failed around the weakened flange section, which meant that the cup– cylinder section could withstand an even greater pressure. The vessel AGC4 failed in the area of the cup–cylinder stress concentration. It was believed that this may have been triggered by the tubing that was incorporated within the structure at this position. The point at which the rupture occurred may have been where the ends of the tubing were.
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8.47 Vessel AGC2 after testing with rupture at the dome–cylinder joint.
The vessel AGC2 failed in the predicted position of the dome–cylinder joint and thus substantiated the claim that there was a high stress concentration in this area, as there was no additional mat reinforcement. The pressure at which the vessel AGC1 failed was 14.13 bar. On first inspection it was thought that the cylindrical section buckled, which caused a circumferential rupture, opposite the line of strain gauges. It was later believed that the circumferential rupture around the weakened flange was caused by the high stress concentrations. This was probably because the flange was skimmed to make sure it lay flat in the tank and, as a consequence, the flange section was thinner than the dome itself. The pressure at which the vessel AGC2 failed was 5.52 bar. It cracked between the cylinder and the dome at the expected area of stress concentration; this vessel was not reinforced with additional mat at this point. The vessel AGC3 failed at a pressure of 8.14 bar. This vessel ruptured around the weakened flange for the same reasons as the model AGC1. The vessel AGC4 failed at a pressure of 10.0 bar. On first inspection, it was believed that the vessel began to fracture around the cylinder and flange, which then caused a radial crack emanating from the connection point between the cylinder and the dome. However, on inspection of the camcorder recording of the collapsed vessel, it was discovered that the vessel began to crack around the dome–cylinder connection joint, the failure emanating from this point. It could also be seen that the tubing was beginning to be forced up within the structure, gradually deforming the vessel. The best way to really appreciate the results obtained from the testing of these dome cups, which were concave to pressure, is to compare them with a similar conventional dome cap that was convex to pressure. Table
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Table 8.26 Collapse pressures Numbers of layers of fibreglass
Dome type
Pressure at collapse (bar)
4 3 (AGC1)
Convex Concave
8.27 14.13
2 (AGC2)
Concave
5.52
2 (AGC3)
Concave
8.14
2 (AGC4)
Concave
10.00
Comments Buckled in dome Failed in flange and not in dome Failed at dome– cylinder joint Failed in flange and not in dome Failed at dome– cylinder joint
8.26 shows the result from the testing of a conventional dome cap which was convex to pressure; this showed a buckling pressure of 8.27 bar. It must also be noted that Ross et al.’s dome157 was made using four layers of fibreglass, whereas the dome cups reported here had fewer layers. When this dome cap collapse pressure is compared with that of the three-layer dome cup AGC1, which failed at 14.13 bar, it can be seen how efficient these dome cups were. This was particularly true when considering that the dome itself did not fail on this vessel, but instead the weakened flange. The other dome cups also had favourable collapse pressures despite having only two layers of matting. It can be seen from Table 8.26 that even the vessel AGC2, whose thickness was not increased at the joint, compared favourably with the four-layered convex dome cap. The experimental strain values obtained from the gauges were plotted against increasing external hydrostatic pressure and the results showed that the strain at fracture of the tensile specimens was considerably larger than the measured bulk strains in models AGC1 to AGC4. The results of the readings taken from the strain gauges appeared to show that different results were occurring for each of the vessels. For AGC1, the gauge results appeared to suggest that the centre T-stack gauges, although increasing as the pressure increased, were not really being significantly affected. Results for vessel AGC2 appear to show an increase of strain that was quite linear with increase in pressure. From the results obtained for the vessel AGC3, it appears that the vessel was behaving in a linear fashion and that little deformation could be identified within the main body of the structure. This could be interpreted that, as the flange of the vessel was
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Pressure vessels
structurally inefficient, the cup–cylinder itself was not close to failure. The results for vessel AGC4 appeared to show that the structure was undergoing major deformation. This can be confirmed from an inspection of the video recording taken. It could clearly be seen that the structure was changing in appearance. This may be attributed to the tubing, which was incorporated within the structure.
8.5.5 Conclusions This investigation has shown that the dome cup ends were structurally more efficient than the dome cap end. In fact, the dome cup AGC1 had a collapse pressure some 70% larger than the conventional dome cap, despite the fact that the latter was some 25% thicker than AGC1. Additionally, AGC1 collapsed at the flange, which was weakened through machining, and not in the dome itself; hence its collapse pressure could have been even higher. The two-layer dome cup AGC2 had fairly good resistance to external pressure, despite the fact that the dome–cylinder joint was not reinforced with additional matting. When this was done for the two-layer dome cup AGC3, the increase in experimental collapse pressure was found to be about 48% higher than that of AGC2. The two-layer dome cup AGC4, which instead of having additional matting at the dome–cylinder joint had an annulus of nylon tubing placed between its two layers of matting, exhibited an increase in collapse pressure of about 81% over that found for AGC2.
8.6
A redesign of the corrugated food can
During the canning process of certain foods,158 it is necessary to place the hot food into a food can and then seal it. When the food cools down, it exerts an internal vacuum in the food can. If such vessels are not ringstiffened, they can fail because of shell instability, caused by this vacuum at pressures that may only be a small fraction of that required to cause axisymmetric yield, as shown in Fig. 8.2. Placing ring stiffeners on a tin can is prohibitively expensive, and one method of improving the circumferential stiffness of these vessels is to circumferentially corrugate them, as suggested by Ross for submarine pressure hulls. For food cans, a typical corrugated configuration is shown in Fig. 8.48. Under an internal vacuum, the failure mode of these vessels is as shown in Fig. 8.2. This mode of failure is known as general instability. This investigation shows that by increasing the cone angle of these vessels to certain optimum values, the buckling resistance of these vessels, under internal vacuums, is considerably increased. The four types of model that were investigated were known as the CA series, the DF series, the MBS series
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8.48 Corrugated food can.
Bulkhead
Bulkhead
Swedges
Lf
Ls
R
RE
RI
Flat
ls
Lf
Axis Lb
8.49 Geometrical details of food cans.
and the MBL series, and details of these models are shown in Fig. 8.49 and Table 8.27.
8.6.1 Material properties The material properties in this study were: E = Young’s modulus = 2 × 105 MPa; v = Poisson’s ratio = 0.3 assumed; σyp = yield stress = 300 MPa for CA series; = 645 MPa for DF series; = 422 MPa for MBS series; = 484 MPa for MBL series.
© Carl T. F. Ross, 2011
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Table 8.27 Geometrical details of vessels (mm) Series
t1
Lf 2
R3
N4
ls5
Ls6
Lb7
RI8
RE9
CA DF MBS MBL
0.35 0.14 0.24 0.27
19 6 5 5
76.36 35.54 49.53 49.55
20 17 14 28
6.9 3.8 3.79 3.8
139.1 68.06 56.32 109.95
177.1 80.06 66.32 119.95
75.58 36.08 49.16 49.12
76.36 36.58 49.59 49.58
1
Shell thickness (measured over flat section of cylinder); 2 length of flat section of cylinder; 3 mean radius of flat section of cylinder; 4 number of swedges; 5 length of individual swedge; 6 length over which there are swedges; 7 distance between bulkheads; 8 mean internal radius of swedge; 9 mean external radius of swedge.
Bleed screw
M10 bolt locating cylinder
Bolts (24) securing pressure vessel lid
‘O’ seals Atmospheric pressure
Water under pressure from hand pump (pressure gauge situated on pump)
Test specimen
8.50 Model in test tank.
8.6.2 Experimental method Three models, per series, were tested to destruction in the test tank shown in Fig. 8.50. From Fig. 8.50, it can be seen that the food cans were subjected to uniform external water pressure. A typical profile of a corrugation is shown in Fig. 8.51, where it can be seen that the profile was of a sinusoidal nature. In the computer analysis, the corrugations were assumed to be of the saw tooth form; previous work on this topic showed that this assumption was quite reasonable. The metrological measurements on the cans were made with a Mitutoyo BN 706 co-ordinate measuring machine and a Renishaw PH9A
© Carl T. F. Ross, 2011
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2.0
mm
1.5
1.0
0.5
0.0
0
1
2 3 x ordinate (mm)
4
5
Swedge profile
8.51 Typical profile of a corrugation for the DF series. Table 8.28 Ovality measurement Series
Out-of-roundness e (mm)
e/t
CA DF MBS MBL
0.2996 0.0574 0.0624 0.1377
0.74 0.41 0.26 0.51
touch-trigger probe. Average values for the out-of-roundness of each series of vessels is given in Table 8.28, where t = wall thickness. To determine the out-of-roundness, a least-squares’ circle was first obtained for each vessel at its midspan. The out-of-roundness value was defined as the maximum difference between the innermost and outermost points on the cylinder from this least-squares’ circle. The ends of each tin can had a push-fit joint, as shown in Fig. 8.48. These joints were waterproofed with the aid of a typical household silicone sealant. It was likely that the end conditions of the vessels were somewhere in between clamped and simply-supported. All 12 vessels failed by general instability, as shown in Fig. 8.52. The experimentally obtained buckling pressures Pexp are given in Table 8.29.
8.6.3 Theoretical analysis The theoretical analysis was carried out with the aid of the finite element conical element of Ross.50 This finite element had four degrees of freedom
© Carl T. F. Ross, 2011
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Pressure vessels
8.52 General instability of circular corrugated cylinders (left to right: CA, DF, MBS and MBL).
Table 8.29 Experimental buckling pressures
Model
Pexp (psi)
Pexp (bar)
CA1 CA2 CA3 DF1 DF2 DF3 MBS1 MBS2 MBS3 MBL1 MBL2 MBL3
31 36 34 44.96 42.06 47.86 57.21 47.86 60.92 40.6 36.8 35.8
2.12 2.48 2.34 3.1 2.9 3.3 3.6 3.3 4.2 2.8 2.5 2.4
Average Pexp (bar)
2.32
3.1
3.7
2.6
per node, making a total of eight degrees of freedom per element, as described in Section 3.2. In the derivation of the stiffness matrices, it was assumed that the w displacement was of cubic form along the meridian of the element and the u and v displacements varied linearly along this meridian. In the computer analysis, all vessels were assumed to be fixed at their top ends and clamped at their bottom ends. By the term clamped, it was assumed that the displacements v0, w0 and θ were zero, but the displacement u0 was free to move axially. It was likely, however, that the ends of the food cans were only partially fixed. The computer program, which used the element of Fig. 3.8 for the analysis, was called SWEDBUCK. It was written in QuickBASIC, and it used the continuous reduction technique of Irons,147 which drastically reduced the number of degrees of freedom remaining.
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Most of the vessels failed inelastically, partly because of their geometries and partly because of the magnitudes of their initial out-of-roundness values. To cater for this, the thinness ratio λ′ was used; this thinness ratio was produced by Ross and Palmer,140 and it was based on the thinness ratio of Windenburg and Trilling.38 The thinness ratio of Windenburg and Trilling was intended for use on unstiffened circular cylinders; it is defined as:
λ′ =
4
( ⎡⎣(L / D )
1 2
b
)
3 / (t 1 / D1 ) ⎤⎦ ×
(σ yp / E )
where D1 = RI + RE; t1 = equivalent wall thickness of shell; σyp = yield stress; E = Young’s modulus. Comparison between experiment and theory is given in Table 8.30, together with values of the plastic knockdown factor, namely Pcr/Pexp, for each of the four series of vessels, where n represents the number of lobes the vessel buckles into. From Table 8.30, it can be seen that as the DF series has a large value for λ′ and a small value for e/tl, the elastic knockdown factor was only 1.15, the reverse applies for vessels with small λ′ and large e/tl. That is, if a vessel is ‘slender’ (large λ′) and has a small e/tl, its plastic knockdown is small. If, however, a vessel is ‘not slender’ (small λ′) and has a large e/tl, its plastic knockdown is large. Other combinations of λ′ and e/ tl, give intermediate values of plastic knockdown and much more experimental work is required. The design process has to calculate Pcr and λ′ for a particular vessel and then, by taking into consideration the value for e/tl, the plastic knockdown factor, namely Pcr/Pexp, is obtained from the design chart, Fig. 8.10, and divided into Pcr to give the inelastic buckling pressure of the vessel. This buckling pressure must then be divided by a large safety factor to give the design pressure. It is considered that the nonlinear finite element of Bosman et al.159 and the nonlinear finite difference method adopted in BOSOR569 are
Table 8.30 Comparison between experiment and theory Series
λ1
e/t
e/t1
Pexp (bar)
Pcr (n) (bar)
Pcr/Pexp
CA DF MBS MBL
2.73 3.57 2.67 3.53
0.74 0.41 0.26 0.51
0.46 0.185 0.185 0.365
2.34 3.1 3.7 2.6
3.38 3.58 5.97 4.39
1.45 1.15 1.61 1.69
© Carl T. F. Ross, 2011
(5) (4) (7) (5)
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Pressure vessels
unsuitable for the present series of vessels, because their out-of-circularity is random, whereas the assumptions made in the non-linear finite element and finite difference methods, is that the out-of-circularity is axisymmetric in nature.
8.6.4 Redesign of food cans The structural efficiency of these vessels may be improved by increasing the cone angle (α) of the corrugations. If this is achieved, there can be considerable weight and cost savings for these vessels. Apart from savings in material costs, the new designs could be said to be more environmentally friendly than those used at present, because less material is being used. The computer program SWEDBUCK was used to analyse all four tin cans. All the dimensions of the vessel were kept the same, except for the cone angles, which were increased in steps of up to a value a little larger than 45°. Plots of the theoretical buckling pressures against various values of cone angle are given in Figs 8.53–8.56. From Figs 8.53–8.56 it can be seen that all four vessels had a maximum buckling resistance at cone angles
10
18
9
16
8
14
7 12 Pcr (bar)
Pcr (bar)
6 10 8
5 4
6 3 4
Current vessel
2 Current vessel
2
1
0
10
20 30 40 50 Cone angle a (degrees)
8.53 Variation of Pcr with α for CA.
0
60
10 20 30 40 Cone angle a (degrees)
8.54 Variation of Pcr with α for DF.
© Carl T. F. Ross, 2011
50
22
22
20
20
18
18
16
16
14
14 Pcr (bar)
Pcr (bar)
Vibration and collapse of novel pressure hulls
12 10
353
12 10
8
8
6
6 Current vessel
4
4
2
2
0
0
40 50 10 20 30 Cone angle a (degrees)
8.55 Variation of Pcr with α for MBS.
Current vessel
10 20 30 40 50 Cone angle a (degrees)
60
8.56 Variation of Pcr with α for MBL.
Table 8.31 New cone angles and wall thicknesses for the redesigned food cans Cone angle (degrees)
Wall thickness (mm)
Model
Old
New
Old
New
Reduction in thickness (%)
CA DF MBS MBL
11.0 14.83 12.78 13.61
43.55 38.35 39.89 44.39
0.35 0.14 0.24 0.27
0.15 0.10 0.14 0.14
57.1 28.6 41.7 48.1
between 38° and 45°. Using the cone angle which related to the maximum buckling pressure for each of the four food cans, each food can was redesigned to have a theoretical buckling pressure of similar or slightly larger value than the original food can. The values for the new cone angles and wall thicknesses are shown in Table 8.31. From Table 8.31, it can be seen
© Carl T. F. Ross, 2011
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Pressure vessels
that the percentage reduction in the wall thicknesses of the redesigned vessels was from 28.6 to 57.1%.
8.6.5 Conclusions The original food cans appeared to fail by general instability. By increasing the cone angles of the corrugations of the original food cans, the wall thicknesses of the vessels were reduced by between 28.6 and 57.1%. The resulting saving in material can be regarded as being environmentally friendly. It is possible that a different shape of corrugation to the saw tooth corrugation may result in even more material savings. Large values of initial out-ofroundness, together with small values of λ′ can cause inelastic buckling at a pressure less than that required to cause elastic buckling for a perfect vessel. More experimental work is required to determine the plastic knockdown of the new vessels for different combination of λ′ and e/tl.
© Carl T. F. Ross, 2011
9 Design of submarine pressure hulls to withstand buckling under external hydrostatic pressure
Abstract: Various methods of calculating the theoretical collapse loads for a pressure vessel under uniform external hydrostatic pressure are presented. The methods are based on various design codes: PD 5500 for vessels under external pressure, and the design charts of Ross of the University of Portsmouth. Current design methodology, PD 5500, was found to be difficult to use and gave inaccurate collapse pressures for some large-scale pressure vessels. Moreover, PD 5500 appeared to be too conservative for one mode of failure and too pessimistic for another mode of failure. A full-scale ‘theoretical’ pressure vessel was used and the described methodologies were applied in its design to see if there were any similarities between the methods. Key words: design charts, PD 5500, submarines, pressure hulls, failure mechanisms.
9.1
Introduction
Under uniform external hydrostatic pressure, a submarine pressure hull can buckle through shell instability or lobar buckling at a pressure (see chapter 1) that may be a fraction of that needed for the same vessel to explode under uniform internal pressure, as shown in Fig. 1.5. This mode of failure is undesirable, as it is structurally inefficient and one way of improving its structural efficiency, is to ring-stiffen it with suitably sized ring stiffeners, spaced at appropriate distances apart. If, however, the ring stiffeners are not strong enough, the entire ring–shell combination can buckle bodily in its flank, through a mode of failure called general instability, as shown in Fig. 1.9. If the ring stiffeners are very strong and the spacing between them is relatively small, then failure can take place through a mode of failure called axisymmetric deformation, where the circular cylinder keeps its circular form while imploding inwards, as shown in fig. 1.8. This mode of failure is more predictable and designers often prefer to ‘design out’ the two instability modes, so that if failure takes place, it fails through this more predictable mode. 355 © Carl T. F. Ross, 2011
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In this chapter, a full-scale model is designed using PD 5500160 and by using Ross’s code.
9.2
The designs
Two ‘full-scale’ structural designs were considered, namely submarine 1 and submarine 2; but only submarine 1 is reported in this chapter; for submarine 2, consult reference 161. Details of these submarines are given in Fig. 9.1.
9.2.1 Submarine 1 The following parameters were used for submarine 1: T = 2″ = 50.8 mm; B = 1″ = 25.4 mm; D = 8″ = 203.2 mm; l = 27″ = 685.8 mm; Lb = 364″ = 9245.6 mm; R = 192″ = 4876.8 mm; E = 3 e 07 psi = 2.07 e 05 GPa; n = 0.3; s = 80 000 psi = 552 MPa; material is HY80 steel.
9.2.2 Submarine 2 The following parameters were used for submarine 2: T = 2″ = 50.8 mm; B = 1″ = 25.4 mm; D = 12″ = 304.8 mm; l = 27″ = 685.8 mm; B X
D
X
Eccentricity of frame T
I
R Axis Lb
9.1 Dimensions of full-scale submarines.
© Carl T. F. Ross, 2011
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357
Lb = 364″ = 9245.6 mm; R = 192″ = 4876.8 mm; E = 3 e 07 psi = 2.07 e 05 GPa; n = 0.3; s = 80 000 psi = 552 MPa; material is HY80 steel.
9.2.3 Designing against shell instability In order to investigate shell instability, the section between adjacent frames was ‘theoretically isolated’ and then the following two methods were employed in order to determine the collapse pressures: Ross et al.’s design chart which makes use of the Von Mises formula,39 Windenburg and Trilling38 (see chapter 3). (b) PD 5500: cylinder thickness calculation.
(a)
For the purpose of presentation, all calculations in this section are for submarine 1.
MISESNP The von Mises collapse pressure was calculated (see chapter 3) using the computer program MISESNP.EXE (Appendix III). The computer program calculates the elastic buckling pressure for a perfect circular cylinder subject to uniform external pressure and simply supported at its ends. The program will also calculate the Windenburg thinness ratio (l); this value will be used in conjunction with the design charts to calculate the plastic knockdown factor (PKD) and thus the actual collapse pressure. The results from this program were: l = 0.711 and Pcr = 25 MPa with 14 lobes (n = 14). After the von Mises theoretical collapse pressure (Pcr) was obtained, the required PKD had to be determined from Ross et al.’s design chart (Fig. 9.2). To use the design chart, we first need to calculate: 1 1 = = 1.4 λ 0.711 From Ross’ design chart of Fig. 9.2, PKD = 2.8. Finally the Pcr (theoretical) is divided by the PKD to give: Pcr (actual ) =
Pcr 25 MPa = = 8.9 MPa = 890 m depth PKD 2.8
From PD 5500’s design chart of Fig. 9.3, the shell instability calculations were carried out.
© Carl T. F. Ross, 2011
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Pressure vessels 1.600 1.400 1.200 1/λ
1.000
Experimental 2006 Experimental 2007 Reynolds Windenburg
0.800 Safe side
0.600 0.400 0.200 0.000 0.000
0.500
1.000
1.500 PKD
2.000
2.500
3.000
Difference in design parameters Δ
9.2 Ross’ design chart for shell instability.
0.7
(a) Cylinders and cones
0.6 0.5 (b) Spheres and dished ends
0.4 0.3 0.2 0.1 0
1
2
3
4 5 6 7 8 Design parameter, K
9
10
11
12
9.3 PD 5500 design chart: (a) cylinders and cones; (b) spheres and dished ends.
From K = Pm Py = 3.1 (Py = yield pressure at mid-bay ≈ 1.4 × 6 × T/R = 8.05 MPa; Pm = von Mises buckling pressure), we calculate the external pressure design pressure design parameter of a cylinder; then we extrapolate the Δ value. For pressure vessel 1, the design parameter was 0.56, i.e.: Δ=
P = 0.56 Py
and Py = 8.05, from the previous equation, therefore: P = 0.56 × Py = 0.56 × 8.05 = 4.51 MPa
© Carl T. F. Ross, 2011
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Thus, it can be seen that a design depth of 450 m, by PD 5500, is too conservative, when dealing with shell instability!
9.2.4 General instability To calculate general instability, three methods below were considered, but only Kendrick Part I,43 is presented in this chapter because Kendrick Part III yields similar results to Kendrick Part I. 1. Kendrick Part I and Kendrick Part III.85 2. PD 5500, which uses Bryant’s formula84 (chapter 4). Kendrick Parts I and III formulae were used to calculate the general instability values of thin walled submarine 1. The vessel that was analysed was ring-stiffened and thus it was difficult to use MISESNP (Appendix III). To overcome this problem an equivalent shell thickness (T′) and equivalent vessel radius (Rf) have to be calculated. These values are then used in the MISESNP computer program to calculate Windenburg’s thinness ratio (l′). From Ross et al.’s design chart (Fig. 9.4), for general instability, l′ = 1.38 therefore: 1 1 = = 0.72 λ ′ 1.38
1/l′
From Ross’ computer program (Appendix IV), namely Kendrick Part I, Pcr (theoretical) = 6.2 MPa (4 lobes). Next, the PKD was extrapolated from Ross et al.’s (Kendrick Part I) design chart, Fig. 9.4; and using the 1/l′ value of 0.72, gives a PKD value of 1.2. The Pcr (theoretical) value was then divided by the PKD to give Pcr (actual). 3.6 3.4 3.2 3.0 2.8 2.6 2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0
Unsafe side
Safe side
Derived from MISES.EXP programme 0
1
2
3
4
5 Pcr /Pexp
6
9.4 Ross’ design chart for general instability.
© Carl T. F. Ross, 2011
7
8
9
10
360
Pressure vessels Pcr (actual ) =
Pcr 6.2 MPa = = 5.2 MPa = 520 m PKD 1.2
From PD 5500, which was very difficult to use, the lowest Pcr value obtained is 6.46 MPa, failing with 4 lobes, corresponding to a maximum diving depth of 646 m; which is positively optimistic!
9.2.5 Axisymmetric deformation Using the boiler formula,1,2 the collapse pressure = P = 5.7 MPa = 570 m. The author has no disagreement with using the boiler formula!
9.3
Conclusions
The findings from the results for pressure vessel 1, appear to show that PD 5500 is too conservative for shell instability, but worse still, too optimistic for general instability. The author does not have any quarrel with PD 5500, as far as the axisymmetric mode of failure is concerned. Moreover, Ross et al.’s design charts are much easier to use and their linear nature make them even easier for the designer to use. In contrast, the design chart of PD 5500 is curved, making ‘life difficult’ for the designer. Calculating the buckling pressures by PD 5500 is made worse because the calculations are very laborious, and because they have to be done for ‘every’ value of ‘n’, the number of circumferential waves that the vessel buckles into. This is necessary to obtain the minimum value of Pcr.
© Carl T. F. Ross, 2011
10 Nonlinear analyses of model submarine pressure hulls using ANSYS
Abstract: Theoretical and experimental analyses are presented of the buckling of seven thin-walled circular cylinders, together with that of a circular cone; all were tested to destruction under uniform external hydrostatic pressure. The mode of failure for three of the cylinders and the cone was plastic axisymmetric collapse, whereas four of the circular cylinders collapsed through nonsymmetric bifurcation buckling or shell instability; also called lobar buckling. All the vessels collapsed under external hydrostatic pressure and for the plastic axisymmetric failures, the vessels imploded inwards while keeping their circular forms throughout their collapse. For nonsymmetric bifurcation buckling, the failure mode of the vessels was lobar buckling or shell instability, where the vessel imploded inwards with evenly spaced waves spaced around its circumference. Key words: finite elements, ANSYS, nonlinear axisymmetric failure, nonlinear shell instability, plastic collapse.
10.1
Introduction
A typical submarine pressure hull consists of a combination of thin-walled circular cylinders, cones and domes, as shown in Fig. 1.4. In this chapter, we will consider both the thin-walled circular cylinder and the thin-walled circular cone. Now under uniform external pressure, such a vessel can implode through shell instability or lobar buckling (see chapter 1), at a fraction of the pressure necessary to cause the same vessel to explode under uniform internal pressure. This mode of failure is shown in Fig. 10.1 and 10.2, and it is an undesirable mode owing to its poor resistance to withstand uniform external hydrostatic pressure. One method of improving its poor resistance to withstanding uniform external hydrostatic pressure is to ring stiffen it, with ring stiffeners spaced suitably apart, as shown in Fig. 10.3. In this instance, if the stiffeners are not strong enough, the vessels can collapse through general instability, as shown in Fig. 1.9 and 10.4. Moreover, the vessel can still fail due to shell instability, if adjacent ring stiffeners are spaced too far apart, but if the stiffeners are strong and closely spaced together, the shell can fail through axisymmetric deformation, as shown in 361 © Carl T. F. Ross, 2011
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Pressure vessels
10.1 Shell instability of cylinders 1 to 3.
10.2 Shell instability for cylinder 7.
Figs 1.8 and 10.5. For axisymmetric deformation the circular cylinder keeps its circular form while imploding inwards. The reasons for these different modes of plastic failure were studied;162 both geometrical and material nonlinearity were considered for both shell instability and axisymmetric deformation. For shell instability, the initial out-of-circularity plays an
© Carl T. F. Ross, 2011
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10.3 Ring-stiffened circular cylinders 4, 5 and 6.
10.4 General instability of ring-stiffened aluminium cylinders 4, 5 and 6.
important role, because the initial out-of-roundness increases nonlinearly with increasing pressure, until parts of the shell become plastic. When this occurs, the tangent modulus of some parts of the shell rapidly decreases and this considerably worsens the situation, until catastrophic failure takes place. For axisymmetric failure, the initial out-of-circularity is less important, but the vessels can still fail through a combination of material and geometrical nonlinearity. Apart from military uses, submarines are gaining popularity in the use of commercial exploitation of the deep seas. For example, Dickens et al.163 have
© Carl T. F. Ross, 2011
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discovered that there are about 10 000 billion tonnes of frozen methane hydrates some 3 km to 11 km below the sea level. The value of this deep sea methane is some 536 times the annual GDP of the USA, or about $1.25 million per person on Earth! Ocean engineers are still not quite sure how to successfully retrieve this methane. A description of the experimental mode of collapse, followed with the theoretical analysis by ANSYS will now be given.
10.2
Experimental analysis
The external pressure was increased by a hand-driven hydraulic pump and the strains noted for each value of pressure. Three of the models were thinwalled circular cones, namely cones A, B and C, which collapsed axisymmetrically as shown in Fig. 10.5. The three other models were thin-walled circular cylinders, namely cylinders 4, 5 and 6, which also collapsed axisymmetrically as shown in Fig. 1.8.5 The vessels were tested to destruction, under uniform external pressure, in a test tank similar to that of Figs 10.6 and 10.7. The material of construction for the models that collapsed axisymmetrically was EN1A mild steel and the load–deflection relationship for this material is given in Ross and Johns.35 The circular cylinders 4, 5 and 6 had electrical resistance strain gauges connected to the inner surfaces of their mid-lengths and a typical pressure–strain distribution for cylinder 4 is shown in Fig. 10.8.
10.5 Axisymmetric collapse of a thin-walled cone.
© Carl T. F. Ross, 2011
Nonlinear analyses of model submarine pressure hulls Scale (ins)
5
3
Outlet to pressure gauge
1 0
365
24 HT bolts 5/8″ B.S.F.
‘O’ rings As below
Model No. 1.
Flexible hose from pump
Oil
1/4″ B.S.F cap head screws every 20°
Centre spindle ‘O’ rings
Closure plate
Pressure-tight cable gland
10.6 The test tank and model.
The experimental collapse pressures for the vessels are shown in Table 10.1. For cones A, B and C, their collapsed bays were of similar size and this is why their experimental collapse pressures were similar in magnitude. For cylinders 4, 5 and 6, cylinder 4 was longer than cylinder 5 and cylinder 5 was longer than cylinder 6. The dimensions of the four cylinders that collapse through shell instability (see chapters 1 and 3), are shown below in Table 10.2. Figure 10.9 shows a schematic diagram of the circular cylinders. The value of N in Table 10.2, represents the number of ring stiffeners present. Models 1, 2 and 3 were made from HE9 WP aluminium alloy and model 7 was made from good quality mild steel. The test tank and the hand-driven hydraulic pump are shown in Fig. 10.7.
© Carl T. F. Ross, 2011
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Pressure vessels
10.7 The test tank and hydraulic pump.
0
Pressure (MPa) 5.52
9.66
Hoop strain/microstrain
–2000 –4000 –6000 –8000
–10000 –12000
10.8 Plot of pressure–strain relationship for cylinder 4.
The inside surface of each circular cylinder was fitted out with a number of electrical resistance strain gauges, including 10 circumferential strain gauges at mid-bay, in the largest bay of each vessel. The purpose of these strain gauges was to study the circumferential deformation pattern, just before buckling. The buckled forms of the vessels are shown in Figs 10.1
© Carl T. F. Ross, 2011
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Table 10.1 Experimental and theoretical collapse pressures
Vessel
Experiment (MPa)
ANSYS (MPa)
Cylinder 4 Cylinder 5 Cylinder 6 Cone A Cone B Cone C
9.724 11.172 13.172 5.97 6.03 6.21
7.457 8.67 10.64 5.68 5.68 5.68
Table 10.2 Geometrical details of the circular cylinders Cylinder no.
L1 (mm)
L (mm)
Lb (mm)
bF (mm)
bf (mm)
d (mm)
N
1 2 3 7
88.90 95.25 63.50 –
114.30 95.25 57.15 254.00
676.28 616.59 400.69 254.00
8.26 10.16 10.16 –
8.26 8.26 8.26 –
15.75 15.75 15.75 –
5.00 5.00 5.00 –
Lb L1
bF
L
bf
d h
L d
bf
h
a
Centre line of cylinder
10.9 Geometrical details of the aluminium alloy circular cylinders; where h = wall thickness = 2 mm; and a = mean radius = 13.1 cm.
and 10.2, where it can be seen that all four vessels collapsed by shell instability. The experimentally obtained collapse pressures (in bar) are model 1 23.10 (8), model 2 24.14 (9), model 3 27.72 (12) and model 7 39.17 (5) (the figures in parentheses represent the number of circumferential waves or lobes (n), that each vessel buckles into).
© Carl T. F. Ross, 2011
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Pressure vessels
10.3
Theoretical analysis
The theoretical analysis was based on the finite element method, using the commercial computer package, namely ANSYS. The method used was a step-by-step incremental method that allowed for both material and geometrical nonlinearity. The element used to model the vessels for both axisymmetric failure and for shell instability was Shell93, an eight-node isoparametric element. The wall of the shell was meshed and the structures were assumed to be fixed at the ‘left’ ends of the computer images. Attached to each ‘right’ end of the computer image, was a circular disc of thickness about 10 times the wall thickness of each vessel. It was considered that this best represented the actual experimental case. A typical mesh is shown in Fig. 10.10. Plots of pressure against axial deflection for cylinders 4 to 6 are given in Figs 10.11 to 10.13. Comparisons with experiment and theory are shown in Table 10.1.
y z
x
Pressure (MPa)
10.10 A mesh for a typical circular cylinder.
15 10 5 0
0
0.2
0.4 0.6 0.8 Axial deflection (mm)
1
10.11 ANSYS pressure–deflection plot for cylinder 4.
© Carl T. F. Ross, 2011
1.2
Pressure (MPa)
Nonlinear analyses of model submarine pressure hulls
369
16 14 12 10 8 6 4 2 0 0
0.2
0.4 0.6 Axial deflection (mm)
0.8
1
Pressure (MPa)
10.12 ANSYS pressure–deflection plot for cylinder 5. 16 14 12 10 8 6 4 2 0
0
0.5
1 Axial deflection (mm)
1.5
2
10.13 ANSYS pressure–deflection plot for cylinder 6. 1 NODAL SOLUTION APR 20 2009 04 : 59 : 04
STEP=1 SUB =999999 TIME =1 USUM (AVG) RSYS=0 DMX =5.798 SMX = 5.798 y z
x MX MN
0
.644229
1.288
1.933
2.577
3.221
3.865
4.51
5.154
5.798
10.14 Screen dump for cylinder 4.
Screen dumps for cylinders 4, 5 and 6 and for cone C are given in Figs 10.14 to 10.17. From the screen dumps of these vesssels it can be seen that all four vessels collapsed in three-hinge mechanisms. The end hinges were hogging hinges and the mid-bay hinges were sagging hinges. When the three hinges formed on each vessel, the effects of the axial pressure caused the
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NODAL SOLUTION
APR 21 2009 03 : 20 : 46
STEP=1 SUB =999999 TIME =1 USUM (AVG) RSYS=0 DMX =3.611 SMX = 3.611 y x
z
MN MX
0
.401209
.802417
1.204
1.605
2.006
2.407
2.808
3.21
3.611
10.15 Screen dump for cylinder 5.
1
NODAL SOLUTION
APR 22 2009 05 : 08 : 05
STEP=1 SUB =999999 TIME =1 USUM (AVG) RSYS=0 DMX =2.921 SMX =2.921 MX
y z
x
MN
0
.649118 1.298 .324559 .973677
1.623
1.947
2.272
2.596
2.921
10.16 Screen dump for cylinder 6.
vessel to eventually collapse catastrophically, in an axial manner; as described in chapter 2. A typical mesh for cylinders 1 to 3 is shown in Fig. 10.18. For the vessels that collapsed through shell instability, the theoretical buckled patterns for cylinders 7 and 1 to 3 are shown in Figs 10.19 to 10.22. The results for the nonlinear and eigen buckling analyses are shown in Table 10.3; the eigen
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NODAL SOLUTION
371
APR 23 2009 01 : 28 : 58
STEP=1 SUB =7 TIME =1 USUM (AVG) RSYS=0 DMX =.114539 SMX =.114539 z
y MX x
MN
0
.012727
.025453
.03818
.050906 .07636 .101813 .063633 .089086 .114539
10.17 Screen dump for cone C.
y z
x
10.18 A typical mesh for cylinders 1 to 3.
buckling results use only linear elastic theory. From Figs 10.19 to 10.22, it can be seen that the complicated plastic hinges formed with ANSYS, to represent the collapsed lobes. In Table 10.3, the constant λ was called the thinness ratio, where λ = {(L/2a)2/(h/2a)3}0.25/(σyp /E)0.5; σyp = yield stress and E = Young’s modulus.
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y z
x
XN
XX
10.19 ANSYS screen dump for cylinder 7.
y z
x
XX XX
10.20 ANSYS screen dump for cylinder 1.
10.4
Conclusions
The results for the nonlinear analysis axisymmetric mode of failure appear to be a little lower than experiment, although still favourable. This was encouraging because the element used, shell93, was not axisymmetric; yet it rightly predicted axisymmetric modes of failure for these vessels. The
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XX
XX y z
x
10.21 ANSYS screen dump For cylinder 2.
XX
x z y
XX
10.22 ANSYS screen dump for cylinder 3.
results for plastic shell instability, show that the ANSYS nonlinear theory compared favourably with experimental results for cylinders 1 to 3, but were less favourable than experimental results for cylinder 7. The reason the nonlinear theory gave less favourable results for cylinder 7 was probably because cylinder 7 collapsed elastically in the experiment and from the screen dump, it appeared that ANSYS found an upper plastic bound. The
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Table 10.3 Experimental and the ANSYS nonlinear and elastic results (buckling pressures for models 1 to 3 and 7)
Cylinder Number 1 2 3 7 a
Von Mises (bar) 25.0 30.6 48.9 42.7
a
(8) (8) (10) (6)
ANSYS Eigen buckling (bar)
ANSYS nonlinear (bar)
Experimental (bar)
λ
32.1 39.9 76.0 37.5
26.5 26.2 27.5 48.5
23.1 24.1 27.7 39.1
1.1842 1.0810 0.8820 1.4240
(8) (9) (10) (6)
(8) (9) (12) (5)
The numbers in parentheses represent the number of lobes (n).
meridional half waves for model 7 was greater than 1, when it should have been 1, which was the case for the experimental mode shown in Fig. 10.2. For the cylinders that collapsed inelastically, namely cylinders 1 to 3, the results of ANSYS nonlinear analysis compared favourably with experimental results, whereas the linear eigen buckling analysis gave results that grossly overestimated the collapse loads as expected. It was interesting to note that the ANSYS screen dumps showed a ‘partial’ plastic hinge collapse mode, similar to that which occurred in the experiments. To apply the ANSYS nonlinear theory to full-scale vessels, more work needs to be done. However, in this chapter the commercial computer package, namely ANSYS, has been shown to be suitable for analysing the plastic axisymmetric collapse for thin-walled circular cylinders and cones under external hydrostatic pressure. The method is also applicable to full-scale submarines, constructed from ductile metals.
© Carl T. F. Ross, 2011
11 Star wars underwater: deep-diving underwater pressure vessels for missile defence systems
Abstract: A conceptual design is presented of an underwater ‘star wars’ missile defence system, that will be more difficult to detect by the enemy than a recently proposed ‘surface’ star wars system. It is suggested that for the proposed structures needed for the underwater star wars system, the material of construction should be a composite and not a metal, as use of the latter for large deep-diving underwater vessels would result in such structures sinking to the bottom of the ocean like stones, owing to the fact that they would have no reserve buoyancy. Composites were shown to have better sound absorption characteristics, thereby making the underwater structures difficult to detect through sonar equipment; carbon nanotubes can also be used. It is proposed that these underwater structures should operate up to a depth of 7.16 miles (11.52 km), as at this depth, all of the oceans’ bottoms can be reached. Key words: star wars, missile defence system, underwater structures, composites, carbon nanotubes, ocean bottoms, diving vessels.
11.1
Introduction
Some three quarters of the Earth’s surface is covered by water and only about 0.1% of the oceans’ bottoms have been explored. Indeed the surface area of the Earth’s surface covered by water is 10 times larger than the surface area of the moon, and the Earth’s surface covered by water is about three times larger than the Earth’s land area. Thus, an interest in underwater star wars is probably more important than an interest in surface star wars. The average depth of the oceans is somewhere between 4000 m and 5000 m and the greatest ocean depth of the oceans is found in the Mariana’s Trench, which is some 7.16 miles (11.52 km) deep. This distance is about 30% larger than the height of Mount Everest and this depth has been conquered by man, just once; in 1960 by Auguste Picard! In that instance the vessel used had a thick-walled spherical shell of diameter 6 ft (1.83 m). Even though this vessel was so small, it had no reserve buoyancy and could 375 © Carl T. F. Ross, 2011
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Pressure vessels
only achieve reserve buoyancy by being attached to an overhead float filled with gasoline. Undersea technology is already used for military purposes, but most large submarines can only dive to a depth of about 1312 ft (400 m). As the maximum depth of the oceans is 7.16 miles (11.52 km), or nearly 29 times the diving depth of a large conventional submarine, the potential of the oceans for military purposes is not being fully exploited and it is the author’s belief that an underwater star wars system would prove far superior to a surface star wars system. The advantages of using an underwater star wars system are: • • • • •
radar does not work underwater; heat-seeking missiles do not ‘work’ underwater; satellite spy cameras for the filming of submarines, operating at depths of 11.52 km, will be ineffective; the surface area of the Earth’s ocean bottoms is about three times larger than the Earth’s land area; the underwater vessels can move around the ocean bottoms without detection more easily than surface ‘vehicles’. The disadvantages of the underwater vessels are:
• • •
underwater vessels can be detected by sonar; it is necessary to supply underwater vessels with food and other provisions; discharge of refuse from underwater vessels can be detected by the enemy.
These disadvantages can, however, be overcome to some extent. For example, to decrease detection by sonar, the hull can be constructed with a material such as ‘S’ glass, which has a sound absorption coefficient as high as the material used for acoustic tiles; this idea was presented164 as part of a conceptual design for a stealth submarine. The shortage of provisions could be overcome by supplying the underwater vessels with provisions (say) every month with the aid of special mini and larger submarines. These supply submarines should have propulsion systems to allow them to hover above the underwater vessels, so that their hatches can mate and form a watertight seal. Potential propulsion systems for these submarines may include water jet propulsion. After the underwater vessels have been supplied with their monthly provisions, they can stealthily move away just above the oceans’ bottoms. Similarly, the discharge of refuse can also take place at monthly intervals. The above arguments appear to show that the advantages of operating an underwater star wars system clearly outweigh the disadvantages.
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377
The design
11.2.1 Hull form The usual shape of a submarine pressure hull is in the form of a ringstiffened circular cylinder, blocked by end caps, as shown by Fig. 1.4. The pressure hull is sometimes surrounded by a hydrodynamic hull, which is in a state of free-flood and is therefore unlikely to fail owing to hydrostatic pressure. The advantages of using a circular cylinder are: • •
• •
a ring-stiffened circular cylinder is a good structure to resist the effects of external hydrostatic pressure; extra space inside the pressure hull can be achieved by making the cylinder longer, so that a circular cylinder of relatively small diameter can have a much larger volume than (say) a sphere of twice the cylinder’s diameter; a circular cylinder is a good hydrodynamic form, better than (say) a spherical form of the same volume; a circular cylindrical shape can be easily docked, better than (say) a spherical shape of the same volume.
The disadvantages of using a circular cylindrical shell for a submarine pressure hull are: •
• •
a cylinder has two ends and if crew members are required to move from the forward end to the aft end, or vice-versa, it may prove difficult because of congestion; hydrostatic and hydrodynamic stability can be a problem for a hull of cylindrical form underwater; the conventional submarine cannot move easily in three dimensions.
For the underwater vessels proposed here, for most of their time underwater, they will be stationary and when they move, they will move slowly. Therefore a very good hydrodynamic form is not a prerequisite. Thus, to incorporate the advantages of using a circular cylinder, and remove the disadvantage that a cylinder has two ends, it is proposed to manufacture the main hulls of these vessels in the forms shown in Figs 11.1 and 11.2. The plan views of the main hull need not necessarily be pure ellipses; they can be more oval, like a racetrack, or of similar forms. If these forms are used, the hull can be constructed in sections, which can then be bolted together, Fig. 11.3, as suggested by NCRE (Smith165) and by the present author for corrugated pressure hulls (Ross133).
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11.1 Underwater space station 1.
11.2. Underwater space station 2.
The proposed design is similar to the design of an underwater drilling rig proposed by Ross and Laffoley-Lane136, and also to a underwater missile launcher (Ross169). However, the present structures have a better hydrodynamic form than those previously suggested, and may also allow better manoeuvrability. They will also make the internal volumes of the submarines much larger, so that they can store more. If water jets are used for manoeuvring and propulsion, the underwater space stations can move three dimensionally, like a helicopter, except that they can also move ‘backwards’. Hydrostatically, stability should prove less of a problem, both on the surface
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11.3. Method of construction of corrugated pressure hulls.
and underwater, because of the shape of the space stations. It is suggested that at the centre of the ellipsoids (plan views), either spherical shells of twice the diameter of the circular cylindrical form are attached, as shown in Fig. 11.1, or another larger circular cylinder, as shown in Fig. 11.2, is attached via walkways. The main purpose of the designs of Figs 11.1 and 11.2 is to give these vessels sufficient volume, so that they can be used as storage devices to supply friendly submarines. Thus, in time of a global war, the friendly submarines can be resupplied without having to return to a land base. The supply vessels can store consumables of every description. Additionally the supply vessels would contain a hospital, so that sick or injured personnel can be attended to without returning to a land base. In the design shown in Fig. 11.2, the large central circular cylinder can be used as a dry dock to repair damaged friendly submarines. This will put them back into action quickly and avoid the danger of returning to the land base; remember Pearl Harbour! Land bases can be more easily targeted today than they were in the days of Pearl Harbour, as mankind now has better radar and satellite spy cameras, and missiles than in those days, including heat-seeking missiles. The above design appears to indicate that if an underwater star wars system is used in preference to a surface star wars system, we are in a ‘winwin’ situation!
11.3
Manpower and living conditions
It is suggested that the required manpower should be about 200 personnel; this is slightly more than the number that are currently used to operate a large military submarine. In peace times, personnel in the underwater space
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stations would be inside them for about 3 months at a time, and, therefore, it will be advisable to give each individual a reasonable amount of space and a good headroom allowance. In current ocean vessels, the average volume allowed per person is about 5 m3. Since personnel on the vessel are required to carry out their work without it causing them any undue stress, a minimum volume of 10 m3 is proposed. This gives a total living quarter requirement of 2000 m3. Canteen and recreation facilities are also required and it is proposed that these are 3000 m3. It is proposed that the hospital space should be about 1000 m3; this makes a total of manpower space requirements of 6000 m3.
11.4
Power requirements
It is suggested that a maximum power rating of 30 MW should be more than adequate to support life and allow the vessel to be operable. The power is required at two fundamental levels; a normal high power level (30 MW) and at an emergency level (several kW), in the event of failure in the primary power system. The selection of the power system would not only be determined by the power level, but also by the endurance time and for this vessel, it would have to be in the region of several years. For the main power requirements, the existing generating sets used on surface vessels are unsuitable, partly because they require large quantities of oxidiser and partly because their exhaust disposal facilities would leave a trace for the enemy to detect. The only suitable source of power generation for this vessel is nuclear power. There are several different types of nuclear generators including: • • • • •
radioisotopic generators; pressurised water reactors (PWR); boiling water reactors; liquid metal fast reactors; thermal system.
Each of these systems has its advantages and disadvantages, but the most suitable reactors are radioisotopic generators, PWR and liquid metal reactors, because these have the smallest cores. Radioisotopic generators are small, but they will have difficulty in generating 30 MW of power. Although liquid metal reactors have the smallest cores, they need to keep the metal molten at all times, even during periods of shutdown. This renders them hazardous, and because of this, they will be unsuitable to power the new underwater vessels. This leaves the PWR as the most suitable for powering the vessel, as it can generate the power, is small, and has been proven safe for submarine usage. Additionally, suitable designs are readily available. A suitable PWR (Haux166), including
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generating sets, etc., would have a size of 3000 m3, and a weight of 2000 tonnes.
11.4.1 Emergency power supply The emergency power supply must be a non-atmospheric system which is normally independent of the main supply. It must be sufficiently large to run emergency life support systems, lighting, rescue and escape operations. Additionally, it must be able to operate some control systems to allow sufficient time for the crew to survive and be rescued, or for the main generator to be repaired. Because the ambient water temperature at 11 520 metres is likely to be between about 0 and 2 °C, there will also be a heating requirement and this may lead to a significant power demand. The emergency power required must be such that it will make the vessel safe in an emergency situation. For example, in addition to providing lighting, etc., the emergency power level must be sufficient to blow out the ballast tanks at a depth of 11.52 km, so that the vessel may return to the surface. The emergency power required will probably be about 60 kW for a period of 5 to 7 days; this can be met by using a large number of batteries, similar to those used on conventional submarines. It must also be remembered that rechargeable batteries will give off hazardous gases, which must be taken into account.
11.5
Environmental control and life support systems
The atmosphere and other factors, such as noise and vibration, within the subsea system must be considered, so that no physiological or psychological performance degradation occurs. Noise should prove less of a problem than with a conventional vessel, because in this instance, a good sound absorbing material such as ‘S’ glass is likely to be used. Although under emergency conditions, limits can be set to which personnel are exposed for short periods to these unwanted conditions, without suffering any adverse effects. Environmental control systems are required to sustain a breathable atmosphere and to maintain the internal climate within a ‘comfort zone’. Logistic support by specialist mini and larger submarines will be required for provision of food, goods, etc. and for the transfer of personnel. The expected crew change will be around 3 months and it is proposed that supplies will come every month.
11.5.1 Atmospheric control The critical aspects of the atmosphere are oxygen supply, carbon dioxide removal and trace containment control, with atmospheric analysis to ensure
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the safety of the environment. Oxygen consumption is dependent on work load and dietary balance (Haux166) see Table 11.1. This equates to an average of 30 l h−1 of O2 per man; this figure is based on extensive data collected from submarines. Conversely, this results in the generation of 25 l h−1 of CO2 per man, and this CO2 must be removed from the atmosphere and the oxygen replaced. Several oxygen replacement systems may be considered including: • • • •
high-pressure (gas) oxygen storage; liquid (cryogenic) oxygen storage; electrolytic oxygen generation; chemical oxygen sources.
The use of electrolytic oxygen generators from water is probably the best method, because there is an available supply of water and it is a well proven technology resulting in high reliability (Haux166). This system does not have the resupply problems of other systems, such as in high-pressure or liquidoxygen storage nor does it have the safety and operational problems. The only drawback of electrolytic oxygen generators is that they require a high level of electrical power, but with a nuclear reactor on board, this will not be a problem. In the event of an emergency, it is suggested that an emergency back-up system of bottled oxygen is kept on board.
11.5.2 Carbon dioxide control The air we breathe contains about 0.03% of CO2 (equivalent to a partial pressure of about 30 Pa) (Haux166). Such a level will be difficult to maintain and the required effort to so do, will not be justified. If the level of CO2 reaches 4%, the atmosphere will prove lethal to humans. Therefore, the system should be capable of maintaining the CO2 level well below that which will impair mental and physical performance. This results in the requirement for maintaining a maximum partial pressure for CO2 of 1500 Pa (Haux166), or 1.5% CO2. There are many appropriate systems currently in existence on both spacecraft and submarines and these depend on absorption and adsorption. Such systems include:
Table 11.1 Oxygen consumption Oxygen consumption (kg (man day)−1) Ideal Normal maximum Normal minimum
0.9 1.6 0.5
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383
metallic absorbents; molecular sieves; monoethanolamine scrubbers; the Bosch reaction; the Sabatier reaction.
Metallic absorbents are currently widely in use, but for large manning levels and long submergence times, they become restrictive, although they would be suitable as an emergency back-up system. The monoethanolamine scrubber is also regenerative and it is currently used in nuclear submarines, although it does have large power requirements and deteriorates with time. The Bosch system can be operated in the Sabatier mode and although it is expensive and complex, it could make an excellent system for a ‘permanent’ system, because it also gives off oxygen.
11.5.3 Contaminant control Because the environment within the new underwater vessels will become sealed, it will become contaminated over a period of time with trace quantities of gaseous and particulate matter emerging from the crew and from the materials and processes within the enclosure. An internal system will therefore be required to control the level of the contaminants, dependent of their type and toxity. Table 11.2 (Haux166) shows a few possible contaminants and their exposure limit, in an enclosed vessel such as the new underwater vessels. There
Table 11.2 Typical contaminant exposure limits
Substance Ammonia Carbon dioxide Carbon monoxide Freon-12 Hydrogen chloride Hydrogen fluoride Mercury Nitric acid Nitrogen dioxide Oil mist Ozone Phosgene Stilbene Sulfur dioxide
Eight-hour weighted average exposure limit (ppm)
Ceiling concentration
50 5000 50 1000 5 ppm 3 0.1 mg m−3 25 5 5 mg m−3 0.1 0.1 0.1 5
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may be many other contaminants resulting from operations such as food preparation and submarine refurbishment. Some of the contaminants will be difficult to detect and remove and therefore it is suggested that the structure is partitioned so that the atmosphere from one section does not contaminate another. The only way for the total removal of all the contaminants within the vessel is to purge the vessel from time to time.
11.5.4 Atmospheric climate The climate of the enclosure is very important and for crew comfort, it must be set so that it does not induce any physiological stresses into the crew. In normal ambient conditions, it is generally accepted that the temperature should be between 18 and 22 °C. Similarly, a relative humidity of between 50 and 65% is pleasant (Haux166). It is therefore proposed that the temperature and humidity in the vessel should be maintained at these levels. Because of the heat generated by all the process equipment, there will be a requirement for a suitable air conditioning system. It is also a good idea for the crew to control the ‘local’ climate in their cabins, etc.
11.6
External requirements
11.6.1 Support legs The main external requirement of the structure is that of a system of legs or base to position the structure in a horizontal position on the seabed. Additionally, it will be ideal if the maximum length of the adjustable legs is such that they are sufficient to keep the submarine’s hull above the mud line. Therefore, any system developed here must be able to take account of the state of the seabed. If the seabed is not flat and horizontal, it will definitely be necessary to have adjustable legs.
11.6.2 Other external requirements There are many other external requirements that are needed for the underwater vessels including: • • • • • •
a sonar system; lighting; cameras; remote operated vehicles (ROVs); a docking system; an escape system.
This list is by no means complete and it will need further investigation.
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11.7
385
Size of elliptical structure
From Fig. 11.1, it can be seen that the structure consists of an outer elliptical structure and several inner spheres, these being joined by connecting tunnels. It is suggested that the internal diameter of the cross-section of the elliptical structure is 10 m. It is suggested that the diameters of the inner spheres are about 20 m; this will correspond to a similar strength as the circular cylinder of similar thickness. This structure should be dockable at many ports. The cross-section of the elliptical structure can be separated by 3 levels; each about 3.33 m apart, on average. The outer major axis of the elliptical structure should be about 100 m in length and the outer minor axis should be about 50 m in width. The internal required volume of the elliptical structure required is likely to be in the region of 18 000 m3.
11.8
Central spherical shell
Based on the structural strength of thick-walled pressure vessels (Case et al.,1 Ross2) and assuming that the wall thickness of the space station’s circular cylindrical shell is to be the same as that of the sphere, then the allowable internal diameter of the sphere can be approximately 20 m. Such a sphere will yield an internal volume of approximately 4190 m3. If a sphere of this diameter is likely to cause a docking problem, then spheres of smaller diameters, which can have smaller wall thicknesses, could be used. To protect the underwater space station, torpedoes will be required. It is suggested that these torpedoes can be launched at any angle, so that they can strike a ship or submarine above, in the same way as surface launched missiles are aimed at aircraft and rockets. It must be emphasised that as the missiles and torpedoes are being launched from great depths, they must be stiffened by rings or corrugations (Ross133) to prevent their hulls from collapsing. Additionally, as these missiles have to be as light as possible, it will be necessary to construct their pressure hulls from composites, as these materials have better strength–weight ratios than metals and low weight is a high premium for missiles. Perhaps, as metal matrix composites have a strength–weight ratio some 30 times greater than high strength steels, they can be used for the hulls of torpedoes and missiles. Ceramic and carbon nanotube composites may also prove suitable. It should be emphasised that if it is required to destroy a building of the enemy, then simply by typing in the zip code of the building into the missile’s internal computer, the building can be ‘taken out’ from the oceans’ bottoms.
11.9
Connecting walkways
It is suggested that there should be many interconnecting walkways. From the calculations regarding volumes, there appears to be enough space in the
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main structure of the underwater space station for the spheres and the peripheral tubing to house all the equipment and storage goods and, because of this, the walkways need only be a means of connecting the various components of the structure together. The walkways will simply be passageways for personnel and for moving equipment; they may also be used for exercise. It is suggested that these walkways should be of internal diameter 7 m.
11.10 Material property requirements Because the structure is to be designed for use up to depths of 11 520 m, then the successful development of such a system will depend on the availability of suitable materials of construction. From previous work on submarines, it is already known that advanced materials with diverse properties will be required. Whereas composites first come to mind, complex alloys may also be used. The materials for underwater pressure vessels must not only be capable of withstanding very high external pressures, but must also have other suitable properties that can withstand the environment, such as: • • • • • • • •
good resistance to corrosion; a high strength–weight ratio (if the wall thickness is too large, the vessel will sink like a stone); good sound absorption qualities; low material costs; appropriate fabrication properties (can the vessel be manufactured ‘easily’ in the chosen material?); suitable pressure hull design; low susceptibility to temperature and fire; long operating life span.
Unfortunately, as with most projects, there is no single material that is best for all the above requirements.
11.11 Choice of material The main materials for the design of submarine pressure hulls are: • • • •
high strength steels; aluminium alloys; titanium alloys; composites.
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11.11.1 General corrosion In the marine environment, corrosion has been extensively studied and many data generated regarding corrosion rates. Hence, it is relatively easy to predict and to compensate for corrosion. The attack of submerged surfaces is governed principally by the rate of diffusion of oxygen through layers of rust and marine organisms. With reference to steel, this amounts to a loss of between 3 to 6 mm per year; it is substantially independent of water temperature and tidal velocity, except that industrial pollution leads to higher rates of corrosion. Certain marine organisms can also generate concentrated cell and sulfur effects (Haux166).
11.11.2 Stress corrosion cracking This is a form of localised failure, which is more severe under the combined action of stress and corrosion than would be expected if the two individual effects were added together. There are many variables affecting the instigation of stress corrosion cracking and amongst these are alloy composition, tensile stress (internal or applied), corrosive environment, temperature and time. There are methods of relieving the internal stress and it is possible to solve the susceptibility of materials to stress corrosion cracking by using fracture mechanics. Thus, although this is a problem, it is one that can be reasonably well predicted.
11.11.3 Other factors Other factors to be taken into account in the choice of material include: • • •
brittle fracture; fatigue fracture; problems induced through fabrication; for example, stresses induced by welding together with the detrimental effects of heat-affected zones.
11.11.4 High-strength steels Table 11.3 shows the properties of some high-strength steels that are popular in submarine construction. HY80 is the most commonly used of the highstrength steels shown in Table 11.3; it is also commonly used for commercial applications including pressure vessels, storage tanks and merchant ships.
11.11.5 Aluminium alloys From Table 11.4, it can be seen that aluminium alloys have a better strength– weight ratio than high-strength steels. Aluminium alloys are attractive as
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Table 11.3 Strength of high-strength steels
Material
Specific density
Young’s modulus (GPa)
Compressive yield strength (MPa)
Heat treatment
HY80 HY100 HY130 HY180
7.8 7.8 7.8 7.8
207 207 207 207
550 690 890 1240
Q Q Q Q
1
& & & &
T1 T T T
Quenched and tempered.
Table 11.4 Strength of aluminium alloys
Material
Specific density
Tensile strength (MPa)
Proof stress 0.2% (MPa)
5086-H1116 6061-T6 7075-T6 7075-T73 L65
2.8 2.8 2.9 2.9 2.8
290 310 572 434 –
207 276 503 400 390
a construction material because of their availability, low cost and fabricability, apart from their high strength–weight ratios. They have the disadvantage of being anodic to most other structural alloys and are therefore vulnerable to corrosion when used in mixed structures. However, these problems can be avoided by special design modifications (SNAME6). It is also difficult or impossible to obtain matching strength in weld metal and base metal and it is therefore necessary for the welds to be thicker than the surrounding base metal or for the welds to be located in light stress areas.
11.11.6 Titanium alloys From Table 11.5, it can be seen that titanium alloys have an even greater strength–weight ratio than aluminium alloys and they are, therefore, an ideal material to be used for the pressure hulls of large submarines. Their main disadvantage is that they are very expensive, being about 5.5 times more expensive than aluminium alloys.
11.11.7 Composites Table 11.6 shows the strength and relative costs of various composites. The most commonly used composite for marine structures such as ships is based
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Table 11.5 Strength of titanium alloys
Material
Specific density
Ultimate tensile strength (MPa)
6-4 alloy (annealed) 6-2-1-1 alloy 6-4 STOA alloy C.P. Grade 2
4.5 4.5 4.5 4.5
896 869 870 345
Yield strength (MPa) 827 724 830 276
Table 11.6 Strength and relative costs of composites
Material GFRP (epoxy/ S-glass unidirectional) GFRP (epoxy/ S-glass filament wound) CFRP (epoxy/HS unidirectional) CFRP (epoxy/HS filament wound) MMC (6061 Al/SiC fibre UD) MMC (6061 Al/alumina fibre UD)
Specific density
Fibre volume fraction
Tensile modulus (GPa)
Compressive strength (MPa)
Relative cost
2.1
0.67
65
1200
1
2.1
0.67
50
1000
3.2
1.7
0.67
210
1200
3.0
1.7
0.67
170
1000
5.1
2.7
0.5
140
3000
11
3.1
0.5
190
3100
15
on glass-fibre reinforced plastic (GFRP). The reason for this is partly that GFRP has a very high strength–weight ratio and its cost is relatively small compared with that of other composites Metal matrix composites (MMC) have many advantages over GFRP and carbon-fibre reinforced composites (CFRP), but at the moment they are still in the development stages and their costs are very high. If a structure is likely to suffer from structural buckling, it is better to use CFRP than GFRP, because the former has a much higher tensile modulus than the latter. However, CFRP is expensive. The cost of HY80 steel per unit weight is in the same region as ‘S’ glass, but as its density is much greater than that of ‘S’ glass, a smaller volume of it than ‘S’ glass costs the same.
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11.11.8 Carbon nanotubes These are still in the development stage, but their strength can be 200 times that of high-tensile steel and their Young’s moduli can be five times that of high-tensile steel. Moreover, their densities can be about 1/6th that of steel.
11.12 Pressure hull designs The general shape of the structure is a circular section cylinder, which is elliptical in plan view, surrounding several spheres, the various structures being connected by interconnecting circular cylindrical walkways. As the structure is of large diameter and it is intended to dive deep, it will be necessary to make the the wall thicknesses very large so that the vessel does not suffer structural buckling. Hence, it will not be necessary to ring stiffen or corrugate the vessel. The vessel will be unstiffened.
11.13 Required wall thickness 11.13.1 Wall thickness calculations Because the wall thicknesses are large compared with the diameters, it will be sufficiently precise to calculate the wall thicknesses of the outer circular cylindrical section, which is elliptical in plan view and the walkways by the Lamé line (Ross2, Case et al.,1). Similarly the wall thickness of the spherical shells can be calculated by standard thick shell theory (Ross2). According to these theories, the calculated wall thicknesses of the circular cylindrical shell of the elliptical structure are given in Table 11.7 for highstrength steel, aluminium alloy, titanium alloy and GFRP, together with the weight/unit length (W) and the buoyancy/unit length (B), where the weight /unit length does not include the weight/unit length of the equipment, goods, etc. From Table 11.7, it can be seen that the only material that possesses reserve buoyancy is ‘S’ glass, and that if any of the other materials are used the vessel will sink like a stone, as their strength–weight ratios are much too low to be used for this vessel. The wall thickness for the sphere is not given, because if the internal diameter is twice that of the circular cylindrical shell, its wall thickness will be of the same order as the cylindrical section. The wall thickness of the walkways will be much smaller than that of the elliptical structure. From Table 11.7, it can be seen that it is virtually impossible to construct this structure in a metal, as the wall thicknesses are much too large. Additionally, even if it were possible to construct the structure in a metal, the structure will have no reserve buoyancy and it will sink like a stone down to the ocean’s bottom. In contrast to these arguments, the structure can be built in GFRP by laying layer upon layer. Construction can be aided by
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Table 11.7 Wall thickness (t) of the circular section of the elliptical structure
Material HY80 steel Aluminium alloy 7075-T6 Titanium alloy 6-4 STOA GFRP composite epoxy/ S-glass
Specific density
‘Yield’ strength (MPa)
External Wall diameter thickness(t) ‘W’ (m) (m) (kg m−1)
7.86 2.9
550 503
14.6 15.2
2.301 2.6
0.7 × 106 0.27 × 106
0.17 × 106 0.19 × 106
4.5
830
13.78
1.39
0.22 × 106
0.15 × 106
2.1
1200
11.8
0.91
0.066 × 106 0.112 × 106
‘B’ (kg m−1)
Table 11.8 Some sound absorption coefficients Material
500 Hz
2000 Hz
Acoustic tiles on solid wall Glass fibre 50 mm resin bonded Marble on solid backing Water, as in swimming pool
0.85 0.70 0.01 0.01
0.65 0.75 0.02 0.02
building the structure in smaller components, which will later be bolted together as described in Section 11.2.1. Additionally, buoyancy calculations on this structure show that it will have adequate reserve buoyancy, so that by the use of buoyancy tanks, it will be possible to raise and lower the structure in the water. Additionally, GFRP has good sound absorption coefficients so that the vessel will be difficult to detect by the enemy and make the noise levels within the vessel tolerable (Table 11.8). From Table 11.8, it can be seen that glass fibre has a sound absorption coefficient as good as an acoustic tile. It should be emphasised that it is possible to use CFRP, but this was not considered as its cost was some five times more than GFRP.
11.14 Conclusions Star wars underwater should prove a more formidable form of defence than star wars on the surface, as the latter system can be detected by radar and satellite spy cameras. Additionally, surface structures can be attacked by
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missiles, including heat-seeking missiles. The hulls of currently available missiles and torpedoes will crush, owing to the water pressure, at comparatively shallow depths of water, thus enhancing the case for star wars underwater. Previous design studies carried out in the 1960s and 1970s show that the present concept can be built with existing technology, except for the hulls of the missiles and torpedoes, as more advances are needed in metal– matrix and ceramic composites to construct the hulls of torpedoes and missiles from these materials, possibly even carbon nanotubes. Problems may occur with the slow build up of contaminants in the atmosphere and from time to time the vessel may need purging. This will be lessened to some extent, as the crew will probably work in two or three rota shifts. Considerations must be made so that the crew does not suffer from physiological and psychological problems. Outside support of the vehicle from specialist mini and other submarines will not be easy. These submarines should have the capability of hovering above the underwater vessels, so that their hatches can engage. This hovering facility can be achieved through water jet propulsion. The use of universally adopted hatch covers for submarines and submersibles should be given much consideration to aid rescue missions. Such vessels should prove suitable for defence purposes, as the enemy will find the vessels very difficult to trace, as their signatures will be miniscule. The plan view of the main hull need not be an elliptical structure, but can be of oval shape, like a racetrack, or of similar form. The use of nuclear power to produce electricity for the vessel should prove quite satisfactory but emergency battery power should also be available. If a GFRP composite is used, the vessel will have sufficient reserve buoyancy to be raised and lowered in the water, even more if carbon nanotubes are adopted as a material of construction. Additionally, the good sound absorption qualities of a GFRP composite will make the vessel very difficult to detect by the enemy and should also make noise levels inside the vessel tolerable. In general, GFRP does not corrode in salt water. However, the building of these new underwater vessels will not be easy.
© Carl T. F. Ross, 2011
12 Vibration of a thin-walled shell under external water pressure using ANSYS
Abstract: Theoretical and experimental investigations carried out on a thin-walled hemi-ellipsoidal prolate dome in air and also under external water pressure are reported. There was good correlation between the experimental and theoretical results. The theoretical investigation was carried out using finite element analysis to model both the structure and the fluid. Two computer programs were used, one of which was the commercial computer program ANSYS and the other was an in-house computer program. For the shell structure, the ANSYS computer program used two different doubly curved thin-walled shell elements, whereas the in-house computer program used a simpler axisymmetric thin-walled shell element. The ANSYS program had the advantage over the in-house program in that it produced excellent graphical displays of the eigenmodes. Key words: finite element analysis, ANSYS, vibrations, external water pressure, prolate dome, eigenmodes.
12.1
Introduction
When a thin-walled shell structure is subjected to periodic forces because of out-of-balance machinery or fluid or other motion, it can vibrate. Such vibrations are usually undesirable as they can cause noise and unpleasant motion, which is off-putting to passengers and crew alike. For military vehicles, such as submarines, the noise can be detected by the enemy through sonar and leaves the vessel open to hostile attack. Additionally, the vibrations can cause the structure to fail at low stress values, owing to fatigue. Thus, it is usually desirable to ensure in the design process that the exciting frequencies do not coincide with the resonant frequencies. In particular, it should be ensured that the exciting frequencies do not occur during the starting and stopping of a machine. Therefore, it is important to predict the resonant frequencies in advance of building the structure. Previous studies (Ross et al.122) showed that the resonant frequencies of thin-walled structures decrease significantly when the structures are submerged in water and, because of this, the effect of the water has to be considered in the design process for submarines. Ross et al.122 developed thin-walled axisymmetric finite element programs, which allowed a very 393 © Carl T. F. Ross, 2011
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accurate prediction of the resonant frequencies underwater. However, the finite element program ANSYS was not used in this study for vibration under external water pressure. It is mainly because of this that the present study was done. For the purpose of an evaluation of ANSYS for this branch of research, a thin-walled prolate hemi-ellipsoidal dome made of a thermosetting plastic called solid urethane plastic (SUP) was used, as described in chapters 5 and 6. The results were compared with both experiments and Ross et al.’s finite element solutions.122 The results of this analysis are relevant for the development of pressure vessels that are used underwater. Such components can be found in a number of situations, including the hulls of submarines as well as oil rigs and pipelines. The analyses are in three parts: (a) vibration in air; (b) vibration with water on the outside; (c) vibration under uniform external water pressure. Each of these analyses are described separately.
12.2
Experimental method
12.2.1 Experimental method of analysing the vibration of pressure vessels In the experiments, the cantilever mode (n = 1) and the eigenmodes having two, three, four and five lobes, on the circumference were determined. These latter eigenmodes corresponded to values of n = 2, 3, 4 and 5, or the number of circumferential waves. Within this chapter, the symbol ‘m’ represents the number of half waves in the flank and the symbol ‘n’ represents the number waves on the circumference. Thus, for the cantilever mode, m = 1 and n = 1.
12.2.2 Test set-up The set-up of the test is shown and described in chapters 5 and 6. The central device of this test is the frequency response analyser (FRA). The FRA was a Solartron 1170 (Schlumberger). The output alternating voltage is used to drive the vibrator. The operator can choose the voltage and the frequency. For the test, a frequency sweep between 100 and 2000 Hz was carried out with a step size of 5 Hz. The input voltage displayed represents the vibration measured by the accelerometer. To excite the dome, two methods were used. For the first method, two electromagnetic shakers and one accelerometer were used, whereas for the second method an electromagnetic shaker was used via a mechanical
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approach, as described in chapters 5 and 6. In the first method, the electromagnetic shakers were connected to the output of the FRA with a voltage of 1.5 V. This transformed the alternating voltage into a magnetic force of the same frequency. The advantage of this method was that it worked without direct contact with the dome, as the shakers only had to be placed very close to the dome’s surface. Because of that, the shakers could be rotated within the dome to change the position of the excitation. This was necessary to determine the eigenmodes once resonance was found. In chapters 5 and 6, a set-up involving the two shakers and a phase switch is used to increase the exciting force. To search the even modes n = 2, 4 and 6 the shakers worked in-phase and to search the uneven modes n = 1, 3 and 5 the phase was switched to 180° out-of-phase. In the mechanical approach, the shaker was attached firmly to the dome by a rubber strap and therefore its position could not be changed as easily during the experiment. The transducer driving it had a voltage of 2.5 V.
12.2.3 Measurement of the resonant frequencies The vibration of the dome was measured using an accelerometer, which was mounted to the dome in line with the shaker. For resonance there was a lobe at the position of the excitation. Therefore, there was an increased displacement and acceleration. The latter was measured by the accelerometer and amplified in a charge amplifier, which was connected to the input of the FRA. When resonance occurred, a peak was visible at the input.
12.2.4 Vibration in water For vibration in water, the testing was carried out in a tank (see chapters 5 and 6) having a depth of 280 mm and a diameter of 167 mm. The tank could be closed with a top and sealed so that testing underwater and external hydrostatic pressure was possible as well. The bottom of the dome was bolted to the top of the tank and placed upside down into the tank.
12.3
Theoretical basis of the finite element method
12.3.1 Vibration analysis underwater The theoretical analysis was via the finite element method, where a pair of coupled matrix equations were merged together and then solved, as described in chapter 6. One equation, namely equation [6.2], was used to represent the vibration of the structure, and the other equation namely equation [6.1], was used to represent the motion of the fluid. Details of these equations are given in chapter 6.
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The theoretical analysis of the vibration of pressure vessels was carried out using a finite element program (CON-FE).28 The program uses a truncated conical axisymmetric element with two nodal circles; each node has four degrees of freedom (i.e. eight per element). The axisymmetric element used for modelling the structure is shown in chapter 3.
12.3.2 The shell elements in ANSYS In ANSYS, the pressure vessels were modelled in three dimensions using small quadrilateral shell elements. The two ANSYS elements used in this study are presented briefly below, namely the four-noded shell element ‘shell63’ (Fig. 12.1) and the eight noded shell element ‘shell93’ (Fig. 12.2). Both elements had six degrees of freedom per node, translation is allowed in all coordinate directions as well as rotation around the coordinate axes. The method of building these elements is now described. The quadrilateral four-noded shell element ‘shell63’ (ANSYS Help > Shell63) has four corner nodes. It uses the following element shape functions to approximate the displacement. Figure 12.3 shows the following function graphically for t = −1. 1 u = [uI (1 − s)(1 − t ) + uJ (1 + s)(1 − t ) + uK (1 + s)(1 + t ) + uL (1 − s)(1 + t )] 4 K
L
z x
y
I
J
12.1 Shell element ‘shell63’.
L
O K
P N z x
y
M
I
J
12.2 Shell element ‘shell93’.
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1 0.8 u(s)
0.6
uI (s, –1) uJ (s, –1)
0.4 0.2 0 –1
0 s
1
12.3 Element shape functions.
An analogous equation is found for v, and s and t are curvilinear coordinates in the planes of the element. As shown, these equations are of first order. Therefore, this shell element represents only a linear surface and displacement between the nodes accurately. It should therefore be avoided in modelling curved structures. The shape functions are C0-continuous, which means the element is able to model plate and thin-shell behaviour, but is not able to consider the effect of bending. In a thin-walled vessel that disadvantage should not affect the results of the analysis. However, in a thicker object it would definitely affect the precision. The other element available is the eight-noded shell element ‘shell93’ (ANSYS 7.1 Help > Shell93). This differs from the simple four-noded shell element described above in having midside nodes on all edges. There are three nodes on each side to describe the deflected shape of the element, and through these three points a parabola is defined. Figure 12.4 shows a two-dimensional graph of the element shape functions in the s direction, where t = −1, and only uI, uJ and uM were different from zero. The element shape function is: 1 u = [uI (1 − s)(1 − t )(− s − t − 1) + uJ (1 + s)(1 − t )( s − t − 1) 4 + uK (1 + s)(1 + t )( s + t − 1) + uL (1 − s)(1 + t )(− s + t − 1)] 1 + [uM (1 − s 2 )(1 − t ) + uN (1 + s)(1 − t 2 ) 2 + uO (1 − s 2 )(1 + t ) + uP (1 − s)(1 − t 2 )] Analogous equations are obtained for v and w. The eigenmodes of vibration have the shape of sine waves; these geometrical conditions can be much more accurately described by a quadratic function than by a linear one. Therefore it was more suitable to use the more sophisticated element with mid-side nodes. Because of that it was
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Pressure vessels 1 0.8
u(s)
0.6 u I (s, –1) u J (s, –1) u M (s, –1)
0.4 0.2 0 –0.2 –1
0
1
s
12.4 Element shape functions.
expected that the eight-noded element produced more accurate results than the four-noded element. In the following analysis, both elements were tested to test whether this assumption was correct. Nevertheless, this element is also based on the thin-shell theory, so it cannot provide a continuous slope on the interelement boundaries. However, in a coupled-fluid structure analysis, ANSYS 7.1 did not allow for the use of elements with mid-side nodes so the four-noded shell element was considered as well. Both elements also allowed a modified shape so that they could be used as a triangular element. In that instance, two corners used the same node. However, it was preferred to keep the quadrilateral shape because it is regarded to be the best compromise between accuracy and efficiency.167
12.3.3 Advantages of ANSYS 7.1 compared with the in-house software There were several reasons why it was desirable to use the finite element software ANSYS, instead of the in-house programs for analysing the vibration of pressure vessels underwater. The first one was that Ross’ program is limited to the analysis of axisymmetric vessels. In contrast to that, the model in ANSYS is built of small doubly-curved shell elements and therefore variable shapes can also be analysed. In practical applications, circular parts are definitely the most common ones, but special applications might require elliptical or other shapes. Secondly, ANSYS offers useful graphical features in the post processing. It is possible to plot the mode shape on the screen. The opportunity to animate the deformed shape is also provided. These aspects are helpful,
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especially to get an overview of more complicated eigenmodes and for the purposes of a presentation. However, to use these new opportunities, it was necessary to verify that ANSYS could predict the resonant frequencies as well as Ross’ program.
12.4
Vibration analysis of a prolate dome in air
The first part of the project dealt with the vibration analysis of a thin-walled prolate dome. This dome had a base diameter of 200 mm and an aspect ratio (AR) of 4, so that it had a height of 400 mm (Fig. 12.5). It had a wall thickness of t = 2 mm and was made of solid urethane plastic. This material had a Young’s modulus of E = 2.9 GPa, density of ρ = 1200 kg m−3 and a Poisson’s ratio of ν = 0.3. The vessel was chosen as a specimen for the finite element analysis, with the finite element program ANSYS, because this dome had already been analysed experimentally by Ross (chapter 5) so that there were data available, with which the results of the following calculations could be compared. In addition, Ross also produced finite element programs for the vibration analysis of axisymmetric structures. Therefore, it was possible to make a comparison between these programs and the commercial software ANSYS, and draw a conclusion about which one was better. In this evaluation, the additional features that each program provides are also considered.
y z
x
12.5 Prolate dome.
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12.4.1 Aspects covered in this analysis In order to create a suitable model that delivered reliable results, the following process was carried out. First, various models for the vibration in air were created in which certain parameters were altered. In doing that, the effect of these parameters on the results of the calculations was determined. With this knowledge, it was possible to produce a suitable model that became the basis for further analyses in water. In detail, the effect of the following changes was tested: – a comparison between four-noded and eight-noded shell elements; and – the influence of the mesh density on the results. The capabilities of ANSYS were investigated in the following manner. At the beginning, several models were created and their results were compared, in order to determine the natural frequencies to which the solution converged. An evaluation was then made of the converged solutions, comparing the calculation both with the experimental results and with the CON-FE results from Ross’ program.
12.4.2 Constraints on the model and symmetry After the geometry of the dome was modelled, constraints had to be applied. In the experiment, the vibration was conducted in a tank. Therefore, the bottom of the dome was firmly clamped to the top of the tank and placed upside down into it. For the finite element model, it was therefore assumed that the conditions at the bottom of the dome were best represented by a fixed boundary condition, which meant that all translational and rotational degrees of freedom of the nodes were set to zero. In the modelling process, the aim was to create a model of the structure that was as simple as possible. However, the model still had to consider all important aspects, so that the structural behaviour was described accurately. For the dome, it was possible not to model the whole vessel but to use the symmetry to reduce the size of model. The aim of the analysis was to determine the resonant frequencies and mode shapes for the cantilever mode and the circumferential modes n = 2, 3, 4 and 5. Therefore, it was sufficient to model only one half of the dome. On the new edge, suitable constraints had to be applied for representing the symmetry. To achieve this, the translation of the nodes in the plane of symmetry was free, as well as the rotation about the normal to the plane. The other three degrees of freedom were fixed. If only the even modes n = 2 and n = 4 were of interest, then it would have been sufficient to model only one quarter of the dome.
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This simplification provided several advantages. Firstly, the half dome only had one half of the elements that were necessary to model the complete dome. Thus, it directly affected the running time of the calculation and the memory required. Furthermore, other modes such as the torsion modes were no longer calculated owing to the restraints, making the postprocessing easier. In order to avoid the triangular version of the shell elements, a hole with a diameter of 1.5 mm was left at the top of the dome. This modification allowed only quadrilateral elements to be used. It was verified that this had almost no effect on the results.
12.4.3 Results using the four-noded element shell63 In the following section, the influence of the number of elements on the results was determined. There were two reasons for this study. In theory a finer mesh provides a more accurate model of the geometry of the structure as well as of the displacement. However, it also means more equations to solve and therefore a longer time for the calculation. Thus, it was desired to find a compromise between a simple mesh and a high degree of precision. As the model of the dome should also be the basis for a further coupled fluid–structural analysis, the number of elements should not be too high. The other reason for doing several analyses with an increasing element density was to make sure that the results converged to a final value when refining the mesh. If that had not been the case there would have been a significant error in the analysis so that the results could not be trusted. To prove convergence, four calculations with different models were made. The first model (dome 1) had 13 four-node ‘shell63’ elements on the flank and 16 elements on the circumference. In the first refinement (dome 2), the number of elements was doubled in both directions. It was especially important to have a sufficient number of elements on the circumference to approximate the displacement of the higher eigenmodes. As there was only one lobe in the flank, the number of elements in that direction was of less importance. Therefore, in the third and fourth model, the number of elements on the flank remained the same and 48 (dome 3) and 64 (dome 4) elements were used on the circumference, respectively. These analyses should show firstly that the results tend to one final value and, secondly, that the difference between the third and the fourth solution was so small that a further refinement was not necessary. In Table 12.1, the results are shown; the relative error compared with the final ANSYS solution of dome 4 is also given. It can be seen that the refinement only had a small effect on the vibration modes n = 1 and n = 2, as there was only a change of 0.6% between dome 1 and dome 4. However, for n = 3 and n = 4, there was a difference of 2.2 and 2.3%, respectively. This result was expected as these eigenmodes have more lobes on the
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Table 12.1 Sensitivity analysis using the four-noded element shell63 Mode number Dome 1 13 on flank 16 on circumference
Dome 2 26 on flank 32 on circumference
Dome 3 26 on flank 48 on circumference
Dome 4 26 on flank 64 on circumference
Frequency (Hz) Relative error to 4 (%) Frequency (Hz) Relative error to 4 (%) Frequency (Hz) Relative error to 4 (%) Frequency (Hz)
n=1
n=2
n=3
n=4
n=5
353.46
487.57
407.96
425.19
518.52
−0.2
−0.6
−2.2
−2.3
−0.9
353.63
489.27
414.35
431.91
520.53
−0.1
−0.3
−0.7
−0.8
−0.5
353.97
490.35
416.45
434.24
522.01
0.0
−0.1
−0.2
−0.2
−0.2
354.10
490.74
417.25
435.31
523.21
circumference and therefore an accurate approximation requires more elements. Nevertheless, Table 12.1 shows that the relative error of dome 3 compared with dome 4 was not more than 0.2%, which proved that the solution converged and that dome 4 had a sufficient element density. In Figs 12.6–12.10, the eigenmodes of vibration for the dome are displayed. Owing to these post-processing features their determination was easy. In the modes n = 2–5 the top of the dome did not deform so this area is of minor importance. m is the number of half waves in the meridional direction.
12.4.4 Results using the eight-noded element shell93 As already explained, an eight-noded element should in theory be more suitable for modelling the curved surface of the dome, because its quadratic shape function can describe the real shape more accurately. Therefore, it was expected that the results for the eight-noded element were closer to the experimental ones. Convergence was proved by comparing the results of three different analyses. The first model (dome 5) had 13 elements on the flank and 16 on the circumference. Then the number of elements on the circumference was doubled in the second calculation (dome 6). Finally, the mesh was refined by using the automatic refinement procedure, giving a model with 40 elements on the flank and 96 elements on the circumference (dome 7).
© Carl T. F. Ross, 2011
Vibration of a thin-walled shell under external water pressure
12.6 Eigenmodes of vibration for dome 4; m = 1, n = 1.
12.7 Eigenmodes of vibration for dome 4; m = 2, n = 2.
12.8 Eigenmodes of vibration for dome 4; m = 1, n = 3.
12.9 Eigenmodes of vibration for dome 4; m = 2, n = 4.
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12.10 Eigenmodes of vibration for dome 4; m = 2, n = 5.
Table 12.2 Sensitivity analysis using the eight-noded element shell93 Mode number Dome 5 13 on flank 16 on circumference
Dome 6 13 on flank 32 on circumference
Dome 7 40 on flank 96 on circumference
Frequency (Hz) Relative error to 7 (%) Frequency (Hz) Relative error to 7 (%) Frequency (Hz)
n=1
n=2
n=3
n=4
n=5
354.04
490.75
419.18
442.11
543.39
0.2
0.2
0.4
1.3
3.3
354.08
490.48
417.74
436.76
526.33
0.2
0.1
0.1
0.0
0.1
353.29
489.96
417.47
436.6
525.95
However, the automatic refinement tool was not used in the analysis of the four-node element because this feature could not be used in the coupled fluid–structure analysis. In Table 12.2, the results of these three calculations are shown. Here dome 7 is used as a reference and the relative error of dome 5 and 6 to it is included. It can be seen that the natural frequencies decreased after refining the mesh. A comparison of the solutions shows that there was a significant difference between dome 5 and dome 6 for the eigenmodes n = 4 and n = 5. In dome 5, the relative error was 1.3 and 3.3%, respectively, for these mode shapes. Therefore, this model was not refined enough to approximate the displacement with sufficient accuracy. However, the next model (dome 6) was much better, as the relative error to the refined model (dome 7) was below 0.2% for all calculated natural frequencies. Thus, the mesh density of dome 6 was considered to be sufficient for a modal analysis.
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12.4.5 Evaluation of modal analysis of the dome In the previous sections, a convergent solution for the vibration analysis of the dome was produced for both the four- and the eight-noded element. The final results were those calculated in using dome 4 for the four-noded element and dome 7 for the eight-noded element. From these data, the accuracy of the modal analysis in ANSYS 7.1 was evaluated. Therefore, the results were compared with the resonant frequencies measured in the experiment and with the finite element calculations using the CON-FE. To evaluate the finite element simulations, the relative error to the experimental results was calculated. The data are given in Table 12.3. It can be seen that all finite element programs are able to predict the experimental results reasonably well. The calculated resonant frequencies were below the ones observed in the experiments. Therefore, the actual dome was stiffer than the finite element model. All finite element solutions were able to predict the right tendency that the resonant frequency of the second mode was higher than that of the third and fourth mode. The results of the ANSYS model (dome 4), which used the four-node element, had a relative error between 3.2 and 4.3% for the vibration modes n = 1 to n = 4. Only the mode n = 5 could not be calculated accurately as it had a relative error of 7.9% compared with the experiment. These results were satisfying as they were close to the experimental results. Nevertheless, dome 7 using the eight-noded element was still closer to the experimental solution than the four-noded element. For the mode n = 5, it produced a
Table 12.3 Comparison between experimental results and CON-FE solution Mode number Experiment Dome 4 Four-noded element Dome 7 Eight-noded element Ross – CON-FE
Frequency (Hz) Frequency (Hz) Relative error to experiment (%) Frequency (Hz) Relative error to experiment (%) Frequency (Hz) Relative error to experiment (%)
n=1
n=2
n=3
n=4
n=5
370 354.10 −4.3
511 490.74 −4.0
431 417.25 −3.2
453 435.31 −3.9
568 523.21 −7.9
353.29 −4.5
489.96 −4.1
417.47 −3.1
436.6 −3.6
525.95 −7.4
351 −5.1
492 −3.7
420 −2.6
439 −3.1
528 −7.0
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maximum relative error of 7.4%, which was 0.5% less than for the fournoded element. Therefore, the assumption that mid-side nodes allowed a more accurate description of the model seemed to be right. However, the difference between the results was smaller than expected, as the biggest difference between both solutions was 2.7 Hz. In theory, it was expected that the CON-FE approximated the displacement more accurately than the finite element models created because it uses more suitable shape functions. However, Table 12.3 shows that the resonant frequencies calculated with the CON-FE were only slightly better than the ANSYS solutions. For the modes n = 2 to n = 5, the CON-FE was about 2 Hz closer to the experiment than the ANSYS calculation using the eight-noded element. The maximum relative error of Ross’ element was 7.0%, which was only 0.5% better than the ANSYS solution.
12.4.6 Conclusion In this chapter, a suitable model for analysing the vibration behaviour of the dome with an AR of four was developed. It was justified to use the symmetry of the vessel in the modelling process to create a smaller model that required less memory and calculation time. The influence of certain parameters, namely the necessary number of elements and the difference between four- and eight-noded shell elements was investigated. As both the four-noded and the eight-noded element produced results with a relative error of less than 10%, the precision was considered to be satisfactory. Up to that value, the difference between these elements and the axisymmetric element CON-FE, which was considered to be very suitable for modelling axisymmetric thin-walled vessels, was small. Therefore, the shell elements shell63 and shell93 and the created models were suitable for a modal analysis of thin-walled vessels. As expected, the eight-noded element with mid-side nodes produced slightly better results because it could approximate the curved geometry better. The sensitivity test showed that the solution converged. It was found that a mesh of 26 four-noded elements on the flank and 64 elements on the circumference was suitable. In using the eight-noded shell element, it was even sufficient to use 13 elements on the flank and 32 on the circumference.
12.5
Vibration analysis of the prolate dome in water
Subsequent to the vibration analysis of the dome in air, the main topic of this study begins in this section: the calculation of the resonant frequencies of the dome in water. The reason for the detailed evaluation of the modal analysis in air was that the model and the results were a basis for the analy-
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sis in water. Knowledge of the suitable mesh density and the relative errors were used for generating the model and evaluating the results.
12.5.1 Previous research Many studies on the vibration of thin-walled pressure vessels have been carried out at the University of Portsmouth by Ross et al.92,122–131 Many vessels including the dome with an AR of 4 that were vibrated in air were also analysed in water. In addition, finite element programs were developed, which allowed the calculation of the resonant frequencies in water. These programs were based on the axisymmetric shell element CON-FE used to analyse the vibration in air. The difference was that now the effect of the fluid around the structure was taken into account. In general, the effect of the water was that the resonant frequencies reduced significantly. In using his finite element programs, Ross was able to predict this effect very accurately. However, the finite element program ANSYS had never been used for analysing the vibration of thin-walled pressure vessels underwater at the University of Portsmouth. Therefore, it was necessary to find out whether this kind of analysis could be carried out using ANSYS. Before starting the actual work on the dome, it was, therefore, necessary to carry out a study of the opportunities that the software offered in this area, and a method of undertaking such an analysis had to be worked out.
12.5.2 Procedure in a coupled fluid structure analysis The first step was to find out which features ANSYS offered for analysing the vibration of thin-walled vessels in water. It was found that the program allows for a coupled acoustic analysis allowing a structure and a fluid to interact with each other. The general procedure was as follows. As before, the first step was to build a model which consisted of two different elements. On the one hand, there were the structural elements as in the vibration in air and, on the other hand, the fluid elements.
12.5.3 Fluid elements ANSYS offers two different fluid elements. There is the two-dimensional fluid element ‘fluid 29’ and the three-dimensional version ‘fluid 30’. In the model of the dome in water, only the latter one could be used. The element can be applied for sound wave propagation and submerged structure dynamics (ANSYS168). It has eight corner nodes with four degrees of freedom per node: translation in the coordinate directions x, y and z and the pressure. However, the translational degrees of freedom are only valid
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on the nodes that are next to the interface with the structural elements. To define the fluid, the user has to define the speed of sound in the fluid and its density. In addition, the sound absorption coefficient MU could be included in defining a value between nought and one for MU. For this instance, it was assumed that there was no sound absorption, thus, the speed of sound in water c was 1490 m s−1 and the density of water ρ was 1000 kg m−3. One important setting for the element was key option 2 with the alternatives ‘structure present’ and ‘structure absent’. This deletes the translational degrees of freedom on all elements that are not next to the structure. The procedure for carrying out an acoustic analysis was worked out in using the ANSYS Help. According to the guidelines the following steps were necessary. At the beginning the finite element model is created consisting of both the structural and the fluid elements. Here, the fluid elements next to the structure should have the key option ‘structure present’. The other fluid elements should have the key option ‘structure absent’. In the coupling of the fluid and structure, it was considered that the structure and the fluid next to it had the same displacement. Therefore, the fluid and the structure used the same nodes at the interface. The fluid element was a brick element with four nodes on each side. Thus, the structure could only be modelled by using the four-noded shell element. Then the boundary conditions were defined. In addition to the constraints applied on the structure in the analysis in air the user has the opportunity to fix the pressure on the outer fluid elements. However, in a tank, the pressure is not fixed as the pressure wave is reflected onto the wall of the tank so that it is not constant there. Finally, the fluid structure interface is generated, and this allows the consideration of both the fluid and the structural element. Firstly, all nodes lying on the interface are selected as well as the fluid elements next to these nodes. Then the fluid structure interface is defined for all selected nodes. → main menu → preprocessor → loads define → loads apply → fluid stru ucture interface → on nodes However, in carrying out the Verification Manual 177, it was found that the GUI command above did not work. It should therefore be replaced by the command: sf , all , fsi. In this instance, the interface created was depicted as a red grid in the model. To check on which nodes the interface was applied the user could choose: utility menu → list → surface → on nodes
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All entities had to be reselected before carrying out the modal analysis.
12.5.4 The finite element model of the dome in the water tank The tank in which the dome was vibrated had a diameter of 284 mm and a depth of 449 mm, but at the top of the tank, the diameter was slightly smaller. The finite element models for the analysis in water were created in the same way as those in air. However, in addition to the nodes for the structural elements, the nodes for the fluid elements were created. The outer nodes of the fluid elements were placed on the wall of the tank to consider the geometrical conditions. As the change in acoustic pressure was small within the fluid only a few rows of fluid elements had to surround the structure to model the conditions accurately. As the fluid was completely surrounded by the tank, the pressure degree of freedom was not fixed on the fluid so that the acoustic pressure was reflected on all walls, which was different from Ross’ program, where the acoustic pressure was set to zero at the bottom of the dome. As before the influence of the number of elements on the results was determined. Therefore, three different models were created. Here, the knowledge of the previous simulations in air could be used where dome 3 and 4 with 26 elements on the flank and 48 and 64 on the circumference, respectively, were found to be suitable. These structural models were used for the first two models (dome 8 and 9) in the tank. Dome 8 was based on dome 3. Between the dome and the sidewall of the tank, three rows of fluid elements were used and, on the top of the dome, five rows of fluid elements were modelled (Fig. 12.11). In this model the top caused a problem. Here, the three fluid elements had to span a distance of 49 mm between the top of the dome and the bottom of the tank. In the model, the biggest element had a height of 10 mm. But as there was a 1.5 mm hole in the top of the model the shortest edge was only 0.1 mm long. Therefore, the cubic fluid element almost adopted the shape of a wedge with an aspect ratio of 100. Finite elements, which differ significantly from their original shape, might cause inaccuracies in the solution. Because of that, ANSYS gave a warning. However, as the top of the dome had very little effect on the natural frequencies in air, it was expected that the fluid elements in this area were of minor importance, too. Nevertheless, this problem was solved in the two following models by increasing the number of fluid elements between the bottom of the tank and the dome from five to ten. Thus, the maximum edge length was only 5 mm for the elements next to the hole. Dome 9 (Fig. 12.12) was based on the model of dome 4 and had four rows of fluid elements on the sidewall. The most refined model (dome 10)
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12.11 Finite element model of dome 8.
12.12 Finite element model of dome 9.
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(Fig. 12.13) was created by refining the mesh in all directions. It had 34 elements on the flank and 100 on the circumference. The structure was surrounded by five rows of fluid elements.
12.5.5 Results of the finite element simulation in water The results of the vibration analysis in the water tank are presented in Table 12.4, which also includes the relative error compared with dome 10. In comparing the first two simulations in water it was observed that the increase in the number of elements had the biggest effect on the higher eigenmodes; in particular, in n = 5, the relative change of 2.3% was significant. Thus, the mesh density was increased in all directions in the third
12.13 Finite element model of dome 10.
Table 12.4 Results of the finite element simulations of the dome in water Mode number Dome 8 Dome 9 Dome 10
Frequency (Hz) Relative error to 10 (%) Frequency (Hz) Relative error to 10 (%) Frequency (Hz)
n=1
n=2
n=3
n=4
n=5
79.9 0.6 79.7 0.4 79.4
115 1.3 114.4 0.8 113.5
118.9 2.1 117.7 1.0 116.5
142 3.1 139.7 1.5 137.7
189.6 4.3 185.3 2.0 181.7
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model and it was expected that the resonant frequencies changed even more. However, the maximum relative change between dome 9 and dome 10 was 2.0% for the eigenmode n = 5. In this complex analysis, this result was satisfying as a proof of convergence as it was not expected that the solution converged as well as it did in air. In Fig. 12.14, the acoustic pressure distribution within the fluid is shown for the modes n = 2 and n = 5. It can be seen that there is a significant increase and drop of pressure at the wall of the tank for the mode n = 2. Therefore, acoustic pressure has a significant influence on the resonant frequencies compared with the situation in an infinite fluid. In the mode n = 5, the acoustic pressure change in the fluid was smaller. In addition, the acoustic pressure plot shows that there was almost no acoustic pressure change at the bottom of the tank so that this area was of minor importance. Thus, it was proved that the complicated situation at the top where the fluid elements differed significantly from their brick shape had no effect on the vibration of the dome.
(a)
(b)
12.14 Acoustic pressure in fluid for (a) n = 2 and (b) n = 5.
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12.5.6 Comparison of finite element solution to experiment and CON-FE To evaluate the coupled acoustic analysis in ANSYS the final solution was compared with the experimental results that were measured by Emile and the finite element solutions using the CON-FE.122 The results and the relative error of both finite element solutions to the experiment are shown in Table 12.5. The comparison with the experiment (Fig. 12.15) shows that it was possible to calculate the resonant frequencies very accurately in ANSYS 7.1 as the error was always below 10%. The maximum relative error occurred for the mode n = 4 and was 3.4%. The prediction of the other resonant frequencies was even better. For the modes n = 2, 3 and 5 the relative error was within 1%. It was of particular interest to test whether the fluid structure analysis in ANSYS 7.1 was a good alternative to the finite element program CON-FE by Ross et al.122 In Table 12.5 it can be seen that the CON-FE produced very accurate results for the modes three, four and five as the relative error was less than 1.3%. However, it had much more difficulty in predicting the Table 12.5 Comparison of finite element solution with experiment and CON-FE Mode number Experiment Dome 10
Frequency (Hz) Frequency (Hz) Relative error to experiment (%) Frequency (Hz) Relative error to experiment (%)
CON-FE
n=1
n=2
n=3
n=4
n=5
79.4 79.4 0.0
114.2 113.5 −0.6
118.8 116.5 −1.9
142.5 137.7 −3.4
183.5 181.7 −1.0
97.8 23.2
118.3 3.6
120.3 1.3
141.4 −0.8
185.3 1.0
Frequency (Hz)
200 150 100
Dome 10 CON-FE
50
Experiment 0
0
1
2 3 4 Number of circumferential lobes n
12.15 Comparison of results in water.
© Carl T. F. Ross, 2011
5
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correct resonant frequency for the cantilever mode. The calculated solution was 23.2% above the experimental result. Here the ANSYS result was much better, as it is shown in Figs 12.15 and 12.16.
12.5.7 Conclusion The previous calculations show that it was possible to simulate the vibration of the dome in the water tank. In building three models with an increasing number of elements it was ensured that the solution converged. The final model (dome 10) could predict the resonant frequencies with a maximum relative error of only 3.4% compared with the experiment. Because of that it can be said that ANSYS 7.1 delivers good results in that type of analysis. It is therefore justified to use the software in the design process of vessels to be used under water that are subject to vibration. The simulation was even closer to the experiment than the Ross program CON-FE.
12.5.8 Precision of the simulation When considering the accuracy of the finite element solutions, one aspect could not be neglected. In air, both ANSYS 7.1 and CON-FE predicted resonant frequencies that were lower than the experimental results. Therefore, if the effect of the fluid is considered correctly, the finite element results in water would also be lower than the experimental ones. However, ANSYS produced results that were almost equal to the experimental results for the eigenmodes n = 1 and n = 2 and only slightly below for the modes n = 3 to n = 5. In the CON-FE solution, the resonant frequencies were even higher than the ANSYS values. This means that the finite element model
25.0 Dome 10 CON-FE
Relative error (%)
20.0 15.0 10.0 5.0 0.0 –5.0
n=1
n=2
n=3
n=4
–10.0 Vibration mode
12.16 Relative error of FE solutions to experiment.
© Carl T. F. Ross, 2011
n=5
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underestimated the effect of the water on the vibration. In this particular instance, this inaccuracy seems to compensate for the error of the vibration in air. Thus, the prediction of the resonant frequencies appeared to be more accurate in water than in air. However, if the finite element model in air was stiffer than the real vessel and therefore delivered resonant frequencies that were too high, the results in water would be much worse. Therefore, further verification is needed before relying on the high accuracy of these results.
12.6
Vibration analysis of the prolate dome under external pressure
In the previous section, the vibration of a pressure vessel in water was analysed. So far it has not been considered that real components such as submarines and oil pipelines are often used at a greater depth where they are subject to the static pressure of the water. This pressure can cause instability so that the structure fails owing to buckling, which is definitely the most important failure mode. However, the external hydrostatic pressure also has an effect on the vibration. It causes a compression of the pressure vessel, which leads to a reduction in stiffness causing the resonant frequencies to decrease. Therefore, the hydrostatic pressure has to be considered when the vibration behaviour under water is analysed in the design process of underwater equipment. In addition, a failure has been observed that is a combination of instability and vibration; this is called dynamic buckling (Ross et al.122). It occurs because the eigenmode for buckling and the corresponding eigenmode for vibration are very similar. Thus, the displacement of the vessel owing to vibration can support buckling even if the external pressure is still below the critical one. As this effect is hard to predict, the design of a pressure vessel must be such that the structure is not excited at the frequency that supports dynamic buckling.
12.6.1 Consideration of pressure in finite element analysis It was necessary to test whether ANSYS could be used for designing real parts by considering the hydrostatic pressure. Such an analysis, which consists of two parts, is carried out in the following way. The expected hydrostatic pressure of the water is applied on the structural elements in the finite element model. The effect of the water pressure is considered in a static analysis. The result of this analysis is a reduced stiffness matrix that takes into account the effect caused by the compression. This new stiffness matrix is then used in the modal analysis to calculate the resonant frequencies.
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12.6.2 Results of the finite element analysis under external pressure The dome made of SUP was tested under external hydrostatic pressure by Ross et al.122–123 (p. 165). In this test, the measurement was carried out at 25, 50, 75 and 95% of the critical buckling pressure Pcr, where Pcr = 0.97 bar. In order to be able to compare the results with the experimentally measured frequencies, the same pressure levels were chosen in the finite element models. The calculations were carried out using dome 9 as it offered sufficient accuracy and an acceptable calculation time. Table 12.6 shows the results of the analyses. It also includes the experimental results as well as the relative error of the finite element solutions. It can be seen that the resonant frequencies of the cantilever mode n = 1 and the mode n = 2 were almost unaffected by the increase of the external hydrostatic pressure. However, in the higher modes, the resonant frequencies dropped significantly in the finite element simulation. The frequency for the eigenmode n = 4 halved from 139.7 to 69.9 Hz from p = 0 to p = 0.95Pcr and, for n = 5, the drop was even greater.
12.6.3 Comparison of the finite element solution with the experiment To evaluate the use of ANSYS in a vibration analysis under pressure, the results were compared with the experimental values in Table 12.6. In Table 12.6 and Fig. 12.17, it can be seen that the finite element results agreed well with the experiment. The resonant frequencies of the eigenmodes n = 1, 2 and 3 could be predicted with a relative error of less than 2%. In n = 4, an inaccuracy occurred at p = 0.5Pcr and p = 0.75Pcr as the error increased to 5%. The only value that was not acceptable was the one of the eigenmode n = 5 for p = 0.95Pcr because ANSYS predicted a frequency which was 19% below the real value. Both in the experiment and in the finite element simulation, it was found that the first buckling mode had five lobes and was therefore similar to the eigenmode of vibration for n = 5. Because the resonant frequency decreased significantly at higher pressures, it has to be ensured that any external excitation is kept at an even lower frequency range.
12.6.4 Conclusion The finite element simulation of the vibration of the dome in water under external hydrostatic pressure agreed well with the experiment, Fig. 12.17. It was found that the resonant frequencies of the higher eigenmodes dropped significantly owing to the external pressure. The finite element analysis
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Table 12.6 Effect of external pressure on natural frequencies in water Pressure (kPa) 0
ANSYS dome 9 Experiment Relative error to experiment (%) ANSYS dome 9 Experiment Relative error to experiment (%) ANSYS dome 9 Experiment Relative error to experiment (%) ANSYS dome 9 Experiment Relative error to experiment (%) ANSYS dome 9 Experiment Relative error to experiment (%)
24.25
48.50
72.75
92.15
n=1
n=2
n=3
n=4
n=5
n=6
79.7 79.4 0
114.4 114.2 0
117.7 118.8 −1
139.7 142.5 −2
185.3 183.5 1
255.5
79.6 79.9 0
113.0 113.6 −1
111.2 111.8 −1
125.3 127.5 −2
163.9 164.0 0
227.9
79.5 79.9 −1
111.6 112.4 −1
104.8 104.8 0
108.9 115.0 −5
138.6 141.5 −2
197.1
79.4 79.2 0
110.1 110.8 −1
96.7 97.7 −1
89.5 94.3 −5
108.0 114.6 −6
160.5
79.3 78.5 1
108.9 109.3 0
90.3 92.2 −2
69.9 71.0 −2
74.2 91.5 −19
123.8
300 p=0 ANSYS p=0 Experiment p=0.25 Pcr ANSYS p=0.25 Pcr experiment p=0.5 Pcr ANSYS p=0.5 Pcr experiment p=0.75 Pcr ANSYS p=0.75 Pcr experiment p=0.95 Pcr ANSYS p=0.95 Pcr experiment
Frequency (Hz)
250 200 150 100 50 0 1
2
3
4
5
6
Number of circumferential lobes n
12.17 Resonant frequencies under external hydrostatic pressure.
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seems to slightly overestimate the effect of the external hydrostatic pressure.
12.7
Conclusions
In the scope of this study, the vibration characteristics of the thin-walled hemi-ellipsoidal prolate dome were analysed successfully by finite element calculations using the software ANSYS and the in-house software written by Ross. Of particular interest was the simulation underwater, as ANSYS fluid elements have not been used in this way before at the University of Portsmouth for this particular problem. Good agreement was found between experiment and theory. Overall, the finite element results for the dome agreed well with the experiment. The effect of the water on resonant frequencies was predicted both in air and water with a relative error of less than 10%. The effect of pressure was remarkable causing the frequencies to decrease by a factor of about three. It was also shown that the effect of external hydrostatic pressure could be considered well in this analysis. The ANSYS results agreed well with the existing finite element programs by Ross. Of particular interest was the fact that the simple in-house package gave results as good as ANSYS. In making the computer models for the dome, the simple in-house package took only a few hours, whereas ANSYS took several weeks. However, the graphics capabilities of the ANSYS post-processor were outstanding. Thus, the two packages complimented each other. ANSYS also has the advantage that it can analyse doubly curved shells such as the hulls and propellers of ships, whereas CON-FE can not. Additionally, ANSYS gave better results for the cantilever mode, where n = m = 1.
© Carl T. F. Ross, 2011
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58. Flugge, W., Stresses in shells, Springer-Verlag, Berlin, 1973. 59. Galletly, G.D., Kruzelecki, J., Moffat, D.G. & Warrington, B., Buckling of shallow torispherical domes subjected to external pressure: A comparison of experiment, theory and design codes, J. Strain Anal., 22, 163–175, 1987. 60. Ross, C.T.F., & Johns, T., The effect of stiffener size on interframe shell instability of ring-reinforced circular cylinders, J. Ship Res., 15, 141–143, 1971. 61. Galletly, G.D. & Aylward, R. W., The influence of end closure shape on the buckling of cylinders under external pressure, Trans. RINA, 117, 255–268, 1975. 62. Bushnell, D., Large deflection elastic–plastic creep analysis of axisymmetric shells, in Numerical solution of non-linear structural problems, ed. R.F. Hartung, ASME, Winter Annual Meeting Detroit, 6, 103, 1973. 63. Walker, A. C. & McCall, S., Buckling tests on stringer-stiffened cylinder models subjected to load combination, Det Norske Veritas, Report 82-0299, 1982. 64. Walker, A.C. & Kwok, M.K., Process of damage in thin-walled cylindrical shells, in Advances in Marine structures, Elsevier Applied Science, 1987. 65. Kwok, M.K., McCall, S. & Walker, A.C., The behaviour of damaged cylindrical shells subjected to static and dynamic loading, in Appl. Solid Mech. – 2, ed. A. S. Tooth & J. Spence, Elsevier Applied Science, pp. 179–209, 1986. 66. Von Kármán, T. & H.-S. Tsien, The buckling of spherical shells by external pressure, in pressure vessel piping design, ASME Collected Papers, 1927–1959. 67. Kloppel, K. & Jungbluth, O., Beitrag zum Durchschlagproblem Dunnwandier Kugelschelen, Der Stahlbau, 1953. 68. Galletly, G.D., Blachut, J. & Kruzelecki, J., Plastic buckling of imperfect hemispherical shells subjected to external pressure, Proc. IMechE., 201, 153–170, 1987. 69. Bushnell, D., BOSOR5–program for buckling of elastic–plastic complex shells of revolution including large deflections and creep, Comp. Struct., 6, 221–239, 1976. 70. Newland, C. N., Collapse of domes under external pressure, Conference on Vessels under Buckling Conditions, IMechE, London, pp. 43–52, Dec. 1972. 71. Blachut, J., Buckling of sharp knuckle torispheres under external pressure, Thin-Walled Struct., 30(1–4), 55–77, 1998. 72. Rajagopalan, K., Finite element buckling analysis of stiffened cylindrical shells, Rotterdam: Balkema, 1993. 73. Guyan, R.J., Reduction of stiffness and mass matrices, AIAA J., 3, 380, 1965. 74. Ross, C.T.F., Vibration and elastic instability of thin-walled conical shells under external pressure, Comp. Struct., 55(1), 85–94, 1995. 75. Kendrick, S.B., Externally pressurised vessels, The stress analysis of pressure vessels and pressure vessel components, ed: S.S. Gill, Pergamon Press, 1970. 76. Ross, C.T.F., Vibration and elastic instability of thin-walled domes under uniform external pressure, Thin-Walled Struct., 26(3), 159–177, 1996. 77. Esslinger, M., & Geier, B., Buckling loads of thin-walled circular cylinders with axisymmetric irregularities, Paper 36, International Conference on Steel Plated Structures, 1974. 78. Malik, F., Morton, J., & Ruiz, C., An experimental investigation into the buckling of cylindrical shells of variable wall thickness under radial external pressure, Exper. Mech., 36, 87–92, 1979.
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79. Ross, C.T.F., Gill-Carson, A. & Little, A.P.F., The inelastic buckling of varying thickness circular cylinders under external hydrostatic pressure, Struct. Eng. Mech., 9(1), 51–68, 2000. 80. Tokugawa, T., Model experiments on the elastic stability of closed and crossstiffened circular cylinders under uniform external pressure, Proc. World Engineering Congress, Tokyo, Vol. 29, Paper No. 651, pp. 249–279, 1929. 81. Levy, M., Mémoire sur un nouveau cas intégrable du problème de l’elastique et l’une de ses applications, J. Math., Pure Appl. (Lionville), Ser. 3–10, 5–42, 1884. 82. Sechler, E.E., Stress distribution in stiffened panels under comparison, J. Aero. Sci., 4, 320, 1937. 83. Sechler, E.E., A preliminary report on the ultimate compressive strength of curved sheet panels, Gugganheim Aeronautical Laboratory California Institute of Technology, Publ. 36, 1937. 84. Bryant, A.R., Hydrostatic pressure buckling of a ring-stiffened tube, NCRE, Report No. R306, October 1954. 85. Kendrick, S., The buckling under external pressure of circular cylindrical shells with evenly spaced, equal strength, circular ring-frames: Part III, NCRE Report, No. R244, Sept. 1953. 86. Kendrick, S., The buckling under external pressure of ring-stiffened circular cylinders, Trans. Royal Inst. Technol., Sweden, 107, 139–156, 1965. 87. Kaminsky, E.L., General instability of ring-stiffened cylinders with clamped edges under external pressure by Kendrick’s method, DTMB Report, No. 855, July 1954. 88. Nash, W. A., General instability of ring-reinforced shells subject to hydrostatic pressure, Proc. 2nd US National Congress on Applied Mech., pp. 359–368, June 1954. 89. Nash, W.A., Hydrostatically loaded structures, Pergamon, Oxford, 1995. 90. Galletly, G.D., Slankard., R.C. & Wenk, E., General instability of ring-stiffened cylindrical shells subject to external hydrostatic pressure: A comparison of theory and experiment, J. Appl. Mech., ASME, 259–266, June 1958. 91. Ross, C.T.F., Vibration and instability of ring-reinforced circular cylindrical and conical shells, J. Ship Res., 20, 22–31, March 1976. 92. Ross, C. T. F., Finite elements for the vibration of cones and cylinders, Int. J. Num. Meth. Eng., 9, 833–845, 1975. 93. Ross, C.T.F., Aylward, W.R. & Boltwood, D.T., General instability of ringreinforced circular cylinders under external pressure, Trans. Royal Inst. Technol., Sweden, 113, 73–82, 1971. 94. Reynolds, T.E. & Blumenberg, W.F., General instability of ring-stiffened cylindrical shells subject to external hydrostatic pressure, DTMB Report No. 1324, June 1959. 95. Seleim, S.S. & Kennedy, J.B., Imperfection sensitivity of stiffened cylinders subjected to external pressure, Comp. Struct., 34(1), 63–69, 1990. 96. Ross, C.T.F., Inelastic general instability of ring-stiffened circular cylinders and cones under uniform external pressure, Int. J. Struct. Eng. Mech., 5(2), 193–207, 1997. 97. Ross, C.T.F., Coalter, B., & Johns, T., Design charts for the buckling of ringstiffened circular cylinders and cones under external hydrostatic pressure, Trans. Royal Inst. Technol., Sweden, 141, Part A, 15–31, 1999.
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98. Ross, C.T.F., Johns, T., & Emile, Vibration of ring-stiffened cones submerged in water, Proceedings of the NUMETA 87 Conference, 2, T18/1–8, 1987. 99. Ross, C.T.F., & Johns, T., Buckling and vibration of ring-stiffened cones under uniform external pressure, Thin-Walled Structures, 6, 321–342, 1988. 100. Ross, C.T.F., Sawkins, D., Thomas, J., & Little, A.P.F., Plastic collapse of circular conical shells under uniform external pressure, Adv. Eng. Software, 30, 631–647, 1999. 101. Ross, C.T.F., Sawkins, D., & Johns, T., Inelastic buckling of thick-walled circular conical shells under external hydrostatic pressure, Ocean Eng., 26, 1297–1311, 1999. 102. Ross, C.T.F., Spahiu, A., Brown, G.X. & Little, A.F.P., Buckling of near-perfect thick-walled circular cylinders under external hydrostatic pressure, J. Ocean Technol., 4(2), 84–103, 2009. 103. Ross, C.T.F., Okoto K.O., & Little, A.P.F., Buckling by general instability of cylindrical components of deep sea submersibles, Paper 300 in Proceedings of the Ninth International Conference on Computational Structures Technology, B.H.V. Topping and M. Papadrakakis, (Editors), Civil-Comp Press, 2008 Stirlingshire, Scotland. 104. Forsberg, K., A review of analytical methods used to determine the modal characteristics of cylindrical shells, NASA Rep., CR-613, 1966. 105. Abdulla, K.M. & Galletly, G.D., Free vibration of cones, cylinders and cone– cylinder combinations, Symposium on Structural Dynamics, Loughborough, 1, Paper No. B.2, 1970. 106. Warburton, G.B., Vibration of thin cylindrical shells, IMechE J. Mech. Eng. Sci., 7, 399–407, 1965. 107. Webster, J.J., Free Vibrations of shells of revolutions using ring finite elements, Int. J. Mech. Sci., 9, 559–570, 1967. 108. Percy, J.H., Pian, T.H., Klein, S. & Navaratana, D.R., Application of matrix displacement method to linear elastic analysis of shells of revolution, AIAA J., 3, 2138–2145, 1965. 109. Lindholm, U. S. & Hu, W.C.L., Non-symmetric transverse vibrations of truncated conical shells, Int. J. Mech. Sci., 8, 561–579, 1966. 110. Williams, J.J., Vibration of cooling-tower shells, Proc. Conf. on Natural Draught Cooling Towers: Ferrybridge and After, Inst. Civil Eng, pp. 37–43, 1967. 111. Forsberg, K., Exact solution for natural frequencies of ring-stiffened cylinders, AIAA/ASME 10th Structures, Structural Dynamics and Materials Conference, New Orleans, USA, pp. 18–30, 1969. 112. Bushnell, D., Analysis of buckling and vibration of ring-stiffened shells of revolution, Int. J. Solids Struct., 6, 157–181, 1970. 113. Weingarten, V.I., Free vibration of ring-stiffened conical shells, AIAA J., 3, 1475–1481, 1965. 114. Ross, C.T.F., & Johns, T., Vibration of submerged hemi-ellipsoidal domes. J. Sound Vibr., 91, 363–373, 1983. 115. Ross, C.T.F. & Johns, T., Vibration of hemi-ellipsoidal axisymmetric domes submerged in water, Proc. IMechE., 200, 389–398, 1986. 116. Zienkiewicz, O.C., & Newton, R.E., Coupled vibrations of a structure submerged in a compressible fluid, International Symposium on Finite Element Techniques, University of Stuttgart, 1969.
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117. Irons, B.M., Role of part-inversion in fluid structure problems with mixed variables, AIAA J., 7, 568, 1970. 118. Jennings, A., Matrix Computation for engineers and scientists, Wiley, 1977. 119. Port, K. F., & Ross, C.T.F., Free vibration of submerged thin-walled domes, Conference on Recent Advances in Structural Dynamics, Univ. of Southampton, 1980. 120. Petyt, M. & Lim, S.P., Finite element analysis of the noise inside a mechanically excited cylinder, IJNME, 13, 109–122, 1978. 121. Wunderlich, W., Private Communication, 1988. 122. Ross, C.T.F., Emile & Johns, T., Vibration of thin-walled domes under external water pressure, J. Sound and Vibr., 114, 453–463, 1987. 123. Ross, C.T.F., Johns, T., & Emile, Dynamic buckling of thin-walled domes under external water pressure, in Applied Solid Mechanics: 2 Conf., ed. A.S. Tooth & J. Spence, Elsevier Applied Science, pp. 211–224, 1987. 124. Ross, C.T.F. & Johns, T., Dynamic buckling of thin-walled domes under external water pressure, Res. Mech., 28, 113–137, 1989. 125. Ross, C.T.F. & Johns, T., Buckling and non-linear vibrations of a thin walled dome under external water pressure, J. Ship Res. 34, 142–148, 1990. 126. Ross, C.T.F., Johns, T. & Stanton, R.M., Vibrations of circular cylindrical shells under external water pressure, Proc. IMechE., J Mech. Eng. Sci., 206, 79–86, 1992. 127. Ross, C.T.F., Johns, T. & Richards, W.D., The vibration of ring-stiffened cones under external water pressure, Proc. IMechE, J. Mech. Eng. Sci., 208, 177–185, 1994. 128. Ross, C.T.F., Johns, T. Beeby, J.P. & May, S.F., Vibration of thin-walled ring-stiffened circular cylinders and cones, J. Thin-Walled Struct., 18, 177–190, 1994. 129. Ross, C.T.F., Haynes, P. & Richards, W.D., Vibration of ring-stiffened circular cylinders under external water pressure, Comp. Struct., 60(6), 1013–1019, 1996. 130. Ross, C.T.F., & Richards, W.D., The vibration of ring-reinforced circular cylinders under external water pressure, Proc. IMechE, J. Mech. Eng. Sci., 212, Part C, 299–306, 1998. 131. Ross, C.T.F., Taylor, M.W., Richards, W.D., & Little, A.P.F., Vibration of varying thickness circular cylinders under external water pressure, Comp. Struct., 73, 453–463, 1999. 132. Ross, C.T.F., Design of dome ends to withstand uniform external pressure, J. Ship Res., 31, 139–143, 1987. 133. Ross, C.T.F., A novel submarine pressure hull design, J. Ship Res., 31, 186–188, 1987. 134. Harris, F. L., Private Communication, Nov. 1977. 135. Anon., Submarine powered by closed-circuit diesel, Ocean Industry, 63, Nov. 1985. 136. Ross, C.T.F., & Laffoley-Lane, G.A., A conceptual design of an underwater drilling rig, SNAME J. Mar. Technol., 35(2), 99–113, April 1998. 137. Ross, C.T.F., A conceptual design of submarine to explore Europa’s oceans, J. Aerospace Eng., July 2007. 138. Yuan, K.-Y., Liang, C.-C., & Ma, Y.-C. Investigation of the cone angle of a novel swedged-stiffened pressure hull, SNAME J. Ship Res., 35, 83–86, 1991.
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139. Liang, C.-C., Yang, M.-F., & Chen, H.-W. Elastic–plastic axisymmetric failure of swedge-stiffened cylindrical pressure hull under external pressure, SNAME J. Ship Res., 37, 176–188, 1993. 140. Ross, C.T.F. & Palmer, A., General instability of swedge-stiffened circular cylinders under uniform external pressure, SNAME J. Ship Res., 37, 68–76, 1993. 141. Ross, C.T.F. & Humphries, M., The buckling of corrugated cylinders under uniform external pressure, J. Thin-Walled Struct., 17, 259–271, 1993. 142. Ross, C.T.F., Apor, G., & Claridge, S.P., Instability of circumferentially corrugated cylinders under uniform external pressure, Proceedings of the ASME/ EJDA International Conference on Structural Dynamics, Vibration and Buckling, 9, PD-81, 249–255, Montpellier, France, 1996. 143. Ross, C.T.F., Lilleland, S.E., Richards, W.D. & Little, A.P.F., Buckling and vibration of circumferentially corrugated cylinders under uniform external water pressure, Proceedings of the ASME International Conference on Engineering Technology, 3, 276–282, Houston, Texas, 1996. 144. Ross, C.T.F. & Waterman, G.A., Inelastic instability of circular corrugated cylinders under external hydrostatic pressure, Ocean Eng., 27, 331–343, 2000. 145. Ross, C.T.F., Waterman, G.A. & Terry, A., Inelastic instability of circular corrugated cylinders under external hydrostatic pressure, Proc. Int. Conf. on Computer Techniques for Civil & Structural Engineering, 197–202, Oxford, Civil-Comp. Press, 1999. 146. Ross, C.T.F., Palmer, M. & Johns, T., Buckling of longitudinally corrugated cylinders under uniform external pressure, Proc. ASME. Int. Conf. On Engineering Technology, 3, 283–289, Houston, Texas. 147. Irons, B.M., Role of a part-inversion on fluid-structure problems with mixed variables AIAA J., 7, 568, 1970. 148. Little, A.P.F. & Ross, C.T.F., The buckling of a carbon-fibre corrugated circular cylinder under external hydrostatic pressure, Proceedings on the International Conference on Computer Techniques for Civil and Structural Engineers, 189– 195, Oxford, Civil-Comp. Press, 1999. 149. Curtis, P.T., CRAG Test methods for the measurement of the engineering properties of fibre-reinforced plastics, RAE, Tech. Rep. 88012, 1988. 150. Curtis, P.T., Gates, J. & Molyneux, C.G., An improved engineering test method for the measurement of the compressive strength of unidirectional carbonfibre composites, DRA Farnborough Tech. Rep. 91031, 1991. 151. Little, A.P.F., & Ross, C.T.F., The vibration of a corrugated carbon fibre cylinder under external water pressure, Proceedings of the International Conference on Developments in Analysis and Design using Finite Element Methods, 119–125, Oxford, Civil Comp. Press, 1999. 152. Ross, C.T.F., & Popken, D., Buckling of tube-stiffened prolate domes under external water pressure, J. Thin-Walled Structures, 22, 159–179, 1995. 153. Khan, R. & Uddin, W., Instability of cup–cylinder compound shell under uniform external pressure, SNAME J. Ship Res., 39(2), 160–165, 1995. 154. Ross, C.T.F., & Rotherham, N., Collapse of inverted hemi-ellipsoidal shell domes under uniform pressure, SNAME J. Ship Res., 36, 378–386, 1992.
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155. Ross, C.T.F., & Etheridge, J., The vibration and instability of tube-stiffened axisymmetric shells under external hydrostatic pressure, 335–342, Civil Comp. Press, Edinburgh, 1998. 156. Ross, C.T.F., & Gill-Carson, A., Collapse of dome cup ends under external hydrostatic pressure, Proceedings of the International Conference on Computer Techniques for Civil and Structural Engineering, 203–210, Oxford, Civil Comp. Press, 1999. 157. Ross, C.T.F., Youster, P. & Sadler, R., The buckling of plastic oblate hemiellipsoidal dome shells under external hydrostatic pressure, Ocean Eng., 28, 789–803, 2000. 158. Ross, C.T.F., A redesign of the corrugated tin can, J. Thin-Walled Struct., 26(3), 179–193, 1996. 159. Bosman, T.N., Pegg, N.G. & Kenning, P.J., Experimental and numerical determination of non-linear overall collapse of imperfect pressure full compartments, International Symposium on Naval Submarines 4, RINA, 11–13 May, 1993, London. 160. British Standards Institution, 2009. PD 5500: 2009, Unfired fusion welded pressure vessels. 161. Ross, C.T.F., Whittaker, T. & Little, A.P.F., Design of submarine pressure hulls under external hydrostatic pressure, Proceedings of the Tenth International Conference on Computational Structures Technology, Valencia, 2010. 162. Ross, C.T.F., Bull, C., Al-Enezi, M. & Little, A.P.F., Geometrical and material non-linear analyses of model submarine pressure hulls, Proceedings of the Tenth International Conference on Computational Structures Technology, Valencia, 2010. 163. Dickens, G.R., Paull, C.K., Wallace, P., & the ODP Leg 164 Scientific Party, Direct measurement of in situ quantities in a large gas-hydrate reservoir, Nature, 385, 30th January, 1997. 164. Ross, C.T.F., The Silent Submarine, Inaugural lecture, University of Portsmouth, Portsmouth, UK, 1992. Also available at http://www.techfaculty.port. ac.uk/CTFR/ or http://homepage.ntlworld.com/Carl.ross/Publications. 165. Smith, C.S., Design of marine structures in composite materials, Elsevier Science Publishers Ltd., UK, 1990. 166. Haux, G., Subsea manned engineering, Balliere Tindall, London, 1981. 167. Ross, C.T.F., Advanced applied finite element methods, Horwood, Chichester, 1998. 168. SAS IP, Inc., ANSYS Help > Documentation > Element Reference > Element Library > Fluid 30, 2003. 169. Ross, C.T.F., A conceptual design of an underwater missile launcher, Ocean Eng., 32, 85–89, 2005.
© Carl T. F. Ross, 2011
Appendix I Computer program for axisymmetric stresses in circular cylinders stiffened by equal-strength ring frames
RINGCYLE.EXE This computer program calculates the axisymmetric deflections and stresses in a circular cylinder under uniform pressure, where the theory described in Section 2.2 is adopted. The program is interactive, and the data should be fed in as follows: L = length between adjacent stiffeners A = mean radius of shell H = wall thickness of cylinder RF = radius of ring centroid BF = width of the web of the stiffener that is in direct contact with the shell PRESS = pressure (positive external) E = Young’s modulus of elasticity NU = Poisson’s ratio
Output x (x = 0 at mid-span)
Deflection w
Hoop stress external internal
Longitudinal stress external internal
A listing for RINGCYLE.BAS is given below: 90 CLS 100 PRINT “COPYRIGHT OF PROF.C.T.F.ROSS”: PRINT 110 PRINT “NOT TO BE COPIED” 130 PRINT “HAVE YOU READ THE APPROPRIATE MANUAL” 140 PRINT : PRINT “BY” 150 PRINT : PRINT “C.T.F.ROSS” 160 PRINT “TYPE Y/N” 170 A$ = INKEY$: IF A$ = “” THEN GOTO 170 180 IF A$ = “Y” OR A$ = “y” THEN GOTO 192 190 PRINT : PRINT “GO BACK & READ IT!”: PRINT : END 428 © Carl T. F. Ross, 2011
Appendix I
429
192 INPUT “TYPE IN THE NAME OF YOUR OUTPUT FILE ”; OUT$ 193 PRINT “IF YOU ARE SATISFIED THAT THIS NAME IS CORRECT, TYPE Y; ELSE N” 194 A$ = INKEY$: IF A$ = “” THEN GOTO 194 195 IF A$ = “Y” OR A$ = “y” THEN GOTO 199 196 IF A$ = “N” OR A$ = “n” THEN GOTO 192 197 GOTO 194 199 OPEN OUT$ FOR OUTPUT AS #2 200 PRINT “STRESSES IN RING-STIFFENED CYLINDERS UNDER COMBINED LATERAL & AXIAL PRESSURE”: PRINT PRINT “UNIFORM STRENGTH STIFFENERS” 210 PRINT : PRINT “EXTERNAL PRESSURE IS SAID TO BE POSITIVE”: PRINT 240 PRINT #2, “STRESSES IN RING-STIFFENED CYLINDERS UNDER COMBINED LATERAL & AXIAL PRESSURE” PRINT #2, “UNIFORM STRENGTH STIFFENERS” INPUT “TYPE IN LENGTH BETWEEN ADJACENT STIFFENERS ”; L INPUT “TYPE IN MEAN SHELL RADIUS ”; A INPUT “TYPE IN SHELL THICKNESS ”; H INPUT “TYPE IN CROSS-SECTIONAL AREA OF RING ”; AF INPUT “TYPE IN RADIUS OF RING CENTROID ”; RF INPUT “TYPE IN THE WIDTH OF THE STIFFENER WEB IN CONTACT WITH THE SHELL ”; BF INPUT “TYPE IN THE PRESSURE (+VE EXTERNAL) ”; PRESS INPUT “TYPE IN YOUNG’S MODULUS ”; E 350 INPUT “TYPE IN POISSON’S RATIO ”; NU PRINT #2, “LENGTH BETWEEN ADJACENT STIFFENERS=”; L PRINT #2, “MEAN SHELL RADIUS=”; A PRINT #2, “SHELL THICKNESS=”; H PRINT #2, “CROSS-SECTIONAL AREA=”; AF PRINT #2, “RADIUS OF RING CENTROID=”; RF PRINT #2, “WIDTH OF THE STIFFENER WEB IN CONTACT WITH THE SHELL=”; BF PRINT #2, “PRESSURE=”; PRESS PRINT #2, “YOUNG’S MODULUS=”; E PRINT #2, “POISSON’S RATIO=”; NU 630 PRINT #2, AL = SQR(SQR(3 * (1 - NU ^ 2) / (A ^ 2 * H ^ 2))) BE = SQR(PRESS * A ^ 3 / (2 * E * H)) + H ^ 2 * NU / (12 * (1 - NU ^ 2)) P = PRESS
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CO = P * A ^ 2 * (1 - .5 * NU) / (E * H) F1 = AL * SQR(1 - AL ^ 2 * BE ^ 2) F2 = AL * SQR(1 + AL ^ 2 * BE ^ 2) G1 = E * (AF / RF ^ 2 + BF * H / A ^ 2) H1 = P * BF * (1 - NU / 2) SYNH = (EXP(.5 * F1 * L) - EXP(-.5 * F1 * L)) * .5 KOSH = (EXP(.5 * F1 * L) + EXP(-.5 * F1 * L)) * .5 SYN = SIN(.5 * F2 * L) KOS = COS(.5 * F2 * L) N1 = -(G1 * CO - H1) * (F1 * KOSH * SYN + F2 * SYNH * KOS) N2 = (G1 * CO - H1) * (F1 * SYNH * KOS - F2 * KOSH * SYN) KOSH = (EXP(F1 * L) + EXP(-F1 * L)) * .5 SYNH = (EXP(F1 * L) - EXP(-F1 * L)) * .5 KOS = COS(F2 * L): SYN = SIN(F2 * L) D1 = (E * H ^ 3 / (12 * (1 - NU ^ 2))) * (2 * F1 * F2 * (F1 ^ 2 + F2 ^ 2) * (KOSH - KOS)) D1 = D1 + .5 * G1 * (F1 * SYN + F2 * SYNH) A1 = N1 / D1 A2 = N2 / D1 C1 = F1 ^ 2 - F2 ^ 2 C2 = 2 * F1 * F2 C3 = A1 * C1 + A2 * C2 C4 = A2 * C1 - A1 * C2 PRINT #2, “X=0 IS AT MID-SPAN & X=”; .5 * L; “ IS AT A FRAME” 1680 REM CALCULATION OF DEFLECTION & STRESS DISTRIBUTIONS 1700 PRINT #2, “ X DEFLECTION HOOPEX HOOPIN LONGEX LONGIN” 1720 FOR I = 0 TO 10 1730 X = I * L / 20 F1X = F1 * X F2X = F2 * X SYNH = (EXP(F1X) - EXP(-F1X)) * .5 KOSH = (EXP(F1X) + EXP(-F1X)) * .5 SYN = SIN(F2X) KOS = COS(F2X) W = A1 * KOSH * KOS + A2 * SYNH * SYN + CO D2W = C3 * KOSH * KOS + C4 * SYNH * SYN HB = E * H * (W / A ^ 2 + NU * D2W) / (2 * (1 - NU ^ 2))
© Carl T. F. Ross, 2011
Appendix I LB = E * H * (NU ^ 2)) HOOP1 = -E * W / HOOP2 = -E * W / LONG1 = -PRESS * LONG2 = -PRESS * 1910 PRINT #2, “ “ ”; HOOP2; “ NEXT I 1960 END
431
* W / A ^ 2 + D2W) / (2 * (1 - NU A A A / A / ”; ”;
PRESS * A * NU / (2 * H) + HB PRESS * A * NU / (2 * H) - HB (2 * H) + LB (2 * H) - LB X; “ ”; W; “ ”; HOOP1; LONG1; “ ”; LONG2
© Carl T. F. Ross, 2011
Appendix II Computer program for axisymmetric stresses in circular cylinders stiffened by unequal-strength ring frames
RINGCYLG.EXE This computer program calculates the axisymmetric deflections and stresses in a circular cylinder stiffened by unequal-strength ring frames. The cylinder can be stiffened or unstiffened, and the lateral pressure can be made to vary along the length of the vessel. The stiffeners can be internal or external, or both, and the wall thickness of the shell can have step variation. The program is based on the theory described in Section 2.2.1. The program is interactive, and the data should be fed in as follows: N = number of ring-stiffeners E = Young’s modulus of elasticity NU = Poisson’s ratio P1 = axial pressure FOR I = 0 TO L(I) H(I) A(I) NEXT I
N = length of the Ith bay = thickness of the Ith bay = mean shell radius in the Ith bay
FOR I = 0 TO N + 1 P(I) = lateral pressure at node I NEXT I FOR I = 0 TO N + 1 B(I) = width of the web of the Ith ring in contact with the shell AF(I) = sectional area of the Ith ring R(I) = centroidal radius of the Ith ring IN(I) = second moment of area of the Ith ring about its z–z axis NEXT I 432 © Carl T. F. Ross, 2011
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Output Bay 0 TO N x Distance from end of bay
Deflection w
Hoop stress external internal
Longitudinal stress external internal
A listing for RINGCYLG.BAS is given below: 90 CLS 100 PRINT “COPYRIGHT OF DR.C.T.F.ROSS”: PRINT 110 PRINT “NOT TO BE COPIED” 130 PRINT “HAVE YOU READ THE APPROPRIATE MANUAL” 140 PRINT : PRINT “BY” 150 PRINT : PRINT “C.T.F.ROSS” 160 PRINT “TYPE Y/N” 170 A$ = INKEY$: IF A$ = “” THEN GOTO 170 180 IF A$ = “Y” OR A$ = “y” THEN GOTO 192 190 PRINT : PRINT “GO BACK & READ IT!”: PRINT : END 192 INPUT “TYPE IN THE NAME OF YOUR OUTPUT FILE ”; OUT$ 193 PRINT “IF YOU ARE SATISFIED THAT THIS NAME IS CORRECT, TYPE Y; ELSE N” 194 A$ = INKEY$: IF A$ = “” THEN GOTO 194 195 IF A$ = “Y” OR A$ = “y” THEN GOTO 199 196 IF A$ = “N” OR A$ = “n” THEN GOTO 192 197 GOTO 194 199 OPEN OUT$ FOR OUTPUT AS #2 200 PRINT “STRESSES IN RING-STIFFENED CYLINDERS UNDER COMBINED LATERAL & AXIAL PRESSURE”: PRINT 210 PRINT : PRINT “EXTERNAL PRESSURE IS SAID TO BE POSITIVE”: PRINT 220 PRINT “NO. OF RING STIFFENERS=”; : INPUT N N10 = (N + 1) * 10 240 PRINT #2, “STRESSES IN RING-STIFFENED CYLINDERS UNDER COMBINED LATERAL & AXIAL PRESSURE” 250 N2 = 2 * N 260 DIM L(N), H(N), A(N), B(N), AF(N), R(N), IN(N), P(N + 1) 270 DIM D(N), AL(N), CO(N), C1(N), C2(N), C3(N), C4(N), F1(N), F2(N), F3(N), F4(N), F5(N), K1(N), K2(N), K3(N), K4(N), K5(N) 280 DIM SH(N), KH(N), SN(N), KS(N) 290 DIM S(N2, N2), W(N2), U(N2)
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DIM X(N10), Y(N10), DEFLECT(N10), HOOPIN(N10), HOOPEX(N10), LONGIN(N10), LONGEX(N10) 300 FOR II = 1 TO N2 310 FOR JJ = 1 TO N2 320 S(II, JJ) = 0 330 NEXT JJ, II 340 INPUT “TYPE IN ELASTIC MODULUS ”; E 350 INPUT “TYPE IN POISSON’S RATIO ”; NU 360 INPUT “TYPE IN AXIAL PRESSURE ”; P1 362 GOSUB 9500 365 IF X$ <> “Y” THEN GOTO 340 370 PRINT #2, “NO. OF RING-STIFFENERS=”; N 380 PRINT #2, “ELASTIC MODULUS=”; E 390 PRINT #2, “POISSON’S RATIO=”; NU 400 PRINT #2, “AXIAL PRESSURE=”; P1 410 PRINT #2, 420 FOR I = 0 TO N 430 PRINT “BAY ”; I; “ LENGTH=”; : INPUT L(I) 440 PRINT “BAY ”; I; “ THICKNESS=”; : INPUT H(I) 450 PRINT “BAY ”; I; “ RADIUS=”; : INPUT A(I) 460 PRINT 470 NEXT I 474 GOSUB 9500 475 IF X$ <> “Y” THEN GOTO 410 480 FOR I = 0 TO N + 1 490 PRINT “LATERAL PRESSURE AT NODE ”; I; “=”; : INPUT P(I) 495 PRINT #2, “LATERAL PRESSURE AT NODE ”; I; “=”; P(I) 500 NEXT I 502 GOSUB 9500 505 IF X$ <> “Y” THEN GOTO 480 510 FOR I = 1 TO N 520 PRINT “B[“; I; ”]=”; : INPUT B(I) 530 PRINT “AF[“; I; ”]=”; : INPUT AF(I) 540 PRINT “R[“; I; ”]=”; : INPUT R(I) 550 PRINT “IN[“; I; ”]=”; : INPUT IN(I) 560 NEXT I 562 GOSUB 9500 565 IF X$ <> “Y” THEN GOTO 510 570 REM OUTPUT 590 PRINT #2, “BAY LENGTH THICKNESS RADIUS” 600 FOR I = 0 TO N
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610 PRINT #2, I; “ ”; L(I); “ ”; H(I); “ ”; A(I) 620 NEXT I 630 PRINT #2, 640 FOR I = 0 TO N + 1 650 PRINT #2, “LATERAL PRESSURE AT NODE ”; I; “=”; P(I) 660 NEXT I 670 PRINT #2, 710 PRINT #2, “GEOMETRICAL PROPERTIES OF THE RINGS” 720 PRINT #2, “ B[I] AF[I] R[I] IN[I]” 730 FOR I = 1 TO N 740 PRINT #2, B(I); “ ”; AF(I); “ ”; R(I); “ ”; IN(I) 750 NEXT I 780 REM END OF OUTPUT 790 REM CALCULATION OF C1 TO C4 800 FOR I = 0 TO N 810 D(I) = E * H(I) ^ 3 / (12 * (1 - NU ^ 2)) 815 PA = P1 820 AL(I) = SQR(SQR((3 - 3 * NU ^ 2) / (A(I) ^ 2 * H(I) ^ 2))) 825 BT = SQR(P1 * A(I) ^ 3 / (2 * E * H(I)) + H(I) ^ 2 * NU / (12 - 12 * NU ^ 2)) 830 CO(I) = A(I) ^ 2 * P(I) * (1 - .5 * NU * PA / P(I)) / (E * H(I)) 835 F1 = AL(I) * SQR(1 - AL(I) ^ 2 * BT ^ 2) 837 F2 = AL(I) * SQR(1 + AL(I) ^ 2 * BT ^ 2) 840 KO(I) = A(I) ^ 2 * (P(I + 1) - P(I)) / (E * H(I) * L(I)) 850 KH(I) = (EXP(F1 * L(I)) + EXP(-F1 * L(I))) * .5: SH(I) = (EXP(F1 * L(I)) - EXP(-F1 * L(I))) * .5 855 SN(I) = SIN(F2 * L(I)): KS(I) = COS(F2 * L(I)) 860 C1(I) = KS(I) * KH(I): C2(I) = SN(I) * SH(I) 870 C3(I) = SN(I) * KH(I): C4(I) = KS(I) * SH(I) 890 NEXT I 900 FOR I = 0 TO N 910 REM CALCULATIONS OF ‘SHELL’ FUNCTIONS 920 GOSUB 1970 925 D = D(I) 930 CN = F1 * F1 - F2 * F2: AN = F1 * F1 - F2 * F2 + 2 * Y1 * F1 * F2
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935 BN = Y1 * CN - 2 * F1 * F2: EN = -F1 * F1 - F2 * F2 940 F1(I) = (D * (AN * C1(I) + BN * C2(I) + S6 * (-F1 * CN / F2 - 2 * F1 * F2) * C3(I) + S6 * EN * C4(I))) 945 GN = -F1 * (F1 * F1 - F2 * F2) / F2 - 2 * F1 * F2: HN = 2 * F1 + S7 * (-F1 * F1 - F2 * F2) 950 F2(I) = (D * (2 * F1 * F2 * Y2 * C1(I) + Y2 * CN * C2(I) + (CN / F2 + S7 * GN) * C3(I) + HN * C4(I))) 955 KN = -F1 * (F1 * F1 - F2 * F2) / F2 - 2 * F1 * F2: LN = -F1 * F1 - F2 * F2 960 F3(I) = (D * (2 * F1 * F2 * Y3 * C1(I) + Y3 * CN * C2(I) + S8 * KN * C3(I) + S8 * LN * C4(I))) 970 F4(I) = (D * (2 * F1 * F2 * Y4 * C1(I) + Y4 * CN * C2(I) + S9 * KN * C3(I) + S9 * LN * C4(I))) 975 MN = Y5 * CN + 2 * CO * F1 * F2 980 F5(I) = (D * ((-CO * CN + 2 * F1 * F2 * Y5) * C1(I) + MN * C2(I) + S0 * KN * C3(I) + S0 * LN * C4(I))) 985 CN = 0: AN = 0: BN = 0: EN = 0: GN = 0: HN = 0: KN = 0: LN = 0: MN = 0 1000 PN = F1 * F1 - 3 * F2 * F2 1002 QN = 3 * F1 * F1 - F2 * F2 1004 RN = F2 * F2 - 3 * F1 * F1 1006 TN = F1 * F1 + F2 * F2 1008 CN = F1 * PN + Y1 * F2 * QN 1010 AN = F2 * RN + Y1 * F1 * PN 1012 BN = F2 * RN - F1 * F1 * PN / F2 1014 K1(I) = (D * (CN * C4(I) + AN * C3(I) + S6 * F1 * (-2 * TN) * C1(I) + S6 * BN * C2(I))) 1016 EN = QN + S7 * F1 * (-2 * TN) 1018 GN = F2 * RN - F1 * F1 * PN / F2 1020 K2(I) = (D * (Y2 * F2 * QN * C4(I) + Y2 * F1 * PN * C3(I) + EN * C1(I) + (F1 * PN / F2 + S7 * GN) * C2(I))) 1022 HN = -2 * TN 1024 KN = F2 * RN - F1 * F1 * PN / F2 1026 K3(I) = (D * (Y3 * F2 * QN * C4(I) + Y3 * F1 * PN * C3(I) + S8 * F1 * HN * C1(I) + S8 * KN * C2(I))) 1028 LN = F2 * RN - F1 * F1 * PN / F2 1030 MN = Y4 * F2 * QN * C4(I)
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1032 K4(I) = (D * (MN + Y4 * F1 * PN * C3(I) + S9 * F1 * HN * C1(I) + S9 * LN * C2(I)) + .5 * P1 * A(I)) 1034 UN = -CO * F1 * PN + Y5 * F2 * QN 1036 VN = -CO * F2 * RN + Y5 * F1 * PN 1038 WN = F2 * RN - F1 * F1 * PN / F2 1040 K5(I) = (D * (UN * C4(I) + VN * C3(I) + S0 * F1 * HN * C1(I) + S0 * WN * C2(I))) 1060 NEXT I 1070 FOR I = 1 TO N 1080 REM CALCULATION OF ‘RING’ FUNCTIONS 1090 GOSUB 1970 1100 D = D(I) 1105 Y6 = D * ((F1 * F1 - F2 * F2) + 2 * Y1 * F1 * F2) 1110 Y7 = D * 2 * Y2 * F1 * F2 1115 Y8 = D * 2 * Y3 * F1 * F2 1120 Y9 = D * 2 * Y4 * F1 * F2 1125 Y0 = D * (2 * Y5 * F1 * F2 - CO * (F1 * F1 F2 * F2)) 1130 E1 = D * F1 * (-2 * (F1 * F1 + F2 * F2)) * S6 1135 E2 = (D * (3 * F1 * F1 - F2 * F2) + D * F1 * (-2 * (F1 * F1 + F2 * F2)) * S7 + .5 * P1 * A(I)) 1140 E3 = D * F1 * (-2 * (F1 * F1 + F2 * F2)) * S8 1145 E4 = D * F1 * (-2 * (F1 * F1 + F2 * F2)) * S9 1150 E5 = D * F1 * (-2 * (F1 * F1 + F2 * F2)) * S0 1200 REM CALCULATION OF LOAD & ‘STIFFNESS’ MATRICES 1210 IF I = 1 THEN GOTO 1360 1220 IF I = N THEN GOTO 1450 1230 S(2 * I - 1, 2 * I - 3) = -F1(I - 1) * A(I - 1) 1240 S(2 * I - 1, 2 * I - 2) = -F2(I - 1) * A(I - 1) 1250 S(2 * I - 1, 2 * I - 1) = -F3(I - 1) * A(I 1) + Y6 * A(I) 1260 S(2 * I - 1, 2 * I) = -F4(I - 1) * A(I - 1) + Y7 * A(I) - E * IN(I) / R(I) 1270 S(2 * I - 1, 2 * I + 1) = Y8 * A(I) 1280 S(2 * I - 1, 2 * I + 2) = Y9 * A(I) 1290 S(2 * I, 2 * I - 3) = -K1(I - 1) * A(I - 1) 1300 S(2 * I, 2 * I - 2) = -K2(I - 1) * A(I - 1) 1310 S(2 * I, 2 * I - 1) = -K3(I - 1) * A(I - 1) + E1 * A(I) + E * AF(I) / R(I)
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1320 S(2 * I, 2 * I) = -K4(I - 1) * A(I - 1) + E2 * A(I) 1330 S(2 * I, 2 * I + 1) = E3 * A(I) 1340 S(2 * I, 2 * I + 2) = E4 * A(I) 1350 GOTO 1530 1360 S(1, 1) = -F3(0) * A(0) + Y6 * A(1) 1370 S(1, 2) = -F4(0) * A(0) + Y7 * A(1) - E * IN(I) / R(I) 1380 S(1, 3) = Y8 * A(1) 1390 S(1, 4) = Y9 * A(1) 1400 S(2, 1) = -K3(0) * A(0) + E1 * A(1) + E * AF(I) / R(I) 1410 S(2, 2) = -K4(0) * A(0) + E2 * A(1) 1420 S(2, 3) = E3 * A(1) 1430 S(2, 4) = E4 * A(1) 1440 GOTO 1530 1450 S(2 * N, 2 * N) = -K4(N - 1) * A(N - 1) + E2 * A(N) 1460 S(2 * N, 2 * N - 1) = -K3(N - 1) * A(N - 1) + E1 * A(N) + E * AF(I) / R(I) 1470 S(2 * N, 2 * N - 2) = -K2(N - 1) * A(N - 1) 1480 S(2 * N, 2 * N - 3) = -K1(N - 1) * A(N - 1) 1490 S(2 * N - 1, 2 * N) = -F4(N - 1) * A(N - 1) + Y7 * A(N) - E * IN(I) / R(I) 1500 S(2 * N - 1, 2 * N - 1) = -F3(N - 1) * A(N 1) + Y6 * A(N) 1510 S(2 * N - 1, 2 * N - 2) = -F2(N - 1) * A(N - 1) 1520 S(2 * N - 1, 2 * N - 3) = -F1(N - 1) * A(N - 1) 1530 W(2 * I - 1) = F5(I - 1) * A(I - 1) - Y0 * A(I) 1540 W(2 * I) = P(I) * B(I) * A(I) * (1 - .5 * NU * P1 / P(I)) + K5(I - 1) * A(I - 1) - E5 * A(I) 1550 NEXT I 1560 PRINT : PRINT “THE STIFFNESS MATRIX IS BEING INVERTED”: PRINT 1570 GOSUB 2190 1580 GOSUB 2470 1590 FOR I = 1 TO N2 1600 W(I) = U(I) 1610 NEXT I 1620 WW(N + 1) = 0: TH(N + 1) = 0
© Carl T. F. Ross, 2011
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1630 WW(0) = 0: TH(0) = 0 1640 FOR I = 1 TO N 1650 WW(I) = W(2 * I - 1) 1660 TH(I) = W(2 * I) 1670 NEXT I 1680 REM CALCULATION OF DEFLECTION & STRESS DISTRIBUTIONS III = -1 1690 FOR I = 0 TO N 1700 PRINT #2, “BAY DEFLECTION HOOPEX HOOPIN LONGEX LONGIN” 1710 GOSUB 1970 1720 FOR J = 0 TO 10 1730 X = .1 * J * L(I) 1740 A1 = WW(I) - CO(I) 1750 A2 = Y1 * WW(I) + Y2 * TH(I) + Y3 * WW(I + 1) + Y4 * TH(I + 1) + Y5 1760 A3 = -F1 * (S6 * WW(I) + (S7 - 1 / F1) * TH(I) + S8 * WW(I + 1) + S9 * TH(I + 1) + S0) / F2 1770 A4 = S6 * WW(I) + S7 * TH(I) + S8 * WW(I + 1) + S9 * TH(I + 1) + S0 1780 R1 = (EXP(F1 * X) + EXP(-F1 * X)) * .5 * COS(F2 * X) 1785 R2 = (EXP(F1 * X) - EXP(-F1 * X)) * .5 * SIN(F2 * X) 1790 R3 = (EXP(F1 * X) + EXP(-F1 * X)) * .5 * SIN(F2 * X) 1795 R4 = (EXP(F1 * X) - EXP(-F1 * X)) * .5 * COS(F2 * X) 1800 DN = A1 * R1 + A2 * R2 + A3 * R3 + A4 * R4 + CO 1805 AN = F1 * F1 - F2 * F2 1810 BN = A2 * AN - 2 * A1 * F1 * F2 1815 CN = A3 * AN - 2 * A4 * F1 * F2 1820 EN = A4 * AN + 2 * A3 * F1 * F2 1825 D2 = (A1 * AN + 2 * A2 * F1 * F2) * R1 + BN * R2 + CN * R3 + EN * R4 1830 AN = 0: BN = 0: CN = 0: EN = O 1835 AN = E * H(I) / (1 - NU * NU) * (DN / A(I) ^ 2 + NU * D2) 1840 BN = E * H(I) / (1 - NU * NU) * (NU * DN / A(I) ^ 2 + D2) 1845 H1 = -E * DN / A(I) - .5 * P1 * A(I) * NU / H(I) + .5 * AN
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1850 H2 = -E * DN / A(I) - .5 * P1 * A(I) * NU / H(I) - .5 * AN 1855 T1 = -.5 * P1 * A(I) / H(I) + .5 * BN 1860 T2 = -.5 * P1 * A(I) / H(I) - .5 * BN 1910 PRINT #2, “ ”; I; “ ”; DN; “ ”; H1; “ ”; H2; “ ”; T1; “ ”; T2 IF I > 0 AND J = 0 THEN GOTO 1920 III = III + 1 IF III > 0 THEN XL = X(III - 1) X(III) = XL + L(I) / 10 X(0) = 0 DEFLECT(III) = DN HOOPEX(III) = H1 HOOPIN(III) = H2 LONGEX(III) = T1 LONGIN(III) = T2 1920 NEXT J 1930 NEXT I GOSUB 10000 1960 END 1970 REM SUBROUTINE VALSYS 1980 AP = SQR(SQR((3 - 3 * NU * NU) / (A(I) * A(I) * H(I) * H(I)))) 1985 BT = SQR(P1 * A(I) ^ 3 / (2 * E * H(I)) + H(I) * H(I) * NU / (12 - 12 * NU * NU)) 1990 CO = P(I) * A(I) * A(I) * (1 - .5 * NU * P1 / P(I)) / (E * H(I)) 1995 F1 = AP * SQR(1 - AP * AP * BT * BT) 2000 F2 = AP * SQR(1 + AP * AP * BT * BT) 2010 S1 = -(F1 * C4(I) - C1(I) * F2 * KS(I) / SN(I) - C1(I) * F1 * KH(I) / SH(I) - F2 * C3(I)) 2015 S2 = -(-F1 * SN(I)) / (F2 * SH(I)) 2020 S3 = -(F2 * KS(I) / SN(I) + F1 * KH(I) / SH(I)) 2025 AN = F1 * C3(I) / F2 - C4(I) 2030 BN = F1 * F1 / F2 + F2 2035 S4 = (AN * F2 * KS(I) / SN(I) + AN * F1 * KH(I) / SH(I) - BN * C2(I)) 2040 CN = C1(I) - 1 2045 S5 = -CO * (-F1 * C4(I) + CN * F2 * KS(I) / SN(I) + CN * F1 * KH(I) / SH(I) + F2 * C3(I)) 2050 S6 = S1 / S4 2055 S7 = S2 / S4
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2060 S8 = S3 / S4 2065 S9 = 1 / S4 2070 S0 = S5 / S4 2075 Y1 = (-C1(I) / C2(I) + S6 * (F1 * KH(I) / (F2 * SH(I)) - KS(I) / SN(I))) 2080 Y2 = (-KH(I) / (F2 * SH(I)) + S7 * (F1 * KH(I) / (F2 * SH(I)) - KS(I) / SN(I))) 2085 Y3 = (1 / C2(I) + S8 * (F1 * KH(I) / (F2 * SH(I)) - KS(I) / SN(I))) 2087 Y4 = (S9 * (F1 * KH(I) / (F2 * SH(I)) - KS(I) / SN(I))) 2090 Y5 = (CO * (C1(I) - 1) / C2(I) + S0 * (F1 * KH(I) / (F2 * SH(I)) - KS(I) / SN(I))) 2180 RETURN 2190 REM SUBROUTINE INVERT MATRIX 2200 FOR II = 1 TO N2 2210 CN = S(II, 1) 2220 S(II, 1) = S(II, 2) 2230 S(II, 2) = CN 2240 NEXT II 2250 FOR UU = 1 TO N2 2260 PK = S(UU, 1) 2270 IF PK = 0 THEN PRINT “THE MATRIX IS SINGULAR” 2280 FOR VV = 1 TO N2 - 1 2290 S(UU, VV) = S(UU, VV + 1) / PK 2300 NEXT VV 2310 S(UU, N2) = 1 / PK 2320 FOR U = 1 TO N2 2330 IF U = UU THEN GOTO 2390 2340 R = S(U, 1) 2350 FOR VV = 1 TO N2 - 1 2360 S(U, VV) = S(U, VV + 1) - R * S(UU, VV) 2370 NEXT VV 2380 S(U, N2) = -R * S(UU, N2) 2390 NEXT U 2400 NEXT UU 2410 FOR II = 1 TO N2 2420 CN = S(1, II) 2430 S(1, II) = S(2, II) 2440 S(2, II) = CN 2450 NEXT II 2460 RETURN 2470 REM SUBROUTINE MATRIX PRODUCT
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2480 FOR I = 1 TO N2 2500 U(I) = 0 2520 NEXT I 2530 FOR I = 1 TO N2 2550 FOR K = 1 TO N2 2560 U(I) = U(I) + S(I, K) * W(K) 2570 NEXT K 2590 NEXT I 2600 RETURN 9500 REM CHECK 9510 PRINT : PRINT “IF YOU ARE SATISFIED WITH THE INPUT OF YOUR LAST BATCH OF DATA, TYPE Y; ELSE TYPE N” 9520 X$ = INKEY$: IF X$ = “” THEN GOTO 9520 9530 IF X$ = “Y” OR X$ = “y” OR X$ = “N” OR X$ = “n” THEN GOTO 9550 9540 GOTO 9520 9550 IF X$ = “y” THEN X$ = “Y” 9560 RETURN 10000 YMIN = 1E+12: YMAX = -1E+12: SMIN = 1E+12: SMAX = -1E+12 FOR II = 1 TO N10 IF DEFLECT(II) > YMAX THEN YMAX = DEFLECT(II) IF DEFLECT(II) < YMIN THEN YMIN = DEFLECT(II) IF LONGIN(II) > SMAX THEN SMAX = LONGIN(II) IF LONGIN(II) < SMIN THEN SMIN = LONGIN(II) DELY = YMAX - YMIN DELS = SMAX - SMIN NEXT II FOR NGRAPH = 1 TO 5 XMIN = X(0): XMAX = X(N10) IF NGRAPH = 1 THEN YDIFF = DELY ELSE YDIFF = DELS SCREEN 9 IF NGRAPH = 1 THEN CONY = 250 ELSE CONY = 150 LINE (0, CONY)-(600, CONY) IF NGRAPH = 1 THEN PRINT “DEFLECTED SHAPE” IF NGRAPH = 2 THEN PRINT “HOOP STRESS(EXTERNAL)” IF NGRAPH = 3 THEN PRINT “HOOPSTRESS(INTERNAL)” IF NGRAPH = 4 THEN PRINT “LONGITUDINAL STRESS(EXTERNAL)” IF NGRAPH = 5 THEN PRINT “LONGITUDINAL STRESS(INTERNAL)” XSCALE = 450 / (X(N10) - X(0)) YSCALE = 225 / YDIFF
© Carl T. F. Ross, 2011
Appendix II FOR II = 0 TO N10 IF NGRAPH = 1 THEN Y(II) = DEFLECT(II) IF NGRAPH = 2 THEN Y(II) = HOOPEX(II) IF NGRAPH = 3 THEN Y(II) = HOOPIN(II) IF NGRAPH = 4 THEN Y(II) = LONGEX(II) IF NGRAPH = 5 THEN Y(II) = LONGIN(II) IF II = 0 THEN GOTO 10200 XI = X(II - 1) * XSCALE: XJ = X(II) * XSCALE YI = Y(II - 1) * YSCALE: YJ = Y(II) * YSCALE LINE (40 + XI, CONY - YI)-(40 + XJ, CONY - YJ) 10200 REM IF II < N10 THEN GOTO 10400 PRINT “TO CONTINUE, PRESS Y”; 10210 A$ = INKEY$: IF A$ = “” THEN GOTO 10210 IF A$ = “Y” OR A$ = “y” THEN GOTO 10300 GOTO 10210 10300 SCREEN 0: WIDTH 80 10400 NEXT II NEXT NGRAPH RETURN
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Appendix III Computer programs for shell instability
MISESNP.EXE This computer program calculates the elastic buckling pressure for a perfect circular cylinder subjected to uniform external pressure and simply supported at its ends. The buckling pressures are based on the DTMB formula and the von Mises formula. The program also calculates the Windenburg thinness ratio. The program is interactive, and the data for it should be fed in as follows: L = unsupported length of cylinder A = mean shell radius H = wall thickness of shell E = Young’s modulus of elasticity NU = Poisson’s ratio SIGMAYP = yield stress
NIORDSON.EXE This program calculates the shell instability buckling pressure for a perfect circular conical shell by an approximate method. E = Young’s modulus of elasticity NU = Poisson’s ratio T = wall thickness R1 = mean shell radius at ‘left’ end of cone R2 = mean shell radius at ‘right’ end of cone Lf = unsupported length of conical shell between adjacent ring stiffeners. A listing for MISESNP.BAS is given below: 100 REM VON MISES FORMULA, DTMB FORMULA & THINNESS RATIO 110 REM PROGRAM BY DR C.T.F.ROSS 120 CLS 130 PRINT : PRINT “VON MISES FORMULA, DTMB FORMULA & THINNESS RATIO” 140 PRINT “PROGRAM BY DR.C.T.F.ROSS” 444 © Carl T. F. Ross, 2011
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150 INPUT “TYPE IN UNSUPPORTED LENGTH OF CYLINDRICAL SHELL ”; L 160 INPUT “TYPE IN MEAN SHELL RADIUS ”; A 170 D = 2 * A 180 INPUT “TYPE IN SHELL THICKNESS ”; H 190 INPUT “TYPE IN YOUNG’S MODULUS OF ELASTICITY ”; E 200 INPUT “TYPE IN POISSON’S RATIO ”; NU 210 INPUT “TYPE IN YIELD STRESS ”; SIGMAYP 215 GOSUB 500 220 C1 = (L / D) ^ 2 230 C2 = (H / D) ^ 3 240 C3 = SIGMAYP / E 250 LAMBDA = SQR(SQR(C1 / C2)) * SQR(C3) 260 PRINT “LAMBDA=”; LAMBDA 270 C1 = (H / D) ^ 2.5 280 C2 = SQR(H / D) 290 C3 = L / D 300 P = 2.6 * E * C1 / (C3 - .45 * C2) 310 PRINT “BUCKLING PRESSURE(DTMB)=”; P 320 P = 1E+20: PI = 3.1415926536# 330 FOR N = 1 TO 200 340 N2 = N * N 350 C1 = PI * A / L 360 NU2 = (1 - NU ^ 2) 370 C2 = (N2 - 1 + C1 ^ 2) ^ 2 380 C2 = H ^ 2 * C2 / (12 * A ^ 2 * NU2) 390 C3 = (N2 * (1 / C1) ^ 2 + 1) ^ 2 400 C4 = N2 - 1 + .5 * C1 ^ 2 410 PCR = E * H / (A * C4) * (1 / C3 + C2) 420 IF P < PCR THEN PRINT “BUCKLING PRESSURE(VON MISES)=”; P; “ LOBES=”; N - 1: GOTO 436 430 P = PCR 435 NEXT N 436 PRINT “Do you want to analyse another vessel ? Type Y or N” 437 INPUT A$: IF A$ = “y” THEN A$ = “Y” IF A$ = “n” THEN A$ = “N” IF A$ = “Y” THEN GOTO 120 ELSE IF A$ = “N” THEN GOTO 440 GOTO 436 440 END
© Carl T. F. Ross, 2011
446
Pressure vessels
500 PRINT “VON MISES FORMULA, DTMB FORMULA & THINNESS RATIO” 510 PRINT “UNSUPPORTED LENGTH OF SHELL=”; L 520 PRINT “MEAN SHELL RADIUS=”; A 530 PRINT “WALL THICKNESS OF CYLINDER=”; H 540 PRINT “YOUNG’S MODULUS=”; E 550 PRINT “POISSON’S RATIO=”; NU 560 PRINT “YIELD STRESS=”; SIGMAYP 570 RETURN A listing for NIORDSON.BAS is given below: 90 CLS : PRINT “Program for calculating the buckling of thin-walled cones”; PRINT “ under uniform external pressure by DR. C.T.F. Ross” INPUT “young’s modulus=”; e INPUT “poissons’ ratio=”; nu INPUT “wall thickness=”; h INPUT “mean radius at left end=”; ra INPUT “mean radius at right end=”; rb INPUT “unsupported length between frames=”; lf alpha = ATN((rb - ra) / lf) am = (ra + rb) / 2 l = lf / (COS(alpha)) ks2 = (COS(alpha)) ^ 2 ks3 = (COS(alpha)) ^ 3 pcr = 1E+30 pi = 3.1415926536# FOR n = 2 TO 20 con = n ^ 2 + .5 * (pi * am / l) ^ 2 - 1 con = e * h * ks3 / (am * con) con1 = 1 / (n ^ 2 * (l / (pi * am)) ^ 2 + 1) ^ 2 con2 = (h ^ 2 / (12 * am ^ 2 * (1 - nu ^ 2) * ks2) * (n ^ 2 + (pi * am / l) ^ 2 - 1) ^ 2) pmc = con * (con1 + con2) IF pcr > pmc THEN pcr = pmc ELSE GOTO 100 NEXT n 100 PRINT “Niordson buckling pressure=”; pcr; “(”; n - 1; “)” dm = 2 * am ks = COS(alpha) ks15 = ks3 / ks2
© Carl T. F. Ross, 2011
Appendix III
447
ks5 = 1 / SQR(ks) pcr = 2.6 * e * (h / dm) ^ 2.5 * ks15 pcr = pcr / ((l / dm) - .45 * (h / dm) ^ .5 * ks5) PRINT “DTMB buckling pressure=”; pcr 200 INPUT “Do you require to do another calculation ? Type Y or N ”; A$ IF A$ = “y” THEN A$ = “Y” ELSE IF A$ = “n” THEN A$ = “N” IF A$ = “Y” THEN GOTO 90 ELSE IF A$ = “N” THEN GOTO 210 GOTO 200 210 END
© Carl T. F. Ross, 2011
Appendix IV Computer programs for general instability
Kendrick Part I: KENDRIC1.EXE This program calculates the general instability buckling pressure for a circular cylinder, stiffened by equal-strength ring frames, spaced at equal intervals and subjected to uniform external pressure. The program is based on Kendrick’s Part I theory, which assumes that the vessel is simply supported at its ends, and fails elastically. The program is interactive, and the data should be fed in as follows: N = number of rings LF = frame spacing LB = bulkhead spacing H = shell thickness A = mean shell radius AF = sectional area of a typical frame IX = second moment of area of a typical frame about its x–x axis EC = eccentricity of frame centroid (e positive inwards) E = Young’s modulus of elasticity NU = Poisson’s ratio NMAX = maximum number of lobes into which the vessel is likely to buckle. (It should be ensured that NMAX is sufficiently large to obtain a minimum value of Pcr.)
Kendrick Part III: KENDPT3.EXE A, H, N, LF, AF, IX, EC, NMAX, E, NU
Bryant: BRYANT.EXE LF, LB, A, H, Ieff, E, NU where Ieff is the effective second moment of area of ring/shell combination.160 448 © Carl T. F. Ross, 2011
Appendix IV
449
A listing for KENDRIC1.BAS is given below: 100 CLS 110 PRINT : PRINT “GENERAL INSTABILITY OF RINGREINFORCED CYLINDERS-KENDRICK PART 1” PRINT “Program by Dr. C.T.F. Ross” 120 INPUT “NUMBER OF RINGS=”; N 130 INPUT “FRAME SPACING=”; LF 140 INPUT “BULKHEAD SPACING=”; LB 150 INPUT “SHELL THICKNESS=”; H 160 INPUT “MEAN RADIUS=”; A 170 INPUT “FRAME AREA=”; AF 180 INPUT “2ND MOA ABOUT HORIZONTAL AXIS=”; IX 190 INPUT “ECCENTRICITY OF FRAME (e +VE INWARDS)=”; EC 200 INPUT “ELASTIC MODULUS=”; E 210 INPUT “POISSON’S RATIO=”; NU 260 K0 = -H * LF / (AF + H * LF) 270 K1 = -AF * LF / (AF + H * LF) 300 W0 = (N + 1) / 2 310 W10 = .5: W11 = .5: W12 = .5: W13 = .5: W14 = .5 320 PI = 3.1415926536# 330 INPUT “MAXIMUM NUMBER OF LOBES=”; NMAX 400 FOR LO = 2 TO NMAX 410 N21 = (LO ^ 2 - 1) 420 N2 = LO ^ 2 430 A10 = (PI * E * W0 / A) * (N21 ^ 2 * IX / A ^ 2 + AF * (N21 ^ 2 * EC ^ 2 / A ^ 2 + 1 + 2 * EC / A - 2 * N2 * EC / A)) 440 A10 = A10 + (E * H ^ 3 / (12 * A * (1 - NU ^ 2))) * (PI ^ 5 * A ^ 2 * W14 / LB ^ 3 + (2 * PI ^ 3 * W12 / LB) * (NU * N21 + N2 * (1 - NU)) + PI * LB * W11 * (12 / H ^ 2 + N21 ^ 2 / A ^ 2)) 450 A11 = PI * LB * W11 * (1 + N2 * K0) - .5 * PI ^ 3 * A ^ 2 * W12 / LB + PI * N2 * K1 * W0 460 A12 = PI * LO * E * AF * W0 * (N2 * EC / A ^ 2 - EC / A ^ 2 - 1 / A) - (PI * LO * E * H / (A * (1 - NU ^ 2))) * (PI ^ 2 * H ^ 2 * (1 - NU) * W12 / (12 * LB) + LB * W11) 470 A13 = -PI * LO * (K1 * W0 + LB * W11 * (1 + K0)) 480 A14 = (PI ^ 2 * E * H / (1 - NU ^ 2)) * (-H ^ 2 * N2 * (1 - NU) * W13 / (12 * A ^ 2) + NU * W10) 490 A15 = PI ^ 2 * A * W10
© Carl T. F. Ross, 2011
450
Pressure vessels
500 A16 = (PI ^ 2 * E * H / (1 - NU ^ 2)) * (.5 * LO * (1 - NU) * W13 * (H ^ 2 / (12 * A ^ 2) - 1) NU * LO * W10) 510 A17 = -.5 * PI ^ 2 * A * LO * W10 520 A18 = PI * N2 * E * AF * W0 / A + (PI * E * H / (1 - NU ^ 2)) * (PI ^ 2 * A * (1 - NU) * W12 * (H ^ 2 / (24 * A ^ 2) + .5) / LB + N2 * LB * W11 / A) 530 A19 = -.5 * PI ^ 3 * A ^ 2 * W12 / LB 540 A20 = (PI * E * H / (1 - NU ^ 2)) * (.25 * N2 * (1 - NU) * LB * (H ^ 2 / (12 * A ^ 2) + 1) / A + .5 * PI ^ 2 * A / LB) 550 A21 = .5 * PI * N2 * (K1 * (N - 1) + LB * K0) 560 A23 = -A16 ^ 2 + A18 * A20 570 A24 = -2 * A16 * A17 + A18 * A21 + A19 * A20 580 A25 = -A17 ^ 2 + A19 * A21 590 A26 = -A12 * A20 + A14 * A16 600 A27 = -A12 * A21 + A13 * A20 + A14 * A17 + A15 * A16 610 A28 = -A13 * A21 + A15 * A17 620 A29 = A12 * A16 - A14 * A18 630 A30 = A12 * A17 + A13 * A16 + A14 * A19 - A15 * A18 640 A31 = A13 * A17 - A15 * A19 650 A32 = A11 * A25 + A13 * A28 + A15 * A31 660 A33 = A10 * A25 + A11 * A24 + A12 * A28 + A13 * A27 + A14 * A31 + A15 * A30 670 A34 = A10 * A24 + A11 * A23 + A12 * A27 + A13 * A26 + A14 * A30 + A15 * A29 680 A35 = A10 * A23 + A12 * A26 + A14 * A29 700 PRESS = 0: PR = 0 710 FOR II = 1 TO 100 720 FUNCT = A32 * PRESS ^ 3 + A33 * PRESS ^ 2 + A34 * PRESS + A35 730 DFUNCTION = 3 * A32 * PRESS ^ 2 + 2 * A33 * PRESS + A34 740 PRESS = PRESS - FUNCT / DFUNCTION 750 IF ABS(PR - PRESS) < .000001 THEN GOTO 900 760 PR = PRESS 770 NEXT II 900 PRINT “LOBES=”; LO, : PRINT “BUCKLING PRESSURE=”; PRESS 910 NEXT LO
© Carl T. F. Ross, 2011
Appendix IV
451
912 PRINT “Do you want to analyse another problem ? Type Y or N”: INPUT A$ IF A$ = “y” THEN A$ = “Y” IF A$ = “n” THEN A$ = “N” IF A$ = “Y” THEN GOTO 100 IF A$ = “N” THEN GOTO 920 GOTO 912 920 END A listing for KENDRIC3.BAS is given below: 90 CLS 100 REM GENERAL INSTABILITY (KENDRICK-PT.3) 110 PRINT : PRINT “GENERAL INSTABILITY OF RINGREINFORCED CYLINDERS UNDER EXTERNAL PRESSURE (KENDRICK PT.3)” 120 PRINT : PRINT “PROGRAM BY DR.C.T.F.ROSS” 130 PRINT DIM A(5, 5), XX(5, 6), VC(5, 5), AM(5), X(5), Z(5), IG(5), VE(5) 140 INPUT “TYPE IN SHELL RADIUS ”; RS 150 INPUT “TYPE IN WALL THICKNESS ”; TH 160 INPUT “TYPE IN THE NUMBER OF FRAMES BETWEEN BULKHEADS ”; FRAMES 170 INPUT “TYPE IN FRAME SPACING ”; LF 180 LB = (FRAMES + 1) * LF 190 INPUT “TYPE IN CROSS-SECTIONAL AREA OF A TYPICAL FRAME ”; AF 200 INPUT “TYPE IN THE 2ND MOMENT OF AREA OF A TYPICAL FRAME ABOUT THE X0 AXIS ”; IX0 210 INPUT “TYPE IN THE ECCENTRICITY OF A TYPICAL FRAME FROM THE MID-SURFACE OF THE SHELL (-VE IF THE FRAMES ARE EXTERNAL) ”; EC 220 INPUT “TYPE IN THE MAXIMUM NUMBER OF LOBES ”; LOBES 225 IF LOBES < 2 THEN PRINT : PRINT “INCORRECT DATA”: GOTO 220 230 INPUT “TYPE IN ELASTIC MODULUS ”; E 235 INPUT “TYPE IN POISSON’S RATIO ”; NU 330 GOSUB 4000 350 GOTO 6020 380 PRINT “THE EIGENVALUES ARE BEING DETERMINED”: PRINT
© Carl T. F. Ross, 2011
452
Pressure vessels
390 GOSUB 510 410 FOR I = 1 TO M1 420 LPRINT “EIGENVALUE=”; AM(I) 425 PCR = 1 / AM(I): PCRIT = PCR * E * TH / (RS * (1 - NU ^ 2)) 430 LPRINT : LPRINT “BUCKLING PRESSURE=”; PCRIT 440 LPRINT : LPRINT “EIGENVECTOR IS” 450 LPRINT 460 FOR J = 1 TO N 470 LPRINT VC(J, I); “ ”; 480 LPRINT : NEXT J 490 LPRINT : NEXT I 495 RETURN 500 GOTO 25000 510 MN = N 520 NN = N 530 GOSUB 1390 540 M = 1 550 FOR I = 1 TO NN 560 VC(I, M) = X(I) 570 XX(I, M) = VC(I, M) 580 NEXT I 590 AM(M) = XM 600 IF M1 < 2 THEN RETURN 610 FOR M = 2 TO M1 620 FOR I = 1 TO NN 630 K4 = ABS(XX(I, M - 1) - 1) 640 IF K4 < .00001 THEN IR = I 650 NEXT I 660 IG(M - 1) = IR 670 FOR I = 1 TO NN 680 XX(MN - I + 1, MN - M + 3) = A(IR, I) 690 NEXT I 700 FOR I = 1 TO NN 710 FOR J = 1 TO NN 720 Z1 = MN - J + 1 730 Z2 = MN - M + 3 740 A(I, J) = A(I, J) - XX(I, M - 1) * XX(Z1, Z2) 750 NEXT J 760 NEXT I 770 FOR I = 1 TO NN 780 IF I = IR THEN GOTO 870 790 IF I > IR THEN K1 = I - 1
© Carl T. F. Ross, 2011
Appendix IV 800 IF I < IR THEN K1 = I 810 FOR J = 1 TO NN 820 IF J = IR THEN GOTO 860 830 IF J > IR THEN K2 = J - 1 840 IF J < IR THEN K2 = J 850 A(K1, K2) = A(I, J) 860 NEXT J 870 NEXT I 880 NN = NN - 1 890 M3 = NN 900 IF M <> MN THEN GOTO 940 910 XM = A(1, 1) 920 X(1) = 1 930 GOTO 950 940 GOSUB 1390 950 FOR I = 1 TO NN 960 XX(I, M) = X(I) 970 NEXT I 980 AM(M) = XM 990 M4 = M - 1 1000 M5 = 1000 - M4 1010 FOR M8 = M5 TO 999 1020 M6 = M3 + 1 1030 M2 = 1000 - M8 1040 M7 = IG(M2) + 1 1050 IF M6 < M7 THEN GOTO 1120 1060 N9 = 1000 - M7 1070 N8 = 1000 - M6 1080 FOR I3 = N8 TO N9 1090 I = 1000 - I3 1100 X(I) = X(I - 1) 1110 NEXT I3 1120 J = IG(M2) 1130 X(J) = 0 1140 SUM = 0 1150 FOR I = 1 TO M6 1160 Z3 = MN - I + 1 1170 Z4 = MN - M2 + 2 1180 SUM = SUM + XX(Z3, Z4) * X(I) 1190 NEXT I 1200 XK = (AM(M2) - XM) / SUM 1210 FOR I = 1 TO M6 1220 X(I) = XX(I, M2) - XK * X(I)
© Carl T. F. Ross, 2011
453
454 1230 1240 1250 1260 1270 1280 1290 1300 1310 1320 1330 1340 1350 1360 1370 1380 1390 1400 1410 1420 1430 1440 1450 1460 1470 1480 1490 1500 1510 1520 1530 1540 1550 1560 1570 1580 1590 1600 1610 1620 1630 1640 1650
Pressure vessels NEXT I SUM = 0 FOR I = 1 TO M6 IF ABS(SUM) < ABS(X(I)) THEN SUM NEXT I FOR I = 1 TO M6 X(I) = X(I) / SUM NEXT I M3 = M3 + 1 IF M2 <> 1 THEN GOTO 1360 FOR I = 1 TO M3 VC(I, M) = X(I) NEXT I NEXT M8 NEXT M RETURN Y1 = 100000! FOR I = 1 TO NN X(I) = 1 NEXT I XM = -100000! FOR I = 1 TO NN SG = 0 FOR J = 1 TO NN SG = SG + A(I, J) * X(J) NEXT J Z(I) = SG NEXT I XM = 0 FOR I = 1 TO NN IF ABS(XM) < ABS(Z(I)) THEN XM = NEXT I FOR I = 1 TO NN X(I) = Z(I) / XM NEXT I IF ABS((Y1 - XM) / XM) > D THEN GOTO 1620 Y1 = XM GOTO 1440 X3 = 0 FOR I = 1 TO NN IF ABS(X3) < ABS(X(I)) THEN X3 = NEXT I
© Carl T. F. Ross, 2011
= X(I)
Z(I)
GOTO 1600
X(I)
Appendix IV
455
1660 FOR I = 1 TO NN 1670 X(I) = X(I) / X3 1680 NEXT I 1690 RETURN 1700 N9 = N - 1 1710 FOR NX = 1 TO N 1720 DI = A(NX, 1) 1730 IF DI = 0 THEN PRINT “MATRIX IS SINGULAR” 1740 FOR NY = 1 TO N9 1750 Y9 = NY + 1 1760 A(NX, NY) = A(NX, Y9) / DI 1770 NEXT NY 1780 A(NX, N) = 1 / DI 1790 FOR NZ = 1 TO N 1800 IF NZ = NX THEN GOTO 1870 1810 O = A(NZ, 1) 1820 FOR NY = 1 TO N9 1830 Y9 = NY + 1 1840 A(NZ, NY) = A(NZ, Y9) - A(NX, NY) * O 1850 NEXT NY 1860 A(NZ, N) = -A(NX, N) * O 1870 NEXT NZ 1880 NEXT NX 1890 RETURN 4000 REM OUTPUT OF INPUT 4020 LPRINT : LPRINT 4030 LPRINT “GENERAL INSTABILITY OF RING-REINFORCED CYLINDERS UNDER UNIFORM EXTERNAL PRESSURE (KENDRICKTRANSACTIONS R.I.N.A.,pp.139-156,1965)” 4040 LPRINT “PROGRAM BY DR.C.T.F.ROSS”: LPRINT 4050 LPRINT “MEAN SHELL RADIUS=”; RS 4060 LPRINT “WALL THICKNESS=”; TH 4070 LPRINT “NUMBER OF FRAMES BETWEEN BULKHEADS=”; FRAMES 4080 LPRINT “FRAME SPACING=”; LF 4090 LPRINT “BULKHEAD SPACING=”; LB 4100 LPRINT “CROSS-SECTIONAL AREA OF A TYPICAL FRAME=”; AF 4110 LPRINT “2nd MOMENT OF AREA OF A TYPICAL RINGSTIFFENER ABOUT THE X0 AXIS=”; IX0 4120 LPRINT “ECCENTRICITY OF THE CENTROID OF A TYPICAL RING-STIFFENER FROM THE MID-SURFACE OF THE SHELL (-ve IF EXTERNAL)=”; EC
© Carl T. F. Ross, 2011
456
Pressure vessels
4130 LPRINT “MAXIMUM NUMBER OF LOBES=”; LOBES 4140 LPRINT “ELASTIC MODULUS=”; E 4150 LPRINT “POISSON’S RATIO=”; NU 4170 RETURN 6020 REM START OF THE MAIN PART OF THE PROGRAM 6030 PI = 3.1415926536#: LAM1 = PI * RS / LB 6040 LAM2 = PI * RS / LF 6050 CON = TH ^ 2 / (12 * RS ^ 2) 6060 AC = TH * LF / (AF + TH * LF) 6070 MU1 = 2 / PI 6080 GAM1 = 1 / ((2 * FRAMES + 3) * PI) 6090 GAM2 = 1 / ((2 * FRAMES + 1) * PI) 6100 BETA = (AC - NU / 2) / (1 - NU ^ 2) 6110 M1 = 1: N = 5 6120 REM DIM A(5, 5), XX(5, 6), VC(N, M1), AM(M1), X(5), Z(5), IG(5), VE(5) 6130 FOR LO = 2 TO LOBES 6140 PRINT : PRINT “NUMBER OF CIRCUMFERENTIAL WAVES=”; LO 6150 RB = (1 - NU ^ 2) * AF * LO ^ 2 / (TH * LF) 6160 RC = (1 - NU ^ 2) * IX0 * (LO ^ 2 - 1) ^ 2 / (TH * RS ^ 2 * LF) + ((1 - NU ^ 2) * AF / (TH * LF)) * (1 - (EC / RS) * (LO ^ 2 - 1)) ^ 2 6170 RBC = -((1 - NU ^ 2) * AF * LO / (TH * LF)) * (1 - (EC / RS) * (LO ^ 2 - 1)) 6180 A(1, 1) = LAM1 ^ 2 + LO ^ 2 * (1 + CON) * (1 - NU) / 2 6190 A(1, 2) = -(LO * LAM1 / 2) * (1 + NU - CON * (1 - NU)) 6200 A(1, 3) = LAM1 * NU - LAM1 * CON * (1 - NU) * LO ^ 2 6210 A(1, 4) = -2 * NU * LAM1 * LO * (MU1 - GAM1 + GAM2) - 2 * LAM2 * LO * (1 - NU) * (GAM1 + GAM2) * (1 - CON) 6220 A(1, 5) = 2 * NU * LAM1 * (MU1 - GAM1 + GAM2) - 4 * CON * LO ^ 2 * LAM2 * (1 - NU) * (GAM1 + GAM2) 6230 A(2, 2) = LO ^ 2 + LAM1 ^ 2 * (1 + CON) * (1 - NU) / 2 + RB 6240 A(2, 3) = -LO * (1 + LAM1 ^ 2 * CON * (1 NU)) + RBC 6250 A(2, 4) = 2 * LO ^ 2 * (MU1 - GAM1 + GAM2) + 2 * LAM1 * LAM2 * (1 - NU) * (GAM1 + GAM2) * (1 + CON)
© Carl T. F. Ross, 2011
Appendix IV
457
6260 A(2, 5) = -2 * LO * (MU1 - GAM1 + GAM2) - 4 * CON * LO * LAM1 * LAM2 * (GAM1 + GAM2) * (1 - NU) 6270 A(3, 3) = 1 + CON * (LAM1 ^ 4 + 2 * LAM1 ^ 2 * (LO ^ 2 - NU) + (LO ^ 2 - 1) ^ 2) + RC 6280 A(3, 4) = -2 * LO * (MU1 - GAM1 + GAM2) - 4 * CON * LO * LAM1 * LAM2 * (GAM1 + GAM2) * (1 - NU) 6290 A(3, 5) = 2 * (MU1 - GAM1 + GAM2) 6300 A(3, 5) = A(3, 5) + CON * (-8 * LAM1 ^ 2 * LAM2 ^ 2 * (GAM1 + GAM2) + 2 * (1 - LO ^ 2) ^ 2 * (MU1 - GAM1 + GAM2) + 8 * NU * LAM2 ^ 2 * (1 - LO ^ 2) * (GAM1 - GAM2)) 6310 A(3, 5) = A(3, 5) + CON * (-2 * NU * LAM1 ^ 2 * (1 - LO ^ 2) * (MU1 - GAM1 + GAM2) + 8 * LO ^ 2 * LAM1 * LAM2 * (GAM1 + GAM2) * (1 - NU)) 6320 A(4, 4) = 3 * LO ^ 2 + 2 * LAM2 ^ 2 * (1 NU) * (1 + CON) 6330 A(4, 5) = -3 * LO - 4 * CON * LAM2 ^ 2 * LO * (1 - NU) 6340 A(5, 5) = 3 + CON * (16 * LAM2 ^ 4 + 3 * (1 LO ^ 2) ^ 2 + 8 * LAM2 ^ 2 * (LO ^ 2 - NU)) 6350 XX(1, 1) = LO ^ 2 - 2 * LO ^ 2 * (1 - AC) / (FRAMES + 1) 6360 XX(1, 2) = LO * LAM1 / 2 - LO * LAM1 * NU * BETA 6370 XX(1, 3) = -LAM1 6380 XX(1, 4) = LO * LAM1 * (MU1 - GAM1 + GAM2) * (1 - 2 * NU * BETA) 6390 XX(1, 5) = -2 * LAM1 * (MU1 - GAM1 + GAM2) 6400 XX(2, 2) = LAM1 ^ 2 / 2 + 2 * LO ^ 2 * (1 + BETA - AC) 6410 XX(2, 3) = -LO * BETA - LO * (1 - AC) * (1 (LO ^ 2 - 1) * (EC / RS)) 6420 XX(2, 4) = 2 * LAM1 * LAM2 * (GAM1 + GAM2) + 4 * LO ^ 2 * BETA * (MU1 - GAM1 + GAM2) 6430 XX(2, 5) = -2 * LO * (MU1 - GAM1 + GAM2) * (AC + BETA - 1) 6440 XX(3, 3) = LO ^ 2 + LAM1 ^ 2 / 2 - 1 6450 XX(3, 4) = -2 * LO * (MU1 - GAM1 + GAM2) * (AC + BETA - 1) 6460 XX(3, 5) = -2 * (MU1 - GAM1 + GAM2) * (1 - LO ^ 2 * AC) + 2 * LAM1 * LAM2 * (GAM1 + GAM2) 6470 XX(4, 4) = 2 * LAM2 ^ 2 + 6 * LO ^ 2 * BETA 6480 XX(4, 5) = -3 * LO * (AC + BETA - 1)
© Carl T. F. Ross, 2011
458
Pressure vessels
6490 XX(5, 5) = -3 * (1 - LO ^ 2 * AC) + 2 * LAM2 ^ 2 6500 FOR II = 1 TO 5 6510 FOR JJ = 1 TO II 6520 A(II, JJ) = A(JJ, II) 6530 XX(II, JJ) = XX(JJ, II) 6540 NEXT JJ 6550 NEXT II 6560 REM THE MATRIX WILL NOW BE INVERTED 6570 PRINT : PRINT “THE MATRIX IS BEING INVERTED”: PRINT 6580 GOSUB 1700 6590 REM (XX)/(A) 6600 FOR II = 1 TO 5 6610 FOR JJ = 1 TO 5 6620 VE(JJ) = 0 6630 FOR KK = 1 TO 5 6640 VE(JJ) = VE(JJ) + A(II, KK) * XX(KK, JJ) 6650 NEXT KK 6660 NEXT JJ 6670 FOR JJ = 1 TO 5 6680 A(II, JJ) = VE(JJ) 6690 NEXT JJ 6700 NEXT II 6710 D = .001 6715 LPRINT : LPRINT “NUMBER OF LOBES=”; LO 6720 GOSUB 380 6730 NEXT LO 6740 END 25000 REM END A listing for BRYANT.BAS is given below: 100 CLS : PRINT “BRYANT’S FORMULA FOR GENERAL INSTABILITY” PRINT “Program by Dr. C.T.F. Ross” INPUT “TYPE IN FRAME SPACING ”, LS INPUT “TYPE IN BULKHEAD SPACING ”, LB INPUT “TYPE IN MEAN RADIUS ”, R INPUT “TYPE IN SHELL THICKNESS ”, T INPUT “2ND M.O.A OF A RING STIFFENER +EFF. BREADTH OF PLATING=”, IC INPUT “TYPE IN YOUNG’S MODULUS ”, E INPUT “TYPE IN POISSON’S RATIO ”, NU
© Carl T. F. Ross, 2011
Appendix IV
459
LAMBDA = 3.14159265# * R / LB KN1 = E * T / R KN2 = LAMBDA ^ 4 PCR = 1E+12 FOR N = 2 TO 100 KN3 = (N ^ 2 - 1 + LAMBDA ^ 2 / 2) * (N ^ 2 + LAMBDA ^ 2) ^ 2 KN4 = (N ^ 2 - 1) * E * IC / (R ^ 3 * LS) PCR2 = KN1 * KN2 / KN3 + KN4 IF PCR < PCR2 THEN GOTO 200 PCR = PCR2 NEXT N 200 PRINT “BUCKLING PRESSURE=”; PCR; “(”; N - 1; “)” IF N = 100 THEN PRINT “ERROR” 210 INPUT “Do you want to do another calculation ? Type Y or N ”, A$ IF A$ = “y” THEN A$ = “Y” ELSE IF A$ = “n” THEN A$ = “N” IF A$ = “Y” THEN GOTO 100 ELSE IF A$ = “N” THEN GOTO 220 GOTO 210 220 END
© Carl T. F. Ross, 2011
Appendix V Conversion tables of imperial units to SI
Table AV.1 Conversion of inches (and powers) to millimetres (and powers) in
mm
in2
mm2
in4
mm4
0.001 0.002 0.0336 0.0355 0.045 0.06 0.073 0.0807 0.095 0.1 0.104 0.252 1 2 3.045 4.067 4.071 8 10 16.2 100
0.0254 0.0508 0.853 0.902 1.143 1.524 1.854 2.0498 2.413 2.54 2.642 6.4 25.4 50.8 77.34 103.3 103.4 203.2 254 411.48 2540
0.06 0.0612 0.1 1
38.71 39.48 64.52 645.2
3.02e × 10−5 4.11e × 10−5 9.794e × 10−5 1.182e × 10−4 1
12.57 17.11 40.75 49.2 0.4162 × 106
460 © Carl T. F. Ross, 2011
Appendix V Table AV.2 Conversion of lbf in−2 to MPa lbf in−2
MPa
1 100 240 4 × 106 6.5 × 106 10.3 × 106 30 × 106
6.895 × 10−3 0.6895 1.6548 27 580 44 818 71 018.5 206 850
Table AV.3 Conversion of lbf s2 in−4 to kg m−3 lbf s2 in−4
kg m−3
2.248 × 10−4 2.4 × 10−4 1.59 × 10−4 7.35 × 10−4
2403 2565 1699 7855
© Carl T. F. Ross, 2011
461
Index
Abdulla and Galletly’s model, 197–8 accelerometer charge amplifier, 330 ACMC element see axisymmetric constant meridional curvature element ALLCUBE, 215 aluminium alloys, 10, 387–8 material properties, 12 strength, 388 ANSYS, 183, 368, 374 thin-walled shell vibration under external water pressure, 393–418 ANSYS 7.1, 398–9, 405 aspect ratio, 40, 130, 145, 335 asymmetrical beam theory, 8 AVMC element see axisymmetric varying meridional curvature element axisymmetric constant meridional curvature element, 36–40 axisymmetric shell element, 37 [B] for ACMC element, 39 longitudinal section through element, 37 axisymmetric deformation, 355, 360, 361–2 aspect ratio 1.5 experimental strains along dome meridian, 44 internal circumferential stress distribution, 46 internal meridional stress distribution, 45 out-of-circularity plot, 47
strain gauge positions and directions, 43 variation in stress with mesh refinement, 42 aspect ratio 3.0 experimental strains along dome meridian, 45 internal circumferential stress distribution, 47 internal meridional stress distribution, 46 out-of-circularity plot, 48 strain gauge positions and directions, 44 variation in stress with mesh refinement, 43 collapse modes cones 7, 8 and 9, 92 cylinders 4, 5 and 6, 94 cylinders 4, 5 and 6, 84–5, 88–9 circular, 89 geometrical dimensions, 88 dome 1 experimental shell section, 71 experimental strain results, 74 hoop stress distribution, 76, 78, 79 meridional stress distribution, 76, 79 strain gauge positions, 73 dome 2 experimental shell section, 71 experimental strain results, 75 hoop stress distribution, 81
463 © Carl T. F. Ross, 2011
464
Index
meridional stress distribution, 77, 80 strain gauge positions, 73 experimental procedure, 85, 89–94 A, B and C collapse models, 90 circular cylinders 4, 5 and 6, 92–4 collapse cone C bottom view, 90 cone 9, 91–2 cone A, B and C collapse pressures, 91 cone C, 89–91 experimental collapse procedures, 92 geometrical details cone 9, 87 cone C, 86 cylinders 4, 5 and 6, 88 matrix axisymmetric constant meridional curvature element, 39 five-node element, 69 four-node element, 67 thick conical shell, 57 thick-walled three-node parabolic element, 62 plastic collapse, 83–5, 86–9 cone 9, 84, 87 cone A, B and C, 86 cone C, 84, 85, 86 cone C geometrical measurements, 85 cones 7, 8 and 9, 87 experimental apparatus, 84 test tank with mode, 89 pressure–hoop strain relationship cylinder 4, 93 cylinder 5, 93 cylinder 6, 94 pressure vessels, 15–99 axisymmetric yield failure, 15 unstiffened circular cylinders and spheres, 15–16 ring-stiffened circular cylinders, 16–30 beam-on-elastic-foundation theory advantages, 30
circular cylinder stiffened by unequally sized rings, 21–7 equilibrium at ith ring, 24 experiment vs theory, 27–30 innermost fibre circumferential stress, 29 innermost fibre longitudinal stress, 29 longitudinal generator deflection, 28 model number 3, 28 model number 3 details, 30 outermost fibre circumferential stress, 29 outermost fibre longitudinal stress, 28 ring-stiffened cylinder, 21 stiffened by equal-size stiffening rings, 18 ring-stiffeners, 77–8, 80–3 ring out-of-plane bending, 82 theoretical plastic analysis, 95–6, 97–8 incremental method of analysis, 95 theoretical plot of pressure vs axial deflection cone 9, 97 cone C, 96 cylinder 4, 97 cylinder 5, 98 cylinder 6, 98 thick-walled cones and domes, 52–77, 78, 79, 80, 81 comparisons between various elements, 70 comparison with tapered dome, 71–2, 73, 74, 75–7, 78, 79, 80, 81 five-node element, 66, 68–70 five-node quartic element, 68 four-node cubic element, 66 four-node element, 64–6, 67 hoop stress distributions for 3-in shell cap, 70 hoop thickness for 9 in. thickness cap, 65 9 in. thickness shell cap, 64
© Carl T. F. Ross, 2011
Index local and global axes, 56 mean hoop stress distribution for 3 in. thickness cap, 63 method of pressuring dome, 72 orthotropic element, 72, 74 parabolic element, 58–64, 65 thick conical shell, 53 three-node parabolic element, 59 thin-walled cones and domes, 30–52, 53 ACMC vs AVMC elements, 40–7, 48 aspect ratio hemi-ellipsoidal domes, 41 axisymmetric constant meridional curvature element, 36–40 axisymmetric shell element, 37 axisymmetric varying meridional curvature element, 34–6 FEA vs Woinowsky-Kreiger, 33–4 hoop forces/unit length for cap, 34 longitudinal section through element, 37 meridional bending moment/unit length for cap, 34 pressure tank with test shell in position, 42 radial deflection of longitudinal generator, 53 spherical shell cap, 33 tapered cylindrical shell element, 47–52 tapered thin-walled circular cylinder, 48 thin-walled conical element, 31 varying meridional curvature element, 35 axisymmetric varying meridional curvature element, 34–6 varying meridional curvature element, 35 BASIC, 33 BCLAM, 304, 314 beam-column effect, 18 beam-on-elastic-foundation theory, 30
465
boiler formula, 360 Bosch reaction, 383 BOSOR5, 144, 351–2 Bryan formula, 101 BRYANT.BAS, 458–9 BRYANT.EXE, 448–51 Bryant’s formula, 359 buckled corrugated circular cylinders, 295 buckling, 117–38, 142–51, 152–3 aspect ratio 0.25 snap-thru failure, 147 theoretical pressure–axial deflection plot (GRP), 151 theoretical pressure–axial deflection plot (SUP), 150 aspect ratio 0.444 theoretical pressure-axial deflection plot (GRP), 152 theoretical pressure-axial deflection plot (SUP), 150 aspect ratio 0.7A lobar failure, 148 theoretical pressure–axial deflection plot (GRP), 152 aspect ratio 0.7B failure in flange, 149 theoretical pressure–axial deflection (GRP), 153 buckling formulae for domes and cones, 142–3 spherical shell cap, 142–3 truncated conical shell, 143 Von Kármán and Tsien’s formula, 142 GRP dome bottom, 145 geometrical details, 146 top, 145 microstrain readings recorded for aspect ratio dome 0.165 MPa pressure, 133 0.483 MPa pressure, 132 orthotropic conical shells, 123–4 truncated conical shell, 124
© Carl T. F. Ross, 2011
466
Index
orthotropic cylinders and cones, 117–24 circular cylindrical element, 119 orthotropic element, 118–23 snap-thru buckling, 144–51, 152–3 aspect ratio 0.44 snap-thru failure, 148 aspect ratio 0.7 theoretical pressure–axial deflection plot (SUP), 151 test tank with dome, 146 thin-walled domes, 124–38 buckling regions position and size, 134 G for a doubly curved axisymmetric element, 129 hemispherical dome lobar buckling, 137 near hemispherical domes, 137–8 ten hemi-ellipsoidal oblate/prolate domes, 130 varying meridional curvature element, 125–30 VMC element, 132, 134–6 illustration, 134 model L2 with rupture, 136 buckling configuration, 171–2 Kendrick Part I, 171 Kendrick Part III, 171 ring-stiffener, 174 buckling pressures, 166, 253, 269, 270 cones and their circular cylindrical shell equivalents, 288 DTMB machined models, 178 hemi-ellipsoidal domes, 149 hemi-ellipsoidal domes (bar), 153 models L1 and L2 number of lobes n, 136 number of lobes for model Number 7, 123 number of lobes n, 133 number of lobes n for oblates domes, 137 orthotropic conical shells, 123 ring-stiffened cones, 185 SUP domes, 131
swedged vessels and ring-stiffened equivalents, 286 theoretical and experimental, 179 three cones, 187 buckling theory, 331 bulk modulus, 14 Bushnell’s cylinder, 202 Bushnell’s ‘Turtle’, 2 illustration, 3 carbon fibre reinforced plastic, 10, 117, 389 corrugated circular cylinder vibration under external hydrostatic pressure, 316–24 CFRP vessel in test tank, 318 corrugated carbon fibre cylinder, 317 experimental method, 316–18 finite element programs, 318–19 n = 3 amplitude plot, 320 n = 3 phase angle plot, 319 results, 319–22 structure–fluid mesh, 321 theoretical analysis, 318 carbon nanotubes, 390 ceramics, 10 CFRP see carbon fibre reinforced plastic circular cylindrical shells experimental method, 244–6, 249–53 details of cylinders and their attachments, 250 method of exciting the vessels under pressure, 252 method of vibrating the cylinders, 253 test rig and pressure tank used, 251 three model cylinders and base plate, 250 vibrating rod position on rubber strap, 252 fluid mesh cylinder 1, 254 cylinder 2, 254 cylinder 3, 255
© Carl T. F. Ross, 2011
Index vibration under external water pressure, 244–6, 249–55, 256, 257, 258 experiment vs theory, 253–4, 255 results for cylinder 1, 256 results for cylinder 2, 257 results for cylinder 3, 258 CMC see constant meridional curvature composites, 388–9 strength and relative costs, 389 computer programs see also specific program axisymmetric stresses in circular cylinders stiffened by equal-strength ring frames, 428–31 stiffened by unequal-strength ring frames, 432–43 general instability, 448–59 shell instability, 444–7 CONEBUCK, 299, 304 CONEBUCKLE, 285 CON-FE, 396, 405, 406, 407 conical element, 229 constant meridional curvature, 125, 229 conversion tables imperial units to SI, 460–1 inches to millimetres (and powers), 460 lbf in−2 to MPa, 461 lbf s2 in−4 to kg m−3, 461 corrosion, 387 corrugated carbon fibre-reinforced plastic cylinder buckling, 303–16 finite element analysis, 304 strain gauge readings for increasing external pressure, 315 strain gauge readings graph, 314 vessel geometry, 304 experimental method, 311–13 out-of-roundness measurements, 312 parts list for assembly drawing, 313 test tank assembly drawing, 313
467
extension of empirical buckling formulae, 305–8 equivalent widths with E = 10 GPa, 306 I(VESSEL) increased by corrugations, 305 I(VESSEL) increased by ringstiffening, 305 0/90/0 lay-up of compound beam, 306 predicted buckling pressures for CFRP vessel, 308 rectangular and trapezoidal beams, 305 second moment of area of a trapezoidal section, 307 extension of thinness ratio, 308–11 geometric details of vessels, 310 predicted buckling pressures for CFRP and hypothetical vessels, 311 predicted experimental buckling pressures using PKD, 312 thinness ratio data for CFRP vessel, 311 thinness ratio data for the hypothetical vessels, 311 corrugated circular cylinders, 293–303 buckling under external hydrostatic pressure, 293–4, 293–303 buckled corrugated circular cylinders, 295 collapsed models, 298 corrugated pressure hull, 294 swedge leg, 300 test tank and model, 298 theoretical buckling pressures, 300 calculation for thinness ratio and PKD, 301–2 design chart, 303 thinness ratios for other series, 302 design chart determination, 299 experimental procedure, 295–7, 298–9 AST series geometrical details, 296
© Carl T. F. Ross, 2011
468
Index
AST series of steel corrugated cylindrical models, 296 collapsed models, 298 cylinder sketch, 296 details of typical corrugation, 297 experimental buckling pressures, 299 tensile test graph, 297 test tank and model, 298 pressure system, 297 previous research on corrugated vessels, 293–4 theoretical analysis, 294–5 thinness ratios calculation, 299–301 corrugated food can, 347 experimental method, 348–9 corrugation for DF series, 349 experimental buckling pressures, 350 general instability of circular corrugated cylinders, 350 model in test tank, 348 ovality measurement, 349 material properties, 347–8 redesign, 346–54 corrugated food can, 347 geometrical details of food cans, 347 geometrical details of vessels, 348 redesign of food cans, 352–4 theoretical analysis, 349–52 experiment vs theory, 351 new cone angles and wall thicknesses, 353 variation of Pcr with α CA, 352 DF, 352 MBL, 353 MBS, 353 corrugated pressure hull, 294 corrugated prolate domes, 290–1 corrugated vessels, 293–4 damping, 220 David Taylor Model Basin formula, 102, 122, 308, 310
deep-diving underwater pressure vessels, 375–92 design charts, 106–10 shell instability machined circular cylinders, 108 soldered and welded circular cylinders, 109 dome cup ends, 335 collapse, 334–46, 338–41 experimental process average material properties, 341 four-layer specimens, 339 material properties, 341 three-layer specimens, 339 two-layer specimens, 339 volume fractions, 341 experimental results, 342–6 collapse pressures, 345 test tank and dome cup, 342 test tank with the vessel ACG3, 343 vessel AGC1 after testing, 343 vessel AGC2 after testing, 344 experimental tests, 342 load vs extension four-layer specimen, 340 three-layer specimen, 340 two-layer specimen, 340 manufacturing GRP dome cups, 335–8 AGC1 dimensions, 337 AGC2 dimensions, 338 AGC3 dimensions, 338 AGC4 dimensions, 338 details of dome cup, 337 dome cup, 337 mould for dome cups, 336 dome cups, 334–46 dome ends, 280–4 domes free vibration, 205, 207–14, 215 large dome, 213 vibrating in air, 208–13 free vibrations in water, 229–35 meshes for dome/fluid, 230 mesh for large SUP dome, 235
© Carl T. F. Ross, 2011
Index vibrations in free flood, 234–5 vibrations in water, 229, 231–3 mesh for dome/fluid small closed tank, AR = 0.44, 238 small closed tank, AR = 1, 238 small closed tank, AR = 2.5, 239 small closed tank, AR = 3, 239 small closed tank, AR = 3.5, 239 small closed tank, AR = 4, 239 method of excitation oblate domes, 208 prolate domes, 208 nodal lobar patterns, 214 plot of frequency against n AR = 1, 233 AR = 4, 233 pressure effects on resonant frequencies of small hemiellipsoidal dome aspect ratio 0.7, 242 aspect ratio 0.25, 240 aspect ratio 0.44, 241 aspect ratio 2.5, 243 aspect ratio 3.0, 244 aspect ratio 3.5, 245 aspect ratio 4.0, 246 resonant frequencies big tank vs small tank, 249 oblate domes (free flood), 232 oblate domes (water external), 231 prolate domes (free flood), 232 prolate domes (water external), 231 resonant frequencies in air oblate domes, 209 prolate domes, 209 resonant frequencies variation aspect ratio 1.0, 210 aspect ratio 1.5, 210 aspect ratio 2.0, 211 aspect ratio 2.5, 211 aspect ratio 3.0, 212 aspect ratio 3.5, 212 aspect ratio 4.0, 213 n for different pressure ratios, 248
469
n for large SUP dome in free flood, 234 SUP dome of aspect ratio 2 : 1, 215 vibration under external hydrostatic pressure, 236–43 exciting oblate domes under pressure, 237 exciting small prolate domes under pressure, 237 large domes, 241–3, 247–9 mesh for fluid and dome L2 in large closed tank, 248 small domes, 236–41 small dome vibration under pressure theory, 247 test tank for vibrating large domes under pressure, 247 DTMB formula see David Taylor Model Basin formula electrolytic oxygen generators, 382 elemental mass matrix, 192, 194, 196, 201 EN1A mild steel, 84, 364 external hydrostatic pressure buckling of corrugated circular cylinders, 293–303 CFRP corrugated circular cylinder vibration, 316–24 collapse of dome cup ends, 334–46 tube-stiffened axisymmetric shells vibration and instability, 324–34 unstiffened and ring-stiffened circular cylinders and cones vibration, 243–75 vibration of domes, 236–43 Faraday’s law, 266 FEM see finite element method Ferrybridge cooling towers, 199–200 finite element method, 111 first ply failure, 309 fluid element, 222–5 fluid–structure interaction matrix, 224 Forsberg’s model, 194–5
© Carl T. F. Ross, 2011
470
Index
Forsberg’s solution, 201–2 Fourier cosine transformation, 18 FRA see frequency response analyser fracture mechanics, 11 free vibration, 192 domes, 205, 207–14, 215 in water, 229–35 equation, 217 ring-stiffened cylinders and cones, 201–5, 206 in water, 221–8 unstiffened circular cylinders and cones, 192–200 frequency amplifier, 329 frequency response analyser, 205, 246, 249, 257–8, 260, 316, 329, 394 Galletly’s model, 216 Gauss points, 33, 40, 61, 130 general instability, 6 see also inelastic general instability circular corrugated cylinders, 350 computer programs, 448–59 design chart Kendrick Part I, 190 Kendrick Part III, 191 RCONEBUR, 189 initial out-of-circularity plots cone 1, 188 cone 2, 188 cone 3, 189 pressure vessels, 165–91 ring-stiffened circular cylinders, 165–79, 180–1 buckling configuration, 171 buckling configuration for ring, 174 buckling pressures for DTMB machined models, 178 circular cylinder P3 collapse pressures, 177 collapse pressures, 178 dimensions of the P-series models, 180 end conditions, 177–9 frame with shell, 176
Galletly’s results, 172–3 Kendrick Part I, 167–71 Kendrick Part III, 171–2 notations for circular cylinder deflections, 168 out-of-circularity plots for model P1, 181 ring–shell combination, 166 ring-stiffener, 168 theoretical and experimental buckling pressures, 179 varying stiffener sizes, 174–7 ring-stiffened conical shells, 184–90 buckling pressures, 185 buckling pressures for the three cones, 187 geometrical details, 187 geometrical stiffness matrix, 176 glass-fibre reinforced plastic (GFRP), 389 glass-reinforced plastic (GRP), 10, 13, 117 dome cups, 335–8 Gunther formula, 17 Guyan reduction, 155, 157 Hencky–von Mises theory, 95 HE9 WP aluminium alloy, 105, 365 higher order elements, 151–63 buckling pressures cone, 158 hemi-ellipsoidal domes, 163 Kendrick’s example, 157 conical shells, 151–8 computer analysis, 157–8 element ALLCUBE, 153–5 element QUQUCUBE, 155–7 Kendrick’s example, 157 simply supported cone, 158 hemi-ellipsoidal domes, 159–63 all-cubic element, 159–61 computational analysis, 162–3
© Carl T. F. Ross, 2011
Index quadratic-quadratic-cubic element, 161–2 three-node varying meridional curvature element, 161 nodal displacement positions conical shells, 154 hemi-ellipsoidal domes, 159 quadratic form, 156 high-strength steels, 387 high-tensile steels, 9, 10 material properties, 11 Holland, 110–11 details and its scale model, 110 illustration, 110 predicted collapse pressures, 111 HY80, 11, 389 hydrodynamic hull, 377 imperial gravitational units, 13 inelastic general instability design chart ANSYS, 183 Kendrick Part I, 184 Kendrick Part III, 185 ring-stiffened circular cylinders, 179, 182–4, 185 P1 and P1/A, 182 P2 and P2/A, 182–4 ring–shell combination for P1 and P1/A, 182 Jacobian matrix, 224 KENDPT3.EXE, 448 KENDRIC3.BAS, 451–8 KENDRIC1.EXE, 448 Kendrick Part I, 167–71, 448 buckling configuration, 171 general instability design chart, 190 ring-stiffened circular cylinders, 167–71 inelastic general instability design chart, 184
471
Kendrick Part III, 171–2, 448 buckling configuration, 171 general instability design chart, 191 ring-stiffened circular cylinders, 171–2 inelastic general instability design chart, 185 Kendrick’s method, 8 Levy’s formula, 165 LILICUBE, 215 Lindolm and Hu’s model, 198–9 lobar buckling, 361 Mariana’s Trench, 9, 13, 14 metallic absorbents, 383 metal matrix composites, 10, 389 miniature submarine, 1 MISESNP.BAS, 444–6 MISESNP.EXE, 357, 359, 444 MMC see metal matrix composites monoethanolamine scrubbers, 383 neutral axis, 306–7 Newton–Raphson iterative process, 170–1 NIORDSON.BAS, 446–7 NIORDSON.EXE, 444 Niordson’s method, 116–17 PD 5500 design chart, 356, 358, 360 PKD factor see plastic knockdown factor PLASCONE, 95 plastic collapse, 83–5 plastic knockdown factor, 312, 357 plastic reduction factor, 105, 294 Poisson ratio, 74 pressure hulls collapse of dome cup ends, 334–46 experimental process, 338–41 experimental results, 342–6 experimental tests, 342 manufacturing GRP dome cups, 335–8
© Carl T. F. Ross, 2011
472
Index
cylindrical body design, 284–90 buckling pressures of cones and their circular cylindrical shell equivalents, 288 cylinder/cone design, 285–6 manufacturing a corrugated submarine pressure hull, 289 ring-stiffened corrugated pressure hull, 289 shell instability of corrugated cylinder, 287–8 swedged shaped pressure hull, 285 truncated conical shell element, 289 tube-stiffened pressure hull, 287 tube-stiffened submarine, 286–7 underwater drilling rig, 290 underwater drilling rig design, 289–90 dome ends design, 280–4 convex to pressure, 284 dome–cylinder under external pressure, 281 external meridional stress in cylinder, 282 external meridional stress in dome, 283 internal hoop stress in cylinder, 283 internal hoop stress in dome, 284 novel design, 280–4 screen dump of the deflected form of the vessel, 282 submarine pressure hull with inverted dome ends, 281 novel designs, 280–92 redesign of corrugated food can, 346–54 experimental method, 348–9 material properties, 347–8 redesign of food cans, 352–4 theoretical analysis, 349–52 ring-stiffened or corrugated prolate domes, 290–1 ring-stiffened prolate dome, 291
submarine for the oceans of Europa, 291–2 robotic submarine, 292 swedged vessels buckling pressure, 286 with ring-stiffeners, 287 with tube-stiffeners, 287 vessel 1 (t = 3 cm), 285 vessel 2 (t = 6 cm), 285 vessel 3 with improved shell instability resistance, 288 vessel 4 with improved shell instability resistance, 288 theoretical and experimental results air, 322 changes in resonant frequency in water, 323 simply supported and clamped ends, 320 water, 322 vibration and collapse, 293–354 CFRP corrugated circular cylinder vibration, 316–24 corrugated carbon fibre-reinforced plastic cylinder buckling, 303–16 tube-stiffened axisymmetric shells, 324–34 pressure vessels axisymmetric buckling circular cylinder, 6 oblate dome under external pressure, 5 axisymmetric deformation, 15–99 axisymmetric yield failure, 15 experimental procedure, 85, 89–94 plastic collapse, 83–5, 86–9 ring-stiffened circular cylinders, 16–30 ring-stiffeners, 77–8, 80–3 theoretical plastic analysis, 95–6, 97–8 thick-walled cones and domes, 52–77, 78, 79, 80, 81 thin-walled cones and domes, 30–52, 53
© Carl T. F. Ross, 2011
Index unstiffened circular cylinders and spheres, 15–16 bulkheads, 8–9 construction materials, 9–13 general corrosion, 10 other factors, 11 possible materials and their problems, 10 stress corrosion cracking, 11 cylinder/cone/dome pressure hulls, 4–7 circular cylinders shell instability, 4 hemi-ellipsoidal prolate lobar buckling, 5 illustration, 4 thin-walled conical shells, 7 under external pressure, 1–14 other vessels that withstand external pressure, 7–8 weakening effect on ring-stiffeners owing to tilt, 8 general instability, 165–91 inelastic, 179–84 ring-stiffened circular cones, 7 ring-stiffened circular cylinders, 6, 165–79 ring-stiffened conical shells, 184–90 material properties, 11–13 aluminium alloys, 12 carbon nanotubes, 12 composites, 12 high tensile steels, 11 titanium alloys, 12 pressure, depth and compressibility, 13–14 compressibility, 14 pressure hull external bulkheads, 9 internal watertight bulkheads, 8 shell instability, 100–64 boundary conditions, 138–41 buckling formulae for domes and cones, 142–3 higher order elements for conical shells, 151–8
473
higher order elements for hemiellipsoidal domes, 159–63 inelastic instability, 144–51, 152–3 offshore drilling rigs legs, 141–2 orthotropic cylinders and cones buckling, 117–24 thin-walled circular cylinders, 100–11 thin-walled conical shells instability, 111–17, 118 thin-walled domes buckling, 124–38 varying thickness cylinders, 163–4 spherical pressure vessel, 1–3 Bushnell’s Turtle, 3 spherical pressure hull, 2 spherical shell lobar buckling, 3 types, 1 pressure vessel shells vibration, 192–220 added virtual mass, 220 damping, 220 domes, 205, 207–14, 215 higher order elements for thinwalled cones, 214–16 higher order elements for thinwalled domes, 216–17 pressure, 217–19 ring-stiffened cylinders and cones, 201–5, 206 unstiffened circular cylinders and cones, 192–200 vibration in water, 221–79 effect of tank size, 275–9 free vibration of domes, 229–35 free vibration of ring-stiffened cones, 221–8 vibration under external hydrostatic pressure domes, 236–43 unstiffened and ring-stiffened circular cylinders and cones, 243–6, 249–75 pressurised water reactors, 380–1
© Carl T. F. Ross, 2011
474
Index
QuickBASIC, 350 QUQUCUBE, 215 radioisotopic generators, 380 Rajagopalan’s idea, 152 Rayleigh–Ritz theory, 169 RCONEBUR, 186, 189 resonance dwell test, 317 RINGCYLE.BAS, 428–31 RINGCYLE.EXE, 428–31 RINGCYLG.BAS, 433–43 RINGCYLG.EXE, 432–43 ring-stiffened circular cones experimental method and results, 268–72 frequency variation with pressure (cone 4), 269 frequency variation with pressure (cone 5), 270 frequency variation with pressure (cone 6), 271 structure and fluid meshes, 272 theoretical buckling pressures, 270 theory vs experiment, 272–5 cone 4, 274 cone 5, 275 zero pressure, 273 vibration under external hydrostatic pressure, 263–75 component parts of measuring device, 267 excitor rod attached to cone, 268 experimental equipment, 263–8 geometrical details of ringstiffened cones, 264 measuring device, cone and tank top, 266 pressure tank layout, 267 ring-stiffened cones, 264 transducer arrangement, 265 ring-stiffened circular cylinders, 16–30, 218 beam-on-elastic-foundation theory advantages, 30
circular cylinder stiffened by unequally sized rings, 21–7 illustration, 21 ring-stiffened cylinder, 21 circumferential stress innermost fibre, 29 outermost fibre, 29 experiment vs theory, 27–30 longitudinal generator deflection, 28 model number 3, 28, 30 general instability, 165–79 inelastic general instability, 179, 182–4 longitudinal stress innermost fibre, 29 outermost fibre, 28 stiffened by equal-size stiffening rings, 18 theoretical and experimental resonant frequencies cylinder 4, 262 cylinder 5, 262 cylinder 6, 262 vibration under external hydrostatic pressure, 255–63 apparatus, 255–8, 259 circuit diagram of electrical and electronic equipment, 260 finite element for the fluid and structure, 261 finite elements, 260 geometrical details of cylinders 4, 5 and 6, 259 procedure, 258, 260 theory vs experiment, 261–3 ring-stiffened cones, 206 free vibration, 201–5, 206 vibration experiments, 205, 206 Weingarten’s conical shells, 204–5 Weingarten’s cylindrical shells, 202–4 free vibration in water, 221–8 annular fluid element crosssection, 223 fluid element, 222–5
© Carl T. F. Ross, 2011
Index out-of-roundness, 225–8 meshes adopted for fluid/structure, 226 underwater vibration tests, 226 variation of frequency with n cone 1, 227 cone 2, 227 cone 3, 228 cones with water inside only, 228 ring-stiffened conical shells general instability, 184–90 buckling pressures, 185 buckling pressures for the three cones, 187 geometrical details, 187 ring-stiffened cylinders free vibration, 201–5, 206 Forsbeg’s solution, 201–2 vibration experiments, 205, 206 Weingarten’s conical shells, 204–5 Weingarten’s cylindrical shells, 202–4 ring-stiffened domes, 290–1 ring-stiffeners, 77–8, 80–3, 168, 174–7, 201 bending strain energy, 167 extensional strain energy, 167 ring out-of-plane bending, 82 Ross’ design chart, 356, 358, 359, 360 Ross’ program, 398–9 Sabatier reaction, 383 S-glass, 376, 381, 389 shell 93, 368, 372 shell element, 222–3 shell instability, 363 boundary conditions, 138–41 end closures, 139–41 models dimensions, 139 out-of-circularity plots for TVR2 at mid-length, 140 test tank with attachments, 141 theoretical vs experimental buckling pressures, 141
475
buckling formulae for domes and cones, 142–3 spherical shell cap, 142–3 truncated conical shell, 143 Von Kármán and Tsien’s formula, 142 computer programs, 444–7 corrugated cylinder, 287–8 design charts, 106–10, 118 machined circular cylinders, 108 machined truncated conical shells design chart, 118 soldered and welded circular cylinders, 109 higher order elements for conical shells, 151–8 buckling pressures for cone, 158 buckling pressures for Kendrick’s example, 157 computer analysis, 157–8 element ALLCUBE, 153–5 element QUQUCUBE, 155–7 Kendrick’s example, 157 nodal displacement positions for conical shells, 154 nodal displacement positions for quadratic form, 156 simply supported cone, 158 higher order elements for hemiellipsoidal domes, 159–63 all-cubic element, 159–61 buckling pressures for hemiellipsoidal domes, 163 computational analysis, 162–3 nodal displacement positions for hemi-ellipsoidal domes, 159 quadratic-quadratic-cubic element, 161–2 three-node varying meridional curvature element, 161 orthotropic cylinders and cones buckling, 117–24 buckling pressures and lobes number for model Number 7, 123
© Carl T. F. Ross, 2011
476
Index
buckling pressures for orthotropic conical shells, 123 circular cylindrical element, 119 orthotropic conical shells buckling, 123–4 orthotropic element, 118–23 truncated conical shell, 124 pressure vessels, 100–64 inelastic instability, 144–51, 152–3 offshore drilling rigs legs, 141–2 varying thickness cylinders, 163–4 snap-thru buckling, 144–51, 152–3 aspect ratio 0.7A lobar failure, 148 aspect ratio 0.7B failure in flange, 149 aspect ratio 0.25 snap-thru failure, 147 aspect ratio 0.44 snap-thru failure, 148 GRP dome (bottom), 145 GRP dome geometrical details, 146 GRP dome (top), 145 hemi-ellipsoidal domes (bar) buckling pressure, 153 hemi-ellipsoidal domes buckling pressure, 149 test tank with dome, 146 theoretical pressure–axial deflection plot aspect ratio 0.7A (GRP), 152 aspect ratio 0.7B (GRP), 153 aspect ratio 0.25 (GRP), 151 aspect ratio 0.444 (GRP), 152 aspect ratio 0.7 (SUP), 151 aspect ratio 0.25 (SUP), 150 aspect ratio 0.444 (SUP), 150 theoretical solutions vs experimental observations Sturm’s model, 104 Widenburg and Trilling’s model, 105 thin-walled circular cylinders, 100–11 buckling pressures for models of Ross and Reynolds, 106
circumferential wave patterns for buckling modes, 102 end connection for Sturm’s models, 103 Holland, 110–11 Sturm’s models, 102–3, 104 test tank with attachments, 107 Von Mises formula, 101–2 Windenburg’s model, 103, 105–6, 107 thin-walled conical shells, 111–17, 118 equivalent cylinder, 117 large apical angle cone axisymmetric buckling, 111 Niordson’s method, 116–17 strain matrix for conical shell element, 114 truncated conical shell element, 112 truncated conical shells, 117, 118 thin-walled domes buckling, 124–38 buckling regions position and size, 134 G for a doubly curved axisymmetric element, 129 hemispherical dome lobar buckling, 137 microstrain readings recorded at a pressure of 0.165 MPa for 3.0 aspect ratio dome, 133 microstrain readings recorded at a pressure of 0.483 MPa for 1.5 aspect ratio dome, 132 model L2 with rupture, 136 near hemispherical domes, 137–8 ten hemi-ellipsoidal oblate/prolate domes, 130 thin-walled domes, 130–2 varying meridional curvature element, 125–30 VMC element, 132, 134–6 Solartron 1170, 394 solid urethane plastic, 40, 130, 325, 394, 416 steels, 9, 10
© Carl T. F. Ross, 2011
Index stiffness matrix, 174 strain matrix, 174 stress corrosion cracking, 11, 387 Sturm’s models, 102–3, 104 end connection, 103 theoretical solutions vs experimental observations, 104 submarine pressure hulls, 355–60 ANSYS pressure–deflection plot cylinder 4, 368 cylinder 5, 369 cylinder 6, 369 ANSYS screen dump cylinder 1, 372 cylinder 2, 373 cylinder 3, 373 cylinder 7, 372 designs, 356–60 axisymmetric deformation, 360 designing against shell instability, 357–9 full-scale submarines dimensions, 356 general instability, 359–60 submarine 1, 356 submarine 2, 356–7 experimental analysis, 364–7 aluminium alloy circular cylinders geometrical details, 367 circular cylinders geometrical details, 367 experimental and theoretical collapse pressures, 367 pressure–strain relationship for cylinder 4, 366 test tank and hydraulic pump, 366 test tank and model, 365 thin-walled cone axisymmetric collapse, 364 non-linear analysis using ANSYS, 361–74 general instability of ring-stiffened cylinders 4, 5, and 6, 363 ring-stiffened circular cylinders 4, 5, and 6, 363 shell instability for cylinder 7, 362
477
shell instability of cylinders 1 to 3, 362 novel designs, 280–92 cylindrical body design, 284–90 dome ends design, 280–4 Europa, 291–2 ring-stiffened or corrugated prolate domes, 290–1 PD 5500 design chart, 358 Ross’ design chart general instability, 359 shell instability, 358 screen dump cone C, 371 cylinder 4, 369 cylinder 5, 370 cylinder 6, 370 theoretical analysis, 368–72, 373 experimental and ANSYS nonlinear and elastic results, 374 mesh for typical circular cylinder, 368 typical mesh for cylinders 1 to 3, 371 SUBPRESC, 319, 324 SUBPRESS, 319 SUP see solid urethane plastic SWEDBUCK, 350, 352 swedged vessels, 285–8 switch box, 329 tank size, 275–9 test tank, 330 thick-walled cones and domes, 52–77, 78, 79, 80, 81 comparisons between various elements, 70 comparison with a tapered dome, 71–2, 73, 74, 75–7, 78, 79, 80, 81 method of pressuring dome, 72 orthotropic element, 72, 74 dome 1 experimental shell section, 71 experimental strain results, 74 hoop stress distribution, 76, 78, 79
© Carl T. F. Ross, 2011
478
Index
meridional stress distribution, 76, 79 strain gauge positions, 73 dome 2 experimental shell section, 71 experimental strain results, 75 hoop stress distribution, 81 meridional stress distribution, 77, 80 strain gauge positions, 73 five-node element, 66, 68–70 hoop stress distributions for 3-in. shell cap, 70 matrix, 69 quartic element, 68 four-node element, 64–6, 67 cubic element, 66 matrix, 67 local and global axes, 56 matrix for thick conical shell, 57 parabolic element, 58–64, 65 hoop thickness for 9 in. thickness cap, 65 9 in. thickness shell cap, 64 matrix for the thick-walled threenode parabolic element, 62 mean hoop stress distribution for 3 in. thickness cap, 63 three-node, 59 thick conical shell, 53 THINCONE, 281 thinness ratio, 299–301, 309, 351, 371 thin-walled cones and domes, 30–52 ACMC vs AVMC elements, 40–7, 48 aspect ratio hemi-ellipsoidal domes, 41 pressure tank with test shell in position, 42 aspect ratio 1.5 experimental strains along dome meridian, 44 internal circumferential stress distribution, 46 internal meridional stress distribution, 45 out-of-circularity plot, 47
strain gauge positions and directions, 43 variation in stress with mesh refinement, 42 aspect ratio 3.0 experimental strains along dome meridian, 45 internal circumferential stress distribution, 47 internal meridional stress distribution, 46 out-of-circularity plot, 48 strain gauge positions and directions, 44 variation in stress with mesh refinement, 43 axisymmetric constant meridional curvature element, 36–40 axisymmetric shell element, 37 [B] for ACMC element, 39 longitudinal section through element, 37 axisymmetric varying meridional curvature element, 34–6 varying meridional curvature element, 35 FEA vs Woinowsky-Kreiger, 33–4 hoop forces/unit length for cap, 34 meridional bending moment/unit length for cap, 34 spherical shell cap, 33 higher order elements, 214–16 tapered cylindrical shell element, 47–52, 53 radial deflection of longitudinal generator, 53 tapered thin-walled circular cylinder, 48 thin-walled conical element, 31 thin-walled prolate dome experimental method, 394–5 analysing the vibration of pressure vessels, 394 resonant frequencies measurement, 395
© Carl T. F. Ross, 2011
Index test set-up, 394–5 vibration in water, 395 finite element model dome 8, 410 dome 9, 410 dome 10, 411 dome in water tank, 409–11 shell63 results using four-noded element, 401–2, 403 sensitivity analysis using the fournoded element, 402 shell element, 396 shell93 results using eight-noded element, 402, 404 sensitivity analysis using the eight nodes element, 404 shell element, 397 theoretical basis of finite element method, 395–9 ANSYS 7.1 vs in-house software, 398–9 element shape functions, 396, 398 shell elements in ANSYS, 396–8 vibration analysis underwater, 395–6 vibration analysis in air, 399–406 aspects covered in analysis, 400 constraints on model and symmetry, 400–1 eigenmodes of vibration for dome, 403–4 evaluation of modal analysis of dome, 405–6 experimental results vs CON-FE solution, 405 prolate dome, 399 vibration analysis in water, 406–15 acoustic pressure in fluid, 412 comparison of finite element solution to experiment and CON-FE, 413–14 comparison of results in water, 413 fluid elements, 407–9 precision of simulation, 414–15
479
previous research, 407 procedure in coupled fluid structure analysis, 407 relative error of FE solutions to experiment, 414 results of finite element simulation in water, 411–12 vibration analysis under external pressure, 415–18 comparison of finite element solution with the experiment, 416, 417 consideration of pressure in finite element analysis, 415 effect of external pressure on natural frequencies in water, 417 resonant frequencies under external hydrostatic pressure, 417 results of finite element analysis, 416 vibration under external water pressure using ANSYS, 393–418 titanium alloys, 10, 388 material properties, 12 strength, 389 Tsai–Hill failure criteria, 308–9, 314 tube-stiffened axisymmetric shells buckling pressures of domes, 333 experimental method, 327–30 buckling experiment, 328 dome out-of-roundness, 327 dome set-up with holder, 329 experimental results for buckling, 328 large tank arrangement with vessel secured for vibration testing, 330 observations of vibrating eigenmodes, 330 out-of-roundness plot, 327 vibration tests, 328–30 internally tube-stiffened domes construction, 325–7 resonant frequencies for tubestiffened dome, 331
© Carl T. F. Ross, 2011
480
Index
theory buckling, 331 vibration, 331 theory vs experiment, 331–4 buckling, 332 vibration, 331–2 values under external pressures and internal pressure bar, 332 vibration and instability, 324–34 dome and reel, 326 dome with stiffeners applied to internal surface, 324 internally tube-stiffened domes construction, 325–7 tube-stiffened submarine, 286–7 underwater drilling rig, 289–90, 290 underwater missile defence system, 375–92 advantages, 376 central spherical shell, 385 choice of material, 386–90 aluminium alloys, 387–8 carbon nanotubes, 390 composite, 388–9 general corrosion, 387 high-strength steels, 387 other factors, 387 strength and relative costs of composites, 389 strength of aluminium alloys, 388 strength of high-strength steels, 388 strength of titanium alloys, 389 stress corrosion cracking, 387 titanium alloys, 388 connecting walkways, 385–6 design, 377–9 corrugated pressure hulls construction method, 379 hull form, 377–9 underwater space station 1, 378 underwater space station 2, 378 disadvantages, 376
environmental control and life support systems, 381–4 atmospheric climate, 384 atmospheric control, 381–2 carbon dioxide control, 382–3 contaminant control, 383–4 oxygen consumption, 382 typical contaminant exposure limits, 383 experimental requirements other external requirements, 384 support legs, 384 manpower and living conditions, 379–80 material property requirements, 386 power requirements, 380–1 different types of nuclear generator, 380 emergency power supply, 381 pressure hull design, 390 required wall thickness, 390–1 sound absorption coefficients, 391 wall thickness calculations, 390–1 wall thickness of circular section of the elliptical structure, 391 size of elliptical structure, 385 unstiffened circular cylinders and cones Abdulla and Galletly’s model, 197–8 cone–cylinder combination, 197 resonant frequencies for cone– cylinder combination, 197 Ferrybridge cooling towers, 199–200 nickel model of the Ferrybridge cooling towers, 200 theoretical resonant frequencies, 200 Forsberg’s model, 194–5 mode shape of cylinder with fixed edges, 195 results corresponding to fundamental frequency n = 4, 195 free vibration, 192–200 circumferential wave pattern, 193 elemental mass matrix, 196 lobar eigenmode of vibration, 193
© Carl T. F. Ross, 2011
Index Lindolm and Hu’s model, 198–9 frequencies for 14.2° conical shell, 198 frequencies for 30.2° conical shell, 198 frequencies for 45.1° conical shell, 199 frequencies for 60.5° conical shell, 199 vibration under external hydrostatic pressure, 243–6, 249–75 Warburton’s model, 195–6 simply supported circular cylinder, 196 variational finite differences, 215 varying curvature elements, 125 varying meridional curvature, 229 VCLAM, 319–20, 322 Verification Manual 177, 408 VFD see variational finite differences VIBCONE, 318 vibration big tank vs small tank resonant frequencies AR = 2.5, 278 AR = 3.0, 277 AR = 3.5, 276 CFRP corrugated circular cylinder, 316–24 CFRP vessel in test tank, 318 corrugated carbon fibre cylinder, 317 experimental method, 316–18 finite element programs, 318–19 n = 3 amplitude plot, 320 n = 3 phase angle plot, 319 results, 319–22 structure–fluid mesh, 321 theoretical analysis, 318 corrugated circular cylinders, 293–303 domes, 205, 207–14, 215 axisymmetric eigenmodes for oblate domes, 207
481
under external hydrostatic pressure, 236–43 large dome, 213 Lobar eigenmodes for prolate domes, 207 nodal lobar patterns, 214 nodal pattern for complex eigenmode, 207 vibrating domes in air, 208–13 effects of pressure, 217–19 hemi-ellipsoidal prolate dome of aspect ratio 2 : 1, 219 large dome, 219 pressure, 217–18 ring-stiffened circular cylinder, 218 higher order elements for thinwalled cones, 214–16 pressure vessel shells, 192–220 effects of added virtual mass, 220 effects of damping, 220 higher order elements for thinwalled domes, 216–17 pressure vessel shells in water, 221–79 effect of tank size, 275–9 free vibration of domes, 229–35 free vibration of ring-stiffened cones, 221–8 testing small dome in large open tank, 279 resonant frequencies Galletly’s model, 216 oblate domes, 216 ring-stiffened circular cylinder, 201 simply supported circular cylinder, 196 Warburton’s circular cylinder, 216 ring-stiffened cylinders and cones, 201–5, 206 block diagram showing the excitation and detection instrumentation, 206
© Carl T. F. Ross, 2011
482
Index
configuration of longitudinal generator of Bushnell’s cylinder, 202 under external hydrostatic pressure, 243–6 Forsberg’s solution, 201–2 method of excitation of ringstiffened cones, 206 ring-stiffened cones, 206 vibration experiments, 205 Weingarten’s conical shells, 204–5 Weingarten’s cylindrical shells, 202–4 theory vs experiment ring-stiffened circular cylinder of rib height 0.095 in, 203 ring-stiffened conical shell of rib height 0.045 in, 204 ring-stiffened conical shell of rib height 0.095 in, 205 ring-stiffened circular cylinder of rib height 0.045 in, 203 tube-stiffened axisymmetric shells instability, 324–34 experimental method, 327–30 internally tube-stiffened domes construction, 325–7 theory, 331 theory vs experiment, 331–4 unstiffened circular cylinders and cones, 192–200 Abdulla and Galletly’s model, 197–8
elemental mass matrix, 196 under external hydrostatic pressure, 243–6 Ferrybridge cooling towers, 199–200 Forsberg’s model, 194–5 Lindolm and Hu’s model, 198–9 Warburton’s model, 195–6 vibration tests, 328–30 vibration theory, 331 Viterbo effect, 17 VMC see varying meridional curvature von Kármán and Tsien’s formula, 142 von Mises formula, 101–2, 308, 357 circumferential wave patterns for buckling modes, 102 von Mises theory, 253 von Sanden formula, 17 Warburton’s circular cylinder, 216 Warburton’s model, 195–6 Weingarten’s conical shells, 204–5 Weingarten’s cylindrical shells, 202–4 Windenburg’s model, 103, 105–6, 107 test tank with attachments, 107 theoretical solutions vs experimental observations for Sturm’s model, 105 Windenburg thinness ratio, 357, 359 Young’s modulus, 74, 78, 95
© Carl T. F. Ross, 2011
