Shell Stability Handbook

SHELL STABILITY HANDBOOK

SHELL STABILITY HANDBOOK Edited by

LARS Å.SAMUELSON The Swedish Plant Directorate, Stockholm, Sweden and

SIGGE EGGWERTZ Bloms Ingenjörsbyrå, Sundbyborg, Sweden

ELSEVIER APPLIED SCIENCE LONDON and NEW YORK

ELSEVIER SCIENCE PUBLISHERS LTD Crown House, Linton Road, Barking, Essex IG11 8JU, England This edition published in the Taylor & Francis e-Library, 2005. “ To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to http://www.ebookstore.tandf.co.uk/.” WITH 9 TABLES AND 192 ILLUSTRATIONS ENGLISH EDITION © 1992 ELSEVIER SCIENCE PUBLISHERS LTD British Library Cataloguing in Publication Data Shell Stability Handbook I. Samuelson, Lars A. II. Eggwertz, Sigge 624.1 ISBN 0-203-21367-X Master e-book ISBN

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PREFACE The Shell Stability Handbook is a handbook for calculation of the carrying capacity of shell structures with respect to buckling. The aim has been to develop a branch independent handbook which gives conservative estimates of the carrying capacity. The safety factors which should be used in a certain case must be taken from the applicable code for the considered structure, e.g. pressure vessel codes, steel structure codes etc. Initiator behind the handbook is Professor Lars Å.Samuelson who already in the seventies during his time at the Aeronautical Research Institute of Sweden felt a strong need for such a handbook. The development of the handbook, which also included substantial research in the field of shell stability, started in 1982. The first Swedish edition was published in 1990. The handbook has been developed by a working group including members from the pressure vessel, the building and the aeronautical branches: Professor Torsten Höglund, Chairman The Royal Institute of Technology, Stockholm

(2, 3.6, 3.7, 6.3.2)

Dr Lars Dahlberg The Swedish Plant Inspectorate, Stockholm

(5.1)

Professor Sigge Eggwertz Bloms Ingenjörsbyrå AB

(3.4, 8)

Mr Gert Hessling The Swedish Plant Inspectorate, Stockholm.

(3.5)

Professor Bernt Johansson Luleå Institute of Technology, Luleå

(3.1, 3.2.1, 3.2.2, 3.3)

Mr Kurt Lundi The Swedish Institute of Steel Construction, Stockholm

Notations

Professor Lars Å.Samuelson The Swedish Plant Inspectorate, Stockholm

(1, 3.2.2, 3.2.3, 3.7, 4, 5, 6, 7, 8)

The work in the group was carried out in close cooperation, but the responsibility for the different chapters was divided as indicated above. A reference group consisting of members from industries, authorities and universities gave technical support during the work: Mr Gunnar Emling, Chairman

The National Swedish Board of Occupational Safety and Health

Mr Kjell Andersson

Hedtank AB

Mr Bengt Andreason

Kockums Marine

Professor Bo Edlund

Chalmers Institute of Technology

Mr Lars Hammarström

Volvo AB

Mr Bengt Hellman

The National Swedish Board of Physical Planning and Building

Mr Gunnar Holmberg

Saab Scania AB

Mr Rolf Jarlås

The Aeronautical Research Institute of Sweden

Mr Karl-Arne Johansson

Amfibieteknik AB

Mr Åke Ringqvist

Flakt Industries AB

Professor Alf Samuels son

Chalmers Institute of Technology

Mr Lars Sjöström

Saab Scania AB

Mr Bengt Skanselid

ÅF Energikonsult Syd AB

Dr Per-Olof Thomasson

The Swedish Institute of Steel Construction

Mr Adam Zdunek

The Aeronautical Research Institute of Sweden

Professor Bengt Åkesson

Chalmers Institute of Technology

The reference group gave valuable support and many contributions during the work including examples from their own practice which increased the usefulness of the handbook. The work was financially supported by: The Swedish Work Environment Fund Bloms Ingenjörsbyrå AB The Swedish Council for Building Research Chalmers Institute of Technology Götaverken Arendal AB Götaverken Mekan AB The Swedish Institute of Plastics end Rubber The National Swedish Board of Physical Planning and Building The Swedish Institute of Steel Construction The Swedish Plant Inspectorate SSAB Association of Swedish Engineering Industries Tyréns Företagsgrupp AB 3K Akustikbyrån

The main financier was The Swedish Work Environment Fund, but all contributions were important for the first publication of the handbook. The figures were, with excellent result, drawn by Mr Anders Samuelson. Lars Dahlberg

CONTENTS PREFACE NOTATIONS

Chapter 1: INTRODUCTION

v ix 1

Chapter 2: DESIGN PHILOSOPHY

33

Chapter 3: ELEMENTARY CASES

55

Chapter 4: EFFECT OF LOCAL LOADS AND DISTURBANCES

163

Chapter 5: INFLUENCE OF TEMPERATURE

193

Chapter 6: STABILITY ANALYSIS BY USE OF NUMERICAL METHODS Chapter 7: SHELLS OF COMPOSITE MATERIALS

212

Chapter 8: EXAMPLES

237

REFERENCES

231

258

APPENDIX Brief review of the theory of stability for shell structures A

266

APPENDIX Elementary cases for stress and deformation analysis of rings B subjected to forces in their own plane

276

APPENDIX Diagrams for analysis of stringer stiffened cylindrical shells C under axial compression

283

F

Force, load.

Fk

Characteristic value of force.

G

Elastic shear modulus.

I

Moment of inertia of cross section.

M

Bending moment.

N

Normal force.

P

Force.

Q

Load.

T

Torsional moment.

V

Shear force, lateral force.

W

Section modulus.

b

Width.

d

Diameter.

e

Eccentricity.

f

Strength (in this handbook the symbol σ has been used synonymously with f).

fk

Characteristic value of strength.

h

Height.

l

Length.

m

Number of halfwaves of the buckling mode in the axial direction.

n

Number of complete waves of the buckling mode in the circumferential direction.

p

Pressure.

r

Radius.

s

Coordinate in the meridional direction.

t

Thickness.

u

Displacement in the x- or s-direction.

v

Displacement in the φ-direction.

w

Displacement in the z-direction (perpendicular to the middle surface of the shell).

x

Coordinate in the axial direction of a global system, Fig. 1.

y

Coordinate in a global system, Fig.1.

z

Coordinate perpendicular to the middle surface of the shell in a local system, Fig. 1.

α

Angle, reduction factor with reference to tolerance class.

β

Rotation of shell element, Fig. 1.23, spherical shell parameter, eqn (3.110).

γ

Load or strength factor (partial coefficient), shear deformation, gravity per unit volume.

γf

Load factor.

γm

Resistance factor related to the material properties.

γn

Resistance factor related to safety class.

γm n

Abbreviation for the product γmγn.

ε

Strain.

η

Reduction factor with reference to initial deflections etc.

λ

Slenderness ratio.

v

Poisson’s ratio.

φ

Angle, co-ordinate in the circumferential direction.

ψ

Angle, coordinate in the meridional direction of a spherical shell.

θ

Meridional rotation, Fig.3.37.

ω

Reduction factor for strength in the elasto-plastic region.

Subscripts R

Resistance, synonomous with carrying capacity.

S

Load effect (sollicitation).

r

Ring (example: Ar), circumferential direction.

x

Coordinate direction.

φ

Coordinate direction.

u

Ultimate strength, at collapse.

el

Buckling quantity according to linear (classical) theory.

elr

Reduced buckling quantity.

s

Stringer, coordinate direction.

b

Bending.

z

Coordinate direction.

y

Coordinate direction.

cr

Critical.

ef

Effective.

eq

Equivalent.

Differentials

NOTATIONS The notations of the Swedish Regulations for Steel Structures are used throughout this handbook. These are very similar to those internationally accepted and used for instance in the ECCS design rules and in the CEN Eurocodes 1 and 3. The symbol σ has been chosen, however, to denote stress whether it refers to the strength of the material or the load effect. The meaning should be evident from the context in all cases.

Coordinate system The most common shell category is, beyond comparison, the shell of revolution which may be composed of cylinders, spherical caps, cones etc. For such shells the symbols shown in Fig. 1 are generally used. In cases where other symbols are introduced, they will be defined in the section where they appear.

Fig. 1. Coordinate systems for cylindrical and axisymmetrical shells. Symbols A

Area.

C

Extensional stiffness, eqn (1.15).

D

Bending stiffness, eqn (1.15).

E

Young’s modulus of elasticity.

Units All formulas in this handbook are independent of the unit system used. SI units are used in some special applications and in the examples . Abbreviations ASME

American Society of Mechanical Engineers.

ASTM

American Standard for Testing Material.

BS

British Standard.

BSK

Swedish Regulations for Steel Structures.

DASt

Deutscher Ausschuss für Stahlbau (German Task Group for Steel Design).

DIN

Deutsche Industrienormen (German industrial code).

ECCS

European Convention for Constructional Steelwork.

EC 1&3

CEN Eurocode 1 for actions on structures and Eurocode 3 for steel structures.

NR

Swedish Building Code (new).

SBN

Swedish Building Code (old).

TKN

Swedish Pressure Vessel Code.

Definitions and concepts Bifurcation, branching into two or more states of equilibrium. The bifurcation load of a perfect structure is commonly called the classical buckling load. Branching point, see Bifurcation. Buckling is a phenomenon giving large deflections perpendicular to the shell surface. It is caused by forces acting in the shell middle surface where they are giving rise to compressive and/or shear stresses. The buckling behavior may be stable or unstable. Characteristic value is a value of the quantity considered which corresponds to a given fractile of its statistical distribution. For the resistance the 5 per cent fractile is normally chosen with the exception for steel materials where a value is chosen approximately equal to the 1 per cent fractile. For loads, the 98 per cent fractile of the distribution of the yearly maximum is usually chosen as representative for an extreme value of a variable load (see 2.4.2). Circumferential direction, see Fig. 1.

Classical buckling theory is a linear theory for analysis of structural stability under the action of a load. It is often based on a number of simplifications which have been introduced in order to obtain solutions in a closed form. The theory assumes that the material is linearly elastic, that the structure has an ideal geometrical shape (straight bar, plane plate) and further that the displacements are small. Collapse means such a decrease of the carrying capacity of a structure that it becomes unsuitable for its purpose. Collapse is characterized by the maximum of the loaddeflection diagram. Ultimate failure is used synonymously with collapse. Conical shell, see Shell. Crown ring, see Fig. 3.55. Cylindrical shell, see Shell. Effective area is a fictitious cross-section area which is used in design calculations instead of the actual area. The effective area can be used in compressed members with cross sections built up from slender elements, where it forms a means of considering the influence of local buckling of the elements on the carrying capacity of the member. The effective area of a slender, rectangular cross-section element can be evaluated either as an effective width multiplied by the actual thickness (see Dubas and Gehri, 1986) or as an effective thickness multiplied by the actual width. Formulas for calculating the effective width, or the effective thickness respectively, have been deduced on the basis of the behavior of slender plates in compression. Equilibrium can be stable, unstable or indifferent. Equilibrium branching (bifurcation) implies that two or more conditions of equilibrium exist at a given load (branching point). Equivalent wall thickness is a concept introduced to describe a shell with varying thickness, or provided with stiffeners, as a homogeneous shell with constant thickness. Failure. In this handbook the word is used as a definition of the state where the maximum load carrying capacity has just been passed under increasing deformation. In this context the word is synonymous with collapse. First-order theory implies that a model is used where the influence of the deflections on the internal forces and moments is neglected. The first-order theory yields first-order forces and moments etc. Generator, see Shell. Imperfection is a term for a deviation from the assumptions on which the classical theory is based. Examples of imperfections in shells are initial buckles, residual stresses and unintended load eccentricity. Initial buckles (often local) are deviations from the ideal geometrical shape in an unloaded shell or an unloaded plate. Interaction formulas are design criteria for such load cases which can easily be divided into simpler cases. Interaction formulas assume that for each one of the simpler load cases a measure of the load effect can be given as well as the corresponding carrying capacity. The terms of the interaction formula contain the ratio between the load effect and the capacity for each single load case. Examples of interaction formulas are found in 3.2.1.8. Limit state, see Serviceability limit state and Ultimate limit state respectively as well as subsection 2.2.4.

Limit state design (method of partial coefficients) is a method for structural design where the safety requirements are considered by introducing statistically determined quantities (characteristic values) and load and resistance factors (partial coefficients). Linear buckling theory (=classical buckling theory) is a theory for the stability of a structure subjected to compressive loads. It assumes that the deformations are small, that the material is a linearly elastic and that internal forces and moments are determined according to first-order theory. Load factor is a factor which takes into account the variability of the load, in addition to the variability considered by the characteristic value. Meridian, see Fig.3.37. Synonym for Generator. Membrane state is a state of loading where only in-plane internal forces in the middle surface of the shell exist. Method of allowable stresses is a method for structural design where the safety requirements are considered by applying a maximum allowable stress for the structure. The stresses in the structure are calculated for given values of loads and load combinations. The allowable stress is determined as a given fraction of the stress which leads to fracture or instability. Mode, shape of buckling deflection. Node line is a line, or curve, formed by the points on the shell which do not move during buckling. Partial coefficient, see Limit state design. Perfect shell is a shell which is assumed to have a perfect geometrical shape, e.g. without initial buckles. It is furthermore assumed to have such material properties that it is in complete accord with the linear theory of elasticity and that it is relieved from residual stresses. The boundary conditions are assumed to agree with a chosen idealized model, such as a hinge without friction, complete clamping, entirely fixed boundary etc. Plastic buckling implies that buckling occurs at a load level where at least part of the structure before or during buckling is subjected to plastic deformation, i.e. strain which does not return to zero on unloading. Polygonal ring, see Fig. 3.55. Postbuckling state. A structural member can, under certain conditions, carry a load which exceeds the buckling load according to the classical theory. This exceedance is a result of redistribution of stresses. When a structural member is subjected to a load which is higher than the buckling load the structure is said to be in a postbuckling state, characterized by loss of stiffness and large deformations. Prebuckling state is the state of the structural member when the load is lower than the buckling load. Residual stresses are stresses present in a structural member which is neither subject to external loads nor to other actions (not even gravity loads). Residual stresses arise from manufacturing procedures such as rolling, flame-cutting, welding, cold forming and straightening. Resistance, (used as a synonym for load carrying capacity) is a measure of the capacity of the structure to carry load. The resistance can be expressed as a load, as an internal cross-sectional quantity (normal force, bending moment, shear force), or as a stress. If not otherwise stated, the maximum resistance is referred to as the load which causes failure of the structure (collapse).

Resistance factor is a factor which takes into account the variability of the resistance in addition to the variability considered by the characteristic value. Safety class refers to a classification of building structures and structural members with regard to the consequences of a failure, considering mainly the risk of human injury. Safety factor is the ratio between the ultimate failure stress and the allowable stress. Second-order theory implies a computer model in which the effect of the deflection on the internal forces and moments is taken into account. The second-order theory yields second-order forces and moments etc. Serviceability limit state is a state where the structure is on the limit of not satisfying one of the requirements prescribed for its function under normal conditions. Shell, a thin-walled body shaped as a curved surface with the thickness measured along the normal to the surface being small compared with the dimensions in the other directions. The shell carries its load mainly by membrane forces.. The middle surface has a finite radius of curvature at each point. The radius of curvature can, however, be infinite in one direction (shell with single curvature, e.g. cylinder). Axisymmetric shells form a particularly important class of shells. They can, depending on the shape of the meridian, be divided into: – cylindrical shells – conical shells – spherical shells (domes, caps) – toroidal shells – torispherical shells. The middle surface of such shells is formed by the points on a so called generator (meridian) when the generator makes one revolution around the axis. The generator may be a straight line or a plane curve defined in the same plane as the axis of rotation, see e.g. Fig. 1. Single-curvature shell, see Shell. Spherical shell, see Shell. Torsional buckling is a stability phenomenon where a bar, loaded in compression, and possibly also in bending, is subject to deflection and simultaneous twisting. Torispherical shell,see Shell. Ultimate limit state is a state where the carrying structural member approaches the limit of some kind of failure.

1 INTRODUCTION 1.1 Scope Stability of structural members of various kinds in buildings, vehicles, tanks, process vessels, silos etc. constitutes one of the factors which must be considered in the design analysis. For straight compressed bars the Euler theory, well known to most engineers, may be applicable under certain conditions. A similar type of theory can be applied for plane plates and shells with results which are more or less accurate. Especially for some types of shells a design based on the classical theory, corresponding to the Euler theory, may imply a significant risk, since the actual carrying capacity may be only a fraction of the value obtained from this theory. The fact that this behavior of shell structures is not widely known and, furthermore, that simple procedures for analysis of the buckling of shells are available only to a very limited extent in design codes, has lead to the development work presented in this Shell Stability Handbook. The objectives of the work can be summarized as follows: – provide practical guidance for determination of the actual resistance of various types of shell elements subject to the most commonly occurring external loads which can cause buckling. – give advise and direction for the interpretation of buckling loads obtained by use of general computer programs based on FEM and similar methods. – discuss possible ways of considering the effect of disturbances and imperfections in geometry and loading etc. on the resistance of a shell loaded in compression. – constitute a supplement to existing design codes and standards. In most countries, shell buckling is treated to a very limited extent. In many cases problems occur which are entirely neglected in these documents. The assessment of structural safety varies today between different codes, which may result in misinterpretations. The Shell Stability Handbook provides routines for determining a safe, lower limit of the resistance of the shell (characteristic value). This resistance can be used as a basis for design. The design value is obtained by dividing the value by a safety factor or a partial coefficient (resistance factor). – form a textbook on shell stability, where different aspects of the basic principles of shell theory and analysis are described. The presentation focuses on the importance of the different parameters which affect the carrying capacity of the shell. Rules for the design of bars, beams and thin plates subjected to buckling are given in most national codes. A comprehensive treatment of plate stability problems was published by the ECCS, see Dubas and Gehri, 1986. Recommendations for design of shell structures were published by ECCS in 1988. The present handbook may be considered as a supplement to the kind of documents mentioned. While the ECCS

Shell stability handbook

2

recommendations are adapted for use in the design of steel structures, the Shell Stability Handbook is directed towards a more general description of the buckling behavior. Applications for different materials are described separately, in particular when the elasto-plastic properties differ. This edition of the handbook is obviously not complete. The intention is to continue the work by developing design recommendations for additional types of shells and loadings. These will be introduced into later editions. The stability problems addressed in this handbook mainly concern shells of metallic materials. Fiber reinforced plastics, reinforced concrete and sandwich shells are used to an increasing extent. Problems occurring with these materials are not treated in detail but reference to relevant literature is made, if available. It should be pointed out that the handbook is mainly concerned with factors which may influence the buckling behavior and, consequently, the buckling resistance. Local effects, such as the design of edge stiffeners around a hole, are considered in the traditional manner according to applicable standards, using the factors prescribed for the critical load case, which may be ultimate failure under internal pressure. Where coupling is possible between e.g. local stresses and the buckling behavior, an assessment of the influence is given. 1.2 Brief historical notes with references to the literature The theory of stability of shells is relatively young but, due to the extensive use of shells as structural elements in modern technical applications—aerospace vehicles, containment vessels, offshore and building structures etc.—very intensive research has been carried out since the thirties. Some of the most important milestones in the development of the shell buckling theory are presented in the following review. 1.2.1 Development of the shell theory Euler’s formulas for determining the critical load of a compressed straight bar were published in the middle of the 18th century. The theory was further developed during the latter half of the 19th century, when the stability of thin plates was also analyzed in an analogous way. A theory of shell buckling was first proposed in the beginning of the 20th century, by Lorenz, 1908, and Timoshenko, 1910, who presented solutions for axially compressed circular cylindrical shells. External lateral pressure on cylindrical shells was treated by Lorenz, 1911, Southwell, 1913, and by von Mises, 1914. More profound studies of the theory of stability for shell structures were, however, not initiated until the thirties. To a large extent, the theoretical and experimental data produced during that time were of a basic nature. The results are, however, still used in the design of shells, although the capabilities for performing accurate numerical analyses have improved drastically since then. Flügge, 1932, 1934, presented an extensive theory for cylindrical shells where he treated various load cases and load combinations. During the thirties the first edition of Timoshenko’s monograph “Theory of elastic stability” was also published, see Timoshenko, 1936, or the revised edition, Timoshenko-Gere, 1961. A linear buckling

Introduction

3

theory for spherical shells was first established by Zoelly, 1915. The stability of conical shells does not seem to have attracted attention during the first decades of the 20th century. The first paper on cones was published by Pflüger, 1937. The first attempts to develop a theory of stability all had in common that a linear theory was used which will be referred to as the classical theory in the text to follow. Donnell, 1934, investigated the influence of second order quantities and found that they could be of great importance in the estimate of the resistance. As a result he developed a nonlinear theory by which he could take second order effects into account, see Donnell and Wan, 1950. Donnell’s equations have, subsequently, been used by a large number of investigators, see e.g. Brush and Almroth, 1975. The nonlinear shell theory leads to very complicated systems of equations. General, closed form solutions of these equations have been achieved only for a few simple cases. In the design of shells against buckling, the methods developed during the thirties are, consequently, still in use. During the course of the years a large number of test series have been carried out in order to verify the theories. A classical work is the NACA Handbook of Structural Stability, Part I– VI, Gerard-Becker, 1957–58, in which all test results known up to that time were collected and compared to current theories. The NACA Handbook has been used as a basis for the design of several generations of aircraft and space vehicles. During the sixties, the awareness of the extreme sensitivity, in some cases, of shell structures to initial imperfections, residual stresses and other disturbances gradually increased, and extensive research work was performed in an effort to find a solution to the buckling problem which would give more realistic results. Many suggestions were made, but few of these lead to any significant improvement. The shape and size of initial deformations vary randomly, and it is now commonly agreed that the design procedure must consider the statistical nature of the disturbances. A comprehensive study of the buckling behavior of shells including their sensitivity to initial deformations was published in a thesis by Koiter, 1945. The research work, which was carried out during World War II, was written in the Dutch language. For that reason it did not become known internationally until the end of the sixties when translations into English were published, Koiter, 1967, 1970. Today it is considered to be a pioneering work. Koiter’s theory for evaluation of the sensitivity of a shell to initial deformations has been of great use in the further development of theories and calculation procedures during the last decade. Since the theory treats the condition at incipient buckling only, it does not enable a quantification of the reduction of the buckling resistance of the shell. It only provides an indication whether the shell is imperfection sensitive or not. Statistical studies of the influence of initial imperfections have been performed by several scientists, see e.g. Hansen, 1977, and Arbocz, 1987, 1991. In these papers, probabilistic methods are introduced in the evaluation of the risk that failure due to buckling will occur at a given load level. During the last years of the sixties, the electronic computer became more commonly available for research and development and several computer programs for buckling analysis were produced during this period. Examples of such programs can be found in : Bushnell, 1972, 1974, Cohen, 1968, Ball, 1968, Zienkiewics, 1971, Almroth et al, 1980, Samuelson, 1975. One of these programs, BOSOR, was developed by Bushnell as a special purpose program for the analysis of buckling and vibration of axisymmetric shells

Shell stability handbook

4

under otherwise rather general conditions. BOSOR determines the buckling load by use of eigenvalue theory and cannot be used to study the effect of nonsymmetric initial imperfections. Other similar programs have been developed, see Ball, 1968, Cohen, 1968, Samuelson, 1975, where nonsymmetric collapse can be evaluated. Use of such programs has made it possible to demonstrate a good agreement between theory and experiments for shells with known initial deformations. Another specialized program is STAGS, see Almroth et al., 1980, which was developed especially for solving general, nonlinear shell problems. It was, therefore, more efficient than many of the Finite Element (FEM) programs which were originally written for solution of linear problems of elasticity and later on were provided with routines for handling the nonlinear behavior. With the fast development of computer systems it has become feasible to solve more and more complicated problems, and today several efficient FEM-systems for nonlinear buckling analysis are available. In spite of this the computer capacity still poses practical limits to the size of the problems which can be analyzed. During the last few years, the development has thus been directed towards establishing efficient routines for numerical analysis of large systems of nonlinear equations with special reference to the behavior of shell structures in the postbuckling state and at failure. An efficient method for integration which is being introduced in many systems for structural analysis, was developed by Riks, 1979. 1.2.2 Current state of the art Shell stability research and development activities are probably even more intense today than ever before. The attention has changed to new fields of application. Optimization with respect to weight and manufacturing cost has become a major design factor. A brief summary of the present situation is given below. 1.2.2.1. Development of the shell theory The fundamental shell theory, the history of which has been briefly reviewed, is no longer subject to further development, since the parameters which govern the buckling process are known and sufficiently well described by existing theories. The emphasis is, today, rather on the development of numerical methods for solving the general shell equations and establishing solution routines for large highly nonlinear problems. The development of FEM-programs for different types of structures, e.g. shells, also proceeded in parallel to the evolution of the shell theory. In many cases it has been possible to solve extremely difficult shell problems by use of FEM-systems based on nonlinear theory. Today, a relatively large number of such systems are available for analysis of stability problems of various kinds. Examples of such systems are given in a collocation published by Fredriksson-Mackerle, 1980. Many shell structures belong to special classes, such as shells of revolution subjected to axisymmetric or nonsymmetric loads. In such cases it may be advantageous to develop special purpose programs, which would be faster and less expensive to use. Examples of such programs have been mentioned above.

Introduction

5

1.2.2.2. Methods of design Despite the fact that powerful computer programs are available for the analysis of buckling of shell structures it is seldom advisable to use these in the design. The reason is that the buckling resistance of the shell is determined to a high degree by other parameters than the nominal geometry and the material properties. Initial imperfections and external disturbances in the loading of a magnitude and form which cannot normally be predicted at the design stage, play a very important role. The design calculations will, thus, under the present circumstances have to be based on methods where the influence of disturbances etc. has been determined by comparison with test results. On the other hand, a subsequent check of the resistance might be performed for the completed structure where the actual geometry can be measured and be properly considered in the analysis. Examples of comparisons between theoretical buckling loads according to the classical theory and experimentally determined values are shown in Fig.1.12. Furthermore, test results are presented in most sections of Chapter 3. The scatter of the test results is often rather large and for that reason it does not appear to be sensible to provide strict recommendations for the buckling analysis. The simple classical theories are, consequently, still used for design calculations together with reduction factors obtained from comparison with experimental results. The classical buckling theory has resulted in solutions for a number of elementary cases, compare Chapter 3, but in practice many shells are built for which analytical solutions do not exist. In such cases the critical load may be calculated by use of a computer program. Such calculations are becoming more frequent as the access to programs and computers is growing steadily. If the buckling analysis is based on linear theory, the computer calculations give results which are equivalent to the classical theory, and it is necessary to interpret them correctly in order to attain an adequate safety with regard to buckling. The progress in this field has not reached very far, however, and general design rules do not yet exist. It does not seem to be feasible to determine theoretically an adequate reduction factor. Guidance may, however, in some cases be obtained by calculating Koiter’s buckling parameter b according to 1.3.3.3. Design with regard to buckling is mostly performed according to the following scheme: – Compute the theoretical buckling stress σel according to the classical theory, using handbook formulas or a computer program. – Define an empirical reduction factor η which represents the reduction of the resistance considering small initial deformations, disturbances in the loading etc. – Define a reduction factor α which reflects the manufacturing process, tolerance level etc. – Calculate the buckling resistance of the elastic shell

Under certain conditions an additional safety factor 0.75 is recommended, e.g. when the test results reveal a large scatter or are too few to form a satisfactory basis for the design. In that case σu is given by

Shell stability handbook

6

If buckling takes place in the elasto-plastic region, i.e, σelr>σy/3, a variable factor ωs, which takes the degree of plasticity into account, is introduced σu=ωsσy where ωs is a function of the slenderness ratio λs according to the expression:

This method of representing the buckling resistance is often used for comparison with experiments. It can be utilized within the plastic region as well as in the elastic region. The function ωs takes on different values depending on whether the extra factor 0.75 is introduced or not. See Chapter 3, eqns (3.10) and (3.37). This scheme of analysis is applied today in many design codes and recommendations and it will also form the basis for design as presented in this handbook. 1.2.2.3. Research activities Primarily, the main contributions to shell stability were made in the aerospace technology, but new applications in other fields have also required efforts within branches such as building construction and offshore. These branches are using other design materials e.g. steel, concrete, glass fiber reinforced plastics (GlRP) the properties of which differ in several respects from those of materials normally used in aircraft and space vehicles. The yield limit of the material, for instance, has a much stronger impact on the resistance of these shells, which, due to the new applications, has lead to an increase in the amount of investigations on plastic buckling. Experimental research is carried out by a number of companies and scientific institutes. Typical applications being investigated are the following (note that the list is only meant to supply a number of examples and does not claim to be complete). Circular cylindrical shells—with or without stiffeners—are studied by several institutions: Plastic buckling of cylinders under axial load was studied at the University of Stuttgart, see Häfner, 1982, Bornscheuer, 1984. The influence on the buckling resistance of a longitudinal weld under lateral external pressure may be considerable as shown in a test series performed at the University of Essen, Stracke-Schmidt, 1984. Plastic buckling of cylindrical shells subjected to edge shear loads has been studied at the University of Liverpool, Galletly-Blachut, 1985a. Buckling of stiffened cylindrical shells (ring and/or stringer stiffened) has attracted an increased interest lately in connection with applications in the offshore industry. Extensive testing has been performed at e.g.: Imperial College, London, Dowling-Harding 1982, and Ronalds-Dowling, 1986, University College, London, Walker et al., 1982, Det Norske Veritas, Valsgård-Steen, 1980,

Introduction

7

Chicago Bridge & Iron Co., Chicago, Miller, 1982a, Technion, Israel, Singer, 1983, Los Alamos Nuclear Research Lab, Baker et al., 1983, University of Stuttgart, Diack, 1983. The influence of disturbing factors on the carrying capacity of shell structures has attracted increased attention during the last few years, compare Knoedel and Schulz, 1985, Teng-Rotter, 1990, Rotter et al, 1991 and Eggwertz-Samuelson, 1991. Finally, the problem of designing by use of computer analysis is being discussed. Suggestions on analysis procedures have been presented by Samuelson, 1987 and Schmidt, 1991. At an international conference, “Buckling of shells in the air, on land and in the sea”, Lyon, France, Sept 1991, Ed. J.F.Jullien, 1991, a large number of different aspects of shell stability were discussed. The proceedings form a valuable state of the art review of the current research activities. 1.2.2.4. Development of codes and recommendations The growing use of shell structures within industrial areas such as steel building, ship building, offshore and nuclear power production has resulted in a need for special codes and recommendations for shell design. At the present time, several working groups are active in the development of design recommendations. The following efforts can be mentioned: – Det Norske Veritas, 1984, has, for a long time, published rules for the design of ships with detailed recommendations for the structural analysis. At the start of the offshore operations rules were also developed for offshore structures. The DNV code includes rules for the design of different types of shell elements. – DASt, 1980, (Deutscher Ausschuss für Stahlbau, Richtlinie 013) treats a large number of shell elements and load cases common in building and construction applications. Rather comprehensive rules are given e.g. for cylindrical containment vessels with varying sheet thickness and subjected to wind loading. The rules can be applied to tanks and silos. Parts of these recommendations have been utilized in this handbook. – DIN 18800, Teil 4, Stahlbauten, Stabilitätsfälle, Beulen von Schalen (in preparation, available in a preliminary edition 1990). – API RP2A , 1982, American Petroleum Institute, Washington D.C. – API-SPEC, 1977, Specification for Fabricated Structural Steel Pipe. – BS 5500, Specification for Unfired Fusion Welded pressure Vessels. – ECCS, 1988, (European Convention for Constructional Steelwork) develops recommendations for design of shells with respect to buckling. The Technical Working Group TWG 8.4, is working continuously to update the contents and introduce new shell elements and load cases. The cooperation with the group and exchange of experience and information etc. has been of direct use in the work on the present handbook. – ASME Code Case N-284, 1980, has approximately the same range of contents as DASt Ri 013. Contacts with the authors of the ASME Code have given valuable input for the handbook. The ASME code is the only code, so far, which gives indications on how the results of computer programs for buckling analysis should be interpreted.

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Many nations have developed their own rules and work is continuing in several organizations. The publications listed above have been studied closely during the preparation of this handbook since they are frequently referred to internationally in the design of shell structures. Although great efforts are being made to improve and extend the present design codes, solutions are still lacking for important cases that occur frequently in practical applications, such as the effect of local or concentrated loads on the shell, influence of cutouts and stiffeners, temperature gradients and variations, deep buckles (dents), or how to perform an analysis of the buckling resistance using a general FEM-program. The discussion of these questions will, hopefully, be of some value in similar cases for which no complete and final answer can be given. 1.3 Theory of stability In the science of applied mechanics three different forms of equilibrium are considered, stable, indifferent and unstable. These conditions can be illustrated simply by a ball on different beddings, see Fig.1.1, a)–c). The stable equilibrium, on a concave spherical surface, implies that a small disturbance (displacement) will bring the ball to return to its original position of equilibrium. In the unstable case, on a convex surface, a small disturbance leads to further increase of the distance from the position of equilibrium. Alternatively it can be stated that in the stable case the potential energy has a minimum in the position of equlibrium. A disturbance gives an increase of the potential energy. In the case of indifferent equilibrium, on a plane surface, the change in potential energy is zero, while it is negative for unstable equilibrium. Variations of the equilibrium conditions may be conceived according to Fig. 1.1, d)–f). The ball may have a stable position d) as long as the disturbance is very small. If the disturbance exceeds a certain limit the equilibrium will become unstable. Case g) illustrates a stable equilibrium state where the stability is extremely sensitive to small external disturbances. In many cases of shell buckling the state of equilibrium will adopt this character where the disturbances are produced by initial deformations, load eccentricities etc.

Fig.1.1. Illustration of the definiton of equilibrium conditions in mechanics. The conditions for stability are of great importance for loaded members and an analysis of the limit of stability can in principle be performed with the same methods as used in

Introduction

9

mechanics. A structure loaded by a system of external forces is remaining in a stable equilibrium if it tries to return to its original state after the introduction of a small disturbance. In the treatment of structural stability a number of conditions or limitations are usually introduced: – the material is linearly elastic, – the structural element has an ideal geometrical shape (straight bar, plane plate, shell with given radii of curvature), – constitutive relationships are based on the assumption of small displacements. A theory based on these assumptions will, subsequently, be referred to as the classical buckling theory. Structural components are designed with regard to several different failure mechanisms. For tension loading the yield, fatigue or ultimate limit of the material is normally critical. Compression may cause buckling necessitating introduction of special methods of analysis. The buckling behavior varies between different types of structural elements, and in many cases the theoretical buckling load turns out to be a very poor estimate of the actual resistance. Plane plates may have a resistance which is considerably higher than the value given by the classical theory—the postbuckling state is stable. For some types of shells the resistance may be much lower than the theoretical buckling load and in most cases, the postbuckling behavior is highly unstable. 1.3.1 Classical buckling theory Brief presentation for bars, plates, shells. A straight bar (column) which is loaded in compression in its longitudinal direction, Fig.1.2, may be subjected to buckling in the lateral direction. The load at which buckling occurs can be estimated for a slender bar by the following expression, where n=1 or, alternatively, lf=l, corresponds to Euler’s buckling case No 2. (1.1) When loaded by an axial force P
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Fig 1.2. Buckling of a straight bar. A plane rectangular plate buckles out of its plane in the manner shown in Fig.1.3. The buckling stress of a thin plate loaded according to Fig 1.3, a) can be obtained as (1.3)

Fig.1.3. Buckling of a plane plate with simply supported edges subjected to in-plane compression and shear, respectively. The derivation of eqn (1.3) is based on the same principles as the theory used for straight bars. The buckling configuration is for this case assumed to be: (1.4)

Introduction

11

Fig.1.4. Buckling coefficient kσ for plates with various aspect ratios and edge restraints. The buckling coefficient kσ depends on the aspect ratio a/b and the edge conditions which are indicated in Fig.1.4. For practical applications it is often sufficient to consider the case with simply supported (hinged) edges and a long plate, where buckling takes place with at least three complete waves in the longitudinal direction, the halfwave length being equal to the width b of the plate. In this case kσ=4 (1.5) provides a good approximation. For other edge conditions the buckling coefficient can be obtained from Fig.1.4. Shells An axially loaded cylindrical shell will, according to the classical theory, buckle at the stress level (1.6) The solution presumes that the axial stress σx is constant throughout the the shell before buckling and that the buckling mode can be described by (1.7)

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Fig.1.5. Buckling of cylindrical shell subjected to axial compression.

The general solution σcr=f(m, n) is given in eqn (A.29) of Appendix A, which represents all possible buckling conditions. If specific wave numbers n (number of buckling waves in circumferential direction) are chosen and σcr is calculated as a function of the wavelength parameter in the longitudinal direction, curves according to Fig. 1.5 are obtained. Each separate curve comprises of one or two minima representing the actual buckling condition. The figure shows that a large number of combinations of the wave numbers m and n will yield the same critical stress, equal, in fact, to the value given by eqn (1.6). The dashed line in the left part of the diagram refers to column buckling of a narrow plate element and the dashed curve to the right represents column buckling of the entire cylinder. Other loading conditions and other types of shells can be treated in an analogous way. The expressions derived for the critical load are in most cases considerably more complicated than eqn (1.6), see Chapter 3. 1.3.2 Characteristic buckling properties. Comparison with test results The buckling properties of various structural elements differ very much from each other, and it is of great importance to be able to estimate the actual ultimate resistance. The objective of the following discussion is to indicate the background of the design rules

Introduction

13

which are presented in Chapter 3. The classical theory, briefly described above, is used as the basis of the design methodology. The resistance of straight bars subjected to buckling may be described by Fig.1.6,a). The load remains nearly constant in the postbuckling range provided buckling occurs within the elastic region. The small increase of the load at very large lateral deflections is hardly of any practical importance since the material will soon receive plastic deformations decreasing the bending stiffness of the bar. This results in a decrease of the resistance with growing deflection, see Fig.1.6,b). No additional resistance above the classical buckling load can be expected. On the contrary, it is common that the classical buckling resistance is not reached due to the fact that actual bars are initially deformed and possibly also contain residual stresses which may reduce the bending stiffness and, thus, the buckling stress. The classical theory assumes that the bar is straight and centrally loaded. If that is not the case, the lateral deflection grows gradually with an increasing load, see Fig.1.6,b). The maximum of the deflection can be written in the following form: (1.8) The actual resistance of the column compared to the theory according to eqn (1.1) has been determined in a large number of tests. Some typical results are shown in Fig.1.7, from which it seems evident that the classical theory gives adequate results for slender columns within the so called elastic region.

Fig.1.6. Deflections vs load of a) straight, and b) initially curved bar subjected to buckling. Very short columns have a resistance equal to the yield load. In columns of medium length, interaction between buckling and yield results in a reduction of the carrying capacity. The resistance can be determined empirically, by testing, or theoretically, where the analysis must include the influence of residual stresses and initial curvature.

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Plane, thin plates often exhibit a considerable resistance also in the postbuckling range as shown in Fig.1.8. The reason is that the load is redistributed towards the longitudinal edges when the plate buckles. The ultimate buckling resistance is reached when the edge stiffeners are no longer stable or the yield limit of the material is passed. In some applications it is considered acceptable that buckling occurs in normal use of the structural member, see e.g. Fig.1.9, showing buckled panels in the fuselage of a Boeing B52 aircraft parked on the ground.

Fig. 1.7. Comparison between test results and theory for axially compressed columns.

Fig.1.8. Relationship between mean stress σx and deflection of an initially curved plate. See VestergrenKnutsson, 1978.

Introduction

15

Fig.1.9. Shear buckles in fuselage shell panels of a Boeing B52 standing on the ground (US Air Force Museum, photo L.Å.Samuelson). The buckling limit is, however, often of interest, especially if the structure is regularly subjected to loads within the postbuckling region and fatigue damage may be expected. Rather comprehensive data from testing of plates under various loads are available in the literature, see e.g. Dubas-Gehri, 1986. Fig.1.10 shows results from tests on steel plates. These have an ultimate resistance which, for slender plates, is higher than the buckling load but for plates with lower slenderness ratios is similar to that of straight columns. The buckling load in the plastic region is estimated by fitting a lower bound curve to the test results. The resistance in the postcritical state is usually determined by use of the effective width model in which it is assumed that the compressive stress in the interior part of the plate does not increase above the buckling limit and that the load is gradually transferred to the edges and their stiffening flanges or profiles, compare Timoshenko-Gere, 1961. As indicated above, most national standards and codes provide design rules for bars and plates subjected to buckling. It may be sufficient here to refer to Dubas and Gehri, 1986.

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Fig.1.10. Comparison between tests and theory for a plane plate loaded by longitudinal compression in its plane. Bornscheuer, 1984. Many types of shells, e. g. spherical caps under external pressure, show a strongly reduced load carrying capacity in the buckled state. Furthermore, shells are often very sensitive to initial imperfections. The buckling stress formula, eqn (1.6) overestimates the resistance, as shown in Fig.1.11. The explanation for this sensitivity may simply be seen as a consequence of the condition that the shell can theoretically reach its high buckling resistance in its ideal shape where the loads are carried by membrane forces. If initial imperfections, or buckles, are present a part of the load must be carried by plate action (bending across the shell wall), which will result in a drastic reduction of the stiffness and, also the resistance of the shell.

Fig 1.11. Buckling behavior of an axially loaded cylindrical shell.

Introduction

17

Experimental results for various types of shells are presented in Fig 1.12. In some cases the ultimate buckling resistance is only a small fraction of the buckling load determined by the classical buckling theory. In some cases the carrying capacity corresponds to a reduction factor which may be as low as 0.1. These stability characteristics of shell structures imply that design against buckling is a delicate task. In several cases, insufficient dimensioning has been the direct cause of collapse.

Fig 1.12. Comparison between theory and test results for a. cylinder under axial load, b. cylinder subjected to torque, BrushAlmroth, 1975 c. stiffened cylinder under axial load, Singer, 1982 d. spherical shell under external pressure, Kollár, 1982.

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1.3.3 Notes on the general buckling theory for shells In recent years numerous articles and books have been written on shell buckling, see the list of references. It is not the intention here to give a comprehensive description of the theory of stability for shells. It is important, however, to understand some of the characteristics which cause the shell stability problem to become so complex and difficult to analyze. A brief summary of the fundamentals of the theory of stability is given, based on “Buckling of Bars, Plates and Shells”, by Brush and Almroth, 1975. 1.3.3.1 Symbols and definitions A cylindrical shell is geometrically defined by its length, l, radius, r, and thickness, t. For description of the shell properties, derivation of the system of equations etc., a local coordinate system oriented in the middle surface of the shell is usually introduced, Fig 1.13. The forces and moments which act on the shell element are denoted by n and m and represent force or moment per unit length. If the shell is a homogeneous sheet of thickness t it follows that

The use of n and m as variables facilitates the derivation of equations, especially if thickness, degree of stiffening, or other factors vary along the coordinate axes. 1.3.3.2 Fundamental equations On the following pages, some of the relationships are discussed which describe the buckling behavior of shell structures. Since the general theory is very complicated it is not practicable to give a complete derivation of the fundamental equations and related methods of solution, and it was, consequently, decided to describe the procedure by means of experiences from experimental investigations and computer analyses. The theory of stability of a cylindrical shell exposed to various types of loads is, moreover, treated in Appendix A, where the deduction of the equations follows the monograph by Brush-Almroth, 1975, which is recommended for a more complete study of shell buckling problems. As mentioned in the discussion in section 1.3.2, the linear theory often provides very poor estimates of the resistance of a shell subjected to buckling. It is, thus, important to analyze in more detail the parameters which are governing the process of buckling.

Introduction

19

Fig 1.13. Definition of coordinate system and force and displacement components for a circular cylindrical shell. A. Cylindrical shell under axial loading, axisymmetric deformation Assume that the cylinder of Fig 1.13 is subjected to an axial load and that the deformation occurs with complete axisymmetry during the loading. A small element of the shell is exposed to the forces shown in Fig 1.14 and the shell is deformed according to Fig 1.15, due to the Poisson effect.

Fig 1.14. Shell element under axisymmetric load and deformation. If the circular boundaries of the shell were free to expand in the radial direction, the diameter would just increase simultaneously with the compression of the cylinder. Fig 1.15 indicates that the shell wall remains straight, and the loading condition may be

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compared to the case of a centrally loaded straight bar. In practice, the end boundaries of the shell are always attached to a stiffener, or the foundation, which reduces the expansion. With some exaggeration the shell takes the shape of a barrel.

Fig. 1.15. Cylindrical shell under axial loading showing the influence of lateral contraction and boundary conditions. As shown in Appendix A the radial displacement function w can be shown to be a damped harmonic wave which in the vicinity of a boundary is written: (1.9) with:

The damping coefficient α thus depends on the load level applied, . The result of the calculation can be illustrated as in Fig.1.16, which shows the radial displacement w and the circumferential force nφ at various load levels.

Introduction

21

Fig 1.16. Stresses and deformations in a circular cylindrical shell under axial load presuming symmetrical deformation. It appears from Fig 1.16 that the bending deflections are, at low load levels, limited to a narrow region close to the boundary. At higher loads this area is extended and the bending mode approaches a pure sine wave. Simultaneously the amplitude is growing and approaches large values for a finite load. It was shown in Appendix A that when the damping term α approaches zero, the critical load according to eqn (1.6) is obtained. At the same time it can be noted that the disturbed zone close to the boundary is given by the wavelength lw of the sine function of eqn (1.9) and may be written (1.10) These formulas are often found in shell problems of various kinds. In particular, eqn (1.6) is considered to be the theoretical value of the buckling stress of a cylindrical shell. But how do these results compare with tests? It has already been pointed out that eqn (1.6) agrees badly with the test results, at least for thin shells. Thicker shells, see Fig.1.17,a), will buckle in an axisymmetric mode (elephant’s foot) and can agree well with theory. If the stress in the circumtferential direction σφ is considered in the analysis, the accuracy will usually be reasonable, since the model in this case reflects the actual behavior in a satisfactory manner. The influence of plastic deformations, however, is not taken into account, implying that the theory gives an increasing overestimate of the buckling resistance with increasing thickness. A thin shell usually buckles into a nonsymmetric pattern as shown in Fig. 1.17b). Since this was not anticipated in the model described above, it may be assumed that a more accurate model allowing nonsymmetrical deformations would give a better estimate of the actual behavior.

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Fig. 1.17. Buckling of a) thick and b) thin circular cylindrical shells, respectively, subjected to axial compression, Samuelson, 1969. B. Axially loaded cylindrical shell, nonsymmetric buckling—Classical theory In the analysis of the deformation of the cylindrical shell referred to above, no assumptions were made about possible, nonsymmetric buckling. The equations described in Appendix A describe the equilibrium conditions of the shell as a function of the load, leading to collapse at the stress level σxel obtained from eqn (1.6). In analogy with the Euler theory for straight columns subjected to compressive loading it is also possible to analyze the stability of a circular cylindrical shell. The simplest case is represented by the classical theory, where it is assumed that the disturbances caused by the boundaries are negligible. The stress distribution in the unbuckled state is thus homogeneous and given by the axial stress σx. Fig.1.18 shows the state of stresses and the deformation in the shell as well as the variation of the radial displacement with the load level. The axially loaded shell receives a radial displacement wo. A small disturbance ∆w is introduced which gives the total displacement w=wo+∆w (1.11)

Introduction

23

Fig.1.18. State of stresses and deformation of a cylindrical shell with classical boundary conditions. This state represents a new equilibrium condition in the shell. The new state is assumed to be nonsymmetric, as shown in Appendix A, and is described by a rather complicated system of differential equations. The state before buckling wo is, however, known and if it is introduced into eqn (A.21) of Appendix A, the terms containing wo will drop out resulting in a homogeneous system in the incremental variable ∆w. L(∆w)=0 (1.12) This system (eqn (A.24) of Appendix A) is solved by eigenvalue analysis. Assume that the buckling deformation may be expressed by: (1.13) Introduction in eqn (1.12) yields a relationship of the following form: (1.14) where the existence of a nonzero solution independent of x and φ requires that f=0. For an axially loaded cylindrical shell this will, according to Appendix A, lead to a buckling stress condition which may be written in the form: (1.15)

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The combination of wave numbers m and n which gives the minimum value of the buckling stress σcr, represents the natural buckling mode of the shell. The corresponding critical stress σxcr is here denoted by σel. Eqn (1.15) is demonstrated graphically in Fig.1.5 for a shell with a given stiffness ratio . Since only the minimum value of the stress is of interest from a design point of view, it is appropriate to draw the result according to Fig.1.19. It is now reasonable to ask if the improved analysis yields a better estimate of the carrying capacity of the cylindrical shell than the calculation according to the preceeding subsection. From Fig.1.19 it is obvious that within a wide range, the buckling stress is independent of the buckling mode, i.e. several different combinations of m and n yield the same buckling load. This buckling load can be shown to have the value: (1.16) i.e. the same value as for the axisymmetric case, eqn (1.6). The inclusion of nonsymmetric buckling deformations in the analysis, consequently, did not lead to an improvement of the theory. Note that linear buckling analysis performed by use of a computer program suffers from the same deficiency, and the result must be judged considering this fact.

Fig.1.19. Buckling stress of a cylindrical shell as a function of geometry and buckling wave numbers m and n for various values of See Flügge, 1934, 1973.

.

Introduction

25

Nonlinear prebuckling state The analysis of the axisymmetric state before buckling of an actual cylindrical shell may be represented by the computer results of Fig.1.16, which are compared with the classical case in Fig.1.18. In particular, it should be observed that a zone with compressive stresses in the circumferential direction is formed in the vicinity of the boundary. Intuitively this zone, with a two way compression state of stress may be anticipated to influence the buckling resistance, Fig.1.20.

Fig.1.20. State of stress close to the boundary of a cylindrical shell under axial compression. Analytical solutions of this problem have been presented in the literature and it has been shown that the result is equal to the classical value multiplied by a constant factor of 0.83. The buckling stress would thus be written: (1.17) This formula provides a somewhat better estimate of the buckling stress than the classical value. Still, it does not represent the influence on the divergence of the experimental results from the classical buckling limit as a function of the slenderness ratio . The remaining difference depends on deviations from the ideal shape and will be further discussed in 1.3.3.3. C. Cylindrical shell under external pressure The case of a cylindrical shell subjected to external radial pressure can be treated in the same way as the axial loading. A certain difference may be noticed, however, for the case with axisymmetric pressure, where a symmetric collapse mode does not exist. Buckling, in this case, always occurs in a nonsymmetrical pattern, and the determination of a critical load and buckling mode can be carried out in exactly the same way as described in the preceding section. The shape of the natural buckling deformation of a cylindrical shell under external pressure is shown for different length to radius ratios in Fig. 1.21. If

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the cylinder is long enough the end reinforcement (or bulkheads etc.) will not influence the behavior and buckling takes place by ovalization similar to the buckling mode of a ring loaded by a uniform radial line load. If the cylinder is short, the length will govern the process and a number of waves will be formed in the circumferential direction. A characteristic feature is that the length of a half wave in the axial direction is equal to the length of the shell. This implies that m=1 is a realistic assumption for the buckling deformation to be introduced in eqn (1.13). The numerical model will, therefore, adopt a mode as shown in Fig.1.21.

Fig.1.21. Buckling modes of cylindrical shells of various lengths and subjected to external pressure. If the derivations are carried out in the same way as for the axial compression case above, the following expression for possible buckling stresses is found: (1.18)

That value of the buckling wave number n which yields the minimum of σφcr represents the natural buckling mode and the corresponding external pressure constitutes the buckling pressure. Graphically, the result can be presented as in Fig.1.22 which shows the theoretical critical stress caused by an external pressure. From the diagram the natural buckling wave number n may also be determined. Diagrams like that shown in Fig.1.22 provide valuable information but are not particularly suitable for design purposes. Several different attempts have, therefore, been made to reduce the results on a form which gives the critical pressure directly as a

Introduction

27

function of the geometrical parameters. The recommendations in Chapter 3 on elementary cases are based on such a model (see eqns (3.32) and (3.34)).

Fig.1.22. Cylindrical shell under external pressure, critical pressure vs for various values of the radius to thickness ratio . See also BrushAlmroth, 1975. D. Nonlinear shell theory The equations used in buckling analysis as presented above, can be characterized as linear since no products of the entering variables are retained. An exception is the theory of axisymmetric load and deformation of subsection 1.3.3.2, where the influence of the axial load on the curvature of the shell was included. This resulted in a nonlinear equation yielding the classical buckling load of the shell. Many of the problems ocurring in practice are, however, much more complicated. Some of these, which have to be considered in a rigorous analysis of the buckling behavior of a shell, will be discussed below. In most design problems it is assumed that the state of deformations and stresses may be described with sufficient accuracy by linear relationships. In Fig.1.23 a plane plate is represented by an element subjected to a normal

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force. If the normal force is caused by an external load of fixed direction along the x-axis, the displacement in this direction will be approximately proportional to the load and a linear solution is satisfactory. If the normal force was governed by a given displacement the result may be misleading for the following reason: The rotation of a beam element according to Fig.1.23 causes displacements in the x-direction which can be determined directly using the thesis of Pythagoras. The strain in the x-direction is described by the nonlinear relationship: (1.19) and it is evident that the solution of the final system of equations will be more complicated. The corresponding expressions for shells will, broadly speaking, be equivalent to eqn (1.19), but in addition the strain components εφ and γxφ have to be considered. The complete system of equations for a linearly elastic cylindrical shell is presented in Fig.1.24, which has been included only in order to demonstrate the complexity of the general shell theory when all significant parameters are retained.

Fig.1.23. Relationships between strains and displacement components of a shell element subjected to moderately large rotations β=w,x. Since the gradient w,x may be assumed to be small it might be rasonable to neglect the squared term of eqn (1.19). In practice, however, it is found in the analysis of shell problems that the nonlinear term will become important even at deflections of a magnitude of half the wall thickness.The problem is discussed further in 1.3.3.3 below. The solution of the general shell equations, the system given in Fig.1.24 forms a special case, can be carried out with various special purpose programs based on different methods of solution. Nonlinear problems are difficult to solve and require experience from the user of the program. In many FEM programs, nonlinear theory has been introduced and very complicated shell problems can be analyzed. Recommendations concerning selection of programs and methods of solution are given in Chapter 6.

Introduction

29

Fig.1.24. Complete system of equations of a shell with regard to second-order effects. From BrushAlmroth, 1975. 1.3.3.3 Sensitivity to initial imperfections A. General requirements for straightness The buckling resistance of bars, plates and shells subjected to compressive loads is to a large extent dependent on the precision of the geometry of the structure. Deviations from

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the nominal shape may, especially for certain types of shells, have a drastic influence on the buckling load. In the design of a column against buckling the normal requirement is that w0≤l/600, corresponding to 5 mm for a 3 m column . Such a column may have a cross-section depth of 100 mm and the deviation is thus only a few per cent of the depth. For shell structures where the wall thickness often is of an order of magnitude of 10 mm, a corresponding tolerance level for local shape deviations is not possible to guarantee. According to section 3.2 the tolerance level for a cylindrical shell is wr≤0.01lr. The reference length lr may in practical applications be of the magnitude of 1000 mm. This means that initial deflections equal to the wall thickness must be accepted. The capability of shells to carry compressive loads is often attributed to their ability to form a homogeneous state of stress. The cylindrical shell in Fig.1.20 showed circumferential bands with compressive and tensile stresses respectively due to boundary clamping and lateral contraction. Assume that the cylindrical shell is substituted by a number of longitudinal strips which are supported by radial spring members. The stiffness of the springs is proportional to the t/r ratio. With perfect axial symmetry, the . If a slight out of length of a half wave of the buckling mode is equal to roundness is present, the compressed strip cannot carry the full load. In the model this is simulated by buckling of the spring members which causes a reduction of the resistance. The following example indicates the great sensitivity to an out of roundness of the shell. For the cylindrical shell in Fig.1.25 the classical buckling stress is:

If such a large out of roundness is assumed that the compressed springs no longer show any resistance, the following extreme condition occurs:

i.e. only 20 per cent of the buckling stress of the perfect cylindrical shell. In reality the buckling stress will mostly fall between these two extreme values. B. Theory of sensitivity to initial deformations A general theory of imperfection sensitivity of shells was developed by Koiter, 1945 (Translations 1967,1970). His starting idea was to study the change of the stress level in the shell at the moment of incipient buckling and, thus, try to predict the severity of the buckling process.

Introduction

31

Fig.1.25. Shell with initial deformations and a simplified beam model to demonstrate their effect on the buckling resistance. Assume that the shell is loaded in such a way that a homogeneous state of stress is developed according to Fig.1.26, and that the buckling load can be determined by classical theory. The theory gives the buckling mode: (1.20)

Fig.1.26. Description of the buckling behavior in the vicinity of the bifurcation point of a cylindrical shell.

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where the amplitude ∆w is indeterminate. Koiter assumed the following expression for the stress level close to the buckling load: (1.21) By studying the variation of the second variation of the elastic potential the constants a and b can be determined and it is thus possible to judge how the shell will behave at the moment of buckling. The various possibilities are shown in Fig.1.27. In most cases a=0, which is understood from the fact that the buckling load usually does not depend on the direction of the bending deflection. Curve a) presents an exception corresponding to asymmetric buckling (prevented in one direction). Curve b) corresponds to a cylindrical shell under axial loading where b<0, indicating that the equilibrium is strongly unstable at the point of bifurcation.

Fig.1.27. Buckling parameters a and b according to Koiter. Curve c) shows a case where b>0, corresponding e.g. to a plane plate subjected to forces in its plane. After buckling the load carrying capacity of the plate increases and the equilibrium is stable. The curve of a straight column is a special case of c) with the value of b close to zero.

2 DESIGN PHILOSOPHY This volume is a handbook and not a code nor a standard. The loads and safety factors which have to be used in the design are stated in national or international codes or standards. The following sections, therefore, are of an informative character but are also intended to provide incentives for harmonization of various kinds of codes. It should be emphasized that this chapter describes models for a rational treatment of safety problems and that the results obtained from these models have been calibrated against experience available from practical applications and tests. Since shell structures are often sensitive to initial imperfections and the scatter in buckling tests is usually fairly high, the safety aspect requires somewhat special attention. The strategy for design against buckling is described in 2.5.3. 2.1 Design procedure The procedure, starting from general information concerning the use, location etc. of the structure to be built, and leading to complete design documents sufficient for the manufacture and erection/installation, is referred to as the design procedure. The course of the design procedure naturally depends on the type of structure, the purchaser, future proprietor etc. In most cases the various phases within the procedure may be described in the following general terms. – The purchaser initiates the project and provides the conditions and general requirements. – The design engineer formulates the conditions and requirements in technical terms guided by the regulations given by authorities. – The designer selects the structural system and materials in cooperation with the purchaser and based on a preliminary design analysis. – The designer performs the design analysis which includes dimensioning by structural analysis, preparation of drawings, specifications and descriptions. – The design documents are approved by the purchaser, authorities and, possibly, by a responsible designer. – The manufacture and erection/installation can be commenced supervised by the purchaser, the authority and the designer. It should be pointed out that all phases of the procedure are of importance in order to arrive at an adequate design implying good quality and acceptable economy. There may be a tendency to underestimate the responsibilities of the designer in the early stage of the procedure. To summarize, the objectives of the design procedure are:

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– to produce design documents (drawings, descriptions,specifications etc.) suitable as a basis for fabrication of the structure, – to verify that the documents are in agreement with the purchaser’s requirements according to the given design conditions and valid regulations, and – to ensure, as far as possible, that the documents specify a structure satisfactory from an economical point of view. 2.2 Requirements on the load carrying structure 2.2.1 Specification of requirements Here, and in the following sections, requirements denote expressions of expectations defined by the purchaser, the future proprietor, utilizers, authorities, etc. concerning the function of the structure. The requirements may to some extent be varied with respect to the balance between quality level and cost. The requirements on a load carrying structure may be specified as follows: – requirements on safety against failure, – requirements on serviceability in normal use, – requirements on durability. 2.2.2 Requirements on safety against failure The concept of failure may imply anything from destruction of a structural element to collapse of the entire structural system. The cause of a failure may be of various kinds and can be classified in three categories. a) Unfavorable combinations of factors affecting the resistance. An unfavorable combination of critical parameters has occurred. These parameters may be interpreted as loads, strength of the material, dimensions, imperfections and minor damages. They possess values which may be extreme, but do not deviate significantly from normal conditions. b) Accidental loads A load not considered in the design has appeared as a single occurrence with such a magnitude that the consequence was failure of the structure. The load may either be of a character entirely different from those considered in the design, or it may be of the same character but of a magnitude not foreseen. c) Gross errors A gross error has been committed in the design work, material production, or construction. A gross error implies that the structure has received some material or geometrical property of a character entirely different from what was intended. The requirement on safety against failure means that the structure shall be designed and fabricated in such a way that the probability of failure becomes sufficiently low. The concept “sufficiently low” also implies that the probability has to be lower the more serious the consequences of a failure would be.

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The measures to be taken to ensure a sufficiently low probability of failure should in principle be adapted to all categories mentioned above. When the cause of afailure is attributed to the first category, the risk of failure can be adequately reduced at the design level by choosing sufficiently large factors of safety. These can also be determined in relation to the consequences of a possible failure. The measures which can be taken against failure occurring because of an essential load not considered in the design, are more difficult to quantify. Some loads of that kind may be known to a certain degree through experience from earlier incidents. This is, for instance, the case with loads arising as a consequence of a collision or an explosion. Other kinds of loads may be possible but so far unknown. A reasonable step may be to design a structure with respect to a few known loads of the kinds mentioned above and further assume that it will also be able to resist other types of loads of a similar category. As a complement, or an alternative, it is possible to select a structure of such a type and perform a detailed design in such a way that the carrying system becomes highly insensitive to local damage, which may arise from loads of the kinds mentioned. Accidental loads may, for example, be caused by impact of various kinds, flood and earthquake.The character of these loads implies that the probability of their occurrence is small. Therefore, they need to be considered only for those types of structures where the consequence of a possible failure may be expected to be very serious. Structures of a vital importance should thus, if possible, be designed according to damage tolerance criteria. Gross errors can, for example, be caused by the designer in miscalculating a wall thickness by a factor of 2, or, welding of a joint or a similar operation may be simply left out in the manufacture of a steel structure. Such errors can not be compensated for by choosing a larger safety factor in the design analysis. Measures to be taken to decrease the frequency of gross errors are: – improved training and information, – improved organization at the building site, – more effective supervision. To summarize it may be stated that the measures which can be taken in order to keep the probability of failure at a low level do not only apply to the choice of safety factors but include,also, training, information, organization and supervision. 2.2.3 Requirements on the serviceability of structures in normal use If a load carrying structural member is, in normal use, subjected to damage or causes damage to other members and, if the damage is unacceptable, the function or serviceability of the structural member can be considered to be unsatisfactory . The damage may be permanent or occasional. The word damage is used here in a wider sense and can be considered to constitute the cause of, for instance, some kind of inconvenience. Examples of permanent damage may be open cracks in the structural member, cracks in other building components, e.g. partition walls, and disturbing permanent deflections of beams. If such damage has occurred and involves inconvenience, it will continue to bring the same or about the same inconvenience until repaired. In this case the requirements given and the measures taken to avoid the inconveniences should be aimed

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at reducing the risk of generation of the damage. In principle, the problem is equivalent to that concerning safety against failure. Even if no well-defined limit exists between these cases, the risk which can be accepted for a minor damage to occur to the structure in normal use, is normally higher, however, than the acceptable risk of failure. This implies that it is, in general, only necessary to consider causes of damage corresponding to category a) of 2.2.2. Examples of occasional damage are cracks in prestressed concrete structures, which are occasionally open, occasional large deflections and vibrations of beams. The inconvenience of such damage will only appear during those periods when the loads (or other actions) occur which cause the damage. The requirements and measures to reduce the inconveniences should in this case be concentrated to the duration of the damage. Vibrations of a certain intensity may be acceptable from a comfort point of view if they appear infrequently and only during short periods of time. On the other hand vibrations of the same intensity may be entirely unacceptable if they act during longer periods. The requirements on the serviceability of a structure in normal use apply, in most cases, to deformations including oscillations and vibrations (considered as time dependent deformations). The inconveniences resulting from large deformations can be that they: – cause damage to other building components, – may convey a feeling of discomfort to people in the building, – can disturb and impair the function of machines, instruments and similar objects supported by the structure, – may be disturbing from an esthetical point of view. Further cases of damage or poor function in normal use may refer to – abrasion, – leakage, e.g. in liquid tanks, – surface finish, e.g. roughness or discoloration etc., It is not possible to express generally valid requirements concerning the function of a structure in normal use by numerical values. The requirements which should be formulated are too much dependent on the situation to which the requirement applies. Usually, the future proprietor/utilizer may establish the requirements after consultation with the design engineer. Moreover, the requirements must be expressed with due regard to the situation. A requirement concerning limitations of the deformations can thus be formulated in one of the following ways: – limitation of absolute values of displacements, – limitation of the mutual displacements between the nodes of e.g. a frame system, – limitation of the deflection of a structural component (e.g. a beam) in proportion to the span, – limitation of the angular deformations of a structural component. Specific recommendations are given in the various national codes concerning limitation of angular deformations in order to avoid damage in adjacent building components. Furthermore recommendations are provided concerning the bending stiffness of floor

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beams required to guarantee that deflections do not cause discomfort for people walking on the floor. 2.2.4 Limit states The requirements on the load carrying function of a structure apply to both safety against failure and to serviceability in normal use. These two requirements are, at least in some cases, quite different in nature and should thus be separated in the formulation of the requirements. This can be achieved by performing the design analysis at two limit states with regard to the function of the structure: – ultimate limit state, which is a state where the structure is at the limit of failure, – serviceability limit state, which is a state where the structure is at the limit of not satisfying the requirements for normal use. The implication of the limit states are illustrated in Fig.2.1, which shows the deflection versus load for a simply supported beam. Serviceability limit state and ultimate limit state are indicated by their upper limits.

Fig.2.1. Illustration of the serviceability and ultimate limit states of a structure. The limit states are thus conceivable states of the structure. The requirements concerning safety against failure are, in principle, formulated such that the probability that any of the possible ultimate limit states shall be exceeded is satisfactorily low. The requirements with regard to serviceability in normal use are established in a corresponding way, or such that the time during which exceedances occur, will be satisfactorily short.

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2.2.5 Safety classes When considering the measures to be taken to obtain sufficient safety against exceeding an ultimate limit state, the consequences of a possible failure must be taken into account. This is often done by assigning structures and structural components to one of three safety classes according to Table 2.1. See applicable national code or standard.

Table 2.1. Safety classes Consequence of failure

Safety class

Comment

Insignificant personal injury

1

Less serious (low safety class)

Some personal injury

2

Serious (normal safety class)

Severe personal injury

3

Very serious (high safety class)

Examples of application of the various safety levels will be given in Chapter 8. The requirements on safety against failure are, finally, dependent on the safety class to which the structure or structural component considered is referred. 2.2.6 Economic considerations on the formulation of requirements Some of the requirements which apply to a structure—in particular those concerning the safety against failure—constitute the requirements of the society. They are given in the national codes and standards and should be regarded as minimum requirements. Therefore, they cannot be modified in an alleviating direction. The remaining requirements are given by the purchaser/future proprietor and utilizer (tenant). This means, that in the early phase of the design procedure, the cost of future maintenance and repair during the service life of the structure have been determined to a certain degree. There are thus good reasons to consider, at an early stage, the formulation of the requirements from an economic point of view. The same ideas can be applied here as those used as basis for definition of safety classes, but with reference to economic consequences. This can be done where the consideration applies to safety against failure, function in normal use and durability. A structural component may, according to Table 2.1, be referred to safety class No 1. If the proper function of the component is critical for the operation of a complete factory, there may be reasons from an economical point of view to choose a higher safety class. 2.3 The design analysis process After the formulation of requirements follows selection of systems and materials.At this point the design analysis begins, which involves a detailed determination of dimensions and strength of structural components. The methods of analysis can often be decided by the designer himself. It is essential that the verification of the structure, with the chosen dimensions and the properties of the materials selected, satisfies the requirements established. The procedure can be described according to Fig.2.2 for a simple case. With the assumptions stated concerning loads, dimensions and material properties, calculation

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models are applied which provide the load effect S (Sollicitation) and carrying capacity R (Resistance). The load effect may be expressed as a section quantity (e.g. a bending moment in a beam) caused by the load. The resistance is the capacity of the structure to resist a load effect of the same kind (the capacity of the beam to transfer a moment). The verification implies that the resistance R has to be higher than the load effect S. The risk of failure (i.e. when R<S) should be sufficiently low. The case described concerns safety against failure, but the procedure of verification that the requirements on the serviceability of the structure in normal use are satisfied will in principle be the same. In many cases the procedure is more complicated. Several different kinds of load effects may act simultaneously and the resistance (e.g. with respect to normal forces or bending moments) may be determined in different ways for different load effects. The verification analysis provides an answer, yes or no. In case the answer is no, the procedure has to be repeated with updated dimensions and material properties.

Fig.2.2. Schematic description of the design analysis process. Applications of the design analysis process will be found in 2.6.1 and 2.6.2. 2.3.1 Methods of verification The quantities which describe the load effect S and the resistance R (e.g. load values F, strength valuesƒand dimensions l) are stochastic variables which can be represented in a simplified manner by frequency curves according to Fig.2.3. The verification consists of demonstrating that the resistance R is greater than the load effect S. This can be done by use of a number of methods. – The safety factor method (method of allowable stresses), – The load factor method with one single load factor (often used in plastic design), – The load and resistance factor method (method of partial coefficients), – Probabilistic methods. The first method has earlier been used, and is still being used in design codes in many countries but it is being replaced by the third method. Within the building sector the load

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and strength factor method was completely accepted in Sweden in 1990. It has also been incorporated into the Danish Pressure Vessel Code, 1986. Probabilistic methods have to be based on statistical data for loads, strength properties etc. which, so far, are available only on a very limited scale. The methods are, therefore, only used in very special cases. The load and resistance factor method and the method of allowable stresses are briefly described below. A more comprehensive discussion of the methods will be found in sections 2.4 and 2.5.

Fig.2.3. Frequency diagrams illustrating schematically the method of partial coefficients and the method of allowable stresses. 2.3.2 The load and resistance factor method The load and resistance factor method (often called the method of partial coefficients) is a verification method which is accepted in many countries. In the following, the method is

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described as it is applied in Sweden. The formulation is very similar to that used in the various Eurocodes, the ECCS and different national codes and standards. The basis is formed by the so called characteristic values: Fk for load fk for strength lk for dimensions where, in most cases, lk is equal to the nominal value, i.e. the value given in drawings and descriptions. The calculation of Fk and fk is indicated in 2.4.2 and 2.5.1. From the characteristic design values are deduced: (2.1) (2.2) (2.3) γf, γm and γn are called partial coefficients. The partial coefficient γf for load is in following referred to as the load factor, and the partial coefficients γm and γn are named resistance factors. ∆l is an additive quantity by which deviations from the ideal dimensions are considered. In most cases ∆l can be set to zero. The partial coefficients are discussed in somewhat more detail in 2.4.3, 2.5.4 and 2.6.1. The design values are used in the calculation models for load effect and resistance and provide the design criteria R(fd,ld)≥S(Fd,ld) (2.4) The load and resistance factor method is illustrated in Fig.2.3 a). Since the load factor can be given different values for different kinds of loads a more consistent design for a low risk of failure can be attained. For example γf=1.0 is adopted for gravity loads and 1.3 for environmental loads, such as snow and wind loads in load combinations. See 2.4.3. 2.3.3 Method of allowable stresses In some design codes the scatter in loads, resistance etc. is covered by one single safety factor s. The verification consists of demonstrating that σ<σall where σ is the stress determined from the load and, for instance when designing against yield failure,

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The safety factor s may vary within rather wide limits (1.3–3.5) depending on what elements of uncertainity have to be considered. In design against buckling, safety factors to the order of magnitude 10 are found in older codes. It should be noted, though, that the analysis in the present handbook provides lower limit values of the carrying capacity with respect to buckling and a safety factor of the order of 1.5 to 2 would be appropriate. The method of allowable stresses is illustrated in Fig.2.3 b). Application of the method for shell structures is discussed in 2.5.2 and 2.5.3. 2.4 Loads and load factors The following discussion on loads is, primarily, applicable to the construction sector, i.e. for buildings, bridge and hydraulic construction, and for scaffoldings in installation and erection, cranes, masts, power-line pylons, lighting posts and similar load carrying structures. The discussion will, however, be of interest also to design engineers working with other types of structures such as cisterns, pressure vessels, tanks, etc. Subsection 2.4.5 gives a brief summary for such constructions. 2.4.1 Classification of actions Actions are in the present publication used as a common name for effects due to forces and deformations. A force effect is primarily caused by external forces on a structure, while the deformation effect is primarily caused by a forced displacement, e.g. a support settlement, change of temperature or humidity. Loads may be classified with respect to their variation with time as a. permanent load

approximately constant in time

b. variable load

other normally occurring loads

b1. static load b2. dynamic load

causes additional forces due to acceleration including resonance

b3. fatigue load

load with so many load cycles that fatigue failure can occur

c. accidental load

e.g. impact, explosion

Loads can also be classified with respect to variation in space d.

fixed load

the load distribution over the structure is uniquely defined

e.

free load

has an arbitrary distribution over the structure possible limits

The duration tq of variable loads, see Fig.2.4, is the time during which the magnitude of the load amounts to at least the value q within the service life ttot of the structure. The relative duration is defined

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It is assumed that the variations of the load are similar during the entire service life ttot. The reduction factor ψ, which defines a normal load value of ψQk, is derived by use of the relative duration. In structures subjected to fatigue loading, the load range, the load level, and the number of load cycles are usually of importance. For the design of steel structures with regard to fatigue, reference is made to national codes or standards. See ECCS, 1985. In addition to the classification considering load variation with time as indicated above, there is a need with certain structures to take into account also long-term loads for the evaluation of creep deformations. Long-term loads are defined as ψ1Qk and can be interpreted as the mean value in time of the load. The reduction factor ψ1 is material dependent. Qk is the characteristic value of a variable load to be defined below. Note that creep of metals at high temperatures is a strongly nonlinear function of the stress level. Stability problems in the creep range are discussed in 5.3.

Fig.2.4. Variation of load with time. 2.4.2 Characteristic loads, normal loads and long-term loads. According to most national codes, loads are defined as follows: – the characteristic value Gk of a permanent load shall be assumed to be the mean value. – the characteristic value Qk of a variable load shall be a value with the probability 0.02 of being exceeded at least once during one year. – the normal value ψQk of a variable load shall be determined considering the relative duration ηq=tq/ttot, – characteristic value Qak of an accidental load shall be determined with respect to the nature of the load. In 2.4.4 it is indicated where Gk, Qk, ψ and Qak for normal loads on buildings, bridges and hydraulic structures are defined. Loads on pressure vessels, tanks etc. are discussed in 2.4.5. If the characteristic value is not available in a load standard, the value of Qk may in

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principle be estimated by use of the following procedure (determination of Gk does usually not present a problem). 1) Several observations, about 50, of the yearly maximum load are available. Fit a reasonable distribution function FQ to measured values and determine Qk from the condition FQ=0.98. 2) A smaller number of observations are available. The problem consists of finding a conservative distribution. A lognormal distribution function complies with this requirement in most cases, and for such a distribution, Qk can be determined by computing: a) the mean value µ of logxi b) the standard deviation σ of logxi c) log Qk=µ+2.05σ, or Qk=exp(µ+2.05σ), where 2.05=Φ−1(0.9 8) and Ф is the distribution function of the standardized normal distribution. 3) No observations of the yearly maximum load are available. In this case it is in principle not possible to determine Qk. The situation is not unusual, however, and it is thus often necessary to make an estimate of Qk. a) Compare with other similar loads for which Qk is known. b) Guess the mean m and the standard deviation s. Adopt Qk=exp(logm+2.05δ) where δ=s/m, compare 2) above. It is normally easier to make a reasonable guess of m and s than to guess directly the 98 per cent fractile. c) Assume Qk to be equal to the physical upper limit of the load. It is sometimes possible to indicate an upper limit. For instance, a reservoir or a tank can only be filled to its capacity. The reduction factor ψ which, in combination with Qk, gives the normal load value ψQk, depends on the relative duration ηq. The normal load (normal load value) appears in load combinations, see below. 2.4.3 Load combinations, design value of the load In general, a load combination can be written: ΣγgGk+γqQk+ΣγψqψQk where γg=load factor for permanent load Gk=characteristic value of a permanent load Qk=characteristic value of a variable load. For accident loads Qk is replaced by Qak γψq=load factor for other variable loads ψQk=normal load. For long-term loads ψQk is replaced by ψ1Qk, see 2.4.1. In Table 2.2, the load factors (partial coefficients) of the ultimate limit state are given for tilting, lifting, and sliding, as well as the load factors applicable in the serviceability limit state. The variable load for which γq=1.3 is referred to as the principle load. A load combination contains only one principal load. The load combination which provides the most unfavorable effect has to be considered.

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Table 2.2. Recommended load factors. Load value

Load factor a,b)

Comments

Permanent loads

Gk

γg=1.0

ultimate limit state in

One variable load

Qk

γq=1.3 c)

general and for

Other variable loads

ψQk

γψq=1.0

tilting, lifting and sliding

Permanent loads

Gk

γg=1.0

serviceability

Variable loads

ψQk

γψq=1.0

limit state

a) γg shall be equal to 0.85 if this load effect is more unfavorable. This does not apply to design for fatigue, however, where γg=1.0 may be used. Loads of the same nature, e.g. gravity loads, may be multiplied by the same load factor. b) If the permanent load is dominating γg shall be taken equal to 1.15 and γq=γψq=0. c) In design with regard to fatigue γ=1.0.

Example Indicate the load combinations in the ultimate limit state which have to be considered in the design analysis of the tank roof in Fig.2.5. The roof shall be designed for gravity load G, snow load S and wind load W. (Other loads may occur but are not included for the sake of simplicity). In general four different alternatives must be investigated: 1.

γgGk+1.3Wk

wind is the principal load

2.

γgGk+1.3Wk+1.0ψSk

wind is the principal load

3.

γgGk +1.3Sk

snow is the principal load

4.

γgGk+1.3Sk+1.0ψWk

snow is the principal load

Fig.2.5. Conical roof subjected to gravity, snow and wind loads. The load factor γg may, according to Table 2.2, assume the values 0.85 and 1.0 respectively. Due to symmetry, only one wind direction has to be investigated, but the wind load may have two different distributions, corresponding to two load cases. The snow load also provides two load cases, either a uniform or a triangular distribution over the roof surface.

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The four alternatives thus result in a large number of possible load combinations. Many of these are not critical, however, and may be sorted out at an early stage. The tank roof will probably be dimensioned by either a) 1.0Gk+1.3Sk or b) 0.85 Gk+1.3 Wk Since two snow and two wind load cases must be examined, a) and b) will result in four load combinations. In an actual situation where Gk, Sk and Wk are known, the number of load combinations may often be further reduced, which implies that an individual structural element normally needs to be examined only for one or a couple of load combinations. In certain types of structures, e.g. an unsymmetrical framework truss, a general application of the rules for selection of design load combinations leads to an overwhelming number of load cases, most of which are critical only for some elements. It should be noted, however, that the designer is free to perform an analysis on the safe side which, in many cases, will lead to a drastic reduction of the load combinations which must be considered. The increase in, for instance, weight that results is often marginal. In the rare cases where the deformation effect is dominating in the ultimate limit state, the load factor γq=1.0 is used instead of 1.3. 2.4.4 Loads on buildings, bridges and hydraulic structures Frequently occurring loads on buildings, bridges and hydraulic structures are given in national or international specifications. Loads on overhead cranes are stated by the suppliers. Loads on power-line pylons are chosen according to special standards, etc. 2.4.5 Loads on pressure vessels and tanks etc. For pressure vessels a design pressure is chosen as the highest allowable service pressure p (plus the pressure caused by the gravity of the contents, if this is higher than 2 per cent of the service pressure) and the pressure ps at which the device for limiting the pressure (safety valve) is activated. When applying the load and resistance factor method on a closed containment vessel it is reasonable to interpret both p and ps as characteristic, variable loads, which gives the design value pd=max (γfp, γfps) Tanks with contents which may cause extensive personal injuries should be referred to safety class 3, implying γf= 1.3 If failure of a tank could lead to very serious consequences, e.g. for tanks in nuclear power plants or offshore structures, the risk of failure should be further reduced. The safety factor 2.0 according to most pressure vessel codes corresponds, approximately, to the product of the factors γf, γm and γn and the additional extra factor of

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1/0.75 introduced in the present publication for elastic buckling of certain types of shells (with extreme sensitivity to imperfections). See, also, ECCS, 1988. The product of the factors is:

Where the extra factor is not included, e.g. for yielding or buckling under external pressure, the product takes the value 1.56. The pressure vessel industry is mainly working with the method of allowable stresses. In the Danish pressure vessel code, however, the load and resistance factor method has been introduced. For an open tank designed such that the liquid will overflow the edge, the liquid pressure pv should, from the safety point of view, be comparable to a gravity load, thus pd=γfpv where γf=1.0 if other loads occur simoultaneously and γf=1.15 if the liquid pressure is the only load, see footnote b, Table 2.2. The motive for choosing γf=1.3 in tanks with safety valves but γf=1.0 (1.15) in an open tank is obviously the risk that a safety valve will not be activated at the rated pressure. Loads in pressure and proof testing can be considered as safe as gravity loads and, therefore, the load factor γf=1.15 should be chosen, compare footnote b, Table 2.2. A partial vacuum is a variable load and, consequently, γf=1.3 shall be used if vacuum is the principal load. If some other load, e.g. the wind pressure is the principal load in a load combination, γf=1.0 should be used for the vaccum. 2.5 Resistance and resistance factors 2.5.1 Assumptions concerning strength properties The material strength properties are the yield and ultimate strength limits in compression and tension, the modulus of elasticity and the shear modulus. Other material properties related to strength are Poisson’s ratio, fatigue strength, fracture toughness, creep properties and thermal expansion. The requirements for design analysis of a structure indicate the strength class of the material to be used. In the analysis, then, various kinds of strength values are introduced which apply to the strength class selected. The strength values introduced in the design analysis are sometimes based on results from tests performed in advance. The producer of the material certifies that the strength properties are according to the requirements specified. Alternatively, the strength properties are checked at the delivery. The procedure used to verify that the strength of the material meets the given requirements normally includes tests with special test specimens and a specified procedure. In certain cases the results of these tests cannot be considered to be directly representative for the strength of the material in the actual structure and, thus, have to be corrected. This may be performed by dividing the strength values obtained in the tests by a number η, normally greater than 1, such that:

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(2.5) The factor η (which should not be mistaken for the reduction factor with respect to buckling) includes, e.g. for concrete, the effect of the casting direction and curing conditions of the actual structure. For wood the influence of the moisture content and the duration of the load are taken into consideration. The characteristic value of strength should be interpreted as a condition for the analysis which refers to the expected results of actual or imagined tests. It thus applies to the strength of the test specimen and not to that of the actual construction. The characteristic value is defined somewhat differently for different materials. For the compressive strength of concrete, as an example, it corresponds to the lower 0.05-fractile of the statistical distribution of the strength. For steel (both structural steel and reinforcement bars) it amounts, approximately, to the 0.01-fractile. The design value for strength should, naturally, be valid for the material of the structure. This means a certain deviation from the basic presentation in 2.3.2 in such a manner that the coefficient η should be entered into eqn (2.2) which translates characteristic values into design values. Furthermore, the consequences of a damage or a failure should be considered. For the resistance term this is achieved by dividing the strength values by a special partial coefficient γn. With this modification the formula for computation of the design value fd from the characteristic value fk becomes (2.6) The value of η depends on factors quite different for different materials, and no generally valid figures can be given. For metals, η=1.0 may be used and η may, therefore, be omitted in eqn (2.6). η for other materials may be found in specific design codes. The partial coefficient γn depends only on limit state and safety class. Typical values for the ultimate limit state are given in Table 2.3. In the serviceability limit state the value 1.0 can be adopted. With reference to 2.2.6 concerning economic consequences of a damage or a failure there may be reasons to choose higher values of γn. This could apply for example to nuclear plants and tanks with highly poisonous contents.

Table 2.3. Values of γn as functions of the safety class. Safety class

γn

1.

Low

1.0

2.

Normal

1.1

3.

High

1.2

By introducing the partial coefficient γm, uncertainties in the strength of the material are taken into consideration as caused by: – the normal scatter of the material strength,

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– the variability of the factor or function η which translates the strength of test specimens into strength of the structure. For practical reasons other factors not directly related to the strength of the material are taken into consideration by γm. Such factors are: – deviations of dimensions and geometry from the nominal values assumed in the design analysis, if such deviations are not considered otherwise, – unreliability of the model of analysis, if kept within reasonable limits. For shell structures, deviations in dimensions and shape and unreliabilities of the model of analysis are treated according to 2.5.2. 2.5.2 Models of analysis The calculations used in the design are based on models by means of which the behavior of the structure is described. The models of analysis may be more or less complicated and provide a more or less accurate description of the function of the structure (see e.g. 1.3). Often a model giving a higher accuracy turns out to be more complicated. In certain cases the nature of the problem demands a more sophisticated model, e.g. for stress analysis in structures subjected to fatigue. Usually, there is an option, however, between different models and the choice has to be made on an economic basis, which applies to the cost of material/construction in relation to the cost of the design analysis. Models of analysis should be considered as approximate descriptions of the function of a structure. Even the most advanced models are thus subject to some uncertainties. With regard to this fact numerical values of coefficients etc. should be chosen in such a way that the model gives results on the safe side. But it is often not feasible to enter such values of the coefficients that the results are conservative in all conceivable cases. Probabilistic aspects may be introduced, choosing the strength coefficients in such a way that the model gives results on the unsafe side only in a small fraction of the cases. This fraction should not exceed 5 per cent. The resulting resistance may thus be interpreted as a characteristic value. 2.5.3 Strategy for design of shell structures In the following chapters of this handbook, methods (models of analysis) are presented for estimation of the resistance of shell structures with regard to buckling. As emphasized in many places, the shell structures are sensitive to deviations from the theoretical shell geometry, implying that the classical theory for a perfect shell in most cases forms a very poor model of analysis. The model must consider the influence of initial buckles, residual stresses, yield etc. (otherwise a very large value of the resistance factor γm, or safety factor s, must be chosen, sometimes to the magnitude 10). In this handbook the aim has been to select models of analysis corresponding to a 5 per cent fractile according to 2.5.2, i.e. the load and resistance coefficients are chosen such that the formulas will give results on the unsafe side in at most 5 per cent of the cases. As an example the procedure is demonstrated for an axially loaded cylindrical shell according to 3.2.1.2.

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The starting value is the classical buckling stress σel of a perfect cylinder. This stress is reduced with regard to initial buckles using the factor η. Deeper buckles and residual stresses are considered by an additional factor α, which depends on the tolerance class for geometrical shape deviations and the method of production . If the shell has a small slenderness ratio, r/t, the buckling stress is also reduced by the factor ωs which takes the influence of plasticity into consideration. See Fig. 2.6.

Fig.2.6. Diagram showing the classical buckling stress σel, allowable stress σall, and the intermediate steps of calculation. The example refers to a cylindrical shell under axial compression where the scatter of the test results is large and an extra reduction factor of 0.75 is recommended. All these reduction factors are mainly based on experience from experimental investigations and theoretical analyses. The large number of tests which have been performed are often not very carefully described and the influence of the various factors affecting the carrying capacity (initial buckles, residual stresses, material properties, boundary conditions) may not be assessed. The separate factors are, therefore, cautiously

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estimated, implying that the formulas will yield results on the unsafe side in much less than 5 per cent of the cases. This may, in particular, be the case for slender shells where the extra factor 0.75 has been introduced to cover the general unreliability of the assumptions. See ECCS, 1988. 2.5.4 Design value of the resistance It appears from 2.5.3 that the estimates of the resistance made in this handbook are chosen in such a way that they correspond to characteristic values, i.e. they overestimate the resistance in 5 per cent of the cases at most. The resistance factor γm, see 2.3.2, therefore, does not have to consider uncertainties in the model of analysis but only the normal scatter of the material strength and deviations of the dimensions, primarily the thickness of the shell. The resistance factors given below are only valid provided that the tolerance limits have been checked. If the yield limit σy is chosen to be the lower yield strength according to national or international standards, a value corresponding to a low fractile is used (apprioximately the 1%-fractile). The resistance factor γm can, then, with reference to the scatter of the material, be given the value: γm=1.0. If the thickness tolerances for sheets and profiled bars are large it may become necessary to increase the factor γm. γm=1.1 should be chosen if a design analysis based on the lower limit of a dimension results in a reduction of more than 6 per cent of the resistance compared to the resistance determined for nominal dimensions. Provided the thickness tolerance of a rolled steel sheet of thickness t>5 mm is at most 5 per cent, the following factors should be applied: γm=1.1 for sheet thickness t<5 mm (2.7) γm=1.0 for sheet thickness t>5 mm (2.8) Alternatively the analysis can be based on a smaller thickness than the nominal, e.g. the minimum value according to the tolerance range. 2.6 Design criteria 2.6.1 The load and resistance factor method The load and resistance factor method is briefly described in 2.3.2. The method is applied in many design specifications and is sometimes referred to as the method of partial coefficients. According to this method the characteristic values of loads and resistance are first determined. Then the design values are obtained by: – multiplying the the characteristic values of the loads by the load factor γf, – dividing the characteristic values of the resistance by the resistance factors γm and γn.

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γn is a coefficient which depends on the consequences of a failure of the structure. In case a failure would not involve a considerable risk of serious personal injury (e.g. failure of a wall tile), γn=1.0 is chosen. Such a structure is referred to safety class 1, see Table 2.1 and 2.3. Structures, the failure of which would imply a high risk of personal injury (e.g. a column in a multistory building), are referred to safety class 3, and γn=1.2 is applied. An intermediate class 2 is defined where the value of the resistance factor is given as γn=1.1. Examples of selection of safety classes and corresponding values of γn are given in applicable standards for building structures. For pressure vessels, large tanks etc. γn=1.2 should be used unless there are reasons for choosing a lower value. The design analysis should verify that the stresses caused by design loads σSd (or section forces Msd) are smaller than the design value of the resistance expressed in terms of the same quantity (σRd, or MRd), i.e. σSd<σRd (2.9) where σSd=stress caused by the load: ΣygGk+ΣγψqψQk (2.10) σu=characteristic resistance , γm=resistance factor considering uncertainties in the material parameters and tolerances for dimensions according to eqn (2.7) or (2.8), γn=coefficient which takes into account the consequences of a failure, see 2.5.1. 2.6.2 Method of allowable stresses A safety factor should consider the unreliability of load assumptions as well as the unreliability of resistance values. Since uncertainties of the methods of analysis (see 2.5.3–2.5.4) are included in the estimation of the resistance, a moderately low safety factor may be chosen, normally 1.5 for normal types of loading. The allowable stress σall is thus determined as (2.11) where σu=the resistance according to this handbook, s=safety factor, normally 1.5. The allowable stress shall be higher than the stress determined from loads without load factors i.e. σ<σall (2.12)

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2.7 Design by testing Since the design analysis of complex shell structures against buckling often leads to extensive computations and the results can be rather unreliable, design based on test results may be attractive. The procedure is applicable primarily for series manufactured objects of such a size and shape that the testing can be performed in full scale, or nearly full-size scale. The test results can then be used directly as starting-value for the resistance after possible adjustments due to deviations in material, dimensions and shape. The testing may apply to such properties which are important for verifying that various requirements are satisfied in the ultimate limit state and the serviceability state, e.g. demands on resistance, ductility and rigidity. They are not meant to be applied for testing of durability properties. It should be pointed out that testing can also be advantageous in order to verify e.g. a model of analysis of a large, complicated structure. The testing may then be carrie d o ut in mo del s and by use of only of a few specimens. Such tests are not included in the concept “design by testing”. In design based on experiments the number of test specimens should be high enough to make a statistical evaluation of the test results possible. 2.7.1 Requirements General advice on the planning of tests, the material properties of the test specimens, dimensions and shape, loading, testing procedure and analysis of results are given in national recommendations which should also provide guidance for the choice of the partial coefficient γmp (resistance factor) corresponding to γm according to 2.6.1. General rules may also be found in various ASTM documents. Some important factors which refer to shell structures in particular are indicated below. Since the buckling resistance of a shell is strongly influenced by geometrical imperfections, at least for some types of shells subjected to particular loads, it is extremely important that the deviations in shape of the test specimen are surveyed carefully and referred to the deviations which are measured or may be expected in the finished structure. The geometric deviations from the nominal shape of the model should be measured according to the directions given in chapter 3, selecting as reference the kind of shell/load which has the closest agreement with the case considered. The clamping of the model in a test rig should simulate the actual boundary conditions, since deviations may be of decisive importance for the results. The material of the model is likely not to be of great significance as long as buckling occurs in the elastic region. A material commonly used in these connections is Mylar or similar plastic materials with a high yield stress. If the finished structure is expected to buckle within the elasto-plastic region, the test specimen should be made of the same material and be manufactured by use of the same techniques as the full size structure. When using plastics it should be noted that deviations in shape must also be simulated to make it possible to translate the test results without modifications.

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The extent of the testing is determined by the accuracy required in the determation of the resistance. For shell structures, a few additional comments should be made: – The scatter of test results can be expected to be higher than normally due to the great influence of deviations in shape. – In the manufacture of the test specimens, systematic shape deviations may be introduced leading to a mean value of the resistance which differs significantly from that of the actual structure. – If the manufacture of the structure includes welding, this should also be simulated in the model. 2.7.2 Evaluation of test results In the analysis of the test results the rules given by for instance ASTM or other generally accepted recommendations shall be applied. Provided that the testing is performed in full scale with specimens taken from the normal production, the rules should apply without limitation. It is of particularly great importance, however, that the test specimens are selected such that the normal variations in quality are covered by the test series.

3 ELEMENTARY CASES 3.1 Introduction In the following compilation of elementary cases, recommendations are given for calculation of the buckling resistance of various shell elements subjected to simple load distributions. The reistance is given as the characteristic value of the load carrying capacity. These values have normally been obtained as lower limit values of test results. Some cases are more sensitive than others to disturbances such as shape deviations from the nominal geometry and deviations from assumed boundary conditions. In the cases where the sensitivity is high, the buckling resistance has been determined with additional caution. The values given for the resistance can be used as basis for design by use of the method of partial coefficients or as allowable loads after division by the safety factor required, see Chapter 2. The rules apply if the tolerance level with regard to initial imperfections etc. is satisfied, or if there are indications that the method of analysis yields conservative results. Shape deviations leading to a lower resistance must be taken into account. Greater deviations than those specified by the tolerances given below are considered by reduction factors indicated in the different sections. Disturbances such as openings and local, concentrated loads may be treated by use of the methods presented in chapter 4. If nothing else is stated in the text, stresses in the shell are computed merely as primary membrane stresses according to linear theory of elasticity and the resistance is expressed with reference to these stresses. Bending stresses and secondary membrane stresses which can appear as a consequence of the deformation of the shell are taken into account implicitly by the reduction factors given for each elementary case. Formulas are given both in a general form and with a value of the Poisson’s ratio v=0.3 introduced. This value may be assumed to apply to metals, such as steels, including stainless steel, and aluminum. 3.2 Cylindrical shells 3.2.1 Unstiffened shells 3.2.1.1 Assumptions The shell shall consist of a complete circular cylinder with radius r and thickness t, supported or stiffened at the edges to maintain its circular form. The following limitations are applied:

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(3.1) (3.2)

Provided , where , the buckling resistance can be determined from the theory of plane plates. The loading shall be entered on the shell in such a way that the stress distribution will be equivalent to the one anticipated in each load case. Load concentrations must be taken into consideration either by load distributing devices or by analysis according to Chapter 4. Geometric deviations from the nominal dimensions of the shell must satisfy the requirements in Table 3.1, where wr denotes the largest shape deviation measured.

Table 3.1. Production methods and tolerance classes for cylindrical shells. Production method

Tolerance class

Tolerance on wr

α

A Hot forming or

1

lr/100

1

stress-relieving

2

lr/50

0.7

B Cold forming or

1

lr/100

0.9

welding

2

lr/50

0.6

In the interval lr/100<wr
Fig.3.1. Method of measuring shape deviations of cylindrical shells. a) with

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a straight ruler which is placed along an arbitrary meridian and between circumferential welds. b) with a circular template (radius=r) which is placed along an arbitrary parallel circle and between longitudinal welds. c) with a straight ruler which is placed across circumferential welds. The length of a ruler and the template shall be , but not longer than 95 percent of the spacing between circumferential and longitudinal welds respectively. When measuring according to c), lr shall not exceed 500 mm. Greater shape deviations than for tolerance class 2 above, but at most twice as much, may be considered by reducing the critical stress according to 3.2.1.2. 3.2.1.2 Uniform axial compression A. Background According to the linear theory of elasticity for a perfect shell with constant thickness, (classical theory) buckling will occur at the stress level (Lorenz, 1908): (3.3) E=Young’s modulus of elasticity r=radius of the middle surface t=thickness v=Poisson’s ratio

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Fig.3.2. Cylindrical shell loaded by a central axial force F giving a uniform compressive stress σx. The classical theory usually implies a strong overrating of the resistance. This is illustrated by Fig.3.3 (Weingarten et al., 1965a), showing experimentally determined buckling stresses divided by the classical value σel and plotted versus r/t. It may be noted that the deviation from the classical theory increases with growing slenderness, and that the scatter of the test results is very large. The reason is that axially compressed cylindrical shells are very sensitive to deviations from the ideal shape. These deviations may originate from the manufacturing process, but may also be due to disturbances in the loading or local variations in stiffness etc. Actual geometrical deviations are usually not given in the test reports, but it can be assumed that they have varied between and within the test series. In many cases the strength of the material is not reported either, but the major part of the tests refer to buckling within the elastic region.

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Fig.3.3. Early tests with axially loaded cylindrical shells (Weingarten et al., 1965a).

Fig.3.4. Resistance against plastic buckling of shells with small shape deviations. Furthermore, Fig.3.4 shows a summary of test results by Ostapenko-Gunzelman, 1978, and Schulz, 1984. These shells were accurately manufactured with shape deviations smaller than those accepted according to tolerance class 1 and they buckled in the plastic

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region. The buckling resistance is related to the yield strength of the material and the slenderness λs is calculated according to eqn (3.12). In both these test series the load was accurately centered by monitoring the stress distribution with strain gauges. In Schulz, 1984, a test series is also reported where the stress distribution was not checked. These results are not included in Fig.3.4. They show a substantially greater scatter as well as a number of improbably low values. The length of the shell has an influence on the carrying capacity only for very short is often used, which may be considered to be a shells. As a reference length, measure of the size of the critical buckles. The influence of the length is illustrated by Fig.3.5 (Weingarten et al., 1965a).

Fig.3.5. Experimental resistance of shells with r/t=400 versus the length of the shell. For very long shells column buckling may be critical for the resistance. If the shell at the same time has a high r/t ratio, say higher than E/(20σy), interaction between column buckling and shell buckling will occur. This interaction may be taken into account by replacing the yield stress by the buckling resistance, σu, in the design analysis against column buckling. B. Design resistance The buckling resistance can be calculated with the following method. The analysis is based on the assumption that the edges of the shell keep their circular shape, e.g. by

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61

introducing a ring beam (see Fig.3.8) with a moment of inertia Ir which, according to Cohen, 1966, shall satisfy the condition: (3.4) Here l denotes the length of the shell. The formula has been deduced from the results of numerical calculations for a shell with

. Larger values of l than

should not be introduced in the formula. If the length is larger than the shell should be provided with ring stiffeners unless the interaction between local buckling and column buckling is considered according to 3.2.1.2A. For elastic conditions the buckling stress is (3.5) where η is an experimentally determined reduction factor of the form (3.6) The reduction factor α takes into account the manufacturing method and tolerances and is given in Table 3.1. The values have been estimated from theoretical calculations, see e.g. Bornscheuer et al., 1983. For larger geometrical shape imperfections than those corresponding to tolerance class 2 the factor α can be obtained from the formula below, where α2 is the value of α for tolerance class 2 according to the table. (3.7) The reduction factor for shells with larger shape deviations than for tolerance class 2 is mainly intended for checking the resistance of manufactured shells, not for design . Under elastic conditions the buckling resistance is determined as (3.8) where σy=the yield stress. For stress levels exceeding one third of the yield stress the buckling behavior is influenced by plastic deformations, and the resistance may be calculated from: (3.9) where the reduction factor ωs2 is given by: (3.10)

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(3.11) (3.12) The reduction factor ωs2 is plotted in Fig.3.6.

Fig.3.6. The reduction factor ωs2 versus the slenderness parameter λs. C. Example A trestle according to the sketch is to be manufactured from two welded tubes. The structure will be designed according to safety class 2 (see Chapter 2). The design loads including load factors are:

Fhd=1.3·70=91 kN

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Fvd=700 kN The length of the leg is Maximum compressive force:

Try tube Ø 600×4.0, SS steel 2132 with σy=350 MPa. The tube is cold formed and welded, and thus tolerance class 2 applies. With regard to shell buckling the reduction factors are: α=0.6 (Table 3.1) (3.6)

σxu=0.71·350=249 MPa The resistance with regard to shell buckling is used in the continued analysis considering column buckling instead of the yield strength of the material. For column buckling the tube is referred to buckling curve c, Eurocode 3, Part 1, 1990. Clamping at the footing and at the connection to the other leg is neglected.

ωc= 0.38 With resistance factor γm=1.1

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Tube Ø 600×4.0 is adequate for the intended purpose. 3.2.1.3 Bending moment For a cylindrical shell which is subjected to a bending moment M according to Fig. 3.7 the stress distribution is given by the beam theory as: (3.13) where y=the distance from the bending axis r=the radius of the shell middle surface (3.14)

In eqn (3.13) compressive stresses have a positive sign.

Fig. 3.7. Cylindrical shell loaded by a bending moment M giving a maximum compressive stress σxb. In the common case where t is small compared to r, the maximum compressive stress in bending is: (3.15) One condition for the technical beam theory to be applicable is that the deformations of the cross section are negligible. As a consequence of the curvature caused by bending, the cylinder section will be ovalized, i.e. I is reduced.

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Introducing the notation σ0 for the bending stress according to eqn (3.15) the ovalization can be calculated with the following procedure. The rules are simplifications of the results by Brazier, 1927, and Pippard, 1932. The following relationship governs the bending stress σ0 referred to σxel, according to eqn (3.3), and the maximum radial displacement w0 (ovalization) (3.16)

For be neglected.

, corresponding to

Within the interval relationship

, the influence of the ovalization may

ratio can be obtained from a linearized (3.17)

The maximum bending stress increases because of the flattening and it can be estimated from: (3.18) Furthermore the radius of curvature increases due to the ovalization. When calculating the buckling stress the radius r of the circular cylinder is replaced by the largest radius of curvature p: (3.19)

is larger than 0.4, the cylinder shall be provided with ring stiffeners to prevent If unstable growth of the ovalization. Also in other cases ring stiffeners can be used to increase the resistance. The spacing between the rings is of importance. Roughly, a ring spacing of 2r will prevent ovalization entirely, i.e. the cylindrical shell can be assumed to retain the shape of its original cross section. At a ring spacing of 20r, the effect may be neglected and the influence of ovalization must be considered according to the formulas given above. The ring stiffeners shall be designed for the bending moment: (3.20)

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where l is the spacing between the rings. This bending moment does not include the effect of a nonuniform pressure distribution over the shell surface. For instance, a wind pressure in the supercritical range (Re>5.105) gives: Mr=0.4pr2l (3.21) where p is the stagnation pressure. Mr is added to the value obtained from eqn (3.20). The stability conditions in pure bending are similar to those for uniform axial compression. However, the stress gradient results in a higher buckling resistance in bending. The procedure described in subsection 3.2.1.2 can be used together with a modification of the reduction factor. (3.22)

where r=p from eqn (3.19), if flattening is considered. Full utilization of the cylindrical shell until the compressive stresses σxb=σy requires for a shell of tolerance class 2. λs≤0.4, corresponding to, approximately, If the shell is designed according to the plastic theory, i.e. assuming full plasticization of the cross section at fully yielded sections, the slenderness has to be further restricted according to: (3.23) 3.2.1.4 Internal pressure When a cylindrical shell, loaded by an axial force or a bending moment, is simultaneously subjected to internal pressure pi, the buckling stress will increase. The internal pressure causes tensile stresses in the circumferential direction of the magnitude (3.24) The stabilizing effect of the tensile stresses may be taken into account in those cases where it is known with certainty that they are acting simultaneously with the longitudinal compressive stresses in the cylinder. One such occasion is when both axial and circumferential stresses are caused by gas or fluid pressure. The action of the internal pressure on the end closures produces a longitudinal stress (3.25) This longitudinal stress will reduce the stress in the axial direction of the shell. If the circumferential stresses are caused by a horizontal pressure from bulk material in a silo it is uncertain whether the horizontal pressure is always present. There is a risk that the bulk

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material may be subjected to arching, which may reduce the tensile stresses locally or even remove them completely. In the cases where it is absolutely clear that the internal pressure is present its effect can be considered by substituting the reduction factor η according to eqn (3.6) by: (3.26)

The increase of the buckling load obtained by eqn (3.26) is based on results presented by Rotter-Seide, 1987. See further Weingarten et al, 1965b. It is obvious that the increase is dependent on the initial imperfections of the shell, and eqn (3.26) is a conservative estimate valid for the unfavorable case that the imperfections are axially symmetric. Besides checking the buckling resistance, the combined action of the biaxial stresses has to be checked according to subsection 3.2.1.8. 3.2.1.5 External pressure A. Constant wall thickness Examples of cylindrical shells subjected to external pressure are pressure/vacuum vessels, tanks and silos under wind loading and submarine structures. For shells under wind loading, the stagnation pressure should be used as characteristic external pressure unless the wave length of the buckles is studied more thoroughly. The shell is assumed to be supported at its ends by stiff bulkheads or rings. If the stiffening consists of a ring beam, the moment of inertia around an axis through the center of gravity of the beam section and parallel with the generator of the shell shall satisfy the conditions in eqn(3.27). (3.27)

Ir=moment of inertia of the stiffener around the GC axis, parallel with the shell, see below, p=external pressure, l=half length of the shell, or for an inner ring, dividing a long shell into sections, the spacing between the rings, rr=radius of the ring centroid, r=radius of the middle surface of the shell, t=shell thickness, σr=pr/t compressive stress in shell and ring, compare 3.2.2.2.

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In the moment of inertia Ir of the ring stiffener a part of the shell wall may be included according to Fig.3.8. The effective width is calculated according to

Fig.3.8. Typical ring stiffeners in a shell under external pressure. In the determination of the buckling resistance, the compressive stress σφ in the circumferential direction is used as design parameter, where (3.28) At the same time, the shell is also subjected to a longitudinal compressive stress as a consequence of the pressure acting on the end closures (3.29) For a long cylindrical shell subjected to external pressure the influence of the end pressure is small and the buckling behavior is of the same character as buckling of a ring beam. The pressure is maintained perpendicular to the shell surface during the buckling deformation, which gives the elastic buckling load (Levy, 1884) (3.30) Expressing the result in terms of the circumferential stress yields: (3.31)

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69

For short or medium length shells, an increase of the buckling stress will result due to the stiffening effect of the end supports. This effect can be taken into account by simplified formulas according to DASt Ri 013, 1980. Distinction is made between short and long shells. Long shells: (3.32) Short shells: (3.33)

In certain cases it is of value to know the buckling mode of the shell, for example if it is intended to introduce stiffeners etc. An estimate of the number of buckling waves is obtained from the following expression: (3.34) The discrepancy between experimental results and the buckling stress according to linear theory of elastic stability is moderate, which is evident from Fig.3.9. The influence of initial shape imperfections has a character differing from that of an axially compressed cylindrical shell. In the latter case local deviations over an area of the same extension as the natural buckling mode seem to be most critical, while for external pressure on a long shell a deviation from the circular shape is the main concern. In short shells the axial compression is of some influence why local buckles may also be critical. Since the influence of shape imperfections is not clearly defined, it is reasonable to require limits on both local buckles and ovality. Local deviations from the nominal shape are chosen according to 3.2.1.1. To these requirements are added the following limits on maximum out-of-roundness (ovality).

Table 3.2. Definition of limits on out-of-roundness for different tolerance classes. Tolerance class

(dmax−dmin)/dave

αη

1

0.005

0.7

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If,

70

0.010

for

0.5

example,

Fig.3.9. Comparison between experiments and theory according to Batdorf for cylindrical shells under external pressure, Bruhn 1965.

With the values of the product αη given in the table, the calculation of the resistance can be performed in a way similar to that given in subsection 3.2.1.2. Thus. σφelr=αη σφel (3.35) Under elastic conditions the resistance is written (3.36) With the slenderness ratio λs defined according to eqn (3.12) the following expression applies within the plastic region (3.37) Eqn (3.36) is equivalent to:

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71

(3.38) Eqns (3.36) and (3.37) are recommended for the cases where the sensitivity to initial deformations is moderate and the experimental basis can be considered satisfactory. In this handbook, two reduction formulas are introduced, viz, those given in 3.2.1.2 and 3.2.1.5. For each case given in the following sections, the calculation of the buckling resistance is referred to one of these reduction formulas. The plasticity reduction factor ωs1 a ccording to eqns (3.37) and (3.38) is presented in Fig.3.10.

Fig.3.10. Reduction factor ωs1 defined by eqns (3.37) and (3.38). B. Example A long tube subjected to external pressure shall be stiffened by rings in such a way that local buckling in the shell sections between the rings occurs about simultaneously with global buckling of the entire length of the tube: NOTE: coinciding buckling loads may lead to an increase in the imperfection sensitivity of the shell, see Tvergaard, 1976. A 10– 20 per cent separation of buckling loads may be recommended.

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r=1000 mm t=10 mm p=200 kN/m2 (2.0 bar) σy=260 MPa s=1.5 (safety factor) E=210000 MPa Try a ring according to the figure: 25t=250>b2 Ar=10·40+8·80+10·2·78=2600 mm2

rr=1000+24.9≈1025 mm=1.025 m

which gives the result lr<5.62 m. In order to achieve an adequate margin to the load carrying capacity of the rings, l=4.5 m is chosen.

The shell is classified as short

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73

(3.33) Tolerance class 1 requires αη=0.7 σφelr=0.7·42.8=30.0 MPa<σy/3 implying: σφu=σφelr=30.0 MPa

C. Varying shell thickness Cylindrical walls of tanks and silos, etc. are usually designed to carry the internal pressure, which in most cases means that the wall thickness will vary along the shell. In service, load cases involving external pressure, e.g. wind loading, may occur, and buckling can thus become critical. For cylindrical shell walls with varying thickness according to Fig.3.11 the following design rules are recommended, where the basic formula in 3.2.1.5A for constant wall thickness has been used, see DASt, 1980, and DIN 18800,1990.

Fig.3.11. Conversion of wall of varying thickness into an equivalent wall consisting of three shell sections. Example: t0=(t1l1+t2l2+t3l3)/l0. First, the following condition is checked for each section of the shell.

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If this is the case, the buckling stress is determined for a long shell according to eqn (3.32) for the shell section considered. If the condition is not satisfied, the calculation of the resistance is performed as follows: The actual structure is replaced by a computational model with three equivalent shells segments where the effective thickness of each of the three segments is calculated according to the example in Fig.3.11. The length l0 is chosen in such a way that it ends at or above the shell section with ti>1.5 tmin. However, it should not exceed l/2. If l0
the shell is considered to be long and the buckling stress is determined by the thickness of the uppermost equivalent shell segment, to and length l0 which are introduced in eqn (3.32). If the condition is not satisfied the buckling stress is calcultated from: (3.39)

The reduction factor β is obtained from Fig. 3.12. The continued analysis follows 3.2.1.5A. One additional condition is, though, that the eccentricity at a circumferential joint does not exceed the thickness of the thinner shell segment.

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75

Fig.3.12. Diagrams for evaluation of the parameter β for a shell with varying thickness. Note that eqn (3.39) is valid also for short shells, the correction factor ξ being omitted. The reason for the omission is that the basis for determining a value of ξ for a cylindrical shell with varying thickness is not sufficient. To be able to take into account the increase of the resistance of very short shell sections an analysis according to Chapter 6 is normally required. Further information on buckling of cylindrical shells with varying wall thickness can be found in Ebner-Schnell, 1961, Malik et al., 1979. 3.2.1.6 Torsional moment When loaded by a torsional moment T, the shell will be subjected to a homogeneous shear stress of the magnitude

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(3.40) If the shell is long, the shear stresses will cause buckling according to the classical theory (Schwerin, 1924), where (3.41) In the case of a short shell, the buckling stress will be underestimated. The problem has been investigated by a number of authors based on somewhat different assumptions. A solution according to Donnell, 1933, can be expressed in the same form as used for plane plates, viz. (3.42) (3.43) (3.44) Entering these values into eqn(3.42) above gives, for v=0.3: (3.45) In Fig.3.13 this solution is shown together with test results. The experimental values show a moderate scatter with a mean value slightly below the theoretical solution. Introducing a reduction factor η=0.65, eqn (3.43) will form a lower limit of the test results. The reduction factor may be considered valid for shells satisfying the requirements of tolerance class 2 according to 3.2.1.1. A basis for estimating a, possibly higher, value corresponding to tolerance class 1 is lacking, but the increase to be gained is likely to be small.

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Fig.3.13. Comparison between the theoretical buckling stress and experimental results, Bruhn, 1965. Z according to Fig.3.9. For design within the plastic region, τelr>τy/3, the same reduction factor ωs1 as in 3.2.1.5 is used, but the slenderness parameter is defined by: (3.46) (3.47) η=0.65 (3.48) (3.49) (3.50) At the transition between the validity ranges of the two equations they will yield slightly different results. Eqn (3.50) entered into eqn (3.42) is a transcription of eqn (3.41). The buckling resistance is finally obtained as: (3.51)

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(3.52) with ωs1 according to eqn (3.37). 3.2.1.7 Transverse shear A cylindrical shell sometimes functions as a beam loaded in the transverse direction, see Fig.3.14. The transverse force gives rise to a shear stress distributed over the cross section according to the function (3.53) The maximum occurs at the neutral axis and is (3.54) Buckling caused by a shear force takes place at a higher stress level than buckling due to a torsional moment since the stress distribution caused by a shear force is not homogeneous. The design analysis can be performed according to subsection 3.2.1.6 provided the maximum shear stress τmax is used as reference.

Fig.3.14. Cylindrical shell supported as a beam and distribution of shear stresses over the cross section.

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Fig.3.15. Ring at the supports of transversally loaded cylindrical shell. The recommended value of the reduction factor is: η=0.75 Normally the shell has to be provided with stiffeners at the supports, see 3.2.1.9 for design recommendations. 3.2.1.8 Combination of load cases If various loads, each of which may cause buckling, are acting simultaneously on a shell, some kind of interaction will occur. The degree of interaction depends on the conformity of the buckling modes caused by the different load distributions. If the buckling modes coincide, it may be expected on theoretical grounds that the interaction is linear, while differing buckling modes should lead to a convex interaction curve or, as an extreme case, to no interaction at all. When combining loads, some of which produce tensile stresses in the shell, two effects will result. One, exemplified in subsection 3.2.1.4, implies that the tensile stresses raise the buckling resistance. The second effect is negative since the effective stress increases and causes early yielding of the material or similar combination effects due to interaction of the stress distributions. If the slenderness of the shell is so small that buckling cannot occur, the resistance is determined by the yield stress of the material. According to the most commonly used criterion, yield will occur due to two orthogonal stress components σ1 and σ2 when the effective stress σef reaches the yield stress σy, i.e. σef2=σ12−σ1σ2+σ22≤σy2 (3.55) The formula can also be written as: (3.56) and may be represented graphically according to Fig.3.16a). It appears from the figure that if σ1 and σ2 have the same sign a strongly convex curve results and yield does not occur until one of the stresses exceeds the yield stress σy (or is equal to σy for stresses of the same magnitude). When the stresses have opposite signs, on the other hand, a relationship is obtained which is only slightly convex. If the resistance is reduced due to buckling, a more or less linear interaction relationship will be obtained, as stated above, for compressive stresses in both directions. Several investigations on plates supported along four edges show that the interaction relationship with growing slenderness (i.e. decreasing ratio between buckling stress and yield stress) changes from a strongly convex to a nearly linear curve as shown in the lower half of Fig.3.16b). See e.g. Narayanan, 1983.

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This conversion can be achieved by multiplying the second term in the interaction formula (3.56) by a correction factor kxφ which is positive for small and negative for large slenderness ratios. If the stress is tensile in one direction and compressive in the other, an interaction relationship according to the upper half of Fig.3.16b) is obtained. For a low slenderness ratio the curve is only slightly convex, while for higher slenderness values, the tensile stresses tend to increase the resistance with regard to buckling, yielding a strongly convex relationship. The resulting relation can, when applied to combinations of loads on a shell, be expressed by eqn (3.57). The stresses resulting from a specific load case are entered with their actual sign and resistance stresses with a positive sign. In addition, when the stresses are of opposite signs, the formula given by (3.58) has to be checked as well. The stresses are referred to a given point on the shell middle surface. If a specific stress component is positive (tension) the corresponding resistance is taken equal to the yield stress, e.g. if σφ>0, then σφu→σy. The conditions given by eqns (3.57) and (3.58) are expressed as ultimate failure criteria. In the design analysis, all stresses expressing the carrying capacity of the shell shall be divided by the applicable resistance factor (γmn), or safety factor (s), and the stresses caused by an external load shall be multiplied by γf. X2+ kxyXY+Y2+kxv|XV|+kyv|YV|+V2≤1 (3.57) where

(σx+σxb)2−(σx+σxb)σφ+σφ2+3(τt+τv)2≤1.2σy2 (3.58) Here: σx

=stress due to a centric compressive force and

σxu

=the corresponding resistance according to 3.2.1.2,

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σxb

=maximum compression stress due to a bending moment and

σxbu

=the corresponding resistance according to 3.2.1.3,

σφ

=circumferential stress due to external pressure and

σφu

=the corresponding resistance according to 3.2.1.5,

τt

=shear stress due to a torsional moment and

τtu

=the corresponding resistance according to 3.2.1.6,

τv

=shear stress due to a transverse force and

τvu

=the corresponding resistance according to 3.2.1.7,

σy

=yield stress.

Fig.3.16 Interaction curves for a) yielding,b) normal stresses in both directions 1 and 2, buckling and yielding, c) shear stress in combination with tension/compression.

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3.2.1.9 Ring stiffeners In Appendix B, formulas are presented for section forces, bending moments and displacements in circular rings subjected to elementary load cases. The formulas are based on the first order theory of elasticity. If compressive forces occur, the formulas are valid only if the buckling load (compare 3.2.1.5) is considerbly higher than the current compressive force, say by a factor of ten. The elementary cases can, with this restriction, be combined with any load case under consideration. The formulas have been compiled from Roark, 1954, and the handbook Bygg, Part T, 1983, and have been extended to better suit the present purpose. 3.2.2 Ring stiffened circular cylindrical shells Cylindrical shells under external pressure are in most cases designed with emphasis on buckling. Some types of thin shells, e.g. tanks which are primarily designed to carry internal pressure, may in certain cases be subjected to vacuum, and may thus be critical from a stability point of view. An increase of the wall thickness may sometimes be the most appropriate way to attain a sufficiently high degree of safety against buckling, but introduction of ring stiffeners can possibly provide a more economical solution of the problem. In the following paragraphs, recommendations are given for the design of ring stiffened cylindrical shells with respect to buckling under external pressure. It is possible to present methods of analysis which yield a very acccurate estimate of the theoretical resistance, but these will necessarily lead to rather extensive computional work. In this chapter, therefore, a simplified basis is given for analysis which agrees well with the directions given in current pressure vessel codes, e.g. ASME, BS, TKN. A more comprehensive theory for the buckling analysis of circular stiffened cylindrical shells is presented in ECCS, 1988. For loading in axial compression, it can be assumed that the shell segments between the rings will buckle as unstiffened shells. This case is thus treated according to 3.2.1.2. 3.2.2.1 Assumptions A cylindrical shell with ring stiffeners is defined according to Fig.3.17. A ring is described by the cross section area Ar and moment of inertia Ir. The ring spacing can vary along the shell and the same may apply to the shell thickness. The design analysis can be accomplished by various methods depending on the degree of stiffening provided by the rings. Where rings are included in order to increase the resistance with regard to buckling the ring stiffness will dominate the expression for the critical pressure, and the stiffeners can be designed according to the theory for rings under radial pressure. This method will be referred to as method No 1. In certain cases, the requirements with regard to safety against buckling of a vacuum vessel may be very nearly satisfied. The introduction of one or a few rings would fulfill the requirements established, and there is consequently a need to be able to design for this case. An extension of the theory for unstiffened shells according to 3.2.1.5 can be applied, providing a simple method to determine the stiffening required. The method of analysis, which gives somewhat conservative results, is denoted in the following as method No. 2.

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Fig. 3.17. Definition of the effective section of a stiffening ring. Ring stiffened cylinder shells are treated in several design codes and the recommendations given in this handbook are in agreement with these in various respects (compare BS 5500, ECCS, 1988, ASME Code Case No 284, DASt Ri 013, DIN 18800, TKN, CN1 and others).

Fig.3.18. Characteristic values of geometric deviations for ring stiffeners. A. Tolerances The shell plate between the rings is supposed to satisfy the tolerance requirements applied according to subsection 3.2.1.5 for an unstiffened cylindrical shell. The out-ofroundness of the rings is assumed to fulfil the demands in Table 3.2. Tolerances for

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angular deviations of the ring, deviations from the nominal junction line to the cylinder etc., are recommended as follows where the measures etc. are defined in Fig.3.18. (3.59) (3.60) u≤0.01r (3.61) B. Buckling modes A cylindrical shell may buckle in a number of different modes: – Local buckling of the rings including buckling of the flanges or the web. It is assumed that the rings have been designed such as to exclude this kind of instability. This can be achieved by choosing the dimensions according to ECCS, 1988. (3.62) (3.63) (3.64) – Local buckling of the shell plate between the rings. This type of buckling is treated according to subsection 3.2.1.5 assuming that the shell is hinged at the stiffeners. – Global buckling, characterized by simultaneous deformation of both the shell and the rings. This buckling mode will be treated in the following. C. Boundary conditions As for buckling of an unstiffened cylindrical shell, the present recommendation is based on the assumption that the radial deformations of the edges of the cylinder are prevented. In design according to method No 1, this condition will be automatically satisfied, since the rings at the ends also have to be designed according to 3.2.2.2. When using method No 2, the requirements given in eqn (3.27) must be considered. 3.2.2.2 Ring design analysis, method No 1 The cylindrical shells between the rings are designed according to subsection 3.2.1.5. In the case of global buckling, the rings are assumed to carry the entire load from the external pressure. The load acting on each separate ring is demonstrated in Fig.3.19. The theoretical buckling stress of the ring is obtained from the following expression provided buckling occurs within the elastic region:

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(3.65) This buckling case should be compared to buckling of a straight column for which design rules are available in several codes, e.g. ECCS, Eurocode 3 and BSK. The buckling curve can be approximated by the expression:

Fig.3.19. Design of ring stiffener against buckling due to external pressure. Definition of loading and effective width when determining Ar and Ir. (3.66) with (3.67) The design condition is: σu≤ωcσy (3.68) Introduction of eqn (3.65) in (3.67) and then entering the result into eqn (3.66), considering that σ=pr/t (this value is based on the assumption that the shell wall between the rings carrie its own load and that the rings will adopt the same stress as the shell wall. The assumption is conservative as long as the shells between the rings have not buckled).

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(3.69) The factor γ=γfγmγn can be replaced by a safety factor s which for pressure vessels should be of the order of 2 (The Swedish pressure vessel code requires s=2). Eqn (3.69) can alternatively be written as (γ→s=2): (3.70) Formula 8:18 in the Swedish Pressure Vessel Code (TKN 87) can be written in a similar form: (3.71) The coefficient 0.083 of eqn (3.70) is lower than the corresponding coefficient in TKN. On the other hand, eqn (3.70) also includes a reduction due to plasticity which should compensate for the difference in a normal case. Method No 1 is thus equivalent to the design method for rings in the current codes. 3.2.2.3 Buckling analysis, method No 2 Method No 1 gives an estimate of the resistance of the shell which is on the safe side, especially for shorter cylindrical shells where the buckling mode involves several waves in the circumferential direction. If it is assumed that the contribution by the rings to the bending stiffness is of the same order of magnitude as the inherent stiffness of the shell, the resistance can be determined by use of the method of analysis presented in subsection 3.2.1.5. It is important to state, however, that the conditions given below must be fulfilled in order to ensure a safe design. A. Model of analysis, constant thickness For a shell with constant thickness and two or more rings uniformly spaced along the meridian according to Fig.3.20, an effective wall thickness tef is defined as: (3.72) (3.73) For the method to be applicable tef shall not exceed 2.0t. The theoretical buckling stress is calculated according to subsection 3.2.1.5, but it is necessary to remember that the stress in the actual shell is higher than that of the model used. In the check on buckling in the plastic region the following relationships are, therefore, to be considered: The stress levels in the equivalent shell and the actual shell, respectively, are denoted by σeq and σact, where σact is the higher value.

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(3.74) In judging if buckling occurs elastically the following condition is applied kσeq≤σy/3 (3.75) where eqn (3.35) is valid. If the condition is not satisfied, buckling takes place in the plastic region and the normal design procedure according to subsection 3.2.1.5 is applicable with the additional information that the slenderness parameter is computed as: (3.76)

Fig.3.20. Cylindrical shell with constant wall thickness and equally spaced ring stiffeners. Analysis model according to method No 2. B. Model of analysis, shell with varying thickness A ring stiffened shell with varying thickness may be designed according to 3.2.1.5. Assume that the cylindrical shell in Fig.3.21 has a carrying capacity with respect to buckling from external pressure which is somewhat too low. The most efficient way to increase the buckling resistance should be to introduce a ring stiffener in the second shell section from the top, for which a model of analysis can be established according to

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Fig.3.21, model No 2. In analogy with the definition of an effective thickness, as introduced above, the following calculation steps may be defined: – Consider the ring, with its participating effective width of the shell, as a separate shell section. The equivalent thickness teq of this section is calculated as shown above: (3.77) – Compute the effective thickness t0 of the upper shell section as in 3.2.1.5. With notations as in the example of Fig.3.21 the following is obtained: (3.78) – Check that the stiffness of the ring does not exceed a value such that following thickness relation holds: t0eq≤tm In case t0eq>tm an approximate result may be obtained by setting t0eq=tm. The result will be on the safe side. – Check that the buckling resistance of the shell between the rings is satisfactory, using the design rules in subsection 3.2.1.5.

Fig.3.21. Thin-walled tank under external pressure. Model for analysis of ring stiffener. It should be observed that the calculation procedure was developed for a specific design case (tanks) and the effects of the introduction of ring stiffeners may sometimes lead to a

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smaller increase of the theoretical buckling resistance than may be expected. Method No 1 will, for such cases, provide a conservative result. In order to achieve an optimum effect of stiffening the use of numerical methods is recommended, e.g. the BOSOR computer program, see Chapter 6. 3.2.2.4 Sample cases A cylindrical wall subjected to external pressure and with dimensions according to Fig.3.22 is provided with ring stiffeners to increase the capacity with regard to buckling. Calculations of the theoretical buckling stress are performed by means of both methods and by BOSOR4 in order to verify the precision of the approximate methods. The operational pressure is 3.0 bar, or 0.30 MPa.

Fig.3.22. Circular cylindrical shell under external pressure. Design analysis of rings. The buckling stress of the unstiffened cylinder shell is obtained from eqn (3.33) (3.33)

corresponding to pel=0.385 MPa. Introducing a reduction factor αη=0.7 according to 3.2.1.5 gives: pu=0.385·0.7=0.27 MPa pall=pu/2=0.135 MPa<pact Ring stiffeners are thus required to be able to allow operation with an external pressure of 0.3 MPa on the cylinder. Method No 1: The method gives directly the requirements on the ring stiffness as stated above: Assume three rings with 1000 mm spacing:

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(3.70)

Try ring dimensions 10×60 mm:

=0.619·106 mm4<0.69·106 Since the calculated value is only marginally lower than that required, a 10×65 mm ring will provide adequate stiffness. Method No 2 Three light rings of flat rolled steel 10×30 mm according to Fig.3.22 give the following buckling pressure. The effective thickness of the stiffened wall and the buckling stress are calculated as follows: Ir=105800 mm4 (lef=125 as in the analysis with Method No 1) (3.72) tef=(12·488000/4000)1/3=11.4 mm (3.73) σel=65 MPa corresponding to pel=0.92 MPa (3.33) Analysis of the buckling pressure pel with the BOSOR4 program resulted in pel=0.38 MPa for the unstiffened and pel=1.18 MPa for the stiffened shell. The agreement is acceptable. For the stiffened shell the approximate method gives slightly conservative results. Comments: Method No 1 yields a conservative design of the rings. A certain overstrength of the rings allows that buckling of the shell wall between the rings can take place without the rings loosing their resistance. This is a design approach to be preferred in case a total collapse would lead to very serious consequences. Fig.3.23 shows a ring stiffened tank which failed due to external pressure. The rings were probably designed according to method No 1, and their resistance was sufficient to limit the buckling damage to the shell segments between the rings. The figure also shows a cylinder with two light rings. Here the failure occurred in a mode where the rings buckled together with the shell wall (global buckling).

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Fig.3.23. Ring stiffened tank which failed due to external pressure. Photo, Sunds AB. The rings are designed to carry the pressure alone. The second picture shows a cylindrical shell with light rings. Photo, The Aeronautical Research Institute of Sweden. 3.2.3 Stringer stiffened circular cylindrical shells Pure unstiffened cylindrical shells are often used in containment and pressure vessels and are mostly designed for internal pressure. In cases where buckling limits the resistance, the most weight economical design will probably be obtained with a thinwalled, stiffened shell. If the shell is loaded by an external pressure, rings are efficient and if the main load is an axial force, longitudinal stiffeners (in aerospace design called stringers) are to be preferred. An example of such stiffened shells is an aircraft fuselage which is built of a very thin shell provided with both rings (ribs) and stringers. Stiffened shells can be designed by means of the classical theory for anisotropic shells where the rigidity of the stiffeners is “smeared out” over the shell surface and the resulting theory is very similar to that of the unstiffened shell which has been presented above. The final result cannot be given in a closed form, however, and numerical methods are needed to determine the critical load. A theory for stiffened shells is given in ECCS, 1988, which is also available as a computer program. Below, some simplified design methods of analysis are presented which are easy to apply and provide reliable results somewhat on the safe side. 3.2.3.1 Assumptions A stringer stiffened cylinder shell is presented in Fig.3.24. It is assumed that the number of stiffeners is ns, and that they are evenly distributed around the circumference (constant

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spacing). If the shell is also provided with ring stiffeners, these are assumed in the following to be stiff enough to guarantee that buckling due to an axial load is limited to the panels between the rings.

Fig.3.24. Stringer stiffened cylindrical shell. In usual cases, stiffened cylindrical shells may buckle in a number of modes, and the design should be based on the following conditions: – Local buckling of ring or stringer cross section: With such occurrences, this may be analyzed according to Dubas and Gehri,1986, Eurocode 3 or BSK, 1987. By introducing limitations according to eqns (3.62–3.64) this kind of instability can be avoided. – Local buckling of shell sections between stiffeners: Under certain conditions this buckling mode may be accepted, provided the deviations from the nominal geomerty do not cause disturbances to the function of the vessel. – Buckling of the stiffened panels between two rings: This buckling mode is assumed to dominate for axial loading, if it is assumed that the rings are so stiff that they do not deform during the buckling process. A requirement of the ring stiffness is given in each separate case. – Global buckling implies that all the stiffeners participate in the buckling deformation. For instance, when failing under external pressure the rings will buckle and the stringers be deflected, and the failure mode may therefore be referred to as global buckling. The various buckling modes are demonstrated in Fig.3.25.

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In this chapter, recommendati ons are given for calculating the resistance of the shell based on simplified assumptions. This leads to simple expressions for the carrying capacity, the results are realistic and somewhat conservative. The more accurate theory according to ECCS, 1988, has been used to establish a number of design diagrams which are given in Appendix C for certain cases which are assumed to occur rather frequently in practical applications.

Fig. 3.25. Possible buckling modes for a ring and stringer stiffened cylindrical shell. As for unstiffened shells, both boundaries are required to be fixed against radial deformations, i.e. the circular cross section must be maintained during buckling. In the case of pure axial loading this is warranted if the boundaries are stiffened by rings of a moment of inertia as defined by eqn (3.4), where, however, t is replaced by an effective thickness tef, which corresponds to the higher effective bending stiffness of the stiffened shell: (3.79) Is is the moment of inertia of a stringer including an effective width bef of the shell wall according to Fig.3.26. Design of the end stiffeners and possible intermediate rings according to eqns (3.4) and (3.62–3.64) will guarantee that buckling takes place in the unstiffened shell between the rings. Thus, global buckling is avoided. For external pressure loading the formulas of eqn (3.27) are valid.

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Fig.3.26. Definition of an effective cross section of a stringer attached to a cylindrical shell wall. 3.2.3.2 Axial loading The compressive stress in the cylindrical shell is obtained from the expression: (3.80) where the effective thickness is given as: (3.81) (3.82) The shell is designed with attention to both local buckling of the shell wall between the stringers and buckling of the stiffened shell panels between the rings/supports. A. Local buckling of panels between longitudinal stiffeners The buckling stress of a plate which is bounded by stringers can be determined by the theory for plane plates. If the curvature of the plate is large, this approach will lead to an underestimation of the buckling load. In that case it is more relevant to use the formulas for a cylindrical shell subjected to axial loading according to 3.2.1.2. The reason why the shell theory is applicable is evident from the results in 1.3.1, where it was shown that the critical stress for a cylindrical shell is nearly independent of the number of waves in the circumferential direction. Consequently, the design against local buckling is carried out as follows:

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a) Compute the buckling stress according to the theory for plane plates. This can be done according to ECCS, 1988, or any other accepted design recommendation. The method given in Bygg K 18, 1985, is given here: (3.83) (3.84) (3.85) b) Compute the buckling stress and resistance of the panels according to the theory for cylindrical shells, 3.2.1.2. c) The highest of the two values of σxu determined according to a) and b) is used in the design against buckling.

Fig.3.27. Buckling stress of a plane plate and a cylindrical shell sector respectively as function of the b/t ratio, E=210000 MPa, σy=350 MPa. The theoretical buckling stress of a cylindrical panel is large compared to the buckling stress of a plane plate supported along four edges. See Fig.3.27.

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Fig.3.28. Carrying capacity of plane and cylindrical panels under axial compression. The circumstances are somewhat different with respect to the ultimate resistance as shown in Fig.3.28. For a plane plate, a postcritical region exists, and the resistance exceeds the buckling stress. This is due to the fact that the panel edges will continue carrying load after buckling has occurred. In a cylindrical shell the resistance is considerably lower than the classical buckling stress, as indicated above. The result is that only in the case when the r/t ratio is small, an increase of the resistance, above that corresponding to plate buckling,will be obtained. For example: r/t must be less than 500 in order to give a higher resistance for the cylindrical shape if b/t≤300. A second example: If b/t=400 and r/t=750 the theoretical buckling stress of an element with cylindrical shape is 40 times higher than that of a plane plate but the cylindrical shape still does not increase the resistance above that of the plane plate. Local buckling of the sheets between the stringer stiffeners does not necessarily mean total collapse of the shell. This appears e.g. from a series of experiments carried out by Groth, 1979, at the Aeronautical Research Institute of Sweden, FFA. Examples from this investigation are presented in Figs 3.29 and 3.30, where both the observed buckling modes and the measured axial displacements versus the load are shown. B. Buckling of stiffened panels between stiff ring Walker et al., 1980, proposed a method for calculation of the buckling load for a stiffened cylindrical shell based on the assumption that the resistance is the sum of the resistances of the stringers and the cylindrical shell determined separately. The expression can be written in the following way, which agrees with the formulation used in ECCS, 1988.

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(3.86)

The following symbols have been introduced: (3.87) (3.88) See also Esslinger, 1973. In the second term of eqn (3.86), the reduction factor η referring to buckling of the cylindrical shell is included. This factor is identical to η according to eqn (3.6). When computing η with that equation it is recommended to enter the effective thickness tef according to eqn (3.81). The resistance σxu corresponding to global buckling is calculated in analogy with 3.2.1.2, but the neduction factor ηs is defined as: (3.89)

In the interval

linear interpolation may be used.

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Fig.3.29. Various buckling modes in stiffened cylindrical shells subjected to axial load, Groth, 1979.

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Fig.3.30. Load-deformation relationships for the stiffened cylindrical shells shown in Fig.3.29. It should be noted that the calculation method presented above does not introduce any restrictions on the number of stiffeners, the length of the shell, or degree of stiffening. In spite of this it gives very good results in most practical applications as shown e.g. by Ellinas et al., 1984. It is clear, however, that eqn (3.86) provides an underestimation of the resistance for longer shells, where the natural buckling mode contains several half waves in axial direction, while the model assumes that buckling takes place in one half wave (Euler case 2). In certain cases the more accurate theory given in ECCS, 1988, will form a better basis for the design of the shell. An additional alternative may be to introduce ring stiffeners with suitable spacing and thus ensure that the stringers will buckle in a shorter wavelength than that corresponding to the natural mode.

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The design of the rings can be made in analogy with the recommendations for the edge stiffeners, see 3.2.1.2B. 3.2.3.3 Bending moment A. Calculation of resistance The resistance of a stiffened shell is calculated in the same way as for axial loading. It is recommended here, though, that the reduction factor η applicable for axial load is used and not the higher value given for an unstiffened shell under pure bending. When calculating the stress level in the shell caused by the bending moment the following formula is valid provided the spacing between the stiffeners is sufficiently close: (3.90) with tef according to eqn (3.81). If the number of stringers is small (3 to 5) a more detailed stress analysis is required. B. Need for ring stiffeners Especially for longer shells, the Brazier effect which was investigated for unstiffened shells in subsection 3.2.1.3, may have a great influence on the resistance. For a stringer stiffened shell this effect can become considerble since the bending stiffness of the shell is much lower in the circumferential than in the longitudinal direction. The conditions indicating the need for ring stiffeners according to 3.2.1.3 are applicable in the present case too, since the bending reflects the higher load level in the stiffened shell. The spacing l between the rings is recommended to satisfy: l≤2r (3.91) 3.2.3.4 External pressure Stringers are used primarily to increase the resistance under axial loading and they contribute to a rather limited degree to the buckling resistance with regard to external pressure. This is, in particular, true if the number of stiffeners is small and the shell is relatively long. It is thus reasonable to design the shell against buckling due to external pressure according to 3.2.1.5 without regard to the stringers The primary effect of the stringers in the present load case is to impose buckling into a mode corresponding to a higher critical load according to Fig.3.31. If the stringer spacing is very close, the buckling mode and the corresponding critical pressure will be influenced as demonstrated in the figure.

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Fig.3.31. Buckling modes of unstiffened and stringer stiffened shells under external pressure. Additional methods to determine the effect of stiffeners will be discussed in Chapters 6 and 8. 3.3 Conical shells 3.3.1 Unstiffened shells 3.3.1.1 Assumptions A conical shell has a middle surface in the form of a complete or truncated axisymmetric cone and is often included in structures composed of different shell elements, e.g. as transition section between cylinders. As in the case of cylindrical shells, the following recommendations are based on the condition that the edges of the shell are stiffened in order to maintain the circular form. The knees at the connections between cones and cylinders may, under certain circumstances, supply sufficient stiffening, see section 3.7, while in other cases extra ring stiffeners are required. The following design rules are based to a large extent on the rules given in section 3.2 Cylindrical shells. Only the additional requirements are given which are needed in order to make the rules of section 3.2 applicable for conical shells. The radius of the cylinder is replaced by the radius of curvature of the conical surface in a plane perpendicular to the meridian. This radius of curvature varies along the cone. For the various load cases the specific radius is defined which is to be used in the analysis. When applying the tolerance rules for cylindrical shells to cones, the variation of the radius of curvature should be considered. This implies that the acceptable deviations from the nominal geometry will also vary along the shell. The rules given in the following are valid for shells with a half apex angle a not exceeding 75°. 3.3.1.2 Uniform axial compression This subsection treats the case where the cone is loaded by a centric axial compressive force F, according to Fig.3.32, causing compressive stresses which vary along the shell as:

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(3.92) If the load is applied according to Fig.3.32 and is carried by membrane stresses according to eqn (3.92), the upper boundary of the shell must have a stiffening ring which is able to carry the horizontal load (3.93)

Fig.3.32. Conical shell loaded by a centric axial compressive force. The stiffness required for the ring beam can be calculated by eqn (3.27) where pl is replaced by nr. The area and moment of inertia of the cross section of the ring can be determined with the effective width from the attached shell wall included in the same manner as for a cylindrical shell, Fig. 3.8. At the bottom end of the shell the conditions are similar, but the ring will be loaded in tension. It should be designed such that the cross section area is sufficient to sustain the . stresses caused by the acting tensile force The solution of the classical buckling problem was given by Seide, 1956. In the evaluation of the resistance of the shell, the state of stress should be calculated at the distance se from the shell boundaries according to: (3.94)

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The analysis is carried out as for a cylindrical shell with the radius (3.95) with notations according to Fig. 3.32. Normally, both ends of the shell have to be checked with respect to buckling. In order to simplify the analysis, se may be assumed, conservatively, to be zero. For a relatively thick shell the upper end may be critical since it is subjected to higher stresses, while for a thin shell, the lower end may be decisive since the resistance decreases faster than the stress caused by an external load. If the shell thickness varies, the carrying capacity has to be checked at each location where the thickness is changed. Normally it is satisfactory to make the check for the part of the shell wall with the smaller thickness. 3.3.1.3 Bending moment In the same manner as for a centric compressive force, the bending moment is assumed to be applied through forces parallel to the axis of the cone, see Fig.3.33 (Weingarten-Seide, 1968). In analogy with the rules given in subsection 3.2.1.3, the highest bending stress is determined as (3.96)

Fig.3.33. Conical shell loaded by a bending moment M. Further notations are given in Fig.3.32. The ring stiffener at the upper edge of the shell is subjected to radial forces according to Fig.3.34. The distributed loads yield a horizontal resultant force which cannot be kept in

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equilibrium by internal forces in the ring only but is balanced by shear stresses in the cone. This leads to a pure membrane state of stress with normal stresses in the meridional direction and shear stresses 90 degrees out of phase with the normal forces. The load case for the ring is thus identical to Case No 9 of Appendix B.

Fig.3.34. Horizontal load on upper ring stiffener for bending moment loading. At a point on the shell surface defined by the angle φ, the stresses are (3.97) (3.98) The maxima of σsb and τ will occur at φ=π/2 and 0 respectively. They shall not exceed the design resistance

according to 3.2.1.2 and

respectively, according to 3.2.1.7.

3.3.1.4 Internal pressure The influence of internal pressure is in principle the same as for a cylindrical shell. The problem was treated theoretically by Seide, 1962a, see also Berkowits et al., 1967, and Schulz, 1981. Experimental results are found in Weingarten et al., 1965b. The longitudinal stresses due to the internal pressure will vary along the shell and will be dependent on how the resultant axial force is carried. For the case given in Fig.3.35, where the cone is provided with end closures, the shell will be entirely subjected to tension: (3.99) (3.100)

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The resistance is checked according to 3.2.1.8.

Fig.3.35. Conical shell under internal pressure.

Fig.3.36. Conical shell loaded by internal hydrostatic pressure. Fig.3.36 shows a case where the cone is subjected to internal hydrostatic pressure and the shell supports the load through axial compressive stresses. The circumferential stress is

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computed using eqn (3.100) for the current pressure pi=γ(h+x) where γ is the volume weight of the liquid. The compression stress in the meridional direction becomes: (3.101) 3.3.1.5 External pressure Only the case of a constant external pressure pe and a constant wall thickness is treated. Furthermore, it is assumed that the shell is provided with end closures. The load case is the same as for internal pressure, see 3.3.1.4, but the stresses have the opposite sign. A solution of the classical buckling load was first given by Pflüger, 1937, later on by Niordson, 1947. The following calculation procedure is based on solutions by Seide, 1958 and 1960. The analysis is performed as for an equivalent cylindrical shell with: (see Fig.3.32) (3.102) (3.103) The calculation gives a value of σelr according to eqn (3.35). This value is transformed to refer to the stress at the larger end of the shell through multiplication by the factor: (3.104) before the slenderness λs of the shell is determined. The resistance computed from ωsσy shall then be compared to the stresses at the larger end, i.e. (3.105) 3.3.1.6 Torsional moment The method of analysis is based on a solution by Seide, 1962b. The rules for cylindrical shells are used noting that the highest shear stresses will be found at the small end of the cone when the shell is loaded by a constant torque. The buckling stress τel is computed from eqn.(3.47) using the radius: (3.106) The buckling stress is then referred to the radius at the small end by the expression (3.107)

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This stress is used for the subsequent calculation of the slenderness parameter λs and the resistance τu according to subsection 3.2.1.6. The resistance is finally compared to the current shear stress at the small end. 3.3.1.7 Shear force If the cone is subjected to a constant shear force, the small end is critical with respect to buckling and the rules in 3.3.1.6 can be applied. Eqn (3.107) is replaced by a linear transformation: (3.108) For a varying shear force, a number of sections will have to be checked. The calculation is, in principle, performed as for a constant shear force, but r1 is replaced by the current radius in the transformation according to eqn (3.108) and current stresses are computed for the section considered. 3.4 Spherical shells 3.4.1 Unstiffened shells 3.4.1.1 General requirements A. Geometry and boundary conditions The shell is assumed to have an axisymmetric geometry, i.e. to form a full spherical dome or cap. A complete sphere supported by a ring at its equator is considered to be composed of two spherical segments. The geometry is defined in Fig.3.37.

Fig.3.37. Geometry and coordinate system for a spherical shell.

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The basic formulas given in 3.4.1.2 are valid when the displacement components at the boundary in the meridional and radial directions u0=w0=0. The boundary is assumed to be simply supported. Clamping, i.e. θψ0=0, usually gives a moderate increase of the buckling load (10–20 per cent). In some cases where the buckling load is low, the effect may, however, be higher. If the edge displacement perpendicular to the axis of revolution, δho=u0cosψ0+w0sinψ0, is not completely prevented but depends on the stiffness of the connecting boundary structure, e.g. a ring beam, a toroidal transition section or a cylindrical shell, see Fig.3.38, the buckling resistance is reduced. In the case where the displacement δho is completely free, the reduction may be considerable, especially for a shell with a steep slope ψ0 at the boundary. Requirements on the cross section area or stiffness of the boundary structure as well as reduction factors are provided in 3.4.1.3. The loading is assumed to be a normal external pressure p. Except for local disturbances close to the boundary a uniform compressive membrane stress is obtained. This stress is the same in all directions of the shell: (3.109)

Fig.3.38. Alternative boundary structures of a spherical shell. In subsections 3.4.1.2 and 3.4.1.3 the pressure p is assumed to be constant over the entire shell surface, while 3.4.1.4 gives information on the buckling resistance when only half of the dome is subjected to a uniformly distributed pressure. If the load is not perpendicular to the shell surface, as for snow loading, the compressive circumferential stress σφ may be less than the meridional stress and may possibly become tensile. An estimate of the increase of the buckling resistance caused by variation of the stress ratio can be made according to Odland, 1981. For a normal pressure p varying slowly and continuously the formulas for constant pressure will give conservative results if the membrane stress is computed from eqn (3.109) with p=pmax. Caution is recommended, however, for domes with large boundary angles subjected to both pressure and suction due to wind, if the wind loading is a dominating action. Resonance between wind gusts and the oscillations of the shell may excite large deformations which may cause a critical reduction of the buckling resistance. A vibration analysis has to be performed for shells with low eigenfrequencies.

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A concentrated radial load interacting with external pressure reduces the buckling strength as indicated in 3.4.1.5. See also 4.1.2. The thickness ratio of the shell is limited to r/t<3000, while the geometry parameter (3.110) is presumed to vary within the interval 7<β<50. Shells with β<7, which are shallow and rather thick, have not been included, while relevant test results from shells with β>50 are not available. The central angle 2ψ0 is to be given in radians, see Fig.3.37. B. Shape deviations Spherical shells subjected to compressive stresses are very sensitive to deviations from their nominal geometric shape. Such deviations shall satisfy the requirements in Table 3.3, where wr denotes the largest deviation measured.

Table 3.3. Tolerance classes and reduction factors for spherical shells. Tolerance class

Max deviation wr

α

1

t/2

1

2

t

0.75

In the interval t/2<wr
Fig.3.39. Measurement of shape deviations in a spherical shell.

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Fig.3.40. Definition of a shape deviation across a welded joint. In cases 1 and 2 the length of the template shall be: (3.111) but in case 1 less than 95 per cent of the spacing between circumferential welds and meridional welds respectively at the measuring points, and in case 2 at least 30t. Greater shape deviations than those given for tolerance class 2, but not more than twice as great, may be considered by reducing the critical stress according to 3.4.1.2. 3.4.1.2 Uniformly distributed external pressure, simply supported edge According to linear (classical) theory for elastic shells with a perfect spherical shape, buckling will occur at the pressure (Zoelly, 1915) (3.112) corresponding to the stress level (3.113) where E=Young’s modulus of elasticity r=radius of middle surface t=shell thickness v=Poisson’s ratio The theoretical buckling stress of a spherical shell thus has exactly the same magnitude as the buckling stress of a cylindrical shell loaded in axial compression, eqn (3.3).

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The classical buckling formula almost always overestimates the resistance considerably even for shells produced with very high precision. This is obvious from Fig.3.41, where the ratio between the experimentally determined buckling stress σexp and the classical value σel has been plotted versus the r/t ratio from a number of investigations on shells made of steel or aluminum alloys. The σexp/σel ratio decreases with increasing slenderness r/t. It has been verified experimentally that the decrease continues at least until r/t=7000, Thurston-Penning, 1966. This divergence from the classical theory depends primarily on the imperfections which arise in the production in the form of initial buckles and residual stresses. A review of available test results shows, however, that the slope of the shell at the boundary (half the central angle of the dome) also affects the magnitude of the reduction, Eggwertz-Samuelson, 1990.

Fig.3.41. Test results σexp/σel versus r/t for steel and aluminum shells. Elastic buckling only. Test results from Klöppel-Jungbluth, 1953, Homewood et al., 1961, Kiernan-Nishida, 1966, Krenzke, 1962, Krenzke-Kiernan, 1963, Costello, 1970a. In Fig.3.42 the σexp/σel ratio from the buckling tests mentioned above have been plotted as a function of the parameter β according to eqn (3.110), which includes the slenderness ratio r/t as well as the slope angle ψ0. Furthermore, two curves have been drawn in the diagram. The upper curve, defined by:

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(3.114) can be interpreted as a lower bound of test results for specimens with a manufacturing accuracy according to tolerance class 1. The lower curve corresponding to the relationship (3.115) forms a lower bound for test specimens which have been estimated to belong to tolerance class 2 see Klönnel-Junghluth, 1953, and Homewood et al, 1961

Fig.3.42. Test results versus the geometry parameter β. In Figs.3.41 and 3.42 only test values from elastic buckling have been included, that is where αησel<σy/3 or σ0.2/3, respectively. Under elastic conditions the following expression gives a value of the critical buckling stress which is on the safe side. (3.116) The reduction factor α accounts for the tolerance class ( manufacturing quality) and is presented in Table 3.3. For larger imperfections than those indicated tolerance class2, where α=α2=0.75, α may be obtained from the following expression

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(3.117) The reduction factor for shells with greater shape deviations than for tolerance class 2 is mainly intended for a resistance check of manufactured shells, not for design purposes. The resistance is calculated according to eqn (3.8) for elastic conditions and eqns (3.9) and (3.10) if buckling occurs in the plastic region. For spherical shells an extra reduction factor of 0.75, compare ECCS, 1988, is applied considering the extreme sensitivity to imperfections and to the great scatter of the test results.

Fig.3.43. Elastic and plastic buckling of spherical shells. Comparison between theory and test results from Klöppel-Jungbluth, 1953, Homewood et al., 1961, Kiernan-Nishida, 1966, Costello-Nishida, 1967, Costello 1970a and 1970b, Krenzke, 1962, Krenzke-Kiernan, 1963, and Parmerter, 1963. Fig.3.43 shows critical buckling stresses determined from experiments, σexp/σy, plotted versus the slenderness parameter λs. When computing λs, the value of σelr was calculated by use of the reduction factor η according to eqn (3.114) for all test series but two, Klöppel-Jungbluth, 1953, and Homewood et al, 1961, where α2 was introduced according to eqn (3.115), see also Eggwertz-Samuelson, 1990.

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Eriksson-Älgestig, 1988, performed a statistical evaluation of the stability properties of spherical caps. The results are included in Figs 3.41–3.44. 3.4.1.3 Requirements on edge rings If the radial deformation of the edge of the cap is not prevented, a severe disturbance of the membrane state of stress develops within a narrow band adjacent to the edge, which will displace outwards (if ψ0<90°). At the edge, a considerable tensile stress will develop in the circumferential direction and high bending stresses in the meridional direction. This implies that the yield stress of the material may be reached, or snap through may occur at an external pressure much lower than that predicted according to 3.4.1.2. In a shallow shell the collapse generally occurs as an axisymmetric snap through. Shells with a large edge angle, on the other hand, will buckle in two complete waves along the edge which is ovalized. Between these extremes buckling may develop gradually from one or more nonsymmetric initial buckles. If the initial deformation is situated close to the edge, the buckling process will be strongly accelerated by the edge deformation. Wang et al., 1966, discovered in an experimental investigation of spherical caps made of plastic material (PVC) that the resistance, if the edge is free to move in the radial direction, is reduced to about one third of the resistance of a shell with hinged edges. These shells had an edge angle ψ0=30°, Wang, 1966, also tested the same caps provided with ring beams with various cross section areas Ar at the boundary. The ratio between the collapse pressure and the classical buckling pressure showed a good correlation with the parameter where Er is Young’s modulus of elasticity for the ring, while E, r and t refer to the shell. Bushnell, 1967b, used the BOSOR computer program to carry out a large number of nonlinear, elastic buckling calculations for spherical caps for various r/t ratios, edge angles ψ0 and edge ring areas Ar. His results agreed well with the test results just mentioned. The calculations were performed for edge rings with a square cross section of the same material as the shell, i.e. Er=E. Bushnell introduced the nondimensional parameter: (3.118) The reduction of the buckling resistance due to the edge deformation was shown to vary mainly with A*, to a lesser extent with the thickness ratio r/t and the edge angle ψ0. It should be mentioned that the calculations were made for spherical caps without initial geometric imperfections. For a shell with an edge angle within the interval 15°<ψ0<60° an edge ring of moderate size could provide a strength almost equivalent to that of a simply supported shell. Shallow and deep caps require an edge ring with a large cross section area or moment of inertia. Bushnell, 1968, also studied ring buckling emanating from the boundary with various numerical methods and was able to verify the great influence of the parameter A* within the interval 60°<ψ0<180° with a minimum of the buckling resistance at ψ0≈100°. The calculations were, however, only made for r/t=100 and an edge ring of square cross section.

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Based on Bushnell’s computer analyses, additional calculations with the BOSOR program performed by J.Cwifeld at the Swedish Plant Inspectorate, SA, and on Wang’s tests, the following recommendations are given: 1. Collapse occurs as a symmetrical snap through. This case is not treated in this handbook, see further Yamamoto-Kokubo, 1984. 2. Provided A*>10 the buckling resistance can be determined by use of the equations in 3.4.1.2. For smaller values of A*, the reduction factor should be reduced to one third of the value given for a simply supported shell. The result is conservative. 3. The buckling resistance can be obtained from the same equations as under point 2, if the edge is provided with a ring beam having a moment of inertia about an axis through the centroid of the section and parallel to the axis of revolution of the shell, which is If this condition is not satisfied a nonlinear, axisymmetric buckling analysis should be carried out to determine the reduction in strength compared to the hinged edge. 4. If a ring beam is introduced at the equator, ψ0=90°, the recommendations of point 3 will be valid. Otherwise a nonlinear buckling analysis must be carried out. 5. If the spherical shell does not have a discrete edge ring but is attached to a cylindrical shell, either directly or through a transition shell of toroidal shape, the edge displacement is usually not sufficiently restricted to allow the assumption of a hinged edge as in 3.4.1.2. Klöppel-Roos, 1956, established a reduction formula for the buckling resistance. Test results for torispherical end closures in cylindrical vessels were presented by Jones, 1962, Bart, 1964, and Galletly et al., 1987. Klöppel-Roos, 1956, used spherical caps/cylinders without a toroidal transition, compare Fig.3.38. Plotting the results from elastic buckling versus the parameter β, a large scatter is obtained which is caused mainly by a variation of the ratio between the knuckle radius rk of the toroidal shell and the radius r of the spherical shell. A lower bound for vessel heads with small rk/r ratios≈5–10 per cent can be described by the relationship (3.119) Introducing this reduction factor in the parameter λs all test points available were plotted in Fig.3.44. Among the test results given by Klöppel-Roos, 1956, those values have been left out which emanate from specimens where the wall thickness of the cylindrical shell is smaller than that of the spherical shell. For spherical vessel heads attached to cylindrical shells (thickness not less than that of the spherical shell) and torispherical shells with a small knuckle radius, this stiffening of

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the edge seems to correspond to A*≈1. In section 3.5 the resistance is given for some standardized end closures with comparatively large knuckle radii rk.

Fig.3.44. Buckling of spherical vessel heads of steel. Comparison between test results and eqn (3.9). Reduction factor for elastic buckling according to eqn (3.119). 3.4.1.4 One-sided load When only one half of a spherical dome is subjected to a uniformly distributed external pressure, a disturbance of the state of membrane stresses is introduced at the boundary between the loaded and the unloaded sectors. Along the load boundary meridians, bending stresses and deformations perpendicular to the shell surface will appear. These local disturbances may be expected to reduce the buckling resistance if they are more serious than those which normally occur due to geometric imperfections and residual stresses. Klöppel-Roos, 1956, studied the buckling resistance of 19 unstiffened spherical caps subjected to external pressure on half of the surface as described above. Test specimens of two different combinations of radius and edge angle were used in the investigation: r=520 mm, ψ0=22.6° and r=250 mm, ψ0=53.1°, respectively. The radius to thickness ratio r/t varied between 300 and 1200. A formula for calculation of the critical buckling pressure of an elastic shell was proposed. It corresponds to the following reduction factor for one-sided loads:

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(3.120) where ψ0 is expressed in degrees. According to this formula the r/t ratio does not influence the magnitude of the reduction factor, which varies with ψ0 only. KlöppelJungbluth, 1953, tested specimens of the same geometry with pressure over the whole cap surface. A comparison between the test results shows that for the shallow shells the buckling strength is higher for one-sided loads when r/t>650. For smaller thickness ratios the one-sided load is critical, however. The corresponding limit for shells with a greater edge angle seems to be at r/t=1400. When r/t=1000 the buckling resistance is some 10 per cent lower for one-sided load, while at r/t=500 the difference is 20 per cent. The experimental results quoted refer to a shell cap where the edge displacement is almost completely prevented. Since deformation of the edge ring results in a reduction of the buckling resistance when the complete shell surface is loaded, it may be expected that the one-sided load will usually not be critical if the edge ring is weak. This conclusion is verified by tests which Klöppel-Roos, 1956, performed on shell caps with the edge welded to a cylindrical shell with a thickness which was varied in the tests. Other distributions over a partly loaded spherical cap may result in lower buckling stresses than for the one-sided load. 3.4.1.5 Concentrated loads A. Single concentrated load If large concentrated loads are applied on a thinwalled spherical shell, stiffeners should be introduced in the meridional or ring direction, or both, in order to distribute the load over a larger part of the shell surface. A point load on an unstiffened shell results primarily in a rapidly growing, nonlinear radial deformation which can become of an order of several times the thickness. In metal shells, the yield limit is reached at the loading point at a fairly low load level. When large plastic deformations have developed, a local yield failure can occur. A buckling collapse may also take place, especially for very shallow caps. The influence of a concentrated load at the apex of a shallow cap with a hinged or clamped edge has been studied theoretically by several authors, see for instance Mescall, 1965, and Bushnell, 1967a. Loo and Evan-Iwanowski, 1964 and 1966, carried out experimental investigations. Wang-Roberts, 1971, treated plastic deformations for aluminum shells both theoretically and experimentally. For an axially supported shell, implying that the edge is free to rotate and to move in the radial direction, a symmetrical snap through occurs at a critical value of the nondimensional point load P*; (3.121) For an elastic material the critical value, according to Mescall, 1965, may be assumed to be a function of the shallowness parameter κ

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(3.122) where (3.123) This relationship is valid within the interval 16<κ2<100 Ellinas et al., 1984, determined the following conservative lower bound based on the test results available (3.124) The experimental results cover the interval 5<κ<15. For plastic collapse, a reasonable estimate is given by Ellinas et al., 1984 (3.125) Shallow spherical caps with clamped edges do not buckle. According to Bushnell, 1967a, the boundary conditions do not influence the buckling resistance when κ≥10. For thickness ratios r/t≥500, nonsymmetric bifurcation buckling occurs at

≈6. For thicker

shells, the critical value of decreases gradually and is some 10 per cent lower at r/t=15. The ultimate strength is not exhausted at this value, however, and the load can be increased further. B. Interaction of concentrated loads and external pressure Interaction between external uniform pressure and a concentrated load at the apex of a shallow spherical dome with a clamped edge has been studied theoretically and experimentally by Loo and Evan-Iwanowski, 1964 and 1966. The test specimens were made of a plastic material, vinyl polyethylene, allowing large elastic deformations. An illustration of the interaction is given in Fig.3.45, which shows test results from a cap with r/t=400 and κ=9.1. Each curve presents the relationship between the nondimensional point load P* and the measured radial deflection at the apex related to the shell thickness, w(0)/t, at a uniform pressure over the shell surface which was varied from an internal pressure p=−0.0017 MPa to an external pressure p=0.0136 MPa. To reach buckling failure, the external pressure must exceed 0.0034 MPa. It was found for a cap with a fixed value of the parameter κ that interaction between a point load and external pressure resulted in buckling at about the same load combinations irrespective of

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which load was first applied. Loo and Evan-Iwanowski, 1966, were therefore able to design interaction diagrams for different κ values.

Fig.3.45. Interaction between external pressure and a point load at the apex of a spherical cap. The nondimensional point load P* is plotted versus the radial deflection at the apex divided by the shell thickness, for various pressures over the shell surface. Ellinas et al., 1984, have presented a simplified interaction diagram, Fig.3.46, where the critical nondimensional point load is given as a function of the pressure p related to the critical elastic buckling pressure ηpel both for shallow caps with a clamped edge and for caps with a steeper edge.The curves are not based on test results of adequate numbers and should not be used for design purposes. Reliable values of load combinations can presently only be obtained from advanced numerical calculations or model tests. For smaller point loads, however, the design recommendations given in section 4.1.2 may be applied.

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Fig.3.46. Simplified interaction diagram. The nondimensional critical point load given as a function of the ratio between the applied external and the critical elastic buckling pressures, 3.4.1.6 Example No 1 A spherical dome, see Fig 3.47, shall be designed according to BSK, safety class 3, considering a permanent load mainly due to the dead weight of the steel shell.

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Fig.3.47. Example No 1. Shell geometry.

Membrane stresses:

At the apex, ψ=0°

At the edge, ψ=45°:

The meridional stress at the edge is governing the buckling behavior:

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Tolerance class 1:

σelr=ησel=0.0979·64.6=6.32 MPa Steel SS1312 with σy=220 MPa: σelr<σy/3 σu=0.75·6.32=4.74 MPa γn=1.2; γm=1.0

For a dominating permanent load: γf=1.1 γfσψ=−1.1·3.56=−3.92 MPa<σd If the horizontal stress component at the edge is to be carried by a ring, the tensile ring force becomes:

Assuming steel SS 1312 with σy=210 MPa (t>0.04 m) the cross section area becomes: Ar≥5.56·1.1·1.2/210=0.0349 m2 According to 3.4.1.3, the area requirement is:

By studying Bushnell, 1967b, it may be concluded that for the dome considered a sufficient magnitude will be A*≥6, which gives Ar=0.424 m2.

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The box section shown has the area Ar=2·(3.5+0.75)·0.05=0.425 m2. The critical value of a point load is obtained as

Plastic collapse:

According to Fig 3.46, buckling of a shell, which is simultaneously subjected to an external pressure giving σ=σelr/2 will occur already at P*=0.5, i.e. Pcr=0.5Et3/r=0.09 MN=90 kN. Point loads greater than 90 kN should be checked by nonlinear numerical analysis. It may also be possible to reduce the shell thickness on the basis of such calculations provided the manufacturing tolerances are fulfilled. It should be pointed out that the weight of the steel dome can be strongly reduced by introducing a large number of stiffening frames in the meridional direction and 2–3 horizontal rings. The wall thickness of the stiffened shell could thus be reduced to approximately 10 mm. Such a design would, however, require an advanced numerical stability analysis where initial buckles of a realistic magnitude are considered. 3.4.1.7 Example No 2 The bottom dome in an open, steel water tank according to Fig.3.48, is to be designed against buckling. above the apex of the dome. The partial coefficient The water level is (see 2.4.5)

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Fig.3.48. Spherical bottom in a water tank Meridional forces

At the apex,

At the edge

Ring forces:

124

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If the shell thickness t=0.02 m the maximum compressive stress becomes r/t=10/0.02=500

Tolerance class 1 requires: α=1.0 σelr=η σel=55.5 MPa<σy/3 σu=0.75·55.5=41.6 MPa Safety class 3:

The dead weight of the steel shell gives an additional contribution to the maximum compressive stress of about 1 per cent.

The method of analysis presupposes an edge ring of the size A*≥10, giving the cross section area:

Examine if omission of the ring beam may result in material saving: Increase the wall thickness to t=0.03 m.

β=18.7 According to 3.4.1.3 the reduction factor η is one third of the value given by eqn (3.114).

σelr=0.0865 378=32.7 MPa<σy/3 σu=0.75·32.7=24.5 MPa

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Maximum compressive stress: σφ=−0.601/0.03=−20.0 MPa γf|σφ|=1.150.20.0=23 MPa>σd The design without an edge ring is estimated to cause an increase of the shell thickness to 32 mm, i.e. ∆t=0.012 m. The surface area of the dome: AD=2πr2(1−cosψ0)=84.2 m2 Increase of steel volume of the dome: ∆V=0.012·84.2=1.01 m3. The ring area required is Ar=0.0894 m2, if the shell thickness is 0.02 m. Taking the effective widths of the cylindrical shell and the spherical shell into account the ring area may be reduced to Ar=0.08 m2. The volume of the ring beam will then become: Vr=2πrψ0Ar=2·π·5·0.08=2.5 m3 i.e. more than twice the saving in the dome. It is, consequently, not economical with respect to material consumption to introduce an edge ring in order to prevent collapse due to buckling. It should be noted, however, that the ring beam will reduce the maximum bending stresses at the edge, as can be shown by a simple hand calculation. If the water level varies considerbly and the number of load cycles during the service life is likely to exceed 1000, the bending stresses should be accurately evaluated and a fatigue analysis be performed. The final result may show that a sturdy ring beam will yield the lowest total cost. 3.5 Pressure vessel heads—torispherical, ellipsoidal and hemispherical 3.5.1 Introduction Convex heads are used, for instance, as end closures in most cylindrical and conical vessels such as pressure vessels, vacuum vessels and small tanks. They are also frequent as end pieces in pipes, steam headers, nozzles etc. Large horizontal steam accumulators and vertical digesters are often designed with hemispherical ends. Some types of heads are standardized in various standards, comprising deep, medium and shallow shells.Typical heads of torispherical shape, see Fig.3.49, consist of a spherical cap and a toroidal knuckle part which ends in a short cylindrical wall section.

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Fig.3.49. Torispherical and ellipsoidal vessel heads respectively. It is a characteristic property of torispherical and most ellipsoidal heads that compressive stresses may develop both for internal and external pressure. When subjected to internal pressure, compressive stresses appear in the circumferential direction in the knuckle, in . For external pressure, the cap is subjected to ellipsoidal heads, though, only if compressive stresses both in the radial and the tangential directions. Buckling may, therefore, result from both internal and external pressure acting on shallow pressure heads. A typical stress distribution and a photo of a head which buckled due to internal pressure are shown in Fig.3.50. The Swedish Pressure Vessel Code, TKN, 1987, presents design rules for dished and hemispherical vessel heads including both internal and external pressure. Similar design recommendations are given in BS 5500 and ECCS, 1988.

Fig.3.50. Typical stress distribution in a vessel head subjected to internal pressure, and buckling mode according to test. Photo AGA-CRYO.

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3.5.2 Torispherical and ellipsoidal vessel heads 3.5.2.1 Uniform internal pressure A. Introduction Torispherical and ellipsoidal heads manufactured of metallic materials and subjected to internal pressure may either buckle or develop plastic collapse. In the plasticization the displacements occur axisymmetrically in one or several plastic zones. Buckling, on the other hand, is a local phenomenon with buckles appearing in the knuckle region of the head in a pattern with short waves in the circumferential direction, see Fig.3.50. Which of these phenomena that will take place depends on many factors such as the ratios between various dimensions of the head (rc/t, rk/rc etc.) and properties of the material, e.g. the yield stress σy and Young’s modulus of elasticity E. It has been reported in the literature that collapse of a steel head usually happens as yield failure for small values of the radius to thickness ratio, rc/t<150, and by elastic buckling for large values of the same ratio, rc/t>750. In an intermediate range, 150
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expression which limits the mean compressive stresses in the cap the following formulas are obtained: (3.126)

(3.127)

(3.128) Eqn (3.126) refers to local elastoplastic buckling and eqn (3.127) to axisymmetric plastic collapse. Eqns (3.126)–(3.128) can be used to estimate the resistance of torispherical heads which satisfy the following restrictions: (1) The material is assumed to be steel with properties in the range, (2) σy≤500 N/mm2, (3)

,

(4) (5) (6) E≈200000 N/mm2 For ellipsoidal heads subjected to internal pressure, the design rules given above should be applicable, provided the different parameters are interpreted as follows. The results will be on the safe side.

rc=a (3.129)

For convex heads of steel, aluminum and copper which are more thickwalled than stated above, rc/t<167, experience from operation and pressure tests has shown that an adequate safety is obtained when applying the design rules given in the various pressure vessel codes. The formulas of current pressure vessel codes cannot always be used, however, for a direct estimate of the resistance (ultimate limit load) since the shape factors given are often independent of the thickness within the range covered by the codes.

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For thinner torispherical and ellipsoidal heads of steel, rc/t>750, and for convex heades of other materials no simple mathematical expressions can be provided for the resistance at the present time. Such heads should be analyzed in each specific case according to Chapter 6, Stability analysis with numerical methods, or 2.7, Design by testing. D. Tolerances Torispherical or ellipsoidal heads of steel which are loaded by internal pressure, and which are designed within the restrictions for use of eqns (3.126) and (3.127), are rather insensitive to geometrical shape deviations. Furthermore, a failure will seldom lead to serious consequences, in particular if elastoplastic buckling, according to eqn (3.126), is the limiting factor for the resistance. The safety factor can therefore be kept low. The formulas should also, in most cases, result in an underestimation of the resistance. The following criteria concerning tolerances and dimensions should, however, be satisfied: (1) The circumference of the head should not differ more than 0.25% from the nominal value. However, a deviation of ±5 mm is allowed for smaller heads (rc≤325 mm). (2) The actual height of the head should not be smaller than the nominal value, neither should it exceed the nominal value by more than 1%, or max 40 mm. (3) The actual knuckle radius rk should be at least equal to the nominal value. (4) The transition between the cap, knuckle and cylindrical parts, respectively, shall be continuous. Eqns (3.126) and (3.127) apply to vessel heads which have been welded together from several plates. For heads manufactured in one piece by pressure turning, the coefficient in the numerator may be increased from 120 to 190. E. Example When cold stretching a tank with torispherical end closures at AGA-CRYO, Göteborg, buckles developed in the closures, as shown in Fig.3.50. The loading was produced by water pressure and amounted to 7.3 bar. The nominal geometry of the head is presented in Fig.3.51. Measurement of the thickness indicated that the head was somewhat thinner in the vicinity of the weld which joins the two halves of the head, see Fig.3.52.

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Fig.3.51. Geometry of sample pressure vessel head subjected to internal pressure. At the cold stretching a pressure of 7.3 bar was applied, which was then maintained during a prescribed period . After a few hours it was found that a buckle had developed close to one of the welded joints according to Fig.3.52. The pressure was subsequently kept over night in order to investigate the long term stability of the buckling process. In the morning buckles had formed in two more of the welds across the knuckle region. An effort to predict theoretically the buckling behavior has been made as follows: The material SS 2333–28 (ASTM 304) has a yield limit (design value) of σy=210 MPa. The material test certificate indicates that, for the current delivery, σy=310 MPa. Introducing these values in 3.5.2.1 gives an estimate of the resistance of the head:

Fig.3.52. Buckling mode of the vessel head in Fig.3.51.

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Plastic buckling, eqn (3.126):

Axisymmetric collapse, eqn (3.127)

The cold stretching at 7.3 bar did not cause an immediate collapse. The design formulas proposed may thus be considered to provide adequate results. Since buckling took place with some delay, it is obvious that a time dependent deformation developed. Also, the buckling deformation in all cases initiated close to the weld where the following parameters may have had an influence: – Due to grinding of the weld, the thickness at the welds has, locally, become somewhat less than in the adjacent parts of the shell wall. – The material properties of the welded zone were not determined and may have deviated from the value given in the material test certificate. – The residual stresses due to welding should have had a magnitude of at least half the yield limit. On loading, a local plasticization will occur and result in small deflections in the vicinity of the weld. These deflections can trigger buckling as shown in Fig.3.52. In the case described simultaneous creep in the welded joint has contributed to the buckling progress. 3.5.2.2 Uniform external pressure A. Introduction In torispherical and ellipsoidal vessel heads which are exposed to external pressure compressive mean stresses occur as a rule in the meridional direction of the entire head. In the circumferential direction, compressive mean stresses are obtained in the spherical cap while the major part of the knuckle is subjected to tensile stresses. In the knuckle, high bending stresses may also be found. The cap can buckle in the elastic or elastoplastic regions or be subjected to snap through as a consequence of large plastic deformations in the cap and the knuckle. Both elastic buckling and snap through lead to dynamic deformations which may have serious consequences. B. Resistance For convex heads with rc/t<167 the pressure vessel codes provide rules for determining the allowable pressure. These rules normally include three formulas which in turn give the allowble pressure with respect to yielding in the knuckle region, elastic buckling of the cap and plastic collapse of the cap.

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In the German code for convex pressure vessel heads, AD-Merkblatt B3, design rules are given for heads with shapes corresponding to SS 482 (Korbbogenböden) and SS 2634 (Klöpperböden). The rules refer to heads satisfying the design condition, , and include two formulas, one providing the allowable pressure with regard to stresses in the knuckle region and the other giving the allowable pressure considering buckling of the spherical part of the head. For the standardized heads SS 482 and SS 2634 the following expression for the characteristic resistance of the knuckle region can be deduced in analogy with the German codes: (3.132) t=thickness of the head, rc=cylindrical radius of the head, see Fig. 3.49, pu=charcteristic resistance of the knuckle region of the head, σy=yield stress, β0=shape factor. The shape factor β0 is a function of the wall thickness and the shape of the head. Values of β0 can be obtained from Fig.3.53 for heads SS 482 and SS 2634 as functions of t/(2rc). It is presumed in Fig.3.53 that the head is not weakened by holes or cutouts and that it has at least the same thickness as the knuckle part. For other wall thicknesses and head shapes the collapse pressure can be calculated according to Chapter 6, Stability analysis with numerical methods. The resistance of the spherical cap is estimated according to the rules in 3.4, where both elastic and plastic effects can be considered. 3.5.3 Hemispherical heads For the analysis of hemispherical heads reference is made to section 3.4, Spherical shells.

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Fig.3.53. Shape factor β0 for standardized pressure vessel heads. 3.6 Tank roofs 3.6.1 Introduction Roof structures may be classified according to their shape, position of the roof sheeting and type of fastening, support conditions and stiffening arrangements as shown in Fig.3.54. The load carrying system may consist of the sheet itself, of sheet and arches (with or without polygonal rings) or of roof trusses. Polygonal rings which can resist compressive forces and which are considered in the structural system are called compression rings. In this section rules are given for roofs of unstiffened shells and of shells stiffened by arches. Since a particular type of structure is treated, the entire design procedure is included. The methods of analysis are based on a Swedish standard but the results are general and are, in fact, similar to those given in the German code DIN 4119. In this section, design methods are given for fixed roofs only. Rules for floating roofs are provided in applicable codes.

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Fig.3.54. Classification of tank roofs. 3.6.1.1 Layout Roofs over small tanks (diameter less than about 12 m) can be designed as conical, unstiffened shells. A cone is simpler to manufacture than a spherical shell (dome) since it has a radius of curvature in one direction only. In larger roofs the sheeting is normally supported by a series of radial arches and polygonal rings according to Fig.3.55. At the top the arches are attached to a crown ring and at the lower boundary to an edge ring. Lattice work is introduced between some of the arches to prevent buckling in the lateral direction. Normally the sheeting is placed loosely on the upper side of the arches and is only fastened to the edge stiffening ring. An internal pressure, therefore, will only act on the sheeting and not on arches, polygonal rings and lattice work.

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Fig.3.55. Tank roof with arches and rings. The sheets are usually overlapping each other and are joined by fillet welding. In most cases originally plane sheets can be used. The conical or spherical shape is thus modelled by a large number of sheets with a single radius of curvature. In certain cases the sheeting is attached below the arches and is then usually intermittently welded to these. The procedure is often used with stainless steel sheets. Some codes may stipulate a minimum nominal sheet thickness. In the present example, tmin=4.0 mm for roofs manufactured from low-alloy steel and tmin=2.5 mm for roofs of austenitic stainless steel. 3.6.1.2 Symbols n=number of half-arches, which should be the least of 9 and twice the radius r in m, m=number of lattice trusses which should be at least 3 and minimum n/6, v= radians=half the angle between two half arches, ψ0=half the opening angle of the roof in radians, 0.14≤ψ0≤0.45 rc=tank radius, defined in Fig.3.56, h=height of roof

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l=length of half arch dome l=rsψ0 cone l=rs tan ψ0, ts=thickness of roof sheeting, tc=thickness of tank wall.

Fig.3.56. Dome and conical roof sections. 3.6.1.3 Design value regarding the yield stress Design methods are given in 3.6.2 to 3 6.8 based on internationally accepted principles. Load and resistance factors according to Chapter 2 are used. In the formulas for the section resistances the design value of the yield stress σyd is utilized. This value is obtained as

where σy=yield stress according to the Swedish Standard (equivalent to ECCS, Eurocode 3 etc), γm=resistance factor according to 2.5.4. For wall thicknesses less than 5 mm, γm=1.1, otherwise γm=1.0, γn=resistance factor which is dependent on the safety class. For tank roofs, safety class 2 is normally selected, yielding γn=1.1. See 2.5.1 and 2.6.1.

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3.6.2 Loads Internal and external overpressures and other loads which may act on the structure, are chosen according to an applicable tank code or similar national or international standard. g1=dead weight of arches and polygonal rings, converted into load per unit length, g2=dead weight of covering plate+insulation+roof sheeting converted into load per unit area, s0=snow load, characteristic value, load per unit area, qk=wind pressure, characteristic value, load per unit area, p1=external overpressure, load per unit area, p2=internal overpressure, load per unit area, pcr=critical load of roof sheeting, load per unit area, P=local point load, from person or similar traffic. The distribution of the snow load over the roof is assumed to be given by (3.131) The evenly distributed snow load 0.8s0 is not governing for design, see Fig.3.57. The distribution of the wind load on the roof is determined by (3.132) See Fig.3.58.

Fig.3.57. Distribution of snow load on dome and conical roofs.

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Fig.3.58. Distribution of wind load. 3.6.3 Roof sheeting The resistance pu of an unstiffened roof can be determined according to 3.3.1 for a conical roof and section 3.4.1 for a domed roof. The resistance can also be obtained using eqns (3.133) and (3.134) which are simplified formulas deduced from those given in 3.3.1 and 3.4.1 In roofs with arches and, possibly, rings the resistance of the sheeting is determined by eqn (3.135). The wall thickness is chosen to satisfy the design condition of eqn (3.139) Unstiffened domed roof: (3.133)

Unstiffened conical roof: (3.134)

Roof with arches (3.135) Here, b is the smaller width of a sheet section bounded by arches and rings, and a is the larger width of the same sector, see Fig. 3.59.

Fig.3.59. Definition of a and b in eqn (3.135).

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The design value for the resistance is (3.136) where γm and γn are the resistance factors to be selected according to 3.6.1.3. The load on the roof is determined as the largest of the following values p=g2+γfs0+p1 (3.137) and p=g2+ψs0+γfp1 (3.138) where γf=1.3 is the load factor for one variable load, ψ=load reduction factor for snow loads. The design condition is p≤pd (3.139) 3.6.4 Edge ring stiffener The greatest tensile force Nt and the greatest compressive force Nc in the edge ring are determined as follows:

Fig.3.60. Edge ring stiffener. For domed roofs: (3.140) (3.141)

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For conical roofs: (3.142) (3.143)

γf=1.3 Nt and Nc shall not exceed the design values: Ntd=Atσyd (3.144) Ncd=Acσyd (3.145) At and Ac=cross section areas of the edge ring in tension and compression respectively. At may include: If At
.

3.6.5 Polygonal ring The bending moment in a polygonal ring will be the greater of the values: (3.146)

(3.147) lp=2 r sin v=length of polygonal ring, r=distance from tank center to polygonal ring, γf=1.3 as stated above, g1=load due to the dead weight of the polygonal ring in load per unit length. Additional loads are given in 3.6.2. The bending moment M shall not exceed the design value: M<Md=Wσyd (3.148)

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W=section modulus of the polygonal ring, σyd=design value for the yield stress of the polygonal ring, see 3.6.1.3. 3.6.6 Arches 3.6.6.1 Dome roof with sheeting fastened at the edge The design formulas given below are based on the assumption that the roof sheeting is fastened to the edge ring stiffener in such a manner that the sheeting prevents the arches from moving upwards on the side where,for instance, the snow load is smaller. The arches may thus be assumed to be loaded by a symmetrical snow load of a magnitude equal to that on the left side in Fig.3.57. Figs.3.61a)–d) demonstrate some results from a design analysis of a tank roof with yielding arches and roof sheeting.The bending moments in the arches depend on the yielding (support stiffness) due to the specific design of the arch supports and the the cross sectional area and fastening of polygonal rings. The forces and moments presented in 3.6.6.1 do not take into account the polygonal rings which have a considerable effect on the moment in the arch when the loading is symmetric. A necessary condition for the following method of calculation to be valid is that

where σyd=design value for the yield stress of the roof sheeting according to 3.6.1.3. A. Loads The loads acting on a half arch are calculated from: G1=g1l (3.149) (3.150) (3.151) (3.152) The arches are unloaded by the critical load Pcr, which the sheeting is able to carry (3.153) where

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Fig.3.61. a) 180° model of tank roof. Within the hatched area the sheeting prevents the arches from deflecting upwards. b) Distribution of snow load on an arch. c) Arch geometry in current example. d) Distribution of moment in arch with yielding and rigid supports respectively. In both cases the sheeting is only fixed to the edge ring stiffener.

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The critical load Pcr is an estimate of the buckling load of a plate supported along four edges by arches and polygonal ring bars (DIN 4119).The support from the arches and the polygonal bars increases the buckling load and provides buckling modes deviating from the buckling deformation of the arches. The arches will thus be able to stiffen the sheeting at the same time as the sheeting is relieving the arches from part of the load.

B. Bending moments and force The moments and forces given below are conservative estimates considering some yielding of the edge ring which constitutes the horizontal support for the arches. (3.154) (3.155) (3.156) (3.157) Rs=support reaction force from symmetric load on an edge ring stiffener. Note that Rs acts as a concentrated load on the cylindrical shell wall and should be analyzed according to section 4.1. Hs=horizontal arch force at the edge ring stiffener, Ms=maximum bending moment in the arch, N=longitudinal force in the arch, γf=load factor=1.3 C. Design conditions The arch shall have a moment of inertia at least corresponding to:

Rs is determined by eqn (3.154), Pcr by eqn (3.153) and b from the definition given in connection with eqn (3.153). (3.158)

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σyd=design value of the yield stress of the arch, A=cross section area of the arch, Ix=moment of inertia of the arch, Wx=section modulus of the arch. Eqn (3.158) yields results on the safe side. 3.6.6.2 Conical roof with sheeting fastened at the edge A. Loads Loads on a half arch: G1=g1l (3.159) (3.160) (3.161) (3.162) where symbols and loads are defined in 3.6.1.2 and 3.6.2. (3.163) where

Pcr is discussed in 3.6.6.1. B. Moments and force Rs=G1+G2+γfQs+P1−0 .85Pcr (3.164) (3.165) (3.166)

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(3.167) where Rs=support reaction from symmetric load on the edge ring stiffener. Note comments in section 3.6.6.1B. Hs=horizontal force at edge ring stiffener, Ms=maximum bending moment in the arch, N=longitudinal force in the arch, γf=load factor=1.3 C. Design conditions The design conditions are equivalent to those given in in 3.6.6.1C. In the application of the different criteria, the variables calculated in 3.6.6.2 shall be used. 3.6.6.3 Dome and conical roofs with roof sheeting not fastened at the edge If the roof sheeting is unable to prevent lifting of the arches, the arch must be designed for an unsymmetrical snow load as shown in Fig.3.57. The snow load is divided into a symmetric part: (3.168) and an antimetric part: (3.169) The symmetric part yields the arch bending moment in a domed roof as: Ms=0.035 (G2+P1−0.85Pcr)·rc+0.05γfQsrc (3.170) and in conical roofs: Ms=0.09Rsrc (3.171) The antimetric part of the load gives a bending moment of: (3.172) approximately valid for both types of roofs. The remaining loads and forces are determined according to 3.6.6.1 and 3.6.6.2, but the condition in eqn (3.158) is replaced by:

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(3.173)

N is determined according to eqns (3.157) or (3.167), Ms according to eqns (3.170) or (3.171) and Ma according to eqn (3.172). The remaining symbols are given in 3.6.6.1C. 3.6.6.4 Dome roof and conical roof with sheeting below the arches For tanks with contents producing corrosive gases it may be appropriate to use stainless sheeting which is placed below the arches and rings for protection. The sheeting may be intermittently welded to the arches. It braces the lower flange against lateral buckling and also prevents the arches from deflecting laterally. The arches may rotate around the lower flange when the upper flange is compressed. A thin sheeting provides no stiffness against the rotation, and the upper flange consequently needs bracing e.g. by polygonal rings. These should be rigidly attached to the arches. The arches can then be designed according to 3.6.6.1, 3.6.6.2 or 3.6.6.4. If no polygonal rings are introduced, it is necessary to replace the yield stress σyd in eqns (3.158) and (3.173) by the stress σcd with regard to lateral buckling of the unbraced flange. The following approximate expression may be used: (3.174) tf=flange thickness, h=height of the arch section, γmγn=resistance factors according to 3.6.1.3. In order to secure stability during erection, polygonal rings and lattice work may be required. 3.6.7 Crown ring The normal force in a crown ring flange can be estimated at (3.175) where N, Ms and εs are determined according to 3.6.6.1 or 3.6.6.2 and hr is defined in Fig.3.62. The force must not exceed the normal force capacity of the ring: (3.176)

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where the nonlinear stress distribution in a strongly curved beam is taken into consideration. lb is the length of a half arch. hr, br and tr are shown in Fig.3.62. Furthermore, it must be verified that (3.177) where Ir=0.5brtr(hr+tr)2 Ix=the moment of inertia of the arch. br>2hr tr>br/15

Fig.3.62. Typical crown ring geometry. 3.6.8 Lattice work The number of lattice works m shall be greater than n/6, and not less than 3. The following symbols are used, see also Fig.3.63. ll=arch length between two node points, lp=length of a polygonal ring sector belonging to a lattice work, lf=length of a diagonal of the lattice work, iy=smallest radius of gyration of a bar in the lattice work, If=smallest moment of inertia of lattice work diagonal. The check of the carrying capacity of the arches according to 3.6.6.1 or 3.6.6.2 is sufficient without particular investigation of torsional-bending buckling if the roof sheeting is attached directly to the arches and if the condition lf/iyl≤150 (3.178)

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is satisfied. If the structure is subjected to external loads during the erection phase and before the sheeting has been mounted, lateral buckling of the arches must be checked, unless lf/iyl≤50 (3.179)

Fig.3.63. Lattice work. Diagonals belonging to the lattice work shall be designed with a minimum bending stiffness of: (3.180)

(3.181) where N is the normal force in an arch according to 3.6.6.1B or 3.6.6.2B and Af is the cross section area of the lattice work diagonal. Polygonal rings, which are included in the lattice work, shall satisfy the same condition, where If, Af and lf refer to the polygonal ring. 3.7 Shells composed of different shell elements A shell structure is usually built up from a number of elements each of which has specific properties with regard to stability. These elements may interact when buckling and the global resistance may be much lower than that which would be predicted if each element is considered separately. Some typical examples of shell combinations are shown in Fig.3.64. In many cases it is not very clear whether a shell composed of shell elements has to be analyzed as a complete structure. The various shell elements can probably be treated separately if the elements are bounded by rings of sufficient stiffness, see Fig.3.65, or if the attachment angle between the shell elements is so large that a similar

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stiffening effect will arise. For small attachment angles, interaction may, clearly, be possible. In the following paragraphs these problems are discussed and recommendations are presented for determining the limits for application of the rules for separate shell elements.

Fig.3.64. Typical shells built up from simple shell elements. 3.7.1 Unstiffened shell combinations Some unstiffened shells are shown in Fig.3.66, including primarily axisymmetric structures which are loaded by external pressure and axial compression. Bending caused by a nonsymmetric wind pressure is also a common load case, and it may be treated, at least approximately, with the theory of axisymmetric loading as described in section 3.2.1.5. Several collapse mechanisms are conceivable and it is important to take all of these into account in the design analysis. In the following paragraphs an attempt will be made to treat these problems systematically and, in the cases where this is feasible, provide a basis for estimating the resistance of the combined shell. 3.7.1.1 Axisymmetric shells subjected to external pressure Vacuum vessels are often composed of cylindrical and conical shells with spherical, ellipsoidal or torispherical end closures. The buckling resistance of the separate shells can be computed using the elementary cases in Chapter 3.

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Provided the angle at the junction between two shell elements is sufficiently large the resistance will be governed by the weakest shell element as demonstrated in the example of Fig.3.67.

Fig.3.65. Stiffened combined shells where the shell elements may be considered separately in the design analysis.

Fig.3.66. Axisymmetric shell structures without ring stiffeners. The results given in the figure suggest that interaction between the shell elements is obtained for attachment angles up to 11 degrees. It is likely that the critical angle depends on several factors such as the length and thickness of the shell elements as well as the clamping conditions at the boundaries. Furthermore, it has been pointed out by several researchers that a shell structure which has been optimized with respect to several types of buckling, will become more sensitive to disturbances of various kinds. An angle of 15

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degrees between cylinder and cone in Fig.3.67 would result in a separation of the global and local buckling stresses of at least 25 per cent which should be acceptable. The following recommendations can be given regarding shells composed of elements joined at more or less steep angles:

Fig.3.67. Cylinder/cone shell combination subjected to external pressure. The diagram shows the ‘global’ buckling stress and the local buckling limit of the cylindrical and conical shell elements versus the angle at the juncture. – Cylinder-cone: Global buckling does not have to be considered if the angle between the meridians of the shell elements exceeds 15 degrees. For angles below this value it is on the safe side to compute the buckling pressure corresponding to global buckling for an equivalent cylindrical shell with a length equal to that of the combined shell according to Fig.3.68. – Cone-cone: The limit angle may also in this case be assumed to be 15 degrees. For an estimate of the critical pressure in global buckling it is conceivable to use various models according to Fig.3.69.

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– Cylinder/cone-torus/sphere: Normally, the buckling pressure of the cylindrical (or conical) shell is critical and interaction in a global buckling mode may be disregarded. On the other hand, the flexibility of the end closure in the vicinity of the attachment to the cylinder (cone) will result in a reduction of the buckling pressure of the shell. This may be taken into consideration by defining a slightly longer shell in the buckling analysis, as indicated in Fig.3.70. Here it is, again, reasonable to set the limit at an angle of 15 degrees, which was shown above to be valid for cylinder-cone transitions. 3.7.1.2 Axisymmetric shells under axial loads This load case is very common, especially in combination with internal or external pressure. Since the buckling load for a cylindrical/conical shell is not primarily dependent on the length of the shell, the buckling stress at the weakest section should be dimensioned by taking buckling into consideration. The elementary cases in Chapter 3 will, consequently, be applicable to each shell element separately. It should be appropriate, also in this case, to avoid coincidence of buckling stresses in adjacent shell elements, since this may cause an increased sensitivity to initial imperfections.

Fig.3.68. Definition of the limit value of the angle between cone and cylinder and definition of an equivalent cylinder. At the junction between two different shell elements, knee loads will develop which may cause high local stresses. It is necessary to consider these stresses in the design in order to avoid large plastic deformations. In some cases a ring stiffener must be introduced at the transition between the shells, see below and examples in Chapter 8. A typical stress

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distribution at a cylinder-cone junction is shown in the example in Fig.3.71. The load combination will lead to plastic deformations at a relatively early stage and subsequently collapse due to yielding. This deformation takes place within a limited region and the failure is likely to be determined entirely by the properties of the material, provided the angle between the shell elements is sufficiently large. Thus, the design has to be checked for adequate safety against plastic collapse according to the applicable code. The problems are discussed in a recent paper by Knoedel, 1991.

Fig.3.69. Definition of an equivalent cylinder, or cone, for estimation of the global buckling stress in a cone to cone shell combination. 3.7.1.3 Axisymmetric shells subjected to bending High slender tanks and process vessels are subjected to bending moments caused by wind pressure. The calculation of the resistance is in this case done in the same manner as for axial loading. The higher value of the reduction factor η according to 3.2.1.3 may, however, be applied. The knee load in the transition zone between the shell elements is also determined in a similar way. It should be noted that these loads do not produce an ovalizing force, since the radial knee loads are balanced by shear forces at sections rotated ±90° from the locations where the maximum normal stresses are found. See further 3.7.2.2.

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Fig.3.70. Determination of the effective length of a cylindrical shell (or cone) with a spherical or a torispherical head.

Fig.3.71. Example of a combined shell under axial loading. 3.7.2 Combined shells ring stiffened at the juncture The following paragraphs concern ring stiffeners at the juncture between the shell elements. The purpose of a ring is to limit the local stresses due to knee loads, and to ensure that buckling will take place within the individual shell elements and, thus, preclude interaction (global buckling). 3.7.2.1 Ring between cylinder and cone under external pressure Recommendations are given in 3.2.1.2 and 3.2.2 for the design of ring stiffeners such that the stiffened section remains circular while any buckling will occur in the shell.

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Application of this design rule in the determination of the ring stiffness required at the junction between two shell elements provides adequate safety against global buckling. The procedure is recommended especially when the angle between the shell elements is small. The result will be conservative, in particular if the angle is large. In such cases it would be feasible to reduce the stiffness of the ring, for example with the help of calculations performed with a suitable computer program. 3.7.2.2 Ring between a cylindrical and a conical shell under axial forces Many silos and tanks consist of a cylindrical container with a conical hopper. See Fig.3.72. It is usually necessary to provide a ring at the transition between the shells. The ring is required for carrying the compressive forces caused by the radial component of the tensile stress in the cone, and together with the cylindrical wall disperse the reaction forces from the legs. The attached conical and cylindrical shells prevent the ring from buckling in or out of the plane of the ring. On the other hand, the free edge of a ring cut from plates can buckle in several waves (10–30) in particular if the br/tr ratio is large.(compare Fig. 1.4)

Fig.3.72. a) Typical silo structure mounted on legs. b) Junction hoppercylinder. c) Buckling mode in the vicinity of the ring stiffener. According to Jumikis-Rotter, 1983, (also in Rotter, 1987), the buckling stress can be determined from (3.182) where

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157

The symbols are defined in Fig.3.73. The effective area of the ring stiffener is obtained as: (3.183)

Fig.3.73. a) Symbols used to define a ring cross section at hopper/cylinder juncture b) Failure mechanism. The normal force in the effective ring is: Nr=Nφhr sinβ (3.184) where Nφh=meridional force in hopper, r=cylinder radius,

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β=half the bottom angle of the hopper, see Fig.3.73. The normal force Nrd shall not exceed: (3.185) where

γmn=γmγn=resistance factor (safety factor), σel=buckling stress according to eqn (3.182), Ar=effective area according to eqn (3.183). If the ring is provided with a flange, the buckling stress will increase, but torsional buckling may still be critical. Normally a thicker plate is chosen. 3.7.2.3 Ring between cylindrical and conical shells subjected to bending moment A pulp tank of cylindrical shape is often mounted on a smaller cylinder with a conical transition See Fig.3.74. Wind loads give a bending moment which results in forces at the juncture between the cylindrical and conical shells. Another example of a combined shell loaded in bending is a column with a stiffened foot as shown in Figs.3.75–3.77. Unless the ring is very stiff, the shear force distribution will be such that the bending moment in the ring becomes close to zero. The ring will thus be loaded mainly by a normal force varying along the circumference according to Fig.3.78. The maximum value of the normal force becomes:

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159

Fig.3.74. Pulp tower—Example of combined shell subjected to bending.

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Fig.3.75. Column with a tubular section and widened footing for moment transfer. This normal force (added to the normal force due to a possible axial load) shall not exceed Nrd according to eqn (3.185) with σel from eqn (3.182) and the effective area Ar from eqn (3.183). The symbols used to define the cross section are indicated in Fig.3.79.

Fig.3.76. Deformations at the transition between a conical and a cylindrical shell loaded in bending.

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161

Fig.3.77. Horizontal load qh and shear force vh with the maximum values qh0 and vh0 acting on the ring. If the cylindrical shell ends at the junction with the conical shell (no skirt), ts=0. For shells which are not too slender, the ring stiffener can be omitted. The normal force Nr must then be smaller than ψArσy/γmn with Ar according to eqn (3.183), in which brtr is obviously equal to zero. The factor ψ was introduced in order to take into consideration that the radial compressive stresses may reduce the buckling resistance against axial compression. Awaiting an investigation of this influence, ψ is assumed to be equal to ωs according to eqn (3.10) with regard to buckling of the cylindrical or the conical shell. Compare the deformation shown in Fig.3.76.

Fig.3.78. Normal force in ring.

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Fig.3.79. Definition of symbols at the cross section. The shear forces vh, which are balancing the qh forces as shown in Fig.3.77, do not have to be added to the transverse force.V if this force is acting in the direction shown in Fig.3.74, since vh gives shear stresses in the opposite direction. If V is acting in the opposite direction, the shear stresses caused by V should be added to the shear stresses due to vh in the cylindrical shell.

4 EFFECT OF LOCAL LOADS AND DISTURBANCES Chapter 3 on elementary cases gives design rules for “pure” shells which are subjected to uniform, distributed loads. In many practical cases, the shell is loaded by local concentrated forces, e.g. from attached piping systems or from the supports. Furthermore, cutouts and local stiffeners may cause disturbances which could initiate collapse. So far very few design recommendations are available and those given below may be interpreted as an attempt to provide rational methods for analysis of these problems. 4.1 Buckling caused by local loads Local loads can be considered to be primary if they are acting parallel to the shell surface and are, more or less directly, causing buckling. Loads which are acting perpendicularly to the shell surface, such as piping forces, may be classified as secondary, since they result in a deformation of the shell surface but not directly in loss of stability. 4.1.1 Buckling caused by forces in the shell middle surface In codes and design recommendations concerned with buckling, so far only complete shell elements with simple geometry and subjected to uniformly distributed loads have been dealt with. In actual structures the load is often unevenly distributed, or concentrated, as at points of support, suspension of platforms and piping attachments etc. In this chapter various types of buckling are discussed. It is not possible to provide general solutions of the problem, since the buckling process almost inevitably becomes highly nonlinear. It turns out, however, that in some cases the resistance can be estimated by use of the elementary cases, provided that the local distribution of the compressive stress is applied in the design analysis. Tests carried out by Gunnarsson—Sandberg, 1987, and Samuelson, 1985, indicate that the method yields realistic results somewhat on the safe side for cylindrical shells under axial loading. Similar problems have been discussed by Öry et al., 1984, and Krupka, 1988, where the attention is more concentrated towards the analysis of the local stresses than on the buckling problem. 4.1.1.1 Cylindrical shell under axial loads Containment vessels of various kinds supported on columns are very common in practice, see Fig.4.1. This type of support leads to compressive stresses in the cylindrical shell wall just above the attachment of the legs, and these stresses may cause buckling. The problem

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becomes particularly intricate if the contained material can develop arching and, thus, cause the total weight of the content to be transferred to the support via the shell wall above the point of support. Collapses caused by this phenomenon are not uncommon. When designing for local loads one of the problems consists of determining the stress distribution in the shell, particularly if the load is applied in a transition region between the cylindrical wall and the end closure. Even in the case of a point load acting on one of the cylinder edges according to Fig.4.2, the stress analysis procedure is not self-evident.

Fig.4.1. Typical column support structures for silos or pressure vessels.

Fig.4.2. Cylindrical shell loaded by “point” loads.

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In many cases it is appropriate to calculate the stress distribution by means of the FEmethod and subsequently estimate the risk of buckling from the results. Such an analysis was performed for a silo/cistern supported by legs and with a geometry shown in Fig.4.3. The loading consists of a fluid pressure acting on the conical end closure. In many practical applications the structure is reinforced by introducing a ring at the cylinder/end closure juncture. This alternative was studied together with two other feasible stiffening geometries. The results are displayed in Fig.4.4 and Table 4.1.

Fig.4.3. FE-model of a silo on columns.

Fig.4.4. Stress distribution in the silo according to Fig.4.3, calculated with linear theory.

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Fig. 4.4 presents a number of interesting results which deviate in some respects from what might be expected. First, A) indicates that the load is spread in a fan shaped pattern to the cylindrical wall and that the angle of the loaded sector is rather narrow. This angle may be estimated at±15 to 20 degrees, which is considerably less than that usually found in plane or three dimensional structures. Secondly, B) shows that the introduction of a ring at the juncture does not have a significant influence on the distribution of the axial stresses in the cylinder. On the other hand, the ring gives a reduction of the circumferential compressive stresses which are caused by the knee load and which may, under extreme circumstances, result in buckling/yielding in the transition region between the cylinder and the end closure. See also section 3.7 concerning buckling of combined shells. An extension of the leg gives, however, a better diffusion of the load into the shell as is evident from C) and D) in Fig. 4.4. Table 4.1 gives the calculated stress levels at a number of points along the generator of the cylindrical wall and compares the results with those obtained if the load is considered to be distributed over a ±15 degree sector as described above. It is obvious that the simplified method for the estimation of the stress level provides a satisfactory approximation of the actual conditions (if the FE-results are assumed to represent the true stress distribution).

Table 4.1. Evaluated stresses according to Fig.4.4 and comparison with the results of the approximate method. Computed stresses σx, MPa Point No Analysis

−1

1

2

3

4

5

6

7

8

Case

Cone

Cylinder

Linear

A

+15

−20

−15

−10

−7

0

+1

+1

+1

FEM

B

+10

−17

−15

−10

−7

−6

−5

−2

0

calcu

C

+10

−3

−4

−4

−6

−6

−5

−3

0

lation

D

+10

−4

−4

−3

−3

−3

−2

−1

−1

Approx

A&B

−22

−15

−10

−8

−6

−5

−4

−3

method

C&D

−5

−5

−4

−4

−10

−8

−6

−5

Stress calculations were also carried out by Gunnarsson and Sandberg, 1987. They demonstrated that stress concentrations will appear at the edges of the supports, particularly if these are wide, see Fig.4.5. This circumstance has not been considered in the following design recommendations since, first, the effect should be less pronounced in practical applications due to a smaller difference in stiffness between support and shell wall and, secondly, the buckling tests did not show a marked effect. Furthermore, rather narrow supports are likely to be chosen for practical reasons. In the introduction, the classical solution was discussed for a cylindrical shell subjected to axial compression. Fig.1.5 indicated that a large number of sets of wave numbers m and n will produce the same buckling stress. The smallest wave length lm in

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the axial direction is of the same order of magnitude as the natural wave length in the . This indicates axisymmetric solution of the prebuckling deformation, i.e. that one buckle should cover an area of the size marked in Fig.4.6, and the buckling load should then be determined by the average stress within this area.

Fig.4.5. Stress distribution in a cylindrical shell with supports of different width. It is reasonable to assume that the critical stress can be defined as the average stress in the area indicated in Fig.4.6 and this value should be compared to that obtained in 3.1.1.2 for a cylindrical shell under a uniformly distributed axial stress σx.

Fig. 4.6. Buckling mode according to the classical theory for a cylindrical shell subjected to local axial loads, compared to test results by Gunnarsson and Sandberg, 1987.

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The method was used by Gunnarsson and Sandberg, 1987, and Samuelson, 1985. Some of the results are presented in Fig.4.7 which provides a comparison between calculated buckling loads and test results. The choice of calculation model can obviously be subject to discussion. The stress diffusion model which will be recommended in the following, gives results which are on the safe side.

Fig.4.7. Comparison between test results and the proposed method of analysis. 4.1.1.2 Local axial loads, design rules A number of load cases may be conceived according to Fig.4.8, which suggests a rule for division of the load in one tension and one compression field (cases b and c) respectively. In case a uniform basic stress σx is present, the shell has to be designed for the sum of the basic and the local stress. Assume that the local compressive force is acting on a width b of the cylindrical shell according to Fig.4.8, and that it is spread out over the shell surface within a sector of ±20 degrees. The stress at a distance x from the support will govern the buckling behavior (4.1) where the stress level is determined from: (4.2)

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σtot=σx+σ0 (4.3) In many cases it is possible that this stress level becomes so high that buckling may occur. The natural measure to be taken in such a situation is to introduce a vertical reinforcement which provides a better transfer of the load into the shell. A simple model of stress diffusion in a stiffened shell is presented in Fig.4.8d. The load which is applied at the lower end of the stiffener will be transferred successively to the shell by shear forces. The opening angle of the sector of diffusion is mainly the same as in an unstiffened shell. If the cross section area of the stiffener is large, the load cannot be fully transferred into the shell in the length ls, implying that part of the force will be carried over to the shell at the upper end of the stiffener where it causes a stress concentration σx2b. The section area As should thus be adjusted to the dimensions t and ls, and the cross section should be tapered towards the upper end.

Fig.4.8. Local stresses in a cylindrical shell caused by concentrated forces in the meridional direction. Note that it is advisable in most cases to finish a stringer at a ring stiffener in order to avoid high local bending in the shell. This is particularly important in structures subjected to fatigue, where the service life may be affected by a factor of 2–5. On the other hand

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the ring will contribute only moderately to the diffusion of the local stresses unless it has an extremely high stiffness in bending and torsion, see Fig.4.4. 4.1.1.3 Local loads in the circumferential direction Very few test results are available for this load case and design recommendations which can be established must, therefore, be conservative. In most cases a buckling analysis by use of numerical methods should be preferred, see Chapter 6. An investigation by Krupka, 1988, treats cylindrical shells supported by saddles. Typical load cases are shown in Fig.4.9, which also indicates the buckling mode for an external pressure. It is reasonable to assume that the stress distribution caused by the local load will contribute to the generation of a buckle if the diffusion area covers a major part of the natural wave length according to the figure. In such a case the following calculation procedure may be applied.

Fig.4.9. Examples of loads causing stresses in the circumferential direction of a cylindrical shell. A. Calculation of the buckling wave length The dominating load is assumed to be an external pressure p. This will cause buckling in the shell in a number of waves, n, in the circumferential direction, and one half wave, m=1, in the meridional direction. The wave number n can be computed from the following equation (3.34) where the half wave length ln in the circumferential direction is obtained as

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171

B. Estimation of the stress distribution For the case of an axial local load, the resulting stress distribution was proven to be limited within a very narrow sector. Similar calculations for a cylindrical shell subjected to local loads in the circumferential direction show that the sector angle is rather of a magnitude ±45 degrees, which implies that a model of analysis according to Fig.4.10 should provide a good approximation of the actual distribution.

Fig.4.10. Definition of stresses in the circumferential direction of a cylindrical shell subjected to a concentrated load. Assuming that the stress at a distance of half a wave length of the buckling mode from the disturbance is representative for the estimation of the buckling load, the following expression is obtained for the stress level: (4.4) where σφ0 is the stress resulting from the external pressure p and b=b0+2y0tan 45° (4.5)

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Fig.4.11. Assumed equivalence between initial imperfections and deflections caused by point loads perpendicular to the shell surface. 4.1.2 Influence of forces perpendicular to the shell surface The shell stiffness is usually very low against forces and moments giving deflections perpendicular to the middle surface. It is, consequently, appropriate to limit the magnitude of such loads in order to minimize the risk of buckling. The criteria on which a limitation of local loads can be founded may be derived from the tolerance requirements given with respect to shape deviations etc. The recommendations may thus be given in the following way: Assume that a force is acting perpendicularly to the shell surface as shown in Fig.4.11. The effect of the resulting deformations, indicated in the figure, is assumed to be equivalent to that of shape deviations of the shell according to Fig.3.1. If it is assumed that the shell is manufac

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173

Fig.4.12. Example of determination of the reduction factor due to a point load acting on a cylindrical, welded shell, tolerance class 1. tured with a tolerance level wr/lr
It is then possible to determine a value of the additional reduction factor a according to Fig.4.12. The probability is low that the deformations caused by the concentrated load will act within the area where the largest initial buckle appears. Consequently, it is appropriate to combine the effects of local loads and initial imperfections as the root mean square value:

The calculation of deflections due to small lateral forces can be carried out in various ways, e.g. by use of an FE-program. Examples of results obtained with such a program are shown in Fig.4.14. A solution of the general shell equations based on Fourierexpansions was presented by Bijlaard, 1954. See further numerical results by Kempner et al., 1955, Forsberg-Flügge, 1966, Bieger, 1976, and Duthie-Tooth, 1984, as well as experimental results by Okubo et al., 1970. In a recent investigation by Samuelson, 1990, the design principle given above was checked against experiments and FE-analyses. It was found that the rules give reasonable, somewhat conservative estimates of the carrying capacity of cylindrical shells subjected to local point forces and bending moments.

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4.1.2.1 Cylindrical shells subjected to local loading Fig.4.13 shows the load cases which can be expected to produce deformations of the category discussed above. A solution of the problem was given by Bijlaard, 1954, using the expansion (4.6)

where the coefficients Cmn depend on geometry, location and distribution of load, material etc.: Cmn=f(r, l, t, x0, φ0, E) (4.7)

Fig.4.13. Forces/moments acting on a cylindrical wall and causing local deflections. A program for calculation of the influence of local loads on the deflection has been written and some computer results of the dimensionless deflection w(P)/lr are presented in Fig.4.14. The curves derived are of a general character and can be utilized for design purposes.

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Fig.4.14. Cylindrical shell subjected to a point load. The dimensionless force is defined as a force giving the maximum deflection w(P)=0.01lr. 4.1.2.2 Spherical shells The design methodology developed for cylindrical shells may also be used for sperical caps according to Samuelson, 1990. A point force acting perpendicularly on the shell surface causes

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Fig. 4.15. Point force which gives a characteristic deflection wr/lr=t/2. a deflection w(P) which may be regarded as an initial deflection, see 3.4.1.1B. The BOSOR4 computer program, Bushnell, 1972, was used to determine a relationship between the point force P* and the characteristic deflection wr/lr=t/2. The result is given in Fig. 4.15. Additional information is provided in 3.4.1.5. 4.1.2.3 Conical shells The basic rules given for cylindrical shells should be applicable also for conical shells provided that the radius of curvature and shell length are defined according to section 3.3. 4.2 Effect of holes and cutouts The pure elementary cases are seldom strictly applicable since it is necessary to provide openings for connecting pipes, inspection doors etc. These cutouts will disturb the state of stress and deformation in the shell and can result in a considerable decrease of the buckling resistance. A rather large number of experimental and theoretical investigations have been carried out in order to study the problems involved but the results have not yet been introduced in design codes and recommendations for analysis of shell stability. The directions given below form an attempt to develop a design method leading to a conservative estimate of the resistance of the shell in the presence of a cutout with or without reinforcement. Our knowledge of the problem is still limited and in such cases where the reliability may be questioned it should be advisable to change the structural design such as to be easier to analyze. 4.2.1 Background The buckling behavior of shell structures with openings has been studied by a number of researchers. Tennyson, 1968, performed a test series with plastic cylinders under axial loading and recording at the same time the development of the buckling mode. Similar test series were carried out by Starnes, 1970, 1972, Toda, 1980a, and Miller, 1982b. The results indicate that small holes have an insignificant effect on the resistance, while larger holes may decrease the carrying capacity severely, see Fig.4.16. Numerical analyses were carried out by Almroth et al., 1973, and Toda, 1980b. Large openings of varying shape were studied experimetally by Montague-Horne, 1981, who also proposed a method of analysis. Similar results are given by Knödel-Schulz, 1985, for cylindrical shells subjected to bending (typical for stacks with a horizontal inlet pipe). Unreinforced openings are seldom used, at least not for large diameter shells. Pipe penetrations and nozzles are providing in themselves a reinforcement of the hole and in cases where it is required, an

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177

extra ring stiffener may be added around the cutout. Some sets of results for cylindrical shells with reinforced holes were given by Miller et al., 1983. Introducing Miller’s results in Fig.4.16 it can be observed that the buckling resistance of the shell is not reduced until the size of the hole is at least twice that of an unstiffened hole when the transition point is just reached, see Fig.4.17. This indicates that simple rules may be given for the design if the hole has a diameter which is small in comparison with the radius of curvature of the shell.

Fig.4.16. Test results from buckling of cylindrical shells with holes according to Tennnyson, 1968, Starnes, 1970, Toda, 1980a, Miller, 1982b. 4.2.2 Recommendations for hole reinforcement A reinforcement has two functions which both work to prevent buckling. First it transfers the shell stresses past the hole and thus replaces the cut out material. Secondly, it stiffens the edge of the hole to prevent local buckling which might initiate a global collapse.

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Fig.4.17. The influence of reinforcements on the buckling resistance according to Miller et al. 1983. It should be underlined that the size of the reinforcement should not be exaggerated, in which case it may absorb too much of the stresses and cause stress concentrations in the shell adjacent to the reinforcement. See further section 4.3. The pressure vessel codes recommend a number of types of ring reinforcements designed primarily for internal pressure, Fig.4.18. Since these reinforcements are also used in practical applications where the dimensions have to be designed with regard to stability, it is necessary to discuss their applicability in the case of buckling. Note that the edge reinforcement of the hole has to be checked according to the requirement in for example ASME, BS, DIN, TKN etc., in addition to the case when buckling is critical. A few examples of common edge reinforcement designs are given in Fig.4.18. The different alternatives are acceptable with respect to buckling provided that the requirements established below are met.

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179

Fig.4.18. Practical shapes of hole reinforcements, according to ASME and TKN. 4.2.3 Design rules The influence of the hole on the resistance of a shell is similar to the effect of initial imperfections. However, the hole provides, from the design point of view, a predictable disturbance, and the lower scatter of the test results also indicates that the effect is deterministic. This circumstance implies that the influence of the initial imperfections is dominating for small holes, while the opening will govern the buckling process for larger hole diameters. It is appropriate, therefore, to define a reduction factor ηh which depends only on the hole and the shell geometry and compute the resistance as follows: (4.8) The procedure is defined in Fig.4.19. It may be questioned if a further reduction is necessary in the transition region η≈ηh. In the development of the design recommendation it was decided that the two reduction factors may be applied independently. This was based on the fact that both η and ηh are defined such as to give conservative results. In addition, the test results given in Fig.4.16 indicate that there is little or no interaction in the transition region.

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Fig.4.19. The reduction factor ηh for a cylindrical shell with a small circular hole. Miller, 1982b, made the same suggestion based on the experimental results. The criteria according to eqn (4.8) will, therefore, be applied below. 4.2.4 Cylindrical shell with a circular hole 4.2.4.1 Cylindrical shell subjected to axial load The background of this case was discussed above. Further basic material is presented in Fig.4.20 after Knödel-Schulz, 1985, concerning cylindrical shells with unreinforced holes. In the figure a reduction curve has been drawn which may be estimated to correspond to the 5 per cent fractile of the resistance. A number of test results fall below this curve, probably because they are not quite representative for the test population due to the shape of the hole, the quality of the test, plastic deformations etc. The following expression agrees well with the curve: (4.9)

The test population from stiffened holes is rather limited as indicated above. According to Miller et al., 1983, and Knödel—Schulz, 1985, the same resistance is obtained, broadly speaking, for a hole stiffened according to current pressure vessel codes, e.g. ASME, BS5500 or TKN-87 as for the shell without a hole. The geometry of the reinforced holes of the latter reference comprises of a number of hole geometries and stiffening

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arrangements which differ from those which are discussed in the present handbook. Design analysis for the case with rectangular cutouts are discussed in 4.2.4.6.

Fig.4.20. Definition of the reduction factor ηh for a cylindrical shell with an unstiffened circular hole and subjected to axial loading. Experimental results according to Knödel and Schulz, 1985. Provided that the cross section area of the reinforcement is approximately equal to the area removed, rht, the resistance is reduced by a fact or ηhs as shown in Fig.4.21, which also includes the factor ηh for unstiffened holes. The curve for stiffened holes was derived by multiplying the abscissa values of the curve for unstiffened holes by a factor 2.5. Available test results indicate that this procedure results in values on the safe side. The curve for reinforced holes ηhs can be represented by the function: (4.10)

The additional reduction factor α2 according to eqn (4.8) may be chosen to be equal to 0.9 both for unstiffened and stiffened holes. The reason is that in both cases it is likely that large residual stresses will be present in the vicinity of the hole. Note that the stress level close to the hole, particularly if it is not reinforced, may be so high that it becomes decisive for the design even if the stability is disregarded.

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If the local stress at the edge of an unstiffened hole exceeds the yield limit, it is assumed that the plastic zone may be simulated by a larger hole radius rhef: rhef=rh+∆rh (4.11) (4.12)

(4.13) The analysis should involve an iterative procedure if the local stress is considerably higher than the yield stress. In most cases a one step calculation should give sufficiently accurate results.

Fig.4.21. Recommended reduction factors for cylindrical shell under axial load with unstiffened and stiffened holes. 4.2.4.2 Cylindrical shell subjected to a bending moment For a shell without holes a higher axial stress is allowed, according to Chapter 3, for loading in pure bending than for a uniform axial load. If there is a hole at the section with the highest compressive stress, the design could be based on the same procedure as for an undisturbed shell. However, the opening causes a certain reduction of the section modulus (if unreinforced), and, in addition, an extra disturbance of the stress field in the vicinity of the hole. Therefore, it seems to be reasonable to design according to the rules in 4.2.4.1 with no benefit from the loading condition and computing the axial reference stress without regard to the hole.

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4.2.4.3 Cylindrical shell under external pressure It is hardly likely that unstiffened holes will be used for this load case. Stiffening of the hole edge is supposed to be designed according to applicable codes, e.g. ASME or TKN87, replacing at least 80 per cent of the cross section area of the hole by an edge reinforcement. If these requirements are fulfilled, the design rules for an undisturbed shell are valid. The extra reduction factor α2 should be applied according to tolerance class 2. 4.2.4.4 Cylindrical shell subjected to a torsional moment and/or a shear force This load case shows only a moderate sensivity to imperfections according to Chapter 3, implying that a design procedure for a shell with a hole similar to the case with external pressure is conceivable. It is possible here, however, that unreinforced holes are used, and in that case a considerable reduction of the resistance may result. For design it is consequently presently recommended to use the following procedure: – For unreinforced holes the reduction factor ηh is chosen from Fig.4.21. – For reinforced holes (according to TKN-87) the resistance is calculated as for an undisturbed shell. However, the reduction factor α2 should be chosen according to at least tolerance class 2. 4.2.4.5 Load combinations When two or more load systems are acting simultaneously the resistance is determined by use of the interaction formulas presented in 3.2.1.8. Furthermore, the stress criteria must consider local stress concentrations in the vicinity of the hole. 4.2.4.6 Noncircular openings Special standards apply for shells with holes of circular or elliptic shape and used as support structure for process vessels, silos etc. Using the recommended alternatives for reinforcements, these hole geometries will result in acceptable stress and strain distributions in the vicinity of the hole with the additional result that the buckling resistance is practically uneffected. Rectangular openings are often desired for functional reasons. The stress distribution around the hole will differ from the situation with a circular hole, and this has to be taken into account when designing the shell against buckling. A simplified procedure is described in the following, which should provide adequate results in most practical applications. A. Estimation of the stress distribution Consider a cylindrical shell wall with a rectangular opening of the geometry shown in Fig.4.22. The wall is assumed to be loaded in such a manner that the axial stress at a considerable distance from the hole is σ0. Provided that the area As of the stiffener only replaces part of the sheet which has been cut out, the shaded area in the figure will be

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almost stress free in the axial direction.This means that the load indicated, σ0at, must be compensated by a corresponding increase of the stress at the upper boundary corner of the opening. The nominal stress at the corner will thus amount to about 2σ0, which will govern the design of the shell with respect to buckling. The stress level decreases with the vertical distance from the hole edge and it is reasonable to assume a linear variation with the height. It seems appropriate to use the stress at the height x0 from the edge, see eqn (4.1), in the design against buckling. The procedure given in 4.1.1.2 can then be applied.

Fig.4.22. Cylindrical shell with a rectangular hole. B. Comments The recommendation presented above is rather crude but it provides results of a satisfactory accuracy as is evident from the example in 8.2. The stress distribution is likely to reach a maximum higher than 2σ0 as shown in Fig.8.8, particularly if a/2r>1/4 and the edge reinforcement is heavier than assumed above. Since the stress is decreasing with the height above the cutout, the result with respect to buckling will be rather close to that indicated by the recommended method of analysis. It should be pointed out that the reinforcement at the upper edge of the hole has a small influence on the stress distribution in the “dead triangle”, indicated in Fig.4.22. A circular upper arch is much more efficient from the point of view of reducing the stress concentration, since it transfers the load σ0at successively to the vertical edge stiffeners.

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Other types of edge reinforcements providing an efficient load transfer between the shell and the stiffener can evidently be conceived. 4.2.5 Conical shells with holes Since conical shells show a buckling behavior very similar to that of cylindrical shells, it is recommended that the design information for cylinders is applied with reference made to section 3.3. Particularly, it is important that the radius r is introduced as the radius of curvature

calculated at the center of the cutout. 4.2.6 Spherical shell with holes

The information available for determining the reduction of the resistance due to a hole is limited, and it is presently difficult to provide reliable rules for the design. It is evident, however, that very small holes have a marginal effect on the buckling limit for a shell with normal imperfections, while a reinforced hole of large dimensions in a spherical dome may change the buckling mode to such an extent that even an increase of the critical pressure may occur. The use of reduction factors ηh and ηhf according to Fig.4.21 should therefore be the most realistic alternative for the present. The radius r in the hole parameter

according to the figure should be interpreted as the radius of the sphere rs. 4.3 Effect of local reinforcements

Reinforcements are used in order to increase the resistance of a structure subjected to an external load system, e.g. stringers in the case of a cylindrical shell under axial loading. Occasionally a local reinforcement is required only, as in the silo installation of Fig.4.23, or in connection with stiffening of the shell wall so as to be able to introduce a local load also shown in the figure. Since shell structures are most frequently fabricated from thin sheets and the stiffeners used may be fairly stocky, the difference in stiffness can cause significant stress concentrations. Some examples are shown in Fig.4.24. The first example is a containment vessel mounted on legs which have been extended some distance along the cylindrical wall in order to effectively diffuse the support reaction into the shell. Since the cross section of the leg is heavy compared to the shell thickness, a considerable part of the force will still be transferred to the shell at the upper end of the leg.

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Fig.4.23. Alternative designs of reinforcements in a silo installation. I) is intended as stiffening against buckling due to the support reaction, II) is a reinforcement at a local load introduction.

Fig.4.24. Stress concentrations in cylindrical shells with local reinforcements. If the stiffener profile is clamped to the foundation as in case II) it will cause a local stress concentration in the shell. If the cross section area is very large the local stresses can be considerable and in the worst case cause buckling as shown in Fig.4.25. A reinforcement located at the mid-height of the cylindrical wall as in Fig.4.24, case III) will cause disturbances at both ends, and it is important to estimate the magnitude of the stress concentration, in order to prevent buckling of the shell.

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Reinforcements around holes and cutouts may sometimes give rise to high stresses which can reduce the buckling resistance as explained in section 4.2. This applies especially if the cutout is large and the dimensions of the edge reinforcement are heavy in comparison with the shell thickness. Even an oversize ring stiffener at a moderately large hole can cause disturbances of the stress pattern which may result in buckling. A blunt end of a stiffener will consequently often involve a local increase of the membrane stresses in the shell. Loading in the form of an axial load or an external/internal pressure, will, furthermore, cause radial displacements of the shell which may result in very high local bending stresses in the vicinity of the end of the stiffener. If the shell is subjected to a pulsating load which can cause fatigue damage, it is particularly important to consider the local stresses. Where appropriate, the reinforcement should be supplemented with a ring which distributes the bending stresses in the circumferential direction, see Fig.4.26. The figure shows examples of deformation modes occurring in typical structures.

Fig.4.25. Example on buckling caused by a heavy local stiffener. 4.3.1 Calculation model Consider a cylindrical shell, as shown in Fig.4.27, subjected to an axial load. The shell is assumed to be clamped at the lower boundary and is provided with a vertical reinforcement which is also fixed to the foundation. The length of the reinforcement is denoted by ls and the cross section area by As. The height of the cylinder is supposed to be much larger than 2ls. If there are more than one stiffners, the spacing is denoted by b. The loading causes a vertical compressive stress σx at the upper boundary of the shell wall according to the figure. The lower boundary is assumed to be fixed in the vertical direction, while the upper boundary is assumed to be uniformly displaced a distance ∆ due to the constant stress σx. As a result of the loading, vertical displacements will take place according to the model indicated in Fig.4.27, and the following hypotheses are made in order to estimate the force acting in the reinforcement.

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Fig.4.26. Cylindrical shell with local reinforcement and subjected to axial loading. Large local deformations and stresses occur around the end of the stiffener. A ring stiffener provides a more even distribution of the bending stresses reducing the risk of fatigue failure. – The reinforcement shortening is denoted by ∆ls. – The force is transferred from the shell to the reinforcement through shear characterized by the shear angle γm at the mid-height of the reinforcement (point B). – The stress in the reinforcement is assumed to vary linearly with the height. – The vertical stress is assumed to decrease to a minimum at the horizontal distance ls≤b/2 from the upper end of the reinforcement. Based on these hypotheses a mean value of the force in the stiffener can be computed as: (4.14) (4.15) (4.16)

(4.17)

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The force Fm represents the mean force in the reinforcement at the height ls/2. The maximum force occurs at the lower end (which may possibly form a plane of symmetry) and will be, approximately: Fmax=2Fm (4.18)

Fig.4.27. Cylindrical shell with a local axial stiffener and a model for calculation of the stress distribution. It can be of interest to examine how large a part of the maximum vertical load in the shell that can be absorbed by the reinforcement. Introduce the notations: (4.19) (4.20) Eqn (4.18) may now be written: (4.21)

As*=0 yields:

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Fmax=0 (4.22)

(4.23)

Fig.4.28. Maximum force in a vertical stiffener versus the stiffener cross section area. ls is the length of the stiffener, however ls≤b/2. The variation of Fmax/Fs shown in Fig.4.28, can be utilized to select an optimum stiffness of the reinforcement The maximum force may be used in the buckling design analysis. 4.3.2 Calculation of the compressive stress in the shell From a stability point of view, the maximum membrane stress in the shell just above the reinforcement is decisive for the design and it is, consequently, of interest to estimate the stress distribution by use of the results derived above. According to the model, Fig.4.27, the strain decreases in the vertical direction from the upper end of the reinforcement to the undisturbed region. The total increase in strain, computed as a mean over the stiffener height ls above the reinforcement is: (4.24)

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The assumption that the variation is linear between zero and maximum yields: (4.25)

(4.26) Introduction of eqn (4.17) into (4.26) yields (4.27)

This stress shall be added to the basic stress σx. If the area of the stiffener is large the stress concentration factor will approach the value 3. Note that a diffusion angle of 45° has been assumed, which is more than was recommended in section 4.1. In this case the assumption will, however, lead to a higher stress level, implying a result on the safe side. Finally, the stress level at a distance x0 from the tip of the reinforcement should form the basis for the buckling design analysis:

(4.28)

4.3.3 Calculation of shear stresses If the spacing between the reinforcements is large there may be a risk of shear buckling of the cylindrical shell. The shear strain γm is obtained from eqn (4.15), where Fm is taken from eqn (4.17). The carrying capacity with respect to shear buckling is checked according to 3.2.1.6 and 3.2.1.7. 4.3.4 Other shell configurations What has been said above is valid for cylindrical shells under axial loading. However, the method should also be applicable in cases of external pressure and for other shell geometries such as conical and spherical shells. If the results of a rough estimate according to the rules above are in doubt it is recommended to perform an FE-analysis of the structure. A linear analysis should be

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adequate for determining the stress level, and the buckling analysis may then be referred to the mean stress within an area of width

from the stress concentration.

4.3.5 Comments The recommendations given above are based on the assumption that the buckling limit is a function of the maximum membrane stress in the shell. This is probably too conservative as indicated in the results for local loading given in Fig.4.7. Similar conclusions may be drawn from a recent investigation, Eggwertz and Samuelson, 1991, where FE analyses and test results showed that the effect of the stress concentration may not be as drastic as indicated by the proposed design method. Interaction with the initial imperfections of the shell wall may, however, occur. This was demonstrated in FE-analyses of silo type structures, see Teng and Rotter, 1990.

5 INFLUENCE OF TEMPERATURE 5.1 Temperature gradient through the shell thickness Pressure and vacuum vessels are occasionally subjected to large temperature differences between the inner and outer surfaces. This will normally not give rise to buckling in itself but, due to local yielding the stiffness of the vessel wall may be reduced which results in a lower buckling resistance with respect to other loads. Some possibly critical cases were discussed by Samuelson and Dahlberg, 1983. The results have been condensed in this chapter in the form of instructions for the analysis. 5.1.1 Shell geometry with prevented bending deformation Examples of shell configurations which will normally not be deformed due to a temperature gradient through the thickness are cylinders, cones and spheres. The temperature difference between the inner and outer surfaces of such a shell will instead give rise to a stress gradient through the shell thickness. For quite realistic temperature differences the stress can reach the yield stress at the surfaces of the shell, and at a following external loading further plasticization will occur. The process is demonstrated in Fig.5.1, which shows the stress distribution in a cylindrical wall subjected to an elevated temperature at the inner surface and loaded by an axial compressive force. The temperature gradient produces compressive stresses at the inner surface and tensile stresses at the outer surface. If the temperature difference is sufficiently large the yield limit is reached which results in a nonlinear stress distribution through the shell thickness. The stress due to the temperature gradient is, for elastic conditions, obtained from (5.1) At the temperature difference ∆Ty incipient plasticization occurs. This temperature difference is derived from eqn (5.1) by entering z=t/2 and σT=σy (5.2) If the shell, in addition, is subjected to external loading, the stress distribution will change. Assume first that the temperature difference is so small that the thermal stress is lower than the yield stress, i.e. ∆Ty>∆T. The stress increment due to external loads can either be so small that the shell remains completely elastic or produce a total stress

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causing plasticization, see Fig. 5.1. The latter case results in a decrease of the shell stiffness.

Fig.5.1. Circular cylindrical shell of elastic-perfectly plastic material subjected to a temperature gradient through the thickness giving stresses σT and an axial compression force producing stresses σF. If ∆Ty<∆T, application of external loads will give a redistribution of the plasticized regions, see the example in Fig.5.1. When buckling occurs the shell wall will be subjected to bending. If the bending takes place as in Fig.5.2, case A, it is conservative to assume that only the elastic part of the shell thickness (te) will contribute to the stiffness. In other parts of the cylinder the bending deformation occurs in the opposite direction, Fig.5.2, case B, where the entire thickness behaves elastically. The earlier case will, however, be decisive for the buckling resistance. The following rules for the design analysis are thus based on the simplification that only the elastic part of the shell thickness will contribute to the shell stiffness when buckling. This model of analysis should provide conservative results, particularly since most materials also exhibit strain hardening. If the shell is assumed not to be deformed by the temperature gradient and the external load will only cause pure membrane stresses in the shell, the following expressions can be derived for the thickness of the elastic part of the section. The following notations are used: σF=stress due to external load. For a cylindrical shell, e.g. σF=−pr/t applies for external pressure, and σF=nx/t for axial load. (5.3) (5.4)

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Fig.5.2. Elastic part of the cross section of a cylindrical shell in bending due to buckling. In case A, te
(5.7)

(5.8) The computed value of te is to be used in the buckling analysis according to Chapter 3. In case the resistance, i.e. the external load at buckling, shall be calculated, an iterative procedure may have to be employed. 5.1.2 Other shell geometries The derivation of expressions for the elastic part of the thickness presupposed that the geometry of the shell prevents deformations due to the temperature gradient through the thickness. The other extreme is exemplified by a plate which will suffer a considerable deflection when exposed to a temperature gradient, thus releasing the thermal stress. In

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practical applications, deflections and stresses will often fall somewhere in between these two extreme cases and it may then become necessary to perform the buckling analysis by use of numerical methods. If a temperature gradient causes mainly large deflections and if it is acting over a limited area of the shell, the deformation may be treated as an initial imperfection. This situation will be discussed more in detail in subsection 5.2.2. 5.2 Nonuniform temperature distribution A variation of the temperature over the surface of a shell may lead to buckling, partly since the temperature differences will cause high stresses, partly because it gives rise to nonuniform deformations which may impair the resistance with respect to external loads. Combinations of these effects are conceivable. 5.2.1 Membrane stresses caused by temperature Typical structures, where stresses of this kind are likely to appear, are boilers and heat exchangers, see Fig.5.3. Due to temperature differences between tubes and flues a high axial stress may occur in the tube, and in the worst case cause buckling. With knowledge of the temperature distribution in the shell, the clamping conditions etc., a stress distribution should be calculated to be used as basis for the buckling analysis. For the fire tube in the example above it is possible to apply 3.2.1.2 directly, or possibly 3.2.1.3. In more complicated situations an analysis using numerical methods according to Chapter 6 is recommended. Thermal loads give rise to a kind of displacement governed buckling. It is sometimes argued that this would lead to different conditions for the buckling analysis than those applied in load governed buckling, but that is not the case. The buckling resistance is determined in exactly the same manner but, the consequences of a loss of stability can be less drastic. In some cases it may be justified to apply a lower safety factor. This has to be considered in each specific case in accordance with the applicable design code. 5.2.2 Deformations caused by nonuniform temperature distributions If local heating of a shell takes place as indicated in Fig.5.4, the heated part will expand and will thus cause a local deflection which can be considered to be equivalent to an initial buckle. The resistance with regard to an external load will then to a great extent depend on the shape and amplitude of the deflection curve. It is reasonable to assume that the recommendations given in Chapter 4 concerning allowable deflections from local forces will be valid.

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Fig.5.3. Heat exchanger in which the cylindrical shell may buckle due to thermal stresses.

Fig.5.4. Change of the shape of a shell subjected to local temperature variations. The calculation of deformations caused by a nonuniform temperature distribution is conveniently carried out by use of an FE-program and a linear analysis is estimated to be adequate. A subsequent buckling, or collapse, analysis must of course be based on a fully nonlinear analysis.

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5.3 Creep buckling Materials which deform gradually with time under constant loading are concrete, polymers and metals. Most metals exhibit creep only at elevated temperatures, while the other materials will often creep at room temperature. A structure exposed to a load which may cause buckling and which is fabricated of a material subjected to creep, can collapse as a result of creep buckling. Creep deformation is, at least for metallic materials, a strongly nonlinear function of the stress in the material and an accurate creep buckling analysis can become very complicated. In certain cases it is possible to make simple estimates based on hypotheses of e.g. “critical strain” with reference to buckling, where the experiences from elastic buckling may be utilized. In more complicated cases it is necessary to carry out a complete collapse analysis considering the variation of the state of stresses and strains with time. Design codes sometimes specify a maximum allowable creep strain, e.g. 1 per cent. This value applies to tensile loading in structures such as pressure vessels and pipe lines subjected to internal pressure. In cases where creep buckling may occur, the critical strain may be considerably less than 1 per cent and quite different criteria have to be introduced as a basis for the design. Creep buckling has been studied to a relatively small extent and the experience from applications in practice is rather limited. Theoretical and experimental investigations have provided sufficient data, however, to allow a reasonably reliable design analysis. 5.3.1 Creep buckling in general Creep buckling exhibits a few peculiarities which, at first sight, may seem somewhat puzzling. Since the creep rate in many cases is proportional to a power of the stress, it might be assumed in the first place that a reduction of the stress, for example by increasing the thickness, would give a drastic increase in the life time. Buckling is also initiated by shape deviations of the shell and by adopting narrow tolerances, the life might be assumed to increase exponentially. In both these cases the increase in life will be less than expected, and it is often difficult to design for an adequate life only by increasing for instance the shell thickness or column bending stiffness. In order to demonstrate these problems some examples are given below, see Samuelson, 1984. First, consider a straight bar subjected to a centric compressive force P. In elastic buckling the theoretical resistance Pel is given by eqn (1.1). If the force P
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where the power n is of the magnitude 3–7, it is understood that the curvature of the bar will grow with time and cause an increase of the bending moment. This leads to a further rise of the creep rate and the deflection rate will accelerate. Fig.5.6 shows a typical creep buckling curve for a bar with rectangular cross section. The marked nonlinear relationship between stress and deformation makes the problem difficult to deal with in a simple and still rigorous manner. Moreover, some of the principles valid in the theory of elastic stability are not applicable in creep buckling:

Fig.5.5. Curved and eccentrically loaded bars respectively.

Fig.5.6. Creep deformation of the bars in Fig.5.5 subjected to equal stresses.

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Fig.5.7. Bar profiles with equal moments of inertia.

Fig.5.8. Bars in Fig.5.7 under the same compressive force. The elastic buckling load is proportional to the moment of inertia I of the bar. Thus, the buckling loads of the two bars in Fig.5.7 are identical. If they are subjected to the same load, see Fig.5.8, the stress level will be higher in the T-section bar and the creep rate will

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become higher. The creep life is considerbly shorter than for the bar with rectangular cross section. Furthermore, the creep life depends on the direction of bending of the Tsection bar.

Fig.5.9. Bars according to Fig.5.7 subjected to the same mean stress. In the case where the two bars are subjected to the same mean stress, the life of the Tsection bar is higher, see Fig.5.9. The initial curvature affects the creep life, see Fig.5.10 which shows the creep buckling curves for a bar with rectangular cross section.

Fig.5.10. Bars with rectangular section according to Fig.5.7 with different values of the initial curvature.

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The diagram reveals that the creep life is a logarithmic function of the amplitude of the initial deflection, which is a result contrary to what might be expected. This proves that the creep buckling life cannot be extended significantly by introducing closer tolerances! In practical applications the creep buckling life is likely to vary only moderately, since the shape deviations are mainly of the same magnitude for various structural elements. Finally, it may be interesting to note that the stress distribution across the bar section becomes highly nonlinear as a result of the creep deformation, as demonstrated in Fig.5.11.

Fig.5.11. Stress distribution in a bar with rectangular section subjected to creep buckling.

Fig.5.12. Creep buckling of cylinder subjected to external pressure.

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In creep buckling of shells the behavior found is often similar to that found for straight bars. Fig.5.12 shows a cylindrical shell subjected to external pressure where the material deforms gradually with time. In this case the load was assumed to be slightly unsymmetric, corresponding to an initial deformation, and the deflection of the shell was computed as a function of time. The result shows that w→∞ at a finite time, which can be denoted the critical time tcr. A cylindrical shell under axial loading behaves somewhat differently as is evident from Fig.5.13. If it is assumed that the shell is perfect it will collapse in an axisymmetric mode corresponding to the solution in eqn (1.9), see also Fig.1.16. In practice small nonsymmetric disturbances are always present, which were simulated in the computer analysis by a slightly nonsymmetric external pressure. Due to these disturbances an unsymmetrical deflection develops. In contrast to the case of external pressure, the nonsymmetric deformations remain small until the critical time is approached. The growth rate increases rapidly just before collapse and the behavior may be interpreted as a bifurcation even if it is rather a collapse. The results given in Fig.5.13 are in good agreement with the experimental behavior which was observed and filmed by Samuelson, 1969.

Fig.5.13. Creep buckling of a cylinder under axial loading. 5.3.2 Design against buckling Very few closed form solutions of creep buckling problems are available in the literature. The designer is mostly left with the options to either perform a rigorous, nonlinear FEanalysis of the structure or to estimate the creep life by means of approximate methods. The design condition is that the creep life must safely exceed the planned service life. Since the life time is strongly dependent on the stress level the required factor on time should be around 3–10, corresponding to a factor of 1.3–1.5 on the mean stress of the structure. This depends on the creep deformation being proportional to σn with 3
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common metallic materials. Furthermore, the scatter of material data is often rather large and, if the values obtained from data tables are based on mean values, it is necessary to take this into account when determining allowable stress levels. When the calculations are carried out by use of the elementary cases, normal values of load and resistance factors, alternatively safety factors, can be used. If the design is based on a complete numerical analysis the question of appropriate reduction factors has to be discussed in each case. It is important that initial deformations, symmetry of the load distributions etc. are considered in a systematic way. If this has been achieved, a factor of tcr>10tall should be desirable. 5.3.2.1 Numerical creep buckling analyses An analysis of the creep progress in a shell under general conditions, including nonsymmetric imperfections and loads, nonlinear stress distribution etc., can only be carried out by use of a computer program. Under these presumptions the realization of the analysis has to be based on the directions given in Chapter 6 concerning nonlinear material properties. Creep deformations will introduce a number of problems, however, which are different from those found in e.g. plastic buckling of shells. Some of these problems will be discussed below. A. Axisymmetric problems Special purpose programs have been developed for axisymmetric shells subjected to symmetric loading, see e.g. BOSOR 5, Bushnell, 1974. The program calculates deformations and stresses (axisymmetric) as functions of time (and load if it is varying simultaneously) and it is sometimes possible to calculate the variation of the buckling load (bifurcation) with time. In certain cases this method is not likely to provide a realistic value of the creep life, for example in the case of a cylindrical shell wall subjected to external pressure, see Fig.5.14. The creep strain will result in a decrease of the diameter of the cylindrical shell, and in the extreme case with very large deformations a “hammock effect” will occur which relieves the stresses in the circumferential direction. A buckling analysis will, therefore, not predict a creep buckling life in the case of uniform external pressure. Test results, and calculations presented in Fig.5.12 above, show that the nonsymmetric initial deformations lead to deflections increasing with time and eventually to a limitation of the creep life.

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Fig.5.14. Creep stresses in a cylindrical shell wall under external pressure in the ideal case of purely axisymmetric deformation. A cylindrical wall subjected to axial loading will develop, with time, a zone with compressive stresses close to the boundary, see subsection 1.3.3.2, and a time dependent buckling analysis may in this case provide a realistic estimate of the life. See Obrecht, 1977. Summarizing, it can be established that creep buckling analysis for axisymmetric problems, including a prediction of a nonsymmetrical buckling mode, may in some cases be carried out by use of special purpose programs. It is important, however, to take into account the limitations of the computer model. B. Axisymmetric shell, nonsymmetric load It was shown in subsections 3.2.1.3 and 3.2.1.5 that these problems can be analyzed approximately for elastic shells by use of linear stability theory. According to subsection 5.3.2.1 this is not feasible in the case of creep buckling. Here it is necessary to solve the complete system of equations for the shell, which can be effected with the aid of general FE-programs or with special purpose programs according to e.g. Samuelson, 1975. See Figs. 5.12 and 5.13. C. General creep buckling problems Shell structures of a general configuration subjected to nonuniform load distributions, local forces etc. have to be analyzed by structural programs which can handle nonlinear material properties and large deformations. The general rules given in Chapter 6 are applicable in this case.

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5.3.2.2 Approximate creep buckling analysis Several efforts have been made to establish a simple method for calculating the critical time in creep buckling, see e.g. Gerard, 1956, and Gerard-Gilbert, 1958. The method which will be recommended in the following is based on the hypothesis that buckling takes place at a critical strain independent of the kind of strain, elastic, plastic or creep. An attempt is made to formulate a general calculation procedure and this is illustrated by a couple of examples. A. Calculation of the critical strain In Chapter 3 the buckling resistance is presented for a large number of elementary cases, where the membrane stress is used as design variable. A given stress level corresponds to a certain strain, which, in the elastic case, can be determined by use of Hooke’s law. The carrying capacity of the shell is given in the elastic region by the following formula, where the reduction factors α and η are defined in Chapter 3 for a number of elementary cases: σelr=αησel (5.10) The extra safety factor 0.75 has been omitted here, since the stress level for creep buckling is lower than that of instantaneous buckling and, furthermore, the assumption of a constant critical strain should give conservative results. It is now assumed that the critical strain can be computed from; (5.11) The definition of a critical stress/strain seems to be natural for those cases where the stress is constant over the shell. In other cases the selection of a reference value is more difficult due to the fact that creep is a highly nonlinear process and the critical time will depend strongly on the stress level. Some recommendations can be given, as follows, but it must be strongly emphasized that the final result may sometimes be fairly uncertain. B. Structures with a stationary state of membrane stress. Examples of shells discussed above are cones or spherical caps according to Fig.5.15.

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Fig.5.15. Examples of shells with a membrane state of stress which does not vary with time.

Fig.5.16. Stress distribution in a circular cylindrical shell under bending moment and subjected to creep, as well as a computed strain in extreme fibres according to Samuelson, 1967. Furthermore the strain rates are given based on the maximum elastic and the steady state stresses.

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In the case of external pressure, the hoop stress close to the lower boundary will determine the buckling limit, and the analysis of the creep buckling process should be based on this stress. To assume that the creep strain rate is a function of the maximum stress, calculated from the linear elastic theory, according to Fig.5.16 can provide a very inaccurate result as is evident from the figure which presents the computed stresses and strains at the extreme fibre of a straight beam subjected to a bending moment. The diagram also shows the creep strains based first on the elastic maximum stress and secondly on the maximum stress at steady state. It is obvious that the selection of reference stress may be of great importance for the computed strain.

Fig.5.17. Straight beam subjected to creep when loaded by a constant bending moment. a) Strain rate versus time, b) Strain rate and maximum . stress in the steady state The variation of the creep rate with time is shown in Fig.5.17 a) with the creep exponent as a parameter. b) gives the stress in the extreme fiber and the creep rate in the steady state situation as functions of the exponent n. In an evaluation of the critical strain for estimation of the creep buckling time it is sufficient to use the stress and strain rate values in the steady state according to Fig.5.17 since the primary creep may often be neglected. D. Creep mode Engineering materials may have significantly different creep properties which require description by use of different material models. The creep rate of metallic materials is, however in many cases adequately described by Norton’s law, which is used here to demonstrate a simplified design strategy for creep buckling analysis. (5.12)

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Provided that the stress level is constant, the total strain can be computed as a function of time using the Hooke-Norton relationship: (5.13) where εpr=primary creep (which is usually neglected). Furthermore, material data tables very seldom supply information on other parameters than B and n. (which often have to be derived by use of the 1 per cent creep stresses) In case the stress level varies during the service life, the total creep strain is the sum of the creep strains at the different load levels (Compare the Palmgren-Miner linear cumulative damage theory in fatigue). The total strain may be expressed as: (5.14)

In case the temperature would also be a variable the material parameters E, B and n will be functions of time. E Calculation of the creep life Provided the stress is constant, the total strain εtot can be determined according to eqn (5.13) When the strain value reaches the critical value according to eqn (5.11), creep buckling is assumed to occur, i.e.: (5.15) Solution of tcr gives: (5.16) Similar expressions can be obtained if other models for the creep properties of the material are employed than Norton’s law, eqn (5.12). With varying stress and/or temperature it is more difficult, however, to derive a closed form expression for the creep buckling life. It is recommended to carry out a direct summation of the creep strain from eqn (5.14), with the design condition: (5.17) It may be tempting to introduce creep models which describe accurately the creep curve obtained from tension test specimens in order to improve the accuracy of the prediction of the creep life of the shell. It is not clear, however, whether existing models are applicable e.g. for a varying stress level or multiaxial stress states. Further research is required within this field to verify available theories or to develop new ones. For the present time, Norton’s hypothesis should be satisfactory, particularly considering the high

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uncertainty concerning the life estimate due to the scatter normally observed in material creep data. The safety requirements are satisfied if (5.18) Example A pressure vessel with a torispherical end closure according to Fig.5.18 is subjected to external pressure at an elevated temperature. Determine the creep life of the vessel:

Fig. 5.18. Pressure vessel subjected to external pressure and elevated temperature within the creep range. The calculation of the allowable service life is performed as follows: a) Cylindrical shell wall: (3.33) σφelr=0 .9·0.7·118=74 MPa εelr=74/180000=0.00041

The safety margin against elastic buckling is satisfactory. Assuming a safety factor of 1.5: (5.16)

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b) End closure (3.113) (3.110) (3.119)

(5.16) The creep life of the end closure will thus be around one third of that of the cylindrical shell wall although the stress levels are equal.

6 STABILITY ANALYSIS BY USE OF NUMERICAL METHODS The rapid development of computers and computer programs during the last few decades has made possible stability analysis of complicated structures by use of numerical methods. Some of the commercially available systems are referenced in Section 1.2 and in the following only general remarks will be made regarding these methods. Examples given in the text will, of course, be based on the use of specific computer codes. One of the main concerns in stability analysis of shell structures is the fact that the actual load carrying capacity of the shell is often only a fraction of the value indicated by the buckling theory. This is true if the stability analysis is based on classical buckling theory. Most computer codes have options for linear bifurcation analysis which gives results equivalent to those of the classical theory. Since use of the computer is becoming increasingly popular in the design of shells, it must be emphasized that it is very important that the results are interpreted correctly in order to arrive at a safe design. In the present chapter, the associated problems are discussed and recommendations are given on how to perform complicated shell buckling analyses. 6.1 Parameters influencing the accuracy of a theoretical prediction A number of factors are important when trying to estimate theoretically the stability limit of a shell structure. This should be evident from the discussions in previous chapters. It is not, therefore, immediately clear how to perform a stability analysis by use of for instance a general FE code. And it is not, in general, possible to define a suitable reduction factor in a case where only some of the important parameters have been considered in the numerical calculation. This dilemma will be demonstrated in the following: Parameters of importance for the load carrying capacity of a shell structure are: * Shell geometry * Type of loading * Initial imperfections * Built-in stresses * Disturbances in the load distribution * Local forces * Local reinforcements causing stress concentrations * Cutouts and local variations in wall thickness * Flexibility of supporting structure * Variations in temperature, etc.

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In most cases a combination of the different parameters will have to be considered in order to arrive at an accurate estimate of the true carrying capacity of the shell. Therefore, it is not possible to provide general rules on how to carry out a rigorous analysis and, in fact, very few attempts in that direction have been noted in the literature. The recommendations given in the present work, therefore, represent a first attempt to provide a rational method for interpretation of numerical stability analyses for shell structures. For obvious reasons a certain amount of conservatism will be aimed for since the choice of appropriate reduction factors may be very difficult. 6.2 Options for stability analysis Several possibilities exist for stability analysis of shells resulting in estimates of the shell carrying capacity of varying accuracy. The options available today will be demonstrated by use of a spherical cap which represents a very common shell element, which is also very sensitive to various types of disturbances. By application of different solution techniques it is possible to simulate the buckling behavior very accurately while for instance the linear buckling analysis may overestimate the collapse pressure by a factor of 2 to 10. The shell geometry, material properties and loading conditions are defined in Fig.6.1.

Fig.6.1. Spherical cap subjected to external pressure. Stability analysis by linear or classical bifurcation theory.

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6.2.1 Linear stability theory Stability analysis by use of bifurcation theory requires that the state of stress in the shell prior to buckling is known, see Section 1.3. Normally, this is achieved by use of the linear theory of stress and strain available in most codes for structural analysis. Linear theory implies that the stress and deflection distributions of the shell do not vary with the load level and it is relatively easy to inspect the conditions of equilibrium for possible adjacent solutions. The inspection procedure is conveniently defined as an eigenvalue problem and the buckling load is given by the bifurcation point. Fig. 6.1 shows the linear solution and the bifurcation point for the sample spherical cap calculated by use of BOSOR4, Bushnell 1972. It may be noted that the linear theory yields a membrane stress which is nearly uniform and that the linear buckling pressure Pel=1.02 bar is very close to the classical value given by eqn.(3.112), Pel=1.0 bar. The carrying capacity pu as defined in 3.4.1.2 is, however, only a fraction of the theoretical estimate, pu=0.16 bar. Linear bifurcation and classical stability theory thus give equivalent, extremely nonconservative results. It is extremely important that the program user applies adequate reduction (knock down) factors to the buckling loads calculated by use of the line bifurcation option. The possibility to use linear bifurcation analysis is not restricted to shells of revolution or other special types of shells. Any type of structure may be analyzed as for instance demonstrated by Brush and Almroth, 1975. Results for a point loaded venetian blind type shell are shown in Fig. 6.2. Again it is evident that the linear bifurcation buckling load is a poor estimate of the true carrying capacity of the shell.

Fig. 6.2. Stability analysis of a shell with a nonhomogeneous prebuckling state of stress according to BrushAlmroth, 1975.

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6.2.2 Nonlinear buckling analysis When the spherical cap, Fig 6.1, is subjected to an increasing pressure, the geometry of the shell will gradually change and the middle surface flattens out. This leads to a decrease in stiffness and the deflections increase faster than predicted by the linear theory, see Fig. 6.3. If the deflections increase it takes higher stresses to maintain equilibrium and the shell becomes even more susceptible to buckling. It is possible to formulate the stability conditions at a given nonlinear prebuckling state based on the tangential stiffness of the shell.

Fig. 6.3. Stability of a spherical cap considering the nonlinear prebuckling deflection. In the example, buckling was found to occur at a load level of 65 per cent of the linear bifurcation pressure as indicated in Fig. 6.3. The nonlinear bifurcation analysis thus appears to provide a better estimate of the carrying capacity of the cap. However, the recommended design value for the pressure pu is still significantly lower than this improved theoretical estimate. 6.2.3 Nonlinear collapse analysis The calculation of the buckling pressure in 6.2.2 was based on nonlinear axisymmetric analysis of the prebuckling state. If the analysis is continued above the bifurcation point, symmetric snap through (collapse) will be predicted at a slightly higher pressure, Fig. 6.4. In the example, the symmetric collapse and nonlinear bifurcation pressures are fairly close. In other cases the difference may be much higher. Fabricated shells always contain initial imperfections which are in general nonaxisymmetric. These disturbances influence the stability behavior of the shell and it is not clear that the ‘nonlinear bifurcation buckling analysis’ as described above is a

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reasonable estimate of the true behavior of a shell of revolution. In general it is necessary to include the nonsymmetric initial imperfections in the analysis. Special purpose programs have been developed for solution of nonlinear collapse problems. Ball, 1968, developed a program based on the general shell equations. He used a Fourier series representation for the circumferential variation of the variables and solved the resulting system of equations by use of FD-technique with respect to the meridional coordinate: w(s)=w0 (s)+w1(s)cosφ+w2(s)cos2φ+…. (6.1)

Fig. 6.4. Symmetric snap through of a perfect spherical cap. Examples given by Samuelson, 1975, show that a small disturbance in the load may give a drastic reduction of the collapse load of the shell. One advantage of the method is that it clearly shows the development of the critical buckling mode. It is also faster than traditional FE-analysis since repeated symmetry conditions are easily considered. In the case of a spherical cap, it may be adequate to study the behavior in the presence of a symmetric imperfection at the apex. This case is conveniently solved by use of BOSOR4 and may provide the information needed to estimate the buckling resistance (collapse load). A couple of calculations for different imperfection amplitudes were carried out for an imperfect cap, Fig.6.5. The results do not deviate noticably from the nonsymmetric solution according to DSOR 06, Samuelson, 1975. This outcome indicates a feasible procedure which sometimes can be used to evaluate the influence of e.g. initial imperfections of a spherical cap. The method cannot, generally, be used to estimate the sensivity to nonsymmetric shape deviations etc. in other types of shells.

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Fig.6.5. Symmetric collapse analysis of an imperfect spherical cap subjected to external pressure. 6.2.4 Actual buckling resistance—test results In design work it is required that the calculated buckling limit is a good estimate of the actual resistance of the shell, i.e. the value which would have been obtained in a realistic experiment. Today, available design information is based on extensive test series as described in previous chapters. These tests were, in most cases, performed with simple shell elements under uniform loading and the results have provided a lower bound curve for the resistance as shown in Fig. 3.43. It should be emphasized that this boundary curve is governing for the design, possibly with some allowance for the influence of the manufacturing tolerances. Irrespective of what computer/design tool is used, the result should approach this curve. However, it may be of interest to evaluate the carrying capacity of a fabricated shell. Then, a rigorous analysis of the shell including modeling of the initial deformations, measured wall thickness etc. should be carried out. This solution should be close to the actual resistance of the shell as shown in Fig.6.6. The diagram displays test results from Yamada-Yamada, 1983, compared to theoretical values computed with regard to measured shape deviations. The agreement is excellent. The results for the spherical cap demonstrated in the previous sections may be summarized in the following way: a) The linear buckling analysis gives results equivalent to those of the classical theory and the full reduction factor η is applicable (or ωs for plastic buckling). b) A nonlinear buckling analysis may provide a considerably more realistic estimate of the buckling load, but the result is still nonconservative as compared to test results. It would be possible to define a ‘nonlinear reduction factor’, but so far no attempts have been made in that direction.

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c) Collapse analyses for the imperfect shell provide more realistic results. It seems to be feasible to define a worst initial buckle which could be used for computing the resistance. Such a calculation will, however, often require considerable computer power as well as experience from similar computations. Consequently the procedure is hardly suited for design purposes. d) In many cases, particularly for shells fabricated from mild steel, the resistance is furthermore reduced due to plastic deformations, which has to be considered in the design. Normally, this reduction is made empirically and based on the classical buckling load of the elastic shell. Experience from buckling analysis in the elastic region and comparisons with experimental results are lacking to a great extent. Such calculations are of value, however, since they contribute to the understanding of the problem.

Fig.6.6. Comparison between results of calculations and tests for spherical caps, according to Yamada-Yamada, 1983. Based on the comments above, it can be stated that the experience concerning buckling analysis is most extensive with respect to linear (classical) theory where rather well defined reduction factors have been established. So far, no adequate strategy for numerical solution of nonlinear problems has been proposed which would provide a satisfactory level of accuracy and thus a well-substantiated reduction factor. Collapse analysis with a nonlinear theory should, consequently, be performed with care and the results be judged considering to which extent the influence of important parameters has been included. In the following, various possibilities to utilize computer programs in stability analysis will be discussed and an attempt will be made to systematize the choice of reduction factors regarding the nature of the problem, selection of calculation method etc.

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6.3 Classification of stability problems Some types of shells subjected to certain types of loading are extremely imperfection sensitive. It is not possible to classify shell stability problems in a fully systematic way and, for instance, if shell structures are analyzed by use of an FE program, different reduction factors would apply in each case. The recommendations given in the following form an attempt to summarize the experience available and give rational rules on how to choose adequate reduction factors to apply to numerically determined buckling loads.

Table 6.1. Recommended procedure for determining reduction factors primarily for axisymmetric shells. Type of shell

Loading

Recommended reference case

Axisymmetric shells Axisymmetric

Closest elementary case according to Chapter 3

Cylinder-cone varying thickness

Mainly axial

η from 3.2.1 or 3.3.1, or possibly section 3.4.

External pressure

See 3.2.1.5 and 3.3.1.5 respectively

Load combinations

See subsection 6.3.2.

Spherical or torispherical

External pressure, Mainly uniaxial loading

See sections 3.4 and 3.5.

Combined shells (cone-cylinder etc.)

Axisymmetric

For each shell element, as above. If buckling involves interaction between different elements the most disadvantageous reference case should be used.

Nonsymmetric

Bending moment etc. treated in resp. elementary case. Pressure distribution, see 3.2.1.5 for ex ample concerning wind load.

Nonsymmetric

Complete nonlinear analysis is recommended.

General

Reference cases lacking, see subsection 6.3.3 concerning recommended calculation procedure.

Nonsymmetric, general shells

6.3.1 Typical shells and loading conditions Shells of revolution are very common in practical applications. Since the theoretical analysis is comparatively simple because of the rotational symmetry, it has been natural to develop special purpose programs for these types of shells. A fully nonlinear analysis of the nonsymmetric collapse behavior mostly requires experience, extensive work and computer analyses. The extra cost may not be motivated at the design stage. In the following table, reference is made to the various cases treated in Chapter 3. The idea is that, when analyzing a general shell, the reduction factor to be applied should be chosen such that it is lower than or equal to that of the reference case. Sometimes the choice is

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obvious, if not it is recommended to choose values on the safe side, preferably related to a similar, more severe, elementary case. 6.3.2 Reduction factor η for load combinations In the analysis of elementary cases, the buckling stress level was determined for different loads separately and load combinations were taken into consideration through interaction formulas. When using a computer program for a buckling analysis, it is of course possible to apply the loads simultaneously. For instance, if a cylindrical shell is subjected to axial compression and external pressure, the buckling limit is a function of the load level and the stress ratio. Since the two types of load require different reduction factors according to 3.2.1.2 and 3.2.1.5, it is not immediately clear how to define an appropriate reduction factor for the combined loading. The treatment of load combinations thus depends on the type of shell and loading. Two methods may be recommended: Method No 1: If the shell geometry and the loading conditions are such that the load can be divided into elementary cases according to 3.2, the interaction relationship between the elastic buckling stresses, σel, can be applied, eqns (3.55–3.58). The result is an interaction curve for the reduced buckling stresses, σelr, where the applicable reduction factors αη and/or ωs are used for the individual load cases. Method No 2: In many cases (possibly most of those where there are reasons to use numerical methods) it is not possible to separate the loading into individual load cases. It is then convenient to use the stress components σxel and σφel corresponding to the buckling load determined in the numerical analysis for the current load condition. The reduced stresses are calculated as 0.75αησxel and αησφel (or ωs2σy or ωs1σy respectively if buckling occurs in the plastic region). The stress combinations may subsequently be checked with the interaction formulas in 3.2.1.8. See 6.3.2.2. 6.3.2.1 Analysis according to method No 1 The method is demonstrated for a cylindrical shell under axial compression in combination with an external pressure according to Fig.6.7. A computer program (e.g. BOSOR 4, Bushnell, 1972) is used to calculate the buckling load for various load combinations. For each of the two load cases the critical stresses σxel0 and σφel0 calculated separately will deviate only marginally from the values obtained with the formulas in Chapter 3. The buckling stresses σxel and σφel for a few load combinations are computed and the results are plotted in a diagram according to Fig.6.7. An interaction curve for the buckling stress according to classical theory is obtained by connecting the points corresponding to the load combinations. According to Chapter 3 the buckling resistance σelr is determined by reducing the buckling stress based on classical theory σel by the factor αη. However, this factor is smaller for axial compression than for external pressure. An interaction curve for σelr is

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obtained by multiplying σxel by αη refered to axial loading and σφel by αη for external pressure. This procedure is indicated by the dashed lines in Fig. 6.7.

Fig.6.7. Interaction relationships for a cylindrical shell under combined loading. The reduction factor αη for combined loading can now be defined as the ratio between the distances a and b in the Fig. 6.7. αη varies with the σx/σφ ratio. The reduced buckling stresses are denoted σxelr and σφelr. If buckling takes place within the elastic region, i.e. if the effective stress : (6.2) the resistance is determined as: σefu=0.75σefelr (6.3) which should be compared with eqn.(3.8) When σefelr>σy/3 plastic buckling is considered by computing the slenderness ratio from: (6.4) and ωs2 according to 3.10 yielding the resistance σefu=ωs2σy (6.5)

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6.3.2.2 Calculation where the loading is not separated in elementary load cases If the loads cannot be separated in elementary cases the following procedure can be applied. The buckling load is calculated and the stresses in the meridional direction σsel and σφel in the circumferential direction are evaluated. See Fig.6.8. Assume that these stresses correspond to fictitious load cases which can be analyzed separately. The reduced stresses σselr and σφel corresponding to the fictitious loads are computed. For instance, σs is assumed to be caused by an axial force, i.e. αη is obtained according to 3.2.1.2, while σφ is assumed to result from external pressure, implying that αη is determined according to 3.2.1.5. The stress combinations are then checked using the interaction formulas in 3.2.1.8, i.e.: (6.6) where

Alternatively, the procedure described in Fig. 6.7 may be applied yielding a more direct evaluation of the interaction curve.

Fig.6.8. Stresses in the meridional and circumferential directions.

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If the shell is subjected to additional stresses, e.g. shear, their influence is considered according to the rules given in 3.2.1.8. 6.3.3 Nonlinear analysis If all governing parameters are known (load distribution, initial deflections, residual stresses etc.) a complete nonlinear collapse analysis should provide the best estimate of the actual resistance of the structure. It is essential, however, that all parameters are taken into account and, consequently, the analysis will require a broad experience from the analyst. Since this is the only possibility in many cases to achieve a close estimate of the resistance, it should be pointed out that experience from similar analyses often is essential. In a nonlinear stability analysis, three different strategies may be followed: – the load is applied successively in steps (increments), the program evaluates deformations and stresses at the end of each increment. – the deformation is prescribed in some part of the structure, the program calculates the deformations in the other parts and the stresses. The load corresponding to the imposed deformation is obtained as a reaction force. – arch length method, which may be simply described as moving along the loaddisplacement curve, yielding both loads and displacements corresponding to an equilibrium state. The last method possesses the capability of computing the resistance beyond the maximum of the P(δ)-curve and may sometimes be preferable in particular if the postbuckling resistance is of interest, see Riks, 1979. Irrespective of the strategy chosen the program has to perform a number of iterations within each load increment in order to find a solution in equilibrium. The operator must indicate a tolerance criterion (e.g. maximum allowable unbalanced forces, or maximum deformation increment between two iterations). The stiffness of the structure can either be updated within each iteration (full Newton) or at the beginning of each load increment and then kept constant during the iterations (modified Newton-Raphson). There are also several methods available for updating the inverted stiffness matrix (of the type BFGS). In general, modified Newton technique requires more iterations, but is least expensive per iteration. The full Newton method is more expensive per iteration but converges faster. A complete calculation procedure for buckling analysis with numerical methods should include the following steps: 1. If the problem is sufficiently close to one of the elementary cases presented in Chapter 3, the design recommendations given for that case should be used. 2. An FE-model should be established and a linear static analysis performed in order to check that the model, loads and boundary conditions are correct. Inspect the state of stress, in particular the ratio between the bending and membrane stresses. Stiffeners should be modeled by shell elements if local buckling can occur, otherwise beam elements are preferable. 3. Perform an eigenvalue analysis and study the form of the lowest eigenmodes. Check if the element grid appears adequate to describe the buckling mode (has a sufficient number of elements per half wave). Select a few nodes which are subjected to large

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deflections to be plotted as functions of the load during the nonlinear analysis. The nodes showing maximum deflections according to the eigenvalue analysis are likely to be relevant in this context. The eigenvalue analysis does not always provide a realistic value of buckling load and buckling mode (Fig.6.2). The result can be used, however, in a preliminary comparison of the relative capacities of different design alternatives. 4. If the ratio between the bending and the membrane stress according to point 2 is very small, the structure will be sensitive to initial imperfections. Relevant initial deviations from the ideal shape should be included, e.g. in the form of the buckling mode according to the linear bifurcation theory and multiplied by a scaling factor. 5. Carry out a geometrically nonlinear analysis, e.g. utilizing the arch length method, and plot load-deflection curves for the nodes selected under point 3. above. 6. Finally, a check is made if the stresses exceed the yield limit for the current load range. If that is the case, the analysis should be repeated including nonlinear material parameters. 6.3.4 Choice of program and structural model When solving structural mechanics problems with numerical methods a discretization of the model must be introduced. Working with the general equations, such as those displayed in Fig.1.24, it is convenient to define the values of the fundamental values (e.g. u, v, w) at a number of discrete nodes and calculate the various derivatives by use of Taylor’s formula. This method is usually referred to as the finite difference method (FDM) and is, for instance, utilized in the BOSOR program. The finite element method (FEM) is basically developed from the general equations, but the discretization occurs at an earlier stage, defining a calculation model which is very similar to the structure itself. The two computational techniques are illustrated in Fig.6.9. Most comprehensive structural computer programs are today based on FEM. FDM is used in some special programs. The final results obtained with the two methods are, however, equivalent, and the discussions below concerning assumptions and analysis results are in general valid for both methods. 6.3.4.1 Choice of program Different options for analysis of buckling problems were described in section 6.2 from which the following recommendations can be summarized: a) An axisymmetric structure subjected to an axisymmetric load may preferably be analyzed by use of a program of the BOSOR type. It is important that nonsymmetric buckling can be predicted in the analysis. Note that the buckling load has to be reduced with regard to initial deformations etc. according to Chapter 3.

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Fig.6.9. Finite difference (FDM) and finite element (FEM) models, applied in shell analyses. b) If the structure according to a) can be subjected to a nonsymmetric load, it is possible to obtain an approximate solution using an (equivalent) axisymmetric buckling analysis. If the shell is loaded in bending or is subjected to a wind induced pressure, the result may be satisfactory provided that reduction factors according to Chapter 3 are applied (membrane stresses are assumed to dominate). If the load distribution produces considerable bending stresses through the shell wall, a general purpose program should be used and a nonlinear analysis applied. c) A shell which is nearly symmetric, e.g. a process tower with a large nozzle, can be analyzed approximatively by axisymmetric theory as described above. The influence of the nozzle may be estimated according to section 4.2. If a more accurate result is required, a collapse analysis by use of nonlinear theory is recommended. d) Nonsymmetric shell structures should always be analysed using nonlinear theory, e.g. with ADINA, ABAQUS, ANSYS, NISA or similar programs, or with special programs which are discussed for instance in 6.2.1. A preliminary design can sometimes be performed by use of elementary cases or calculations according to a) above. Earlier experience from testing and/or similar collapse analyses can not be too strongly emphasized. 6.3.4.2 Modeling Irrespective of which method of analysis is used in the determination of the buckling load by numerical methods the model must be capable to describe with sufficient accuracy:

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a) The state of stress and deformation in the prebuckling state in a bifurcation analysis. An error in the membrane stress in a region which will be critical in buckling leads to an error in the theoretical buckling load. In cases where buckling occurs in the plastic region it is important that the growth of the plastic deformations is modeled accurately. b) In a bifurcation analysis, the model should be able to reproduce the buckling mode with sufficient accuracy. This implies that a certain number of elements or mesh points are required to describe half a buckling wavelength. If the wavelength is small the required model may become extensive. c) In a nonlinear collapse analysis the requirements according to both a) and b) should in general be fulfilled. General rules hardly exist for determining suitable element sizes. These depend to a large extent on the accuracy of specific elements. Directions may sometimes be found in the manuals of the different programs. Some simple pieces of advice are given below, and it is recommended that the program user always makes a parametric study of the influence of the element size on the result, if the accuracy should be in doubt. Bushnell, 1972, provides some rules for the choice of the minimum number of mesh points along the shell when using the FD-method, e.g.: (6.7) The same requirement should be valid also for FE-models if n denotes the number of nodes. Since the element formulations vary considerably it is impossible to establish strict rules. It is likely that satisfactory results can be achieved with an accurate element using fewer nodes than indicated by eqn (6.7). Use the recommendations of the program supplier, if available. Fig.6.10 shows the buckling modes of a spherical shell analyzed by use of BOSOR4. The assumed number of mesh points along the shell was about one point per halfwave in the first, and five in the second case. The plots clearly indicate that the lower number is insufficient to define the buckling mode of the shell. It is usually possible to check if the model is appropriate by inspecting the result. If it is found, for example, that the deformation is oscillating between positive and negative values at consecutive mesh points, the model grid is too coarse. 6.3.4.3 Selection of type of element Many FE-codes offer a wide selection of elements and it may be a frustrating task to select a suitable element from a catalogue which contains several elements which all appear to be suitable for buckling analysis. Elements are developed for specific purposes and the accuracy of the solution may depend on the choice of an appropriate element. It appears that each FE-system is built up around specific types of elements, the properties of which are suitable for the problems which the system was designed to solve. Therefore, rather few comparisons and recommendations exist concerning elements for the analysis of shell structures, and the user, consequently, must build up his own

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experience through parametric studies and comparative analyses with elements of different types and sizes.

Fig.6.10. Spherical shell analyzed with program BOSOR 4, I) showing model with too few mesh points, while II) shows a suitable model. An experience from the development of the BASIS FE-program at the Aeronautical Research Institute of Sweden, FFA, may serve as an example, even if the element types offered by current program systems would hardly produce such large errors. In the development of a 4-node plate element based on four triangular elements according to Fig.6.11, studies were made of the accuracy of the elements. The deformations of a ring beam built up from 4-node elements and from traditional triangular elements respectively were calculated. The results showed that δ4≈4δ3, which indicated that the ∆-elements provided a very poor accuracy in this case. When using triangular elements, the problem was solved through increasing the number of elements. In a normal analysis it was difficult, however, to estimate the accuracy of the result due to limitations with respect to the size of the model and the computer costs. Convergence studies were, therefore, seldom carried out. Unfortunately, it must be pointed out that some of these difficulties still remain today, particularly in connection with large nonlinear problems where it is sometimes necessary to use large models which brings the problem close to the limit of a reasonable size and cost. A few recommendations concerning choice of appropriate types of shell elements can still be made as follows: – FE-programs which include plate and shell elements often recommend specific element types e.g. for nonlinear analysis of shell structures. As an example the SOLVIA (ADINA) system recommends, in the first place, a 16 node shell element and secondly a 9 node element. Several FE-programs are offering 8-node elements as suitable for shell analysis, see e.g. ABAQUS. – In the analysis of a shell problem it is important that the shape of the element allows an adaption to the middle surface of the shell since, otherwise, the model will generate bending stresses as a consequence of local deviations in the geometry. An example is the use of plane triangles to model a cylindrical shell subjected to external pressure, see Fig.6.12 a). The triangular elements will furthermore produce a skew stiffness matrix. The model given in Fig.6.12 b) will produce different values of the deflection

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when the load is applied at corner 1 or 2, i.e. the element generates grid dependent properties.

Fig.6.11. 4-nodes plate element builtup from triangular elements. – Similar errors can occur in a model including a transition from a coarse to a fine element grid, see Fig.6.12 c). The stiffness matrix will include disturbances due to the lack of symmetry of the skew elements. The effect of these disturbances are comparable to those caused by initial deformations and might possibly be used to initiate nonsymmetric deflections in a nonlinear collapse analysis. Such a procedure would require considerable experience, however, and it is therefore important to minimize the influence of grid dependent errors in order to be able to make a safe estimate of the influence of the actual initial deformations of the structure. – In some FE-programs the normals to the element surface are used in formulating the stiffness matrix . It is important that the normals of two elements which are connected to each other coincide at the connection point, see Fig.6.12 d). – Coupling to other elements, e.g. beams, normally requires that the stiffness matrices are formulated analogously for all elements and, in particular, that nonlinear terms are included. – Solid elements can sometimes be appropriate to use. It should be observed, however, that the aspect ratios in most cases should be kept close to one (a/b≈b/c≈1) to guarantee a sufficiently accurate solution. This implies that the model may become very large, in particular since several elements may be required through the thickness.

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Fig.6.12. Subdivision in elements for analysis of shell problems. Note that the examples given have an explanatory character and that the element density needs to be increased considerably in practical cases. a) Modeling with plane triangular elements. b) Resulting skew bending of beam modeled from triangular elements. c) Transition between fine and coarse meshes in an element grid of a cylindrical shell. The skew elements cause disturbances in the stiffness matrix. d) Modeling with straight element sides can result in skew elements and an ambigous definition of surface normals at adjoining nodes. – It may be assumed in general, that the computing accuracy increases if the element size is decreased. However, this does not neccessarily apply to certain types of elements, see Cook, 1981. – A correct description of the boundary conditions is necessary to produce a relevant result of the computer analysis. This is particularly true in cases where the buckling analysis is performed on a smaller part which has been cut out from a structure. – It is essential to emphasize the importance of reanalyzing the case with different mesh sizes in order to to verify that the accuracy is adequate. When refining the element grid a better estimate of the buckling load of the shell should be achieved. It is possible, however, that a first refinement of the grid tends to show that the model has already converged, but that a further refinement results in a considerable change of the buckling load. This may be due to the fact that the grids of the first models, 1 and 2,

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were so coarse that they were not able to represent the critical buckling mode, but that the third model did. – Parametric studies give an increased experience with regard to the program system and nonlinear analyses which is of value in subsequent numerical studies of similar problems. – Furthermore, it is sometimes advisable to make comparative studies of simple, similar shells of a geometry for which analytical solutions are available.

7 SHELLS OF COMPOSITE MATERIALS New materials such as fiber reinforced plastics, pure polymers etc. are used to an increasing extent in load carrying structures, including shells. Typical examples are shell support structures (skirts) in process towers, tanks and pressure vessels, ships, stacks and scrubbers. Fiber reinforced plastics provide a good potential for weight optimization through variation of the layup angles and stacking sequence of the fibre material as is done in reinforced concrete structures. Furthermore, by stiffening the shell or designing it as a double layer shell, i.e. a sandwich structure, the buckling resistance can be increased considerably. In a general case, the design of composite material shells becomes very complicated, because the relationship between forces and strains cannot be described in the same simple manner as for metals where Hooke’s law is applicable. In general, the following relations are assumed to be valid for a composite shell. Shear deformations in the xz- and yz-planes are neglected as well as normal stresses in the local z-direction in agreement with the assumptions normally made in shell theory. See Jones, 1975 (p154). (7.1)

This set of equations may be compared to the relation normally valid for single layer shells of metallic materials: (7.2)

In many cases, several of the coefficients in the matrices A–D will vanish. With a symmetric layup of the laminate, the coupling between normal forces and bending moments is eliminated. A multi-layer (0°/+60°/−60°) laminate is quasi isotropic, and the theories for buckling analysis of elastic shells can be applied in the form presented in this handbook. In addition to the requirements which apply to buckling, the criteria which refer to the particular properties of the composite materials such as fracture mechanisms, nonlinear material properties, creep etc. must be considered. The resistance of double layer shells is

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also governed to a high degree by the shear stiffness of the core which must be considered in the design. The analysis of such shells by use of numerical methods must be performed with caution since elements provided in current FEM-systems do not always take this degree of freedom into account. The use of an inadequate theory (e.g. an element neglecting transverse shear deformation) might under unfavorable conditions lead to an overestimation of the carrying capacity by a factor of 2–4. The numerical methods for buckling analysis of composite structures, mainly made of fiber reinforced plastics and similar materials, are relatively well developed, see further Chapter 6. Note, however, the comment given above concerning double layer (sandwich) shells. See for instance Olsson, 1981. 7.1 Buckling analysis of composite shells 7.1.1 Basic assumptions A general treatment of composite shells with properties according to eqn (7.1) is far too complicated to be approached in this edition of the handbook. Some of the properties specific for composite shells will be commented on, however, and directions will be given for design with respect to buckling using the stability data available for isotropic shells. The influence of anisotropy will be demonstrated in examples. 7.1.1.1 Material properties In order to be able to apply the isotropic shell theory, the material properties must be approximately the same in all directions in the plane of the shell surface. This requirement can be considered to be satisfied if the composite material is characterized by: – short fibers, randomly orientated in all directions. – cloth/mat/filament wound fibers/prepreg, symmetrically layed-up with respect to the middle surface and orientated in the (0°, +60°, −60°) directions. The number of laminas should exceed 12. – other types of shells, e.g. steel shells with GRP lining, sandwich shells with a stiff core, etc. Other properties will be discussed where appropriate. 7.1.1.2 Boundary conditions The recommendations given for the individual cases in Chapter 3, are valid in applicable parts also for composite shells. Note, however, that the required moment of inertia of the ring stiffener as defined in eqn (3.4), is valid if the shell and ring are made of the same material. In the case of a GRP shell with an end stiffener made of steel, the required minimum moment of inertia of the ring, Ir, becomes:

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(7.3) Other combinations of shell and ring materials may be treated in a similar way. The same condition applies to eqn (3.27) where, furthermore, σ0.2 should be substituted for σy if σ0.2 is defined. Otherwise it is recommended to set σy=σu/2, where σu should be interpreted as the characteristic ultimate strength of the laminate in compression. 7.1.1.3 Shape deviations—tolerances Metal structures design often has the advantage that the sheet thickness is uniform and the material is nearly homogeneous with only small deviations in stiffness and strength. The corresponding parameters of a GRP shell may vary strongly depending on the fabrication method. It is hardly feasible to state strict tolerance limits since the experience available from analyses and experiments is limited. Based on data for metal shells and an estimation of the possible influence from the specific properties of the composite material, in particular fiber reinforced plastics, on the buckling resistance, the following recommendations may be given: – Parameters which influence the buckling limit: * geometrical shape deviations * shell thickness * relative fiber volume * fiber orientations—variations over the shell surface. – Geometrical shape deviations must be kept within the tolerance limits given for the differen shell geometries in Chapter 3. For a cylindrical shell under axial load, tolerance limits are given in 3.2.1.1, which are assumed to determine the value of the reduction factor η. – Strength and stiffness material parameters are strongly interdependent and it is easier to establish requirements on the stiffness rather than on thickness etc. Assume that the shell has been designed with a nominal thickness t and a nominal modulus of elasticity E. The buckling resistance may be severely reduced if the local variations in each of these parameters are considerble. The quantities which govern the buckling strength are the extensional stiffness Et and the bending stiffness Et3 (neglecting the influence of Poisson’s ratio). These vary with thickness and fiber contents and the variations are also depending to a high degree on the manufacturing procedure. Recommended values of the reduction factor α, related to the manufacturing process are given in Table 7.1 7.1.1.4 Material properties The distinct yield limit of most metallic materials is used in many codes and recommendations as a basis for design and as a reference stress also in the case of buckling. Composite material shells often exhibit a more or less elastic behavior until the

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ultimate limit is reached implying that a design method based on the yield limit is irrelevant.

Table 7.1. Manufacturing methods and tolerance classes for a composite shell (GRP or similar material). Fabrication method Ak Extrusion with short fibres in plastic matrix

Tolerance class

Tolerance for wr/l

Tolerance for Et

α

1

1/100

0–10%

0.8

−5–25%

0.6

0–10%

0.5

−5–25%

0.4

0–5%

0.9

−5–10%

0.8

0–5%

0.6

−5–10%

0.5

0–5%

1.0

−5–10%

0.9

0–5%

0.7

−5–10%

0.6

2 Bk Manual layup of cloth or prepreg airhardened

1 2

Ck Fiber wound or prepreg shells hardened in autoclave

1 2

1/50 1/100 1/50 1/100 1/50

Fig.7.1 shows a typical relation between stress and strain in a composite material. The curve is continuous and slightly curved all the way until the ultimate stress is reached. Several proposals have been presented in the literature for analysis of the buckling behavior of shells manufactured of such a material. The secant modulus Es and the tangent modulus Et are likely to be significant parameters, where Es would be representative of the prebuckling state, while Et should determine the buckling behavior. A realistic parameter for determining the theoretical buckling stress could be based on an effective modulus Eef (see Gerard-Becker, 1957, Becker,1958): (7.4) Since Eef varies with the load level, the buckling resistance has to be determined through iteration.

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Fig.7.1. Stress-strain curves for metals and typical composite materials. 7.1.2 Buckling analysis, elementary cases 7.1.2.1 Isotropic material properties Provided that the shell material properties are essentially isotropic, as discussed above, the theoretical buckling stress can be determined according to Chapter 3. Reduction factors are chosen according to Table 7.1. If possible, the estimate of the resistance of the shell (allowable stress) should be based on measured minimum values of thickness and stiffness of the finished shell. 7.1.2.2 Anisotropic material properties One of the advantages of the composite materials is, clearly, that the material properties may be chosen such that an optimum design results. Consequently, it is likely that an anisotropic fiber reinforced shell may be stronger with respect to buckling than an isotropic shell, and the material will be more efficiently utilized. In aircraft and space applications, composite materials have been used for several years and the development within this branch is still strong. Provided that the properties required are known for the various directions of a laminate the layup angles can be optimized using special purpose programs, see for instance Thor-Lundemo, 1976, or Tsai-Hahn, 1980 In order to achieve an optimum buckling resistance it is necessary to carry out additional calculations in an iterative process. General routines for such a process are not yet commersially available but investigations on special types of elements have been reported, compare Stehlin-

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Holsteinson, 1971. Trials with various types of laminates must usually be made manually and suitable types of laminate selected on the basis of several calculations. Methods of analysis for anisotropic shells have been developed and many computer program are today including composite elements. BOSOR 4, Bushnell, 1972, contains routines both for shells with general stiffness properties and for laminated shells, and thus provides an ideal tool for buckling analysis of composite material structures. A disadvantage with the system is, however, that the material properties must be linear. On the other hand, certain problems of material nonlinearity may be treated in BOSOR 5 (Bushnell, 1974).An example is given below where BOSOR 4 is used to compute the resistance of a GRP shell. Example: A cylindrical shell with geometry as shown in Fig.7.2 is loaded in axial compression. It is assumed that the shell wall is built up from a number of unidirectional layers oriented in the ±φ°, or alternatively, 0°, 90° directions. The layup angle is to be optimized for maximum resistance. Material data for the cylinder shell are assumed as: E11=30000 MPa E22=300 MPa G12=90 MP The BOSOR4 program was used to calculate the buckling stress and the buckling mode for three different fiber orientations. The results are included in Fig.7.2. It is evident that the buckling limit is strongly dependent on the layup angles. In extreme cases still greater differences may be anticipated.

Fig.7.2. Buckling stress and buckling mode of a fiber wound cylinder under an axial load.

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8 EXAMPLES This chapter describes a number of practical applications of the recommendations on stability given in the handbook. The cases are reproduced with kind permission by different companies. The calculations are carried out with particular references to the relevant sections of Chapters 3–6 to facilitate the follow up of the various steps in the computations. 8.1 Pulp tower Pulp storage towers according to Fig.8.1 have been built at a large number of pulp mills in Sweden. As a consequence of their special geometry, they are to a large extent designed with regard to buckling criteria. Since their dimensions are often very large, the demands on safety against collapse must be high. As a matter of fact, the Swedish Plant Inspectorate, SA, discovered in an analysis of a pulp tower that the requirements concerning buckling were not directly applicable for this type of structure, and this led to a research project which has resulted in the present handbook. 8.1.1 Geometry and material The geometry of the tower is defined in Fig.8.1, which specifies the measures of sections in detail. Material data: Lower cylinder and cone: Steel sheeting DAN 350 WD, σy=350 MPa. Upper cylinder: The five lowest sheets with the height 10.4 m: SS 2172, σy=310 MPa. The following two sheets, height 4.4m: SS 1412, σy=260MPa. Subsequently one sheet, height 2.2 m: SS 1312, σy=220 MPa, and the uppermost five sheets, height 10.0 m: SS 2343–28, σ0.2=220 MPa. Roof: Roof sheeting and arches: SS 1412, σy=260 MPa. Ring stiffener and crown ring, SS 1312, σy=220 MPa.

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Fig.8.1. Assembly drawing of pulp tower with measures and material specification. 8.1.2 Loading assumptions The loading on the tower includes the dead weight of the shell structures, weight and fluid pressure from the paper pulp, wind effects and snow load. The dead weight, mainly from sheeting and beams of steel, γs=77 kN/m3, has a marginal influence except for the roof, but will be included for completeness.

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Lower cylinder

G1=130 kN

Conical transition

G2=290 kN

Upper cylinder

G3=680 kN

Roof, sheeting, arches, edge ring

G4=100 kN

Total dead weight

G=1200 kN

Load factor γf=1.0 The unit weight of the paper pulp, γf=10 kN/m3, gives a vertical bulk loading on the floor and on the lower cylinder, as well as a fluid pressure on the cylindrical and conical shells. The total volume of the tank is 4120 m3. The weight of the paper pulp Gk=41.2 MN, 13.5 MN of which is taken by the floor and 27.7 MN by the lower cylinder. This loading is considered to be permanent. In load combinations where it is dominating, the load factor γf=1.15 is applied, in other cases γf=1.0. The Swedish building code requires design against wind loading corresponding to a characteristic stagnation pressure qk=1.0 kN/m2 at the 30 m level. The wind load causes axial stresses due to bending, which are added to the stresses caused by the paper pulp and the weight of the structure. These additional stresses are so small, however, that they can be neglected. In an empty or partially filled tank, compressive stresses and bending stresses will occur in the circumferential direction due to wind loading. Also, ring oscillations (ovalling) caused by variations in the wind pressure must be investigated. Since the upper cylindrical shell is sufficiently stiffened at both boundaries, bending stresses and ring oscillations are not critical in the present case. Cantilever beam vibrations of the tower due to vortex-excitation or wind gusts have also been investigated, the results are omitted for the same reason. The resistance of the upper cylinder under constant external pressure equal to the stagnation pressure qk will be calculated below. The load factor to be applied is then γf=1.3. The basic value of the snow load is in this case s0=2.5 kN/m2, with a shape factor µ=0.8. The characteristic snow load on the roof is thus sk=µs0=2.0 kN/m2. In load cases where the snow load is the critical variable load the load factor γf=1.3 is used. Where the snow load is not dominating in the design of the shells, only the common snow load s=ψsk=0.7·2.0=1.4 kN/m2 is considered. S=π ·6.52·1.4=190 kN=the total snow load γf=1.0 or 0. 8.1.3 Design check 8.1.3.1 Lower cylindrical shell Design load case: 1.0G+1.15Gk=1.2+1.15·27.7=33.1 MN

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Since the weight of the paper pulp Gk is the dominating load and this is multiplied by the load factor γf=1.15, snow load and wind are omitted. Including load factor, the axial stress of the lowest sheet becomes

Internal fluid pressure pi=hγp=352 kN/m2 gives the circumferential stress σφ. Load factor γf=1.0 yields

The carrying capacity for axial compression is obtained according to 3.2.1.2:

Reduction factor according to eqn (3.6):

The lower cylindrical shell has a number of cutouts with edge stiffeners designed according to the Swedish Pressure Vessel Code TKN (see also ASME). In 4.2.4.1 the hole parameter was introduced:

Eqn (4.10) gives the reduction factor for the cylindrical shell with a reinforced hole:

This means that no further reduction of the buckling resistance due to the hole is necessary. In a welded structure of tolerance class 1, the additional reduction factor is α=0.9. σelr=0.9·0.536·900=434 MPa>σy/3

According to eqn (3.10)

The characteristic buckling resistance σxu=ωs2σy=0.598·350=209 MPa

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According to eqn (3.26), the internal fluid pressure pi will increase the magnitude of the reduction factor:

The reduction factor due to the hole should not be affected, however, by internal pressure and hence the ultimate stress σxu is maintained. The resistance factors are taken as γn=1.2; γm=1.0

The shell is also checked for interaction according to eqns (3.57) and (3.58). Eqn (3.57) gives:

A check will also have to be made with a circumferential stress σφ determined from the internal pressure pi multiplied by the load factor γf=1.3. It is obvious, however, that the sum will still be less than 1.0. The yield limit criterion according to eqn (3.58) gives:

The structure thus possesses an adequate margin against collapse, provided that the edges of the holes are stiffened according to valid design codes. 8.1.3.2 Conical shell The lower part of the cone is critical with regard to buckling. As for the lower cylindrical shell, the design load gives the following vertical reaction force at the lower boundary: 1.0·(G−G1)+1.15 Gk=1.07+1.15·27.7=32.9 MN The compressive stress in the meridional direction is obtained from eqn (3.101)

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where σsg is the meridional stress caused by the dead weight of the mantle sheets. It varies nonlinearly with the height of the cone. It is assumed here that it has the same value in the lower part as that at the lower boundary. At the lower boundary, x=5.2 m, r=r1=3.5 m

Since the stresses in the conical shell are mainly due to the dead weight of the paper pulp and are not notably influenced by a variable fluid pressure when overfilling occurs, a local factor γf=1.15 should be appropriate. σs=58.0·1.15+2.52=69.0 MPa. The same result is obtained when starting from the support reaction at the lower boundary

The tensile membrane stresses in the circumferential direction is:

At the lower boundary:

At the boundary it is necessary to introduce a ring stiffener with the cross-section area 40× 350 mm in order to absorb the compressive circumferential force Nr. Nr=69.1·0.025·3.5 sin 30°=3.02 MN

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Fig.8.2. Ring stiffener at the juncture between the lower cylinder and the cone. The effective area of the ring includes a part of the cylinder, see 3.2.1.5A:

and of the conical shell:

The effective area of the ring Ar=0.04·0.35+0.131·0.032+0.124·0.025=0.0213 m2.

According to eqn (3.27) the moment of inertia Ir must have at least the following magnitude to prevent ring buckling.

The centroid of the effective ring area is situated at a distance e=0.12 m outside the inner surface of the cylindrical shell. The moment of inertia about the vertical axis through the centroid is: Iref=2.7·10−4 m4>1.52·10−4 m4 The buckling resistance of the cone is determined at a distance se from the lower boundary. r=3.5+0.592 sin 30°=3.80 m

The reduction factor is calculated according to eqn (3.6):

a=0.9 σselr=0.9·0.500·719=324 MPa>σy/3

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Introducing the resistance factors γn=1.2, γm=1.0,

At the section r=3.80 m, x=5.2−0.592 cos30°=4.69 m, the meridional stress becomes:

=40.5·1.15·1.215+2.25=58.8 MPa<149 MPa. The internal fluid pressure should increase the resistance against buckling but the effect is probably marginal and is normally not considered for conical shells. The tensile stress in the circumferential direction amounts to

The interaction check can be made in a similar way as in 8.1.3.1 for the lower cylinder. It is obvious that the margins are adequate also in the conical shell. For the sake of completeness the stresses are also determined at the distance se=0.59 m from the upper boundary. r=6.5−0.30=6.20 m; x=0.51 m

The buckling stress at this section is considerably lower than in the vicinity of the lower boundary, since the radius is larger and the sheet thickness is smaller. Due to the low compressive stress, however, the margin against buckling is higher. 8.1.3.3 Upper cylinder The highest vertical stresses will occur for the load combination of dead weight of the roof and the cylindrical shell and snow load on the roof. For evenly distributed snow with the shape factor µ=0.8 the characteristic snow load on the entire roof becomes: Sk=πr2sk=π.6.52·2.0=265 kN. Including the weight of the roof, G4=100 kN, the total design load acting on the cylindrical shell becomes:

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245

γgG4+γfSk=1.0·100+1.3·265=450 kN which gives an axial stress in the uppermost sheet with t=4 mm:

The alternative nonuniformly distributed snow load will, locally, result in a higher value of σx, which, furthermore, increases by an amount of 0.08 MPa per m downwards due to the dead weight of the sheets. The highest stress in the sheets of thickness t=4 mm is estimated at σx<5 MPa. The resistance is calculated in the following, considering the stainless steel material properties:

α=0.9 σxelr=0.9·0.20·74=13.3 MPa<σy/3 σxu=0.75·13.3=10.0 MPa

Load concentrations below the attachment points of the roof arches will have to be considered, however, according to 4.1.1.2. Local stiffening may be required. The primary load on the upper cylinder is judged to be wind loading on an empty cistern, which may cause buckling in the circumferential direction. According to 3.2.1.5, wind load on a cylindrical shell with a varying pressure intensity around the circumference can be replaced in the buckling analysis by a constant pressure equal to the characteristic stagnation pressure qk. This pressure is assumed here to be constant, qk=1.0 kN/m2, for the entire height of the cylindrical shell. The thickness variation of the shell wall and the equivalent shell consisting of three sheet sections are shown in Fig.8.3.

t0=4.52 mm tm=7.98 mm tu=11.33 mm

Fig.3.12 gives the buckling coefficient β=1.00. Since

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the buckling stress is determined by eqn (3.39): (3.39)

Fig.8.3. Definition of model, upper cylindrical shell. Tolerance class 1: αη=0.7 gives:

From the external pressure p=γfqk=1.3·1.0=1.3 kN/m2 the circumferential stress of the uppermost equivalent wall section is obtained.

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247

This exceeding of σφd could possibly be accepted with reference to the fact that the cistern is generally at least partially filled. The probability that a 50-year wind should occur when the tank is empty is, consequently, small. However, a ring stiffener was attached at a distance of 7.0 m from the upper boundary. Calculation of the buckling pressure of the cylinder with BOSOR 4 gave σφel=2 .15 MPa in good agreement with the result presented above. When a stiffening ring, 10×50 mm, was included in the analysis at the level 7.0 m below the roof, the buckling stress increased to σφel=2.72 Mpa. 8.1.3.4 Roof structure The roof geometry is shown in Fig.8.4. The number of half-arches of IPE 160 is n=22 with the following cross-section data

Fig.8.4. Roof geometry. Conical roof sheeting, placed below the arches, ts=4 mm Slope angle, ψ0=12° rs=rc/sinψ0=6.5/0.208=31.3 m Height, h=rctan 12°=1.38 m Length of half-arch, l=rstanψ0=6.65 m Thickness of cistern sheeting, tc=4 mm Dead weight of arch, g1=0.155 kN/m Dead weight of cover plate, insulation, roof sheeting, g2=0.50 kN/m2 Internal vacuum, p1=0 Internal overpressure, p2=0 Edge ring stiffener (at eaves)

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Cross-section area, steel plate 15×180=2700 mm2 Effective width of cylindrical shell,

In compression bcc=12·0.004=0.048 m From the conical sheeting: bsc=12·0.004=0.048 m The effective area of the ring stiffener: Art=2.7·10−3+0.056·0.004+0.124·0.004=3.42·10−3 m2 Arc=2.7·10−3+0.048·0.004+0.048·0.004=3.08·10−3 m2 Maximum tensile force in the ring, eqn (3.142):

qld=0.8·1.3·2.5=2.6 kN/m

Maximum compressive force in the ring, eqn (3.143):

where

Tension is critical

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249

Arches The following procedure applies according to eqns(3.159)−(3.162) G1=g1l=0.155·6.65=1.03 kN

The arches are relieved by the load which the sheeting is able to carry, eqn (3.163), where:

Moments and forces, eqns (3.164)–(3.167)

Ms=0.09Rsrc=0.09·16.5·6.5=9.7 kNm

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Check of the design conditions according to 3.6.6.1C:

Crown ring Normal force in the crown ring flange, Fig.8.5:

Fig.8.5. Typical crown ring geometry. Carrying capacity with respect to a normal force:

Additional conditions

The flange thickness required is

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251

tr=24 mm. br and hr may be somewhat reduced, but tr should have a value close to 20 mm. 8.2 Cylindrical shell with large openings A cylindrical shell was to be built to carry a 400 metric ton silo according to Fig. 8.6. In the design, a number of mistakes were made which caused failure of the shell at a load of approximately half the design level. The example is included in order to demonstrate that it is extremely important to interpret the results of a stability analysis in a correct way.

Fig.8.6. Cylindrical skirt according to original proposal and in actual design. According to the regulations for design of supporting structures, a sheet thickness of t≈7– 8 mm would be required. The design engineer wished, however, to optimize the structure and asked a consultant firm to carry out a buckling analysis with numerical methods (BOSOR 4) for the sheet thickness t=5 mm. The calculations gave the result σel=205 MPa, which should be compared to the classical theory, eqn. (3.3) of Chapter 3, yielding 210 MPa, i.e. practically the same value (note the warnings in Chapter 6). The maximum stress in service was estimated at 42 MPa (corresponding to the load of 400 tons), implying a nominal ‘safety factor’ against buckling of almost 5. According to the rules of section 3.2 the allowable load on the shell should have been computed as follows: (3.6) α is assumed to equal 0.9 σelr=0.9·0.314·210=59 MPa

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σu=0.75·59=44 MPa (3.8) If a safety factor of 1.5 is adopted the allowable stress is obtained: σall=30 MPa (Analogously, t=7 mm gives σall=48 MPa, in good agreement with the applicable code requirements). Obviously the designer overestimated tha carrying capacity of the skirt on the basis of the theoretical buckling stress and reduced the wall thickness to t=4 mm. In addition, two large openings were cut out, stiffened by steel profiles and a ring at the upper boundary as shown in Fig.8.6. The cross-section area of the stiffeners was probably not of the same magnitude as that of the sheet cut away. The resistance of the skirt can be estimated at: a) Complete shell, t=4 mm and no cutouts

b) By cutting out holes for the entrances, the stresses are increased in the lower part of the skirt. This increase is, however, not of interest here, since the critical stresses occur in other parts of the shell. c) The shell wall above the cutouts is inefficient according to section 4.3 and a nominal stress concentration by a factor 2 close to the vertical edge stiffeners is obtained, i.e. σmax≈2.3=106 MPa This stress is likely to have caused local buckling, leading to deflection of the tower with further increase of the load and complete collapse as result. d) Assuming that 1.25σelr corresponds to the mean value of the actual buckling stress the following ratio gives the collapse load:

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253

Fig 8.7. Results of computations for skirt according to Fig.8.6.

Fig.8.8. FE-model with computed deformations and stresses in the skirt according to Fig.8.6. The silo was reported to collapse at a load of approximately 250 tons, i.e. in good agreement with the result obtained above. The various computed values are shown in Fig.8.7, where the scatter of the data obtained in buckling tests is included (Fig.3.3). Furthermore, a FE-analysis was performed in order to check the stress distribution in the actual shell with cutouts. The results are shown in Fig.8.8, from which it is evident that the maximum stress in the shell just above the end of the vertical stiffener amounts to 215 MPa while the mean stress without cutouts is 100 MPa. The stress concentration, 2.15, is in good agreement with the value calculated above. The simplified models of analysis thus appears to form a rational basis for the design of structures with large openings.

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8.3 Analysis of a spherical roof buckling incident During the final work on a cryogenic storage tank according to Fig.8.9. the inner roof collapsed and formed a large inward buckle covering approximately one quarter of the total roof area. The cause of the incident has not been completely explained but, there are indications that an overpressure was applied during the final filling of the insulation. The filling was done by use of a fan blowing the insulation powder into the outer tank through two ports at the outer rim of the roof and ventilating through two ports on the other side of the tank and through the manhole at the top. The filling system had a capacity of 0.5 bar overpressure and if the ventilating ports were filled, there is a fair chance that an overpressure would build up at the filling ports. Since this would happen when the insulation filling would be almost complete, the pressure distribution would be nonuniformly distributed over the inner roof.

Fig.8.9. Geometry of cryogenic tank roof and extension of damage caused by buckling. An investigation was carried out in order to a) verify that an external overpressure of the order of 0.5 bar could have caused the incident and b) to calculate the internal pressure required to cause a ‘snap-back’ of the buckled roof.

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255

A. Buckling of the inner roof. The roof was designed for an external pressure of 0.025 bar, which should be compared to the classical buckling pressure of 0.46 bar. The reduction factor applied is 0.08.

Fig.8.10. FE-model of the tank roof under an assumed nonsymmetric external pressure. It is reasonable to believe that buckling may have been caused by a local pressure of the order of 0.1–0.2 bar. The FE-model used is not quite adequate according to the recommendations given in Chapter 6. On the other hand, neither the pressure distribution nor the pressure amplitude are known and an improved model would, thus, not provide a better result. It may be noted, though, that the pressure required to cause buckling was well within the capacity of the filling system. B. Snap-back of the buckled roof. Two models of the buckled roof were analyzed. In the first case, an axisymmetric model was subjected to a uniform internal pressure, Fig.8.11, and the symmetric collapse pressure was calculated. The theoretical buckling pressure was 0.115 bar. This result indicates that an internal pressure of the order of 0.1 bar would be sufficient to snap the shell back in its original form provided local plastic deformations had not developed.

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Fig.8.11. Axisymmetric model of buckled shell Since the actual buckle was not axisymmetric, an FE-model of the buckled shell was developed according to Fig.8.12. The smooth transition between the original shell geometry and the buckle was not modeled and the result may, therefore, be somewhat too low. The analysis resulted in a collapse (snap-back) pressure of approximately 0.08 bar according to the figure which is as close as could be expected to the axisymmetric result of 0.115 bar.

Fig. 8.12. FE-model of the buckled roof structure.

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c) Repair work The tank was first pressurized to 0.054 bar. No snap-back of the buckle resulted at that level. Then, pullies were attached to the buckled plate and 2–4 point forces of the order of 100–1000 N were applied. The shell then snapped back intermittently into its original shape and the entire process was finished in three steps.

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TIMOSHENKO, S. and GERE, J.M.: Theory of elastic stability. New York 1961. TODA, S.: Experimental investigation on the effects of elliptic cutouts on the buckling of cylindrical shells loaded by axial compression. Trans. Japan Society Aero Space Sciences, Vol 23, No 59, 1980a, pp 57–63. TODA, S.: Some considerations on the buckling of the thin cylindrical shells with cutouts. Trans. Japan Society Aero Space Sciences, Vol 23, No 60, 1980b, pp 104–112. TSAI, S.W. and HAHN, H.T.: Introduction to Composite Materials. Technomic Publishing Co., Westport, CT, pp. 217–276. TVERGAARD, V.: Buckling behaviour of plate and shell structures. Theoretical and Applied Mechanics, W.T.Koiter, ed. North-Holland Publishing Company, 1976. WALKER, A.C., ANDRONICOU, A. and SRIDHARAN, S.: Experimental investigation of the buckling of stiffened shells using small scale models. Buckling of Shells in Offshore Structures, ed: J.E.Harding et al, Granada, London 1982, pp 45–71. WALKER, A.C. and SRIDHARAN, S.: Analysis of the behaviour of axially compressed stringerstiffened cylindrical shells. Proc. ICE, Part 2 69, 1980, pp 447–472. VALSGÅRD, S. and STEEN, E.: Simplified strength analysis of narrow-panelled stringer-stiffened cylinders under axial compression and lateral load. Det Norske Veritas, Report 80–0590, 1980. WANG, L.R.L.: Effects of edge restraint on the stahility of spherical caps. AIAA Journal, Vol 4, No 4, April 1966, pp 718–719. WANG, L.R.L., Rodriques-Agrait, L. and Litle, W.A.: Effect of boundary conditions on shell buckling. Journal Eng Mechanics Division, ASCE, Vol 92, No EM 6, Dec 1966, pp 101–116. WANG, S.S.K. and Roberts, S.B.: Plastic buckling of pointloaded spherical shells. ASCE, Journal Engineering Mechanics Division, Vol 97, No EM 1, Febr 1971, pp 77–93. WEINGARTEN, V.I. and Seide, P.. Buckling of thin-walled truncated cones. NASA Space Vehicle Criteria (Structures), NASA SP-8019, Washington DC, 1968. WEINGARTEN, V.I., MORGAN, E.J. and SEIDE, P.: Elastic stability of thin-walled cylindrical and conical shells under axial compression. AIAA Journal, Vol 3, 1965a, pp 500–505. WEINGARTEN, V.I., MORGAN, E.J. and SEIDE, P.: Elastic stability of thin-walled cylindrical and conical shells under combined internal pressure and axial compression. AIAA Journal, Vol 3, 1965b, pp 1118–1125. VESTERGREN, P. and KNUTSSON, L.: Theoretical and experimental investigation of the buckling and postbuckling characteristics of flat carbon fibre reinforced plastic (CFRP) panels subjected to compression or shear loads. ICAS XI Congress. Proc., Vol 1, Lisbon 1978. YAMADA, M.—YAMADA, S.: Agreement between theory and experiment on large-deflection behaviour of clamped shallow spherical shells under external pressure. IUTAM Symposium on Collapse, ed: J.M.T.Thomson and G.W.Hunt, Cambridge, 1983. YAMAMOTO, Y. and KOKUBO, K.: Effects of geometrical imperfections and boundaries on the buckling strength of a spherical shell. Computers & Structures, Vol 19, No 1–2, 1984, pp 285– 290. ZIENKIEWICZ, O.C.: The Finite Element method in engineering science. McGraw-Hill, London, 1971. ZOELLY, R.: Über ein Knickungsproblem an der Kugelschale. Dissertation, Zürich 1915. ÖRY, H., Reimerdes, H.G. und Tritsch, W.: Beitrag zur Bemessung der Schalen von Metallsilos. Der Stahlbau, Jahrgang 53, Heft 8, 1984, S 243–248.

Appendix A Brief review of the theory of stability for shell structures A1 Cylindrical shells A1.1 Axisymmetric loading The cylindrical shell given in Fig.1.13 is subjected to a uniform axial compression qx and it is assumed that the deflections are symmetric and small compared to the wall thickness. Forces and bending moments acting on a shell element are defined in Fig. A.1.

Fig. A.1. Forces and moments acting on a cylindrical shell deforming axisymmetrically. The radial deformation is denoted by w. Since qx is the only force acting in the meridional direction it follows that: nx=−qx (A.1) Force equilibrium in the radial direction yields, see Samuelson, 1969 (A.2) Moment equilibrium requires

Appendix A

267

(A.3) Introduction of (A.3) into (A.2) yields: (A.4) Compatibility, the relation between deflections and strains may be expressed by: (A.5) (A.6) (A.7) The material properties are assumed to follow Hooke’s law: (A.8)

and finally, equilibrium across the wall thickness: (A.9)

The final result may be written in the following form: (A.10) This equation yields the radial deflection w of the cylindrical shell subjected to a uniform axial compression of qx. A few interesting results may be deduced from the equation even though it was developed for the static prebuckling situation only. eqn.(A.10) is of the same form as that of a straight beam-column on an elastic foundation. The supporting force from the foundation is in the cylinder case supplied by the stresses in the circumferential direction. One difference between the two problems is represented by the right hand term in eqn.(A.10) which stems from the effect of Poisson’s

Appendix A

268

ratio. If the cylinder wall is unrestrained, a radial deflec-tion w results from application of an axial load according to Fig. A.2. (A.11) The boundaries, on the other hand, are not free to expand and the state of stress and strains in the vicinity of the edge will be disturbed. This disturbed state is also shown in Fig. A.2. The solution may be obtained from eqn.(A.10) and may be written in the following form in the case of clamped edges, (w=w’=0):

Fig.A.2. Effect of the boundary conditions on the radial deflections of a circular cylindrical shell under axial compression. (A.12)

where: (A.13)

(A.14)

Appendix A

269

The solution, eqn.(A.12), is a damped harmonic function as long as the damping coefficient α≥0. The value α=0 may be interpreted as a stability limit above which the deflections become large and the cylinder loses its carrying capacity.

(A.15)

Fig. A.3 Maximum deflection as a function of the axialcompression for the cylinder in Fig.A.2. This is the formula found in handbooks usually found in handbooks for the buckling load of cylinders under axial compression. The solution, eqn.(A.12), is plotted in Fig.1.16 where the cicumferential force nφ is included. nφ is calculated from the following expression: (A.16) wmax and nφ are shown in Fig. A.3. In practical applications, this buckling mode rarely appears. The reasons for this will be discussed later. However, it may be noted that thick-walled shells often exhibit an elephant’s foot buckling mode while thin-walled shells mostly buckle in a nonsymmetric pattern as shown in Fig.1.17. It is possible to suppress the non-symmetric buckling mode through application of an internal pressure. Bushnell, 1985, demonstrated that even very thin-walled cylinders could be observed to buckle in an axisymmetric mode similar to the solution given by eqn.(A.12). Provided buckling occurs in the elastic range, the buckling load may be found to approach the classical value given by eqn.(A.15). Practically, these favorable conditions may seldom be utilized.

Appendix A

270

A1.2 Cylinder under axial compression, nonsymmetric buckling In most cases the buckling mode is nonsymmetric, at least for thin-walled shells. This might be one reason why experimental buckling loads do not agree with the classical value of the buckling stress given by Eq.(A.15), compare Fig.1.12,a). Another reason might be the fact that shells are always imperfect, but it is also possible that the buckling behavior is governed mainly by the geometrical properties of the shell such as length, radii, thickness and boundary conditions. In order to be able to predict the load resulting in a nonsymmetric buckling mode, it is necessary to develop the appropriate set of stability equations. This can be done in different ways, here the method used by Brush-Almroth, 1975, will be summarized. Since it is not the intention to describe the stability theory in detail, only a brief summary will be given. As in the axisymmetric case, eqn.(A.10), the equations of equilibrium for nonsymmetric deformations of a shell element according to Fig. A.4 may be written in the form:

(A.17)

(Here the notation used for differentiation is: f,x=df/dx) The equations corresponding to eqns.(A.5–A.9) may in their linear form be written as:

(A.18)

(A.19)

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271

Fig. A.4. Definition of coordinate system and force components Eliminating the shear forces v and the bending moments m from eqns (A.19) yields the following comparatively simple form of the general equations: rnx,x+nxφ,φ=0 rnxφ,x+nφ,φ=0 (A.20)

(A.21) If all nonsymmetric terms are excluded, eqn.(A.20) is reduced to the same form as (A.10). Note however that derivation of the two systems was based on different coordinates. Eqn (A.20) defines an equilibrium state which may be represented by the fundamental, or primary, solution u0,v0,w0 shown in Fig. 1.18. Now assume that a neighboring solution exists characterized by the deflection increments ∆u,∆v,∆w: u→u0+∆u v→v0+∆v (A.22) w→w0+∆w Forces and bending moments are expressed in the same manner: (A.23)

Appendix A

272

Insertion of eqns.(A.22) and (A.23) into (A.2) yields a set of equations containing both the u0 and ∆u variables. But since u0 satisfies eqn.(A.20) exactly, all terms including the primary solution drop out and a set of equations in the deflection increments ∆u results:

(A.24)

Here the following notations are used:

These are the Donnell stability equations for a circular cylindrical shell. Obviously the third equation is independent of equations one and two and the problem may thus be solved by use of this single equation. Once the buckling mode represented by ∆w is known the ∆u and ∆v components are readily obtained from the two first equations. The terms nx0, nφ0 and nxφ0 included in eqn.(A.24) represent the membrane forces acting in the shell middle surface in the primary, or pre-buckling, state and they are the forces which may cause instability due to buckling. In the present example, it is assumed that only an axial force nx0 is acting on the shell: (A.25) Then, Eq. No 3 in (A.24) becomes: (A.26) If it is possible to find a solution to this equation, satisfying at the same time the boundary conditions, then a theoretical estimate of the critical load of the cylinder would be obtained. The prebuckling state will still be axisymmetric but the buckling bode is ingeneral non-symmetric. Assume a solution of the following form: (A.27) This equation satisfies the boundary conditions: (A.28)

Appendix A

273

Introduction of the assumed mode shape (A.28) in (A.26) yields the following expression which must be zero for a non-zero solution of Eq. to exist. (A.29) Minimization of (A.29) with respect to the wave numbers m and n yields the minimum value of the critical axial force qxel=minqxcr. The wave numbers m and n corresponding to the minimum load represent the buckling mode of the cykinder. Graphically eqn.(A.29) may be presented as in Fig.1.5 if it is assumed that m is a contious variable. For a given value of the wave length parameter l/mr there exists an n value corresponding to the curve providing the lowest buckling load. The combination of m and n values which corresponds to an absolute minimum of the buckling load represents the buckling mode. (A.30) (A.31) Minimization of qx as a function of b leads to: bmin=2[3(1−v2)]½r/t for l/mr≤3 (A.32) Introduction of eqn.(A.32) into eqn.(A.30) yields: (A.33) This is exactly the same expression as was deduced for a circular cylinder subjected to symmetric collapse, eqn(A.15). The more general theory of nonsymmetric bifurcation buckling thus fails to give a solution which compares better with experimental results. This is a consequence of the fact that the theory is linear and cannot consider for instance initial imperfections. Possible improvements are discussed below. A1.3 Cylinder under external pressure The equation system according to (A.20) also applies to the load cases of external pressure and torsion. If only an external pressure is applied, the following will result: nφ0=−pr (A.34) which, introduced into (A.24) gives: (A.35)

Appendix A

274

The same type of buckling mode as for the case of axial compression may be assumed but, since deflection in the meridional direction is rather characterized by a half sine wave, compare Fig.1.21, it is appropriate to set m=1 in eqn. (A.27) (A.36)

It is convenient to present the results graphically. Assume a specific value for the slenderness parameter r/t. Then, let the dimensionless length parameter l/πr vary and pcr may be plotted as a function of the shell length l. If only the parts of the curves representing the minimum value of the critical pressure, Pel, are retained, Fig.1.22 is obtained. The diagram yields, directly, the theoretical buckling pressure and the buckling mode. A1.4 Nonlinear shell equations The set of equations for the rotationally symmetric case containes terms of the form nxβ,x. In the general case these terms indicate the nonlinearity of the problem. The rigogous theory of shell stability is, however, even more complicated as will be demonstrated in the following: A more comprehensive presentation may be found in Brush-Almroth, 1975. The general equilibrium equations according to (A.17) and (A.24) are valid for a shell subjected to small rotations β In most cases, in particular if the collapse load of the shell is to be calculated, it is necessary to include the second order effects in the evaluation of the strains, see Fig.1.23. This leads to the following nonlinear set of kinematic equations: Introduction of (A.37) into the equilibriun conditions leads to a set of highly nonlinear equations which are much more complex than the linearized set represented by eqns.(A.24). An example is shown above in Fig.1.24. (A.37)

It should be evident that the general set of nonlinear shell equations may only be solved explicitely in a few very restricted cases. In fact, the problem is so complicated that it may not even be feasible to write special purpose programs for the solution of common practical problems. Several general computer codes have, however, been developed incorporating all essential aspects of nonlinearity of the shell stability problem. It is, therefore, not necessary for the engineer to solve the complicated mathematical problem of shell

Appendix A

275

stability, but knowledge about the puzzling buckling behavior of some shells is essential in the evaluation of the theoretical results. Different programs and methods of solution are discussed in Chapter 6 above.

Appendix B Elementary cases for stress and deformation analysis of rings subjected to forces in their own plane Formulas are presented for forces, moments and deformations in circular rings. These formulas are based on the first order theory of elasticity. If compressive forces occur, the formulas are only applicable if the buckling stress of the ring (compare 3.2.1.5) is substantially higher than the current compressive stress, say by a factor of 10. With this provision the elementary cases may be combined to solve a large number of different load cases. The major part of the formulas were derived from Roark, 1954, and the handbook ‘Bygg’. The following symbols are used, see Fig.B1. r=radius to the centroid of the ring section w=radial deformation, positive outwards, assuming that the center E is fixed wA, wB, wC=radial deformations at points A, B, C EA=extensional stiffness of the ring EI=bending stiffness of the ring, possible effective width of shell mantle is calculated according to 32.15 M=bending moment in the ring, positive when the curvature of the ring decreases N=normal force in the ring, positive in tension V=lateral shear force, positive according to the figure F=concentrated load p=distributed load, per length unit

Fig.B1. Definition of coordinate system and directions of deflections and forces in a plane ring subjected to forces in its own plane.

Appendix B

Case No 1

Case No 2

Case No 3

277

Appendix B

Case No 4

278

Appendix B

Case No 5

Case No 6

279

Appendix B

Case No 7

280

Appendix B

Case No 8

281

Appendix B

Case No 9

282

Appendix C Diagrams for analysis of stringer stiffened cylindrical shells under axial compression Stringer stiffeners will in some cases provide a considerable increase of the resistance of cylindrical shells subjected to axial stresses. The stress level in the shell is reduced due to the increased cross section area, at the same time as the buckling stress is raised with the increase in bending stiffness of the wall. The design charts presented are based on comprehensive calculations using the general theory for orthotropic cylindrical shells with plate stiffeners with the aspect ratio h/t=5 and 10. Both internal and external stringers are included, which is of great importance, since the resistance may differ considerably, see Esslinger, 1973. The design charts were established assuming that the stiffeners are so closely spaced that the stiffness properties of the shell can be considered to be continuous in the two directions. This assumption requires that the number of stringers ns>3.5n, where n is the buckling wave number in the circumferential direction. This requirement does not necessarily have to be adhered to very strictly, but if the design parameters fall outside given limits a certain caution is recommended. The design charts can be utilized as shown in the following example: A cylindrical shell with r=3000, l=3800, t=4 mm is to be designed for a total vertical load F=4 MN. The shell material is assumed to be steel with the yield limit σy=220 MPa. External stiffeners may be used. The classic buckling stress of the unstiffened shell is:

σu=0.75·0.9·0.285·168=32.3 MPa

a) r/t=750 and l/r=1.3 give, according to Fig.Ca, a) the buckling wave number n=13, ns>3.5 n=>ns=45, but the number ns=40 should be acceptable. b) Assume As/(bt)=0.35 and hs/t=10. Diagram c) yields: σsel=1.9σel=319 MPa c) Reduction with respect to the influence of length according to Diagram d) gives with l/r= 1.3 and reduction factor η=0.65:

Appendix C

284

σselr=0.65·0.8·319=166 MPa>σy/3, i.e. buckling in the plastic region.

The stress in the stiffened shell is reduced to

The stiffening is thus more than adequate. It is likely that the load requirement could be satisfied by a stiffening ratio of

. This may be checked as demonstrated above.

Appendix C

285

Fig.Ca. Design chart for calculating the theoretical buckling load of a cylindrical shell with external stringers. a) Buckling wave number in the circumferential direction, ns should exceed 3.5n. b) Theoretical buckling stress cylinder with stringers,

of and r=l.

Appendix C

286

c) As in b) but for

.

d) Correction for length. .

Fig.Cb. Design chart for calculating the theoretical buckling load of a cylindrical shell with internal stringers.

Appendix C

287

a) Buckling wave number in the circumferential direction, ns should exceed 3.5n. b) Theoretical buckling stress

and r=l.

cylinder with stringers, c) As in b) but for d) Correction for length. .

of

.