Vibration of Mindlin Plates Programming the p-Version Ritz Method
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Vibration of Mindlin Plates Programming the p-Version Ritz Method
K.M. LlEW School of Mechanical and Production Engineering Nanyang Technological University, Nanyang Avenue Singapore C.M. WANG Department of Civil Engineering The National University of Singapore, Kent Ridge Singapore Y. XIANG School of Civic Engineering and Environment The University of Western Sydney, Nepean, Kingswood Australia
S. KlTlPORNCHAl Department of Civil Engineering The University of Queensland, Brisbane Australia
1998 ELSEVIER Amsterdam - Lausanne - New York - Oxford Shannon - Singapore - Tokyo
-
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FOREWORD Having been involved myself with the study of plate vibrations for some four decades, I am especially pleased to see this monograph which is based entirely upon the Mindlin Theory. A tremendous amount of research has taken place on plate vibrations. My own monograph, completed in 1967 and published in 1969, presented frequencies and mode shapes taken from approximately 500 references. Since then at least 2000 additional relevant publications have appeared; however, the vast majority of these, on the order of 90 percent I would estimate, consider only thin plates. The present book deals with thick plates, although thin plates are also accounted for. A plate is typically considered to be thin when the ratio of its thickness to representative lateral dimension (e.g., circular plate diameter, square plate side length) is 1/20 or less. In fact, most plates used in practical applications satisfy this criterion. This usually permits one to use classical, thin plate theory to obtain a fundamental (i.e., lowest) frequency with good accuracy. However, the second frequency of a plate with thickness ratio of 1/20, determined by thin plate theory, will not be accurate. It will be somewhat too high. And the higher frequencies will typically be much too h i g h - too high to be of practical value. The inaccuracies described above are largely eliminated by use of the Mindlin theory, for it does include the effects of additional plate flexibility due to shear deformation, and additional plate inertia due to rotations (supplementing the translational inertia). Both effects decrease the frequencies. There are still other effects not accounted for by the Mindlin theory (e.g., stretching in the thickness direction, warping of the normals to the midplane), but these are typically unimportant for the lower frequencies until very_ thick plates are encountered. For such situations a three-dimensional analysis should be used. Some of my students and I have made such analyses during the past three decades. As for all boundary value problems, a few exact solutions for plate vibration problems exist in rectangular and polar coordinates, applicable to some cases of rectangular, circular, annular and sectorial plates. But for the vast majority of problems, including these shapes, approximate solutions must be found. The present monograph uses the well-known Ritz method exclusively. Displacements are assumed in the form of algebraic polynomials which satisfy the geometric boundary conditions, a procedure that has been effectively used for at least a half century. If this is done properly, then one can approach the exact frequencies and mode shapes as closely as desired as sufficient polynomial terms are utilized. The present monograph lays out the Mindlin theory briefly, including the appropriate equations in polar, rectangular, and skew coordinates. In each chapter it is shown how the procedure (Ritz method, with algebraic polynomial displacement functions) may be applied straightforwardly to solve a host of free vibration problems. In a series of researches during the past decade, resulting in numerous published papers, the authors have developed further the algebraic equation manipulation procedures previously initiated by others (most notably
vi Professor Yoshihiro Narita of Sapporo, Japan), which results in the rather general computer programs displayed throughout this book. The authors go one step further by presenting extensive numerical results - frequencies for circular, annular, sectorial, elliptical, triangular, parallelogram and trapezoidal plates, all according to the Mindlin theory. This serves several purposes: It permits one to verify the correctness of the computer programs listed, and of one's utilization of them. 2.
Considerable new, previously unpublished, frequency data are presented. The effects of shear deformation and rotary inertia may be readily seen. The tables typically give frequencies for thin, as well as thicker, plates.
.
Using the accurate, benchmark data presented, one may determine the accuracy of other approximate methods, especially finite element codes which take various forms.
I congratulate the authors for having taken the time and effort to produce this work. It should be useful to many persons.
Arthur Leissa Columbus, Ohio March 1, 1998
vii
PREFACE Ever since Chladni in 1787 observed nodal pattems on square plates at their resonant frequencies, there has been a tremendous research interest in the subject of plate vibrations. To date, abundant thin plate vibration solutions based on the Kirchhoff plate assumptions are available in the literature. A good reference source on this subject may be found in the monograph entitled "Vibration of Plates" by Professor Leissa of The Ohio State University. This invaluable document, initially published by NASA in 1969 and recently reprinted by the Acoustical Society of America in 1993 due to a great demand, presents mainly vibration results based on the Kirchhoffplate theory. This classical plate theory, however, overpredicts all the vibration frequencies for thick plates, and the higher frequencies for thin plates, as it neglects the effects of transverse shear deformation and rotary inertia. This shortcoming of the Kirchhoff theory forced researchers to develop more refined plate theories. As a result, we now have many such theories ranging from the first-order shear deformation plate theory of Mindlin to higher-order plate theories such as the one proposed by Reddy. Recent advances into this subject of plate vibration have focused more on these shear deformable plates that are somewhat complicated to analyse. This trend in plate research is further fueled by the availability of powerful digital computers that permit large numbers of variables to be processed within a relatively short time. Over the last several years, the four authors have jointly conducted research into the analysis of vibrating Mindlin plates as a collaborative project between Nanyang Technological University, The National University of Singapore, and The University of Queensland. The research was prompted by the fact that there is a dearth of vibration results for Mindlin plates when compared to classical thin plate solutions. To generate the vibration results, the authors have successfully employed the Ritz method for general plate shapes and boundary conditions. The Ritz method, once thought to be awkward for general plate analysis, can be automated through suitable trial functions (for displacements) that satisfy the geometric plate boundary conditions a p r i o r i . This work has been well-received by academicians and researchers, as indicated by the continual requests of the authors' papers and the Ritz software codes. The present monograph is written with the view to share this socalled p-Ritz method for the vibration analysis of Mindlin plates and its software codes with the research community. To the authors' knowledge, the monograph contains the first published Ritz plate software codes of its kind. Since it is a voluminous task to provide engineers and researchers with vibration solutions of Mindlin plates of various shapes and boundary conditions, the software codes listed in this monograph enable easy generation of the required results. The classical plate solutions can be readily computed from the software by setting the plate thickness to a small value. As it is more convenient to handle certain plate shapes in their natural coordinate systems, four versions of the p-Ritz software are given. The software code VPRITZP1 is based on a one-dimensional polar coordinate system for solving axisymmetric plate problems. VPRITZP2 allows the analysis of nonaxisymmetric plates in polar coordinates. VPRITZRE is based on the rectangular Cartesian coordinate system, and the software can be used to analyse any plate shape whose edges are defined by polynomial functions. Finally VPRITZSK is based on the skew coordinate system that is expedient for handling
viii parallelogram plates and plates having parallel oblique edges. Although the software codes are written for isotropic plates, they may be readily modified for laminated plates and for complicating effects such as initial stress effects, foundation effects, etc. The programs can also be modified to accommodate the bending and buckling analyses of Mindlin plates. The authors would like to thank Professor A.W. Leissa for providing useful comments and for writing the Foreword to this book, Dr C.W. Lim for generating the vibration mode shapes, Dr K.K. Ang for useful comments on the software code, Mr W.H. Traves for proofreading the manuscript, and finally to our wives for their patience and support.
K.M. Liew Nanyang Technological University, Singapore C.M. Wang The National University of Singapore, Singapore Y. Xiang The University of Westem Sydney Nepean, Australia S. Kitipornchai The University of Queensland, Australia
ix
CONTENTS FOREWORD PREFACE
V
vii
INTRODUCTION
1.1 Background of vibration 1.2 Plate vibration 1.3 About this monograph MINDLIN PLATE T H E O R Y AND RITZ M E T H O D
5
2.1 Mindlin plate theory 2.1.1 Displacement components 2.1.2 Strain-displacement relations 2.1.3 Stress resultant-displacement relations 2.1.4 Energy functionals 2.1.5 Governing equations of motion 2.1.6 Boundary conditions 2.2 Relations between Kirchhoff and Mindlin plates 2.2.1 Reduction of Mindlin theory to Kirchhoff 2.2.2 Frequency relationship for a class of plates 2.3 Shear correction factor 2.4 Ritz method 2.4.1 Preliminary remarks 2.4.2 Application of Ritz method to Mindlin plates
5 6 7 7 8 9 12 15 15 16 23 24 24 25
F O R M U L A T I O N IN P O L A R COORDINATES
3.1 Introduction 3.2 Energy functionals 3.3 Eigenvalue equation 3.3.1 Circular and annular plates 3.3.2 Sectorial and annular sectorial plates 3.4 Computer program 3.4.1 Software code: VPRITZP1 3.4.2 Sample files for VPRITZP1 3.4.3 Software code:VPRITZP2 3.4.4 Sample files for VPRITZP2 3.5 Benchmark checks 3.5.1 Annular plates 3.5.2 Sectorial plates 3.5.3 Annular sectorial plates
27 27 33 35 35 37 40 40 54 56 71 74 74 74 76
F O R M U L A T I O N IN R E C T A N G U L A R COORDINATES
89
4.1 Introduction 4.2 Energy functionals 4.3 Eigenvalue equation
89 92 93
4.4 Computer program 4.4.1 Software code: VPRITZRE 4.4.2 Input file 4.4.3 Output file 4.5 Benchmark checks 4.5.1 Isosceles triangular plates 4.5.2 Trapezoidal plates 4.5.3 Elliptical plates
96 96 108 113 118 118 118 118
F O R M U L A T I O N IN S K E W C O O R D I N A T E S
133
5.1 5.2 5.3 5.4 5.5
Introduction Skew coordinates transformation Energy functional in skew coordiantes Eigenvalue equation Computer program 5.5.1 Software code: VPRITZSK 5.5.2 Sample files 5.6 Benchmark checks
133 133 134 135 137 138 153 156
PLATES W I T H C O M P L I C A T I N G E F F E C T S 6.1 Introduction 6.2 Initial inplane stresses 6.3 Elastic foundations 6.4 Stiffeners 6.5 Nonuniform thickness 6.6 Line/curved/loop internal supports 6.7 Point supports 6.8 Mixed boundary conditions 6.9 Reentrant corners 6.10 Perforated plates 6.11 Sandwich construction
165
165 165 167 169 173 175 176 177 178 179 179
REFERENCES
183
RELEVANT REFERENCE BOOKS
189
APPENDIX I - GAUSSIAN QUADRATURE SUBROUTINES
191
A P P E N D I X II - S U B R O U T I N E S F O R M A T H E M A T I C A L O P E R A T I O N S ON POLYNOMIALS
197
SUBJECT INDEX
201
CHAPTER ONE
INTRODUCTION
1.1
BACKGROUND OF VIBRATION
Vibration, in mechanics, is the to and fro motion of an object. Examples of vibration abound in nature as nearly everything vibrates; though some vibrations may be too low or too weak for detection. Vibration can be felt by the sense of touch when a vehicle passes by or felt by our eardrums when a guitar string is being plucked. Large vibrations occur during earthquakes and when the ocean level rises and falls causing tides. Vibration can be exploited for useful tasks, such as the use of a vibrator to massage the body, to compact loose soil, to increase the workability of wet concrete and to shake sugar, pepper and salt from their containers. On the other hand, vibration can cause discomfort for people and problems for machines. Too much vibration can cause people to loose concentration and to fall sick. In machines, vibration causes wear and tear and can even cause the malfunctioning of the machine. In view of the intensive use of structural components in various engineering disciplines, especially in the aerospace, marine and construction sectors, a thorough understanding of their vibratory characteristics is of paramount importance to design engineers in order to ensure a reliable and lasting design. The negligence of considering vibration as a design factor can lead to excessive deflections and failures. An unforgettable incident showing the destructive nature of vibration is the dramatic collapse of Tacoma Narrows Bridge in 1940 that was captured on film. Other failures due to wind induced vibrations include collapse of chimneys, water tanks, transmission towers, etc. There are also failures triggered by seismic shocks leading to large-scale destruction of cities such as the 1995 Kobe earthquake. The vibration design aspect is even more important in micromachines such as electronic packaging, micro-robots, etc. because of their enhanced sensitivities to vibration.
1.2
PLATE VIBRATION
The study of plate vibration dates back to the early eighteenth century, with the German physicist, Chladni (1787), who observed nodal patterns for a flat square plate. In his experiments on the vibrating plate, he spread an even distribution of sand which formed regular patterns as the sand accumulated along the nodal lines of zero vertical displacements upon induction of vibration. In the early 1800s, Sophie Germain, a French mathematician, obtained a differential equation for transverse deformation of plates by means of calculus of variations. However, she made the error of neglecting the strain energy due to the warping of the plate midplane. The correct version of the governing differential equation, without its derivation, was found posthumously among Lagrange's notes in 1813. Thus, Lagrange has been credited as being
2
Sec. 1.2 Plate Vibration
the first to use the correct equation for thin plates. Navier (1785-1836) derived the correct differential equation of rectangular plates with flexural resistance. Using trigonometric series introduced by Fourier around that time, Navier was able to readily determine the exact bending solutions for simply supported rectangular plates. Poisson (1829) extended Navier's work to circular plates. The extended plate theory that considered the combined bending and stretching actions of a plate has been attributed to Kirchhoff (1850). His other significant contribution is the application of the virtual displacement method for solving plate problems. Lord Rayleigh (1877) presented a theory to explain the phenomenon of vibration which to this day has been used to determine the natural frequencies of vibrating structures. Based on the plate assumptions made by Kirchhoff (1850) and Rayleigh's theory, early researchers used analytical techniques to solve the vibration problem. For example, Voigt (1893) and Carrington (1925) successfully derived the exact vibration frequency solutions for a simply supported rectangular plate and a fully clamped circular plate, respectively. Ritz (1909) most probably was one of the early researchers to solve the problem of the freely vibrating plate which does not have an exact solution. He showed how to reduce the upper bound frequencies by including more than a single trial (admissible) function and performing a minimization with respect to the unknown coefficients of these trial functions. The method became known as the Ritz method. Improving on the Kirchhoff plate theory, Hencky (1947) and Reissner (1945) proposed a first order shear deformation plate theory to cater to thick plates where the effect of transverse shear deformation is significant and thus cannot be neglected. Mindlin (1951) presented a variational approach for deriving the governing plate equation for free vibration of first-order shear deformable plates and incorporated the effect of rotary inertia. The first order shear deformation plate theory of Mindlin, however, requires a shear correction factor to compensate for the error due to the assumption of a constant shear strain (and thus constant shear stress) through the plate thickness that violates the zero shear stress condition at the free surfaces. The correction factors not only depend on material and geometric parameters but also on the loading and boundary conditions. The study of Wittrick (1973) indicates that it may be impossible, in general, to obtain the shear correction factors for a general orthotropic plate. A more refined plate theory that does away with the shear correction factor is that proposed by Reddy (1984, 1997). His third-order shear deformation theory ensures that the zero shear stress condition at the free surfaces of the plate is satisfied at the outset. Even higher-order theories have also been proposed, but the tractability of solving the equations become too difficult to warrant the relatively small improvement in the accuracy of the plate solutions. Owing to the technological advances in recent years, plate elements are commonly selected as design components in many engineering structures because of their ability to resist loads by two-dimensional structural action. With the evolution of light plate-structures, tremendous research interests in vibration of plates are generated. From what has been done, plates of almost any conceivable shape, support and loading conditions have been investigated. Along with this, various analytical and numerical methods have been proposed. Of these, the finite element method is the most commonly used because of its versatility to handle any plate shape and boundary conditions. The use of the finite element method started around the mid of 1950s. In 1956, Turner, Clough, Martin and Topp introduced the method, which allows the numerical solution of complex plate and shell problems in an efficient way. Numerous contributions in this field are also due to Argyris (1960) and Zienkiewicz (1977). In recent years, especially, the Ritz method has been shown to be an efficient alternative to other numerical techniques for the free vibration analysis of plates of arbitrary shape and
Chap. 1 Introduction
3
boundary conditions. This is made possible by using a set of geometrically generated Ritz functions that automatically satisfy the geometric boundary conditions. The development of the Ritz method for a single general plate brings us a step closer towards the ultimate goal of generalizing the Ritz method for the analysis of complicated plate-structures.
1.3
ABOUT THIS MONOGRAPH
This monograph provides software codes for the free vibration analysis of thick plates. The computer programs were written based on the p-Ritz method. The Mindlin plate theory was employed to incorporate the effects of transverse shear deformation and rotary inertia. Chapter 2 presents the plate theory of Mindlin and its underlying assumptions. The goveming plate equations and the boundary conditions were derived using the Hamilton's principle. Also highlighted herein is the existence of an exact relationship between the frequencies of the classical thin (Kirchhoff) plates and Mindlin plates of polygonal shape and simply supported for all the straight edges. The relationship allows one to obtain the vibration frequencies of Mindlin plates from widely available Kirchhoff plate solutions. The shear correction factor required in the Mindlin plate theory is discussed. The implementation of Mindlin plate theory into the Ritz method is detailed. Chapter 3 gives the Ritz formulation for plate analysis in a polar coordinate system. Such a polar coordinate formulation is expedient for circular, annular and sectorial plates. Some exact solutions for circular and annular Mindlin plates are given for validation purposes. The software codes VPRITZP 1 and VPRITZP2 are provided with illustrative input and output files. The former code is designed for the analysis of axisymmetric plates and the latter for non-axisymmetric plates. Tables of vibration frequencies are also given for sectorial and annular sectorial plates with the view to providing benchmark results. Chapter 4 details the Ritz formulation in a rectangular coordinate system. This coordinate system is the most commonly used as it can readily handle almost any plate shape. The software code VPRITZRE is given with illustrative examples. Benchmark vibration frequencies are also presented for isosceles triangular plates, trapezoidal plates and elliptical plates. Chapter 5 fumishes the Ritz formulation in a skew coordinate system. This coordinate system is useful for plates with oblique parallel edges. The software code VPRITZSK is given with an example. Vibration frequencies for skew plates with various boundary conditions are also presented. Finally, Chapter 6 discusses the treatment of various complicating effects such as inplane stresses, an elastic foundation, presence of stiffeners, non-uniform thickness, line/curved/loop intemal supports, point supports, mixed boundary conditions, re-entrant comers, perforations and sandwich construction.
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CHAPTER TWO
MINDLIN PLATE THEORY AND RITZ METHOD
2.1
MINDLIN PLATE TIIEORY
In the well-known classical thin plate or Kirchhoffplate theory for vibration, the following assumptions have been made (Kirchhoff 1850):
9 No deformation occurs in the midplane of the plate; 9 Transverse normal stress is not allowed; 9 Normals to the undeformed midplane remain straight and normal to the deformed midplane and unstretched in length; and 9 The effect of rotary inertia is negligible. The assumption regarding normals to the midplane remaining normal to the deformed plane amounts to neglecting the effect of transverse shear deformation. This effect, together with the rotary inertia effect, become important when the plate is relatively thick or when accurate solutions for higher modes of vibration are desired. Wittrick (1987) pointed out that excluding edge effects, the error in the Kirchhoff plate theory is O(hZ/fl 2) where h is the thickness and /,t is a typical half-wavelength of the vibrating plate. If the Kirchhoff plate theory is used, the frequency responses are overpredicted. A more refined plate theory is thus necessary for thick plate analysis. There have been many thick (shear deformable) plate theories proposed with the implicit objective of reducing the error to less than O(h 2/t~2 ). Reissner (1944, 1945) proposed the simplest thick plate theory by introducing the effect of transverse shear deformation through a complementary energy principle. Unlike Reissner's work, Mindlin (1951) presented a first-order theory of plates where he accounted for shear deformation in conjunction with a shear correction factor. In this theory, the first two Kirchhoff assumptions are maintained. To allow for the effect of transverse shear deformation, the theory relaxes the normality assumption so that
Normals to the undeformed midplane remain straight and unstretched in length but not necessarily normal to the deformed midplane. This assumption implies a non-zero transverse shear strain, but it also leads to the statical violation of zero shear stress at the free surfaces since the shear stress becomes constant through the plate thickness. To compensate for this error, Mindlin (1951) proposed a shear correction factor Ic2 to be applied to the shear force. Besides, Mindlin (1951) modified the fourth assumption so that the
6
Sec. 2.1 Mindlin Plate Theory 9 Effect o f rotary inertia is included
In the literature, vibrating plates based on the first-order shear deformation plate theory assumptions are widely referred to as Mindlin plates. 2.1.1
DISPLACEMENT COMPONENTS
In the Mindlin plate theory, the displacement components are assumed to be given by: u(x, y , z , t ) = zNx(x, y,t )
(2.1a)
v(x, y , z , t ) = Zp'y (x, y,t)
(2.1b)
~(x, y , z , t ) = w(x, y,t)
(2.1c)
where t is the time, u, v are the inplane displacements, w the transverse displacement and ~x, ~y the bending rotations of a transverse normal about the y and x axes, respectively, as shown in Fig. 2.1. The notation that Nx represents the rotation about the y-axis and vice versa may be confusing to some and in addition they do not follow the right-hand rule. However, these notations will be used herein because of their extensive use in the open literature. Note that by setting r = - d w / d x and p,y = - dw/dy, the Kirchhoff plate theory may be recovered. For higher-order plate theories, higher-order polynomials are used in the expansion of the displacement components through the thickness of plate (see, for example, papers by Nelson and Lorch 1974, Lo et al. 1977, Levinson 1980, Reddy 1984, Lim et al. 1989). Notable among them is the one proposed by Reddy (1984, 1997) who derived the displacement field by imposing zero transverse shear strain condition at the free surfaces a priori to the expanded inplane displacements up to the third power for the thickness coordinate. Based on this displacement field, he derived variationally consistent equations and boundary conditions for the so-called Reddy (third-order shear deformation) plate theory.
c%v ax
~x
LZ x Fig. 2.1 Rotations of the normals
Chap. 2 M i n d l i n P l a t e T h e o r y a n d Ritz M e t h o d
7
2.1.2 S T R A I N - D I S P L A C E M E N T RELATIONS In view of Eq. (2.1), the linear components of the engineering (non-tensorial) strains can be expressed as c3~,x
~x~ = z ~ &
s
:Z
(2.2a)
(~/'Y Oy
(2.2b)
ezz = 0
(2.2c)
}'xy : Z
+ OW
7 " = ~'x + Y yz "-- ~bCy "[-
(2.2d) (2.2e)
aX OW
(2.20
cry
where c a , Cyy,Czz, are the normal strains and Yxy, Yxz, Yyz, the shearing strains. 2.1.3 STRESS R E S U L T A N T - D I S P L A C E M E N T RELATIONS Based on the above strain-displacement relations and assuming a plane stress distribution in accordance with Hooke's law, the stress-resultants are obtained by integrating the stresses: M xx =
J-h~2
Crxxz dz =
d-h~2
I_V 2
(2.3a)
M y y = J-h~2
=D
M ~y =
o ' y y z d z = d-h~2
1-v
+v
(2.3b)
f h/2 f h/2 %, z dz = G T ~yz dz l-h~2 9 ,l-h~2
D(I- v) &,x +
J-h/2 rxz d z = K G h ~'x +
(2.3c)
(2.3d)
8
Sec. 2.1 Mindlin Plate Theory =
"['yz
.l-h~2
dz =
tf2Gh
~l'y Jr-
(2.3e)
where Mx~, Myy, Mxy are the bending moments per unit length of plate, Qx,Qy the transverse shear forces per unit length of plate, Crx~,Oyythe normal stresses, rxy,rxz,ryz the shear stresses, h is the plate thickness, E the modulus of elasticity, G = E/[2(1 + v)] the shear modulus, v the Poisson ratio, D = Eh3/[120-v2)] the flexural rigidity and tr 2 the shear correction factor to compensate for the error in assuming a constant shear stress throughout the plate thickness. A discussion of the shear correction factor will be given in the sequel. The stresses, bending moments and shear forces are shown in Fig. 2.2 in their positive senses. 2.1.4 ENERGY FUNCTIONALS The strain energy functional U due to bending of the Mindlin plate is given by
6r B6dV
u=lf -
2
(2.4)
v
Z
Z
,xy I ' [ ' ~ _ l
"
O'xx
i i
Mxy ~QY.
Y x
O'nn
J'k "~_
x "
Mxx+
i~
- ~~
? xx a~ /
M xv+ Y
C3Mx" " dx OX
/ ....
~,,"xy
Q d x ~
9 ~M -
xu
yy l
\ ~
Ox
~
I
x
Y
C3Mxy 092
ely
0 +"~Y d~
,, C3Myy . Myy"]- Oy ay
Fig. 2.2 Stresses, bending moments and shear forces of a plate
where c r = {Cx~eyy YxyYxzYyz}, V is the plate volume and the material property matrix B for an isotropic elastic plate is given by
Chap. 2 Mindlin Plate Theory and Ritz Method vE
E l_v 2
1 --V 2
E 1--V 2
B
0
0
0
0
0
0
__
9
(2.5) 0
0
K.2G
0
G
K.2G
symmetrical
The substitution of Eqs. (2.2) and (2.5) into Eq. (2.4), and after the integration over the plate thickness, yields U = -1 2
D
Ogx + e?gy v3c ~
- 2(1 - v) O~,x C)~y v3c 03,
1
O~x
4
@
+
d~y &
A
+ tc2Gh ~ +
+ ~y +
dA
(2.6)
where A is the plate area, and dA = dxdy. The kinetic energy T of the vibrating Mindlin plate is given by T =1 IpI(--~-/2 2
+
("~/2
+
(-~/21
(2.7)
dV
v
where t denotes time, p the mass density (per unit volume). By substituting Eqs. (2.1a-c) into Eq. (2.7), and integrating through the thickness dimension, the kinetic energy T may be expressed as
--&
+--~ [k, Ot )
( Ot )
dA
(2.8)
The Lagrangian FI is thus given by 1-I = T - U 2.1.5
(2.9)
GOVERNING EQUATIONS OF MOTION
The governing equations of motion may be derived from the Hamilton's principle which requires 6 I,/'f YIdt = 0
(2.10)
10
Sec. 2.1 Mindlin Plate Theory
where 6 is the variational operator. Note that Hamilton's principle states that of all the paths of admissible configurations that the body can take as it moves from configuration "i" at time ti to configuration "f" at time t I , the path that satisfies Newton's second law at each instant during the interval is the path that extremizes the time integral of the Lagrangian during the interval. The substitution of Eqs. (2.6), (2.8), (2.9) into Eq. (2.10) yields
- ~
2
Oy
-tf2Gh
+
Ox
Nx +
Oy
+
d~Y'x }'-~x
Ox -}- ~ll'y q-
6~bry +
----+Owc36wPhZCONx C36Nx F any c36NY)}dAdt=0 +__oh Ot c3t 12 ~. 3t c3t c3t c3t
(2.11)
Performing integration by parts, Eq. (2.11) results in
9{ It i +
fi 2o2 3 ~x
D
6~x +
3 Ny
d ~x
6Ny + v
3 ~y 6~x
i
D(1 - v) (c72Nx s s 2 ~/)rx 021//y 61//'x ] 2 ~ & 2 61/]'x nt- O~X2 6~]'y nt" ~ & 6[pCy nt- O'~&
)
- x 2Gh Nx rNx
ONXfw+Ny rNy _ fiw+ ax Of ox
\
02W
~
d2W
- ~----r~w+-g~,,--g-~w - ph--~-fw-
fdt i +
6Ny + v
D
12
Ot 2
6NxdY-
x
)
J 6Nx +
c3t2
/t
6Ny dAdt
6Nydx- v
OX
6Nydx + v O~y &&dy) @
D(1- v) ( aNx ONx any oN, ) 6NxdX+ 6Nydy 6NxdX+ fig/y@ 2 -~ Oy --~-x c3x (2.12)
where F is the boundary path.
Chap. 2 Mindlin Plate Theory and Ritz Method
11
By grouping the terms in the foregoing functional with respect to the variation terms, we have tz ~
I ti
{I 2 D
0 ~x OX 2
+v
~'y
"t-
Oxdy
- Ph---~30Ot 2 2 6p'x + D .~_
2
Ox2
~t-
Ox
D
fJti
~r
+
-- I~ 2 G h
02~x OY2 +
My Ow -- tfZGh I//x + Oxdy -~x
,5,
Oy
~Oey +
-
12
+
Ox ~
dy+v
Iv)
2
cTY2 + v c3cc32
c3xdy
+ tc2Gh C3~x + - -
D(1
~)
dy-
-ph
2
+ - D C~y dx+vCT~xdx + c~ cgx + tr2Gh p,'xdy+---~dy-~tydx-
fiw dAdt
Oy
dx+
c3x
dy+
2
6~bCy
Ot2
dx 5~x
dy 6gy
dx 6w dt = 0
(2.13)
Equating the coefficients of the variation terms to zero for the functional over the plate area and assuming free harmonic motion, the following three goveming equations of motion (after omitting the factor e i~ ) are obtained: D(1-V)v 2
( l + v ) 0(I)
D(1 v ) z
(l+v)0cI)
2
~ x + - - - - - t r Ox
v~,~+
~
2
(
~x)
(
~)
Gh ~x +
~ - , c ~ C h ~ty +
tc2 Gh(V2 w + ~ ) = -phco 2w
ph3co2
= - 12
ph3
~x
2
= - 12 coV/~
(2.14a)
(2.14b)
(2.14c)
where the right hand sides of Eqs. (2.14a) and (2.14b) are the rotary inertia terms, co is the angular frequency of vibration in radians per second, V 2(,) = 8 2(,)/c3x 2 + 0 2(*)/c~ 2 is the two-dimensional Laplacian operator and
~ = D ( ONxcgx+ c~Y)c~= M~x(I+v)+Myy
(2.15)
12
Sec. 2.1 Mindlin Plate Theory
The auxiliary function 9 may be referred to as the Mindlin Marcus-moment (or moment sum). Marcus (1932) proposed the moment sum so as to convert the fourth-order differential plate equation of classical theory into two Poisson equations. The governing equations of motion may also be expressed in terms of the stressresultants [defined by Eqs. (2.3a-e)] as follows: OMx,
OMxy
-Qx
OMyy
0)14xy
Oy
Ox
~ +
OQx
2.1.6
+
-ay
ph 3
= - ~
ph 3
= - ~
cO2gx
(2.16a)
cO2gy
(2.16b)
12
OQy +~=-phco
2 w
(2.16c)
BOUNDARY CONDITIONS
For boundary conditions, the line integral of Eq. (2.13) is set to zero, i.e.
@
1
+
2
Oy
+~
@ Ox
6~ydy-~
2
ox j Oy
+
Ox
6~xdx
(2.17)
In view of the stress-resultants expressions given in Eqs. (2.3a-e), Eq. (2.17) leads to
+ OxaWdy - O, awdx }dt = O
(2.18)
Considering a portion of the path F, as shown in Fig. 2.3, the rectangular coordinates (x, y) are related to the normal and tangential coordinates (n, s) by dx = - sin 0 ds dy = cos0 ds
(2.19a) (2.19b)
In view of Eq. (2.19), Eq. (2.18)can be further written as
~t,tl ~ {M= cos 08gx + Myy sin 08gy + M~y cos 08gy + M,y sin 06g + Qx cos ONv + Qy sin ONv}dsdt = 0
(2.20)
Chap. 2 Mindlin Plate Theory and Ritz Method
13
The bending rotations can be expressed in terms of their normal-tangential counterparts by the following relations g x = ~. cos 0 - ~"s sin 0
(2.21a)
y = ~. sin 0 + ~/s cos 0
(2.21b)
where the subscripts n, s denote the normal and tangential directions, respectively, gs the rotation of the midplane normal in the tangent plane sz to the plate edge and g , the rotation of the midplane normal to the edge.
Fig. 2.3 Rectangular coordinates and normal-tangential
By substituting Eq. (2.21) into Eq. (2.20), one obtains
sT / ooos0(oosO .-sinO
+<,sinO(sin
+cos
+ Mxy [cos O(sin Oa~n + cos Oag, ) + sin O(cos O6"g. - sin Oags )1 + Qx cos 0fiw + Qy sin Ofiw}dsdt = 0
(2.22)
The relationships between the bending moments/shear forces in the n-s and x-y coordinate systems are given by M,, = M~x COS 2 0
M,,s = ( ~
"~- M
yy sin 2 0 + 2Mxy sin 0 cos 0
-M~)sin0c~
~~ 0 - s i n ~ 01
(2.23a) (2.23b)
14
Sec. 2.1 Mindlin Plate Theory Q, = Qx cos 0 + Qy sin 0
(2.23c)
where M,n is the bending moment normal to the edge, M,, the twisting moment at the edge, and Q, the shear force in the normal direction n to the plate edge. In view of Eqs. (2.23a-c), Eq. (2.22) may be simply expressed as
tf ~ {Mnn61Yn+ Mn,81//, + Qn6W}dsdt -- 0 It/
(2.24)
Equation (2.24) implies that along the boundary of the plate,
Either JM,,
or
is specified
(2.25)
[Q, Equation (2.25) requires the specification of three boundary conditions at the plate edge for Mindlin plates. The common edge conditions for Mindlin plates are given as follows.
9 Free edge (F) For this type of edge condition = Mn s
cTn
+v
Os
=0
= D(1-v)(cTN, ~ + cTigsl=0 2 c~ cTn
Qn= tf2Gh(~n+--~1 =0
(2.26a) (2.26b) (2.26c)
9 Simply supported edge (S and S*) There are two kinds of simply supported edges for Mindlin plates. The first kind (S), which is referred to as the hard type simple support, requires
mnn =0,
Us =0, w = 0
(2.27a-c)
The second kind (S*), commonly referred to as the soft type simple support, requires M,, =0, M,s =0, w = 0
(2.28a-c)
Clamped edge ( C) This type of edge condition requires ~, = O, ~s = O, w = 0
(2.29a-c)
Chap. 2 Mindlin Plate Theory and Ritz Method 2.2
15
RELATIONS BETWEEN KIRCHHOFF AND MINDLIN PLATES
2.2.1. REDUCTION OF MINDLIN THEORY TO KIRCHHOFF In the Mindlin plate theory, there are three displacement variables, i.e. gx, gy, w, whereas the Kirchhoff (or classical thin) plate theory [Timoshenko and Woinowsky-Krieger 1959] involves only a single displacement variable ~ . The latter theory can be deduced from the Mindlin plate theory by setting
c~ g/x = - ~ v3c
(2.30a)
c3~ ~'= c,3, tr 2 --~ oo
(2.30b) (2.30c)
and neglecting the rotary inertia term. In Eqs. (2.30a) and (2.30b), ~ denotes the transverse deflection of the Kirchhoff plate. Using Eqs. (2.30a-c) and neglecting the rotary inertia effect, Eq. (2.11) reduces to the Lagrangian FI for Kirchhoffplates,
1 I JDI(02]~
02W~2_2(l_v) [d21~ d&221~ ( d2142] 2
A - ph~)2~, 2 t dA
(2.31)
where o5 is the natural frequency of the Kirchhoff plate. Using variational calculus, the governing differential equation of motion for the Kirchhoffplate is given by '
DV4~,- ph(_~2]&,= 0
(2.32)
Unlike the three boundary conditions used to specify the condition of a plate edge in the Mindlin plate theory, only two are required in the Kirchhoff plate theory. The usual boundary conditions for Kirchhoff plates are given by
9 Free edge (F) For this type of edge condition, M,, = - D
02]41n + v On2
2 "-"
Osz
= 0,
(2.33a)
" =O, "~ + ~C~"s = - D k [ 02~" + (2 - V)0=Wn1 = 0 1I,
cTs
On k On2
Os2
(2.33b)
16
Sec. 2.2 Relations Between Kirchhoff and Mindlin Plates v
where M,, is the bending moment normal to the Kirchhoff plate edge, V, the effective shear force, Q, the shear force in the direction n normal to the plate edge, and M,, the twisting moment at the edge. Simply supported edge (S) For this type of edge condition, M,, = O, ~ = 0
(2.34a,b)
Clamped edge ( C) For this type of edge condition, a%
- - = O, ~ = 0
(2.35a,b)
c,'n
2.2.2. FREQUENCY RELATIONSHIP FOR A CLASS OF PLATES There exists an exact relationship between the natural frequencies of Mindlin plates and those of the corresponding Kirchhoff plates. This relationship is, however, restricted to a class of polygonal plates in which all the straight edges are S-simply supported [see boundary conditions as defined by Eqs. (2.27a-c)]. By differentiating Eqs. (2.14a) and (2.14b) with respect to x and y, respectively, then summing them up and noting Eq. (2.14c), one obtains
V2(i) -- K.2Gh((I) ~ -.1-.V 2 W) O
1
=--
~
/oh 3o) 2 -@
(2.36)
O
The substitution of the expression of (I) from Eq. (2.14c) into Eq. (2.36) yields V 4 w ..}_/
ph nt" ph3 I
K2Gh
w q_Ph ( Ph3(.02
12D) (-O2V2
) co 2w = 0
(2.37)
Equation (2.37) may be factored to give
(~72 + yl XV2 + Y2)W=0
(2.3S)
where 7'j = -~ K 2Gh + 12D ) c~ +(-1) s -~ tc-i-Gh 12D)
(2.39)
D
Alternatively, Eq. (2.38) may be written as two second order equations given by V
--}-yi)W = W
(V 2 -{- y j >
-- 0
(2.40a) (2.40b)
Chap. 2 Mindlin Plate Theory and Ritz Method
17
where i = 1 if j = 2 and vice versa. For the hard type of simply supported (S) polygonal plate, the boundary conditions are given by Eqs. (2.27a-c). Since along the straight edge ~, = 0 implies that c3~, /Os = 0, then together with the condition M,, = 0, one may deduce that c3g//On = 0. In view of this fact and Eq. (2.14c), the boundary conditions may be given as w = O, 9 = O, V 2w = O, ~ = 0 on the boundary.
(2.41a-d)
The following governing equation for the vibrating Kirchhoff plate is given by Eq. (2.32) or may be obtained from Eq. (2.37) by setting tc 2 ~ oo and omitting the rotary inertia
term [=(ph3 go2V2w)/(12D) ]
+yX -y>-o
(2.42)
y2 = Ph~)______~ 2
(2.43)
D For a simply supported polygonal Kirchhoff plate, the deflection and the Kirchhoff Marcus moment are zero at the boundary, i.e. ]~ -" 0, V 21,~ = 0
on the
(2.44a,b)
boundary.
As pointed out by Conway (1960), and later proven by Pnueli (1975), the frequency solutions of the fourth order differential equation (2.42) and the boundary conditions given by Eq. (2.44) are the same as those given by solving simply the following second order differential equation, (V 2 + )7> = 0
(2.45)
and the boundary condition if, = 0. Owing to the mathematically similarity of Eqs. (2.40b) and (2.41d) with Eqs. (2.45) and (2.44a), it follows that the aforementioned vibration Mindlin plate problem is analogous to the vibration Kirchhoff plate problem. Thus, for a given simply supported, polygonal plate Yj = )7
(2.46)
The substitution of Eqs. (2.39) and (2.43) into Eq. (2.46) furnishes the frequency relationship between the two kinds of plates,
C02N= - -
ph
-
1+
1+
ff)Nh2
-i2
ff_)Nh2
l
1 + ~ tc2(l-v)
+
~
tc (1 - v)
--
3tr
2Gff)N2
(2.47)
18
Sec. 2.2 Relations Between Kirchhoff and Mindlin Plates
where N = 1,2,...., corresponds to the mode sequence number. If the rotary inertia effect is neglected, it can be shown that the frequency relationship simplifies to ~2
^2
coN
coN "--
ff)Nh 2
1+ 6(1:v)-K2
~ph
(2.48)
D
where cb is the frequency of Mindlin plate without the rotary inertia effect. This frequency value cb is greater than its corresponding co but smaller than o3. Graphical representations of the frequency relationships given by Eqs. (2.47) and (2.48) are shown in Fig. 2.4, where v=0.3 and tr 2 = 5 / 6 have been assumed. By nondimensionalizing the circular frequency using p,h and D , the curves shown in Fig. 2.4 become independent of the plate shape! Note that as one moves along the curves away from the origin, the plate gets thicker or the frequency value becomes higher. It is clear from the figure that when the frequencies are low (lower modes of frequency or thin plates), the Mindlin solutions are close to the Kirchhoff solutions. When the plate thickness increases and for higher mode frequencies, the Mindlin solutions decrease relative to the Kirchhoff solutions. The effect of rotary inertia is also shown in the same figure and it can be seen that this effect becomes significant for high frequency values. Although the relationship given in Eq. (2.47) is exact only for polygonal plates with straight edges, it has been shown by Wang (1994) that the relationship provides reasonably accurate frequencies for Mindlin plates from Kirchhoff solutions even when the simply supported edges are curved. This relationship enables a quick deduction of simply supported Mindlin plate frequencies from the abundant Kirchhoff plate vibration solutions. It may be used as a basic form in which approximate formulas may be developed for predicting the Mindlin plate frequencies for other boundary conditions. Moreover, the exact relationship provides a useful means to check the validity, convergence and accuracy of numerical results and software. Note that Wang et al. (1998) have also established a similar frequency relationship between Kirchhoff plates and Reddy plates. Conway (1960) pointed out the analogies between the vibration problem of the Kirchhoff plate, the buckling problem of Kirchhoff plate and the vibration problem of uniform prestressed membranes. Owing to these existing analogies, one may choose to substitute into Eq. (2.47) the buckling solution for the corresponding simply supported Kirchhoff plate under hydrostatic inplane load, or the frequency of the corresponding uniformly prestressed membrane, instead of the Kirchhoff plate frequency. In other words, o3N may take any one of the following expressions:
_
NN
_(~NIUl D
(2.49)
-i ph
where N N is the buckling load for the N-th mode,
(/)N
the N-th frequency of the vibrating
prestressed membrane, p the mass density per unit area of membrane and T the uniform tension per unit length of membrane.
Chap. 2 Mindlin Plate Theory and Ritz Method
Kirchhoff plate \
5 tt~
/ ~
19
Mindlin plate (without rotary ia) 9
4
ID
E
3 //"~.... 2
o
Mindlin plate ~ ( w i t- h " '" ary inertia)
1
.E t
i
i
J
3
4
5
6
.,..~
0
1
2
Kirchhoff frequency parameter, o3 V'(phS/D)
Fig. 2.4 Frequency relationships between Mindlin and Kirchhoff plates
Owing to the importance of having exact solutions for checking the convergence and accuracy of numerical results, highly accurate simply supported Kirchhoff (classical thin) plate frequencies for various polygonal shapes are presented in Tables 2.1-2.4. These Kirchhoff solutions when used together with Eq. (2.46) or Fig. 2.3 provide benchmark Mindlin plate vibration results for analysts. The accurate results for simply supported circular plates, annular plates, sectorial plates and annular sectorial plates are also presented in Tables 2.4 to 2.7, respectively as they may be used to generate the corresponding Mindlin results quite accurately.
Sec. 2.2 Relations Between Kirchhoff and Mindlin Plates
20
Table 2.1 Frequencies o f triangular and rectangular Kirchhoff plates
with simply supported edges Frequency Parameters ~a 2 = ffg.a2~ p h / D
Plate Shapes
d/a
b/a
Mode Sequence Number
Triangle
1 1/4
~ 1 ~d a~x
1/2
2/5 1/2 2/3 1.0 2/~/3 2.0 2/5 1/2 2/3 1.0 2/'43 2.0
23.75 27.12 33.11 46.70 53.78 101.5 23.61 26.91 32.72 45.83 52.64 98.57
12 40.80 49.47 65.26 100.2 115.9 195.8 40.70 49.33 65.22 102.8 122.8 197.4
31415[ 60.54 75.18 88.68 117.2 134.7 275.0 60.55 76.29 87.38 111.0 122.8 256.2
70.33 78.27 106.1 171.8 198.1 315.6 69.78 76.30 106.0 177.3 210.5 335.4
83.39 107.4 141.2 197.0 229.2 427.7 83.42 108.0 142.5 199.5 228.1 394.8
Rectangle
0
,J v!
Source 6 101.8 117.4 156.4 220.4 252.0 464.2 101.5 116.4 154.8 203.4 228.1 492.5
Liew (1993a)
Leissa (1969)
where m and n are the number of half waves
Chap. 2 Mindlin Plate Theory and Ritz Method
21
Table 2.2 Frequencies of parallelogram Kirchhoffplates with simply supported edges Plate Shape
F r e q u e n c y Parameters ~ 2 "-- O)nb - 24 p h / D
a/b
fl
M o d e Sequence N u m b e r
Parallelogram 1.0
|..q
/_2
1.5
,,J vl
a
I 2
1
2.0
Source
3 1 4 1 5 ] 6
15 ~
20.87
48.20
56.12
79.05
104.0
108.9
Liew, Xiang,
30 ~
24.96
52.63
71.87
83.86
122.8
122.8
Kitipornchai,
45 ~
35.33
66.27
100.5
108.4
140.8
168.3
and Wang
60 ~
66.30
105.0
148.7
196.4
213.8
250.7
(1993)
15 ~
15.10
28.51
46.96
49.76
61.70
75.80
30 ~
18.17
32.49
53.48
58.02
76.05
78.61
45 ~
25.96
42.39
64.80
84.18
93.31
107.5
60 ~
48.98
70.51
96.99
127.3
162.3
171.1
15 ~
13.11
20.66
33.08
44.75
50.24
52.49
30 ~
15.90
23.95
36.82
52.64
56.63
63.26
45 ~
23.01
32.20
46.21
63.50
82.08
83.00
60 ~
44.00
56.03
72.79
92.80
117.4
151.7
Table 2.3 Frequencies of symmetrical trapezoidal Kirchhoffplates with simply supported edges Plate Shape
F r e q u e n c y Parameters ~ 2
i/
r~
I
1/5
3.336
4.595
6.860
10.19
10.23
11.53
Liew and
2/5
2.198
3.479
5.499
5.789
7.737
9.027
Lim
3/5
1.654
3.066
3.728
5.394
6.037
6.156
(1993)
4/5
1.356
2.833
2.879
4.560
5.086
5.192
1.5
t
a
J
2
1.0
C
F
--
c/b
M o d e Sequence N u m b e r
1
Symmetric Trapezoid
=
a/b
2.0
2
I
3
4
I
Source
5
6
1/5
6.158
7.269
9.507
12.85
17.30
20.42
2/5
3.703
5.175
7.390
9.272
10.63
14.22
3/5
2.636
4.313
5.971
6.575
9.787
10.05
4/5
2.089
3.802
4.494
6.121
7.257
8.000
1/5
9.919
10.76
13.17
16.51
20.99
26.74
2/5
5.351
7.575
9.633
12.83
13.69
17.07
3/5
3.680
5.973
8.255
8.398
11.41
14.48
4/5
2.856
5.053
6.187
7.452
10.32
10.64
w,
Sec. 2.2 Relations Between Kirchhoff and Mindlin Plates
22
Table 2.4 Frequencies of regular polygonal and circular Kirchhoffplates with simply supported edges Plate Shape Frequency Parameters ~b 2 = ~n b 2~]ph/D Numbe of
Regular Polygon
Source
Mode Sequence Number 2 3 4 5
6
52.6
122
122
210
228
228
Liew (1993a)
19.7
49.3
49.3
79.0
98.7
98.7
Leissa (1973)
28.9
73.0
73.2
130
130
151
22.2
57.8
57.8
102
102
117
1
ffn+l( ~ R ) l_ln+l ( ~ R ) _ 2~
Liew and Lam (1991b)
Leissa (1969)
where Jn and I n are, respectively the Bessel and the modified Besselfunctionsof the first kind of order n
Table 2.5 Frequencies of annular Kirchhoffplates with simply supported edges Frequency Parameters ~ 2 = Nnb 2~]ph /D
Plate Shape
Number of Nodal Diameters n and Number of Nodal Circles s, (n, s)
Annular
9~iii:i:ii::
a/b 0.1 0.3 0.5 0.7 0.9
(0, o) 1 ( 1 , 0 ) 1 ( 2 , 0 ) [ ( 3 , 0 ) [ ( o , 14.5 21.1 40.0 110 988
16.7 23.3 41.8 112 988
25.9 30.2 47.1 116 993
40.0 42.0 56.0 122 998
[
Source
1
1) 1(1,1) 51.7 81.8 159 439 3948
56.5 84.6 161 441 3948
Vogel and Skinner (1965)
Chap. 2 Mindlin Plate Theory and Ritz Method
23
Table 2.6 Frequencies of sectorial Kirchhoffplates with simply supported edges Frequency Parameters ~ 2 = ~,b 2~/ph/D
Plate Shape
Sectorial
Mode Sequence N u m b e r 30 ~ 45 ~
]
3
I
Source
1
2
4
5
97.82
183.7
277.9
288.2
412.0
I
431.3
6
56.67
121.5
148.5
205.6
256.3
277.9
Xiang, Liew and
60 ~
39.78
94.38
97.82
168.5
177.4
183.7
Kitipornchai
90 ~
25.43
56.67
69.95
97.82
121.5
134.1
(1993)
Table 2. 7 Frequencies of annular sectorial Kirchhoffplates with simply supported edges Frequency Parameters y-b2 =co,b -~ 2 4ph/D
Plate Shape
Annular Sectorial
a/b
p
0.2
30 ~
11
0.4
0.5
2
3
45 ~
97.82 56.73
183.8 122.3
277.9 148.5
60 ~
40.16
97.48
90 ~
27.17
56.73
30 ~ 45 ~
98.75 60.31
195.5 148.2
60 ~
46.28
90 ~
36.19
30 ~
103.3 68.34
45 ~
2.3
Source
Mode Sequence N u m b e r
I
I
6
4
5
288.5 210.1
413.9 256.3
431.3 277.9
Xiang, Liew and
97.82
177.4
179.9
183.8
Kitiporn-
78.68
97.82
122.3
148.5
ehai
277.9 148.7
336.3 260.4
430.5 277.8
529.4 286.1
(1993)
98.75
131.7
177.5
195.5
269.7
60.31
98.75
120.0
148.2
148.7
228.6 150.8
278.2 189.7
427.0 278.2
438.8 283.5
539.8 387.8
60 ~
55.97
103.3
176.1
178.8
228.6
278.3
90 ~
47.15
68.34
103.3
150.7
166.5
189.7
SHEAR CORRECTION FACTOR
Mindlin (1951) pointed out that for an isotropic plate, the shear correction factor x 2 depends on the Poisson ratio v and it m a y vary from tc 2 = 0.76 for v = 0 to tr 2 = 0.91 for v = 0.5. Following Mindlin's suggestion of equating the angular frequency of the first antisymmetric mode of thickness-shear vibration according to the exact three-dimensional theory to the corresponding frequency according to his theory, it can be shown that the shear correction factor is given by the following cubic equation:
Sec. 2.4 Ritz Method
24
(K.2)3 8(,v2)2 8(2- v)K"2 8 + - ~ =0 1-v 1-v
(2.50)
For examples, if v =0.3, then K 2 =0.86 and if v =0.176, then K "2 -- 7 " / ' 2 / 1 2 . On the other hand, comparing Mindlin plate equations for the constitutive shear forces with the ones proposed by Reissner (1945), who assumed a parabolic variation of the shear stress distribution, the implicit shear correction factor of Reissner takes the value of 2 5 tc = 6
(2.51)
Based on an analytical vibration solution of three-dimensional, simply supported, rectangular, isotropic plate, Wittrick (1987) performed a calibration of the Mindlin shear correction factor. He proposed that the shear correction factor be given by 2 5 tc = ~ 6-v
(2.52)
Wittrick's shear correction factor gives a value of 0.877 for v = 0.3 that corresponds closely to the value of 0.88 observed earlier by Srinivas et al. (1970) and Dawe (1978). It appears that the Wittrick shear correction factor is the best to date as it has a simple form and allows for the effect of Poisson's ratio.
2.4
RITZ METHOD
2.4.1
PRELIMINARY REMARKS
The Mindlin plate vibration problem may be solved using either the energy functional (Eq. 2.9) or the governing partial differential equations (2.14a-c). Both can be attempted using standard analytical and numerical techniques. Among the techniques available are the finite element method, the boundary element method, the finite difference method, the differential quadrature method, the collocation method, the Galerkin method and the Ritz method. In this monograph, the Ritz method is adopted due to its simplicity in numerical implementation. The Ritz approximate approach (Ritz 1909) is a generalized version of the Rayleigh method (Rayleigh 1877) which came into existence more than a century ago. The Rayleigh method is based on the principle that a system vibrating in one of its natural modes interchanges its energy completely between its potential and kinetic forms assuming no energy dissipation. By using a trial function for the mode shapes, that satisfies at least the geometric boundary conditions, and assuming simple harmonic motion, the equalization of maximum potential energy and the maximum kinetic energy yields the vibration frequencies. The resulting frequency is an upper bound solution, unless an exact eigenfunction of free vibration for the trial function is assumed. Ritz (1909) generalized the Rayleigh method by assuming a set of admissible trial functions, each having independent amplitude coefficients. He showed that a closer upper bound for the frequency could be achieved by minimizing the energy functional H with respect to each of the coefficients. Ritz demonstrated his method on a completely free square plate for which no exact solution is possible.
Chap. 2 Mindlin Plate Theory and Ritz Method 2.4.2
25
A P P L I C A T I O N OF RITZ M E T H O D TO M I N D L I N P L A T E S
In this monograph, the Ritz approximate procedure is employed to the vibration analysis of Mindlin plates. In the Ritz method, we approximate the displacement function, say ~tt(x,y) by a finite linear combination of the form m
~:~(X,y) ,~,E Ci~i(X, y)
(2.53)
i=l
where ~bi(x,y ) are the approximate functions which individually satisfies at least the geometric boundary conditions. The unknown coefficients q are obtained by determining the extremum of the energy functional YI can-/
= O; i = 1, 2,...,m
(2.54)
which yields a set of homogeneous equations expressed in terms of the coefficients q. Thus, the problem is reduced to the following eigenvalue and eigenvector problem: (K - A2M){c} = 0
(2.55)
where the stiffness matrix K is given by 3U K =~
(2.56)
and the mass matrix M is given by c7/" M =~ ge
(2.57)
According to the Ritz method, the limit of the approximations in Eq. (2.53) for m--~ oo is the exact solution of 9t(x, y) if the system of chosen functions ~b/(x,y) satisfies the following conditions (Bazant and Cedolin 1991, Reddy 1997)" 9 Functions ~b,.(x, y) are linearly independent. 9 Functions ~bi (x, y) form a complete system of functions. 9 Functions ~bi (x, y) satisfy the kinematic boundary conditions. It is not necessary for the chosen functions qki(x,y) to satisfy the static boundary conditions. However, if they do, the approximations are sometimes much better. It is impractical to assume an infinite value of m and thus convergence studies are usually carried out to establish the finite value of m for achieving the desired order of accuracy. The accuracy and the rate of convergence of the Ritz method depend on the choice of finite number of trial functions used in the series representing the displacement field of the
26
Sec. 2.4 Ritz Method
plate. Generally, the method is able to furnish accurate eigenvalues provided an adequate number of terms is included in the series. Usually, analysts find a balance between numerical accuracy and computational economy. Probably the most commonly used trial functions are the products of eigenfunctions of vibrating beams (Young 1950, Warburton 1954, Durvasula 1968, Leissa 1973, Dawe and Roufaeil 1980, Roufaeil and Dawe 1982, Dickinson and Li 1982, Narita and Leissa 1990); the degenerated beam functions (Bassily and Dickinson 1975, 1978); the spline functions (Mizusawa 1986); and the beam characteristic orthogonal polynomials (Bhat 1985, Dickinson and B lasio 1986). Many other Ritz functions can also be found in Leissa's monograph (Leissa 1969). Of these functions, the set of beam characteristic orthogonal polynomials yields the best results for rectangular plates having any combinations of boundary conditions. However, their application to arbitrarily shaped plates with different combinations of edge conditions may not be convenient. Thus many intensive investigations have been focused on the use of two-dimensional polynomials (Bhat 1987, Laura et al. 1989) associated with appropriate basic functions (Liew 1990, Liew and Wang 1992, 1993) in the vibration analysis of plates. The use of such latter trial functions has enabled the Ritz method to be automated for plates of general shape and boundary conditions. Furthermore, computational accuracy may be enhanced since the polynomial functions permit differentiation and integration processes to be carried out in an exact manner. In this monograph, the Ritz method which employs such polynomial functions is referred to as the p-version Ritz method. The letter p indicates the use of polynomials as opposed to other kinds of functions.
CHAPTER THREE
FORMULATION IN POLAR COORDINATES
3.1
INTRODUCTION
When dealing with plates of circular peripheries, it is expedient to use polar coordinates in the formulation. An early paper on the vibration of moderately thick plates in polar coordinates is due to Mindlin and Deresiewicz (Mindlin and Deresievicz 1954, Deresievicz and Mindlin 1955). They considered only plates with free edges. Irie, Yamada and Aomura (1980), Irie, Yamada and Takagi (1982), and Huang, McGee and Leissa (1994) extended Mindlin and Deresiewicz's work to circular, annular and sectorial plates with various circular edge conditions. These researchers solved the three governing equations of motion in ~r, ~0 and w by recasting them into three harmonic equations involving three potentials 191,192,193 as suggested earlier by Mindlin (1951). These potentials are defined by: ~1 [fir = ( 0 " 1 - 1 ) W + ( r T 2
-1)
~2
8Z
1 c~ 3 +--~
~o = (or1 - 1) --1 c~ 1 +(o- 2 -1) 1 cqDz
zoo
(3.1a)
z,r
cqD3
zoo
(3.1b)
oz
= 191 + 192
(3.1c)
where (6~, 612)
2(3"2,3`2 )
(3.2a)
or'' cr2 = [ rzA212 6(1 -v v)tr) 1~= 62 2 (1 -
~12' ~2 _- T
~ q- 6(1 - v)K 2 +
2
632 = ( l - v )
_w w=-.
R' Z =
, "t'=
)2 4 ]
6(1 -- v)tr 2
+ ~-
v, 21
12
r
12
r2
(3.2c)
r z
h
,d,=coR 2
(3.2b)
~_~
9
(3.2d-g)
in which w is the transverse deflection, r the radial coordinate, R the plate radius, h the plate thickness, co the circular frequency, D the flexural rigidity, p the mass density per unit
27
Sec. 3.1 Introduction
28
volume and the subscripts r,O denote the quantities in the radial and circumferential directions, respectively. Note that 2 is the frequency parameter. Based on these nondimensionalized potentials, the governing equations of vibrating Mindlin plates, in polar coordinates, can now be expressed as
(V 2 + 62 )|
=0
(3.3a)
(V 2 + 6 2 ) 0 2 = 0
(3.3b)
(V 2 + 6 2 ) 0 3 : 0
(3.3c)
where the Laplacian operator 1 6(.) 1 62(,) v' (o) = 6'(-) ~ ~ ++ z~ Z 6% 602
(3.4)
The solutions to Eq. (3.3) are (~)1 = A 1 J n (~'IX) cO$
|
nO + B1Y, (6,Z)COS nO
= AzJ, ( 6 2 Z ) C ~
(3.5a) (3.5b)
nO + BEYn (6zZ)COS nO
(9 3 = A3J , (632") sin nO + B3Yn ( 6 3 Z ) sin nO
(3.5c)
where Ai and Bi are the arbitrary constants, J , (.) and I1, (.) are the Bessel functions of the first and second kinds of order n, respectively and n corresponds to the number of nodal diameters. The boundary conditions of a circular edge of a Mindlin plate are: or
~=0
Q. =
tf 2Gh ~S~y + l//r J = O
r
(3.6a)
"/
~//r " - 0
or
D 61/Sr+__ Nr+ Mrr =--RE 6j7( /~ ---~
~o = 0
or
M,. o =
D[l-v)II(6~r
R t , - - 2 - ) L z t , 0o - ~ o
=0 )
6!6o 1
+ hA
(3.6b)
=o
(3.6c)
The classical periphery conditions of circular Mindlin plates at the edge r = R (or Z = 1) are given by the following combinations of the boundary conditions in Eq. (3.6)"
Mrr
=
Mr|
=
Qr = 0 , for free edges
(3.7a)
M rr = Mr| = w = 0 , for simply supported edges (soft type)
(3.7b)
M rr = Ikto = w = 0 ,
for
simply supported edges (hard type)
(3.7c)
~bCr -- ~ff0 = ~ = 0 ,
for
clampededges
(3.7d)
For solid circular plates with the foregoing classical boundary conditions, the constants B/ are set equal to zero in Eqs. (3.5) so as to avoid infinite displacements, slopes
Chap. 3 Formulation in Polar Coordinates
29
and bending moments at r = 0. The natural frequency parameter 2 can then be determined by substituting Eqs. (3.1) and (3.5) into Eq. (3.7). This results in the determinantal equation given by CI1 C12 C21 C22 C31 C32
I
C13 C23 = 0 C33
(3.8)
For a circular plate with free edges, the elements of the determinant of Eq. (3.8) are
Cli --" (0"i --1)[Jn w(8i)-k- VJtn(8i ) - vn2Jn (8i)]
(3.9a)
C2i -~ -2n(o" i --1)[J; (8i ) -- Jn (8i )]
(3.9b)
C3i =o'iJ'n(8i) C,3 =n(1-v)[J;(83)-J,,(83)]
(3.9c)
C23 = -J~ (63) + J'n (63) - n2 J, (63) c3~ = nJ. (8~)
(3.9d) (3.9e) (3.90
For a circular plate with simply supported edges (soft type),
Cli ~- (0"i --l~J: (8i)'F viS: (8i)- vn2Jn (8i)
(3.10a)
C2i
(3.lOb)
=-2n(cri-l~J'n(6i)-Jn(8i)
C3i "-- Jn (8i)
(3.10c) (3.10d)
C:3 = - J ; (6'3) + J~ (S 3) -- n2j,, (6 3)
(3.10e)
C33 = 0
(3.100
For a circular plate with simply supported edges (hard type), C~; = (ori -1)[J: (8i)+ vJ'. (8i)- vn2j. (8/)]
c:, = nO,-OJ.(4) C3i = Jn (6i) !
C23 = J. (63) C33 = 0
(3.1 la) (3.11b) (3.11c) (3.1 ld) (3.1 le) (3.110
For a circular plate with clamped edges,
c,i--0/- :)J'. (8,) c:i = . ( o / - 1)J.(~/)
(3.12a) (3.12b)
Sec. 3.1 Introduction
30 C3i "-" J,, ((~i ) C13 =
(3.12c)
nJ. (6 3)
(3.12d)
C23 = J'. (~3)
(3.12e)
C33 -" 0
(3.12f)
where i = 1,2.
Table 3.1 Fre, luencyparameters 2 of circularplates with free edge(v :
0.3)
- h/R
0 9.003 38.443 87.750 156.82
0.05 8.969 37.787 84.443 146.76
0.10 8.868 36.041 76.676 126.27
0.15 8.710 33.674 67.827 106.40
0.20 8.505 31.111 59.645 90.059
0.25 8.267 28.605 52.584 76.936
20.475 59.812 118.96 197.87
20.260 58.215 112.98 182.27
19.711 54.257 99.935 152.75
18.917 49.341 86.235 126.05
17.978 44.434 74.331 105.03
16.979 39.948 64.462 88.312
5.358 35.260 84.366 153.31
5.330 34.598 81.185 143.56
5.278 33.033 73.875 123.77
5.205 30.942 65.510 104.47
5.114 28.668 57.722 88.530
5.008 26.427 50.956 75.651
12.439 53.008 111.95 190.69
12.311 51.537 106.41 175.92
12.064 48.227 94.531 147.99
11.722 44.116 81.930 122.49
11.214 39.960 70.862 102.27
10.866 36.110 61.613 86.155
21.835 73.543 142.43 231.03
21.492 71.799 133.63 209.86
20.801 64.891 115.96 172.45
19.871 58.043 98.446 140.26
18.816 51.545 83.801 115.57
17.724 45.819 71.918 96.464
33.495 96.755 175.74 274.25
32.766 92.172 162.64 245.20
31.270 82.722 137.95 197.06
29.334 72.464 114.96 157.77
27.255 63.253 96.513 128.42
25.221 55.458 81.864 102.98
47.378 122.57 211.79 320.30
46.031 115.47 193.27 281.76
43.255 101.48 160.36 221.72
39.831 87.201 131.42 175.01
36.351 74.982 108.98 140.81
33.113 64.981 91.436 112.96
Tables 3.1-3.3 present some values of the natural frequency parameters for circular plates with free, soft simply supported and clamped edges, respectively. Note that the
Chap. 3 Formulation in Polar Coordinates
Poisson's ratio is taken as v =0.3 and the shear correction factor
31 as
K "2
=it'2/12.
The
eigenvalues of thin plates are also given under the column headed by h/R = 0 in the Tables. In the tables, n refers to the number of nodal diameters while s denotes the number of nodal circles, excluding the boundary circle. Axisymmetric vibration solutions of thick circular plates that include the effects of geometric nonlinearity, may be obtained from the paper by Kanaka Raju and Venkateswara Rao (1976) who have used the finite element method to solve the problem.
Table 3.2 Frequency parameters A of circular plates with soft simply supported edge (v = 0.3) n
s
~ = h/R
0 4.935 29.720 74.156 138.32
0.05 4.925 29.323 71.756 130.35
0.10 4.894 28.240 65.942 113.57
0.15 4.844 26.715 59.062 96.775
0.20 4.777 24.994 52.514 82.766
0.25 4.696 23.254 46.775 71.603
13.898 48.479 102.77 176.80
13.784 47.411 98.238 164.11
13.510 44.691 87.994 139.27
13.109 41.174 76.847 116.17
12.620 37.537 66.946 97.873
12.080 34.127 58.701 83.782
25.613 70.117 134.30 218.20
25.215 67.875 126.68 199.36
24.313 62.552 110.66 165.02
23.079 56.212 94.501 135.12
21.687 50.126 80.950 112.43
20.270 44.751 70.104 95.391
39.957 94.549 168.68 262.49
39.023 90.509 156.90 235.95
36.962 81.526 133.77 190.77
34.319 71.610 112.00 153.68
31.547 62.675 94.597 126.53
28.900 55.147 81.102 -
56.842 121.70 205.85 309.61
55.034 115.12 188.71 273.71
51.158 101.37 157.20 216.47
46.495 87.219 129.33 171.90
41.908 75.137 107.94 140.25
37.762 65.341 91.772 -
76.203 151.52 245.78 359.53
73.097 141.54 221.94 312.52
66.647 121.90 180.84 242.09
59.358 102.94 146.48 189.82
52.587 87.491 121.02 153.65
46.738 75.354 103.10 -
97.995 183.95 288.41 412.22
93.072 169.59 256.46 352.25
83.214 142.95 204.63 267.61
72.724 118.71 163.46 207.46
63.462 99.729 133.87 165.85
55.757 85.204 112.18 -
32
See. 3.1 Introduction
Analytical solutions for annular plates and sectorial plates can be found, respectively, in the papers by Irie, Yamada and Takagi (1982) and Huang, McGee and Leissa (1994). The sectorial plates treated by Huang, McGee and Leissa (1994) are, however, restricted to simply supported, radial edges.
Table 3.3 Frequency parameters 2 of circular plates with clamped edge (v = 0.3) r =h/R 0 10.216 39.771 89.104 158.18
0.05 10.145 38.855 84.995 146.40
0.10 9.941 36.479 75.664 123.32
0.15 9.629 33.393 65.551 102.09
0.20 9.240 30.211 56.682 85.571
0.25 8.807 27.253 49.420 73.054
21.260 60.829 120.08 199.05
21.002 58.827 112.98 181.21
20.232 53.890 97.907 148.70
19.116 48.002 82.861 120.84
17.834 42.409 70.473 100.12
16.521 37.550 60.748 84.801
34.877 84.583 153.82 242.72
34.258 80.933 142.68 217.30
32.406 72.368 120.55 174.05
29.890 62.929 100.01 139.22
27.214 54.557 83.937 114.24
24.670 47.650 71.715 96.109
51.030 115.02 190.30 289.18
49.782 105.03 173.97 254.56
46.178 91.712 143.50 199.36
41.618 78.077 117.02 157.30
37.109 66.667 97.152 128.02
33.083 57.624 82.414 -
69.666 140.11 229.52 338.41
67.420 130.95 206.69 292.85
61.272 111.74 116.69 224.61
54.038 93.368 133.90 175.12
47.340 78.733 110.16 141.53
41.657 67.493 92.895 -
90.739 171.80 271.43 390.39
87.022 158.53 240.70 332.05
77.454 132.31 190.03 249.79
66.960 108.75 150.66 192.70
57.793 90.747 122.99 155.78
50.331 77.264
114.21
108.45
206.07 316.00 445.09
187.63 275.85 372.07
94.527 153.30 213.48 274.89
80.252 124.17 167.29 210.07
68.396 102.70 135.65 160.62
59.063 86.939
Chap. 3 Formulation in Polar Coordinates 3.2
33
ENERGY FUNCTIONALS
The strain energy functional in Eq. (2.6) and the kinetic energy functional in Eq. (2.8) can be expressed in polar coordinates as: U = -1 1 P / z l J { D I(__~]2 + 2 v C~r (___~_c~,+Vr ) 2 ~--fl/2 r o3" +--5r
+~
+ ~2r 2 ~o - j
&
+ ~c2Gh
+Nr
+-7 --~ + rN~
=2 j-p/2
phc~
w2 + -i2
00 r drdO
+ ~2
(3.13)
r drdO
(3.14)
where w = w(r, 0 ) is the transverse deflection, gr = [Prr(r, 0) the bending slope in the radial plane, Ve = ~/o (r, 0) the bending slope in the circumferential plane, R the plate radius and (r,O) the radial coordinates. The definitions of a and fl are shown in Figs. 3.1a-d for the cases of circular, annular, sectorial and annular sectorial plates, respectively. In order to set the limits of integration f r o m - 1 to +1, the polar coordinates (r,O) are transformed into a set of nondimensionalized coordinates (~,r/) by R
r = --[(1 + a ) - ( 1 - a ) g ] 2 0 = -~flr/ 1
(3.15a) (3.15b)
In view of the nondimensional terms in Eqs. (3.2c-e) and by substituting Eqs. (3.15a,b) into Eqs. (3.13) and (3.14)
=if, r, 2 J_~ J_~ 1
+~5+
2
2 (l-a)
( 2~ ) +2 Vr o37 r2
-
+--4v 0~
( 1 - a ) Z Og a o~7
1-1/( We 2Z 2
+~
( l -- a ) -0~
+ ~r
2Z O~o ( l - a ) 3~ at"
-ZT
--~
20~r]21 a cTq +
ZWo
Z d~d rl
(3.16)
(3.17)
Sec. 3.2 Energy Functionals
34
For circular and annular plates, the foregoing energy functionals may be reduced to a single variable since fl = 2~r and by using the standard separation of variable functions: ~(~, r/) : ~(~') cos nr/ 2 nr/ ~prr(~, iT) : ~)rr (~) COS 2
(3.18a) (3.18b)
g/o (~:, r/) = ~o (~:) sin nr/ 2
(3.18c)
where n is the number of circumferential nodal diameters.
(a) Circular plate a = 0, fl=2rc
(b) Annular plate a r 0, fl= 27c
SffSS~Sff
(c) Sectorial plate a=0, fl~0
(d) Annular sectorial plate
a , O , fl~O
Fig. 3.1 Definitions of a and flfor various plate shapes
Chap. 3 Formulation in Polar Coordinates
35
As a result of using Eqs. (3.18a-c), the strain and kinetic energy functionals can be expressed as
~;i[/~ (1 - a) $1 ~r~ ~r~s 0r ) ~+ ~(1 --4~ a)j~f $1 W (',~"
U =
1/~
+ - ~ Sl
+
~:~
v2
S2n ~//0 -Jr"~/r
/
)~1_~( ~~o~ )~1 "1---2Z2 82 ~7"0-- (1 -
S, (1- a) 0~: + ~ r
II~I~176 ~4
2n [pro + ~/'r
/ + ~1 S:
a)
0~
n~+
+_~-~~S~r+S~o~}~ ~
-
--an ~ffr
Z~o
Z d~
(3.19)
(3.20)
where
S1
= I2, [ 1,
when n = 0 when n > 0
(3.21 a)
$2
= IO, [ 1,
when n = 0 when n > 0
(3.21b)
The Lagrangian is thus given by H =T- U
3.3
(3.22)
EIGENVALUE EQUATION
3.3.1 CIRCULAR AND ANNULAR PLATES For axisymmetric plates, the energy functionals in the one-dimensional form given in Eqs. (3.19) and (3.20) are preferred for the analysis over the two-dimensional form given by Eqs. (3.16) and (3.17) because of their simplicity in numerical implementation. In this case, the pversion Ritz functions may be assumed to be given by: w(~:) =(f--~l
ciqkw](~-l)r'(~-Ot)r2
(3.23a)
~ffr(~)--( N.2=l~tdi~bri(~ - l)r3 (~ - a)r'
(3.23b)
~0(') =(Ni~=leiqk~
(3.23c)
Sec. 3.3 Eigenvalue Equation
36 where Ns,s=l,2,3
is the degree of the one-dimensional polynomial; ci,di,e i are the
unknown coefficients and the powers yk,k = 1,2,..., 6 take on the value of either 0 or 1 depending on the boundary conditions 1,2,...,6
(3.24a)
if the edge (outer or inner) is flee.
9
Yk = 0 , k =
"
Y, = 0, k = 3,4; Yk = l , k =1,2,5,6 if the edge (outer or inner) is simply supported. Yk = 1, k = 1,2,...,6 if the edge (outer or inner) is clamped.
9
(3.24b) (3.24c)
Note that in the case of circular plates, a is set to a very small number (10 -7) and the inner edge is made free. Applying the Ritz method,
i= 1,...,N s
(5;~ci ' o~di ' GQei
(3.25)
The substitution of Eqs. (3.19) to (3.23) into Eq. (3.25) yields the following eigenvalue equation
ii.c ...'c"..el
. .cellf!=fit
(3.26)
where the elements of the stiffness submatrices [K] are:
k~ -
6(1-v)K2 I i { aZ S18~bwc3~b7 1 - a r~ (1 - a) 0~ 0~ aX
2. . . . .
}d~:; (3.27a)
i = 1,2,..., N 1; j = 1,2 ..... N 1 k~d
6(1- V)K"2 f l a/If
=
~
CQ~7 r
~oTSl~-<
de;
i = 1,2,..., N1; j - 1,2,..., N 2 kce" = 6(1 -2 v)tc r-
(3.27b)
I i (1-a)Zn2 2S2~b;w~bj~d~:," (3.27c)
i = 1,2,..., N 1; j = 1,2,..., N 3 k~/ =
4
+
2-2
(l-v)(1-a)
2ctz
SIZ~[~; -I- S 1 - - ~ ; ~ j
z
2 rr n S2~bi ~bj +
i = 1,2,..., N
2; j
V a [S 1 Or gO;
--2
= 1,2,..., N
8r
2
+
+
S1
~;
8r
r]}
,~bj
04 04 d~:; (3.27d)
Chap. 3 Formulation in Polar Coordinates ae~f
0r
1-a
r o
37
rO
(1-v)(1-a)
1-v 0r + - - nS2r r d~' ; 2 8~'J i = 1,2,...,N2; j = 1,2,...,N 3
kijee -- I i { 1 4 a
(3.27e)
[ 6 Z a ( 1 - v ) x2 oCy + 1 - v r 1 6 2 l a x ( l - v ) 0 r 1 7 6 1 6 2 ~ r2 Z 2(1- a) c3~ c3~' 1-a aZ
2,-,~o~o
+ ~ n
a1r i r
-
i = 1,2,..., N
a(1-v) I 0r176 $2 r + 4 L or (3.270
3 , j = 1,2,..., N 3
and the elements of the mass submatrices [M] are: cc ~~ a ( 1 - a ) Z S l ~ ; ~ ; m,j = 4
d4;
i = 1,2,.. N,; j = 1,2 ..... N 1 "'
(3.28a)
i = 1,2,..., N1; j = 1,2,..., N
2
(3.28b)
N 3
(3.28c)
muCd = 0", ce
.
m~/ = 0;
i = 1,2,..., N 1, j = 1,2,.
dd
?"
r
.
mo = -v2 ~ I f a ( 1 -4a ) Z a l e / Cj d~:, m,jae = 0",
..,
.
i = 1,2,... N 2, j = 1,2,...,N 2 i = 1,2,..., N 2",j = 1,2,..., N 3
ee r2 i ' a ( l - a ) Z S z r 1 7 6 1 6 2 1 7 6 m#. = -~ 4
i =1,2,.. N 3;j=1,2,.. N 3 "' "'
(3.28d) (3.28e) (3.28f)
The eigenvalue equation (3.26) may be solved using any standard eigenvalue solver for the natural frequencies (eigenvalues) and mode shapes (eigenvectors) for circular and annular plates. 3.3.2
SECTORIAL
AND ANNULAR
SECTORIAL
PLATES
For the sectorial and annular sectorial plates, the transverse deflection function and the rotations may be approximated by the following two-dimensional polynomial functions Pi q
W(~,~7)-- Z Z Cmr
(~, ~7)
(3.29a)
q=0 i=o P2 q
~[/r(~,~7) : Z Z dmr r (~,
~7)
(3.29b)
V/o(~,rl) = y'~_, % 0 ~ (~, r/)
(3.29c)
q=0 i=0 P3 q
q=0 i=0
Sec. 3.3 Eigenvalue Equation
38
where p , , s = 1,2,3 is the degree set of the complete polynomial space; c,, unknown coefficients, the subscript m is determined by m=
(q + 1)(q + 2) -i 2
din, e m
are the
(3.30)
and the total numbers of c,, d , , e, are Ark,k=-1,2,3 which are dependent on the degree set of polynomial space ps, given by Nk = (Pk + 1)(Pk + 2) 2
(3.31)
The functions ~w, ~r, ~m o consist of (3.32) The basic functions ~b~w, r
~o in Eq. (3.11) are given as follows:
(3.33)
where
ne
is the number of edges of the plate and Ej (~', r/) is the boundary equation of thej-th
supporting edge. yj is dependent on the supporting edge condition, such that w {~ free(F); YJ = simply supported (S) or clamped (C) {~ free (F)or simply supported (S)in the O-direction; Y~ = simply supported (S) in the r - direction or clamped (C) o {~ free (F)or simply supported (S)in the r-direction; YJ = simply supported (S) in the 0 - direction or clamped (C)
(3.34a) (3.34b) (3.34c)
Applying the Ritz method, (3.35) OCm ' tS~lm ' ~ e m
The substitution of Eqs. (3.16), (3.17) and (3.18) into Eq. (3.26) yields the following eigenvalue equation
K CC K Cd Kda
Chap. 3 Formulation in Polar Coordinates
39
.eI IMccMe"Mcellf!lfit
(3.36)
M dd M d,
_/~2
K de Kee
=
M e~
where the elements of the stiffness submatrices [K] are:
ko
=
~ ~ + .-1 - ot O~ 8~ aZ
q~2
Or? O,?
Jd~dv;
i = 1,2 .... , N1; j = 1,2,..., N~
ca I ' I i 6(1- v)xZ ~ d~bw r k~ = v2 2 c?~ qk; d~drl;
(3.37a)
i = 1,2,...,N~;j = 1,2,...,N 2 (3.37b)
i = 1,2,..., N~; j = 1,2,..., N 2 ce
~'I'6(1-v)xZ(1-a)ZC?qkwqkOd~drl;
k~/ =
r2
2
07?
i = 1,2,..., N1; j = 1,2,..., N 3 ~ k~aa = IC f { c ~ ( 1 -4 ~ [ 6v () It -cr2
(3.37c)
Z ~ '/~;r + I O'/qk;r 1 + ctZ d~ ( d~b~ 1-ct d~ 84
+ ( 1 - v ) ( 1 - a ) 8~br d~b~ + ~va I qkr d~b, + dqkr qk;rl } d~dq; 2aZ drI dr1 2 8r 8r (3.37d)
i = 1,2,..., N2; j = 1,2,..., N 2
ae f ' Jfl' "L f C3Of ko = v d~
8rl
+
1- a 8qk~ (1- v)(1- a) dqkf Of 2Z 8rl 4Z dq
1- v O~b. O~b~
+ -2
drl d~
J
d~du;
i = 1,2,..., N2; j = 1,2,..., N 3 6Za(1 - v)~; 2 qk.o~bo+ rz
ee I i{ E
k~ =
1- a 4
(3.37e)
qbo~bo + c~X(1-v) O~b~ 0~b~ Z
1- a dq~/~dq~~ a(1- v) [~bo d~b~ + aZ
dr/
c3q
4
~
1
o/o
2(l-a)
d~: d~:
d~:
i = 1,2,..., N 3 ; j = 1,2,..., N 3
(3.37f)
and the elements of the mass submatrices [M] are:
cc ~l l ) a ( 1 - a ) Z ~b~qkT.d~d,?; m~ = 4 m(/Cd = 0;
i = 1,2,..., N~;j = 1,2,..., N 1
(3.38a)
i = 1,2,..., N1; j = 1,2,..., N 2
(3.38b)
Sec. 3.4 Computer Program
40
ce
m~/ = 0;
dd m~/ = m o'Cte=
ee
m~ =
.
i = 1,2,..., N1, j = 1,2,.
IlIllr2 a(1-a)Z~i 12 4 1 0;
.., N
(3.38c)
3
r ~jr d~dr],
i = 1,2,..., N2; j = 1,2,..., N 2
(3.38d)
i = 1,2,..., N2 j", = 1,2,...,
(3.38e)
N 3
I~l I~l Z'20~(1 -- O~)2' ~O #O d~dr]; 12 4 i = 1,2,..., N
3; j =
1,2 .....
(3.380
N 3
The resulting eigenvalue Eq. (3.36) may be solved using any standard eigenvalue solver for the natural frequencies (eigenvalues) and mode shapes (eigenvectors) for sectorial and annular sectorial plates.
3.4
COMPUTER PROGRAM
Two FORTRAN programs have been written, based on the foregoing polar coordinate formulations. One program, referred to as VPRITZP1, is for determining the natural frequencies of circular and annular plates and it is based on the one-dimensional eigenvalue equation (Eqs. 3.26-3.28). The other, referred to as VPRITZP2, is to compute the natural frequencies of sectorial and annular sectorial plates and it is based on the two-dimensional eigenvalue equation (Eqs. 3.36-3.38). The listings of these programs are given below. Note that the eigenvalue solver adopted for generating the illustrative sample solutions is the subroutine RSG of EISPACK (Smith et al. 1974). The source code of this subroutine is not given herein as it is readily available. The analyst may use any other standard eigenvalue solver if desired. 3.4.1
SOFTWARE CODE: VPRITZP1
************************************************************ * * *
p-VERSION RITZ METHOD FOR VIBRATION OF CIRCULAR AND ANNULAR MINDLIN PLATES BASED ON POLAR COORDINATES SYSTEM
* * *
*:,g********* ***** :'g******* ********* $************** :0*** ******** MAIN VPRITZP 1 IMPLICIT DOUBLE PRECISION (A-H,O-Z) PARAMETER (NTERM=40,NTOTAL=3*NTERM,N 1=NTERM, $ N2=NTOTAL,NTAB=25) DIMENSION S(18,N 1,N 1),COEF(NTAB) DOUBLE PRECISION $ K(NZ,NZ),K 1(NZ,NZ),W(NZ),Z(NZ,NZ),FV 1(N2), $ FV2(N2) DOUBLE PRECISION E,LX,LY,H,NU,KAPPA,RHO INTEGER P0,P 1,P2
Chap. 3 Formulation in Polar Coordinates
* * * * * * * * * * * * * * * * * * * *
N T E R M = the number of polynomial terms used in the Ritz functions; N T O T A L = 3 * N T E R M = the total number of degrees of freedom; N1 = NTERM; N2 = NTOTAL; S(18,N 1 ,N 1) stores the 18 integrated values generated by subroutine BASICINT; K(N2,N2) = linear stiffness matrix; K1 (N2,N2) = mass matrix; W(N2) - frequency parameter; Z(N2,N2), FV 1(N2), FV2(N2) are working matrices required by EISPACK; LX, LY = inner and outer radii; H = plate thickness; E = Modulus of elasticity (any relative number can be taken); NU = Poisson's ratio; RHO = plate density per unit volume; K A P P A = shear correction factor; P0 = lower value of degree of polynomials; P2 = upper value o f degree of polynomials; P1 = increment step o f degree of polynomials. OP EN(30,FILE='INPUT.DAT') OPEN( 15,FILE='OUTPUT.DAT ')
* INPUT.DAT stores the input information; * OUTPUT.DAT prints the output information; WRITE(15,*)
$
WRITE(15,*) ' p-Version Ritz Method for Vibration of Mindlin Plates' WRITE(15,*) WRITE(15,*) WRITE(15,*) WRITE(15,*) WRITE(6,*)'Read input data from file INPUT.DAT' WRITE(6,*) READ(30,*)E WRITE(15,'(1X,"Modulus of elasticity E - ",F13.5)')E READ(30,*)RHO WRITE(15,'(1X,"Plate density per unit volume RHO = ", F11.5)')RHO READ(30,*)NU
41
Sec. 3.4 Computer Program
42
WRITE(15,'(1X,"Poisson ratio NU = ",F8.5)')NU READ(30,*)KAPPA WRITE(15,'(1X,"Shear correction factor KAPPA = ",F8.5)')KAPPA READ(30,*)LX WRITE(15,'(1X,"Inner radius ALPHA*R = ",F8.5)')LX READ(30,*)LY WRITE(15,'(1X,"Outer radius R = ",F8.5)')LY READ(30,*)H WRITE(15,'(1X,"Plate thickness h = ",F8.5)')H READ(30,*)NN WRITE(15,'(1X,"The number of circumferential waves n = ",I5)')NN READ(30,*)P0,P2,P 1 WRITE(15,'(1X,"Polynomial terms change from NK =", I3," to",I3," with step",I3)')P0,P2,P 1 WRITE(15,*) $
'
WRITE(15,*) WRITE(15,*) WRITE(15,'(1X,"Plate cutout ratio ALPHA = ", F8.5)') LX/LY WRITE(15,'(1X,"Plate thickness to outer radius ratio h/R = ", $ F8.5)') H/LY WRITE(15,'(1X,"The number of circumferential waves n = ", $ I5)')NN WRITE(15,*) $
,
t
WRITE(15,*) WRITE(15,*) WRITE(15,*) NMAX=P2 CC=LX/LY CALL STANDVB 1(E,RHO,NU,KAPPA,LX,LY,H,P0,P 1,P2,S, COEF,CC,NTAB,NMAX,K,K1,W,Z,FV1,FV2,NN) CLOSE(15) CLOSE(30) WRITE(6,*)'End of running the program'
Chap. 3 Formulation in Polar Coordinates
STOP END
$ $
SUBROUTINE STANDVB 1(E,RHO,NU,KAPPA,LX,LY,H, P0,P1,P2,S, COEF,CC,NTAB,NMAX,K,K1,W,Z,FV1,FV2,NN)
* This subroutine STANDVB 1 is the main subroutine to calculate the frequency * parameters IMPLICIT DOUBLE PRECISION (A-H,O-Z) INTEGER P0,P 1,P2 DIMENSION S(18,NMAX,NMAX),COEF(NTAB) DOUBLE PRECISION K(NMAX*3,NMAX*3), $ K1 (NMAX*3,NMAX*3), $ W(NMAX*3),Z(NMAX*3,NMAX*3), $ FV1 (NMAX*3), FV2(NMAX*3) DOUBLE PRECISION LX,LY,KAPPA,NU MNL=NMAX WRITE(6,*)'Generate the basic integrated matrix' WRITE(6,*) CALL BASICINT(MNL,NTAB,S,COEF,CC) WRITE(6,*)'NTAB = ',NTAB DO 20 ICASE=P0,P2,P1 M=ICASE WRITE(6,'(1X,"Solve eigenvalue equation for NK = ",I3)')ICASE WRITE(6,*) WRITE(15,'(1X,"Polynomial terms NK = ",I3)')ICASE N=M L=M IF(NN.EQ.0) L=0 D=E*H** 3/12.0/( 1.0-NU** 2) CALL STFF(M,N,L,MNL,LX,LY,H,E,NU,KAPPA,RHO,D, S,K,K1,W,Z,FV 1,FV2,NN) * Call subroutine STFF to form and solve the eigenvalue equation.
10
ITRR=I DO 10 ITR= 1,3*m IF(W(ITR).EQ.NAN) GOTO 10 W(ITR)=DSQRT(W(ITR)*(LY)**4*(RHO*H/D)) ITRR=ITRR+ 1 CONTINUE WRITE(15,'(1X,"The first six frequency parameters"
43
Sec. 3.4 ComputerProgram
44
20
" corresponding to n = 0 are:")') WRITE(15,'(1X,6F 13.3)') (W(ITR),ITR= 1,6) WRITE(6,'(1X,6F13.3)') (W(ITR),ITR=I,6) WRITE(15,*) CONTINUE RETURN END
SUBROUTINE CLEAR(A,M,N) * This subroutine CLEAR initializes the two dimensional array (A). IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION A(M,N)
20 10
DO 20 I=I,M DO 10 K = 1,N A(I,K)=0.0 CONTINUE CONTINUE RETURN END
SUBROUTINE STFF(M,N,L,MNL,LX,LY,H,E,V,KAPPA,RHO,D, S,K,K1,W,Z,FV1,FV2,NN) * This subroutine STFF forms the linear stiffness and mass matrices and * solves the generalized eigenvalue problem using EISPACK. IMPLICIT DOUBLE PRECISION (A-H,K,O-Z) DIMENSION K(M+N+L,M+N+L),K1 (M+N+L,M+N+L), S(18,MNL,MNL) DIMENSION W(M+N+L),Z(M+N+L,M+N+L), FV 1(M+N+L),FV2(M+N+L) DOUBLE PRECISION LX,LY,KAPPA INTEGER S 1,$2 IF(NN.EQ.O) THEN Sl=2 S2=0 END IF IF(NN.GT.0) THEN Sl=l S2=l
Chap. 3 Formulation in Polar Coordinates
END IF PI=3.1415926 MI=M+N+L CALL CLEAR(K,M1,M1) CALL CLEAR(K1,M1,M1) H 1=KAPPA*E'H/2.0/(1.0+V)
10 20
DO 20 I=I,M DO 10 J=I,M K(I,J)=PI* H 1*(S1 * S (3,I,J)+SZ*NN** 2" S (1 ,I,J)) K1 (I,J)=PI* S 1*RHO*H*LY**2*S(2,I,J) CONTINUE CONTINUE
30 40
DO 40 I=I,M DO 30 J=I,N K(I,J+M)=PI*H 1*S 1*LY* S (4,I,J) CONTINUE CONTINUE
50 60
DO 60 1= 1,M DO 50 J=I,L K(I,J+M+N)=-PI*H1 *S2*LY*NN*S(5,I,J) CONTINUE CONTINUE
70 80
DO 80 I=I,N DO 70 J=I,N K(I+M,J+M)=PI* D* (S 1* S(8,I,J)+NU* S 1* (S(9,I,J)+S(10,I,J))+ (S 1+(1.0-NU)/2.0*NN**2)* S(6,I,J)+ H 1/D*S 1*LY**2* S (7,I,J)) K1 (I+M,J+M)=PI*RHO*H** 3" LY** 2" S 1/12.0" S(7,I,J) CONTINUE CONTINUE
DO 100 1= 1,N DO 90 J=I,L K(I+M,J+M+N)=PI D $2 (NU NN LY*S(12,I,J)+ $ (1.0+(1.0-NU)/2.0)*NN*LY* S(11,I,J)$ (1.0-NU)/2.0*NN*LY*S(13,I,J)) CONTINUE 90 100 CONTINUE DO 120 I=I,L DO 110 J=I,L
45
Sec. 3.4 Computer Program
46
K(I+M+N,J+M+N)=PI*D* $2" ((NN**2+ ( 1.0-NU)/2.0)* S ( 14,I,J)+ ( 1.0-NU)/2.0* S ( 16,I,J)+ H 1/D*LY**2* S(15,I,J)( 1.0-NU)/2.0* (S ( 17,I,J)+S (18,I,J))) K1 (I+M+N,J+M+N)=PI*RHO*H**3*LY**2* $ $2/12.0"S(15,I,J) 110 CONTINUE 120 CONTINUE $ $ $ $
DO 140 I=I,M1 DO 130 J=I,M1 K1 (J,I)=K1 (I,J) K(J,I)=K(I,J) 130 CONTINUE 140 CONTINUE CALL RSG(M 1,M 1,K,KI,W,0,Z,FV 1,FV2,IERR) * Call subroutine RSG from EISPACK to solve the eigenvalue equation. IF(IERR.GT.0) THEN WRITE(6,*) $ 'MATRICES ARE ILL-CONDITIONED' $ 'IN THE EIGENVALUE FUNCTION' STOP END IF RETURN END
SUBROUTINE BASICINT(NTERM,NTAB,S,COEF,CC) * This subroutine BASICINT is to generate the integrated values. IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION S(18,NTERM,NTERM) DIMENSION COEF(NTAB) COMMON/C3/NSIDE * S(18,NTERM,NTERM) is an array which stores the * 18 integrated values; CALL TABLE(COEF,NTAB,CC) * Call subroutine TABLE to generate the following * integrated one-dimensional array * along the plate radius:
Chap. 3 Formulation in Polar Coordinates
1/4 ~c I
1!
d~:
C
~N
B 2
* where CC is the cutout ratio ct; * NSIDE = number of edges of a plate 9 CALL POLYCOMP1D * Call POLYCOMP1D to form the polynomial terms. MNL=NTERM M=NTERM N=M L=M READ(30,*)NSIDE CALL SS S(M,N,L,MNL,S,NTAB,COEF) * Call subroutine SSS to form the S(18,NTERM,NTERM) matrix 9 RETURN END SUBROUTINE TABLE(COEF,NTAB,CC) * This subroutine TABLE is to integrated one-dimensional array * along the plate radius:
[1/~ C
* where CC is the cutout ratio or; IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION COEF(NTAB) COEF(1)=LOG(1.O)-LOG(CC)
47
Sec. 3.4 Computer Program
48
DO 10 I=2,NTAB COEF(I) = 1.0/DFLOAT(I- 1)*( 1.0-CC** (I- 1)) CONTINUE
10
RETURN END
SUBROUTINE FF 1XY(FXY,IR,IS,N) * This subroutine FF1XY forms the boundary conditions IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION FXY(*),IR(*),IS (*),A2(10),IR2(10),IS2(10) COMMON/C3/NSIDE WRITE(6,*)'NSIDE= ',NSIDE READ(30,*) N1 DO 10 I-1,N1 READ(30,*) FXY(I),IR(I) CONTINUE
10
DO 30 J= 1,NSIDE- 1 READ(30,*) N2 DO 20 I=I,N2 READ(30,*) AZ(I),IR2(I) CONTINUE CALL MULTI(FXY,IR,IS,N 1,A2,IR2,IS2,N2,NM) N1 = N M CONTINUE
20
30
N=N1 RETURN END
$ $ $ $
SUBROUTINE CALCULUS (FXYM,IRM,ISM,NM, FXYN,IRN,ISN,NN, FXYSI,IRSI,ISSI,NSI, FXYSJ,IRSJ,ISSJ,NSJ, II,SUM,NTAB,COEF)
* This subroutine CALCULUS performs the differentiation * and integration. For further details refer to * Xiang, Wang and Kitipomchai (1995). IMPLICIT DOUBLE PRECISION (A-H,O-Z)
Chap. 3 Formulation in Polar Coordinates
$ $ $
DIMENSION FXYM(*),IRM(*),ISM(*),FXYN(*),IRN(*),ISN(*), FXYSI(*),IRSI(*),ISSI(*),FXYSJ(*),IRSJ(*),IS SJ(*), FXYSII( 100),IRSII( 100),ISSII( 100),FXYSJJ(100), IRSJJ( 100),ISSJJ(100) DIMENSION COEF(NTAB) NSII=I NSJJ=I FXYSII(1)=I.0 IRSII(1)=0 ISSII(1)=0 FXYSJJ(1)=I.0 IRSJJ(1)=0 ISSJJ(1)=0 CALL MULTI(FXYSII,IRSII,ISSII,NSII,FXYM,IRM,ISM,NM,N1) CALL MULTI(FXYSJJ,IRSJJ,IS SJJ,NSJJ,FXYN,IRN,ISN,NN,N2) CALL MULTI(FXYSII,IRSII,IS SII,N1,FXYSI,IRSI,ISSI,NSI,N11) CALL MULTI(FXYSJJ,IRSJJ,IS SJJ,NZ,FXYSJ,IRSJ,ISSJ,NSJ,N22) NI=N11 N2=N22 IF(II.EQ. 1.0R.II.EQ.6.OR.II.EQ. 11.0R.II.EQ. 14) THEN CALL SIMPLE(FXYSII,IRSII,ISSII,N 1,N3) CALL SIMPLE(FXYSJJ,IRSJJ,IS SJJ,N2,N4) CALL MULTI(FXYSII,IRSII,ISSII,N3,FXYSJJ,IRSJJ,IS SJJ,N4,N5) CALL CALCU(FXYSII,IRSII,IS SII,N5,SUM,II,2,NTAB,COEF) GOTO 10 ENDIF IF(II.EQ.2.OR.II.EQ.7.OR.II.EQ. 15) THEN CALL SIMPLE(FXYSII,IRSII,ISSII,N 1,N3) CALL SIMPLE(FXYSJJ,IRSJJ,ISSJJ,N2,N4) CALL MULTI(FXYSII,IRSII,IS SII,N3,FXYSJJ,IRSJJ,ISSJJ,N4,N5) CALL CALCU(FXYSII,IRSII,ISSII,N5,SUM,II, 1,NTAB,COEF) GOTO 10 ENDIF IF(II.EQ.3.OR.II.EQ.8.OR.II.EQ. 16) THEN CALL DIFXY(FXYSII,IRSII,ISSII,N 1,1,0,N3) CALL DIFXY(FXYSJJ,IRSJJ,ISSJJ,N2,1,0,N4) CALL MULTI(FXYSII,IRSII,IS SII,N3,FXYSJJ,IRSJJ,IS SJJ,N4,N5) CALL CALCU(FXYSII,IRSII,ISSII,N5,SUM,II, 1,NTAB,COEF) GOTO 10 ENDIF IF(II.EQ.4) THEN CALL DIFXY(FXYSII,IRSII,IS SII,N 1,1,0,N3) CALL SIMPLE(FXYSJJ,IRSJJ,IS SJJ,N2,N4)
49
Sec. 3.4 Computer Program
50
CALL MULTI(FXYSII,IRSII,IS SII,N3,FXYSJJ,IRSJJ,IS SJJ,N4,N5) CALL CALCU(FXYSII,IRSII,IS SII,N5,SUM,II, 1,NTAB,COEF) GOTO 10 ENDIF IF(II.EQ.5) THEN CALL SIMPLE(FXYSII,IRSII,IS SII,N 1,N3) CALL SIMPLE(FXYSJJ,IRSJJ,ISSJJ,N2,N4) CALL MULTI(FXYSII,IRSII,IS SII,N3,FXYSJJ,IRSJJ,IS SJJ,N4,N5) CALL CALCU(FXYSII,IRSII,ISSII,N5,SUM,II,0,NTAB,COEF) GOTO 10 ENDIF IF(II.EQ.9.OR.II.EQ. 13.OR.II.EQ. 17) THEN CALL SIMPLE(FXYSII,IRSII,ISSII,N 1,N3) CALL DIFXY(FXYSJJ,IRSJJ,ISSJJ,N2,1,0,N4) CALL MULTI(FXYSII,IRSII,IS SII,N3,FXYSJJ,IRSJJ,ISSJJ,N4,N5) CALL CALCU(FXYSII,IRSII,IS SII,N5,SUM,II,0,NTAB,COEF) GOTO 10 ENDIF IF(II.EQ. 10.OR.II.EQ. 12.OR.II.EQ. 18) THEN CALL DIFXY(FXYSII,IRSII,IS SII,N 1,1,0,N3) CALL SIMPLE(FXYSJJ,IRSJJ,IS SJJ,N2,N4) CALL MULTI(FXYSII,IRSII,IS SII,N3,FXYSJJ,IRSJJ,IS SJJ,N4,N5) CALL CALCU(FXYSII,IRSII,IS SII,N5,SUM,II,0,NTAB,COEF) GOTO 10 ENDIF 10
RETURN END SUBROUTINE CALCU(F,IR,IS,N,SUM,II,IND,NTAB,COEF)
* This subroutine CALCU calculates the integrated values of * a polynomial function. IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION F(*),IR(*),IS(*) DIMENSION COEF(NTAB) SUM=0.0 DO 10 I= 1,N IF(IND.EQ.0) FAC-COEF(IR(I)+ 1+ 1) IF(IND.EQ. 1) FAC=COEF(IR(I)+ 1+2) IF(IND.EQ.2) FAC=COEF(IR(I)+ 1)
Chap. 3 Formulation in Polar Coordinates
10
IF(DABS(F(I)).LE. 1.D-100.OR.F(I).EQ.0.) GOTO 10 SUM=SUM+F(I)*FAC CONTINUE RETURN END
SUBROUTINE POLYCOMP1D * This subroutine POLYCOMP1D forms the polynomial terms * in symbolic form. IMPLICIT DOUBLE PRECISION (A-H,O-Z) COMMON/C1/FXY(21),IRR(21),ISS(21)
10
DO 10 11=0,20 FXY(II+I)=I.0 IRR(II+I)=I1 ISS(II+I)=0 CONTINUE RETURN END
SUBROUTINE FFXY(I,FXY,IRR,ISS,N) * This subroutine FFXY assigns the i-th term of the polynomial function to * {FXY,IRR,ISS,1 }. IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION FXY( 1),IRR( 1),IS S ( 1) COMMON/C1/F(21),IR(21),IS(21) N=I FXY(1)=F(I) IRR(1)=IR(I) ISS(1)=IS(I) RETURN END
SUBROUTINE S S S(M,N,L,MNL,S,NTAB,COEF) * This subroutine SSS forms the integrated terms. IMPLICIT DOUBLE PRECISION (A-H,K,O-Z)
51
52
Sec. 3.4 Computer Program
$ $ $ $
DIMENSION S(18,MNL,MNL),FXYM(100),IRM(100),ISM(100), FXYN(100),IRN(100),ISN(100), FXYL(100),IRL(100),ISL(100), FXYSI( 100),IRSI( 100),ISSI(100), FXYSJ( 100),IRSJ( 100),ISSJ(100) DIMENSION COEF(NTAB) CALL FF 1XY(FXYM,IRM,ISM,NM) CALL FF 1XY(FXYN,IRN,ISN,NN) CALL FF 1XY(FXYL,IRL,ISL,NL)
10 20 30
DO 30 I=I,M CALL FFXY(I,FXYSI,IRSI,ISSI,NSI) DO 20 J=I,M CALL FFXY(J,FXYSJ,IRSJ,ISSJ,NSJ) DO 10 II= 1,3 CALL CALCULUS(FXYM,IRM,ISM,NM, FXYM,IRM,ISM,NM, FXYSI,IRSI,ISSI,NSI, FXYSJ,IRSJ,ISSJ,NSJ, II,SUM,NTAB,COEF) S(II,I,J)=SUM CONTINUE CONTINUE CONTINUE
40 50
DO 50 I=I,M CALL FFXY(I,FXYSI,IRSI,ISSI,NSI) DO 40 J=I,N CALL FFXY(J,FXYSJ,IRSJ,ISSJ,NSJ) II=4 CALL CALCULUS(FXYM,IRM,ISM,NM, FXYN,IRN,ISN,NN, FXYSI,IRSI,ISSI,NSI, FXYSJ,IRSJ,ISSJ,NSJ, II,SUM,NTAB,COEF) S(II,I,J)=SUM CONTINUE CONTINUE DO 70 I=I,M CALL FFXY(I,FXYSI,IRSI,ISSI,NSI) DO 60 J=l,t CALL FFXY(J,FXYSJ,IRSJ,ISSJ,NSJ) II=5 CALL CALCULUS(FXYM,IRM,ISM,NM, FXYL,IRL,ISL,NL, FXYSI,IRSI,ISSI,NSI, FXYSJ,IRSJ,ISSJ,NSJ,
Chap. 3 Formulation in Polar Coordinates
II,SUM,NTAB,COEF) 60 70
S(II,I,J)=SUM CONTINUE CONTINUE
DO 100 I=I,N CALL FFXY(I,FXYSI,IRSI,ISSI,NSI) DO 90 J= 1,N CALL FFXY(J,FXYSJ,IRSJ,ISSJ,NSJ) DO 80 II=6,10 CALL CALCULUS(FXYN,IRN,ISN,NN, FXYN,IRN,ISN,NN, FXYSI,IRSI,ISSI,NSI, FXYSJ,IRSJ,ISSJ,NSJ, II,SUM,NTAB,COEF) S(II,I,J)=SUM 80 CONTINUE 90 CONTINUE 100 CONTINUE DO 130 I=I,N CALL FFXY(I,FXYSI,IRSI,ISSI,NSI) DO 120 J=I,L CALL FFXY(J,FXYSJ,IRSJ,IS SJ,NSJ) DO 110 II=l 1,13 CALL CALCULUS(FXYN,IRN,ISN,NN, FXYL,IRL,ISL,NL, FXYSI,IRSI,ISSI,NSI, FXYSJ,IRSJ,ISSJ,NSJ, II,SUM,NTAB,COEF) S(II,I,J)=SUM 110 CONTINUE 120 CONTINUE 130 CONTINUE DO 160 I=I,L CALL FFXY(I,FXYSI,IRSI,ISSI,NSI) DO 150 J=I,L CALL FFXY(J,FXYSJ,IRSJ,IS SJ,NSJ) DO 140 II= 14,18 CALL CALCULUS (FXYL,IRL,ISL,NL, FXYL,IRL,ISL,NL, FXYSI,IRSI,ISSI,NSI, FXYSJ,IRSJ,ISSJ,NSJ, II,SUM,NTAB,COEF)
S(n,U)=SUM 140 CONTINUE 150 CONTINUE 160 CONTINUE
53
See. 3.4 Computer Program
54
RETURN END
************************************************************
* * * *
The subroutines PLUS, MULTI, DIFXY and SIMPLE are presented in Appendix II. They perform mathematical operations on the polynomial functions (for details, refer to Xiang, Wang and Kitipornchai, 1995b)
3.4.2 SAMPLE FILES FOR VPRITZP1 3.4.2.1 Sample Problem To illustrate the use of VPRITZP1, consider an annular plate of thickness h / R = 0.1, inner
radius a = 0.2 and outer radius R (= 1), modulus of elasticity E (- 1), mass density p (= 1), Poisson's ratio v = 0.3 and shear correction factor tc 2 = 5/6. It is to be noted that as the obtained frequency parameter is nondimensional, the radius, the modulus of elasticity and the mass density may be arbitrarily set to unity. Its inner edge is C-clamped while its outer edge is S-simply supported as shown in Fig. 3.2.
(a)
(b)
Fig. 3.2 Annular plate problem Adopting the nondimensional coordinate ~', the basic functions for this plate are given by ~1w = (~ _ 1) 1 (~ _ a ) 1
(3.39a)
~b( = (~' - 1) ~ (~ - a ) ~
(3.39b)
~b~ = (~ - I ) ' (~ - a )
(3.39c)
1
Chap. 3 F o r m u l a t i o n in P o l a r Coordinates
55
Note that the basic functions are formed from the product of the equations of inner and outer plate peripheries raised to appropriate powers as described by Eqs. (3.23). For this sample p-version Ritz analysis, the range of polynomial terms used is N k = 8. The sample input data file is given below:
3.4.2.2 Sample Input Data File 1NPUT.DAT 1. 1. 0.3 0.86667 0.2 1. 0.1 0 8,8,1 lower
modulus of elasticity mass density Poisson ratio shear correction factor inner radius, txR outer radius, R plate thickness, h number of circumferential waves; value of polynomial terms, upper value of polynomial terms and increment step change of polynomial terms (note that since the lower value is set to be the same as the upper value, only one value of degree of polynomial, in this case 10, is adopted).
2
Number of sides
2 1.,1 -1 .,0
Number of terms in the boundary equation (~:- 1)1 of outer periphery for ~b~w The 1st number is the coefficient and 2nd the power of ~ for the first term; The 1st number is the coefficient and 2nd the power of ~ for the 2nd term;
2 Number of terms in the boundary equation (~:- a)' of inner periphery for ~b~w 1.,1 The lst number is the coefficient and 2nd the power of ~: forthe firstterm; -0.2,0 The 1st number is the coefficient and 2nd the power of ~ for the 2nd term; 1
1.,0
Number of terms in the boundary equation (~:- 1)~ = 1 of outer periphery for ~b( The 1st number is the coefficient and 2rid the power of ~: for the 2nd term;
2 Number of terms in the boundary equation (~:- a)' of inner periphery for ~b( 1., 1 The 1st number is the coefficient and 2nd the power of ~ for the first term; -0.2,0 The 1st number is the coefficient and 2nd the power of ~ for the 2nd term;
2 1.,1 -1 .,0
Number of terms in the boundary equation (4'- 1)1 of outer periphery for ~bl~ The 1st number is the coefficient and 2nd the power of 4 for the first term; The 1st number is the coefficient and 2nd the power of 4 for the 2nd term;
2
Number of terms in the boundary equation ( ~ - a)' of inner periphery for ~bf The 1st number is the coefficient and 2nd the power of ~: for the first term;
1.,1
Sec. 3.4 Computer Program
56
-0.2,0 The 1st number is the coefficient and 2nd the power of ~' for the 2nd term; 3.4.2.3 Sample Output Data File
OUT.DAT p-Version Ritz Method for Vibration of Mindlin Plates
Modulus of elasticity E = 1.00000 Plate density per unit volume RHO = 1.00000 Poisson ratio NU = 0.30000 Shear correction factor KAPPA = 0.86667 Inner radius ALPHA*R = 0.20000 Outer radius R = 1.00000 Plate thickness h = 0.10000 The number of circumferential waves n = 0 Polynomial terms change from NK = 8 to 8 with step 1
Plate cutout ratio ALPHA = 0.20000 Plate thickness to outer radius ratio h/R = 0.10000 The number of circumferential waves n = 0
The polynomial terms NK = 8 The first six frequency parameters corresponding to n = 0 are: 20.669 64.985 124.005 192.079 275.971 368.003
3.4.3
S O F T W A R E CODE: V P R I T Z P 2 ************************************************************
* p - V E R S I O N R I T Z M E T H O D F O R V I B R A T I O N OF * * CIRCULAR AND ANNULAR SECTORIAL MINDLIN PLATES * * B A S E D ON P O L A R C O O R D I N A T E S S Y S T E M *
MAIN VPRITZP2 IMPLICIT DOUBLE PRECISION (A-H,O-Z) PARAMETER (NDEGREE=20,NTERM=(NDEGREE+ 1)*(NDEGREE+2)/2, NTOTAL-3 *NTERM,N 1=NTERM,N2=NTOTAL,NTAB =51 ) DIMENSION S(21 ,N 1,N 1),COEF0(NTAB,NTAB),COEF 1(NTAB,NTAB), COEF2(NTAB,NTAB) DOUBLE PRECISION K(N2,N2),K1 (N2,N2), W(N2),Z(N2,N2),FV 1(N2),FV2(N2) DOUBLE PRECISION LX,LY,H,BETA,NU,KAPPA INTEGER P0,P 1,P2
Chap. 3 Formulation in Polar Coordinates
* * * * * * * * * * * * * * * * * * * * * * *
N D E G R E E = the number of degree of polynomial terms used in the Ritz functions N T E R M = the number of polynomial terms; N T O T A L = 3 * N T E R M = the total number of degrees of freedom; N1 = NTERM; N2 = NTOTAL; S(21,N1,N 1) stores the 21 integrated values generated by by subroutine BASICINT; K(N2,N2) = linear stiffness matrix; K1 (N2,N2) = mass matrix; W(N2) = frequency parameter; Z(N2,N2), FV 1(N2), FV2(N2) are working matrices required by EISPACK; LX, LY = inner and outer radii; H = plate thickness; E = Modulus of elasticity (any relative number can be taken); NU = Poisson ratio; RHO = plate density per unit volume; K A P P A = shear correction factor; BETA = sectorial angle; P0 = lower value o f degree of polynomials; P2 = upper value o f degree of polynomials; P1 = increment step o f degree of polynomials. OPEN(30,FILE='INPUT.DAT ') OPEN( 15,FILE='OUTPUT.DAT ')
* INPUT.DAT stores the input information; * OUTPUT.DAT prints the output information; WRITE(15,*) v*******************************************************,
$
WRITE(15,*) ' p-Version Ritz Method for Vibration of Mindlin Plates' WRITE(15,*) WRITE(15,*) '*******************************************************'
WRITE(15,*) WRITE(15,*) WRITE(6,*)'Read input data from file INPUT.DAT' WRITE(6,*) READ(30,*)E WRITE(15,'(1X,"Modulus of elasticity E = ",F13.5)')E READ(30,*)RHO WRITE(15,'(1X,"Plate density per unit volume RHO = ",
57
Sec. 3.4 ComputerProgram
58 F11.5)')RHO
READ(30,*)NU WRITE(15,'(1X,"Poisson ratio NU = ",F8.5)')NU READ(30,*)KAPPA WRITE(15,'(1X,"Shear correction factor KAPPA = ",F8.5)')KAPPA READ(30,*)LX WRITE(15,'(1X,"Inner radius ALPHA*R = ",F8.5)')LX READ(30,*)LY WRITE(15,'(1X,"Outer radius, R = ",F8.5)')LY READ(30,*)H WRITE( 15 ,'(1X,"Plate thickness h = ",F8.5)')H READ(30,*)BETA WRITE(15,'(1X,"Sectorial angle BETA (in degrees) = ",F8.5)')BETA READ(30,*)P0,P2,P 1 WRITE(15,'(1X,"Degree of polynomials changes from p =", I3," to",I3," with step",I3)')P0,P2,P 1 WRITE(15,*) $
,
WRITE(15,*) WRITE(15,*) WRITE(15,'(1X,"Plate cutout ratio ALPHA = ",F8.5)') LX/LY WRITE(15,'(1X,"Plate thickness to radius ratio h/R = ",F8.5)') H/LY WRITE(15,'(1X,"Sectorial angle BETA = ",F8.5)')BETA WRITE(15,*) $
'
WRITE(15,*) WRITE(15,*) WRITE(15,*) NMAX=(P2+ 1)* (P2+2)/2
$ $
CALL STANDVB 1(E,RHO,NU,KAPPA,LX,LY,H, BETA,P0,P 1,P2,S,COEF0,COEF 1,COEF2, NTAB,NMAX,K,K1,W,Z,FV1,FV2) CLOSE(15) CLOSE(30) WRITE(6,*)'End of running the program'
Chap. 3 Formulation in Polar Coordinates
STOP END SUBROUTINE STANDVB 1(E,RHO,NU,KAPPA,LX,LY,H, $ BETA,P0,P 1,P2,S,COEF0,COEF 1,COEF2, $ NTAB,NMAX,K,K1,W,Z,FV1,FV2) * This subroutine STANDVB 1 is the main subroutine * to calculate the frequency parameters IMPLICIT DOUBLE PRECISION (A-H,O-Z) INTEGER P0,P 1,P2 DIMENSION S(21,NMAX,NMAX),COEF0(NTAB,NTAB), $ COEF 1(NTAB,NTAB),C OEF2 (NTAB,NTAB) DOUBLE PRECISION $ K(NMAX*3,NMAX*3),KI(NMAX*3,NMAX*3), $ W(NMAX* 3),Z(NMAX* 3,NMAX* 3),FV 1(NMAX* 3), $ FV2(NMAX*3) DOUBLE PRECISION LX,LY,KAPPA,NU MNL=NMAX WRITE(6,*)'Generate the basic integrated matrix' WRITE(6,*) CALL BASICINT(MNL,NTAB,S,COEF0,COEF 1,COEF2) DO 20 ICASE=P0,P2,P1 M=(ICASE+ 1)* (ICASE+2)/2 WRITE(6,'(1X,"Solve eigenvalue equation for p = ",I3)')ICASE WRITE(6,*) WRITE(15,'(1X,"The degree of polynomials p - ",I3)')ICASE N=M L=M D=E*H** 3/12.0/(1.0-NU** 2) CALL STFF(M,N,L,MNL,LX,LY,H,BETA,E,NU,KAPPA, RHO,D,S,K,K1,W,Z,FV 1,FV2) * Call subroutine STFF to form and solve the eigenvalue equation.
10
20
DO 10 ITR- 1,6 W(ITR)=DSQRT(W(ITR)*(LY)**4*(RHO*H/D)) CONTINUE WRITE(15,'(1X,"The first six frequency parameters are:")') WRITE(15,'(1X,6F 13.3)') (W(ITR),ITR- 1,6) WRITE(15,*) CONTINUE RETURN
59
See. 3.4 Computer Program
60 END
SUBROUTINE CLEAR(A,M,N) * This subroutine CLEAR initializes the two dimensional array (A). IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION A(M,N)
10 20
DO 20 I=I,M DO 10 K=I,N A(I,K)=0.0 CONTINUE CONTINUE
RETURN END
SUBROUTINE STFF(M,N,L,MNL,LX,LY,H,BETA 1,E,V,KAPPA, RHO,D, S,K,K 1,W,Z,FV 1,FV2) * This subroutine STFF forms the linear stiffness and mass matrices * and solves the generalized eigenvalue problem using EISPACK. IMPLICIT DOUBLE PRECISION (A-H,K,O-Z) DIMENSION K(M+N+L,M+N+L),K1 (M+N+L,M+N+L), S(21,MNL,MNL) DIMENSION W(M+N+L),Z(M+N+L,M+N+L),FV 1(M+N+L), FV2(M+N+L) DOUBLE PRECISION LX,LY,KAPPA BETA=3.1415926/180.0*BETA 1 GM=(LY-LX)/2.0 BT=BETA/2.0 MI=M+N+L CALL CLEAR(K,M1,M1) CALL CLEAR(K1,M1,M1) H 1=KAPPA* E'H/2.0/(1.0+V) DO 20 I=I,M DO 10 J=I,M K(I,J)=B T/GM* H 1* S (2,I,J)+GM/BT* H 1* S(3,I,J) K1 (I,J)=BT*GM*RHO*H* S( 1,I,J)
Chap. 3 Formulation in Polar Coordinates
10 20
CONTINUE CONTINUE
30 40
DO 40 I=I,M DO 30 J=I,N K(I,J+M)=H 1*BT* S(4,I,J) CONTINUE CONTINUE
50 60
DO 60 I=I,M DO 50 J=I,L K(I,J+M+N)=H 1*GM* S(5,I,J) CONTINUE CONTINUE
70 80
DO 80 I=I,N DO 70 J=I,N K(I+M,J+M)=H1 *BT*GM* S(6,I,J)+D*BT*GM* S(7,I,J)+ D* B T/GM* S (8,I,J)+D* ( 1.0-V)* GM/B T* S (9,I,J)/2.0+ D*V*BT* S(10,I,J)/2.0+D*V*BT* S(11 ,I,J)/2.0 K1 (I+M,J+M)=BT*GM*RHO*H**3/12.0" S(6,I,J) CONTINUE CONTINUE
DO 100 1= 1,N DO 90 J=I,L K(I+M, J+M +N)=D *V* S( 12,I,J)+D* GM* S( 13 ,I,J)$ D*( 1.0-V)* GM* S ( 14,I,J)/2.0+D* ( 1.0-V)* S( 15,I,J)/2.0 90 CONTINUE 100 CONTINUE DO 120 I=I,L DO 110 J= 1,L K(I+M+N,J+M+N)=H 1*BT* GM* S (16,I,J)+ D*( 1.0-V)*BT* GM/2.0* S ( 17,I,J)+ D*( 1.0-V)*BT/GM/2.0* S ( 18,I,J)+ D* GM/BT* S(19,I,J)D*( 1.0-V)*BT* S(20,I,J)/4.0D*(1.0-V)*BT* S(21 ,I,J)/4.0 K1 (I+M+N,J+M+N)=BT*GM*RHO*H** 3/12.0" S(16,I,J) 110 CONTINUE 120 CONTINUE DO 140 I=I,M1 DO 130 J=I,M1 KI(J,I)=KI(I,J) K(J,I)=K(I,J) 130 CONTINUE 140 CONTINUE
61
Sec. 3.4 Computer Program
62
CALL RSG(M 1,M 1,K,K 1,W,0,Z,FV 1,FV2,IERR) * Call subroutine RSG from EISPACK to solve the eigenvalue equation. IF(IERR.GT.0) THEN WRITE(6,*) 'MATRICES ARE ILL-CONDITIONED' 'IN THE EIGENVALUE FUNCTION' STOP END IF RETURN END
SUBROUTINE BASICINT(NTERM,NTAB,S,COEF0, COEF1,COEF2) * This subroutine BASICINT is to generate the integrated values. IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION S(21,NTERM,NTERM) DIMENSION COEF0(NTAB,NTAB),COEF 1(NTAB,NTAB), COEF2(NTAB,NTAB) COMMON/C3/NSIDE * S(21,NTERM,NTERM) is an array which stores * the 21 integrated values; OPEN(20,FILE='COEF0.TAB ') READ(20,*)((COEF0(I,J),J =I,NTAB),I =I,NTAB) CLOSE(20) OP EN( 20,FILE-'C OEF 1.TAB') READ(20,*)((COEFI(I,J),J =I,NTAB),I =I,NTAB) CLOSE(20) OPEN(20,FILE='COEF2.TAB ') READ(20,*)((COEF2(I,J),J =I,NTAB),I-1,NTAB) CLOSE(20) * COEF0(NTAB,NTAB) is a matrix which stores the integrated value * of the following matrix over the plate domain:
1
7
" ~~
9
i.
77 m - 1
dA
Chap. 3 Formulation in Polar Coordinates
* COEF1 (NTAB,NTAB) is a matrix which stores the integrated value * of the following matrix over the plate domain:
1
1
7?
~:
~7
"'"
477 m-1
.
..
.
- [ ( + a ) - (1 - a)~'.l 2.,1
9
L~ -'
77 m-I
" ' '
o
~-' ~ '
dA
.
~m-'~m-'
* COEF2(NTAB,NTAB) is a matrix which stores the integrated value * of the following matrix over the plate domain:
,
1
_[tl
+ a) - (1 - a)~']
2 t~l
1
r/
...
r/m-1
~
~J7
"'"
~J7 m-1
i
i
..
"
~,m-1 ~,m-,r/ ...
dA
~,,,-lr/m-,
* COEF0, COEF1 and COEF2 are stored respectively * in data files COEF0.TAB, COEF1.TAB and COEF2.TAB * which can be generated from Gaussian quadrature or any mathematical * software package such as M A T H E M A T I C A (Wolfram 1991) * and MAPLE V (Char et al. 1992). A program GAUSSIAN.F * that does the integrations is given in Appendix I. * NSIDE = number of edges of a plate; CALL P O L Y C O M P * Call P O L Y C O M P to form the polynomial terms. MNL=NTERM M=NTERM N=M L=M READ(30,*)NSIDE CALL SSS(M,N,L,MNL,S,NTAB,COEF0,COEF 1,COEF2) * Call subroutine SSS to form the S(21,NTERM,NTERM) matrix. RETURN END
SUBROUTINE FF 1XY(FXY,IR,IS,N)
63
Sec. 3.4 Computer Program
64
* This subroutine FF1XY forms the boundary conditions IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION FXY(*),IR(*),IS(*),A2(10),IR2(10),IS2(10) COMMON/C3/NSIDE WRITE(6,*)'NSIDE= ',NSIDE READ(30,*) N1 DO 10 I=I,N1 READ(30,*) FXY(I),IR(I),IS(I) CONTINUE
10
DO 30 J=I,NSIDE-1 READ(30,*) N2 DO 20 I=l,N2 READ(30,*) A2(I),IR2(I),IS2(I) CONTINUE CALL MULTI(FXY,IR,IS,N1,A2,IR2,IS2,N2,NM) N1 =NM CONTINUE
20
30
N=N1 RETURN END
$ $ $ $
SUBROUTINE CALCULUS(FXYM,IRM,ISM,NM, FXYN,IRN,ISN,NN, FXYSI,IRSI,ISSI,NSI, FXYSJ,IRSJ,ISSJ,NSJ, II,SUM,NT AB, COEF0, COEF 1,COEF2)
* This subroutine CALCULUS performs the differentiation * and integration. For further details, refer to * Xiang, Wang and Kitipomchai (1995). IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION FXYM(*),IRM(*),ISM(*),FXYN(*),IRN(*),ISN(*), $ FXYSI(*),IRSI(*),ISSI(*),FXYSJ(*),IRSJ(*),ISSJ(*), $ FXYSII( 100),IRSII( 100),IS SII(100), $ FXYSJJ(100),IRS JJ( 100),ISSJJ(100) DIMENSION COEF0(NTAB,NTAB),COEF 1(NTAB,NTAB), $ COEF2(NTAB,NTAB) NSII=I NSJJ=I FXYSII(1)=I.0
Chap. 3 Formulation in Polar Coordinates
IRSII(1)=0 ISSII(1)=0 FXYSJJ(1)=I.0 IRSJJ( 1)=0 ISSJJ(1)=0 CALL MULTI(FXYSII,IRSII,ISSII,NSII,FXYM,IRM,ISM,NM,N1) CALL MULTI(FXYSJJ,IRSJJ,IS SJJ,NSJJ,FXYN,IRN,ISN,NN,N2) CALL MULTI(FXYSII,IRSII,ISSII,N 1,FXYSI,IRSI,ISSI,NSI,N 11) CALL MULTI(FXYSJJ,IRSJJ,IS SJJ,N2,FXYSJ,IRSJ,ISSJ,NSJ,N22) NI=N11 N2=N22 IF(II.EQ. 1.0R.II.EQ.6.OR.II.EQ. 16) THEN CALL SIMPLE(FXYSII,IRSII,ISSII,N 1,N3) CALL SIMPLE(FXYSJJ,IRSJJ,IS SJJ,N2,N4) CALL MULTI(FXYSII,IRSII,ISSII,N3,FXYSJJ,IRSJJ, ISSJJ,N4,N5) CALL CALCU(FXYSII,IRSII,ISSII,N5,SUM,II, 1, NTAB,COEF0,COEF 1,COEF2) GOTO 10 ENDIF IF(II.EQ.2.OR.II.EQ.8.OR.II.EQ. 18) THEN CALL DIFXY(FXYSII,IRSII,IS SII,N 1,1,0,N3) CALL DIFXY(FXYSJJ,IRSJJ,ISSJJ,N2,1,0,N4) CALL MULTI(FXYSII,IRSII,ISSII,N3,FXYSJJ,IRSJJ,ISSJJ,N4,N5) CALL CALCU(FXYSII,IRSII,ISSII,N5,SUM,II, 1, NTAB,COEF0,COEF 1,COEF2) GOTO 10 ENDIF IF(II.EQ.3.OR.II.EQ.9.OR.II.EQ. 19) THEN CALL DIFXY(FXYSII,IRSII,IS SII,N 1,0,1 ,N3) CALL DIFXY(FXYSJJ,IRSJJ,ISSJJ,N2,0,1,N4) CALL MULTI(FXYSII,IRSII,IS SII,N3,FXYSJJ,IRSJJ, ISSJJ,N4,N5) CALL CALCU(FXYSII,IRSII,ISSII,N5,SUM,II,2, NTAB,COEF0,COEF 1,COEF2) GOTO 10 ENDIF IF(II.EQ.4) THEN CALL DIFXY(FXYSII,IRSII,ISSII,NI,I,0,N3) CALL SIMPLE(FXYSJJ,IRSJJ,ISSJJ,N2,N4) CALL MULTI(FXYSII,IRSII,ISSII,N3,FXYSJJ,IRSJJ, ISSJJ,N4,N5) CALL CALCU(FXYSII,IRSII,ISSII,N5,SUM,II, 1, NTAB,COEF0,COEF 1,COEF2) GOTO 10
65
Sec. 3.4 ComputerProgram
66 ENDIF
IF(II.EQ.5) THEN CALL DIFXY(FXYSII,IRSII,ISSII,N1,0,1,N3) CALL SIMPLE(FXYSJJ,IRSJJ,ISSJJ,N2,N4) CALL MULTI(FXYSII,IRSII,ISSII,N3,FXYSJJ,IRSJJ, ISSJJ,N4,N5) CALL CALCU(FXYSII,IRSII,ISSII,N5,SUM,II,0, NTAB, COEF0,COEF 1,COEF2) GOTO 10 ENDIF IF(II.EQ.7.OR.II.EQ. 17) THEN CALL SIMPLE(FXYSII,IRSII,ISSII,N 1,N3) CALL SIMPLE(FXYSJJ,IRSJJ,ISSJJ,N2,N4) CALL MULTI(FXYSII,IRSII,IS SII,N3,FXYSJJ,IRSJJ, ISSJJ,N4,N5) CALL CALCU(FXYSII,IRSII,ISSII,N5,SUM,II,2, NTAB,COEF0,COEF 1,COEF2) GOTO 10 ENDIF IF(II.EQ. 10.OR.II.EQ.20) THEN CALL SIMPLE(FXYSII,IRSII,ISSII,N1,N3) CALL DIFXY(FXYSJJ,IRSJJ,ISSJJ,N2,1,0,N4) CALL MULTI(FXYSII,IRSII,IS SlI,N3,FXYSJJ,IRSJJ, ISSJJ,N4,N5) CALL CALCU(FXYSII,IRSII,ISSII,N5,SUM,II,0, NTAB,COEF0,COEF 1,COEF2) GOTO 10 ENDIF IF(II.EQ. 11.0R.II.EQ.21) THEN CALL DIFXY(FXYSII,IRSII,IS SII,N 1,1,0,N3) CALL SIMPLE(FXYSJJ,IRSJJ,ISSJJ,N2,N4) CALL MULTI(FXYSII,IRSII,ISSII,N3,FXYSJJ,IRSJJ, ISSJJ,N4,N5) CALL CALCU(FXYSII,IRSII,ISSII,N5,SUM,II,0, NTAB,COEF0,COEF 1,COEF2) GOTO 10 ENDIF IF(II.EQ. 12) THEN CALL DIFXY(FXYSII,IRSII,ISSII,NI,I,0,N3) CALL DIFXY(FXYSJJ,IRSJJ,ISSJJ,N2,0,1,N4) CALL MULTI(FXYSII,IRSII,ISSII,N3,FXYSJJ,IRSJJ, ISSJJ,N4,N5) CALL CALCU(FXYSII,IRSII,ISSII,N5,SUM,II,0, NTAB,COEF0,COEF 1,COEF2)
Chap. 3 Formulation in Polar Coordinates GOTO 10 ENDIF IF(II.EQ. 13) THEN CALL SIMPLE(FXYSII,IRSII,ISSII,N1,N3) CALL DIFXY(FXYSJJ,IRSJJ,ISSJJ,N2,0,1,N4) CALL MULTI(FXYSII,IRSII,ISSII,N3,FXYSJJ,IRSJJ, ISSJJ,N4,N5) CALL CALCU(FXYSII,IRSII,ISSII,N5,SUM,II,2, NTAB,COEF0,COEF 1,COEF2) GOTO 10 ENDIF IF(II.EQ. 14) THEN CALL DIFXY(FXYSII,IRSII,ISSII,N1,0,1,N3) CALL SIMPLE(FXYSJJ,IRSJJ,IS SJJ,N2,N4) CALL MULTI(FXYSII,IRSII,IS SII,N3,FXYSJJ,IRSJJ, ISSJJ,N4,N5) CALL CALCU(FXYSII,IRSII,IS SII,N5,SUM,II,2, NTAB,COEF0,COEF 1,COEF2) GOTO 10 ENDIF IF(II.EQ. 15) THEN CALL DIFXY(FXYSII,IRSII,IS SII,N 1,0,1 ,N3) CALL DIFXY(FXYSJJ,IRSJJ,ISSJJ,N2,1,0,N4) CALL MULTI(FXYSII,IRSII,ISSII,N3,FXYSJJ,IRSJJ, ISSJJ,N4,N5) CALL CALCU(FXYSII,IRSII,ISSII,N5,SUM,II,0, NTAB,COEF0,COEF 1,COEF2) GOTO 10 ENDIF 10
RETURN END
$
SUBROUTINE CALCU(F,IR,IS,N,SUM,II,IND,NTAB, COEF0,COEF 1,COEF2)
* This subroutine CALCU calculates the integrated values of * a polynomial function. IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION F(*),IR(*),IS(*) DIMENSION COEF0(NTAB,NTAB),COEF 1(NTAB,NTAB), COEF2(NTAB,NTAB) SUM=0.0
67
Sec. 3.4 Computer Program
68
10
DO 10 I=I,N IF(IND.EQ.0) FAC=COEF0(IR(I)+ 1,IS(I)+ 1) IF(IND.EQ. 1) FAC=COEF 1(IR(I)+ 1,IS(I)+ 1) IF(IND.EQ.2) FAC=COEFZ(IR(I)+ 1,IS(I)+ 1) IF(DABS(F(I)).LE. 1.D-100.OR.F(I).EQ.0.) GOTO 10 SUM=SUM+F(I)*FAC CONTINUE RETURN END
SUBROUTINE POLYCOMP * This subroutine POLYCOMP forms the polynomial terms * in symbolic form. IMPLICIT DOUBLE PRECISION (A-H,O-Z) COMMON/C1/FXY(231),IRR(231),ISS(231)
10 20
DO 20 I1=0,20 DO 10 I2=0,I1 K-(II+1)*(II+2)/2-(II-I2) FXY(K)=1.0 IRR(K)=II-I2 ISS(K)=I2 CONTINUE CONTINUE RETURN END SUBROUTINE FFXY(I,FXY,IRR,IS S,N)
* This subroutine FFXY assigns the i-th term of the polynomial function to * {FXY,IRR,ISS, 1}. IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION FXY( 1),IRR( 1),IS S( 1) COMMON/C1/F(231),IR(231),IS(231) N=I FXY(1)=F(I) IRR(1)=IR(I) ISS(1)=IS(I) RETURN
Chap. 3 Formulation in Polar Coordinates
END SUBROUTINE SSS(M,N,L,MNL,S,NTAB,COEF0,COEF1,COEF2) * This subroutine SSS forms the integrated terms. IMPLICIT DOUBLE PRECISION (A-H,K,O-Z) DIMENSION S(21 ,MNL,MNL),FXYM(100),IRM(100),ISM(100), $ FXYN(100),IRN(100),ISN(100), $ FXYL( 100),IRL(100),ISL(100), $ FXYSI( 100),IRSI( 100),ISSI(100), $ FXYSJ( 100),IRSJ( 100),ISSJ(100) DIMENSION COEF0(NTAB,NTAB),COEF1(NTAB,NTAB), $ COEF2(NTAB,NTAB) CALL FF 1XY(FXYM,IRM,ISM,NM) CALL FF 1XY(FXYN,IRN,ISN,NN) CALL FF 1XY(FXYL,IRL,ISL,NL)
10 20 30
DO 30 I=I,M CALL FFXY(I,FXYSI,IRSI,ISSI,NSI) DO 20 J=I,M CALL FFXY(J,FXYSJ,IRSJ,ISSJ,NSJ) DO 10 II=l,3 CALL CALCULUS(FXYM,IRM,ISM,NM, FXYM,IRM,ISM,NM, FXYSI,IRSI,ISSI,NSI, FXYSJ,IRSJ,ISSJ,NSJ, II,SUM,NTAB,COEF0,COEF1,COEF2) S(II,I,J)=SUM CONTINUE CONTINUE CONTINUE
40 50
DO 50 I=I,M CALL FFXY(I,FXYSI,IRSI,ISSI,NSI) DO 40 J-1,N CALL FFXY(J,FXYSJ,IRSJ,ISSJ,NSJ) II=4 CALL CALCULUS(FXYM,IRM,ISM,NM, FXYN,IRN,ISN,NN, FXYSI,IRSI,ISSI,NSI, FXYSJ,IRSJ,ISSJ,NSJ, II,SUM,NTAB,COEF0,COEF 1,COEF2) S(II,I,J)=SUM CONTINUE CONTINUE
69
70
60 70
Sec. 3.4 Computer Program
DO 70 I=I,M CALL FFXY(I,FXYSI,IRSI,IS SI,NSI) DO 60 J= 1,L CALL FFXY(J,FXYSJ,IRSJ,ISSJ,NSJ) 11=5 CALL CALCULUS(FXYM,IRM,ISM,NM, FXYL,IRL,ISL,NL, FXYSI,IRSI,ISSI,NSI, FXYSJ,IRSJ,ISSJ,NSJ, II,SUM,NT AB, COEF0, COEF 1,COEF2) S(II,I,J)=SUM CONTINUE CONTINUE
DO 100 I=I,N CALL FFXY(I,FXYSI,IRSI,IS SI,NSI) DO 90 J=I,N CALL FFXY(J,FXYSJ,IRSJ,IS SJ,NSJ) DO 80 II=6,11 CALL CALCULUS(FXYN,IRN,ISN,NN, FXYN,IRN,ISN,NN, FXYSI,IRSI,ISSI,NSI, FXYSJ,IRSJ,ISSJ,NSJ, II,SUM,NTAB,COEF0,COEF 1,COEF2) S(II,I,J)=SUM CONTINUE 80 CONTINUE 90 100 CONTINUE DO 130 I=I,N CALL FFXY(I,FXYSI,IRSI,IS SI,NSI) DO 120 J=I,L CALL FFXY(J,FXYSJ,IRSJ,IS SJ,NSJ) DO 110 I1=12,15 CALL CALCULUS (FXYN,IRN,ISN,NN, FXYL,IRL,ISL,NL, FXYSI,IRSI,ISSI,NSI, FXYSJ,IRSJ,ISSJ,NSJ, II, SUM,NTAB,COEF0,COEF 1,COEF2) S(II,I,J)=SUM 110 CONTINUE 120 CONTINUE 130 CONTINUE DO 160 I=I,L CALL FFXY(I,FXYSI,IRSI,IS SI,NSI) DO 150 J=I,L CALL FFXY(J,FXYSJ,IRSJ,IS SJ,NSJ)
Chap. 3 Formulation in Polar Coordinates
71
DO 140 II=16,21 CALL CALCULUS(FXYL,IRL,ISL,NL, FXYL,IRL,ISL,NL, FXYSI,IRSI,ISSI,NSI, FXYSJ,IRSJ,ISSJ,NSJ, II,SUM,NTAB,COEF0,COEF1,COEF2) S(II,I,J):SUM 140 CONTINUE 150 CONTINUE 160 CONTINUE RETURN END ************************************************************
* * * *
The subroutines PLUS, MULTI, DIFXY and SIMPLE are presented in Appendix II. They perform mathematical operations on the polynomial functions (for details, refer to Xiang, Wang and Kitipomchai, 1995b)
3.4.4 SAMPLE FILES FOR VPRITZP2 3.4.4.1 Sample Problem To illustrate the use of VPRITZP2, consider an annular sectorial plate of thickness h / R =0.15, inner radius a R =0.5R and outer radius R (= 1), sectorial angle , 8 = 4 5 ~ modulus of elasticity E (= 1), mass density p (= 1), Poisson's ratio v = 0.3 and shear correction factor ~:2 = 0.86667. Its inner circumferential side AB is C-clamped, radial side BC F-free, outer cirumferential side CD C-clamped and radial side DA S-simply supported as shown in Fig. 3.3a. Adopting the nondimensional coordinates (~:,r/), the annular sectorial domain is transformed into a square domain as shown in Fig. 3.3b. As a result, the basic functions for this plate are given by 1)~(~: - 1)*(77-1) 1
(3.40a)
q~l = (~: + 1) 1 (/7 +
1)~(~: - 1) 1 (17 - 1) 1
(3.40b)
~ f = ( ~ + 1) 1 (17 +
1)~(~: - 1) 1 (/7 - 1)~
(3.40c)
q~lw = (~: + 1) 1 (17 +
Note that the basic functions are formed from the product of the equations of transformed square sides AB, BC, CD and DA, raised to appropriate powers as described by Eqs. (3.23). For this sample p-version Ritz analysis, the degree of polynomials used is p = 10. The sample input data file is given below: 3.4.4.2 Sample Input Data File INPUT.DAT
Sec. 3.4 Computer Program
72 ,
1. 0.3 0.86667 0.5 1. 0.15 45.0 10,10,1
modulus of elasticity mass density Poisson ratio shear correction factor inner radius, aR outer radius, R plate thickness, h sectorial angle, /3 lower value of degree of polynomials, upper value of degree of polynomials and increment step change of degree of polynomials (note that since the lower value is set to be the same as the upper value, only one value of degree of polynomial, in this case 10, is adopted).
D
D
A 1
<.') p
c
c
r
I B
F B
1
.L
1
~l-~
(a)
.Ic v I
(b)
Fig. 3.3 (a) Annular sectorial plate problem and (b) transformed plate domain 4
Number of sides
2 Number of terms in the boundary equation (~ + 1)1 of side AB for ~b? 1., 1,0 The 1st number is the coefficient, 2nd the power of ~ and 3rd the power of r/ for the 1st term 1 .,0,0 The 1st number is the coefficient, 2nd the power of ~' and 3rd the power of r/ for the 2nd term Number of terms in the boundary equation (r/+ 1)~ = 1 of side BC for ~b? 1 1.,0,0 Number of terms in the boundary equation (~'- 1)1 of side CD for ~b?
Chap. 3 Formulation in Polar Coordinates
1 .,1,0 -1 .,0,0 2 Number of terms in the boundary equation (r/- 1)' of side DA for q~y 1 .,0,1 -1 .,0,0 2 Number of terms in the boundary equation (~ + 1)' of side AB for ~( 1 .,1,0 1.,0,0 1 Number of terms in the boundary equation (r/+ 1)~ = 1 of side BC for 0( 1.,0,0 2 Number of terms in the boundary equation (~ - 1)' of side CD for q~( 1 .,1,0 -1 .,0,0 2 Number of terms in the boundary equation (r/- 1)' of side DA for ~1 1 .,0,1 -1 .,0,0 2 Number of terms in the boundary equation (~' + 1)' of side AB for ~o 1 .,1,0 1 .,0,0 1 Number of terms in the boundary equation (r/+ 1)~ = 1 of side BC for ~o 1 .,0,0 2 Number of terms in the boundary equation (~:- 1)' of side CD for 0 ~ 1 .,1,0 -1 .,0,0 Number of terms in the boundary equation (7/- 1)~ = 1 of side DA for q~o 1 1 .,0,0
3.4.4.3 Sample Output Data File OUTPUT.DAT p-Version Ritz Method for Vibration of Mindlin Plates
73
74
Sec. 3.5 Benchmark Checks
Modulus of elasticity E = 1.00000 Plate density per unit volume RHO = 1.00000 Poisson ratio NU = 0.30000 Shear correction factor KAPPA = 0.86667 Inner radius ALPHA*R = 0.50000 Outer radius R = 1.00000 Plate thickness h = 0.15000 Sector angle BETA (in degree) = 45.00000 Degree of polynomials changes from p = 10 to 10 with step 1
Plate length to width ratio a/b = 0.50000 Plate thickness to length ratio h/b = 0.15000 Sector angle BETA = 45.00000 The degree of polynomials p = 10 The first six frequency parameters are: 60.002 79.299 124.391 126.300
145.915
180.502
and their corresponding mode shapes are shown in Fig. 3.4. These mode shapes are determined by the Ritz functions where the coefficients are the eigenvectors. The contour plots and the three-dimensional displays of the mode shapes are generated using TECPLOT (Amtec Engineering, 1996)
3.5
BENCHMARK CHECKS
With a view to providing benchmark checks on the validity of computer code, this section presents some accurate frequency results for annular, sectorial and annular sectorial plates. These results have been either obtained analytically or computed from the foregoing p-version Ritz programs. 3.5.1 ANNULAR PLATES Tables 3.4 to 3.6 present some results for annular plates with all combinations of inner and outer edge conditions and h/R = 0.2. These results have been analytically obtained by Irie, Yamada and Takagi (1982). 3.5.2 SECTORIAL PLATES Tables 3.7 to 3.12 present numerical results for sectorial plates with various apex angles and six different combinations of edge conditions as shown in Fig. 3.5. These results have been generated using VPRITZP2 with the inner radius to outer radius ratio taken as a very small value of 0.00001. The shear correction factor tc 2 = 5/6 has been adopted. Additional results may be found in Xiang, Liew and Kitipornchai (1993).
Mode 1
Mode 2
Mode 3
Mode 4
60.002
79.299
124.391
126.300
Fig. 3.4. First six mode shapes o f CFCS annular sectorial plate
Mode 5
145.915
(•
= 0.5, fl = 45 ~
Mode 6
180.502
and h / R = O.15)
76
Sec. 3.5 Benchmark Checks
C
9
,.
C
C
C
.
S C
Fig. 3.5 Boundary conditions o f sectorial plates considered
3.5.3
ANNULAR SECTORIAL PLATES
Tables 3.13 to 3.18 present numerical results for annular sectorial plates with various apex angles, cut-out ratios a and six combinations of edge conditions as shown in Fig. 3.6. These results have been computed from VPRITZP2. The shear correction factor K " 2 " - 5/6 has been used in the computation. Some more results may be obtained from Xiang, Liew and Kitipornchai (1993).
Chap. 3 Formulation in Polar Coordinates
S
77
0g
F
CQ c
s ,; II
Qc
C
/
,t : s
s ,; I
C
II
,
Fig. 3.6 Boundary conditions of annular sectorial plates considered
Sec. 3.5 Benchmark Checks
78
Table 3.4 Frequencyparameters /l o f annularplates with free inner edge (h/R = 0.2 and v = 0 . 3 ) B.C.
F-F
n
s
0
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
1 2 0
F-S
1 2 0
F-C
1 2
0.1 8.30 31.23 17.75 42.93 5.03 28.39 4.70 24.94 12.61 36.52 21.76 49.77 9.22 30.20 17.58 41.05 26.72 53.91
0.3 7.89 39.57 15.13 43.17 4.61 26.63 4.53 30.49 11.19 34.80 20.71 45.73 10.35 36.77 15.87 40.18 25.33 49.74
Table 3.5 Frequency parameters A o f annular plates with inner edge (h/R = 0.2 and v = 0 . 3 ) B.C. n s 0.1 0.3 3.26 3.33 0 1 17.35 25.89 2 2.18 3.23 1 S-F 1 20.29 27.87 2 5.16 5.74 2 i 1 28.93 33.52 2 12.45 18.21 0 1 38.34 56.08 2 14.64 19.98 1 S-S 1 41.66 57.60 2 22.16 25.22 2 1 50.96 62.11 2 18.07 25.68 0 1 43.43 61.56 2 20.11 27.04 1 S-C 1 46.53 62.87 2 27.23 31.32 2 1 55.17 66.81 2 I
I
I
I
I
I
i
0.5 8.55 64.01 13.77 65.32 4.00 23.64 4.91 48.56 9.95 50.22 18.56 55.00 15.40 55.27 17.94 56.57 24.33 60.44
0.7 10.89 131.1 15.27 131.4 3.29 24.19 6.49 97.41 10.90 98.04 18.86 99.90 32.44 99.31 33.35 99.85 36.08 101.5
simply supported
0.5 4.01 44.64 4.65 45.85 7.42 49.35 31.87 90.64 33.04 91.39 36.49 93.64 41.62 95.27 42.41 95.93 44.85 97.89
0.7 5.86 92.39 7.63 92.99 11.79 94.78 70.06 162.3 70.70 162.7 72.60 163.9 80.34 171.7 80.78 172.1 82.13 173.2
Chap. 3 Formulation in Polar Coordinates
79
Table 3. 6 Frequency parameters 2 o f annular plates with clamped inner edge (h/R = 0.2 and v = 0.3) B.C.
n
s
0 C-F
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
1 2 0
C-S
1 2 0
C-C
C~
0.1 3.92 19.05 2.84 21.31 5.22 29.13 14.07 40.06 15.60 42.84 22.32 51.33 19.84 44.91 21.18 47.59 27.44 55.53
1 2
0.3 6.14 29.76 5.79 31.20 6.89 35.68 22.44 59.38 23.52 60.62 27.32 64.42 30.04 64.23 30.77 65.36 33.67 68.81
0.5 11.46 49.27 11.43 50.25 12.14 53.15 38.36 93.78 39.12 94.45 41.51 96.46 48.31 97.39 48.73 98.02 50.18 90.90
0.7 28.05 93.37 28.13 93.96 28.63 95.71 76.76 172.2 77.26 172.5 78.78 173.5 90.20 172.3 90.41 172.7 91.12 173.7
Table 3. 7 Frequency parameters 2 for SSS sectorial plates(v = 0.3)
fl
h/R
Mode sequence number
1
2
3
4
5
6
30 ~
.001 0.10 0.20
97.82 84.43 64.81
183.7 144.0 100.8
277.9 200.1 132.0
288.2 205.8 135.0
412.0 268.9 166.7
431.3 278.0 168.2
45 ~
.001 0.10 0.20
56.67 51.70 42.70
121.5 101.9 75.83
148.5 120.8 87.29
205.6 157.8 108.7
256.3 187.9 125.3
277.9 200.1 132.0
60 ~
.001 0.10 0.20
39.78 37.22 32.03
94.38 81.81 63.13
97.82 84.43 64.81
168.5 134.1 95.15
177.4 139.9 98.53
183.7 144.0 100.8
90 ~
.001 0.10 0.20
25.43 24.34 21.87
56.67 51.70 42.70
69.95 62.63 50.35
97.82 84.43 64.81
121.5 101.9 75.83
134.1 110.8 81.30
Sec. 3.5 Benchmark Checks
80
Table 3.8 Frequency parameters A of SSF sectorial plates (v = 0.3) h/R
Mode sequence number
1
2
3
4
5
6
122.4 101.4 74.99
176.8 137.5 96.89
211.5 160.4 109.1
320.0 222.0 141.1
329.5 225.4 144.0
30 ~
,001 9.10 9.20
47.81 43.49 36.44
45 ~
,001 0.10 0.20
22.14 20.99 18.90
73.36 64.72 51.44
82.25 71.27 55.88
142.1 115.8 83.85
176.8 137.5 96.89
181.7 141.1 98.41
60 ~
.001 0.10 0.20
12.66 12.21 11.39
47.81 43.49 36.44
52.78 48.01 39.80
102.7 86.72 65.97
111.6 94.40 70.85
122.4 101.4 74.99
90 ~
.001 0.10 0.20
5.504 5.386 5.183
22.14 20.99 18.90
34.99 32.77 28.45
47.81 43.49 36.44
73.36 64.72 51.44
82.25 71.30 55.89
Table 3. 9 Frequency parameters /~ of CCS sectorial plates (v = 0.3) h/R Mode sequence number 1 2 3 4 5
6
30 ~
,001 3.10 0.20
165.4 117.6 76.35
272.5 175.3 109.2
384.9 226.4 135.2
397.0 233.6 140.9
544.9 293.0 172.4
561.8 298.8 173.7
45 ~
.001 0.10 0.20
90.75 73.40 52.43
169.2 124.3 83.21
200.4 142.1 92.45
266.3 178.8 113.9
322.8 206.4 128.1
347.5 218.3 134.5
60 ~
.001 0.10 0.20
61.29 52.72 40.15
125.9 99.06 69.60
129.3 101.0 70.24
209.7 150.8 99.80
219.1 155.8 102.0
225.8 159.5 104.1
90 ~
.001 0.113 0.2C
37.26 33.91 27.85
72.76 62.46 47.50
88.54 74.40 55.52
118.5 95.74 68.61
144.2 112.9 79.11
159.3 122.9 85.08
Chap. 3 Formulation in Polar Coordinates
81
Table 3.10 Frequency parameters 2 o f CCC sectorial plates (v = 0.3)
fl
h/R
Mode sequence number
1
2
3
4
5
6
30 ~
.001 0.10 0.20
188.2 128.6 81.58
300.0 183.9 111.2
417.4 235.3 137.3
429.7 240.9 142.2
578.8 299.1 172.7
600.5 305.2 174.8
45 ~
.001 0.10 0.20
107.9 83.80 57.81
191.2 133.6 85.98
224.2 151.9 95.81
293.4 187.1 115.7
352.1 214.3 129.8
377.8 226.7 136.7
60 ~
.001 0.10 0.20
75.63 62.52 45.62
145.1 108.6 72.86
148.8 111.0 74.28
234.0 159.8 101.9
243.4 165.0 104.9
250.7 168.3 106.3
90 ~
.001 0.10 0.20
48.78 42.80 33.38
87.77 72.09 52.23
104.8 84.03 59.39
136.9 105.3 72.43
164.5 122.4 82.05
180.7 132.3 87.82
fl
Table 3.11 Frequency parameters 2 of SSC sectorial plates ( v : O . 3 ) h/R Mode sequence number 1 2 3 4 5
6
30 ~
.001 0.10 0.20
114.2 93.61 68.17
206.0 153.0 102.9
304.0 207.4 133.4
315.9 213.7 136.2
445.0 275.5 166.7
464.0 284.1 168.6
45 ~
.001 0.10 0.20
69.66 60.71 47.04
140.1 111.5 78.87
168.2 129.6 89.82
229.5 166.8 110.5
282.1 195.9 126.8
304.0 207.4 133.4
60 ~
.001 0.10 0.20
51.02 45.82 36.85
111.0 91.59 66.76
114.2 93.61 68.17
190.2 143.6 97.50
198.7 148.4 100.7
206.0 153.0 102.9
90 ~
.001 0.10 0.20
34.87 32.23 27.05
69.66 60.71 47.04
84.58 72.32 54.64
114.2 93.61 68.17
140.1 111.5 78.87
153.8 120.7 84.22
Sec. 3.5 Benchmark Checks
82
Table 3.12 Frequency parameters A of SCC sectorial plates (v = 0.3) h/R Mode sequence number 1 2 3 4 5
6
30 ~
.001 0.10 0.20
148.7 111.0 74.28
250.6 168.3 106.3
358.6 222.0 135.7
370.4 226.8 138.1
509.1 286.3 169.4
530.1 295.2 174.2
45 ~
.001 0.10 0.20
87.77 72.09 52.23
164.5 122.4 82.05
195.4 141.1 93.07
260.3 176.6 112.5
316.2 205.4 128.4
340.2 217.3 135.0
60 ~
.001 0.10 0.20
62.82 54.04 41.11
126.7 99.71 69.56
131.9 102.7 71.40
211.1 151.3 99.35
220.0 156.8 102.8
229.1 161.0 104.7
90 ~
.001 0.10 0.20
41.72 37.47 30.14
78.37 66.36 49.72
94.91 78.21 56.87
125.1 99.50 70.35
151.8 116.9 80.51
167.8 126.5 85.82
Chap. 3 Formulation in Polar Coordinates
83
Table 3.13 Frequency parameters A of SSSS annular sectorial plates (v = 0.3) /3 h/R Mode sequence number 1 2 3 4 5 6
0.2
0.4
30 ~
.001 O. 10 0.20
97.82 84.43 64.81
183.8 144.0 100.8
277.9 200.1 132.0
288.5 206.0 135.1
413.9 269.8 167.7
431.3 278.0 168.6
45 ~
.001 0.10 0.20
56.73 51.75 42.73
122.3 102.4 76.15
148.5 120.8 87.29
210.1 160.4 110.0
256.3 187.9 125.3
277.9 200.1 132.0
60 ~
.001 0.10 0.20
40.16 37.53 32.24
97.48 84.05 64.47
97.82 84.43 64.81
177.4 139.9 98.53
179.9 141.3 99.12
183.8 144.0 100.8
90 ~
.001 0.10 0.20
27.17 25.86 23.01
56.73 51.75 42.73
78.68 69.39 54.68
97.82 84.43 64.81
122.3 102.4 76.15
148.5 120.8 87.29
30 ~
.001 0.10 0.20
98.75 85.11 65.24
195.5 151.4 105.0
277.9 200.1 132.0
336.3 231.1 148.4
430.5 278.2 171.1
529.4 321.4 172.9
45 ~
.001 0.10 0.20
60.31 54.69 44.76
148.2 120.4 86.90
148.7 120.9 87.36
260.4 190.2 126.6
277.8 200.1 132.0
286.1 204.7 134.2
60 ~
.001 0.10 0.20
46.28 42.81 36.12
98.75 85.11 65.24
131.7 108.9 79.95
177.5 140.0 98.56
195.5 151.4 105.0
269.7 195.2 129.0
.001 0.10 0.20 .001 0.10 0.20
36.19 33.96 29.36 103.3 88.52 67.39
60.31 54.69 44.76 228.6 171.6 116.2
98.75 85.11 65.24 278.2 200.2 132.0
120.0 100.6 74.77 427.0 275.9 171.5
148.2 120.4 86.90 438.8 281.5 174.6
148.7 120.9 87.36 539.8 326.3 174.7
45 ~
.001 0.10 O.2O
68.34 61.25 49.29
150.8 122.3 88.18
189.7 147.6 102.7
278.2 200.2 132.0
283.5 203.1 133.5
387.8 256.9 161.6
60 ~
.001 0.10 0.20
55.97 51.02 42.06
103.3 88.52 67.39
176.1 138.9 97.69
178.8 140.8 99.04
228.6 171.6 116.2
278.3 200.2 132.0
90 ~
.001 0.10 0.20
47.15 43.52 36.56
68.34 61.25 49.29
103.3 88.52 67.39
150.7 122.3 88.18
166.5 132.6 94.00
189.7 147.6 102.7
90 ~
30 ~
0.5
Sec. 3.5 Benchmark Checks
84
Table 3.14 Frequency parameters A of CCCC annular sectorial plates (v = 0.3) ,B h/R Mode sequence number 1 2 3 4 5 6
0.2
0.4
30 ~
9001 0.10 0.20
188.2 128.6 81.58
300.0 184.0 111.2
417.4 235.3 137.3
429.5 241.0 142.3
577.9 299.4 173.0
599.5 305.2 174.8
45 ~
9001 0.10 0.20
107.9 83.81 57.83
191.2 133.8 86.18
224.2 151.9 95.81
294.2 188.4 116.7
352.1 214.3 129.8
377.7 226.7 136.7
60 ~
9001 O. 10 0.20
75.69 62.62 45.74
146.3 109.7 73.74
148.8 111.0 74.28
240.9 164.4 104.4
243.5 165.2 105.4
250.7 168.3 106.3
90 ~
9001 0.10 0.20
50.28 43.98 34.22
87.82 72.14 52.29
113.9 89.97 62.81
136.9 105.3 72.43
165.3 123.0 82.49
195.4 141.1 93.07
30 ~
.001 0.10 0.20
188.3 128.9 81.92
305.0 188.3 114.2
417.3 235.3 137.3
460.9 259.5 153.2
596.1 305.4 174.9
672.0 341.5 194.3
45 ~
.001 0.10 0.20
111.2 86.33 59.65
219.0 149.9 95.15
224.3 152.0 95.95
354.7 216.6 131.3
377.7 226.4 136.4
380.8 229.5 138.7
60 ~
.001 0.10 0.20
85.24 69.25 49.80
150.0 111.8 74.93
194.2 136.5 87.76
243.5 165.0 104.9
265.9 176.2 110.8
357.9 218.8 132.9
.001 0.10 0.20 .001 0.10 0.20
69.83 57.90 42.35 192.1 131.7 83.87
95.70 77.06 55.12 342.1 206.5 123.7
138.9 106.4 73.11 417.2 235.5 137.4
179.7 127.8 82.45 570.4 301.2 174.6
195.5 141.1 93.19 599.9 308.7 177.2
207.8 145.0 93.46 728.2 351.2 199.4
45 ~
.001 0.10 0.20
125.5 95.10 64.50
227.4 153.8 97.10
280.2 179.1 109.5
378.3 226.9 136.9
387.2 230.1 138.3
517.3 281.0 164.4
60 ~
.001 0.10 0.20
106.1 82.35 56.83
159.3 116.9 77.70
245.7 166.2 104.3
263.6 170.1 105.6
317.5 198.8 121.7
357.3 222.3 135.9
90 ~
.001 0.10 0.20
95.21 74.62 51.57
114.9 88.51 61.03
150.5 112.4 76.03
201.1 143.7 94.39
253.1 164.2 100.6
267.5 176.3 108.6
90 ~
30 ~
0.5
Chap. 3 Formulation in Polar Coordinates
85
Table 3.15 Frequency parameters 2 of SFSF annular sectorial plates (v = 0.3) fl
0.2
0.4
Mode sequence number 1
2
3
4
5
6
30 ~
.001 0.10 0.20
47.81 43.49 36.44
122.4 101.4 74.98
176.8 137.5 96.89
211.4 160.3 109.1
319.4 221.7 141.0
329.5 225.4 144.0
45 ~
.001 0.10 0.20
22.13 20.98 18.89
73.20 64.57 51.33
82.25 71.27 55.88
140.9 115.0 83.32
176.8 137.5 96.89
181.7 141.1 98.41
60 ~
.001 0.10 0.20
12.63 12.17 11.34
47.81 43.49 36.44
52.21 47.48 39.41
102.7 86.71 65.97
108.7 92.19 69.40
122.4 101.4 74.98
90 ~
.001 0.10 0.20
5.295 5.142 4.917
22.13 20.98 18.89
33.71 31.62 27.53
47.81 43.49 36.44
73.20 64.57 51.33
79.63 69.77 54.71
30 ~
.001 0.10 0.20
47.76 43.44 36.41
120.4 99.97 74.10
176.8 137.5 96.89
199.9 152.6 104.7
306.5 212.0 134.8
329.5 225.4 144.0
45 ~
.001 0.10 0.20
21.87 20.72 18.65
69.09 61.24 48.97
82.24 71.27 55.88
132.3 107.4 78.24
176.8 137.5 96.89
181.1 140.7 98.21
60 ~
.001 0.10 0.20
12.16 11.70 10.88
47.76 43.44 36.41
48.12 43.81 36.46
102.7 86.71 65.97
106.3 88.67 66.79
120.4 99.97 74.10
.001 0.10 0.20 001 9 0.10 0.20
4.725 4.577 4.352 47.48 43.20 36.21
21.88 20.72 18.65 114.9 95.88 71.42
30.85 28.52 24.59 176.8 137.5 96.89
47.76 43.44 36.41 201.0 151.3 103.5
69.09 61.24 48.97 328.4 224.9 142.3
82.24 71.27 55.88 353.4 235.1 143.8
45"
.001 0.10 0.20
21.33 20.21 18.19
65.97 58.15 46.40
82.14 71.19 55.82
145.8 115.8 83.55
176.5 137.5 96.60
176.8 137.9 96.89
60 ~
.001 0.10 0.20
11.60 11.16 10.37
46.71 41.94 34.57
47.48 43.20 36.21
102.6 86.66 65.94
114.9 95.88 71.42
124.3 101.7 75.57
90 ~
.001 0.10 0.20
4.365 4.223 4.001
21.33 20.21 18.19
30.51 27.72 23.52
47.48 43.20 36.21
65.97 58.15 46.40
82.14 71.19 55.82
90 ~
30 ~
0.5
h/R
Sec. 3.5 Benchmark Checks
86
Table 3.16 Frequency parameters ~ of SCSC annular sectorial plates (v = 0.3) P
0.2
0.4
Mode sequence number 1
2
3
4
5
6
30 ~
.001 0.10 0.20
114.2 93.61 68.17
206.3 153.0 103.0
304.0 207.4 133.4
317.9 214.1 136.3
452.4 277.0 167.7
464.0 284.1 169.1
45 ~
.001 0.10 0.20
70.25 60.96 47.13
144.2 113.0 79.42
168.2 129.6 89.82
243.4 171.5 112.3
282.0 195.9 126.8
304.0 207.4 133.4
60 ~
.001 0.10 0.20
53.38 47.09 37.42
114.2 93.61 68.17
121.6 96.62 68.81
198.7 148.4 100.7
206.3 153.0 102.2
217.9 155.1 103.0
90 ~
.001 0.10 0.20
41.81 36.98 29.59
70.25 60.96 47.13
106.5 84.94 60.47
114.2 93.61 68.17
144.2 113.0 79.42
168.2 129.6 89.82
30 ~
.001 0.10 0.20
119.4 95.87 69.01
238.3 165.7 108.0
304.0 207.4 133.4
407.1 246.1 150.2
465.3 284.3 171.1
575.4 331.5 173.7
45 ~
.001 0.10 0.20
84.59 69.44 51.07
169.6 130.0 89.95
199.1 140.7 91.93
296.7 200.2 128.4
304.0 207.4 133.4
366.2 224.2 136.9
60 ~
.001 0.10 0.20
73.62 60.74 44.58
119.4 95.87 69.01
186.3 132.3 86.07
199.3 148.6 100.7
238.3 165.7 108.0
304.2 207.4 132.2
.001 0.10 0.20 .001 0.10 0.20
66.67 55.31 40.39 135.5 103.91 72.42
84.59 69.44 51.07 300.3 192.2 119.9
119.4 95.87 69.01 305.8 207.8 133.5
169.6 126.5 81.87 487.0 289.4 172.9
177.6 130.0 89.95 541.0 294.3 174.7
199.1 140.7 91.93 575.4 331.5 175.6
45 ~
.001 0.10 0.20
107.5 83.39 58.02
178.8 133.5 91.25
269.4 174.0 107.8
305.6 207.8 133.5
346.4 218.2 135.9
475.0 278.7 163.4
60 ~
.001 0.10 0.20
98.92 77.08 53.32
135.5 103.9 72.42
205.8 150.7 101.5
259.0 167.8 103.5
300.3 192.2 119.9
303.2 207.8 133.5
90 ~
.001 0.10 0.2O
93.31 73.13 50.39
107.5 83.39 58.02
135.5 103.9 72.42
178.0 133.5 91.25
236.7 163.6 100.3
251.9 169.0 107.8
90 ~
30 ~
0.5
h/R
Chap. 3 Formulation in Polar Coordinates
87
Table 3.17 Frequency parameters 2 of CSCS annular sectorial plates (v = 0.3) fl
0.2
0.4
Mode sequence number
1
2
3
4
5
6
30 ~
.001 0.10 0.20
165.4 117.6 76.35
272.5 175.3 109.2
384.9 226.4 135.2
396.8 233.6 140.9
540.2 292.9 172.5
561.5 298.8 173.7
45 ~
.001 0.10 0.20
90.75 73.40 52.42
169.2 124.3 83.17
200.4 142.1 92.45
266.2 178.6 114.3
322.6 206.4 128.1
347.5 218.3 134.5
60 ~
.001 0.10 0.20
61.28 52.69 40.11
125.8 98.93 69.76
129.3 101.0 70.24
210.1 152.1 101.8
219.1 155.8 102.0
225.7 159.5 104.1
90 ~
.001 0.10 0.20
37.18 33.82 27.84
72.76 62.46 47.51
89.66 75.77 57.27
118.5 95.74 68.61
144.2 113.0 79.31
167.5 130.7 90.06
30 ~
.001 0.10 0.20
165.3 117.5 76.28
272.5 175.9 110.8
384.9 226.4 135.2
401.9 244.2 150.6
560.7 298.8 173.8
581.8 327.8 183.1
45 ~
.O01 0.10 0.20
90.70 73.33 52.62
175.7 131.1 89.86
200.4 142.1 92.50
306.6 207.1 129.3
322.7 209.4 134.5
347.5 218.3 135.2
60 ~
.001 0.10 0.20
62.04 53.51 41.24
129.3 101.1 70.44
143.4 114.0 81.50
219.0 155.8 102.0
229.0 162.8 107.5
277.6 197.3 129.5
.001 0.10 0.2O .001 O. 10 0.20
41.49 37.90 31.52 165.4 117.6 76.77
73.74 63.40 48.56 283.2 186.5 119.5
118.1 95.72 68.79 384.9 226.4 135.3
123.7 102.3 75.32 465.6 282.0 172.4
159.4 125.1 88.24 562.3 300.4 175.8
174.0 132.0 90.15 685.3 344.4 182.1
45 ~
.001 0.10 0.20
92.89 75.52 54.92
200.6 142.4 93.05
206.1 153.1 104.1
332.6 215.2 134.6
347.5 218.3 135.5
398.6 258.9 162.0
60 ~
.001 0.10 0.20
67.41 58.34 45.30
130.8 102.4 71.83
182.7 141.3 98.35
219.4 156.1 102.3
251.4 178.5 117.7
329.2 214.0 133.9
90 ~
.001 0.10 0.20
50.63 45.93 37.76
78.83 67.74 51.98
120.8 97.70 70.39
168.4 132.4 90.73
173.9 133.3 94.22
196.8 150.2 103.3
90 ~
30 ~
0.5
h/R
Sec. 3.5 Benchmark Checks
88
=
Q
0.2
0.4
Table -3.18 Fret; myparameters R of SSCC annular sectorialplates (v = 0.3) Mode sequence number P 1 2 3 4 5 6 358.6 .001 148.7 250.6 370.3 508.2 529.6 0.10 222.0 11 1.0 168.3 226.8 286.7 295.2 30" 0.20 74.28 106.3 135.7 138.2 170.0 174.2
45"
.001 0.10 0.20
87.77 72.09 52.24
164.6 122.5 82.28
195.4 141.1 93.07
261.2 177.8 113.7
316.2 205.4 128.4
340.2 217.3 135.0
60"
.001 0.10 0.20
62.84 54.07 41.21
127.5 100.5 70.51
132.0 102.8 71.40
216.6 155.9 102.5
220.2 156.8 103.0
229.3 161.1 104.8
90"
.001 0.10 0.20
42.55 38.24 30.85
78.46 66.42 49.74
100.9 83.08 60.31
125.2 99.50 70.35
152.6 117.4 80.79
181.5 135.4 91.48
30"
.oo1 0.10 0.20
148.9 111.1 74.58
256.5 172.8 109.9
358.6 222.1 135.7
404.1 246.8 150.6
529.9 295.3 174.3
605.0 332.3 186.3
45"
.001 0.10 0.20
90.13 73.99 53.78
187.4 137.8 91.79
195.8 141.1 93.16
318.3 207.1 129.6
337.8 217.2 135.0
340.3 218.9 136.5
.001 60"
0.10
0.20
68.99 58.97 44.64
131.6 102.9 71.77
165.9 125.4 84.89
220.3 156.9 102.8
238.8 167.3 108.6
316.4 209.1 131.3
,001 0.10 0.20 .001 0.10 0.20
54.82 48.21 37.63 152.2 113.5 76.40
82.74 69.55 5 1.75 289.5 191.2 120.3
126.2 100.1 70.77 358.7 222.1 135.9
151.5 116.7 79.86 502.8 289.9 172.7
181.0 135.3 91.11 535.3 298.0 176.2
182.9 135.8 91.58 649.6 341.6 187.9
45"
.001 0.10 0.20
99.96 81.01 58.16
197.1 142.2 93.92
237.7 165.8 106.9
338.7 217.4 135.1
344.7 219.3 136.4
453.0 270.7 163.0
60"
.001 0.10 0.20
82.85 69.15 50.91
137.3 106.5 73.91
220.8 157.0 101.9
222.4 157.7 103.3
277.8 187.9 119.5
330.0 215.0 134.7
90"
.001 0.10 0.20
71.83 61.11 45.72
94.62 77.88 56.67
132.7 104.2 73.06
184.2 137.1 92.43
210.4 151.0 98.50
234.3 164.6 106.5
90"
30"
0.5
-
--
CHAPTER FOUR
FORMULATION IN RECTANGULAR COORDINATES
4.1
INTRODUCTION
In this chapter, the vibration analysis of Mindlin plates is formulated in terms of rectangular Cartesian coordinates. Such a coordinate system is commonly used for plate shapes with straight edges. However, for the case of skew (or parallelogram) plates, the skew coordinate system is preferred, as will be discussed in Chapter 5. The most common plate shape with straight edges used in practical applications is that of the rectangular planform, which is also the most intensively studied by researchers. Mindlin (1951), and Mindlin, Schacknow and Deresiewicz (1956) investigated the free flexural vibration of rectangular Mindlin plates with simply supported edge conditions and also with one pair of parallel edges free while the other pair simply supported. They solved the problem in a manner similar to that of their earlier work on circular plates, i.e. by recasting the three goveming equations (2.8) into three harmonic equations involving three potentials | 1~)2'1~3 defined as: gx = (or, - 1 ) a @
+ (o-2 - 1)a ~ 2 + ~~)3
~5"~1
Vy =(0-~-1)-~--+ (or2 -1) =|
(4.1a)
~}2 - a ~ ~3
(4.1b)
+|
(4.1c)
where (62,6"?) r
6(1 v)tc 2 12
=
2(62,6?)
(4.2a)
632(1 - v)
,/.2 6(1-v)tc 2 +
r2 )~ 41
6(1_ v)tc2
~-~
+-~5-
(4.2b)
(4.2c)
89
Sec. 4.1 Introduction
90
2w
N=~;b
2x
2y .
b
h
[,oh
~: = ~r/=--b---,a ; a = - ; a r = b" 3" = wb2 ~ ' D '
(4.2d-i)
in which w is the transverse deflection, a the length, b the width, h the plate thickness, co the circular frequency, D the flexural rigidity, p the mass density per unit volume and the subscripts x, y denote the quantities in the x and y directions, respectively. The origin of the rectangular Cartesian axes is located at the centroid of the rectangular plate. 3, is the frequency parameter. The types of motion generated by O1,O 2 and O 3 are flexural, thickness-shear and thickness-twist, respectively. Based on these nondimensionalized potentials, the governing equations of vibration of Mindlin plates in rectangular Cartesian coordinates can now be expressed as three harmonic equations (V 2 + 6~ )|
=0
(4.3a)
(V 2 +622)O2 = 0
(4.3b)
(V 2 +632)03 = 0
(4.3c)
where the Laplacian operator V2 (@) = a2 a 2 (0) a 2 (0) 0,~ + &-----W
(4.4)
One set of solutions to Eq. (4.3) is O 1 = A1 sin 61~$'sin ~'1~7 0 2 = A2 sin ~924'sin q2r/ 03 = A3 cos ~3~:cos q3r]
(4.5a) (4.5b) (4.5c)
provided that (4.6a)
(4.6b) (4.6c) For hard-type S-simply supported plates, the boundary conditions are:
w=O, Ny=o, Mxb = ( a dNx + v dNy)
--
2D
~=0,
Nx =0,
d~
on~
=_+1
Myb= (aNy+ a v aNx) o n r / = +_1 2D
cTq
(4.7a)
dr/
d~'
Substituting Eqs. (4.1) and (4.5) into Eq. (4.7), one obtains
(4.7b)
Chap. 4 F o r m u l a t i o n in R e c t a n g u l a r Coordinates
miz
oQi =
i = 1,2,3 i = 1,2,3
~'i = ni;rt',
91
(4.8a) (4.8b)
where m i and n i are integers. Substituting Eq. (4.8) into Eq. (4.6), and solving for the frequency parameter 2, one obtains
A~ =
36mv,a{E / 2/1 ff:i
+(-1) i
1 +--~-- 1 + -------------~ (1- v)~c
1+
12
1 + ~ (1 - v)tr 2
-
18(1 - v)tr 2 '
i=1,2
(4.9a)
A23 = 72(1 7:i-v)x2 ( 1 + -lZr2 7 ~~- /)
(4.9b)
~"2 = 7/.4[(ami)2 +(ni)2~
(4.10)
where
A is the frequency parameter of the corresponding simply supported Kirchhoff plate (Timoshenko and Woinowsky-Krieger 1959, p. 335). Note that the frequency relationship given by Eq. (4.9a) checks out with the one given in Eq. (2.47) of Chapter 2. For a given aspect ratio, cr and mode order, the frequencies of the three types of motion satisfy the inequalities 2~ < 223 _<222
(4.11)
As can be seen, the analytical vibration solutions consist of three independent families of modes for the simply supported boundary condition. However, mode couplings were observed by Mindlin, Schacknow and Deresiewicz (1956) for the case of rectangular plates with one pair of free parallel edges and one pair of simply supported edges. Following the work of Mindlin, different methods of analysis have been used by researchers to solve vibrating Mindlin plates with various boundary conditions, and other complicating effects such as in-plane stresses, elastic foundations, point supports, elastically restrained edges and composite materials. For example, a recent series of papers by Liew, Xiang and Kitipornchai (1994a-d) treated some of these complicating effects for rectangular plates. A survey paper by Liew, Xiang and Kitipornchai (1995b) has covered a quite comprehensive literature on the vibration studies of Mindlin plates of various shapes.
Sec. 4.2 Energy Functionals
92
4.2
ENERGYFUNCTIONALS
Consider a general polygonal plate with maximum length a and maximum width b. Adopting the rectangular coordinate system as shown in Fig. 4.1a, the strain energy due to bending of the plate is given by U=
!s({/1 D
2
+ x2Gh
~x
cgx
+c~y c~
"Jr-
+
/
-2(l-v)
~ry ..1-
c~
1 C~x+ 4 c~ cgx
dxdy
(4.12)
and the kinetic energy T may be expressed as T = ~phco 1 2
w~ d_ _i.~..
(4.13)
q_ ~bCy ~ax+
A
/2
F
I
a/2
[
a/2
1
(a)
J.
1
(b)
Fig. 4.1 General plate showing the maximum dimensions in (a) original and (b) transformed coordinate systems
For generality and convenience, the coordinates are normalised with respect to the maximum length dimensions and the following nondimensional terms are introduced: 2x 2y. h _ 2w b = ~ ; a r/=--b-, r=-~-," w= --b--" a = - ; a
2 = ('~
lPh ~ D
(4.1 4)
The normalisation of the coordinates allows the transformation of the plate dimensions into that given by Fig. 4.lb. Using Eq. (4.14), Eqs. (4.12) and (4.13) can be expressed as:
Chap. 4 Formulation in Rectangular Coordinates
--
a
2
c~:
+
Or/
- 2(1 - v)
A +
U/x + a
4v 2
2
-i-'~ W
+ ~y +
"~"~ 8 (~bCx "JI-~//y
E a
O~: Or/ d~drl
4
Or/
93
c7~: (4.15)
(4.16)
A in
which A is the normalised area of the plate. The Lagrangian is given by _
_
m
1-I=U - T
4.3
(4.17)
EIGENVALUE EQUATION
For the Mindlin plate vibration problem, the p-version Ritz functions in rectangular coordinates for approximating the displacement field are defined as Pl q
W(~,~7): Z Z Cmr (~,~7)
(4.18a)
~tx (~,rl) : Z Z dm~bx (~, r])
(4.18b)
q=O i=0 P2 q
q=0 i=0 P3 q
~y (~,rl) = ~_~.em~bY (~,rl)
(4.18c)
q=O i=0
where ps, s = 1,2,3 is the degree set of the complete polynomial space; Ci, di and ei are the unknown coefficients, the subscript m is determined by m = (q + 1)(q + 2) _ i
(4.19)
and the total numbers of Ci, di and ei are Ark, k=-1,2,3 which are dependent on the degree set of polynomial space ps, given by
Nk = (Pk + 1)(Pk + 2) 2 The functions ~w, ~x, Cy consist of
(4.20)
Sec. 4.3 Eigenvalue Equation
94
(r (4, ~), Ox (~, v), C (4, rl)) =
~i17q-i(~1 (~,~7),r (~,17),~( (~,~7))
(4.21)
The basic functions ~b~w, ~b(, ~by in Eq. (4.21) are given as follows"
(C (4, v), 0,x (~, v), 0( (4, v)) = (4.22)
where
is the number of plate edges and Z j(~,rl) the boundary equation of the j-th
ne
supporting edge. yj is dependent on the supporting edge condition, such that w {~free(F); YJ =
(4.23a)
simply supported (S and S*) or clamped (C)
x
{~ free ( F ) o r simply supported (S*)or (S)in the y-direction;
7'j =
simply supported (S) in the x - direction or clamped (C)
(~ free (F) or simply supported (S*) or (S) in the x - direction; YY = simply supported (S) in the y - direction or clamped (C)
(4.23b)
(4.23c)
The hard type of simply supported boundary condition (S) cannot be implemented if the edge is not parallel or perpendicular to the coordinate axes. This drawback can, however, be remedied by introducing point restraints to ensure that the rotation ~s = 0 along the edge. Details are given in Section 6.7 of Chapter 6. Applying the Ritz method,
en)_(o,o,o)
(4.24)
OCm ' dt/, ' cTe"
where m = 1,2,..., N k. The substitution of Eqs. (4.15)-(4.17) and (4.18) into Eq. (4.24) yields the following eigenvalue equation
I KC~ KCa KC~ MC~ MCa M ~ Kdd Kde _ 2 2 M d~ Mde Kee M ee
II
c
0
lf:tf00t =
where the elements of the stiffness submatrices [K] and mass submatrices [M] are:
(4.25)
Chap. 4 Formulation in Rectangular Coordinates k cc = 6 ( 1 - e ) K 2 Y
ko
4r 2
__
4r 2
II[ a _ ,4 i=1,2 ~
_
0~bW0~b) ~ ~ + ~ 0~' O~
10~b/W0~b;l d~drl; a dr/ dr/ (4.26a)
..... N1; j = 1, 2,..., N 1 x
d~ Cj
95
d ~ d ~ 9o
A
(4.26b)
i = 1 , 2 ..... N~; y = 1, 2,..., N 2 ce
ko =
6(1 4T ~y)K'2 2 I-I
I 1 ~ ?- - t 1 -G-C;
A i = 1, 2,..., N1; j = l , 2 ..... N 3
ad ! I [ d~x d();" 1 - v d()x d ~ kij = a ~ ~ + ~ ~+--
16(1-v)x'2
x]
O;Oj d4d,7 ; (4.26d)
i = 1, 2,..., N2; j = 1, 2,..., N 2
k ~i =
_
v
d~ c~
t
2
d ~d rl ;
cgrl d ~
(4.26e)
i = 1, 2,..., N2; j = 1, 2,..., N 3 dO y ~dO y q_ a 1l- v ~ dO,. ee .._ f f l l ~ y dOf nt. -1 6 ( 1 - v ) x 2 ku .I.I L _ a cTr/ 057 2 d~: d~: a 4"t"2 A i = 1, 2,..., N3; j = 1, 2,..., N 3 m,j - 16a
_
(4.26c)
O?O;l dr ; (4.260
9
A i = 1, 2,..., NI; j = I , 2,...,N 1 m,7~a =0",
i = 1, 2,..., N 1", j = 1, 2,..., N 2
ce
.
.
i=l, 2,...,N1, j = l , 2,.. N 3
m,j = 0 ;
(4.27a) (4.27b) (4.27c)
qkxqkjd~drl;
m~/ - 48a
A i = 1, 2,..., N2 ; j = 1, 2,..., N 2 m~/ae=0", mo = 4 8 a
(4.27d)
i = 1, 2,. .., N 2 ", j = 1, 2,. .., N 3
(4.27e)
_ ~/y ~y d~drl ; A i = 1, 2,..., N3; j = 1, 2,..., N 3
(4.270
The resulting eigenvalue Eq. (4.25) m a y be solved using any standard eigenvalue solver for the respective eigenvalue and eigenvectors.
Sec. 4.4 Computer Program
96 4.4
COMPUTER
PROGRAM
A computer program, based on the foregoing rectangular Cartesian coordinate formulation, computes the natural frequencies of plates with edges defined by polynomial functions. Equations of plate edges that are not in polynomial form must be approximated by polynomials if the program is to be used. Although the eigenvalue equation is solved using EISPACK subroutine RSG (Smith et al. 1974) for the results given in this monograph, any other standard eigenvalue solver may be used. The program given below does not include the eigenvalue subroutine solver as it is freely available. The program in its present form is unable to handle the following situations: (1) plates with an S-type simply supported edge that is not parallel or perpendicular to the adopted coordinate axes, (2) plates with reentrant comers formed by simply supported and clamped edges, and (3) mixed boundary conditions on the same edge. Modifications to the software for handling these aforementioned situations are discussed in Sections 6.7-6.9 of Chapter 6. 4.4.1
SOFTWARE CODE: VPRITZRE
************************************************************
* * *
p - V E R S I O N R I T Z M E T H O D F O R V I B R A T I O N OF MINDLIN PLATES BASED ON RECTANGULAR CARTESIAN COORDINATES SYSTEM
,,,**,*******,**********************************************
MAIN VPRITZRE IMPLICIT DOUBLE PRECISION (A-H,O-Z) PARAMETER (NDEGREE=20,NTERM=(NDEGREE + 1)* $ (NDEGREE+2)/2, $ NTOTAL=3 *NTERM,N 1=NTERM,N2=NTOTAL) DIMENSION S(13,N 1,N 1),COEF(65,65) DOUBLE PRECISION K(N2,N2),K1 (N2,N2),W(N2), $ Z(N2,NZ),FVI(N2), $ FVZ(N2) DOUBLE PRECISION LX,LY,H,E,NU,RHO,KAPPA INTEGER P0,P 1,P2 * NDEGREE = the number of degree of polynomial terms used * in the Ritz functions; * NTERM = the number of polynomial terms; * NTOTAL = 3*NTERM = the total number of degrees of freedom; * N1 = NTERM; * N2 = NTOTAL. * S(13,N 1,N 1) stores the integrated values generated * by subroutine BASICINT; * K(N2,N2) = linear stiffness matrix; * K1 (N2,N2) = mass matrix; * W(N2) -- frequency parameter; * Z(N2,N2), FVI(N2), FV2(N2) are working matrices * required by EISPACK; * LX, LY = plate dimensions along x and y;
* * *
Chap. 4 Formulation in Rectangular Coordinates * * * * * * * *
H = plate thickness; E = Modulus of elasticity (any relative number can be taken); NU = Poisson ratio; RHO = plate density per unit volume; KAPPA = shear correction factor; P0 = lower value of degree of polynomials; P2 = upper value of degree of polynomials; P 1 = increment step of degree of polynomials. OPEN(30,FILE='INPUT.DAT') OPEN( 15,FILE='OUTPUT.DAT ')
* INPUT.DAT stores the input information; * OUTPUT.DAT prims the output information; $
$ $
WRITE(15,*)
t*******************************************************,
WRITE(15,*)'p-Version Ritz Method for Vibration' ' o f Mindlin Plates' WRITE(15,*) WRITE(15,*)
t*******************************************************t
WRITE(15,*) WRITE(15,*) WRITE(6,*)'Read input data from file INPUT.DAT' WRITE(6,*)
READ(30,*)E WRITE(15,'(1X,"Modulus of elasticity E = ",F13.5)')E READ(30,*)RHO WRITE(15,'(1X,"Plate density per unit volume RHO = ", F 11.5)')m-IO READ(30,*)NU WRITE(15,'(1X,"Poisson ratio NU = ",F8.5)')NU READ(30,*)KAPPA WRITE(15,'(1X,"Shear correction factor KAPPA = ", F8.S)')KAPPA READ(30,*)LX WRITE(15,'(1X,"Maximum length of plate a = ",F8.5)')LX READ(30,*)LY WRITE( 15,'(1X,"Maximum width of plate b - ",F8.5)')LY READ(30,*)H WRITE(15,'(1X,"Plate thickness h = ",F8.5)')H
97
Sec. 4.4 Computer Program
98
READ(30,*)P0,P2,P 1 WRITE(15,'(1X,"Degree of polynomials changes from p =", I3 ," to", I3 ," with step",I3 )')P0,P2,P 1 WRITE(IS,*) !
!
WRITE(15,*) WRITE(15,*) WRITE(15,'(1X,"Plate length to width ratio a/b = ", F8.5)') LX/LY WRITE(15,'(1X,"Plate thickness to length ratio tvb = ", $ F8.5)') H/LY WRITE(15,*) $
$
'
WRITE(15,*) WRITE(15,*) WRITE(15,*)
NMAX=(P2+ 1)* (P2+2)/2 NTAB=65
$ $
CALL STANDVB 1(E,RHO,NU,KAPPA,LX,LY,H, P0,P 1,P2,S,COEF, NT AB ,NMAX,K,K 1,W,Z,FV 1,FV 2)
CLOSE(15) CLOSZ(30) WRITE(6,*)'End of running the program' STOP END
$ $
SUBROUTINE STANDVB 1(E,RHO,NU,KAPPA,LX,LY,H, P0,P 1,P2,S,COEF, NT AB ,NMAX,K,K 1,W,Z,FV 1,FV 2)
* This subroutine STANDVB 1 is the main subroutine to calculate * the frequency parameters IMPLICIT DOUBLE PRECISION (A-H,O-Z) INTEGER P0,P 1,P2 DIMENSION S(13,NMAX,NMAX),COEF(NTAB,NTAB) DOUBLE PRECISION K(NMAX*3,NMAX*3),
Chap. 4 Formulation in Rectangular Coordinates
$ $ $
K1 (NMAX*3,NMAX*3), W(NMAX*3),Z(NMAX*3,NMAX*3), FV 1(NMAX* 3),FV2 (NMAX* 3) DOUBLE PRECISION LX,LY,KAPPA,NU MNL=NMAX WRITE(6,*)'Generate the basic integrated matrix' WRITE(6,*) CALL BASICINT(MNL,NTAB,S,COEF) DO 20 ICASE=P0,P2,P1 M=(ICASE+ 1)* (ICASE+2)/2 WRITE(6,'(1X,"Solve eigenvalue equation for p = ",I3)')ICASE WRITE(6,*) WRITE(15,'(1X,"The degree of polynomials p = ",I3)')ICASE N=M
L=M D=E*H** 3/12.0/( 1.0-NU** 2) CAL L STFF (M,N,L,MNL,LX,LY,H,E,NU ,KAPP A,D, S,K,KI,W,Z,FV 1,FV2,RHO) * Call subroutine STFF to form and solve the eigenvalue equation.
10
DO 10 ITR= 1,6 W(ITR)=DSQRT(W(ITR)* (LY)**4* (RHO*H/D)) CONTINUE
WRITE(15,'(1X,"The first six frequency parameters are:")') WRITE(15,'(1X,6F 13.3)') (W(ITR),ITR= 1,6) WRITE(15,*) 20 CONTINUE RETURN END
SUBROUTINE CLEAR(A,M,N) * This subroutine CLEAR initializes the two dimensional array (A). IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION A(M,N) DO 20 I=I,M DO 10 K=I,N
99
1O0
10 20
Sec. 4.4 Computer Program A(I,K)=0.D0 CONTINUE CONTINUE RETURN END
SUBROUTINE STFF(M,N,L,MNL,LX,LY,H,E,V,COEF,D, S,K,K1,W,Z,FV 1,FV2,RHO) * This subroutine STFF forms the linear stiffness and mass matrices * and solves the generalized eigenvalue problem using EISPACK. IMPLICIT DOUBLE PRECISION (A-H,K,O-Z) DIMENSION K(M+N+L,M+N+L),K1 (M+N+L,M+N+L), S(13,MNL,MNL) DIMENSION W(M+N+L),Z(M+N+L,M+N+L), FV 1(M+N+L),FV2(M+N+L) DOUBLE PRECISION LX,LY MI=M+N+L CALL CLEAR(K,M1,M1) CALL CLEAR(K1,M1,M1) H 1=COEF *E* H/2.0/(1.0+V) DO 20 I=I,M DO 10 J=I,M K(I,J)=H 1*LY/LX* S(1 ,I,J)+H 1*LX/LY* S (2,I,J) K1 (I,J)=LX*LY*RHO*H* S(13,I,J)/4.0 CONTINUE 10 20 CONTINUE DO 40 I=I,M DO 30 J=I,N K(I,J+M)=HI*LY*S(3,I,J)/2.0 CONTINUE 30 40 CONTINUE DO 6O I-1,M DO 50 J=I,L K(I,J+M+N)=HI*LX*S(4,I,J)/2.0 CONTINUE 50 60 CONTINUE DO 80 I=I,N DO 70 J=I,N
Chap. 4 Formulation in Rectangular Coordinates
K(I+M,J+M)=D*LY/LX* S (5,I,J)+( 1.0-V)*D/2.0* LX/LY* S(6,I,J)+ H1 *LX*LY*S(7,I,J)/4.0 K1 (I+M,J+M) = 1.0/12.0" LX* LY* RHO* H** 3 * S (7,I,J)/4.0 70 CONTINUE 80 CONTINUE $
DO 100 I=I,N DO 90 J=I,L K(I+M,J+M+N)=V*D* S(8,I,J)+( 1.0-V)*D/2.0* S(9,I,J) 90 CONTINUE 100 CONTINUE DO 120 I=I,L DO 110 J=I,L K(I+M+N,J+M+N)=D*LX/LY* S(10,I,J)+ $ (1.0-V)*D/2.0*LY/LX* S(11 ,I,J)+ $ H1 *LX* LY* S ( 12,I,J)/4.0 K1 (I+M+N,J+M+N)= 1.0/12.0* LX* LY*RHO* $ H**3*S(12,I,J)/4.0 110 CONTINUE 120 CONTINUE DO 140 I=I,M1 DO 130 J=I,M 1 KI(J,I)-KI(I,J) K(J,I)=K(I,J) 130 CONTINUE 140 CONTINUE CALL RSG(M 1,M 1,K,KI,W,0,Z,FV1,FV2,IERR) * Call subroutine RSG from EISPACK to solve the eigenvalue equation. IF(IERR.GT.0) THEN WRITE(6,*)'Matrices are ill-conditioned' 'in the eigenvalue function' STOP END IF RETURN END
SUBROUTINE BASICINT(NTERM,NTAB,S,COEF) * This subroutine BASICINT is to generate the integrated values. IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION S(13,NTERM,NTERM)
101
Sec. 4. 4 Computer Program
102
DIMENSION COEF(NTAB,NTAB) * S(13,NTERM,NTERM) is an array which stores * the 13 integrated values; OPEN(20,FILE='COEF.TAB ') * COEF(NTAB,NTAB) is a matrix which stores the integrated * value of the following matrix over the plate domain:
i
4 ~-1
/7
"'"
/7m-1
47~ 9 9
"'" ~ ,
477 m-1 , ,
4 m-l~7
"'"
4 m-l~7 m-1
dA
* COEF(NTAB,NTAB) is stored in a data file COEF.TAB * which can be generated from Gaussian quadrature or any mathematical * software package such as MATHEMATICA (Wolfram 1991) and * MAPLE V (Char et al. 1992). A program GAUSSIAN.F that does * the integrations is given in Appendix I. * NSIDE = number of edges of a plate; READ(20,*)((COEF(I,J),J =I,NTAB),I =I,NTAB) CLOSE(20) CALL POLYCOMP * Call POLYCOMP to form the polynomial terms. MNL=NTERM M=NTERM N=M L=M
READ(30,*)NSIDE CALL S S S(M,N,L,MNL,S,NTAB,COEF,NSIDE) * Call subroutine SSS to form the S(13,NTERM,NTERM) matrix. RETURN END SUBROUTINE FF 1XY(FXY,IR,IS,N,NSIDE)
Chap. 4 Formulation in Rectangular Coordinates
* This subroutine FF1XY forms the boundary conditions IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION FXY(*),IR(*),IS (*),A2(10),IR2(10),IS2(10) READ(30,*) N1 DO 10 I=I,N1 READ(30,*) FXY(I),IR(I),IS(I) 10 CONTINUE DO 30 J=I,NSIDE- 1 READ(30,*) N2 DO 20 I=I,N2 READ(30,*) A2(I),IR2(I),IS2(I) 20 CONTINUE CALL MULTI(FXY,IR,IS,N 1,A2,IR2,IS2,N2,NM) N1 =NM 30 CONTINUE N=N1 RETURN END
$ $ $ $
SUBROUTINE CALCULUS(FXYM,IRM,ISM,NM, FXYN,IRN,ISN,NN, FXYSI,IRSI,ISSI,NSI, FXYSJ,IRSJ,ISSJ,NSJ, II,SUM,NTAB,COEF)
* This subroutine CALCULUS performs the differentiation and * integration. For further details, refer to Xiang, Wang and * Kitipomchai (1995). IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION FXYM(*),IRM(*),ISM(*), $ FXYN(*),IRN(*),ISN(*), $ FXYSI(*),IRSI(*),ISSI(*), $ FXYSJ(*),IRSJ(*),ISSJ(*), $ FXYSII( 100),IRS II( 100),ISSII(100), $ FXYS JJ( 100),IRS JJ( 100),ISSJJ(100) DIMENSION COEF(NTAB,NTAB) NSII=I NSJJ=I FXYSII(1)=I.0 IRSII(1)=0 ISSII(1)=0
103
Sec. 4.4 Computer Program
104 FXYSJJ(1)=I.O IRSJJ(1)=O ISSJJ(1)=O
CALL MULTI(FXYSII,IRSII,ISSII,NSII,FXYM,IRM,ISM,NM,N1) CALL MULTI(FXYSJJ,IRSJJ,IS SJJ,NSJJ,FXYN,IRN,ISN,NN,N2) CALL MULTI(FXYSII,IRSII,IS SII,N 1,FXYSI,IRSI,IS SI,NSI,N 11) CALL MULTI(FXYSJJ,IRSJJ,ISSJJ,N2,FXYSJ,IRSJ,ISSJ,NSJ,N22) NI=N11 N2=N22 IF(II.EQ. 1.0R.II.EQ.5.OR.II.EQ. 11) THEN CALL DIFXY(FXYSII,IRSII,IS SII,N 1,1,0,N3) CALL DIFXY(FXYSJJ,IRSJJ,ISSJJ,N2,1,0,N4) ENDIF IF(II.EQ.2.OR.II.EQ.6.OR.II.EQ. 10) THEN CALL DIFXY(FXYSII,IRSII,IS SII,N 1,0,1 ,N3) CALL DIFXY(FXYSJJ,IRSJJ,ISSJJ,N2,0,1,N4) ENDIF IF(II.EQ.3) THEN CALL DIFXY(FXYSII,IRSII,ISSII,N 1,1,0,N3) CALL SIMPLE(FXYSJJ,IRSJJ,ISSJJ,N2,N4) ENDIF IF(II.EQ.4) THEN CALL DIFXY(FXYSII,IRSII,IS SII,N 1,0,1 ,N3) CALL SIMPLE(FXYSJJ,IRSJJ,ISSJJ,N2,N4) ENDIF IF(II.EQ. 13.OR.II.EQ.7.OR.II.EQ. 12) THEN CALL SIMPLE(FXYSII,IRSII,ISSII,N 1,N3) CALL SIMPLE(FXYSJJ,IRSJJ,IS SJJ,N2,N4) ENDIF IF(II.EQ.8) THEN CALL DIFXY(FXYSII,IRSII,ISSII,NI,I,0,N3) CALL DIFXY(FXYSJJ,IRSJJ,ISSJJ,N2,0,1,N4) ENDIF IF(II.EQ.9) THEN CALL DIFXY(FXYSII,IRSII,IS SII,N 1,0,1,N3) CALL DIFXY(FXYSJJ,IRSJJ,ISSJJ,N2,1,0,N4) ENDIF CALL MULTI(FXYSII,IRSII,ISSII,N3,FXYSJJ,IRSJJ,IS SJJ,N4,N5) CALL CALCU(FXYSII,IRSII,ISSII,N5,SUM,II,NTAB,COEF)
Chap. 4 Formulation in Rectangular Coordinates
RETURN END SUBROUTINE CALCU(F,IR,IS,N,SUM,II,NTAB,COEF) * This subroutine CALCU calculates the integrated values of * a polynomial function. IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION F(*),IR(*),IS(*) DIMENSION COEF(NTAB,NTAB) SUM=0.0 DO 10 I=I,N IF(DABS(F(I)).LE.1.0D-100.OR.F(I).EQ.0.0) GOTO 10 SUM=SUM+F(I)* COEF(IR(I)+ 1,IS(I)+ 1) 10 CONTINUE RETURN END SUBROUTINE POLYCOMP * This subroutine POLYCOMP forms the polynomial terms * in symbolic form. IMPLICIT DOUBLE PRECISION (A-H,O-Z) COMMON/C1/FXY(231),IRR(231),ISS(231) DO 20 I1=0,20 DO 10 I2=0,I1 K=(I1 + 1)*(I1 +2)/2-(I 1-I2) FXY(K)=I.0 IRR(K)=II-I2 ISS(K)=I2 CONTINUE 10 20 CONTINUE RETURN END SUBROUTINE FFXY(I,FXY,IRR,ISS,N) * This subroutine FFXY assigns the i-th term of the polynomial function to * {FXY,IRR,ISS,1 }. IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION FXY( 1),IRR( 1),IS S ( 1)
105
Sec. 4. 4 Computer Program
106
COMMON/C1/F(231),IR(231),IS(231) N=I FXY(1)=F(I) IRR(1)=IR(I) ISS(1)=IS(I) RETURN END SUBROUTINE SSS(M,N,L,MNL,S,NTAB,COEF,NSIDE) * This subroutine SSS forms the integrated terms. IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION S(13,MNL,MNL),FXYM(100),IRM(100),ISM(100), $ FXYN(100),IRN(100),ISN(100), $ FXYL(100),IRL(100),ISL(100), $ FXYSI( 100),IRSI( 100),ISSI(100), $ FXYSJ( 100),IRSJ( 100),ISSJ(100) DIMENSION COEF(NTAB,NTAB) CALL FF 1XY(FXYM,IRM,ISM,NM,NSIDE) CALL FF 1XY(FXYN,IRN,ISN,NN,NSIDE) CALL FF 1XY(FXYL,IRL,ISL,NL,NSIDE) DO 30 I=I,M CALL FFXY(I,FXYSI,IRSI,ISSI,NSI) DO 20 J=I,M CALL FFXY(J,FXYSJ,IRSJ,ISSJ,NSJ) II=13 CALL CALCULUS(FXYM,IRM,ISM,NM, FXYM,IRM,ISM,NM, FXYSI,IRSI,ISSI,NSI, FXYSJ,IRSJ,ISSJ,NSJ, II,SUM,NTAB,COEF) S(II,I,J)=SUM DO 10 II=l,2 CALL CALCULUS(FXYM,IRM,ISM,NM, FXYM,IRM,ISM,NM, FXYSI,IRSI,ISSI,NSI, FXYSJ,IRSJ,ISSJ,NSJ, II,SUM,NTAB,COEF) S(II,I,J)=SUM 10 CONTINUE 20 CONTINUE 30 CONTINUE DO 50 I=I,M
Chap. 4 Formulation in Rectangular Coordinates
CALL FFXY(I,FXYSI,IRSI,ISSI,NSI) DO 40 J=I,N CALL FFXY(J,FXYSJ,IRSJ,ISSJ,NSJ) II=3 CALL CALCULUS(FXYM,IRM,ISM,NM, $ FXYN,IRN,ISN,NN, $ FXYSI,IRSI,ISSI,NSI, $ FXYSJ,IRSJ,ISSJ,NSJ, $ II,SUM,NTAB,COEF) S(II,I,J)=SUM 40 CONTINUE 50 CONTINUE DO 70 I-1,M CALL FFXY(I,FXYSI,IRSI,ISSI,NSI) DO 60 J=I,L CALL FFXY(J,FXYSJ,IRSJ,ISSJ,NSJ) II=4 CALL CALCULUS(FXYM,IRM,ISM,NM, $ FXYL,IRL,ISL,NL, $ FXYSI,IRSI,ISSI,NSI, $ FXYSJ,IRSJ,ISSJ,NSJ, $ II,SUM,NTAB,COEF) S(II,I,J)=SUM 60 CONTINUE 70 CONTINUE DO 100 I=I,N CALL FFXY(I,FXYSI,IRSI,ISSI,NSI) DO 90 J-1,N CALL FFXY(J,FXYSJ,IRSJ,ISSJ,NSJ) DO 80 II=5,7 CALL CALCULUS(FXYN,IRN,ISN,NN, FXYN,IRN,ISN,NN, FXYSI,IRSI,ISSI,NSI, FXYSJ,IRSJ,ISSJ,NSJ, II,SUM,NTAB,COEF) S(II,I,J)=SUM CONTINUE 80 90 CONTINUE 100 CONTINUE DO 130 I=I,N CALL FFXY(I,FXYSI,IRSI,ISSI,NSI) DO 120 J=I,L CALL FFXY(J,FXYSJ,IRSJ,ISSJ,NSJ) DO 110 II=8,9 CALL CALCULUS(FXYN,IRN,ISN,NN, FXYL,IRL,ISL,NL,
107
Sec. 4.4 Computer Program
108
FXYSI,IRSI,ISSI,NSI, FXYSJ,IRSJ,ISSJ,NSJ, II,SUM,NTAB,COEF) S(II,I,J)=SUM 110 CONTINUE 120 CONTINUE 130 CONTINUE DO 160 1= 1,L CALL FFXY(I,FXYSI,IRSI,IS SI,NSI) DO 150 J=I,L CALL FFXY(J,FXYSJ,IRSJ,ISSJ,NSJ) DO 140 II=10,12 CALL CALCULUS (FXYL,IRL,ISL,NL, FXYL,IRL,ISL,NL, FXYSI,IRSI,ISSI,NSI, FXYSJ,IRSJ,ISSJ,NSJ, II,SUM,NTAB,COEF) S(II,I,J)=SUM 140 CONTINUE 150 CONTINUE 160 CONTINUE
RETURN END ************************************************************
* The subroutines PLUS, MULTI, DIFXY and SIMPLE are presented in * Appendix II. They perform mathematical operations on the polynomial * functions (for details, refer to Xiang, Wang and Kitipornchai, 1995b). 4.4.2
I N P U T FILE
The input file INPUT.DAT requires the following information: (1) modulus of elasticity E, (2) mass density of the plate p , (3) Poisson's ratio v, (4) shear correction factor tr 2 , (5) plate maximum length to width ratio a/b, (6) plate thickness to width ratio h/b, (7) degree of polynomial function p, and (8) shape and boundary conditions of plate. The manner in which the program reads in the plate shape defined by n e (= number of plate sides) polynomial functions is explained below. Consider a plate edge defined by an N term two-dimensional polynomial function,
f(~,rl)=a,~r'~7 s' +a2~r~rl s~ +...+aN~"N~7 s~
(4.28)
in which a j, j = 1,2,..., N are real values, r and s are positive integers. Using the FORTRAN language, this function can be represented by a real one-dimensional array A(N) and two integer one-dimensional arrays IR(N), IS(N),
Chap. 4 Formulation in Rectangular Coordinates
109
D O U B L E PRECISION A(N) INTEGER IR(N), IS(N) in which A ( j ) = a s stores the coefficient of the j-th term of the function f(~:,/7), while
IR(j) =rj and I S ( j ) = s j store the powers of ~ and /7 of the j-th term, respectively, with j = 1,2 .... , N . For example, (4.29)
f(~,/7) = 3 + 2~: - 12.5~:2/73
is a three-term two-dimensional function which is to be define in the F O R T R A N language. The arrays A, 1R and IS are then (4.30a-c) (4.30d-f) (4.30g-i)
A(1) = 3.0, IR(1) - 0, IS(l) - 0, A ( 2 ) = 2 . 0 , I R ( 2 ) = 1, I S ( 2 ) = 0 , A(3) =-12.5, IR(3) = 2, IS(3)= 3.
4.4.2.1 Input Data File- Sample 1 As the first illustration of the way in which the plate shape and boundary condition are read, consider a rectangular plate of length a = 2, width b = 1, thickness h / b = 0.15, modulus of elasticity E (= 1), mass density p (= 1), Poisson's ratio v = 0.3 and shear correction factor tr 2 = 0.86667. Its side AB is S-simply supported, side BC F-free, side CD C-clamped and side DA S-simply supported as shown in Fig. 4.2a. Note that the length and width dimensions of the rectangular plate shown in Fig. 4.2b have been normalized by a and b, respectively. In view of the boundary conditions, the basic functions for this plate are given by 1) ~(~: - 1) 1 ( 1 7 - 1 ) 1
(4.31a)
~b~x = (~: + 1) 0 (r/+ 1) 0 (~ - 1) 1 (/7 - 1) 1
(4.31b)
~Y = ( ~ + 1) 1 (/7 + 1) ~ (~' - 1) 1 ( / 7 - 1 )
(4.31c)
q~lw = (~: + 1) 1 (17 +
~
D
A
D
A L
/2
jc
F BE,
a/2
a/2
~l c
B
v|
Fig. 4.2 SFCS plate
v i-.,,
r|
110
Sec. 4. 4 Computer Program
Note that the basic functions are formed from the product of the equations of sides AB, BC, CD and DA, raised to appropriate powers as described by Eqs. (4.22) and (4.23). For this sample p-version Ritz analysis, the range of the degree of polynomials used is p = 2 to 10. The sample input data file is given below:
INPUT.DAT .
1. 0.3 0.86667 2. 1. 0.15 2,10,1
modulus of elasticity mass density Poisson ratio shear correction factor maximum length, a maximum width, b plate thickness, h lower value of degree of polynomials, upper value of degree of polynomials and increment step change of degree of polynomials (note that if only one value of degree of polynomial is desired, then set the lower value to be the same as the upper value). Number of sides
2 1 .,1,0 1 .,0,0
Number of terms in the boundary equation (~' + 1)' of side AB for 01w The 1st number is the coefficient, 2nd the power of ~ and 3rd the power of 77 for the 1st term The 1st number is the coefficient, 2nd the power of ~' and 3rd the power of 7/ for the 2nd term
1 1.,0,0
Number of terms in the boundary equation (r/+ 1)~ = 1 of side BC for ~w
2 1 .,1,0 -1 .,0,0
Number of terms in the boundary equation (~'- 1)1 of side CD for ~1w
2 1 .,0,1 -1 .,0,0
Number of terms in the boundary equation (7?- 1)' of side DA for ~w
1 1 .,0,0
Number of terms in the boundary equation (~ + 1)~ = 1 of side AB for ~x
1 1.,0,0
Number of terms in the boundary equation (r/+ 1)~ = 1 of side BC for Cx
Chap. 4 Formulation in Rectangular Coordinates
111
2 1 .,1,0 -1 .,0,0
Number of terms in the boundary equation (~'- 1)' of side CD for ~b~x
2 1 .,0,1 -1 .,0,0
Number of terms in the boundary equation (r/- 1)1 of side DA for ~b1
2 1 .,1,0 1.,0,0
Number of terms in the boundary equation (~' + 1)1 of side AB for ~by
1 1.,0,0
Number of terms in the boundary equation (r/+ 1)~ = 1 of side BC for ~Y
2 1 .,1,0 -1 .,0,0
Number of terms in the boundary equation (~:- 1)1 of side CD for q~Y
1 1l ,0,0
Number of terms in the boundary equation ( q - 1)~ = 1 of side DA for ~Y
4.4.2.2 Input Data File- Sample 2 For the second illustration, consider a right-angled isosceles triangular plate of thickness h/b =0.15, modulus of elasticity E (= 1), mass density p (= 1), Poisson's ratio v =0.3 and
shear correction factor tc 2 =0.83333. Its side AB is S*-simply supported, side BC Cclamped and side CA C-clamped as shown in Fig. 4.3. Az.,, b/2
q-
1-4
2 =0
C
b/2 L ~ ' "
2 =0
~' + 1 =0.,
BL i-~
a
1 .I.
., Fig. 4.3 S*CC isosceles
yl-,,
I
.I r|
112
Sec. 4.4 Computer Program The basic functions for this plate are given by ~b,w = (~' + 1)' (q - 0.5~' + 0.5)' (r/+ 0.5~' - 0.5)' ~b~x = (~' + 1)~ ( q - 0.5~, "4- 0.5)1 (~'] Jr- 0.5~: -- 0.5) l
(4.32b)
~by =(~: + 1)~
(4.32c)
0.5~: nt- 0.5)1 (17 Jr- 0.5~ -- 0.5) 1
(4.32a)
Note that the basic functions are formed from the product of the equations of sides AB, BC, and CA, raised to appropriate powers as described by Eqs. (4.22) and (4.23). For this sample p-version Ritz analysis, the degree of polynomials used is p - 10. The sample input data file INPUT.DAT is given below:
INPUT.DAT
1. 1. 0.3 0.83333 0.5 1. O. 15 10,10,1
modulus of elasticity mass density Poisson's ratio shear correction factor maximum length, a maximum width, b plate thickness, h degree of polynomials adopted is p = 10,
3
Number of sides
2 1 .,1,0
Number of terms in the boundary equation (~' + 1)1 of side AB for The 1st number is the coefficient, 2nd the power of ~ and 3rd the power of r/ for the 1st term The 1st number is the coefficient, 2nd the power of ~' and 3rd the power of r/ for the 2nd term
1.,0,0 3 1.,0,1 -0.5,1,0 0.5,0,0
Number of terms in the boundary equation (r/- 0.5~: + 0.5)' of side BC for 0Y
3 1 .,0,1 0.5,1,0 -0.5,0,0
Number of terms in the boundary equation (7/+ 0.5~ - 0.5) 1 of side CA for 07
1
Number of terms in the boundary equation (~ + 1)~ = 1 of side AB for ~b(
1.,0,0
Chap. 4 Formulation in Rectangular Coordinates 3 1 .,0,1 -0.5,1,0 0.5,0,0
Number of terms in the boundary equation (r/- 0.5~: + 0.5)' of side BC for ~?
3 1 .,0,1 0.5,1,0 -0.5,0,0
Number of terms in the boundary equation (r/+ 0.5~' - 0.5)' of side CA for ~1
1 1 .,0,0
Number of terms in the boundary equation (~ + 1)~ = 1 of side AB for ~by
3 1 .,0,1 -0.5,1,0 0.5,0,0
Number of terms in the boundary equation (r/- 0.5~ + 0.5)' of side BC for r
3 1 .,0,1 0.5,1,0 -0.5,0,0
Number of terms in the boundary equation (r/+ 0.5~: - 0.5)' of side CA for CY
4.4.3
113
O U T P U T FILE
The output file OUTPUT.DAT prints the following information: (1) modulus of elasticity E, (2) mass density p , (3) Poisson ratio v, (4) shear correction factor tc ~ , (5) plate maximum length to width ratio a/b, (6) plate thickness to width ratio h/b, (7) degree of polynomial functions p, and (8) frequency parameters 2 = cob 2~/ph/D. Based on the two foregoing sample input data files, the output files are presented below. 4.4.3.1 Output File for Input Data File - Sample 1
The output file OUTPUT.DAT for the SFCS rectangular plate, as described in Section 4.4.2.1, is given below:
OUTPUT.DAT
p-Version Ritz Method for Vibration of Mindlin Plates
114
Sec. 4. 4 Computer Program
Modulus of elasticity E = 1.00000 Plate density per unit volume RHO = 1.00000 Poisson ratio NU = 0.30000 Shear correction factor KAPPA = 0.86667 Maximum length of plate a = 2.00000 Maximum width of plate b = 1.00000 Plate thickness h = 0.15000 Degree of polynomials changes from p = 2 to 10 with step 1 Plate length to width ratio a/b = 2.00000 Plate thickness to length ratio h/b = 0.15000
The degree of polynomials p = 2 The first six frequency parameters are: 4.999 18.143 20.292 38.776
52.628
59.922
The degree of polynomials p = 3 The first six frequency parameters are: 4.976 13.302 18.169 30.716
37.579
55.882
The degree of polynomials p = 4 The first six frequency parameters are: 4.960 13.213 17.872 25.425
26.487
45.902
The degree of polynomials p = 5 The first six frequency parameters are: 4.949 13.115 17.783 25.150
26.037
38.262
The degree of polynomials p = 6 The first six frequency parameters are: 4.943 13.092 17.763 24.631
25.857
37.617
The degree of polynomials p = 7 The first six frequency parameters are: 4.940 13.080 17.754 24.581
25.813
37.128
The degree of polynomials p = 8 The first six frequency parameters are: 4.939 13.075 17.751 24.552
25.793
37.043
The degree of polynomials p = 9 The first six frequency parameters are: 4.939 13.072 17.750 24.542
25.785
37.002
The degree of polynomials p = 10 The first six frequency parameters are:
Chap. 4 Formulation in Rectangular Coordinates 4.938
13.071
17.749
24.537
25.782
115
36.988
and their corresponding mode shapes are shown in Fig. 4.4. The frequencies compare well with the accurate ones obtained by Claasen and Thorne (1962) for the doubly antisymmetric modes of CFCF thin plates (see Table 4.43 in Leissa 1969).
4.4.3.2 Output File for Input Data File- Sample 2 The output file OUTPUT.DAT for the S*CC isosceles triangular plate, as described in Section 4.4.2.2, is given below:
OUTPUT.DAT p-Version Ritz Method for Vibration of Mindlin Plates Modulus of elasticity E = 1.00000 Plate density per unit volume RHO = 1.00000 Poisson ratio NU = 0.30000 Shear correction factor KAPPA = 0.83333 Maximum length of plate a = 0.50000 Maximum width of plate b = 1.00000 Plate thickness h = 0.15000 Degree of polynomials changes from p = 10 to 10 with step 1 Plate length to width ratio a/b = 0.50000 Plate thickness to length ratio h/b = 0.15000 The degree of polynomials p = 10 The first six frequency parameters are: 88.295 133.279 160.659 185.476
206.478
and their corresponding mode shapes are shown in Fig. 4.5.
239.056
4~
t%
Fig. 4. 4. First six mode shapes of SFCS rectangular plate (a / b = 2, h / b = O. 15)
4~
~.
t~
E" c~ ~~
Fig. 4.5. First six mode shapes of S*CC right angled isosceles triangularplate with base length b and h/b = 0.15
118 4.5
Sec. 4.5 Benchmark Checks
BENCHMARK CHECKS
With the objective of providing benchmark results for validating computer code, this section presents accurate frequency parameters for a few selected plate shapes. The vibration results have been computed from the p-version Ritz program and also from other sources using various computational techniques. 4.5.1 ISOSCELES TRIANGULAR PLATES Consider an isosceles triangular plate with an apex angle of fl for eight different boundary conditions as shown in Fig. 4.6. Tables 4.1-4.8 present the first six vibration frequencies for these plates with various apex angles, v = 0.3 and ic = zc2/12. The designation CS*S*, for example, means that the equal sides are simply supported (soft) and the other side is clamped. Note that the fundamental frequencies of S*S*S* thin plates (i.e. h / b = 0.001) given in Table 4.1 agree well with the accurate solutions obtained by Conway and Farnham (1965) using the point-matching technique. For further vibration results, refer to the paper by Kitipornchai, Liew, Xiang and Wang (1993). 4.5.2 TRAPEZOIDAL PLATES Consider the symmetrical trapezoidal plates as shown in Fig. 4.7. Tables 4.9 to 4.14 present the first six vibration frequencies for various dimensions of plate, v = 0.3 and K "2 = 5/6. Further details of the treatment of such trapezoidal plates are given in the paper by Kitipornchai, Xiang, Liew and Lim (1994). 4.5.3 ELLIPTICAL PLATES Consider the elliptical plates with three different edge conditions F, S*, and C as shown in Fig. 4.8. Tables 4.15 to 4.17 present the first six vibration frequencies for various dimensions of plate, v = 0.3 and Ir 2 = 5 / 6. Note that the data for clamped, thin, elliptical plates in Table 4.17 compares well with the frequencies given on pp. 37 and 38 of Leissa's monograph (Leissa 1969). Further details of the treatment of such elliptical plates and semi-elliptical plates are given in the paper by Wang, Xiang and Kitipornchai (1995).
Chap. 4 Formulation in Rectangular Coordinates
S*
119
F W" S*
b/2 ~,~" S*
,k~" C
v A
b/2
S*
I
~
i-~
C
C
I
F
r I
F
C
Fig. 4. 6 Considered boundary conditions of isosceles triangular plates
F L~
r b
C
F
;,"...r
C
_, v
F
S
S
S
C
F
C
C
S
C
S
S
C
I"
Fig. 4. 7 Considered boundary conditions of symmetrical trapezoidal plates
Sec. 4.5 Benchmark Checks
120
F
S*
b
I I"
a
L 0 Wl
Fig. 4.8 Considered boundary conditions of elliptical plates
Table 4.1 Frequency parameters 2 o r s * s ' s * isosceles triangular plates
30
60
90
h/b ~.001 9.05 0.10 0.15 0.20 }.001 0.05 0.10 0.15 0.20 ).001 0.05 0.10 0.15 0.20
1 28.12 27.42 26.10 24.50 22.81 52.63 50.46 46.62 42.31 38.16 98.71 92.11 81.43 70.87 61.72
2 52.60 50.68 46.97 42.72 38.59 122.8 114.4 99.65 85.33 73.39 197.4 177.7 147.2 121.1 101.2
Mode sequence number 3 4 78.67 82.50 75.15 78.54 68.05 70.75 60.32 62.42 53.29 54.97 122.8 210.5 114.4 190.0 99.65 156.6 85.33 128.4 73.39 107.1 256.6 335.6 225.6 288.6 181.3 224.4 146.3 176.9 120.7 143.6
5 118.6 111.1 97.26 83.57 72.07 228.1 204.4 166.9 135.8 112.7 396.5 332.9 252.6 196.3 157.7
6 122.1 114.4 99.90 85.70 73.76 228.1 204.4 166.9 135.8 112.7 495.9 404.4 298.1 228.0 161.0
Chap. 4 Formulation in Rectangular Coordinates
121
Table 4.2 Frequency parameters A of CS*S* isosceles triangular plates h/b
30
60
90
0.001 0.05 0.10 0.15 0.20 0.001 0.05 0.10 0.15 0.20 0.001 0.05 0.10 0.15 0.20
1 32.55 31.69 29.90 27.69 25.42 66.16 62.83 56.39 49.45 43.25 131.6 119.5 99.53 81.87 68.30
2 58.72 56.28 51.45 46.02 40.92 142.8 131.2 110.1 91.29 76.65 242.2 211.3 165.4 130.7 106.4
Mode sequence number 3 4 85.83 90.04 81.68 85.07 73.14 75.46 63.96 65.53 55.93 56.94 143.4 236.9 131.3 209.5 110.8 166.9 92.30 133.5 77.85 109.6 308.9 392.8 260.5 324.1 197.1 241.3 152.8 185.1 123.2 148.4
5 128.4 118.9 102.0 86.27 73.45 255.3 224.2 176.8 140.5 114.8 460.2 369.5 266.3 201.1 159.2
6 131.1 122.0 105.0 88.78 75.65 255.6 224.4 177.4 141.4 115.9 566.9 440.2 308.8 230.7 167.3
Table 4.3 Frequency parameters 2 of S*CC isosceles triangular plates h/b
30
60
90
0.001 0.05 0.10 0.15 0.20 0.001 0.05 0.10 0.15 0.20 0.001 0.05 0.10 0.15 0.20
1 47.62 45.76 41.58 36.80 32.36 81.62 76.65 66.79 56.83 48.53 146.8 132.7 108.6 87.96 72.57
2 78.10 73.95 65.03 55.81 47.99 165.0 148.6 120.6 97.26 80.11 263.1 224.9 171.3 132.8 106.9
Mode sequence number 3 4 109.5 113.7 101.7 105.7 86.58 90.04 72.19 75.34 60.76 63.65 165.3 265.1 149.4 229.7 122.1 177.4 99.27 139.0 82.31 112.7 330.0 421.0 277.7 340.3 145.0 244.9 160.0 184.7 128.5 146.5
5 156.3 142.1 117.0 95.63 79.67 284.2 244.4 187.1 145.9 118.0 484.8 385.3 273.8 205.6 162.7
6 159.4 145.0 119.0 96.82 80.24 284.8 244.6 187.3 146.1 118.3 593.8 459.9 320.0 238.0 180.6
Sec. 4.5 Benchmark Checks
122
Table 4.4 Frequencyparameters A of CCC isosceles triangular plates
/~
0
30
60
90
h/b 0.001 0.05 0.10 0.15 0.20 0.001 0.05 0.10 0.15 0.20 0.001 0.05 0.10 0.15 0.20
1 53.80 51.55 46.35 40.55 35.32 99.02 91.86 77.79 64.59 54.19 187.6 164.4 127.9 100.2 81.05
2 85.70 80.61 69.81 59.07 50.20 189.0 167.5 132.3 104.7 85.26 315.5 260.9 190.3 144.0 114.4
Mode sequence number 3 4 118.4 122.8 109.5 113.2 92.17 94.88 76.15 78.29 63.59 65.47 189.0 295.2 167.5 250.8 132.3 188.6 104.7 145.3 85.26 116.8 389.6 486.0 313.7 378.4 223.2 262.1 166.7 194.1 131.4 152.7
5 169.5 150.5 121.7 98.27 81.12 315.2 265.1 197.0 150.8 120.7 556.1 423.2 289.0 212.9 167.2
6 170.3 153.4 124.2 99.90 82.18 315.2 265.1 197.0 150.8 120.7 674.8 495.6 330.6 241.1 188.4
Table 4.5 Frequency parameters A of FS*S* isosceles triangular plates
30
60
90
h/b 0.001 0.05 0.10 0.15 0.20 0.001 0.05 0.10 0.15 0.20 0.00] 0.05 0.10 0.15 0.20
1 13.55 13.31 12.91 12.40 11.83 16.08 15.65 15.02 14.26 13.45 19.60 18.65 17.48 16.20 14.87
2 35.08 33.97 31.98 29.63 27.21 57.62 55.10 50.64 45.49 40.40 69.24 65.09 58.30 50.82 43.59
Mode sequence number 3 4 49.56 60.49 47.98 57.99 44.69 53.26 40.78 47.98 36.84 42.96 68.3 120.8 64.8 112.7 58.85 97.95 52.31 83.25 46.20 70.50 117.1 161.5 107.5 147.2 92.24 124.2 76.97 106.1 63.40 85.39
5 91.67 86.77 77.47 67.80 59.25 148.0 136.0 115.7 96.45 80.30 204.1 181.1 145.8 113.8 87.84
6 92.87 87.96 78.41 68.42 59.51 148.0 136.5 117.0 98.71 83.94 291.9 253.6 195.5 146.6 110.4
Chap. 4 Formulation in Rectangular Coordinates
123
Table 4. 6 Frequency parameters ~ of FCC isosceles triangular plates
/~
0
30
60
90
h/b 0.001 0.05 0.10 0.15 0.20 0.001 0.05 0.10 0.15 0.20 0.001 O.05 0.10 0.15 O.2O
1 30.29 29.45 27.53 25.13 22.69 40.01 38.62 35.66 32.15 28.74 58.18 55.47 50.15 44.33 39.02
2 56.51 53.98 48.42 42.31 36.81 95.82 89.47 77.53 65.60 55.71 127.1 116.4 98.02 81.24 68.05
Mode sequence number 3 4 77.72 87.78 73.58 82.56 64.78 71.88 55.61 61.19 47.68 52.31 101.8 172.8 94.62 155.9 80.86 127.4 67.61 103.3 56.92 85.39 179.7 232.2 160.8 202.2 129.5 158.3 103.5 124.7 84.26 101.2
5 125.7 115.4 97.20 80.74 67.48 193.8 173.0 138.9 111.4 91.23 291.4 249.3 190.5 148.1 119.2
6 126.6 116.6 97.83 80.74 67.92 197.6 175.6 140.1 111.7 91.48 377.6 309.8 226.4 171.3 135.5
Table 4. 7 Frequency parameters 2 of CFF isosceles triangular plates h/b
30
60
90
0.001 0.05 0.10 0.15 0.20 0.001 0.05 0.10 0.15 0.20 0.001 0.05 0.10 0.15 0.20
1 1.975 1.969 1.957 1.942 1.924 8.922 8.822 8.646 8.413 8.137 25.29 24.67 23.51 22.06 20.53
2 8.558 8.482 8.332 8.118 7.860 35.09 33.85 31.41 28.46 25.45 71.88 68.11 60.70 52.73 45.55
Mode sequence number 3 4 13.45 20.81 13.08 20.48 12.44 19.78 11.63 18.82 10.74 17.72 38.48 89.60 37.25 84.82 34.81 75.40 31.83 64.97 28.81 55.30 105.9 168.5 97.89 153.1 84.26 126.4 71.06 103.2 60.21 85.77
5 33.39 32.06 29.69 26.84 23.88 93.12 87.08 76.15 65.53 56.93 191.8 171.3 139.4 112.8 93.05
6 38.50 37.57 35.48 32.82 30.03 107.8 100.5 87.02 73.95 63.21 263.1 226.4 175.7 138.0 112.2
Sec. 4.5 Benchmark Checks
124
Table 4.8 Frequency parameters 2 of FCF isosceles triangular plates
/~0 30
60
90
h/b
0110 0.15 0.20 O.OOl 0.05 0.10 0.15 0.20 O.OOl 0.05 0.10 0.15 0.20
1 6.352 6.302 6.200 6.066 5.906 8.922 8.822 8.646 8.413 8.137 12.33 12.15 11.83 11.42 10.93
2 17.39 17.05 16.38 15.51 14.56 35.09 33.85 31.41 28.46 25.45 46.91 44.91 41.12 36.67 32.20
Mode sequence number 3 4 30.50 35.04 29.69 34.09 27.97 32.08 25.81 29.61 23.54 27.09 38.48 89.60 37.25 84.82 34.81 75.40 31.83 64.97 28.81 55.30 65.35 112.3 61.25 104.0 53.90 89.47 46.17 75.21 39.39 62.96
5 57.76 55.56 50.96 45.72 40.74 93.12 87.08 76.15 65.53 56.93 152.9 139.3 116.1 94.62 76.84
6 62.81 59.97 54.24 47.92 42.14 107.8 100.5 87.02 73.95 63.21 198.6 175.3 140.1 110.6 88.40
Chap. 4 Formulation in Rectangular Coordinates
125
Table 4. 9 Frequency parameters A of FCFF trapezoidal plates c/a
I
B/a 1.0
0.2 2.0
1.0 0.4 2.0
1.0 0.6 2.0
1.0 0.8 2.0
h/a 0.001 0.10 0.20 0.001 0.10 0.20 0.001 0.10 0.20 0.001 0.10 0.20 0.001 0.10 0.20 0.001 0.10 0.20 0.001 0.10 0.20 0.001 0.10 O.20
I
!
1 5.148 5.053 4.866 1.306 1.294 1.278 4.418 4.356 4.215 1.076 1.104 1.093 3.987 3.938 3.822 0.964 0.991 0.981 3.701 3.649 3.549 0.892 0.913 0.905
2 22.15 20.29 17.17 6.183 6.065 5.803 16.66 15.46 13.46 5.808 5.708 5.470 12.89 12.08 10.73 5.603 5.322 4.874 10.33 9.733 8.778 4.500 4.283 3.972
Mode sequence number 3 4 24.39 57.68 22.86 49.98 19.91 39.08 9.544 15.87 8.903 15.26 7.848 13.95 23.05 45.84 21.73 40.63 18.99 33.03 7.251 15.46 6.817 14.89 6.136 13.63 22.34 38.87 21.10 34.89 18.45 28.94 5.609 15.27 5.513 14.54 5.287 13.00 21.82 34.27 20.62 31.05 18.03 26.06 5.464 13.53 5.378 12.77 5.158 11.57
5 61.85 54.56 43.15 23.37 21.25 17.91 60.63 53.68 42.56 18.23 17.12 14.98 49.02 44.45 36.84 15.52 14.73 13.48 36.09 33.43 28.76 15.15 14.61 13.38
78.73 67.04 51.42 30.29 28.39 24.62 64.34 56.40 44.68 29.91 28.07 24.35 60.36 53.38 42.26 28.20 25.99 22.44 59.47 52.26 41.53 25.26 23.46 20.50
Sec. 4.5 Benchmark Checks
126
Table 4.10 Frequency parameters • of S*FS*F trapezoidal plates c/a
b/a 1.0
0.2 2.0
1.0 0.4 2.0
1.0 0.6 2.0
1.0 0.8 2.0
h/a 0.001 O. 10 0.20 0.001 0.10 0.20 0.001 0.10 0.20 0.001 0.10 0.20 0.001 0.10 0.20 0.001 0.10 0.20 0.001 0.10 0.20 0.001 0.10 0.20
!
i
i
I
I
1 14.66 13.86 12.60 12.92 12.36 11.40 13.80 13.13 12.01 12.42 11.91 11.03 12.79 12.28 11.34 11.83 11.40 10.61 12.80 12.40 11.55 12.36 11.96 11.15
2 48.20 42.96 35.32 28.83 26.62 23.13 37.21 33.99 28.85 24.42 22.78 20.09 26.64 24.54 21.31 19.92 18.86 16.96 21.88 20.08 17.52 16.86 16.12 14.72
Mode sequence number 3 4 86.21 91.23 67.42 77.22 46.66 59.12 46.32 66.73 41.70 58.41 34.72 44.88 50.41 63.40 45.35 54.61 37.33 43.06 36.64 46.19 33.57 42.09 28.66 35.18 46.93 47.66 40.63 43.24 32.99 35.95 26.70 36.14 24.83 32.50 21.77 27.34 43.01 49.41 37.92 44.96 31.27 37.36 22.69 33.07 21.00 29.98 18.42 25.48
5 114.1 94.18 69.49 79.36 63.65 46.69 98.46 80.55 59.94 49.61 44.67 37.02 80.11 69.24 53.88 44.55 40.82 34.32 67.36 59.47 47.49 47.28 42.71 35.25
146.9 116.1 74.64 90.16 76.59 58.92 99.84 82.56 62.16 63.71 55.56 44.16 84.76 71.13 54.85 50.62 44.60 36.45 81.56 69.62 54.10 47.89 43.30 36.23
Chap. 4 Formulation in Rectangular Coordinates
127
Table 4.11 Frequency parameters A of S*CS*C trapezoidal plates c/a
b/a 1.0
0.2 2.0
1.0 0.4 2.0
1.0 0.6 2.0
1.0 0.8 2.0
h/a 0.001 0.10 0.20 0.001 0.10 0.20 0.001 0.10 0.20 0.001 0.10 0.20 0.001 0.10 0.20 0.001 0.10 0.20 0.001 0.10 0.20 0.001 0.10 0.20
1 45.84 40.76 33.15 26.60 24.82 21.64 38.21 34.29 28.39 22.51 21.22 18.85 33.60 30.34 25.25 18.82 17.89 16.14 30.78 27.94 23.29 15.87 15.14 13.81
2 92.99 76.59 57.03 44.67 40.23 33.23 80.11 66.73 50.00 35.69 32.63 27.68 74.33 62.43 46.79 29.45 27.13 23.35 63.52 55.88 44.65 25.84 23.95 20.76
Mode sequence number 3 4 108.6 155.6 89.10 118.5 65.60 82.44 65.47 74.77 57.04 65.09 45.10 51.00 91.23 140.1 77.03 107.9 58.44 74.96 51.62 66.85 45.77 59.06 37.20 47.12 75.78 118.0 65.66 95.00 51.23 69.24 44.51 58.93 39.85 52.89 32.72 43.02 71.25 103.4 60.19 84.26 45.15 62.04 40.88 50.80 36.89 46.34 30.47 38.52
5 180.5 136.0 93.31 89.28 75.27 57.26 143.7 113.0 80.36 71.75 61.61 47.99 134.1 104.1 72.26 64.47 56.03 43.97 122.0 100.7 70.94 60.85 53.31 42.04
196.2 146.7 100.1 107.6 89.35 66.48 170.3 131.6 91.99 91.29 77.66 59.34 145.5 116.3 83.44 75.52 65.91 51.85 131.1 102.2 74.39 61.44 54.71 46.12
Sec. 4.5 Benchmark Checks
128
Table 4.12 Frequency parameters 2 of S*S*CC trapezoidal plates
c/a
0.2
0.4
0.6
0.8
h/a 0.001 O. 10 1.0 0.20 0.001 0.10 2.0 0.20 0.001 0.10 1.0 , 0.20 0.001 0.10 2.0 1 0.20 0.001 0.10 1.0 0.20 0.001 0.10 2.0 0.20 0.001 0.10 1.0 0.20 0.001 0.10 2.0 0.20
b/a
1 48.51 42.76 34.16 31.83 29.07 24.33 40.21 36.02 i 29.48 28.07 25.87 , 21.94 34.03 30.84 25.74 24.39 22.69 19.52 29.86 27.28 23.07 20.85 19.58 i 17.11
2 94.75 78.29 58.06 51.22 45.24 36.07 76.78 65.16 49.88 42.13 37.83 30.83 67.92 56.79 45.43 34.13 31.12 26.00 63.33 55.04 43.06 28.56 26.33 22.46
Mode sequence number 3 4 113.5 154.0 91.29 118.9 65.85 83.13 73.20 83.57 62.54 70.37 47.84 52.97 98.21 130.3 80.80 103.7 59.48 74.27 57.65 75.90 50.47 64.78 39.82 49.62 83.75 121.3 70.50 97.64 53.07 70.37 46.80 64.34 41.79 56.17 34.02 44.33 70.87 103.7 60.95 84.82 46.94 62.52 41.16 59.01 37.21 52.13 30.83 41.66
5 186.2 138.6 94.25 98.14 81.05 59.81 151.0 116.8 81.68 76.34 65.16 49.73 122.1 98.02 70.62 69.05 59.74 46.10 117.1 95.13 68.86 61.31 53.85 42.20
202.7 148.0 99.40 117.9 95.00 68.55 179.8 134.8 91.92 98.27 81.68 61.03 157.6 121.5 84.26 86.77 73.14 54.86 135.5 107.5 76.15 71.63 61.79 47.56
Chap. 4 Formulation in Rectangular Coordinates
129
Table 4.13 Frequencyparameters A of S'S'S'S* trapezoidalplates
c/a
b/a 1.0
0.2 2.0
1.0 0.4 2.0
1.0 0.6 2.0
1.0 0.8 2.0
h/a 0.001 0.10 0.20 0.001 0.10 0.20 0.001 0.10 0.20 0.001 O. 10 0.20 0.001 0.10 0.20 0.001 0.10 0.20 0.001 0.10 0.20 0.001 0.10 0.20
1 37.74 34.31 29.15 23.61 22.14 19.69 30.80 28.15 24.39 20.44 19.31 17.39 25.64 23.54 20.67 17.38 16.54 15.08 22.12 20.43 18.11 14.60 13.96 12.86
2 79.67 68.05 53.22 40.70 37.08 31.37 63.90 55.39 44.59 32.76 30.22 26.13 55.96 49.34 40.34 26.33 24.43 21.48 51.82 46.26 38.19 22.23 20.71 18.42
Mode sequence number 3 4 96.57 135.0 81.30 107.8 62.03 78.79 60.56 69.81 53.57 61.30 43.37 48.90 82.88 113.4 71.25 93.24 55.51 69.87 47.05 63.33 42.25 56.26 35.17 45.42 69.99 105.1 61.34 87.65 48.90 66.10 38.81 56.04 35.19 49.76 29.86 40.71 58.48 88.97 52.08 74.96 42.46 56.71 34.56 49.73 31.65 45.41 27.20 37.91
5 164.5 128.4 91.04 83.44 71.57 55.68 132.1 106.8 78.10 64.72 56.55 45.48 105.7 88.09 66.66 56.72 51.06 41.88 101.1 85.58 65.09 51.82 46.56 38.50
6 179.7 138.3 96.89 101.5 85.45 64.84 158.7 125.2 89.28 86.90 73.89 57.28 138.4 112.0 81.43 72.95 64.03 50.85 117.9 98.08 73.07 59.45 53.20 43.35
Sec. 4.5 Benchmark Checks
130
Table 4.14 Frequency parameters 2 of CCCC trapezoidal plates B/a
c/a
1.0 0.2 2.0
1.0 0.4 2.0
1.0 0.61 2.0
1.0 0.8
!
2.0
h/a 0.001 0.10 0.20 0.001 0.10 0.20 0.001 0.10 0.20 0.001 0.10 0.20 0.001 0.10 0.20 0.001 0.10 0.20 0.001 0.10 0.20 0.001 0.10 0.20
1 71.25 59.21 43.42 45.74 40.20 31.37 57.96 49.59 37.57 40.14 35.75 28.37 47.50 41.71 32.65 34.72 31.33 25.33 40.49 36.18 29.03 29.41 26.91 22.20
2 125.0 95.76 65.53 68.24 57.46 42.69 97.89 78.85 56.26 56.40 48.62 37.05 83.82 69.49 50.91 45.52 40.16 31.54 77.09 64.72 48.02 37.03 33.40 27.11
Mode sequence number 3 4 146.5 190.6 109.3 136.0 73.20 89.10 93.18 104.9 75.52 83.69 54.14 58.87 125.0 154.0 96.45 116.3 66.16 78.92 73.83 93.12 61.81 75.84 45.72 54.93 105.0 140.9 83.75 108.5 58.99 74.64 58.16 75.40 50.37 63.96 38.86 48.34 87.27 122.1 71.88 96.07 52.14 67.42 49.45 67.54 43.82 58.41 34.83 45.04
5 227.6 155.8 99.46 121.1 94.50 65.72 183.4 132.2 86.83 95.19 77.22 55.10 146.1 110.8 75.46 85.51 70.56 51.13 135.1 104.9 72.51 75.46 63.46 46.75
245.7 165.9 105.4 142.8 108.3 73.51 215.0 150.5 97.33 115.4 91.80 65.35 186.2 135.0 88.97 98.21 81.18 59.69 158.0 118.9 80.24 86.46 71.44 51.92
Chap. 4 Formulation in Rectangular Coordinates
hlb
alb
1
1 I 2
3
alb 1
2
0.001 0.05 0.10 0.15 0.001 0.05 0.10 0.15 0.001 0.05 0.10 0.15
4 21.43 21.11 20.46 19.58 6.668 6.636 6.568 6.468 3.000 2.992 2.980 2.956
5 21.43 21.11 20.46 19.58 10.54 10.39 10.15 9.836 6.936 6.824 6.668 6.472
Mode sequences 6 7 49.72 36.01 35.47 48.24 45.28 34.03 32.05 41.64 16.92 22.01 16.74 21.61 20.86 16.33 19.87 15.76 7.688 13.68 7.648 13.43 13.04 7.556 7.424 12.54
8 49.72 48.24 45.28 41.64 27.76 27.36 26.40 25.09 14.47 14.34 14.04 13.61
9 81.88 78.84 71.96 64.00 31.51 30.94 29.65 27.94 21.76 21.32 20.54 19.53
4 19.74 19.58 19.11 18.42 13.20 13.10 12.84 12.46 12.04 11.93 11.70 11.38
5 55.56 54.04 50.52 46.12 23.64 23.25 22.43 21.34 17.82 17.56 17.04 16.34
Mode sequences 6 7 102.4 55.56 54.04 97.28 50.52 86.84 75.76 46.12 38.32 46.12 45.08 37.43 35.51 42.56 39.30 33.06 34.57 25.28 33.75 24.79 23.81 32.07 29.95 22.54
8 102.4 97.28 86.84 75.76 57.48 55.68 51.80 47.12 43.76 42.84 40.56 37.57
9 118.8 113.0 100.1 86.56 62.76 60.68 56.12 50.72 45.76 44.48 41.76 38.48
hlb
0.001 0.05 0.10 0.15 0.001 0.05 0.10
3
I I
0.15 0.001 0.05 0.10 0.15
131
Sec. 4.5 Benchmark Checks
132
a/b
Table 4.17 Frequency parameters 2 for clamped (C) elliptical plates.
h/b
0.001 0.05 0.10 0.15 0.001 0.05 0.10 0.15 0.001 0.05 0.10 0.15
4 40.84 39.77 37.00 33.48 27.38 26.81 25.31 23.33 25.24 24.74 23.39 21.60
5 85.00 80.72 71.12 60.92 39.50 38.42 35.72 32.35 31.82 31.05 29.08 26.55
Mode sequences 6 7 85.00 139.5 80.72 128.9 71.12 108.2 60.92 89.16 55.96 69.84 54.00 66.72 49.32 59.56 43.88 51.64 40.08 50.16 38.96 48.52 36.13 44.52 32.65 39.83
8 139.5 128.9 108.2 89.16 76.96 73.60 65.92 57.48 62.20 59.84 54.28 48.00
159.0 146.0 121.1 98.88 88.04 83.36 73.12 62.48 66.68 63.80 57.08 49.60
CHAPTER FIVE
FORMULATION IN SKEW COORDINATES
5.1
INTRODUCTION
Skew (or parallelogram) plates find many applications, such as in wings of aircrafts, fins of missiles, and slabs for overhead bridges that are orientated at an angle over a highway. For the analysis of such a plate shape, it is more convenient to use the skew coordinate system. For example, the hard type S-simple support can be readily implemented in the p-version Ritz method for the oblique edges which would not be possible if the rectangular coordinate system was used without modifications. Kanaka Raju and Hinton (1980) presented vibration results for rhombic Mindlin plates with various combinations of simply supported and clamped edges. A rhombic plate is a special case of the skew plate with equal side lengths. They used the finite element method with a nine-noded Lagrangian quadrilateral isoparametric plate element in their analysis. A similar problem was solved by Ganesan and Nagaraja Rao (1985) using a variational approach. More recently, Liew, Xiang, Kitipornchai and Wang (1993) used the Ritz method for the vibration analysis of such plates and generated extensive genetic frequency results. This p-version Ritz method as formulated in skew coordinates system is presented below.
5.2
SKEWCOORDINATES TRANSFORMATION
Consider the skew Mindlin plate with length a, oblique width b and skew angle/3 as shown in Fig. 5.1. Defining the rectangular coordinate system (x,y) and the skew coordinate system (~,y) as depicted in the figure, it is clear from simple geometry that they are related by
Y
Ay
D
~/y, ~r 9
r"
9 X,X
Vl~
Vl
Fig. 5.1 Skew coordinates and plate dimensions
133
Sec. 5.3 Energy Functional in Skew Coordiantes
134
X= x - y tan /3 y = ysec fl
(5.1a) (5.1b)
The Mindlin bending rotations Vx, gty can also be expressed in the skew coordinates as
q/x (X, y) = q/~(~, y) cos fl ~bCy(x, y) = - ~ (~, y) sinfl + ~y(Y, y)
(5.2a) (5.2b)
in which the directions of ~tz and gy are defined in Fig. 5.1. The partial derivatives between the rectangular and skew coordinates are related by
,r
o(.)
d(.)
(5.3a)
d(.) tan 13 + d(-) sec/3
(5.3b)
cy
5.3
ENERGY FUNCTIONAL IN SKEW COORDINATES
For generality and convenience, the following nondimensional terms have been adopted: 2y b ~ = 2 ~ _ 1; r/ =--if-- 1; a =--', a a
2" =
h " b-'
W =
2w " --if-'
2
= cob 2
~ph D
(5.4a-e)
Using Eqs. (5.1)-(5.4), the strain energy functional in Eq. (2.6) and the kinetic energy in Eq. (2.8) may be expressed in nondimensional skew coordinates (~,rl) as:
-
-
4
j, j
{ /
-1
sec 2fl a
d~:
- 2(1- v) a ~ &'~ 0~' a57 + 6(14r 2 -v)~ 2
--
c~
-asin/3
+
dr/
3'
1 &,~ + a &'~ Oq 0~ + ~ - a tan fl 0~
d~drI
I: Iill l---W2+ ~2(W2- 2 s i n p r 1 6 2
The Lagrangian is given by
d~:
4
cos fl ~'~ + a
- sin fl ~ + sec/3
T = 122
-sinfl
(5.5) ~
)} d~d~7
(5.6)
Chap. 5 F o r m u l a t i o n in S k e w C o o r d i n a t e s
135
FI=U -T
5.4
(5.7)
EIGENVALUE EQUATION
For the Mindlin plate vibration problem, the p-version Ritz functions in skew coordinates for approximating the displacement field are defined as Pl q
w(~, 7?) = ~ ~ c~r 2 (~, r/) q=Oi=o P2 q ~x (~, 17) = Z Z dm~X (~' 17) q=o i=o p3 q ~ry (~, 17) = Z Z e m ~y (~'/7)
(5.8a) (5.8b)
(5.8C)
q=0 i=0
where Ps, s = 1,2,3 is the degree set of the complete polynomial space; ci, d i and e i are the unknown coefficients, the subscript m is determined by m=
(q + 1)(q + 2) -i 2
(5.9)
and the total numbers of ci, d i and e i are Nk, k=-1,2,3 which are dependent on the degree set of polynomial space Ps, given by Nk =
(Pk + 1)(Pk + 2) 2
(5.10)
The functions ~b,~,~bx , ~bm y consist of
<~ (~,]7), r (~,]7), ~Y (~,JT) I "- ~i]Tq-i <~1 (~,]7), ~; (~,]7), ~( (~,]7) I
(5.11)
The basic functions ~b~w, ~bl, ~by in Eq. (5.11) are given as follows:
(Ow(~, ~), Ox(4, e), Oy(4, v)) = (5.12)
where the equations of the skew edges AB, BC, CD and DA are, respectively, given by Z, = (r/+ 1), Z2 = (~ - 1), Z3 = (r/- 1), Z4 = (~ + 1) The power ,/j in Eq. (5.12) is dependent on the supporting edge condition, where
(5.13a-d)
Sec. 5.4 Eigenvalue equation
136
w {~ free(F); 7"j = simply supported (S and S*) or clamped (C)
(5.14a)
x {~ free (F)or simply supported (S*)or (S)in the y-direction; 7'; = simply supported (S) in the x - direction or clamped (C)
(5.14b)
= ~0 free (F)or simply supported (S*)or (S)in the x-direction; simply supported (S) in the y - direction or clamped (C)
(5.14c)
7'Y
Applying the Ritz method,
an
an
an) =(o, o, o)
(5.15)
~c~' Odm ' Oem
where m= 1, 2, ..., N k. The substitution of Eqs. (5.5)-(5.8) into Eq. (5.15) yields the following eigenvalue equation
IIKCCeKee . IMcc eMCellf! M :fit
(5.16)
where the elements of the stiffness submatrices [K] and mass submatrices [M] are: 2 sec/3 ~ f l l k~i - 6(1-v)x24r r
D
O ~O~b/w O~:~ ~O~: cT~b~ b ~+ a1ac~bw dr/ / dr/
c3~ ----~-+~~cgrI 0~
d~drl;
(5.17a)
i = 1, 2,..., N1; j = 1,2 ..... N~ k cd = 6(1-2 4v)tr r 2 ~ - ' ; 110~bw x 0~: L ~o;-~lsinfl 1 d~bw~.~bjX] , d~dq ; tj
(5.17b)
i = 1, 2,..., N~; j = 1, 2,..., N 2 k~i = ce
6(a--v)K2f1~l[a63qjW 0~ w Y] d~drl; 4r 2 ~--~--~jy -sin fl---~-qkj (5.17c)
i = 1,2 ..... N1; j = 1, 2,..., N3 k~/ =secfl
a c3~: c3~:
F1a cos 2/3 tan2fl+
6 ( c~7 1 - 4v&bx&b}) c~7 ) a d-~ : z +d 1- ~ -sinp /&bX&b} -7-2a 4/.2
dr/ dr/
COS2/~ ~;~; 1 d~dT];
Chap. 5 Formulation in SkewCoordinates
137
i = 1,2,...,N2; j = 1,2 ..... N 3
k~i=cos,B~_,~_lI(tan2fl+v)OqkxOqkf --~---+#q -
(5.17d)
tanzfl+--~cgqcT~l-v) cTqkxOqky
( Oqkx O~bfld~b x O~bf.) 16(1-v)tC2sinfl#i#jld~d~7 ' + x y .
tan fl sec fl a
d~:
d~:
O: Or] Or]
or
4"t"2
i = 1 , 2 ..... N2; j = l , 2 ..... N 3
ku sec,b,~ ;1 [ 1 63~bYcT~bY+ a cos ee =
--
_
fl tan fl + -l-v)8~ -~ d~:yd~f d~:
2 ( 2
sin fl( ~O~b[~ O~bf + ~Oqk[~ Oqkf ) + 16 (1- v)K2 cos : ~ O4
~
~
04
(5.17e)
a
~y0y
4r ~
J d4a,7;
i = 1, 2,..., N3; j = 1, 2,..., N 3
cc = 16or 1 cosp f
m~/
mo.ca=0", ce = O"
mU
dd =
mu
1
(5.18a)
i = 1, 2,..., N~; j = 1, 2,..., N 2
(5.18b)
i=1, 2,..., N1; j = l ,
(5.18c)
~1 ; ~
ae - - - - ~ sin fl cos fl mu - 48a i=1, muee = ~COS480~ fl
7 0 7 dCd~;
i = 1, 2,..., N~; j = 1, 2,..., N~
~.2
~cosfl 48c~
(5.170
2,..., N 3
#X#~d~d~7;
i=1, 2,..., N2; j = l ,
~i xOjy d ~ d 1 7 ,
2,...,N2;
j=l,
2,..., N 2
(5.18d)
.
2,..., N 3
(5.18e)
2,..., N 3
(5.180
~b,.Y~bfd~d~7; i=1, 2,..., N3; j = l ,
The resulting eigenvalue Eq. (5.16) may be solved using any standard eigenvalue solver for the respective natural frequencies (eigenvalues) and mode shapes (eigenvectors).
5.5
COMPUTER
PROGRAM
The computer program VPRITZSK, as given below, has been developed on the basis of the foregoing skew coordinate formulation. It can be used to compute the natural frequencies of skew plates with any combination of boundary conditions. It should be noted that this program cannot handle mixed boundary conditions on the same plate edge without modification. Users may refer to Section 6.8 of Chapter 6 for the details of modification.
138 5.5.1
* *
Sec. 5.5 Computer Program SOFTWARE
p-VERSION
CODE: VPRITZSK
RITZ METHOD FOR VIBRATION OF MINDLIN PLATES BASED ON SKEW COORDINATES SYSTEM
MAIN VPRITZSK IMPLICIT DOUBLE PRECISION (A-H,O-Z) P A R A M E T E R (NDEGREE=20,NTERM=(NDEGREE+ 1)*(NDEGREE+2)/2, $ N T O T A L = 3 * N T E R M , N 1=NTERM,N2=NTOTAL) DIMENSION S(24,N 1,N 1),COEF(65,65) DOUBLE PRECISION K(N2,N2),KI(N2,N2),W(N2),Z(N2,N2),FV 1(N2), $ FV2(N2) DOUBLE PRECISION LX,LY,H,BETA,E,NU,RHO,KAPPA INTEGER P0,P 1,P2 * * * * * * * * * * * * * * * * * * * * *
N D E G R E E = the number of degree of polynomial terms used in the Ritz functions NTERM = the number of polynomial terms; NTOTAL = 3*NTERM = the total number of degrees of freedom; N1 = NTERM; N2 - NTOTAL. S(13,N 1,N 1) stores the integrated values generated by by subroutine BASICINT; K(N2,N2) = linear stiffness matrix; K1 (N2,N2) = mass matrix; W(N2) = frequency parameter; Z(N2,N2), FV 1(N2), FV2(N2) are working matrices required by EISPACK; LX, LY = plate dimensions along x and y; H = plate thickness; BETA = the skew angle; E = Modulus of elasticity (any relative number can be taken); NU = Poisson ratio; RHO = plate density per unit volume; KAPPA = shear correction factor; P0 = lower value of degree of polynomials; P2 = upper value of degree of polynomials; P 1 = increment step of degree of polynomials; OPEN(30,FILE='INPUT.DAT') OPEN( 15 ,FILE='OUTPUT.DAT ') WRITE(15,*)
$
WRITE(15,*) ' p-Version Ritz Method for Vibration of Mindlin Plates' WRITE(15,*)
* *
Chap. 5 Formulation in Skew Coordinates
WRITE(15,*) $
***************************************************************
WRITE(15,*) WRITE(15,*) WRITE(6,*)'Read input data from file INPUT.DAT' WRITE(6,*) READ(30,*)E WRITE(15,'(1X,"Modulus of elasticity E = ",F 13.5)')E READ(30,*)RHO WRITE(15 ,'( 1X,"Plate density per unit volume RHO - ",F 11.5)')RHO READ(30,*)NU WRITE(15,'(1X,"Poisson ratio NU = ",F8.5)')NU READ(30,*)KAPPA WRITE(15,'(1X,"Shear correction factor KAPPA = ",F8.5)')KAPPA READ(30,*)LX WRITE(15,'(1X,"Maximum length of plate a = ",F8.5)')LX READ(30,*)LY WRITE(15,'(1X,"Maximum width of plate b - ",F8.5)')LY READ(30,*)H WRITE(15,'(1X,"Plate thickness h = ",F8.5)')H READ(30,*)BETA WRITE(15,'(1X,"Plate skew angle BETA (in degree)= ",F8.5)')BETA READ(30,*)P0,P2,P 1 WRITE(15,'(1X,"Degree of polynomials changes from p =",I3," to", I3," with step",I3)')P0,P2,P 1 WRITE(15,*) !
WRITE(15,*) WRITE(15,*) WRITE(15,'(1X,"Plate length to width ratio a/b = ",F8.5)') LX/LY WRITE(15,'(1X,"Plate thickness to length ratio h/b = ",F8.5)') H/LY WRITE(15,'(1X,"Plate skew angle BETA = ",F8.5)') BETA WRITE(15,*) !
WRITE(15,*) WRITE(15,*) WRITE(15,*)
!
139
Sec. 5.5 Computer Program
140
NMAX=(P2+ 1)* (P2+2)/2 NTAB=65 CALL STANDVB 1(E,RHO,NU,KAPPA, LX,LY,H,BETA,P0,P 1,P2,S,COEF, NTAB,NMAX,K,K 1,W,Z,FV 1,FV2)
CLOSE(15) CLOSE(30) WRITE(6,*)'End of running the program' STOP END
SUBROUTINE STANDVB 1(E,RHO,NU,KAPPA,LX,LY,H,BETA,P0,P 1,P2,S, COEF,NTAB,NMAX,K,K 1,W,Z,FV 1,FV2) * This subroutine STANDVB 1 is the main subroutine to calculate the frequency * parameters. IMPLICIT DOUBLE PRECISION (A-H,O-Z) INTEGER P0,P 1,P2 DIMENSION S(13,NMAX,NMAX),COEF(NTAB,NTAB) DOUBLE PRECISION K(NMAX*3,NMAX*3),K1 (NMAX*3,NMAX*3), W(NMAX*3),Z(NMAX*3,NMAX*3), FV 1(NMAX* 3),FV2(NMAX* 3) DOUBLE PRECISION LX,LY,KAPPA,NU MNL=NMAX WRITE(6,*)'Generate the basic integrated matrix' WRITE(6,*) CALL BASICINT(MNL,NTAB,S,COEF)
10
DO 20 ICASE=P0,P2,P1 M=(ICASE+ 1)* (ICASE+2)/2 WRITE(6,'(1X,"Solve eigenvalue equation for p = ",I3)')ICASE WRITE(6,*) WRITE(15,'(1X,"The degree of polynomials p = ",I3)')ICASE N=M L=M D=E*H** 3/12.0/(1.0-NU**2) CALL STFF(M,N,L,MNL,LX,LY,H,E,NU,KAPPA,D, S,K,K 1,W,Z,FV 1,FV2,RHO,BETA) DO 10 ITR-1,6 W(ITR)-D SQRT (W(ITR)* (LY)**4" (RHO* H/D)) CONTINUE
Chap. 5 Formulation in Skew Coordinates
20
WRITE(15,'(1X,"The first six frequency parameters are:")') WRITE(15,'(1X,6F 13.3)') (W(ITR),ITR= 1,6) WRITE(15,*) CONTINUE RETURN END
SUBROUTINE CLEAR(A,M,N) * This subroutine CLEAR initializes the two dimensional array (A). IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION A(M,N)
10 20
DO 20 I=I,M DO 10 K=I,N A(I,K)=0.0 CONTINUE CONTINUE RETURN END
SUBROUTINE STFF(M,N,L,MNL,LXX,LYY,H,E,V,COEF,D, SW,K,K 1,W,Z,FV 1,FV2,RHO,B ETA) * This subroutine STFF forms the linear stiffness and mass matrices and solves the * generalized eigenvalue problem using EISPACK. IMPLICIT DOUBLE PRECISION (A-H,K,O-Z) DIMENSION K(M+N+L,M+N+L),K1 (M+N+L,M+N+L), SW(Z4,MNL,MNL) DIMENSION W(M+N+L),Z(M+N+L,M+N+L),FV 1(M+N+L),FV2(M+N+L) DOUBLE PRECISION LX,LY,LXX,LYY LX=LXX/2.0 LY=LYY/2.0 CT=BETA* 3.141592654/180.0 MI=M+N+L CALL CLEAR(K,M 1,M 1) CALL CLEAR(K1,M1,M1) H 1-COEF* E* H/2.0/(1.0+V)
141
142
Sec. 5.5 Computer Program
10 20
DO 20 I=I,M DO 10 J=I,M K(I,J)=L Y/LX *DCOS(CT) *H 1/(DCOS(CT))* *2 * SW( 1,I,J)+ LX/LY/DCOS(CT)*HI*SW(2,I,J)+ DTAN(CT)* (-H 1)* (SW(3,I,J)+S W(4,I,J)) CONTINUE CONTINUE
30 40
DO 40 I=I,M DO 30 J=I,N K(I,J+M)=LY*H 1* SW(5,I,J)-LX*D S1N(CT)*H 1* SW(6,I,J) CONTINUE CONTINUE
50 60
DO 60 I= 1,M DO 50 J=I,L K(I,J+M+N)=-LY*D SIN(CT)*H 1* SW(7,I,J)+LX*H 1* SW(8,I,J) CONTINUE CONTINUE
70 80
DO 80 I=I,N DO 70 J=I,N K(I+M,J+M)=LY/LX*D/DCOS(CT)*SW(9,I,J)+ LX/LY*D*DCOS(CT)*((DTAN(CT))**2+(1.0-V)/2.0)* SW(10,I,J)-D*DTAN(CT)* (SW(11 ,I,J)+ SW(12,I,J))+LX*LY*DCOS(CT)*H 1*SW(13,I,J) CONTINUE CONTINUE
90 100
DO 100 I=I,N DO 90 J=I,L K(I+M,J+M+N)=-LY/LX*D*DTAN(CT)* SW(14,I,J)+ D*DCOS(CT)*(1.0/(DCOS(CT))**2-(1.0-V))* SW(15,I,J)+D*DCOS(CT)*((DTAN(CT))**2+ ( 1.0-V)/2.0)* SW( 16,I,J)-LX/LY* D* DTAN(CT)* SW(17,I,J)-LX*LY*DSIN(CT)*DCOS(CT)*H 1* SW(18,I,J) CONTINUE CONTINUE
$ $ $ $ 110 120
DO 120 I=I,L DO 110 J=I,L K(I+M+N,J+M+N)=LX/LY*D/DCOS(CT)* SW(19,I,J)+ LY/LX*D*DCOS(CT)*((DTAN(CT))**2+ ( 1.0-V)/2.0)* SW(Z0,I,J)-D*DTAN(CT)* (SW(21 ,I,J)+SW(ZZ,I,J))+LX* LY* DCOS(CT)*Hl*SW(23,I,J) CONTINUE CONTINUE
$ $ $ $ $
Chap. 5 Formulation in Skew Coordinates
DO 140 1= 1,M DO 130 J=I,M K1 (I,J)=LX*LY*DCOS(CT)*H*RHO*SW(Z4,I,J) 130 CONTINUE 140 CONTINUE DO 160 I=I,N DO 150 J=I,N KI(I+M,J+M)=LX*LY*DCOS(CT)*(H**3*RHO/12.0)*SW(13,I,J) 150 CONTINUE 160 CONTINUE DO 180 I= 1,N DO 170 J=I,L K1 (I+M,J+M+N)=-LX*LY*DSIN(CT)*DCOS(CT)* $ (H**3 *RHO/12.0)* SW(18,I,J) 170 CONTINUE 180 CONTINUE DO 200 I=I,L DO 190 J=I,L K1 (I+M+N,J+M+N)=LX*LY*DCOS(CT)*(H**3*RHO/12.0)* $ SW(Z3,I,J) 190 CONTINUE 200 CONTINUE DO 220 I=I,M1 DO 210 J=I,M1 KI(J,I)=KI(I,J) K(J,I)=K(I,J) 210 CONTINUE 220 CONTINUE CALL RSG(M 1,M 1,K,KI,W,0,Z,FV 1,FV2,IERR) * Call subroutine RSG from EISPACK to solve the eigenvalue equation. IF(IERR.GT.0) THEN WRITE(6,*)'Matrices are ill-conditioned in the eigenvalue function' STOP END IF RETURN END
SUBROUTINE BASICINT(NTERM,NTAB,S,COEF)
143
Sec. 5.5 Computer Program
144
* This subroutine BASICINT is to generate the integrated values.
IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION S (24,NTERM,NTERM),COEF(NTAB,NTAB) COMMON/C3/NSIDE
* S(13,NTERM,NTERM) is an array which stores the integrated values; OPEN(20,FILE='COEF.TAB') * COEF(NTAB,NTAB) is a matrix which stores the integrated value of the following * matrix over the plate domain: 1]
~n ,
9
A
* * * *
{ -'
{'-'n
''"
'l] m-1
... ,o
"
,
~n ~-' 9 ,
dA
~m-l,lqm-1
COEF(NTAB,NTAB) is stored in a data file COEF.TAB which can be generated from Gaussian quadrature or any mathematical software package such as MATHEMATICA (Wolfram 1991) and MAPLE V (Char et al. 1992). A program GAUSSIAN.F that does the integrations is given in Appendix I.
* NSIDE = number of edges of a plate; READ(20,*)((COEF(I,J),J =I,NTAB),I =I,NTAB) CLOSE(20) CALL POLYCOMP * Call POLYCOMP to form the polynomial terms. MNL=NTERM M=NTERM N=M L=M READ(30,*)NSIDE CALL S S S(M,N,L,MNL,S,NTAB,COEF) * Call subroutine SSS to form the S(13,NTERM,NTERM) matrix. RETURN END
Chap. 5 Formulation in Skew Coordinates
SUBROUTINE FF 1XY(FXY,IR, IS,N) * This subroutine FF 1XY forms the boundary conditions IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION FXY(*),IR(*),IS (*),A2(10),IR2(10),IS2(10) COMMON/C3/NSIDE READ(30,*)N1 DO 10 I=I,N1 READ(30,*)FXY(I),IR(I),IS(I) CONTINUE
10
DO 30 J=I,NSIDE-1 READ(30,*)N2 DO 20 I=I,N2 READ(30,* )AZ(I),IR2(I),IS 2(I) CONTINUE CALL MULTI(FXY,IR,IS,N1,A2,IR2,IS2,N2,NM) N1 =NM CONTINUE
20
30
N=N1 RETURN END
$ $ $ $
SUBROUTINE CALCULUS (FXYM,IRM,ISM,NM, FXYN,IRN,ISN,NN, FXYSI,IRSI,ISSI,NSI, FXYSJ,IRSJ,ISSJ,NSJ, II,SUM,NTAB,COEF)
* This subroutine CALCULUS performs the differentiation and integration. For further * details, refer to Xiang, Wang and Kitipomchai (1995). IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION FXYM(*),IRM(*),ISM(*),FXYN(*),IRN(*),ISN(*), FXYSI(*),IRSI(*),ISSI(*),FXYSJ(*),IRSJ(*),ISSJ(*), FXYSII( 100),IRSII( 100),ISSII(100), FXYS JJ( 100),IRSJJ( 100),ISSJJ(100) DIMENSION COEF(NTAB,NTAB) NSII=I NSJJ=I FXYSII(1)=I.O
145
Sec. 5.5 ComputerProgram
146 IRSII(1)=0 ISSII(1)-0 FXYSJJ(1)=I.0 IRSJJ(1)=0 IS SJJ( 1)=0
CALL MULTI(FXYSII,IRSII,IS SII,NSII,FXYM,IRM,ISM,NM,N1) CALL MULTI(FXYSJJ,IRSJJ,IS SJJ,NSJJ,FXYN,IRN,ISN,NN,N2) CALL MULTI(FXYSII,IRSII,IS SII,N1 ,FXYSI,IRSI,IS SI,NSI,N 11) CALL MULTI(FXYSJJ,IRSJJ,IS SJJ,N2,FXYSJ,IRSJ,ISSJ,NSJ,N22) NI=N11 N2=N22 IF(II.EQ. l) THEN CALL DIFXY(FXYSII,IRSII,ISSII,N l, 1,0,N3) CALL DIFXY(FXYSJJ,IRSJJ,ISSJJ,N2,1,0,N4) GOTO 10 ENDIF IF(II.EQ.2) THEN CALL DIFXY(FXYSII,IRSII,ISSII,N1,0,1,N3) CALL DIFXY(FXYSJJ,IRSJJ,ISSJJ,N2,0,1,N4) GOTO 10 ENDIF IF(II.EQ.3) THEN CALL DIFXY(FXYSII,IRSII,ISSII,N 1,1,0,N3) CALL DIFXY(FXYSJJ,IRSJJ,ISSJJ,N2,0,1,N4) GOTO 10 ENDIF IF(II.EQ.4) THEN CALL DIFXY(FXYSII,IRSII,IS SII,N 1,0,1 ,N3) CALL DIFXY(FXYSJJ,IRSJJ,ISSJJ,N2,1,0,N4) GOTO 10 ENDIF IF(II.EQ.5) THEN CALL DIFXY(FXYSII,IRSII,ISSII,N l, 1,0,N3) N4=N2 GOTO 10 ENDIF IF(II.EQ.6) THEN CALL DIFXY(FXYSII,IRSII,IS SII,N 1,0,1 ,N3) N4=N2 GOTO 10 ENDIF
Chap. 5 Formulation in Skew Coordinates
IF(II.EQ.7) THEN CALL DIFXY(FXYSII,IRSII,IS SII,N 1,1,0,N3) N4=N2 GOTO 10 ENDIF IF(II.EQ.8) THEN CALL DIFXY(FXYSII,IRSII,IS SII,N 1,0,1 ,N3) N4=N2 GOTO 10 ENDIF IF(II.EQ.9) THEN CALL DIFXY(FXYSII,IRSII,ISSII,N 1,1,0,N3) CALL DIFXY(FXYSJJ,IRSJJ,ISSJJ,N2,1,0,N4) GOTO 10 ENDIF IF(II.EQ. 10) THEN CALL DIFXY(FXYSII,IRSII,ISSII,N 1,0,1,N3) CALL DIFXY(FXYSJJ,IRSJJ,ISSJJ,N2,0,1,N4) GOTO 10 ENDIF IF (II.EQ. 11) THEN CALL DIFXY(FXYSII,IRSII,IS SII,N l, 1,0,N3) CALL DIFXY(FXYSJJ,IRSJJ,ISSJJ,N2,0,1,N4) GOTO 10 ENDIF IF(II.EQ. 12) THEN CALL DIFXY(FXYSII,IRSII,ISSII,N 1,0,1,N3) CALL DIFXY(FXYSJJ,IRSJJ,ISSJJ,N2,1,0,N4) GOTO 10 ENDIF IF(II.EQ.13) THEN N3=N1 N4=N2 GOTO 10 ENDIF IF(II.EQ. 14) THEN CALL DIFXY(FXYSII,IRSII,IS SII,N 1,1,0,N3) CALL DIFXY(FXYSJJ,IRSJJ,IS SJJ,N2,1,0,N4) GOTO 10 ENDIF IF(II.EQ. 15) THEN
147
Sec. 5.5 Computer Program
148
CALL DIFXY(FXYSII,IRSII,IS SII,N 1,1,0,N3) CALL DIFXY(FXYSJJ,IRSJJ,ISSJJ,N2,0,1,N4) GOTO 10 ENDIF IF(II.EQ. 16) THEN CALL DIFXY(FXYSII,IRSII,IS SII,N 1,0,1 ,N3) CALL DIFXY(FXYSJJ,IRSJJ,ISSJJ,N2,1,0,N4) GOTO 10 ENDIF IF(II.EQ. 17) THEN CALL DIFXY(FXYSII,IRSII,IS SII,N 1,0,1 ,N3) CALL DIFXY(FXYSJJ,IRSJJ,ISSJJ,N2,0,1,N4) GOTO 10 ENDIF IF(II.EQ. 18) THEN N3=N1 N4=N2 GOTO 10 ENDIF IF(II.EQ. 19) THEN CALL DIFXY(FXYSII,IRSII,IS SII,N 1,0,1 ,N3) CALL DIFXY(FXYSJJ,IRSJJ,IS SJJ,N2,0,1,N4) GOTO 10 ENDIF IF(II.EQ.20) THEN CALL DIFXY(FXYSII,IRSII,IS SII,N 1,1,0,N3) CALL DIFXY(FXYSJJ,IRSJJ,IS SJJ,N2,1,0,N4) GOTO 10 ENDIF IF(II.EQ.21) THEN CALL DIFXY(FXYSII,IRSII,ISSII,NI,I,0,N3) CALL DIFXY(FXYSJJ,IRSJJ,IS SJJ,N2,0,1 ,N4) GOTO 10 ENDIF IF(II.EQ.22) THEN CALL DIFXY(FXYSII,IRSII,IS SII,N 1,0,1,N3) CALL DIFXY(FXYSJJ,IRSJJ,IS SJJ,N2,1,0,N4) GOTO 10 ENDIF IF(II.EQ.23) THEN N3=N1
Chap. 5 Formulation in Skew Coordinates
N4=N2 GOTO 10 ENDIF IF(II.EQ.24) THEN N3=N1 N4=N2 GOTO 10 ENDIF 10
CALL MULTI(FXYSII,IRSII,ISSII,N3,FXYSJJ,IRSJJ,IS SJJ,N4,N5) CALL CALCU(FXYSII,IRSII,ISSII,N5,SUM,II,NTAB,COEF) RETURN END
SUBROUTINE CALCU(F,IR,IS,N, SUM,II,NTAB,COEF) * This subroutine CALCU calculates the integrated values of a polynomial function. IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION F(*),IR(*),IS(*) DIMENSION COEF(NTAB,NTAB)
10
SUM=0.0 DO 10 I=I,N IF(DABS(F(I)).LE. 1.D-100.OR.F(I).EQ.0.) GOTO 10 SUM=SUM+F(I)* C OEF(IR(I)+ 1,IS(I)+ 1) CONTINUE RETURN END SUBROUTINE POLYCOMP
* This subroutine POLYCOMP forms the polynomial terms in symbolic form. IMPLICIT DOUBLE PRECISION (A-H,O-Z) COMMON/C1/FXY(231),IRR(231),ISS(231)
10 20
DO 20 11=0,20 DO 10 I2=0,I1 K=(I1 + 1)*(I 1+2)/2-(I 1-I2) FXY(K)=1.0 IRR(K)=II-I2 ISS(K)=I2 CONTINUE CONTINUE
149
Sec. 5.5 ComputerProgram
150
RETURN END
SUBROUTINE FFXY(I,FXY,IRR,IS S,N) * This subroutine FFXY assigns the i-th term of the polynomial function to * {FXY,IRR,ISS,1 }. IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION FXY( 1),IRR( 1),IS S( 1) COMMON/C1/F(231),IR(231),IS(231) N=I FXY(1)=F(I) IRR(1)=IR(I) ISS(1)=IS(I) RETURN END
SUBROUTINE SSS(M,N,L,MNL, S,NTAB,COEF) * This subroutine SSS forms the integrated terms. IMPLICIT DOUBLE PRECISION (A-H,K,O-Z) DIMENSION S(24,MNL,MNL),FXYM(100),IRM(100),ISM(100), FXYN(100),IRN(100),ISN(100),FXYL(100),IRL(100),ISL(100), FXYSI( 100),IRSI( 100),ISSI(100), FXYS J(100),IRSJ(100),IS SJ(100) DIMENSION COEF(NTAB,NTAB) CALL FF 1XY(FXYM,IRM,ISM,NM) CALL FF 1XY(FXYN,IRN,ISN,NN) CALL FF 1XY(FXYL,IRL,ISL,NL) DO 30 I=I,M CALL FFXY(I,FXYSI,IRSI,IS SI,NSI) DO 20 J= 1,M CALL FFXY(J,FXYSJ,IRSJ,ISSJ,NSJ) DO 10 II-1,4 CALL CALCULUS(FXYM,IRM,ISM,NM, FXYM,IRM,ISM,NM, FXYSI,IRSI,ISSI,NSI, FXYSJ,IRSJ,ISSJ,NSJ, II,SUM,NTAB,COEF) S(II,I,J)=SUM
Chap. 5 Formulation in Skew Coordinates
10 20 30
CONTINUE CONTINUE CONTINUE
40 50 60
DO 60 I=I,M CALL FFXY(I,FXYSI,IRSI,ISSI,NSI) DO 50 J=I,N CALL FFXY(J,FXYSJ,IRSJ,IS SJ,NSJ) DO 40 II=5,6 CALL CALCULUS(FXYM,IRM,ISM,NM, FXYN,IRN,ISN,NN, FXYSI,IRSI,ISSI,NSI, FXYSJ,IRSJ,ISSJ,NSJ, II,SUM,NTAB,COEF) S(II,I,J)=SUM CONTINUE CONTINUE CONTINUE
70 80 90
DO 90 I= 1,M CALL FFXY(I,FXYSI,IRSI,ISSI,NSI) DO 80 J=I,L CALL FFXY(J,FXYSJ,IRSJ,IS SJ,NSJ) DO 70 II=7,8 CALL CALCULUS(FXYM,IRM,ISM,NM, FXYL,IRL,ISL,NL, FXYSI,IRSI,ISSI,NSI, FXYSJ,IRSJ,ISSJ,NSJ, II,SUM,NTAB,COEF) S(II,I,J)=SUM CONTINUE CONTINUE CONTINUE
DO 120 I=I,N CALL FFXY(I,FXYSI,IRSI,ISSI,NSI) DO 110 J-1,N CALL FFXY(J,FXYSJ,IRSJ,IS SJ,NSJ) DO 100 II=9,13 CALL CALCULUS (FXYN,IRN,ISN,NN, FXYN,IRN,ISN,NN, . FXYSI,IRSI,ISSI,NSI, FXYSJ,IRSJ,ISSJ,NSJ, II,SUM,NTAB,COEF) S(II,I,J)=SUM CONTINUE 100 CONTINUE 110 120 CONTINUE
151
152
Sec. 5.5 Computer Program
DO 150 I=I,N CALL FFXY(I,FXYSI,IRSI,IS SI,NSI) DO 140 J=I,L CALL FFXY(J,FXYSJ,IRSJ,ISSJ,NSJ) DO 130 II-14,18 CALL CALCULUS(FXYN,IRN,ISN,NN, FXYL,IRL,ISL,NL, FXYSI,IRSI,ISSI,NSI, FXYSJ,IRSJ,ISSJ,NSJ, II,SUM,NTAB,COEF) S(II,I,J)=SUM CONTINUE 130 140 CONTINUE 150 CONTINUE DO 180 I=I,L CALL FFXY(I,FXYSI,IRSI,IS SI,NSI) DO 170 J=I,L CALL FFXY(J,FXYSJ,IRSJ,ISSJ,NSJ) DO 160 II=l 9,23 CALL CALCULUS(FXYL,IRL,ISL,NL, FXYL,IRL,ISL,NL, FXYSI,IRSI,ISSI,NSI, FXYSJ,IRSJ,ISSJ,NSJ, II,SUM,NTAB,COEF) S(II,I,J)=SUM CONTINUE 160 CONTINUE 170 180 CONTINUE
$ $ $ $ 190 200
DO 200 I=I,M CALL FFXY(I,FXYSI,IRSI,IS SI,NSI) DO 190 J=I,M CALL FFXY(J,FXYSJ,IRSJ,ISSJ,NSJ) 11=24 CALL CALCULUS (FXYM,IRM,ISM,NM, FXYM,IRM,ISM,NM, FXYSI,IRSI,ISSI,NSI, FXYSJ,IRSJ,ISSJ,NSJ, II,SUM,NTAB,COEF) S(II,I,J)=SUM CONTINUE CONTINUE RETURN END
Chap. 5 Formulation in Skew Coordinates
153
* The subroutines PLUS, MULTI, DIFXY and SIMPLE are presented in Appendix II. * They perform mathematical operations on the polynomial functions (for details, refer to * Xiang, Wang and Kitipomchai, 1995b).
5.5.2 SAMPLE FILES To illustrate the use of the foregoing computer program, consider a skew plate of length a 2.0, oblique width b = 1.0, thickness h/b = 0.10, skew angle fl= 30 ~ modulus of elasticity E (=1), mass density p (=1), Poisson's ratio v=0.3 and shear correction factor tc 2 = 0.86667. Its side AB is C-clamped, side BC F-free, side CD S-simply supported and side DA C-clamped as shown in Fig. 5.2. Note that the length and width dimensions of the skew plate shown in Fig. 5.2 have been normalized by a and b, respectively. The basic functions for this plate are given by
~w = (~: + 1)1 (17 + 1) ~ (~: - 1) 1 ( / 7 - 1 )
1
~b~x = ( ~ + 1) 1 (/7 + 1) ~ (~: - 1) ~ (/7 - 1) 1 ~1y - ( ~ + 1) 1 (/7 + 1) ~ ( ~ - 1) ~ ( / 7 - 1 ) ~
Note that the basic functions are formed from the product of the equations of sides AB, BC, CD and DA, raised to appropriate powers as described by Eqs. (4.22) and (4.23). For the p-version Ritz analysis, the degree of polynomials used is p = 14. The sample input and output data files are given below for a SCCF skew plate vibration problem.
A y/ ///
BL
A
/////////////D
a
I"
Ic
D
i B
rl
"1"
Fig. 5.2 Skew plate example 5.5.2.1 Sample Input Data File INPUT.DAT 1. 1. 0.3
modulus of elasticity mass density Poisson's ratio
"1
Sec. 5.5 Computer Program
154
0.86667 2. 1. 0.10 30. 6,14,2
shear correction factor length, a oblique width, b plate thickness, h skew angle, fl (in degrees) lower value of degree of polynomials, upper value of degree of polynomials and increment step change of degree of polynomials Number of sides
2 1 .,1,0 1.,0,0
Number of terms in the boundary equation (~ + 1) 1 of side AB for ,~ The 1st number is the coefficient, 2nd the power of ~, and 3rd the power of q for the 1st term The 1st number is the coefficient, 2nd the power of V, and 3rd the power of q for the 2nd term
1 1.,0,0
Number of terms in the boundary equation (r/+ 1)~ = 1 of side BC f o r , ~'
2 1 .,1,0 -1 .,0,0
Number of terms in the boundary equation (~,-1) 1 of side CD for ,~'
2 1 .,0,1 -1 .,0,0
Number of terms in the boundary equation (r/- 1)1 of side DA for ,~'
2 1 .,1,0 1 .,0,0
Number of terms in the boundary equation (~ + 1)' of side AB for , ;
1 1.,0,0
Number of terms in the boundary equation (r/+ 1)~ = 1 of side BC for , ;
1 1.,0,0
Number of terms in the boundary equation ( ~ - 1)~ = 1 of side CD for q~i~
2 1 .,0,1 -1 .,0,0
Number of terms in the boundary equation (r/- 1)1 of side DA for , ;
2 1 .,1,0 1.,0,0
Number of terms in the boundary equation (~ + 1)' of side AB for ,Y
Chap. 5 Formulation in Skew Coordinates
1 1.,0,0
Number of terms in the boundary equation (q + 1)~ = 1 of side BC for ~Y
2
Number of terms in the boundary equation (~ -1)' of side CD for ~Y
1 .,1,0 -1 .,0,0
Number of terms in the boundary equation ( q - 1)' of side DA for ~Y 2 1 .,0,1 -1 .,0,0 5.5.2.2 Sample Output Data File
OUTPUT.DAT
p-Version Ritz Method for Vibration of Mindlin Plates
Modulus of elasticity E = 1.00000 Plate density per unit volume RHO = 1.00000 Poisson ratio NU = 0.30000 Shear correction factor KAPPA = 0.86667 Maximum length of plate a = 2.00000 Maximum width of plate b = 1.00000 Plate thickness h = 0.10000 Plate skew angle BETA (in degree) = 30.00000 Degree of polynomials changes from p = 6 to 14 with step 2
Plate length to width ratio a/b = 2.00000 Plate thickness to length ratio h/b = 0.10000 Plate skew angle BETA = 30.00000
The degree of polynomials p = 6 The first six frequency parameters are: 8.250 16.966 29.121 30.852 The degree of polynomials p = 8
40.996
48.301
15 5
156
Sec. 5.6 Benchmark Checks
The first six frequency parameters are: 8.231 16.886 28.822 30.660
40.429
43.276
The degree of polynomials p = 10 The first six frequency parameters are: 8.225 16.870 28.781 30.638
40.333
42.951
The degree of polynomials p = 12 The first six frequency parameters are: 8.223 16.865 28.773 30.632
40.311
42.923
The degree of polynomials p = 14 The first six frequency parameters are: 8.221 16.862 28.771 30.629
40.303
42.919
and their corresponding mode shapes are shown in Fig. 5.3.
5.6
BENCHMARK CHECKS
Using the software VPRITZSK, some results for skew plates with various boundary conditions as shown in Fig. 5.4 have been generated. Tables 5.1 to 5.6 give the first six vibration frequency parameters 2 for SSSS, SCSC, CCCC, FSFS, FCFC, and CFFF skew plates with a/b=l.O, 2.0, h/b = 0.001, 0.10, 0.20 and fl = 0~ 15~ 30 ~ 45 ~ 60 ~ .
c~
c~
Fig. 5.3. First six mode shapes o f S C C F skew plate (a / b = 2 and h / b = O. 1O)
...j
Sec. 5.6 Benchmark Checks
158
f A 1
_
_
~
/f//////////////////
f//////////////////t
F
Fig. 5.4 Skew plates with various boundary conditions
Chap. 5 Formulation in Skew Coordinates
Table 5.1 Frequency parameters 2 o f SSSS skew plates a/b
h/b
.001
1.0
0.10
0.20
.001
2.0
0.10
0.20
flo 0 15 30 45 60 0 15 30 45 60 0 15 30 45 60 0 15 30 45 60 0 15 30 45 60 0 15 30 45 60
.
1 19.74 20.87 24.96 35.33 66.30 19.06 20.11 23.88 33.11 58.51 17.43 18.32 21.44 28.75 47.03 12.34 13.11 15.90 23.01 44.00 12.06 12.80 15.45 22.05 40.66 11.36 12.01 14.34 19.92 34.46
2 49.35 48.20 52.63 66.27 105.0 45.45 44.47 48.24 59.55 89.62 38.08 37.37 40.11 48.10 67.94 19.74 20.66 23.95 32.20 56.03 19.06 19.91 22.96 30.38 50.63 17.43 18.15 20.68 26.63 41.64
Mode sequence number 3 4 49.35 78.96 56.12 79.05 71.87 83.86 100.5 108.4 148.7 196.4 45.45 69.72 51.16 69.79 64.05 73.56 86.19 91.83 120.4 151.6 38.08 55.01 42.20 55.05 51.17 57.56 65.75 69.24 86.70 104.9 32.08 41.95 33.08 44.75 36.82 52.64 46.21 63.50 72.79 92.80 30.35 39.07 31.25 41.50 34.56 48.24 42.70 57.27 64.37 80.22 26.63 33.35 27.34 35.17 29.91 40.11 36.03 46.52 51.24 61.86
159
(v = 0.3) 5 98.70 104.0 122.8 140.8 213.8 84.94 88.91 102.6 115.3 160.5 64.96 67.50 76.07 83.72 109.3 49.35 50.23 56.63 82.08 117.4 45.45 46.20 51.59 72.06 97.74 38.08 38.63 42.51 56.53 72.89
98.70 108.9 122.8 168.3 250.7 84.94 92.53 102.6 133.7 183.5 64.96 69.78 76.07 94.63 122.6 49.35 52.49 63.26 83.00 151.7 45.45 48.11 57.07 72.85 118.2 38.08 40.01 46.37 57.08 84.76
Sec. 5.6 Benchmark Checks
160
Table 5.2 Frequency parameters 2 of SCSC skew plates (v = 0.3) a/b
h/b
I
.001 ,
1.0
0.10
0.20
.001
2.0
0.10
0.20
/~~ 0 15 30 45 6o
0 15 30 45 60 0 15 30 45 60 0 15 30 45 60 0 15 30 45 60 0 15 30 45 60
Mode sequence number
1
2
3
4
5
28.95 30.69 36.96 52.49 97.27 26.65 28.11 33.26 45.36 76.19 22.32 23.37 27.02 35.19 54.14 13.69 14.51 17.44 24.73 45.79 13.26 14.03 16.77 23.43 41.86 12.27 12.93 15.28 20.81 35.10
54.75 56.70 64.27 83.59 137.6 49.06 50.55 56.34 70.59 106.2 39.76 40.55 44.10 52.84 73.26 23.65 24.70 28.48 37.59 62.53 22.37 23.32 26.66 34.48 54.74 19.67 20.43 23.04 29.02 43.59
69.32 74.62 93.00 123.3 188.4 59.12 63.12 76.37 98.24 136.8 44.47 47.24 55.77 69.76 90.80 38.70 39.82 44.18 55.22 85.37 35.56 36.53 40.18 49.13 71.89 29.67 30.39 33.02 39.28 54.32
94.58 94.05 100.8 137.4 238.7 78.69 78.64 83.34 105.6 166.3 58.37 58.45 61.20 73.33 107.9 42.59 45.50 55.77 74.18 110.4 39.54 42.03 50.65 64.42 89.95 33.59 35.44 41.54 49.66 65.25
102.2 112.0 137.7 168.0 270.4 86.72 93.21 110.2 127.2 176.8 65.58 69.23 78.65 86.88 111.8 51.68 55.10 63.69 84.87 137.9 47.07 49.89 56.28 73.83 107.6 38.87 40.86 44.44 57.42 75.73
129.1 137.3 142.7 193.5 301.4 101.2 107.3 112.2 144.4 197.9 70.30 74.01 78.93 97.27 124.9 58.66 59.14 66.83 91.48 164.6 52.09 52.57 59.33 77.73 126.9 41.34 41.74 47.37 58.81 86.62
Chap. 5 Formulation in Skew Coordinates
161
Table 5.3 Frequency parameters A o f CCCC skew plates (v
=0.3)
Mode sequence number
a/b
h/b
.001
1.0
0.10
0.20
.001
2.0
0.10
0.20
fl~ 0 15 30 45 60 0 15 30 45 60 0 15 30 45 60 0 15 30 45 60 0 15 30 45 60 0 15 30 45 60
,
!
1
2
3
4
5
35.98 38.19 46.09 65.65 121.8 32.49 34.30 40.61 55.31 91.74 26.46 27.69 31.89 41.05 61.45 24.58 26.22 32.19 47.32 92.46 22.77 24.18 29.19 41.25 72.67 19.20 20.22 23.75 31.67 50.02
73.39 72.90 81.60 106.5 177.8 61.94 61.43 67.40 83.67 124.0 46.14 45.70 49.11 58.25 79.57 31.83 33.58 39.87 55.62 101.8 29.11 30.56 35.69 47.84 79.07 24.15 25.17 28.65 36.42 54.33
73.39 82.62 105.2 148.3 231.9 61.94 68.68 84.20 110.5 153.4 46.14 50.30 59.36 74.44 96.38 44.77 46.63 53.36 70.08 118.1 40.14 41.61 46.83 59.12 90.16 32.39 33.35 36.70 44.22 61.61
108.2 109.6 119.2 157.2 291.7 86.78 87.55 93.48 116.4 183.4 61.94 62.25 65.36 76.89 113.0 63.33 64.93 71.86 89.78 141.0 54.90 56.33 61.35 73.75 105.0 41.36 43.09 46.50 53.75 70.83
131.6 139.0 165.0 196.8 305.4 102.2 106.8 122.4 139.2 189.4 70.57 73.08 81.55 90.96 113.9 63.98 68.70 85.09 112.6 172.6 55.27 58.49 70.12 89.44 122.4 43.01 44.09 50.57 63.36 80.83
132.2 145.2 165.3 229.5 358.5 103.2 111.3 122.5 156.6 213.2 71.53 76.17 82.51 99.58 129.3 71.07 75.58 90.03 126.4 216.4 60.37 63.67 74.12 96.98 142.4 45.24 47.28 53.75 65.57 91.42
Sec. 5.6 Benchmark Checks
162
Table 5. 4 Frequency parameters ~ o f FSFS skew plates (v = 0.3)
Mode sequence number a/b
h/b ,
.001
1.0
0.10
0.20
.001
2.0
0.10
0.20
flo
1
2
3
4
5
0 15 30 45 60 0 15 30 45 60 0 15 30 45 60 0 15 30 45 60 0 15 30 45 60 0 15 30 45 60
9.632 10.20 12.15 16.43 25.47 9.439 9.939 11.63 15.15 21.95 8.977 9.395 10.76 13.47 18.25 2.379 2.501 2.911 3.773 5.617 2.364 2.476 2.847 3.608 5.140 2.331 2.432 2.761 3.415 4.655
16.13 16.48 17.71 20.46 26.88 15.38 15.65 16.61 18.71 23.32 14.07 14.27 14.96 16.45 19.48 6.880 6.708 6.534 6.651 7.267 6.614 6.475 6.324 6.411 6.874 6.228 6.121 5.994 6.051 6.374
36.72 35.97 36.02 39.65 53.85 33.84 33.41 33.69 36.94 47.81 29.10 28.89 29.25 31.65 38.50 9.632 10.50 13.05 15.66 17.84 9.439 10.20 12.38 14.96 16.72 8.977 9.592 11.35 13.71 15.00
38.94 41.43 49.41 59.75 72.82 36.33 38.36 44.56 52.24 60.86 31.22 32.66 36.89 41.90 46.47 16.13 15.66 15.22 17.90 27.23 15.38 14.99 14.61 16.38 23.67 14.07 13.78 13.46 14.44 19.58
46.73 50.67 61.34 78.96 100.2 42.76 45.64 53.51 66.66 83.44 35.89 37.77 43.00 51.82 62.08 21.82 23.43 26.44 26.76 31.02 20.93 22.31 25.02 25.22 28.06 18.96 20.03 22.26 22.33 24.12
70.74 66.31 65.57 81.31 136.4 62.09 59.05 58.76 70.39 104.8 49.45 47.60 47.53 54.90 73.29 26.37 27.81 27.79 36.19 42.76 24.97 26.22 26.06 32.78 38.59 22.32 23.19 22.91 27.76 32.02
Chap. 5 Formulation in Skew Coordinates
163
Table 5. 5 Frequency parameters ~ of FCFC skew plates (v = 0.3) Mode sequence number
a/b
h/b
.001
1.0
0.10
0.20
.001
2.0
0.10
0.20
fl~
1
2
3
4
5
0 15 30 45 60 0 15 30 45 60 0 15 30 45 60 0 15 30 45 60 0 15 30 45 60 0 15 30 45 60
22.17 23.34 27.40 36.45 57.82 20.61 21.59 24.91 32.01 46.61 17.50 18.19 20.51 25.34 35.20 5.509 5.725 6.443 7.925 10.84 5.393 5.593 6.255 7.596 10.10 5.116 5.290 5.854 6.963 8.937
26.40 27.33 30.55 38.07 58.23 23.99 24.71 27.19 32.83 47.86 19.84 20.33 22.05 25.92 35.81 8.989 9.210 10.02 12.02 17.24 8.531 8.727 9.430 11.10 15.35 7.733 7.896 8.481 9.870 13.27
43.59 44.82 49.50 61.78 96.86 38.50 39.54 43.38 52.82 76.26 31.18 31.97 34.74 40.89 54.37 15.19 15.91 18.26 23.09 31.99 14.53 15.15 17.15 21.05 28.32 13.06 13.52 14.99 17.77 22.52
61.17 64.50 73.91 86.79 116.6 52.61 54.97 61.70 70.98 89.63 39.75 41.16 45.36 51.93 63.69 20.59 20.59 21.29 23.99 33.24 19.23 19.26 19.89 22.18 28.67 16.77 16.83 17.38 19.19 24.17
67.18 70.36 80.93 102.6 147.1 56.97 59.06 65.96 80.23 109.8 42.76 44.03 48.29 57.49 76.58 27.36 29.23 34.71 37.87 47.62 25.67 27.30 31.99 34.43 42.15 22.65 23.86 26.76 28.72 34.27
79.82 79.20 85.22 113.9 194.4 68.39 67.43 71.18 88.65 132.9 52.99 51.31 53.01 61.54 83.84 29.86 31.49 34.90 44.63 62.27 27.73 29.07 32.00 39.45 52.37 23.56 24.55 26.93 31.51 39.85
Sec. 5. 6 Benchmark Checks
164
Table 5. 6 Frequency parameters A o f CFFF skew plates (v = 0.3) a/b
h/b
.001
0.10
1.0
0.20
.001 i
2.0
I
0.10
0.20
flo 0 15 30 45 60 0 15 30 45 60 0 15 30 45 60 0 15 30 45 60 0 15 30 45 60 0 15 30 45 60
,
1 3.471 3.584 3.931 4.511 5.276 3.431 3.536 3.858 4.387 5.049 3.338 3.434 3.719 4.171 4.719 3.493 3.637 4.060 4.676 5.431 3.457 3.594 3.986 4.547 5.171 3.368 3.493 3.842 4.318 4.837
2 8.508 8.792 9.413 11.25 16.10 8.058 8.228 8.870 10.53 14.89 7.340 7.489 8.055 9.524 13.20 5.352 5.536 6.250 8.294 14.44 5.192 5.366 6.050 7.982 13.53 4.915 5.071 5.692 7.434 12.17
Mode sequence number 3 4 21.28 27.19 22.24 27.61 25.30 25.93 26.99 31.56 30.65 45.60 20.07 25.48 20.84 24.64 23.23 24.27 24.76 28.26 27.07 38.32 17.54 22.44 18.05 21.59 19.51 21.34 20.78 23.55 22.09 29.03 10.18 19.07 10.40 19.27 11.23 20.08 13.36 22.24 19.26 27.79 9.747 18.08 9.949 18.28 10.72 19.04 12.65 20.93 18.02 25.23 8.993 16.26 9.178 16.43 9.850 17.08 11.51 18.56 15.96 21.50
5 30.95 34.18 41.33 50.71 59.32 28.23 30.77 37.03 44.95 51.42 23.84 25.96 30.57 36.30 41.05 21.84 22.85 25.64 29.68 37.27 20.62 21.42 23.48 26.47 32.05 17.96 18.47 19.66 21.46 24.99
54.18 52.07 50.63 59.08 80.74 47.48 46.12 45.24 51.55 67.27 38.26 37.47 37.00 40.44 47.05 24.67 25.97 30.28 35.37 47.71 23.05 24.19 28.11 32.30 40.31 19.92 20.80 23.92 27.29 31.65
CHAPTER SIX
PLATES WITH COMPLICATING EFFECTS
6.1
INTRODUCTION
This chapter considers plates with complicating effects such as the presence of initial inplane stresses, elastic foundations, stiffeners, nonuniform thickness, line/curved internal supports, point supports, mixed boundary conditions, reentrant corners, perforations and sandwich construction. Guidelines for the treatment of these complicating effects in the rectangular Cartesian coordinates are given below. They allow the users to readily modify the computer program VPRITZRE, as given in Chapter 4, for the inclusion of these complicating effects in the analysis. The treatment of these complicating effects in polar coordinates and in skew coordinates can be readily obtained from suitable coordinate transformation relations.
6.2
INITIAL INPLANE STRESSES
Consider a plate subjected to isotropic inplane stresses cr as shown in Fig. 6.1.
Fig. 6.1 Initially uniform stressedplate The inclusion of the effect of these uniform stresses that remain constant during vibratory motion can be readily done by augmenting the total energy functional with the potential energy V of the inplane stresses
V= ~ oh (V2w)dA
(6.1)
A 165
Sec. 6.2 Initial lnplane Stresses
166
where h is the plate thickness, A the plate area and w the transverse deflection. Here positive stresses imply tensile stresses while negative stresses imply compressive stresses. In view of Eq. (6.1), a geometric stiffness matrix G must be added to the eigenvalue equation. The eigenvalue equation becomes
IKCCKCdKce
I
K da
K ,~e -/12 K ee
IMCC MceI ccO011f!t fit M ed M de +
0 0
D
M ee
=
(6.2)
0
where the elements of the geometric matrix are given by guc c =
!i[
a
3~C
F
a~~
1
d~d~7;
A
for i = 1,2,...,N1; j = 1,2,...,N 1
(6.3)
Thus, the subroutine in the computer programs that evaluates the elements of the matrices need to include Eq. (6.3) for solving the vibration problem of uniformly stressed plates. Further, this modification enables the program to solve the elastic buckling problem of Mindlin plates under isotropic inplane stresses when the frequency parameter k is set to zero. The lowest positive eigenvalue of crh b 2/D corresponds to the critical buckling stress. The buckling problem of Mindlin plates has been investigated by Herrmann and Armenakas (1960), Brunelle and Robertson (1974) for rectangular plates; Hong, Wang and Tan (1993) for circular plates; Kitipomchai, Xiang, Wang and Liew (1993), Wang, Kitipornchai, Xiang and Liew (1993), Xiang, Wang and Kitipomchai (1995a) for skew plates; and Wang, Xiang, Kitipomchai and Liew (1994) for regular polygonal, annular and elliptical plates; and Wang (1995) for general polygonal plates. If the frequencies of initially stressed Mindlin plates are to be determined from solving the goveming equations of motion, only Eq. (2.14c) of the set of equations given in Eqs. (2.14a-c) needs to be modified to: ic2 G h ( V 2w + ~ ) = - p h o ) 2w - o h V 2w
(6.4)
As in Section 2.2 of Chapter 2, one may obtain an exact relationship between the frequencies of simply supported, initially stressed, polygonal Mindlin plates and the corresponding frequencies of Kirchhoff plates. Following the same procedure given in Section 2.2.2, this frequency relationship is found to be 2 ON =
6x'~G (.~ _ 4 . ~ _ ~ ) ,oh 2
1
.~ = l + ~ & N h 2 12
2
1+
K'2(1- v)
(6.5) (6.6a)
Chap. 6. Plates with ComplicatingEffects 32
:
3K.2G (bN C3N 1+
+-~--
167
(6.6b)
where co is the natural frequency of the simply supported, initially stressed, polygonal Mindlin plate, ~ the frequency of the corresponding simply supported, polygonal Kirchhoff plate without initial stresses, and the subscript N = 1,2,..., denotes the mode sequence. The relationship furnishes directly the Mindlin plate frequencies upon supplying the Kirchhoff plate frequencies. A similar relationship was derived by Irschik (1985), which expresses the Mindlin plate frequencies in terms of the frequencies of the corresponding prestressed membranes instead. The presence of compressive inplane stresses decreases the natural frequencies while tensile inplane stresses increases the frequencies. Research studies on the vibration of initially stressed Mindlin plates have been carried out by Roufaeil and Dawe (1982), and Liew, Xiang and Kitipornchai (1993d) for rectangular plates; Liew, Xiang, Kitipornchai and Wang (1994) for annular plates; and Xiang, Liew, Kitipornchai and Wang (1994) for triangular plates.
6.3
ELASTIC FOUNDATIONS
Consider a plate resting on Pasternak foundation (Pasternak 1954). In addition to the wellknown Winkler foundation springs, the Pasternak model takes into account the shear interaction between the spring elements. This is accomplished by connecting the ends of the springs to the plate with incompressible vertical elements that deform only by transverse shear as shown in Fig. 6.2. The foundation medium is assumed to be linear, homogeneous and isotropic. If the effects of damping and inertia force in the foundation are neglected, the effect of this elastic two-parameter foundation can be incorporated by augmenting the strain energy given in Eq. (2.6) with Up
Plate
w
~ ~ ~
~~~~~'-
Shear layer Gb
,w,/
f///////////////////~//~//~/////////////////////////////A
Subgradek
Fig. 6.2 Plate resting on Pasternak foundation
lssE
U~ = ~
g~'~ +Gu
A
(6.7)
168
Sec. 6. 3 Elastic Foundations
where k is the modulus of subgrade reaction for the foundation and G b the shear modulus of the subgrade. If G b is viewed as a constant membrane tension, the Pastemak model becomes the Filonenko-Borodich model, while if G o = 0, the model reduces to the Winkler model (Horvath 1989). In view of Eq. (6.7), the eigenvalue equation is modified to include the elements of the Pastemak stiffness matrix K F, i.e.
/I +
e1
eel/f:t: f:t0
(6.8)
where kFU =
k~b?~b~. + G b a ~
_
040r
A
~
a~~
d~drl;
for i = 1,2..... , N l ; j = 1,2,..., N 1
(6.9)
while the rest of the stiffness elements and the mass elements are given in Eqs. (4.26a-f) and (4.27a-f). Thus, the subroutine in the computer programs that evaluates of the stiffness matrix needs a recoding to include Eq. (6.9) for this foundation effect. If the frequencies of Mindlin plates resting on Pastemak foundation are to be computed from solving the equations of motion, only Eq. (2.14c) of the set of goveming equations given in Eqs. (2.14a-c) is to be modified to: (6.10) As in Section 2.2 of Chapter 2, one may obtain an exact relationship between the frequencies of simply supported, polygonal Mindlin plates on Pastemak foundation and the corresponding frequencies of Kirchhoff plates. Following the same procedure as described in Section 2.2.2, this frequency relationship is found to be 2
6 ~ G (.~
~o~= Ph ~ -~3 =1+
"~ 4
=
12K2 G
3~c2G
2_~
~-~/.~ +
-~
ff)u h2
)
(6.11a)
4
1 + ~c2(1- v) + K"2Gh
+o5 N + +o5 N 1+ ~--~ I,---D tc2 G h te2 G h
(6.11b)
(6.1 lc)
where o is the natural frequency of the simply supported Mindlin plate resting on Pastemak foundation, ~ the frequency of the corresponding simply supported Kirchhoff plate without elastic foundation, the subscript N = 1,2,..., denotes the mode sequence. The relationship fumishes directly the Mindlin plate frequencies upon supplying the Kirchhoff plate frequencies. A similar relationship was derived by Irschik (1985) but the Mindlin plate
Chap. 6. Plates with Complicating Effects
169
frequencies are instead expressed in terms of the frequencies of the corresponding prestressed membranes. Studies on vibration of Mindlin plates on elastic foundation include that of Hinton and A1-Janabi (1987) who considered simply supported, rectangular Mindlin plates on Winkler foundation and that of Xiang, Wang and Kitipornchai (1994) who treated the same problem but with the more general Pasternak foundation.
6.4
STIFFENERS
Consider a Mindlin plate reinforced with stiffeners of arbitrary orientation as shown in Fig. 6.3a. The stiffeners are constructed from the same plate material. The stiffeners are placed symmetrically with respect to the plate midplane. To include the effect of transverse shear deformation in the stiffener, Engesser's column theory (1891) associated with consideration of torsion has been used to derive the strain energy (Usr) and the maximum kinetic energy (T~.r) for the stiffener.
y,r/
1
.,
I"
a
~1 "l
Section 1-1
x,~
Fig. 6.3a Plate with stiffeners
The presence of the stiffeners may be included in the Ritz formulation by augmenting the strain and kinetic energy functionals of the plate. The strain energy of the r-th stiffener is given by (Liew, Xiang, Kitipornchai and Meek 1995)
Sec. 6.4 Stiffeners
170
Usr - - 7
EIr cOs2 )~r
+ lf r G A r COS~r
-
C~x + sin ~ r &
~0rx -1t-
sinflrCOSfl~(CTlffx
COS ~ r ( ~ / x
&
+sinflr
~t'y nt-
Y/+sin r I r
+ Gflr 2
C3~Yl - sin 2 ~r - ~ 1 } 1
sec fl~ dx
(6.12)
y=f(x)
while the kinetic energy for the r-th stiffener is given by Zsr : "~0) 12 p
If'[ArW e"~-Ir (~/2x "]- ~[[y2)]2 Iy=f(x) seC~rd x
(6.13)
where Ar is the cross-sectional area of the r-th stiffener, Ir the second moment of area of r-th stiffener, Jr the torsional constant of the r-th stiffener, K"r the shear coefficient of the r-th stiffener, fir the angle of the r-th stiffener with respect to the x-axis (see Fig. 6.3b), x 0 and x~ are the start and end points of the r-th stiffener in the x-direction, and y = f ( x ) is the equation of the line of stiffener orientation. Note that if the angle fl approaches + ~r/2, the value secfl in Eqs. (6.13a,b) approaches infinity. In this case, the integration of Eqs. (6.13a,b) should be changed to be along the y-direction. The total strain energy and kinetic energy for n s stiffeners are, respectively given by ns U S : ~ Usr r=l ns TS -" Z Lr
(6.14a)
(6.14b)
r=l
In view of Eq. (6.13), the eigenvalue equation of a stiffened Mindlin plate is given by K cc + K s
K ca + KCsd
K ce + K s
K aa+Kas a
K ae+Kas e
II .... 1 I cc K ee +
M cc + M s
_ 22
K ee s
M ca + M ca s Maa + Maa s
M ce + M s Ma~ + M sa~
cellf:tiO0 t
M ee + M ee s
c
0
=
(6.15)
Chap. 6. Plates with Complicating Effects
171
y, rl The r-the stiffener
gx
~.'3~r
Y
Y
~/y Xo
X1
Fig. 6.3b Location of r-th stiffener and coordinate system
The elements o f the submatrices K s are given b y
~.i ~ ;~[~ r=l
r
0~
,s~n2~r,~/~;
0~
0( COS/~r
~
+sinrlOiw< OwO l]l I
~
+
~
~
d~;
rl=g(,r for i = 1,2 .... , N1; j = 1,2,... , N 1
ksijdi -"
"z ;, E .,Wx..,w x]l ~r
COS ~r - - ~ ~ j
r=l
-1-- - s i n ]~r a W
(6.16a)
d4;
~j
r/=g(~)
for i = 1,2 .... ,N~ ; j = 1 , 2 , . . . , N 2
ksgii =
~r
r=l
sin
fl, --~-~Y +
(6.16b)
~~q~Y a COS fir
f o r i = 1 , 2 .... , N l ; j = 1 , 2
d~; O~7 ..... N 3
r/=g(r (6.16c)
Sec. 6. 4 Stiffeners
172 kso'i
=
r=
o
~r
dg+
COS~ r ~ ; # ;
~ ~
r=l
+
~r
r/=g(~)
r=l
+ c~
o
a
COS3 ~r ~ ~
fir sin ]~r d~:
&
de
; o~sin2flrCOS/~r ~-~ ~-~ "~ lifo
tSg COS/~r &
0%/7
d4:
+sin3fl'~, d~' dr/ ~=g(r for i = 1,2,..., N 2 ; j" = 1,2,..., N 2
kdji= ~1 s 9 t SI1 sin fl,~biX~b;Xll r=
o
(6.16d)
d$:
r/=g(~)
r=l
o
0-~ 0-~
+ l s i n 2 f l . COSflr(Or ~ ~ dCf + ~ dr~ OCf) +sin3 ~r 0~ x dr a 0r & 8q c~ 8v &
d~:; rl=g(4:)
for i = 1,2 ..... N2; j = 1,2 .... ,N 3
ee~-~~~l~ISlnj~r s ! " 2 r=l I$ ~o r a ~OS~r
ks~ii =
+ +s r=l
#/y ~;
][ r/=g(~)
d~ +
s
(6.16e)
fJfl
r=l
o
F[
9t r a sin 2 cos ft, ~
1 sin4 fir O~b/YOq}y ( O#Y O#YO~bY)I [ ---+ sin3 fir o~7 ~ +
re' 9t rl 1 r a sin93 ,B, -Oqki - ~ v- -8qb ~ y + --sin 2 L
~r
a
v dqbf i~rCOS/~r -O~bi --037 &7
de rl=g(4')
+COS'rSin'rC'"'III
~=g(r
for i = 1,2,..., N3; j = 1,2,..., N 3 The elemems of the submatrices M s are given by mCsCiji=
'j;' .E1 Fr
r=l
o
_
a COS~r
qjwqj;
It
d~;
(6.160
Chap. 6. Plates with Complicating Effects for i = 1,2,..., N1; j = 1,2,..., N 1 ca = 0; mso, ce mso~ = 0; dd
ms~ii =
mes~'i= "Z r=l
(6.17a)
for i = 1,2 . . . . . N l ; j = 1,2 .... , N 2
(6.17b)
for i = 1,2,..., N1 ; j = 1,2,..., N 3
(6.17c)
Zn ~j~lFri I 1 1 COS~ r r=] o
ae = 0; mso.i
173
~=g(~)
for i = 1,2,..., N 2; j = 1,2,..., N 2 for i = 1,2 . . . . , N 2 ; j = 1,2 .... , N 3
(6.17d) (6.17e)
for i = 1,2,...,N3; j = 1,2,...,N 3
(6.170
~Jf'U II1O, COSfl 1 r o
where S _ l~r GArb
-
D
F
'~}~r
EIr r GJr A PAr b2 =bD,~r = D ' Fr = T '
Pfr F/----..~-
(6.18a-e)
Vibration studies on Mindlin plates with stiffeners were done by Liew, Xiang, Kitipornchai and Lim (1994) for rectangular plates, and Liew, Xiang, Kitpornchai and Meek (1995) for arbitrarily shaped plates.
6.5
NONUNIFORMTmCKNESS
For a Mindlin plate with a thickness distribution defined by h(x,y), the strain energy is obtained by substituting Eqs. (2.2) and (2.5) into Eq. (2.4). The resulting strain energy functional is the same as that given in Eq. (2.6) except that now the flexural rigidity D = E[h(x,y)] 3/[12(1-v2)] instead of being a constant. Similarly the kinetic energy is given by Eq. (2.8) with h = h(x, y) instead of a constant and h is to be placed in the integrand. Following the p-version Ritz procedure, the eigenvalue equation for the nonuniform thickness Mindlin plate is obtained
ilKcc
Kee Kee Mcc McellI! Mee fit
(6.19)
where the elements of the stiffness submatrices [K] and mass submatrices [M] are as follows:
. . . k o. 4r 2
.
. A
[
h a d~ d~
+
a ~
dr1
d~drl ;
i=1, 2,..., N1; j = l ,
2,..., N 1
(6.20a)
Sec. 6.5 Nonuniform Thickness
174
m,;d = 0 ; me' = 0 ;
A
m,: = 0 ;
A
and
i = l , 2 ,..., N , ; j = 1 , 2 ,..., N 2
(6.20b)
i = l , 2 ,..., N , ; j = 1 , 2,..., N ,
(6.20~)
i = l , 2 ,..., N,; j = l , 2,..., N,
(6.20d)
i = l , 2,..., N,; j = 1 , 2 ,..., N,
(6.20e)
i = l , 2 ,..., N,; j = 1 , 2,..., N ,
(6.200
i = l , 2 ,..., N , ; j = l , 2,..., N, i = l , 2,..., N,; j = 1 , 2 ,..., N 2
(6.21a) (6.21b)
i = l , 2 ,..., N , ; j = 1 , 2 ,..., N,
(6.21~)
i = l , 2 ,..., N,; j = 1 , 2 ,..., N,
(6.21d)
i = l , 2 ,..., N,; j = 1 , 2 ,..., N,
(6.21e)
i = l , 2,..., N,; j = 1 , 2,..., N,
(6.210
Chap. 6. Plates with Complicating Effects
ho
r0=m, b
175
h(4,V)
h = ~ , ho
(6.22)
where h 0 is the uniform thickness of the reference plate that has the same volume as the varying thickness plate.
6.6
LINE/CURVED/LOOP INTERNAL SUPPORTS
Consider a plate with internal line/curved/loop supports as shown in Fig. 6.4a. The supports impose a zero deflection (w= 0) along its length. The geometric constraint due to the supports can be readily handled by the Ritz method through the modification of the basic function for the transverse deflection to include the equations of the line/curved/loop supports as follow:
(6.23)
where the underlined term is the product of the equations of the line/curved/loop internal supports with Aj being the equation of the j-th support and n s the number of these supports. The eigenvalue equations presented in the earlier chapters remain unchanged.
Line support
Curved support
Loop support
/
(a)
(b)
(c)
Fig. 6.4 Plates with (a) internal line, (b) curved and (c) loop supports
It is to be remarked that the above modification applies to ring/loop supports or line/curved supports whose ends must terminate at the plate pheriphery. When the supports whose ends do not meet this condition, see for example Fig. 6.5, the use of the basic function in Eq. (6.3) leads to an erroneous constraint of zero deflection on the extrapolated or interpolated portions of the supports as shown by the dashed lines in Fig. 6.5. For the latter
Sec. 6. 7 Point Supports
176
support condition, it is proposed that a series of point constraints along the support length be introduced to simulate the line/curved support. The imposition of these point constraints will be discussed in Section 6.7.
Internal line support
Internal c u ~ e d support
Point supports to simulate internal supports
Fig. 6.5 Simulation of internal supports using point supports
Studies on the vibration of Mindlin plates with line/curved/loop internal supports have been made by Liew, Xiang and Kitipornchai (1993b,c) for rectangular plates with line/ring supports; Liew, Xiang, Wang and Kitipornchai (1993) for circular and annular plates with ring supports; Xiang, Kitipornchai, Liew and Wang (1994) for skew plates with oblique line supports; and Liew, Kitipornchai and Xiang (1995) for annular sectorial plates with radial line and circumferential arc supports.
6.7
POINT SUPPORTS
Consider a Mindlin plate with point supports located at the plate edges or internally as shown in Fig. 6.6. These point supports constrained the transverse deflection to be zero at their locations, i.e.
W ( ~ i , 17i ) = 0
(6.24)
where i = 1,2 .... ,n c and n c is the number of point supports. These point constraints can be incorporated into the energy functional by using the Lagrangian multiplier method. The augmented Lagrangian is given by nc
FI = U - T + ~
i=1
Liw(~g,rli )
(6.25)
Chap. 6. Plates with C o m p l i c a t i n g Effects
where
Li
177
is the i-th Lagrangian multiplier. This yields the frequency equation as K CC K Cd
K ce
H
K da
K de
0
K ee
0
M CC _2 2
0 M ad
0
0
c
0
0
d
M ee
0
0
0 =
0
(6.26)
0
where the elements of the H submatrix are given by w
tj -- ~j (~i'/']i)'
i = 1,2.... ,nc, j = 1,2,...,N,
(6.27)
9 upport
Fig. 6. 6 Plate with point supports
For more information on this complicating effect, the reader may refer to the paper by Kitipornchai, Xiang and Liew (1994). It is to be remarked that the difficulty of implementing the S-simple support for oblique line and curved edges in the rectangular coordinates system may be overcome by using S*-simple support and point constraints on the tangential rotation ~, on the concerned edges. The reader may refer to the paper by Xiang, Wang and Kitipornchai (1994) where it was demonstrated how the S-simple support was implemented for vibration of elliptical Mindlin plates.
6.8
MIXED BOUNDARY CONDITIONS
Consider a Mindlin plate with mixed boundary conditions on its edge/edges as shown in Fig. 6.7. The p-version Ritz functions with the equations of the plate edges raised to appropriate powers run into difficulties with such mixed boundary conditions. To overcome the problem,
Sec. 6.9 Reentrant Corners
178
it is proposed that the plate edge with mixed boundary condition be simulated by suitable point restraints. For example,
Mixed edtge condition
/
Clamped point support to simulate clamped edge portion
/
Simple point support to simulate simply supported edge portion (a)
(b) Fig. 6. 7 Plate with mixed edges
6.9
REENTRANT CORNERS
The p-Ritz functions involving the boundary equations in general fail to handle plates with reentrant comers formed from simply supported and clamped edges. The reason being that the boundary equations of the concemed edges cut into the plate domain and impose a zero deflection or a zero rotation as shown in Fig. 6.8a. To avoid this, it is proposed that these edges be made free and a series of point restraints be introduced to simulate the support conditions as shown in Fig. 6.8b. There are, however, in some situations the p-version Ritz functions may be used for plates with reentrant comers. For example, a sectorial plate with sectorial angle greater or equal to 180 degrees has a reentrant comer. The use of polar coordinates in Chapter 3 overcome this reentrant comer problem in the sectorial plate problem which would otherwise exist for p-Ritz functions in rectangular coordinates. Another example is when the reentrant comer, formed by two edges of the same type of support conditions, is right angled. In this situation, the super-elliptical function (Wang, Wang and Liew 1994) which can approximate the rectangular shape at high powers may be used to as the basic function for the two edges. It should, however, be noted that if the stress (moment) singularities are severe at the re-entrant comers, then the use of smooth polynomial functions with the Ritz method may lead to a bad convergence. For such severe stress singularities, one must add proper comer (singularity) functions to the polynomials to get good results. Some of the papers where this is demonstrated are those by McGee et al. (1992a, 1992b), Leissa et al. (1993a, 1993b) and
Chap. 6. Plates with Complicating Effects
179
Huang et al. (1995). The same comer functions may be used with the Mindlin plates to accelerate the convergence of the frequencies.
Reentrant
'1
I
I
I ~////////_//////////~fi.
~/////////////////~fi
(a)
(b)
Fig. 6.8 Plate with reentrant corner
6.10
PERFORATED PLATES
The same Ritz software may be used for the vibration analysis of plates with arbitrarily shaped holes as shown in Fig. 6.9. All that one has to do is to minus the total energy contributed by the perforation as shown by Lim and Liew (1995). This problem is straightforward if the boundary of the perforation is free. However, in the case of simply supported or clamped perforated edge, it is necessary that the perforation shape be defined by a single continuous function when using the Ritz method in such a way. When the perforation shape cannot be defined by a continuous function, the point constraint method as described in Sections 6.7 to 6.9 may be adopted as illustrated by Liew (1993b).
6.11
SANDWICH CONSTRUCTION
Consider a sandwich plate whose core thickness is hc and the thickness of each facing is h/ as shown in Fig. 6.10. The modulus of elasticity E, Poisson ratio v and shear modulus G of the core and facings will be identified with subscripts c a n d f respectively. Based on the Mindlin plate theory and assuming that the deformations are continuous through the plate thickness, the strain energy due to bending of such a sandwich Mindlin plate is given by
U = 1 2
D~
{I
&
A
+
- 2(1 - v~) c~
,l x y121} +
dx
@
4@
dx
Set. 6.11 Sandwich Construction
180 + ~c2G~h~
+DI
{/
~x +
+ ~'y +
/2
a~x +aSFY - 2 ( 1 - v l )
E x l/ x y/21) +
(6.28)
Fig. 6. 9 Perforated plate
Fig. 6.10 Sandwich Mindlin plate
and the kinetic energy T is given by
A (6.29)
Chap. 6. Plates with Complicating Effects
181
where
EcI ffXI__z h3c 2 ( 3h2 3hchi ~ D~ =---~C2'cl-v Di = l - v } ' Ic =-i-2-' Ii =-~h i ( +) 4 h + ) 2
(6.30)
Using calculus of variations, the Euler equations of the Lagrangian ( U - T ) yields the following three equations of motion for sandwich Mindlin plates: 02 ~t'y
(6.31a)
2
d 2~l,X
+-[(1 - v c 2
-
+
-tcZ(Gchc +2GfhfI~y + ~ l = - ( p c l c
+Pflf)o)Z~'y
tr 2(Gchc + 2Gih i ~0~ + V2w)=-(pchc + 2pih I }02w
(6.31b)
(6.31c)
where ~s--
~//x
dx
~;;~//Y
+~--
c?y
Mx~+Myy (l+vc)Oc+O+vl)Df
(6.32)
Following the same procedure as in Section 2.2, one can obtain an exact relationship between the sandwich Mindlin plate frequencies cos, with the corresponding Kirchhoff plate frequencies cok for general polygonal shaped plates. The relationship is given below, co~=X,
1 + (1+ X2)X~
(6.33)
where
See. 6.11 Sandwich Construction
182
1E2cchc 1
Z1 = -~
pclc+ p i l i
Z2 = K2(Gch~+ 2Gih i
,o~-~+ p i I i
I, p~I~ + p i I i ] Z3 = ,och~+ 2plh i A more detailed derivation of the above relationship is given in Wang (1996).
(6.34b)
(6.34b)
(6.34b)
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Mindlin, R.D., Schacknow, A. and Deresiewicz, H. (1956). "Flexural vibrations of rectangular plates," Trans. ASME Journal of Applied Mechanics, 23, 430-436. Mizusawa, T. (1986). "Natural frequencies of rectangular plates with free edges," Journal of Sound and Vibration, 105, 451-459. Nair, P.S. and Durvasula, S. (1973). "Vibration of skew plates," Journal of Sound and Vibration, 26, 1-19. Narita, Y. and Leissa, A.W. (1990). "Buckling studies for simply supported symmetrically laminated rectangular plates," International Journal of Mechanical Sciences, 32, 909924. Nelson, R.B. and Lorch, D.R. (1974). "A refined theory for laminated orthotropic plates," Trans. ASME, Journal of Applied Mechanics, 41,177-183. Pasternak, P.L. (1954). "On a new method of analysis of an elastic foundation by means of two foundation constants," Gos. Izd. Lit. po Strait IArkh, Moscow, Russia (in Russian). Pnueli, D. (1975). "Lower bounds to the gravest and all higher frequencies of homogeneous vibrating plates of arbitrary shape," Trans. ASME, Journal of Applied Mechanics, 42, 815-820. Poisson, S.D. (1829). "L'Equilibre et le Mouvement des Corps Elastiques," Mem. Acad. Roy. Des Sci. de L 'Inst, France, Ser. 2, Vol. 8, 357. Rayleigh, J.W. (1877). Theory of Sound, Macmillan Vol. 1 (reprinted by Dover Publications, 1945). Reddy, J.N. (1984). "A simple higher-order theory for laminated composite plates," Trans. ASME,Journal of Applied Mechanics, 51,745-752. Reddy, J.N. (1997). Mechanics of Laminated Composite Plates: Theory and Analysis, CRC Press, Boca Raton, Florida. Reissner, E. (1944). "On the theory of bending of elastic plates," Journal of Mathematical Physics, 23, 184-191. Reissner, E. (1945). "The effect of transverse shear deformation on the bending of elastic plates," Journal of Applied Mechanics, 12, A69-A77. Ritz, W. (1909). "Uber eine neue Methode zur L6sung gewisser Variationsprobleme der mathematischen Physik," Journal ]Kr Reine und Angewandte Mathematik, 135, 1-61. Roufaeil, O.L. and Dawe, D.J. (1982). "Rayleigh-Ritz vibration analysis of rectangular Mindlin plates subjected to membrane stresses," Journal of Sound and Vibration, 85, 263-275. Smith, B.T., Boyle, J.M., Garbow, B.S., Ikebe, Y., V.C. Klema and Moler, C.B. (1974). Matrix Eigensystem Routines -- EISPACK Guide, Springer-Verlag, New York, U.S.A. Srinivas, S.R., Joga Rao, C.V. and Rao, A.K. (1970). "An exact analysis for vibration of simply supported homogeneous and laminated thick rectangular plates," Journal of Sound and Vibration, 12, 187-199. Timoshenko, S.P. and Woinowsky-Krieger, S. (1959). Theory of Plates and Shells, McGraw Hill, New York. Turner, M.J., Clough, R.W., Martin, G.C. and Topp, L.J. (1965). "Stiffness and deflection analysis of complex structures," Journal of Aeronautical Sciences, 23, 805-823. Vogel, S.M. and Skinner, D.W. (1965). "Natural frequencies of transversely vibrating uniform annular plates," Journal of Applied Mechanics, 32, 926-931. Voigt, W. (1893). "Bemerkungen zu dem Problem der transversalen Schwingungen recteckiger Platten," Nachr. Ges. Wiss. (Gottingen), 6, 225-230. Wang, C.M. (1994). "Natural frequencies formula for simply supported Mindlin plates," Trans. ASME, Journal of Vibration and Acoustics, 116, 536-540.
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Wang, C.M. (1995). "Allowance for prebuckling deformations in buckling load relationship between Mindlin and Kirchhoff simply supported plates of general polygonal shape," Engineering Structures, 17, 413-418. Wang, C.M. (1996). "Vibration frequencies of simply supported polygonal sandwich plates via Kirchhoff solutions," Journal of Sound and Vibration, 190, 255-260. Wang, C.M., Kitipomchai, S., Xiang, Y. and Liew, K.M. (1993). "Stability of skew Mindlin plates under isotropic in-plane pressure," Journal of Engineering Mechanics, ASCE, 119, 393-401. Wang, C.M., Wang, L. and Liew, K.M. (1994). "Vibration and buckling of super elliptical plates," Journal of Sound and Vibration, 171, 301-314. Wang, C.M., Xiang, Y., Kitipomchai, S. and Liew, K.M. (1994). "Buckling solutions for Mindlin plates of various shapes," Engineering Structures, 16, 119-127. Wang, C.M., Xiang, Y., and Kitipomchai, S. (1995). "Vibration frequencies of elliptical and semi-elliptical Mindlin plates," Structural Engineering and Mechanics, 3, 35-48. Wang, C.M., Kitipronchai, S. and Reddy, J.N. (1998). "Relationship between vibration frequencies of Reddy and Kirchhoff polygonal plates with simply supported edges," Trans. ASME, Journal of Vibration and Acoustics, in press. Warburton, G.B. (1954). "The vibration of rectangular plates," Proceedings of the Institution of Mechanical Engineers, Series A, 168, 371-384. Wittrick, W.H. (1973). "Shear correction factors for orthotropic laminates under static load," Journal of Applied Mechanics, 40(1), 302-304. Wittrick, W.H. (1987). "Analytical, three-dimensional elasticity solutions to some plate problems, and some observations on Mindlin's plate theory," International Journal of Solids and Structures, 23, 441-464. Wolfram, S. (1991). Mathematica - a System for Doing Mathematics by Computer, 2nd Edition, Addison-Wesley, California, U.S.A. Xiang, Y., Liew, K.M. and Kitipomchai, S. (1993). "Transverse vibration of thick annular sector plates," Journal of Engineering Mechanics, ASCE, 119, 1579-1599. Xiang, Y., Kitipomchai, S., Liew, K.M. and Wang, C.M. (1994). "Vibration of Mindlin skew plates with oblique line supports," Journal of Sound and Vibration, 178, 535-551. Xiang, Y., Wang, C.M. and Kitipornchai, S. (1994). "Exact vibration solution for initially stressed Mindlin plates on Pasternak foundations," International Journal of Mechanical Sciences, 36, 311-316. Xiang, Y, Liew, K.M., Kitipomchai, S. and Wang, C.M., (1994). "Vibration of triangular Mindlin plates under isotropic inplane compression," Acta Mechanica, 102, 123-135. Xiang, Y., Wang, C.M. and Kitipornchai, S. (1995a). "Buckling of skew Mindlin plates subjected to in-plane shear loadings," International Journal of Mechanical Sciences, 37, 1089-1101. Xiang, Y., Wang, C.M. and Kitipomchai, S. (1995b). "FORTRAN subroutines for mathematical operations on polynomial functions," Computers and Structures, 56, 541551. Young, D. (1950). "Vibration of rectangular plates by the Ritz method," Trans. ASME, Journal of Applied Mechanics, 17, 448-453. Zienkiewicz, O.C.(1977) The finite element method, 3rd Edition, McGraw-Hill, United Kingdom.
RELEVANT REFERENCE BOOKS
Hinton, E. (1987). Numerical Methods and Software for Dynamic Analysis of Plates and Shells, Pineridge Press, Swansea, U.K. Hinton, E. and Owen, D.R.J. (1977). Finite Element Programming, Academic Press, London. Huang, H.C. (1989). Static and Dynamic Analyses of Plates and Shells, Springer, Berlin. Jaeger, L.G. (1964). Elementary Theory of Elastic Plates, Macmillian, New York. Leissa, A.W. (1969). Vibration of Plates, U.S. Government Printing Office, NASA SP-160, reprinted by the Acoustical Society of America (1993). Mansfield, E.H. (1964). The Bending and Stretching of Plates, Macmillan, New York. Marguerre, K. and Woemle, H.T. (1969). Elastic Plates, Ginn/Blaisdell, Massachusetts. McFarland, D., Smith, B.L., and Benhart, W.D. (1972). Analysis of Plates, Spartan Books, New York. Meirovitch, L. (1967). Analytical Methods in Vibrations, Macmillian, New York. Morley, L.S.D. (1963). Skew Plates and Structures, Macmillian, New York. Panc, V. (1975). Theories of Elastic Plates, Noordhoff, Leyden, Netherlands. Plantema, F.J. (1966). Sandwich Construction, The Bending and Buckling of Sandwich beams, Plates and Shells, John Wiley & Sons, New York. Reddy, J.N. (1984). Energy and Variational Methods in Applied Mechanics, John-Wiley & Sons, New York. Reddy, J.N. (1997). Mechanics of Laminated Composite Plates: Theory and Analysis, CRC Press, Boca Raton, Florida. Reddy, J.N. and Miravete, A. (1995). CompositeLaminates, CRC Press, Boca Raton, Florida. Shames, I.H. and Dym, C.L. (1985). Energy and Finite Element Methods in Structural Mechanics, McGraw-Hill, New York. Szilard, R. (1974). Theory and Analysis of Plates, Classical and Numerical Methods, Prentice-Hall, Englewood Cliffs, New Jersey. Timoshenko, S.P. and Woinowsky-Krieger, S. (1959). Theory of Plates and Shells, McGraw Hill, New York. Timoshenko, S.P. and Woinowsky-Krieger, S. (1970). Theory of Plates and Shells, McGrawHill, Singapore. Troitsky, M.S. (1976). Stiffened Plates - Bending, Stability and Vibrations, Elsevier, New York. Ugural, A.C. (1981). Stresses in Plates and Shells, McGraw-Hill, New York. Way, S. (1962). "Plates," in Handbook of Engineering Mechanics, McGraw-Hill, New York. Yu, Y.Y. (1996). Vibrations of Elastic Plates, Springer-Verlag, New York. Zienkiewicz, O.C. (1977). Thefinite element method, 3~aEdition, McGraw-Hill, London.
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APPENDIX I GAUSSIAN QUADRATURE SUBROUTINES
* This program GAUSSIAN.F calculates the integrated value of the following * matrix over area A using the Gaussian quadrature method:
*
/O /
J
A
1 ~ .
1"1 .
~'1] .
" ' "
"'"
qm-1 ~'1]m-1
.
~ m - I ~m-l~] ...
dA ~m-1q m-1
and stores the data in a data file called COEF.TAB MAIN GAUSS IMPLICIT DOUBLE PRECISION (A-H, O-Z) INTEGER M,N,NBOUND PARAMETER (M=65,N=40,NBOUND=20) DIMENSION W(N,Z),YLENGTH(N),YY1 (N) DIMENSION COEF(M,M),IXCOEF(M,M),IYCOEF(M,M) DIMENSION TTT(M,M) DIMENSION Y 1(NBOUND),IY 1X(NBOUND),Y2(NBOUND),IY2X(NBOUND) OPEN(20,FILE='INPUT.DAT ') READ(20,*)MM * Read in the dimension of the matrix to be integrated READ(20,*)NN * Read in the number of Gaussian points to be used CALL GA(MM,NN,NBOUND,W,YLENGTH,YY 1,COEF,IXCOEF,IYCOEF, Y 1,IY 1X,Y2,IY2X,TTT) STOP END
SUBROUTINE GA(M,N,NBOUND,W,YLENGTH,YY1,COEF,IXCOEF, IYC OEF,Y 1,IY 1X,Y2,IY2X,TTT) * This subroutine GA is to generate the integrated values of the matrix 191
Appen. I Gaussian Quadrature Subroutines
192
IMPLICIT DOUBLE PRECISION (A-H, O-Z) DIMENSION DIMENSION DIMENSION DIMENSION * * * * * * * * * * *
W(N,2),YLENGTH(N),YY 1(N) COEF(M,M),IXCOEF(M,M),IYCOEF(M,M) TTT(M,M) Y 1(NBOUND),IY 1 X ( N B O U N D ) , Y 2 ( N B O U N D ) , I Y 2 X ~ O U N D )
M = Dimension of the integrated matrix N = Number of Gaussian points NBOUND = Mximun number of terms in the function of integral limits W(N,2) = Gaussian points and weight coefficients YLENGTH(N) = Half width of the area at Gaussian points along x YY1 (N) = Starting y-coordinates at Gaussian points along x COEF(M,M) = Integrated matrix IXCOEF(M,M) = Index of power over x IYCOEF(M M) = Index of power over y ( Y I , I Y 1 X ) = Function of lower integral limit (Y2,IY2X) = Function of upper integral limit OPEN(UNIT= 10, FILE='GAUSSPNT.DAT') READ(10,*) ((W(I,J),J=I,2),I=I,N) CLOSE(10)
* Read in the Gaussian points and weight coefficients
10 20
DO 20 I=I,M DO 10 J= 1,M COEF(I,J)=0. IXCOEF(I,J)=I-1 IYCOEF(I,J)=J- 1 CONTINUE CONTINUE
* Form matrix before integrating READ(20,*)NPARTS * Read in the number of integral parts
30
DO 90 II=I,NPARTS READ(20,*)X1,X2 READ(20,*)NY1 DO 30 I I - I , N Y 1 READ (20,*)Y 1(I 1),IY 1X(I 1) CONTINUE READ(20,*)NY2 DO 40 I2=1,NY2 READ(20,*)Y2(I2),IY2X(I2)
Appen. I Gaussian Quadrature Subroutines 40
CONTINUE Read in the integral limits along the x and y directions CALL XYLENGTH(XLENGTH,YLENGTH,YY1,N,W,X1,X2,Y 1,IY1X,NY1, $ Y2,IY2X,NY2) DO 60 I=I,M DO 50 J=I,M CALL GAUSSVALUE(W,N,X1,XLENGTH,YY1, $ YLENGTH,TTT(I,J),IXCOEF(I,J),IYCOEF(I,J)) 50 CONTINUE 60 CONTINUE DO 80 II=I,M DO 70 I2=l,M C O EF (I 1,I2)=C OEF (I 1,I2)+ TTT (I 1,I2 ) 70 CONTINUE 80 CONTINUE 90 CONTINUE *
OPEN(30,FILE='COEF.TAB ') WRITE(30,*) ((COEF(I,J),J=I,M),I= 1,M) RETURN END
SUBROUTINE XYLENGTH(XLENGTH,YLENGTH,YY 1,N,W, X1,X2,Y 1,IY1X,NY 1,Y2,IY2X,NY2) * This subroutine XYLENGTH calculates the half lengths along the x and y direcitons * over the integral domain IMPLICIT DOUBLE PRECISION (A-H, O-Z) DIMENSION Y 1(MY 1),IY 1X(NY 1),Y2 (NY2),IY2X(NY2),W(N,2), YLENGTH(N),YY 1(N) EXTERNAL FUNCTION F 1D XLENGTH=(X2-X 1)/2.
10
20 30
DO 3 0 I = 1, N XT=XLENGTH*W(I, 1)+(X1 +XLENGTH) YYI(I)=0. DO 10 K-1,NY1 YYI(I)=YYI(I)+F1D(YI(K),IY1X(K),XT) CONTINUE YY2-0. DO 20 K-1,NY2 YY2=YY2+F 1D(Y2(K),IY2X(K),XT) CONTINUE YLENGTH(I) = (YY2-YY 1(I))/2. CONTINUE
193
Appen. I Gaussian Quadrature Subroutines
194
RETURN END
SUBROUTINE GAUSSVALUE(W,N,X 1,XLENGTH,YY1,YLENGTH, F,IFX,IFY) * This subroutine GAUSSVALUE sums the Gaussian integration value IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION W(N,2),YLENGTH(N),YY 1(N) EXTERNAL FUNCTION F2D FF=I.0 F=0.0
10 20
DO 2 0 I = 1, N XT=XLENGTH* W(I, 1)+(X 1+XLENGTH) SUM=0.0 DO10J=I,N YT -- YLENGTH(I)*W(J, 1) + (YY 1(I)+YLENGTH(I)) SUM=SUM+W(J,2)*F2D(FF,IFX,IFY,XT,YT) CONTINUE F=F+W(I,2)* SUM*YLENGTH(I)*XLENGTH CONTINUE RETURN END
DOUBLE PRECISION FUNCTION F1D(F,I,X0) IMPLICIT DOUBLE PRECISION (A-H, O-Z) IF(I.EQ.0.AND.X0.EQ.0.D0) THEN TEMPI=I.0 ELSE TEMPI=X0**I ENDIF F1D=F*TEMP1 RETURN END
DOUBLE PRECISION FUNCTION F2D(F,I,J,X0,Y0) IMPLICIT DOUBLE PRECISION (A-H, O-Z)
Appen. I Gaussian Quadrature Subroutines IF(I.EQ.0.AND.X0.EQ.0.0) THEN TEMPI=I.0 ELSE TEMPI=X0**I ENDIF IF(J.EQ.O.AND.YO.EQ.O.O) THEN TEMP2=I.0 ELSE TEMP2=Y0**J ENDIF F2D=F*TEMP 1*TEMP2 RETURN END
INPUT.DAT
3 20
The dimension of the integrated matrix, M The number of Gaussian points The number of integral domain parts
-1 .,0.
The integral limits X1 (lower) and X2 (upper) for the first part
1 0.,0
The number of terms of function Y l(x) The 1st number is the coefficient and 2nd the power o f x for the 1st term
2 2.,1 2.,0
The number of terms of function Y2(x) The 1st number is the coefficient and 2nd the power ofx for the 1st term The 1st number is the coefficient and 2nd the power o f x for the 2nd term
.,1.
The integral limits X 1 (lower) and X2 (upper) for the second part
1 0.,0
The number of terms of function Y l(x) The 1st number is the coefficient and 2nd the power ofx for the 1st term
2 -2.,1 2.,0
The number of terms of function Y2(x) The 1st number is the coefficient and 2nd the power o f x for the 1st term The 1st number is the coefficient and 2nd the power ofx for the 2nd term
COEF.TAB
2.000000000000000
1.333333333333333
1.333333333333333
195
196
Appen. I Gaussian Quadrature Subroutines
5.551115123125783E-17 -5.551115123125783E-17 0.00000000000000E+00 0.3333333333333334 0.1333333333333333 8.88888888888888 E-02
APPENDIX II SUBROUTINES FOR MATHEMATICAL OPERATIONS ON POLYNOMIALS
************************************************************************
* The following subroutines PLUS, MULTI, DIFXY and SIMPLE perform mathematical * operations on the polynomial functions (for details, refer to Xiang, Wang and * Kitipomchai, 1995)
SUBROUTINE PLUS(A 1,IR1 ,IS 1,N 1,A2,IR2,IS2,N2,NP) IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION A1 (*),IR1 (*),IS 1(*),A2(*),IR2(*),IS2(*) II=l DO 30 J=I,N2 IND=0 DO 10 I=I,N1 IF(IR1 (I).EQ.IR2(J).AND.IS 1(I).EQ.ISZ(J)) THEN A1 (I)=A 1(I)+AZ(J) IND=I GOTO 20 ENDIF 10 CONTINUE IF(IND.EQ.1) GOTO 30 20 A1 (NI+II)=A2(J) IR1 (NI+II)=IR2(J) IS I(NI+II)=IS2(J) NP=N 1+II II=II+l IND=0 30 CONTINUE CALL SIMPLE(AI,IRI,IS 1,NP,NS) NP-NS RETURN END
197
Appen. H Mathematical Operations on Polynomials
198
SUBROUTINE MULTI(A 1,IRI,IS 1,N1,A2,IR2,IS2,N2,NM) IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION A1 (*),IR1 (*),IS 1(*),A2(*),IR2(*),IS2(*),AW(500),IRW(500), ISW(S00),AA(S00),IRR(500),ISS(500) DO 101= 1,500 AA(I)=0.0 IRR(I)=O ISS(I)=0 10 CONTINUE NM=0 DO 30 J=I,N2 DO 20 I=I,N1 AW(I)=AZ(J)*AI(I) IRW(I)=IR2(J)+IR 1(I) ISW(I)=ISZ(J)+IS 1(I) 20 CONTINUE CALL PLUS(AA,IRR,IS S,NM,AW,IRW,ISW,N1,NP) NM=NP 30 CONTINUE DO 40 I=I,NM A1 (I)=AA(I) IR1 (I)=IRR(I) IS I(I)=ISS(I) 40 CONTINUE CALL SIMPLE(A1 ,IR1 ,IS 1,NM,NS) NM=NS RETURN END
SUBROUTINE DIFXY(A,IR,IS,N,IX,IY,ND) IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION A(*),IR(*),IS(*) DO 20 I=I,IX DO 10 J=I,N A(J)=A(J)*IR(J) IR(J)=IR(J)- 1 CONTINUE 10 20 CONTINUE
Appen. II Mathematical Operation for Polynomials DO 40 I=I,IY DO 30 J=I,N A(J)=A(J)*IS(J) IS(J)=IS(J)-I 30 CONTINUE 40 CONTINUE CALL SIMPLE(A,IR,IS,N,ND) RETURN END
SUBROUTINE SIMPLE(A,IR, IS,N,NS) IMPLICIT DOUBLE PRECISION (A-H, O-Z) DIMENSION A(*),IR(*),IS(*),AW(500),IRW(500),ISW(500) II=0 DO 20 I=I,N IF(DABS(A(I)).LE. 1.0D-a 0) GOTO 20 II=II+l AW(II)=A(I) IRW(II)=IR(I) ISW(II)=IS(I) DO 10 J=I+ 1,N IF(IR(I).EQ.IR(J).AND.IS(I).EQ.IS(J)) THEN AW(II)=AW(II)+A(J) IRW(II)=IR(I) ISW(II)=IS(I) A(J)=0.0 END IF 10 CONTINUE 20 CONTINUE NS=II DO 30 I=I,NS A(I)=AW(I) IR(I)=IRW(I) IS(I)-ISW(I) 30 CONTINUE END
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SUBJECT INDEX
Bending rotations Boundary conditions of clamped edge of free edge of simply supported edge Buckling Closed form solution Coordinate system Polar rectangular Cartesian skew Displacement components Effect of shear deformation Effect of rotary inertia Eigenvalue equation Energy functionals in rectangular coordinates in polar coordinates in skew coordinates Equations of motion Gaussian quadrature subroutines Geometric stiffness matrix Hamilton's principle Initial stresses Internal support Kirchhoff plate Lagrangian Lagrangian multiplier Mass matrix Mindlin plate Mode shapes Nonuniform thickness Pasternak foundation Perforated plate Plate shapes Annular Annular sectorial Circular Elliptical Rectangular
6, 13, 134 14, 14, 14, 18,
16, 28 15, 28 16, 17, 28, 90 166,
27 89 134 6 2,5, 169 2, 5, 15, 19 25, 36, 38, 95, 136, 166, 168, 170, 173, 177 9 35 135 15 191 166 9 166 175 5, 6, 15, 91,166, 168, 181 9, 35, 93, 135, 181 176 25, 37, 94, 136, 173 5, 28, 90, 166 75, 116, 117, 158 173 167 179 27, 40 27, 56, 71 27, 40 166, 177 24, 26, 89 201
202
Subject Index
Sectorial 27, 56 Skew 133 Trapezoidal 21 Triangular 20 Plate vibration 1, 18, 24, 93, 135 Polynomial 6, 26, 36, 38, 93, 135, 179 Prestressed membrane 18, 167 Reentrant comer 178 Ritz functions 3, 26, 35, 38, 93, 135, 178 Ritz method 2, 24, 36, 38, 94, 136, 175, 180 Sandwich plate 179 Shear correction factor 2, 5, 23 Skew coordinates 133 Stiffener 169 Stiffness matrix 25, 36, 39, 95, 136, 173 Strain 8 Strain-displacement relations 7 Stress 7 Stress resultants-displacement relations 7 Vibration frequency relationship Winkler foundation 167
