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University Microfilms International A Bell & Howell Information Company 300 North Zeeb Road, Ann Arbor, Ml 48106-1346 USA 313/761-4700 800/521-0600 Order Number 9105122
Thick plate theories, with applications to vibration
Hanna, Nagy Fouad, Ph.D.
The Ohio State University, 1990
Copyright ©1991 by Hanna, Nagy Fouad. All rights reserved.
UMI 300 N. Zeeb Rd. Ann Arbor, MI 48106 Thick Plate Theories, with Applications to Vibration
DISSERTATION
Presented in Partial Fulfillment of the Requirements for
the Degree Doctor of Philosophy in the Graduate
School of The Ohio State University
By N. F. HANNA, RSir. M.Sc.
* » * * *
The Ohio State University
1990
Dissertation Committee: Approved by
Prof. O. G. McGee
Prof. D. A. Mendelsohn ProLA. W.LE1SSA Department of Engineering Mechanics DEDICATION
To my Lord Jesus Christ ACKNOWLEDGEMENTS
I would like to thank most my adviser, Professor A. W. Leissa for suggesting the topic and advising me throughout its development
Special gratitude and sincere regards are extended to Professor O. G. McGee and
Professor D. Mendelsohn for being on the reading committee.
Finally, greatest sincere thanks and appreciation go to my family for their love, support, understanding and encouragement without which this study would never have been completed. VITA
August 16, 1957 Bom, Cairo, EGYPT
June 1979 B.S. in Civil Engineering Ain Shams University, EGYPT
August 1985 M.S. in Civil Engineering The Ohio State University Columbus, Ohio, USA
1981 - 1983 Teacher Assistant, Civil Engineering, Helwan Univer sity, Cairo, EGYPT
1981 - 1983 Structural engineer, Sabbour Consulting firm, Cairo, EGYPT
1985 - Present Graduate Teacher Assistant, Math Department, The Ohio State University, Columbus, Ohio, USA
FIELDS OF STUDY
Major Field : Engineering Mechanics.
Mechanics : Strength of Materials, Continuum Mechanics, Composite Materials, Vibrations of Continuous media.
Mathematics : Linear Algebra, Tensors, Calculus of Variations, Calculus.
Numerical Analysis : Numerical Integration, Numerical Methods of Linear Algebra, Ritz Method, Finite Element Method. TABLE OF CONTENTS
PAGE
D e d ica tio n ...... ii
ACKNOWLEDGEMENTS...... iii
VITA ...... iv
LIST OF FIGURES...... vii
LIST OF TABLES ...... ix
CHAPTER
I. INTRODUCTION ...... 1
II. LITERATUREREVIEW ...... 3
2.1 Introduction ...... 3 2.2 Classical Plate Theory (Thin Plates):- ...... 4 2.3 First Order Shear Deformation Theories ...... 7 2.3.1 Reissner's Approach ...... 8 2.3.2 Mindlin's A pproach ...... 11 2.3.3 Ambartsumyan's Approach ...... 18 2.4 Higher-Order Shear Deformation T heories ...... 24 2.4.1 Higher-Order Theories using Reissner's Approach .. . 25 2.4.2 Higher-Order Theories using Mindlin's Approach .. . 32 2.4.3 Higher-Order Theories using Ambartsumyan's Approach ...... 44 2.5 Discrete I.ayer T heories ...... 47 2.6 Two Dimensional Analysis using Finite Elements ...... 50 2.7 Three Dimensional Analysis ...... 54 2.7.1 Exact Solutions ...... 54
- v - 2.7.2 Approximate Solutions ...... 56 2.8 Other Reviews of the L iterature ...... 57 2.9 Other Ref erences ...... 58
m . ENERGY FUNCTIONALS FOR FREE VIBRATIONS OF RECTANGULAR PLATES ...... 59
3.1 Classical Plate T h e o ry ...... 60 3.2 Mindlin's T h e o ry ...... 61 3.3 Modified Kant T h eo ry ...... 63
TV. DIFFERENTIAL EQUATIONS AND BOUNDARY CONDITIONS FOR MODIFIED KANT TH EO RY ...... 68
V. FREE VIBRATION ANALYSIS OF ISOTROPIC HOMOGENEOUS RECTANGULAR PLATES USING THE RITZ M ETH O D ...... 76
5.1 Mindlin Theory : ...... 76 5.2 Modified Kant T h eory ...... 79
VI. RESULTS FOR FREE VIBRATIONS OF COMPLETELY FREE RECTANGULAR THICK PL A T E S...... 82
6.1 Introduction ...... 82 6.2 Symmetry Consideration ...... 83 6.3 Convergencc Study ...... 86 6.4 Comparison with Classical Plate Theory ...... 92 6.5 Thick Plate Deformations ...... 98 6.6 Effect of Varying Poisson's Ratio upon the Frequency Param eter...... I ll 6.7 Effect of Aspect Ratio and Thickness-to-Width Ratio .... 117
VII. DISCUSSION...... 120
7.1 Summary and Conclusions ...... 120 7.2 Recommendations for Further Research ...... 121
APPENDICES
A. MODE SHAPES FOR POISSON'S RATIO 0 3 122 B. MODE SHAPES FOR OTHER POISSON'S RATIOS...... 198
REFERENCES ...... 222
- vii -
it LIST OF FIGURES
Rectangular plate (cartesian coordinates). . 6 General coordinates used in the analysis 77
SS-2 a/b=l, h/b=0.5 ...... 109
SS-5 a/b=l, h/b=0.^ ...... 110
Mode shape coordinates ...... 122
Mode shape SS-1 , a/b =1 ...... 123
Mode shape SS-2 , a/b =1 ...... 124
Mode shape SS-3 , a/b = 1 ...... 125
Mode shape SS-4 , a/b =1 ...... 126
Mode shape SS-5 , a/b =1 ...... 127
Mode sha])e SS-6 , a/b =1 ...... 128
Mode shape SS-7 , a/b =1 ...... 129
Mode shape SS-8 , a/b =1 ...... 130
Mode shape SS-9 , a/b =1 ...... 131
Mode shape SS-10, a/b = 1 ...... 132
Mode shape SS-11 , a/b = 1 ...... 133
Mode shape SA-1 , a/b =1 ...... 134
Mode shape SA-2 , a/b =1 ...... 135
Mode shape SA-3 , a/b =1 ...... 136
Mode shape SA-4 , a/b =1 ...... 137 A.17 Mode shape SA-5 , a/b -= ] ...... 138
A.18 Mode shape SA-6 , a/b = 1 ...... 139
A.l 9 Mode shape SA-7 , a/b = 1 ...... 140
A.20 Mode shape SA-8 , a/b = 1 ...... 141
A.21 Mode shape SA-9 , a/b =1 ...... 142
A.22 Mode shape SA-10 , a/b =1 143
A.23 Mode shape SA-11 , a/b =1 144
A.24 Mode shape AA-1 , a/b = 1 ...... 145
A.25 Mode shape AA-2 , a/b = 1 ...... 146
A.26 Mode shape AA-3 , a/b = 1 ...... 147
A.27 Mode shape AA-4 , a/b = 1 ...... 148
A.28 Mode shape AA-5 , a/b = 1 ...... 149
A.29 Mode shape AA-6 , a/b = 1 ...... 150
A.30 Mode shape AA-7 , a/b = 1 ...... 151
A.31 Mode shape AA-8 , a/b = 1 ...... 152
A.32 Mode shape AA-9 , a/b = 1 ...... 153
A.33 Mode shape AA-10 , a/b = 1 ...... 154
A.34 Mode shape SS-1 , a/b = 2 155
A.35 Mode shape SS-2 , a/b = 2 ...... 156
A.36 Mode shape SS-3 , a/b = 2 157
A.37 Mode shape SS-4 , a/b = 2 158
A.38 Mode shape SS-5 , a/b = 2 ...... 159
A.39 Mode shape SS-6 , a/b = 2 160
A.40 Mode shape SS-7 , a/b = 2 161
A.41 Mode shape SS-8 , a/b = 2 162
A.42 Mode shape SS-9 , a/b = 2 163
- ix - A43 Mode shape SS-10, a/b = 2 ...... 164
A44 Mode shape SS-11 , a/b - 2 ...... 165
A.45 Mode shape SA-1 , a/b = 2 ...... 166
A.46 Mode shape SA-2 , a/b = 2 ...... 167
A.47 Mode shape SA-3 , a/b = 2 ...... 168
A.48 Mode shape SA-4, a/b = 2 ...... 169
A.49 Mode shape SA-5, a/b = 2 ...... 170
A.50 Mode shape SA-6, a/b = 2 ...... 171
A.51 Mode shape SA-7 , a/b = 2 ...... 172
A.52 Mode shape SA-8, a/b = 2 ...... 173
A.53 Mode shape SA-9, a/b = 2 ...... 174
A.54 Mode shape SA-10, a/b = 2 ...... 175
A.55 Mode shape SA-11 , a/b = 2 ; . . 176
A.56 Mode shape AS-1 , a/b = 2 ...... 177
A.57 Mode shape AS-2 , a/b = 2 ...... 178
A.58 Mode shape AS-3 , a/b = 2 ...... 179
A.59 Mode shape AS-4, a/b = 2 ...... 180
A.60 Mode shape AS-5 , a/b = 2 ...... 181
A.61 Mode shape AS-6, a/b = 2 ...... 182
A.62 Mode shape AS-7 , a/b = 2 ...... 183
A.63 Mode shape AS-8, a/b = 2 ...... 184
A.64 Mode shape AS-9 , a/b = 2 ...... 185
A.65 Mode shape AS-10, a/b = 2 ...... 186
A.66 Mode shape AS-11 , a/b = 2 ...... 187
A.67 Mode shape AA-1 , a/b = 2 ...... 188
A.68 Mode shape AA-2 , a/b = 2 ...... 189
- x - Mode shape AA-3 , a/b = 2 190
Mode shape AA-4 , a/b = 2 191
Mode shape AA-5 , a/b = 2 192
Mode shape AA-6, a/b = 2 193
Mode shape AA-7 , a/b = 2 194
Mode shape AA-8 , a/b = 2 195
Mode shape AA-9 f a/b = 2 196
Mode shape AA-10 , a/b = 2 197
Mode shape SS-4 , a/b = 1 199
Mode shape SS-5 , a/b = 1 200
Mode shape SS-6 , a/b = 1 201
Mode shape SS-7 , a/b = 1 202
Mode shape SS-8 , a/b = 1 203
Mode shape SS-9 , a/b = 1 204
Mode shape SS-10 , a/b = 1 205.
Mode shape SS-11 , a/b = 1 206
Mode shape SA-4 , a/b = 1 207
Mode shape SA-5 , a/b = 1 208
Mode shape SA-6 , a/b = 1 209
Mode shape SA-7 , a/b = 1 210
Mode shape SA-8 , a/b = 1 . 211
Mode shape SA-9 , a/b = 1 . 212
Mode shape SA-10, a/b = 1 213
Mode shape SA-11 , a/b = 1 214
Mode shape AA-4 , a/b = 1 . 215
Mode shape AA-5 , a/1) = 1 . 216 B.19 Mode shape AA-6, a/b = 1 ...... 217
B.20 Mode shape AA-7, a/b = 1 ...... 218
B.21 Mode shape AA-8 , a/b = 1 ...... 219
B.22 Mode shape AA-9 , a/b = 1 ...... 220
B.23 Mode shape AA-10 , a/b =1 ...... 221
- xii - LIST OF TABLES
TABLE PAGE
1. Comparison among the first order shear deformation theories ...... 23
2. Comparison among linear high-order shear deformation theories 45
3. Combination of the polynomial terms used in displacement expressions ...... 84
4. Conevrgence study for Ritz method based on Mindlin's theory h/b=0.1 ...... 88
5. Convergence study for Ritz method based on Mindlin's theory h /b = 0 .5 ...... 89
6. Convergence study for Ritz method based on modified Kant theory h/b=0.1 ...... 90
7. Convergence study for Ritz method based on modified Kant theory h /b = 0 .5 ...... 91
8. Comparison of the nonzero frequency parameters with classical plate theory a/b = l ...... 94
9. Comparison of the nonzero frequency parameters with classical plate theory a/b=2 ...... 96
10. Normalized eigenvectors for thick (h/b=0.5), square (a/b=l) plates from the modified Kant theory (first distinct terms of 36 for the four generalized displacements) SS-modes ...... 100
11. Normalized eigenvectors for thick (h/b=0.5), square (a/b=l) plates from the modified Kant theory (first distinct terms of 36 for the four generalized displacements) SA-modes ...... 103
12. Normalized eigenvectors for thick (h/b=0.5), square (a/b=l) plates from the modified Kant theory (first distinct terms of 36 for the four generalized displacements) AA-modes ...... 106
13. Effect of Poisson's ratio on the frequency parameter. Case of a/b=l , h/b=0.1 113
- xiii - 14. Effect of Poisson's ratio on the frequency parameter. Case of a/b=l , h/b=0.5 ...... 114
15. Effect of Poisson's ratio on the frequency parameter. Case of a/b= 2, h/b=0.1 ...... 115
16. Effect of Poisson's ratio on the frequency parameter. Case of a/b=2, h/b=0.5 ...... 116
17. Effect of a/b and h/b on the frequency parameter (Mindlin) ...... 118
18. Effect of a/b and h/b on the frequency parameter (modified Kant) ...... 119
- xiv - CHAPTER 1
INTRODUCTION
The analysis of plates has attracted the interest of researchers for a long time ago.
Basically, the plate problem is a three dimensional problem, but the idea of approximat
ing it with a two dimensional problem has tempted a lot of researchers.
Starting with the Kirchhoff-Love theory, many different theories were developed
to include the effect of different factors such as shear deformations, rotary inertia, nor
mal strains, surrounding enviroment, etc. In their theory, Kirchhoff and Love assumed
that normals to the midsurface of the plate before deformation remain straight and
normal 1o the midsurf ace after deformation. This theory has long served as an impor
tant base for structural analysis and design.
However, this theory is limited to studying the vibrations of plates with inodes
having waves of short length. As the plate thickness increases, comparison of the
Kirchhoff-Love theory (often called classical plate theory) with the three- dimensional analysis of the plate problem shows significant inaccuracy in the solutions. This inaccu
racy provided the motive for many researchers to develop different approaches to solve
the different plate problems (static, dynamic, stability). The main issue for this work was to include the effects of shear deformation and rotary inertia to develop the so called thick plate theories.
In the present work, a literature review is made of the various approaches dealing with thick plates, and subsequently, regroup them according 1o the underlying assump tions used. Taken together, two shear deformation plate theories are adopted in the
present anlysis, that is, a first order one by Mindlin, and a higher order one by Kant
and modified by the author.
Because physical edge conditions which are completely free are the only ones for
which no difficulty exists in providing consistent mathematical boundary conditions,
the completely free, thick, rectangular plate was chosen for a comparison problem.
Moreover, for this problem there is a lack of numerical results for free vibration fre
quencies and mode shapes in the published literature.
The Ritz method is combined with both Mindlin and modified Kant theories to
determine the natural frequencies of the vibration problem. The displacements are
expressed in terms of simple algebraic polynomials, which are mathematicaly complete.
Studies are made to check the convergence of the solution and to examine the effecl of the different parameters upon the frequency.
in the present work, the second chapter is devoted to a comprehensive literature
review, and the rest of the work provides an explanation of the method used and pro
vides numerical results. CHAPTER II
LITERATURE REVIEW
2.1 INTRODUCTION
The improvement of the Kirchhoff-Love theory (classical plate theory) to permit the analysis of thick plates has been the interest of many researchers. The major assumptions separating thin and thick plate theories are shear deformation and rotary inertia. Much research has been done to include these effects in the theory of plates.
The objective of this chapter is to categorize this research into groups according to the approach adopted in each work and to compare the underlying assumptions governing plate theories.
Generally, the plate theory works can be divided into the following major groups:-
1. Two-dimensional analysis which does not involve the finite element method :-
This group includes all works which develop a shear delormation theory and
solve the plate problem either analyticaly or numerically using a continuum
methods of analysis. This continuum methods may be divided into different sub
groups (first-order plate theories and higer-order shear deformation plate theo
ries) as will be seen when they will be discussed in later sections [1-164]
2. Two-dimensional analysis using finite element:- This group includes all works
which develop numerical solutions using the finite element method or a similar
procedure, such as the finite slrip or finite layer methods [165-205] 3. Three-dimensional analysis:- The plate problem is considered as three-
dimensional [206-241]
4. Others:- This group includes all works written in languages other than English
for which il was difficult for the author to understand all details; yet they are
related to the topic of thick plate deformation.
First the classical plate theory will be discussed briefly so that it will be easy to
compare the different thick plate theories to it. In subsequent sections the thick plate
theories will be discussed.
In all sections, discussion of the theories will be based on cartesian coordinates (for
rectangular plates) and it is uderstood that the theory can be transformed to any other coordinate system by means of the proper transformation methods. Thus, each group of references will include work done with rectangular plates as well as any other shapes
(circular, skew, triangular, etc.).
2 2 CLASSICAL PLATE THEORY (THIN PLATES):-
The assumptions made by KirchholT to establish the first base of two-dimensional plate theory will be first discussed briefly. No references are included for this well- known theory.
Considering cartesian coordinates (Fig. 2.1), the basic assumptions used in this theo ry are:-
1. Displacement components are small compared with the thickness of the plate.
2. Normals to the midsurface of the plate before deformation remain straight and
normal 1o the midsurface after deformation.
3. Transverse normal strain ez, is negligible. 4. Transverse normal stresses oa are small compared to the inplane normal stress
es o xx,o and can be neglected.
These assumptions can be expressed mathematically through the following dis placement field:-
u(x,y,z) = uo- z w M (21)
v(x,y,z) = vo - z way (22)
w(x,y,z) = wo (23) where u ,v , w are the displacement components in the x ,y, z directions respectively, and u0, v0 are the in plane displacement components of the middle surface. un, v0and w0 are functions in x and y only. (A comma followed by a subscript indicates a derivative with respect to the variable following the comma).
The corresponding strain-displacement relations are:-
u = u —z w 'xx a ax axx (2.4)
V„ = V — ZW '.v.v •> ay ayy (2.5) II 'zz = * c (2.6)
+ + — \ v u ,y v .x= u ay v ax 2 z w axy (2.7)
;Xz = u a. + w .x = —w ax + w ax = 0 (2.8)
V, + W = — W + W yz ~ a. ,y ay ay = 0 (2.9) One can see that the effect of transverse shear and normal strains
( c H, e xz and e vz ) are all equal to zero, which is consistent with assumptions (2) and
(3). The effect of shear deformation increases as the plate thickness (h) increases rela tive to the plate width (bj. Comparison with three dimensional solutions shows that the classical plate theory is applicable up to b /h ^2 0 . I-or b/h<20 thick plate theories that include shear deformation, should be uiili/ed. 6
4 1
*
Figure 2.1: Rectangular plate (cartesian coordinates). The classical plate theory underestimates the deflection (in the flexure problem)
and overestimates the natural frequencies (in the vibration problem) because it overes
timates the rigidity of the plate (that is, it neglects the flexibility due to shear). For
these reasons researchers began to look for other theories which give more precise
results for thick plates, as well as being applicable to thin plates. These theories are dis
cussed ih the following sections.
2 3 FIRST ORDER SHEAR DEFORMATION THEORIES
[1- 122]
This theory is the first approximation of thick plate theory. The difference
between the classical plate theory and the first order shear deformation theory is in
replacing only assumption (2) (on page 4)by the following assumption :-
(2) Normals to the midsurface of the plate before deformation remain straight (but not
necessary normal) to the midsurface after deformation. (Some theories even relax this assumption and assume that normals are neither straight nor normal afler deforma tion.) A1J other assumptions remain the same.
Three major approaches have been used by researchers to develop this theory. The first one is based upon an assumed inplane normal and shear stress field as well as dis placement field. The second approach starts with an assumed displacement field, while the third one starts with an assumed stress field. The first approach was first adopted by Reissner [1-16], and has been followed by many others [17-34] . Hencky [35] was the first one to use the second approach and then Mindlin [36] used the same approach
(although usually known by Mindlin's name) and then used by other researchers for different applications [37-118]. The third approach was established by Ambartsumyan [119] and used by others [120-122]. The differences between these approaches is clari
fied by discussing each one individually in detail.
23.1 Reissner's Approach
[1-34]
Reissner [l] began the development of his theory by assuming linearly distributed
inplane normal and shear stresses along the thickness of the plate as follows:-
a™ = -T-TP> (2.10) h /6 h/2
7 cryvyv = h -2/6 Z-T h/2 7 7
7 ^ = ~P ~ T7T 12.12) J hV6 h/2 where Ms , Mv are bending couples, and Mxy is the twisting couple (per unit length).
Using equations (2.10) - (2.12) , and solvingthe differential equations of equilibrium for the rest of the stress components, Reissner got the following expressions:-
'--•sM ■-(&)"’] t2i3)
“V* = 2h/3 [ 1 ~ (• h / 2 -) ] (2"l4)
(2-15) where Qx , Qy are the shear stress resultants and q is the external transverse pressure
(positive in tension) applied at the top surface of the plate (z = h /2 ) . Equations (2.13) -
(2.15) satisfy the following conditions at the top and bottom surfaces of the plate:-
a z7 ( x , y , h/2 ) = q , cr?j ( x , .v , -h/2 ) = 0 (2.1 b)
crxz ( x , y , ± h/2 ) = 0 , crw ( x . v , ± h/2 ) = 0 (2.17) and the stress resultants satisfy:- Q„, + «»„ *= -q 2 (.18 )
o , = M „ + M „ v (2.19)
Then Reissner applied Casligliano's theorem of least work combined with the
Lagrangian multiplier method of the calculus of variations to lake into consideration equations (2.18) - (2.20). In mathematical expressions this takes the following form:-
sn = 6 j Te f 2P I {tr- + + ~ 2v[
+ O ’y y * rn ] + 2( 1 +*/) [o*y + + < r2yj ]} dx dy dz
~ J I I ^ u + ^ v + ^ w j d s d z -*K 2 XOry
0 it + f f >( Q „ + Qy.v + + \ < x ,y ) ( mxa + msv>. - qx ) *?>*?)
+ Af ( x . y ) ( MxyA + My v - Qv ) ] dx dy } = 0 (2.21) where II is the total energy and 8 indicates the first variation, o n , o^, ando^ are the boundary stresses, and u, v and w are the boundary displacements which are assumed
(to satisfy the previously slated assumptions) as follows:
u = z0 x (2.22)
v = z0y (2.23)
w = (2.24) where w , /3X and j3 represeni Ihe rate of change of displacement and rotation quanti ties through the plate thickness, in equation (2.21) A;t, Au, Ac are Lagrange multipliers.
Integrating by parts and evaluating the variations, Reissner concluded that:- 10
xa = W (2.25)
Xb = Px (2.26)
Ac = Py (2.27) for any part of the plate. Subsequently, he obtained the following differential equa tions:-
Q ,-^ V !Qs + Brf ^ 7 q,=-D (V2W)5 (2.28)
% - W + kKT^) % ~ ~ D < ^ >, (2-2q> where V 2 = ( ),xx + ( )yv
Equations (2.28), (2.29) with equation (2.18) are the governing equations in Reiss- ner's theory. They are sixth order defferenlial equations requiring three boundary con ditions to be satisfied at each edge. Once these equations are solved for w ,QX and 0 V
(the variables in this theory), these values may be substituted inlo the following equa tions to determine the resl of 1he stress resultants
m, = - D < + - > + t - iofr^TT q (230)
Mv = - D ( W.y, + - > + T Q.v..v - Tofr^T) “ (23‘ >
M„y = -D ( 1 - p ) + 0 „ ) (2.32)
Then substituting the values of the stress resultants in equations (2.10) to (2.15) , the stresses can be determined at any point in the plate.
In this theory, /3X and /J represent quantities which arc equivalent to but not iden tical with components of change of slope (rotations) of the norma) to Ihe undcforincd middle surface, while w is a weighted average, taken over the thickness, of the tran sverse displacements of the points of the plate. In terms of the actual displacements u , v and w , /3X , /3y and w are defined by:
(2.33)
(2.34)
(2.35)
Reissner also showed that the governing equations (2.18), (2.28) and (2.29) can be reduced to two differential equations interms of w and a stress function 4>, such that :-
(2.36)
(2.37)
It was shown that, by projx;r changes, this theory can be extended to solve many different problems such as (he flexure of orthotropic, anisotropic and laminated plate, which was done by Reissner [2-13] and others [22-24] , or to study the problems of plates resting on elastic foundations [18,27,32,33] or vibrating plates [14,22,28]. Also it can be applied to problems dealing with polar coordinates [14-16,20,31-34].
232, Mindlin's Approach
[35-118]
In contrast to Reissner's theory, Mindlin [36] developed his theory by assuming a displacement field. Although Hencky [35] was the first to adopt this approach, it was named after Mindlin partly because; his theory is more comprehensive and partly because llencky's work was written in German.
Based upon the modified assumptions of the shear deformation theory mentioned at the beginning of section 2.3, Mindlin assumed that the inplane displacements are lin 12 ear functions of the thickness coordinate z while the transverse displacement is z inde
pendent This can be expressed in the following forms
u(x,y,z,t) = z^s(x,y ,t) (2.38)
v(x,y,z,t) = z
w(x,y ,z,t) = w0(x,y ,t) (240)
where i/rx, ilry are the actual rotations of the normals to the midsurface in the xandy directions, respectively, and different from wx,wy (rotations in classical plate theory)
by the shear strain. This is different than Reissner's displacement field, which leads to quantities proportional to but not identical with the rotations and to a weighted average across the thickness of the deflections of all points of the plate which lie on a normal to the middle surface.
The coresponding strain-displacemenl relations are
e >:\ = z 0 ru ,’ e yy = z \ ^y.yb
6=0zz
exz = M w o,x * €yz = ^ y + W o.y
€x.v=z(^,y + ^y.x) (2.41)
Equations (2.41) show that the strains e xs, e vy and e xy are linearly distributed through the plate thickness, while 6^ = 0 (as in classical plate theory), and cxzand cvz are constant which is not the case in reality. To overcome this problem, Mindlin intro duced a shear correction factor k2 such that:
G' = k2 G (2.42) where G is Ihe shear modulus, G' is the modified shear modulus. In this work, Mindlin chose the value tt2/\ 2 for k2, which is very close to the one chosen by Reissner
( k2 = 5/fa) in the general theory [2] which leads to the equations given before in the 13
case of isotropic plates.
To get the governing differential equations, Mindlin used the differential equilibri
um equations of the theory of elasticity. By integrating through the thickness after
mullipying the first two equations by z and using the following stress resultant defini
tions :
Mx=D( +„ + *,,> •
= f ( ] — ) ( *xJ + 'As , )
Qs = k2Gh(^s + w0x) , Qy = k2Gh(i/ry + w0y) (243)
where:
D = — 12(1 -v2)
' v = Poisson's ratio
E = Young's modulus,
Mindlin got the following governing equations :•
-y [(1 -v) + (1+^T J-k 'O h (^ x + w0x) = p -^ i/^ u (2.44)
y [ ( 1 - v)V2^v + ( l+v)rv]-k 2Gh(0v + w0y) = *y u (245)
k2G h( V 2 wQ+T)+ q = ph w0u (2.46)
w here:
r = ^.x + ^v.y (247) and p is the mass density j)er unit volume. Equations (2.44) - (2.46) with the suitable
boundary conditions for each problem represeni the problem statement of a typical vibration problem. 14
Other researchers [37-79] showed that Mindlin's theory can be applied to different
problems with the proper changes. It can also be applied for circular or skew, as well as
rectangular plates [80-115].
In order to account for the midplane extensions in the case of inplane forces or unsymmetrical laminated plates, the displacement field (2.38) - (2.39) is modified :
u(x,y ,z) = u0(x,y) + z^x(x,y) (2.48)
v(x,y,z) = v0(x,y) + zi/ry(x,y) (2.49)
w(x,y ,z) = wQ(x,y ) (2.50) where u0,v0 are the midplane extensions in the xandy directions, respectively, as before.
Following the same procedure as Mindlin, Narasimhamurthy [116] developed a theory which did not Use Mindlin's correction factor, k2 . Narasimahaniurlhy's gov erning equations [116] are based upon the following displacement field :
u = _ z w o,x + (z+JjT ) e x“ 8(X + 2G) ( 1 - P ~ )q-* (2’5U
v = ~ - o , + (z+f ^ )0y-8(l^G T(l-1 7 )q, (2’52)
w = w „ + " h [l^_i]v2 [4^_8^_5 ] a 0 8 (1 - 1/) h2 h2 h2 6
- £(\2 + 2G)(z-l4 + 8 (2*53) where 0x,0y,wo are functions of x ,y and t only, and similar to i/fx,tfv,w0 used in
Mindlin's theory, and A = 9XX + 0y y which is analogous to T in Mindlin's equations.
For free vibrations (q = 0 ), the strains are given by :
(2-54) 15
£yy = - zw o.yy + ( z - - ^ 7 ,e v.y (2'55)
t2-56)
- ( I l l - ) A | (2.57) h2 3h 6
)A ] (2.58) h2 3h 6 * J
V - - : 2 z * w +<*■- r f l ) < K, + ) (2.59) 3 n The coresponding stress resultants are given by :
M. = — D( "Wo^j. + v w0 yv) + + (2.60)
My — D(wftyy + ■'«„.„) + |D (8 yv + vflM) (2.6))
Mxy = - D (1 - » ) w ,^ + 1 D () - y ) ( 9W + 6y,x) (2.62)
0 . = - jD ( V2-w0)x + + | Gh«, (2.63)
Oy= - f D ( V zw0)>v + | ^ A y + |G h e y (2.64)
Using equations (2.60) - (2.64) and following Mindlin's steps, Narasimhamurthy obtained the following governing equations : where
(2.68 )
0S, 0y and w0 can be determined from the above equations with the appropriate bound ary conditions.
The set of equations (2.65) to (2.67) is of sixth order, which requires satisfying three boundary conditions at each edge. Thus this theory is of the same order as Mind lin's and Reissner's theories.
Levinson [117] in his work tried also to develop a theory which does not depend upon an unknown coefficient. He chose his displacement field to be:
u(x,y,z,t) = zi/fx + z3 0 x (2.69)
v(x,y,z,t) = z^y + z30v (2.70)
w ( x , y , z , t ) = w() (2.71) where \ltx, \Uy, d>s, To satisfy the condition of stress-free top and bottom surfaces, equations (269) - (2.71) were reduced to : u =z[0x- i l _ (^ + Wos)] (2.72) 3 h v=zl'',y-T7I(*y + woly>] (2.73) 3 n w = w. (2.74) The strains are given by : €kx = Z I ^ . x - “ 7 T ( ^x.x + W 0,xx) ^ (2.75) 3 n 37 e =z[ iti —----- (0 + w . )] (2.76) yy ,v.y _ , 2 v y.y o.yy7 J 3 n £zz = ° (2.77) 4z €Xz = (1- T 1 - )(^ + wo.x > (2.78) h 4z € v, = ( l } ( + W(U > (2.79) e xy = z (0 T x.y +0 ^y.x )------2 (0 r x,y +0 r y,x + 2 w_ O.xy7 ) (2.80) 3 n and the stress resultants are: (2-81) M x = f I 4 ( K* +V'l'y,y)- { W0.xx + v W0,yy > 1 Mv = j [ 4 ( 0 y>>. + I7 0 XiX ) - ( WQ yy + p WQ sx ) ] (2.82) M xy = — ( 2 ( 0 N >. + 0 y_x ) ~ Wasy ] (2.83) Qx = fU2 h(0x + wOA) (2.84) 18 Qy-§OhC+y + wa,) (235) Then following the same procedure as before, Levinson arrived at the following governing equations: (236) (237) (2.8 8) where T is given by equation (2.47). This set of equations is of sixth order as before. Levinson compared his theory with Mindlin's and found that it gives the same results if the correction faclor in Mindlin's is taken as 5/6 . This theory gives better results than Mindlin's, although it is of the same order, because the terms of higher power in z . The same theory developed by Levinson was applied to laminated composite plates by Librescu et al [118]. 2 3 3 Ambartsumyan's Approach [119-122] Ambartsumyan [119] derived a general theory for anisotropic plates which was of order higher than six, but the order is reduced to six if the case of isotropic plates is considered. Ambartsumyan's theory is based on the following assumed transverse shear stresses and normal strains: (T \7. ( x ,y ,z ) = X i( x ,y ) + h X,( - x , v ) + f . I( z )( x,v ) (2 .8 9 ) 19 o’yzCx,y,z) = Y1(x,y) + -|-Y2(x,y) + f2(z)0(x,y) (2.90) <^ = 0 (2.91) where X,(x,y)= j[o-x2(x,y,h/2)-crxz(x,y,-h/2)] X2 (x,y) = Substituting (2.89) and (2.90) into the third equilibrium equation from the theory of elasticity and solving for oz , Ambartsumyan obtained : ff.-JfU .y)-!,,*,, - Jm #,-*(X u +Y w) - Jj. crz = Z+ at z = h/2 , crz = Z at z = — h/2 (2.94) and given by X(x,y) = K > s + K ^ y + |-(X2x + Y2y) + Zj (2.95) where J0.= f f.(z)dz (2.96) Jo Kr = j [ J« Z1 = 1(Z+-Z ') (2.98) Z2 = Z+ + Z~ (2.99) 20 Using (2£9) and (2.90), the transverse shear strains can be determined a s: • » = £ ( x i + f* 2 + f.* ) (2.100) «K= 5 - ( Y. + f Ya + f2* ) (2J01) Integrating (2.91) , (2.100) and (2.101) with respect to z , the displacements u,v and w , are determined as follows: = + (2.102) v(,,y,z) = -zww + ^.* + |.Y , + I ^ . Y 2 (2.103) w(x,y ,z) = w0(x,y) (2.104) The remaining strain components are determined from the displacements, using strain- displacement relations, and they equal: + + (2.105) J 2 e ~ — 7. w + -22. \h + — Y + — — Y {*> 106) V.v «-vv ^ G ..v (j 1 i.x T 2 h G 2-.v )' S = “ 2z wo,.y + f - t y + -G t * * + + + 2hGs£f(x» + v*> (2-l07) The stress resultants for this theory are given by : h! N*= 24(Tr2r)[(2- ,')x^ +‘'Y«] (2-108) (2.109) 21 Mx :01c +»”»*„)+> , (frjr) *,+J. ( ) *» M , = - D < + " wo.x,)+JX f r v ) ^ +J j ( 1 ^ 7 ) ^ + ^ [ C t 3 7 ) x..« + ( | ^ ) y W] MxS = - D (l- ‘')W0.x.v + J3*,y + J4^ + ^ ( Xl, + Yu ) (2.1133 Qx = J5<6 + hX, (2.114) Qy = J60 + hY, (2.115) where h'2 J3= | z J 01(z)dz (2.116) *> 2 J4 = zJ02(z)dz ' (2.117) (z)dz (2.118) '• ■ J / ' J6= ^ f 2(z)dz (2.119) Following the same procedure as before, Ambartsumyan derived his governing equations. For the case of shear stress free surfaces: II II II II X * (2.120) to to *> f . ( Z ) = f ,( Z )= — ( —------7~ ) (2.121) 1 2 2 4 j = j = L ( ! n L - i L ) (2.122) 01 2 4 3 22 • Js=,*= f? (2-123) these governing equations become : D{ w_ +w . -JL -0 - i L LLlHltf, -jLil+Zlw, } O.xxx O.xyy j q q “ sx 2 0 G ** 20 G ** + •0 -^ = 0 (2.124) D(w„ +w„ --iL* -hLl±£20 ) 0.yyy O.xxy 1 0 G ^,yy 20 G •Itx 20 G ’** + | ^ = 0 (2.125) ^ .>v = ~ h3 H Z2 (2.126) where Z2 is given by equation (2.99) . The same approach was adopted by other researchers [120-122] to solve other problems. Comparing the different first order shear deformation theories mentioned in this seel ion, it can be concluded that they have the same order of governing differential equations, and the difference among them mainly in the procedure to get the governing system of equations and in the use of the shear correction factor. Table 1 shows a brief comparison among the different theories. 23 Table 1 Comparison among the first order shear deformation theories Transverse Transverse Shear Theory shear norma] Theory B.C. correction strain strain variables factor €xz-€vz €z » 5 Reissner quadratic 0 W ,Q x .Qy 3 6 2 IT Mindlin constant 0 3 W0 ’* s ’*y 12 Narasima- quadratic not zero W0 ’M y 3 none hamurthy Levinson quadratic 0 W0 * * s ’*y 3 hone Ambart quadratic 0 Wo,0,0 3 none sumyan (variable, in general) 24 2A HIGHER-ORDER SHEAR DEFORMATION THEORIES [123-157] Many theories are developed in much the same ways as the first order shear defor mation theories but the order of the set of differential equations obtained is higher than 6 (This is why these theories are called higher order theories). In the present sec tion these theories will be examined. The general assumptions used by these theories corresponding to the classical plate assumptions mentioned before are relaxed to the following : 1. Displacement components are small compared with the thickness of the plate. 2. Normals to the midsurface of the plate before deformation which are straight are not necessarily normal nor straight after deformation. 3. The transverse normal strain t z , in general, is not negligible. 4. Transverse normal stresses generally arc not negligible. Thus, only the first assumption listed on page 4 is retained. This is required for the theories to be linear. Assumptions 3 and 4 above are seen to be changed from those of the first order shear deformation theories. Again as in first order theories, these theories can be categorized according to the approach used. The first is based on assumed stress as well as displacement fields (Reiss- ner's approach). The second starts with an assumed displacement field (Mindlin's approach) and the third is using an assumed stress field (Ambartsumyan's approach). These different approaches are discussed briefly through the different theories in the following subsections. At the conclusion, comparisons will be made among the major theories. 25 2.4.1 Higher-Order Theories using Reissner's Approach [123-131] Iyengar et al [123] developed a higher order theory, which is an extension of Reiss ner's theory [1,2] by considering higher order terms in both the stress and the displace ment fields as follows: the stresses are, 0\xx=12M xZ + Rsf ]+Pxf2 i2 .\2 1 ) trzz = f (-1 + 3z-4z3) + s fs+ Tf6 (2.129) ^ | Qx ^1 - 4 z" ^ + Ax f 3 + Bx f4 (2.130) Cr.Vz = |O y (1- 4z2) + AVf3 + Byr4 %> = 12Mxyz + Rxyf| +Psyf, (2.132) and the displacement are, u = zjBs + z20x + z (2.133) v = z/3y + z2ey + z30v (2.134) w = w + zw ,+z2w, (2.135) where Mj, R^, Pu, A ,, Bj, S , T , ^ , 0j, ^ , W; are functions in x and y only and f 1 are functions of z and given by : 2 f^-lSd-nz2) f2 = -420z(l-^l) (2.136) f =15z(l-4z2) f =-ii(i-244 z2 + 80z4 ) (2.137) f ^ - J r d - S ^ + lfcz4) f = ^ - z( 1 - 8 z2 + 16 z4) (2.138) 3 16 " 4 26 Using Reissner's variational principle, Iyengar et al obtained the following eight eenth order set of differential equations in the nine independent variables Qx,QytBJ.,By,Ax,Ay,W,w, and w2: O - — V2Q + — V 2B + —7- 1— r T + — 7—-----rQ Vs 10 6 x 6(l-i/) -x 10(1-*/) *x = -D ( V 2w')v (2.139) ,X 0v-^7rV20v+ l V2B +77^ r T + 1 y 10 6 y 6(l — */) *y io (i-i/)M-y = —D( V 2 w ')*y (2.140) Qu + 0 M = - q (2.141) B> - -5 7 V 2 B, + - ± - V; o„ - m r . ) T, - 90 x 1400 90(1 -v ) -x 1400(l-i/) -x = - ^ ( V 2w2))! (2.142) By_ 90V By + 1400 V 90(1-1/) T.> ~ 1400( 1 - V ) q->' + Bv.y * T (2.144) A« - i v2 A» -42(Tr^s.» = -|r(v:!w.).» (2A45) V i f y2 Ar - 42TTT17) S, ■ -f' ) , (2.146) Avs + Ayy = S (2.147) where w' = w + -^y- (2.148) Solving the nine differential equations simultaneously and applying the projter boundary conditions, a problem can be solved for a rectangular plate subjected to later 27 al load q(x,y). Speare and Kemp [124] developed a higher order theory from Reissner's theory by repeated differentiation and back substitution of equations (2.28) and (2.29) in them selves and equations (2.30) to (2.32) The same set of equations were obtained by Baluch and Voyiadjis [125] . Baluch and Voyiadjis considered the following approximation which corresponds to the thin plate theory : q = DV4W (2.149) + (2.150) Qy = - D (Wyyy + Wxsy) (2.151) Mx = - D ( W sx + v W yy) (2.152) My = ~ D ( W yy + v Wxx) (2.153) M •v =D (1—i>)w »*y (2.154) By approximating V 2q . V2QX and V2Qv . using (2.149) to (2.151) and substitut ing in Reissner's equations (2.28) , (2.29) , and (2.30) to (2.32) , they obtained a sixth order differential equation in w as follows: ^ w + W 7 = 7 T v ‘ w = d tzl55) 2 Q„— D < + > - w frrr D ("•»»=+2 w» ™+ 5 (2J 56) <3.v = - D < + > - W T ^ T D ( w -vyOT + 2 + 5 C2-1 5 7 ) M, = - D My = - D < w „. +v ) - 1(2- f ) Wwyjr + 2 W „yy +v Wss„ K2.159) Mw - D ( 1 - f ) Wkv + i l l ! i ( + Wsvvy) (2.160) 28 By solving equation (2.155) for w , the stress resultants can be determined using equations (2.156) - (2.160) after substituting for w( with V n= (V2)1^2 ) • Repeating the same procedure but using equations (2.155) to (2.160) instead of (2.149) to (2.154) for approximation, Baluch and Voyiadjis [125] developed a higher order theory (eighth order) by obtaining expressions similar to equations (2.155) to (2.160). But the first equation is replaced by the following eighth order equation in w : (2.161) Then they obtained a tenth order theory by repealing the same procedure, yielding an equation for w as : (2-*/)h2 J V8W + |~ ( 2 — */)h2 10(1-*/) 1 0 ( 1 -*/) = JL (2.162) D Expressions for the stress resultants are similar 1o equations (2.156) to (2.160) , but with higher order terms. Solving equation (2.162) for w and substituting it into the expressions for the stress resultants the latter values can be obtained for any given problem. Expression for the stress resultants are nol explicitly given here because they are very lengthy. They can be found in reference [125] Using a different procedure, but starling with assumed stress and displacement fields, Kuznetsov et al [126] developed a twelfth order theory for laminated anisotropic plates and flat shells. They took the variations of the displacements and shear stresses in the thickness of the plate through the forms: u ( x , v . z , t ) = u{) ( x, y , x ) + f j ( 7.) ( x , y , t ) (2.163) v ( x , v , z , t ) = v 0 ( x, y , t ) + f , ( z ) yv ( x , y , t ) (2.164) w(x,y ,z,t ) = w0( x,y ,t ) + f3( z)yz( x,y ,t) (2.165) 29 O’yj (x,y,z,t) = g2(z)Py(x,y,t) (2.167) o'2z(x,y,z,t) = g3(z)Pz(x,y,t) (2.168) where P, can be related to yj through the stress-strain relations. The choice of the func tion f, ( z ) depends upon the fact that displacements should be continuous at the bound aries between the laminae and their derivatives should be piecewise continuous, because the strain components are discontinuous at those boundaries. On the other hand gi ( z ) should be continuous and satisfy the conditions of zero shear and transverse nor mal stresses al the top and the bottom surfaces of the plate (which is needed for free vibrations). Using Hamilton's principle, differential equations of twelfth order are obtained in terms of the variables u0, v0,w0,yx,yy and y2. The theory was applied to a cantilev er square carbon-plastic plate to determine the natural frequencies, and results were compared with the classical theory and experimental results. Different thickness-to- length ratios are considered ( 0 < h/a ^ 1 ) with one layer of carbon-plastic. Then a rectangular plate (a/b = 2.4) made of seven alternating layers of high-modulus braided- cord glass fabric and LU-2 carbon ribon with h/b = 0.029 was considered to study the dependence of the natural frequencies on the mode order. The agreement between this theory and the experimental results was good. The agreement was better than with the results from classical plate theory. Using the same idea of applying a mixed method (assumed stresses and displace ments), Vlasov [127] was the first lo use the method of initial functions (MIF). Others [128-131] applied the same melhod for different applications. For example, Celep [130] applied this method to the free vibration of a circular plate with clamped edges. Then 3 0 Iyengar and Pandaya [131] used the same approach to analyze the response of ortho tropic thick rectangular plates under dynamic loads. Then they applied the theory to the problem of a freely vibrating, simply supported rectangular plate and compared results with both Mindlin's and Ambartsumyan's theories. The results of the M1F meth od were closer to Mindlin's theory than to Ambartsumyan's theory. Vlasov started his derivation in the usual way with the differential equations of equilibrium and the following stress-displacement relations: % = «•„ = ° < “, + V (2.170) Expressions for ayy, oa , oM and oyz can be obtained by interchanging u,v,w and x.y.z . Eliminating axx,ayy and axy from equations (2.169) and (2.170) and the equilibrium equations, Vlasov obtained the following set of equations : U^ = - W x + X (2.171) V •z = — W »y + Y (2.172) w , = - 7 ^ 7 ( u .- + V + 2T7^ T z (2173) Yy-c (2.174) * = ” i — v U xv * y — ^V * xt* T i “— — v V vv ^ — T1 ~— — v Z *yv ~ b (2*175) X =-l±^V -(U + _i—UKs)--ii-z -a (2.176) z 1 — j/ *yy l — j/ l — v ,x where l) = Gu V =G v W=Gw (2.177) x = °-« Y= 2 U — U0 + z ( )0 + — (L ^ )0 + ------.... (2.179) V-V 0 + z(V ,)0 +£- W - W 0 + l(W J)0+ i.(W al)0 + . (2.181) X = X0 + z(Xi )0 + i-(X J I )0 + . (2.182) Y = Y0 + zCYi )0 + £-(YJ , ) 0 + . (2.183) Z = Z0 + z(Z^)0 + ^ -(Z ^ ) 0 + .., (2.184) where U0,V0,W{),— ,Z0 are the initial functions ( in x and y only); that is, the values of the functions U , V ."W Z in the initial coordinate plane z = 0 . Equations (2.171) to (2.176) combined with the appropriate boundary conditions are the governing equa tions for Vlasov's theory, liy successive differentiation ol equations (2.171) - (2.176) and substitution into equations (2.179) - (2.184) , Vlasov showed that U,V,W ,...... ,Z can be expressed in terms of U0,V0,W0 Z0 (the initial functions) as follows : u o u V„ V 0 w = [L ] Wo (2.185) X u Xo Y Y Z 0 Zo where L(j arc* linear differential operators which can be expressed as infinite series in terms of z, or interms of trigonometric functions. For example, Ln in infinite series and in closed form is given, respectively, by : 32 (2.186) where (2.187) with this method, the generalized coordinates are a mixture of geometric variables (U0, V0, W0) and static variables ( X0, Y0,Z0 ). This is why this method is considered as using Reissner's approach, which is based on assumed stresses as well as displace ments. 2.4.2 Higher-Order Theories using Mindlin's Approach [132-156] Higher order theories can be determined from Mindlin's theory by considering high-order terms in z in the displacement field. Different theories can be determined in this way, which is considered the most popular way because of its directness and clari ty. In general the displacement field can be expressed as a power series in z . In what follows, tlie different theories available will be examined by giving explicitly the dis placement field used. The governing equations will be given for some of these theories (only linear) which are of some importance to the author's interest, because there are many theories and the equations are either lengthy or not available in most of these references. Schmidt [132] developed a nonlinear theory by considering nonlinear terms in the strain-displacement relations. In his theory the displacements take the forms : (2.188 ) (2 .1 8 9 ) 33 w(x,y,z,t) = w0 (2.190) where u0, v0, w0, 0X and 9y are functions in x and y only. Following the same procedure as Mindlin, Schmidt obtained a set of tenth order nonlinear differential equations which requires five boundary conditions at each edge. By linearizing these equations, he obtained the same equations obtained by Reissner. Schmidt applied his theory to an infinite strip plate (x=0,a;y=±oo and z—± h/2) subjected to sinusoidal loading q = q 0sin 2L£2L at z = —h/2 . a Krishna Murty [133] developed a general Mth degree theory by assuming the fol lowing displacements: M u(x,y,z,t) = u0 —zwQs+ £ptl0n (2.191) n=i M v(x,y,z,t) = v0- z w 0v+ £ p ^ n (2.192) n= I w (x ,y ,z,t) = w0 (2.193) where uo,vo,wo,0n and i//n are functions of x ,y and t only, and pn to satisfy the con dition of zero shear stress al the lop and bottom surfaces is given by : n = ^ . ' _ 2 n+j.f (2|94) where £=z/h . A zero-order (M=0) theory (or approximation) corresponds to classical plate theory and a first order (M=l) corresponds to Mindlin's theory. Krishna Murty substituted his equations into the usual strain-displacement and slress-strain relations and used Hamil ton's principle to obtain a set of differential equations having order which depends on M and given by : 34 D [ W0.xxxx+ 2 (" + 2 S )w 0,yyxx + W0,yyyy] + ^ h W0,tt “ ^ W0,**« + W0 ,y J M + 4*1T'TtB n { (“n,xxx f ) vn,yyy+ <£r '+ (v + * 2g)($ '' ^axyy ’axxy"+ \p ^ )} — ii'^n,ytt pB (\b ^axtty + d> J )~| n-1 = q (2.195) M T B n I W 0,xxx+ ( l / + 2 g ) W 0.xyy J - P Bn W 0.xtt+ D T E m „ [ ^ x x n-1 + gd>b *>.yy + ( v + 6'^i),xyJg ) $ "1 — rgA 5 mn“n d> — r pE mn^attJ M TBntW0.yyy+^ + 2g^W0.yxxl - P BnW0,ytl + Z {TEm„[^yy ne J + gb “ \ b n.xx +(v+g)0 6 n,xy J 1 -rg o A nin^n ib - p E^ nm 0*iui •» ) = 0 (2.197)> j .where m = 1,2, — M _ E G T = ~ £ = “ 1 - v ‘ T h ’ B_. =~ — £ f p zp . . dz = / 2p"'p"dz A n ,n = J P,n^P.udZ (2-19^ ■■ i For numerical application, Krishna Murty considered the cases when M = 0,1,2 and 3 for a simply supported rectangular plate and compared the calculated frequency from the different cases (see table I ,[133] ). In other work, Krishna Murty [134] adopted the following displacement field : u(x,y,z,t) = -zw bs-p0 (2.199) v(x,y,z,t) = — z -wby — p 0 (2.200) w(x,y,z,t) = wb + ws (2 .201) where ws,wb,0 and 0 are functions in x ,y and t only, wb is the partial deflection due 4z2 to bending, ws is the partial deflection due to shear and p = z( 1 — — - ). 3h*1 Applying Hamilton's principle, using the usual stress-strain and strain-displacement relations, Krishna Murty obtained a tenth order set of differential equations, which requires five boundary conditions. One of the benefits of this theory is is the ability to obtain four different vibration modes, namely bending, thickness-shear and two inplane modes. In recent works, Krishna Murty [135,136] developed two different theories for composite plates. In the first work [135] he used the following displacement field : U ( X , V , Z ) = U() + Z + z2 tf>^ + Z 0x (2.202) V ( x , y , z ) = V0 + z 0y + z2 0 + -L 0 (2.203) w(x,y ,z) = w0 (2.204) where again u0,v0, w0, 0X, ,0 ,0X,0y,0X and 0y are functions of x and y only. The same displacement field was also used previously by Reddy [137,138,139], but Reddy in [138] and [139] used the nonlinear strain-displacement relations (Von Karman strains). Both Krishna Murty and Reddy applied the conditions of zero transverse shear stress at the top and bottom surfaces of the plate and reduced the displacement field to : (2.205) (2.206) (2 .2 0 7 ) 36 Reddy applied the principle of virtual displacements to develop his governing equa tions; on the other hand, Krishna Murty used Hamilton's principle. For isotropic plates the following set of equations can be obtained (from Reddy): 8Gh . (A + 2G)h3 4 , . ph3 ,5 V ' W» + ■ V 4 Wo + P » wftu - H - (w to u + Wowtt 3 (-' 3V5C)”J [<»»»« + (*,.ro + W ] - + +^ r (*u«++,J«) =0 (2 a » ) .4(A + 2Q)h3( +w ) _ 8Gh. + l£Slw _*2£* 315 °-xxx o -w 15 °-x 315 0>xtt 15 x + m x + 2G)hi( )+ iZ G h !^ - i l £ i = 0 (2.209) 315 x-xx >-x>' 315 x-vv 315 x-tt iiA±2G)hl(w +w )-MiLw +i£iilw -l£h. 3 1 5 0,xx.v O.yyy 1 5 W0,y + 3 , 5 W 0,yu 1 5 Vy + -17Lxt ? G )h (,/, + ,/, )+ lZ5JL^ _il£hl^ = 0 (2.210) 315 xxv '•>> 315 v-vv 315 v-u This se1 of equations is of eighth order and requires four boundary conditions at each edge. For general (i.e. unsymmelrically laminated) composite plates the order increases to twelfth order and requires six boundary conditions per edge. Krishna Murty showed that by introducing the transformation ^s + wo.x = -<£ * 'J'y + w o,y= ~ ^ (2.211) equations (2.205) - (2.207) reduce to : u = uo-zw ox-p 0 (2.212) v = v()-7.w()y-pi/» (2.213) w = w„ (2.214) 37 A -2 where p=z( 1 ) , which is the same set of equations he used before . This dis- 3h placement field constrains the transverse shear strains to be zero at all points where the inplane displacement is zero, such as those on fixed edges. This situation is unrealistic because the transverse shear effect is quite significant near such points. Krishna Murty overcame this situation by introducing the new variable ws in (2.212) - (2.214) such that: w = wb + ws (2.215) Then equations (2.212) - (2.214) reduce exactly to equations (2.199) - (2.201). Krishna Murty compared his theory with Reddy's, Mindlin's and the exact solution for a cross- ply square plate simply supported at all edges. Krishna's results are the closest to the exact solution (elasticity solution). Equations (2.212) - (2.214) were used by Bhimaraddi [140]. He employed Hamil ton's principle to derive his governing equations for composite plates. Then he applied the theory to sludy the free vibrations of simply supported square plates, both homoge neous and laminated. In later work, Krishna Murty [l 36] chose the following displacements: u(x,y,z,t) = — z w0x — p, 0 (2.216) v(x,y ,z,t) = -zw 0y-p,^ (2.217) w(x,y,z,t) = w0 + p,w2 (2.218) where (2.219) Using the principle of virtual displacements, Krishna Murty derived his governing equations (tenlh order). Then he applied the theory to an infinite, 4-ply (0/90/90/0) 3 8 laminate strip ( x=0,L and y = ± oo ) simply supported at x = constant and subjected to sinusoidal load at z = ± h/2: q=q0sin-^2- ' (2.220) The theory gave results very close to the elasticity solution. Kant [141] developed a higher-order shear deformation theory (twelfth order) based on a higher order displacement model and three dimensional Hooke's law. The displacement model chosen by Kant w as: u(x,y ,z) = z0x +z30x (2.221) v(x,y,z) = z0y + z 30y (2.222) w(x,y,z) = wo + z20z (2.223) Using the principle of stationary potential energy, Kant derived the following set of twelfth order differential equations: 3 3 Gh V ‘ wf ,-pli w .)ti + Gh + ^-V 20I+^-0iu = O (2.224) h3(x+2G2( + ) + im + 2 o )((> +0 , 1 2 v x.sx x.xx y.xy h3(2A —G) ^ s -G h (w 0iX + ^s) 12 iL * L if, + £!!)_ 0 - £LiL_$ — £ iL _ tj, — P h.. g = o ( 2 P 2 5 ) 12 '■■yy 80 x-yy 4 x 12 xu 80 *•« 39 + Ji^2£2(.w+.w , + i!c^zC2^_oh(ww+V ^ *,» + T T '*U " °> ~ V - ^ V ■ 0 C2226) 5!<2vt2G)(* + + hj(2A-30) _Ghi( + , ) 8 0 Z,X 4 0 ,X V X y Gh5 . , Gh7 _ 9Ghs _ phs . _ ph7 _ n ro 80 x-v>’ 448 . x-yy 80 x 80 x-u 448 (2.227) h5U + 2G)( ) + h^X ± lG )(e +e ) 80 r x,xy ^y ,v y / 4 4 8 x,xy y,yy j. hs(2\ —3G) ^ Gh3 , , , ^ + ------80------" — (w o.y + ^y) £2l_ xf, + £Jl_e - —G h- e - $ - £ii_ 0 = 0 (2.228) so y-xx 448 y-xx 80 y so y*u 448 y-tl + - - f r 3G)(«~-V + ±it±±222* " t t v ' w° " ^ + t t w° « + ■ 0 Whitney and Sun [142] cxlended the shear deformation theory developed by Paga- no and Whitney [44] for laminated composite plates to include the first symmetric thickness shear and thickness stretch modes by including higher-order terms in the dis- 40 placement expression. The displacement field chosen was : 2 u (x ,y ,z,t) = U0 + Z^x + y (2.230) 2 v ( x ,y , z , t ) = v o + ziAy + y 0 y (2.231) w (x ,y ,z,t) = w 0 + zi/fz (2.232) where z is measured from the midplane of the laminate. Using Hamilton's principle, Whitney and Sun [142] derived a set of governing differential equations (for composite plates) of sixteenth order, which requires specify ing eight boundary conditions at each edge (the governing equations are not given explicitly here because they are very lengthy and can be found in [142]). To get better results they introduced correction factors kj(i = l — 5) similar to that used by Mindlin for isotropic plates. They chose the following values: 2 (2.233) s 15 where the k, are used as follows (2.234) (2.235) (2.236) Another higher order theory was derived by Nelson and Lorch [143]. Their dis placements are: u ( x , y , z, t) = u() + z \(ts + -i (2.237) v(x,y,z,t) = v0 + zi/fy + -l w ( x , y . z, t) = w()4 z ^ + 7.2^ (2.239) 41 By means of Hamilton's principle, they obtained for laminated orthotropic plates a set of differential equations of eighteenth order (governing equations are not given in their published work explicitly) which requires one to specify nine boundary conditions at each edge. Although using high order terms, Nelson and Lorch introduced correc tion factors similar to Mindlin's for isotropic plates and determined in the same way. They applied their theory to isotropic plates, an orthotropic homogeneous plate and an unsymmetric, four-layer, laminate plate, and compared the frequencies with those obtained by Mindlin's theory. One of the highest order shear deformation theories was derived by Lo et al for isotropic plates [144] and for laminated plates [145,146]. Lo et al included more high order terms in their displacement field, which was : u ( x , y , z ) = u0 + z v(x,y,z) = v0 + z \f/y + z w ( x ,y , z ) = w 0 + z ^ + z2 They used the principle of stationary jxMenlial energy to derive the set of govern ing differential equations of twenty second order (These governing equations are not given explicitly in terms of the generalized displacements in their work). This requires one to specify eleven boundary conditions al each edge. In the first part of their work they applied their theory to an infinite, isotropic plate subjected to sinusoidal load. Then in the second part they applied it to doubly infinite angle-ply laminate and doubly infi nite bidirectional laminate. Doong [147,148] combined Lo's theory with a perturbation technique to study the vibration and stability of isotropic and laminated initially stressed plates. The incremen tal displacement in the perturbation technique takes the same form of the displacement model assumed by Lo et a l . 42 In polar coordinates, Wilson and Boresi [149] derived their higher order theory. They assumed the following displacement field for axisymmetric deformation : u(r,z) — uQ(r) + — ■ (r ) (2.243) w(r,z) = w 0 (r) + 4r-w.(r) h 1 + (4 h r ^ w2^r^2 (2.244) where u is the radial displacement component. Applying the condition of zero shear stress at the top and bottom surfaces ( z = ± h/2 ) of the plate, equations (2.243) and (2.244) reduce to : u = u0-z( w0 r + w2 r) (2.245) w = w0 + ( ^ )2 w 2 (2.246) where w , = — (1 + v )(r2 — a2) -P— and a is the radius of the plate. . 4 Eh .. Applying the principle of stationary potential energy leads to a set of governing equations in terms of the generalized coordinates u0and w0. They appliedtheir theory 1o clamped circular plate sub jected to uniform tarnsverse load. In forms very similar to Vlasov's displacements, Prokopov [150,151] assumed dis placements ( u , v , w ) in lerms of trigonometric differential operators which can be expressed in terms of series of and z . Prokopov's displacement field takes the following form: u = cu - —; -z ■ x sD + su, — —— r d Ds0 , (2.247) (> 2(i/-2) x 0 1 4 ( v — l ) (2.248) v = cv» “ 2(7=27 sDA + sv- ' 4Ti^T7 dDv6. (2.249) w = cw« + in ^2 T xu;>o + 1 ) S0' where 43 oo ,2n 2n .-n c = coszDxy = £ *21 (2250) ( 2 n )! s = (2251) ,n 2n +3 r-.il s —zc d = (2252) (2253) where Dx,Dy,Dxy as in Vlasov's case, and again u0,v0,w 0,u, ,Vj and w, are func tions in x and y only. Applying the principle of minimum potential energy, Propokov obtained differential equations and boundary conditions expressed in the forms of series in powers of the thickness of the plate and the independent variables U0*VO'WO>Ul *vi and w i • With a similar approach, bul using a slightly different displacement field, Igarashi et al (152,153] obtained a similar theory. They analyzed the deflection of simply sup ported, elamjied and free plates, but no numerical results were given in 1152]. In [153] numerical results were presented in the form of graphs for the bending stress distribu tion. They compared their results with those of the classical theory, theory of Ambart sumyan, Lo et al and exact theories. The higher the number of terms in the power series they took (n-approximation), the better the results were. Following the same idea, other researchers [154-156] derived other theories based upon similar displacement fields expressed as power seriesras Prokopov used. 44 2 A 3 Higher-Order Theories using Ambartsumyan's Approach [119,157] In addition to Ambartsumyan's general theory [119] for anisotropic plates which is of tenth order, Rehfield and Valisetty [157] developed another theory following the same procedure. They obtained the displacement expressions by integrating the follow ing relations: W.z = = ^sx+ ^yy+ ^zz (2.254) U^ + W,x = €« = a55Crxz (2255) V.z + W.y = €yz = a44CTyz (2.256) where ai} are elastic constants of the plate material. Neglecting the oa term in equation (2.254) and integrating, Rehfield obtained the following displacement field: N. 2 M 3 I.X Z 1,X 7. + S j l ( b2z _ i L ) — + (2.257) 3i h 2 I 6 N. 2 3 I..V Z Z v = V -zWy + h J L - + (2.258) h 2 I 6 N. w (2.259) = w + a3i( i f 2 + T T ) i = 1’2 where Q, = »5S Ox <2.260) 0 , = Q, (2.261) Substituting into the strain-displacement and stress-strain relations, the stresses can be determined in terms of the slress resultants and the generalized displacements. Then fol lowing Reissner's procedure, the governing equations were derived which are of tenth order (these equations are not given in their published work). Five boundary conditions are required for each edge in order to solve a given problem. Comparison among the different theories can be found in Table 2. 45 Table 2 Comparison among linear high-order shear deformation theories Transverse Transverse Shear Theory shear normal Theory B.C. correction strain strain variables factor €sz ’ €yz €z Iyengar fourth order not zero V A x ’ A y 9 none W,Wj,w2 Speare quadratic not zero W ,Q x ,Qy 4 none Baluch quadratic not zero w,Qx,Qy 5 none Kuzentov variable not zero U0 ,Y0 )W0 6 none ?K> V ? * Vlasov series not zero u0,v 0,w0 series none X0 ,Y 0 ,Z 0 Krishna M=1 quadratic 0 uo«vo>wo ^ i ^ i 3 none 0 Murty M=2 fourth order UO’V0 *WO^ l’^l 5 none [133] M=3 sixth order 0 uo*vc wo 7 none K. Murty quadratic 0 w b,ws,<£,^ 5 none [134,135] K. M. [136] quadratic not zero W0,W 5 none 46 Table 2 (cont.) Transverse Transverse Shear Theory shear normal Theory B.C. correction strain strain variables factor €xz ’ €yz €z Kant quadratic not zero 6 none WO’^ ’*y’ 0x’0y 2 7T Whitney and linear not zero 8 U0’V0>W0’^x 12 ’ 7r 2 Sun * y ’* x '* y 15 quadratic not zero 9 Nelson and U0*V0’W0 ’^ x ’^ K*i ' Lorch 0 z. U0 ,V0 ,W0 ,” z,6 ’ x ,9 * y Prokopov series not zero U0 ’V0 ’W0 series none ul-v l-w l Rehfield cubic not zero U,V,W,Qx,Qy 5 none Nx,Ny,Mx,Mv 47 2.5 DISCRETE LAYER THEORIES [158-164] All theories mentioned in the previous sections, when applied to laminated compos ite plates, consider the laminae as smeared into one thick lamina, and evaluate the effective stiffness of the entire plate in terms of the stiffnesses of each lamina. These theories do not give good results for stresses (although they are typically accurate for displacements and vibration frequencies) because of the use of polynomial displacement functions over the entire thickness of the layered plate, as the displacement derivatives should have discontinuities at the boundaries between the layers. To overcome this dif ficulty, some researchers [158-163] developed different theories based on considering each lamina separately. The displacements are assumed to take similar forms as before within each lamina. Then the displacements should satisfy the continuity conditions between adjacent lamina, which gives more equations than the usual differential equa tions for the smeared case. Sun and Whitney [158] considered that the displacement in each layer be given by : uk = uk ( x . y , t ) — zk v k = vk(x,y,t) - zki/rk(x,y,t) (2.263) wk = w 0 (x,y ,t) (2.264) where Uq , v £ are the inplane displacements at the midsurface of the kth layer, and zk is the z coordinate measured from the midsurface of the klh layer. It was also assumed that w is constant through all layers; i.e. w is constant through the plate thickness. Using equations (2.202) - (2.204) , Sun and Whitney calculated the strain energy Uk and the kinetic energy Tk for each layer. The total strain energy and kinetic energy is the sum of those; of each layer : 48 U = £uk (2265) k fk T = £ T k (2.266) k Then, using Hamilton's principle, they derived the governing equations in the usual way which leads to a set of six order differential equations for each layer in terms of the dispalcements and rotations (i.e 4N+1 equations overall, where N is the total num ber of layers). Besides the governing equations Sun and Whitney [158] used three methods to satisfy the continuity conditions between adjacent layers: • Theory I : Only the displacements are continuous (k-* 1) . (k+l) (k) _ 1 . ,k .1 , , (k < 1) and v -v = - \ \ l i y + - h where hk is the thickness of the kth layer. • Theory II: Displacements and shear stresses ( oxz and oy/) are both continuous = (2.269) and c£ ( w y - i//k)) + cJJJ ( w s - ^ ) = C44I)(w.y-*y+1}) + C45'>( } (2-270) in addition to (2.267) and (2.268), where cjk> are the elastic constants of the kth layer. • Theory III: Displacements are continuous and bending rotations are constants i.e. straight but not normal after deformation. Applying these constraints reduces the number of equations tremendously. For example, for theory II the number of equations is reduced from 4N+1 to 2N+3 (instead 4 variables for each layer and one constant for all layers, one gets 2 variables for each layer and 3 constants for the whole plate). Whitney and Sun compared between their theories and the exact solution of the one way harmonic vibration of the two-layered plate of infinite extent where the layers are made of homogeneous, elastic and isotropic materials, and also three-layered plate symmetrically stacked with respect to the midsurface. The second theory gave the best results. The same approach was adopted by Seide [159] and Green and Naghdi [160] . On the other hand, Pagano [161,162] instead of assuming the displacements, assumed that the inplane stresses in each layer take the form : N 12 M —- + --X Z (2.272) N 12 M .v _JL + ------z (2.273) N. 12M cr. z (2.274) S.v h h 3 which are the same relations adopted by Reissner [l] for homogeneous plates. Then he solved the equilibrium equations for the rest of the stresses and applied Reissner's varia tional principle to get the field equations which contain the constitutive equations and equilibrium equations to be satisfied in each layer. Besides the field equations, Pagano's stresses satisfy the interface relations (continuity or prescribed tractions) and the boundary conditions. Although this theory permits the treatment of discontinuous inter faces, Pagano reported that it could not handle more than six layers due to the large number of equations involved in the solution. 5 0 To overcome this difficulty Pagano and Soni [163] developed another theory. They considered the laminated plate to be divided into local and globed parts. In the local part they used the same theory developed by Pagano [161]. In the global part they asssumed the following displacement field : u = u 0 + z«/fs (2.275) v = v0 + z tfty (2.276) w = w0 + z x//z + y 0z (2.277) where u0, v0, w0, , */iy, r/;2 and Comparing classical plate theory, smeared laminate plate theories of Ihe first order (Levinson [117] ) and higher order (Lo et al [145] ), a discrete layer theory (Seide [ 15(l] ), and the exact elasticity solution, Bert [lt>4] concluded that discrete layer theories give the closest results to elasticity solutions in predicting shear stress distribution. However, it must, be pointed that the discrete layer theories typically require using sets of differ ential equations of much higher order than the smeared theories. it* - , 2.6 TWO DIMENSIONAL ANALYSIS USING FINITE ELEMENTS [165-205] In all the theories of the previous sections a set of governing differential equations and boundary conditions were derived. They are then solved analytically to obtain exact, closed form solutions (e.g. all rectangular plates with at least two opposite edges simply-supported), or solved by one of the known approximate methods (e.g.,Galerkin, 51 collocation, finite difference methods). In the present section, all works included solve the plate problem numerically using the finite element method. Finite elements is prob ably the most popular and attractive numerical method used for plate problems [165-198]. Other similar methods used are the finite layer method [199] and the finite strip method [200-205]. The finite element method is considered one of the most powerful numerical meth ods available to analyze dynamic, as well as static, plate problems. It permits applica tions to structures that are composed of different types of structural components such as beams, plates, shells, etc. . It also permits the discription of complicated three- dimensional geometrical configurations, as well as two-dimensional ones. In this section only the two-dimensional plate theory will be considered. In order to sel up a finite element solution procedure, weighted residual methods (such as Galerkin's) may be used. This involves direct use of the governing differential equations and does not require the existence of energy functionals. On the other hand the variational principle methods are more preferred by finite element researchers because they provide automatically the correct number of boundary conditions. Varia tional methods require a construction of a functional such that its variation vanishes if and only if the field equations together with the boundary conditions are satisfied. As before, works using the finite element method can be categorized according to the plate theory used to set up the finite element formulation. Some researchers [165-168] formulated their finite element models based on Reissner's theory [l] to study different problems. Some used rectangular elements [165,167] . Others used either tri angular elements [166] or sectorial elements [168]. Other researchers [169-189] used a f inite element method based on either Mindlin's theory [36] or Whitney [43] and then Pagano and Whitney's theory [44], which is the 5 2 Mindlin theory extended to include heterogeneous anisotropic plates. This finite element model is used to solve various plate problems such as and large amplitude free vibra tions for rectangular as well as circular plates [173,175,176,183,186,187], vibration of corner supported plates [181], and nonlinear dynamic responce problems [171]. To obtain better results, some researchers [190-193] followed the same approach discribed above to generate shear deformation theories of higher order than Mindlin's and Reissner's to set up their finite element formulations. Reddy and Phan [189] used the same theory Reddy used before [137] , eqs. (2.205) to (2.207) to study static, dynamic and stability problems of composite plates. They used a rectangular element with five kinematical degrees of freedom at each node (three displacements and two bending rotations). Putcha and Reddy [191] used the same theory, except that mixed degrees of free dom were taken at each node (five generalized displacements and six stress resultants). Kozma and Ochoa [192] used a finite element method based on high order theory with the following displacement field : u(x,y,z) = uo + z0x + z2tf>x (2.278) v(x,y,z) = v0 + zv/»y + z2 w(x,y,z) = w0 (2.280) where again u0,v 0,w 0,ilrx,i/fy,d>J. and To account for inplane extensions of the midsurface of composite plates Kanl and Pandaya [193] used almost the same displacement field vised before by Kant (eqs. (2.221) to (2.223) ) with two additional terms ( u„, v„ ) and ignored the Iransvcrse nor mal strains as follows: 5 3 u(x,y,z) = u0 + z^x + z30. (2.281) v(x,y,z) = vD + z*l/y + z30y (2.282) w(x,y,z) = wq (2.283) This displacement field is the same one used by Reddy [189] (eqs. (2.205) - (2.207) ) with : (2.284) (2.285) Equations (2.205) - (2.207) are determined by applying the zero shear stress condition at the top and the bottom faces of the plate. Kant used equations (2.281) - (2.283) and introduced the zero stress condition later in the shear rigidity matrix. Sometimes, when shear deformation effects are included, some difficulties arise such as the rotation of surface normals for different layers of a laminated plate being different. This is not satisfied by the conventional finite element displacement method. To avoid these difficulties some researchers [194-198] used the hybrid-stress finite ele ment method. Spilker [197] explained briefly the hybrid-stress method as follows : ’The hybrid-stress model is based on a modified complementary energy statement in which equilibrating intra-element stresses and independently, intra-element or element boundary displacements are interpolated in terms of stress parameters and nodal displacements, respectively.” He applied the method to thin as well as Ihick laminated plates [197] and isotropic plates [ 1 9 8 ] . 54 2.7 THREE DIMENSIONAL ANALYSIS 1206-241] In the previous sections the problem of a thick plate was transformed from a three dimensional problem into a two dimensional one by expressing either displacements or stresses as power series in the thickness coordinate z and functions in the plane coordi nates x and y . The governing equations obtained by each theory were expressed in terms of these functions and they are independent of z . Exact solutions for the thick plate problem can be obtained as a three dimensional problem; i.e., all the unknowns in the governing equations will be functions in x ,y and z (if cartesian coordinates are used). Also, the three dimensional problem can be solved approximately. The two methods are discussed generally in the following sec tions. 2.7.1 Exact Solutions [206-233] To solve the three dimensional problem, one may begin by considering the dynam ic equilibrium equations of the theory of elasticity expressed in terms of the displace ments u , v and w as follows (for an isotropic plate}. (2.286) e,v = pv.. (2.287) (2.288) where e = u A + v + w 7 Variables separable solutions are assumed in the forms: u = 4>(z)U(x)a;(y) e'"’1 (2.280) 55 v = ^(z)V(x)|8(y)ei“t (2390) w = X(z)WCx)y(y)ei6,t (2.291) such that the functions U,V,W and a,j3,y satisfy certain boundary conditions at the faces x = 0,a and y = 0,b , respectively. Then one obtains a set of differential equa tions in terms of (closed forms) are possible only for certain combinations of boundary conditions. Considering, at least, two opposite edges simply supported always provides closed form solutions. Different definitions for simply supported edges have been considered by several researchers to solve various problems. For example, at an edge x = constant, some researchers [210,213,214,216,222] used the following definitions: •w = 0 , v = 0 and ctxx = 0 (2.292) others [211] used a different definition : \v = 0 , v 7 = 0 and Clamped edges are usually defined by ; u = v = w = 0 (2.294) On the other hand, for an axisyinmetric static problem in cylindrical coordinates it is easier to follow another simple approach which starts by choosing a stress function V2 V24,s=0 (2.295) where (2.296) and the stresses are defined by : a = ( vV24>-4>>rr)i2 (2.297) CT0 = U V 24>--U r)z (2.298) (2.299) crrz = [ ( l - ,) V 2$-4>s ]r (2.300) Then the three dimensional equilibrium equations are exactly satisfied. Also the dis placements can be determined in terms of . The only problem is to choose 2.72. Approximate Solutions [234-241] Different approximate, three dimensional methods were found in the literature. One of these approaches is set forth by Donnell [234] and then used by others [235-237] . This approach, in general, is developed by starting with expressions suggested by classi cal thin plate theory for the stresses as functions of the loading, and calculating addi tional terms, so as to bring the results as close as possible to those of the three dimen sional theory of elasticity. bromine and 1-eissa [238] used the associated-pcriodidy extension of Fourier analy sis to obtain a solution of the classical three-dimensional elasticity problem of free vibration of the completely free rectangular parallelepiped. Numerical results are pre 57 sented for the frequency spectrum of plane-strain vibrations of completely free rectan gles according to two dimensional elasticity. Jones [215] and Srinivas and Rao [216] adopted the same procedure explained in the previous section but they did not satisfy the boundary conditions exactly. They used the following conditions :- w = 0 , tsv=0 and Noor [239] used a mixed finite difference method to solve the problem after reduc ing the governing equation to six first-order ordinary differential equations in terms of the thickness coordinate by means of Fourier series. Sonoda and Horikawa [240] considered the solution of a thick rectangular plate stiffened with beams as consisting of two parts, (a) a particular solution which is derived from three dimensional elasticity to express regiorously the .distributions of stresses and displacements under surface loadings for all simply supported plates, and (b) a complementary solution derived from the extended Reissner theory to meet the conditions of two opposite free edges. Leissa and Zhang [241] used the Ritz method to determine the free vibration fre quencies and mode shapes for rectangular parallelepipeds which are completely fixed on one face and completely free on the other five faces. 2.8 OTHER REVIEWS OF THE LITERATURE [242-247] Related reviews were made by others [242-247] to cover specific points of interest. Dzhanelidze [242] covered in his review the theory of bending of thin and thick plates only in the USSR prior lo 1950. He was interested only in the problem of bending as a particular part of the plate problem, not in the theory of thick plates in general. 5 8 Leissa [243-246] on the other hand collected many references related to different topics of plates (thin) in general, and thick plates were considered in the part of the complicating effects. The theory was not discussed in detail, but was mentioned by means of the references which included shear deformation and rotary inertia in the development of the theory. Reissner [247] described in general (without going through the equations of each theory in detail) the important theories of thick plates (first order and higher order). He did not cover all theories which were presented earlier in this chapter, but he discussed the basic theories developed for thick plates. 2.9 OTHER REFERENCES [248-281] All references in this section are included because they are related to the subject of thick plates. However, they are not discussal because the author had some difficulty in understanding the information contained because they were written in languages other than those known by the author (English and Arabic). CHAPTER III ENERGY FUNCTIONALS FOR FREE VIBRATIONS OF RECTANGULAR PLATES In this chapter, the energy functionals required to study the free vibrations of iso tropic homogeneous rectangular plates will be derived for the three theories to be used in this work. The energy functionals will be expressed in terms of the generalized dis placements according to the assumed displacement field of each theory separately. In laler chapters, these functionals will be used to apply the Ritz method lo study the free vibration for different cases of rectangular plates. In general, to study the free vibrations of rectangular plates only two energy func tionals are required; namely, kinetic energy and potential energy (in case of free vibra tion with no externally applied static forces it is reduced to the strain energy). The kinetic energy T , in general, is expressed as: (3.1) where u t, v , and w t are the velocity components in the x , y and z directions, respec tively, and p is the mass density as before (mass/volume). On the other hand, the strain energy U is given by : (3.2) where ojjt e h are the stresses and strains as before, and the summation convention for repeated indices is implied. - 59- 6 0 In the following sections energy expressions will be obtained in terms the general ized displacements for each theory to be used. 3.1 CLASSICAL PLATE THEORY As mentioned before in the previous chapter, the displacement field assumed by the classical plate theory is : u = -zw 0x (3.3) v = ”zwo,y (3-4) w = w0 (3.5) Substituting equations (3.3) - (3.5) into the strain-displacement relations, the strains are found in terms of w0 : 6 , = -ZW. ' € = — 2 ZW„ \x O.xx xy O.xy €,v.v - - 7'wo,„ «„ - 0 = 0 4ji = 0 (3.W using the stress-strain relations, and ignoring the transverse normal stresses ( a 2 = 0 ), the following relations are obtained : _ _ zE / . x ^ - "77377 (Wo- + ‘' W°'” ) o-xy = -2zGw0xy = o-yz = 0 (3.7) To gel the kinetic and strain energy functionals in terms of w0 , we substitute equations (3.3) to (3.7) into the general expressions of the energy functionals (3.1) and (3.2.) as follows: Kinetic energy: T = 2 f xf yf w <*»)2 + z (w°o* )2 + (w <*)2 1 dzdydx (3,8) Integrating along .the thickness and ignoring the rotary inertia terms, we get: T ~ 2 J J [ h(wc >2 ]dydx (3.9) Strain energy: U = — / / / fa € +a e +a e + + (ryz6yz]dzdydx (3.10) u 'if J,f,[ 77T7T(+^ } (' 2 w-'} + 77^77( + * wo,xx ) ( - z w0yy ) - 2 z G W0 Xy ( - 2 z W0 Xy ) ] dz dy dx (3.11) Integrating along the thickness, U becomes: U = T / x / y[ ( W°’sx ^ + ( W °Wr )2 + 2 V W0.xx W 0,yy + 2 ( 1 “ v } C W 0,xy )2 ] d V d x (3.12) w here: D - — S i i - and (3.13) 12(1- a 2) » 2 Equations (3.9) and (3.12) represent the energy functionals of the classical plate theory for the free vibration of rectangular plates. 3.2 MINDLIN S THEORY Following the same procedure as in classical plate theory, Mindlin derived the energy functional expressions by starting with the following displacement field (men tioned in Chapter 2 ): 62 u = z«/»x (3.14) v = z w = w0 (3.16) where , ifry are the rotations due to bending in the x and y directions, respectively (rotation due to bending ( t/rs ) = total rotation w0x - rotation due to shear deforma tions ). In contrast to the total rotation adopted by the classical plate theory which ignored the effect of shear deformation. The corresponding strain-displacement relations are 6«,“ l(*w + + » .x ) e„ = ° t *. = +, + W.W <3-n > and (again, ignoring oz ) the stress-displacement relations are cr = ——— ) (l-v2) v-v O’ = --- —--- (l/r + ) yy ( 1 — v ) v’v x’s = zG (^,y + ’/'ylX) ° \ z = k2G(^x + W 0,x) o-yz = k2G(*v + w0y) (3.18) where k2 is the shear correction factor. Then the energy functionals can be derived as follows: Kinetic energy : T = f / / J | z2 ( Integrating along the thickness: 63 T = 2 f x / y[ ^ [(^x,t)2 + ^y.t)2j + h (w o.t)2] dy dx (3.20) the first two terms are the rotary inertia terms which are ignored by the classical plate theory. Strain energy: U = I f f f | - ^ E, [ U Xib ) + (d +i/ih )C 0 )1 2 j s J y j ( ( 1 - 1 / ) vy.y'vvx.x/ vvy,y v v y.y J + z! G(*,ly + *y s)! + k2G[(*, + Woai)2 + (* y + wfty)2 ]Jdzdydx (3.21) Integrating through the thickness and combining terms: 3 3 [ —-E - — [(0 )2 + (\p )2 + 2 v\b \b ]+ ^Jl_(+ +J, )2 12(1-i/2) x,x y,y x,xvy-y 12 + k 2 G h [ ( ^rs + w0 x )2 + ( \f/y + w0 y )2 ] J dy dx (3,22) then using (3.13) we finally get: u = J f K / [ Dt< >2 + < * „ >2 + 2 » ♦„♦M+ ± ( i - » ) < > 21. + k2Gh[((//x + w0x)2+(i//y + w0y)2]jdydx (3.23) Equations (3.20) and (3.23) represent the energy functionals expressions for Mind- lin's theory. * 3 3 MODIFIED KANT THEORY To develop his theory, Kant used the following displacement field (equations (2.221) to (2.223)): u = z. 0^. + z3 6s v = zif/y + z3 ey \v = w0 + z2d>z (3.24) 6 4 which can be considered as the displacement field used by Lo et al (equations (2.240) to (2.242) ) and repeated here : u = uQ + z 0x + z2 0s + z3 0x v = vo + z ^ y + z20y + z3ey w = w0 + z ^z + z2<£2 (3.25) but reduced to be applied for the case of free transverse vibration of homogeneous iso tropic plates. In this case, there is no extension in the midsurface of the plate so u0 and v0 in the first two equations of (3.25) can be ignored. Also, because the interest is only in the flexural vibration of the plate, both u and v should be odd and w should be even in terms of z . Thus, d>x , Furthermore, equations (3.24) can be reduced once more by applying the condi tions of zero shear stress (and consequently zero shear strain) at the top and bottom of the plate. Doing that, 0X and 0y are expressed in terms of the rest of the variables as follows: (3.26) (3.27) Substituting these into equations (3.24), (3.28) (3.29) w = W0 + z2 The corresponding strains arc : 65 6“ = (z-7 7 )*“ "7 ?Wta~T *- (3 3 l) e" = (z" fF )<,w" IF w €„ = 2z e,v - (z “ ip -} (^ +**■J ~ 7 7 ' * ' < « " ¥ ■ < W (3 -3 4 ) «» =(> -* £ > (* , + ’'..> (3.35) n ^ = (1- ^ « * y + '”a,3 (3.36) f i i and the stresses (without ignoring ou as before ) are: °» = (z- ^Sh T )l - y [ ( X + 2G)0ixx + \ <^yy ] + 2zX0z (3.37) ^ = _ £ [ (X + 2G)0iyy + X^xs] + 2zX0z (3.38) = X (z --^ -)(< x + ^yy)--i 4 A V 2wo- 4 ^ V 30z 3h ^ y,y 3h 3 + 2z(X + 2G)0z (3.39) ",v * G (z - i3 4 h 2 ) ( *.y v + K,r -v.x ) - 3-^ h 2 4 G wavv o.xy - ^ 3 4 G 4>iX v (3.40) 6 6 (342) Following the same prooedure as before, the energy functionals of the modified Kant theory are found to be: Kinetic energy: T = — r f AZiil [C^, )2 + (0 )2]-IIlI[0 vr + 0 w ] 2 J , J y 315 w y* 315 w 0,xt y* °->a — ^ ^ [lb 0 +0 0 1 + J L .[( vi )2 + ( w )*1 315 w V , y.tv\v tJT 252 0.xt 0,yi' J ly dx (343) + h ( w 0i,)2 +-76r w o ,A t + Strain energy: u " i f ,/,{«>->{ 336 MX w — ■ ^ [ 0 w + 0 w„ 1 - 24 h [0 0 +0 0 1 315 *x.x Ojcx v y.y 0,yyJ 315 1 “ x.x Tz.xx t *y,y ▼'z.y.v 1 v X 408 . , + 42 ^ W0.xx ^r,xx + W 0,yy ^z,yy ^ (1 —2 v) I 315 *-x v v 24 h* 315 ^ *^x'x U ° + ^ W 0.*x W 0.yy + ^ I " ’tax + "o.y.v **xx 3+W ^ + 16 (0, +0 )0 — + w_ ) 0 —^L(0 +0 ')d> 5 T x.* v y,y 1 5 0.xx ttyy'^at 5 ^z.xx ^vy w 67 102 (ip +\p ) - - ^ - w 0 - 0 + ( ^ y + *y.*> 315 *-y ** 315 °>xy 315 ^y + —— (w )2s2, + -2— h2 ...w. J 0 +, h4 -•— (0 )2 21 toy 21 toy *z,xy ]5g z .x y + ^-Gh[(0x + wOx)2 + (0y + wOy)2]}dydx (344) Equations (343) and (3.44) represent the energy functionals for this theory. In later chapter these energy functionals will be used with Ritz method for the analysis of the free vibrations of rectangular thick plates. CHAPTER IV DIFFERENTIAL EQUATIONS AND BOUNDARY CONDITIONS FOR MODIFIED KANT THEORY To obtain both the differential equations that govern the thick plate problem and the necessary boundary conditions explicitly, Hamilton's principle is used. This principle can be expressed (for the case of free vibrations) as : 5 J/ (( T --U U )d t = 0 (4.1) h where 8 indicates the first variation. Substituting eqations (3.1) and (3.2) into this, it becomes: V hA2 h h 2 — J £ J p [(u t )2 + (v t )2 + (w t )2 JdAdzdt - J £ J* [ ov 8 6 .^ ] dA dzdt = 0 (4.2) Substituting equations (3.28) to (3.30) and equations (3.31) to (3.36) into equation (4.2) and integrating through the thickness yields : 17h3 r,,. c, i 8 h3 [ 2 1b 80 + 20 80 1------r 80 w _ + 0 8w„ + 80 w 3 1 5 x,t x,t yA y.x 315 f- x,t 0,x' x,r 0,,it y.i o,yt 2 h5 + % ,t 5 w o.yt ] “ 31J I S^z,xt + S^y.t K yx + fv ., + [ 2 W °-x‘ 6 W 0.xi + 2 W 0.y, 6 w C).y, J+2h " o , 8w - 6 8 - 69 + 504 [ + W0,xt ^z.x t + Swo ,y A y t + W0,yt'®**yt 3 i. 3 + ——4032 ■"-[20 w 80 z,xt +20 ^z,yt 80 ^z,ytJ ] + g— 1 [8w„ wo,vv z.t0 + 0,tw v 80z,tJ ] dA dt - 4 8 * ,« + 2M.&* 3h 2 x r x,x ^ jj 2 x O.xx 3 ” z,xx + (Mv v --i-P _ . •> v )8 ' ” v.v 0 ----- . —7 P V 8 w O.w „ -----2-80 “ T; 3h 2 y r y,y y 0,yy 3 '"'z.yy + + (M - P ) (80 +80 ) — P 8w„ - —P 80 dAdt *y ^ h 2 xy ,y y 3 h 2 xy y 3 x y ^ w = 0 (4.3) in which use was made of the following stress-resultant definlions hr (Mx,My,Mz,Mxy) = J jo - x, (Px’Py P*y> = j ^ x ’Vy’Vxy^3^ (Qx.Qy)= J dZ --Th/2-Th 1/2 (s. (4.4) -T ff/2 Integrating by parts and collecting terms, one obtains: ~17ph 0 + - p-- w, + -£-^-0 + M - — P - Q 3 15 x tl 3 1 5 3 1 5 v*.x« x x ,,2 x.s v.\ + — S + M — 4 P 80x h2 x xy'v 3 h 2 vv,y 70 dl£5l* +l£Slw +£hl« +M __4_P _ 0 [ 315 y*tt 315 0'^ 315 ^ y-y 3 h 2 r y.y v y + 4rS + M —P I 80 2 y xy.x 2 Jj2 *y,x y + r —phw — 6i? A — +£il_w 4- Ph m L 0,tt 12 ttt 315 x-xtt 252 °'xxu 1008 ™z*xxtt _ 4- P^ v/ 4- 0 4-n —-d_S 315 y-y11 252 0,yytt 1008 z-yy*1 v *>* h2 x,x + ^ P 4- _fL. p j. 8 p ”1 *w + Qy.y“ J 7 Sy.y ,y 3 h 2 x,xx 3 h 2 y,yy 3 h 2 xy,sy .[ :i£ jL. 0 - £ l L w _ £ i L ^ + 4- ,p 2L_ 0 L 80 12 O’11 315 x'xU 1008 °>xstt 4032 ^ z’xxu - .PJ/ 4- P^ w 4- P^ a —2M +i.p 315 1008 o-yy 11 4032 “z,yytt .z 3 x.xx + J Py.yy+ | Pxy.xy] ^ dx dy dt t, + I / ( t 1 7 ^ 8 W 0,x + 8<*>z,x 1 - ( M x - A - Px ) S^x - ( Mx, - A - Pxv } 8*y 4- r £ p 2 l 0 — £iL w — — - 0 — Q + — s — 4 P — 8 p 1 8w [ 315 *-u 252 °-xtt 1008 ^ x h2 * 3h2 x 3h y,y + [ 315 ^x’u ~ 1008 W°>Ktt ~~ 4032 ~ T Pxx T ^ y 1 8<^z I dy dt —* Jv )/|tj x I ^ l 3 p ‘5W°’y + ^ y]" (My" ^ 7Py)8,/'y_(Mj:y" ^ 2Pxy)8'/'x 4- [ 4 P h 0 — B l L w — _pJL_ 0 —n 4- s — p - 8 p 1 S v.' [ 315 v-u 252 0vtt 1008 ^ v h 2 v 3 h 2 vv 3 h 2 xy,x 0 +1 315 ^vu ~ 1008 W°yu ” 4032 ^y"- ~ 1 P>' v ~ "3 Px>‘x ] S(^z I dx dt which gives the foliowing differential equations : 0v x.x +0 v y,y - h —(S2 v x.x +T S y.y ) + —— 3 ^ 2 v(P x.xx T +2P* xy.xy T +P r y,y )J 3 3 3 = ph W. + — —(h + —( \Sf + \h ) — ^ ^ ( y 4" w ) 0,u J 2 *>tt 315 x.xtt vy.ytt ' 252 O.xxtt O.yytt ' --£*L.(0 +0 ) 1008 z>xstt z-yy11 (4.6) M + M - Q - —l- ( p +P — 3S )= _ 1 pJl x,x xy.y v x ^ ^2 x.x xy,y J l,x / 315 ^x.tt 315 w0.xtt (4.7) 315 M +M —O — (P + P —3S )- j7ph 3 . _ 4p__ ' -h^ xy.x y.y ^y 3- ,h 2 2 ' * xy.xxy.x ’ ' y.yy.y ~ "y ' 315325 V'y.tt’’y.tt 3J315 - W "O.yu £ * 1 0 (4.8) 315 z--vu — (P +2P +P )-2 M = £ iL .0 + £ iL .w + £iL.(0 2 x'xx xy.xy y.y z 80 ^ 12 °-u 315 ^x.xu ^y.yu ) p h 5 h7 (4.9) 1008 ^ W°’xxu + w o-yyu ^ ~ 4032 ^ ^ z-xxtt + ^ .y y u ^ and the corresponding boundary conditions. Along x = constant the following must be prescribed: 72 p, or d> or ( £ ^ -\h — w - d> - l p - i - P ) (410) z 315 *•“ 1008 °*xtt 4032 z-xu 3 ss 3 *.v.y v ,u; Along y = constant the following must be prescribed: W° °r ^ 315 252 W°-ytt 1008 ~ + Sy ~ 7 h ^ Py,y ~ 3h2 Pxy,x^ * , « or ' “ i - j j r V Py or ('^ T w 0 or xv. - -ph 0 - —P - — P ) (4.11) z 315 y-u 1008 ^ 4032 ^ 3 y*y 3 xy-x At the corners p,vvv or ( -^ 3h2 T wo + 3T^) (4J2) must be given. To get the stress-resultants in terms of the generalized displacements, one substitutes equations (3.37) to (3.42) into equations (4.4) and gets the following : Mx = I 5 f (X+2 G^x.x + ^ y ,y l" fo 1 U + 2 G)W 0.xx + XW 0,yy 1 “ ^ 0 1 (X+2G)^ x + A^y.v ] + My = T5 1U + 2G )xpyy + x ^ 1 “ lo 1 (x + 2G} w°,v.v+ x wo.xx] - ^ r (x+2G)^yv+A0,v.v] + ^ 0 : z 7 3 M = +0 ) _ 2i_ v 2w„ — v 2^ + — (\ + 2G)0 2 15 y** yy«y 60 0 240 6 " M = £ * ! ( * * )--^ L w 0 - ° i L 0 xy 15 x.y^y.x 30 °-xy 120 z-xv P* = ^ 5 [ -15W [u + 20)* - + ^ )+is!r*‘ P» = h 1 (X + 2 G )'»’yJ. + ,l^ ) - -^6 [ (x + 2G )'> 'o „+ x '>'t o ) p»y - W (V - J - W wo.«v ' W *« 0, = |oh(*, + wM) Qy= |o h (* J + w0y) s, = i o h ( * ,+ w IU) Sy = i r G1’U .v + "'».r) ‘4.13) Substituting equations (4.13) into equations (4.0) to (4.9) , one obtains the govern ing differential equations in terms of the generalized displacements as follows: {-!£h v*+ hi(x±2Gi 4 J l_ £hi(_ jl_ +j ! _ )}Wft 15 252 et2 252 dx 2 0 t 2 Qy2 a t2 0 + {_ 8G h fl _ 4h3(X + 2G) ( fl3 fl3 } , 4ph3 fl3 } . 15 ax 315 ax3 axay2 315 axdt2;Vx , 8 Gh fl 4h3(X + 2G) / fl3 fl3 x 4ph 3 fl3 } if/ ^ a it « _ 71 < 2 ' ” Y 151 fly 31571 a y 3 flx2fly ' 315 flyflt + {h5U + 2G) 4 _ h 3X 2 + ph3 fl2 _ phS ( fl4 fl4 )} 1008 V 30 V 12 a t -’ 1(X)8 flx 2Qt 2 fly 2flt 2 ' = 0 (4 .1 4 ) 74 {_ 4hl_(X + 2 G ) f j L + _6 3 ) _ 8 G h __a_ + 4 p h 3 Q3 ? 3 1 5 dx 3 exdy 2 15 ax 3 1 5 axat 2 0 + / 17h*U + 2G) a2 + 17Gh3 fl2 _ 8Gh _ 17ph3 Q2 , . 315 dx2 315 d y 2 15 315 0 t2 + {nh^x+G2_§l_} 315 axfiy y + {-hL^±2G)(-Sl+ ..) + liLA JL + £11..J.., )4> = 0 315 ex3 axdy 2 15 *x 315 axat 2 ’ (4.15) {_ 4h (x+2G)/ aJ aJ)_ soh a | 4Ph3 3 a „3 , 315ax2ay ay 3 15 ®y 315 a y a t2 0 + {nV u + G2_ai_} 315 e x a y + { 17h3° a2 . I7 h3( X + 2G ) a2 _ 8G h _ 17ph3 a2 i ^ 315 0 X2 315 d y 2 15 315 a t 2 + {--• h3(-A±2G)(__a4_ + a3.) + 2h3x a + a 3 } ^ = 0 315 ax2ay ay3 15 ^y 315 ayat2 (4.16) { h5(x+2G) v4 _ x_h1^2 + ph^ _af_ _ phs ( a4 ^ a4 )} 1008 30 12 gt2 1008 ax2at2 Q y2 d t2 ° + {_ h5(x+ 2G)( a3 + a3 )+ 2h3x a Ph5 a3 } . 31^ ax 3 a x a y 2 ^ x 31^ a x a t 2 x + { — h (x + 2G)^ a 3 + a ) + 2hlx _a_ ph5 a3 } * 315 ay3 ax2ay 15 9y 315 ayat 2 y . r h7( X + 2G ) _4 _ hlx _~2 h3( X + 2G ) ph^ 4032 60 3 80 aat* r ’ - ph7 ( a4 + a4 — )} 0Z = 0 (4.17) 4032 ex2at2 ay28f: 75 Equations (4.6) to (4.9) or equations (4.14) to (4.17) combined with the boundaiy conditions (4.10) to (4.12) give a complete set of equations which represent the modi fied Kant theory. CHAPTER V FREE VIBRATION ANALYSIS OF ISOTROPIC HOMOGENEOUS RECTANGULAR PLATES USING THE RITZ METHOD The focus of this chapter is to combine the Ritz method with the Mindlin and mod ified Kant theories to develop a two dimensional analysis of the free vibration of iso tropic homogeneous rectangular thick plates. The procedure adopted here is based on plates having general boundary conditions at x = x0 and y = y 0 (Fig. 5.1) and the other two edges are free. In both theories the displacement are represented by algebraic poly nomials, because these polynomials are more adaptable to different boundary condi tions, and they form a mathematically complete set of functions. In the next sections the Ritz method is applied using the two theories. 5.1 MINDLIN THEORY : The generalized displacements for free vibration analyis are taken as : w 0 = W sin tot (5.3) - 7b- 77 Figure 5.1: General coordinates used in the analysis 78 where is the circular frequency and : ^ X = £ £ a . . x iyj (5.4) i_i0 j-io K L * , = Z Z bklxky' (5.5) k-is, i-i0 £ Z^n^y" C5.6) m -m 0 n-jip Substituting those into the energy functionals developed before, the maximum kinetic and strain energy are seen to be, respectively: xj y» T™» = £ f- j / -JJt^ + 'Pp + hW2 dy dx (5.7) xo J’o xi U-“ 4 j j { Dl(1'>.‘)' + (V !+2' + { ' 1 - :" ) ( K y + * y , I xo y0 + k2G h [( +W x )2 + ( + W y )2 ] }dy dx (5.8) The Ritz method is based on minimizing the functional LmhX = Tmax — Uma!. . The minimizing equations can be written a s : 9L —7r~~ = 0 i = i0 ...... 1 ; j = j0 J (5.9) Oaij 9L k = k_ K ; 1 = 1 . , .....L (5.10) 9bk| 9L — m = m0 , M ; n = nQ N (5.11) 0 cm» yielding a set of homogeneous, linear algebraic equations of order equal to (I—i0 + l ) (J—j0 +J ) + (K — k0+ 1 )(L-10+J ) + (M -m0+ l)(N -n 0+ l) in the unknowns ajjt bkj and c 1(lli. These equations can be represented by : 79 ( K - f l 2M)q = 0 (5.12) where: K = the matrix developed by minimizing Umax (Stiffness matrix) M = the matrix developed by minimizing Tmas (Mass matrix ) q = the unknown coefficient vector (eigenvector). and Cl = is the frequency parameter (eigenvalue). xi For a nontrivial solution the determinant of the coefficient matrix is set equal to zero, and its roots f t2 are the squares of the eigenvalues (nondimensional frequency parameters) of the problem. The complete solution of the problem is evaluated by using a mathematics software provided by The Ohio State University Instruction and Research Computer Center. This software uses the Cholesky factorization method to determine both eigenvalues as well as the eigenvectors. 5.2 MODIFIED KANT THEORY In this theory, the same displacements in Mindlin's theory are used in addition to one more displacement given by : = R sin o> t (5.13) where * = z £ d, , s '’5'’ (5.14) I' I,0<1-(1„ and following the same procedure, the maximum energy functionals are found to be: Maximum kinetic energy: 81 + -^Gh[(¥x + Wx)2 + (¥y + Wy)2]}dydx (5.16) The minimizing equations will take the same form as before but with higher order equal to the order of Mindlin's theory + ( P — p0 +1 )(Q —q0+ l ) . The Ritz method requires satisfaction of the geometric boundary conditions only. With the proper choice of the variables x0,x ,,y 0,y, ,i0,j0,k0,l0,m 0 and n0 ( and p0,q0 in the case of modified Kant theory) one can consider many problems with different boundary conditions. In this work, the case of completely free rectangu lar plate is considered, and the next chapter is devoted to this purpose. CHAPTER VI RESULTS FOR FREE VIBRATIONS OF COMPLETELY FREE RECTANGULAR THICK PLATES 6.1 INTRODUCTION Despite the large number of works written on the subject of the vibration of thick plates and the different boundary conditions considered, one finds that a relatively small number of works dealt with the case of completetly free thick plate, and these used three-dimensional formulations, rather than thick plate theory. In their work, Hutchinson and Zillmer [220] considered the three dimensional problem of the paralle lepiped. They provided some curves for the first three frequencies of problems with different aspect and thickness-to-width ratios (no explicit numbers are given for those frequncies). Also, Fromme and Leissa [238] provided numerical results only for the two-dimensional, infinite strip (plane strain). In spite of the large number of papers written on plate vibrations using two dimen sional, thick plate theory (especially Mindlin's theory), these theories have yet to be applied to the completely free plate. In this work, the Ritz method explained in the last chapter is used to obtain the natural frequencies for the free vibrations of the completely free rectangular thick plates to fill part of the gap in the literature in this area. -82- 83 6-2 SYMMETRY CONSIDERATION One advantage of applying the Ritz method to study the free vibration of the com pletely free plate is that the displacement functions chosen are not required to satisfy any boundary conditions. The boundary conditions to be satisfied at a free edge are forced boundary conditions (not geometric) and the Ritz method , as mentioned before, requires the satisfaction of geometric boundary conditions only. In order to make use of the symmetry in the problem x0,x ,,y 0 and y t in (Fig. 4.1) one should take the following values (that is, assuming the origin of the coordinate system to be at the center of the plate): x0 = —a/2 x, = a/2 y0 = -b /2 y,=b/2 '(6.1) and the set of i0,j0,k0,l0,m0 and n0 (also p0 , .q0 in the case of modified Kant theory) should take the values of 0 or 1 according to the symmetry class under consideration (The symmetry is considered with respect to the x and y axes). This results in four symmetry classes; doubly symmetric (SS which means symmetric with respect to both x and y axes, respectively), doubly antisymmetric (AA), symmetricTantisymmetric (SA) and antisymmetric-symmetric (AS). These four classes may be obtained separately by considering either the even terms or the odd terms of the polynomials in the displace ment expressions **= Z Z vV i = O or 1 j = O or I £ i k=Oorl l = Oor 1 = I L c >'">■" tu s Oor 1 it “ o nt I 84 R - Z L Table 3 shows the proper combinations of indices for each symmetry class. The Rite method then can be applied as described in the last chapter. Two computer pro grams were wriiien (one for each theory) to determine both the eigenvalues (frequency parameters) and the eigenvectors. The eigenvectors are used to determine the displace ments (eigenfunctions) which arc used to draw the mode shapes for each mode. Table 3 Combination of the polynomial terms used in displacement expressions Symm. Displacements class direction ** W/R X odd (i«=l,3,J) even (k«0,2,..K) even (m/p*0,2..M/P) SS y even (j-0,2^J) odd (l-l,3 .X ) :ven (n/q-0,2,~N/Q) X even odd odd AA y odd even odd X even odd odd SA y even odd even X odd even even AS y odd even odd The programs are developed such that the effect of the following parameters the natural frequencies can be analyzed: 1. Aspect ratio (a/b) 2. Thickness-to-width ratio (h/b) 3. Material properties (Poisson's ratio v) 6 3 CONVERGENCE STUDY In order to determine how many polynomial terms one should take in each dis placement expression, a convergence study is made. In this study, only one symmetry class is considered (SS) and only one material ( i> = 0.3 ) is used. Previous experience [244] has shown that the rest of the modes or different materials converge in similar ways. The convergence study is for square plates (a/b = 1). 2 Tables 4 to 7 show the convergence of the frequency parameter Q = h for both Mindlin and modified Kant theories for both thickness-to-width ratios ( h/b = 0.1 and 0.5). The first column in each table lists the number of terms used in each of the polynomial in both coordinate directions. Thus, for example, in table 4, a (2,2,5) solution uses 2x2=4 non-zero terms for and ^ y, and 5x5=25 terms for W, yielding 33 model degrees of freedom and a corresponding eigenvalue determinant size. In addition to the frequencies listed in Tables 4 to 7, there is one zero frequency for ihe SS modes, which corresponds fo rigid body translation in the z-direction. It is observed that the natural frequencies of thicker plates ( h/b=0.5 ) converge faster than the thinner ones ( h/b = 0.J ), for both theories. It should be noted here that the convergence study is made for the first (lowest) ten nonzero modes in one symmetry class and not for the absolute first ten modes of the problem. On this basis, considering 108 terms in Mindlin's theory and 144 terms in modified Kant's theory were adequate to give good accuracy (up to at least four signifi cant figures) for the tenth mode of the complete frequency spectrum. The number of terms are equal to the number of degrees of freedom (D.O.F) of the problem, which is equal to the size of the mass and stiffness matrices of the eigenvalue problem. The chosen number of terms to be used for subsequent results is equal to 3 r2 (Mindlin) or 4r2 (modified Kant) where r is the number of polynomial terms in x and y directions ( r = 6 ); that is, parametric studies will be made using 108 (Mindlin) and 144 (modified Kant) terms. Table 4 Convergence study for Ritz method based on Mindlin's theory Case o f a/b>l , h/b-0.1 y * « 0 3 my. Det. Doubly symmetric modes (non zero frequencies) terms size SS-1 SS-2 SS-3 SS-4 SS-5 1,1 S 27 632758 8.78040 49.97668 6431606 6435373 2.25 33 5.76917 7.13682 19.94385 4459804 4752147 3,3,5 43 5.73289 7J05929 16.96905 3131735 32.80162 3.3.6 54 5.73289 7J05929 16.96904 3131641 32.80157 33.7 67 5.73289 7J05929 16.96904 3131623 32.80157 4 ,4 3 57 ‘ 5.73193 7J05751 16.90975 29.78757 31.11366 4.4.6 68 5.73193 7.05751 16.90975 29.78683 31.11255 4.4.7 81 5.73193 7J05751 16.90975 29.78682 31.11255 5 3 3 75 5.73164 7J05667 16.85321 29.75070 3157781 5 3 .6 86 5.73164 755667 16.85314 29.73732 3155928 5 3 .7 99 5.73164 7J05667 16.85314 29.73730 3155927 6 .6 3 97 5.73160 755654 1634486 29.74759 3157538 6,6,6 108 5.73160 755654 1634485 29.73481 3155744 7 .7 3 123 5.73160 755654 16.84485 29.73481 3155744 7,7,7 147 5.73160 7J05653 1634408 29.73449 3155751 Poly. Det. Doubly symmetric modes (non zero frequencies) terms size SS-6 SS-7 SS-8 SS-9 SS-10 1.15 27 7958840 7958840 10056138 10720609 10741205 2A S 33 58.93563 6157210 96.11434 9753925 98.12987 3 3 5 43 4258756 44 j66091 7759470 9359436 9557897 3 5 5 54 4258324 4456086 7759273 9151622 9317143 35.7 67 4258235 4456086 7759263 9151635 9354079 4 .4 5 57 3852555 4057595 6057577 7037951 7238872 4.4,6 68 3852511 4057513 6056922 6957578 7155621 4.4,7 81 3852504 4057510 60.06911 6956623 7154564 5 5 5 75 3853226 4050850 59.74518 67.17541 6959548 5 5 5 86 3851870 3955220 5959803 6343767 64.86983 55.7 9 9 3851869 39.95219 5959775 6340399 64.82781 6 5 5 97 3850975 39.95188 5957174 6757969 6950790 6 ,6 5 108 38.19419 39.92653 5959985 6358190 6449552 7,75 123 3851887 39.94989 5956567 6757585 6950614 7.7.7 147 38.19252 39.92461 5959232 62.80882 64.17024 Table S Convergence study for Ritz method based on Mindlin's theory VIV E Case of a/b-1 , h/b-0.5 , v = 0 5 Poly. Det. Doubly symmetric modes (nan zero frequencies) terms size SS-1 SS-2 SS-3 SS-4 SS-S 1,1,5 27 4.11257 5.11522 9.99151 9.99534 10.83828 2,2,5 33 3.77246 451959 8.80632 949294 9.98596 3,3*5 43 3.76351 4.48768 7.90888 9.14866 9.74808 3,3,6 54 3.76351 4.48768 7.90888 9.14863 9.74802 3,3,7 67 3.76351 4.48768 7.90888 9.14863 9.74802 4,4,5 57 3.76334 4.48736 7.89046 9.13572 9.72759 4.4.6 68 3.76334 4.48736 7.89046 9.13567 9.72751 4,4.7 81 3.76334 4.48736 7.89046 9.13567 9.72751 5,5,5 75 3.76334 4.48734 7.89014 9.13558 9.72714 55,6 86 3.76334 448734 749014 9.13554 9.72704 55,7 99 3.76334 4.48734 749014 9.13553 9.72704 6,6 5 97 3.76334 448734 7.89013 9.13558 9.72714 6,6,6 108 3.76334 448734 7.89013 9.13554 9.72704 7,7 5 123 3.76334 448734 7.89013 9.13558 9.72714 7,7,7 147 3.76334 448734 7.89013 9.13553 9.72701 Poly. Det. Doubly symmetric modes (non zero frequencies) terms size SS-6 SS-7 SS-8 SS-9 SS-10 1,15 27 1456805 14.80729 15.81768 15.81768 20.01228 2 4 5 33 10.77842 1451991 1453366 14.79638 15.76950 3,35 43 1055008 12.93272 14J03180 14.14499 14.96476 33.6 54 1054995 12.93241 14.02080 14.13377 14.95939 33.7 67 1054995 12.93241 14.02066 14.13363 14.95932 4 ,4 5 57 1051375 12.84242 13.71491 13.92813 1450634 4,4,6 68 1051354 1234197 13.70503 13.91583 1450057 4,4,7 81 1051354 1244197 13.70492 13.91569 1450049 5 5 5 75 1051268 1244103 1347077 13.90500 1452945 5 5 5 86 1051236 12.84040 1345782 13.88830 1452072 55.7 99 1051236 12.84040 1345766 13.88810 1452060 6 ,65 97 1051267 1244103 1346983 13.90445 1452802 6.6.6 108 1051235 12.84039 1345640 13.88731 1451891 7.75 123 1051267 1244103 1346982 13.90445 1452801 7.7,7 147 1051235 12.84037 13.65618 13.88704 1451875 Table 6 Convergence study for Ritz method based on Modified Kant's theory n - s a i V f Case of a/b-1 , h/b-0.1 , v=03 Poly. Det. Doubly symmetric modes (non zero frequencies) terms size SS-1 SS-2 SS-3 SS-4 SS-5 1,15.1 28 6.43893 856762 4057109 5256433 52.74526 2 2 5 2 37 5.76827 7.13571 1951516 39.94402 42.18487 3 .3 5 2 47 5.74264 7.07806 17.13701 32.22955 34.00761 3,3,5,3 52 5.73729 7.06764 174)4362 314)8531 32.62337 3 3 5 ,4 59 5.73729 7.06764 17.04336 314)7826 32.61302 4,45,3 66 5.73630 7J06S65 16.96190 30.17158 3157172 5 5 5 .3 84 5.73600 7J06479 16.96182 30.13175 3153568 4,45.4 73 5.73630 74)6563 16.96182 29.96175 3130956 55.5,4 91 5.73600 74)6477 16.90471 29.92652 3137732 5 5 5 5 100 5.73600 74)6477 16.90468 29.92453 3127415 6,6,6,6 144 5.73596 74)6464 16.89630 29.90915 3125429 7,7,7,7 196 Numerical difficulties Poly. Det. Doubly symmetric modes (non zero frequencies) terms size SS-6 SS-7 SS-8 SS-9 ' SS-10 1.15,1 28 64.93243 64.93868 83.07482 9226684 9252162 2 3 5 3 37 52.08625 53.42904 81.96991 82.15822 8325810 3 2 5 2 47 4257614 4537836 73.16227 79.95018 8155250 3 3 5 2 52 41.70927 43.80512 7223483 79.89050 8152778 3,35.4 59 41.70519 43.79128 72.13618 7958345 8120130 4 ,4 5 3 66 38.87129 4059094 6156724 73.19584 75.75858 5 5 5 .3 84 38.76137 4058981 61.16817 70.19848 72.78202 4,45,4 73 3859341 4037938 6052838 7150289 7359162 5 5 5 ,4 91 38.49904 4029169 6039093 68.77205 70.95009 5 5 -5 5 100 38.49627 4028707 6038117 6838227 7055058 6,6,6,6 144 38.45874 4023868 6023361 63.85361 6534018 7,7,7,7 196 Numerical difficulties Table 7 Convergence study for Ritz method based on Modified Kant's theory Case of a/b-1 , h/b-0.5 , v*=03 , Poly. Det. Doubly symmetric modes (non zero frequencies) terms size SS-1 SS-2 SS-3 SS-4 SS-5 1.13.1 28 3.89359 439593 9.08164 10.10866 11.03731 2 3 3 3 37 3.79920 436172 8.64762 9.24376 9.97008 3,3,5,2 47 3.79837 434707 8.07660 930422 9.94918 3,33,3 52 3.79306 433827 8.05277 9.19109 9.90652 3.33.4 59 . 3.79165 433638 8.04444 9.18096 9.90510 4 ,4 3 3 66 3.79280 433792 8D4103 9.18386 9.90347 5 3 3 .3 84 3.79279 433791 8.04068 9.18368 9.90310 4,43.4 73 3.79148 433616 8.03481 9.17589 9.90254 5 3 3 .4 91 3.79148 433615 8.03462 9.17579 9.90220 5 3 3 3 100 3.79142 433610 8.03431 9.17536 9.90207 6,6,6,6 144 3.79140 433607 8.03428 9.17522 9.90195 | 7,7,7.7 196 3.79140 433607 8.03428 9.17520 9.90194 Poly. Det. Doubly symmetric modes (non zero frequencies) terms size SS-6 SS-7 SS-8 SS-9 SS-10 1.13.1 28 1539746 1548987 16.05571 1630426 2233205 2,23,2 37 10.66782 1434679 1436388 1531670 1644981 3 3 3 .2 47 1034213 1341418 1433136 1432226 1533860 3 3 3 3 52 10.61809 1333822 1430618 1447720 1540861 333,4 59 10.61191 1332152 13.94203 14.19068 15.14812 4.43,3 66 10.60921 13.19703 13.78982 1437266 14.81297 5 3 3 .3 84 1030851 13.19259 13.74415 1434975 14.77764 4.43.4 73 1030633 13.19263 13.72237 14.13940 14.64608 5 3 3 ,4 91 10.60577 13.18840 1338773 14.12754 1439792 5 3 3 3 100 1030570 13.18789 1338410 14.12048 1438805 6,6,6.6 144 1030542 13.18692 1336449 14.09287 1437113 7,7,7,7 196 1030542 13.18691 1336433 14.09262 1437096 92 6A COMPARISON WITH CLASSICAL PLATE THEORY Because the lack of numerical values in the literature, the only source which can be compared with is the classical plate theory [282] . The natural frequency parame ters determined by this theory are independent of the thickness of the plate, i.e. the theory gives the same values of go a2 whatever the thickness of the plate is E h (because both shear deformation and rotary inertia are ignored). Table 8 shows the comparison of the results of the two theories with the classical plate theory for the square plate with Poisson's ratio 0.3 . The results of classical plate theory are obtained by considering 6 terms in each direction (36x36 determinant) and data from the other theories is as mentioned before (108x108 Mindlin, 144x144 modi fied Kant). The classical plate frequencies agree closely with those obtained by Leissa and Narila [282] using 16x16 polynomial solution. 11 is observed that the classical plate theory results are considerably higher than the results from the oiher two theories, and that the results from modified Kanl are slightly higher than the Mindlin results. Thus, it appears that the first order shear deformation theories of Mindlin's overcorrect the natural frequencies of the free vibration problem. This was also observed in the three-dimensional study by Leissa and Zhang [241]. It is also noticed that the difference between the results of each theory and the classical plate theory increases in the higher modes. That is because the effect of shear deformation significantly increases in the higher modes. Actually, one can categorized mode shapes into flexural modes and thickness-shear modes for Mindlin's theory. In modified Kant theory, a third type of modes occur, in addition to those two types, called the thickness-flexible modes. The first type is called flexural because the dominating faclor in these modes is deformation due to bending. On the other hand, shear deforma tion is strongest in the second type. The third type occurs only in modified Kant theory 93 because this theory allows for thickness flexibility through the extra term in the expression of w . One can distinguish between these different types of modes by looking at the eigenvectors (next section is devoted for that). Usually, the first type occurs with the fundamental modes (lowest ones) and then becomes the second type with mixed modes in between the two types. The third type occurs with very high modes and only with modified Kant theory. Table 8 Comparison of the nonzero frequency parameters with classical plate theory a/b-1 v * 0 3 Symm Mode number class h/b Theory 1 3 3 4 5 - classical 5.930* 7344 19372 35439 37054 Mindlin 5.732 7057 16.845 29.735 31057 SS 0.1 mod. Kant 5.736 7065 16.896 29.909 31354 Mindlin 3.763 4487 7390 9.136 9.727 0 3 mod. Kant 3.791 4536 8034 9.175 9.902 - classical 10531* 13488 31.914 39.785 60008 Mindlin 9555 16.739 26497 32520 46305 SA 0.1 mod. Kant 9571 16.793 26525 32.826 46524 Mindlin 5397 6328 8533 9934 10.721 0 3 mod. Kant 5353 6.949 8581 10015 11012 • clam’cal 4076* 20961 23353 46353 61.783 Mindlin 3348 18333 20459 36377 46383 AA 0.1 mod. Kant 3351 18390 20543 36517 47390 Mindlin 3550 6454 8310 9.782 10.867 0 5 mod. Kant 2561 6513 8513 4 9390 10.958 * Agree to four significant figures with 16 terms nlutiaas of [282] 95 Table 8 (cant.) Symm Mode number class h /b Theory 6 7 8 9 10 • - classical 48.875 50.987 85.093 90.160 92.197 Mindlin 38.114 39.927 59600 63.082 64496 SS 0.1 mod. Kant 38459 40539 60534 63854 65540 Mindlin 10512 12480 13656 13.887 14519 0 5 mod. Kant 10.605 13.187 13.665 14693 14571 - classical 65.190 73525 94545 110.196 120575 Mindlin 48574 53682 65.149 72632 77628 SA 0.1 mod. Kant 48.809 54528 65.956 73576 78.734 Mindlin 11413 12.808 13.982 14881 15477 0 5 mod. Kant 11589 12.967 14.108 14.883 15.746 - classical 64.754 88535 90555 130593 133.650 Mindlin 49.071 61529 62.917 82.820 84583 AA 0.1 mod. Kant 49545 62.005 63661 84.035 85510 Mindlin 11490 12671 13.716 14560 15.999 0 5 mod. Kant 11538 12575 13.934 14553 16.127 Table 9 ^ Comparison of the nonzero frequency parameters with classical plate theory a/b-2 v « 0 3 fi = ^ Symm Mode number class h/b Theory 1 2 3 4 5 - classical 6.495 26.633 35.929 43531 80.902 Mindlin 6.430 25.665 34.012 40.175 70.874 SS 0.1 mod. Kant 6.432 25.691 34.073 40556 71.125 Mindlin 5.413 16583 19557 21.749 31.779 0 5 mod. Kant 5.434 16.647 19.754 22.093 32528 - classical 18.038 31572 58550 61528 106595 Mindlin 17522 29.764 53.122 55.771 90.856 SA 0.1 mod. Kant 17538 29.803 53562 55.947 91572 Mindlin 11.961 17.982 25.862 26.875 33586 0 5 mod. Kant 12.084 18.184 26593 27571 33579 - classical 17597 48580 74.164 89562 103542 Mindlin 16.787 44.446 67.172 78539 89578 AS 0.1 mod. Kant 16.802 44547 67596 78.744 89.708 Mindlin 11526 22554 27520 28.909 33.814 0 5 mod. Kant 11.400 22.968 27.808 29531 34503 - classical 8.042 30.716 71.113 78.901 107.855 Mindlin 7.709 28.705 63587 70.729 92.165 AA 0.1 mod. Kant 7.713 28.747 63.789 70.982 92591 Mindlin 5.712 17.028 26587 29.194 33508 0 5 mod. Kant 5.733 17512 26.705 29.761 33.953 Table 9 CcooO 97 Symm Mode number class h/b Theory 6 7 8 9 10 - classical 90.868 138.386 146.070 160428 202.067 Mindlin 80.134 114.233 122465 131361 157.938 SS 0.1 mod. Kant 80.508 114.892 123344 132335 159315 Mindlin 32450 35.550 39.689 40327 41308 0 3 mod. Kant 33.159 35.842 40.310 41.074 41.910 - classical 128.638 149.101 175.890 180394 231.747 .Mindlin 109.337 124313 140.868 144.813 176.664 SA 0.1 mod. Kant 109.985 125.091 141.844 145.887 178311 Mindlin 34.984 38304 39318 41315 43.757 0.5 mod. Kant 35.262 39.151 40.108 41.873 44353 - classical 131.232 193375 238.902 247476 257380 Mindlin 109.209 151.719 172454 186496 194.025 AS 0.1 mod. Kant 109.810 152.855 174.732 188338 195.963 Mindlin 38.706 39403 40.053 43.744 45.101 0 3 mod. Kant 39410 39.942 40.722 44362 45.744 - classical 141.061 159314 234359 244331 272457 Mindlin 117.525 129.002 177.965 87341 203332 AA 0.1 mod. Kant 118.226 129.839 179476 89.105 205302 Mindlin 35.724 39378 41380 43372 44.869 0.5 mod. Kant 35.997 39.924 41341 44J095 45.747 98 6.5 THICK PLATE DEFORMATIONS The way a thick plate deforms in each mode can be determined by looking at the eigenvectors (i.e. looking at the relative magnitudes of the w0,i/Jx,\|fy and d>z terms). This section is devoted to this purpose. Tables 10 to 12 show the eigenvectors for a thick (h/b = 0.5), square (a/b=l) plate with i'=0.3 for the different symmetry classes (only modified Kant theory is considered). The first (and the seventh) column shows the mode number (excluding the zero-frequency modes), the second (and the eighth) column shows the exponents of the polynomial terms in the displacement expressions (r for x, and s for y). The remaining columns show the 15 largest terms (out of 36 each) for ^ , W and R (a*, b‘ ,c* and d* , respectively ). In order to understand the process only the SS-modes are considered. In this case the displacements are given by : u = Zr Z , « Z-i4 3 h > , a'-+1+ 1’s ‘ 3h rs 3 " )xr'V} v = T TK ?.-i4)b v-v-' _(i4sc‘ -4-sd LxV } r . 3h ,-’1 3h ,s 3 '« • w = £ rs + z2d rs } x' ys (6.3) t‘ S To obtain a thickness-shear mode, both u and v should be large relative to w . This can be obtained by large a and b* terms and small c and d* terms. On the other hand, if all terms are large (u,v and w are of the same order of magnitude) or if the c and d terms are relatively large compared to a and b* terms (w is larger than u and v), one obtains a flexural mode. A thickness-flexible mode (in modified Kant theory only) can be obtained only if the d' terms are large compared to the remaining terms. 99 Referring to Table 10 and Fig. 6.1 and 6.2, the first mode (second including the zero frequency mode) has a*,b* and c* terms of the same order; this is why one obtains a flexural mode. Looking at the fourth mode (fifth including the zero frequency mode) one observes a' and b* terms of magnitudes similar to the first mode, but c and d' terms are relatively small compared to a‘ and b‘ . Thus in this case one obtains a thickness-shear mode (Fig. 6.2). The thickness-flexible modes are accompanied always by very high frequencies and can not be obtained within the first ten modes. The same argument can be repeated for the other symmetry classes. Table 10 Normalized eigenvectors for thick (h/b-05), square Ca/b-1) plates from the modified Kant theory (first distinct terms of 36 for , ¥ y ,W and R) SS-modes m-mode number (excluding the zero frequency mode) • • t • m rs C r» m rs * r+Ij M+l ® n ® r*U 00 -1.000 1000 0.000 OOOO 00 0 3 9 8 0398 1000 *0.196 0.049 02 0.035 -0.082 -0.834 0058 02 0 0 5 3 •0076 -0 3 8 5 04 -0.002 0.015 0.076 •0013 04 -0.011 0.011 0.065 -0.008 0.004 06 0.001 •0.005 -0.005 0006 06 0.000 -0.003 -0.004 0.000 0.000 0.000 •0002 08 0.000 0.000 0.000 •0.001 08 -0.485 -0.049 20 0.082 -0.035 0.834 •0058 20 -0.076 0.053 0.016 •0.016 0.000 OOOO 22 0004 0.004 -0.013 •0.005 22 -0.001 1 24 0.004 0.000 0.005 OOOl 2 24 OOOO 0.002 0.000 26 0.002 0.000 0.000 -OOOl 26 OOOO •OOOl 0.000 0.001 40 -0.015 0002 -Oj076 0013 40 0.011 -O O ll 0.065 -0.008 42 0.000 0 0 0 4 -0.005 -OOOl 42 0002 0.000 OOOO -0.001 44 0.000 0.000 0.000- OOOO 44 -OOOl •OOOl OOOO OOOl 60 0005 -OOOl 0.005 •0006 60 •0 003 OOOO -0 0 0 4 0.004 0.001 62 0.000 -0002 0.000 OOOl 62 -OOOl OOOO OOOO 80 0.000 0.000 0.000 0002 80 OOOO OOOO 0.000 -0.001 00 0.243 0.243 0.951 -0.186 00 -1 0 0 0 1000 OOOO OOOO 02 -0.580 0.116 •0.954 0.130 02 0.837 -0.151 0044 -0 088 04 0.154 •0.004 0.138 -0033 04 -0 0 5 3 0 0 0 3 -0.036 -0.010 06 -0.008 •0005 -OjOIO 0009 06 0009 0 0 0 3 OOOO •0 0 0 6 08 0.003 0.000 0.000 -0002 08 OOOO •OOOl OOOO 0.002 20 0.116 -0.580 -0.954 0.130 20 0.151 -0337 -0.044 0.088 22 -0002 •0002 1JOOO -0078 22 •0 020 0 0 2 0 OOOO OOOO 3 24 -0004 -0018 •0.153 0027 4 24 -0025 0007 0.118 0 0 0 4 26 -0003 0007 0.000 •0009 26 0011 -0004 •0021 -0 002 40 -0.004 0.154 0.138 -0033 40 -00 0 3 0053 0 0 3 6 -0 0 1 0 42 -0018 •0004 -0.153 0027 42 -00 0 7 0025 •0.118 0.008 44 0009 0 0 0 9 0023 -0u017 44 0 0 0 2 -0002 OOOO OOOO 60 -0005 -0008 -0.010 0009 60 •0003 -0009 OOOO 0.006 62 0007 •0.003 0013 •0009 62 0 0 0 4 -0.011 0021 -0.004 80 OOOO 0.003 0.000 -0002 80 OOOl OOOO OOOO •0.002 101 Table 10 (cont.) * • • • • m m rs * t*U **»♦! C r» d’n rs * r+M k r.1+1 Cr» 00 0.835 -0.835 0.000 0.000 00 0687 06 8 7 -0.764 *0.173 02 -0.184 0088 -1.000 0.121 02 -0054 -0.092 1600 •0.091 04 -0.014 0011 0443 -0.063 04 0024 0.009 -0.478 0.061 06 0.006 -0.008 -0.079 0.018 06 -0.006 0.001 0.093 -0612 08 0.000 0.001 0.007 -0.005 08 0002 -OjOOI -0.009 0.002 20 -0.088 0.184 1.000 -0.121 20 -0092 -0.054 1600 •0.091 22 0.072 -0072 0.000 0.000 22 -0003 -0.003 •0677 0617 5 24 -0.015 OOOl -0.024 0.009 6 24 0023 -0.014 0.021 -0603 26 0.000 0.002 0.006 -0.004 26 -0007 0.002 •0.002 0.000 40 -0.011 0014 -0443 0.063 40 0.009 0.024 -0478 0.061 42 -0.001 0015 0.024 -0.009 42 -0014 0.023 0.021 •0.003 44 •0.003 0003 OOOO 0.000 44 -0002 -0.002 -0.008 0600 60 0.008 •0006 0079. -0.018 60 OOOl •0.006 0693 -0612 62 •0.002 OOOO •0.006 0004 62 0002 -0007 •0602 0600 80 -0.001 OOOO -0.007 0005 80 •OOOl 0.002 -0 609 0602 00 -0.171 -0.171 0331 -0015 00 -0778 0.778 0 6 0 0 0.000 02 0.673 •0082 -0.706 0095 02 06 33 -0.138 -16 0 0 0329 04 •0.283 0024 0338 -0 049 04 •0614 0 6 4 2 0.749 -0.124 06 0.036 •0005 •0071 0013 06 0218 -0.002 •0324 0611 08 -0.002 OOOl 0.008 -0.002 08 -0025 0.000 0 6 3 0 0605 20 -0.082 Oj673 •0.706 0095 20 0.138 -0.633 1600 -0329 22 -0.066 •0066 1000 -0.177 22 0703 -0.703 0 6 0 0 0600 7 24 0.042. •0007 -0360 0.064 8 24 •0340 0.133 0 3 9 8 0612 26 0002 •0002 0061 -0 020 26 0009 -0027 •0684 0630 40 0.024 •0283 0338 -0049 40 •0042 0614 -0.749 0.124 42 -0007 0042 -0360 0064 42 •0133 0 3 4 0 -0 398 •0612 44 0001 OOOl a n o •0018 44 0073 -0 073 0 6 0 0 0600 60 -0.005 0036 •0071 0013 60 0002 -0318 0324 •0611 62 0.007 0002 0061 •0020 62 0027 ■0009 0.084 •0630 80 OOOl •0002 0008 •0002 80 OjOOO 0 0 2 5 •0.030 -0.005 102 Table 10 (cant.) • • • m m IS ® r» d‘„ xs C, C TU 00 -0510 -0 510 •0567 0 5 2 3 00 •0581 0581 0.000 0.000 02 0557 •0.065 0.921 -0578 02 0546 -0539 I jOOO •0 518 04 •0.066 0.101 -0.891 0.165 04 -0593 0.179 -0.836 0.153 06 •0.006 -0.021 0584 -0.034 06 0.131 -0.035 0 5 4 6 -05 3 9 08 0.004 0.000 -0.041 0.000 08 -0.016 0.002 -0.034 0.003 20 •0.065 0251 0.921 -0578 20 0539 -0546 -IJOOO 0 5 1 8 22 -0.141 -0.141 1.000 -0.128 22 0538 -0538 0.000 0 5 0 0 9 24 0.163 -0.035 -0594 0.050 10 24 •0.166 0.046 0 5 6 5 -0.015 26 -0.051 0.008 0.082 -0.002 26 0.003 -0.010 -0.084 0.023 40 0.101 -0.066 -0.891 0.165 40 -0.179 0593 0.836 -0.153 42 -0.035 0.163 -0594 0.050 42 -0.046 0.166 -0 5 6 5 0 5 1 5 44 -0.006 -0.006 0.088 -0.009 44 0.047 -0.047 0 5 0 0 0.000 60 -0.021 -0.006 0584 -0.034 60 0.035 -0.131 •0546 0539 62 0.008 -0.051 0.082 -0.002 62 0.010 -0.003 0.084 -0.023 80 0.000 0.004 -0.041 0.000 80 •0.002 0.016 0434 -0.003 Table 11 Normalized eigenvectors for thick (h/b-05X square fa/b-1) dates from the modified Kant theory (first distinct terms of 36 for , ¥ y ,W and R) SA-xnodes m-mode number (excluding the zero frequency mode) • 0 • 0 * m is ^ r+1,* m rs * r , b r+lj+1 c r*M ^ r+l,* * r , ^ r+M+1 00 -0.061 1.000 0.773 •0.132 00 0484 0.012 1600 -0.144 0.256 -0.094 -0.972 0068 02 0.052 •0.045 •0.150 0.018 02 -0.001 04 -0.084 0.020 0.111 -0017 04 •0611 0.002 0636 06 0.001 -0.006 -0.007 0007 06 0.002 0.001 -0.003 0.000 0.000 08 -0.001 0.001 0.000 •0002 08 0.000 0.000 0.000 0.060 20 -0.254 -0.100 0.035 0.009 20 0.186 0.026 -0464 22 0.057 -0.003 0.076 •0.011 22 0.047 0.013 0.003 -0.005 1 24 -0.005 0.000 -0.005 0.002 2 24 -0.009 0.004 -0.003 -0.001 26 0.005 0.001 -0.001 OOOO 26 0600 -0.002 0.000 0.001 40 0.012 -0.005 •0.008 0002 40 -0.044 -0.011 0670 -0609 060 0 42 0.012 0.005 •0.004 •0003 42 -0605 0601 0600 0601 44 -0003 -0.002 -0.002 0004 44 -0601 •0604 0601 60 -0.003 -0.001 0.Q01 •0002 60 0.006 0601 -0605 0602 62 0.002 0.000 -0.001 0002 62 0601 -0.001 0 6 0 0 0601 0 6 0 0 80 0002 •0.001 0.000 OOOl 80 -0601 0600 0600 00 1.000 0.121 0238 -0.115 00 -0.742 -0316 1600 •0319 02 -0.863 0.067 -0.004 0054 02 -0.151 -0.066 -0.194 0651 04 0.125 -0.010 -0.099 0002 04 0.032 0609 0681 -0607 0.004 06 -0.008 0.001 -0.021 OOOl 06 0603 -0.002 -0.011 08 OOOO 0.000 -0002 •OOOl 08 0.000 0601 0.002 -0602 20 0.149 0.099 -0211 0012 20 0344 0329 -0.877 0.122 •0228 -0.020 0229 -0019 22 0648 0635 0.028 -0.018 22 0.000 3 24 0.048 -0.006 -0049 0009 4 24 •0635 0604 -0.030 26 -0.010 0.002 0008 -0006 26 -0601 0600 0606 •0602 40 -0.017 0.022 0021 -0005 40 -0646 -0645 0336 -0.029 42 0.012 -0.005 •0026 0610 42 .-0613 -0.006 0604 0602 44 0.007 0.001 0.018 -0611 44 0612 -0607 0611 0 6 0 0 60 OOOO 0.000 •6.011 OOOO 60 0605 0604 •0633 -0608 62 -0.005 0.000 0004 -0004 62 0.004 0601 •0601 0 6 0 0 80 0.000 0.001 -0.001 0600 80 -0603 0600 0603 -0603 104 Table 11 (coat.) • » m m rs a r, cV i , m IS • r , k r+l,rM Cr*M ** r+M 00 •0.001 0358 -0368 0.058 00 -0068 I jOOO 0.891 -0.187 02 0.108 -0.172 1.000 -0.147 02 0324 0.181 -0399 0.000 04 -0.066 0.003 -0.304 0.051 04 -0.059 -0.050 -0.128 0.003 06 0.008 0.006 0.046 -0014 06 -0.012 0.009 0.038 -0.013 08 0.000 -0.001 -0.004 0.003 08 0.001 -0.002 -0.006 0.005 20 -0.180 -0.106 0365 •0032 20 -0.478 -0.780 -0.950 0.162 22 0.190 0.078 -0.459 0.056 22 -0.132 -0.069 0385 -0.032 5 24 -0.021 0.002 0.100 -0.019 6 24 0.170 0.006 -0.024 0.001 26 0.001 -0.003 -0.012 0.007 26 -0.007 -0.002 -0.011 0.012 40 0.010 •0.016 -0.039 0.007 40 0334 0.176 0372 -0.044 42 -0.031 -0.007 0.087 -0.012 42 -0.046 0.001 -0.173 0.016 44 0.006 -0.007 -0.019 0.008 44 -0.016 0.018 •0.003 •0.001 60 0.002 0.003 0.002 -0.001 6 0 . -0.034 -0.016 -0038 0.010 62 0003 OjOOO -0.009 0.002 62 0 0 0 7 -0.002 0.024 -OJ004 80 0.000 0.000 0.000 0.000 80 -0.001 0.002 0.003 -0.003 00 -0.707 I jOOO 0317 •0.058 00 0.407 -0.797 -0.401 0.084 02 I jOOO 0.181 -0.040 0.058 02 -03 6 4 0.141 I jOOO -0.177 04 -0.276 -0.050 -0.157 0.020 04 0.188 0.013 -0399 0.087 06 0.043 0.009 0.058 0.004 06 -0.054 -0.009 0.162 •0316 08 -0004 •0.002 •0.008 -0.004 08 0.007 0.001 -0.021 0.000 20 0.406 -0.780 -0395 0.039 20 0 3 7 8 03 7 3 0316 -0.056 22 -0324 -0.069 0390 -0.090 22 -0 3 1 7 -0.055 -0399 0.102 7 24 0.031 0.006 -0.105 Oj032 8 24 0 3 7 5 -0.009 0.096 -0.046 26 -0.016 -0.002 0.011 -0.012 26 -0.094 0.005 -0.024 0 3 1 0 40 -0.098 0.176 Oj084 -OJOll 40 -0.052 -0.133 -0.113 0.017 42 0028 0.001 -0.104 0.023 42 0.001 OJ017 0.145 -0.024 44 0.010 0.018 OJ037 -0.008 44 -0.005 -0.007 -0047 0.018 60 0.016 •0.016 •0.012 0.004 60 0.009 0.012 0.020 -0.006 62 -0.010 -0.002 0.013 -0.008 62 0.009 0.000 -0.027 0.014 80 -0.003 0.002 0.001 -0.001 80 0.000 0.000 •0.002 0.002 105 Table 11 (cant.) • • • • IS m rs m ■w k r+l^+l c’f i , * r , ** *•»*♦! C’r*I, 00 -0.781 0358 -0 604 -0383 00 0.164 0647 -0.157 0.127 02 •0.029 -0307 0.131 -OjOIO 02 0 3 0 6 -0615 1600 -0379 04 -0.044 0.107 -0 314 0.036 04 •0.391 0369 -0794 0.180 06 0.023 -0.018 0.072 -0.006 06 0.132 -0048 0334 -0.050 08 -0.004 0.001 -0.010 -0.001 08 •0.020 0603 •0632 0.006 20 1.000 -0.067 1600 -OJ024 20 •0.486 •0562 -0631 -0.005 22 0.385 0.030 0308 -0.046 22 0.699 0.142 -0313 0641 9 24 -0.175 -0.039 -0.022 0.013 10 24 •0619 -0.077 0364 -0.021 26 0.027 0.004 -0.003 •OOOl 26 0.120 0.013 -0673 0.013 40 -0366 0.002 -0335 0.074 40 0.159 0.176 0649 •0.010 42 -0.058 0.049 -0.098 0.016 42 -0.018 0645 0014 •0.005 44 0.041 •0.001 0.026 -OOll 44 0.023 0.007 -0017 -0.010 60 0.087 -0.007 0.129 -0019 60 -0.033 -0648 -0014 0605 62 -0.010 •0.012 0.019 -0007 62 -0.025 0.003 0008 •0.012 80 -0.013 0.002 -0.015 0002 80 0.004 0606 0002 -0.001 106 Table 12 Normalized eigenvectors for thick (h/b-(X5X square (a/b-1) plates from the modified Kant theory (first distinct terms of 36 for »w and R) AA-modes ro-mode number (excluding the zero frequency mode) • » e • m IS d r*U+l m rs •V. c r+M*! ** rM.rU * r.s+1 Cr+M*l OOOO 00 -0.788 •0.788 1.000 -0011 00 -1000 1.000 0.000 -0.462 0038 02 0.082 0.004 -0.037 0005 02 0034 0/475 -0009 0.003 OOOO 04 -0030 -0053 0.060 -0.011 04 0.007 0.004 06 0.001 OOOl 0.000 OOOO 06 OOOl OOIO -0 0 0 4 0.000 0.000 0.000 OOOO 08 OOOO -0.003 OOOO •0.001 08 •0.038 20 0.004 0082 •0.037 0005 20 -0/475 •0034 0/462 0.000 0.000 22 0.006 0006 -0.003 -0003 22 0.073 -0.073 1 -0.004 .0002 -0.001 OOOl 2 24 OOOO -0006 -0.006 0003 24 -0.002 26 -0.001 •0002 OOOO -OOOl 26 0.001 OOOO OOOl 40 -0.009 0007 0003 OOOO 40 0053 0 0 3 0 •0 060 OOll 42 0.002 -0004 •OOOl OOOl 42 0006 OOOO 0 0 0 6 •0003 OOOO 0.001 -0002 44 •0004 0004 OOOO OOOO 44 0.000 -0004 60 0.001 OOOl OOOO OOOO 60 •0010 -OOOl 0 0 0 4 62 0.002 -OOOl OOOO -OOOl 62 OOOO -OOOl •OOOl 0002 80 0.000 OOOO OOOO OOOO 80 0003 OOOO OOOO OOOl 00 0352 0352 1000 -0758 00 0.944 -0.944 OOOO OOOO 02 •0.069 0070 -0355 0046 02 0097 1000 -0.879 0071 04 -0.004 -0032 0057 -0006 04 •0066 -0.095 0 3 2 8 -0029 06 0.001. 0003 -0.005 OOOO 06 0005 0013 -0031 0010 08 0.000 OOOO 0.000 OOOO 08 -OOOl -0007 0 0 0 2 -0003 20 0.070 •0069 •0355 0046 20 -1000 -0097 0.879 •0071 22 0.036 0036 0058 -0009 22 0322 -0322 OOOO OOOO 3 24 -0.009 OOOl •0006 -0002 4 24 •0025 0 0 0 5 0 0 0 2 0008 •0003 26 OjOOI .0003 OOOO 0002 26 0006 •0009 -0 0 0 2 40 •0032 -0004 0057 •0006 40 0 0 9 5 0 0 6 6 -0 3 2 8 00 2 9 42 0.001 •0009 -0006 -0002 42 •0005 0 0 2 5 •0 002 •0008 44 OOOO OOOl OOOO 0004 44 •0006 0 0 0 6 OOOO OOOO 60 0.003 OOOl -0U005 OOOO 60 •0013 -0005 0031 -0010 62 0003 OOOl OOOO 0002 62 00 0 9 •0.006 0002 0003 80 •0.002 OOOO 0.000 OOOO 80 0007 OOOl -0 0 0 2 0003 107 Table 12 (cant.) • * • ** r+M+l m IS * M+l C r+l#tl ^ r*l4+l m IS * M+l ^r+U C r+to+l 00 -0.774 -0.774 1.000 -0.160 00 I jOOO -1.000 0.000 0.000 02 0.267 -0.018 •0.532 0.077 02 -0630 -0422 -0.163 0094 04 -0.043 -0.047 0.166 -0.013 04 0.156 0.097 0.062 -0016 06 0.007 0.007 -0.025 0.004 06 -0.012 -0.002 -0.007 0007 08 -0.001 -OOOl 0.003 -0.002 08 0.000 •0.003 0.001 -0.003 20 -0.018 0267 -0.532 0.007 20 0422 0.630 0.163 -0.094 22 0.148 0.148 0.006 -0.022 22 -0.331 0.331 0.000 0.000 5 24 -0.064 0.007 -0.001 -0.006 6 24 0.019 -0.058 -0.041 0.001 26 0.006 0.008 0.000 0.003 26 -0.007 0.008 0.010 -0.004 40 -0.047 -0.043 0.166 -0.013 40 -0.097 -0.156 •0.062 0.016 42 0.007 -0.065 -0.001 -0.006 42 0.057 -0.019 0.041 -0.001 44 0.005 0.005 -0.002 0.012 44 0.000 0.000 0.000 0.000 60 0.007 0.007 -0.025 0.004 60 0.002 0.012 . 0.007 -0.007 62 0.008 0.006 0.000 0.003 62 0.008 0.007 -0010 0.004 80 -0.001 •OOOl 0.003 0.002 80 0.003 0.000 -0.001 0.003 00 0.486 0486 1.000 •0.265 00 0.716 -0.716 0.000 OOOO 02 -0.549 0003 -0.702 0.116 02 -0.314 1.000 0.402 •0069 04 0.142 0.123 0.150 •0.031 04 0.038 -02 7 5 -0.188 0.030 06 •0.020 -0.027 -0.016 0.004 06 •0.002 0.035 0.037 -0.008 08 0.001 0.002 0.001 0.000 08 0.000 -0.001 -0.004 0002 20 0.003 -0.549 -0.702 0.116 20 -I jOOO 0214 -0402 0069 22 -0.099 0099 0.473 0.037 22 0418 -0418 0.000 0.000 7 24 0.068 *0050 -0.098 0.014 8 24 -0.041 0.124 0.032 -0.006 26 -0.005 0.007 0.011 -0.002 26 -0601 -0.016 -0.008 0002 40 0.123 0.142 0.150 •0.031 40 0275 -0.038 0.188 -0.030 42 •0.050 0.068 -0.098 0.014 42 •0.124 0.041 -0.032 0.006 44 0.008 0008 0.021 -0014 44 Oj020 -0.020 OOOO OOOO 60 -0.027 •0020 -0.016 0.004 60 -0.035 0.002 -0.037 0008 62 0.007 0005 0.011 -0.002 62 0016 OOOl 0.008 -0.002 80 0.002 0.001 0.001 0.000 80 0.001 0.000 0.004 -0002 108 Table 12 (cant.) 9 * 9 • IS m xs * r#+l Cr+U+I m * nr+1 ^ r+M ® rH#*l ^ r+l^+l 00 ■0229 ■0229 1000 •0.111 00 0.932 -0.932 0.000 0.000 02 0251 ■0256 -0.849 0.135 02 -1000 0.766 -0249 0.015 04 -0.096 0.191 0235 •0.042 04 0224 -0233 0.146 -0J024 06 0.011 -0.035 -0.033 0009 06 -0.052 0.030 •0.039 0.008 08 -0.001 OOOl 0003 •0.002 08 0.005 -0.002 0.005 -OOOl 20 -0.256 0.351 •0.849 0.135 20 -0.766 1.000 0249 -0.015 22 -0.049 -0.044 0.658 -0.109 22 0.673 -0.673 0.000 0.000 9 24 0.035 -0.060 •0.168 0.032 10 24 -0.170 0.171 0.002 -0.001 26 -0.007 0.008 0022 -0.008 26 0021 -0.026 -OOOl •0.002 40 0.191 •0.096 0235 -0.042 40 0233 -0224 -0.146 0024 42 -0.060 0.035 -0.168 0.032 42 -0.171 0.170 -0.002 0.001 44 0.002 0002 0.042 -0012 44 0.035 -0.035 OOOO 0.000 60 -0.035 OOll -0.033 0.009 60 -0.030 0.052 0039 •0.008 62 0.008 -0007 0.022 -0.008 62 0.026 -0.021 0.001 0002 80 0.001 •0.001 0003 -0.002 80 0.002 -0.005 -0 005 OOOl SECOND MODE (MOD. KANT) 109 l ~ \ x ■ a/2 1 y - b/2 f y - b/4 II 7 x ■ a/4 —■ ■— ny = 0.0 i x = 0.0 7j1 1 y - -b/4 1 I_-Jx ■ —a/4 LZIy ■ —b/2 L_J x ■ >*a/2 at y «= const. at x ■* const. mode shape at z ■ h/2 I) - S.791 Figure 6.1: SS-2 a/b=l, h/b*0.5 FIFTH MODE (MOD. KANT) 110 C U C Z l y — b/2 x — a/2 y — b/4 x ■ a/4 y = 0.0 x * 0.0 -b/4 x ■ —a/4 const. const. mode shape at z — h/2 0 - 0.176 Figure 6.2: SS-5 a/b-1, h/b-0.5 I ll 6.6 EFFECT OF VARYING POISSON'S RATIO UPON THE FREQUENCY PARAMETER In this section, studies are made for plates with different aspect ratios (a/b =1,2) and thickness-to-width ratios (h/b = 0.1 , 0.5). The frequency parameter considered ( Q = ^ ay j -g-) is independent of Poisson's ratio, which helps to study its effect. Tables 13 to 16 give the first ten nonzero frequencies as well as the zeros (rigid body motion modes) in each symmetry class for all cases considered. In the case of a/b = 1 , only SA-modes are considered because AS-modes are the same as SA-modes rotaded 90° for square plates. Only the AA-modes do not contain a zero frequency mode. It is observed that Poisson's ratio does not affect the frequency parameter in a sys tematic way. Sometimes it is increasing, sometimes it is decreasing, and other times it is non-monotonic. However, the differences between the Mindlin and modified Kant theo ry frequencies are seen to become quite large for some of the higher frequencies when ?’ = 0.499 (a nearly incompressible material). It is also noticed that although Poisson's ratio may affect the frequency, signifi cantly, it does not influence the corresponding mode shapes as much. Appendix (A) shows contour plots of the first ten nonzero frequency mode shapes as well as the zero frequency ones for both a/b=l and 2, but for Poisson's ratio 0.3 and h/b = 0.1 only. Appendix (B) shows the only mode shapes significantly affected by varying Poisson's ratio. Two Poisson's ratios are used in Appendix (B), 0.0 and 0.499, for all symmetry classes of the square plate (0.499 was used instead of 0.5, because 1 — 2 v appears in the denominator of certain terms in the modified Kant strain energy functional, yielding finite, but indeterminate forms which the computer program could not deal with when v = 0.5. 112 Positive displacement contours in the mode shapes plots of Appendices A and B have solid lines, negative contour lines are dashed. Nodal lines (lines of zero displace ments) are shown as heavier solid lines in the contour plots. In general the effect of Poisson's ratio upon the mode shapes occur as a change in the curvature (or the pattern) of the nodal lines and consequently the rest of the con tour lines. 113 Table 13 Effect of Poisson’s ratio on th e^e^uency parameter. Case of a/b-1 , 0499 Symm v>=0.0 v * 0 3 V 0 ModJCant class Mindlin ModJCant Mindlin ModKant Mindlin OO OO OjO OjO OO OO 6258 6259 5.732 5.736 5486 5495 6258 6259 7057 7065 7.889 7.921 18.117 18.137 16.845 16.896 16.484 16639 30670 30.172 . 30202 29.735 29.909 29.979 33917 SS 30.172 30202 31J0S7 31254 32.800 40373 40444 38.114 38459 38302 39.736 40.064 40.726 40.796 39.927 40239 40415 63240 63404 59600 60234 59.196 63086 64.768 64.911 63082 63.854 62646 66264 64.768 64.911 64496 65340 66.119 72.189 OO OjO 0 0 0 0 OO 0 0 10336 10344 9655 9671 9336 9371 16398 16406 16.739 16.793 17552 17.777 28.076 28.116 26497 26625 26253 26.758 33.024 33J065 32620 32326 33029 33913 SA 46.4S8 46527 46205 46624 46.836 48.907 50.972 51084 48374 48.809 48680 51317 55413 55529 53682 54228 54.012 57367 66.761 66.918 65.149 65.956 65.126 69386 76.928 77.158 72632 73576 72.139 78478 81308 81563 77628 78.734 77239 84.905 4344 4 3 4 6 3348 3351 3602 3606 19.830 19348 18333 18390 17630 17.774 19.948 19.967 20459 20543 21327 21680 38618 38687 36377 36617 36.129 37282 48.805 48387 46383 47290 46237 47325 AA 48331 48.912 49071 49545 50321 52393 64226 64.994 61329 62005 61236 65663 65425 65590 62317 63661 62.877 67510 86.865 87.101 82.820 84035 80.962 86369 86.877 87.112 84283 85510 83448 91453 114 Table 14 Effect of Poisson's ratio on thc^ragJCTiGy parameter. Case of a/b-1 , 0499 Symm */ = 0.0 v*=03 Mindlin ModJCant class Mindlin ModJCant Mindlin ModJCant OO OO 0 0 OO OO 0.0 4.173 4.177 5763 5791 5556 3.606 4.173 4.177 4487 4536 4.720 4.978 8.769 8.830 7590 8034 7501 7.953 9.899 . 9.967 9.136 9.175 5730 8.769 9514 SS 10.923 10.966 9.727 9.902 9.149 10.923 10.966 10512 10505 10543 11.771 13.938 14039 12480 15187 15185 13044 15.215 15032 13556 13565 12434 13268 15215 15032 13587 14093 15348 14498 15524 15584 14219 14571 13502 14.966 OjO 0 0 OO OjO 0 0 0 0 5754 5780 5 2 9 7 5353 5 0 3 9 5 1 3 6 7587 7414 6.828 5949 5 5 3 6 5918 9562 9524 5633 8581 5181 8245 10.994 10.958 9.934 10015 9 4 2 5 9.760 SA 11569 11543 10.721 11012 10203 10.909 12548 12.720 11413 11589 10440 11442 14J071 14088 12.808 15967 15129 15387 15547 15655 15982 14.108 15123 13526 15761 . 15696 14.881 14483 14485 14.947 15751 15767 15477 15746 14580 15385 3J020 3031 2 5 5 0 2561 2468 2487 7519 7570 6454 6513 6024 5094 5726 5773 5310 5513 5018 8518 11047 11045 9.782 9 4 9 0 9.137 9348 12286 12252 10567 10958 10.161 10510 AA 12596 12445 11490 11538 10989 11036 13.905 14020 12571 12475 12046 12490 15474 15515 15716 13934 12455 13274 15481 15571 14260 14553 13513 14521 17.170 17218 15999 15127 15009 15229 Table IS Effect of Poisson's ratio on the^frajuency parameter. Case of a/b -2 ** 0499 Symm * « 0 O * ■ 0 3 Mindlin ModJCant claat ModJCant Mindlin ModJCant 0 0 OO OO 0 0 0 0 0 0 6406 6406 6430 6432 6472 6475 25031 25035 75665 25691 . 26676 26956 33467 33477 34012 34073 35319 . 35403 41575 41.704 40375 40356 41353 41524 ss 74391 74.968 70874 71325 70598 71516 79075 79.141 •0334 •0508 83.160 85128 119547 119.723 11433>3 114692 114387 117509 120589 . 120607 122465 123344 127519 131973 132.139 132303 131661 132535 136340 141509 162.749 163031 157938 159315 160465 168564 OUO 0 0 0 0 0 0 0 0 0 0 17410 17413 17522 17538 17595 17.732 29.942 29955 26764 29603 y ia a a 30951 53.981 54008 53322 52609 53339 56970 57021 55.771 55947 58422 ' 59434 SA 95335 96449 91372 •0904 72557 106823 106930 109337 109985 111967 115097 123623 123.753 124313 125091 128598 132698 145442 145658 140668 141644 146782 148446 147581 147616 144613 145687 148454 156783 113091 183450 176664 176211 176935 186682 0 0 0 0 0 0 0 0 0 0 0 0 11497 16506 16787 16602 15970 15995 46475 46506 11116 44547 43597 43.961 •5590 65624 67372 67396 70619 71554 79668 79.741 78439 76744 •0438 •1.763 AS 90487 90583 •9378 •9.706 70985 73026 113439 113596 109309 109610 110.732 111766 159356 159552 151.719 152356 158569 177957 178644 1724S4 174.732 170949 189.731 185833 186.110 186496 186338 193271 209094 195293 195620 194025 195963 199.949 214914 6711 6714 7.709 7.713 7316 7321 30706 30723 26705 26747 27642 27933 66174 63587 66789 • 62531 61123 69329 59373 70729 70982 74338 75368 94647 94959 92365 •2591 74018 95963 AA 119393 119535 117525 118326 118537 121625 134594 134608 129002 129639 130301 134602 187051 187428 177965 179476 176167 186.947 186230 186519 187341 189405 190330 206687 207073 703737 205302 207.768 222320 Table 16 Effect of Poison's ratio on the frequoicy parameter. Case of a/b-2 n/D-co Symm »-OuD •>*03 •»*0499 ModJCant dut Mindlin ModJCant Mindlin ModJCant Mindlin OuD 0 0 0 0 0 0 0 0 0 0 5487 5490 5413 5434 5365 5419 16693 16.709 16483 16547 . 16339 . 16.944 20219 20378 ; 19457 19.754 19441 20555 23044 23.151 21.749 22093 21371 23042 31A87 SS 3S410 31.779 32328 30147 3S463 35.733 32450 33.159 30963 33084 34.934 39294 35550 35342 33373 35.139 43494 43365 39A89 40310 37573 39.765 44.166 44299 40527 41074 39032 41579 44236 44384 41308 41310 40503 44374 0 0 0 0 0 0 0 0 0 0 0 0 12443 12465 11361 12084 11626 11978 18492 18336 17382 18284 17314 18.754 28243 2SJ62 26393 24.752 26058 29027 29310 36375 27471 ymn 28472 SA 37454 37486 33386 33479 31355 31569 38423 38.700 34384 y ty t 33559 43081 43397 38504 39.151 36.140 37354 432S2 43588 39318 40108 17349 40351 43464 44052 41515 41373 40305 42543 47387 47440 43.757 44553 41324 44067 0 0 0 0 0 0 0 0 0 0 OO 12.758 12302 11326 11400 10623 10.769 24960 25095 22654 22368 21401 21597 29348 29656 27320 27308 26224 27522 31386 31.775 28309 29331 27560 28690 AS 34791 37030 33314 34403 32327 yifu 41351 42043 38.706 39410 36.791 37528 43.975 43333 39403 39342 37218 39253 44327 44418 40722. fftM 41281 49032 49322 43.744 44362 41317 43441 49367 50208 45.101 45.744 42573 44386 4505 6S25 5.712 5.733 f » t 5353 18472 18352 17028 17312 16085 16474 29308 29.762 26287 26.705 24.723 25348 31042 31327 19294 29.761 28301 29.722 36053 36390 yf »0t 33353 3L718 33395 AA 40.162 40331 35.724 35997 SUM 43.762 43906 39378 39324 37073 38951 45936 45395 41380 41541 39038 40119 47402 47A36 43372 44095 41312 43.144 49344 49S29 44369 45.747 42.788 45.179 117 6.7 EFFECT OF ASPECT RATIO AND THICKNESS-TO-WIDTH RATIO In order to study the effect of both a/b and h/b ratios another form of the fre quency parameter is considered. The new form is independent of a and h and includes b only, which is kept fixed during this study. The new frequency parameter ( D* ) is obtained as follows (a /b )2 k E (a/b)2 (6.4) Tables 17 to 18 shows the first ten nonzero frequencies as well as the zero ones for Poisson's ratio =0.3 . The two theories are considered and they agree for all results. By keeping h/b( and b ) fixed and changing a/b from 1 to 2 (increasing both the fiexibilily and the inertia of the plate), the frequency decreases. This effect is agreed upon by both theories for all modes of the different symmetry classes. On the other hand, if.one changes h/b from 0.1 1o 0.5 and keeps a/b(.andb) fixed (increasing Ihc rigidily of the plate more than the inertia), the circular frequency usu ally increases. The only exceptions are in the case of the square, the last frequency in the SA modes and the last two in the AA modes. It is also noticed that the modified Kant theory gives higher frequencies than Mindlin's in every case. Table 17 Effect of a/b and h/b on the frequency parameter (Mindlin) 1/ = 0 .3 = ii/b*2 Symm. »/b»l k/b-0.1 k/W 15 k/b-0.1 t,' fc/b-05 OjO OO 0 0 OO 05732 14815 01608 06766 07057 24435 06416 24604 19845 39450 08503 24321 2.9735 45680 14044 37186 1.7719 39724 SS 3.1057 48635 34114 53560 34034 44563 39927 _ 64400 34558 44438 59600 58280 34616 4.9611 53082 59435 34915 94659 54496 74095 39485 31635 0 0 0 0 OO 0 4 09655 44485 04381 14951 14739 34140 07441 52478 34497 43165 33381 52328 34620 49670 33943 53S94 SA 46205 53605 53714 42608 43374 57065 37334 43730 53682 54040 51078 49130 05149 59910 SS217 49023 y y n 14405 56303 31894 74628 57385 44166 34696 0 0 0 4 . 04197 34158 U 117 58318 • S u m 35793 34150 19690 55136 AS m m 35320 42368 3.7302 i f M t SA SA 37930 49254 43114 • 45624 34680 35376 04848 33250 01927 07140 14333 34270 07176 51285 30459 41550 39897 52984 30377 44910 39682 55493 ' 44883 • 54335 53041 41635 39381 4 i4 ft« AA 49071 59450 41339 53355 S32S1 49098 5SS80 44491 31725 04917 i«s 04820 73300 34215 34283 79995 50908 36086 119 Table 18 Effect of a/b and h/b on the frequency parameter (modified Kant) v - 0.3 o" = Oib Ii/b«2 Symm. «/b»1 k/b-0.1 k/M>5 CliK k/b-0.1 BifeOS 0.0 0 0 , 0.0 OO 06793 05736 IJ935 03608 07065 23680 06423 30809 40170 03518 34693 I6B96 *7616 25909 45875 10064 44510 77781 40410 SS XI254 4.1449 SS495 53025 >0137 40239 *5935 34723 44803 40234 48325 >0811 50388 43154 70*65 33133 *1343 15340 73855 >9104 53388 00 00 00 0 0 15105 05671 >4765 04385 10793 >4745 *7451 33730 24625 43405 13316 *3866 JJS 26 urns 13987 *4339 SA 44624 *5060 33811 43849 44809 *7945 *7496 44078 *4228 *1273 44939 6J956 70540 >5461 *0135 2JS76 74415 34472 *3341 74734 74730 44553 *5691 00 00 04201 84230 11137 *4710 few CUK 84849 *4760 19616 *6664 AS m 3J427 43004 *7453 49263 SA SA *8214 49921 43683 *0903 4.7085 *5453 44991 *7180 03451 23305 *1921 *7166 34390 33565 *7187 *3515 40543 43565 84947 *3381 44617 49450 17746 *7301 47290 *4990 *3148 43441 AA 49545 *7690 *1557 .44996 02005 *4375 *3460 49905 03661 *9670 44869 *1926 ffHC . 73765 4.7276 . *5119 ■5510 *0635 *1326 *7184 CHAPTER VII DISCUSSION 7.1 SUMMARY AND CONCLUSIONS In this work a comprehensive literature review has been caried out to identify the various shear deformation theories available in the published literatureThe large num ber of theories uncovered by this study have been examined and organized according to fundamental approaches used. Then, one of those approaches was chosen and applied by making use of two theories, a first order one (Mindlin's) and a higher order theory (modified Kant). The two theories were combined with the Ritz method to establish a two dimen sional plate theory. This theory is used to study the free vibration of the completely free thick plate. A computer program was written for each theory and studies were made to examine the convergence and the effect of various parameters on the natural frequencies of the problem. As a result of these studies, the following conclusions can be m ade: 1. Modified Kant theory gives higher frequencies than Mindlin's theory, but both sets of frequencies are lower than those given by the classical plate theory. Thus, the first order shear deformation theories, like Mindlin's, apparently over correct the classical theory natural frequencies. Then, higher order theories recorrect the frequency by relaxing some of the restrictions pul by the first order theories. - 120- 121 2. In both theories, the convergence of the frequency parameter was faster for thick plates than for the thinner ones. 3. Poisson's ratio significantly affects the natural frequencies for all modes, and the mode shapes for some modes (especially higher modes). 4. Thicker plates have higher frequencies than thinner plates with the same length and width (same aspect ratio, different thickness-to-width ratio). 5. The longer the plate, the less the frequency is, provided that both width and thickness are kept constant (same thickness-to-width ratio and different aspect ratio). 6. The modified Kant high order shear deformation theory is recommended for thick plate problems with h/a > 0.1 , or if a large number of accurate frequen cies are needed (to obtain the higher frequencies more accurately). 12. RECOMMENDATIONS FOR FURTHER RESEARCH The method and theories used here can be applied to study different problems with various boundary conditions. An interesting example to be considered is the free vibra tion of a cantilever thick plate. It can also be extended to study the forced vibration as well as buckling (stability) of thick plates. With the proper adjustments, the method used in the current work can be modified to consider different types of plate materials. Laminated and anisotropic plates are good examples for such problems. It is generally known that shear deformation effects are more important in laminated plates, than in isotropic, homogeneous ones. Whether or not the thickness-stretch function 4>r of the modified Kant theory is also important for such materials may be worthy of investigation. Appendix A MODE SHAPES FOR POISSON'S RATIO 03 In all the figures shown in this appendix the following is constant h /b = 0.1 , v - 0 3 , a = n £ and the coordinates shown below are used: i 1 * * a/2 N .O a/2 Figure A.1: . Mode shape coordinates. Mode shapes are shown by means of plate cro6s-sections fatm at constant values of x and y , and by normalized contour plots of w. Maximum and minimum values of ±1 are shown there by blade dots. -122- FIRST MODE (MINDLIN) y * b/2 x — a/2 ■ ‘ E y ■ b/4 x ■ a/4 » ■ E y « 0.0 x ■* 0.0 ■ i E y ■= —b / 4 x *■ —a / 4 I U1-,...... ) I-...... 3 y « —b/2 x — —a/2 at y = const. at x = const. mode shape at z * h/2 o - o.ooo Figure A2: Mode shape SS-1, a/b -1 124 SECOND MODE (MINDLIN) b / 2 a / 2 b / 4 a / 4 0.0 0.0 £ - b / 4 - a / 4 c 3 y - b / 2 x —a / 2 . a t y c o n s t . a t x ; c o n s t . mode shape at z = h/2 0 .«= 5.732 Figure A.3: Mode shape SS-2, a/b - 1 125 THIRD MODE (MINDLIN) b / 2 a / 2 b / 4 a / 4 0.0 0.0 J C - b / 4 - a / 4 y - b / 2 - a / 2 a t y c o n s t . a t x c o n s t . mode shape at z = h/2 o «= 7.057 Figure AA: Mode shape SS-3 , a/b - 1 126 FOURTH MODE (MINDLIN) b / 2 a / 2 3 C b / 4 a / 4 3 C y - 0 .0 0.0 3 C - b / 4 - a / 4 3 . y - b / 2 X - a / 2 a t y c o n s t . a t x c o n s t . mode shape at z = h/2 0 « 16.645 Figure A.5: Mode shape SS4, a/b - 1 FIFTH MODE (MINDLIN) y - b /2 y « b/4 x ■= a / 4 y - 0 .0 x = 0 .0 X « - a / 4 - b / 2 —a/ at x = const. mode shape at z = h/2 0 «= 29.735 Figure A& Mode shape SS-5, a/b - 1 128 SIXTH MODE (MINDLIN) y - b / 2 x ^ a / 2 y - b /4 x •= a / 4 y «= 0 .0 x - 0 .0 - b / 4 - a / 4 y - - b / 2 x » —a / 2 at y = const. at x = const. o o \N )) ( i a o o mode shape at z *= h/2 fl - 31.057 Figure A.7: Mode shape SS-6, a/b - 1 129 SEVENTH MODE (MINDLIN) x «= a / 2 y «= b / 4 x ■= a / 4 y -= 0 .0 x - 0 .0 - b / 4 - a / 4 y - b / 2 x « - a / 2 at y = const. at x — const. (-oUyi mode shape at z = h/2 0 ■= 36.194 Figure A& Mode shape SS-7, a/b - 1 130 EIGHTH MODE (MINDLIN) y - b/2 x «= a/2 ( 7 ■77 y - b/4 x * a/4 y - 0.0 X * 0.0 l — - 3 r y -= -b/4 X B - a/4 L y -b/2 x -a/2 at y ; const. at x : const. mode shape at z = h/2 0 - 39.927 Figure A.9: Mode shape SS-8, a/b - 1 131 NINTH MODE (MINDLIN) y - b /2 x « a/2 y « b/4 x ■ a/4 y - 0.0 x «= 0.0 y - -b/4 x ■ —a/4 y ■ —b/2 x ■ —a/2 at y = const. at x — const. mode shape at z = h/2 O « 59.600 Figure A.1Q: Mode shape SS-9, a/b - 1 132 TENTH MODE (MINDLIN) ■n r— ----/ y - b/2 x - a/2 [ ■ "1 \ 1 y -= b/4 x = a/4 L r 7 y «= 0.0 x - 0.0 i " 1 1 1 y *= -b/4 x ■ —a/4 f'T ^ "1 ^ y - -b/2 x — —a/2 at y = const. at x = const. \V\TV* '7 T (C & m b mode shape at z *= h/2 n « 63.062 Figure A.11: Mode shape SS-10, a/b - l Figure A.12: 134 FIRST MODE (MINDLIN) x ■ a/2 x * a/4 x - 0.0 x ■ —a/4 y « —b/2 X * —a/2 at y = const. at x = const. mode shape at z *= h/2 (1 « o.ooo Figure A.13: Mode shape SA-1, a/b - 1 135 SECOND MODE (MINDLIN) y - b/2 x ■ a/2 x ■ a/4 3 C y «= 0.0 x ■= 0.0 3 y « -b/4 x -» —a/4 y - -b/2 x - -a/2 at y = const. at x = const. E E ^ x \ mode shape at z = h/2 O = 9.655 Figure A.14: Mode shape SA-2, a/b - 1 136 THIRD MODE (MINDLIN) I- — - I—ar —1 I - - r z 3 x = a/4 0.0 “ ft/4 E -b/2 X -a/2 at y = const. at x const. TT^7! x z II II 11 A j 11 i o. 4 o I k 4 ° i II V Ill II IIH H i 1 L ± 1 mode shape at z = h/2 O « 16.739 Figure A.I5: Mode shape SA-3, a/b - 1 137 FOURTH MODE (MINDLIN) x > -S ?2 y - b/4 x -= a/4 0.0 x *= 0.0 *b/4 x «= —a/4 const. at x = const. a rI m v i !f,r.ix \\ mode shape at z *= h/2 f) - 26.497 Figure A.1& Mode shape SA-4, a/b - 1 FIFTH MODE (MINDLIN) y - b/2 x - a/2 y - b/4 x ■* a/4 y = 0.0 x «= 0.0 y -= -b/4 const. const. ^ — — C C - 5 t S ) mode shape at 2 = h /2 n = 32.620 Figure A.17: Mode shape SA-S, a/b - 1 139 SIXTH MODE (MINDLIN) b/2 a/2 r ~ ------^ r _ i y * b/4 x “= a/4 1 - \ i I y - 0.0 x -= 0.0 < "-1 l l y - -b/4 x - -a/4 I------1 y - -b/2 x * —a/2 at y = const. at x = const. \ \ 1 I mode shape at z *= h/2 n = 46.205 Figure A.1& Mode shape SA-6 „ a/b - 1 140 SEVENTH MODE (MINDLIN) 3 y - b/2 a/2 3 C □ y “ b /4 x «= a/4 c y •= o.o x <= 0.0 c r y * -b/4 x = —a/4 c a c : y « -b/2 X —a/2 at y = const. at x : const. [ O S ) ) [ C |) ( S ^ ) i ) ( @ § ] (.C mode chape at z = h/2 0 « 48.374 Figure A. 19: Mode shape SA-7, a/b - 1 141 EIGHTH MODE (MINDLIN) b/2 a/2 b/4 a/4 0.0 0.0 3 C —b/4 -a/4 y B -b/2 X -a/2 at y = const. at x const. !fr\) LL^M mode 6hape at z = h/2 0 « 53.682 Figure A.2Q: Mode shape SA-8, a/b - 1 142 NINTH MODE (MINDLIN) b/2 x « a/2 □ C y « b/4 x - a/4 x - 0.0 3 C y - -b/4 x = —a/4 const. mode shape at z- = h/2 (1 * 65.149 Figure A.21: Mode shape SA-9, a/b - 1 Figure A.22: 144 ELEVENTH MODE (MINDLIN) y - b/2 x - a/2 y - b/4 X “ ft/4 3 C y - 0.0 x - 0.0 y - -b /4 x * —a/4 y -b/2 -a/2 at y const. at x »= const. mode shape at z = h/2 Cl - 77.629 Figure AJ23: Mode shape SA-11, a/b - 1 145 FIRST MODE (MINDLIN) b/2 a/2 y - b /4 x — a/4 D L y - 0.0 x - 0.0 y - -b/4 x «= —a/4 y «= —b/2 x — -a/2 at y = const. at x = const. - 7 " r ~ f i y ; i ^ / / / ('A ^ \ \ . 1 / / / mode shape at z = h/2 0 - 3.B4B Figure A2A: Mode shape AA -1, a/b - 1 i 146 SECOND MODE (MINDLIN) b/2 a/2 □ b/4 a/4 1 L y « 0.0 0.0 c 3 C 3 -b/4 -a/4 3 C y -b/2 -a/2 at y const. at x = cOnst. ! S\ f v \ ( rr/ r ^ l mode shape at z = h/2 n « 16.333 Figure A.25: Mode shape AA-2, a/b - 1 147 THIRD MODE (MINDLIN) L . :____ _ l I ...... -_____ y = 0.0 x «= 0.0 at y = const. at x = const. f -0. 3665 J I 0. ^665 \ _____^ r ^ s . \ o. 3665 1 | -0. ^665^^* v \ W \ V mode shape at z = h/2 0 *= 20.459 Figure A .26: Mode shape AA-3, a/b - 1 148 FOURTH MODE (MINDLIN) y - b /s x * a/2 y * b/4 x *= a/4 3 C y - 0.0 x « 0.0 -b/4 x ■= —a/4 y - -b/2 x ■= —a/2 at y — const. at x = const. 0 = 36.377 Figure A.27; Mode shape AA-4, a/b - I 149 FIFTH MODE (MINDLIN) y * b/2 x * a/2 ci rr -• ,3 c-.z.— 3 y = b/4 x «= a/4 cr: i r ■ - i y = o.o x — o.o y «= -b/4 x « -a/4 y « -b/2 x « -a/2 at y = const. at x = const. -o. k o o b j ) ^ io . koeg i § M mode shape at z * h/2 0 r 46.BB3 Figure A2& Mode shape AA-S, a/b - 1 SIXTH MODE (MINDLIN) y - b /2 a/2 y = b/4 x « a/4 y - 0.0 x = 0.0 y *= -b/4 x = —a/4 -b /2 const. const. (Sy) J v O - y / /\ \ J ^ ^971 ^9 7 ^ — v o \\ /\v § ^ i N h i / S v . A ^ l mode shape at z = h/2 0 = 49.071 Figure A29: Mode shape AA-6, a/b - 1 151 SEVENTH MODE (MINDLIN) y - b/2 x « a/2 y = b/4 1 c y - 0.0 x *= 0.0 y - -b/4 x «= —a/4 d i y -b/2 x -a/2 at y ; const. at x - const. M i m A i i # i mode shape at z = h/2 n « 61.329 Figure A30t Mode shape AA-7, a/b - l 152 EIGHTH MODE (MINDLIN) y » b/2 x « a/2 r —------J C - - — 3 y * b/4 x ■* a/4 1 | I' ' " I TTTTI y «= 0.0 x = o.o £ y c — , - 3 y -= -b /4 x ■* —a/4 ------3 y -= -b/2 x * —a/2 at y = const. at x — const. 1 ^ - ) c \I ku^ J f e mode shape at z *= h/2 0 *= 62.817 Figure A31: Mode shape AA-8, a/b — 1 153 NINTH MODE (MINDLIN) - - — - Z j y -= b/2 x ■= a/2 [— i I . y «= b/4 x = a/4 1 1 i 1 ... 1 y = 0.0 x - 0.0 1 ■ 1 i 1 y = -b/4 x *= —a/4 y - -b/2 x *= —a/2 et y = const. at x — bonst. {P ^xs^.N\ y T -6.iz5Q^ C* ^ ^ 7 » / f i t i mode shape at z * h/2 n = 82.820 Figure A.32 Mode shape AA-9, a/b - 1 154 TENTH MODE (MINDLIN) » - * C y -= b/2 x — a/2 y = b/4 x *= a/4 y * 0.0 x *= 0.0 ) - - . 3 C y « —b/4 x « —a/4 r “~ i c y -= —b/2 x « -a/2 at y = const. at x = const. P p ^ © o i mode shape at z * h/2 n - 84.283 Figure A .3 3: Mode shape AA-10. a/b - 1 » 155 FIRST MODE (MINDLIN) y = b/2 x — a/2 y = b/4 x = a/4 y ■= 0-0 x «= 0.0 y ■= —b/4 x *= —a/4 3 r : i y ■ —b/2 x ■» —a/2 at y = const. at x — const. mode shape at z = h/2 n - o.ooo Figure A34: Mode shape SS-1, a/b — 2 156 SECOND MODE (MINDLIN) y ® b/2 a/2 y -= b/4 a/4 y « 0.0 0.0 c y - -b/4 -a/4 y - -b/2 x «= -a/2 at y = const. at x * const. u t am ini INI m i urn iiii mode shape at z = h/2 n = 6.430 Figure A35: Mode shape SS-2, a/b - 2 157 THIRD MODE (MINDLIN) mode shape at z - h/2 n « 25.665 Figure A.36: Mode shape SS-3, a/b » 2 158 FOURTH MODE (MINDLIN) EE y b/2 x - ft/2 b/4 y - 0.0 x -= 0.0 y b/4 -b/2 x ■= —ft/2 at y « const. at x = const. mode shape at z = h/2 0 *= 34.012 Figure A.37: Mode shape SS4, a/b - 2 159 FIFTH MODE (MINDLIN) x ■= a /2 y - b /4 x = ei/4 y « 0.0 x = 0.0 y « - b /4 x f= —ft/4 x - -ft/2 at x = const. mode shape at z = h/2 0 « 40.175 Figure A.3& Mode shape SS-3, a/b - 2 SIXTH MODE (MINDLIN) y - b/2 X = - e / 2 y - b/4 x *= a/4 x - 0.0 y *= — b/4 K *= —a/4 x “ =ro/5? at x = const. f t f t 3 mode shape at z = h/2 0 « 70.874 Figure A.39: Mode shape SS-6, a/b - 2 SEVENTH MODE (MINDUN) * /4 0.0 - b / 4 - a / 4 y *r —b / 2 x ■= —a/ 2 at y = const. at x * const. ill M F T f l ! ii >11I II ii 11[. &■! 1-:.ke|i|!«>. 5701 fealJ:: |\ u II , 11 1 / a \ 'I 11 ... 'L h mode shape at z - h/2 D * 80.134 Figure A 40: Mode shape SS-7, a/b » 2 162 EIGHTH MODE (MINDLIN) e : y ■= b /2 x ■= a / 2 3 C y = b/4 x « a / 4 y ~ 0.0 -b/4 x ■= —a/ 4 y - -b/2 at y * const. mode shape at z = h/2 O - 114.233 Figure A41: Mode shape SS-8, a/b - 2 163 NINTH MODE (MINDLIN) m y - b/2 x - a / 2 g y * b/4 x = a/4 3 y - 0.0 x = 0.0 g y * -b/4 x *= —a/4 y - -b/2 x *= - a / 2 at y = const. at x = const H i ^ fezea^l'^>rS s^ 6 • mode shape at z = h/2 0 « 122.465 Figure AA2: Mode shape SS-9 ,a/b -2 164 TENTH MODE (MINDLIN) y - V 2 x ■= a/2 y - b/4 x = a/4 y - 0.0 x « 0.0 y = -b/4 x = —a/4 y - -b/2 x = -a/g at y = const. at x = const. VO & D ) 1 v. ED) CCl ( C o . 6036^^) J ^ ^ ^ mode shape at 2 = h/2 0 « 131.651 Figure A43: Mode shape SS-10, a/b - 2 165 ELEVENTH MODE (MINDLIN) y - b/2 y «= b / 4 y - 0.0 X *= 0.0 y b —b/ 4 x ■= —a/4 y - -b/2 x ** —a/2 at y = const. at x = const. mode 6hape at z = h/2 n - 157.938 Figure AAA: Mode shape SS-11, a/b - 2 166 FIRST MODE (MINDLIN) t y « b / 2 ■X “ U/2 e / 4 y » b / 4 y - 0 .0 x * 0.0 —a / 4 y - - b / 4 y « —b/2 x «= - a / 2 at y *= const. at x = const. mode shape at z * h/2 n -= 0.000 Figure A45: Mode shape S A -1 , a /b - 2 167 SECOND MODE (MINDLIN) y - b / 2 a/2 y B b/4 a/4 y * 0 .0 0.0 y *= —b / 4 -a/4 3 y «= —b / 2 x -a/2 at y = const. at x const. II II II I mode shape at z = h/2 0 « 17.522 Figure A46: Mode shape SA-2. a/b ■ 2 168 THIRD MODE (MINDLIN) b/2 y b/4 y ■ 0.0 x = 0.0 y ■ -b/4 x * —a/4 y - -b/2 x « —a/2 at y = const. at x = const. ( c r — mode shape at z = h/2 n - 29.764 Figure A47: Mode shape SA-3, a/b - 2 169 FOURTH MODE (MINDLIN) y - b/2 x «= a / 2 3 C y « b / 4 x *= a / 4 D C y « 0.0 x - 0.0 C 3 C y - - b / 4 x « —e/ 4 y -b /2 x -a/2 at y : const. at x : const. mode shape at z - h/2 n « 53.122 Figure AA& Mode shape SA-4, a/b - 2 170 FIFTH MODE (MINDLIN) y « b/4 x ■= a/4 f - 1 x «= 0.0 x *= —a/4 at x — const. mode shape at z = h/2 0 « 55.771 Figure AA9: Mode shape SA-5. a/b - 2 171 SIXTH MODE (MINDLIN) y - b/2 a/2 3 y *= b/4 x *= a/4 y - 0.0 x - 0.0 y -= -b/4 x *= —a/4 y -b/2 x = -a/2 at y : const. at x = const. mode shape at z = h/2 n - 90.656 Figure A.5ft Mode shape SA-6. a/b - 2 172 SEVENTH MODE (MINDLIN) e / 2 y - b / 2 y «= b / 4 e/4 y = 0 .0 0.0 • a / 4 y — —b / 4 y «= —b / 2 x «= —a / 2 at y = const. at x = const. e o i ii ii a i I kU, I II II II II K mode shape at z = h/2 0 «= 109.337 Figure A.51: Mode shape SA-7, a/b - 2 173 EIGHTH MODE (MINDLIN) y - Wz £ y *= b/4 ( = = r- — y « 0.0 x *= 0.0 y *= -b/4 y -= -b/2 x «= -a/2 at y = const. at x — const. ------ mode shape at z *= h/2 n « 124.313 Figure A.52: Mode shape SA-8, a/b - 2 174 NINTH MODE (MINDLIN) s i y - b/2 x = a /2 y - b / 4 x ** a / 4 £ : y = 0.0 x = 0.0 y = - b / 4 y - —b/2 X - a / 2 at y = const. a t x : c o n s t . mode shape at z = h / 2 fi « 140.666 Figure A.53: Mode shape SA-9, a/b - 2 TENTH MODE (MINDLIN) y = b/2 x = e/2 y = b/4 x ■= e/4 y - 0.0 x = 0.0 y = -b/4 x * —e/4 y b/2 x -= —e/2 at y = const. at x = const. mode shape at z * h/2 0 « 144.813 Figure A.54: Mode shape SA-10, a/b - 2 176 ELEVENTH MODE (MINDLIN) y * b /2 x “ a/2 L 1 1 y - b/4 x ■» a/4 LT | 1 y *= 0.0 x » 0.0 c r .. i y - -b/4 x ■ -a/4 ------3 L y - -b/2 x - -a/2 at y = const. at x * const. 1ISIsisHi <3 iIffiSII M1 mode shape at z - h/2 n « 176.664 Figure A.55: Mode shape SA-11, a/b - 2 177 FIRST MODE (MINDLIN) L--.. . £==-w -----— ...... __j y - b/2 x *= a/2 |------| ^ | y « b/4 x «= a/4 r——— I 1 *-----— —1. ------J y - 0.0 X - 0.0 1------3 £==■ - « = i = s * = = ———— y *» —b/4 x ■= —a/4 y ■= —b/2 x « -a/2 at y = const. at x = const. mode shape at z = h/2 n - o.ooo Figure A.56: Mode shape AS-1, a/b - 2 178 SECOND MODE (MINDLIN) y = b/2 a / 2 y n b /4 a / 4x y - 0.0 y - - b / 4 x - - f t / 4 y - - b / 2 x -«i/2 a t y = c o n s t . a t x ; c o n s t . W i « mode shape at z > = h /2 n = 16.787 Figure A.57: Mode shape AS-2, a/b - 2 179 THIRD MODE (MINDLIN) i — — j E ... y *= —b/4 x ** —a/4 y * -b/2 x *= —a/2 at y — const. at x = const. mode shape at z — h/2 n «= 44.446 Figure A.58: Mode shape AS-3 , a/b - 2 180 FOURTH MODE (MINDLIN) x *= a/2 y = b/4 x = a/4 y = 0.0 x *= 0.0 -b/4 x = —a/4 y - -b/2 x*Ts -a/2 at y = const. at x = const. mode shape at z = h/2 I> « 67.172 Figure A.59: Mode shape AS-4, a/b - 2 181 FIFTH MODE (MINDLIN) lI-SSSSJM mode shape at z = h/2 n «= 76.439 Figure A.60: Mode shape AS-5, a/b - 2 182 SIXTH MODE (MINDLIN) y ** b/2 a / 2 y - b /4 a / 4 0.0 0.0 c - b / 4 - a / 4 y - - b / 2 x * —a/ 2 at y = const. at x = const. mode 6hape at z * h/2 n - B9.276 Figure A.61: Mode shape AS-6, a/b - 2 183 SEVENTH MODE (MINDLIN) at y = const. at x = const. L@ JII P> s ® mode shape at z = h/2 0 *= 109.209 Figure A.62: Mode shape AS-7, a/b - 2 184 EIGHTH MODE (MINDLIN) y - b /2 x - a /2 y - b /4 x «= a /4 y - 0.0 x = 0.0 f ~ y — - b / 4 x *= —.a /4 f c y -b /2 x —a/2 a t y ; const. a t x const. mode shape at z = h/2 0 - 151.719 Figure A.63: Mode shape AS-8, a/b - 2 185 NINTH MODE (MINDLIN) x * a /2 y = b /4 x = a /4 y « 0.0 x *= 0.0 y *= - b / 4 x «= —a/4 s 3 > x « —a/2 const. at x = const. n U M mode shape at z = h/2 Q - 172.454 Figure A.64: Mode shape AS-9, a/b - 2 186 TENTH MODE (MINDLIN) y - b /2 a /2 y = b/4 x = a /4 y = 0.0 x = 0.0. y = -b/4 x *= —a/4 y -b/2 at y const. - ^ k q j* ------3 5 3 3 3 mode shape at z = h/2 O *= 166.496 Figure A.65: Mode shape AS-10, a /b -2 ELEVENTH MODE (MINDLIN) y = b /2 e/2 3 C y = b /4 x = a /4 y - 0.0 x = 0.0 3 L y *= - b / 4 x « —a/4 y - -b/2 X -a/2 at y = const. at x const. mode shape at z = h/2 n - 194.025 Figure A.66: Mode shape AS-11 , a/b - 2 188 FIRST MODE (MINDLIN) y «= b /2 a/2 b/4 a/4 J y x 0.0 —.— ^ _ mode shape at z * h /2 0 ■= 7.709 Figure A.67: Mode shape AA-1, a/b - 2 SECOND MODE (MINDLIN) y “ —b/4 x ■= —a/4 y — —b/2 x -* —a/2 at y = const. at x = const. V ^ / V ___✓ l 1^ 0 V (( ( 1 \\ \\W ( f ! mode shape at z = h/2 n « 28.705 Figure A.6& Mode shape AA-2, a/b - 2 190 THIRD MODE (MINDLIN) y «= b/2 x *= a/2 3 C y = b / 4 x a/4 3 C y = 0.0 x «= 0.0 y - -b/4 x ■= —a/4 y - -b /2 x ■= —a/2 at y = const. at x = const. mode shape at z = h/2 0 - 63.5B7 Figure A.69: Mode shape AA-3, a/b - 2 191 FOURTH MODE (MINDLIN) V - b/2 x « a/2 3 £ y - b/4 x *= a/4 3 C y - 0.0 x «= 0.0 3 C 3 y *= —b/4 x *= —a/4 y -= -b/2 x « —a/2 at y = const. at x = const. -— ) mode shape at z = h/2 O «= 70.729 Figure A.70t Mode shape AA-4, a/b - 2 FIFTH MODE (MINDLIN) at y = const. at x — const. 1 g ) € (SS))_ ( g mode shape at z * h/2 n = 82.165 Figure A.71: Mode shape AA-5, a/b - 2 193 SIXTH MODE (MINDLIN) y - b /2 x *= a /2 y - b /4 x *= a /4 D C y * 0.0 x * 0.0 y « - b / 4 x ** —a/4 y - - b / 2 x «= —a/2 at y = const. at x = const. mode shape at 2 = h /2 11 «= 117.525 Figure A.72: Mode shape AA-6. a/b - 2 194 SEVENTH MODE (MINDLIN) y - t /2 x = a /2 y «= b /4 x *= a /4 1 L y - 0.0 x = 0.0 y «= - b / 4 x *= —a/4 y - - b / 2 x *= —a/2 at y = const. at x = const. mode shape at z = h/2 n = 129.002 Figure A.73: Mode shape AA-7, a/b - 2 195 EIGHTH MODE (MINDLIN) y - b /2 x - a / 2 y = b /4 x = a / 4 y = 0.0 x - 0.0 y = - b / 4 x *= —a/ 4 y - - b / 2 x - a / 2 at y = const. a t x : c o n s t. mode shape at z = h/2 n « 177.965 Figure A.74: Mode shape AA-8, a/b - 2 1 % NINTH MODE (MINDLIN) B y - b/ 2 y - b /4 D L y = 0.0 x = 0.0 c y = - b / 4 x = —a/4 y = - b / 2 X '«= —.a /2 at y = const. at x sb c o n s t. mods Bhape at z = h/2 n - 1B7.341 Figure A.75: Mode shape AA-9.a/b- 2 197 TENTH MODE (MINDLIN) x *= a/2 y - b/4 x ■= a/4 1 C y -= 0.0 x - 0.0 y «= -b /4 x «= —a/4 y - -b /2 x ■= —a/2 at y = const. at k = const. T < iS ^ t s 20 (2K 2) (2 2 ) tf23j!55) <2 mode shape at z = h/2 0 * 203.232 Figure A.76: Mode shape AA-10, a/b - 2 Appendix B MODE SHAPES FOR OTHER POISSON'S RATIOS In all the figures shown in this appendix the following is constant: h /b = 0.1 s/T n c and the coordinates as used before. Mode shapes shown are those which are significant ly different than those for i>=0.3 found in Appendix A. 199 FOURTH MODE (MINDLIN) v * s W i ( r ^ \ - 0 . k 7 « / j i ; n n n \M l . y mode shape at z * h/2 v * 0.0 16.117 mode shape at z ■ h/2 v - 0.409 O - 16.404 Figure B.1: Mode shape SS-4, a/b - 1 200 f if t h m o d e (MINDLIN) mode shape at 2 = h /2 v - 0.0 0 - 30.172 P . Ja322, mode shape at z ■ h/2 v - 0.409 0 « 29.979 Figure R2: Mode shape SS-5, a/b - 1 • 201 SIXTH MODE (MINDLIN) PM )) ) ((||\( 'ft i\\ \\ //'"A m m $ h m mode shape at z = h /2 v - 0.0 O - 30.172 mode shape at z ■ h/2 v - 0.400 0 - 32.800 Figure B.3: Mode shape SS-6. a/b ■» 1 202 SEVENTH MODE (MINDLIN) mode shape at z = h/2 v - 0.0 (2 - 40.373 L 547' mode shape at z * h/2 v - 0.490 0 - 38.302 Figure R4: Mode shape SS-7, a/b - 1 203 EIGHTH MODE (MINDLIN) p. 3S7E 3. 3875, O. 3875 mode shape at z = h/2 v - o.o 0 - 40.726 w 0. 3592 mode shape at z * h/2 : 0.400 n 40.415 Figure B.5: Mode shape SS-8, a/b - 1 204 NINTH MODE (MINDLIN) mode shape a t z — h/2 v - o.o 0 - 63.240 mode shape at z * h/2 v « 0.499 0 « 59.196 Figure B.6: Mode shape SS-9, a/b - 1 205 TENTH MODE (MINDLIN) mode shape at z = h/2 v * o.o 0 - 64.768 mode shape at z * h/2 v - 0.499 O - 62.646 Figure B.7: Mode shape SS-10, a/b - 1 206 ELEVENTH MODE (MINDLIN) mode shape at z = h/2 v m o.o n - 64.768 .OlSSI mode shape at z ■= h/2 v - 0.499 0 - 66.119 Figure R& Mode shape SS-1I, a/b - 1 207 FOURTH MODE (MINDLIN) W ^ ^ / j i\U *J) w ///'„'N M I SSI J | to. fesii | i ' l v w / u , W f . mode shape at z = h/2 v - o.o (1 - 28.076 mode shape at z * h/2 v - 0.489 fl - 26.253 Figure B.9: Mode shape SA-4, a/b « 1 FIFTH MODE (MINDLIN) 208 ^ ■—"urC-—— C c s "I All II II III f n \ 5 ~ 5 j C c s ii= = = ^ r\ mode shape at z = h/2 v - o.o 0 - 33.024 c S d o c C l mode shape at z «= h/2 v « 0.499 0 « 33.029 Figure RIO: Mode shape SA-5, a/b - 1 SIXTH MODE (MINDLIN) II II III III II II II mode shape at z * h/2 v « o.o n - 46.458 mode shape at z * h/2 v « 0.400 n - 46.836 Figure B.11: Mode shape SA-6 , a/b - 1 210 SEVENTH MODE (MINDLIN) V(? */ f C j ^ 2 * y ------ ( { o . k s a l ) I _ _1 ( ) mode shape at z *= h/2 v - o.o £1 - 50.672 V mode shape at z *= h/2 V - 0.489 (} - 4B.6B0 Figure B.12: Mode shape SA-7, a/b - 1 EIGHTH MODE (M1NDLIN) M III mode shape at z - h/2 v - o.o n * 55.413 P f o i a i f f l M i mode shape at z * h/2 v - 0.499 n ■ 54.012 Figure E13: Mode shape SA-8, a/b * 1 212 NINTH MODE (MINDLIN) 652 i S S > 492 C S 5 3 314 S 5 D ( _^ ~-3~~ —e .i B52 ,— _ . mode shape at z = h/2 v - o.o Q - 66.761 m i i^ 2 = § S s s i r G ^ 3 $ mode shape at z « h/2 v - 0.499 n - 65.126 Figure B.14: Mode shape SA-9, a/b - 1 213 TENTH MODE (MINDLIN) mode shape at z =* h/2 v - o.O n - 76.028 5 ^ H i i § i p ® f i f t mode shape at z *= h/2 v - 0.499 n - 72.139 Figure E l5: Mode shape SA-10, a/b - 1 214 ELEVENTH MODE (MINDLIN) v^o3ibij\(^ (® \& ~ S ) f(£I2) (io. jioao) ( C m ode shape at z <= h/2 V - 0.0 n - 81.308 (L l^ qs^ J (\j(| ( § £ § ) ^(( S IS B > (S E S /2 ((oiB p gd -j mode shape at 2 * h /2 v - 0.499 n « 77.239 Figure B.16: Mode shape SA-11 ,a/b - 1 215 FOURTH MODE (MINDLIN) ■\v^37 1 1 (Cj^i 1 0 , mode shape at z = h/2 v m o.o 0 * 38.616 i 7 ~ ^ \ 1 1 0 0 mode 6hape at z ■> h/2 v - 0.499 n - 36.129 Figure B.17: Mode shape AA-4. a/b - 1 216 FIFTH MODE (MINDLIN) p ® M | C-o. S8ijJ j N. V M _/ / S \K sbj/// m t i t f mode shape at z = h/2 v - o.o fl - 48.805 l\ 1 \> V f j A s w \1 i ^ S l mode shape at z = h/2 v - 0.490 n - 46.237 Figure B.1& Mode shape AA-5, a/b - 1 217 SIXTH MODE (MINDLIN) ^ 30^ 6 ) > C ^ -jO. mode shape at z = h/2 v — o.o n ■ 48.831 mode shape at z * h/2 v - 0.49B n - 50.321 Figure B. 19: Mode shape A A -6, a /b - 1 218 SEVENTH MODE (MINDLIN) | f @ \ 7 ® i I f e mode 6hape at z - h/2 v - o.o O - 64.626 IlK|Pltjp ~ ) /[(^ i mode shape at z ■* h/2 v - 0.498 U * 61.236 Figure R20: Mode shape AA-7, a/b - 1 219 EIGHTH MODE (MINDLIN) ' 0 ^ 7 "g ^ \ i k i o.o n 65.425 1’j^ j yj((m SSf(W«YVs3 0 / z mode shape at z *= h/2 V - 0.499 0 - 62.877 Figure R21: Mode shape A A -8 , a /b - 1 NINTH MODE (MINDLIN) 220 J B M*&sr _ \ / -m3. IMad «oafey < s^ 3 m mode Bhape atIJH 2 - h /2 v - 0.0 O -- 66.665 5 > \ / — />sXoCi*5£i 1 Ilkmode shape at z ** h/2 V - 0.468 (1 - 60.962 Figure B.22: Mode shape AA-9, a/b - I 221 TENTH MODE (MINDLIN) W P i A*1 ~ ( l l s f GO " ' i p0 j —P— 0 N®) ^ N®) w k ) I S I J ^ . mode shape at 2 — h /2 v ■ 0.0 0 - B6.B77 1) © Ilf mode shape at z — h/2 v « 0.499 0 - B3.44B Figure R23: Mode shape AA-JO, a/b - 1 REFERENCES 1. ReissnerJI,The effect of transverse shear deformation on the bending of elastic plates, J, Appl. Mech., Trans. ASME 6 7 , A69-A77 (1945). 2. Reissner, E., On bending of elastic plates, Quart Appl. Math. 5 , 55-68 (1947). 3. Reissner, E, Finite deflection of sandwich plates, J. Aero. Sci. 15 , 435-440 (July 1949). 4. Reissner, E, On generalized two-dimensional plate theory. Part 1, Intl. J.Solids Struc. 5,5, 525-532 (May 1969). 5. Reissner, E, On generalized two-dimensional plate theory. Part 2, Intl. J. Solids Struc. 5,6, 629-637 (June 1969). 6. Reissner, E., A consistent treatment o f transverse shear deformations in laminated anisotropic plates, AIAA J. 10 , 5, 716-718, Tech. notes (May 1972). 7. Reissner, E, On transverse bending of.plates, including the effect of transverse shear deformation , Intl. J. Solids Struc. 11,5, 569-573 (May 1975). 8. Reissner, E., On the theory of transverse bending of elastic plates, Intl. J. Solids Struc. 12 , 8, 545-554 (1976). 9. Reissner, E, Transverse bending of laminated anisotropic plates, J. Engrg. Mech. Div. , Proc. ASCE 102 , EM3, 559-563, Tech. notes (June 1976). 10. Reissner, E., Note on the effect of transverse shear deformation in lami nated anisotropic plates, Computer Methods Appl. Mech. Engrg. 20 , 2, 203-209 (Nov. 1979). 11. Reissner, E., A note on the bending of plates including the effects of transverse shearing and normal strains, Z Angew. Math. Phys. 32 , 6, 764-767,Brief repots (Nov. 1981). 12. 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