Thin-Walled Structures g (1989) 273-293

Buckling and Vibrations of Shear-Flexible Orthotropic Plates Subjected to Mixed Boundary Conditions

Dimi t r i o s K a r a m a n l i d i s & Vikas P r a k a s h

Department of Civil Engineering. University of Rhode Island. Kingston. Rhode Island 02881. USA

(Received 28 July 1988; revised version received 5 December 1988: accepted 12 December 19SS)

ABSTRACT

An approximate technique is presented for the analysis of buckling and vibrations of free-form, orthotropic, shear-flex'ible ('Mindlin ') plates subject to mixed boundary conditions. The method falls into the catego~" of Ra.vleigh/ Ritz-techniques: however, by using Lagrangian multipliers to 'relax" the geometric boundary conditions, the selection of appropriate trial functions is made considerably simpler. Accuracv and reliabili O' of the proposed technique is demonstrated on the basis of several sample problems.

1 INT R ODUCTION

The study of buckling and/or vibration of shear-flexible (so-called Mindl in t) plates has attracted considerable attention over the past two decades. In one of the early studies, Srinivas and Rao-" presented an exact three-dimensional elasticity analysis for the stability of an all-around simply-supported rectangular plate. For the same plate, Levinson 3 studied the free vibrations also using a three-dimensional elasticity approach. The stability of Levy-type plates (i.e. two simply-supported opposite sides and the other two subject to arbitrary boundary conditions) was studied by BruneUe. 4 Rao et al. 5 and Luo, 6 among others, studied the buckling of Mindlin plates using the finite element method,

273 Thin-Walled Structures 0263-8231/89/$03"50© 1989 Elsevier Science Publishers Ltd, England. Printed in Great Britain

274 Dimitrios Karamanlidis. l, Tkas Prakash

whereas in a paper by Benson and Hinton 7 the finite strip method was utilized. More recently. Roufaeil and Daweso studied the buckling and vibration of Mindlin plates using both the finite strip method and the classical Rayleigh-Ritz method. Yet another technique, namely the collocation method, was utilized in a paper by Mikami and Yoshimura *'1 who studied the flexural vibrations of rectangular Mindlin plates.

In all of the above cited work. the scope was quite limited due to the simplicity of the boundary conditions considered. More general ('mixed') boundary conditions were considered in two recent papers by Sakiyama and Matsuda. ~t, t2 In both papers, solutions for the pertinent partial differential equations were obtained by translating the same into integral equations and applying numerical integration.

In the present paper, an alternative approximate method is proposed for studying the buckling and vibrations of orthotropic Mindlin plates subjected to mixed boundary conditions. More specifically, a Rayleigh/ Ritz-U'pe technique is developed which, however, is not based on the classical principle of min imum potential energy but a modification thereof. In this modified variational equation, the boundary conditions have been "relaxed" using Lagrangian multipliers. Thus, selecting proper ('permissible') trial functions becomes a relatively easy task. The usefulness and accuracy of the proposed method is demonstrated on the basis of several sample problems.

2 MATHEMATICAL FORMULATION

The vibration problem of a free-form, orthotropic, shear-flexible (Mindlin) plate (see Fig. 1) subjected to arbitrary, boundary conditions shall be considered first. To obtain the pertinent equations of motion, Hamilton's principle can be employed, which in absence of potential- free forces states that

6f~,/Ldt = 0 (1)

provided that at t = to and t = t~ all variations are equal to zero. In the foregoing. L = T - U represents the Lagrangian while T and U denote the kinetic and deformation energy of the shear-flexible plate, respectively, i.e.

T = ~rl,w~,dA + ~rl2(Ox.t+O~.t)dA (2)

Buckling and vibrations of orthotropic plates 275

Z~W

Fig. 1. Kinematic and gtalic variables for shear-flexible plate.

fA 1 U = ~ eTEed,4 (3)

whereA is the area of the plate; r/I and 02 represent the translational and rotational masses per unit area, respectively; and, w, 0x and 0y are the midsurface displacements. Furthermore,

ryy -0x y E = Kxy ---- Oy. y - - Ox. x

Yxz w. ~ + 0y ]

Yyz w. y - Ox /

276 Dimitrios Karamanlidis, ~Tkas Prakash

Dtl DIz 0 0 0 / DI_~ D,_, 0 0 0

E = 0 0 D~3 0 0

0 0 0 D~ 0

0 0 0 0 D~

where Dij represent the stiffness parameters of an orthotropic plateJ 3 while (...),p denotes differentiation with respect to the variable p (i.e. x ,y or t). Equation (1) is valid under the provision that the geometric boundary conditions are satisfied a priori, i.e.

w-~-~ = 0: 0n-On = 0: 0~-0~ = 0 onCu (6) Concerning the notation used above. (7Z..) indicates a specified variable, Cu is the part of the boundary OA on which displacements are specified whereas the rotations 0, and 0~ are defined in Fig. 2.

In order to discretize eqn (1), i.e. obtain the equations of motion for a system with a finite number of degrees-of-freedom, the following approximations are used:

W =

M

A,~e~m(x,y)coswt = OVacosogt m = 0

N

Ox = Z Bnxn(x,y)cosogt = xTbcosogt (7) n = 0

K

Oy = Z CklIIk(X'y)cos O)t = I / / T e c o s 09 /

k = 0

where Am, B, and Ck are yet undetermined constants and o9 denotes a plate eigenfrequency. The trial functions ~m(x,y), Xn(x,y) and I~'k(X,y ) must be chosen so as to fulfil eqns (6). a formidable task in the case of a plate subjected to mixed boundary conditions.

For the solution of non-academic problems it is, therefore, necessary to modify eqn (1) so that the independent variables w, 0x and 0y no longer have to exactly satisfy eqns (6). This is accomplished by introducing eqns (6) into eqn (1) by means of Lagrangian multipliers, viz.

6 Ldt - < 6 (~ [A(w - ~ ) +/~(0n - On) a t ~ u

+ P(Os - O0]ds > d t = 0 (8)

Buckling and vibrations of orthotropic plates 277

s 0

Ol Y I w /

./-'c3A /

Fig. 2. Variables at the boundary of the plate.

VX = C 0 5 ~

Vy = sin Ol

In order to discretize eqn (8) it is now necessary to use along with the approximations for w, 0x and 0y, i.e. eqns (7), the following approxi- mations for the Lagrangian multipliers A., ~ and p:

fc )~ (w- ~)ds = Z " q = l

aq~,q(Wq -- Wq) = ~,TGa -- AVg

R

fc la(O. - 0. )ds = Z ctd~,(O., - 0.r) = U v + u v Foe - uTf u r = l

(9)

P

(~ p(O, - O,)ds = Z apPp(O~P - 0,p) = pTHbb + pTH¢c - p-r~ p = l

where q = 1, 2 . . . . . Q; r = 1, 2 . . . . . R and p = 1, 2 . . . . . P represent Gaussian stations, and a represents the pertinent weight factors.

Introducing eqns (7) and (9) into eqn (8) and taking the variations with respect to the independent variables a, b, e, A,/~ and p yields the algebraic system of equations

°i/(i )/o/ K¢, - to-Me,. 0 -F¢ r ~ = ' - H ~ (10) 0 0

(symmetric) o \ - ~ /

27S Dimitrios Karamanlidis. ~Tkas Prakash

where the matrices are defined in the Appendix. The o9 for which the determinant of the coefficient matrix in eqn (10) vanishes represents the eigenfrequencies of the plate. In the present study, these eigenvalues were obtained on the basis of a determinant search approach.

Equation (8) can be modified quite easily so as to include additional effects. For instance, in order to study the free vibrations of plates subjected to in-plane forces and resting on a one-parameter ( 'Winklef) elastic foundation the following modification is necessary:

1 1- 1 1 u=f eEeaA+f. kw'-aA+f,. , 5N~, w.owBdA ( !1 )

where N~a (a, fl = 1, 2) are the in-plane forces and k represents the foundation modulus. The study of buckling phenomena follows the same pattern as the free-vibration problem and will not be outlined in detail here.

3 APPROXIMATIONS

In the present study, all boundary integrals in eqn (9) were evaluated using six Gaussian stations whereas the displacement variables w. 0~ and 0, in the interior of the plate were approximated by means of double F:ourier series, i.e.

3.1 Kirchhoff plate

I J

w(x,y) = Z Z A°sin(inx/a)sin(jny/b) (12) i = I j = I

3.2 Shear-f lexible plate

w(x, y) = I J

E Z AO sin(inx/a) sin(jny/b) i = 1 j = l

Ox(x, y) =

I J

Z E Brj sin(inx/a)cos(jzry/b) i = l j = l

(13)

I J

Cij cos(inx/a) sin (]ny/b) i = l j = [

Buckling and vibrations of orthotropic plates 279

Clearly, in the case of a rectangular all-around simply-supported ('Navier') plate, these trial functions are exact for they satisfy the geometric and static boundary conditions along with the equilibrium equations in the interior of the plate.

4 NUMERICAL RESULTS

The proposed method has been implemented into a computer code and test studies were carded out for a variety of plate problems. Numerical results are presented subsequently for rectangular plates only. Unless otherwise stated, in all cases considered the following data were used:

Isotropic material Young's modulus E = 1.0 Poisson's ratio v = 0.3

Orthotropic material Ex = 20.83;Ey -- 10.94 Vxy = 0.44; Vy x = 0.23 Gy z = 6.19; Gxz = 3.71

densi typ = 1-0 shear correction factor k = 5/6 a = b = lO;h = a/lO (all units are consistent)

4.1 Buckling analysis

The first study carried out in order to evaluate the performance of the proposed method is concerned with the buckling of rectangular plates subjected to various boundary conditions. Table 1 shows the obtained results for the buckling loads (Tx)c~it of a square plate with two opposite sides simply supported while the other two are either (a) simply- supported (case 1), (b) one fixed, one simply-supported (case 2), (c) both fixed (case 3), or (d) half-fixed, half simply-supported (case 4).

For cases 1 and 3, numerical results have previously been reported by Sakiyama and Matsuda 12 and Harik and Ekambaram) 4 respectively. For comparison purposes, their results are listed in Table 2 along with those obtained herein as well as the exact 3-D elasticity solution by Srinivas and Rao.-'

As an additional test, the buckling of isotropic shear-flexible plates with h -- b/lO0, a varying aspect ratio a/b and with different boundary conditions was studied. The obtained numerical results are listed in Tables 3 and 4.

4.2 Free vibrations

The second study is concerned with free-vibration plate problems. First, cases 1-4 were considered and Table 5 shows the numerical values for the

280 Dimitrios Karamanlidis. Vikas Prakash

TABLE 1 Buckling Loads for a Square Plate Loaded in x-Direction (Cases I-4)

I . . . . . . . . . i I - - - - "

[i casel ii case2 i i a 3

~'--~-i--i--:l : i T ~ i / i i iTi , ~ I I I 1 . . . . . . . . . i i:l l'Txl I I f i l l Isotropic Kirchhoff Isotropic shear-flexible Orthotropic Kirchhoff Orthotropic shear-flexible

0.0370 0-050 0.06 0.06 0.0341 0.041 0-05 0.05 0.5099 0.699 1.06 0.95 0.6099 0.7199 0-909 0-89

TABLE 2 Stability Parameter TxaZ#r'-D for Cases I and 3

Case 1 Case 3 (isotropic shear-flexible) (isotropic Kirchhof~

Exact 3-74 - - Ref. 15 3-83 - - Ref. 16 - - 6.90 Present 3-77 6-97

first nine eigenfrequencies whereas Figs 3-5 display the pertinent modes of vibration (isotropic shear-flexible plate only). Whenever applicable. the results for the eigenfrequencies were compared with existing reference solutions 2" 3. 13. ~5. 16 and found to be in excellent agreement.

In the final part of the present study, the effects of elastic foundation and in-plane loading were investigated. The static problem was solved first and the obtained results for an isotropic shear-flexible plate are displayed graphically in Figs 6-9. The vibration problem of plates resting on an elastic foundation and subject to in-plane loading is of considerable practical interest, however, only few solutions 17 seem to have been reported thus far.

In the present study, the loading was assumed to be biaxial and uniformly distributed, i.e. Tx = Ty = 2rt'-D/a 2. The foundation parameter was chosen to be k -- D/a 4 whereas the dimensions of the plate were assumed to be a = b = 1 and h = 0-1. Cases 1-4 were analyzed by the

Buckling and vibrations of orthotropic plates 28 !

TABLE 3 Buckling Loads for Rectangular Plates Loaded

in x-Direction (lsotropic. Shear-Flexible)

a/b

b b/2 b/2

. ]

1 t 0-4 - - - - 12.39 0.5 - - u 9.29 0.6 - - 10.78 7.46 0.7 11.064 9.24 6.30 0.8 9.405 8.08 5-59 0.9 7.744 7.14 5.08 1.0 7.190 6.58 4.81 I. I 6-638 6.14 4-65 1.2 6.080 5.80 4.59 !.3 5.750 5.64 4.54 !-4 5.532 5.47 4.54 i-5 5.432 5.42 I-6 5.420 5.36 4.48 1-7 5-420 5.36 1-8 5-310 5.25 4.37 1.9 5-200 5.08 2.0 4.970 4.87 4.26 2.25 4.750 4.65 4.20 2.5 4-600 4.51 4.18

proposed method and the obtained results for the dimensionless frequency parameter o9*, -- ogm,ab v/Ph/D are listed in Table 6. It is seen that the influence of shear deformation effects is quite significant. Finally, Figs 10-12 display three-dimensionally the first three vibration modes for an isotropic shear-flexible plate (cases 2-4).

5 CONCLUSIONS

In this paper, an approximate method has been outlined that can be used to analyze the buckling and vibration of shear-flexible orthotropic plates

282 Dimitrios Karamanlidis, ~'ikas Prakash

TABLE 4 Buckling Loads for Rectangular Plates Simply-Supported along Three Sides (Isotropic,

Shear-Flexible)

b

P

,I

, : T x

a/h c/a = 1.00 c/a = 0-75 c/a = 0.50 c/a = 0-25 c/a = 0.00

0.5 7.03 7.03 6.91 6.75 6.25 0.6 6.14 6.14 5.97 5.69 5.14 0.7 5.75 5-75 5.50 5.20 4.52 0.8 5.64 5.64 5.36 4.86 4.20 0.9 5.75 5.75 5.36 4.75 4.09 1.0 5.92 6.02 5.47 4.64 4.04 1. I 6.25 6.36 5.47 4.64 4.04 1.2 6.02 6.02 5.25 4.60 4.04 1.3 5.75 5.75 5-03 4.59 4.14 1.4 5.58 5.58 4.81 4.75 4.32 1.5 5.48 5.43 4.70 4.64 4.48 1.6 5.47 5.42 4.59 4.40 4.37 1.7 5.46 5.42 4.54 4.40 4.20 1.8 5-53 5.41 4.48 4.37 4.20 1.9 5.58 5.35 4.38 4.37 4.15 2.0 5.50 5.20 4.38 4.37 4.09 2. I 5.42 5.14 4.38 4.42 4.04 2.25 5.25 5.03 4.38 4.42 4.04 2.5 5.14 4.86 4.40 4.37 4.04

subjected to mixed b o u n d a r y condi t ions . The basic idea o f the m e t h o d consists in in t roduc ing the geomet r ic b o u n d a r y cond i t ions via Lagrang ian mult ipl iers into the pe r t inen t var ia t iona l equat ion . Thus , select ing the appropr i a t e trial func t ions becomes a relatively s imple mat te r regardless o f the shape o f the plate and the boundary , cond i t ions it is subjected to. On the basis o f the o b t a i n e d results for some representa t ive examples , it can be c o n c l u d e d that the p r o p o s e d m e t h o d represents a s imple but reliable, h ighly accura te a n d c o m p u t a t i o n a l l y efficient tool for the solut ion o f a wide variety, o f plate b e n d i n g problems.

Buckling and vibrations of orthotropic plates 283

t~

e.

I ........ ]

J

¢,q

I i ..........

o

m

e~

e-

.6 e .

¢- °~

e~ .e

"6 ¢ -

O

°~

ca .u

e ~

o e .

°~

°~

284 Dimitrios Karamanlidis, l"ikas Prakash

Fig. 3. Vibration modes for isotropic shear-flexible plate (case 21.

~"r'T';-;7l

Fig. 4. Vibration modes for isotropic shear-flexible plate (case 3).

Buckling and vibrations of orthotropic plates 285

~, • ~,, .~

Fig. 5. Vibration modes for isotropic shear-flexible plate (case 4).

TABLE 6 Eigenvalues for Square Plates on Elastic Foundation under Uniform In-Plane Loading

Compressive loading

Case 1 Case 2 Case 3 Case 4

a) b) a) b) a) b) a) b) o9~'n 1.00 0-96 12.37 10. 19 21.41 18-01 17.31 13-38 o9~_, 38.23 33.37 41.32 35.80 45.70 38-50 42.20 36-18 o9*_, 68-39 58.14 77.52 74.43 88-57 75.51 80.79 74.53

Tensile loading

Case 1 Case 2 Case 3 Case 4

a) b) a) b) a) b) a) b) o97't 27.92 27.63 31.58 30.57 36-51 34.36 33.87 31.56 o9~'2 58.39 54.69 63.62 56-67 64-27 58.65 64-01 56-67 co* 88.28 79.81 96.33 84.43 1 0 6 . 2 4 90.37 98.64 85.75

a)isotropic Kirchhoff. b)Isotropic shear-flexible.

286 Dimitrios Karamanlidis. Vikas Prakash

7.

\ \ ×

\ x \

/~" °~ I o ~"

e .

d.

e -

° ~ t L

\

\ x

Buckling and vibrations of orthotropic plates 287

/ /

,\X ~\X

288 Dimitrios Karamanlidis. lqkas Prakash

Oy I 1% I I i l

l c a s e _ ! " c a s e 2

/ / / f x /

I Oy & ey i

cose 3 I c a s e &

9,

/ / /

~x / / x

Fig. 8. Isotropic shear-flexible square plate on elastic foundation: variation of rotation Oy = G(x. y).

R E F E R E N C E S

1. Mindlin, R. D., Infuence of rotary inertia and shear on flexural motions of isotropic, elastic plates. J. AppL Mech.. 23 (1951) 431-6.

2. Srinivas, S. & Rao, A. K., Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates. Int. J. Solid~ Struct.. 6 (1970) 1463-81.

3. Levinson, M., Free vibrations of a simply supported, rectangular plate: an exact elasticity solution. J. Sound Vibr., 98 (1985) 289-98.

4. Brunelle, E. J., Buckling of transversely isotropic Mindlin plates. A/AA J.. 9 (1971) 1018-22.

5. Rao, G. V., Venkatarama, J. & Raju, K. K., Stability of moderately thick rectangular plates using a high precision triangular finite element. Comp. Struct.. 5 (1975) 257-9.

6. Luo, J.-W., A hybrid/mixed model finite element analysis for buckling of moderately thick plates. Comp. Struct.. 15 (1982) 359-64.

7. Benson, P. R. & Hinton, E., A thick finite strip solution for static, free vibration and stability problems. Int. J. num. Meth. Eng., l0 (1976) 665-78.

Buckling and vibrations of orthotropic plates 289

0

0

0

r- -

~o

0 x

0

6

6

o l ~ I I + ~ l ~ l i l ~ l ~ I ~ I ~ I o I ~ I i l ~ l ~

o

e-

ll I I ~ I ~ I ~ I ' o " E

II

II

l a l I l i l l l l l I I i I I I I I i ,

( O / ?ob. 0 [ ) M 'NOI1031330

( O / ~ °b" OL) M 'NOt1337J30

o

6

° x Ox

0 0

6

0 0

0 6

o o 6

( 0 / ~ob, OL ) M 'NOI1~3733Q

o

,w

~J

e-

C

,,-j

P.

c-

o

L

e-

r.. ,o

N

290 Dimitrios Karamanlidis. Vikas Prakash

I _ 3 _ i _I .,J

T ~ - ~ ""T T f--I---I-~, 'T Tx ~

~ (1.1 .~.,.,..,,.,....~..... • - . - . . . . . . . . / ~

Fig. 10. Vibration modes for isotropic shear-flexible plate on elastic foundation under in-plane loading (case 2).

Buckling and vibrations of orthotropic plates 291

t l l I _T I i i l -~

, / \,, /

, J ,./

Y >,"

Fig. I 1. Vibration modes for isotropic shear-flexible plate on elastic foundation under in-plane loading (case 3).

_9. Dimitrios Karamanlidis. Vikas Prakash

1

T; / ,?

;.::.

Fig. 12. Vibration modes for isotropic shear-flexible plate on elastic foundation under in-plane loading (case 4).

8. Roufaeil, O. L. & Dawe, D. J., Vibration analysis of rectangular Mindlin plates by the finite strip method. Comp. Struct.. 12 (1980) 833-42.

9. Dawe, D. J. & Roufaeil, O. L., Buckling of rectangular Mindlin plates. Comp. Struct.. 15 (1982) 461-71.

10. Mikami, C. & Yoshimura, J., Application of the collocation method to vibration analysis of rectangular Mindlin plates. Comp. Struct.. 18 (1984) 425-31.

11. Sakiyama, T. & Matsuda, H., Free vibration of rectangular Mindlin plate with mixed boundary conditions. J. Sound Vibr.. 113 (1987) 208-14.

12. Sakiyama, T. & Matsuda, H., Elastic buckling of rectangular Mindlin plate with mixed boundary conditions. Comp. Struct.. 25 (1987) 801-8.

13. Troitsky, M. S., Stiffened Plates." Bending, Stability and Vibrations, Elsevier, Amsterdam, 1976.

Buckling and vibrations of orthotropic plates 293

14. Harik, I. E. & Ekambaram, R., Seminumerical solution for buckling of rectangular plates. Computers Structures. 23 (1986) 649-55.

15. Leissa, A. W., Vibrations of Plates, NASA SP-160, 1969. 16. Warburton, G. B., The Dynamical Behaviour of Structures, 2nd Edition,

Pergamon Press, Oxford. 1976. 17. Chert. L.-W. & Doong, J.-L., Vibrations of an initially stressed transversely

isotropic circular thick plate. Int. J. Mech. Sci.. 26 (1984) 253-63.

APPENDIX

The matrices appearing in eqn (10) are defined as follows:

K~ = fA (D44¢x¢"r* + Ds/p'YdPxy)dxdy

Kab = - fA D55~P'Y~'T dxdy

K~ = fA D4"t¢'xvrdxdy

Kbb = fA (D22X'YX"rY + D33X'xX'Tx + D55zZr)cbcdy

K~ = - fa (DI2X'Yt//"rx + D33Xx~T'y)dxdy

K¢¢ = f4 (Dit V.x~,.r, + D331Pt.ylg.T + D44Vv/r)dxdy