Thermal post-buckling of thin simply supported orthotropic square plates

Thermal post-buckling of thin simply supported orthotropic square plates

Composite Structures 12 (1989) 149-154 Technical Note Thermal Post-buckling of Thin Simply Supported Orthotropic Square Plates ABSTRACT Thermal post...

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Composite Structures 12 (1989) 149-154

Technical Note Thermal Post-buckling of Thin Simply Supported Orthotropic Square Plates

ABSTRACT Thermal post-buckling behaviour of orthotropic square plates with simply supported edges has been studied in this note using the Rayleigh-Ritz method. The formulation and the solution for the linear buckling temperature and post-buckling temperature are presented in closed form together with a set of numerical results.

1 INTRODUCTION In flight, m o d e r n aerospace structures are subjected to severe thermal loads. These structures, assembled from simple structural components such as bars, plates, etc., are subjected to compressive stress fields due to thermal loads especially when their ends or edges are respectively restrained axially or in-plane. A n understanding of the thermal buckling and post-buckling behaviour of such structural components is essential for efficient design. Boley and Weiner I have discussed the thermal buckling of beams and plates. Also they presented a few results on the thermal post-buckling of isotropic rectangular plates. Earlier studies by the authors considered the thermal post-buckling of columns e-4 and circular and square plates 5-7 made of isotropic materials. Both closed form solutions and finite element methods were used to predict the post-buckling behaviour of these structural elements. While closed form solutions are elegant and simple to use by the designers, finite element solutions provide very accurate predictions and were used to validate the approach used for the closed form solutions. However, similar solutions are not readily available for the thermal 149 Composite Structures 0263-8223/89/$03.50 © 1989 Elsevier Science Publishers Ltd, England. Printed in Great Britain


K. Kanaka Ralu, G. Venkateswara Rao

post-buckling behaviour of orthotropic plates, although a few recent studies by Tauchert 8 and Chen et al. 9 have considered thermal buckling of laminated plates. In this note, solutions are developed for the postbuckling behaviour of simply supported, square orthotropic plates made of (a) Kevlar epoxy, (b) glass epoxy and (c) carbon epoxy, using the Rayleigh-Ritz method.

2 F O R M U L A T I O N AND SOLUTION For an orthotropic square plate of side a, the strain energy, U, is given by:


U = ~

[C.(ux + ~WxJ ,. 2,2 + C22(Vy + ~Vy) ~. 2,2

+ 2c12(ux + ~w~)(vy + ~w~) + c66(Uy + Vx + Wx Wy)2 + Dll wz~, + D2zw2y +2DlzwxxWyy + 4 D 6 6 ~ y ] d x d y


in which Ex h Cll =

1 - Vxyl.,'yx


C22 = 1 -

C11h2 D~1 - - 12 D22_

C22h 2


Vxy Vyx

C12 = lzyxCll

DI2 = P y x D l l

C66 = Gxy h

D66 -

Gxy h 3


and Ex, Ey are Young's moduli in the x and y coordinate directions, Vxy and Vyx are the Poisson ratios, Gxyis the shear modulus and u, v and w are the in-plane and transverse displacements. Subscripts x and y denote differentiation with respect to spatial coordinates x and y, respectively, h is the thickness of the plate. The work done, W, due to a constant temperature differential, T, is: W =


[Nx I¢'2 + Ny w 2 ] dr, dy

where mx=

ExhT(a~ + Uyx a s) 1 - V~y Vyx


Thermal post-buckling of orthotropic square plates



Ny =

EyhT(ay + vxyax) 1 - V~y lpyx

are the stress resultants, and ax and ay are the coefficients of linear thermal expansion in the x and y directions. For a simply supported square plate, the admissible displacement distribution functions satisfying the boundary conditions are w=Asin

~rx sin ~Y a a

u = B sin

2rrx sin ~ a a

v = C sin

-/'/~2 ,fix sin- J a a


with A , B, C being the u n d e t e r m i n e d coefficients. Following the Rayleigh-Ritz m e t h o d , i.e. minimising the total potential energy with respect to A, B and C after performing the integrations, the linear critical t e m p e r a t u r e Tcr and the t e m p e r a t u r e TNL in the post-buckling range can be obtained. T h e linear critical t e m p e r a t u r e Tcr is given by Tcr -

"n'2h2[1 + fl + 2Vyx+ 4Gt] 12axa2[1 + fl~t + Vyx + VyxOl]


where ~ = ay/Ctx, fl = Ey/Ex and G1 = Gxy(1 - Vxy Uyx)/E~. Further, from the formulation above the ratio TNL/Tcr is obtained as


TNL _ 1 + Tcr ~


F4 + 3(Vyx+al)F2 J

in which

F2 =~(Vyx+G1 )-9*'rg(4fl+Gl+Glfl+-~ --) 32(Vyx + G1) F3 = 1 + [3 + 2Vyx + 4G~,

F4 = ~[1


F5 = 36~2fl-32(Vyx + G1)(1 + f l + GI - Vyx)

K. Kanaka Raju, G. Venkateswara Rao


and A is the deflection at the centre. These expressions for Tcr and T N L / T c r can be reduced to the case of plates with isotropic material properties by substituting E = Ex = E y ; v = Vxy = Vvx; a = a x = Oly and G = E~ 2(1 + v) and, accordingly, /3 = 1 and ~ -- 1. The results match exactly with those of Ref. 7 for thin plates. Also, by proper reduction, the Tcr expression in eqn (4) above can be seen to be the same as that given by Tauchert 8 for orthotropic plates.

3 N U M E R I C A L RESULTS AND DISCUSSION Numerical results are presented for three types of orthotropic materials, Kevlar, glass and carbon epoxy. The thermoelastic properties of these materials are given in Table 1. Using the expressions (4) and (5), the buckling and post-buckling temperatures have been evaluated for square plates made of these materials and are given in Table 2. Corresponding results for an isotropic plate are also included in Table 2. TABLE 1 T h e r m o - e l a s t i c P r o p e r t i e s of Uni-directional L a m i n a e


Kevlar/epoxy Glass/epoxy Carbon/epoxy

Ex Ey G ~y (kglmm e) (kglmm 2) (kglmme) 5530 5493 13 020

370 1830 601.4

94.8 880 280.7


0.34 0.25 0.314




- 5 . 3 2 x 10 -6 6.3 x 10 -6 - 0 . 4 7 x 10 -6

42.4 x 10 -6 20-52 × 10 -6 36.98 x 10 -6

With the exception of Kevlar epoxy plates, the linear buckling temperatures given indicate that buckling arises as a consequence of heating. For Kevlar epoxy plates, however, buckling arises from cooling. This type of p h e n o m e n o n was earlier observed by Whitney and Ashton.10 The nature of buckling, be it due to heating or cooling, depends on the sign (positive or negative) and magnitude of the ratio of the two stress resultants Nx and Ny in x and y directions (the ratio can be evaluated directly from the material properties considered), as ax is negative for some of these materials. The temperature ratios T N L / T c r presented in Table 2 for the central deflection to thickness ratio ( A / h ) varying between 0-0 and 1.0, indicate that in all cases the post-buckling temperatures are of same order and 2-3-2-5 times the linear critical temperature for A / h = 1.0, with again the results for Kevlar epoxy corresponding to cooling.

Thermal post-buckling of orthotropic square plates



Tcr and TNL/TerValues for Simply Supported Square Orthotropic Plates Subjected to Uniform Temperature Differential (T)


TNL/Tcr Kevlar/epoxy

0"0 0.2 0.4 0.6 0.8 1-0 Tcr

1-0000 1.0582 1.2327 1-5236 1.9308 2.4544 3-1358a



1.0000 1.0514 1.2055 1-4624 1-8221 2.2845 1-1085a

~rhe multiplying factor is: ~'he multiplying factor is:

1.0000 1.0547 1.2188 1.4923 1-8752 2-3676 0.0410~

Isotropic 1-0000 1-0561 1.2246 1-5053 1.8984 2-4037 2-0 b

7t2h2 12ax(1 + Vyx~)a2 7r2h2 12a(1 + v)a 2

4 CONCLUDING REMARKS With appropriate displacement distribution functions, the present m e t h o d for thermal post-buckling analysis may be extended to plates with other shapes and b o u n d a r y conditions.

REFERENCES 1. Boley, B. A. & Weiner, J. H. Theory of ThermalStresses. John Wiley & Sons, New York, 1960. 2. Venkateswara Rao, G. & Kanaka Raju, K. Thermal post-buckling of columns, AIAA J., 22(6) (1984) 850-1. 3. Kanaka Raju, K. & Venkateswara Rao, G. Thermal post-buckling of tapered columns. A I A A J., 22(10) (1984) 149%1501. 4. Kanaka Raju, K. & Venkateswara Rao, G. Finite element analysis of thermal post-buckling of tapered columns. Computers and Structures, 19(4) (1984) 617-20. 5. Kanaka Raju, K. & Venkateswara Rao, G. Thermal post-buckling of circular plates. Computers and Structures, 18(6) (1984) 117%82. 6. Kanaka Raju, K. & Venkateswara Rao, G. Thermal post-buckling of a square plate resting on an elastic foundation by finite element method. Computers and Structures, 28(2) (1988) 195--9.


K. Kanaka Raju, G. VenkateswaraRao

7. Kanaka Raju, K. & Venkateswara Rao, G. Thermal post-buckling of thick simply supported circular and square plates, Res Mechanica (submitted). 8. Tauchert, T. R. Thermal buckling of thick antisymmetric angle-ply laminates. Journal of Thermal Stresses, 10 (1987) 113-24. 9. Lien-Wen Chen & Lei-yi Chen. Thermal buckling of laminated cylindrical plates. Composites and Structures, 8 (1987) 189-205. 10. Whitney, J. M. & Ashton, J. E. Effect of environment on the elastic response of layered composite plates. AIAA Journal, 9(9) (1971) 1708-13.

K. Kanaka Raju & G. Venkateswara Rao Structural Design & Analysis Division, Structural Engineering Group, Vikram Sarabhai Space Centre, Trivandrum - 695 022, India