THERMOMECHANICAL POSTBUCKLING OF MULTILAYERED COMPOSITE PANELS WITH CUTOUTS

Ahmed K. Noor*, James H. Stames, Jr.** and Jeanne M. Peters*** NASA Langley Research Center

Hampton, Virginia

Abstract

The results of a study of the detailed thermomechanical postbuckling response characteristics of flat unstiffened composite panels with central circular cutouts are presented. The panels are subjected to combined temperature changes and applied edge loading (or edge displacements). The analysis is based on a first-order shear deformation plate theory. A mixed formulation is used with the fundamental unknowns consisting of the generalized displacements and the stress resultants of the plate. The postbuckling displacements, transverse shear stresses, transverse shear strain energy density, and their sensitivity coefficients are evaluated. The sensitivity coefficients measure the sensitivity of the postbuckling response to variations in the different lamination and material parameters of the panel. Numerical results are presented showing the effects of the variations in the hole diameter, laminate stacking sequence, fiber orientation, and aspect ratio of the panel on the thermomechanical postbuckling response and its sensitivity to changes in panel parameters.

Notation

[A], [Bl, [Dl, [&I matrices of the extensional, coupling, bending and transverse shear stiffnesses of the panel

EL, ET elastic moduli of the individual layers in the direction of fibers and normal to it, respectively

m linear flexibility matrix of the panel (see equation (B2) in Appendix II)

GLT r GTT shear moduli of the individual layers in the plane of fibers and normal to it, respectively

~@Z)I vector of nonlinear terms of the panel (see equation (1))

h total thickness of the panel (HI vector of stress resultant

paramem

*Professor of Aerospace Structures and Applied Mechanics, and Director, Center for Computational Structures Technology, University of Virginia, Fellow AIAA. **Head, Aircraft Structures Branch, Fellow AIAA. ***Senior Programmer Analyst, Center for Computational Structures Technology, University of Virginia.

Release C: "This paper is declared a work of the U.S. Government and is not subject to copyright protection in the United States."

global linear structure matrix (see equation (1) and (B2) in Appendix n) side length of the panel bending stress resultants subvectors of nonlinear terms (see equation (B3) in Appendix 11)

in-plane (extensional) stress resultants applied in-plane edge shear stress resultant vectors of in-plane and bending stress resultants (see equation (Al) in Appendix I) vectors of thermal forces and moments in the panel (see equation (Al) in Appendix I) total number of layers in the panel total axial force at the edge of the panel applied edge displacement critical value of qe thermal strain and edge displacement parameters

associated with {$"I, {$2'],

respectively transverse shear stress resultants vector of transverse shear stress resultants (see equation (Al) in Appendix I)

vectors of normalized thermal and mechanical strains

matrices of the extensional and transverse shear stiffnesses of the kth layer of the plate (referred to the XI, x2, x3 coordinate system) linear strain displacement matrices associated with the free nodal displacements, (XI, and the constrained (prescribed nonzero) edge displacements, qz {%I uniform temperature change critical value of T displacement components in the coordinate directions (see Fig. 1) total strain energy of the panel

transverse shear strain energy

respectively Cartesian coordinate system (x3 normal to the middle plane of the panel) vector of free (unknown) nodal displacements normalized vector of constrained (prescribed nonzero) edge displacements response vector of the panel which includes both (H) and {X) coefficients of thermal expansion of the individual layers in the direction of fibers and normal to it, respectively vector of coefficients of thermal expansion of the kth layer of the panel (referred to the x,, x2, x3 coordinate system) transverse shear strains in the panel vector of transverse shear strain components of the panel (see equation (Al) in Appendix I) vector of extensional strain components of the panel (see equation (Al) in Appendix I) transverse shear strain components thermal strain subvector (see equation (B4) in Appendix 11) fiber orientation angle of an individual layer vector of bending strain components of the panel (see equation (Al) in Appendix I) major Poisson's ratio of an individual layer lamination and material parameters of the panel transverse shear stresses in the panel rotation components of the middle plane of the panel

i, j = 1 to the total number of degrees of freedom (free nodal displacements and stress resultant parameters) in the model

I, J = 1 to the total number of free nodal displacement components in the model (components of the vector (XI)

I' = 1 to the total number of stress-resultant parameters in the modal (components of the vector {HI)

P = 1 to the total number of material and lamination parameters considered

L = direction of fibers T = transverse direction T = thermal

Superscripts

t matrix transposition r iteration cycle k layer

Considerable attention has been devoted to the study of buckling and postbuckling of isotropic panels with cutouts. More recently, a number of studies considered the buckling of composite panels with cutouts. These studies included both experimental investigations as well as approximate analytical and numerical studies (see, for example, Refs. 1 to 7). Except for Refs. 5, 6 and 7, all the cited references considered only mechanical loading. Very few studies considered the postbuckling behavior of composite plates with cutouts (see, for example, Refs. 8 and 9). Because of the increasing use of fibrous composite materials in flight-vehicle structures subjected to elevated temperatures, an understanding of the thermomechanical postbuckling response and associated failure mechanisms of composite panels with cutouts is needed. Moreover, a study of the sensitivity of the postbuckling response to variations in the material and lamination parameters of these panels is needed to provide an indication of the effects of changes in these parameters on the panel response.

The present study focuses on understanding the detailed postbuckling response characteristics of multilayered composite panels with cutouts, and subjected to combined mechanical and thermal loads. The sensitivity of these response characteristics to variations in lamination and geometric parameters is also considered in the study. The unstiffened flat panels, with central circular holes, considered in the study consist of a number of perfectly bonded layers and are symmetrically laminated with respect to the middle plane. The individual layers are assumed to be homogeneous and anisotropic. A plane of thermoelastic symmetry exists, at each point of the panel, parallel to the middle plane. The loading consists of a combination of a uniform temperature change and either an applied edge shortening or an applied in-plane edge shear loading. The material properties are assumed to be independent of temperature.

Mathematical Formulation

Governing Finite Element Eauations

The analytical formulation is based on a first-order shear deformation, von-Karman type plate theory with the effects of large displacements, average transverse shear deformation through-the-thickness, and laminated anisotropic material behavior included. A linear Duhamel- Neumann type, constitutive model is used and the material properties are assumed to be independent of temperature. The thermoelastic constitutive relations used in the

present study are given in Appendix I. A total Lagrangian formulation is used in which the vane1 deformations, at different values of the applied loading, are referred to the original undeformed configuration. The panel is discretized by using two-field mixed finite element models. The stress resultants are allowed to be discontinuous at interelement boundaries in the model. The sign convention for generalized displacements and stress resultants for the model is shown in Fig. 1. The external loading consists of a uniform temperature change To (independent of the coordinates xl, x2 and x3); and either an applied edge displacement qe or an applied in-

plane edge shear loading k. The governing finite element equations describing the

large deflection postbuckling response of the panel can be written in the following compact form:

where [El is the global linear structural matrix which includes the flexibility and the linear strain-displacement matrices; (2 ) is the response vector which includes both unknown (free) nodal displacements and stress-resultant parameters; (@z)} is the vector of nonlinear terms; ql and q2 are thermal strain and edge displacement (or edge

-0) loading) parameters; (Q ) is the vector of normalized

1 2 ) . thermal strains; and (Q ) IS the vector of normalized mechanical loads (or mechanical strains). The form of the

arrays [d, (@z)}, (dl)] and (3'1 is described in Appendix 11.

The standard approach for the solution of equation (1) is to fix the value of one of the two parameters ql and q2 and to vary the other or to choose a functional relationship between q, and q2 which is dependent on a single parameter q. In either case, the solution corresponding to the chosen combination of q1 and q2 (which is effectively dependent on a single parameter) constitutes a curve on the equilibrium surface of the panel. The details of the computational procedure used in determining the prebuckling response, stability boundary and the postbifurcation equilibrium configurations, corresponding to specified values of the parameters q1 and q2 is described in Ref. 10 and is not reproduced herein.

Sensitivity of the Postbuckling Response to Variations in Lamination and Material Parameters

Sensitivity coefficients can be used to study the sensitivity of the postbuckling response to variations in the different material and lamination parameters of the panel. The expressions for the sensitivity coefficients used in the present study are given subsequently. The derivatives of the postbuckling response with respect to the lamination and material parameters of the panel, kt, are obtained by differentiating equation (1). The resulting

linear algebraic equations have the following form:

Note that the matrix on the left-hand-side of equation (2) is identical to that used in the Newton-Raphson iterative process (see Ref. (10)). Therefore, if the Newton-Raphson technique is used, the evaluation of the sensitivity coefficients requires the generation of the right-hand-side of equation (2), and a forward-reduction/back-substitution operation only (no decomposition of the left-hand-side

matrix is required). The explicit form of la:) - is given . s ,

in Appendix 11.

In order to reduce the cost of generating the stability boundary, postbifurcation equilibrium configurations and the sensitivity coefficients, multiple-parameter reduction methods have been developed for substantially reducing the number of degrees of freedom used in the initial discretization (see Refs. 1 1 to 13). The methods are based on successive applications of the finite element method and the classical Rayleigh-Ritz technique. The finite element method is used to generate a few global approximation vectors (or modes) for approximating each of the nonlinear equations, equation (1); and the equations for the sensitivity coefficients, equation (2). The Rayleigh-Ritz technique is then used to generate the reduced sets of equations in the amplitudes of these modes. An effective set of modes for approximating the nonlinear equations was found to be the path derivatives of the response quantities with respect to the parameters ql and q2. The modes used in approximating the equations for the sensitivity coefficients include both the path derivatives and their derivatives with respect to kt. The equations used in evaluating the path derivatives, and their derivatives with respect to hf are obtained by successive differentiation of the original nonlinear equations, equation (l), with respect to ql, q2 and hf. The left-hand-side matrix in these equations is the same as that of equation (2). The details of applying reduction methods to the generation of the stability boundary, and the postbifurcation equilibrium path, are given in Refs. 11 and 12; and their application to the evaluation of the sensitivity coefficients is described in Ref. 13.

The most time-consuming steps of the procedure are those associated with operating on the original, full system of equations, namely, evaluation of the basis vectors and generation of an initial (or improved) nonlinear solution. Recently, a procedure has been developed to reduce the size of the model used in evaluating the basis vectors. The procedure is described in Refs. 14 and 15 and is based on decomposing each of the vectors into symmetric and antisymmetric components. It is particularly effective for panels with symmetric geometry but unsymmetric response.

Numerical Studies

To study the effects of variations in the hole size, laminate staking sequence and fiber orientation on the thermomechanical postbuckling characteristics and their sensitivity coefficients, several panel postbuckling problems were solved. The loading on the panels consisted of a uniform temperature change T, an applied edge shortening q,, and an applied in-plane edge shear - loading N12. For each problem, the derivatives of the postbuckling response with respect to different material and lamination parameters were evaluated. The material properties and geometric characteristics for the panels considered in the present study are given in Fig. 1. The material properties, the fiber orientation and the stacking sequence selected are those typical of composite panels considered for high-performance aircraft. The two types of boundary conditions considered are listed in Fig. 1. The first type has the edge dispIacement ul prescribed at xl = + L1/2 and the second type has the edge in-plane shear loading prescribed along the four edges. In all the boundary conditions considered, the transverse displacement w is restrained along all the edges.

Three parameters were varied, namely, the hole diameter d, the fiber orientation angle, and the laminate stacking sequence. Both quasi-isotropic and anisotropic panels were considered. The fiber orientation, stacking sequence and the designation of the quasi-isotropic panels Q1,Q2 and 4 3 and the anisotropic panels Al, A2 and A3 are shown in Table 1. Mixed finite element models were used for the discretization of each panel. Biquadratic shape functions were used for approximating each of the generalized displacements, and bilinear shape functions were used for approximating each of the stress resultants. The characteristics of the finite element model are given in Ref. 16. For each panel, the multiple parameter reduction methods outlined in Refs. 11, 12 and 13 were used in determining the stability boundary, tracing the postbuckling response, and evaluating the sensitivity coefficients. The effects of the parameters varied on the stability boundaries for the panels considered are described in Ref. 7. Typical results for the postbuckling and sensitivity studies are presented in Figs. 2 to 13 for square sixteen-layer panels and are described subsequently.

Postbuckling Reswnse

The postbuckling responses of both initially stress- free and thermally stressed, sixteen-layer quasi-isotropic panels are shown in Figs. 2 to 5. Plots of the edge displacement and edge shear, versus the transverse displacement at point c, w,, and the total strain energy U, for different values of the temperature change, T, are shown in Figs. 2 and 3 for the quasi-isotropic panels Q 1, 4 2 and 43, and in Figs. 4 and 5 for the anisotropic panels Al, A2 and A3. The edge displacement q, is normalized by the length L and the thickness h; the edge shear is normalized by L, ET and h; the transverse displacement w, is normalized by h; and the total strain energy is

normalized by L, ET and h. The temperature change T is normalized by the maximum value of the critical temperature Tc,ba,, in the absence of mechanical loads. An examination of Figs. 2 to 5 reveals:

1. For quasi-isotropic panels, the stacking sequence has little effect on the total strain energy U for small values of the thermal prestress (produced by the temperature change T) and small values of d/L. As T and/or d/L increase, the effect of the stacking sequence becomes more pronounced. For a given edge shortening qe (or applied edge loading il 2), the corresponding values of the strain energy U in panels Q1 and 4 2 are not much different, and are larger than the corresponding value for panel 43.

2. For anisotropic panels, the fiber orientation has a very significant effect on the total strain energy. For a given q, (or il 2), the value of the total strain energy U in panel A1 by far exceeds the corresponding values of U in panels A2 and A3. The value of U in panel A3 is less than the corresponding value for panel A2, with the difference increasing with increasing T.

Surface plots are presented in Figs. 6 and 7 depicting the effects of thermal prestress (produced by T) and hole diameter d/L on the total strain energy U in the postbuckling range for panels Q1 and Al. The two loading cases with q, and 4, are shown. Each load is normalized by its critical value (associated with the bifurcation buckling of the panel).

The results in Figs. 6 and 7 show that in the initial postbuckling stage, the total strain energy increases rapidly with the increase in the value of T for both q, and - Nl ,. However, in the advanced postbuckling stage, an increase in the value of T can cause a decrease in U. This phenomenon is very pronounced for quasi-isotropic panels Q1 with d/L 6 0.3, and subjected to & ,.

The through-the-thickness distributions of the transverse shear stresses 031 and the transverse shear strain energy density USh for the quasi-isotropic panels Q1,

subjected to edge shortening and in-plane shear z,,. are shown in Figs. 8 and 9. The expression of the transverse shear strain energy density is:

where 2~~~ are the transverse shear strain components.

As can be seen from Figs. 8 and 9, for small hole diameters, d/L < 0.3, the changes in d/L can have a pronounced effect on both the magnitude and distribution of 031 and Ush. Even the locations of the maximum values of ogl and Ush in the thickness direction change with the changes in d/L.

Sensitivitv Studia

Sensitivity analyses were conducted to identify which material and lamination parameters most affect the postbuckling response. The sensitivity of the total strain energy U to variations in the five material properties EL, E , GLT, aL and a~ and the four fiber angles +45", -45", 0" and 90" are shown in Figs. 10 and 11 for the quasi- isotropic panels and in Figs. 12 and 13 for the anisotropic panels. The applied end shortening case is shown in Figs. 10 and 12, and the applied edge shear case is shown in Figs. 11 and 13. The sensitivity coefficients with respect

au au to the material properties and fiber angles, - and -, are ax ae L normalized by multiplying by and -

qh4 E,. h4' respectively, where h refers to eaih of the material properties.

An examination of Figs. 10 to 13 reveals:

1. For both the quasi-isotropic and anisotropic panels, the largest normalized sensitivity coefficients are associated with the material parameters aT. ET and EL. The magnitude of the sensitivity coefficients increases with increasing d/L and the fiber angles 0=+45O and -45". Exceptions to this observation are the quasi-isotropic

- au . panels subjected to N, ,, where the largest value of - is ae

au associated with 0 = +45" and 0" and the magnitude of - aE L

decreases with an increase in d/L.

2. For quasi-isotropic panels, the magnitude of the sensitivity coefficients, associated with the material properties, for panels Q1 and 4 2 are not much different and are larger than those for panel 43. However, the sensitivity coefficients with respect to the fiber angles are larger for panels Q1, than the corresponding ones for panels Q1 and 42.

3. For the anisotropic panels, the magnitudes of the au au corresponding values of - and - for panels A2 and A3 ax ae

are not much different and are considerably smaller than the corresponding values for panels Al. Exceptions to

au this observation are the magnitudes of - (0 = 45" and ae -45") for panels with d/L I 0.3 in the advanced postbuckling stage.

Concluding Remarks

An analytical study is made of the thermomechanical postbuckling response of flat unstiffened composite panels with central circular cutouts. The panels considered consist of a number of perfectly bonded layers and symmetric lamination stacking sequences with respect to the middle plane. The loading consists of a combined

uniform temperature change and either an applied edge displacement or an applied in-plane edge shear loading. The analysis is based on a first-order shear deformation, von-Karman type nonlinear plate theory with the effects of large displacements, average transverse shear deformation through-the-thickness, and laminated anisotropic material behavior included. A linear Duhamel-Neumann type constitutive model is used and the material properties are assumed to be independent of temperature. A total Lagrangian formulation is used in which the panel deformations, at different values of the applied loading, are referred to the original undeformed configuration. The panel is discretized by using two-field mixed finite element models with the fundamental unknowns consisting of the nodal displacements and stress-resultant parameters. The stress resultants are allowed to be discontinuous at interelement boundaries.

An efficient multiple-parameter reduction method is used for determining the stability boundary and postbuckling response. The reduction method is also used for evaluating the sensitivity coefficients that measure the sensitivity of the buckling and postbuckling responses to variations in the different lamination and material parameters of the panel. Numerical results are presented that show the effects of variations in the hole diameter, the fiber orientation angle, and the laminate stacking sequence of the panels on their thermomechanical postbuckling response, and their sensitivity coefficients.

The lamination parameters (viz., stacking sequence and fiber orientation) can significantly affect the relative stiffnesses of the panel. The numerical results presented show a complex interaction between the relative stiffnesses of the panel, the hole diameter, the boundary conditions and the loading.

The work of the first and third authors was partially supported by NASA Cooperative Agreement NCCW-0011 and by NASA Grant No. NAG-1-1162. The material properties were supplied by the Aircraft Division of Northrop Corporation. The numerical studies were performed on the CRAY Y-MP computer at NASA Ames Research Center.

References

Nemeth, M. P., "Buckling Behavior of Compression- Loaded Symmetrically Laminated Angle-Ply Plates with Holes," AIAA Journal, Vol. 26, No. 3, March 1988, pp. 330-336. Owen, V. and Klang, E. C., "Shear Buckling of Specially Orthotropic Plates with Centrally Located Cutouts," presented at the Eighth DOD/NAS NFAA Conference on Fibrous Composites in Structural Design, Norfolk, VA, Nov. 28-Dec. 1, 1989. Srivatsa, K. S. and Krishna Murty, A. V.. "Stability of Laminated Composite Plates with Cutouts," Computers and Structures, Vol. 43, No. 2, 1992, pp. 273-279.

Jones, K. M. and Klang, E. C., "Buckling Analysis of Fully Anisotropic Plates Containing Cutouts and Elastically Restrained Edges," in Proceedings of the 33rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, April 13-15, 1992, Dallas, TX, A Collection of Technical Papers, Part 1, Structures I, pp. 190-200. Chang, J. S. and Shiao, F. J., "Thermal Buckling Analysis of Isotropic and Composite Plates with a Hole," Journal of Thermal Stresses, Vol. 13, 1990, pp. 315-332. Chen, W. J., Lin, P. D. and Chen, L. W., "Thermal Buckling Behavior of Composite Laminated Plates with a Circular Hole," Composite Structures, Vol. 18, 1991, pp. 379-397. Noor, A. K., Starnes, Jr., J. H. and Peters, J. M., "Thermomechanical Buckling of Multilayered Composite Panels with Cutouts," Proceedings of the 34th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, April 19-21, 1993, La Jolla, CA. Nemeth, M. P., "Buckling and Postbuckling Behavior of Square Compression-Loaded Graphite- Epoxy Plates with Circular Cutouts," NASA TP- 3007, August 1990. Lee, H. H. and Hyer, M. W., "Postbuckling Failure of Composite Plates with Holes," in Proceedings of the 33rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, April 13-15, 1992, Dallas, TX, A Collection of Technical Papers, Part 1, Structures I, pp. 201-21 1.

10. Noor, A. K., Starnes, J. H., Jr. and Peters, J. M., "Thermomechanical Buckling and Postbuckling of Multilayered Composite Panels," Journal of Composite Structures, Vol. 23, 1993, pp. 233-251.

11. Noor, A. K. and Peters, J. M., "Multiple-Parameter Reduced Basis Technique for Bifurcation and Postbuckling Analyses of Composite Plates," International Journal for Numerical Methods in Engineering, Vol. 19, 1983, pp. 1783-1803.

12. Noor, A. K. and Peters, J. M., "Recent Advances in Reduction Methods for Instability Analysis of Structures," Computers and Structures, Vol. 16, No. 1-4, Jan. 1983, pp. 67-80.

13. Noor, A. K. and Peters, J. M., "Reduced Basis Technique for Calculating Sensitivity Coefficients of Nonlinear Structural Response," AIAA Journal, Vol. 30, No. 7, July 1992, pp. 1840-1847.

14. Noor, A. K. and Peters, J. M., "Buckling and Postbuckling Analyses of Laminated Anisotropic Structures," International Journal for Numerical Methods in Engineering, Vol. 27, 1989. pp. 383- 401.

15. Noor, A. K. and Peters, J. M., "Potential of Mixed Formulations for Advanced Analysis Systems," Computers and Structures, Vol. 35, 1990, pp. 369- 380.

16. Noor, A. K. and Andersen, C. M., "Mixed Models and ReducedJSelective Integration Displacement Models for Nonlinear Shell Analysis," International Journal for Numerical Methods in Engineering, Vol. 18, 1982, pp. 1429-1454.

17. Jones, R. M., Mechanics of Composite Materials, McGraw Hill, New York, 1975.

18. Tsai, S. W. and Hahn, H. T., Introduction to Composite Materials, Technomic Publishing Co., Westport, CT, 1980.

19. Padovan, J., "Anisotropic Thermal Stress Analysis," Thermal Stresses I, ed. by R. B. Hetnarski, Elsevier Science Publishers, Amsterdam, 1986, pp. 143-262.

20. Bert, C. W., "Analysis of Plates," Vol. 7 - Structural Design and Analysis, Part I, ed. by C. C. Chamis, Composite Materials, Academic, New York, 1975, pp. 149-206.

21. Noor, A. K. and Tenek, L. H., "Stiffness and Thermal Coefficients for Composite Laminates," Journal of Composite Structures, Vol. 21, No. 1, 1992, pp. 57- 66.

Ap~endix I - Thermoelastic Constitutive Relations for the Laminate

The thermoelastic model used in the present study is based on the following assumptions:

1) The laminates are composed of a number of perfectly bonded layers.

2) Every point of the laminate is assumed to possess a single plane of thermoelastic symmetry parallel to the middle plane.

3) The material properties are independent of temperature.

4) The constitutive relations are described by lamination theory, and can be written in the following compact form:

where [N}, (MI, (Q} and [E), (u) , (y) are the vectors of extensional, bending and transverse shear stress resultants and strain components of the laminate given by:

and

The matrices [A], [Bl, [Dl and [A,] contain the extensional, coupling, bending and transverse shear stiffnesses of the laminate which can be expressed in terms of the layer stiffnesses as follows:

where [d "' and [q] "' are the extensional and transverse shear stiffnesses of the kth layer (referred to the x,, x,, x3 coordinate system); [I] is the identity matrix; hk and hkSl are the distances from the top and bottom surfaces of the kth layer to the middle surface; and NL is the total number of layers in the laminate. The expressions for the different

coefficients of the matrices [iS] and [q] "' in terms of the material and geometric properties of the constituents (fiber and matrix) are given in Refs. 17 and 18.

The vectors of thermal effects, IN,] and {M,], are given by:

where ( a ] is the vector of coefficients of thermal expansion (referred to the coordinates xl, x2 and x3 - see, for example, Refs. 19 and 20).

A~Iendix I1 - Form of the Arravs in the Governing Discrete Eauations of the Panel

The governing discrete equations of the panel, equation (I), consist of both the constitutive relations and the equilibrium equations. The response vector, {Z], can be partitioned into subvectors of stress-resultant parameters, (HI , and free (unconstrained) nodal displacements, {XI, as follows:

The different arrays in equations (1) and (2) can be partitioned as follows:

for the applied edge loading case, and

for the applied edge displacement case.

where IF] is the linear flexibility matrix; [S1] and [S,] are the linear strain-displacement matrices associated with the free nodal displacements, {XI, and the constrained . .

(prescribed noniero) edge displacements, a (7); (PI is

the vector of applied edge forces; (k (x, %j) and

(N (H, X, 2)) are the subvectors of nonlinear terms: { E ~ } is the subvector of normalized thermal strains; 0 is a null matrix or vector; and superscript t denotes transposition. The explicit form of ( E ~ ] is given in Ref. 10.

For the purpose of obtaining analytic derivatives with respect to lamination parameters (e.g., fiber orientation

a [ ~ l angle of different layers), it is convenient to express - a 4

in terms of - gFT1 as fol~ows: 3 4

are given in Ref. The explicit forms of -

Analytic expressions are given in Ref. 21 for the laminate stiffnesses [A], [B], [Dl, [A,]; the vectors of thermal effects {NT ) and {MT ); and their derivatives with respect to each of the material properties and fiber orientation angles.

Table 1 Composite panels considered in the numerical studies

Quasi-isotro~ic Panels Anisotro~icPanels

Panel Laminate Stacking Panel Laminate Stacking No. Sequence No. Seauence

Q1 [+45/0/9012, A1 [f45Ids

Q2 [+45/0/-45/90]2s A2 [+45/902l2,

EL=l 53. GPa ET=9.62 GPa GL+.93 GPa G,=3.23 GPa vLT=0.32 aL=l .8x1 0-810C aT=23.6x1 06/"C Thickness of individual

layers=1.32~1 0-4 m

Boundary Conditions At x, = f Ll2

w = q 1 = 0

For Type 1 At x, = f Ll2

u, = f qJ2 u2=W=@1=$2=o

For Type 2 At x, + Ll2

U1=W=@1=@2=o

Figure 1 - Panels considered in the present study and sign convention for generalized displacements, stresses and stress resultants.

473

3 6 9 12

U L / (E, h4)

3 6 9 12

U L / (E, h4)

5 10 15 20

U L 1 (E, h4) Figure 2 -Effect of hole diameter and stacking sequence on the postbuckling response of thermally-stressed square

panels with central circular cutout. Sixteen-layer quasi-isotropic panels with central circular cutout subjected to edge displacement q,.

25 50 75 100 125

U L 1 (E, h4)

10 20 30 40 50

U L I (E, h4)

O 10 20 30 40 50

U L 1 (E, h4) Figure 3 - Effect of hole diameter and stacking sequence on the postbuckling response of thermally-stressed square

panels with central circular cutout. Sixteen-layerquasi-isotropic panels subjected to in-plane shear loading N, 2.

475

3 6 9 12

U L / (E, h4)

4 8 12 16

U L / (E, h4)

I I I I I I

O 10 20 30 40 50

U L I (E, h4) Figure 4 - Effect of hole diameter on the postbuckling response of thermally-stressed square panels with central

circular cutout. Sixteen-layer anisotropic panels subjected to edge displacement q,.

476

0 35 70 105 140

U L I (E, h4)

30 60 90 120

U L I (E, h4)

50 100 150 200

U L I (E, h4) Figure 5 - Effect of hole diameter on the postbuckling response of thermally-stressed square panels with central

circular cutout. Sixteen-layer anisotropic panels subjected to in-plane shear loading Rl2.

477

a) Combined q, and T

b) Combined N,, and T

Figure 6 - Surface plots depicting the effect of the thermal prestress and the hole diameter on the postbuckling response for sixteen-layer square quasi-isotropic panel Q1.

1 1 m a

2.3 1 .9 .8 .7

a) Combined qe and T .6 .5 .4

b) Combined N,, and T

Figure 7 - Surface plots depicting the effect of the thermal prestress and the hole diameter on the postbuckling response for sixteen-layer square anisotropic panel A1 .

Figure 9 - Through-the-thickness distributions of transverse shear stresses 03, and transverse shear strain energy density Ush at wc/h=3. Thermally stressed sixteen-layer anisotropic panel A1 subjected to in-plane shear loading NI2, T=0.5 T,, Through-the-thickness distribution shown is for the point marked with x.

479

-~ -

031 I E r U* I Er Figure 8 - Through-the-thickness distributions of transverse shear stresses 031 and transverse shear strain energy density Ush

at wJh=3. Thermally stressed sixteen-layer quasi-isotropic panel Q1 subjected to edge displacement q,, T=0.5 T,,. Thickness distribution shown at the point marked with x.

O32 I ET "Sh I ET Figure 9 - Through-the-thickness distributions of transverse shear stresses 03, and transverse shear strain energy density Ush

at w,/h=3. Thermally stressed sixteen-layer anisotropic panel A1 subjected to in-plane shear loading NI2, T=0.5 To Thickness distribution shown at the point marked with x.

Figure 10 - Effect of hole diameter on the normalized sensitivity coefficients of the total strain energy U, with respect to material characteristics and fiber orientation angles of the individual layers. Thermally stressed, sixteen-layer quasi-isotropic panels with central circular cutout subjected to edge displacement q,.

Figure 1 1 - Effect of hole diameter on the normalized sensitivity coefficients of the total strain energy U, with respect to material characteristics and fiber orientation angles of the individual layers. Thermally stressed, sixteen-layer quasi-isotropic panels with central circular cutout subjected to in-plane shear loading NI2.

Figure 12 - Effect of hole diameter on the normalized sensitivity coefficients of the total strain energy U, with respect to material characteristics and fiber orientation angles of the individual layers. Thermally stressed, sixteen-layer anisotropic panels with central circular cutout subjected to edge displacement q,.

Figure 13 - Effect of hole diameter on the normalized sensitivity coefficients of the total strain energy U, with respect to material characteristics and fiber orientation angles of the individual layers. Thermally stressed, sixteen-layer anisotropic panels with central circular cutout subjected to in-plane shear loading NI2.