THERMOMECHANICAL BUCKLING AND POSTBUCKLING OF MULTILAYERED COMPOSITE PANELS

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

Harnpton, Virginia

A study is made of the thermomechanical buckling and postbuckling responses of flat unstiffened composite panels. The panels are subjected to combined temperature change and applied edge displacement. The analysis is based on a first-order shear deformation, von- Karman type nonlinear plate theory. A mixed formulation is used with the fundamental unknowns consisting of the generalized dsplacements and the stress resultants of the plate. An efficient multiple-parameter reduction method is used in conjunction with mixed finite element models, for determining the stability boundary and postbuckling response. The reduction method is also used for evaluating the sensitivity coefficients which 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 showing the effects of variations in the laminate stacking sequence, fiber orientation, number of layers and aspect ratio of the panels on their thermomechanical buckling and postbuckling responses and their sensitivity coefficients.

Nomenclature

matrices of the extensional, coupling, bending and transverse shear stiffnesses of the panel lamination and material parameters of the panel elastic moduli of the individual layers in the direction of fibers and normal to it, respectively linear flexibility matrix of the panel, see equations (B2) - Appendix 11 shear moduli of the individual layers in the plane of fibers and normal to it, respectively

vector of nonlinear terms of the panel, see equations (1) vector of stress resultant paramem total thickness of the panel

*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.

global linear structure matrix, see equations (1) and (B2) - Appendix II

geometric stiffness matrices of the panel, see equations (3), (B6) and (B7) - Appendix I1 side lengths of the panel in the x, and x2 coordinate directions bending stress resultants

subvectors of nonlinear terms, see equations (B3) - Appendix 11 in-plane (extensional) stress resultants number of layers in the panel vectors of in-plane and bending stress resultants, see equations (Al) - Appendix I vectors of thermal forces and moments in the panel, see equations (Al) - Appendix I shape functions used in approximating each of the stress resultants total axial force at the edge of the panel transverse shear stress resultants vector of transverse shear stress resultants

vectors of normalized thermal and mechanical strains

matrices of the extensional and transverse shear stiffnesses of the kth layer of the plate (referred to xl, x2, x3 coordinate system) applied edge displacement critical value of q, thermal strain and edge displacement parameters

associated with ($'), {$I),

I respectively linear strain displacement matrices associated with the free nodal displacements, (X) , and the constrained (prescribed nonzero) edge displacements, q3 (2.) uniform temperature change critical value of To total strain energy of the panel

transverse shear strain energy density displacement components in the coordinate directions, see Fig. 1 vector of free (unknown) nodal displacements

normalized vector of constrained (prescribed nonzero) edge displacements Cartesian coordinate sysem (x3 normal to the middle plane of the panel) 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 xl, x2, x3 coordinate system) vector of transverse shear strain components of the panel, see equations (A 1) - Appendix I transverse shear strains in the

panel vector of extensional strain components of the panel, see equations (Al) - Appendix I thermal strain subvector, see equations (B4) - Appendix 11, and equations (C4) - Appendix III fiber orientation angles of the individual layers vector of bending strain components of the panel, see equations (Al) - Appendix I major Poisson's ratio of the individual layers 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 smss resultant parameters) in the model

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

I' = 1 to the total number of stress-resultant parameters in the model (components of the vector (H))

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

' I .I w!mx (X))

3 = 1 to the total number of shape functions used in approximating each of the stress resultants (within individual elements)

L = direction of fibers T = transverse direction T = thermal p = 1,2

&werscri~u t denotes matrix transposition r denotes iteration cycle k denotes layer

Introduction

Although considerable literature has been devoted to the buckling and postbuckling analyses of laminated composite plates subjected to mechanical loads, investigations of the postbuckling response of composite plates subjected to thermal, or combined thermal and mechanical loads, are rather limited in extent. Reviews of the recent contributions to the subject are contained in Refs. 1-6. Because of the increasing use of fibrous composite materials in flight-vehicle structures subjected to elevated temperatures, an understanding of the thermomechanical postbuckling response of multilayered anisotropic panels is desirable. Moreover, a study of the sensitivity of the buckling and postbuckling responses 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 structural response.

The present study focuses on understanding the detailed buckling and postbuckling response characteristics of multilayered composite panels subjected to combined mechanical and thermal loads and the sensitivity of these response characteristics to variations in lamination and geometric parameters. The unstiffened flat panels considered 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. At each point of the panel a plane of thermoelastic symmetry exists parallel to the middle plane. The loading consists of a combination of a uniform temperature change and an applied edge displacement. The material properties are assumed to be independent of temperature. The thermoelastic constitutive relations for the panel are given in Appendix I.

Mathematical Formulation

Governing Finite Element Equations

The analytical formulation is based on a fist- 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 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. The stress resultants are allowed to be disconti~uous 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 an applied edge displacement qe, and a uniform temperature change To (independent of the coordinates xl, x2 and x3).

The governing finite element equations describing the large deflection postbuckling response of the panel can be written in the following compact form:

. - where [a is the global linear structure matrix which includes the flexibility and the linear strain-displacement matrices; (Z) is the response vector which includes both unknown (free) nodal displacements and stress-resultant

parameters; (E(Z)J is the vector of nonlinear terms; ql and q2 are thermal strain and edge displacement parameters;

($I) is the vector of normalized thermal strains; and

($)I is the vector of normalized mechanical strains. The 4 2 )

form of the arrays [El, (@z)\, (dl)) and (Q ) is des- cribed in Appendix 11.

The standard approach for the solution of equations (1) is to fix the value of one of the two parameters ql and Q and to vary the other; or to choose a functional relationship between q, and Q which is dependent on a single parameter q. In either case, the solution corresponding to the chosen combination of ql and q2 (which is effectively dependent on a single parameter) constitutes a curve on the equilibrium surface of the panel.

The prebuckling responses (generalized displacements and stress resultants), associated with the thermal strain and applied edge dispIacements, respectively, are given by the following set of linear equations with two right-hand sides (one corresponding to ql=l, qz=O, and the other to ql=O, w l ) :

where subscripts 1 and 2 refer to the response vectors associated with the thermal loading and applied edge displacement, respectively.

Identification and Determination of Stabilitv Boundary

For certain combinations of the two parameters ql, q2, an instability (or bifurcation) occurs. The totality

of the critical (or bifurcation) points in the ql-q2 space constitutes the stability boundary which separates regions of stability and instability.

If prebuckling deformations are neglected, the equations that determine the stability boundary for the panel can be cast into the form of a linear algebraic eigenvalue problem as follows:

[a + i [Ell + i [E21) (2) = 0

where 61. &) represents a critical combination of the load

parameters ql, e; (2) is the associated modal response

vector; and [El] and [E2] are the geometric stiffness

matrices. The explicit forms of [El] and [&] are given in Appendix 11.

Post-Bifurcation Equilibrium Configurations

Finding the post-bifurcation equilibrium configurations corresponding to specified values of the parameters ql and q2 is accomplished by solving the nonlinear system of algebraic equations, equations (I), using an incremental-iterative technique such as the Newton-Raphson method. The recursion formula for the rth iteration can be written in the following form:

and (zJ("l)=( Z) + {AZ)"' (5)

where (AZ)"' is the change in the response vector during the rth iteration cycle; and the range of i and j is 1 to the total number of free nodal displacements and stress resultant parameters in the model.

Sensitivitv of the Postbucklinrr Res~onse to Variations in Lamination and Material Parametea

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, dl, are obtained by differentiating equations (1). The resulting linear algebraic equations have the following form:

Note that the matrix on the left-hand-side of equations (6) is identical to that used in the Newton-Raphson iterative process (see equations (4)). Therefore, if the Newton- Raphson technique is used, the evaluation of the sensitivity coefficients requires the generation of the right-

hand-side of equations (6), and a forward-reductionback- substitution operation only (no decomposition of the left-

hand-side matrix is required). The explicit form of

is given in Appendix 11.

A~plication of Multiple-Parameter Reduction Metho& and Other Recent Com~utational Enhancements

In order to reduce the cost of generating the stability boundary, post-bifurcation 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. 7, 8 and 9). 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 linear eigenvalue problem, equations (3); the nonlinear equations, equations (1); and the equations for the sensitivity coefficients, equations (6). 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 linear eigenvalue problem and the nonlinear equations was found to be the path derivatives of the response quantities with respect to the parameters ql and Q. The modes used in approximating the equations for the sensitivity coefficients include both the path derivatives and their derivatives with respect to dl. The equations used in evaluating the path derivatives, and their derivatives with respect to dl are obtained by successive differentiation of the original nonlinear equations, equations (I), with respect to ql, Q and a. The left-hand-side matrix in these equations is the same as that of equations (4). The details of applying reduction methods to the generation of the stability boundary, and the post-bifurcation equilibrium path are given in Refs. 7 and 8; and their application to the evaluation of the sensitivity coefficients is described in Ref. 9. The application of reduction methods involves the following basic steps:

a) Generation of the initial basis vectors, at ql=q2=0, and formation of the reduced linear stiffness mamx and the geometric stiffness matrices.

b) Determination of the stability boundary using the reduced eigenvalue problem.

c) Generation of a nonlinear solution in the vicinity of the stability boundary. This step is accomplished by selecting a suitable value for a typical displacement component, and then using the associated eigenmode as a predictor and obtaining a corrected solution by using Newton-Raphson iterative technique.

d) Evaluation of the path derivatives, and their derivatives with respect to dl. Then the reduced equations which approximate equations (1) and (6) are formed.

e) Generation of the approximate solutions, associated with different values of the parameters ql and Q

using an incremental-iterative approach, in conjunction with the reduced equations. For each pair of ql and q2 at which the nonlinear solution is obtained, the sensitivity derivatives are generated using the reduced equations, which approximate equations (6).

f) Determination of the error resulting from the use of the approximate reduced equations. Whenever the error exceeds a prescribed tolerance, an iterative procedure is used to generate an improved solution, a new (updated) set of basis vectors (path derivatives and their derivatives with respect to dl), and a new set of reduced equations.

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. 10 and 11 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 !h!&gi

To study the effect of variations in the geometric and lamination parameters of composite panels on their stability boundary, postbuckling response and sensitivity coefficients, several buckling and postbuckling problems of panels were solved. For each problem, the derivatives of the various response quantities with respect to the different material and lamination parameters were evaluated. The panels were subjected to a uniform temperature change To, and an applied edge displacement, q,. Four parameters were varied, namely, the fiber orientation, laminate stacking sequence, number of layers in the laminate and the panel aspect ratio. Both quasi- isotropic and anisotropic panels were considered. The fiber orientation, stacking sequence and the designation of the panels used in the present study are shown in Table 1. Panels with 8, 16,24 layers, and with aspect ratios L2/L1 = 1.0,0.8,0.5 were considered. The material properties, geometric characteristics and boundary conditions are given in Fig. 1. The numerical results were obtained

using a 16x16 grid of mixed finite elements 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 total number of stress-resultant parameters was 8192, and the total number of nonzero generalized displacement degrees of freedom was 5053. The characteristics of the finite

element model are given in Ref. 12. For each panel, the multiple-parameter reduction methods outlined in the preceding section were used in determining the stability boundary, generating the postbuckling response, and evaluating the sensitivity coefficients. Typical results are presented in Figs. 2 to 12 and in Tables 2-4, and are described subsequently.

Buckling Response

All the bifurcation buckling modes of the panels considered exhibit either inversion symmetry or antisymmetry. The inversion symmetry is characterized by the following relations (see Ref. 13):

For inversion antisymmetry the right-hand-sides of equations (7) are multiplied by a minus sign.

The stability boundaries for quasi-isotropic panels Q1 and 4 2 are shown in Fig. 2. Panels with 8, 16 and 24 layers, and with aspect ratios L2/L1 = 1.0 and 0.5 are considered. In the figure, the applied edge displacement q, is normalized by the width L2 and the thickness h and the temperature change To is normalized by the transverse coefficient of thermal expression a ~ , L2 and h. Note that the Q1 panels, with adjacent +45" and -45" layers, have higher buckling loads and critical temperatures than the corresponding 4 2 panels. with nonadjacent +45" and -45" layers. The differences are more pronounced for the thin, 8-layer panels than for the thicker panels. For the Q1 panels, the stability boundary moves inward as the number of layers NL increases. An opposite trend is observed for the 4 2 panels. As the number of layers increases beyond 24, the stability boundary becomes insensitive to the relative locations of the +45" and -45" layers.

The stability boundaries for the anisotropic panels Al, A2 and A4 are shown in Fig. 3. Panels with 8, 16 and 24 layers, and with aspect ratios L2/L1 = 1.0 and 0.5 are considered. Two different forms of the interaction curves are shown, q, versus To; and the total edge force, N,, versus To. The total edge force N, is normalized by E T , L2 and h. The A1 panels, with +45" layers, have higher critical temperatures than the other panels. However, the critical values of N, for the A1 panels are lower than those for the corresponding A2 and A3 panels (with combinations of +45", -45" and 90" layers). The critical temperatures for the A2 and A3 panels are lower than those for all the other anisotropic panels. The N,-To interaction curves for panels A2 and A3 are nearly horizontal, indicating that the total edge force at buckling is insensitive to the particular combination of q, and To. The effect of the relative locations of the +45" and -45" layers (i.e., either adjacent or nonadjacent) on the stability

boundary of anisotropic panels is similar to that for quasi- isotropic panels.

The effect of the stacking sequence, number of layers and aspect ratio on the number of half waves associated with the lowest buckling mode is shown in Table 2. Three cases are considered: To* (applied edge displacement only), T0=0.5 T, and To=Tc, (no applied edge displacement). The threedigit entries in Table 2 refer to the number of half waves in the lowest buckling modes for each of the three cases, in the order listed above. For square panels (L2& = 1.0) the lowest buckling modes of all laminates have only one half wave in the x1 and x2 directions. For rectangular quasi-isotropic panels, the effect of the stacking sequence on the number of half waves associated with the lowest buckling mode is only noticeable for the thin, 8-layer panels. For rectangular anisotropic panels the effect of the stacking sequence is noticeable for 8, 16 and 24-layer panels. However, the relative locations of the +45" and -45" layers in the anisotropic 24-layer panels does not affect the number of half waves in the buckling mode. The effect of fiber orientation on the mode shapes associated with the lowest critical temperature change To with edge displacement q,=O, and the critical edge displacement q, with temperature change To* is shown in Fig. 4. For panels with aspect ratio L2/L1 = 0.8, the dependence of the number of half waves in the x1 direction on the fiber orientation is clearly demonstrated in Fig. 4. The replacement of the 90" layers by 0" layers changes the number of half waves from 2 to 1.

m u c k l i n y Response

The postbuckling responses of initially stressed, 16-layer quasi-isotropic Q1 panels are shown in Figs. 5 and 6. Plots of the temperature change To versus the transverse displacement, w,, and the total strain energy, U, for different values of the applied edge displacement q, are shown in Fig. 5. The temperature change To is normalized by ar, L2 and h; the transverse displacement, w,, is normalized by h; and the total strain energy U is normalized by ET. L2 and h. As to be expected, the transverse displacement, w,, and the total strain energy, U, associated with a given value of the temperature change To, increase with the increase in the initial value of the edge displacement q,.

The through-the-thickness distributions of the transverse shear stresses T~ and the transverse shear strain

2 energy density Ush = 1 C ~ 3 p x y3p (where y3p are the

2 p=1

transverse shear strains) for panels subjected to an initial value of q, = 0.5 q,,,, are shown in Fig. 6. The two values of To at which the distributions of ~ 3 1 and Ush are

shown correspond to initial (ar To L%' = 15.2) and

advanced (ar To ~?/h ' = 137.1) postbuckling stages. The

1 locations at which T3i and Ush are computed are given in Table 3. For isothermal problems, it was observed that delamination starts at the location of maximum transverse shear strain energy density USh (see Ref. 14). Figure 6 shows that the distributions of ~3~ and Ush as well as the location of their maximum values change as the temperature increases in the postbuckling range, which may affect the location of failure initiation in the panels. Although the To - w, and To - U plots were found to be fairly insensitive to the relative locations of the 45O and -45" layers, the thickness distributions and magnitudes of T,, and USh are strongly dependent on the location of these layers (results not shown).

The postbuckling responses of initially stressed, 16-layer anisotropic panels Al, A3 and A5 are shown in Figs. 7 to 9. Plots of To versus w, and U, for different values of qc are shown in Fig. 7. As for quasi-isotropic panels, To is normalized by aT, L2 and h; w, is normalized by h; and U is normalized by ET, L2 and h. For panels A1 and A3, w, and U associated with a given value of To increase with the increase in the initial value of q,. The total strain energy for panel A5 is insensitive to the initial value of q,. The global postbuckling response was found to be insensitive to the relative locations of the 4 5 " and -45" layers (results not shown).

The through-the-thickness distributions of ~ 3 ,

and USh for panels Al, A3 and A5 when subjected to an initial value of q, = 0.5 qe.cr are shown in Fig. 8, immediately after buckling, and in Fig. 9 in the advanced postbuckling stage. The locations at which 231 and Ush are computed are given in Table 3. Examination of Figs. 8 and 9 reveals that as in the case of quasi-isotropic panels, the distribution of T,, and USh changes as the temperature increases. Immediately after buckling, panels A3 and A5 have lower values of U* than panels A2 and A4 (results for A2 and A4 not shown). The same trend continues in the postbuckling range for panel A5 but changes for panel A3.

Sensitivitv Studies

Sensitivity analyses were conducted to identify which material parameters most affected structural response. The sensitivity of the total strain energy U to variations in the five material properties EL, ET, GLT, m, and a T and the four fiber angles 45". -45", 0" and 90' are shown in Fig. 10 for the quasi- isotropic panels and in Fig. 11 for the anisotropic panels Al, A3 and A5. As can be seen from Fig. 10, for quasi- isotropic panels, the total strain energy is very sensitive to variations in a ~ . ET and EL. In the advanced postbuckling state U is very sensitive to variations in the +45" and -45" fiber angles. Figure 11 shows that the total strain energy U for anisotropic panels is much more sensitive to variations in the +45Oand -45' fiber angles than all the other fiber angles. Also, U is very sensitive

to variations in a~ and ET, and is relatively insensitive to variations in GLTand a ~ . For panels A1 and A5, U is somewhat sensitive to variations in EL.

For the 16-layer quasi-isotropic panel Q1, normalized contour plots for the sensitivity coefficients of the generalized displacements, w, $1, $2, with respect to the elastic modulus EL, coefficient of thermal expansion OIT and the fiber angles 45" and 0" are shown in Fig. 12. The sensitivity coefficients are shown after buckling ( a ~ TO ~ $ / h ~ = 15.2) and at higher temperature (aT To

= 137.1). The normalizing factors are listed in Table 4. The normalized contour plots for the sensitivity coefficients of the generalized displacements after buckling, shown in Fig. 12a, are similar to the corresponding ones of the generalized displacements associated with the buckling mode. However, at higher temperatures (e.g.. Fig. 12b) the sensitivity coefficients with respect to the material parameters and fiber angles differ from each other, and are different from those associated with the buckling mode.

Concluding Remarks

An analytical study is made of the buckling and postbuckling responses of flat unstiffened composite panels subjected to combined temperature change and applied edge displacement. The panels considered consist of a number of perfectly bonded layers and symmetric lamination stacking sequences with respect to the middle plane. The analysis is based on a first-order shear deformation, von-Karrnan-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-Neuman-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 laminate stacking sequence, fiber orientation, number of layers and aspect ratio of the panels on their thermomechanical buckling and postbuckling responses, and their sensitivity coefficients.

On the basis of the numerical studies the following observations and conclusions can be made:

1. The laminate stacking sequence can change the number of half waves in the buckling mode of rectangular panels. This result is particularly true for anisotropic panels with aspect ratios less than 0.8.

2. For panels with adjacent +45" and -45" layers, the critical values of the applied displacement q, and the temperature change To are higher than for panels with non-adjacent +45" and -45" layers. Also, for panels with adjacent +45" and -45" layers, the maximum values of the transverse shear stresses, z~,, and transverse shear strain energy density, Ush, in the postbuckling range are lower than those for the corresponding panels with nonadjacent +45" and -45" layers. The differences between the critical values of q, and To, for panels with adjacent and nonadjacent k45" layers, decrease with the increase in the number of layers.

3. The relative locations of the +45" and -45" layers do not have much effect on either the global postbuckling response, or the qualitative behavior of the sensitivity derivatives. This effect is particularly true for laminates with 16 or more layers. However, the thickness distributions of the transverse stresses and transverse shear strain energy are very sensitive to the stacking sequence used.

4. The through-the-thickness distribution of the transverse shear stresses and transverse shear strain energy density changes significantly after buckling as the temperature increases.

5. The total strain energy in the postbuckling range is very sensitive to variations in the material parameters aT and ET. It is somewhat sensitive to variations in EL, and is insensitive to variations in GLT and CYL.

Acknowledrrement

The work of the first and third authors was partially supported by NASA Cooperative Agreement NCCW-0011 and by NASA Grant No. NAG-1-1 162. The numerical studies were performed on the CRAY Y-MP computer at NASA Ames.

References

1. Huang, N. N. and Tauchert, T. R., "Postbuckling Response of Antisymmetric Angle-Ply Laminates to Uniform Temperature Loading," Acta Mechanica, Vol. 72, 1988, pp. 173-183.

2. Chen, L. W. and Chen, L. Y., "Thermal Postbuckling Analysis of Laminated Composite Plates by the Finite Element Method," Composite Structures, Vol. 12, 1989, pp. 257-270.

3. Librescu, L. and Souza, M. A., "Postbuckling Behavior of Shear Deformable Flat Panels Under the

Complex Action of Thermal and In-Plane Mechanical L o a d s , " P r o c e e d i n g s of t h e AIAA/ASME/ASCE/AHS/ASC 32nd Structures, Structural Dynamics and Materials Conference, AIAA Paper No. 91-0913,1991.

4. Tauchert, T. R., "Thermally Induced Flexure, Buckling and Vibration of Plates," Applied Mechanics Reviews, Vol. 44, No. 8, August 1991, pp. 347-360.

5. Noor, A. K. and Peters, J. M., "Thermomechanical Buckling of Multilayered Composite Plates." Journal of Engineering Mechanics, ASCE (to appear).

6. Meyers, C. A. and Hyer, M. W., "Thermal Buckling and Postbuckling of Symmetric Composite Plates," Proceedings of the 5th Conference of the American Society for Composites, June 12-14, 1990, East Lansing, MI, Technomic Publishing Co., 1990, pp. 439-448.

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

8. Noor, A. K. and Peters, J. M., "Recent Advances in Reduction Methods for Instability Analysis of Structures," Computers and Structures, Vol. 16, 1-4, 1983, pp. 67-80.

9. Noor, A. K. and Peters, J. M., "Reduced Basis Technique for Calculating Sensitivity Coefficients of Nonlinear Structural Response," AIAA Journal (to appear).

10. 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.

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

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

13. Noor, A. K. and Camin, R. A., "Symmetry Considerations for Anisotropic Shells," Computer Methods in Applied Mechanics and Engineering, Vol. 9, 1976, pp. 317-335.

14. Noor, A. K., Starnes, J. H., Jr. and Waters, W. A., Jr., "Postbuckling Response Simulations of Laminated Anisotropic Panels," Journal of the Aerospace Division, ASCE (to appear).

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

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

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

18. 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.

19. Noor, A. K. and Tenek, L. H., "Stiffness and Thermal Coefficients for Composite Laminates," Journal of Composite Structures (to appear).

APPENDIX 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 ) , ( K ) , ( Y ) are the vectors of extensional, bending and transverse shear stress resultants and strain components of the laminate given by:

(N) '=[NI N~ N ~ ~ ] (A21

(M)'=[MI M2 ~ 1 2 1 ( A 3

(Q) '=[QI 421 (A41

(&}'=[El E2 2~121 (As)

(K}'= [KI K2 2~121 (A@

and ( y ) ' = [ 2 ~ 1 3 2 2 3 1 ( 0

The matrices [A], [B], [Dl and [&I 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 [a (" and [d are the extensional and transverse shear kffnesses of the kth layer (referred to the x,, x2, x3 coordinate system); [I] is the identity matrix; hk and h,-, 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

(k) coefficients of the matrices [a and [GI in terms of the material and geometric properties of the constituents (fiber and matrix) are given in Refs. 15 and 16.

The vectors of thermal effects, (NT] and (MT), are given by:

[ a 6 ) ( a y ) [ 1 x31 m3 (AIO)

where ( a ) is the vector of coefficients of thermal expansion (referred to the coordinates xl, x2 and x3 - see, for example, Refs. 17 and 18).

APPENDIX I1 - FORM OF THE ARRAYS IN THE GOVERNING DISCRETE EQUATIONS

OF THE PANEL

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

The different arrays in equations (I), (3), (4) and (6) can be partitioned as follows:

where m is the linear flexibility matrix; [S1] and [SJ are the linear strain-displacement matrices associated with the free nodal displacements, (X) , and the constrained

(prescribed nonzero) edge displacements,

(M(x, %)) a n d ( k ( ~ , X, %,)I are the subvectors of nonlinear terms; (ET) is the subvector of normalized thermal strains; a 0 refers to a null matrix or vector; and superscript t denotes transposition.

For the purpose of obtaining analytic derivatives with respect to lamination parameters (e.g., fiber orientation angle of different layers), it is convenient to

8~1 8~1- ' express - in terms of - as follows: adl adl

XFI- The explicit forms of - and adl . .

Appendix III.

APPENDIX I11 - EXPLICIT FORMS OF

a[F]-'/a&, (ET) AND a(e~)/adc

It is convenient to partition the matrices CF1-l

and a[F]-'/adt into blocks and to partition the vectors (ET) and a ( ~ ~ ) / a d c into subvectors.

The expression of a typical block (i', j') of

a[F]-'/a& is given by:

A] Dl 0 - j = 1.. NjP & [:Bl [D] 0 ] - ad(

0 0 [&I 0 )

where

Ni* and Ny are the shape functions used in approximating each of the stress resultants; and a(") is the element domain.

The expression of a typical partition i' of the thermal strain vector (ET) is given by:

The expression of a typical partition i' of the

vector - is given by: (a),. where

(C5)

Analytic expressions are given in Ref. 19 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. Anisotro~' Ouasl-lsotro~lc 1C

Panel Laminate Stacking Panel Laminate Stacking No. Seauence No. Seauence Q 1 [A 45/0/901,, A1 [ f 4 5 l ~ ~ 4 2 [ ~ ~ / O / ~ ~ / ~ o ~ , , A2 [ f 45/90JnS 4 3 [ ~ ~ / ~ 0 / ~ ~ / 0 ] , , A3 [ ~ ~ / ~ o / ~ ~ / ~ ~ I n s

A4 [f 45/02Ins A5 [~5/0/-45/0l,s

Note: n=l, 2 or 3 corresponds to the total number of layers NL.=8,16 or 24 respectively.

Table 2 - Effect of aspect ratio and lamination parameters on the number of half waves in the lowest buckling modes of the composite panels listed in Table 1.

Composite Panels

L24l NL Q1 42 43 A1 A2 A3 A4 A5

Notes: 1) For L2/L1 = 1.0, the lowest buckling modes of all laminates have only one half wave in the xl and x2 directions.

2) The three entries for each laminate refer to the three cases To=O (applied edge displacement only), T0=0.5 T, and To=T,, (no applied edge displacements), respectively.

Table 3 - Locations of through-the-thickness distributions of T31 and Ugh shown in Figs. 6.8 and 9. Initially stressed, 16-layer panels with L2/Ll = 1.0 and qe=0.5 q,,,.

Composite Panels Location

0 1 A1 A3 A5 - - - (see Fig. 6) (see Figs. 8 and 9)

a) Initial Postbuckling

-0.469 -0.28 1 -0.03 1

-0.03 1 -0.094 -0.469

b) at wJh = 2.5

-0.219 -0.469 -0.469

-0.469 -0.03 1 -0.28 1

Table 4 - Maximum absolute values of the sensitivity coefficients of the generalized displacements with respect to elastic modulus EL, coefficient of thermal expansion, aT, and the fiber angles 45' and 0'. Sixteen-layer quasi-

isotropic panel Q 1, L2/LI= 1 .O, qe=0.5 qe,=.

Sensitivity a ~ T, ~ $ h ~ = 15.2 aT To ~ $ h ~ = 137.1 Coefficient

a) &=EL

6.88 1 .OO

0.17 0.032

0.17 0.037

b) d f = a ~

7.40 1.13

0.18 0.036

0.18 0.042

c) dt = 45" fiber angle

3.08 0.77

0.077 0.040

0.075 0.045

d) dc = O" fiber angle

0.56 0.35

0.014 0.033

0.014 0.022

Th~ckness of individual layers = 1 . 2 7 ~ 1 0 - ~ m

force = Nt At x2 = + L2/2. W = $ l = o

W

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

I Panel A2 1

I Panel A4 1

L2iL1 = 1 .O L2/L1 = 0.5

a) q, - To interaction curves

Figure 3 - Effect of number of layers and fiber orientation on the stability boundary for anisotropic panels Al , A2 and A4. a) q,-To - interaction curves; b) N,-To - interaction curves.

Figure 2 - Effect of number of layers and stacking sequence on the stability boundary for quasi- isotropic panels Q1 and 42.

1 Panel A4 1 24 r

L21LI = 1 .O L21L1 = 0.5

b) Nt - To ~nteract~on curves

Figure 3 - Concluded

Figure 4 - Effect of fiber orientation of anisotropic panels on the mode shapes associated with the critical edge displacement and critical temperature. Sixteen-layer panels with hJLI=0.8.

Figure 5 - Thermal postbuckling response of initially stressed panels. Initially stressed, 16-layer quasi- isotropic Q1 panels shown in Fig. 1. q,,,Jh = 0.0296, L2/Ll = 1.0.

Figure 6 - Through-the-thickness distributions of transverse shear stresses 731 and transverse shear strain energy density Us,,. Initially stressed, 16- layer quasi-isotropic Q1 panel with L2/Ll=l.0, 9.74.5 qe.w

Figure 7 - Thermal buckling response of initially stressed panels. Sixteen-layer anisotropic panels shown in Fig. 1 and Table 1, L2/L1 = 1.0.

Figure 8 - Through-the-thickness distributions of transverse shear stresses 'C3i and transverse shear strain energy density Ush in the initial postbuckling range. Initially stressed, 16-layer anisotropic panels with L,/L1=l.O, qe=0.5 q ,,,.

Figure 9 - Through-the-thickness distributions of transverse shear stresses 'C31 and transverse shear strain energy density USh at wJh = 2.5. Initially stressed, 16-layer anisotropic panels with L2/Ll=1.0, qe=0.5 q,,,.

Figure 10 - Normalized sensitivity coefficients of the transverse displacement w, and the total strain energy, U, with respect to material characteristics and fiber orientation angles of individual layers. Initially stressed, 16-layer quasi-isotropic Q1 panel shown in Fig. 1 subjected to uniform temperature increase, q, = 0.5 q,,, , L A 1 = 1 .O.

Figure 11 - Normalized sensitivity coefficients of the total strain energy, U, with respect to material characteristics and fiber orientation angles of individual layers. Initially stressed. 16-layer anisotropic panels shown in Fig. 1 subjected to uniform temperature increase, q, = 0.5 q,,,, , L2/L1 = 1.0.

ax ax %/ladl l max

..--.. rfl > \ / I \ \ , I , \ ( ( ( 1 , I 1 )

I , I t / / / /

/ , , , ,, I ! , / I / , I ,

' I , -- ,* , - - -, __- - -

2 (a) ~ T T , - = 15.2 2

Figure 12 - Normalized contour plots for the sensitivity coefficients with respect to EL, aT, and fiber angles

0 = 45" and 0", at a~ To ~ ~ 2 / h ~ = 15.2 and 137.1, 16-layer quasi-isotropic panel Q1, L2/LI=l.0, q,=0.5 q,,,. Spacing of contour lines is 0.2, and dashed lines denote negative contours. Locations of maximum absolute values are identified with X, and normalizing factors are listed in Table 4.

Copyright @ 1992 by the American Institute of Aeronautics and Astronautics, Inc. No copyright is asserted in the

United States under Title 17, U.S. Code. The U.S. Govern- ment has a royalty-free license to exercise all rights under the copyright claimed herein for Governmental pdrposes.

All other rights are reserved by the copyright owner. 1068