Approximate Solutions for Unsymmetrically Laminated Plates

Abstract

In a current article, Whitney [1] has presented exact solutions for certain problems of the bending of unsymmetrically laminated plates. This note considers an approximate method of solution for such plates which, although approximate, yields solutions that compare favorably with those presented by Whitney. Furthermore, since the approximate method reduces the problem to an equivalent anisotropic bending problem, the techniques that have been developed for the analysis of orthotropic and anisotropic plates can be directly utilized. It should be noted that the possibility of using the approximate analysis discussed herein was first suggested in [2] on the basis of experimental evidence. The present note presents an explanation of why this approximation does not introduce large errors, and presents numerical comparisons with Whitney's exact solutions. BASIS OF THE APPROXIMATE ANALYSISThe total potential energy of a general laminated plate can be written in terms of the geometric midplane strains ( e ) and plate curvatures ( k ) as follows:where the elements of the matrices are defined such that the stress resultants [N] and moments [M] are given as and a superscript T denotes a transpose. The potential energy expression (1) exhibits coupling between midplane strains and plate curvatures through the coupling matrix [B]. If the constitutive Equation (2) is solved to give [e] in terms of [N] and [k], and then this expression is substituted into (1), an expression for the potential energy in terms of the stress resultants and plate curvatures can be obtained:

Postbuckling Formulation for Symmetric Laminated Panels with Various Edge Support Conditions

“…These results are compared with established data of isotropic plates of Rhodes [15], and composite plates of Chai and Khong [19], Chai [22], and Chai et al [23], to verify its Tables 2-4 compare the present results for the respective CCCC, CCSS, and CSCS cases with those obtained by Chai [22], who used a 144-term trigonometric series function as de ection function to solve the buckling solution, so his results can be deemed "accurate. "…”

“…Also included in the comparison are the results obtained by Chai and Khong [19], whose nite-strip method, FSM, for buckling has been extensively validated in their article. From Tables 2 and 3 it can be seen that the present results agree very well with the established Table 2 Effect of aspect ratio a b on the buckling coef cient k for the case CCCC Present Chai [22] % diff.…”

The Analysis of Unbalanced Cross-Plied Elliptic Plates Under Uniform Pressure

“…The &dquo;approximate&dquo; solutions are obtained by solving Equations (20) with m12 = m21 = 0. This solution procedure has been presented previously by Reissner and Stavsky [5] and Ashton [6]. The bending-membrane coupling is ignored in the mathematical coupling of the equilibrium and compatibility equation.…”

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“…Therefore, a reduced bending stiffness method is used here as an approximation method (Ashton 1969). The method reduces the problem to an equivalent anisotropic bending problem without bending-extensional coupling.…”