Approximate elasticity solution for laminated anisotropic finite cylinders

Abstract

This paper deals with the static response of the axisymmetric problem of arbitrarily laminated, anisotropic cylindrical shells of finite length using three-dimensional elasticity equations. The closed cylinder is simply supported at both ends. The highly coupled partial differential equations (PDEs) are reduced to ordinary differential equations (ODEs) with variable coefficients by choosing the solution composed of trigonometric functions along the axial direction. Through dividing each layer into thin laminas, the variable coefficients in ODEs become constants, and the resulting equations can be solved exactly. Numerical examples are presented for ( - 45/0 deg) and ( - 45/457 - 45 deg) laminations under sinusoidal normal loading on the outer surface and uniform internal pressure. From the present study, it is found that, although the general behavior is similar to that of isotropic shells, the coupling is obvious in general, and the shear effect is very important in the edge region. Moreover, the initial curvature effect plays an essential role, especially in stress distributions. TATIC and dynamic responses of composite laminated closed cylindrical shells and curved panels have received wide attention in recent years. Because of the anisotropy in composites and the presence of curvature in shell structures, obtaining exact three-dimensional elasticity solutions for lami- nated closed cylinders and open panels subjected to general loading and arbitrary boundary conditions becomes a chal- lenging task. The mathematical complexity in analyzing three- dimensional elasticity equations usually makes exact solutions difficult to obtain. However, certain problems in which a three-dimensional approach can be used still exist. Most of these problems can be solved by assuming the solution to be composed of trigonometric functions in the axial and circumferential directions. The main reason is that the partial differential equations (PDEs) governing three-di- mensional problems can be reduced to one-dimension al ordi- nary differential equations (ODEs) with variable coefficients. The solution for the resulting ODEs can be obtained by intro- ducing the displacement potential function. Usually, this method is used with isotropic and transversely isotropic mate- rials, whereas the Frobenius method is used with orthotropic materials. When the three-dimensional elasticity solutions are available, they are very useful in evaluating the accuracy of approximate results, e.g., in the case of two-dimensional shell theories. For the static problem, Flugge and Kelkar1 and Yao2 ob- tained an exact solution for closed isotropic long cylinders under general two-dimensional surface traction. Using the Frobenius method, Srinivas3 developed an exact three-dimen- sional solution for orthotropic finite cylinders with simply supported conditions. However, the numerical results are given for free vibration only. Varadan and Bhaskar4 also performed the static stress analysis using the procedures pro- posed by Srinivas. Pagano5 obtained the stress field for a homogeneous, anisotropic closed cylinder under two-dimen- sional surface loads in which the problems are independent of the axial coordinate. However, numerical results are reported only for a single orthotropic layer. Recently, Ren6'7 presented an exact solution for simply supported laminated cross-ply circular cylindrical panels of infinite and finite length in the