Justification for a homogeneous model of a laminated composite in problems of optimization of thin-walled shells functioning for stability

Abstract

The question of the limiting number of layer elements in a stack -- N* -- is of practical importance to the engineering design of laminated structures functioning for stability (the number of layers N > i), since for N ~ N* it is possible to treat a multilayer stack as macrohomogeneous, which fundamentally simplifies the formulation and solution of the corresponding optimization problem of a multilayer structure. It is technically exceedingly complicated to solve this problem in general form (for laminated structures of arbitrary geometry which are composed of anisotropic layers having different physicomechanical properties and designed for operation under different loading conditions) due to the known polyvariant nature of the design situations. However, qualitative estimates can be obtained by the particular example of solving the indicated problem for a laminated cylindrical shell composed of orthotropic layers made out of the same material and functioning for stability in the following simplest loading conditions: static axial compression, external pressure, and natural radial oscillations. This paper is devoted to this task. i. Let us consider a multilayer thin cylindrical shell of constant total thickness h with a radius R of the medial surface and length I. In the general case the elastic and orthotropic layer elements of the shell are assumed to be macrohomogeneous with a constant thickness h/N. Upon deformation, the layers remain elastic and function together without slipping relative to each other (rigid contact). The principal orthotropy axes of the layers form the angles +~ with the x and y axes of the (x, y, z)-coordinate system associated with the shell. The equilibrium equations of the shell referred to the imdeformed state for the loading conditions under consideration and are of the form [i]