Approximate solution of the finite cylinder problem using Legendre polynomials

Abstract

Governing equations of elasticity for small deformations of a hollow cylinder of finite length are examined to define suitable approximate theories. Each dependent variable in the problem is represented as a series expansion in Legendre polynomials in the radial coordinate; such representations permit the field equations to be exactly reduced to two-dimensional sets of equations by separation of variables. Attention is directed toward establishment of a logical approach to truncation of the series; important variables for approximate theories of any order are established by consideration of the strain energy expression, and a truncation scheme is postulated. To establish the validity of the truncation scheme and hence of the resulting approximate theories, the problem of an infinitely long thick cylinder under axisymmetric band loading is considered. Comparisons with exact solutions and with well-known shell theories are obtained; results indicate that the new approximate theories yield predictions in better agreement with exact solutions than previous efforts in this area.