Nonlinear, buckling and postbuckling analysis of shells of revolution

Abstract

This thesis presents a series of developments leading to an efficient and accurate finite element analysis for nonlinear, buckling and postbuckling behaviour of thin elastic shells of revolution. An accurate isoparametric doubly-curved axisymmetric shell element is employed in these developments, with circumferential variations of loads, deformations and internal forces all represented by Fourier series. The work represents an important step towards the eventual development of a so-called 'advanced analysis' for stability design of steel shell structures in which imperfections, residual stresses and material plasticity are all realistically modelled. A review of the existing literature reveals that while there have been many studies on the finite element analysis of shells of revolution, only a limited number of analyses have been developed for nonlinear shells of revolution under arbitrary loads. A number of deficiencies are identified in existing studies. Among these are the uncertainty concerning the accuracy of existing nonlinear shell theories, the inability to trace the descending part of the load-deflection path of unsymmetrically loaded shells when the pseudo-load concept is used, and the lack of efficient and powerful techniques to predict postbuckling responses with mode switching. The present developments overcome these existing deficiencies and result in a computer analysis which is both accurate and efficient. A study on suitable nonlinear strain-displacement relations for numerical analysis of complex branched shells is first described. Such relations are a basic building block in any numerical nonlinear, buckling and postbuckling analysis. A number of nonlinear strain-displacement relations have been developed in the past for thin shells. Most of these theories were developed in the pre-computer era for analytical studies when simplicity was emphasized more than accuracy. With the availability of greatly increased computing power in recent years, accuracy rather than simplicity is given more emphasis. A new set of nonlinear strain-displacement relations for thin shells of general form developed directly from the nonlinear elastic theory of three-dimensional solids is thus presented in this thesis. In this new theory, all nonlinear terms, large and small, are retained. This new theory is found to reduce to that of Rotter and Jumikis and others for shells of revolution. This new theory is compared with other nonlinear shell theories both analytically and numerically. Shells and plates on elastic foundations are found in many practical applications. A large deflection analysis of axisymmetric shells and plates on a nonlinear tensionless elastic foundation is next presented in this thesis. Through the use of discrete data points, any form of nonlinear elastic foundation behaviour can be easily modelled. The analysis is applied to investigate the behaviour of shallow spherical shells subjected to a central concentrated load on tensionless linear elastic foundations. A number of insightful conclusions regarding the behaviour of such structure-foundation systems are drawn. The numerical results for shells are believed to be the first correct results which may be useful in benchmarking results from other sources in the future. A new finite element formulation is then presented for the nonlinear and collapse (limit point buckling) analysis of elastic doubly-curved segmented and branched shells of revolution subject to arbitrary loads. The circumferential variations of all quantities are described by truncated Fourier series with an appropriate number of harmonic terms. Coupling between different harmonics is dealt with directly rather than by the use of pseudo-loads. This coupled harmonics approach allows an easy implementation of the arc-length method. As a result, the post-buckling load-deflection path can be traced efficiently and accurately. The results from the present study are independently verified using ABAQUS, while those from other studies are found to be inaccurate in general. Post-bifurcation buckling analysis of shells of revolution subject to axisymmetric loads is a key to understanding their postbuckling behaviour and imperfection sensitivity. Two efficient and accurate methods are thus developed to trace the postbuckling equilibrium path: the direct method and the load-disturbance method. In the direct method, an automated solution procedure is employed to determine the bifurcation buckling load and the bifurcation buckling mode. Then, a displacement increment parallel to this bifurcation buckling mode is introduced. In the load-disturbance method, small non-symmetric loads in appropriate harmonic modes are added so that the post-bifurcation buckling analysis is converted into a nonlinear non-symmetric analysis. The advantage of the direct method is that the bifurcation load and the postbuckling path in its vicinity can be determined precisely. On the other hand, the load-disturbance method is able to follow mode switching and interaction which cannot be handled by the direct method as presented here. Numerical results obtained here lead to some important conclusions concerning postbuckling analysis and behaviour. Finally, the finite element formulation for perfect shells under arbitrary loading mentioned earlier is extended to shells with non-symmetric imperfections. The initial imperfections are also arbitrary and expanded into truncated Fourier series. As a special case of this analysis, the imperfection-disturbance method is presented for tracing complex postbuckling paths involving mode switching and interaction. In addition, the imperfection sensitivity of a number of problems is examined using the developed analysis.