Response stability and catastrophoes of stochastically parametered structural systems

Abstract

Vibrational and stability behaviour of structural systems having stochastically distributed material properties and subjected to stochastic loadings in space are the concerns of this thesis. The Young’s modulus and mass density are treated as stochastic. A general loading pattern, spatially distributed with a stochastic description and of conservative or nonconservative nature, is considered. Analytical solutions as well as solutions by a general stochastic finite element method are developed. The fluctuations of material property values about their mean values are considered to constitute independent stochastic fields in space. Each of the stochastic fields is a onedimensional, univariate, homogeneous, real stochastic field in space. The statistical description of these fields is given in terms of means, variances, and scale of fluctuations. The firstorder perturbation analysis is employed. The approach is to split the problem into two: the averaged problem and the perturbed problem. The stochastic perturbations are applied to the averaged problem so that each characteristic of the structural system can be expanded about the characteristic of the averaged problem. This allows direct application of the results to the catastrophic classifications of structural systems. The need for analysing problems of mechanics in a probabilistic setting is stressed in the introductory chapter. Some practical examples of material property uncertainties are also illustrated. It can be seen that the magnitudes of these practical fluctuations are well within the limits of firstorder perturbation analysis. Chapter 2 deals with the bifurcation and snapthrough instability of stochastic systems. A brief review of bifurcation instability is presented. Snapthrough instability of a shallow stochastic truss is considered. Since this is a snapthrough instability problem, bounds for the variance of snapthrough load are derived. Chapter 3 is concerned with the dynamical behaviour of systems in general and the dynamic method of stability investigation. Vibration analysis is performed for nonconservatively loaded structures and the statistics of eigensolutions are derived. Concepts from the frequency domain are applied. Approaches are suggested for the kinetic method of stability analysis. The eigensolution statistics are transformed to yield critical load statistics. The effects of mass stochasticity in divergent systems are derived and discussed. Chapter 4 gives the numerical method to analyse the vibration and stability behaviour of stochastic structural systems. A new version of the stochastic finite element method is developed which overcomes the limitations of existing approaches. Both variational and weighted residual methods are used. As a generalisation, random eigenvalue problems are solved. Expressions for the stochastic description of eigensolutions are derived. Solutions to the free vibration of beams, beamcolumns, and the buckling of columns are obtained by specialising the general solution. Numerical examples are illustrated. The superiority of the present method over existing ones is demonstrated. The thesis concludes with a summary of the investigations made and discusses the application of the present work to catastrophe theory. Possible areas for further investigation are identified.