Convex Models of Uncertainty in Applied Mechanics

Recognition of the need to introduce the ideas of uncertainty in a wide variety of scientific fields today reflects in part some of the profound changes in science and engineering over the last decades. Nobody questions the ever-present need for a solid foundation in applied mechanics. Neither does anyone question nowadays the fundamental necessity to recognize that uncertainty exists, to learn to evaluate it rationally, and to incorporate it into design.This volume provides a timely and stimulating overview of the analysis of uncertainty in applied mechanics. It is not just one more rendition of the traditional treatment of the subject, nor is it intended to supplement existing structural engineering books. Its aim is to fill a gap in the existing professional literature by concentrating on the non-probabilistic model of uncertainty. It provides an alternative avenue for the analysis of uncertainty when only a limited amount of information is available. The first chapter briefly reviews probabilistic methods and discusses the sensitivity of the probability of failure to uncertain knowledge of the system. Chapter two discusses the mathematical background of convex modelling. In the remainder of the book, convex modelling is applied to various linear and nonlinear problems. Uncertain phenomena are represented throughout the book by convex sets, and this approach is referred to as convex modelling.This book is intended to inspire researchers in their goal towards further growth and development in this field.

Table of Contents:
1. Probabilistic Modelling: Pros and Cons. Preliminary considerations. Probabilistic modelling in mechanics. Reliability of structures. Sensitivity of failure probability. Some quotations on the limitations of probabilistic methods. 2. Mathematics of Convexity. Convexity and Uncertainty. What is convexity? Geometric convexity in the Euclidean plane. Algebraic convexity in Euclidean space. Convexity in function spaces. Set-convexity and function-convexity. The structure of convex sets. Extreme points and convex hulls. Extrema of linear functions on convex sets. Hyperplane separation of convex sets. Convex models. 3. Uncertain Excitations. Introductory examples. The massless damped spring. Excitation sets. Maximum responses. Measurement optimization. Vehicle vibration. Introduction. The vehicle model. Uniformly bounded substrate profiles. Extremal responses on uniformly bounded substrates. Duration of acceleration excursions on uniformly bounded substrates. Substrate profiles with bounded slopes. Isochronous obstacles. Solution of the Euler-Lagrange equations. Seismic excitation. Vibration measurements. Introduction. Damped vibrations: full measurement. Example: 2-dimensional measurement. Damped vibrations: partial measurement. Transient vibrational acceleration. 4. Geometric Imperfections. Dynamics of thin bars. Introduction. Analytical formulation. Maximum deflection. Duration above a threshold. Maximum integral displacements. Impact loading of thin shells. Introduction. Basic equations. Extremal displacement. Numerical example. Buckling of thin shells. Introduction. Bounded Fourier coefficients: first-order analysis. Bounded Fourier coefficients: second-order analysis. Uniform bounds on imperfections. Envelope bounds on imperfections. Estimates of the knockdown factor. First and second-order analyses. 5. Concluding Remarks. Bibliography. Index.