Computational concepts in structural stability

Abstract

The present essay contains a general structural stability theory for discretized structural systems. Instabilities are essential constituents of nonlinear structural responses, the computational assessment of which in a modern treatment is exclusively based on incremental-iterative (step-wise) numerical techniques, applied to the tangential equation of motion. The paper derives this fundamental equation as first variation of the nonlinear equation of motion in its standard form and its phase projection. Further, it transforms the principle of virtual work for arbitrary nonlinear (Kelvin-Voigt) continua into its incremental variant and finally into the consistent tangential equation of motion. Its various applications then are demonstrated to classes of time-independent and time-dependent, unstable structural responses, for which suitable numerical instruments are outlined. The derived algorithms are based on the concepts of Lyapunow exponents and Poincare multipliers which are introduced as universal stability measures. Qualitative and quantitative convergence properties of perturbations in the phase space enable the proper establishment of stability definitions. The validity of the received concepts is illustrated by several examples.