Equivalent Linearization Method IN NONLINEAR FEM

Abstract

A finite element formulation for a class of nonlinear viscoelastic continua subjected to stochastic excitation is developed by means of a stochastic equivalent linearization technique. The nonlinear viscoelastic properties of the continua are modeled in terms of the constitutive equation which linearly involves the hysteretic tensor and in terms of the auxiliary equation which describes a nonlinear relationship among the strain and hysteretic tensors and their time derivatives. This auxiliary equation is linearized with the aid of a stochastic linearization technique which minimizes the expected value of the square of the difference between the nonlinear and linearized auxiliary equations. The linearized boundary value problem then involves a set of two linearization coefficients which are functions of space and time. With the aid of the weighted residuals method, the boundary value problem developed for the continuum is transformed into that in the finite element formulation. In integrating the equation of motion representing the discrete model, an iterative upgrading of the values of these linearization coefficients is performed simultaneously for all the finite elements in the first time interval until a convergence criterion is satisfied. The iterative process is then repeated in the time intervals that follow, until the entire time interval for which the dynamic analysis is performed is covered. The proposed finite element analysis produces a covariance function and hence a variance function of the nodal displacement and velocity, and of the element strain and stress components, among other response quantities. The variance function developed on the basis of the present analysis and those constructed by means of the Monte Carlo technique show a reasonable agreement.