Stochastic Analysis of Nonlinear Systems

The present chapter is devoted to the probabilistic analysis of the random response of nonlinear structural systems exposed to random excitation with special attention to earthquake action. The random system response may be due to random excitation, to random system properties, to random boundary conditions. The nonlinear character of the response is mainly due to the nonlinear materials properties and to the effect of large displacements (the so-called P-D effect). The first pioneering studies on this topic took place in the 1960s and 1970s (VanMarcke et al. 1970; Iwan 1973; Atalik and Utku 1976; Spanos 1976), when equivalent linearization techniques were used, taking advantage of the availability of first digital computers. The attention is limited herein to systems having deterministic properties, including deterministic boundary conditions, considering those types of nonlinearity that can really occur during earthquakes. After a brief review of the types of nonlinear behavior that can be expected, the available methods for the analysis of the response are discussed. They can be classified into analytical and simulation methods. Among the first ones are considered in the following: the Fokker-Planck-Kolmogorov (FPK) equation, the equivalent linearization, the perturbation method, and the stochastic averaging. The simulation methods based on Monte Carlo techniques are then presented.