Weighted Integral SFEM Including Higher Order Terms

The inclusion of the higher order terms in the Taylorʹs series expansion of the status variable is presented in the formulation for stochastic analysis with the weighted integral method. Generally, in almost all the numerical formulations, omission of the higher order terms is introduced due partly to the complexities of deriving the appropriate simple equations for stochastic analysis or due to the large amount of additional computation time and memory requirement. In this study, the Lagrangian remainder is included in the expansion of the status variable with respect to the mean value of the random variables, which results in simple and efficient formulas for stochastic analysis in the weighted integral method. In the resulting equation, only the "proportionality coefficients" are introduced; thus, no additional computation time or memory requirement is needed. Various examples are investigated to show the efficiency and appropriateness of the suggested formula. The results obtained by the improved weighted integral method equations proposed in this study are reasonable and are in good agreement with those of the Monte Carlo simulation.