Application of an implicit and explicit integration rules

This paper concerns the application of the generalized trapezoidal implicit integration rule and the automatic substepping explicit integration rule. By varying the value of α, the generalized trapezoidal rule (e.g. Δε^p = λ[(1 – α)R_n + αR_n+1]) represents a few commonly used integration schemes. It is applied to an advanced bounding surface type of sand model. Its performance including convergence and integration accuracy, with various values of α, is investigated by using both consistent and continuum tangent stiffness operators. The automatic substepping explicit integration rule is characterized with robustness, and its robustness is demonstrated in a finite element analysis of footing behavior by using extraordinarily large loading increments. Its application to the yield vertex non-coaxial model is also discussed.