A time-stepping method of seismic response analysis for structures using differential quadrature rule

A numerical time-stepping method is developed for thesolution of equations governing the motion of a MDOF system subjected toearthquake induced ground acceleration on the basis of the differentialquadrature (DQ) rule. Initially the history of the seismic ground motion andthe structural dynamic responses are divided into a sequence of timeintervals, and the DQ rule is used within each discrete time step, thus thefinal response of velocity and acceleration during the interval may beexpressed as a weighted linear sum of the initial condition and the responseof displacement at the sampling grid points. It is assumed that the groundacceleration over the time step increases linearly with time, and theformulae for the quadrature solution of the differential equations of motionare deduced so that the history of the dynamic response could be calculatedin the time-stepping procedure using the DQ method. A two-story shear framestructure is employed for numerical illustration, and results from numericalanalysis show that the proposed quadrature procedure can still achieve anexcellent analysis for the time history of the seismic response even whenthe length of the time step is selected to be a little larger.