Accurate representation of external force in time history analysis

It is usually thought that the integration time step should be small enough to represent properly the variation of the dynamic loading with respect to time. However, there is no evaluation criterion that can be used to determine whether the external force is accurately represented. In this paper, criteria for accurate representation of external force are proposed based on analytical results. It is found that amplitude distortion both in the transient response and the steady-state response for each time step is closely related to the step discretization error of external force. In fact, for a negligible period distortion, an amplitude distortion will be less than 5% if the relative step discretization error is constrained to be less than 5% at each time step for the Newmark explicit method, Fox-Goodwin method, and linear acceleration method while for the constant average acceleration method it must be less than 2.5%. This criterion leads to the need of using of eight or more integration time steps to accurately represent a complete cycle of a harmonic loading for the Newmark explicit method, Fox-Goodwin method, and linear acceleration method while for the constant average acceleration method 12 or more integration time steps are required.