| The time-history analysis of realistic structures can often be computationally quite expensive. The main reason for the high volume of computation is the fact that the time interval must usually be very small. The factors that limit the time interval fall into three categories: (1) The input may involve very rapid variations, as in some earthquake acceleration records or in pressures for blast loading. (2) Stiffness of members can undergo rapid or discontinuous changes because of plastic behavior or unloading. (3) Models may have to include many degrees of freedom. The higher modes that are present may have very short periods, which require very short time intervals for stability and convergence of the calculation and for accuracy.;A coordinated set of methods is developed to handle each of these limitations, finally permitting the use of fairly large time intervals while preserving high accuracy.;The first limitation is overcome by introducing integrated displacements as new variables in integrated equations of motion and also by using a more accurate integration method for ordinary differential equations than is usual. These techniques together with a simple interpolation procedure make it possible to pass right through points of yield and unloading without restarting time intervals at those points. The ordinary calculations require such restarting or very small time intervals to preserve accuracy over large time. In this way, the second limitation is removed.;The third limitation is handled by treating both relaxed constraints and small masses by what is essentially an extension of the idea of static condensation. The higher modes are really not considered dynamically, but the additional degrees of freedom do properly modify the lower modes and their frequencies. The calculation is carried out without finding modes and frequencies explicitly. An unusual but simple perturbation method is used in which all the perturbations that can be obtained accurately are found by direct numerical integration of the perturbation equations. Not all of the degrees of freedom enter the perturbation equations as dynamical variables; the others are eliminated algebraically. |
