## Structural Response by Step-by-Step Integration Methods

The search for ways to represent the true nonlinearity of structures goes back to Renaissance times, and present theories are the result of approximately two hundred years of steady development.

W. McGuire, 1994

The response spectrum method presented in the foregoing chapter is an effective and convenient means of evaluating the seismic response of structures. Still, the method suffers from some shortcomings that make it diffi cult, if not impossible, to obtain the exact response of a system. Among these shortcomings are (a) the need to make the assumption that the system’s mode shapes are also orthogonal with respect to the system’s damping matrix and (b) having to use an approximate modal combination rule to estimate the system’s maximum response. More importantly, its application is limited to the analysis of linear elastic systems. Fortunately, alternative methods are available that offer the possibility of evaluating a system’s seismic response without any of such shortcomings. These alternative methods are numerical procedures that involve (a) dividing the total solution time into a series of small time intervals or steps (see Figure 11.1), and (b) integrating the pertinent equations of motion in a sequential manner using the initial and loading conditions at the beginning of each time interval to determine the response of the system at the end of the interval. That is, they involve marching step-by-step along the time axis to integrate the equations of motion and to determine the response of the system at a sequence of specifi c times. The advantage they offer is that by working with small time intervals it is possible to make suitable assumptions in regard to the variation or characteristics of the response within each of such time intervals without introducing signifi cant errors. Another advantage is that any desired degree of accuracy may be attained by simply reducing the size of the time interval. Furthermore, nonlinear behavior (see Section 6.5.2) may be incorporated into the analysis by merely assuming that the structural properties remain constant during each time interval and reformulating them from one time interval to the next in accordance to an established load-deformation relationship and the solution obtained at the end of the previous time interval.