Principal/Secondary Upper Bound and Equivalent Static Solution in Seismic Design

Abstract

The second author has recently proposed an innovative method to calculate upper bounds on the seismic response of structures with stiffness uncertainty. It has been shown that there exist two upper bounds on the seismic response, the principal upper bound (PUB) and the secondary upper bound (SUB), and two associated modes, the principal mode (PM) and the secondary mode (SM), which correspond to the two most unfavorable stiffness distributions. The PM and SM have similar shapes to the conventional first and second modes yet are not identical to them. Although a complete but non-unique set of modes can be generalized by the orthogonality condition from the PM and SM, the higher order modes (third, fourth, and so on) have zero modal participation factors and thus do not participate in modal response. The seismic design of a complicated multi-degree-of-freedom system can therefore be made on the basis of a simplified two-degree-of-freedom system. Through several examples in this paper, it is further shown that the difference between the PUB and SUB is just the equivalent static solution (ESS) if the acceleration response spectrum is taken as the input ground motion. A formula for calculating the PUB and SUB of the seismic bending moment for high-rise buildings is provided.