Envelopes for Seismic Response Vectors. I: Theory

Abstract

In the design or analysis of structures for seismic loads, the effects of forces acting simultaneously in a member must be considered. The most common example is in the interaction of bending moments and axial load in columns. The usual response spectrum method provides the maximum values of individual responses, but the critical combination of these responses may not involve any of these maxima. In this paper, a response-spectrum-based procedure for predicting the envelope that bounds two or more responses in a linear structure is developed. It is shown that, for an assumed orientation of the principal axes along which the ground motion components are uncorrelated, this envelope is an ellipsoid. For the case when the orientation of the principal axes is unknown, a “supreme” envelope is derived, which corresponds to the most critical orientation of the axes. The response envelope can be superimposed on a capacity curve to determine the adequacy of a given design. In a companion paper, the results of a numerical study are presented to illustrate the accuracy and significance of this method.