Keywords: seismic, bidirectional ground motion, random vibration theory, structures, algebra of distributions, stochastic method.

Determining the probable seismic input motion at a location is important for the design of new buildings, determining the withstand capability of existing buildings either by analysis or testing and when researching buildings damaged by previous events. An important consideration in such evaluations is the determination of the magnitude of the resultant of the combination of the bidirectional horizontal seismic input motions. This aspect has recently attracted renewed attention in the USA as a result of a change in the National Earthquake Hazards Reduction Program 2009 provisions and has resulted in a number of papers on the topic.

For most cases it is the seismic shear waves that are significant and when they reach the surface they are measured by modern recording instruments as acting along orthogonal tri-axis. The tri-axis of the instrument is positioned so that one axis is vertical and the other two axes are in the horizontal plane. In practice it is unlikely that the horizontal measuring axes will align with the principal axes from the event and will instead be formed from components from each principal input direction. As a result, the maximum ground accelerations could be underestimated if based on either the maxima of the measuring axes or the geometric mean.

This paper proposes a new approach based to random vibration theory and the algebra of distributions to determine the probable magnitude of the resultant horizontal seismic input motion. The method depends on the key parameters of the moment magnitude, the hypocentral distance and the stress drop. They provide sufficient information to derive the frequency content, the spectral ground accelerations and the duration at the hypocentral distance. From these parameters the resultant motion can be determined for a particular probability of exceedance, thereby providing a measure of the uncertainty. The approach is compatible with stochastic based source to site models which are becoming the method of choice for defining modern ground motion prediction equations. The effects of the resultant of the bidirectional seismic inputs are considered in general and then in the context of structures with different plan profiles and their vulnerability to the resultant seismic input are determined.

The results show that for a typical peak factor of 3.0 that if both horizontal components have the same magnitude, then basing input components on the geometric mean can result in an underestimate, so to can the maximum direction approach but to a lesser extent. However if spatial SRSS is applied to components each based on the geometric mean, even when assumed as low as 0.8 of the true maximum, then the result is close to that from the probabilistic approach in this paper.

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