Environmental contour analysis in earthquake engineering

In many engineering problems the uncertainty in the design and analysis process is dominated by uncertainty concerning the environmentally induced loading on the system. One such type of system is a structure located in high seismic zones, where there is a high level of uncertainty associated with the ground acceleration. In this study the demand caused by the environmental loads is statistically characterized in terms of magnitude, site-to-source distance and attenuation error at a specified structural period. Through the use of a reliability-based procedure known as the Inverse-First Order Reliability Method all of the combinations of the random loading variables that produce a response spectrum with the specified return period may be identified. These infinite number of combinations produce an Environmental Contour that may be derived for the two, three, or four-dimensional case. Because these contours represent an infinite number of combinations of environmental loading random variables, and in turn, a family of response spectra with the same return period, one need only search the Environmental Contour for the response spectrum producing the peak spectral acceleration at the structural period of interest. This study presents a general multi-variate framework focusing on the derivation of these two, three, and four-dimensional Environmental Contours. Initially, the magnitude and site-to-source distance are assumed to be statistically independent. Two and three-dimensional Environmental Contours are derived for this case and critical response spectra for three different, commonly used return periods are examined. Then a hypothetical example in which the site-to-source distance is assumed LogNormal and dependent on magnitude, is used as a basis for discussion for a generic California site. A qualitative discussion between the assumption of independence and dependence between magnitude and site-to-source distance is addressed. Finally, the peak spectral accelerations for the dependent two, three, and four-dimensional cases are compared to one another and the possible consequences associated with increasing the complexity of the environmental contour model analyzed and discussed.