Rotation-invariant measures of earthquake response spectra
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Rotation-invariant measures of earthquake response spectra R. Rupakhety · R. Sigbjörnsson
Received: 17 April 2013 / Accepted: 22 June 2013 © Springer Science+Business Media Dordrecht 2013
Abstract A new procedure for combining the response spectra of two horizontal components of recorded ground motion is presented. The presented formulation accounts for different orientations of accelerometer sensors and derives the maximum and the expected (mean) horizontal response spectra at a site, both of which are invariant to rotation of sensor axes. The maximum response spectrum is derived as the peak resultant response of single degree of freedom oscillators subjected to the as-recorded ground acceleration. The expected spectrum is derived by projecting the displacement response (due to as-recorded motion) along two orthogonal axes to a principal axes in which the displacement responses are uncorrelated. This property is used to formulate an approximation for the expected response spectrum over all possible sensor orientations. A large set of accelerometric data from Europe and the Middle East is used to demonstrate the applicability of the proposed response spectral measures. Keywords analysis
Earthquake response spectra · Rotation invariance · Principal component
1 Introduction Earthquake response spectra play an important role in seismic hazard assessment, and in seismic design of structures. Ground motion prediction equations are commonly used to describe response spectra in seismic hazard assessment. The most commonly used prediction equations are derived for response spectra corresponding to horizontal shaking of ground inferred from accelerometric records on orthogonal directions. Accelerometric data provide
R. Rupakhety · R. Sigbjörnsson (B) Earthquake Engineering Research Centre, University of Iceland, 800 Selfoss, Iceland e-mail: [email protected] R. Sigbjörnsson Department of Structural Engineering, Norwegian University of Science and Technology, 7491 Trondheim, Norway
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Bull Earthquake Eng
ground shaking on three orthogonal directions—two horizontal and a vertical. Response spectra corresponding to the horizontal components are combined in various ways to establish a single response measure (see Douglas 2003, and references therein). Currently, the most common approach is to use the geometric mean of the response spectra of two as-recorded horizontal components. This geometric mean measure has been advocated on the basis of the observation that it results in the smallest residual error in empirically calibrated prediction equations (Beyer and Bommer 2006). Apart from this purely statistical argument, there is apparently no physical reason why the geometric mean of the two as-recorded horizontal components could be a good representation of possible motion at the site. Furthermore, this measure of ground motion is clearly dependent on the orientation of the recording sensor, meaning that it represents only a sample from a range of possible values for different orientations of the sensor. There
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