Damping‐dependent correlations between response spectral ordinates

Abstract

Correlations between response spectral ordinates at different periods are used in several seismic hazard computations, such as for the construction of conditional mean spectra and conditional spectra. Conventionally, these correlations have been computed and reported only for a damping ratio of 5%; however, structures may have damping ratios substantially lower or higher than 5%. Therefore, in those cases, one requires correlations of spectral ordinates at different periods but having the same damping ratios that are different than 5%, correlations of spectral ordinates at the same period but having different damping ratios, or the general case of correlation between spectral ordinates of two oscillators having different damping ratios and different periods. This work computes such damping-dependent correlations by using the NGA-West2 ground motion database. In general, it is found that correlations increase as the damping ratio of any of the two spectral ordinates increases and as the ratio of periods of vibration of the two oscillators departs from one. A nonlinear regression model is fitted to the resulting damping-dependent correlations to simplify future computations. Finally, the use of the new damping-dependent correlations is illustrated by computing example conditional spectra for damping ratios differing from 5%. The results show that using 5%-damped correlations for the construction of condition mean spectra, overestimates spectral ordinates for damping ratios lower than 5% and underestimates spectral ordinates for damping ratios higher than 5%.

Damping‐dependent correlations between horizontal‐horizontal, horizontal‐vertical, and vertical‐vertical pairs of spectral accelerations

“…The calculated σ ln SA ( T , ξ ) needs to be partitioned for inter‐ and intra‐event residual analysis. Since the inter‐ and intra‐event standard deviations for ξ ≠ 5% are not directly available from the GMPEs and DSFs, it is assumed that the standard deviations of the inter‐event, intra‐event, and total residuals have the following proportionality relationship for different damping ratios, such that τ ln SA ( T , ξ ) and ϕ ln SA ( T , ξ ) can be calculated via 3 : $$$$\begin{equation}\frac{{{\tau }_{\ln SA}\left( {T,\xi } \right)}}{{{\tau }_{\ln SA}\left( {T,5\% } \right)}} = \frac{{{\phi }_{\ln SA}\left( {T,\xi } \right)}}{{{\phi }_{\ln SA}\left( {T,5\% } \right)}} = \frac{{{\sigma }_{\ln SA}\left( {T,\xi } \right)}}{{{\sigma }_{\ln SA}\left( {T,5\% } \right)}}\end{equation}$$$$where τ ln SA ( T , 5%), ϕ ln SA ( T , 5%), and σ ln SA ( T , 5%) are directly provided by a GMPE; σ ln SA ( T , ξ ) is obtained from Equation ().…”

“…Since the correlation between IMs is usually quantified by analyzing the normalized residuals, 39,45,55 the normalized total residual δ ij from Equation () is partitioned into the normalized inter‐ and intra‐event residuals η i and ε ij using the mixed‐effects regression method 28,49 . For a given GMPE and its associated standard deviations, the maximum likelihood estimate of the normalized inter‐event residual η i can be expressed as 3,40 : $$$$\begin{equation}{\eta }_i = \frac{{\sum_{j = 1}^n {{{\sigma _{\ln IM}^{\left( j \right)}{\delta }_{ij}} \mathord{\left/ {\vphantom {{\sigma _{\ln IM}^{\left( j \right)}{\delta }_{ij}} {{{\left( {\phi _{\ln IM}^{\left( j \right)}} \right)}}^2}}} \right. \kern-\nulldelimiterspace} {{{\left( {\phi _{\ln IM}^{\left( j \right)}} \right)}}^2}}} }}{{{\tau }_{\ln IM}\left[ {{1 \mathord{\left/ {\vphantom {1 {\tau _{\ln IM}^2}}} \right.$$…”

“…For comparison and validation purposes, Figure 6(A) compares the median 5%‐damped H‐H correlations with the existing correlation models for SA H ‐SA H (in Baker and Jayaram, 39 and Poulos and Miranda 3 ), SA H ‐PGA H (in Bradley, 55 and Baker and Bradley 40 ), and SA H ‐PGV H (in Bradley, 50 and Baker and Bradley 40 ). Note that the correlation results from Baker and Bradley updated in their GitHub repository 40 are used.…”

“…As one of the most representative intensity measures (IMs), the 5%‐damped (pseudo) spectral acceleration at an oscillator period ( T ), denoted as SA ( T , 5%), has been widely used to represent the characteristics of earthquake ground motions. Yet, the spectral accelerations with other damping ratio ( ξ ) values can be of interest for different types of structures 1–3 . As are related to the energy dissipation, the damping ratios can be larger than 5% for structures with base‐isolation systems and energy‐dissipation devices (e.g., ref.…”

“…24, 39–42), and few efforts have been made on the horizontal‐vertical (H‐V) and vertical‐vertical (V‐V) pairs 43–46 . Regarding structures with different damping ratios, only a non‐5%‐damped correlation model proposed by Poulos and Miranda 3 is available for the H‐H pairs of SAs, in which a single GMPE was used for the correlation calculation and the associated uncertainty was not considered.…”