Executive Summary
A novel way of constructing the rotation dependent IMs, such as the RotDxx and GMRotDxx spectra with “xx” denoting the percentile is developed. The framework leads to a computational cost saving of more than 90% when compared with the conventional approach. Proposed framework is particularly useful in case of large-scale construction such as the development of GMPEs. Rotation independent IMs such as the RotIxx and GMRotIxx spectra are likely to be sensitive on the maximum period selected in the process of construction. This study proposes a new definition for the same (and denoted as RotÎxx and GMRotÎx) that nearly eliminates the sensitivity and show better similitude with the target rotation dependent spectra. In other words, RotÎxx and GMRotÎx appear to be better representative of the rotation independent IM.
Source-to-site distance measure plays a pivotal role in the seismic ground motion characterization, especially, for structural design. Among various other measures, Joyner-Boore distance (Rjb) is often preferred for the construction of next generation ground motion predictive equations (GMPE). This part of the research is aimed to construct the Rjb for PESMOS and COSMOS database representing the earthquake triggered in Indian sub-continent. By definition, Rjb is the shortest distance from the site to the horizontal projection of the rupture plane. Although Rjb appears to be an unambiguous parameter that can be constructed completely based on geometry, real scenario is too complex on account of the uncertainty involved in the required information including the rupture geometry, location of hypocentre on rupture plane, strike and dip etc. A vector algebra-based approach is proposed for estimating epicentral distance and azimuth. Next, a set of empirical relationships is proposed to estimate the rupture plane from the moment magnitude using a dataset of 354 earthquakes based on tectonic settings and focal mechanisms. Further, a step-by-step process of computing Rjb is developed considering the uncertainty in the location of hypocenter on the rupture plane. Finally, the process is extended to account for the uncertainty in available information of strike and/or dip. The proposed framework is assessed against a total of 4247 records from PEER database with Rjb reported based on geometry and location of rupture plane. The framework is applied to compute Rjb associated with PESMOS (474 records) and COSMOS (148 records) database, and the results are expected to serve as a valuable resource while constructing GMPEs of shallow focused earthquakes. The appendix consisting of these results is available online.
In addition to the issues with distance measure, another internal issue is magnitude of completion. An important step in seismic hazard analysis is investigating the completeness of available data. Out of the various methods proposed by several researchers, Stepp’s method is one of the most commonly used methods for completeness analysis. However, some drawbacks are identified in this method, which results in erroneous estimation of the completeness period. A 2-point moving average based representation is proposed that works directly on the mean annual rate of occurrence instead of the estimation of its variance. The proposed representation not only increases the sensitivity in identifying the missing events but also enables in detecting possible translation of any event into the adjacent magnitude intervals. Several globally recognized seismic catalogues are analysed to demonstrate the efficacy of the proposed methodology.
Ground motion prediction equations (GMPEs) play a critical role in seismic hazard analysis. However, the conventional methods for developing GMPEs, which rely on functional forms and assumptions like homoscedasticity, can introduce biases. To address this issue, an alternative approach called the Consistent Spectral Shape Approach (CSS) is proposed in this project. The CSS framework works directly on the available database to capture the underlying physics of the problem and overcomes the limitations of both parametric and non-parametric methods. The proposed approach first clusters (defined as bins) the database using a set of independent variables. Next, the logarithmic mean and variance spectrum are estimated in each cluster. This construction adequately accounts for the effect of inter- and intra- event variability. Next, the framework decouples the logarithmic mean spectrum into spectral shape and normalizing factor. This decoupling enables the recreation of a proxy of the underlying database using very few parameters. This proxy database is used further for interpolation given a set of independent variables. This process is free from functional form bias as it is directly constructed from the database. The proxy of the ground motion recorded within a cluster does not significantly sacrifice the physics of the problem. Additionally, working with clusters also remove the requirement of heteroscedasticity assumption. In other words, the physics of the problem is preserved by capturing the attenuation of PGA without any initial assumptions on the functional form. Next, the spectral shape is brought into the picture to understand the interaction between different definitions of intensity measures. As discussed, the spectral shape is nearly invariant of the definitions of spectral accelerations (RotD50, RotD100, Geo-mean, GMRotD50, and GMRotD100). This allows the development of GMPE by constructing that of PGA for different definitions to capture the underlying physics completely. The CSS-GMPEs proposed in this part of the work are constructed in line with the requirements of routine seismic design. Information regarding only three independent variables (M-R-Soil category) is generally available with the structural designers. Hence, the proposed CSS-GMPEs are purposefully degenerated to account only for these three independent variables. Nevertheless, if required, the proposed framework can include other variables and/or refine the spacing between the clusters, given a robust database is available without losing the generality. However, higher order Lagrangian interpolation are required in such cases to implement with/without modification for nonorthogonal mapping.
The performance of CSS-GMPEs is assessed against those reported in NGA-West2 project. Two different yardsticks are considered for the comparison: overall root mean square error (RMSE) and similitude of spectral shape. Only RotD50 is considered in this assessment. In the case of overall RMSE, all six GMPEs perform equally well for PEER and virgin Indian dataset. However, the CSS GMPE utilizes significantly smaller number of independent variables. Next, A step-by-step framework is proposed for assessing the similitude of the spectral shape based on a recorded database. The proposed CSS GMPE is found to capture the spectral shape better than other existing GMPEs for the PEER dataset. It is noteworthy that unlike NGA-West2 database, the CSS-GMPE does not considers focal mechanism as an independent variable during this assessment. Further, the CSS-GMPEs are calibrated for the Himalayan region. The data corresponding to Rock site for Himalayan region is not yet available. Therefore, the GMPEs are modified for Medium and Hard soil in Himalayan region. These CSS-GMPEs modified for Himalayan region are expected to be a valuable tool for site-specific seismic hazard analysis.
While the GMPEs developed using the CSS approach present the marginal distribution, a novel framework is proposed for constructing a correlation model of spectral acceleration directly from the database. The proposed framework first numerically constructs the correlation structures contingent on the multidimensional clusters (bins). These numerically constructed correlation structures are next idealized accounting for any trends that may exist against the seismological parameters. This framework is implemented using the NGA-West2 database for five definitions of intensity measure (RotD100, RotD50, Geo-mean, GMRotD100, and GMRotD50). The correlation structure is observed to be independent of the definitions of the intensity measure. An idealized correlation model using a few parameters is proposed accounting for the trends observed against the seismological parameters. The existing correlation models (independent of seismological parameters) based on the NGA-West2 database (or its subsets) do not represent the correlation structure with sufficient accuracy.
The state-of-the-art seismic standards recommend using the RotD100 definition as the maximum direction shaking to account for the directionality. However, the conservatism associated with this definition is well recognized. This part of research first introduces the Rotated RotD100 spectrum as an improved measure of maximum direction shaking. It defines the spectrum corresponding to the orientation that maximizes the spectral ordinate at a specific conditioning period among all possible rotations (0–180°). One possibility of constructing the target spectrum consistent with Rotated RotD100 is using the probability distribution of spectral orientation associated with the spectral ordinate. The underlying marginal distribution is developed using data from NGA West2 and Himalayan region datasets. The distribution parameters are observed to be consistent across geographical regions and do not show any strong dependency on the seismological parameters. Correlation models associated with spectral orientation have also been developed to define the joint distribution. A step-by-step framework is proposed for constructing a target spectrum consistent with the Rotated RotD100 definition of maximum direction shaking.
An alternate perspective for constructing Rotated RotD100 spectrum is also envisioned. Unlike the previous approach, which is independent of seismological parameters, a novel framework is proposed for modifying the RotD100 CSS-GMPE to Rotated RotD100 GMPE. The proposed framework is demonstrated with a subset of the NGA-West2 database. The modification factor is first numerically constructed, followed by its idealization for smoothness and practical convenience. Proposed GMPE is compared with GMPEs for conventional IM (Geo-mean and RotD100) highlighting its unique ability to maximize response at the period of interest while still representing a realistic single-component ground motion.
In line with the developments, a total of two approaches are proposed for constructing target spectrum consistent with this definition: i) Using the distribution of spectral orientation associated with the RotD100 spectral ordinate; and ii) Using the modification factor proposed for converting GMPE associated with RotD100 to Rotated RotD100. The first approach proposes a framework based on simulation of spectral orientation and construct the Rotated RotD100 target spectrum if a RotD100 target spectrum is specified. Alternatively, following the latter, one can perform the seismic hazard analysis with the to come up with the target spectrum using the CSS-GMPEs and correlation model.
Next, the projects propose framework for selection and scaling of ground motion suit to match the target spectra consistent with the maximum direction shaking and critical orientation. As the target spectrum consistent maximum direction shaking was proposed later, geometric mean spectrum was used first for developing the framework. Key contribution in this part of the research is to consider the positively correlated ground motion suit from a record library. Selection and scaling of ground motion suite(s) followed by response history analysis form the main stem of pre-processing for the seismic design and performance assessment of structures. Ground motion selection (and scaling) frameworks reported in the prior art considered randomly crossing target spectra while accounting for the possible correlation or covariance matrix. This random crossing is likely to underestimate the failure probability owing to the possible positive correlation of EDP with the intensity measures. In this part of the work, a framework for selection and scaling of the positively correlated ground motion suite is proposed. The framework is aimed to increase the variance of resulting realization of EDP and which in turn minimizes the otherwise possible underestimation of failure probability. The framework requires only a vector of conditional marginal mean and a vector of conditional marginal variance for generating the target non-crossing fractile spectra that minimizes the negative correlation of the selected ground motions. The proposed framework also adopts a different selection scheme in comparison to the previous studies but uses by- and -large similar error metrices for the quality assessment. Possible impact of the positively correlated ground motion suite on EDP hazard and novelty involved in the proposed framework are demonstrated through an example of ground motion selection exercise followed by the seismic performance assessment of a structure. The proposed framework utilizing non-crossing fractile spectra as the target offers a complete remedy to the selection and scaling of ground motion suite(s) for seismic performance assessment and enables capturing the propagation of epistemic uncertainty in PSHA.
Next, the formulation is developed for critical orientation. By definition, the critical orientation for an orthogonal ground motion pair is the direction that maximizes the response of any particular EDP of interest. In fact, this critical orientation of a horizontal pair of ground motion varies for different EDPs in the same structure. However, considering critical orientation associated with all the EDPs is not possible due to the high computational cost associated with the non-linear time history analysis. Hence, if possible, maximizing the modal response of a few dominant modes is expected to account for the effect of critical orientation in an approximate sense and most importantly regardless of the EDPs. In this part of the research, a formulation is derived from the first principle for calculating a critical orientation that maximizes the modal response. Given this formulation, it may be sufficient to target the first two translational modes while seeking the maximum response from a selected ground motion pair. Nevertheless, to improve the accuracy, it is proposed that the structure should be analysed by applying the ground excitation in the following four orientations: i) critical orientation associated with the first translational mode; ii) critical orientation associated with the second translational mode; iii) 0-90 case: ‘a-0’ is along the x-axis and ‘a-90’ along the y-axis; and iv) 90-0 case: ‘a-90’ is along the x-axis and ‘a-0’ along the y-axis. Here, x- and y-axes denote the principal axes of the structure; while ‘a-0’ and ‘a-90’ are the ground motion components rotated along and normal to the direction of arrival, respectively. This proposal is shown to capture the maximum response reasonably well, even with inelastic excursion. Overall, the proposed recommendations are deemed fit for all practical purposes.
Finally, the Application of maximum direction shaking in seismic performance assessment is demonstrated. Two approaches are proposed, one with critical orientation and an additional approach that directly accounts for directionality in Hazard analysis.
Approach-1, Based on critical orientation (irrespective of faults that are triggered): In this approach, the hazard analysis is performed, and the disaggregation results are generated to find the governing causal rupture scenarios. However, the faults over which this scenario occurs are not identified and hence for the selected ground motion suites, the analysis is performed accounting for the critical orientation. A ten-storied building located at a site in the West Garo Hills (Meghalaya) district is considered for application. The vulnerability assessment is performed by applying the selected ground motion pairs that are consistent with the critical orientation. In comparison to the conventional recommendations, the incorporation of critical orientation is observed to have a considerable effect. This part of the research shows the importance of considering the maximum direction shaking while conducting a performance assessment of a structure.
Approach-2, Accounting for directionality in Hazard analysis (considering faults that are triggered): Spatially varying ground motion (SVGM) filed is characterized by its phase, amplitude and frequency variations over an extended area, and has the potential to induce additional internal forces in the structures when compared to the spatially uniform ground motion (SUGM). This part of research first aims to develop a comprehensive framework that integrates the conditional simulation of multicomponent spatially varying ground motion field (accounting for the site-specific epistemic uncertainties) with the seismic performance assessment of a structure. Subsequently, this work demonstrates its application to a medium-span reinforced concrete highway bridge subjected to six-component SVGM field. This is to offer dual objectives: i) understanding the influence of SVGM field on different engineering demand parameters (EDPs); and ii) influence of multicomponent excitation on the EDPs. PSHA at the site (Golaghat, Assam) is carried out for selection and scaling of the ground motion suite considering local geological conditions and accounting for the various possible sources of epistemic uncertainties. The conditional simulation of 6C-SVGM field is carried out using an evolutionary power spectral density-based framework and considering the coherency and site-specific effects. Two types of bridges, namely, one simply supported and another 4-span continuous, are considered in this part of research. Nonlinear time history analyses of the bridges are carried out while considering various combinations of translational and rotational components of ground motion and the results from SVGM are compared with that using SUGM.
In summary, this project lays the groundwork for state-of-the-art seismic design and performance assessment practices. By integrating the proposed tools and methodologies, the reliability involved in the performance assessment practices is expected to be considerably improved, leading to safer predictions.
Publication out of this Project:
- Vats, F., & Basu, D. (2021). A computationally efficient framework for rotation dependent and rotation independent intensity measures. Earthquake Engineering & Structural Dynamics, 50(6), 1562-1575.
PDF - Vats, F., Basu, D. On the construction of Joyner-Boore distance (Rjb) for PESMOS and COSMOS databases. J Seismol (2023). https://doi.org/10.1007/s10950-022-10129-1
PDF - Gurjar, N., & Basu, D. (2024). Revisiting Stepp’s method for the completeness of regional seismic catalogues. Journal of Seismology, 28(4), 1055-1086. https://doi.org/10.1007/s10950-024-10231-6
PDF - Vats, F. and Basu, D., Alternate method to develop ground motion prediction equations: Calibration over Himalayan region. Soil Dyn. Earthq. Eng., 2024, 176, 108312.
PDF - Vats, F., & Basu, D. (2025). A novel cluster-based framework for developing correlation model and its implementation for spectral acceleration. Soil Dynamics and Earthquake Engineering, 188, 109056. https://doi.org/10.1016/j.soildyn.2024.109056
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