Conditional Mean Spectrum Research Articles

Seismic fragility analysis of RC frame structures based on IDA analysis and machine learning

This study introduces a machine learning-based methodology designed to rapidly predict the seismic responses of reinforced concrete (RC) frame structures. The research focuses on three types of RC frame structures: low-rise, multi-story, and small high-rise buildings. Ground shaking records are selected according to the conditional mean spectrum (CMS). A sample database, constructed via Incremental Dynamic Analysis (IDA), facilitates the prediction of structural responses using ground shaking intensity and structural details as inputs. Concurrently, the study performs a feature importance analysis of the model. Machine learning algorithms, including integrated learning and neural networks, are utilized to predict the seismic responses of the RC frame structures. This methodology also assists in evaluating the seismic fragility of these structures. The results show that the discrepancy between the neural network-based seismic fragility assessments and the IDA results is minimal, indicating a high degree of accuracy in the proposed methodology. Among the characteristic parameters, Average Spectral Acceleration (AvgSa) is identified as the most significant. This methodology serves as a valuable tool for the rapid prediction of seismic responses in RC frame structures, demonstrating substantial practical value.

A modal conditional mean spectrum for nonlinear structural response time-history analysis of tall buildings to consider higher mode effects

A modal conditional mean spectrum for nonlinear structural response time-history analysis of tall buildings to consider higher mode effects

Seismic assessment of a long-term lunar habitat

The establishment of secure earth-independent long-term lunar habitats has been envisioned by numerous government agencies and private companies. Recent advancements in assessing seismic hazards caused by shallow moonquakes have highlighted the importance of incorporating this phenomenon into the design of robust and resilient lunar structures. However, further research is required to explore lunar habitat design that considers seismic loads. This paper proposes assessing the structural response of a lunar habitat made of sulfur concrete covered with a regolith layer. The numerical model of the structure is subjected to gravitational, internal pressure and seismic loads. The seismic analysis of the structure is carried out using spectral and nonlinear time history methods. Conditional mean spectra for shallow moonquakes with return periods of 75, 475, 970 and 2475 years are used in the seismic analysis. The records used for the temporal analyses were ground motions that agree with a preliminary seismic hazard on the Moon. The results of temporal analyses reveal that shallow moonquakes with return periods greater than 475 years can lead to the loss of the global stability of the structure. Consequently, the findings imply that seismic loads have the potential to impose unacceptable demands on lunar structures constructed from in-situ materials like sulfur concrete. Hence, it is imperative to incorporate seismic considerations in the design process for developing resilient and long-term lunar habitats.

The Generation of the Target Aftershock Spectrum Based on the Conditional Mean Spectrum of Aftershocks

Numerous studies have examined the responses of various structures to the mainshock–aftershock (MS–AS) ground motion, and the MS–AS ground motions are very important as the input. Therefore, in the absence of aftershock information, it is particularly critical to construct a reasonable MS–AS seismic sequence. This paper aims to provide a new reasonable method for generating the target aftershock response spectrum, which can be used to select or artificially simulate aftershock ground motion, given the seismic information of the main shock. Firstly, the magnitude, fault size, and location of the aftershock are determined. Then, other parameters required for the aftershock ground motion prediction equation (GMPE) are calculated. Subsequently, the correlation of the spectral shape to the MS–AS ground motion is used to modify the response spectrum predicted using the GMPE to obtain the conditional mean spectrum of aftershocks (CMSA). Finally, the relative errors of the predicted spectrum via the ASK14 model and CMSA are compared for four different assumptions. The results show that the simulated aftershock parameters and the actual ones accord well, and the relative errors of the CMSA can be controlled within 20%. Meanwhile, the discrete property of the target aftershock response spectrum is closer to the real recorded response spectrum.

Correlation and Joint Distribution of Spectral Accelerations of Vertical Ground Motions

ABSTRACT This paper estimates a correlation model and joint distribution model of spectral accelerations of vertical ground motions based on the NGA-West2 ground motion database. The correlation model calculated via BC16 vertical GMPE in this study is compared with those published models via other vertical NGA-West2 GMPEs. It is verified that the calculated residuals of vertical ground motions at multiple periods follow the multivariate normal distributions via both a qualitative validation method and a set of quantitative validation methods. The correlation model and the joint distribution model of vertical spectral accelerations are the theoretical basis for research on the joint occurrence of spectral accelerations of vertical ground motions, and an example of applied research on joint occurrence of spectral accelerations of vertical ground motions is studied: the conditional mean spectrum of vertical ground motions. Finally, the effects of different vertical correlation models and ground motion prediction equations on vertical conditional spectra are investigated for wide application of the correlation model and the joint distribution model of spectral accelerations of vertical ground motions.

Generating ground motions using the Fourier amplitude spectrum

AbstractIn this paper, a simplified method aimed at generating ground motions records using a Fourier amplitude spectrum model obtained by a response spectrum is presented. The use of white noise with specific conditions on the variance and inter‐frequency correlation allows to get a realistic variability of the Fourier amplitude spectrum. Moreover, a two‐corner frequency model is defined from empirical ground‐motion data and a filter is applied to capture the attenuation at high frequencies. The generated series of stochastic ground motions accurately match the mean value of the target response spectrum. The procedure was tested on an Italian site and results showed excellent matching in terms of mean and dispersion values with either the median obtained from ground motion prediction equations (GMPE) and conditional mean spectra (CMS). The advantage of the proposed methodology is that time‐histories can be generated with low computational effort, they do not need scaling or frequency content adjustments and are spectrum compatible with a given target spectrum.

Generation of Site-Specific Accelerograms and Response Spectra Involving Sampling Information from Borehole Records

This paper is aimed at serving the needs of structural engineering designers of an important structure (or a group of structures located on the same site) who is seeking guidance on how to obtain accelerograms and/or derive response spectra that accurately represent the site subsoil conditions as informed by the borelogs. The presented site-specific seismic action model may be used to replace the default seismic action model stipulated for the designated site class. Presented in this article is a procedure for generating soil surface motions in an earthquake, and their associated site-specific response spectra, taking into account details of the soil layers. Dynamic site response analyses are involved. The conditional mean spectrum methodology is employed for selecting and scaling accelerograms for obtaining input motion on bedrock. The selection depends on the natural period of both the site and the structure. Multiple borelogs taken from within the same site are analysed to identify the critical soil column models without having to conduct site response analysis on every borelog. The technique for simplifying the soil layers utilising the shear strain profile is introduced to further cut down on the time of analyses. The procedures described in this article have been written into a web-based program that is freely accessible to engineering practitioners.

Joint Occurrence of Spectral Accelerations Between Horizontal and Vertical Ground Motions

ABSTRACT This paper estimates the joint occurrence of spectral accelerations (SA) between horizontal (HGM) and vertical ground motions (VGM) based on the NGA-West2 ground motion database, which includes the correlation model and joint distribution model. In this study, the correlation model calculated via CB14 horizontal ground motion prediction equation (GMPE) and BC16 vertical GMPE is obtained. It is verified that the joint distribution of residuals follows a multivariate normal distribution via both a set of quantitative validation methods and a qualitative validation method. Conditional mean spectra of VGMs conditioned on SAs of HGMs are studied as an application of joint occurrence of SA V–SA H.

A new framework for ground motion selection for structural seismic assessment

Performance-based structural seismic assessment generally requires the selection of ground motions wherein the target acceleration response spectrum (Sa) should be matched in a desired period range. The pros and cons of the commonly used uniform hazard spectrum (UHS), conditional mean spectrum (CMS) and conditional spectrum (CS) are thoroughly examined. Aiming at combining the merits and overcoming the challenges of these methods, a new framework is proposed to construct a series of conditional target spectra for selecting ground motions. The response spectrum is expressed as the product of peak ground acceleration (PGA) and the normalized spectra (β = Sa/PGA) which can be smoothed by four spectral shape parameters. The proposed framework provides the expected median and covariance of the spectral shape parameters conditioned on specified design values of PGA and PGV/PGA (peak ground velocity/PGA). Efficient step-by-step procedures are developed for the application of the proposed alternative approach, and a sample application is illustrated and compared with the CS approach. It is demonstrated that the proposed methodology is capable of incorporating conventional probabilistic seismic hazard analysis (PSHA), considering spectral shape and spectral variability of realistic ground motions while avoiding the choice of the conditioning period. Site class effects can also be incorporated into the proposed approach.

Site-Specific Response Spectra and Accelerograms on Bedrock and Soil Surface

This paper is aimed at serving the needs of structural engineering researchers who are seeking accelerograms that realistically represent the time histories of earthquake ground in support of their own investigations. Every record is identified with a specific earthquake scenario defined by the magnitude–distance combination and site conditions; the intensity of the presented records is consistent with ultimate limit state design requirements for important structures in an intraplate region. Presented in this article are accelerograms that were generated on the soil surface of two example class Ce sites and two example class De sites based on site response analyses of the respective soil column models utilizing bedrock excitations as derived from the conditional mean spectrum (CMS) methodology. The CMS that were developed on rock sites were based on matching with the code spectrum model stipulated by the Australian standard for seismic actions for class Be sites at reference periods of 0.2, 0.5, 1 and 2 s for return periods ranging from 500 to 2500 years. The reference to Australian regulatory documents does not preclude the adoption of the presented materials for engineering applications outside Australia. To reduce modeling uncertainties, the simulation of the soil surface ground motion is specific to the site of interest and is based on information provided by the borelogs. The site-specific simulation of the strong motion is separate to the CMS-based accelerogram selection–scaling for obtaining the bedrock accelerograms (utilizing strong motion data provided by the PEER). The decoupling of the two processes is a departure from the use of the code site response spectrum models and has the merit of reducing modeling uncertainties and achieving more realistic representation of the seismic actions.

Damping‐dependent correlations between response spectral ordinates

AbstractCorrelations 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%.

Selecting Ground Motions for Flexible Structures Based on Multiple Conditional Mean Spectra: A Case Study on Cable-Stayed Bridges

ABSTRACT For flexible structures, this study proposes multiple conditional mean spectrum (MCMS) to account for the scattered structural modes and the piecewise conditional mean spectrum (PCMS) for the higher mode effect. A flexible cable-stayed bridge is used for a case study. Several scenario earthquakes are identified responsible for the maximum responses of the bridge. The envelope of structural responses under different scenario earthquakes would be used as the design seismic effect. Adopting uniform hazard spectrum to selection ground motions would for most cases amplify the structural seismic responses, especially for tower bending moment and shear force.

A recurrent-neural-network-based generalized ground-motion model for the Chilean subduction seismic environment

This paper proposes a deep learning-based generalized ground motion model (GGMM) for interface and intraslab subduction earthquakes recorded in Chile. A total of ∼7000 ground-motion records from ∼1700 events are used to train the proposed GGMM. Unlike common ground-motion models (GMMs), which generally consider individual ground-motion intensity measures such as peak ground acceleration and spectral accelerations at given structural periods, the proposed GGMM is based on a data-driven framework that coherently uses recurrent neural networks (RNNs) and hierarchical mixed-effects regression to output a cross-dependent vector of 35 ground-motion intensity measures (denoted as IM). The IM vector includes geometric mean of Arias intensity, peak ground velocity, peak ground acceleration, and significant duration (denoted as Iageom, PGVgeom, PGAgeom, and D5-95geom, respectively), and RotD50 spectral accelerations at 31 periods between 0.05 and 5 s for a 5 % damped oscillator (denoted as Sa(T)). The inputs to the GGMM include six causal seismic source and site parameters, including fault slab mechanism, moment magnitude, closest rupture distance, Joyne-Boore distance, soil shear-wave velocity, and hypocentral depth. The statistical evaluation of the proposed GGMM shows high prediction power with R2 > 0.7 for most IMs while maintaining the cross-IM dependencies. Furthermore, the GGMM is carefully compared against two state-of-the-art Chilean GMMs, showing that the proposed GGMM leads to better goodness of fit for all periods of Sa(T) compared to the two considered GMMs (on average 0.2 higher R2). Finally, the GGMM is implemented to select hazard-consistent ground motions for nonlinear time history analysis of a sophisticated finite-element model of a 20-story steel special moment-resisting frame. Results of this analysis are statistically compared against those for hazard-consistent ground motions selected based on the conditional mean spectrum (CMS) approach. In general, it is observed that the drift demands computed using the two approaches cannot be considered statistically similar and the GGMM leads to higher demands.

Evaluation of Vector Hazard for Conditional Mean Spectrum with Different Definitions of Multivariate Exceedance Rate

ABSTRACT The conditional mean spectrum (CMS) and its generalized forms, like the generalized CMS (GCMS), and the CMS conditioned at the logarithm-average spectral acceleration over a period range (CMS-AvgSa), are anchored on a conditional period (or periods) with a target hazard level. This study evaluated the vector hazard of multiple spectral accelerations (Sa) at different periods for CMS, GCMS, and CMS-AvgSa. By invoking the invariance property of scalar disaggregation and Copula function, we developed the computational methods for three types of multivariate exceedance rate (MER): multivariate simultaneous exceedance rate (MERs), multivariate arbitrary exceedance rate (MERa), and Kendall-function-based MER (MERk). Taking the cities of Xi’an and Kunming (China) as examples, scalar probabilistic seismic-hazard analysis (PSHA), and disaggregation were performed and CMS was built at different conditional periods (T* = 0.05, 0.2, 1.0, and 2.0 s, respectively). The MERk computation results lies between the other two extreme cases (MERa and MERs), which are either too strict or too tolerant compared with the anchored hazard level. It is found that when the earthquake disaggregation characteristics not differ markedly between T* and other periods in CMS, the calculated MERk results would be roughly consistent with the anchored hazard level at Sa(T*). The conclusion holds true when different Sa(T) were considered in the calculation of MERk . On the other hand, the calculated MERa and MERs would be significantly influenced by the choice of Sa(T) and the results are not comparable with the anchored hazard level. We also evaluated the MERk for GCMS and CMS-AvgSa, which were constructed using mean scenario earthquake from vector-valued PSHA and disaggregation. Comparing with the UHS, the MERk of GCMS and CMS-AvgSa are more close to the anchored hazard level within a certain period range. Although no benchmark exists for vector hazard level, the MERk is likely to be a promising approach for measuring the hazard over a vector of intensity measures in target spectrum.

Post-earthquake seismic capacity estimation of reinforced concrete bridge piers using Machine learning techniques

This study analytically explored a parameterized set of 20 reinforced concrete bridge piers which share several geometrical and material properties commonly found in Canadian bridges. A nonlinear fiber-based modelling approach with a proposed material strength degradation scheme is developed using the OpenSEES platform. A multiple conditional mean spectra approach is used to select a suite of 50 mainshock-aftershock (MS-AS) ground motion records for the selected site in Vancouver, British Columbia. Nonlinear time history analysis is performed for mainshock-only and mainshock-aftershock excitations, and static pushover analysis is also performed in lateral and axial directions for the intact columns, as well as in their respective post-MS and post-AS damaged states. Using the resulting data, a framework for post-earthquake seismic capacity estimation of the bridge piers is developed using machine learning regression methods, where several candidate models are tuned using an exhaustive grid search algorithm approach and k-fold crossvalidation. The tuned models are fitted and evaluated against a test set of data to determine a single best performing model using a multiple scorer performance index as the metric. The resulting performance index suggests that the decision tree model is the most suitable regressor for capacity estimation due to this model exhibiting the highest accuracy as well as lowest residual error.

Preliminary approach to assess the seismic hazard on a lunar site

The passive seismic network deployed on the Moon during the Apollo missions operated for eight years and allowed the observation of seismic activity. More than 12500 seismic events were registered, where 28 were classified as shallow moonquakes with moment magnitudes up to 4.1. Seismic events of this nature pose a significant risk to future long-term lunar habitats; thus, these events must be carefully studied and considered in the seismic design of these structures. This paper proposes a preliminary seismic hazard assessment imposed by shallow moonquakes. The hazard assessment is performed using the Probabilistic Seismic Hazard Analysis (PSHA) methodology, considering previous studies and theories regarding the seismic environment of the Moon. The study zone covers ∼860 km2 of the Taurus-Littrow Valley, containing the Apollo 17 landing site and the Lee-Lincoln lobate scarp as the considered seismic source. The seismic hazard is quantified in terms of peak ground acceleration (PGA) and spectral acceleration (5% damped pseudo-acceleration, PSA). Seismic hazard deaggregation scenarios, Uniform Hazard Spectra (UHS) for different hazard levels, and a Conditional Mean Spectrum (CMS) for a target period of 0.2 s are obtained to quantify the seismic hazard on a specific site on the Moon. The developed seismic hazard assessment provides a preliminary approach for realistic scenarios to conduct structural designs that ensure the seismic performance of fully operational long-term lunar structures.

Sensitivity of the conditional period selection in the structural response using the CMS as target spectrum

A framework is presented for assessing the sensitivity of typical engineering demand parameters (EDP) to the conditional period selection when using conditional mean spectra (CMS) as targets for ground-motion selection in a performance-based seismic evaluation. The framework consists of computing a suite of CMS targets anchored at conditioning periods within a period range of interest to discretize the demand at a given hazard level, as represented by a uniform hazard spectrum (UHS). Ground motions are selected and scaled for the CMS suite and the associated UHS. The envelope of the median responses from the CMS suite is compared with the median response from the UHS. The framework is instrumental in identifying the conditioning period ( T*) range for estimating CMS to capture the maximum median responses at the hazard level of interest. It also helps to characterize the relative difference in responses between using CMS targets and a UHS. The implementation of the methodology is illustrated by evaluating response quantities such as displacement-, acceleration, and force-based EDPs of four reinforced concrete moment frame structures of different heights under three levels of increasing hazard. Results confirm that the conditioning period used for ground-motion selection has a significant impact on the seismic response of displacement-based EDPs, and the sensitivity of the response varies with building height. For other EDPs, like maximum base shear and story acceleration, the results vary. Based on a limited-size ground motion set typically used in practice, the results indicate that UHS-targeted ground motions do not necessarily yield greater demand in comparison with using the CMS for estimating peak story drifts. For maximum floor accelerations, however, the CMS did produce smaller responses.

Weighted Average Conditional Mean Spectrum (WACMS): A bridge between common CMS and Uniform Hazard Spectrum (UHS)

In order to accurately predict structural response subject to ground motion excitation in the hazard level of interest, it is necessary to determine appropriate target spectrum. Conditional Mean Spectrum (CMS) conditioned on occurrence of a target spectral acceleration (Sa) at a single period is an alternative of the overly conservative Uniform Hazard Spectrum (UHS). The first-mode period is usually advised as conditioned period to create CMS, but it is not always appropriate. A Weighted Average Conditional Mean Spectrum (WACMS) is newly introduced as a more suitable target spectrum to long-period structures and practical guidelines for use are presented. WACMS is the weighted average of CMS suite conditioned on occurrence of target Sa at first several periods, the sum of whose mass participation factors exceeding 90%. The significant advantage is that it can consider hazard levels at multiple periods and use mass participation factor as the weight. CMS and UHS can be integrated with WACMS. WACMS is equivalent to CMS when one period is considered while almost overlaps UHS when all periods are considered. That is, WACMS bridges the gap between CMS and UHS. In this paper, three different structures designed by Chinese code are utilized as examples to verify this method. Results show that WACMS is more appropriate as a target spectrum to predict mean inter-storey drift ratio (IDR) consistent with Chinese code, compared to CMS conditioned on occurrence of Sa at the first-mode period. Meanwhile, WACMS increases the rate of exceeding IDR limit and IDR dispersion.

On the seismic performance evaluation of dam-foundation-reservoir system for the effect of frequency content and foundation flexibility

The present paper investigates the effect of earthquake frequency content and foundation flexibility on the performance of the Koyna dam-foundation-reservoir system. Ground motions that match the conditional mean spectrum (CMS) are adopted for the seismic analysis. The concrete damaged plasticity model is used to represent the dam's nonlinear behavior; however, the foundation is assumed to be linear elastic. To study the foundation flexibility effect on the seismic performance of the dam, three different cases which comprise different elastic modulus of the foundation are considered. The damage indices, based on the crack length and dissipation energy, are evaluated for the performance assessment of the dam-foundation-reservoir system. The ground motion parameters such as predominant period, mean period, and effective ground period are chosen in order to explore the earthquake frequency content effect on the performance of dam-foundation-reservoir system. Results indicate that foundation flexibility has a significant effect on the dam. Further, the frequency content has a substantial impact on the dam's seismic response compared to the intensity and duration of ground motion. It can be concluded that the mean period is a good indicator of the earthquake frequency content and has a strong correlation with the damage parameters.

Using conditional mean spectra and uniform hazard spectra for seismic analysis of bridges

Older bridges are often not adequately designed for earthquakes. To enable a structural verification, new methods such as conditional spectra and conditional mean spectra have been developed to estimate seismic hazards more accurately. To quantify the advantage of the latter method for regions with low and moderate seismic hazards, the seismic internal forces of four Swiss existing bridges were calculated and compared based on conditional mean spectra and uniform hazard spectra. The examples chosen were two reinforced concrete and two steel bridges, one of which was used for road traffic and one for railway traffic. The bridge selection was intended to show the validity of the results for different bridge types. On average, the computations using conditional mean spectra showed lower internal forces than when using uniform hazard spectra. However, the difference was smaller than expected. In addition, practical application in an engineering office would be very time-consuming with today's software solutions. Nevertheless, the results also showed that the differences at component level can be significant.