A ground motion selection and modification method through stratified sampling
 445 Downloads
Abstract
In this article a ground motion selection and modification (GMSM) method is presented, suitable for the probabilistic seismic assessment of the damage state and collapse potential of building structures. The objective is to predict the probability distribution of the engineering demand parameters in a future earthquake event, provided that the spectral acceleration at the fundamental period of the structure is given. The GMSM method uses a vectorvalued intensity measure that incorporates the Normalized Spectral Area parameter. Through stratified sampling on the Normalized Spectral Area, optimised ground motion suites are formed. Its advantage over other GMSM methods is that it implicitly matches the multivariate distribution of the response spectrum in the region of the structure elongated period, and it adopts an unbiased estimator of the response central tendency. The GMSM method is applied in the probabilistic response assessment of a firstmode dominated multidegreeoffreedom system that represents a tenstorey building. Substantial reduction in the computational work is achieved, compared to another method.
Keywords
Ground motion selection method Intensity measure Normalized spectral area Nonlinear response Probabilistic seismic assessment1 Introduction
A ground motion selection and modification (GMSM) method is presented, suitable for the probabilistic seismic assessment of the damage state and collapse potential of building structures. The objective is to predict the probability distribution of the engineering demand parameters (EDPs) when the structure is subjected to a ground motion having a spectral acceleration \(S_{a} \left( {T_{1} } \right)\) at the fundamental period of the structure, \(T_{1}\). Its advantage over other GMSM methods is that it implicitly matches the multivariate distribution of the response spectrum in the region of the structure elongated period, and it adopts an unbiased estimator of the EDP central tendency.
Ground motion intensity is expressed using the vectorvalued intensity measure (IM) \(\left\langle S_{a} \left( {T_{1} } \right), S_{dN} \left( {T_{1} ,T_{1}^{{\prime }} } \right) \right\rangle\) (Theophilou 2014; Theophilou et al. 2017), where \(T_{1}^{{\prime }}\) is an approximation of the elongated period of the structure due to inelasticity effects. Optimized suites of ground motions are obtained from a large ground motion dataset, through stratified sampling on \(S_{dN} \left( {T_{1} ,T_{1}^{{\prime }} } \right)\). The optimized suites are used in the dynamic analysis of the structure, resulting in a response prediction that is optimized, compared to using a less efficient IM, or to random sampling. An optimized response prediction implies that a reduced number of ground motions is required to obtain the same level of prediction accuracy, or, conversely, an improved accuracy is achieved when the same number of ground motions is used. The concept is that stratified sampling on IM results in an optimized replication of the central tendency and the dispersion of the IM as well as the associated EDPs, provided that there is sufficiently high correlation between the IM and the EDP.
The proposed GMSM method is applied to the dynamic analysis of a multidegreeoffreedom (MDOF) structure with high participation of the fundamental mode, which is assumed to represent a real tenstorey building structure designed to Eurocode 2 (2004a) and Eurocode 8 (2004b) for ductility class ‘High’.
2 Motivation and background
Probabilistic seismic response assessment is of interest both in the design of new structures, and in the assessment of existing structures. In the design of new structures the goal is to ensure that the safety level required by the building codes is attained, while in the assessment of existing structures the goal is to quantify the inherent safety level.
Dynamic responsehistory analysis has various advantages over other methods of evaluating structural response, i.e. it is more accurate, it enables the explicit evaluation of the response at every time step, and it can be used in cases where other methods are unsuitable, such as with structures having complex configuration and complex nonlinear response. However, there are two significant impediments in adopting dynamic responsehistory analysis for meeting the above safety objectives. First, the dynamic responsehistory analysis of MDOF structures requires substantial computational work, especially when nonlinear behaviour is considered. Even though computers are becoming continuously more sophisticated in terms of processing power and memory resources, the sophistication and complexity of the mathematical models representing the structural system are also evolving. The specification of appropriately small suites of ground motions, consequently resulting in decreased computational work, will always be a topic of great interest and utility. Second, although the number of recorded ground motions is continuously increasing, there is still relative scarcity of high intensity records, which are of most interest. These two challenges call for the need to develop GMSM methods, to predict the true response with sufficient accuracy and efficiency.
A convenient way of conveying the seismic intensity of the earthquake scenario to the structural engineer is through the spectral ordinate at the fundamental period of the structure, such as \(S_{a} \left( {T_{1} } \right)\). One category of GMSM methods aim at predicting the central tendency of the structural response, given \(S_{a} \left( {T_{1} } \right)\). Such methods are the response spectrum matching (e.g. Eurocode 8 2004b; ICB 2015), which can be facilitated with software tools such as REXEL (Iervolino et al. 2010), the Conditional Mean Spectrum matching proposed by Baker (2011), the genetic algorithm selection and scaling proposed by Naeim et al. (2004), the selection based on a vector of record properties identified by proxy proposed by WatsonLamprey and Abrahamson (2006), the selection based on a precedence list proposed by Azarbakht and Dolsek (2007), the selection using a harmony search algorithm proposed by Kayhan et al. (2011), the scaling to target scenario with epsilon preservation and selection based on median dispersion minimization proposed by Ay and Akkar (2012), and the integrated software environment coupling ground motion selection with structural analysis proposed by Katsanos and Sextos (2013).
The proposed GMSM method predicts, in addition to the central tendency, the dispersion of the response. A method consistent with this objective is the FEMA P58 (2012) procedure. A weakness of this method is that by selecting all records to have essentially the same shape, the multivariate nature of the response spectrum distribution (Jayaram and Baker 2008) is suppressed. Jayaram et al. (2011) illustrate through application examples that GMSM methods focusing on matching the response spectrum shape underestimate the response central tendency and dispersion. The semiautomated procedure proposed by Kottke and Rathje (2008), matches the central tendency and the dispersion of the entire suite. Since the procedure matches the median response spectrum to the target response spectrum, the response spectrum shape of the individual records within a suite could exhibit significant variation. However, this procedure does not match the multivariate distribution of the response spectrum.
Some GMSM methods have been proposed that consider, directly or indirectly, the multivariate nature of the response spectrum distribution (Jayaram and Baker 2008). Shome et al. (1998) suggested forming bins of records based on magnitude and distance, and then forming record suites through random sampling and scaling them to \(S_{a} \left( {T_{1} } \right)\); this method is used herein for comparison with the proposed GMSM. Goulet et al. (2007) selected records to match the deaggregation results in terms of magnitude, distance, and epsilon (Baker and Cornell 2005), and subsequently scaled them to the target \(S_{a} \left( {T_{1} } \right)\). Jayaram et al. (2011) proposed an algorithm for selecting suites of records to match simulated response spectra that have a specified mean, variance, and correlation between any two periods. Wang (2011) proposed a similar method, with the difference that the response spectra are conditioned on magnitude and distance, rather than \(S_{a} \left( {T_{1} } \right)\). Bradley (2010) proposed the concept of the Generalized Conditional Intensity Measure, which is a vectorvalued IM that contains a multitude of different IMs; records are selected using the algorithm proposed by Bradley (2012) such that the empirical distribution function of the suite matches the target IM distribution.
The proposed GMSM method is a contribution to the methods that match the multivariate nature of the response spectrum distribution.
3 Intensity measure
Due to the normalization to \(S_{d} \left( {T_{1} } \right)\), the \(S_{dN} \left( {T_{1} ,T_{1}^{{\prime }} } \right)\) value does not change with scaling. In this way, \(S_{dN} \left( {T_{1} ,T_{1}^{{\prime }} } \right)\) captures the effect of the excitation spectral characteristics (i.e. frequency content) on the response. Thus, it is a measure of the intensity that affects the inelastic response associated with period elongation. In turn, the degree of period elongation depends on the frequency content, which is unique for each ground motion. Hence, the purpose of integrating the response spectrum is to capture the elongated period within appropriately estimated bounds. In this context, \(S_{dN} \left( {T_{1} ,T_{1}^{{\prime }} } \right)\) can be seen as a descriptor of the local response spectrum shape between periods \(T_{1}\) and \(T_{1}^{'}\).
Theophilou et al. (2017) review a number of similar vectorvalued IMs proposed by other researchers.
4 Specification of largesample datasets
5 Methodology
The GMSM method aims to replicate the true inelastic response distribution of the structure, for a given \(S_{a} \left( {T_{1} } \right)\). In the present section the GMSM method is presented, and in the following section the sample statistics are described. Use is made of the IM and EDP distributions, appropriately transformed so that they are normally distributed, and denoted as follows: \(IMT\) is defined as the \(S_{dN} \left( {T_{1} ,T_{1}^{'} } \right)\) element of \(\varvec{IM}\) transformed so that its distribution is normal; \(EDPT\) is defined as a scalar element of \(\varvec{EDP}\) transformed likewise.

Step 1: Formation of ground motion largesample dataset

Initially, dataset \(G\) is formed, having a sufficiently large size (e.g. \(N_{gm} \ge 30\)). Ground motions selected have seismological parameters (such as magnitude, and sourcetosite distance) consistent with the earthquake scenario. It is strongly recommended to check the distribution of the dataset with respect to a ground motion prediction model representative of the expected earthquake.

Step 2: Ground motion normalization to \(S_{a} \left( {T_{1} } \right)\)

All ground motions in dataset \(G\) are normalized to \(S_{a} \left( {T_{1} } \right)\). An upper limit can be imposed to the scale factor, e.g. 3 as proposed by Shome et al. (1998) or 4 by Iervolino and Cornell (2005), beyond which scaling is considered inappropriate (i.e. the frequency characteristics of the scaled motion are not consistent with its amplitude); ground motions exceeding this limit should be discarded.

Step 3: Determination of \(T_{1}\) and \(T_{1}^{{\prime }}\)

The integration interval periods \(T_{1}\) and \(T_{1}^{{\prime }}\) are determined, which are then used in the evaluation of \(S_{dN} \left( {T_{1} ,T_{1}^{{\prime }} } \right)\). \(T_{1}^{{\prime }}\) represents the ‘ultimate’ elongated period and can be calculated using the simplified procedure proposed by Theophilou et al. (2017), or the procedure proposed in FEMA 440 (2005). Alternatively, the optimum \(T_{1}^{{\prime }}\) can be evaluated by first carrying out dynamic analyses on a singledegreeoffreedom system using all ground motions in dataset \(G\). Then, through regression analysis for a range of candidate \(T_{1}^{{\prime }}\) values, the final \(T_{1}^{{\prime }}\) value is selected as the one with the highest correlation between \(IMT\) and \(EDPT\).

Step 4: Calculation of \(S_{dN} \left( {T_{1} ,T_{1}^{{\prime }} } \right)\)

\(S_{dN} \left( {T_{1} ,T_{1}^{{\prime }} } \right)\) is calculated for each ground motion using Eq. (2).

Step 5: Evaluation of \(IMT\) mean and variance

The \(IMT\) mean and variance of dataset \(G\) are evaluated.

Step 6: Partition of \(IMT\) distribution into \(N_{s}\) strata

The \(IMT\) domain is partitioned into \(N_{s}\) strata, such that all strata have equal probability of occurrence. It is strongly recommended that at least \(N_{s} = 5\) strata are used.

Step 7: Formation of optimized suites

In the last step, optimized suites are formed through ‘stratified sampling’, by selecting \(N_{i}\) ground motions from each stratum. The total number of selected ground motions is \(N_{g,s} = N_{s} \cdot N_{i}\).
6 Estimation of response distribution through stratified sampling
6.1 Statistical dependence between intensity and response
6.2 Stratified sampling on \(IMT\)
The theoretical distribution of \(IMT\) is normal with mean \(\mu_{IMT}\) and standard deviation \(\sigma_{IMT}\). The area under the curve is divided into \(N_{s}\) strata, which have equal probability of occurrence. Random variable \(IMT_{i}\) represents variable \(IMT\) within stratum \(i\). Within any stratum \(i\) the theoretical mean is denoted as \(\mu_{IMTi}\), and the theoretical standard deviation as \(\sigma_{IMTi}\). The sampling fraction of stratum \(i\), \(w_{IMTi}\), is equal to \(1/N_{S}\) for all strata.
The estimator of the sample mean \(IMT_{i}\), \(\overline{IMT}_{i}\), is an unbiased estimator of \(\mu_{{IMT_{i} }}\), on the assumption of simple random sampling within stratum \(i\) (e.g. Ang and Tang 1975). Consequently, the estimator of the sample mean \(IMT\), \(\overline{IMT}\), is an unbiased estimator of \(\mu_{IMT}\) (e.g. Cochran 1977).
6.3 EDPT statistics
The standard error of the mean \(EDPT\), \(\sigma_{{\overline{EDPT,s} }}\), is (Cochran 1977)
6.4 Comparison to random sampling
Stratified sampling on \(IMT\) nearly always results in a standard error of the mean that is lower than that obtained through random sampling (Cochran 1977). In this section it is shown that, as a consequence, the standard error of the mean \(EDPT\) is also lower, by a degree that depends on \(\rho\).
Plotted in Fig. 1 is \(\sigma_{{\overline{EDPT,s} }} /\sigma_{{\overline{EDPT,r} }}\), for \(\left \rho \right\) values from 0 to 1.0.
6.5 Size of suites
The size of the suites plays an important role in the accuracy of \(IMT\) and \(EDPT\) estimation. Theophilou (2014) applied the method to a singledegreeoffreedom system assumed to represent an idealized structure. Using suites of \(N_{g,s} = 8\) ground motions (\(N_{s} = 8\), \(N_{i} = 1\)) resulted in a standard error in the mean \(IMT\) of < 2%, and in a standard error in the mean \(EDPT\) of approximately 10%. The variance of \(IMT\) was overestimated by about 10%, and the variance of \(EDPT\) by about 10%. This estimation accuracy is deemed acceptable. Using suites of \(N_{g,s} = 5\) ground motions (\(N_{s} = 5\), \(N_{i} = 1\)) resulted in less accurate, but still acceptable estimation.
Further research could lead to further optimization of the method with respect to the \(N_{s}\) and \(N_{i}\) combinations used, e.g. choosing between (\(N_{s} = 4\), \(N_{i} = 2\)) and (\(N_{s} = 8\), \(N_{i} = 1\)).
6.6 Multivariate distribution of response spectrum
The objective of the proposed GMSM method is to match the multivariate distribution of the response spectra between \(T_{1}\) and \(T_{1}^{{\prime }}\). This is achieved in an implicit manner, by matching the dispersion of \(S_{dN} \left( {T_{1} ,T_{1}^{{\prime }} } \right)\), given that the response spectra are conditioned on \(S_{a} \left( {T_{1} } \right)\). The match is implicit, in the sense that instead of matching the multivariate distribution of the response spectra over a range of discrete periods, the dispersion of the scalar \(S_{dN} \left( {T_{1} ,T_{1}^{{\prime }} } \right)\) is matched.
7 Application to a building structure
The GMSM method was applied in the probabilistic response assessment of a firstmode dominated MDOF system, representative of a real building structure.
7.1 Structure description
The structure examined is a tenstorey reinforced concrete building, used for residential occupancy. Each storey has a rectangular floor plan of dimensions 24 m × 16 m, and a height of 3.2 m. The columns are located in a 5 × 5 array, spaced 6 m apart in the longitudinal direction, and 4 m apart in the transverse direction. The structure was designed to Eurocode 2 (2004a) and Eurocode 8 (2004b) for ductility class ‘High’. Further details of the structure are provided in Theophilou (2014).
7.2 Finite element model
A twodimensional finite element model was developed, representing one interior frame in the longitudinal direction of the structure. The model was analysed using the computer program OpenSees version 2.4.0 (PEER 2013). Beams and columns were represented as elastic beamcolumn elements. At the locations of the beam and column joints, and at the location of the column and foundation joints, the beam and column frame elements are connected to the respective nodes by zerolength ‘nonlinear spring’ elements that allow rotation only. The momentrotation relationship assigned to the nonlinear springs is obtained using the modified IbarraKrawinkler hysteresis model (Lignos and Krawinkler 2012a). The nonlinear spring parameters were determined using the empirical relationships and values given by Haselton and Deierlein (2007), and by Lignos and Krawinkler (2012b). The hysteresis rules are consistent with the modified CloughJohnston model (Clough and Johnston 1966; Mahin and Lin 1983), with the exception that they are modified to conform with the multilinear backbone curve. The finite element model considered secondorder (PDelta) effects.
7.3 Natural modes
Natural modes of structure
Mode  Period (s)  Mass participation ratio (%) 

1  1.452  81.0 
2  0.479  9.9 
3  0.276  3.7 
7.4 Ground motion largesample dataset
Ground motion records
Earthquake  Station  M  R (km)  Azimuth/direction 

Loma Prieta  CDMG 47379  6.93  28.64  000 
18/10/1989—00:05  Gilroy Array #1  090  
Victoria, Mexico  Cerro Prieto  6.33  33.73  045 
09/06/1980—03:28  315  
Coalinga  CDMG 46175  6.36  33.52  045 
02/05/1983—23:42  Slack Canyon  315  
San Fernando  CDMG 126  6.61  24.19  111 
09/02/1971—14:00  Lake Hughes #4  201  
Duzce, Turkey  Lamont 531  7.14  27.74  000 
12/11/1999  090  
Kozani, Greece  ITSAK 99999  6.40  18.27  L 
13/05/1995—08:47  Kozani  T  
Irpinia, Italy  ENEL 99999  6.20  22.29  000 
23/11/1980—19:35  Bagnoli Irpinio  270  
Whittier Narrows  CDMG 24399  5.99  19.56  000 
01/10/1987—14:42  Mt Wilson  090  
Basso Tirreno  Milazzo  6.00  34  NS 
15/04/1978—23:33  EW  
Montenegro  Hercegnovi Novi  6.90  65  NS 
15/04/1979—06:19  Pavicic School  EW  
Tabas, Iran  9102 Dayhook  7.35  20.63  LN 
16/09/1978  TR  
Umbria Marche  AssisiStallone  6.00  21  NS 
26/09/1997  EW  
North Palm Springs  CDMG 12206 Silent Valley  6.06  20.70  000 
08/07/1986  090  
Loma Prieta  USGS 1032 Hollister  6.93  49.52  270 
18/10/1989—00:05  360  
ChiChi, Taiwan  CWB 99999 TCU045  7.62  77.50  N 
20/09/1999  E  
Northridge  USC 90059 Burbank Howard  6.69  23.18  060 
17/01/1994—12:31  330  
San Fernando  USGS 266 Pasadena  6.61  39.17  180 
09/02/1971—14:00  270  
Whittier Narrows  USC 90017  5.99  28.48  075 
01/10/1987—14:42  LA Wonderland  165  
Northridge  USGS 5080  6.69  19.19  270 
17/01/1994—12:31  Monte Nido  360  
Irpinia, Italy  Auletta  6.90  33.10  000 
23/11/1980—19:34  270 
To ensure that the dataset of ground motions is a representative sample of the population, its statistics were compared to the Boore and Atkinson (2008) ground motion prediction model. Epsilon was evaluated in the period range 0.5–2.0 s and was found to match very well the standard normal distribution, which is the theoretical distribution of epsilon.
7.5 Ground motion suites
One dataset of 2000 optimized suites was formed through stratified sampling, using \(N_{s} = 8\) strata, sampling \(N_{i} = 1\) ground motion from each stratum, resulting in \(N_{g,s} = 8\) ground motions per suite. For comparison, another dataset of 2,000 suites was formed through random sampling, using \(N_{g,r} = 8\) ground motions per suite.
The “Random” sampling method referred to herein, corresponds to the method described in PEER Report 2009/01 (Haselton et al. 2009) as “\(S_{a} \left( {T_{1} } \right)\) Scaling with Bin Selection” proposed by Shome et al. (1998). With this GMSM method a bin of ground motions is first selected with moment magnitude and distance criteria. Then a suite of ground motions is formed through random sampling without replacement, and scaling to \(S_{a} \left( {T_{1} } \right)\).
7.6 Incremental dynamic analysis
Incremental dynamic analysis (Vamvatsikos and Cornell 2002) was carried out by incrementing \(S_{a} \left( {T_{1} } \right)\) up to 1.2 g. \(S_{dN} \left( {T_{1} ,T_{1}^{{\prime }} } \right)\) is evaluated by integration of the elastic displacement spectrum from \(T_{1} = 1.45\,{\text{s}}\) to \(T_{1}^{{\prime }} = 2.90\,{\text{s}}\), i.e. \(T_{1}^{{\prime }} = 2.0T_{1}\). The upper intensity limit corresponds to the highest \(S_{a} \left( {T_{1} = 1.0\,{\text{s}}} \right)\) found on seismic hazard maps (e.g. Southern California) with probability of exceedance of 2% in 50 years. At this probability of exceedance, safety against collapse is evaluated. The ground motions used at each intensity increment were scaled to the target \(S_{a} \left( {T_{1} } \right)\). The response parameters investigated were the ParkAng overall structural damage index (Park and Ang 1985), \(OSDI\), and the maximum interstorey drift ratio, \(MIDR\).
7.7 Regression analysis
The probability distribution of \(S_{dN} \left( {1.45,2.90} \right)\) was found to conform well to the normal distribution, and to the lognormal distribution, using Lilliefors (1967) test at a significance level of 5%. The normal distribution was assumed in the regression analysis, as it resulted in the highest correlation with the response parameters. Similarly, by Lilliefors (1967) test it was found that the probability distributions of the response parameters \(OSDI\), and \(MIDR\), conform well to the lognormal distribution, at a significance level of 5%.
It can be observed that, in general, \(\rho\) is low (in the range of 0.2–0.3) at the low nonlinearity levels (at intensity \(S_{a} \left( {T_{1} } \right)\) between 0.2 and 0.6 g), and moderate (in the range of 0.5–0.6) at the high nonlinearity levels (at intensity \(S_{a} \left( {T_{1} } \right)\) between 1.0 and 1.2 g).
7.8 Response sample statistics
7.9 Response prediction
The computational work needed using the proposed GMSM method, expressed as number of ground motions required to obtain the same standard error as through random sampling, was reduced to about 72% at the moderate nonlinearity levels, and to about 50% at the high nonlinearity levels.
7.10 Elongated period estimation
After the application of the GMSM method in the response prediction of the MDOF system, in which the elongated period was assumed to be equal to \(T_{1}^{{\prime }} = 2.0T_{1}\) for practical purposes (Theophilou et al. 2017), the elongated period observed was investigated. The elongated period is the predominant period of oscillation when the structure enters the inelastic range and was estimated from the response power spectrum.
8 Differentiation to other methods
Goulet et al. (2007) suggested selecting records to match the deaggregation results in terms of magnitude, distance, and epsilon (Baker and Cornell 2005), and subsequently scaling them to the target \(S_{a} \left( {T_{1} } \right)\). In their method the spectral shape (i.e. epsilon) is assumed to be dependent on the magnitude and distance, thus a sufficiently large number of records (e.g. in their example 34) is required to cover all magnitudedistance combinations. The difference of the present GMSM method is that the selection criteria of magnitude and distance are relaxed. Thus, the method is applicable when the normalized spectral shape between \(T_{1}\) and \(T_{1}^{{\prime }}\) (i.e. Normalized Spectral Area) does not change significantly with magnitude and distance, as supported by the findings of Shome et al. (1998). In this way the number of required records for the same magnitude and distance ranges is reduced.
Jayaram et al. (2011) proposed a method that initially generates simulated acceleration response spectra, which collectively match the central tendency, dispersion, and multivariate distribution of the target response spectrum. Subsequently, ground motions are selected that match the simulated response spectra. As Jayaram et al. (2011) states, the suite of selected ground motions resulting from the main procedure may deviate slightly from the target central tendency and dispersion. For this reason, a supplementary “greedy” procedure is specified to replace the ground motions onebyone and thus minimize the residuals.
Bradley (2012) proposed an algorithm that generates random simulations of response spectra from the conditional multivariate distribution of intensity measures, obtained from the Generalized Conditional Intensity Measure (Bradley 2010). The method considers a multitude of intensity measures (including spectral acceleration), in contrast to the Jayaram et al. (2011) method that considers only spectral acceleration. It can also utilize the full seismic hazard disaggregation probability. Bradley (2012) also discusses the issue of variability between the matching ground motion suites, particularly for small suites, and proposed selecting the suite with the lowest residual between the empirical and target conditional distribution.
Both the Jayaram et al. (2011) method and the Bradley (2012) method place stringent requirements over the matching spectral shape. As Bradley (2012) states, finding a ground motion with identical spectral shape to the simulation is generally not possible. For this reason, ground motion selection is based on minimizing the spectral shape deviation from the simulations. The proposed GMSM method adopts a different approach, in which selection is based on \(S_{dN} \left( {T_{1} ,T_{1}^{{\prime }} } \right)\), which is a scalar parameter descriptive of the spectral shape, rather than the actual spectral shape. By limiting the matching period range between \(T_{1}\) and \(T_{1}^{{\prime }}\), the number of matching ground motions is higher. In addition, it adopts an unbiased estimator of the EDP central tendency, and it has been found to match the EDP dispersion well, thus avoiding the problem of residuals associated with the Jayaram et al. (2011) and Bradley (2012) methods.
9 Concluding remarks
A GMSM method has been proposed with which optimized suites of ground motions can be sampled, suitable for a probabilistic seismic response assessment of building structures. The key aim of the method is to estimate with reasonable accuracy the central tendency and the dispersion of the EDPs. It can be used in cases wherein seismic intensity can be defined in terms of \(S_{a} \left( {T_{1} } \right)\). Its advantage over other GMSM methods is that it implicitly matches the multivariate distribution of the response spectrum in the region of the structure elongated period. It also adopts an unbiased estimator of the EDP central tendency, and has been found to match the EDP dispersion well.
Ground motion intensity is expressed using the vectorvalued IM \(\left\langle S_{a} \left( {T_{1} } \right), S_{dN} \left( {T_{1} ,T_{1}^{{\prime }} } \right) \right\rangle\). Optimized suites of ground motions are formed by partitioning the distribution of \(S_{dN} \left( {T_{1} ,T_{1}^{{\prime }} } \right)\) into \(N_{s}\) strata, and then selecting \(N_{i}\) ground motions from each stratum. The proposed method replicates the mean and variance of \(IMT\), whereas the standard error is reduced, compared to random sampling. At the same time the mean and the variance of \(EDPT\) are also replicated. The advantage over random sampling is that when there is high enough correlation between \(IMT\) and \(EDPT\), stratified sampling results in a reduced standard error in the mean \(EDPT\). Consequently, an optimized response prediction is achieved.
The method was applied in the analysis of a firstmode dominated MDOF system, representing a tenstorey building frame designed to Eurocode 2 (2004a) and Eurocode 8 (2004b) for ductility class High. The central tendency of the response was found to have excellent conformity to the largesample dataset, while some discrepancy in the dispersion estimation was observed. Expressing the 95% percentile response as 95% prediction interval it was found that the highest ‘Efficiency Index’ achieved was 0.43 for \(OSDI\), and 0.52 for \(MIDR\). The computational work needed to obtain the same accuracy as through random sampling was reduced to about 72% at the moderate nonlinearity levels, and to about 50% at the high nonlinearity levels. The elongated period, estimated using the power spectrum method, was found to be in reasonable agreement with the assumed elongated period at the higher nonlinearity levels.
In conclusion, the proposed GMSM method is most efficient in predicting response at moderate to high nonlinearity levels, due to the significant reduction in the standard error of the \(EDPT\), which is a result of the sufficiently high correlations observed between the proposed \(IMT\) and the \(EDPT\). It is therefore suitable for firstmode dominated structures when the probabilistic assessment focuses on limit states associated with moderate to severe damage and collapse.
Notes
Acknowledgements
The author would like to express his sincere gratitude to Prof Marios K. Chryssanthopoulos and to Prof Andreas J. Kappos, who provided valuable feedback to this article.
References
 Ambraseys N, Smit P, Sigbjornsson R, Suhadolc P, Margaris B (2002) InternetSite for European StrongMotion Data, European Commission, ResearchDirectorate General, Environment and Climate Programme. http://smbase.itsak.gr/. Accessed 19 Sept 2012
 Ang AH, Tang WH (1975) Probability concepts in engineering planning and design: volume I—basic principles. Wiley, New YorkGoogle Scholar
 Ay BO, Akkar S (2012) A procedure on ground motion selection and scaling for nonlinear response of simple structural systems. Earthq Eng Struct Dyn 41(9):1693–1707. https://doi.org/10.1002/eqe.1198 CrossRefGoogle Scholar
 Azarbakht A, Dolsek M (2007) Prediction of the median IDA curve by employing a limited number of ground motions. Earthq Eng Struct Dyn 36(15):2401–2421. https://doi.org/10.1002/eqe.740 CrossRefGoogle Scholar
 Baker JW (2011) Conditional mean spectrum: tool for ground motion selection. J Struct Eng Am Soc Civ Eng 137(3):322–331. https://doi.org/10.1061/(ASCE)ST.1943541X.0000215 CrossRefGoogle Scholar
 Baker JW, Cornell CA (2005) A vectorvalued ground motion intensity measure consisting of spectral acceleration and epsilon. Earthq Eng Struct Dyn 34(10):1193–1217. https://doi.org/10.1002/eqe.474 CrossRefGoogle Scholar
 Boore DM, Atkinson GM (2008) Groundmotion prediction equations for the average horizontal component of PGA, PGV, and 5%damped PSA at spectral periods between 0.01 s and 10.0 s. Earthq Spectra 24(1):99–138. https://doi.org/10.1193/1.2830434 CrossRefGoogle Scholar
 Bradley BA (2010) A generalized conditional intensity measure approach and holistic ground motion selection. Earthq Eng Struct Dyn 39(12):1321–1342. https://doi.org/10.1002/eqe.995 Google Scholar
 Bradley BA (2012) A ground motion selection algorithm based on the generalized conditional intensity measure approach. Soil Dyn Earthq Eng 40:48–61. https://doi.org/10.1016/j.soildyn.2012.04.007 CrossRefGoogle Scholar
 Clough RW, Johnston SB (1966) Effect of stiffness degradation on earthquake ductility requirements. In: Proceedings of the Japan earthquake engineering symposiumGoogle Scholar
 Cochran WG (1977) Sampling techniques, 3rd edn. Wiley, New YorkGoogle Scholar
 European Committee for Standardization (2004a) BS EN 199211:2004. Eurocode 2—design of concrete structures. Part 11: general rules and rules for buildings. British Standards InstitutionGoogle Scholar
 European Committee for Standardization (2004b) BS EN 19981:2004. Eurocode 8—design of structures for earthquake resistance. Part 1: general rules, seismic actions and rules for buildings. British Standards InstitutionGoogle Scholar
 Federal Emergency Management Agency (2005) Improvement of nonlinear static seismic analysis procedures, Report FEMA 440, Washington DCGoogle Scholar
 Federal Emergency Management Agency (2012) Seismic performance assessment of buildings, volume 1—methodology, Report FEMA P581, Washington DCGoogle Scholar
 Goulet CA, Haselton CB, MitraniReiser J, Beck JL, Deierlein GG, Porter KA, Stewart JP (2007) Evaluation of the seismic performance of a codeconforming reinforced concrete frame building—from seismic hazard to collapse safety and economic losses. Earthq Eng Struct Dyn 36(13):1973–1997. https://doi.org/10.1002/eqe.694 CrossRefGoogle Scholar
 Haselton CB, Deierlein GG (2007) Assessing seismic collapse safety of modern reinforced concrete moment frame buildings. Report No. 156. John A. Blume Earthquake Engineering Research Center, Stanford UniversityGoogle Scholar
 Haselton CB, Baker JW, Bozorgnia Y, Goulet CA, Kalkan E, Luco N, Shantz T, Shome N, Stewart JP, Tothong P, WatsonLamprey J, Zareian F (2009) Evaluation of ground motion selection and modification methods: predicting median interstory drift of buildings. PEER Report 2009/01, Pacific Earthquake Engineering CentreGoogle Scholar
 Iervolino I, Cornell CA (2005) Record selection for nonlinear seismic analysis of structures. Earthq Spectra 21(3):685–713. https://doi.org/10.1193/1.1990199 CrossRefGoogle Scholar
 Iervolino I, Galasso C, Cosenza C (2010) REXEL: computer aided record selection for codebased seismic structural analysis. Bull Earthq Eng 8(2):339–362. https://doi.org/10.1007/s1051800991461 CrossRefGoogle Scholar
 International Code Council (2015) International building code. International Code Council, Inc., Country Club Hills, ILGoogle Scholar
 Jayaram N, Baker JW (2008) Statistical tests of the joint distribution of spectral acceleration values. Bull Seismol Soc Am 98(5):2231–2243. https://doi.org/10.1785/0120070208 CrossRefGoogle Scholar
 Jayaram N, Lin T, Baker JW (2011) A computationally efficient groundmotion selection algorithm for matching a target response spectrum mean and variance. Earthq Spectra 27(3):797–815. https://doi.org/10.1193/1.3608002 CrossRefGoogle Scholar
 Katsanos EI, Sextos GS (2013) ISSARS: an integrated software environment for structurespecific earthquake ground motion selection. Adv Eng Softw 58:70–85. https://doi.org/10.1016/j.advengsoft.2013.01.003 CrossRefGoogle Scholar
 Kayhan AH, Korkmaz KA, Irfanoglu A (2011) Selecting and scaling real ground motions using a harmony search algorithm. Soil Dyn Earthq Eng 31(7):941–953. https://doi.org/10.1016/j.soildyn.2011.02.009 CrossRefGoogle Scholar
 Kottke A, Rathje EM (2008) A semiautomated procedure for selecting and scaling recorded earthquake motions for dynamic analysis. Earthq Spectra 24(4):911–932. https://doi.org/10.1193/1.2985772 CrossRefGoogle Scholar
 Lignos DG, Krawinkler H (2012a) Sidesway collapse of deteriorating structural systems under seismic excitations. Rep. No. 177, JA Blume Center, Department of Civil and Environmental Engineering, Stanford UniversityGoogle Scholar
 Lignos DG, Krawinkler H (2012b) Development and utilization of structural component databases for performancebased earthquake engineering. J Struct Eng Am Soc Civ Eng 139(8):1382–1394. https://doi.org/10.1061/(ASCE)ST.1943541X.0000646 CrossRefGoogle Scholar
 Lilliefors HW (1967) On the KolmogorovSmirnov test for normality with mean and variance unknown. J Am Stat Assoc 62:399–402CrossRefGoogle Scholar
 Mahin SA, Lin J (1983) Construction of inelastic response spectra for single degree of freedom systems. UCB/EER83/17, Earthquake Engineering Research Center, University of California at BerkleyGoogle Scholar
 Naeim F, Alimoradi A, Pezeshk S (2004) Selection and scaling of ground motion time histories for structural design using genetic algorithms. Earthq Spectra 20(2):413–426. https://doi.org/10.1193/1.1719028 CrossRefGoogle Scholar
 Pacific Earthquake Engineering Research Center (2005) NGA Database. University of California at BerkleyGoogle Scholar
 Pacific Earthquake Engineering Research Center (2013) OpenSees software version 2.4.0. The Regents of the University of California. http://opensees.berkeley.edu/index.php. Accessed 4 July 2016
 Park YJ, Ang AHS (1985) Mechanistic seismic damage model for reinforced concrete. J Struct Eng Am Soc Civ Eng 111(4):722–739. https://doi.org/10.1061/(ASCE)07339445(1985)111:4(722) CrossRefGoogle Scholar
 Shome N, Cornell CA, Bazzuro P, Carballo JE (1998) Earthquakes, records and nonlinear responses. Earthq Spectra 14(3):469–500. https://doi.org/10.1193/1.1586011 CrossRefGoogle Scholar
 Theophilou AI (2014) A ground motion selection and modification method suitable for probabilistic seismic assessment of building structures. Ph.D. Thesis, University of Surrey, UKGoogle Scholar
 Theophilou AI, Chryssanthopoulos MK, Kappos AJ (2017) A vectorvalued ground motion intensity measure incorporating normalized spectral area. Bull Earthq Eng 15(1):249–270. https://doi.org/10.1007/s1051801699597 CrossRefGoogle Scholar
 Vamvatsikos D, Cornell CA (2002) Incremental dynamic analysis. Earthq Eng Struct Dyn 31(3):491–514. https://doi.org/10.1002/eqe.141 CrossRefGoogle Scholar
 Walpole PE, Myers RH, Myers SL, Ye K (2007) Probability and statistics for engineers and scientists. Pearson Prentice Hall, Upper Saddle RiverGoogle Scholar
 Wang G (2011) A ground motion selection and modification method capturing response spectrum characteristics and variability of scenario earthquakes. Soil Dyn Earthq Eng 31(4):611–625CrossRefGoogle Scholar
 WatsonLamprey JA, Abrahamson NA (2006) Selection of ground motion time series and limits on scaling. Soil Dyn Earthq Eng 26(5):477–482. https://doi.org/10.1016/j.soildyn.2005.07.001 CrossRefGoogle Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.