Vector-valued Intensity Measures Research Articles

A Computationally Efficient Algorithm for Constructing Effective Vector-Valued Seismic Intensity Measures for Engineering Structures

ABSTRACT Seismic intensity measures (IMs) quantify the severity of ground motions and their impacts on structures. They play a vital role in many aspects of earthquake engineering. This paper proposes a novel method, namely the express iteration method (EIM), for constructing effective vector-valued IMs based on dozens of existing scalar ones given a specific engineering structure or a class of them. Taking advantage of the sophisticated while efficient mapping between scalar IMs and engineering demand parameters (EDPs) via a machine learning model, EIM iteratively eliminates less important scalar IMs from a pool of candidates to find the most effective combinations for a vector-valued IM and achieves superior computational efficiency by avoiding updating the nonlinear mapping during the process. Taking a base-isolated structure and its non-isolated counterpart for a demonstrating case study, the performance of the vector-valued IMs determined by EIM is compared with those by other existing methods in the literature for the task of selecting the most unfavorable ground motions. The results show that EIM prioritizes records with the largest peak inter-story drift PIDs and thus leads to the smallest subset that imposes most severe structural damage, while its computational cost was two orders of magnitude smaller as compared to the existing methods of similar effectiveness. Such superior performance can also be expected in all tasks that involve vector-valued IMs, including but not limited to multi-dimensional fragility analysis, incremental dynamic analysis, and real-time seismic damage prediction.

Seismic fragility assessment of aqueduct bent structures subjected to mainshock-aftershock sequences

This study aims to assess the seismic fragility of the aqueduct bent structure subjected to mainshock-aftershock (MS-AS) ground motion sequences by double incremental dynamic analysis method. The finite element model of the aqueduct bent structure is developed in OpenSees platform and numerical simulation technique is validated through physical test results. A series of nonlinear dynamic time-history analysis is carried out on the aqueduct bent structure to obtain its seismic response under different ground motion intensities. Furthermore, the incremental dynamic analysis (IDA) data cluster and median lines of the damage measure of the aqueduct bent structure are constructed. The advanced probabilistic seismic demand model (PSDM) of the aqueduct bent structure subjected to MS-AS ground motion sequences is proposed. Based on the proposed PSDM and the predefined damage states, the fragility surfaces of the aqueduct bent structure are also developed by adopting the vector-valued intensity measures considering the mainshock intensities (IMm) and aftershock intensities (IMa). Meanwhile, for the sake of comparation, the aftershock fragility curves of the aqueduct bent structure under different mainshock intensities are drawn, which emphasizes the importance of aftershocks in the seismic risk assessment. The analysis results indicate that the proposed piecewise probabilistic seismic demand model can better fit the logarithmic relationship between the aftershock intensities and the median of the reinforcement longitudinal tensile strain. The structural fragility probability increase with the increase of mainshock and aftershock intensities. It turns out that the mainshock intensities play a leading role in the damage probability of the aqueduct structure subjected to MS-AS ground motion sequences, the aftershock can cause additional damage to the aqueduct bent structure and reduce its seismic capacity. The meaningful results can provide supports for seismic optimization design, resilience improvement and post-earthquake repair-demand of the aqueduct structure in Southwest China.

Vector-Valued Intensity Measures for Seismic Risk Assessment of Base-Isolated Liquid Storage Tanks

The present study proposes vector-valued intensity measures (IMs), consisting of two IM parameters, for improved and efficient seismic risk assessment of liquid storage tanks base isolated with laminated rubber bearing (LRB) and lead rubber bearing or New Zealand base-isolation system (NZ system). To assess the seismic risk of the base-isolated liquid storage tanks, their seismic fragility was obtained using stripe analysis. The joint hazard of the selected IMs was obtained using vector-valued probabilistic seismic hazard analysis. It is shown that the vector-valued intensity measure consisting of peak ground acceleration (PGA) and spectral acceleration (Sa) at the isolation period (Tb) is efficient for predicting the peak base shear demand in the liquid storage tanks base isolated with LRB. Similarly, the vector-valued intensity measure consisting of PGA and peak ground velocity (PGV) is efficient for predicting the peak base shear in the liquid storage tanks base isolated with the NZ system. This study also presents parametric investigations on the effect of thicknesses of the tank wall (tw) and the slenderness ratio (S) of the tank on the seismic risk of the base-isolated liquid storage tanks.

Vector-Valued Intensity Measures Based on Spectral Shape to Predict Seismic Fragility Surfaces in Reinforced Concrete Buildings

Although some studies have been conducted to compute fragility surfaces of buildings using vector-valued seismic intensity measures (IMs), in all the cases, the first component of the vector usually is the spectral acceleration at first mode of vibration of the structure Sa(T1). In this study, fragility surfaces of three reinforced concrete buildings (RCB) subjected to narrow-band ground motions obtained from soft soil of Mexico City are computed considering vector-valued IMs based not only on Sa(T1), but also the velocity V(T1), pseudo-velocity Sv(T1), and normalized input energy by the mass EI/m(T1) as the first component. As second component of the vector-valued IMs, the Peak Ground Acceleration (PGA), Peak Ground Velocity (PGV), effective duration (tD), earthquake damage potential (ID) and four Np spectral shape-based parameters obtained through spectral acceleration (NpSa), velocity (NpV), pseudo-velocity (NpSv), and input energy (NpEI) have been analyzed. In order to obtain fragility surfaces, Multinomial Logistic Regression (MLR) was applied. It is observed that those vector-valued IMs based on the spectral shape proxies were more efficient to predict the probability of failure of RCB. For this reason, it is important to consider spectral shape vector-valued IMs in order to reduce uncertainty in the structural response of buildings under earthquakes. Thus, the use of two parameters instead of a single intensity measure improves the efficiency. Moreover, the fragility surfaces can be used for the seismic risk evaluation of buildings.

Scores: an algorithm for records selection to employ in seismic risk and resilience analysis

Modern approaches for the assessment of seismic behavior of structures (i.e. PBEE) are often based on nonlinear dynamic analyses. In this respect, the selection of ground motion records is of paramount importance, because of the record-to-record variability of the structural response. In the past, selection of natural records was often performed based on the compatibility between the response spectra and a target spectrum, usually derived from a PSHA or a code spectrum. Spectral matching can be considered the most commonly proposed earthquake record selection, currently adopted by many seismic codes. The main problem of this approach is that the record-to-record variability can result excessive, obtaining a considerable scatter in the non-linear response. For these reasons, it is usually preferable to synthetize the seismic hazard properties by using a scalar or vector-valued Intensity Measure (IM). Nevertheless, the selection of efficient, sufficient and practical IMs leads to different approaches, without converging toward a unified approach. Consequently, aim of this paper is to propose a novel algorithm for records selection to employ in risk analysis, which is able to control the dispersion without resorting to the artificial scaling of the accelerograms and thus easily employed in three-dimensional structural analysis. It is based on the idea to control the record-to-record variability rather than to reduce it by scaling the accelerograms. The proposes methodology has been implemented in the software SCoReS and used for the risk assessment of a RC frame structures.

A Simplified Method for Performing Vector-Valued Probabilistic Seismic Hazard Analysis

ABSTRACT For a seismic risk analysis, the seismic hazard results are usually combined with fragility functions to calculate the probability of seismic losses. Because vector-valued ground-motion intensity measures (IMs) are increasingly used by researchers and practicing engineers to evaluate fragility functions, seismic hazard calculations in terms of vector-valued IMs are required, which are determined by vector-valued probabilistic seismic hazard analysis (PSHA). However, the large computational cost associated with vector-valued PSHAs impedes the application of vector-valued IMs in performing a seismic risk analysis. The study presents a simplified method for conducting vector-valued PSHAs. This simplified method is developed using proposed equivalent earthquakes that are derived from conventional seismic hazard deaggregation. Formulas for performing vector-valued PSHAs using the simplified method are also proposed. Based on a numerical example, this study demonstrates the accuracy and efficiency of the proposed method.

Optimal selection of scalar and vector-valued intensity measures for improved fragility analysis in cross-fault hydraulic tunnels

Seismic intensity measures (IMs) are crucial because they are significantly connected to earthquake hazards and structural responses. It means that selecting appropriate IMs is necessary for underground structures under unfavourable earthquake hazard conditions. The seismic design code and dynamic analysis have conventionally employed the peak ground acceleration (PGA) as an optimal IM, mainly referring to the seismic design code of aboveground structures. However, seismic investigation of underground structures using PGA is still controversial. Simultaneously, directly employing the PGA as an ideal IM may increase the uncertainty associated with ground motions, which in turn increases the uncertainty for probabilistic seismic demand analysis of hydraulic tunnels. As a result, it is important to understand the roles played by different IMs while analyzing the fragility of hydraulic tunnels. This work aims to conduct a comprehensive analysis to identify the optimal scalar- and vector-valued IMs for fragility investigation of cross-fault hydraulic tunnels. Therefore, the 1080 complete nonlinear dynamic time history analysis by surrounding rock-concrete lining-fluid interaction system is conducted to express the earthquake-induced drift ratio of a cross-fault hydraulic tunnel by incremental dynamic analysis. Subsequently, a refinement process is performed to determine the optimal scalar-valued IMs by comparing the correlation, efficiency, practicality, and proficiency of the 7 examined IMs. Meanwhile, the probabilistic seismic demand model of optimum vector-valued IMs is also developed and compared, followed by the three scalar IMs. Eventually, the difference between the fragility curves of the tunnel produced using the optimal scalar- and vector-valued IM is compared. The results demonstrate that compared with scalar two-dimensional fragility curves, the vector three-dimensional fragility surface may increase the reliable probability and minimize the uncertainty in analyzing the structural response hazard curve. The generated vector fragility surface can also estimate the seismic fragility of identical cross-fault hydraulic tunnels in an approximative manner.

Identification of optimal ground-motion intensity measures for assessing liquefaction triggering and lateral displacement of liquefiable sloping grounds

This article aims at identifying the optimal scalar- and vector-valued intensity measures (IMs) for predicting liquefaction triggering and the consequence of liquefaction (i.e. permanent deformation of sloping grounds), respectively. A total of 169 ground motions were selected from the NGA-West2 database. Using the selected ground motions as input, one-dimensional dynamic analyses were conducted for generic sloping ground soil profiles consisting of a loose-sand liquefiable layer implemented in OpenSees. The excess pore water pressure ratio and the liquefaction-induced lateral displacement were regarded as the engineering demand parameters (EDPs) of interest. A total of 20 scalar-IMs and 21 vector-IMs representative of various characteristics of seismic loading were evaluated as candidate IMs. The criteria for identifying optimal IMs include consideration for the efficiency, sufficiency, and predictability of the given IMs. An error propagation approach was employed to evalute the overall uncertainty of EDPs under scenario earthquakes using different IMs as predictor variables. Based on the results of extensive comparative analyses, acceleration spectrum intensity (ASI), and the vector of peak ground acceleration and spectral acceleration at 0.3 s (peak ground acceleration (PGA) and Sa(0.3 s), respectively) are identified as the optimal scalar- and vector-IM for liquefaction triggering evaluation, respectively. However, Arias intensity ( Ia) and vectors of [ Ia, cumulative absolute velocity] and [PGA, Ia] are identified as the optimal scalar- and vector-IM for predicting the liquefaction-induced lateral displacements. The optimal IMs identified could be used in performance-based liquefaction hazard evaluations, to assess the liquefaction triggering and consequence of liquefiable sites.

Multi-Regression Analysis to Enhance the Predictability of the Seismic Response of Buildings

Several methodologies for assessing seismic risk extract information from the statistical relationship between the intensity of ground motions and the structural response. The first group is represented by intensity measures (IMs) whilst the latter by engineering demand parameters (EDPs). The higher the correlation between them, the lesser the uncertainty in estimating seismic damage in structures. In general, IMs are composed by either a single (scalar-based IMs) or a group of features of both the ground motion and the structure (vector-valued IMs); the latter category provides higher efficiency to explain EDPs when compared to the first one. This paper explores how to find new vector-valued IMs, which are highly correlated with EDPs, by means of multi-regression analysis. To do so, probabilistic nonlinear dynamic analyses have been performed by considering a seven-story reinforced concrete building as a testbed. At a first stage, 30 scalar-based IMs have been correlated with 4 EDPs (i.e., 120 groups of IM-EDP pairs have been studied). Afterwards, the structural responses have been classified as elastic, inelastic and a combination of both. It has been analyzed how efficiency behaves when making these classifications. Then, 435 vector-valued IMs have been created to enhance the predictability of the scalar EDPs (i.e., 1740 groups of IM-EDP pairs have been analyzed). Again, the most efficient IMs have been identified. Sufficiency, which is another statistical property desired in IMs, has also been examined. Results show that the efficiency and sufficiency to predict the structural response increase when considering vector-valued IMs. This sophistication has important consequences in terms of design or assessment of civil structures.

Seismic vulnerability assessment of tall pier bridges under mainshock-aftershock-like earthquake sequences using vector-valued intensity measure

In most current seismic investigations on bridges with tall piers, only mainshock actions are considered without incorporating the effect of mainshock-aftershock (MA) sequences. This paper investigates the seismic fragility and the associated direct financial losses of tall pier bridges subjected to MA sequences using the vector-based intensity measure considering both main- and after-shocks. Numerical model of a typical RC tall pier bridge is developed and validated through shake table test results. Nonlinear time-history analyses are extensively conducted to obtain the seismic responses of the bridge under MA sequences, which are further utilized for the development of fragility surfaces. The analytical results show that in the MA sequences, the contribution of the aftershocks to the peak structural seismic demands is basically negligible unless their intensities exceeding critical values. While the structural vulnerabilities increase with both the intensities of main- and after-shocks, the role of mainshocks is observed more substantial. On the other hand, however, by totally neglecting the effect of the aftershocks, the failure probability and financial loss of the tall bridge would be underestimated, particularly when strong aftershocks occur following mainshocks.

Performance of Vector-valued Intensity Measures for Estimating Residual Drift of Steel MRFs with Viscous Dampers

Performance of Vector-valued Intensity Measures for Estimating Residual Drift of Steel MRFs with Viscous Dampers

Analyzing the performance of different seismic demand models in RC moment-resisting frames

The relationship between seismic characteristics, known as the intensity measure, and the structural demand parameter of the structure, called the seismic demand model and has been assumed to be linear on a logarithmic scale in classical studies. Because of the critical role of the seismic demand model in reliability analysis of structures and reducing uncertainties, in the present study, the performance of different demand models is compared to the simple linear model. Different criteria such as sufficiency, efficiency, and correlation are considered in this analysis. The effect of different scalar or vector-valued intensity measures, pulse-like earthquake records, structural damage level, structural demand parameters, and height of structures on the performance of the demand models with the help of relative entropy concepts is estimated. The results indicate the necessity of nonlinear demand models, especially at high structural damage levels. Therefore the second-order demand model is more suitable due to simplicity rather than other nonlinear demand models.

A Novel Structure-Pulse Coupled Model for Quantifying the Column Ductility Demand under Pulse-Like GMs

ABSTRACT A novel model is constructed to investigate the coupling effect of pulse period (Tp ) and peak pulse velocity (Vp ) on structural demands with the varied fundamental period (T1 ) under pulse-like ground motions. The Gaussian function is proposed to quantitatively describe the bell-shape curve of column ductility demand in terms of ln(T1 /Tp ). Further investigation reveals that the two critical parameters of Gaussian function, i.e. the height of the curve’s peak and the position of the center of the peak, are significantly influenced by Vp , and analytically illustrated by Power function and Boltzmann function, respectively. verification of the proposed two-dimensional structure-pulse coupling model exhibits its accuracy and feasibility in predicting the seismic demand conditioned on vector-valued intensity measure ([Vp , ln(T1 /Tp )]) under pulse-like ground motion. Moreover, the coupling model can be used to identify and quantify the response regularity, for instance, the phenomenon that the center position of the peak shifts from 1.0 to 0.5 by increasing Vp is analytically captured in this study. Lastly, the coupling model is also capable to identify the unfavorable range of structural parameters, which is quite practical for near-fault seismic design and risk assessment.

Optimal selection of scalar and vector-valued seismic intensity measures based on Gaussian Process Regression

Intensity measure (IM) represents the power of ground motion, and its ability to describe the characteristics of ground motion plays a vital role in seismic risk and damage assessment. Therefore, the selection of optimal IM has always been one of the focuses of researchers in seismic engineering. The performance metrics based on regression or non-parametric methods, including efficiency, sufficiency, proficiency, and practicality, have been widely applied to select the optimal IM in past studies. This paper proposes a new procedure for performance evaluation of IM based on the Gaussian Process Regression (GPR), which can deal with the linear or nonlinear demand-IM relationship, and scalar or vector-valued IM. Two novel criteria, including G-Precision and G-Sufficiency, have also been developed to present the prediction accuracy and sufficiency of IM, combined with the concept of generalization performance in machine learning to update the existing metrics. A practical algorithm called Sequential Floating Feature Selection (SFFS) is proposed to automatically find the optimal vector-valued IM from a set of candidates. The proposed method can be integrated with multi-dimensional seismic fragility analysis to determine the optimal IM input, either scalar or vector-valued. Finally, the proposed procedure is demonstrated and discussed on a three-span continuous girder bridge.

Synthesis of a vector-valued intensity measure for improved prediction of seismic demands in Inter-Story-Isolated (ISI) buildings subjected to near fault ground motions

An optimal Intensity Measure (IM) is key to accurate prediction of the Engineering Demand Parameters (EDPs) in structures subjected to seismic excitations. This study synthesizes a structure-specific, vector-valued IM for assessing the EDPs in Inter-Story-Isolated (ISI) buildings under near-fault (pulse and non-pulse type) ground motions. The synthesis is inspired by the comparable modal masses in the fundamental and another higher (generally second) mode in ISI buildings. The synthesis accommodates the dual mode participation by employing the respective pseudo-spectral accelerations Spa and the spectral shape (‘epsilon’ε) parameters in the proposed IM as components. A suite of pulse and non-pulse type motions are employed for the synthesis and subsequently fitting an IM vs. EDP regression model. Thereafter, the “Efficiency” and “Sufficiency” analysis are conducted for validation. The EDPs are obtained by non-linear dynamic analyses on alternative building frames with High Damping Rubber Bearing (HDRB) based ISI systems. A Principal Component Regression (PCR) is employed for accommodating the multi-collinearity in the IM. The choice of the Principal Components (PCs) adopts a leave-one-out cross-validation approach. The proposed IM is shown to offer higher Efficiency and Sufficiency compared with the existing non-structure-specific (Peak Ground Acceleration, Velocity, Displacement, Arias Intensity, Cumulative Absolute Velocity, Characteristic Intensity, Specific Energy Density etc.) and structure-specific (Acceleration/Velocity Spectrum Intensity, Housner Intensity, Pseudo-spectral acceleration at fundamental time period etc.) IMs. Higher Efficiency is also noted under the pulse type motions comparing the non-pulse ones. The IM is more efficient in predicting the Inter-Story Drift Ratios (IDRs), Total Drift Ratios (TDRs) than the Peak Floor Accelerations (PFA) and Bearing Displacements (PBD). The proposed IM is shown to be unbiased and robust.

An extended probabilistic demand model with optimal intensity measures for seismic performance characterization of isolated bridges under coupled horizontal and vertical motions

Intensity Measures (IMs) provide the relationship between Engineering Demand Parameters (EDPs) and seismic hazard characteristics for different structures and hence, their role in performance-based bridge design is significant. A few studies have investigated the optimal IMs for bridges under horizontal ground motions, however, detailed studies to explore the optimal IMs for isolated-bridges under the coupled vertical and horizontal records are very rare. This paper presents a procedure for the selection of optimal IMs for seismic-isolated bridges under the combined strong Horizontal Component (HC) and Vertical Component (VC) seismic excitations. Soil-Structure-Interaction (SSI) effects, the high level of uncertainties for soil properties, uncertainties for structural and geotechnical issues and advanced plasticity model for non-liquefied soil are included to explore the optimal vector-valued IMs. Four individual situations are considered to investigate the geometry effects on the selected optimal IMs. Optimal IMs criteria including efficiency, practicality, proficiency and sufficiency are investigated and developed to achieve a set of optimal vector-valued IMs for critical EDPs: drift ratio, pile-cap displacement and bearing displacement, which are affected by both HC and VC in isolated Soil-Pile-Bridge (SPB) systems. The results show that velocity-related IMs: peak ground velocity (PGVH), Housner spectrum intensity (HIH) and root mean square of velocity (VRMSH) are optimal IMs as representative of HCs. In addition, structure-dependent spectral IMs: vertical spectral acceleration at T = 0.2 s ( $${\text{S}}_{\text{a}0.2}^{\text{V}}$$ ) and square-root-of-the-sum-of-square of vertical spectral acceleration at the first and second vertical periods ( $${\text{S}}_{\text{a}}^{\text{V}}\left({\text{T}}_{\text{s}}\right)$$ ) are the appropriate optimal IMs as representative of VCs. However, the displacement-related IM, peak ground displacement of HCs (PGDH), may be treated as optimal IMs for SPB system supported by inclined pile foundations to investigate pile-cap displacement. In addition, the sufficiency term is investigated extensively with respect to seismological parameters.

Scalar- and vector-valued vulnerability analysis of shallow circular tunnel in soft soil

This paper presents a scalar- and vector-valued vulnerability analysis of shallow circular tunnel in soft soil. Numerous dynamic analyses were firstly performed to capture seismic response of the tunnel lining under transversal seismic shaking. Subsequently, a series of 15 intensity measures (IMs) are selected and tested to find the optimal IM used in the scalar-valued vulnerability analysis. The selected IMs are all tested based on their correlation, efficiency, practicality and proficiency, according to the regression analyses between the IMs and the damage measure (DM) for the examined tunnel. The peak ground acceleration (PGA) at the ground surface was demonstrated to be optimal IM, followed by peak ground velocity (PGV) and acceleration spectrum intensity (ASI). The scalar-valued vulnerability function was then developed in terms of optimal IM (PGA). Finally, the vulnerability functions based on vector-valued IMs were developed to obtain the exceeding probability of various damage states as a function of two IMs (PGA, ASI). This study reveals that the use of vector-valued IMs in the vulnerability analysis can significantly improve the correlation and efficiency with the seismic demand DM compared to the scalar-valued IM, thus reducing the uncertainties in evaluating earthquake risk. The scalar-valued vulnerability curves cannot represent the effect of a second IM on the seismic behavior of the tunnel, and it may either underestimate or overestimate the damage probability. Hence, the use of the second IM in vulnerability analysis and the development of vector-valued vulnerability functions lead to more proper evaluations of seismic risk.

The role of intensity measures on the accuracy of seismic fragilities

Seismic fragilities are the probability that structural response of a system overcomes specified limit values for given seismic intensity measures. These curves are frequently defined as functions of single/multiple ordinates of the pseudo-acceleration response spectrum. Recently it was reported that this approach can lead to inaccurate estimation of the structural performance for complex non-linear multi-degree of freedom systems. This preliminary paper investigates the accuracy of fragility curves for an actual building struck by the 2016 Italian earthquake. Numerical analyses of a multi-degree of freedom structural system subjected to this earthquake are performed, considering both linear and non-linear behaviour, and used to construct scatter plots of intensity measures versus the corresponding actual maximum of the response time history. Finally, an example of the fragility curves estimated using different intensity measures is presented for both linear and nonlinear structural behaviour. It is shown that these fragilities have very large variability depending on the selected scalar or vector-valued intensity measure.

Assessment Supporting the Use of Outcropping Rock Evolutionary Intensity Measures for Prediction of Liquefaction Consequences

This study evaluates a variety of intensity measures (IMs) for predicting the liquefaction-induced residual settlement and tilt of shallow-founded structures. We use data from both numerical and physical (centrifuge) models of soil-foundation-structure systems. The relative quality of these IMs is quantified in terms of efficiency, sufficiency, and predictability. We consider both scalar and vector-valued IMs and evaluate the relative performance of IMs recorded at different locations (outcropping rock, within rock, far-field, and foundation) from nonlinear and equivalent-linear simulations. Cumulative absolute velocity (CAV) at outcropping rock is the optimum IM for predicting foundation settlement, while either outcropping rock CAV, peak ground velocity, or peak incremental ground velocity is optimum for predicting permanent foundation tilt. Vector IMs offer improvements to efficiency and sufficiency but may be impractical to predict.

New vector-valued intensity measure for predicting the collapse capacity of steel moment resisting frames with viscous dampers

Viscous Dampers (VDs), as a passive energy dissipation system, are increasingly applied for the seismic design and retrofit of structures. In general, probabilistic seismic assessment of structures with VDs is performed by traditional scalar Intensity Measures (IMs) such as pseudo-spectral acceleration at the first mode period of the structure, Sa(T1). In this paper, the seismic collapse of steel Special Moment Resisting Frames (SMRFs) with VDs is assessed by 18 vector-valued IMs. The first component of the vector-valued IMs is Sa(T1) and their second components are un-scalable. Incremental Dynamic Analyses (IDAs) are performed to compute the seismic collapse capacity, Sacol, values of steel SMRFs with VDs. Then, the efficiency and sufficiency of the IMs for the prediction of the collapse capacity of the structures are studied, and (Sa(T1), SaRatioM-D) is proposed as an advanced vector-valued IM that includes the spectral shape and Ground Motion (GM) duration effects. The results indicate that (Sa(T1), SaRatioM-D) possesses high efficiency and sufficiency, and can reliably predict the collapse capacity of steel SMRFs with VDs.