Intensity measures for the seismic response prediction of mid-rise buildings with hysteretic dampers

Engineering Structures 102 (2015) 278–295

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Intensity measures for the seismic response prediction of mid-rise buildings with hysteretic dampers J. Donaire-Ávila a, F. Mollaioli b, A. Lucchini b, A. Benavent-Climent c,⇑ a

Department of Structural Mechanics and Hydraulic Engineering, University of Granada, Spain Department of Structural and Geotechnical Engineering, Sapienza University of Rome, Italy c Department of Mechanical Engineering, ETSII, Polytechnic University of Madrid, Spain b

a r t i c l e

i n f o

Article history: Received 10 February 2015 Revised 13 August 2015 Accepted 14 August 2015

Keywords: Intensity measures Seismic response prediction Hysteretic dampers Ordinary records Pulse-like near-fault records

a b s t r a c t Dampers are energy-dissipating devices widely applied for new and existing structures in earthquake prone areas. Among the different types of devices, hysteretic dampers are particularly popular due to their simplicity, economy and low cost. Although many studies have focused on ordinary buildings for evaluating the predictive capability of the different intensity measures (IMs), those dedicated to structures with dampers are scarce. The objective of this paper is to evaluate the capability of the most commonly used IMs to predict the seismic response of frame structures with hysteretic dampers, having low-to-moderate height (less than about 12 stories) and low height-to-width aspect ratios (less than approximately 3). To this end, a 6-story reinforced concrete (RC) frame structure designed to fulfill the old Italian seismic codes was retrofitted with hysteretic dampers. The dampers were designed for two different scenarios depending on the distance to the fault (i.e. near and far field ground motions). Two sets of accelerograms, consisting of ordinary and pulse-like near-fault records, are used in the analyses. Modified versions of existing IMs are also proposed, with the intention of improving the correlations between the considered IMs and response quantities. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction In the conceptual framework introduced in Performance-Based Earthquake Engineering, PBEE [1], the characterization of strong ground motion by means of suitable intensity measures (IMs) is a crucial element in the analysis of seismic risk of structures. It is particularly important to define optimal IMs that are capable of describing the probability that a structure would exceed a given limit state, usually represented by an Engineering Demand Parameter (EDP), at a designated site [2,3]. In PBEE, the selection of an optimal IM—an intermediate variable between ground motion hazard and structural demand estimates—for representing ground motion uncertainty is clearly a key issue to be addressed. The use of a specific IM in seismic risk analysis should correspond to the local or global damage of a given structural system. The stronger the correlation between the predicted EDP and the adopted IM, the more precise the result of a probabilistic risk assessment. A number of concepts and quantities can be considered when appraising the suitability of an IM in representing the dominant features of ground shaking. Optimum ⇑ Corresponding author. Tel.: +34 913 363135. E-mail address: [email protected] (A. Benavent-Climent). http://dx.doi.org/10.1016/j.engstruct.2015.08.023 0141-0296/Ó 2015 Elsevier Ltd. All rights reserved.

intensity measures are therefore defined in terms of sufficiency, efficiency, scaling robustness, predictability (through a probabilistic seismic hazard analysis) and practicality [3–5]. The first two of these properties are of particular importance for the present study. Efficiency refers to the total variability of an EDP for a given IM. A highly efficient IM calls for fewer ground motions and numerical analyses to achieve a desired level of confidence in the EDP response. In turn, sufficiency describes the extent to which the IM is statistically independent of ground motion characteristics such as magnitude and distance; that is, sufficiency renders the structural demand measure regardless of the earthquake scenario. Hence, when using a sufficient IM, a comprehensive groundmotion record selection is not needed, though the same accuracy in seismic structural performance estimation is achieved. The two properties may be quantified via statistical analysis of the response of a structure for a given set of records. Several alternative IMs and their predictive capability have been put forth and evaluated in previous research efforts. It was shown that the optimal IM to be used in the seismic response prediction depends, in general, on the specific type of structure considered, and on the specific response quantity of interest. For buildings, building aggregates and structures, some modifications of the Peak Ground Acceleration (PGA) and Spectral acceleration

J. Donaire-Ávila et al. / Engineering Structures 102 (2015) 278–295

at the fundamental period, Sa(T1), have been proposed [3,6,7]. The aim is twofold: to improve the predictive efficiency of the IM for every limit damage state of a given structure, while also accounting for IM computability through a ground-motion hazard analysis without any other prediction equation. Some studies show that good intensity measures can be derived from a vector-valued IM consisting of Sa(T1) and spectral values at other periods, and of Sa(T1) and e (defined as the number of standard deviations by which ln Sa(T1) diverges from its predicted mean value as obtained from a ground motion attenuation relationship) [8–10]. For regular structures and buildings where most of the mass participates in the first mode, spectral acceleration (Sa) and/or spectral displacement (Sd) may be the preferred IMs. Moreover, previous research has demonstrated, that in some cases, Sa(T1) and related parameters may not be good predictors; spectrum-based scalar IMs (energy- and velocity-derived) are in general better correlated to different deformation EDPs, both for ordinary and pulse-like ground motions [11–13]. The good predictive capabilities of energy-based parameters are linked to the amplitude, frequency content and duration of the motion, as well as the properties of the structure. Energy input spectra can be used to develop IMs that would explicitly account for higher mode influence [8] and the elongation of periods of vibration caused by damage [14]. Recent studies focused on ground motion prediction equations use input energy equivalent velocities [15–17] to overcome the problem of IM computability in ground-motion hazard analysis. When designing earthquake-resistant structures [18,19], the energy input in its relative (EIr) or absolute form (EIa)—or the respective equivalent velocity form, VEIr or VEIa— would be the reference IMs used to obtain EDP values. For practical purposes, the energy that contributes to damage [19], expressed by an equivalent velocity, VD, is considered as the IM when designing structures with hysteretic dampers [18,20,21]. This IM takes into account only the energy dissipated by plastic deformations and the vibrational energy. It is usually obtained from empirical equations that provide the ratio VD/VEIr [22–24]. The present study aims to shed new light on some still unclear aspects of IMs and the seismic response prediction of frame structures with hysteretic dampers that have low-to-moderate height and low height-to-width aspect ratios. For this kind of structures, the contribution of vibration modes higher than the fundamental one, and the contribution of bending beam behavior are both negligible. To date, analyses have involved many different types of buildings [14,25,26], but only a few studies have focused on structures with hysteretic dampers [27–29]. In particular, the objectives set forth here were:

279

ones. Because of the substantial effort required for their assessment, vector IMs were not used. Spectral derived IMs based on energy concepts were evaluated only in the elastic range. Elastic input energy, and derived parameters, correlate well to the nonlinear response of MDOF structures [13,14]. Parameters based on the inelastic behavior of structures might improve the structural response prediction. However, as our focus was on the probabilistic seismic hazard and risk analyses of structures, using hystereticbased parameters would have meant developing as many ground motion prediction equations (GMPEs) as the parameters used to characterize the hysteretic behavior of the structure and its particular hysteretic model. Moreover, input energy is held to be an effective tool in seismic design, as it is a very stable parameter that hardly depends on the hysteretic properties of the structure (e.g. see [30]). 2. Case studies 2.1. Studied buildings The case studies involved a three-bay frame extracted from a 6-story reinforced concrete building with an aspect ratio of 21/18  1, hereafter called the main structure (MS), retrofitted with hysteretic dampers. The dampers were installed in parallel at each story level with the MS, forming a ‘‘flexible-stiff mixed structure” [19]. The MS constitutes the flexible part and the hysteretic dampers form the stiff part. The MS, designed according to a past code DM 96 [31], is representative of existing ordinary buildings located in a zone of high seismicity (‘‘zone 1” of DM 96 [31]). It is worth emphasizing that the bare frames (i.e. without dampers) were first designed using the old seismic code DM’96, and then this frame underwent seismic upgrading with hysteretic dampers to withstand three levels of seismic hazard associated with earthquakes of moment magnitude ranging between 4.2 and 7.1. A representation of the 6-story reinforced concrete frame with the hysteretic dampers is depicted in Fig. 1, which includes column sections (CS) and beam sections (BS). All the columns of the frame have the same section dimensions. Moreover, all the columns of each story have the same reinforcement. The beams


To investigate the predictive capability of IMs with respect to EDPs related to damage in the hysteretic dampers (dissipated energy by plastic deformations), and more specifically to EDPs that can describe damage in the main frames of the building (maximum inter-story displacement, maximum chord rotation in beams and columns, maximum dissipated energy over all stories by plastic deformations or maximum floor acceleration).
To evaluate the different predictive capability of the IMs when ordinary or pulse-like near-fault ground motions are applied to the RC frame structure with dampers. To this end, the response of a building of moderate height and low aspect ratio, having two different hysteretic damper designs, was studied. First, the non-linear dynamic response of the two different structures with hysteretic dampers subjected to two different sets of ground motions (ordinary and pulse-like records) was analyzed. Scalar IMs often used to predict the response of fixed-base buildings were used, together with newly proposed

Fig. 1. Schematic representation of the frame structure with hysteretic dampers.

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of the frame also have the same section dimensions, though in this case the reinforcement varies in each floor for exterior and interior column-beam connections. Details of the column and beam sections included in Fig. 1 are shown in Table 1, where b and h are the width and the height, respectively, of the cross section; Ntop, Nbot and Nlat are the number of longitudinal bars located at the top, bottom and the lateral sides of the section, respectively; and /top, /bot and /lat are the counterpart rebar diameters. The concrete strength was 25 MPa and the yield stress of the steel 440 MPa. The masses of the floors were 43,065, 43,065, 43,065, 42,540, 42,540, 33,600 kg (ordered from bottom to top). The periods of the first three modes of vibration of the fixed-base frames of the MS modes were obtained with a reduced cracked stiffness of the structural elements (equal to half the initial elastic stiffness according to the ACI code [32]). Further, their corresponding modal damping ratios were calculated according to the Rayleigh damping model, fixing the first and second damping ratio, nn, at 0.05. Both the periods and damping ratios are given in Table 2. For the frame-MS, one damper is considered at the central span of each floor. Two different designs for the dampers were obtained depending on the type of ground motion considered (i.e. near- and far-field). The hysteretic dampers were designed by applying an energy-based procedure within the framework of PerformanceBased Earthquake Engineering. For the first damper design, designated D5-12, the distance fault, Df, from the assumed location of the structure to the epicentre was considered in a range of 5 6 Df 6 12 km, where the effects of the near-field ground motions are significant. For the second damper design, D25, the Df = 25 km corresponded to far field earthquakes. The method considers three different seismic hazard levels (SHLs); under each one the structure must fulfill a performance level (PL). In this study, the SHLs used correspond to the range of moment magnitude (Mw): (i) 4:2 6 M w < 5:2 for SHL-1; (ii) 5:2 6 M w < 6:2 for SHL-2 and (iii) 6:2 6 M w 6 7:1 for SHL-3. For each SHL, the index AEI proposed by Decanini and Mollaioli [33] is employed to characterize the energy input that contributes to damage spectra used to design the dampers, expressed as equivalent velocity, VD. The performance levels (PLs) considered for an ordinary building were: (i) Operational (O) for SHL-1 (the structure does not need to be repaired); (ii) Life Safety (LS) for SHL-2 (the structure needs to be repaired and it is feasible) and (iii) Near Collapse (NC) for SHL-3 (the structure can suffer severe damage and repairing it would not be economically feasible). The level of damage permitted on the structure for each PL is in agreement with its definition. Therefore, for PL-O the damage is permitted only in dampers, for PL-LS the damage is permitted in both the MS and the dampers without attaining the degradation zone of the materials (strain softening part of the backbone curve) and finally, for PL-NC damage is possible in all elements of the structure without collapse. These PLs are taken into account in the design stage through the limit inter-story drift values, IDlim, permitted in each case, which are taken from

Table 1 Section details of the 6-story frame building: dimensions and reinforcement. Section

BSext4-6 BSint4-6 BSext1-3 BSint1-3 CS4-6 CS1-3

Dimensions

Reinforcement

b (m)

h (m)

Ntop

/top (mm)

Nbot

/bot

Nlat

/lat (mm)

0.3 0.3 0.3 0.3 0.3 0.3

0.6 0.6 0.6 0.6 0.45 0.45

3 3 4 4 4 6

20 20 20 20 20 20

2 2 3 3 4 6

20 20 20 20 20 20

– – – – 1 1

– – – – 20 20

Table 2 Modal periods (Ti) and modal damping ratios (ni) for the 6-story frame with and without hysteretic dampers. Frame

T1 (s)

T2 (s)

T3 (s)

n1

n2

n3

6-story frame-MS D5-12 D25

1.17 0.68 0.70

0.40 0.23 0.24

0.24 0.13 0.14

0.05 0.05 0.05

0.05 0.05 0.05

0.070 0.073 0.071

Table 3 Design parameters for hysteretic dampers. Design

Df (km)

SHL-1

SHL-2

SHL-3

D5-12

5-12

AEI (cm2/s) VD (cm/s) IDlim (%)

6000 55 0.60

18,000 95 1.50

75,000 195 2.50

D25

25

AEI (cm2/s) VD (cm/s) IDlim (%)

2500 35 0.50

10,000 71 1.50

50,000 162 2.50

SEAOC [34]. Table 3 summarizes the values of the parameters used for designing the dampers in each seismic scenario (far field and near fault ground motion) and under each SHL. Fig. 2 shows the flow chart of the design method, where it can be seen that the dampers are first designed for the most critical SHL (i.e. SHL-3) and later checked for the rest of SHLs. The characteristic of the dampers obtained are summarized in Table 4, where for each story ‘‘i”: (i) Ki is the ratio between the lateral stiffness of the dampers with respect to that of the frame-MS; (ii) fai, sai and ai correspond to the yield shear force coefficients of the frame-MS, dampers and mixed flexible-stiff frame structure, respectively. The yield shear force coefficient is defined as the ratio between the yield shear force and the upward cumulative weight which is born by the story. The value of this coefficient for the first story (bolded in Table 4) is known as the base shear coefficient. As can be observed, the lateral strength for structures designed for near-field earthquakes is notably higher than the one obtained when the structure is designed for far-field earthquakes. Therefore, the frame-MS could satisfy the performance levels prescribed by modern seismic codes under a far-field ground motion, but could not under a near-fault ground motion. In the latter case, the lateral strength of the MS at each story level is about 15% smaller than the required value. The reason is that the equivalent number of cycles, neq, is smaller for the former than for the latter [35]. neq represents the number of plastic cycles at the maximum value of displacement that the structure must develop in order to dissipate the hysteretic energy. For general SDOF systems it is defined [35] by neq = Eh/[Qy(du  dy)], where Eh is the hysteretic energy dissipated by the system, and Qy, dy and du its yield shear force, yield displacement and maximum displacement, respectively. neq depends on both the hysteretic behavior of the structure and the characteristics of the ground motion, and several expressions have been proposed in the literature for its prediction [19,35]. The value of neq directly affects the required strength on the dampers: the larger neq is, the smaller the strength required of the dampers. The vibration periods and modal damping ratios of the frame-MS with dampers are reported in Table 2. It is worth noting that the period of structure D25 is smaller than that of the MS because the former has dampers (which notably increase the stiffness) while the latter does not. However, the base shear force coefficient of D25 is only slightly larger than that of MS. The reason is that the dampers of model D25 were calculated for far-field ground motions. This type of earthquake introduces the energy to the building gradually, and the structure can develop a large number of cycles of plastic deformation to dissipate the energy. As a result, the lateral strength to be provided by the dampers is relatively small.

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Fig. 2. Final stage of the damper design for D5-12 and D25, SHL-3 being the most critical.

The response of the structure to this set of records is expected to range continuously between elastic behavior and collapse.

Table 4 Strength and lateral stiffness of the hysteretic dampers. Story

6 5 4 3 2 1

RC

D5-12

f i

a

Ki

s i

ai

Ki

D25 s i

ai

0.596 0.568 0.531 0.484 0.431 0.374

3.50 3.50 3.50 3.50 3.50 3.50

0.104 0.086 0.074 0.065 0.058 0.052

0.700 0.654 0.605 0.549 0.489 0.426

3.00 3.00 3.00 3.00 3.00 3.00

0.001 0.001 0.001 0.001 0.001 0.001

0.597 0.569 0.532 0.485 0.432 0.375

a

a

2.2. Ground motion database In this study, 138 earthquake ground motions (GMs), obtained from the Next Generation of Attenuation project database [36], are used as input for non-linear dynamic analyses on buildings with hysteretic dampers. In order to highlight the effect of pulsetype motions, the set of GMs is divided into two subgroups: (i) 79 ordinary ground motions, OGMs, with closest distance ranging from 0.34 km to 87.87 km, and magnitude from 5.74 to 7.9; (ii) 59 pulse-like near-fault ground motions, PLGMs, with closest distance ranging from 0.07 km to 20.82 km, and magnitude from 5 to 7.62. Selected time histories are recorded on soil classified as type C or D, according to the NEHRP site classification based on the preferred Vs30 values. The horizontal component—following Mollaioli et al. [14]—considered for OGMs is the component having larger spectral acceleration at the fundamental period of the structure; for PLGMs, it is the fault-normal rotated component.

2.3. Non-linear modelling and analysis The response of the selected case studies is evaluated via nonlinear dynamic analyses run in OpenSees 2.4.4 [37]. The models of the structures were built using Beam with Fiber-Hinges Elements for modelling beams and columns of the frame, Elastic Beam and Column Elements for braces and ZeroLength Elements as spring elements for the hysteretic dampers. The cyclic behavior of the springs used for the hysteretic dampers was characterized with the Gioffré-Menegotto-Pinto steel model [38]. This model is characterized by an isotropic strain hardening. The Bauschinger effect is considered, which consists of the decay of the yield stress–strain threshold when the material has previously yielded in the opposite domain of loading. If the material is first subjected to plastic deformation in one direction, this effect makes it ‘‘soften” upon the inversion of the loading. The backbone curve is defined by three parameters: (i) yield strength; (ii) initial elastic stiffness; and (iii) the strain-hardening ratio between the post-yield tangent and initial elastic tangent. The remaining parameters are aimed to control the hysteretic behavior, being especially important the parameter that sets the degree of intensity of the Bauschinger effect. The parameters that control its behavior were calibrated with the results of tests [39]. On the other hand, the constitutive law used for the concrete fibers of the Fiber-Hinge Elements was able to capture stiffness deterioration and strain softening effects. Consequently, the modelization

Table 5 Non structure-specific intensity measures considered in this study. Notation

Name

Non-structure-specific intensity measures Acceleration-related PGA Peak Ground Acceleration

Definition

€ g ðtÞj PGA ¼ max ju € g ðtÞ ¼ acceleration time history u R p tf u € 2g ðtÞdt AI ¼ 2g 0

AI

Arias Intensity [41]

CAV

Cumulative Absolute Velocity [42]

CAV ¼

Ia

Compound Acc.-Related IM [43]

Ic

Characteristic Intensity [44]

Ia ¼ PGA  td ; td ¼ t2  t1 ; t1 : instant 5% AI; t 2 : t 2 : instant 95% AI pffiffiffiffiffi Ic ¼ ðarms Þ3=2 td qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi R t 2 € g ðtÞ2 dt arms ¼ t1d t1 u

Velocity-related PGV

Peak Ground Velocity

tf ¼ total duration R tf € 0 jug ðtÞjdt 1=3

PGV ¼ max ju_ g ðtÞj u_ g ðtÞ ¼ velocity time history

FI

Fajfar Intensity [45]

FI ¼ PGV  t0:25 d

Iv

Compound Vel.-Related IM [43]

CAD

Cumulative Absolute Displacement

Iv ¼ PGV2=3  td Rt CAD ¼ 0f ju_ g ðtÞjdt

IV SED

Incremental Velocity [46] Specific Energy Density

Displacement –related PGD Id

Peak Ground Displacement Compound Disp.-Related IM [43]

CAI

Cumulative Absolute Impulse

ID

Incremental Displacement [46]

1=3

SED ¼

R tf 0

2

½u_ g ðtÞ dt

PGD ¼ max jug ðtÞj 1=3

Id ¼ PGD  td Rt CAI ¼ 0f jug ðtÞjdt

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Table 6 Structure-specific intensity measures considered in this study. Name

Definition

Structure-specific intensity measures Spectral Sa Spectral acceleration at T

Spa ðTÞ Spa ¼ 5% damp: pseudo acc: spectrum Rt € g ðtÞu_ r ðtÞdt EIr ¼  0f u € g ðtÞ ¼ ground acceleration time history u u_ r ðtÞ ¼ relative velocity time history of aðn ¼ 5%; TÞ SDOF Rt R € t ðtÞdug ¼ 0tf u € t ðtÞu_ g ðtÞdt EIa ¼ 0f u u_ g ðtÞ ¼ ground velocity time history € t ðt ¼ absolute acceleration time history of aðn ¼ 5%; TÞ SDOF u

EIr

Relative Input Energy [47] at T

EIa

Absolute input energy [47] at T

Integral ASI

Acceleration Spectrum Intensity

VSI

Velocity Spectrum Intensity

IH

Housner Intensity [48]

VEIrSI

Relative input equivalent velocity spectrum intensity

VEIaSI

Absolute input equivalent velocity spectrum intensity

MASI

Modified ASI

MASIEC8

Modified ASI (EC8)

MVSI

Modified VSI

MVSIEC8

Modified VSI (EC8)

MIH

Modified IH

MIHEC8

Modified IH (EC8)

MVEIrSI

Modified VEIrSI

MVEIrSIEC8

Modified VEIrSI (EC8)

MVEIaSI

Modified VEIaSI

MVEIaSIEC8

Modified VEIaSI (EC8)

ASI ¼

used for the RC members was able to capture important features such as cyclic softening. The masses are concentrated at the nodes representing the beam-column joints, and the stiffness of the floors is modelled with rigid diaphragm constraints. A Rayleigh damping proportional to the mass and tangent stiffness matrix is used, the modal damping ratios being those given in Table 2. The effects of geometric nonlinearities are not considered in the analyses. For the maximum interstory drifts attained in the prototypes investigated in this study, the P–D effects can be regarded negligible according to Eurocode 8 [40].

(a)

1.8 1.6

D5-12 D25

0:1 Spa dT R 2:5 VSI ¼ 0:1 Sv dT Sv ¼ 5% damp: velocity spectrum R 2:5 IH ¼ 0:1 Spv dT Spv ¼ 5% damp: pseudo velocity R 3:0 pffiffiffiffiffiffiffiffiffi V EIr SI ¼ 0:1 2EIr dT R 3:0 pffiffiffiffiffiffiffiffiffi V EIa SI ¼ 0:1 2EIa dT R 1:25T MASI ¼ 0:5T Spa dT R 2:0T MASIEC8 ¼ 0:2T Spa dT R 1:25T MVSI ¼ 0:5T Sv dT R 2:0T MVSIEC8 ¼ 0:2T Sv dT R 1:25T MIH ¼ 0:5T Spv dT R 2:0T MIH EC8 ¼ 0:2T Spv dT R 1:25T MVEIr SI ¼ 0:5T V EIr dT R 2:0T MVEIr SIEC8 ¼ 0:2T V EIr dT R 1:25T MVEIa SI ¼ 0:5T V EIa dT R 2:0T MVEIa SIEC8 ¼ 0:2T V EIa dT

3. Intensity measures and engineering demand parameters 3.1. Intensity measures In this study, the intensity measures under investigation from the literature are categorized into two groups: (i) non-structurespecific IMs calculated directly from ground motion time histories, given in Table 5; and (ii) structure-specific IMs obtained from response spectra of ground motion time histories depending on the period of the structure, defined in Table 6. The first group of IMs is further classified into three categories: acceleration-related,

(b)

1.8

D5-12 D25

1.6

1.4

1.4

MRDRσε

1.2 1.0 0.8

1.2 1.0 0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0 PGA AI CAV Ia Ic PGV FI Iv CAD IV SED PGD Id CAI ID Sa EIr EIa ASI VSI IH VEIrSI VEIaSI MASI MASIEC8 MVSI MVSIEC8 MIH MIHEC8 MVEIrSI MVEIrSIEC8 MVEIaSI MVEIaSIEC8

MRDRσε

R 0:5

PGA AI CAV Ia Ic PGV FI Iv CAD IV SED PGD Id CAI ID Sa EIr EIa ASI VSI IH VEIrSI VEIaSI MASI MASIEC8 MVSI MVSIEC8 MIH MIHEC8 MVEIrSI MVEIrSIEC8 MVEIaSI MVEIaSIEC8

Notation

Fig. 3. Standard error of residuals re obtained in the IMs-MRDR regression: (a) OGMs and (b) PLGMs.

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1.8

D5-12 D25

(b)

1.8

1.4

1.2

1.2

MIDRσε

1.4

1.0 0.8

D5-12 D25

1.6

1.0 0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0 PGA AI CAV Ia Ic PGV FI Iv CAD IV SED PGD Id CAI ID Sa EIr EIa ASI VSI IH VEIrSI VEIaSI MASI MASIEC8 MVSI MVSIEC8 MIH MIHEC8 MVEIrSI MVEIrSIEC8 MVEIaSI MVEIaSIEC8

MIDRσε

1.6

PGA AI CAV Ia Ic PGV FI Iv CAD IV SED PGD Id CAI ID Sa EIr EIa ASI VSI IH VEIrSI VEIaSI MASI MASIEC8 MVSI MVSIEC8 MIH MIHEC8 MVEIrSI MVEIrSIEC8 MVEIaSI MVEIaSIEC8

(a)

Fig. 4. Standard error of residuals re obtained in the IMs-MIDR regression: (a) OGMs and (b) PLGMs.

Table 7

Table 8

re values of the most efficient IMs for predicting MIDR.

re values of the most efficient IMs for predicting MFA.

MIDR re

MFA re OGMs

PLGMs

Most efficient IMs

D5-12

D25

D5-12

D25

MIHEC8 MASIEC8 VSI MVSIEC8 MVEIaSIEC8

0.21 0.26 0.31 0.22 0.29

0.20 0.29 0.24 0.24 0.26

0.23 0.30 0.29 0.31 0.36

0.22 0.30 0.23 0.30 0.27

velocity-related and displacement-related IMs. The second group is further sorted into two groups: IMs obtained from the response spectral ordinate at certain periods and from the integration of response spectra over a defined period range. The PGA (Peak Ground Acceleration), PGV (Peak Ground Velocity), PGD (Peak Ground Displacement), IV (incremental velocity), ID (incremental displacement), and AI (Arias intensity) are included in the first group. PGA, PGV and PGD are the most common time domain parameters of strong ground motion. IV is the area under the maximum acceleration pulse, while ID is the area under the maximum velocity pulse [46]. As proposed by Arias [41], AI accounts for duration and amplitude but does not reflect the frequency content. AI tends to overestimate the intensity of long duration motions with high amplitude and a broad range of frequency content. The other IMs of the first group considered in the study are presented in Table 5.

1.8

D5-12 D25

(b)

D25

D5-12

D25

0.17 0.20 0.21 0.24 0.23 0.24 0.21 0.27 0.26 0.19 0.20 0.19 0.23

0.18 0.17 0.17 0.28 0.27 0.25 0.30 0.30 0.26 0.21 0.24 0.20 0.31

0.16 0.18 0.19 0.28 0.29 0.22 0.19 0.30 0.22 0.22 0.24 0.22 0.26

0.15 0.16 0.18 0.24 0.25 0.20 0.18 0.26 0.24 0.19 0.20 0.18 0.23

D5-12 D25

1.6 1.4

1.2

1.2

0.8

D5-12

1.8

1.4

1.0

PLGMs

Sa (5% damped pseudo acceleration spectral value at T), EIa (5% damped absolute input energy spectral value at T) and EIr (5% damped relative input energy spectral value at T) [47] are considered as structure-specific IMs, where T is the main period of the mixed flexible-stiff frame (Table 2). IMs evaluated by integration of the structural response in a given period range can explicitly account for higher mode effects as well as period lengthening due to structural softening. Those considered in this study, which are classified as the second group of investigated IMs, are: ASI

MFAσε

MFAσε

1.6

1.0 0.8

0.6

0.6

0.4

0.4

0.2

0.2

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(a)

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OGMs

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Most efficient IMs

Fig. 5. Standard error of residuals re obtained in the IMs-MFA regression: (a) OGMs and (b) PLGMs.

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D5-12 D25

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1.6 1.4

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PGA AI CAV Ia Ic PGV FI Iv CAD IV SED PGD Id CAI ID Sa EIr EIa ASI VSI IH VEIrSI VEIaSI MASI MASIEC8 MVSI MVSIEC8 MIH MIHEC8 MVEIrSI MVEIrSIEC8 MVEIaSI MVEIaSIEC8

0.6

Fig. 6. Standard error of residuals re obtained in the IMs-sgmax regression: (a) OGMs and (b) PLGMs.

1.8

D5-12 D25

(b)

1.8

1.2

1.0

1.0

η σ

s avg ε

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η σ

s avg ε

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(a)

Fig. 7. Standard error of residuals re obtained in the IMs-sgavg regression: (a) OGMs and (b) PLGMs.

Table 9 Regressions of the sgmax obtained for D5-12 and D25 buildings: re values of the most efficient IMs.

g

s max

re

Most efficient IMs

MVEIaSIEC8 MIHEC8 MVSIEC8 VSI

OGMs

PLGMs

D5-12

D25

D5-12

D25

0.25 0.31 0.34 0.42

0.32 0.43 0.43 0.42

0.33 0.36 0.36 0.42

0.34 0.46 0.46 0.44

(Acceleration Spectrum Intensity), VSI (Velocity Spectrum Intensity), IH (Housner Intensity [48]), VEIrSI, and VEIaSI (relative and absolute Input Equivalent Velocity Spectrum Intensity, respectively). The definitions of all these IMs are given in Table 6. In this paper, ASI was used as proposed by Von Thun et al. [49]. The main difference between VSI [49], and IH [48] is that the absolute velocity spectrum is used for computing VSI, while the pseudo velocity spectrum is used for IH. VEIrSI, and VEIaSI are parameters obtained from integration of the energy response spectra in the period range 0.1–3.0 sec, deemed as more appropriate for near-fault signals [30,33]. Two modified versions of the second group of structure-specific IMs are also considered (see Table 6). The first version of the IMs (identified by the letter ‘‘M” at the beginning of the IM name that is modified) was obtained from the existing ones by changing the

period range of integration into 0.50T–1.25T. A similar modification was also considered by Avsßar and Özdenmir [50] when investigating the capability of ASI, VSI and IH in predicting the seismic response of base-isolated bridges. The second version is based on the former, but varying the period range of integration to 0.20T– 2.0T, according to the European standard Eurocode 8 (EC8) Part 1 [40], to select records having a good agreement between the acceleration response spectrum and the one given by the code. Therefore, the influence of the higher vibration modes and the lengthening of the main period are taken into account. The second modified IMs are designated by adding the acronym EC8 at the end of the first modified version name. 3.2. Engineering demand parameters The EDPs considered in this study are the following:
Maximum Inter-story Drift Ratio (MIDR), namely, the maximum value of the peak inter-story drift ratio (drift normalized by the story height) over all stories.
Maximum Roof Drift Ratio (MRDR), namely, the ratio of the peak lateral roof displacement (with respect to the base) to the building height.
Maximum Floor Acceleration (MFA), namely, the maximum value of the peak floor absolute acceleration over all stories of superstructures.

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(a) 1.8

D5-12 D25

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(b) 1.8

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1.4

1.2

1.2

1.0 σ

0.8

0.6

η

0.6

f max ε

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η

f max ε

D5-12 D25

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0.4

0.2

0.2

0.0

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PGA AI CAV Ia Ic PGV FI Iv CAD IV SED PGD Id CAI ID Sa EIr EIa ASI VSI IH VEIrSI VEIaSI MASI MASIEC8 MVSI MVSIEC8 MIH MIHEC8 MVEIrSI MVEIrSIEC8 MVEIaSI MVEIaSIEC8

0.4

Fig. 8. Standard error of residuals re obtained in the IMs-fgmax regression: (a) OGMs and (b) PLGMs.

Table 10 Regressions of the fgmax obtained for D5-12 and D25 buildings: re values of the most efficient IMs.

re

Most efficient IMs

MIHEC8 VSI IH MVEIaSIEC8

OGMs

PLGMs

D5-12

D25

D5-12

D25

0.54 0.66 0.73 0.51

0.64 0.71 0.72 0.62

0.64 0.71 0.72 0.62

0.56 0.57 0.78 0.68


Maximum damage of the dampers (sgmax), namely, the maximum value of the damage observed over all the dampers of the frame. The damage of the damper of the story ‘‘i”, is represented by the dimensionless coefficient sgi, defined as the ratio between the hysteretic energy dissipated by the damper (stiff part), sEh,i, to the product of its yield force, sQy,i by its yield displacement, sdy,i.
Average damage of the dampers (sgavg), namely, the average damage over all the dampers of the frame.
Maximum story damage of the reinforced concrete frame (fgmax), namely, the maximum value of the damage observed over all stories of the frame-MS. The damage of the story ‘‘i” of the frame-MS is evaluated by the dimensionless coefficient fg,i, defined as the ratio between the hysteretic energy dissipated by the RC frame (flexible part), fEh,i, to the product of its yield shear force, fQy,i and its yield displacement, fdy,i.

(a) 1.8

D5-12 D25

4. Regression analyses 4.1. Predictive models The predictive models are based on regression analysis of the response of the structure under a set of unscaled ground motions (cloud analysis). Regression were used with this cloud of data to compute the conditional mean and standard deviation of the EDP

(b)

1.8

1.2

1.2

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1.0

MCRσε

1.4

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D5-12 D25

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MCRσε

1.6

MIDR, which is shown to be closely related to local damage, instability, and story collapse, is one of the most widely used EDPs for evaluating structural damage. MRDR gives a good correlation to the overall structural damage. MFA reflects the level of non-structural damage of the buildings. MCR and MBR are good variables to measure the level of local damage on the RC structure. sgi is a key index for structures with hysteretic dampers to measure the damage concentrated on these devices, noting that the dampers were designed taking into account an optimal shear force distribution for the structure, such that sgi = sg = Const. Finally, fgi is an index used to determine the level of damage on each story of the main structure and, furthermore, the damage concentration level over it.

PGA AI CAV Ia Ic PGV FI Iv CAD IV SED PGD Id CAI ID Sa EIr EIa ASI VSI IH VEIrSI VEIaSI MASI MASIEC8 MVSI MVSIEC8 MIH MIHEC8 MVEIrSI MVEIrSIEC8 MVEIaSI MVEIaSIEC8

g

f max


Maximum column chord rotation (MCR), namely, the maximum value of the chord rotation over all columns of the structure.
Maximum beam chord rotation (MBR), namely, the maximum value of the chord rotation over all beams of the structure.

Fig. 9. Standard error of residuals re obtained in the IMs-MCR regression: (a) OGMs and (b) PLGMs.

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D5-12 D25

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D5-12 D25

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MBRσε

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PGA AI CAV Ia Ic PGV FI Iv CAD IV SED PGD Id CAI ID Sa EIr EIa ASI VSI IH VEIrSI VEIaSI MASI MASIEC8 MVSI MVSIEC8 MIH MIHEC8 MVEIrSI MVEIrSIEC8 MVEIaSI MVEIaSIEC8

(a)

Fig. 10. Standard error of residuals re obtained in the IMs-MBR regression: (a) OGMs and (b) PLGMs.

Table 11 Regressions of the MCR for D5-12 and D25 buildings: re values of the most efficient IMs.

ejIM ¼ aM þ bM  M; ejIM ¼ aR þ bR  R

MCR re Most efficient IMs

MIHEC8 VSI MVSIEC8 MVEIaSIEC8

OGMs

PLGMs

D5-12

D25

D5-12

D25

0.25 0.31 0.27 0.30

0.23 0.26 0.26 0.27

0.26 0.30 0.35 0.36

0.30 0.28 0.39 0.30

Table 12 Regressions of the MBR for D5-12 and D25 buildings: re values of the most efficient IMs. MBR re Most efficient IMs

MIHEC8 VSI IH IV MASIEC8 MVSIEC8 MVEIaSIEC8

was calculated and plotted against M or R, and the following predictive models were adopted for correlating e|IM with magnitude (M) and distance (R):

OGMs

PLGMs

D5-12

D25

D5-12

D25

0.20 0.27 0.30 0.29 0.25 0.21 0.26

0.20 0.23 0.24 0.27 0.29 0.24 0.24

0.24 0.27 0.33 0.43 0.31 0.32 0.33

0.27 0.25 0.26 0.40 0.36 0.35 0.28

ð2Þ

Once the slope coefficients bM and bR are estimated, the sufficiency can be directly measured by the p-value. The p-value is defined as the likelihood of detecting a slope coefficient equal to or greater than the estimated bM or bR (absolute) value, if the underlying (true) value of bM or bR is in fact zero [53]. In other words, the null hypothesis that M or R have no influence on residuals of EDP (i.e. bM = 0 or bR = 0) is posed, and it is tested by estimating the probability (p-value) of obtaining bM P 0 or bR P 0 if the null hypothesis is true. Small p-values indicate that the null hypothesis is not true and therefore M or R have a significant influence on IM, i.e. IM is not sufficient. Generally, IM is considered sufficient when the p-value is higher than 0.05. A smaller p-value of b indicates a less sufficient IM, M or R having a significant influence on the residuals of EDP. 4.2. Evaluation results The range of structure response covered by the selected records, expressed in terms of the maximum inter-story drift ratio MIDR, varies from 0.07% to 4.36%, with very few exceptions that reached 6%.

ð1Þ

4.2.1. Efficiency The efficiency analysis results for the different EDPs are reported in the sections below. The most efficient IMs, that is, those with re less than or equal to 0.30, are highlighted in tables. Their values are bolded if they are efficient for both designs, D5-12 and D25, under OGMs and/or PLGMs.

where ln (a) and b are model parameters. Among the properties usually adopted for evaluating the predictive capabilities of an IM (e.g., see Tothong and Luco [52]), this study took into account sufficiency and efficiency. An IM is sufficient when the conditional probability distribution of EDP given IM is independent of the other parameters involved in computing the seismic hazard (e.g. the magnitude and closest distance considered). An efficient IM yields little variability of the predicted EDP for a given IM level. The standard error of residuals re is considered as a measure of the predictive efficiency of the IM. IMs resulting in EDP standard errors of the order of 0.20–0.30 would have good efficiency, while the range 0.30–0.40 is reasonably suitable. The regression residuals e|IM, in turn, are used for evaluating the IM sufficiency. To this end, e|IM

4.2.1.1. Prediction of MIDR and MRDR. Figs. 3 and 4 report the re values obtained in the regressions on the MRDR and MIDR, respectively, of the two structures D5-12 and D25. By looking at the re values for both, the following conclusions can be drawn. Similar trends are observed with MRDR and MIDR. The MIDR is more efficiently predicted when ordinary records rather than pulse-like near-fault records are used. MRDR and MIDR are better predicted with structure-specific intensity measures than with nonstructure-specific intensity measures, for both D5-12 and D25 designs, and for both ordinary and pulse-like ground motions. As for the former IMs, better predictions are seen for the coefficients with integral format, including a range of spectral values and therefore taking into account the lengthening of the main period

given IM. Previous researchers (e.g., Cornell et al. [51]) found a linear relationship (determined using regression) between the logarithms of the two variables EDP-IM. Therefore, all regressions are performed according to the following functional form:

ln ðEDPÞ ¼ lnðaÞ þ b  lnðIMÞ

J. Donaire-Ávila et al. / Engineering Structures 102 (2015) 278–295

(a)

(b)

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D5-12 D25

MRDR p-value

D5-12 D25 0.1 0.05

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PGA AI CAV Ia Ic PGV FI Iv CAD IV SED PGD Id CAI ID Sa EIr EIa ASI VSI IH VEIrSI VEIaSI MASI MASIEC8 MVSI MVSIEC8 MIH MIHEC8 MVEIrSI MVEIrSIEC8 MVEIaSI MVEIaSIEC8

PGA AI CAV Ia Ic PGV FI Iv CAD IV SED PGD Id CAI ID Sa EIr EIa ASI VSI IH VEIrSI VEIaSI MASI MASIEC8 MVSI MVSIEC8 MIH MIHEC8 MVEIrSI MVEIrSIEC8 MVEIaSI MVEIaSIEC8

MRDR p-value

287

Fig. 11. P-values of the regressions of e|IM residuals of the MRDR prediction with R: (a) OGMs and (b) PLGMs.

(a)

(b)

MRDR p-value

1

D5-12 D25

MRDR p-value

1

D5-12 D25

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PGA AI CAV Ia Ic PGV FI Iv CAD IV SED PGD Id CAI ID Sa EIr EIa ASI VSI IH VEIrSI VEIaSI MASI MASIEC8 MVSI MVSIEC8 MIH MIHEC8 MVEIrSI MVEIrSIEC8 MVEIaSI MVEIaSIEC8

0.01

0.1 0.05

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0.1 0.05

Fig. 12. P-values of the regressions of e|IM residuals of the MRDR prediction with M: (a) OGMs and (b) PLGMs.

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1 D5-12 D25

MIDR p-value

0.1 0.05

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PGA AI CAV Ia Ic PGV FI Iv CAD IV SED PGD Id CAI ID Sa EIr EIa ASI VSI IH VEIrSI VEIaSI MASI MASIEC8 MVSI MVSIEC8 MIH MIHEC8 MVEIrSI MVEIrSIEC8 MVEIaSI MVEIaSIEC8

MIDR p-value

D5-12 D25

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(a)

Fig. 13. P-values of the regressions of e|IM residuals of the MIDR prediction with R: (a) OGMs and (b) PLGMs.

owing to plastic deformations or the influence of higher vibration modes. Among these, the modified integral intensity measures with integration limits according to EC8 lead to lower deviations for the predictions of MRDR and MIDR. Proving most efficient are the measures related with the velocity spectrum and the energy input for both ordinary and pulse-like ground motion records, such as MIHEC8, VSI, MVSIEC8

and MVEIaSIEC8. Nonetheless, there are significant differences between ordinary records and pulse-like near-fault records for the structure D5-12, with higher values of re when the latter records are used. Notwithstanding, MIHEC8 and MASIEC8 are the IMs most efficiently predicting MRDR and MIDR, with low values of deviations for both OGM and PLGM records. Among the nonstructure-specific IMs, PGV and IV (velocity-related IMs) are the

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(a)

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D5-12 D25

MIDR p-value

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PGA AI CAV Ia Ic PGV FI Iv CAD IV SED PGD Id CAI ID Sa EIr EIa ASI VSI IH VEIrSI VEIaSI MASI MASIEC8 MVSI MVSIEC8 MIH MIHEC8 MVEIrSI MVEIrSIEC8 MVEIaSI MVEIaSIEC8

MIDR p-value

D5-12 D25

Fig. 14. P-values of the regressions of e|IM residuals of the MIDR prediction with M: (a) OGMs and (b) PLGMs.

Table 13 Sufficiency analysis of the MIDR for D5-12 and D25 buildings: p-value of the most sufficient IMs. MIDR p-value OGMs

PLGMs

Most sufficient IMs Distance

Distance

Magnitude

D5-12 D25

D5-12 D25

D5-12 D25

D5-12 D25

0.36 0.05 0.01 0.07 0.37 0.46 0.19 0.27 0.15 0.12

0.69 0.89 0.11 0.79 0.05 0.09 0.22 0.11 0.75 0.08

0.73 0.89 0.28 0.02 0.48 0.02 0.01 0.02 0.06 0.17

0.22 0.25 0.07 0.17 0.01 0.12 0.31 0.15 0.97 0.10

0.44 0.03 0.07 0.11 0.16 0.69 0.33 0.45 0.49 0.34

0.23 0.21 0.79 0.96 0.25 0.13 0.24 0.11 0.22 0.79

0.60 0.18 0.07 0.00 0.97 0.03 0.01 0.02 0.02 0.04

0.60 0.11 0.89 0.68 0.36 0.21 0.42 0.23 0.14 0.97

most efficient predictors, but with higher deviations. The values for re obtained for the most efficient IMs for MIDR are reported in Table 7. 4.2.1.2. Prediction of MFA. The results obtained in the regressions on MFA show that structure-specific IMs are in general the most efficient IMs for this EDP, and in particular those with an integral format. Yet among the non-structure-specific IMs, the most

(a)

4.2.1.4. Prediction of fgmax. The results obtained for fgmax are depicted in Fig. 8, where the values of efficiency are seen to be

(b)

1

1

D5-12 D25

D5-12 D25

MFA p-value

0.1 0.05

0.01

PGA AI CAV Ia Ic PGV FI Iv CAD IV SED PGD Id CAI ID Sa EIr EIa ASI VSI IH VEIrSI VEIaSI MASI MASIEC8 MVSI MVSIEC8 MIH MIHEC8 MVEIrSI MVEIrSIEC8 MVEIaSI MVEIaSIEC8

MFA p-value

4.2.1.3. Prediction of sgmax and sgavg. Figs. 6 and 7 depict the results obtained with sgmax and sgavg, respectively, for D5-12 and D25 designs. The results for sgmax and sgavg are seen to be very similar. This is a consequence of the design method to calculate the dampers, using an optimal yield-shear force coefficient distribution in order to arrive at an even distribution of the damage for the dampers s gi ¼ s gD ¼ constant. Of all the IMs, the structure-specific kind shows better efficiency than the non-structure type, especially for integral parameters. The values of re obtained with the most efficient IMs for sgmax are reported in Table 9. Only MVEIaSIEC8 is efficient for both OGM and PLGM records; meanwhile MIHEC8 and MVSIEC8 can be considered as IMs with acceptable efficiency for PLGM records, the values of re obtained for them (=0.46) being slightly higher than the upper limit considered in this study (=0.40).

0.1 0.05

0.01

PGA AI CAV Ia Ic PGV FI Iv CAD IV SED PGD Id CAI ID Sa EIr EIa ASI VSI IH VEIrSI VEIaSI MASI MASIEC8 MVSI MVSIEC8 MIH MIHEC8 MVEIrSI MVEIrSIEC8 MVEIaSI MVEIaSIEC8

Ic MIHEC8 MVEIrSI CAD IV PGD Id ID Sa EIr

Magnitude

efficient ones pertain to the acceleration-related group. This is clearly seen in the plots of Fig. 5. The re values obtained for the most efficient IMs are reported in Table 8. As observed for MIDR, the use of modified IMs for MFA leads to improved predictions. Moreover, the IMs efficiencies for ordinary and pulse-like nearfault records are almost the same. It should be stressed that MIHEC8 is one of the most efficient IMs for MFA as well as for MIDR and MRDR.

Fig. 15. P-values of the regressions of e|IM residuals of the MFA prediction with R: (a) OGMs and (b) PLGMs.

J. Donaire-Ávila et al. / Engineering Structures 102 (2015) 278–295

(a)

(b)

1

1

D5-12 D25 0.1

MFA p-value

0.1 0.05

0.01

0.05

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PGA AI CAV Ia Ic PGV FI Iv CAD IV SED PGD Id CAI ID Sa EIr EIa ASI VSI IH VEIrSI VEIaSI MASI MASIEC8 MVSI MVSIEC8 MIH MIHEC8 MVEIrSI MVEIrSIEC8 MVEIaSI MVEIaSIEC8

PGA AI CAV Ia Ic PGV FI Iv CAD IV SED PGD Id CAI ID Sa EIr EIa ASI VSI IH VEIrSI VEIaSI MASI MASIEC8 MVSI MVSIEC8 MIH MIHEC8 MVEIrSI MVEIrSIEC8 MVEIaSI MVEIaSIEC8

MFA p-value

D5-12 D25

289

Fig. 16. P-values of the regressions of e|IM residuals of the MFA prediction with M: (a) OGMs and (b) PLGMs.

Table 14 Sufficiency analysis of the MFA for D5-12 and D25 buildings: p-value of the most sufficient IMs. MFA p-value OGMs

PLGMs

Most sufficient IMs Distance

Distance

Magnitude

D5-12 D25

D5-12 D25

D5-12 D25

D5-12 D25

0.17 0.17 0.83 0.84 0.76 0.15 0.99 0.68 0.36

0.63 0.46 0.37 0.29 0.25 0.33 0.17 0.60 0.29

0.41 0.12 0.10 0.11 0.20 0.11 0.76 0.01 0.23

0.93 0.95 0.30 0.19 0.00 0.01 0.03 0.55 0.02

0.45 0.38 0.64 0.90 0.87 0.30 0.53 0.60 0.85

0.46 0.35 0.56 0.45 0.44 0.28 0.18 0.83 0.12

0.53 0.62 0.77 0.80 0.05 0.78 0.24 0.24 0.78

0.95 0.90 0.27 0.19 0.00 0.03 0.03 0.54 0.01

higher than 0.40 in all cases. This is a consequence of the local effect of damage in the RC structure, unlike the distribution of fgmax. Table 10 offers the most efficient IMs, their values ranging between 0.50 and 0.80.

4.2.1.5. Prediction of MCR and MBR. Below, Figs. 9 and 10 show the results obtained for MCR and MBR. These EDPs represent the damage occurring in the RC elements of the main structure, related

(a)

(b)

1

p-value

0.1

η

s max

0.05

0.01

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p-value η

1

D5-12 D25

D5-12 D25

s max

4.2.2. Sufficiency The sufficiency of the IMs is expressed through the p-value of the hypothesis contrast for the slope coefficient, b, obtained from the regression analysis between the regression residuals of the IMs-EDPs relationships of each structure (D5-12 and D25), e|IM, and either the distance (R) or the magnitude (M). The influence of ordinary and pulse-like ground motions are considered separately for each case. The p-value of 0.05 is the significance level considered in this study. For lower values, the null hypothesis is refused, the b coefficient has a non-negligible value, and therefore the IM bears a significant relationship with the variables studied (R or M). The IMs with p-values greater or equal to 0.05 for R and M, regardless of the design or record (OGMs or PLGMs), will be considered as the most sufficient. These IMs, and those having

0.1 0.05

0.01

PGA AI CAV Ia Ic PGV FI Iv CAD IV SED PGD Id CAI ID Sa EIr EIa ASI VSI IH VEIrSI VEIaSI MASI MASIEC8 MVSI MVSIEC8 MIH MIHEC8 MVEIrSI MVEIrSIEC8 MVEIaSI MVEIaSIEC8

PGA ASI MASI MIH Ic Sa MASIEC8 MVSI MVSIEC8

Magnitude

with MIDR. The trends of the results obtained for MCR and MBR are similar when the same structure (D5-12 or D25) is subjected to the same kind of records (ordinary or pulse-like ground motions). The most efficient IMs correspond to integral coefficients, as in the previous EDPs analyzed. Namely, MIHEC8, VSI, MVSIEC8 and MVEIaSIEC8 give similar residual values for both ordinary and pulse-ground motions records. Tables 11 and 12 respectively show the values of re obtained for the most efficient IMs for MCR and MBR. MIHEC8 is seen to be the most efficient IM for both EDPs, as well as for both kind of records. Yet for MBR the list of IMs having optimal efficiency values with OGMs is considerably larger.

Fig. 17. P-values of the regressions of e|IM residuals of the sgmax prediction with R: (a) OGMs and (b) PLGMs.

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(a)

(b)

1

D5-12 D25

s max

0.05

η

0.01

0.01

PGA AI CAV Ia Ic PGV FI Iv CAD IV SED PGD Id CAI ID Sa EIr EIa ASI VSI IH VEIrSI VEIaSI MASI MASIEC8 MVSI MVSIEC8 MIH MIHEC8 MVEIrSI MVEIrSIEC8 MVEIaSI MVEIaSIEC8

PGA AI CAV Ia Ic PGV FI Iv CAD IV SED PGD Id CAI ID Sa EIr EIa ASI VSI IH VEIrSI VEIaSI MASI MASIEC8 MVSI MVSIEC8 MIH MIHEC8 MVEIrSI MVEIrSIEC8 MVEIaSI MVEIaSIEC8

η

0.1

p-value

p-value

0.1 0.05

s max

1

D5-12 D25

Fig. 18. P-values of the regressions of e|IM residuals of the sgmax prediction with M: (a) OGMs and (b) PLGMs.

(a)

(b)

1

D5-12 D25

1

D5-12 D25

p-value

p-value

0.1

s avg

η

0.01

0.1 0.05

0.01

PGA AI CAV Ia Ic PGV FI Iv CAD IV SED PGD Id CAI ID Sa EIr EIa ASI VSI IH VEIrSI VEIaSI MASI MASIEC8 MVSI MVSIEC8 MIH MIHEC8 MVEIrSI MVEIrSIEC8 MVEIaSI MVEIaSIEC8

PGA AI CAV Ia Ic PGV FI Iv CAD IV SED PGD Id CAI ID Sa EIr EIa ASI VSI IH VEIrSI VEIaSI MASI MASIEC8 MVSI MVSIEC8 MIH MIHEC8 MVEIrSI MVEIrSIEC8 MVEIaSI MVEIaSIEC8

η

s avg

0.05

Fig. 19. P-values of the regressions of e|IM residuals of the sgavg prediction with R: (a) OGMs and (b) PLGMs.

(b)

1

1

D5-12 D25

p-value

0.1

η

s avg

0.05

0.01

PGA AI CAV Ia Ic PGV FI Iv CAD IV SED PGD Id CAI ID Sa EIr EIa ASI VSI IH VEIrSI VEIaSI MASI MASIEC8 MVSI MVSIEC8 MIH MIHEC8 MVEIrSI MVEIrSIEC8 MVEIaSI MVEIaSIEC8

η

s avg

p-value

D5-12 D25

0.1 0.05

0.01

PGA AI CAV Ia Ic PGV FI Iv CAD IV SED PGD Id CAI ID Sa EIr EIa ASI VSI IH VEIrSI VEIaSI MASI MASIEC8 MVSI MVSIEC8 MIH MIHEC8 MVEIrSI MVEIrSIEC8 MVEIaSI MVEIaSIEC8

(a)

Fig. 20. P-values of the regressions of e|IM residuals of the sgavg prediction with M: (a) OGMs and (b) PLGMs.

sufficiency for R and M for just one kind of records (OGMs or PLGMs), are considered to be the most sufficient of all. Their values are bolded in tables. 4.2.2.1. Sufficiency of IMs for MIDR and MRDR. The results of sufficiency analyses for MIDR and MRDR are reported in Figs. 11–14.

As can be observed, the trend is similar in both EDPs. The most sufficient IMs are included in Table 13, where Ic is seen to be the most sufficient overall, and the only one sufficient for both kinds of records. Nonetheless, it is not an efficient IM, with values (6 0.46) near the threshold of re 6 0:40. As can be seen, there are more IMs with sufficiency level for OGMs than for PLGMs.

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For OGMs, the IMs with enough sufficiency and efficiency are IV (re < 0:33), Sa (re < 0:35) and EIr (re < 0:41). For PLGMs, MIHEC8 is one of the most efficient for MIDR and sufficient only for this

Table 15 Sufficiency analysis of sgmax for D5-12 and D25 buildings: p-value of the most sufficient IMs.

Table 16 Sufficiency analysis of fgmax for D5-12 and D25 buildings: p-value of the most sufficient IMs.

g

f max

p-value

p-value OGMs

OGMs

PLGMs

Most sufficient IMs Distance

PGV Iv SED FI CAD IV PGD Id ID Eir Ear IH VEIrSI VEIaSI MVEIaSIEC8

(a)

Magnitude

Distance

Magnitude

D5-12 D25

D5-12 D25

D5-12 D25

D5-12 D25

0.96 0.45 0.24 0.87 0.07 0.31 0.20 0.10 0.09 0.57 0.79 0.40 0.40 0.67 0.36

0.94 0.35 0.21 0.20 0.33 0.35 0.91 0.72 0.82 0.48 0.70 0.76 0.09 0.08 0.29

0.64 0.80 0.56 0.71 0.07 0.83 0.43 0.20 0.21 0.66 0.81 0.78 0.76 0.48 0.14

0.55 0.71 0.04 0.13 0.04 0.21 0.23 0.55 0.79 0.01 0.11 0.44 0.51 0.02 0.49 0.01 0.52 0.01 0.13 0.01 0.17 0.32 0.24 0.88 0.13 0.74 0.24 0.41 0.28 0.09

0.95 0.12 0.17 0.05 0.44 0.10 0.64 0.02 0.00 0.49 0.18 0.58 0.05 0.78 0.02 0.98 0.06 0.84 0.00 0.06 0.09 0.60 0.49 0.77 0.87 0.02 0.77 0.02 0.79 0.06

Magnitude

Distance

Magnitude

D5-12 D25

D5-12 D25

D5-12 D25

D5-12 D25

0.21 0.87 0.48 0.69 0.89 0.19 0.73 0.14 0.37 0.29 0.90 0.08 0.14 0.37 0.60 0.19

0.79 0.06 0.62 0.76 0.38 0.61 0.68 0.86 0.07 0.82 0.92 0.34 0.06 0.06 0.41 0.28

0.67 0.53 0.58 0.26 0.19 0.82 0.27 0.02 0.42 0.53 0.01 0.01 0.07 0.46 0.52 0.09

0.28 0.55 0.37 0.11 0.00 0.03 0.04 0.73 0.00 0.00 0.85 0.29 0.01 0.00 0.00 0.41

0.80 0.64 0.30 0.72 0.11 0.14 0.15 0.14 0.50 0.46 0.62 0.64 0.74 0.72 0.45 0.14

0.79 0.06 0.62 0.74 0.50 0.83 0.24 0.66 0.07 0.74 0.63 0.29 0.06 0.08 0.58 0.21

0.29 0.10 0.05 0.05 0.76 0.21 0.00 0.00 0.86 0.75 0.01 0.01 0.03 0.32 0.69 0.02

(b)

1

1 D5-12 D25

0.1

0.1

p-value

0.05

0.05

f max

0.01

0.01

PGA AI CAV Ia Ic PGV FI Iv CAD IV SED PGD Id CAI ID Sa EIr EIa ASI VSI IH VEIrSI VEIaSI MASI MASIEC8 MVSI MVSIEC8 MIH MIHEC8 MVEIrSI MVEIrSIEC8 MVEIaSI MVEIaSIEC8

PGA AI CAV Ia Ic PGV FI Iv CAD IV SED PGD Id CAI ID Sa EIr EIa ASI VSI IH VEIrSI VEIaSI MASI MASIEC8 MVSI MVSIEC8 MIH MIHEC8 MVEIrSI MVEIrSIEC8 MVEIaSI MVEIaSIEC8

η

p-value η

IV Ic VSI EIa MVEIaSIEC8 MVEIaSI MVEIrSIEC8 MVEIrSI VEIaSI IH EIr PGD Iv FI PGV Ia

0.23 0.38 0.56 0.59 0.53 0.01 0.61 0.63 0.67 0.00 0.00 0.00 0.70 0.21 0.01

D5-12 D25

f max

PLGMs

Most sufficient IMs Distance

Fig. 21. P-values of the regressions of e|IM residuals of the fgmax prediction with R: (a) OGMs and (b) PLGMs.

(a)

(b)

1

p-value

0.1

η

f max

0.05

0.01

PGA AI CAV Ia Ic PGV FI Iv CAD IV SED PGD Id CAI ID Sa EIr EIa ASI VSI IH VEIrSI VEIaSI MASI MASIEC8 MVSI MVSIEC8 MIH MIHEC8 MVEIrSI MVEIrSIEC8 MVEIaSI MVEIaSIEC8

p-value η

f max

1

D5-12 D25

D5-12 D25

0.1 0.05

0.01

PGA AI CAV Ia Ic PGV FI Iv CAD IV SED PGD Id CAI ID Sa EIr EIa ASI VSI IH VEIrSI VEIaSI MASI MASIEC8 MVSI MVSIEC8 MIH MIHEC8 MVEIrSI MVEIrSIEC8 MVEIaSI MVEIaSIEC8

g

s max

kind of records. It is not sufficient for OGMs, given the R for structures designed in far field (p-value = 0.03). MVEIrSI is sufficient, though it is not efficient (re > 0:40).

Fig. 22. P-values of the regressions of e|IM residuals of the fgmax prediction with M: (a) OGMs and (b) PLGMs.

0.72 0.05 0.08 0.74 0.01 0.88 0.39 0.32 0.00 0.30 0.24 0.30 0.02 0.00 0.01 0.52

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(a)

(b)

1

1

D5-12 D25

MCR p-value

0.1 0.05

0.01

0.1 0.05

0.01

PGA AI CAV Ia Ic PGV FI Iv CAD IV SED PGD Id CAI ID Sa EIr EIa ASI VSI IH VEIrSI VEIaSI MASI MASIEC8 MVSI MVSIEC8 MIH MIHEC8 MVEIrSI MVEIrSIEC8 MVEIaSI MVEIaSIEC8

PGA AI CAV Ia Ic PGV FI Iv CAD IV SED PGD Id CAI ID Sa EIr EIa ASI VSI IH VEIrSI VEIaSI MASI MASIEC8 MVSI MVSIEC8 MIH MIHEC8 MVEIrSI MVEIrSIEC8 MVEIaSI MVEIaSIEC8

MCR p-value

D5-12 D25

Fig. 23. P-values of the regressions of e|IM residuals of the MCR prediction with R: (a) OGMs and (b) PLGMs.

(b)

1

1

D5-12 D25

MCR p-value

0.1 0.05

0.01

PGA AI CAV Ia Ic PGV FI Iv CAD IV SED PGD Id CAI ID Sa EIr EIa ASI VSI IH VEIrSI VEIaSI MASI MASIEC8 MVSI MVSIEC8 MIH MIHEC8 MVEIrSI MVEIrSIEC8 MVEIaSI MVEIaSIEC8

MCR p-value

D5-12 D25

0.1 0.05

0.01

PGA AI CAV Ia Ic PGV FI Iv CAD IV SED PGD Id CAI ID Sa EIr EIa ASI VSI IH VEIrSI VEIaSI MASI MASIEC8 MVSI MVSIEC8 MIH MIHEC8 MVEIrSI MVEIrSIEC8 MVEIaSI MVEIaSIEC8

(a)

Fig. 24. P-values of the regressions of e|IM residuals of the MCR prediction with M: (a) OGMs and (b) PLGMs.

Table 17 Sufficiency analysis of MCR for D5-12 and D25 buildings: p-value of the most sufficient IMs.

included in Table 14 (values bolded) show a good level of efficiency. MIHEC8, with good levels of efficiency and sufficiency for MIDR, is efficient but not sufficient for this kind of EDP, due to the magnitude.

MCR p-value OGMs

PLGMs

Most sufficient IMs Distance

Ic Ia PGV CAD IV PGD Id ID Sa EIr EIa VEIaSI

Magnitude

Distance

Magnitude

D5-12 D25

D5-12 D25

D5-12 D25

D5-12 D25

0.31 0.05 0.22 0.11 0.38 0.56 0.26 0.37 0.26 0.17 0.20 0.52

0.27 0.58 0.19 0.87 0.52 0.28 0.48 0.28 0.17 0.59 0.45 0.07

0.96 0.35 0.14 0.01 0.61 0.03 0.01 0.02 0.03 0.10 0.80 0.19

0.59 0.12 0.00 0.29 0.04 0.14 0.32 0.16 0.51 0.31 0.02 0.00

0.40 0.07 0.08 0.16 0.15 0.81 0.42 0.56 0.46 0.29 0.33 0.14

0.07 0.21 0.42 0.76 0.96 0.31 0.46 0.27 0.05 0.65 0.78 0.05

0.23 0.06 0.35 0.00 0.61 0.02 0.01 0.02 0.00 0.01 0.16 0.57

0.20 0.82 0.00 0.87 0.93 0.22 0.40 0.24 0.04 0.48 0.91 0.00

4.2.2.2. Sufficiency of IMs for MFA. The results of the sufficiency analyses for MFA are reported in Figs. 15 and 16. There is generally a good level of sufficiency for all IMs, except for some velocity and displacement related non-structure IMs and for the efficient MVEIaSIEC8. Table 14 includes the figures of the most sufficient IMs. For both kinds of records, PGA, ASI, MASI and MIH are the most sufficient and have good levels of efficiency. For OGMs, all the IMs

4.2.2.3. Sufficiency of IMs for sgmax and sgavg. The analysis of sufficiency carried out reflects a similar trend for sgmax and sgavg, as can be seen in Figs. 17–20. The IMs with a good sufficiency level for sgmax are included in Table 15. Only the intensity measure PGV is sufficient for both OGM and PLGM records, though is not efficient for either. Moreover, there is a set of IMs attaining the sufficiency level only for one kind of record (bolded in Table 15), but none of them are efficient. A greater number of IMs with the sufficiency level is observed for OGMs than for PLGMs. Among the former, only MVEIaSIEC8 is efficient for both OGM and PLGM records. However, this IM is not sufficient with PLGMs records, except for M in the D25 prototype. Furthermore, MIHEC8 and MVSIEC8, which were considered to have acceptable efficiency levels for PLGMs records, show good sufficiency levels for R with OGMs, but not for PLGM records. 4.2.2.4. Sufficiency of IMs for fgmax. Depicted in Figs. 21 and 22 are the results obtained for fgmax. The IMs with the most sufficiency levels are extracted in Table 16. Those with a sufficiency level for both OGMs and PLGMs are IV, Ic, VSI and EIa. Among these, VSI is the only IM included in Table 10, with minimum values of re, though exceeding the limits for an acceptable efficiency. The rest

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(a)

(b)

1

1

D5-12 D25

MBR p-value

0.1 0.05

0.01

0.1 0.05

0.01

PGA AI CAV Ia Ic PGV FI Iv CAD IV SED PGD Id CAI ID Sa EIr EIa ASI VSI IH VEIrSI VEIaSI MASI MASIEC8 MVSI MVSIEC8 MIH MIHEC8 MVEIrSI MVEIrSIEC8 MVEIaSI MVEIaSIEC8

PGA AI CAV Ia Ic PGV FI Iv CAD IV SED PGD Id CAI ID Sa EIr EIa ASI VSI IH VEIrSI VEIaSI MASI MASIEC8 MVSI MVSIEC8 MIH MIHEC8 MVEIrSI MVEIrSIEC8 MVEIaSI MVEIaSIEC8

MBR p-value

D5-12 D25

Fig. 25. P-values of the regressions of e|IM residuals of the MBR prediction with R: (a) OGMs and (b) PLGMs.

(a)

(b)

1

1 D5-12 D25

MBR p-value

0.1 0.05

0.01

0.1 0.05

0.01

PGA AI CAV Ia Ic PGV FI Iv CAD IV SED PGD Id CAI ID Sa EIr EIa ASI VSI IH VEIrSI VEIaSI MASI MASIEC8 MVSI MVSIEC8 MIH MIHEC8 MVEIrSI MVEIrSIEC8 MVEIaSI MVEIaSIEC8

PGA AI CAV Ia Ic PGV FI Iv CAD IV SED PGD Id CAI ID Sa EIr EIa ASI VSI IH VEIrSI VEIaSI MASI MASIEC8 MVSI MVSIEC8 MIH MIHEC8 MVEIrSI MVEIrSIEC8 MVEIaSI MVEIaSIEC8

MBR p-value

D5-12 D25

Fig. 26. P-values of the regressions of e|IM residuals of the MBR prediction with M: (a) OGMs and (b) PLGMs.

Table 18 Sufficiency analysis of MBR for D5-12 and D25 buildings: p-value of the most sufficient IMs.

Table 19 IMs corresponding to each EDPs with the best values of efficiency and sufficiency. EDPs

IMs for OGMs Efficiency

MBR p-value

MRDR and MIDR OGMs

Ic Ia CAD IV PGD Id ID Sa EIr EIa

Magnitude

Distance

D5-12 D25

D5-12 D25

D5-12 D25

0.35 0.05 0.08 0.36 0.50 0.22 0.30 0.20 0.13 0.16

0.42 0.83 0.95 0.19 0.16 0.32 0.17 0.32 0.26 0.16

0.99 0.34 0.01 0.61 0.02 0.01 0.02 0.03 0.09 0.79

0.54 0.10 0.27 0.03 0.14 0.33 0.16 0.55 0.27 0.01

0.37 0.06 0.17 0.09 0.87 0.46 0.61 0.38 0.25 0.28

0.17 0.40 0.95 0.42 0.16 0.26 0.13 0.16 0.98 0.82

0.30 0.08 0.00 0.75 0.02 0.01 0.02 0.00 0.01 0.22

0.29 0.99 0.84 0.71 0.20 0.37 0.21 0.06 0.62 0.70

of the IMs included in Table 16 are sufficient only for OGMs, but none are efficient.

4.2.2.5. Sufficiency of IMs for MCR. The results obtained for MCR are depicted in Figs. 23 and 24. The best IMs are Ia and Ic, which are reported with bold figures in Table 17 for both OGMs and PLGMs.

MIHEC8

PGA ASI MASI MIH

PGA ASI MASI MIH

PGA ASI MASI MIH

MVEIaSIEC8

MVEIaSIEC8 PGV

MVEIaSIEC8

MVEIaSIEC8 PGV

g

VSI*

VSI EIa

VSI*

VSI EIa

MCR and MBR

Ic** Ia** EIa VEIaSI

Ic Ia EIa VEIaSI

Ic** Ia**

Ic Ia

g

s max

and sgavg

f max

*

MIHEC8 MASIEC8

PGA ASI MASI MIH

MFA

**

Sufficiency

IV Sa EIr

Magnitude

D5-12 D25

Efficiency

MIHEC8 MASIEC8 IV Sa EIr

PLGMs

Most sufficient IMs Distance

IMs for PLGMS Sufficiency

Values 6 0.71. Values 6 0.51.

Though these IMs are not efficient, the values attained by the standard deviation are not far from the limit of acceptable efficiency, with re 6 0:46 and re 6 0:51 for Ic and Ia, respectively. The rest

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of IMs included in Table 17 have a sufficiency level only for OGMs. Among these IMs, some have acceptable efficiency, that is re 6 0:40, such as PGV, IV, Sa, EIa and VEIaSI. 4.2.2.6. Sufficiency of IMs for MBR. The results obtained for MBR are depicted in Figs. 25 and 26. As for MCR, the best IMs in this case are Ia and Ic, which are reported as bold figures in Table 18 for both OGMs and PLGMs. Furthermore, the values attained by the standard deviation are again far from the limit of acceptable efficiency, with re 6 0.44 and re 6 0.46 in this case for Ic and Ia, respectively. The remaining IMs have a sufficient level for OGMs, also being efficient IV, Sa, EIr and EIa. It can be seen that the IMs with sufficiency are nearly the same as for MCR. Therefore, both serve as a reference as to the local damage on RC elements. 5. Summary and conclusions The objective of this study was to select the IMs that best predict the seismic response of RC frame structures with hysteretic dampers, having low-to-moderate height and low aspect ratio, according to the potential predictive model proposed. The response of the structure is represented through the variables MRDR and MIDR, related with the lateral displacements of the structure, sgmax and sgavg corresponding to the dissipated energy of the dampers, and fgmax, MCR and MBR related with the damage on the RC structure. In order to attain this objective, one 6-story frame RC structure was retrofitted with hysteretic dampers designed for far-field (D25) and closest distance field (D5-12) seismic actions. Non-linear time history analyses with ordinary and pulse-like ground motion records were carried out on both designs. The prediction capability of a large number of IMs was evaluated. Some of the integral parameters were modified in their integration limits in two different ways, one of them according to the criteria of Eurocode 8. The main conclusions obtained from this study are:
MIHEC8 and MASIEC8 are the best IMs for predicting the MRDR and MIDR for both kind of records, OGMs and PLGMs. Nonetheless, only the first IM is sufficient for PLGMs, and only with structures designed in near-field for OGMs.
IV, Sa and EIr are the only IMs for predicting the MRDR and MIDR with acceptable efficiency and sufficiency level for OGMs.
PGA, ASI, MASI and MIH are the IMs for predicting MFA with the most efficiency and, furthermore, with sufficiency for both OGMs and PLGMs. Among these, MIH has in general the lower values of standard deviation for both kind of records, and therefore is the most efficient.
For the EDPs of sgmax and sgavg, only MVEIaSIEC8 shows good efficiency levels for both OGM and PLGM records, and good sufficiency levels except for M in prototype D25. Furthermore, PGV is the only IM with an acceptable sufficiency level for both kind of records, but not the same degree for efficiency.
For the EDP of fgmax, there is no efficient IM, either for OGMs or PLGMs. VSI is the IM having the lowest deviations with respect to the predictive model (re 6 0:71) and could be taken into account for less accurate predictions. Furthermore, VSI is sufficient for both kind of records, OGMs and PLGMs.
Ic and Ia are the IMs showing efficiency not far from acceptable levels (re 6 0:46 and re 6 0:51, respectively) and good values of sufficiency for both MCR and MBR, taking into account both OGM and PLGM records. For OGMs is particular, the IMs with the best levels of efficiency and sufficiency are EIa and VEIaSI. Table 19 summarizes the IMs selected as the most efficient and/ or sufficient for the EDPs considered in this study. As can be seen, a great number of these IMs are energy-based parameters. Clearly,

EIr, EIa and VEIaSI belong to this typology. Others, such as VSI, MIH, MVEIaSIEC8 and MIHEC8, are velocity-response integral parameters, being directly related with the energy that contributes to damage of a structure (Housner [18]) when it is expressed in the equivalent velocity form. According to the results obtained, for predicting the MIDR, the best option among all the IMs considered is IV, Sa or EIr for OGMs, and MIHEC8 for PLGMs. For MFA, the IMs selected is MIH for both kinds of records. For sgmax and sgavg, MVEIaSIEC8 can be considered as an efficient but not complete sufficient IM and PGV as the best sufficient IM. For fgmax the IM selected is VSI which is sufficient but not efficient although it is one of the IM with best predictive capacities between all them, not far from the acceptable levels. Finally, Ic can be considered as an adequately efficient and sufficient IM for MCR and MBR for all records considered in this study. The IMs suggested above should be used in any direction in the actual design process based on a probabilistic approach. Finally, it is important to remark that these conclusions are not general but hold only for structures similar to those considered in the present study, namely, medium-rise buildings having reinforced concrete frame structures and hysteretic dampers. A larger set of buildings, characterized by different heights and structural typologies, should be considered in future research to confirm and generalize the obtained results. Acknowledgments This work has been partially carried out under the program DPC-Reluis 2014–2016. The financial support of the Italian Ministry of the Instruction, University and Research (MIUR) is also acknowledged. This research received also the financial support of the local government of Spain, Consejería de Innovación, Ciencia y Tecnología, Junta de Andalucía, Project PE2012 TEP12 2429, and from the European Union (Fonds Européen de Dévelopment Régional). This support is gratefully acknowledged. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect those of the sponsor. References [1] Moehle J, Deierlein GG. A framework methodology for performance-based earthquake engineering. In: Proceedings of the 13th world conference on earthquake engineering, Vancouver, Canada; 2004 [Paper No. 679]. [2] Cornell CA, Krawinkler H. Progress and challenges in seismic performance assessment. PEER Center News, vol. 3(2); 2000. [3] Luco N, Cornell CA. Structure-specific scalar intensity measures for near-fault and ordinary earthquake ground motions. Earthq Spectra 2007;23(2):357–92. [4] Shome N, Cornell CA, Bazzurro P, Carballo JE. Earthquakes, records, and nonlinear responses. Earthq Spectra 1998;14(3):469–500. [5] Padgett JE, Nielson BG, DesRoches R. Selection of optimal intensity measures in probabilistic seismic demand models of highway bridge portfolios. Earthquake Eng Struct Dyn 2008;37:711–25. [6] Shome N, Cornell CA. Probabilistic seismic demand analysis of nonlinear structures (RMS Program). Report No. RMS35. Stanford University; 2014 [320pp]. [7] Cordova PP, Deirlein GG, Mehanny SSF, Cornell CA. Development of a twoparameter seismic intensity measure and probabilistic assessment procedure. In: Proceedings of the 2nd U.S.–Japan workshop on performance-based earthquake engineering of reinforced concrete building structures, Sapporo, Hokkaido, Japan; 2000. [8] Luco L, Manuel L, Bazzurro P. Correlation of damage of steel moment-resisting frames to a vector-valued ground motion parameter set that includes energy demands. Report prepared for U.S.G.S., Grant No. 03HQGR0057; February2005. [9] Baker JW, Cornell CA. Spectral shape, epsilon and record selection. Earthq Eng Struct Dyn 2006;35(9):1077–95. [10] Baker JW, Cornell CA. Vector-valued intensity measures incorporating spectral shape for prediction of structural response. J Earthq Eng 2008;12(4):534–54. [11] Riddell R. On ground motion intensity indices. Earthq Spectra 2007;23 (1):147–73. [12] Jayaram, N, Mollaioli F, Bazzurro P, De Sortis A, Bruno S. Prediction of structural response of reinforced concrete frames subjected to earthquake

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