soil-structure interaction effect on fragility curve of 3d...

Research ArticleSoil-Structure Interaction Effect on Fragility Curve of3D Models of Concrete Moment-Resisting Buildings

Ali Anvarsamarin 1 Fayaz Rahimzadeh Rofooei 2 andMasoud Nekooei1

1Department of Civil Engineering Science and Research Branch Islamic Azad University Tehran Iran2Civil Engineering Department Sharif University of Technology Tehran Iran

Correspondence should be addressed to Fayaz Rahimzadeh Rofooei rofooeisharifedu

Received 21 August 2017 Revised 23 January 2018 Accepted 27 February 2018 Published 19 April 2018

Academic Editor Sara Muggiasca

Copyright copy 2018 Ali Anvarsamarin et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper presents the probabilistic generation of collapse fragility curves for evaluating the performance of 3D reinforced concrete(RC) moment-resisting building models considering soil-structure interaction (SSI) by concentration on seismic uncertaintiesIt considers collapse as the loss of lateral load-resisting capacity of the building structures due to severe ground shaking andconsequent large interstory drifts intensified by 119875-Δ effects as well as the strength and stiffness deterioration of their lateral loadcarrying systemsThe estimation of the collapse performance of structures requires the relation between the intensitymeasure (IM)and the probability of collapse that is determined using the generated collapse fragility curves Considering a number of 6- 12-and 18-story 3D RC moment-resisting buildings two scalar IMs are employed to estimate their collapse fragility curve On theother hand the effect of the site soil type on the collapse fragility curves was taken into account by considering the soil-structureinteraction According to the obtained results adopting the average of spectral acceleration (Saavg) intensity measure is moreefficient in capturing the effect of the inherent uncertainties of the strong ground motions on the structural response parametersIn addition considering the SSI for soil type D with shear-wave velocity of 180ms to 360ms reduces the median of intensitymeasure (IM = Sa(1198791)) of fragility curve in 6- 12- and 18-story buildings by 492 2226 and 2303 respectively

1 Introduction

According to a conventional definition the collapse of build-ings during or within short time after earthquake excitationis due to the loss of their structural system integrity thatin turn is caused by generation of large deformation orforce demands in their primary structural components Thestochastic nature of the strong ground motions as well as thefact that no numerical approach can exactly model all thefeatures of complex structural behavior makes the seismiccollapse analysis of structures very difficult Probabilistictechniques on the other hand have introduced the combi-nation of variable possible sources in collapse assessmentprocess as well as collapse potential of available buildingsin the form of collapse probability [1ndash4] The probabilisticestimation ofmaximum story drift demands by considering a9-story moment-resisting frame building subjected to severeground motions was improved by Stoica et al [5] They

concluded that the three-parameter log-normal distributiondescribes maximum story drifts at higher values of spectralacceleration more rationally statistically adding values toreplace truncated data points provides better fitness in maxi-mum drift demand prediction when the structure is close tothe onset of dynamic instability and the least-squares fittingof the log-normal distribution yields parameters that providean improved fit compared to that from maximum likelihoodestimation and the method of moments Current seismicdesign codes do not take into account duration effects and aremainly based on spectral acceleration This fact is because ofthe lack of actual recorded for long duration ground motionand debates about whether artificial records can take place ofrecorded motions In the last few years many long durationground motions have been recorded after events such asTohokuThis led to few researches on the effect of consideringground motion duration on buildings damage and risk ofcollapse (eg [6])

HindawiShock and VibrationVolume 2018 Article ID 7270137 13 pageshttpsdoiorg10115520187270137

2 Shock and Vibration

In general the probabilistic approaches for assessment ofthe buildings collapse potential use the very basic assumptionthat the effects originated from the uncertainties existing inthe earthquake recordsrsquo characteristics are independent ofthe uncertainties in the numerical modeling and predictionof seismic hazard level A performance target in collapselevel can be considered as a tolerable collapse probabilityin a given hazard level Currently collapse fragility curve isused as one of the most essential probabilistic approachesto assess a structure collapse measurement A set of IDAanalyses can play a vital role in determining estimationparameters and in turn determining collapse fragility curveThe seismic collapse safety of the older RC frames whichwas designed and constructed prior to the 1970s was usedto study the effect of two structural parameters such asthe transverse reinforcing ratio and the flexural strengthratio of the column to the beam As a result of a studyit was found out that the two ratios that interacted witheach other must be considered to appreciate the collapse riskof a building [7] The experience of recent decade revealedthat in big heavy and rigid buildings founded on soft andstiff soils SSI effects are of high importance in terms ofthe buildings stability In buildings founded on soft and stiffsoil SSI increases the fundamental period of system anddecreases the base shear forceThis study uses 6- and 12-storyRC buildings with intermediate moment frames system aswell as an 18-story RC building with special moment framesystem to evaluate SSI effects on the determination of thedemand and capacity of collapse and finally to determine thestructural fragility curves Finally by studying the dispersionof IDA curves and the extent of the results dependencyon seismic records properties the appropriate IM will beselected among two mentioned scalar IMs Soil-structureinteraction (SSI) can alter the performance of structure totallyincluding its dynamic characteristics response maximaand more importantly distribution of nonlinear responsethrough structure where the accurate calculation of it is vitalfor the performance evaluation

2 Global Collapse

The study of global collapse was triggered by considering 119875-Δ effects on seismic response Although hysteresis modelsconsidered positive postyielding stiffness the structure tan-gent stiffness became negative under large 119875-Δ effects whichin turn lead to the systemic collapse Structural performanceassessment needs a numerical criterion According to FEMA2000 the IDA approach is proposed to estimate the structuralcapacity of the buildings In addition FEMA2000 is classifiedas the point of occurrence of collapse based on the followingconditions numerical nonconvergence in structural analysisalgorithm and the occurrence of a slope equal to 20 of theinitial elastic slope in the IDA curve and exceedance of themaximum internal drift ratio above 10 Jalayer and Cornellused IDA concept to estimate total dynamic instability capac-ity of a regular RC structure [8] Robust fragility assessmentfor CloudAnalysis considering the global dynamic instabilityand IDA can also be found in provided detail by Jalayer et al[9] In addition their research includes details about how a

modified version of the Cloud Analysis can officially handlethe existence of records that trigger collapse Haselton andDierlin evaluated the collapse risk of 30 4-story buildingswith special moment frame designed in accordance withASCE9-02 According to their results likelihood of structuralcollapse for earthquakes with a return period of 2475 yearsranges from 3 to 20 with a mean of 11 Their studyrevealed that the minimum essential base shear determinedby ASCE7-02 is an important element guaranteeing almosta fixed collapse risk for buildings with different heights [10]Haselton et al studied the collapse capacity of 1- to 20-storybuildings with special moment frames Their study showedthat for a 2-in-50 years earthquake the mean collapseprobability in the studied buildings is about 11 [11] Lignoset al used the results of a collapse test conducted on a 4-story steel moment frame on shaking table of E-DefenceLaboratory and derived the essential parameters required formodeling the collapse reliability of buildings According totheir results the difference in collapse capacity in regularplan buildings is not significant between 3D and 2D models[12] Rajeev and Tesfamariam demonstrated fragility basedseismic vulnerability of structures with consideration of softstory (SS) and quality of construction (CQ) on three- five-and nine-story RC frames designed prior to the 1970s Theydeveloped probabilistic seismic demand model (PSDM) forthose gravity load designed structures using the nonlinearfinite element analysis considering the interactions betweenSS and CQ Result of their study showed the sensitivity ofthe model parameter to the interaction of SS and CQ [13]Palermo et al evaluated the efficiency of current modelingtechniques in predicting the collapse capacity of the studiedstructuresThe results of numerical modeling were comparedwith real samples of LrsquoAquila earthquake in Italy Accordingto the study modern techniques have better capability ofpredicting the collapse capacity of structures [14]

The median collapse intensity can be visualized throughthe concept of incremental dynamic analysis (IDA) in whichindividual groundmotions are scaled to increasing intensitiesuntil the structure reaches a collapse point [15] To considerthe uncertainties of seismic intensity measure predictionsthis method normalizes an earthquake record in a mannerto cover a wide range of intensity measure On the otherhand acceptable numbers of seismic records are requiredto take the uncertainties of the frequency content andspectral shape of earthquakes into account The relationshipbetween engineering demand parameter (EPD) registered inan IDA and one or more seismic intensity measures (IMs)which describe the normalized earthquake records appliedto the structure is called IDA curve Collapse assessmentof structures based on IDA curves necessitates the selectionof fit IM and a fit seismic demand (DM) used to constructthese IDA curves as well as variability in the set of groundmotions considered in the analysis [16] Intensity measureshould be selected in a manner to be connected to the mainrecord using a scale factor PGA PGV and elastic spectralacceleration in fundamental period of a structure for 5 ofcritical damping are themost conventional indices which canbe used as scalar or vector indices The selection of an IMdepends on the efficiency in terms of seismic intensity and on

Shock and Vibration 3

Monitoring sections

Element

Steel fiber Concrete fiber

Section

X

Y

Y

Z

Z

Figure 1 Modeling flexural members using fiber command

0 Axial strain0

Com

pres

sive s

tress

11

Confined concrete

Unconfined concrete

Assumed forcover concrete

First hoopfracture

Ec

EＭ＝

fcc

fc0

Figure 2 Determining the compressive strength of confined con-crete in accordance with Mander model

the sufficiency in terms of the number of earthquake recordsGenerally if the selected IM can reduce the dispersion ofIDA results it will have a better efficiency In additionif the selected IM can reduce the dependency of demandand seismic capacity distributions on the record selection itwill provide a better sufficiency Generally the aim of thisfeature is to reduce the dependency of results on recordsspecifications including magnitude and distance from faultIf there are no near field earthquake-caused directivity effectsin seismic records selecting Sa(1198791 5) for moderate heightstructures where the first mode is the governingmode will besufficient for describing the primary specifications of groundmotion in structural responses [17] In this way by estimatingseismic demand and capacity sufficient and appropriatedata of structure response can be obtained using fewerrecords (ranging from 10 to 20 earthquake records) with nodependency of results on record intensity and fault distanceSpectral acceleration average (Saavg) was introduced as thegeometric mean of the acceleration response spectra pointsof seismic record at 5 of damping [18 19]

Eads et al showed that Saavg is generally more efficientand more sufficient and provides more stable collapse riskestimates than Sa(1198791) after evaluating the efficiency and suf-ficiency of a similar IM for collapse prediction using nearly700moment-resisting frame and shearwall structures of vari-ous heights [20] A recasting of conditional spectrum recordselection which is based on Saavg over a period range asthe conditioning IM was presented by Kohrangi et al [21]

Two sets of ground motions consisting of ordinary and near-fault records were used for evaluating the IMs on 2D RCframe structures with fixed base and isolated at the baseThe most suitable IMs for predicting the considered differ-ent demand parameters and types of structure were identifiedin terms of both efficiency and sufficiency [22] Luco andCornell introduced several new scalar IMs for performanceassessment of three steel frames of moderate-to-long periodsubjected to the 31 near-source andor 59 ordinary groundmotions records These IMs were compared by describingthe ldquoefficiencyrdquo and ldquosufficiencyrdquo of an IM that can be deter-mined by nonlinear dynamic analysis of the building underthe selected ground motion records and linear regressionanalysis One of the considered IMs has been found to berelatively efficient and sufficient for the near-source as wellas the ordinary earthquake records [23]

By reviewing the dispersion of IDA curve and the depen-dency of results on seismic records properties this studyselects the best fit IM among Saavg(119879119894 5) and Sa(1198791 5)

3 Numerical Modeling

The open system for earthquake engineering simulation(OpenSees) program [24] is used for numerical modelingand analysis of the considered structural models The forcebased element (FBE) and displacement based element (DBE)are two techniques by OpenSees software for modeling thenonlinear behavior of different structural members In thisstudy nonlinearBeamColumn command is adopted tomodelthe structural members using the fiber section which isbased on the force formulation and considers the spread ofplasticity along the element [25]This is the most economicaland accurate approach to investigate the seismic behavior ofRC structures Figure 1 shows the unidirectional steel andconcrete layers in flexural members [26 27] Accuracy of thesolution in FBE can be improved by increasing either thenumber of integration points or the number of elements

Concrete02 material commands are used for concretemodeling in OpenSees software However RCmembers havebeen generally confined using stirrups which leads to theenhanced stress-strain constitutive behavior Therefore thecover and core concrete can bemodeled as differentmaterialsusing the same material type but different stress and straincharacteristics and different material tags The compressivestrength and strain of core concrete are determined usingMander-Priestley model [28] (Figure 2) in order to promote


modeling accuracy The aforementioned model is used asa general model to take confinement effects into accountin different columns Lateral reinforcements have differenttypes such as circular spiral and rectangular stirrup with orwithout tie This project considered the structural memberssections as rectangular sectionswith tie stirrup exposed to theunidirectional loads This issue has been studied widely withregard to the stress-strain relationship of confined concreteusing Manderrsquos equation

119891119888 = 119891119888119888119883119903119903 minus 1 + 1198832 119883 = 120576119888120576119888119888 120576119888119888 = [119877(

1198911198881198881198911198880 minus 1) + 1] 1205761198880 119903 = 119864119888119864119888 minus 119864119904119890119888 (1)

where 119883 represents ratio of strain to strain at maximumstress 119891119888119888 represents the maximum stress of confined con-crete 119903 is the ratio of the primary module of elasticity tothe difference of the primary and secondary modules ofelasticity and 119877 is an empirical parameter obtained fromdifferent tests This model suggests 119877 = 3 and 119877 = 6 forhigh strength concrete and typical concrete respectively Thefollowing equation gives the maximum stress of confinedconcrete based on Mander relation

119891119888119888 = 119891119888119900(2254radic1 + 794 1198911119891119888119900 minus 2 1198911119891119888119900 minus 1254) 1198911 = 12119870119890120588119904119891119910ℎ

119864119904119890119888 = 119891119888119888120576119888119888 (2)

The effective confinement coefficient 119870119890 is a very importantparameter It shows the effectiveness of different lateralstirrups Mander et al introduced different relations fordifferent lateral reinforcements especially circular and spiralstirrups in order to calculate119870119890

119870119890 = 1 minus 119896119878119863101584010158401 minus 120588119888119888 (3)

where 120588119888119888 represents the ratio of longitudinal reinforcementarea to core concrete section area 120588119904 is the ratio of lateralconfining lateral reinforcements volume to the confinedcore concrete volume 119891119910ℎ is the yielding stress of lateralreinforcements and 119896 is 05 for spiral and 10 for stirrupreinforcement

The stress-strain curves of confined core concrete instructural element sections were determined using KSU-RCprogram [29] and Mander relationship Rayleigh dampingis used in modeling of structures It should be noted thatto avoid numerical instability or nonconvergence Steel02command is used for reinforcing bars modeling in OpenSeessoftware In Steel02 command the strain-hardening ratio isconsidered to be equal to 001

31 Soil-Structure Model One of the issues that can affectthe seismic response of structures is soil-structure interac-tion (SSI) which has been investigated by many researches(eg [30ndash32]) Depending on the stiffness of the soil theseismic performance and energy dissipation of structurescan significantly be changed by considering the SSI effectsKhoshnoudian et al showed that soil flexibility increasesthe dynamic instability of structural system and collapsestrength reduction factor highly decreases by increasingnondimensional frequency [30] Shakib and Homaei showedthat SSI reduces the structural seismic capacities ductiledeformation and life-safety confidence level [31] In additionthey showed that SSI increases the drift demand Behnamfarand Banizadeh showed that with a flexible base the locationof maximum drift moves to the first story where the mostsevere vulnerability is observed [32] They showed that SSIchanges the pattern of distribution of vulnerability especiallyfor the beams of shear wall buildings and increases theseismic vulnerability on soft soils

Due to the importance of preventing the collapse ofstructures under severe groundmotions and the considerableeffects of SSI on the structural seismic response the effectsof SSI on seismic collapse capacity of RC moment-resistingbuildings are investigated in the present study The soil-structure model is then completed by attaching the soil-foundation element at the base of the superstructure Todefine the coefficients of this element taking also into accountsoil nonlinearity established parameters of the cone modelare modified based on the equivalent linear approach Tomodel the soil effect under the structure the cone model(monkey-tail model) was used with presented modificationsFigure 3 shows the schematic model intended for soil-foundation element based on cone model concept [33]

Table 1 presents the properties of the cones and discrete-element models representing a rigid rectangular foundationwith area 1198600 and area moment of inertia about the axisof rotation 1198680 (for torsional rotation 1198680 is polar moment ofinertia) on surface of homogeneous half-space In this Table] is Poissonrsquos ratio of the soil 119881119904 is the shear-wave velocityof the soil 119881119901 is the 119875-wave velocity of the soil 120588 is densityand119866 is the shear modulus of the soil It should be noted thatstiffness coefficients of foundation in Table 1 were calculatedfor rectangle shaped foundations (119871 times 119861 119871 gt 119861)

To model substructure using the monkey-tail model twonodes at the same location have been defined at base levelUsing spring stiffness independent of frequency and taking adamping coefficient into account the frequency dependenceof the interaction is the easiest way to consider the effectsof SSI The nodes have been connected by multiple uniaxialmaterial objects based on conemodel All degrees of freedomin first node and vertical degree of freedom in second nodehave been constrained The springs and dampers have beenconnected from the first node to the second node using azero-length element and the vertical degree of freedom wasignored In addition themasses of separatedmodel have beenapplied to the second node The soil Poissonrsquos coefficient soildamping ratio and soil density as the essential parametersfor soil modeling were considered to be 033 005 and2000 kgm3


Z0

r0

CC

0M0

M

C

1K

P0 U0

ΔM

K

ΔM

Vp

Figure 3 Cone model layout and equivalent lumped elements for soil replacement model

Table 1 Con model of properties of the rectangular foundation on a homogeneous half-space [33]

Motion Horizontal Vertical Rocking Torsion

Equivalent radius (1199030) radic1198600120587 radic1198600120587 4radic41198680120587 4radic21198680120587Aspect ratio 11991101199030 1205878 (2 minus 120592) 1205874 (1 minus 120592)( 119881119881119904 )

2 912058732 (1 minus 120592)( 119881119881119904 )2 912058732

Poissonrsquos ratio All values ] le 13 13 lt ] le 12 ] le 13 13 lt ] le 12 All values

Wave velocity (V) 119881119904 119881119901 2119881119904 119881119901 2119881119904 119881119904Added mass 0 0 24 (120592 minus 13) 12058811986001199030 0 12 (120592 minus 13) 12058811986011986801199030 0

Lumped-parametermodel

119870119909 = 1198661198612 minus 120592 [34 ( 119871119861)065 + 12] 119870119909119909 = 11986611986131 minus 120592 [04 ( 119871119861) + 01]119870119910 = 1198661198612 minus 120592 [34 ( 119871119861)

065 + 04 119871119861 + 08] 119870119910119910 = 11986611986131 minus 120592 [04 ( 119871119861)24 + 0034]

119870119911 = 1198661198611 minus 120592 [155 (119871119861)075 + 08] 119870119911119911 = 1198661198613 [053 (119871119861)

245 + 051]119862 = 120588 sdot 119881 sdot 1198600 119862120579 = 120588 sdot 119881 sdot 1198680 1198621015840120579 = 212058501205960 1198701205791198621015840 = 212058501205960 1198701198981015840 = 12058501205960119862 119872120579 = 120588119868011991101198981015840120579 = 12058501205960119862120579

32 The Considered Structural Models The selection ofthe building is based on the different period range whichincluded mid-to-high-rise buildings on soil type DThree 6-12- and 18-story 3D reinforced concrete moment-resistingbuildings with the fundamental periods of 086 138 and197 second were considered and designed according to theACI 318-08 and ASCE 7-10 code requirements [34 35]These structural models are assumed to be of administrativebuildings type with the same plan dimensions located on soiltype D according to the ASCE 7-10 code [35] The structuralsystem of the 6- and 12-story buildings is intermediatemoment-resisting buildings while the 18-story building isconsidered as special moment-resisting building The spec-ified compressive strength of concrete is 240 kgcm2 in the 6-and 12-story buildings while it is 280 kgcm2 in the 18-storybuilding In all models the ultimate strength of longitudinalbars is 5000 kgcm2 ETABS (2013) was used to design thestructural models [36] The studied buildings have a 30m times18m rectangle plan with a story height of 32m and spansof 6m In designing the building models the story drift

Table 2 Gravity loads

Load type Story RoofDead load (Kgm2) 520 520Live load (Kgm2) 250 150Perimeter walls (Kgm) 570 300

ratios were limited to values specified by the considered codeFigure 4 shows the structural models and typical plan ofbuildings were considered in this study The applied gravityloads to the structural models are presented in Table 2

33 Validation of the Mathematical Model for Soil-StructureInteraction To evaluate the validity of the employed conemodel for SSI the variation of the fundamental periods ofthe 12-story building model due to change in soil shear-wavevelocity is presented in Table 3 As expected an increasein soil shear-wave velocity reduces the fundamental periodof the structural model Also one should observe that the


A

B

C

D

E

F

1

2

3

4

(a)

A

B

C

D

E

F

1

2

3

4

(b)

2

3

4

1

C

D

E

F

B

A

(c)

1

2

3

4

A B C D E F

STAIR

6m6m6m6m6m

6m

6m

6m

(d)

Figure 4 Structural elevation and plan (a) 6-story building (b) 12-story building (c) 18-story building and (d) typical plan

fundamental period of the rigid based 12-story buildingbecomes equal to the case with SSI for the soil shear-wavevelocity of 2000ms and above

34 The Earthquake Records Used for Parametric Studies 23pairs of far field earthquake records have been selected fromPEER [37] that have mostly been used in ATC-55 (FEMA-440) project [38] Table 4 shows the characteristics of theselected earthquake records that have been registered onstiff soil (soil type D with shear-wave velocity of 180ms to360ms) resulting from events with a magnitude of 62 to73 and fault distance of 212 to 507 km The selected recordswere normalized according to the ASCE 7-10 code [35] beforebeing used in the extensive nonlinear dynamic time historyanalyses

4 Analysis and Results

Following the selection and normalization of the earthquakerecords and preparation of the structural models the IDAanalyses were conducted using OpenSees program underthe applied bidirectional seismic excitations In the processof modeling two-way eccentricities of 5 of the building

Table 3 The fundamental period of 12-story building for differentsoil shear wave velocities

119881119904 (msec) 256 2000 No SSIPeriod (sec) 143463 13758 13753

dimensions are considered by proper shifting of the masscenters of each story In calculation of the stiffness coeffi-cients the correction factor for embedment was consideredequal to 1 The IDA curves were developed considering bothscalar intensity measures (IMs) that is Saavg(1198791 119879119899) andSa(1198791 5) The spectral acceleration average (Saavg) that isdescribed by (4) is defined as the geometric mean of thepseudospectral acceleration of an earthquake record over acertain range of periods Figure 5 shows how to calculateSaavg for a given earthquake record response spectrum [18]

Saavg (11989611198791 11989621198791 1198961198991198791) = ( 119899prod119894=1

119878119860 (1198961198941198791))1119899 (4)

where 1198791 is the structure fundamental period and 1198961 to 119896119899are constants specifying lower and upper bounds used to


Table 4 Characteristics of 23 pairs of far field earthquake records for the numerical analyses

Number Date Earthquake name Record name Magnitude (Ms) Station number PGA (g) 119881119904 (ms)

1 011794 Northridge NORTHRMUL279 67 90013 0516 356NORTHRMUL009 0416

2 011794 Northridge NORTHRLOS270 67 90057 0482 309NORTHRLOS0 041

3 011794 Northridge NORTHRHOL90 67 24303 0358 256NORTHRHOL360 0231

4 11121999 Duzce Turkey DuzceBOL090 73 Bolu 0822 326DuzceBOL0 0728

5 101579 Imperial Valley IMPVALLH-DLT352 69 6605 0351 275IMPVALLH-DLT262 0238

6 101579 Imperial Valley IMPVALLH-EL1230 69 5058 038 196IMPVALLH-EL140 0364

7 101579 Imperial Valley IMPVALLH-CHI012 69 6621 027 256IMPVALLH-CHI282 0254

8 112487 Superstition Hills(B) SUPERSTB-CAL315 66 5061 0247 208SUPERSTB-CAL225 018

9 81799 Kocaeli Turkey KOCAELIDZC270 78 Duzce 0358 276KOCAELIDZC180 0312

10 10011987 Whittier Narrows WHITTIERA-BIR180 57 90079 0299 276WHITTIERA-BIR090 0243

11 062892 Landers LANDERSYER270 74 22074 0245 354LANDERSYER360 0152

12 062892 Landers LANDERSCLW-TR 74 23 0417 271LANDERSCLW-LN Coolwater 0283

13 101889 Loma Prieta LOMAPCAP000 71 47125 0529 289LOMAPCAP090 0443

14 101889 Loma Prieta LOMAPGO3000 71 47381 0555 350LOMAPGO3090 0367

15 101889 Loma Prieta LOMAPSLC360 71 1601 0278 289LOMAPSLC270 0194

16 112487 Superstition Hills(B) SUPERSTB-ICC000 66 1335 0358 192SUPERSTB-BICC090 0258

17 112487 Superstition Hills(B) SUPERSTB-POE270 66 Poe Road 0446 208SUPERSTB- POE360 03

18 042592 Cape Mendocino CAPEMENDRIO360 71 89324 0549 312CAPEMENDRIO270 0385

19 092099 Chi-Chi Taiwan CHICHICHY101-N 76 CHY101 044 259CHICHICHY101-W 0353

20 042484 Morgan Hill MORGANHD4165 61 1656 0098 256MORGANHD4255 0092

21 02091971 San Fernando SFERNPEL090 66 135 LA 021 316SFERNPEL180 0174

22 05021983 Coalinga COALINGAH-C05270 65 36227 0147 256COALINGAH-C05360 0131

23 011695 Kobe KOBESHI000 69 Shin-Osaka 0243 256KOBESHI090 0212


k1T1

k2T1

knT11 2 3 40

Period (sec)

0

005

01

015

02

025

03

035

IM

Figure 5 Schematic approach to calculate the average spectralacceleration from the response spectrum of a seismic record

0308153659

IDRs

rss-

Saav

g1M

ean

IDRm

ax-S

aavg

1Mea

nID

Rsrs

s-Sa

avg1

Txi

IDRm

ax-S

aavg

1Txi

IDRs

rss-

Saav

g1Tz

iID

Rmax

-Saa

vg1T

ziID

Rsrs

s-Sa

avg2

Mea

nID

Rmax

-Saa

vg2M

ean

IDRs

rss-

Saav

g2Tx

iID

Rmax

-Saa

vg2T

xiID

Rsrs

s-Sa

avg2

Tzi

IDRm

ax-S

aavg

2Tzi

IDRs

rss-

Saav

g3M

ean

IDRm

ax-S

aavg

3Mea

nID

Rsrs

s-Sa

avg3

Txi

IDRm

ax-S

aavg

3Txi

IDRs

rss-

Saav

g3Tz

iID

Rmax

-Saa

vg3T

ziID

Rsrs

s-Sa

avg4

Mea

nID

Rmax

-Saa

vg4M

ean

IDRs

rss-

Saav

g4Tx

iID

Rmax

-Saa

vg4T

xiID

Rsrs

s-Sa

avg4

Tzi

IDRm

ax-S

aavg

4Tzi

IDRs

rss-

Saav

g5M

ean

IDRs

rss-

Saav

g5M

ean

IDRs

rss-

Saav

g5Tx

iID

Rsrs

s-Sa

avg5

Txi

IDRs

rss-

SaBa

vg5T

ziID

Rsrs

s-Sa

avg5

Tzi

IDRs

rss-

Sa(T

1)ID

Rmax

-Sa(

T1)

IDRs

rss-

SaM

ean

IDRm

ax-S

aMea

nID

Rsrs

s-Sa

srss

IDRm

ax-S

asrs

s

0001020304050607080910

Engineering demand parameter-intensity measure (EDP-IM)

Logarithmic standard deviation of D|IM for various EDP-IM cases

Disp

ersio

n (D|IM

)

Figure 6 Dispersion values for various EPD-IM scenarios in 12-story building with fixed base

determine the averaging points in the elastic accelerationspectrum of the considered record The coefficients 1198961 and119896119899 respectively introduce the effect of different modes onthe dynamic behavior of structure as well as the effect ofnonlinear behavior of the structural system that can betaken into account through the increased effective period ofdamaged system The minimum number between the lowerand upper bound for 1198961 to 119896119899 is 10 [18] Also 1198961 and 119896119899 areselected such that the response dispersion is minimized Toselect the best averaging range different values for 1198961 and 119896119899were evaluated based on the study conducted by Bianchiniet al [18] It should be noted that the mentioned coefficientsvary from 02 to 2

In this study two engineering demand parameters (EDP)are considered maximum internal drift ratio between twohorizontal directions (IDRx or IDRz) and the maximumsquare root of the sum of the squares (SRSS) of IDRx andIDRz (IDRsrss) Also 18 intensitymeasures (IMs) are chosenAccording to the various engineering demand parametersand intensity measures presented in Notations 36 EDP-IM scenarios are generated to perform the incrementaldynamic analysis The dispersion of IDA curves for differentmentioned EDP-IM scenarios is estimated for the selectedstructures Observing dispersion is the best tool to selectof EDP-IM scenario for IDA curves A simple logarithmiclinear regressionmodel is applied to offer an efficient EDP-IMscenario using the estimation of dispersion [22] For example

the dispersion range of 02ndash03 is normally considered asan appropriate efficiency while the range of 03ndash04 is stillrationally acceptable [39] For all considered buildings fixed-base or SSI configurationwith different number of the storiesthe minimum dispersion was observed for the EDP-IMscenario which corresponded to the SRSS(IDR)-Saavg and1198961 = 05 119896119899 = 2 For instance Figure 6 shows dispersionof EDP-IM scenarios (12-story structural model with a two-way eccentricity of 5 and fixed base) which are listed inNotations

The fragility curves of the studied building models weredetermined for the following two cases

(A) EDP = IDRSRSS and IM = Sa (1198791 120585 = 005)(B) EDP = IDRSRSS and IM = Saavg (119879119911 120585 = 005)

41 Determination of Collapse Fragility Curves for the StudiedBuildings To extract the occurrence probability of collapsefrom IDA results the so-called fragility curves are usedCollapse fragility curve can be considered as a cumulativedistribution function (CDF) of a stochastic variable namelycollapse capacity (Sac) Ibarra and krawinkler showed thatSac points follow a log-normal distribution that is ln(119878119886119888) rarr119873(120578119862 120573119877119862) where 120578119862 and 120573119877119862 are median collapse capacityand dispersion of collapse capacity values due to differentearthquake records which are numerically equal to thestandard deviation of collapse capacity values [2] For a givenhazard level like 119875119877 corresponding spectral accelerationcan be obtained using seismic hazard curves and collapseprobability can be calculated from (5) where 120578119862 and 120573119877119862are median and standard deviation of log-normal cumulativedistribution function respectively

119875 (119862 | 119878119875119877119886 ) = Φ( ln (119878119875119877119886 ) minus ln (120578119862)120573119877119862 ) (5)

Figures 7(a) and 7(b) present the fragility curves of the stud-ied buildings under two-way eccentricities of the mass centercorresponding to 5 of building dimensions and under thesimultaneous effects of the horizontal components of theselected earthquake records with the two aforementionedIMs for the cases with fixed base and considering SSI effects

It should be noted that the higher collapse capacity ofthe 6-story structure is because of its lower vulnerability to119875-Δ effects in comparison with the high-rise structures Indesign of this structure an overestimation was considered torepresent the design of administrative buildings In additionthe 6- and 12-story structures aremoderatemoment-resistingbuildings (where the 18-story structure is special moment-resisting building) that increases the design base shear andleads to the higher collapse capacity values

Since the above curves are in the form of log-normalcumulative distribution function with median (120578119862) and stan-dard deviation (120573119877119862) parameters their values are summa-rized in Table 5

It is worth mentioning that stiffness of the 6-storybuilding is high and the structural fundamental period is lowtherefore the soil-structural interaction is not considerable


0

01

02

03

04

05

06

07

08

09

1C

olla

pse p

roba

bilit

y

05 1 15 2 25 3 35 4 45 5 55 6 65 7 75 80Sa(T1 = 005) (g)

6F no SSI Sa(T1)6F with SSI Sa(T1)12F no SSI Sa(T1)

12F with SSI Sa(T1)18F no SSI Sa(T1)18F with SSI Sa(T1)

(a)

05 1 15 2 25 3 35 4 45 5 550Saavg(Tzi = 005) (g)

0

01

02

03

04

05

06

07

08

09

1

Col

laps

e Pro

babi

lity

6F no SSI Saavg(Tzi)6F with SSI Saavg(Tzi)12F no SSI Saavg(Tzi)12F with SSI Saavg(Tzi)18F no SSI Saavg(Tzi)18F with SSI Saavg(Tzi)

(b)

Figure 7 Fragility curves of the studied buildings in two states fixed base and considering SSI (a) IM = Sa(1198791 120585 = 005) (b) IM =Saavg(119879119911119894 120585 = 005)

Table 5 Mean and standard deviation of fragility curves of the studied buildings in different states

Type of support CDF parameters IM = Sa(1198791 120577 = 005) IM = Saavg(119879119911119894 120577 = 005)6-story 12-story 18-story 6-story 12-story 18-story

No SSI Median (120578) 33406 21869 10397 26408 14185 08287Standard deviation (120573RC) 04318 04344 03867 03338 03082 03307

With SSI Median (120578) 31761 17000 08003 25861 11515 06552Standard deviation (120573RC) 04366 04852 04669 03271 02822 04439

Error estimation of fixed-based assumption () 492 2226 2303 207 1882 2093

According to Table 5 the IM values corresponding to the col-lapse probability of 50 (median spectral acceleration capac-ity) decrease respectively by 492 2226 and 2303 for6- 12- and 18-story buildings for IM = Sa(1198791 120577 = 005) andby 207 1882 and 2093 for IM = Saavg Therefore basedon the observed reduction percentage in this study the effectof SSI on themedian amount of the fragility curve for 18-storybuilding is higher than the 12-story building In additionselecting Saavg instead of Sa(1198791 120577 = 005) can decrease thestandard deviation of the fragility curves In other wordsSaavg is more efficient than Sa(1198791 120577 = 005) to be used asan IM

42 Mean Annual Frequency (MAF) of Collapse Based on119878119886(1198791 120577 = 005) The mean annual frequency (MAF) ofcollapse is derived by integration of the building collapsefragility curve relative to the seismic hazard curve defined as

120582119862 = int119875119862|119909 1003816100381610038161003816119889120582119868119872 (119909)1003816100381610038161003816 (6)

Closed-form solution introduced by Jalayer was used toestimate the MAF of collapse [40] The closed-form solutionis shown in (7) where 119896 is seismic hazard curve slope in thelog-log domain and 120578119862 and 120573119877119862 are median and standarddeviation of collapse fragility curve respectively

120582119862 = [120582Sa (120578C)] [exp (1211989621205732119877119862)] (7)

To determine the MAF of collapse for buildings seismichazard curve is required To this end the seismic hazard curvewas derived from Probabilistic Seismic Hazard Analysis(PSHA) corresponding to fundamental period of studiedbuildings In addition seismic hazard curve can be estimatedusing a linear relation in log-log domain shown as follows

120582119878119886 = 1198960119878119886minus119896 (8)

The collapse probability of the building due to seismic actionhas been recommended to be less than 2 in a 50-year


Data points from PSHA

0001822(Sa)^(minus308327)vicinity of (Sa) = 00004

0001304(Sa)^(minus21715)

100E minus 04

100E minus 03

100E minus 02

100E minus 01

100E + 00A

nnua

l pro

babl

ity o

f exc

eeda

nce

02 04 06 08 1 12 14 16 18 2 220Sa(T1 = 086) (g)

Fitted ko middot (Sa)minusK using data points in the

Fitted ko middot (Sa)minusK using all data points

(a)



00004664 (Sa)^(minus22293)

10E minus 04

10E minus 03

10E minus 02

10E minus 01

10E + 00

Ann

ual p

roba

blity

of e

xcee

danc

e

02 04 06 08 1 120Sa(T1 = 138) (g)



(b)



000007252 (Sa)^(minus23342)

10E minus 04

10E minus 03

10E minus 02

10E minus 01

10E + 00

Ann

ual p

roba

blity

of e

xcee

danc

e

01 02 03 04 05 060Sa(T1 = 197) (g)



(c)

Figure 8 Seismic hazard curve for the location (a) period (1198791 = 086 sec) of the 6-story building (b) period (1198791 = 138 sec) of the 12-storybuilding and (c) period (1198791 = 197 sec) of the 18-story building

period with 95 confidence for this purpose by SACFEMAguidelines [41] This objective translates into limiting theMAF of collapse to 00004 (with 95 confidence) assuminga Poisson process for collapse occurrence The SACFEMAguidelines do not include a target that addresses collapse at aspecific hazard level In this section the 1198960 and 119896 parameterswere obtained by fitting a linear regression model in log-logdomain forwhole hazard curve and a local zone in the vicinityof 120582(119878119886) = 00004 (Figure 8) These estimated values for 119896and 1198960 parameters were used to estimate the MAF of collapseusing the closed-form solution of (7) Figures 8(a)ndash8(c) showthe data points representing the seismic hazard curve forthe location of the studied buildings in the high-hazardlevel region of Tehran The blue colored curves show linearregression models using all data points and the red colored

curves show linear regression models using a local zone inthe vicinity of 00004 hazard level

Table 6 shows the values of mean annual frequency(MAF) of collapse estimated using directly applying thenumerical integration to (6) and using the closed-formsolution of (7) for the three studied structural modelsAdditionally for spotting the most important zone of theintegrand in (7) the calculation of the MAF of collapse usingthe closed-form solution has been obtained by fitting thelinear regression model to both whole hazard curve and alocal zone in the vicinity of 00004 hazard level and the resultsare shown in Table 6

As seen in Table 6 solving the closed-form solution of (7)using the linear regressionmodel to whole hazard curve over-estimates the MAF of collapse in comparison with numerical


Table 6 Comparison between MAF of collapse obtained from (6) and (7) in studied building with fixed base

Number ofstories

Numericalintegrationequation (6)

Closed-form solution equation (7)Fitted 1198960(Sa)minus119896 using datapoints in the vicinity of120582(Sa) = 00004

Fitted 1198960(Sa)minus119896 using alldata points of hazard

curve6-story 12882 times 10minus4 10724 times 10minus4 14751 times 10minus412-story 11436 times 10minus4 93398 times 10minus5 13024 times 10minus418-story 8785 times 10minus5 6911 times 10minus5 99589 times 10minus5

integration equation (6) However solving the closed-formsolution of (7) using the linear regressionmodel to local zoneof vicinity of 120582(Sa) = 00004 underestimates the MAF ofcollapse compared to using numerical integration equation(6) The difference between the results obtained from (6) or(7) is construed as the relative accuracy of the closed-formsolution which is developed for solution of (6) by (7)

In Table 6 it is surprising that unlike values derivedfrom fragility curves the collapse probability of the high-story buildings decreases by taking seismic hazard valuesinto account in MAF of collapse calculations whereas thiscollapse probability will be high if such seismic hazard valuesare ignoredTherefore high-rise structures have less collapseprobability The reduction of structurersquos fundamental periodincreases the seismic hazard and on the other hand decreasesthe likelihood of exceeding the collapse performance levelcaused This is the reason behind the increase of MAF ofcollapse in short-story buildings

5 Conclusions

Assuming nonlinear behavior for steel and concretematerialsand a two-way eccentricity of mass center correspondingto 5 of the building dimension at every side this studymodeled three 6- 12- and 18-story RC moment-resistingbuildings Incremental dynamic analysis (IDA) was con-ducted to take the uncertainties of earthquake records intoaccount The buildings performance was studied for fixed-base and SSI consideration using seismic demand proba-bilistic analysis In addition the efficiency of two scalar IMswas evaluated It could be concluded that selecting Saavgreduces the uncertainty due to record-to-record variabilityin seismic behavior study and increases the results reliabilityThis is because it considers the effects of high modes ondynamic behavior of inelastic structural system as well asthe effect of system response on nonlinear region which isaccompanied with increased effective period of the damagedsystem It should be noted that no fixed period range isselected and it can be different from structure to structureAdditionally in 6-story building (short structure) fixed-base assumption leads to an error range by 207ndash492 inthe IM value corresponding to the collapse probability of50 which is negligible compared to SSI effects Moreoverin 12- and 18-story buildings (high-rise buildings) fixed-base assumption leads to the overestimation range of IMvalue by 1882ndash2226 and 2093ndash2309 in the log-normal cumulative distribution function of fragility curves

compared to SSI effects This difference cannot be neglectedThe MAF of collapse for the studied buildings were obtained(numerical integration and solving the closed-form solution)by integrating the structuresrsquo collapse fragility curve overthe seismic hazard curve for the location of the structuresAccordingly it is observed that unlike fragility curve valuesconsidering seismic hazard for calculation of mean annualfrequency of collapse decreases the collapse probability of thehigh-story buildings whereas this probability is higher whenno seismic hazard is considered in fragility curves

The level of the accuracy of the estimatedMAF of collapseusing closed-form solution depends on the linear regressionmodel whereas it was obtained over the whole hazard curveor a vicinity of 120582(Sa) = 00004 Therefore using all datapoints of hazard curve in linear regression model leads tothe overestimation of the MAF of collapse In other wordsthe MAF of collapse by solving the closed-form solutionusing all the data points of hazard curve was overestimatedin comparison with numerical integration

Notations

Engineering Demand Parameters (EDP) and IntensityMeasures (IMs) Considered Used in This Study

IDRmax Maximum internal drift ratio between twohorizontal directions (IDRx or IDRz)

IDRsrss Maximum square root of the sum of thesquares of internal drift ratio in 119909 and 119910axes direction

Sa(1198791) Elastic spectral acceleration infundamental period of a structure for 5of critical damping

Sasrss Square root of the sum of the squares ofthe pseudospectral acceleration overfundamental and second periods of astructure for 5 of critical dampingradicSa(1198791)2 + Sa(1198792)2

SaMean Geometric mean of the pseudospectralacceleration over fundamental and secondperiods a structure for 5 of criticaldampingradicSa(1198791) sdot Sa(1198792)

Saavg1Tzi Spectral acceleration average of anearthquake recorded over a certain rangeof fundamental period corresponding to1198961 = 02 and 119896119899 = 2


Saavg1Txi Spectral acceleration average of anearthquake recorded over a certain rangeof second mode period corresponding to1198961 = 02 and 119896119899 = 2

Saavg1Mean Geometric mean of Saavg1Txi andSaavg1TziradicSaavg1Txi times Saavg1Txi













Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] C A Cornell F Jalayer R O Hamburger and D A FoutchldquoProbabilistic basis for 2000 SAC federal emergency man-agement agency steel moment frame guidelinesrdquo Journal ofStructural Engineering vol 128 no 4 pp 526ndash533 2002

[2] L F Ibarra and H Krawinkler Global Collapse of Frame Struc-tures under Seismic Excitations Pacific Earthquake EngineeringResearch Center Berkeley CA USA 2005

[3] C BHaseltonA B Liel B SDean JHChou andGGDeier-lein ldquoSeismic collapse safety and behavior of modern rein-forced concrete moment frame buildingsrdquo in Structural Engi-neering Research Frontiers pp 1ndash14 2007

[4] F Zareian and H Krawinkler ldquoAssessment of probability ofcollapse and design for collapse safetyrdquo Earthquake Engineeringamp Structural Dynamics vol 36 no 13 pp 1901ndash1914 2007

[5] M Stoica R A Medina and R H McCuen ldquoImproved proba-bilistic quantification of drift demands for seismic evaluationrdquoStructural Safety vol 29 no 2 pp 132ndash145 2007

[6] R Chandramohan J W Baker and G G Deierlein ldquoImpactof hazard-consistent ground motion duration in structuralcollapse risk assessmentrdquo Earthquake Engineering amp StructuralDynamics vol 45 no 8 pp 1357ndash1379 2016

[7] P H Galanis and J P Moehle ldquoDevelopment of collapse indi-cators for risk assessment of older-type reinforced concretebuildingsrdquo Earthquake Spectra vol 31 no 4 pp 1991ndash20062015

[8] F Jalayer and C A Cornell ldquoA technical framework for proba-bility-based demand and capacity factor (DCFD) seismic for-matsrdquo RMS 2003

[9] F Jalayer H Ebrahimian AMiano GManfredi andH SezenldquoAnalytical fragility assessment using unscaled ground motionrecordsrdquo Earthquake Engineering amp Structural Dynamics vol46 no 15 pp 2639ndash2663 2017

[10] C B Haselton and G G Deierlein ldquoAssessing seismic collapsesafety o f modern reinforced concrete moment-frame build-ingsrdquo PEER Report Pacific Earthquake Engineering ResearchCenter College of Engineering University of California Berke-ley CA USA 2007

[11] C B Haselton A B Liel G G Deierlein B S Dean and J HChou ldquoSeismic collapse safety of reinforced concrete buildingsI Assessment of ductile moment framesrdquo Journal of StructuralEngineering vol 137 no 4 pp 481ndash491 2011

[12] D G Lignos T Hikino Y Matsuoka andM Nakashima ldquoCol-lapse assessment of steel moment frames based on E-defensefull-scale shake table collapse testsrdquo Journal of Structural Engi-neering (United States) vol 139 no 1 pp 120ndash132 2013

[13] P Rajeev and S Tesfamariam ldquoSeismic fragilities for reinforcedconcrete buildings with consideration of irregularitiesrdquo Struc-tural Safety vol 39 pp 1ndash13 2012

[14] M Palermo R R Hernandez S Mazzoni and T TrombettildquoOn the seismic behavior of a reinforced concrete building withmasonry infills collapsed during the 2009 Lrsquoaquila earthquakerdquoEarthquake and Structures vol 6 no 1 pp 45ndash69 2014

[15] D Vamvatsikos and C Allin Cornell ldquoIncremental dynamicanalysisrdquo Earthquake Engineering amp Structural Dynamics vol31 no 3 pp 491ndash514 2002

[16] R Villaverde ldquoMethods to assess the seismic collapse capacityof building structures State of the artrdquo Journal of StructuralEngineering vol 133 no 1 pp 57ndash66 2007

[17] N Shome and C A Cornell ldquoProbabilistic seismic demandanalysis of non-linear structuresrdquo Report No RMS-35 StanfordUniversity Stanford CA USA 1999

[18] M Bianchini P P Diotallevi and J W Baker ldquoPrediction ofinelastic structural response using an average of spectral accel-erationsrdquo in Proceedings of the 10th International Conference onStructural Safety and Reliability (ICOSSAR09) pp 13ndash17 OsakaJapan 2009

[19] M Kohrangi P Bazzurro and D Vamvatsikos ldquoVector andscalar IMs in structural response estimation Part I Hazardanalysisrdquo Earthquake Spectra vol 32 no 3 pp 1507ndash1524 2016


[20] L Eads E Miranda and D G Lignos ldquoAverage spectral accel-eration as an intensity measure for collapse risk assessmentrdquoEarthquake Engineering amp Structural Dynamics vol 44 no 12pp 2057ndash2073 2015

[21] M Kohrangi P Bazzurro D Vamvatsikos and A SpillaturaldquoConditional spectrum-based ground motion record selectionusing average spectral accelerationrdquo Earthquake Engineering ampStructural Dynamics vol 46 no 10 pp 1667ndash1685 2017

[22] H Ebrahimian F Jalayer A Lucchini F Mollaioli and GManfredi ldquoPreliminary ranking of alternative scalar and vectorintensity measures of ground shakingrdquo Bulletin of EarthquakeEngineering vol 13 no 10 pp 2805ndash2840 2015

[23] N Luco and C A Cornell ldquoStructure-specific scalar intensitymeasures for near-source and ordinary earthquake groundmotionsrdquo Earthquake Spectra vol 23 no 2 pp 357ndash392 2007

[24] OpenSeesOpen System for Earthquake Engineering SimulationOpenSees University of Berkeley Berkeley CA USA 2011

[25] V Terezic Force-based element vs Displacement-based elementPacific Earthquake Engineering Research Center (PEER) Uni-versity of California Berkeley CA USA 2011

[26] E Spacone F C Filippou and F F Taucer ldquoFibre beam-columnmodel for non-linear analysis of RC frames part I Formu-lationrdquo Earthquake Engineering amp Structural Dynamics vol 25no 7 pp 711ndash725 1996

[27] E Spacone F C Filippou and F F Taucer ldquoFibre beamndashcolumnmodel for non-linear analysis of rc frames part ii applicationsrdquoEarthquake Engineering amp Structural Dynamics vol 25 no 7pp 727ndash742 1996

[28] J B Mander M J N Priestley and R Park ldquoObserved stress-strain behavior of confined concreterdquo Journal of Structural Engi-neering vol 114 no 8 pp 1827ndash1849 1988

[29] KSU-RC KSU-RC Moment-Curvature Force and InteractionAnalysis of Reinforced Concrete Member V 1011 Inc KansasState University Kansas USA

[30] F Khoshnoudian E Ahmadi M Kiani and M H TehranildquoDynamic instability of Soil-SDOF structure systems under far-fault earthquakesrdquo Earthquake Spectra vol 31 no 4 pp 2419ndash2441 2015

[31] F Behnamfar andM Banizadeh ldquoEffects of soil-structure inter-action on distribution of seismic vulnerability in RC structuresrdquoSoil Dynamics and Earthquake Engineering vol 80 pp 73ndash862016

[32] H Shakib and F Homaei ldquoProbabilistic seismic performanceassessment of the soil-structure interaction effect on seismicresponse of mid-rise setback steel buildingsrdquo Bulletin of Earth-quake Engineering vol 15 no 7 pp 2827ndash2851 2017

[33] J P Wolf ldquoCone models as a strength of materials approach tofoundation vibrationrdquo in Proceedings of the 10th European Con-ference on Earthq Eng A A Balkea Ed pp 1ndash10 ViennaAustria 1994

[34] ACI Committee American Concrete Institute and Interna-tional Organization for Standardization Building code require-ments for structural concrete (ACI 318-08) and commentaryAmerican Concrete Institute 2008

[35] ASCE 7-10 Minimum Design Loads for Buildings and OtherStructures American Society of Civil Engineers Reston VAUSA 2010

[36] ETABS Structural Analysis Program Computers and StructuresInc Berkeley CA USA 2013

[37] Pacific Earthquake Engineering Research Center (PEER) PEERNext Generation Attenuation (NGA) Database 2013 httpsngawest2berkeleyedu

[38] Federal Emergency Management Agency (FEMA) ldquoImprove-ment of nonlinear static seismic proceduresrdquo Report NoFEMA-440 Federal Emergency Management Agency Wash-ington DC USA

[39] F Mollaioli A Lucchini Y Cheng and G Monti ldquoIntensitymeasures for the seismic response prediction of base-isolatedbuildingsrdquo Bulletin of Earthquake Engineering vol 11 no 5 pp1841ndash1866 2013

[40] F Jalayer Direct probabilistic seismic analysis implementingnon-linear dynamic assessments [PhD dissertation] StanfordUniversity 2003

[41] Federal EmergencyManagement Agency (FEMA) ldquoState of theart report on performance prediction and evaluation of steelmoment-frame buildingsrdquo Report No FEMA-355F FederalEmergencyManagement AgencyWashington DC USA 2000

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In general the probabilistic approaches for assessment ofthe buildings collapse potential use the very basic assumptionthat the effects originated from the uncertainties existing inthe earthquake recordsrsquo characteristics are independent ofthe uncertainties in the numerical modeling and predictionof seismic hazard level A performance target in collapselevel can be considered as a tolerable collapse probabilityin a given hazard level Currently collapse fragility curve isused as one of the most essential probabilistic approachesto assess a structure collapse measurement A set of IDAanalyses can play a vital role in determining estimationparameters and in turn determining collapse fragility curveThe seismic collapse safety of the older RC frames whichwas designed and constructed prior to the 1970s was usedto study the effect of two structural parameters such asthe transverse reinforcing ratio and the flexural strengthratio of the column to the beam As a result of a studyit was found out that the two ratios that interacted witheach other must be considered to appreciate the collapse riskof a building [7] The experience of recent decade revealedthat in big heavy and rigid buildings founded on soft andstiff soils SSI effects are of high importance in terms ofthe buildings stability In buildings founded on soft and stiffsoil SSI increases the fundamental period of system anddecreases the base shear forceThis study uses 6- and 12-storyRC buildings with intermediate moment frames system aswell as an 18-story RC building with special moment framesystem to evaluate SSI effects on the determination of thedemand and capacity of collapse and finally to determine thestructural fragility curves Finally by studying the dispersionof IDA curves and the extent of the results dependencyon seismic records properties the appropriate IM will beselected among two mentioned scalar IMs Soil-structureinteraction (SSI) can alter the performance of structure totallyincluding its dynamic characteristics response maximaand more importantly distribution of nonlinear responsethrough structure where the accurate calculation of it is vitalfor the performance evaluation

2 Global Collapse

The study of global collapse was triggered by considering 119875-Δ effects on seismic response Although hysteresis modelsconsidered positive postyielding stiffness the structure tan-gent stiffness became negative under large 119875-Δ effects whichin turn lead to the systemic collapse Structural performanceassessment needs a numerical criterion According to FEMA2000 the IDA approach is proposed to estimate the structuralcapacity of the buildings In addition FEMA2000 is classifiedas the point of occurrence of collapse based on the followingconditions numerical nonconvergence in structural analysisalgorithm and the occurrence of a slope equal to 20 of theinitial elastic slope in the IDA curve and exceedance of themaximum internal drift ratio above 10 Jalayer and Cornellused IDA concept to estimate total dynamic instability capac-ity of a regular RC structure [8] Robust fragility assessmentfor CloudAnalysis considering the global dynamic instabilityand IDA can also be found in provided detail by Jalayer et al[9] In addition their research includes details about how a

modified version of the Cloud Analysis can officially handlethe existence of records that trigger collapse Haselton andDierlin evaluated the collapse risk of 30 4-story buildingswith special moment frame designed in accordance withASCE9-02 According to their results likelihood of structuralcollapse for earthquakes with a return period of 2475 yearsranges from 3 to 20 with a mean of 11 Their studyrevealed that the minimum essential base shear determinedby ASCE7-02 is an important element guaranteeing almosta fixed collapse risk for buildings with different heights [10]Haselton et al studied the collapse capacity of 1- to 20-storybuildings with special moment frames Their study showedthat for a 2-in-50 years earthquake the mean collapseprobability in the studied buildings is about 11 [11] Lignoset al used the results of a collapse test conducted on a 4-story steel moment frame on shaking table of E-DefenceLaboratory and derived the essential parameters required formodeling the collapse reliability of buildings According totheir results the difference in collapse capacity in regularplan buildings is not significant between 3D and 2D models[12] Rajeev and Tesfamariam demonstrated fragility basedseismic vulnerability of structures with consideration of softstory (SS) and quality of construction (CQ) on three- five-and nine-story RC frames designed prior to the 1970s Theydeveloped probabilistic seismic demand model (PSDM) forthose gravity load designed structures using the nonlinearfinite element analysis considering the interactions betweenSS and CQ Result of their study showed the sensitivity ofthe model parameter to the interaction of SS and CQ [13]Palermo et al evaluated the efficiency of current modelingtechniques in predicting the collapse capacity of the studiedstructuresThe results of numerical modeling were comparedwith real samples of LrsquoAquila earthquake in Italy Accordingto the study modern techniques have better capability ofpredicting the collapse capacity of structures [14]

The median collapse intensity can be visualized throughthe concept of incremental dynamic analysis (IDA) in whichindividual groundmotions are scaled to increasing intensitiesuntil the structure reaches a collapse point [15] To considerthe uncertainties of seismic intensity measure predictionsthis method normalizes an earthquake record in a mannerto cover a wide range of intensity measure On the otherhand acceptable numbers of seismic records are requiredto take the uncertainties of the frequency content andspectral shape of earthquakes into account The relationshipbetween engineering demand parameter (EPD) registered inan IDA and one or more seismic intensity measures (IMs)which describe the normalized earthquake records appliedto the structure is called IDA curve Collapse assessmentof structures based on IDA curves necessitates the selectionof fit IM and a fit seismic demand (DM) used to constructthese IDA curves as well as variability in the set of groundmotions considered in the analysis [16] Intensity measureshould be selected in a manner to be connected to the mainrecord using a scale factor PGA PGV and elastic spectralacceleration in fundamental period of a structure for 5 ofcritical damping are themost conventional indices which canbe used as scalar or vector indices The selection of an IMdepends on the efficiency in terms of seismic intensity and on


Monitoring sections

Element


Section

X

Y

Y

Z

Z


0 Axial strain0

Com

pres

sive s

tress

11

Confined concrete

Unconfined concrete


First hoopfracture

Ec

EＭ＝

fcc

fc0











119891119888 = 119891119888119888119883119903119903 minus 1 + 1198832 119883 = 120576119888120576119888119888 120576119888119888 = [119877(

1198911198881198881198911198880 minus 1) + 1] 1205761198880 119903 = 119864119888119864119888 minus 119864119904119890119888 (1)



119864119904119890119888 = 119891119888119888120576119888119888 (2)


119870119890 = 1 minus 119896119878119863101584010158401 minus 120588119888119888 (3)








Z0

r0

CC

0M0

M

C

1K

P0 U0

ΔM

K

ΔM

Vp





2 912058732 (1 minus 120592)( 119881119881119904 )2 912058732





065 + 04 119871119861 + 08] 119870119910119910 = 11986611986131 minus 120592 [04 ( 119871119861)24 + 0034]

119870119911 = 1198661198611 minus 120592 [155 (119871119861)075 + 08] 119870119911119911 = 1198661198613 [053 (119871119861)








A

B

C

D

E

F

1

2

3

4

(a)

A

B

C

D

E

F

1

2

3

4

(b)

2

3

4

1

C

D

E

F

B

A

(c)

1

2

3

4

A B C D E F

STAIR

6m6m6m6m6m

6m

6m

6m

(d)









Saavg (11989611198791 11989621198791 1198961198991198791) = ( 119899prod119894=1

119878119860 (1198961198941198791))1119899 (4)





























k1T1

k2T1

knT11 2 3 40

Period (sec)

0

005

01

015

02

025

03

035

IM


0308153659

IDRs

rss-

Saav

g1M

ean

IDRm

ax-S

aavg

1Mea

nID

Rsrs

s-Sa

avg1

Txi

IDRm

ax-S

aavg

1Txi

IDRs

rss-

Saav

g1Tz

iID

Rmax

-Saa

vg1T

ziID

Rsrs

s-Sa

avg2

Mea

nID

Rmax

-Saa

vg2M

ean

IDRs

rss-

Saav

g2Tx

iID

Rmax

-Saa

vg2T

xiID

Rsrs

s-Sa

avg2

Tzi

IDRm

ax-S

aavg

2Tzi

IDRs

rss-

Saav

g3M

ean

IDRm

ax-S

aavg

3Mea

nID

Rsrs

s-Sa

avg3

Txi

IDRm

ax-S

aavg

3Txi

IDRs

rss-

Saav

g3Tz

iID

Rmax

-Saa

vg3T

ziID

Rsrs

s-Sa

avg4

Mea

nID

Rmax

-Saa

vg4M

ean

IDRs

rss-

Saav

g4Tx

iID

Rmax

-Saa

vg4T

xiID

Rsrs

s-Sa

avg4

Tzi

IDRm

ax-S

aavg

4Tzi

IDRs

rss-

Saav

g5M

ean

IDRs

rss-

Saav

g5M

ean

IDRs

rss-

Saav

g5Tx

iID

Rsrs

s-Sa

avg5

Txi

IDRs

rss-

SaBa

vg5T

ziID

Rsrs

s-Sa

avg5

Tzi

IDRs

rss-

Sa(T

1)ID

Rmax

-Sa(

T1)

IDRs

rss-

SaM

ean

IDRm

ax-S

aMea

nID

Rsrs

s-Sa

srss

IDRm

ax-S

asrs

s

0001020304050607080910



Disp

ersio

n (D|IM

)








119875 (119862 | 119878119875119877119886 ) = Φ( ln (119878119875119877119886 ) minus ln (120578119862)120573119877119862 ) (5)






0

01

02

03

04

05

06

07

08

09

1C

olla

pse p

roba

bilit

y

05 1 15 2 25 3 35 4 45 5 55 6 65 7 75 80Sa(T1 = 005) (g)



(a)

05 1 15 2 25 3 35 4 45 5 550Saavg(Tzi = 005) (g)

0

01

02

03

04

05

06

07

08

09

1

Col

laps

e Pro

babi

lity


(b)









120582119862 = int119875119862|119909 1003816100381610038161003816119889120582119868119872 (119909)1003816100381610038161003816 (6)


120582119862 = [120582Sa (120578C)] [exp (1211989621205732119877119862)] (7)


120582119878119886 = 1198960119878119886minus119896 (8)





0001304(Sa)^(minus21715)

100E minus 04

100E minus 03

100E minus 02

100E minus 01

100E + 00A

nnua

l pro

babl

ity o

f exc

eeda

nce

02 04 06 08 1 12 14 16 18 2 220Sa(T1 = 086) (g)



(a)



00004664 (Sa)^(minus22293)

10E minus 04

10E minus 03

10E minus 02

10E minus 01

10E + 00

Ann

ual p

roba

blity

of e

xcee

danc

e

02 04 06 08 1 120Sa(T1 = 138) (g)



(b)



000007252 (Sa)^(minus23342)

10E minus 04

10E minus 03

10E minus 02

10E minus 01

10E + 00

Ann

ual p

roba

blity

of e

xcee

danc

e

01 02 03 04 05 060Sa(T1 = 197) (g)



(c)








Number ofstories







5 Conclusions




Notations

























References













































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Monitoring sections

Element


Section

X

Y

Y

Z

Z


0 Axial strain0

Com

pres

sive s

tress

11

Confined concrete

Unconfined concrete


First hoopfracture

Ec

EＭ＝

fcc

fc0











119891119888 = 119891119888119888119883119903119903 minus 1 + 1198832 119883 = 120576119888120576119888119888 120576119888119888 = [119877(

1198911198881198881198911198880 minus 1) + 1] 1205761198880 119903 = 119864119888119864119888 minus 119864119904119890119888 (1)



119864119904119890119888 = 119891119888119888120576119888119888 (2)


119870119890 = 1 minus 119896119878119863101584010158401 minus 120588119888119888 (3)








Z0

r0

CC

0M0

M

C

1K

P0 U0

ΔM

K

ΔM

Vp





2 912058732 (1 minus 120592)( 119881119881119904 )2 912058732





065 + 04 119871119861 + 08] 119870119910119910 = 11986611986131 minus 120592 [04 ( 119871119861)24 + 0034]

119870119911 = 1198661198611 minus 120592 [155 (119871119861)075 + 08] 119870119911119911 = 1198661198613 [053 (119871119861)








A

B

C

D

E

F

1

2

3

4

(a)

A

B

C

D

E

F

1

2

3

4

(b)

2

3

4

1

C

D

E

F

B

A

(c)

1

2

3

4

A B C D E F

STAIR

6m6m6m6m6m

6m

6m

6m

(d)









Saavg (11989611198791 11989621198791 1198961198991198791) = ( 119899prod119894=1

119878119860 (1198961198941198791))1119899 (4)





























k1T1

k2T1

knT11 2 3 40

Period (sec)

0

005

01

015

02

025

03

035

IM


0308153659

IDRs

rss-

Saav

g1M

ean

IDRm

ax-S

aavg

1Mea

nID

Rsrs

s-Sa

avg1

Txi

IDRm

ax-S

aavg

1Txi

IDRs

rss-

Saav

g1Tz

iID

Rmax

-Saa

vg1T

ziID

Rsrs

s-Sa

avg2

Mea

nID

Rmax

-Saa

vg2M

ean

IDRs

rss-

Saav

g2Tx

iID

Rmax

-Saa

vg2T

xiID

Rsrs

s-Sa

avg2

Tzi

IDRm

ax-S

aavg

2Tzi

IDRs

rss-

Saav

g3M

ean

IDRm

ax-S

aavg

3Mea

nID

Rsrs

s-Sa

avg3

Txi

IDRm

ax-S

aavg

3Txi

IDRs

rss-

Saav

g3Tz

iID

Rmax

-Saa

vg3T

ziID

Rsrs

s-Sa

avg4

Mea

nID

Rmax

-Saa

vg4M

ean

IDRs

rss-

Saav

g4Tx

iID

Rmax

-Saa

vg4T

xiID

Rsrs

s-Sa

avg4

Tzi

IDRm

ax-S

aavg

4Tzi

IDRs

rss-

Saav

g5M

ean

IDRs

rss-

Saav

g5M

ean

IDRs

rss-

Saav

g5Tx

iID

Rsrs

s-Sa

avg5

Txi

IDRs

rss-

SaBa

vg5T

ziID

Rsrs

s-Sa

avg5

Tzi

IDRs

rss-

Sa(T

1)ID

Rmax

-Sa(

T1)

IDRs

rss-

SaM

ean

IDRm

ax-S

aMea

nID

Rsrs

s-Sa

srss

IDRm

ax-S

asrs

s

0001020304050607080910



Disp

ersio

n (D|IM

)








119875 (119862 | 119878119875119877119886 ) = Φ( ln (119878119875119877119886 ) minus ln (120578119862)120573119877119862 ) (5)






0

01

02

03

04

05

06

07

08

09

1C

olla

pse p

roba

bilit

y

05 1 15 2 25 3 35 4 45 5 55 6 65 7 75 80Sa(T1 = 005) (g)



(a)

05 1 15 2 25 3 35 4 45 5 550Saavg(Tzi = 005) (g)

0

01

02

03

04

05

06

07

08

09

1

Col

laps

e Pro

babi

lity


(b)









120582119862 = int119875119862|119909 1003816100381610038161003816119889120582119868119872 (119909)1003816100381610038161003816 (6)


120582119862 = [120582Sa (120578C)] [exp (1211989621205732119877119862)] (7)


120582119878119886 = 1198960119878119886minus119896 (8)





0001304(Sa)^(minus21715)

100E minus 04

100E minus 03

100E minus 02

100E minus 01

100E + 00A

nnua

l pro

babl

ity o

f exc

eeda

nce

02 04 06 08 1 12 14 16 18 2 220Sa(T1 = 086) (g)



(a)



00004664 (Sa)^(minus22293)

10E minus 04

10E minus 03

10E minus 02

10E minus 01

10E + 00

Ann

ual p

roba

blity

of e

xcee

danc

e

02 04 06 08 1 120Sa(T1 = 138) (g)



(b)



000007252 (Sa)^(minus23342)

10E minus 04

10E minus 03

10E minus 02

10E minus 01

10E + 00

Ann

ual p

roba

blity

of e

xcee

danc

e

01 02 03 04 05 060Sa(T1 = 197) (g)



(c)








Number ofstories







5 Conclusions




Notations

























References













































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




119891119888 = 119891119888119888119883119903119903 minus 1 + 1198832 119883 = 120576119888120576119888119888 120576119888119888 = [119877(

1198911198881198881198911198880 minus 1) + 1] 1205761198880 119903 = 119864119888119864119888 minus 119864119904119890119888 (1)



119864119904119890119888 = 119891119888119888120576119888119888 (2)


119870119890 = 1 minus 119896119878119863101584010158401 minus 120588119888119888 (3)








Z0

r0

CC

0M0

M

C

1K

P0 U0

ΔM

K

ΔM

Vp





2 912058732 (1 minus 120592)( 119881119881119904 )2 912058732





065 + 04 119871119861 + 08] 119870119910119910 = 11986611986131 minus 120592 [04 ( 119871119861)24 + 0034]

119870119911 = 1198661198611 minus 120592 [155 (119871119861)075 + 08] 119870119911119911 = 1198661198613 [053 (119871119861)








A

B

C

D

E

F

1

2

3

4

(a)

A

B

C

D

E

F

1

2

3

4

(b)

2

3

4

1

C

D

E

F

B

A

(c)

1

2

3

4

A B C D E F

STAIR

6m6m6m6m6m

6m

6m

6m

(d)









Saavg (11989611198791 11989621198791 1198961198991198791) = ( 119899prod119894=1

119878119860 (1198961198941198791))1119899 (4)





























k1T1

k2T1

knT11 2 3 40

Period (sec)

0

005

01

015

02

025

03

035

IM


0308153659

IDRs

rss-

Saav

g1M

ean

IDRm

ax-S

aavg

1Mea

nID

Rsrs

s-Sa

avg1

Txi

IDRm

ax-S

aavg

1Txi

IDRs

rss-

Saav

g1Tz

iID

Rmax

-Saa

vg1T

ziID

Rsrs

s-Sa

avg2

Mea

nID

Rmax

-Saa

vg2M

ean

IDRs

rss-

Saav

g2Tx

iID

Rmax

-Saa

vg2T

xiID

Rsrs

s-Sa

avg2

Tzi

IDRm

ax-S

aavg

2Tzi

IDRs

rss-

Saav

g3M

ean

IDRm

ax-S

aavg

3Mea

nID

Rsrs

s-Sa

avg3

Txi

IDRm

ax-S

aavg

3Txi

IDRs

rss-

Saav

g3Tz

iID

Rmax

-Saa

vg3T

ziID

Rsrs

s-Sa

avg4

Mea

nID

Rmax

-Saa

vg4M

ean

IDRs

rss-

Saav

g4Tx

iID

Rmax

-Saa

vg4T

xiID

Rsrs

s-Sa

avg4

Tzi

IDRm

ax-S

aavg

4Tzi

IDRs

rss-

Saav

g5M

ean

IDRs

rss-

Saav

g5M

ean

IDRs

rss-

Saav

g5Tx

iID

Rsrs

s-Sa

avg5

Txi

IDRs

rss-

SaBa

vg5T

ziID

Rsrs

s-Sa

avg5

Tzi

IDRs

rss-

Sa(T

1)ID

Rmax

-Sa(

T1)

IDRs

rss-

SaM

ean

IDRm

ax-S

aMea

nID

Rsrs

s-Sa

srss

IDRm

ax-S

asrs

s

0001020304050607080910



Disp

ersio

n (D|IM

)








119875 (119862 | 119878119875119877119886 ) = Φ( ln (119878119875119877119886 ) minus ln (120578119862)120573119877119862 ) (5)






0

01

02

03

04

05

06

07

08

09

1C

olla

pse p

roba

bilit

y

05 1 15 2 25 3 35 4 45 5 55 6 65 7 75 80Sa(T1 = 005) (g)



(a)

05 1 15 2 25 3 35 4 45 5 550Saavg(Tzi = 005) (g)

0

01

02

03

04

05

06

07

08

09

1

Col

laps

e Pro

babi

lity


(b)









120582119862 = int119875119862|119909 1003816100381610038161003816119889120582119868119872 (119909)1003816100381610038161003816 (6)


120582119862 = [120582Sa (120578C)] [exp (1211989621205732119877119862)] (7)


120582119878119886 = 1198960119878119886minus119896 (8)





0001304(Sa)^(minus21715)

100E minus 04

100E minus 03

100E minus 02

100E minus 01

100E + 00A

nnua

l pro

babl

ity o

f exc

eeda

nce

02 04 06 08 1 12 14 16 18 2 220Sa(T1 = 086) (g)



(a)



00004664 (Sa)^(minus22293)

10E minus 04

10E minus 03

10E minus 02

10E minus 01

10E + 00

Ann

ual p

roba

blity

of e

xcee

danc

e

02 04 06 08 1 120Sa(T1 = 138) (g)



(b)



000007252 (Sa)^(minus23342)

10E minus 04

10E minus 03

10E minus 02

10E minus 01

10E + 00

Ann

ual p

roba

blity

of e

xcee

danc

e

01 02 03 04 05 060Sa(T1 = 197) (g)



(c)








Number ofstories







5 Conclusions




Notations

























References













































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Z0

r0

CC

0M0

M

C

1K

P0 U0

ΔM

K

ΔM

Vp





2 912058732 (1 minus 120592)( 119881119881119904 )2 912058732





065 + 04 119871119861 + 08] 119870119910119910 = 11986611986131 minus 120592 [04 ( 119871119861)24 + 0034]

119870119911 = 1198661198611 minus 120592 [155 (119871119861)075 + 08] 119870119911119911 = 1198661198613 [053 (119871119861)








A

B

C

D

E

F

1

2

3

4

(a)

A

B

C

D

E

F

1

2

3

4

(b)

2

3

4

1

C

D

E

F

B

A

(c)

1

2

3

4

A B C D E F

STAIR

6m6m6m6m6m

6m

6m

6m

(d)









Saavg (11989611198791 11989621198791 1198961198991198791) = ( 119899prod119894=1

119878119860 (1198961198941198791))1119899 (4)





























k1T1

k2T1

knT11 2 3 40

Period (sec)

0

005

01

015

02

025

03

035

IM


0308153659

IDRs

rss-

Saav

g1M

ean

IDRm

ax-S

aavg

1Mea

nID

Rsrs

s-Sa

avg1

Txi

IDRm

ax-S

aavg

1Txi

IDRs

rss-

Saav

g1Tz

iID

Rmax

-Saa

vg1T

ziID

Rsrs

s-Sa

avg2

Mea

nID

Rmax

-Saa

vg2M

ean

IDRs

rss-

Saav

g2Tx

iID

Rmax

-Saa

vg2T

xiID

Rsrs

s-Sa

avg2

Tzi

IDRm

ax-S

aavg

2Tzi

IDRs

rss-

Saav

g3M

ean

IDRm

ax-S

aavg

3Mea

nID

Rsrs

s-Sa

avg3

Txi

IDRm

ax-S

aavg

3Txi

IDRs

rss-

Saav

g3Tz

iID

Rmax

-Saa

vg3T

ziID

Rsrs

s-Sa

avg4

Mea

nID

Rmax

-Saa

vg4M

ean

IDRs

rss-

Saav

g4Tx

iID

Rmax

-Saa

vg4T

xiID

Rsrs

s-Sa

avg4

Tzi

IDRm

ax-S

aavg

4Tzi

IDRs

rss-

Saav

g5M

ean

IDRs

rss-

Saav

g5M

ean

IDRs

rss-

Saav

g5Tx

iID

Rsrs

s-Sa

avg5

Txi

IDRs

rss-

SaBa

vg5T

ziID

Rsrs

s-Sa

avg5

Tzi

IDRs

rss-

Sa(T

1)ID

Rmax

-Sa(

T1)

IDRs

rss-

SaM

ean

IDRm

ax-S

aMea

nID

Rsrs

s-Sa

srss

IDRm

ax-S

asrs

s

0001020304050607080910



Disp

ersio

n (D|IM

)








119875 (119862 | 119878119875119877119886 ) = Φ( ln (119878119875119877119886 ) minus ln (120578119862)120573119877119862 ) (5)






0

01

02

03

04

05

06

07

08

09

1C

olla

pse p

roba

bilit

y

05 1 15 2 25 3 35 4 45 5 55 6 65 7 75 80Sa(T1 = 005) (g)



(a)

05 1 15 2 25 3 35 4 45 5 550Saavg(Tzi = 005) (g)

0

01

02

03

04

05

06

07

08

09

1

Col

laps

e Pro

babi

lity


(b)









120582119862 = int119875119862|119909 1003816100381610038161003816119889120582119868119872 (119909)1003816100381610038161003816 (6)


120582119862 = [120582Sa (120578C)] [exp (1211989621205732119877119862)] (7)


120582119878119886 = 1198960119878119886minus119896 (8)





0001304(Sa)^(minus21715)

100E minus 04

100E minus 03

100E minus 02

100E minus 01

100E + 00A

nnua

l pro

babl

ity o

f exc

eeda

nce

02 04 06 08 1 12 14 16 18 2 220Sa(T1 = 086) (g)



(a)



00004664 (Sa)^(minus22293)

10E minus 04

10E minus 03

10E minus 02

10E minus 01

10E + 00

Ann

ual p

roba

blity

of e

xcee

danc

e

02 04 06 08 1 120Sa(T1 = 138) (g)



(b)



000007252 (Sa)^(minus23342)

10E minus 04

10E minus 03

10E minus 02

10E minus 01

10E + 00

Ann

ual p

roba

blity

of e

xcee

danc

e

01 02 03 04 05 060Sa(T1 = 197) (g)



(c)








Number ofstories







5 Conclusions




Notations

























References













































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



A

B

C

D

E

F

1

2

3

4

(a)

A

B

C

D

E

F

1

2

3

4

(b)

2

3

4

1

C

D

E

F

B

A

(c)

1

2

3

4

A B C D E F

STAIR

6m6m6m6m6m

6m

6m

6m

(d)









Saavg (11989611198791 11989621198791 1198961198991198791) = ( 119899prod119894=1

119878119860 (1198961198941198791))1119899 (4)





























k1T1

k2T1

knT11 2 3 40

Period (sec)

0

005

01

015

02

025

03

035

IM


0308153659

IDRs

rss-

Saav

g1M

ean

IDRm

ax-S

aavg

1Mea

nID

Rsrs

s-Sa

avg1

Txi

IDRm

ax-S

aavg

1Txi

IDRs

rss-

Saav

g1Tz

iID

Rmax

-Saa

vg1T

ziID

Rsrs

s-Sa

avg2

Mea

nID

Rmax

-Saa

vg2M

ean

IDRs

rss-

Saav

g2Tx

iID

Rmax

-Saa

vg2T

xiID

Rsrs

s-Sa

avg2

Tzi

IDRm

ax-S

aavg

2Tzi

IDRs

rss-

Saav

g3M

ean

IDRm

ax-S

aavg

3Mea

nID

Rsrs

s-Sa

avg3

Txi

IDRm

ax-S

aavg

3Txi

IDRs

rss-

Saav

g3Tz

iID

Rmax

-Saa

vg3T

ziID

Rsrs

s-Sa

avg4

Mea

nID

Rmax

-Saa

vg4M

ean

IDRs

rss-

Saav

g4Tx

iID

Rmax

-Saa

vg4T

xiID

Rsrs

s-Sa

avg4

Tzi

IDRm

ax-S

aavg

4Tzi

IDRs

rss-

Saav

g5M

ean

IDRs

rss-

Saav

g5M

ean

IDRs

rss-

Saav

g5Tx

iID

Rsrs

s-Sa

avg5

Txi

IDRs

rss-

SaBa

vg5T

ziID

Rsrs

s-Sa

avg5

Tzi

IDRs

rss-

Sa(T

1)ID

Rmax

-Sa(

T1)

IDRs

rss-

SaM

ean

IDRm

ax-S

aMea

nID

Rsrs

s-Sa

srss

IDRm

ax-S

asrs

s

0001020304050607080910



Disp

ersio

n (D|IM

)








119875 (119862 | 119878119875119877119886 ) = Φ( ln (119878119875119877119886 ) minus ln (120578119862)120573119877119862 ) (5)






0

01

02

03

04

05

06

07

08

09

1C

olla

pse p

roba

bilit

y

05 1 15 2 25 3 35 4 45 5 55 6 65 7 75 80Sa(T1 = 005) (g)



(a)

05 1 15 2 25 3 35 4 45 5 550Saavg(Tzi = 005) (g)

0

01

02

03

04

05

06

07

08

09

1

Col

laps

e Pro

babi

lity


(b)









120582119862 = int119875119862|119909 1003816100381610038161003816119889120582119868119872 (119909)1003816100381610038161003816 (6)


120582119862 = [120582Sa (120578C)] [exp (1211989621205732119877119862)] (7)


120582119878119886 = 1198960119878119886minus119896 (8)





0001304(Sa)^(minus21715)

100E minus 04

100E minus 03

100E minus 02

100E minus 01

100E + 00A

nnua

l pro

babl

ity o

f exc

eeda

nce

02 04 06 08 1 12 14 16 18 2 220Sa(T1 = 086) (g)



(a)



00004664 (Sa)^(minus22293)

10E minus 04

10E minus 03

10E minus 02

10E minus 01

10E + 00

Ann

ual p

roba

blity

of e

xcee

danc

e

02 04 06 08 1 120Sa(T1 = 138) (g)



(b)



000007252 (Sa)^(minus23342)

10E minus 04

10E minus 03

10E minus 02

10E minus 01

10E + 00

Ann

ual p

roba

blity

of e

xcee

danc

e

01 02 03 04 05 060Sa(T1 = 197) (g)



(c)








Number ofstories







5 Conclusions




Notations

























References













































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





























k1T1

k2T1

knT11 2 3 40

Period (sec)

0

005

01

015

02

025

03

035

IM


0308153659

IDRs

rss-

Saav

g1M

ean

IDRm

ax-S

aavg

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s-Sa

avg1

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rss-

Saav

g1Tz

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Rmax

-Saa

vg1T

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vg2M

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IDRs

rss-

Saav

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IDRs

rss-

Saav

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IDRs

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Saav

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IDRs

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Saav

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Sa(T

1)ID

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IDRs

rss-

SaM

ean

IDRm

ax-S

aMea

nID

Rsrs

s-Sa

srss

IDRm

ax-S

asrs

s

0001020304050607080910



Disp

ersio

n (D|IM

)








119875 (119862 | 119878119875119877119886 ) = Φ( ln (119878119875119877119886 ) minus ln (120578119862)120573119877119862 ) (5)






0

01

02

03

04

05

06

07

08

09

1C

olla

pse p

roba

bilit

y

05 1 15 2 25 3 35 4 45 5 55 6 65 7 75 80Sa(T1 = 005) (g)



(a)

05 1 15 2 25 3 35 4 45 5 550Saavg(Tzi = 005) (g)

0

01

02

03

04

05

06

07

08

09

1

Col

laps

e Pro

babi

lity


(b)









120582119862 = int119875119862|119909 1003816100381610038161003816119889120582119868119872 (119909)1003816100381610038161003816 (6)


120582119862 = [120582Sa (120578C)] [exp (1211989621205732119877119862)] (7)


120582119878119886 = 1198960119878119886minus119896 (8)





0001304(Sa)^(minus21715)

100E minus 04

100E minus 03

100E minus 02

100E minus 01

100E + 00A

nnua

l pro

babl

ity o

f exc

eeda

nce

02 04 06 08 1 12 14 16 18 2 220Sa(T1 = 086) (g)



(a)



00004664 (Sa)^(minus22293)

10E minus 04

10E minus 03

10E minus 02

10E minus 01

10E + 00

Ann

ual p

roba

blity

of e

xcee

danc

e

02 04 06 08 1 120Sa(T1 = 138) (g)



(b)



000007252 (Sa)^(minus23342)

10E minus 04

10E minus 03

10E minus 02

10E minus 01

10E + 00

Ann

ual p

roba

blity

of e

xcee

danc

e

01 02 03 04 05 060Sa(T1 = 197) (g)



(c)








Number ofstories







5 Conclusions




Notations

























References













































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



0

01

02

03

04

05

06

07

08

09

1C

olla

pse p

roba

bilit

y

05 1 15 2 25 3 35 4 45 5 55 6 65 7 75 80Sa(T1 = 005) (g)



(a)

05 1 15 2 25 3 35 4 45 5 550Saavg(Tzi = 005) (g)

0

01

02

03

04

05

06

07

08

09

1

Col

laps

e Pro

babi

lity


(b)









120582119862 = int119875119862|119909 1003816100381610038161003816119889120582119868119872 (119909)1003816100381610038161003816 (6)


120582119862 = [120582Sa (120578C)] [exp (1211989621205732119877119862)] (7)


120582119878119886 = 1198960119878119886minus119896 (8)





0001304(Sa)^(minus21715)

100E minus 04

100E minus 03

100E minus 02

100E minus 01

100E + 00A

nnua

l pro

babl

ity o

f exc

eeda

nce

02 04 06 08 1 12 14 16 18 2 220Sa(T1 = 086) (g)



(a)



00004664 (Sa)^(minus22293)

10E minus 04

10E minus 03

10E minus 02

10E minus 01

10E + 00

Ann

ual p

roba

blity

of e

xcee

danc

e

02 04 06 08 1 120Sa(T1 = 138) (g)



(b)



000007252 (Sa)^(minus23342)

10E minus 04

10E minus 03

10E minus 02

10E minus 01

10E + 00

Ann

ual p

roba

blity

of e

xcee

danc

e

01 02 03 04 05 060Sa(T1 = 197) (g)



(c)








Number ofstories







5 Conclusions




Notations

























References













































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





0001304(Sa)^(minus21715)

100E minus 04

100E minus 03

100E minus 02

100E minus 01

100E + 00A

nnua

l pro

babl

ity o

f exc

eeda

nce

02 04 06 08 1 12 14 16 18 2 220Sa(T1 = 086) (g)



(a)



00004664 (Sa)^(minus22293)

10E minus 04

10E minus 03

10E minus 02

10E minus 01

10E + 00

Ann

ual p

roba

blity

of e

xcee

danc

e

02 04 06 08 1 120Sa(T1 = 138) (g)



(b)



000007252 (Sa)^(minus23342)

10E minus 04

10E minus 03

10E minus 02

10E minus 01

10E + 00

Ann

ual p

roba

blity

of e

xcee

danc

e

01 02 03 04 05 060Sa(T1 = 197) (g)



(c)








Number ofstories







5 Conclusions




Notations

























References













































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




Number ofstories







5 Conclusions




Notations

























References













































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



















References













































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


soil-structure interaction effect on fragility curve of 3d...

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