performance based seismic engineering -...

15. Performance Based Seismic Engineering 757

Chapter 15

Performance Based Seismic Engineering

Farzad Naeim, Ph.D., S.E.Vice President and Director of Research and Development, John A. Martin & Associates, Inc., Los Angeles, California

Hussain Bhatia, Ph.D., P.E.Senior Research Engineer, John A. Martin & Associates, Inc., Los Angeles, California

Roy M. Lobo, Ph.D., P.E.Senior Research Engineer, John A. Martin & Associates, Inc., Los Angeles, California

Key words: Seismic Performance; Performance Based Design; Seismic Demand; Capacity; ADRS Spectrum; TargetDisplacement; Performance Objectives; Push-over Analysis; Capacity Spectrum; Static Analysis; NonlinearAnalysis; Damage Control; Life safety, Collapse Prevention; Immediate Occupancy

Abstract: Performance based seismic engineering is the modern approach to earthquake resistant design. Rather thanbeing based on prescriptive mostly empirical code formulations, performance based design is an attempt topredict buildings with predictable seismic performance. Therefore, performance objectives such as life-safety, collapse prevention, or immediate occupancy are used to define the state of the building following adesign earthquake. In one sense, performance based seismic design is limit-states design extended to coverthe complex range of issues faced by earthquake engineers. This chapter provides a basic understanding ofthe promises and limitations of performance based seismic engineering. The state-of-the-art methodologiesand techniques embodied in the two leading guidelines on this subject (ATC-40 and FEMA 273/274) areintroduced and discussed. Numerical examples are provided to illustrate the practical applications of themethods discussed.

758 Chapter 15


15.1 INTRODUCTION

The promise of performance-based seismicengineering (PBSE) is to produce structureswith predictable seismic performance. To turnthis promise into a reality, a comprehensive andwell-coordinated effort by professionals fromseveral disciplines is required.

Performance based engineering is not new.Automobiles, airplanes, and turbines have beendesigned and manufactured using this approachfor many decades. Generally in suchapplications one or more full-scale prototypesof the structure are built and subjected toextensive testing. The design andmanufacturing process is then revised toincorporate the lessons learned from theexperimental evaluations. Once the cycle ofdesign, prototype manufacturing, testing andredesign is successfully completed, the productis manufactured in a massive scale. In theautomotive industry, for example, millions ofautomobiles which are virtually identical intheir mechanical characteristics are producedfollowing each performance-based designexercise.

What makes PBSE different and morecomplicated is that in general this massivepayoff of performance-based design is notavailable. That is, except for large-scaledevelopments of identical buildings, eachbuilding designed by this process is virtuallyunique and the experience obtained is notdirectly transferable to buildings of other types,sizes, and performance objectives. Therefore,up to now PBSE has not been an economicallyfeasible alternative to conventional prescriptivecode design practices. Due to the recentadvances in seismic hazard assessment, PBSEmethodologies, experimental facilities, andcomputer applications, PBSE has becomeincreasing more attractive to developers andengineers of buildings in seismic regions. It issafe to say that within just a few years PBSEwill become the standard method for design anddelivery of earthquake resistant structures.

In order to utilize PBSE effectively andintelligently, one need to be aware of the

uncertainties involved in both structuralperformance and seismic hazard estimations.We discuss these issues first before exploringthe philosophies and detailed requirements ofthe two most prominent PBSE guidelinesavailable today. These guidelines are generallyreferred to by their short names: ATC-40(15-1)

and FEMA-273/274(15-2,15-3).

15.2 UNCERTAINTIES INSEISMIC DESIGN ANDPERFORMANCE

Every structural system is designed to havea seismic capacity that exceeds the anticipatedseismic demand. Capacity is a complexfunction of strength, stiffness and deformabilityconjectured by the system configuration andmaterial properties of the structure.

A key requirement of any meaningful PBSEexercise is the ability to assess seismic demandsand capacities with a reasonable degree ofcertainty. The recent popularity of PBSE hasbrought many state-of-the-art analysis anddesign techniques into the mainstream ofearthquake engineering practice. Furthermore, ithas opened the door for a multi-disciplinaryapproach to seismic design which involvesdevelopers and building officials as well asengineers and earth-scientists. These are verypositive developments which are bound toimprove the quality of earthquake resistantconstruction.

The mere desire to produce structures withpredictable seismic performance does not byitself, however, turn PBSE into a reality. Manyuncertainties and gaps of knowledge have to bedealt with before PBSE turns from a promiseinto a reality. Structural engineering practicehas been able to produce structures which witha few notable exceptions (i.e., welded steelmoment frame structures during the 1994Northridge earthquake) generally exceedperformance expectations postulated by routinedesign analysis. Our capability to estimate theultimate seismic capacities and failure loadsassociated with a structure, however, at least

760 Chapter 15

outside the academic research settings is fairlylimited and not up to the standards needed for areliable prediction of seismic performance.

For example, following the Northridgeearthquake, the Applied Technology Councilconducted a survey of 530 buildings whichwere located within 300 meters of strong-motion recording sites(15-4). From the total of530 buildings which were located in the areasof strong shaking (San Fernando Valley, SantaMonica, and West Los Angeles) with peakground acceleration in their vicinity rangingfrom 0.15g to 1.78g, only 10 (less than two-percent) showed heavy damage, a total of 78buildings (about 15-percent) showed moderatedamage and 340 (64-percent) were marked byinsignificant damage (Figure 15-1). If responseof these buildings were predicted by standarddesign analysis techniques, a far worse picturewould have been predicted.

Crandell(15-5) performed a similarstatistically-based study of the seismicperformance of residential buildings locatedwithin a 10-mile radius of the Northridgeearthquake epicenter (Figure 15-2). Threehundred forty one of the 375 randomly selected

homes were surveyed and although more than90 percent of the homes in the sample were oldand built prior to the 1971 San Fernando Valleyearthquake the cases of moderate to highdamage were infrequent (less than 2-percent).Most occurrences of serious damage werelocated in foundation systems and wereassociated with localized site conditions such asliquefaction, fissuring, and hillside slopefailures. Here again, design analysis wouldhave predicted much larger damage percentagethan the 2-percent number reported by Crandell.

Large uncertainties also exist in ourestimates of design ground motion. Forexample, median estimates of spectralaccelerations for a magnitude 7.0 event atrupture distance of 10 km obtained fromvarious attenuation relations can vary by asmuch as 50 percent(15-6). If the uncertaintiesassociated with other source and regionalvariables are also considered, the variancecould be significantly larger. Most attenuationrelations are updated every few years (Figure15-3), indicating that there are still many thingsto be learned about the generation andpropagation of earthquake ground motion.

ATC-38 Damage Database

None19%

Insignificant64%

Moderate15%

Heavy2%

Figure 15-1. Damage State in 530 Buildings within 15 km of epicenter Surveyed After the 1994 Northridge Earthquake


Sam

ple

Siz

e

No

Dam

age

Low

Dam

age

Mod

erat

e D

amag

e

Hig

h D

amag

e

Foundation

Foundation to Walls

Walls

Roof

0

50

100

150

200

250

300

350

Figure 15-2. Description of Damage During the 1994 Northridge Earthquake to Single Family Dwellings Within a 10 MilesRadius of the Epicenter (data from Crandell, 1997)

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00

UNDAMPED NATURAL PERIOD (SEC)

PS

EU

DO

-RE

LAT

IVE

VE

LOC

ITY

(F

T/S

EC

)

CAMPBELL (1993)

CAMPBELL (1991R)

CAMPBELL (1990)

CAMPBELL (1989)

Figure 15-3. Evolution of a Typical Attenuation Relation (Spectral velocity estimates are shown for a magnitude 7.0 eventat 5.0 km for a strike-slip fault)

762 Chapter 15

Another source of uncertainty is criticalshortage of recorded earthquake ground motionwhere they are needed most. Despite thetremendous growth in the number of earthquakerecords during the past decade, the number ofrecordings from large earthquakes close by.Figure 15-3(15-7) shows a bivariate histogram ofhorizontal components recorded in north andcentral America categorized by magnitude andepicentral distance, indicating practically norecord of M >7.5 at distances less than 20 km.All of the data for M>8 records come from a

single event (Mexico, 1985). Clearly, this isone of the areas where more information isneeded for performance based design

Since PBSE is inherently multi-disciplinaryin nature, further educational efforts are also ofvital importance in bringing PBSE to fruitionby developing a common understanding ofissues and a common PBSE language andvocabulary. Only a broad multi- disciplinaryapproach can succeed in reduction ofuncertainties, knowledge gaps, and commonmisunderstandings.

Figure 15-4. Distribution of Magnitude and Distance among Available Earthquake Records for North and Central America,1933-1994 (M>5.5; PGA>0.05g)


15.3 ATC-40

15.3.1 Introduction

Seismic Evaluation and Retrofit of ConcreteBuildings(15-1) commonly referred to as ATC-40was developed by the Applied TechnologyCouncil (ATC) with funding from theCalifornia Safety Commission. Although theprocedures recommended in this document arefor concrete buildings, they are applicable tomost building types. This document provides apractical guide to the entire evaluation andretrofit process using performance-basedobjectives. Although it is not intended for thedesign of new buildings, the analyticalprocedures described in this document arecertainly applicable.

ATC-40 recommends the following stepsfor the entire process of evaluation and retrofit:1. Initiation of a Project: Determine the

primary goal and potential scope of theproject.

2. Selection of Qualified Professionals: Selectengineering professionals with ademonstrated experience in the analysis,design and retrofit of buildings inseismically hazardous regions. Experiencewith PBSE and non-linear procedures isalso needed.

3. Performance Objective: Choose aperformance objective from the optionsprovided for a specific level of seismichazard.

4. Review of Building Conditions: Perform asite visit and review drawings.

5. Alternatives for Mitigation: Check to see ifthe non-linear procedure is appropriate orrelevant for the building underconsideration.

6. Peer Review and Approval Process: Checkwith building officials and consider otherquality control measures appropriate toseismic evaluation and retrofit.

7. Detailed Investigations: Perform a non-linear static analysis if appropriate.

8. Seismic Capacity: Determine the inelasticcapacity curve also known to pushovercurve. Covert to capacity spectrum.

9. Seismic Hazard: Obtain a site specificresponse spectrum for the chosen hazardlevel and convert to spectral ordinates(ADRS(15-8,15-9,15-10), see Section 15.3.6)format.

10. Verify Performance: Obtain performancepoint as the intersection of the capacityspectrum and the reduced seismic demandin spectral ordinates (ADRS) format. Checkall primary and secondary elements againstacceptability limits based on the globalperformance goal.

11. Prepare Construction Documents: Detailretrofit to conform to code requirementsand get analysis and design peer-reviewedand submit for plan check.

12. Monitor Construction Quality.The performance-based roots of ATC-40 are

essentially the same as FEMA-273and FEMA-274, NEHRP Guidelines for the SeismicRehabilitation of Building(15-2, 15-3) andSEAOC’s Vision 2000: Performance-BasedSeismic Engineering of Buildings (1995)(15-11).

15.3.2 Performance Objectives

A performance objective has two essentialparts – a damage state and a level of seismichazard. Seismic performance is described bydesignating the maximum allowable damagestate (performance level) for an identifiedseismic hazard (earthquake ground motion). Aperformance objective may includeconsideration of damage states for severallevels of ground motion and would then betermed a dual or multiple-level performanceobjective.

The target performance objective is splitinto Structural Performance Level (SP-n, wheren is the designated number) and Non-structuralPerformance Level (NP-n, where n is thedesignated letter). These may be specifiedindependently, however, the combination of thetwo determines the overall BuildingPerformance level.

764 Chapter 15

Structural Performance Levels are definedas:• Immediate Occupancy (SP-1): Limited

structural damage with the basic verticaland lateral force resisting system retainingmost of their pre-earthquake characteristicsand capacities.

• Damage Control (SP-2): A placeholder fora state of damage somewhere betweenImmediate Occupancy and Life Safety.

• Life Safety (SP-3): Significant damagewith some margin against total or partialcollapse. Injuries may occur with the risk oflife-threatening injury being low. Repairmay not be economically feasible.

• Limited Safety (SP-4): A placeholder for astate of damage somewhere between LifeSafety and Structural Stability.

• Structural Stability (SP-5): SubstantialStructural damage in which the structuralsystem is on the verge of experiencingpartial or total collapse. Significant risk ofinjury exists. Repair may not be technicallyor economically feasible.

• Not Considered (SP-6): Placeholder forsituations where only non-structural seismicevaluation or retrofit is performed.

Non-structural Performance Levels aredefined as:• Operational (NP-A): Non-structural

elements are generally in place andfunctional. Back-up systems for failure ofexternal utilities, communications andtransportation have been provided.

• Immediate Occupancy (NP-B): Non-structural elements are generally in placebut may not be functional. No back-upsystems for failure of external utilities areprovided.

• Life Safety (NP-C): Considerable damageto non-structural components and systemsbut no collapse of heavy items. Secondaryhazards such as breaks in high-pressure,toxic or fire suppression piping should notbe present.

• Reduced Hazards (NP-D): Extensivedamage to non-structural components butshould not include collapse of large andheavy items that can cause significantinjury to groups of people..

• Not Considered (NP-E): Non-structuralelements, other than those that have aneffect on structural response, are notevaluated.

Combinations of Structural and Non-structural Performance Levels to obtain aBuilding Performance Level are shown in Table15-1.

15.3.3 Nonlinear Static Procedures

In Nonlinear Static Procedure, the basicdemand and capacity parameter for the analysisis the lateral displacement of the building. Thegeneration of a capacity curve (base shear vsroof displacement Figure 15-5) defines thecapacity of the building uniquely for anassumed force distribution and displacementpattern. It is independent of any specific seismicshaking demand and replaces the base shearcapacity of conventional design procedures. Ifthe building displaces laterally, its responsemust lie on this capacity curve. A point on thecurve defines a specific damage state for thestructure, since the deformation for allcomponents can be related to the globaldisplacement of the structure. By correlatingthis capacity curve to the seismic demandgenerated by a specific earthquake or groundshaking intensity, a point can be found on thecapacity curve that estimates the maximumdisplacement of the building the earthquakewill cause. This defines the performance pointor target displacement. The location of thisperformance point relative to the performancelevels defined by the capacity curve indicateswhether or not the performance objective ismet.


Thus, for the Nonlinear Static Procedure, astatic pushover analysis is performed using anonlinear analysis program for an increasingmonotonic lateral load pattern. An alternative isto perform a step by step analysis using a linearprogram. The base shear at each step is plottedagain roof displacement. The performance pointis found using the Capacity SpectrumProcedure[15-8,15-9,15-10] described in subsequentsections. The individual structural componentsare checked against acceptability limits thatdepend on the global performance goals. Thenature of the acceptability limits depends onspecific components. Inelastic rotation istypically one of acceptability parameters forbeam and column hinges. The limits oninelastic rotation are based on observation fromtests and the collective judgement of thedevelopment team.

15.3.4 Inelastic Component Behavior

The key step for the entire analysis isidentification of the primary structural

elements, which should be completely modeledin the non-linear analysis. Secondary elements,which do not significantly contribute to thebuilding’s lateral force resisting system, do notneed to be included in the analysis.

Figure 15-5. Building Capacity Curve

In concrete buildings, the effects ofearthquake shaking are resisted by verticalframe elements or wall elements that are

Table 15-1. Combinations of Structural and Non-structural Levels to form Building Performance Levels (15-1)

Building Performance LevelsStructural Performance Levels

Non-structuralPerformance Levels

SP-1ImmediateOccupancy

SP-2DamageControl(Range)

SP-3Life Safety

SP-4LimitedSafety

(Range)

SP-5StructuralStability

SP-6Not

Considered

NP-AOperational

1-AOperational

2-A NR NR NR NR

NP-BImmediateOccupancy

1-BImmediateOccupancy

2-B 3-B NR NR NR

NP-CLife Safety

1-C 2-C 3-CLife Safety

4-C 5-C 6-C

NP-DReducedHazards

NR 2-D 3-D 4-D 5-D 6-D

NP-ENotConsidered

NR NR 3-E 4-E 5-EStructuralStability

NotApplicable

Legend

Commonly referenced Building Performance Levels (SP-NP)

Other possible combinations of SP-NP

NR Not recommended combinations of SP-NP

Roof Displacement, δR

BaseShearV

ImmediateOccupancy

LifeSafety

StructuralStability

GlobalCapacityCurve

PossiblePerformancePoint

766 Chapter 15

connected to horizontal elements (diaphragms)at the roof and floor levels. The structuralelements may themselves comprise of anassembly of elements such as columns, beam,wall piers, wall spandrels etc. It is important toidentify the failure mechanism for theseprimary structural elements and define theirnon-linear properties accordingly. Theproperties of interest of such elements arerelationships between the forces (axial, bendingand shear) and the corresponding inelasticdisplacements (displacements, rotations, drifts).Earthquakes usually load these elements in acyclic manner as shown in Figure 15-6a. Formodeling and analysis purposes, theserelationship can be idealized as shown in Figure15-6b using a combination of empirical data,theoretical strength and strain compatibility.

Using the component load-deformation dataand the geometric relationships amongcomponents and elements, a global model ofthe structure relates the total seismic forces on abuilding to it overall lateral displacement togenerate the capacity curve. During thepushover process of developing the capacitycurve as brittle elements degrade, ductileelements take over the resistance and the resultis a saw tooth shape that helps visualize theperformance. Once the global displacementdemand is estimated for a specific seismichazard, the model is used to predict theresulting deformation in each component. TheATC 40 document provides acceptability limitsfor component deformations depending on thespecified performance level.

15.3.5 Geotechnical effects

The deformation and movement of thefoundations of a building can significantlyaffect the seismic response and performance ofstructures. As the structural components arerepresented by non-linear load-displacementrelationships, analogous techniques compatible

and consistent with the general methodologyshould be used for the effects of thefoundations.

The response parameters of foundationelements are dependent on structural as well asgeotechnical components. Spread footingselements, for example, might consist of a rigidstructural plate component model of theconcrete footing bearing on soil represented bygeotechnical components with appropriateforce-displacement properties. Some genericmodels for typical foundation elements andacceptance criterion for structural componentsof the foundations are provided in ATC-40.

There is a large degree of uncertaintyassociated with both strength and stiffness ofthe geotechnical components. Thus, ATC-40recommends enveloping analysis to determinethe sensitivity of seismic performance tofoundation behavior (See Figure 15-8).Guidance in provided for representativeproperties of normally encountered soilmaterials that are based on limited initialinvestigations in ATC-40. If the analysis showssensitivity to foundation behavior than moredetailed investigations and tests of geotechnicalproperties may be warranted.

Geotechnical properties are very ductile andfailure is rarely encountered. Thus, deformationlimits of geotechnical components are notexplicitly defined. However, deformation ofgeotechnical components may affect thedeformation and acceptability of components inthe superstructure. It should also be noted thatgeotechnical components tend to accumulateresidual displacements. This tendency mayaffect the acceptability of a structure for higherperformance objectives such as ImmediateOccupancy. Soil structure interaction also hasbeneficial affects such as lower demands onstructural members due to base rotation, lowerforces due to uplift and damping effects thatreduce demand on the superstructure.


-18 -14 -10 -6 -2 2 6 10 14 18

Top Displacement (inches)

-110

-90

-70

-50

-30

-10

10

30

50

70

90

110

Late

ral L

oad

(kip

s)

Displacement Ductility Factor-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

-45 -35 -25 -15 -5 5 15 25 35 45

Top Displacement (cm)

-450

-350

-250

-150

-50

50

150

250

350

450

Late

ral L

oad

(KN

)

Figure 15-6. Idealized Component Force-Deformation Relationships

A

BC

D

Idealizedcomponentbehavior

BackbonecurveF

D A

B

C,D

E

F

D

A

B,C,D

E

F

D

(a) Backbone curve from actual hysteretic behavior

(b) Idealized component behavior from backbone curves(15-1)

Ductile(deformation controlled)

Semi-ductile

Brittle(force controlled)


Figure 15-7. Shallow Foundation Model(15-1)

Figure 15-8. Basic Force-Displacement Envelope for Geotechnical Components(15-1)

Structural Components

Distributed Vertical Geotechnical Properties• Vertical bearing properties of soil• Component spacing along footing length

b. Element Model for Analysis

Soil

Spread Footing

Column/WallGrade Beams

a. Spread Footing Foundation

Horizontal Geotechnical Component• Passive properties against side of footing• Friction properties at bottom of footing

Envelope

KstiffKflexible

Force Q

Displacement, d

Actual Behavior

Upper Qc

Lower Qc


15.3.6 Capacity Spectrum Method

One of the methods used to determine theperformance point is the Capacity SpectrumMethod(15-8,15-9,15-10), also known as theAcceleration-Displacement Response Spectramethod (ADRS). The Capacity SpectrumMethod requires that both the capacity curveand the demand curve be represented inresponse spectral ordinates. It characterizes theseismic demand initially using a 5% dampedlinear-elastic response spectrum and reduces thespectrum to reflect the effects of energydissipation to estimate the inelasticdisplacement demand. The point at which thecapacity curve intersects the reduced demandcurve represents the performance point at whichcapacity and demand are equal.

To convert a spectrum from the standard Sa(Spectra Acceleration) vs T (Period) formatfound in the building codes(15-13) to ADRSformat, it is necessary to determine the value ofSdi (Spectral Displacement) for each point onthe curve, Sai,.Ti This can be done with theequation:

gSaT

Sd ii

i 2

2

4π= (15-1)

Standard demand response spectra contain arange of constant spectral acceleration and asecond range of constant spectral velocity, Sv.Spectral acceleration and displacement atperiod Ti are given by:

SvT

SdSvT

gSa ii

i

i ππ

2,

2 == (15-2)

The capacity spectrum can be developedfrom the pushover curve by a point by pointconversion to the first mode spectralcoordinates. Any point Vi (Base Shear), δi (RoofDisplacement) on the capacity (pushover) curveis converted to the corresponding point Sai, Sdi

on the capacity spectrum using the equations:

Figure 15-9. Response Spectrum Conversion(15-1)

1

/

αWV

Sa ii = (15-3)

)( ,11 roof

ii PF

Sdφ

δ×

= (15-4)

Where α1 and PF1 are the modal masscoefficient and participation factors for the firstnatural mode of the structure respectively. φ1,roof

is the roof level amplitude of the first mode.The modal participation factors and modalcoefficient are calculated as:

=∑

∑

=

=n

iii

n

iii

gw

gwPF

1

21

11

1

/)(

/)(

φ

φ(15-5)

Sa

Sai

TTiTo

To

Ti

Sd

Sa

Sdi

Standard Format (Sa vs T)

ADRS Format (Sa vs Sd)

770 Chapter 15

=

∑∑

∑

==

=

n

iii

n

ii

n

iii

gwgw

gw

1

21

1

2

11

1

/)(/

/)(

φ

φα (15-6)

Where wi is the weight at any level i.As displacement increase, the period of the

structure lengthens. This is reflected directly inthe capacity spectrum. Inelastic displacementsincrease damping and reduce demand. TheCapacity Spectrum Method reduces the demandto find an intersection with the capacityspectrum, where the displacement is consistentwith the implied damping.

Figure 15-10. Capacity Spectrum Conversion(15-1)

The damping that occurs when the structureis pushed into the inelastic range can be viewedas a combination of viscous and hystereticdamping. Hysteretic damping can berepresented as equivalent viscous damping.Thus, the total effective damping can beestimated as:

05.0+= oeff λββ (15-7)

Where βo is the hysteretic damping and 0.05 isthe assumed 5% viscous damping inherent in

the structure. The λ-factor (called κ-factor inATC-40) is a modification factor to account forthe extent to which the actual buildinghysteresis is well represented by the bilinearrepresentation of the capacity spectrum (SeeTable 15-3 and Figure 15-11).The term βo can be calculated using:

So

Do E

E

πβ

4

1= (15-8)

Where ED is the energy dissipated by dampingand ESo is the maximum strain energy. Thephysical significance is explained in Figure 15-11.

Figure 15-11. Derivation of Energy dissipated byDamping(15-1)

To account for the damping, the responsespectrum is reduced by reduction factors SRA

and SRV which are given by

12.2

)ln(68.021.31 eff

S

A BSR

β−== (15-9)

BaseShear- V

Roof Displacement - δ

Capacity Curve

Capacity Spectrum

Vi, δi,Roof

SpectralAcceleration- Sa

Spectral Displacement - Sd

Sai,Sdi

Spectral Displacement

SpectralAcceleration

Sapi

Say

Sdy

Kinitial

Keffective

Bilinear representation ofCapacity Spectrum

Capacity Spectrum

ESo = Maximum Strain Energy= Area of hatched triangle= Sapi Sdpi / 2

Sdpi

ED = Energy dissipated by damping= Area enclosed by hysteresis loop= Area of shaded parallelogram


65.1

)ln(41.031.21 eff

L

V BSR

β−== (15-10)

Both SRA and SRV must be greater than or equalto allowable values in Table 15-4.

The elastic response spectrum (5% damped)is thus reduced to a response spectrum withdamping values greater than 5% criticallydamped (See Figure 15-12). Note, the limits ofthe spectral reduction factors are arbitrary andneed farther study.

Table 15-2. Structural Behavior Types(15-1)

ShakingDuration1

EssentiallyNew

Building2

AverageExisting

Building3

PoorExisting

Building4

Short Type A Type B Type CLong Type B Type C Type C

1. See Section 4.5.2 of ATC-40 for criterion.2. Buildings whose primary elements make up an

essentially new lateral system and little strength orstiffness is contributed by non-complying elements.

3. Building whose primary elements are combination ofexisting and new elements, or better than averageexisting systems.

4. Buildings, whose primary elements make up non-complying lateral force systems with poor andunreliable hysteretic behavior.

Table 15-3. Values for Damping Modification Value, λStructuralBehavior Type

ββββo

(percent)λλλλ

Type A 25.16≤

25.16≥

1.0

1.13 – 0.51(Say Sdpi-Sdy

Sapi)/Sapi Sdpi

Type B 25≤

25≥

0.67

0.845 – 0.446(SayS dpi-Sdy

Sapi)/Sapi Sdpi

Type C Any Value 0.33

Table 15-4. Minimum Allowable Value for SRA andSRV

[15-1]

StructuralBehavior Type

SRA SRV

Type A 0.33 0.50Type B 0.44 0.56Type C 0.56 0.67

Figure 15-12. Reduced Response Spectrum(15-1)

There are three procedures described inATC-40 to find the performance point. Themost transparent and most convenient forprogramming is Procedure A. To find theperformance point using Procedure A thefollowing steps are used:1. A 5% damped response spectrum

appropriate for the site for the hazard levelrequired for the performance objective isdeveloped and converted to ADRS format.

2. The capacity curve obtained from the non-linear analysis is converted to a capacityspectrum using Equations 15-3 and 15-4.

3. A trial performance point Sapi, Sdpi isselected. This may be done using the equaldisplacement approximation (See Figure15-13) or on the basis of engineeringjudgement.

4. A bilinear representation of the capacityspectrum is developed such that the areaunder the capacity spectrum and thebilinear representation is the same. In thecase of a saw-tooth capacity spectrum, thebilinear representation must be based on thecapacity spectrum that makes up the portionof the composite capacity spectrum wherethe performance point Sapi, Sdpi occurs.

5. The spectral reduction factors SRA and SRV

are computed using Equations 15-9 and 15-10 and the demand spectrum is reduced asshown in Figure 15-12. The reduceddemand spectrum is plotted together withthe capacity spectrum.



2.5CA

CV/TSRA x 2.5CA

SRV x CV/T

ElasticResponseSpectrum 5%Damped

Reduced Response Spectrum


6. If the reduced demand spectrum intersectsthe capacity spectrum at Sapi, Sdpi or if theintersection point Sdp is within 5% of Sdpi,then this point represents the performancepoint.

7. If the intersection point does not lie withinacceptable tolerance (5% of Sdpi or other)then select another point and repeat Steps 4to 7. The intersection point obtained in Step6 can used as the starting point for the nextiteration.

Procedure B is also an iterative method tofind the performance point, which uses theassumption that the yield point and the postyield slope of the bilinear representation,remains constant. This is adequate for mostcases, however, in some cases this assumptionmay not be valid. Procedure C is graphicalmethod that is convenient for hand analysis.

15.3.7 Checking Performance at ExpectedMaximum Displacement

Once the performance point Sap, Sdp (whichare in spectral ordinates) is found, the baseshear (Vp) and roof displacement (δp) at theperformance point are found using Equation 15-3 and 15-4. The following steps should be usedin the performance check:

1. For the global building response, verifya. The lateral force resistance has not

degraded by more than 20% of the peakresistance.

b. The lateral drift limits satisfy the limitsgiven in the Table 15-5.

2. Identify and classify the different elementsin the building in the following types:beam-column frames, slab-column frames,solid walls, coupled walls, perforated walls,punched walls, floor diaphragms andfoundations.

3. Identify all primary and secondaryelements.

4. For each element type, identify the criticalcomponents and actions to check asdetailed in Chapter 11 of ATC-40.

5. The strength and deformation demands atthe performance point should be equal to orless than the capacities detailed in Chapter11 of ATC-40.

6. The performance of secondary elements(such as gravity load carrying members notpart of the lateral load resisting system) arereviewed for acceptability for the specifiedperformance level.

7. Non-structural elements are checked for thespecified performance level.

Figure 15-13. Capacity Spectrum Procedure A to Determine Performance Point



2.5CA

CV/T

SRA x 2.5CA

SRV x CV/T

Elastic ResponseSpectrum 5% Damped

Reduced Response Spectrum

Sdy Sdpi

Say

Sapi

Bilinearrepresentation ofcapacity

Capacity Spectrum

Intersection point of reduced demandspectrum and capacity spectrum

Sdp


15.3.8 Other Considerations

Other considerations that should be notedare1. Torsion: For 3D models, the lateral load

should be applied at the center of mass ofeach floor and the displacement plotted onthe capacity curve should be for the centerof mass for the roof. Use of 2D modelsshould be limited to building where thetorsional effects are sufficiently small suchthat the maximum displacement at anypoint is not more than 120% of thedisplacement at the center of mass.

2. For structure with long fundamental modes,higher mode effects may be more critical.Pushover analysis should be performed foradditional mode shapes usingcorresponding force distributions.

Table 15-5. Deformation Limits(15-1)

Performance LimitInterstory

Drift LimitImmediateOccupancy

DamageControl

LifeSafety

StructuralStability

MaximumTotal Drift

0.01 0.01 –0.002

0.02 0.03Vi/Pi

MaximuminelasticDrift

0.005 0.005 –0.015

NoLimit

NoLimit

15.3.9 Example

An example is provided of the procedure todetermine the performance point using theCapacity Spectrum Method. This examplereworked from numbers provided in the ATC-40 document.

15.3.9.1 Building DescriptionThe example building is a seven-story

reinforced concrete building. The total weight

of the building is 10,540 kips. The pushovercurve determined for the building is given isTable 15-6. The pushover (capacity) curve isconverted into a capacity spectrum usingEquation 15-3 and 15-4. The demand for thebuilding for the performance level desired isdetermined to be Soil Type SD with CA and CV

being 0.44 and 0.64 respectively. The demandspectrum is converted to ADRS format usingEquation 15-1.

The demand and capacity spectrum areplotted together as shown in Figure 15-14.Using an equal displacement approximation,the first trial performance point Sap1, Sdp1 isselected. A bilinear representation is developedsuch that the area under the capacity spectrumis the same as the area under the bilinear curve.Thus:

Sap1 = 0.36g Sdp1 = 5.5 inSay = 0.31g Sdy = 2.35 in

%11.14

5)(7.63

11

11

=

+−

=pp

pypy

eff SdSa

SaSdSdSaλβ

A λ of 0.33 is used for structural behaviortype C from Table 15-3. Thus, the spectralreduction factors are calculated from Equations15-9 and 15-10 as:

665.012.2

)11.14ln(68.021.3 =−=ASR

742.065.1

)11.14ln(41.031.2 =−=VSR

Table 15-6. Conversion of Pushover Curve to Capacity Spectrum (15-1)

Point V(kips)

δδδδR

(in)V/W PF1.φφφφ1,roof αααα1111 Sa

(g)Sd(g)

T (sec)

A 2200 2.51 0.209 1.31 0.828 0.254 1.92 0.88B 2600 3.60 0.247 1.28 0.800 0.309 2.81 0.96C 2800 5.10 0.266 1.35 0.770 0.346 3.78 1.06D 3000 10.90 0.285 1.39 0.750 0.380 7.84 1.45

PF1 and α1 change because the mode shape is changing as yielding occurs

774 Chapter 15

Using the spectral reduction factors, thedemand spectrum is reduced as per Figure 15-12. The reduced spectrum is plotted togetherwith the capacity spectrum and the intersectionpoint is found (See Figure 15-15). The demandspectrum intersects the capacity spectrum at aspectral displacement of 6.1 inches. As thisdisplacement is not with 5% of the first trialdisplacement of 5.5 inches.

A new trial performance point must bechosen and the process repeated. The secondtrial point may be chosen as the intersectionfrom the previous iteration. However, in thisexample, the second trial performance point ischosen by engineering judgement at a spectraldisplacement of 5.9 inches. A new bilinearrepresentation is constructed and the processrepeated:

Sap2 = 0.365g Sdp2 = 5.9 inSay = 0.305g Sdy = 2.3 in

%37.14

5)(7.63

22

22

=

+−

=pp

pypy

eff SdSa

SaSdSdSaλβ

The new spectral reduction factors arecalculated from Equations 15-9 and 15-10 as:

659.012.2

)37.14ln(68.021.3=

−=ASR

738.065.1

)37.14ln(41.031.2=

−=VSR

A new reduced demand spectrum is plottedand a new intersection point is obtained. Asseen in Figure 15-17, the intersection point is ata spectral displacement of 6.0 inches. As thisintersection is within 5% of the second trialpoint, the demand spectral displacement is 6.0inches.

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Spectral Displacement (inches)

Sp

ectr

al A

ccel

erat

ion

(g

)T = 0.5 secs

T = 1.5 secs

T = 1.0 secs

T = 2.0 secs

Capacity Spectrum Curve

Equal Displacement Approx

First Trial Performance Point

Sd p1Sd y

Figure 15-14. Determination of the First Trial Performance Point


0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15


Sp

ectr

al A

ccel

erat

ion

(g

)T = 0.5 secs

T = 1.5 secs

T = 1.0 secs

T = 2.0 secs

Reduced Demand Spectrum

First Trial Performance Point

Intersection Point

Figure 15-15. Determination of Intersection Point and Comparison with the First Trial Performance Point

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15


Sp

ectr

al A

ccel

erat

ion

(g

)

T = 0.5 secs

T = 1.5 secs

T = 1.0 secs

T = 2.0 secs


Second Trial Performance Point

Sd p2Sd y

Figure 15-16. Determination of Second Performance Point

776 Chapter 15

The actual roof displacement at theperformance point is calculated from Equation15-4. The modal participation factor is used bylinear interpolation from Table 15-6.

inches 1.80.6351 1,1

=×=×⋅=

.

SdPF prooft φδ

Similarly, the base shear can be found from thespectral acceleration at the performance pointby using Equation 15-3. The modal masscoefficient can be found by linear interpolationfrom Table 15-6.

277.0365.076.0

/ 1

=×=×= pp SaWV α

The element capacities are checked for thebuilding at this performance point as detailed inSection 15.3.7.

15.3.10 Recent Advances in the CapacitySpectrum Method

In recent publications it has been reportedby Chopra and Goel(15-14,15-15) that the CapacitySpectrum Method as described in ATC-40 doesnot produce conservative estimates of inelasticpeak displacements when compared to inelasticresponse spectrum analysis. It has also beenreported that the ATC-40 procedures aredeficient relative to even the elastic designspectrum in the velocity and displacementsensitive regions of the spectrum. An improvedmethod has been suggest by Chopra andGoel(15-15) which makes use of inelastic spectrausing any of three Ry-µ-T equations (Newmarkand Hall(15-16), Krawinkler and Nassar(15-17) andVidic, Fajfar and Fischinger(15-18)). In thisimproved Capacity Spectrum Method, thecapacity and the constant ductility designspectra are plotted in ADRS format. Thecapacity spectrum intersects the demandspectrum for several values of ductility µ. The

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15


Sp

ectr

al A

ccel

erat

ion

(g

)

T = 0.5 secs

T = 1.5 secs

T = 1.0 secs

T = 2.0 secs


Second Trial Performance Point

New Intersection Point

New Reduced Demand Spectrum

Figure 15-17. Determination of Final Performance Point


deformation at the performance point is givenby the one intersection point where the ductilityfactor calculated from the capacity spectrummatches the value associated with theintersected demand spectrum.

Another method for determining theperformance point is suggested by Fajfar(15-19).Here the ductility demand is determined usingthe equal displacement rule and the inelasticdesign spectra. Another variant of the CapacitySpectrum method called the Yield PointSpectra(15-20) has recently been suggested. Herethe yield displacement is plotted on the abscissainstead of the spectral displacement and Ry-µ-Trelations or exact computations are used insteadof equivalent viscous damping.

15.4 FEMA 273 and 274

15.4.1 Introduction

NEHRP Guidelines for the SeismicRehabilitation of Buildings (FEMA-273)(15-2)

and the associated commentary (FEMA-274)(15-

3) was developed by the Building SeismicSafety Council (BSSC) with subcontractorsAmerican Society of Civil Engineering (ASCE)and the Applied Technology Council (ATC)with the funding provided by the FederalEmergency Management Agency (FEMA). Theprimary purpose of FEMA-273 was to providetechnically sound and nationally acceptableguidelines for the seismic rehabilitation ofbuildings. Although the document was writtenwith the objective of performance based retrofitof existing structures, the procedures describedtherein are equally applicable for new design.Unlike the ATC-40 document, theserecommendations are applicable to all buildingmaterials and define acceptability limits forlinear as well as non-linear analysis.

The basic procedure is similar to thatrecommended in ATC-40. The owner decidesthe performance object that needs to beachieved. The engineer then designs the retrofitor new structure to achieve the performanceobjective. The definitions of the basic

performance levels are similar to those definedin ATC 40 (See Section 15.3.2).

FEMA-273 defines ground motion hazardlevels in a probabilistic basis. Four groundmotion hazard levels are defined

Earthquake Probability Mean Returnof Exceedence Period (years)50% in 50 years 7220% in 50 years 225BSE-1 10% in 50 years 474BSE-2 2% in 50 years 2,475

Where BSE is the Basic Safety Earthquake. Thebroad range of performance objectivesrecommended for a given earthquake hazardlevels are shown in Table 15-7

Table 15-7. Rehabilitation Objectives(15-2)

Building Performance Levels

Ope

rati

onal

Lev

el(1

-A)

Imm

edia

teO

ccup

ancy

Lev

el(1

-B)

Lif

e Sa

fety

Per

form

ance

Lev

el(3

-C)

Col

laps

eP

reve

ntio

nP

erfo

rman

ceL

evel

50%/50yrs

a b c d

20%/50yrs

e f g h

BSE-110%/50yrs

i j k l

Ear

thqu

ake

Haz

ard

Lev

el

BSE-22%/50yrs

m n o p

k+p = Basic Safety Objectivek+p+any of a, e, i or m; or b, f, j, or n = EnhancedObjectiveso = Enhanced Objectivesk alone or p alone = Limited Objectivec, g, d, h = Limited Objectives

From Table 15-7, it is clear that FEMA-273specifies a two-level design to achieve theBasic Safety Objective (BSO), Life SafetyPerformance Level for BSE-1 demands andCollapse Prevention Level for BSE-2 demands.However, for new structures it is possible tocontrol ductility and configuration of the design

778 Chapter 15

to an extent that will permit those structuresdesigned to achieve Life Safety PerformanceLevel for a BSE-1 level earthquake to alsoavoid collapse for much larger events.

Two sets of earthquake hazard maps aredistributed with FEMA-273 and 274. One setprovide key response acceleration for theMaximum Considered Earthquake (MCE)which in most areas represents a 2%/50 yearsexceedence level. The other uses 10%/50 yearsexceedence probability. Thus, it is possible toobtain a BSE-1 and BSE-2 level spectra fromthese maps.

15.4.2 Mathematical Modeling

FEMA-273 provides four analysisprocedures for systematic design andrehabilitation of buildings. The Linear Static(LSP) and Linear Dynamic Procedures (LDP)are linearly elastic analysis, which may includegeometric non-linearity. Also some materialnon-linearity is also introduced by use ofcracked properties for concrete and masonrycomponents even though the analysis is linear.In the Nonlinear Static (NSP) and NonlinearDynamic Procedures (NDP) material non-linearity is included in the analysis.

15.4.2.1 Basic AssumptionsIn general, a three dimensional analysis

consisting of an assembly of elements andcomponents is recommended. Three-dimensional analysis is required when thebuilding has plan irregularities and whentorsional effects cannot be ignored or indirectlycaptured.

For buildings with flexible diaphragms, thediaphragms may be individually modeled andanalyzed as two-dimensional assemblies ofcomponents and elements or three-dimensionalmodels with flexible elements.

Explicit modeling of connections is notrequired if the connection is stronger than theconnected components or when the deflectionof the connection does not cause a significant

increase in the relative deformation between theconnected components.

15.4.2.2 Horizontal TorsionIn addition to the actual eccentricities

between the centers of mass and centers ofrigidity, a additional accidental torsionalmoment should be included which may beproduced by including a horizontal offset in thecenters of mass equal to a minimum of 5% ofthe horizontal dimension at a given floor level.

For buildings with rigid diaphragms, theeffects of torsion must be included when themaximum displacement at any point in adiaphragm exceeds the average displacement inthat diaphragm by more than 10%. For linearanalysis, the effect of accidental torsion isamplified by a factor Ax:

2

max

2.1

=

avg

xAδ

δ(15-11)

Where δmax and δavg are the maximum andaverage displacements in a diaphragm. Ax isgreater than 1 and not greater than 3.

If η =δmax/δavg is greater than 1.5, then athree-dimensional analysis is required. For two-dimensional analysis subject to this limitation,the effect of torsion can included for LSP andLDP by increasing the design forces anddisplacement by η. For NSP, the targetdisplacement is increased by η and for NDP theamplitude of the ground acceleration record isincreased by η.

15.4.2.3 Primary and Secondary ElementsPrimary elements are key parts of the

seismic framing system required in the designto resist earthquake effects. These must beevaluated to resist earthquake forces as well asgravity loads if required. Secondary elementsare not designed to be part of the lateral forceresisting system but must be evaluated to ensurethey can simultaneously sustain earthquakeinduced deformation and gravity loads.

For linear analysis procedures, thesecondary elements must not constitute more


than 25% of the total stiffness of the primaryelements at any level and may not be includedin the analysis. For nonlinear procedures, thestiffness of the primary as well as the secondaryelements must be included in the model.Additionally, the stiffness of non-structuralelements must not exceed 10% of the totallateral stiffness of any story. If this is exceeded,then the non-structural elements must beincluded in the model.

15.4.2.4 Deformation and ForceControlled Elements

Elements can be classified as eitherdeformation controlled or force controlled. Adeformation controlled element is one that hasan associated deformation that is allowed toexceed yield value, that is, the maximumassociated deformation of the element is limitedby the ductility of the element. A forcecontrolled element is one where the maximumassociated displacement is not allowed toexceed yield value. Elements with limitedductility shall be considered to be forcecontrolled. See Table 15-8 for calculation ofelement capacities used to compare withdemands.

15.4.2.5 Stiffness and StrengthAssumptions

Element and component stiffness propertiesand strength assumptions for most materialtypes are provided in FEMA-273. Guidelinesfor structural and foundation elements are alsoprovided. These are similar to those provided inATC-40.

15.4.2.6 Foundation ModelingFoundation modeling assumptions are

similar to ATC-40 (See Section 15.3.5). Thefoundation system may be included in themodel for analysis with stiffness and dampingproperties as defined in Chapter 4 of FEMA-273. Otherwise, unless specifically prohibited,the foundation may be assumed to rigid and notincluded in the model.

Table 15-8. Calculation of Element Capacities(15-2)

Parameter DeformationControlled

ForceControlled

Linear ProceduresExisting MaterialStrength

Mean value withallowance forstrain hardening

Lower bound(Mean – StdDev)

Existing Capacity mκ QCE κ QCE

New MaterialStrength

Mean value Specified value

New Capacity QCE QCE

Nonlinear ProceduresDeformationCapacity –Existing Element

κ xDeformationlimit

N/A

DeformationCapacity – NewElement

Deformationlimit

N/A

Strength Capacity– ExistingElement

N/A κ QCL

Strength Capacity– New Element

N/A κ QCL

κ = Knowledge factorm = Demand Modifier for expected ductilityQCE = Expected StrengthQCL = Lower Bound Estimate of Strength

15.4.2.7 DiaphragmsDiaphragms transfer earthquake induced

inertial loads to the vertical elements of theseismic framing system. Connection betweenthe diaphragms and the vertical elements of thelateral load resisting system must havesufficient strength to transfer the maximumcalculated inertial loads. Diaphragms may beflexible, stiff or rigid. Flexible diaphragms arethose where the maximum lateral deformationof the diaphragm is more than twice the averageinter-story drift of the story below thediaphragm. Rigid diaphragms are those wherethe maximum lateral deformation of thediaphragm is less than half the average inter-story drift of the associated story. Diaphragmsthat are neither rigid nor flexible can beconsidered to be stiff.

Mathematical models of buildings with stiffor flexible diaphragms must consider the effectof diaphragm flexibility. For buildings withflexible diaphragms at each floor level, the

780 Chapter 15

vertical lines seismic framing may be designedindependently with seismic masses assigned onthe basis of tributary areas.

15.4.2.8 P-Delta EffectsFor linear procedures, at each story the

quantity θi shall be computed for each directionof response as follows:

ii

iii hV

P δθ = (15-12)

Where Pi is the portion of the total weight ofthe structure including dead, permanent line and25% of the transient live loads acting on thecolumns and load bearing walls. Vi is the totalcalculated shear force, hi is the story height andδi is the lateral drift in the direction underconsideration at story i.

For linear procedures, the story drifts δi

must be increased by 1/(1- θi) for evaluation ofthe stability coefficient, θi. Therefore, theprocess is iterative. If the stability coefficient, θi

is less than 0.1, the static P-Delta effects aresmall and can be ignored. If the stabilitycoefficient, θi is greater than 0.33, the structureis unstable. If it lies between 0.1 and 0.33 thanthe seismic forces at level i must be increasedby 1/(1- θi).

For non-linear procedures, these secondorder effects must be directly included in themodel by use of geometric stiffness of allelements subject to axial loads. Dynamic P-Delta effects are included in the LSP and NSPby use of Coefficient C3 (See Section 15.4.3.1and 15.4.3.3).

15.4.2.9 Soil Structure InteractionSoil Structure Interaction (SSI) may modify

the seismic demand on the structure. To includeSSI, one may use the effective fundamentalperiod and effective damping ratios of thefoundation-structure system to compute seismicdemand or explicitly model SSI. SSI effectsshall not be used to reduce component andelement actions by more than 25%.

15.4.2.10 Multidirectional EffectsBuildings should be designed for seismic

forces in any horizontal direction. For regularbuildings, seismic displacements and forcesmay be assumed to act non-concurrently in thedirection of each principle axis of the building.For buildings with plan irregularities andbuildings with intersecting elements,multidirectional effects must be considered. Anacceptable procedure is use of 100% of theseismic force in one horizontal direction and30% of the seismic force in the perpendiculardirection. Alternately SRSS may be used tocombine forces in orthogonal directions.

Vertical excitation of horizontal cantileversand pre-stressed elements must be considered.Vertical shaking characterized by a spectrumwith ordinates equal to 67% of those of thehorizontal spectrum is acceptable where site-specific data is not available.

15.4.2.11 Load CombinationsThe component gravity loads to be

considered for combination with seismic loadsare:When effects of gravity and seismic loads areadditive:

)(1.1 SLDG QQQQ ++= (15-13)

When the effects of gravity counteract seismicloads

DG QQ 9.0= (15-14)

Where QD, QL and QS are dead, live and snowloads respectively. Effective live loads may beassumed to be 25% of the unreduced live loadbut not less than measured live loads. Effectivesnow loads are 70% of the full design snowloads or an approved percentage by a regulatoryagency.

Combination with earthquake loads isdiscussed in subsequent sections. Note suchload combinations are relevant for linearanalysis. Non-linear analysis is not conduciveto checking both of the above load


combinations and therefore only the criticalload combination (by inspection) may be used.

15.4.3 Analysis Procedures

15.4.3.1 Linear Static ProcedureIn this procedure a linear elastic model is

used in the analysis with an equivalent dampingthat approximates values expected for loadingnear the yield point. A pseudo-lateral load iscomputed as shown in the following section andapplied to the model. The resulting forces anddisplacements in the elements are then checkedagainst capacities modified to account forinelastic response demands.

15.4.3.1.1 Pseudo Lateral LoadTo compute the pseudo lateral load, the

fundamental period must be first determined.The period may be determined by one of thefollowing methods:1. Eigenvalue value analysis of the building.

For buildings with flexible diaphragms, themodel must consider representation ofdiaphragm flexibility unless it can beshown that the effects of the omission willnot be significant.

2. Use of the following equation

4/3

nt hCT = (15-15)

Where T is the fundamental period inseconds under the direction underconsideration and hn is the height above thebase to the roof.Ct = 0.035 for steel moment resistingframes.Ct = 0.030 for moment resisting framesystem of concrete and eccentrically bracedsteel frames.Ct = 0.020 for all other framing systems.Ct = 0.060 for wood buildings.

3. For one-story buildings with flexiblediaphragms:

5.0)078.01.0( dwT ∆+∆= (15-16)

Where ∆w and ∆d are in-plane wall anddiaphragm displacements in inches due to alateral loads in the direction underconsideration equal to the weight tributaryto the diaphragm. For multiple spandiaphragms, a lateral load equal to thegravity weight tributary to the span underconsideration can be applied to each span tocalculate a separate period for eachdiaphragm span. The period so calculatedthat maximizes the pseudo lateral load is tobe used for the design of all walls anddiaphragm spans in the building.The total pseudo lateral load, V in a given

horizontal direction is determined as

WSCCCV a321= (15-17)

WhereC1 = Modification factor to relate expectedmaximum inelastic displacements todisplacements calculated for the linear elasticresponse. C1 can be calculated as in Section15.4.3.3.4 with the elastic base shear substitutedfor Vy. Alternatively C1 may be calculated asfollowsC1=1.5 for T < 0.10 secsC1=1.0 for T ≥ T0 secsLinear interpolation can be used to calculate C1

for intermediate value of T.T = Fundamental period of the building in thedirection under consideration. For SSI, theeffective fundamental period should be used.T0 = Characteristic period of the responsespectrum, defined as the period associated withthe transition from the constant accelerationsegment of the spectrum to the constantvelocity segment of the spectrumC2 = Modification factor to represent the effectof stiffness degradation and strengthdeterioration on the maximum displacementresponse. Values for different framing fordifferent performance levels are listed in Table15-9. Linear interpolation can be used tocalculate C2 for intermediate value of T.C3 = Modification factor to represent theincreased displacement due to dynamic P-Deltaeffect. This effect is in addition to P-Delta

782 Chapter 15

described in Section 15.4.2.8. For values of θ less than 0.1, C3 may be set equal 1.0. Forvalues of θ greater than 0.1, C3 shall becalculated as 1+5(θ −0.1)/Τ. The maximumvalue of θ for all stories shall be used tocalculate C3.Sa = Response spectrum acceleration at thefundamental period and damping ratio of thebuilding in the direction under consideration.W = Total dead load and anticipated live loadas indicated below:• In storage and warehouse occupancies, a

minimum of 25% of the floor live load,• The actual partition weight or minimum

weight of 10 psf of floor area, whichever isgreater,

• The applicable snow load,• The total weight of permanent equipment

and furnishings.

Vertical distribution of the base shear V isdone by the following:

VCF vxx = (15-18)

Table 15-9. Values of Modification Factor C2(15-2)

T=0.1 second T ≥ T0 secondsPerformanceLevel

FramingType 1

FramingType 2

FramingType 1

FramingType 2

ImmediateOccupancy

1.0 1.0 1.0 1.0

LifeSafety

1.3 1.0 1.1 1.0

CollapsePrevention

1.5 1.0 1.2 1.0

Framing Type 1 = Structures in which more than 30% ofthe story shear any level is resisted by components orelements whose strength and stiffness deteriorate duringthe design earthquake. Such elements and componentsinclude: ordinary moment-resisting frames, concentricallybraced frames, frames with partially restrainedconnections, tension only braced frames, unreinforcedmasonry walls, shear-critical walls and piers, or anycombination of the above.Framing Type 2 = All frames not assigned to FramingType 1

∑=

=n

i

kii

kxx

vx

hw

hwC

1

(15-19)

k = 1.0 for T ≤ 0.5 second = 2.0 for T ≥ 2.5 secondLinear interpolation is used to estimate valuesof k for intermediate values of T. Cvx is thevertical distribution factor, V is the pseudolateral load from Equation 15-17, wi is theweight of level i, wx is the weight of thebuilding of any level x, hi is height from thebase to floor level i and hx is height from thebase to floor level x.

Floor diaphragms are designed to resist theinertial forces developed at the level underconsiderations and the horizontal forcesresulting from offsets or changes in stiffness inthe vertical seismic framing elements above andbelow the diaphragm. The diaphragm inertialforce Fpx at level x is given by

∑∑=

=

=n

xin

xii

xipx

w

wF

CCCF

321

1(15-20)

Where Fi is the lateral load applied at floorlevel i as given by Equation 15-18.

The base shear, vertical distribution andforces on the diaphragms for the LSP is notunlike current codes, however force levels andacceptance criterion for the elements in thelateral load resisting systems depend on thedesired performance level.

15.4.3.1.2 Acceptance Criteria to satisfyPerformance Point requirements

The design forces shall be calculated as perthe following:For Deformation-Controlled Elements -

EGUD QQQ ±= (15-21)

For Force-Controlled Elements -


JCCC

QQQ E

GUF

321

±= (15-22)

321 CCC

QQQ E

GUF ±= (15-23)

Where QUD and QUF are the demands due togravity and earthquake forces for deformationand force controlled elements respectively. QE

is the demand due to the earthquake forcesdescribed in the previous section and J is theforce delivery reduction factor given by:

XSSJ += 0.1 (15-24)

J cannot exceed 2 and SXS is the short periodspectral acceleration parameter for the designspectrum. Alternately, J can be taken as thesmallest demand capacity ratio of thecomponents in the load path delivering force tothe component in question.

The capacities of elements must be checkedagainst the demands as follows:For Deformation-Controlled elements -

UDCE QQm ≥κ (15-25)

For Force-Controlled elements -

UFCL QQ ≥κ (15-26)

Where QCE and QCL are the expected and lowerbound strength of the element or componentrespectively. m is the demand modifier toaccount for the deformation associated withdemand at the selected performance level. κ isthe knowledge factor to account for uncertaintyin capacity evaluations. A value of 0.75 is usedfor κ when only a minimum knowledge isavailable and a value of 1.0 can be used whencomprehensive knowledge is available for theelement or component in question.

The capacities that need to be checkedagainst demands for each element type andmaterial are listed in Chapters 5 to 8 in FEMA-

273 together with the demand modifiers, m, foreach performance level.

15.4.3.2 Linear Dynamic ProcedureThe basis, modeling approaches and

acceptance criterion for the Linear DynamicProcedure (LDP) is similar to those describedfor LSP. The main exception is that theresponse is obtained from either a linearlyelastic response spectrum or a time-historyanalysis. As with LSP, LDP will producedisplacements that are approximately correct,but will produce inertial forces that exceedthose that would be obtained in a yieldingbuilding.

The response spectrum method uses peakmodal responses calculated from an eigenvalueanalysis of a mathematical model. The timehistory method involves a time-step by time-step evaluation of the building response using adiscretized record or synthetic record as basemotion input. In both the methods, only modescontributing significantly to the response needto be considered. In the response spectrumanalysis, modal responses are combined usingrational methods to estimate total buildingresponse quantities.

15.4.3.2.1 Ground MotionThe ground motion can be characterized by

either a linearly elastic response spectrumwhich may be site specific or a groundacceleration time history which may berecorded or synthesized. In both cases, theground motion must be appropriately scaled toreflect the hazard level that is associated withthe performance level desired (See Table 15-7)

15.4.3.2.2 Response Spectrum MethodAll significant modes must be included in

the response spectrum analysis such that at least90% seismic mass participation is achieved ineach of the building’s principle directions.Modal damping must reflect the dampinginherent in the building at the deformationlevels less than yield deformation.

The peak member forces, displacements,story forces, shears and base reactions for each

784 Chapter 15

mode should be combined using SRSS (squareroot sum of squares) or CQC (completequadratic combination). It should also be notedthat the directivity of the forces is lost in theresponse spectrum analysis and therefore thecombination of forces must reflect this loss.

Multidirectional effects should also beinvestigated when using the response spectrumanalysis.

15.4.3.2.3 Time History MethodAll the requirements for response spectrum

analysis are also identical for the time historyanalysis. Response parameters are computed foreach time history analysis. If 3 pairs of timehistories are used, the maximum response of theparameter of interest shall be used for thedesign. If seven or more pairs of time historiesare used, the average response (of the maximumof each analysis) of the parameter of interest isto be used.

Multidirectional effects can be accounted byusing a three dimensional mathematical modeland using simultaneously imposed pairs ofearthquake ground motions along each of thehorizontal axes of the building.


The acceptance criterion for LDP is similarto that described for LSP. However, alldeformations and force demands obtained fromeither the response spectrum or the time historyanalysis must be multiplied by the product ofthe modification factors C1, C2 and C3. Forcedemands on elements of the floor diaphragmneed not be increased by these factors. Theseismic forces on the diaphragm obtained in theanalysis must not be less than 85% than thoseobtained in LSP (See Equation 15-20).

15.4.3.3 Nonlinear Static ProcedureIn the Nonlinear Static Procedure (NSP) the

nonlinear load-deformation characteristics ofindividual elements and components aremodeled directly. The mathematical model ofthe building is subjected to monotonicallyincreasing lateral load until a target

displacement is reached or the buildingcollapses. The target displacement is intendedto represent the maximum displacement likelyto be experienced during the design earthquake.The nonlinear effects are directly included inthe model and therefore the calculated inertialforces are reasonable approximations of thoseexpected during the design earthquake.

The target displacement can be calculatedby any procedure that accounts for nonlinearresponse on displacement amplitude as well asdamping effects at the performance point. Onesuch procedure called the DisplacementCoefficient Method is described in FEMA 273.ATC-40 also includes this method as analternative method of finding the performancepoint. The advantage of this method over theCapacity Spectrum procedure is it simplicity.

The modeling requirements for NSP aresimilar to those described in ATC-40. Thepushover analysis is performed and a curverelating the base shear force and the lateraldisplacement of the control node are establishedbetween 0 and 150% of the target displacement,δt. Acceptance criterion is based on the forcesand deformation corresponding to thedisplacement of the control node equal to δt.

The analysis model must be sufficientlydiscretized to represent the load-deformationresponse of each element or component.Particular attention needs to be paid toidentifying locations of inelastic action alongthe length of element or component. Thus, localmodels of elements or assemblages of elementsneed to be studied before embarking on theglobal models.

15.4.3.3.1 Control NodeThe control node is usually the center of

mass of the roof of the building. The top of thepenthouse should not be considered to be theroof. As the displacement of the control node iscompared with the target displacement, thechoice of the control node is very important.

15.4.3.3.2 Lateral Load PatternsThe lateral load should be applied to

building in profiles that approximately bound


the likely vertical and horizontal distribution ofthe inertial force in an earthquake. At least twovertical distributions of lateral loads must beconsidered with NSP. Note use of only one loadpattern may not identify potential deficienciesin the building.

The two lateral load patterns that arerecommended are1. Uniform Load Pattern: Here the lateral load

may be represented by values of Cvx asgiven by Equation 15-19.

2. Modal Pattern: Here the lateral load patternis consistent with story shear distribution ina response spectrum analysis where there isat least 90% mass participation and theappropriate ground motion is used.

Other appropriate load patterns substantiated byrational analysis may be substituted for theabove.

15.4.3.3.3 Period DeterminationThe effective fundamental period, Te in the

direction considered can be computed using thepushover curve obtained in the NSP. A bilinearrepresentation of the pushover curve isconstructed to estimate the effective lateralstiffness, Ke, and the yield strength of thebuilding, Vy. The effective lateral stiffness canbe taken as the secant stiffness calculated at abase shear force equal to 60% of the yieldstrength (See Figure 15-18).

The effective fundamental period, Te iscomputed as:

e

iie K

KTT = (15-27)

Where Ti and Ki are the initial elasticfundamental period in seconds and initialstiffness of the building in the direction underconsidered.

It is obvious that to determine the effectivefundamental period, Te, and the targetdisplacement, δt, the pushover curve for thebuilding is needed.

Figure 15-18. Calculation of Effective Stiffness Ke(15-2)

15.4.3.3.4 Target DisplacementUsing the Displacement Coefficient Method

the target displacement can be computed as:

gT

SCCCC eat 2

2

3210 4πδ = (15-28)

WhereC0 = Modification factor to relate the spectraldisplacement and likely building roofdisplacement. C0 can be calculated using one ofthe following1. The first modal participation factor at the

level of the control node.2. The modal participation factor at the level

of the control node calculated using a shapevector corresponding to deflected shape ofthe building at the target displacement.

3. The appropriate value from Table 15-10.C1 = Modification factor to relate maximuminelastic displacements to displacementscalculated for linear elastic response. C1 may becalculated as follows:

Table 15-10. Values for Modification Factor C0(15-2)

Number of Stories Modification Factor1

1 1.02 1.23 1.35 1.4

10+ 1.51. Linear interpolation should be used to calculateintermediate values

Roof Displacement

Base Shear

Vy

0.6Vy

δy

Ke

Ki

Bilinear representation ofPushover Curve

Pushover Curve

δi

αKe

786 Chapter 15

C1=1.0 for Te ≥ T0

C1=[1.0 + (R-1) T0/Te]/R for Te < T0

Values for C1 need not exceed those given forLSP (See Section 15.4.3.1.1) and in no case isC1 taken less than 1.0.T0 = Characteristic period of the responsespectrum, defined as the period associated withthe transition from the constant accelerationsegment of the spectrum to the constantvelocity segment of the spectrum.R = Ratio of the elastic strength demand tocalculated yield strength coefficient. R can becomputed as

0

1/ CWV

SR

y

a= (15-29)

Where W is the dead weight and anticipated liveas computed for LSP (See Section 15.4.3.1.1)and Vy is the yield strength determined from thebilinear representation of the pushover curve(See Figure 15-18).C2 = Modification factor to represent the effectof hysteresis shape on the maximumdisplacement response. Values of C2 can beobtained from Table 15-9.C3 = Modification factor to represent increaseddisplacements due to dynamic P-Delta effects.For buildings with positive post-yield stiffness,C3 can be set equal to 1.0. For buildings withnegative post yield stiffness C3 is given as

eT

RC

2/3

3

)1(0.1

−+=

α(15-30)

Where α is the ratio of post-yield stiffness toeffective elastic stiffness (See Figure 15-18). C3

need not exceed values calculated for LSP (SeeSection 15.4.3.1.1).Sa = Response spectrum acceleration at theeffective fundamental period, Te and dampingratio for the building in the direction underconsideration.

For buildings with flexible diaphragms ateach floor level, a target displacement can becalculated for each line of vertical framing.

Equation 15-28 can be used to determine thistarget displacement using the effectivefundamental period of the line of verticalframing. The general procedures described forNSP are to be used for each line of verticalframing with masses assigned to themathematical model on the basis of tributaryarea.

For stiff diaphragms, which are neither rigidnor flexible, any rational procedure can be usedto determine target displacements. Anacceptable procedure is to multiply the targetdisplacement obtained from Equation 15-28 bythe ratio of the maximum displacements at anypoint on the roof to the displacements of thecenter of mass of the roof, both computed by aresponse spectrum analysis of a 3-D model ofthe building using a design response spectrum.The target displacement thus computed may notbe less than those obtained from Equation 15-28 assuming rigid diaphragms. No vertical lineof framing can have displacements less than thetarget displacement. The target displacementshould also be modified as per Section 15.4.2.2to account for system torsion.

Diaphragms are designed for forcescomputed in LSP (See Section 15.4.3.1.1) orLDP (See Section 15.4.3.2.4)


For deformation-controlled elements, themaximum deformation demand must be lessthan expected deformation capacity. Proceduresfor computing expected deformation capacityare specified in Chapters 5 to 8 of FEMA-273for various elements and materials.

For force-controlled elements, the maximumdesign forces must be less than the lower boundstrengths QCL. Procedures for computing thelower bound strengths are also specified inChapters 5 to 8 of FEMA-273 for variouselements and materials.

15.4.3.4 Nonlinear Dynamic ProcedureThe Nonlinear Dynamic Procedure (NDP)

uses a dynamic time history analysis of anonlinear mathematical model. The basis,


modeling approaches and acceptance criterionfor the NDP are similar to those of the NSP.With the NDP the design displacements are notestablished using a target displacement, butdetermined directly through the dynamic timehistory analysis. As the analysis can be verysensitive to characteristics of individual groundmotions, it is advisable to perform the analysiswith more than one ground motion. Groundmotions used for the analysis and the analysisprocedure should be similar to those used inLDP (See Section 15.4.3.2).

It should be noted that the volume of datagenerated in NDP is enormous and it is difficultto condense the data to useful performancebased design information. Sensitivity analysisto various parameters is also a prerequisite forNDP analysis. Thus, NDP must only be usedwith caution for very important, irregular andunusual structures.

15.4.4 Example

An example is provided of an analysis ofexisting building using NSP.

15.4.4.1 Building DescriptionThe example building is a reinforced

concrete structure located in California. Thebuilding was constructed circa 1962. Thestructure is irregular in plan, with a footprintsimilar to a compressed "H". The structure hasbeen divided into the East, West, and CentralWings, as illustrated in Figure 15-19.

The building is situated on a site that slopesto the west. The structure has a total of sevenlevels, plus two small penthouses. The slopingsite introduces significant complexities to thestructure. The upper five levels are essentiallyabove grade. The West Wing is a total of sevenlevels tall, two of which are partially below orbelow grade, depending on the slope of the site.The East Wing is five levels tall, with a partialbasement. A portion of the first level is belowgrade, due to the sloping site.

Vertical loads are resisted by one-wayconcrete slabs spanning to reinforced concretebeams and girders. Thicker slabs are used in

some areas, including a 17-inch thick"sonovoid" slab, a cast-in-place concrete slabwith voids. The sonovoid slabs are located atthe ground and first floor. The slabs, beams,and girders are supported by tied and spirallyreinforced concrete columns and concretebearing walls. The columns rest on spreadfootings, with continuous footings under theperimeter and interior walls.

There are some unusual features in thevertical load-carrying system. Along the northand south exterior walls and the Central Wing,vertical loads are carried by concrete columnsoutside the building envelope. At the secondlevel, columns are discontinuous and aresupported by transfer girders. At the First Floor,the Central Wing relies on massive concreteframes to resist vertical loads.

Figure 15-19. 3-D Linear Model of Example Building

The lateral force-resisting system of theexample building consists of the concrete floorand roof slabs, acting as rigid diaphragms andreinforced concrete shear walls. The majority ofthe shear walls are concentrated around theelevator shafts and stair wells, with additionalwalls internally and on the building exterior.There are numerous vertical discontinuities inthe interior shear walls, especially below thefirst floor. Most of the shear walls are in theEast and West Wings.

West Wing

East Wing

Central Wing

788 Chapter 15

15.4.4.2 Performance ObjectiveIn keeping with project requirements, the

linear as well as nonlinear analysis andrehabilitation design focused on the BasicSafety Objective. In the nonlinear staticanalysis, the building is pushed to the targetdisplacement for the BSE-1 and BSE-2 levelearthquakes.

15.4.4.3 Mathematical ModelingThe nonlinear analysis of the example

building was performed using NLPUSH, thenonlinear module to SAP2000. The concreteshear walls were modeled using columnelements. P-M interaction diagrams weregenerated for each column element. Thecolumn elements have stiffness in the strongaxis computed based on the stiffness of theactual wall. Weak axis stiffness was assumed tobe negligible. As NLPUSH requires theinteraction surface to be input for bothdirections of bending, the wall is assumed tohave the same moment capacity in bothdirections of strong axis bending. The gravityframes have been identified as secondaryelements, and representative frames have beenexplicitly modeled to monitor the demands onthe gravity load-carrying system. Thediaphragms have been assumed to be rigid.

Potential failures in shear and flexure areconsidered in the analytical model. The walland column elements have flexural hinges inputat the top and bottom of the element at adistance of 0.05 times the element length fromeach end. Shear hinges are input at mid-heightof the element. Because of numericalconvergence problems, the column and wallelements had to be split into three segmentswith one hinge per segment. The flexure hingesare assigned to the top and bottom segments,and the shear hinge to the central segment. Wallelements with flanges are uncoupled and treatedas separate walls, with the effective flangewidth assigned individually to the two walls.

Beams and coupling beams are modeled asframe elements with flexure or shear hingesdepending which is the governing mode offailure. Full height walls spanning between

walls or columns are connected by stiffunyielding elements.

Values for effective stiffness of thestructural elements for the initial analysis aretaken from Table 6-4 of FEMA 273. Thestiffness for walls is the cracked stiffness, witha flexural rigidity of 0.5EcIg. The columns areassumed to be in compression with a flexuralstiffness 0.7EcIg. The beams are non-prestressed and have an initial stiffness of0.5EcIg. The shear stiffness is included forcolumns, beams and walls as 0.4EcAw.

The mathematical model of the building wassubjected to monotonically increasing lateralforces until either the target displacement isreached or until the model became unstable.Because the building is not symmetric aboutany plane, the lateral loads were independentlyapplied in both positive and negative directions.

The relationship between the base shear andlateral force was established for displacementsranging between 0 and 150% of δt, where δt

corresponds to the target displacement for theBSE-1 earthquake. Two lateral load patternswere applied to the structure. The uniform loadpattern was applied using lateral loads that areproportional to the mass at each floor. Thedynamic load pattern was applied, using alateral load pattern similar to the story sheardistribution calculated by combining the modalresponses from a response spectrum analysiswith sufficient number of modes to capture90% of the mass. Foundation flexibility wasnot expected to be a significant factor in thenonlinear analysis of the building.

15.4.4.4 Target DisplacementThe mapped short period response

acceleration parameter, SS and the modifiedmapped response acceleration parameter at onesecond period, S1, for the given site are obtainedfrom the maps provided with FEMA 273. Thesemaps are the Probabilistic Earthquake GroundMotion maps for California/Nevada for the 0.2seconds and 1.0 second Spectral ResponseAcceleration (5% of Critical Damping) with10% probability of exceedence in 50 years. Thevalues obtained for the example site are:


Figure 15-20. 3-D Nonlinear Model of Example Building

Figure 15-21. Typical Force Deformation Curve forMembers Controlled by Flexure

Figure 15-22. Typical Force Deformation Curve forColumns Controlled by Shear

SS = 1.5g and S1 = 0.75g

These values adjusted for Site Class C fromTables 2-13 and 2-14 of FEMA-273 give thedesign short period spectral responseacceleration parameter, SXS and design spectralresponse acceleration parameter, SX1 as:

g975.00.175.0

g5.10.15.1

1 =×==×=

X

XS

S

S

The period T0 of the general responsespectrum curve at an effective damping of 5%is:

seconds 65.05.1

975.0

1

10 ===

BS

BST

XS

SX

Where BS and B1 are 1.0 from Table 2-15 ofFEMA-273.

790 Chapter 15

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00

Roof Displacement (inches)

Bas

e S

hea

r (k

ips)

0.6VY

V Y

Figure 15-23. Pushover Curve for the Positive East-WestDirection Loading (Uniform Pattern)

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0.00 1.00 2.00 3.00 4.00 5.00 6.00

Roof Displacement (inches)

Bas

e S

hea

r (k

ips)

0.6VY

VY

Figure 15-24. Pushover Curve for the Positive North-South Direction Loading (Uniform Pattern)

The period of the building is less than 0.65,thus the spectral acceleration, Sa for the sitefalls in the constant acceleration part of thespectrum, and is equal to 1.5g.

The target displacement is calculated using:

C0 = 1.3 from Table 15-10, as the lower level isvery stiff compared with the rest of thestructure.C2 = 1.0 from Table 15-9 for framing Type 2.C3 = 1.0 for positive post yield stiffnessassumed.W = 38,064 kipsTi = 0.65 secondsTe = 0.41 seconds in East-West direction = 0.46 seconds in North-South direction

For the East-West Direction for Vy = 7,200 lbsfrom Figure 15-23:

6.1

3.1

1

064,38/200,7

5.11

/ 0

=

×==CWV

SR

y

a

49.1

1.6

1

41.0

65.0)11.6(1

1)1(1 0

1

=

−+=

−+=

RT

TRC

e

This value is reduced to the maximum value ofC1 in Section 15.4.3.1.1, which is 1.28(interpolated for Te = 0.41 seconds). Thus:

inches 11.4 4

0.411.5111.281.3

4

2

2

2

2

3210

=

×××××=

=

g

gT

SCCCC eat

π

πδ

Similarly for the North-South direction:Vy = 6,400 lbs from Figure 15-24R = 6.86C1 = 1.33δt = 5.37 inches

Thus, using Equation 15-28, the targetdisplacements for the North-South and East-West directions was determined to be 4.11inches, and 5.37 inches respectively. Thepushover analysis has be continued for 1.5times the target displacements for collapseprevention

15.4.4.5 Analysis ResultsPushover analyses were performed for the

positive and negative North-South and East-West directions of the building. The pushovercurves were not able to achieve the targetdisplacement even for the Life Safety


acceptance criteria for BSE-1 in the East-Westand North-South directions.

The maximum displacement reached and thetype and number of hinges formed for thevarious pushover analyses performed wasrecovered. From the results of the pushoveranalyses, it was seen that the Modal pattern ismore detrimental to this building as morenumber of hinges were formed for a givendisplacement level compared to the Uniformpattern. This also goes to show that the lowerfloors of this building are relatively strongerthan the upper floors. However this building inits existing configuration was unable to achieveits target displacement. The building could onlybe pushed to a displacement of 2.8” in thenegative East-West direction and 4.34” in thenegative North-South direction.

The analyses also revealed a number ofcolumns supporting walls above to haverotations beyond collapse. Many of the wallsand beams also had plastic rotations beyond theLife Safety requirement at the targetdisplacement. Some of the columns in thecentral wing had shear failures under theuniform pattern for push in the East-Westdirection. Clearly, this building does not meetthe acceptance criteria of the basic safetyobjective, and therefore needs retrofit.

15.5 Conclusions

The principal advantage of PBSE is that thechoice of performance goals lies with the ownerwho can decide the acceptable damage state.The engineer can also convey to the owner abetter understanding of the expected damagestate. PBSE does not eliminate the risksassociated with uncertainties in groundmotions, material properties, element behavioror geotechnical properties. However, itprovides a new technique to removeunnecessary conservatism for some parametersand discover unidentified deficiencies forothers. If implemented correctly andcompetently, PBSE can produce a design that ismore reliable than traditional procedures.

One very useful characteristic of the ATC-40 and FEMA 273/274 documents is that theyprovide a step-by-step approach for PBSE.This is an important first step towards abuilding code implementations of performancebased design.

There are some weaknesses that need to beaddressed with additional research. Threebroad areas need work:1. A more reliable and conservative

methodology, which is widely accepted,needs to be developed for establishing theperformance point. More accurateequations need to be developed to find theeffective damping or equivalent ductilityused to reduce the design response spectrato levels consistent with observed structuralbehavior.

2. More sophisticated computer analysisprograms are needed which can dononlinear analysis of concrete/masonry/plywood shear walls, concrete and steeljoints, confined concrete sections, etc.There is also a need to reduce the data to afinite number of parameters than can beused for design.

3. The element capacities and deformationslimits for various performance levels arecurrently based on engineering judgment orrelatively small number of experiments.More experimental and theoretical work isneeded to establish reliable elementcapacities and deformation limits for givenperformance objectives.

REFERENCES

15-1 Applied Technology Council (1996), SeismicEvaluation and Retrofit of Concrete Buildings,ATC-40, Volume 1 and 2, Report No. SSC 96-01,Seismic Safety Commission, Redwood City, CA.

15-2 Federal Emergency Management Agency (1997),NEHRP Guidelines for the Seismic Rehabilitationof Buildings, FEMA-273, Washington, D.C.

15-3 Federal Emergency Management Agency (1997),NEHRP Commentary on the Guidelines for theSeismic Rehabilitation of Buildings, FEMA-274,Washington, D.C.

792 Chapter 15

15-4 King, S.A. and Rojhan, C. (1997), "ATC-38Database on the Performance of Buildings NearStrong-Motion Recordings," Proceedings ofNorthridge Earthquake Research Conference,CUREe, Los Angeles, August.

15-5 Crandell, J.H. (1997), "Statistical assessment ofResidential Construction Damage by theNorthridge Earthquake," Proceedings of NorthridgeEarthquake Research Conference, CUREe, LosAngeles, August.

15-6 Naeim, F. and Kelly, J.M. (1999), Design ofSeismic Isolated Structures – From Theory toPractice, John Wiley & Sons, New York.

15-7 Naeim, F. (1998), “Earthquake Ground Motionsand Performance Based Design”, PerformanceBased Seismic Engineering Invitational Workshop,Earthquake Engineering Research Institute, SanDiego, California.

15-8 Freeman, S.A., Nicoletti, J.P. and Tyrell, J.V.,1975, “Evaluation of Existing Buildings forSeismic Risk: A Case Study of Pudget SoundNaval Shipyard, Bremerton, Washington,”Proceedings of U.S. National Conference ofEarthquake Engineers, Berkeley, EarthquakeEngineering Research Institute.

15-9 Freeman, S.A., 1998, “Development and use ofCapacity Spectrum Method,” Paper No. 269,Proceedings of the 6th U.S. National Conference ofEarthquake Engineering, Seattle, Washington.

15-10 U.S. Army, 1986, Seismic Design Guidelines forEssential Buildings, Departments of the Army(TM5-809-10-1), Navy (NAVFAC P355.1), andthe Air Force (AFM88-3), Washington, DC.

15-11 Structural Engineers Association of California(SEAOC), 1995, Vision 2000: Performance-BasedSeismic Engineering of Buildings, Sacramento,California.

15-12 International Code Council, 2000, InternationalBuilding Code 2000.

15-13 International Conference of Building Officials,1997, Uniform Building Code, Whittier, CA.

15-14 Chopra, A.K. and Goel R.K., 1999, Capacity-Demand-Diagram Methods for Estimating SeismicDeformation of Inelastic Structures: SDF Systems,Pacific Earthquake Engineering Research Center,PEER-1999/02, University of California, Berkeley,California.

15-15 Chopra, A.K. and Goel R.K., 2000, “Capacity-Demand-Diagram Methods Based on InelasticDesign Spectrum,” Earthquake Spectra, Volume15, Number 4, EERI, Oakland, California.

15-16 Newmark, N.M., and Hall,W.J., 1982, EarthquakeSpectra and Design, Earthquake EngineeringResearch Institute, Berkeley, California.

15-17 Krawinkler, H. and Nassar, A.A., 1992, “SeismicDesign based on Ductilities and CumulativeDamage Demands and Capacities,” in Nonlinear

Seismic Analysis and Design of ReinforcedConcrete Buildings, P. Fajfar and J. Krawinkler,Editors., Elsevier Applied Science, New York.

15-18 Vidic, T., Fajfar, P. and Fischinger, M., 1994,“Consistent Inelastic Design Spectra: Strength andDisplacement,” Earthquake Engineering andStructural Dynamics 23(5).

15-19 Fajfar, P., 2000, “A Nonlinear Analysis Method forPerformance Based Seismic Design,” Accepted forPublication in Earthquake Spectra, EERI, Oakland,California.

15-20 Aschheim M., Black, E.F., 2000, “Yield PointSpectra for Seismic Design and Rehabilitation,”Earthquake Spectra, Volume 16, Number 2, EERI,Oakland, California.

15-21 Cormartin, C.D., Niewiarowski, Freeman, S.A. andTurner, F.M., 2000, “Seismic Evaluation andRetrofit of Concrete Buildings; A PracticalOverview of the ATC-40 Document,” EarthquakeSpectra, Volume 16, Number 1, EERI, Oakland,California.

15-22 Chai, W. and Guh, J., 1999, “Performance-BasedDesign of Concrete Shear Wall Buildings,”Proceedings of 1999 SEAOC Convention,Structural Engineers Association of California,Santa Barbara, California.