Seismic Design of Reinforced Concrete Structurescivilwares.free.fr/The Seismic Design...

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Chapter 10

Seismic Design of Reinforced Concrete Structures

Arnaldo T. Derecho, Ph.D.Consulting Strucutral Engineer, Mount Prospect, Illinois

M. Reza Kianoush, Ph.D.Professor, Ryerson Polytechnic University, Toronto, Ontario, Canada

Key words: Seismic, Reinforced Concrete, Earthquake, Design, Flexure, Shear, Torsion, Wall, Frame, Wall-Frame,Building, Hi-Rise, Demand, Capacity, Detailing, Code Provisions, IBC-2000, UBC-97, ACI-318

Abstract: This chapter covers various aspects of seismic design of reinforced concrete structures with an emphasis ondesign for regions of high seismicity. Because the requirement for greater ductility in earthquake-resistantbuildings represents the principal departure from the conventional design for gravity and wind loading, themajor part of the discussion in this chapter will be devoted to considerations associated with providingductility in members and structures. The discussion in this chapter will be confined to monolithically castreinforced-concrete buildings. The concepts of seismic demand and capacity are introduced and elaboratedon. Specific provisions for design of seismic resistant reinforced concrete members and systems arepresented in detail. Appropriate seismic detailing considerations are discussed. Finally, a numerical exampleis presented where these principles are applied. Provisions of ACI-318/95 and IBC-2000 codes are identifiedand commented on throughout the chapter.

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10. Seismic Design of Reinforced Concrete Structures 465

10.1 INTRODUCTION

10.1.1 The Basic Problem

The problem of designing earthquake-resistant reinforced concrete buildings, like thedesign of structures (whether of concrete, steel,or other material) for other loading conditions,is basically one of defining the anticipatedforces and/or deformations in a preliminarydesign and providing for these by properproportioning and detailing of members andtheir connections. Designing a structure to resistthe expected loading(s) is generally aimed atsatisfying established or prescribed safety andserviceability criteria. This is the generalapproach to engineering design. The processthus consists of determining the expecteddemands and providing the necessary capacityto meet these demands for a specific structure.Adjustments to the preliminary design maylikely be indicated on the basis of results of theanalysis-design-evaluation sequencecharacterizing the iterative process thateventually converges to the final design.Successful experience with similar structuresshould increase the efficiency of the designprocess.

In earthquake-resistant design, the problemis complicated somewhat by the greateruncertainty surrounding the estimation of theappropriate design loads as well as thecapacities of structural elements andconnections. However, informationaccumulated during the last three decades fromanalytical and experimental studies, as well asevaluations of structural behavior during recentearthquakes, has provided a strong basis fordealing with this particular problem in a morerational manner. As with other developingfields of knowledge, refinements in designapproach can be expected as more informationis accumulated on earthquakes and on theresponse of particular structural configurationsto earthquake-type loadings.

As in design for other loading conditions,attention in design is generally focused on thoseareas in a structure which analysis and

experience indicate are or will likely besubjected to the most severe demands. Specialemphasis is placed on those regions whosefailure can affect the integrity and stability of asignificant portion of the structure.

10.1.2 Design for Inertial Effects

Earthquake-resistant design of buildings isintended primarily to provide for the inertialeffects associated with the waves of distortionthat characterize dynamic response to groundshaking. These effects account for most of thedamage resulting from earthquakes. In a fewcases, significant damage has resulted fromconditions where inertial effects in the structurewere negligible. Examples of these latter casesoccurred in the excessive tilting of severalmultistory buildings in Niigata, Japan, duringthe earthquake of June 16, 1964, as a result ofthe liquefaction of the sand on which thebuildings were founded, and the loss of anumber of residences due to large landslides inthe Turnagain Heights area in Anchorage,Alaska, during the March 28, 1964 earthquake.Both of the above effects, which result fromground motions due to the passage of seismicwaves, are usually referred to as secondaryeffects. They are distinguished from so-calledprimary effects, which are due directly to thecausative process, such as faulting (or volcanicaction, in the case of earthquakes of volcanicorigin).

10.1.3 Estimates of Demand

Estimates of force and deformation demandsin critical regions of structures have been basedon dynamic analyses—first, of simple systems,and second, on inelastic analyses of morecomplex structural configurations. The latterapproach has allowed estimation of force anddeformation demands in local regions ofspecific structural models. Dynamic inelasticanalyses of models of representative structureshave been used to generate information on thevariation of demand with major structural aswell as ground-motion parameters. Such aneffort involves consideration of the practical

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range of values of the principal structuralparameters as well as the expected range ofvariation of the ground-motion parameters.Structural parameters include the structurefundamental period, principal member yieldlevels, and force—displacement characteristics;input motions of reasonable duration andvarying intensity and frequency characteristicsnormally have to be considered.

A major source of uncertainty in the processof estimating demands is the characterization ofthe design earthquake in terms of intensity,frequency characteristics, and duration of large-amplitude pulses. Estimates of the intensity ofground shaking that can be expected atparticular sites have generally been based onhistorical records. Variations in frequencycharacteristics and duration can be included inan analysis by considering an ensemble ofrepresentative input motions.

Useful information on demands has alsobeen obtained from tests on specimenssubjected to simulated earthquake motionsusing shaking tables and, the pseudo-dynamicmethod of testing. The latter method is acombination of the so-called quasi-static, orslowly reversed, loading test and the dynamicshaking-table test. In this method, the specimenis subjected to essentially statically appliedincrements of deformation at discrete points,the magnitudes of which are calculated on thebasis of predetermined earthquake input and themeasured stiffness and estimated damping ofthe structure. Each increment of load after theinitial increment is based on the measuredstiffness of the structure during its response tothe imposed loading of the precedingincrement.

10.1.4 Estimates of Capacity

Proportioning and detailing of criticalregions in earthquake-resistant structures havemainly been based on results of tests onlaboratory specimens tested by the quasi-staticmethod, i.e., under slowly reversed cycles ofloading. Data from shaking-table tests and frompseudo-dynamic tests have also contributed tothe general understanding of structural behavior

under earthquake-type loading. Design anddetailing practice, as it has evolved over the lasttwo or three decades, has also benefited fromobservations of the performance of structuressubjected to actual destructive earthquakes.

Earthquake-resistant design has tended to beviewed as a special field of study, not onlybecause many engineers do not have to beconcerned with it, but also because it involvesadditional requirements not normally dealt within designing for wind. Thus, while it isgenerally sufficient to provide adequatestiffness and strength in designing buildings forwind, in the case of earthquake-resistant design,a third basic requirement, that of ductility orinelastic deformation capacity, must beconsidered. This third requirement arisesbecause it is generally uneconomical to designmost buildings to respond elastically tomoderate-to-strong earthquakes. To survivesuch earthquakes, codes require that structurespossess adequate ductility to allow them todissipate most of the energy from the groundmotions through inelastic deformations.However, deformations in the seismic forceresisting system must be controlled to protectelements of the structure that are not part of thelateral force resisting system. The fact is thatmany elements of the structure that are notintended as a part of the lateral force resistingsystem and are not detailed for ductility willparticipate in the lateral force resistantmechanism and can become severely damagedas a result. In the case of wind, structures aregenerally expected to respond to the designwind within their “elastic” range of stresses.When wind loading governs the design (drift orstrength), the structure still should comply withthe appropriate seismic detailing requirements.This is required in order to provide a ductilesystem to resist earthquake forces. Figure 10-1attempts to depict the interrelationshipsbetween the various considerations involved inearthquake-resistant design.


Figure 10- 1. Components of and considerations inearthquake-resistant building design

10.1.5 The Need for a Good DesignConcept and Proper Detailing

Because of the appreciable forces anddeformations that can be expected in criticalregions of structures subjected to strong groundmotions and a basic uncertainty concerning theintensity and character of the ground motions ata particular site, a good design concept isessential at the start. A good design conceptimplies a structure with a configuration thatbehaves well under earthquake excitation anddesigned in a manner that allows it to respondto strong ground motions according to apredetermined pattern or sequence of yielding.The need to start with a sound structuralconfiguration that minimizes “incidental” andoften substantial increases in member forcesresulting from torsion due to asymmetry orforce concentrations associated withdiscontinuities cannot be overemphasized.Although this idea may not be met with favorby some architects, clear (mainly economic)benefits can be derived from structuralconfigurations emphasizing symmetry,regularity, and the avoidance of severediscontinuities in mass, geometry, stiffness, orstrength. A direct path for the lateral (inertial)forces from the superstructure to anappropriately designed foundation is verydesirable. On numerous occasions, failure totake account of the increase in forces anddeformations in certain elements due to torsionor discontinuities has led to severe structural

distress and even collapse. The provision ofrelative strengths in the various types ofelements making up a structure with the aim ofcontrolling the sequence of yielding in suchelements has been recognized as desirable fromthe standpoint of structural safety as well asminimizing post-earthquake repair work.

An important characteristic of a good designconcept and one intimately tied to the idea ofductility is structural redundancy. Sinceyielding at critically stressed regions andsubsequent redistribution of forces to lessstressed regions is central to the ductileperformance of a structure, good practicesuggests providing as much redundancy aspossible in a structure. In monolithically castreinforced concrete structures, redundancy isnormally achieved by continuity betweenmoment-resisting elements. In addition tocontinuity, redundancy or the provision ofmultiple load paths may also be accomplishedby using several types of lateral-load-resistingsystems in a building so that a “backup system”can absorb some of the load from a primarylateral-load-resisting system in the event of apartial loss of capacity in the latter.

Just as important as a good design conceptis the proper detailing of members and theirconnections to achieve the requisite strengthand ductility. Such detailing should aim atpreventing nonductile failures, such as thoseassociated with shear and with bond anchorage.In addition, a deliberate effort should be madeto securely tie all parts of a structure that areintended to act as a unit together. Becausedynamic response to strong earthquakes,characterized by repeated and reversed cyclesof large-amplitude deformations in criticalelements, tends to concentrate deformationdemands in highly stressed portions of yieldingmembers, the importance of proper detailing ofpotential hinging regions should command asmuch attention as the development of a gooddesign concept. As with most designs but moreso in design for earthquake resistance, wherethe relatively large repeated deformations tendto “seek and expose,” in a manner of speaking,weaknesses in a structure—the proper fieldimplementation of engineering drawings

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ultimately determines how well a structureperforms under the design loading.

Experience and observation have shown thatproperly designed, detailed, and constructedreinforced-concrete buildings can provide thenecessary strength, stiffness, and inelasticdeformation capacity to perform satisfactorilyunder severe earthquake loading.

10.1.6 Accent on Design for StrongEarthquakes

The focus in the following discussion willbe on the design of buildings for moderate-to-strong earthquake motions. These casescorrespond roughly to buildings located inseismic zones 2, 3 and 4 as defined in theUniform Building Code (UBC-97).(10-1) Byemphasizing design for strong ground motions,it is hoped that the reader will gain anappreciation of the special considerationsinvolved in this most important loading case.Adjustments for buildings located in regions oflesser seismic risk will generally involverelaxation of some of the requirementsassociated with highly seismic areas.

Because the requirement for greater ductilityin earthquake-resistant buildings represents theprincipal departure from the conventionaldesign for gravity and wind loading, the majorpart of the discussion in this chapter will bedevoted to considerations associated withproviding ductility in members and structures.

The discussion in this chapter will beconfined to monolithically cast reinforced-concrete buildings.

10.2 DUCTILITY INEARTHQUAKE-RESISTANT DESIGN

10.2.1 Design Objective

In general, the design of economicalearthquake resistant structures should aim atproviding the appropriate dynamic andstructural characteristics so that acceptable

levels of response result under the designearthquake. The magnitude of the maximumacceptable deformation will vary dependingupon the type of structure and/or its function.

In some structures, such as slender, free-standing towers or smokestacks or suspension-type buildings consisting of a centrally locatedcorewall from which floor slabs are suspendedby means of peripheral hangers, the stability ofthe structure is dependent on the stiffness andintegrity of the single major element making upthe structure. For such cases, significantyielding in the principal element cannot betolerated and the design has to be based on anessentially elastic response.

For most buildings, however, andparticularly those consisting of rigidlyconnected frame members and other multiplyredundant structures, economy is achieved byallowing yielding to take place in somecritically stressed elements under moderate-to-strong earthquakes. This means designing abuilding for force levels significantly lowerthan would be required to ensure a linearlyelastic response. Analysis and experience haveshown that structures having adequate structuralredundancy can be designed safely to withstandstrong ground motions even if yielding isallowed to take place in some elements. As aconsequence of allowing inelastic deformationsto take place under strong earthquakes instructures designed to such reduced forcelevels, an additional requirement has resultedand this is the need to insure that yieldingelements be capable of sustaining adequateinelastic deformations without significant lossof strength, i.e., they must possess sufficientductility. Thus, where the strength (or yieldlevel) of a structure is less than that whichwould insure a linearly elastic response,sufficient ductility has to be built in.

10.2.2 Ductility vs. Yield Level

As a general observation, it can be statedthat for a given earthquake intensity andstructure period, the ductility demand increasesas the strength or yield level of a structuredecreases. To illustrate this point, consider two


vertical cantilever walls having the same initialfundamental period. For the same mass andmass distribution, this would imply the samestiffness properties. This is shown in Figure 10-2, where idealized force-deformation curves forthe two structures are marked (1) and (2).Analyses(10-2, 10-3) have shown that the maximumlateral displacements of structures with thesame initial fundamental period and reasonableproperties are approximately the same whensubjected to the same input motion. Thisphenomenon is largely attributable to thereduction in local accelerations, and hencedisplacements, associated with reductions instiffness due to yielding in critically stressedportions of a structure. Since in a verticalcantilever the rotation at the base determines toa large extent the displacements of points abovethe base, the same observation concerningapproximate equality of maximum lateraldisplacements can be made with respect tomaximum rotations in the hinging region at thebases of the walls. This can be seen in Figure10-3, from Reference 10-3, which shows resultsof dynamic analysis of isolated structural wallshaving the same fundamental period (T1 = 1.4sec) but different yield levels My. The structureswere subjected to the first 10 sec of the east—west component of the 1940 El Centro recordwith intensity normalized to 1.5 times that ofthe north—south component of the same

record. It is seen in Figure 10-3a that, except forthe structure with a very low yield level (My =500,000 in.-kips), the maximum displacementsfor the different structures are about the same.The corresponding ductility demands,expressed as the ratio of the maximum hingerotations, θmax to the corresponding rotations atfirst yield, θy, are shown in Figure 10-3b. Theincrease in ductility demand with decreasingyield level is apparent in the figure.

Figure 10-2. Decrease in ductility ratio demand withincrease in yield level or strength of a structure.

Figure 10-3. Effect of yield level on ductility demand. Note approximately equal maximum displacements for structureswith reasonable yield levels. (From Ref. 10-3.)

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A plot showing the variation of rotationalductility demand at the base of an isolatedstructural wall with both the flexural yield leveland the initial fundamental period is shown inFigure 10-4.(10-4) The results shown in Figure10-4 were obtained from dynamic inelasticanalysis of models representing 20-storyisolated structural walls subjected to six inputmotions of 10-sec duration having differentfrequency characteristics and an intensitynormalized to 1.5 times that of the north—southcomponent of the 1940 El Centro record.Again, note the increase in ductility demandwith decreasing yield level; also the decrease inductility demand with increasing fundamentalperiod of the structure.

The above-noted relationship betweenstrength or yield level and ductility is the basisfor code provisions requiring greater strength(by specifying higher design lateral forces) formaterials or systems that are deemed to haveless available ductility.

10.2.3 Some Remarks about Ductility

One should note the distinction betweeninelastic deformation demand expressed as aductility ratio, µ (as it usually is) on one hand,and in terms of absolute rotation on the other.An observation made with respect to onequantity may not apply to the other. As anexample, Figure 10-5, from Reference 10-3,

Figure 10-4. Rotational ductility demand as a function of initial fundamental period and yield level of 20-story structuralwalls. (From Ref. 10-4.)


shows results of dynamic analysis of twoisolated structural walls having the same yieldlevel (My = 500,000 in.-kips) but differentstiffnesses, as reflected in the lower initialfundamental period T1 of the stiffer structure.Both structures were subjected to the E—Wcomponent of the 1940 El Centro record. Eventhough the maximum rotation for the flexiblestructure (with T1 = 2.0 sec) is 3.3 times thatof the stiff structure, the ductility ratio for thestiff structure is 1.5 times that of the flexiblestructure. The latter result is, of course, partlydue to the lower yield rotation of the stifferstructure.

Figure 10-5. Rotational ductility ratio versus maximumabsolute rotation as measures of inelastic deformation.

The term “curvature ductility” is also acommonly used term which is defined as

rotation per unit length. This is discussed indetail later in this Chapter.

Another important distinction worth notingwith respect to ductility is the differencebetween displacement ductility and rotationalductility. The term displacement ductility refersto the ratio of the maximum horizontal (ortransverse) displacement of a structure to thecorresponding displacement at first yield. In arigid frame or even a single cantilever structureresponding inelastically to earthquakeexcitation, the lateral displacement of thestructure is achieved by flexural yielding atlocal critically stressed regions. Because of this,it is reasonable to expect—and results ofanalyses bear this out(10-2, 10-3, 10-5)—thatrotational ductilities at these critical regions aregenerally higher than the associateddisplacement ductility. Thus, overalldisplacement ductility ratios of 3 to 6 mayimply local rotational ductility demands of 6 to12 or more in the critically stressed regions of astructure.

10.2.4 Results of a Recent Study onCantilever Walls

In a recent study by Priestley and Kowalsky(10-6) on isolated cantilever walls, it has beenshown that the yield curvature is not directlyproportional to the yield moment; this is incontrast to that shown in Figure 10-2 which intheir opinions leads to significant errors. In fact,they have shown that yield curvature is afunction of the wall length alone, for a givensteel yield stress as indicated in Figure 10-6.The strength and stiffness of the wall varyproportionally as the strength of the section ischanged by varying the amount of flexuralreinforcement and/or the level of axial load.This implies that the yield curvature, not thesection stiffness, should be considered thefundamental section property. Since wall yieldcurvature is inversely proportional to walllength, structures containing walls of differentlength cannot be designed such that they yieldsimultaneously. In addition, it is stated that walldesign should be proportioned to the square of

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wall length, L2, rather than the current designassumption, which is based on L3 .

It should be noted that the above findingsapply to cantilever walls only. Further researchin this area in various aspects is currentlyunderway at several institutions.

M 1

M 2

M

M 3

y

Figure 10-6. Influence of strength on moment-curvaturerelationship (From Ref. 10-6).

10.3 BEHAVIOR OFCONCRETE MEMBERSUNDER EARTHQUAKE-TYPE LOADING

10.3.1 General Objectives of MemberDesign

A general objective in the design ofreinforced concrete members is to so proportionsuch elements that they not only possessadequate stiffness and strength but so that thestrength is, to the extent possible, governed byflexure rather than by shear or bond/anchorage.Code design requirements are framed with theintent of allowing members to develop theirflexural or axial load capacity before shear orbond/anchorage failure occurs. This desirablefeature in conventional reinforced concretedesign becomes imperative in design forearthquake motions where significant ductilityis required.

In certain members, such as conventionallyreinforced short walls—with height-to-widthratios of 2 to 3 or less—the very nature of theprincipal resisting mechanism would make ashear-type failure difficult to avoid. Diagonalreinforcement, in conjunction with horizontaland vertical reinforcement, has been shown toimprove the performance of such members (10-7).

10.3.2 Types of Loading Used inExperiments

The bulk of information on behavior ofreinforced-concrete members under load has‘generally been obtained from tests of full-sizeor near-full-size specimens. The loadings usedin these tests fall under four broad categories,namely:

1. Static monotonic loading—where load inone direction only is applied in increments untilfailure or excessive deformation occurs. Datawhich form the basis for the design ofreinforced concrete members under gravity andwind loading have been obtained mainly fromthis type of test. Results of this test can serve asbases for comparison with results obtained fromother types of test that are more representativeof earthquake loading.

2. Slowly reversed cyclic (“quasistatic”)loading—where the specimen is subjected to(force or deformation) loading cycles ofpredetermined amplitude. In most cases, theload amplitude is progressively increased untilfailure occurs. This is shown schematically inFigure 10-7a. As mentioned earlier, much of thedata upon which current design procedures forearthquake resistance are based have beenobtained from tests of this type. In a few cases,a loading program patterned after analyticallydetermined dynamic response(10-8) has beenused. The latter, which is depicted in Figure 10-7b, is usually characterized by large-amplitudeload cycles early in the test, which can produceearly deterioration of the strength of aspecimen.(10-9) In both of the above cases, theload application points are fixed so that themoments and shears are always in phase—acondition, incidentally, that does not alwaysoccur in dynamic response.


This type of test provides the reversingcharacter of the loading that distinguishesdynamic response from response tounidirectional static loading. In addition, therelatively slow application of the load allowsclose observation of the specimen as the testprogresses. However, questions concerning theeffects of the sequence of loading as well as thephase relationship between moment and shearassociated with this type of test as it is normallyconducted need to be explored further.

3. Pseudo-dynamic tests. In this type of test,the specimen base is fixed to the test floor whiletime-varying displacements determined by anon-line computer are applied to selected pointson the structure. By coupling loading rams witha computer that carries out an incrementaldynamic analysis of the specimen response to apreselected input motion, using measuredstiffness data from the preceding loadingincrement and prescribed data on specimenmass and damping, a more realistic distributionof horizontal displacements in the test structureis achieved. The relatively slow rate at whichthe loading is imposed allows convenientinspection of the condition of the structureduring the progress of the test.

This type of test, which has been usedmainly for testing structures, rather thanmembers or structural elements, requires afairly large reaction block to take the thrustfrom the many loading rams normally used.

4. Dynamic tests using shaking tables(earthquake simulators). The most realistic testconditions are achieved in this setup, where aspecimen is subjected to a properly scaled inputmotion while fastened to a test bed impelled bycomputer-controlled actuators. Most currentearthquake simulators are capable of impartingcontrolled motions in one horizontal directionand in the vertical direction.

The relatively rapid rate at which theloading is imposed in a typical dynamic testgenerally does not allow close inspection of thespecimen while the test is in progress, althoughphotographic records can be viewed after thetest. Most currently available earthquakesimulators are limited in their capacity to small-scale models of multistory structures or near-full-scale models of segments of a structure oftwo or three stories. The difficulty of viewingthe progress of damage in a specimen as theloading is applied and the limited capacity ofavailable (and costly) earthquake simulators hastended to favor the recently developed pseudo-dynamic test as a basic research tool for testingstructural systems.

The effect of progressively increasing lateraldisplacements on actual structures has beenstudied in a few isolated cases by means offorced-vibration testing. These tests haveusually been carried out on buildings orportions of buildings intended for demolition.

Figure 10-7. Two types of loading program used in quasi-static tests.

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10.3.3 Effects of Different Variables onthe Ductility of ReinforcedConcrete Members

Figure 10-8 shows typical stress—straincurves of concrete having different compressivestrengths. The steeper downward slope beyondthe point of maximum stress of curvescorresponding to the higher strength concrete isworth noting. The greater ductility of the lower-strength concrete is apparent in the figure.Typical stress-strain curves for the commonlyavailable grades of reinforcing steel, withnominal yield strengths of 60 ksi and 40 ksi, areshown in Figure 10-9. Note in the figure thatthe ultimate stress is significantly higher thanthe yield stress. Since strains well into thestrain-hardening range can occur in hingingregions of flexural members, stresses in excessof the nominal yield stress (normally used inconventional design as the limiting stress insteel) can develop in the reinforcement at theselocations.

Figure 10-8. Typical stress-strain curves for concrete ofvarying compressive strengths.

Rate of Loading An increase in the strainrate of loading is generally accompanied by anincrease in the strength of concrete or the yieldstress of steel. The greater rate of loadingassociated with earthquake response, ascompared with static loading, results in a slightincrease in the strength of reinforced concretemembers, due primarily to the increase in the

yield strength of the reinforcement. Thecalculation of the strength of reinforcedconcrete members in earthquake-resistantstructures on the basis of material propertiesobtained by static tests (i.e., normal strain ratesof loading) is thus reasonable and conservative.

Figure 10-9. Typical stress-strain curves for ordinaryreinforcing steel.

Confinement Reinforcement The AmericanConcrete Institute Building Code Requirementsfor Reinforced Concrete, ACI 318-95(10-10)

(hereafter referred to as the ACI Code),specifies a maximum usable compressive strainin concrete, εcu of 0.003. Lateral confinement,whether from active forces such as transversecompressive loads, or passive restraints fromother framing members or lateralreinforcement, tends to increase the value of εcu.Tests have shown that εcu, can range from0.0025 for unconfined concrete to about 0.01for concrete confined by lateral reinforcementsubjected to predominantly axial (concentric)load. Under eccentric loading, values of εcu forconfined concrete of 0.05 and more have beenobserved.(10-11, 10-12,10-13)

Effective lateral confinement of concreteincreases its compressive strength anddeformation capacity in the longitudinaldirection, whether such longitudinal stressrepresents a purely axial load or thecompressive component of a bending couple.


In reinforced concrete members, theconfinement commonly takes the form oflateral ties or spiral reinforcement covered by athin shell of concrete. The passive confiningeffect of the lateral reinforcement is notmobilized until the concrete undergoessufficient lateral expansion under the action ofcompressive forces in the longitudinaldirection. At this stage, the outer shell ofconcrete usually has reached its useful loadlimit and starts to spall. Because of this, the netincrease in strength of the section due to theconfined core may not amount to much in viewof the loss in capacity of the spalled concretecover. In many cases, the total strength of theconfined core may be slightly less than that ofthe original section. The increase in ductilitydue to effective confining reinforcement,however, is significant.

The confining action of rectangular hoopsmainly involves reactive forces at the corners,with only minor restraint provided along thestraight unsupported sides. Because of this,rectangular hoops are generally not as effectiveas circular spiral reinforcement in confining theconcrete core of members subjected tocompressive loads. However, confinement inrectangular sections can be improved usingadditional transverse ties. Square spirals,because of their continuity, are slightly better

than separate rectangular hoops.The stress—strain characteristics of

concrete, as represented by the maximumusable compressive strain εcu is important indesigning for ductility of reinforced concretemembers. However, other factors also influencethe ductility of a section: factors which mayincrease or diminish the effect of confinementon the ductility of concrete. Note the distinctionbetween the ductility of concrete as affected byconfinement and the ductility of a reinforcedconcrete section (i.e., sectional ductility) asinfluenced by the ductility of the concrete aswell as other factors.

Sectional Ductility A convenient measure ofthe ductility of a section subjected to flexure orcombined flexure and axial load is the ratio µ ofthe ultimate curvature attainable withoutsignificant loss of strength, φu , to the curvaturecorresponding to first yield of the tensionreinforcement, φy. Thus

Sectional ductility, y

u

φφµ =

Figure 10-10, which shows the strains andresultant forces on a typical reinforced concretesection under flexure, corresponds to thecondition when the maximum usablecompressive strain in concrete, εcu is reached.The corresponding curvature is denoted as the

Figure 10-10. Strains and stresses in a typical reinforced concrete section under flexure at ultimate condition.

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ultimate curvature, φu.. It will be seen in thefigure that

dku

cuu

εφ =

where kud is the distance from the extremecompression fiber to the neutral axis.

The variables affecting sectional ductilitymay be classified under three groups, namely:(i) material variables, such as the maximumusable compressive strain in concrete,particularly as this is affected by confinement,and grade of reinforcement; (ii) geometricvariables, such as the amount of tension andcompression reinforcement, and the shape ofthe section; (iii) loading variables, such as thelevel of the axial load and accompanying shear.

As is apparent from the above expressionfor ultimate curvature, factors that tend toincrease εcu or decrease kud tend to increasesectional ductility. As mentioned earlier, amajor factor affecting the value of εcu is lateralconfinement. Tests have also indicated that εcu

increases as the distance to the neutral axisdecreases, that is, as the strain gradient acrossthe section increases(10-14, 10-15) and as themoment gradient along the span of the memberincreases or as the shear span decreases.(10-16, 10-

17) (For a given maximum moment, the momentgradient increases as the distance from the pointof zero moment to the section considereddecreases.)

The presence of compressive reinforcementand the use of concrete with a high compressivestrength,a as well as the use of flanged sections,tend to reduce the required depth of thecompressive block, kud, and hence to increasethe ultimate curvature φu. In addition, thecompressive reinforcement also helps confinethe concrete compression zone and, incombination with adequate transversereinforcement, allows the spread of the inelasticaction in a hinging region over a longer lengththan would otherwise occur, thus improving the

a The lower ductility of the higher-strength (f′c >5000 psi ),however, has been shown to result in a decrease insectional ductility, particularly for sections with lowreinforcement indexes. (10-18)

ductility of the member.(10-19) On the other hand,compressive axial loads and large amounts oftensile reinforcement, especially tensilereinforcement with a high yield stress, tend toincrease the required kud and thus decrease theultimate curvature φu.

Figure 10-11 shows axial-load—moment-strength interaction curves for a reinforced-concrete section subjected to a compressiveaxial load and bending about the horizontalaxis. Both confined and unconfined conditionsare assumed. The interaction curve provides aconvenient way of displaying the combinationsof bending moment M and axial load P which agiven section can carry. A point on theinteraction curve is obtained by calculating theforces M and P associated with an assumedlinear strain distribution across the section,account being taken of the appropriate stress—strain relationships for concrete and steel. Foran ultimate load curve, the concrete strain at theextreme compressive fiber, εc is assumed to beat the maximum usable strain, εcu while thestrain in the tensile reinforcement, εs, varies. Aloading combination represented by a point onor inside the interaction curve can be safelyresisted by the section. The balance point in theinteraction curve corresponds to the conditionin which the tensile reinforcement is stressed toits yield point at the same time that the extremeconcrete fiber reaches its useful limit ofcompressive strain. Points on the interactioncurve above the balance point representconditions in which the strain in the tensilereinforcement is less than its yield strain εy, sothat the strength of the section in this range isgoverned by failure of the concrete compressivezone. For those points on the curve below thebalance point, εs > εy. Hence, the strength of thesection in this range is governed by rupture ofthe tensile reinforcement.

Figure 10-11 also shows the variation of theultimate curvature φu (in units of 1/h) with theaxial load P. It is important to note the greaterultimate curvature (being a measure of sectionalductility) associated with values of P less thanthat corresponding to the balance condition, forboth unconfined and confined cases. Thesignificant increase in ultimate curvature


resulting from confinement is also worth notingin Figure 10-11b.

In the preceding, the flexural deformationcapacity of the hinging region in members wasexamined in terms of the curvature at a section,φ, and hence the sectional or curvature ductility.Using this simple model, it was possible toarrive at important conclusions concerning theeffects of various parameters on the ductility ofreinforced concrete members. In the hingingregion of members, however, the curvature canvary widely in value over the length of the“plastic hinge.” Because of this, the totalrotation over the plastic hinge, θ, provides amore meaningful measure of the inelasticflexural deformation in the hinging regions ofmembers and one that can be related directly toexperimental measurements. (One can, ofcourse, speak of average curvature over thehinging region, i.e., total rotation divided bylength of the plastic hinge.)

Shear The level of shear present can have amajor effect on the ductility of flexural hingingregions. To study the effect of this variable,controlled tests of laboratory specimens havebeen conducted. This will be discussed furtherin the following section.

10.3.4 Some Results of Experimental andAnalytical Studies on the Behaviorof Reinforced Concrete Membersunder Earthquake-Type Loadingand Related Code Provisions

Experimental studies of the behavior ofstructural elements under earthquake-typeloading have been concerned mainly withidentifying and/or quantifying the effects ofvariables that influence the ability of criticallystressed regions in such specimens to performproperly. Proper performance means primarilypossessing adequate ductility. In terms of the

Figure 10-11. Axial load-moment interaction and load-curvature curves for a typical reinforced concrete section withunconfined and confined cores.

478 Chapter 10

quasistatic test that has been the most widelyused for this purpose, proper performancewould logically require that these criticalregions be capable of sustaining a minimumnumber of deformation cycles of specifiedamplitude without significant loss of strength.

In the United States, there is at present nostandard set of performance requirementscorresponding to designated areas of seismicrisk that can be used in connection with thequasi-static test. Such requirements would haveto specify not only the minimum amplitude(i.e., ductility ratio) and number of deformationcycles, but also the sequence of application ofthe large-amplitude cycles in relation to anysmall-amplitude cycles and the permissiblereduction in strength at the end of the loading.

As mentioned earlier, the bulk ofexperimental information on the behavior ofelements under earthquake-type loading hasbeen obtained by quasi-static tests usingloading cycles of progressively increasingamplitude, such as is shown schematically inFigure 10-7a. Adequacy with respect toductility for regions of high seismicity hasusually been inferred when displacementductility ratios of anywhere from 4 to 6 orgreater were achieved without appreciable lossof strength. In New Zealand,(10-20) momentresisting frames are designed for a maximumductility, µ, of 6 and shear walls are designedfor a maximum ductility of between 2.5 to 5.Adequate ductile capacity is considered to bepresent if all primary that are required to resistearthquake-induced forces are accordinglydesigned and detailed.

In the following, some results of tests andanalyses of typical reinforced-concretemembers will be briefly reviewed. Whereappropriate, related code provisions, mainlythose in Chapter 21 of the ACI Code(10-10) arealso discussed.

Beams Under earthquake loading, beamswill generally be most critically stressed at andnear their intersections with the supportingcolumns. An exception may be where a heavyconcentrated load is carried at someintermediate point on the span. As a result, thefocus of attention in the design of beams is on

these critical regions where plastic hinging cantake place.

At potential hinging regions, the need todevelop and maintain the strength and ductilityof the member through a number of cycles ofreversed inelastic deformation calls for specialattention in design. This special attention relatesmainly to the lateral reinforcement, which takesthe form of closed hoops or spirals. As might beexpected, the requirements governing thedesign of lateral reinforcement for potentialhinging regions are more stringent than thosefor members designed for gravity and windloads, or the less critically stressed parts ofmembers in earthquake-resistant structures. Thelateral reinforcement in hinging regions ofbeams is designed to provide (i) confinement ofthe concrete core, (ii) support for thelongitudinal compressive reinforcement againstinelastic buckling, and (iii) resistance, inconjunction with the confined concrete, againsttransverse shear.

In addition to confirming the results ofsectional analyses regarding the influence ofsuch variables as concrete strength,confinement of concrete, and amounts and yieldstrengths of tensile and compressivereinforcement and compression flangesmentioned earlier, tests, both monotonic andreversed cyclic, have shown that the flexuralductility of hinging regions in beams issignificantly affected by the level of shearpresent. A review of test results by Bertero(10-21)

indicates that when the nominal shear stress

exceeds about cf ′3 , members designed

according to the present seismic codes canexpect to suffer some reduction in ductility aswell as stiffness when subjected to loadingassociated with strong earthquake response.When the shear accompanying flexural hinging

is of the order of cf ′5 or higher, very

significant strength and stiffness degradationhas been observed to occur under cyclicreversed loading.

The behavior of a segment at the supportregion of a typical reinforced-concrete beamsubjected to reversed cycles of inelasticdeformation in the presence of high shear(10-22,


10-23) is shown schematically in Figure 10-12. InFigure 10-12a, yielding of the top longitudinalsteel under a downward movement of the beamend causes flexure—shear cracks to form at thetop. A reversal of the load and subsequentyielding of the bottom longitudinal steel is alsoaccompanied by cracking at the bottom of thebeam (see Figure 10-l2b). If the area of thebottom steel is at least equal to that of the topsteel, the top cracks remain open during theearly stages of the load reversal until the topsteel yields in compression, allowing the topcrack to close and the concrete to carry somecompression. Otherwise, as in the more typicalcase where the top steel has greater area thanthe bottom steel, the top steel does not yield incompression (and we assume it does notbuckle), so that the top crack remains openduring the reversal of the load (directedupward). Even in the former case, completeclosure of the crack at the top may be preventedby loose particles of concrete that may fall intothe open cracks. With a crack traversing theentire depth of the beam, the resisting flexuralcouple consists of the forces in the tensile andcompressive steel areas, while the shear alongthe through-depth crack is resisted primarily bydowel action of the longitudinal steel. Withsubsequent reversals of the load andprogressive deterioration of the concrete in thehinging region (Figure 10-12c), the through-depth crack widens. The resulting increase intotal length of the member due to the openingof through-depth cracks under repeated loadreversals is sometimes referred to as growth ofthe member.

Where the shear accompanying the momentis high, sliding along the through-depth crack(s)can occur. This sliding shear displacement,which is resisted mainly by dowel action of thelongitudinal reinforcement, is reflected in apinching of the associated load—deflectioncurve near the origin, as indicated in Figure 10-13. Since the area under the load—deflectioncurve is a measure of the energy-dissipationcapacity of the member, the pinching in thiscurve due to sliding shear represents adegradation not only of the strength but also theenergy-dissipation capacity of the hinging

region. Where the longitudinal steel is notadequately restrained by lateral reinforcement,inelastic buckling of the compressivereinforcement followed by a rapid loss offlexural strength can occur.

Figure 10-12. Plastic hinging in beam under high shear.(Adapted from Ref. 10-31.)

Figure 10-13. Pinching in load-displacement hysteresisloop due to mainly to sliding shear

Because of the significant effect that shearcan have on the ductility of hinging regions, ithas been suggested(10-24) that when two or moreload reversals at a displacement ductility of 4 ormore are expected, the nominal shear stress incritical regions reinforced according to normal

480 Chapter 10

U.S. code requirements for earthquake-resistant

design should be limited to 6 cf ′ . Results of

tests reported in Reference 10-24 have shownthat the use of crossing diagonal or inclinedweb reinforcement, in combination withvertical ties, as shown in Figure 10-14, caneffectively minimize the degradation ofstiffness associated with sliding shear.Relatively stable hysteretic force—displacement loops, with minimal or nopinching, were observed. Tests reported inReference 10-25 also indicate the effectivenessof intermediate longitudinal shearreinforcement, shown in Figure 10-15, inreducing pinching of the force—displacementloops of specimens subjected to moderate levels

of shear stresses, i.e., between 3 cf ′ and

6 cf ′ .

Figure 10-14. Crossing diagonal web reinforcement incombination with vertical web steel for hinging regionsunder high shear. (Adapted from Ref. 10-24)

As mentioned earlier, a major objective inthe design of reinforced concrete members is tohave the strength controlled by flexure ratherthan shear or other less ductile failuremechanisms. To insure that beams develop theirfull strength in flexure before failing in shear,ACI Chapter 21 requires that the design forshear in beams be based not on the factoredshears obtained from a lateral-load analysis butrather on the shears corresponding to themaximum probable flexural strength, Mpr, thatcan be developed at the beam ends. Such aprobable flexural strength is calculated byassuming the stress in the tensile reinforcement

to be equal to 1.25fy and using a strengthreduction factor φ equal to 1.0 (instead of 0.9).This is illustrated in Figure 10-16 for the caseof uniformly distributed beam. The use of thefactor 1.25 to be applied to fy is intended to takeaccount of the likelihood of the actual yieldstress in the steel being greater (tests indicate itto be commonly 10 to 25% greater) than thespecified nominal yield stress, and also inrecognition of the strong possibility of strainhardening developing in the reinforcementwhen plastic hinging occurs at the beam ends.

Figure 10-15. Intermediate longitudinal webreinforcement for hinging regions under moderate levelsof shear.

0.1and25.1onbased2

2

==

−+

=

++

=

φyspr

u

Bpr

AprB

c

u

Bpr

AprA

c

ffM

lW

l

MMV

lW

l

MMV

Figure 10-16. Loading cases for shear design of beamsuniformly distributed gravity loads


ACI Chapter 21 requires that when theearthquake-induced shear force calculated onthe basis of the maximum probable flexuralstrength at the beam ends is equal to or morethan one-half the total design shear, thecontribution of the concrete in resisting shear,Vc, be neglected if the factored axialcompressive force including earthquake effectsis less than Ag cf ′ /20, where Ag is the gross area

of the member cross-section. In the 1995 NewZealand Code,(10-26) the concrete contribution isto be entirely neglected and web reinforcementprovided to carry the total shear force in plastic-hinging regions. It should be pointed out thatthe New Zealand seismic design code appearsto be generally more conservative thancomparable U.S. codes. This will be discussedfurther in subsequent sections.

Columns The current approach to the designof earthquake-resistant reinforced concrete rigid(i.e., moment-resisting) frames is to have mostof the significant inelastic action or plastichinging occur in the beams rather than in thecolumns. This is referred to as the “strongcolumn-weak beam” concept and is intended tohelp insure the stability of the frame whileundergoing large lateral displacements underearthquake excitation. Plastic hinging at bothends of most of the columns in a story canprecipitate a story-sidesway mechanism leadingto collapse of the structure at and above thestory.

ACI Chapter 21 requires that the sum of theflexural strengths of the columns meeting at ajoint, under the most unfavorable axial load, beat least equal to 1.2 times the sum of the designflexural strengths of the girders in the sameplane framing into the joint. The mostunfavorable axial load is the factored axialforce resulting in the lowest correspondingflexural strength in the column and which isconsistent with the direction of the lateral forcesconsidered. Where this requirement is satisfied,closely spaced transverse reinforcement need beprovided only over a short distance near theends of the columns where potential hingingcan occur. Otherwise, closely spaced transversereinforcement is required over the full height ofthe columns.

The requirements associated with the strongcolumn-weak beam concept, however, do notinsure that plastic hinging will not occur in thecolumns. As pointed out in Reference 10-5, abending-moment distribution among framemembers such as is shown in Figure 10-17,characterized by points of inflection locatedaway from the mid-height of columns, is notuncommon. This condition, which has beenobserved even under static lateral loading,occurs when the flexural mode of deformation(as contrasted with the shear—beam componentof deformation) in tall frame structuresbecomes significant and may also arise as aresult of higher-mode response under dynamicloading. As Figure 10-17 shows, a majorportion of the girder moments at a joint isresisted (assuming the columns remain elastic)by one column segment, rather than beingshared about equally (as when the points ofinflection are located at mid-height of thecolumns) by the column sections above andbelow a joint. In extreme cases, such as mightresult from substantial differences in thestiffnesses of adjoining column segments in acolumn stack, the point of contraflexure can beoutside the column height. In such cases, themoment resisted by a column segment mayexceed the sum of the girder moments. Inrecognition of this, and the likelihood of thehinging region spreading over a longer lengththan would normally occur, most buildingcodes require confinement reinforcement to beprovided over the full height of the column.

Tests on beam-column specimensincorporating slabs,(10-27, 10-28) as in normalmonolithic construction, have shown that slabssignificantly increase the effective flexuralstrength of the beams and hence reduce thecolumn-to-beam flexural strength ratio, if thebeam strength is based on the bare beamsection. Reference 10-27 recommendsconsideration of the slab reinforcement over awidth equal to at least the width of the beam oneach side of the member when calculating theflexural strength of the beam.

482 Chapter 10

Figure 10-17. Distribution of bending moments incolumns at a joint when the point of inflection is locatedaway from mid-height.

Another phenomenon that may lead toplastic hinging in the columns occurs in two-way (three-dimensional rigid) frames subjectedto ground motions along a direction inclinedwith respect to the principal axes of thestructure. In such cases, the resultant momentfrom girders lying in perpendicular planesframing into a column will generally be greaterthan that corresponding to either girderconsidered separately.(10-5) ( except for certaincategories of structures and those with certainirregularities, codes allow consideration ofdesign earthquake loads along each principalaxes of a structure separately, as non-concurrentloadings.) Furthermore, the biaxial momentcapacity of a reinforced-concrete column underskew bending will generally be less than thelarger uniaxial moment capacity. Tests reportedin Reference 10-28 indicate that where bi-directional loading occurs in rectangularcolumns, the decrease in strength of the columndue to spalling of concrete cover, and bonddeterioration along the column longitudinal barsat and near the corner can be large enough toshift the hinging from the beams to thecolumns. Thus, under concurrent bi-directionalloading, columns in two-way frames designedaccording to the strong column-weak beam

concept mentioned above can either yieldbefore the framing girders or start yieldingimmediately following yielding of the girders.

It is worth noting that the 1985 report ofACI-ASCE Committee 352 on beam-columnjoints in monolithic reinforced concretestructures(10-29) recommends a minimumoverstrength factor of 1.4, instead of the 1.2given in ACI 318-95, for the flexural strengthof columns relative to that of beams meeting ata joint when the beam strength is based only onthe bare beam section (excluding slab). Adesign procedure (capacity design), based onthe work of Paulay,(10-13,10-30) that attempts tominimize the possibility of yielding in thecolumns of a typical frame due to the factorsdescribed in the preceding paragraph has beenadopted in New Zealand.(10-26) The avowedpurpose of capacity design is to limit inelasticaction, as well as the formation of plastichinges, to selected elements of the primarylateral-force-resisting system. In the case offrames, the ideal location for plastic hingeswould be the beams and the bases of the first orlowest story columns. Other elements, such ascolumns, are intended to remain essentiallyelastic under the design earthquake bydesigning them with sufficient overstrengthrelative to the yielding members. Thus elementsintended to remain elastic are designed to havestrengths in the plastic hinges. For all elements,and particularly regions designed to developplastic hinges, undesirable modes of failure,such as shear or bond/anchorage failures, areprecluded by proper design/detailing. Thegeneral philosophy of capacity design is nodifferent from that underlying the currentapproach to earthquake-resistant design foundin ACI Chapter 21, UBC-97 and IBC-2000. Theprinciple difference lies in the details ofimplementation and particularly in therecommended overstrength factors. Forexample, the procedure prescribes overstrengthfactors of 1.5 or greater(10-13,10-32) fordetermining the flexural strength of columnsrelative to beams. This compares with the 1.2factor specified in ACI Chapter 21. In capacitydesign, the flexural strength of T or inverted-Lbeams is to be determined by considering the


slab reinforcement over the specified width(depending upon column location) beyond thecolumn faces as effective in resisting negativemoments. It is clear from the above that theNew Zealand capacity design requirements callfor greater relative column strength than iscurrently required in U.S. practice. A similarapproach has also been adopted in the CanadianConcrete Code of Practice, CSA StandardA23.3-94.(10-33) Reference 10-13 gives detailedrecommendations, including worked outexamples, relating to the application of capacitydesign to both frames and structural wallsystems.

To safeguard against strength degradationdue to hinging in the columns of a frame, codesgenerally require lateral reinforcement for bothconfinement and shear in regions of potentialplastic hinging. As in potential hinging regionsof beams, the closely spaced transversereinforcement in critically stressed regions ofcolumns is intended to provide confinement forthe concrete core, lateral support of thelongitudinal column reinforcement againstbuckling and resistance (in conjunction with theconfined core) against transverse shear. Thetransverse reinforcement can take the form ofspirals, circular hoops, or rectangular hoops, thelast with crossties as needed.

Early tests(10-34) of reinforced concretecolumns subjected to large shear reversals hadindicated the need to provide adequatetransverse reinforcement not only to confine theconcrete but also to carry most, if not all, of theshear in the hinging regions of columns. Thebeneficial effect of axial load—a maximumaxial load of one-half the balance load was usedin the tests—in delaying the degradation ofshear strength in the hinging region was alsonoted in these tests. An increase in columnstrength due to improved confinement bylongitudinal reinforcement uniformlydistributed along the periphery of the columnsection was noted in tests reported in Reference10-35. Tests cited in Reference 10-32 haveindicated that under high axial load, the plastichinging region in columns with confinementreinforcement provided over the usuallyassumed hinging length (i.e., the longer section

dimension in rectangular columns or thediameter in circular columns) tends to spreadbeyond the confined region. To prevent flexuralfailure in the less heavily confined regions ofcolumns, the New Zealand Code(10-20) requiresthat confining steel be extended to 2 to 3 timesthe usual assumed plastic-hinge length whenthe axial load exceeds 0.25φ cf ′ Ag, where φ =

0.85 and Ag is the gross area of the columnsection.

The basic intent of the ACI Code provisionsrelating to confinement reinforcement inpotential hinging regions of columns is topreserve the axial-load-carrying capacity of thecolumn after spalling of the cover concrete hasoccurred. This is similar to the intentunderlying the column design provisions forgravity and wind loading. The amount ofconfinement reinforcement required by theseprovisions is independent of the level of axialload. Design for shear is to be based on thelargest nominal moment strengths at the columnends consistent with the factored design axialcompressive load. Some investigators,(10-5)

however, have suggested that an approach thatrecognizes the potential for hinging in criticallystressed regions of columns should aimprimarily at achieving a minimum ductility inthese regions. Studies by Park and associates,based on sectional analyses(10-32) as well astests,(10-36, 10-37) indicate that although the ACICode provisions based on maintaining the load-carrying capacity of a column after spalling ofthe cover concrete has occurred areconservative for low axial loads, they can beunconservative for high axial loads, withparticular regard to attaining adequate ductility.Results of these studies indicate the desirabilityof varying the confinement requirements for thehinging regions in columns according to themagnitude of the axial load, more confinementbeing called for in the case of high axial loads.

ACI Chapter 21 limits the spacing ofconfinement reinforcement to 1/4 the minimummember dimension or 4 in., with no limitationrelated to the longitudinal bar diameter. TheNew Zealand Code requires that the maximumspacing of transverse reinforcement in thepotential plastic hinge regions not exceed the

484 Chapter 10

least of 1/4 the minimum column dimension or6 times the diameter of the longitudinalreinforcement. The second limitation isintended to relate the maximum allowablespacing to the need to prevent prematurebuckling of the longitudinal reinforcement. Interms of shear reinforcement, ACI Chapter 21requires that the design shear force be based onthe maximum flexural strength, Mpr , at eachend of the column associated with the range offactored axial loads. However, at each columnend, the moments to be used in calculating thedesign shear will be limited by the probablemoment strengths of the beams (the negativemoment strength on one side and the positivemoment strength on the other side of a joint)framing into the column. The larger amount oftransverse reinforcement required for eitherconfinement or shear is to be used.

One should note the significant economy,particularly with respect to volume of lateralreinforcement, to be derived from the use ofspirally reinforced columns.(10-32) The saving inthe required amount of lateral reinforcement,relative to a tied column of the same nominalcapacity, which has also been observed indesigns for gravity and wind loading, acquiresgreater importance in earthquake-resistantdesign in view of the superior ductileperformance of the spirally reinforced column.Figure 10-18b, from Reference 10-38, showsone of the spirally reinforced columns in thefirst story of the Olive View Hospital buildingin California following the February 9, 1971San Fernando earthquake. A tied corner columnin the first story of the same building is shownin Figure 10-18c. The upper floors in the four-story building, which were stiffened by shearwalls that were discontinued below the second-floor level, shifted approximately 2 ft.horizontally relative to the base of the first-story columns, as indicated in Figure 10-18a.

Beam—Column Joints Beam-column jointsare critical elements in frame structures. Theseelements can be subjected to high shear andbond-slip deformations under earthquakeloading. Beam-column joints have to be

designed so that the connected elements canperform properly. This requires that the jointsbe proportioned and detailed to allow thecolumns and beams framing into them todevelop and maintain their strength as well asstiffness while undergoing large inelasticdeformations. A loss in strength or stiffness in aframe resulting from deterioration in the jointscan lead to a substantial increase in lateraldisplacements of the frame, including possibleinstability due to P-delta effects.

The design of beam-column joints isprimarily aimed at (i) preserving the integrity ofthe joint so that the strength and deformationcapacity of the connected beams and columnscan be developed and substantially maintained,and (ii) preventing significant degradation ofthe joint stiffness due to cracking of the jointand loss of bond between concrete and thelongitudinal column and beam reinforcement oranchorage failure of beam reinforcement. Ofmajor concern here is the disruption of the jointcore as a result of high shear reversals. As inthe hinging regions of beams and columns,measures aimed at insuring proper performanceof beam-column joints have focused onproviding adequate confinement as well asshear resistance to the joint.

The forces acting on a typical interior beam-column joint in a frame undergoing lateraldisplacement are shown in Figure 10-19a. It isworth noting in Figure 10-19a that each of thelongitudinal beam and column bars is subjectedto a pull on one side and a push on the otherside of the joint. This combination of forcestends to push the bars through the joint, acondition that leads to slippage of the bars andeven a complete pull through in some testspecimens. Slippage resulting from bonddegradation under repeated yielding of thebeam reinforcement is reflected in a reductionin the beam-end fixity and thus increased beamrotations at the column faces. This loss in beamstiffness can lead to increased lateraldisplacements of the frame and potentialinstability.


(a)

(b) (c)Figure 10-18. Damage to columns of the 4-story Olive View Hospital building during the February 9, 1971 San Fernando,California, earthquake. (From Ref. 10-38.) (a) A wing of the building showing approximately 2 ft drift in its first story. (b)Spirally reinforced concrete column in first story. (c) Tied rectangular corner column in first story.

486 Chapter 10

Figure 10-19. Forces and postulated shear-resistingmechanisms in a typical interior beam-column joint.(Adapted from Ref. 10-32.) (a) Forces acting on beam-column joint. (b) Diagonal strut mechanism. (c) Trussmechanism.

Two basic mechanisms have beenpostulated as contributing to the shearresistance of beam—column joints. These arethe diagonal strut and the joint truss (ordiagonal compression field) mechanisms,shown in Figure 10-19b and c, respectively.After several cycles of inelastic deformation inthe beams framing into a joint, the effectivenessof the diagonal strut mechanism tends todiminish as through-depth cracks start to open

between the faces of the column and theframing beams and as yielding in the beam barspenetrates into the joint core. The joint trussmechanism develops as a result of theinteraction between confining horizontal andvertical reinforcement and a diagonalcompression field acting on the elements of theconfined concrete core between diagonalcracks. Ideally, truss action to resist horizontaland vertical shears would require bothhorizontal confining steel and intermediatevertical column bars (between column cornerbars). Tests cited in Reference 10-39 indicatethat where no intermediate vertical bars areprovided, the performance of the joint is worsethan where such bars are provided.

Tests of beam-column joints(10-27,10-40,10-41) inwhich the framing beams were subjected tolarge inelastic displacement cycles haveindicated that the presence of transverse beams(perpendicular to the plane of the loadedbeams) considerably improves joint behavior.Results reported in Reference 10-27 show thatthe effect of an increase in joint lateralreinforcement becomes more pronounced in theabsence of transverse beams. However, thesame tests indicated that slippage of columnreinforcement through the joint occurred withor without transverse beams. The use ofsmaller-diameter longitudinal bars has beensuggested (10-39) as a means of minimizing barslippage. Another suggestion has been to forcethe plastic hinge in the beam to form away fromthe column face, thus preventing highlongitudinal steel strains from developing in theimmediate vicinity of the joint. This can beaccomplished by suitably strengthening thesegment of beam close to the column (usually adistance equal to the total depth of the beam)using appropriate details. Some of the detailsproposed include a combination of heavyvertical reinforcement with cross-ties (seeFigure 10-14), intermediate longitudinal shearreinforcement (see Figure 10-15),(10-42) andsupplementary flexural reinforcement andhaunches, as shown in Figure 10-20.(10-32)

The current approach to beam—columnjoint design in the United States, as contained inACI Chapter 21, is based on providing


sufficient horizontal joint cross-sectional areathat is adequately confined to resist the shearstresses in the joint. The approach is basedmainly on results of a study by Meinheit andJirsa(10-41) and subsequent studies by Jirsa andassociates. The parametric study reported inReference 10-41 identified the horizontal cross-sectional area of the joint as the mostsignificant variable affecting the shear strengthof beam—column connections. Althoughrecognizing the role of the diagonal strut andjoint truss mechanisms, the current approachdefines the shear strength of a joint simply interms of its horizontal cross-sectional area. Theapproach presumes the provision ofconfinement reinforcement in the joint. In theACI Chapter 21 method, shear resistancecalculated as a function of the horizontal cross-sectional area at mid-height of the joint iscompared with the total horizontal shear acrossthe same mid-height section. Figure 10-21shows the forces involved in calculating theshear at mid-height of a typical joint. Note thatthe stress in the yielded longitudinal beam barsis to be taken equal to 1.25 times the specifiednominal yield strength fy of the reinforcement.

The ACI-ASCE Committee 352Recommendations(10-29) have added arequirement relating to the uniform distributionof the longitudinal column reinforcementaround the perimeter of the column core, with amaximum spacing between perimeter bars of 8in. or one-third the column diameter or thecross-section dimension. The lateralconfinement, whether from steel hoops orbeams, and the distributed vertical columnreinforcement, in conjunction with the confinedconcrete core, provide the necessary elementsfor the development of an effective trussmechanism to resist the horizontal and verticalshears acting on a beam—column joint. Resultsof recent tests on bi-directionally loadedbeam—column joint specimens(10-28) confirmthe strong correlation between joint shearstrength and the horizontal cross-sectional areanoted by Meinheit and Jirsa.(10-41)

Some investigators(10-13, 10-32, 10-39) havesuggested that the ACI Chapter 21 approachdoes not fully reflect the effect of the different

variables influencing the mechanisms ofresistance operating in a beam-column joint andhave proposed alternative expressions based onidealizations of the strut and joint trussmechanisms.

Figure 10-20. Proposed details for forcing beam hingingaway from column face(10-26). See also Fig. 10-15. (a)Supplementary flexural reinforcement. (b) Haunch. (c)Special reinforcement detail.

To limit slippage of beam bars throughinterior beam-column joints, the ACI-ASCECommittee 352 Recommendations call for aminimum column dimension equal to 20 timesthe diameter of beam bars passing through thejoint. For exterior joints, where beam barsterminate in the joint, the maximum size ofbeam bar allowed is a No. 11 bar.

488 Chapter 10

Figure 10-21. Shear force at mid-height of beam-columnjoint- ACI Chapter 21 design practice.

When the depth of an exterior column is notsufficient to accommodate the requireddevelopment length for beam bars, a beam stubat the far (exterior) side of the column,(10-32)

such as is shown in Figure 10-22, can be used.Embedding the 90o beam bar hooks outside ofthe heavily stressed joint region reduces thestiffness degradation due to slippage andimproves the overall performance of theconnection.

Figure 10-22. Exterior beam stub for anchoring beam bars

Slab—Column Connections By omittingconsideration of the reinforced concrete flatplate in its provisions governing the design ofstructures in high-seismic-risk areas, ACIChapter 21 essentially excludes the use of sucha system as part of a ductile frame resisting

seismic loads in such areas. Two-way slabswithout beams, i.e., flat plates, are, however,allowed in areas of moderate seismic risk.

The flat plate structure is an economical andwidely used form of construction in non-seismic areas, especially for multistoryresidential construction. Its weakest feature, asis well known, is its vulnerability to a punchingshear failure at the slab-column junctions. Thecollapse of a number of buildings using such asystem during the 1964 Anchorage, Alaska andthe 1967 Caracas, Venezuela earthquakes, aswell as several buildings using waffle slabsduring the September 1985 Mexicanearthquake,(10-43, 10-44) clearly dramatized thisvulnerability. Although a flat plate may bedesigned to carry vertical loads only, withstructural walls taking the lateral loads,significant shears may still be induced at theslab-column junctions as the structure displaceslaterally during earthquake response.

Tests on slab—column connectionssubjected to reversed cyclic loading(10-45, 10-46)

indicate that the ductility of flat-slab—columnconnections can be significantly increasedthrough the use of stirrups enclosing bands offlexural slab reinforcement passing through thecolumns. Such shear-reinforced bandsessentially function as shallow beamsconnecting the columns.

Structural Walls Reinforced concretestructural walls (commonly referred to as shearwalls), when properly designed, representeconomical and effective lateral stiffeningelements that can be used to reduce potentiallydamaging interstory displacements inmultistory structures during strong earthquakes.The structural wall, like the vertical steel trussin steel buildings, has had a long history of usefor stiffening buildings laterally against windforces. The effectiveness of properly designedstructural walls in reducing earthquake damagein multistory buildings has been welldemonstrated in a number of recentearthquakes.

In earthquake-resistant design, theappreciable lateral stiffness of structural wallscan be particularly well utilized in combinationwith properly proportioned coupling beams in


coupled wall systems. Such systems allowconsiderable inelastic energy dissipation to takeplace in the coupling beams (which arerelatively easy to repair) at critical levels,sometimes even before yielding occurs at thebases of the walls.

Attention in the following discussion will befocused on slender structural walls, i.e., wallswith a height-to-width ratio greater than about2.0, such as are used in multistory buildings.These walls generally behave like verticalcantilever beams. Short or squat walls, on theother hand, resist horizontal forces in theirplane by a predominantly truss-typemechanism, with the concrete providing thediagonal compressive strut(s) and the steelreinforcement the equilibrating vertical andhorizontal ties. Tests on low-rise wallssubjected to slowly reversed horizontalloading(10-47) indicate that for walls with height-to-width ratios of about 1.0 , horizontal andvertical reinforcement are equally effective. Asthe height-to-width ratio of a wall becomessmaller, the vertical reinforcement becomesmore effective in resisting shear than thehorizontal steel.(10-48)

In the following discussion, it will beassumed that the isolated structural wall isloaded by a resultant horizontal force acting atsome distance above the base. Under such aloading, flexural hinging will occur at the baseof the wall. Where the wall is designed andloaded so that it yields in flexure at the base, asmight be expected under strong earthquakes, itsbehavior becomes a function primarily of themagnitude of the shear force that accompaniessuch flexural hinging as well as thereinforcement details used in the hinging regionnear the base. Thus, if the horizontal force actshigh above the base (long shear arm), it willtake a lesser magnitude of the force to produceflexural hinging at the base than when the pointof application of the load is close to the base(short shear arm). For the same value of thebase yield moment, the moment-to-shear ratioin the former case is high and the magnitude ofthe applied force (or shear) is low, while in thelatter case the moment-to-shear ratio is low andthe applied shear is high. In both cases, the

magnitude of the applied shear is limited by theflexural yield strength at the base of the wall.

In this connection, it is of interest to notethat dynamic inelastic analyses of isolatedwalls(10-4) covering a wide range of structuraland ground motion parameters have indicatedthat the maximum calculated shear at the baseof walls can be from 1.5 to 3.5 times greaterthan the shear necessary to produce flexuralyielding at the base, when such shear isdistributed in a triangular manner over theheight of the wall, as is prescribed for design inmost codes. This is shown in Figure 10-23,which gives the ratio of the calculatedmaximum dynamic shear, Vdyn

max, to theresultant of the triangularly distributed shearnecessary to produce flexural yielding at thebase, VT, as a function of the fundamentalperiod T1 and the available rotational ductilityµa

r . The input accelerograms used in theanalyses had different frequency characteristicsand were normalized with respect to intensityso that their spectrum intensity (i.e., the areaunder the corresponding 5%-damped velocityresponse spectrum, between periods 0.1 and 3.0sec) was 1.5 times that of the N-S component ofthe 1940 El Centro record. The results shown inFigure 10-23 indicate that a resultant shearforce equal to the calculated maximum dynamicshear need not be applied as high as two-thirdsthe height of the wall above the base to produceyielding at the base. Figure 10-24, also fromReference 10-4, shows the distance (expressed

as the ratio dyny VM max/ ) from the base at which

the resultant dynamic force would have to act toproduce yielding at the base, as a function ofthe fundamental period and the availablerotational ductility of the wall. The ordinate onthe right side of the figure gives the distanceabove the base as a fraction of the wall height.Note that for all cases, the resultant dynamicforce lies below the approximate two-thirdspoint associated with the triangular loadingspecified in codes.

490 Chapter 10

Figure 10-23. Ratio dynVmax /VT as a function of T1 and

arµ . -20 story isolated structural walls. (From Ref. 10-4.)

These analytical results suggest not onlythat under strong earthquakes the maximumdynamic shear can be substantially greater thanthat associated with the lateral loads used todesign the flexural strength of the base of thewall, but also, as a corollary, that the moment-to-shear ratio obtained under dynamicconditions is significantly less than that impliedby the code-specified distribution of designlateral loads. These results are importantbecause unlike beams in frames, where thedesign shear can be based on the maximumprobable flexural strengths at the ends of themember as required by statics (see Figure 10-16), in cantilever walls it is not possible todetermine a similar design shear as a functionof the flexural strength at the base of the wallusing statics alone, unless an assumption ismade concerning the height of the appliedresultant horizontal force. In the capacity designmethod adopted in New Zealand as applied tostructural walls,(10-13,10-49) the design base shearat the base of a wall is obtained by multiplyingthe shear at the base corresponding to the code-

specified forces by a flexural overstrengthfactor and a "dynamic shear magnificationfactor”. The flexural overstrength factor in thiscase represents the ratio of flexural overstrength(accounting for upward deviations from thenominal strength of materials and other factors)to the moment due to the code-specified forces,with a typical value of about 1.39 or higher.Recommended values for the dynamic shearmagnification factor range from 1.0 for a one-story high wall to a maximum of 1.8 for walls6-stories or more in height.

Figure 10-24. Ratio Y = My/dynVmax as a function of T1

andarµ - 20 story isolated structural walls. (From Ref. 10-4.)

Tests on isolated structural walls(10-50,10-51)

have shown that the hinging region, i.e., theregion where most of the inelastic deformationoccurs, extends a distance above the baseroughly equal to the width of the wall. Theductility of the hinging region at the base of awall, like the hinging region in beams andcolumns, is heavily dependent on thereinforcing details used to prevent earlydisruption of critically stressed areas within theregion. As observed in beams and columns,tests of structural walls have confirmed the


effectiveness of adequate confinement inmaintaining the strength of the hinging regionthrough cycles of reversed inelasticdeformation. The adverse effects of high shears,acting simultaneously with the yield moment,on the deformation capacity of the hingingregion of walls has also been noted in tests.

Early tests of slender structural walls understatic monotonic loading(10-52) have indicatedthat the concentration of well-confinedlongitudinal reinforcement at the ends of thewall section can significantly increase theductility of the wall. This is shown in Figure10-25 from Reference 10-52. This improvementin behavior resulting from a concentration ofwell-confined longitudinal reinforcement at theends of a wall section has also been observed in

tests of isolated walls under cyclic reversedloading.(10-50, 10-51) Plain rectangular walls, nothaving relatively stiff confined boundaryelements, are prone to lateral buckling of thecompression edge under large horizontaldisplacements.(10-50, 10-52)

Figure 10-26 shows a sketch of the region atthe base of a wall with boundary elements aftera few cycles of lateral loading. Several modesof failure have been observed in the laboratory.Failure of the section can occur in flexure byrupture of the longitudinal reinforcement or bya combination of crushing and sliding in aweakened compression flange. Alternatively,failure, i.e., loss of lateral-load-resistingcapacity, can occur by sliding along a near-horizontal plane near the base (in rectangular-

Figure 10-25. Moment-curvature curves for statically loaded rectangular walls as a function of reinforcementdistribution.(10-52)

492 Chapter 10

section walls especially) or by crushing of theweb concrete at the junction of the diagonalstruts and the compression flange (in walls withthin webs and/or heavy boundary elements).

Figure 10-26. Moment-curvature curves for staticallyloaded rectangular walls as a function of reinforcementdistribution.(10-54)

Since walls are generally designed to beunder-reinforced, crushing in the usual senseassociated with monotonic loading does notoccur. However, when the flanges areinadequately confined, i.e., with thelongitudinal and lateral reinforcement spacedfar apart, concrete fragments within the cores ofthe flanges that had cracked in flexure underearlier cycles of loading can be lost insubsequent loading cycles. The longitudinalbars can buckle under compression and whensubsequently stretched on reversal of theloading can rupture in low-cycle fatigue. It isalso worth noting that because of theBauschinger effect (i.e., the early yielding,reflected in the rounding of the stress—straincurve of steel, that occurs during load reversalsin the inelastic range and the consequentreduction in the tangent modulus of the steelreinforcement at relatively low compressive

stresses), the compression steel in memberssubjected to reversed cycles of inelastic loadingtends to buckle earlier than in comparablemonotonically loaded specimens.

As in beams and columns, degradation ofstrength and ductility of the hinging region ofwalls is strongly influenced by the magnitude ofthe shear that accompanies flexural yielding.

High shears ( > 6 cf ′ ), when acting on a web

area traversed by crisscrossing diagonal cracks,can precipitate failure of the wall by crushing ofthe diagonal web struts or a combinedcompression—sliding failure of thecompression flange near the base. Shear in thehinging region is resisted by severalmechanisms, namely, shear-friction along anear-horizontal plane across the width of thewall, dowel action of the tensile reinforcementand to a major extent (as in beams) by shearacross the compression flange. After severalcycles of load reversals and for moderatemoment-to-shear ratios, the flexural cracksbecome wide enough to reduce the amount ofshear carried by shear friction. As suggested byFigure 10-26, the truss action that develops inthe hinging region involves a horizontal (shear)component of the diagonal strut that acts on thesegment of the compression flange close to thebase. If the compression flange is relativelyslender and inadequately confined, the loss ofcore concrete under load reversals results in aloss of stiffness of this segment of thecompression flange. The loss of stiffness andstrength in the compression flange or itsinability to support the combined horizontal(shear) component of the diagonal strut and theflexural compressive force can lead to failure ofthe wall.

Thus confinement of the flanges of walls,and especially those in the hinging region, isnecessary not only to increase the compressivestrain capacity of the core concrete but also todelay inelastic bar buckling and, together withthe longitudinal reinforcement, prevent loss ofthe core concrete during load reversals (the so-called “basketing effect”). By maintaining thestrength and stiffness of the flanges,confinement reinforcement improves the sheartransfer capacity of the hinging region through


the so-called “dowel action” of the compressionflange, in addition to serving as shearreinforcement. As in beams, the diagonaltension cracking that occurs in walls and theassociated truss action that develops inducestensile stresses in the horizontal webreinforcement. This suggests the need forproper anchorage of the horizontalreinforcement in the flanges.

Where high shears are involved, properlyanchored crossing diagonal reinforcement in thehinging regions of walls, just as in beams,provides an efficient means of resisting shearand particularly the tendency toward slidingalong cracked and weakened planes.

A series of tests of isolated structural wallspecimens at the Portland CementAssociation(10-50, 10-51) have provided someindication of the effect of several importantvariables on the behavior of walls subjected toslowly reversed cycles of inelasticdeformations. Some results of this investigationhave already been mentioned in the preceding.Three different wall cross-sections wereconsidered in the study, namely, plainrectangular sections, barbell sections withheavy flanges (columns) at the ends, andflanged sections with the flanges having aboutthe same thickness as the web. In the following,results for some of the parameters consideredwill be presented briefly.

1. Monotonic vs. reversed cyclic loading. Inan initial set of two nominally identicalspecimens designed to explore the effect of loadreversals, a 15% decrease in flexural strengthwas observed for a specimen loaded by cyclesof progressively increasing amplitude ofdisplacement when compared with a specimenthat was loaded monotonically. Figures 10-27aand 10-28a show the corresponding load—deflection curves for the specimens. Acomparison of these figures shows not only areduction in strength but also that the maximumdeflection of the wall subjected to reversedloading was only 8 in., compared to about 12in. for the monotonically load specimen,indicating a reduction in deflection capacity ofabout 30%. Figure 10-28b, when compared

with Figure 10-27b, shows the more severecracking that results from load reversals.

2.Level of shear stress. Figure 10-29 showsa plot of the variation of the maximumrotational ductility with the maximum nominalshear stress in isolated structural wallspecimens reported in References 10-50 and10-51. The decrease in rotational ductility withincreasing values of the maximum shear stresswill be noted. The maximum rotation used indetermining ductility was taken as that for thelast cycle in which at least 80% of the previousmaximum observed load was sustainedthroughout the cycle. The yield rotation wasdefined as the rotation associated with theyielding of all of the tensile reinforcement inone of the boundary elements.

The presence of axial loads—of the order of10% of the compressive strength of the walls—increased the ductility of specimens subjectedto high shears. In Figure 10-29, the specimenssubjected to axial loads are denoted by opensymbols. The principal effect of the axial loadwas to reduce the shear distortions and henceincrease the shear stiffness of the hingingregion. It may be of interest to note that forwalls loaded monotonically,(10-52) axialcompressive stress was observed to increasemoment capacity and reduce ultimate curvature,results consistent with analytical results fromsectional analysis.

3. Section shape. As mentioned earlier, theuse of wall sections having stiff and well-confined flanges or boundary elements, asagainst plain rectangular walls, not only allowsdevelopment of substantial flexural capacity (inaddition to being less susceptible to lateralbuckling), but also improves the shearresistance and ductility of the wall. In wallswith relatively stiff and well-confined boundaryelements, some amount of web crushing canoccur without necessarily limiting the flexuralcapacity of the wall. Corley et al.(10-53) point outthat trying to avoid shear failure in walls,particularly walls with stiff and well-confinedboundary elements, may be a questionabledesign objective.

494 Chapter 10

(a)

(b)

Figure 10-27. (a) Load-deflection curve of monotonically loaded specimen. (b) view of specimen at +12 in. topdeflection.(10-53)


Figure 10-28. (a) Load-deflection curve of specimen subjected to load cycles of progressively increasing amplitude. (b)

View of specimen at +8 in. top deflection. (10-53)

496 Chapter 10

Figure 10-29. Variation of rotational ductility withmaximum average shear stress in PCA isolated walltests(10-51).

Thus, although ACI Chapter 21 limits themaximum average shear stress in walls to

10 cf ′ (a value based on monotonic tests)

with the intent of preventing web crushing, webcrushing occurred in some specimens subjectedto shear stresses only slightly greater than

7 cf ′ . However, those specimens where web-

crushing failure occurred were able to developdeformations well beyond the yield deformationprior to loss of capacity.

4.Sequence of large-amplitude load cycles.Dynamic inelastic analyses of isolated walls(10-8)

have indicated that in a majority of cases, themaximum or a near-maximum response toearthquakes occurs early, with perhaps only oneelastic response cycle preceding it. Thiscontrasts with the loading program commonlyused in quasi-static tests, which consists of loadcycles of progressively increasing amplitude.To examine the effect of imposing large-amplitude load cycles early in the test, twonominally identical isolated wall specimenswere tested. One specimen was subjected toload cycles of progressively increasingamplitude, as were most of the specimens inthis series. Figure 10-30a indicates thatspecimen B7 was able to sustain a rotationalductility of slightly greater than 5 through three

repeated loading cycles. The second specimen(B9) was tested using a modified loadingprogram similar to that shown in Figure 10-7b,in which the maximum load amplitude wasimposed on the specimen after only one elasticload cycle. The maximum load amplitudecorresponded to a rotational ductility of 5. Asindicated in Figure 10-30b, the specimen failedbefore completing the second load cycle.Although results from this pair of specimenscannot be considered conclusive, they suggestthat tests using load cycles of progressivelyincreasing amplitude may overestimate theductility that can be developed under what maybe considered more realistic earthquakeresponse conditions. The results do tend toconfirm the reasonable expectation that anextensively cracked and “softened” specimensubjected to several previous load cycles oflesser amplitude can better accommodate largereversed lateral deflections than a virtuallyuncracked specimen that is loaded to near-capacity early in the test. From this standpoint,the greater severity of the modified loadingprogram, compared to the commonly usedprogressively increasing-amplitude loadingprogram, appears obvious.

5. Reinforcement detailing. On the basis ofthe tests on isolated walls reported inReferences 10-50 and 10-51, Oesterle et al.(10-54)

proposed the following detailing requirementsfor the hinging regions of walls:

• The maximum spacing of transversereinforcement in boundary elements should be5db, where db is the diameter of the longitudinalreinforcement.• Transverse reinforcement in the boundaryelement should be designed for a shear

Vnb = Mnb/1.5 lb ,

whereMnb = nominal moment strength of boundary

elementlb =width of boundary element (in the plane

of the wall)


(a)

(b)

Figure 10-30. Comparison of behavior of isolated walls subjected to different loading histories. (10-53)

(a) specimen subjected to progressively increasing load amplitudes (see Fig. 10-7a). (b) Specimen subjected to loadinghistory characterized by large-amplitude cycles early in loading (see Fig. 10-7b).

498 Chapter 10

• No lap splices should be used for cross-ties insegments of boundary elements within thehinging region.

• A recommendation on anchoring horizontalweb reinforcement in the boundary elements,such as is shown in Figure 10-31a, has beenadopted by ACI Chapter 21. For levels of shear

in the range of 5 cf ′ to 10 cf ′ , the study

indicates that alternate 90° and 135° hooks, asshown in Figure 10-31b, can be used.

Figure 10-31. Alternative details for anchorage ofhorizontal web reinforcement in boundary elements.(10-54)

(a) detail for walls subjected to low –to-moderate stresslevels. (b) Detail for walls subjected to high shear stresslevels.

The specimens tested in this series hadspecial confinement reinforcement only over alength near the base equal to the width of thewall, i.e., the approximate length of the hingingregion. Strain readings as well as observationsof the general condition of the walls afterfailure showed that significant inelasticity anddamage were generally confined to the hingingregion. In view of this, it has been suggestedthat special confinement reinforcement forboundary elements need be provided only overthe lengths of potential hinging regions. Theseare most likely to occur at the base and at points

along the height of the wall wherediscontinuities, associated with abrupt andsignificant changes in geometry, strength, orstiffness, occur.

Coupled Walls As mentioned earlier, adesirable characteristic in an earthquake-resistant structure is the ability to respond tostrong ground motion by progressivelymobilizing the energy-dissipative capacities ofan ascending hierarchy of elements making upthe structure.

In terms of their importance to the generalstability and safety of a building, thecomponents of a structure may be grouped intoprimary and secondary elements. Primaryelements are those upon the integrity of whichdepend the stability and safety of the entirestructure or a major part of it. In this categoryfall most of the vertical or near-verticalelements supporting gravity loads, such ascolumns and structural walls, as well as long-span horizontal elements. Secondary elementsare those components whose failure wouldaffect only limited areas or portions of astructure.

The strong column-weak beam designconcept discussed earlier in relation to moment-resisting frames is an example of an attempt tocontrol the sequence of yielding in a structure.The “capacity design” approach adopted inNew Zealand which, by using even greaterconservatism in the design of columns relativeto beams, seeks to insure that no yieldingoccurs in the columns (except at their bases)—is yet another effort to achieve a controlledresponse in relation to inelastic action. Bydeliberately building in greater flexural strengthin the primary elements (the columns), thesedesign approaches force yielding and inelasticenergy dissipation to take place in thesecondary elements (the beams).

When properly proportioned, the coupled-wall system can be viewed as a furtherextension of the above design concept. Bycombining the considerable lateral stiffness ofstructural walls with properly proportionedcoupling beams that can provide most of theenergy-dissipative mechanism during response


to strong ground motions, a better-performingstructural system is obtained. The stiffness ofthe structural wall makes it a desirable primaryelement from the standpoint of damage control(by restricting interstory distortions), while themore conveniently repairable coupling beamsprovide the energy-dissipating secondaryelements. Figure 10-32a shows a two-wallcoupled-wall system and the forces acting at thebase and on a typical coupling beam. The totaloverturning moment at the base of the coupledwall = M1 + M2 + TL. A typical distribution ofthe elastic shear force in the coupling beamsalong the height of the structure due to astatically applied lateral load is shown in Figure10-32b. Note that the accumulated shears ateach end of the coupling beams, summed overthe height of the structure, are each equal to theaxial force (T) at the base of the correspondingwall. The height to the most critically stressedcoupling beam tends to move downward as thecoupling-beam stiffness (i.e., the degree ofcoupling between the two walls) increases.

Figure 10-32. Laterally loaded coupled wall system. (a)Forces on walls at base. (b) Typical distribution of shearsin coupling beams over height of structure.

In a properly designed earthquake-resistantcoupled-wall system, the critically stressedcoupling beams should yield first—before thebases of the walls. In addition, they must becapable of dissipating a significant amount of

energy through inelastic action. Theserequirements call for fairly stiff and strongbeams. Furthermore, the desire for greaterlateral-load-resisting efficiency in the systemwould favor stiff and strong coupling beams.However, the beams should not be so stiff orstrong flexurally that they induce appreciabletension in the walls, since a net tension wouldreduce not only the yield moment but also theshear resistance of the wall (recall that amoderate amount of compression improves theshear resistance and ductility of isolated walls).This in turn can lead to early flexural yieldingand shear-related inelastic action at the base ofthe tension wall. Dynamic inelastic analyses ofcoupled-wall systems(10-56) have shown, andtests on coupled-wall systems under cyclicreversed loading(10-57) have indicated, that whenthe coupling beams have appreciable stiffnessand strength, so that significant net tension isinduced in the “tension wall”, a major part ofthe total base shear is resisted by the“compression wall” (i.e., the wall subjected toaxial compression for the direction of loadingconsidered), a situation not unlike that whichoccurs in a beam.

The design of a coupled-wall system wouldthen involve adjusting the wall-to-couplingbeam strength and stiffness ratios so as to strikea balance between these conflictingrequirements. A basis for choosing anappropriate beam-to-wall strength ratio,developed from dynamic inelastic response dataon coupled-wall systems, is indicated inReference 10-58. The Canadian Code forConcrete, CSA Standard A23.3-94(10-33),recommends that in order to classify as a fullyeffective coupled wall system, the ratio

TLMM

TL

++ 21

must be greater than 2/3. Those

with lower ratios are classified as partiallycoupled wall system in which the coupled wallsystem are to be designed for higher seismicdesign forces (14% greater) due to their loweramount of energy dissipation capacity due toreduced coupling action. Once the appropriaterelative strengths and stiffness have beenestablished, details to insure adequate ductilityin potential hinging regions can be addressed.

500 Chapter 10

Because of the relatively large shears thatdevelop in deep coupling beams and thelikelihood of sliding shear failures underreversed loading, the use of diagonalreinforcement in such elements has beensuggested (see Figure 10-33). Tests by Paulayand Binney(10-59) on diagonally reinforcedcoupling beams having span-to-depth ratios inthe range of 1 to 1½ have shown that thisarrangement of reinforcement is very effectivein resisting reversed cycles of high shear. Thespecimens exhibited very stable force—deflection hysteresis loops with significantlyhigher cumulative ductility than comparableconventionally reinforced beams. Tests byBarney et al.(10-60) on diagonally reinforcedbeams with span-to-depth ratios in the range of2.5 to 5.0 also indicated that diagonalreinforcement can be effective even for theselarger span-to-depth ratios.

Figure 10-33. Diagonally reinforced coupling beam.(Adapted from Ref. 10-59.)

In the diagonally reinforced couplingsbeams reported in Reference 10-60, nosignificant flexural reinforcement was used.The diagonal bars are designed to resist bothshear and bending and assumed to function attheir yield stress in both tension andcompression. To prevent early buckling of thediagonal bars, Paulay and Binney recommendthe use of closely spaced ties or spiral bindingto confine the concrete within each bundle ofdiagonal bars. A minimum amount of“basketing reinforcement,” consisting of twolayers of small-diameter horizontal and vertical

bars, is recommended. The grid should providea reinforcement ratio of at least 0.0025 in eachdirection, with a maximum spacing of 12 in.between bars.

10.4 CODE PROVISIONS FOREARTHQUAKE-RESISTANT DESIGN

10.4.1 Performance Criteria

In recent years, the performance criteriareflected in some building code provisions suchas IBC-2000(10-61) have become more explicitthan before. Although these provisionsexplicitly require design for only a single levelof ground motion, it is expected that buildingsdesigned and constructed in accordance withthese requirements will generally be able tomeet a number of performance criteria, whensubjected to earthquake ground motions ofdiffering severity. The major framework of theperformance criteria is discussed in the reportby the Structural Association of CaliforniaVision 2000 (SEAOC, 1995).(10-62) In thisreport, four performance levels are defined andeach performance level is expressed as thedesired maximum level of damage to a buildingwhen subjected to a specific seismic groundmotion. Categories of performance are definedas follows:1. fully operational2. operational3. life-safe4. near collapse

For each of the performance levels, there isa range of damage that corresponds to thebuilding’s functional status following aspecified earthquake design level. Theseearthquake design levels represent a range ofearthquake excitation that have definedprobabilities of occurrence over the life of thebuilding. SEAOC Vision 2000 performancelevel definition includes descriptions ofstructural and non-structural damage, egresssystems and overall building state. Alsoincluded in the performance level descriptions


is the level of both transient and permanent driftin the structure. Drift is defined as the ratio ofinterstory deflection to the story height.

The fully operational level represents theleast level of damage to the building. Except forvery low levels of ground motion, it is generallynot practical to design buildings to meet thisperformance level.

Operational performance level is one inwhich overall building damage is light.Negligible damage to vertical load carryingelements as well as light damage to the lateralload carrying element is expected. The lateralload carrying system retains almost all of itsoriginal stiffness and strength, with minorcracking in the elements of the structure isexpected. Transient drift are less than 0.5% andthere is inappreciable permanent drift. Buildingoccupancy continues unhampered.

Life-safe performance level guidelinesinclude descriptions of damage to contents, aswell as structural and non-structural elements.Overall, the building damage is described asmoderate. Lateral stiffness has been reduced aswell as the capacity for additional loads, whilesome margin against building collapse remains.Some cracking and crushing of concrete due toflexure and shear is expected. Vertical loadcarrying elements have substantial capacity toresist gravity loads. Falling debris is limited tominor events. Levels of transient drift are to bebelow 1.5% and permanent drift is less than0.5%.

Near collapse performance includes severeoverall damage to the building, moderate toheavy damage of the vertical load carryingelements and negligible stiffness and strength inthe lateral load carrying elements. Collapse isprevented although egress may be inhibited.Permissible levels of transient and permanentdrift are less than 2.5%. Repair of a buildingfollowing this level of performance may not bepractical, resulting in a permanent loss ofbuilding occupancy.

In the IBC-2000 provisions, the expectedperformance of buildings under the variousearthquakes that can affect them are controlledby assignment of each building to one of thethree seismic use groups. These seismic use

groups are categorized based on the type ofoccupancy and importance of the building. Forexample, buildings such as hospitals, powerplants and fire stations are considered asessential facilities also known as post-disasterbuildings and are assigned as seismic use groupIII. These provisions specify progressivelymore conservative strength, drift control,system selection, and detailing requirements forbuildings contained in the three groups, in orderto attain minimum levels of earthquakeperformance suitable to the individualoccupancies.

10.4.2 Code-Specified Design LateralForces

The availability of dynamic analysisprograms (see References 10-63 to 10-68) hasmade possible the analytical estimation ofearthquake-induced forces and deformations inreasonably realistic models of most structures.However, except perhaps for the relativelysimple analysis by modal superposition usingresponse spectra, such dynamic analyses, whichcan range from a linearly elastic time-historyanalysis for a single earthquake record tononlinear analyses using a representativeensemble of accelerograms, are costly and maybe economically justifiable as a design tool onlyfor a few large and important structures. Atpresent, when dynamic time-history analyses ofa particular building are undertaken for thepurpose of design, linear elastic response isgenerally assumed. Nonlinear (inelastic) time-history analyses are carried out mainly inresearch work. However, non-linear pushoverstatic analysis can be used as a design tool toevaluate the performance of the structure in thepost-yield range of response. Pushover analysisis used to develop the capacity curve, illustratedgenerally as a base shear versus top storydisplacement curve. The pushover test showsthe sequence of element cracking and yieldingas a function of the top story displacement andthe base shear. Also, it exposes the elementswithin the structure subjected to the greatestamount of inelastic deformation. The forcedisplacement relationship shows the strength of

502 Chapter 10

the structure and the maximum base shear thatcan be developed. Pushover analysis, which isrelatively a new technology, should be carriedout with caution. For example, when theresponse of a structure is dominated by modesother than the first mode, the results may notrepresent the actual behavior.

For the design of most buildings, reliancewill usually have to be placed on the simplifiedprescriptions found in most codes(10-1) Althoughnecessarily approximate in character-in view ofthe need for simplicity and ease of application-the provisions of such codes and the philosophybehind them gain in reliability as design guideswith continued application and modification toreflect the latest research findings and lessonsderived from observations of structuralbehavior during earthquakes. Code provisionsmust, however, be viewed in the properperspective, that is, as minimum requirementscovering a broad class of structures of more orless conventional configuration. Unusualstructures must still be designed with specialcare and may call for procedures beyond thosenormally required by codes.

The basic form of modern code provisionson earthquake-resistant design has evolvedfrom rather simplified concepts of the dynamicbehavior of structures and has been greatlyinfluenced by observations of the performanceof structures subjected to actual earthquakes.(10-

69) It has been noted, for instance, that manystructures built in the 1930s and designed onthe basis of more or less arbitrarily chosenlateral forces have successfully withstoodsevere earthquakes. The satisfactoryperformance of such structures has beenattributed to one or more of the following(10-70,

10-71): (i) yielding in critical sections of members(yielding not only may have increased theperiod of vibration of such structures to valuesbeyond the damaging range of the groundmotions, but may have allowed them todissipate a sizable portion of the input energyfrom an earthquake); (ii) the greater actualstrength of such structures resulting from so-called nonstructural elements which aregenerally ignored in analysis, and thesignificant energy-dissipation capacity that

cracking in such elements represented; and (iii)the reduced response of the structure due toyielding of the foundation.

The distribution of the code-specified designlateral forces along the height of a structure isgenerally similar to that indicated by theenvelope of maximum horizontal forcesobtained by elastic dynamic analysis. Theseforces are considered service loads, i.e., to beresisted within a structure’s elastic range ofstresses. However, the magnitudes of thesecode forces are substantially smaller than thosewhich would be developed in a structuresubjected to an earthquake of moderate-to-strong intensity, such as that recorded at ElCentro in 1940, if the structure were to respondelastically to such ground excitation. Thus,buildings designed under the present codeswould be expected to undergo fairly largedeformations (four to six times the lateraldisplacements resulting from the code-specifiedforces) when subjected to an earthquake withthe intensity of the 1940 El Centro.(10-2) Theselarge deformations will be accompanied byyielding in many members of the structure, and,in fact, such is the intent of the codes. Theacceptance of the fact that it is economicallyunwarranted to design buildings to resist majorearthquakes elastically, and the recognition ofthe capacity of structures possessing adequatestrength and ductility to withstand majorearthquakes by responding inelastically tothem, lies behind the relatively low forcesspecified by the codes. These reduced forcesare coupled with detailing requirementsdesigned to insure adequate inelasticdeformation capacity, i.e., ductility. Thecapacity of an indeterminate structure to deformin a ductile manner, that is to deform wellbeyond the yield limit without significant lossof strength, allows such a structure to dissipatea major portion of the energy from anearthquake without serious damage.


10.4.3 Principal Earthquake-DesignProvisions of ASCE 7-95, IBC-2000, UBC-97, and ACI Chapter 21Relating to Reinforced Concrete

The principal steps involved in the design ofearthquake-resistant cast-in-place reinforcedconcrete buildings, with particular reference tothe application of the provisions of nationallyaccepted model codes or standards, will bediscussed below. The minimum design loadsspecified in ASCE 7-95, Minimum designLoads for Buildings and Other Structures(10-72)

and the design and detailing provisionscontained in Chapter 21 of ACI 318-95,Building Code Requirements for ReinforcedConcrete,(10-10) will be used as bases for thediscussion. Emphasis will be placed on thoseprovisions relating to the proportioning anddetailing of reinforced concrete elements, thesubject of the determination of earthquakedesign forces having been treated in Chapters 4and 5. Where appropriate, reference will bemade to differences between the provisions ofthese model codes and those of related codes.Among the more important of these is the IBC-2000(10-61) which is primarily a descendant ofATC 3-06(10-73) and the latest edition of theRecommended Lateral Force Requirements ofthe Structural Engineers Association ofCalifornia (SEAOC-96).(10-74)

The ASCE 7-95 provisions relating toearthquake design loads are basically similar tothose found in the 1997 Edition of the UniformBuilding Code (UBC-97)(10-1). The currentUBC-97 earthquake design load requirementsare based on the 1996 SEAOCRecommendations (SEAOC-96). Except forminor modifications, the design and detailingrequirements for reinforced concrete membersfound in UBC-97 (SEAOC-96) and IBC-2000are essentially those of ACI Chapter 21.

Although the various code-formulatingbodies in the United States tend to differ inwhat they consider the most appropriate form inwhich to cast specific provisions and in theirjudgment of the adequacy of certain designrequirements, there has been a tendency for thedifferent codes and model codes to gradually

take certain common general features. Andwhile many questions await answers, it cangenerally be said that the main features of theearthquake-resistant design provisions in mostcurrent regional and national codes have goodbasis in theoretical and experimental studies aswell as field observations. As such, they shouldprovide reasonable assurance of attainment ofthe stated objectives of earthquake-resistantdesign. The continual refinement and updatingof provisions in the major codes to reflect thelatest findings of research and fieldobservations(10-75) should inspire increasingconfidence in the soundness of theirrecommendations.

The following discussion will focus on theprovisions of ASCE 7-95 and ACI Chapter 21,with occasional references to parallel provisionsof IBC-2000 and UBC-97 (SEAOC-96).

The design earthquake forces specified inASCE 7-95 is intended as equivalent staticloads. As its title indicates, ASCE 7-95 isprimarily a load standard, defining minimumloads for structures but otherwise leaving outmaterial and member detailing requirements.ACI Chapter 21 on the other hand, does notspecify the manner in which earthquake loadsare to be determined, but sets down therequirements by which to proportion and detailmonolithic cast-in-place reinforced concretemembers in structures that are expected toundergo inelastic deformations duringearthquakes.

Principal Design Steps Design of areinforced concrete building in accordance withthe equivalent static force procedure found incurrent U.S. seismic codes involves thefollowing principal steps:

1. Determination of design “earthquake”forces:

• Calculation of base shear corresponding tothe computed or estimated fundamentalperiod of vibration of the structure. (Apreliminary design of the structure isassumed here.)

• Distribution of the base shear over theheight of the building.

504 Chapter 10

2. Analysis of the structure under the (static)lateral forces calculated in step (1), as well asunder gravity and wind loads, to obtain memberdesign forces and story drift ratios. The lateralload analysis, of course, can be carried out mostconveniently by using a computer program foranalysis.

For certain class of structures having plan orvertical irregularities, or structure over 240 feetin height, most building codes require dynamicanalysis to be performed. In this case, ASCE 7-95 and IBC-2000 require that the designparameters including story shears, moments,drifts and deflections determined from dynamicanalysis to be adjusted. Where the design valuefor base shear obtained from dynamic analysis(Vt) is less than the calculated base shear (V)determined using the step 1 above, these designparameters is to be increased by a factor ofV/Vt.

3. Designing members and joints for themost unfavorable combination of gravity andlateral loads. The emphasis here is on thedesign and detailing of members and theirconnections to insure their ductile behavior.

The above steps are to be carried out in eachprincipal (plan) direction of the building. Mostbuilding codes allow the design of a structure ineach principal direction independently of theother direction on the assumption that thedesign lateral forces act non-concurrently ineach principal direction. However, for certainbuilding categories which may be sensitive totorsional oscillations or characterized bysignificant irregularities and for columnsforming part of two or more intersecting lateral-force-resisting systems, orthogonal effects needto be considered. For these cases, the codesconsider the orthogonal effects requirementsatisfied if the design is based on the moresevere combination of 100 percent of theprescribed seismic forces in one direction plus30 percent of the forces in the perpendiculardirection.

Changes in section dimensions of somemembers may be indicated in the design phaseunder step (3) above. However, unless therequired changes in dimensions are such as to

materially affect the overall distribution offorces in the structure, a reanalysis of thestructure using the new member dimensionsneed not be undertaken. Uncertainties in theactual magnitude and distribution of the seismicforces as well as the effects of yielding inredistributing forces in the structure wouldmake such refinement unwarranted. It is,however, most important to design and detailthe reinforcement in members and theirconnections to insure their ductile behavior andthus allow the structure to sustain withoutcollapse the severe distortions that may occurduring a major earthquake. The code provisionsintended to insure adequate ductility instructural elements represent the majordifference between the design requirements forconventional, non-earthquake-resistantstructures and those located in regions of highearthquake risk.

Load Factors, Strength Reduction Factors,and Loading Combinations Used as Bases forDesign Codes generally require that thestrength or load-resisting capacity of a structureand its component elements be at least equal toor greater than the forces due to any of anumber of loading combinations that mayreasonably be expected to act on it during itslife. In the United States, concrete structures arecommonly designed using the ultimate-strengthb method. In this approach, structuresare proportioned so that their (ultimate)capacity is equal to or greater than the required(ultimate) strength. The required strength isbased on the most critical combination offactored loads, that is, specified service loadsmultiplied by appropriate load factors. Thecapacity of an element, on the other hand, isobtained by applying a strength-reductionfactor φ to the nominal resistance of theelement as determined by code-prescribedexpressions or procedures or from basicmechanics.

Load factors are intended to take account ofthe variability in the magnitude of the specified

b Since ACI 318-71, the term “ultimate” has been dropped,so that what used to be referred to as “ultimate-strengthdesign” is now simply called “strength design.”


loads, lower load factors being used for types ofloads that are less likely to vary significantlyfrom the specified values. To allow for thelesser likelihood of certain types of loadsoccurring simultaneously, reduced load factorsare specified for some loads when considered incombination with other loads.

ACI 318-95 requires that structures, theircomponents, and their foundations be designedto have strengths not less than the most severeof the following combinations of loads:

+++

++±

±++

=

1)-(10 1.4T) 1.7L 0.75(1.4D

1.4F)or (1.7H 0.9D

l.4F)or (1.7H 1.7L 1.4D

1.43E)(1.3Wor 0.9D

l.87E)]or(1.7W 1.7L 0.75[l.4D

1.7L 1.4D

U

where

U = required strength to resist the factored loads

D = dead loadL = live loadW = wind loadE = earthquake loadF = load due to fluids with and maximum

heights well-defined pressuresH = load due to soil pressure

T = load due to effects of temperature, shrinkage, expansion of shrinkage compensating concrete, creep, differential settlement, or combinations thereof.

ASCE 7-95 specifies slightly different loadfactors for some load combinations, as follows:

+++++++

+++++++

=

E) 1.0or W 1.3 ( D 0.9

S 0.2 L 0.5 E 1.0 D 1.2

R)or Sor 0.5(L 0.5L 1.3W D 1.2

W)0.8or (0.5L R)or Sor 1.6(L 1.2D

R)or Sor 0.5(L H)1.6(L T)F1.2(D

D 1.4

r

r

r

U

(10-2)

whereLr = roof live loadS = snow loadR = rain load

For garages, places of public assembly, and allareas where the live load is greater than 100lb/ft2, the load factor on L in the third, fourth,and fifth combinations in Equation 10-2 is to betaken equal to 1.0.

For the design of earthquake-resistantstructures, UBC-97 uses basically the same loadcombinations specified by ASCE 7-95 asshown in Equation 10-2.

IBC-2000 requires that the loadcombinations to be the same as those specifiedby ASCE 7-95 as shown in Equation 10-2.However, the effect of seismic load, E, isdefined as follows:

E = ρ QE + 0.2 SDS DE = ρ QE - 0.2 SDS D (10-3)

whereE = the effect of horizontal and vertical

earthquake-induced forces,SDS = the design spectral response

acceleration at short periodsD = the effect of dead loadρ = the reliability factorQE = the effect of horizontal seismic forces

To consider the extent of structural redundancyinherent in the lateral-force-resisting system,the reliability factor, ρ, is introduced forbuildings located in areas of moderate to highseismicity. This is basically a penalty factor forbuildings in which the lateral resistance islimited to only few members in the structure.The maximum value of ρ is limited to 1.5.

The factor 0.2 SDS in Equation (10-3) isplaced on the dead load to account for theeffects of vertical acceleration.

For situations where failure of an isolated,individual, brittle element can result in the lossof a complete lateral-force-resisting system orin instability and collapse, IBC-2000 has aspecific requirement to determine the seismicdesign forces. These elements are referred to ascollector elements. Columns supporting

506 Chapter 10

discontinuous lateral-load-resisting elementssuch as walls also fall under this category. Theseismic loads are as follows:

E = Ωo QE + 0.2 SDS DE = Ωo QE - 0.2 SDS D (10-4)

where Ωo is the system overstrength factorwhich is defined as the ratio of the ultimatelateral force the structure is capable of resistingto the design strength. The value of Ωo variesbetween 2 to 3 depending on the type of lateralforce resisting system.

As mentioned earlier, the capacity of astructural element is calculated by applying astrength reduction factor φ to the nominalstrength of the element. The factor φ is intendedto take account of variations in materialstrength and uncertainties in the estimation ofthe nominal member strength, the nature of theexpected failure mode, and the importance of amember to the overall safety of the structure.For conventional reinforced concrete structures,ACI 318-95 specifies the following values ofthe strength reduction factor φ:

0.90 for flexure, with or without axial tension

0.90 for axial tension

0.75 for spirally reinforced memberssubjected to axial compression, withor without flexure

0.70 for other reinforced members (tiedcolumns) subjected to axialcompression, with or without flexure(an increase in the φ value formembers subjected to combinedaxial load and flexure is allowed asthe loading condition approaches thecase of pure flexure)

0.85 for shear and torsion0.70 for bearing on concrete

ACI Chapter 21 specifies the followingexception to the above values of the strength-

reduction factor as given in the main body ofthe ACI Code:

For structural members other than joints, avalue φ = 0.60 is to be used for shear when thenominal shear strength of a member is less thanthe shear corresponding to the development ofthe nominal flexural strength of the member.For shear in joints, φ = 0.85.

The above exception applies mainly to low-rise walls or portions of walls betweenopenings.

Code Provisions Designed to InsureDuctility in Reinforced Concrete Members

The principal provisions of ACI Chapter 21will be discussed below. As indicated earlier,the requirements for proportioning and detailingreinforced concrete members found in UBC-97(SEAOC-96) and IBC-2000 are essentiallythose of ACI Chapter 21. Modifications to theACI Chapter 21 provisions found in UBC-97and IBC-2000 will be referred to whereappropriate.

Special provisions governing the design ofearthquake-resistant structures first appeared inthe 1971 edition of the ACI Code. Theprovisions Chapter 21 supplement or supersedethose in the earlier chapters of the code and dealwith the design of ductile moment-resistingspace frames and shear walls of cast-in-placereinforced concrete.

ACI 318-95 does not specify the magnitudeof the earthquake forces to be used in design.The Commentary to Chapter 21 states that theprovisions are intended to result in structurescapable of sustaining a series of oscillations inthe inelastic range without critical loss instrength. It is generally accepted that theintensity of shaking envisioned by theprovisions of the first seven sections of ACIChapter 21 correspond to those of UBC seismiczones 3 and 4. In the 1983 edition of the ACICode, a section (Section A.9; now section 21.8)was added to cover the design of frames locatedin areas of moderate seismic risk, roughlycorresponding to UBC seismic zone 2. Forstructures located in areas of low seismic risk(corresponding to UBC seismic zones 0 and 1)


and designed for the specified earthquakeforces, very little inelastic deformation may beexpected. In these cases, the ductility providedby designing to the provisions contained in thefirst 20 Chapters of the code will generally besufficient.

A major objective of the design provisionsin ACI Chapter 21, as well as in the earlierchapters of the code, is to have the strength of astructure governed by a ductile type of flexuralfailure mechanism. Stated another way, theprovisions are aimed at preventing the brittle orabrupt types of failure associated withinadequately reinforced and over-reinforcedmembers failing in flexure, as well as withshear (i.e., diagonal tension) and anchorage orbond failures. The main difference betweenChapter 21 and the earlier chapters of the ACICode lies in the greater range of deformation,with yielding actually expected at criticallocations, and hence the greater ductilityrequired in designs for resistance to majorearthquakes. The need for greater ductilityfollows from the design philosophy that usesreduced forces in proportioning members andprovides for the inelastic deformations that areexpected under severe earthquakes by specialductility requirements.

A provision unique to earthquake-resistantdesign of frames is the so-called strong column-weak beam requirement. As discussed inSection 10.3.4 under “Beam—Column Joints,”this requirement calls for the sum of theflexural strengths of columns meeting at aframe joint to be at least 1.2 times that of thebeams framing into the joint. This is intended toforce yielding in such frames to occur in thebeams rather than in the columns and thuspreclude possible instability due to plastichinges forming in the columns. As pointed outearlier, this requirement may not guarantee non-development of plastic hinges in the columns.The strong column-weak beam requirementoften results in column sizes that are larger thanwould otherwise be required, particularly in theupper floors of multistory buildings withappreciable beam spans.

1. Limitations on material strengths. ACIChapter 21 requires a minimum specifiedconcrete strength cf ′ of 3000 lb/in.2 and a

maximum specified yield strength ofreinforcement, fy of 60,000 lb/in.2. These limitsare imposed with a view to restricting theunfavorable effects that material propertiesbeyond these limits can have on the sectionalductility of members. ACI Chapter 21 requiresthat reinforcement for resisting flexure andaxial forces in frame members and wallboundary elements be ASTM 706 grade 60low-alloy steel intended for applications wherewelding or bending, or both, are important.However, ASTM 615 billet steel bars of grade40 or 60 may be used provided the followingtwo conditions are satisfied:

(actual fy) ≤ (specified fy) ± 18,000 lb/in.2

25.1 actual

stress tensileultimate actual

y

≥f

The first requirement helps to limit the increasein magnitude of the actual shears that candevelop in a flexural member beyond thatcomputed on the basis of the specified yieldstress when plastic hinges form at the ends of abeam. The second requirement is intended toinsure reinforcement with a sufficiently longyield plateau.

In the “strong column-weak beam” frameintended by the code, the relationship betweenthe moment capacities of columns and beamsmay be upset if the beams turn out to havemuch greater moment capacity than intended bythe designer. Thus, the substitution of 60-ksisteel of the same area for specified 40-ksi steelin beams can be detrimental. The shear strengthof beams and columns, which is generallybased on the condition of plastic hinges forming(i.e., My acting) at the member ends, maybecome inadequate if the actual momentcapacities at the member ends are greater thanintended as a result of the steel having asubstantially greater yield strength thanspecified.

508 Chapter 10

2.Flexural members (beams). These includemembers having a clear span greater than fourtimes the effective depth that are subject to afactored axial compressive force not exceedingAg cf ′ /10, where Ag is the gross cross-sectional

area. Significant provisions relating to flexuralmembers of structures in regions of highseismic risk are discussed below.

(a) Limitations on section dimensionswidth/depth ≥ 0.3

×+≤

≥

)beam ofdepth (5.1

supporting ofwidth

.in 10

columnwidth

(b) Limitations on flexural reinforcement ratio(see also Figure 10-34):

=

y

c

y

f

f

f

'3

member of bottom and

both topat bars continuous two

200/

minρ

ρmax = 0.025

The minimum steel required can be waived ifthe area of tensile reinforcement at everysection is at least one-third greater thanrequired by analysis.

(c) Moment capacity requirements:At beam endsMy

+ ≥ 0.5My-

At any section in beam spanMy

+ or My- ≥ 0.25 (My

max at beam ends)

Figure 10-34. Longitudinal reinforcement requirements for flexural members


(d) Restrictions on lap splices: Lap splicesshall not be used

(1) within joints,(2) within 2h from face of support, where h

is total depth of beam,(3) at locations of potential plastic hinging.

Lap splices, where used, are to beconfined by hoops or spiralreinforcement with a maximum spacingor pitch of d/4 or 4 in.

(e) Restrictions on welding of longitudinalreinforcement: Welded splices and mechanicalconnectors may be used provided:

(1) they are used only on alternate bars ineach layer at any section;

(2) the distance between splices of adjacentbars is ≥ 24 in.

(3) Except as noted above, welding ofreinforcement required to resist loadcombinations including earthquakeeffects is not permitted. Also, thewelding of stirrups, ties, inserts, or othersimilar elements to longitudinal bars isprohibited

(f) Development length requirements forlongitudinal bars in tension:

(1) For bar sizes 3 through 11 with astandard 90° hook (as shown in Figure10-35) in normal weight concrete, thedevelopment length

≥. 6

8

'65

in

d

f

df

l b

c

by

dh

(db is bar diameter).

(2) When bars are embedded in lightweight-aggregate concrete, the developmentlength is to be at least equal to thegreater of 10db, 7.5 in. or 1.25 times thevalues indicated above.

(3) The 90° hook shall be located within theconfined core of a column or boundaryelement.

(4) For straight bars of sizes 3 through 11,the development length,

ld ≥ 2.5 x (ldh for bars with 90° hooks) ,when the depth of concrete cast in onelift beneath the bar is ≤ 12 in., or ld ≥ 3.5× (ldh for bars with 90° hooks) if theabove mentioned depth is > 12 in.

Figure 10-35. Development length for beam bars with 90o

hooks.

(5) If a bar is not anchored by means of a90° hook within the confined columncore, the portion of the required straightdevelopment length not located withinthe confined core shall be increased by afactor of 1.6.

(6) When epoxy-coated bars are used, thedevelopment lengths calculated above tobe increased by a factor of 1.2. However,for straight bars, with covers less than3db or clear spacing less than 6db, afactor of 1.5 to be used.

(g) Transverse reinforcement requirements forconfinement and shear: Transversereinforcement in beams must satisfyrequirements associated with their dual functionas confinement reinforcement and shearreinforcement (see Figure 10-36).

(1) Confinement reinforcement in the formof hoops is required:

510 Chapter 10

(i) over a distance 2d from faces ofsupport (where d is the effectivedepth of the member);

(ii) over distances 2d on both sides ofsections within the span whereflexural yielding may occur due toearthquake loading.

(2) Hoop spacing:(iii) First hoop at 2 in. from face of

support.(iv) Maximum spacing

××

≤

.12

)(24

) (8

4/

in

barshoopofdiameter

barallongitudinsmallestofdiameter

d

Figure 10-36. Transverse reinforcement limitations forflexural members. Minimum bar size- #3

(3) Lateral support for perimeterlongitudinal bars where hoops arerequired: Every corner and alternatelongitudinal bar shall be supported bythe corner of a hoop with an includedangle 135°, with no longitudinal barfarther than 6 in. along the tie from sucha laterally supported bar. Where thelongitudinal perimeter bars are arrangedin a circle, a circular hoop may be used.

(4) Where hoops are not required, stirrupswith seismic hooks at both ends with aspacing of not more than d/2 to beprovided throughout the length of themember.

(5) Shear reinforcement—to be provided soas to preclude shear failure prior todevelopment of plastic hinges at beamends. Design shears for determiningshear reinforcement are to be based on acondition where plastic hinges occur atbeam ends due to the combined effectsof lateral displacement and factoredgravity loads (see Figure 10-16). Theprobable flexural strength, Mpr

associated with a plastic hinge is to becomputed using a strength reductionfactor φ = 1.0 and assuming a stress inthe tensile reinforcement fs = 1.25fy.

(6) In determining the required shearreinforcement, the contribution of theconcrete, Vc, is to be neglected if theshear associated with the probableflexural strengths at the beam ends isequal to or greater than one-half the totaldesign shear and the factored axialcompressive force including earthquakeeffects is less than Ag cf ′ /20.

(7) The transverse reinforcement providedmust satisfy the requirements forconfinement or shear, whichever is morestringent.

Discussion:

(a) Limitations on section dimensions: Theselimitations have been guided by experiencewith test specimens subjected to cyclicinelastic loading.

(b) Flexural reinforcement limitations: Becausethe ductility of a member decreases withincreasing tensile reinforcement ratio, ACIChapter 21 limits the maximumreinforcement ratio to 0.025. The use of alimiting ratio based on the “balancedcondition” as given in the earlier chaptersof the code, while applicable to membersloaded monotonically, fails to describeconditions in flexural members subjected to


reversals of inelastic deformation. Thelimiting ratio of 0.025 is based mainly onconsiderations of steel congestion and alsoon limiting shear stresses in beams oftypical proportions. From a practicalstandpoint, low steel ratios should be usedwhenever possible. The requirements of atleast two continuous bars top and bottom,refers to construction rather than behavioralrequirements.

The selection of the size, number, andarrangement of flexural reinforcementshould be made with full consideration ofconstruction requirements. This isparticularly important in relation to beam-column connections, where constructiondifficulties can arise as a result ofreinforcement congestion. The preparationof large-scale drawings of the connections,showing all beam, column, and jointreinforcements, will help eliminateunanticipated problems in the field. Suchlarge-scale drawings will pay dividends interms of lower bid prices and a smooth-running construction job. Reference 10-76provides further recommendations onreinforcement detailing.

(c) Positive moment capacity at beam ends: Toallow for the possibility of the positivemoment at the end of a beam due toearthquake-induced lateral displacementsexceeding the negative moment due to thegravity loads, the code requires a minimumpositive moment capacity at beam endsequal to 50% of the corresponding negativemoment capacity.

(d) Lap splices: Lap splices of flexuralreinforcement are not allowed in regions ofpotential plastic hinging since such splicesare not considered to be reliable underreversed inelastic cycles of deformation.Hoops are mandatory for confinement oflap splices at any location because of thelikelihood of loss of the concrete cover.

(e) Welded splices and mechanical connectors:Welded splices and mechanical connectorsare to conform to the requirements given inChapter 12 of the ACI 318-95. A majorrequirement is that the splices develop at

least 125% of the specified yield strength ofthe bar.As mentioned earlier, the welding ofstirrups, ties, inserts, or other similarelements to longitudinal bars is notpermitted.

(f) Development length: The expression for ldh

given above already includes thecoefficients 0.7 (for concrete cover) and0.80 (for ties) that are normally applied tothe basic development length, ldb. This is sobecause ACI Chapter 21 requires that hooksbe embedded in the confined core of acolumn or boundary element. Theexpression for ldh also includes a factor ofabout 1.4, representing an increase over thedevelopment length required forconventional structures, to provide for theeffect of load reversals.

Except in very large columns, it isusually not possible to develop the yieldstrength of a reinforcing bar from theframing beam within the width of a columnunless a hook is used. Where beamreinforcement can extend through acolumn, its capacity is developed byembedment in the column and within thecompression zone of the beam on the farside of the connection (see Figure 10-34).Where no beam is present on the oppositeside of a column, such as in exteriorcolumns, the flexural reinforcement in aframing beam has to be developed withinthe confined region of the column. This isusually done by means of a standard 90°hook plus whatever extension is necessaryto develop the bar, the development lengthbeing measured from the near face of thecolumn, as indicated in Figure 10-35. Theuse of a beam stub at the far (exterior) sideof a column may also be considered (seeFigure 10-22). ACI Chapter 21 makes noprovision for the use of size 14 and 18 barsbecause of lack of sufficient information onthe behavior at anchorage locations of suchbars when subjected to load reversalssimulating earthquake effects.

(g) Transverse reinforcement: Because theductile behavior of earthquake-resistant

512 Chapter 10

frames designed to current codes ispremised on the ability of the beams todevelop plastic hinges with adequaterotational capacity, it is essential to insurethat shear failure does not occur before theflexural capacity of the beams has beendeveloped. Transverse reinforcement isrequired for two related functions: (i) toprovide sufficient shear strength so that thefull flexural capacity of the member can bedeveloped, and (ii) to insure adequaterotation capacity in plastic-hinging regionsby confining the concrete in thecompression zones and by providing lateralsupport to the compression steel. To beequally effective with respect to bothfunctions under load reversals, thetransverse reinforcement should be placedperpendicular to the longitudinalreinforcement.

Shear reinforcement in the form ofstirrups or stirrup ties is to be designed forthe shear due to factored gravity loads andthe shear corresponding to plastic hingesforming at both ends of a beam. Plastic endmoments associated with lateraldisplacement in either direction should beconsidered (Figure 10-16). It is important tonote that the required shear strength inbeams (as in columns) is determined by theflexural strength of the frame member (aswell as the factored loads acting on themember), rather than by the factored shearforce calculated from a lateral loadanalysis. The use of the factor 1.25 on fy forcalculating the probable moment strength isintended to allow for the actual steelstrength exceeding the specified minimumand also recognizes that the strain inreinforcement of sections undergoing largerotations can enter the strain-hardeningrange.

To allow for load combinations notaccounted for in design, a minimumamount of web reinforcement is requiredthroughout the length of all flexuralmembers. Within regions of potentialhinging, stirrup ties or hoops are required.

A hoop may be made of two pieces ofreinforcement: a stirrup having 135° hookswith 6-diameter extensions anchored in theconfined core and a crosstie to close thehoop (see Figure 10-37). Consecutive tiesare to have their 90° hooks on oppositesides of the flexural member.

Figure 10-37. Single and two-piece hoops

3.Frame members subjected to axial loadand bending. ACI Chapter 21 makes thedistinction between columns or beam—columns and flexural members on the basis ofthe magnitude of the factored axial load actingon the member. Thus, if the factored axial loaddoes not exceed Ag cf ′ /10, the member falls

under the category of flexural members, theprincipal design requirements for which werediscussed in the preceding section. When thefactored axial force on a member exceedsAg cf ′ /10, the member is considered a beam—

column. Major requirements governing thedesign of such members in structures located inareas of high seismic risk are given below.

(a) Limitations on section dimensions:shortest cross-sectional dimension ≥ 12 in.(measured on line passing throughgeometric centroid);

4.0dimensionlarperpendicu

dimensionshortest ≥


(b) Limitations on longitudinal reinforcement:

ρmin = 0.01, ρmax = 0.06

(c) Flexural strength of columns relative tobeams framing into a joint (the so-called“strong column-weak beam” provision):

∑∑ ≥ ge MM5

6(10-5)

where

∑Me = sum of the design flexural strengthsof the columns framing into joint. Columnflexural strength to be calculated for thefactored axial force, consistent with thedirection of the lateral loading considered,that results in the lowest flexural strength

∑Mg =sum of design flexural strengths ofbeams framing into joint

The lateral strength and stiffness ofcolumns not satisfying the above requirementare to be ignored in determining the lateralstrength and stiffness of the structure. Suchcolumns have to be designed in accordancewith the provisions governing members notproportioned to resist earthquake-inducedforces, as contained in the ACI section 21.7.However, as the commentary to the Codecautions, any negative effect on the buildingbehavior of such non-conforming columnsshould not be ignored. The potential increase inthe base shear or of torsional effects due to thestiffness of such columns should be allowedfor.(d) Restriction on use of lap splices: Lap

splices are to be used only within themiddle half of the column height and are tobe designed as tension splices.

(e) Welded splices or mechanical connectorsfor longitudinal reinforcement: Weldedsplices or mechanical connectors may beused at any section of a column, providedthat:(1) they are used only on alternate

longitudinal bars at a section;

(2) the distance between splices along thelongitudinal axis of the reinforcement is≥ 24 in.

(f) Transverse reinforcement for confinementand shear: As in beams, transversereinforcement in columns must provideconfinement to the concrete core and lateralsupport for the longitudinal bars as well asshear resistance. In columns, however, thetransverse reinforcement must all be in theform of closed hoops or continuous spiralreinforcement. Sufficient reinforcementshould be provided to satisfy therequirements for confinement or shear,whichever is larger.

(1) Confinement requirements (see Figure10-38):

– Volumetric ratio of spiral or circular hoopreinforcement:

−

≥

yh

c

ch

g

yh

c

s

f

f

A

A

f

f

'145.0

'12.0

ρ (10-6)

fyh = specified yield strength of transverse reinforcement, inlb/in.2

Ach = core area of column section,measured to the outside oftransverse reinforcement, in in.2

– Rectangular hoop reinforcement, total cross-sectional area, within spacings:

−

≥

yh

c

ch

gc

yh

cc

f

f

A

Ash

f

fsh

'13.0

' 0.09

Ash (10-7)

where

hc = cross-sectional dimension of column core, measured center-to-center of confining reinforcements = spacing of transverse reinforcement

514 Chapter 10

measured along axis of member, in in.smax = min ¼(smallest cross-sectional dimension of member),4 in. maximum permissible spacing in plane of cross-section between legs of overlapping hoops or cross ties is 14 in.(2) Confinement reinforcement is to be

provided over a length l0 from eachjoint face or over distances l0 on bothsides of any section where flexuralyielding may occur in connection withlateral displacements of the frame,where

l0

≥.18

)(6/1

in

memberofspanclear

memberofddepth

UBC-97 further requires thatconfinement reinforcement be providedat any section of a column where thenominal axial strength, φ Pn is less thanthe sum of the shears corresponding to

the probable flexural strengths of thebeams (i.e., based on fs = 1.25fy and φ =1.0) framing into the column above thelevel considered.

(3) over segments of a column not providedwith transverse reinforcement inaccordance with Eqs. (10-6) and (10-7)and the related requirements describedabove, spiral or hoop reinforcement isto be provided, with spacing notexceeding 6 × (diameter of longitudinalcolumn bars) or 6 in., whichever is less.

(4) Transverse reinforcement for shear incolumns is to be based on the shearassociated with the maximum probablemoment strength, Mpr, at the columnends (using fs = 1.25 fy and φ = 1.0)corresponding to the range of factoredaxial forces acting on the column. Thecalculated end moments of columnsmeeting at a joint need not exceed thesum of the probable moment strengthsof the girders framing into the joint.However, in no case should the design

Figure 10-38. Confinement requirements for column ends.


shear be less than the factored sheardetermined by analysis of the structure.

(g) Column supporting discontinued walls:Columns supporting discontinued shearwalls or similar stiff elements are to beprovided with transverse reinforcement overtheir full height below the discontinuity (seeFigure 10-39) when the axial compressiveforce due to earthquake effects exceedsAg cf ′ /10.

The transverse reinforcement in columnssupporting discontinued walls be extendedabove the discontinuity by at least thedevelopment length of the largest verticalbar and below the base by the same amountwhere the column rests on a wall. Where thecolumn terminates in a footing or mat, thetransverse reinforcement is to be extendedbelow the top of the footing or mat adistance of at least 12 in.

Discussion:

(b) Reinforcement ratio limitation: ACIChapter 21 specifies a reduced upper limitfor the reinforcement ratio in columns fromthe 8% of Chapter 10 of the code to 6%.However, construction considerations willin most cases place the practical upper limiton the reinforcement ratio ρ near 4%.Convenience in detailing and placingreinforcement in beam-column connectionsmakes it desirable to keep the columnreinforcement low. The minimum reinforcement ratio isintended to provide for the effects of time-dependent deformations in concrete underaxial loads as well as maintain a sizabledifference between cracking and yieldmoments.

Figure 10-39. Columns supporting discontinued wall.

516 Chapter 10

)M(M5

6)M(M br

prblpr

cbpr

ctpr +≥+

Figure 10-40. Strong column-weak beam framerequirements.

(c) Relative column-to-beam flexural strengthrequirement: To insure the stability of aframe and maintain its vertical-load-carrying capacity while undergoing largelateral displacements, ACI Chapter 21requires that inelastic deformations begenerally restricted to the beams. This is theintent of Equation 10-5 (see Figure 10-40).As mentioned, formation of plastic hinges atboth ends of most columns in a story canprecipitate a sidesway mechanism leading tocollapse of the story and the structure aboveit. Also, as pointed out in Section 10.3.4under “Beam—Column Joints,” compliancewith this provision does not insure thatplastic hinging will not occur in thecolumns. If Equation 10-5 is not satisfied at a joint,columns supporting reactions from such ajoint are to be provided with transversereinforcement over their full height.Columns not satisfying Equation 10-5 are tobe ignored in calculating the strength andstiffness of the structure. However, sincesuch columns contribute to the stiffness ofthe structure before they suffer severe lossof strength due to plastic hinging, theyshould not be ignored if neglecting themresults in unconservative estimates of designforces. This may occur in determining thedesign base shear or in calculating theeffects of torsion in a structure. Columns notsatisfying Equation 10-5 should satisfy theminimum requirements for members not

proportioned to resist earthquake-inducedforces, discussed under item 6 below.

(f)Transverse reinforcement for confinementand shear: Sufficient transversereinforcement in the form of rectangularhoops or spirals should be provided tosatisfy the larger requirement for eitherconfinement or shear.

Circular spirals represent the mostefficient form of confinementreinforcement. The extension of such spiralsinto the beam—column joint, however, maycause some construction difficulties.

Rectangular hoops, when used in place ofspirals, are less effective with respect toconfinement of the concrete core. Theireffectiveness may be increased, however,with the use of supplementary cross-ties.The cross-ties have to be of the same sizeand spacing as the hoops and have to engagea peripheral longitudinal bar at each end.Consecutive cross-ties are to be alternatedend for end along the longitudinalreinforcement and are to be spaced nofurther than 14 in. in the plane of the columncross-section (see Figure 10-41). Therequirement of having the cross-ties engagea longitudinal bar at each end would almostpreclude placing them before thelongitudinal bars are threaded through.

Figure 10-41. Rectangular transverse reinforcement incolumns.


In addition to confinementrequirements, the transverse reinforcementin columns must resist the maximum shearassociated with the formation of plastichinges at the column ends. Although thestrong column-weak beam provisiongoverning relative moment strengths ofbeams and columns meeting at a joint isintended to have most of the inelasticdeformation occur in the beams of a frame,the code recognizes that hinging can occurin the columns. Thus, the shearreinforcement in columns is to be based onthe shear corresponding to the developmentof the probable moment strengths at theends of the columns, i.e., using fs = 1.25 fy

and φ = 1.0. The values of these endmoments —obtained from the P-Minteraction diagram for the particularcolumn section considered—are to be the

maximum consistent with the range ofpossible factored axial forces on the column.Moments associated with lateraldisplacements of the frame in bothdirections, as indicated in Figure 10-42,should be considered. The axial loadcorresponding to the maximum momentcapacity should then be used in computingthe permissible shear in concrete, Vc.

(g) Columns supporting discontinued walls:Columns supporting discontinued shearwalls tend to be subjected to large shearsand compressive forces, and can beexpected to develop large inelasticdeformations during strong earthquakes;hence the requirement for transversereinforcement throughout the height of suchcolumns according to equations (10-6) and(10-7) if the factored axial force exceeds Ag

cf ′ /10

h

MMVV

bpr

tpr

bt

+==

Figure 10-42. Loading cases for design of shear reinforcement for columns.

518 Chapter 10

4. Beam-column connections. In conventional reinforced-concretebuildings, the beam-column connectionsusually are not designed by the structuralengineer. Detailing of reinforcement withinthe joints is normally relegated to adraftsman or detailer. In earthquake resistantframes, however, the design of beam-columnconnections requires as much attention as thedesign of the members themselves, since theintegrity of the frame may well depend onthe proper performance of such connections.Because of the congestion that may resultfrom too many bars converging within thelimited space of the joint, the requirementsfor the beam—column connections have tobe considered when proportioning thecolumns of a frame. To minimize placementdifficulties, an effort should be made to keepthe amount of longitudinal reinforcement inthe frame members on the low side of thepermissible range.

The provisions of ACI Chapter 21 dealingwith beam-column joints relate mainly to:(a) Transverse reinforcement for confinement:

Minimum confinement reinforcement, asrequired for potential hinging regions incolumns and defined by Equations 10-6 and10-7, must be provided in beam-columnjoints. For joints confined on all four sidesby framing beams, a 50% reduction in therequired amount of confinementreinforcement is allowed, the requiredamount to be placed within the depth of theshallowest framing member. In this case,the reinforcement spacing is not to exceedone-quarter of the minimum memberdimension nor 6 in. (instead of 4 in. fornon-confined joints). A framing beam isconsidered to provide confinement to ajoint if it has a width equal to at least three-quarters of the width of the column intowhich it frames.

(b) Transverse reinforcement for shear: Thehorizontal shear force in a joint is to becalculated by assuming the stress in thetensile reinforcement of framing beamsequal to 1.25fy (see Figure 10-21). The

shear strength of the connection is to becomputed (for normal-weight concrete) as

=

casesother allfor

'12

sides opposite twoon

or sides threeon confined jointsfor

'15

sidesfour all on confined jointsfor

'20

jc

jc

jc

c

Af

Af

Af

V

φ

φ

φ

φ

whereφ = 0.85 (for shear)Aj = effective (horizontal) cross-sectional area of joint in a plane parallel to the beam reinforcement generating the shear forces (see Figure 10-43)

Figure 10-43. Beam-column panel zone.

As illustrated in Fig. 10-43, the effective area,Aj, is the product of the joint depth and theeffective width of the joint. The joint depth istaken as the overall depth of the column(parallel to the direction of the shearconsidered), while the effective width of thejoint is to be taken equal to the width of the


column if the beam and the column are of thesame width, or, where the column is wider thanthe framing beam, is not to exceed the smallerof:– beam width plus the joint depth, and– beam width plus twice the least column

projection beyond the beam side, i.e. thedistance x in Fig. 10-43.

For lightweight concrete, Vc is to be taken asthree-fourths the value given above for normal-weight concrete.(c) Anchorage of longitudinal beam

reinforcement terminated in a column mustbe extended to the far face of the confinedcolumn core and anchored in accordancewith the requirements given earlier fordevelopment lengths of longitudinal bars intension and according to the relevant ACIChapter 12 requirements for bars incompression.Where longitudinal beam bars extendthrough a joint ACI Chapter 21 requiresthat the column depth in the direction ofloading be not less than 20 times thediameter of the largest longitudinal beambar. For lightweight concrete, thedimension shall be not less than 26 timesthe bar diameter.

Discussion:

(a) Transverse reinforcement for confinement:The transverse reinforcement in a beam-column connection helps maintain thevertical-load-carrying capacity of the jointeven after spalling of the outer shell. It alsohelps resist the shear force transmitted by theframing members and improves the bondbetween steel and concrete within the joint.

The minimum amount of transversereinforcement, as given by Equations 10-6and 10-7, must be provided through the jointregardless of the magnitude of the calculatedshear force in the joint. The 50% reduction inthe amount of confinement reinforcementallowed for joints having beams framing intoall four sides recognizes the beneficialconfining effect provided by these members.

(b) Results of tests reported in Reference 10-41indicate that the shear strength of joints isnot too sensitive to the amount of transverse(shear) reinforcement. Based on theseresults, ACI Chapter 21 defines the shearstrength of beam-column connections as afunction only of the cross-sectional area ofthe joint, (Aj) and cf ′ (see Section 10.3.4

under “Beam-Column Joints”).When the design shear in the joint

exceeds the shear strength of the concrete,the designer may either increase the columnsize or increase the depth of the beams. Theformer will increase the shear capacity of thejoint section, while the latter will tend toreduce the required amount of flexuralreinforcement in the beams, withaccompanying decrease in the sheartransmitted to the joint. Yet anotheralternative is to keep the longitudinal beambars from yielding at the faces of thecolumns by detailing the beams so thatplastic hinging occurs away from the columnfaces.

(c) The anchorage or development-lengthrequirements for longitudinal beamreinforcement in tension have been discussedearlier under flexural members. Note that lapsplicing of main flexural reinforcement is notpermitted within the joint.

5. Shear Walls. When properly proportionedso that they possess adequate lateralstiffness to reduce inter-story distortionsdue to earthquake-induced motions, shearwalls or structural walls reduce thelikelihood of damage to the non-structuralelements of a building. When used withrigid frames, walls form a system thatcombines the gravity-load-carryingefficiency of the rigid frame with thelateral-load-resisting efficiency of thestructural wall. In the form of coupled wallslinked by appropriately proportionedcoupling beams (see Section 10.3.4 under“Coupled Walls”), alone or in combinationwith rigid frames, structural walls provide alaterally stiff structural system that allowssignificant energy dissipation to take place

520 Chapter 10

in the more easily repairable couplingbeams. Observations of the comparativeperformance of rigid-frame buildings andbuildings stiffened by structural wallsduring earthquakes(10-77) have pointed to theconsistently better performance of the latter.The performance of buildings stiffened byproperly designed structural walls has beenbetter with respect to both life safety anddamage control. The need to insure thatcritical facilities remain operational after amajor tremor and the need to reduceeconomic losses from structural andnonstructural damage, in addition to theprimary requirement of life safety (i.e., nocollapse), has focused attention on thedesirability of introducing greater lateralstiffness in earthquake-resistant multistorybuildings. Where acceleration-sensitiveequipment is to be housed in a structure, thegreater horizontal accelerations that may beexpected in laterally stiffer structuresshould be allowed or provided for.

The principal provisions of ACI Chapter21 relating to structural walls and diaphragmsare as follows (see Figure 10-44):

(a) Walls (and diaphragms) are to be providedwith shear reinforcement in two orthogonaldirections in the plane of the wall. Theminimum reinforcement ratio for bothlongitudinal and transverse directions is

0025.0 ≥== ncv

svv

A

A ρρ

where the reinforcement is to be continuousand distributed uniformly across the sheararea, and

Acv = net area of concrete section, i.e., product of thickness and width of wall section

Asv = projection on Acv of area of shear reinforcement crossing the plane of Acv

Figure 10-44. Structural wall design requirements.


ρn = reinforcement ratio corresponding to plane perpendicular to plane of Acv

The maximum spacing of reinforcement is18 in. At least two curtains of reinforcement,each having bars running in the longitudinaland transverse directions, are to be providedif the in-plane factored shear force assigned

to the wall exceeds 2Acv cf ′ . If the

(factored) design shear force does not exceed

Acv cf ′ , the shear reinforcement may be

proportioned in accordance with theminimum reinforcement provisions of ACIChapter 14.

(b) Boundary elements: Boundary elements areto be provided, both along the verticalboundaries of walls and around the edges ofopenings, if any, when the maximumextreme-fiber stress in the wall due tofactored forces including earthquake effects

exceeds 0.2 cf ′ . The boundary members

may be discontinued when the calculatedcompressive stress becomes less than

0.15 cf ′ . Boundary elements need not be

provided if the entire wall is reinforced inaccordance with the provisions governingtransverse reinforcement for memberssubjected to axial load and bending, asgiven by Equations 10-6 and 10-7.Boundary elements of structural walls are tobe designed to carry all the factored verticalloads on the wall, including self-weight andgravity loads tributary to the wall, as well asthe vertical forces required to resist theoverturning moment due to factoredearthquake loads. Such boundary elementsare to be provided with confinementreinforcement in accordance with Equations10-6 and 10-7.Welded splices and mechanical connectionsof longitudinal reinforcement of boundaryelements are allowed provided that:

1) they are used only on alternate longitudinalbars at a section;

2) the distance between splices along thelongitudinal axis of the reinforcement is ≥24 in.

The requirements for boundary elements inUBC-97 and IBC-2000 provisions which areessentially similar are much more elaborate anddetailed in comparison with ACI-95. In thesetwo provisions , the determination of boundaryzones may be based on the level of axial, shear,and flexural wall capacity as well as wallgeometry. Alternatively, if such conditions arenot met, it may be based on the limitations onwall curvature ductility determined based oninelastic displacement at the top of the wall.Using such a procedure, the analysis should bebased on cracked shear area and moment ofinertia properties and considering the responsemodification effects of possible non-linearbehavior of building. The requirements ofboundary elements using these provisions arediscussed in detail under item (f) below.

(c) Shear strength of walls (and diaphragms):For walls with a height-to-width ratio hw/lw

≥ 2.0, the shear strength is to be determinedusing the expression:

( )ynccvn ffAV ρφφ += '2

whereφ = 0.60, unless the nominal shear strength provided exceeds the shear corresponding to development of nominal flexural capacity of the wallA cv= net area as defined earlierhw = height of entire wall or of segment of wall consideredlw= width of wall (or segment of wall) in direction of shear force

For walls with hw/lw < 2.0, the shear may bedetermined from

( )yncccvn ffAV ραφφ += '

where the coefficient αc varies linearly froma value of 3.0 for hw/lw = 1.5 to 2.0 for hw/lw

522 Chapter 10

= 2.0. Where the ratio hw/lw <2.0 , ρv can notbe less than ρn.

Where a wall is divided into severalsegments by openings, the value of the ratiohw/lw to be used in calculating Vn for anysegment is not to be less than thecorresponding ratio for the entire wall. The nominal shear strength Vn of all wallsegments or piers resisting a common lateral

force is not to exceed 8Acv cf ′ where Acv is

the total cross-sectional area of the walls.The nominal shear strength of any individualsegment of wall or pier is not to exceed

10Acp cf ′ where Acp is the cross-sectional

area of the pier considered.(d) Development length and splices: All

continuous reinforcement is to be anchoredor spliced in accordance with provisionsgoverning reinforcement in tension, asdiscussed for flexural members.

Where boundary elements are present, thetransverse reinforcement in walls is to beanchored within the confined core of theboundary element to develop the yield stress intension of the transverse reinforcement. Forshear walls without boundary elements, thetransverse reinforcement terminating at theedges of the walls are to be provided withstandard hooks engaging the edge (vertical)reinforcement. Otherwise the edgereinforcement is to be enclosed in U-stirrupshaving the same size and spacing as, andspliced to, the transverse reinforcement. Anexception to this requirement is when Vu in the

plane of the wall is less than Acv cf ′ .

(e) Coupling beams: UBC-97 and IBC-2000provide similar guidelines for coupling beamsin coupled wall structures. For coupling beamswith ln/d≥ 4, where ln = clear length of couplingbeam and d = effective depth of the beam,conventional reinforcement in the form of topand bottom reinforcement can be used.However, for coupling beams with ln/d< 4 , and

factored shear stress exceeding 4 cf ′ ,

reinforcement in the form of two intersectinggroups of symmetrical diagonal bars to be

provided. The design shear stress in coupling

beams should be limited to 10φ cf ′ where φ =

0.85.(f) Provisions of IBC-2000 and UBC-97

related to structural walls: These provisionstreat shear walls as regular memberssubjected to combined flexure and axialload. Since the proportions of such wallsare generally such that they function asregular vertical cantilever beams, the strainsacross the depth of such members (in theplane of the wall) are to be assumed to varylinearly, just as in regular flexuralmembers, i.e., the nonlinear straindistribution associated with deep beamsdoes not apply. The effective flange widthto be assumed in designing I-, L-, C- or T-shaped shear wall sections, i.e., sectionsformed by intersecting connected walls,measured from the face of the web, shallnot be greater than (a) one-half the distanceto the adjacent shear wall web, or (b) 15percent of the total wall height for theflange in compression or 30 percent of thetotal wall height for the flange in tension,not to exceed the total projection of theflange.

Walls or portions of walls subject to anaxial load Pu> 0.35 P0 shall not beconsidered as contributing to theearthquake resistance of a structure. Thisfollows from the significantly reducedrotational ductility of sections subjected tohigh compressive loads (see Fig. 10-11(b)).

When the shear Vu in the plane of the

wall exceeds Acv cf ′ , the need to develop

the yield strength in tension of thetransverse reinforcement is expressed in therequirement to have horizontalreinforcement terminating at the edges ofshear walls, with or without boundaryelements, anchored using standard hooksengaging the (vertical) edge reinforcementor alternatively, having the vertical edgereinforcement enclosed in “U” stirrups ofthe same size and spacing as, and spliced to,the horizontal reinforcement.


Shear Wall Boundary Zones - Thedetailing requirements for boundary zones,to be described subsequently, need not besatisfied in walls or portions of walls where

≤otherwise 05.0

sections walllsymmetrica

lly geometricafor 10.0

'

'

cg

cg

ufA

fA

P

and either

'3 or 0.1 cwwuuu

u fhlVlV

M ≤≤

where lw is the length of the entire wall in thedirection of the shear force, and hw is the heightof the wall.

Shear walls or portions of shear walls notmeeting the above conditions and having Pu <0.35 Po (so that they can be considered ascontributing to the earthquake resistance of thestructure) are to be provided with boundaryzones at each end having a length varyinglinearly from 0.25lw for Pu = 0.35Po to 0.15lw

for Pu = 0.15Po, with a minimum of 0.15lw andare to be detailed as will be described.

Alternatively, the requirements of boundaryzones not meeting the above conditions may bebased on the determination of the compressivestrain levels at wall edges using cracked sectionproperties. Boundary zone detailing, however,is to be provided over the portions of the wallwhere compressive strains exceed 0.003. It isimportant to note that compressive strains arenot allowed to exceed 0.015.

For shear walls in which the flexural limitstate response is governed by yielding at thebase of the wall, the total curvature demand(φ t) can be obtained from:

yppw

it llh

φφ +−

∆=

)2/(

where∆i = inelastic deflection at the top of the wall

= (∆t - ∆y)∆t = total deflection at the top of the wall equal ∆M, using cracked section properties, or may be taken as 2∆M , using gross section properties. ∆y = displacement at the top of wall corresponding to yielding of the tension reinforcement at critical section, or may be taken as

(M′n/ME) ∆E , where ME equals unfactored moment at critical section when top of wall is displaced ∆E . M’n is nominal flexural

strength of critical section at P′u.hw = height of the walllp = height of the plastic hinge above

critical section and which shall be established on the basis of substantiated test data or may be alternatively taken at 0.5lw

φy = yield curvature which may beestimated at 0.003/lw

If φt is less than or equal to 0.003/c′u,boundary zone details as defined below are notrequired. c′u is the neutral axis depth at P′u andM′n. If φt exceeds 0.003/c′u , the compressivestrains may be assumed to vary linearly overthe depth c′u , and have maximum value equalto the product of c′u and φt .

The use of the above procedure is furtherdiscussed with the aid of the design example atthe end of this Chapter.

Shear wall boundary zone detailingrequirements. When required as discussedabove, the boundary zones in shear walls are tobe detailed in accordance with the followingrequirements:(1) Dimensional requirements:

(a) The minimum section dimension of theboundary zone shall be lw/16.

(b) Boundary zones shall extend above theelevation where they are required adistance equal to the developmentlength of the largest vertical bar in theboundary zone. Extensions of theboundary zone lateral reinforcementbelow its base shall conform to the samerequirements as for columns terminating

524 Chapter 10

on a mat or footing. However, thetransverse boundary zone reinforcementneed not extend above the base of theboundary zone a distance greater thanthe larger of lw or Mu/4Vu.

(c) Boundary zones shall have a minimumlength of 18 inches (measured along thelength) at each end of the wall orportion of wall.

(d) In I-, L-, C- or T-section walls, theboundary zone at each end shall includethe effective flange width and shallextend at least 12 in. into the web.

(2) Confinement Reinforcement:(a) All vertical reinforcement within the

boundary zone shall be confined byhoops or cross-ties having a steel cross-sectional area

Ash> 0.09 h fc′ / fyh

(b) Hoops and cross-ties shall have avertical spacing,

×<

zone)boundary within

bar rticallargest ve of(diameter 6

in. 6

maxS

(c) The length-to-width ratio of the hoopsshall not exceed 3; and all adjacenthoops shall be overlapping.

(d) Cross-ties or legs of overlapping hoopsshall not be spaced farther apart than 12in. along the wall.

(e) Alternate vertical bars shall be confinedby the corner of a hoop or cross-tie.

(3) Horizontal reinforcement:(a)All horizontal reinforcementterminating within a boundary zoneshall be anchored as described earlier,

i.e., when Vu > Acv cf ′ , horizontal

reinforcement are to be provided withstandard hooks or be enclosed in U-stirrups having the same size andspacing as, and spliced to, thehorizontal bars.

(b)Horizontal reinforcement shall notbe lap spliced within the boundaryzone.

(4) Vertical reinforcement:(a) Vertical reinforcement shall be

provided to satisfy all tension andcompression requirementsindicated by analysis. (Note againthat, in contrast to earlier editionsof the code, there is no longer thestipulation of rather arbitraryforces that “boundary elements”,and hence the vertical steelreinforcement in these, are to bedesigned for.)

(b) Area of vertical reinforcement,

>×

zoneboundary the

of edge eachat bars 5 No. Two

zone)boundary of(area 0.005

vA

(c) Lap splices of verticalreinforcement within theboundary zone shall be confinedby hoops and crossties. Thespacing of hoops and crosstiesconfining lap-spliced verticalreinforcement shall not exceed 4in.

Discussion:

(a) The use of two curtains of reinforcement inwalls subjected to significant shear (i.e., >2Acv fc′) serves to reduce fragmentation andpremature deterioration of the concrete underload reversals into the inelastic range.Distributing the reinforcement uniformlyacross the height and width of the wall helpscontrol the width of inclined cracks.

(b) ACI Chapter 21 allows calculation of theshear strength of structural walls using acoefficient αc = 2.0. However, advantagecan be taken of the greater observed shearstrength of walls with low height-to-widthratios hw/lw by using an αc value of up to 3.0for walls with hw/lw = 1.5 or less.


The upper bound on the average nominalshear stress that may be developed in any

individual segment of wall (10 cf ′ ) is

intended to limit the degree of shearredistribution among several connected wallsegments. A wall segment refers to a part ofa wall bounded by openings or by anopening and an edge.

It is important to note that ACI Chapter21 requires the use of a strength-reductionfactor φ for shear of 0.6 for all members(except joints) where the nominal shearstrength is less than the shear correspondingto the development of the nominal flexuralstrength of the member. In the case ofbeams, the design shears are obtained byassuming plastic end momentscorresponding to a tensile steel stress of1.25fy (see Figure 10-16). Similarly, for acolumn the design shears are determined notby applying load factors to shears obtainedfrom a lateral load analysis, but fromconsideration of the maximum probablemoment strengths at the column endsconsistent with the axial force on thecolumn. This approach to shear design isintended to insure that even when flexuralhinging occurs at member ends due toearthquake-induced deformations, no shearfailure would develop. Under the aboveconditions, ACI Chapter 21 allows the use ofthe normal strength-reduction factor forshear of 0.85. When design shears are notbased on the condition of flexural strengthbeing developed at member ends, the coderequires the use of a lower shear strength-reduction factor to achieve the same result,that is, prevention of premature shear failure.

As pointed out earlier, in the case ofmultistory structural walls, a conditionsimilar to that used for the shear design ofbeams and columns is not so readilyestablished. This is so primarily because themagnitude of the shear at the base of a(vertical cantilever) wall, or at any levelabove, is influenced significantly by theforces and deformations beyond theparticular level considered. Unlike the

flexural behavior of beams and columns in aframe, which can be considered as close-coupled systems (i.e., with the forces in themembers determined by the forces anddisplacements within and at the ends of themember), the state of flexural deformation atany section of a structural wall (a far-coupledsystem) is influenced significantly by thedisplacements of points far removed fromthe section considered. Results of dynamicinelastic analyses of isolated structural wallsunder earthquake excitation(10-3) also indicatethat the base shear in such walls is stronglyinfluenced by the higher modes of response.

A distribution of static lateral forcesalong the height of the wall essentiallycorresponding to the fundamental moderesponse, such as is assumed by mostcodes,(10-1) will produce flexural yielding atthe base if the section at the base is designedfor such a set of forces. Other distributions oflateral forces, with a resultant acting closerto the base of the wall, can produce yieldingat the base only if the magnitude of theresultant horizontal force, and hence the baseshear, is increased. Results of the study ofisolated walls referred to above,(10-3) whichwould also apply to frame—shear-wallsystems in which the frame is flexiblerelative to the wall, in fact indicate that for awide range of wall properties and inputmotion characteristics, the resultant of thedynamic horizontal forces producingyielding at the base of the wall generallyoccurs well below the two-thirds-of-total-height level associated with the fundamental-mode response (see Figure 10-24). Thiswould imply significantly larger base shearsthan those due to lateral forces distributedaccording to the fundamental mode response.The study of isolated walls mentioned aboveindicates ratios of maximum dynamic shearsto “fundamental-mode shears” (i.e., shearsassociated with horizontal forces distributedaccording to the fundamental-moderesponse, as used in codes) ranging from 1.3to 4.0, the value of the ratio increases with

526 Chapter 10

increasing fundamental period (see Figure10-23).

(c) Since multistory structural walls behaveessentially as vertical cantilever beams, thehorizontal transverse reinforcement is calledupon to act as web reinforcement. As such,these bars have to be fully anchored in theboundary elements, using standard 90°hooks whenever possible.

(d) ACI Chapter 21 uses an extreme-fibercompressive stress of 0.2fc′, calculated usinga linearly elastic model based on grosssections of structural members and factoredforces, as indicative of significantcompression. Structural walls subjected tocompressive stresses exceeding this valueare generally required to have boundaryelements.

Figure 10-45 illustrates the conditionassumed as basis for requiring that boundaryelements of walls be designed for all thegravity loads (W) as well as the verticalforces associated with overturning of thewall due to earthquake forces (H). Thisrequirement assumes that the boundaryelement alone may have to carry all thevertical (compressive) forces at the criticalwall section when the maximum horizontalearthquake force acts on the wall. Underload reversals, such a loading conditionimposes severe demands on the concrete inthe boundary elements; hence therequirement for confinement reinforcementsimilar to those for frame memberssubjected to axial load and bending.Diaphragms of reinforced concrete, such asfloor slabs, that are called upon to transmithorizontal forces through bending and shearin their own plane, are treated in much thesame manner as structural walls.

6. Frame members not forming part of lateral-force-resisting system. Frame members that arenot relied on to resist earthquake-induced forcesneed not satisfy the stringent requirementsgoverning lateral-load-resisting elements. Theserelate particularly to the transversereinforcement requirements for confinementand shear. Non-lateral-load-resisting elements,

whose primary function is the transmission ofvertical loads to the foundation, need complyonly with the reinforcement requirements ofACI Chapter 21, in addition to those found inthe main body of the code.

Figure 10-45. Loading condition assumed for design ofboundary elements of structural walls.

The 1994 Northridge earthquake caused thecollapse or partial collapse of at least twoparking structures that could be attributedprimarily to the failure of interior columnsdesigned to gravity loads only. Following theexperience, the requirements for framemembers not proportioned to resist forcesinduced by earthquake motions have beenextensively rewritten for the ACI 95 code. Aflow chart is provided in Figure 10-46 for easein understanding the new provisions. Therequirements are as follows:

A special requirement for non-lateral-load-resisting elements is that they be checked foradequacy with respect to a lateral displacementrepresenting the expected actual displacementof the structure under the design earthquake.For the purpose of this check, ACI Chapter 21uses a value of twice the displacementcalculated under the factored lateral loads, or2×1.7 = 3.4 times the displacement due to thecode-specified loads. This effect is combinedwith the effects of dead or dead and live loadwhichever is critical. If Mu and Vu for anelement of gravity system are less than the


corresponding nominal values, that element isgoing to remain elastic under the designearthquake displacements. If such an element isa beam (Pu≤ Ag fc′/10), it must conform tosection 2 described earlier for minimumlongitudinal reinforcement requirements. Inaddition, stirrups spaced at no more than d/2must be provided throughout the length of themember. If such an element is a column, it mustconform to some of the requirements listedunder sections 2 and 3 for longitudinal andshear reinforcement. In addition, similarrequirements for cross-ties under section 3(f),discussion, must be met. Also ties at amaximum spacing of so must not exceed sixtimes the smallest longitudinal bar diameter,nor 6 in. Further, if Pu> 0.35 Po, the amount oftransverse reinforcement provided must be noless than one-half that required by 3(f).

If Mu and Vu for an element of gravitysystem exceeds the corresponding nominalvalues, then it is likely to become inelasticunder the design earthquake displacements.Also if deformation compatibility is notchecked, this condition will be assumed to bethe case. In that case, the structural materialmust satisfy the requirements described insection 1 and splices of reinforcement mustsatisfy section 2(e). If such an element is abeam (Pu ≤ Ag fc′ /10), it must conform tosections 2(b), and 2(g)- (5) and (6). In addition,the stirrups at no more than d/2 must beprovided throughout the length of the member.If it is a column, it must be provided with fullductile detailing in accordance with section3(f), 3(g), and 4(a) as well as sections 2(g)-(5)and (6).

7. Frames in regions of moderate seismicrisk. Although ACI Chapter 21 does not define“moderate seismic risk” in terms of acommonly accepted quantitative measure, itassumes that the probable ground-motionintensity in such regions would be a fraction ofthat expected in a high-seismic-risk zone, towhich the major part of Chapter 21 isaddressed. By the above description, an area ofmoderate seismic risk would correspondroughly to zone 2 as defined in UBC-97(10-1) and

ASCE 7-95.(10-72) For regions of moderateseismic risk, the provisions for the design ofstructural walls given in the main body of theACI Code are considered sufficient to providethe necessary ductility. The requirements inACI Chapter 21 for structures in moderate-riskareas relate mainly to frames and are containedin the last section, section 21.8.

The same axial compressive force (Ag fc′ /l0)used to distinguish flexural members fromcolumns in high-seismic-risk areas also appliesin regions of moderate seismicity.(a) Shear design of beams, columns, or two-

way slabs resisting earthquake effects: Themagnitude of the design shear is not to beless than either of the following:(1) The sum of the shear associated with

the development of the nominalmoment strength at each restrained endand that due to factored gravity loads, ifany, acting on the member. This issimilar to the correspondingrequirement for high-risk zones andillustrated in Figure 10-16, except thatthe stress in the flexural tensilereinforcement is taken as fy rather than1.25fy.

(2) The maximum factored shearcorresponding to the design gravity andearthquake forces, but with theearthquake forces taken as twice thevalue normally specified by codes.Thus, if the critical load combinationconsists of dead load (D) + live load (L)+ earthquake effects (E), then thedesign shear is to be computed from

U = 0.75[1.4D + 1.7L + 2(1.87E)]

(b) Detailing requirements for beams: Thepositive moment strength at the face of ajoint must be at least one-third the negativemoment capacity at the same section. (Thiscompares with one-half for high-seismic-risk areas.) The moment strength—positiveor negative—at any section is to be no lessthan one-fifth the maximum momentstrength at either end of a member. Stirrupspacing requirements are identical to thosefor beams in high-seismic-risk areas.

528 Chapter 10

However, closed hoops are not requiredwithin regions of potential plastic hinging.It should be noted that lateral reinforcementfor flexural framing members subjected tostress reversals at supports to consist ofclosed ties, closed stirrups, or spiralsextending around the flexural reinforcementas required according to chapter 7 of ACI318-95.

(c) Detailing requirements for columns: Thesame region of potential plastic hinging (lo)as at the ends of columns in a region ofhigh seismicity is defined at each end of acolumn. The spacing of ties within theregion of potential plastic hinging must notexceed the smallest of 8 times the diameterof the smallest longitudinal bar enclosed;24 times the diameter of the tie bar; or One-half the smallest cross-sectional dimensionof the column, and 12 in. Outside the regionof potential plastic hinging, the spacingmust not exceed twice the above value. Thefirst tie must be located at no more than halfthe above spacing from the joint face.(e) Detailing requirements for two-way

slabs without beams: As mentionedearlier, requirements for flat plates inACI Chapter 21 appear only in thesection relating to areas of moderateseismic risk. This suggests that ACIChapter 21 considers the use of flatplates as acceptable components of thelateral-load-resisting system only forareas of moderate seismicity.

Specific requirements relating to flat-plateand flat-slab reinforcement for frames inmoderate-risk zones are given in ACI Chapter21 and illustrated in the correspondingCommentary.

10.5 DESIGN EXAMPLES —REPRESENTATIVEELEMENTS OF A 12-STORY FRAME - SHEARWALL BUILDING

10.5.1 Preliminaries

A significant part of the damage observed inengineered buildings during earthquakes hasresulted from the effects of major structuraldiscontinuities that were inadequately providedfor. The message here is clear. Unless properprovision is made for the effects of majordiscontinuities in geometry, mass, stiffness, orstrength, it would be prudent on the part of theengineer to avoid such conditions, which areassociated with force concentrations and largeductility demands in localized areas of thestructure. Where such discontinuities areunavoidable or desirable from the architecturalstandpoint, an analysis to obtain estimates ofthe forces associated with the discontinuity isrecommended. IBC-2000(10-61) providesguidelines for estimating design forces instructures with various types of vertical andplan irregularities.

In addition to discontinuities, majorasymmetry, with particular regard to thedisposition in plan of the lateral-load-resistingelements, should be avoided whenever possible.Such asymmetry, which can result in asignificant eccentricity between the center ofstiffness and the center of mass (and hence ofthe resultant inertial force), can produceappreciable torsional forces in the structure.Torsional effects can be critical for cornercolumns or end walls, i.e., elements located farfrom the center of stiffness.

Another important point to consider in thepreliminary design of a structure relates to theeffectiveness of the various lateral-load-resisting components, particularly where thesediffer significantly in deformation capacity.Efficient use of structural components wouldsuggest that the useful range of deformation of


No Yes

No Yes

No Yes

No Yes

Figure 10-46. Requirements for frame members not proportioned to resist forces induced by earthquake motions.

Computed and combined with

effects of 1.05D + 1.28L or 0.9D,

whichever is critical, resulting in Mu,Vu

1

2(e)

2(b)

Stirrups @ d/2 or less throughout

the length of the member

2(b)

2(g)-(5),(6)

Stirrups @ d/2 or less throughout

the length of the member

Relevant Subsection of section 10.4.3 in this Chapter

1. Limitation on material strength2(b). Limitation on flexural reinforcement ratio2(e). Welded splices and mechanically connected reinforcement2g-(5),(6). Shear reinforcement3(b). Limitations on longitudinal reinforcement3(f). Transverse reinforcement for confinement and shear3(g). Columns supporting discontinued walls4(a). Transverse reinforcement for confinement

3(b)

2(g)-(5),(6)

Req’t for crossties 3(f), discussion

S ≤ So for full height

So ≤ 6db (smallest long. Bar) ≤ 6 in.

3(f) ,3(g)

2(g)-(5),(6)

4(a)

Mu > φMn

orVu> φVn

Pu > 0.35Po

10

'cg

u

fAP >

10

'cg

u

fAP >

Not ComputedMoments and shears due to 2

times displacements resulting from

factored lateral forces

Amount of transverse

reinforcement ≥ 1/2 that required

by 3(f)

530 Chapter 10

the principal lateral-load-resisting elements in astructure be of about the same magnitudewhenever practicable. This is illustrated inFigure 10-47a, which shows load—deformationcurves of representative elements (1) and (2) ina structure. Such a design allows all theresisting elements to participate in carrying theinduced forces over the entire range ofdeformation. In Figure 10-47b, the resistingelements (1) and (2) not only possess differentinitial stiffnesses but, more importantly, exhibitdifferent ductilities (not ductility ratios) ordeformation capacities. In such a case, which istypical of a frame—shear-wall structure, thedesign should be aimed at insuring that themaximum probable deformation or lateraldisplacement under dynamic conditions doesnot exceed the deformation capacity ∆2 ofelement (2); or, if the maximum expecteddeformation could exceed ∆2 , then element (1)should be so designed that it can support theadditional load that may come upon it whenelement (2) loses a considerable part of its load-carrying capacity. It is worth noting that,generally, the lateral displacements associatedwith full mobilization of the ductility of rigid(open) frames are such that significantnonstructural damage can be expected. For this

reason, the building codes limit the amount ofdeformation that can be tolerated in thestructure.

The need to tie together all the elementsmaking up a structure or a portion of it that isintended to act as a unit cannot beoveremphasized. This applies to thesuperstructure as well as foundation elements.Where a structure is divided into different partsby expansion joints, as when the various partsdiffer considerably in height, plan size, shape,or orientation, a sufficient gap should beprovided between adjacent parts to prevent theirpounding against each other. To avoidpounding between adjacent buildings or parts ofthe same building when vibrating out of phasewith each other, a gap equal to the square rootof the sum of the squares (SRSS) of themaximum lateral deflections (considering thedeflection amplification factors specified inbuilding codes) of the two structures under thedesign (code-specified) lateral forces, or theSRSS of the maximum deflections of the twostructures as indicated by a dynamic analysis,would be desirable.

A good basis for the preliminary design ofan earthquake-resistant building is a structureproportioned to satisfy the requirements for

Figure 10-47. Relative deformation capacity in lateral-load-resisting elements in structure


gravity and wind loads. The planning andlayout of the structure, however, must beundertaken with due consideration of thespecial requirements for earthquake-resistantdesign. Thus, modifications in bothconfiguration and proportions to anticipateearthquake-related requirements should beincorporated at the outset into the basic designfor gravity and wind. Essential to the finisheddesign is particular attention to details that canoften mean the difference between a severelydamaged structure and one with only minor,repairable damage.

10.5.2 Example Designs of Elements of a12-Story Frame-Shear WallBuilding

The application of the earthquake-resistantdesign provisions of IBC-2000 with respect todesign loads and those of ACI 318-95(10-10)

relating to proportioning and detailing ofmembers will be illustrated for representativeelements of a 12-story frame—shear wallbuilding located in seismic zone 4. The use ofthe seismic design load provisions in IBC-2000,is based on the fact that it represents the moreadvanced version, in the sense of incorporatingthe latest revisions reflecting current thinking inthe earthquake engineering profession.

The typical framing plan and section of thestructure considered are shown in Figure 10-48ac and b, respectively. The columns andstructural walls have constant cross-sectionsthroughout the height of the building. The floorbeams and slabs also have the same dimensionsat all floor levels. Although the dimensions ofthe structural elements in this example arewithin the practical range, the structure itself ishypothetical and has been chosen mainly forillustrative purposes. Other pertinent designdata are as follows:

Service loads — vertical:• Live load:

c Reproduced, with modifications, from Reference 10-78,with permission from Van Nostrand ReinholdCompany.

Basic, 50 lb/ft2.Additional average uniform load to

allow for heavier basic load on corridors, 25 lb/ft2.

Total average live load, 75 lb/ft2.Roof live load = 20 lb/ft2

• Superimposed dead load:Average for partitions 20 lb/ft2.Ceiling and mechanical 10 lb/ft2.Total average superimposed dead

load, 30 lb/ft2.Material properties:• Concrete:

fc′ = 4000 lb/in.2 wc = 145 lb/ft3.• Reinforcement:

fy = 60 ksi.

Determination of design lateral forcesOn the basis of the given data and the

dimensions shown in Figure 10-48, the weightsthat may be considered lumped at a floor level(including that of all elements located betweentwo imaginary parallel planes passing throughmid-height of the columns above and below thefloor considered) and the roof were estimatedand are listed in Tables 10-1 and 10-2. Thecalculation of base shear V, as explained inChapter 5, for the transverse and longitudinaldirection is shown at the bottom of Tables 10-1and 10-2. For this example, it is assumed thatthe building is located in Southern Californiawith values of Ss and S1 of 1.5 and 0.6respectively. The site is assumed to be class B(Rock) and the corresponding values of Fa andFb are 1.0. On this basis, the design spectralresponse acceleration parameters SDS and SMI

are 1.0 and 0.4 respectively. At this level ofdesign parameters, the building is classified asSeismic Group D according to IBC-2000. Thebuilding consist of moment resisting frame inthe longitudinal direction, and dual systemconsisting of wall and moment resisting framein the transverse direction. Accordingly, theresponse modification factor, R, to be used is8.0 in both directions.

532 Chapter 10

Calculation of the undamped (elastic) naturalperiods of vibration of the structure in thetransverse direction (N-S)As shown in Figure 10-49 using the storyweights listed in Table 10-1 and memberstiffnesses based on gross concrete sections,yielded a value for the fundamental period of1.17 seconds. The mode shapes and thecorresponding periods of the first five modes ofvibration of the structure in the transversedirection are shown in Figure 10-49. Thefundamental period in the longitudinal (E-W)direction was 1.73 seconds. The mode shapeswere calculated using the Computer ProgramETABS(10-66), based on three dimensionalanalysis. In the computer model, the floors wereassumed to be rigid. Rigid end offsets wereassumed at the end of the members to reflectthe actual behavior of the structure. Theportions of the slab on each side of the beamswere considered in the analysis based on theACI 318-95 provisions. The structure wasassumed to be fixed at the base. The twointerior walls were modeled as panel elementswith end piers (26x26 in.). The correspondingvalues of the fundamental period determinedbased on the approximate formula given inIBC-2000 were 0.85 and 1.27 seconds in the N-S and the E-W directions respectively.However, these values can be increased by 20%provided that they do not exceed thosedetermined from analysis. On this basis, thevalue of T used to calculate the base shearswere 1.02 and 1.52 seconds in the N-S and theE-W directions respectively.

The lateral seismic design forces acting atthe floor levels, resulting from the distributionof the base shear in each principal direction arealso listed in Tables 10-1 and 10-2.

For comparison, the wind forces and storyshears corresponding to a basic wind speed of85 mi/h and Exposure B ( urban and suburbanareas), computed as prescribed in ASCE 7-95,are shown for each direction in Tables 10-1 and10-2.

Lateral load analysis of the structure alongeach principal direction, under the respectiveseismic and wind loads, based on three

dimensional analysis were carried out assumingno torsional effects.

Figure 10-48. Structure considered in design example. (a)Typical floor framing plan. (b) Longitudinal section

Figure 10-49. Undamped natural modes and periods ofvibration of structure in transverse direction


Table 10-1. Design Lateral Forces in Transverse (Short) Direction (Corresponding to Entire Structure).

story Seismic forces Wind forces

FloorLevel

Height,hx, ft

hxk

k=1.26

weight,wx,kips

wx hxk

ft-kips

x103

Cvx Lateral

force,F

xkips

Story

shear

ΣFx, kips

wind

pressure

lbs/ft2

lateral

force

Hx, kips

Story shear

ΣHx, kips

Roof 148 543 2100 1140 0.162 208.8 208.8 21.1 23.0 23.0

11 136 488 2200 1073 0.152 196.0 404.8 20.9 45.6 68.9

10 124 434 2200 955 0.135 174.0 578.8 20.5 44.8 113.4

9 112 382 2200 840 0.120 154.7 733.5 20.2 44.1 157.5

8 100 331 2200 728 0.103 132.8 866.3 19.8 43.2 200.7

7 88 282 2200 620 0.088 113.4 979.7 19.4 42.4 243.1

6 76 234 2200 515 0.073 94.1 1073.8 18.9 41.3 284.4

5 64 189 2200 415 0.059 76.1 1149.9 18.4 40.2 324.6

4 52 145 2200 320 0.045 58.0 1207.9 17.8 38.9 363.5

3 40 104 2200 230 0.033 42.5 1250.4 17.1 37.3 400.8

2 28 67 2200 147 0.021 27.1 1277.5 16.2 35.4 436.2

1 16 33 2200 72 0.010 12.9 1290.4 14.9 38.0 474.2

Total - 26,300 7055 - 1290.4 - - 474.2 -

Calculation of Design Base Shear in Transverse (Short) Direction

Base shear, V= CS W where 0.1 SD1 I < CS = IR

SDS

/ <

)/(1

IRT

SD

SDS = 2/3 SMS, where SMS = Fa SS = 1.0 × 1.5 = 1.5 and SD1 = 2/3 SMI

where SMI = Fv S1 = 1.0 × 0.6 = 0.6; SDS = 1.0, SD1 = 0.4; R=8; I=1.0;T=CT hn3/4 = 0.02 × (148)3/4 =0.849 sec; T can be increased

by a factor of 1.2 but should be less than the calculated value (i.e. 1.17 sec). T∴ = 0.849 × 1.2 =1.018<1.17

0.1 × 0.4 < CS = 1/8

0.1 <

)1/8(018.1

4.0

0.04 < CS = 0.125 < 0.0491 ∴ use CS = 0.0491V = 0.0491 x 26,300 = 1290.4 kips

534 Chapter 10

Table 10-2. Design Lateral Forces in Longitudinal Direction (Corresponding to Entire Structure).

Seismic forces Wind forces

FloorLeve

l

Height,

hx, ft

hxk

k=1.51

story

weight,

wx, kips

wx hxk ft-

kips x103

Cvx Lateral

force, Fx,

kips

Story

shear

ΣFx, kips

wind

pressure

lbs/ft2

lateral

force

Hx, kips

Story

shear

ΣHx,

kips

Roof 148 1893 2100 3975 0.178 154.5 154.5 17.2 6.8 6.8

11 136 1666 2200 3665 0.164 142.4 296.9 17.0 13.5 20.3

10 124 1449 2200 3188 0.142 123.3 420.2 16.6 13.1 33.4

9 112 1243 2200 2734 0.122 105.9 526.1 16.3 12.9 46.3

8 100 1047 2200 2304 0.103 89.4 615.5 15.9 12.6 58.9

7 88 863 2200 1899 0.085 73.8 689.3 15.5 12.3 71.2

6 76 692 2200 1522 0.068 59.0 748.3 15.0 12.0 83.2

5 64 534 2200 1174 0.052 45.1 793.4 14.5 11.5 94.7

4 52 390 2200 858 0.038 33.0 826.4 13.9 11.0 105.7

3 40 263 2200 578 0.026 22.6 849.0 13.2 10.5 116.2

2 28 153 2200 337 0.015 13.0 862.0 12.3 9.7 125.9

1 16 66 2200 145 0.006 5.2 867.2 11.0 10.2 136.1

Total - 26,300 22,379 - 867.2 - - 136.1 -

In longitudinal direction, Ct (for reinforced concrete moment resisting frames) = 0.03;T = Ct (hn)

3/4 = (0.03) (148) = 1.27; T can be increased by a factor of 1.2,∴ T = 1.2 × 1.27 = 1.524 < 1.73

0.1 × 0.4 < CS = 1/8

0.1 <

)1/8(524.1

4.0

0.04 < CS = 0.125 < 0.0329∴ use CS = 0.0329V = 0.033 × 26,300 = 867.2 kips


(a) Lateral displacements due to seismic andwind effects: The lateral displacements dueto both seismic and wind forces listed inTables 10-1 and 10-2 are shown in Figure10-50 . Although the seismic forces used toobtain the curves of Figure 10-50 areapproximate, the results shown still serve todraw the distinction between wind andseismic forces, that is, the fact that theformer are external forces the magnitudesof which are proportional to the exposedsurface, while the latter represent inertialforces depending primarily on the mass andstiffness properties of the structure. Thus,while the ratio of the total wind force in thetransverse direction to that in thelongitudinal direction (see Tables 10-1 and10-2) is about 3.5, the corresponding ratio

for the seismic forces is only 1.5. As aresult of this and the smaller lateralstiffness of the structure in the longitudinaldirection, the displacement due to seismicforces in the longitudinal direction issignificantly greater than that in thetransverse direction. By comparison, thedisplacements due to wind are about thesame for both directions. The typicaldeflected shapes associated withpredominantly cantilever or flexurestructures (as in the transverse direction)and shear (open-frame) buildings (as in thelongitudinal direction) are evident in Figure10-50. The average deflection indices, thatis, the ratios of the lateral displacement atthe top to the total height of the structure,are 1/5220 for wind and 1/730 for seismic

Figure 10-50. Lateral displacements under seismic and wind loads.

536 Chapter 10

loads in the transverse direction. Thecorresponding values in the longitudinaldirection are 1/9350 for wind and 1/590 forseismic loads. It should be noted that theanalysis for wind was based on uncrackedsections whereas that for seismic was basedon cracked sections. The use of crackedsection moment of inertia is a requirementby IBC-2000 for calculation of drift due toearthquake loading. However, under windloading, the stresses within the structure inthis particular example are within theelastic range as can also be observed fromthe amount of lateral deflections. As aresult, the amount of cracking within themembers is expected to be insignificant.However, for the case of seismic loading,the members are expected to deform wellinto inelastic range of response under thedesign base shear. To consider the effectsof cracked sections due to seismic loads,the moments of inertia of beams, columnsand walls were assumed to be 0.5, 0.7 and0.5 of the gross concrete sectionsrespectively.

(b) Drift requirements: IBC-2000 requires thatthe design story drift shall not exceed theallowable limits. In calculating the driftlimits, the effect of accidental torsion wasconsidered in the analysis. On this basis,the mass at each floor level was assumed todisplace from the calculated center of massa distance equal to 5% of the buildingdimension in each direction. Table 10-3shows the calculated displacements and thecorresponding story drifts in both E-W andN-S directions. To determine the actualstory drift, the calculated drifts wereamplified using the Cd factor of 6.5according to IBC-2000. These increaseddrifts account for the total anticipated driftsincluding the inelastic effects. Theallowable drift limit based on IBC-2000 is0.025 times the story height whichcorresponds to 3.6 in. and 4.8 in. at atypical floor and first floor respectively.The calculated values of drift are less thanthese limiting values. It is to be noted thatusing IBC-2000 provisions, it is permissible

to use the computed fundamental period ofthe structure without the upper boundlimitation when determining the story driftslimits. However, the drift values shown arebased on the calculated values of thefundamental period based on the codelimits. Since the calculated drifts are lessthan the allowable values, further analysisbased on the adjusted value of period wasnot necessary. In addition, the P-∆ effectneed not to be considered in the analysiswhen the stability coefficient as defined byIBC-2000 is less than a limiting value. Forthe 12-story structure, the effect of P-∆ wasfound to be insignificant.

(c) Load Combinations: For design anddetailing of structural components, IBC-2000 requires that the effect of seismicloads to be combined with dead and liveloads. The loading combinations to be usedare those prescribed in ASCE-95 asillustrated in Equation (10-2) except thatthe effect of seismic loads are according toIBC-2000 as defined in Equation (10-3).

To consider the extent of structuralredundancy inherent in the lateral-force-resisting system, the reliability factor, ρ, isdefined as follows for structures in seismicdesign category D as defined by IBC-2000:

xArmax

202 −=ρ

wherermax = the ratio of the design story shear

resisted by the single elementcarrying the most shear force in thestory to the total story shear, for agiven direction of loading. For shearwalls, rmax is defined as the shear inthe most heavily loaded wallmultiplied by 10/lw , divided by thestory shear (lw is the wall length)

Ax = the floor area in square feet of thediaphragm level immediately abovethe story


When calculating the reliability factorfor dual systems such as the frame wallstructure in the N-S direction, it can bereduced to 80 percent of the calculated valuedetermined as above. However, this valuecan not be less that 1.0.

In the N-S direction, the most heavilysingle element for shear is the shear wall.Table 10-4 shows the calculated values for rover the 2/3 height of the structure. Themaximum value of r occurs at the base ofthe structure where the shear walls carrymost of the shear in the N-S direction. Onthis basis, the maximum value of ρdetermined was 1.0.The load combinations used for the designbased on ρ= 1.0 and SDS=1.0 by combining

Table 10-4.Element story shear ratios for redundancyfactor in N-S direction.

Vi x story10/Lw shear

StoryLevel

Vi = shearforce in

wallri

8 189 78 886 0.097 234 97 980 0.106 275 114 1074 0.115 317 131 1150 0.114 359 149 1208 0.123 408 169 1250 0.142 448 185 1278 0.151 570 236 1290 0.18

xArmax

202 −=ρ

0.1minbut 99.01826618.0

202 ==

××−= ρρ

Table 10-3. Lateral displacements and Inerstory drifts Due to Seismic Loads (in.).

E-W Direction N-S DirectionStory displacement drift drift × displacement drift drift ×Level Cd

*Cd

*

Roof 3.03 0.07 0.45 2.43 0.19 1.24

11 2.96 0.12 0.78 2.24 0.20 1.30

10 2.84 0.16 1.04 2.04 0.21 1.37

9 2.68 0.20 1.30 1.83 0.23 1.50

8 2.48 0.24 1.56 1.60 0.24 1.56

7 2.24 0.27 1.76 1.36 0.24 1.56

6 1.97 0.28 1.82 1.12 0.23 1.50

5 1.69 0.31 2.02 0.89 0.23 1.50

4 1.38 0.32 2.08 0.66 0.22 1.43

3 1.06 0.33 2.15 0.44 0.18 1.17

2 0.73 0.34 2.21 0.26 0.15 0.98

1 0.39 0.39 2.54 0.11 0.11 0.72

* Cd = 6.5

538 Chapter 10

equations (10-2) and (10-3) are as follows:

±+±

++=

E

E

r

Q 1.0 D0.7

L0.5 Q 1.0 D1.4

L0.5 L1.6 D1.2

U (10-8)

The 3-D structure was analyzed using theabove load combinations. The dead and liveloads were applied to the beams based ontributary areas as shown in Figure 10-51. Theeffect of accidental torsion was also consideredin the analysis.

To protect the building against collapse,IBC-2000 requires that in dual systems, themoment resisting frames be capable to resist atleast 25% of prescribed seismic forces. For thisreason, the building in the N-S direction wasalso subjected to 25% of the lateral forcesdescribed above without including the shear

walls.An idea of the distribution of lateral loads

among the different frames making up thestructure in the transverse direction may beobtained from Table 10-5, which lists theportion of the total story shear at each levelresisted by each of the three groups of frames.The four interior frames along lines 3, 4,5, and6 are referred to as Frame T-1, while the FrameT-2 represents the two exterior frames alonglines 1 and 8. The third frame, T-3 representsthe two identical frame-shearwall systemsalong lines 2 and 7. Note that at the top (12th

floor level), the lumped frame T-1 takes 126%of the total story shear. This reflects the factthat in frame-shear-wall systems of averageproportions, interaction between frame and wallunder lateral loads results in the frame“supporting” the wall at the top, while at thebase most of the horizontal shear is resisted by

2 6 '

22'

W

W

45 °

W

In te rio rB eam s

E x te rio rB eam s

D = 3 .5 2 k /ft

W = 1 .6 4 k /ftLW = 0 .4 4 k /ftr

W = 1 .7 6 k /ftDW = 0 .8 2 k /ftL

W = 0 .2 2 k /ftr

Transverse

Beam

s

L o ng itu d in a l B eam s

Figure 10-51. Tributary area for beam loading.


the wall. Table 10-5 indicates that for thestructure considered, the two frames with wallstake 90% of the shear at the base in thetransverse direction.

To illustrate the design of two typical beamson the sixth floor of an interior frame, theresults of the analysis in the transverse directionunder seismic loads have been combined, usingEquation 10-8, with results from a gravity-loadanalysis . The results are listed in Table 10-6.Similar values for typical exterior and interiorcolumns on the second floor of the sameinterior frame are shown in Table 10-7.Corresponding design values for the structuralwall section at the first floor of frame on line 3(see Figure 10-48) are listed in Table 10-8. The

last column in Table 10-8 lists the axial load onthe boundary elements (the 26 × 26-in, columnsforming the flanges of the structural walls)calculated according to the ACI requirementthat these be designed to carry all factored loadson the walls, including self-weight, gravityloads, and vertical forces due to earthquake-induced overturning moments. The loadingcondition associated with this requirement isillustrated in Figure 10-45. In both Tables 10-7and 10-8, the additional forces due to the effectsof horizontal torsional moments correspondingto the minimum IBC-2000 -prescribedeccentricity of 5% of the building dimensionperpendicular to the direction of the appliedforces have been included.

Table 10-5. Distribution of Horizontal Seismic Story Shears among the Three Transverse Frames.

Frame T-1 Frame T-2 Frame T-3Story (4 interior frames) (2 exterior frames) (2 interior frames with shear walls)Level

Story % of Story % of Story % of

Totalstoryshear,

shear total shear total shear Totalkips

Roof 263.6 126 102.1 49 -156.9 -75 208.8

11 228.5 56 90.3 22 86.0 21 404.8

10 259.9 45 101.9 18 216.8 37 578.8

9 282.5 39 110.4 15 340.6 46 733.5

8 303.6 35 117.3 14 445.4 51 866.3

7 317.3 32 123.6 13 538.8 55 979.7

6 324.0 30 125.6 12 624.2 58 1073.8

5 320.0 28 124.0 11 705.9 61 1149.9

4 303.2 25 117.9 10 786.8 65 1207.9

3 269.6 22 104.4 8 876.4 70 1250.4

2 225.1 18 86.4 7 966.0 75 1277.5

1 96.0 7 34.8 3 1159.6 90 1290.4

540 Chapter 10

Table 10-6. Summary of design moments for typicalbeams on sixth floor of interior transverse frames alonglines 3 through 6 (Figure 10-48a).

−±−±+−++

=)89(0.17.0

)89(0.15.04.1

)89(5.06.12.1

cQD

bQLD

aLLD

U

E

E

r

Design moment, ft-kipsBEAM AB

A Midspan of AB B9-8 a -76 +100 -202Sides way

to right+91 +83 -326

9-8b Sides way

to left-213 +85 -19

Sides wayto right

+127 +35 -2299-8c Sides way

to left-177 +37 +79

Design moment, ft-kipsBEAM BC

B Midspan of BC C9-8 a -144 +92 -144Sides way

to right-41 +77 -282

9-8b Sides way

to left-282

+77 -41

Sides wayto right

+110+33 -213

9-8c Sides way

to left-213 +33 +110

It is pointed out that for buildings located inseismic zones 3 and 4 (i.e., high-seismic-riskareas), the detailing requirements for ductilityprescribed in ACI Chapter 21 have to be meteven when the design of a member is governedby wind loading rather than seismic loads.

2.Design of flexural member AB. The aim isto determine the flexural and shearreinforcement for the beam AB on the sixthfloor of a typical interior transverse frame. Thecritical design (factored) moments are showncircled in Table 10-6. The beam has dimensionsb = 20 in. and d = 21.5 in. The slab is 8 in.thick, cf ′ = 4000 lb/in.2 and yf = 60,000 lb/in.2

In the following solution, the boxed-insection numbers at the right-hand margincorrespond to those in ACI 318-95 .

(a) Check satisfaction of limitations onsection dimensions:

5.21

20=depth

width

= 0.93 > 0.3 O.K 21.3.1.3 21.3.1.4

=+=×+

≤≥

=

O.K. in. 58.25 1.5(21.5) 26

beam of depth 1.5

column suuporting of (width

O.K. in. 10

. 20 inwidth

Table 10-7. Summary of design moments and axial loads for typical columns on second floor of interior transverse framesalong lines 3 through 6 (Figure 10-48a).

−±

−±+

−++

=

)89(0.17.0

)89(0.15.04.1

)89(5.06.12.1

cEQD

bEQLD

arLLD

U

Exterior Column A Interior Column BAxial load, Moment, ft-kips Axial load, Moment, ft-kips

kips Top Kips kips Top Bottom

9-8 a -1076 -84 +94 -1907 +6 -12Sides way to

right -806 -33 +25 -1630 +73 -1089-8 b

Sides way toleft -1070 -110 +134 -1693 -94 +119

Sides way toright -280 +8 -20 -698 +79 -111

9-8 cSides way to

left -544 -69 +88 -760 -88 +116


(b)Determine required flexuralreinforcement:

(1) Negative moment reinforcement atsupport B: Since the negativeflexural reinforcement for bothbeams AB and BC at joint B will beprovided by the same continuousbars, the larger negative moment atjoint B will be used. In the followingcalculations, the effect of anycompressive reinforcement will beneglected. From C = 0.85fc′ba = T=Asfy,

ss

c

s AA

bf

Aa 882.0

)20)(4)(85.0(

60

85.0 '===

( )2/adfAMM ysnu −=≤ φφ[ ])882.0)(5.0(5.21)60)(90.0()12)(326( ss AA −×=−

2

2

in. 64.3

03.16476.48

=

=+−

s

ss

A

or

AA

Alternatively, convenient use may bemade of design charts for singlyreinforced flexural members withrectangular cross-sections, given in

standard references. (10-79) Use five No. 8bars, As=3.95 in.2 This gives a negativemoment capacity at support B of φMn =351 ft-kips.

Check satisfaction of limitations onreinforcement ratio:

0092.0

)5.21)(20(

95.3

=

==bd

Asρ21.3.2.1

0033.0200

min ==>yf

ρ

0032.0000,60

40003'3min ===>

y

c

f

fρ

and <ρmax = 0.025 O.K.

(2) Negative moment reinforcement at support A:

Mu = 213 ft-kips As at support B, a = 0.882As. Substitution into

)2/( adfAM ysu −= φ

yields As = 2.31 in.2. Use three No. 8bars, As = 2.37 in.2 This gives anegative moment capacity at support Aof φMn = 218 ft-kips.

Table 10-8. Summary of design loads on structural wall section at first floor level of transverse frame along line 2 (or 7)(Figure 10-48a).

−±

−±+

−++

=

)89(0.17.0

)89(0.15.04.1

)89(5.06.12.1

cEQD

bEQLD

arLLD

U

Design forces acting on entire structural wall

AxialLoad, kips

Bending(overturning)

Moment, ft-kips

Horizontalshear,kips

Axial load#

on boundaryelement,

kips

9-8 a -5767 Nominal Nominal -28849-8 b -5157 30469 651 -39639-8 c -2293 30469 651 -2531

# Based on loading condition illustrated in Figure 10-45 @ bending moment at base of wall

542 Chapter 10

(3)Positive moment reinforcement atsupports: A positive moment capacityat the supports equal to at least 50% ofthe corresponding negative momentcapacity is required, i.e., 21.3.2.2

kipsftM u −== 1092

218 A)support at (min

which is less than M+max = 127 ft-kips at

A (see Table 10-6), but greater than therequired Mu

+ near midspan of AB (=100ft-kips).

min Mu+ (at support B for both spans AB and

BC) kipsft −== 1762

351

Note that the above required capacity isgreater than the design positivemoments near the mid-spans of bothbeams AB and BC.

Minimum positive/negativemoment capacity at any section alongbeam AB or BC = 351/4 =87.8 ft-kips.

(4) Positive moment reinforcement at mid-span of beam AB- to be madecontinuous to supports: (with aneffective T-beam section flange width =52 in.)

( )( )( ) ss

c

ysA

A

bf

fAa 339.0

52485.0

60

85.0

'===

Substituting into

( )( )

−==

212127

adfAM ysu φ

yields As (required) = 1.35 in.2. Similarly,corresponding to the required capacity at

support B, +uM = 163 ft-kips, we have As

(required) =1.74 in.2. Use three No. 7 barscontinuous through both spans. As = 1.80in.2 This provides a positive momentcapacity of 172 ft-kips.Check:

0042.0)5.21)(20(

8.1 ==ρ

0033.0200

min ==>yf

ρ O.K. 10.5.1

000,60

400033min =

′=>

y

c

f

fρ

(c) Calculate required length of anchorage offlexural reinforcement in

exterior column:

Development length

≥.6

8

65/ '

in

d

fdf

l b

cby

dh21.5.4.1

(plus standard 90º hook located in confinedregion of column). For the No. 8 (top) bars (bend radius, measured oninside of bar, bd3≥ = 3.0 in.),

( )( )

( )( )

=

=

≥. 6

0.80.18

. 15400065

0.1000,60

in

in

in

ldh

For the No. 7 bottom bars (bend radius

bd3≥ = 2.7 in.), ldh = 13 in.Figure 10-52 shows the detail of flexuralreinforcement anchorage in the exteriorcolumn. Note that the development lengthldh is measured from the near face of thecolumn to the far edge of the vertical 12-bar-diameter extension (see Figure 10-35).

Figure 10-52. Detail of anchorage of flexuralreinforcement in exterior column


(d) Determine shear-reinforcementrequirements: Design for shearscorresponding to end moments obtained byassuming the stress in the tensile flexuralreinforcement equal to 1.25fy and a strengthreduction factor φ = 1.0, plus factoredgravity loads (see Figure 10-16). Table 10-9shows values of design end shearscorresponding to the two loading cases to beconsidered. In the table,

WU = 1.2 WD + 1.6 WL = 1.2 × 3.52 + 1.6 × 1.64 = 6.85 kips/ft

ACI Chapter 21 requires that thecontribution of concrete to shear resistance,Vc, be neglected if the earthquake-inducedshear force (corresponding to the probableflexural strengths at beam ends calculatedusing 1.25fy instead of fy and φ = 1.0) isgreater than one-half the total design shearand the axial compressive force includingearthquake effects is less than Ag f′c /20.

21.3.4.2For sidesway to the right, the shear at

end B due to the plastic end moments in thebeam (see Table 10-9) is

kipsVb 4.3520

477230 =+= A

Wu230 ’k

which is approximately 50% of the totaldesign shear, Vu = 69.6 kips. Therefore, thecontribution of concrete to shear resistancecan be considered in determining shearreinforcement requirements.At right end B, Vu = 69.6 kips. Using

kipsdbfV wcc 4.541000

)5.21)(20(400022 ' ===

we have

4.5485.06.69 ×−=−= cus VVV φφ 11.1.1 kips4.23=

kipsVs 5.27=Required spacing of No. 3 closed stirrups(hoops), since Av (2 legs) = 0.22 in.2:

( )( )( )5.27

5.216022.0==s

yv

V

dfAs 11.5.6.2

= 10.3 in.

Maximum allowable hoop spacing withindistance 2d = 2(21.5) = 43 in. from faces ofsupports:

Table 10-9. Determination of Design Shears for Beam AB.

)(,2

kipslw

l

MMV u

Bpr

Apr

u ±+

=Loading

A B

A

Wu230 ’k

A B

W u2 3 0 ’ k

4 7 7 ’ k

1.1 69.6

A B

W u

2 9 9 ’k

2 3 0 ’k

60.7 7.8

Shear Diagram

A B

6 0 .7

1 .1

7 .8

6 9 .6

544 Chapter 10

==×==

×==

=

in. 12

in. 9 24(0.375) bars) hoop of (dia. 24

in. 78(0.875)

bar) long.smallest of dia.(8

. 4.54/5.214/

max

ind

s

21.3.3.2

Beyond distance 2d from the supports,maximum spacing of stirrups:

. 75.102/max inds == 21.3.3.4

Use No. 3 hoops/stirrups spaced as shownin Figure 10-53. The same spacing, turnedaround, may be used for the left half ofbeam AB.

Where the loading is such that inelasticdeformation may occur at intermediatepoints within the span (e.g., due toconcentrated loads at or near mid-span), thespacing of hoops will have to be determinedin a manner similar to that used above forregions near supports. In the presentexample, the maximum positive momentnear mid-span (i.e., 100 ft-kips, see Table

10-6) is much less than the positive momentcapacity provided by the three No. 7continuous bars (172 ft-kips). 21.3.3.1

(e) Negative-reinforcement cut-off points: Forthe purpose of determining cutoff points forthe negative reinforcement, a momentdiagram corresponding to plastic endmoments and 0.9 times the dead load willbe used. The cut-off point for two of thefive No. 8 bars at the top, near support B ofbeam AB, will be determined.With the negative moment capacity of asection with three No. 8 top bars equal to218 ft-kips (calculated using fs = fy = 60 ksiand φ = 0.9), the distance from the face ofthe right support B to where the momentunder the loading considered equals 218 ft-kips is readily obtained by summingmoments about section a—a in Figure 10-54 and equating these to -218 ft-kips. Thus,

21860

2.34778.513

−=−− xx

Figure 10-53. Spacing of hoops and stirrups in right half of beam AB


Solution of the above equation gives x = 5.1ft. Hence, two of the five No. 8 bars nearsupport B may be cut off (noting that d =21.5 in.> l2db = 12 × 1.0=12 in.) at

12.10.3

ft 7.0say 9.612

5.211.5 ftdx =+=+

from the face of the right support B. With ldh

(see figure 10-35) for a No. 8 top bar equalto 14.6 in., the required development lengthfor such a bar with respect to the tensileforce associated with the negative momentat support B is ld = 3.5 ldh = 3.5 × 14.6/12 =4.3 ft < 7.0 ft. Thus, the two No. 8 bars maybe cut off 7.0 ft from the face of the interiorsupport B. 21.5.4.2

At end A, one of the three No. 8 bars mayalso be cut off at a similarly computeddistance of 4.5 ft from the (inner) face of theexterior support A. Two bars are required torun continuously along the top of the beam.

21.3.2.3

Figure 10-54. Moment diagram for beam AB

(f)Flexural reinforcement splices: Lap splicesof flexural reinforcement should not beplaced within a joint, within a distance 2dfrom faces of supports, or at locations ofpotential plastic hinging. Note that all lap

splices have to be confined by hoops orspirals with a maximum spacing or pitch ofd/4, or 4 in., over the length of the lap.21.3.2.3

(1) Bottom bars, No. 7: The bottom barsalong most of the length of the beammay be subjected to maximum stress.Steel area required to resist themaximum positive moment nearmidspan of 100 ft-kips (see Table 10-6),As = 1.05 in.2 Area provided by thethree No. 7 bars = 3 (0.60) = 1.80 in.2,so that

0.271.105.1

80.1

)(

)( <==requireds

provideds

A

A

Since all of the bottom bars will bespliced near midspan, use a class Bsplice. 12.15.2

Required length of splice = 1.3 ld ≥ 12 in. where

+=

b

trc

ybd

d

kcf

fdl

αβγλ'40

312.2.3

whereα = 1.0 (reinforcement location factor)β = 1.0 (coating factor)γ = 1.0 (reinforcement size factor)λ = 1.0 (normal weight concrete)

31.22

875.0375.05.1 =++=c (governs)

(side cover, bottom bars)

or( )

84.32

875.0375.05.1220

2

1 =

−+−=c in.

(half the center to center spacing of bars)

sn

fAk yttr

tr 1500=

where

546 Chapter 10

Atr = total area of hoops within the spacing sand which crosses the potential planeof splitting through the reinforcementbeing developed (ie. for 3#3 bars)

fyt = specified yield strength of hoops = 60,000 psis = maximum spacing of hoops = 4 in.n = number of bars being developed alongthe plane of splitting = 3

1.130.41500

000,60)11.03(=

××

×=trk

5.290.3875.0

1.131.2>=

+=

+

bd

trkc, use 2.5

9.245.2

1

4000

000,60875.0

40

3=

×=∴ dl in.

Required length of class B splice = 1.3 ×24.9 = 32.0 in.

(2) Top bars, No. 8: Since the mid-span portionof the beam is always subject to a positive

bending moment (see Table 10-6), splices in thetop bars should be located at or near midspan.Required length of class A splice = 1.0 ld.

For No. 8 bars,

+=

b

trc

ybd

d

kcf

fdl

αβγλ'40

3

where α = 1.3 (top bars), β = 1.0, γ = 1.0,and λ = 1.0

375.22

0.1375.05.1 =++=c in. (governs)

( )81.3

2

0.1375.05.1220

2

1 =

−+−=c in.

ktr= 1.1

5.30.1

1.1375.2 =+=+

b

tr

d

kc>2.5 use 2.5

in. 37.05.2

3.1

4000

600000.1

40

3 ==∴ xld

Required length of splice = 1.0 ld = 37.0 in.(g) Detail of beam. See Figure 10-55.

Figure 10-55. Detail of reinforcement for beam AB.


3. Design of frame column A. The aim hereis to design the transverse reinforcement for theexterior tied column on the second floor of atypical transverse interior frame, that is, one ofthe frames in frame T-1 of Figure 10-48. Thecolumn dimension has been established as 22in. square and, on the basis of the differentcombinations of axial load and bendingmoment corresponding to the three loadingconditions listed in Table 10-7, eight No. 9 barsarranged in a symmetrical pattern have beenfound adequate.(10-80,10-81) Assume the samebeam section framing into the column asconsidered in the preceding section.

22' ./000,60./4000 inlbfandinlbf yc ==From Table 10-7, Pu(max) = 1076 kips:

( ) ( ) ( )kips

fAkipsP cg

u 19410

422

101076max

2'

==>=

Thus, ACI Chapter 21 provisions governingmembers subjected to bending and axial loadapply. 21.4.1 (a)Check satisfaction of vertical reinforcement

limitations and moment capacityrequirements:(1) Reinforcement ratio: 06.001.0 ≤≤ ρ

( )( )( ) 0165.0

2222

0.18 ===g

st

A

Aρ O.K.

21.4.3.1(2) Moment strength of columns relative to

that of framing beam in transverse direction(see Figure 10-56)

Figure 10-56. Relative flexural strength of beam andcolumns at exterior joint transverse direction.

( ) ( )beamsMcolumnsM ge 5

6≥ 21.4.2.2

From Section 10.5.2, item 2, −nMφ of

the beam at A is 218 ft-kips, which maybe mobilized during a sidesway to theleft of the frame. From Table 10-7, themaximum axial load on column A atthe second floor level for sidesway tothe left is Pu = 1070 kips. Using the P-M interaction charts given in ACI SP-17A,(10-81) the moment capacity of thecolumn section corresponding to Pu =φPn = 1070kips, fc′ = 4 ksi, fy = 60 ksi, γ= 0.75 (γ = ratio of distance betweencentroids of outer rows of bars todimension of cross-section in thedirection of bending, and ρ = 0.0165 isobtained as φMn = Me = 260 ft-kips).With the same size column above andbelow the beam, total moment capacityof columns = 2(260) = 520 ft-kips.Thus,

( )( )5

2186

5

6520 =>=∑ ge MM

= 262 ft-kips O.K.

(3) Moment strength of columns relative tothat of framing beams in longitudinal direction(see Figure 10-57): Since the columnsconsidered here are located in the center portionof the exterior longitudinal frames, the axialforces due to seismic loads in the longitudinaldirection are negligible. (Analysis of thelongitudinal frames under seismic loadsindicated practically zero axial forces in theexterior columns of the four transverse framesrepresented by frame on line 1 in Figure 10-48)Under an axial load of 1.2 D + 1.6 L + 0.5 Lr =1076 kips, the moment capacity of the columnsection with eight No. 9 bars is obtained asφMn= Me = 258 ft-kips. If we assume a ratio forthe negative moment reinforcement of about0.0075 in the beams of the exterior longitudinalframes (bw = 20 in., d = 21.5 in.), then

( )( )( )5.21200075.0≈= dbA ws ρ = 3.23 in.2

548 Chapter 10

Assume four No. 8 bars, As = 3.16 in. Negativemoment capacity of beam:

( )( )( )( )( ) in

wbcf

yfsAa 79.2

20485.0

6016.3'85.0

===

−== −−

2a

dfAMM ysgn φφ

( )( )( ) 12/)39.15.21(6016.390.0 −== 286 ft-kips

Figure 10-57. Relative flexural strength of beam andcolumns at exterior joint longitudinal direction.

Assume a positive moment capacity of thebeam on the opposite side of the column equalto one-half the negative moment capacitycalculated above, or 143 ft-kips. Total momentcapacity of beams framing into joint inlongitudinal direction, for sidesway in eitherdirection:

∑ −=+= kipsftM g 429143286

( ) kipsftM e −==∑ 5162582

( ) kipsftM g −==> ∑ 5154295

6

5

6

O.K. 21.4.2.2(b) Orthogonal effects: According to IBC-

2000, the design seismic forces arepermitted to be applied separately ineach of the two orthogonal directionsand the orthogonal effects can beneglected.

(c) Determine transverse reinforcementrequirements:

(1) Confinement reinforcement (seeFigure 10-38). Transversereinforcement for confinement isrequired over a distance l0 fromcolumn ends, where

≥0l ( )

=×=

=

.18

.206

1210

6

1

)( .22

in

inheightclear

governsinmemberofdepth

21.4.4.4

Maximum allowable spacing of rectangularhoops:

( )

( )

===

governsin

ins

.4

.5.5422

column of dimensionsmallest41

max

21.4.4.2

Required cross-sectional area ofconfinement reinforcement in theform of hoops:

−

≥

yh

c

ch

gc

yh

cc

sh

f

f

A

Ash

f

fsh

A'

'

13.0

09.0

21.4.4.1

where the terms are as defined for Equation10-6 and 10-7. For a hoop spacing of 4 in., fyh =60,000 lb/in.2, and tentatively assuming No. 4bar hoops (for the purpose of estimating hc andAch)’ the required cross-sectional area is

( )( )( )( )

( )( )( )

=

−

=≥

(governs) 50.0

000,60

40001

361

4845.1843.0

44.0

000,60

40005.18409.0

2

2

in

inAsh 21.4.4.3

No. 4 hoops with one crosstie, as shown inFigure 10-58, provide Ash = 3(0.20) = 0.60in.2


Figure 10-58. Detail of column transverse reinforcement.

(2) Transverse reinforcement for shear: Asin the design of shear reinforcement forbeams, the design shear in columns isbased not on the factored shear forcesobtained from a lateral-load analysis,but rather on the maximum probableflexural strength, Mpr (with φ = 1.0 andfs = 1.25 fy), of the member associatedwith the range of factored axial loads onthe member. However, the membershears need not exceed those associatedwith the probable moment strengths ofthe beams framing into the column.

If we assume that an axial forceclose to P = 740 kips (φ = 1.0 and tensilereinforcement stress of 1.25 fy,corresponding to the “balanced point’ onthe P-M interaction diagram for thecolumn section considered – whichwould yield close to if not the largestmoment strength), then thecorresponding Mb = 601 ft-kips. Bycomparison, the moment induced in thecolumn by the beam framing into it inthe transverse direction, with Mpr = 299ft-kips, is 299/2 = 150 ft-kips. In thelongitudinal direction, with beamsframing on opposite sides of the column,we have (using the same steel areasassumed earlier),Mpr (beams) = M-

pr (beam on one side) +M+

pr (beam on the other side) = 390 +195 = 585 ft-kips, with the momentinduced at each end of the column =585/2 =293 ft-kips. This is less than Mb

= 601 ft-kips and will be used to

determine the design shear force on thecolumn. Thus (see Figure 10-42),Vu = 2 Mu/l = 2(293)/10 = 59 kips

using, for convenience,

bdfV cc'2=

( )( )

kips 541000

5.192240002 ==

Required spacing of No. 4 hoops with Av

= 2(0.20) = 0.40 in.2 (neglectingcrossties) and

( ) : 8.14/ kipsVVV cus =−= φφ

( )( )( )( ).6.31

8.14

5.19600.22 in

V

dfAs

s

yv ===

11.5.6.2

Thus, the transverse reinforcementspacing over the distance l0 = 22 in. near thecolumn ends is governed by the requirementfor confinement rather than shear.

Maximum allowable spacing of shearreinforcement: d/2 = 9.7 in. 11.5.4.1

Use No. 4 hoops and crossties spacedat 4 in. within a distance of 24 in. fromthe columns ends and No. 4 hoopsspaced at 6 in. or less over theremainder of the column.

(d) Minimum length of lap splices forcolumn vertical bars:ACI Chapter 21 limits the location of

lap splices in column bars within themiddle portion of the member length,the splices to be designed as tensionsplices. 21.4.3.2As in flexural members, transversereinforcement in the form of hoopsspaced at 4 in. (<d/4 =19.5/4 = 4.9 in.)is to be provided over the full length ofthe splice. 21.3.2.3

Since generally all of the column barswill be spliced at the same location, aClass B splice will be required. 12.15.2

The required length of splice is 1.3ld where

550 Chapter 10

+=

b

trc

ybd

d

kcf

fdl

αβγλ'40

3

where α = 1.0, β = 1.0, γ = 1.0, and λ = 1.0

6.22

128.15.05.1 =++=c in. (governs)

or ( )

2.42

128.15.05.1222

2

1 =

−+−=c in.

ktr= 0.2341500

000,60)2.03(

1500=

××××=

sn

fA yttr

1.4128.1

0.26.2 =+=+

b

tr

d

kc>2.5 use 2.5

in. 32.15.2

0.1

4000

000,60128.1

40

3 ==∴ xld

Thus, required splice length = 1.3(32.1) =42 in. Use 44-in, lap splices.

(e) Detail of column. See Figure 10-59.

Figure 10-59. Column reinforcement details.

4.Design of exterior beam—columnconnection. The aim is to determine thetransverse confinement and shear-reinforcementrequirements for the exterior beam-columnconnection between the beam considered initem 2 above and the column in item 3. Assumethe joint to be located at the sixth floor level.

(a) Transverse reinforcement for confinement:ACI Chapter 21 requires the same amount ofconfinement reinforcement within the jointas for the length l0 at column ends, unlessthe joint is confined by beams framing intoall vertical faces of the column. In the lattercase, only one-half the transversereinforcement required for unconfined jointsneed be provided. In addition, the maximumspacing of transverse reinforcement is(minimum dimension of column)/4 or 6 in.(instead of 4 in.).21.5.2.121.5.2.2In the case of the beam-column joint

considered here, beams frame into only threesides of the column, so that the joint isconsidered unconfined.

In item 4 above, confinement requirementsat column ends were satisfied by No. 4 hoopswith crossties, spaced at 4 in.(b) Check shear strength of joint: The shear

across section x-x (see Figure 10-60) ofthe joint is obtained as the differencebetween the tensile force at the topflexural reinforcement of the framingbeam (stressed to 1.25fy) and thehorizontal shear from the column above.The tensile force from the beam (three No.8 bars, As = 2.37 in.2) is

(2.37)(1.25)(60) = 178 kips

Figure 10-60. Horizontal shear in exterior beam-columnjoint.

An estimate of the horizontal shear from thecolumn, Vh can be obtained by assuming that


the beams in the adjoining floors are alsodeformed so that plastic hinges form at theirjunctions with the column, with Mp(beam) =299 ft-kips (see Table 10-9, for sidesway toleft). By further assuming that the plasticmoments in the beams are resisted equally bythe columns above and below the joint, oneobtains for the horizontal shear at the columnends

( )

kipsheightstory

beamMV p

h 2512

299 ===

Thus, the net shear at section x-x of joint is178 -25 = 153 kips. ACI Chapter 21 gives thenominal shear strength of a joint as a functiononly of the gross area of the joint cross-section,Aj, and the degree of confinement provided byframing beams. For the joint considered here(with beams framing on three sides),

jcc AfV '15φφ =

( )( )( )( )

1000

2240001585.0 2

=

kipsVkips u 153390 =>= O.K.

21.5.3.1 9.3.4.1

Note that if the shear strength of theconcrete in the joint as calculated above wereinadequate, any adjustment would have to takethe form (since transverse reinforcement abovethe minimum required for confinement isconsidered not to have a significant effect onshear strength) of either an increase in thecolumn cross-section (and hence Aj) or anincrease in the beam depth (to reduce theamount of flexural reinforcement required andhence the tensile force T).(c) Detail of joint. See Figure 10-61. (Thedesign should be checked for adequacy in thelongitudinal direction.)

Note: The use of crossties within the jointmay cause some placement difficulties. Torelieve the congestion, No. 6 hoops spaced at 4in. but without crossties may be considered asan alternative. Although the cross-sectional areaof confinement reinforcement provided by No.6 hoops at 4 in. (Ash = 0.88 in.2) exceeds therequired amount (0.59 in.2), the requirement of

section 21.4.4.3 of ACI Chapter 21 relating to amaximum spacing of 14 in. between crosstiesor legs of overlapping hoops (see Figure 10-41)will not be satisfied. However, it is believedthat this will not be a serious shortcoming inthis case, since the joint is restrained by beamson three sides.

Figure 10-61. Detail of exterior beam-column connection.

5. Design of interior beam-columnconnection. The objective is to determinethe transverse confinement and shearreinforcement requirements for theinterior beam-column connection at thesixth floor of the interior transverseframe considered in previous examples.The column is 26 in. square and isreinforced with eight No. 11 bars.

The beams have dimensions b = 20 in. and d =21.5 in. and are reinforced as noted in Sectionitem 2 above (see Figure 10-55).

552 Chapter 10

(a) Transverse reinforcement requirements (forconfinement): Maximum allowable spacingof rectangular hoops,

( )

( )

===governsin

ins

.6

.5.64/26

columnofdimensionsmallest4

1

max

21.5.2.221.4.4.2

For the column cross-section considered andassuming No. 4 hoops, hc = 22.5 in., Ach = (23)2

= 529 in.2, and Ag = (26)2 = 676 in.2. With ahoop spacing of 6 in., the required cross-sectional area of confinement reinforcement inthe form of hoops is

( )( )( )( )

( )

( )( )( )

=

−=

−

=

=

≥

2

'

2

'

75.0

000,60

40001

529

6765.2263.0

13.0

81.0

000,60

40005.22609.009.0

in

f

f

A

Ash

governsin

f

fsh

Ayh

c

ch

gc

yh

cc

sh

21.4.4.1

Since the joint is framed by beams (having

widths of 20 in., which is greater than 4

3 of

the width of the column, 19.5 in.) on all foursides, it is considered confined, and a 50%reduction in the amount of confinementreinforcement indicated above is allowed.Thus, Ash(required) ≥ 0.41 in.2.No. 4 hoops with crossties spaced at 6 in.o.c. provide Ash = 0.60 in.2. (See Note at endof item 4.)

(b) Check shear strength of joint: Following thesame procedure used in item 4, the forcesaffecting the horizontal shear across asection near mid-depth of the joint shown inFigure 10-62 are obtained:

(Net shear across section x-x) = T1 + C2 - Vh

=296 + 135 –59= 372 kips = Vu

Shear strength of joint, noting that joint isconfined:

jcc AfV '20φφ =

( )( ) ( )

1000

2640002085.0 2

= 21.5.3.1

= 726 kips > Vu = 372 kips O.K.

Figure 10-62. Forces acting on interior beam-column joint.


6.Design of structural wall (shear wall).The aim is to design the structural wall sectionat the first floor of one of the identical frame-shear wall systems. The preliminary design, asshown in Figure 10-48, is based on a 14-in.-thick wall with 26-in. -square vertical boundaryelements, each of the latter being reinforcedwith eight No. 11 bars.

Preliminary calculations indicated that thecross-section of the structural wall at the lowerfloor levels needed to be increased. In thefollowing, a 14-in.-thick wall section with 32 ×50-in. boundary elements reinforced with 24No. 11 bars is investigated, and otherreinforcement requirements determined.

The design forces on the structural wall atthe first floor level are listed in Table 10-8.Note that because the axis of the shear wallcoincides with the centerline of the transverseframe of which it is a part, lateral loads do notinduce any vertical (axial) force on the wall.

The calculation of the maximum axial forceon the boundary element corresponding toEquation 10-8b, EQ1.0 L0.5 D1.4 ±+ , Pu = 3963kips, shown in Table 10-8, involved thefollowing steps: At base of the wall:

Moment due to seismic load (from lateralload analysis for the transverse frames), Mb =32,860 ft-kips.

Referring to Figure 10-45, and noting theload factors used in Equation 10-8a of Table10.8,

W = 1.2 D + 1.6 L + 0.5 Lr

= 5767 kipsHa = 30,469 ft-kips

d

HaWCv +=

2

kips396322

469,30

2

5157 =+=

(a) Check whether boundary elements arerequired: ACI Chapter 21 (Section21.6.2.3) requires boundary elements to beprovided if the maximum compressiveextreme-fiber stress under factored forcesexceeds '2.0 cf , unless the entire wall isreinforced to satisfy Sections 21.4.4.1

through 21.4.4.3 (relating to confinementreinforcement).

It will be assumed that the wall willnot be provided with confinementreinforcement over its entire height. For ahomogeneous rectangular wall 26.17 ftlong (horizontally) and 14 in. (1.17 ft)thick,

( )( ) 43

.. 174712

17.2617.1ftI an ==

( )( ) 26.3017.2617.1 ftAg ==

Extreme-fiber compressive stress under Mu

= 30,469 ft-kips and Pu = 5157 kips (seeTable 10-8):

( )( )1747

217,26469,30

6.30

51572/

..

+=+=an

wu

g

uc I

hM

A

Pf

= 397 ksf = 2.76 ksi > 0.2 'cf = (0.2)(4)

= 0.8 ksi.

Therefore, boundary elements are required,subject to the confinement and specialloading requirements specified in ACIChapter 21.

(b) Determine minimum longitudinal andtransverse reinforcement requirements forwall:(1) Check whether two curtains of

reinforcement are required: ACI Chapter21 requires that two curtains ofreinforcement be provided in a wall if thein-plane factored shear force assigned to

the wall exceeds '2 ccv fA , where Acv is

the cross-sectional area bounded by theweb thickness and the length of section inthe direction of the shear forceconsidered. 21.6.2.2

From Table 10-8, the maximum factored shear force on the wall at the first floor level is Vu = 651 kips:

( )( )( )1000

40001217.261422 ' ×=ccv fA

= 556 kips< Vu = 651 kips

554 Chapter 10

Therefore, two curtains ofreinforcement are required.

(2) Required longitudinal and transversereinforcement in wall:

Minimum required reinforcement ratio,

0025.0 ≥== ncv

svv A

A ρρ (max.

spacing = 18 in.) 21.6.2.1With Acv = (14)(12) = 168 in.2, (per footof wall) the required area ofreinforcement in each direction per footof wall is (0.0025)(168) = 0.42 in.2/ft.Required spacing of No. 5 bars [in twocurtains, As = 2(0.31) = 0.62 in.2]:

( ) ( )( ) .18.7.171242.0

31.02ininrequireds <==

(c) Determine reinforcement requirements forshear. [Refer to discussion of shear strengthdesign for structural walls in Section 10.4.3,under “Code Provisions to Insure Ductilityin Reinforced Concrete Members,” item 5,paragraph (b).] Assume two curtains of No.5 bars spaced at 17 in. o.c. both ways. Shearstrength of wall( 266.517.26148 >==ww lh ):

+= ynccvn ffAV ρφφ '2

where φ = 0.60 Acv = (14)(26.17×12) = 4397 in.2

( )

( )( ) 0037.01214

31.02 ==nρ

Thus,

( )( ) ( )( )[ ]1000

000,600037.040002439760.0 +=nVφ

[ ]kips919

1000

2225.1262.2638 =+=

kipsVu 651=> O.K.

Therefore, use two curtains of No. 5 barsspaced at 17 in o. c. in both horizontal andvertical directions. 21.7.3.5

(d) Check adequacy of boundary element actingas a short column under factored vertical

forces due to gravity and lateral loads (seeFigure 10-45): From Table 10-8, themaximum compressive axial load onboundary element is Pu = 3963 kips.21.5.3.3With boundary elements having dimensions32 in.×50 in. and reinforced with 24 No. 11bars,

Ag = (32)(50) = 1600 in.2

Ast = (24)(1.56) = 37.4 in.2

ρst = 37.4/1600 = 0.0234ρmin = 0.01 < ρst < ρmax = 0.06 O.K.21.4.3.1Axial load capacity of a short column:

( ) ( )[ ]stystgcn AfAAfP +−= '85.080.0max φφ= (0.80)(0.70)[(0.85)(4)(1600 - 37.4)+(60)(37.4)]= (0.56)[5313+ 2244] = 4232 kips > Pu =3963 kips O.K. 10.3.5.2

(e) Check adequacy of structural wall section atbase under combined axial load and bendingin the plane of the wall: From Table 10-8,the following combinations of factored axialload and bending moment at the base of thewall are listed, corresponding to Eqs. 10-8a,b and c:

9-8a: Pu = 5767 kips, Mu small9-8b:Pu = 5157 kips, Mu= 30,469 ft-kips9-8c: Pu = 2293 kips, Mu= 30,469 ft-kips

Figure 10-63 shows the φPn-φ Mn interactiondiagram (obtained using a computerprogram for generating P-M diagrams) for astructural wall section having a 14-in.-thickweb reinforced with two curtains of No. 5bars spaced at 17 in o.c. both ways and 32in.×50-in. boundary elements reinforced

with 24 No. 11 vertical bars, with 'cf =

4000 lb/in.2, and fy = 60,000 lb/in.2 (seeFigure 10-64). The design loadcombinations listed above are shown plottedin Figure 10-63. The point marked arepresents the P-M combinationcorresponding to Equation 10-8a, withsimilar notation used for the other two loadcombinations.


0

5000

10000

15000

20000

25000

0 20000 40000 60000 8000 0 100000

Bendin g Mom ent Capacity, φ M n (ft-kip s)

φ

M ax. A llo wableAxia l Load =14,123 k ip s

B alanced Po int(M b,P b)

14123

9 -8 a9 -8 b

9 -8 c

B en d i ng M o m e n t C a p ac ity, M ( ft-k ip s )n

Axi

al L

oad

Cap

acity

,

P (k

ips)

n

Figure 10-63. Axial load-moment interaction diagram forstructural wall section.

Figure 10-64. Half section of structural wall at base.

It is seen in Figure 10-63 that the threedesign loadings represent points inside theinteraction diagram for the structural wallsection considered. Therefore, the section isadequate with respect to combined bendingand axial load.Incidentally, the “balanced point” in Figure10-63 corresponds to a condition where thecompressive strain in the extreme concretefiber is equal to εcu = 0.003 and the tensile

strain in the row of vertical bars in theboundary element farthest from the neutralaxis (see Figure 10-64) is equal to the initialyield strain, εy = 0.00207.

(f) Determine lateral (confinement)reinforcement required for boundaryelements (see Figure 10-64): The maximumallowable spacing is

===

)governs(.4

.84/32

)elementboundary of

dimensionsmallest (4/1

max

in

ins

21.6.6.221.4.4.2

(1) Required cross-sectional area ofconfinement reinforcement in shortdirection:

−

≥

yh

c

ch

gc

yh

cc

sh

f

f

A

Ash

f

fsh

A'

13.0

'09.0

21.4.4.1

Assuming No. 5 hoops and crosstiesspaced at 4 in. o.c. and a distance of 3 in.from the center line of the No. 11 verticalbars to the face of the column, we have

hC = 44 + 1.41 + 0.625 = 46.04 in. (for shortdirection),

Ach= (46.04 + 0.625)(26 + 1.41 + 1.25) =1337 in.2

=

−

=>

2

2

. 72.0

)60

4)(1

1337

)50)(32()(04.46)(4)(3.0(

)(10.1

)60/4)(04.46)(4)(09.0(

in

governsinAsh

(required in short direction).With three crossties (five legs, including

outside hoops),

556 Chapter 10

Ash (provided) = 5(0.31) = 1.55 in.2 O.K.(2) Required cross-sectional area of

confinement reinforcement in longdirection:

hc = 26 + 1.41 + 0.625 = 28.04 in.

(for long direction),Ach = 1337 in.2

=

=≥

2

2

in.0.44

1)(4/60) -68.04)(1.19(0.3)(4)(2

(governs) in. 0.67

(4/60) (28.04) (0.09)(4)

shA

(required in long direction).With one crosstie (i.e., three legs, including

outside hoop),

Ash (provided) = 3(0.31) = 0.93 in.2 O.K.(g) Determine required development and splice lengths:

ACI Chapter 21 requires that all continuousreinforcement in structural walls be anchored orspliced in accordance with the provisions forreinforcement in tension.21.6.2.4

(1) Lap splice for No. 11 vertical bars inboundary elements (the use of mechanicalconnectors may be considered as analternative to lap splices for these largebars): It may be reasonable to assume that50% or less of the vertical bars are splicedat any one location. However, anexamination of Figure 10-63 suggests thatthe amount of flexural reinforcementprovided–mainly by the vertical bars in theboundary elements–does not represent twicethat required by analysis, so that a class Bsplice will be required. 12.15.2Required length of splice = 1.3 ld where ld

= 2.5 ldh 12.15.1and

( )

==

==≥

.6

.12)41.1(88

)(.21400065

)41.1)(000,60(

'65/

in

ind

governsin

fdf

l

b

cby

dh

21.5.4.2

Thus the required splice length is(1.3)(2.5)(21) = 68 in.

(2) Lap splice for No. 5 vertical bars inwall “web”: Here again a class B splicewill be required. Required length ofsplice = 1.3 ld , whre ld = 2.5 ldh, and

( )

==

==≥

.6

.0.5)625.0(88

)(.9400065

)625.0)(000,60(

'65/

in

ind

governsin

fdf

l

b

cby

dh

Hence, the required length of splice is(1.3)(2.5)(9) = 30 in.Development length for No. 5 horizontalbars in wall, assuming no hooks are usedwithin the boundary element: Since it isreasonable to assume that the depth ofconcrete cast in one lift beneath ahorizontal bar will be greater than 12 in.,the required factor of 3.5 to be applied tothe development length, ldh, required for a90° hooked bar will be used [Section10.4.3, under “Code Provisions Designed toInsure Ductility in Reinforced-ConcreteMembers”, item 2, paragraph (f)]:

21.5.4.2ld = 3.5 ldh , where as indicated above, ldh =9.0 in. so that the required developmentlength ld = 3.5(9) = 32 in.

This length can be accommodated withinthe confined core of the boundary element,so that no hooks are needed, as assumed.However, because of the likelihood of largehorizontal cracks developing in theboundary elements, particularly in thepotential hinging region near the base of the


wall, the horizontal bars will be providedwith 90° hooks engaging a vertical bar, asrecommended in the Commentary to ACIChapter 21 and as shown in Figure 10-64.Required lap splice length for No. 5horizontal bars, assuming (wherenecessary) 1.3 ld = (1.3)(32) = 42 in.

(h) Detail of structural wall: See Figure 10-64. It will be noted that the No. 5vertical-wall “web” reinforcement,required for shear resistance, has beencarried into the boundary element. TheCommentary to ACI Section 21.6.5specifically states that the concentratedreinforcement provided at wall edges(i.e. the boundary elements) for bendingis not be included in determining shear-reinforcement requirements. The areaof vertical shear reinforcement locatedwithin the boundary element could, ifdesired, be considered as contributingto the axial load and bending capacity.

(i) Design of boundary zone using UBC- 97 and IBC-2000 Provisions:

Using the procedure discussed inSection 10.4.3 item 5 (f), the boundaryzone design and detailing requirementsusing these provisions will bedetermined.(1) Determine if boundary zonedetails are required:Shear wall boundary zone detailrequirements to be provided unless Pu ≤0.1Ag f′c and either Mu/Vulu ≤ 1.0 or Vu ≤3 Acv cf ′ . Also, shear walls with Pu >

0.35 P0 (where P0 is the nominal axialload capacity of the wall at zeroeccentricity) are not allowed to resistseismic forces.Using 26 inch square columns; 0.1Ag f′c

= 0.1 × (14 × 19.83 × 12 + 2 × 262 ) × 4= 1873 kips < Pu = 3963 kips. Using 32× 50 columns also results in the value of

0.1Ag f′c to be less than Pu.Therefore, boundary zone details arerequired.

Assume a 14 in. thick wall section with32 × 50 in. boundary elementsreinforced with 24 No. 11 bars as usedpreviously. Also, it was determined that2#5 bars at 17 in. spacing is needed asvertical reinforcement in the web. Onthis basis, the nominal axial loadcapacity of the wall (P0) at zeroeccentricity is:P0 = 0.85 f′c (Ag –Ast) + fy Ast

= 0.85 × 4 × (6195-82.68) + (60 ×82.68) = 25,743 kipsSince Pu = 3963 kips = 0.15 P0 < 0.35 P0

= 9010 kips, the wall can be consideredto contribute to the calculated strengthof the structure for resisting seismicforces.Therefore, provide boundary zone ateach end having a distance of 0.15 lw =0.15 × 26.17 × 12 = 47.1 in. On thisbasis, a 32×50 boundary zone asassumed is adequate.Alternatively, the requirements forboundary zone can be determined usingthe displacement based procedure. Assuch, boundary zone details are to beprovided over the portion of the wallwhere compressive strains exceed0.003. The procedure is as follows:Determine the location of the neutralaxis depth, c′u.From Table 10-8, P′u = 5767 kips; thenominal moment strength, M′n ,corresponding to P′u is 89,360 k-ft (seeFigure 10-63). For 32 × 50 in. boundaryelements reinforced with 24 #11 bars,c′u is equal to 97.7 in. This value can bedetermined using the straincompatibility approach.From the results of analysis, the elasticdisplacement at the top of the wall, ∆E isequal to 1.55 in. using gross sectionproperties and the correspondingmoment, M′n, at the base of the wall is30,469 k-ft (see Table 10-8). From theanalysis using the cracked sectionproperties, the total deflection, ∆t, at top

558 Chapter 10

of the wall is 15.8 in. (see Table 10-3, ∆t

= 2.43 × Cd = 2.43 × 6.5 = 15.8in.), also∆y = ∆E M′n/M′E = 1.55 × 89,360/30,469= 4.55 in.The inelastic deflection at the top of thewall is:∆i = ∆t - ∆y = 15.8 – 4.55 = 11.25in.Assume lp = 0.5 lw = 0.5 × 26.17 × 12 =157 in., the total curvature demand is:

510176.5

1217.26

003.0

157)2/15712148(

25.11

−×=×

+×−×

=tφ

Since φt is greater than 0.003/c′u =0.003/97.7=3.07×10-5 , boundary zonedetails are required. The maximumcompressive strain in the wall is equalto φ t c′u = 5.176 × 10-5 × 97.7 = 0.00506which is less than the maximumallowable value of 0.015. In this case,boundary zone details are required overthe length,

.8.397.9700506.0

003.07.97 in=

×−

This is less than the 50 in. lengthassumed. Therefore, the entire length ofthe boundary zone will be detailed forductility.(2) Detailing requirements:

Minimum thickness:

=lu/16= O.K. . 32. 5.1016

24)1216(inin <=−×

Minimum length = 18 in. < 50 in. O.K.The minimum area of confinement

reinforcement is:

yh

ccsh f

fshA

'09.0=

Using the maximum allowable spacingof 6db = 6 x 1.41 = 8.46 in. or 6 in.(governs), and assuming #5 hoops andcrossties at a distance of 3 in. from thecenter line of #11vertical bars to theface of the column, we have

hc = 44 + 1.41 + 0.625 = 46.04

2in. 66.160

404.46609.0 =×××=shA

With four crossties (six legs, includingoutside hoops), Ash provided = 6 (0.31)= 1.86 in.2 O.K.Also, over the splice length of thevertical bars in the boundary zone, thespacing of hoops and crossties must notexceed 4 in. In addition, the minimumarea of vertical bars in the boundaryzone is 0.005×322 = 5.12 in.2 which ismuch less than the area provided by24#11 bars. The reinforcement detail inthe boundary zone would be verysimilar to that shown previously inFigure 10-64.

REFERENCES

The following abbreviations will be used to denotecommonly occurring reference sources:

• Organizations and conferences:

EBRI Earthquake Engineering Research InstituteWCEE World Conference on Earthquake EngineeringASCE American Society of Civil EngineersACI American Concrete InstitutePCA Portland Cement AssociationPCI Prestressed Concrete Institute

• Publications:

JEMD Journal of Engineering Mechanics Division,ASCE

JSTR Journal of the Structural Division, ASCEJACI Journal of the American Concrete Institute

10-1 International Conference of Building Officials,5360 South Workman Mill Road, Whittier, CA90601, Uniform Building Code. The latest editionof the Code is the 1997 Edition.

10-2 Clough, R. W. and Benuska, K. L., “FHA Study ofSeismic Design Criteria for High-Rise Buildings,”Report HUD TS-3. Federal HousingAdministration, Washington, Aug. 1966.

10-3 Derecho, A. T., Ghosh, S. K., Iqbal, M., Freskakis,G. N., and Fintel, M., “Structural Walls in


Earthquake-Resistant Buildings, Dynamic Analysisof Isolated Structural Walls—Parametric Studies,”Report to the National Science Foundation, RANN,Construction Technology Laboratories, PCA,Skokie, IL, Mar. 1978.

10-4 Derecho, A. T., Iqbal, M., Ghosh, S. K., Fintel, M.,Corley, W. G., and Scanlon, A., “Structural Wallsin Earthquake-Resistant Buildings, DynamicAnalysis of Isolated Structural Walls—Development of Design Procedure, Design ForceLevels,” Final Report to the National ScienceFoundation, ASRA. Construction TechnologyLaboratories, PCA, Skokie, IL, July 1981.

10-5 Park, R. and Paulay, T., Reinforced ConcreteStructures, John Wiley & Sons, New York, 1975.

10-6 Priestley, M.J.N. and Kowalsky, M.J., “Aspects ofDrift and Ductility Capacity of RectangularCantilever Structural Walls”, Bulletin of the NewZealand National Society for EarthquakeEngineering, Vol. 31, No. 2, 1998.

10-7 Paulay, T., “Earthquake-Resisting Walls—NewZealand Design Trends,” JACI, 144—152, May—June 1980.

10-8 Derecho, A. T., Iqbal, M., Fintel, M., and Corley,W. G., “Loading History for Use in Quasi-staticSimulated Loading Test,” Reinforced ConcreteStructures Subjected to Wind and EarthquakeForces, ACI Special Publication SP-63, 329—344,1980.

10-9 Oesterle, R. G., Aristizabal-Ochoa, J. D., Fiorato,A. E., Russell, H. G., and Corley, W. G.,“Earthquake-Resistant Structural Walls—Tests ofIsolated Walls—Phase II,” Report to the NationalScience Foundation, ASRA, ConstructionTechnology Laboratories, PCA, Skokie, IL, Oct.1979.

10-10 American Concrete Institute, Detroit, Michigan,“Building Code Requirements for ReinforcedConcrete—ACI 318-95.” The latest edition of thecode is the 1995 Edition.

10-11 Iyengar, K. T. S. R., Desayi, P., and Reddy, K. N.,“Stress—Strain Characteristics of ConcreteConfined in Steel Binders,” Mag. Concrete Res. 22,No. 72, Sept. 1970.

10-12 Sargin, M., Ghosh, S. K., and Handa, V. K.,“Effects of Lateral Reinforcement upon theStrength and Deformation Properties of Concrete,”Mag. Concrete Res. 75—76, June—Sept. 1971.

10-13 Paulay, T. and Priestley, M.J.N., Seismic Design ofReinforced Concrete and Masonry Buildings, JohnWiley & Sons, New York, 1992.

10-14 Sturman, G. M., Shah, S. P., and Winter, G.,“Effects of Flexural Strain Gradients onMicrocracking and Stress—Strain Behavior ofConcrete,” Title No. 62-50, JACI, July 1965.

10-15 Clark, L. E., Gerstle, K. H., and Tulin, L. G.,“Effect of Strain Gradient on the Stress—Strain

Curve of Mortar and Concrete,” Title No. 64-50,JACI, Sept. 1967.

10-16 Mattock, A. H., “Rotational Capacity of HingingRegions in Reinforced Concrete Beams,” Proc.Intl. Symposium on Flexural Mechanics ofReinforced Concrete, ASCE, 1965, 143—181,1965. Also PCA Development Dept. Bulletin 101.

10-17 Corley, W. G., “Rotational Capacity of ReinforcedConcrete Beams,” JSTR Proc. 92 (STS), 121—146,Oct. 1966. Also PCA Development Dept. Bulletin108.

10-18 Naaman, A. E., Harajli, M. H., and Wight, J. K.,“Analysis of Ductility in Partially PrestressedConcrete Flexural Members,” PCIJ., 64—87,May—June 1986.

10-19 Bertero, V. V. and Fellippa, C., “Discussion of‘Ductility of Concrete,’ by Roy, H. E. H. andSozen, M. A.,” Proc. Intl. Svmp. on FlexuralMechanics of Reinforced Concrete, ASCE, 227—234, 1965.

10-20 Standard Association of New Zealand, Code ofPractice for General Structural Design and DesignLoadings for Buildings—NZS 4203:] 984,Wellington, 1992.

10-21 Bertero, V. V., “Seismic Behavior of StructuralConcrete Linear Elements (Beams and Columns)and Their Connections,” Proc. of the A. IC. A. P-C.E. B. Symposium on Structural Concrete underSeismic Actions, Rome, I, 123—212, 1979.

10-22 Popov, E. P., Bertero, V. V., and Krawinkler, H.,“Cyclic Behavior of Three Reinforced ConcreteFlexural Members with High Shear,” Report No.EERI 72-5, Univ. of California, Berkeley, Oct.1972.

10-23 Brown, R. H. and Jirsa, J. 0., “Shear Transfer ofReinforced Concrete Beams Under ReversedLoading,” Paper No. 16, Shear in ReinforcedConcrete, Vol. 1, ACI Publication SP-42, 347—357, 1974.

10-24 Bertero, V. V. and Popov, E. P., “HystereticBehavior of R. C. Flexural Members with SpecialWeb Reinforcement,” Proc. U.S. NationalConference on Earthquake Engineering— 1975,Ann Arbor, MI, 316-326, 1975.

10-25 Scribner, C. F. and Wight, J. K., “Delaying ShearStrength Decay in Reinforced Concrete Membersunder Large Load Reversals,” Report UMEE 78R2,Dept. of Civil Engineering, Univ. of Michigan,Ann Arbor, 1978.

10-26 Standards Association of New Zealand, Code ofPractice for the Design of Concrete Structures,NZS 3101, Part 1:1995, Wellington, 1995.

10-27 Ehsani, M. R. and Wight, J. K., “Effect ofTransverse Beams and Slab on Behavior ofReinforced Beam-to-Column Connections,” JACI82, No. 2, 188—195, Mar.-Apr. 1985.

560 Chapter 10

10-28 Leon, R. and Jirsa, J. 0., “Bidirectional Loading ofR. C. Beam—Column Joints,” EERI EarthquakeSpectra 2. No. 3, 537—564, May 1986.

10-29 ACI—ASCE Committee 352, “Recommendationsfor Design of Beam—Column Joints in MonolithicReinforced Concrete Structures,” ACIJ. Proc. 82,No. 3, 266—283, May—June 1985.

10-30 Paulay, T., “Deterministic Design Procedure forDuctile Frames in Seismic Areas,” Paper No. 15,Reinforced Concrete Structures Subjected to Windand Earthquake Forces, ACI Publication SP-63,357—381, 1980.

10-31 Paulay, T., “Developments in Seismic Design ofReinforced Concrete Frames in New Zealand,”Can. J. Civil Eng. 8, No. 2, 91—113, June 1981.

10-32 Park, R., “Ductile Design Approach for ReinforcedConcrete Frames,” EERI Earthquake Spectra 2,No. 3, 565—619, May 1986.

10-33 CSA Standard A23.3-94, “Design of ConcreteStructures”, Canadian Standards Association, 1994.

10-34 Wight, I. K. and Sozen, M. A., “Strength Decay ofRC Columns under Shear Reversals,” JSTR 101,No. STS, 1053—1065, May 1975.

10-35 Sheikh, S. and Uzumeri, S. M., “Strength andDuctility of Tied Concrete Columns,” JSTR 106,No. STS, 1079—1102, May 1980.

10-36 Park, R., Priestley, M. J. N., and Gill, W. D.,“Ductility of Square-Confined Concrete Columns,”JSTR 108. No. 5T4, 929—950, Apr. 1982.

10-37 Priestly, M. J. N. and Park, R., “Strength andDuctility of Concrete Bridge Columns underSeismic Loading,” A CI Strut’turalJ., 61—76,Jan.—Feb. 1987.

10-38 Jennings, P. C. (ed.), “Engineering Features of theSan Fernando Earthquake, February 9, 1971,”Earthquake Engineering Research Laboratory,California Institute of Technology. Pasadena, June1971.

10-39 Paulay, T., Park, R., and Priestley, M. J. N.,“Reinforced Concrete Beam—Column Joints underSeismic Actions,” JA CI Proc. 75, No. 11, 585—593, Nov. 1978.

10-40 Hanson, N. W. and Conner, H. W., “SeismicResistance of Reinforced Concrete Beam—ColumnJoints,” JSTR 93, 5T5, 533—560, Oct. 1967.

10-41 Meinheit, D. F. and Jirsa, J. 0., “Shear Strength ofR/C Beam—Column Connections,” JSTR 107,5Th, 2227—2244, Nov. 1982.

10-42 Abdel-Fattah, B. and Wight, J. K., “Study ofMoving Beam Plastic Hinging Zones forEarthquake-Resistant Design of R/C Buildings,”ACI Structural J., 31—39, Jan—Feb. 1987.

10-43 Rosenblueth, E. and Meli, R., “The 1985Earthquake: Causes and Effects in Mexico City,”ACI Concrete Int. 8, No. 5, 23—34, May 1986.

10-44 Mitchell, D., Adams, J., DaVall, R. H., Lo, R. C.,and Weichert, “Lessons from the 1985 Mexican

Earthquake,” Can. J. Civil Eng. 13, No. 5, 535—557, 1986.

10-45 Carpenter, J. E., Kaar, P. H., and Corley W. G.,“Design of Ductile Flat Plate Structures to ResistEarthquakes,” Proc. 5th WCEE, Rome, 1973.

10-46 Symonds, D. W., Mitchell, D., and Hawkins, N.M., “Slab—Column Connections Subjected toHigh Intensity Shears and Transferring ReversedMoments” SM 76-2, Division of Structures andMechanics, Univ. of Washington, Oct. 1976.

10-47 Cardenas, A. E., Russell, H. G., and Corley, W. G.,“Strength of Low-Rise Structural Walls,” PaperNo. 10, Reinforced Concrete Structures Subjectedto Wind and Earthquake Forces, ACI PublicationSP-63, 221—241, 1980.

10-48 Barda, F., Hanson, J. M., and Corley, W. G.,“Shear Strength of Low-Rise Walls with BoundaryElements,” Reinforced Concrete Structures inSeismic Zones, ACI Publication SP-53, 149—202,1977.

10-49 Paulay, T., “Seismic Design Strategies for DuctileReinforced Concrete Structural Wall”, Proc. ofInternational Conference on Buildings with LoadBearing Concrete Walls in Seismic Zones, Paris,1991.

10-50 Oesterle, R. G., Fiorato, A. E., Johal, L. S.,Carpenter, J. E., Russell, H. G., and Corley, W. G.,“Earthquake-Resistant Structural Walls—Tests ofIsolated Walls,” Report to the National ScienceFoundation, Portland Cement Association, Nov.1976.

10-51 Oesterle, R. G., Aristizabal-Ochoa, J. D., Fiorato,A. E., Russell, H. G. and Corley, W. G.,“Earthquake Resistant Structural Walls—Tests ofIsolated Walls—Phase II,” Report to the NationalScience Foundation, Portland Cement Association,Oct. 1979.

10-52 Cardenas, A. and Magura, D. D., “Strength ofHigh-Rise Shear Walls— Rectangular CrossSection,” Response of Multistory ConcreteStructures to Lateral Forces, ACI Publication SP-36, American Concrete Institute, 1973.

10-53 Corley, W. G., Fiorato, A. E., and Oesterle, R. G.,“Structural Walls,” Paper No. 4, SignificantDevelopments in Engineering Practice andResearch, Sozen, M. A. (ed.), ACI Publication SP-72, 77—130, 1981.

10-54 Oesterle, R. R., Fiorato, A. E., and Corley, W. G.,“Reinforcement Details for Earthquake-ResistantStructural Walls,” ACI Concrete mt., 55—66, Dec.1980.

10-55 Paulay, T., “The Design of Ductile ReinforcedConcrete Structural Walls for EarthquakeResistance,” EFRI Earthquake Spectra 2, No. 4,783—823, Oct. 1986.

10-56 Saatcioglu, M., Derecho, A. T., and Corley, W. G.,“Dynamic Inelastic Response of Coupled Walls as


Affected by Axial Forces,” Non-linear Design ofConcrete Structures, Proc. of CSCE—ASCE—ACI—CEB International Symposium, Univ. ofWaterloo, Ontario, 639—670, Aug. 1979.

10-57 Shiu, K. N., Takayanagi, T., and Corley, W. G.,“Seismic Behavior of Coupled Wall Systems,”JSTR 110, No. 5, May 1051—1066, 1984.

10-58 Saatcioglu, M., Derecho, A. T., and Corley, W. G.,“Parametric Study of Earthquake-Resistant CoupleWalls,” JSTR 113, No. 1, 141—157, Jan. 1987.

10-59 Paulay, T. and Binney, J. R., “DiagonallyReinforced Coupling Beams of Shear Walls,” PaperNo. 26, Shear in Reinforced Concrete, ACIPublication SP-42, Vol. 2, 579—598, 1974.

10-60 Barney. G. B., Shiu, K. N., Rabbat, B., Fiorato, A.E., Russell, HG and Corley, W. I., “Behavior ofCoupling Beams under Load Reversal,” PCA Res.& Dcv. Bulletin No. 68, 1980.

10-61 International Code Council (2000), InternationalBuilding Code 2000, Virginia.

10-62 Vision 2000, “Performance Based SeismicEngineering of Buildings”, Structural EngineersAssociation of California (SEOAC), Sacramento,California, 1995.

10-63 “Integrated Finite Element Analysis and Design ofstructures, SAP 2000”, Computers and Structures,Inc. Berkeley, California, 1997.

10-64 “NASTRAN (NASA Structural AnalysisComputer System)”, Computer SoftwareManagement and Information System (COSMIC),Univ. of Georgia, Athens, Georgia.

10-65 “ANSYS—Engineering Analysis Systems,”Swanson Analysis Systems, Inc., Houston, PA.

10-66 Habibullah, A., “Three Dimensional Analysis ofBuilding Systems, ETABS”, Version 6.2,Computers and Structures Inc., Berkeley,California, 1997.

10-67 Parkash, V. and Powell, G.H. “DRAIN-2DX: AGeneral Purpose Computer program for DynamicAnalysis of Inelastic Plane Structures”, EarthquakeEngineering Research Center, University ofCalifornia, Berkeley, CA, 1992.

10-68 Valles, R.E., Reinhorn, A.M., Kunnath, S.K., Li, C.and Madan, A., “IDARC Version 4.0: A ComputerProgram for the Inelastic Damage Analysis ofBuildings”, Report No. NCEER-96-0010, NationalCenter for Earthquake Engineering Research, StateUniversity of New York at Buffalo, NY, 1996.

10-69 Clough, R. W., “Dynamic Effects of Earthquakes,”Trans. ASCE 126, Part II, Paper No. 3252, 1961.

10-70 Blume, J. A., “Structural Dynamics in Earthquake-Resistant Design,” Trans. ASCE 125, Part I, PaperNo. 3054, 1960.

10-71 Berg, G. V., “Response of Multistory Structures toEarthquakes,” Paper No. 2790, JEMD, Apr. 1961.

10-72 Minimum Design Loads for Buildings and otherStructures (ASCE 7-95), a revision of

ANSI/ASCE 7-93, American Society of CivilEngineers, New York, 1996.

10-73 Applied Technology Council, “TentativeProvisions for the Development of SeismicRegulations for Buildings,” ATC Publication 3-06,U.S. Government Printing Office, Washington, 505pp., 1978.

10-74 Seismology Committee, Structural EngineersAssociation of California (SEAOC), RecommendedLateral Force Requirements and commentary, Dec.15, 1996.

10-75 Earthquake Engineering Research Institute,“Reducing Earthquake Hazards: Lessons Learnedfrom Earthquakes,” EERI Publication No. 86-02,Nov. 1986.

10-76 ACI Committee 315, “Details and Detailing ofConcrete Reinforcement (ACI 315-80),” JACI 83,No. 3, 485—512, May—June 1986.

10-77 Fintel, M., “Ductile Shear Walls in Earthquake-Resistant Multistory Buildings,” JACI 71, No. 6,296—305, June 1974.

10-78 Derecho, A. T., Fintel, M., and Ghosh, S. K.,“Earthquake-Resistant Structures,” Chapter 12,Handbook of Concrete Engineering, 2nd Edition,M. Fintel (ed.), Van Nostrand Reinhold, 411—513,1985.

10-79 ACI Committee 340, “Design Handbook inAccordance with the Strength Design Method ofACI 318- 89: Vol. 1—Beams, One-Way Slabs,Brackets, Footings, and Pile Caps” (ACI 340.1R-84), Publication SP-17(84), American ConcreteInstitute, 1984.

10-80 Concrete Reinforcing Steel Institute, CRSIHandbook, Schaumburg, IL. The latest edition ofthe handbook is the 1998 Edition.

10-81 ACI Committee 340, Design Handbook inAccordance with the Strength Design Method ofACI 318-89: Vol. 2—Columns, Publication SP-17A(90), American Concrete Institute, Detroit,Michigan, 1990.

562 Chapter 10

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