ACI STRUCTURAL JOURN'AL .. TECHNICAL PAPER

Lateral Stiffness of Concrete Shear Walls for Tall Buildings

The present ACI 318-11 code provisions on the effective stiffnessof shear wall, as well as several other standards and technicalpapers, are reviewed. A series of nonlinear analyses are performedfor shear wall under wind and gravity loading considering well-known constitutive relations for concrete, including the tension-stiffening behavior. The results are compared with AC1 318-11and other references. It is demonstrated that the ACI provisionsfor shear walls can be overly conservative for a range of behaviortypical of wind loading. Two methods and formulas are proposedfor the effective stiffness of shear wall suitable for design officepractice. The proposed methods are capable of predicting theeffective stiffness of the shear walls for different loading intensities,as could be related to serviceability or ultimate states. This couldfacilitate the performance-based design of shear wall structuresunder wind loading.

INTRODUCTIONOne of the most important factors in the design of tall

buildings is evaluating the lateral movement inducedby wind or earthquakes and limiting it to the acceptablethreshold for serviceability and other limit state criteria.This requires a fairly accurate assessment of the building'slateral stiffness, among other factors, especially for anefficient design. In concrete buildings, the cracking ofstructural elements, namely walls, columns, beams, andslabs exposed to tensile stresses induced by lateral loads isone of the key factors affecting the lateral stiffness of thebuilding. Thus, the proper evaluation of the lateral stiffnesshas a direct impact on the performance and economy of thestructure by affecting the sizes of the primary elements ofthe lateral system-that is, shear walls, columns, and beams.Currently, the ACI 318-111 stiffness modifier provisions ofChapter 10 are the most widely used guidelines in the U.S.for shear walls. The findings of this study show that the ACIprovisions for shear wall stiffness are overly conservativefor wind applications.

ACI 318-11,1 Section 10.10.4.1, provides specificrecommendations for stiffness adjustment factors for varioustypes of members, including shear walls. The main focus ofthe ACI section is the evaluation of the second-order effecton slender columns. In the absence of any other guidelinesas an unintended consequence, however, these provisionshave been widely used by engineers for evaluating thestiffness of the structures under wind and seismic loads,which can be highly conservative, as demonstrated in thisstudy. The ACI 318-111 provisions are primarily basedon work by MacGregor and Hage,2 where they originallysuggest factors of O.4Ig for beams and 0.8Ig for columns. Itwas further refined later by MacGregoil to 0.35Ig and 0.7Igfor beams and columns, respectively, and to 0.7Ig and 0.35Igfor walls at uncracked and cracked states, respectively,all at the factored load level. For the serviceability state,the stiffness modifiers are increased by 43% to arrive at

factors of 1.0 and 0.5 for uncracked and cracked conditions,respectively. ACI 318-111 also provides alternate provisionsbased on work by Khuntia and Ghosh4,5 on columns andbeams, which are permitted to be applied to shear walls.

Khuntia and Ghosh4,5 provide a comprehensive accountof various parameters influencing the stiffness of columnsand beams and suggest a formula for calculating IelIgfor columns and beams. In a second paper, Khuntia andGhosh4•5 compared their proposed formula with previous testresults in the form of axial load and moment relationships,which includes the second-order effect.

With respect to the evaluation of shear wall stiffness,there are significant variations among recommendations byvarious international codes and research papers, which maystem from the effect of different types of loadings (seismic,wind, and gravity); the considered limit state (strengthversus service); and the influence of other parameters, suchas the effect of gravity load, concrete tensile strain softening,reinforcing bar percentage, and so on.

In 1971, ACI 3186 adopted Branson's7 formula for theeffective moment of inertia Ie, which was originally developedfor a nonprestressed concrete beam, as shown in Eq. (1).

where ir is the modulus of rupture; S is the section modulus;Ier is the cracked moment of inertia; Mer is the crackingmoment capacity of the section; Ma is the applied moment;and Ig is the gross moment of inertia. To account for thecompressive effect from axial load in prestressed members,Mer is adjusted per Eq. (2) in Branson and Heinhrich.7

Kordina8 suggested a stiffness modifier of 0.6 + 15(p + p')for the service state and 0.2 + 15(p + p') for the ultimate statefor frame elements with a rectangular cross section underbending and axial compression for service and ultimatestates, where p is the tension reinforcement ratio; and p' isthe compression reinforcement ratio.

Paulay and Priestley9 reported a simplified IelIg value atthe first yield of the steel reinforcement subjected to seismic

ACI Structural Journal, V. 108, No.6, November-December 2011.MS No. S-2010-322:R3 received October 19, 2010, and reviewed under Institute

pUblication policies. Copyright © 2011, American Concrete Institute. All rightsreserved, including the making of copies unless permission is obtained from thecopyright proprietors. Pertinent discussion including author's closure, if any, will bepublished in the September-October 2012 ACI Structural Journal if the discussion isreceived by May 1,2012.

ACI member Ahmad Rahimian is Chief Executive of WSP Cantor Seinuk, New York.He received his PhD from the Polytechnic Institute of New York University, Brooklyn,NY. He is Chair of ACI Committee 375, Perfonnance-Based Design of ConcreteBuildings for Wind Loads.

Ee /~,It Er--:- 0

t I

II

: I, I, I

--/'eL ~ _

Fig. i-Concrete constitutive relation in compressionand tension.

loading, as shown in Eq. (3). This is a modified versionof I.JIg suggested by Priestley and HartlO for a reinforcedmasonry wall.

I =[14.5 +faJIe F !' g.v e

where folic' is the ratio of axial stress to the concrete strength;and Fy is the steel yield strength in ksi. FEMA 2731\ suggestsan I.JIg factor of 0.8 and 0.5 for uncracked and cracked shearwalls, respectively.

Ibrahim and Adebar12 suggest a formula for calculatingthe effective moment or inertia ielIg at strength level (ultimate)for shear walls. The procedure considers the effect of the axialstress and a trilinear load-deformation curve (uncracked,cracked, and plastic) for the evaluation of the effectivemoment of inertia. For the cracked segment, they suggest anupper and lower bound on the extent of the cracking-lightlycracked or severely cracked due to cyclic loading. The upperbound considers the effect of the tension-stiffening zone,whereas the lower bound ignores this.

The Canadian Code13 provides a simplified equation, asshown in Eq. (4), for the effective stiffness of shear wallsapplicable to the entire wall as a function of the ratio of axialstress to the concrete strength. This provision is intended toaccount for shear wall stiffnesS. under-seismic loading."

I =[0.6+ fa JIe , gfc

All of these references suggest applying the adjustmentfactor over the entire height of the wall, which is inherentlyapproximate, as the cracking state is dependent on theshear wall geometry and load profile. While a single factor

for the entire length might be appropriate for a prismaticelement, it may become excessively approximate for abuilding shear wall system with all its customary geometricvariations and complexities.

Bazant and Oh \4 provided a general nonlinear model forthe analysis of the curvature and deformation of concretebeams. The approach is derived using a constitutive lawfor concrete and steel, considering the tensile stresses inthe cracked concrete. The approach provides rotation anddeformation behavior consistent with Branson's? formulafor reinforced concrete beams, as shown in Eq. (1).

Vecchio and Collins \5 suggested a modified compressionfield theory for concrete elements subject to in-plane stresses,considering the tensile strength between the cracks usingexperimental values for the average stress-strain relation-ships of the cracked concrete. Their findings from 30 testsshow that significant tensile stresses were observed betweenthe cracks, even at high values of tensile strain.

Orakcal and Wallace16 provided a detailed comparison ofexperimental results for several shear wall load tests withnonlinear analysis, where reasonably good agreementswere observed. The hysteretic force-deformation relationsby Chang and.. Mander\? were· used for modeling thecompressive zone; Belarbi and HSU'S18relations were usedfor the tension stiffening.

RESEARCH SIGNIFICANCEThe use of shear walls in high-rise buildings for resisting

wind and seismic effects is common. The modeling criteriaof the shear wall stiffness could have an appreciable effecton the overall performance and economy of the project.The focus of this study is to outline the main concerns withthe current state of the standards and practice, present thedifferences by numerical examples, and provide recom-mendations for the effective stiffness of shear walls. Thiscould facilitate and improve the current practice of theperformance-based design of shear wall structures underwind loading.

NONLINEAR ANALYSISA nonlinear program is developed and a series of nonlinear

shear wall analyses are performed, considering well-knownconcrete constitutive relationships and a linear elastic-plastic model for steel reinforcing. The results are comparedwith those of several test results, referenced papers, andstandards. In addition, two methods for calculating theeffective shear wall stiffness are presented that are suitablefor design off..ce practices.

A series of nonlinear analyses of a shear wall under axialgravity and lateral wind load is performed. Figure 1 showsthe constitutive relations as expressed in Eq. (5) to (8), whichare used in the nonlinear analysis. The discussion in this

-paper "is'Hillite<.rto rectanguiar wa1Isectlons:'Th'e axl:iOoad""-'is considered to be due to gravity and to remain stationary.The moment is caused by the wind load and is consideredto vary from zero to the ultimate design value with a loadfactor of 1.6 per ACI 318-11.\ It is assumed that the plane ofthe shear wall cross section remains plane, which is knownas the Bernoulli-Navier assumption. For steel reinforcing, abilinear elastic-plastic stress-strain relationship is employed;the strain in the steel and concrete is assumed to be equal-aperfect bond.

The concrete stress-strain relation for the compressionzone is based on Young's19 formula, as shown in Eq. (5),

where it has a linear ascending zone whereby the initialtangent modulus Eo is almost equal to Ee, which is typical ofhigh-strength concrete.

n = Ec·Eoo f:

where Ee is the concrete modulus of elasticity; and Eo isthe compressive strain at peak stress. For the tension zone,Belarbi and Hsu's 18 stress-strain relation is shown inEq. (7) and (8).

where Eer is the tensile strain at rupture, as defined inthe following

E =fcrcr E

c

Concrete Secant Modulus of Elasticity and ItsEffect on Effective Sectional Stiffness

Fig. 2-Concrete secant modulus of elasticity and its effecton effective cross section.

Case No. w. kip/ft falfc' (elh)max pmax= A,IAc

1 1.0 0.10 0.81 0.0060

2 1.0 0.20 0.43 0.0021

3 1.5 0.10 1.17 0.0124

4 1.5 0.20 0.65 0.0085

Figure 2 diagrammatically shows the effect of the secantmodulus of elasticity of concrete-in the compression andtension zones--on the effective cross-sectional area. Thenumerical nonlinear analyses were solved by using Newton'smethod for systems of simultaneous nonlinear equations.

Whereas the ACI 435-9520 recommended values for themodulus of rupture of fr = 7.5..Jf/ psi (0.62..Jf/ MPa) andtensile strength of j,' = 6.7....Jf/ psi (0.62..J!c' MPa) are wellknown, Belarbi and HSU18and Vecchio and Collins15 recom-mend a tensile strength of approximately fr' = 4..J!c' psi RESULTS OF NONLINEAR STUDY(0.62....JF/ MPa). While Orakcal and Wallacel6 used the ';' T t f . 1 t d' rf d t 'fy

J( )1 wo se so numenca s u Ies are pe orme : one 0 ven= 4..Jf/ psi (0.62..J!c'MPa) in their analysis of a shear wall the analytical procedure presented herein against availabletest, however, they explained that a better calibration on the test data, and the second to perform a parametric study ofstiffness of the walls can be achieved by usingfr'= 6..Jf/ psi a high-rise shear wall structure. Examples 1 and 2 compare(0.50..Jf/ MPa), which is also consistent with the recommen- the result of the analysis against previous test data for beamsdation made by Carreinrand-ehu?LIn this~tucly;-the-tensile---·--(momentinabsence ofaxiaUoad) and she&rwalls(momeI1t_cracking strength of!cr= 6..Jf/ psi (0.50....Jf/ MPa) is used. and axial load combined). Example 3 is a parametric study

The nonlinear analysis is set up to track the strain and of a 40:story shear wal~ considering a wide rang.e of lateralstress at each fiber along the depth of the shear wall section. and aXIal loads and remforcements, as shown m ~able 1.Integration methods are used to com ute the internal forces, Refer to Exa~ple~ 1,2, .and.3 for complete explanatIO?s.

.. . . . p. . The followmg IS a hIghlIght of some of the key IssueseqUllIbnum condItIOn, secant modulus of elaStlClty Ese" and d . h . d' E 13Th h. .. encountere m t e parametnc stu y m xamp e. e s eareffectlve ~ross-se.ctIOnal WIdth as repres~nted by Esctb;V' wall is first designed according to ACI 318-111 to obtainw.here bw IS the WIdth.of the ~all cross s~ctlon, as s~own m the flexural/axial reinforcing; subsequently, a nonlinearFIg. 2. The cross-sectIOnal stlffness Ese'! IS then obtamed by analysis is performed to evaluate the effective stiffness andintegration as the product of the secant modulus of elasticity deformation at various stages of loading.and moment of inertia. The effective moment of inertia at Figure 3 shows the IjIg from the nonlinear analysiseach level was then computed per Eq. (10). versus e/h for Case 2 in Example 3. The result is also

where it has a linear ascending zone whereby the initialtangent modulus Eo is almost equal to En which is typical ofhigh-strength concrete.

E .Eon =_c_o f:

where Ee is the concrete modulus of elasticity; and £0 isthe compressive strain at peak stress. For the tension zone,Belarbi and HSU'S18 stress-strain relation is shown inEq. (7) and (8).

where Eer is the tensile strain at rupture, as defined inthe following

Whereas the ACI 435-9520 recommended values for themodulus of rupture of fr = 7.5~fc' psi (O.62~fc' MPa) andtensile strength of.ft' = 6.7~fc' psi (O.62~fc' MPa) are wellknown, Belarbi and HSU18 and Vecchio and Collins15 recom-mend a tensile strength of approximately .ft' = 4~fc' psi(O.62~fc' MPa). While Orakcal and Wallace16 used the.ft'= 4~fc' psi (O.62~fc' MPa) in their analysis of a shear walltest, however, they explained that a better calibration on thestiffness of the walls can be achieved by using.ft' = 6~fc' psi(O.50~fc' MPa), which is also consistent with the recommen-dation made by Carreira and Chu.21 In this study, the tensilecracking strength offcr= 6~fc' psi (O.50~fc' MPa) is used.

The nonlinear analysis is set up to track the strain andstress at each fiber along the depth of the shear wall section.Integration methods are used to compute the internal forces,equilibrium condition, secant modulus of elasticity Ese" andeffective cross-sectional width as represented by Esctbw,

where bw is the width of the wall cross section, as shown inFig. 2. The cross-sectional stiffness EseJ is then obtained byintegration as the product of the secant modulus of elasticityand moment of inertia. The effective moment of inertia ateach level was then computed per Eq. (10).

Concrete Secant Modulus of Elasticity and ItsEffect on Effective Sectional Stiffness

- -----------\-------~

Fig. 2-Concrete secant modulus of elasticity and its effecton effective cross section.

Case No. w, kip/ft fa/fc' (elh)",a., Pnuu= A,IAc

1 1.0 0.10 0.81 0.0060

2 1.0 0.20 0.43 0.0021

3 1.5 0.10 1.17 0.0124

4 1.5 0.20 0.65 0.0085

Figure 2 diagrammatically shows the effect of the secantmodulus of elasticity of concrete-in the compression andtension zones-on the effective cross-sectional area. Thenumerical nonlinear analyses were solved by using Newton'smethod for systems of simultaneous nonlinear equations.

RESULTS OF NONLINEAR STUDYTwo sets of numerical studies are performed: one to verify

the analytical procedure presented herein against availabletest data, and the second to perform a parametric study ofa high-rise shear wall structure. Examples 1 and 2 comparethe result of the analysis against previous test data for beams(moment in absence of axial load) and shear walls (momentand axial load combined). Example 3 is a parametric studyof a 40-story shear wall considering a wide range of lateraland axial loads and reinforcements, as shown in Table 1.Refer to Examples 1, 2, and 3 for complete explanations.

The following is a highlight of some of the key issuesencountered in the parametric study in Example 3. The shearwall is first designed according to ACI 318-111 to obtainthe flexural/axial reinforcing; subsequently, a nonlinearanalysis is performed to evaluate the effective stiffness anddeformation at various stages of loading.

Figure 3 shows the I.JIg from the nonlinear analysisversus e/h for Case 2 in Example 3. The result is also

compared with several references: ACI 318-11,1 theCanadian Code, l3 Branson and Heinhrich,7 Khuntia andGhosh,4.5 Paulay and Priestley,9 and Ibrahim and Adebar.12 Itshould be noted that ACI 318-111 provides effective stiffnessvalues through its stiffness modifiers at serviceabilityand ultimate condition; Branson and Heinhrich 7 provideeffective stiffness at any intensity of loading and the balanceof the references suggest effective stiffness values only at theultimate load. All the aforementioned references consider

Ultimate

~1

the effect of the reinforcing only after crack initiation, whichbasically is to account for the cracked moment of inertia Ienignoring the effect of reinforcing in the calculation of thegross moment of inertia. For consistency in the proposedanalysis procedure, however, the effect of the reinforcingis accounted at all stages, including in the uncracked state.Considering the effect of reinforcing at the uncrackedsection, the /iIg value would be larger than 1.0. The effectof reinforcing at the uncracked section can be appreciable(refer to Khuntia and Ghosh4).

In Fig. 3, the ACI 318-111 criteria seem to be accuratein the uncracked stage only if the effect of reinforcing onthe uncracked section is ignored; however, Fig. 3 showsthat the ACI 318-111 criteria underestimate the post-cracking stiffness. The Canadian Codel3 and Ibrahim andAdebarl2 fairly reflect the ultimate load condition. However,it should be noted that the curves from the nonlinear analysisshown in Fig. 3 are only at the base of the wall and not theaverage effect on the wall, whereas all the other referencesrepresent the average effect on the entire height of the wall.

The I.JIg curves for the upper levels of the shear wall areshown in Fig. 4. As expected, the magnitude of crackingat upper levels is different, and beyond a certain level, nocracking is observed. Therefore, the average I.JIg value forthe entire height of the wall would be larger than the value forthe lowest levels; however, reinforcing can playa significantrole in the I.JIg at the post-cracking stage, as shown in Fig. 5.

Figure 5(a) shows the progression of the effect of crackingon the I.JIg in a series of curves along the height of the shearwall as the lateral load factor increases from zero to 1.6.The analysis is performed at 10 segments along the heightof the shear wall and the curves in Fig. 5(a) are obtainedby connecting the 10 data points. Figure 5(b) shows thecorresponding profile of the longitudinal reinforcement forthe shear wall at Fig. 5(a). The dashed lines in Fig. 5(a) showthe I/lg for a specific lateral load factor, as indicated on thegraph ranging from zero to 1.6.

Figure 5(c) shows the l/lg profile based on an increasedreinforcement, as shown in Fig. 5(d). The reinforcement atthe base is identical to Fig. 5(b). A review of Fig. 5(a) and(c) shows that the lowest I.JIg could occur in locations other

o Pl'lult'ly.Priestley

Itxeohirn8. AdebstUpperBouod

x lbrahim&Adeb;,r

lower Bound- Khunlia.Ghosh

10.6.~ Progression of cracking.x ffeet as the lateral load

increases

----IeAg-L-S

_. -·-Ie.4g-L-9

_. - --IeAg-l-13

-IeJIg--l.17

-JeAg-L-21

-1e~g-L-25

-IeAg.L.29

0.80 0.90 -IeAg-l-33

IF=1.6 • ILF='.4 "",.'

LF=1.2 ~ /LF=1.0 : 1 '•••

LF=O.8 _:----I--l::: ...1LF=O to 0.6 _ .....--,-.-~

.A.t Shear wall Base (l-1)1.20--- ---- -------- ..--~ r - ._--- -- -~---1

!

EM.; * ProgresSion of cracking% effect as the lateral load

increases

Fig. 5--(a) IelIg ratio along height; (b) reinforcing percentage; (c) IelIg ratio along height;and (d) reinforcing percentage.

than the maximum moment, depending on the reinforcingbar percentage. This phenomenon can also be observed ingraphs for upper-levels 5 through 13 in Fig. 4.

Figure 6(a) shows the cracked zone under an applied lateraland vertical load. It also shows the position of the centroidalong the height of the wall due to cracking of the wall. Theeccentricity between the shifted position at the crack zoneand position of the applied load, which is considered to be atthe centroid of the uncracked section, creates a counteractingmoment, thus reducing the effect of the moment caused bythe lateral load. It should be noted that the effect of the shiftof the centroid has been considered in the nonlinear analysisand the proposed equivalent linear methods.

For comparison, Fig. 6(b) demonstrates the influence ofthe shift of the centroid on the lei Ig values along the heightof the shear wall. The dashed line in Fig. 6(b) shows thebehavior when this effect is not accounted for; the solid lineshows when the shift in the centroid is considered.

PROPOSED EQUIVALENT LINEAR METHODSTwo methods suitable for design office practice for

obtaining the effective shear ,wall stiffness Iellg are proposedherein. The difference between the two approaches is themethod of consideration of the cracking effect in the analysis.Both methods can be applied within a finite element analysis;however, in the first approach, the effect of cracking isaddressed locally to the wall segments at each level of thebuilding where tensile stresses exceed the concrete tensilestrength fer' This method, which is called the "explicitmethod," is load-dependent and the effect of change in thelocation of the centroid is accounted for explicitly.

In the second method, which is called the "implicit method,"the stiffness adjustment factor is applied to the entire sectionof the shear wall at each specific level. Because the stiffnessmodifier factor Iellg is applied over the entire shear wallcross section in this method, the centroid of the crackedsection is not adjusted, as it remains at the same locationas the gross section. Therefore, the effect of the axial loadeccentricity will not be explicitly modeled. As a result, the leiIg adjustment factor is obtained so that it implicitly accountsfor this effect.

EXPLICIT METHODIn this method, the appropriate stiffness modifier is applied

locally to each segment of the wall under consideration ata particular level of the building where the axial stressesexceed };;r. The lateral load from each direction needs to bestudied independently as a separate load condition. The axialstiffness of the elements where the tensile stress exceedsfer can be adjusted per the proposed 'A coefficient, as perEq. (12). As a first step, the shear walls should be analyzedand designed for the forces using the gross properties,as the'amount of reinforcement is required for the following steps.

Because the simplified procedure is based on a linear analysis,the Eser. as shown in Eq. (11), should be based on the tensilestress-strain relation from the linear analysis, as depictedin Fig. 7. Therefore, by substituting}; from Eq. (8), Cefrom Eq. (7), and Cer from Eq. (9) into Eq. (11), the secantmodulus Ese, for linear analysis can be obtained, as shownin Eq. (12).

I06 I

I,0.4 .....:::-

//

/

".-0.2 •••••,\

Fig. 6-(a) Effect of axial load-centroid eccentricity; and(b) effect of axial load on IelIg along shear wall height.(Note: 1ft = 304 mm.)

II

II

II,

I

-J0.0020

where};, is the shear wall tensile stress from the linearanalysis. The stiffness adjustment factor 'A, as shown inEq. (13), should be applied as a stiffness modifier to thefinite element membrane stiffness in the cracked zonealong the axis parallel to the wall height (in ETABS andSAP2000 software, depending on the finite element localaxis, it could be fzz) as follows

------ A-- -E'A-_e __ eAe Ee

where p is the reinforcing percentage in the affected area;Ae is the concrete area in the affected area; and Es is themodulus of elasticity of the steel reinforcing.

0.8E~

0.6 -

0.4

0.2 Graph for Fig. 6 in Reference 4 for columnwith I% reinforcement and fc=4000psi

0.00.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70

falfc

Fig. 8-IelIg versus fa/f~ from Eq. (16) and Fig. 6 ofReference 4. (Note: 1 psi = 0.0069 MPa.)

Cl~ 0.400

E

Fig. 9-IelIg versus reinforcement percentage based onrigorous method, proposed Eq. (20), and PCl equation.

IMPLICIT METHODIn this method, for each level, an overall stiffness

adjustment factor for the entire shear wall cross section willbe obtained and the Ijlg factor will be applied to the wallelement in a linear analysis model.

Two simplified formulas for the Ijlg ratio are proposedto account for the crack effect, considering the effectof reinforcing, tension stiffening, and axial stress. Thedifference between the formulas is due to the variablesemployed. The first method, called Implicit Method I, isbased on overall sectional parameters such as elh, reinforcingbar percentage, and so on. This method is more suitable forhand calculations or simple computer analysis. The secondmethod, called Implicit Method II, is more suitable for linearfinite element analysis and is based on maximum wall stressand reinforcing values at each level.

Implicit Method IThe proposed Eq. (15) shows the Ij Ig obtained by regression

analysis. Equation (16) shows the effect of the reinforcing

and concrete axial strain on the gross moment of inertia, asshown by Ignl, where Ig is the gross moment of inertia of theconcrete section without accounting for reinforcement.

Ie = Ier + [Ignl_ Ier J.k

I I I I Ig g g g

where Ignl is the gross moment of inertia, considering theeffect of reinforcement and axial strain on the modulus oft<lasticity; and Zj accounts for. the influence of the concretemodulus under compressive axial strain, where

where d' is the distance between the center of tension andcompression reinforcement. The equations are obtained forrectangular cross sections with concentrated axial/flexuralreinforcement at each end. For uniform continuous flexuralreinforcement arrangement, d' can be considered as 0.5h andp = p' = A/(2Ae) in Eq. (16).

Figure 8 shows the ljIg from Eq. (16) versusfa1fc' with theIj Ig graph from Fig. 6 of Reference 4 for elh = 0.1. Both graphsare for 1% reinforcement, as described in Reference 4. Thespread between the two lines is attributed to different concreteconstitutive relations used in Reference 4 and in this paper-the Hognestad parabola4 versus Young, 19 respectively.

Equation 19 shows the proposed Ie,lIg equation, which isobtained parametrically by modifying the PCI22 formula,as shown in Eq. (18)., to account for- the-effect of -thecompression reinforcement.

Ier = 12np(l-1.67 FP)(d I h)2 ; peI22 equation for Ier (18)Ig

I-..E:.. = 10.5np(1- 0.84np + 0.42n'p'Ig

-0.95FP + 0.20~n'p')(d I h)2

Ier =10.5np(I-0.42p(n+l)-0.75v!nP)(dlh)2 (20)Ig

Figure 9 shows the Icrl1g for a rectangular cross sectionwith equal reinforcement at each end (p = p') based onthe proposed Eq. (20) and compares that with the rigorouscomputation and the PCI22 formula. The tension-stiffeningeffect on the moment of inertia is accounted by the factor k I,

as shown in Eq. (21), which is obtained by parametric study.

k =((elh)cr +fa)k2

I (e I h) f:

k = z ( (e I h) 1)2 0; and 'S:22 2 ( I h)e cr

(elh)cr is the eccentricity at the inception of cracking in thepresence of axial load stress fa. This, in concept, is similar tothe Branson and Heinhrich 7 Mer in the presence of axial load.

Implicit Method IIThe proposed Eq. (25) is more suitable for application

in analysis programs, where elh is not readily available;however, extreme tensile and compressive stresses fromlinear analysis can easily be obtained. The !JIg is derived byregression analysis.

The tension-stiffep,ing effect ()n the moment of inertia isaccounted by the factor k3, as shown inEq. (26),wiJ.lchlsobtained by parametric study.

k4

= (I+ (fa,.f;- (cr)JO.65 (l_f;- fc, .~)fc fc !r + fc Z3

1.60

1.40

1.20

"C 1.000.s:a; 0.80:;:01 0.60~

0.40

------ •....~---- -~.-

~~. ./. -

......•0.20

./

*"0.00

./

0.00 0.20 0.40 0.60 0.80 1.00 1.20

.::-.- -~ . -~-;'~--~ 0.80 ---- ---~ •• \7-~-----to.so ~

0.40 -.- - .~~ . -*"~-

*"0.00 *"

0.00 0.20

Fig. IO-(a) IelIgfrom Method I versus nonlinear analysis;and (b) IelIgfrom Method II versus nonlinear analysis.

Z = (1- /, - fcr )l.l (_p_)O.4 + 4 fa 'S:13 /, + fc 0.006 f:

The Ijlg from Methods I and II were calculated andcompared with the results of the nonlinear analysis for thewall in Example 3 for the indicated range of parameters.

Using the shear wall in Example 3, a wide range ofparametric studies was performed for 90 combinationsof lateral loads, axial load for falfe' from 0 to 0.3, andreinforcing for p = p' from 0.001 to 0.012 at 10 segmentsalong the height of the wall. This provides a total of 900 datapoints for the entire height of the wall. The reinforcementswere obtained for combinations of ultimate lateral andvertical ,loads using ACI 318-11 design provisions.Figure lO(a) shows the ratio of Iellg of the nonlinear analysisto that of Method I, and Fig. lO(b) shows the ratio of Ijlg ofthe nonlinear analysis to that of Method II. For each method,the' mean value of tile ratfoofI;lI;fo tnat'onhenoiili"riearanalysis is 0.96, and the standard deviation is 0.04. In thisstudy, a total of 260 data points exhibited cracking. For thecases that exhibited cracking, the mean values of the ratioof Ijlg for Methods I and II to that of the nonlinear analysisare 0.94 and 0.92, respectively; and the standard deviation is0.05 and 0.06, respectively.

NUMERICAL STUDIES AND COMPARISONSWITH TEST RESULTS

To validate the nonlinear analysis procedure forExamples I and 2, compare the results of the nonlinear

lil.90 40.0~a.-0~ 30.0o..J

~J!l 20.0j

0.00.00

Fig. ii-Comparison of nonlinear analysis with test resultfor Beam A-3 by Bresler and Scordelis.23 (Note: 1 in.25.4 mm; 1 kip = 4.45 kN; 1 ft = 304 mm.)

'[ 25.0g1Ua:: 20.0-0~oS~3

15.0 --I• Pax:: O.07Ac f'c

10.0 !--~--------+~--50 J - -rr:w~~-0.0

0.00 0.10 0.20 0.30 0.40 0.50 0.60

Fig. i2-Comparison of nonlinear analysis with test resultfor Rectangular Wall RW2 by Orakcal and Wallac~.16(Note:1 in. = 25.4 mm; 1 kip = 4.45 kN.)

analysis with test results by Bresler and Scordelis23 for asimply supported beam and Orakcal and Wallacel6 for ashear wall under axial and lateral load. Example 3 examinesa shear wall of rectangular shape for a series of axial andlateral loads, resulting in a range of reinforcement and e/h.The results are compared with various codes and references.

Example 1Figure 11 compares the result of the nonlinear analysis

with the Bresler and Scordelis23 test result for Beam A-3.

dI- •~:. ;,+4 ,1:\

L

The test is for a simply supported beam under midspanpoint load.

Example 2Figure 12 compares the result of the flexural component of

the nonlinear analysis with Orakcal and Wallace's16 test forWall RW2, a rectangular shear wall under axial and lateralload. The graph shows the first cycle of the hysteretic response.

Example 3A slender shear wall, as shown in Fig. 13, is designed

for the lateral and gravity loads shown in Table 1 perACI 318-11 provisions. Subsequently, nonlinear analysesare performed according to the procedure presented herein.The results are compared with several references, includingACI318-11.

The shear wall has a height of 400 ft (121.9 m) and a crosssection of 40 ft x 14 in. (12.2 m x 355.6 mm). A concretestrength !c' of 6000 psi (41 MPa) is considered. Axial andflexural reinforcing are clustered at each end of the wallusing Fy = 60,000 psi (414 MPa) reinforcing. The shear wallis designed and analyzed for the four load cases shown inTable 1. The ACI 318-111 load combination of 0.9D ± 1.6Wis used for the design of reinforcing. A minimum reinforcingratio at each end is considered to be 0.10%. For comparativestudy, the range of parameters considered in this example is

To obtain a realistic reinforcement profile, the shear wall foreach case is separately designed according toACI 318-11 loadcombinations. After obtaining the reinforcement along theheight of the shear wall for each case, a nonlinear analysisis performed for monotonically increasing lateral loadfrom zero to a load factor of 1.6. The cracking of the cross

----~~_._----~_.~-_._---~~~~--~~~~~

I-"",,,,.,,,iI

Ana¥sis I~ •• -Branson iI .• AC1316-Service!i ,i • A031•. Ll"''''1I 0 Ca_Code

! Pa""'-"'iI. 0 ",,,,,,, •• _,.

""'" Sou"".• - -" -···111 I x "'_&A""';

~ ~-I

0.000.00 0.10 0.20 0.30 0.010 0.50 0.60 0.10 0.80 0.50 1_-_ =~h I~ elh __ ~I-'-~-' I

Effective Moment of Inertiafa/fc=O.10, p=p'=.OO6

Effective Moment of Inertiafalfc=O.10, p=p'=.0124

!--N:>ninearI Anattsis

!.... Branson

I .•. AC018-Service

I • A03'''lJ1''''

:::~~I"'ohm&A_jlWef Bound

tlf8htn& Adeba IlCMIerBound

""""""""'hi

Service----.... ----.;

1.20 ----~~~~ --......'~:....... .....;,.,

~o.80

~ 0.60

0.60 oeo

elh

Effective Moment of Inertiafalfc=O.20, p=p '=.0021

A03' •.•.••• " i,Cona<f•• Code I

-0--- ,~ 0 PatJay·Ai!st!ey I

! ~ 0.60 - ,,----- - - - + bfatWn&A~

~ Q """'-,

:: - --- ~---~ .-------~- 1-:-=~~II~ Q~ Ql0 Q15 ~ ~ Q~ ~ ~ ~ Q~

e/h 1_. __ ,~ ,--I _----1

Effective Moment of Inertiafalfc=O.20, p=p'=.0085

I.• Aa318-Service I

__ ~Itimate..~ 1__

~:: ---=--.- ~.::.~.---'---."-.=-t--~ 0.60 - ----~-. --- -~

o Pa".""""'jb'ahin & Adeba

""'" Bouod"'ohm&A_1Lower Bound t

- IIJutia-Qlosh

0.10 1---_. -Method-I

0.30 0.40

e/h

Case#4

Fig, 14-Comparison oJIefIgJrom nonlinear analysis with other reJerencesJor Cases 1 to 4,

section along the height of the wall is calculated at eachstage of the load, and then the Ijlg and load-deformationcurves are obtained.

Figure 14 shows the change in I.IIg as lateral loadincreases, which is represented by the variable elh. TheIjlg curve is compared with the ACI 318-111 provisions,the Canadian Code,13 Branson and Heinhrich,7 Khuntiaand Ghosh,4.5 Paulay and Priestley,9 and Ibrahim andAdebar,I2 The nonlinear analysis shows an Ijlg of greaterthan 1,0 for the precrack state due to the consideration of thereinforcing effect. To maintain consistency in comparisonamong different references, however, the Ig is calculatedbased on the gross section moment of inertia of the concretesection only.

This study shows that the ACI 318-11 provisions signifi-cantly underestimate the shear wall lateral stiffness in thepost-cracked state, whereas the Canadian .Codel3 provi-sion more accurately represents the ultimate condition. Itshould be noted that the effective stiffness values of Paulayand Priestley,9 Ibrahim and Adebar,12 and Khuntia andGhosh4,5 are only at the ultimate load or strength level.

Figure 14 also compares the Ijlg obtained from the twoimplicit methods with the nonlinear analysis. The graphsshow fairly consistent results.

Figure 15 shows the load-deformation curves forCases 1 to 4 at the top of the shear wall and compares thosewith the results of the explicit method using the ETABS finiteelement analysis for service and ultimate load conditions atload factors of 1.0 and 1.6, respectively. '

CONCLUSIONSA series of nonlinear analyses for shear walls is performed

using concrete constitutive relations by Young19 for concretein compression and Belarbi and HSUl8 for concrete in tension,

The nonlinear analyses and empirical methods are checkedagainst a series of test data. A series of numerical examplesof a shear wall with various loading conditions and rein-forcing is performed, The results are compared with variousstandards and references. In general, the results betweenvarious references are widely scattered, underestimating theshear wall stiffness at some loading condition or throughout.

Three different empirical procedures for the calculationof the effective stiffness of shear walls are proposed. Themethods .are specifically tailored to be suitable for designoffice practice, The proposed methods provide a practicalapproach for evaluating the Ijlg at various loadings, whichmay be associated with service or strength limit states. Themethods are all suitable for performance-based design, asthe shear wall stiffness can be obtained for any specificloading intensity related to any wind event.

ACKNOWLEDGMENTSThe assistance of S. Towfighi and B. Sullivan, along with their valuable

comments after reviewing the manuscript, is appreciated.

NOTATIONAc gross concrete cross-sectional area neglecting reinforcementA, effective cross-sectional areaA,IAc ratio of effeclive to gross cross-sectional areaAs area of steel reinforcement at lension zoneAs' area of steel reinforcement at compression zone

crEeE,Eset

Eoeelh(elh)erFy

faicic'ic,f,ItIt'ItlHhI

t8l

'EO -----1<.

120

tOO

~---0'"

; _Noninear

0'"Analysis

0.'" , &pEl01" ,t!!hod

000OlXXl 0005 on" 0.05

-------_.-----'---Il-~earl

AnaJysis j

. _ .. ~ 1 II ' &pEl !

"'!hod II !,~---

O.1XI5 0.01) 0.05

Drift Index

I--I AnalysIS

I

I' =O.Oti

Fig. 15-Load-defiection curve for top of shear wall using nonlinear analysis andexplicit method.

width of shear walldistance from extreme· compression fiber to centroid oftension reinforcementdistance between center of tension and compression reinforcementmodulus of elasticity of concretemodulus of elasticity of steel reinforcingsecant modulus of elasticity of concreteconcrete tangent modulus at zero stressMIP, eccentricity of axial loadMIPh, eccentricity ratioMe'! Ph, eccentricity ratio for Meryield strength of steel reinforcingaxial stress due to axial loadconcrete maximum compressive stressconcrete compressive strengthcracking strength of concretemodulus of rupture of coiicrete' .- - -- -. --concrete maximum tensile stresstensile strength of concretemaximum tensile stress from linear analysisoverall height of shear walloverall depth of membermoment of inertia of cross sectioncracked moment of inertia of cross sectioneffective moment of inertia of cross sectionratio of effective to gross moment of inertiagross moment of inertia of concrete cross section neglectingreinforcementgross moment of inertia of concrete cross section consideringreinforcement and nonlinear effect of secant concrete secantmodulus, Ig,Eel E"t

Ig,

k,k2

k3k4

MaMernnon,

SWeZl

Z2 ~

Z3

Ee

EerEo

"App'

gross moment of inertia of concrete cross section consideringreinforcementfactor for tension-stiffening effect for Ie in Implicit Method Idecay power factor for k,factor for tension-stiffening effect for Ie in Implicit Method IIfactor for tension-stiffening effect for Ie in Implicit Method IIapplied moment capacity of sectioncrack moment capacity of sectionratio of modulus of elasticity of steel to concreteparameter for Young's'9 stress strain equationn-lsection modulus of cross sectionweight density of concretefactor to account for effect of change in concrete modulus underaxial loadfactor to account for influence_ of reinforcement -in tension--stiffening effect for Ie in Implicit Method Ifactor to account for influence of reinforcement in tension-stiffening effect for Ie in Implicit Method IIconcrete strainconcrete tensile strain at crackconcrete compressive strain at peak stressstiffness adjustment factor for explicit methodtension reinforcement ratio, A,IAe

compression reinforcement ratio, A:IAe

REFERENCES1. ACI Committee 318, "Building Code Requirements for Structural

Concrete (ACI 318-11) and Commentary," American Concrete Institute,Farmington Hills, MI, 2008, 503 pp.

2. MacGregor, J. G., and Hage, S. v., "Stability Analysis and Designof Concrete Frames," Journal of the Structural Division, ASCE, V. 103,No. 10, Oct. 1977, pp. 1953-1970.

3. MacGregor, J., "Design of Slender Columns-Revisited," ACIStructural Journal, V. 90, No.3, May-June 1993, pp. 302-309.

4. Khuntia, M., and Ghosh, S. K., "Flexural Stiffness of ReinforcedConcrete Columns and Beams: Analytical Approach," ACI StructuralJournal, V. 101, No.3, May-June 2004, pp. 351-363.

5. Khuntia, M., and Ghosh, S. K., "Flexural Stiffness of ReinforcedConcrete Columns and Beams: Experimental Verification," ACI StructuralJournal, V. 101, No.3, May-June 2004, pp. 364-374.

6. ACI Committee 318, "Building Code Requirements for ReinforcedConcrete (ACI 318-71)," American Concrete Institute, Farmington Hills, MI,1971,78 pp.

7. Branson, D. E., and Heinhrich, T., "Unified Procedure for Predictingthe Deflection and Centroidal Axis Location of Partially Cracked Nonpre-stressed and Prestressed Concrete Members," ACI Structural Journal, V. 79,No.2, Feb. 1982, pp. 119-130.

8. Kordina, K., "Discussion No. 3-Cracking and Crack Control,"Proceedings of the International Conference on Planning and Designof Tall Buildings, V. III, Lehigh University, Bethlehem, PA, Aug. 1972,pp.721-722.

9. Paulay, T., and Priestley, M. J. N., Seismic Design of Reinforced Concreteand Masonry Buildings, John Wiley & Sons, Inc., New York, 1991,768 pp.

10. Priestley, M. J. N., and Hart, G. C., "Design Recommendation forthe Period of Vibration of Masonry Wall Building," Research Report SSRP89/05, University of California, San Diego and Los Angeles, CA, 1989,46pp.

II. FEMA 273, "NEHRP for Seismic Rehabilitation of Buildings,"Federal Emergency Management Agency, Washington, DC, 1997, pp. 6-12.

12. Ibrahim, A. M. M., and Adebar, P., "Effective Flexural Stiffness ofConcrete Walls in High-Rise Buildings," 2000 ACI Spring Convention,San Diego, CA, Mar. 2000.

13. CSA A23.3-04, "Design of Concrete Structures," Canadian Stan-dards Association, Mississauga, ON, Canada, 2004, 240 pp.

14. BaZant, Z. P., and Oh, B. H., "Deformation of Progressively CrackingReinforced Concrete Beams," ACI Structural Journal, V. 81, No.3, May-June 1984, pp. 268-278.

15. Vecchio, F. J., and Collins, M. P., ''The Modified Compression-FieldTheory for Reinforced Concrete Elements Subject to Shear," ACI JOURNAL,Proceedings V. 83, No.2, Mar. 1986, pp. 219-231.

16. Orakcal, K., and Wallace, J. W., "Flexural Modeling of ReinforcedConcrete Walls-Experimental Verification," ACI Structural Journal,V. 103, No.2, Mar.-Apt. 2006, pp. 196-206.

17. Chang, G. A., and Mander, J. B., "Seismic Energy BasedFatigue Damage Analysis of Bridge Columns: Part I-Evaluation ofSeismic Capacity," NCEER Technical Report No. NCEER-94-0006,State University of New York, Buffalo, NY, 1994, p. 3-3.

18. Belarbi, H., and Hsu, T. C. c., "Constitutive Laws of Concrete inTension and Reinforcing Bars Stiffened by Concrete," ACI StructuralJournal, V. 91, No.4, July-Aug. 1994, pp. 465-474.

19. Young, L. E., "Simplifying Ultimate Flexural Theory by Maximizingthe Moment of the Stress Block," ACI JOURNAL,Proceedings V. 57, No. II,Nov. 1960, pp. 549-556.

20. ACI Committee 435, "Control of Deflection in Concrete Structures(ACI 435-95) (Reapproved 2008)," American Concrete Institute,Farmington Hills, MI, 2008, 88 pp.

21. Carreira, D. J., and Chu, K. H., "Stress-Strain Relationship for PlainConcrete in Tension," ACI Structural Journal, V. 83, No. I, Jan.-Feb. 1986,pp.21-28. '

22. PCI Design Handbook, fourth edition, PrecastlPrestressed ConcreteInstitute, Chicago, IL, 1992.

23. Bresler, B., and Scordelis, A. C., "Shear Strength of Reinforced ConcreteBearns," ACIJOURNAL,Proceedings V. 60, No. I, Jan. 1963, pp. 51-74.