Engineering Structures 29 (2007) 2262–2273www.elsevier.com/locate/engstruct

Three dimensional analysis of reinforced concrete frames with cracked beamand column elements

Cengiz Dundar∗, Ilker Fatih Kara

Department of Civil Engineering, Cukurova University, 01330, Adana, Turkey

Received 20 July 2006; received in revised form 20 November 2006; accepted 21 November 2006Available online 28 December 2006

Abstract

In the design process of reinforced concrete buildings, the serviceability limit state for lateral drift is an important design criterion that mustbe satisfied to prevent large second-order P–delta effects. Deformation control is also important to ensure the serviceability requirements. Foraccurate determination of the deflections, cracked members in the reinforced concrete structure need to be identified along with their effectiveflexural and shear rigidities. In this study, a computer program has been developed using the rigid diaphragm model for the three dimensionalanalysis of reinforced concrete frames with cracked beam and column elements. ACI, CEB and probability-based effective stiffness modelsare used for the effective moment of inertia of the cracked members. In the analysis, shear deformations, which can be large following crackdevelopments, are taken into account and the variation of the shear rigidity due to cracking is considered by reduced shear stiffness models.The computer program is based on an iterative procedure which is subsequently verified experimentally through a series of reinforced concreteframe tests. A parametric study is also carried out on a four-story, three-dimensional reinforced concrete frame. The iterative analytical procedurecan provide an accurate and efficient prediction of deflections of reinforced concrete structures due to cracking under service loads. The mostsignificant feature of the proposed procedure is that the variations in the flexural stiffness of beams and columns can be observed explicitly. Theprocedure is efficient from the viewpoint of computational time and convergence rate, and is also more direct than the finite element method.c© 2006 Elsevier Ltd. All rights reserved.

Keywords: Three dimensional analysis; Reinforced concrete frames; Effective shear modulus; Effective moment of inertia; Deflections

1. Introduction

When designing reinforced concrete structures, a designermust satisfy not only strength requirements but alsoserviceability requirements. To ensure the serviceabilityrequirements of tall reinforced concrete buildings, it isnecessary to accurately assess the deflection under lateral andgravity loads. In recent years high-rise and slender structureshave been constructed using high-strength steel and concrete.Therefore, the serviceability limit state for lateral drift becomesa much more important design criterion and must be satisfiedto prevent large second-order P–delta effects. In addition, thecontrol of the deformation in reinforced concrete beams is alsoimportant to ensure serviceability requirements. Due to thelow tensile strength of concrete, cracking, which is primarily

∗ Corresponding author. Tel.: +90 322 3386762; fax: +90 322 3386702.E-mail address: dundar@cu.edu.tr (C. Dundar).

0141-0296/$ - see front matter c© 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2006.11.018

load dependent, can occur at service loads and reduce theflexural and shear stiffness of reinforced concrete members. Foraccurate determination of the deflections, cracked members inreinforced concrete structures need to be identified and theireffective flexural and shear rigidities determined.

Cracked states in reinforced concrete beam elements canbe considered by several methods [1,2]. These methods takeinto account the constitutive relationships of both steel andconcrete together with the bond–slip relationship. However, dueto the complexities of the actual behavior of the reinforcedconcrete frame and the cumbersome computations to be carriedout, these methods can not easily be adopted by designengineers. Several nonlinear finite element procedures havealso been developed for computing the deflections in reinforcedconcrete structures. These procedures can differ in severalaspects: the types of finite elements used, the constitutivemodels adopted, and the methods of stiffness evaluation andmesh discretization [3]. Modeling the stiffness reduction dueto cracking using nonlinear finite element methods has been

C. Dundar, I.F. Kara / Engineering Structures 29 (2007) 2262–2273 2263

performed by many researchers. The procedures concerningthe types of finite elements and mesh discretization canbe grouped as microelement and macroelement approaches.With the microelement approach, the structure is divided intomany small finite element including the two dimensional andthree dimensional elements modeling concrete, bar elementsmodeling steel, and discrete crack representation modeling forthe cracking. With the macroelement approach, each elementrepresents both steel and concrete and the local phenomena areincorporated into a constitutive model that is used to obtainthe stiffness matrix of a macro element [4–8]. On the otherhand most, if not all, of these analyses are expensive and timeconsuming, especially for large-scale tall reinforced concretestructures.

Spiliopoulos and Lykidis used three-dimensional solidelements based on a smeared fixed crack model to predict thedynamic response of reinforced concrete structures [9]. Theresults of the developed procedure were found to be in goodagreement with those of experiments available in the literature.

The analysis of reinforced concrete frames with crackedmembers can also be performed by some commercial FEMsoftware packages such as LUSAS [10] which includesadvanced treatment of reinforced concrete behavior. Theprogram considers the cracking effect in the members throughcrack models such as the smeared crack model using finiteelement methods. In the present study, the stiffness matrixmethod has been employed taking into account cracking effectswith effective stiffness models. The reduction of shear rigiditiesafter the development of cracks is also considered by employingreduced shear stiffness models available in the literature.

In the design of tall reinforced concrete structures, themoments of inertia of the beams and columns are usuallyreduced at the specified ratios to compute the lateral driftby considering the cracking effects on the stiffness of thestructural frame. The gross moment of inertia of columns isgenerally reduced to 80% of their uncracked values while thegross moment of inertia of beams is reduced to 50%, withoutconsidering the type, history and magnitude of loading, and thereinforcement ratios in the members [11].

A simplified and computationally more efficient method forthe analysis of two dimensional reinforced concrete frameswith beam elements in the cracked state was developed byTanrikulu et al. [12]. ACI [13] and CEB [14] model equations,which consider the contribution of tensile resistance of concreteto flexural rigidities by moment–curvature relationships, wereused to evaluate the effective moment of inertia. Sheardeformations were also taken into account in the formulationand the variation of shear rigidity due to cracking wasconsidered by reduced shear stiffness models. In the analysis,cracking was considered only for beam elements, hence,the linear elastic stiffness equation was used for columns.However, for accurate determination of lateral deflections oftall reinforced concrete structures, it is important to considerthe effects of concrete cracking on the stiffness of the columns.Therefore, the reduction of flexural and shear rigidities in thecolumns due to cracking should also be taken into account.

Two iterative analytical procedures were developed forcalculating the lateral drifts in tall reinforced concretestructures [3]. A general probability-based effective stiffnessmodel was used to consider the effects of concrete cracking.Analytical procedures named the direct effective stiffness andload incremental methods were based on the proposed effectivestiffness model and iterative algorithm with linear finite elementanalysis. The variation of shear rigidity due to cracking was notconsidered in the analysis. Whereas, after the development ofcracks, shear deformations, can be large and significant. Hence,the reduction of shear rigidities due to cracking should also beincluded for improving the results of the analysis.

In practice, the analysis of reinforced concrete frames areusually carried out by linear elastic models which either neglectthe cracking effect or consider it by reducing the stiffness ofmembers arbitrarily. It is also quite possible that the design oftall reinforced concrete structures on the basis of linear elastictheory may not satisfy serviceability requirements. Therefore,an analytical model which can include the effects of concretecracking on the flexural and shear stiffness of the members andaccurately assess the deflections would be very useful. In thepresent study, a computer program has been developed usinga rigid diaphragm model for the three dimensional analysisof reinforced concrete frames with cracked beam and columnelements. In the analysis, the stiffness matrix method is appliedto obtain the numerical solutions, and the cracked memberstiffness equation is evaluated by including the uniformlydistributed and point loads on the member. In obtaining theflexibility influence coefficients a cantilever beam model is usedwhich greatly simplifies the integral equations for the case ofpoint loads. In the program, the variation of the flexural rigidityof a cracked member is evaluated by ACI, CEB and probability-based effective stiffness models. Shear deformation effects arealso taken into account and reduced shear stiffness is consideredby using effective shear modulus models [15–17]. The resultshave been verified with the experimental results available in theliterature. Finally a parametric study is carried out on a four-story, three-dimensional reinforced concrete frame.

2. Models used for the effective flexural stiffness of acracked member

ACI and CEB models which consider the effect of crackingand participation of tensile concrete between cracks have beenproposed to define the effective flexural behavior of reinforcedconcrete cracked sections. In the ACI model the effectivemoment of inertia is given as

Ieff =

(Mcr

M

)m

I1 +

[1 −

(Mcr

M

)m]I2, for M ≥ Mcr (1a)

Ieff = I1, for M < Mcr (1b)

where m = 3. This equation was first presented byBronson [18] with m = 4 when Ieff is required for thecalculation of curvature in an individual section. In the CEB

2264 C. Dundar, I.F. Kara / Engineering Structures 29 (2007) 2262–2273

model Ieff is also defined in the following form:

Ieff =

[β1β2

(Mcr

M

)2 1I1

+

(1 − β1β2

(Mcr

M

)2)

1I2

]−1

,

for M ≥ Mcr (2a)

Ieff = I1, for M < Mcr (2b)

in which β1 = 0.5 for plain bars and 1 for high bondreinforcement; β2 = 1 for the first loading and 0.5 for theloads applied in a sustained manner or in a large number ofload cycles [19].

In Eqs. (1) and (2), I1 and I2 are the moments of inertia of thegross section and the cracked transformed section, respectively,M is the bending moment, Mcr is the moment corresponding tothe flexural cracking considered. The cracking moment, Mcr iscalculated by the program using the following equation:

Mcr =( fr + σv) I1

yt(3)

where σv is the axial compressive stress, fr is the flexuraltensile strength of concrete, and yt is the distance from centroidof gross section to extreme fiber in tension.

In addition to the ACI and CEB models, the probability-based effective stiffness model [3], which accounts for theeffects of concrete cracking with stiffness reduction, has beenconsidered. In the probability-based effective stiffness model,the effective moment of inertia is obtained as the ratio of thearea of the moment diagram segment over which the workingmoment exceeds the cracking moment Mcr to the total area ofthe moment diagram in the following form (Fig. 1).

Auncr = A1 + A2 =

∫M(x)<Mcr

M(x) (4a)

Acr = A3 =

∫M(x)≥Mcr

M(x) (4b)

A = Acr + Auncr (4c)

Puncr [M(x) < Mcr] =Auncr

A(4d)

Pcr [M(x) ≥ Mcr] =Acr

A(4e)

Ieff = Puncr I1 + Pcr I2 (4f)

where Acr is the area of moment diagram segment over whichthe working moment exceeds the cracking moment Mcr and Ais the total area of the moment diagram. In the same equation,Pcr and Puncr are the probability of occurrence of crackedand uncracked sections, respectively. The applicability of theproposed model was also verified with frame test results [3].

In the literature [20–22], the effective moment of inertiagiven by ACI and CEB is the best among the commonlyaccepted simplified methods for the estimation of instantaneousdeflection. Although the ACI and CEB models are usuallyconsidered for beams, in the present study these models arealso used for columns considering the axial force in thedetermination of the cracking moment. In the analysis the axial

Fig. 1. Cracked and uncracked regions of the simply supported beam elementsubjected to point and uniformly distributed loads.

force in the beams is also considered for the calculation of thecracking moment, no matter if it is relatively small.

In the computer program developed in the present study, theaforementioned models are used for the effective moment ofinertia of the cracked section.

3. Models used for the effective shear stiffness of concrete

Shear deformation can be large and significant after thedevelopment of cracks in a frame structure and thus can be ofpractical importance in the design of structures. The effectiveshear modulus of concrete due to cracking is given by severalmodels in the literature.

Cedolin and dei Poli [15] observed that a value of Gc linearlydecreasing with the fictitious strain normal to the crack wouldgive better predictions for beams failing in shear, and suggestedthe following equation

Gc = 0.24Gc(1 − 250ε1), for ε1 ≥ εcr (5a)

Gc = Gc, for ε1 < εcr (5b)

where Gc is the elastic shear modulus of uncracked concrete,ε1 is the principal tensile strain normal to the crack and εcr isthe cracking tensile strain.

Al-Mahaidi [16] recommended the following hyperbolicexpression for the reduced shear stiffness Gc to be employedin the constitutive relation of cracked concrete

Gc =0.4Gc

ε1/εcr. (6)

Yuzugullu and Schnobrich [17] used a constant value of Gcfor the effective shear modulus

Gc = 0.25Gc, for deep beams (7a)

Gc = 0.125Gc,

for shear wall and shear wall-frame systems. (7b)

In the computer program developed in the present study,these models can be considered for the effective shear modulusof cracked concrete.

In this study, since the rigid diaphragm model is considered,cracking occurs only in the beam element due to flexuralmoments in the local y direction. On the other hand, the columnelement cracks due to the flexural moments in the local y and zdirections. Hence Ieff, Mcr, M , I1, I2, ε1 and ε2, are the valuesrelated to the flexure in the local y and z directions.

C. Dundar, I.F. Kara / Engineering Structures 29 (2007) 2262–2273 2265

Fig. 2. A typical member subjected to point and uniformly distributed loads.

4. Basic equations and formulation of the problem

In this section, the flexibility influence coefficients of amember will first be obtained, and then using compatibilityconditions and equilibrium equations, the stiffness matrix andthe load vector of a member with some regions in the crackedstate will be evaluated.

A typical member subjected to point and uniformlydistributed loads, and positive end forces with correspondingdisplacements, are shown in Fig. 2. A cantilever model is usedfor computing the relations between nodal actions and basicdeformation parameters of a general space element (Fig. 3). Thebasic deformation parameters of a general space element maybe established by applying unit loads in turn in the directionsof 1–3 and 7–9. Then, the compatibility conditions give thefollowing equation in matrix form:

f11 0 0 0 0 00 f22 f23 0 0 00 f32 f33 0 0 00 0 0 f77 f78 00 0 0 f87 f88 00 0 0 0 0 f99

P1P2P3P7P8P9

=

d1d2d3d7d8d9

(8)

in which, fi j is the displacement in i-th direction due to theapplication of unit loads in j-th direction, and can be evaluatedby means of the principal of virtual work as follows:

fi j =

∫ L

0

(Mzi Mz j

Ec Ieffz+

Myi My j

Ec Ieffy+

Vyi Vy j

Gc As

+Vzi Vz j

Gc As +

Mbi Mbj

Gc I0+

Ni N j

Ec A

)dx . (9)

In Eq. (9), Mzi , Mz j , Myi , My j , Vzi , Vz j , Vyi , Vy j , Mbi , Mbj ,Ni and N j are the bending moments, shear forces, torsionalmoments and axial forces due to the application of unit loadsin i-th and j-th directions, respectively, Ec and I0 denote themodulus of elasticity of concrete and torsional moment ofinertia of the cross section, s and A are the shape factor andthe cross sectional area, respectively.

The three dimensional member stiffness matrix is obtainedby inverting the flexibility matrix in Eq. (8) and using theequilibrium conditions.

The member fixed-end forces for the case of point anduniformly distributed loads can be evaluated by means of

Fig. 3. A cantilever model for computing the relations between the nodalactions and basic deformation parameters.

compatibility and equilibrium conditions as follows

P10 = P20 = P30 = P40 = P50 = P60 = P90 = P110 = 0(10a)

P70 = −( f88 f70 − f78 f80)/( f77 f88 − f78 f87) (10b)P80 = −( f77 f80 − f78 f70)/( f77 f88 − f78 f87) (10c)P100 = −(q L + P + P70) (10d)

P120 = −(q L2/2 + P(L − a) + P70L + P80) (10e)

where fi0 (i = 7, 8) is the displacement in i-th direction dueto the application of span loads which can be obtained by usingthe principal of virtual work in the following form

fi0 =

∫ L

0

(Myi M0

Ec Ieffy+

Vzi V0

Gc As)

dx (11)

where M0 and V0 are the bending moment in local y directionand shear force in local z direction due to the span loads.Finally, the member stiffness equation can be obtained as

kd + P0 = P (12)

where k (12 × 12) is the stiffness matrix, d (12 × 1) is thedisplacement vector, P0 (12 × 1) is the fixed end force vectorand P (12×1) is the total end force vector of the member. SinceEq. (12) is given in the member coordinate system (x, y, z),it should be transformed to the structure coordinate system(X, Y, Z).

In the rigid diaphragm model, member equations are firstobtained and then by considering the contributions which comefrom each element, the system stiffness matrix and system loadvector are assembled. Finally, the system displacements andmember end forces are obtained by solving the system equation.This procedure is repeated step by step in all iterations.

The flexibility influence coefficient can now be evaluated bymeans of Eqs. (9) and (11), with the following expressions ofmoment and shear forces obtained from the application of unitand span loads.

M2(x) = x; V2(x) = 1 (13a)M3(x) = −1; V3(x) = 0 (13b)M7(x) = −x; V7(x) = 1 (13c)M8(x) = −1; V8(x) = 0 (13d)M9(x) = −1; V9(x) = 0 (13e)

M0(x) =

−

qx2

20 ≤ x ≤ a

−qx2

2− P(x − a), a < x ≤ L

(13f)

V0(x) =

{qx, 0 ≤ x ≤ a

qx + P, a < x ≤ L

}. (13g)

2266 C. Dundar, I.F. Kara / Engineering Structures 29 (2007) 2262–2273

If the ACI and CEB models are used for the effective momentof inertia of the cracked members, the flexibility influencecoefficient can be evaluated using Eqs. (9), (11) and (13) asfollows

f22 =1Ec

∫ L

0

x2

Ieffzdx +

sA

∫ L

0

1

G cdx (14a)

f23 =1Ec

∫ L

0

(−x)

Ieffzdx (14b)

f33 =1Ec

∫ L

0

1Ieffz

dx (14c)

f77 =1Ec

∫ L

0

x2

Ieffydx +

sA

∫ L

0

1

Gcdx (14d)

f78 =1Ec

∫ L

0

xIeffy

dx (14e)

f88 =1Ec

∫ L

0

1Ieffy

dx (14f)

f70 =q

2Ec

∫ L

0

x3

Ieffydx +

qsA

∫ a

0

x

Gcdx

+PEc

∫ L

a

x(x − a)

Ieffydx +

sA

∫ L

a

P

Gcdx (14g)

f80 =q

2Ec

∫ L

0

x2

Ieffydx +

PEc

∫ L

a

(x − a)

Ieffydx . (14h)

On the other hand, if the probability-based effective stiffnessmodel is considered for the effective flexural stiffness of thecracked members, the flexibility influence coefficients can beobtained as

f22 =L3

3Ec Ieffz+

sA

∫ L

0

1

G cdx (15a)

f23 = −L2

2Ec Ieffz(15b)

f33 =L

Ec Ieffz(15c)

f77 =L3

3Ec Ieffy+

sA

∫ L

0

1

Gcdx (15d)

f78 =L2

2Ec Ieffy(15e)

f88 =L

Ec Ieffy(15f)

f70 =q L4

8Ec Ieffy+

qsA

∫ a

0

x

Gcdx +

P3Ec Ieffy

× (L3+ a3/2 − 3aL2/2) +

sA

∫ L

a

P

Gcdx (15g)

f80 =q L3

6Ec Ieffy+

PEc Ieffy

(L2/2 + a2/2 − aL). (15h)

It should be noted that, since the member has cracked anduncracked regions, the integral operations in Eqs. (14) and (15)

Fig. 4. Cracked and uncracked regions of the member.

will be performed in each region. In general, the member hasthree cracked regions and two uncracked regions, as seen inFig. 4.

In the cracked regions where M > Mcr, Ieff and Gcvary with M along the region. Hence, the integral values inthese regions should be computed by a numerical integrationtechnique. The variation of effective moment of inertia andeffective shear modulus of concrete in the cracked regionsnecessitate the redistribution of the moments in the structure.Therefore, an iterative procedure should be applied to obtainthe final deflections and internal forces of the structure.

5. Computer program

A general purpose computer program developed in thepresent study on the basis of an iterative procedure is codedin the FORTRAN 77 language. The flow chart of the solutionprocedure of the program is given in Fig. 5. In the solutionprocedure, using a procedure that evaluates the flexibilitycoefficients by means of the end forces provided by theprevious step is not convergent in many cases. If the beamsand columns are cracked in both positive and negative momentregions, it is indicated that both these regions will be moreflexible in the next step; as a consequence, the moment willdecrease either at midspan or at the end of the member,with a consequent decrease of the flexibility. The alternateincrease and decrease of the flexibility generates a generallynon-convergent procedure. Therefore a procedure considering,in each step, the value of the end forces evaluated as the meanvalue of the forces of all previous steps have been used in thepresent study [20]. This procedure decreases the fluctuations inthe flexibility and accelerates the convergence of the algorithm.

In the program, the following equation is used as theconvergence criterion∣∣∣∣∣ Pn

i − Pn−1i

Pni

∣∣∣∣∣ ≤ ε. (16)

In Eq. (16) n is the iteration number, ε is the convergencefactor and Pn

i (i = 1, 12) is the end force of each member ofthe structure for the n-th iteration.

As seen in the flow chart of the program, the analyticalprocedure initially considers all the beams and columns to

C. Dundar, I.F. Kara / Engineering Structures 29 (2007) 2262–2273 2267

Fig. 5. Solution procedure of the program.

be uncracked and performs the linear elastic analysis of thestructure. The cracked members are then identified and theirflexural and shear stiffness are reduced by the effective stiffnessmodels available in the literature. Finally, an iterative procedureis applied until the convergence criterion is less than thepredefined tolerance.

Due to space limitation, the listing of the computer programis not given in the paper. A PC version and the manual of theprogram can be obtained free of charge from the authors uponrequest.

6. Verification of iterative procedure by experimentalresults and a four story frame example

In this section, three examples are presented. The first twoexamples are taken from the literature to verify the applicabilityof the analytical procedure. The last example is the applicationof the proposed analytical method on a three-dimensional, four-story reinforced concrete frame.

6.1. Example 1

In this example, the test results given by Vecchio andEmara [6] for a two story reinforced concrete frame have beencompared with the results of the present computer program.This reinforced concrete frame was designed with a center to

center span of 3500 mm and a story height of 2000 mm (Fig. 6).All members of the frame were 300 mm wide by 400 mmdeep and similarly reinforced with four No 20 deformed bars asboth tensile and compressive steel. The testing setup involvedfirst applying a total axial load of 700 kN to each columnwhich was maintained throughout the test. The lateral load(Q) was then monotonically applied until the ultimate capacityof the frame was achieved. The reinforced concrete frame ismodeled by four columns and two beam elements as shownin Fig. 6. The cross-sections of the members, the axial loadsand the span are also shown in the figure. In the analysis,ACI and probability-based effective stiffness models are usedfor the effective moment of inertia and Al-Mahaidi’s modelis used for the shear modulus of concrete in the crackedregions. The flexural stiffness reduction of beams and columnswith increasing lateral loads are also obtained by using aprobability-based effective stiffness model. In computing theflexural tensile strength and modulus of elasticity of concrete,the following equations (ACI Code Eq. [23]) are also used.

Ec = 4730√

fc (N/mm2) (17a)

fr = 0.62√

fc (N/mm2) (17b)

in which, fc is the compressive strength of concrete.In order to determine the applicability of proposed

procedure, the comparison between the test and theoreticalresults for the lateral deflection of joint 5 obtained by the linearanalysis, cracking analysis and the layered model developedby Vecchio and Emara [6], is presented in Figs. 7 and 8.The numerical results obtained from the present study are ingood agreement with the test results for applied loads withinthe 270 kN load level with maximum discrepancies of 11%.The results of Vecchio and Emara’s layer model agree wellwith the test results especially after 82% of the ultimate load.However, the theoretical model developed in the present studyshows an improved prediction over Vecchio and Emara’s layermodel [6] for loads up to 270 kN. The proposed analyticalmethod also predicts the lateral deflection in the serviceabilityloading range with good accuracy, and beyond this range thedifferences between the numerical and test results increase withthe increase in the lateral loads.

Fig. 9 presents a comparison of the top deflections usingthe different models for the effective moment of inertia of thecracked members. As seen from the figure, although differentmodels have been used for the effective flexural stiffness, theresults are very close to one another.

The theoretical results of the cracking sequence and theflexural stiffness reductions of beams and columns with respectto the lateral applied load is also shown in Fig. 10. It can beseen from the figure that the beams at the first and secondstories, B1 and B2, crack first and then both columns at the firststory, C1 and C2, crack, respectively. Finally the two columnsat the second story, C3 and C4, start to crack, respectively. FromFig. 10 it is shown that at 82% of the ultimate lateral load, thetwo columns at the first story have 55 and 58%, respectively,of their gross moment of inertia, and two beams at the firstand second stories have 45 and 47% of their uncracked values.

2268 C. Dundar, I.F. Kara / Engineering Structures 29 (2007) 2262–2273

Fig. 6. Two story reinforced concrete frame model tested by Vecchio and Emara [6] (dimensions in mm).

Fig. 7. Comparison between experimental and analytical results of the lateraldeflection of joint 5.

Fig. 8. Comparison of predicted deflections at joint 5 obtained by present andother analytical studies.

The columns of the second story have also 60 and 65% of thegross moment of inertia. The results show that considering thecracking effect on the stiffness of structural frames by assigningan 80% reduced moment of inertia to all the columns and a 50%reduced moment of inertia to all the beams does not alwaysguarantee a conservative prediction of the lateral drift.

Figs. 11 and 12 present the process of convergence byvariations of deflection and flexural stiffness of each memberwith number of iterations at the 270 kN load level. As seenfrom the figures, both the deflection and the flexural stiffnessconverge toward a solution after three iterations.

Fig. 9. Numerical comparison of lateral deflection obtained by various modelsfor the effective flexural stiffness.

Fig. 10. Flexural stiffness reduction of beams and columns with respect toincreasing lateral loads.

Fig. 11. Convergent process of lateral deflection of joint 5 with the iterationnumbers at 270 kN loading stage.

C. Dundar, I.F. Kara / Engineering Structures 29 (2007) 2262–2273 2269

Fig. 12. Convergent process of flexural stiffness of each member with iterationnumbers at 270 kN loading stage.

Fig. 13. Theoretical influence of shear deformation on lateral deflection offrame.

Fig. 13 also shows the influence of shear deformation on thetotal lateral deflection (joint 5) of the reinforced concrete frame.It can be seen that the contribution of the shear deformationto the total lateral deformation of the frame increases withincreasing lateral loads, such as 10% for 82% of the ultimateload and 12% for the ultimate lateral load.

6.2. Example 2

Further verification of the proposed analytical method hasalso been conducted by comparison with results of anotherexperimental program [3]. As seen in Fig. 14, a two-storyreinforced concrete frame was designed with a center-to-center

span of 3000 mm, a first story height of 1170 mm and asecond story height of 2000 mm. This frame is modeled byfour columns of 250 × 375 mm and two beam elements of250 × 350 mm cross sections. The reinforcing steel in thebeams and columns, the span and the loads are also shownin the figure. The test procedure involved first applying atotal axial load of 200 kN to each column and maintainingthis load throughout the test. The lateral load (Q) was thenmonotonically applied until the ultimate capacity of the framewas achieved. In the analysis, Ieff is predicted using the ACI andprobability-based effective stiffness models and Gc is evaluatedusing Al-Mahaidi’s model.

The comparison between the experimental and theoreticalresults of the lateral deflection of joint 6 is presented in Fig. 15.It is seen that the deflections calculated by the developedcomputer program using the ACI model agree well with the testresults for applied lateral loads up to approximately 78% of theultimate load (i.e., at the value of approximately 155 kN). Whenthe lateral load is beyond this load level, the difference betweenthe experimental and theoretical results becomes significant. Onthe other hand, the results of the proposed analytical procedurewhich considers the variation of shear rigidity due to crackinggives a better prediction of deflections than the other onepresented in [3].

Using the probability-based effective stiffness and ACImodels for the effective flexural rigidity the theoretical lateraldeflections of joint 6 are also obtained by the computer programas shown in Fig. 16. It should be noted that different modelsprovide quite similar results.

The variation of the flexural stiffness of beams and columnswith respect to the lateral applied load is also shown in Fig. 17.As seen from the figure, the beams of the first and second storiescrack first and then two columns on the lateral loading side,C4 and C2, start to crack followed by, in the final stage, thecracking of both columns on the opposite loading side, C3 andC1. Fig. 17 also shows that when the lateral load reaches 78% ofthe ultimate lateral load, the beams at the first and second storieshave 45 and 47%, respectively, of the gross moment of inertia,and the two columns at the second story have 55 and 60% of

Fig. 14. Dimensions of reinforced concrete frame model (dimensions in mm), (Chan et al. [3]).

2270 C. Dundar, I.F. Kara / Engineering Structures 29 (2007) 2262–2273

Fig. 15. Comparison among linear elastic method and analytical andexperimental results for the lateral deflection at joint 6.

Fig. 16. Comparison of lateral deflections obtained, by various models for theeffective moment of inertia.

Fig. 17. Stiffness reduction of each member versus lateral loads.

Fig. 18. Convergence process of lateral deflection of joint 6 with the iterationnumbers at 155 kN loading stage.

their uncracked values. The other two columns at the first storyhave also 53 and 60% of their gross moment of inertia.

Figs. 18 and 19 present the convergence process of thedeflection and flexural stiffness of each member with number

Fig. 19. Convergence process of flexural stiffness of each member withiteration numbers at 155 kN loading stage.

Fig. 20. Theoretical influence of shear deformation on the lateral deflection offrame.

of iterations at the 155 kN load level. As seen from the figures,both the deflection and the flexural stiffness converge toward asolution after three iterations.

Fig. 20 shows the contribution of the shear deformationeffect in the overall lateral displacement (joint 6). As seenfrom the figure, the percentage of shear deformation in the totaldeflection increases with increasing lateral loads, such as 14%for 78% of the ultimate lateral load.

6.3. Example 3

In the last example, the four-story reinforced concrete frameshown in Fig. 21 is analyzed by the developed computerprogram. This three-dimensional reinforced concrete frameis subjected to lateral loads at each story master point anduniformly distributed loads on the beams. The dimensionsof the members and the loads are given in Table 1. In theanalysis, the effective moment of inertia is evaluated by theACI and probability-based effective stiffness models and theeffective shear modulus is predicted by Al-Mahaidi’s model.In this example, firstly, the lateral loads acting at each storymaster point, which are expressed in terms of the values ofP , are increased while the intensity of uniform loads (q =

30 kN/m) remain constant. The variation of the maximumrelative lateral displacement of the second floor with the lateralload, when cracking is considered and not considered for beamsand columns, are shown in Fig. 22. As seen from the figure,the differences in the maximum relative lateral displacement ofthe second floor between the two cases increase with increasinglateral loads. The difference becomes significant at higher loadssuch as 70% for P = 200 kN. Fig. 23 also shows the variation

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Table 1Dimensions of the members and loads applied to the frame

First-story Second-story Third-story Fourth-story

Dimensions of beam 300 ∗ 500 mm 300 ∗ 500 mm 300 ∗ 500 mm 300 ∗ 500 mmDimensions of column 500 ∗ 500 mm 500 ∗ 500 mm 500 ∗ 500 mm 500 ∗ 500 mmUniformly distributed loads (kN/m) q q q 0.8 ∗ qLateral loads acting at master points in X direction P 2P 3P 2.5P

Fig. 21. Four-story reinforced concrete frame (dimensions in m).

Fig. 22. The variation of the maximum relative lateral displacement of thesecond floor with increasing lateral loads.

of the lateral displacement of joint 1, which displayed the fastestdeparture from linearity with respect to the lateral applied load.Similar results are also obtained for the variations in the relativelateral displacement of the second floor.

The flexural stiffness reductions of various members withrespect to the lateral applied load are shown in Fig. 24. As seenfrom the figure, when the stiffness of beams are reduced to 50%of their uncracked stiffness, the two first-story columns havereduced to 57–61% of their uncracked stiffness.

Fig. 25 also shows the theoretical influence of sheardeformation on the maximum relative lateral displacement ofthe second floor. The shear deformations contribute up toapproximately 10% of the total lateral displacements.

In this example, the intensity of uniform load is also variedfrom 10 to 45 kN/m while the lateral load (P = 100 kN)

Fig. 23. The variation of the lateral displacement of joint 1 with respect to thelateral applied load.

Fig. 24. Flexural stiffness reductions of beams and columns with respect to thelateral applied load.

Fig. 25. Theoretical influence of shear deformation on the lateral deflection ofa three dimensional reinforced concrete frame.

remained constant. Fig. 26 shows the lateral displacementof joint 1 predicted by linear analysis and crack analysis.Increasing only the uniformly-distributed loads and keepingthe lateral loads constant caused an increase in the lateraldisplacement of the reinforced concrete frame. The beams startcracking with increasing uniform loads. Thus, the crackingeffect in these members causes the reduction of the overalllateral stiffness which in turn results in an increase in the lateraldeflection of the reinforced concrete structures.

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Fig. 26. The variation of the lateral displacement of joint 1 with uniform load.

7. Conclusions

An iterative procedure has been developed to analyzethree dimensional reinforced concrete frames with crackedbeam and column elements. The proposed analytical procedureprovides nonlinear force–deformation relationships for thecracked members.

The variation of the flexural rigidity of a cracked memberhas been evaluated by using the ACI, CEB and probability-based effective stiffness models. Shear deformations, which canbe large after the development of cracks and be of practicalimportance in the design and behavior of the structure, are alsotaken into account in the analysis. The variation of shear rigidityin the cracked regions of the member has also been consideredby employing various reduced shear stiffness models availablein the literature.

The capability and the reliability of the proposed procedurehave been tested by means of comparisons with the theoreticaland experimental results available in the literature. Thenumerical results of the analytical procedure have been found tobe in good agreement with the test results for applied loads up toapproximately 78% of the ultimate load capacity of the frame.The analytical procedure not only predicts the deflections toa value of load equal to approximately 78% of the ultimateload with good accuracy, but it also can give an estimation ofbehavior at approximately 85% of the ultimate load with anacceptable degree of accuracy. On the other hand, beyond thisload level there is a large difference between the theoretical andexperimental results. Material nonlinearity may also becomesignificant in the behavior of reinforced concrete frames if theload is beyond the serviceability loading range.

In the analysis, flexural stiffness reductions of beams andcolumns with respect to the lateral applied load can be obtainedby the developed program using the probability-based effectivestiffness model for the effective moment of inertia. This isthe most significant feature of the proposed procedure andthe variations in the flexural stiffness reductions of beams andcolumns in the reinforced concrete structure can be observedexplicitly. This feature can minimize the uncertainty of theflexural stiffness of members in reinforced concrete frames andalso provide design engineers with significant information onthe consequence of cracking in members.

The theoretical deflections of reinforced concrete structureshave also been evaluated using different effective flexural

stiffness models. It should be noted that different modelsprovide quite similar results.

The numerical results of the flexural stiffness reductions ofbeams and columns involved in the testing frame indicate thatconsidering the cracking effect on the stiffness of a structuralframe by assigning an 80% reduced moment of inertia to all thecolumns and a 50% reduced moment of inertia to all the beamsdoes not always guarantee a conservative prediction of thelateral drift. However, the flexural stiffness reductions of beamsand columns are also dependent on the values of uniformly-distributed and span loads on the members.

The stiffness matrix method is applied to obtain thenumerical solutions of the proposed procedure. This procedureis efficient from the viewpoints of computational effort andconvergence rate, and is also more direct than the finite elementmethod.

The numerical results of the analytical procedures indicatethat the proposed procedure, which considers the variation ofshear rigidity after the development of cracks, gives betterpredictions of deflections than other analytical procedures. It istherefore important to consider the variation of shear rigidityin the cracked regions of members in order to obtain moreaccurate results.

In the present study, cracking effects in the reinforcedconcrete shear walls have not been considered. Stiffnessreductions due to cracking in the shear walls have to beconsidered comprehensively. The effect of various height-to-width ratios, vertical load levels, and reinforcement ratiosshould be considered in the analysis. This can be achieved byusing nonlinear finite element methods or by integrating theprobability-based effective stiffness model with finite elementanalysis [24].

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