Three dimensional analysis of reinforced concrete frames with cracked beam and column elements

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Three dimensional analysis of reinforced concrete frames with cracked beam and 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 2006

Available 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 must be satisfied to prevent large second-order P–delta effects. Deformation control is also important to ensure the serviceability requirements. For accurate determination of the deflections, cracked members in the reinforced concrete structure need to be identified along with their effective flexural and shear rigidities. In this study, a computer program has been developed using the rigid diaphragm model for the three dimensional analysis of reinforced concrete frames with cracked beam and column elements. ACI, CEB and probability-based effective stiffness models are used for the effective moment of inertia of the cracked members. In the analysis, shear deformations, which can be large following crack developments, 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 concrete frame tests. A parametric study is also carried out on a four-story, three-dimensional reinforced concrete frame. The iterative analytical procedure can provide an accurate and efficient prediction of deflections of reinforced concrete structures due to cracking under service loads. The most significant feature of the proposed procedure is that the variations in the flexural stiffness of beams and columns can be observed explicitly. The procedure 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 designer must satisfy not only strength requirements but also serviceability requirements. To ensure the serviceability requirements of tall reinforced concrete buildings, it is necessary to accurately assess the deflection under lateral and gravity loads. In recent years high-rise and slender structures have been constructed using high-strength steel and concrete.

Therefore, the serviceability limit state for lateral drift becomes a much more important design criterion and must be satisfied to prevent large second-order P–delta effects. In addition, the control of the deformation in reinforced concrete beams is also important to ensure serviceability requirements. Due to the low 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).

load dependent, can occur at service loads and reduce the flexural and shear stiffness of reinforced concrete members. For accurate determination of the deflections, cracked members in reinforced concrete structures need to be identified and their effective flexural and shear rigidities determined.

Cracked states in reinforced concrete beam elements can be considered by several methods [1,2]. These methods take into account the constitutive relationships of both steel and concrete together with the bond–slip relationship. However, due to the complexities of the actual behavior of the reinforced concrete frame and the cumbersome computations to be carried out, these methods can not easily be adopted by design engineers. Several nonlinear finite element procedures have also been developed for computing the deflections in reinforced concrete structures. These procedures can differ in several aspects: the types of finite elements used, the constitutive models adopted, and the methods of stiffness evaluation and mesh discretization [3]. Modeling the stiffness reduction due to cracking using nonlinear finite element methods has been

0141-0296/$ - see front matter c2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.engstruct.2006.11.018

performed by many researchers. The procedures concerning the types of finite elements and mesh discretization can be grouped as microelement and macroelement approaches.

With the microelement approach, the structure is divided into many small finite element including the two dimensional and three dimensional elements modeling concrete, bar elements modeling steel, and discrete crack representation modeling for the cracking. With the macroelement approach, each element represents both steel and concrete and the local phenomena are incorporated into a constitutive model that is used to obtain the stiffness matrix of a macro element [4–8]. On the other hand most, if not all, of these analyses are expensive and time consuming, especially for large-scale tall reinforced concrete structures.

Spiliopoulos and Lykidis used three-dimensional solid elements based on a smeared fixed crack model to predict the dynamic response of reinforced concrete structures [9]. The results of the developed procedure were found to be in good agreement with those of experiments available in the literature.

The analysis of reinforced concrete frames with cracked members can also be performed by some commercial FEM software packages such as LUSAS [10] which includes advanced treatment of reinforced concrete behavior. The program considers the cracking effect in the members through crack models such as the smeared crack model using finite element methods. In the present study, the stiffness matrix method has been employed taking into account cracking effects with effective stiffness models. The reduction of shear rigidities after the development of cracks is also considered by employing reduced shear stiffness models available in the literature.

In the design of tall reinforced concrete structures, the moments of inertia of the beams and columns are usually reduced at the specified ratios to compute the lateral drift by considering the cracking effects on the stiffness of the structural frame. The gross moment of inertia of columns is generally reduced to 80% of their uncracked values while the gross moment of inertia of beams is reduced to 50%, without considering the type, history and magnitude of loading, and the reinforcement ratios in the members [11].

A simplified and computationally more efficient method for the analysis of two dimensional reinforced concrete frames with beam elements in the cracked state was developed by Tanrikulu et al. [12]. ACI [13] and CEB [14] model equations, which consider the contribution of tensile resistance of concrete to flexural rigidities by moment–curvature relationships, were used to evaluate the effective moment of inertia. Shear deformations were also taken into account in the formulation and the variation of shear rigidity due to cracking was considered 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 of tall reinforced concrete structures, it is important to consider the effects of concrete cracking on the stiffness of the columns.

Therefore, the reduction of flexural and shear rigidities in the columns due to cracking should also be taken into account.

Two iterative analytical procedures were developed for calculating the lateral drifts in tall reinforced concrete structures [3]. A general probability-based effective stiffness model was used to consider the effects of concrete cracking.

Analytical procedures named the direct effective stiffness and load incremental methods were based on the proposed effective stiffness model and iterative algorithm with linear finite element analysis. The variation of shear rigidity due to cracking was not considered in the analysis. Whereas, after the development of cracks, shear deformations, can be large and significant. Hence, the reduction of shear rigidities due to cracking should also be included for improving the results of the analysis.

In practice, the analysis of reinforced concrete frames are usually carried out by linear elastic models which either neglect the cracking effect or consider it by reducing the stiffness of members arbitrarily. It is also quite possible that the design of tall reinforced concrete structures on the basis of linear elastic theory may not satisfy serviceability requirements. Therefore, an analytical model which can include the effects of concrete cracking on the flexural and shear stiffness of the members and accurately assess the deflections would be very useful. In the present study, a computer program has been developed using a rigid diaphragm model for the three dimensional analysis of reinforced concrete frames with cracked beam and column elements. In the analysis, the stiffness matrix method is applied to obtain the numerical solutions, and the cracked member stiffness equation is evaluated by including the uniformly distributed and point loads on the member. In obtaining the flexibility influence coefficients a cantilever beam model is used which greatly simplifies the integral equations for the case of point loads. In the program, the variation of the flexural rigidity of a cracked member is evaluated by ACI, CEB and probability- based effective stiffness models. Shear deformation effects are also taken into account and reduced shear stiffness is considered by using effective shear modulus models [15–17]. The results have been verified with the experimental results available in the literature. 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 a cracked member

ACI and CEB models which consider the effect of cracking and participation of tensile concrete between cracks have been proposed to define the effective flexural behavior of reinforced concrete cracked sections. In the ACI model the effective moment of inertia is given as

I_eff = Mcr

M m

I₁+

1− Mcr

M m

I₂, for M≥ M_cr (1a) Ieff =I1, forM <Mcr (1b) where m = 3. This equation was first presented by Bronson [18] with m = 4 when Ieff is required for the calculation of curvature in an individual section. In the CEB

modelIeffis also defined in the following form:

I_eff =

"

β1β2

Mcr

M 2 1

I1

+ 1−β1β2

Mcr

M 2!

1 I2

#−1

,

forM ≥Mcr (2a)

I_eff=I₁, forM <M_cr (2b) in which β1 = 0.5 for plain bars and 1 for high bond reinforcement; β2 = 1 for the first loading and 0.5 for the loads applied in a sustained manner or in a large number of load cycles [19].

In Eqs.(1)and(2),I1andI2are the moments of inertia of the gross section and the cracked transformed section, respectively, Mis the bending moment,Mcris the moment corresponding to the flexural cracking considered. The cracking moment, Mcris calculated by the program using the following equation:

M_cr= (fr +σv)I1

y_t (3)

where σv is the axial compressive stress, f_r is the flexural tensile strength of concrete, andy_tis the distance from centroid of 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 the effects of concrete cracking with stiffness reduction, has been considered. In the probability-based effective stiffness model, the effective moment of inertia is obtained as the ratio of the area of the moment diagram segment over which the working moment exceeds the cracking momentM_crto the total area of the moment diagram in the following form (Fig. 1).

A_uncr=A₁+A₂= Z

M(x)<Mcr

M(x) (4a)

A_cr=A₃= Z

M(x)≥Mcr

M(x) (4b)

A=A_cr+A_uncr (4c)

Puncr[M(x) <Mcr]= A_uncr

A (4d)

Pcr[M(x)≥Mcr]= A_cr

A (4e)

I_eff= P_uncrI₁+P_crI₂ (4f) whereAcris the area of moment diagram segment over which the working moment exceeds the cracking moment McrandA is the total area of the moment diagram. In the same equation, Pcr and Puncr are the probability of occurrence of cracked and uncracked sections, respectively. The applicability of the proposed model was also verified with frame test results [3].

In the literature [20–22], the effective moment of inertia given by ACI and CEB is the best among the commonly accepted simplified methods for the estimation of instantaneous deflection. Although the ACI and CEB models are usually considered for beams, in the present study these models are also used for columns considering the axial force in the determination of the cracking moment. In the analysis the axial

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

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

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

3. Models used for the effective shear stiffness of concrete Shear deformation can be large and significant after the development of cracks in a frame structure and thus can be of practical importance in the design of structures. The effective shear modulus of concrete due to cracking is given by several models in the literature.

Cedolin and dei Poli [15] observed that a value ofGclinearly decreasing with the fictitious strain normal to the crack would give better predictions for beams failing in shear, and suggested the following equation

G_c=0.24G_c(1−250ε1), forε1≥εcr (5a)

G_c=G_c, forε1< εcr (5b)

whereGc is the elastic shear modulus of uncracked concrete, ε1 is the principal tensile strain normal to the crack andεcris the cracking tensile strain.

Al-Mahaidi [16] recommended the following hyperbolic expression for the reduced shear stiffnessGc to be employed in the constitutive relation of cracked concrete

Gc=0.4G_c

ε1/εcr. (6)

Yuzugullu and Schnobrich [17] used a constant value ofG_c for the effective shear modulus

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

G_c = 0.125G_c,

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 modulus of cracked concrete.

In this study, since the rigid diaphragm model is considered, cracking occurs only in the beam element due to flexural moments in the localydirection. On the other hand, the column element cracks due to the flexural moments in the localyandz directions. HenceI_eff,M_cr,M,I₁,I₂,ε1andε2, are the values related to the flexure in the localyandzdirections.

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 a member will first be obtained, and then using compatibility conditions and equilibrium equations, the stiffness matrix and the load vector of a member with some regions in the cracked state will be evaluated.

A typical member subjected to point and uniformly distributed loads, and positive end forces with corresponding displacements, are shown inFig. 2. A cantilever model is used for computing the relations between nodal actions and basic deformation parameters of a general space element (Fig. 3). The basic deformation parameters of a general space element may be established by applying unit loads in turn in the directions of 1–3 and 7–9. Then, the compatibility conditions give the following equation in matrix form:















f₁₁ 0 0 0 0 0

0 f₂₂ f₂₃ 0 0 0

0 f₃₂ f₃₃ 0 0 0

0 0 0 f₇₇ f₇₈ 0

0 0 0 f₈₇ f₈₈ 0

0 0 0 0 0 f₉₉



























 P₁ P₂ P₃ P₇ P₈ P₉















=













 d₁ d₂ d₃ d₇ d₈ d₉















(8)

in which, f_{i j} is the displacement in i-th direction due to the application of unit loads in j-th direction, and can be evaluated by means of the principal of virtual work as follows:

f_{i j} = Z L

0

M_ziM_{z j}

E_cI_effz +M_yiM_{y j}

E_cI_effy +V_yiV_{y j} G_cA s +V_ziV_{z j}

GcA s+M_biM_bj

G_cI₀ + N_iN_j E_cA

dx. (9)

In Eq.(9), M_zi, M_{z j}, M_yi,M_{y j},V_zi,V_{z j},V_yi,V_{y j}, M_bi,M_bj, N_i and N_j are the bending moments, shear forces, torsional moments and axial forces due to the application of unit loads ini-th and j-th directions, respectively, E_c and I₀denote the modulus of elasticity of concrete and torsional moment of inertia of the cross section,s and A are the shape factor and the cross sectional area, respectively.

The three dimensional member stiffness matrix is obtained by inverting the flexibility matrix in Eq. (8) and using the equilibrium conditions.

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

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

compatibility and equilibrium conditions as follows P10=P20= P30=P40= P50=P60= P90=P110=0

(10a) P₇₀= −(f₈₈f₇₀− f₇₈f₈₀)/(f₇₇f₈₈− f₇₈f₈₇) (10b) P80= −(f77f80− f78f70)/(f77f88− f78f87) (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 ini-th direction due to the application of span loads which can be obtained by using the principal of virtual work in the following form

fi0= Z L

0

M_yiM₀

E_cI_effy + V_ziV₀ GcAs

dx (11)

whereM₀andV₀are the bending moment in localydirection and 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 the displacement vector, P₀(12×1) is the fixed end force vector andP(12×1) is the total end force vector of the member. Since Eq. (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 first obtained and then by considering the contributions which come from each element, the system stiffness matrix and system load vector are assembled. Finally, the system displacements and member 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 by means of Eqs.(9) and(11), with the following expressions of moment and shear forces obtained from the application of unit and 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)=







−q x²

2 0≤x≤a

−q x²

2 −P(x−a), a<x ≤L







(13f)

V₀(x)=

q x, 0≤x≤a q x+P, a<x≤ L

. (13g)

If the ACI and CEB models are used for the effective moment of inertia of the cracked members, the flexibility influence coefficient can be evaluated using Eqs. (9), (11) and(13) as follows

f₂₂= 1 Ec

Z L 0

x² Ieffz

dx+ s A

Z L 0

1 Gc

dx (14a)

f23= 1 E_c

Z L 0

(−x)

I_effz dx (14b)

f33= 1 E_c

Z L 0

1

I_effz dx (14c)

f₇₇= 1 Ec

Z L 0

x² Ieffy

dx+ s A

Z L 0

1

G_c dx (14d)

f78= 1 Ec

Z L 0

x Ieffy

dx (14e)

f88= 1 E_c

Z L 0

1

I_effy dx (14f)

f₇₀ = q 2E_c

Z L 0

x³ Ieffy

dx+qs A

Z a 0

x G_cdx + P

E_c Z L

a

x(x−a) I_effy dx+ s

A Z L

a

P Gc

dx (14g)

f₈₀= q 2E_c

Z L 0

x² Ieffy

dx+ P E_c

Z L a

(x−a) Ieffy

dx. (14h)

On the other hand, if the probability-based effective stiffness model is considered for the effective flexural stiffness of the cracked members, the flexibility influence coefficients can be obtained as

f22= L³ 3E_cI_effz + s

A Z L

0

1 Gc

dx (15a)

f₂₃= − L² 2EcIeffz

(15b) f₃₃= L

E_cI_effz (15c)

f77= L³ 3E_cI_effy + s

A Z L

0

1

G_cdx (15d)

f78= L²

2E_cI_effy (15e)

f₈₈= L EcIeffy

(15f) f₇₀ = q L⁴

8EcIeffy

+qs A

Z a 0

x

G_cdx+ P 3EcIeffy

×(L³+a³/2−3a L²/2)+ s A

Z L a

P Gc

dx (15g)

f80= q L³

6E_cI_effy + P

E_cI_effy(L²/2+a²/2−a L). (15h) It should be noted that, since the member has cracked and uncracked 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 has three cracked regions and two uncracked regions, as seen in Fig. 4.

In the cracked regions where M > Mcr, Ieff and Gc

vary with M along the region. Hence, the integral values in these regions should be computed by a numerical integration technique. The variation of effective moment of inertia and effective shear modulus of concrete in the cracked regions necessitate the redistribution of the moments in the structure.

Therefore, an iterative procedure should be applied to obtain the final deflections and internal forces of the structure.

5. Computer program

A general purpose computer program developed in the present study on the basis of an iterative procedure is coded in the FORTRAN 77 language. The flow chart of the solution procedure of the program is given in Fig. 5. In the solution procedure, using a procedure that evaluates the flexibility coefficients by means of the end forces provided by the previous step is not convergent in many cases. If the beams and columns are cracked in both positive and negative moment regions, it is indicated that both these regions will be more flexible in the next step; as a consequence, the moment will decrease either at midspan or at the end of the member, with a consequent decrease of the flexibility. The alternate increase and decrease of the flexibility generates a generally non-convergent procedure. Therefore a procedure considering, in each step, the value of the end forces evaluated as the mean value of the forces of all previous steps have been used in the present study [20]. This procedure decreases the fluctuations in the flexibility and accelerates the convergence of the algorithm.

In the program, the following equation is used as the convergence criterion

P_iⁿ−P_iⁿ⁻¹ P_iⁿ

≤ε. (16)

In Eq.(16)n is the iteration number,εis the convergence factor andP_iⁿ(i =1,12) is the end force of each member of the structure for then-th iteration.

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

Fig. 5. Solution procedure of the program.

be uncracked and performs the linear elastic analysis of the structure. The cracked members are then identified and their flexural and shear stiffness are reduced by the effective stiffness models available in the literature. Finally, an iterative procedure is applied until the convergence criterion is less than the predefined tolerance.

Due to space limitation, the listing of the computer program is not given in the paper. A PC version and the manual of the program can be obtained free of charge from the authors upon request.

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

In this section, three examples are presented. The first two examples are taken from the literature to verify the applicability of the analytical procedure. The last example is the application of 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 and Emara [6] for a two story reinforced concrete frame have been compared 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 mm deep and similarly reinforced with four No 20 deformed bars as both tensile and compressive steel. The testing setup involved first applying a total axial load of 700 kN to each column which was maintained throughout the test. The lateral load (Q)was then monotonically applied until the ultimate capacity of the frame was achieved. The reinforced concrete frame is modeled by four columns and two beam elements as shown inFig. 6. The cross-sections of the members, the axial loads and the span are also shown in the figure. In the analysis, ACI and probability-based effective stiffness models are used for the effective moment of inertia and Al-Mahaidi’s model is used for the shear modulus of concrete in the cracked regions. The flexural stiffness reduction of beams and columns with increasing lateral loads are also obtained by using a probability-based effective stiffness model. In computing the flexural tensile strength and modulus of elasticity of concrete, the following equations (ACI Code Eq. [23]) are also used.

E_c=4730p

f_c (N/mm²) (17a)

fr =0.62p

fc (N/mm²) (17b)

in which, f_cis the compressive strength of concrete.

In order to determine the applicability of proposed procedure, the comparison between the test and theoretical results for the lateral deflection of joint 5 obtained by the linear analysis, cracking analysis and the layered model developed by Vecchio and Emara [6], is presented in Figs. 7 and 8.

The numerical results obtained from the present study are in good agreement with the test results for applied loads within the 270 kN load level with maximum discrepancies of 11%.

The results of Vecchio and Emara’s layer model agree well with the test results especially after 82% of the ultimate load.

However, the theoretical model developed in the present study shows an improved prediction over Vecchio and Emara’s layer model [6] for loads up to 270 kN. The proposed analytical method also predicts the lateral deflection in the serviceability loading range with good accuracy, and beyond this range the differences between the numerical and test results increase with the increase in the lateral loads.

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

The theoretical results of the cracking sequence and the flexural stiffness reductions of beams and columns with respect to the lateral applied load is also shown inFig. 10. It can be seen from the figure that the beams at the first and second stories, B1 and B2, crack first and then both columns at the first story, C1 and C2, crack, respectively. Finally the two columns at the second story, C3 and C4, start to crack, respectively. From Fig. 10it is shown that at 82% of the ultimate lateral load, the two columns at the first story have 55 and 58%, respectively, of their gross moment of inertia, and two beams at the first and second stories have 45 and 47% of their uncracked values.

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 lateral deflection of joint 5.

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

The columns of the second story have also 60 and 65% of the gross moment of inertia. The results show that considering the cracking effect on the stiffness of structural frames by assigning an 80% reduced moment of inertia to all the columns and a 50%

reduced moment of inertia to all the beams does not always guarantee a conservative prediction of the lateral drift.

Figs. 11 and 12 present the process of convergence by variations of deflection and flexural stiffness of each member with number of iterations at the 270 kN load level. As seen from the figures, both the deflection and the flexural stiffness converge toward a solution after three iterations.

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

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

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

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

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

Fig. 13also shows the influence of shear deformation on the total lateral deflection (joint 5) of the reinforced concrete frame.

It can be seen that the contribution of the shear deformation to the total lateral deformation of the frame increases with increasing lateral loads, such as 10% for 82% of the ultimate load and 12% for the ultimate lateral load.

6.2. Example 2

Further verification of the proposed analytical method has also been conducted by comparison with results of another experimental program [3]. As seen in Fig. 14, a two-story reinforced concrete frame was designed with a center-to-center

span of 3000 mm, a first story height of 1170 mm and a second story height of 2000 mm. This frame is modeled by four columns of 250 ×375 mm and two beam elements of 250 × 350 mm cross sections. The reinforcing steel in the beams and columns, the span and the loads are also shown in the figure. The test procedure involved first applying a total axial load of 200 kN to each column and maintaining this load throughout the test. The lateral load (Q) was then monotonically applied until the ultimate capacity of the frame was achieved. In the analysis,Ieffis predicted using the ACI and probability-based effective stiffness models andGcis evaluated using Al-Mahaidi’s model.

The comparison between the experimental and theoretical results of the lateral deflection of joint 6 is presented inFig. 15.

It is seen that the deflections calculated by the developed computer program using the ACI model agree well with the test results for applied lateral loads up to approximately 78% of the ultimate load (i.e., at the value of approximately 155 kN). When the lateral load is beyond this load level, the difference between the experimental and theoretical results becomes significant. On the other hand, the results of the proposed analytical procedure which considers the variation of shear rigidity due to cracking gives a better prediction of deflections than the other one presented in [3].

Using the probability-based effective stiffness and ACI models for the effective flexural rigidity the theoretical lateral deflections of joint 6 are also obtained by the computer program as shown in Fig. 16. It should be noted that different models provide quite similar results.

The variation of the flexural stiffness of beams and columns with respect to the lateral applied load is also shown inFig. 17.

As seen from the figure, the beams of the first and second stories crack first and then two columns on the lateral loading side, C4 and C2, start to crack followed by, in the final stage, the cracking of both columns on the opposite loading side, C3 and C1.Fig. 17also shows that when the lateral load reaches 78% of the ultimate lateral load, the beams at the first and second stories have 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]).

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

Fig. 16. Comparison of lateral deflections obtained, by various models for the effective 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 iteration numbers at 155 kN loading stage.

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

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

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

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

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

Fig. 20 shows the contribution of the shear deformation effect in the overall lateral displacement (joint 6). As seen from the figure, the percentage of shear deformation in the total deflection 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 frame shown in Fig. 21 is analyzed by the developed computer program. This three-dimensional reinforced concrete frame is subjected to lateral loads at each story master point and uniformly distributed loads on the beams. The dimensions of the members and the loads are given in Table 1. In the analysis, the effective moment of inertia is evaluated by the ACI and probability-based effective stiffness models and the effective shear modulus is predicted by Al-Mahaidi’s model.

In this example, firstly, the lateral loads acting at each story master point, which are expressed in terms of the values of P, are increased while the intensity of uniform loads (q = 30 kN/m) remain constant. The variation of the maximum relative lateral displacement of the second floor with the lateral load, when cracking is considered and not considered for beams and columns, are shown inFig. 22. As seen from the figure, the differences in the maximum relative lateral displacement of the second floor between the two cases increase with increasing lateral loads. The difference becomes significant at higher loads such as 70% forP =200 kN.Fig. 23also shows the variation

Table 1

Dimensions 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 mm

Dimensions of column 500∗500 mm 500∗500 mm 500∗500 mm 500∗500 mm

Uniformly distributed loads (kN/m) q q q 0.8∗q

Lateral loads acting at master points inXdirection 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 the second floor with increasing lateral loads.

of the lateral displacement of joint 1, which displayed the fastest departure from linearity with respect to the lateral applied load.

Similar results are also obtained for the variations in the relative lateral displacement of the second floor.

The flexural stiffness reductions of various members with respect to the lateral applied load are shown inFig. 24. As seen from the figure, when the stiffness of beams are reduced to 50%

of their uncracked stiffness, the two first-story columns have reduced to 57–61% of their uncracked stiffness.

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

In this example, the intensity of uniform load is also varied from 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 the lateral applied load.

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

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

remained constant. Fig. 26 shows the lateral displacement of joint 1 predicted by linear analysis and crack analysis.

Increasing only the uniformly-distributed loads and keeping the lateral loads constant caused an increase in the lateral displacement of the reinforced concrete frame. The beams start cracking with increasing uniform loads. Thus, the cracking effect in these members causes the reduction of the overall lateral stiffness which in turn results in an increase in the lateral deflection of the reinforced concrete structures.

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

7. Conclusions

An iterative procedure has been developed to analyze three dimensional reinforced concrete frames with cracked beam and column elements. The proposed analytical procedure provides nonlinear force–deformation relationships for the cracked members.

The variation of the flexural rigidity of a cracked member has been evaluated by using the ACI, CEB and probability- based effective stiffness models. Shear deformations, which can be large after the development of cracks and be of practical importance in the design and behavior of the structure, are also taken into account in the analysis. The variation of shear rigidity in the cracked regions of the member has also been considered by employing various reduced shear stiffness models available in the literature.

The capability and the reliability of the proposed procedure have been tested by means of comparisons with the theoretical and experimental results available in the literature. The numerical results of the analytical procedure have been found to be in good agreement with the test results for applied loads up to approximately 78% of the ultimate load capacity of the frame.

The analytical procedure not only predicts the deflections to a value of load equal to approximately 78% of the ultimate load with good accuracy, but it also can give an estimation of behavior at approximately 85% of the ultimate load with an acceptable degree of accuracy. On the other hand, beyond this load level there is a large difference between the theoretical and experimental results. Material nonlinearity may also become significant in the behavior of reinforced concrete frames if the load is beyond the serviceability loading range.

In the analysis, flexural stiffness reductions of beams and columns with respect to the lateral applied load can be obtained by the developed program using the probability-based effective stiffness model for the effective moment of inertia. This is the most significant feature of the proposed procedure and the variations in the flexural stiffness reductions of beams and columns in the reinforced concrete structure can be observed explicitly. This feature can minimize the uncertainty of the flexural stiffness of members in reinforced concrete frames and also provide design engineers with significant information on the consequence of cracking in members.

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

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

The numerical results of the flexural stiffness reductions of beams and columns involved in the testing frame indicate that considering the cracking effect on the stiffness of a structural frame by assigning an 80% reduced moment of inertia to all the columns and a 50% reduced moment of inertia to all the beams does not always guarantee a conservative prediction of the lateral drift. However, the flexural stiffness reductions of beams and columns are also dependent on the values of uniformly- distributed and span loads on the members.

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

The numerical results of the analytical procedures indicate that the proposed procedure, which considers the variation of shear rigidity after the development of cracks, gives better predictions of deflections than other analytical procedures. It is therefore important to consider the variation of shear rigidity in the cracked regions of members in order to obtain more accurate results.

In the present study, cracking effects in the reinforced concrete shear walls have not been considered. Stiffness reductions due to cracking in the shear walls have to be considered comprehensively. The effect of various height-to- width ratios, vertical load levels, and reinforcement ratios should be considered in the analysis. This can be achieved by using nonlinear finite element methods or by integrating the probability-based effective stiffness model with finite element analysis [24].

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