An efficient tension-stiffening model for nonlinear

Engineering Structures 30 (2008) 2069–2080 www.elsevier.com/locate/engstruct

An efficient tension-stiffening model for nonlinear analysis of reinforced concrete members Renata S.B. Stramandinoli a,∗ , Henriette L. La Rovere b a Civil Engineering Department, COPEL, Parana, Brazil b Civil Engineering Department, Federal University of Santa Catarina, Brazil

Received 7 November 2006; received in revised form 16 October 2007; accepted 27 December 2007 Available online 25 February 2008

Abstract A constitutive model for reinforced concrete elements that takes into account the tensile capacity of the intact concrete between cracks, effect known as tension-stiffening, is proposed in this paper. In the model, the tensile stress–strain curve of concrete displays an exponential decay in the post-cracking range, defined by a parameter that depends on the reinforcement ratio and on the steel-to-concrete modular ratio. This parameter was derived taking as a basis the CEB tension-stiffening model. The model was implemented into a computational program that allows for nonlinear finite element analysis of reinforced concrete beams. The numerical results obtained by the program compared extremely well with several experimental results from simply supported beams tested under 4-point bending that displayed a dominant flexural behavior. Extension of the model to members subjected to combined flexural and shear is also presented. c 2008 Elsevier Ltd. All rights reserved.

Keywords: Tension-stiffening; Finite elements; Reinforced concrete beams; Nonlinear analysis

1. Introduction It is well known that the intact concrete between cracks can still carry tensile stresses after the onset of cracking in reinforced concrete (R/C) elements due to the bond between the reinforcing bars and the surrounding concrete. This effect, known as tension-stiffening, was neglected in the past since it does not significantly affect the ultimate strength of the reinforced concrete members. Since the 70s, however, the tensile behavior of concrete was introduced in the analysis of load-deflection characteristics of R/C elements, and since the 80s in design code recommendations for service load level. It is also important to consider tension-stiffening when evaluating the serviceability of existing R/C structures. The tension-stiffening effect depends on several factors, such as member dimensions, reinforcement ratio, rebars diameters, and the materials elastic modulus and strength. This effect occurs until yielding of the longitudinal reinforcement takes place, and it tends to increase as the reinforcement ratio of the member decreases. ∗ Corresponding address: Rua Pasteur, 413 ap.1502 – Batel, CEP: 80250080, Curitiba, Paraná, Brazil. Tel.: +55 41 91742486; fax: +55 41 33523090. E-mail addresses: [email protected] (R.S.B. Stramandinoli), [email protected] (H.L. La Rovere).

c 2008 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter doi:10.1016/j.engstruct.2007.12.022

Several models to represent the tension-stiffening effect have already been proposed, ranging from simple to very refined models of great degree of complexity. One of the simplest models, but yet very well accepted by designers for beam deformation calculations, is the model of Branson [1] which considers an equivalent inertia for the cracked beam section. The model developed by Cosenza [2] can also be quoted as a simplified tension-stiffening model. Quite a few models that modify the constitutive equation of steel or concrete after cracking have also been proposed for nonlinear finite element analysis of reinforced concrete structures. Among the models that modify the steel constitutive equation it can be quoted: Gilbert and Warner [3], Choi and Cheung [4] and the CEB manual design [5] model; and among those that modify the concrete constitutive equation: Scanlon and Murray [6], Lin and Scordelis [7], Collins and Vecchio [8], Stevens et al. [9], Balakrishnan and Murray [10], Massicotte et al. [11]. There are more complex models based on the bond-slip mechanism and on localized phenomena such as those of: Floegl and Mang [12], Gupta and Maestrini [13], Wu et al. [14], Russo and Romano [15], Choi and Cheung [16] and Kwak and Song [17]. These complex models, also known as “microscopic models”, depend on a series of parameters that

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R.S.B. Stramandinoli, H.L. La Rovere / Engineering Structures 30 (2008) 2069–2080

Notation Ac Ae f As d E ccr E ci Es f cc f ct fy h ` ∆` N Ncr Ny n sh x α ε εcr εs1 εs2 εs1r εs2r εsm εy ∆εs ∆εsmax ρ ρef ρeq θ σct σs2 σsr

concrete area effective concrete area reinforcement area usual effective depth secant elastic modulus of concrete in the postcracking range elastic modulus of concrete before cracking elastic modulus of the reinforcing steel concrete compressive strength concrete tensile strength reinforcing steel yield stress nominal depth of the beam member length total extension of a reinforced concrete member of length ` axial force axial force at the onset of cracking axial force at yielding of reinforcement steel-to-concrete modular ratio plastic-to-elastic modular ratio neutral axis depth exponential decay parameter of the tension-stiffening model strain in the R/C member strain corresponding to concrete tensile strength strain in the reinforcement in State I strain in the reinforcement in State II strain in the reinforcement in State I corresponding to stress σsr strain in the reinforcement in State II corresponding to stress σsr average strain in the reinforcing steel reinforcing steel yield strain contribution of the concrete in tension between cracks maximum variation between the strains εs1 and εs2 member reinforcement ratio effective reinforcement ratio equivalent reinforcement ratio angle between the x-direction and the principal direction 1. concrete tensile stress stress in the reinforcement at a cracked section under the applied load stress in the reinforcement calculated on the basis of a cracked section, where the maximum stress in the concrete under tension is equal to f ct

are usually difficult to obtain, requiring specific experiments for each particular member, hence they are not usually applied to full-scale problems. Generally, the models that modify the constitutive equation of concrete, in which the descending branch of the tensile

stress–strain curve of concrete is modified to take into account the tension-stiffening effect in an average way, are more widely used. These so-called “macroscopic” models are easier to implement and, by being simpler than the “microscopic” ones, they can be readily applied to analyze full-scale structures. However, most macroscopic models oversimplify the tensionstiffening effect by considering only one equation to describe the post-cracking range of the tensile stress–strain curve, independently of the member reinforcement ratio and material properties. Seeking a simple model that can represent more realistically the tension-stiffening effect and at the same time be easily implemented into finite element programs, we introduce in this work a novel tension-stiffening model. The proposed model uses an exponential decay curve to describe the post-cracking range of the tensile stress–strain law of concrete. The exponential decay parameter (α) is a function of the member reinforcement ratio (ρ) and of the steel-to-concrete modular ratio (n = E s /E c ), and is derived taking as basis the CEB [5] tension-stiffening model. Gupta and Maestrini [13] have also utilized an exponential curve to formulate a simplified tension-stiffening model and have obtained good correlation with experimental results. A brief review of the CEB [5] model is initially presented. Next the model is proposed for members under direct tension and verified by comparison with other tension-stiffening models and also with experimental results obtained from pull-out tests on R/C bars. The proposed tension-stiffening model is then extended to members under bending, assuming that all cracks are orthogonal to the reinforcement. This model is implemented into a finite element program called ANALEST that allows for nonlinear analysis of reinforced concrete beams and plane frames. Application of the model to simply supported R/C beams tested under 4-point bending in different research laboratories are presented in the sequence. In all the examples, shear deformation, stirrups contribution, and geometric nonlinearities are neglected. A simplified approach to further extend the tension-stiffening model to members under combined bending and shear is also described. Concluding remarks are given at the end of the work. 2. The CEB manual design model [5] The CEB model developed for reinforced concrete members subjected to tension considers the tension-stiffening effect through an increase in stiffness of the reinforcement. The cracking mechanism of a reinforced concrete member subjected to uniaxial tension can be observed in Fig. 1. An equation for determining the stress–strain curve for the reinforcement is proposed in terms of an average strain, which lies between the strain of an uncracked section (State I) and that of a totally cracked section (State II): εsm =

∆` = εs2 − ∆εs `

(1)

where ∆` is the total extension of a reinforced concrete member of length ` subjected to an axial tensile force N which is


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An expression for the average strain can be obtained by substituting Eq. (2) into Eq. (1): σsr σsr εsm = εs2 − ∆εs max = εs2 − (εs2r − εs1r ) ∴ σs2 σs2 σsr εs2r σsr + εs1r ∴ εsm = εs2 1 − εs2 σs2 σs2 " 2 2 # σsr σsr εsm = εs1 + 1 − εs2 (3) σs2 σs2

Fig. 1. Cracking mechanism of a reinforced concrete member subjected to uniaxial tension: (a) reinforcement stress; (b) bond stress; (c) concrete stress (CEB [5]).

Fig. 2. Stress–strain curve for the reinforcement (CEB [5]).

greater than the force Ncr which produces the first crack; ∆εs represents the contribution of concrete between cracks which follows a hyperbolic relationship approaching the line εs2 asymptotically for stresses in excess of σsr . In the CEB Manual Design [5], the following expression for ∆εs , based on experimental results, is proposed: σsr ∆εs = ∆εs max σs2

(2)

where: σsr is the stress in the reinforcement calculated on the basis of a cracked section, where the maximum stress in the concrete under tension is equal to f ct ; σs2 is the stress in the reinforcement at a cracked section under the applied load; and ∆εsmax is the maximum variation between the strains εs1 and εs2 which occurs at the beginning of the cracking process. The definition of all these parameters can be better observed in Fig. 2.

where: εs1r is the strain in the reinforcement in State I (uncracked section) corresponding to stress σsr ; εs2r is the strain in the reinforcement in State II (totally cracked section without any concrete contribution) corresponding to stress σsr . Eqs. (1)–(3) were derived for pure tension, however, as indicated in the CEB Manual [5], they are also valid for flexure. Eq. (3) was derived assuming monotonic loading and high-bond bars, but it was further modified to take into account cyclic loading and the use of smooth bars. This model presents a consistent theory to represent the average post-cracking behavior of a reinforced concrete member under tension. Since the proposed constitutive equation is based on experimental results, it is also an accurate model. However, as it can be observed from Eqs. (1)–(3), it is difficult to be implemented into a finite element code, since σs2 cannot be explicitly obtained from the average strain εsm . A simplification of this model was proposed later on CEBFIP Model Code-90 [18] where a trilinear curve was utilized to represent the “stress–average strain” relationship for the reinforcement. This curve is an approximation of the original curve shown in Fig. 2, with a bilinear branch adopted after cracking instead of a continuous curve. D’Avila and Campos Filho [19] proposed a trilinear curve for the tensile constitutive equation of concrete in the post-cracking range, based on this simplified model from CEB-FIP MC-90. A continuous curve, however, is more desirable for computational implementation into nonlinear finite element codes. 3. Proposed model A novel tension-stiffening model that modifies the tensile constitutive equation of concrete is proposed in the following. The model uses an explicit formulation for the concrete stress–strain curve and thus can be easily implemented into a finite element code. Some features of the CEB manual design [5] model instead of the CEB-FIP MC-90 [18] model are utilized, therefore a continuous stress–strain curve is obtained for the concrete in the post-cracking range providing numerical stability in nonlinear finite element analysis of R/C members. Concrete is assumed to behave like a linear-elastic material until its tensile strength is reached, so that a straight line defines initially the stress–strain curve. In the post-cracking range, an exponential decay curve is adopted until yielding of reinforcement takes place, and is defined by the following

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equation: σct = f ct e

where: −α

ε εcr

(4)

where, f ct is the concrete tensile strength and εcr is the corresponding strain; α is an exponential decay parameter to be determined. In the absence of experimental results, the expression given by CEB-FIP MC-90 [18] can be used to estimate f ct : fct (MPa) = 1.4

fcc (MPa) − 8 10

2/3

.

(5)

An alternative way for determining an expression for α would be by adjusting the experimental results from reinforced concrete bars subjected to direct tension, by varying the specimen longitudinal reinforcement ratio (ρ) and material properties. Many authors have conducted parametric studies to investigate the influence of ρ and of material properties (fracture energy or specimen diameter-to-length ratio, tensileto-bond strength ratio, etc. . . ), on tension-stiffening. Amongst all properties, the reinforcement ratio is the one that has shown the greatest influence on the tension-stiffening effect (Hegemier et al. [20]). Hence, instead of adjusting the experimental data, an expression for the parameter α, defined as a function of the member reinforcement ratio (ρ) and of the steel-to-concrete modular ratio (n = E s /E c ), is derived in this paper, taking as basis the CEB [5] model. The same concept of average deformation (εsm ) and its definition given by Eq. (3) is adopted. The concrete stress–strain curve is determined through the analysis of a reinforced concrete member subjected to direct tension. The contribution of concrete between cracks can be observed from the graphs displayed in Fig. 3. In this figure, point “a” represents the onset of cracking; “b” defines the point where the strain in the reinforcement for State II reaches the strain at yield (εs2 = ε y ); and “c” the point where the strain in the member reaches the strain at yield (ε = ε y ). Additionally, it is assumed that the strain in the reinforcement is equal to the strain in the surrounding concrete. Thus, in the linear-elastic range and before the onset of cracking, the strain in the R/C member can be determined by: ε=

N E s As + E ci Ac

(6)

where: E s is the elastic modulus of the reinforcing steel; E ci is the elastic modulus of concrete before cracking; As is the reinforcement area; and Ac is the concrete area. After cracking, the strain in the concrete between points “a” and “b” in the curve shown in Fig. 3(b) is calculated by Eq. (3): " 2 # 2 σsr σsr εs1 + 1 − εs2 (7) εc = εs = ε= σs2 σs2

σs2 =

N As

and

σsr =

(1 + nρ) f ct ρ

in which ρ is the reinforcement ratio equal to As /(Ac ). The strain in the member after cracking is: ε=

N E s As + E ccr Ac

(8)

where, E ccr is an equivalent elastic modulus of concrete in the postcracking range, defined by the secant modulus: E ccr =

σct ε

(9)

which varies according to the cracking level in the member. Substituting Eqs. (7) and (9) into Eq. (8), the stress in the concrete can be obtained by: σct =

N − εE s As = σs2 ρ − εE s ρ. Ac

(10)

For tracing the concrete stress–strain curve between points “a” and “b”, the values of ε are initially calculated using Eq. (6), by varying the applied force N from Ncr (the axial force at the onset of cracking) up to N y (the axial force at yielding of reinforcement). From the obtained values for ε, the stress in concrete is then calculated by means of Eq. (10). In the descending branch where εs2 > ε y (from point “b” to point “c”), a straight line is found until ε reaches the strain at yielding of reinforcement, where the stress in concrete drops to zero. Seeking an expression to determine the exponential decay parameter α, several concrete stress–strain curves were initially traced using the procedure described above, by varying the material properties f ct and f y , and by selecting different values of (nρ) for fixed values of f ct and f y . An exponential curve, as defined by Eq. (4), was then adjusted to each traced stress–strain curve. It was then observed that (nρ) was the most important property to define the exponential decay parameter. Hence, several exponential curves were fitted to the traced concrete stress–strain curves by varying only (nρ), and a value of α was found for each corresponding value of (nρ). The curve fitting was made using the Mathcad 2001 program for normalized concrete stress (σct / f ct ) versus strain curves. Fig. 4 illustrates an example of curve fitting obtained for (nρ) = 0.2, yielding an exponential decay parameter α = 0.069. An expression for the exponential decay parameter was then derived by using all the obtained values of α for different values of (nρ), as shown in Fig. 5. The best fit curve achieved was a third degree polynomial (COR = 0.996), described by the following equation: α = 0.017 + 0.255 (nρ) − 0.106 (nρ)2 + 0.016 (nρ)3 .

(11)

This tension-stiffening model was derived for R/C members subjected to direct tension. Extension of the model to R/C beams under bending, can be done by employing the effective area concept that corresponds to the tensile zone in the member


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Fig. 3. Tension-stiffening effect in a reinforced concrete member under tension: (a) Applied force × strain curve for the R/C member, (b) Concrete stress–strain curve.

0.1h, the above expression for the effective area simplifies to: b.h Aef ∼ . (13) = 4 Therefore, for R/C beams, an effective reinforcement ratio, expressed as ρef =

As Ae f

(14)

must be employed in Eq. (11).

Fig. 4. Curve fitting for tensile stress–strain curve of concrete obtained for nρ = 0.2.

Fig. 5. Curve fitting to obtain the exponential decay parameter (α) expression.

section. An equation for this effective area is suggested in the CEB-FIP MC-90: Ae f = 2.5b (h − d)
0.1 the tension-stiffening effect becomes very small. The beams analyzed are: – VRE tested by Ferrari [27], at the Federal University of Santa Catarina, Brazil. – VT1 and VT2 tested by Beber [28], at the Federal University of Rio Grande do Sul, Brazil; – VB6 and VC3 tested by Juvandes [29], at the University of Porto, Portugal; Figs. 10–13 illustrate the beam dimensions, reinforcing detailing, the tests set-up showing the load application and support positions, and the finite element meshes adopted for the beam models.

Two analyses were performed with ANALEST program for each example, one considering the proposed tension-stiffening model and the other without tension-stiffening consideration. In all analyses, geometric nonlinearities are neglected and the Newton–Raphson Method (tangent stiffness) was employed. Confinement provided by stirrups is disregarded, with the Hognestad parabola being adopted for the compressive constitutive law of concrete. The material properties used in the finite element models are condensed in Table 1. Comparison between FE analyses and experiments are shown in Figs. 14–17, for the beams VRE, VT1 and VT2, VB6 and VC3, respectively, in terms of total applied vertical load versus mid-span vertical displacement graphs. A close agreement between numerical and experimental results is observed for the beam VRE, in both the elastic and the post-cracking range of the beam. The yield load

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Fig. 12. Tested beam (VB6) geometry and reinforcement; load application and support positions (Juvandes [29]).

Fig. 13. Tested beam (VC3) geometry and reinforcement; load application and support positions (Juvandes [29]). Table 1 Material properties utilized in the finite element analyses of beam examples Concrete Beam

f cm (MPa)

f tm (MPa)

εo

Tension-stiffening n ρeff (%)

α

VRE

30.7

2.95

0.0020

6.19

1.33

0.037

VT1/VT2

33.5

2.62

0.0020

6.39

1.50

0.040

VB6

37.9

2.90

0.0023

5.37

4.47

0.072

VC3

20.7

1.60

0.0020

9.00

3.80

0.093

predicted analytically was 34 kN, while in the experiment the measured value was between 33 and 35 kN. When no strain hardening is considered, no convergence of the iterative procedure could be achieved in the analysis after yielding of the steel reinforcement, either using the Newton–Raphson or the Arc-length Method. By adopting a small strain hardening coefficient for the longitudinal reinforcement, sh = 0.01, three more load increments of 0.5 kN could be applied after yielding, reaching an ultimate displacement of 30 mm. For the analysis without considering the tension-stiffening effect (NO T.S.), a much more flexible response is observed, however yielding of reinforcement could be captured and an horizontal load-displacement threshold is displayed, as it can be seen in Fig. 14. In the experimental test, Ferrari reported yielding of

Steel φ (mm)

f y (MPa)

E s (GPa)

s.h.

6 8 6 10 3 8 12.5

767.5 545.8 738 565 192 497 507

210 210 214.8 214.8 174 195 184.6

0.016 0.01 0.016 0.000 0.001 0.0042 0.0014

reinforcement and large deflections of the beam near to failure, as it can be observed from Fig. 14; however, the instruments have been removed from the specimen prior to failure, thus hindering the measurement of ultimate displacement. For the beams VT1 and VT2, it can be observed from Fig. 15 that the FE model can capture very well the ascending branch of the curve and approaches well the experimental curves after the onset of cracking when tension-stiffening of concrete is considered. The numerical model shows a slightly stiffer response until a total applied load of 30 kN is reached, but beyond that load a close agreement to the experimental curves is observed. For the analysis without tension-stiffening (NO T.S.) consideration, the finite element model shows a much more flexible response as compared to the experimental

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Fig. 14. Comparison between numerical (ANALEST) and experimental results for the beam VRE.

Fig. 15. Comparison between numerical (ANALEST) and experimental results for the beams VT1/VT2.

Fig. 17. Comparison between numerical (ANALEST) and experimental results for the beam VC3.

numerical value obtained at failure by the program ANALEST was 46 kN. An excellent agreement can be observed from Fig. 16 by comparison between the numerical analysis and the experimental test on the beam VB6. The numerical model is only a little stiffer at the beam elastic range, before the onset of cracking, region more susceptible to instrumentation imprecision. For the analysis without considering tensionstiffening (NO T.S.), a more flexible response is again observed, showing the importance of considering the tension-stiffening effect in the beam post-cracking behavior. The finite element model predicted very well the ultimate load, but a smaller value for the corresponding displacement was obtained by the ANALEST program. For the beam VC3, the comparison between analysis and experiment shows an excellent agreement in the elastic range (see Fig. 17), but beyond that, for loads higher than 25 kN, the numerical model becomes slightly stiffer than the experimental model. In this example, the analytical curves obtained considering and not considering tension-stiffening (NO T.S.) are very close, since for this beam the elastic modulus of concrete is low, yielding a high value for the modular ratio n, which results in a high value for the α parameter, and therefore the tension-stiffening effect becomes very small. 8. Extension of the TS model to 2D constitutive models

Fig. 16. Comparison between numerical (ANALEST) and experimental results for the beam VB6.

curves. The onset of yielding of reinforcement was accurately captured by the finite element model, corresponding to a total applied load of 44 kN. However, the post-yielding response of the beams and the ultimate displacement could not be measured experimentally, since the instruments have been removed from the specimens to avoid damage. The ultimate total load measured experimentally was 47 kN, while the

In order to extend the TS model to allow for the use in 2D constitutive models, by taking into account the angle between the cracks and the reinforcing bars, a simplified approach is introduced. The tensile constitutive equation for concrete is defined in the principal direction 1, hence in Eq. (4) σc1 should be used instead of σct , and the α parameter is calculated from Eq. (11) by using an equivalent reinforcement ratio, ρeq , defined by: ρeq = ρx cos2 θ + ρ y sin2 θ

(15)

where: ρx is the reinforcement ratio in the x-direction; ρ y is the reinforcement ratio in the y-direction and θ is the angle between x-direction and principal direction 1.


Fig. 18. 2D cracking parameters.

These parameters are illustrated in Fig. 18. With this modification the TS model can also be applied to beams (or planar elements) subjected to combined bending and shear. This modified TS model was implemented more recently into the ANALEST program, by considering the Timoshenko beam theory in the FE formulation, which takes shear deformation into account. Each section layer is then considered under a biaxial stress state, and the 2D constitutive model developed by Collins and Vecchio [8] is adopted. This Timoshenko beam element is more suitable for members where significant shear cracks develop, and it has shown excellent correlation with experimental results from 3-point bending tests on beams with low transverse reinforcement ratio (Bresler and Scordelis [30]) and also from plane frame tests (Vecchio and Balopolou [31]; Ernst et al. [32]). 9. Conclusions In this work a novel tension-stiffening model for reinforced concrete members is developed. An exponential decay curve is utilized for the concrete tensile stress–strain curve in the postcracking range. The exponential decay parameter (α) is defined as a function of the reinforcement ratio (ρ) and of the steelto-concrete modular ratio (n), and is derived taking as a basis the CEB [5] tension-stiffening model. The model is initially validated by comparison with other tension-stiffening models and also with experimental pull-out tests on R/C bars. The proposed tension-stiffening model is implemented into a finite element program, named ANALEST, that allows for nonlinear analysis of reinforced concrete beams that display a dominant flexural behavior. The program is then applied to analyze several beams tested under 4-point bending in different research laboratories. Comparison between numerical and experimental results, in terms of load-displacement curves, showed very good agreement. The analyses showed that the tension-stiffening effect plays an important role in the post-cracking behavior, especially for the beams with α ≤ 0.072. The proposed tensionstiffening model proved to be very efficient allowing both, easy implementation and numerical stability and, additionally, representing the concrete tension-stiffening effect in reinforced concrete members with different reinforcing and modular ratios. The proposed tension-stiffening model can also be extended to planar members when the crack angle with respect to the longitudinal rebars is different from 90◦ by means of an equivalent reinforcing ratio. The ANALEST program has

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already been extended for planar members under combined flexural and shear (Stramandinoli [33]), by using a Timoshenko beam element in conjunction with the 2D smeared and rotatingcracked model, proposed by Collins and Vecchio [8], and the modified TS model proposed in this work. Very good results have then been obtained by comparison with experimental testing on beams and planar frames that showed significant shear distress. For those examples of beams under combined bending and shear where flexural cracks are dominant, the Bernoulli beam element still gives very good results in the postcracking range, at service loads. The ANALEST program is currently been extended to also allow for the analysis of 3D reinforced concrete frames. Acknowledgments The authors gratefully acknowledge the scholarship granted by Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnológico (CNPq) to the first author of this paper. Thanks are also due to researchers Juvandes, L.F.P.; Beber A.J. and Ferrari, V.J. for making available the experimental data used in this work. References [1] Branson DE. Design procedures for computing deflection. ACI Journal 1968;65(8):730–42. [2] Cosenza E. Finite element analysis of reinforced concrete elements in a cracked state. Computers and Structures 1990;36(1):71–9. [3] Gilbert RI, Warner RF. Tension stiffening in reinforced concrete slabs. Journal of the Structural Division ASCE 1978;104(2):1885–900. [4] Choi CK, Cheung SH. A simplified model for predicting the shear response of reinforced concrete membranes. Thin-Walled Structures 1994;19:37–60. [5] CEB. Cracking and deformation. Bulletin d’information N ◦ 158. Paris, France; 1985. [6] Scanlon A, Murray DW. Time-dependent reinforced concrete slab deflections. Journal of the Structural Division 1974;100(8):1911–24. [7] Lin CS, Scordelis AC. Nonlinear analysis of RC shells of general form. Journal of the Structural Division ASCE 1975;101(3):523–38. [8] Collins MP, Vecchio FJ. The modified compression-field theory for reinforced concrete elements subjected to shear. ACI Journal 1986;83(2): 219–31. [9] Stevens NJ, Uzumeri SM, Collins MP. Analytical modeling of reinforced concrete subjected to monotonic and reversed loadings. Report. Canada: University of Toronto; 1987. [10] Balakrishnan S, Murray DW. Concrete constitutive model for NLFE analysis of structures. ASCE Journal of Structural Engineering 1988; 114(7):1449–66. [11] Massicote B, Elwi AE, MacGregor JG. Tension-stiffening model for planar reinforced concrete members. ASCE Journal of Structural Engineering 1990;106(11):3039–58. [12] Floegl H, Mang HA. Tension stiffening concept based on bond slip. Journal of the Structural Division ASCE 1982;108(12):2681–701. [13] Gupta A, Maestrini SR. Tension-stiffness model for reinforced concrete bars. ASCE Journal of Structural Engineering 1990;116(3):769–91. [14] Wu Z, Yoshikawa H, Tanabe T. Tension stiffness model for cracked reinforced concrete. ASCE Journal of Structural Engineering 1991; 117(3):715–32. [15] Russo G, Romano F. Cracking response of RC members subjected to uniaxial tension. ASCE Journal of Structural Engineering 1992;118(5): 1172–90. [16] Choi CK, Cheung SH. Tension stiffening model for planar reinforced concrete members. Computers and Structures 1996;59(1):179–90.

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