efficient tension stiffening model for concrete fem

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Engineering Structures ( ) –www.elsevier.com/locate/engstruct

An efficient tension-stiffening model for nonlinear analysis of reinforcedconcrete members

Renata S.B. Stramandinolia,∗, Henriette L. La Rovereb

a Civil Engineering Department, COPEL, Parana, Brazilb Civil Engineering Department, Federal University of Santa Catarina, Brazil

Received 7 November 2006; received in revised form 16 October 2007; accepted 27 December 2007

Abstract

A constitutive model for reinforced concrete elements that takes into account the tensile capacity of the intact concrete between cracks, effectknown as tension-stiffening, is proposed in this paper. In the model, the tensile stress–strain curve of concrete displays an exponential decay in thepost-cracking range, defined by a parameter that depends on the reinforcement ratio and on the steel-to-concrete modular ratio. This parameterwas derived taking as a basis the CEB tension-stiffening model. The model was implemented into a computational program that allows fornonlinear finite element analysis of reinforced concrete beams. The numerical results obtained by the program compared extremely well withseveral experimental results from simply supported beams tested under 4-point bending that displayed a dominant flexural behavior. Extension ofthe 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 canstill carry tensile stresses after the onset of cracking in rein-forced concrete (R/C) elements due to the bond between the re-inforcing bars and the surrounding concrete. This effect, knownas tension-stiffening, was neglected in the past since it does notsignificantly affect the ultimate strength of the reinforced con-crete members. Since the 70s, however, the tensile behavior ofconcrete was introduced in the analysis of load-deflection char-acteristics of R/C elements, and since the 80s in design coderecommendations for service load level. It is also important toconsider tension-stiffening when evaluating the serviceabilityof existing R/C structures. The tension-stiffening effect dependson several factors, such as member dimensions, reinforcementratio, rebars diameters, and the materials elastic modulus andstrength. This effect occurs until yielding of the longitudinal

∗ Corresponding address: Rua Pasteur, 413 ap.1502 – Batel, CEP: 80250-080, Curitiba, Parana, Brazil. Tel.: +55 41 91742486; fax: +55 41 33523090.

E-mail addresses: [email protected](R.S.B. Stramandinoli), [email protected] (H.L. La Rovere).

reinforcement takes place, and it tends to increase as the rein-forcement ratio of the member decreases.

Several models to represent the tension-stiffening effect havealready been proposed, ranging from simple to very refinedmodels of great degree of complexity. One of the simplestmodels, but yet very well accepted by designers for beamdeformation calculations, is the model of Branson [1] whichconsiders an equivalent inertia for the cracked beam section.The model developed by Cosenza [2] can also be quoted as asimplified tension-stiffening model.

Quite a few models that modify the constitutive equationof steel or concrete after cracking have also been proposedfor nonlinear finite element analysis of reinforced concretestructures. Among the models that modify the steel constitutiveequation it can be quoted: Gilbert and Warner [3], Choi andCheung [4] and the CEB manual design [5] model; and amongthose that modify the concrete constitutive equation: Scanlonand Murray [6], Lin and Scordelis [7], Collins and Vecchio [8],Stevens et al. [9], Balakrishnan and Murray [10], Massicotteet al. [11]. There are more complex models based on thebond-slip mechanism and on localized phenomena such asthose of: Floegl and Mang [12], Gupta and Maestrini [13], Wu

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

Please cite this article in press as: Stramandinoli RSB, La Rovere HL. An efficient tension-stiffening model for nonlinear analysis of reinforced concrete members.Engineering Structures (2008), doi:10.1016/j.engstruct.2007.12.022

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Interesante explicación del modelo. Emplearemos el modelo lineal.

ARTICLE IN PRESS2 R.S.B. Stramandinoli, H.L. La Rovere / Engineering Structures ( ) –

Notation

Ac concrete areaAe f effective concrete areaAs reinforcement aread usual effective depthEcr

c secant elastic modulus of concrete in the post-cracking range

E ic elastic modulus of concrete before cracking

Es elastic modulus of the reinforcing steelfcc concrete compressive strengthfct concrete tensile strengthfy reinforcing steel yield stressh nominal depth of the beam` member length∆` total extension of a reinforced concrete member

of length `

N axial forceNcr axial force at the onset of crackingNy axial force at yielding of reinforcementn steel-to-concrete modular ratiosh plastic-to-elastic modular ratiox neutral axis depthα exponential decay parameter of the

tension-stiffening modelε strain in the R/C memberεcr strain corresponding to concrete tensile strengthεs1 strain in the reinforcement in State Iεs2 strain in the reinforcement in State IIεs1r strain in the reinforcement in State I correspond-

ing to stress σsrεs2r strain in the reinforcement in State II correspond-

ing to stress σsrεsm average strain in the reinforcing steelεy reinforcing steel yield strain∆εs contribution of the concrete in tension between

cracks∆εsmax maximum variation between the strains εs1 and

εs2ρ member reinforcement ratioρef effective reinforcement ratioρeq equivalent reinforcement ratioθ angle between the x-direction and the principal

direction 1.σct concrete tensile stressσs2 stress in the reinforcement at a cracked section

under the applied loadσsr stress in the reinforcement calculated on the basis

of a cracked section, where the maximum stressin the concrete under tension is equal to fct

et al. [14], Russo and Romano [15], Choi and Cheung [16] andKwak and Song [17]. These complex models, also known as“microscopic models”, depend on a series of parameters thatare usually difficult to obtain, requiring specific experiments for

each particular member, hence they are not usually applied tofull-scale problems.

Generally, the models that modify the constitutive equationof concrete, in which the descending branch of the tensilestress–strain curve of concrete is modified to take into accountthe tension-stiffening effect in an average way, are more widelyused. These so-called “macroscopic” models are easier toimplement and, by being simpler than the “microscopic” ones,they can be readily applied to analyze full-scale structures.However, most macroscopic models oversimplify the tension-stiffening effect by considering only one equation to describethe post-cracking range of the tensile stress–strain curve,independently of the member reinforcement ratio and materialproperties.

Seeking a simple model that can represent more realisticallythe tension-stiffening effect and at the same time be easilyimplemented into finite element programs, we introduce in thiswork a novel tension-stiffening model.

The proposed model uses an exponential decay curve todescribe the post-cracking range of the tensile stress–strainlaw of concrete. The exponential decay parameter (α) is afunction of the member reinforcement ratio (ρ) and of thesteel-to-concrete modular ratio (n = Es/Ec), and is derivedtaking as basis the CEB [5] tension-stiffening model. Guptaand Maestrini [13] have also utilized an exponential curveto formulate a simplified tension-stiffening model and haveobtained 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 tensionand verified by comparison with other tension-stiffeningmodels and also with experimental results obtained frompull-out tests on R/C bars. The proposed tension-stiffeningmodel is then extended to members under bending, assumingthat all cracks are orthogonal to the reinforcement. Thismodel is implemented into a finite element program calledANALEST that allows for nonlinear analysis of reinforcedconcrete beams and plane frames. Application of the model tosimply supported R/C beams tested under 4-point bending indifferent research laboratories are presented in the sequence. Inall the examples, shear deformation, stirrups contribution, andgeometric nonlinearities are neglected. A simplified approachto further extend the tension-stiffening model to members undercombined bending and shear is also described. Concludingremarks are given at the end of the work.

2. The CEB manual design model [5]

The CEB model developed for reinforced concrete memberssubjected to tension considers the tension-stiffening effectthrough an increase in stiffness of the reinforcement. Thecracking mechanism of a reinforced concrete member subjectedto uniaxial tension can be observed in Fig. 1.

An equation for determining the stress–strain curve for thereinforcement is proposed in terms of an average strain, whichlies between the strain of an uncracked section (State I) and thatof a totally cracked section (State II):

εsm =∆`

`= εs2 − ∆εs (1)


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ARTICLE IN PRESSR.S.B. Stramandinoli, H.L. La Rovere / Engineering Structures ( ) – 3

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

where ∆` is the total extension of a reinforced concrete memberof length ` subjected to an axial tensile force N which isgreater than the force Ncr which produces the first crack;∆εs represents the contribution of concrete between crackswhich follows a hyperbolic relationship approaching the lineεs2 asymptotically for stresses in excess of σsr.

In the CEB Manual Design [5], the following expression for∆εs, based on experimental results, is proposed:

∆εs = ∆εs maxσsr

σs2(2)

where:σsr is the stress in the reinforcement calculated on the basis

of a cracked section, where the maximum stress in the concreteunder tension is equal to fct;

σs2 is the stress in the reinforcement at a cracked sectionunder the applied load; and

∆εsmax is the maximum variation between the strains εs1and εs2 which occurs at the beginning of the cracking process.The definition of all these parameters can be better observed inFig. 2.

An expression for the average strain can be obtained bysubstituting Eq. (2) into Eq. (1):

εsm = εs2 − ∆εs max

(σsr

σs2

)= εs2 − (εs2r − εs1r)

(σsr

σs2

)∴

εsm = εs2

[1 −

εs2r

εs2

(σsr

σs2

)]+ εs1r

(σsr

σs2

)∴

εsm =

(σsr

σs2

)2

εs1 +

[1 −

(σsr

σs2

)2]

εs2 (3)

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.

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

Eqs. (1)–(3) were derived for pure tension, however, asindicated in the CEB Manual [5], they are also valid for flexure.Eq. (3) was derived assuming monotonic loading and high-bondbars, but it was further modified to take into account cyclicloading and the use of smooth bars.

This model presents a consistent theory to representthe average post-cracking behavior of a reinforced concretemember under tension. Since the proposed constitutiveequation is based on experimental results, it is also an accuratemodel. However, as it can be observed from Eqs. (1)–(3), itis difficult to be implemented into a finite element code, sinceσs2 cannot be explicitly obtained from the average strain εsm.A simplification of this model was proposed later on CEB-FIP Model Code-90 [18] where a trilinear curve was utilizedto represent the “stress–average strain” relationship for thereinforcement. This curve is an approximation of the originalcurve shown in Fig. 2, with a bilinear branch adopted aftercracking instead of a continuous curve. D’Avila and CamposFilho [19] proposed a trilinear curve for the tensile constitutiveequation of concrete in the post-cracking range, based on thissimplified model from CEB-FIP MC-90. A continuous curve,however, is more desirable for computational implementationinto nonlinear finite element codes.

3. Proposed model

A novel tension-stiffening model that modifies the tensileconstitutive equation of concrete is proposed in the following.The model uses an explicit formulation for the concretestress–strain curve and thus can be easily implemented intoa finite element code. Some features of the CEB manualdesign [5] model instead of the CEB-FIP MC-90 [18] model areutilized, therefore a continuous stress–strain curve is obtainedfor the concrete in the post-cracking range providing numericalstability in nonlinear finite element analysis of R/C members.

Concrete is assumed to behave like a linear-elastic materialuntil its tensile strength is reached, so that a straight linedefines initially the stress–strain curve. In the post-crackingrange, an exponential decay curve is adopted until yieldingof reinforcement takes place, and is defined by the following


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equation:

σct = fcte−α

(ε

εcr

)(4)

where,fct 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 fct:

fct(MPa) = 1.4(

fcc(MPa) − 810

)2/3

. (5)

An alternative way for determining an expression for α

would be by adjusting the experimental results from reinforcedconcrete bars subjected to direct tension, by varying thespecimen longitudinal reinforcement ratio (ρ) and materialproperties. Many authors have conducted parametric studiesto investigate the influence of ρ and of material properties(fracture energy or specimen diameter-to-length ratio, tensile-to-bond strength ratio, etc. . . ), on tension-stiffening. Amongstall properties, the reinforcement ratio is the one that has shownthe greatest influence on the tension-stiffening effect (Hegemieret al. [20]).

Hence, instead of adjusting the experimental data, anexpression for the parameter α, defined as a function of themember reinforcement ratio (ρ) and of the steel-to-concretemodular ratio (n = Es/Ec), is derived in this paper, taking asbasis the CEB [5] model.

The same concept of average deformation (εsm) andits definition given by Eq. (3) is adopted. The concretestress–strain curve is determined through the analysis of areinforced concrete member subjected to direct tension. Thecontribution of concrete between cracks can be observed fromthe graphs displayed in Fig. 3. In this figure, point “a”represents the onset of cracking; “b” defines the point wherethe strain in the reinforcement for State II reaches the strainat yield (εs2 = εy); and “c” the point where the strain in themember reaches the strain at yield (ε = εy).

Additionally, it is assumed that the strain in thereinforcement is equal to the strain in the surrounding concrete.Thus, in the linear-elastic range and before the onset ofcracking, the strain in the R/C member can be determined by:

ε =N

Es As + E ic Ac

(6)

where:Es is the elastic modulus of the reinforcing steel;E i

c is the elastic modulus of concrete before cracking;As is the reinforcement area; andAc 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):

εc = εs = ε=

(σsr

σs2

)2

εs1 +

[1 −

(σsr

σs2

)2]

εs2 (7)

where:

σs2 =N

Asand σsr =

(1 + nρ) fct

ρ

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

ε=

N

Es As + Ecrc Ac

(8)

where,Ecr

c is an equivalent elastic modulus of concrete in the post-cracking range, defined by the secant modulus:

Ecrc =

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 − εEs As

Ac= σs2ρ − εEsρ. (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 atthe onset of cracking) up to Ny (the axial force at yieldingof reinforcement). From the obtained values for ε, the stressin concrete is then calculated by means of Eq. (10). In thedescending branch where εs2 > εy (from point “b” to point“c”), a straight line is found until ε reaches the strain at yieldingof reinforcement, where the stress in concrete drops to zero.

Seeking an expression to determine the exponential decayparameter α, several concrete stress–strain curves were initiallytraced using the procedure described above, by varying thematerial properties fct and fy , and by selecting differentvalues of (nρ) for fixed values of fct and fy . An exponentialcurve, as defined by Eq. (4), was then adjusted to each tracedstress–strain curve. It was then observed that (nρ) was the mostimportant property to define the exponential decay parameter.

Hence, several exponential curves were fitted to the tracedconcrete stress–strain curves by varying only (nρ), and a valueof α was found for each corresponding value of (nρ). Thecurve fitting was made using the Mathcad 2001 program fornormalized concrete stress (σct/ fct) versus strain curves. Fig. 4illustrates 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 thenderived by using all the obtained values of α for different valuesof (nρ), as shown in Fig. 5. The best fit curve achieved wasa third degree polynomial (COR = 0.996), described by thefollowing 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 memberssubjected to direct tension. Extension of the model to R/Cbeams under bending, can be done by employing the effectivearea concept that corresponds to the tensile zone in the member



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–straincurve.

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

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

section. An equation for this effective area is suggested in theCEB-FIP MC-90:

Ae f = 2.5b (h − d) <b(h − x)

3(12)

where,h is the nominal depth of the beam, d is the usual effective

depth, and x is the neutral axis depth.Recalling that in R/C beams under bending the relationship

between nominal and effective depth is usually given by h−d ∼=

0.1h, the above expression for the effective area simplifies to:

Aef ∼=b.h

4. (13)

Therefore, for R/C beams, an effective reinforcement ratio,expressed as

ρef =As

Ae f(14)

must be employed in Eq. (11).

4. Comparison of the proposed model with othertension-stiffening models

To verify the validity of the proposed model, correlationstudies with other tension-stiffening models are initiallyconducted. The proposed model is compared to some refinedtension-stiffening models, such as those of Gupta andMaestrini [13], Kwak and Song [17], and also the CEB [5]model that was utilized for deriving the proposed model.Comparison is made in terms of tensile stress–strain curvesof concrete obtained from R/C bars under direct tension byselecting two typical values of nρ: (i) nρ = 0.2, the samevalue utilized in the previous example which yielded α = 0.064for the exponential decay parameter in the proposed model(see Eq. (11)); and (ii) nρ = 0.4, yielding α = 0.103 fromequation. (11). The resulting normalized stress versus straincurves obtained by the different models for the two examplescan be seen in Fig. 6.

Analyzing the graphs of Fig. 6(a) and (b) it can be observedthat, as expected, the proposed model curves approach theones given by the CEB [5] model. For nρ = 0.4 the twocurves are almost coincident, and small differences are foundfor nρ = 0.2, due to the fact that a continuous curve wasderived in the proposed model from discrete points, obtainedusing the CEB [5] model for a given nρ. The proposed modelalso displays fairly good agreement with the models proposedby Gupta and Maestrini [13] and Kwak and Song [17].

Next, the model is compared to two simplified modelscommonly employed in finite element analysis of R/C



Fig. 6. Comparison of different tension-stiffening models: (a) nρ = 0.2; (b) nρ = 0.4.

Fig. 7. Comparison of tension-stiffening models: (a) Collins and Vecchio [8] and bilinear model [21]; (b) the simplified models and the proposed model for differentvalues of nρ.

members: Collins and Vecchio [8] and a bilinear model(Figueiras [21]), shown in Fig. 7(a). Three values of nρhave been selected for the proposed model application. Acomparison of the tensile stress–strain curves given by theproposed and other simplified models is shown in Fig. 7(b).As it can be noted from this figure, the simplified modelsare not able to consider the effect of different reinforcementratios on tension-stiffening. This fact shows the advantage ofa model such as the present, which takes into account thiseffect. Therefore, our model combines an accuracy comparableto more refined models with the ease of implementation as othersimplified models.

5. Comparison of the proposed model with experiments —pull-out tests

To further verify the validity of the proposed model,comparisons with experiments are also performed. Two pull-outtests on reinforced concrete bars have been selected. The firstone, V3, tested by Rostasy et al., apud Massicotte et al. [11],

uses a bar of 6 m length and cross-section dimensions of30 cm × 50 cm. The longitudinal steel ratio is 0.67%, andthe material properties are as follows: fct = 1.17 MPa; Ec =

10 GPa; fy = 526 MPa; Es = 197 GPa. The resulting value forα from Eq. (11) is 0.049. Fig. 8 shows the stress (MPa) versusstrain (h) curves, obtained experimentally and numericallyusing Eq. (4).

The second test was conducted by Hwang and Riskalla, apudGupta and Maestrini [13], in a study of tension members. Thecase chosen for comparison is example number 7, which uses abar of 76.2 cm length and cross-section of 17.8 cm × 30.5 cm.The longitudinal steel ratio is 1.476%, and the materials havethe following properties: fct = 2.62 MPa; Ec = 27.8 GPa;fy = 469 MPa; Es = 199 GPa. The resulting value for α fromEq. (11) is 0.043. Fig. 9 shows the stress (MPa) versus strain(h) graph, obtained experimentally and numerically using Eq.(4).

It can be observed from Figs. 8 and 9 that the proposed TSmodel can reproduce very well the experimental behavior. The



Fig. 8. Comparison between numerical and experimental results for V3member tested by Rostasy et al.

Fig. 9. Comparison between numerical and experimental results for # 7member tested by Hwang and Riskalla.

model shows only a slightly stiffer behavior at the end of thetests, near the onset of yielding in the reinforcing steel.

6. Finite element model for R/C beams

The proposed tension-stiffening model was introduced intoa nonlinear finite element model, which was then implementedinto a computational program called ANALEST. The programallows for nonlinear analysis of reinforced concrete beams, andboth geometric and material nonlinearity can be considered.A 2D finite element bar with two external nodes and threedegrees of freedom each, and one internal node with onlyone axial degree of freedom is employed (Mari, [22]). Asshown by Chan [23], the internal node needs to be includedto properly capture the variation of the neutral axis positiondue to material nonlinearities. Three Gauss points have beenused for the numerical integration of the stiffness matrix andthe restoring force vector along the length of the element.The fiber model is employed, discretizing the element sectioninto layers, with each layer assumed to be under a uniaxialstress-state. The finite element model allows for the completeresponse of a reinforced concrete member under bending,from its elastic phase, passing by cracking of the concreteand yielding of reinforcement, until collapse is reached. Themodel neglects shear deformation and is limited to members

that display dominant bending behavior, with orthogonal cracksto the longitudinal reinforcement. Either the Hognestad [24]parabola or the constitutive equations given by the CEB-FIPMC-90 [18] can be employed for the compressive stress–strainlaw of concrete. Confinement provided by stirrups can alsobe considered, by using a modified compressive law (Manderet al. [25]). The proposed tension-stiffening model is utilizedto represent the tensile behavior of concrete, with the slightmodification in the stress–strain curve shown in Fig. 3(b): theconcrete stress drops to zero directly when the strain in themember reaches the strain at yield of reinforcement, ε = εy ,hence a straight line orthogonal to the horizontal axis, betweenpoints b and c, defines the cut-off. This modification simplifiesthe numerical implementation and does not significantly alterthe overall behavior of the R/C member. It is assumed thatthe reinforcing steel is an elasto-plastic material, modeled by abilinear stress–strain curve. To avoid convergence problems andoscillations in the numerical iterative process, a parabolic curveis fitted between the elastic and plastic branches of the bilinearstress–strain curve, between 0.8 and 1.2εy . Strain hardening ofthe reinforcing steel may or may not be considered, through theuse of a coefficient sh , which is the plastic-to-elastic modularratio (sh = 0 for perfectly plastic steel). Either the classicNewton–Raphson iterative Methods (load control) or the Arc-length Method of Riks in conjunction with Newton iterativeprocedures can be employed to solve the nonlinear equilibriumequations. The convergence criterion is based on the ratio ofthe norm of residual forces to the norm of applied forces andreactions. Several parametric studies have been conducted byLa Rovere et al. [26] to investigate the effect of mesh size,number of layers for element section discretization, variationin material properties, etc., in the flexural behavior of R/Cmembers.

Some conclusions from these studies are reproduced here:(1) no dependence on mesh size was observed when tension-stiffening is included, showing objectivity of results; (2) fewerelements are needed to model the members when the internalnode with axial degree of freedom is included; (3) 12 layersare usually sufficient to discretize the member section; (4)the tension-stiffening parameter α usually plays a significantrole in the post-cracking response of the member; and (5) nosignificant difference is observed when different equations (theHognestad parabola or the equation from CEB-FIP MC-90) areemployed to describe the compressive constitutive behavior ofconcrete.

7. Comparison of FE analyses with experiments — beamtests

To investigate the reliability of the finite element model andANALEST program, comparisons with experimental tests onreinforced concrete beams have been performed. Since the FEmodel is limited to members that display dominant bendingbehavior, beams subjected to 4-point bending that displayeda typical flexural failure, without significant development ofshear cracks, were selected for the comparison. Four beamstested by different researchers from different countries were



Fig. 10. Tested beam (VRE) geometry, load application and support positions (Ferrari [27]).

Fig. 11. Tested beams (VT1 and VT2) geometry and reinforcement; load application and support positions (Beber [28]).

selected, taking also as basis a certain variation on the α

parameter (ranging from 0.037 to 0.094), in order to validatethe proposed tension-stiffening model. Larger values of α

parameter need not to be considered, since when α > 0.1 thetension-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 andsupport positions, and the finite element meshes adopted forthe beam models.

Two analyses were performed with ANALEST program foreach example, one considering the proposed tension-stiffeningmodel and the other without tension-stiffening consideration.In all analyses, geometric nonlinearities are neglected and theNewton–Raphson Method (tangent stiffness) was employed.Confinement provided by stirrups is disregarded, with theHognestad parabola being adopted for the compressiveconstitutive law of concrete.

The material properties used in the finite element models arecondensed in Table 1. Comparison between FE analyses andexperiments are shown in Figs. 14–17, for the beams VRE, VT1and VT2, VB6 and VC3, respectively, in terms of total appliedvertical load versus mid-span vertical displacement graphs.

A close agreement between numerical and experimentalresults is observed for the beam VRE, in both the elasticand the post-cracking range of the beam. The yield load



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 1Material properties utilized in the finite element analyses of beam examples

Concrete Tension-stiffening SteelBeam fcm (MPa) ftm (MPa) εo n ρeff (%) α φ (mm) fy (MPa) Es (GPa) s.h.

VRE 30.7 2.95 0.0020 6.19 1.33 0.037 6 767.5 210 0.0168 545.8 210 0.01

VT1/VT2 33.5 2.62 0.0020 6.39 1.50 0.040 6 738 214.8 0.01610 565 214.8 0.000

VB6 37.9 2.90 0.0023 5.37 4.47 0.072 3 192 174 0.0018 497 195 0.0042

VC3 20.7 1.60 0.0020 9.00 3.80 0.093 12.5 507 184.6 0.0014

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

reinforcement and large deflections of the beam near to failure,as it can be observed from Fig. 14; however, the instrumentshave been removed from the specimen prior to failure, thushindering the measurement of ultimate displacement.

For the beams VT1 and VT2, it can be observed fromFig. 15 that the FE model can capture very well the ascendingbranch of the curve and approaches well the experimentalcurves after the onset of cracking when tension-stiffening ofconcrete is considered. The numerical model shows a slightlystiffer response until a total applied load of 30 kN is reached,but beyond that load a close agreement to the experimentalcurves is observed. For the analysis without tension-stiffening(NO T.S.) consideration, the finite element model shows amuch more flexible response as compared to the experimental



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

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

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

curves. The onset of yielding of reinforcement was accuratelycaptured by the finite element model, corresponding to a totalapplied load of 44 kN. However, the post-yielding responseof the beams and the ultimate displacement could not bemeasured experimentally, since the instruments have beenremoved from the specimens to avoid damage. The ultimatetotal load measured experimentally was 47 kN, while the

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

numerical value obtained at failure by the program ANALESTwas 46 kN.

An excellent agreement can be observed from Fig. 16by comparison between the numerical analysis and theexperimental test on the beam VB6. The numerical modelis only a little stiffer at the beam elastic range, before theonset of cracking, region more susceptible to instrumentationimprecision. For the analysis without considering tension-stiffening (NO T.S.), a more flexible response is again observed,showing the importance of considering the tension-stiffeningeffect in the beam post-cracking behavior. The finite elementmodel predicted very well the ultimate load, but a smallervalue for the corresponding displacement was obtained by theANALEST program.

For the beam VC3, the comparison between analysis andexperiment shows an excellent agreement in the elastic range(see Fig. 17), but beyond that, for loads higher than 25kN, the numerical model becomes slightly stiffer than theexperimental model. In this example, the analytical curvesobtained considering and not considering tension-stiffening(NO T.S.) are very close, since for this beam the elastic modulusof concrete is low, yielding a high value for the modular ratio n,which results in a high value for the α parameter, and thereforethe tension-stiffening effect becomes very small.

8. Extension of the TS model to 2D constitutive models

In order to extend the TS model to allow for the use in 2Dconstitutive models, by taking into account the angle betweenthe cracks and the reinforcing bars, a simplified approach isintroduced. The tensile constitutive equation for concrete isdefined in the principal direction 1, hence in Eq. (4) σc1 shouldbe used instead of σct, and the α parameter is calculated fromEq. (11) by using an equivalent reinforcement ratio, ρeq, definedby:

ρeq = ρx cos2 θ + ρy sin2 θ (15)

where: ρx is the reinforcement ratio in the x-direction;ρy is the reinforcement ratio in the y-direction andθ 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 bendingand shear. This modified TS model was implemented morerecently into the ANALEST program, by considering theTimoshenko beam theory in the FE formulation, which takesshear deformation into account. Each section layer is thenconsidered under a biaxial stress state, and the 2D constitutivemodel developed by Collins and Vecchio [8] is adopted. ThisTimoshenko beam element is more suitable for members wheresignificant shear cracks develop, and it has shown excellentcorrelation with experimental results from 3-point bending testson beams with low transverse reinforcement ratio (Bresler andScordelis [30]) and also from plane frame tests (Vecchio andBalopolou [31]; Ernst et al. [32]).

9. Conclusions

In this work a novel tension-stiffening model for reinforcedconcrete members is developed. An exponential decay curve isutilized for the concrete tensile stress–strain curve in the post-cracking range. The exponential decay parameter (α) is definedas a function of the reinforcement ratio (ρ) and of the steel-to-concrete modular ratio (n), and is derived taking as a basisthe CEB [5] tension-stiffening model. The model is initiallyvalidated by comparison with other tension-stiffening modelsand also with experimental pull-out tests on R/C bars. Theproposed tension-stiffening model is implemented into a finiteelement program, named ANALEST, that allows for nonlinearanalysis of reinforced concrete beams that display a dominantflexural behavior. The program is then applied to analyzeseveral beams tested under 4-point bending in different researchlaboratories. Comparison between numerical and experimentalresults, in terms of load-displacement curves, showed very goodagreement. The analyses showed that the tension-stiffeningeffect plays an important role in the post-cracking behavior,especially for the beams with α ≤ 0.072. The proposed tension-stiffening model proved to be very efficient allowing both,easy implementation and numerical stability and, additionally,representing the concrete tension-stiffening effect in reinforcedconcrete members with different reinforcing and modularratios. The proposed tension-stiffening model can also beextended to planar members when the crack angle with respectto the longitudinal rebars is different from 90◦ by means ofan equivalent reinforcing ratio. The ANALEST program has

already been extended for planar members under combinedflexural and shear (Stramandinoli [33]), by using a Timoshenkobeam element in conjunction with the 2D smeared and rotating-cracked model, proposed by Collins and Vecchio [8], and themodified TS model proposed in this work. Very good resultshave then been obtained by comparison with experimentaltesting on beams and planar frames that showed significantshear distress. For those examples of beams under combinedbending and shear where flexural cracks are dominant, theBernoulli beam element still gives very good results in the post-cracking range, at service loads. The ANALEST program iscurrently been extended to also allow for the analysis of 3Dreinforced concrete frames.

Acknowledgments

The authors gratefully acknowledge the scholarship grantedby Conselho Nacional de Desenvolvimento Cientıfico eTecnologico (CNPq) to the first author of this paper. Thanksare also due to researchers Juvandes, L.F.P.; Beber A.J. andFerrari, V.J. for making available the experimental data usedin this work.

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