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Finite element modeling of concrete structures reinforced with internal and external fibrereinforced polymers1 Ali Nour, Bruno Massicotte, Emre Yildiz, and Viacheslav Koval
Abstract: Externally bonded fibre-reinforced-polymer (FRP) laminates and fabrics have been successfully used for strengthening damaged or deficient concrete members, whereas internal FRP reinforcements are becoming an efficient alternative to steel reinforcement, particularly in corrosive environments. Despite the enormous progress that has been observed in the last decade, further research is still required to consolidate recent developments and expand the scope of application of FRPs for structural uses. Nonlinear finite element analysis combined with laboratory testing constitutes an efficient approach for pursuing this objective. The scope of this paper is to illustrate, through a selection of a wide variety of typical applications, the contribution of a refined three-dimensional (3-D) constitutive model for investigating the nonlinear response of concrete structures reinforced with internal and external FRPs. The analyses are carried out using a general and portable constitutive concrete model implemented as a user-defined subroutine at Gauss integration point level in commercial finite element software. The constitutive law follows a 3-D hypoelastic approach that models the nonlinear behaviour of concrete using a scalar damage parameter that accounts for the anisotropic behaviour of partially confined concrete and the inelastic volume expansion upon reaching the peak strength. In tension, the model adopts a macroscopic approach that is directly integrated into the concrete law. It simulates implicitly the reinforcing bar – concrete interaction using tension-stiffening factors modified according to the nature of reinforcement that vary as a function of the member strain. The applications include results of well-known test series published in the literature on beams with external and internal FRP reinforcement, slabs with internal reinforcements, bond failure analysis of external FRP, and the effect of confinement on the behaviour in compression of circular and square elements. The paper demonstrates the ability of the concrete model to correctly simulate the behaviour of structural elements reinforced with FRPs at service load level and reproduce failure mechanisms and loads that are consistent with the experimental observations. Key words: constitutive model, nonlinear analysis, finite element, reinforced concrete, glass-fibre-reinforced polymer (GFRP), carbon-fibre-reinforced polymer (CFRP), strengthening, steel. Résumé : Les laminés et les tissus en polymères renforcés de fibres (« FRP ») fixés sur les faces externes ont été utilisés avec succès pour renforcer des éléments de béton endommagés ou défectueux alors que les renforcements « FRP » internes commencent à remplacer efficacement un renforcement en acier, particulièrement dans les environnements corrosifs. Malgré les grands progrès réalisés au cours de la dernière décennie, une recherche plus approfondie est nécessaire afin de faire la synthèse des récents développements et d’élargir l’utilisation des « FRP » dans les structures. Une analyse non linéaire par éléments finis combinée aux essais en laboratoire constitue un moyen efficace pour atteindre cet objectif. Cet article illustre, grâce à des exemples provenant d’un très grand nombre d’utilisations typiques diverses, la contribution d’un modèle constitutif tridimensionnel perfectionné à l’étude de la réponse non linéaire des structures de béton armé de « FRP » internes et externes. Les analyses ont été réalisées en utilisant un modèle constitutif du béton général et universel en tant que sous-programme défini par l’usager au point d’intégration Gauss dans un logiciel commercial d’analyse par éléments finis. La loi constitutive suit une approche hypoélastique tridimensionnelle qui représente le comportement non linéaire du béton par l’utilisation d’un paramètre de dommages scalaire qui tient compte du comportement anisotrope du béton lorsque partiellement confiné et de l’expansion inélastique de volume lors de l’atteinte de la contrainte maximum. En tension, le modèle adopte une approche macroscopique directement intégrée dans la loi régissant le béton. Il simule implicitement l’interaction tige de renforcement-béton en utilisant des facteurs de raidissement en tension modifiés selon la nature du renforcement ; ces facteurs varient en fonction de la contrainte dans les éléments. Les utilisations comprennent les résultats de séries d’essais bien connues, publiées dans la littérature, sur des
Received 25 November 2005. Revision accepted 20 September 2006. Published on the NRC Research Press Web site at cjce.nrc.ca/ on 12 May 2007. A. Nour, B. Massicotte,2 E. Yildiz, and V. Koval. Département de génie civil, géologique et mines, École Polytechnique de Montréal, CP 6079, Station Centre-ville, Montréal, QC H3C 3A7, Canada. Written discussion of this article is welcomed and will be received by the Editor until 31 July 2007. 1 2
This article is one of a selection of papers in this Special Issue on Intelligent Sensing for Innovative Structures (ISIS Canada). Corresponding author (e-mail:
[email protected]).
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doi: 10.1139/L06-140
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341 poutres munies de renforcement en « FRP » externe et interne, des dalles munies de renforcement interne, d’analyse de la rupture dans les « FRP » fixés sur les faces externes et l’effet du confinement sur le comportement en compression des éléments circulaires et carrés. Le présent article démontre la capacité du modèle à simuler correctement le comportement des éléments structuraux renforcés de « FRP » lors d’une utilisation à la charge de service et à reproduire les mécanismes et les charges de défaillance concordant bien avec les observations expérimentales. Mots-clés : modèle constitutif, analyse non linéaire, élément fini, béton armé, polymère renforcé de fibres de verre (« GFRP »), polymère renforcé de fibres de carbone (« CFRP »), renforcement, acier. [Traduit par la Rédaction]
1. Introduction During their service life, civil infrastructures may need structural repair or strengthening for several reasons, such as design and construction defaults, climatic or chemical agent damage, corrosion, accidental overloading or impact, fires, or earthquakes. Modifications to the loading level, design forces, or code specifications may require the strengthening of critical load-carrying members. Generally, the complete replacement of deficient structures is very expensive, and their repair or strengthening is often considered as the most economical solution. In recent years fibre-reinforced-polymer (FRP) composites have become an alternative for strengthening and repairing deficient concrete structures. Externally bonded FRP fabric and laminates have been successfully used for strengthening reinforced concrete beams, slabs, or columns. They represent promising means for remedying structural deficiencies and for enhancing the performance of civil engineering structures (Nitereka and Neale 1999). Steel has been used for more than a century as an effective and cost-efficient internal reinforcement for concrete structures. When it is not prone to ion attack, steel reinforcement can last for decades without any noticeable deterioration. However, concrete members in corrosive environments or exposed to deicing salts are sensitive to reinforcement corrosion, which often leads to significant diminution of their strength and thereby affects structure safety. Costs related to repair or replacement of structures damaged by corrosion is very high and occupies a large portion of bridge owner budget (El-Salakawy et al. 2003). Most solutions adopted to protect reinforcing steel, such as galvanization, concrete additives, epoxy coating, or cathodic protection, generally have limited success (Benmokrane et al. 2000). Internal FRP reinforcement is an alternative material to replace conventional steel bars. Fibre-reinforced polymers present outstanding characteristics, such as durability, because of their high corrosion resistance, high strength-to-weight ratio, and advantageous fatigue resistance. In addition, FRP reinforcements exceed the strength and can match stiffness properties of steel. Savings in maintenance usually largely compensate for the higher initial cost of FRPs. In the last decade, significant progress on the use of FRPs as construction materials has been observed. Recent codes and standards have introduced specifications for the design or the strengthening of members with internal or external FRPs, respectively. Despite these significant advances, more research is still required to expand the scope of utilisation of FRPs and relax some limitations imposed on their use owing to the lack of scientific knowledge.
Laboratory tests on structural elements or on reduced-scale structures are essential for observing the actual behaviour and failure modes. Testing is, however, expensive and timeconsuming and often limits the pace of research progress. Existing structures cannot be tested at ultimate failure, whereas because of limitations of testing equipment, scale effects on large structures cannot be observed experimentally. This situation has strongly encouraged the development of advanced analytical methods capable of representing the behaviour of concrete structures internally and (or) externally reinforced by composite materials under all possible loading conditions. In this context, recourse to numerical strategies based on the finite element method could enable correct prediction of structural behaviour up to ultimate failure and capture with fidelity the main complexities associated with material nonlinearities, namely concrete confinement, lap splice strength, concrete postcracking, and the interaction between concrete and composite or steel reinforcements. Bouzaiene and Massicotte (1997) developed a threedimensional constitutive model that can simulate the behaviour of concrete under multiaxial stress conditions. The model is based on the hypoelastic approach for which the behaviour of concrete is modeled using an equivalent uniaxial stress–strain curve combined with a compression scalar damage parameter. The model accounts for anisotropy, elastic modulus degradation under loading–unloading, and inelastic volume expansion. The original model was developed as an academic finite element software. Within the scope of the Center of Excellence ISIS Canada project 2.3.4, the model was improved and modified to make it portable to most commercial nonlinear finite element software. The objective was to provide a powerful constitutive model that could be made available to other users. Massicotte et al. (2007) introduced a new tension-stiffening model to the original law of Bouzaiene and Massicotte and made the constitutive model portable. The proposed tensionstiffening model, inspired by CEB-FIP (1990) and Winkler et al.’s (2004) recommendations, adopts a macroscopic approach that modifies the concrete law. The model integrates variable tension-stiffening factors defined as a function of the member strain, as used by many investigators to account for steel–concrete interaction (Fields and Bischoff 2004). The postcracking modeling of concrete follows the smeared crack approach proposed by Rachid (1968), which constitutes an efficient representation of the cracked concrete in nonlinear finite element analysis. The term portable means that the constitutive model is implemented as a user-defined subroutine, at Gauss integration point level, into finite element software. For the present study, the model was introduced into each of the two components of the
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Fig. 1. Concrete model: (a) concrete in compression; (b) concrete in tension.
general-purpose finite element software ABAQUS (Hibbitt et al. 2004) (i.e., ABAQUS standard (STD) and ABAQUS explicit (EXP) (Massicotte et al. 2007)). Experimental results aimed at defining tension-stiffening factors for internal FRP reinforcement are scarce. The high elastic modulus of carbon-fibre-reinforced polymers (CFRPs) would presumably lead to relationships similar to those of steel reinforcement. Reinforcing bars made of glass-fibre-reinforced polymers (GFRPs) have stiffness that is lower than steel and thereby require appropriate formulation for describing the associated tension-stiffening factors. Bischoff and Paixao (2004) indicated that concrete members reinforced with GFRP bars exhibit greater tension stiffening than those reinforced with steel bars. Moreover, concrete members reinforced with GFRP develop wider and deeper cracks than members reinforced with steel. This paper presents the main features of the concrete model developed at École Polytechnique (Bouzaiene and Massicotte 1997; Massicotte et al. 2007) and the recent modifications implemented to the model to enable analysis of concrete members reinforced with FRPs. The paper illustrates the performance of the model for capturing and predicting the behaviour of concrete members and structural elements reinforced with internal and external FRPs. The examples presented in the paper are taken from well-known tests, published the literature, that have been specifically selected for illustrating the capabilities and versatility of the model for simulating a wide variety of structural behaviour.
2. Material models 2.1. Concrete For simulating the behaviour of concrete under multiaxial loading, Bouzaiene and Massicotte (1997) developed a threedimensional hypoelastic constitutive model that accounts for the anisotropic behaviour of concrete, elastic modulus degradation under loading–unloading, the transition point that separates brittle and ductile behaviour of concrete under increasing confinement, and the volume increase observed in
concrete upon approaching ultimate strength. The behaviour of concrete is modeled using an equivalent uniaxial curve, a compression scalar damage parameter λ, and either the five parameters of Willam and Warnke (1975) or the four parameters of Hseih et al. (1982) failure surfaces. The scalar damage parameter provides an invariant measure of the evolution of irreversible damage in concrete, which overcomes the difficulties encountered in the application of other hypoelastic models (Darwin and Pecknold 1977; Elwi and Murray 1979). Only an overview is presented in this paper; details on the model are available in Bouzaiene and Massicotte (1997). The damage evolution of concrete in compression is described by a scalar function in relation to the major compressive stress by means of an equivalent uniaxial curve. A realistic representation of the cumulative damage, along any stress path, is obtained by the following incremental relationship: 1 dλ = dε d [1] λ= d εmax load path
≈
εeq − σ /Ec εc − σc /Ec
for
0≤λ≤1
where dλ stands for the damage parameter increment, εeq is the uniaxial equivalent strain increment, σ is the compression stress increment according to the major axis, εc is the uniaxial equivalent strain at peak load, σc is the major compressive stress at failure, and Ec is the material elastic modulus. Equation [1] expresses the inelastic stain increment dεd and the total inelastic d . For an elastic behaviour of concrete, the strain at failure εmax parameter λ is set equal to zero and equals one at the peak. The concept of equivalent damage allows the evaluation of the increment and the total equivalent strain for a nonproportional loading as follows (Fig. 1a): [2]
σ − σ˜ Et = ε˜ eq (λ) + εeq
εeq = εeq
where σ˜ stands for the effective compression stress obtained
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Fig. 2. Constitutive relations for steel and fibre-reinforced-polymer (FRP) reinforcements.
from the equivalent uniaxial curve corresponding to the accumulated damage at the previous step, Et is the stiffness modulus corresponding to λ, and σ is the major compression stress. One also notes that the uniaxial stress–strain curve of Saenz (1964) is adopted for its convenience. In compression, the rapid increase of Poisson ratio is related to the opening of microcracks in the direction of the unconfined
[3]
υ + λ (1 − υ ) σi − σj for σi < 0 0 0 υij = σc υ0 otherwise
and
where υ0 stands for the initial Poisson ratio and σi and σj take the values of principal stresses. In tension, concrete cracks are generally characterized by the occurrence of microscopic discontinuities that are transformed into discrete cracks with a total loss of resistance after reaching a certain strain value. For modeling tensile rupture, the fundamental concept of energy equivalence is adopted. It is combined with the smeared crack technique, which assumes that cracks are uniformly distributed in the concrete mass. This enables modeling of the crack effect using a stress–strain relation, called a softening curve. As suggested by Feenstra and De Borst (1996), concrete softening in tension is modeled using a simple function defined as follows (Fig. 1b): ε − εe [4] σt = ft exp − εa for which [5]
εa = (Gf /h)ft
and εe = ft /Ec
where Gf is the fracture energy consumed by the crack per unit of crack surface and h is the equivalent length for which the displacement due to crack opening is uniformly distributed. As stated by Bažant and Oh (1983) and Oliver (1989), the use of the equivalent length h in the finite element calculations leads to results that are insensitive with regard to the global
principal axes, interpreted also as inelastic volume expansion. This asymmetrical behaviour suggests the use of an unsymmetrical Poisson ratio, where for each principal plane the expansion effect is limited to the lowest confined axis. For a triaxial state, the effective Poisson ratio, a function of the damage parameter λ and the gradient of the principal stresses, is expressed in the following form according to Bouzaiene and Massicotte (1997):
σi < σj
structural response upon mesh refinement and corresponds to a representative dimension of the mesh size. When the tensile strength criterion is exceeded, a microcrack band perpendicular to the principal direction in tension develops. The material gradually loses its integrity, leading to the degradation of material properties interpreted as tension damage. As in compression, tension damage evolution is described by means of a scalar parameter d, which represents the degradation of material properties due mainly to crack propagation. Thus, once the crack is initiated, the elastic modulus Ec is reduced using the damaged material modulus Ed as follows: [6]
Ed = (1 − d)Ec
where d = 0 corresponds to undamaged material and d = 1 to completely damaged material. The parameter d is estimated as proposed in Crisfield and Wills (1989). Therefore, with this formulation, the damage parameters in compression λ and in tension d are totally coupled during the calculation of the degraded elastic modulus and the residual tensile stress (Massicotte et al. 2007). 2.2. Steel and fibre-reinforced-polymer reinforcements Steel reinforcement is assumed to behave in an elastic-plastic manner or as a strain hardening material when applicable (Fig. 2a). The yield strength is represented by fys . As shown by Fig. 2b, FRP reinforcements are assumed to be linear elastic © 2007 NRC Canada
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Fig. 3. Constitutive relations accounting for the tension stiffening effect.
with brittle fracture in tension. The ultimate tensile strength of the material is represented by fyFRP , while the corresponding strain at failure is εuFRP . When FRPs are used as external reinforcement, each lamina is considered as an orthotropic layer in a plane stress condition, but the lamina is transversely isotropic, where the properties of the FRP are nearly the same in the direction perpendicular to the fibres. In ABAQUS, FRPs are modeled as linear elastic materials. 2.3. Tension-stiffening modeling The presence of steel or internal FRP reinforcement necessitates the consideration of bar–concrete interaction. The bar– concrete adherence allows the concrete located between cracks to resist tensile stresses, thereby reducing the average reinforcement stress level compared to its magnitude at the crack. This bar–concrete interaction phenomenon results in a gain in rigidity, also called tension stiffening. A simple way to account for this local phenomenon is to integrate the bar–concrete inter-
[9]
action in a global dimension by modifying the stress–strain relationship of the material, either the reinforcing bar or the concrete. The tension-stiffening model described and validated in detail in Massicotte et al. (2007) follows the approach of the original law of Bouzaiene and Massicotte (1997), which associates tension stiffening with the concrete law. This model is inspired from recommendations by CEB-FIP (1990) and Winkler et al. (2004) and integrates the tension-stiffening factors used by many investigators to account for bar–concrete interaction. It is also combined with the smeared crack approach (Rachid 1968) in the context of nonlinear finite element analysis. The bond between reinforcing bars and the surrounding concrete is assumed to be perfect because the local effects associated with the rebar–concrete interface, such as bond slip and dowel action, are considered implicitly by the tension-stiffening model. For a given strain, the equivalent concrete stress σc,TSE attributed to tension stiffening can be described as the difference between the average bar stress σbar of the reinforced concrete member and the stress of the bar σbar,II at the crack, determined as follows (Winkler et al. 2004): [7] σc,TSE = ρeff σbar − σbar,II for which [8]
ρeff = Abar /Ac,eff
where ρeff is the effective reinforcement ratio, Abar is the bar area, and Ac,eff is the concrete area involved in the stiffening, computed according to CEB-FIP (1990) recommendations. Thus, as shown in Fig. 3 the stress–strain relationship is defined by Winkler et al. (2004) and Massicotte et al. (2007)
ε = ft 1 − [1 − β (ε)] for ε < ε1 ε1 for ε1 < ε < ε2 σt (ε) = β(ε)ft εbar,y − ε for ε2 < ε < εbar,y = β(ε)ft εbar,y − ε2
where ε1 and ε2 are defined as follows (ε1 < ε2 < εbar,y ): 1 1 + ε1 = [1.3 − β (ε)] ft Ebar ρeff Ec [10] 1 1 ε2 = εbar,y − β (ε) ft + Ebar ρeff Ec In eqs. [9] and [10] Ebar is the reinforcing bar modulus of elasticity, εbar,y corresponds to the strain at yielding for steel reinforcements or the ultimate strain for composite bars, and β is the tension-stiffening factor that accounts for bar–concrete interaction. The tension-stiffening relationship (eq. [9]) is established for the case of cracks orthogonal to the bar direction. In real structures cracks have an unspecified orientation with the reinforcing bar direction. Therefore, for steel and composite bars not orthogonal to the direction normal to the crack, the
model adopts an equivalent reinforcement ratio, which generates the same stiffening contribution as that produced by the actual reinforcement, as suggested in Massicotte et al. (1990). The tension-stiffening relationship expressed by eq. [9] and proposed by Massicotte et al. (2007) considers the tensionstiffening factor β(ε) as a function of the member strain. This formulation allows the introduction of various empirical models proposed in the literature by proceeding to a direct modulation with the various sections of the stress–strain relationship (eq. [9]). For the case of steel reinforcement, a sensitivity analysis was performed to select the most appropriate tension-stiffening factor. It was determined that the Fields and Bischoff (2004) model, expressed as follows, provides the best results (Massicotte et al. 2007): [11]
βs (ε) = exp [−800 (ε − εe )]
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no experimental evidence is available to quantify this effect. Other phenomena such as thermal cracking or alkali–aggregate reaction could also influence this parameter. It is assumed in this paper that analyses are carried out for sound concrete.
3. Nonlinear finite element analysis
Tension stiffening for FRP rebars with an elastic modulus in the range of steel could probably be modeled satisfactorily with eq. [11], although, to the authors’ knowledge, no experimental data of direct tension tests are available. In the case of GFRP reinforcing bars, limited information is available in the scientific literature regarding the tension-stiffening factor β(ε). Bischoff and Paixao (2004) observed that GFRP-reinforced concrete members exhibit greater tension stiffening than equivalent steelreinforced concrete elements for any given axial strain value. Moreover, concrete members reinforced with GFRPs develop wider and deeper cracks than members reinforced with steel or CRFP rebars. They proposed the following expression for the tension-stiffening factor applicable to GFRP or any type of reinforcement: [12]
βG (ε) = exp [−1100 (ε − εe ) (Ebar /200)]
where Ebar is the modulus of elasticity of the composite bar in gigapascals. Bischoff and Paixao (2004), Fields and Bischoff (2004), and Bischoff (2001, 2005) have demonstrated that bar finish does not influence tension stiffening when member response is considered. They showed that the bar elastic modulus is the main parameter that affects the member response in term of forceelongation properties, as is considered in the formulation. Furthermore, they indicated that the effect of the surface characteristics (sand coated, ribbed, etc.) have an effect on crack spacing and crack width only. They concluded that their formulation (eq. [12]) is independent of the surface characteristics of GFRP bars. However, they also indicated that further tests are needed to confirm their findings. For this reason, with the adopted formulation, the bond is implicitly considered by the tension-stiffening factor β(ε) and does not appear in the stress– strain relationship. Figure 4 synthesizes the tension-stiffening factors β(ε) for both steel- and GFRP-reinforced members. It is well depicted that GFRP-reinforced concrete exhibits greater tension stiffening than the steel reinforced concrete. In the subsequent analyses, eq. [11] is used for simulating the steel–concrete interaction, and eq. [12] is used for GFRP reinforcement. It is worthwhile to mention that the influence of concrete age on β is an interesting concern. To the authors’ knowledge
The constitutive model is implemented as a user-defined subroutine, at Gauss integration point level. For the results presented in this paper, the model has been introduced in the two components of the general purpose finite element software ABAQUS (Hibbitt et al. 2004), i.e., ABAQUS/Standard (STD) and ABAQUS/Explicit (EXP). The formulation, element types, and solution strategies for solving nonlinear problems are quite different in these two software components. The concrete model is written in FORTRAN and is included in the UMAT subroutine for the analyses with STD and the VUMAT subroutine for the analyses with EXP. In this paper, all analyses were carried out using STD only, the comparison with EXP solutions being beyond the scope of the paper. More details on that matter are provided in Massicotte et al. (2007). Analysis with STD represents the classical version of the general finite element software and uses a tangential stiffness approach that is unconditionally stable. The two principal methods available for the resolution of nonlinear problems are the traditional Newton–Raphson method (Static General), and the Riks arc length method (Static Riks). STD calls the subroutine UMAT at each increment and iteration of the Newton–Raphson solving process, for each Gauss integration point of all elements of the mesh. The finite element software proves the concrete model subroutine with the updated strain tensor. With these data, the subroutine updates the stress tensor and the solution-dependent state variables that contain the damage history for each integration point of the finite element model. The subroutine also returns the Jacobean matrix ∂σ /∂ε at the Gauss integration level of the current iteration, which is used to assemble the tangent stiffness matrix. At each Gauss integration point, the concrete models uses two damage vectors, WA01 and WA02. The first contains to the updated information at the end of the last converged step, whereas the second includes the damage variables at the current iteration. For damage in each of the three principal directions, the model uses a damage index called NCR, defined according to the following: • NCR= 0, undamaged material • NCR= 1, damaged material in tension with partial crack opening • NCR= 2, damaged material in tension with reduced tension stiffening (or softening), but the crack still active • NCR= 3, damaged material in tension and the crack do not transmit any tensile stress • NCR= –1, compression material failure • NCR= –2, compression material failure, but is in compression softening • NCR= –3, compression material failure with a total loss of resistance
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Fig. 5. Description of M’Bazaa (1995) and Chicoine (1997) beams (adapted and revised from M’Bazaa (1995) and Chicoine (1997)). CFRP, carbon-fibre-reinforced polymer.
4. Applications 4.1. General Three-dimensional finite element analyses are carried out to examine several aspects of the complex behaviour of reinforced concrete (RC) structural elements internally or externally reinforced with FRPs. The constitutive model is validated by means of a variety of well-known tests, taken from the literature, for demonstrating the reliability and versatility of the proposed concrete model. Basically, the model requires only the compressive strength of concrete and the yield strength of steel or the elastic modulus and ultimate strength of FRPs. For concrete, actual values can be provided, but the model computes default properties in the absence of data. For all the applications presented in this paper, only default material data were used. It is worthwhile to note that the constitutive concrete model was already validated by Bouzaiene and Massicotte (1997) and by Massicotte et al. (2007) for plain concrete and reinforced concrete members, respectively. Concrete is modeled using 8-node 3-D solid elements. The internal reinforcements by steel and FRP bars are modeled using 2-node embedded bar formulation in the isoparametric concrete elements. With this feature of ABAQUS, the reinforcing bars are treated as integral parts of the concrete element to determine the total internal resisting forces that are directly added to those of concrete. External FRP reinforcements are modeled using multilayered 4-node shell elements. It is worthwhile to mention that in ABAQUS STD analyses are performed using 8-Gauss integration points for solid elements and 4-Gauss integration points for shell elements, with 2-Gauss integration points over the thickness. For capturing the post-peak behaviour of concrete via the STD solution strategy, all analyses were performed using the Riks arc length method. 4.2. Rectangular beam externally strengthened with carbon-fibre-reinforced polymer This section presents the modeled behaviour of rectangular beams strengthened in flexure with bonded unidirectional carbon fibre laminates. The experimental work was carried out at Sherbrooke University by M’Bazaa (1995) and Chicoine
Table 1. Material properties for M’Bazaa (1995) and Chicoine (1997) beams. CFRP, carbon-fibre-reinforced polymer. Properties Concrete fc (MPa) 45
Ec (MPa) 30 200
Steel Es (MPa) 200 000
fys (MPa) 440
εc –0.0035
Composite (CFRP) E11 (MPa) E22 (MPa) G12 (MPa) ν12 fyCFRP (MPa) εuCFRP 82 000 34 750 3 720 0.25 1378 0.0168
Fig. 6. Load–deflection curves, validation.
(1997), followed by a numerical investigation presented by Nitereka and Neale (1999).The geometry and loading conditions reported by Chicoine (1997) are shown in Fig. 5, and the different material properties are summarized in Table 1.
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Fig. 7. Cards showing details of the finite elements model: (a) material failure indices for Chicoine beam; (b) stress card for the composite; (c) stress card for the longitudinal steel bars and the stirrups.
The beam was strengthened with three CFRP layers for a total thickness of 0.9 mm. In M’Bazaa (1995) the beam exhibited an early FRP delamination failure mode, a problem eliminated by Chicoine (1997) by applying U-shaped composite anchors at the ends of the beam. The anchorage was not modeled, but its presence allows the assumption in the analysis of a perfect bond between the reinforced concrete beam and the composite. Using symmetry, only half of the beam is analyzed. Figure 6 shows the load–deflection curves of the control reinforced concrete beam without composite (M’Bazaa 1995) and the strengthened beam with CFRPs (Chicoine 1997). In both tests failure was caused by concrete crushing. For the reference beam, failure occurred after significant steel reinforcement yielding, while the second beam exhibited an important increase of its load-carrying capacity with less ductility compared to the reference beam without CFRP debonding. The modeled responses of the two beams reproduce accurately the global experimental measurements and observed failure mechanisms. First, the significant load-carrying capacity and additional stiffness provided by the CFRP laminated were accurately simulated by the numerical analysis. At early loading stages, the same response was obtained for the two beams, an indication of the small contribution of the composite. However, as cracking progressed and yielding of the steel reinforcement occurred upon increasing the applied load, the contribution of the composite became more important and its effect was accurately depicted by the analytical model. Figure 7a shows the material failure indices at failure. Crack-
Fig. 8. Load–deflection curves, prediction.
ing (in dark gray and light gray in the print version of this article; in red and yellow in the Web version) is predominant in the central portion of the beam, whereas a local compression failure by crushing appears in the vicinity of the applied loads. This is in accordance with the experimental observation reported by Chicoine (1997).Figure 7b shows the longitudinal stress card for the composite. Also, it is possible to capture with fidelity the effect of steel stirrups on the performance of external strengthening. In ABAQUS, as for the longitudinal bars, the stirrups
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Fig. 9. Description of Grace et al. (1998) beams (adapted and revised from Grace et al. (1998)). GFRP, glass-fibre-reinforced polymer.
are rigorously modeled using truss elements embedded in the concrete mass (Fig. 7c). Figure 8 depicts the predicted load–deflection curves for the additional cases of 1 and 2 layers of CFRP, presented along with the model response of the two beams of Fig. 6. The analyses illustrate the effect of the number of layers on the load-carrying capacity, which corresponds to a decrease of the midspan deflection attributed mainly to the additional stiffness brought by the composite. In the case of two CFRP layers, the numerical model predicted the same failure mode as for three layers, but with only one layer the failure is caused by the rupture of the composite. 4.3. Rectangular beam internally reinforced with glass-fibre-reinforced polymer This section compares the model prediction and the experimental observations for a rectangular beam internally reinforced with GFRP bars selected from the testing program carried out by Grace et al. (1998). The geometry and loading conditions are illustrated in Fig. 9. The analysis of two beams is presented: a control beam reinforced with conventional steel bars and a composite beam that used GFRP bars for both flexural reinforcement and stirrups (specimens sb-st and gb-gt in the original paper, respectively). The control specimen used high-strength steel reinforcement with 650 MPa yielding strength and modulus of elasticity of 205 GPa. The mean GFRP bar elastic modulus was 41.8 GPa, and the mean tensile strength was 1100 MPa. Symmetry was used to analyze only half the beam. For tension stiffening, eq. [11] is used for the steel reinforcements for modeling the bar–concrete interaction, whereas eq. [12] was adopted for the analysis of the beam reinforced with GFRP. Figure 10 illustrates the load–deflection curves corresponding to the reference and composite beams at midspan. In both cases, numerical results agree satisfactory with the experimental measurements. Grace et al. (1998) observed large deflections for the composite beam due to the low stiffness of the GFRP bars in comparison with the control beam. The authors attributed this difference in deflection to the large deformation
Fig. 10. Load–deflection response.
of the GFRP stirrups. Furthermore, they reported that after steel reinforcement yielding, failure developed in the control beam with extensive propagation of flexural cracks in the central portion of the span. In spite of the presence of steel stirrups, the diagonal cracks propagate up to supports, leading to a flexural failure mode. In the composite beam, the failure developed with the extensive propagation of flexural cracks in the central portion of the span, accompanied by a large shear deformation attributed to the low elastic modulus of the GFRP stirrups. For both beams, the model reproduced with fidelity the observed failure mechanisms. Since GFRP reinforcement has no yielding point, the behaviour of a beam with composite bars was quite different than conventionally reinforced concrete members. First, in the vicinity of the loading point, the compression stress of concrete at the end of the analysis exceeded at several Gauss points the compressive strength of concrete, indicating that the failure mechanism is of a shear-compression type, which is in accordance with experimental observations. Furthermore, the GFRP bars and stirrups did not reach their tensile strength (fyGFRP = 1100 MPa). The maximum tensile stress in the longitudinal bars at failure predicted by the model is
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Fig. 11. Description of slab specimens: (a) Benmokrane et al. (2004) slabs; (b) El-Sayed et al. (2005) slabs (adapted and revised from Benmokrane et al. (2004) and El-Sayed et al. (2005)). GFRP, glass-fibre-reinforced polymer.
about 920 MPa, while the corresponding maximum value in the stirrups is 260 MPa. The high tensile strains in the GFRP bars associated with these stress levels lead to nearly zero tensionstiffening stress in the concrete, indicating that concrete is not contributing to the resistance mechanism that leads to a shear failure. 4.4. One-way concrete slabs reinforced with glass-fibrereinforced-polymer bars This section presents the results of tests of two series of oneway concrete slabs internally reinforced with GFRP bars; these tests were performed at Sherbrooke University. The first set of
slabs was selected from the experimental program carried out by Benmokrane et al. (2004) and the second from the El-Sayed et al. (2005) experimental program. The geometry and loading conditions of the selected slabs are illustrated in Fig. 11.Result presentation adopts the same notation for the slabs as reported in the original papers. For the first group, the model predictions are compared with the experimental results for two slabs: specimen S-GGB, reinforced with two double mats of GFRP bars, and control specimen S-ST, reinforced with conventional steel bars. For the second series, the selected slabs SG1, SG2, and SG3 refer to specimens having the same GFRP reinforcing bars in all directions except the bottom GFRP reinforcement in
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Table 2. Reinforcing bar properties of Benmokrane et al. (2004) slabs. GFRP, glass-fibre-reinforced polymer. Reinforcement Modulus of elasticity bar type (GPa)
Tensile strength (MPa)
GFRP Steel
775 465 (yielding)
45 200
Fig. 12. (a) Moment–deflection curves. (b) Load–deflection curves.
Table 3. Reinforcing bars properties of El-Sayed et al. (2005) slabs. GFRP, glass-fibre-reinforced polymer. Reinforcement Modulus of elasticity bar (GPa)
Tensile strength (MPa)
GFRP No. 16 GFRP No. 22
597 540
40 40
the main direction, as shown by Fig. 11b. In both test series, all slabs were 3100 mm long and 1000 mm wide, and specimen thickness of the first and the second groups were 150 and 200 mm, respectively. The main properties of the reinforcing bars are summarized in Tables 2 and 3. Owing to symmetry, only one quarter of the slabs is analyzed. Results obtained from the numerical analysis are presented in terms of moment–deflection (Fig. 12a) for Benmokrane et al. (2004) slabs, and in terms of of load–deflection (Fig. 12b) for El-Sayed et al. (2005) slabs. For all cases the numerical solutions are globally in good agreement with the experimental values. For the first group (Benmokrane et al. 2004) shown in Fig. 12a, the experimental moment–deflection curve for the control slab (S-ST) is trilinear with a constant strain ascending branch attributed to the reinforcement strain hardening. The authors observed that after steel reinforcement yielding, failure developed with extensive propagation of flexural cracks in the central portion of the span and towards the supports. For the composite slab (S-GGB), the experimental moment–deflection curve is bilinear. The authors indicated that the GFRP slab presented an appreciably higher load-carrying capacity compared with the control specimen. Furthermore, they attributed the 20% larger deflection of the GFRP slab observed at ultimate to the low stiffness of the composite material. For the composite slab, the model predicted a failure governed by concrete crushing, as several Gauss points in the vicinity of the applied load exhibited compression failure (NCR = –3), while the GFRP bars were still elastic. This prediction is in accordance with the concrete crushing reported by the authors. One notes that the transition between the uncracked and the fully cracked states and the stiffness in the post-cracking condition are well reproduced by the tension-stiffening models adopted for steel and GRFP reinforcements. In Fig. 12b related to the second test series, the measured experimental load–deflection curves for SG1, SG2, and SG3 slabs are bilinear. As reported by the authors (El-Sayed et al. 2005), the flexural stiffness of the slabs increases with the GFRP reinforcement percentage. For the slab SG1, failure develops with extensive propagation of flexural cracks in the central portion of the slab up to the supports. On the other hand, the slabs SG2 and SG3 exhibited the same behaviour as slab S-GGB (Benmokrane et al. 2004), and the observed failure was caused by concrete crushing in the vicinity of load application.
For this test series the numerical model correctly predicted the failure modes of the three slabs. For slab SG1, fc was not exceeded at any Gauss points in compression, while in the central portion of the slab several Gauss points exceeded their tensile strength and were unable to transmit any tensile stress (NCR = 3), indicating that the failure is of diagonal tension. In this case, the predicted maximum tensile strain in the GFRP bars is approximately 1.15%. For slabs SG2 and SG3, fc was exceeded at several Gauss points, as NCR = –3 in the vicinity of the loading point at ultimate, which indicates a failure by concrete crushing while GFRP was still active. The tensile strains in the longitudinal bars at failure in the model are 0.95% and 0.79% for SG2 and SG3 slabs, respectively. 4.5. Analysis of damaged beams externally reinforced with carbon-fibre-reinforced polymer This section presents the analysis of damaged beams externally strengthened in flexure with bonded unidirectional CFRP laminates. The tests were carried out by Koval and Massicotte (2007) at École Polytechnique de Montréal in a project aimed at studying the improvement of the concrete–CFRP interface as well as investigating the performance of repair techniques for
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Fig. 13. Description of damaged beams externally reinforced by CFRP (adapted and revised from Koval and Massicotte 2007): (a) schematic drawing; (b) test set-up for MC1O specimen; (c) damaged zone for MC1O specimen; (d) damaged zone for MB1O specimen; (e) damage repaired zone.
damaged beams. This study follows a previous experimental investigation (Folcher et al. 2003) in which premature debonding due to the flexional rigidity discontinuity between the damaged zone and the sound concrete was observed. The main scope of this numerical example is to examine the ability of the model to capture local phenomena such as the influence of local repair of damaged zones on the global behaviour of the beams. The geometry and loading conditions are shown in Fig. 13, and material properties are summarized in Table 4. In this application, three RC beams are analyzed. As shown in Fig. 13, these beams are externally strengthened by two sheets of CFRP laminate (2900 mm × 50 mm × 1.2 mm). The un-
damaged beam MA1O is considered as the reference. Beams MB1O and MC1O have initial imperfections reproducing the effect of damage that could have been caused by an impact or corrosion of prestressing strands. To simulate the damaged zone, the formwork was fabricated to keep the central bottom part of the beams without concrete.The longitudinal T13 strands were cut in the centre to simulate a 150 mm discontinuity. Concrete surface in the repaired zone was roughened according to construction practice. Figure 13 illustrates the dimensions of the damaged zone. Two repair methods were considered. In beam MB1O additional reinforcement was added in the damage zone to eliminate the flexional rigidity discontinuity. Each
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Can. J. Civ. Eng. Vol. 34, 2007 Table 4. Material properties for Koval and Massicotte (2007) beams.
Fig. 14. Load–deflection curves.
Properties Old concrete fc (MPa) 52
Ec (MPa) 34 887
Reparation concrete fc (MPa) Ec (MPa) 54 31 000
εc –0.003
ν 0.23
ft (MPa) 2.1
εc –0.0035
ν 0.22
ft (MPa) 2.3
10M steel bars Es (MPa) fys (MPa) 200 000 400 T13 strand Es (MPa) fys (MPa) 205 600 1860 Composite (CFRP) E (MPa) ν fyCFRP (MPa) 207 000 0.25 2400
εuCFRP 0.016
cut T13 strand was overlapped on a length of 150 mm by one 10M reinforcing steel bar. Moreover, a 50 mm × 50 mm steel wire mesh with 9 mm2 wires was placed around the overlapping bars. For comparison with the Folcher et al. (2003) study, the repair concrete in the damaged zone of MC1O specimen was not reinforced. The observed failure mode for the two repaired beams was initiated by the debonding of the repair concrete along the inclined faces of the repaired zones (Fig. 13a), which was then followed by the composite debonding. For the reference specimen, failure was governed by the CFRP debonding. Therefore, it is important to correctly model the concrete–composite as well as old-to-new concrete contacts. In the model, all surface contacts are modeled withABAQUS/ Standard, which considers contact between two bodies in terms of two surfaces that may interact. The order in which the two surfaces are specified defines the slave and master surfaces. Two algorithms are available: the small-sliding contact algorithm and the finite-sliding contact algorithm. For small-sliding contact problems the contact area is calculated from the undeformed shape of the model and is thus kept constant throughout the analysis. In this formulation, contact pressures are calculated according to an invariant contact area. The modeled behaviour is different from that in finite-sliding contact problems, where the contact area and contact pressures are calculated according to the deformed shape of the model. For this application the analyses performed clearly indicated that the finite sliding formulation is the most appropriate for reproducing the observed behaviour. In the Interaction module of ABAQUS, the strategy adopted for the concrete–composite contact modeling uses the beam as the master surface and the composite as the the slave surface. Similarly, for the concrete– concrete contact, the old concrete is selected as the master surface and the repair concrete as the slave surface. Contact properties are specified for tangential and normal behaviour. Another important aspect of modeling the damaged beams is related to the definition of the tension-stiffening properties of the repair concrete. For beams MB1O and MC1O, different
concrete properties were defined using eq. [8], which accounts for the presence of strands in both specimens and additional reinforcement for specimen MB1O. The results presented in Fig. 14 illustrate the comparison between the two repair techniques. For the reference beam MA1O, the measured load-carrying capacity was 90 kN, while for the repaired beams MB1O and MC1O, the measured strength was 71 and 50.3 kN, respectively. The tests clearly showed that the repairing technique adopted for specimen MB1O is more efficient compared with that of specimen MC1O, which exhibited premature failure similar to that observed by Folcher et al. (2003). Using symmetry, only half of the beams were modeled. For the three beams, the numerical solutions are in good agreement with the experimental results for predicting the maximum load and failure mechanism. The analytical obtained results showed that the model captured with fidelity the influence of local repair of the damaged zones on the global behaviour of the beams. 4.6. Concrete columns confined with fibre composite sheets The scope of this section is to examine the model performance for predicting the increase in strength and the associated ductility of concrete columns confined with fibre composite sheets. Two test series on circular and square concrete columns strengthened with FRP are analyzed. The first test series deals with circular column specimens tested by Demers and Neale (1994, 1999) and the associated numerically investigation by Deniaud and Neale (2006). The specimens consisted of 300 mm long and 150 mm diameter plain concrete cylinders strengthened with CFRP sheets with an average thickness of 0.34 mm and a tensile strength of 380 N/mm, with a corresponding ultimate strain of 1.6%. The results of the analyses are compared in Fig. 15 to test measurements corresponding to two types of concrete. In all cases the numerical solution agrees well with the experimental data. Based on the comparison of numerical predictions using various concrete models reported by Deniaud and Neale (2006), the results obtained herein are consistent with the experiment for all types of concrete and compare advantageously with the © 2007 NRC Canada
Nour et al. Fig. 15. Load–strain curves response for plain concrete circular columns: (a) 32 MPa concrete; (b) 44 MPa concrete. CFRP, carbon-fibre-reinforced polymer.
353 Fig. 16. Load–strain curves for square reinforced concrete column specimens S1 and S3.
5. Conclusions The scope of this paper was to investigate by finite elements the nonlinear response of concrete structures reinforced with internal and external FRP. A general and portable constitutive concrete model is used. It is implemented as a user-defined subroutine at Gauss integration point level into the generalpurpose finite element software ABAQUS. This paper demonstrated the ability of the proposed concrete model to correctly simulate different kind of composite reinforcements (internal and (or) external), and its reliability was confirmed by means of several tests covering various types of structural elements taken from the literature. The numerical predictions obtained for load–displacement responses and failure mechanisms agreed well with the experimental values and observations. This study is evidence that this concrete model can be used with confidence to simulate and estimate loadcarrying capacity of reinforced concrete structures, reinforced or not with FRPs. The model can be used in conjunction with experimental testing in research environments or as a predictive tool for actual structures.
elastoplastic model predictions recommended by Deniaud and Neale regarding the specimen with three carbon sheets. In the second test series, RC square columns specimens S1 and S3 tested by Pessiki et al. (2001) were selected. Both specimens are 1830 mm high and 457 mm wide, reinforced with eight 22 mm longitudinal steel bars and 9.5 mm steel stirrups spaced at 356 mm. Specimen S1 had no composite and is considered as the control column, whereas specimen S3 was jacketed with three plies of GFRP composite. Each GFRP ply had a tensile strength of 383 N/mm, an elastic modulus of 21.6 kN/mm, and an ultimate strain of 1.9%. Concrete had a compressive strength of fc = 26.4 MPa with εc = 0.0021. Figure 16 illustrates that the model predictions are consistent with the experimental data. The model reproduced correctly the increase in strength caused by the presence the three GFRP plies. Failure in column specimen S3 was initiated by the rupture of the composite in tension; this was also correctly predicted by the model.
Acknowledgements The authors thank the Network of Centres of Excellence ISISCanada and the Natural Sciences and Engineering Research Council of Canada (NSERC) for their financial support.
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