Numerical simulation of squat reinforced concrete wall strengthened by FRP composite material

Abstract

The advanced design rules and the latest known earthquakes, have imposed a strengthening of reinforced concrete structures. Many research works and practical achievements of the application of the external reinforcement by using FRP composite materials have been particularly developed in the recent years. This type of strengthening seems promising for the seismic reinforcement of buildings. Among of the components of structures that could affect the stability of the structure in case of an earthquake is the reinforced concrete walls, which require in many cases a strengthening, especially in case where the diagonal cracks can be developed. The intent of this paper is to present a numerical simulation of squat reinforced concrete wall strengthened by FRP composite material (carbon fiber epoxy). The intent of this study is to perform finite element model to investigate the effects of such reinforcement in the squat reinforced concrete walls. Taking advantage of a commercial finite element package ABAQUS code, three-dimensional numerical simulations were performed, addressing the parameters associated with the squat reinforced concrete walls. An elasto-plastic damage model material is used for concrete, for steel, an elastic-plastic behavior is adopted, and the FRP composite is considered unidirectional and orthotropic. The obtained results in terms of displacements, stresses, damage illustrate clearly the importance of this strengthening strategy.

Keywords

simulation / strengthening / reinforced concrete wall / squat wall / FRP composite material / damage / Abaqus

Introduction

The last known earthquakes have imposed an evolution of seismic design codes in the building [ 1]. For decades the building and structures were dimensioned in terms of capacity, which classified them as resistant structures, but particularly vulnerable [ 2, 3]. In addition, many structural elements do not meet the current requirements, hence they need to be strengthened or rehabilitated. Strengthening or rehabilitation of reinforced concrete structures by external bonding of FRP composite materials is emerging as one of the most promising technique in construction. The Strengthening consists to increase the mechanical performance of structures, while, the rehabilitation consists to offset the loss of the rigidity and the strength. Earthquake resistant structures should be provided with lateral and vertical seismic-resisting force systems, which are able to transfer the forces of the structural elements to the foundations.

Reinforced concrete walls are the main lateral resisting elements in buildings because of their high stiffness in the plane [ 4]. Reinforced concrete walls can be divided into three groups based on their aspect ratio (height divided by length). When the aspect ratio is greater than 2.0, they are called slender reinforced concrete walls; when the aspect ratio is less than 1.0, they are called squat reinforced concrete walls; when the aspect ratio is between 1.0 and 2.0, they are called transition reinforced concrete walls. For slender reinforced concrete walls, the failure is mainly governed by flexure, on the other hand, for squat reinforced concrete walls; the failure is essentially governed by shear. For transition walls, the failure is governed by flexure and shear.

The intent of this present paper is to study the behavior of squat reinforced concrete walls strengthened by carbon epoxy composite. To achieve this objective, ABAQUS/ Explicit software has been used, and seven reinforced concrete squat walls were simulated [ 5]. They were tested under horizontal monotonic loading; this allows to compare and to evaluate the contribution of the external strengthening on the behavior of squat reinforced concrete walls in terms of damage, strength and displacement. The comparison between the obtained results and those of literature [ 6– 9] showed that the proposed simulation can take into account the local and global behaviors that occur during the loading.

Finite element method

Finite element method supplied a new way to study reinforced concrete walls by computer, which can help the researchers to analyze and to complete the experimental results, and to have a better understanding of the experiments. The modeling of the reinforced concrete walls is carried out in three dimensions by taking advantage of a commercial finite element package ABAQUS code. The simulation of the behavior of squat reinforced concrete using Abaqus requires the careful selection of the proper geometry and material modeling parameters because of the variety of idealizations as the complex behavior of concrete. Three dimensions simulations can reproduce accurate results that are as close as possible to the real behavior than the two or one dimension analyses. Before starting the simulation of such problem, it is necessary to think about the choice of the following parameters: the behavior of materials, the applied load, the boundary conditions, the contact behavior, the modeling of the reinforcement, and geometrical nonlinearity. The results and time of calculation can vary considerably by choosing these parameters differently. The following describes the modeling approaches and techniques used in this study.

Model description

To confirm the applicability of the proposed model which are used in this study, the experimental results from one of the reinforced concrete wall specimens tested by Ref. [ 10] is used. The wall was 750 mm length, 750 mm high, 70 mm thickness, and monolithic-ally connected to an upper and lower beam. The upper beam (1150 mm long, 150 mm deep and 200 mm thickness) functioned as both the element through which axial and horizontal loads were applied to the walls and as a cage for anchorage of the vertical bars. The lower beam (1150 mm long, 300 mm deep and 200 mm thick) is used to clamp the specimens down to the laboratory floor [ 10]. The performed simulations lead to highlight the contribution of the strengthening of reinforced concrete walls by FRP composite materials. For this, seven squat reinforced concrete walls are tested under horizontal monotonic loading. The first reinforced concrete wall is the wall tested by (Lefas, SW13); it is considered as witness specimen and noted CM (Fig. 1). The five other models are reinforced with FRP bands of composite materials (carbon fiber and epoxy) stuck on the external surface: a band along the circumference of the web and one other band according to the compression struts (diagonal). The band stuck in the compression struts inclined by 45° from the horizontal plane. The bands of FRP composite have a width of 100mm and a minimal thickness of 1mm (Fig. 2(b)). To investigate the influence of the carbon fiber orientation on the behavior of squat reinforced concrete walls, the carbon fibers of FRP composite material are provided in the direction of 0°,30°,45°,60°90° degrees from the horizontal (Fig. 2(b)). The seventh wall is strengthened by the bonding of the FRP composite material over its entire external surface; the carbon fibers are oriented following the 45-degree direction and the model is noted TSM (total strengthening model) (Fig. 2(a)).

Finite element model

In this study, nonlinear analysis is used using the finite element analysis software ABAQUS. The concrete wall is modeled using linear solid elements (HEX8, ABAQUS elements C3D8/C3D8R, Lagrangian formulation). In the ABAQUS software, there are two possible ways to model reinforcement bars in 3D reinforced concrete elements. Reinforcement bars can be modeled either as an embedded rebar layer or as discrete elements. The first method is better for the regular distribution of the reinforcements. However, considering the arrangement of the reinforcements in boundary elements, we can't use this method to model the reinforcements in this study. In the second method, there are two ways to model the discrete reinforcements: the first way is to model the reinforcement like a solid element and to introduce a law contact between concrete element's surfaces and reinforcement element's surfaces, but this approach leads to a very expensive calculation in terms of time because of the large number of mesh elements that can be generated, especially around the voids reserved to reinforcements in the concrete elements. Therefore, it is better to model discrete reinforcements in one dimension by using truss elements (T3D2) or beam elements (B31).

The effects associated with the reinforcement-concrete interface, like bond slip and dowel action, are considered in this study by introducing some “tension stiffening” into the concrete modeling. External FRP composite is modeled using 4-node shell elements with orthotropic behavior. In all simulations a perfect contact is considered between the concrete and the external FRP (Fig. 2(c)). The low beam of the wall is considered as fixed and no translation or rotation can occur, this is, to simulate the boundary conditions applied in the experimental test (Fig. 1(b)). Moreover, in order to simulate the loading conditions on the wall: a vertical pressure load is applied to the top beam to simulate the vertical load, and a lateral load is applied at the left top beam to simulate the lateral loading (Fig. 1(b)).

Material model for concrete

An elasto-plastic damage model is used to describe the nonlinear material properties of concrete [ 5]. It bases on the classical continuum plasticity-damage theory and it assumes that the two main failure mechanisms are the tensile cracking and the compression crushing of the concrete material. The model was developed by Ref. [ 11] and elaborated by Ref. [ 12] and has been implemented in the Abaqus code [ 5] under the name of CONCRETE DAMAGED PLASTICITY MODEL (CDP). This model provides a general capability for the analysis of concrete structures under static or dynamic loading.

The strain rate is decomposed in elastic and plastic parts as shown in following:

(1) \dot{\epsilon }={\dot{\epsilon }}^{el}+{\dot{\epsilon }}^{pl}.

The stress-strain relation is given in a matrix form by

(2) \sigma =(\mathrm{1}-d)D_{\mathrm{0}}^{el}:(\epsilon -{\epsilon }^{pl})=D^{el}:(\epsilon -{\epsilon }^{pl}).

In Eq. (2), D₀^el represents elastic stiffness matrices, d represents the scalar damage variance and D^el represents the degradation stiffness matrix.

The effective stress is given by the following equations:

(3) \tilde{\sigma }=D_{\mathrm{0}}^{el}(\epsilon -{\epsilon }^{pl}),

(4) \tilde{\sigma }={\displaystyle \frac{\sigma }{(\mathrm{1}-d)}}.

The CDP model uses the loading function f proposed by Ref. [ 11] with the modifications proposed by Ref. [ 12] to take into account the different evolution of strength under tension and compression (Figs. 11 and 12). This function is given by the following equation:

(5)

Where, parameter a depends on the ratio of the biaxial and uniaxial compressive strengths (s_b0/ s_c0), \gamma is a parameter depends on K_c which is the ratio of the second stress invariant on the tensile meridian to that on the compressive meridian and its value must be 0.5<K_c≤1.0 (Fig. 3). The function \beta is a function of the plastic variables hardening (Eq. (8)) with {\sigma }_t , {\sigma }_c are the uniaxial stress in tensile and compression, respectively, defined by the users, \bar{p} is the hydrostatic pressure stress and \bar{q} is the Von Mises equivalent stress.

(6) {\tilde{\epsilon }}^{pl}=\left(\begin{array}{l}{\tilde{\epsilon }}_t^{pl}\\ {\tilde{\epsilon }}_c^{pl}\end{array}\right),

(7) a={\displaystyle \frac{{\sigma }_{b\mathrm{0}-}{\sigma }_{c\mathrm{0}}}{\mathrm{2}{\sigma }_{b\mathrm{0}-}{\sigma }_{c\mathrm{0}}}},

(8) \beta \left({\tilde{\epsilon }}^{pl}\right)={\displaystyle \frac{{\sigma }_c{\tilde{\epsilon }}_c^{pl}}{{\sigma }_t{\tilde{\epsilon }}_t^{pl}}}(\mathrm{1}-a)-(\mathrm{1}+a),

(9) r={\displaystyle \frac{\mathrm{3}(\mathrm{1}-K_c)}{\mathrm{2}{K}_c-\mathrm{1}}}.

This model assumes non-associated potential plastic flow. The flow potential G used for this model is the modified Drucker-Prager model (hyperbolic function) and given by the following equation:

(10) G=\sqrt{q^{\mathrm{2}}+{(\epsilon {\sigma }_{t\mathrm{0}}\tan \psi )}^{\mathrm{2}}}-p\tan \psi ,

where, {\sigma }_{t\mathrm{0}} is the uniaxial tensile stress at failure, \psi is the angle of dilatation measured in the p-q plane the hydrostatic pressure stress, \epsilon is an eccentricity of the plastic potential surface, \bar{p} and \bar{q} is the hydrostatic pressure stress and Von Mises equivalent stress, respectively, as mentioned before.

In addition, a visco-plastic regularization technique has been added to the CDP Model to permit stresses to be outside of the yield surface by using a viscosity parameter m.

The damage is caused by cracking or crushing under tensile or compressive loading conditions. Thus, tensile d_t as well as compressive d_c parts constitute the total damage d.

(11) (\mathrm{1}-d)=(\mathrm{1}-s_t{d}_c)(\mathrm{1}-s_c{d}_t),

where, s_t and s_c are function of stress state introduced to represent the restitustion of the stiffness.

(12) s_t=\mathrm{1}-{\omega }_t{r}^{\ast }\left({\bar{\sigma }}_{\mathrm{11}}\right),\mathrm{0}\le {\omega }_t\le \mathrm{1},

(13) s_c=\mathrm{1}-{\omega }_c{r}^{\ast }\left({\bar{\sigma }}_{\mathrm{11}}\right),\mathrm{0}\le {\omega }_c\le \mathrm{1},

Where,

(14)

The weight factors w_t and w_c control the stiffness recovery in tension and compression during the cyclic loading. Figure 5 shows the compression stiffness restitution effect of w_c parameter and Fig. 6 shows the full stiffness recovery.

To define the CDP model in Abaqus, the following parameters should be described:

The stress-strain curves in compression, in tension and the damage-strain curves, to define the uniaxial behavior.

For multi-axial behavior the parameters that should be defined are: he ratio of initial equibiaxial compressive yield stress to initial uniaxial compressive yield stress, the ratio of the second deviatoric stress invariant on the tensile meridian to that on the compressive meridian, the eccentricity and the dilation angle parameters for the flow potential.

The relation between the uniaxial stress, dented by {\sigma }_x and the corresponding scalar plastic strain, denoted by {\epsilon }_x^p , is assumed as Ref. [ 12].

(15) {\sigma }_x=f_{x\mathrm{0}}[(\mathrm{1}+a_x)Exp(-b_x{\epsilon }_x^p)-a_x\exp (-\mathrm{2}{b}_x{\epsilon }_x^p)],

where f_x0 is the initial yield stress, defined as the maximum stress without damage, a_x and b_x are two parameters defined to reproduce the experimental curve shape. The localization of deformations, especially in the tensile behavior, leads significant convergence problems of the finite element model. To solve partially the problems and to keep objectivity regarding the size of the mesh, Hillerborg method [ 13] (was chosen as regularization method. In this approach, the evolution of stress-strain curve depends on the size of the finite element. The dissipated energy at cracking is then kept constant when the size of the element changes by refining. The energy density of cracking g_fx is related to the cracking energy G_fx by the following Eqs. (13) and (14):

(16) g_{fx}={\displaystyle \frac{G_{fx}}{l_c}}.

And the relation between the scalar damage and the corresponding scalar plastic is defined by the following equation:

(17) \mathrm{1}-D_t=\exp (-c_t{k}_t).

The experimental value of compressive concrete strength, the crackling energy and the element size of the adopted mesh are used to generate the uniaxial response of concrete by using the equations above. The parameters define the multi-axial response cannot be defined experimentally, for this the default value given by Abaqus is used. The Table 1 presents the calculated values using to generate the uniaxial response and the values adapted to the multi-axial response.

The Figs. 7 through 10 represent the uniaxial curves injected in the Abaqus code to take into account the nonlinear behavior of concrete

Material model for reinforcement

The constitutive behavior of reinforcements are modeled using an elastic plastic model. The parameters used to define this model are: Young modulus (E), yield stress (s_y), ultimate stress (s_u) and the Poisson’s ratio (u). The values of these parameters are presented in Table 2.

Material model for FRP composite material

FRP composites are materials that consist of the two constituents. The constituents are combined at a macroscopic level and are not soluble in each other. One constituent is the reinforcement fiber, which is embedded in the second constituent, a continuous polymer called the matrix [ 15]. The reinforcing fibers, which are typically stiffer and stronger than the matrix, take up to 70% of the compound volume. The FRP composites are anisotropic materials; that's mean that, their properties are not the same in all directions. Figure 10 shows a schematic of FRP composites. As shown in Fig. 11, the unidirectional lamina has three mutually orthogonal planes of material properties (12, 13, and 23 planes). The 123 coordinate axes are referred to as the principal material coordinates, where the 1 direction is in the same direction of the fibers, and the 2 and 3 directions are perpendicular to the 1 direction. Therefore, it is considered as an orthotropic material. This isotropic material is also transversely isotropic, where the properties of the FRP composites are nearly the same in any direction perpendicular to the fibers. Thus, the properties in the 2 direction are the same as those in the 3 direction. So, this material is so-called an especially orthotropic material (E1= E2, u23= u32). The stress-strain relationship for a unidirectional composite can be written as follows:

(18) \left(\begin{array}{l}{\epsilon }_{\mathrm{1}}\\ {\epsilon }_{\mathrm{2}}\\ {\epsilon }_{\mathrm{3}}\\ {\gamma }_{\mathrm{23}}\\ {\gamma }_{\mathrm{13}}\\ {\gamma }_{\mathrm{12}}\end{array}\right)=\left[\begin{array}{llllll}\mathrm{1}/E_{\mathrm{1}}& -{\nu }_{\mathrm{21}}/E_{\mathrm{2}}& -{\nu }_{\mathrm{31}}/E_{\mathrm{3}}& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ -{\nu }_{\mathrm{12}}/E_{\mathrm{1}}& \mathrm{1}/E_{\mathrm{2}}& -{\nu }_{\mathrm{32}}/E_{\mathrm{3}}& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ -{\nu }_{\mathrm{13}}/E_{\mathrm{1}}& -{\nu }_{\mathrm{23}}/E_{\mathrm{2}}& \mathrm{1}/E_{\mathrm{3}}& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{1}/G_{\mathrm{23}}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{1}/G_{\mathrm{13}}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{1}/G_{\mathrm{12}}\end{array}\right]•\left(\begin{array}{l}{\sigma }_{\mathrm{1}}\\ {\sigma }_{\mathrm{2}}\\ {\sigma }_{\mathrm{3}}\\ {\tau }_{\mathrm{23}}\\ {\tau }_{\mathrm{13}}\\ {\tau }_{\mathrm{12}}\end{array}\right).

The input data needed to model the FRP composite material in Abaqus are: The overall thickness of the plate, the orientation of the fiber direction in the plate, elastic modulus of the FRP composite in three directions (E1, E2 and E3), shear modulus of the FRP composite in three planes (G 12, G 23 and G 13), major Poisson's ratio in three planes (u12, u23 and u13). Assuming the 1 direction is the same as the fiber direction, while the 2 and 3 directions are perpendicular to the 1 direction. We can find those input data by using the following mixing law:

(19) E_{\mathrm{1}}=V_f{E}_f+E_m(\mathrm{1}-V_f),

(20) {\nu }_{\mathrm{12}}={\nu }_f{V}_f+{\nu }_m(\mathrm{1}-V_f),

(21) E_{\mathrm{2}}={\displaystyle \frac{E_m{E}_f}{V_f{E}_m+(\mathrm{1}-V_f)E_f}},

(22) {\nu }_{\mathrm{12}}=E_{\mathrm{2}}\left[V_f{\displaystyle \frac{{\nu }_f}{E_f}}+(\mathrm{1}-{\nu }_f){\displaystyle \frac{{\nu }_m}{E_m}}\right],

(23) G_{\mathrm{12}}={\displaystyle \frac{G_m{G}_f}{V_f{G}_m+(\mathrm{1}-V_f)G_f}},

(24) G_{\mathrm{23}}={\displaystyle \frac{E_{\mathrm{23}}}{\mathrm{2}(\mathrm{1}+{\nu }_{\mathrm{23}})}},

(25) G_f={\displaystyle \frac{E_f}{\mathrm{2}(\mathrm{1}+{\nu }_f)}},

(26) G_f={\displaystyle \frac{E_m}{\mathrm{2}(\mathrm{1}+{\nu }_m)}}.

The failure of the FRP composite material is defined by specifying a Tsai-Hill failure theory [ 16]. The input data require for this failure is: the tensile and compressive stress limits, t1 and c1, in the 1 direction; the tension and compressive stress limits, t₂ and c₂, in the 2 direction; and shear strength (maximum shear stress) in the 1, 2 plane.

Verification of the finite element model

The validity of the proposed material constitutive models for steel, concrete and FRP were verified by testing against experimental data. The results of the verification study, Figs. 12 and 13, demonstrate that the numerical model fitted with acceptable accuracy the experimental results of the reference wall. Moreover, damage patterns in the FE model fitted with the experimental results of the reference wall. It is worthwhile to mention that the concrete damaged plasticity model does not have the notion of cracks developing at the material integration point. However, in order to show the cracking patterns, it’s possible to show the maximum principal plastic strain in finite element analysis because it can be assumed that the cracks initiate at points where the tensile equivalent plastic strain is greater than zero.

Results and discussion

The classical model squat RC wall (CM) is strongly damaged; whether in its base or overall in its web. We note a failure by sliding shear at the bottom, characterized by the horizontal crack at the base of the wall (Fig. 14 CM). This failure is obtained by progressive buckling of the vertical reinforcements under the action of bending and shearing. The web of a wall becomes deformed in rhombus and allow a failure by shear action, and it is illustrated by the crack of the concrete in a diagonal direction (the diagonals “red” bands. Figure 14 CM) and by buckling of the reinforcements along these cracks.

The stacked FRP bands on the wall allow the elimination of the slip failure occurred at the base in the classical model, and reduced considerably the damage on the web of the wall. All the lateral force in the classical model (CM) is transferred through the concrete by compressive, tensile struts, aggregate interlock and by dowel action. The web crushing failure occurs when the compressive stress exceeds the average compressive strength in the concrete strut. The FRP composite leads to transfer some part of the force. As a result, the force carried by the compressive and tensile struts is reduced.

In the five models (SM0°, SM30°, SM45°, SM60°, SM90°) strengthened with FRP composite, we found that the model with the orientation of the carbon fibers at 45°(SM45°) gives the best result compared to other models (SM0°, SM30°, SM45°, SM60°, SM90°), this means that 45° is the most efficient angle for a direct transfer of some part of the stresses. The TSM model (total strengthening model), where the FRP composite material is distributed over all the surface of the wall and the carbon fibers are oriented in 45° direction relative to the horizontal, shows that the behavior of the wall is strongly modified, the diagonal tension cracks and the crushing of concrete struts are not developed on the web of the wall for the same loading, this means that the FRP composite bands have confined all the concrete of the wall.

The load-displacement curves of Fig. 16 confirm that: the FRP reinforced concrete squat wall models have more powerful behavior than the RC squat wall without FRP composite. A gain of 20% to 70% of load is noted.

Conclusion

The finite element model of the seven squat RC walls in this study can provide a wide range of information that can be useful for studying the behavior of squat reinforced concrete walls. The Finite element model in this study has supplied a new way of studying reinforced concrete walls by computer, which will help the researchers to get a better understanding of the behavior of walls.

During this numerical simulation, we highlighted the effects of the external reinforcement by FRP composite on squat reinforced concrete walls. The use of the reinforcement of FRP composite proves effective in many aspects (load-displacement, tensile damage...). The evaluation of the damages can be used to highlight when the contribution of the FRP composite is interesting beyond of a certain critical loading allowing an important cracking of the structure. From this critical loading and on the scale of the element, the FRP composite controls the damage of the structure and modify the mechanisms of cracking by ensuring a redistribution of the efforts in the elements. The global results for the squat RC wall is the increasing of the capacity of displacement, the reduction of damages in a meaningful way.

The comparison of the six strengthening models with different orientation angle of carbon fibers shows that 45° is the most performing angle to make work optimally the FRP composite. This implies that the 45° angle is the nearest direction of the principal developed stresses.

It should be noted that 45° is the most performing angle, only in the case of the geometry considered in this study, where the aspect ratio (height divided by length) is equal to 1.

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