1 Introduction

Predicting load-deformation behavior, cracking pattern, strength, and stiffness of shear-critical reinforced concrete (RC) members has been the focus of a lot of research over the last century [1, 2]. However, complex nature of diagonal tension accompanied by the formation of new cracks as well as closing and propagating preexisting cracks has deterred researchers to achieve an analytical and mathematical solution.

A part of the literature in this field contains the studies aimed to obtain cracking shear strength or ultimate shear strength of RC structural elements regardless their complete load-deformation response. These studies are usually based upon simplified physical models—considering classical mechanics of material [3–11] or fracture mechanics [12–16] and experimental data fitting. Therefore, each of these models is applicable for a certain range of specifications that they crystalized and calibrated for. Another part of the literature contains the studies aimed to achieve full load-deformation response of reinforced concrete in a general 2D or 3D loading condition. To achieve such response, one needs to consider constitutive, equilibrium, and compatibility equations, simultaneously. These studies can be divided into smeared crack and discrete crack approaches. The former category models the cracks by applying an equivalent theory of continuum mechanics and assumes the cracks as rotating or fixed [17–31]. The difference of these procedures is mainly due to, first, the different constitutive laws they adopted, and second, the numerical approach that they used to solve the aforementioned equations. The latter category is based on fracture mechanics principles and models the cracks as geometric discontinuities [32–36].

This paper reports the second part of a study aiming to capture the full load-deformation response of RC beams. In the first part, a set of constitutive laws for normal strength concrete and embedded steel bars was developed to predict the response of cracked reinforced concrete subjected to in-plane stresses [28]. An interesting property of these stress–strain relations is to account the interactive effects of concrete and embedded bars on each other. In this model, the amount of reinforcement ratio is supposed to affect average stress–strain relation of cracked concrete in tension and compression. On the other hand, the model accounts for the effect of concrete on the average behavior of reinforcing bars. This model is categorized as an orthotropic smeared rotating crack model. Orthotropic models present uniaxial stress–strain relations in the axes of orthotropy.

The aim of this paper is to implement the aforementioned constitutive laws into finite element method, FEM, and to validate the model with experimental data. The numerical algorithm used here is based on the procedure proposed by Vecchio [19] and it is an iterative, secant-stiffness formulation. Simplicity of implementing due to applying some modifications on linear FEM and possibility of using low-order elements is two salient characteristics of this approach.

To assess the robustness of the proposed method, it is implemented on several reinforced concrete shear-critical beams. The results are corroborated with the experimental data. In addition, for the sake of comparison, the results of modified compression field theory, MCFT, [18, 19] and its extension disturbed stress field model, DSFM, [20–22] are included in the paper. It is observed from the results that the proposed approach provides load-deformation behavior, cracking pattern, and failure mode prediction with an excellent agreement with the experimental data; furthermore, it has more accuracy than MCFT and DSFM. More interestingly, in beams containing no transverse reinforcement and subjected to a high shear span-to-depth ratio in which MCFT, DSFM, and other prevailing methods show a considerable deficiency, the present study provides good predictions.

2 Materials and Methods

2.1 Constitutive Laws

The basic information needed for analyzing every structure is stress–strain relations under various conditions of loading, and reinforced concrete structures are not exceptional. In this paper, the stress–strain relations proposed in a recent research [28] are utilized for nonlinear finite element modeling of reinforced concrete beams. This approach is based on smeared crack concept, and it is an orthotropic model which considers different uniaxial stress–strain relations in the axes of orthotropy. In what follows, a detailed description of compressive and tensile stress–strain relation is presented.

2.1.1 Average Stress–Strain Relationship of Steel

Since it is assumed in many studies that reinforcing bars can only transmit axial forces, a bilinear uniaxial stress–strain relationship, as shown in Fig. 1a, is adopted to model the behavior of embedded reinforcing steels in the concrete. However, the behavior of an embedded bar in concrete is different from a bare one. When a crack initiates, the concrete fails to carry tensile stresses at the location of the crack; therefore, the tensile stress of reinforcing steel is more than its value in midway between two cracks where concrete resists moderate amount of tension. Hence, steel yielding initiates from a section in which the concrete is cracked, and then it propagates along its length by increasing the amount of loads. As a result, after the occurrence of the first yielding in steel, stiffness reduces gradually in the average stress–strain of the embedded steel in concrete until the whole bar yields. It has been showed that if a stress–strain curve of the bare steel reinforcement is used for analyzing behavior of reinforced concrete, the results will be considerably overestimated [24].

Fig. 1
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a Bilinear behavior of steel bars; b proposed trilinear behavior of steel bars

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In this research, a trilinear stress–strain relationship, as shown in Fig. 1b, is utilized to model the reinforcing bars. The first line of this stress–strain curve has the slope of E s, which is steel modulus of elasticity, up to a critical point corresponding to the initial yielding of steel. Then, it continues with a reduced slope until the whole reinforcement yields at the average stress of f yield which is the yield stress of a bare steel bar. Then, the stress remains constant until failure. The strains corresponding to the initial yielding and yielding of the whole steel are equal to 0.8ε yield and 4ε yield, respectively, where ε yield is the yield strain of the bare steel bars.

2.1.2 Average Compressive Stress–Strain Relationship of Concrete

It is now evident that concrete in compression subjected to transverse tensile strain has less strength and stiffness than a uniaxially compressed concrete. This phenomenon is called compression softening, and it is quantified by incorporating softening coefficients into stress–strain curve of concrete [18, 37, 38]. A second-degree parabola, such as Hognestad curve [39], is widely used function for ascending portion of the basic compressive stress–strain curve which is used in this study. The descending part of the curve is also a parabola limited by ε f, which is the strain when stress falls to zero (Fig. 2). Based on Kent and Park research [40], the following relation for this strain is proposed [28]:

Fig. 2
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Proposed compressive average stress–strain model for concrete

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$$\varepsilon_{f} = \frac{{\left( {f_{c}^{'} + 7} \right)\varepsilon_{c}^{'} - 0.042}}{{f_{c}^{'} - 7}}$$
(1)

where \(f_{\text{c}}^{{\prime }}\) is uniaxial compressive strength of the concrete in MPa, and \(\varepsilon_{\text{c}}^{{\prime }}\), which is a negative quantity, is the strain of the concrete cylinder at the point corresponding to \(f_{\text{c}}^{{\prime }}\). If psi unit is used for stresses, 7 and 0.042 should be replaced by 1000 and 6, respectively. The suggested curve for falling branch reflects the phenomenon of low strength concrete has low-slope descending portion. As shown in Fig. 2, the last part of the compression curve is assumed to be a plateau at \(0.2f_{\text{c}}^{{\prime }}\).

It should be noted that the employed model is calibrated for reinforced concrete specimens 890 × 890 mm2 square with 70 mm thickness (35 inch × 35 inch × 2.75 inch) [28]. To use the constitutive laws for elements with different sizes, a localization limiter should be employed [33]. There are several kinds of localization limiters, and the simplest method which is employed in this study is crack band method, CBM [34]. CBM proposes a correlation between the element size and the constitutive laws in a way that the total fracture energy of concrete G F remains constant. This is performed by adjusting the value of ε f with the element size. For this purpose, the value of ε f should be decreased by increasing the size of element. In Fig. 3, the shaded area is equal to G f/h 0, where h 0 is the size of element (in this study 890 mm or 35 inch). Thus, for a square element with different sizes of h, the value of ε f is modified as follows, and it is denoted by \(\bar{\varepsilon }_{f}\):

Fig. 3
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Fracture energy of compressive concrete

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$$\bar{\varepsilon }_{f} = \frac{{h_{0} }}{h}\left( {\varepsilon_{f} - \varepsilon_{1} } \right) + \varepsilon_{1}$$
(2)

where ε 1 is the plastic strain corresponding to the peak point of stress–strain relation, as shown in Fig. 3.

The previous study [28] showed that maximum attainable compressive stress in concrete and its corresponding strain increase by increasing the reinforcement ratio. To consider this phenomenon in stress–strain curve of concrete, two modification factors, including α and µ, were introduced to adjust the values of \(f_{\text{c}}^{{\prime }}\) and \(\varepsilon_{\text{c}}^{{\prime }}\) and denoted by \(f_{c}^{{\prime \prime }}\) and \(\varepsilon_{c}^{{\prime \prime }}\), respectively:

$$f_{c}^{{\prime \prime }} = \alpha f_{c}^{{\prime }} , \alpha = 1 + 0.03\left( {100\rho_{x} } \right)^{2} \left( {100\rho_{y} } \right)^{2}$$
(3)
$$\varepsilon_{c}^{{\prime \prime }} = \mu \varepsilon_{c}^{{\prime }} , \mu = 1 + 0.04\left( {100\rho_{x} } \right)^{2} \left( {100\rho_{y} } \right)^{2}$$
(4)

where ρ x and ρ y are reinforcement ratios of the orthogonally reinforced concrete panel in x- and y-directions, respectively.

To describe the softening effect of the basic curve which was defined above, a factor proposed by Vecchio and Collins [18] in the following form is applied to the curve to modify the maximum attainable compressive stress:

$$\beta = \frac{1}{{0.8 - 0.34\varepsilon_{c1} /\varepsilon_{c}^{{\prime }} }} \le 1$$
(5)

in which ε c1 is the principal tensile strain of concrete. As a result, the compression curve of reinforced concrete panels will be obtained in the form of Fig. 2 and formulated as follows:

$$f_{c2} = \beta f_{c}^{{\prime \prime }} \left[ {\left( {\frac{{\varepsilon_{c2} - \varepsilon_{c}^{{\prime \prime }} }}{{\varepsilon_{c}^{{\prime \prime }} }}} \right)^{2} - 1} \right], \frac{{\varepsilon_{c2} }}{{\varepsilon_{c}^{{\prime \prime }} }} \le 1$$
(6a)
$$f_{c2} = \beta f_{c}^{{\prime \prime }} \left[ {\left( {\frac{{\varepsilon_{c2} - \varepsilon_{c}^{{\prime \prime }} }}{{\varepsilon_{f}^{{\prime \prime }} - \varepsilon_{c}^{{\prime \prime }} }}} \right)^{2} - 1} \right], \frac{{\varepsilon_{c2} }}{{\varepsilon_{c}^{{\prime \prime }} }} > 1$$
(6b)

where f c2 and ε c2 represent the average principal compressive stress and strain in cracked concrete, respectively, and ε c1 is the coexisting tensile strain.

2.2 Average Tensile Stress–Strain Relationship of Concrete

Neglecting concrete tensile stiffness can result in significant overestimating of the post-cracking deformation in reinforced concrete structures [18]. An experimental investigation to examine the cracking behavior of reinforced concrete panels conducted by Wollrab et al. [41]. According to their work, it can be concluded that reinforcement spacing does not have a significant impact on the post-cracking contribution of concrete, while increasing the reinforcement ratio has a clear influence, and it makes the post-peak branch be steeper; as a result, the average contribution of concrete decreases. In addition, the result of their experiment showed that there are three distinct branches in tensile stress–strain curve of concrete: (1) a linearly ascending branch of uncracked concrete; (2) a crack formation phase; (3) the descending branch with stable crack pattern. According to this conceptual model, the average tensile stress–strain relation of cracked concrete is depicted in Fig. 4 and branches of this curve are formulated as follows:

$$f_{c1} = E_{c} \varepsilon_{c1 } \quad \varepsilon_{c1} < \varepsilon_{cr}$$
(7a)
$$f_{c1} = f_{cr} \quad \varepsilon_{cr} < \varepsilon_{c1} < \varepsilon_{cr}^{'}$$
(7b)
$$f_{c1} = \frac{{f_{cr} }}{{1 + \sqrt {k\varepsilon_{c1} } - \sqrt {k\varepsilon_{cr}^{'} } }} \quad \varepsilon_{cr}^{'} < \varepsilon_{c1}$$
(7c)

where f c1 represents the average tensile stress in cracked concrete; E c is the elastic modulus of concrete in tension which can be taken as \(2f_{\text{c}}^{{\prime }} /\varepsilon_{\text{c}}^{{\prime }}\); f cr is the average tensile stress of concrete in the crack formation phase; and k is defined as follows:

$$k = 300 + 250\left( {100\rho_{x} } \right)\left( {100\rho_{y} } \right).$$
(8)

The previous research [28] showed that \(f_{c}^{{\prime }}\) and reinforcement ratio are two parameters influencing the value of f cr and the following relation in MPa was proposed:

$$f_{cr} = 0.3\alpha^{'} \sqrt {f_{c}^{'} + 8}$$
(9)

where α′ is interestingly equal to α as defined in Eq. (3). For psi units, 0.3 and 8 should be replaced by 3.6 and 1150, respectively. In Eq. (7a, 7b, 7c), ε cr is the average tensile strain at which concrete initiates cracking, and \(\varepsilon_{\text{c}}^{{\prime }}\) is the strain corresponding to the end of crack formation phase. The experimental database illustrates that \(\varepsilon_{\text{c}}^{{\prime }}\) has a direct relationship with the reinforcement ratio, and the following correlation is proposed for this parameter [27]:

$$\varepsilon_{cr}^{'} = \eta \varepsilon_{cr} , \eta = 1 + 6\left( {100\rho_{x} } \right)\left( {100\rho_{y} } \right).$$
(10)
Fig. 4
figure 4

Proposed tensile average stress–strain model for concrete

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2.3 Nonlinear Analysis Procedure

In this study, it is aimed to apply the above-mentioned constitutive laws, obtained from the previous research based on some available test results of RC panels [28], for the analysis of RC beams. Therefore, each element in the FEM mesh is considered to be a membrane element, as shown in Fig. 5. The reinforcement and forces on each element in x- and y-directions are delineated in Fig. 5. To construct global and local stiffness matrices of elements and the whole structure, two other sets of equations are needed: (1) compatibility equations and (2) equilibrium equations. Assuming that steel bars carry no shear stresses, the equilibrium equations can be written as follows:

Fig. 5
figure 5

FEM model of RC elements and its forces

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$$f_{x} = f_{cx} + \rho_{x} f_{sx}$$
(11a)
$$f_{y} = f_{cy} + \rho_{y} f_{sy}$$
(11b)

where f x and f y are longitudinal and transverse stresses applied in the x- and y-directions, respectively; f cx and f cy are the corresponding stresses in concrete; and f sx and f sy are the corresponding values for reinforcement.

Assuming that there is no slip between reinforcement and concrete, the compatibility leads to the same longitudinal and transversal average strain for both steel and concrete. Another assumption considered in this model is that the principal stresses and strains have the same axes. The described constitutive laws along with equilibrium and compatibility provide a set of nonlinear equation which should be solved numerically.

2.4 Finite Element Procedure

A reinforced concrete structure, such as a beam or shear wall, under a given loading can be considered as an assemblage of some membrane elements. Thus, simultaneous solving the equilibrium, compatibility, and nonlinear constitutive laws governing all the elements yields the load-deformation response of the whole structure. Fortunately, with the current advances in the finite element methods, this task is feasible easily in an approximate manner with a desirable accuracy. An efficient FEM algorithm for the nonlinear analysis of reinforced concrete structures is suggested by Vecchio [16] which is briefly explained as follows. Based on this procedure, some modifications are made in linear elastic FEM to incorporate nonlinear constitutive laws. Therefore, the stress–strain relationships presented previously in this study can be utilized in examining nonlinear behavior of reinforced concrete structures.

In the finite element method, material stiffness matrix [D] is employed to relate the stresses to strains:

$$\left\{ f \right\} = \left[ D \right]\left\{ \varepsilon \right\}$$
(12)

where \(\{ f\} = \left\langle {f_{\text{x}} \left| {f_{\text{y}} } \right|v_{\text{xy}} } \right\rangle\) represents the stresses vector and \(\{ \varepsilon \} = \left\langle {\varepsilon_{\text{x}} |\varepsilon_{\text{y}} |\gamma_{\text{xy}} } \right\rangle\) is the strain vector according to plain stress theory, in which f x and f y are normal stresses in x- and y-directions, respectively, v xy shear stress, and ε x, ε y, and γ xy are the corresponding strains.

The material stiffness matrix is defined by combining component stiffness matrices using appropriate transformation to incorporate directional dependence of materials. According to the material model, cracked concrete element is considered as an orthotropic material with its principal axes corresponding to the direction of principal compressive and tensile strains. Moreover, the effect of Poisson’s ratio can be neglected after cracking; therefore, the concrete material stiffness matrix with respect to principal axes 1 and 2 can be stated in terms of \(\bar{E}_{c1} = f_{c1} /\varepsilon_{c1}\), \(\bar{E}_{c2} = f_{c2} /\varepsilon_{c2}\), and \(\bar{G}_{c} = \bar{E}_{c1} \bar{E}_{c2} /(\bar{E}_{c1} + \bar{E}_{c2} )\) which are secant moduli of cracked concrete. For each reinforcing bar, the material stiffness matrix is defined in terms of \(\bar{E}_{sx} = f_{sx} /\varepsilon_{x}\) and \(\bar{E}_{sy} = f_{sy} /\varepsilon_{y}\) which are the secant moduli of reinforcement in x- and y-directions.

Having determined the material stiffness matrix [D], the stiffness matrix of element [k] can be evaluated as follows:

$$\left[ k \right] = \mathop \int \nolimits \left[ B \right]^{T} \left[ D \right]\left[ B \right]{\text{d}}V.$$
(13)

Now, the steps of a nonlinear analysis are as follows. At first, topological properties of the structure (e.g., node coordinates, element indices, support conditions, etc.) and material properties (e.g., concrete and steel stiffness matrices, reinforcement orientation, etc.) are determined. Next, nodal loads and distributed loads are input and form a nodal force vector {R}. In the next step, secant material stiffness of each material is calculated (i.e., \(\bar{E}_{c1}\), \(\bar{E}_{c2}\), \(\bar{E}_{sx}\), and \(\bar{E}_{sy}\)), and [D] is computed. The element stiffness matrices [k] are calculated and assembled in the global stiffness matrix of the structure [K]. Then, the structure stiffness matrix is inverted, and joint displacements {r} are found as follows:

$$\left\{ r \right\} = \left[ K \right]^{ - 1} \left\{ R \right\}.$$
(14)

Using the joint displacement, the element strains and stresses can be determined as follows:

$$\left\{ \varepsilon \right\} = \left[ B \right]\left\{ r \right\}$$
(15)
$$\left\{ f \right\} = \left[ D \right]\left\{ \varepsilon \right\}.$$
(16)

Knowing the strains and stresses of each element, new material stiffness matrices [D] are calculated, and it is used for the next iteration. This procedure is repeated until convergence is achieved.

3 Results and Discussion

To assess the proposed model, the results of some experimental tests on beams are compared with the present model. The test experiments conducted by Bresler and Scordelis [42] and their replicates by Vecchio and Shim [43], including a set of beams with a wide range of span-to-depth ratios and longitudinal and transversal reinforcement ratios which encompass various types of failure, are selected to evaluate the predictions of the proposed model by comparing the results.

3.1 Characteristics of Experimental Specimens

A four series of three beams tested by Bresler and Scordelis [42] named OA, A, B, and C have a different longitudinal reinforcement ratio, transverse reinforcement ratio, length of span, cross sectional size, and concrete compressive (and consequently tensile) resistance. All the beams are simply supported having a single-concentrated load at the mid-span. The span-to-depth ratios vary from 3.3 to 5.8. The ratio of shear reinforcements ranged from 0 to 0.002. Detailed properties of the beams are presented in Table 1. The longitudinal reinforcements of beams are all No. 4 bars, while shear reinforcement in the form of stirrups is No. 2 bars. OA beams have no shear reinforcement. To impose the shear failure mode to the beams, all the beams are constructed with high ratios of longitudinal bars. Geometrical and mechanical properties of concrete, longitudinal bars, and transverse bars are included in Table 2. The maximum aggregate size used in the beams is 20 mm. All the beams were loaded by a monotonic load control mechanism at the mid-span.

Table 1 Cross-sectional properties of Bresler–Scordelis beams
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Table 2 Material properties of Bresler–Scordelis beams
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3.2 Corroborating the Results with Experimental Database

A 2D nonlinear finite element analysis based on the discussed procedure is performed to study the load-deformation behavior and crack propagation of Bresler–Scordelis beams. Because of the symmetry of beams, only a half of beams are modeled. FEM mesh is constructed using four-node quadrilateral elements of 80 mm × 80 mm (3.15 inch × 3.15 inch). Longitudinal and transverse reinforcements are modeled in a smeared manner. Schematic view of beam models along with smeared reinforcement percent for beams without stirrups is presented in Fig. 6 and Table 3, and the ones for beams with stirrups are depicted in Fig. 7 and Table 4. The models are studied under displacement-control monotonic loading by imposing 0.5 mm (0.02 inch) displacement steps on OA and A beams and 1.0 mm (0.04 inch) steps on B and C beams.

Fig. 6
figure 6

FEM model of without stirrup beams of Bresler–Scordelis

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Table 3 Longitudinal reinforcement of no stirrup beams of Bresler–Scordelis in different zones
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Fig. 7
figure 7

FEM model of with stirrup beams of Bresler–Scordelis

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Table 4 Reinforcement of with stirrup beams of Bresler–Scordelis in different zones (top: ρ x %, bottom: ρ y %)
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In Fig. 8, load-deformation curves of the present study are compared with the results of the experimental tests25 and also with the results of MCFT [28]. From this figure, it can be concluded that the proposed method predicts nonlinear behavior of the beams with an acceptable accuracy for both beams with and without stirrups.

Fig. 8
figure 8figure 8

Comparison of load-deformation curves of the present study, MCFT [23], and the tests [43]

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To delineate the fracture mode of the beams, a graphical ability is added to the FEM code which shows the shear-critical cracks in a different color seen darker in Fig. 9. In addition, the Gaussian points at which compressive failure occurs are shown by bold points. According to Fig. 9, the predicted cracking patterns and failure modes of the beams are in a good agreement with the experimental test results, as shown in Fig. 10.

Fig. 9
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Predicted crack pattern of the Bresler–Scordelis beams

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Fig. 10
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Cracking pattern and failure mode for experimental tests [43] (D-T diagonal tension, V-C shear-compression, F–C flexure-compression)

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In Table 5, the ultimate loads, P u , of the present study are compared with those of the experimental tests, MCFT [23] and DSFM [20–22]. DSFM is a newer version of MCFT aiming to reduce the deficiency of MCFT in predicting load-deformation behavior of RC beams with low stirrups or without stirrups. According to Table 5, the ratio of prediction to experimental ultimate loads has a mean of 1.02 and covariance of 5 % for the proposed method, while these values for DSFM approach are 1.15 and 11 %, and for MCFT, they are 1.42 and 55 %, respectively. As seen in Table 5, the strength of beams containing no stirrups is significantly underestimated by MCFT.

Table 5 Ultimate load obtained by the present study, MCFT [23], DSFM [43], and the tests [43]
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In Table 6, computational results for the mid-span displacements at peak load, δ 0, of the present study are compared with those of the experimental tests, MCFT and DSFM. According to this table, the ratio of prediction to experimental ultimate displacements has a mean of 0.98 and covariance of 9 % for the proposed method, while these values for MCFT are 0.97 and 40 % and for DSFM approach are 0.95 and 24 %, respectively.

Table 6 Mid-span deflection at peak load obtained by the present study, MCFT [23], DSFM [43], and the tests [43]
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4 Conclusions

Recently developed interactive constitutive laws for normal strength concrete and reinforcing steel were integrated with a nonlinear finite element procedure to predict the shear behavior of reinforced concrete structures. The constitutive laws were used to obtain the secant stiffness of elements which is needed for nonlinear analysis. Conclusions derived from the present work can be summarized as follows:

  1. 1.

    Interactive effects of concrete and reinforcing bars on the average stress–strain relations of each other are an important phenomenon that must be considered in constitutive laws.

  2. 2.

    The response of shear-critical reinforced concrete beams with or without stirrups can be predicted with a good accuracy using the proposed interactive constitutive laws.

  3. 3.

    The nonlinear finite element formulation based on secant-stiffness approach can provide acceptable results, and low-order elements can be used for the analysis which makes the procedure very efficient and simple.

  4. 4.

    Corroboration with experimental data, including shear-critical beams with a wide range of properties, showed that the model can predict the cracking patterns, shear capacity, load-deformation response, and failure mode with an excellent agreement with reality.

  5. 5.

    The accuracy of response prediction based on the interactive constitutive laws is more than MCFT and DSFM when they are compared with the experimental test results.