Evaluation of Smeared Constitutive Laws for Tensile Concrete to Predict the Cracking of RC Beams under Torsion with Smeared Truss Model

In this study, the generalized softened variable angle truss-model (GSVATM) is used to predict the response of reinforced concrete (RC) beams under torsion at the early loading stages, namely the transition from the uncracked to the cracked stage. Being a 3-dimensional smeared truss model, the GSVATM must incorporate smeared constitutive laws for the materials, namely for the tensile concrete. Different smeared constitutive laws for tensile concrete can be found in the literature, which could lead to different predictions for the torsional response of RC beams at the earlier stages. Hence, the GSVATM is used to check several smeared constitutive laws for tensile concrete proposed in previous studies. The studied parameters are the cracking torque and the corresponding twist. The predictions of these parameters from the GSVATM are compared with the experimental results from several reported tests on RC beams under torsion. From the obtained results and the performed comparative analyses, one of the checked smeared constitutive laws for tensile concrete was found to lead to good predictions for the cracking torque of the RC beams regardless of the cross-section type (plain or hollow). Such a result could be useful to help with choosing the best constitutive laws to be incorporated into the smeared truss models to predict the response of RC beams under torsion.


Introduction
In the second half of the last century, the Space Truss Analogy (STA) was successively refined in order to better predict the response of structural concrete beams under torsion. Nowadays, modern truss-based models can be considered reliable, comprehensive and unified analytical models. They are able to simulate the complex 3-dimensional features of the torsional phenomenon, including the nonlinear behavior and the interaction between the material components of the beam in all loading stages. Models based on the STA constitute the basis models for most codes of practice to establish the design procedures for torsion and still continue to be improved and extended [1,2].
A STA-based model assumes that a reinforced concrete (RC) beam under torsion behaves like a cracked thin tube, where the external torque is resisted through a transversal circulatory shear flow. The tube is modeled with a spatial truss, which includes longitudinal and transverse steel reinforcement under tension interacting with inclined concrete struts under compression. The model satisfies the three Navier's principles of the mechanics of materials, namely, stress equilibrium, strain compatibility and constitutive laws.
Among the STA-based models that have been developed, one of the most commonly used and extended is the Variable-Angle Truss Model (VATM), which was originally proposed by Hsu and Mo in 1985 [3]. This model incorporated for the first time smeared constitutive laws, or smeared stress (σ)-strain (ε) relationships, for both tensile steel reinforcement embedded in concrete and compressive concrete. Such constitutive laws (for concrete in compression and steel reinforcement in tension), different proposals of smeared constitutive laws for tensile concrete can be found in the literature. To the best of the authors' knowledge, no previous study was found with the aim of checking such constitutive laws in smeared truss models, in order to evaluate which features allow the model to give the best predictions for the low loading stages. Usually, researchers working with smeared truss models use their own smeared constitutive laws or choose them based on the proposals from other studies.
In this study, the GSVATM is used to check some proposed smeared constitutive laws for tensile concrete found in the literature. The GSVATM was the chosen model because, as previously stated, it is able to predict the full response of the RC beams under torsion for all loading stages. In addition, this model was proposed by the corresponding author [16] and has also been successfully used in previous studies [2,[17][18][19]21]. The chosen parameters to be studied are the cracking torque and the corresponding twist. The theoretical predictions of such parameters are compared with the experimental results from several reported tests on RC beams under torsion. Only RC beams with rectangular sections are studied because they constitute the current solution used in practice. In addition, the number of reported experimental results in the literature for such beams is much higher than for other typologies such as PC beams or beams with a flanged cross-section.

The Generalized Softened Variable Angle Truss-Model
For the sake of the readers of this article, a brief description of the GSVATM is presented. The GSVATM was initially proposed for RC plain beams under torsion [16]. Recently, the model was extended and unified for RC hollow beams under torsion [18]. Details about the assumptions of the model, the derivation of the equations and the justification of the calculation solution procedure can be found [16,18].
According to the GSVATM, a cracked RC thin beam element under a vertical shear force V, which induces a shear flow q in the cross-section, is modeled with a smeared plain truss analogy, as illustrated in Figure 1. The behavior of the RC thin beam is governed by Equations (1) to (5). The smeared plain truss incorporates inclined concrete struts (with compressive force C) with an angle α to the longitudinal axis, and perpendicular concrete ties (with tensile force T). The corresponding stress fields are denoted by σ c 2 and σ c 1 , respectively. The meanings of the parameters are (see Figure 1): R is the resultant force, β is the angle of R to the force C, γ is the angle of R to the longitudinal axis, t c is the width of the cross-section and d v is the distance between centers of the longitudinal bars.
An equivalent cracked RC hollow beam under a torque M T , as illustrated in Figure 2, is modeled as the union of four thin beam elements as in Figure 1. Each thin beam constitutes a wall of the RC hollow beam. As a result of this union, the torque M T induces a circulatory shear flow q and the beam can be modeled with a smeared spatial truss analogy. The center line of the circulatory shear flow q coincides with the center line of the walls. The behavior of the RC hollow beam is governed by equilibrium equations, Equations (6) to (8), and compatibility equations, Equations (9) to (12). If γ = α + β > 90 • , Equation (7) must be multiplied by (−1). The previous equations account for the strain gradient along the walls' thickness due to the bidirectional opposite curvatures induced by bending ( Figure 3).
An equivalent cracked RC hollow beam under a torque , as illustrated in 2, is modeled as the union of four thin beam elements as in Figure 1. Each thin be stitutes a wall of the RC hollow beam. As a result of this union, the torque in circulatory shear flow and the beam can be modeled with a smeared spatial tru ogy. The center line of the circulatory shear flow coincides with the center lin walls. The behavior of the RC hollow beam is governed by equilibrium equation tions (6) to (8), and compatibility equations, Equations (9) to (12). If γ = α + Equation (7) must be multiplied by (−1). The previous equations account for th gradient along the walls' thickness due to the bidirectional opposite curvatures by bending (Figure 3). An equivalent cracked RC hollow beam under a torque , as illustrated in Figur 2, is modeled as the union of four thin beam elements as in Figure 1. Each thin beam con stitutes a wall of the RC hollow beam. As a result of this union, the torque induces circulatory shear flow and the beam can be modeled with a smeared spatial truss anal ogy. The center line of the circulatory shear flow coincides with the center line of th walls. The behavior of the RC hollow beam is governed by equilibrium equations, Equa tions (6) to (8), and compatibility equations, Equations (9) to (12). If γ = α + β > 90° Equation (7) must be multiplied by (−1). The previous equations account for the strain gradient along the walls' thickness due to the bidirectional opposite curvatures induced by bending ( Figure 3).  Figure 2. RC hollow beam element [18].

•
if t > 0.91t c,cr,plain the RC hollow beam has a "thick wall". Then, the beam is recalculated considering the real cross-section (hollow). For the RC beams under torsion, average stresses σ c 2 (Equation (30)) and σ c 1 (Equation (31)) are computed for the concrete strut and tie, respectively, accounting for the section type through the correction coefficient η (Equations (32) to (35)). This simplification is assumed because the real stress diagrams along the effective wall's thickness t c are not uniform due to the strain gradient ( Figure 3). The coefficients k c 2 and k c 1 are computed from the numerical integration of the smeared σ-ε relationships.  (35) To solve the nonlinear procedure of the GSVATM, an algorithm incorporating a trial-and-error technique was implemented using the programming language Delphi (see flowchart in Figure 4) [16,18]. For each iteration, the input parameter ε c 2s = 2ε c 2 (strain at the outer fiber of the concrete strut) is incremented in order to compute each solution point to draw the theoretical M T -θ curve. The calculation procedure ends when the assumed failure strains for the materials is reached, either for concrete in compression (ε cu ) or for steel reinforcement in tension (ε su ). In this study, European code Eurocode 2 was used to define the conventional failure strains for the materials.

Smeared Constitutive Laws for Tensile Concrete
This section presents eight smeared σ-ε relationships for tensile concrete proposed in previous studies (laws l1 to l8), so that they can be implemented in the GSVATM and checked (Section 4). In a previous study, it was showed that these relationships are suitable to be implemented in smeared truss models, such as the GSVATM, to account for the contribution of the tensile concrete [27].
Some of the presented smeared σ-ε relationships for tensile concrete were proposed based on the experimental results from concrete panels under shear. In such cases, the average stress σ c 1 in the tensile concrete after cracking (ε c 1 > ε cr ) is usually obtained from the equilibrium of the stress fields applied to the panels by separating the average stresses in both the tensile steel reinforcement and the tensile concrete. The other smeared σ-ε relationships for tensile concrete were proposed by refining the previous ones in order to improve the predictions of the used smeared models.
For all presented smeared σ-ε relationships for tensile concrete, two equations are written. The first one aims to model the tensile behavior of the concrete before cracking and is equal for all smeared constitutive laws: The second equation aims to model the tensile behavior of the concrete after cracking, and accounts for the tension softening (the influence of the cracks) and the tension stiffening (the retention of concrete tensile stress due to the interaction with steel reinforcement).  As presented in Section 2, parameters ε cr and E c are computed according to Equations (24) and (25), which apply for all the presented smeared σ-ε relationships. Further, for all presented equations, the symbology was adapted to the same one used in the previous section.

Law l1-Cervenka in 1985
In 1985, Cervenka proposed a smeared model for cracked RC panels. In this model, the author implemented the following equation for the descending branch of the smeared σ-ε relationships for tensile concrete [28]: Parameter c is the average tensile strain (ε c 1 ) for which the principal tensile stress can be considered null. The author observed that c ranges between 0.004 and 0.005. For this study, the average value (0.0045) was considered. The exponent k 2 is related with the curvature shape of the descending branch of the σ-ε curve after the peak tensile stress. Cervenka proposed to consider k 2 = 0.5.

Law l2-Vecchio and Collins in 1986
In 1986, based on several experimental results from RC panels under shear performed at the University of Toronto, Vecchio and Collins proposed the smeared model called Modified Compression Field Theory. For this model, the following postpeak smeared σ-ε relationship for tensile concrete was proposed [29]:

Law l3-Hsu in 1991
In 1991, Hsu [30] proposed an efficient algorithm for his softened truss model theory to analyze the nonlinear behavior of concrete membrane elements. For this model, a refined version of the postpeak smeared σ-ε relationship for tensile concrete from Vecchio and Collins in 1986 [29] was proposed:

Law l4-Belarbi and Hsu in 1994
Based on experimental studies on RC panels under shear performed at the University of Houston, Belarbi and Hsu in 1994 [24] proposed Equation (40) for the descending branch of the smeared constitutive law for tensile concrete.

Law l5-Collins and Colaborators in 1996
In 1996, Collins et al. [31] proposed a postpeak smeared constitutive law for tensile concrete slightly different from the one proposed by Vecchio and Collins in 1986 [29]:

Law l6-Vecchio in 2000
The Disturbed Stress Field Model for RC was proposed by Vecchio in 2000 [32]. For this model, the author proposed a somewhat more complicated postpeak smeared constitutive law for tensile concrete, in order to account more precisely for the tension stiffening. The author proposed two equations, with a maximum condition, to also account indirectly for the level of reinforcement ratio (Equations (42) to (45)). When a low (high) reinforcement ratio exists, tension softening (stiffening) is more relevant.
Parameter ε ts represents the terminal strain, which depends on the fracture energy (G f ), assumed to be constant and equal to 75 N/m by Vecchio, and also on half of the distance between cracks (L r ). Parameter c t can be simply considered equal to 200 for small members or for members incorporating steel reinforcement grids with very small spacing, and 500 for large members. For this study, L r was infered from the experimental data of the used reference beams (Section 4).
Parameter M (in "mm" units) accounts for the effective tensile concrete area around the rebars (A c ) and for the rebars' diameter (φ). For this study, A c was computed considering the effective thickness of the concrete tie (t c ), which is computed from the GSVATM.

Law l8-Stramandinoli and Rovere in 2008
In 2008, for the nonlinear analysis of reinforced concrete members, Stramandinoli and Rovere proposed equations for the postpeak smeared constitutive law for tensile concrete [34] (Equations (48) to (50)). The law accounts directly for the longitudinal reinforcement ratio ρ.

Comparison between the Smeared Constitutive Laws
For comparison, Figure 5 illustrates the smeared σ-ε curves for tensile concrete for each of the proposals presented in the previous subsections. The curves were computed considering the same arbitrary and typical cross-section with current materials.
presented. The highlighted point in the curves (with marker ""•) corresponds to the effective cracking torque, which is reached for a strain 1 c cr ε ε > , i.e., in the descending branch of the smeared σε curve for tensile concrete. This explains why different smeared σ-ε curves for tensile concrete incorporated in the GSVATM will lead to different coordinates for the cracking torque (cracking torque and corresponding twist).

Comparison with experimental results
For this study, the experimental results of 103 RC beams tested under torsion were collected from the literature. Both RC beams with plain and hollow rectangular cross section were considered. These beams were selected based on criteria related to minimum requirements from codes of practice (for instance, the beams should incorporate a minimum torsional reinforcement, the spacing between rebars should not exceed the maxi- After the peak stress, namely for the descending branch, Figure 5 shows high variability between the σ-ε curves. In spite of the peak stress coincides for all the curves, it should be noted that the referred variability will influence the calculation of the cracking torque and corresponding twist with the GSVATM. This is because, as previously referred, the tensile stress σ c 1 computed from Equation (31) represents an average stress since the real stress diagram along the effective tie's thickness is not uniform due to the strain gradient ( Figure 3). The representative concrete tensile stress in the GSVATM (σ c 1 ) does not coincide with the maximum tensile stress. Hence, the strain ε c 1 corresponding to the effective cracking torque in the M T -θ curve computed with the GSVATM does not coincide with the strain ε cr corresponding to the peak stress in the smeared σ-ε curves for tensile concrete. This is illustrated in Figure 6, where an example of M T -θ and corresponding σ-ε curves for tensile concrete, computed with the GSVATM, are presented. The highlighted point in the curves (with marker " EVIEW 12 torque and corresponding twist with the GSVATM. This is because, as previously referred, the tensile stress ε ε > , i.e., in the descending branch of the smeared σε curve for tensile concrete. This explains why different smeared σ-ε curves for tensile concrete incorporated in the GSVATM will lead to different coordinates for the cracking torque (cracking torque and corresponding twist). ") corresponds to the effective cracking torque, which is reached for a strain ε c 1 > ε cr , i.e., in the descending branch of the smeared σ-ε curve for tensile concrete. This explains why different smeared σ-ε curves for tensile concrete incorporated in the GSVATM will lead to different coordinates for the cracking torque (cracking torque and corresponding twist).

Comparison with Experimental Results
For this study, the experimental results of 103 RC beams tested under torsion were collected from the literature. Both RC beams with plain and hollow rectangular cross section were considered. These beams were selected based on criteria related to minimum requirements from codes of practice (for instance, the beams should incorporate a minimum torsional reinforcement, the spacing between rebars should not exceed the maximum allowed, etc.) in order to ensure a typical behavior under torsion. A detailed discussion on such applied criteria can be found in [21]. For the RC plain beams, the data were collected from the following studies: Hsu in 1968 [35], McMullen and Rangan in 1978 [36], Koutchkali and Belarbi in 2001 [37], Fang and Shiau in 2004 [38], and Peng and Wong in 2011 [39]. For RC hollow beams, the following studies were consulted: Hsu in 1968 [35], Lampert and Thürlimann in 1969 [40], Leonhardt and Schelling in 1974 [41], Bernardo and Lopes in 2009 [42], and Jeng in 2015 [26]. Table A1 in Appendix A summarizes the main properties for each reference beam. In Table 1, "P" and "H" stand for "plain" and "hollow" cross-section, respectively. For all the reference beams from Table A1, the experimental values of the cracking torque (M exp Tcr ) and corresponding twist (θ exp cr ) were obtained from the data or graphs given by the authors [26,[35][36][37][38][39][40][41][42]. Such values are presented for each reference beam in Tables A2-A4 (see Appendix A). The torsional response of all the reference beams was computed using the GSVATM, for each of the smeared σ-ε relationships for the tensile concrete presented in Section 3 (laws l1 to l8). From the obtained theoretical M T -θ curves, the theoretical coordinates of the cracking point, i.e., the cracking torque (M thli Tcr , with i = 1 to 8) and corresponding twists, i.e., the cracking twists (θ thli cr , with i = 1 to 8), were obtained. Such values are also presented for each reference beam in Tables A2-A4 (see Appendix A). In addition, the ratios between the experimental to the theoretical values are also presented for each reference beam (M exp Tcr /M thli Tcr and θ exp cr /θ thli cr , with i = 1 to 8). Figure 7 presents, as an example, a graph with the experimental and theoretical M T -θ curves, computed for each smeared constitutive law for tensile concrete, for reference beam N-20-20 [38]. Figure 7 confirms that the coordinates of the cracking point, namely the cracking torque, as well as the postcracking response, highly depends on the used smeared constitutive law for the tensile concrete. The influence of the used smeared constitutive law is residual at the ultimate stage, namely for the maximum torque.
presented for each reference beam in Tables A2-A4 (see Appendix A). In addition, the ratios between the experimental to the theoretical values are also presented for each ref-  θ curves, computed for each smeared constitutive law for tensile concrete, for reference beam N-20-20 [38]. Figure 7 confirms that the coordinates of the cracking point, namely the cracking torque, as well as the postcracking response, highly depends on the used smeared constitutive law for the tensile concrete. The influence of the used smeared constitutive law is residual at the ultimate stage, namely for the maximum torque.
the average value ( x ) and the coefficient of variation ( (%) 100 / cv s x = × , with s being the sample standard deviation). Table 1 also presents separately the results for plain (P) and hollow (H) beams. This is because some studies showed that noticeable differences exist between the response of plain and hollow beams under torsion for the low loading stages, namely for the transition between the uncracked and the cracked stage [26].    Table 1 also presents separately the results for plain (P) and hollow (H) beams. This is because some studies showed that noticeable differences exist between the response of plain and hollow beams under torsion for the low loading stages, namely for the transition between the uncracked and the cracked stage [26]. Table 1 shows that, for the RC plain beams, the smeared constitutive laws l1, l2, l4, l5 and l6 allow us to predict the cracking torque M Tcr (with 0.95 < x < 1.05) very well and with a very acceptable degree of dispersion (cv < 13%). Among those models, the smeared constitutive law l4 from Belarbi and Hsu (1994) [24] is the best (with x = 1.00 and cv = 11.35%). For the RC hollow beams, this constitutive law gives the better average value x = 1.03, although the degree of dispersion is high (cv = 32.17%). The higher difficulty of reliably predicting the cracking torque for the RC hollow beams, when compared with the RC plain beams, was also observed and discussed in previous studies [18,26,27]. In particular, the RC hollow beams are more sensitive to the high variability of concrete tensile strength, which highly influences the cracking torque. When all beams are considered together, the smeared constitutive laws l2, l4, l6 and l7 give the best results with x ≈ 1.00, although the degree of dispersion is higher (cv < 23%) due to the influence of the results for the RC hollow beams. In general, it can be stated that the smeared constitutive law l4 from Belarbi and Hsu (1994) [24] allows us to best predict the cracking torque, regardless of the cross-section type. This constitutive law has been widely used by authors in previous studies [9,[16][17][18][19]23,26]. The results from Table 1 confirm the validity of such studies having chosen this smeared constitutive law for tensile concrete.
Regarding the twist corresponding to the cracking torque (θ cr ), Table 1 shows that, in general, there is a higher difficulty in obtaining a good prediction of this parameter. The constitutive laws l3 and l8 give the best average values for both the RC plain beams (0.95 < x < 1.05) and also for all the RC beams together (x ≤ 1.10). However, the dispersion of these results is high (cv > 25%). The results are the worst for the RC hollow beams, which was also reported in previous studies [17,18,25,27]. One possible explanation for this is that the experimental twists are very small until the end of the uncracked stage. Hence, experimental limitations related to the accurate measurement of the twists at this stage are expected. However, since the cracking twist is not very important for design, the previously reported worst results can also be considered not very important. Figure 8 presents, for each smeared constitutive law (l1 to l8), scatter graphs showing the experimental versus the theoretical values for the cracking torque. Similar graphs are not presented for the cracking twist because of the high dispersion of the results previously reported. In the graphs, different markers were used to distinguish the results regarding the cross-section type, namely " Materials 2021, 14, x FOR PEER REVIEW 15 not presented for the cracking twist because of the high dispersion of the results previously reported. In the graphs, different markers were used to distinguish the results regarding the cross-section type, namely "⬛" for RC plain beams and "⬜" for RC hollow beams. " for RC plain beams and " Materials 2021, 14, x FOR PEER REVIEW 15 not presented for the cracking twist because of the high dispersion of the results previously reported. In the graphs, different markers were used to distinguish the results regarding the cross-section type, namely "⬛" for RC plain beams and "⬜" for RC hollow beams. " for RC hollow beams.
rials 2021, 14, x FOR PEER REVIEW 15 not presented for the cracking twist because of the high dispersion of the results previously reported. In the graphs, different markers were used to distinguish the results regarding the cross-section type, namely "⬛" for RC plain beams and "⬜" for RC hollow beams.   Table 1, namely the higher dispersion of the results for the RC hollow beams.

Conclusions
In this study, the GSVATM was used to check some proposed smeared constitutive laws for tensile concrete found in the literature in order to predict the response of the RC beams under torsion for the low loading stage; namely the transition from the uncracked stage to the cracked stage. As referred to in the introduction section, the smeared model GSVATM is simpler than the other, more complex models for the RC beams under torsion. In addition, it was also validated in several previous studies. Hence, the GSVATM was considered to be sufficiently simpler and reliable to evaluate the smeared constitutive laws for tensile concrete. From the obtained results, the following conclusions can be drawn: (1) The different proposals for the smeared constitutive law for tensile concrete analyzed in this study lead to high differences in the shape of the postpeak descending branch of the corresponding smeared σ-ε curves; (2) The obtained results confirm that the predicted response of the RC beams under torsion, for the transition from the uncracked stage to the cracked stage highly depends on the smeared constitutive law for tensile concrete incorporated into the model; (3) The predictions for the cracking torque of the RC plain beams are better than the same ones for the RC hollow beams for which higher variability of the results is observed, as also reported in previous studies; (4) Regardless of the used smeared constitutive law for tensile concrete, the cracking twist is not very well predicted. Namely, higher variability of the results is observed, as also reported in previous studies; (5) Among the studied smeared constitutive laws for tensile concrete, the one proposed by Belarbi and Hsu in 1994 allows us to reliably predict the cracking torque of the RC beams under torsion, regardless of the cross-section type (plain or hollow). This result confirms the validity of several previous studies having incorporated this constitutive law in the used smeared truss models.
Finally, the authors consider that the results obtained in this study, using the smeared model GSVATM as reference model, can be extrapolated and could be useful to other smeared models for the RC beams under torsion. It must also be pointed out that additional solutions of experiments on the different failure mechanisms and related suitable approaches for the identification process for the parameters of relations of concrete are greatly needed and should be further studied, namely for the cracking of the RC beams under torsion.