Prediction of Shear Strength of Reinforced High-Strength Concrete Beams Using Compatibility-Aided Truss Model

: This study proposes an analytical model applicable to the shear analysis of reinforced high-strength concrete beams. The proposed model satisﬁes the equilibrium and compatibility conditions and constitutive laws of the materials. The proposed model is based on the ﬁxed angle theory and allows the principal stress to rotate as the load increases, so that the RC beams can be analyzed more realistically. High-strength material models were used in the proposed model to consider the characteristics of high-strength concrete. The concrete shear contribution at crack surfaces was calculated from Mohr’s circle. The proposed model considers the effect of bending moment on shear by reducing the amount of longitudinal reinforcement resisting shear. To verify the accuracy of the proposed model, a total of 64 experimental results were collected from the literature. A comparison with previous experimental results conﬁrmed that the proposed model can be predicted relatively accurately with an average of 0.98 and a coefﬁcient of variation of 12.1%.


Introduction
As structures are becoming taller and larger, the development and use of high-strength materials are becoming a trend in the construction field. In the past, the application of concrete to structures such as high-rise buildings was difficult because of the disadvantages of high self-weight and large volume; however, with the fabrication of high-strength concrete, it is now widely used in structures.
Reinforced concrete (RC) beams are structural members that transmit loads by flexure, but most of them exhibit a combination of flexure and shear. To ensure safety in structures, most structural design standards induce ductile failure of RC beams. To induce flexural ductile failure before brittle shear failure, accurate prediction of the flexural and shear behaviors of RC beams is required. A simple method of predicting flexural behavior using the equilibrium and compatibility conditions and the material laws was already developed, and its accuracy is significantly high. However, the prediction of the shear behavior is challenging because the shear is related to the non-homogeneity and tensile properties of concrete.
High-strength concrete behaves more brittle than normal-strength concrete, and the increase of tensile strength of concrete is significantly lower than that of compressive strength. In addition, the cracks are close to straight lines and have the property of penetrating the aggregates. In RC beams, shear is transmitted by compression and tension in concrete and shear transfer at crack surfaces as well as steel reinforcement. Therefore, as the compressive strength of concrete increases, the compressive, tensile, and cracking properties of concrete have a significant effect on shear strength and shear deformation.
Since the 1980s, studies on the shear of reinforced high-strength concrete beams have been actively conducted. In 1986, Elzanaty et al. [1] conducted a shear test of reinforced high-strength concrete beams with and without shear reinforcement and evaluated the applicability of the ACI building code. In the same year, Ahmad et al. [2] experimentally evaluated the shear capacity of high-strength RC beams without shear reinforcement and proposed a shear strength equation. In 1989, Johnson and Ramirez [3] conducted a study on the minimum shear reinforcement of high-strength concrete beams. In 1992, Sarsam and Al-Musawi [4] compared the accuracy of each country's design codes for high-strength beams with shear reinforcement. After that, experimental studies to evaluate the shear strength of reinforced high-strength concrete beams were continued by many researchers. Since the 2000s, more complex experimental studies such as shear strength tests of highstrength concrete beams with FRP bars instead of steel bars or various types of fibers have been conducted [5][6][7][8][9][10]. Most of these studies focus on the shear strength prediction based on experiments.
When the shear strength of RC members is estimated by using only the equilibrium condition, the shear deformation of the RC members cannot be predicted, and the accuracy is lowered if the experimental range is exceeded. The development of a shear analytical model that can consider both shear strength and deformation began with the compression field theory (CFT) developed by Collins and Michael [11], and the meaningful high accuracy was demonstrated by employing the modified compression field theory (MCFT) developed by Vecchio and Collins [12] and the rotating angle softened truss model (RA-STM) developed by Hsu et al. [13,14]. Early compatibility-aided truss models are also called rotation angle theory because of the assumption that the principal stress and crack direction are the same. Subsequently, the fixed angle theory was developed to fix the crack angle to be closer to the actual characteristics of RC structures [15,16].
In RC beams, the difference between the longitudinal and transverse reinforcement ratios is generally large. When the difference in both directional rebar ratios is large, the direction of the principal stress changes significantly from the initial crack direction as the load increases. The truss model based on the fixed angle theory may overestimate the experimental results if this is not considered. Lee et al. [17,18] developed a transformation angle truss model (TATM) that separated the crack angle and principal stress direction and was successfully applied to predict the shear behavior of RC beams and columns. However, TATM has been applied only to normal-strength concrete and has not been applied to high-strength concrete. The purpose of this study is to extend the analytical model TATM to be applicable to high-strength concrete. Material models applicable to high-strength concrete are used in the proposed analytical model. In particular, shear transfer at crack surfaces is induced from the Mohr's circle. To verify the accuracy of the proposed model, the existing experimental results collected from the literature and calculated results by ACI building code are used. Figure 1a shows a typical simply supported RC beam subjected to a concentrated load. It is assumed that the RC beam resists the external shear force with a truss mechanism consisting of a concrete compression strut and longitudinal and transverse steel bars. The stress state of a shear critical RC element located in the shear critical section separated by d es from the maximum moment position can be expressed as shown in Figure 1b. In this study, the effective shear depth, d es , is taken as 0.9d [17]. Figure 2 shows the stress state and coordinates of the RC element. The land t-directions are the coordinate systems on which external loads are applied, and the mand n-directions are the coordinate systems related to the crack direction. Furthermore, the 1-and 2-directions represent the principal stress coordinate system. As shown in Figure 2, the equilibrium of the RC element can be expressed as follows, based on the assumption that the reinforcing bars transmit only axial stress.

Equilibrium Conditions
where σ l and σ t are the applied stresses acting in the land t-directions, respectively, τ lt is the applied shear stress in the l-t coordinate system, σ c l and σ c t are the concrete resistances in the land t-directions, respectively, ρ l and ρ t are the steel ratios in the land t-axes, respectively, f l and f t are the steel stresses in the land t-axes, respectively, τ c lt is the shear resistance of concrete in the l-t coordinate system.     The resistance of concrete can be expressed as follows: where σ c m and σ c n are the normal stresses of concrete in the mand n-axes, respectively, τ c mn is the shear stress developed along concrete diagonal cracks, α is the initial crack angle due to applied external load. After the initial crack is caused by an external load, the directions of the principal stresses changes as the load increases, depending on the state of the materials. As shown in Figure 3a, the proposed model transforms the stresses in the principal stress directions to the stresses in the mand n-directions using the transformation angle, β, as follows.
σ c n = σ c 2 sin 2 β + σ c 1 cos 2 β where σ c 1 and σ c 2 are the principal tensile and compressive stresses of concrete, respectively, and β is the difference in the angles in the crack and principal compression directions.
to the stresses in the m-and n-directions using the transformation angle, β , as follows.

Compatibility Conditions
The compatibility conditions for strains in the shear critical RC element depicted in Figure 1a are based on the assumption that the deformation of the concrete and rebar are the same. According to the Mohr's strain circle presented in Figure 3b, the compatibility conditions can be expressed as: where l ε and t ε are the average normal strains in the l-and t-axes, respectively, lt γ and mn γ are the average shear strains in the l-t and m-n coordinate systems, respectively, and m ε and n ε are the average normal strains of concrete in the m-and n-axes, respectively. As shown in Figure 3b, the strain in the m-and n-directions can be obtained by converting the strains in the 1-and 2-directions as follows.
where 1 ε and 2 ε are the principal tensile and compressive strains, respectively.

Compatibility Conditions
The compatibility conditions for strains in the shear critical RC element depicted in Figure 1a are based on the assumption that the deformation of the concrete and rebar are the same. According to the Mohr's strain circle presented in Figure 3b, the compatibility conditions can be expressed as: ε t = ε m sin 2 α + ε n cos 2 α − γ mn sin α cos α (10) where ε l and ε t are the average normal strains in the land t-axes, respectively, γ lt and γ mn are the average shear strains in the l-t and m-n coordinate systems, respectively, and ε m and ε n are the average normal strains of concrete in the mand n-axes, respectively. As shown in Figure 3b, the strain in the mand n-directions can be obtained by converting the strains in the 1-and 2-directions as follows.
where ε 1 and ε 2 are the principal tensile and compressive strains, respectively.

Constitutive Laws
This study used the principal compressive stress-strain relationship, as shown in Figure 4a, proposed by Collins and Porasa [19] for high-strength concrete as follows. (15) where f p is the peak stress of concrete (=ν f c ), f c is the uniaxial strength of cylinder concrete (positive), ν is the softening coefficient of cracked concrete, ε p is the strain at peak stress (=νε 0 ), ε 0 is the strain at peak stress of uniaxial cylinder concrete (negative), n is the curve fitting factor, and k = 1 for the ascending branch and k = 0.67 + f c /62 for the descending branch. The web of the RC beam undergoes compression softening because it is in a state of biaxial stress that is simultaneously subjected to compressive and tensile stresses. In this study, the softening coefficient was applied to both the stress and strain, as shown in Equation (14).
where p f is the peak stress of concrete (= ' c f ν ), ' c f is the uniaxial strength of cylinder concrete (positive), ν is the softening coefficient of cracked concrete, p ε is the strain at peak stress (= 0 νε ), 0 ε is the strain at peak stress of uniaxial cylinder concrete (negative), n is the curve fitting factor, and k = 1 for the ascending branch and k = ' 0.67 / 62 c f + for the descending branch. The web of the RC beam undergoes compression softening because it is in a state of biaxial stress that is simultaneously subjected to compressive and tensile stresses. In this study, the softening coefficient was applied to both the stress and strain, as shown in Equation (14). The softening coefficient, the strain at the peak stress of the cylinder concrete, and the elastic modulus of the concrete used in this study are as follows [19]: The concrete tensile stress-strain relationship proposed by Vecchio and Collins [12] are used as follows, as shown in Figure 4b.
1 500 The softening coefficient, the strain at the peak stress of the cylinder concrete, and the elastic modulus of the concrete used in this study are as follows [19]: The concrete tensile stress-strain relationship proposed by Vecchio and Collins [12] are used as follows, as shown in Figure 4b.
where ε cr is the cracking strain of the concrete (= f cr /E c ) and f cr is the tensile strength of the concrete (=0.33 f c (MPa)). The truss model based on the fixed angle has the advantage of being able to directly consider the shear transfer owing to the aggregate interlock. Unlike normal-strength concrete, high-strength concrete has smoother crack surfaces and penetrates the aggregate rather than bypassing it. The Mohr's stress and strain circles shown in Figure 3 can be used to derive the following equation.
Based on Equation (21), τ c mn − γ mn relationship is summarized as follows.
For the stress-strain relationship of reinforcing bars, the following equation proposed by Belarbi and Hsu [20] was used, but was limited to not exceed the yield strength of steel bars, as shown in Figure 4c.

Consideration of Bending Moment
Unlike RC elements, RC beams are generally subjected to flexure and shear simultaneously. The method of applying the proposed truss model developed for the reinforced high-strength concrete element to RC beams is the same as that applied in a previous study [17] for reinforced normal-strength concrete beams. For details, please refer to [7], and this paper briefly introduces the method.
To consider the effect of the bending moment, the remainder of the longitudinal steel ratio, except for the reinforcement ratio used for flexural resistance, is assumed to resist shear. Based on this assumption, the following equation can be derived: where ρ lt is the total longitudinal steel ratio and ρ l f is the longitudinal steel ratio resisting bending moment. According to the assumption in Equation (25), there is no axial longitudinal stress due to the external force in the shear critical element of RC beams without axial force. Therefore, the initial crack angle, α, is 45 • and σ l and σ t are zero.
The steel ratio that resists the bending moment, M s , occurring in the shear critical section can be obtained as follows: where f ly is the yield strength of longitudinal steel bar, jd is the lever arm, b and d are the width and effective depth of the beam section, respectively, V is the shear force, a is the shear span. Equation (26) is substituted into Equation (25) and jd is assumed to be d es ; thus, the steel ratio, ρ l , applied to the proposed model is as follows: where A lt is the area of tension reinforcement. The contribution of compression reinforcement was ignored in this study based on the studies conducted by Wilby [21], Al-Alusi [22], and Taub and Neville [23]. Figure 5 shows the calculation procedure for predicting the shear behavior of reinforced high-strength concrete beams. To find all solutions, a progressively increasing ε 2 is first selected, and then ε 1 and γ mn are assumed sequentially. The transformation angle, β, can be calculated as follows based on the Mohr's strain circle presented in Figure 3b.

Calculation Procedure
is first selected, and then 1 ε and mn γ are assumed sequentially. The transformation angle, β , can be calculated as follows based on the Mohr's strain circle presented in Figure   3b.

Prediction of Shear Strength of Reinforced High-Strength Concrete Beams
To verify the accuracy of the proposed analytical model for reinforced high-strength concrete beams, 64 experimental results [24][25][26][27][28][29][30][31][32][33] with 90 • stirrups were collected. All collected specimens were failed by shear before 90% of their predicted flexural yield strength. The specimens have a simple beam that receives a concentrated load and a beam that receives an anti-symmetric load. The shape of the cross-section is rectangular-and T-types. The collected specimens have a concrete compressive strength of 42.9~108.7 MPa, a beam width of 125~300 mm, an effective depth of 198~420 mm, a tension reinforcement ratio of 0.00228~0.00454, a shear reinforcement ratio of 0.001~0.0171, yield strengths of tension and shear reinforcement of 406.2~953.2 and 255.0~1431.4 MPa, respectively, and a shear span ratio of 1.7~5. All specimens with a shear span ratio of less than 2.5 appear to be subjected to anti-symmetrical moments. It is noted that a direct compression arch action is not formed, and the truss model is applicable. Table 1 presents the details of collected specimens and the comparison of the experimental results with the analytical ones. In Table 1, the analytical results by ACI 318-19 [34] were calculated according to the ACI shear provisions that the values of f c greater than 8.3 MPa shall be permitted in the RC beams satisfying minimum shear reinforcement. The shear strength of concrete was calculated using the detailed formula recommended by ACI 318-19. In addition, the yield strength of steel bars was not limited to evaluate the applicability of ACI 318-19. As shown in Table 1 and Figure 6, the analytical results by ACI 318-19 conservatively predicted the experimental results with a mean of 1.28 and a coefficient of variation (COV) of 16.4%. In contrast, the proposed analytical model predicted the shear strength of 64 specimens well with a mean of 0.98 and a COV of 12.1%. Figure 7 shows the prediction results of the proposed model on test variables. As shown in Figure 7, the proposed analytical model predicts the experimental results well without being affected by the compressive strength of the concrete, tension and shear reinforcement ratios, and η (= ρ t f ty /ρ l f ly ). This means that the material models used in the proposed model are applicable to the shear analysis of reinforced high-strength concrete beams. [Note] Refs. [24,25], ([27] S7-Series), [33]: simple beams with 3-point load and rectangular section. Ref. [26], ([27] S1~S5, S8), [32]: simple beams with 4-point load and rectangular section. Refs. [28,29]: simple beams with 3-point load and rectangular and T-sections. Refs. [30,31]: restrained beams with anti-symmetrical moments and rectangular section.  Table 2 shows the prediction results of the analytical models. TM-NM and TM-FA are the same as the proposed analytical model, but the former does not consider the moment effect, and the latter assumes that the transformation angle, β , is zero. As shown in Table 2, TM-NM, which did not consider the moment effect of the beam, overestimated  Table 2 shows the prediction results of the analytical models. TM-NM and TM-FA are the same as the proposed analytical model, but the former does not consider the moment effect, and the latter assumes that the transformation angle, β , is zero. As shown in Table 2, TM-NM, which did not consider the moment effect of the beam, overestimated the experimental results with an average of 0.76. Figure 8a shows that TM-NM tends to  Table 2 shows the prediction results of the analytical models. TM-NM and TM-FA are the same as the proposed analytical model, but the former does not consider the moment effect, and the latter assumes that the transformation angle, β, is zero. As shown in Table 2, TM-NM, which did not consider the moment effect of the beam, overestimated the experimental results with an average of 0.76. Figure 8a shows that TM-NM tends to overestimate the experimental results as the shear span-to-depth ratio increases. On the other hand, the proposed model was hardly affected by the concrete compressive strength and support conditions, and shear span-to-depth ratio, as shown in Table 2 and Figure 8b. other hand, the proposed model was hardly affected by the concrete compressive strength and support conditions, and shear span-to-depth ratio, as shown in Table 2 and Figure 8b.   Table 2, the TM-FA with the angle β fixed to zero greatly overestimated the experimental results with an average of 0.86 for the shear strength of 64 specimens. As shown in Figure 9a, this phenomenon becomes clearer when η is less than 0.2. That is, as η decreases, the difference between the crack direction and the principal stress direction increases. However, it can be seen from Figures 7d and 9b that the proposed model applying the transformation angle β is not significantly affected by η . Therefore, the proposed shear analytical model can be reasonably used to predict the shear strength of reinforced high-strength concrete beams.   Table 2, the TM-FA with the angle β fixed to zero greatly overestimated the experimental results with an average of 0.86 for the shear strength of 64 specimens. As shown in Figure 9a, this phenomenon becomes clearer when η is less than 0.2. That is, as η decreases, the difference between the crack direction and the principal stress direction increases. However, it can be seen from Figures 7d and 9b that the proposed model applying the transformation angle β is not significantly affected by η. Therefore, the proposed shear analytical model can be reasonably used to predict the shear strength of reinforced highstrength concrete beams. other hand, the proposed model was hardly affected by the concrete compressive strength and support conditions, and shear span-to-depth ratio, as shown in Table 2 and Figure 8b.   Table 2, the TM-FA with the angle β fixed to zero greatly overestimated the experimental results with an average of 0.86 for the shear strength of 64 specimens. As shown in Figure 9a, this phenomenon becomes clearer when η is less than 0.2. That is, as η decreases, the difference between the crack direction and the principal stress direction increases. However, it can be seen from Figures 7d and 9b that the proposed model applying the transformation angle β is not significantly affected by η . Therefore, the proposed shear analytical model can be reasonably used to predict the shear strength of reinforced high-strength concrete beams.

Conclusions
In this study, a shear analytical model was proposed to predict the shear strength of reinforced high-strength concrete beams. The analytical results of the proposed model were compared with the experimental results, and the following conclusions can be obtained.
(1) The proposed shear analytical model was extended for application to high-strength concrete. The material models, including the shear transfer model by aggregate interlock, were replaced with those suitable for high-strength concrete. A total of 61 experimental results were well predicted, with a mean of 0.98 and a COV of 12.1%, without being affected by the concrete compressive strength.
(2) A comparison with the experimental results confirmed that the analytical results were barely affected by various shear-span-to depth ratios. Thus, the proposed method for considering the bending moment effect can be used for reinforced high-strength concrete beams. (3) RC beams generally exhibit a large difference between the longitudinal and transverse reinforcement ratios. To consider this property of the RC beams, the proposed model based on the fixed angle theory allows the principal stresses and strains to rotate as the load increases. When 64 experimental results were predicted using the same model as the proposed analytical model but the principal stresses did not rotate, the accuracy was significantly reduced. This result confirmed that the proposed transformation angle system is applicable to reinforced high-strength concrete beams.