DIAGONAL TENSION FAILURE OF RC BEAMS WITHOUT STIRRUPS

The shear failure of reinforced concrete beams is one of the fundamental problems in civil engineering; however, the diagonal tension strength of reinforced concrete (RC) beams without stirrups is still in question. This paper focuses on the prediction of diagonal cracking strength of RC slender beams without stirrups. In slender beams, flexural cracks develop in the tension zone prior to a diagonal cracking. Using the basic principles of mechanics, but cracking included, and theory of elasticity, a diagonal cracking strength equation is proposed for both normal and high strength concrete beams. The proposed equation, the requirements of six codes of practice and seven equations proposed by different researchers are compared to the experimental results of 282 beams available in the literature. It is found that the predictions from the proposed equation are in good agreement with the experimental results.


Introduction
The results of experiments show that the shear failure of reinforced concrete (RC) slender beams without stirrups is always governed by diagonal tension failure mode rather than compression failure mode. During the last 40 years, researchers have made several attempts to predict the shear strength of RC beams based on mainly experimental results and statistical studies. With respect to the various empirical formulas, considerable differences exist as a result of the following factors: the uncertainty in assessing the influence of complex parameters in a simple formula; the scatter of the selected test results due to inappropriate tests being considered; the poor representation of some parameters in tests; and finally, the concrete tensile strength often not being evaluated from control specimens. These issues limit the validity of empirical formulas, and increase the necessity for rational models and theoretically justified relationships (ASCE-ACI445 1999). It is believed that an analytical formula is more satisfactory than an empirical formula, as it provides physical insight into the phenomenon (Gastebled, May 2001).
To determine the minimum amount of stirrups and to obtain the shear strength of RC beams with stirrups, it is necessary to know the diagonal cracking shear strength of RC slender beams. In case of a slender beam with shear span-to-depth ratio a/d > 2.5, inclined tensile cracking develops in the direction perpendicular to the principal tensile stress axis when the principal tensile stress within the shear span exceeds the tensile strength of concrete. Taylor (1960) has indicated that the diagonal cracking stage is not clearly defined in the experimental beams where the crack formed is close to the applied load because the development of the inclined cracks is gradual. Since the diagonal cracking load is very sensitive to the judgment of the observer and the location of the initiating flexural crack, experimental values scatter significantly (Bazant, Kazemi 1991). Diagonal cracking shear strength is defined in Mphonde and Frantz (1984) tests as the shear load when the critical crack becomes inclined and crosses at mid-depth. The inclined shear strength, therefore, obviously is affected by the observer's judgment and also is sensitive to the actual location of the initiating flexural crack. Mphonde and Frantz (1984) tests on beams without stirrups have shown that the ratio of the shear causing inclined cracking to the measured shear strength ranges from 0.74 to 0.97 and is very unpredictable. Therefore, it is difficult to determine the value of the diagonal tension stress and the cracking load in a RC beam because the distributions of shear and flexural stresses are not known precisely. Furthermore, the crack initiation load is not proportional to the failure load and it can be much smaller or only slightly smaller depending on the beam size and other factors (Bazant, Kazemi 1991). The contribution of this study is to present an equation for predicting the diagonal cracking strength of RC slender beams without stirrups. The proposed equation, the requirements of six codes of practice and seven equations proposed by different researchers for either cracking or ultimate shear strength are compared to the experimental diagonal cracking shear strengths available in the literature.
To determine the minimum amount of stirrups and to obtain the shear strength of RC beams with stirrups, it is necessary to know the diagonal cracking strength of RC slender beams. The requirements of several codes and methods of prediction of the shear strength are based on the experimental results of normal strength concrete. In design, using these equations may not be appropriate, and verifications and modifications may be required for the evaluation of shear strength of high strength concrete beams. In this study, using the basic principle of mechanics and calibrating against the factors of the effective depth and slenderness ratio, an equation for predicting diagonal cracking strength is proposed for both normal and high strength concrete beams.

Existing shear strength models
A number of equations proposed by various codes and researchers are considered. These are ACI318 Building Code (2008), based on experimental results of numerous beams; Turkish Building Code (TS500 2000), based on the adaptation of ACI Code simplified equation; CSA Code (1994), based on the modified compression field theory; NZS (1995);EN 1992EN -1-1:2004EN (2004; CEB-FIP90 (1993) model code equation, introduced empirically; Zsutty's equation (Zsutty 1971), deduced by multiple regression analysis; Okamura's equation (Okamura, Higai 1980), developed empirical equation from experimental data; Bazant's equation (Bazant, Kim 1984), based on non-linear fractures mechanics considering the size effect; Kim's equation (Kim, Park 1996), based on basic shear transfer mechanisms, a modified Bazant's size effect law and test data; Collins' equation (Collins, Kuchma 1999), resulting from an enhancement of the modified compression field theory based on a hypothesis that crack spacing causes size effect; Rebeiz's equation (Rebeiz 1999), obtained from multiple regression analysis; and Khuntia's equation (Khuntia, Stojadinovic 2001), based on basic principles of mechanics and parametric study of experimental data. All the equations considered within the scope of this study are summarized in Table 1. These equations are applied to a database consisting of 282 specimens so that the results of the equations can be compared to the test results.
Based on the test results, Jelic et al. (1999) reported that dowel action cannot be considered as a viable component in the shear mechanism of a cracked reinforced concrete beam section without stirrups. The main factor resisting the applied shear force is the shear resistance of the uncracked concrete. According to Zararis and Papadakis (2001), and Kotsovos and Pavlovic (1998) the compression zone of intact concrete prevents shear-slip of the crack surfaces. Reinhardt and Walraven (1982) reported that the tension zone damaged by the flexural cracks does not significantly contribute to the shear resistance of the beams. Normally, dowel action is not very significant in members without stirrups, since the maximum shear in a dowel is limited by the tensile strength of the cover concrete supporting the dowel (Bauman, Rüsch 1970). Therefore, the aggregate interlock along the crack surfaces and the dowel action of longitudinal reinforcement do not significantly contribute to the shear strength of the beams (Kotsovos, Pavlovic 1998). According to Jelic et al. (1999), the dowel action of longitudinal rein-forcement placed in one layer can be neglected for RC beams without stirrups. In the present study, dowel action is neglected for simplicity and conservatism.

The prediction of diagonal cracking strength of beams without stirrups
The shear failure of reinforced concrete members without stirrups initiates in the form of diagonal cracks, which later propagates through the beam web, when the principal tensile stress within the shear span exceeds the tensile strength of concrete. For a RC slender beam where a/d is greater than 2.5, the shear strength at section is primarily concerned with the effective shear depth of critical diagonal crack and the tensile strength of concrete. The effective shear depth, based on Khuntia's equation (Khuntia, Stojadinovic 2001), is obtained using the depth of neutral axis and the compressive strain in concrete. The diagonal cracking strength equation can be obtained by using a number of simplifying assumptions, and is based on the basic principles of mechanics, but cracking included, and the theory of elasticity. The proposed equation is compared with the test results of high-strength concrete (HSC) beams with compressive strength of 55 c f ≥ MPa, and normal-strength concrete (NSC) beams with lower c f values reported in the literature. In 1902 Mörsch derived the shear stress distribution for a RC beam containing flexural cracks. Mörsch predicted that shear stress would reach its maximum value at the neutral axis and would then remain constant from the neutral axis down to the flexural steel (Cladera 2003). The value of this maximum shear stress would be: where: w b is the web width; z is the flexural lever arm. According to Zink (2000), the stress distribution for o τ shown in Fig. 1a is neither able to describe the shear cracking force in quality nor in quantity. New approaches (Khuntia, Stojadinovic 2001;Zink 2000) were proposed to adopt the influence of the components in the cracked flexural tension zone more accurately. Khuntia and Stojadinovic (2001) modeled the shear stress distribution as a parabola with an integral factor of 2/3 over the effective shear depth with the maximum value at the neutral axis (Fig. 1d). The shear capacity of beams without stirrups, o V , carried in the uncracked compressive zone, is determined by integrating the shear stress and is expressed as follows: where 1 c is the effective shear depth and t f is the tensile strength of concrete.
The shear capacity controlled by tension is normally less than that controlled by compression, regardless of the magnitude of flexural deformation. This is because the tensile strength of concrete is much less than the compressive strength (Park et al. 2006). According to Khuntia  Okamura and Higai (1980) ( Bazant and Kim (1984) 3 5 Kim and Park (1996) /3 3/8 Collins and Kuchma (1999) ( ) Khuntia and and Stojadinovic (2001), the shear stress distribution is modeled as parabolic over the effective shear depth with the maximum value at the neutral axis such that the maximum shear stress over the effective cross section equals to , where w b is the width of section and c 1 is the effective shear depth. The shear failure of RC members without stirrups initiate when the principal tensile stress within the shear span exceeds the tensile strength of concrete and a diagonal crack propagates through the beam web. Mathematically: where t f is the tensile strength of concrete and  (2003)); b) cross-section X-X; c) distribution of concrete stresses (Khuntia, Stojadinovic 2001); d) shear stress distribution (Khuntia, Stojadinovic 2001); e) longitudinal strain distribution is the shear force. The procedure proposed by Khuntia and Stojadinovic (2001) for the shear strength of RC members without stirrups, at the design section under the effect of the factored bending moment M u and axial load P u , calculates the effective shear depth c 1 , using the method of satisfaction of strain compatibility and equilibrium conditions: where c is the depth of neutral axis, c ε is the compressive strain in concrete, which is taken as 0.002, and cr ε is the cracking strain value in concrete, which is taken as the ratio of the tensile strength of concrete t f to its modulus of elasticity c E : Since concrete is relatively weak and brittle in tension, cracking is expected when significant tensile stress is induced in a RC member (ACI224 1992). The tensile strength of plain concrete t f , ranges from about 0.25 to 0.50 c f (Nilson, Darwin 1997;Paulay, Priestley 1992;Carreira, Chu 1986 where c f is the compressive strength of concrete in MPa and c/d is the ratio of neutral axis depth to effective depth, which is the positive root of the second order equation given by Eq. Eq. (7) does not capture the effects of slenderness and size on the diagonal cracking strength, which were considered by many researchers and codes. The diagonal cracking strength of RC beams decreases with increasing member depth and slenderness. In order to take into account the effects of slenderness and size, Eq. (7) is calibrated to the test results available in the literature.

Calibration by comparison with the effect of slenderness and size
In order to obtain more accurate diagonal cracking strength, the principal shear strength equation is modified by the factors of the effective depth and slenderness ratio (a/d). Based on the principal shear strength v o carried by the compression zone, the influence of the slenderness and size can be considered with the factors ( / ) k a d and (1/ ) k d . The diagonal cracking strength causing shear tension failure can be written as follows: where ( / ) k a d and (1/ ) k d are the coefficient of slenderness and size effect, respectively.

Slenderness effect on diagonal cracking strength
The weak influence of a/d is neglected in some design formulas (ACI318 2008; TS500 2000; CSA 1994; NZS 1995; EN 1992-1-1:2004 2004; Collins, Kuchma 1999; Arslan 2005Arslan , 2008. Tension stiffening causes a minor influence of a/d, which is described with a coefficient ( / ) k a d in some other design formulas (CEB-FIP90 1993;Zsutty 1971;Okamura, Higai 1980;Bazant, Kim 1984;Kim, Park 1996;Rebeiz 1999;Khuntia, Stojadinovic 2001). CEB-FIP90 equation is proposed to determine the relationship between a/d and the diagonal cracking strength, leading to the following expression for slender beams: . For a slenderness ratio ( ) / 3.0 a d = , a complete crack formation can be observed when shear failure occurs. Therefore, the coefficient of slenderness ratio is set to 1.0 for ( ) The exponent in the formula has a relatively small effect on the coefficient of slenderness ratio. The cracking shear strength decreases 21% even a/d is increased to 6.0 from 3.0. According to the Okamura and Higai (1980) Kim and Park (1996) Fig. 2. A new design expression is proposed for the diagonal cracking strength based on the principal shear strength for slender beams by considering the slenderness effect.

Size effect on diagonal cracking strength
The influence of the effective depth d is neglected in some design formulas (ACI318 2008;TS500 2000;NZS 1995;Zsutty 1971;Rebeiz 1999). However, in generally, size effect on the cracking shear strength is significant and is described with a coefficient k(1/d) in some other design formulas (CSA 1994;EN 1992EN -1-1:2004EN 2004CEB-FIP90 1993;BS 8110 1997;Okamura, Higai 1980;Bazant, Kim 1984;Kim, Park 1996;Collins, Kuchma 1999). The CSA Code (1994) includes a term to account for the size effect in its simplified shear design expression but does not take the reinforcing steel ratio, ρ, into account. This shows the concern of this code regarding the size effect phenomenon. When the effective depth d is quite large, the CSA Code (1994) equation considers an over strong asymptotic size effect , which is contrary to the point of view that the linear elastic fracture mechanics size effect for very large beam depths (Bazant, Yu 2005).
BS 8110 (1997) equation relates the cracking shear strength to the size effect as follows: Based on the regression analysis, c v is taken as proportional to ( ) 0.25 400 / d in this study to identify the size effect, which is similar to BS 8110 equation.

Proposed diagonal cracking strength equation for RC slender beams without stirrups
Based on the principal shear strength v o carried in the compression zone, considering the influence of parameters; the slenderness ratio (a/d) and size effect (1/d), the diagonal cracking strength of RC slender beams without stirrups can be expressed as follows: Eq. (12) is proposed for RC beams with a shear span to depth ratio ( ) / a d equal to or greater than 2.5.

Evaluation of proposed equation
The diagonal cracking strength equation was applied to the 282 specimens that had been tested by 22 researchers. These specimens were subjected to single-or two-point loads at mid-span. The specimens have a broad range of design parameters: 0.47 ≤ ρ ≤ 5.01 (%), 2.50 ≤ a/d ≤ 8.52, 6.1 ≤ f c ≤ 53.9 MPa and 41 ≤ d ≤483 mm for NSC and 0.33≤ ρ ≤ 6.64 (%), 2.50 ≤ a/d ≤ 6.00, 56.5 ≤ f c ≤ 91.8 MPa, and 184 ≤ d ≤ 822 mm for HSC. The effects of concrete compressive strength, slenderness ratio and flexural reinforcement ratio on the proposed diagonal cracking strength of RC slender beams without stirrups are discussed below. Fig. 3 compares the proposed diagonal cracking strength obtained from Eq. (12) with experimental results for NSC beams (Taylor 1960;Bazant, Kazemi 1991;Mphonde, Frantz 1984;Moody et al. 1954;Diaz de Cossio, Siess 1960;Van den Berg 1962;Taylor, Brewer 1963;Bresler, Scordelis 1963;Mathey, Watstein 1963;Mattock 1969;Krefeld, Thurston 1966;Cho 2003;Cladera, Mari 2005) and HSC beams (Mphonde, Frantz 1984;Van den Berg 1962;Cho 2003;Cladera, Mari 2005;Ahmad et al. 1986;Elzanaty et al. 1986;Kwak et al. 2002;Shah, Ahmad 2007;Sneed, Ramirez 2010). The mean values (MV) and the standard deviations (SD) of the ratio of the experimental cracking shear strength to the proposed diagonal cracking strength are 1.108 and 0.118 for NSC, 1.012 and 0.153 for HSC, respectively.   According to Leonhardt and Walter (1962), if the flexural reinforcement ratio is kept constant, the mode of failure of rectangular RC beams without stirrups depends on the slenderness ratio. If the reinforcement ratio is greater than approximately 1.8%, shear failure is more critical than flexural failure for slenderness ratios between 1 and 7. Based on test results, ACI Building Code provisions are nonconservative for low values of the flexural reinforcement ratio, ρ, and, therefore, unsafe for beams without stirrups (Ahmad et al. 1986). According to TS500 (2000), the amount of the flexural reinforcement ratio is limited within the range of ρ ≤ 2.0%. However; the strength of members with low reinforcing ratios was rarely investigated in the past and is often overestimated in the present codes (ASCE-ACI445 1999). As shown in Fig. 5, the test results of the cracking shear strength of HSC slender beams with low reinforcing ratios are very limited (ρ < 1.0%), consequently further research is required to verify the proposed equations for HSC beams. Since the test data for HSC members are very limited, further research is required to verify the proposed equations for HSC beams. Table 2 summarizes the comparisons of the predictions obtained from the proposed equation, ACI318 Building Code (2008), TS500 (2000), CSA Code (1994), NZS (1995), EN 1992-1-1:2004(2004, CEB-FIP90 (1993), Zsutty's equation (Zsutty 1971), Okamura's equation (Okamura, Higai 1980), Bazant's equation (Bazant, Kim 1984), Kim's equation (Kim, Park 1996), Collins' equation (Collins, Kuchma 1999), Rebeiz's equation (Rebeiz 1999), and Khuntia's equation (Khuntia, Stojadinovic 2001) with the test results available in the literature. The resulting coefficient of variation (COV) of the ratio of the experimental value (NSC) to the prediction from the proposed equation is 27% of those obtained for CSA Code prediction, 37% of those obtained for NZS Code prediction, 52% of those obtained for Bazant's

Conclusions
On the basis of results obtained in this study, the following conclusions are drawn: 1. It can be seen that the proposed diagonal cracking strength equation (Eq. (12) for RC slender beams results in the lowest coefficient of variation (COV) for the ratio of experimental value to the predicted value for NSC and HSC beams. Hence Eq. (12) provides better results than six codes of practice and seven equations proposed by different researchers for the prediction of diagonal cracking strength of RC beams with NSC and HSC. However, further research is required to verify the proposed equation since the test data for HSC members is very limited.
2. The mean value (MV) of the experimental cracking shear strength to the proposed diagonal cracking strength is 1.108 for NSC and 1.012 for HSC beams. Therefore, it can explain that the contribution of dowel action to the diagonal cracking strength provides additional conservation.
3. The predictions by the proposed equation for the shear strength of test beams are relatively better, whereas ACI318, CSA, EN 1992-1-1:2004, Collins' equation is excessively conservative for most of the test results and the NZS and Bazant's equations give unsafe results for slender beams.
4. The ratio of the experimental to the proposed diagonal cracking strength is not significantly influenced by increasing a/d, ρ and f c , but it is important to note that test data are not homogeneous.