Statistical Prediction Equations for RC Deep Beam Without Stirrups
Abstract
Reinforced concrete (RC) deep beams mainly fail in shear, brittle and sudden in nature can lead to calamitous consequences. Thence, it is critical to determine the shear characteristics of RC deep beams accurately due to involving many parameters at same time. Some of the recent researches have shown that current equations for predicting ultimate shear strength are nonconservative when applied to high strength concrete (HSC) beams as well as some of design codes provisions. There are different approaches for analyzing the behavior of beams in shear.
In this paper, a semiempirical approach is adopted in which a database of existing experimental and literature results of deep beams, d > 300 mm & d < 300 mm, failing in shear under two point loads statically at midspan, was constructed. The database, 725 deep beams, was used to propose two simplified shear equations using multiple regression analysis, IBMSPSSStatistics, to find out and evaluate the most important factors affecting the ultimate shear strength formulating them in a suitable predictive equation for the ultimate shear strength of deep beams without web reinforcement (stirrups). The test database covers a wide range of individual parameters as: cylindrical concrete compressive strength (20 ≤ f ′ _{ c } ≤ 104 MPa), longitudinal main steel reinforcement ratio (0.17% ≤ ρ _{ s }% ≤ 6.64%), effective depth of deep beams “d” (127–1000 mm), shear span to effective depth ratio (1 ≤ a/d ≤ 2.5), Beam width “b”; b/d < 1, where all database from literature were based on two point loading.
This paper concluded that, the proposed equations seem to predict the ultimate shear strengths well, conservative and give lower coefficient of variation “COV” and smaller range of results when compared with the available methods of design; ECP2032007, ACI 31814, CSA, and BS8110 codes and other equations proposed by Sudheer Reddy, Zararis, Bazant, Zsutty and Shah.
Keywords
Deep beams Predictive design equation Shear strength Tension reinforcement ratio Shear span to effective depth ratio1 Introduction
Normally, Reinforced concrete has extensively use in the construction industry around the world. Concrete is heterogeneous in nature and that cause difficulty in the calculation of stresses, in addition to the presence of reinforcement steel make the situation more complex.
As well known, among researchers in constructions and concrete technology field, that the shear capacity of high strength reinforced concrete (HSRC) beams unlike that of normal strength reinforced concrete (NSRC) do not increase, in the same proportion, as the compressive strength of concrete increase due to brittle behavior of the High Strength Concrete (HSC).
The use of deep beams in many applications is going faster than developing new design equations in code provisions for shear design. In spite of availability of massive of tested NSRC & HSRC deep beams, but it is seemed that the database still need more specimens to can understand the complex shear mechanism of deep beam especially with HSC. Therefore, the current equations proposed empirically by most of the design codes for shear strength of RC beam are less conservative as compared to those of NSRC beams. This major observation by the authors is the focus of this research.
Several researchers discovered that failure mode is dependent on (a/d) substantially. Berg (1962) found increase in shear capacity with decreasing of (a/d) ratio. Nevertheless, Ferguson (1956) explained what Berg shown, increased resistance to diagonal tension in case of small (a/d), a local loading effect because of the thrust of load transfer directly to supports through concrete strut, where diagonal cracking load exceed gradually with increasing shear span for concrete compressive strength up to 28 MPa. Kani (1966) summarized that, f′_{ c } between (17–34 MPa), a decrease in relative flexural strength with increase in (a/d) ≤ 2.5. Most of studies mentioned above had done on nonpractical beam size causing unconservative in results used in formulate new predictive design equation.
The test data of thirtysix reinforced concrete beams carried out in this study, besides results of test beams obtained from the literature by other investigators used in a multiple regression analysis to formulate predictive equations for deep beams taking into account the main factors affecting the ultimate shear strengths.
A detailed comparison performed between the experimental values of the ultimate shear strengths obtained from this study and those of the literature and the estimated values using the proposed equations and the other predictive equations proposed by other investigators.
2 Experimental Program
2.1 Specimens
Specimen details
No  Beam designation  h (mm)  a (mm)  d  a/d  a_{h}  a_{r}  S_{s}  L (mm)  ρ_{s} (%)  f′_{cu}, MPa 

B1  B7002r1  700  1224  660  2  500  326  200  3600  0.73  74.8 
B2  B7002r2  700  1224  660  2  500  326  200  3600  1.21  74.8 
B3  B7002r3  700  1224  660  2  500  326  200  3600  1.83  74.8 
B4  B7001.5r1  700  918  660  1.5  1100  332  800  3600  0.73  74.8 
B5  B7001.5r2  700  918  660  1.5  1100  332  800  3600  1.21  74.8 
B6  B7001.5r3  700  918  660  1.5  1100  332  800  3600  1.83  74.8 
B7  B7001r1  700  1600  660  1  674  326  1000  3600  0.73  74.8 
B8  B7001r2  700  1600  660  1  674  326  1000  3600  1.21  74.8 
B9  B7001r3  700  1600  660  1  674  326  1000  3600  1.83  74.8 
B10  B4002r1  400  670  360  2  1000  330  800  3000  0.73  77.35 
B11  B4002r2  400  670  360  2  1000  330  800  3000  1.21  77.35 
B12  B4002r3  400  670  360  2  1000  330  800  3000  1.83  77.35 
B13  B4001.5r1  400  502.5  360  1.5  1300  348  1000  3000  0.73  77.35 
B14  B4001.5r2  400  502.5  360  1.5  1300  348  1000  3000  1.21  77.35 
B15  B4001.5r3  400  502.5  360  1.5  1300  348  1000  3000  1.83  77.35 
B16  B4001r1  400  335  360  1  1600  365  1000  3000  0.73  77.35 
B17  B4001r2  400  335  360  1  1600  365  1000  3000  1.21  77.35 
B18  B4001r3  400  335  360  1  1600  365  1000  3000  1.83  77.35 
2.2 Material Properties
A local readymix supplier provided the desired concrete for tested beams. Two patches, for two sets of beams, of same concrete mix was placed in wooden forms and vibrated to ensure the concrete workability. From the same patches, concrete cylinders 150 × 300 mm are casted for standard concrete tests, and cured at room temperature for 28 days. The concrete strength was monitored by compression testing of the cylinders; 3, 7 and 28 days. The concrete strength is ranged in between (73–77) MPa with an average value of 75 MPa at 28 days age. High strength deformed bars 10, 12, 14, 18 and 20 mm and of 765, 650, 670, 670, and 670 MPa proof strengths respectively were used for tensile steel. Plain bars 8 mm used in compression zone just to hold the stirrups, which used to prevent bearing failure occur, at supports area in place during casting.
2.3 Test Setup
3 Equations Predicting Ultimate Shear Strength
Sudheer Reddy.
4 Regression Analysis
4.1 Introduction to Regression Analysis
Regression analysis is described as the study of relationship between variables. Regression analysis is used to predict a continuous dependent variable from a number of independent variables.
A linear relationship between two random variables refers to simple linear regression and the linear relationship between more than two random variables is called multiple linear regression. When the relationship is sought between a single predictor and the response variable, it is called simple linear regression. Relationships involving more than one variable are of the form. For this research, multiple linear regression is adopted.
A multiple regression equation involves more than one regression coefficient to describe the data. The best way to choose between alternative regression coefficients is to compare the errors of prediction associated with different linear regression equations. Errors of prediction are defined as the differences between the observed values of the independent variable and the predicted values for that variable obtained using a given regression equation and the observed values of the independent variable.
SPSSIBM .v20 is powerful software that solves many statistical problems with ease. This software is frequently used in area of mathematics, statistics, economics, and engineering. This is a general purpose statistical computing system, designed especially for students and researchers. SPSSIBM is designed to be used interactively. This means that a command is immediately carried out and the results are shown. For the purpose of this research, regression analysis has been carried out using this software.
For the shear database, the response variable is the ultimate shear strength, predicted by code equations and other investigators, and the predictors are the concrete compressive strength f′_{c}, longitudinal main steel reinforcement ratio ρ _{ s }%, and Effective depth of beams d, shear span to effective depth ratio a/d, and Beam width, b.
4.2 Factors Influencing Ultimate Shear Strength
There are different approaches for analyzing the behavior of beams in shear. In this study, a semiempirical approach is adopted in which shear data collected from the past literature is analyzed and an equation is generated which can be used to predict the ultimate shear strength of a high strength concrete beam. This is done by using regression analysis using a variety of parameters and beam properties.
4.3 Data Sources

Cylindrical concrete compressive strength (20 ≤ fc′ ≤ 104 MPa).

Longitudinal main steel reinforcement ratio (0.17% ≤ ρ _{ s } % ≤ 6.64%).

Effective depth of deep beams, d, (127–1000 mm) & slender beams (70–2000 mm).

Shear span to effective depth ratio (1 ≤ a/d ≤ 2.5).

Beam width, b; b/d < 1, All database from literature were based on twopoint loading.
4.4 Need for Statistical Analysis
One of the statistical methods for modeling the association among two or more random variables and the predictors using a linear equation is regression analysis. The effect of one variable upon another can be investigated using regression analysis. A linear relationship between two random variables refers to simple linear regression and the linear relationship between more than two random variables is called multiple linear regressions. For this research, multiple linear regressions are adopted.
In the study of HSC beams in shear, experimental data collected from previous studies involving different parameters affecting the ultimate shear strength is collected. This analysis also gives an assessment of “statistical significance” of the estimated relationships, that is, the degree of confidence that the true relationship is close to the estimated relationship.
5 Proposed Equations and Codes Comparison: Results and Discussions
5.1 Proposed Design Formulas for Ultimate Shear Strength
Modified equations are proposed to predict the ultimate shear strength based on the statistical analysis of the abovementioned test data. Regression analysis yielded the following:
5.2 Comparisons of Test Results of Deep Beams with Codes and the Proposed Equations
To evaluate the three proposed models, nine wellknown expressions for computing the ultimate beam shear strength have been selected for comparison with respect to the overall number 743 of beams.
The experimental results were used to check the validity of the shear strength equations given by Egyptian Code “ECP2032010”, American Code “ACI 31814”, Canadian Code “CSA”, British Standard “BS8110” and some interested equations by other investigators. The test data specimens, here considered, have been collected from almost 240 papers, Ramadan (2015).
There is something have to say, before analyzing and evaluating the proposed equations, about choosing a certain depth 300 mm for the deep beams as a basic depth, Author’s definition for the current study, or as Shuraim (2014) named it as basic shear strength; the intersection of the shear demand curve and the shear resistance curve. Where, at beams/300 mm, the behavior of deep beams very different from beams < 300 mm in shear resistance mechanism. Iguro et al. (1984), on RC beams under uniformly distributed loading with beam depth varying between 100 mm and 3000 mm without shear reinforcement have revealed that as the effective depth of beam increases the shear strength decreases sequentially. As the depth of the beam increases, Rao and Prasad (2010), from 600 mm to 1200 mm, the diagonal cracking strength decreases with increasing the beam depth. Small size beams exhibited improved shear ductility, also as the depth of the beam increases, the beams exhibit relatively brittle failure about 20–25% decrease in the diagonal cracking strength was observed when the beam depth increases from 600 mm to 1200 mm. For (1.0 ≤ a/d ≤ 3.0), Kim and Park (1996) proposed a model for estimation of the shear strength of RC deep beams in which d ≥ 250 mm. Carmona and Ruiz (2014), Beam failure occurs when a flexural crack reaches a certain depth, which call critical depth, Zararis and Papadakis (2001). It depends on the crack position and on the boundary and loading conditions.
Acceptable results are obtained from the application of proposed Eqs. (12) and (13) on the 743 tested beams. According to the statistical analysis for measured to proposed values of deep beams ≥ 300, <300; the mean value of the observed shear strength to the predicted shear strength is 1.05, 1.07 with a standard deviation (S.D.) of 0.36, 0.26 and coefficient of variation (COV) is 0.35, 0.24 respectively. Based on the statistical analysis for measured values of deep beams ≥ 300, <300; the mean standard error of estimate (M.S.E) is found to be 0.17, 0.22 and the correlation coefficient (r) is 0.8, 0.78 respectively.
The horizontal line at (1) represents the perfect correspondence between experimental and computed shear strength values. Therefore, the closer to the line, the points are, the more accurate the shear strength prediction is. The thinner the width of the strip including the points is, the greater the prediction uniformity.
 1.
 2.
The scatter is greater in the other predicted code equations or others except ECP20310 and CSA Eqs.
 3.
From the points of view of safety (ѵ_{u,test}/ѵ_{u, pred.} > 1) the proposed equations are the best among all other equations considered in this work.
The performance of building codes and the proposed models were compared with the experimental results compiled from the literature. The comparison was assessed with six statistical parameters, which are commonly used for shear prediction models by researchers (Machial et al. 2012; Slater et al. 2012).
The proposed equations for predicting shear strength without web reinforcement were more conservative than the other equations. Comparing other equations ECP, ACI CSA, BS, SIP, Zararis, Bazant, Zsutty, and Shah in the value of AAE were 79, 86, 89, 61, 564, 157, 150, 139 and 114% higher than proposed equation “Ramadan1”. Except equations of codes, the others were more unconservative and unsafe. In addition, CSA had very high SD (2.29), r (0.02), COV (75%), and AAE (53%) which were higher than the ACI and BS values.
SIP proposed model was the least accurate in its prediction since it had the highest AAE value (186%). On the contrary, British model (BS8110) was the most conservative model where it’s average PF and SD values were 1.76 and 1.99, respectively. In case of other models, in spite of high SD, COV, and AAE values, the models still underestimated the shear strength since their average PF is upper one, Therefore, it was clearly observed that these models (i.e., SIP, Zararis, Bazant, Zsutty, and Shah) were poor in predicting the shear strength of deep beams without web reinforcement. After comparing all the models, it can be concluded that the proposed equations “Ramadan1” and “Ramadan2” predicted the shear strength more accurately and safely than the other models. Since both PF (1.05) and Correlation coefficient (r) are very close to unity and AAE value (28%) is the lowest one than the codes and other equations, so, this make “Ramadan1” and “Ramadan2” highly conservative.
6 Conclusions
An experimental investigation on the shear mechanism of reinforced concrete beams without web reinforcement made of HSC is presented. Two equations were proposed for predicting the ultimate shear strengths depending on the studied test variables. A detailed comparison was made between the experimental values of the ultimate shear strengths obtained from this study and from the literature and the estimated values using the proposed equations in this study and the other predictive equations proposed by other investigators.

HSRC deep beams without stirrups exhibit very brittle behavior.

The failure mode was significantly altered by changing the beam depth, where sufficient ductility was achieved in small size beams, and relatively very high brittleness was observed in large size beams.

The ultimate loads increase as shear span to depth ratio (a/d) decreases, this is because cracks form in the shear regions at places of high moments which are towards the applied concentrated loads, as shear span to depth ratio (a/d) decreases this distance also decreases, and the slope of the cracks become more steep.

The shear carrying capacity of tested deep beams was observed to increase in a slightly rate between (10–50%) with the increase in ρ_{s}% ratio, but in case of h = 700 mm, a/d = 2 and 1.5 a gradual decrease except some anomalies of the tested beams.

The main steel percentage increase, the increase in values of V_{test/}V_{pred.} was, measured to predicted shear strength values of codes and comparative equations, mainly recorded due to the dowel action which improves with the amount of longitudinal steel crossing the cracks. Hence, it may be noted that the tensile reinforcement significantly affects the deflection of a beam, thus this is the most important parameter in controlling deflections of HSC beams.

Two equations have been proposed for predicating the ultimate shear strength of reinforced concrete deep beams, for d ≥ 300 mm and d < 300 mm based on the multiple regression analysis for the test data of 743 reinforced concrete beams with and without stirrups failed in shear obtained from this study and from other previous investigators including the major factors affecting the ultimate shear strength.

The different design equations considered in this study do not accurately reflect the increase in shear capacity of beams with shorter shear spans (a/d = 1.5). Most of the design models are excessively conservative, and the code predictions only seem to be more accurate as a/d increases beyond a value of 2.0.

The ACI code prediction overestimates the ultimate shear strength of deep beams with overall depth 700 mm. Thus, it can be expected that the ACI code prediction regarding the ultimate shear strength of deep beam is conservative for HSRC deep beams with overall depth 400 and 700 mm.

SIP, Zararis, Bazant, Zsutty, and Shah were poor in predicting the shear strength of deep beams without web reinforcement. Comparing all the models, it could be concluded that the proposed equations “Ramadan1” and “Ramadan2” predicted the shear strength more accurately and safely than the other models since both PF (1.05) and Correlation coefficient (r) are very close to one and AAE value (28%) is the lowest one than the codes and other equations, so, this make “Ramadan1” and “Ramadan2” highly conservative.

ACI, CSA, and Shah models, in spite of high COV, and AAE values, the models still underestimated the shear strength since their average PF is upper one. It was clearly observed that these models (i.e., BS, SIP, Zararis, Bazant, and Zsutty) were poor in predicting the shear strength of slender beams without web reinforcement.
Notes
Acknowledgements
The authors would like to acknowledge the financial support from the Center of Excellence for Research in Engineering Materials (CEREM) at the College of Engineering—King Saud University. The help of engineers and technicians in the Center of Excellence for Concrete Research and Testing (CoECRT) and in the structural laboratory is highly appreciated.
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