# Experimental and Numerical Study on Ultimate Shear Load Carrying Capacity of Corroded RC Beams

Authors: Zhanzhan Tang; Zhixiang He; Zheng Chen; Lingkun Chen; Hanyang Xue; Hanqing Zhuge
Journal: Advances in Civil Engineering
 (2021)

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## Abstract

For an RC beam, the strength of steel rebar, the bonding strength between the concrete and reinforcement, and the bite action between the aggregates will deteriorate significantly due to corrosion. In the present study, 10 RC beams were designed to study the impact of corrosion on the shear bearing capacity. The mechanism of corrosion for stirrups and longitudinal bars and their effects were analyzed. Based on the existing experimental data, the correlation between the stirrup corrosion factor and the cross section loss rate was obtained. An effective prediction formula on the shear bearing capacity of the corroded RC beams was proposed and validated by the experimental results. Moreover, a numerical analysis approach based on the FE technique was proposed for the prediction of the shear strength. The results show that corrosion of the reinforcements could reduce the shear strength of the RC beams. The corrosion of stirrups can be numerically simulated by the reduction of the cross section. The formulae in the literature are conservative and the predictions are very dispersed, while the predictions by the proposed formula agree very well with the experiment results.


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## Body

## 1. Introduction

Most of the infrastructures, e.g., buildings and bridges, are exposed to the natural environment for decades and unavoidably be corroded due to the corrosive environments around them. The corrosion of a reinforced concrete (RC) beam always leads to the strength loss of the reinforcement, the degradation of material mechanical properties, the descending of the bond strength between the reinforcements and concrete, and the decrease of biting force between the inclined crack of the aggregates, which are the main factors decreasing the durability of an RC structure. At present, most studies focused on the deterioration of bending strength of RC beams caused by corrosion, rather than the decrease of the shear strength. Therefore, it is very significant to study the influence of corrosion on the shear performance of the RC beams and to propose effective formulae to facilitate the shear bearing capacity evaluation process.A series of studies have been carried out on the shear failure mechanism of the corroded RC beams. Corrosion is a continuous process, and the extension of cracks produced by structural corrosion is related to the loss of cross section [1–3]. Usually, the corrosion of reinforcement induces the crack initiation and propagation, and in turn, the development of cracks accelerates the corrosion [4]. Corrosion cracking also leads to the decrease of the bond strength between the reinforcement and concrete [5]. Khan et al. [6], Ye et al. [7], Zhao et al. [8], and Xu et al. [9] conducted experimental studies on the shear strength of corroded beams using the rapid electrochemical corrosion method. However, the effect of the longitudinal reinforcement corrosion was not involved. Sola et al. [3], Higgins et al. [10], Huo [11], and Wang et al. [12] investigated experimentally the shear performance of RC beams with consideration of the longitudinal reinforcement corrosion and the reduction of bond strength, while stirrup corrosion was ignored in these studies. Dai et al. [13] and Xue et al. [14] discussed the effect of longitudinal reinforcement corrosion on the shear performance of RC beams through a targeted method. The abovementioned studies focused on the influences of either the longitudinal reinforcement corrosion or the stirrup corrosion. However, the combined effect of them on the shear bearing capacity was not analyzed.Various improved prediction formulae and theoretical models were proposed after the establishment of the classical truss model, e.g., the variable angle truss model, the fixed pressure field theory, the truss-arch model, the pull-rod model, the limit equilibrium theory, the plasticity theory, and the statistical analysis. Based on the limit equilibrium theory, Xu et al. [9] analyzed the contribution proportions of the concrete and reinforcement to shear bearing capacity of the RC beams, in which the correlation coefficient of stirrup corrosion was introduced. El-Sayed [15] and Alaskar et al. [16] proposed a shear bearing capacity evaluation model for the corroded RC beams, in which the effective width of the member and the reduction of reinforcement cross section were considered. Zhao et al. [8] introduced the fitting empirical coefficient Pv to reduce the shear bearing capacity of noncorroded RC beams using the equivalent truss theory. Li et al. [17] considered the effective width of the geometrical dimension of the member after corrosion and the reduction of the yield strength of the longitudinal reinforcements when evaluating the shear bearing capacity of the corroded RC beams. Yu et al. [18] proposed a precise prediction formula which includes the angle of the critical inclined crack and the reduction of the cross section. However, most of the proposed formulae ignore either the impact of the longitudinal reinforcement corrosion or the deterioration of the stirrup and the decrease of the bonding strength. Therefore, the predictions are very conservative. Although the prediction of the formula proposed by Yu et al. [18] agrees well with the experiments, the method is very complicated when applying in engineering practice. Finite element analysis was carried out to study the mechanical performance of RC beams in recent years. Hawileh et al. [19] examined the critical parameters that influence the effectiveness of side-bonded EB-FRP systems through a newly developed finite element (FE) model. Naser et al. [20] discussed an advanced finite element simulation as a mean to understand and predict the performance of FRP-strengthened structures.In this study, the influences of stirrups and longitudinal reinforcement corrosion on the shear bearing capacity of RC beams were experimentally and theoretically studied. A short-time test was adopted to facilitate the experiment, in which the longitudinal reinforcements were wrapped by a thin Teflon insulated film with perfect nonadhesion property to simulate the nonbonding effect, and the axial rust swelling crack was simulated by filling an acrylic plate. A more practical formula for the shear bearing capacity evaluation of corroded RC beams was proposed, whose effectiveness was validated by the experimental results. Moreover, numerical analysis based on the FE method was carried out and compared with the test results.


## 2. Structural Tests and Discussions

### 2.1. Shear Strength Test

A series of experimental studies on the shear bearing capacity of corroded RC beams have been carried out in the literature. However, the time costs of these experiments are extremely high; thus, in this study, a short-time test was adopted. The mechanical cutting method was adopted to simulate the stirrup corrosion, the wrapping of insulated Teflon film was adopted to simulate the unbonding effect of the corroded longitudinal reinforcements, and the filling of acrylic plate was adopted to simulate the rust expansion cracks. Sufficient bending reinforcements were utilized so that only shear failure could occur. The steel and concrete materials specified by the Japan JIS-G-3112-2004 standard [21] and the Japan Architectural Institute JASS5 standard [22] were used. In the test, 10 beams were designed. The specimens were divided into two groups, in which 2 specimens were designed for the stirrup corrosion and 2 specimens were designed with consideration of the bonding strength and 1 ordinary RC beam for comparison purpose in Group A. In Group B, 2 specimens were designed for simulating the combined effects of bond strength and rust swelling crack and 2 specimens were designed to simulate rust swelling crack and 1 ordinary beam for comparison. Table 1 lists the design parameters of the specimens, where η1 and ηw are the cross section loss rates due to corrosion of the longitudinal reinforcements and stirrups, respectively. s and λ are the stirrup spacing and the shear span ratio, respectively.Table 1
Specimen design parameters.
GroupNo.Dimension (mm)s (mm)λCorrosionη1 (%)ηw (%)AA1125 × 200 × 15001252.94No corrosion00A2125 × 200 × 15001252.94Stirrup corrosion025.2A3125 × 200 × 15001252.94Stirrup corrosion044.2A4125 × 200 × 15001252.94Longitudinal bar corrosion23.10A5125 × 200 × 15001252.94Longitudinal bar corrosion41.10BB1150 × 235 × 22001802.93No corrosion00B2150 × 235 × 22001802.93Longitudinal bar corrosion20.00B3150 × 235 × 22001802.93Longitudinal bar corrosion26.00B4150 × 235 × 22001802.93Longitudinal bar corrosion44.10B5150 × 235 × 22001802.93Longitudinal bar corrosion69.40The extent of the bond strength degradation can be determined by the corrosion level [23]. The unbonded longitudinal reinforcements can be converted to a mass loss rate. The relationship between the reduction factor of bond strength ξ and the mass corrosion rate of the reinforcement ηm is [24](1)ξ=1.0+0.125×ηm,ηm≤2.0%,1.59×ηm−0.35,ηm>2.0%.Similarly, the simulated rust swelling crack can be transformed into the corresponding area loss rateηs (ηs = ΔAsm/As, where ΔAsm and As are the average area loss and the original area of reinforcement, respectively). The relationship between ΔAsm and the corrosive crack width w is [25](2)w=0.1916ΔAsm+0.164.Before the structural test, material tests were conducted. Figure1(a) shows the tensile test of the reinforcements, in which a 50 kN universal testing machine was used. Four types of reinforcements with different diameters, including D6 (SD295), D10 (SD295), D13 (SD345), and D22 (SD490), were tested. The specimen ID of D6 indicates the nominal diameter of the steel bar, and SD295 in the parenthesis represents the strength grade of steel bar, which indicates the yield point or 0.2% guaranteed strength is greater than 295 MPa. Standard tensile tests were carried out on the steel bars in the same batch. Different mass loss rates of 0%, 20%, and 40% were considered by the cutting treatment for D6 steel bars. Table 2 shows the mechanical properties of the tested reinforcements. As it can be seen, the effective elastic modulus, the effective yielding strength, and ultimate strength decrease due to the cross section loss. Moreover, ordinary Portland cement was used to get a target compressive strength of 21 MPa for the concrete. Standard material tests were conducted using the cylinder blocks (ϕ100 mm × 200 mm), and the measured compressive and tensile strengths are 24.1 MPa and 2.28 MPa, respectively. Figure 1(b) shows the shear strength test of the corroded RC beams, in which a 2 MN mechanical universal testing machine and a TDS-7130 (version 1.3) recorder were used. The midspan vertical displacement, the longitudinal strain at the midspan of the beam bottom, and the stirrup strain in the shear-compression zone were measured during the test.Figure 1
Material and structural tests. (a) Tensile tests of reinforcements. (b) Shear bearing capacity tests of beams.
(a)(b)Table 2
Mechanical properties of reinforcements.
TypeGradeEffective yielding strength (MPa)Effective ultimate strength (MPa)Effective elastic modulus (GPa)D6-0% mass loss ratesSD295356499193D6-20% mass loss ratesSD295305418155D6-40% mass loss ratesSD29522029797D10-0% mass loss ratesSD295374519200D13-0% mass loss ratesSD345391565199D22-0% mass loss ratesSD490533712192


### 2.2. Effect of Corrosion on Shear Bearing Capacity

A typical shear failure mode was observed from the tests. The shear bearing capacity declines due to the corrosion of the stirrup and the longitudinal reinforcements. Figure2 shows the relationship between the shear bearing capacity Vu of each beam and the corrosion ratio of the steel bars, in which the effects of the corrosion ratios of the stirrup ηw and the longitudinal reinforcement η1 were revealed. It can be seen from the tests of Group A that the shear bearing capacities of A2 and A3 beams with corroded stirrups decrease remarkably. At the early stage of the corrosion, the shear capacity decreases slightly. As the corrosion level becomes severe, the shear strength decreases significantly. The shear bearing capacity of A3 beam with a stirrup mass loss rate of 44.2% is 62.35 kN, which is 21.4% lower than that of the noncorroded beam. Likewise, the shear strength of A5 beam with a longitudinal reinforcement corrosion rate of 41.1% is only 58.25 kN, which is 26.6% lower than its original strength. The tests of Group B show that except for beam B4, the shear bearing capacity decreases with the increase of the corrosion level.Figure 2
Shear strength versus the corrosion ratio. Test results of (a) Group A and (b) Group B.
(a)(b)


## 2.1. Shear Strength Test

A series of experimental studies on the shear bearing capacity of corroded RC beams have been carried out in the literature. However, the time costs of these experiments are extremely high; thus, in this study, a short-time test was adopted. The mechanical cutting method was adopted to simulate the stirrup corrosion, the wrapping of insulated Teflon film was adopted to simulate the unbonding effect of the corroded longitudinal reinforcements, and the filling of acrylic plate was adopted to simulate the rust expansion cracks. Sufficient bending reinforcements were utilized so that only shear failure could occur. The steel and concrete materials specified by the Japan JIS-G-3112-2004 standard [21] and the Japan Architectural Institute JASS5 standard [22] were used. In the test, 10 beams were designed. The specimens were divided into two groups, in which 2 specimens were designed for the stirrup corrosion and 2 specimens were designed with consideration of the bonding strength and 1 ordinary RC beam for comparison purpose in Group A. In Group B, 2 specimens were designed for simulating the combined effects of bond strength and rust swelling crack and 2 specimens were designed to simulate rust swelling crack and 1 ordinary beam for comparison. Table 1 lists the design parameters of the specimens, where η1 and ηw are the cross section loss rates due to corrosion of the longitudinal reinforcements and stirrups, respectively. s and λ are the stirrup spacing and the shear span ratio, respectively.Table 1
Specimen design parameters.
GroupNo.Dimension (mm)s (mm)λCorrosionη1 (%)ηw (%)AA1125 × 200 × 15001252.94No corrosion00A2125 × 200 × 15001252.94Stirrup corrosion025.2A3125 × 200 × 15001252.94Stirrup corrosion044.2A4125 × 200 × 15001252.94Longitudinal bar corrosion23.10A5125 × 200 × 15001252.94Longitudinal bar corrosion41.10BB1150 × 235 × 22001802.93No corrosion00B2150 × 235 × 22001802.93Longitudinal bar corrosion20.00B3150 × 235 × 22001802.93Longitudinal bar corrosion26.00B4150 × 235 × 22001802.93Longitudinal bar corrosion44.10B5150 × 235 × 22001802.93Longitudinal bar corrosion69.40The extent of the bond strength degradation can be determined by the corrosion level [23]. The unbonded longitudinal reinforcements can be converted to a mass loss rate. The relationship between the reduction factor of bond strength ξ and the mass corrosion rate of the reinforcement ηm is [24](1)ξ=1.0+0.125×ηm,ηm≤2.0%,1.59×ηm−0.35,ηm>2.0%.Similarly, the simulated rust swelling crack can be transformed into the corresponding area loss rateηs (ηs = ΔAsm/As, where ΔAsm and As are the average area loss and the original area of reinforcement, respectively). The relationship between ΔAsm and the corrosive crack width w is [25](2)w=0.1916ΔAsm+0.164.Before the structural test, material tests were conducted. Figure1(a) shows the tensile test of the reinforcements, in which a 50 kN universal testing machine was used. Four types of reinforcements with different diameters, including D6 (SD295), D10 (SD295), D13 (SD345), and D22 (SD490), were tested. The specimen ID of D6 indicates the nominal diameter of the steel bar, and SD295 in the parenthesis represents the strength grade of steel bar, which indicates the yield point or 0.2% guaranteed strength is greater than 295 MPa. Standard tensile tests were carried out on the steel bars in the same batch. Different mass loss rates of 0%, 20%, and 40% were considered by the cutting treatment for D6 steel bars. Table 2 shows the mechanical properties of the tested reinforcements. As it can be seen, the effective elastic modulus, the effective yielding strength, and ultimate strength decrease due to the cross section loss. Moreover, ordinary Portland cement was used to get a target compressive strength of 21 MPa for the concrete. Standard material tests were conducted using the cylinder blocks (ϕ100 mm × 200 mm), and the measured compressive and tensile strengths are 24.1 MPa and 2.28 MPa, respectively. Figure 1(b) shows the shear strength test of the corroded RC beams, in which a 2 MN mechanical universal testing machine and a TDS-7130 (version 1.3) recorder were used. The midspan vertical displacement, the longitudinal strain at the midspan of the beam bottom, and the stirrup strain in the shear-compression zone were measured during the test.Figure 1
Material and structural tests. (a) Tensile tests of reinforcements. (b) Shear bearing capacity tests of beams.
(a)(b)Table 2
Mechanical properties of reinforcements.
TypeGradeEffective yielding strength (MPa)Effective ultimate strength (MPa)Effective elastic modulus (GPa)D6-0% mass loss ratesSD295356499193D6-20% mass loss ratesSD295305418155D6-40% mass loss ratesSD29522029797D10-0% mass loss ratesSD295374519200D13-0% mass loss ratesSD345391565199D22-0% mass loss ratesSD490533712192


## 2.2. Effect of Corrosion on Shear Bearing Capacity

A typical shear failure mode was observed from the tests. The shear bearing capacity declines due to the corrosion of the stirrup and the longitudinal reinforcements. Figure2 shows the relationship between the shear bearing capacity Vu of each beam and the corrosion ratio of the steel bars, in which the effects of the corrosion ratios of the stirrup ηw and the longitudinal reinforcement η1 were revealed. It can be seen from the tests of Group A that the shear bearing capacities of A2 and A3 beams with corroded stirrups decrease remarkably. At the early stage of the corrosion, the shear capacity decreases slightly. As the corrosion level becomes severe, the shear strength decreases significantly. The shear bearing capacity of A3 beam with a stirrup mass loss rate of 44.2% is 62.35 kN, which is 21.4% lower than that of the noncorroded beam. Likewise, the shear strength of A5 beam with a longitudinal reinforcement corrosion rate of 41.1% is only 58.25 kN, which is 26.6% lower than its original strength. The tests of Group B show that except for beam B4, the shear bearing capacity decreases with the increase of the corrosion level.Figure 2
Shear strength versus the corrosion ratio. Test results of (a) Group A and (b) Group B.
(a)(b)


## 3. Shear Bearing Capacity Evaluation of Corroded RC Beams

### 3.1. Models in the Codes

In the Chinese and American specifications, the shear resistances of the longitudinal reinforcements and the concrete are separately considered for the RC beams, indicating the classical truss model is adopted. The shear strength prediction formula of RC beams under a concentrated load stipulated by the Chinese specifications is [26](3)Vcs=1.75λ+1.0ftbwh0+h0sAswfyw,where Vcs is the shear strength of the beam, ft is the concrete tensile strength, bw and h0 are the width and effective height of the cross section, respectively, Asw is the cross section area of the stirrup, and fvw is the yield strength of the stirrup.The shear strength prediction formula of an RC beam with stirrups in the American standard is [27](4)Vcs=φVc+Vs=φ0.17ωfc′bwh0+Aswfywh0s,where φ is the strength reduction coefficient of 0.85, Vc and Vs are shear strength provided by concrete and stirrups, respectively, ω is the concrete correction coefficient of 1.0, and fc is the compressive strength of the standard concrete cylinder.The variable angle truss model is adopted by the European standard, in which the concrete is treated as a diagonal compressive bar. The angle between the equivalent diagonal bar and the longitudinal axis of the beam is variable within a prescribed range [28]. Figure 3 shows the variable angle truss model, where θ is the inclination angle between the effective compressive rod of the concrete and the longitudinal axis and α is the angle between the effective tension rod caused by the stirrups and the longitudinal axis. Fcs is the compression force of the equivalent compression rod caused by concrete, and Fsw is the tensile force of the equivalent tension rod caused by the stirrup. z (=0.9h0) is the inner lever arm corresponding to the maximum bending moment in the element under consideration.Figure 3
Variable angle truss model in Eurocode 2.According to the specifications,cotθ ranges from 1.0 to 2.5, and the shear strength of the RC beam can be predicted by(5)VRd=VRd,c+VRd,s=1βCRd,ck100ρlfck1/3bwh0+1β0.9h0fywAswscotθ+cotαsinα,β=0.25,av<0.5h0,av2h0,0.5h0≤av≤2h0,1,av>2h0,where VRd is the shear strength of RC beams, VRd,c and VRd,s are shear strength provided by concrete and stirrups, respectively. CRd,c=0.18/γc, in which γc (=1.5) is the material coefficient of concrete. k = 1+200/h0 ≤ 2, and ρ1 is the longitudinal reinforcement ratio. fck is the characteristic compressive strength of concrete, fck = fcu,k/1.226 [29]. fcu,k is the standard value of the cubic concrete compressive strength. β is the coefficient of the shear span ratio, and av is the horizontal distance between the concentrated load point and the bearing support.The partial factor of the shear resistance contributed by the stirrups is taken as 1.0 in the Chinese and American specifications, indicating the angle between the diagonal compressive bar and the longitudinal axis is a constant value of 45°. Table3 shows the shear strength errors between the experimental results and the predictions by the formulae in the standards [29], where Kp is the ratio of the experimental strength to the prediction by the formulae and μkp and σkp are, respectively, the average value and discrete coefficient of Kp. It is found that the prediction of the European standard is more accurate due to the consideration of the angle θ, while the predictions based on the Chinese and American standards are very conservative.Table 3
Statistical parameters ofKp.
ParametersChinese standardAmerican standardEuropean standardμkp1.44031.84030.9697σkp0.23190.30080.2297


## 3.1. Models in the Codes

In the Chinese and American specifications, the shear resistances of the longitudinal reinforcements and the concrete are separately considered for the RC beams, indicating the classical truss model is adopted. The shear strength prediction formula of RC beams under a concentrated load stipulated by the Chinese specifications is [26](3)Vcs=1.75λ+1.0ftbwh0+h0sAswfyw,where Vcs is the shear strength of the beam, ft is the concrete tensile strength, bw and h0 are the width and effective height of the cross section, respectively, Asw is the cross section area of the stirrup, and fvw is the yield strength of the stirrup.The shear strength prediction formula of an RC beam with stirrups in the American standard is [27](4)Vcs=φVc+Vs=φ0.17ωfc′bwh0+Aswfywh0s,where φ is the strength reduction coefficient of 0.85, Vc and Vs are shear strength provided by concrete and stirrups, respectively, ω is the concrete correction coefficient of 1.0, and fc is the compressive strength of the standard concrete cylinder.The variable angle truss model is adopted by the European standard, in which the concrete is treated as a diagonal compressive bar. The angle between the equivalent diagonal bar and the longitudinal axis of the beam is variable within a prescribed range [28]. Figure 3 shows the variable angle truss model, where θ is the inclination angle between the effective compressive rod of the concrete and the longitudinal axis and α is the angle between the effective tension rod caused by the stirrups and the longitudinal axis. Fcs is the compression force of the equivalent compression rod caused by concrete, and Fsw is the tensile force of the equivalent tension rod caused by the stirrup. z (=0.9h0) is the inner lever arm corresponding to the maximum bending moment in the element under consideration.Figure 3
Variable angle truss model in Eurocode 2.According to the specifications,cotθ ranges from 1.0 to 2.5, and the shear strength of the RC beam can be predicted by(5)VRd=VRd,c+VRd,s=1βCRd,ck100ρlfck1/3bwh0+1β0.9h0fywAswscotθ+cotαsinα,β=0.25,av<0.5h0,av2h0,0.5h0≤av≤2h0,1,av>2h0,where VRd is the shear strength of RC beams, VRd,c and VRd,s are shear strength provided by concrete and stirrups, respectively. CRd,c=0.18/γc, in which γc (=1.5) is the material coefficient of concrete. k = 1+200/h0 ≤ 2, and ρ1 is the longitudinal reinforcement ratio. fck is the characteristic compressive strength of concrete, fck = fcu,k/1.226 [29]. fcu,k is the standard value of the cubic concrete compressive strength. β is the coefficient of the shear span ratio, and av is the horizontal distance between the concentrated load point and the bearing support.The partial factor of the shear resistance contributed by the stirrups is taken as 1.0 in the Chinese and American specifications, indicating the angle between the diagonal compressive bar and the longitudinal axis is a constant value of 45°. Table3 shows the shear strength errors between the experimental results and the predictions by the formulae in the standards [29], where Kp is the ratio of the experimental strength to the prediction by the formulae and μkp and σkp are, respectively, the average value and discrete coefficient of Kp. It is found that the prediction of the European standard is more accurate due to the consideration of the angle θ, while the predictions based on the Chinese and American standards are very conservative.Table 3
Statistical parameters ofKp.
ParametersChinese standardAmerican standardEuropean standardμkp1.44031.84030.9697σkp0.23190.30080.2297


## 4. Proposed Formula for Prediction of Shear Bearing Capacity

As the variable angle truss theory is more effective, the contributions of the concrete and stirrups to the shear resistance are discussed in this section. In the theory, the concrete between the diagonal cracks is treated as a pressure rod to resist the compressive force, while the stirrups are equivalent to an upper chord, and the bottom longitudinal bars are considered as a lower chord to resist the tensile force. In order to ensure that the stirrups can reach their yield strength before the concrete crushing, the EC2 stipulates that 21.8° <θ < 45° (1.0 ≤ cotθ ≤ 2.5) [28]. Supposing that α = 90°, then(6)cotθ=αcvv1fcdρwfyw−1,v1=0.6,fck≤60MPa,0.9−fck200>0.5,fck>60MPa,where αcv is a coefficient considering the stress state of the compression rod, and αcv = 1 for nonprestressed structures. ρw is the stirrup ratio, v1 is the reduction factor with consideration of the concrete cracking strength, and fcd (=αccfck/γc) is the design value of the concrete compression force in the direction of the longitudinal member axis. αcc is the influencing factor of the long-term action effect of concrete, which can be taken as 0.85 [28].As surface spalling of the concrete may occur due to severe corrosion of the reinforcements, the width of the beam should be reduced. According to the suggestions of Higgins et al. [10], the effective width bwc can be calculated by(7)bwc=bw,ηw≤30%,bw−2c+dsw+s5.5,ηw>30%,s≤5.5c,bw−5.5sc+dsw2,ηw>30%,s>5.5c,where c is the thickness of the concrete cover and dsw is the diameter of the stirrups. The relationship between the mass loss rate ηm and the area loss rate ηs is(8)ηs=0.015+0.97ηm,ηm≤10%,0.062+0.95ηm,ηm>10%.A theoretical prediction formula for shear strength evaluation of the corroded RC beams is proposed, in which the factors ofϕ and γ to, respectively, consider the corrosion impacts of the longitudinal reinforcements and the stirrups are involved:(9)VRc=ϕVRd,c+γVRd,s=1βϕCRd,ck100ρlfck1/3bwch0+γ0.9h0fywcAswscotθ,where VRc is the shear strength of corroded RC beams, bwc and fywc (=fyw1−1.077ηs/1−ηs) are, respectively, the effective width of the section and the nominal yield strength of the steel bars after corrosion. cotθ can be calculated using equation (6), in which ρw and fyw can be replaced by ρwc and fywc in consideration of corrosion. The corrosion factors of ϕ can be obtained by [11](10)ϕ=1.0,ηl≤5%,1.098−1.96ηl,ηl>5%.As the conditions of the stirrups have a significant effect on the shear performance of the corroded RC beams, the correlation between parametersγ and ηw is discussed. Thus, 10 tests in this study and 118 tests in the literature are selected for further analysis [9–12, 30–34]. Table 4 lists the summary of the experimental results on shear strength of the RC beams with corroded stirrups, in which fc is the design value of the concrete axial compressive strength and N is the number of tests in each literature.Table 4
Summary of experimental data on shear capacity of corroded beams with stirrups.
Data sourceNBeam no.fc (MPa)bw (mm)h0 (mm)ρl (%)ρw (%)fyw (MPa)Asw (mm2)s (mm)λThis study10A1∼A514.31251703.60.4136063.41252.94B1∼B514.31502052.50.2336063.41802.93Xu et al. [9]21A/B/A2∼C215.06∼16.251202001.920.32275571502.00A3∼C3/A5∼C515.06∼16.251202001.920.32275571502.00C/A1∼A315.06∼16.251202001.920.32275571501.00A4∼C4/A6∼C615.06∼16.251202001.920.32275571501.00Higgins et al. [10]88RA/8RD17.582545211.90.1965851012032.0410RA∼10RD20.042545211.90.1575851012542.0412RA/12RD17.762545211.90.135851013052.04Huo [11]14L1∼L149.61001701.940.44324661501.50Wang et al. [12]10BC2.0-1∼518.29∼26.741501501.290.31441.5571502.00BC3.0-1∼518.29∼26.741501501.290.31441.5572003.00Xia et al. [30]18A-0∼A-512.971202002.680.253321.867.21001.50B-0∼B-512.971202002.680.19321.867.21001.50C-0∼C-512.971202002.620.475463.9921501.50Yang et al. [31]81.5-1∼1.5-421.271502602.620.4754821011501.50L1-1∼L1-421.271502602.620.5614821011501.00Wang et al. [32]70#∼6#16.91∼18.322504552.340.447337.9662001.74Yu [33]8JL-L0∼JL-L511.02∼12.831802602.340.447280571202.22HJL-L1/L3/L511.02∼12.831802601.630.12280571202.22Lu et al. [34]24X1/X219.492002752.150.19/0.1433957150/2002/2.5X3/X419.492002752.150.2/0.25373101250/2003/3.5X5/X619.492002752.150.19/0.1445857150/2002/2.5X7/X819.492002752.150.2/0.25433101250/2003/3.5X9/X1019.492002752.150.19/0.1447657150/2002/2.5X11/X1219.492002752.150.2/0.25524101250/2003/3.5Y1/Y223.862002752.150.19/0.1433957150/2002/2.5Y3/Y423.862002752.150.2/0.25373101250/2003/3.5Y5/Y623.862002752.150.19/0.1445857150/2002/2.5Y7/Y823.862002752.150.2/0.25433101250/2003/3.5Y9/Y1023.862002752.150.19/0.1447657150/2002/2.5Y11/Y1223.862002752.150.2/0.25524101250/2003/3.5Figure4 shows the collection of the experimental data and the fitting relation between the parameters γ and ηw. The experimental results show that the shear strength is almost unaffected when the corrosion rate is relatively small. Therefore, γ is taken as 1.0 in this study when ηw is less than 10%, indicating the shear strength of the corroded RC beam can retain the original strength as the noncorroded one. When the corrosion rate becomes larger, the shear strength decreases gradually, and a linear fitting relation between parameters γ and ηw is adopted:(11)γ=1.0,ηw≤10%,1.076−0.76ηw,ηw>10%.Figure 4
Correlation betweenγ and ηw.


### 4.1. Verification of Prediction Formula

The ratioKp (=Vu,exp/Vu) between the experimental strength Vu,exp and the predicted value Vu is obtained. Meanwhile, the mean value and the dispersion coefficient of the ratio are calculated. Figure 5 shows the comparison between the experimental and the predicted shear strengths using different theoretical models, where μkp is the mean value of the ratio Kp and σkp is the corresponding coefficient of variation. The predicted strengths obtained from the theoretical models are different from each other. The formula proposed by Yu is based on the modified pressure field theory, in which the influence of corrosion on the angle of critical inclined crack, geometric reduction, and the stirrup ratio is considered. Therefore, the predictions are in a good agreement with the test values. The mean value of KP is 1.0785 with a dispersion coefficient of 0.2834. However, complicated iterative computation is required in Yu’s model, making it very difficult for engineering practice. Huo’s model is based on the truss-arch theory, and the accuracy of the prediction is lower than Yu’s model. The model proposed by El-Sayed is based on the American standard, which considers the reduction of the cross section and the effective shear section caused by corrosion. Nevertheless, the inclination of the crack in this model is assumed as a fixed value of 45°, so the predictions are rather conservative. The strength is a sum of the shear resistances of the stirrup and the concrete in the Xu and Li’s models. The angle between the baroclinic bar and the longitudinal axis is also a constant value, and the corrosion of the longitudinal reinforcement is not considered. Thus, conservative predictions are obtained. Noticeably, the average value of Kp is 1.0027 with a dispersion coefficient of 0.172 in the proposed model, indicating the highest precision with a relatively small dispersion can be achieved by the proposed formula in this study.Figure 5
Comparison between the experimental and theoretical shear strength of the corroded RC beams. (a)Vu1 predicted by Xu’s model. (b) Vu2 predicted by Huo’s model. (c) Vu3 predicted by Yu’s model. (d) Vu4 predicted by Li’s model. (e) Vu5 predicted by El-Sayed’s model. (f) Vd predicted by the proposed model in this study.
(a)(b)(c)(d)(e)(f)


### 4.2. Numerical Simulation of Shear Capacity of Corroded Beams

Finite element analysis is carried out to study the shear failure of the corroded RC beam in this section. Figure6 shows the three-dimensional FE model of the RC beam. Commercial software ABAQUS was adopted for the numerical analysis. The concrete beam was modeled by 3D solid element (C3D8). The longitudinal reinforcements and stirrups were modeled by truss elements (T3D2). The plastic-damage constitutive model was adopted for the simulation of the concrete material, and the bilinear plastic constitutive model was used for the mechanical behavior of the reinforcements [35–37]. The cross section reduction method was adopted for the simulation of the reinforcement corrosion [38–41]. The coupling interaction between the longitudinal reinforcement and the concrete was utilized to control they were fully bonded or disconnected [42]. The meshing of the structure was carried out using the structured meshing method during the FE modelling.Figure 6
FE model of the corroded RC beam. (a) Framework of the reinforcement. (b) Mesh of the model.
(a)(b)Figure7 shows the comparison of the load-deflection curves obtained from the numerical and the test results. It can be seen that the numerical results of both the test beams in Group A and Group B (FEM) are in good agreement with the experimental results (EXP). The effect of the corrosion can be well simulated by the section reduction of the rebar and the coupling interaction between the reinforcement and concrete beam.Figure 7
Comparison between the numerical and experimental results. Load-deflection curves of beams in (a) Group A and (b) Group B.
(a)(b)128 models of the corroded RC beams in the literature with stirrups were created, and the shear bearing capacity was computed. Figure8 shows the comparison between the numerical and the experiment results of the shear strength. Electrochemical corrosion or other fast corrosion methods were adopted by most of tests in the literature. Therefore, the corroded reinforcement was computed using the cross section reduction technique during the numerical analysis. As it can be seen, the mean value of the ratio Kp (=VFEM/Vu,exp) is 1.0160 and the corresponding dispersion coefficient is 0.1301. Kp is higher than the mean value predicted by the proposed formula, but the dispersion coefficient is lower, indicating that the numerical results using the FE method are more accurate than the proposed theoretical method.Figure 8
Comparison between numerical and experimental results for 128 specimens.Figure9 shows the comparison of the numerical results and the experimental or theoretical results. It can be found that it is feasible to evaluate the shear strength of the corroded beam by the cross section reduction method and the interaction technique. Moreover, the effectiveness of the proposed formula is verified by the numerical analysis.Figure 9
Comparison of the numerical results and the experimental or theoretical results. Ratios of (a)Vu,exp/VFEM and (b)Vu,exp/Vd.
(a)(b)


## 4.1. Verification of Prediction Formula

The ratioKp (=Vu,exp/Vu) between the experimental strength Vu,exp and the predicted value Vu is obtained. Meanwhile, the mean value and the dispersion coefficient of the ratio are calculated. Figure 5 shows the comparison between the experimental and the predicted shear strengths using different theoretical models, where μkp is the mean value of the ratio Kp and σkp is the corresponding coefficient of variation. The predicted strengths obtained from the theoretical models are different from each other. The formula proposed by Yu is based on the modified pressure field theory, in which the influence of corrosion on the angle of critical inclined crack, geometric reduction, and the stirrup ratio is considered. Therefore, the predictions are in a good agreement with the test values. The mean value of KP is 1.0785 with a dispersion coefficient of 0.2834. However, complicated iterative computation is required in Yu’s model, making it very difficult for engineering practice. Huo’s model is based on the truss-arch theory, and the accuracy of the prediction is lower than Yu’s model. The model proposed by El-Sayed is based on the American standard, which considers the reduction of the cross section and the effective shear section caused by corrosion. Nevertheless, the inclination of the crack in this model is assumed as a fixed value of 45°, so the predictions are rather conservative. The strength is a sum of the shear resistances of the stirrup and the concrete in the Xu and Li’s models. The angle between the baroclinic bar and the longitudinal axis is also a constant value, and the corrosion of the longitudinal reinforcement is not considered. Thus, conservative predictions are obtained. Noticeably, the average value of Kp is 1.0027 with a dispersion coefficient of 0.172 in the proposed model, indicating the highest precision with a relatively small dispersion can be achieved by the proposed formula in this study.Figure 5
Comparison between the experimental and theoretical shear strength of the corroded RC beams. (a)Vu1 predicted by Xu’s model. (b) Vu2 predicted by Huo’s model. (c) Vu3 predicted by Yu’s model. (d) Vu4 predicted by Li’s model. (e) Vu5 predicted by El-Sayed’s model. (f) Vd predicted by the proposed model in this study.
(a)(b)(c)(d)(e)(f)


## 4.2. Numerical Simulation of Shear Capacity of Corroded Beams

Finite element analysis is carried out to study the shear failure of the corroded RC beam in this section. Figure6 shows the three-dimensional FE model of the RC beam. Commercial software ABAQUS was adopted for the numerical analysis. The concrete beam was modeled by 3D solid element (C3D8). The longitudinal reinforcements and stirrups were modeled by truss elements (T3D2). The plastic-damage constitutive model was adopted for the simulation of the concrete material, and the bilinear plastic constitutive model was used for the mechanical behavior of the reinforcements [35–37]. The cross section reduction method was adopted for the simulation of the reinforcement corrosion [38–41]. The coupling interaction between the longitudinal reinforcement and the concrete was utilized to control they were fully bonded or disconnected [42]. The meshing of the structure was carried out using the structured meshing method during the FE modelling.Figure 6
FE model of the corroded RC beam. (a) Framework of the reinforcement. (b) Mesh of the model.
(a)(b)Figure7 shows the comparison of the load-deflection curves obtained from the numerical and the test results. It can be seen that the numerical results of both the test beams in Group A and Group B (FEM) are in good agreement with the experimental results (EXP). The effect of the corrosion can be well simulated by the section reduction of the rebar and the coupling interaction between the reinforcement and concrete beam.Figure 7
Comparison between the numerical and experimental results. Load-deflection curves of beams in (a) Group A and (b) Group B.
(a)(b)128 models of the corroded RC beams in the literature with stirrups were created, and the shear bearing capacity was computed. Figure8 shows the comparison between the numerical and the experiment results of the shear strength. Electrochemical corrosion or other fast corrosion methods were adopted by most of tests in the literature. Therefore, the corroded reinforcement was computed using the cross section reduction technique during the numerical analysis. As it can be seen, the mean value of the ratio Kp (=VFEM/Vu,exp) is 1.0160 and the corresponding dispersion coefficient is 0.1301. Kp is higher than the mean value predicted by the proposed formula, but the dispersion coefficient is lower, indicating that the numerical results using the FE method are more accurate than the proposed theoretical method.Figure 8
Comparison between numerical and experimental results for 128 specimens.Figure9 shows the comparison of the numerical results and the experimental or theoretical results. It can be found that it is feasible to evaluate the shear strength of the corroded beam by the cross section reduction method and the interaction technique. Moreover, the effectiveness of the proposed formula is verified by the numerical analysis.Figure 9
Comparison of the numerical results and the experimental or theoretical results. Ratios of (a)Vu,exp/VFEM and (b)Vu,exp/Vd.
(a)(b)


## 5. Conclusions

The shear bearing capacity of the corroded RC beams was experimentally and numerically studied. A short-time test was adopted to facilitate the experimental study, and a more practical formula for the shear strength evaluation was proposed and validated by the experimental data. Moreover, numerical analysis based on the FE method was carried out and compared with the test results. The following conclusions can be drawn:(1)
Stirrup corrosion has a significant effect on the shear strength of the RC beams, and the shear strength decreases gradually along with the increase of the corrosion rate. The axial rust swelling crack and the degradation of the bond strength due to longitudinal reinforcement corrosion can also decrease the shear strength. Moreover, the wrapping of the thin Teflon insulated film on the longitudinal reinforcements is an effective way to simulate the nonbonding effect, and the filling of an acrylic plate is feasible for the simulation of rust swelling crack in the short-time test.(2)
Most of the existing theoretical models for the shear strength estimation of the corroded RC beams are conservative and the predicted results are very dispersed, while the models with relatively higher accuracy are complicated in application. As the corrosion effects of the longitudinal reinforcement and the stirrup are both considered, the predictions by the proposed formula in this study are in good agreement with the experimental results, and the formula is very simple and easy for engineering application.(3)
The numerical results based on the FE technique agree very well with the experimental results, which indicates that the corrosion effect can be well simulated by the cross section reduction of the reinforcements and the coupling interaction between the reinforcement and the concrete beam. Moreover, the proposed prediction formula is validated by the numerical simulations.



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*Source: 4848483-2021-09-27.xml*