# Relationship between Corrosion of Reinforcement and Surface Cracking Width in Concrete

Authors: Jian Cao; Liangfang Liu; Shangchuan Zhao
Journal: Advances in Civil Engineering
 (2020)

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## Abstract

The durability of structure cannot be guaranteed when a corrosion expansion crack reaches the surface of the reinforced concrete member. In this paper, firstly, based on the existing theoretical model of steel corrosion degree, the calculation process of the model and the determination of the relevant parameters in the model were analyzed and discussed. Secondly, the stiffness reduction factor of concrete in the model was calculated according to the existing experimental data, and the engineering formula of the steel corrosion degree was established, which was related to the surface crack width of reinforced concrete. Moreover, the experiments of steel bar corrosion were carried out with different components of surface crack width, in which the parameters of the bar diameter, concrete protection layer thickness, and water-cement ratio were taken into consideration. The experimental phenomena and results were further analyzed and discussed. Finally, comparing with the experimental data, the engineering formula presented in the paper was validated. The results show that the calculated values by the engineering formula are in better agreement with the experimental values than those by the existing model, which provide a theoretical basis for further study on the durability limit state of the structure.


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## Body

## 1. Introduction

Steel bar in a reinforced concrete structure located in coastal areas are bounded to different degrees of corrosion due to chloride ions, resulting in cracking and peeling of the concrete cover, which is badly endangering the reliability of the structure. Although there are different views on how to define the structural durability limit state, it is affirmed that the cracking of the concrete cover induced by the steel corrosion is one of the durability limit states of the reinforced concrete structure [1].According to the serviceability limit states of structure, there is a limit for the presence of surface crack width of the concrete component. However, it is possible that the structure of security or durability has been destroyed before the limit is reached. Present studies mainly focus on the two stages of the off time of steel bar and the crack initiation time of the concrete protective layer, and the calculation equations of the time of the steel bar corrosion and crack expansion have also been deduced [2–4]. However, the investigation on the corrosion rate of steel after corrosion cracking is still insufficient. However, in the tests, the researchers found that the corrosion degree has a significant effect on the component surface crack width, and it is in relevance with the parameters including the concrete strength grade, component size, thickness of concrete protection layer, and diameter of the steel [5–8]. There is a lack of contrast and regularity among the results due to different methods and conditions of the tests. In the aspect of numerical simulation, the finite element models established by Du et al. [9] and Chen and Leung [10] simulated the corrosion process of the component surface, and the evolution law of the corrosion expansion stress was calculated by the numerical analysis method. However, in the process of finite element modeling, certain assumptions on the properties and constitutive relation of materials were required to put forward, on which the test data were insufficient for comparison, so the results of numerical calculation were not validated. In the aspect of the theoretical model, Cao et al. [11] discussed the interaction between the crack propagation and the corrosion medium propagation and presented a theoretical model of the corrosion rate of the steel bar during the coupling process of micro- and macrochloride corrosion. Based on the size effect and reliability theory, taking into account the actual size of the structure and physical environment, Papakonstantinou and Shinozuka [12] proposed the calculation model of the corrosion rate of reinforcing steel bar in the large size structure. According to Bossio et al. [13, 14], three- and four-layer models for concrete cracking were proposed, as well as a model to correlate surface cracks with internal corrosion depending on many parameters (distance of cracks on external surface, concrete cover, bar dimeter, and accounting for different reliabilities of these data at inspection time). Paul et al. [15–18] investigated on accelerated chloride-induced corrosion in cracked reinforced strain-hardening cement-based composites and reinforced mortar beams, both in loaded and unloaded states. Li et al. [19–22] carried out a series of theoretical and experimental studies on the relationship between the surface crack width and rust expansion rate, combining with concrete material mechanics and elastic mechanics theory. They provided an analytical model for corrosion-induced crack width in the concrete structure when taking the softening characteristics of concrete after cracking into consideration. Moreover, Lau et al. [23] proposed a new methodology for predicting the time to corrosion-induced concrete cracking based on fracture mechanics criteria. Nevertheless, there are still disadvantages because of the complex propagation behaviors of corrosive cracks in the concrete and insufficient knowledge on the corrosive mechanism of concrete, which presents the complex calculation process in the existing model and significant discreteness of the computing results, as well as the difficulties to determine the parameters in the model. Recently, the effects of nonuniform rebar corrosion are concerned and accurate performance evaluation can be achieved by experimental studies [24–27].As one of the basic parameters that can reflect the applicability of the structure, the width of the corrosion expansion crack can be easily obtained in the detection of the actual structure. Thus, on the premise that the structure safety is not destroyed, which has a great significance to adopt nondestructive detection on surface crack width of measurement units for tracking the corrosive crack propagation behavior and grasping the dynamic rules of reinforcement corrosion in concrete. In this paper, firstly, based on the existing studies, the calculation process and related parameters of the existing model [19] were analyzed and discussed. Secondly, in order to simplify the calculation of the model [19], the existing test data were utilized for the regression calculation of stiffness degradation factor in the model [19], and the engineering formula of steel corrosion degree was presented in the paper. Moreover, considering the parameters including the bar diameter, concrete cover thickness, and water-cement ratio, tests concerning the relationship between the steel corrosion degrees and crack width were conducted, and the phenomena and results of the tests were analyzed and discussed. Finally, the engineering formula presented in the paper was compared and validated by the test data, and suggestions on its applicability were proposed.


## 2. Materials and Methods

### 2.1. Li-CQ’s Model

Based on the material mechanics and elasticity theory of concrete, Li et al. [19, 20] analyzed the expansion cracking region of the damaged protective layer of concrete resulted from steel corrosion, considering concrete with embedded steel bars to be a thick wall cylinder, as shown in Figures 1(a)–1(c).Figure 1
Schematic of crack evolution: (a) reinforced concrete considered to be a thick wall cylinder; (b) expansion stress caused by corrosion products; (c) concrete cover cracking.
(a)
(b)
(c)In Figure1(a), D is the diameter of reinforcement bar; C is the thickness of the concrete cover; the inner and outer radii of the thick wall cylinder are a = (D + 2d0)/2 and b = C + (D + 2d0)/2; r means the distance from any point in the thick walled cylinder of the concrete protection layer to the center of the steel bar, which is the function of the radial coordinate; d0 is the thickness of the annular layer of concrete pores (that is, a pore band) at the interface between the bar and concrete; and d0 is usually constant once concrete has hardened.In Figure1(b), P1(t) is the applied load at time t via the transverse (radial) component of the bond stress between the bar and concrete caused by the products of corrosion of the reinforcing steel bar and ds(t) is the thickness of a ring of the corrosion product formed as the corrosion propagates in concrete at time t, which can be obtained by the following equation:(1)dst=WrusttπD+2d01ρrust−αrustρst,where αrust is a coefficient related to the type of corrosion products, ρrust is the density of corrosion products, ρst is the density of steel, and Wrustt is the mass of corrosion products at time t. Then, the degree of steel corrosion can be expressed by(2)θ=WrusttW×100%,where W is the initial mass of steel bar and θ is the degree of steel corrosion (%).According to Bažant and Planas [28], the stiffness reduction factor α is dependent on the average tangential strain εθ¯ throughout [a, r0], which is represented as follows:(3)α=ftexp−γεθ¯−εθc¯Eefεθ¯,where ft is the tensile strength of concrete, εθc¯ denotes the average tangential cracking strain, and γ is a material constant, which can be defined between 5000 and 10000. According to [29], the approximate value of γ is 7500.After initiation, the compressive stress of the thick wall cylinder of the concrete protection layer is increasing with the continuous growth of corrosion products. The crack divides the thick wall cylinder into two coaxial cylinders: inner cracked and outer uncracked ones, as shown in Figure1(c). According to the elastic theory of concrete, as the result of the corrosion crack in concrete protective layer arising, the governing equation for the displacement u(r) of any point r in the concrete protective layer should satisfy the following equation [19]:(4)d2urdr2+1r⋅durdr−αurr2=0,in which u is the displacement of the point O along the s-axis, as shown in Figure 2.Figure 2
Model for calculation of initial crack width.The solution to equation (4) can be expressed as(5)ur=c1r0rα+c2r0r−α.At present, the boundary conditions of equation (5) require that(6)c11+νcbα−1−c21−νcb−α−1=0,(7)c1aα+c2a−α=dst,where νc is Poisson’s ratio of concrete in equation (6). c1 and c2 can be obtained by solving equations (6) and (7) simultaneously as follows:(8)c1=1−νcaα1−νca2α+1−νcb2αdst,(9)c2=1−νcaαb2α1−νca2α+1−νcb2αdst.According to Li et al. [19], the corresponding average tangential strain εθ¯ can be indicated as follows:(10)εθ¯=1b−a∫abεθrdr=1b−a∫abc1rα−1+c2r−α−1dr=bα−aαc1+c2/abααb−a.The average tangential strain over the cracked surfaceεθc¯ can be determined as follows:(11)εθc¯=1b−a∫abc1r+c2rr2dr.Consequently, the stiffness reduction factorα can be indicated as(12)α=ftexp−γbα−aαc1+c2/abα/αb−a−1/b−a∫abc1r+c2r/r2drEefbα−aαc1+c2/abα/αb−a.Afterward, by substituting equations (8) and (9) into equation (12), the relationship between α and ds(t) can be calculated.Withc1 and c2 known, the tangential strain at b can be determined from equation (5), that is,(13)εθb=c1bα−1+c2b−α−1.In Figure2, evidently, in accordance with the geometric relationship between the width and the angle of the crack on the surface of the concrete component, the crack width wc on the surface of the concrete cylinder should be(14)wc=2πbεθb−εθe,mb,where εθe,mb is the maximum elastic strain at r = b and equal to the following equation [30]:(15)εθe,mb=σθ,mb−νcσrbEef,where σθ,m(b) and σr(b) are the maximum tangential stress and radial stress at r = b, respectively, and Eef is the effective elastic modulus of concrete, in which Eef = Ec/(1 + φcr) and φcr is the coefficient of concrete creep.On account ofσr(b) = 0 and σθ,m(b) = ft, the crack width wc is finally given by(16)wc=2πbc1bα−1+c2b−α−1−ftEef=4πdst1−νca/bα+1+νcb/aα−2πbftEef,where the essential variables are the thickness of corrosion products ds(t) and the stiffness reduction factor α. ds(t) is directly related to the corrosion degree, as shown in equation (1), and α is related to concrete geometry and property. Obviously, with the accumulation of corrosion products, the crack width increases with time. This makes sense both theoretically and practically, as experienced and observed [6–8, 31].In [32, 33], the values of basic variables in equations (1) and (16) had been provided, where d0 = 12.5 µm, αrust = 0.57, ρrust = 3600 kg/m3, and ρst = 7850 kg/m3. Parameters a, b, D, C, ft, Ec, vc, and φcr can be determined by the size of the component and the performance of concrete materials.Therefore, the calculation process of Li-CQ’s model includes the following steps:(1)
Determine the basic parameters of reinforced concrete members, and the relationship between the steel corrosion product thicknessds(t) and concrete stiffness reduction factor α can be calculated in association with equations (8), (9), and (12).(2)
Substitute the relationship withds(t) and α to equation (16), and the relationship between the crack width wc and ds(t) can be obtained.(3)
Substituteds(t) represented by wc to equation (1), and the relationship between wc and Wrustt can be established. Furthermore, the model of the corrosion degree of steel bar can be calculated by equation (2).


### 2.2. Engineering Formula considering the Reduction Coefficient of Stiffness

In Li-CQ’s model, the calculation process of parametersds(t) and α is an iterative process, which is more complicated and inconvenient. In addition, when the difference method is used to calculate equation (12), the accuracy of calculation results could not be guaranteed if the difference step is large, while the computation burden will be doubled if the difference step is small. Therefore, parameters in the Li-CQ’s model should be simplified for engineering applications.In equation (16), parameters D, C, ft, and Ec can be determined by the size of the current concrete members and concrete material properties. The value of Poisson’s ratio vc is 0.18 in common. Concrete creep coefficient φcr can be obtained by CEB-FIP Model Code 2010 [33].For concrete stiffness reduction factorα, considering the influence of the corrosion time of steel bar, the data in the literature [20–22, 34–38] were applied to calculate the simplified calculation of α′(t), and the nonlinear regression equation can be indicated as follows:(17)α′t=0.01137638×t365−2.198836253+0.1235655576adj.R2=0.974,where α′(t) denotes modified concrete stiffness reduction factor and t is the time of steel corrosion, day as the unit.Consequently, substituting equations (16) and (17) into equation (1), the modified parameter of the mass of corrosion products Wrustαt can be proposed as follows:(18)Wrustαt=wc+2πbft/Eef⋅D+2d0⋅1−νca/bα′t+1+νcb/aα′t4⋅1/ρrust−αrust/ρst.Thus, the modified calculation of the steel corrosion degree can be expressed as(19)θα=WrustαW×100%.


### 2.3. Experimental Programs

Experiments were designed to investigate the influence on the relationship between corrosion degree of steel bar and corrosion-induced crack width by parameters including water-cement ratio, the thickness of concrete cover, and diameter of steel bar, and the engineering formula presented above was verified through the experiment.


#### 2.3.1. Mix Design

In the experiments, materials including ordinary Portland cement (with 60.6 MPa of the 28-day standard compressive strength), continuous grading gravel with diameters from 5 to 25 mm, medium sand with diameters from 0.08 to 2.0 mm, as well as a kind of naphthalene-based superplasticizer were used to produce concrete. The yield strength of steel bar is 335 MPa. Experiments [39] indicate that a certain amount of chloride mixed with concrete can significantly reduce the resistivity of the concrete and distinguish steep corrosion speed because of different mixed concretes. Under this circumstance, not only the steel corrosion degree is accelerated but also an appropriate corrosion current density is applied. The mix proportion is listed in Table 1.Table 1
Mix proportion of concrete (kg/m3).
No.
W
/
C
Water : cement : sand : macadam : water-reducing agent : sodium chloride
S1
0.3
126 : 378 : 586 : 1244 : 8.4 : 4.2
S2
0.4
168 : 378 : 549 : 1168 : 8.4 : 4.2
S3
0.5
210 : 378 : 514 : 1092 : 8.4 : 4.2


#### 2.3.2. Specimen Design

To achieve the experimental objective, the size of the specimen is 300 mm × 200 mm × 250 mm, as shown in Figure3.Figure 3
Size of the specimen (unit: mm).In Figure3, C1, C2, and C3 represent the thickness of concrete cover of the diameter steel bar of 12 mm, 20 mm, and 25 mm, respectively. The thickness of the concrete cover is 30 mm, 40 mm, and 50 mm, respectively. The predetermined corrosive cracks width on the member surface is set to 0.2 mm, 0.4 mm, 0.7 mm, and 1.1 mm. The orthogonal design method [40] of L9 (34) was adopted in the tests. The levels of test parameters are shown in Table 2, in which each number includes two specimens.Table 2
The levels of test parameters.
Parameter level no.
W
/
C
Thickness of concrete protection layer (mm)
Diameter of steel bar (mm)
Preset crack width in the surface of concrete (mm)
1
0.3
30
12
0.2/0.4/0.7/1.0
2
0.3
40
20
3
0.3
50
25
4
0.4
30
25
0.2/0.4/0.7/1.0
5
0.4
40
12
6
0.4
50
20
7
0.5
30
20
0.2/0.4/0.7/1.0
8
0.5
40
25
9
0.5
50
12In addition, the same batches of concrete specimens were indwelled to determine the basic 28 d mechanical properties of concrete materials: 6 groups of elasticity modulus specimens, 3 groups of cube compressive strength specimens, and 3 groups of splitting tensile strength specimens.


#### 2.3.3. Specimen Preparation

Firstly, the steel bars of different diameters were cut to 300 mm in length, the steel rust on the steel surface was cleaned, and the initial weight of steel was recorded. The molds were manufactured according to Figure3, and the steel bars were fixed in the reserved holes of the molds. Next, fresh concrete was poured into the molds and was vibrated compaction. The molds were removed after 24 hours; both ends of the steel bars were cleaned up and then welded with wires and sealed by epoxy resins. Finally, the test specimens and referenced specimens were put into the curing room together and conducted for testing after 28 days of standard curing. The formed reinforced concrete specimens are shown in Figure 4.Figure 4
The formed reinforced concrete specimens.


#### 2.3.4. Testing Procedure

In the accelerated steel corrosion test, the specimens were soaked in 5% NaCl solution and the steel bars were connected with each other in parallel via wires reinforced by the voltage of 60 V until the cracking on the specimen surface was developing into the preset width. After breaking the specimens in batches, corroded steels were soaked in hydrochloric acid solution next and then were washed clean after the surface acidic substance was neutralized by calcium carbonate solution. Finally, the corrosive steel reinforcements were weighted by the electronic balance accurate to 0.1 g, and the corrosion degrees of the steel bars were calculated. Experimental procedures are shown in Figures5 and 6.Figure 5
Energized accelerate steel corrosion test.Figure 6
Specimen breakages.


#### 2.3.5. Phenomena and Analysis of Experiment

In the experiment, it can be observed that (1) with the progress of energization and steel corrosion, it appeared heating was done within the specimen and the surface temperature was rising. The reason can be explained that concrete generates more heat, while current gets through the concrete in the condition of continuous power as it is a poor conductor with large resistance. (2) A small amount of bubbles and damp phenomenon could be observed at the concrete near the cathodic protection, and there were drops of water generated at reinforced ends. After energization for some days, there were cracks on the surface on the reinforced concrete specimen, and overflow rust obviously appeared in the position of cracks and the end of steel bars, which was caused by the concrete pores around the steel bars which was filled with the corrosion products generated due to continuous steel corrosion. (3) The larger the concrete water-cement ratio was, the earlier the crack appeared, and the greater the corrosion degree of steel was when cracks appeared. The thickness of concrete cover also had certain influence on the development patterns of crack. (4) After breaking the specimens, the longitudinal ribs of steels were usually the origin of the crack. This is because the stress concentration was arising at the longitudinal ribs of steel, resulting in large expansion force applied on concrete surrounding. (5) The longitudinal cracks, with uniform width, were formed after specimen cracking since the reinforced specimens were in uniform corrosion when they were accelerated to be corroded, so did the cracks.


## 2.1. Li-CQ’s Model

Based on the material mechanics and elasticity theory of concrete, Li et al. [19, 20] analyzed the expansion cracking region of the damaged protective layer of concrete resulted from steel corrosion, considering concrete with embedded steel bars to be a thick wall cylinder, as shown in Figures 1(a)–1(c).Figure 1
Schematic of crack evolution: (a) reinforced concrete considered to be a thick wall cylinder; (b) expansion stress caused by corrosion products; (c) concrete cover cracking.
(a)
(b)
(c)In Figure1(a), D is the diameter of reinforcement bar; C is the thickness of the concrete cover; the inner and outer radii of the thick wall cylinder are a = (D + 2d0)/2 and b = C + (D + 2d0)/2; r means the distance from any point in the thick walled cylinder of the concrete protection layer to the center of the steel bar, which is the function of the radial coordinate; d0 is the thickness of the annular layer of concrete pores (that is, a pore band) at the interface between the bar and concrete; and d0 is usually constant once concrete has hardened.In Figure1(b), P1(t) is the applied load at time t via the transverse (radial) component of the bond stress between the bar and concrete caused by the products of corrosion of the reinforcing steel bar and ds(t) is the thickness of a ring of the corrosion product formed as the corrosion propagates in concrete at time t, which can be obtained by the following equation:(1)dst=WrusttπD+2d01ρrust−αrustρst,where αrust is a coefficient related to the type of corrosion products, ρrust is the density of corrosion products, ρst is the density of steel, and Wrustt is the mass of corrosion products at time t. Then, the degree of steel corrosion can be expressed by(2)θ=WrusttW×100%,where W is the initial mass of steel bar and θ is the degree of steel corrosion (%).According to Bažant and Planas [28], the stiffness reduction factor α is dependent on the average tangential strain εθ¯ throughout [a, r0], which is represented as follows:(3)α=ftexp−γεθ¯−εθc¯Eefεθ¯,where ft is the tensile strength of concrete, εθc¯ denotes the average tangential cracking strain, and γ is a material constant, which can be defined between 5000 and 10000. According to [29], the approximate value of γ is 7500.After initiation, the compressive stress of the thick wall cylinder of the concrete protection layer is increasing with the continuous growth of corrosion products. The crack divides the thick wall cylinder into two coaxial cylinders: inner cracked and outer uncracked ones, as shown in Figure1(c). According to the elastic theory of concrete, as the result of the corrosion crack in concrete protective layer arising, the governing equation for the displacement u(r) of any point r in the concrete protective layer should satisfy the following equation [19]:(4)d2urdr2+1r⋅durdr−αurr2=0,in which u is the displacement of the point O along the s-axis, as shown in Figure 2.Figure 2
Model for calculation of initial crack width.The solution to equation (4) can be expressed as(5)ur=c1r0rα+c2r0r−α.At present, the boundary conditions of equation (5) require that(6)c11+νcbα−1−c21−νcb−α−1=0,(7)c1aα+c2a−α=dst,where νc is Poisson’s ratio of concrete in equation (6). c1 and c2 can be obtained by solving equations (6) and (7) simultaneously as follows:(8)c1=1−νcaα1−νca2α+1−νcb2αdst,(9)c2=1−νcaαb2α1−νca2α+1−νcb2αdst.According to Li et al. [19], the corresponding average tangential strain εθ¯ can be indicated as follows:(10)εθ¯=1b−a∫abεθrdr=1b−a∫abc1rα−1+c2r−α−1dr=bα−aαc1+c2/abααb−a.The average tangential strain over the cracked surfaceεθc¯ can be determined as follows:(11)εθc¯=1b−a∫abc1r+c2rr2dr.Consequently, the stiffness reduction factorα can be indicated as(12)α=ftexp−γbα−aαc1+c2/abα/αb−a−1/b−a∫abc1r+c2r/r2drEefbα−aαc1+c2/abα/αb−a.Afterward, by substituting equations (8) and (9) into equation (12), the relationship between α and ds(t) can be calculated.Withc1 and c2 known, the tangential strain at b can be determined from equation (5), that is,(13)εθb=c1bα−1+c2b−α−1.In Figure2, evidently, in accordance with the geometric relationship between the width and the angle of the crack on the surface of the concrete component, the crack width wc on the surface of the concrete cylinder should be(14)wc=2πbεθb−εθe,mb,where εθe,mb is the maximum elastic strain at r = b and equal to the following equation [30]:(15)εθe,mb=σθ,mb−νcσrbEef,where σθ,m(b) and σr(b) are the maximum tangential stress and radial stress at r = b, respectively, and Eef is the effective elastic modulus of concrete, in which Eef = Ec/(1 + φcr) and φcr is the coefficient of concrete creep.On account ofσr(b) = 0 and σθ,m(b) = ft, the crack width wc is finally given by(16)wc=2πbc1bα−1+c2b−α−1−ftEef=4πdst1−νca/bα+1+νcb/aα−2πbftEef,where the essential variables are the thickness of corrosion products ds(t) and the stiffness reduction factor α. ds(t) is directly related to the corrosion degree, as shown in equation (1), and α is related to concrete geometry and property. Obviously, with the accumulation of corrosion products, the crack width increases with time. This makes sense both theoretically and practically, as experienced and observed [6–8, 31].In [32, 33], the values of basic variables in equations (1) and (16) had been provided, where d0 = 12.5 µm, αrust = 0.57, ρrust = 3600 kg/m3, and ρst = 7850 kg/m3. Parameters a, b, D, C, ft, Ec, vc, and φcr can be determined by the size of the component and the performance of concrete materials.Therefore, the calculation process of Li-CQ’s model includes the following steps:(1)
Determine the basic parameters of reinforced concrete members, and the relationship between the steel corrosion product thicknessds(t) and concrete stiffness reduction factor α can be calculated in association with equations (8), (9), and (12).(2)
Substitute the relationship withds(t) and α to equation (16), and the relationship between the crack width wc and ds(t) can be obtained.(3)
Substituteds(t) represented by wc to equation (1), and the relationship between wc and Wrustt can be established. Furthermore, the model of the corrosion degree of steel bar can be calculated by equation (2).


## 2.2. Engineering Formula considering the Reduction Coefficient of Stiffness

In Li-CQ’s model, the calculation process of parametersds(t) and α is an iterative process, which is more complicated and inconvenient. In addition, when the difference method is used to calculate equation (12), the accuracy of calculation results could not be guaranteed if the difference step is large, while the computation burden will be doubled if the difference step is small. Therefore, parameters in the Li-CQ’s model should be simplified for engineering applications.In equation (16), parameters D, C, ft, and Ec can be determined by the size of the current concrete members and concrete material properties. The value of Poisson’s ratio vc is 0.18 in common. Concrete creep coefficient φcr can be obtained by CEB-FIP Model Code 2010 [33].For concrete stiffness reduction factorα, considering the influence of the corrosion time of steel bar, the data in the literature [20–22, 34–38] were applied to calculate the simplified calculation of α′(t), and the nonlinear regression equation can be indicated as follows:(17)α′t=0.01137638×t365−2.198836253+0.1235655576adj.R2=0.974,where α′(t) denotes modified concrete stiffness reduction factor and t is the time of steel corrosion, day as the unit.Consequently, substituting equations (16) and (17) into equation (1), the modified parameter of the mass of corrosion products Wrustαt can be proposed as follows:(18)Wrustαt=wc+2πbft/Eef⋅D+2d0⋅1−νca/bα′t+1+νcb/aα′t4⋅1/ρrust−αrust/ρst.Thus, the modified calculation of the steel corrosion degree can be expressed as(19)θα=WrustαW×100%.


## 2.3. Experimental Programs

Experiments were designed to investigate the influence on the relationship between corrosion degree of steel bar and corrosion-induced crack width by parameters including water-cement ratio, the thickness of concrete cover, and diameter of steel bar, and the engineering formula presented above was verified through the experiment.


### 2.3.1. Mix Design

In the experiments, materials including ordinary Portland cement (with 60.6 MPa of the 28-day standard compressive strength), continuous grading gravel with diameters from 5 to 25 mm, medium sand with diameters from 0.08 to 2.0 mm, as well as a kind of naphthalene-based superplasticizer were used to produce concrete. The yield strength of steel bar is 335 MPa. Experiments [39] indicate that a certain amount of chloride mixed with concrete can significantly reduce the resistivity of the concrete and distinguish steep corrosion speed because of different mixed concretes. Under this circumstance, not only the steel corrosion degree is accelerated but also an appropriate corrosion current density is applied. The mix proportion is listed in Table 1.Table 1
Mix proportion of concrete (kg/m3).
No.
W
/
C
Water : cement : sand : macadam : water-reducing agent : sodium chloride
S1
0.3
126 : 378 : 586 : 1244 : 8.4 : 4.2
S2
0.4
168 : 378 : 549 : 1168 : 8.4 : 4.2
S3
0.5
210 : 378 : 514 : 1092 : 8.4 : 4.2


### 2.3.2. Specimen Design

To achieve the experimental objective, the size of the specimen is 300 mm × 200 mm × 250 mm, as shown in Figure3.Figure 3
Size of the specimen (unit: mm).In Figure3, C1, C2, and C3 represent the thickness of concrete cover of the diameter steel bar of 12 mm, 20 mm, and 25 mm, respectively. The thickness of the concrete cover is 30 mm, 40 mm, and 50 mm, respectively. The predetermined corrosive cracks width on the member surface is set to 0.2 mm, 0.4 mm, 0.7 mm, and 1.1 mm. The orthogonal design method [40] of L9 (34) was adopted in the tests. The levels of test parameters are shown in Table 2, in which each number includes two specimens.Table 2
The levels of test parameters.
Parameter level no.
W
/
C
Thickness of concrete protection layer (mm)
Diameter of steel bar (mm)
Preset crack width in the surface of concrete (mm)
1
0.3
30
12
0.2/0.4/0.7/1.0
2
0.3
40
20
3
0.3
50
25
4
0.4
30
25
0.2/0.4/0.7/1.0
5
0.4
40
12
6
0.4
50
20
7
0.5
30
20
0.2/0.4/0.7/1.0
8
0.5
40
25
9
0.5
50
12In addition, the same batches of concrete specimens were indwelled to determine the basic 28 d mechanical properties of concrete materials: 6 groups of elasticity modulus specimens, 3 groups of cube compressive strength specimens, and 3 groups of splitting tensile strength specimens.


### 2.3.3. Specimen Preparation

Firstly, the steel bars of different diameters were cut to 300 mm in length, the steel rust on the steel surface was cleaned, and the initial weight of steel was recorded. The molds were manufactured according to Figure3, and the steel bars were fixed in the reserved holes of the molds. Next, fresh concrete was poured into the molds and was vibrated compaction. The molds were removed after 24 hours; both ends of the steel bars were cleaned up and then welded with wires and sealed by epoxy resins. Finally, the test specimens and referenced specimens were put into the curing room together and conducted for testing after 28 days of standard curing. The formed reinforced concrete specimens are shown in Figure 4.Figure 4
The formed reinforced concrete specimens.


### 2.3.4. Testing Procedure

In the accelerated steel corrosion test, the specimens were soaked in 5% NaCl solution and the steel bars were connected with each other in parallel via wires reinforced by the voltage of 60 V until the cracking on the specimen surface was developing into the preset width. After breaking the specimens in batches, corroded steels were soaked in hydrochloric acid solution next and then were washed clean after the surface acidic substance was neutralized by calcium carbonate solution. Finally, the corrosive steel reinforcements were weighted by the electronic balance accurate to 0.1 g, and the corrosion degrees of the steel bars were calculated. Experimental procedures are shown in Figures5 and 6.Figure 5
Energized accelerate steel corrosion test.Figure 6
Specimen breakages.


### 2.3.5. Phenomena and Analysis of Experiment

In the experiment, it can be observed that (1) with the progress of energization and steel corrosion, it appeared heating was done within the specimen and the surface temperature was rising. The reason can be explained that concrete generates more heat, while current gets through the concrete in the condition of continuous power as it is a poor conductor with large resistance. (2) A small amount of bubbles and damp phenomenon could be observed at the concrete near the cathodic protection, and there were drops of water generated at reinforced ends. After energization for some days, there were cracks on the surface on the reinforced concrete specimen, and overflow rust obviously appeared in the position of cracks and the end of steel bars, which was caused by the concrete pores around the steel bars which was filled with the corrosion products generated due to continuous steel corrosion. (3) The larger the concrete water-cement ratio was, the earlier the crack appeared, and the greater the corrosion degree of steel was when cracks appeared. The thickness of concrete cover also had certain influence on the development patterns of crack. (4) After breaking the specimens, the longitudinal ribs of steels were usually the origin of the crack. This is because the stress concentration was arising at the longitudinal ribs of steel, resulting in large expansion force applied on concrete surrounding. (5) The longitudinal cracks, with uniform width, were formed after specimen cracking since the reinforced specimens were in uniform corrosion when they were accelerated to be corroded, so did the cracks.


## 2.3.1. Mix Design

In the experiments, materials including ordinary Portland cement (with 60.6 MPa of the 28-day standard compressive strength), continuous grading gravel with diameters from 5 to 25 mm, medium sand with diameters from 0.08 to 2.0 mm, as well as a kind of naphthalene-based superplasticizer were used to produce concrete. The yield strength of steel bar is 335 MPa. Experiments [39] indicate that a certain amount of chloride mixed with concrete can significantly reduce the resistivity of the concrete and distinguish steep corrosion speed because of different mixed concretes. Under this circumstance, not only the steel corrosion degree is accelerated but also an appropriate corrosion current density is applied. The mix proportion is listed in Table 1.Table 1
Mix proportion of concrete (kg/m3).
No.
W
/
C
Water : cement : sand : macadam : water-reducing agent : sodium chloride
S1
0.3
126 : 378 : 586 : 1244 : 8.4 : 4.2
S2
0.4
168 : 378 : 549 : 1168 : 8.4 : 4.2
S3
0.5
210 : 378 : 514 : 1092 : 8.4 : 4.2


## 2.3.2. Specimen Design

To achieve the experimental objective, the size of the specimen is 300 mm × 200 mm × 250 mm, as shown in Figure3.Figure 3
Size of the specimen (unit: mm).In Figure3, C1, C2, and C3 represent the thickness of concrete cover of the diameter steel bar of 12 mm, 20 mm, and 25 mm, respectively. The thickness of the concrete cover is 30 mm, 40 mm, and 50 mm, respectively. The predetermined corrosive cracks width on the member surface is set to 0.2 mm, 0.4 mm, 0.7 mm, and 1.1 mm. The orthogonal design method [40] of L9 (34) was adopted in the tests. The levels of test parameters are shown in Table 2, in which each number includes two specimens.Table 2
The levels of test parameters.
Parameter level no.
W
/
C
Thickness of concrete protection layer (mm)
Diameter of steel bar (mm)
Preset crack width in the surface of concrete (mm)
1
0.3
30
12
0.2/0.4/0.7/1.0
2
0.3
40
20
3
0.3
50
25
4
0.4
30
25
0.2/0.4/0.7/1.0
5
0.4
40
12
6
0.4
50
20
7
0.5
30
20
0.2/0.4/0.7/1.0
8
0.5
40
25
9
0.5
50
12In addition, the same batches of concrete specimens were indwelled to determine the basic 28 d mechanical properties of concrete materials: 6 groups of elasticity modulus specimens, 3 groups of cube compressive strength specimens, and 3 groups of splitting tensile strength specimens.


## 2.3.3. Specimen Preparation

Firstly, the steel bars of different diameters were cut to 300 mm in length, the steel rust on the steel surface was cleaned, and the initial weight of steel was recorded. The molds were manufactured according to Figure3, and the steel bars were fixed in the reserved holes of the molds. Next, fresh concrete was poured into the molds and was vibrated compaction. The molds were removed after 24 hours; both ends of the steel bars were cleaned up and then welded with wires and sealed by epoxy resins. Finally, the test specimens and referenced specimens were put into the curing room together and conducted for testing after 28 days of standard curing. The formed reinforced concrete specimens are shown in Figure 4.Figure 4
The formed reinforced concrete specimens.


## 2.3.4. Testing Procedure

In the accelerated steel corrosion test, the specimens were soaked in 5% NaCl solution and the steel bars were connected with each other in parallel via wires reinforced by the voltage of 60 V until the cracking on the specimen surface was developing into the preset width. After breaking the specimens in batches, corroded steels were soaked in hydrochloric acid solution next and then were washed clean after the surface acidic substance was neutralized by calcium carbonate solution. Finally, the corrosive steel reinforcements were weighted by the electronic balance accurate to 0.1 g, and the corrosion degrees of the steel bars were calculated. Experimental procedures are shown in Figures5 and 6.Figure 5
Energized accelerate steel corrosion test.Figure 6
Specimen breakages.


## 2.3.5. Phenomena and Analysis of Experiment

In the experiment, it can be observed that (1) with the progress of energization and steel corrosion, it appeared heating was done within the specimen and the surface temperature was rising. The reason can be explained that concrete generates more heat, while current gets through the concrete in the condition of continuous power as it is a poor conductor with large resistance. (2) A small amount of bubbles and damp phenomenon could be observed at the concrete near the cathodic protection, and there were drops of water generated at reinforced ends. After energization for some days, there were cracks on the surface on the reinforced concrete specimen, and overflow rust obviously appeared in the position of cracks and the end of steel bars, which was caused by the concrete pores around the steel bars which was filled with the corrosion products generated due to continuous steel corrosion. (3) The larger the concrete water-cement ratio was, the earlier the crack appeared, and the greater the corrosion degree of steel was when cracks appeared. The thickness of concrete cover also had certain influence on the development patterns of crack. (4) After breaking the specimens, the longitudinal ribs of steels were usually the origin of the crack. This is because the stress concentration was arising at the longitudinal ribs of steel, resulting in large expansion force applied on concrete surrounding. (5) The longitudinal cracks, with uniform width, were formed after specimen cracking since the reinforced specimens were in uniform corrosion when they were accelerated to be corroded, so did the cracks.


## 3. Results

### 3.1. Mechanical Properties of Concrete

According to the Ordinary Concrete Mechanical Performance Test Method [41], the specimens for compression strength are 100 mm × 100 mm × 100 mm cubes, the specimens for modulus of elasticity are 100 mm × 100 mm × 300 mm prisms, and the specimens for splitting tensile strength are 150 mm × 150 mm × 150 mm cubes, respectively. The experimental procedures of the mechanical properties of concrete are shown in Figures 7–11.Figure 7
Compression machine and data acquisition.Figure 8
Test of concrete cube compressive strength.Figure 9
Strength of concrete prism.Figure 10
Elastic modulus of concrete.Figure 11
Splitting tensile strength of concrete.The experimental results of the mechanical properties of concrete at 28 days are shown in Table3.Table 3
Experimental results of mechanical properties of concrete at 28 days.
No.
W
/
C
Experimental results
Cube compressive strength (MPa)
Elastic modulus (GPa)
Splitting tensile strength (MPa)
S1
0.3
58.22
51.13
3.86
S2
0.4
47.34
44.23
3.27
S3
0.5
34.90
37.27
2.88


### 3.2. Relationship between Cracking Width on a Concrete Surface and Steel Corrosion Degree

#### 3.2.1. Effect ofW/C

In the condition of three kinds of water-cement ratio (0.3, 0.4, and 0.5), the diameter of steel bars is 12 mm, 20 mm, and 25 mm, respectively, and the relationship between crack width on concrete surface and steel corrosion degree is shown in Figure12.Figure 12
Effect ofW/C on the relationship between cracking width and steel corrosion degree for different reinforcement diameters: (a) D = 12 mm; (b) D = 20 mm; (c) D = 25 mm.
(a)
(b)
(c)Regarding the average test value as the object of discussion, in Figures12(a)–12(c), it indicates that when the diameter of steel bars is 12 mm, 20 mm, and 25 mm, respectively, reinforcement corrosion degree raises with the increase in the crack width on the concrete surface with the same water-cement ratio. When the diameter of steel bar is 12 mm and the crack width on concrete surface is approximately less than 0.6 mm, the rebar corrosion degree increases firstly and then decreases as the water-cement ratio rises. When the crack width on the concrete surface is approximately more than 0.6 mm, the rebar corrosion degree decreases firstly and then increases as the water-cement ratio rises. When the diameter of the steel bar is 20 mm, the rebar corrosion degree increases firstly and then decreases with the rising water-cement ratio. When the diameter of the steel bar is 25 mm and the crack width on the concrete surface is approximately less than 0.8 mm, the steel corrosion degree decreases with the rising water-cement ratio. When the crack width on the concrete surface is approximately more than 0.8 mm, the steel corrosion degree decreases firstly and then increases with the rising water-cement ratio.Take the steel diameter of 12 mm as an example. In Figure12(a), when the water-cement ratio is 0.3, 0.4, and 0.5, respectively, the corrosion degree is 2.718%, 3.161%, and 2.886%, and the width of cracks on the component surface is 0.225 mm, 0.255 mm, and 0.27 mm correspondingly. When the corrosion degree is 3.610%, 5.104%, and 3.471%, the width of cracks on the component surface is corresponding to 0.40 mm, 0.475 mm, and 0.425 mm, respectively. The steel corrosion degree increases firstly and then decreases as the water-cement ratio rises. When the corrosion degree is 6.05%, 5.454%, and 5.209%, respectively, the width of cracks on the component surface is 0.72 mm, 0.755 mm, and 0.72 mm correspondingly. The steel corrosion degree decreases with the increase in water-cement ratio. When the corrosion degree is 7.905%, 7.507%, and 8.892%, the width of cracks on the component surface is 1.11 mm, 1.075 mm, and 1.26 mm correspondingly.


#### 3.2.2. Effect of the Diameter of Steel bar

When the diameter of the steel bar is 12 mm, 20 mm, and 25 mm, the thickness of concrete covers is 30 mm, 40 mm, and 50 mm, respectively, and the relationship between crack width on concrete surface and steel corrosion degree is shown in Figure13.Figure 13
Effect of the diameter of steel bar on the relationship between cracking width and steel corrosion degree for different thickness of concrete covers: (a)C = 30 mm; (b) C = 40 mm; (c) C = 50 mm.
(a)
(b)
(c)Take the average test value as the object of discussion. In Figures13(a)–13(c), it indicates that when the thickness of concrete covers is 30 mm, 40 mm, and 50 mm, respectively, reinforcement corrosion degree rises with the increase in the crack width on a concrete surface with the same diameter of steel bar. For the same thickness of the concrete cover, the corrosion degree of the steel bar rises with the increase in the diameter of the steel bar under a certain crack width.Take the thickness of concrete cover of 12 mm as an example. In Figure13(b), when the diameter of steel bars is 12 mm, 20 mm, and 25 mm respectively, the corrosion degree is 3.161%, 1.507%, and 1.071%, and the width of cracks on the component surface is 0.255 mm, 0.26 mm, and 0.28 mm correspondingly. When the corrosion degree is 5.104%, 1.976%, and 1.236%, the width of cracks on the component surface is corresponding to 0.475 mm, 0.44 mm, and 0.40 mm. When the corrosion degree is 5.454%, 2.643%, and 1.891%, the width of cracks on the component surface is 0.755 mm, 0.76 mm, and 0.705 mm, respectively. When the corrosion degree is 7.507%, 3.661%, and 3.437%, the width of cracks on the component surface is 1.075 mm, 1.12 mm, and 1.345 mm, respectively.


#### 3.2.3. Effect of the Thickness of Concrete Cover

When the thickness of the concrete cover is 30 mm, 40 mm, and 50 mm, the water-cement ratio is 0.3, 0.4, and 0.5, respectively, and the relationship between crack width on concrete surface and steel corrosion degree is shown in Figure14.Figure 14
Effect of thickness of concrete cover on the relationship between cracking width and steel corrosion degree for different water-cement ratios: (a)W/C=0.3; (b) W/C=0.4; (c) W/C=0.5.
(a)
(b)
(c)Take the average test value as the object of discussion. In Figures14(a)–14(c), it indicates that when the water-cement ratio is 0.3, 0.4, and 0.5, respectively, reinforcement corrosion degree rises with the increase in the crack width on a concrete surface with the same thickness of the concrete cover. For the same water-cement ratio, the corrosion degree of the steel bar rises with the increase in the diameter of the steel bar under a certain crack width. When the water-cement ratio is 0.3, the steel corrosion degree decreases firstly and then increases as the thickness of the concrete cover increases. When the water-cement ratio is 0.4, the steel corrosion degree increases firstly and then decreases as the thickness of the concrete cover increases. When the water-cement ratio is 0.5, the steel corrosion degree decreases firstly and then increases as the thickness of the concrete cover increases, which is the same as when the water-cement ratio is 0.3.Take the water-cement ratio equal to 0.5 as an example. In Figure14(c), when the thickness of concrete cover is 30 mm, 40 mm, and 50 mm, respectively, the corrosion degree is 1.613%, 1.071%, and 2.886%, and the width of cracks on the component surface is 0.25 mm, 0.28 mm, and 0.27 mm correspondingly. When the corrosion degree is 1.713%, 1.236%, and 3.471%, the width of cracks on the component surface is 0.435 mm, 0.40 mm, and 0.425 mm, respectively. When the corrosion degree is 2.944%, 1.891%, and 5.209%, the width of cracks on the component surface is corresponding to 0.765 mm, 0.705 mm, and 0.72 mm. When the corrosion degree is 3.865%, 3.437%, and 8.892%, the width of cracks on the component surface is 1.235 mm, 1.345 mm, and 1.26 mm correspondingly.


## 3.1. Mechanical Properties of Concrete

According to the Ordinary Concrete Mechanical Performance Test Method [41], the specimens for compression strength are 100 mm × 100 mm × 100 mm cubes, the specimens for modulus of elasticity are 100 mm × 100 mm × 300 mm prisms, and the specimens for splitting tensile strength are 150 mm × 150 mm × 150 mm cubes, respectively. The experimental procedures of the mechanical properties of concrete are shown in Figures 7–11.Figure 7
Compression machine and data acquisition.Figure 8
Test of concrete cube compressive strength.Figure 9
Strength of concrete prism.Figure 10
Elastic modulus of concrete.Figure 11
Splitting tensile strength of concrete.The experimental results of the mechanical properties of concrete at 28 days are shown in Table3.Table 3
Experimental results of mechanical properties of concrete at 28 days.
No.
W
/
C
Experimental results
Cube compressive strength (MPa)
Elastic modulus (GPa)
Splitting tensile strength (MPa)
S1
0.3
58.22
51.13
3.86
S2
0.4
47.34
44.23
3.27
S3
0.5
34.90
37.27
2.88


## 3.2. Relationship between Cracking Width on a Concrete Surface and Steel Corrosion Degree

### 3.2.1. Effect ofW/C

In the condition of three kinds of water-cement ratio (0.3, 0.4, and 0.5), the diameter of steel bars is 12 mm, 20 mm, and 25 mm, respectively, and the relationship between crack width on concrete surface and steel corrosion degree is shown in Figure12.Figure 12
Effect ofW/C on the relationship between cracking width and steel corrosion degree for different reinforcement diameters: (a) D = 12 mm; (b) D = 20 mm; (c) D = 25 mm.
(a)
(b)
(c)Regarding the average test value as the object of discussion, in Figures12(a)–12(c), it indicates that when the diameter of steel bars is 12 mm, 20 mm, and 25 mm, respectively, reinforcement corrosion degree raises with the increase in the crack width on the concrete surface with the same water-cement ratio. When the diameter of steel bar is 12 mm and the crack width on concrete surface is approximately less than 0.6 mm, the rebar corrosion degree increases firstly and then decreases as the water-cement ratio rises. When the crack width on the concrete surface is approximately more than 0.6 mm, the rebar corrosion degree decreases firstly and then increases as the water-cement ratio rises. When the diameter of the steel bar is 20 mm, the rebar corrosion degree increases firstly and then decreases with the rising water-cement ratio. When the diameter of the steel bar is 25 mm and the crack width on the concrete surface is approximately less than 0.8 mm, the steel corrosion degree decreases with the rising water-cement ratio. When the crack width on the concrete surface is approximately more than 0.8 mm, the steel corrosion degree decreases firstly and then increases with the rising water-cement ratio.Take the steel diameter of 12 mm as an example. In Figure12(a), when the water-cement ratio is 0.3, 0.4, and 0.5, respectively, the corrosion degree is 2.718%, 3.161%, and 2.886%, and the width of cracks on the component surface is 0.225 mm, 0.255 mm, and 0.27 mm correspondingly. When the corrosion degree is 3.610%, 5.104%, and 3.471%, the width of cracks on the component surface is corresponding to 0.40 mm, 0.475 mm, and 0.425 mm, respectively. The steel corrosion degree increases firstly and then decreases as the water-cement ratio rises. When the corrosion degree is 6.05%, 5.454%, and 5.209%, respectively, the width of cracks on the component surface is 0.72 mm, 0.755 mm, and 0.72 mm correspondingly. The steel corrosion degree decreases with the increase in water-cement ratio. When the corrosion degree is 7.905%, 7.507%, and 8.892%, the width of cracks on the component surface is 1.11 mm, 1.075 mm, and 1.26 mm correspondingly.


### 3.2.2. Effect of the Diameter of Steel bar

When the diameter of the steel bar is 12 mm, 20 mm, and 25 mm, the thickness of concrete covers is 30 mm, 40 mm, and 50 mm, respectively, and the relationship between crack width on concrete surface and steel corrosion degree is shown in Figure13.Figure 13
Effect of the diameter of steel bar on the relationship between cracking width and steel corrosion degree for different thickness of concrete covers: (a)C = 30 mm; (b) C = 40 mm; (c) C = 50 mm.
(a)
(b)
(c)Take the average test value as the object of discussion. In Figures13(a)–13(c), it indicates that when the thickness of concrete covers is 30 mm, 40 mm, and 50 mm, respectively, reinforcement corrosion degree rises with the increase in the crack width on a concrete surface with the same diameter of steel bar. For the same thickness of the concrete cover, the corrosion degree of the steel bar rises with the increase in the diameter of the steel bar under a certain crack width.Take the thickness of concrete cover of 12 mm as an example. In Figure13(b), when the diameter of steel bars is 12 mm, 20 mm, and 25 mm respectively, the corrosion degree is 3.161%, 1.507%, and 1.071%, and the width of cracks on the component surface is 0.255 mm, 0.26 mm, and 0.28 mm correspondingly. When the corrosion degree is 5.104%, 1.976%, and 1.236%, the width of cracks on the component surface is corresponding to 0.475 mm, 0.44 mm, and 0.40 mm. When the corrosion degree is 5.454%, 2.643%, and 1.891%, the width of cracks on the component surface is 0.755 mm, 0.76 mm, and 0.705 mm, respectively. When the corrosion degree is 7.507%, 3.661%, and 3.437%, the width of cracks on the component surface is 1.075 mm, 1.12 mm, and 1.345 mm, respectively.


### 3.2.3. Effect of the Thickness of Concrete Cover

When the thickness of the concrete cover is 30 mm, 40 mm, and 50 mm, the water-cement ratio is 0.3, 0.4, and 0.5, respectively, and the relationship between crack width on concrete surface and steel corrosion degree is shown in Figure14.Figure 14
Effect of thickness of concrete cover on the relationship between cracking width and steel corrosion degree for different water-cement ratios: (a)W/C=0.3; (b) W/C=0.4; (c) W/C=0.5.
(a)
(b)
(c)Take the average test value as the object of discussion. In Figures14(a)–14(c), it indicates that when the water-cement ratio is 0.3, 0.4, and 0.5, respectively, reinforcement corrosion degree rises with the increase in the crack width on a concrete surface with the same thickness of the concrete cover. For the same water-cement ratio, the corrosion degree of the steel bar rises with the increase in the diameter of the steel bar under a certain crack width. When the water-cement ratio is 0.3, the steel corrosion degree decreases firstly and then increases as the thickness of the concrete cover increases. When the water-cement ratio is 0.4, the steel corrosion degree increases firstly and then decreases as the thickness of the concrete cover increases. When the water-cement ratio is 0.5, the steel corrosion degree decreases firstly and then increases as the thickness of the concrete cover increases, which is the same as when the water-cement ratio is 0.3.Take the water-cement ratio equal to 0.5 as an example. In Figure14(c), when the thickness of concrete cover is 30 mm, 40 mm, and 50 mm, respectively, the corrosion degree is 1.613%, 1.071%, and 2.886%, and the width of cracks on the component surface is 0.25 mm, 0.28 mm, and 0.27 mm correspondingly. When the corrosion degree is 1.713%, 1.236%, and 3.471%, the width of cracks on the component surface is 0.435 mm, 0.40 mm, and 0.425 mm, respectively. When the corrosion degree is 2.944%, 1.891%, and 5.209%, the width of cracks on the component surface is corresponding to 0.765 mm, 0.705 mm, and 0.72 mm. When the corrosion degree is 3.865%, 3.437%, and 8.892%, the width of cracks on the component surface is 1.235 mm, 1.345 mm, and 1.26 mm correspondingly.


## 3.2.1. Effect ofW/C

In the condition of three kinds of water-cement ratio (0.3, 0.4, and 0.5), the diameter of steel bars is 12 mm, 20 mm, and 25 mm, respectively, and the relationship between crack width on concrete surface and steel corrosion degree is shown in Figure12.Figure 12
Effect ofW/C on the relationship between cracking width and steel corrosion degree for different reinforcement diameters: (a) D = 12 mm; (b) D = 20 mm; (c) D = 25 mm.
(a)
(b)
(c)Regarding the average test value as the object of discussion, in Figures12(a)–12(c), it indicates that when the diameter of steel bars is 12 mm, 20 mm, and 25 mm, respectively, reinforcement corrosion degree raises with the increase in the crack width on the concrete surface with the same water-cement ratio. When the diameter of steel bar is 12 mm and the crack width on concrete surface is approximately less than 0.6 mm, the rebar corrosion degree increases firstly and then decreases as the water-cement ratio rises. When the crack width on the concrete surface is approximately more than 0.6 mm, the rebar corrosion degree decreases firstly and then increases as the water-cement ratio rises. When the diameter of the steel bar is 20 mm, the rebar corrosion degree increases firstly and then decreases with the rising water-cement ratio. When the diameter of the steel bar is 25 mm and the crack width on the concrete surface is approximately less than 0.8 mm, the steel corrosion degree decreases with the rising water-cement ratio. When the crack width on the concrete surface is approximately more than 0.8 mm, the steel corrosion degree decreases firstly and then increases with the rising water-cement ratio.Take the steel diameter of 12 mm as an example. In Figure12(a), when the water-cement ratio is 0.3, 0.4, and 0.5, respectively, the corrosion degree is 2.718%, 3.161%, and 2.886%, and the width of cracks on the component surface is 0.225 mm, 0.255 mm, and 0.27 mm correspondingly. When the corrosion degree is 3.610%, 5.104%, and 3.471%, the width of cracks on the component surface is corresponding to 0.40 mm, 0.475 mm, and 0.425 mm, respectively. The steel corrosion degree increases firstly and then decreases as the water-cement ratio rises. When the corrosion degree is 6.05%, 5.454%, and 5.209%, respectively, the width of cracks on the component surface is 0.72 mm, 0.755 mm, and 0.72 mm correspondingly. The steel corrosion degree decreases with the increase in water-cement ratio. When the corrosion degree is 7.905%, 7.507%, and 8.892%, the width of cracks on the component surface is 1.11 mm, 1.075 mm, and 1.26 mm correspondingly.


## 3.2.2. Effect of the Diameter of Steel bar

When the diameter of the steel bar is 12 mm, 20 mm, and 25 mm, the thickness of concrete covers is 30 mm, 40 mm, and 50 mm, respectively, and the relationship between crack width on concrete surface and steel corrosion degree is shown in Figure13.Figure 13
Effect of the diameter of steel bar on the relationship between cracking width and steel corrosion degree for different thickness of concrete covers: (a)C = 30 mm; (b) C = 40 mm; (c) C = 50 mm.
(a)
(b)
(c)Take the average test value as the object of discussion. In Figures13(a)–13(c), it indicates that when the thickness of concrete covers is 30 mm, 40 mm, and 50 mm, respectively, reinforcement corrosion degree rises with the increase in the crack width on a concrete surface with the same diameter of steel bar. For the same thickness of the concrete cover, the corrosion degree of the steel bar rises with the increase in the diameter of the steel bar under a certain crack width.Take the thickness of concrete cover of 12 mm as an example. In Figure13(b), when the diameter of steel bars is 12 mm, 20 mm, and 25 mm respectively, the corrosion degree is 3.161%, 1.507%, and 1.071%, and the width of cracks on the component surface is 0.255 mm, 0.26 mm, and 0.28 mm correspondingly. When the corrosion degree is 5.104%, 1.976%, and 1.236%, the width of cracks on the component surface is corresponding to 0.475 mm, 0.44 mm, and 0.40 mm. When the corrosion degree is 5.454%, 2.643%, and 1.891%, the width of cracks on the component surface is 0.755 mm, 0.76 mm, and 0.705 mm, respectively. When the corrosion degree is 7.507%, 3.661%, and 3.437%, the width of cracks on the component surface is 1.075 mm, 1.12 mm, and 1.345 mm, respectively.


## 3.2.3. Effect of the Thickness of Concrete Cover

When the thickness of the concrete cover is 30 mm, 40 mm, and 50 mm, the water-cement ratio is 0.3, 0.4, and 0.5, respectively, and the relationship between crack width on concrete surface and steel corrosion degree is shown in Figure14.Figure 14
Effect of thickness of concrete cover on the relationship between cracking width and steel corrosion degree for different water-cement ratios: (a)W/C=0.3; (b) W/C=0.4; (c) W/C=0.5.
(a)
(b)
(c)Take the average test value as the object of discussion. In Figures14(a)–14(c), it indicates that when the water-cement ratio is 0.3, 0.4, and 0.5, respectively, reinforcement corrosion degree rises with the increase in the crack width on a concrete surface with the same thickness of the concrete cover. For the same water-cement ratio, the corrosion degree of the steel bar rises with the increase in the diameter of the steel bar under a certain crack width. When the water-cement ratio is 0.3, the steel corrosion degree decreases firstly and then increases as the thickness of the concrete cover increases. When the water-cement ratio is 0.4, the steel corrosion degree increases firstly and then decreases as the thickness of the concrete cover increases. When the water-cement ratio is 0.5, the steel corrosion degree decreases firstly and then increases as the thickness of the concrete cover increases, which is the same as when the water-cement ratio is 0.3.Take the water-cement ratio equal to 0.5 as an example. In Figure14(c), when the thickness of concrete cover is 30 mm, 40 mm, and 50 mm, respectively, the corrosion degree is 1.613%, 1.071%, and 2.886%, and the width of cracks on the component surface is 0.25 mm, 0.28 mm, and 0.27 mm correspondingly. When the corrosion degree is 1.713%, 1.236%, and 3.471%, the width of cracks on the component surface is 0.435 mm, 0.40 mm, and 0.425 mm, respectively. When the corrosion degree is 2.944%, 1.891%, and 5.209%, the width of cracks on the component surface is corresponding to 0.765 mm, 0.705 mm, and 0.72 mm. When the corrosion degree is 3.865%, 3.437%, and 8.892%, the width of cracks on the component surface is 1.235 mm, 1.345 mm, and 1.26 mm correspondingly.


## 4. Discussion

### 4.1. Validation of the Engineering Formula

The result of the test, considering the water-cement ratio is 0.3, 0.4, and 0.5, respectively, and in which the diameter of the steel bar was 25 mm, was analyzed. Calculation value of Li’s model and the engineering formula were compared with the test results, as shown in Figure15.Figure 15
Comparison between Li’s model and the engineering formula with the test results (water-cement ratio is 0.3, 0.4, and 0.5; diameter of steel bar is 25 mm).As shown in Figure15, for different water-cement ratios, when the crack width ranges from 0.2 mm to 0.4 mm, the average value of the deviation between the calculated engineering formula and experimental values is 4.27%, whereas that between the calculated Li’s model and experimental values is 8.65%. When the crack width ranges from 0.4 mm to 0.7 mm, the average value of the deviation between the calculated engineering formula and experimental values is 6.49%, whereas that between the calculated Li’s model and experimental values is 9.65%. When the crack widths range between 0.7 mm and 1.0 mm, the average value of the deviation between the calculated engineering formula and experimental values is 3.83%, whereas that between the calculated Li’s model and experimental values is 7.20%. When the crack width ranges from 1.0 mm to 1.4 mm, the average value of the deviation between the calculated engineering formula and experimental values is 6.37%, whereas that between the calculated Li’s model and experimental values is 10.64%. The above analyses show that, with the increase in the cracking width on the component surface, the calculation values by the engineering formula presented in this paper, comparing with that by Li’s model, are in good agreement with experimental value and can be applied to predict the corrosion degree of steel bar after concrete cracking on the component surface.


### 4.2. Applicability Analysis of Engineering Formula

In the practical engineering structure, the reinforced concrete structures are often subjected to load. The internal pore structure and the expansion of microcracks are closely related to the current stress state of the component. The time-dependent characteristics of concrete materials and the diffusion degree of corrosive ions in concrete will affect the corrosion of steel bars. In addition, the existence of stirrups or distributed reinforcement near the main reinforcement in the component will not only change the process of electrochemical corrosion [42] but also influence and restrict the expansion process of crack induced by corrosion. Consequently, it is necessary to carry out further research on the effect of load and the actual distribution of reinforcement in the structure on the process of cracking.


## 4.1. Validation of the Engineering Formula

The result of the test, considering the water-cement ratio is 0.3, 0.4, and 0.5, respectively, and in which the diameter of the steel bar was 25 mm, was analyzed. Calculation value of Li’s model and the engineering formula were compared with the test results, as shown in Figure15.Figure 15
Comparison between Li’s model and the engineering formula with the test results (water-cement ratio is 0.3, 0.4, and 0.5; diameter of steel bar is 25 mm).As shown in Figure15, for different water-cement ratios, when the crack width ranges from 0.2 mm to 0.4 mm, the average value of the deviation between the calculated engineering formula and experimental values is 4.27%, whereas that between the calculated Li’s model and experimental values is 8.65%. When the crack width ranges from 0.4 mm to 0.7 mm, the average value of the deviation between the calculated engineering formula and experimental values is 6.49%, whereas that between the calculated Li’s model and experimental values is 9.65%. When the crack widths range between 0.7 mm and 1.0 mm, the average value of the deviation between the calculated engineering formula and experimental values is 3.83%, whereas that between the calculated Li’s model and experimental values is 7.20%. When the crack width ranges from 1.0 mm to 1.4 mm, the average value of the deviation between the calculated engineering formula and experimental values is 6.37%, whereas that between the calculated Li’s model and experimental values is 10.64%. The above analyses show that, with the increase in the cracking width on the component surface, the calculation values by the engineering formula presented in this paper, comparing with that by Li’s model, are in good agreement with experimental value and can be applied to predict the corrosion degree of steel bar after concrete cracking on the component surface.


## 4.2. Applicability Analysis of Engineering Formula

In the practical engineering structure, the reinforced concrete structures are often subjected to load. The internal pore structure and the expansion of microcracks are closely related to the current stress state of the component. The time-dependent characteristics of concrete materials and the diffusion degree of corrosive ions in concrete will affect the corrosion of steel bars. In addition, the existence of stirrups or distributed reinforcement near the main reinforcement in the component will not only change the process of electrochemical corrosion [42] but also influence and restrict the expansion process of crack induced by corrosion. Consequently, it is necessary to carry out further research on the effect of load and the actual distribution of reinforcement in the structure on the process of cracking.


## 5. Conclusions

Undoubtedly, it is of practical significance to carry out fast and nondestructive determination of corrosion degree of steel bars for reinforced concrete structures in service for reasonable and accurate assessment on structural durability, determination of the maintenance and reinforcement measures, and redesign of the structure based on the reasonable and accurate assessment for structure durability. In this paper, the theoretical and experimental investigation of the relationship between crack width and steel corrosion degree of reinforced concrete members was carried out, and the conclusions can be obtained as follows:(1)
Li-CQ’s model is established on the basis of elasticity theory. Although it can be used to describe the mechanism of corrosion expansion process in concrete caused by tensile force leading to cracking of concrete cover, its calculation process is more complex and difficult to apply, which presents obvious deviation from test results.(2)
The experiment shows that the higher the water-cement ratio is, the earlier the crack appears, and the greater the corrosion degree of the steel bar is when the crack on the component surface appears. The greater the thickness of the concrete cover is, the greater the corrosion degree of the steel bar is when the crack appears. For the same water-cement ratio and thickness of the concrete cover, the corrosion degree of the steel bar decreases with the increase in the steel bar’s diameter under a certain crack width.(3)
The engineering formula of the steel corrosion degree is presented by means of the simplified calculation of the stiffness reduction factor of the concrete. The comparison shows that the calculated values of the engineering formula are in good agreement with the experimental results.



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*Source: 7936861-2020-01-22.xml*