Prediction of Corrosion-Induced Longitudinal Cracking Time of Concrete Cover Surface of Reinforced Concrete Structures under Load

Reinforced concrete (RC) structures suffer from different types of loads during service life, and the corrosion characteristics of steel bars embedded in concrete under load are different from those under non-load. In this paper, when the interface between steel bars and concrete (IBSC) cracked and the concrete cover surface (CCS) cracked, the effects of load on the critical corrosion depth of steel bars were analysed based on the thick-walled cylinder model, and a prediction model for the corrosion-induced longitudinal cracking (CLC) time (i.e., initiation cracking time) of the CCS of RC structures under load was proposed. Finally, the influence of load on the CLC time of CCS was discussed on the basis of the proposed prediction model. The results showed that the load had a significant effect on the critical corrosion depth of steel bars when the IBSC cracked induced by corrosion, while the influence of load on the critical corrosion depth of steel bars when the CCS cracked induced by corrosion was not obvious. When the CCS cracks induced by corrosion under load, the influence of the rust-filling layer on the critical corrosion depth of steel bars was larger than that of the load. With the increase in load, the CLC time of CCS decreased. The calculated values of the proposed prediction model were in reasonable agreement with the experimental values, which can provide a reference for durability evaluation and service life prediction of RC structures and lay the foundation for the investigation of the corrosion depth of steel bars in concrete under load.


Introduction
In coastal, marine, or other chloride environments, chloride ion erosion is the main cause of steel corrosion in reinforced concrete (RC) structures, and corrosion further leads to the degradation of the durability of RC structures [1,2]. The chloride ions in the environment reach the surface of steel bars through the pores and micro-cracks inside the concrete, and their concentration is continuously accumulated on the reinforcement surface. When the chloride ion concentration reaches a threshold, the dense passive film on the surface of the steel bars is destroyed, and then, with the presence of oxygen, the local corrosion is activated. Subsequently, the steel bars undergo electrochemical corrosion, that is, the steel dissolution reaction of the anode and the oxygen consumption reaction of the cathode occur simultaneously on the surface of the steel bars [3]. Generally, chloride ion erosion will cause non-uniform corrosion of steel bars, which is serious on the side of steel bars facing the concrete cover surface (CCS). Non-uniform corrosion can significantly reduce the strength and ductility of steel bars [4,5] and then lead to the degradation of the load-bearing capacity, stiffness, and ductility of RC structures [6]. There are many factors affecting the corrosion of steel bars in concrete, such as the water-binder ratio or concrete strength, mineral admixture, cement type, concrete cover thickness, reinforcement type and diameter, temperature and relative humidity, chloride ion concentration, external load, crack width, structures suffer from different loads, and the loads affect the corrosion characteristics of steel bars embedded in concrete [35][36][37]. In contrast to that without considering load, the load can shorten the initial corrosion time, change the corrosion rate and the corrosion distribution, and cause local pitting of steel bars in RC structures [38][39][40], resulting in serious corrosion-induced cracking of the concrete cover [41,42]. Therefore, the influence of load should be considered when establishing the corrosion-induced cracking model of the concrete cover. However, to the best knowledge of the authors, the study of the corrosion-induced cracking model considering the effect of the load is severely inadequate.
To fill this research gap, in this paper, based on the thick-walled cylinder model, the effect of the load was considered to analyse the critical corrosion depth of steel bars when the IBSC cracked and the CCS cracked. Then, a prediction model for the corrosion-induced longitudinal cracking (CLC) time (i.e., initiation cracking time) of the CCS of RC structures under load was proposed, and the rationality of the prediction model was verified by the experimental data of Wang et al. [43]. Finally, based on the proposed prediction model, the influence of load on the CLC time of CCS was analysed.

Interaction between Ribbed Steel Bars and Surrounding Concrete under Load
The bonding force between ribbed steel bars and surrounding concrete consists of three parts: the chemical adhesive force between the cement gel and the surface of steel bars, the friction force between steel bars and concrete, and the mechanical interlocking force between the transverse ribs of steel bars and concrete. Among them, the mechanical interlocking force is the main part of the bonding force. The oblique extrusion force between the transverse ribs of steel bars and the surrounding concrete forms slip resistance. The component force of the oblique extrusion force along the axial direction of the steel bars makes the concrete between two transverse ribs bend and shear as a cantilever beam, which causes oblique cracks inside the concrete. The radial component of the oblique extrusion force makes the surrounding concrete like a pipe wall under internal pressure, resulting in hoop tensile stress inside the concrete, which causes radial cracks in the surrounding concrete. After the ribbed steel bars are subjected to a force, the distribution of cracks near the transverse ribs is shown in Figure 1.
surrounding concrete. Some researchers have established corrosion-induced cracking models of the concrete cover considering the effect of the rust-filling layer, while the effect of the load was not considered in their models [30,31]. During the service stage, RC structures suffer from different loads, and the loads affect the corrosion characteristics of steel bars embedded in concrete [35][36][37]. In contrast to that without considering load, the load can shorten the initial corrosion time, change the corrosion rate and the corrosion distribution, and cause local pitting of steel bars in RC structures [38][39][40], resulting in serious corrosion-induced cracking of the concrete cover [41,42]. Therefore, the influence of load should be considered when establishing the corrosion-induced cracking model of the concrete cover. However, to the best knowledge of the authors, the study of the corrosion-induced cracking model considering the effect of the load is severely inadequate.
To fill this research gap, in this paper, based on the thick-walled cylinder model, the effect of the load was considered to analyse the critical corrosion depth of steel bars when the IBSC cracked and the CCS cracked. Then, a prediction model for the corrosioninduced longitudinal cracking (CLC) time (i.e., initiation cracking time) of the CCS of RC structures under load was proposed, and the rationality of the prediction model was verified by the experimental data of Wang et al. [43]. Finally, based on the proposed prediction model, the influence of load on the CLC time of CCS was analysed.

Interaction between Ribbed Steel Bars and Surrounding Concrete under Load
The bonding force between ribbed steel bars and surrounding concrete consists of three parts: the chemical adhesive force between the cement gel and the surface of steel bars, the friction force between steel bars and concrete, and the mechanical interlocking force between the transverse ribs of steel bars and concrete. Among them, the mechanical interlocking force is the main part of the bonding force. The oblique extrusion force between the transverse ribs of steel bars and the surrounding concrete forms slip resistance. The component force of the oblique extrusion force along the axial direction of the steel bars makes the concrete between two transverse ribs bend and shear as a cantilever beam, which causes oblique cracks inside the concrete. The radial component of the oblique extrusion force makes the surrounding concrete like a pipe wall under internal pressure, resulting in hoop tensile stress inside the concrete, which causes radial cracks in the surrounding concrete. After the ribbed steel bars are subjected to a force, the distribution of cracks near the transverse ribs is shown in Figure 1. After the ribbed steel bars are subjected to a force, the surrounding concrete produces oblique extrusion force p on transverse ribs, as shown in Figure 2. In the figure, θ is the angle between the concrete sliding surface near the transverse ribs and the axis of ribbed steel bars, and μ is the friction coefficient between the ribbed steel bars and concrete. By decomposing the oblique extrusion force p and friction force μp on the concrete sliding surface along the axis and radial directions of the ribbed steel bars, the bond stress τ and the radial pressure q1 can be obtained, as written in Equations (1) and (2), respectively. After the ribbed steel bars are subjected to a force, the surrounding concrete produces oblique extrusion force p on transverse ribs, as shown in Figure 2. In the figure, θ is the angle between the concrete sliding surface near the transverse ribs and the axis of ribbed steel bars, and µ is the friction coefficient between the ribbed steel bars and concrete. By decomposing the oblique extrusion force p and friction force µp on the concrete sliding surface along the axis and radial directions of the ribbed steel bars, the bond stress τ and the radial pressure q 1 can be obtained, as written in Equations (1) and (2), respectively.  Tepfers [44] found that the angle θ as shown in Figure 2 was 30°-40°, and Xu [45] found that the angle ranged from 10° to 40°, with an average of 25°. The friction coefficient μ between the ribbed steel bars and concrete increases with the corrosion degree of ribbed steel bars, and if the ribbed steel bars are not corroded, the friction coefficient can be taken as 0.3 [45]. Guo and Shi [46] showed that the steel bars changed from a non-corrosion state to a corrosion state, and the variation range of the friction coefficient was 0.2-0.6. Zhao and Xiao [47] confirmed that the friction coefficient between the concrete and rolled steel was 0.25-0.35. According to the above research results, θ is taken as 30° and μ is taken as 0.35 in this paper.
The relationship between the bond stress τ and the radial pressure q1 can be derived from Equations (1) and (2): The bond stress as shown in Figure 2 can be determined by establishing the mechanical equilibrium equation along the axis direction of the ribbed steel bars: where R is the radius of the ribbed steel bars, and σs is the tensile stress of the ribbed steel bars.
Substituting Equation (4) into Equation (3), the radial pressure can be obtained as: To calculate dσs/dx in Equation (5), it is first necessary to derive the equation between the tensile stress of the ribbed steel bars and the bending moment, and then take the derivative of x on both sides of the equation to obtain dσs/dx. The detailed calculation process is as follows: (1) During the service stage, before the load-induced cracking of RC members, the full cross-section of RC members is subjected to force. When calculating the tensile stress of ribbed steel bars under the bending moment, the cross-section of the steel bars needs to Tepfers [44] found that the angle θ as shown in Figure 2 was 30 • -40 • , and Xu [45] found that the angle ranged from 10 • to 40 • , with an average of 25 • . The friction coefficient µ between the ribbed steel bars and concrete increases with the corrosion degree of ribbed steel bars, and if the ribbed steel bars are not corroded, the friction coefficient can be taken as 0.3 [45]. Guo and Shi [46] showed that the steel bars changed from a non-corrosion state to a corrosion state, and the variation range of the friction coefficient was 0.2-0.6. Zhao and Xiao [47] confirmed that the friction coefficient between the concrete and rolled steel was 0.25-0.35. According to the above research results, θ is taken as 30 • and µ is taken as 0.35 in this paper.
The relationship between the bond stress τ and the radial pressure q 1 can be derived from Equations (1) and (2): The bond stress as shown in Figure 2 can be determined by establishing the mechanical equilibrium equation along the axis direction of the ribbed steel bars: where R is the radius of the ribbed steel bars, and σ s is the tensile stress of the ribbed steel bars. Substituting Equation (4) into Equation (3), the radial pressure can be obtained as: To calculate dσ s /dx in Equation (5), it is first necessary to derive the equation between the tensile stress of the ribbed steel bars and the bending moment, and then take the derivative of x on both sides of the equation to obtain dσ s /dx. The detailed calculation process is as follows: (1) During the service stage, before the load-induced cracking of RC members, the full cross-section of RC members is subjected to force. When calculating the tensile stress of ribbed steel bars under the bending moment, the cross-section of the steel bars needs to be transformed to the concrete section. For RC beams with a rectangular cross-section, the transformed cross-section is shown in Figure 3, and the area of the transformed cross-section can be calculated by Equation (6).
where A 0 is the area of cross-section after transformation, n 1 denotes the ratio of the elastic modulus of tensile steel bars to that of concrete, n 2 refers to the ratio of the elastic modulus of the compressive steel bars to that of concrete, A s is the area of cross-section of tensile steel bars, A s is the area of cross-section of compressive steel bars, and b and h are the width and height of the cross-section, respectively.
where A0 is the area of cross-section after transformation, n1 denotes the ratio of the elastic modulus of tensile steel bars to that of concrete, n2 refers to the ratio of the elastic modulus of the compressive steel bars to that of concrete, As is the area of cross-section of tensile steel bars, s A′ is the area of cross-section of compressive steel bars, and b and h are the width and height of the cross-section, respectively.  The height of the compression zone of the transformed cross-section can be determined by the equal area moment of the tension zone and compression zone to the neutral axis of the cross-section: where x0 is the height of the compression zone, and h0 is the effective height of the crosssection.
After the height of the compression zone is obtained, the moment of inertia I0 of the transformed cross-section can be expressed as: Therefore, before load-induced cracking of RC beams, the tensile stress of ribbed steel bars under bending moment is: where M is the bending moment.
During the service stage, after the load-induced cracking of RC beams, the concrete in the tension zone of the cross-section exits work, and only the steel bars in the tension zone are subjected to force. For RC beams with a rectangular cross-section, the tensile The height of the compression zone of the transformed cross-section can be determined by the equal area moment of the tension zone and compression zone to the neutral axis of the cross-section: where x 0 is the height of the compression zone, and h 0 is the effective height of the cross-section. After the height of the compression zone is obtained, the moment of inertia I 0 of the transformed cross-section can be expressed as: Therefore, before load-induced cracking of RC beams, the tensile stress of ribbed steel bars under bending moment is: where M is the bending moment. During the service stage, after the load-induced cracking of RC beams, the concrete in the tension zone of the cross-section exits work, and only the steel bars in the tension zone are subjected to force. For RC beams with a rectangular cross-section, the tensile stress of steel bars in the tension zone of the cross-section can be calculated according to the method of the transformed cross-section, or calculated according to the method stipulated in China Standard (GB 50010) [48]. The latter method is adopted in this paper, as written in Equation (10).
(2) For the RC beams with a rectangular cross-section, the bending moment is often different along the length of beams under load. Taking the RC beams subjected to thirdpoint concentrated load (i.e., F/2) as an example (Figure 4), the bending moment can be calculated according to Equation (11).
where M(x) is the bending moment at the distance x from the left support of beams under the third-point concentrated load, and l1 is 1/3 of the calculated span of RC beams. Substituting Equation (11) into Equations (9) and (10), respectively, the tensile stress of ribbed steel bars can be obtained, and then Equation (5) can be rewritten as: Before load-induced cracking of RC beams: After load-induced cracking of RC beams:  Substituting Equation (11) into Equations (9) and (10), respectively, the tensile stress of ribbed steel bars can be obtained, and then Equation (5) can be rewritten as: Before load-induced cracking of RC beams: After load-induced cracking of RC beams: This paper only gives the calculation formulas of radial pressure when the RC beams with rectangular cross-sections are subjected to third-point concentrated loads. Similarly, the calculation method of radial pressure can be established for beams under other types of loads.
For RC structures under load, before the corrosion of steel bars, there is only radial pressure between steel bars and surrounding concrete. The thick-walled cylinder model [44,49,50] is usually adopted to calculate the interaction between steel bars and the concrete cover, as shown in Figure 5. In this figure, C is the thickness of the concrete cover, d is the diameter of steel bars, and q 1 is the radial pressure. For RC structures under load, before the corrosion of steel bars, there is only radia pressure between steel bars and surrounding concrete. The thick-walled cylinder mode [44,49,50] is usually adopted to calculate the interaction between steel bars and the concrete cover, as shown in Figure 5. In this figure, C is the thickness of the concrete cover d is the diameter of steel bars, and q1 is the radial pressure. Along the longitudinal direction of the steel bars, the concrete cover may be in uncracked, partially cracked, or completely cracked states under radial pressure. Thi paper only discusses the longitudinal cracking of the concrete cover under the coupled effects of load and steel corrosion; hence, the transverse cracking of the concrete cover i Along the longitudinal direction of the steel bars, the concrete cover may be in uncracked, partially cracked, or completely cracked states under radial pressure. This paper only discusses the longitudinal cracking of the concrete cover under the coupled effects of load and steel corrosion; hence, the transverse cracking of the concrete cover is not considered and is out of the scope of this paper. Assuming that the longitudinal cracks do not generate in concrete cover under radial pressure, that is, the concrete cover is in the elastic deformation stage ( Figure 5). Therefore, the elastic theory can be adopted to calculate the stress distribution in the concrete cover under radial pressure. According to the theory of elasticity [51], the hoop stress at any position r x (R ≤ r x ≤ b) of the thick-walled cylinder under radial pressure can be expressed as: where b is the outer radius of the thick-walled cylinder, b = d/2 + C.
Equation (14) can be used to judge the longitudinal cracking of the concrete cover: if σ θ (R) < f t , no cracking, and if σ θ (R) ≥ f t and σ θ (b) < f t , partial cracking. After the corrosion of the steel bars, the surrounding concrete does not crack immediately. Corrosion products accumulate at the IBSC and continue to exert corrosion expansion pressure q 2 on the surrounding concrete. Therefore, the time when longitudinal cracks appear on the CCS under the combined action of corrosion expansion pressure q 2 and radial pressure q 1 is the longitudinal cracking time of CCS under the coupled effects of load and steel corrosion. The total radial pressure q calculated by Equation (15) can be directly applied to the inner wall of the thick-walled cylinder, and then the elasticity and damage mechanics theories are adopted to derive the time for the appearance of longitudinal cracks on the CCS.

Rust Filling Model
During the corrosion-induced cracking process of the concrete cover, corrosion products accumulate at the IBSC forming the rust layer and filling the pores of surrounding concrete forming the rust-filling layer. The process of corrosion products filling the pores of surrounding concrete is affected by many factors, such as the position of the aggregate and the distribution of pores. Zhao et al. [29,30,32] and Wu [31] found that the average thickness of the rust-filling layer increased with the thickness of the rust layer until the thickness of the rust layer reached a critical value, that is, the corrosion products were constantly filling the pores of surrounding concrete during the formation of the rust layer. The relationship between the average thickness of the rust-filling layer and the thickness of the rust layer is shown in Figure 6 and can be expressed by Equation (16) [29,30].
where T CL is the thickness of the rust layer at the IBSC, T CP denotes the average thickness of the rust-filling layer; T max CP is the maximum value of T CP , T cr CL is the T CL corresponding to T max CP , and k T is the ratio of T max CP to T cr CL . To consider the influence of the rust-filling layer in the corrosion-induced cracking model of the concrete cover, the rust-filling layer should be converted to the equivalent rust layer as shown in Figure 7, and the thickness of the equivalent rust layer can be expressed as [30]: where T CL,pore is the thickness of the equivalent rust layer, T CP denotes the thickness of the rust-filling layer, and γ is the porosity of concrete around the steel bars, which can be approximately calculated as [30]: where W/C is the water-cement ratio of concrete, and α is the hydration degree of concrete.
layer until the thickness of the rust layer reached a critical value, that is, the corrosion products were constantly filling the pores of surrounding concrete during the formation of the rust layer. The relationship between the average thickness of the rust-filling layer and the thickness of the rust layer is shown in Figure 6 and can be expressed by Equation (16) [29,30].
Where TCL is the thickness of the rust layer at the IBSC, C P To consider the influence of the rust-filling layer in the corrosion-induced cracking model of the concrete cover, the rust-filling layer should be converted to the equivalent rust layer as shown in Figure 7, and the thickness of the equivalent rust layer can be expressed as [30]: Where TCL,pore is the thickness of the equivalent rust layer, TCP denotes the thickness of the rust-filling layer, and γ is the porosity of concrete around the steel bars, which can be approximately calculated as [30]: where W/C is the water-cement ratio of concrete, and α is the hydration degree of concrete. Generally, the average thickness of the rust-filling layer does not reach the maximum average thickness before the CCS cracks [30,31], as shown in Figure 6. Therefore, when establishing a prediction model for the cracking of CCS, only the case of C L (16) is considered. The thickness of the equivalent rust layer can be determined according to Equation (17): It can be found from Equation (19) that the thickness of the equivalent rust layer is related to the type of concrete. After the corrosion of the steel bars, assuming that Vrust is the total volume of rust, and Vsteel is the volume lost by the steel bars due to corrosion, the actual rust volume expansion ratio n can be defined as [30]: Generally, the average thickness of the rust-filling layer does not reach the maximum average thickness before the CCS cracks [30,31], as shown in Figure 6. Therefore, when establishing a prediction model for the cracking of CCS, only the case of T CL < T cr CL in Equation (16) is considered. The thickness of the equivalent rust layer can be determined according to Equation (17): It can be found from Equation (19) that the thickness of the equivalent rust layer is related to the type of concrete. After the corrosion of the steel bars, assuming that V rust is the total volume of rust, and V steel is the volume lost by the steel bars due to corrosion, the actual rust volume expansion ratio n can be defined as [30]: The total volume of rust V rust is the sum of the volume of rust filling the concrete pores V rust,CP and the volume of rust layer V rust,CL : However, during the corrosion process of steel bars, only the rust layer exerts extrusion force on the surrounding concrete, which causes the concrete cover to crack. Therefore, the nominal rust volume expansion ratio n 0 can be defined as [30]: The relationship between the actual rust volume expansion ratio and the nominal rust volume expansion ratio is [30]: The following equation can be derived from Equations (19) and (23): Equation (24) shows that when n = n 0 , the volume of rust filled into the pores of the concrete around the steel bars is zero, and n 0 /n decreases with the increase of concrete porosity γ, that is, the volume of rust filled into the pores of concrete around the steel bars increases with concrete porosity. It can be seen from the above analysis that n 0 should be used instead of n when establishing the corrosion-induced cracking model of the concrete cover. However, for different types of concrete, before calculating n 0 , the values of γ and k T need to be determined.

Non-cracking Stage Model of Concrete Cover
For the thick-walled cylinder model, as shown in Figure 8, when the IBSC does not crack under the total radial pressure q, the stress and strain at any point in the concrete cover can be calculated by the elastic theory. In Figure 8, δ cc and δ r are, respectively, the radial deformations of the concrete and rust layers at the IBSC, d ρ is the residual diameter of the steel bars after corrosion, and d 1 denotes the diameter of the steel bars after the free expansion of rust.
Stress components at any point in the concrete cover under the total radial pressure can be formulated as [13,30]: where σ r and σ θ are the radial stress and the hoop stress in the concrete cover at a distance r x from the center of the steel bar, respectively, and b = R + C.
Strain components at any point in the concrete cover under the total radial pressure can be derived as [13,30]: where ε r and ε θ are the radial strain and the hoop strain in concrete cover at the distance r x from the center of the steel bar, respectively, and E c and v c are the elastic modulus and Poisson's ratio of concrete, respectively. Equation (24) shows that when n = n0, the volume of rust filled into the pores of the concrete around the steel bars is zero, and n0/n decreases with the increase of concrete porosity γ, that is, the volume of rust filled into the pores of concrete around the steel bars increases with concrete porosity. It can be seen from the above analysis that n0 should be used instead of n when establishing the corrosion-induced cracking model of the concrete cover. However, for different types of concrete, before calculating n0, the values of γ and kT need to be determined.

Non-cracking Stage Model of Concrete Cover
For the thick-walled cylinder model, as shown in Figure 8, when the IBSC does not crack under the total radial pressure q, the stress and strain at any point in the concrete cover can be calculated by the elastic theory. In Figure 8, δcc and δr are, respectively, the radial deformations of the concrete and rust layers at the IBSC, dρ is the residual diameter of the steel bars after corrosion, and d1 denotes the diameter of the steel bars after the free expansion of rust. Stress components at any point in the concrete cover under the total radial pressure can be formulated as [13,30]: Where σr and σθ are the radial stress and the hoop stress in the concrete cover at a distance rx from the center of the steel bar, respectively, and b = R + C.
Strain components at any point in the concrete cover under the total radial pressure can be derived as [13,30]: where εr and εθ are the radial strain and the hoop strain in concrete cover at the distance rx from the center of the steel bar, respectively, and Ec and vc are the elastic modulus and Poisson's ratio of concrete, respectively. According to the elastic theory, the radial displacement u r at any point in the concrete cover can be derived by ε θ = u r /r x : The critical state of cracking at the IBSC is defined as the hoop strain of concrete reaching ultimate tensile strain: where ε t and f t are the ultimate tensile strain and tensile strength of concrete, respectively.
When the IBSC cracks, the critical value q inner of total radial pressure q can be obtained on the basis of Equations (27) and (28): According to the deformation coordination conditions of the concrete and rust layer at the IBSC, the following equation can be obtained [13,30]: where R 1 is the radius of steel bars after the free expansion of rust. The calculation methods of parameters in Equation (30) are [13,30]: where η s is the mass loss of steel bars, E r is the elastic modulus of rust, which can be taken as 100 MPa [52], and v r is the Poisson's ratio of rust, which can be taken as 0.25 [52]. When the IBSC cracks, the relationship between the critical total radial pressure q inner and critical mass loss η inner s of steel bars can be derived from Equations (29) to (31): The critical total radial pressure q inner can be derived from Equation (32): Substituting x 1 = (n 0 − 1)η inner s + 1 into Equation (33), and assuming the following equations can be obtained: The algebraic cubic equation as shown in Equation (34a) has three roots. However, only one analytic real root for Equation (34a) is useful, which can be calculated as [53]: If Equation (34a) has no real roots, a real root can be approximately obtained as the value of x 1 through the MATLAB program, and then the critical mass loss of steel bars can be determined as: Through the critical mass loss of steel bars, the critical corrosion depth of steel bars can be derived as: where δ inner steel is the critical corrosion depth of steel bars when the IBSC cracks. According to the calculation model of the critical corrosion depth of steel bars established above, to analyse the influence of load on the critical corrosion depth of steel bars when the IBSC cracks, an example is adopted. Assuming that the length of an RC beam is 2300 mm, its calculated length is 2100 mm, the size of its cross-section is 150 (width) × 250 (height) mm, the strength grade of concrete is C30, the tensile steel bars are two HRB335 ribbed steel bars with a diameter of d = 18 mm and the yield strength is f y = 335 MPa, the thickness of the concrete cover is C = 30 mm, and the actual rust volume expansion ratio is n = 2. The parameters of concrete are tensile strength f t = 2.01 MPa, compressive strength f c = 20.1 MPa, elastic modulus E c = 30 GPa, and Poisson's ratio v c = 0.2. The range of n/n 0 is about 1-1.2 [31], and 1.0 and 1.2 are taken in this example.
For the RC beam under a third-point concentrated load, the radial pressure q 1 occurs at the IBSC, which causes hoop tensile stress in the concrete cover, as shown in Equation (14). Assuming that the hoop tensile stress at the position r x = R of the concrete cover caused by the radial pressure is 0, 0.2, and 0.3 times the tensile strength of concrete, the tensile stress level δ T = 0, 0.2, and 0.3, meanwhile, the IBSC does not crack [54]. Figure 9 shows the critical corrosion depth of steel bars when the IBSC cracks are induced by corrosion under load. It can be seen from Figure 9 that the load has a significant effect on the critical corrosion depth of the steel bars. The greater the load, the smaller the critical corrosion depth, and for the two cases of n/n 0 = 1.0 and n/n 0 = 1.2, when the tensile stress level δ T increases from 0 to 0.3, the critical corrosion depth is reduced by 67.9% and 59.4%, respectively. Therefore, the influence of load needs to be considered when establishing the corrosion-induced cracking model of the concrete cover. Figure 9 also depicts that when n/n 0 = 1.2, the critical corrosion depth is significantly larger than that of n/n 0 = 1.0, which is because, the larger the n/n 0 , the more rust filled into the pores of concrete around steel bars, and the rust-filling layer does not produce squeezing force on the IBSC. As a result, the more serious the corrosion of steel bars, the larger the critical corrosion depth of steel bars. For the RC beam under a third-point concentrated load, the radial pressure q1 oc at the IBSC, which causes hoop tensile stress in the concrete cover, as shown in Equa (14). Assuming that the hoop tensile stress at the position rx = R of the concrete co caused by the radial pressure is 0, 0.2, and 0.3 times the tensile strength of concrete, tensile stress level δT = 0, 0.2, and 0.3, meanwhile, the IBSC does not crack [54]. Figu shows the critical corrosion depth of steel bars when the IBSC cracks are induced corrosion under load. It can be seen from Figure 9 that the load has a significant effec the critical corrosion depth of the steel bars. The greater the load, the smaller the cri corrosion depth, and for the two cases of n/n0 = 1.0 and n/n0 = 1.2, when the tensile st level δT increases from 0 to 0.3, the critical corrosion depth is reduced by 67.9% and 59 respectively. Therefore, the influence of load needs to be considered when establish the corrosion-induced cracking model of the concrete cover. Figure 9 also depicts when n/n0 = 1.2, the critical corrosion depth is significantly larger than that of n/n0 = which is because, the larger the n/n0, the more rust filled into the pores of concrete aro steel bars, and the rust-filling layer does not produce squeezing force on the IBSC. A result, the more serious the corrosion of steel bars, the larger the critical corrosion de of steel bars.

Partial Cracking Stage Model of Concrete Cover
After the IBSC cracks under the total radial pressure, the cracks propagate radiall the surface of the cylinder. Therefore, the thick-walled cylinder can be divided into coaxial cylinders. The inside one is a cracked cylinder and the outside one is an in cylinder, as depicted in Figure 10. In the figure, Rc represents the radius of the inter between the cracked cylinder and the intact cylinder.

Partial Cracking Stage Model of Concrete Cover
After the IBSC cracks under the total radial pressure, the cracks propagate radially to the surface of the cylinder. Therefore, the thick-walled cylinder can be divided into two coaxial cylinders. The inside one is a cracked cylinder and the outside one is an intact cylinder, as depicted in Figure 10. In the figure, R c represents the radius of the interface between the cracked cylinder and the intact cylinder. The cracked concrete inside the thick-walled cylinder is assumed to be non-uniform orthotropic linear elastic material, and the outside intact concrete is assumed to be isotropic linear elastic material [33]. Then, the stress component and strain component at any point in the intact concrete can be derived from elastic theory [55]: Stress components: where c R q is the radial pressure at the interface between cracked and intact cylinders.
Strain components: For the cracked cylinder, assuming that the cracks are uniformly distributed on the circumference of the cracked cylinder and propagate outwards radially, the damage degree of concrete along the direction of cracks is different, which near the steel bars is larger than that of concrete far away from the steel bars. The relationship between the radial stress and hoop stress at any point in the cracked cylinder is [55]: Based on the Mohr-Coulomb failure criterion [13] and Mazar's damage model [56], the damage variable D is adopted to describe the damage degree of the cracked cylinder along the radial direction. Equation (40) can be rewritten as [57]: where φ is the internal friction angle of concrete, and c is the cohesive strength of concrete. The cracked concrete inside the thick-walled cylinder is assumed to be non-uniform orthotropic linear elastic material, and the outside intact concrete is assumed to be isotropic linear elastic material [33]. Then, the stress component and strain component at any point in the intact concrete can be derived from elastic theory [55]: Stress components: where q R c is the radial pressure at the interface between cracked and intact cylinders. Strain components: For the cracked cylinder, assuming that the cracks are uniformly distributed on the circumference of the cracked cylinder and propagate outwards radially, the damage degree of concrete along the direction of cracks is different, which near the steel bars is larger than that of concrete far away from the steel bars. The relationship between the radial stress and hoop stress at any point in the cracked cylinder is [55]: Based on the Mohr-Coulomb failure criterion [13] and Mazar's damage model [56], the damage variable D is adopted to describe the damage degree of the cracked cylinder along the radial direction. Equation (40) can be rewritten as [57]: where ϕ is the internal friction angle of concrete, and c is the cohesive strength of concrete.
The internal friction angle of concrete decreases slightly with increasing concrete strength, which can be approximately taken as 37 • , and the cohesive strength of concrete can be determined as [30,58]: where f c is the compressive strength of concrete. The damage variable D can be derived by Mazar's damage equation [59]: where A t and B t are the coefficients of Mazar's damage model, 0.7 < A t < 1 and 10 4 < B t < 10 5 . Substituting Equation (41) into Equation (40) yields [13,30]: The boundary conditions are expressed as: Substituting Equation (45a) into Equation (44) yields: Substituting Equation (45b) into Equation (39) results in: Substituting Equation (45b) and Equation (46) into Equation (44), the total radial pressure can be derived as [13,30]: Equation (48) can be simplified to Equation (29) when R c = R. Hence, the calculated results obtained at the non-cracking stage and partial cracking stage of the concrete cover are continuous. To simplify the calculation, the cracked cylinder is divided into N concentric rings of equal thickness, as depicted in Figure 11. For each ring, the damage variable D can be considered constant. The radial displacement u r in the cracked cylinder should satisfy the following equation [30,60,61]: The radial displacement at any point in the cracked cylinder can be obtained from Equation (49), which is expressed as [13,30]: where C 1 and C 2 are the constants.
where C1 and C2 are the constants.
According to the geometric equation between displacement and strain, the radial and hoop strains at any point in the cracked cylinder can be derived as [13,30]: Assuming that the thickness of each ring in the cracked cylinder is ΔR, the total thickness of the cracked cylinder is NΔR. For the Nth ring shown in Figure 11 adjacent to the uncracked concrete, its damage variable D N is close to 0. Therefore, it can be assumed that the damage variable of the whole Nth ring is 0, and the constants N  (50) can be derived according to the strain continuity condition of the two rings at the contact surface. The damage variable D N−1 of the (N−1)th ring can be determined by Equation (43), and the parameter εθ in Equation (43) takes the hoop strain on the inner surface of the Nth ring. As a result, the radial and hoop strains at any point in the (N−1)th ring can also be calculated according to Equation (51). Similarly, the constants and strains from the (N−2)th ring to the 1th ring can be calculated. The above-detailed analysis process can be referred to in the literature [13]. According to the above analysis, the radial deformation of the concrete at the IBSC is equal to the radial displacement of the inner surface of the 1th ring, that is: where D 1 is the damage variable of 1th ring, which can be calculated according to Equation (43). According to the geometric equation between displacement and strain, the radial and hoop strains at any point in the cracked cylinder can be derived as [13,30]: Assuming that the thickness of each ring in the cracked cylinder is ∆R, the total thickness of the cracked cylinder is N∆R. For the Nth ring shown in Figure 11 adjacent to the uncracked concrete, its damage variable D N is close to 0. Therefore, it can be assumed that the damage variable of the whole Nth ring is 0, and the constants C N 1 and C N 2 in Equation (50) can be determined according to the condition that Equation (51) and Equation (39) are equal at the position of r x = R c . When the constants in Equation (50) are determined, the radial and hoop strains at any point in the Nth ring can be calculated according to Equation (51). Since the inner surface of the Nth ring is the outer surface of the (N−1)th ring, for the (N−1)th ring, the constants C N−1 1 and C N−1 2 in Equation (50) can be derived according to the strain continuity condition of the two rings at the contact surface. The damage variable D N−1 of the (N−1)th ring can be determined by Equation (43), and the parameter ε θ in Equation (43) takes the hoop strain on the inner surface of the Nth ring. As a result, the radial and hoop strains at any point in the (N−1)th ring can also be calculated according to Equation (51). Similarly, the constants and strains from the (N−2)th ring to the 1th ring can be calculated. The above-detailed analysis process can be referred to in the literature [13]. According to the above analysis, the radial deformation of the concrete at the IBSC is equal to the radial displacement of the inner surface of the 1th ring, that is: where D 1 is the damage variable of 1th ring, which can be calculated according to Equation (43). At the partial cracking stage of the concrete cover, the mass loss and corrosion depth of the steel bars can be derived according to Equations (30)- (37). When the cracks develop on the surface of the thick-walled cylinder, i.e., R c = R + C, the critical corrosion depth of the steel bars can be determined as: where η surface s is the critical mass loss of steel bars when the CCS cracks.
To analyse the influence of load on the critical corrosion depth of steel bars when the CCS cracks are induced by corrosion, the same example as Section 4 is used. The internal friction angle ϕ of concrete is 37 • , the cohesion strength c = 5.01 MPa of concrete is calculated according to Equation (42), ∆R is 1 mm, and the values of A t and B t are 0.7 and 10 4 , respectively. Zhao et al. [13] found that when the cracks propagated to 0.8 times the thickness of the concrete cover, the corrosion expansion force at the IBSC reached the maximum, and at this time, even if the corrosion process of steel bars stopped, the cracks could spontaneously extend to the CCS. Therefore, this paper assumes that when the radius of the interface between the cracked cylinder and intact cylinder is R c = R + 0.8C, the CCS cracks. Assuming that the hoop tensile stress at the position r x = R of concrete cover caused by the radial pressure is 0, 0.2, and 0.3 times the tensile strength of concrete, that is, the tensile stress level δ T = 0, 0.2, and 0.3, and meanwhile, the IBSC does not crack [54]. The range of n/n 0 is about 1-1.2 [31], and 1.0, 1.05, and 1.1 are taken in this example. When the CCS cracks are induced by corrosion under load, the critical corrosion depth of steel bars is shown in Figure 12.

Prediction Model of CLC Time of CCS under Load
According to the definition of the nominal rust volume expansion ratio n0 de in Section 3, when the CCS cracks, the thickness of the rust layer is: Liu and Weyers [16] found that after the steel bars in concrete were corrod total thickness of the rust layer when the corrosion time was t could be expressed  Figure 12 illustrates that the load affects the critical corrosion depth of the steel bars when the CCS cracks, but the effect is not significant. The larger the load, the smaller the critical corrosion depth of the steel bars. The larger the n/n 0 , the more rust filled into the pores of concrete around steel bars, resulting in a larger critical corrosion depth of the steel bars. For the three cases of δ T = 0, 0.2, and 0.3, when the n/n 0 increases from 1.0 to 1.1, the critical corrosion depth of steel bars increases by 30.8%, 30.2%, and 30.6%, respectively. For the three cases of n/n 0 = 1.0, 1.05, and 1.1, when the tensile stress level δ T increases from 0 to 0.3, the critical corrosion depth of the steel bars decreases by 4.6%, 4.8%, and 4.7%, respectively. Apparently, the value of n/n 0 , that is, the influence of the rust-filling layer on the critical corrosion depth of the steel bars is greater than that of the load. Compared with the analysis results of Section 4, it can be found that the influence of the load on the critical corrosion depth of the steel bars when the IBSC cracks is larger than that of when the CCS cracks.

Prediction Model of CLC Time of CCS under Load
According to the definition of the nominal rust volume expansion ratio n 0 described in Section 3, when the CCS cracks, the thickness of the rust layer is: Considering the thickness of the equivalent rust layer, when the CCS cracks, the total thickness of the rust layer is: Liu and Weyers [16] found that after the steel bars in concrete were corroded, the total thickness of the rust layer when the corrosion time was t could be expressed as: where α rust is the constant related to the type of corrosion products, which can be taken as 0.57 [16]; ρ rust is the density of corrosion products, which can be taken as 3600 kg/m 3 [16]; ρ st denotes the density of the steel bars and is taken as 7850 kg/m 3 ; W rust (t) means the total amount of corrosion products at corrosion time t; and i corr (t) represents the current density at corrosion time t.
When the current density of steel corrosion is a constant, the CLC time of CCS under load can be directly derived according to Equations (55) and (56): When the current density of steel corrosion changes with time, the relationship between current density and time needs to be determined according to the measured values. Then, the relationship is substituted into Equation (56), followed by integration with respect to t from 0 to t surface cr . Finally, the CLC time of CCS under load can be determined according to Equations (55) and (56).

Model Validation
For the model established in Section 5, the rationality has been verified by Zhao et al. [30] when the influence of the load is not considered. This section focuses on verifying the rationality of the model considering the effect of load. Wang et al. [43] recorded the CLC time (i.e., initiation cracking time) of the CCS of recycled concrete beams under load as listed in Table 1, and in their test, the load applied to recycled concrete beams was 0.2, 0.4, and 0.6 times the flexural capacity, respectively. It can be found that the CLC time of the CCS of recycled concrete beams tends to decrease with the increase of the applied load. Due to the load borne by RC structures during the service stage not being larger than 0.6 times the flexural capacity [62], it can be concluded that the recycled concrete beams in the test of Wang et al. [43] are all in the service stage. It should be noted that Equation (57) is established based on the uniform corrosion of steel bars, whereas the corrosion of steel bars under load has obvious pitting corrosion characteristics. However, the electrochemical corrosion method is adopted in the corrosion test of Wang et al. [43]; hence, the corrosion of steel bars in recycled concrete beams under load is mainly uniform corrosion, which has been discussed in detail by Wang et al. [43]. Therefore, the experimental values of Wang et al. [43] can be adopted to verify the rationality of Equation (57). When the Equation (57) is adopted to calculate the CLC time of CCS under load, the values of the parameters in Equation (57) are as follows: α rust = 0.57, ρ rust = 3600 kg/m 3 , ρ st = 7850 kg/m 3 , R = 6 mm, n = 2, i corr = 0.01 mA/mm 2 , n/n 0 = 1.1, C = 23 mm, b = 29 mm, v c = 0.2, ϕ = 37 • , A t = 0.7, B t = 10 4 , ∆R = 0.5 mm, and assuming R c = 6 + 0.8 × 23 ≈ 25 mm when the CCS cracks. The comparison of the calculated values from Equation (57) and the experimental values are listed in Table 1. It can be found that the calculated values are slightly larger than the experimental values, but the two are basically consistent. There are two main reasons for this phenomenon: on the one hand, in the test of Wang et al. [43], the steel bars embedded in recycled concrete beams are not completely rusted uniformly, and there are many local pits on the surface of steel bars, and it is well known that pitting corrosion of steel bars can shorten the CLC time of CCS; on the other hand, it is difficult to accurately determine the values of n and n/n 0 , and the analysis results of Sections 4 and 5 show that the value of n/n 0 has an effect on the critical corrosion depth of steel bars, i.e., the larger the n/n 0 , the larger the critical corrosion depth, which in turn affects the CLC time of CCS under load, hence the n/n 0 = 1.1 is taken in Equation (57), which may be greater than the actual value of n/n 0 for the recycled concrete beams in the test of Wang et al. [43].

Parametric Analysis
There are some factors affecting the CLC time (i.e., initiation cracking time) of CCS, such as concrete tensile strength, concrete cover thickness, reinforcement diameter, current density, rust property, etc. [14,49,60]. Based on the prediction model established in Section 6, this section analyses the influence of load, n, and, n/n 0 on the CLC time of CCS. The RC beam as shown in Section 4 is adopted, the values of parameters are listed in Sections 4 and 5, and the current density i corr = 0.01 mA/mm 2 . Assuming that the hoop tensile stress at the position r x = R of concrete cover caused by load is 0, 0.1, 0.2, and 0.3 times the tensile strength of concrete, the tensile stress level δ T = 0, 0.1, 0.2, and 0.3. The range of the actual rust volume expansion ratio n is about 2-4 [13,14], and 2, 3, and 4 are adopted in this section. The range of n/n 0 is about 1-1.2 [31], and 1.0, 1.1, and 1.2 are taken in this section. The influence of load, n, and n/n 0 on the CLC time of CCS is shown in Figure 13.  [13,14], and 2, 3, and 4 are adopted in this section. The range of n/n0 is about 1-1.2 [31], and 1.0, 1.1, and 1.2 are taken in this section. The influence of load, n, and n/n0 on the CLC time of CCS is shown in Figure 13. Figure 13 shows that, with the increase of n or load, the CLC time of CCS decreases, while the CLC time increases obviously with the n/n0. For example, in the case of n = 3 and n/n0 = 1.2, when δT increases from 0 to 0.3, the CLC time decreases from 397.42 h to 355.04 h; while in the case of n = 4 and δT = 0.30, when n/n0 increases from 1.0 to 1.2, the CLC time increases from 142.78 h to 243.80 h. In order to further investigate the effect of load on CLC time, the influence of load on the CLC time of CCS can be calculated quantitatively according to Figure 13, as listed in Table 2. It can be found from Table 2 that the values of n and n/n0 affect the influence of load on the CLC time. When n/n0 remains unchanged, the greater the n, the more obvious the load can shorten the CLC time, that is, the percentage reduction in CLC time caused by load increases with n. For instance, in the case of n/n0 = 1.1, when n increases from 2 to 4, the percentage reduction in CLC time caused by the load is increased by 178.5%. However, when n remains unchanged, the percentage reduction in CLC time caused by the load decreases with the increase of n/n0. For instance, for the case of n = 3, when n/n0 increases from 1.0 to 1.2, the percentage reduction in CLC time caused by the load is decreased by 27.2%.

Conclusions
This paper studied the effect of load on the CLC time of CCS of RC structures, and the following conclusions can be drawn: (1) The load has a significant effect on the critical corrosion depth of steel bars when the IBSC cracks induced by corrosion, and for the n/n0 = 1.0 and 1.2, when the tensile stress level increases from 0 to 0.3, the critical corrosion depth is reduced by 67.9% and 59.4%, respectively;  Figure 13 shows that, with the increase of n or load, the CLC time of CCS decreases, while the CLC time increases obviously with the n/n 0 . For example, in the case of n = 3 and n/n 0 = 1.2, when δ T increases from 0 to 0.3, the CLC time decreases from 397.42 h to 355.04 h; while in the case of n = 4 and δ T = 0.30, when n/n 0 increases from 1.0 to 1.2, the CLC time increases from 142.78 h to 243.80 h. In order to further investigate the effect of load on CLC time, the influence of load on the CLC time of CCS can be calculated quantitatively according to Figure 13, as listed in Table 2. It can be found from Table 2 that the values of n and n/n 0 affect the influence of load on the CLC time. When n/n 0 remains unchanged, the greater the n, the more obvious the load can shorten the CLC time, that is, the percentage reduction in CLC time caused by load increases with n. For instance, in the case of n/n 0 = 1.1, when n increases from 2 to 4, the percentage reduction in CLC time caused by the load is increased by 178.5%. However, when n remains unchanged, the percentage reduction in CLC time caused by the load decreases with the increase of n/n 0 . For instance, for the case of n = 3, when n/n 0 increases from 1.0 to 1.2, the percentage reduction in CLC time caused by the load is decreased by 27.2%.