Experimental Study on Bond-Slip Behavior between Corroded I-Shaped Steel and Concrete in Subsea Tunnel

Degradation of the bond between I-shaped steel and concrete due to the corrosion of I-shaped steel significantly affects the durability of steel reinforced concrete (SRC) structures. This study carried out the accelerated corrosion test and push-out test to study the bond-slip behavior and characteristics considering the corrosion of I-shaped steel, and test results indicated that: (1) The performance degradation of the bond-slip accelerated when the corrosion ratio reached 12%. (2) The corrosion failure pattern of SRC experienced slip phase and destruction phase in the rising stage. (3) Based on the principle of minimum potential energy, the bond stress was obtained only with the load and the displacement in the free end and the loading end. (4) Meanwhile, a new bond-slip degradation model was developed using the interface damage theory. Finally, the proposed model agreed with the experimental results.


Introduction
The subsea tunnels are often constructed by drilling and blasting method. For example, the subsea tunnels in Norway were all constructed by drilling and blasting method [1], and the Japanese Seikan Tunnel (53.8 km) was also constructed by drilling and blasting method [2]. In China, the drilling and blasting method was used in Xiang'an Submarine Tunnel (5.9 km) in Xiamen [3], and Jiaozhou Bay Submarine Tunnel (6.17 km) in Qingdao [4]. The composite lining composed of initial support and secondary lining is often used in the subsea tunnel support system constructed by drilling and blasting method. Since SRC has several advantages over the traditional reinforced concrete (RC): these include high bearing capacity, high rigidity, good ductility and energy dissipation [5,6], I-shaped steel is generally installed in the initial support in the soft rock section of the subsea tunnel. Shotcrete employed for initial support is sprayed onto the substrate at a high speed through a pneumatic hose or pipe under the action of air pressure and compacted instantaneously [7]. Compared to ordinary shotcrete, shotcrete has better Cl − penetration properties, and Wang et al. [8] attributed these to the porosity generated during shooting and the "looseness" of the microstructure due to the fast formation of ettringite and C-S-H gel. Therefore, the I-shaped steel corrosion in the initial support of subsea tunnels occurred easily due to the electrochemical action caused by chloride ions invasion. When the cumulative concentration of chloride ions reaches a certain level, the passive film of steel will be activated and the electrochemical reaction will happen [9,10]. The electrochemical reaction of significantly improve bond strength. Wang et al. [31] conducted the steel-concrete specimens test at a high temperature. The results showed that the ultimate slip load and residual load of the shaped steel concrete were significantly lower than those at room temperature, and the decline was fastest in the range from 20 to 200 • C.
In this investigation, with respect the effects of corrosion ratio of I-shaped steel, this paper adopted accelerated corrosion test and push-out test to obtain bond-slip behavior and bond-slip degradation constitutive model between corroded I-shaped steel and concrete. It not only has practical application value in guiding the design, construction and maintenance of support system of subsea tunnel, but also enriches and develops the bond-slip degradation theory of corroded steel reinforced concrete structure.

Accelerated Corrosion Test
It is well known that natural chloride corrosion of I-shaped steel embedded in concrete takes up to several years or more, which makes it difficult to carry out in a laboratory. Thus, the corrosion of I-shaped steel is accelerated by accelerated corrosion test. According to Faraday's law, as expressed by Equation (1).
where t is the estimated corrosion age (s), F is Faraday's constant (96,500 C/mol), z is the ionic charge number (z = 2) of iron, m is the target mass loss of corroded reinforcement (g), M is the molar mass of iron (56 g/mol), i is the corrosion current density (A/cm 2 ), and S is the superficial area of I-shaped steel within the corrosion region (cm 2 ). Although the corrosion mass can be directly obtained from Equation (1), the electrolytic corrosion test is affected by a variety of interference factors, and the theory is quite different from the practice. Therefore, this paper determines the quantitative relationship between the corrosion mass ( m) and the corrosion time (t) through the accelerated corrosion test of I-shaped steel concrete with five small samples.

Preparation for Testing
The main device used in accelerated corrosion test was the electrolytic cell composed of direct-current power supply, ammeter, wire, 5% NaCl solution, copper sheet, as shown in Figure 1. and residual load of the shaped steel concrete were significantly lower than those at room temperature, and the decline was fastest in the range from 20 to 200 °C. In this investigation, with respect the effects of corrosion ratio of I-shaped steel, this paper adopted accelerated corrosion test and push-out test to obtain bond-slip behavior and bond-slip degradation constitutive model between corroded I-shaped steel and concrete. It not only has practical application value in guiding the design, construction and maintenance of support system of subsea tunnel, but also enriches and develops the bond-slip degradation theory of corroded steel reinforced concrete structure.

Accelerated Corrosion Test
It is well known that natural chloride corrosion of I-shaped steel embedded in concrete takes up to several years or more, which makes it difficult to carry out in a laboratory. Thus, the corrosion of I-shaped steel is accelerated by accelerated corrosion test. According to Faraday's law, as expressed by Equation (1).
where t is the estimated corrosion age (s), F is Faraday's constant (96500 C/mol), z is the ionic charge number (z = 2) of iron, △m is the target mass loss of corroded reinforcement (g), M is the molar mass of iron (56 g/mol), i is the corrosion current density (A/cm 2 ), and S is the superficial area of I-shaped steel within the corrosion region (cm 2 ). Although the corrosion mass can be directly obtained from Equation (1), the electrolytic corrosion test is affected by a variety of interference factors, and the theory is quite different from the practice. Therefore, this paper determines the quantitative relationship between the corrosion mass (△m) and the corrosion time (t) through the accelerated corrosion test of I-shaped steel concrete with five small samples.

Preparation for Testing
The main device used in accelerated corrosion test was the electrolytic cell composed of directcurrent power supply, ammeter, wire, 5% NaCl solution, copper sheet, as shown in Figure 1. According to the actual design of Xiang'an Subsea Tunnel, the specimens adopted NO.18 Ishaped steel with equal length. The mass proportion of concrete was water: gravel: sand: cement = 1:3.5:2.7:1.8. The type of concrete was C25. The parameters of I-shaped steel and concrete were shown in Table 1. The specific size of the specimens was shown in Figure 2, the cross section was a square According to the actual design of Xiang'an Subsea Tunnel, the specimens adopted NO.18 I-shaped steel with equal length. The mass proportion of concrete was water:gravel:sand:cement = 1:3.5:2.7:1.8. The type of concrete was C25. The parameters of I-shaped steel and concrete were shown in Table 1.
The specific size of the specimens was shown in Figure 2, the cross section was a square section of 40 cm × 40 cm, I-shaped steel was arranged symmetrically along the central axis of the cross section. section of 40 cm × 40 cm, I-shaped steel was arranged symmetrically along the central axis of the cross section.

Testing Procedure
The accelerated corrosion test was carried out in the following steps: Step 1: 5% NaCl solution was configured in the electrolytic cell. The specimens were placed into the electrolytic cell and soaked for two days to make the salt water infiltrate into the specimens and contact with the I-shaped steel inside; Step 2: Connecting the electrolytic circuit, as shown in Figure 1. Note that direct-current power supply was used here; Step 3: Opening the power supply and adjusting the current, the control current was 5A; Step 4: Maintain the current constant, electrify to the original design time, take out the components, chisel out I-shaped steel (seen in Figure 3a); Step 5: The corrosion mass of each specimen was determined by measuring the weight of the Ishaped steel after removing the rust (seen in Figure 3b) with 12% hydrochloric acid solution for pickling and with 3% sodium carbonate solution neutralization.

Testing Procedure
The accelerated corrosion test was carried out in the following steps: Step 1: 5% NaCl solution was configured in the electrolytic cell. The specimens were placed into the electrolytic cell and soaked for two days to make the salt water infiltrate into the specimens and contact with the I-shaped steel inside; Step 2: Connecting the electrolytic circuit, as shown in Figure 1. Note that direct-current power supply was used here; Step 3: Opening the power supply and adjusting the current, the control current was 5A; Step 4: Maintain the current constant, electrify to the original design time, take out the components, chisel out I-shaped steel (seen in Figure 3a); Step 5: The corrosion mass of each specimen was determined by measuring the weight of the I-shaped steel after removing the rust (seen in Figure 3b) with 12% hydrochloric acid solution for pickling and with 3% sodium carbonate solution neutralization.

The Relationship between the Mass Loss and Conducting Time
According to accelerated corrosion test, Table 2 could be obtained.  Table 2 was represented by curve, as shown in Figure 4. It can be seen from Figure 4, under constant current (5A), the relationship between the mass loss and the corrosion time was linear, and the test results were consistent with Faraday's law. Therefore, we obtained the relationship between the mass loss and the conducting time in our accelerated corrosion tests, as expressed by Equation (2).

The Relationship between the Mass Loss and Conducting Time
According to accelerated corrosion test, Table 2 could be obtained.  Table 2 was represented by curve, as shown in Figure 4.

The Relationship between the Mass Loss and Conducting Time
According to accelerated corrosion test, Table 2 could be obtained.  Table 2 was represented by curve, as shown in Figure 4. It can be seen from Figure 4, under constant current (5A), the relationship between the mass loss and the corrosion time was linear, and the test results were consistent with Faraday's law. Therefore, we obtained the relationship between the mass loss and the conducting time in our accelerated corrosion tests, as expressed by Equation (2)  It can be seen from Figure 4, under constant current (5A), the relationship between the mass loss and the corrosion time was linear, and the test results were consistent with Faraday's law. Therefore, we obtained the relationship between the mass loss and the conducting time in our accelerated corrosion tests, as expressed by Equation (2). where m is the mass loss of corroded I-shaped steel (kg), t is the conduction time (h). Note that the premise of Equation (2) is that the control current is 5A. According to Equation (2), the corrosion mass can be obtained from the corrosion time, and then the corrosion ratio can be obtained by dividing the total mass (Equation (3)), which lays a foundation for the subsequent bond-slip degradation test.

Testing Method and Specimen Preparation
The specimens produced in the test were the same as those produced in accelerated corrosion test. There were 6 groups in this test, one group of I-shaped steel was not corroded, the other 5 groups were corroded (all the current was 5A), and the mass loss was obtained from the conducting time according to Equation (2), and then the corrosion ratio was obtained by Equation (3), as shown in Table 3.

Setup for Push-Out Test
The push-out test was conducted using a custom-built loading setup with pressure testing machine with a maximum capacity of 5000 kN, as shown in Figure 5. All the specimens were loaded under loading speed control with a constant loading rate of 40 kN/min, per 20 kN was one level in the loading process, until the bond was broken. The lower end of the concrete was fixed on the test bench by the support and the upper end was free. When loading, the steel back board pushed down the I-shaped steel, and the lower end of the concrete bear the load from the pressure testing machine. Therefore, the upper end of the I-shaped steel was the loading end and the lower end was the free end. Two displacement meters with accuracy of 0.01 mm were used to monitor the slip displacement at both the free end and loading end of the specimens. where △m is the mass loss of corroded I-shaped steel (kg), t is the conduction time (h). Note that the premise of Equation (2) is that the control current is 5A. According to Equation (2), the corrosion mass can be obtained from the corrosion time, and then the corrosion ratio ρ can be obtained by dividing the total mass (Equation (3)), which lays a foundation for the subsequent bond-slip degradation test.

Testing Method and Specimen Preparation
The specimens produced in the test were the same as those produced in accelerated corrosion test. There were 6 groups in this test, one group of I-shaped steel was not corroded, the other 5 groups were corroded (all the current was 5A), and the mass loss was obtained from the conducting time according to Equation (2), and then the corrosion ratio was obtained by Equation (3), as shown in Table 3. The push-out test was conducted using a custom-built loading setup with pressure testing machine with a maximum capacity of 5000 kN, as shown in Figure 5. All the specimens were loaded under loading speed control with a constant loading rate of 40 kN/min, per 20 kN was one level in the loading process, until the bond was broken. The lower end of the concrete was fixed on the test ben

Fracture Failure Process
At the initial stage of loading, the displacement sensors at the loading and free ends were basically unchanged due to the smaller load, which meant that there was almost no relative slip generated inside. As the applied load continued to increase, the slip at the loading end increased slowly and the cracks at the end face developed rapidly. The cracks began to appear from the two ends of the flange, and rapidly expanded from the inside to the outside with the increase of the load, as shown in Figure 6. When the applied load reached or just exceeded the bond strength of the specimens, the load readings of the pressure testing machine had fallen back rapidly; the pressure testing machine could not continue loading.

Fracture Failure Process
At the initial stage of loading, the displacement sensors at the loading and free ends were basically unchanged due to the smaller load, which meant that there was almost no relative slip generated inside. As the applied load continued to increase, the slip at the loading end increased slowly and the cracks at the end face developed rapidly. The cracks began to appear from the two ends of the flange, and rapidly expanded from the inside to the outside with the increase of the load, as shown in Figure 6. When the applied load reached or just exceeded the bond strength of the specimens, the load readings of the pressure testing machine had fallen back rapidly; the pressure testing machine could not continue loading.

The Relationship between Bond Failure Load and Corrosion Ratio
Through the six groups of push-out test, the scatter plot of corrosion ratio and bond failure load can be obtained, as shown in Figure 7. Through the observation and analysis of Figure 7, with the increase of the corrosion ratio, the bond failure load of the specimens presents a decreasing trend. When the corrosion ratio is less than 12%, the trend decreases linearly. When the corrosion ratio is greater than 12%, the decline rate increases suddenly. Therefore, the corrosion ratio of 12% is a turning point for accelerating degradation of bond-slip performance.

The Relationship between Bond Failure Load and Corrosion Ratio
Through the six groups of push-out test, the scatter plot of corrosion ratio and bond failure load can be obtained, as shown in Figure 7.

Fracture Failure Process
At the initial stage of loading, the displacement sensors at the loading and free ends were basically unchanged due to the smaller load, which meant that there was almost no relative slip generated inside. As the applied load continued to increase, the slip at the loading end increased slowly and the cracks at the end face developed rapidly. The cracks began to appear from the two ends of the flange, and rapidly expanded from the inside to the outside with the increase of the load, as shown in Figure 6. When the applied load reached or just exceeded the bond strength of the specimens, the load readings of the pressure testing machine had fallen back rapidly; the pressure testing machine could not continue loading.

The Relationship between Bond Failure Load and Corrosion Ratio
Through the six groups of push-out test, the scatter plot of corrosion ratio and bond failure load can be obtained, as shown in Figure 7. Through the observation and analysis of Figure 7, with the increase of the corrosion ratio, the bond failure load of the specimens presents a decreasing trend. When the corrosion ratio is less than 12%, the trend decreases linearly. When the corrosion ratio is greater than 12%, the decline rate increases suddenly. Therefore, the corrosion ratio of 12% is a turning point for accelerating degradation of bond-slip performance. Through the observation and analysis of Figure 7, with the increase of the corrosion ratio, the bond failure load of the specimens presents a decreasing trend. When the corrosion ratio is less than 12%, the trend decreases linearly. When the corrosion ratio is greater than 12%, the decline rate increases suddenly. Therefore, the corrosion ratio of 12% is a turning point for accelerating degradation of bond-slip performance. (1) Slip phase: Once loading begins, the loading end and free end also begin to slip and develop stably. However, the slip value at the loading end is significantly greater than that at the free end. In this stage, the slip value of the loading end is about 0.04-0.07mm per 20 kN, while that of the free end is about 0.01-0.03 mm. The curve at this stage is approximately linear until the load reaches about 85% of the ultimate load.

Load-Slip Curve
(2) Destruction phase: When the load is close to 85% of the ultimate load, the sliding development of the loading end and the free end is accelerated. At this time, the slip value of the loading end is about 0.07-0.12 mm per 20 kN, while that of the free end is about 0.02-0.04 mm. When the applied load is close to the ultimate load, longitudinal cracks will suddenly appear at the loading end, or the existing cracks caused by the original corrosion will develop rapidly and widen. When the load reaches the ultimate load, bond failure occurs.  Through the observation and analysis of Figure 8, the P-S curve of the corroded steel reinforced concrete in the rising segment can be roughly divided into the following two stages: (1) Slip phase: Once loading begins, the loading end and free end also begin to slip and develop stably. However, the slip value at the loading end is significantly greater than that at the free end. In this stage, the slip value of the loading end is about 0.04-0.07mm per 20 kN, while that of the free end is about 0.01-0.03 mm. The curve at this stage is approximately linear until the load reaches about 85% of the ultimate load.
(2) Destruction phase: When the load is close to 85% of the ultimate load, the sliding development of the loading end and the free end is accelerated. At this time, the slip value of the loading end is about 0.07-0.12 mm per 20 kN, while that of the free end is about 0.02-0.04 mm. When the applied load is close to the ultimate load, longitudinal cracks will suddenly appear at the loading end, or the existing cracks caused by the original corrosion will develop rapidly and widen. When the load reaches the ultimate load, bond failure occurs.

Distribution of Bond Stress along Specimen Length
In the push-out test, the internal bond stress test was not carried out due to corrosion, but the distribution law of bond stress along anchorage length was obtained by analyzing the push-out test based on the energy method.
In order to analyze the relative slip between I-shaped steel and concrete, I-shaped steel is taken as the detached body, and its stress is shown in Figure 9a. Take any I-shaped steel element from the detached body, and its stress is shown in Figure 9b.

Distribution of Bond Stress along Specimen Length
In the push-out test, the internal bond stress test was not carried out due to corrosion, but the distribution law of bond stress along anchorage length was obtained by analyzing the push-out test based on the energy method.
In order to analyze the relative slip between I-shaped steel and concrete, I-shaped steel is taken as the detached body, and its stress is shown in Figure 9a. Take any I-shaped steel element from the detached body, and its stress is shown in Figure 9b. It is assumed that the bond stress of the I-shaped steel is the same everywhere at the same anchorage depth. Equation (4) can be obtained from the static equilibrium condition [32] of I-shaped steel.
( ) ( ) where τ(x) is the bond stress between I-shaped steel and concrete at x, Ps(x) is the axial force of Ishaped steel at x, Cs is the perimeter of I-shaped steel section. It is assumed that the longitudinal deformation of I-shaped steel at x is Ss(x), and the relationship between axial force, elastic deformation and longitudinal strain of concrete is obtained by Equation (5), according to Hooke's law [32].
where εs(x) is axial strain of I-shaped steel at x, σs(x) is axial stress of I-shaped steel at x, Es is elastic modulus of I-shaped steel, As is cross-sectional area of I-shaped steel. Substitute Equation (5) into Equation (4) to get Equation (6).
Similarly, the concrete element at x is taken as the element body, and Equation (7) is obtained.
where Sc(x) is the longitudinal deformation of concrete at x, Ec is elastic modulus of concrete, Ac is cross-sectional area of concrete. It is assumed that the bond stress of the I-shaped steel is the same everywhere at the same anchorage depth. Equation (4) can be obtained from the static equilibrium condition [32] of I-shaped steel.
where τ(x) is the bond stress between I-shaped steel and concrete at x, P s (x) is the axial force of I-shaped steel at x, C s is the perimeter of I-shaped steel section. It is assumed that the longitudinal deformation of I-shaped steel at x is S s (x), and the relationship between axial force, elastic deformation and longitudinal strain of concrete is obtained by Equation (5), according to Hooke's law [32].
where ε s (x) is axial strain of I-shaped steel at x, σ s (x) is axial stress of I-shaped steel at x, E s is elastic modulus of I-shaped steel, A s is cross-sectional area of I-shaped steel. Substitute Equation (5) into Equation (4) to get Equation (6).
Similarly, the concrete element at x is taken as the element body, and Equation (7) is obtained.
where S c (x) is the longitudinal deformation of concrete at x, E c is elastic modulus of concrete, A c is cross-sectional area of concrete.
In the state of use, the deformation of I-shaped steel and concrete can be regarded as elastic, so the relative slip of them at x can be obtained by their displacement difference, as expressed by Equation (8).
S(x) = S s (x) − S c (x) (8) According to the test results in reference [24,33], it is deduced that the bond stress of steel reinforced concrete presents an exponential distribution along the anchorage length, which can be expressed by Equation (9).
τ(x) = Ae Bx (9) where parameters A and B are undetermined coefficients. Substitute Equation (9) into Equations (6) and (7) to obtain two second-order constant coefficient differential equations, when K s and K c are respectively represented by Equations (10) and (11), the general solution expression is expressed by Equation (12).
where C 1 , C 2 , C 3 and C 4 are undetermined coefficients. Substitute Equation (12) into Equation (8), the expression of relative slip between I-shaped steel and concrete can be obtained, as expressed by Equation (13).
In the push-out test of steel reinforced concrete, the slip value S(0) at the loading end can be measured by setting a displacement meter at the loading end. Therefore, the difference between coefficient C 1 and C 3 in Equation (13) can be calculated by the slip test value at the loading end, as expressed by Equation (14).
For linear elastic materials, the necessary and sufficient condition for allowable displacement S(x) to be true displacement is that the total potential energy of the system is minimum [34] according to the principle of minimum potential energy, simultaneously satisfying Equation (15).
where C i is the unknown quantity in Equation (12), i = 2,4. The total potential energy of the system consists of two components [34], as expressed by Equation (16).
where U is the strain energy of the whole structure, W is the external potential energy of the whole structure.
For the integral members, the strain energy of I-shaped steel and concrete is expressed by Equation (17).
The external potential energy is expressed by Equation (18).
Substitute Equations (17) and (18) into Equation (16) to get the total potential energy of the system, as expressed by Equation (19).
According to Equation (15), C 2 and C 4 can be obtained, as expressed by Equation (20).
Therefore, under the action of external load P, the relative slip at any section of the steel reinforced concrete can be obtained, as expressed by Equation (21).
By substituting the anchorage length l of I-shaped steel into Equation (21), the relational expressions of slip S(0) at loading end, slip S(l) at free end and bond stress coefficient A and B can be obtained, as expressed by Equation (22).
Based on balance condition of forces [32], Equation (23) can be obtained.
According to Equation (23), the relationship between external load P and coefficients A and B can be obtained, as expressed by Equation (24).
The relative slip S(0) of the loading end and S(l) of the free end are measured by setting a displacement meter at both ends of the I-shaped steel ( Figure 5). Based on the above analysis, under the premise that the geometrical and physical parameters of the specimen are certain, the concrete parameters A and B of the bond stress distribution curve can be obtained by Equations (22) and (24), which only need the corresponding information of the external load P and the slip S(0) and S(l) at both ends of the specimens.
Since the energy analysis method adopted in this paper is based on the linear elastic theory, Equation (21) is only applicable to the linear elastic stage of bond-slip relationship. From the research results [35], it can be seen that in the bond-slip curve of steel reinforced concrete, before the applied load reaches 80% of the bond failure load, the bond stress increases linearly with the slip, that is, the bond-slip relationship is in the linear elastic stage. Therefore, it can be considered that Equation (21) is suitable for calculating the local slip of steel reinforced concrete with a push-out load less than 80% of the bond failure load.
According to the measured P-S curve in the push-out test of steel reinforced concrete and the parameters of the specimens (Table 4), the distribution of bond stress calculated by the Equations (22) and (24) in this paper is shown in Figure 10. Table 4. Characteristics of push-out specimens. which only need the corresponding information of the external load P and the slip S(0) and S(l) at both ends of the specimens. Since the energy analysis method adopted in this paper is based on the linear elastic theory, Equation (21) is only applicable to the linear elastic stage of bond-slip relationship. From the research results [35], it can be seen that in the bond-slip curve of steel reinforced concrete, before the applied load reaches 80% of the bond failure load, the bond stress increases linearly with the slip, that is, the bond-slip relationship is in the linear elastic stage. Therefore, it can be considered that Equation (21) is suitable for calculating the local slip of steel reinforced concrete with a push-out load less than 80% of the bond failure load.

Cross-Sectional Area
According to the measured P-S curve in the push-out test of steel reinforced concrete and the parameters of the specimens (Table 4), the distribution of bond stress calculated by the Equations (22) and (24) in this paper is shown in Figure 10. Table 4. Characteristics of push-out specimens.

Study on Interface Damage of Corroded I-Shaped Steel and Concrete
In the study of damage mechanics, it is necessary to select reasonable variables to describe the damage state of materials and structures. The interface damage between I-shaped steel and concrete due to corrosion cannot be obtained by simple test means, so the area of interface damage cannot be simply used to determine the damage variable. Wu et al. [36,37] used the bond strength of the interface to define the damage variable, as expressed by Equation (25).
where Pu is the limit load of I-shaped steel concrete before interface damage, u P − is the limit load of Ishaped steel concrete after interface damage, A is the surface area of I-shaped steel wrapped by concrete, here is A = 289200 mm 2 .
Through the above tests and Equation (25), the damage variable D is calculated in Table 5. According to Table 5, the relationship between damage variable D and corrosion ratio is drawn (Figure 11), and polynomial curve fitting is performed. Finally, the quadratic polynomial curve is selected, as expressed by Equation (26)

Study on Interface Damage of Corroded I-Shaped Steel and Concrete
In the study of damage mechanics, it is necessary to select reasonable variables to describe the damage state of materials and structures. The interface damage between I-shaped steel and concrete due to corrosion cannot be obtained by simple test means, so the area of interface damage cannot be simply used to determine the damage variable. Wu et al. [36,37] used the bond strength of the interface to define the damage variable, as expressed by Equation (25).
where P u is the limit load of I-shaped steel concrete before interface damage, − P u is the limit load of I-shaped steel concrete after interface damage, A is the surface area of I-shaped steel wrapped by concrete, here is A = 289,200 mm 2 .
Through the above tests and Equation (25), the damage variable D is calculated in Table 5. According to Table 5, the relationship between damage variable D and corrosion ratio is drawn (Figure 11), and polynomial curve fitting is performed. Finally, the quadratic polynomial curve is selected, as expressed by Equation (26) and the correlation coefficient reach 0.99. D = 11.78ρ 2 + 0.7395ρ (26)

Bond-Slip Degradation Constitutive Model Based on Damage Theory
Bond stress τ is generally represented by the average value along the bond length, which can be calculated by Equation (27).
where P is push-out force, A is the surface area of I-shaped steel wrapped by concrete, in this test A = 289200mm 2 .

Bond-Slip Constitutive Model between Non-Corroded I-Shaped Steel and Concrete
Based on the research results of many scholars [26,29,35], a complete τ-s curve can be obtained, as shown in Figure 12. Three characteristic slip value were defined in τ-s curve: ① The turning point slip Sl in the rising segment corresponding to the control point A,(Sl, 0.5τu); ② Limit state slip Su; ③ Initial slip values Sr for residual stage. Due to the limitations of experimental conditions, only the rising segment OA of the τ-s curve was obtained in this experiment, and the descending segment AB and the horizontal residual stage BC were not obtained. Therefore, only the rising segment OA was analyzed in this paper. The rising segment OA is generally described by hyperbola [35], as expressed by Equation (28).

Bond-Slip Degradation Constitutive Model Based on Damage Theory
Bond stress τ is generally represented by the average value along the bond length, which can be calculated by Equation (27).
where P is push-out force, A is the surface area of I-shaped steel wrapped by concrete, in this test A = 289,200 mm 2 .

Bond-Slip Constitutive Model between Non-Corroded I-Shaped Steel and Concrete
Based on the research results of many scholars [26,29,35], a complete τ-s curve can be obtained, as shown in Figure 12. Three characteristic slip value were defined in τ-s curve: 1 The turning point slip S l in the rising segment corresponding to the control point A,(S l , 0.5τ u ); 2 Limit state slip S u ; 3 Initial slip values S r for residual stage. Due to the limitations of experimental conditions, only the rising segment OA of the τ-s curve was obtained in this experiment, and the descending segment AB and the horizontal residual stage BC were not obtained. Therefore, only the rising segment OA was analyzed in this paper.

Bond-Slip Degradation Constitutive Model Based on Damage Theory
Bond stress τ is generally represented by the average value along the bond length, which can be calculated by Equation (27).
where P is push-out force, A is the surface area of I-shaped steel wrapped by concrete, in this test A = 289200mm 2 .

Bond-Slip Constitutive Model between Non-Corroded I-Shaped Steel and Concrete
Based on the research results of many scholars [26,29,35], a complete τ-s curve can be obtained, as shown in Figure 12. Three characteristic slip value were defined in τ-s curve: ① The turning point slip Sl in the rising segment corresponding to the control point A,(Sl, 0.5τu); ② Limit state slip Su; ③ Initial slip values Sr for residual stage. Due to the limitations of experimental conditions, only the rising segment OA of the τ-s curve was obtained in this experiment, and the descending segment AB and the horizontal residual stage BC were not obtained. Therefore, only the rising segment OA was analyzed in this paper. The rising segment OA is generally described by hyperbola [35], as expressed by Equation (28). The rising segment OA is generally described by hyperbola [35], as expressed by Equation (28).
where a = s u −2s l τ u (s u −s l ) , b = s u s l τ u (s u −s l ) . Through the above data of Figure 8a, the parameters a = 0.102, b = 0.669 of the non-corroded I-shaped steel and concrete can be obtained. The bond-slip constitutive model between non-corroded I-shaped steel and concrete is proposed, as expressed by Equation (29). τ = S 0.102S + 0.669 (29) The curve fitted by Equation (29) is compared with the test curve, as shown in Figure 13. It can be seen that the formula fitting curve is in good agreement with the test curve. Therefore, it is feasible to use this formula to express the constitutive relationship of bond-slip between non-corroded I-shaped steel and concrete. . .

S S
τ = + 0 102 0 669 (29) The curve fitted by Equation (29) is compared with the test curve, as shown in Figure 13. It can be seen that the formula fitting curve is in good agreement with the test curve. Therefore, it is feasible to use this formula to express the constitutive relationship of bond-slip between non-corroded Ishaped steel and concrete.
Where τ is average bond stress, S is relative slip, D is damage variable, ρ is corrosion ratio. The curve fitted by Equation (30) is compared with the test curve, as shown in Figure 14. In order to reflect the reliability of the formula, the relative error δ is defined, as expressed by Equation (31).
Through data processing and analysis, the range of relative error is about −12%~5%, it indicates that the formula fitting curve is in good agreement with the test curve. Therefore, it is feasible to use this formula to express the bond-slip degradation constitutive model between corroded I-shaped steel and concrete.
( ) where δ is relative error, T τ is the bond stress obtained by test, F τ is the bond stress obtained by Equation (30).

Bond-Slip Degradation Constitutive Model between Corroded I-Shaped Steel and Concrete
Using the constitutive model of bond-slip between non-corroded I-shaped steel and concrete, the degenerate constitutive relationship with damage variable of bond-slip between corroded I-shaped steel and concrete is proposed, as expressed by Equation (30). τ = (1 − D) S 0.102S + 0.669 = (1 − 11.78ρ 2 − 0.7395ρ) S 0.102S + 0.669 (30) Where τ is average bond stress, S is relative slip, D is damage variable, ρ is corrosion ratio. The curve fitted by Equation (30) is compared with the test curve, as shown in Figure 14. In order to reflect the reliability of the formula, the relative error δ is defined, as expressed by Equation (31). Through data processing and analysis, the range of relative error is about −12%~5%, it indicates that the formula fitting curve is in good agreement with the test curve. Therefore, it is feasible to use this formula to express the bond-slip degradation constitutive model between corroded I-shaped steel and concrete.
where δ is relative error, τ T is the bond stress obtained by test, τ F is the bond stress obtained by Equation (30).