3.2. Analysis of Bond Performance Data of Reinforced Concrete
In order to study the effects of three factors on the ultimate slip and bond strength of specimens, the linear fitting of the test results of specimens with two factors consistent and one factor inconsistent is carried out according to the control variable method. The relationship between the ultimate bonding performance and three factors is analyzed. The factors include the corrosion rate of stirrups, the number of freeze-thaw cycles and the stirrup ratio
ρsv. The
ρsv of the specimen can be calculated through Equation (6):
where
Asv is the area of stirrup; the value of
n is 2;
b is the width of the specimen, and
s is stirrup spacing.
Based on the experimental data in
Table 4 and
Table 5, the influence of stirrup corrosion rate on bonding properties can be revealed by studying the variation of coefficients in these formulas as follows:
When the corrosion rate of the stirrup is low, the corrosion product fills between reinforcement and concrete, and the longitudinal reinforcement is subjected to the annular compressive stress, which increases the friction force between reinforcement and concrete contact surface, thus improving the bonding force. In addition, the small corrosion rate would involve many rust pits on the stirrup surface. When the longitudinal reinforcement and stirrup work together, these rust pits can increase the friction coefficient of the overall concrete [
18].
Table 4 and
Table 5 reflect the effect of the corrosion rate of stirrups as a single variable on the ultimate bond strength and ultimate slip. When the corrosion rate is not more than 5%, the slope of the expression is positive, indicating that slight corrosion of stirrups can enhance the bond strength of reinforced concrete. When the corrosion rate exceeds 5%, the slopes of the expressions are all negative and the numerical value is large, indicating that as the stirrup corrosion further deepens, the bonding performance of reinforced concrete deteriorates significantly. This law is consistent with the one discovered by Zhou et al. [
24], which to some extent validates the accuracy of the experiments conducted in this study.
When the stirrup spacing is set at 150 mm, slight corrosion can significantly improve the bonding performance of specimens frozen and thawed 15 times. However, when the stirrup spacing is set at 100 mm, slight corrosion promotes the bonding performance of specimens without freeze–thaw treatment more obviously. The intercept of the equation represents the original bond strength and the ultimate slip of uncorroded stirrups. By comparing the intercept distance of specimens with different stirrup spacing and freeze–thaw times, it can be found that the increase in the number of freeze–thaw cycles significantly affects the bond strength and ultimate slip between the reinforcement and the concrete, which can be explained by that freeze–thaw cycles can cause micro-cracks to expand within the structure and intersect with the reinforcement. Furthermore, the reduction of the stirrup spacing can increase the bond strength and ultimate slip [
15]. This is mainly due to the fact that when the stirrup spacing is small, the densification of the stirrups inhibits the cracking of the concrete and the reduction of the bond strength between the reinforcement and the concrete [
31].
Table 6 reflects the effect of freeze–thaw times as a single variable on ultimate bond strength and ultimate slip. It is obviously seen that the slopes of bond strength and slip are constantly negative, which indicates that the bond performance of reinforced concrete deteriorates gradually with the increase in freeze–thaw times. The ultimate bond stress of specimens with different freeze–thaw cycles is shown in
Figure 8. It can be concluded that the ultimate bond strength decreases by 6.1% on average after 15 freeze–thaw cycles, and it decreases by 25.8% on average after 30 freeze–thaw cycles, meaning that when the freeze–thaw cycles are performed 15 times, the bond strength does not decrease significantly, while it decreases rapidly when the cycles are performed 30 times.
The reduction in bond strength can be explained by the following aspects. The bond strength between reinforcement and concrete is mainly composed of chemical adhesive force, machine bite force and friction force. Repeated freeze–thaw cycles would destroy the chemical adhesive force of the interface between steel bar and concrete. In addition, the repeated freeze–thaw process means that the pore structure in concrete bears repeated fatigue load, which not only leads to the appearance of micro-cracks, but also reduces the strength of concrete and the relative dynamic modulus of elasticity, further resulting in the decrease in bond strength. Furthermore, with the increase in freeze–thaw cycle numbers, micro-cracks continue to extend and eventually intersect with longitudinal reinforcement (e.g., C100-15-n shown in
Figure 9), and the confinement effect of concrete on longitudinal reinforcement decreases. At the same time, as the strength of concrete decreases, the circumferential compressive stress of concrete and stirrup on core concrete decreases, which reduces the friction force and mechanical bite force, and further results in a reduction of ultimate bond strength [
21].
The arrangement of stirrups would restrain the transverse deformation of concrete during the pull-out of steel bars and this binding force would increase the friction force of concrete on the longitudinal bars and offset the annular tensile stress of the longitudinal bars to the concrete during the tension, thus delaying the crack development. Moreover, the configurations of stirrups could change the constitutive properties of reinforced concrete and improve its tensile strength. According to the data analysis in
Table 7, it can be observed that when the stirrup ratio is a single variable, the slopes of the expressions of ultimate bond strength and ultimate slip are positive, so both the ultimate bond strength and ultimate slip increase with the increase in stirrup ratio.
Finally, the range of the coefficients in these formulas with the different corrosion rates, number of freeze–thaw cycles and stirrup ratio are listed in
Table 8, so as to sum up the variation of the coefficients and give a more simplified formula. The
X in equations represents
ω,
N and
ρsv, respectively. The equations fitted by the actual tested data of specimens could contribute to improving the model of the bond between the concrete and steel bar.
3.3. Empirical Formula of Ultimate Bond Strength
According to the discussion in the previous section, the thickness of the protective layer and the spacing of stirrups affect the bonding force without considering the external environmental impact. If the bond strength is affected by the external environment, such as chloride ion corrosion, corrosion rate and freeze–thaw cycle times, the empirical formula of bond strength could be referred to as the ultimate strength Equation (7) of steel mixtures proposed by Zhao and Ma [
15]:
where
τst is the bonding stress of the steel bar, and
τcon is the bonding stress of concrete.
The data of the above tables are systematically integrated referring to the empirical formula of bond strength, and the deduced formula of bond strength is Equation (8):
In this formula, kcon = −0.07, τ0 = 9.75, A = 0.94 − 1.84ω, k0 = 26.81 − 18.24ω, where ω is the corrosion rate, N is the number of freeze–thaw cycles, and ρsv is the stirrup ratio.
The comparison of the measured data of the experiment with the calculation results from the theoretical deduction Equation (8) is shown in
Table 9. It can be found that the average error between them is 5.52%, showing that with the consideration of the influence of freeze–thaw cycles, corrosion of stirrups and stirrup ratio, the bond strength derived by the formula is in good agreement with the actual data.
3.4. Analysis of Constitutive Model of Bond–Slip Curve
It is found that when the degree of corrosion is low, the ultimate bond strength and ultimate slip slightly increases. As the degree of corrosion continues to increase, the concrete protective layer is cracked. When the corrosion rate is high, as the corrosion rate increases, the amount of slip decreases. First, when the corrosion rate of stirrups is low, the protective layer of concrete has no cracks. As the stirrups rust, their corrosion products fill in the gap between the steel bar and concrete, and as the gap is filled, the corrosion products apply hoop compressive stress to the longitudinal steel bar. The stress increases the friction between the steel bar and the concrete contact surface, thereby increasing the bonding force. Second, when the stirrup corrosion rate is low, there are many rust pits on the surface of the stirrup. When the longitudinal reinforcement and the stirrup work together, these rust pits can increase the coefficient of friction between the reinforcement and the concrete. When the corrosion rate continues to rise, the cracks on the concrete surface continue to develop, the damage to the protective layer is intensified, the hoop compressive stress of the stirrups to the longitudinal reinforcement continues to decrease, and the reduction of friction leads to the decrease in the final bond strength.
The damage of concrete in the repeated freeze–thaw cycles belong to the scope of physical changes. The volume of the small gap in concrete increases when it is refrozen, and cracks begin to appear when reaching the limit of concrete. According to many research results, repeated freeze–thaw cycles are equivalent to the continuous loading and unloading in the concrete interior, and each cycle would cause damage to the durability of the material so that the tiny cracks continue to merge and expand into larger cracks. Taking the specimens with 100 mm stirrup spacing and 5% stirrup corrosion rate as an example,
Figure 10 represents
τu-
Su diagrams of different freeze–thaw cycle control groups under the same stirrup distribution and stirrup corrosion rate. The initial slope of the rising part of the curve represents the bond stiffness. The steeper the slope of the curve, the greater the bonding stiffness. When the number of freeze–thaw cycles increases from 0 to 15 and further to 30, the diagram becomes smoother and the bond stiffness is smaller, which indicates that freeze–thaw cycles can cause a significant reduction in the durability of concrete.
Also, it can be seen in
Figure 11 that with the same stirrup ratio and corrosion rate, the ultimate bond strength decreases with the increase in freeze-thaw cycle number. Additionally, it can also be seen that with the increase in the number, the coverage area under
τ-
S curve is smaller, indicating that the bonding performance deteriorates more, and the energy dissipation capacity and the ductility decrease.
This study combines empirical formulas with experimental results. Based on the model established by the existing scholars [
32,
33] and combined with the experimental data of this test, a segmented bond–slip model under monotonic loading is established, considering concrete freeze–thaw cycles, stirrup ratio and corrosion rate of stirrups. The entire bond–slip process is divided into three stages: ascending stage, descending stage and residual stress stage. With reference to
Figure 11, specific stages are described as follows:
The first stage is the ascending stage, which is usually before the mass emergence of concrete through cracks. At the initial stage of test loading, concrete and longitudinal bars are in a tight bond stage, and the bond–slip curve generally increases linearly, and there is no slip at the free end. As the load continues to increase, the bond–slip curve presents a non-linear growth trend, and the slip increases rapidly. The peak value of bond stress is near the middle and rear parts of the steel bar, and the load will reach its peak soon.
The second stage is the descent stage. At this stage, the bond–slip curve decreases rapidly, and longitudinal cracks appear on the concrete surface. At the same time, the concrete around the longitudinal reinforcement is cracked, and the gripping force, friction force and mechanical bite force rapidly decrease, and the slip increases rapidly.
The third stage is the residual stress stage. Due to the restraint of stirrups, the steel bar pulls out slowly from the concrete, accompanied by a small amount of concrete residue on the surface of the steel bar. The load gradually stabilizes and the displacement slowly increases.
In summary, the three stages of the bond–slip curve of reinforced concrete are shown in
Figure 12.
According to Equations (9)–(11), it can be seen that four basic parameters can determine the slip constitutive model so that the process can be described as a whole. The four basic parameters are the ultimate bond stress (τu), the ultimate slip (Su), the residual bond stress (τr) and the residual slip (Sr).
The first is the ultimate bond stress (τu). τu is derived from the deduced formula of bond strength in Equation (8).
The second is the peak slip value (
Su). According to the fitting results of test data, the ultimate slip of specimens can be calculated by Equation (12):
where
ρsv is the stirrup ratio, and
c/
d is the ratio of the thickness of the protective layer to the diameter of reinforcement
The third is residual bond stress (
τr), the value of residual bond stress (
τr) is proportional to the ultimate bond strength. When the material is fixed, the residual stress is generally a fixed value. Therefore, according to the research of domestic and foreign scholars [
29,
34,
35,
36], the functional relationship between the residual stress and the ultimate bond strength is studied. The peak bond stress and residual bond stress are listed in
Table 10.
By analyzing the data in
Table 10 and taking the average value of
τr/
τu as the reference basis, Equation (13) for calculating the bond strength of residual stress can be obtained:
The fourth is residual slip (
Sr). According to the test data and the bond–slip curve of the specimens, the ratio of residual slip to ultimate slip is given in Equation (14):
In the above, the three stages of the bond–slip constitutive model were analyzed, and the relation formulas of bond and slip were used.
3.5. Effect of Anchorage Position on Bond–Slip Constitutive Relation
In order to find a more accurate constitutive model describing the bond stress–strain relationship between reinforcement and concrete, it is necessary to find out the variation of bond strength and ultimate slip of reinforced concrete, caused by different anchorage positions of reinforcement. Methods for calculating different bond stress at different anchorage locations are shown in
Figure 13. In the groove of the side ribs of the longitudinal reinforcements, a total of eight steel strain gauges are attached, and the distance between each two steel strain gauges is 25 mm.
Assuming that the strain values measured at points 1 and 2 are
ε1 and
ε2, then
σ1 = E
s∙
ε1,
σ2 = E
s∙
ε2. If the cross-sectional area of the reinforcing bar is equal, the calculation is carried out by Equation (15).
where
A is the side area of the selected element and
As is the cross-section area of the longitudinal bar.
The accumulated total bond stress is different from the average bond stress. According to the principle of reverse distribution, the residual average bond stress is distributed to each section, then the bond stress at each anchorage position can be obtained and the bond stress curve can be drawn.
Since some specimens are in the repeated freeze–thaw cycles, the temperature is too low, which causes water to enter the specimens and leads to the failure of the strain gauge. Therefore, the bond–slip curve of the anchorage section is mainly composed of three specimens: C150-0-0, C150-5-0 and C150-5-15. According to
Figure 13, bond stress and slip position have the following rules:
With the change of anchorage position, the bond stress on the longitudinal reinforcement can be regarded as a parabola. The bond stress in the middle is the largest, while the stress at the loading end and the free end is the smallest.
When the tension is 20 kN, the peak value of bond stress is near the loading end, and most of the stress is balanced at the loading end. Therefore, the bond stress near the free end gradually decreases.
As the load increases, the stress at each anchorage section increases. When the adhesion force at the loading end cannot be balanced, a local tension crack appears with local slip. It can be seen that the peak value of bonding force is generally 0.6la away from the loading end.
When the bonding force at the free end cannot offset the external load, cracks appear at the entire contact surface of the reinforcing steel and concrete simultaneously. The bond stress at each point decreases rapidly, and the steel bar is pulled out slowly.
Comparing
Figure 14a,b, it can be observed that when the stirrup corrosion rate is low, the peak value is enhanced and the stirrup rusts slightly, increasing the restraint effect of concrete on the longitudinal reinforcement. Comparing
Figure 14b,c, it can be found that with the increase in the number of freeze–thaw cycles, the curve becomes steeper and the peak value further decreases.