In recent years, with the frequent accidents worldwide caused by overloading of bridges and environmental erosion, the fatigue damage of bridges is becoming more and more serious. A large number of experimental studies on concrete beams show that [1] - [4] , the failure characteristics of concrete beam under fatigue load generally behave as one or part of the tensioned main reinforcement fracture due to fatigue, and the bearing capacity of reinforced concrete beam after the failure will be significantly reduced. For the concrete beams without fatigue failure, the average strain of the cross-section is still in accordance with the assumption of plane section. The height of the compression zone in the cross-section is basically the same (related to the ceiling load). The concrete in the compression area is still in the elastic stage, and the fatigue elastic modulus of concrete decays continuously. The width of the concrete crack in the tension zone increases, resulting in the decrease of the flexural rigidity of the concrete beam, the increase of the residual deformation and the increase of the tensile steel stress. Therefore, the working state and remaining bearing capacity of concrete bridges without fatigue damage have become the research focus of the bridge sector.

In terms of assessing the bearing capacity of concrete bridges, China has formed a more perfect standard system, usually through structural detection combined with the calculation and analysis. The calculation method of bearing capacity adopted in the Technical Code for the Detection and Evaluation of Urban Bridges (CJJ/T 233-2015) is based on the test results of the bridge material and appearance, but the bearing capacity calculation of structures with more serious damage is still a difficult problem. The method of calculating the bearing capacity of the bridge is based on the test results of the bridge inspection and assessment. The “scoring” method is adopted by The Testing and Evaluation Procedures of Highway and Bridge Carrying Capacity (JTG/T J21-2011), which mainly depends on the bridge damage classification, the refinement and accuracy of scoring standards as well as the experience and technical level of detection personnel, and subject to the influence of subjective factors. Besides, the generally considered reliable method-load test is often used to verify the calculation result. However, the traditional static load test can only evaluate the bridge performance under the “test load”, which is to evaluate the normal bearing capacity according to the change of the structural rigidity or the cross-sectional stiffness, and the bearing capacity is indirectly reflected by the elastic behavior. Thus, a contradiction emerges between conventional static load test and the actual bridge failure test; some bridges which are judged to have a decreasing bearing capacity or fail to meet the requirements by the conventional static load test, are tested to have the same bearing capacity as before by actual bridge failure test. For example, Zhang Jianren [5] carried out an on-site failure test for a 43-year long-term overloaded RC beam subjected to over-limit vehicle loading. The results show that cumulative fatigue damage has no significant effect on the static load-carrying capacity of the bridge. Structural response analysis shows that overload makes the mechanical behavior of the structure changes from elastic to plastic. Yu Zhiwu [6] and Sun Xiaoyan [7] also found through the experimental study that, when the fatigue failure doesn’t happen on concrete beams, the corresponding fatigue residual capacity is almost unchanged. Therefore, they believed that fatigue cumulative damage mainly reduces the deformation properties of the structure, but has little influence on the bearing capacity of the structure, however, they don’t further analyze the reason.

In terms of theoretical analysis, the present research mainly focuses on the analysis of structural damage accumulation and life prediction based on material damage. Wang Chunsheng [8] proposed an evaluation method for fatigue life of concrete bridges based on S-N curves and fracture mechanics. Zhu Jinsong [9] and Wang Qing [10] established the whole process analysis method of concrete bridges fatigue failure. The application process of the above-mentioned analysis method requires to know how the load works, but the functioning course of the actual bridge is often not clear, and the structure or the cross-section stress redistribution after the damage, and the effect of nonlinear coupling of fatigue, creep, temperature and shrinkage during service add more complexity to the evaluation of the cumulative damage degree of the existing structure concrete, the result obtained by simplified prediction of fatigue cumulative damage analysis is often different from the actual results. Therefore, in view of the urgent need of the engineering community to assess the cumulative damage of existing concrete bridges, an evaluation method that can directly determine the cumulative damage degree and residual bearing capacity is urgently needed.

In this paper, according to the measured elastic modulus of concrete after damage, the calculation method of equivalent strain and residual strain as well as the evaluation method of concrete strength after damage are put forward. Then, the structural analysis method is adopted for the evaluation of the existing stress, strain state and residual bearing capacity of the bridge.

According to the uniqueness assumption proposed by Sinha [11] , the relationship between the load and the deformation will remain the same regardless of the repeated load history of the concrete, so long as the residual deformation is the same and the same repeated load is applied. This hypothesis has now been confirmed by many experiments, so the monotone loading stress-strain relationship can be used to show the fatigue envelope of concrete. The compressive stress-strain curves of concrete under fatigue loading are shown in Figure 1 . On the basis of the constitutive relations of concrete under uniaxial compression provided by Code for Concrete Structure Design (herein after referred to as Code), the complexity of the energy dissipation law is deducted, that is to say, the loading deformation modulus E_f and unloading elasticity modulus E_r after concrete damage is assumed to be in the linear mode as shown in Figure 1 .

Many concrete fatigue tests show [12] - [14] that the damage and failure of concrete are internal microcracks extending to macroscopic cracks. When the crack length reaches a certain critical length, it will expand unsteadily until it breaks. And under various time-varying effects, the uniform internal damage of concrete is also caused by the expansion of micro-cracks inside the concrete, which results in deterioration of the mechanical properties of concrete [15] . The elastic modulus and the residual strain can reflect the microscopic damage mechanism of the concrete macroscopically, and have nothing to do with the loading history, which is a good choice for the concrete structure assessment. For various strength concrete, the development of fatigue residual strain is more stable and representative [13] [16] , but it cannot be tested directly due to the time-varying effect, in consequence, the deformation modulus E_f cannot be measured either, only the elastic modulus of unloading concrete E_r can be tested on-site, however, there’s still some limitations for the application of E_r [17] . It can be seen that the residual strain ε_p is related to the unloaded elastic modulus E_r and the static equivalent strain ε_r in Figure 1 . Therefore, how to obtain the residual strain according to the unloaded elastic modulus has become the key to evaluate the cumulative damage of concrete.

Now the residual strain of concrete is basically analyzed by the test statistic method. The Berkeley loading and unloading model [18] illustrates the relationship between the residual strain ε_p and fatigue cumulative strain ε_r as shown in formula (1). The statistical formula shown in formula (2) is also derived by Guo Zhenhai [19] . The residual strain calculated by the Berkeley loading and unloading model is close to that calculated by the Guo Zhenhai model which is slightly larger.

$\begin{array}{l}\left[\frac{{\epsilon }_{\text{r}}}{{\epsilon }_{c,r}}\right]<2\frac{{\epsilon }_p}{{\epsilon }_{c,r}}=0.145\times {\left[\frac{{\epsilon }_{\text{r}}}{{\epsilon }_{c,r}}\right]}^2+0.13\times \left[\frac{{\epsilon }_{\text{r}}}{{\epsilon }_{c,r}}\right]\\ \left[\frac{{\epsilon }_{\text{r}}}{{\epsilon }_{c,r}}\right]\ge 2\frac{{\epsilon }_p}{{\epsilon }_{c,r}}=0.707\times \left[\frac{{\epsilon }_{\text{r}}}{{\epsilon }_{c,r}}-2\right]+0.834\end{array}$ (1)

$\frac{{\epsilon }_p}{{\epsilon }_{c,r}}=0.247\times {\left[\frac{{\epsilon }_{\text{r}}}{{\epsilon }_{c,r}}\right]}^{1.77}$ (2)

Holmen [12] believes that “concrete ultimate strain can be used as criteria for concrete fatigue failure”, a large number of fatigue test results also verify his point. Therefore, the maximum strain of concrete under fatigue loading is equivalent to the strain corresponding to the maximum stress of the monotone loading softening zone [20] , such as ε_un in Figure 1 , but this value is related to the maximum stress σ_max corresponding to the softening zone of the concrete and is difficult to accurately determine. “The fatigue residual stress ε_p is equal to 0.4ε_c,r” is generally recognized as the practical failure criterion for fatigue failure [2] [20] . The actual structure of the concrete stress is not uniform, the site of damage is often local. And structures which break this limit usually have serious cracking and can no longer be effectively used. Therefore, this failure criterion is also used in this paper. The relationship between residual strain and cumulative strain as shown in formula (3) is presented. The comparison with Berkeley loading-unloading model and oversea model is shown in Figure 2 . It can be seen that the residual strain curve calculated by the relational expression is between the curve of the Berkeley loading-unloading model and the Guo Zhenhai model, which is close to the curve of Guo Zhenhai model, and also proves the rationality of the relational expression. When the concrete residual strain ε_p = 0.4ε_c,r, the corresponding cumulative strain is 1.3ε_c,r according to the formula (3). When the concrete reaches the peak strain ε_c,r, the corresponding ε_p = 0.25ε_c,r.

$\frac{{\epsilon }_p}{{\epsilon }_{c,r}}=0.186\times {\left[\frac{{\epsilon }_{\text{r}}}{{\epsilon }_{c,r}}\right]}^2+0.067\times \left[\frac{{\epsilon }_{\text{r}}}{{\epsilon }_{c,r}}\right]\left[\frac{{\epsilon }_{\text{r}}}{{\epsilon }_{c,r}}\right]\le 1.3$ (3)

Then combined with formula (3), the static equivalent strain analysis model of the cumulative damage of compressive concrete is established. The residual strain can be calculated by static equivalent strain and the stress and the strength of concrete after damage can be further inferred. Specific test and analysis steps are as follows:

Step 1: Using Non-destructive testing methods such as ultrasonic, blast wave or core sample method to test and analyze the mechanical parameters of non-destructive concrete in representative concrete member and parts, including the concrete compressive strength f_c,r, the concrete elastic modulus E_c and the peak compressive strain ε_c,r.

Step 2: The uniaxial compression stress-strain curve can be made according to the actual compressive strength of concrete f_c,r and elastic modulus E_c, the formula is shown below:

$\sigma =\left(1\text{-}{d}_c\right)E_c\epsilon$ (4)

where σ, ε are the compressive resistance and compressive strain of the concrete; d_c is the evolutionary parameter of the compression damage of the concrete. For details, see C.2.4 of the Code.

Step 3: The unloading elastic modulus E_r of the critical point of the structural concrete (usually the most unfavorable force section of the stressed member) can be measured by using the non-destructive testing method or the static loading and strain testing system; for example, by applying the static load to the structural concrete, the stress increment and strain increment of the concrete are obtained, and the unloading elastic modulus E_r is calculated according to formula (5).

$E_{\text{r}}=\frac{\Delta \sigma }{\Delta \epsilon }$ (5)

In the formula, $\Delta \sigma$, $\Delta \epsilon$ are the stress increment and strain increment of concrete under static loading respectively.

Step 4: The relationship between the unloaded elastic modulus E_r and the static equivalent strain ε_r is established according to Figure 1 .

$E_{\text{r}}=\frac{\left(1\text{-}{d}_c\right)E_c{\epsilon }_{\text{r}}}{{\epsilon }_r-{\epsilon }_{\text{p}}}$ (6)

Step 5: According to the relationship between residual strain and fatigue cumulative strain given by formula (3) and formula (6), the static equivalent strain ε_r can be calculated.

Step 6: Calculate the static equivalent stress σ_r corresponding to ε_r according to equation (7), then the residual strain value ε_p of the concrete is obtained by formula (8):

${\sigma }_r=\left(1\text{-}{d}_c\right)E_c{\epsilon }_{\text{r}}$ (7)

${\epsilon }_{\text{p}}={\epsilon }_{\text{r}}-{\sigma }_r/E_r$ (8)

Step 7: The residual strength of concrete after damage can be deduced according to the concrete stress-strain curve and static equivalent strain ε_r: when ε_r ≤ ε_c,r, the concrete strength remains to be f_c,r; when ε_r > ε_c,r, the concrete strength decreases, the static equivalent stress σ_r calculated by formula (7) is the residual strength of the concrete after damage.

Step 8: The practical damage indices D_p and D_ε calculated based on the residual strain and static equivalent strain of concrete are shown in Equations (9) and (10), respectively. The comparison of D_p and D_ε applicability is shown in Figure 3 . D_ε increases linearly with the increasing strain, but D_p can reflect the accelerated development of the damage when the strain is increasing, therefore D_p index is more reasonable.

$D_{\text{p}}=\frac{{\epsilon }_{\text{p}}}{0.4{\epsilon }_{cr}}$ (9)

$D_{\epsilon }=\frac{{\epsilon }_{\text{r}}}{1.3{\epsilon }_{cr}}$ (10)

Feng Zeying [24] carried out load failure test on 8 old T-shaped beams in 2004 that had been in use for 35 years in Mang Niu River Bridge, Liaoning Province, China. The cross-section and three-point static loading test diagram of the beams was shown in Figure 5 . The cross-section geometric size of the beams and the measured compressive strength of the concrete were listed in Table 1 , main bearing bar in beams was 8Φ28 + 4Φ16, which measured mean yield strength was 335 MPa. The damage of 8 old beams was mainly manifested by vertical through cracks near the mid-span bending area, and the measured crack characteristics under dead load were shown in Table 2 .

Although the old ordinary reinforced concrete beams had serious cumulative fatigue damage, the ultimate bending capacity calculated based on the measured material data and the actual section size is very close to the text breaking moment in Table 3 . Among them, the maximum deviation rate of beams is only 7.3%, indicating that the cumulative fatigue damage has little effect on the ultimate bearing capacity of reinforced concrete beams.

Considering the accurate loading value of the test beam at yielding and the essentially linear load-deflection curve before yielding, these yielding load values are selected to verify the rationality of the evaluation method in this paper. Among these beams, 2#, 3#, 6# and 7# exhibit bending cracks with a relative height of less than 0.6, indicating that the concrete in the compression zone is primarily in an elastic working state with minimal residual strain effect. On the other hand, beams 1#, 4#, 5# and 8# show significant decline, allowing for calculation of their equivalent additional load under dead load based on residual strain analysis. The calculated load values at yield deviate from test results by less than 5%. This simplified evaluation method meets engineering accuracy requirements and is particularly suitable for quantitatively assessing bearing capacity in reinforced concrete beams with severe fatigue damage.

Based on the above analysis, the ultimate load-carrying capacity of reinforced concrete bridges after fatigue damage has not decreased or decreased very little. And the reason why it is generally believed that the bearing capacity of the structure decreases under the fatigue damage is that the structural concrete strain, residual strain are increasing under fatigue effect, so that the “equivalent dead load moment” of the cross section is greater than the dead load moment, in consequence, the actual live load bending moment is reduced.

1) According to the practical relationship among concrete stress-strain curve, residual strain and accumulated strain, the calculation method of concrete equivalent static strain and residual strain based on unloading elastic modulus is put forward, and the specific testing and analysis steps of concrete structure site are detailed described, which is applicable for the acquisition of fatigue cumulative strain and residual strain of structural concrete.

2) A proposed analysis method for reinforced concrete bridges involves considering the effect of residual strain on the “equivalent bending moment under constant load”, and it’s applicable for the working state rapid evaluation of existing reinforced concrete bridges.

3) The main reason why the residual capacity of Reinforced Concrete Bridge is basically invariable or declining after fatigue damage is analyzed, and the practical evaluation method of residual capacity of structure is given. It points out that the main reason is that the equivalent strain or residual strain of structural concrete increases under the influence of fatigue, resulting in the decline of the actual live load level that the bridge can bear.

The research on the evaluation method of bearing capacity of reinforced concrete bridges after fatigue damage is discussed above, which is applicable for the in-situ evaluation of concrete bridges. This simplified evaluation method meets engineering accuracy requirements and is particularly suitable for quantitatively assessing bearing capacity in reinforced concrete beams with severe fatigue damage.

The influence of longitudinal reinforcement ratio, concrete strength and prestressing effect is not analyzed in this paper. In addition, the concrete residual strain analysis method and the damage index can be used to evaluate the deformation performance of concrete bridge, and the analysis and evaluation of the deformation behavior after damage are worthy of further study.