## 2 Methodology
### Equivalent Fatigue Damage Constitution
#### 2.3.3 Fatigue Residual Elasticity Modulus  
For cyclic loading involving inelastic deformation, two types of elasticity modulus, i.e., tangent stiffness and secant stiffness, should be differentiated. Fatigue residual secant stiffness, the slope of the straight line connecting the two points representing the maximum and minimum fatigue loading in a given loading cycle, is smaller than fatigue residual tangent stiffness, the tangent slope of the initial straight part of the loading branch. The degradation process of fatigue residual stiffness is exhibited in Figure 12[30]. In Figure 12, \(E_{0}\) is the initial stiffness; \(E_{\text{n}}\) is the fatigue residual stiffness after \(n\) cycles, considering plastic flow and microcrack damage. The maximum fatigue strain consists of three parts: \(\varepsilon_{\text{e}}\) is the elastic part calculated as \(\sigma_{\text{max}}/E_{0}\); \(\varepsilon_{\text{d}}\) is the inelastic strain induced by microcrack damage evaluated by subtracting \(\varepsilon_{\text{e}}\) from \(\sigma_{\text{max}}/E_{\text{n}}\); and \(\varepsilon_{\text{pl}}\) is the inelastic strain induced by plasticity, which equals \(\varepsilon_{\text{max}}^{\text{n}}-\varepsilon_{\text{e}}-\varepsilon_{ \text{d}}\). The fatigue residual stiffness could be defined as in Equation (26):  
\[E_{\text{n}}=E_{0}/(1+(\varepsilon_{\text{d}}/\varepsilon_{\text{e}})). \tag{26}\]  
Assuming that \(\varepsilon_{\text{pl}}\) is coaxial with \(\varepsilon_{\text{d}}\) and \(\delta=\varepsilon_{\text{pl}}/\varepsilon_{\text{d}}\)[30; 36], Equation (26) could be rewritten as Equation (27):  
\[E_{\text{n}}=\frac{E_{0}(1+\delta)}{1+\delta+(\varepsilon_{\text{d}}+ \varepsilon_{\text{pl}})/\varepsilon_{\text{e}}}. \tag{27}\]  
Figure 12: Sketch of fatigue residual stiffness degradation after \(n\) loading cycles between fatigue stresses of \(\sigma_{\text{min}}\) and \(\sigma_{\text{max}}\). The maximum fatigue strain \(\varepsilon_{\text{max}}^{\text{n}}\) is divided into three parts. Since \(\epsilon_{\text{pl}}\) and \(\epsilon_{\text{d}}\) cannot be directly measured, substituting \(\epsilon_{\text{max}}^{\text{n}}=\epsilon_{\text{e}}+\epsilon_{\text{pl}}+ \epsilon_{\text{d}}\) and \(\epsilon_{\text{e}}=\sigma_{\text{max}}/E_{0}=f_{\text{r}}S_{\text{max}}/E_{0}\) into Equation (27), it could be rewritten as Equation (28):  
\[E_{\text{n}}=\frac{E_{0}(1+\delta)}{\delta+E_{0}\epsilon_{\text{max}}^{\text{n }}/f_{\text{r}}S_{\text{max}}}. \tag{28}\]  
Now, think about the last loading cycle just before fatigue failure. Please note that the maximum strain of concrete at fatigue failure \(\epsilon_{\text{max}}^{\text{f}}\) could be obtained by substituting \(\sigma_{\text{max}}\) into Equation (22), the descending branch of the monotonic constitutive equation. Based on Equations (26)-(28), and assuming the tangent or secant modulus follows a similar proportion, the elasticity modulus at fatigue failure can be calculated as in Equation (29):  
\[E_{\text{f}}=\frac{E_{0}f_{\text{r}}S_{\text{max}}(1+\delta)}{\delta f_{\text{ r}}S_{\text{max}}+E_{0}\epsilon_{\text{max}}^{\text{f}}}. \tag{29}\]  
\(E_{0}\) is the initial stiffness; \(f_{\text{r}}\) is the static strength; \(S_{\text{max}}\) is the maximum stress level; \(\epsilon_{\text{max}}^{\text{f}}\) is the maximum fatigue strain at fatigue failure. \(\Delta\) is the ratio of plasticity-induced inelastic strain to inelastic strain induced by damages such as microcracks and microvoids, which ranges [1; 1.5] according to experimental results, and 1.2 is taken in this study [30; 36]. Please note that two types of stiffness exist; their difference grows as the fatigue loading cycles increase. Therefore, when calculating the fatigue residual tangent stiffness at fatigue failure \(E_{\text{t}}^{\text{f}}\), the initial stiffness \(E_{0}\) should be taken as its tangent value \(E_{\text{t}}^{0}\). In contrast, the initial secant stiffness \(E_{\text{s}}^{0}\) should be used to evaluate fatigue residual secant stiffness at fatigue failure \(E_{\text{s}}^{\text{f}}\).