## 4 Development of Restoring Force Model
### Defining Hysteretic Rule
#### 4.2.1 Rule of Strength and Stiffness Degradation  
(1) Cycle degradation index  
The cyclic degradation index was used to reflect the cyclic degradation effect of the mechanical properties of corroded steel frame beam specimens in this paper [41], namely:  
\[\beta_{i}=[E_{i}/(E_{\mathrm{t}}-\sum_{j=1}^{i}E_{j})]^{c} \tag{16}\]  
Figure 12: Comparing the calculation value with the test value. (**a**) B-1; (**b**) B-2; (**c**) B-3; (**d**) B-4; (**e**) B-5; (**f**) B-6. where,  
\[E_{t}=\Lambda\cdot M_{y} \tag{17}\]  
\[\Lambda=495\cdot\left(\frac{h}{t_{w}}\right)^{-1.34}\cdot\left(\frac{b_{f}}{2 \cdot t_{f}}\right)^{-0.595}\cdot\left(\frac{c_{unit}^{2}\cdot f_{y}}{355} \right)^{-0.360} \tag{18}\]  
where \(c\) is the rate of the cyclic degradation, \(1\leq c\leq 2\), in this paper it is equal to 1.5, \(E_{i}\) is the energy dissipated by the specimen during the ith cycle loading, \(\sum\limits_{j=1}^{i}E_{j}\) is the cumulative dissipative energy of the specimen before the ith cycle loading, \(E_{t}\) is the ability of energy dissipation of specimen, \(\Lambda\) is the cumulative plastic angles of beam specimens, \(M_{y}\) is the effective yield moment, in this paper, it is equal to \(1.17M_{p}\), \(h\) is the web height, \(b_{f}\) is the flange plate width, \(c_{unit}^{2}\) is the unit conversion factor, which is equal to 1 when using millimeters and megapascals, and the rest of the parameter definitions are the same as before. Considering the influence of corrosion damage, the cumulative plastic angles of beam specimens \(\Lambda\) can be rewritten by combining Equation (2):  
\[\Lambda=495\cdot\left(\frac{h}{t_{w}}\right)^{-1.34}\cdot\left(\frac{b_{f}}{2 \cdot t_{f}}\right)^{-0.595}\cdot\left[\frac{c_{unit}^{2}\cdot(1-0.9721D_{w}) f_{y}}{355}\right]^{-0.360} \tag{19}\]  
(2) Strength degradation rules  
According to the test results, the yield load decreases with the increase of the displacement amplitude and the number of cycles, and the peak load of the hysteresis loop decreases with the increase of the number of cycles under the same displacement amplitude after the specimen enters the elastoplastic stage. The strength degradation rules can be defined as:  
\[P_{y,i}^{\pm}=(1-\beta_{i})P_{y,i-1}^{\pm} \tag{20}\]  
\[P_{j,i}^{\pm}=(1-\beta_{i})P_{j,i-1}^{\pm} \tag{21}\]  
where, \(P_{y,i}^{\pm}\), \(P_{y,i-1}^{\pm}\) are the yield load of the specimen at the time of the ith and i-1 th cyclic loading respectively, \(P_{j,i}^{\pm}\), \(P_{j,i-1}^{\pm}\) are the peak load of the specimen under the ith and i-1 th cycle loading at jth stage displacement respectively. The superscript \"\(\pm\)\" indicates the loading direction, where \"\(+\)\" is positive loading and \"\(-\)\" indicates reverse loading. The specific strength degradation rules are shown in Figure 13.  
(3) Stiffness degradation rules  
The unloading stiffness of the specimen is also degraded continuously under the action of repeat cyclic loading, which can be described by the following formula:  
\[K_{u,i}=(1-\beta_{i})K_{u,i-1} \tag{22}\]  
where \(K_{u,i}\), \(K_{u,i-1}\) are the unloading stiffness of specimen during the ith and i-1 th cycle loading, respectively. In addition, the hysteresis curves obtained from the test shows that the reloading stiffness is degraded since the last cycle unloading stiffness, and the degradation law of the reloading stiffness can be characterized as:  
\[K_{r,i}=(1-\beta_{i})K_{u,i-1} \tag{23}\]  
where \(K_{r,i}\) is the reloading stiffness of specimen during the ith cycle loading. The specific stiffness degradation rules are shown in Figure 12:0  
In summary, according to the established skeleton curves and hysteresis rules, the restoring force model of corroded steel frame beams in the acidic atmospheric environment is shown in Figure 12:1 The specific calculation steps are obtained as follows:  
Step 1: The feature points of the skeleton curve for the steel beam with different degrees of corrosion were calculated according to the Equations (5)-(15).