## 4 Development of Restoring Force Model  
### Defining Skeleton Curve  
#### 4.1.1 Simplified Skeleton Curve and Feature Points  
Based on the above test results and analysis, a trilinear skeleton curve model is used in this paper, and the positive and negative skeleton curves are assumed to be symmetric, as shown in Figure 11. The points A (the yield point), B (the peak point), and C (the limit point) will be calculated later in this section, and then the skeleton curve model can be determined. Figure 10: Skeleton curve of specimens. The specific calculation methods for each feature point of the corroded member are as follows:  
(1) Yield point A  
The yield load \(P_{\rm y}\) of the specimen is:  
\[P_{\rm y}=\frac{M_{\rm e}}{L}=\frac{W_{\rm n}}{L}f_{\rm y} \tag{5}\]  
where \(M_{\rm e}\) is the bottom moment of the beam end when calculating the elastic limit, \(W_{\rm n}\) is the section modulus of H-section steel beam, \(L\) is the length of steel beam, and \(f_{\rm y}\) is the measured yield strength of steel. Considering the influence of corrosion damage, Equation (2) is substituted into Equation (5) to obtain the yield load of the steel beam \(P_{\rm y}^{\prime}\) as follows:  
\[P_{\rm y}^{\prime}=\frac{W_{\rm n}}{L}f_{\rm y}^{\prime}=\frac{W_{\rm n}}{L}(1- 0.9721D_{\rm w})f_{\rm y} \tag{6}\]  
In a similar way, based on the theory of material mechanics, the yield displacement of the corroded beam \(\Delta_{\rm y}^{\prime}\) is derived as follows:  
\[\Delta_{\rm y}^{\prime}=\lambda\frac{P_{\rm y}^{\prime}L^{3}}{3E_{\rm s}^{ \prime}I}=\lambda\frac{(1-0.9721D_{\rm w})f_{\rm y}L^{3}}{3(1-0.9019D_{\rm w}) E_{\rm s}I} \tag{7}\]  
where \(I\) is the section moment of inertia, \(E_{\rm s}\) is the elastic modulus of the steel material, and \(\lambda\) is the correction factor that mainly considers the stiffness decrease of the support beam caused by the deformation or sliding and results in the actual displacement of the test specimen when the yielding is significantly larger than the theoretical displacement. (2) Peak point B  
Due to the existence of the hardening stage of the steel beam, the flat section assumption is no longer true, and it is difficult to derive the peak point bending moment \(M_{\rm max}\) through a theory that would exceed the plastic limit moment of the beam end \(M_{\rm p}\). Therefore, in the combined test results, the \(M_{\rm max}\) can be taken as:  
\[M_{\rm max}=1.2M_{\rm p} \tag{8}\]  
where,  
\[M_{\rm p}=W_{\rm p}f_{\rm y} \tag{9}\]  
\[W_{\rm p}=Bt_{\rm f}(H-t_{\rm f})+\frac{1}{4}(H-2t_{\rm f})^{2}t_{\rm w} \tag{10}\]  
where \(W_{\rm p}\) is the plastic section modulus of the H-shaped steel beam, \(B\), \(H\) are the section width and height of H steel, respectively, \(t_{\rm f}\) is the flange plate thickness, and \(t_{\rm w}\) is the web thickness, the rest of the parameter definitions are the same as before. The peak load \(P_{\rm max}\) of the beam specimen is:  
Figure 11: Simplified skeleton curve. \[P_{\rm max}=\frac{M_{\rm max}}{L} \tag{11}\]  
Considering the influence of corrosion damage, Equation (2) is substituted into Equation (11) to obtain,  
Figure 10:0  
where \(M^{\prime}_{\rm max}\), \(P^{\prime}_{\rm max}\) are the peak moment and the load after corrosion, respectively. Statistical analysis was performed on the test results, and the expressions of peak displacement \(\Delta^{\prime}_{\rm m}\) is obtained by the regression analysis considering the influence of corrosion:  
Figure 10:1  
(3) Ultimate point C  
The ultimate load can take 85% of the peak load, namely:  
Figure 10:2  
where \(P^{\prime}_{\rm u}\) is the ultimate load of the beam when considering the influence of the corrosion damage. The relationship between the ratio of the ultimate displacement and the yield displacement as a function of the weight loss rate is as follows:  
Figure 10:3  
where \(\Delta^{\prime}_{\rm u}\) is the ultimate displacement when considering the influence of the corrosion.