# Bending Behavior of Corroded H-Shaped Steel Beam in Underground Environment
## 3 Analysis of Results
### Simplified Calculation of Residual Bearing Capacity of Corroded H-Shaped Steel Beams
#### Calculation Process  
According to assumptions (1)-(3), the cross-section of the H-shaped steel beams after corrosion is shown in Figure 15.  
Figure 14: Stress-strain relationship of corroded steel.  
Figure 13: Ductility index and energy dissipation index of corroded steel beams: (**a**) ductility index; (**b**) energy dissipation index.  
According to Standard for design of steel structures (GB 50017-2017) [30], the yield bending moment and full plastic bending moment of an H-shaped steel beam can be calculated using Equations (8) and (9), respectively.  
\[M_{\rm y}=\frac{I_{\rm x}}{y_{\rm max}}f_{\rm y}=W_{\rm x}f_{\rm y} \tag{8}\]  
\[M_{\rm P}=W_{\rm P}f_{\rm y} \tag{9}\]  
Here, \(M_{\rm y}\) is the yield bending moment, \(I_{\rm x}\) is the section inertia moment, \(y_{\rm max}\) is the distance from the x-axis to the edge of the section, \(W_{\rm x}\) is the elastic section modulus, \(f_{\rm y}\) is the yield strength of the steel, and \(W_{\rm P}\) is the plastic section modulus.  
After the steel beam is corroded, the relationships between the thickness of its flange and web and the corrosion rate are expressed by Equations (10) and (11), respectively. Additionally, the sectional inertia moment can be calculated using Equation (12).  
\[t_{\rm wc}=(1-\eta)t_{\rm w} \tag{10}\]  
\[t_{\rm fuc}=(1-\eta)t_{\rm f} \tag{11}\]  
\[I_{\rm wc}=\frac{1}{12}(1-\eta)t_{\rm w}h_{\rm w}^{3}+\frac{1}{12}b_{\rm f}[( 1-\eta)t_{\rm f}]^{3}+\frac{1}{4}b_{\rm f}(1-\eta)t_{\rm f}[h_{\rm w}+(1-\eta )t_{\rm f}]^{2} \tag{12}\]  
In Equations (10)-(12), \(t_{\rm w}\) is the web thickness of the uncorroded steel beam, \(h_{\rm w}\) is the web height of the uncorroded steel beam, \(b_{\rm f}\) is the flange width of the uncorroded steel beam, \(t_{\rm f}\) is the thickness of the uncorroded steel beam, and \(I_{\rm wc}\) is the sectional inertia moment of a corroded steel beam.  
The distance from the \(x\)-axis to the farthest edge of the section of a corroded steel beam can be calculated using Equation (13).  
\[y_{\rm max,c}=0.5h_{\rm w}+(1-\eta)t_{\rm f} \tag{13}\]  
The elastic section modulus of a corroded H-shaped steel beam can be calculated using Equation (14) and the plastic section modulus can be obtained using Equation (15).  
\[W_{\rm x}=\frac{(1-\eta)t_{\rm w}h_{\rm w}^{3}+b_{\rm f}[(1-\eta)t_{\rm f}]^{ 3}+3b_{\rm f}(1-\eta)t_{\rm f}[h_{\rm w}+(1-\eta)t_{\rm f}]^{2}}{6h_{\rm w}+ 12(1-\eta)t_{\rm f}} \tag{14}\]  
\[W_{\rm P}=\frac{1}{2}(1-\eta)t_{\rm w}h_{\rm w}^{2}+b_{\rm f}(1-\eta)t_{\rm f }[h_{\rm w}+(1-\eta)t_{\rm f}] \tag{15}\]  
The yield strength of the corroded steel can be calculated using Equation (6). Then, the yield moment of a corroded H-shaped steel beam can be calculated using Equations (6), (8) and (14),  
Figure 15: Simplified cross-section of corroded H-shaped steel beams.  
and the ultimate moment of a corroded steel beam can be calculated using Equations (6), (9) and (15). According to the yield moment and ultimate moment, the yield load and ultimate load of a corroded H-shaped steel beam can be easily obtained. The calculation results are summarized in Table 7. Notably, the calculated yield load values are slightly smaller than the measured values, and the ratio of the former to the latter is between 1.03 and 1.15. The calculated ultimate load values are slightly smaller than the measured values, and the ratio of the former to the latter is between 1.07 and 1.23. These results demonstrate the effectiveness of the simplified calculation method and this model can provide reference for evaluating the bearing capacity of corroded H-shaped steel beam.