## 5 Strength Check and Fatigue Properties of Seven-Wire Strands in Hangers under Lateral Bending
### The Fatigue Properties of Seven-Wire Strands under Lateral Bending  
Based on the previous analysis, it is known that the ultimate tensile strength of seven-wire strands is weakened when the seven-wire strands experience lateral bending deformation. The fatigue properties of seven-wire strands may be weakened by a decrease in the ultimate tensile strength, according to [38]. The fatigue properties of seven-wire strands are one of the main influential factors for the fatigue life of hangers. Therefore, an investigation of the properties of seven-wire strands under lateral bending is conducted. The S-N curve is a typical method to reflect the fatigue properties of materials, so we decide to study the S-N curve of seven-wire strands under lateral bending [39; 40; 41]. The S-N curve of the seven-wire strands with 1860 MPa ultimate tensile strength under 1050 MPa mean stress is [42]:  
\[\log_{10}N=13.84-3.5\lg S_{a} \tag{21}\]  
where \(N\) is the number of cycles to failure and \(S_{a}\) is the cyclic stress amplitude (\(S_{a}=\frac{S_{max}-S_{max}}{2}\)). When the ultimate tensile strength of the seven-wire strands is assumed as the function of the lateral bending angle, the S-N curve of the seven-wire strand under lateral bending can be deduced based on Equation (21). For Equation (21), when \(N\) is \(2\times 10^{6}\) or \(2\times 10^{7}\), the corresponding value of \(S_{a}\) is described as endurance limit \(\sigma_{a}\). When the stress amplitude \(S_{a}\) is lower than \(\sigma_{a}\), N is bigger than \(2\times 10^{6}\) or \(2\times 10^{7}\) and the number of cycles to failure is regarded as infinity. In this paper, we take \(2\times 10^{6}\) as the value of N to obtain the endurance limit \(\sigma_{a}\) of seven-wire strands. Thus, when \(N=2\times 10^{6}\), \(\sigma_{a}\) is 143 MPa. According the Goodman equation [43] (Equation (22)), when \(S_{m}\) is the mean stress whose value is 1050 MPa; \(\sigma_{a}\) is 143 MPa; \(S_{u}\) is 1860 MPa, the following can be obtained: \(\sigma_{-1}\) = 328 MPa. \[\sigma_{a}=\sigma_{-1}(1-\frac{S_{m}}{S_{u}}) \tag{22}\]  
Furthermore, the \(\sigma_{a}\) under \(N=2\times 10^{6}\) and \(S_{m}=1050\) MPa is obtained:  
\[\sigma_{a}=328(1-\frac{S_{m}}{S_{u}}) \tag{23}\]  
According to the ideal S-N curve, the slope of the S-N curve of materials with different ultimate tensile strengths is constant, and the intercept of the S-N curve is:  
\[b=\log 2\times 10^{6}+3.5\log S_{a}. \tag{24}\]  
The S-N curve of the seven-wire strands under the mean stress \(S_{m}=1050\) MPa is  
\[\log N=b-3.5\log S_{a} \tag{25}\]  
where \(b\) is  
\[b=\log 2\times 10^{6}+3.5\log(328(1-\frac{1050}{S_{u}}))=15.1+3.5\log(1-\frac{10 50}{S_{u}}). \tag{26}\]The function of the ultimate tensile strength of the seven-wire strands under lateral bending is \(y=-14.03x+1800\). The S-N curve of the seven-wire strands under lateral bending can be obtained by Equations (25) and (26):  
\[\log N=15.1+3.5\log(\frac{1}{S_{a}}-\frac{1050}{S_{a}(-14.03\theta+1800)}) \tag{27}\]  
where \(N\) is the number of loading cycles to failure for the seven-wire strands under lateral bending; \(S_{a}\) is the stress amplitude; \(S_{m}\) is the mean stress; and, \(\theta\) is the lateral bending angle of the seven-wire strands. When a value is specified for stress amplitude \(S_{a}\), Equation (27) can be plotted to show the influence of lateral bending \(\theta\) on the number of loading cycles to failure for the seven-wire strands (\(N\)). Therefore, a suitable value 2 MPa is specified for stress amplitude \(S_{a}\) (according to [44; 45], the modes of monitoring data for the stress amplitude of hangers of many arch bridges are close to 2 MPa). The value 2 MPa for stress amplitude \(S_{a}\) is only used to provide the necessary parameter for Equation (22) and it does not represent the actual stress amplitude. Figure 20 shows the curve of the relationship between lateral bending \(\theta\) and the number of loading cycles to failure of the seven-wire strands (\(N\)).