## 3 Study on the Main Bending Form of Hangers
### Theoretical Formulation for the Bending Angle and Maximum Bending Stress of Hangers
According to [15], the Young's modulus of the hangers that consist of seven-wire strands is mainly determined by the seven-wire strands and can be the same as that of the seven-wire strands. The moment of inertia of the hangers can be considered as the moment of inertia for all the seven-wire strands in the hangers, because the PE pipe and compression wrapping layer do not have significant bending stiffness and can be ignored in the process of calculating the bending stiffness of hangers. The study in [21] provides a method for calculating the moment of inertia for the seven-wire strands in hangers. This method uses the number of seven-wire strands in hangers, the radius of the seven-wire strands, and the transverse strain of the seven-wire strands to calculate the moment of inertia for the seven-wire strands in the hangers with good accuracy. Therefore, the moment of inertia of the \(i\)-th hanger can be calculated, as follows, according to the above method:  
\[\Delta L_{i}=\alpha\Delta_{t}L_{i} \tag{1}\]1  
where \(r\) is the radius of the seven-wire strands; \(C_{n}\) is the coefficient of the moment of inertia of the cross-section; and, \(\varepsilon^{\prime}\) is the transverse strain of the seven-wire strands in the hangers. \(C_{n}\) and \(\varepsilon^{\prime}\) can be calculated by the following formulas:  
\[\Delta L_{i}=\alpha\Delta_{t}L_{i} \tag{1}\]2  
\[\Delta L_{i}=\alpha\Delta_{t}L_{i} \tag{1}\]3  
where \(n\) is the number of the seven-wire strands in the hangers; \(v\) is the Poisson's ratio of the seven-wire strands; \(T_{i}\) is the tension force of the \(i\)-th hanger; and, \(A\) is the sum of the section area of all seven-wire strands in the hangers. According to the slope-deflection equations [22; 23], the bending moment at the end of the \(i\)-th hanger caused by the bending angles \(\theta_{1}\) and \(\theta_{2}\) are, respectively,  
\[\Delta L_{i}=\alpha\Delta_{t}L_{i} \tag{1}\]4  
\[\Delta L_{i}=\alpha\Delta_{t}L_{i} \tag{1}\]5where \(T_{i}\) is the tension force of the \(i\)-th hanger. The maximum bending stress that the seven-wire strand bears in \(i\)-th hanger caused by \(\theta_{1}\) is  
\[\Delta L_{i}=\alpha\Delta_{t}L_{i} \tag{1}\]6  
where \(y_{\max}\) is the distance from the center of the hanger to the outermost steel strand (as shown in Figure 9). In the same way, the maximum bending stress that the seven-wire strand bears in \(i\)-th hanger caused by \(\theta_{2}\) is  
\[\Delta L_{i}=\alpha\Delta_{t}L_{i} \tag{1}\]7  
The maximum bending stress that the seven-wire strand bears in \(i\)-th hanger caused by \(\theta_{1}\) and \(\theta_{2}\) is:  
\[\Delta L_{i}=\alpha\Delta_{t}L_{i} \tag{1}\]8  
where \(y_{\max}\) is the distance between the seven-wire strand at the outer edge of the hangers between the centers of the hangers. The theoretical deduction above is based on the assumption that the hangers are simplified as beams and the curvature of hangers is not considered. In fact, the stiffness of hangers would decrease with the increase of length, so the theoretical deduction above would be more precise when applied for short hangers [24]. Thus far, the formulas for the bending angle and maximum bending stress of the hangers are obtained. The finite modeling method (FEM) in next section further verified the accuracy of these formulas. Validation of the Theoretical Formulations by FEM and Defining the Main Form of the Bending of Hangers  
In this section, the FEM is used to verify the accuracy of the above theoretical formulation. Figure 10 shows a finite element model of a TAB with a floating deck structure in Southeast China [23]. The main span of the bridge is 105 m. There are 12 pairs of hangers for the bridge, and all of the hangers are fixed at both ends. Each hanger is made of 19 \(\Phi\)15.2 seven-wire strands. The radius of the seven-wire strands in the hangers is 7.6 mm [25]. The distance between the seven-wire strand at the outer edge of the hangers between the centers of the hangers (\(y_{max}\)) is 91 mm.