## 3 Study on the Main Bending Form of Hangers  
### Theoretical Formulation for the Bending Angle and Maximum Bending Stress of Hangers  
The theoretical formulation for the lateral bending angles and maximum bending stress of hangers is conducted here. Presently, bearings that allow for bridge decks to have longitudinal deformation of their bridges and rotate (such as rubber bearings) are widely used in HTABs and TABs with floating deck structures [13]. Under the effect of temperature and the temperature gradient, bridge decks can produce longitudinal deformation and rotate. The longitudinal deformation of bridges is related to the temperature changes of the whole bridge, and the rotation of bridge decks is related to the temperature gradient throughout the bridge sections. Based on the analysis above, the lateral bending angle of hangers consists of two parts. One is the bending angle (\(\theta_{1}\)) that is caused by the longitudinal deformation of the bridge decks (as shown in Figures 5 and 6), and the other part is the bending angle (\(\theta_{2}\)) that is caused by the rotation of the bridge decks [19] (shown in Figure 7). We theoretically deduce the formulas to describe the bending  
Figure 4: Skewback of HTABs and TABs with a floating deck structure. angle caused by the longitudinal deformation of bridge decks and the bending angle caused by the rotation of the bridge decks, according to the analysis above. Firstly, the bending angle that is caused by the longitudinal deformation of bridge decks is discussed. The displacement of the lower anchorage zone of the \(i\)-th hanger on the right side of the bridge center is  
\[\Delta L_{i}=\alpha\Delta_{t}L_{i} \tag{1}\]  
where \(L_{i}\) is the distance between the bridge center and the \(i\)-th hanger; \(\Delta t\) is the change in the temperature of the whole bridge deck; and, \(\alpha\) is the thermal expansion coefficient of the bridge deck [20]. Figure 5: Lateral bending angles caused by the longitudinal deformation of bridge decks. Figure 6: Lateral bending deformation of the hangers under the longitudinal deformation of bridge decks. The lateral bending angle \(\theta_{1}\) of the hangers caused by the longitudinal deformation of the bridge decks (as shown in Figure 6) of the \(i\)-th hanger on the right side of the center of the bridge is  
\[\theta_{1}=\frac{\Delta L_{i}}{H_{i}} \tag{2}\]  
where \(H_{i}\) is the length of the \(i\)-th hanger. Subsequently, the lateral bending angle \(\theta_{2}\) of the hangers that is caused by the rotation of the bridge decks is further studied (as shown in Figure 7). It is assumed that the temperature at the top of the bridge decks is \(T_{1}\), and the temperature at the bottom of the bridge decks is \(T_{2}\). According to Figure 8, the following can be obtained:  
\[d\theta_{2}=\frac{\alpha(T_{1}-T_{2})dx}{h} \tag{3}\]  
where \(h\) is the height of the bridge deck. Figure 8: Rotation angle of the bridge decks. Figure 7: Bending angles caused by the rotation of the bridge decks. After integrating the above equation, the rotation angle of the bridge decks is  
\[\theta_{2}=\frac{\alpha(T_{1}-T_{2})L}{2h} \tag{4}\]  
where \(L\) is the distance \(L_{i}\) between the \(i\)-th hanger and the center line of the bridge. The rotation angle \(\theta_{2}\) is the bending angle of the \(i\)-th hanger that is caused by the rotation of the bridge decks. Thus, the total bending angle of the hangers is  
\[\theta=\theta_{1}+\theta_{2}. \tag{5}\]  
The maximum bending stress is further studied based on the formula for the bending angle of the hangers derived above. According to the analysis in Section 2.1, the bending stiffness of the hanger that consists of seven-wire strands and can be calculated by the bending theory of beams, so the bending stiffness of the \(i\)-th hanger is  
\[\Delta L_{i}=\alpha\Delta_{t}L_{i} \tag{1}\]0  
where \(E_{i}\), \(I_{i}\) are, respectively, the Young's modulus and the moment of inertia of the \(i\)-th hanger.