A semi-analytical form-finding method of the 3D curved cable considering its flexural and torsional stiffnesses in suspension bridges

During the configuration transformation, the 3D curved main cable's torsion increased the construction control's complexity in the suspension bridge with the spatial cable system. The calculation of the torsional angle of the main cable must consider the cable's flexural and torsional stiffnesses. However, most form-finding methods adopted the assumption of the ideal flexible cable, failing to obtain the torsional angle of the main cable. Hence, this study proposed a semi-analytical form-finding method for the 3D curved cable that considered the main cable's flexural and torsional stiffnesses. This method was based on the Kirchhoff-Love rod theory, which provided a feasible framework for analyzing the torsion in the main cable. Firstly, the Euler angles were incorporated to facilitate the parameterization of finite rotation and the transformation of static equilibrium differential equations. Then the governing equations of the target configuration were established using the geometric compatibility conditions and the static equilibrium conditions. Moreover, the equations were numerically solved through the finite difference method and the Levenberg-Marquardt method. Finally, four examples were employed to verify the feasibility and accuracy of the proposed method, including two cantilever beams and two suspension bridges with curved 3D cable and 2D cable, respectively. The results of the case study demonstrated that the torsion in the main cable was primarily caused by bidirectional bending rather than internal torque. This torsion was observed under the influence of the cable's self-weight and non-eccentric hanger forces. Furthermore, it was observed that the maximum torsion did not occur at the mid-span point. The position of the inflection point was found to be highly sensitive to the orientation constraints imposed at the fixed end of the cable.