# A Novel Shape Finding Method for the Main Cable of
###### Abstract
[[https://doi.org/10.3390/app1106444](https://doi.org/10.3390/app1106444)]([https://doi.org/10.3390/app1106444](https://doi.org/10.3390/app1106444))  
**CEssentially, the gravity stiffness steams from the geometric nonlinearity, which strongly depends on the configuration and internal force of the main cable. However, the shape of the main cable should be prescribed accounting for the effects of all external forces, which is always unable to be amended during the construction [2]. The configuration or target shape for the main cable at the initial equilibrium state under dead loads due to the self-weights of the bridge should be predetermined at the preliminary design stage. A shape finding or form finding process is therefore conducted to determine the shape and internal forces of the main cable through minimizing the dead-load-induced deformation of the bridge. Moreover, the increase of the main cable dimension and the reduction of the cable design safety factor require a more refined estimation of the cable configuration and internal forces in the structural components [3]. In the past several decades, the theory of the shape finding for the main cable of suspension bridge has received intensive attention. For a given cable segment of specified unstrained length, cross section area, Young's elastic modulus, and density, there are many kinds of relations between node force and node displacement/position, e.g., a link element considering sag effect [4; 5], a two-node curved cable element [6; 7], and various two-dimensional [8] and three-dimensional catenary cable elements [9; 10; 11; 12]. The segmental parabola theory assumes the shape of the main cable to be a parabolic curve under uniform load [4; 13]. This approximation ignores the concentrated forces transferred from the hangers, which is only acceptable for medium and small span suspension bridges. The segmental catenary theory allows one to achieve the analytical solutions of each cable segment shape. The equations describing the analytical relations between the axial forces and the strained/unstrained lengths of a cable segment under the action of the self-weight can be found in many pioneering studies, e.g., [14]. These equations were then widely employed to develop various shape finding approaches for the main cable of the suspension bridge, such as the initial force method or segmental catenary method (SCM) [3; 15; 16; 17; 18], the target configuration under dead load (TCUD) method [19], the improved TCUD method [20], the Generalized TCUD method [21], the coordinate iteration method [22], and the perturbation approach [23; 24]. The unstrained length of each main cable segment in these methods is unknown and solved in the successive nonlinear equations using the nonlinear finite element iteration [15; 16; 25; 26; 27; 28; 29]. Generally, at least two loop layers are required for calculating the unstrained length and axial force of each catenary segment before determining the shape of the main cable. To avoid a loop over each element, the unstrained length of each cable is treated as an unknown parameter in formulating a tangential stiffness matrix in the TCUD method [19] and the Generalized TCUD method [21]. Shape finding process based on the analytical solution of each catenary element is mostly exact. However, its theoretical derivation is complex, and the achievement of the algorithm requires high programing skills. Moreover, most of them are restricted to two-dimensional (2D) parallel main cables. The three-dimensional (3D) spatial main cables, which are intended to enhance the lateral stiffness of the suspension bridge, have been employed in some recent projects, such as the Yongjong Grand Bridge in Korea, Jiangdong Bridge in China, the New Oakland Bay Bridge in the USA, and Halogaland Bridge in Norway. [PERSON] et al. [25] extended the 2D segmental catenary approach to 3D by iterating the axial component of the main cable to achieve a target sag-to-span ratio. [PERSON] et al.