# An Improved Analytical Algorithm on Main Cable System of Suspension Bridge  
[PERSON]  
1 School of Civil Engineering, Changsha University of Science and Technology, Changsha 410114, China; [EMAIL_ADDRESS] (C.L. ); [EMAIL_ADDRESS] (Y.L. ); [EMAIL_ADDRESS] (H.K. ); [EMAIL_ADDRESS] (C.D. ); [EMAIL_ADDRESS] (H.L.) 2 School of Civil Engineering, Dalian University of Technology, Dalian 116024, China; [EMAIL_ADDRESS] (Correspondence: [EMAIL_ADDRESS] or [EMAIL_ADDRESS]; Tel. : +86-180-0846-6632; Fax: +86-0731-8525-6006 2  
iteration method [14; 15], the target configuration under dead loads (TCUD) related methods [9], the optimization method [16; 17], and the combined method [18]. Above mentioned various form-finding approaches are generally into three categories: (1) the simplified approach; (2) the Finite Element (FE)-based approach; and (3) the analytical method. The simplified method assumes that the load acts uniformly along the span of the main cable, which follows a parabolic shape [2; 17; 19]. To account for a cable's sag effect, [PERSON] proposed the equivalent modulus of elasticity for a parabolic cable [20]. The simplicity of [PERSON]'s formula has made it widely used not only in the research field, but also for the practical designs of suspension bridges. Owing to its simplicity, this approach has been adopted by several investigators [21; 22; 23], and has been proved to be sufficient for some cases. Namely, when a cable has relatively high stress and small length, the [PERSON] equivalent modulus approach could achieve a good result. However, the parabolic approximation becomes inaccurate for cables with a large sag-to-span ratio (\(>\)1/8), which experience self-weight along the length of the cable and concentrated forces from the hangers. To improve the accuracy and facilitate nonlinear analyses of suspension bridges, various FE-based approaches have been developed. In these approaches, most of the finite element packages are still lack of suitable cable elements. A sagging cable is often simulated as two-node element, multi-node element, and curved element with rotational degrees of freedom [24; 25; 26]. The two-node element is only suitable for modeling the cables with high pretension and small length [27; 28], and equivalent modulus are used to account for the sag effect. For cables with large sag, a series of straight elements is used to model the curved geometry of cables. The multi-node element is based on the higher order polynomials for the interpolation functions [29; 30]. The tangent stiffness matrix and nodal force vector are obtained while using the iso-parametric formulation. These elements give accurate results for cables with small sag. For cable element with large sag, it is necessary to use a large number of elements to model the curved geometry of cable. Therefore, it causes computational costs. These FE-based approaches identify the target configuration of main cable via updating nodal positions and internal tension of cable elements based on nonlinear structural analysis. However, these FE-based approaches elevate the computational effort, and their convergence depends to a large extent on the assumed initial cable configuration and forces. The alternative approach is based on exact analytical expressions for the elastic catenary, since the equilibrium configuration of a hanging cable is a catenary in nature. This method was originally proposed by [PERSON] and [PERSON] [31] and was later extensively developed [32; 33; 34; 35; 36]. In particular, there are various catenary-type analytical elements available, which can be used to model large sag cables in suspension bridges:  
(1) _Inextensible catenary elements_: The cable elements adopted are infinitely stiff in the axial direction and cannot experience any increment of length. In practice, computer applications that are based on this type of element encounter severe difficulties, solving procedures tend to experience large numerical instability, causing a very difficult or even impossible convergence.