# Form-Finding Analysis of the Rail Cable Shifting System of Long-Span Suspension Bridges
## 2 Mechanical Characteristics of the RCS System under the Non-Loading Condition
### Mechanical Model of the RCS System under Non-Loading Condition  
During the computation of the conventional suspension bridge construction process, the parameters of main cable are usually solved by the segmental catenary method. The sling force is determined from the completed bridge state, with backward disassembly to the cable finish state. The internal force, line shape, unstressed cable length, and sling length of the unloaded cable can be directly solved through iteration [34]. In the computation and analysis of conventional double-layer cable systems and cable dome structures, the hanger is often considered as being of rigid handling, or it is assumed to have infinite sling stiffness to ensure that the vertical displacements of the load-bearing cable and the stabilization cable are the same. This is inconsistent with the actual situation. In the finite element analysis of single-layer flexible cables, the two-node straight element, the two-node curve element [35], and the multi-node isoparametric element [36], among others, are usually used. Although there are certain limitations and deviations in the computation accuracy, computational efficiency, and applicable range of each element, the computational results in respective adaptive range can reach high accuracy and can be applied to engineering practice [37].  
To simplify the form-finding analysis of the RCS process under the non-loading condition, the analytical method is used to establish the global mechanical model of the main cable, sling, and rail cable and to solve the equation, under the condition that the sling force is unknown. The basic assumptions of the mechanical analysis of flexible cables are introduced and the elastic elongation of the sling is considered. The main cable segment is simulated by a segmental catenary element, and the rail cable segment is simulated by a two-node straight pole element. The analytical results are compared with the finite element results and model test results.  
The non-loading condition is defined as the status after the erection of the main cable and sling, the installation of the saddle support and the rail cable, and the tensioning of the rail cable are completed. To simplify the computational model for the system under the non-loading condition, the following assumptions are introduced [34]:  
1. The flexible cable can only be tensed but not be pressed or bent.
2. The stress-strain of the flexible cable is consistent with [PERSON]'s theorem.
3. The cross-sectional area before deformation is used for the calculation of the tensile stiffness of the main cable, rail cable, and sling before and after stress.
4. The friction between the rail cable and the saddle is ignored, and the sling does not tilt after the rail cable tension is completed.  
Then, a mechanical analytical model for the shifting system under the non-loading condition is established as shown in Figure 2, where z represents the main cable, g represents the rail cable, and d represents the sling. To describe the analysis process, an inter-segmental model is extracted for analysis, as shown in Figure 3.