Shape-finding for spatial cables and hangers in multiple-tower suspension bridge

Spatial main‑cable suspension bridges exhibit favorable lateral stiffness, torsional resistance, and spanning capacity. However, their complex structural configuration introduces challenges in shape‑finding for the completed state, which refers to determining the equilibrium profile of the cable-hanger system (CHS) under self‑weight and permanent external loads in the completed state. Based on the segmental catenary theory, this study establishes the geometry and mechanical relationships between and within catenaries of the CHS in which the hangers are rigorously modeled as catenaries rather than simplified as concentrated forces. Accurate calculation of hanger force is essential for determining the transverse mechanical behavior and geometric configuration of spatial main cables. The known and unknown catenary parameters of the cable segments and hangers during the shape‑finding process are clarified, and solution procedures are developed. By utilizing the mechanical equilibrium and geometric compatibility conditions at each cable node of the controlling span, the equations for a CHS are established and solved via a modified interactive matrix method. Subsequently, the solution is extended to other spans by incorporating the mechanical equilibrium conditions at the tower tops. A spatial triple-tower suspension bridge is analyzed as a numerical example to demonstrate the shape‑finding results. Based on the unstressed length and coordinate results, a finite element method (FEM) was established. The internal forces difference between proposed method and nonlinear FEM results are less than 1% and the coordinate error are less than less than 0.1 mm.