An Initial Equilibrium State Analysis of Cable Structure
by Newton-Raphson Method
Most of framed structures have their initial stiffness in undeformed state and initial lengths of members are determined from an undeformed geometry. However, the undeformed geometry of cable cannot be defined because its stiffness is caused by tension. So either the unstrained length or the tension of cable must be calculated from the deformed geometry. Determining either unstrained length or tension which satisfies the given geometry in equilibrium is defined as initial equilibrium state analysis.
Both successive iteration and Newton-Raphson method are presented to solve the compatibility conditions and the sag constraint of the elastic catenary cable which is suspended between two fixed points with self-weight. However, stiffness method is required for the analysis of the more complicated cable structure such as suspension bridge or cable-stayed bridge.
Incremental formulation based on stiffness method is proposed. Nodal coordinates and the unstrained lengths of cable elements are taken as unknowns in incremental equations. The cable stiffness equations are derived from the compatibility conditions to construct the equilibrium equations. Since equilibrium equations are indeterminate, geometric constraints as many as cable elements must be given. The geometric constraints prescribe the coordinates of the control DOFs which are selected to satisfy the given geometry. Nodal coordinates and unstrained lengths are updated simultaneously by solving the incremental equations which consist of equilibrium equations and geometric constraints. The increments of nodal coordinates are not displacements but the change of geometry due to changes of the unstrained lengths of cables.
For the analysis of suspension bridge and cable-stayed bridge, modeling methods are presented and the control DOFs are defined. Two numerical examples are performed to demonstrate the validity and the effectiveness of the proposed method compared with previous studies based on successive iteration.
Initial equilibrium state analysis, Newton-Raphson Method, Successive iteration, Elastic catenary cable, Geometric constraint, Control DOF