On the well-posedness and efficient approximation for the classical Melan equation in suspension bridges

RESUMO

The classical Melan equation modeling suspension bridges is considered. We first study the explicit expression and global properties of the analytical solution for the simplified ``less stiff'' model, based on which we derive an a priori estimate for the original classical Melan equation and establish, under an explicit load condition, that it has a unique solution with nonnegative integral under downward live loads, thereby showing the uniqueness of the corresponding deflection curve in the engineering setting. We also develop an efficient iterative approximation method by taking the solution of the simplified ``less stiff'' model as the first iterate, and prove its geometric convergence with explicit error estimates. The applicability and computational efficiency of the method are demonstrated through calculations for two actual bridges, which also quantify the influence of the nonlinear nonlocal term on the solution and clarify the relationship between the simplified and original models. Several engineering observations are verified and explained, and some related open problems are suggested.