Existence, uniqueness, and approximability of solutions to the classical Melan equation in suspension bridges

TL;DR

This paper investigates the mathematical properties of the classical Melan equation for suspension bridges, establishing existence, uniqueness, and approximation methods, and demonstrating their practical relevance through engineering examples.

Contribution

It introduces a monotone iterative technique for solving the Melan equation, providing new insights into its solutions and their computational approximation.

Findings

Explicit solutions for simplified models are derived.

The iterative method converges for the original Melan equation.

Applications to real bridge design are demonstrated.

Abstract

The classical Melan equation modeling suspension bridges is considered. We first study the explicit expression and the uniform positivity of the analytical solution for the simplified ``less stiff'' model, based on which we develop a monotone iterative technique of lower and upper solutions to investigate the existence, uniqueness and approximability of the solution for the original classical Melan equation.The applicability and the efficiency of the monotone iterative technique for engineering design calculations are discussed by verifying some examples of actual bridges. Some open problems are suggested.