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

Jinxiang Wang

TL;DR

This work develops a rigorous, constructive analysis of the classical Melan equation for suspension bridges by first solving the explicit linear nonlocal model with a less stiff approximation. It proves a maximum principle and uniform positivity, then introduces a monotone lower-upper solutions scheme to establish existence, uniqueness (in certain parameter regimes), and approximability for the original nonlinear nonlocal problem. The method yields convergent, extremal solutions that bracket the true solution and, under suitable conditions, converges to a unique solution, with practical validation on bridge-like parameters. It also discusses the applicability and limitations for real bridges, offering guidance for engineering design and posing open questions for extending the framework. Overall, the paper provides a rigorously grounded, constructive approach to Melan-type models that complements traditional engineering approximations.

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.

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

TL;DR

This work develops a rigorous, constructive analysis of the classical Melan equation for suspension bridges by first solving the explicit linear nonlocal model with a less stiff approximation. It proves a maximum principle and uniform positivity, then introduces a monotone lower-upper solutions scheme to establish existence, uniqueness (in certain parameter regimes), and approximability for the original nonlinear nonlocal problem. The method yields convergent, extremal solutions that bracket the true solution and, under suitable conditions, converges to a unique solution, with practical validation on bridge-like parameters. It also discusses the applicability and limitations for real bridges, offering guidance for engineering design and posing open questions for extending the framework. Overall, the paper provides a rigorously grounded, constructive approach to Melan-type models that complements traditional engineering approximations.

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.
Paper Structure (8 sections, 157 equations, 4 figures)

This paper contains 8 sections, 157 equations, 4 figures.

Table of Contents

  1. Introduction and historical overview
  2. The analytical solution for "less stiff" model (1.5), (1.2)
  3. The positivity of the analytical solution for "less stiff" model (1.5), (1.2) and a Maximum principle
  4. Monotone iterative technique for classical Melan equation (1.4),(1.2)
  5. The typical application of the monotone iterative technique
  6. Applicability to actual bridges
  7. Conclusions and open problems
  8. Appendix: The derivation of classical Melan equation

Figures (4)

  • Figure 1: Beam (blue) sustained by a cable (red) through parallel hangers.
  • Figure 2: The first and second iterations for problem (5.7).
  • Figure 3: The first and second iterations for problem (6.19).
  • Figure 4: The first iteration for problem (6.27).