A singular elliptic problem related to the membrane equilibrium equations

Abstract

This paper presents a mathematical problem related to the equilibrium analysis of a membrane with rigid and cable boundaries for the so-called prestressing phase. The idea of using membranes in civil engineering applications such as footbridges, a new technology being developed in Spain, implies higher structural responsibility and more accurate analysis procedures. The membrane is represented by a regular surface with negative gaussian curvature, and its boundary by a set of regular curves the curvature of which depends on the structural elements, rigid or cable. Equilibrium is directly expressed by means of boundary differential problems in terms of the membrane shape and its stress tensor. The membrane–cable interface equilibrium requires taking into account a singular condition that makes the problem more difficult. Therefore, starting from the equilibrium equations, two mathematical approaches can be considered: a direct problem and a dual problem. The direct problem is defined and analysed, and its principal qualitative aspects are studied in this paper. Moreover, a numerical solution procedure is proposed to obtain practical results.

Keywords:

• membrane
• footbridge
• rigid boundary
• cable boundary
• elliptic problem

2010 AMS Subject Classifications:

• 35J25
• 74K15
• 74S05

Acknowledgements

This paper has been partially supported by the Project MTM2010-16401 of the Ministerio de Economía y Competitividad/FEDER of the Gobierno de España and the Consejería de Economía, Innovación, Ciencia y Empleo of the Junta de Andalucía.