A mixed finite-element finite-difference method to solve the equilibrium equations of a prestressed membrane having boundary cables

ABSTRACT

This paper proposes a mathematical method to solve the equilibrium equations of a membrane with rigid and cable boundaries for the so-called prestressing phase. More exactly, no constrained thrust network approach is implemented and the problem is studied starting directly from its continuous formulation. In this sense, the membrane is represented by a regular surface with negative Gaussian curvature, and its boundary by a set of regular curves, whose curvature depends on the structural elements, rigid or cable. The equilibrium is 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 which makes the problem more complex. Therefore, the corresponding mathematical problem is defined and analysed, and its principal qualitative aspects are discussed in this paper. Moreover, a numerical resolution procedure is proposed and applied.

KEYWORDS:

• Membrane
• rigid boundary
• cable boundary
• elliptic problem
• finite difference
• finite element

AMS SUBJECT CLASSIFICATIONS:

• 35J25
• 74G15
• 74S05
• 74S20

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author gratefully acknowledges Sardinia Regional Government for the financial support (P.O.R. Sardegna, F.S.E. 2007-2013). He is a member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). This paper is also supported by the research group FQM315-Análisis teórico y Numérico de modelos de las ciencias experimentales, of the Department of Mathematics of the University of Cadiz (Spain).