By: Piotr Szablewski

Technical University of Łódź
Department of Technical Mechanics and Informatics
ul. Żeromskiego 116, 90-543 Łódź, Poland
E-mail: piotr.szablewski@p.lodz.pl

Abstract

This paper deals with a specific method of examine the state of equilibrium of a flat textile structure. This kind of structure is modelled as an inextensible elastica loaded with its dead weight and axial force. The elastica represents, as an example, a longitudinal section of a fabric. It is assumed that the elastica rests on a flat, immovable base. We considered only those forms of deformed elastica where it’s the two ends were supported by pivot bearings, and the tangent at those points lay on the immovable supporting plane. In the analysis, the shape of the deflection curve was determined for a given axial force, and it was examined whether a given position is stable or unstable. The analysis was carried out on the basis of the energetic method, by examining potential energy of the system. The investigations can be used for simulation of fabric buckling, folding and another applications of textile mechanics.

Keywords

stability, fabrics, textile mechanics, elastica, deflection curve, states of equilibrium

1. Assumptions of model and initial equations

Let us consider a flat fabric resting on a fixed, immovable base (Figure 1). Under the action of compressive forces, folds arise on its surface, which remain there due to the occurrence of friction forces.

Depending on the friction force quantity, those folds will remain or disappear after the action of deforming forces. In order to enable a thorough examination of stability of the deformed fabric, a substitute model was assumed which was limited to its deformed shape, i.e. to the fold. To generalize our further considerations, the term "elastica" in place of "fabric" is introduced hereafter.

Let heavy elastica be loaded with the axial force P and continuous load q in the coordinate system as in Figure 2. The elastica rests on a flat, fixed base and is supported on both ends by pivot bearings. It is inextensible so it cannot change its length l under the influence of loads acting on it.

However, it is subject to Hooke's law while being bent, and the known relation for the bending moment M is applicable to it