Abstract
The
aim of this paper is to examine the state of equilibrium of a flat textile
structure loaded by compression forces and deadweight. Only those forms of
deformed structure were considered where its two ends were supported by pivot
bearings, and lie on the immovable supporting plane. Such structure may be a
flat textile structure (e.g. fabric) and above-mentioned boundary conditions
result directly from behavior under specified loading. In the analysis, shape
of the deflection curve was determined for a given axial force and deadweight,
and it was examined whether a given position is stable or unstable. Two shape
parameters occurring in the analysis are used for simulation of different
shapes of bending curve in the middle of compression. The analysis was made on
the basis of the energetic method, by examining potential energy of the system.
Results may be used for different algorithms and programs for simulation of
fabric buckling, folding and for another application of textile mechanics.
1. Assumptions
of model and initial equations
Let
us consider a flat textile structure of length l as its longitudinal
section loaded by compression force P and deadweight q as in
Figure 1. The structure lies on immovable supporting plane and is
supported on both ends by pivot bearings in points A and B. However, it is
subject to Hooke's law while being bent, and the known relation for the bending
moment 
is
applicable to it, where is the radius of curvature, and EI means the
bending rigidity. In this case, the existence of the rigid base causes the
limitation of the y coordinate. It must be greater or equal to zero for
each value of the arc coordinate s, which is measured along the
deflection curve (
).The
boundary conditions for this load scheme are the following:
Let
us consider the infinitesimal section of the structure presented in
Figure 2. The structure is inextensible, thus
. Therefore we obtain the following
geometrical condition
.
Writing the elementary equations of equilibrium for section from Figure 3,
next multiplying by the appropriate virtual displacements x, y and, adding the sides and integrating within the limits from
0 to l, we obtain
