Geometric nonlinear analysis of tensioned membranes using the Positional Finite Element Method

Resumo

This paper describes a total Lagrangian formulation of the Finite Element Method based on positions
and its application to the analysis of tensioned membranes. In these structures, the geometric nonlinearity is very
pronounced due to large displacements and the lack of flexural stiffness of the elements; therefore, the equilibrium
must be evaluated at the current configuration and the geometric stiffness plays an important role in the analysis.
The use of nodal positions as main variables allows a direct consideration of the geometric nonlinearity. For the
positional description of membrane elements, two mappings are employed: one for the initial configuration and
other for the current configuration, resulting in a simple chain rule to calculate the deformation gradient. These
mappings are defined in such a way that results in square and invertible mapping gradients for the membrane
element in three-dimensional space. In the form finding stage, the dynamic relaxation technique is employed
to find an initial prestressed configuration for further loaded analysis. Numerical examples for isotropic fabrics
are presented to demonstrate the applicability of the positional formulation in the assessment of stiffness and
displacements in this category of problems.