Optimal Design of Fibre Reinforced Membrane Structures

TL;DR

This paper presents a novel approach for designing optimally stiff fibre-reinforced membrane structures by formulating a bi-level optimization problem, deriving a membrane model from 3D elasticity, and solving it with finite element methods.

Contribution

It introduces a new membrane modeling approach from 3D elasticity and formulates the optimal design as a bi-level minimization problem with convex fiber density optimization.

Findings

Numerical solutions for statically determined and undetermined structures.

Optimal fiber orientation aligns with principal stresses.

The fiber density optimization problem is convex and solvable via optimality criteria.

Abstract

A design problem of finding an optimally stiff membrane structure by selecting one-dimensional fiber reinforcements is formulated and solved. The membrane model is derived in a novel manner from a particular three-dimensional linear elastic orthotropic model by appropriate assumptions. The design problem is given in the form of two minimization statements, reminiscent of a Nash game. After finite element discretization, the separate treatment of each of the two minimization statements follows from classical results and methods of structural optimization: the stiffest orientation of reinforcing fibers coincides with principal stresses and the separate selection of density of fibers is a convex problem that can be solved by optimality criteria iterations. Numerical solutions are shown for two particular configurations. The first for a statically determined structure and the second for a statically undetermined one. The latter shows related but non-unique solutions.