Nonsmooth geometry and collapse of flexible structures under smooth loads

TL;DR

This paper investigates how combined smooth loads can cause non-smooth behaviors and collapse in flexible structures, revealing critical geometries and collapse mechanisms through analysis, simulation, and experiments.

Contribution

It demonstrates the occurrence of non-smooth solutions in flexible strings under combined loads and identifies critical collapse geometries, extending understanding beyond straight cable optimization.

Findings

Non-smooth solutions occur under certain load ratios.

Two critical collapse geometries are identified.

Analytical results are validated by simulations and experiments.

Abstract

While static equilibria of flexible strings subject to various load types (gravity, hydrostatic pressure, Newtonian wind) is well understood textbook material, the combinations of the very same loads can give rise to complex spatial behaviour at the core of which is the unilateral material constraint prohibiting compressive loads. While the effects of such constraints have been explored in optimisation problems involving straight cables, the geometric complexity of physical configurations has not yet been addressed. Here we show that flexible strings subject to combined smooth loads may not have smooth solutions in certain ranges of the load ratios. This non-smooth phenomenon is closely related to the collapse geometry of inflated tents. After proving the nonexistence of smooth solutions for a broad family of loadings we identify two alternative, critical geometries immediately preceding the collapse. We verify these analytical results by dynamical simulation of flexible chains as well as with simple table-top experiments with an inflated membrane.