Bridging a gap: A heavy elastica between point supports

TL;DR

This work analyzes a heavy elastica suspended on two point supports under a uniform body force, first in the frictionless case and then including Coulomb friction. Using asymptotic (outer/inner) analysis and numerical boundary-value problem solutions, the authors derive a regime diagram in the load–gap plane and show that a critical half-gap $a_{\infty}=1/e$ exists below which the beam can withstand arbitrarily large loads without slipping; for larger gaps, a finite maximum load $f_{\max}$ emerges. Friction raises $f_{\max}$ and shifts $a_{\infty}$, with inner-outer analyses revealing how contact-layer dynamics near the supports govern stability. Experiments with slender rods validate the qualitative shape transitions and the existence of the critical regime, supporting implications for filtration where fiber retention depends on pore size relative to fiber length. Overall, the paper advances understanding of load-bearing in point-supported elastic systems and provides a framework for predicting failure loads in practical fiber filtration contexts, including frictional effects.

Abstract

We study the deformation and slip-through of a heavy elastic beam suspended above two point supports and subject to an increasing body force -- an idealized model of a fibre trapped in the pores of a filter as flow strength increases, for example. Using both asymptotic and numerical techniques, we investigate the behaviour of the beam under increasing body force and the maximum force that can be supported before it must slip between the supports. We quantify this maximum body force as a function of the separation between the two supports. Surprisingly, we show the existence of a critical separation below which the beam can withstand an arbitrarily large body force, even in the absence of friction. This is understood as the limit of a catenary between the supports that is connected to (and supported by the tension in) a vertically hanging portion outside the supports. We explore how frictional forces impact the deformation and load-bearing capacity of the beam and show that our results are consistent with laboratory experiments.

Figures (11)

• Figure 1: Sketch of the setup for the point-supported heavy elastica (PSHE) of length $2L$ under the action of a body force $\hat{f}$ per unit length and supported by points located a distance $2\hat{a}$ apart.
• Figure 2: Plots of the PSHE shape for various values of the half-gap, $a$, and increasing force, $f$. The dashed curves represent the small-deformation shape, $y(x)/f$, or (equivalently) the shape predicted with $f=1$ by the small-deformation theory. Solid curves show the numerically-computed shapes for: (a) $a=0.8$, $\log_{10}f=\{-2,-1,0\}$. (b) $a=0.2$, $\log_{10}f~=~\{0,1,2,3,4,5\}$. The different shapes illustrated are referred to as '$u$'-shape (as in (a)) or '$m$' and '$n$'-shape (as in (b)).
• Figure 3: Plots of the force, $f$, against the contact arc length, $s^*$, for half-gaps $a~=~\{0.2,0.4,0.5,0.6,0.8,0.9185,0.95\}$.
• Figure 4: (a) Sketch of the shape of the PSHE near the supports in the limit $f \gg 1$. (b) Plots of the numerically determined tension, $t(s)/f$, as a function of arc length $s$ for half-gap $a~=~0.36$ (solid curves), for $\log_{10}(f)~=~\{4, 5, 6, 7\}$, alongside $T(s)$ from the outer solution (\ref{['eqn2:t_outer_c']}) and (\ref{['eqn2:t_outer_o']}) (dashed curve).
• Figure 5: (a) Plot of the contact arc length, $s^*$, in the outer solution, (\ref{['eqn2:arc_length_cat']}), against half-gap, $a \leq 1/\mathrm{e}$, (\ref{['eqn2:a_inner_sol']}). (b) Plot of the (rescaled) potential energy, $U_g/f$, (\ref{['eqn2:inner_sol_energy']}), against half-gap, $a$, (\ref{['eqn2:a_inner_sol']}). The solid curves represent the branch of solutions with $s^*< s^*(a_{\infty})$, and dashed curves those with $s^*>s^*(a_{\infty})$. The limiting point $a=1/\mathrm{e}$ is shown by the vertical dotted line.