Beaded catenary

We study gravity-driven sag in a “slack” beaded rod (a necklace) where finite-size beads can freely slide along an inextensible thread. In the wide-span, low-friction limit with effectively uniform weight per length $w$ , the centerline follows the classic catenary $y = a \cosh{x/b}$ , with $a = 1/b$ . As the span narrows, however, ‘beadiness’, or the fact that mass is carried by mobile, finite-size beads rather than a continuous filament, drives systematic departures from the classic shape.

Beaded catenary with a wide span.

For narrower spans the necklace apex is more round, i.e. not as pointed. For a better fit, we can switch to a more general model, where $a \neq 1/b$ . This formulation is known as the ‘flattened’, ‘transformed’, or ‘two-nosed’ catenary, and is used to describe the minimal shape of structures under non-uniform vertical loads.

Identical catenary as above, now with a narrower span. Lines are fits for classic form (yellow) and “two-nosed” form (blue).

For even narrower spans, the flattened catenary fails to describe the necklace shape. Experimentally, the profile transitions from a “U” to a tear-drop form characteristic of an elastic rod. We therefore use an elastic-catenary (heavy-elastica) model that couples bending and gravity through a stiffness parameter $K$ . Fitting $K$ to our profiles shows an increase in the effective stiffness with bead size. Interpreting for beaded assemblies is the next step. Two features distinguish this system from the continuum limit:

Beads can slide on the thread, allowing mass to redistribute along the span
Finite curvature introduces bead–thread contact and associated friction.

We observe small, systematic deviations from the smooth heavy-elastica fit at the shortest spans and treat $K$ as an effective parameter here; developing a micro–to–macro map from bead radius/shape, thread properties, and friction to $K$ is left for future work.

Beaded catenaries fitted with different catenary models. First image: classic (yellow) and generalized “flattened” form (blue). Second image: elastic catenaries fitted with different stiffness parameter K for three different bead radii.

As a further probe of the influence of beads in catenary structures, we examine the large-radius limit. When the bead radius increases to where the cross-section approaches a flat element, our experiments no longer yield a smooth family of heavy-elastica profiles. Instead, the necklace clusters and fractures, interrupting the continuity of the centerline. After fracture, multiple mechanically admissible configurations emerge, characterized by discrete cluster sizes rather than a single continuous shape. In this regime, $K$ is no longer sufficient to organize the shapes; the outcome becomes state-dependent (history, contact geometry, and clustering), marking a qualitative departure from the continuum descriptions that succeed for round, freely sliding beads.

Catenaries made with cylindrical (flat-faced) beads.

← Back to :

Beaded metamaterials