Grid-beam mechanics: from elastica to Timoshenko

The shape of a bent slender beam is classically governed by the Euler–Bernoulli elastica equation. In contrast, thick beams experience shear deformations, as described by Timoshenko. Here, we describe the mechanical properties of a novel class of elastic structures, grid-beams, featuring a shearable internal architecture. These beams consist of an array of long, parallel ribbons regularly interconnected by transverse columns. Depending on the design of the junctions between the longitudinal ribbons and the columns, the structure can accommodate shear up to a tunable shear angle. Beyond this threshold, further shear generates a restoring torque. As a result, the grid-beam exhibits unique bending behavior. For instance, the curvature of a sagging grid-beam assumes a bell-shaped profile, characterized by a striking change in curvature. While geometric constraints require that the columns and ribbons remain mutually parallel, the orientation of the columns introduces an additional degree of freedom. To model this behavior, we propose a modified elastica equation that incorporates internal shear within the beam. When column orientation is unconstrained, the model must include an internal torque balance to fully define the system. Experimental realizations with various configurations are compared to theoretical predictions. Moreover, the kinematics of the grid-beams can be actively controlled by dynamically varying the boundary conditions, suggesting potential applications as adaptive actuators.