Three dimensional elastic beam frames: rigid joint conditions in variational and differential formulation

TL;DR

This paper develops a mathematical framework for three-dimensional elastic beam frames with rigid joints, analyzing the self-adjoint differential operators and their symmetry-based decompositions to understand oscillation modes.

Contribution

It introduces a method to generate joint conditions from geometric descriptions and analyzes the operator's decomposition in symmetric frames, including special planar cases.

Findings

Differential operator is self-adjoint.

Operator decomposes into simpler parts in planar frames.

Symmetry-based decomposition captures specific oscillation modes.

Abstract

We consider three-dimensional elastic frames constructed out of Euler--Bernoulli beams and describe a simple process of generating joint conditions out of the geometric description of the frame. The corresponding differential operator is shown to be self-adjoint. In the special case of planar frames, the operator decomposes into a direct sum of two operators, one coupling out-of-plane displacement to angular (torsional) displacement and the other coupling in-plane displacement with axial displacement (compression). Detailed analysis of two examples is presented. We actively exploit the symmetry present in the examples and decompose the operator by restricting it onto reducing subspaces corresponding to irreducible representations of the symmetry group. These ``quotient'' operators are shown to capture particular oscillation modes of the frame.