On the geometry of load paths

Abstract

This thesis develops a novel framework for defining and analyzing structural load paths using variational principles. Load paths describe the internal transmission of forces within a structure and are fundamental to understanding structural behavior, optimization, and design. Traditional optimization methods like SIMP and level set techniques typically rely on stress approximations, often yielding impractical or non-intuitive results. However, approximations based on internal loads have proven to be more accurate and may illustrate more meaningful approaches. This work formulates load paths as geodesics governed by the stress field, drawing analogies from differential geometry. Two types of formulation are discussed, the Lagrangian and the Hamiltonian. The Lagrangian formulation interprets load paths as curves that minimize a Lagrangian functional, analogous to minimizing the action in mechanics, while the Hamiltonian formulation introduces complementary insights via energy conservation and contravariant stress tensors. First, examples are given for each system separately, while application to an ESAVE wing structure is used to demonstrate how these load path formulations can guide structural design. Numerical solutions are obtained using finite element analysis and differential equation solvers, followed by design optimization using MSC.Nastran. The study also draws analogies with the Schwarzschild metric to further interpret geodesic behavior in structural systems. Ultimately, this approach offers a physically intuitive and computationally tractable means to identify meaningful load paths, potentially enhancing structural optimization methods and practical design workflows.