3D Numerical Cross-Section Analysis of a Tapered Beam Slice

Cross-section analysis is an important tool used to recover stresses and strains in a structure at specific cross-sections of arbitrary geometries, without the need for a full 3D model. This is particularly essential for large-scale structures such as aircrafts, wind turbine blades, etc. where making a full model can be computationally very expensive or impractical. The majority of currently available cross-section analysis frameworks are based on stepwise prismatic assumptions, which are hardly suited for the analysis of tapered beams. In fact, high-fidelity stress analysis obtained from analytical and full 3D models shows that predictions of stepwise prismatic approximations can significantly deviate from the correct solution of tapered beams. In this work, a prismatic 3D cross-section analysis method is extended to analyze a symmetrically tapered finite cross-section slice. In this study, the cross-section slice is discretized with 8-node and 20-node solid elements. The boundary conditions are applied as six constraint equations via the Lagrange multiplier method. The external nodal forces acting on the cross-section faces are obtained from the equivalent tractions induced by the cross-section forces. The developed numerical model is validated against the exact analytical solutions of a wedge as well as commercial finite element (FE) software COMSOL and it is shown that the numerically predicted displacement and stress fields agree well with those provided by the wedge's analytical solution and the FE COMSOL results. This work contributes to the advancement of high-fidelity numerical tapered cross-section analysis methods with an application for many engineering structures.