An Explicit, Compact, and Rotation-Agnostic Formulation for Equivalent Nodal Loads for Point-Loaded Beams

In time-marching dynamical simulations, treatment of contact forces in deformable bodies represented by finite element meshes requires a compromise between simulation fidelity and computational costs. External forces directly evaluated at the mesh nodes offer better computational performance at the cost of modelling fidelity. Alternatively, externally applied span-wise member loads can be distributed to the mesh nodes through the element's shape functions. The shape functions enable the development of nodal forces and torques that produce consistent deformations to a load applied along the span of the element. For deformable bodies, represented using finite element methods, which undergo large gross translational and angular motions in time-marching simulations, constant reorientation of the shape function matrices is required for each evaluation of force distribution. This can be computationally expensive for larger and stiffer systems. In this work, a compact and orientation-agnostic ‘equivalent’ formulation for expressing nodal loads is presented. The formulation is compared against the consistent formulation for nodal forces of an arbitrarily oriented Euler-Bernoulli beam. Isolated benchmarking tests and sample application tests indicate nearly identical force distributions in time-marching simulations for the consistent and equivalent formulations. Evaluated computational performance metrics reveal that the equivalent formulation results in run-time performance gains. However, the significance of the gains is proportional to the fraction of computational workload associated with the force distribution.