A closed-form model for nonlinear spatial deflections of rectangular beams in intermediate range

Modeling the nonlinear load-displacement relations for flexible beams has been a key objective in compliant mechanisms research. There have been several practically useful methods for modeling planar deflections, but less work has been done in modeling spatial deflections. This work proposes the load-displacement relations for rectangular beams by solving the nonlinear governing differential equations of the beams using the power series method and then simplifying the solution by Taylor series expansion and truncation. The solution is validated to be accurate by comparing with two commercial finite element software packages, ANSYS and Abaqus. This comparison shows that this approach is capable of capturing the relevant geometric nonlinearities in the intermediate deflection range defined as 10% of the beam length. The load-displacement relation offers a useful and parameterized tool for understanding the constraint (i.e. stiffness and motion) behavior of rectangular cross-section beams and generating compliant mechanism designs with nonlinear kinetostatic behaviors.