A unified beam theory with strain gradient effect and the von Kármán nonlinearity

Abstract

In this paper, the governing equations and finite element formulations for a microstructure‐dependent unified beam theory with the von Kármán nonlinearity are developed. The unified beam theory includes the three familiar beam theories (namely, Euler‐Bernoulli beam theory, Timoshenko beam theory, and third‐order Reddy beam theory) as special cases. The unified beam formulation can be used to facilitate the development of general finite element codes for different beam theories. Nonlocal size‐dependent properties are introduced through classical strain gradient theories. The von Kármán nonlinearity which accounts for the coupling between extensional and bending responses in beams with moderately large rotations but small strains is included. Equations for each beam theory can be deduced by setting the values of certain parameters. Newton's iterative scheme is used to solve the resulting nonlinear set of finite element equations. The numerical results show that both the strain gradient theory and the von Kármán nonlinearity have a stiffening effect, and therefore, reduce the displacements. The influence is more prominent in thin beams when compared to thick beams.