An accurate and refined beam model fulfilling the shear and the normal stress traction condition

Abstract

This paper presents an analytical refined beam model taking into account the axial, the shear and the transverse normal stress distributions. Based on the equilibrium equations in local form and an iterative procedure for the solution of the bipotential equation, the beam differential equations for the axial and the transverse motion are derived. From a mathematical point of view the formal structure of the obtained differential equations is identical to the Timoshenko beam equations but with increased accuracy. This is achieved by splitting the ansatzfunctions for the displacement field into three weight-averaged degrees of freedom (for the axial and the transverse deflection and the rotation) and residual deflections, so-called higher order terms which depend on the state of stress. It is shown that the shear traction condition is always fulfilled and even the normal traction condition may hold. Furthermore an analogy to the Timoshenko beam equations is given and it is shown that the shear correction coefficient depends on Poisson’s ratio. Finally analytical results of the refined beam theory are compared to several analytical beam theories and two-dimensional finite element results for statically determinate and indeterminate beam configurations.