Finite Element Method for Solids and Structures A Concise Approach

In this chapter we consider the finite element formulation for bending of slender bodies under a tensile or compressive axial force. In order to capture the effect of axial force we look at the force and moment equilibrium in the deformed configuration, but still assuming small translational displacement and rotation of the cross-section. In the finite element formulation, it is shown that the effect of axial force on bending manifests as an effective bending stiffness. It will be shown that the finite element formulation of a slender body under compressive axial force results in a matrix equation for eigenvalue analysis from which we can determine the static buckling load and the buckling mode. Subsequently, we consider the finite element formulation for vibration analysis of slender bodies to investigate the effect of axial force on the natural frequencies and modes. Finally, we introduce the finite element formulation of slender bodies subjected to a compressive follower force in which the direction of the applied force is always parallel to the body axis in the deformed configuration.