Timoshenko beam theory with extension effect and its stiffness equation for finite rotation

The nonlinear effect of axial extension is taken into account in the elastic Timoshenko beam theories and corresponding stiffness equations for finite rotation. Elastic buckling problems of a column and a beam on an elastic foundation are solved to examine the effect. This effect enhances the critical load of a column, and plays an important role in the buckling loads of higher modes. In the formulation of theories, two constitutive models for the shear resultant force are introduced, leading to two different buckling formulas, one of which corresponds to Engesser's formula and the other to the so-called modified one. The stiffness equations for these two models with finite rotation are derived by superposition of the rigid finite rotation and substantial deformation of a finite element. In order to show the consistency between the theories and stiffness equations, the FEM solutions with finite rotation are compared with direct solutions of the differential equations by numerical integration. A few more illustrative examples are solved to ensure the potentiality of the FEM solutions with finite rotation.