Introduction to Finite Element Analysis and Design, 2nd Edition

Chapter 3Finite Element Analysis of Beams and Frames

In chapter 1, the finite element equations of a truss were obtained using the direct stiffness method. Similar direct methods for beams are possible but quite complicated, and such methods are impossible for plates and two‐dimensional and three‐dimensional solids. In chapter 2, we introduced the Galerkin method and the principle of minimum potential energy for different engineering problems. In this chapter, we will formally derive the finite element equations for beams using the energy method. The same finite element equation can be obtained using the principle of virtual work.¹

In chapter 2, we learned that in the finite element method, the displacements in an element are interpolated using an expression of the form , in which {N(x)} is the column vector of shape functions, and {q} is the vector of nodal displacements or in general nodal degrees of freedom (DOFs). In the case of beam finite element, the nodal DOFs include the vertical (or transverse) deflection as well as the rotation (or slope). Using this interpolation scheme, the stiffness matrix and applied load vector are derived and solved for the nodal DOFs.

After a review of the elementary beam theory in section 3.1, we will first present the Rayleigh‐Ritz method in section 3.2. The formal development of the interpolation functions for the beam finite elements is ...