3. Euler–Bernoulli Beams and Frames

This chapter treats the simple or Euler–Bernoulli beam member. This beam theory, also called the shear-rigid theory, assumes that the shear forces do not contribute to the beam deflection. The three basic equations of continuum mechanics, i.e. the kinematics relationship, the constitutive law and the equilibrium equation, are summarized as well as the describing partial differential equation. A general solution for constant material and geometrical properties is presented. Furthermore, typical boundary conditions and the internal reactions are briefly mentioned. The finite element formulation is focused on beam elements with two nodes under the assumption of constant material and geometrical properties. Furthermore, the beam element is superimposed with a rod element to form a generalized beam element, which considers both elongation and deflection. The post-computation, which is based on the nodal results, is treated in detail. The chapter concludes with the spatial arrangement of beam elements in a plane to form a plane frame structure.