# ## Plane Beam Approximations: Introduction

Many of the large standing towers, bridges, and buildings owe their existence to some simplified versions of the equilibrium equations applied to slender, line-like structures. The first beam approximation is represented in the Euler Bernoulli’s beam theory, which originated sometime in the eighteenth century. This approximation has been so successful that it is still the basic approximation used in modern structural analysis taught in current structural analysis courses. In this chapter, three different beam approximations are presented. The first is the the Euler Bernoulli’s beam approximation which relies on assuming plane sections perpendicular to the neutral axis of the beam remain plane and perpendicular to the neutral axis after deformation. The second beam approximation is the Timoshenko’s beam approximation which assumes that the cross sections perpendicular to the neutral axis before deformation stay plane, but not necessarily perpendicular to the neutral axis after deformation. These beams are termed: “shear flexible” and because they allow more deformation, the model predicts slightly more deformation than the Euler Bernoulli beam model. The third beam approximation is the beams under axial loading. All these models are presented under the umbrella of small deformations and for isotropic linear elastic material models.