On Mathematic Modeling of Elastic Beam

The traditional model of beams and the Cosserat model of elastic rods are compared in this paper as two different modeling approaches for one-dimensional elastic body. The three-dimensional motion of a beam is analyzed considering all influence factors, such as the tensile or compressed deformation of the centerline, the shear deformation and the inertia of rotation of the cross section, and the loads on the ends. The stability conditions and free frequency of lateral vibration of a beam are derived on the basis of exact Cosserat model. The Bernoulli beam, Timoshenko beam, and Kirchhoff rod are simplifications of exact Cosserat model under different conditions.