Classical Plane Linearized Beam Theories

Abstract

We speak of a classical plane linearized beam as being a classical linearized beam fulfilling the following assumptions. The cross section geometry remains the same for all cross sections, the motion is restricted to a plane, the reference configuration is straight and the material of the continuous body is described by a linear elastic material law. These assumptions on the motion of the beam and material law enable us to formulate statements which are not easily accessible for a more general configuration of a beam. One key point is that we are able to arrive at a fully induced beam theory where the integration of the stress distributions over the cross sections can be performed analytically. Hence, we recognize relations between the generalized internal forces and the three-dimensional stress field of the Euclidean space. This allows to apply concepts of the theory of strength of materials to beams which is of vital importance to solve engineering problems. In order to achieve such a connection, we restate the generalized internal forces for the plane linearized beam. The restriction to small displacements allows us to start from the internal virtual work formulated with the linearized strain. Afterwards, we proceed in a similar way as in the previous chapters. We state the constrained position field of the beam and apply it to the virtual work which leads us consequently to the boundary value problem of the beam. Using the solutions of the boundary value problem and non-admissible virtual displacements, it is possible to access in a further step the constraint stresses of the beam which guarantee the restricted kinematics of the beam. The outline of the chapter is as follows. In Sect. 7.1 we repeat the principle of d’Alembert–Lagrange for linear elasticity and introduce an elastic constitutive law for the impressed stresses. In Sects. 7.2–7.4 the equations of motion and the plane stress distribution of the plane linearized Timoshenko, Euler–Bernoulli and Kirchhoff beam are determined.