From the Euler-Bernoulli beam to the Timoshenko one through a sequence of Reddy-type shear deformable beam models of increasing order

A sequence of elastic Reddy-type shear deformable beams of increasing (odd) order is envisioned, which starts with the Euler-Bernoulli beam (first order) and terminates with the Timoshenko beam (infinite order). The kinematics of the generic beam, including the warping mode of the cross sections, is specified in terms of three deformation variables (two curvatures, one shear angle), work-conjugate of as many stress resultants (two bending moments, one shear force). The principle of virtual power is used to determine the (static) equilibrium equations and the boundary conditions. The equations relating the bending moments and shear force to the curvatures and shear angle are also reported. Two governing differential equations (equivalent to a deflection differential equation of the sixth order) are provided, along with their general solution to within six unknown constants; the solutions for the Euler-Bernoulli and Timoshenko beams are recovered for first and infinite orders, respectively. A principle of minimum total potential energy is shown to hold for the generic beam. Extended forms of the classical Navier formula for the computation of the (axial) normal stress, and of the Jourawski formula for the computation of the “equilibrium” shear stress, are provided. A simple cantilever beam is used for an illustrative application.