In this work we consider solutions for the EulerBernoulli and Timoshenko theories of beams in which material behavior may be elastic or inelastic. The formulation relies on the integration of the local constitutive equation over the beam cross section to develop the relations for beam resultants. For this case we include axial, bending and shear effects. This permits consideration in a direct manner of elastic and inelastic behavior with or without shear deformation.A finite element solution method is presented from a three-field variational form based on an extension of the Hu-Washizu principle to permit inelastic material behavior. The approximation for beams uses equilibrium satisfying axial force and bending moments in each element combined with discontinuous strain approximations. Shear forces are computed as derivative of bending moment and, thus, also satisfy equilibrium. For quasi-static applications no interpolation is needed for the displacement fields, these are merely expressed in terms of nodal values. The development results in a straight forward, variationally consistent formulation which shares all the properties of so-called flexibility methods. Moreover, the approach leads to a shear deformable formulation which is free of locking effects -identical to the behavior of flexibility based elements.The advantages of the approach are illustrated with a few numerical examples. IntroductionThe development of computational models for beam bending problems dates from the earliest days of structural analysis and the literature is too extensive to fully cite here. Mike Crisfield considered solution of beam problems from many perspectives as indicated in approaches contained in his books [1-3] and papers with co-workers [4][5][6][7][8]. Most of his work was for finite displacement applications -many using co-rotational formulations for which he was well known. Here we would like to remember him for his pioneering work in this field of endeavor.In a displacement formulation, the nonlinear straindisplacement relations are postulated and polynomial interpolation functions are used for the displacement approximation [9][10][11]. Because the postulated displacement interpolation functions are approximate in nonlinear material and geometric behavior, each structural member needs to be discretized into several elements in order to capture the actual variation of deformations along its axis. This fine discretization results in a large number of degrees of freedom in the final numerical model of the structure, thus, reducing the computational efficiency of this approach. Alternatively, higher order displacement interpolation functions can be used. This approach results in several internal degrees of freedom that need to be condensed out during the element state determination. Even so, mesh discretization is often required for accuracy. With both of these approaches numerical instabilities are not uncommon, particularly under cyclic loading conditions. Problems with the displacement formulation of beam elements encou...

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