Semi-analytical finite elements in the higher-order theory of beams

The paper presents semi-analytical finite elements leading to the development of a higher-order theory for beams. An analysis of existing beam theories reveals their semi-analytical character. The semi-analytical finite element (SFE) method proposed assumes the cross-section of a solid beam to be an assemblage of certain semi-analytical elements. The SFE method is based on independent approximations of three-dimensional fields of displacements and stresses. The method reflects the multi-level nature of beams where conventional finite elements are transferred to the level of beam equations for heuristic considerations. A mathematical formulation of the linear theory of a solid beam involves the equations of compatibility, equilibrium and constitutive relationships, i.e. the elastic law as well as static and kinematic boundary conditions. Compatibility conditions are derived directly by using finite element approximations, while equilibrium equations, constitutive equations and boundary conditions may be derived by applying stationarity of a modified Hellinger-Reissner functional. This derivation may be also defined by straightforward application of a generalised virtual work formulation. The equations of a beam are formulated in terms of generalised variables-integral resultants of stresses and strains as well as of nodal displacements. The definition of generalised variables is assigned to the nodes of the finite element mesh. Bending, warping and other higher-order deformation modes are described automatically. Finally, the set of governing equations includes mixed algebraic-differential operators. Their explicit expressions depend on one of the element types-isoparametric, subparametric or superparametric. The versatility of SFE is demonstrated in practical applications involving well-known solutions. The classical Euler-Bernoulli and Timoshenko theories are derived using a single one-dimensional element. The beams with a thin-walled cross-section are described as an assemblage of semi-analytical elements. On the basis of a higher-order theory, the conventional semi-analytically based finite elements (SABFE) are derived. Their basic relations are also presented. Some numerical examples of the applications to thin-walled beams illustrate the basic features. Possible extensions of SFE to dynamics, geometric non-linearity and shells are briefly discussed.