General finite element description for non-uniform and discontinuous beam elements

Abstract

The theory of generalized functions is used to address the static equilibrium problem of Euler-Bernoulli non-uniform and discontinuous 2-D beams. It is shown that if simple integration rules are applied, the full set of response variables due to end nodal displacements and to in-span loads can be derived, in a closed form, for most common beam profiles and arbitrary discontinuity parameters. On this basis, for finite element analysis purposes, a non-uniform and discontinuous beam element is implemented, for which the exact stiffness matrix and the fixed-end load vector are derived. Upon computing the nodal response, no numerical integration is required to build the response variables along the beam element.

Bending and vibration of a discontinuous beam with a curvic coupling under different axial forces

“…In engineering design, however, analytical or semianalytical methods are more direct and effective for assessing the influences of parameters on the bending and vibration of discontinuous beams. Failla et al [21][22][23][24][25] presented a systematic study on the deformation and vibration of a Euler-Bernoulli discontinuous beam based on Green's functions, in which closed-form solutions were obtained. However, the effects of axial loads, shearing, and gyroscopic moments were not included in their work.…”

A Model for the T-shaped Beams with Straight Haunches for Uniformly Distributed Load

“…The geometrical properties of the cross section of the beam in general form are: (6) (7) where: y g is the center of gravity of the cross section, I x is the moment of inertia at a distance "x" in function of "d x ", A sx is the shear area at a distance "x" in function of "d x ". The geometrical properties for the second cross section of the beam are: (11) 12…”

“…In these tables the bending deformations are considered only and the shear deformations are neglected, and also the length-height relationship of the beam is not considered in definition of the various stiffness factors, simplifications that can lead to significant errors in the determination of the stiffness factors. There are important contributions in the non-prismatic beams analysis that are based on the theory of Euler-Bernoulli beams as Just [2], Schreyer [3], Medwadowski [4], Brown [5], Banerjee and Williams [6], Eisenberger [7,8], Tena-Colunga [9], Failla and Santini [10], Failla and Impollonia [11]. The new design aids are found in appendix B of the book of "Analysis of structures with matrix methods" that replace to the old PCA tables, and the tables provided the fixed-end moments, carryover, and stiffness factors for beams of sections in shape of "I" and "T" [12].…”

Structural analysis of non-prismatic beams: Critical issues, accurate stress recovery, and analytical definition of the Finite Element (FE) stiffness matrix

“…Failla and Impollonia [29] developed a method based on the theory of generalized functions to solve non-uniform and discontinuous beams in static analysis, resulting particularly useful in sensitivity, damage identification, and optimization. Trinh and Gan [67] derived shape functions for a linearly tapered Timoshenko solid beam element starting from the Hamilton principle.…”