A novel one dimensional beam model for analysis of prismatic thin-walled beams with deformable cross sections is introduced and a novel cross section mode determination procedure, which leads to the three dimensional beam displacement modes, is derived. The first order beam model for linear analysis includes: shear deformations related to both Timoshenko and Mindlin-Reissner type shear deformations, the warping effects of torsion, cross section distortion with related warping effects, as well as the Poisson effect with transverse displacements due to normal stress. The generality of the model allows it to handle open, closed and multi-cell cross sections with branched walls. The cross section analysis procedure leads to two types of beam displacement modes referred to as distortional beam modes and fundamental beam modes, with exponential and polynomial variations along the beam axis, respectively. It turns out that each of the beam deformation modes consists of a sum of one to four cross section displacement fields each with an individual axial variation. The displacement modes can facilitate the formulation of an advanced thin-walled beam element. The beam displacement modes will be illustrated for an open and a closed cross section.
Using energy principles, a thin-walled beam element is introduced for the analysis of beams with deformable crosssections that are prone to distortion. The beam element is based on previously attained semi-analytical displacement solution modes of an advanced thin-walled beam model. The first-order beam element for linear analysis handles shear deformations related to both Timoshenko and Mindlin-Reissner type deformations, warping effects of torsion, crosssection distortion including associated warping effects, as well as the transverse displacement effect from normal stress. The formulation can handle both open and closed cross-sections without special attention. The formulation of the displacement solution modes and the stiffness integration of the products of the advanced displacement modes using the Hadamard product are described. The paper also presents the transformations between modal degrees of freedom and element displacement degrees of freedom. Four examples show the beam element capabilities and good agreement with results obtained using the shell and solid elements of a commercial finite element program. The kinematic assumptions that the thin-walled beam model accommodates leads to local shear stress transfer at corners. This transfer of shear stresses is not normally seen in thin-walled beam formulations or shell models. However, the shear transfer is verified through examination of a finite element model using solid elements.
In modern structural steel frame design, the modelling of joints between beams and columns are based on very simple assumptions. The joints are most often assumed to behave as a perfect hinge or as a rigid joint. This means that in the overall static analysis relative rotations and changes in the moment curves due to joint deformations are neglected. This simplification eases the modelling but it is at the cost of losing a detailed understanding of the behaviour of the joint. This happens even though the European code has introduced the so-called component method in order to determine the rotational stiffness of a connection. Based on a modelling of any beam-to-column joint using finite shell elements and springs for single components such as bolts, it is the primary hypothesis that it is possible to formulate a generalized connection model with few degrees of freedom related to a relevant set of deformation modes. This hypothesis is based on the idea of modal decomposition performed in generalized beam theories (GBT). The question is -is it possible to formulate an eigenvalue problem with a solution corresponding to mode shapes for the deformation of the joint by using the finite element model and some type of GBT beam elements? It is believed that this is possible. The paper will address our investigations and show the progress of our research.
For decades, engineers have assessed and analysed steel frames using simple joints between beams and columns. These joints are often based on oversimplified assumptions using hinges or a direct transfer of beam displacements without any relative displacements. More seldom is the use of spring models that allow relative beam and column displacements at the joints. This despite the standardised component method approach, which can be used to determine the rotational spring stiffness of the relative rotation in a joint. This paper gives a background overview of essential developments in joint modelling and generalised thin-walled beam modelling, including torsional, distortional and related warping effects. For particular situations, some recent proposals for joint models can be applied to joints between thin-walled beams. On this basis, this paper presents a novel idea and a generic methodology that allows the interface between an extended number of generalised beam displacement modes and joints that are modelled using shell elements. The main novelty is the idea to transform from standard degrees of freedom of the interface into a reduced number of beam displacement mode related degrees of freedom. Thus, the number of degrees of freedom of the joint can be reduced to the corresponding total sum of beam modes that have been chosen for the modelling of each of the connected beam elements. The total number of degrees of freedom used for modelling the complete framework will depend on the selected number of modes in each beam element and on the number of extra internal modes chosen in the joint models. For enhanced structural analysis with advanced beam elements and joints that allow relevant distortions and built-in refined connection components, it is believed that this methodology will enable the full detailed analysis of large steel frameworks with a reasonable number of degrees of freedom.
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