Shell-to-Beam Numerical Homogenization of 3D Thin-Walled Perforated Beams

Abstract

Determining the geometric characteristics of even complex cross-sections of steel beams is not a major challenge nowadays. The problem arises when openings of various shapes and sizes appear at more or less regular intervals along the length of the beam. Such alternations cause the beam to have different stiffnesses along its length. It has different bending and shear stiffnesses at the opening point and in the full section. In this paper, we present a very convenient and easy-to-implement method of determining the equivalent stiffness of a beam with any cross-section (open or closed) and with any system of holes along its length. The presented method uses the principles of the finite element method (FEM), but does not require any formal analysis, i.e., solving the system of equations. All that is needed is a global stiffness matrix of the representative volumetric element (RVE) of the 3D representation of a beam modeled with shell finite elements. The proposed shell-to-beam homogenization procedure is based on the strain energy equivalence, and allows for precise and quick determination of all equivalent stiffnesses of a beam (flexural and shear). The results of the numerical homogenization procedure were compared with the existing analytical solution and experimental results of various sections. It has been shown that the results obtained are comparable with the reference results.

Parametric Optimization of Thin-Walled 3D Beams with Perforation Based on Homogenization and Soft Computing

“…Therefore, it is greatly desirable to use the numerical techniques which are able to reduce the number of problem degrees of freedom while maintaining the accuracy of modelling. Such property can be achieved by adopting the numerical technique proposed in Staszak et al [ 18 ].…”

“…In this paper, the shell-to-beam homogenization technique [ 18 ] was used as part of an optimization problem in structural engineering to model the representative volume element (RVE) stiffness of thin-walled 3D perforated beams. The general question considered in the study was: what should a cross-section design of a thin-walled 3D perforated beam from a Z, C or Σ profile be for a simply supported beam with an evenly distributed load to achieve a specific load capacity due to bending moment and maximum displacement.…”

“…The parametrized models were used to build the representative volume element (required in the numerical homogenization technique [ 18 ]) for a particular set of parameters used in the optimization framework. The technique is based on the finite element method approach, but does not require formal computational stress analyses, i.e., solving a system of equations.…”

“…Consequently, many homogenization techniques have emerged. There are homogenization methods based on the deformation energy [ 11 ], the periodic homogenization [ 12 , 13 ] or based on the equivalence of the deformation energy [ 14 , 15 , 16 , 17 , 18 ]. The paper [ 17 ] presents a method of determining the stiffness of perforated corrugated board, which is based on the homogenization method presented by Garbowski and Gajewski [ 15 ].…”

“…It is a universal method used not only for cardboard homogenization, but also for engineering structures. For instance, it was used for the homogenization of the cross-section of prefabricated Filigran slabs in the work of Staszak et al [ 16 ] and for thin-walled profiles with holes [ 18 ]. On the other hand, in [ 19 ] the method based on the matrix eigenvectors and the main vectors of the state transfer matrix for the lattice beam-like structure was used.…”