The need for quick and easy deflection calculations of various prefabricated slabs causes simplified procedures and numerical tools to be used more often. Modelling of full 3D finite element (FE) geometry of such plates is not only uneconomical but often requires the use of complex software and advanced numerical knowledge. Therefore, numerical homogenization is an excellent tool, which can be easily employed to simplify a model, especially when accurate modelling is not necessary. Homogenization allows for simplifying a computational model and replacing a complicated composite structure with a homogeneous plate. Here, a numerical homogenization method based on strain energy equivalence is derived. Based on the method proposed, the structure of the prefabricated concrete slabs reinforced with steel spatial trusses is homogenized to a single plate element with an effective stiffness. There is a complete equivalence between the full 3D FE model built with solid elements combined with truss structural elements and the simplified homogenized plate FE model. The method allows for the correct homogenization of any complex composite structures made of both solid and structural elements, without the need to perform advanced numerical analyses. The only requirement is a correctly formulated stiffness matrix of a representative volume element (RVE) and appropriate formulation of the transformation between kinematic constrains on the RVE boundary and generalized strains.

Sandwich structures are widely used in construction, as well as in the aviation, spaceship, and electronics industries. The interesting result, among others, is the fact that individual layers can be freely selected to meet the planned requirements. In the case of sandwich structures in construction, they must meet the requirements of load-bearing capacity, thermal, and acoustic insulation, and additionally, they must be resistant to biological and chemical corrosion. The paper presents calculation algorithms for Hoff’s three-layer panels. In the first case, the well-known and proven method of finite differences in variation terms was used, assuming actual geometrical and material parameters. In the second case, the numerical homogenization method of the layered panel was used, replacing the stiffnesses of individual layers with a homogeneous equivalent plate with substitute stiffness corrected in shearing by an analytically derived shear correction factor. A comparative analysis of the results of the calculations with the use of both approaches was carried out. A good agreement between the displacement values and the calculated cross-sectional forces was obtained. On this basis, it can be assumed that the static analysis of a slab by simplified methods using numerical homogenization with an analytical shear correction factor is appropriate and can be applied to layer structures.

As long as non-contact digital printing remains an uncommon standard in the corrugated packaging industry, corrugated board crushing remains a real issue that affects the load capacity of boxes. Crushing mainly occurs during the converting of corrugated board (e.g., analog flexographic printing or laminating) and is a process that cannot be avoided. However, as this study shows, it can be controlled. In this work, extended laboratory tests were carried out on the crushing of double-walled corrugated board. The influence of fully controlled crushing (with a precision of ±10 μm) in the range from 10 to 70% on different laboratory measurements was checked. The typical mechanical tests—i.e., edge crush test, four-point bending test, shear stiffness test, torsional stiffness test, etc.—were performed on reference and crushed specimens. The residual thickness reduction of the crushed samples was also controlled. All empirical observations and performed measurements were the basis for building an analytical model of crushed corrugated board. The proven and verified model was then used to study the crushing effect of the selected corrugated board on the efficiency of simple packages with various dimensions. The proposed measurement technique was successfully used to precisely estimate and thus control the crushing of corrugated board, while the proposed numerical and analytical techniques was used to estimate the load capacity of corrugated board packaging. A good correlation between the measured reduced stiffness of the corrugated cardboard and the proposed analytical predictive models was obtained.

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.

The production of thin-walled beams with various cross-sections is increasingly automated and digitized. This allows producing complicated cross-section shapes with a very high precision. Thus, a new opportunity has appeared to optimize these types of products. The optimized parameters are not only the lengths of the individual sections of the cross section, but also the bending angles and openings along the beam length. The simultaneous maximization of the compressive, bending and shear stiffness as well as the minimization of the production cost or the weight of the element makes the problem a multi-criteria issue. The paper proposes a complete procedure for optimizing various open sections of thin-walled beam with different openings along its length. The procedure is based on the developed algorithms for traditional and soft computing optimization as well as the original numerical homogenization method. Although the work uses the finite element method (FEM), no computational stress analyses are required, i.e., solving the system of equations, except for building a full stiffness matrix of the optimized element. The shell-to-beam homogenization procedure used is based on equivalence strain energy between the full 3D representative volume element (RVE) and its beam representation. The proposed procedure allows for quick optimization of any open sections of thin-walled beams in a few simple steps. The procedure can be easily implemented in any development environment, for instance in MATLAB, as it was done in this paper.

The use of layered or hollow floors in the construction of buildings obviously reduces the self-weight of the slab, and their design requires some expertise. In the present work, a sensitivity analysis and numerical homogenization were used to select the most important characteristics of bubble deck floors that have a direct or indirect impact on their load capacity. From the extensive case study, conclusions were drawn regarding the optimal selection of geometry, materials, and the arrangement and size of air voids in such a way as to ensure high stiffness of the cross-section and at the same time maximally reduce the self-weight of the slabs. The conducted analyses showed that the height of the slab and the geometry of the voids had the greatest impact on the load-bearing capacity. The concrete class and reinforcement used are of secondary importance in the context of changes in load-bearing capacity. Both the type of steel and the amount of reinforcement has a rather small or negligible influence on the bubble deck stab stiffness. Of course, the geometry of the voids and their arrangement and shape have the greatest influence on the drop in the self-weight of the floor slabs. Based on the presented results of the sensitivity analysis combined with numerical homogenization, a set of the most important design parameters was ordered and selected for use in the optimization procedure.