Filters
Optimal Design of Bubble Deck Concrete Slabs: Serviceability Limit State
In engineering practice, one can often encounter issues related to optimization, where the goal is to minimize material consumption and minimize stresses or deflections of the structure. In most cases, these issues are addressed with finite element analysis software and simple optimization algorithms. However, in the case of optimization of certain structures, it is not so straightforward. An example of such constructions are bubble deck ceilings, where, in order to reduce the dead weight, air cavities are used, which are regularly arranged over the entire surface of the ceiling. In the case of these slabs, the flexural stiffness is not constant in all its cross-sections, which means that the use of structural finite elements (plate or shell) for static calculations is not possible, and therefore, the optimization process becomes more difficult. This paper presents a minimization procedure of the weight of bubble deck slabs using numerical homogenization and sequential quadratic programming with constraints. Homogenization allows for determining the effective stiffnesses of the floor, which in the next step are sequentially corrected by changing the geometrical parameters of the floor and voids in order to achieve the assumed deflection. The presented procedure allows for minimizing the use of material in a quick and effective way by automatically determining the optimal parameters describing the geometry of the bubble deck floor cross-section. For the optimal solution, the concrete weight of the bubble deck slab was reduced by about 23% in reference to the initial design, and the serviceability limit state was met.
Optimal Design of Bubble Deck Concrete Slabs: Sensitivity Analysis and Numerical Homogenization
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.
Numerical Homogenization of Single-Walled Corrugated Board with Imperfections
Numerical homogenization is an excellent tool for the quick simplification of complex structures with a model that is much simpler and, at the same time, accurately reflects the mechanical behavior of the original model. Corrugated cardboard modeling, with all geometrical nuances preserved, is a complicated and time-consuming process. The transfer of a full 3D model of corrugated board composed of two flat layers and a corrugated middle layer to one layer only, with substitute elastic parameters, greatly simplifies this process. Because the individual layers of corrugated cardboard are made of paper with a grammage in the range of 80–200 g/m2, i.e., very thin plates, they are slightly buckled even in the initial configuration. These imperfections affect the equivalent parameters that are obtained in the homogenization process. This paper presents an approach of taking into account these imperfections when creating a simplified model. The numerical homogenization method based on the equivalence of elastic energy between a representative volumetric element (i.e., a part of a full 3D model) and an equivalent plate were applied. Different shapes of imperfections were analyzed in order to account for the buckling modes, notably for a specific unit deformation and curvature. Finally, one form of initial imperfections was proposed, which most accurately reflects the decrease in all plate stiffnesses.
Shell-to-Beam Numerical Homogenization of 3D Thin-Walled Perforated Beams
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.
Optimal Design of Double-Walled Corrugated Board Packaging
Designing corrugated board packaging is a real challenge, especially when the packaging material comes from multiple recycling. Recycling itself is a pro-ecological and absolutely necessary process, but the mechanical properties of materials that are processed many times deteriorate with the number of cycles. Manufacturers are trying to use unprecedented design methods to preserve the load-bearing capacity of packaging, even when the material itself is of deteriorating quality. An additional obstacle in the process of designing the structure of paper packaging is the progressive systematic reduction of the grammage (the so-called lightweight process) of corrugated cardboard. Therefore, this research presents a critical look at the process of optimal selection of corrugated cardboard for packaging structures, depending on the paper used. The study utilizes analytical, simplified formulas to estimate the strength of cardboard itself as well as the strength of packaging, which are then analyzed to determine their sensitivity to changes in cardboard components, such as the types of paper of individual layers. In the performed sensitivity analysis, numerical homogenization was used, and the influence of initial imperfections on the packaging mechanics was determined. The paper presents a simple algorithm for the optimal selection of the composition of corrugated cardboard depending on the material used and the geometry of the packaging, which allows for a more conscious production of corrugated cardboard from materials derived, e.g., from multiple recycling.
Non-Local Sensitivity Analysis and Numerical Homogenization in Optimal Design of Single-Wall Corrugated Board Packaging
The optimal selection of the composition of corrugated cardboard dedicated to specific packaging structures is not an easy task. The use of lighter boards saves material, but at the same time increases the risk of not meeting the guaranteed load capacity. Therefore, the answer to the question “in which layer the basis weight of the paper should be increased?” is not simple or obvious. The method proposed here makes it easy to understand which components and to what extent they affect the load-bearing capacity of packages of various dimensions. The use of numerical homogenization allows for a quick transformation of a cardboard sample, i.e., a representative volume element (RVE) into a flat plate structure with effective parameters describing the membrane and bending stiffness. On the other hand, the use of non-local sensitivity analysis makes it possible to find the relationship between the parameters of the paper and the load capacity of the packaging. The analytical procedures presented in our previous studies were used here to determine (1) the edge crush resistance, (2) critical load, and (3) the load capacity of corrugated cardboard packaging. The method proposed here allows for obtaining a comprehensive and hierarchical list of the parameters that play the most important role in the process of optimal packaging design.
Influence of humidity and temperature on mechanical properties of corrugated board - Numerical investigation
Paper is a material whose mechanical properties are highly dependent on humidity and temperature, naturally building the relationship between the stiffness and strength of corrugated board and changing weather conditions. In this paper, attention is focused on the dependence of the physical properties of the cardboard on changes in humidity and temperature, which undergo dynamic fluctuations both during the production of corrugated board and during its storage. Two techniques were used to test this effect, namely numerical homogenization and global sensitivity analysis. Both methods were implemented to determine the theoretical relationships between the change in humidity and/or temperature in each layer of corrugated board and its global bending, compression, and shear stiffness. The procedure was used to analyze different types of 5-ply and 3-ply cardboard. The obtained results allowed the authors to build a complete map of the relationship between the change in humidity of selected layers and the strength characteristics of the full assembly.
Influence of Imperfections on the Effective Stiffness of Multilayer Corrugated Board
There are many possible sources of potential geometrical inaccuracies in each layer of corrugated board during its manufacture. These include, among others, the processes of wetting the corrugated layers during profiling, the process of accelerated drying, the gluing process, and any mechanical impact of the pressure rollers on the cardboard. Work taking into account all the above effects in numerical modeling is not well described in the literature. Therefore, this article presents a simple and practical procedure that allows us to easily account for geometric imperfections in the calculation of the effective stiffness of corrugated board. As a main tool, the numerical homogenization based on the finite element method (FE) was used here. In the proposed procedure, a 3D model of a representative volumetric element (RVE) of a corrugated board is first built. The numerical model can include all kinds of geometrical imperfections and is used to calculate the equivalent tensile and bending stiffnesses. These imperfections were included in the 3D numerical model by appropriate modeling of individual layers, taking into account their distorted shape, which was obtained on the basis of a priori buckling analysis. This paper analyzes different types of buckling in order to find the most representative one. The proposed procedure is easy to implement and fully scalable.
Effective Stiffness of Thin-Walled Beams with Local Imperfections
Thin-walled beams are increasingly used in light engineering structures. They are economical, easy to manufacture and to install, and their load capacity-to-weight ratio is very favorable. However, their walls are prone to local buckling, which leads to a reduction of compressive, as well as flexural and torsional, stiffness. Such imperfections can be included in such components in various ways, e.g., by reducing the cross-sectional area. This article presents a method based on the numerical homogenization of a thin-walled beam model that includes geometric imperfections. The homogenization procedure uses a numerical 3D model of a selected piece of a thin-walled beam section, the so-called representative volume element (RVE). Although the model is based on the finite element method (FEM), no formal analysis is performed. The FE model is only used to build the full stiffness matrix of the model with geometric imperfections. The stiffness matrix is then condensed to the outer nodes of the RVE, and the effective stiffness of the cross-section is calculated by using the principle of the elastic equilibrium of the strain energy. It is clear from the conducted analyses that the introduced imperfections cause the decreases in the calculated stiffnesses in comparison to the model without imperfections.
Inverse-based multi-step numerical homogenization for mechanical characterization of converted corrugated board
Review on homogenization of corrugated materials. State-of-the-art in modeling of corrugated board
Corrugated materials, particularly corrugated board, form the backbone of contemporary packaging due to their light weight and high-strength properties. The application of numerical homogenization techniques to model and predict the mechanical behavior of these materials has evolved significantly, enabling refined structural design and optimization. This review examines advances in the homogenization of corrugated structures, with an emphasis on analytical, numerical, and experimental approaches as applied to corrugated board. Developments in the theoretical modeling of key mechanical properties, such as elasticity, bending, and shear stiffness, are highlighted, alongside methods for predicting structural responses under varying loading conditions. Efforts to optimize structural design through homogenization and the integration of digital tools, including artificial intelligence, are also discussed. Additionally, challenges in adapting homogenization models to account for environmental factors such as humidity and temperature, which impact mechanical properties, are analyzed. The review concludes by outlining future research directions and opportunities for bridging theoretical advances with practical applications in corrugated material design and usage.
Elastic Analysis of Three-Layer Concrete Slab Based on Numerical Homogenization with an Analytical Shear Correction Factor
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.