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Parametric Optimization of Thin-Walled 3D Beams with Perforation Based on Homogenization and Soft Computing
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.
Estimation of the Compressive Strength of Cardboard Boxes Including Packaging Overhanging on the Pallet
In this study, a numerical investigation was conducted on a verified packaging model, which sticks out beyond the pallet base area, which will evidently weaken its load-bearing capacity. This could lead to damage of the protected goods transported within this packaging. It might also result in the unnecessary overengineered design of the packaging, particularly when the potential for overhanging is anticipated beforehand, but its exact extent is not known. The article analyzed hundreds of cases, varying in terms of packaging dimensions (from 150 mm up to 600 mm), the extent of protrusion beyond the edge of the pallet (from 1% to 50% of box dimensions) and the use of various corrugated boards (B-, C-, EB- and BC-flute), in order to assess the decrease in the load-bearing capacity of the packaging compared to reference packaging, which was not overhanging on a pallet. For instance, it appeared that the decrease in the load-bearing capacity of the packaging when overhanging was insensitive to the corrugated cardboard material used. Additionally, the decrease in box strength was rapid while overhanging, even for a small value of overhanging.
Investigating the Effect of Perforations on the Load-Bearing Capacity of Cardboard Packaging
The impact of perforation patterns on the compressive strength of cardboard packaging is a critical concern in the packaging industry, where optimizing material usage without compromising structural integrity is essential. This study aims to investigate how different perforation designs affect the load-bearing capacity of cardboard boxes. Utilizing finite element method (FEM) simulations, we assessed the compressive strength of packaging made of various types of corrugated cardboards, including E, B, C, EB, and BC flutes with different heights. Mechanical testing was conducted to obtain accurate material properties for the simulations. Packaging dimensions were varied to generalize the findings across different sizes. Results showed that perforation patterns significantly influenced the compressive strength, with reductions ranging from 14% to 43%, compared to non-perforated packaging. Notably, perforations on multiple walls resulted in the highest strength reductions. The study concludes that while perforations are necessary for functionality and aesthetics, their design must be carefully considered to minimize negative impacts on structural integrity. These findings provide valuable insights for designing more efficient and sustainable packaging solutions in the industry.
Influence of Analog and Digital Crease Lines on Mechanical Parameters of Corrugated Board and Packaging
When producing packaging from corrugated board, material weakening often occurs both during the die-cutting process and during printing. While the analog lamination and/or printing processes that degrade material can be easily replaced with a digital approach, the die-cutting process remains overwhelmingly analog. Recently, new innovative technologies have emerged that have begun to replace or at least supplement old techniques. This paper presents the results of laboratory tests on corrugated board and packaging made using both analog and digital technologies. Cardboard samples with digital and analog creases are subject to various mechanical tests, which allows for an assessment of the impact of creases on the mechanical properties of the cardboard itself, as well as on the behavior of the packaging. It is proven that digital technology is not only more repeatable, but also weakens the structure of corrugated board to a much lesser extent than analog. An updated numerical model of boxes in compression tests is also discussed. The effect of the crushing of the material in the vicinity of the crease lines in the packaging arising during the analog and digital finishing processes is taken into account. The obtained enhanced computer simulation results closely reflect the experimental observations, which prove that the correct numerical analysis of corrugated cardboard packaging should be performed with the model taking into account the crushing.
A Simplified Dynamic Strength Analysis of Cardboard Packaging Subjected to Transport Loads
The article presents a simplified method for determining the strength of corrugated board packaging subjected to dynamic transport loads. The proposed algorithm consists of several calculation steps: (1) a static analysis of the compressive strength of the package, (2) an analysis of random vibrations in the frequency domain used to determine the resonance frequencies and (3) a dynamic analysis of the package loaded with computed resonant frequencies. For this purpose, numerical models of the static compression test of the packaging before and after the dynamic analysis of the package subjected to general transport loads were developed. In order to validate the model, laboratory packaging compression tests were also performed for samples of boxes using three-layer cardboard. Due to this, it was possible to verify the numerical simulation results of the compression tests for several box geometries. This, in turn, allowed for the development of a method based on dynamic and post-dynamic (static) numerical analyses, permitting a high-accuracy determination of the resistance of the selected packaging to vibrations and dynamic loads. The results of the (experimentally validated) numerical analysis proved the usefulness of the simplified method presented herein for precise estimation of the load capacity of various packages dynamically loaded during transport.
Parametric Study of the Numerical Model of a Bolted Connection of Steel Structure for Photovoltaic Panels
In the face of the reality that unexpectedly mobilized the governments of most central European countries (including Poland), the development of renewable energy sources (RES) seems to be an important direction. Therefore, both wind parks and solar farms will be constructed at double speed for energetic independence. This urgency makes the market of producers of structures for mounting solar panels also need to adapt quickly to the new situation. New constructions adapted to quick assembly with the use of nutless screw connections seem to be one of the best solutions. These structures must not only be easy and quick to install but also durable, which makes the connections resistant to cyclical loads. The speed of assembly of the substructure can be achieved precisely with the help of nutless connections, but their durability should be carefully analyzed. This article presents parametric analyses of the numerical model of this type of connection. The selection of appropriate numerical models for simulation is of key importance in the fatigue strength analysis of bolted connections. This article investigates two different models used in numerical fatigue analyses performed in the Abaqus FEA and FE-Safe program, namely, traditional bolt with nut and innovative self-tapping nutless bolt. Extended parametric analyses of both numerical models were carried out, which ultimately allowed optimization of the fatigue capacity of the connection.
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.
Corrugated board packaging with innovative design for enhanced durability during transport
Laboratory tests were conducted on innovatively designed corrugated board packaging under random vertical vibrations. The innovative designs had reinforced critical corner zones and lid–base interfaces through geometry modifications that increased double-wall regions. A total of 25 packaging variants, differentiated in structure, layer configuration (three-layer and five-layer boards), and surface finish (with and without coatings) were evaluated. The experimental study included box compression tests (BCT) and random vibration tests according to international standards (ISO 12048:1994 and ISO 13355:2016), simulating storage and transportation conditions. All packages were assessed before and after random vibration tests to determine the influence of dynamic loads on structural load-bearing capacity. Unlike previous studies limited to static testing, this work evaluated combined vibration and compression effects under standardized dynamic loading conditions for packaging with relatively low probability of being dropped. Furthermore, it was shown that the innovative design of corrugated board transport packaging presents higher static load capacity after random vibration testing in terms of column compression strength, indicating that no reduction in box strength was observed during simplified transport simulation under pure one-direction dynamic loading. The findings contribute to the optimization of high-durability packaging solutions tailored for the growing demands of complex logistics chains.
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
On the use of artificial intelligence in predicting the compressive strength of various cardboard packaging
AbstractArtificial intelligence is increasingly used in various branches of engineering. In this article, artificial neural networks are used to predict the crush resistance of corrugated packaging. Among the analysed packages were boxes with ventilation openings, packages with perforations and typical flap boxes, which make the proposed estimation method very universal. Typical shallow feedforward networks were used, which are perfect for regression problems, mainly when the set of input and output parameters is small, so no complicated architecture or advanced learning techniques are required. The input parameters of the neural networks are selected so as to take into account not only the material used for the production of the packaging but also the dimensions of the box and the impact of ventilation holes and perforations on the load capacity of individual walls of the packaging. In order to maximize the effectiveness of neural network training process, the group of input parameters was changed so as to eliminate those to which the sensitivity of the model was the lowest. This allowed the selection of the optimal configuration of training pairs for which the estimation error was on the acceptable level. Finally, models of neural networks were selected, for which the training and testing error did not exceed 10%. The demonstrated effectiveness allows us to conclude that the proposed set of universal input parameters is suitable for efficient training of a single neural network model capable of predicting the compressive strength of various types of corrugated packaging.
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.
Sensitivity Analysis of Open-Top Cartons in Terms of Compressive Strength Capacity
Trays in which fruit and vegetables are transported over vast distances are not only stored in extreme climatic conditions but are also subjected to long-term loads. Therefore, it is very important to design them correctly and select the optimal raw material for their production. Geometric parameters that define the shape of the packaging may also be optimized in the design process. In this work, in order to select the most important parameters that affect the load capacity of a tray, sensitivity analysis was used. A sensitivity analysis is often the first step in the process of building artificial-intelligence-based surrogates. In the present work, using the example of a specific tray’s geometry, the selection of starting parameters was carried out in the first step, based on the Latin hypercube sampling method. In the next step, local sensitivity analyses were performed at twenty selected starting points of the seventeen-dimensional space of the selected parameters. Based on the obtained results, it was possible to select the parameters that have a significant impact on the load capacity of the tray in the box compression test and whose influence is negligible or insignificant.
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.