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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.
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.
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.
Modified Compression Test of Corrugated Board Fruit Tray: Numerical Modeling and Global Sensitivity Analysis
This article presents a modified configuration of the box compression test (BCT), which reflects the actual behavior of the vegetable or fruit trays during transport and storage. In traditional load capacity tests, trays are treated as classic transport boxes, i.e., they are compressed between two rigid plates, which does not take into account the specific geometry of this type of packaging. Both the boundary conditions and the loads acting on the tray were modified. The paper presents the concept of a new test, as well as numerical models and a sensitivity analysis of the modified BCT to the basic geometrical dimensions of the tray. The conducted research clearly shows that the proposed configuration of the load-bearing capacity test of a tray is closer to the actual operation of the packaging. As a result, most of the parameters that are not active under the conditions of the classical BCT become more important in the new configuration, which corresponds to the observations on the real performance of the packaging.
Inverse-based multi-step numerical homogenization for mechanical characterization of converted corrugated board
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.
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.
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.
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.
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.
Bending stiffness of unsymmetrical multilayered corrugated board: Influence of boundary conditions
In laboratory practice, several standards for testing the bending stiffness of corrugated board are used. There are often cases of tests where the results depend on the way the sample is placed on the supports. The problem arises when the board is five-ply (with two corrugated layers with different corrugation heights) or when the board has asymmetrically selected papers on the flat layers. This article focuses on the problem related to boundary conditions, with particular attention to the local effects of the support of the sample. Because the cardboard layers, both flat and corrugated, have a small thickness, a slight deformation of the papers can always be observed at the point of contact between the sample and the support, which affects the readings of the measured stiffness. The paper presents theoretical and numerical analyses showing how much the method of supporting the sample affects the measured bending stiffness of various samples. Numerical observations were compared with the results of analyses presented by other scientists as well as with experimental results.
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.