Role of Transverse Shear Modulus in the Performance of Corrugated Materials

Optimization Design of the Bending-Vibration Resistance of Magnetorheological Elastomer Carbon Fibre Reinforced Polymer Sandwich Sheets

An optimization design of the bending-vibration resistance of magnetorheological elastomer carbon fibre reinforced polymer sandwich sheets (MECFRPSSs) was studied in this paper. Initially, by adopting the classical laminate theory, the Reddy's high-order shear deformation theory, the Rayleigh-Ritz method, etc., an analytical model of the MECFRPSSs was established to predict both bending and vibration parameters, with the three-point bending forces and a pulse load being considered separately. After the validation of the model was completed, the optimization design work of the MECFRPSSs was conducted based on an optimization model developed, in which the thickness, modulus, and density ratios of magnetorheological elastomer core to carbon fibre reinforced polymer were taken as design variables, and static bending stiffness, the averaged damping, and dynamic stiffness parameters were chosen as objective functions. Subsequently, an artificial bee colony algorithm was adopted to execute single-objective, dual-objective, and multi-objective optimizations to obtain the optimal design parameters of such structures, with the convergence effectiveness being examined in a validation example. It was found that it was hard to improve the bending, damping, and dynamic stiffness behaviours of the structure simultaneously as the values of design variables increased. Some compromised results of design parameters need to be determined, which are based on Pareto-optimal solutions. In further engineering application of the MECFRPSSs, it is suggested to use the corresponding design parameters related to a turning point to better exert their bending-vibration resistance.

Compressive Strength of Corrugated Paperboard Packages with Low and High Cutout Rates: Numerical Modelling and Experimental Validation

The finite element method is a widely used numerical method to analyze structures in virtual space. This method can be used in the packaging industry to determine the mechanical properties of corrugated boxes. This study aims to create and validate a numerical model to predict the compression force of corrugated cardboard boxes by considering the influence of different cutout configurations of sidewalls. The types of investigated boxes are the following: the width and height of the boxes are 300 mm in each case and the length dimension of the boxes varied from 200 mm to 600 mm with a 100 mm increment. The cutout rates were 0%, 4%, 16%, 36%, and 64% with respect to the total surface area of sidewalls of the boxes. For the finite element analysis, a homogenized linear elastic orthotropic material model with Hill plasticity was used. The results of linear regressions show very good estimations to the numerical and experimental box compression test (BCT) values in each tested box group. Therefore, the numerical model can give a good prediction for the BCT force values from 0% cutout to 64% cutout rates. The accuracy of the numerical model decreases a little when the cutout rates are high. Based on the results, this paper presents a numerical model that can be used in the packaging design to estimate the compression strength of corrugated cardboard boxes.

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.

Compression Strength Estimation of Corrugated Board Boxes for a Reduction in Sidewall Surface Cutouts-Experimental and Numerical Approaches

Corrugated cardboard boxes are generally used in modern supply chains for the handling, storage, and distribution of numerous goods. These packages require suitable strength to maintain adequate protection within the package; however, the presence and configuration of any cutouts on the sidewalls significantly influence the packaging costs and secondary paperboard waste. This study aims to evaluate the performance of CCBs by considering the influence of different cutout configurations of sidewalls. The compression strength of various B-flute CCB dimensions (200 mm, 300 mm, 400 mm, 500 m, and 600 mm in length, with the same width and height of 300 mm), each for five cutout areas (0%, 4%, 16%, 36%, and 64%) were experimentally observed, and the results were compared with the McKee formula for estimation. The boxes with cutout areas of 0%, 4%, 16%, 36%, and 64% showed a linear decreasing tendency in compression force. A linear relationship was found between compression strength and an increase in cutout sizes. Packages with 0% and 4% cutouts did not show significant differences in compression strength (p < 0.05). Furthermore, this study shows a possible way to modify the McKee estimation for such boxes after obtaining empirical test data since the McKee formula works with a relatively high error rate on corrugated cardboard boxes with sidewall cutouts. Utilizing the numerical and experimental results, a favorable estimation map can be drawn up for packaging engineers to better manage material use and waste. The results of the study showed that the McKee formula does not appropriately estimate the box compression strength for various cutout sizes in itself.

The Effect of Side Wall Cutout Sizes on Corrugated Box Compression Strength in the Function of Length-to-Width Ratios—An Experimental Study

Packaging made from corrugated cardboard is a widely used solution in modern supply chains for the handling, storage and distribution of goods. These packages are required to maintain adequate protection conditions; however, in many cases, the cardboard box dimensions, handles and/or ventilation holes, quality and their configuration could compromise its protection strength. This study observes and evaluates the performance of corrugated cardboard boxes made with B-flute boards by considering different cutout sizes from the side walls (0%, 20%, 40%, 60% and 80%) in various box length–width ratios of 200 mm, 300 mm, 400 mm, 500 mm and 600 mm in length and a constant 300 mm width and height. Box compression tests were performed in a laboratory, and results were compared with mathematical statistics. In each cutout case, the maximum compression force was observed with the box with dimensions of 400 × 300 × 300 mm. The measurement results showed that the 1.33 length-to-width ratio has the best maximum compression force result. The statistical tests showed that there is no significant difference between the 0% and 20% cutout groups.

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 Impregnation with Modified Starch of a Paper Core on Bending of Wood-Based Honeycomb Panels in Changing Climatic Conditions

The main objective of the study was to determine the effect of impregnation of the paper core with acetylated starch on the mechanical properties and absorbed energy in the three-point bending test of wood-based honeycomb panels under varying temperatures and relative air humidity conditions. Nearly six hundred beams in various combinations, three types of facings, three core cells geometries, and two paper thicknesses were tested. The experiment results and their statistical analysis prove a significant relationship between the impregnation of paper with modified starch and mechanical properties. The most effective in absorbing energy, the honeycomb panels, consisted of a core with a wall thickness of 0.25 mm and a particleboard facing.

Transverse Vibration of Viscoelastic Sandwich Structures: Finite Element Modeling and Experimental Study

In the present work, the nonlinear vibration behavior of elastic-viscoelastic-elastic sandwich (EVES) beams is studied. A finite element (FE) equation taking intoaccount the transverse compression deformation of the viscoelastic core for the EVES beams is derived. In order toaccurately characterize the frequency-dependent feature of the viscoelastic materials layer, athird-order seven-parameter Biot model isused. A 2-node 8-DOF element is established to discretize the EVES beams. The experimental testing onEVES beams validates the numerical predication of the FE model. Numerical and analytical investigations are carried on a series of EVES beams with different thicknesses. The results indicate that the presented FE model has better accuracy in predicting the natural frequency of the sandwich beams, and in predicting damping, the accuracy is related to the thickness of each layer. The results of this paper have important reference values for the design and optimization of the viscoelastic sandwich structure.

Efficient Design of Thin Wall Seating Made of a Single Piece of Heavy-Duty Corrugated Cardboard

Corrugated cardboard has waved cores with small flutes that prevent the use of detailed numerical models of whole structures. Many homogenization methods in the literature overcome this drawback by defining equivalent homogeneous plates with the same mechanical behaviour at a macro-mechanical scale. However, few homogenization works have considered complete structures, focusing mainly on beams or plates. For the first time, this study explores the application of homogenization approaches to larger structures as an aid in their design process. We also considered triple-wall boards rather than single- and double-wall configurations commonly addressed in the literature. To this end, we adapted the homogenization methods proposed by Talbi and Duong to analyze thin-walled stools made of triple-wall corrugated cardboard. Using a progressive design process, we performed an efficient stool design by removing material zones with lower stresses, with 35% less material, 35% lower vertical deflections, and 66% lower stresses than the initial design. Unlike other corrugated cardboard stools, this design comprises just one folded piece instead of three, thus saving storage space. These results demonstrate the utility of homogenization techniques as an aid in the design process of whole structures made of corrugated cardboard. Further research will consider buckling analysis.

New Edge Crush Test Configuration Enhanced with Full-Field Strain Measurements

The standard edge crush test (ECT) allows the determination of the crushing strength of the corrugated cardboard. Unfortunately, this test cannot be used to estimate the compressive stiffness, which is an equally important parameter. This is because any attempt to determine this parameter using current lab equipment quickly ends in a fiasco. The biggest obstacle is obtaining a reliable measurement of displacements and strains in the corrugated cardboard sample. In this paper, we present a method that not only allows for the reliable identification of the stiffness in the loaded direction of orthotropy in the corrugated board sample, but also the full orthotropic material stiffness matrix. The proposed method uses two samples: (a) traditional, cut crosswise to the wave direction of the corrugated core, and (b) cut at an angle of 45°. Additionally, in both cases, an optical system with digital image correlation (DIC) was used to measure the displacements and strains on the outer surfaces of samples. The use of a non-contact measuring system allowed us to avoid using the measurement of displacements from the crosshead, which is burdened with a large error. Apart from the new experimental configuration, the article also proposes a simple algorithm to quickly characterize all sought stiffness parameters. The obtained results are finally compared with the results obtained in the homogenization procedure of the cross-section of the corrugated board. The results were consistent in both cases.

Estimation of the Compressive Strength of Corrugated Board Boxes with Shifted Creases on the Flaps

In the modern world, all manufacturers strive for the optimal design of their products. This general trend is recently also observed in the corrugated board packaging industry. Colorful prints on displays, perforations in shelf-ready-packaging and various types of ventilation holes in trays, although extremely important for ergonomic or functional reasons, weaken the strength of the box. To meet the requirements of customers and recipients, packaging manufacturers outdo each other with new ideas for the construction of their products. Often the aesthetic qualities of the product become more important than the attention to maintaining the standards of the load capacity of the packaging (which, apart from their attention-grabbing functions, are also intended to protect transported products). A particular flaps design (both top and bottom) and its influence on the strength of the box are investigated in this study. An updated analytical–numerical approach is used here to predict the strength of packaging with various flap offsets. Experimental results indicated a significant decrease in the static load-bearing capacity of packaging in the case of shifted flap creases. The simulation model proposed in our previous work has been modified and updated to take into account this effect. The results obtained by the model presented in this paper are in satisfactory agreement with the experimental data.

Crushing of Double-Walled Corrugated Board and Its Influence on the Load Capacity of Various Boxes

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.

Numerical Homogenization of Multi-Layered Corrugated Cardboard with Creasing or Perforation

The corrugated board packaging industry is increasingly using advanced numerical tools to design and estimate the load capacity of its products. This is why numerical analyses are becoming a common standard in this branch of manufacturing. Such trends cause either the use of advanced computational models that take into account the full 3D geometry of the flat and wavy layers of corrugated board, or the use of homogenization techniques to simplify the numerical model. The article presents theoretical considerations that extend the numerical homogenization technique already presented in our previous work. The proposed here homogenization procedure also takes into account the creasing and/or perforation of corrugated board (i.e., processes that undoubtedly weaken the stiffness and strength of the corrugated board locally). However, it is not always easy to estimate how exactly these processes affect the bending or torsional stiffness. What is known for sure is that the degradation of stiffness depends, among other things, on the type of cut, its shape, the depth of creasing as well as their position or direction in relation to the corrugation direction. The method proposed here can be successfully applied to model smeared degradation in a finite element or to define degraded interface stiffnesses on a crease line or a perforation line.

Crushing of Single-Walled Corrugated Board during Converting: Experimental and Numerical Study

Corrugated cardboard is an ecological material, mainly because, in addition to virgin cellulose fibers also the fibers recovered during recycling process are used in its production. However, the use of recycled fibers causes slight deterioration of the mechanical properties of the corrugated board. In addition, converting processes such as printing, die-cutting, lamination, etc. cause micro-damage in the corrugated cardboard layers. In this work, the focus is precisely on the crushing of corrugated cardboard. A series of laboratory experiments were conducted, in which the different types of single-walled corrugated cardboards were pressed in a fully controlled manner to check the impact of the crush on the basic material parameters. The amount of crushing (with a precision of 10 micrometers) was controlled by a precise FEMat device, for crushing the corrugated board in the range from 10 to 70% of its original thickness. In this study, the influence of crushing on bending, twisting and shear stiffness as well as a residual thickness and edge crush resistance of corrugated board was investigated. Then, a procedure based on a numerical homogenization, taking into account a partial delamination in the corrugated layers to determine the degraded material stiffness was proposed. Finally, using the empirical-numerical method, a simplified calculation model of corrugated cardboard was derived, which satisfactorily reflects the experimental results.

Full-Field Measurements in the Edge Crush Test of a Corrugated Board-Analytical and Numerical Predictive Models

This article focuses on the derivation of simplified predictive models for the identification of the overall compressive stiffness and strength of corrugated cardboards. As a representative example an unsymmetrical 5-ply sample (with E and B flute) was used in this study. In order to exclude unreliable displacement measurement in the standard edge crush test, virtual strain gauges were used. Video extensometry was employed to collect measurements from the outer surfaces of the sample on both sides. Additional data allowed real force-displacement curves to be obtained, which were used in the validation procedure. To emulate the experimental results, besides a simple analytical model, a 3D numerical model fully reflecting the geometry of the corrugated board, based on the finite elements method was also built. In both cases good agreement between the experimental results and the analytical and numerical calculations was observed. This proved that the proposed analytical model can be successfully used to determine the overall stiffness and compressive strength of corrugated board, provided that the geometry and properties of all the layers of the board are known. The simple model presented in this work enables quick and reliable design and prototyping of new assemblies without the need to manufacture them.

Mechanical Behavior Modeling of Containers and Octabins Made of Corrugated Cardboard Subjected to Vertical Stacking Loads

The aim of this paper is to characterize the mechanical behavior of corrugated cardboard boxes using simple models that allow an approach to the load capacity and the deformation of the boxes. This is very interesting during a box design stage, in which the box does not exist yet. On the one hand, a mathematical model of strength and deformation of boxes with different geometry is obtained from experiments according to the Box Compression Test and Edge Crush Test standards. On the second hand, a finite element simulation is proposed in which only the material elastic modulus in the compression direction is needed. For that, corrugated cardboard sheets are glued to build billets for testing, and an equivalent elastic modulus is obtained. This idea arises from the fact that the collapse of the box is given by the local bucking of the corrugated cardboard panels, due to the slenderness itself, and the properties in the compression direction are predominant. As a result, the numerical models show satisfactory agreement with experiments, concluding that it is an adequate methodology to simulate in a simple and efficient way this type of boxes built with corrugated cardboard.

Determination of Transverse Shear Stiffness of Sandwich Panels with a Corrugated Core by Numerical Homogenization

Knowing the material properties of individual layers of the corrugated plate structures and the geometry of its cross-section, the effective material parameters of the equivalent plate can be calculated. This can be problematic, especially if the transverse shear stiffness is also necessary for the correct description of the equivalent plate performance. In this work, the method proposed by Biancolini is extended to include the possibility of determining, apart from the tensile and flexural stiffnesses, also the transverse shear stiffness of the homogenized corrugated board. The method is based on the strain energy equivalence between the full numerical 3D model of the corrugated board and its Reissner-Mindlin flat plate representation. Shell finite elements were used in this study to accurately reflect the geometry of the corrugated board. In the method presented here, the finite element method is only used to compose the initial global stiffness matrix, which is then condensed and directly used in the homogenization procedure. The stability of the proposed method was tested for different variants of the selected representative volume elements. The obtained results are consistent with other technique already presented in the literature.

Estimation of the Compressive Strength of Corrugated Cardboard Boxes with Various Perforations

This paper presents a modified analytical formula for estimating the static top-to-bottom compressive strength of corrugated board packaging with different perforations. The analytical framework is based here on Heimerl’s assumption with an extension from a single panel to a full box, enhanced with a numerically calculated critical load. In the proposed method, the torsional and shear stiffness of corrugated cardboard, as well as the panel depth-to-width ratio is implemented in the finite element model used for buckling analysis. The new approach is compared with the successful though the simplified McKee formula and is also verified with the experimental results of various packaging designs made of corrugated cardboard. The obtained results indicate that for boxes containing specific perforations, simplified methods give much larger estimation error than the analytical–numerical approach proposed in the article. To the best knowledge of the authors, the influence of the perforations has never been considered before in the analytical or analytical–numerical approach for estimation of the compressive strength of boxes made of corrugated paperboard. The novelty of this paper is to adopt the method presented to include perforation influence on the box compressive strength estimation.

Manufacturing and Characterization of Highly Environmentally Friendly Sandwich Composites from Polylactide Cores and Flax-Polylactide Faces

This work focuses on the manufacturing and characterization of highly environmentally friendly lightweight sandwich structures based on polylactide (PLA) honeycomb cores and PLA-flax fabric laminate skins or facings. PLA honeycombs were manufactured using PLA sheets with different thicknesses ranging from 50 to 500 μm. The PLA sheets were shaped into semi-hexagonal profiles by hot-compression molding. After this stage, the different semi-hexagonal sheets were bonded together to give hexagonal panels. The skins were manufactured by hot-compression molding by stacking two Biotex flax/PLA fabrics with 40 wt% PLA fibers. The combined use of temperature (200 °C), pressure, and time (2 min) allowed PLA fibers to melt, flow, and fully embed the flax fabrics, thus leading to thin composite laminates to be used as skins. Sandwich structures were finally obtained by bonding the PLA honeycomb core with the PLA-flax skins using an epoxy adhesive. A thin PLA nonwoven was previously attached to the external hexagonal PLA core, to promote mechanical interlock between the core and the skins. The influence of the honeycomb core thickness on the final flexural and compression properties was analyzed. The obtained results indicate that the core thickness has a great influence on the flexural properties, which increases with core thickness; nevertheless, as expected, the bonding between the PLA honeycomb core and the skins is critical. Excellent results have been obtained with 10 and 20 mm thickness honeycombs with a core shear of about 0.60 and facing bending stresses of 31–33 MPa, which can be considered as candidates for technical applications. The ultimate load to the sample weight ratio reached values of 141.5 N·g⁻¹ for composites with 20 mm thick PLA honeycombs, which is comparable to other technical composite sandwich structures. The bonding between the core and the skins is critical as poor adhesion does not allow load transfer and, while the procedure showed in this research gives interesting results, new developments are necessary to obtain standard properties on sandwich structures.

Estimation of the Compressive Strength of Corrugated Cardboard Boxes with Various Openings

This paper presents mixed analytical/numerical method for estimating the static top-to-bottom compressive strength of corrugated packaging with different ventilation openings and holes, in which the torsional and shear stiffness of corrugated cardboard as well as the panel depth-to-width ratio are included. Analytical framework bases on Heimerls assumption with a modification to a critical force, which is here computed by a numerical algorithm. The proposed method is compared herein with the successful McKee formula and is verified with the large number of experiment results of various packaging designs made of different qualities of corrugated cardboard. The results show that, for various hole dimensions or location of openings in no-flap and flap boxes, the estimation error may be reduced up to three times than in the simple analytical approach.

The Influence of Symmetrical Boundary Conditions on the Structural Behaviour of Sandwich Panels Subjected to Torsion

The paper discusses the influence of load and support conditions on the behaviour of sandwich panels subjected to torsion. 3-D numerical models are presented, in which various boundary conditions have been defined. The case of the load causing the concentrated torque in the span is analyzed, and the load definition affects the structural response. The numerical results were compared with the results obtained for the analytical beam model, which included both free torsion and secondary warping torsion. The conditions under which the models achieve a high agreement between the results were determined, but the significant sensitivity of the solution was also indicated. In each case of the considered load and boundary conditions, the structural response shows appropriate symmetry.

Torsional and Transversal Stiffness of Orthotropic Sandwich Panels

In the present work, an analytical equation describing the plate torsion test taking into account the transverse shear stiffness in sandwich plates is derived and numerically validated. Transverse shear becomes an important component if the analyzed plate or shell is thick with respect to the in-plane dimensions and/or its core has significantly lower stiffness than the outer faces. The popular example of such a sandwich plate is a corrugated cardboard, widely used in the packaging industry. The flat layers of a corrugated board are usually made of thicker (stronger) material than that used for the corrugated layer, the role of which is rather to keep the outer layers at a certain distance, to ensure high bending stiffness of the plate. However, the soft core of such a plate usually has a low transverse shear stiffness, which is often not considered in the plate analysis. Such simplification may lead to significant calculation errors. The paper presents the generalization of the Reissner’s analytical formula, which describes the torsional stiffness of the plate sample including two transverse shear stiffnesses. The paper also presents the implementation of the numerical model of the plate torsion test including the transverse shear stiffnesses. Both approaches are compared with each other on a wide range of material parameters and different aspect ratios of the specimen. It has been proved that both analytical and numerical formulations lead to an identical result. Finally, the performance of presented formulations is compared with other numerical models using commercial implementation of various Reissner–Mindlin shell elements and other analytical formulas from the literature. The comparison shows good agreement of presented theory and numerical implementation with other existing approaches.

The Role of Buckling in the Estimation of Compressive Strength of Corrugated Cardboard Boxes

This paper presents analytical methods for estimating the static top-to-bottom compressive strength of simple corrugated packaging, in which the torsional and shear stiffness of corrugated cardboard as well as the panel depth-to-width ratio are included. The methods are compared herein with a basic and more detailed buckling description with the successful McKee formula, which is over fifty years old but still widely used among packaging designers and quality control departments. Additionally, the assumptions and applied simplifications used in the literature are analyzed, and the limits of applicability of different versions of the selected methods are checked. Finally, all approaches are verified with the experiment results of various packaging designs made of corrugated cardboard. The results show that, for certain proportions of dimensions of simple flap boxes, simplified methods give an even two times larger estimation error than the analytical approach proposed in the paper. Furthermore, it is evidenced that including all flexural, torsional and shear stiffnesses in the buckling force estimation gives a very precise prediction of the box compressive strength for the full range of package dimensions.