Application of beam on elastic foundation to the interaction between a corrugated box and pallet deckboard

Corrugated boxes are ubiquitous in shipping and warehousing logistics. In physical distribution, corrugated boxes are often shipped in a unit load form where the interaction between the components determines the effectiveness and safety of the overall system. When lower stiffness pallets are used to support the corrugated boxes, the compression strength of boxes is reduced due to the uneven support conditions caused by the deforming top deckboards of the pallet. In this study, a modification of the principle of beam on elastic foundation was used to predict the effect of pallet deck stiffness on the performance of a corrugated box. In the model, the corrugated box acts as the elastic foundation, and the deckboard is represented as the beam. Pallet deck stiffness, pallet connection stiffness, and package stiffness are required model inputs. The resulting model was capable of predicting the normalized distribution of forces along the boxes' length sidewall but was not capable of predicting the compression strength of the box at failure.     

The effect of pallet top deck stiffness on the compression strength of asymmetrically supported corrugated boxes

During unitized shipment, the components of unit loads are interacting with each other. During floor stacking of unit loads, the load on the top of the pallet causes the top deck of the pallet to bend, which creates an uneven top deck surface resulting in uneven or asymmetrical support of the corrugated boxes. This asymmetrical support could significantly affect the strength of the corrugated boxes, and it depends on the top deck stiffness of the pallet. This study is aimed at investigating how the variations of pallet top deck stiffness and the resulting asymmetric support affect corrugated box compression strength. The study used a scaled-down unit load compression test on quarter-scale pallet designs with different deckboard thicknesses using four different corrugated box designs. Pallet top deck stiffness was determined to have a significant effect on box compression strength. There was a 27%–37% increase in box compression strength for boxes supported by high-stiffness pallets in comparison with low-stiffness pallets. The fact that boxes were weaker on low-stiffness pallets could be explained by the uneven pressure distribution between the pallet deck and bottom layer of boxes. Pressure data showed that a higher percentage of total pressure was located under the box sidewalls that were supported on the outside stringers of low-stiffness pallets in comparison with high-stiffness pallets. This was disproportionately loading one side of the box. Utilizing the effects of pallet top deck stiffness on box compression performance, a unit load cost analysis is presented showing that a stiffer pallet can be used to carry boxes with less board material; hence, it can reduce the total unit load packaging cost.     

Investigation of the effect of column stacked corrugated boxes on load bridging using partial four‐way stringer class wooden pallets

Pallets are the foundation of the global packaging supply chain. They provide a way to store and transport products in an efficient manner. The load capacity of pallets greatly depends on the type of packages carried by the pallet; however, current pallet design methods do not consider the effect of packages on the load carrying capacity of the pallet. This results in excessive use of materials which reduces the sustainability of unit loads, drives costs up, and creates potential safety issues. The objective of this study was to investigate the effect of corrugated box size and headspace on pallet deflection and stress distribution on the top of the pallet as a function of pallet stiffness across multiple pallet support conditions. Data analysis identified that box size had a significant effect on the deflection of the pallet. This effect was only significant for warehouse racking across the width and length support conditions. As much as a 53% reduction in pallet deflection was observed for high stiffness pallets supporting corrugated boxes with 25.4‐mm headspace when the size was increased from small to large. The redistribution of vertical compression stresses towards the supports as a function of the increasing box size was observed. The increased concentration of compression stresses on top of the supports and the resulting lower pallet deflection could significantly increase the actual load carrying capacity of some pallet designs. The effect of box headspace was significant in some scenarios but inconsistent; thus, more investigation with a larger sample size is recommended.     

Investigation into the load bridging effect for block class pallets as a function of package size and pallet stiffness

Pallets and corrugated boxes are ubiquitous in the global supply chain. However, the interactions that exist between the boxes and pallets are ignored during the pallet design process resulting in the over designing of pallets and the waste of raw materials. The goal of this research is to understand how pallet performance is affected by headspace, box size, and base design across multiple support conditions using block class wooden pallets. Headspace and base design had no effect on pallet deflection for the experimental weights used throughout testing. However, the effect of box size was significant on pallet deflection across multiple support conditions. The effect was greatest for lower stiffness pallets and low‐stiffness support conditions (racking across the width; RAW) with up to a 50% reduction in pallet deflection observed by switching from small to large boxes on a very low‐stiffness pallet. Evaluation of pressure mat data showed an increase in the redistribution of pressure away from the center of the pallet and toward the supports as box size increased. The redistribution of pressure toward the supports is known as load bridging and validates the observed reduction in pallet deflection as a function of box size. The results indicate that incorporating this effect of packages into current pallet design practices could result in more effective and cheaper pallet designs.     

The effect of pallet connection stiffness,deck stiffness and static load level on the resonant response of pallet decks to vibration frequencies occurring in the distribution environment

This paper summarizes the results of a preliminary study of the relationship between pallet design and the resonant response of pallet decks to sinusoidal vibration. Sine sweep frequency vibration tests between 3 Hz and 50 Hz were conducted to determine the effect of pallet deckboard stiffness, joint stiffness and the static load level on the resonant response of pallet decks. As the stiffness of the pallet joints and deckboards increases, the resonant frequency of the unit load increases and transmissibility decreases. As the static load on the pallet increases the resonant frequency of the unit load decreases and the transmissibility increases. The survivability of a unitized product, sensitive to vibration, during shipping and distribution is affected by the pallet design. Pallets designed using rigid connections and stiffer decks will reduce damage to vibration‐sensitive products. Copyright © 1999 John Wiley & Sons, Ltd.     

Box Compression Strength: A Crippling Approach

A crippling analysis method has been utilized to estimate the compression strength paperboard boxes. Crippling analysis is typically utilized in the aerospace industry to predict the compressive failure strength of thin, slender structures with complex cross‐sectional geometry. This type of analysis is investigated, because crippling is a simple, predictive method that can provide very good estimates of the compressive failure strength of thin, slender structures. This preliminary study investigates the possibility of applying crippling analysis to estimate paperboard box compression strength by comparing experimental and theoretical results for box compression tests of milk and cigarette boxes in various loading scenarios and with various materials. This preliminary study shows that the crippling method provides results which are almost as accurate as pre‐existing methods, although significant work remains to verify the validity and applicability of the crippling approach for paper‐based boxes. Copyright © 2015 John Wiley & Sons, Ltd.     

瓦楞纸箱抗压强度的研究进展

目的综述国内外对瓦楞纸箱抗压强度的研究进展,探讨瓦楞纸箱的发展趋势。方法以查阅文献等形式,了解国内外瓦楞纸箱抗压强度的研究现状。论述对瓦楞纸箱力学性能的研究,讨论抗压强度经验公式的修订、抗压强度影响因素及提高方法。结论抗压强度的研究为瓦楞纸箱生产企业和包装设计者提供了必要的实验数据。轻量化、定量化、节约资源和降低成本等是瓦楞纸箱科学发展的要求,也为全球瓦楞纸箱的发展指明了方向。     

基于刚度-强度-重力的蜂窝纸板优化设计

分析了蜂窝纸板轻量型优化设计的刚度、强度约束函数和重量目标函数,找出了制箱或制托盘用蜂窝纸板结构优化设计方法,并用图解法举例说明了如何根据包装的具体需要优选蜂窝纸板结构.     

Static and Dynamic Strength of Paperboard Containers Subjected to Variations in Climatic Conditions

The variability in climatic conditions during product distribution, especially across large distances, can be significant and is well known to affect the mechanical properties of many packaging materials. As the use of environmentally friendly materials, such as paperboard and bio‐cushions, increases, the challenge associated with overcoming the effects of extremes in temperature and relative humidity in the distribution chain becomes critical. To date, in the case of paperboard boxes, this is dealt with by accounting for the loss of static (compression) strength with increasing relative humidity. However, no method exists to address the dynamic loads induced by vehicle shocks and vibrations especially for configurations that involve stacked boxes and where the vibration intensity within the stack is influenced by the dynamic characteristic of the boxes themselves. In such scenarios, it is the variation in the stiffness of the box as a function of environmental conditions and dynamic load that needs to be established. This paper describes the evaluation of the fatigue resistance of paperboard boxes subjected to random excitation and compares the results with those obtained from quasi‐static compression tests under various environmental conditions. Results reveal a lack of correlation between the static and dynamic tests. This finding is attributed to changes in internal damping of the paperboard box samples which, when reduced, results in increased dynamic force. The paper concludes that static testing alone is insufficient to establish the fatigue resistance of stacked packaging subject to variations in climatic conditions. Copyright © 2017 John Wiley & Sons, Ltd.     

Flexural stiffness of selected corrugated structures

Top‐to‐bottom compression strength of a corrugated fibreboard box is partly dependent on the load‐carrying ability of central panel areas of the box. The ability of these central panel areas to resist a bending force from loading may increase the stacking strength of the box. The difference in the compression strengths of boxes that have identical dimensions and were fabricated with identical components but different flute types, is primarily caused by flexural stiffness of the box panels. Top‐to‐bottom compression strength of boxes can be accurately predicted by flexural stiffness measurements and edge crush test (ECT) of the combined boards. This study was carried out to analyse the flexural stiffness, to measure bending force and bending deflection by a four‐point bending test for various corrugated fibreboards, and to provide the major constructional factors which play a role in improving the compression strength of the box. Copyright © 2004 John Wiley & Sons, Ltd.     

瓦楞纸板托盘结构及性能研究

目的探讨瓦楞纸板托盘结构与性能的关系。方法托盘由铺板和垫块或纵梁粘合而成,铺板为1块3A楞瓦楞纸板或2块AB楞瓦楞纸板粘合而成,垫块或纵梁通过瓦楞纸板层叠粘合或卷绕成型,利用抗弯和抗压试验测试托盘抗弯强度和抗压强度,并分析托盘力学性能。结果托盘抗弯性能主要与铺板瓦楞纸板层数有关,1块3A楞瓦楞纸板铺板的抗弯性能优于2块AB楞瓦楞纸板粘合铺板;托盘抗压性能与垫块或纵梁结构有关,层叠粘合成型的垫块或纵梁的抗压性能优于回字卷绕成型的,抗压强度可达到33.48 k N;B楞瓦楞纸板层叠粘合成型的垫块或纵梁的抗压强度优于AB楞的;垫块的抗压强度优于纵梁。结论通过对瓦楞纸板托盘结构、抗弯性能、抗压性能的研究与测试,可为实际应用提供参考和依据。     

Finite element analysis of vent/hand hole designs for corrugated fibreboard boxes

Corrugated fibreboard is an economical and efficient material for fabricating shipping containers that are widely used for the distribution, transportation and storage of goods. Corrugated fibreboard is usually considered to be an orthotropic material because the principal fibre directions, machine direction (MD) and cross‐machine direction (CD), are identical to the fibres in paperboard, which has apparent directional property differences. The purpose of this study is to investigate the principal design parameters of ventilation holes and hand holes in the faces of the shipping container, (corrugated fibreboard boxes), using finite element analysis (FEA). Various designs of ventilation holes were studied with respect to stress distribution and stress level. It was found that the appropriate pattern and location of the ventilation holes were vertical oblong‐shaped and symmetrically positioned within a certain extent of distance to the right and left from the centre of the front and rear faces of the boxes. On the other hand, the appropriate location and pattern of the hand holes were a short distance from the centre to the top of the boxes on both side faces. The appropriate pattern was a modified shape, such as the radius of curvature of both sides in horizontal oblong. The pattern and location of both the ventilation holes and the hand holes determined by the FEA simulation generally agreed well with laboratory experimental results. The decrease in compression strength of the box could be minimized with identical area of the ventilation holes if the length of the major axis of the ventilation hole is less than 1/4 of the depth of the box and the ratio of the minor axis to the major axis is 1/3.5–1/2.5, provided that even‐numbered holes are located symmetrically. Copyright © 2006 John Wiley & son, Ltd.     

The Influence of Package Size and Flute Type of Corrugated Boxes on Load Bridging in Unit Loads

Shipping pallets often are designed with the assumption that the payload carried is flexible and uniformly distributed on the pallet surface. However, packages on the pallet can act as a series of discrete loads, and the physical interactions among the packages can add stiffness to the pallet/load combination. The term ‘load bridging’ has been used to describe this phenomenon. The study reported in this paper investigated the relationships of package size, corrugated flute type and pallet stiffness to load bridging and the resulting unit‐load deflection. The experimental results indicated that an increase in box size changed the unit‐load deflection by as much as 75%. Flute type was found to impact load bridging and the resulting unit‐load deflection. Changing the corrugated box flute type from B‐flute or BC‐flute to E‐flute reduces the unit‐load deflection by as much as 40%. Also, experimental data indicates that the effect of package size and corrugated board flute type on pallet deflection is the greatest for low stiffness pallets. The results provide information that can be used to design unit loads that use material more efficiently. Copyright © 2017 John Wiley & Sons, Ltd.     

Estimation of the compression strength of corrugated fibreboard boxes and its application to box design using a personal computer

Many papers have been published on the compression strength of corrugated fibreboard boxes, using such formulae as Kellicut's equation and McKee's equation for the calculation. These equations, however, require known values of the strength of linerboard or corrugated fibreboard, they do not include the influence of moisture content and they are inadequate in the case of wrap-around boxes. The present author measured the mechanical properties of a large number of fibreboard boxes, and has derived a statistical formula useful for estimating the compression strength of a box based on its specifications — grade of corrugated fibreboard, size of box, type of box, printed area and moisture content. The calculation gives fairly good agreement with experimental results. The estimation technique has further been converted into a personal computer program, which renders the design of corrugated fibreboard boxes an easier task.     

压痕形状对瓦楞纸箱抗压强度影响的试验

目的 在不同形状的压痕条件下,对瓦楞纸箱进行抗压试验,研究纸箱的变形情况和抗压能力。方法 首先设计无压痕纸箱、一字型压痕箱、八字型压痕箱以及菱形压痕箱;其次将各种压痕形状下的纸箱,利用纸箱抗压试验机进行空箱抗压实验,记录各自的最大压溃力;最后对实验数据进行分析,明确抗压强度与压痕形状的关系。结果 不同压痕形状对瓦楞纸箱的抗压强度有不同程度的影响,其中菱形影响最大。菱形压痕通过阻碍纸箱变形趋势可提高瓦楞纸箱的抗压强度。结论 在瓦楞纸箱侧板上通过施加阻碍纸箱工字型变形的压痕（如菱形压痕）,可以增加瓦楞纸箱的抗压强度,对瓦楞纸箱的生产设计具有参考意义。     

家电托盘货物单元的共振性能实验研究

家电产品的托盘货物单元采用多层堆码时,各层货箱的共振频率不同于单一货箱的。介绍了托盘集装单元的动力学模型,通过扫频振动实验和自功率谱分析,确定了各层货箱的共振频率,探讨了其固有频率与堆码高度之间的关系。实验结果表明:在一定振动频率范围内,对于多层堆码的高粘弹性货箱,底层货箱的固有频率高于顶层货箱的,各层货箱堆码高度对其共振频率有影响。     

管式预粘合纸箱结构参数与抗压强度的相关关系

目的研究管式预粘合纸箱结构参数与抗压强度的相关关系,为优化纸箱结构设计以及提高其抗压强度提供依据。方法首先设计普通、四棱台、花形锁等3种代表性管式预粘合纸箱,进而确定3种纸箱影响抗压强度的关键结构参数。其次将各结构参数依次作为自变量,其余结构参数作为常量,利用纸箱抗压试验机进行空箱抗压试验,记录各自的最大压溃力。最后对实验数据进行分析,明确抗压强度与结构参数间的相关关系。结果 3种纸箱的抗压强度均随着周长的增大而增大,其抗压强度并非全部随高度的增大而降低。考虑周长因素时,普通箱的抗压强度最佳;考虑高度因素时,花形锁箱的抗压强度最佳;考虑箱型因素时,当普通箱长宽比接近1.0,四棱台箱下上底比在1.2~1.4范围内,花形锁箱的啮合点和花形锁圆心的连线与水平线所成夹角接近30°时,抗压强度最佳。因结构参数的影响,管式预粘合纸箱的抗压强度整体上显著优于标准开槽纸箱。结论预粘合纸箱的关键结构参数对其抗压强度有显著影响,在设计时应给予足够的重视。     

基于SolidWorks的托盘结构有限元分析及优化设计

托盘结构设计一直采用经验设计方法,其结构的合理性有待研究。通过SolidWorks对托盘进行了建模和参数化设计,并运用其集成的Simulation仿真分析功能对所建立的模型进行了有限元分析。通过与托盘试验原型测试对比,验证了分析的正确性和有效性。最后通过SolidWorks的优化功能对托盘模型进行了优化设计,得到了以安全系数为约束条件的托盘上连板的最优厚度。该方法可行、有效,为托盘结构分析和优化设计提供了一条新的思路。     

THREE DIMENSIONAL PALLETIZATION OF MIXED BOX SIZES

The pallet loading problem has historically been addressed in two-dimensions by attempting to maximize the pallet area utilized in each loading layer. This paper investigates a constrained version of three-dimensional pallet loading problem with mixed box sizes. This loading method allows many boxes of various sizes to be placed onto the same pallet. A restriction is placed on the number of boxes of each size that can be loaded. The modeling procedure presented converts the three-dimensional pallet loading problem into a standard mixed 0-1 integer programming model. The solution procedure for the formulated model is also described.     

大长宽比对纸箱抗压能力影响的研究与分析

对获取纸箱抗压能力的3种主要技术手段:试验测试方法、经典公式计算方法和计算机仿真模拟方法的优缺点进行了对比分析。选择0201型BC楞瓦楞纸箱为对象,分别采取试验测试和经典公式计算,在纸板材料、纸箱高度、周长相同的条件下,对长宽比从1～3范围变化的纸箱抗压能力进行了计算和测试。结果表明:长宽比从1变化到3时,纸箱的抗压强度先升再降;当长宽比为1.6左右时,抗压强度达到最大值。试验结果和利用经典抗压能力计算结果的对比分析表明,计算结果均较大地偏离了试验结果。由于不能充分考虑到纸板加工质量、纸箱成型工艺、温湿度及其它因素的影响,经典抗压能力计算公式均是偏保守的。