compression testing to simulate real-world stresses

Copyright © 2010 John Wiley & Sons, Ltd.

Compression Testing to Simulate Real-World Stresses

By Benjamin Frank,1*† Michael Gilgenbach1‡ and Michael Maltenfort2§

1Packaging Corporation of America, Mundelein, IL, USA2Department of Mathematics, Truman College, Chicago, IL, USA

SUMMARY

Unit load testing of boxes reported in the literature typically uses empty boxes to explore the impact and interactions among box confi gurations, pallet support and other environmental parameters. However, this approach leads to failure in the weakest box in the unitized structure, while in the fi eld, failure almost always occurs in the bottom box, which may or may not be the weakest. We fi nd in this paper that math-ematically, numerically and experimentally, forcing box failure to the bottom results in higher test values. While this occurs naturally for boxes in use in the fi eld, it is an interaction which to date has been over-looked by researchers examining box performance in the lab. The impact on box estimation can be on the order of 5% or more, which can be as signifi cant as some of the environmental factors we are working to quantify, and which can have signifi cant cost implications. To improve the assessment used in the industry to account for the impact of a box’s ‘in use environment’ on its performance, we need further testing on confi gurations where the boxes are loaded. Copyright © 2010 John Wiley & Sons, Ltd.

Received 8 January 2010; Accepted 2 March 2010

KEY WORDS: corrugated; compression; unitizing; testing; boxes

INTRODUCTION

One of the primary goals of laboratory testing of boxes is to provide an assessment of a box’s strength and likely performance in its service environment. The typical approach to predicting performance relates actual box strength in the service environment to the strength of a single box as determined by a dynamic compression test (BCT). We measure or estimate the BCT and then reduce it by a factor associated with the environmental and unitizing conditions that the box is expected to experience. While this approach is well accepted across the industry,1 in practice our characterization and under-standing of the infl uence of unitizing conditions is insuffi ciently defi ned for this approach to be considered precise.

A box’s structural strength and stability comes disproportionately from its corners and corner structure.2–5 The maximum warehouse stacking strength comes when all corners align and are sup-ported by the underlying pallet structure. There have been various studies of the infl uence of overhang and pallet gapping,6–10 but the results across these studies are inconsistent. It is unclear if the incon-sistencies arise from different testing methods or different box sizes, or if the results are actually consistent within a large testing variability for each study (typically unreported). The impact of stack-ing pattern specifi cally is somewhat better understood, although most of the reported results seem to fall back on the work of Ievans.10 Ievans assumes that all the different factors are independent, but

PACKAGING TECHNOLOGY AND SCIENCEPackag. Technol. Sci. 2010; 23: 275–282

Published online 18 June 2010 in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/pts.898

* Correspondence to: B. Frank, Packaging Corporation of America, 250 S. Shaddle Ave., Mundelein, IL 60060, USA.E-mail: [email protected]†Manager, Materials Optimization and Development. ‡Manager, Package Testing.§Assistant Professor of Mathematics.

276 B. FRANK, M. GILGENBACH AND M. MALTENFORT

Copyright © 2010 John Wiley & Sons, Ltd. Packag. Technol. Sci. 2010; 23: 275–282 DOI: 10.1002/pts

there is evidence in the literature11 that the actual performance is greater than would be assumed by straight multiplication of the negative impacts of different environmental conditions. Furthermore, even pure column stacking loses some strength relative to the BCT measured on a single box.6,10 While column stacking is the most stable static alignment for boxes, it is often not the best alignment for dynamic or transit situations.12

All the data available in the literature appear to use empty boxes in stack testing. However, when empty boxes are tested in a stack (e.g. a column stack of three or fi ve boxes), it is often not the bottom box that fails. By contrast, in practical applications of boxes in use, it is the bottom box that nearly always fails in a stack. As a result, one can legitimately ask whether the BCTs measured for stacked confi gurations in the literature truly model and can be used to characterize the real-world fi eld failures.

In a BCT test on a stack of boxes, each box in the stack experiences the same compression forces during the test. The test effectively ‘polls’ all the boxes, fi nds the weakest one, causes failure there fi rst and reports that as the strength of the stack. In the case where that box is not the bottom box, the stack strength that is reported is lower than the strength that one would see in the fi eld, since in practice the load is higher on the bottom box than those above it in the stack due to the contents of the boxes. It is almost always the bottom box that collapses under the dead load of the boxes above. One might thus presume that BCT testing on a stack of boxes is subject to a type of ‘sampling error’ that would lead to erroneously low results, an error that raises questions about the accuracy and applicability of the literature to date.

In this paper we explore the impact of that sampling error in detail. We start by developing a closed form expression for the mathematical difference one would expect between effective box strength in use (the average strength of the bottom box in a unit), and the value one would get in a BCT test of a stack of boxes where the weakest of the boxes is allowed to fail. We next present the results of numerical simulation work, consistent with the closed form expression for a stack of boxes, expanding and confi rming that analysis. We see that allowing the weakest of the boxes in the stack to fail, and calling that the compression strength of the confi guration, can have a substantial impact on our assess-ment of the strength required. We then report the results of limited testing on confi gurations of loaded boxes, boxes containing contents that effectively apply extra load to the boxes below in the stack and thus shift the failure point to the bottom of the stack. Comparing those results both to testing of empty boxes and to our numerical simulations, we can identify changes to our testing and evaluation proto-cols that would be required to produce accurate assessments of the interactions of boxes in unit loads. These changes are needed in future studies if we are to meet the goal of further reduction in packag-ing coupled with confi dence in package performance.

MATHEMATICAL APPROACH

Closed form equation

To assess the differences that would arise in the measured compression strength of a stack of boxes, we need to be able to compare the strength of the weakest box in a stack to the strength of the bottom box in a stack. A given lot of boxes can be described as having some average strength m (the mean strength) and some variation around that strength described by the standard deviation s. If we assume that the boxes have strengths that vary normally around the mean, then the population of all boxes can be described using the normal distribution function:

f xs

ex m

s( ) =−

−( )1

2 22

2

2

π (1)

and the probability of having a given box xi with a given strength can be found by integrating. For example,

P x k f q dqik

>( ) = ( )∞

∫ (2)

gives us the probability of a given box having a strength greater than some arbitrary value k, where f(q) is the normal distribution function described above (Equation 1).

SIMULATION OF REAL-WORLD STRESSES 277


Each box in a stack of n boxes will have a slightly different strength, and the strength of the weakest box w is the minimum of the group of n boxes

w x x xn= ( )min , , . . . ,1 2 (3)

This is essentially the strength measured by a BCT test of a stack where the weakest box fails. The minimum w is more than an arbitrary value k only when each xi is more than k. So, to calculate the probability that w is greater than k, we multiply the (equal) probabilities for each xi being greater than k. The probability that w is less than k is simply one minus the probability that it is greater. Thus, we can conclude that

P w k P w k f q dqk

n

<( ) = − >( ) = − ( )( )∞

∫1 1 (4)

The probability distribution function of w is the derivative of Equation (4) (after changing k to x). Plugging Equation (1) into (4) and taking the derivative we get

n

se e dq

n

x m

s

q m

sx

n

2 2 22 2

12

2

2

2

π( )⎛

⎝⎜

⎞

⎠⎟

−−( ) −

−( )∞

−

∫ (5)

which we can simplify by substituting tq m

s=

− to get

n

se e dt

n

x m

s

tn

x ms2 2

2 2

12

2

2

π( )⎛

⎝⎜⎞

⎠⎟−

−( )−∞

−

−∫ (6)

If we could assure that the bottom box always failed, then the average strength at failure would simply be m, the average strength of the population of boxes, since boxes randomly end up at the bottom of the stack. Therefore, the difference between m and the average of w is the size of the error we make when we test a stack of empty boxes in which the failure occurs randomly in the stack, compared to the true failure in the fi eld where the bottom box fails.

In general, we mathematically evaluate the average value in a group or population by integrating xf(x) across the population:

xf x dx( )−∞

∞

∫ (7)

where for a single box f(x) is described by Equation (1), and for w, the minimum of a stack of n boxes, the function f(x) is described by Equation (6). So, to get the average (or typical value) of the minimum of a group of n boxes, we must evaluate the rather unpleasant-looking expression

n

sxe e dt dx

n

x m

s

tn

x ms2 2

2 2

12

2

2

π( )⎛

⎝⎜⎞

⎠⎟−∞

∞ −−( )

−∞−

∫ ∫ − (8)

We can make this expression more tractable (though less appealing) by substituting yx m

s=

− to get

nm

e e dt dyns

ye en

y t

y

n

n

y

2 222 2

1

22

2 2 2

π π( )⎛

⎝⎜⎞

⎠⎟+

( )−

−∞

∞ −∞−

−

−∞

∞

∫ ∫ ∫−−∞

−

∫⎛

⎝⎜⎞

⎠⎟t

y

n

dt dy2

2

1

(9a,b)

The fi rst integral (9a) can be evaluated directly to get 2 2π( )n

n, so that the whole fi rst term reduces

to just m, the population mean. The second integral (9b) must be solved using integration by parts, and results in an expression that only depends on n, the number of boxes in the stack. Putting these together, the average value one would expect from the minimum of n boxes can be expressed as

mn n s

e e dt dyn

yt

y

n

−−( )

( )⎛

⎝⎜⎞

⎠⎟−

−∞

∞ −∞−

∫ ∫1

2 22

22

2

π (10)



For n = 1, a stack of one box, the second term drops out and we just get m, the mean value of the whole population. This is just what we would expect. For n = 2, stacks of two boxes, the expression can be solved explicitly, and we get m

sm s− ≈ −

π0 564. . The measured value of the stack when

the weakest box fails is lower than the population average by an amount depending on the variation in the population. This confi rms that there is an impact of measuring the weakest box, instead of the bottom box. For stacks n > 2, the integral of Equation 10 cannot be solved in closed form but can be evaluated numerically. Table 1 indicates the coeffi cients of the standard deviation for n from 1 to 10.

Numerical simulation

We can also approach the problem of the impact of stacking using ‘brute force’ and the power of computers to perform multiple calculations. Using a typical spreadsheet program, we can create a list of normally distributed values xi that forms a population with a given average and standard deviation. We can imagine for example that this is a list of the individual compression strengths of all the boxes (e.g. 10 000) from a given production run. We then form sub-samples from this list by taking groups of n boxes sequentially, where as above n corresponds to the number of boxes in the stack, and identify the lowest value (lowest strength box) in each sub-sample grouping as well as the last value (bottom box) in each grouping. So, if we have stacks of fi ve (n = 5) and an overall population of 10 000, for example, we now have 2000 values for the lowest value and 2000 values for the last value from our original set of 10 000 values. When we select fi ve boxes, in ∼1/5 of the cases the lowest and last value will be the same, but otherwise the last value will be higher than the lowest value.

We can then generate averages of those two sub-populations. The average and standard deviation of the ‘last value’ group are the same as the average and standard deviation of the entire list, assuming a large enough population n. By comparing this average with the value that we would get from the imaginary stack test that polls the weakest box (the lowest value sub-population), we can generate an estimate of the impact on our BCT measurement from any test where a box other than the bottom one fails consistently.

This impact depends on the variation in the population of boxes, since if there is more variation, the minimum value would tend to be lower (a wider spread over which we sample the lowest strength box.) Calculating the impact for various stack heights and population variations (Figure 1) shows clearly that as the stack height increases or as the variation increases, the ‘effective BCT’ reported by considering the failure of the weakest box in the stack falls progressively further from the average box compression strength of the population as a whole. This is in agreement with the calculations of Equation 10, and the slopes of the lines in the fi gure are consistent with the coeffi cients in Table 1.

The typical fractional variation of a population of boxes in a BCT test (assuming boxes without signifi cant fabrication defects, tested in a reliable, conditioned environment) is on the order of 4–6%, and we see that for those typical boxes we might estimate the effective box strength in a reasonably sized stack of empty boxes to be 6–8% lower than the actual level of performance. That difference

Table 1. Factors multiplying the standard deviation s for compression of a stack of n boxes calculated numerically from Equation (10).

n Coeffi cient of s

1 0 2 0.564 3 0.846 4 1.029 5 1.163 6 1.267 7 1.352 8 1.424 9 1.48510 1.539



in performance estimation might cause the packaging engineer to choose a liner or a medium one grade (or more) higher than needed when designing the box, at an added cost that may be on the order of several cents per box.

EXPERIMENTAL RESULTS

The impact of box loading

To explore this shift in practice we used three different sets of boxes, each made from a different board combination and with varying physical properties and dimensions (Table 2). Each set of boxes, identifi ed here as Box A, Box B and Box C, was supplied from independent, single runs of material. Comparing the compression of individual boxes with the compression of stacked boxes (Table 3), we see that a stack of three boxes was 6–15% weaker than boxes tested individually, while a stack of fi ve was about 16–20% weaker. Consistent with the work of Kellicutt6 and Ievans,10 these losses are larger than we would expect given just the sampling impact graphed above due to losses that occur in load transfer between individual boxes in a stack. (Where we report more than one result throughout this report, it indicates that the same confi guration was retested at a separate time in the box-evaluation process, using a fresh subset of boxes. Paired results are almost always statistically similar at >95% confi dence.)

The location of the failure in each set of tests is also noted in Table 3, with the fi rst number cor-responding to the bottom box in the stack and the last number to the top box in the stack. So, for the fi rst set of testing on stacks of three boxes of Box A, fi ve stacks of the 10 stacks tested failed in the bottom box, three in the middle box and two in the top box. When more than one box in a stack failed concurrently at the peak load, they are each counted as an appropriate fraction of the failure. So, for the fi rst set of testing on stacks of three boxes of Box B, two stacks of ten failed in the bottom box, four in the middle box and two in the top box, and one stack exhibited failures at both in the bottom and in the middle box when the stack failure was identifi ed.

Figure 1. Calculated change in measured box compression strength of a stack of boxes relative to the strength measured on a single box, for variation in stack height and variation in

population strength.



It is possible to force failure to the bottom box by placing weight in the boxes above. Load was added in the form of a bag of road salt placed on a plywood board in each box. The inclusion of a plywood spreader distributed the load within the box and avoided the problem of the bag applying uneven or point pressures to one part of the box below. The effective load on the collapsing box is then the sum of the load from the compression tester and the weight applied in the boxes above the one that fails. For Box A we placed 40 lb in each of the two boxes above the bottom box. In the three-high confi guration the two top boxes each had 40 lb, while in the fi ve-high confi guration boxes two and three had added weight, effectively loading the bottom box with 80 lb more than the top (or the top three). We can then adjust the compression tester readings by adding 80 lb to get a measure of the effective BCT on the bottom box. We report both the actual measurements and the adjusted values (the actual measurements plus 80 lb to account for the additional loading) in Table 4

Forcing the failure to the bottom of the stack increases the strength measured by 4% for a stack of three and 8% for a stack of fi ve. The percentages are calculated relative to the strength measured on individual boxes. These values are in the range of our model (Figure 1) for boxes with a fractional variation (standard deviation/average, or s/m) around 5%, similar to the values observed in the single

Table 3. Compression values (lb) for individual boxes and stacks of boxes. How many times a box in a given position in the stack fails at peak load is also noted, with the fi rst number corresponding to the

bottom box in the stack and the last number to the top box in the stack (see text.)

Box A Box B Box C

ValuesFailurelocation Values

Failurelocation Values

Failurelocation

Individual 667 ± 18, 682 ± 32

– 826 ± 31, 865 ± 48

– 1148 ± 47 –

Stack of three 572 ± 19, 578 ± 28

5/3/2, 2/4/4

785 ± 52, 798 ± 37

2.5/4.5/3, 2.5/6/1.5

1043 ± 47 2.5/3.5/4

Stack of fi ve 538 ± 57 3/1/2/0/0 707 ± 36 1.33/2.33/0/0.5/0.83 959 ± 60 4/0.5/0.5/2/3

Table 4. A comparison of load at failure for stacks of empty boxes (Box A) and boxes with contents applying load to the boxes below. Note the shift in failures to the bottom boxes in the stack sand the

corresponding increase in applied load at failure.

Empty Loaded

Difference(%)Values

Failurelocation Measured Adjusted

Failurelocation

Stack of three 572 ± 19, 578 ± 28

5/3/2, 2/4/4

522 ± 37 602 ± 37 10/0/0 4

Stack of fi ve 538 ± 57 3/1/2/0/0 512 ± 17 592 ± 17 4/1/0/0/0 8

Table 2. Physical properties measured on the three boxes used in this study. Edge crush test (ECT) measurements followed the TAPPI T811 standard.

Box A Box B Box C

Nominal combination (lb/msf) 35-23-56 56-33-56 69-36-69Caliper (mils) 145.0 ± 2.4 168.8 ± 0.5 173.0 ± 3.1ECT (lb/in.) 41.8 ± 1.9 56.7 ± 4.6 71.2 ± 3.2Length (in.) 18.25 16.25 17Width (in.) 13.375 12.5 13.5Depth (in.) 7.5 10.5 12



Table 6. A comparison of peak load measured on empty and loaded stacks in different confi gurations.

Empty Loaded

Difference(%)Values

Failurelocation Values

Failurelocation

Box A, Stack of three, on pallet 525 ± 36 5.5/2/2.5 560 ± 40 8/2/0 5

Box B, unit load, column 5703 ± 210 5961 ± 150 5

Box B, unit load, interlocked 3278 ± 132 3676 ± 186 12

box testing. We can subtract these differences from those measured on the difference between columns and individual boxes, to fi nd that the impact of column stacking these boxes is about 10–12%. This is very much in line with expectations based on common understanding of the infl uence of column stacking on compression.1,6,10 Given these observations, it may be that for column compression the factors in Table 1 could be used to ‘correct’ the results in the literature and improve the accuracy of our estimations. Further experimentation would be required to verify this approach.

We performed similar experiments for Box B and Box C (Table 5). For Box B we also used 40 lb salt bags as internal loading. Due to the strength of Box C, we used 50 lb salt bags, and for the fi ve-high stack we loaded all four boxes above the bottom. However, for Box C we still did not apply enough extra load to consistently force failure to the box at the bottom of the stack. We see in Table 5 that failures did not always occur in the bottom for box C, and, thus, while we found a shift in measured strength for Box B in general agreement with expectations, for Box C the values with and without added load are statistically similar.

Beyond simple columns

Boxes are not typically used in small stacks on fi rm platens. Of additional concern in unit load testing is the interaction of the assembly of boxes with any supporting structure (e.g. the pallet) as well as the interaction of boxes with each other (for example, in interlocked or hybrid stacking patterns). In fact, the full dynamics of unit load performance on a typical shipping platform remains one of the most signifi cant areas for potentially advancing our knowledge of box requirements. If the bottom box does not fail, an interaction impacting failure dynamics in the fi eld may be missed. We were able to probe these interactions in a limited number of confi gurations using the boxes described above.

We tested three-high columns of Box A stacked on the corner of a pallet. When oriented with the length of the box along the length direction of the pallet, all four corners were supported by deck boards. In this confi guration, forcing the failure to the bottom raises the average strength measured in line with our observations above (Table 6), though we fi nd an added loss (∼40 lb, ∼7%) likely due to interactions with the pallet itself.

Table 5. A comparison of load at failure for stacks of empty boxes and boxes with contents applying load to the boxes below.

Empty Loaded

Difference(%)Values

Failurelocation Adjusted

Failurelocation

Box B Stack of three 785 ± 52, 798 ± 37

2.5/4.5/3, 2.5/6/1.5 807 ± 54 9/.5/.5 2

Box B Stack of fi ve 707 ± 36 1.33/2.33/0/0.5/0.83 754 ± 41 5/0/0/0/0 6

Box C Stack of three 1043 ± 47 2.5/3.5/4 1058 ± 71 7.5/2.5/0 1

Box C Stack of fi ve 959 ± 60 4/0.5/0.5/2/3 951 ± 93 8.5/1/0/0/0.5 0



We had suffi cient numbers of Box B specimens to test both loaded and unloaded boxes in both seven-down column and interlocked confi gurations using a fl at plate as the support base (no pallet). The need for 21 boxes in each ‘unit’ signifi cantly limited the number of replicates of each confi gura-tion. While accurately capturing the failure location also becomes a challenge for this type of large assembly of boxes, we again observed that the failures shift towards the lower boxes in the units, and the effective strength measured increased, when the upper boxes included contents that applied load to the boxes below. While the empty and loaded box values are not statistically different for the column confi guration, they are for interlocked boxes. Additional testing may have produced more robust results, but unit load testing exhausted our Box B supply. Thus, while it is clear that in more typical unit load confi gurations there is an impact when boxes are tested with contents, and that this impact is likely signifi cant enough to call into question conclusions made regarding the magnitude of the impact of different factors when empty boxes are tested (stacking pattern, pallet interaction, etc.), it is not clear if the values in Table 1 accurately represent the difference that arises from testing empty boxes. More work is needed here as well.

CONCLUSIONS

We fi nd that mathematically, numerically, and experimentally, forcing box failure to the bottom results in higher test values. While this occurs naturally for boxes in use in the fi eld, it is an interaction which to date has been overlooked by researchers examining box performance in the lab. The factors pro-vided above provide a sense of scale as to the magnitude of the impact that arises from ignoring the need to test loaded boxes. To improve the factors used in the industry to account for the impact of a box’s ‘in use environment’ on its performance, we need further testing on confi gurations where the boxes are loaded, even though it is more time-consuming and harder to do such testing. The impact on box estimation can be on the order of 5% or more, which can be as signifi cant as some of the environmental factors we are working to quantify, and which can have signifi cant cost implications.

REFERENCES

1. Fibre Box Handbook. Fibre Box Association. http://www.fi brebox.org [accessed 24 March 2010]. 2. Urbanik TJ, Frank B. Box compression analysis of world-wide data spanning 46 years. Journal of Wood and Fiber Science

2006; 38(3): 399–416. 3. Kutt H, Mithel BB. Structural strength characteristics of containers. TAPPI Journal 1969; 52(9): 1683–1688. 4. Kim E. Loss in Compression Strength of Corrugated Containers as a Function of Stacking Pattern and Dynamic Compres-

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ers. TAPPI Journal 1962; 45; republished in Maltenfort GG, Performance and Evaluation of Shipping Containers. Jelmar Publishing: New York, 1989; 141–143.

7. Singh SP. Instability of stacked pallet loads due to misalignment. Journal of Testing and Evaluation 1999; 27(5): 349–354. 8. Monaghan J, Marcondes J. Overhang and pallet gap effects on the performance of corrugated fi berboard boxes. Transac-

tions of the ASAE 1992; 35(6): 1945–1947. 9. Koning JW, Moody RC, Slip Pads. Vertical alignment increase stacking strength 65%. Boxboard Containers 1966; 74(4).10. Ievans UI. The Effect of warehouse mishandling and stacking patterns on the compression strength of corrugated boxes.

TAPPI Journal 1975; 58(8): 108–111.11. DiSalvo MH. Interactive Effects of Palletizing Factors on Fiberboard Packaging Strength. Master’s Thesis, San Jose State

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compression testing to simulate real-world stresses

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