J. Pack. Sci. Technol. Vol. 5 No. 3 (1996)

Effect of Temperature/Relative Humidity Changes on Compression
Strength of Corrugated Containers for Food Products:
How should it be modeled and evaluated?
Chiaki MURAOThe effect of cyclic environment on the compression strength of corrugated containers which are filledwith carton boxes is investigated. The compression strength characteristics at constant temperature andrelative humidity conditions are tested under various cyclic environmental conditions. These observations
give that there is the suitable time accelerated effect for the real stacking life of those in cyclic environmental condition. The cyclic environmental model conditions for compression strength are suggested toevaluate which is the optimum design at constant environmental condition.
Keywords: Corrugated container, Carton box, Compression strength, Constant environmental condition, Cyclic environmental condition, Stacking life, Optimum design

tive of the real life situation in which corrugated

1. Introduction

containers are used.
There were several studies on the compression

The compression strength characteristics of corrugated containers filled with carton boxes were

creep characteristics of corrugated containers
made from various materials in cyclic RH environ-studied at constant temperature and relative humidity conditions by using mathematical model in thements6)-13). These results indicated large varieties
associated with variation in board composition. Con-previous papers1)-5).
Compression test samples were provided insequently, generalized criteria for the compression

advance with being conditioned at 20℃,

creep characteristics of corrugated containers in

65%RH/40℃, 90%RH for more than 48 hr without
top loads. Compression creep tests were carried out

cyclic RH environments shall not be derived fromin the same constant environments of conditioning.

these results.
In this paper, the mathematical model and its

It is well known that the stacking life of corru-

analytical results1)-5) in constant environment are

gated containers is reduced by exposure to highevaluated to match up to the broad cyclic environ-

RH. Because most warehouses do not have controlled RH environments, cyclic RH is representa-

ments, and meet the failure reports of commercially sold commodities for many years.

LIFETECHNO CORP., 10-14, Kitaueno-1, Taitoh-ku, Tokyo, 110 Japan-191-

Environmental Model Conditions for Compression Strength

are tested. One is wrap around type. The other is

2. Materials and Methods
2.1 Materials and packaging styles
Corrugated containers filled with two kinds ofcarton boxes are tested in this study. One is small
size. The other is medium size.
Filling patterns of carton boxes in corrugatedcontainers (Fig. 1) are vertical style and perpendicular style. Two kinds of corrugated containers

regular slotted type. Combinations of carton boxes
and corrugated containers (Table 1 ) are provided toinvestigate the adaptability of results obtained in constant temperature and RH environments studies2)-5)
for cyclic environments.
Namely, model I, II and III are the packaging
styles of commercially sold commodities. Model Ivar. is improved on model I for cost reduction bymodifying the board composition of corrugated containers associated with distributing compression
loads uniformly to the carton boxes2),3). Model IIvar. is improved on model II by reinforcementof compression strength with optimizing top clearance4),5).
Model III-var.l is improved on model III
by reinforcement of compression strength with modifying the filling patterns of carton boxes2),3). Moreover, model III-var.2 is improved on model III-var.l
for cost reduction by modifying the board composition of corrugated containers.

2.2 Methods
Cyclic temperature and RH environments are car--192-

J. Pack. Sci. Technol. Vol. 5 No. 3 (1996)

Table 1 Specifications of test samples

Model

I

Filling patternof carton boxes

Spec, ofcarton boxes

Spec. ofcorrugated containers

130×30×165mm

394×309×168mm

Coated paper

KNN180/SCP180/KNN180g/m2

Type 1

A/F, Wrap around typewith inner joint flap
at top panel

2310g/m

I -var.

Type 1

130×30×165mm
Coated paper
310g/m2

II

Type 2110×86×176mm
447×227×181mm
Coated paper
KNN220/SCP180/ KNN220g/ m2

Type 2

A/F, Wrap around typewith outer joint flap
at side panel

III

Type 3450g/mA/F, Regular slotted type

110×86×176mm

447×227×190mm

Coated paper
450g/m2

KNN220/SCP180/ KNN220g/m2
A/F, Regular slotted type

117×30×141mm
302×245×315mm
Coated paper
KNN180/SCP160/KNN180g/m2
2

Type 4

117×30×141mm
302×245×315mm
Coated paper
KNN180/SCP160/KNN180g/m2
310g/m2

III -var.2

Type 4

0

6mm

15mm

12mm

A/F, Regular slotted type310g/mIII -var. 1

0394×309×168mm
CN 170/SCP125/CN170g/m2

2

II -var.

Top
clearance

A/F, Regular slotted type

117×30×141mm
302×245×294mm
Coated paper
CN170/SCP125/CN170g/m2
310g/m2
A/F, Regular slotted type11mm
adjustedby pads
11mm

ried out with program-controllable environmental
test chamber (Fig. 2) that is controlled the temperature to ±0.3°C and the RH to ±2.5% for set point.
Compression loads are applied through iron plates
adjusted to the anticipated stacking loads on thecommodity. That are 1177N to model I, 1187N tomodel II and 745N to model III.
Compression creep tests are made on two patterns of box arrangement. One is single corrugatedcontainer. The other is 2-tier model (Fig. 3) thatFig.3 2-tier model for the evaluation ofcompression strength loss

- 1 9 3 -Environmental Model Conditions for Compression Strength

demonstrate the most simple form of compression
strength loss during warehousing 3 ) .

hr(20°C, 65%RH).
(6) ll.5 hr (20°C, 65%RH)-1 hr (40°C, 90%RH)-ll.5
hr (20°C, 65%RH).

2.3 Cyclic environmental conditions
Two kinds of environmental condition" are set
u p t o :(a) 20°C, 65%RH(standard condition)(b) 40°C, 90%RH(high temperature and high humidity condition).
The compression creep tests are carried out inthe following 6 patterns of 24 hr period cyclic condition which starts and ends in standard condition.
(1) 4 hr cycle of 20°C, 65%RH and 40°C, 90%RH.
(2) 6 hr cycle of 20°C, 65%RH and 40°C, 90%RH.
(3) 8 hr cycle of 20°C, 65%RH and 40°C, 90%RH.
(4) 10 hr (20°C, 65%RH)-4 hr (40°C, 90%RH)-10 hr(20°C, 65%RH).
(5) ll hr (20°C, 65%RH)-2 hr (40°C, 90%RH)-ll

2.4 Evaluation
After 24 hr period test finished, the failure extent
of carton boxes filled in corrugated containers isobserved. The results of observation are ranked to
4 grades:grade 1 = nothing of failure.
grade 2 = failure to a lesser extent.
grade 3 = failure to a medium extent.
grade 4 = failure to a greater extent.
3. Results and Discussion
The model I, II and III used in this study havebeen commercially sold commodities for more than

Table 2 Effect of various cyclic environmental condition tests
for model I and I -var. on single corrugated container- 1 9 4 -

J. Pack. Sci. Technol. Vol. 5 No. 3 (1996)

Table 3 Effect of various cyclic environmental condition tests for model II and II -var on
single corrugated container
Cyclicenvironmental
condition

Model

Total

Distribution ratio of

number of
cartonboxes

each failure grade
gradel

(4)

II

10

(5)

II
II -var.

10
10

0.20
0.70

(6)

II
II -var.

10
10

0.20
1.0

grade2

grade30.20

grade4
0.80
0.80

0.30
0.40

0.40

Table 4 Effect of cyclic environmental condition No.5 (5) test for model III , III-var.land
III -var.2 on single corrugated container

Cyclicenvironmental
condition

(5)

Model

III
III -var.1
III -var.2

Total
number of
carton
boxes

40
40
40

Distribution ratio of
each failure grade
gradelgrade2

grad e3

0.90
1.0
0.97

0.05

0.05

grade40.03

tainers during warehousing.

10 years. Failure does not have taken place for

The results of cyclic environmental tests formodel I and III in the meantime, but has taken placemodel II and II-var. (Table 3) reveal that:for model II once for all in the high temperatureand high RH environments of summer season.

(a) the compression creep strength of model IIvar.1 is greater than that of model II in test con-

3.1 Single corrugated container
The results of cyclic environmental tests formodel I and I-var. (Table 2) reveal that:(a) the compression creep strength of model I-var.
is greater than that of model I in test condition
No.2 (2)～No.5 (5).
(b) the test condition No.5 (5) and No.6 (6) provide the suitable time accelerated effect, corresponding to the stacking life of corrugated con-dition No.5 (5) and No.6 (6).
(b) the test condition No.5 (5) and No.6 (6) provide the suitable time accelerated effect, corresponding to the stacking life of corrugated containers during warehousing.
The results of cyclic environmental tests formodel III, III-var.l and III-var.2 (Table 4) reveal that:(a) the compression creep strength of model IIIvar.-l and that of model III-var.2 are nearly-195-

Environmental Model Conditions for Compression Strength

Table 5 Effect of cyclic environmental condition No,5(5) and No.6(6) tests for model I
and I -var. on 2-tier model
Cyclicenvironmental
condition

Model

Total
number of
cartonboxes

Distribution ratio of
each failure grade
grade 1

grad e2(5)I
I -var.

90
90

0.91
1.0

0.09

(6)

I

90

0.91

0.09

grade3

grade4

Table 6 Effect of cyclic environmental condition No,5(5) and No.6(6) tests for model II
and n -var. on 2-tier model
Cyclicenvironmental
condition

(5)

(6)

Model

Total

Distribution ratio of

number of
cartonboxes

each failure grade
grade1

grade2

grade3

grade4

0.10

0.07

0.40

II

30

0.43

II -var.

30

0.93

II
II -var.

30

0.60

0.03

30

0.93

0.07

0.07
0.34

0.03

equal and greater than that of model III in test

tainers during warehousing.

c o n d i ti o n N o . 5 (5 ) .

The results of cyclic environmental tests for(b) the test condition No.5 (5) provides the suitablemodel II and II-var. (Table 6) reveal that:time accelerated effect, corresponding to the(a) the compressive creep strength loss of model

stacking life of corrugated containers during
warehousing.

II is greater than that of model II-var. in test
condition No.5 (5) and No.6 (6).
(b) the test condition No.5 (5) and No.6 (6) pro-

3.2 2-tier model
The results of cyclic environmental tests formodel I and I-var. (Table 5) reveal that:(a) the compressive creep strength loss of model I
is greater than that of model I-var. in test condition No.5 (5).
(b) the test condition No.5 (5) and No.6 (6) provide the suitable time accelerated effect, corresponding to the stacking life of corrugated con-vide the suitable time accelerated effect, corresponding to the stacking life of corrugated containers during warehousing.
The results of cyclic environmental tests formodel III and III-var.2 (Table 7) reveal that:(a) the compressive creep strength loss of model
III is greater than that of model III-var.2 in test
condition No.5 (5).
(b) the test condition No.5 (5) provides the suitable-196-

J. Pack. Sci. Technol. Vol. 5 No. 3 (1996)

Table 7 Effect of cyclic environmental condition No,5(5) test for model III and III-var2. on
2 - t i e r
m o d e l

Cyclicenvironmental
condition

(5)

Total
Model

Distribution ratio of
each failure grade

number of
cartonboxes

grade1

grade2

grade3

grade4

III

120

0.86

0.10

0.02

0.02

III -var.2

120

0.99

0.01

time accelerated effect, corresponding to theextent of carton boxes.

stacking life of corrugated containers during
warehousing.

x1 = the distribution ratio of carton boxes to theg r ad e 1 fai lu r e .
x2 = the distribution ratio of carton boxes to the

3.3 Criteria
There are some essential requisites as criteria forlong term phenomena are performed.
Namely,(a) the test methods for final evaluation shouldhave suitable time accelerated effect.
(b) clear judgement should be achieved by thesecriteria.
The following steps are proposed to satisfy theserequisites.

g r ad e 2 fai lu r e .
x3 = the distribution ratio of carton boxes to theg r ad e 3 fai lu r e .
x4= the distribution ratio of carton boxes to theg r a d e 4 f ai l u r e .
Σxi= 1.0
S t e p 3
The composite summation of weighted values tothe failure extent of carton boxes is defined as foll o w s .

S t e p 1
Next values are multiplied to the distribution

W = w

ratio of each 4 grade having been ranked according
to the failure extent of carton boxes. The multipli-

+ w

2

(2)

w h e r eer are 0 to grade 1,1 to grade 2, 3 to grade 3 andW = the composite sum of weighted values to the

5 to grade 4.
S t e p 2

failure extent of carton boxes for stacked corrugated containers.

The summation of weighted values to the failure

w1 = the sum of weighted values to the failuree x t e n t o f c a r t o n b o x e s a r e c a r r i e d o u t.
w = x 2 + 3 x 3 + 5 x 4

1extent of carton boxes for the single corrugated containers test.

(1)

w2 = the sum of weighted values to the failure
extent of carton boxes for the 2-tier modelt e s t .
S t e p 4
The compression strength for corrugated con-Where
w = the sum of weighted values to the failure-197-

Environmental Model Conditions for Compression Strength

Table 8 Evaluation for the compression strength of filled corrugated containers
by using the composite sum of weighted values to the failure extend of
carton boxes on the cy clic environme ntal condition tests

Model

Cyclic environmental
condition No.5 (5)
w1

w2

I
I –var.

0.69
0.07

II
II –var.
III
III -var. 1
III -var. 2

Cyclic environmental
condition No.6 (6)

W

w1

w2

W

0.09

0.78

0.49

0.09

0.58

0

0.07

0*

0*

0*

4.00

2.31

6.31

3.20

1.20

4.40

0.30

0.21

0.51

0

0.07

0.07

0.20
0*

0.26
0*

0.46
0*

0*

0*

0*

0.03

0.01

0.04

0*

0*

0*

* Estimated value.
- Measurements are omitted.

tainers is determined by using guarantee index e asfollows.

Acknowledgement
W ≦ ε

(3)
The author would like to thank Dr. T. Yano, pro-where

fessor at the Yokohama National University, and N.

ε= the guarantee index for nothing of commodi-

Tachikawa, executive managing director at theties failure during warehousing.

Ajinomoto Co., Inc., for their encouragement andAccording to the above procedures, the results

valuable comments for this paper and H. Uehara,
staff at the Rengo Co., Ltd., for the measurements.

of calculations (Table 8) by using the data of Table
1~7 in the test condition No. 5 (5) and No. 6 (6)References
1) C. Murao, J. Pack. Sci. Technol., 4(4), 287(1995)
2) C. Murao, J. Pack. Sci. Technol., 4(4), 295(1995)
3) C. Murao, J. Pack. Sci. Technol., 5(1), 32(1996)
4) C. Murao, J. Pack. Sci. Technol., 5(1), 43(1996)
5) C.MuraoJ.Pack.Sci.Technol.,5(2) , 99(1996)
6) V. L Byrd, Tappi, 55(2), 247(1972)
7) V. L. Byrd, Tappi, 55(ll), 1612(1972)
8) V. L. Byrd and J. W. Koning, Jr., Tappi, 61(6),35 (1978)9) V. L. Byrd, Tappi, 67(7), 86(1984)

reveal that;(a) the results of cyclic environmental test condition No. 5 (5) satisfy the essential requisites
being necessary to criteria and coincide withthe failure results of commercially sold commodities for more than 10 years.
(b) the suggested optimum values in the test condition No. 5 (5) are ε=1.0-198-

J. Pack. Sci. Technol. Vol. 5 No. 3 (1996)

10) V. L. Byrd, Tappi, 69(10), 98(1986)

ping Containers" (Edited by G. G. Maltenfort),

11) C.H. Leake.Tappi, 71 (10), 71(1988)
12) A. Boonyasarn, B. R. Harte, D. Twede and J. L.
Lee, Tappi, 75(10), 79(1992)
13) U. I. Ievans, "Performance and Evaluation of Ship-

Jelmar Publishing Co., Inc., pl56(1989)(Received 20 June 1995)(Accepted 29 March 1996)

食品外装段ボール箱圧縮強さに及ぼす温湿度環境の影響：モデル化と評価法
村尾千秋*
カートン個箱入り外装段ボール箱圧縮強さに及ぼす、サイクリック温湿度環境の影響を検討した。
一定温湿度環境で測定したカートン個箱入り外装段ボール箱圧縮強さ特性を、各種サイクリック環境
条件化で確認した。これらの確認実験の結果、現実の段ボール箱の積付寿命に対応する、適度の時間促進効果を有するサイクリック環境モデル条件が存在することが判明した。このようにモデル化したサイクリック環境条件で圧縮強さを確認すれば、一定環境条件化のテストにより設計された外装段
ボール箱圧縮強さの最適設定の妥当性を評価できることが判明した。
キーワード：外装段ボール箱、カートン個箱、圧縮強さ、一定環境条件、サイクリック環境条件、
積付寿命、最適条件*（株）ライフテクノ（〒110東京都台東区北上野1-10-14）-199-

J. Pack. Sci. Technol. Vol. 5 No. 3 (1996)

Method of Optimum Design for Compression Strength of CorrugatedContainers for Food Products

Chiaki MURAO

The structure of the mathematical model for the compression strength of corrugated containers with
contents is investigated. The characteristics of 3-dimensional curved surface which consists of 3 state variables in the model are revealed. The top clearance and the board composition of corrugated containers
filled with carton boxes are two controllable factors. The graphical method of optimum design for those isdemonstrated. The flow chart of optimum design for the compression strength of corrugated containers
with contents is proposed by using this method and the cyclic environmental model condition. It is indicated that the theme of this study is similar to the bimatrix game of game theory.
Keywords: Corrugated container, Carton box, Compression strength, Mathematical model, Optimum
design, Graphical method, Top clearance, Board composition

1. Introduction
The mathematical model was proposed for thecompression strength of corrugated containers with
contents in the previous paper1). Furthermore, thecharacteristics of the model was described in detailfor the various carton boxes of contents2),3).
Top clearance means the clearance between the
top panel of corrugated containers and contents.
The correlative relationship between the top clearance and the coefficients of the model was
explained4),5).

ate which was the optimum design under the constant environmental condition 6 ) .
In this paper, the structure of the model is analyzed in the first half. The graphical method of optimum design for the compression strength of corrugated containers filled with carton boxes are shownby using this analyzed results.
Finally, the flow chart of optimum design is proposed, which is capable of adapting not only to carton boxes but also to various contents.

2. Structure of Mathematical Model

Moreover, the cyclic environmental model conditions for the real stacking life were studied to evalu-

The mathematical model which was studied in

LIFETECHNO CORP., 10-14, Kitauenol, Taitoh-ku, Tokyo, 1 10 Japan- 2 0 0 -Method of Optimum Design for Compression Strengtharound type is only used in the case of the compres-the previous papers1)-5) was shown as follows.

sion strength of contents is sufficient to the stacking
C

=B+αD

(1)

C

=βA

(2)

loads. Therefore, the compression strength designof corrugated containers is omitted.
The above mentioned two factors to design are
expressed as follows.

where
C = the compression strength of filled corrugatedδ{} = the initial top clearance of filled corrugatedcontainers.

containers without stacking loads.

B = the compression strength of contents which

δ()i = the i-th value of δ() which is constituted of nare filled in corrugated containers.

cases of various top clearances.

D = the compression strength of empty corrugated

M = the board composition of corrugated containers.

containers.
A = the anticipated stacking loads to the bottom

Mj = the j-th case of M which is consisted of m casesof various composition.

corrugated containers in storage.
α = coefficient which means the contribution of

The equations of the model, corresponding to the
combination of δ()i and Mj, are shown as follows.

empty corrugated containers to the compression strength of filled corrugated containers.
β = coefficient which means compensating factor
for the compression strength loss during warehousing being defined by 2-tier model 3),5) .

Cij = B+αijDj(3)

Cij=βijA(4)

The dimension and the board composition of carton boxes will be determined at the commodities

where

designing.

Cij = C which corresponds to the combination of δ()ia n dTherefore, B is a given factor in the equation (1).

M j .

D j = D which corresponds to M j .

A, which is determined by distribution condition,

αij = α which corresponds to the combination of δ()iis also given factor similarly in the equation (2).

and Mij.

The length and width of corrugated containers

βij= β which corresponds to the combination of δ()iand Mj.

are given by the filling pattern of carton boxes.
Two factors still remain to design corrugatedcontainers. Namely, one is the height which isequivalent to the top clearance of corrugated con-

It is supposed that Dj is not affected by δ()i whichtainers and another is the composition of the corrugated board.

varies within a practical limit. Moreover, we should

The type of corrugated containers is restricted
only the regular slotted one in this paper. Wrap-

Mｊ has an allowable lower limit to support the stacking loads.

notice that δ()I has a perfect degree of freedom but-201-

J. Pack. Sci. Technol. Vol. 5 No. 3 (1996)

The theme of this study is described that it is todesign the minimum D j by the optimum combina-

constant B/A are straight lines and intersect at apoint of B/A of β i j -axis.

tion of αij and βij. The combination of αij and βij iscorresponding to δ()i and Mj. Therefore, the theme is[Corollary 6-2] The contours of same S Dj at various B/A are parallel straight lines.

eqivalent to design the minimum Dj by the optimum[Corollary 6-3] The slopes of straight lines are

combination of δ ( ) i and M j .
The equation (3) and (4) give

S

Dj

=

A(βij-

2/

where

B/A)/αij(5)[Corollary 6-1] means Corollary-1 of the equation(6).

If δ()i and Mj are not optimum design, the follow-

The standardization of the equation (5) givesS

Dj, equal to A/Dj.

D j = D j /0.5A=2(β i j -B/A)/ α i jing equations are presented instead of the equat i o n ( 4 ) .

( 6 )

Cij=βijKA

(7)

where
S

Dj = the state variable derived from the standard-whereization of the equation (5) by Dj = 0.5A, which

K = t h e d e v ia t i o n f a c t o r w h i c h e x p r e s s e s t h ecorresponds to D j at B/A=0.5, α i j =1.0 and
βij=l.0.

degree of deviation from the optimum design.

The equation (5) and (6) provide the 3-dimen-

It is shown by the follows.
If C i j ＞optimum C i j , then K>1.
If C ij ＞optimum C ij , then O<K<1.

sional curved surface to the model. Equation (6)gives the followings.

Comparing the equation (7) to (4), we indicate
the following characteristics.

If βij=constant, then S Dj is proportional to l/αij.
If αij=constant, then S Dj is linear to βij.
Similarly, equation (6) gives the sensitivity of S Dj
corresponding to variation of αij and βij as follows.
If β i j = constant, then[Corollary 7] KA in the equation (7) is equivalent to A in the equation (4).

The equation (7) shows that the imaginary stacking loads of the non-optimum design are K times

If α i j = constant, thenbigger as compared with the real stacking loads ofthe optimum design.
The equation (3) and (7) give

It is indicated that the contours of S Dj, which areprojected to βij-αij plane, show the following characteristics.

Dｊ= KA (βij- B/KA)/αij

The standardization of the equation (8) gives[Corollary 6-1] The contours of variousS

D j at-202-(8)

Method of Optimum Design for Compression StrengthS

Dj, = Dj/0.5KA = 2(βij-B/KA)/αij.

Considering those results, the variables are set(9)

as the range of αij and βij to αij= 0.6-1.2 and βij= 1.0~4.0 in this study. β ij = 4.0 corresponds to the casewhereof empty corrugated containers of which theSwidth/length ratio of dimension is small, for exam-

D j = the state variable derived from the standardization of the equation (8) by Dj = 0.5KA, which

ple 0.5.

corresponds to Dj at B/KA=0.5, αij=1.0, and βij =1.0.

Fig. 1 shows the 3-dimensional curved surface ofthe equation (6) of which B/A = 0.50.
Fig. 2 shows the case of B/A = 0.25 and 0.75.
Fig.1 and 2 reveal the followings.

Replacing A of [corollary 6-1, 6-2, 6-3] by KA, wehave a generalized form of corollary.

1) S Dj is maximum at αij= 0.6 and βij = 4.0 and minimum at α ij = 1.2 and β ij = 1.0. Those are monotone increasing curved surfaces from a point of

3. Graphical Method of Optimum Design

minimum S D j to a point of maximum S D j .
The graphical method of optimum design is2) If βij is small, then S Dj is also small. If βij increas-

investigated for the corrugated containers filled withcarton boxes.

es, then S Dj increases too. In the1 same βij section,

3.1 Graph of mathematical model
The studies on the characteristics of αij and βij forthe corrugated containers filled with carton boxes
were made under the constant environmental condi-

3) If B/A increases, then the curved surface moves

tion of 20°C, 65%RH/40°C, 90%RH2)-5). Those gave
the following results.
1) The range of αij is 0≦ αij ≦ 1.2.
2) The function form of αij is the monotone increasing function of δ()i within the range of 0 ≦αij≦
1.0.
3) There is a large probability that the upper limit of
αij increases to 1.2.

if α ij decreases, then S D j increases rapidly.
to the positive direction of β ij -axis by the increment of B/A.
The matters to take care are summarized as follows.
1 ) S ma l l β i j i s t h e f i r s t .
2) Large αij is the second. Considering the function
form of βij to δ()i, βij has the various level of lower
limit in accordance with the packaging style.
Therefore, if it is difficult to keep βij small, thenlarge αij is important.
3) If the increment of B/A is 0.5, then the decrement of S D j is approximately 1.0~2.0.

4) The range of βij is 1.0 ≦ βij <4.0.
5) The function form of βij could not be described inunitary one.
The following two forms of βij were suggested.
① βij is the monotone increasing function of β()i.
② βij has a minimum within the practical range ofβ()i.

Fig. 3 shows the contour map of S D j , which is
projected to βij-αij plane, in the case of B/A=0.50.
If B/A=0.75, then the contour lines of S D j , whichis shown in Fig. 3, moves to the positive direction ofthe βij-axis by 0.25. If B/A = 0.25, then the contour
lines of S Dj moves to the negative direction of the βij-

-203-

J. Pack. Sci. Technol. Vol. 5 No. 3 (1996)

axis

Fig.2 3-dimensional curved surface of the model (B/A=0.25, 0.75)- 2 0 4 -

by

0.25.

3.2 Graphical method of optimum
design for δO
The graphical method of optimum
design for δ ( ) which is the initial top
clearance of corrugated containers
filled with carton boxes is proposed.
The studies was made on the characteristics of coefficients a and (3 inthe equation (1) and (2) 2 ) - 5 ) . The dataof those are applied to explain theprocedures on the graphical methodof optimum design.
Fig. 4 shows the filling patterns of
the carton boxes in corrugated containers. Table 1 shows the specifications of test samples.
3.2.1 Determination of environmental conditions and creep time
for the measurement of B
The contour map of S D j is used to
the graphical method. A is given.
T h u s w e s h o u l d d e t e r mi n e b y w h a t
conditions B is measured. The condition for measurement of B is determined to 5 min. creep time in 40°C,
90%RH by the following study.
( 1 ) C a s e I
A=1187 N, B=4 12N, C=1177 N
then C/A=0.99, B/A=0.35
where
A = the anticipated stacking loads tothe bottom corrugated containers in 3 high stacks of palletizin g.

Method of Optimum Design for Compression Strength

rugated containers which is classified as
C a s e I d u r i n g d i s t r i b u t i o n , f a i lu r e t a k eplace once for all in the high temperatureand th e high relative hu midity environmen ts.
By considering C/A=1.0 as the approximate index for optimum design, then theabove C/A=0.99 reflects accurately the
results of a long term distribution.
(2) Case II
A=745N, B=294N, C=843N
then C/A=1. 13, B/A=0.39
0.5

The commodities classified as Case II

1.0

also have been sold for more than 10
Fig.3 Contour map of sDi protected to pij-cxij plane (B/A = 0.5)

years. Failure does not have taken placein the term. Considering C/A=1.0 is theapproximate index for optimum design,then the board composition of the corrugatedcontainers classified as Case II is over specifications
to some extent for compression strength. Therefore,
the ab ove C/A=1.13 also reflects accurately the
results of a long term distribution.

3.2.2 Optimum design procedure for δ O
The procedure in optimu m desig n for δ ( ) are
shown as follows.
Step lPlot the set of the measured value of α ij and β ij
corresponding to the variation of δ ( ) i to the β ij -α i j
B = the compression strength of carton boxes filledin corrugated containers at 40°C, 90%RH and 5
min. creep time conditions.
C = the compression strength of corrugated containers filled with carton boxes at the same
conditions of B.
According to the survey of the failure of the cor-plane, and draw the locus curve.
Step 2Give the tangent to the locus curve from a pointof B/A on the βij-axis.
The touch point of the tangent to the locus curve
gives the suggested optimum δ ( ) (δ ( ) , o p t ).
Step 3- 2 0 5 -

J. Pack. Sci. Technol. Vol. 5 No. 3 (1996)

Table 1 Spe cifications of test sa mple s
Case

I

II

Filling patternof carton boxes

Type lType 2

Spec. of carton
boxes

Spec, of corrugated containers

110×86×176mm
Coated paper
450g/m2

447×227×190mm
KNN220/SCP180/KNN220g/m2
A/F
Regular slotted type

117×30×141mm
Coated paper
310g/m2

302X245X315mm
KNN180/SCP160/KNN180g/m2
A/F
Regular slotted typeThe slope of the tangent gives the suggested
minimu m S D j (D o p t ).
Fig. 5 shows the graphical method of finding
δ(),opt for the Case I. The points of δ()1, δ()2, δ()3, δ()4and δ()5 indicate the set of measured αij and βij corresponding to the variation of δ() which are 6, 9, 12, 15
and 20mm. B/A is expressed as 0.35.
The δ O,o p t is approximately 14mm and the D o p t is
2.3.
Fig. 6 shows the same procedure to the Case II.
A vriation of δ() are 5, 8, ll, 14 and 20mm. From Fig.
6, it can be stated that if B/A is expressed by 0.39,
then the δ(),opt is approximately 13mm and the Dopt is1.2. B/A = 0.39 for the Case II corresponds to
C/A=1.13 in accordance with the previous section
study. If β ij =1.0 is the index for optimum design,t h e n K = 1 .1 3 i s d e r i v e d f r o m t h e e q u at i o n ( 7 ) .

the δ(),opt is approximately 13mm. If K is 0.5, equal toReplacing A by KA in accordance with the corollary

B/KA=0.78, then the δ(),opt is approximately 12mm.

(7), B/KA=0.35 can be obtained instead of

It is revealed that a large variation of K or themeasurement error of B exerts an influence to a

B/A=0 .39.
Therefore, it is correct that the tangent for βij=1.0

lesser extent on δ(),opt for the Case II. It is suggested

intersects a point of B/KA=0.35 on the βij-axis.

that there is no significant difference for the Case I

Fig. 7 shows how variation of K exerts an influence on (),opt. If K is infinity, equal to B/KA=0, thenby the observation of the locus curve in Fig. 5.
Finally, the fixed value of B/A, for example 0.5, is- 2 0 6 -Method of Optimum Design for Compression Strength

The intersection point of contours at the βij-axis
moves from B/A to B/KA in accordance with the[corollary 7,6-1]. The slope of contours is multiplied
by K in accordance with the [corollary 7, 6-3].
Consequently, the slope of the tangent touching
the locus curve, equivalent to Dopt, is shown as follows.
If K>1, then Dopt is over K times bigger as compared with that of the optimum design at K=1.0.
If O<K<1, then Dopt is K times smaller as compared with that of the optimum design.
Thus the variation of K or the measurement errorof B exerts an significant influence on Dopt.

3.3 Optimum design procedure for M
The optimum design for M can be obtained byusing the cyclic environmental model condition forcompression strength of corrugated containers in
the p revious pap er 6 ) .
The procedure in the optimum design of M are
shown as follows.
Step lDetermine the appropriate M and find out theapproximate δ(),opt.
Step 2
Test the combination of the M and the approximate δ(),opt by the cyclic environmental model condition6).

If W>1.0, then the higher compression strength ofM.
If W<0.1, then the lower compression strength of M.
wherecapable of using for finding out the approximate
value of δ(), opt in the Case I and II.
How does the variation of K or the measurement
error of B exert an influence on Dopt?

- 2 0 7 -W= the composite sum of weighted values to thefailure extent of carton boxes for stacked corrugated containers6) .

J. Pack. Sci. Technol. Vol. 5 No. 3 (1996)- 2 0 8 -Method of Optimu m Desi gn for Compression Strength

S

t

e

p

3

not be concluded that there is only one equilibrium

Back to step 1 untill W reaches to the range of0 . 1 ≦ w ≦ 1 . 0

point for all cases of corrugated containers with contents. There is a possibility to have two or threeequilibrium points. It can be expected that the func-

It is interpreted that W>1 corresponds to O<K<1
and W<0.1 corresponds to K>1 in this paper.
The value of K depends on the distribution
p r o c e s s .
Therefore, the above mentioned relationship of
W to K varies similarly according to the distribution

tion forms of the locus curve of δ()i more clearly
revealed by the accumulation of data.
The theme of this study is similar to the problem
of game theory which is a bimatrix game of αij andβij7) . In the case of multi-equilibrium points, the strategy of game theory is to be reffered.

process. Deviation of S D j from D opt is attributed tothe error of K, B or initial deviation of M from theoptimum board composition. This can be correctedby the procedure in step 2. Finally, δ(),opt and Dopt are
obtained by the recycle of the step 1 and 2.

Acknowledgment
The author would like to thank Dr. T. Yano, professor at the Yokohama National University, and N.
Tachikawa, executive managing director at the4.Conclusion

Ajinomoto Co., Inc., for their encouragement andFig. 8 shows the flow chart of optimum design forthe compression strength of corrugated containers
with contents. The equations and corollaries are
established by not only carton boxes but also various contents. It is suggested that the methodologyof this study is capable of being adapted to the corrugated containers with various contents.
The function forms of various locus curve of δ{}ishall be ensured by further data accumulation. Itmeans that αij and βij for the various styles of packaging and various board compositions shall be measured.
As the results of the data accumulation, a numberof trial cycle to reach the optimum design shall bedecreased, for example 1 to 3 cycles. From thegraphical method of this study, there is only one
equilibrium point foe the Case I and II. But it does

-209-

valuable comments for this paper and H. Uehara,
staff at the Rengo Co., Ltd., for the measurements.

References
1) C. Murao, J. Packaging Sci. & Technol. Japan, 4(4),287 (1995)
2) C. Murao, J. Packaging Sci. & Technol. Japan, 4(4),295 (1995)
3) C. Murao, J. Packaging Sci. & Technol. Japan, 5(1),32 (1996)
4) C. Murao, J. Packaging Sci. & Technol. Japan, 5(1),43 (1996)
5) C. Murao, J. Packaging Sci. & Technol. Japan, 5(2),99 (1996)
6) C. Murao, J. Packaging Sci. & Technol. Japan 5(3),191 (1996)
7) T. Nishida, "Game theory" JUSE Press, pl40, (1992)(Received 21 September 1995)(Accepted 24 May
1996)