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Lecture 14:Lecture 14:Design of paper and board

packages

Design of paper and board packaging: stacking, analytical methods.

Software such asBillerud Box Design, EUPS,

Model PACK & Korsnäs

After lecture 14 you should be able to

• describe the theoretical foundation for, and use, the most important analytical expressions for box compressionimportant analytical expressions for box compression strength/resistance

• describe the theoretical foundation for, and use,analytical approaches for determination of the bending stiffness of paperboard and corrugated board

• qualitatively discuss the influence of non perfect stacking• qualitatively discuss the influence of non-perfect stacking

• acknowledge the use of different types of computer software for prediction of the stacking strength of packages

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Literature• Pulp and Paper Chemistry and Technology - Volume 4,

Paper Products Physics and Technology, Chapter 10Paper Products Physics and Technology, Chapter 10

• Paperboard Reference Manual, pp. 119-128

• Fundamentals of Packaging Technology, Chapter 15

• Handbook of Physical Testing of Paper, Chapter 11

The design procedure

• Theoretical predictionsL b t t ti• Laboratory testing

• Full-scale testing

Design – Implement – Test!!

Not different from the automotive or many other types of industries!

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Loads during transport and storage

• Transport between manufacturer, wholesaler and retailer by different types of vehicles (truck retailer by different types of vehicles (truck, railcar, aircraft, ship etc.)

• Reloading by, for example, forklifts

• Many time-consuming manual operations at wholesalers and retailers

• Varying climate conditions (temperature and moisture)

Stacking of boxesStatic compression strength (BCT/BCR)

Top-load compression of the most stressed package.

Beldie, 2001

Most stressedpackage

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Methods for determination of box compression strength/resistance• Laboratory and service testing

+ Closest to reality and reliable+ Closest to reality and reliable- Time consuming and expensive to do parametric investigations

• Empirical analytical calculations+ Quick to use with acceptable accuracy in many applications- Models approximate and less useful for parametric studies

• Numerical simulations of box deformation based on the finite element method (FEM)

+ In general high accuracy and easy to do parametric investigations+ Understanding of deformation and damage mechanisms - Not straight-forward to use and still not fully developed for every paper and

board application

Box Compression Test (BCT)Box Compression Resistance (BCR)

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Cartonboard boxes

Box compression resistance of rectangular box

Consider a box subjected to compression loading due to stacking.

1. At low load levels, the load is evenly distributed along the perimeter of the box

2. At a certain load the panels of the box buckle in a characteristic way

3. At the corners of the box the corners themselves prevent buckling of the panels

4 L d i th i il i d b ll4. Load is then primarily carried by small zones at the corners of the box

5. Failure of the box finally occurs by compressive failure at the corners

Grangård (1969, 1970) shows that the compression strength of CARTONBOARD boxes (the BCT-value) correlate well with the strength of laboratory tested panels.

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Buckling of paperboard boxes

Observation:In-plane stiffness of panel is in

0,5 %

St r ain

300 mm

400 mmIn plane stiffness of panel is in general much larger than bending stiffness

Panel 1: This panel wants to buckle, i.e. the panel would like to deform in the x1-direction.

1

2

Bulge20 mm300

mm

Panel 2: The in-plane deformation of this panel is small, i.e. this panel will not deform very much in the x1-direction.

x1

x2

x3

Consequently, close to the corners Panel 1 cannot deform in the x1-direction, and the corners will remain primarily vertical.

Ultimate load Ultimate load (based on the yield stress in compression) of a simply supported isotropic plate subjected to uniform compressive loadingTimoshenko (1936)

2sc

c 23(1 )

t EP

π σ

υ=

−

ultimate load of buckled panelplate thickness

cPt

=

=

sc

plate thicknessPoisson's ratio

= in-plane Young's modulus YIELD STRESS IN COMPRESSION

t

Eυ

σ

==

=

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Modifications for an anisotropic plate

• Introduce the geometric mean of the bending stiffness

b b bMD CDS S S=

bending stiffness

• Introduce the bending stiffness per unit width, Sb, instead of Young’s modulus, E, and the panel thickness, t

• Neglect influence of Poisson’s ratio

3

12b EtS =

SCT• Replace σsc by the short span compression strength Fc

SCT per unit width

• THEN FOR A PANEL:

scσ →SCT

cFt

c 2 SCT b bc MD CDP F S Sπ=

Panel Compression Resistance

cP

Grangård (1969, 1970)

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Short Span Compression Strength

0 7 mm

SCTcF

0.7 mm

BOX compression resistanceCartonboard boxes

Grangård’s formula: SCTP k F S= b b bMD CDS S S=Grangård s formula:

The constant k, that is introduced instead of 2π,may vary depending on the dimensions of the box and the design (type of box).This constant needs to be determined through

bcP k F S= MD CDS S S

This constant needs to be determined through extensive testing.The quality of the creases will also affect k.

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A comment on fibre orientation and mechanical properties

Board dried with 2 % stretch in MD and free drying in CD

Corrugated board containers

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Stacking strength of corrugated board boxes (15 RSC boxes)

• Mean box compression strength, 5764 N

• Maximum, 6420 N• Minimum, 5100 N• Standard deviation, 374 N• Coefficient of variation• Coefficient of variation,

6.5 %

Analysis of typical load-deformation curve

A. Any unevenness in the box is l ll d t T lilevelled out. Top crease lines begin to roll.

B. The steepest corners of the box start to take load.

C. Sub-peak caused by small-scale yielding of one of the fold crease lines.

D. Buckling of long panels.E. Maximum load. Collapse of

box corners and buckling of short panels.

F. Localized stability

Load versus deformation for an A-flute RSC-box using fixed platens.Load versus deformation for an A-flute RSC-box using fixed platens.

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Usefulness of box compression strength

• Boxes are tested individually.If boxes are stacked in patterns other than column the full strength potential will not be realized.

• Climatic conditions may degrade box compression strength.

• Creep affects the results considerably.y

• The box may be subjected to dynamic loading, such as vibrations, that will accelerate failure.

BOX compression strengthMcKee’s formula

0,75 0,25 0,5P F S Zβ=c cP F S Zβ=Pc = Box compression strengthFc = Compressive strength of plane panel (ECT)S = Geometric mean of MD and CD bending stiffnessZ = Perimeter of box

b bMD CDS S

β = Empirical constant

Note! The exponentials in the equation above can have slightly different values for different types of boxes!

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The McKee modelTheoretical foundation

Semi-empirical approach for description of the post-buckling behaviour

b

,

c

CR

c

PPFc b

===

=

ultimate strength of the panelbuckling load for simply supported plate

edgewise compression strength of panel (ECT)constants

( ) 1b bc c CRP c F P −=

The McKee modelTheoretical foundation

Buckling load for thin orthotropic panel

212 MD CDCR CR

S SP kW

=

where

2 2 2

2 2 212CR

r nk Kn r

π ⎛ ⎞= + +⎜ ⎟

⎝ ⎠

t

W⎝ ⎠

1/ 4

MD

CD

S trS W

⎛ ⎞= ⎜ ⎟⎝ ⎠

n is related to the buckling pattern

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1. The parameter K is a complex function of

The McKee modelApproximations

several corrugated board and liner para-meters, but the value K = 0,5 was adopted by McKee without further notice.

2. The parameter was set to 1,17 from practical measurements

( )1/ 4MD CDS S

from practical measurements.3. The panel width was related to the

perimeter Z by W = Z/4, i.e. a square box.

( ) ( ) ( )12 2 2 1 14bb b b b b b

c c MD CDP c F S S Z kπ−− − −=

Simplified expression for total box load

( ) ( ) ( )c c MD CD

where k is a modified buckling coefficient.

Further simplifications: 1 1,33 when 0,76− = ≈bk bfor boxes with depth-to-perimeter values 0,143≥

( )1 2 1b

b b b bP F S S Z−

−( ) 2 1b b b bc c MD CDP aF S S Z −=

Evaluation of constants a and b for A-, B- and C-flute RSC-boxes yields in SI-units:

( )0,250,75 0,5375 b b

c c MD CDP F S S Z=

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Comments on McKee’s formula• The constants evaluated for typical U.S. boxes in the

early 1960searly 1960s

• It assumes that the boxes are square, but modification for the effect of aspect ratio exists.

• It predicts maximum load, but not deformation.

• Influence of transverse shear is ignored. Examining boxes during failure often reveals a pattern that suggestsboxes during failure often reveals a pattern that suggests the presence of shear near the corners (leaning flutes).

1. Global buckling

Failure in corrugated board panels

1. Global buckling

2. Failure initiated by local buckling in the corner regions of the concave side of a panel

3. Multi-axial stress state! Nordstrand (2004)

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Influence of box perimeter and height on BCT-value

Why linear?

Box compression strength/N

Height/mm

Perimeter/mm

Why flat?

Micromechanical models

Tensile stiffness:=⎧

⎨=⎩

b

EA EBtE Et per unit width

Bending stiffness:

3

312

12

⎧= =⎪⎪

⎨⎪ =⎪⎩

b

BtS EI E

tS E per unit width

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Micromechanical models of corrugated board

linertlinertt t

0coreE ≈ α – take-up factor

In-plane stiffness of corrugated board panels

0MD

flutingcore flutingCD CD

core

Et

E Et

α

≈

⎛ ⎞= ⎜ ⎟

⎝ ⎠

α take up factortfluting – fluting thicknesstcore – core thickness

,,, , liner topliner bottomliner bottom liner topMD MD MD

ttE E E

t t⎛ ⎞⎛ ⎞

= + ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

,,, , liner topliner bottomliner bottom core liner topcoreCD CD CD CD

t t

tt tE E E Et t t

⎝ ⎠ ⎝ ⎠⎛ ⎞⎛ ⎞ ⎛ ⎞= + + ⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠ ⎝ ⎠

Rules of mixture from parallel model for lamellar composites

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Simplified expressions for the bending stiffness of corrugated board panelsA first order approximation in both MD and CD neglects the influence of the medium. However, the medium should give an appreciable

2 2 2

2 2 2liner

liner liner linert t tI Bt Bt Bt

⎛ ⎞⎛ ⎞ ⎛ ⎞= + = ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

contribution to the bending stiffness, particularly in CD.

}{2 2 2

Steadman2 2 2

linerb linerliner b

t t tS E t E S⎛ ⎞ ⎛ ⎞ ⎛ ⎞

= = = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

Steiner’s theorem!

}{2 2 2⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

• More advanced models exist, but they are cumbersome to use, andcannot be considered to be part of a fundamental course on packaging materials. Needs to be implemented into easy-to-use software.

• Numerical calculation of the bending stiffness is of course also possibleand explored in the scientific literature.

EUPS

EUPSEuropean standard for defining the strength characteristics of corrugated packaging. The End Use Performance Standard, EUPS, is based on studies of supply chain requirements. It provides comprehensive performance criteria that can be applied when selecting corrugated board.http://www.bfsv.de/Eups/Website/eups_website/frameie.html.

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EUPSBending Stiffness Calculations

Bending Stiffness Calculation

Single wall board :

Corrugated Board: Liner Specific: Fluting Specific:

Wall: Inner liner: Inside fluting:Flute Height: 3,66 mm Tensile Stiffness, CD 425 kN/m Tensile Stiffness, CD 345 kN/m

Flute Pitch: 7,95 mm Tensile Stiffness, MD 1150 kN/m Thickness 184 μm

Take-up factor: 1,42 (cal.) Thickness 165 μm

Outer liner:Tensile Stiffness, CD 425 kN/m

Tensile Stiffness, MD 1150 kN/m

Thickness 165 μm

Predicted Geometrical Mean of Bending Stiffness: 5,4 (Nm)

(Disregarded w hen Double flute boards are calculated)

Double wall board :

Wall: Middle Liner: Outside Fluting:Flute Height: 2,5 mm Tensile Stiffness, CD 425 kN/m Tensile Stiffness, CD 345 kN/m

Flute Pitch: 6,5 mm Tensile Stiffness, MD 1150 kN/m Thickness 184 μm

Take-up factor: 1,31 (cal.) Thickness 165 μm

Predicted Geometrical Mean of Bending Stiffness: 16,8 (Nm)

(Disregarded w hen Single w all boards are calculated)

Stacking - Alternative load casesRoll cage

The corrugated board boxes are1

6

4

3

5

2

• not stacked perfectly on top of each other

• stacked incorrectly

• leaning

• stacked on other products than

8

7

6

11

9 10

• stacked on other products than boxes

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Ranking of load cases

”Average” number of loaded vertical box panels

”4” ”3” ”2” ”0”

4 3,5 3 2,5

Safe and risky load casesIn average 4-2,5 loaded vertical panels

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2 1 5 1 0

Critical load casesIn average 2-0 loaded vertical panels

2 1,5 1 0

Distribution of load cases for a sample containing 290 boxes

25% 100%

10%

15%

20%

40%

60%

80%

FrekvensAck. frekvens

0%

5%

0 0,5 1 1,5 2 2,5 3 3,5 4 el. obel.

antal belastade sidopaneler (ABS-tot)

0%

20%

194 rent belastade lådor100% = 290 lådor

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BCT-value of paperboard boxesStaplingsstyrka två kapslar i höjd

(rätt, förskjuten 6 mm längs, förskjuten 6 mm längs och åt 250

BCT NStacking strength for two boxes on top of each other

(correct stacking and displaced 6 mm in different directions)sidan)

100

150

200

250

medelvärdestandardavvikelse.

average

standard dev.

0

50

1 2 3förskjutningsmönsterstacking pattern

Product – package interactionInteraction between packages

P P

δ δ

Primary packaging Secondary packaging

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Interaction between packagesInfluence of head space

PP

δδ

Company relates software for analysis of box compression strength

In general, paper and packaging companies have in-house developed software for box compression analysis.In general, paper and packaging companies have in-house developed software for box compression analysis.

• Billerud Box Design– CD

• SCA (based on analyses using the Finite Element Method)

• Korsnäs

so t a e o bo co p ess o a a ys sso t a e o bo co p ess o a a ys s

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Billerud Box DesignCD

Software from Korsnäs

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ModelPACK by Innventia AB

Board Properties

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Box Types

RSC 0201

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Fruit tray

Material Properties

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Correction CoefficientsStorage time, moisture etc.

After lecture 14 you should be able to• describe the theoretical foundation for and use the most

important analytical expressions for box compression strength

• describe the theoretical foundation for and use analytical approaches for determination of the bending stiffness of paperboard and corrugated board

• qualitatively discuss the influence of non-perfect stackingq y p g

• acknowledge the use of different types of computer software for prediction of packaging stacking strength