2/22/2010 1 Lecture 13: Design of paper and board packages Stacking shocks climate loading Stacking, shocks, climate loading, analytical methods, computer based design tools After lecture 13 you should be able to • use the most important analytical expressions for box compression strength • describe analytical approaches for determination of the bending stiffness of paperboard and corrugated board panels • qualitatively discuss the influence of non-perfect stacking • perform simple design of cushioning materials • describe how heat transfer mechanisms influences product protection
• use the most important analytical expressions for box compression strengthp g
• describe analytical approaches for determination of the bending stiffness of paperboard and corrugated board panels
• qualitatively discuss the influence of non-perfect stacking• perform simple design of cushioning materials• describe how heat transfer mechanisms influences
product protection
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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
Not different from for example 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 vehiclesof vehicles
• Reloading by i.e. forklifts• Many time consuming manual operations
at wholesalers and retailers• Varying climate conditions (temperature
and moisture)
From Jamialahmadi, Trost, Östlund, 2009
EXAMPLE: Stacking of boxesStatic compression load
Top-load compression of the most stressed package in the pallet.
Most stressedpackage
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Methods for determination of box compression strength
• Laboratory and service testing+ 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)(Will be discussed in more detail in next lecture)
+ In general high accuracy and easy to do parametric investigations- Not straight-forward to use and still not fully developed for every
paper and board application
Box Compression Test (BCT)
Determination of the maximum load that a rectangular box can carry.
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Paperboard cartons
Box compression strength of rectangular boxesConsider a box subjected to compressive loading due to stacking.
1 At small loadings the load is evenly1. At small loadings, 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. Load is then primarily carried by small zones at the corners of the boxzones at the corners of the box
5. Failure of the box finally occurs by compressive failure at the corners
Grangård (1969, 1970) show that the compression strength of PAPERBOARD 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 general much larger than bending stiffnessIn 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.
Panel 2: The in-plane deformation of this panel is small, i.e. this panel will not deform very much in the x1-direction.
1
2
x1
x2
x3Consequently, close to the corners Panel 1 cannot deform in the x1-direction, and the corners will remain primarily vertical.
Buckling of simply supported isotropic plate subjected to uniform compressive loadingTimoshenko (1936)
2sc
c 23(1 )
t EP
π σ
υ=
−
ultimate strength of buckled panelplate thickness
cPt
==
sc
plate thicknessPoisson's ratio
= in-plane Young's modulusyield stress in compression
t
Eυ
σ
==
=
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Modifications for an anisotropic plates
• Introduce the geometric mean of the bending stiffnessI t d th b di tiff
b b bMD CDS S S=
3• Introduce the bending stiffness per unit width, Sb, instead of Young’s modulus E and the panel thickness t
• Consider influence of Poisson’s ratio to be negligible
• Replace σsc by the short span
3
12b EtS =
SCTcFReplace σsc by the short span
compression strength (SCT) per unit width
• THEN FOR A PANEL:
scσ → c
t
c 2 SCT b bc MD CDP F S Sπ=
Short Span Compression Strength
SCT
0 7 mm
SCTcF
0.7 mm
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BOX compression strengthPaperboard boxes
Grangård’s formula: SCTP k F 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=
This constant needs to be determined through
extensive testing.
The quality of the crease will also strong affect k.
A comment on fibre orientation and mechanical properties
Board dried with 2 % stretch in MD and free drying in CD
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Corrugated board containers
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 %
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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.
• Boxes are tested individually. If boxes are stacked in patterns other
Usefulness of box compression strength
boxes are stacked in patterns other than a columns the full strength potential will not be realized.
• Climatic conditions may degrade box compression strength.
• Creep will affect the results considerably.
• The box may be subjected to• The box may be subjected to dynamic loading, such as vibrations, that will accelerate failure.
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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
The McKee model
Semi-empirical approach for description of the post-buckling behaviour
( ) 1b bP F P −( ) 1 bZ c CRP c F P=
,
Z
CR
c
PPFc b
==
==
ultimate strength of the panelbuckling load for simply supported plate
edgewise compression strength of panel (ECT)constants
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The McKee modelBuckling 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
The McKee modelApproximations
1. The parameter K is a complex function of 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.
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Simplified expression for total box load
( ) ( ) ( )12 2 2 1 14bb b b b b b
c MD CDP c F S S Z kπ−− − −= ( ) ( ) ( )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≥
( )12 1
−−
bb b b bP F S S Z( ) 2 1−= b b b b
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 MD CDP F S S Z
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).
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Influence of box perimeter and height on BCT-value
Box compression strength/N
Height/mm
Perimeter/mm
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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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
Micromechanical models of corrugated board
linertlinertt t
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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 core liner topcoreCD CD CD CD
t t
tt tE E E Et t t
⎝ ⎠ ⎝ ⎠⎛ ⎞⎛ ⎞ ⎛ ⎞= + + ⎜ ⎟⎜ ⎟ ⎜ ⎟
⎝ ⎠⎝ ⎠ ⎝ ⎠
Rules of mixture from parallel model for lamellar composites
Simplified expressions for the bending stiffness of corrugated board panels
A 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.
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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
Ranking of load cases
”Average” number of loaded vertical box panels
”4” ”3” ”2” ”0”
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4 3,5 3 2,5
Safe and risky load casesIn average 4-2,5 loaded vertical panels
2 1 5 1 0
Critical load casesIn average 2-0 loaded vertical panels
2 1,5 1 0
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Distribution of load cases for a sample containing 290 boxes25% 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
BCT-value of paperboard boxes
Staplingsstyrka 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
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Product – package interactionInteraction between packages
P P
δ δ
Primary packaging Secondary packaging
Interaction between packagesInfluence of head space
PP
δδ
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Company relates software for analysis of box compression strength
In general, paper companies have in-house developed software for box compression analysis.
In general, paper companies have in-house developed software for box compression analysis.
• Optipack from Korsnäs– http://www.korsnas.com/en/Products/Services/Korsnas-Packaging-
Performance-Service/OptiPack/#
• Billerud Box Design– CD
• SCA (based on analyses using the finite element method)
compression analysis.compression analysis.
• SCA (based on analyses using the finite element method)• EUPS
– (European 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.)
Predicted Geometrical Mean of Bending Stiffness: 16,8 (Nm)
(Disregarded w hen Single w all boards are calculated)
Design against shocks
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Design of shock absorbance/damping materials
A. No dampingB. Incorrect dampingC. Correct damping
A B C
Drop testing
• Drop the product against an elastic foundation p p g(Winkler foundation)
• Measure the acceleration (retardation) (expressed as a multiple of the acceleration due to gravity, g) [alternatively measure the force during the drop test]
• By successively increasing the stiffness of the• By successively increasing the stiffness of the shock absorber/damping material, the value at which the product fails can be determined.
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Drop testing - II
• In general, different values will be obtained g ,depending on the orientation of the product.
• The durability against shocks is measured in multiples of g.– For electronic devices, for example, this value is
typically 20-80g.• The packaging price is increasing very quickly
if the durability value is below 20g.
Drop and impact testing of packaging
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Loads during transport and handling
• Shocks– Drop a package– Movement of package during vehicle
transportation• Overturn a package• It is in practise impossible to estimate a
design drop height that a package possibly can be subjected to during handling.
Loads during transport and handling
• Design must be based on experienceDesign must be based on experience– Low weight products are in general treated
less carefully than heavy products.– Package stacked on pallets are in general
subjected to lower drop height than single packagingpackaging.
• Typically design values are 0,3 – 1,0 m.
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Typical designs of cushioning
The mechanical properties of damping materials
a = retardation (in g)h = drop heighta = retardation (in g)h = drop height
aTch
=h = drop heightT = thickness of damping
material
h = drop heightT = thickness of damping
material
h
mghWV
=
W W = impact energy (strain energy) per unit volume
m = massg = constant of gravityV = volume of damping material
W = impact energy (strain energy) per unit volume
m = massg = constant of gravityV = volume of damping material
W
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Optimal damping factor
• The cushioning factor varies depending on how the h i l b h i f h i l i ili dmechanical behaviour of the material is utilised.
• The cushioning factor is dependent of the load rate (through h)
• There is an optimal impact energy for the cushioning material.– For lower values is the material not used efficiently– For higher values is the material thickness not high enough
• A good damping material has a value of the cushioning factor c not below 2-3 m/s2
Corrugated board“Typical” dampening factor
- Damaged after one shock- Small working region
Wmin-Wmax- Hygroscopic
+ Low price
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Foam“Typical” dampening factor
+ Can be tailor made to different shapes
+ Good damping properties+ Can be obtained in
different stiffness
- Damaged during shock loading. Will lose some of its damping properties.
Foam particles“Typical” dampening factor
+ Can be used as filler+ Packing density can vary
- Not as good damping properties as homogeneous materials
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Example of cushioning factor
c = damping factorCushioning factor
Design for box performance in a given environment
• Simulate given environmental conditions• Test box performance using a realistic
load
• Compare with, for example, design against metal fatigue in the vehicle industry
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Design for heat isolation of packaging
The physical problem is to prevent heatThe physical problem is to prevent heat transport.
1. Isolate by use of an isolation material2. Avoid that the product is exposed top p
heat (fans, cooling systems etc.)
Principles for heat transport
Conduction (sv. värmeledning)Through bodies
Convection (sv. konvektion)At solids/fluid or fluids/fluids interfaces
Radiation (sv. strålning)
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Heat transport
warm warmcold
Conduction (värmeledning) through bodies depends on temperature gradienttemperature gradient
Radiation (strålning) depends on temperature
Convection (konvektion) at solids/fluid or fluids/fluids interfacesdepends on surface roughness and mixing
ConductionFourier’s law
Fourier's law is an empirical law.The rate of heat flow, dQ/dt, through a homogenous solid is directly proportional to the area A of the section at right angles to theproportional to the area, A, of the section at right angles to the direction of heat flow, and to the temperature difference along the path of heat flow, dT/dx i.e.
λ= heat conductivitycoefficient
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ConvectionHeat transfer from the solid surface to the fluid can be described by Newton's law of cooling. It states that the heat transfer, dQ/dt, from a solid surface of area A at a temperature Tw to a fluid offrom a solid surface of area A, at a temperature Tw, to a fluid of temperature T, is:
α = heat transfer coefficientα heat transfer coefficient
Heat transfer coefficient
Combining conduction and convection gives the total heat transfer coefficient (värmeövergångstal) k as:heat transfer coefficient (värmeövergångstal) k as:
2 1( )dQ kA T Tdt
= −
1 2
11 1m
m m
k dα λ α
=+ +∑
1 2m m
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Radiation
4dQ ATε
Emitted energy-rate
4Q ATdt
εσ=
8 2 45,67 10 W/m K (Stefan-Bolzmann constant)0 1 (emissivity, for a non-black body)σ
ε
−= ⋅≤ ≤
F bj t i l th di ti h b t bj t d ll i
T is the absolute temperature in K
For object in an enclosure the radiative exchange between object and wall is
( )4 4
object-wall object object wall=dQ F A T Tdt
σ −
For concentric bodies with Aobject<< Awall,the geometry factor Fobject-wall is εobject.
After lecture 13 you should be able to• use the most important analytical expressions
for box compression strengthp g• describe analytical approaches for determination
of the bending stiffness of paperboard and corrugated board panels
• qualitatively discuss the influence of non-perfect stacking
• perform simple design of cushioning materials• describe heat transfer mechanisms that
