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Lecture 13: Design of paper and board packages 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 comppgression 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
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Literature
• Pulp and Paper Chemistry and Technology - Volume 4, Paper 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 predictions • Laboratory testing • Full-scale testing
Design – Implement – Test!!
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 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 boxes Static compression load
Top-load compression of the most stressed package in the pallet.
Most stressed package
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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 boxes Consider a box subjected to compressive loading due to stacking.
1. 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 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-In pplanelane ststiffnessiffness ooff papanelnel iiss in gegeneralneral mmuchuch llargerarger tthanhan bebendingnding ststiffnessiffness
Panel 1: This panel wants to buckle, i.e. the
panel would like to deform in the x1-direction.
1 Panel 2: The in-plane deformation of this panel is small, i.e. this panel will not deform 2 very much in the x1-direction. x3 Consequently, close to the corners Panel 1
cannot deform in the x1-direction, and the x2 corners will remain primarily vertical.
x1
Buckling of simply supported isotropic plate subjected to uniform compressive loading Timoshenko (1936)
2 πσtEsc Pc = 3(1−υ 2 )
Pc = ultimate strength of buckled panel t = plate thickness υ = Poisson's ratio E = in-plane Young's modulus
σ sc = yield stress in compression
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Modifications for an anisotropic plates
• Introduce the geometric mean of the SSSbbb= bending stiffness MDCD 3 • Int rod uce th e b endi ng stiff ness per b Et b S = unit width, S , instead of Young’s 12 modulus E and the panel thickness t • Consider influence of Poisson’s ratio to be negligible F SCT • Replace σsc by the short span σ → c compression strength (SCT) per unit sc t width
SCT b b • THEN FOR A PANEL: PFSSc = 2π cMDCD
Short Span Compression Strength
SCT Fc
07mm0.7 mm
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BOX compression strength Paperboard boxes
SCT Grangård’s formula: PkFS= c b
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 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 , 6.5 %
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Analysis of typical load-deformation curve
A. Any unevenness in the box is llldtTlevelled 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 Load versus deformation for an A- short panels. flute RSC-box using fixed platens. F. Localized stability
Usefulness of box compression strength
• Boxes are tested individually. If 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 dynamic loading, such as vibrations, that will accelerate failure.
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BOX compression strength McKee’s formula
0,75 0,25 0,5 PFSZc = β c
Pc = Box compression strength
Fc = Compressive strength of plane panel (ECT) S = Geometric mean of MD and CD bending stiffness bb SSMDCD Z = Perimeter of box β = Empirical constant
The McKee model
Semi-empirical approach for description of the post-buckling behaviour
b 1−b PFPPZ = c(FPcCR)
PZ = ultimate strength of the panel
PCR = buckling load for simply supported plate
Fc = edgewise compression strength of panel (ECT) cb, = constants
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The McKee model
Buckling load for thin orthotropic panel SS Pk=12 MDCD CR CR W 2 t π 22⎛⎞rn 2 where kKCR =++⎜⎟222 12 ⎝⎠nr 1/4 W ⎛⎞S t r = ⎜⎟MD ⎝⎠SWCD
n is related to the buckling pattern
The McKee model Approximations
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. 1/4 2. The parameter () SS MD CD was set to 1,17 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 1−b Pc= 4π 22− bb F SSbb Z211 b− k− b ( ) ( cMDCD) ( ) where k is a modified buckling coefficient. Further simplifications: kb1−b =≈1, 33 when 0, 76 for boxes with depth-to-perimeter values ≥ 0,143 1−b PFP = aFSSZb SSbb Z21 b− cMDCD( ) Evaluation of constants a and b for A-, B- and C-flute RSC- boxes yields in SI-units: 0,25 PFSSZ= 375 0,75 bb 0,5 cMDCD( )
Comments on McKee’s formula
• The constants evaluated for typical U.S. boxes in the early 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 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
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
⎧EA= EBt Tensile stiffness: ⎨ b ⎩EEt= per unit width ⎧ Bt 3 SEIE== ⎪ 12 ⎨ Bending stiffness: t3 ⎪SEb = per unit width ⎩⎪ 12
Micromechanical models of corrugated board
t t tliner tliner
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In-plane stiffness of corrugated board panels
core EMD ≈ 0 α – take-up factor
core⎛⎞t fluting fluting tfluting – fluting thickness EECD= α ⎜⎟ CD ⎝⎠tcore tcore – core thickness
liner,, bottom⎛⎞tliner, bottom liner top ⎛⎞tliner, top EEMDMD=+⎜⎟ E MD⎜⎟ ⎝⎠tt⎝⎠
liner,, bottom⎛⎞tliner, bottom core⎛⎞tcore liner top ⎛⎞tliner, top EECD=++ CD⎜⎟ E CD⎜⎟ E CD ⎜⎟ ⎝⎠tt⎝⎠ ⎝⎠ 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 contribution to the bending stiffness, particularly in CD.
222 liner ⎛⎞ttt ⎛⎞ ⎛⎞ Steiner’s IBtBtBt=+=liner⎜⎟ liner ⎜⎟ liner ⎜⎟ theorem! ⎝⎠22 ⎝⎠ ⎝⎠ 2
22 2 b liner ⎛⎞ttliner ⎛⎞ ⎛⎞ t SEt====liner⎜⎟ E b ⎜⎟{Steadman} S ⎜⎟ ⎝⎠22 ⎝⎠ ⎝⎠ 2
• More advanced models exist, but they are cumbersome to use, and cannot 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 possible and explored in the scientific literature.
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Stacking - Alternative load cases Roll cage
The corrugated board boxes are 1 • not stacked perfectly on top of each other
3 • stacked incorrectly 2 5 4 • leaning
6 • stacked on other products than 7 boxes
8 9 10
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Ranking of load cases
”Average” number of loaded vertical box panels
”4” ”3” ”2” ”0”
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Safe and risky load cases In average 4-2,5 loaded vertical panels
4 3,5 3 2,5
Critical load cases In average 2-0 loaded vertical panels
2 151,5 1 0
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Distribution of load cases for a sample containing 290 boxes
25% 100%
20% 80%
15% 60%
Frekvens Ack. frekvens
10% 40%
5% 20%
0% 0% 0 0,5 1 1,5 2 2,5 3 3,5 4 el. obel. 194 rent belastade lådor antal belastade sidopaneler (ABS-tot) 100% = 290 lådor
BCT-value of paperboard boxes
Stacking strengthStaplingsstyrka for two boxestvå kapslar on top i höjd of each other BCT N (correct(rätt, förskjuten stacking and6 mm displaced längs, förskjuten 6 mm in different6 mm längs directions) och åt 250 sidan)
200 medelvärdeaverage standardavvikelse.standard dev.
150
100
50
0 123 stackingförskjutningsmönster pattern
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Product – package interaction Interaction between packages
P P
δδ Primary packaging Secondary packaging
Interaction between packages Influence of head space
P P
δ δ
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Company relates software for analysis of box compression strength
In general, paper companies have in-house developed software for box cocompressionmpression aanalysis.nalysis. • 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) •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.) – http://www.bfsv.de/Eups/Website/eups_website/frameie.html
Optipack
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Billerud Box Design
EUPS
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EUPS Bending 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)
Design against shocks
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Design of shock absorbance/damping materials
A. No damping B. Incorrect damping C. Correct damping
ABC
Drop testing
•Droppp the product a gainst an elastic foundation (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 shock absorber/damping material, the value at which the product fails can be determined.
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Drop testing - II
•In gg,eneral, different values will be obtained 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 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 packaging. • Typically design values are 0,3 – 1,0 m.
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Typical designs of cushioning
The mechanical properties of damping materials aT c = a = retardation (in g) h h = drop height T = thickness of damping material
WW W = impact energy (strain energy) per unit volume mgh W = m = mass V g = constant of gravity V = volume of damping material
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Optimal damping factor
• The cushioning factor varies depending on how the mechan ica l be hav iour o f t he mater ia l is ut ilise d. • 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
+ Low price - Damaged after one shock - Small working region Wmin-Wmax - Hygroscopic
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Foam “Typical” dampening factor
+ Can be tailor made to - Damaged during shock different shapes loading. Will lose some of + Good damping properties its damping properties. + Can be obtained in different stiffness
Foam particles “Typical” dampening factor
+ Can be used as filler - Not as good damping + Packing density can vary properties as homogeneous materials
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Example of cushioning factor
Cushioning factor c = damping 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 heat transport.
1. Isolate by use of an isolation material 2. Avoid that the ppproduct is exposed to 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 cold warm
Conduction (värmeledning) through bodies depends on temperature gradient Radiation (strålning) depends on temperature Convection (konvektion) at solids/fluid or fluids/fluids interfaces depends on surface roughness and mixing
Conduction Fourier’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 the direction of heat flow, and to the temperature difference along the path of heat flow, dT/dx i.e.
λ= heat conductivity coefficient
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Convection
Heat 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 of temperature T, is:
α = heat transfer coefficient
Heat transfer coefficient
Combining conduction and convection gives the total heat transfer coefficient (värmeövergångstal) k as:
dQ 1 =−kA() T T k = 21 11d dt ++∑ m α12m λαm
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Radiation
Emitted energy-rate dQ = εσ AT 4 dt T is the absolute temperature in K
σ =⋅5,67 10−824 W/m K (Stefan-Bolzmann constant) 0≤≤ε 1 (emissivity, for a non-black body)
For obj ec t i n an enc losure the ra diati ve exc hange be tween o bjec t an d wa ll is
dQ 44 =FATTσ − dt object-wall object( object wall )
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 comppgression 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 heat transfer mechanisms that influences product protection
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