After lecture 13 you should be able to

Transcription

1 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 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 1

2 Literature Pulp and Paper Chemistry and Technology - Volume 4, Paper Products Physics and Technology, Chapter 10 Paperboard Reference Manual, pp 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! 2

3 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 3

4 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. 4

5 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. 5

6 Buckling of paperboard boxes Observation: In-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 x 1 -direction. 2 1 Panel 2: The in-plane deformation of this panel is small, i.e. this panel will not deform very much in the x 1 -direction. x 3 Consequently, close to the corners Panel 1 x 2 cannot deform in the x 1 -direction, and the corners will remain primarily vertical. x 1 Buckling of simply supported isotropic plate subjected to uniform compressive loading Timoshenko (1936) P c πt = 2 σ E sc 2 3(1 υ ) P c t = ultimate strength of buckled panel = plate thickness υ = Poisson's ratio E = in-plane Young's modulus σ = yield stress in compression sc 6

7 Modifications for an anisotropic plates Introduce the geometric mean of the bending stiffness Introduce the bending stiffness per unit width, S b, instead of Young s modulus E and the panel thickness t Consider influence of Poisson s ratio to be negligible Replace σ sc by the short span compression strength (SCT) per unit width S = S S b b b MD CD b Et S = 12 3 SCT F σ sc c t THEN FOR A PANEL: P π F S S c = 2 SCT b b c MD CD Short Span Compression Strength SCT F c 07mm 0.7 7

8 BOX compression strength Paperboard boxes Grangård s formula: SCT P = k Fc Sb 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 8

9 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 % 9

10 Analysis of typical load-deformation curve Load versus deformation for an A- flute RSC-box using fixed platens. A. Any unevenness in the box is levelled ll out. Top crease lines begin to roll. B. The steepest corners of the box start to take load. C. Sub-peak caused by smallscale 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 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. 10

11 BOX compression strength McKee s formula P = β F S Z c 0,75 0,25 0,5 c P c = Box compression strength F c = Compressive strength of plane panel (ECT) S = Geometric mean of MD and CD bending stiffness Z = Perimeter of box β = Empirical constant S b MD S b CD The McKee model Semi-empirical approach for description of the post-buckling behaviour P P F Z CR c cb, = constants ( ) 1 b b 1 b P Z = c Fc PCR = ultimate strength of the panel = buckling load for simply supported plate = edgewise compression strength of panel (ECT) 11

12 The McKee model Buckling load for thin orthotropic panel where k P CR CR = S S W MD CD 12kCR π r n = n r r S MD = SCD /4 t W K W t 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 parameters, but the value K = 0,5 was adopted by McKee without further notice. 2. The parameter ( S ) 1/4 MD SCD 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. 12

13 Simplified expression for total box load b 2 2b b b b 2b 1 1 b = ( 4π ) ( 1 c) ( MD CD ) P c F S S Z k where k is a modified buckling coefficient. 1 b = 1,33 when b 0, 76 Further simplifications: for boxes with depth-to-perimeter values 0,143 k ( ) b b b b 2 b 1 c MD CD P = af S S Z Evaluation of constants a and b for A-, B- and C-flute RSCboxes yields in SI-units: b b P= 375F S S Z ( ) 0,25 0,75 0,5 c MD CD 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). 13

14 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) 14

15 Micromechanical models Tensile stiffness: Bending stiffness: EA = EBt E = Et per unit width b 3 Bt S = EI = E 12 3 b t S = E per unit width 12 Micromechanical models of corrugated board t t liner t t liner 15

16 In-plane stiffness of corrugated board panels core E 0 α take-up factor E MD t α core fluting CD = tcore E fluting CD t fluting fluting thickness t core core thickness liner, bottom tliner, bottom liner, top tliner, top EMD = EMD + EMD t t liner, bottom tliner, bottom core tcore t liner, top liner, top ECD = ECD + ECD + ECD t 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 contribution to the bending stiffness, particularly in CD liner t t t I = Btliner + Btliner = Btliner Steiner s theorem! liner b liner t t t S = E tliner = Eb = { Steadman} = S 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. 16

17 Stacking - Alternative load cases Roll cage The corrugated board boxes are 1 not stacked perfectly on top of each other stacked incorrectly leaning 6 7 stacked on other products than boxes Ranking of load cases Average number of loaded vertical box panels

18 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 ,

19 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 0,5 1 1,5 2 2,5 3 3,5 4 el. obel. 194 rent belastade lådor antal belastade sidopaneler (ABS-tot) 100% = 290 lådor 0% BCT-value of paperboard boxes BCT N 250 Stacking strength Staplingsstyrka for two boxes två kapslar on top i höjd of each other (correct (rätt, förskjuten stacking and 6 mm displaced längs, förskjuten 6 mm in different 6 mm längs directions) och åt sidan) 200 medelvärde average standardavvikelse. dev stacking förskjutningsmönster pattern 19

20 Product package interaction Interaction between packages P P δ Primary packaging δ Secondary packaging Interaction between packages Influence of head space P P δ δ 20

21 Company relates software for analysis of box compression strength In general, paper companies have in-house developed software for box compression analysis. Optipack from Korsnäs 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.) Optipack 21

22 Billerud Box Design EUPS 22

23 EUPS Bending Stiffness Calculations Bending Stiffness Calculation Single wall board : Corrugated Board: Liner Specific: Fluting Specific: Wall: Inner liner: Inside fluting: Tensile Stiffness, Tensile Stiffness, Flute Height: 3,66 mm CD 425 kn/m 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 Tensile Stiffness, MD Thickness 425 kn/m 1150 kn/m 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: Tensile Stiffness, Tensile Stiffness, Flute Height: 2,5 mm CD 425 kn/m 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 23

24 Design of shock absorbance/damping materials A. No damping B. Incorrect damping C. Correct damping A B C Drop testing Drop the product against 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. 24

25 Drop testing - II In general, 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 25

26 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. 26

27 Typical designs of cushioning c = The mechanical properties of damping materials at h a = retardation (in g) h = drop height T = thickness of damping material W W mgh = V W = impact energy (strain energy) per unit volume m = mass g = constant of gravity V = volume of damping material 27

28 Optimal damping factor The cushioning factor varies depending on how the mechanical 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/s 2 Corrugated board Typical dampening factor + Low price - Damaged after one shock - Small working region W min -W max - Hygroscopic 28

29 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 29

30 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 30

31 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 product 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) 31

32 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 32

33 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 = ka ( T 2 T 1) dt k = 1 1 dm α λ α 1 m m 2 33

34 Radiation σ = 0 ε 1 (emissivity, for a non-black body) Emitted energy-rate dq = εσ AT dt , W/m K (Stefan-Bolzmann constant) For object in an enclosure the radiative exchange between object and wall is 4 T is the absolute temperature in K 4 4 ( ) dq = F A σ T T dt object-wall object object wall For concentric bodies with A object << A wall, the geometry factor F object-wall is ε object. 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 heat transfer mechanisms that influences product protection 34