Design of paper and board packages

Lecture 14: Design of paper and board packages Design of paper and board packaging: stacking, analytical methods. Software such as Billerud 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 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 acknowledge the use of different types of computer software for prediction of the stacking strength of packages 1

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 testingti Full-scale testing Design Implement Test!! Not different from the automotive or many other types of industries! 2

Loads during transport and storage Transport between manufacturer, wholesaler and 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 boxes Static compression strength (BCT/BCR) Top-load compression of the most stressed package. Beldie, 2001 Most stressed package 3

Methods for determination of box compression strength/resistance 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) + 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) 4

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. Load is then primarily il 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. 5

Buckling of paperboard boxes St rain Observation: In-plane stiffness of panel is in general much larger than bending stiffness 300 mm 300 mm 400 mm Bulge 20 mm 0,5 % 2 1 Panel 1: This panel wants to buckle, i.e. the panel would like to deform in the x 1 -direction. x 3 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 2 x 1 Consequently, close to the corners Panel 1 cannot deform in the x 1 -direction, and the corners will remain primarily vertical. Ultimate load (based on the yield stress in compression) of a 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 load of buckled panel = plate thickness υ = Poisson's ratio E = in-plane Young's modulus σ = YIELD STRESS IN COMPRESSION sc 6

Modifications for an anisotropic plate 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 Neglect influence of Poisson s ratio Replace σ sc by the short span compression strength F SCT c per unit width THEN FOR A PANEL: c = 2 S = S S b b b MD CD 3 b Et S = 12 σ sc F c t SCT P π F S S SCT b b c MD CD Panel Compression Resistance P c Grangård (1969, 1970) 7

Short Span Compression Strength SCT F c 07mm 0.7 BOX compression resistance Cartonboard boxes Grangård s formula: SCT P= k Fc Sb b b b = S = S MD S CD 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 creases will also affect k. 8

A comment on fibre orientation and mechanical properties Board dried with 2 % stretch in MD and free drying in CD Corrugated board containers 9

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 % 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 10

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. The box may be subjected to dynamic loading, such as vibrations, that will accelerate failure. 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 Note! The exponentials in the equation above can have slightly different values for different types of boxes! 11

The McKee model Theoretical foundation Semi-empirical approach for description of the post-buckling behaviour P c P F CR c cb, = constants b 1 b c = ( c) CR P c F P = ultimate strength of the panel = buckling load for simply supported plate = edgewise compression strength of panel (ECT) The McKee model Theoretical foundation Buckling load for thin orthotropic panel where k CR P CR = S S W MD CD 12kCR 2 2 2 2 π r n = + + 12 n r r S MD = SCD 2 2 2 1/4 t W K W t n is related to the buckling pattern 12

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. Simplified expression for total box load b 2 2b b b b 2b 1 1 b c = c MD CD ( 4π ) ( ) ( ) 1 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 ( ) 2 1 1 b b b b 2 b 1 c = 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 c MD CD 13

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). 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

Influence of box perimeter and height on BCT-value Why linear? Box compression strength/n Height/mm Why flat? Perimeter/mm 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 15

Micromechanical models of corrugated board t t liner t t liner 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 16

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. 2 2 2 liner t t t I = Btliner + Btliner = Btliner 2 2 2 Steiner s theorem! 2 2 { } 2 b liner t liner t t S = E tliner = Eb = Steadman = S 2 2 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. EUPS 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. 17

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) Stacking - Alternative load cases Roll cage The corrugated board boxes are 1 not stacked perfectly on top of each other 3 4 2 5 stacked incorrectly leaning 6 7 stacked on other products than boxes 8 9 10 11 18

Ranking of load cases Average number of loaded vertical box panels 4 3 2 0 Safe and risky load cases In average 4-2,5 loaded vertical panels 4 3,5 3 2,5 19

Critical load cases In average 2-0 loaded vertical panels 2 15 1,5 1 0 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% 20

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. 150 100 50 0 1 2 3 stacking förskjutningsmönster pattern Product package interaction Interaction between packages P P δ Primary packaging δ Secondary packaging 21

Interaction between packages Influence of head space P P δ δ Company relates software for analysis of box compression strength In general, paper and packaging companies have in-house developed software for box compression o analysis. a s Billerud Box Design CD SCA (based on analyses using the Finite Element Method) Korsnäs 22

Billerud Box Design CD Software from Korsnäs 23

ModelPACK by Innventia AB Board Properties 24

Box Types RSC 0201 25

Fruit tray Material Properties 26

Correction Coefficients Storage 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 stacking acknowledge the use of different types of computer software for prediction of packaging stacking strength 27