Stability And Collapse Of Corrugated Board - Lth.se

Chapter 2Corrugated board2.1HistoryIn 1856 two Englishmen, Healy and Allen, received the first patent oncorrugated paper. With a simple hand-driven machine they produced acorrugated paper that was used as a lining in hats. The American Albert L Jones attached a plane sheet to the corrugated paper and patentedthe technique in which heat was used to corrugate paper in 1871. Thisproduct was used to protect fragile goods as bottles. Then in 1874 theAmerican Oliver Long patented the concept of strengthening the corrugated paper by adding another facing. The first Swedish manufacturingof corrugated board was started in Malmö by Carl Th Norén in 1905.The improvement of the machines made it possible to produce corrugatedboard of higher quality and in the 1920’s the boxes made of corrugatedboard started to compete with the ones made of wood. [1, 2]2.2ManufacturingCorrugated board consists of one or several layers of corrugated paperwhich is glued on or in between plane sheets of paper, see figure 2.1.The manufacturing process can roughly be divided into two parts. Thewet part, where the fluting is corrugated between two rolls and then gluedonto the liner, see figure 2.2, and the dry part, where heat is applied todry the corrugated board. A problem in the manufacturing of corrugated5

6CHAPTER 2. CORRUGATED BOARDa)b)c)d)Figure 2.1: Different types of corrugated board: a) Single face. b) Singlewall. c) Double wall. d) Triple wall.board is when the moisture contents in the different layers are out ofbalance. Then the corrugated board can deform in a buckling shape oras a dip in the facings between the corrugations. These phenomena arecalled warp and washboard respectively [3].Corrugated boardLinerFlutingLinerFigure 2.2: The manufacturing of a single wall corrugated board.As well as there are different types of panels there are different fluteprofiles, see table 2.11 .Corrugated board is often considered to be the packaging material of the1The inconsequence in profile notation, for profiles B and C, in regard to the height,is due to the order in which the two profiles were invented.

2.3. TODAY’S FIELD OF APPLICATION7Table 2.1: Different flute profiles.Flute profile Height (mm) Number of .5550G&Nfuture. There are many advantages and very few disadvantages. Someof them are: Low weight. Saves money when transporting. Can be entirely customised for the purpose. It is strong and stiff compared to its weight. Easy to handle. Easy to print. Fully recyclable. In Sweden 98 % of the produced corrugated boardis recycled [4].- Very sensitive to humidity.2.3Today’s field of applicationCorrugated board is mainly used in packaging and can be customisedentirely for its purpose. If a moisture-repellent membrane is applied onthe surface the box can contain wet products, see figure 2.3. In the fishindustries this is useful, because the box is only used once and does notneed to be washed or transported back after delivery. Instead, the boxcan be recycled immediately after it has served its purpose. An exampleof a new type of application is pallets, on which goods can be stapled.

8CHAPTER 2. CORRUGATED BOARDFigure 2.3: A box made of corrugated board containing wet products, inthis case fish.2.4Future developmentThe development of flute profiles with very small wave-heights, microflute, has made corrugated board to a strong competitor to cardboard.A new product area, is e.g. consumer packages. Here the high stiffnessof the thin corrugated board type, in comparison with the small amountof material used, makes it a favourable alternative to cardboard.

Chapter 3TheoryThis chapter describes theory used in chapter 4 and should be consideredonly as a brief introduction.3.1The principles of a sandwich structureThe studied corrugated board consists of a core, the fluting, and of twofacings, the liners. As for a typical sandwich material, the purpose of thefacings are to carry normal stresses resulting from in-plane deformationand curvature of the board. The purpose of the core is to carry shearstresses and to keep the facings apart. Since the core is supposed tostabilize the facings it must posses sufficient rigidity against deformationin planes normal to the facings. It is these two properties that gives riseto the outstanding strength and stiffness characteristics, compared to thelow weight, of a sandwich structure. The distance between the facingsalso affects the stiffness properties of the sandwich. The larger distance,the stiffer the composite will be in bending. The layers can be very weakwhen separated, but together they create a stiff composite.3.2Constitutive properties of paperSince paper is made of oriented wood fibres the stiffness and strengthproperties are anisotropic. Commonly the fibre orientation is approxi-9

10CHAPTER 3. THEORYmately symmetric. This means that the stiffness properties can be assumed to be orthotropic, i.e. three symmetry planes for the elastic properties can be found. Therefore, the constitutive relation, the relationbetween stresses and strains, for paper is assumed to be: νyx1 νzx 000ExEyEz xσx ν νzy000 σy y Exxy E1y νxz νyz E1z σz z 000EyEz Ex (3.1)1 τxy γxy 00000 Gxy γxz 0000 G1xz 0 τxz γyzτyz00000 G1yzSymmetry leads toνyxνxy ,ExEyνxzνzx ,ExEzνyzνzy EyEz(3.2)Now only nine unknown parameters remain, Ex , Ey , Ez , νxy , νxz , νyz ,Gxy , Gxz , Gyz . Generally, these parameters have to be measured. However, some of the parameters are not straightforward to measure, due tothe small dimension in the thickness direction of paper. The in-planeproperties, Ex and Ey , can fairly easy be obtained by standard tests, e.g.stress-strain curves. For the rest of the parameters, estimations can beused to obtain approximate values.According to [5] isEx(3.3)200a good approximation for the Young’s modulus in the out-of-plane direction. The shear moduli are approximated according to [6, 7] withpGxy 0.387 Ex Ey(3.4)Gxz Ex /55Gyz Ey /35Ez The values of νxy , νxz and νyz are set according to [14]. In additionto the approximations in (3.4), are values of Gxz and Gyz , suggested in[14], also used in the analyses. A trial and error procedure has also beenperformed, in which different values of Gxz and Gyz have been tested, inorder to get results that are as similar as possible to the tests.

3.3. BUCKLING OF A STRUCTURAL CORE SANDWICH3.311Buckling of a structural core sandwichA slender structure carries its load by axial or membrane action, seefigure 3.1, rather than by bending action. The out-of-plane deformationis usually very small at load levels below the critical buckling load. Whenthe critical load is reached, the structure often deforms rapidly and thein-plane stiffness decreases.Figure 3.1: Plate loaded in-plane.Corrugated board can buckle in two different ways, globally or locally.Local buckling occurs when the board facings buckles between the corrugations and global is when the entire board buckles, see figure 3.2.The theory is the same in the two cases, it is only the geometry of thestructure and the boundary and loading conditions that are different.a)Figure 3.2: a) Global andb)b) local buckling of a corrugated board.

123.4CHAPTER 3. THEORYEigenvalue buckling predictionEigenvalue buckling analysis is often used to predict the critical bucklingload and failure mode of a structure. In the general eigenvalue buckling problem the critical load is given when the stiffness matrix becomessingular, so that (3.5) has nontrivial solutions.Kν 0(3.5)K is the tangent stiffness matrix when the loads are applied and ν arethe nontrivial displacement solutions. In Abaqus [8], an incrementaltechnique is used in the eigenvalue buckling problem:(K 0 λi K ) ν i 0(3.6)K 0 is the initial stiffness matrix, K is the differential stiffness matrixdue to the applied load, λi are the eigenvalues and ν i are the corresponding eigenvectors, i.e. the shape of the buckling modes. The criticalbuckling load is obtained by multiplying the applied load with the lowesteigenvalue, λ1 . The eigenvectors ν i are normalized so that the maximumdisplacement component has a magnitude of 1.0. The direction, i.e. thesign, of the buckling displacement is not found by the eigenvalue analysis.In order to stipulate which direction to use in the postbuckling analysis,there are two possibilities. One is to perform a nonlinear postbucklinganalysis on a geometrically perfect panel and from this get the correctdirection. This is possible if the panel consists of a non-symmetric layup of individual layers, i.e. a coupling exists between membrane andbending action. The other is to study in which direction a real panelbuckles, when exposed to a compression test. The results from the twopossibilities, might not be similar if the real panel have an imperfectionlarge enough to affect which buckling mode the panel gets when loaded.They will also differ if the panel is forced to buckle in a certain mode bythe testing equipment, e.g. by asymmetric loading conditions.

3.5. NONLINEAR POSTBUCKLING ANALYSIS3.513Nonlinear postbuckling analysisThe nonlinear geometry response due to large deformations, requires anincremental technique to capture the complex load and displacement performance during the analysis, see figure 3.3. In a displacement controlledanalysis, the displacement, a1 , is prescribed in the first iteration in eachincrement and then the internal forces, f int , are calculated at this newload level. The difference between the total applied load, P , and theinternal forces are the out-of-balance forces, ψ, for this iteration. Thenthe largest out-of-balance force at any degree of freedom is comparedto the out-of-balance tolerance and if it is less then the structure is inequilibrium. If not, then these out-of-balance forces are the load in thefollowing iterations until equilibrium is reached for this increment.Abaqus uses the full Newton Raphson method, i.e the model’s stiffnessmatrix is updated in each iteration and the system of equations are solvedfor each iteration in this nonlinear analysis. This is very time consuming,since the tangential stiffness has to be formed and assembled in each iteration but the advantage is a very fast convergence. Abaqus also has anautomatic increment control, so if the solution appears to diverge thenit will restart the increment with a smaller increment length and if thesolution converges easily then the increment length is increased for thenext increment.Load PKt01P2fint1fintKt12ψ2ψ1Displacement0a1 a1a2aFigure 3.3: Graph illustrating the equilibrium iterations of an increment.

14CHAPTER 3. THEORYIteration scheme for a displacement controlled analysis using the fullNewton-Raphson method.- Initiation of quantities 0 , Kt , a0- For increment inc 1,2, . . . ,Nmax- Set boundary condition to load the body- Set ψ i 0- Set ψ norm tolerance- Iterate while ψ norm tolerance- Calculate the tangential stiffness matrix, Kt- Calculate ai from Kt ai ψ i- Set boundary condition to zero- Calculate i B ai- Calculate internal forces fint- Calculate out-of-balance forces ψ i 1 fint- Calculate ψ norm ψ i 1norm- End iteration- Accept quantities , σ, a, Fint- End increment3.6Damped postbucklingAbaqus may run into difficulties when local buckling of the corrugatedboard starts to occur. This problem and a method of solving this, isdescribed in the Abaqus manual [8]. ”If the instability is localized, therewill be a local transfer of strain energy from one part of the model toneighbouring parts and global solution methods do not work. This class ofproblems has to be solved either dynamically or with the aid of (artificial)damping; for example, by using dashpots. Abaqus provides an automaticmechanism for stabilizing unstable static problems through the automaticaddition of volume proportional damping to the model”.

3.7. FAILURE CRITERIA OF THE FACINGS15If the STABILIZE parameter is included in the input file, then viscousforcesF υ cM ȧ(3.7)are added to the global equilibrium equations.P f int F υ 0(3.8)M is an artificial mass matrix, c is a damping coefficient and ȧ is thevector of nodal velocities. While the buckling is stable, the viscous energydissipated is very small, but when local instabilities occur and the localvelocities increase then the strain energy is dissipated by the applieddamping. The damping coefficient, c, is calculated so that the dissipatedenergy is a small fraction of the strain energy in the first increment.The value of this fraction, called the dissipation intensity, is by default2.0 10 4 , but should be specified by the user so that the influence ofthe applied damping is as small as possible.3.7Failure criteria of the facingsOne of the most commonly used criterion for material failure of paperloaded in-plane is the Tsai-Wu tensor polynomial criterion [9]. The relations between the stresses in the Cartesian coordinate system and thestresses in the spherical coordinate system areσ11 σ R sin φ cos θσ12 σ R sin φ sin θσ22 σ R cos φ0 φ π, 0 θ 2π(3.9)where σ R is the length of the stress vector σ, see figure 3.4.The Tsai-Wu failure criterion in a spherical coordinate system is thenF11 n211 F22 n222 F66 n212 2F12 n11 n22 Rσtw 2 R 1 0,(F1 n11 F2 n22 ) σtwRσtw 0(3.10)

16CHAPTER 3. THEORYσ22φ(σ11 , σ12 , σ22 )σRσ12θσ11Figure 3.4: The stress vector in the Cartesian coordinate system.wheren11 sin φ cos θn12 sin φ sin θn22 cos φ(3.11)andF1 1111111 , F2 , F11 , F22 , F66 2Xt XcYt YcXt XcYt YcT(3.12)XtXcYtYcT Tensile strength in MDCompressive strength in MDTensile strength in CDCompressive strength in CDShear strengthRσtwis the failure stress radius for material failure.The tensile and compressive strength for both MD and CD must be determined by experimental tests. Because of the difficulties to determinethe shear strength, T , and the equibiaxial strength, F12 , by experimentaltests some approximations are used that have been proven to be reasonable for paper [10, 11].pwhere f 0.36(3.13)F12 f F11 F22pwhere α 0.78(3.14)T α Xc Yc

3.7. FAILURE CRITERIA OF THE FACINGS17The critical stress for structural failure [12], i.e. when the facings becomesinstable due to local buckling, can be calculated withP33Ri 1 gi(3.15)σcr P22260a hλ ( c2 n11 2a µ n12 a2 n22 ) 27i 1 hiRσcr 0, [λ, µ] xcrwhere h is the thickness of the facing, 2λ is the buckling wavelength, µ isthe inclination of nodal lines and a is the wavelength of the corrugatedcore, see figure 3.5. The coefficients gi and hi are described in [13].Ris the failure stress radius for structural failure and the numericalσcrvalue of this parameter can be found by an unconstrained minimizationprocedure. A failure index can then be calculated asΦ (σ11 , σ12 , σ22 ) kσkσfR(3.16)where R R, σcrσfR min σtw(3.17)2λµaa1Figure 3.5: Buckle subjected to transverse shear.

18CHAPTER 3. THEORY

Chapter 4Finite element analysisIn this chapter are the finite element analyses of the detailed model andthe simplified model described as well as the post-processor developed inMatlab for visualization of the local buckling.4.1Detailed modelThis section describes how the corrugated paper board is modelled bythe finite element method with veritable geometry.4.1.1Geometry modellingThe board dimensions are given by the available test equipment. Thetest rig is designed to fit in a quadratic panel of 400 mm width. Thefluting is shaped approximately as a sine wave and has a wavelength of7.77 mm. The distance between the two liners are 3.68 mm as shown infigure 4.1.0.264Liner "200 KL"0.2633.68Fluting "150 SC"Liner "200 WTK"0.2697.77(mm)Figure 4.1: Dimensions of the corrugated paper board.19

20CHAPTER 4. FINITE ELEMENT ANALYSISIn order to translate the geometry to a FE model some basic assumptions has to be made. Of course, the fluting cannot be modelled with aperfect sine shape, figure 4.2 and in order to avoid constraint equations,the contact nodes between fluting and liner are modelled without offset,figure 4.3. Because the fluting and liner are connected to the same node,the bending stiffness of the fluting to liner connections will be overestimated. The coordinates of the nodes for the fluting are adjusted so thatthe length of the fluting elements will be almost the same.Figure 4.2: Shape of the fluting.3.68 mmFigure 4.3: Model with and without offset.4.1.2Material modellingThe material behaviour is orthotropic, as seen in section 3.2, and approximated as linear-elastic. This approximation is made because nononlinear material model suitable for paper was available, and to reducethe computation time. The FEA is taking large deformations in accountand if the material behaviour also would be nonlinear, the number of iterations would have increased significantly. This would, having approximately 400000 degrees of freedom in the model, give an unendurablylong computation time.The material parameters, Ex and Ey are from experimental measurements. Ez and Gxy are calculated according to (3.3) and (3.4), respectively and the Poisson’s ratios are set according to [14]. These parameters, which are the same for all analyses, are presented in table 4.1. Since

4.1. DETAILED MODEL21the shear moduli, Gxz and Gyz , are difficult to estimate, three analyseswere performed with different values of these parameters. The different values used in the analyses were, values suggested by Nordstrand[14], (No. 3), values calculated with Baum’s approximations (3.4), (No.1), and values adjusted to fit the load versus out-of-plane displacementcurves from the tests, (No. 2). The values of Gxz and Gyz for the threeanalyses are presented in table 4.2. The tensile and compressive strengthparameters in table 4.3 are determined from experimental tests and areused for the failure criteria.Table 4.1: Material parametersLiner200 WTKEx (GPa)7.604.02Ey (GPa)0.038Ez (GPa)0.34νxy0.01νxz0.01νyzGxy (GPa)2.140.26

Keywords: FEM, nite element analysis, corrugated board, buckling, postbuckling, tests, strength, collapse, packages The cover picture illustrates the local deformation pattern of the facing and . Concerning the simpli ed analysis, the collapse load predicted by material failure will overestimate the strength with 16.3 %. For struc-