Refined nonlinear finite element models for corrugated fiberboards

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Composite Structures 87 (2009) 321–333

Refined nonlinear finite element models for corrugated fiberboards

Rami Haj-Ali *, Joonho Choi, Bo-Siou Wei, Roman Popil, Michael Schaepe

Georgia Institute of Technology, Atlanta, GA 30332-0355, United States

Available online 12 February 2008

Abstract

A refined nonlinear finite element modeling approach is presented for the analysis of corrugated fiberboard material and structuralsystems. The anisotropic and nonlinear material stress–strain behavior of the linerboards and fluting medium layers of the corrugatedfiberboard composite system is modeled using orthotropic material model with Hill’s anisotropic plasticity. Uniaxial tensile tests are con-ducted separately for the linear and medium fiberboards to generate their stress–strain curves in the cross and machine directions (CDand MD) orientations that can be used to calibrate the anisotropic plasticity model for each of the corrugated board layers. The com-bined material and structural modeling approach includes both material and geometric nonlinear effects. Once the material nonlinearbehavior has been calibrated, we simulate the response of several corrugated material systems subjected to Tappi-type edge crush test(ECT) using a clamping fixture. An alternative non-standard ECT geometry having larger free span length is also examined. Simulationsare also performed for multi-wall corrugated systems during ECT. Failure in the FE models is monitored using the Tsai–Wu anisotropicfailure criterion calibrated from the uniaxial tests of the layers. Refined parametric models that explicitly incorporate glue-line charac-teristics are also generated in order to examine their effect on the overall response. The proposed refined modeling approach yields goodpredictions for the overall mechanical behavior and the ultimate failure for a wide range corrugated systems.� 2008 Elsevier Ltd. All rights reserved.

Keywords: Nonlinear finite element (FE); Corrugated board; Single-wall; Multi wall; Glue line

1. Introduction

Corrugated core fiberboard has an efficient geometricdesign that enhances the stability and the strength of thestructure. Corrugated boxes are extensively used for stor-ing and the transportation of goods. The analysis and pre-diction of the stacking compression load capacity ofcorrugated boxes are essential especially through simula-tions. McKee et al. [1] developed a critical compressioncritical load equation for corrugated plates. They proposeda semi-empirical equation on post-buckling behavior andmaterial failure. This conventional semi-analytical formulacan be calibrated from ECT tests (e.g., Tappi T839-PM [2])and used to predict for the buckling of corrugated boxesunder compression. However, mechanical-based nonlinearsimulations, such as the finite element (FE) method, arenow sufficiently mature and capable for the analysis of dif-

0263-8223/$ - see front matter � 2008 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compstruct.2008.02.001

* Corresponding author.E-mail address: [email protected] (R. Haj-Ali).

ferent performance aspects of corrugated fiberboards.Urbanik [3,4] proposed a nonlinear material model to pre-dict the compressive strength of the panels and containers.He used a three-parameter hyperbolic tangent equationthat characterized by the stress–strain relationship todescribe the material nonlinearity. The corrugated struc-ture in his study was taken as an effective homogeneousplate. Their material nonlinearity theory predicted moreconservative buckling loads for low width panels andyielded a more accurate and sensitive compressive strengthprediction than those using linear material properties.

More refined FE models have been used to describe thecomplicated geometry and flute-liner joints using shell-based elements. The effect of the transverse-shear stiffnesson the compression behavior of corrugated panels has beeninvestigated. Nordstand et al. [5] used a sandwich modelfor corrugated panels including transverse shear stiffnessof the layers. The effective shear modulus, Gxz, was derivedfrom curved beam theory for the entire cross sectionincluding liners and a medium. The rigid liner assumption

322 R. Haj-Ali et al. / Composite Structures 87 (2009) 321–333

gave the upper limit of the Gxz, and then it was reduced bythe elastic liner models. The Gyz, was derived based on thedeformation of the core. Different types of corrugatedshapes were investigated in their study. It was found theflute shape had more influence on Gxz than Gyz, where xis the MD-direction. Nordstrand and Carlsson [6] investi-gated the transverse shear using different test methodsincluding ASTM block shear test, three-point bend (TPB)test. The FE analysis is used to simulate the TPB andshowed a good agreement with the experiments. Five differ-ent geometries of boards including four single-wall and adouble-wall boards were investigated for block shear test.They found that the experimental results were significantlyless than those obtained from curved beam theory. Theyattributed this difference to the experiments having delam-inations and buckling which were not accounted for in theanalytical predictions. Also, the values from TPB werelower than those from block shear test. The differencescould be due to the local denting at the supports in TPB.Similar results were discovered in the biaxial loading car-ried out by Patel et al. [7]. They investigated the cylindricalcorrugated board subjected to biaxial stresses. Axial com-pression, torsion and external pressure were applied inde-pendently and in combination to the specimens. Their FEcompression analysis was performed using the Tsai–Wufailure criterion in order to predict the failure of the speci-mens. The FE models gave good predictions to the collapsefailure compared to the experimental results with only uni-axial loadings. However, the FE models over-predicted thecollapse loads for the multiaxial combined loading. Gil-christ et al. [8] used FE models to simulate the four-pointbending and pure twisting test for C-flute board. The non-linear stress–strain curve of liner and medium in MD andCD orientation was obtained by fitting the experiment datausing a hyperbolic tangent equation, and the parametersrequired in the finite element models were calibrated fromthe regression. The simulation results in their work showedfairly good correlations with the experiments but underes-timated the stiffness of the board. This may be attributed tothe lack of modeling the glue bounding the liners and med-ium. Beldie [9] pointed out the effect of the formation ofcreases due to compression failure and how it controlsthe collapse load. A numerical model using ABAQUS�

FE code for corrugated paperboard was proposed andshowed good comparisons between experiments andnumerical models to validate the simulations. The Hill’sorthotropic yield criterion was used to describe the materialplasticity. An approach to model the paperboard creaseswas presented in the last part of this report. A force–dis-placement relationship was obtained from compressingcrease tests. Based on this relationship, a user-defined ele-ment was applied to model the creases. The numericalmodels considering the crease gave more accurate results.Therefore, it was necessary to take into account the creaseinto the FE models of corrugated paperboards. In the sec-ond part of their report, Beldie et al. [10] performed numer-ical studies of paperboard packages under compressive

loading using ABAQUS� FE code. They considered thelayers as anisotropic elastic–plastic materials using Hill’sorthotropic yield criterion. The FE models had threeorthotropic layers considering linerboards and fluted med-ium. Their study represented three sets of experiments andanalysis models: compression of panel, package segments,and a whole package box. Good comparison was shownbetween the numerical result and experiment for a panelwith a height to width ratio of two or higher. However,the numerical results showed the models are stiffer thanthose of experimental results for the package segmentsand the whole package cases. This was because the numer-ical model they developed did not consider the behavior ofcreases of a paperboard package.

To analyze the buckling load of the sandwich structure,Nordstand [11] treated the corrugated core as a homoge-neous linear-elastic layer. He used FE analysis to calculatethe buckling load and collapse load of the sandwich panel.The Tsai–Wu failure criterion was used in the analysis ofthe collapse load. Three different parametric studies wereinvestigated in this study: first, the influence on collapseload due to the initial imperfections and the transverseshear stiffness; second, how the slenderness and the asym-metry affect the strength of the panel; third, how theeccentric loading reduces the collapse load. He concludedthat the collapse load was not sensitive to small imperfec-tions but could reduce 40% with large imperfections. Theasymmetry could induce an eccentric moment to the paneland decrease the collapse load. The increase of the slen-derness ratio could also decrease the collapse load. Eccen-tric loading was applied to a symmetric panel in thisstudy, and it showed the eccentric loading reduced the col-lapse load of the panel. In a later work, Nordstrand[12,13] provided an analytical solution of the critical buck-ling load and failure load of a simply supported corru-gated board. The corrugated board was also assumed tobe an effective panel with effective material properties.The stress components of the panel were used incorpo-rated with the Tsai–Wu criterion to predict the failure.Furthermore, the analytical model was capable of predict-ing the applied load versus out-of-plane displacementcurve. His analytical model and experiments showed 15–20% difference of the buckling load and load–displacementcurve. This could be due to the corrugated panel exhibits anonlinear behavior and local buckling at high stress level.Another explanation may be because the model did notaccount for the effective transverse shear of the board.In the same year, he developed an explicit equation thataccounts for the transverse shear effect, which was notincluded in the classical sandwich buckling theory, inorder to solve the buckling load of the corrugated board.FE analysis with first order transverse shear deformationwas conducted, and had a good agreement with the ana-lytical solution. However, large error was observed com-pared with the experiments. This could be attributed tothe nonlinear behavior of the paper. Allansson and Svard[14] performed two refined and simplified FE analyses for

R. Haj-Ali et al. / Composite Structures 87 (2009) 321–333 323

a corrugated board panel and compare the results withexperiments. The refined model included two liners andone medium while the simplified model used an effectivecore representing the corrugated core. The ABAQUS�

FE code was used for analysis and Matlab� was utilizedfor post-process to display load–deformation curves,stress, and strain failure index. Nonlinear material behav-ior was not included and linear-elastic anisotropic materialproperties were used and Tsai–Wu criterion was appliedto predict the failure of the corrugated board panel. Goodagreement of the load–displacement curves between exper-iments and FE simulation was shown. Urbanik [15]applied the edge crush test (ECT) model developed byJohnson and Urbanik [16] to calculate the edge crushstrength and bending stiffness for different flute geome-tries. He proposed a model to have an optimized flute pro-file. Several geometry parameters (e.g., length of the flank,angle of the warp and take-up factor) were taken intoaccount. In addition, the balance cost, runnability,strength and stiffness were studied to optimize the fluteprofile. Biancolini and Brutti [17] also used FE modelsfor linearized eigenvalue buckling analysis of corrugatedboard packages. They used elastic material propertiesfrom experimental results for numerical analysis andshowed that the measured ECT results are in good agree-ment with those predicted by FE models using eigenvaluebuckling analysis. An equivalent in-plane and bendingstiffness were also developed from a rectangular corru-gated board model. Their BCT model was created usingthis equivalent element. Good agreement was obtainedbetween measured BCT load and the numerical resultsusing buckling analysis. Urbanik and Saliklis [18] per-formed various FE analyses to determine a simplified for-mula for corrugated box panels under axial compression.The commercial ANSYS� FE code was used to calculatethe buckling load. Material nonlinearities and initial geo-metric imperfections were taken into account. Their platemodel was subjected to a uniform displacement on the topedge and the other edges were taken as simply supported.A simplified failure formula for corrugated box panel wasgenerated. Rahman and Abubakr [19] applied differentadhesive modulus to the linear material FE models andcalculate the buckling load. They varied the modulus ofadhesive from one to twenty times large as that of liner-board. A model included one defective glue line was alsoinvestigated in their study. Later, Popil et al. [20] experi-mentally investigated the effect of different adhesive levelsand materials for the glue lines. Refined 3D FE simula-tions were also reported. The later two studies assume thatthe glue lines share portion of the compressive load withliners and medium. Higher adhesive stiffness or adhesivelevel will increase the loading capacity. Rahman et al.[21] used ANSYS� FE code to investigate the compressivedeflection of C-flute under humidity cycles. The liners andflute were modeled using shell elements. Their modelsaccounted for the coefficients of moisture diffusion andexpansion and four constants in the creep constitutive

law. They performed the transient moisture analysis tocalculate the relative humidity for the entire fiberboardas a function of time, and then obtained the hygroexpan-sion due to the change of relative humidity. The hygroex-pansion response was superimposed to the creep responsein order to determine the overall structural behavior.Their method showed a very good agreement with theexperiments and can be used to reduce the experimentaleffort. The effect of the adhesive between liners and med-ium has been investigated utilizing FE method.

This study introduces refined FE models including thedetailed meshed liners and corrugated core structure. Non-linear material properties of the liners and medium areaccounted for in the FE models. These are calibrated byuniaxial tension tests separately for the liner and mediumfiberboards. Those tests are performed mechanically withaxial extension measurements. The compression responseis estimated from the uniaxial tension tests. Shell-basednonlinear finite element modeling is employed. ExtensiveECT tests, for Tappi and in-house coupons were conductedfor a wide range of corrugated paperboards. The refinedFE modeling with experimentally-calibrated nonlinearmaterial behavior of the medium-liners constitute an effec-tive material and structural modeling approach andshowed good prediction capabilities. Damage degradationdue to crease formation or crushing has not beenaccounted for once the Tsai–Wu anisotropic failure crite-rion has indicated material failure.

2. Material property tests and nonlinear FE parametercalibration

The required material properties needed to calibrate thematerial constitutive models within the nonlinear FE simu-lations were obtained from uniaxial stress–strain curves fora selected unbleached southern softwood kraft linearboardand neutral sulphite semi-chemical (NSSC) medium layers.Tensile uniaxial tests were performed using an InstronTM

unidirectional loading cell. The specimens were straightcoupons of 1 in. (2.54 cm) width and 7 in. (17.78 cm) long.The tensile tests included four different categories of speci-mens: liners with MD-direction, liners with CD-direction,flat medium with MD direction, and flat medium withCD-direction. Figs. 1 and 2 illustrate the measuredstress–strain relationship of the liner board specimensunder both MD and CD tension. The FE simulation useda bilinear elastic–plastic relation by fitting the experimentaldata. The compressive yielding in the FE model wasassumed to be 60% of the measured tensile yield stress fol-lowing typical reported stress ratios for fiberboards in theliterature, e.g. Nordstrand [13].

Table 1 shows the measured and assumed yielding ten-sion–compression stresses and strains of the linerboardand medium for both directions. Table 2 shows elasticmoduli and Poisson’s ratio of the liner and medium usedin this paper. The in-plane elastic properties were obtainedby the experiments. The in-plane shear modulus G12 was

0 0.005 0.01 0.015 0.02 0.025

strain (mm/mm)

0

10

20

30

40

50

60

70

stre

ss (

MP

a)Experimental data

Bilinear tensionBilinear compression

(a) Liner MD-direction

0 0.01 0.02 0.03 0.04 0.05

strain (mm/mm)

0

10

20

30

40

stre

ss (

MP

a)

Experimental data

Bilinear tensionBilinear compression

(b) Liner CD-direction

Fig. 1. Stress–strain relationship of 205 g/m2 linerboard.

0 0.004 0.008 0.012 0.016 0.02

strain (mm/mm)

0

10

20

30

40

stre

ss (

MP

a)

Experimental dataBilinear tensionBilinear compression

(a) Medium MD-direction

0 0.005 0.01 0.015 0.02 0.025 0.03

strain (mm/mm)

0

5

10

15

stre

ss (

MP

a)

Experimental dataBilinear tensionBilinear compression

(b) Medium CD-direction

Fig. 2. Stress–strain relationship of 126 g/m2 NSSC medium.

Table 1Yield stress and strain of linerboard and medium

Tension Compression

ry (MPa) ey ry (MPa) ey

Liner MD 38.2 0.008 22.92 0.0048Liner CD 18.4 0.009 11.02 0.0053Medium MD 15.7 0.0035 9.39 0.0021Medium CD 8.06 0.0050 4.84 0.003

Y ðrÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiF ðr22 � r33Þ2 þ Gðr33 � r11Þ2 þ Hðr11 � r22Þ2 þ 2Lr2

2

q

324 R. Haj-Ali et al. / Composite Structures 87 (2009) 321–333

calculated using Eq. (1) [6]. The out-of-plane elastic modu-lus was calculated by dividing the elastic modulus E33 inmachine direction by 250 [9].

G12 ¼ 0:387�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE11 � E22

pð1Þ

Hill’s anisotropic plasticity model is used for the aniso-tropic yielding and hardening behavior in this study. Hill’spotential function is a generalized quadratic function of thestresses which can be written as

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 þ 2Mr2

31 þ 2Nr212 ð2Þ

Table 2Material properties of linerboard and medium

E11 (MPa) E22 (MPa) E33 (MPa) m12 m13 m23 G12 (MPa) G13 (MPa) G23 (MPa)

Liner 4775 2088 19.1 0.18 0.01 0.01 1222 166.28 137.28Medium 4472 1612 17.9 0.18 0.01 0.01 1039 226.34 198.72

R. Haj-Ali et al. / Composite Structures 87 (2009) 321–333 325

where F, G, H, L, M, and N are constants obtained bymaterial tests in different orientations. They are defined by

F ¼r2

11y

2

1

r222y

þ 1

r233y

� 1

r211y

!ð3aÞ

G ¼r2

11y

2

1

r233y

þ 1

r211y

� 1

r222y

!ð3bÞ

F 1 ¼1

rðþÞMD

þ 1

rð�ÞMD

; F 2 ¼1

rðþÞCD

þ 1

rð�ÞCD

; F 11 ¼ �1

rðþÞMDrð�ÞMD

; F 22 ¼ �1

rðþÞCDrð�ÞCD

; F 66 ¼1

T 2ð5bÞ

T ¼ 0:78

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirð�ÞMDrð�ÞCD

q; F 12 ¼ f �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiF 11F 22

p; f � ¼ �0:36 ð5cÞ

Table 4Material properties used in the Tsai–Wu failure criterion (stress units:MPa)

rðþÞMD rð�ÞMD rðþÞCD rð�ÞCD T(S) f* [9]

Linerboards 59.36 �23.82 27.97 �16.30 15.37 �0.36Medium 30.16 �16.52 12.30 �9.56 9.80 �0.36

H ¼r2

11y

2

1

r211y

þ 1

r222y

� 1

r233y

!ð3cÞ

L ¼r2

11y

2

1

r223y

!; M ¼

r211y

2

1

r213y

!; N ¼

r211y

2

1

r212y

!

ð3dÞ

where r11y, r22y, r33yr12y, r13y and r23y are the yield stressvalues for each directions. Table 3 shows both linerboardand medium tension/compression yield stresses. Only theassumed compression stresses were used in the calibrationof Hill’s plasticity model. Calibrating this plasticity modelrequires obtaining additional yield stresses not tested inthis study. These include the in-plane shear stress, r12y,and the out-of-plane yield stresses, r33y, r13y, and r23y.We followed Beldie’s [19] assumed relation for the out-of-plane normal yield stress and the reported shear stres-ses written as

r33y :4

r211yr

222y

>1

r233y

� 1

r211y

þ 1

r222y

!" #2

ð4aÞ

r12y ¼ 2:1 MPa; r13y ¼ 0:024 MPa; r23y ¼ 0:024 MPa

ð4bÞ

The Tsai–Wu anisotropic and plane-stress failure criterionwas used in this study to determine the ultimate failure at a

Table 3Material parameters used for Hill’s anisotropic plasticity model (unit:MPa, mm)

r11y r22y r33y

Liner compression 22.92 11.02 7.52Liner tension 38.2 18.4 12.55Medium compression 9.391 4.836 3.23Medium tension 15.65 8.06 5.38

point in the linerboard and medium. The equation for theT–W failure criterion can be expressed as

F 1r11 þ F 2r22 þ F 11r211 þ F 22r

222 þ F 66r

212 þ 2F 12r11r22 6 1:0

ð5aÞwhere the coefficients were determined similarly to Beldie[9] by the uniaxial ultimate strength using:

where rðþÞMD is the ultimate tensile stress in MD-direction;

rð�ÞMD is the ultimate compressive stress in MD-direction;

rðþÞCD is the ultimate tensile stress in CD-direction; rð�ÞCD isthe ultimate compressive stress in CD-direction.

Table 4 lists the ultimate stresses and other parametersused in the calibration of the Tsai–Wu failure criterionfor both linerboard and medium layers. Those values wereobtained by compression and tension tests performed usingthe standard STFI-SCT (short span compression test) andaxial strip tension, respectively.

3. Experiments and FE simulations for single-wall

corrugated boards

The material properties needed for the FE simulationsare listed in the previous section. These can be describedas the ‘‘in situ” material properties. This section deals withstructural FE modeling and simulations of single-wall cor-rugated boards along with their edge crush test (ECT) inorder to examine the full-range of the overall mechanicalresponse predicted by the FE models. Towards that goal,four different commercial single-wall corrugated boards(A-, B-, C-, and E-flute) were subject to ECT tests and sim-ulated with refined FE models. Fig. 3 shows a typical crosssection of the corrugated FE mesh. A special code waswritten to automatically generate FE mesh for a general

Fig. 3. General flute geometry used in FE simulations.

Table 5Four classes of flute geometries (typical ranges from Steadman [22]) (units:mm)

A-flute B-flute C-flute E-flute

Typical flute spacing (k) 8.3–10 6.1–6.9 7.1–8.3 3.2–3.6Typical flute height (h) 4.67 2.46 3.61 1.15Measured flute spacing (k) 8.6 6.0 7.9 3.4FE model (k) 10.0 6.35 8.50 3.63FE model (h) 4.60 2.46 3.50 1.15

326 R. Haj-Ali et al. / Composite Structures 87 (2009) 321–333

system of two linerboards and fluting medium given basicgeometric properties. The sine function was used to idealizeflute wave between the two liners. Table 5 lists the pitchlength (k) and the flute height (h) for the four differentgeometries. The geometries used in the FE models maynot be exactly the same as those measured from the speci-men due to the idealized FE geometry parameters (e.g.,sine wave and pitch).

Two types of ECT tests were performed in this study.Fig. 4a and b shows the two ECT test specimens and fix-tures, respectively. The dimensions for the non-standardspecimen, denoted as IPST, are 1.5 in. wide and 2 in.height. The IPST coupon edges were dipped into moldswith resin to fully bond them and create a small flat strepfor load application, as shown in Fig. 4a. Popil et al. [23]initially used these specimens to study the onset of liner-board buckling during ECT tests. The TAPPI-T839 testspecimens were conducted with 2 in. by 2 in. and a 0.4in. of an unclamped free spacing. The four different flutegeometries in Table 5 were used in the FE simulations ofthe two types of ECT tests. The 8-node reduced integration

Fig. 4. ECT specimens a

three-dimensional quadratic shell element (S8R) was usedfor liner and flute paperboard. The area near the jointbetween linerboards and flute has relatively more elementscompared to other parts on the surface of the linerboards.At this stage of the FE simulations, a fully bonded contactbehavior was assumed at the line of nodes (joint) betweenthe medium and linerboards. Fixed boundary DOF condi-tions were used for bottom coupon edge nodes. Similarboundary conditions were also used at the top edge exceptfor the axial DOFs where all the edge nodes were con-strained to have the same shortening displacement. TheTAPPI T839 test uses a steel frame clamping fixture to holdthe corrugated board specimen, Fig. 4b, and prevent ordelay buckling in order to allow for crushing of the0.4 in. free span. The FE TAPPI ECT models use 10 psiuniform pressure on the linerboard surfaces that are in con-tact with the steel frame clamping surfaces (as it is recom-mended by TAPPI through a normal spring). Geometryand material nonlinearity are considered in the FE models.The Tsai–Wu failure criterion was employed in order todetermine the failure initiation and estimate the ECT ulti-mate load of the corrugate boards. Three failure states weredefined in this study and schematically illustrated in Fig. 5.The first failure state indicates the initiation stage and it isidentified by examining the failed points (values larger than1) in the contour plots of the Tsai–Wu failure criterion. Thesecond failure state is called the ultimate failure of the cor-rugated board. This state is identified when failed region onthe linerboards have reached the full length of the pitchlength. A third state was also investigated where it is plot-ted at twice the failure initiation. This third state came in

nd fixtures for tests.

Fig. 5. Schematic illustration of three failure states (stages) determinedbased on the Tsai–Wu contour plots and identified on a typical load-shortening curve.

R. Haj-Ali et al. / Composite Structures 87 (2009) 321–333 327

most cases after the second failure state (failure wavelengthequal to pitch length). The Tsai–Wu contours increase andprogress with continued deformations. It was observed thatconnected failure regions fully develop at the third failedstate with links established from both sides of the coupon.Fig. 6 illustrates the normalized load versus end-shorteningdisplacement response of A-flute ECT IPST repeatedexperiments and their FE simulation. The experimentaldata in the form of axial deformation contained axial rigidbody motion due to the continuous crease formation.Therefore the raw experimental exhibits larger shorteningthan the FE results where progressive crease formationand wrinkles were not accounted during the analysis.Therefore, in order to compare the FE predicted results(without crease formation) with the experimental data, anexperimental shortening factor (ESF) is used for each setof ECT tests. The ESF is calculated by initially plottingthe FE results with the non-scaled normalized experimental

0 0.4 0.8 1.2 1.6 2

Normalized end-shortening (in/in) (%)

0

10

20

30

40

50

60

70

Nor

mal

ized

forc

e (lb

/in)

Experimental dataNonlinear FE analysisContinued FEA withoutdamage degradation

Tsai-Wu Out-of-plane displ.

Tsai-Wu

Tsai-Wu

1.000.830.670.500.330.170.00

0.300.180.06-0.06-0.18-0.30

Tsa

i-Wu

Out

-of-

plan

e di

spl.

(mm

)

Estimated ultimate failure

2x displ. of failure initiation

Failure initiation

ESF = 0.38

Out-of-plane displ.

Out-of-plane displ.

Fig. 6. Normalized load end-shortening response of A-flute single-wallIPST ECT FE nonlinear simulation with several repeated tests.

data. Then, we proceed to identify all the non-scaled exper-imental data points having up to 0.25% of end-shortening.A liner regression is then performed for the later set ofexperimental data to identify the slope in this non-scaleddata set. The ESF factor is then calculated as the ratio ofthe slope of the experimental data over the slope of theFE results. The ESF is then used to scale all experimentaldata by multiplying it with this factor to guarantee a matchin the initial slope between the FE and experimentalresults. The axial displacement of the experiments andthe FE simulations conform to each other while the forceremains unaffected. Fig. 7 shows the normalized loadend-shortening responses of A-flute single-wall TAPPI-T839 ECT FE nonlinear simulation with two repeatedtests. It is interesting to note that the FE nonlinear simula-tion do predict the peak load and the drop afterwards.However, since the FE is only nonlinear without progres-sive damage, the dashed line in the figure illustrates theresults without damage degradations. Examining theTsai–Wu contour plots, it is noted that initiation (first stageof failure) occurs immediately after the peak load as thelast point on the solid load-shortening response curve.The fully developed wavelength failure (second stage) fol-lows and is very close to initiation. This illustrates theimmediate crushing behavior of this ECT test which doesnot allow post-buckling type structural response todevelop. The failure contours are associated with out-of-plane displacement contours to identify local inter-flutebuckling that is usually associated with the second stageof failure response. Fig. 8 shows the normalized loadend-shortening response of B-flute single-wall IPST typeECT experimental results for several repeated tests. It is

0 0.5 1 1.5 2 2.5 3 3.5

Normalized end-shortening (in/in) (%)

0

10

20

30

40

50

60

70

Nor

mal

ized

forc

e (lb

/in)

Experimental dataNonlinear FE analysisContinued FEA withoutdamage degradation

Tsai-Wu

Tsai-Wu

1.000.830.670.500.330.170.00

Tsa

i-Wu

Out

-of-

plan

edi

spl.

(mm

)

2x displ. of failure initiation

Failure initiation

ESF = 1.0

0.190.110.04-0.04-0.11-0.19

Out-of-plane displ.

Estimated ultimate failure

Out-of-plane displ.Tsai-Wu

Out-of-plane displ.

Fig. 7. Normalized load end-shortening response of A-flute single-wallTAPPI-T839 ECT FE nonlinear simulation with two repeated tests.

0 0.4 0.8 1.2 1.6 2

Normalized end-shorting (in/in) (%)

0

10

20

30

40

50

60

70

80

90

Nor

mal

ized

forc

e (lb

/in)

Experimental dataNonlinear FE analysisContinued FEA without damage degradation

Tsai-Wu Out-of-plane displ.

Tsai-Wu

Tsai-Wu

1.000.830.670.500.330.170.00

Tsa

i-Wu

Out

-of-

plan

e di

spl.

(mm

)

Estimated ultimate failure

2x displ. of failure initiation

Failure initiation

ESF = 0.47

0.110.070.02-0.02-0.07-0.11

Out-of-plane displ.

Out-of-plane displ.

Fig. 8. Normalized load end-shortening response of B-flute single-wallIPST ECT FE nonlinear simulation with several repeated tests.

0 0.5 1 1.5 2 2.5

Normalized end-shorting (in/in) (%)

0

10

20

30

40

50

60

70

Nor

mal

ized

forc

e (lb

/in)

Experimental dataNonlinear FE analysisContinued FEA withoutdamage degradation

Tsai-Wu Out-of-plane displ.

Tsai-Wu Out-of-plane displ.

Tsai-Wu Out-of-plane displ.

1.000.830.670.500.330.170.00

Tsa

i-Wu

Out

-of-

plan

e di

spl.

(mm

)

Estimated ultimate failure

2x displ. of failure initiation

Failure initiation

ESF = 0.300.110.070.02-0.02-0.07-0.11

Fig. 9. Normalized load end-shortening response of B-flute single-wallTAPPI-T839 ECT FE nonlinear simulation with several repeated tests.

0 0.4 0.8 1.2 1.6 2

Normalized end-shorting (in/in) (%)

0

10

20

30

40

50

60

70

Nor

mal

ized

forc

e (lb

/in)

Experimental dataNonlinear FE analysisContinued FEA withoutdamage degradation

Tsai-Wu Out-of-plane displ.

Tsai-Wu Out-of-plane displ.Tsai-Wu Out-of-plane displ.

1.000.830.670.500.330.170.00

0.300.180.06-0.06-0.18-0.30

Tsa

i-Wu

Out

-of-

plan

e di

spl.

(mm

)

2x displ. of failure initiationFailure initiation

ESF = 0.51

Estimated ultimate failure

Fig. 10. Normalized load end-shortening response of C-flute single-wallIPST ECT FE nonlinear simulation with several repeated tests.

328 R. Haj-Ali et al. / Composite Structures 87 (2009) 321–333

clearly seen that there is a wide scatter in the experimentalresults which may indicate the imperfection sensitivity ofthe ECT test with B-flute corrugated material system.The scatter in the experimental data is not limited to thepeak load but also in the post-buckling response region.The FE nonlinear simulation did not include imperfectionof the linerboards for post-buckling analysis which usuallyconducted using first few eigenmodes. In addition, we usedisplacement control to simulate the response in the FEresults and therefore do not expect to generate the full-range of post-buckling behavior should it occur. The lateris a fully structural response which can be investigatedusing an arc-length type solution scheme. This FE typesimulation is recommended for long corrugated panels orboxes and considered beyond the scope of this paper.Fig. 9 includes the normalized load end-shorteningresponse of B-flute single-wall TAPPI-T839 ECT repeatedtests along with the FE nonlinear simulation. Similar to theTAPPI test of the A-flute, Fig. 7, the failure initiation isvery closely predicted immediately after the peak loading,plotted as the last point on the solid response curve. How-ever, the second stage of failure in the current B-flute issomewhat delayed and is near the 1% normalized axial dis-placement (average strain) which is closer to the third fail-ure stage. In addition the second stage of failure in the B-flute is not associated with large out-of-plane displacementas in Fig. 7 for A-flute. Fig. 10 shows the normalized loadend-shortening response of C-flute single-wall IPST ECTrepeated tests. These have relatively less scatter than theprevious experiments with IPST type geometry. The FEsimulation without imperfection or damage degradationis close to the experimental response up to the peak load.

The estimated ultimate state of failure (second state) is rel-atively further away from the initiation state after 0.8% ofaverage axial strain. Fig. 11 shows the same results for C-flute however using the TAPPI T893 ECT. The experimen-tal result in this case is very close to the IPST wider freespan type specimen. Figs. 12 and 13 show the experimentaland FE simulation results for the IPST and TAPPI E-flutecoupons, respectively. Since the E-flute has the smallest

0 0.4 0.8 1.2 1.6 2

Normalized end-shorting (in/in) (%)

0

10

20

30

40

50

60

70

80

Nor

mal

ized

forc

e (lb

/in)

Experimental dataNonlinear FE analysisContinued FEA withoutdamage degradation

1.000.830.670.500.330.170.00

0.300.180.06-0.06-0.18-0.30

Tsa

i-Wu

Out

-of-

plan

e di

spl.

(mm

)

Estimated ultimate failure

2x displ. of failure initiation

Failure initiation

Tsai-Wu

Tsai-Wu Out-of-plane displ.

Tsai-Wu Out-of-plane displ.

ESF = 0.49

Out-of-plane displ.

Fig. 11. Normalized load end-shortening response of C-flute single-wallTAPPI-T839 ECT FE nonlinear simulation with an experimental result.

0 0.4 0.8 1.2 1.6 2

Normalized end-shorting (in/in) (%)

0

10

20

30

40

50

60

70

Nor

mal

ized

forc

e (lb

/in)

Experimantal dataNonlinear FE analysisContinued FEA without damage degradation

Tsai-Wu Out-of-plane displ. Tsai-Wu Out-of-plane displ.

1.000.830.670.500.330.170.00

Tsa

i-Wu

Out

-of-

plan

edi

spl.

(mm

)

2x displ. of failure initiationFailure initiation

2.001.200.40-0.40-1.20-2.00

ESF = 0.33

Fig. 12. Normalized load end-shortening response of E-flute single-wallIPST ECT FE nonlinear simulation with three repeated tests.

0 0.5 1 1.5 2 2.5

Normalized end-shorting (in/in) (%)

0

10

20

30

40

50

60

70

80

Nor

mal

ized

forc

e (lb

/in)

Experimental dataNonlinear FE analysisContinued FEA withoutdamage degradation

Tsai-Wu Out-of-plane displ.

Tsai-Wu Out-of-plane displ.

1.000.830.670.500.330.170.00

Tsa

i-Wu

Out

-of-

plan

edi

spl.

(mm

)

2x displ. of failure initiation

Failure initiation

ESF = 0.51

Tsai-Wu Out-of-plane displ.

Estimated ultimate failure

1.000.830.670.500.330.170.00

Fig. 13. Normalized load end-shortening response of E-flute single-wallTAPPI T839 ECT FE nonlinear simulation with two repeated tests.

Table 6Comparing ultimate loads for IPST ECT single-wall (unit: lb/in)

Flute geometry FE failure initiation Experiment FE estimatedultimate failure

IPST A-flute 41.2 44.8 39.3B-flute 48.5 40.2 49.3C-flute 47.5 40.9 46.0E-flute 38.9 35.3 34.0

T839 A-flute 44.8 44.4 44.0B-flute 42.8 52.1 53.1C-flute 49.4 44.0 52.7E-flute 48.0 40.3 41.1

R. Haj-Ali et al. / Composite Structures 87 (2009) 321–333 329

overall thickness, it is expected to have interaction betweenthe axial compression and out-of-plane displacements. Thisis mainly expressed by the post-peak dramatic load reduc-tion. The FE models in both cases predict larger responseand peak loads with similar overall behavior. This can beexplained due to the perfect geometry and lack of damagedegradations in the simulations. The experimental averageultimate load and the estimated failure load from the FE

simulations (first and second stage failure) are shown inTable 6 for all cases. The FE predicted failure point inthe proposed method is closer to the TAPPI ECT thanthe IPST. This can be explained by the larger scatter inthe IPST test results along with the fact that the IPST spec-imens are more sensitive to geometry imperfections thatlead to larger out-of-plane displacements.

4. Double-wall boards

This section deals with the ECT simulations and testsfor double-wall corrugated fiberboards using the previousIPST geometry. Towards that goal, hybrid double-wallECT specimens were constructed and FE models weredeveloped for AB-flute, AC-flute, BC-flute and CE-flute.The IPST experiments were conducted to verify the predic-tion capability of the FE models calibrated previously forthe linerboard and flute layers. The double-wall corrugatedboards consisted of different flute size. For example, AB-

330 R. Haj-Ali et al. / Composite Structures 87 (2009) 321–333

flute paperboard was made by an A-flute and a B-flute cor-rugates glued by a shared liner. Fig. 14a shows the AB-fluteFE model where the A-flute is in the front and the B-flute isin the back side. A typical deformed ECT response is plot-ted in Fig. 14b with the Tsai–Wu criterion applied sepa-rately for the linerboard and medium layers. Fig. 15shows the FE simulations and experimental results for anAB-flute multi-wall ECT geometry. The failure initiationand estimated ultimate failure stages are assumed in a sim-ilar manner to the single-wall corrugated boards analyzed

Fig. 14. FE model and si

0 0.2 0.4 0.6 0.8 1

Normalized end-shortening (in/in) (%)

0

20

40

60

80

100

120

140

Nor

mal

ized

forc

e (lb

/in)

Experimantal dataNonlinear FE analysisContinued FEA without damage degradation

1.000.830.670.500.330.170.00

Tsa

i-Wu

Failure initiation

A-flute side

B-flute side

Failure of A-flute

A-flute side B-flute side

ESF = 0.26

Fig. 15. Normalized load end-shorting response of AB-flute double-wallIPST ECT FE simulation with several repeated tests.

in the previous section. The failure contours shown inFig. 15 are for both sides, respectively. Failure initiationoccurs first in the A-flute side with relatively minor failurein the B-flute side immediately after the peak overall load-shortening curve. The six repeated experimental resultshave minimal scatter and are very repeatable in all stagesof the load–displacement curve. Fig. 16 shows the FE sim-ulations and experimental results for the AC-flute double-wall geometry. The failure initiation first occurs also inthe A-flute side. However, the damage degradation is also

mulation of AB-flute.

0 0.2 0.4 0.6 0.8 1.2

Normalized end-shortening (in/in) (%)

0

20

40

60

80

100

120

140

Nor

mal

ized

forc

e (lb

/in)

Experimantal dataNonlinear FE analysisContinued FEA without damage degradation

1.000.830.670.500.330.170.00

Tsa

i-Wu

ESF = 0.37

Failure initiation

A-flute side

C-flute side

Failure of C-flute

A-flute side C-flute side

1

Fig. 16. Normalized load end-shorting response of AC-flute double-wallIPST ECT FE simulation with several repeated tests.

0 0.2 0.4 0.6 0.8 1 1.2

Normalized end-shortening (in/in) (%)

0

20

40

60

80

100

120

140

Nor

mal

ized

forc

e (lb

/in)

Experimantal dataNonlinear FE analysisContinued FEA withoutdamage degradation

1.000.830.670.500.330.170.00

Tsa

i-Wu

ESF = 0.25

Failure initiation

C-flute side E-flute side

Failure of C-flute

C-flute side

E-flute side

Fig. 18. Normalized load end-shorting response of CE-flute double-wallIPST ECT FE simulation with several repeated tests.

Table 7Comparing ultimate loads for IPST ECT single-wall (unit: lb/in)

Flute geometry FE failure initiation Experiment FE estimatedultimate failure

AB-flute 74.0 73.1 77.9AC-flute 74.3 81.3 76.2BC-flute 85.6 75.1 87.1CE-flute 84.7 81.6 94.8

R. Haj-Ali et al. / Composite Structures 87 (2009) 321–333 331

evident and relatively higher in magnitude in the C-fluteside. In fact, the rate of damage in the C-flute side is fasterand the C-flute reaches the estimated failure stage prior orclose to the A-flute side. Fig. 17 shows the FE simulationsand experimental results for the BC-flute double-wallgeometry. Consistent reputable experimental ECT resultsare obtained in this case. The failure initiation also occursat the knee area immediately after the change in slope ofthe effective load–deformation curve. The estimated failurestages first occur in the C-flute side, while the B-flute sidehas relatively minor damage. The FE simulation showsthe failure regions to occur axially at the inter-fluting andtend to develop along the joints between the linerboardand the medium in the C-flute side. Finally, Fig. 18 showsthe FE simulations and experimental results for the CE-flute double-wall geometry. The failure initiation and esti-mated ultimate failure first occur in the C-flute side only.The E-flute side shows less inter-fluting buckling and dam-age due to the thin thickness of the E-flute-linerboard. It isinteresting to note that most of the double-wall ECT testshave significantly less scatter than the single-wall experi-ments. Furthermore, the ESF factors in the double-wallcoupons are very close except for the AC-flute where ithas more scatter in the experimental results. This indicatesthat proposed ESF is effective in allowing better correlationof the FE simulations especially that the FE modeling didnot include damage degradation and the crease formation.

The ultimate loads of the multi-wall corrugated boardswere determined using the same method in single-wall anal-yses. Table 7 shows the comparison of the ECT ultimateload between experiments and FE simulations at both ini-tiation and estimated ultimate states.

0 0.2 0.4 0.6 0.8 1 1.2

Normalized end-shortening (in/in) (%)

0

20

40

60

80

100

120

140

Nor

mal

ized

forc

e (lb

/in)

Experimantal dataNonlinear FE analysisContinued FEA withoutdamage degradation

1.000.830.670.500.330.170.00

Tsa

i-Wu

Failure initiation

B-flute side C-flute side

Failure of C-flute

B-flute side

C-flute side

ESF = 0.27

Fig. 17. Normalized load end-shorting response of BC-flute double-wallIPST ECT FE simulation with several repeated tests.

5. Parametric glue line 3D models

In this section, 3D continuum elements were addedbetween the medium and liners in order to examine the glueeffect on the overall performance. Previous FE modelsdeveloped in this study did not take into account the gluelines between the liners and the flute. Instead, the linerand flute were fully bonded in these models. Fig. 19 showstwo different glue joints in a single-wall linerboard. The fig-ure illustrates the different nature of the glue joint due tothe manufacturing bonding sequence in the single-faceand double-back side of the corrugated board. The sin-gle-face joint side was created with an applied pressure of1000 psi, while the double-back side had only 1 psi ofapplied pressure. The effect of the pressure applicationcan be seen in Fig. 19 where the single-glue joint is moreflattened and thinner to the relatively bulkier double-backjoint. Therefore, the difference of the geometry of two gluejoints should be taken into account in the refined localanalysis. Fig. 20 shows ECT FE single-wall models with3D glue elements for the C-flute board. The glue geometryparameters used in our study are illustrated in shown inFig. 20a for both glue joints. The elements used in the

Fig. 19. Glue line joints at single-face and double-back of the corrugatedboard.

Table 8Glue geometry used in FE models

Type Double-back glue joint Single-face glue joint

a (mm) b (mm) a � b (mm2) c (mm) d (mm) c � d (mm2)

I 1.50 0.20 0.30 1.50 0.10 0.150II 0.80 0.10 0.08 0.60 0.03 0.018III 0.30 0.10 0.03 1.00 0.03 0.030

0 0.4 0.8 1.2 1.6

Normalized end-shortening (in/in) (%)

0

15

30

45

60

Nor

mal

ized

forc

e (lb

/in)

Experimental dataModel without 3-D glue lineModel with type-I glue lineModel with type-II glue lineModel with type-III glue line

ESF = 0.58

Fig. 21. Normalized load–displacement response of experiments and FEsimulations for different glue levels.

332 R. Haj-Ali et al. / Composite Structures 87 (2009) 321–333

FE coed for the liner and flute were quadrilateral shell ele-ments (S8R) similar to the previous sections. The glue linewas assumed isotropic and 3D solid 20-node quadraticbrick elements (C3D20R) were used for the glue. TheYoung’s modulus and Poisson’s ratio were 400 MPa and0.3, respectively. The total element number used in thismodel was 15,024, and the node number was 79,840. Forthe model without 3D glue line elements, the connectionsbetween the flute and the liners were ideally fixed by directnodal equivalence. Three different glue geometries weresimulated in this parametric study. Table 8 shows the sizeof the simulated FE glue lines with the parameters shownin Fig. 20a. The first glue line geometry (type-I) is the larg-est of the three studied. The glue line front area is in adecreasing order for the double-back joint.

Fig. 21 shows the normalized load end-shorteningresponse of the different glue geometry compared to themodel without glue lines (perfect bond) and to the previousECT experimental data closer to the type-I geometry. Theultimate load of the type-I is similar to the model withoutglue lines and to the ECT tests but the stiffness around ulti-

Fig. 20. Finite element ECT mo

mate load of type-I is slightly lower and the strain at theultimate load is relatively larger. Type-II and III give sim-ilar responses with a lower bound predicted ultimate load.We find about 15% difference in ECT when changing the

dels with 3D glue elements.

R. Haj-Ali et al. / Composite Structures 87 (2009) 321–333 333

adhesive volumes from type-I to type-III. In addition,higher glue levels suppress linerboard buckling whichtranslate in higher ECT.

6. Conclusions

This study introduces a refined nonlinear material andstructural modeling approach using nonlinear finite ele-ments with experimentally calibrated constitutive modelsfor the layers. The FE models gave good ECT responseup to and including failure for a wide range of corrugatedboards and geometries. The calibrated constituents wereused to simulate the ECT response of single-wall ECT,multi-wall ECT and to study the effect of the glue line vol-ume. The Tsai–Wu anisotropic failure criterion wasemployed and can be effective in determining the failureof the material and the different stages in failure progres-sion for the structure. Good overall predictions for the ulti-mate failure can be obtained from the FE models. Thecalibrated FE models can be used to simulate and designnew single and multi-wall corrugated structures includingother non-traditional cross-sectional shapes.

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