ou

, F.x 53

es R0, C

e anultsd restruAccf th

computationally demanding than FEA, and appears viable for prediction of the twist stiffness of corru-

strucmadet numbcommuble-w

ing. The principal material directions of each layer are denoted byMD (machine-direction), CD (cross-direction) and ZD (thickness-direction). The CD of the liners and web core layers is orientedparallel to the corrugations.

The concept of determining the twist stiffness of a sandwichstrip specimen under torsion was rst introduced by McKinlay[1]. He constructed a twist tester, patented in 1990 [2], where a

termine thouble web

In addition, nite element analysis (FEA) and experiments aducted to validate the predictions. Parametric analysis of thstiffness of corrugated board as a function of the core out-of-planeshear moduli is presented.

2. Twist test

This work focuses on the twist stiffness of single and double-wall boards. This test was rst proposed as a shear test of thinplates by Ndai in 1968 [8]. For this specimen, torque is achieved

Corresponding author. Tel.: +46 018 471 3026.

Composite Structures 110 (2014) 715

Contents lists availab

S

sevE-mail address: hepadrian@gmail.com (A. Hernndez-Prez).that the material is considered orthotropic. During the manufac-turing process the paper web is stretched along the direction ofmanufacture (machine direction) which results in further stiffen-

tion (FOSD) theory.In this study, the FOSD solution is used to de

stiffness of corrugated boards with single and d0263-8223/$ - see front matter 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.compstruct.2013.11.006e twistcores.

re con-e twistSingle-wall (SW) corrugated board is a regular sandwich, consist-ing on three layers, viz, two at linerboards bonded to a sinewave shaped web core. Double-wall (DW) corrugated board con-sists of two layers of corrugated web bonded to three at linersheets, one in the center separating the two corrugated layers,and two at the outer surfaces. The liners and corrugated websconsist on paper layers made from cellulose bers approximately13 mm long, aligned in the plane of the layers in such a manner

icted on the web core during the corrugation and board assemblyprocesses is delamination. Such damage is detrimental to the out-of-plane stiffness and strength of the corrugate panel. The twist re-sponse of corrugated board panels may also be determined using aquasi-static test method, called the sandwich plate twist test, seeMure [4], Pommier and Poustis [5] and Carlsson et al. [6]. Recently,Hernndez-Prez et al. [7] developed a Fourier series solution forthe sandwich plate twist specimen using rst order shear deforma-1. Introduction

Corrugated board is a sandwichcore glued to at sheets (liners), allboard is manufactured with differenon the packaging application. Twoable boards are the single and dogated board. 2013 Elsevier Ltd. All rights reserved.

ture consisting of webfrom paper. Corrugateder of layers dependingon commercially avail-all corrugated boards.

strip of corrugated board is clamped at both ends and under tor-sional oscillations by the aid of a counter weight. The twist stiff-ness of the board is calculated from the natural frequencyaccording to the harmonic equation of the torsional pendulum.An important feature of this test is that the twist stiffness is verysensitive to factors such as damage of the core, face/core adhesionand shape of the web core [13]. A common source of damage in-Finite element analysisPlate theory

between the torsional stiffness predictions by analytical and numerical approaches and test results isfound for the range of single and double-wall boards examined. The FOSD solution is signicantly lessAnalysis of twist stiffness of single and d

A. Hernndez-Prez a,, R. Hgglund b, L.A. Carlsson caDepartment of Engineering Sciences, The Angstrm Laboratory, Uppsala University, Bob SCA R&D Center, Box 716, SE-851 21 Sundsvall, SwedencDepartment of Ocean and Mechanical Engineering, Florida Atlantic Universiy, 777 GladdCentro de Investigacin Cientca de Yucatn, A.C. Unidad de Materiales, Calle 43 # 13

a r t i c l e i n f o

Article history:Available online 20 November 2013

Keywords:Twist stiffnessCorrugated boardsTorsion

a b s t r a c t

The twist stiffness of singlmation (FOSD) theory. Reslarge range of torsion loadeenized core. In addition, asented by shell elements.transverse shear moduli o

Composite

journal homepage: www.elble-wall corrugated boards

Avils d

4, SE-751 21 Uppsala, Sweden

oad, Boca Raton, FL 33431, USAol. Chuburn de Hidalgo, C.P. 97200 Mrida, Yucatn, Mexico

d double-wall corrugated board is analyzed using rst order shear defor-are compared to nite element analysis (FEA) and dynamic test data for actangular board specimens. The FOSD approach and FEA employ a homog-ctural nite element model was developed where the web core is repre-ording to FOSD analysis, the twist stiffness is linearly dependent on thee web core along both principal directions of the core. Good agreement

le at ScienceDirect

tructures

ier .com/locate /compstruct

by two end couples produced by application of two concentratedforces (P/2) at diagonally opposite corners of the panel with theother two corners pin supported, see Fig. 1. For sandwich panels,a very important deformation mode is transverse shear deforma-tion of the core. To analyze the twist stiffness of a sandwich panel,a solution based on rst order shear deformation (FOSD) theoryhas recently been derived [7]. This solution utilizes plate stiffness-es dened in layered plate theory [9], which are calculated fromthe effective elastic properties of each layer in the sandwich panelas explained in Appendix A. On the other hand, denition of platestiffnesses are given in Appendix B. FEA of the twist test specimenis conducted using two types of models, one with a structural mod-el of the web core, and the other with a homogenized core.

Fig. 1 shows single and double-wall boards panels under twistloading. The specimens are 25 mm wide and 105 mm long.

Chalmers [10] considered a twist loaded specimen as shown inFig. 2, and dened the torsional stiffness (twist stiffness) DQM as,

DQM Tahb 1

where T is the torque applied by the concentrated forces (P/2) and his the angle of twist at each end of the specimen, see Fig. 2. a and bare the length and width of the loaded area of the specimen denedby the rectangle formed by the four loading and support pins. In or-der to maintain a linear elastic response, the angle h must be small(tanh h) and it can be approximated by,

elements to represent the wave-shaped cores, see Fig. 3. The facesheets and core are connected by common nodes at the utingcrests. Elastic properties of the face and web sheets are listed inTables 3 and 4.

Table 1Lay-ups and ply thicknesses of single-wall (SW) boards. L = liner, W = web.

Board Ply number/material/thickness (mm) Core height (mm)

Top face Web Bottom face

SW1 L1 (0.243) W1 (0.240) L1 (0.243) 3.6SW2 L2 (0.228) W2 (0.220) L2 (0.228) 3.6SW3 L3 (0.215) W2 (0.220) L3 (0.215) 3.6SW4 L4 (0.185) W5 (0.181) L4 (0.185) 3.6SW5 L2 (0.228) W4 (0.220) L3 (0.215) 2.54

8 A. Hernndez-Prez et al. / Composite Structures 110 (2014) 715h d=b 2According to Figs. 1 and 2 the magnitude of the torque applied

at the ends of the corrugated boards is given by,

T Pb=2 3By substituting Eqs. (2) and (3) into (1), the twist stiffness DQM

becomes,

DQM P a b2d 4Fig. 1. Plate twist test for board specimens. (a) Single-wall and (b) double-wall.Notice that the units of the twist stiffness DQM are in Nm. Eq. (4)is used to determine the twist stiffness of single and double-wallboards by the FOSD and FEA approaches.

3. Materials and specimens

A total of 14 boards were considered, 7 single-wall and 7 dou-ble-wall boards. Tables 1 and 2 provide the specic combinationof liners and web cores considered in the analysis of the singleand double-wall boards, Fig. 1. In these tables L refers to liner, Wto web and the number next to W and L species the liners andcores as specied in Tables 3 and 4.

All constituent layers of the boards are considered orthotropicwith in-plane elastic stiffnesses (E1, E2 and G12) listed in Tables 3and 4. Since the out-of-plane stiffnesses of the constituent papersheets have negligible inuence on the twist stiffness of the board,they were assumed to be the same for all papers considered here,i.e., E3 = 37 MPa, G23 = 75 MPa and G13 = 133 MPa [1114]. Further-more, an in-plane Poissons ratio (m12) of 0.43 was assumed for allpapers [15]. Although m12 may vary somewhat, the results from theanalysis are very little inuenced by such variations.

3.1. Structural nite element model of panels

A structural FEA model of the twist specimen was implementedin ANSYS [16]. The web core layers were assumed to be sinusoidalwith wavelengths of 7.7 and 6.41 mm. The lay-ups and coreheights are specied in Tables 1 and 2. The specimen length (a)was 105 mm and the width (b) was 25 mm, see Fig. 1. Each layerwas modeled by 8-node quadrilateral isoparametric shell elements(SHELL93) [16,17]. In total, 23,040 elements were employed for thesingle-wall specimens, and 26,400 elements for the double-wallspecimens. The elements are at, making necessary to use small

Fig. 2. Deformed shape of a single-wall board loaded in torsion.SW6 L4 (0.185) W3 (0.181) L4 (0.185) 2.54SW7 L5 (0.170) W3 (0.181) L5 (0.170) 2.54

posiTable 2Lay-ups and ply thicknesses for double-wall (DW) boards. L = liner, W = web.

Board Ply number/material/thickness (mm)

1 2 3

DW1 L6 (0.369) W4 (0.221) L5 (0.17)DW2 L2 (0.228) W4 (0.221) L5 (0.17)DW3 L4 (0.185) W3 (0.181) L7 (0.181)DW4 L8 (0.16) W3 (0.181) L7 (0.181)DW5 L9 (0.21) W4 (0.221) L7 (0.181)DW6 L10 (0.24) W4 (0.221) L4 (0.185)DW7 L11(0.185) W3 (0.181) L7 (0.181)

Table 3Elastic stiffnesses (GPa) of the face sheets.

Face E1 E2 G12

L1 7.32 2.63 1.70L2 7.42 2.65 1.72L3 6.00 2.56 1.52L4 5.95 2.60 1.52L5 5.77 2.59 1.50L6 6.52 2.46 1.55L7 4.93 1.78 1.15L8 8.13 3.00 1.91L9 7.62 2.86 1.81L10 7.71 2.75 1.78L11 8.24 4.00 2.22

A. Hernndez-Prez et al. / ComTorsion is applied to the model by application of two verticalpoint loads, each of a magnitude of 0.5 N, at the top liner as shownin Fig. 2. Corner supports, see Fig. 2, were simulated by imposingzero transverse displacement at the nodes of the bottom liner atthe corners. The twist stiffness DQM was determined from Eq. (4)using the nodal displacement (d) at the panel corner (a/2,b/2).

3.2. Finite element model of panels with homogenized core

In addition to the structural modeling of the web core, sand-wich specimen models with a homogenized core were analyzed.In this approach the complex shape of the web core is replacedwith a homogeneous layer with equivalent mechanical stiffness,see Appendix A. Finite element analysis of the homogenized corru-gated board panels was conducted in ANSYS [16] using isopara-metric solid brick (hexahedron) elements (SOLID95) as shown inFig. 4. This element is dened by 20 nodes having three degreesof freedom per node: translations in the nodal x, y, and z directions.The material properties of the homogenized core material used inthe FEA are listed in Table 5. The orthotropic material directionscorrespond to the element coordinate directions. The specimensmodeled were 105 mm long and 25 mm wide. The model em-ployed 4 through thickness elements for both the liners andhomogenized core layer, 30 elements along the width and 60elements along the length. In total, 21,600 elements and 36,000elements were employed for the single and double-wall speci-mens. Torsion was accomplished by application of two point loadsof 0.5 N at two corner nodes at diagonally opposite locations, see

Table 4Elastic stiffnesses (GPa) of web cores.

Web E1 E2 G12

W1 5.67 2.06 1.32W2 5.71 2.05 1.32W3 4.92 1.77 1.14W4 5.74 2.06 1.33W5 4.92 1.77 1.14Core height (mm)

4 5 2 4

W2 (0.221) L6 (0.369) 2.54 3.6W2 (0.221) L2 (0.228) 2.54 3.6W5 (0.181) L3 (0.215) 2.54 3.6W5 (0.181) L3 (0.215) 2.54 3.6W5 (0.181) L3 (0.215) 2.54 3.6W2 (0.221) L6 (0.369) 2.54 3.6W5 (0.181) L4 (0.185) 2.54 3.6

te Structures 110 (2014) 715 9Fig. 1. The pin supports were modeled by constraining two cornernodes in the z direction. The twist stiffness DQM was calculatedfrom Eq. (4). Table 5 lists the elastic properties of the web coresobtained from the homogenization procedure presented inAppendix A.

3.3. FOSD approach

The twist stiffness of corrugated board panels is also deter-mined from recently developed solutions based on FOSD theory[7] and classical laminate plate theory (CLPT) analysis of the twisttest specimen employing Fourier series. The FOSD solution isbriey outlined in this section whereas the CLPT solution is out-lined in Appendix C. A more detailed derivation of both solutionscan be found in reference [7]. Because layered plate theory is lim-ited to homogeneous panels, the homogenized elastic properties ofthe web core listed in Table 5 were used. A loading function q(x,y)to represent the two applied forces (Fig. 1) is expressed in the formof double sine Fourier series [9],

Fig. 3. Structural nite element model of the corrugated cardboard beams. (a)Single-wall board and (b) double-wall board.

where w is the deection, wx and wy are the midplane rotationsaround the x and y axes and Wmn, Amn and Bmn are Fourier coef-cients. Introducing Eq. (7) into the plate equilibrium equationsand solving for the Fourier coefcients Amn, Bmn and Wmn, an alge-braic system of equations is formed,

AmnBmnWmn

264

375

H11 H12 H13H12 H22 H23H13 H23 H33

264

3751 0

0qmn

264

375 8

where qmn is given by Eq. (6) and the parameters Hij (i, j = 1, 2, 3) aregiven by,

H11 D11 mpa 2

D66 npb 2

kA55; H12 D12D66 mpa np

b

9a;b

posite Structures 110 (2014) 715Fig. 4. Finite element models of homogenized board specimens. (a) Single-wall

10 A. Hernndez-Prez et al. / Comqx; y XMm1

XNn1

qmnsinmpxa

sinnpyb

5

where m and n are integers (m, n = 1, 2, . . . , 40), M and N are thenumber of terms taken in the series, a and b are the specimen lengthand width. qmn are the Fourier coefcients given by [7],

qmn32Pmnp2e2

cosmpa

ae2

cosmp

2

h icos

npb

be2

cosnp

2

6

where P is the total applied load and e is the side-length of a smallarea (e/2 e/2) over which the corner load is assumed to be uni-formly distributed.

The plate equilibrium equations and the displacement condi-tions, i.e. zero transverse displacement at the corner supportsand maximum deections at the two load application points, aresatised by the following expressions proposed for the deectionand midplane rotations [7,9],

wx; y XMm1

XNn1

Wmnsinmpxa

sinnpyb

7a

wxx; y XMm1

XNn1

Amncosmpxa

sinnpyb

7b

wyx; y XMm1

XNn1

Bmnsinmpxa

cosnpyb

7c

board and (b) double-wall board.

Table 5Elastic properties of homogenized web cores (MPa).

Web E1 E2 G12 G23 G13

W1 0.88 32 1.03 31.3 13.4W2 0.70 29 0.81 27.1 11.7W3 0.98 23 1.61 30.2 18.5W4 1.97 32 3.29 46.0 26.7W5 0.34 20 4.92 16.1 7.45H13 kA55 mpa

; H22 D66 mpa 2

D22 npb 2

kA44 9c;d

H23 kA44 npb

; H33 kA55 mpa 2

kA44 npb 2

9e; f

where k is the shear correction factor (k = 1). Dij and Aij are thebending and transverse stiffness dened in laminated plate theory[9,18], see Appendix B.

Concentrated loads of magnitude 5 N were applied to the panelat two corners and the corner deection d =w(a/2,b/2) was deter-mined from Eq. (7a). A converged solution for dwas obtained using40 Fourier terms in the FOSD solution (M = N = 40 in Eq. (7)). Thetwist stiffness DQM was determined from Eq. (1). The extensionaland shear stiffnesses Aij and bending stiffnesses Dij of the boards re-quired for the analysis (Eq. (9)) were obtained from Eqs. (B.1) and(B.2) for the three-layer single and ve-layer double-wall boards(see Appendix B) with homogenized elastic properties of the webcore listed in Table 5. The plate stiffnesses for single and double-wall panels are listed in Tables B.1 and B.2.

3.4. Experimental testing

Specimens extracted from large single and double-wall corru-gated board panels (Tables 1 and 2) were loaded in torsion usingthe dynamic stiffness tester (DST) model Kurutest beta [3]. This de-vice produces torsion by slightly rotating one end through a smallangle of twist, see Fig. 5. The rotating clamp includes a counterweight connected with a thin wire to achieve a torsion pendulum.When the clamp is released, the oscillations yield a damped sinewave curve where the angular frequency, x, is monitored by anoptical electronic pickup [3]. To determine DQM for a specic board,the DST is rst calibrated. The calibration is conducted by compar-ing several DQM data points to corresponding squared angular fre-quencies (x2) obtained from the oscillation pulse data. In thisFig. 5. Schematic of the dynamic stiffness tester (DST) [10].

work, a linear relationship between DQM and x2 was found for allcorrugated boards examined, with a slope of 53.4 Nmm s2 (notshown).

4. Results and discussion

4.1. Comparison between measured and predicted twist stiffnesses

Fig. 6 shows the twist stiffness DQM measured by the DST, calcu-lated from FOSD, homogenized FEA and structural FEA for thesingle-wall (Fig. 6a) and double-wall (Fig. 6b) corrugated boards.Overall, the twist stiffness predictions are in good agreement withthe measured data. The analytical FOSD results are also in overallagreement with the numerical FEA prediction. Fig. 6a shows, how-ever, that the homogenized FEA results for the single-wall boardsSW1, SW2 and SW3 slightly exceed the experimental and struc-tural FEA results. For the thinnest single-wall boards SW5, SW6and SW7 the structural FEA underpredicts DQM due to the indenta-tion deformation. The results show that DQM increases withincreasing transverse shear (A44 and A55) and the D66 stiffness(Tables B.1 and B.2).

For double-wall boards, Fig. 6b, the homogenized FEA is inexcellent agreement with experiments. The FOSD and structuralFEA slightly underpredicts the experimental results. It is also ob-served that twist stiffness DQM of double-wall boards increases inproportion to the in-plane elastic moduli of the liners, see Tables

2 and 3. The slight difference between DQM predictions and mea-surements observed in some boards might be due to variationson the elastic properties of commercial made paperboards and lo-cal damage of the web core inicted during pressing operations.For most single-wall and double-wall boards, however, the resultsin Fig. 6 show that the FOSD approach predicts the twist stiffnesswith good accuracy.

5. Parametric studies

5.1. Inuence of D66

D66 is a torsional stiffness element dened in classical plate the-ory which is proportional to the in-plane shear modulus G12.Analysis of the inuence of D66 on DQM is conducted on thedouble-wall boards listed in Table 2. D66 stiffnesses used in thisanalysis were obtained from Eq. (B.2) with the thickness and

A. Hernndez-Prez et al. / Composite Structures 110 (2014) 715 11Fig. 6. Twist stiffness DQM measured and predicted by FEA and FOSD for corrugatedboards. (a) Single-wall and (b) double-wall.in-plane shear modulus of each layer of the double-wall boardsexamined (see Table 2). Fig. 7 shows DQM as function of D66 forthe range of double-wall boards examined. It is observed in Fig. 7that the solution based on CLPT predicts a linear relation betweenthe torsional stiffness DQM and D66. CLPT approach provides a twiststiffness (DQM) which is much larger than that obtained from theFOSD solution. This behavior is because CLPT does not considertransverse shear deformation. Therefore, the CLPT solution pro-vides an upper bound for DQM. On the other hand, as discussedearlier, experimental measurements and predictions of DQM byFEA and FOSD theory are in good agreement. These results showthat DQM is very weakly dependent on D66 due to extensive out-of-plane shear deformation.

5.2. Effect of transverse shear moduli

The inuence of the transverse shear moduli of the core, (G13)cand (G23)c, on the twist stiffness is examined using the FOSD ap-proach. The twist stiffness, DQM, was calculated for the SW2 andSW5 single-wall boards for a range of transverse shear moduli val-ues. Thus, the shear moduli G13 and G23 were varied while all otherproperties and dimensions were kept constant. In this analysis, G13and G23 were reduced over a range from 0% to 75% of their initialvalues (SW2 board has (G13)c = 11.7 MPa and (G23)c = 27.1 MPawhereas SW5 board has (G13)c = 26.7 MPa and (G23)c = 46 MPa).The twist stiffness results are normalized with the original value(D0QM). Fig. 8a shows that the twist stiffness decreases in proportionto the reduction of the core shear modulus in the 13 planeFig. 7. Torsional stiffness DQM of double-wall boards as function of D66.

((G13)c). For example, the SW2 board experiences a reduction of18% of DQM when (G13)c is reduced by 25%. The results shown inFig. 8a demonstrate that the transverse shear modulus, G13, has alarge effect on the twist stiffness. Fig. 8b shows that the transverseshear modulus in the 23 plane ((G23)c) similarly has a large inu-ence on the twist stiffness.

5.3. Effect of specimen aspect ratio

The twist test specimen dimensions are a = 10.5 cm andb = 2.5 cm. An analysis of the specimen aspect ratio, here denedas b/awas conducted. The aspect ratio for the test specimen is thusb/a = 0.238. The length of the panel along the CD (a) was kept con-stant at a = 105 mm and the width (b) was varied from 5 to 60 mmto obtain aspect ratios in the range of 0.047 < b/a < 0.571. For thisanalysis the CLPT and FOSD solutions were employed, and weagain considered the SW2 and SW5 boards. Fig. 9 shows the twiststiffness predictions from the FOSD and CLPT solutions vs. b/a. Forthe SW2 board, Fig. 9a, the FOSD solution predicts a practicallyconstant torsional stiffness for specimens with aspect ratiosb/a > 0.3. On the other hand, CLPT provides an upper bound thatis very far from the FOSD results at small aspect ratios. Similar re-sults were obtained for the SW5 board, Fig. 9b. The two approachesconverge for aspect ratios greater than 0.5 because the transverseshear deformation becomes less important for large panels. Theresults shown in Fig. 9 for the FOSD approach indicate that thetwist stiffness DQM is quite insensitive to changes in the panelgeometry, which is an advantage for experimental studies.

5.4. Effect of specimen asymmetry

In order to investigate the inuence of the board asymmetry onthe twist stiffness of a single-wall board, the top and bottom facesheet thicknesses (h1 and h3) were varied. The sum of the face thick-nesses h1 + h3, was kept constant at 0.456 mm. An asymmetryparameter, ap, is dened as ap = (h1 h3)/(h3 + h1). Hence for a sym-metric board ap = 0. Fig. 10 shows the twist stiffness,DQM calculated

Fig. 9. Twist stiffness as function of the specimen aspect ratio (b/a). (a) SW2 boardand (b) SW5 board. a = 105 mm.

12 A. Hernndez-Prez et al. / Composite Structures 110 (2014) 715Fig. 8. Inuence of transverse shear moduli on twist stiffness of the SW2 and SW5boards (a) (G13)c and (b) (G23)c.Fig. 10. Inuence of thickness asymmetry on twist stiffness for SW2 board.ap = (h1 h3)/(h3 + h1) and h1 + h3 = 0.456 mm.

from FOSD theory, normalized with the twist stiffness for a sym-metric board, D0QM , plotted vs. the asymmetry parameter (ap). Theresults in Fig. 10 show that the twist stiffness varies in a parabolicmanner with respect to ap. The maximum stiffness occurs forsymmetric boards, ap = 0. For ap = 0.5 the twist stiffness decreases,but only by about 0.5% from its original value. Hence, the twiststiffness is not strongly inuenced by the asymmetry of the board.

ux 1; uy 0; uz 0; hy 0; at x k A:2b

Fig. A.2. Schematic of loading cases applied in the homogenization procedure ofcorrugated core. The arrows indicate deformation mode.

A. Hernndez-Prez et al. / Composite Structures 110 (2014) 715 136. Conclusions

The twist stiffness DQM of single and double-wall corrugatedboards has been examined by nite element analysis (FEA), classi-cal laminate plate theory (CLPT) and rst order shear deformation(FOSD) theory. The predictions were compared with test results fora large range of boards. Twist stiffness predictions by both FEA anda FOSD solution were found in good agreement, and bothapproaches agree with experimental data for the range of corru-gated boards examined. Parametric analysis showed signicantinuence of both transverse shear moduli along the machine(G13)c and cross direction (G23)c on the twist stiffness. The inuenceof D66 on the twist stiffness was small. The CLPT solution providesan upper bound for the twist stiffness DQM. The twist stiffness of ac-tual panels fell far below the upper bound because of extensivetransverse shear deformation. Analysis of board asymmetry re-vealed that asymmetry of the board does not strongly inuencethe twist stiffness. The FOSD analysis further reveals that DQM isquite insensitive to panel aspect ratio. It is concluded that the ana-lytical solution for the plate twist specimen using FOSD is a viableapproximation for the calculation of the twist stiffness of corru-gated boards.

Acknowledgment

A. Hernndez-Prez thanks the DS Smith Plc. company fornancial support of the post-doc position.

Appendix A. Homogenization of corrugated board

The analysis of a sandwich panel with a corrugated core isgreatly simplied if the web core is replaced by an equivalenthomogenous core. The homogenization procedure determineseffective elastic constants of the web core from the geometry,thickness and elastic properties of the web material. Several ap-proaches for homogenization of structural cores have been pre-sented e.g. [11,19]. In this study we selected a method based onnite element analysis.

For an orthotropic material the strainstress relation is given by[12],

e1e2e3c23c13c12

2666666664

3777777775

1=E1 m21=E2 m31=E3m12=E1 1=E2 m32=E3m13=E1 m23=E2 1=E3

0 0 00 0 00 0 0

0 0 00 0 00 0 0

1=G23 0 00 1=G13 00 0 1=G12

2666666664

3777777775

r1r2r3s23s13s12

2666666664

3777777775

A:1Fig. A.1. Illustration of homogenization of single-wall corrugated board.where subscripts 1, 2 and 3 refer to the perpendicular directions ofthe material symmetry, see Fig. A.1. The shear moduli (G) and Pois-son ratios (m) of the paper are calculated using engineering estima-tions [20,21]. A Poissons ratio (m12) of 0.43 was assumed for the facesheets [15]. Because the paper layers are thin, a plane stress condi-tion can be assumed in the 13 and 23 planes. Therefore, smallvalues of Poissons ratios m13 and m23 were assumed (0.01). Twocommercial B-ute and C-ute web cores were considered. TheB-ute has a height of 2.54 mm with 47 3 corrugations each0.3 m, whereas the C-ute has a height of 3.6 mm with 39 3 cor-rugations each 0.3 m.

Effective orthotropic stiffnesses of the web core are obtainedusing a homogenization model where six fundamental loadingcases are applied to a unit cell of the web core, Fig. A.2. These casesare required in order to establish the six basic orthotropic exten-sional and shear stiffnesses (Ex, Ey, Ez) and (Gxy, Gxz, Gyz). The effec-tive Poissons ratio mxy of the web core was assumed to be 0.05 forall cases.

The unit cell of the web core has the dimensions (k k hf ) asshown in Fig. A.3. For simplicity, the length of the unit cell in thedirection perpendicular to the waves was selected equal to thewave length (k) of the corrugations. For each of the six cases, a unitdisplacement is applied on one of the surfaces in order to obtainthe desired deformation mode. To complete the analysis, condi-tions for the displacements (ux, uy, uz) and rotations (hx, hy) arespecied. With reference to the loading cases shown in Fig. A.2.The boundary conditions for the unit cell are given by,

Ex:

ux 0; uy 0; uz 0; hy 0; at x 0 A:2aFig. A.3. Geometry considered in the homogenization procedure.

theory for the plate twist specimen

A solution based on CLPT for the sandwich plate twist specimenhas recently been developed [7]. In this appendix a brief outline ispresented. For the solution based on CLPT, the loading functionq(x,y) is given by Eq. (5), where q(x,y) represents four point loadsapplied at the specimen corners. The deection function w(x,y) isgiven by Eq. (7a) since it represents a physically admissible dis-placement of the plate twist specimen. Because CLPT does not con-sider transverse shear in the equilibrium equations, the Fouriercoefcients Wmn are in this case expressed in terms of the in-planestiffnesses [7],

Wmn qmnD11 mpa

4 2D12 4D66 mpa 2 npb 2 D22 npb 4C:1

References

[1] McKinlay PR. Analysis of the strain eld in a twisted sandwich panel withapplication to determining the shear stiffness of corrugated board. In:Proceedings of the 10th fundamental research symposium of the Oxford andCambridge series. Oxford UK, September 1993.

[2] McKinlay PR. Shear stiffness tester. United States Patent Number 4958 522.September 25, 1990.

[3] Chalmers IR. Method and apparatus for testing of shear stiffness in board.United States Patent Number 7 621 187 B2. November 24, 2009.

[4] Mure M. Corrugated board new method: anticlastic rigidity (In French).

SW7 101 81 0.94

posiEy:

ux 0;uy 0;uz 0; at y 0 A:3a

ux 0;uy 1;uz 0; at y k A:3b

hy 0;uz 0; at z 0 A:3cEz:

ux 0;uy 0;uz 0; hy 0; at z 0 A:4a

uz 1; at z hf A:4bGxy:

uy 0; at x 0 A:5a

uy 1; at x k A:5b

hxy 0 hxy k; coupling A:5c

hyx 0 hyx k coupling A:5dGxz:

ux 0;uz 0; at z 0 A:6a

ux 1;uz 0; at z hf A:6b

hyx 0 hyx k; coupling A:6cGyz:

uy 0; at z 0 A:7a

uy 1; at z hf A:7b

hxy 0 hxy k; coupling A:7cThe model is solved using linear nite element analysis with iso-parametric 8-node shell elements in ANSYS 11.0 [16]. The effectiveelastic moduli of the web cores are calculated from Eq. (A.1), bysolving for the stiffnesses from the displacements applied in Eqs.(A.2)(A.7) and the reactions (stresses) obtained from FEA.

Appendix B. Denition of the elements of the plate stiffnessmatrices

The elements of the plate stiffness matrices are given by [18],

Aij XNn1

Qijnzn zn1 B:1

Dij 13XNn1

Qijnz3n z3n1 B:2

where Aij (i, j = 1, 2, 6) and Dij are the elements of the extensionaland bending stiffness matrices, N is the total number of plies, n isthe nth ply and z is the thickness coordinate. Qijn is the ij elementof the stiffness matrix for ply n [18]. For all board specimens exam-ined here, the principal axes (1, 2) of the plies coincide with the xand y-axes of the panel. Hence Qij = Qij (on axis). For each n plythe stressstrain relation then becomes,

r1r2

2664

3775

Q11 Q12 0

Q12 Q22 0

2664

3775

e1

e2

2664

3775 B:3

14 A. Hernndez-Prez et al. / Coms12 n 0 0 Q66 n c12 nThe transverse shear stiffness A44 and A55 are given by [9],A44 XNn1

G23nzn zn1 B:4a

A55 XNn1

G13nzn zn1 B:4b

where G13 and G23 are the transverse shear moduli of the plate inthe 13 and 23 material planes. The shear moduli (G13 and G23)of the web core were determined using the homogenization analy-sis described in Appendix A. Tables B.1 and B.2 list the plate stiff-nesses of single-wall and double-wall boards.

Appendix C. A solution based on classical laminated plate

Table B.2Plate stiffnesses for double-wall boards.

Board A44 (kN/m) A55 (kN/m) D66 (Nm)

DW1 278 217 12.9DW2 261 191 8.48DW3 172 130 6.54DW4 172 134 6.78DW5 217 163 7.63DW6 273 209 11.1DW7 178 135 7.41Table B.1Plate stiffnesses for single-wall boards.

Board A44 (kN/m) A55 (kN/m) D66 (Nm)

SW1 148 111 3.05SW2 132 105 2.87SW3 129 90 2.38SW4 85 65 2.02SW5 150 123 1.37SW6 104 86 1.05

te Structures 110 (2014) 715Revue A.T.I.P. 1986;40:32530.[5] Pommier JC, Poustis J. Rsistance au gerbage des caisses aujourdhui Mc Kee et

demain (in French). Revue A.T.I.P. 1987;41:399402.

[6] Carlsson LA, Nordstrand T, Westerlind B. On the elastic stiffness of corrugatedcore sandwich. J Sand Struct Mater 2001;3:25367.

[7] Hernndez-Prez A, Avils F, Carlsson LA. First-order shear deformationanalysis of the sandwich plate twist specimen. Sand Struct Mater2012;14:22945.

[8] Ndai A. Die elastischen Platten (In German). Berlin, Germany: Springer; 1968.[9] Whitney JM. Structural analysis of laminated anisotropic plates. Lancaster, PA:

Techmonic; 1987.[10] Chalmers IR. A new method for determining the shear stiffness of corrugated

boards. Appita J 2006;59:35761.[11] Biancolini ME. Evaluation of equivalent stiffness properties of corrugated

board. Compos Struct 2005;69:3228.[12] Agarwal BD, Broutman LJ. Analysis of performance of ber composites. New

York, NY: John Wiley & Sons; 1990.[13] Persson K. Material model for paper: experimental and theoretical aspects.

Master thesis. Lund, Sweden: Lund University; 1991.

[14] Allansson A, Svrd B. Stability and collapse of corrugated board. Master thesis.Lund, Sweden: Lund University; 2001.

[15] Nordstrand T. On buckling loads for edge-loaded orthotropic plates includingtransverse shear. Compos Struct 2004;65:16.

[16] ANSYS 11.0. Swanson analysis systems. Houston, PA; 2007.[17] Bathe KJ. Finite element procedures in engineering analysis. Newark,

NJ: Prentice-Hall; 1982.[18] Hyer MW. Stress analysis of ber-reinforced composite materials. Boston, MA:

McGraw Hill; 1998.[19] Talbi N, Batti A, Ayad R, Guo YQ. An analytical homogenization model for nite

element modelling of corrugated cardboard. Compos Struct 2009;88:2809.[20] Baum GA, Brennan DC, Habeger CC. Orthotropic elastic constants of paper.

Tappi J 1981;64:97101.[21] Mann RW, Baum GA, Habeger CC. Determination of all nine orthotropic elastic

constants for machine-made paper. Tappi J 1980;63:1636.

A. Hernndez-Prez et al. / Composite Structures 110 (2014) 715 15

Analysis of twist stiffness of single and double-wall corrugated boards1 Introduction2 Twist test3 Materials and specimens3.1 Structural finite element model of panels3.2 Finite element model of panels with homogenized core3.3 FOSD approach3.4 Experimental testing

4 Results and discussion4.1 Comparison between measured and predicted twist stiffnesses

5 Parametric studies5.1 Influence of D665.2 Effect of transverse shear moduli5.3 Effect of specimen aspect ratio5.4 Effect of specimen asymmetry

6 ConclusionsAcknowledgmentAppendix A Homogenization of corrugated boardAppendix B Definition of the elements of the plate stiffness matricesAppendix C A solution based on classical laminated plate theory for the plate twist specimenReferences