-
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: [email protected] (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