DEGREE PROJECT, IN , SECOND LEVELSOLID MECHANICS
STOCKHOLM, SWEDEN 2015
Modelling of failure mechanisms forcorrugated board
MOHAMED HAIDAR
KTH ROYAL INSTITUTE OF TECHNOLOGY
SCHOOL OF ENGINEERING SCIENCES
Modelling of failure mechanisms forcorrugated board
Mohamed Haidar
Master thesis, January 2015School of Engineering SciencesDepartment of Solid Mechanics
KTH Royal Institute of TechnologySE-100 44 Stockholm, Sweden
Abstract
The present work describes the construction of a semi-analytical model forprediction of buckling loads in simply supported corrugated paperboard pan-els. The model accounts for transverse shear, due to the weakness of thecore in such plates compared to the facings. This was done utilizing energyrelations and first order transverse shear. The panel was homogenised us-ing laminate theory. A detailed model using FEM was derived in order tovalidate the predictive capabilities of the analytical model. Experimentaltesting was performed to estimate the accuracy of both theoretical models,and assess the limitation of the analytical model. All modes of analysisshowed good agreement for cubic boxes. Further investigation into expand-ing the scope of the analytical model was carried out and commented on.
Keywords: paper, corrugated paperboard, FEM, experiments, testing,transverse shear, buckling, energy relations, Reissner-Mindlin
Sammanfattning
Detta arbete beskriver framtagandet av en semianalytisk modell for berakningav bucklingslaster hos fritt upplagda wellpappaneler. Modellen tillater trans-versell skjuvning eftersom skillnaden i styvhet mellan mittskiktet och tackskikteni wellpapp ar stor. Detta uppnaddes genom anvandandet av energimetodoch forsta ordningens transversell skjuvning. Wellpappanelen homogeniser-ades med laminatteori. En detaljerad modell for paneler togs fram genomanvndning av FEM for att kunna validera resultat erhallna med den semian-alytiska modellen. Slutligen utfordes experimentella tester for att uppskattanoggrannheten hos bade den semianalytiska och FEM baserade modellen.Alla tre analyserna visade god samstamning for kubiska lador. Forslag pa vi-dare arbete for att utoka den semianalytiska modellens anvandningsomradepresenteras.
Nyckelord: papper, wellpapp, FEM, experiment, transversell skjuvning,buckling, energimetoder, Reissner-Mindlin
i
Preface
The work in this report was carried out in the spring and summer of 2014at Innventia on behalf of BillerudKorsnas. The report represents my masterthesis for M.Sc.-degree in Vehicle Engeneering with specialization in SolidMechanics at KTH.
I would like to greatly thank my supervisor at Innventia, Dr. JohanAlfthan for his invaluable support and guidance. Without him this projectcould not have been finished in the specified time frame. I also would liketo show my gratitude to my supervisor at KTH, Prof. Soren Ostlund, andBillerudKorsnas, M.Sc Magnus Bjorklund, for their support.
Finally, I would like to give my love to my wife, children and family foralways being there for me.
Stockholm, November 2014
Mohamed Haidar
ii
Contents
Abstract i
Sammanfattning i
Preface ii
1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Aim of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 FEM 52.1 Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 62.4 Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3 Analytical Model 103.1 Laminate theory . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Plate theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 Energy relations . . . . . . . . . . . . . . . . . . . . . . . . . 133.4 Buckling load . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4 Experimental Study 184.1 Test procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5 Results 225.1 FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.2 Analytical model . . . . . . . . . . . . . . . . . . . . . . . . . 225.3 Experimental verification . . . . . . . . . . . . . . . . . . . . 235.4 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
6 Discussion 25
1. Introduction
1.1 Background
History
The first corrugated paperboard material was produced in 1856 [1] and was
used as a liner for tall hats. In 1871 the first patent for a boxboard material
used for shipping was issued in New York. The application was wrapping
of fragile glass bottles, which in turn was packed and shipped in wooden
boxes. At the turn of the century paperboard was starting to see usage
as a standalone shipping solution. This was because of the invention of
corrugated boxboard with liner sheets on both sides thereby making the
boards much stronger.
By the mid-1950s corrugated paperboard containers had surpassed wood
boxes as a transport packaging material and was starting to see usage in
transportation of fruit and produce. Using boxboard instead of wood al-
lowed for the goods to be transported from the farms to the retailers with
significantly less damage, improving the return on investment, and opening
up export markets.
Manufacturing
Corrugated paperboard is manufactured from paper sheets. The operation
from paper sheet to corrugated paper board is achieved in one continuous
machine line. First the paperboard is formed and assembled, requiring wet
conditions, after which the paperboard is dried.
The rolls of paper sheets are fed into large, high precision machines
called corrugators, see Figure 1.1. The middle sheet of the board, the flute,
is softened by hot steam and corrugated between two rolls, attaining the
desired shape. Adhesive starch is applied to the tips of the corrugations,
1
and the flute is glued together with one of the liners, the outer edges of the
corrugated paperboard. Lastly the second liner is glued to the other side of
the board.
Figure 1.1: Schematic diagram of a corrugator.
Properties
Corrugated paperboard comes in varying shapes and sizes, see Figure 1.2.
The profile of the corrugated medium plays a major role for the bending
stiffness of the board, therefore also in the stacking strength of the box.
Figure 1.2: Board profile
Paperboards are divided into different ranges, denoted by a character A
to N, with regards to the height of the flute profile. The A profile represents
the highest flute and is used for the most demanding packaging needs. More
common are the B and C profiles, while D profile flutes and beyond in general
are less chosen for their load baring capabilities, but instead for printability
and cosmetic reasons.
The success of corrugated paperboard is attributed to the fact that the
material has many good properties and few limitations. It is very flexible
2
and can be tailor made for most tasks. It has a very high strength and
stiffness to weight and cost ratio. It has low weight and is fully recyclable.
When used to make boxes the paperboard can be pre shaped. This makes
transportation and storage of the boxes cheap and final assembly can take
place on site of product packaging.
Two major disadvantages of corrugated paperboard are sensitivity to
moisture and low usage cycles. Moisture sensitivity is lowered by spraying
the surface of the liners with moisture resistant coating, but this drives up
the production costs.
1.2 Aim of thesis
Description
In the paper industry, generally, the production chain is split into multiple
areas. It is common for a company to only operate in one of those areas,
therefore handling only one part in the chain. The reason for this is the
different scales of production in each area. Companies that make paper
strive to anticipate the needs of companies that use the paper to make
boxes. BillerudKorsnas has developed their BoxLab software to be able to
be involved with the box makers in the development of their products. Given
the packaging demand of the box manufacturer BoxLab can be utilized to
optimize paper material suggestions.
BillerudKorsnas is interested in expanding the BoxLab software to more
accurately model the behaviour of corrugated paperboard boxes. This will
be achieved if transverse shear is introduced into the model. Transverse
shear is normally disregarded in plate analysis because the effects are in
general very small. An exception to this is when the core of the plate is very
weak in comparison to the edges, a condition that applies to corrugated
paperboard. The two edge layers of a corrugated paperboard have high in-
plane strength and the corrugated medium is much weaker. Its primary role
is separating the two liners and holding them in place, thereby giving the
board good bending stiffness.
3
Problem formulation
The aim of the project was an analytical (or semi-analytical) homogenized
model, of a corrugated board panel, that takes the efficient behaviour of the
corrugated medium including transverse shear into account. The homog-
enized model needed to be verified by comparing the result to those of a
detailed model of a corrugated paperboard analysed using FEM, and also
to experiments done at the facilities of BillerudKorsnas . The models and
experiments must show sufficient agreement.
The task was divided into the following activities:
1. Detailed model of corrugated board (FEM).
2. Homogenization of detailed model.
3. Analytical presentation of buckling of panels.
4. Experimental verification.
Major assumptions
In order to reduce the complexity of the project some assumptions were
made. Paper was regarded as a linear elastic medium with orthotropic prop-
erties. Furthermore influence of time dependent effects e.g. creep, moisture
and inertia were disregarded.
With regards to the analytical model only the first buckling mode was
estimated.
4
2. FEM
A detailed model was developed for evaluation in the finite element software
Abaqus. Matlab was used to generate the geometry and write the input file
for the Abaqus standard solver.
The model was based on previous work done at Innventia [2].
2.1 Core
The corrugated core profile, see Figure 2.1, was modelled by interchange-
ably using a half period sinus function for the shape of the curved parts,
and straight lines where the core was in contact with the liners. Numerical
Figure 2.1: Parametric representation of the corrugated core geometry.
values for flute hight Hf wavelength Lw and the length of contact between
core and liner a were taken from [2]. The parameters Hf and Lw are indus-
trial standards while a is an estimation based on corrugated board samples.
Values for B and C profile flute are given in Table 2.1.
Profile Hf Lw a
B 2.6 6.667 0.8C 3.6 7.692 1
Table 2.1: Flute parameters for B and C profile paperboard
5
The contact between core and liner was modelled as rigid with no con-
sideration for glue and imperfections.
2.2 Geometry
In order to raise computational efficiency only one quarter of the corrugated
board was used. This is due to symmetry in the geometry and loading case
at the middle lines of the board in both MD and CD. This was achieved
using the Abaqus commands YSYMM and XSYMM respectively.
The liners and core were each modelled as shell elements, i.e. two di-
mensional structures allowing for in-plane loading and bending.
The quarter panel was of width 140 mm, and height 140 mm while the
thickness of the plate was calculated by the preprocessor using the thickness
of the liners, tc, and the hight of the flue Hf .
2.3 Boundary conditions
In order to apply the load in the middle plane of the corrugated board
new virtual nodes were introduced [3]. The coordinate system defining the
degrees of freedom is represented in Figure 2.2 where the displacements in
Figure 2.2: Degrees of freedom used.
the respective directions are denoted u1, u2, u3, and the rotations u4, u5, u6.
The translational degrees of freedom for the virtual middle plate were
obtained using mean values for the respective node pair of the top and
bottom facings as
6
um1 =ut1 + ub1
2(2.1)
um3 =ut3 + ub3
2(2.2)
where m is the node of the middle plane, and t and b stand for the top
and bottom liners, respectively.
For the rotational degree of freedom u5, see Figure 2.3, it is assumed that
the rotation is small and the distance between the facings H is constant. It
then follows from the geometry that
um5 =ut1 − ub1H
(2.3)
Figure 2.3: Rotation.
Equations (2.1) and (2.3) give
ut1 − um1 −H
2um5 = 0
ub1 − um1 +H
2um5 = 0
(2.4)
and Equation (2.2) gives
ub3 + ut3 − 2um3 = 0 (2.5)
and the load is applied on the virtual nodes along the edges of the middle
plane. The equations for displacement u2 and rotation u4 is derived similarly
by replacing u1 and u5 in Equations (2.1) and (2.3).
7
The corrugated paperboard panel of a packaging box is commonly mod-
elled as a simply supported plate at the boundaries. This allows rotation
and in plane translation but does not allow translation out of plane at the
boundary. In Abaqus this was modelled by locking the displacements of the
boundary nodes on the virtual middle plane in the 2, 5 and 6 directions.
2.4 Buckling
An often used method for obtaining the buckling load is solving the following
linear eigenvalue problem
Kν = 0 (2.6)
where K is the stiffness matrix for the loaded geometry and ν is a vector
containing the non-trivial solutions for the displacements.
Abaqus uses an incremental method where Equation (2.4) is modified to
(K0 + λiK∆)νi = 0 (2.7)
where K0 is the initial stiffness matrix, K∆ is the incremental stiffness
matrix, λi is a vector containing the eigenvalues for the buckling problem
and νi is the eigenvector for buckling mode i. The process is repeated for
the desired number of buckling modes.
The Abaqus command BUCKLE was used, but limited to only find
the first buckling mode. In the Abaqus post processor, see Figure 2.4, the
buckling mode can be viewed. The eigenvalue is simply multiplied with the
reference load to obtain the buckling load for the given mode. The reference
load used was 1 N. To obtain the buckling load for the complete corrugated
board, the buckling load of the quarter panel is doubled, to account for that
the edge of the full size panel is twice as long.
8
Figure 2.4: The first buckling mode in Abaqus where the deformation field isrepresented by a colour gradient ranging from smallest (blue) to largest (red)
9
3. Analytical Model
Consider a plate of corrugated paper board. The layers of the plate are
made of paper material, which is anisotropic in nature. The liners making
up the top and bottom surfaces of the plate have strong in-plane properties,
giving the plate good stiffness against compression. The core, made of a
corrugated paper sheet, is much weaker and has the primary function of
separating the liners and holding them in place. This gives the composite
plate high bending stiffness resulting in a structure that can carry a high load
with regards to its weight. Because of the difference in stiffness between the
constituents of the plate, transverse out-of-plane shear will have a significant
contribution to the deformation.
The corrugated paperboard, being a sandwich structure with inert or-
thotropic properties, should be homogenized if an analytical model is to be
constructed. This is achieved using laminate theory resulting in a solid plate
model with effective properties.
To include the effects of transverse shear Reissner-Mindlin [4, 5] plate
theory was applied. This addition to the classical plate theory places less
constraints on the bending of the plate and introduces transverse out-of-
plane deformation.
Attention was then turned to formulating an energy equation for the
plate under unilateral loading. To further simplify the model, an investiga-
tion of the possibility of uncoupling the in-plane mode of deformation from
the out of plane shear was conducted.
Using the Rayleigh-Ritz approximation, the buckling load was obtained.
10
3.1 Laminate theory
The in-plane stiffness relations of an orthotropic plate are formulated using
Hooke’s law in stiffness form asσx
σy
τxy
=
Qxx Qxy 0
Qyx Qyy 0
0 0 Qss
εx
εy
γxy
(3.1)
where
Qxx =Ex
1− νxyνyx
Qxy =νxyEy
1− νxyνyx
Qyx =νxyEx
1− νxyνyx
Qyy =Ey
1− νxyνyxQss = Gxy
(3.2)
For linear elastic orthotropic materials the number of material parame-
ters is reduced byExEy
=νxyνyx
(3.3)
and, for paper, approximations for the in-plane shear modulus and Poisson’s
ratio are given by [6]
Gxy =
√ExEy
2(1 +√νxyνyx)
(3.4)
√νxyνyx = 0, 293 (3.5)
Using (3.5) with (3.3) and (3.4) gives
νxy = 0.293
√ExEy
(3.6)
Gxy = 0.387√ExEy (3.7)
For the effective properties of the core layer, an approximation [7] was
11
made as followsEx ≈ 0
Ey =αEy,ctc
(3.8)
where Ey,c is the in-plane Young’s modulus of the core sheet, tc is the thick-
ness of the core sheet and α is the take-up factor describing the ratio between
the arc length and the wavelength of the corrugation pitch.
If the corrugated core is symmetric, [8] show that Gxy can be set to zero
and the contribution of the corrugated cores to shear and twisting stiffnesses
can be neglected.
A corrugated paperboard is made up of multiple layers of paper based
plates. Assuming that each layer is orthotropic, uniformly thick and do not
move relative to other layers, the laminate can be represented in a coordinate
system with the Z-direction pointing downwards, see Figure 3.1. Here,
Figure 3.1: Laminate coordinate system and ordering.
t =n∑k=1
tk
z0 = − t2
zk = zk−1 + tk
(3.9)
The structure can then be homogenized. Effective in-plane properties
for the laminate are found by formulating the equations of motion for the
12
system leading, in compact matrix notation, to(N
M
)=
(A B
B D
)(ε0
κ
)(3.10)
where ε0 is the vector containing the membrane strains in the plane of the
laminate and κ is the vector containing the curvatures, while N and M rep-
resent the external forces and moments acting on the laminate. Furthermore
A =
n∑i=1
Qi(zi − zi−1)
B =1
2
n∑i=1
Qi(z2i − z2i−1)
D =1
3
n∑i=1
Qi(z3i − z3i−1)
(3.11)
For symmetric laminates the coupling between tension and bending vanishes,
i.e., B = 0.
3.2 Plate theory
In Reissner-Mindlin plate theory cross-sections to the middle plane does
not necessarily remain orthogonal after deformation, see Figure 3.2. This
is an expansion of the classical Kirchhoff plate theory and allows for shear
deformation.
3.3 Energy relations
Total potential energy for a plate under the influence of external loading is
U = Ue + UN (3.12)
where Ue is the elastic energy in the plate defined by
Ue =1
2
∫Vσijεij dV ∀ i, j ∈ x, y, z (3.13)
13
Figure 3.2: Mindlin-Reissner plate.
and
UN =1
2
∫ANy
(dw
dy
)2
dA (3.14)
is the potential energy, due to external work, in the unilateral loading case,
where Ny is the applied load and w is the deflection in the Z-direction.
The elastic energy can be separated into in-plane elastic energy and
transverse elastic energy, allowing for the two parts to be examined sepa-
rately.
Ue,plane =1
2
∫Vσxεx + σyεy + τxyγxy dV (3.15a)
Ue,trans =1
2
∫Vτxzγxz + τyzγyz dV (3.15b)
In order to use the compact effective properties of laminate theory for-
mulated in Equation (3.10), Equation (3.15a) is rewritten as
Ue,plane =1
2
∫A
∫ h2
−h2
σxεx + σyεy + τxyγxy dzdA
=1
2
∫A
∫ h2
−h2
σx(ε0x + zκx
)+ · · ·+ τxy
(ε0xy + zκxy
)dzdA
=1
2
∫ANxε0x +Mxκx + · · · dA
=1
2
∫AN · ε0 +M · κ dA
(3.16)
14
where ε0 is the strain in the middle plane of the plate and κ is the curvature
through the thickness of the plate.
From (3.10)
N = Aε0 +Bκ⇒ ε0 = A−1(N −Bκ)
M = Bε0 +Dκ⇒M = BA−1(N −Bk) +Dκ(3.17)
Inserting (3.17) into (3.16) yields
Ue,plane =1
2
∫AN ·A−1N −N ·A−1Bκ+
+BA−1N · κ+(D−BA−1B
)κ · κ dA
(3.18)
The first term in the integral of (3.18) is the pure membrane state and since
this is the energy in the plate before buckling it is ignored in this analysis.
The second and third terms are equal, after rewriting, and cancels each
other. The only term that remains is the fourth term reducing (3.18) to
Ue,plane =1
2
∫A
(D−BA−1B
)Tκ · κ dA (3.19)
Turning the attention back to the transverse energy, Equation (3.15b)
can be modified, using τ = Gγ, as
Ue,trans =1
2
∫A
∫ h2
−h2
τxzγxz + τyzγyz dzdA
=1
2
∫Akh(Gxzγ
2xz +Gyzγ
2yz
)dA
(3.20)
with k being a shear correction factor to account for the shear field through-
out the thickness not being purely linear. Different values for k are sug-
gested [9] and in the present analysis k = 56 was used.
The transverse shear moduli can be calculated explicitly using the method
described in [10], but given the scope of this analysis tabulated values will
be used. Furthermore [2] showed that in order to account for the damage
in the corrugation process, the in-plane Young’s modulus for the corrugated
layer was scaled down by a factor of four. Tabulated values are of sufficient
accuracy for B-profile boards but will start to diverge for other profiles. This
will be further elaborated in the discussion.
15
3.4 Buckling load
An approximate method for obtaining the buckling load is the Rayleigh-
Ritz procedure [11]. Considering a plate with length L and height W , a
coordinate system is placed with the origin in the middle point of the plate,
the X-axis coincides with MD and the Y-axis coincides with CD. Kinematic
assumptions for the transverse deflection and rotation fields are introduced
fulfilling all kinematic boundary conditions
ω = q1 cos(πxL
)cos(πyW
)βx = q2 sin
(πxL
)cos(πyW
)+
dw
dx
βy = q3 cos(πxL
)sin(πyW
)+
dw
dy
(3.21)
Using Equation (3.21), the curvature vector κ is introduced as
κ =
−dβxdx
−dβydy
−dβydx −
dβxdy
(3.22)
Combining Equations (3.22), (3.20), (3.19) and (3.14) the total potential
energy is
U =1
2
∫ L2
−L2
∫ W2
−W2
((D−BA−1B
)Tκ · κ + (3.23)
kh
(Gxz
(−βx +
dω
dx
)2
+Gyz
(−βx +
dω
dx
)2)
+
Nb
(dω
dy
)2)
dxdy
By solving the following system of equations
U = 0
dU
dqi= 0
(3.24)
16
the buckling load is obtained by
Nb = Ny (3.25)
Since the system (3.24) have multiple solutions, due to the direction of
deflection being undefined at the bifurcation point, one of the constants qi
needs to be prescribed. This is best done by giving q1 a value, which in
essence tells the model which way to buckle.
17
4. Experimental Study
In order to verify the analytical model, experimental testing was performed.
The experiments were carried out at the BillerudKorsnas Gruvon facility.
The aim in the planning of the tests was to get as close as possible to a real
world application scenario. It was important to investigate the limitation of
the model so the experiments also included combinations of geometry and
materials that would fall outside of the models predictability range.
Figure 4.1: Compression machine used to preform BCT on corrugated containers.
18
4.1 Test procedure
BCT
The standard BCT (Box compression test), see Figure 4.1, was used to
predict the collapse load of a box. BCT is a common method [12] for testing
corrugated containers where stacking strength is a key performance factor.
In a BCT, a box is placed in a compression machine between two rigid plates.
The plates are then pushed together with a constant velocity applying a top
down compressive force on the box. The force is then plotted against the
deformation in a graph, see Figure 4.2.
Figure 4.2: BCT test series.
The graph represents the compression curves of a test series. The peak
in force represents the collapse load of the box. A test series of 6 to 8 tests
per box geometry was carried out and the mean collapse load was used.
Panel Buckling Load
Since the analytical model is based on the buckling load of a panel, and the
BCT results represent the collapse load of a box. The box collapse load is
directly related to the panel collapse load [13]. For a box with a square
footprint the collapse load of a panel is obtained from
19
Ncr =Ncr,box
N(4.1)
where Fcr is the collapse load of a panel, Ncr,boxis the collapse load of
the box and N is the number of vertical panels in the box.
In order to accurately determine the relationship between Ncr and the
buckling load, Nb, a post-buckling analysis has to be made. For the purpose
of this study, the following approximation [7] was used
Nb =Ncr
1.6(4.2)
The approximation is valid with sufficient accuracy for panels with B-
profile flute. As other profile flutes are modelled the error will increase. This
is further discussed in the discussion.
Material and Geometry
Four different corrugated boards were tested. The boards were denoted B2,
B4, C2 and E2. The boards were in turn composed of combinations of four
different types of paper materials. The properties for each paper material is
presented in Table 4.1.
Name t (mm) Emd (GPa) Ecd (GPa) Basis weight (g/m2)
170Bkl 0.22 7.57 3.03 170140Bf 0.18 6.8 2.38 140120Bf 0.21 6.94 2.39 120135Pw 0.165 7.57 3.03 135
Table 4.1: Properties for each paper material.
For each material, the thickness t, Young’s modulus in MD Emd, Young’s
modulus in CD, Ecd, and the basis weight were obtained from BillerudKo-
rsnas.
20
The ply composition of each board is presented in Table 4.2.
Board Ply 1 Ply 2 Ply 3
B2 170Bkl 120Bf 170BklC2 170Bkl 140Bf 170BklB4 135Pw 135Pw 135PwE4 135Pw 135Pw 135Pw
Table 4.2: Board composition.
Three different regular slotted box geometries were tested, see Table 4.3.
Only the B2 and C2 boards were tested for all geometry combinations.
Geometry Width Length Height
1 280 280 2802 200 200 2803 200 280 280
Table 4.3: Box geometries (mm).
4.2 Equipment
To manufacture the boxes used in the test a high precision, laser guided,
corrugated board cutting machine was utilized. A CAD model of the final
box was loaded into the machine. Large plies of corrugated board were
placed on the table surface of the cutter one by one. The plies were fixed
on the surface using vacuum. Cutting and scoring was then done by the
automatic cutting head resulting in a pre-shaped box. The assembly of the
final box was done by hand using adhesive and tape.
The box was placed in a tensile testing machine, see Figure 4.1, between
two rigid plates. The bottom plate was fixed and mounted on a load cell
with a capacity of 20 kN. The top plate was mounted on a joint and could
rotate freely. The load was applied by a prescribed displacement of the
upper plate downwards on the top of the box. Care was taken to place the
box as centred as possible using graded scales.
21
5. Results
5.1 FEM
The FEM analysis was only done for corrugated board profiles B2 and C2,
see Table 4.2 and Table 4.1 for board parameters. The result are presented
in Table 5.1.
Board Nb (N)
B2 486C2 616
Table 5.1: Results of FEM analysis where Nb is the buckling load of a panel.
5.2 Analytical model
The results for the analytical analysis are presented in Table 5.2.
Box geometry Board Nb (N)
1B2 492C2 654B4 489
2B2 417C2 616E1 412
3B2 445C2 693
Table 5.2: Results of analytical analysis using the semi-analytical model to predictthe buckling load.
22
5.3 Experimental verification
Results for the experiments are presented in Table 5.3. The box collapse load
Ncbox represents the mean value for each test series. Using Equations (4.1)
and (4.2) the buckling loads for a board, represented in the last column of
Table 5.3, were obtained.
Box geometry Board Ncbox (F) Nb (N)
1B2 3098 484C2 3769 588B4 2872 448
2B2 2674 417C2 3256 508E1 1774 277
3B2 2853 371C2 3302 430
Table 5.3: Results of experiments.
5.4 Comparison
A graph showing a comparison between the results from the semi-analytical
model, FEM and experiments is presented in Figure 5.1 for B and C pro-
file corrugated boards and cubic boxes. The results show excellent agree-
ment between the semi-analytical model, FEM analysis and experiments for
the indicated board profiles and geometries, with the analytically obtained
buckling load for B2 profile board falling within a 1.5% margin of the exper-
imental results. The margin of error is slightly higher for C2 profile boards,
where the difference between the analytical model and experiments is 11%.
For the non-cubic boxes and E4 profile the discrepancy between the ana-
lytical model and experiments in general becomes too large, see Figures 5.2
and 5.3. The only viable combination is a B2 profile board in box geometry
2, where the results for the analytical model and the experiments are almost
identical. This will be further discussed in Chapter 6.
23
Figure 5.1: Comparison of board buckling load between all methods of analysisfor B2 and C2 board in cubic boxes, geometry 1, respectively.
Figure 5.2: Comparison of board buckling load between analytical analysis andexperiments for B2, C2 and E4 boards in boxes of geometry 2.
Figure 5.3: Comparison of board buckling load between analytical analysis andexperiments for B2 and C2 boards in boxes of geometry 3.
24
6. Discussion
The results indicate that the analytical model is sufficiently valid, when
compared to FEM and experimental testing, for cubic box geometries of B
and C-profile corrugated board in cubic box geometries. It is of no surprise
that the model gets less accurate when the board geometry starts to differ
substantially from B-profile. The reason for this is that the shear moduli
used are based on tabulated values calculated for a B-profile board. The
transverse shear modulusGxz is very sensitive to the profile of the corrugated
core, and differ with an order of magnitude between the stiffest and the most
flexible core [10]. Table 6.1 lists tabulated values for the shear modulus
coefficient Γ∗, witch was used in [10] to calculate Gxz, for different types of
core profiles. The parameter Γ∗ is directly proportional to Gxz. For a higher
level of accuracy across all corrugated board profiles, explicit calculation of
the shear moduli would have to be built into the analytical model.
Circular Circular-straight Sinusoidal Triangular
Γ∗103 0.43 0.961 2.75 8.72
Table 6.1: Gxz ∼ Γ∗.
Another limitation is the panel dimensions. The model predictions are
based on the first buckling mode. This is only valid for quasi-quadratic pan-
els. When one of the dimension becomes larger then the other, it becomes
more likely for the panel to buckle with higher order buckling modes. In the
experimental testing the non-cubic geometries were sufficiently elongated for
the second buckling mode, see Figure 6.1.
To expand the analytical model so that higher order buckling is captured,
different kinematic assumptions in (3.21) should be made [14]. The task of
finding the buckling load becomes underdetermined and use of, for example,
linear programming (e.g. Simplex-method [11]), becomes necessary.
25
Figure 6.1: Second buckling mode.
The experimental testing measured the collapse load of a box, while
the analytical model predicted the buckling load of a panel. To compare
the results, equation (4.2) was used. This is a good enough approximation
for B-flute panels. For the full spectrum of panel profiles a post-buckling
analysis has to be made. The post-buckling regime is complex and sensitive
to properties of the plate like slenderness and eccentricity [7].
26
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