1
Finite Element Analysis of Cut-Outs
Advanced Materials Subjected to
Compressive Buckling Load
Hakim S. Sultan Aljibori
W.P. Chong
Engineering Department, College of Applied Sciences-SoharSultanate of Oman
Email: [email protected]
Mechanical Eng. Department, Faculty of Engineering,
University of Malaya- Malaysia
Abstract—A further study is carried out to continue the
previous research, at which the work has been published in
Journals of Materials and Design 31 (2010) 466-474 (available
at Science Direct- Elsevier). Experiment work has been
performed on the composite materials plates by using
INSTRON machine. A simulation technique has been
performed in this present research, but a new research
element is added, that is finite element analysis was performed
simultaneously. In present research, numerical study of the
buckling response of compression-loaded quasi-static six
layers woven cross-ply [0º/90º/0º]s laminated composite
materials plates cutout is presented. The objective of this
study is to obtain the buckling load with the effect of cutout
sizes and cutout shapes numerically and compare that with
experimental work. The experiment buckling loads are
compared with the results obtained by ANSYS simulation
program. Both experimental and numerical results are having
a very good agreement. Results have shown that the size of
cutout is getting larger; the buckling load of the plate is
decreasing. Findings are also showing that the cutout area,
cutout ratio (D/W), fiber weight, plate thickness and fiber type
of the plate has a significant effect to the buckling load of the
cross-ply laminated composite plates.
Keywords: Composite materials, cutouts laminated plate,
buckling load, compression, and cross-ply laminated.
I. INTRODUCTION1
Rapid growth and huge demands from the transportation
and manufacturing industries are making most of the
products are fabricated by the advanced materials
especially fiber reinforced composite materials, to meet the
requirements of advanced design, cost saving, physical and
chemical properties for the safety reasons. By now, our
society faces the high-price oil and high-price raw materials
crisis causing a high impact in the transportation industries,
“This work was supported in part by the Mechanical Engineering Department University of Malaya”.
Author Hakim S. Sultan Aljibori, Engineering Department, College of
Applied Sciences-Sohar- Sultanate of Oman (e-mail:
[email protected]
Author W.P. Chong, Mechanical Eng. Dept, Faculty of Engineering,
University of Malaya- Malaysia
one practical solution is to reduce the structural weight but
maintain or increase the payload efficiency at the same time
[1]. It caused the use of composite materials are in widely
and fast developing progress in some industries like
aerospace, marine, automotive, biomedical, civil, military
services and etc., due to their excellent mechanical
properties such as high impact-strength (strength-to-weight
ratio), high stiffness-to-weight ratio, high tear resistant,
ease to handle and low fabrication cost [2-4]. Those
industries have driven much of the development of our
most sophisticated composite systems nowadays.
Cut-outs are provided in structural components for
ventilation, stability maneuverability and sometime to
lighten the structure. For instance, aircraft components like
wing, spar, fuselage and ribs, cutouts are necessary for
accessing, inspection, electric and hydraulic lines, fuel lines
as well as to reduce the overall weight of the aircraft. Cutouts are often found in composite structures [4]. During
operation, these structural elements may experience
compressive loads and thus lead to buckling. Hence,
structural instability becomes a major concern in safe and
reliable design of the composite materials.
Woven fabric fiber is a way of weaving by interlacing the
fiber thread of the weft and warp on a loom [5]. Many
researchers have carried out the buckling behavior of
laminated plates experimentally. Hakim et al.,
Ghannadpour et al. and P. Nemeth [5-9] investigated the
buckling behavior of the laminated composite plates due to
circular/elliptical cutouts under compression. Both studies
have showed that the effects of cutout size, shape, plate
aspect ratio, fiber orientations and stacking sequences, and
boundary conditions have a significant effect on the
buckling behavior of the plates.
Züleyha Aslan & Sahin M. [10] examined the effects of
delaminations size on the critical buckling load and
compressive failure load of E-glass/epoxy composite
laminates with multiple large delaminations experimentally
and numerically. They discovered that the longest and nearsurface delamination size influences the buckling load and
compressive failure load of composite laminates.
Compressive buckling analysis on metal-matrix composite
(MMC) plates with central square holes was performed by
Fig. 1: The determination of the buckling load [14]
L. Ko [11]. Study showed that by increasing the hole sizes,
compressive buckling strengths of the perforated MMC
plates could be considerably increased under certain
boundary conditions and aspect ratios. M. F. Altan & M. E.
Kartal [12] also investigated the changes of buckling
coefficients of symmetrically laminated reinforced concrete
plates with central rectangular hole under biaxial static
compression loadings.
x
[0º/90º/0º] s
l
e
II. PROCEDURE AND BUCKLING LOAD OF LAMINATED
t
COMPOSITE PLATES
In the work of Laurin et al. [13], it has been derived that the
mathematic model of buckling of laminated composite
plate. The buckling load of a thin composite cross-ply
laminated plate with a ratio R = l/w, simply supported and
subjected to a compressive loading is presented. The
buckling load per unit length is given by
N xcr 
2 
A
2
2
4
 m A11  2 R  A12  2 A66   222 R 
l 
m

2
(1)
d
d
Plate thickness
(t) in mm:
E-glass (600 g)
= 3.230
E-glass (400 g)
= 2.320
Carbon (400 g)
= 1.625
Kevlar-29 (200
g) = 2.200
Six layers
w
Where l is the plate length, w is the plate width, m is the
buckling mode, A is the bending stiffness of the laminate
given by the following relation:
A
 3 h
1
3
k 1

 hk3  Q k
(2)
Fig. 2 Geometry of laminated composite material plate
k
Where the subscript k indicates the kth layer from the top of
laminate, hk and hk+1 are respectively the distance from the
mid-plan to the bottom and the top of the kth layer and Qk is
the stiffness expressed in the laminate axis of the kth layer.
The first-buckling mode is the one that leads to the
buckling load and thus depends on the ratio A11/A22.
Laminate composite plates constituted by plies of the sane
orientation but with different stacking sequences could
exhibit different A11/A22 ratios (and thus different buckling
loads). Buckling load is the compressive load which is just
sufficient to bend the plate slightly, the plate will remain
straight after the load is releasing. For the determination of
buckling load graphically, the last point on the curve where
just left from the straight line and the value of this point on
the y-axis is called as the buckling load as illustrated in
Figure 1.
III. DETERMINATION OF THE MECHANICAL
PROPERTIES OF LAMINATED COMPOSITE PLATE
The mechanical properties of laminated composite plates
are obtained under quasi-static loading conditions
according ASTM standards. Average mechanical properties
of the laminated composite materials obtained from the
experiment results are listed in Table 1. In this table, Ex, Ey
and Ez are Young’s modules corresponding to x, y and z
planes; Gxy, Gyz and Gxz are the shear modules
corresponding to x-y, y-z and x-z planes, respectively; and
Prxy, Pryz and Prxz are the corresponding Poisson’s ratios.
1. TABLE1 MECHANICAL PROPERTIES OF LAMINATED
COMPOSITE
Ex
Ey
Ez
Prxy=Pryz=Prxz Gxy=Gyz=Gxz
Fiber
(GPa)
(GPa)
(GPa)
(GPa)
type
E-glass
(400
gram)
Carbon
(400
gram)
Kevlar29
(200
gram)
72.40
7.10
7.10
0.22
2.40
224.00
14.00
14.00
0.20
14.00
61.00
4.20
4.20
0.35
2.90
IV. NUMERICAL BUCKLING ANALYSIS
In the present study, buckling analysis is carried out for the
laminated composite plates with and without central
circular/square cutouts. These plates are analyzed by using
3
ANSYS package. The buckling load is determined by
solving for eigen-values and the corresponding
eigenvectors represents the buckled mode shape. Figure 3
shows the boundary conditions and typical finite element
mesh as same as experimental conditions. As the element
type, SHELL Layered 99 element with six degree of
freedom is selected, and structural elastic with orthotropic
material is chosen. Real constant was defined by entering
the values of number of layers, fiber angle orientations and
ply’s thickness. The sketch of real constant model is shown
in Figure 4. Mechanical properties are shown in the Table
1. To simulate clamped loaded edges, the displacement UX,
UY, UZ and the rotations ROTX, ROTY, ROTZ of all
nodes at edges are set equal to zero. To mesh the shell
model, free and mapped meshing with “concatenating”
operation is used. Better meshing skill is important to yield
better and accurate results. The meshing of the test
specimens are illustrated in Figure 5 by the ANSYS
program. To apply loading on the top of the shell model, a
unit pressure is applied along the upper nodes. As to
perform eigen-buckling, pre-bucking system is switched on
and extraction mode is operated before proceeding to final
result. Finally, the buckling load of the first buckling mode
is always as our favorite result, normally.
Pressure
UX, UZ, ROTX, ROTY,
ROTZ = 0
Free edge
Free edge
UX, UY, UZ,
ROTX, ROTY, ROTZ
=0
Fig. 3 Boundary, loading and meshing conditions of laminated composite
0º
90º
0º
Ply 6
0º
Ply 5
90º
Ply 4
0º
Ply 3
Ply
thickne
ss
Ply 2
Ply 1
Fig.4: Number of plies, fiber orientations and thickness of composite
Fig. 5: The meshing of test specimens (a) fine plate, (b) plate with
central circular cutout, and (c) plate with central square cutout
V. RESULTS AND DISCUSSION
In this present work, the buckling load results are obtained
experimentally and numerically for cross-ply [0º/90º/0º]s
laminated composite plates with and without central
circular/square cutouts, of which including the E-glass fiber
400 gram types, carbon fiber 400 gram types and Kevlar-29
fiber 200 gram types; except the E-glass fiber 600 gram
types which is performed experimentally only. The
experimental results produced the load-displacement curve
of each of the specimens, while the numerical results are
obtained from the nodal solution graphics simulated by
ANSYS program. The experimental and numerical
buckling loads for all the test specimens are summarized in
Table 2. It is seen from this table, the agreement between
the experiment and finite element prediction is showing
good. For E-glass fiber 400 gram type plates, the
differences are in between 0.03 – 2.47 %; for carbon fiber
400 gram type plates, the differences are found in the range
0.52 – 2.70 %. The differences for the Kevlar-29 200 gram
type plates are slightly higher, dropped at 1.47 – 8.24 %,
although these are acceptable. Also, it is observed that the
buckling load is greatly depending on its cutout size
(preferable in term of cutout ratio, d/w). As cutout ratio
increases, the buckling load is decreasing. It is also found
that the buckling load of the plate with central circular
cutout is higher than the plate with central square cutout. In
additional, for the same cutout size’s plate, the carbon fiber
plate always has the highest buckling load, following by the
E-glass fiber plate and Kevlar-29 fiber plates. The buckling
load of the plate is affected by their fiber weight and
thickness as well.
Figures 6–8, from (a) – (c) show the
nodal solution graphics for the cross-ply laminated
composite plates, with and without central circular/square
cutouts, respectively. All the results of the specimens is
tabulate in Table 2
2. TABLE 2 SUMMARY OF THE EXPERIMENTAL AND
NUMERICAL BUCKLING LOAD OF LAMINATED
COMPOSITE PLATE
Experime NumeriCut-ntal
cal
Differn
Plate
Cutout
out
buckling
Buckli-ces
type
size
ratio
load
ng load
(%)
(kN)
(kN)
Without
0
1.15260
cutout
ECircular d =
0.40
1.05771
glass
16 mm
fiber
Circular d =
(600
0.65
0.98738
26 mm
gram)
Circular d =
0.90
0.81208
36 mm
Without
0
0.41074
0.41052
0.05
cutout
Circular d =
0.40
0.38389
0.39102
1.82
16 mm
Circular d =
0.65
0.36483
0.35604
2.47
26 mm
ECircular d =
0.90
0.30101
0.29784
1.06
glass
36 mm
fiber
Square
(400
(16×16)
0.40
0.36566
0.36576
0.03
gram)
mm
Square
(26×26)
0.65
0.34423
0.34860
1.25
mm
Square
(36×36)
0.90
0.27114
0.26994
0.44
mm
Without
0
0.46187
0.45948
0.52
cutout
Circular d =
0.40
0.43523
0.42504
2.40
16 mm
Circular d =
0.65
0.40524
0.39930
1.49
26 mm
Circular d =
Carbo
0.90
0.35037
0.34134
2.65
36 mm
n fiber
Square
(400
(16×16)
0.40
0.42087
0.40980
2.70
gram)
mm
Square
(26×26)
0.65
0.39215
0.38670
1.41
mm
Square
(36×36)
0.90
0.32165
0.31374
2.52
mm
Without
0
0.34362
0.33684
2.01
cutout
Circular d =
0.40
0.27785
0.28200
1.47
16 mm
Circular d =
0.65
0.25000
0.26004
3.86
26 mm
Keval
Circular d =
0.90
0.20195
0.22008
8.24
r-29
36 mm
fiber
Square
(200
(16×16)
0.40
0.25490
0.26688
4.49
gram)
mm
Square
(26×26)
0.65
0.23943
0.25266
5.24
mm
Square
(36×36)
0.90
0.19718
0.20220
2.48
mm
a
b
c
Fig. 6: Nodal solutions for E-glass fiber (400 gram) plates. (a) plate
without cutout; (b) plates with central circular cutout d = 36 mm; (C) plates
with central square cutout sizes (36 × 36) mm.
5
VI. EFFECT OF CUTOUT SHAPE
a
Due to the design requirements and philosophy, different
cutout shape may be used. Effect of cutout shape can be
seen clearly that, for the same fiber type and same cutout
size, the plate with central circular cutout has higher
buckling load than the one with central square cutout,
nearly about 1.02 – 1.11 times. This behavior is at least
partially explained by noting that, the central cutout area
which was subtracted from the plate is the main cause.
With the same cutout diameter or width, the area of square
is higher than the area of circle. Therefore, the more area is
subtracted, the more losing of mass in the central of plate, it
will experience a larger loss in central bending stiffness,
tends to buckle at lower loading.
VII. EFFECT OF CUTOUT SIZE
b
c
Cutout size, normally preferable referring to cutout ratio,
d/w, where d is the cutout width or diameter, and w is the
plate width. Many researchers tend to use this term to
investigate the relationship in between of the buckling load
and cutout ratio. In this study, it is clear to show that the
plate has no cutout or small cutout has higher strength than
the plate has larger cutout. It is because a loss of mass (less
material) in the center is a loss of the interfacial bond
between the matrix and fibers. The interface is responsible
for transmitting the loading from the matrix to fibers, which
contribute the greater portion of composite strength. As a
result, as cutout size is getting bigger means the bonding is
getting loosing, it will reduce the strength of plate.
Inherently, it associated with the central located cutout is a
loss in bending stiffness in the central region of a plate that
grows in important as the central cutout size increases. As
the more losing in bending stiffness caused by the
increasing of central cutout size will result in reduction of
buckling strength.
VIII. VALIDATION OF BUCKLING LOAD AND
CUTOUT RATIO
Fig.7 Nodal solutions for arbon fiber (400 gram) plates, (a) plate without
cutout; (b) plates with central circular cutout (d = 36 mm); (c) plates with
central square cutout sizes (36 × 36) mm
Agreement between the experimental buckling loads and
the finite element buckling load predictions is having a very
good tolerance. Figures 9 and 10 present the comparisons
of buckling load vs. cutout ratio between the experimental
and numerical results for all the specimens. It is seen that
the results obtained from numerical buckling analysis close
to experimental buckling results. As mentioned before, the
buckling load of the plate decreases while the cutout ratio is
increasing. In the same Figures, the comparison also has
been made between the present results with the numerical
results investigated by Tercan M. & Aktaş M. [14]. Tercan
M. & Aktaş M. had investigated the buckling behavior of 1
× 1 rib knitting laminated plates with cutouts, which the
plates are made from five layers glass fabrics and CY225
epoxy resin, with length L = 80 mm and width w = 25 mm.
Their experimental works including the boundary and
loading conditions are similar with the present research,
therefore heir numerical results based on the same
experiment set up are validated to compare. It is observed
that the laminated composite plates manufactured by
Tercan M. & Aktaş M. higher buckling load than the plates
in this present research at low cutout ratios, it dropped
evidently below the present research’s buckling loads of
carbon and E-glass plates at higher cutout ratios. It is due to
the different materials in used in the research and their
different properties influences, with the different plate
manufacturing process and cutout shapes (The eccentricity
of the plates are fixed at e/w = 0.4 and varying different
cutout ratios by Tercan M. & Aktaş M. in their numerical
simulations). The most important evidence and similarity
between the comparisons is that all the plots are showing
the same trend in the buckling load vs. cutout ratio curve,
noting that the buckling load of the plate decreases while
the cutout ratio is increasing.
IX. CONCLUSION
Buckling load ( kN )
Buckling load ( kN )
An experimental and finite element analysis study of the
buckling behavior of the woven composite fiber/epoxy
laminated plates, with and without central circular/square
cutouts subjected to quasi-static compressive loading has
been presented. Four different fiber types’ laminated
composite plates are fabricated to test experimentally; they
are E-glass fiber 600 gram & 400 gram, carbon fiber 400
gram and Kevlar-29 fiber 200 gram. Circular cutout sizes
are fixed at d = 16 mm, 26 mm and 36 mm, and the square
cutout sizes are fixed at (16×16) mm, (26×26) mm, and
(36×36) mm. Eigenbuckling is introduced in this study to
solve the buckling load by ANSYS simulation program.
From the results and discussion, the effects of fiber weight,
plate thickness, fiber type, cutout size and shape on the
0.60
buckling load of the laminated composite plates are welldiscussed. In addition, the following conclusions can be
summarized as below:
Tercan M. & Aktaş M.[4]
0.50
a) For all cases of symmetrical laminates, the cross-ply
Carbon fiber 400 gram
[0˚/90˚/0˚]s laminates, this type of ply configuration is
0.40
capable of absorbing large amount of energy before
E-glass fiber 400 gram
fracture.
b)
As the central cutout size increases, the cutout
0.30
ratio
increases
proportionally, the buckling load of
Kevlar-29 fiber 200 gram
laminated composite plate will be reduced.
0.20
c) A loss of mass in the centre of the laminated composite
plate will lose the interfacial bonding of its matrix and
Experimental
Numerical
0.10
fibers. The strength of plate will reduce as the bonding is
losing.
Tercan M. & Aktaş M. (fixed at plate’s e/w ratio at 0.4) [4] d) More fiber weight gain more matrixes during
0.00
composition,
resulting in the increasing of plate thickness.
0
0.2
0.4
0.6
0.8
1
Cutout ratio, d/w
Thicker plate produces great stiffer strength to buckling at
higher load.
e) The reducing of central bending stiffness of the
Fig. 9 Comparisons of buckling load vs. cutout ratio for composite plates
with/without central circular
laminated composite plate caused by the increasing of
cutouts
central cutout size yields a reduction in buckling resistance
of the plate.
f) In same fiber type and cutout size plates, the one with
central circular cutout has higher buckling strength than the
0.60
one with central square cutout.
g) Carbon fibers give the best mechanical properties than
Tercan M. & Aktaş M.[4]
0.50
E-glass fibers and Kevlar-29 fibers. Having the highest
buckling load, great modulus and strength in low weight.
Carbon fiber 400 gram
0.40
h) The low compressive strength of Kevlar-29 is due to
the anisotropic properties and low shear stiffness. Although
E-glass fiber 400 gram
0.30
its compressive strength is low, it undergoes a huge
Kevlar-29 fiber 200 gram
displacement without heavy damage and crack before its
0.20
failure. This prove that Kevlar-29 is good capable in
shearing.
0.10
Tercan M. & Aktaş M. (fixed at plate’s e/w ratio at 0.4)[4]
ACKNOWLEDGMENT
0.00
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Cutout ratio, d/w
Fig.10 Comparisons of load vs. cutout ratio for laminated composite
plates with/without central square cutouts
1
The authors would like to thank the University of Malaya,
(Malaysia) and college of Applied Sciences-Sohar
(Sultanate of Oman) for the financial support for
experimental work and Finite element analysis.
7
REFERENCES
[1] R.M. O’Higgins, M.A. McCarthy, C.T. McCarthy. Comparison of
open hole tension characteristics of high strength glass and carbon fibrereinforced composite materials. J Compos Sci Technol 2008; 68: 27702778.
[2] F. Laurin, N. Carrere, J. –F. Maire. Laminated composite structures
subjected to compressive loading: A material and structural buckling
analysis. J Compos Struct 2007; 80: 172-182.
[3] Duosheng Xu, Rajamohan Ganesan, Suong V. Hoa. Buckling analysis
of tri-axial woven fabric composite structures. Part I: Non-linear finite
element formulation. J Compos Struct 2005; 67:37-55.
[4] Payal Jain, Ashwini Kumar. Postbuckling response of square
laminates with a central circular/elliptical cutout. J Compos Struct 2004;
65:179-185.
[5] Hakim S. Sultan Aljibori, W. P. Chong, T. M. I Mahlia, W. T.
Chong, Prasetyo Edi, Haidar Al-qrimli, Irfan Anjum, R. Zahari. Laoddisplacement behavior of glass fiber/epoxy composite plates with circular
cut-outs subjected to compressive load. J Mater & Design 2010; 31: 466474.
[6] S. A. M. Ghannadpour, A. Najafi, B. Mohammadi. On the buckling
behavior of cross-ply laminated composite plates due to circular/elliptical
cutouts. J Compos Struct 2006; 75:3-6.
[7] Nemeth, M.P. Buckling behavior of compression-loaded
symmetrically laminated angle-ply plates with holes. AIAA Journal 1988;
26 (3):330-336.
[8] Nemeth, M.P. Buckling and post buckling behavior of square
compression-loaded graphite-epoxy plates with circular cutouts. NASA
1990; TP-3007.
[9] Nemeth, M.P. Buckling and postbuckling behavior of laminated
composite plates with a cutout. NASA 1996; TP-3587.
[10] [10] Züleyha Aslan, Mustafa Şahin. Buckling behavior and
compressive failure of composite laminates containing multiple large
delaminations. J Compos Struct 2009; 89: 382-390.
[11] William L. Ko. Anomalous buckling characteristics of laminated
metal-matrix composite plates with central square holes. NASA/TP-1998206559.
[12] Mehmet Fatih Altan, Murat Emre Kartal. Investigation of buckling
behavior of laminated reinforced concrete plates with central rectangular
hole using finite element method. J Mater & Design 2009; 30:2243-2249.
[13] Laurin, F., Carrere, N. and Maire, J. –F. (2007). Laminated
Composite Structures Subjected to Compressive Loading: A Material and
Structural Buckling Analysis. Comp Struct, 80: 172 – 182.
[14] Tercan, Mevlüt., and Aktas. (2009). Buckling Behavior of 1 × 1 Rib
Knitting Laminated Plates With Cutouts. Comp Struct, 89: 245 – 252.