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EFFECT OF PLY STACKING SEQUENCE ON STRUCTURAL RESPONSE OF SYMMETRIC COMPOSITE LAMINATES
Mosfequr Rahman Department of Mechanical Engineering
Georgia Southern University Statesboro, GA, USA
Saheem Absar Department of Mechanical Engineering
Georgia Southern University Statesboro, GA, USA
FNU Aktaruzzaman Department of Mechanical Engineering
Georgia Southern University Statesboro, GA, USA
Abdur Rahman Visiting Assistant Professor
Department of Electrical Engineering, Georgia Southern University
Statesboro, GA, USA N.M. Awlad Hossain Associate Professor
Department of Engineering and Design Eastern Washington University
Cheney, WA
ABSTRACT In this work, the effect of ply stacking sequence on the
structural response of multi-ply unidirectional fiber-reinforced
composite laminates was evaluated using finite element
analysis. The objective of this study was to develop a
computational model to analyze the stress response of
individual plies in a composite laminate for a given stacking
sequence. A laminated composite plate structure under tensile
loading was modeled in ANSYS. Stress profiles of the
individual plies were obtained for each lamina. An Epoxy
matrix with both unidirectional Graphite and Kevlar fibers was
considered for the model. Three dimensional sectioned shell
elements (SHELL181) were used for meshing the model.
Several sets of stacking sequences were implemented,
symmetrical to the mid-plane of the laminate. Symmetric
stacking configurations of 6 layers stacked in ply angles of
[0/45/-45]s, [0/60/-60]s, [0/45/90]s, and an 8-layered
arrangement of [0/45/60/90]s were modeled for the analysis.
The layer thickness was maintained at 0.1 mm. The results were
compared against an analytical model based on the generalized
Hooke’s law for orthotropic materials and classical laminate
theory. A numerical formulation of the analytical model was
implemented in MATLAB to evaluate the constitutive
equations for each lamina. The stress distributions obtained
using finite element analysis have shown good agreement with
the analytical models in some of the cases.
NOMENCLATURE Vf , Vm Volume fraction of fiber and matrix
E1, E2, E3 Elastic modulus in longitudinal and
transverse direction of fiber axis
G1, G2, G3 Shear modulus in longitudinal and transverse
direction of fiber axis
ν Poisson’s Ratio
Stress
ε Strain
INTRODUCTION Composite materials can be defined as heterogeneous
materials which feature the combination of the best aspects of
dissimilar constituents [1]. Each constituent maintains its
mechanical and physical properties and contributes towards
sharing applied structural loads. The main advantages of
composite materials are their high strength and stiffness per
unit weight. In unidirectional fiber-reinforced composites, the
constituent phases consist of a fiber phase (such as glass,
aramid, carbon) and a matrix phase composed of a polymer,
metal or ceramic. Due to their high aspect ratio and fewer
defects along axial direction, fibers provide high strength and
stiffness. The reinforcement phase thus provides the desired
strength of the composite. The matrix (continuous phase)
maintains the fibers in the proper orientation and spacing and
1 Copyright © 2014 by ASME
Proceedings of the ASME 2014 International Mechanical Engineering Congress and Exposition IMECE2014
November 14-20, 2014, Montreal, Quebec, Canada
IMECE2014-37217
protects them from abrasion and environmental effects. The
matrix also transmits loads from the matrix to the fibers
through shear loading at the interface. Fiber-reinforced
composite laminates are prepared by stacking single sheets of
continuous fibers in different orientations to obtain the desired
strength and stiffness properties [1].
Due to the anisotropic nature of the material properties of
the constituent fibers, the modeling of composite structures
differs from conventional materials. Also the stacking sequence
of the layers and orientation of the fibers affect the structural
response of the composite. Due to the costs and complexities
involved in the processing of composite materials it would be
beneficial to predict the performance of such materials before
manufacturing. Thus, computational analysis of the structural
response of composite structures can provide significant insight
into their behavior and expected performance in various
loading conditions and material properties.
Laminated plate theories provide a suitable approximation
for analyzing laminated composite plates. Zhang et al have
discussed several laminated plate theories presented by various
authors [2]. A review of various equivalent single layer and
layerwise laminated plate theories was presented by Reddy and
Robbins [3]. Liu and Li presented an overall comparison of
laminated theories based on displacement hypothesis [4],
including shear deformation theories, layerwise theories,
generalized Zigzag theories, and the proposed global–local
double-superposition theories. Reviews of laminated and
sandwich plates have been presented by Altenbach [5].
Displacement and stress-based refined shear deformation
theories of isotropic and anisotropic laminated plate was
reviewed by Ghugal and Shimpi [6]. A review of shear
deformation plate and shell theories was presented by Reddy
and Arciniega [7], a selective review and survey of the theories
with emphasis on estimation of transverse/interlaminar stresses
in laminated composites was given by Kant and Swaminathan
[8].
The laminated plate theories can be broadly divided into
the following two categories [2]:
(a) Equivalent single layer (ESL) theories, including
a. Classical lamination theory (CLT)
b. The first-order shear deformation theory
(FSDT)
c. Higher-order shear deformation theories
(HSDT)
d. Layer-wise lamination theory (LLT)
(b) Continuum-based 3D elasticity theory.
The modeling of laminated composites differs from
conventional materials in the sense that the constitutive
equations of each lamina are orthotropic in nature. These
equations of each element depend on the kinematic
assumptions of the shell theory used [9]. Typically, composites
are modeled as plates or shells. The material is assumed to be in
a state of plane stress, i.e., the stresses and strains in through-
the-thickness directions are ignored. The thickness dimension is
also assumed to be much smaller than the length and width
dimensions [10]. These assumptions help simplify the solution
of the 3-D governing equations of the system to 2-D.
Thus, in this work a meso-scale level approach is used to
model the laminates, based on the aforementioned
considerations. The lamina stacking sequence, orthotropic
elastic properties of the materials, thickness and fiber
orientation are supplied to the computational model to obtain
the required stress distributions.
The method of rule of mixtures is used to obtain the
orthotropic material properties of the laminate, for some
volume fraction of fiber in a resin matrix. For a composite
material consisting of a fiber volume fraction of Vf and matrix
volume fraction of Vm, the orthotropic elastic properties along
the longitudinal and transverse directions of the laminate are
obtained using the following relations [11]:
𝑉𝑓 + 𝑉𝑚 = 1
𝐸1 = 𝐸𝑓𝑉𝑓 + 𝐸𝑚𝑉𝑚
1
𝐸2=𝑉𝑓
𝐸𝑓+𝑉𝑚𝐸𝑚
𝐸3 = 𝐸2
𝜈12 = 𝜈𝑓𝐸𝑓 + 𝜈𝑚𝐸𝑚
𝜈23 = 𝜈12 ×𝐸2𝐸1
𝜈13 = 𝜈12;
1
𝐺12=𝑉𝑓
𝐺𝑥𝑦 +𝑉𝑚𝐺𝑥𝑦
𝐺23 =𝐸2
2(1 + 𝜈23)
𝐺13 = 𝐺12
(1)
Generalized Hooke’s Law for Orthotropic Materials The stress-strain relationship of an orthotropic material can
be obtained using the generalized Hooke’s Law:
𝜎𝑖 = 𝐶𝑖𝑗𝜖𝑗
𝜖𝑖 = 𝑆𝑖𝑗𝜎𝑗
𝑆𝑖𝑗 = [𝐶𝑖𝑗]−1
Cij is the stiffness matrix and Sij is the compliance matrix.
The complete equation with elastic constants,
(2)
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[ 𝜖1𝜖2𝜖3𝛾23𝛾13𝛾12]
=
[ 1/𝐸1 −𝜈21/𝐸2 −𝜈31/𝐸3 0 0 0−𝜈12/𝐸1 1/𝐸2 −𝜈32/𝐸3 0 0 0−𝜈13/𝐸1 −𝜈23/𝐸3 1/𝐸3 0 0 00 0 0 1/𝐺23 0 00 0 0 0 1/𝐺31 00 0 0 0 0 1/𝐺12]
[ 𝜎1𝜎2𝜎3𝜏23𝜏13𝜏12]
Which can be written in compliance form as,
[ 𝜎1𝜎2𝜎3𝜏23𝜏13𝜏12]
=
[ 𝐶11 𝐶12 𝐶13 0 0 0𝐶21 𝐶22 𝐶23 0 0 0𝐶31 𝐶32 𝐶33 0 0 00 0 0 𝐶 0 00 0 0 0 𝐶 00 0 0 0 0 𝐶 ]
[ 𝜖1𝜖2𝜖3𝛾23𝛾13𝛾12]
Considering plane stress: 𝜎3 = 0, 𝜏23 = 𝜏23 = 0, and
removing constants associated with these stress
components,
[
𝜎1𝜎2𝜏12] = [
𝐶11 𝐶12 0𝐶21 𝐶22 00 0 𝐶
] [
𝜖1𝜖2𝛾12]
Where,
𝐶11 =𝐸1
(1 − 𝜈12𝜈21)
𝐶12 =𝜈12𝐸2
(1 − 𝜈12𝜈21)
𝐶22 =𝐸2
(1 − 𝜈12𝜈21)
𝐶 = 𝐺12
(3)
(4)
(5)
Classical Laminate Theory
The classical laminate theory is used to obtain stiffness
matrices for an orthotropic laminate in the Cartesian coordinate
system. The following kinematic assumptions are considered
according to the Kirchoff Hypothesis, for a thin laminate with a
small deflection in the transverse direction [12, 13]:
Straight lines normal to the mid-surface of the plate remain
straight after deformation.
Normals remain normal to mid-surface of plate.
Normals remain unstretched (thickness of plate remains
unchanged).
Perfect bonding between layers is also assumed:
There are no gaps or flaws between the layers.
No-slip conditions exist at the bonding regions, preventing
slippage of lamina.
The elastic constants of an orthotropic ply (1,2) can be
transformed to that of laminate axes (x,y) by using axis
transformation euation [13] (eqns. 6).
𝑄11 = 𝐶11𝑚 + 2(𝐶11 + 2𝐶 )𝑚
2𝑛2 + 𝐶22𝑛
𝑄22 = 𝐶11𝑛 + 2(𝐶12 + 2𝐶 )𝑚
2𝑛2 + 𝐶22𝑚
𝑄12 = (𝐶11 + 𝐶22 − 4𝐶 )𝑚2𝑛2 + 𝐶12(𝑚
+ 𝑛 )
𝑄 = (𝐶11 + 𝐶22 − 2𝐶12 − 2𝐶 )𝑚2𝑛2 + 𝐶 (𝑚
+ 𝑛 )
𝑄1 = (𝐶11 − 𝐶12 − 2𝐶 )𝑛𝑚3 + (𝐶12 − 𝐶22 + 2𝐶 )𝑚𝑛
3
𝑄2 = (𝐶11 − 𝐶12 − 2𝐶 )𝑚𝑛3 + (𝐶12 − 𝐶22 + 2𝐶 )𝑛𝑚
3
Where, 𝑚 = cos 𝜃 and 𝑛 = sin 𝜃
[
𝜎𝑥𝜎𝑦𝜏𝑥𝑦] = [
𝑄11 𝑄12 0𝑄21 𝑄22 00 0 𝑄
] [
𝜖𝑥𝜖𝑦𝛾𝑥𝑦]
The general constitutive equation relating the loads (N) and
moments (M) on a laminate to the strains and curvature of
the plate is written as:
[𝑁𝑖𝑀𝑖] = [
𝐴𝑖𝑗 𝐵𝑖𝑗𝐵𝑖𝑗 𝐷𝑖𝑗
] [𝜖𝑖𝜅𝑖]
(6)
(7)
(8)
Here, 𝐴𝑖𝑗 is the extensional or membrane stiffness, 𝐷𝑖𝑗 is
the flexural or bending stiffness and 𝐵𝑖𝑗 refers to the coupling
between membrane and bending behavior. Summating terms of
each layer k at a distance of h from the mid-plane,
𝐴𝑖𝑗 =∑(𝑄𝑖𝑗)𝑘(ℎ𝑘 − ℎ𝑘−1)
𝑛
𝑘=1
For laminates symmetric to the mid-plane, the B and D
terms are ignored. The constitutive equation then reduces
to,
[𝑁𝑖] = [𝐴𝑖𝑗][𝜖𝑖]
(9)
(10)
After obtaining the strains from Eqn (10), the stresses in
each lamina are then obtained by,
[
𝜎𝑥𝜎𝑦𝜏𝑥𝑦] = [
𝐴11 𝐴12 0𝐴21 𝐴22 00 0 𝐴
] [
𝜖𝑥𝜖𝑦𝛾𝑥𝑦]
(11)
MODELING AND FINITE ELEMENT ANALYSIS A surface body with the geometry of a unit square
representing the composite laminate was modeled using in
ANSYS. The plate was constrained against displacements in all
directions in the lower left corner. The left edge and bottom
edges were constrained along the X and Y directions
respectively. A load of 100kN was applied along the X-
direction on the right edge of the plate, and a 50kN load was
applied along the Y axis on the top edge.
3 Copyright © 2014 by ASME
Fig 1: Model of plate
The model was meshed using SHELL181 elements.
SHELL181 is suitable for analyzing thin to moderately-thick
shell structures. It is a four-noded element with six degrees of
freedom at each node: translations in the x, y, and z directions,
and rotations about the x, y, and z axes. It is well-suited for
linear, large rotation, and/or large strain nonlinear applications.
SHELL181 can be used for layered applications for modeling
composite shells or sandwich construction. The accuracy in
modeling composite shells is governed by the first-order shear-
deformation theory.
The shell section commands allow for layered shell
definition. Options are available for specifying the thickness,
material, orientation, and number of integration points through
the thickness of the layers [14]. The storage of all layer data
was enabled in the element options to enable processing results
of each of the laminate layers.
Four sets of stacking sequences were implemented for the
plate model. The layers were stacked symmetrical to the mid-
plane of the laminate. Laminates consisting of 6 layers stacked
in ply angles of 0/45/-45, 0/60/-60, 0/45/90 degrees, and an 8-
layered laminate of 0/45/60/90 degrees symmetrical to the mid-
plane were modeled for the analysis. The layer thickness was
maintained at 0.1 mm.
The composite materials used for the analysis and their
corresponding mechanical properties are specified in Table 1.
Graphite and Kevlar were considered as fiber reinforcements in
an epoxy matrix. The material strength values are provided in
Table 2 [11].
Table 1: Composite material properties
Composite
Fiber
volume
Elastic
Modulus
Poisson’s
Ratio
Shear
Modulus
fraction,
Vf
(GPa) (GPa)
E1 E2 ν12 ν23 G12 G23
Graphite/
Epoxy 0.67 138.0 8.96 0.30 0.019 7.10 4.57
Kevlar49/
Epoxy 0.60 76.0 5.50 0.34 0.025 2.30 2.82
Table 2: Composite strength
Composite
Tensile
Strength
(MPa)
Compressive
Strength
(MPa)
Shear
Strength
(MPa)
X Y X Y XY
Graphite/
Epoxy 1450 51.7 1450 206 93.0
Kevlar49/
Epoxy 1400 12.0 235.0 53.0 34.0
The different stacking arrangements used for the analysis
are listed in Table 3. Figs 3A-3D show the maximum stress
values along the X direction in each ply in a Graphite/Epoxy
(G/E) laminate with the stacking arrangements shown in Table
3.
Table 3: Stacking arrangements
Type
Symmetric ply
stacking with angles
S1 0/45/-45
S2 0/45/90
S3 0/60/-60
S4 0/45/60/90
RESULTS AND DISCUSSION The stress components along X and Y direction along with
the shear stress in the XY plane was obtained for each layer of
the laminates using ANSYS. Only the maximum stress values
of each ply were considered for plotting the stress profiles. The
maximum value of the X, Y and XY direction stresses were
obtained from the nodal results for each ply. The stress
response was also calculated using the classical laminate theory
(Eqns 1-11) using a numerical code developed in MATLAB.
The finite element results were found to show good agreement
with the theoretical results. Some of the results have also shown
inconsistencies with the theoretical model. These
inconsistencies can be attributed to probable nonlinearities in
the finite element solutions. Also, the simplified assumptions
used for the classical laminate theory regarding bending and
shear-extension coupling of laminate layers may also have had
an effect on the varying results.
The stress distributions along the X axis are symmetrical
for ech of the stacking configurations. The outer plies have
been found to carry the maximal loads in the laminate.
50kN
100kN
4 Copyright © 2014 by ASME
Fig 3a: X-direction stress for S1 (G/E)
Fig 3b: X-direction stress for S2 (G/E)
Fig 3c: X-direction stress for S3 (G/E)
Fig 3d: X-direction stress for S4 (G/E)
The longitudinal or X direction stresses are largely carried
along the outermost layers which are aligned at 0 degrees,
along the fiber direction. For laminates with balanced inner ply
angles of 60° (S3), the stress distribution profile appears similar
compared to those having 45° angles (S1) (Figs 3a, 3c). But the
S3 configuration carries higher stress on the outer layers than
the S2 arrangement.
The distribution of the maximum stress values obtained
along the Y axis or transverse direction for Graphite-Epoxy
(G/E) laminates are shown in Figs 4a-4d. The stress response
along the transverse axis is opposite to that of the longitudinal
response for each laminate configuration. The transverse or Y
direction stresses are carried by the layers adjacent to the mid-
planes. The S1 and S3 arrangements display similar stress
profiles in this case (Figs 4a, 4c). The S2 configuration shows a
smooth parabolic stress distribution (Fig 4b). Addition of an
intermediate ply angle of 60° creates an irregular parabolic
pattern as shown in Fig 4d, for the S4 arrangement.
Fig 4a: Y-direction stress for S1 (G/E)
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Fig 4b: Y-direction stress for S2 (G/E)
Fig 4c: Y-direction stress for S3 (G/E)
Fig 4d: Y-direction stress for S4 (G/E)
The distribution of the maximum shear stress values
obtained along the XY plane for Graphite-Epoxy (G/E)
laminates are shown in Figs 5a-5d. The stresses are carried by
the inner layers, excluding the outer 0° layers. For the balanced
stacking arrangements of S1 and S3, the inner layers develop
stresses along opposite directions with the same magnitudes
(Figs 5a, 5c). For the unbalanced laminates (S2 and S4), the
maximum stress values are obtained at the layers adjacent to
the outermost layers (Figs 5b, 5d).
Fig 5a: XY shear stress for S1 (G/E)
Fig 5b: XY shear stress for S2 (G/E)
Fig 5c: XY shear stress for S3 (G/E)
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Fig 5d: XY shear stress for S4 (G/E)
The distributions of the maximum stress values obtained
along the X axis or longitudinal direction for Kevlar-Epoxy
(K/E) laminates are shown in Figs 6a-6d. The stress distribution
profiles are similar to those of the Graphite-Epoxy (G/E)
laminates. Stress profiles of balanced plies of S1 and S3
arrangements appear similar, but with a higher range in the case
of S3 (Figs 6a, 6c).
Fig 6a: X-direction stress for S1 (K/E)
Fig 6b: X-direction stress for S2 (K/E)
Fig 6c: X-direction stress for S3 (K/E)
Fig 6d: X-direction stress for S4 (K/E)
The stress profiles of arrangements S2 and S4 (Figs 6c, 6d)
are different compared to similar configurations for G/E
laminates (Figs 3c, 3d). In this case, for S4 stress profile is
more evenly distributed among the layers compared to that of
S2.
The distribution of the maximum stress values obtained
along the Y axis or transverse direction for Kevlar-Epoxy (K/E)
laminates are shown in Figs 7a-7d. These profiles are also
inverted compared to those of the X direction stress profiles for
similar corresponding stacking arrangements. The stresses are
carried by the inner layers, similar to the results obtained for
the G/E laminates.
The Y direction stresses for K/E laminates for
arrangements S2 and S4 are also similar to the G/E laminate
results. The stress profile for S2 is bell shaped showing a more
even distribution compard to that of S4 (Figs 7b, 7d).
7 Copyright © 2014 by ASME
Fig 7a: Y-direction stress for S1 (K/E)
Fig 7b: Y-direction stress for S2 (K/E)
Fig 7c: Y-direction stress for S3 (K/E)
Fig 7d: Y-direction stress for S4 (K/E)
The distribution of the maximum in-plane (XY) shear
stress values obtained along the for Graphite-Epoxy (G/E)
laminates are shown in Figs 8a-8d. Similar to the G/E
laminates, the stresses are mainly carried only by the inner
layers. Also for the balanced stacking arrangements of S1 and
S3, the inner layers develop stresses along opposite directions
with the almost the same magnitudes. For the unbalanced
laminates (S2 and S4), the maximum stress values are obtained
at the layers adjacent to the outermost layers.
Fig 8a: XY-shear stress for S1 (K/E)
Fig 8b: XY-shear stress for S2 (K/E)
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Fig 8c: XY-shear stress for S3 (K/E)
Fig 8d: XY-shear stress for S4 (K/E)
In Figs 9a-9d, contour plots of total deformation of the G/E
laminate for different stacking sequences are shown. For
arrangements S1 and S2, the deformation profiles from low to
high are rotated clockwise towards the bottom right of the plate
(Figs 9a, 9b). For S2 and S4, similar behavior is observed (Figs
9c, 9d).
Fig 9a: Total deformation contours for S1 (G/E)
Fig 9b: Total deformation contours for S2 (G/E)
Fig 9c: Total deformation contours for S3 (G/E)
Fig 9d: Total deformation contours for S4 (G/E)
CONCLUSION A finite element computational model for predicting the
structural properties of multi-ply composite laminates has been
developed using ANSYS. Different materials with orthotropic
properties and various ply stacking configurations were chosen
for modeling the fiber-reinforced composites. The results were
found to be in good agreement with an analytical model based
on thin plate theory. In the next phase of this work, further
investigations will be carried out with loading conditions such
as flexural loads and fatigue or cyclic loading. The thin plate
9 Copyright © 2014 by ASME
approximation will also be tested for other geometric
configurations such as curved shells and surface features such
as stress risers.
REFERENCES [1] Roylance, D., 2000., Laminated Composite Plates.,
MIT., Cambridge, USA, pp 1-17.
[2] Zhang, Y.X. and Yang, C.H., 2009, “Recent
Developments in Finite Element Analysis for Laminated
Composite Plates,” Composite Structures, 88, pp. 147-157.
[3] Reddy, J.N. and Robbins, D.H., 1994, “Theories And
Computational Models for Composite Laminates,” Appl
Mech Rev, 47, pp. 147–69.
[4] Liu, D.S. and Li, X.Y., 1996, “An Overall View of
Laminate Theories Based on Displacement Hypothesis,” J
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[5] Altenbach, H., 1998, “Theories for Laminated and
Sandwich Plates, A Review,” Mech Compos Mater, 34(3),
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[6] Ghugal, Y.M. and Shimpi, R.P., 2001, “A Review of
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[8] Kant T, Swaminathan K., 2000, “Estimation of
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[9] Barbero, E.J., 2007, Finite Element Analysis of
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[11] Staab, G, 1999, Laminar Composites, Butterworth
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[12] Reddy, J. N., 2007, Theory and analysis of elastic
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[14] SAS IP Inc, 2012, “ANSYS Mechanical APDL
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USA.
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