effect of ply stacking sequence on structural response of

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


FNU Aktaruzzaman Department of Mechanical Engineering


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)


[ 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.


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


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)


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)


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).


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)


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


approximation will also be tested for other geometric

configurations such as curved shells and surface features such

as stress risers.

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[3] Reddy, J.N. and Robbins, D.H., 1994, “Theories And

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effect of ply stacking sequence on structural response of

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