Design and Analysis of Laminated Composite Materials

Design and Analysis of Laminated Composite Materials
Design and Analysis of Laminated Composite
Materials
Thesis submitted in partial fulfillment of the requirements for the Degree of
Bachelor of Technology (B. Tech.)
In
Mechanical Engineering
By
BHAGYASHREE SUNA
Roll No. 110ME0335
Under the Guidance of
Prof. J. Srinivas
NATIONAL INSTITUTE OF TECHNOLOGY
ROURKELA 769008, INDIA
0
NATIONAL INSTITUTE OF TECHNOLOGY
ROURKELA 769008, INDIA
Certificate of Approval
This is to certify that the thesis entitled Design and Analysis of Laminated Composite
Materials submitted by Miss Bhagyashree Suna has been carried out under my supervision in
partial fulfillment of the requirements for the Degree of Bachelor of Technology in Mechanical
Engineering at National Institute of Technology, NIT Rourkela, and this work has not been
submitted elsewhere before for any other academic degree/diploma.
-----------------------------------------Dr. J. Srinivas
Associate Professor
Department of Mechanical Engineering
National Institute of Technology, Rourkela
Rourkela-769008
Date:
1
Acknowledgement
I would like to express my deep sense of gratitude and indebtedness to Dr. J. Srinivas, Associate
Professor, Department of Mechanical Engineering, NIT Rourkela, my supervisor, whose
invaluable encouragement, suggestions, and support leads to make successful completion of the
thesis work. His meticulous guidance at each phase of this thesis has inspired and helped me
innumerable ways.
I would also like to show my sincere thanks to Prof. K. P. Maity, Professor and Head of the
Department, Mechanical Engineering; I offer my thanks to all my friends and especially those
involved in helping me to give basic ideas in writing graphic user interfaces and use of ANSYS
software in laminated composite analysis. I am grateful to my parents and all my teachers who
indirectly helped me through-out.
Bhagyashree Suna
2
Abstract
Composite materials have interesting properties such as high strength to weight ratio, ease of
fabrication, good electrical and thermal properties compared to metals. A laminated composite
material consists of several layers of a composite mixture consisting of matrix and fibers. Each
layer may have similar or dissimilar material properties with different fiber orientations under
varying stacking sequence. There are many open issues relating to design of these laminated
composites. Design engineer must consider several alternatives such as best stacking sequence,
optimum fiber angles in each layer as well as number of layers itself based on criteria such as
achieving highest natural frequency or largest buckling loads of such structure. Analysis of such
composite materials starts with estimation of resultant material properties. Both classical theory
and numerical methods such as finite element modeling may be employed in this line. Further,
these estimated properties are to be used for computing the dynamic properties of the members
made-up of these materials as equivalent isotropic members. At this level, a Graphic User
Interface (GUI) device is developed with MATLAB programming to interactively create a user
friendly environment for computing overall material properties using classical laminate theory.
User can enter the number of layers and layer orthotropic properties and the back end program
calculates the extension, bending and coupling stiffness matrices and further it estimates the
overall elastic constants, Poisson ratios and density. The result will be displayed in the front end
interface boxes. The obtained constants are validated with an ANSYS model, where the laminate
stacking sequence is built and the member is subjected to a uniform strain at free end, while the
reaction stress at the fixed end is predicted. The developed interface simplifies the design process
to some extent. The dynamic analysis in terms of fundamental natural frequency and critical
buckling load is illustrated by using these overall material constants as a later part of analysis.
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Contents
Content
Page
1.Introduction
1.1 Laminated composite Structures
1.2 Literature Review
1.3 Objectives
5
7
14
2. Mathematical Modeling
2.1 Generalized Hook‟s Law
2.2 Force, moment relations
2.3 Effective elastic constants
2.4 FEM of laminated composite
2.5 Dynamic analysis
16
17
19
21
23
3. Methodology
3.1 Development of GUI
3.2 Overall material properties using ANSYS
3.3 Calculation of natural frequencies
26
27
29
4. Results & Discussion
4.1 Graphical User Interface (GUI) functionality
4.2 Development of finite element model in ANSYS
4.3 Natural frequencies and Critical Buckling loads
30
32
34
5. Conclusions
35
References
Appendix-A MATLAB program developed
Appendix-B Use of Shell elements in ANSYS
36
39
50
4
Chapter 1
Introduction
Laminated composite materials are extensively used in aerospace, defense, marine, automobile,
and many other industries. They are generally lighter and stiffer than other structural materials. A
laminated composite material consists of several layers of a composite mixture consisting of
matrix and fibers. Each layer may have similar or dissimilar material properties with different
fiber orientations under varying stacking sequence. Because, composite materials are produced
in many combinations and forms, the design engineer must consider many design alternatives. It
is essential to know the dynamic and buckling characteristics of such structures subjected to
dynamic loads in complex environmental conditions. For example, when the frequency of the
loads matches with one of the resonance frequencies of the structure, large translation/torsion
deflections and internal stresses occur, which may lead to failure of structure components. The
structural components made of composite materials such as aircraft wings, helicopter blades,
vehicle axles and turbine blades can be approximated as laminated composite beams.
1.1 Laminated Composite Structures
A laminate is constructed by stacking a number of laminas in the thickness (z) direction. Each
layer is thin and may have different fiber orientation. The fiber orientation, stacking
arrangements and material properties influence the response from the laminate. The theory of
lamination is same whether the composite structure may be a plate, a beam or a shell. Fig.1.1
shows a laminated plate or panel considered in most of the analysis. The following assumptions
are made in formulations: (i) The middle plane of the plate is taken as the reference plane. (ii)
The laminated plate consists of arbitrary number of homogeneous, linearly elastic orthotropic
layers perfectly bonded to each other. (iii) The analysis follows linear constitutive relations i.e.
5
obeys generalized Hooke's law for the material. (iv)The lateral displacements are small
compared to plate thickness. (v) Normal strain in z-direction is neglected.

y
z
x
Fig.1.1 Plate
As shown in Fig.1.2, laminated beams are made-up of many plies of orthotropic materials and
the principal material axes of a ply may be oriented at an arbitrary angle with respect to the xaxis. In the right-handed Cartesian coordinate system, the x-axis coincides with the beam axis
and its origin is on the mid-plane of the beam. The length, breadth and thickness of the beam are
represented by L, b and h, respectively.
y
z
x
h
L
Fig.1.2 Beams
In practical engineering applications, laminated shells of revolution may have different
geometries based mainly on their curvature characteristics such as cylindrical shells, spherical
6
shells and conical shells. The composite shell of revolution is composed of orthotropic layers of
uniform thickness as shown in Fig.1.3. A differential element of a laminated shell shown with
orthogonal curvilinear coordinate system located on the middle surface of the shell. The total
thickness of the shell is h.
z
x
z


x
Fig.1.3 Shell (cylindrical)
1.2 Literature Review
This section brief-outs the various earlier works done in the area of laminated composite
material. These are grouped under four broad headings. More recently, Hajianmaleki[1]
presented a review of analysis of laminated composite structures used in recent decades.
Laminated Beams
Many authors analyzed the laminated beam structures.
Yildirim [2] used stiffness method for the solution of the purely in-plane free vibration problem
of symmetric cross-ply laminated beams. The rotary inertia, axial and transverse shear
deformation effects are considered in the mathematical model by the first-order shear
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deformation theory. A total of six degrees of freedom, four displacements and two rotations are
defined for an element. The exact in-plane element stiffness matrix of 6×6 is obtained based on
the transfer matrix method. The element inertia matrix consists of the concentrated masses. The
sub-space iteration and Jacobi‟s methods are employed in the solution of the large-scale general
eigenvalue problem.
Jun et al. [3] introduced a dynamic finite element method for free vibration analysis of generally
laminated composite beams on the basis of first-order shear deformation theory. The influences
of Poisson effect, couplings among extensional, bending and torsional deformations, shear
deformation and rotary inertia are incorporated in the formulation. The dynamic stiffness matrix
is formulated based on the exact solutions of the differential equations of motion governing the
free vibration of generally laminated composite beam.
Gurban and Gupta [4] analyzed the natural frequencies of composite tubular shafts using
equivalent modulus beam theory (EMBT) with shear deformation, rotary inertia and gyroscopic
effects has been modified and used for the analysis. The modifications take into account effects
of stacking sequence and different coupling mechanisms present in composite materials. Results
obtained have been compared with that available in the literature using different modeling. The
close agreement in the results obtained clearly show that, in spite of its simplicity, modified
EMBT can be used effectively for rotor-dynamic analysis of tubular composite shafts.
Yegao et al.[5] presented a general formulation for free and transient vibration analysis of
composite laminated beams with arbitrary lay ups and any boundary conditions. A modified
variational principle combined with a multi-segment partitioning technique is employed to derive
the formulation based on a general higher order shear defomation theory. The material coupling
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for bending-stretching, bending-twist, and stretching twist as well as the poison‟s effect are taken
into account.
Shell Structures
Qu et al. [6] introduced a variational formulation for predicting the free, steady-state and
transient vibrations of composite laminated shells of revolution subjected to various
combinations of classical and non-classical boundary conditions. A modified variational
principle in conjunction with a multi-segment partitioning technique was employed to derive the
formulation based on the first-order shear deformation theory.
Xiang et al.[7] studied a simple yet accurate solution procedure based on the Haar wavelet
discretization method (HWDM) is applied to the free vibration analysis of composite laminated
cylindrical shells subjected to various boundary conditions. The Reissner–Naghdi‟s shell theory
is adopted to formulate the theoretical model. The initial partial differential equations (PDE) are
first converted into system of ordinal differential equations by the separation of variables. Then
the discretizations of governing equations and corresponding boundary conditions are
implemented by means of the HWDM, which leads to a standard linear eigenvalue problem.
Plates
Sahoo and Singh [8] proposed a new trigonometric zigzag theory for the static analysis of
laminated composite and sandwich plates. This theory considers shear strain shape function
assuming the non-linear distribution of in-plane displacement across the thickness. It satisfies the
shear-stress-free boundary conditions at top and bottom surfaces of the plate as well as the
continuity of transverse shear stress at the layer interfaces obviating the need of an artificial
shear correction factor.
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Rarani et al. [9] used analytical and finite element methods for prediction of buckling behavior,
including critical buckling load and modes of failure of thin laminated composites with different
stacking sequences. A semi-analytical Rayleigh–Ritz approach is first developed to calculate the
critical buckling loads of square composite laminates with SFSF (S: simply-support, F: free)
boundary conditions. Then, these laminates are simulated under axially compression loading
using the commercial finite element software, ABAQUS. Critical buckling loads and failure
modes are predicted by both eigenvalue linear and nonlinear analysis.
Alnefaie [10] developed a 3D-FE model of delaminated fiber reinforced composite plates to
analyse their dynamics. Natural frequencies and modal displacements are calculated for various
case studies for different dimensions and delamination characteristics. Numerical results showed
a good agreement with available experimental data. A new proposed model shows enhancement
of the accuracy of the results.
Sino et al. [11] worked on the dynamic instability of an internally damped rotating composite
shaft. A homogenized finite element beam model, which takes into account internal damping, is
introduced and then used to evaluate natural frequencies and instability thresholds. The influence
of laminate parameters: stacking sequences, fiber orientation, transversal shear effect on natural
frequencies and instability thresholds of the shaft are studied. The results are compared to those
obtained by using equivalent modulus beam theory (EMBT), modified EMBT and layerwise
beam theory (LBT).
Optimization issues and dynamics
Topal[12] presented a multiobjective optimization of laminated cylindrical shells to maximize a
weighted sum of the frequency and buckling load under external load. The layer fiber orientation
is used as the design variable and the multi-objective optimization is formulated as the weighted
10
combinations of the frequency and buckling under external load. The first order shear
deformation theory is used for the finite element formulation of the laminated shells. Five shell
configurations with eight layers are considered as candidate designs. The modified feasible
direction method (MFD) is used as optimization routine. Finally, the effect of different weighting
ratios, shell aspect ratio, shell thickness-to-radius ratios and boundary conditions on the optimal
designs is investigated and the results are compared.
Topal and Uzman[13] proposed a multiobjective optimization of symmetrically angle-ply square
laminated plates subjected to biaxial compressive and uniform thermal loads. The design
objective is the maximization of the buckling load for weighted sum of the biaxial compressive
and thermal loads. The design variable is the fiber orientations in the layers. The performance
index is formulated as the weighted sum of individual objectives in order to obtain optimal
solutions of the design problem. The first-order shear deformation theory (FSDT) is used in the
mathematical formulation of buckling analysis of laminated plates.
Roos and Bakis [14] analysed the flexible matrix composites which consist of low modulus
elastomers such as polyurethanes which are reinforced with high-stiffness continuous fibers such
as carbon. This fiber–resin system is more compliant compared to typical rigid matrix
composites and hence allows for higher design flexibility. Continuous, single-piece FMC
driveshafts can be used for helicopter applications. Authors employed an optimization tool using
a genetic algorithm approach to determine the best combination of stacking sequence, number of
plies and number of in-span bearings for a minimum-weight, spinning, and misaligned FMC
helicopter driveshaft. In order to gain more insight into designing driveshafts, various loading
scenarios are analyzed and the effect of misalignment of the shaft is investigated. This is the first
time that a self-heating analysis of a driveshaft with frequency- and temperature-dependent
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material properties is incorporated within a design optimization model. For two different
helicopter drivelines, weight savings of about 20% are shown to be possible by replacing
existing multi-segmented metallic drivelines with FMC drivelines.
Sadr and Bargh [15] studied the fundamental frequency optimization of symmetrically laminated
composite plates using the combination of Elitist- Genetic algorithm(E-GA) and finite strip
method(FSM). The design variables are the number of layers, the fiber orientation angles, edge
conditions and plate length/width ratios.
Kayikci and Sonmez [16] studied and optimized the natural frequency response of symmetrically
laminated composite plates. An analytical model accounting for bending–twisting effects was
used to determine the laminate natural frequency. Two different problems, fundamental
frequency maximization and frequency separation maximization, were considered. Fiber
orientation angles were chosen as design variables. Because of the existence of numerous local
optimums, a global search algorithm, a variant of simulated annealing, was utilized to find the
optimal designs. Results were obtained for different plate aspect ratios. Effects of the number of
design variables and the range of values they may take on the optimal frequency were
investigated. Problems in which fiber angles showed uncertainty were considered. Optimal
frequency response of laminates subjected to static loads was also investigated.
Khandan et al.[17] researched and added an extra term to the optimisation penalty function in
order to consider the transverse shear effect. This modified penalty function leads to a new
methodology whereby the thickness of laminated plate is minimised by optimizing the fiber
orientations for different load cases. Therefore the effect of transverse shear forces is considered
in this study.
12
Montagnier and Hochard [18] studied the optimisation of hybrid composite drive shafts
operating at subcritical or supercritical speeds, using a genetic algorithm. A formulation for the
flexural vibrations of a composite drive shaft mounted on viscoelastic supports including shear
effects is developed. In particular, an analytic stability criterion is developed to ensure the
integrity of the system in the supercritical regime. Then it is shown that the torsional strength can
be computed with the maximum stress criterion. A shell method is developed for computing
drive shaft torsional buckling. The optimisation of a helicopter tail rotor driveline is then
performed. This study yielded some general rules for designing an optimum composite shaft
without any need for optimisation algorithms.
Rocha et al. [19] presented a genetic algorithm combining two types of computational
parallelization methods, resulting in a hybrid shared/distributed memory algorithm based on the
island model using both Open MP and MPI libraries. In order to take further advantage of the
island configuration, different genetic parameters are used in each one, allowing the
consideration of multiple evolution environments concurrently. To specifically treat composite
structures, a three-chromosome variable encoding and special laminate operators are used. The
resulting gains in execution time due to the parallel implementation allow the use of high fidelity
analysis procedures based on the Finite Element Method in the optimization of composite
laminate plates and shells. Two numerical examples are presented in order to assess the
performance and reliability of the proposed algorithm.
Abadi and Daneshmehr [20] developed the buckling analysis of composite laminated beams
based on modified coupled stress theory. By applying principle of minimum potential energy and
considering two different beam theories ,i.e, Euler-Bernouli and Timoshinko beam theories,
13
governing equations, boundary and initial conditions are derived for micro composite laminated
beam.
Apalak et al. [21] carried-out the layer optimization for achieving maximum fundamental
frequency of laminated composite plates under any combination of the three classical edge
conditions. The optimal stacking sequences of laminated composite plates were searched by
means of Genetic Algorithm. The first natural frequencies of the laminated composite plates with
various stacking sequences were calculated using the finite element method. Genetic Algorithm
maximizes the first natural frequency of the laminated composite plate defined as a fitness
function (objective function).
In addition to above, numerous conference articles and textbooks emphasize the analysis issues
of composite laminated structures.
1.3 Objectives
Based on the above literature, it is observed that enormous works concentrated on the classical
lamination theory along with micro-mechanics models for material analysis. There is a necessity
to develop interactive software which generalizes the analysis procedure at least in computing
the elastic properties of overall laminated composite material. The following are the objectives
of the present work:
1. Develop computer program to predict the overall elastic properties of multilayer composite
material of given number of layers, stacking sequence and elastic constants of each layer.
2.Validate the elastic data with finite element modeling. 3. Implement the program in a graphic
user interface. 4. Using equivalent modulus beam theory, estimate the natural frequencies and
critical buckling load of beams.
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The organization of the thesis is as follows: Chapter-2 gives mathematical modeling including
classical lamination theory, finite element modeling and concept of equivalent modulus beam
theory to compute natural frequencies and buckling loads. Chapter-3 presents a brief
methodology of the present work. Chapter-4 illustrates the approach with numerical examples.
Brief conclusions and scope for future work is presented in chapter-5.
-----
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Chapter2
Mathematical Modeling
This chapter presents the mathematical analysis used in conventional classical lamination theory
and other analysis issues.
2.1 Hooke’s Law
Generalized Hooke‟s law for orthotropic material is given by:
{}=[Q]{}
(2.1)
[Q] is called material stiffness matrix. For plane stress conditions, we can write for each layer:
 11   Q11 Q12
  Q
Q 22
 22   12
0
 23    0
   0
0
 13  
 12   0
0
0
0
0
0
Q 44
0
0
Q 55
0
0
0   11 
0   22 
  
0   23 

0   13 
 
Q 66   12 
The elastic constants in the principal material coordinate system are: Q11=
Q12=
 12 E 2
,
1   12 21
Q22=
E2
,
1   12 21
(2.2)
E1
,
1   12 21
Q44=G23, Q55=G13, Q66=G12.
Here, E1, E2, G12, G23, G13 and 12 are engineering parameters of the nth layer (lamina) in the
laminate obtained from rule of mixtures. The transformed stress-strain relations for each lamina
can be written as:
 xx   Q11 Q12
  
 yy   Q 21 Q 22
0
  yz    0
   0
0
 xz  
 xy   Q16 Q 26
0
0
0
0
Q 44
Q 45
Q 45
Q55
0
0
Q16   xx 
 
Q 26   yy 
 
0   yz 

0   xz 
 
Q66   xy 
(2.3)
The transformed reduced stiffness terms are given as follows:
16
c 2

[Q]  [T] [Q][T] , where [T]= s 2
 cs

1
1
 2cs 

c
2cs  with c=cos() and s=sin(). Also  is fiber
 cs (c 2  s 2 )

s2
2
orientation angle for kth lamina with respect to x-axis of the beam.
2.2 Force, moment relations
Fiber-reinforced composite consisting of multiple layers of material is called laminate. Each
layer is thin and may have a different fiber orientation. Two laminates may have the same
number of layers and the same fiber angles but the two laminates may be different because of the
arrangement of the layers. Figure 2.1 shows a global Cartesian coordinate system and a general
laminate consisting of N layers.
z
Nth layer
x
hk
y
zk-1
zk
h
y
z
L
1st layer
Fig.2.1 Geometry and coordinates of laminate
The laminate thickness is denoted by h and the thickness of a kth layer is hk. The origin of the
thickness coordinate, designated z, is located at the laminate geometric mid-plane. The geometric
midplane may be within a particular layer or at an interface between layers. The laminate extends
in the z direction from −h/2 to +h/2. The layer at the most negative location is as layer 1, the next
layer in as layer 2, the layer at an arbitrary location is layer k, and the layer at the most positive z
position is layer N. The locations of the layer interfaces are denoted by a subscripted z; The first
17
layer is bounded by locations z0 and z1, the second layer by z1 and z2, the kth layer by zk−1 and zk,
and the Nth layer by zN−1 and zN.
The important assumption of classical lamination theory is that each point within the volume of a
laminate is in a state of plane stress. Therefore, stresses can be computed if we know the strains
and curvatures of the reference surface. Given the force and moment resultants, we want to
calculate the stresses and strains through the thickness as well as the strains and curvatures on the
reference surface. We also want to do this by computing the laminate stiffness matrix. The force
resultants Nx, Ny, and Nxy can be shown to be related to the mid-plane strains 0 and curvatures 0
at the reference surface by the following equation:
 N x   A11 A12

 
 N y   A12 A 22
 N  A
 xy   16 A 26
A16    x0   B11
 
A 26    y0    B12


0 

A 66  

 xy   B16
B12
B 22
B 26
B16    x0 


B 26    y0 

0 
B66  
 xy 
(2.4)
Similarly, the moment resultants Mx, My, and Mxy can also be shown to be related to the strains
and curvatures at the reference surface by the following equation:
 M x   B11

 
 M y    B12
M   B
 xy   16
B12
B 22
B 26
B16    x0   D11
 
B 26    y0    D12


0 

B66  

 xy   D16
D12
D 22
D 26
D16    x0 
 
D 26    y0 

0 
D 66  
 xy 
(2.5)
where matrix [A], [B] and [D] are given by
A ij   ( Qij ) k z k  z k 1 
N
(2.6)
k 1
Bij 
D ij 

1 N
2
2
 ( Qij ) k z k  z k 1
2 k 1

1 N
3
3
 ( Qij ) k z k  z k 1
3 k 1


(2.7)
(2.8)
The eqs. can be combined as
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  x0 
 Nx 
 0
N 
y 
 y 
0 
 N xy  [ A] [ B]  xy



 
 0 
M
[
B
]
[
D
]

 x 
 x 
 My 
  y0 


 0 
 M xy 
 xy 
(2.9)
Note that the inverse of [A] matrix relates strains with forces according to {}=[a]{N}, where
[a]=[A]-1. A laminate is symmetric if for every layer to one side of the laminate reference surface
(with a specific thickness, specific material properties and specific fiber orientation), there is
another layer at the same distance on the opposite side of the reference surface (with the same
thickness, material properties and fiber orientation). If the laminate is not symmetric, then it is
referred to as an unsymmetric laminate. For a symmetric laminate, all the elements of the [B]
matrix are identically zero. In engineering design, the stacking sequence of laminated plates is
designed to be symmetric and balanced to avoid unpredictable warp deflections.
2.3 Effective elastic constants
We assume that the effective material properties of each ply can be expressed in terms of a
micromechanical model using rule of mixture:
E11=VfE11f+VmEm
f
f
Vf
 2f E m / E 22
  m2 E 22
/ E m  2 f  m
Vm
1
 f  m  V f Vm
f
E 22 E 22 E
V f E 22  Vm E m
(2.10)
(2.11)
Vf
V
1
 f  mm (ij=12,13 and 23)
Gij Gij G
(2.12)
ν12=Vf νf+Vm νm
(2.13)
ρ=Vf ρf+Vm ρm
(2.14)
where E11f, E22f, G12f, G13f, G23f, νf, and ρf are the Young‫׳‬s moduli, shear moduli, Poisson‫׳‬s ratio
and mass density, respectively, of the fiber, while Em, Gm, νm and ρm are the corresponding
19
properties for the matrix, respectively. V f and V m are the fiber and matrix volume fractions and
must satisfy the unity condition of V f+V m=1.
The concept of effective elastic constants for the laminate is useful to idealize the system as
equivalent isotropic material. These constants are the effective extensional modulus in the x
direction E x , the effective extensional modulus in the y direction E y , the effective Poisson‟s
ratios  xy and  yx and the effective shear modulus in the x-y plane G xy . The effective elastic
constants are usually defined when considering the in-plane loading of symmetric balanced
laminates. The following three average laminate stresses are defined:

1 h /2
 x dz
h  h /2
(2.15)
y 
1 h /2
 y dz
h  h /2
(2.16)
 xy 
1 h /2
 xy dz
h  h /2
(2.17)
Using the above relations we obtain the relation between the average stresses and resultant forces
as follows:-
x 
1
Nx
h
(2.18)
y 
1
Ny
h
(2.19)
 xy 
1
N xy
h
(2.20)
Hence, the laminate compliance matrix of 3×3 size can be defined as follows [20]:-
 x0   a h a h
0   x 
11
12
 0  
 
0   y 
 y    a21h a22 h
 0  0
0
a66 h   xy 
 xy  
 
(2.21)
20
By comparing with stress-strain relations, we obtain the elastic constants for the laminate as
follows:Ex 
1
a11h
Ey 
1
Gxy 
a22 h
1
a66 h
(2.22)
(2.23)
(2.24)
 xy  
a12
a11
(2.25)
 yx  
a12
a22
(2.26)
We can observe that  yx and  xy are not independent of each other and their reciprocity relation
can be shown as below:
 xy  Ex
 yx
EY
(2.27)
2.4 Finite element modeling of laminated composite
Finite element model of composite laminated structure discretizes the entire thickness along the
linear direction into number of elements. Often 2D-modeling is sufficient for getting accurate
results. The shell elements are the famous 2D discretisation elements. A shell element has nnodes with each node having 6 DOFs. The present work FEM is merely employed for the
verification of elastic constants obtained from classical beam theory. In addition the stresses and
strain at the each layer and the interfaces can be addressed. An eight-nodded quadrilateral C0
continuous isoparametric shell element (SHELL 281) with six-degrees-of-freedom per node (ux,
uy, uz, x, y, z) is employed. The generalized displacements included are expressed as:
21
n
   N i i
i 1
(2.28)
where, Ni is the shape function associated with the node i and n is the number of nodes per
element, which is eight in present study. The strain vector {ε} can be expressed in terms of δ
containing nodal degrees of freedom as,
{εbk}=[Bbk]{δ}
{γsk}=[Bsk]{δ},
(2.29)
where [B] is the strain–displacement matrix in the Cartesian coordinate system. The [B] matrix
can be divided in two parts, one which contains the bending terms and other containing the shear
terms. ANSYS SHELL281 is suitable for analyzing thin to moderately-thick shell structures.
SHELL281 may be used for layered applications for modeling composite shells or sandwich
construction. The accuracy in modeling composite shells is governed by the first-order sheardeformation theory (usually referred to as Mindlin-Reissner shell theory). Fig.2.2 shows the
element description.
Fig.2.2 Eight node SHELL element
The element is defined by shell section information and by eight nodes (I, J, K, L, M, N, O and
P). The shell section commands allow for layered shell definition. Options are available for
22
specifying the thickness, material, orientation and number of integration points through the
thickness of the layers. You can designate the number of integration points (1, 3, 5, 7, or 9)
located through the thickness of each layer when using section input. SHELL281 includes the
effects of transverse shear deformation.
2.5 Dynamic analysis
Modal analysis of composite structure will be carried out as well as the natural frequencies and
mode shapes of homogenized equivalent isotropic characteristics with different boundary
conditions are obtained. For a beam structure, we can write axial and transverse displacements as
u  u0  z
v  z
w  w0
(2.30)
Here,  and  are rotations of normal to mid-plane. As for a beam Ny=Nxy=My=0, the
constitutive equations of laminate from classical laminate theory becomes
 N x   A11

 
 M y    B11

 D
 M xy   16
B11
D11
D16
0
B16   x 



D11   x0 
D66   xy0 
 
(2.31)
Transverse shear force / unit length
Qxz  A55 xz
(2.32)
h/2
where A55  k  h / 2 Q55 dz with k as shear correction factor and Q55  G13 cos  G23 sin  .
Total strain energy of beam
V
1 L
( N x x0  M x x  M xy xy  Qxz  xz )bdz
2 0

1 L  
=   EI  y
2  0  x


2
2
2
2

L


   
 v
 

 w
 
   z  dx   GA 





dx




z
y



 x
 
 x  
0


 x
 

(2.33)
23
where,
N
 Ri4  Ri41 


4


EI=  E xi  
i 1
(2.34)
N
GA=  Gxyi  ( Ri2  Ri21 )
(2.35)
i 1
are the homogenized flexural and torsional rigidities and Ri and Ri-1 are external and internal
radius of layer i.
Total kinetic energy is
T
1 L h/2
y
v
w
[( )2  ( )2  ( )2 ]bdzdx


0

h
/
2
2
t
t
t
=   Av 2  w 2  I D y2  z2   I P  2   z y  y z dx 
2
1 L

0

(2.36)
Here, ID and IP are transverse and polar mass moments of inertia of the shaft. A simply supported
composite shaft is shown in Fig.2.3.
Fig.2.3 Laminated composite simply-supported beam
In present work, the fundamental natural frequency of simply supported beam is computed for
only Euler-Bernouli beam model as follows:
 n2 
 n2 
where  2 =
 2 n 4 4
2
- simply supported beam
 2 (  n ) 4
2
-- cantilever beam
(2.37)
EI
with E= E x obtained from effective elastic constant result. If first natural
A
frequency is required n=1 and l=1.8751. Also, I and A are moment of inertia and cross-section.
24
Chapter 3
Methodology
The following methodology is being adopted to carry out the above mentioned objectives:
1. An interactive interface is created using GUI in MATLAB to compute the overall
laminate properties of the composite.
2. Using ANSYS the overall material properties are computed and tried to validate with
classical theory.
3. Using these equivalent properties of the composite the natural frequency computations
are done.
Fig.3.1 shows present methodology.
Enter each layer data
Compute all stiffness matrices and elastic
constants
Develop the finite element model of n layers
in ANSYS and find elastic constants
No
Is
coincidence?
Yes
Find
dynamic
properties
stop
Fig.3.1 Flowchart of present methodology
25
3.1 Development of GUI
The steps employed for the calculation of overall laminate properties using MATLAB by
creating the GUI are given as follows:
Program Description
The program makes use of the user input elastic properties (E1, E2, G12, NU12) of a single ply
material in its given principal directions (1, 2, and 3) as well as the ply geometry and also tacking
sequence to constructs the [A], [B] and [D] matrices of a laminated fiber-reinforced composite.
Using these [A], [B] and [D] matrices, it finds the overall laminate elastic properties (Ex, Ey,
Gxy, NUxy, etc). A fundamental understanding of composite laminates and its basic theories are
expected in understanding the concepts and results presented.
System requirements
The program needs a computer that has MATLAB installed in it. The M-file
LaminateAnalyzer.m can be opened and executed using the run command which will pop up the
Graphical User Interface. The relevant Matlab code is presented at the appendix.
Program Functionality
Currently the program enclosed is incapable of analyzing symmetric-balanced laminates of
varying thicknesses about the mid-plane. While the laminates with asymmetric stacking
sequences with uneven number of plies can be analyzed with modification of the program.
Variables
Ex , Ey ,E1 , E2 - Young‟s Modulus in the x- , y-, 1- , and 2- directions respectively NUxy , U12
– Poisson‟s ratios for x-y and 1-2 directions Gxy – Shear modulus referred to the x- and y- axes
ETAsx , ETAsy – Shear coupling coefficients. Fig.3.2 shows the GUI environment.
26
Fig. 3.2 Interface environment
3.2 Overall material properties using ANSYS
In order to predict elastic modulus in longitudinal direction, the laminated model is loaded in
axial tension by applying a small normal displacement at one side and fully restraining the other
side. For example, the x = 0 end is constrained in the axial direction (x direction) and free to
move in the lateral directions as shown in Fig.3.3. The free edges are constrained to their
respective normal directions in order to allow contraction of the model due to tension. An axial
displacement, equivalent to the approximate less than 10% of total length of plate, is applied to
all nodes on the end surface (x = L), where L is the length of the laminated structure. The
displacement and reaction forces are calculated on the data collection surfaces (i.e. constraint
27
area). Then, the values of these displacements and reaction forces are employed to evaluate the
effective elastic properties of the composites.
Fig.3.3 Methodology of predicting longitudinal modulus
The reaction forces calculated on constraint surface gives average stress on that surface. A far
field uni-axial displacement is applied to the finite domain only in the longitudinal direction (the
x-direction). The modulus of the laminated composite is estimated using the displacement and
average stress result at data collection surfaces normal to the x-axis by the following formula:
EX 
x
L


 x ( U )avg
where U is the displacement applied and the average value of stress on a surface is given by

1
 x (( z  0), x, y )dxdy
A A
where A is the area of constrained surface at L=0 with the help of FEM results of average stress
can be evaluated for the laminated composite. The laminated structure is subjected to uniform
extension within the linear region of the stress -strain curve.
We use
28
y  
 xy
Ex
 x   xy
L b

L
b
or
 xy 
b / b
L / L
where L and b are the length and width of laminated structure respectively.
3.3 Calculation of natural frequencies
Then the static and dynamic analysis of the composite is done as follows:
The dynamic analysis of laminated composite rotating shaft with the given fiber orientations and
different boundary fixations, are investigated analytically. The equivalent modulus beam theory
is employed for simply-supported and cantilever boundary conditions. The equivalent stiffness
and mass of laminated composite laminate that have been derived by using the lamination theory
is used here to compute the natural frequencies.
In equivalent modulus beam theory (EMBT), the equivalent longitudinal and in-plane
shear modulii are determined using Classical Laminate Theory (CLT). These modulii are used to
calculate shaft natural frequencies using beam theory in the same manner as that for isotropic
shafts. Following are limitations of this theory: In multilayered composite shaft, different layers
(plies) have different contributions to the overall stiffness of the shaft depending on their
locations from the mid-plane. For unbalanced configuration, shear–normal and bending–twisting
couplings are present. These couplings affect significantly shaft natural frequencies. However,
these effects are not incorporated in EMBT formulation. In unsymmetric configurations,
bending–stretching coupling is present which affects the shaft natural frequencies. This effect is
also not included in the EMBT formulation.
29
Chapter 4
Results and Discussion
This chapter deals with the output results for some existing cases in literature.
4.1 Graphical User Interface (GUI) functionality
In order to illustrate the methodology, we considered two cases of laminates. In the first case,
carbon-epoxy with 8 layers having a stacking sequence of [0/45/-45/90]s is considered.
Inputs:
Units can be selected based on user requirement whether in GPa or Psi for stress, and mm or
inches for length.
Ply Elastic Properties: E-1, E-2, G-12, and NU-12 (in the relevant units)
Laminate Characteristics:
a. Orientations: enter the angles of orientations of each independent ply starting from the bottom
to top; make sure to separate each angle by a space or a comma.
b. Symmetry: leave this as zero if the laminate is asymmetric or enter the value of symmetry.
Ex: (0/90)
c. Aspect ratio: give the aspect ratio of each independent ply.
d. Total Height: Enter the total height of the laminate (not the ply).
The program computes the stiffness matrices and overall elastic properties based on the relations
given in earlier chapter. Fig.4.1 shows the screen-shot of the GUI where the data is entered
accordingly.
Outputs:
There are 3 different outputs that portray the overall characteristics of the laminate being
analyzed as seen from Fig.4.1.
30
These are: ply data, [A], [B] and [D] matrices and overall elastic properties.
Fig. 4.1 The Carbon/Epoxy example carried out in the GUI Interface
It is seen that last 4 constants are not accounted due to transverse isotropic approximation. Also,
we can observe in the [A] matrix, A16=A26=0 with [B]=0, indicating balanced-symmetric
laminate condition.
Case-2: Boron-epoxy composite
The following properties of the layer are considered:
-------------------------------------------------------------------------------------------------------------------Material
density(kg/m3) E11 (GPa)
E22(GPa) G12(GPa)
12
tply
Narmaco 5505
1965
24.1
0.36
1mm
211
6.9
Fig.4.2 shows the outputs computed from the GUI platform.
31
Fig.4.2 Second example of 10-layered graphite epoxy composite
Obviously longitudinal modulus is 109.164 GPa.
4.2 Development of finite element model in ANSYS
In order to verify the longitudinal modulus obtained from classical lamination theory, a finite
element model is developed in ANSYS software. Fig.4.3 shows the screen shot of material data
entered for case-1 (Carbon-epoxy) and Fig.4.4 shows the corresponding ply data (thickness and
orientation). A plate is created and is meshed with these elements (SHELL 281) and further
applied with a constant displacement at one end and other end is fixed. The reaction forces
(stresses) are computed at fixed end and finally the longitudinal modulus and Poisson ratio are
estimated. Similarly other elastic constants can be computed. As in the present work, we
employed only EMBT, other moduli are not required to validate.
32
Fig.4.3 Screen shot of material data
Fig.4.4 Screen shot showing ply data
Table-1 shows the comparison of the longitudinal modulus obtained from classical theory and
using finite element modeling with ANSYS.
Table-1 Longitudinal Elastic modulus obtained by GUI and through ANSYS
Constant computed
Classical lamination
Finite element model
theory (GUI)
(ANSYS)
Ex (GPa)
69.69
72.43
0.293
0.356
xy
33
4.3 Natural frequencies and Critical Buckling loads
To utilize the longitudinal modulus data for frequency computations, first case (carbon-epoxy) is
only illustrated. Substituting the following values in eqs.(2.37) as: E=Ex=69.69 GPa,
I=bh3/12=0.00000032
kgm4,with
b=0.004m,
h=8
mm,
l=1
m,
ρ=1680
kgm-3
and
A=area=0.00024m2, we get the natural frequencies and crucial buckling load as follows:
n =600 kHz for simply supported beam
n =19.446 kHz
for cantilever beam
The critical buckling load can also be calculated as:
Pcr  EI
2
l2
=249 kN
The fiber orientation in one of the layer of balanced-symmetrical laminate is varied and
corresponding changes in elastic modulus are noted and the fundamental natural frequency
variation of cantilever beam is observed as shown in Fig.4.5.
Effect of fiber orientation in one of the ply
Natural frequency (rad/s)
30000
25000
20000
15000
10
20
30
40
Stacking angle (degrees)
Fig.4.5 Variation of first frequency with fiber orietation
It has drastic effect on natural frequencies. The same thing can be illustrated for case-2 also.
34
Chapter5
Conclusions
In the present work, general classical lamination theory has been employed to predict the
stiffness matrices connecting the forces and strains as well as moments and curvatures. The
methodology was generalized by using a graphic user interface (GUI). At the back of this, the
code employs all the classical relations and finally displays the stiffness matrices as well as the
overall elastic constants of the laminate. As a next step, these values were validated using finite
element modeling in commercial ANSYS software.
The concept of equivalent modulus beam theory introduced early 1990s for EulerBernoulli beams has been employed to obtain the fundamental natural frequencies for two end
conditions. This method involves calculating the eigenvalues of the isotropic Bernoulli beam,
using the longitudinal modulus of the composite material computed with the classical laminate
theory. This theory is applicable for symmetric balanced laminates, but the EMBT approach has
proved to have some limitations in the case of unbalanced and unsymmetrical laminates. The
EMBT does not take into account the ply location relative to the axis when dealing with
multilayered unsymmetrical laminates. Also, EMBT does not take shear–normal coupling in to
account in the case of unbalanced laminates, or bending– stretching and bending–twisting
coupling in that of unsymmetrical laminates. So, as a future scope, a more generalized solution
approach based on finite element modeling can be also planned to display as a contour plot in
GUI.
35
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---
38
APPENDIX-A
MATLAB program developed
Now-a-days, MATLAB graphic user interface (called GUIDE) is one of the popular means of
projecting the front-end data. Here, is the program developed in this work:
function varargout = LaminateAnalyzer(varargin)
%LAMINATEANALYZER M-file for LaminateAnalyzer.fig
%
LAMINATEANALYZER, by itself, creates a new LAMINATEANALYZER or raises the existing
%
singleton*.
%
%
H = LAMINATEANALYZER returns the handle to a new LAMINATEANALYZER or the handle to
%
the existing singleton*.
%
%
LAMINATEANALYZER('Property','Value',...) creates a new LAMINATEANALYZER using the
%
given property value pairs. Unrecognized properties are passed via
%
varargin to LaminateAnalyzer_OpeningFcn. This calling syntax produces a
%
warning when there is an existing singleton*.
%
%
LAMINATEANALYZER('CALLBACK') and LAMINATEANALYZER('CALLBACK',hObject,...) call the
%
local function named CALLBACK in LAMINATEANALYZER.M with the given input
%
arguments.
%
%
*See GUI Options on GUIDE's Tools menu. Choose "GUI allows only one
%
instance to run (singleton)".
%
% See also: GUIDE, GUIDATA, GUIHANDLES
% Edit the above text to modify the response to help LaminateAnalyzer
% Begin initialization code - DO NOT EDIT
gui_Singleton = 1;
gui_State = struct('gui_Name',
mfilename, ...
'gui_Singleton', gui_Singleton, ...
'gui_OpeningFcn', @LaminateAnalyzer_OpeningFcn, ...
'gui_OutputFcn', @LaminateAnalyzer_OutputFcn, ...
'gui_LayoutFcn', [], ...
'gui_Callback', []);
if nargin && ischar(varargin{1})
gui_State.gui_Callback = str2func(varargin{1});
end
if nargout
[varargout{1:nargout}] = gui_mainfcn(gui_State, varargin{:});
else
gui_mainfcn(gui_State, varargin{:});
end
% End initialization code - DO NOT EDIT
% --- Executes just before LaminateAnalyzer is made visible.
function LaminateAnalyzer_OpeningFcn(hObject, eventdata, handles, varargin)
% This function has no output args, see OutputFcn.
% hObject handle to figure
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% varargin unrecognized PropertyName/PropertyValue pairs from the
%
command line (see VARARGIN)
% Choose default command line output for LaminateAnalyzer
handles.output = hObject;
% Update handles structure
39
guidata(hObject, handles);
% UIWAIT makes LaminateAnalyzer wait for user response (see UIRESUME)
% uiwait(handles.figure1);
% --- Outputs from this function are returned to the command line.
function varargout = LaminateAnalyzer_OutputFcn(hObject, eventdata, handles)
% varargout cell array for returning output args (see VARARGOUT);
% hObject handle to figure
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Get default command line output from handles structure
varargout{1} = handles.output;
% --- Executes on button press in commandSubmit.
function commandSubmit_Callback(hObject, eventdata, handles)
% hObject handle to commandSubmit (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
set(handles.tableABD,'Data',eye(6))
% ---Define Inputs
% Ply Properties
E1 = str2num(get(handles.inputE1,'String'));
E2 = str2num(get(handles.inputE2,'String'));
G12 = str2num(get(handles.inputG12,'String'));
NU12 = str2num(get(handles.inputNU12,'String'));
% Symmetry
sym = str2num(get(handles.inputSym,'String'));
% Total Height
htotal= str2num(get(handles.inputTotHeight,'String'));
% Independent ply orientations
O = str2num(get(handles.inputPlyOri,'String'));
% Independent ply Aspect ratios
AR = str2num(get(handles.inputPlyAR,'String'));
% --- Error statements
if (isempty(E1))
errordlg('Please enter a numerical value for E1.','Missing Parameter')
end
if (isempty(E2))
errordlg('Please enter a numerical value for E2.','Missing Parameter')
end
if (isempty(G12))
errordlg('Please enter a numerical value for G12.','Missing Parameter')
end
if (isempty(NU12))
errordlg('Please enter a numerical value for NU12.','Missing Parameter')
end
if (isempty(AR))
errordlg('Please enter values separated by spaces or comas for the Independent Ply Aspect Ratios.','Missing Parameter')
end
if (isempty(O))
errordlg('Please enter values separated by spaces or comas for the Independent Ply Orientations.','Missing Parameter')
end
if (isempty(htotal))
errordlg('Please enter a numerical value for Total Height.','Missing Parameter')
end
% Counting number of rows and columns in AR and O inputs for comparison
%display(AR);
[ARrows,ARcols] = size(AR(1,:));
[orows,ocols] = size(O(1,:));
%Composite Laminate Analyzer Users Manual
% AR and O number count needs to be the same
if ocols~=ARcols
errordlg('Number of entries in the ply orientations field should equal the number of entries in aspect ratios field.','Check Independent
Ply Orientation/Aspect Ratio fields')
end
40
% =====END OF INPUT DATA CHECK=====
% =====START OF PREPROCESSING STATE=====
% --- Developing the complete laminate geometry
% Getting NU21
NU21 = (NU12*E2)/E1 ;
% Considering symmetry input counting the total number plies in laminate
if sym==0
plycount = ocols;
else
plycount= ocols*2*(sym);
end
% Considering symmetry input developing the the bottom half of the laminate
% orientations(O) and aspect ratios(AR)
if sym >= 2
Oini = O;
ARini = AR;
for i= 2:sym
O = [O Oini];
AR = [AR ARini];
end
end
% Applying symmetry(if needed) and developing the complete O and AR array
if sym>0
k=0;
for i = (plycount/2)+1 : plycount
AR(1,i) = AR(1, i-(1+2*k));
O(1,i) = O(1, i-(1+2*k));
k=k+1;
end
end
% Finding the Height(H) of each ply using the AR and total height (htotal)
sumAR = sum(AR);
H = htotal*(AR/sumAR);
% Developing the Z matrix
Z=zeros(1,plycount+1);
hindex = -(htotal)/2 ;
for i=1:plycount+1
if i==1
Z(:,i) = hindex;
else
Z(:,i) = Z(:,i-1)+ H(i-1);
end
% Rounding small values of Z to zero
if abs(Z(:,i)) <1.0e-4
Z(:,i)=0;
end
end
%Composite Laminate Analyzer Users Manual
% =====END OF PREPROCESSING STATE=====
% =====START OF CALCULATIONS=====
% Finding the components of Q matrix in the principal directions
% reference: Engineering mechanics of composites, second edition
% Daniel and Ishai P77, eq. 4.56
Q11 = E1/(1- NU12*NU21) ;
Q22 = E2/(1- NU12*NU21) ;
Q12 = (NU21*E1)/(1- NU12*NU21) ;
Q66 = G12;
% note that Q21 = Q12
Qp = [ Q11, Q12 , 0 ;
Q12, Q22 , 0 ;
0 , 0 , Q66 ];
% Intializing ABD as a 3x3 zero matrix
A = zeros(3,3) ;
B = zeros(3,3) ;
D = zeros(3,3) ;
% Calculating the A, B, D Matrices for each ply
Qp(3,3) = Qp(3,3)*2 ;
for l = 1 : plycount
thetar = (O(l)/180)*pi;
% define "m" and "n" as
41
m = cos(thetar);
n = sin(thetar);
% 2D Transformation matrix T(Daniel and Ishai P76) :
T = [m^2 , n^2 , 2*m*n ;
n^2 , m^2 , -2*m*n ;
-m*n , m*n , m^2 - n^2 ];
% Correcting for engineering strain-true strain(Daniel and Ishai P79)
Q = T\Qp*T ;
Q(:,3) = Q(:,3)*.5 ;
for i=1:3
for j=1:3
A(i,j)= A(i,j) + (Q(i,j))*( Z(l+1) - Z(l) );
B(i,j)= B(i,j) + 0.5* ((Q(i,j))*( (Z(l+1))^2 - (Z(l))^2 ));
D(i,j)= D(i,j) + (1/3)*((Q(i,j))*( (Z(l+1))^3 - (Z(l))^3 ));
end
end
end
% Filter to round close to zero numbers to zero
for i=1:3
for j=1:3
if abs(A(i,j))< 1.0e-4
A(i,j) = 0;
end
if abs(B(i,j))< 1.0e-4
B(i,j) = 0;
end
if abs(D(i,j))< 1.0e-4
D(i,j) = 0;
end
end
%Composite Laminate Analyzer Users Manual
end
%Finding AB-BD matrix and its inverse
ABDmatrix = [A B ; B D];
abcdmatrix = inv(ABDmatrix);
a = abcdmatrix(1:3, 1:3);
% Define matrix with ply orientations as row one and corresponding height
% as row two for display purposes
OHmatrix = [O;H];
% Find Overall laminate elastic properties
Ex = 1/((htotal)*a(1,1));
Ey = 1/((htotal)*a(2,2));
Gxy = 1/((htotal)*a(3,3));
NUxy = -a(2,1)/a(1,1);
NUyx = -a(1,2)/a(2,2);
ETAsx = a(1,3)/a(3,3);
ETAxs = a(3,1)/a(1,1);
ETAys = a(3,2)/a(2,2);
ETAsy = a(2,3)/a(3,3);
% =====END OF CALCULATIONS=====
% =====START OUTPUT RESULTS TO GUI=====
% Display the OHmatrix in GUI table
set(handles.tableOH,'Data',OHmatrix)
% Display the [A B ; B D] matrix in GUI table
set(handles.tableABD,'Data',ABDmatrix)
% Display the overall Laminate properties in Gui
set(handles.outputEx,'String',Ex);
set(handles.outputEy,'String',Ey);
set(handles.outputGxy,'String',Gxy);
set(handles.outputNUxy,'String',NUxy);
set(handles.outputNUyx,'String',NUyx);
set(handles.outputETAxs,'String',ETAxs);
set(handles.outputETAsx,'String',ETAsx);
set(handles.outputETAys,'String',ETAys);
set(handles.outputETAsy,'String',ETAsy);
% =====END OUTPUT RESULTS TO GUI=====
guidata(hObject, handles);
% =====END OF SCRIPT=====
42
function inputPlyOri_Callback(hObject, eventdata, handles)
% hObject handle to inputPlyOri (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of inputPlyOri as text
%
str2double(get(hObject,'String')) returns contents of inputPlyOri as a double
% --- Executes during object creation, after setting all properties.
function inputPlyOri_CreateFcn(hObject, eventdata, handles)
% hObject handle to inputPlyOri (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have a white background on Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
function inputSym_Callback(hObject, eventdata, handles)
% hObject handle to inputSym (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of inputSym as text
%
str2double(get(hObject,'String')) returns contents of inputSym as a double
% --- Executes during object creation, after setting all properties.
function inputSym_CreateFcn(hObject, eventdata, handles)
% hObject handle to inputSym (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have a white background on Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
%% =====START OF INPUT ARGUMENTS=====
% --- Input Ply orientations for each ply
function inputPlyAR_Callback(hObject, eventdata, handles)
% hObject handle to inputPlyAR (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of inputPlyAR as text
%
str2double(get(hObject,'String')) returns contents of inputPlyAR as a double
% --- Executes during object creation, after setting all properties.
function inputPlyAR_CreateFcn(hObject, eventdata, handles)
% hObject handle to inputPlyAR (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have a white background on Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
43
function inputTotHeight_Callback(hObject, eventdata, handles)
% hObject handle to inputTotHeight (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of inputTotHeight as text
%
str2double(get(hObject,'String')) returns contents of inputTotHeight as a double
% --- Executes during object creation, after setting all properties.
function inputTotHeight_CreateFcn(hObject, eventdata, handles)
% hObject handle to inputTotHeight (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have a white background on Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
function inputE1_Callback(hObject, eventdata, handles)
% hObject handle to inputE1 (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of inputE1 as text
%
str2double(get(hObject,'String')) returns contents of inputE1 as a double
% --- Executes during object creation, after setting all properties.
function inputE1_CreateFcn(hObject, eventdata, handles)
% hObject handle to inputE1 (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have a white background on Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
function inputE2_Callback(hObject, eventdata, handles)
% hObject handle to inputE2 (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of inputE2 as text
%
str2double(get(hObject,'String')) returns contents of inputE2 as a double
% --- Executes during object creation, after setting all properties.
function inputE2_CreateFcn(hObject, eventdata, handles)
% hObject handle to inputE2 (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have a white background on Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
function inputG12_Callback(hObject, eventdata, handles)
44
% hObject handle to inputG12 (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of inputG12 as text
%
str2double(get(hObject,'String')) returns contents of inputG12 as a double
% --- Executes during object creation, after setting all properties.
function inputG12_CreateFcn(hObject, eventdata, handles)
% hObject handle to inputG12 (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have a white background on Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
function inputNU12_Callback(hObject, eventdata, handles)
% hObject handle to inputNU12 (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of inputNU12 as text
%
str2double(get(hObject,'String')) returns contents of inputNU12 as a double
% --- Executes during object creation, after setting all properties.
function inputNU12_CreateFcn(hObject, eventdata, handles)
% hObject handle to inputNU12 (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have a white background on Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
% --- Executes on selection change in popupmenuStress.
function popupmenuStress_Callback(hObject, eventdata, handles)
% hObject handle to popupmenuStress (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
handles.outputUnits1='';
guidata(hObject, handles);
switch get(handles.popupmenuStress,'Value')
case 1
set(handles.outputUnits1,'String','GPa');
case 2
set(handles.outputUnits1,'String','psi');
end
% Hints: contents = cellstr(get(hObject,'String')) returns popupmenuStress contents as cell array
%
contents{get(hObject,'Value')} returns selected item from popupmenuStress
% --- Executes during object creation, after setting all properties.
function popupmenuStress_CreateFcn(hObject, eventdata, handles)
% hObject handle to popupmenuStress (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: popupmenu controls usually have a white background on Windows.
45
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
% --- Executes on selection change in popupmenuLength.
function popupmenuLength_Callback(hObject, eventdata, handles)
% hObject handle to popupmenuLength (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
handles.outputUnits2='';
guidata(hObject, handles);
switch get(handles.popupmenuLength,'Value')
case 1
set(handles.outputUnits2,'String','mm');
case 2
set(handles.outputUnits2,'String','inches');
end
% Hints: contents = cellstr(get(hObject,'String')) returns popupmenuLength contents as cell array
%
contents{get(hObject,'Value')} returns selected item from popupmenuLength
% --- Executes during object creation, after setting all properties.
function popupmenuLength_CreateFcn(hObject, eventdata, handles)
% hObject handle to popupmenuLength (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: popupmenu controls usually have a white background on Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
function outputEx_Callback(hObject, eventdata, handles)
% hObject handle to outputEx (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of outputEx as text
%
str2double(get(hObject,'String')) returns contents of outputEx as a double
% --- Executes during object creation, after setting all properties.
function outputEx_CreateFcn(hObject, eventdata, handles)
% hObject handle to outputEx (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have a white background on Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
function outputEy_Callback(hObject, eventdata, handles)
% hObject handle to outputEy (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of outputEy as text
%
str2double(get(hObject,'String')) returns contents of outputEy as a double
46
% --- Executes during object creation, after setting all properties.
function outputEy_CreateFcn(hObject, eventdata, handles)
% hObject handle to outputEy (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have a white background on Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
function outputGxy_Callback(hObject, eventdata, handles)
% hObject handle to outputGxy (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of outputGxy as text
%
str2double(get(hObject,'String')) returns contents of outputGxy as a double
% --- Executes during object creation, after setting all properties.
function outputGxy_CreateFcn(hObject, eventdata, handles)
% hObject handle to outputGxy (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have a white background on Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
function outputNUxy_Callback(hObject, eventdata, handles)
% hObject handle to outputNUxy (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of outputNUxy as text
%
str2double(get(hObject,'String')) returns contents of outputNUxy as a double
% --- Executes during object creation, after setting all properties.
function outputNUxy_CreateFcn(hObject, eventdata, handles)
% hObject handle to outputNUxy (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have a white background on Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
function outputNUyx_Callback(hObject, eventdata, handles)
% hObject handle to outputNUyx (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of outputNUyx as text
%
str2double(get(hObject,'String')) returns contents of outputNUyx as a double
% --- Executes during object creation, after setting all properties.
47
function outputNUyx_CreateFcn(hObject, eventdata, handles)
% hObject handle to outputNUyx (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have a white background on Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
function outputETAxs_Callback(hObject, eventdata, handles)
% hObject handle to outputETAxs (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of outputETAxs as text
%
str2double(get(hObject,'String')) returns contents of outputETAxs as a double
% --- Executes during object creation, after setting all properties.
function outputETAxs_CreateFcn(hObject, eventdata, handles)
% hObject handle to outputETAxs (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have a white background on Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
function outputETAsx_Callback(hObject, eventdata, handles)
% hObject handle to outputETAsx (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of outputETAsx as text
%
str2double(get(hObject,'String')) returns contents of outputETAsx as a double
% --- Executes during object creation, after setting all properties.
function outputETAsx_CreateFcn(hObject, eventdata, handles)
% hObject handle to outputETAsx (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have a white background on Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
function outputETAys_Callback(hObject, eventdata, handles)
% hObject handle to outputETAys (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of outputETAys as text
%
str2double(get(hObject,'String')) returns contents of outputETAys as a double
% --- Executes during object creation, after setting all properties.
function outputETAys_CreateFcn(hObject, eventdata, handles)
% hObject handle to outputETAys (see GCBO)
48
% eventdata reserved - to be defined in a future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have a white background on Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
function outputETAsy_Callback(hObject, eventdata, handles)
% hObject handle to outputETAsy (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of outputETAsy as text
%
str2double(get(hObject,'String')) returns contents of outputETAsy as a double
% --- Executes during object creation, after setting all properties.
function outputETAsy_CreateFcn(hObject, eventdata, handles)
% hObject handle to outputETAsy (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have a white background on Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
% --- Executes during object deletion, before destroying properties.
function outputEy_DeleteFcn(hObject, eventdata, handles)
% hObject handle to outputEy (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
--------------
49
APPENDIX-B
USE OF SHELL ELEMENTS IN ANSYS
The FEM modeling is done in ANSYS and the steps for the same are given below:First start ANSYS: choose “Mechanical APDL Product Launcher” in the ANSYS menu and
define job name and work directory. Make sure the “ANSYS” simulation environment and
“ANSYS Academic Teaching Advanced” license are chosen.
By “Preprocessor > Element Type > Add” we select the element to use as the 8‐node shell
element SHELL281. Alternatively, we could use SHELL181, but for this plane example with
constant stress state, the element choice is of less importance.
The following options for the element are changed: Key option 3 (Integration option) is set to
“Full w/ incompatible modes” instead of the default "Reduced integration”, and Key-option 8
(Storage of layer data) is changed to “All layers”.
Next material properties are defined: “Main Menu > Preprocessor > Material Props > Material
Models > Structural > Linear> Elastic > Orthotropic” and you enter the material data.
As default you enter the major Poisson's ratio (PRXY, etc.). It is a good idea also to define the
density (1470 kg/m3) for the laminate.
Using the menu “Main Menu > Preprocessor > Sections > Shell > Lay‐up > Plot Section” we get
the following plot (Fig.B1):
50
Fig.B1 Stacking layout
Using “Main Menu > Preprocessor > Meshing > MeshTool” we can select a given area and
change the definition of area attributes (including the element coordinate system) and with this
section laminated area has been meshed for FE analysis as shown in Fig.B2.
Fig.B2 Meshed-plate
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