“Free Vibration Analysis of the Laminated Composite

“Free Vibration Analysis of the Laminated Composite
“Free Vibration Analysis of the Laminated Composite
Beam with Various Boundary Conditions”
A THESIS SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE AWARD OF
Master of Technology
In
Machine Design and Analysis
By
Yogesh Singh
Roll No: 210ME1141
Department of Mechanical Engineering
National Institute of Technology
Rourkela
2012
“Free Vibration Analysis of the Laminated Composite
Beam with Various Boundary Conditions”
A THESIS SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE AWARD OF
Master of Technology
In
Machine Design and Analysis
By
Yogesh Singh
Roll No: 210ME1141
Under the Guidance of
Prof. N.KAVI
Department of Mechanical Engineering
National Institute of Technology
Rourkela
2012
NATIONAL INSTITUTE OF TECHNOLOGY
ROURKELA
CERTIFICATE
This is to certify that the thesis entitled, “Free Vibration Analysis of the Laminated
Composite Beam with Various Boundary Conditions” by Yogesh Singh in partial
fulfillment of the requirements for the award of Master of Technology Degree in
Mechanical Engineering with specialization in “Machine Design & Analysis” at
the National Institute of Technology, Rourkela is an authentic work carried out by
him under my supervision and guidance.
To the best of my knowledge the matter embodied in the thesis has not been
submitted to any other University/ Institute for the award of any Degree or
Diploma.
Prof.N.KAVI
Date:
Dept. of Mechanical Engineering
National Institute of Technology
Rourkela-769008
i
ACKNOWLEDGEMENT
Successful completion of this work will never be one man’s task. It requires hard
work in right direction. There are many who have helped to make my experience
as a student a rewarding one.
In particular, I express my gratitude and deep regards to my thesis guide
Prof. N.KAVI, for his valuable guidance, constant encouragement and kind cooperation throughout period of work which has been instrumental in the success of
thesis.
I also express my sincere gratitude to Prof.K.P.Maity, Head of the Department,
Mechanical Engineering, for providing valuable departmental facilities.
I would like to thank Dr. Subrata Kumar Panda, Assistant Professor, NIT
Rourkela, for his guidance and constant support.
I would like to thank to all of my colleagues and friends for their inspiration and
help.
I thank all the member of the Department of Mechanical Engineering and the
Institute who helped me by providing the necessary resources and in various other
ways for the completion of my work.
I would like to thanks my parents for their encouragement, love and friendship.
I would like to thank to all those who are directly or indirectly supported me in
carrying out this thesis work successfully.
Finally, I thanks god for everything.
Yogesh Singh
Roll No.210ME1141
Department of Mechanical Engineering
National Institute of Technology
Rourkela
ii
ABSTRACT
A dynamic finite element approach for free vibration analysis of generally laminated composite
beams is introduced on the basis of first order shear deformation theory. The effect of Poisson
effect, bending and torsional deformations, couplings among extensional, shear deformation and
rotary inertia are comprised in the formulation. The dynamic stiffness matrix is defined based on
the exact solutions of the differential equations of motion governing the free vibration of
generally laminated composite beam. The influences of Poisson effect, material anisotropy,
slender ratio, shear deformation and boundary condition on the natural frequencies of the
composite beams are analyze in detail by specific carefully favored examples. The natural
frequencies and mode shapes of numerical results are presented and, whenever possible,
compared to those previously published solutions in order to describe the correctness and
accuracy of the present approach.
Free vibration analysis of laminated composite beams is carried out using higher order shear
deformation theory. Two-node, finite elements of eight degrees of freedom per node, based on
the theories, are presented for the free vibration analysis of the laminated composite beams in
this project work. Numerical results have been computed for various ply orientation sequence
and number of layers and for various boundary conditions of the laminated composite beams and
compared with the results of other higher order theories available in literature. The comparison
study shows that the present considered higher order shear deformation theory forecast the
natural frequencies of the laminated composite beams better than the other higher order theories
considered.
For considered examples, the coding of the formulation of first order shear deformation theory
and higher order shear deformation theory (two-node , finite elements of eight degree of freedom
per node) done by the help of MATLAB and ANSYS 12 software package.
Keywords: - shear deformation, slender ratio, natural frequencies, higher order shear deformation
theory, laminated composite beam and free vibration.
iii
CONTENTS
Description
Page no.
Certificate
İ
Acknowledgement
ii
Abstract
iii
Contents
iv
List of figures
vi
List of tables
viii
Chapter 1.Introduction
1.1 Definitions
1
1.2 Fibres
2
1.2.1 Glass fibres
3
1.2.2 Carbon fibres
3
1.2.3 Aramid fibres
4
1.2.4 Boron fibres
4
1.2.5 Ceramic fibres
4
1.3 Polymeric matrix
5
1.3.1 Polyster resins
5
1.3.2 Epoxy resins
5
1.3.3 Vinyl ester resins
6
1.3.4 Phenolic resins
6
iv
1.4 Application of composites
6
1.5 The Timoshenko Beam Theory
7
1.6 Present Investigation
8
Chapter 2 Literature Review
9
2.1 Outline of the present work
17
Chapter 3 Theory and Formulation
3.1 First order shear deformation theory
18
3.1.1 Dynamic finite element formulation
3.2 Higher order shear deformation theory
25
35
3.2.1 Analytical solution of the equation of motion
43
3.2.2 Finite element formulation
45
3.2.3 Derivation of stiffness and consistent mass matrices
48
3.2.4 Equations of motion
50
Chapter 4 Computer Program
4.1 Computer program for the laminated composite beam
51
4.2 Using MATLAB to design and analyze composite laminate beam
51
4.3 Using ANSYS (12) to design and analyze composite laminate beam
54
4.5 Procedure in modeling ANSYS
57
Chapter 5 Results and Discussion
64
Chapter 6 Conclusion and Scope for the future work
94
Chapter 7 References
95
v
List of figures
Description
Page no.
Figure 1 Geometry of a laminated composite beam element
18
Figure 2 Sign convention for positive normal force, Shear force, bending moment and torque 30
Figure 3 Boundary conditions for displacements and forces of composite beam
30
Figure 4 Geometry and co-ordinate system of the beam
35
Figure 5 Flow chart for MATLAB
53
Figure 6 SHELL99 Geometry
54
Figure 7 Flow chart for ANSYS
57
Figure 8 Preference menus on ANSYS
58
Figure 9 Defined element types
58
Figure 10 Add degree of freedom
59
Figure 11 Real constants set number
59
Figure 12 Define material model behaviors
60
Figure 13 Create key points in active co-ordinate system
60
Figure 14 Mesh tool
61
Figure 15 New analyses
62
Figure 16 Modal Analysis
62
Figure 17 Solution status
63
Figure 18 Natural frequencies (Hz) for clamped-clamped (glass-polyester laminated composite
beam) (45°/45/°45°/45°)
68
Figure 19 First mode shape of glass-polyester laminated beam (45°/45/°45°/45°)
69
Figure 20 Second mode shape of glass-polyester laminated beam (45°/45/°45°/45°)
69
Figure 21 Third, fourth and fifth mode shapes
70
vi
Figure 22 Natural frequencies (Hz) for clamped-free (glass-polyester laminated composite
beam) (45°/45/°45°/45°)
71
Figure 23 First to five mode shapes
72
Figure 24 Natural frequencies (Hz) for simply supported-simply supported (glass-polyester
laminated composite beam) (45°/45/°45°/45°)
73
Figure 25 First to five modes shapes
74
Figure 26 Natural frequencies (Hz) for simply clamped-simply supported (glass-polyester
laminated composite beam) (45°/45/°45°/45°)
74
Figure 27 First to five mode shapes
75
Figure 28 Natural frequencies (Hz) of clamped-clamped supported (graphite-epoxy laminated
composite beam) (30°/50°/30°/50°).
77
Figure 29 First to fifth mode shapes
78
Figure 30 Natural frequencies (Hz) of clamped-free supported (graphite-epoxy laminated
composite beam) (30°/50°/30°/50°).
79
Figure 31 First to fifth mode shapes
80
Figure 32 Natural frequencies (Hz) of clamped-free supported (T300/976 graphite-epoxy
laminated composite beam) (30°/-60°/30°/-60°).
83
Figure 33 First to fifth mode shapes
84
Figure 34 First five normal mode shapes of clamped-free laminated composite beam for
example 2.
87
Figure 35 First five normal mode shapes of simply supported- simply supported laminated
composite beam for example 2.
88
Figure 36 First five normal mode shapes of clamped-free laminated composite beam for
example 3.
89
vii
List of Tables
Description
Page no.
Table 1 Classification of FRP composite materials .
2
Table 2 SHELL99 Real Constants.
56
Table 3 Natural frequencies (in Hz) of glass-polyester laminated composite beam.
(45°/45/°45°/45°)
Table
4
Natural
67
frequencies
(Hz)
of
graphite-epoxy
laminated
composite
.(30°/50°/30°/50°)
Table
5
Natural
beam
76
frequencies
(Hz)
of
graphite-epoxy
(30°/50°/30°/50°).
laminated
composite
beam.
81
Table 6 Natural frequencies (Hz) of T300/976 graphite-epoxy laminated composite beam. (30°/60°/30°/-60°)
83
Table 7 Natural frequencies (Hz) of AS4/3501-6 graphite-epoxy laminated composite beam.
(0°/90°/0°/90°)
85
Table 8 Natural frequencies (Hz) of T300/976 graphite-epoxy laminated composite beam 30°/60°/30°/-60°).considering shear deformation ignored .
86
Table 9 Comparison of non- dimensional natural frequencies of a simply – simply supported
laminated composite beam. (0/0/90/90/0/0)
91
Table 10 Comparison of non- dimensional natural frequencies of a clamped-free supported
laminated composite beam. (0/90/90/0)
92
viii
Chapter 1
INTRODUCTION
INTRODUCTION
1.1 DEFINITIONS [56]:A composite material is defined as a material system which consists of a mixture or a
combination of two or more distinctly different materials which are insoluble in each other and
differ in form or chemical composition.
Thus, a composite material is labeled as any material consisting of two or more phases. Many
combinations of materials termed as composite materials, such as concrete, mortar, fiber
reinforced plastics, fibre reinforced metals and similar fibre impregnated materials.
Two- phase composite materials are classified into two broad categories: particulate composites
and fibre reinforced composites. Particulate composites are those in which particles having
various shapes and sizes are dispersed within a matrix in a random fashion. Examples as mica
flakes reinforced with glass, lead particles in copper alloys and silicon carbon particles in
aluminium.
Particulate composites are used for electrical applications, welding, machine parts and other
purposes.
Fibre reinforced composite materials consists of fibres of significant strength and stiffness
embedded in a matrix with distinct boundaries between them. Both fibres and matrix maintain
their physical and chemical identities, yet their combination performs a function which cannot be
done by each constituent acting singly. fibres of fibre reinforced plastics (FRP) may be short or
continuous. It appears obvious that FRP having continuous fibres is indeed more efficient.
Classification of FRP composite materials into four broad categories has been done accordingly
to the matrix used. They are polymer matrix composites, metal matrix composites, ceramic
matrix composites and carbon/carbon composites. Polymer matrix composites are made of
thermoplastic or thermoset resins reinforced with fibres such as glass, carbon or boron. A metal
matrix composite consists of a matrix of metals or alloys reinforced with metal fibres such as
boron or carbon. Ceramic matrix composites consist of ceramic matrices reinforced with ceramic
1
fibres such as silicon carbide, alumina or silicon nitride. They are mainly effective for high
temperature applications.
Table 1 Classification of FRP composite materials [1]
Matrix type
Fibres
Matrix
Polymer
E-glass
Epoxy
S-glass
Polyimide
Carbon(graphite)
Thermoplastics
Kevlar
Polyester
Boron
Polysulfone
Boron
Aluminium
Carbon (graphite)
Magnesium
Silicon carbide
Titanium
Alumina
Copper
Silicon carbide
Silicon carbide
Alumina
Alumina
Silicon nitride
Glass ceramic
Metal
Ceramic
Silicon nitride
Carbon
Carbon
Carbon
Of all the types of composites discussed above, the most important is the fibre reinforced
composites this is form the application point of view. This project is deal with fibre reinforced
polymer matrix composite materials.
1.2 Fibres[56] :Materials in fibre form are stronger and stiffer than that used in a bulk form. There is a likely
presence of flaws in bulk material which affects its strength while internal flaws are mostly
absent in the case of fibres. Further , fibres have strong molecular or crystallographic alignment
2
and are in the shape of very small crystals. Fibres have also a low density which is
disadvantageous.
Fibres is the most important constituent of a fibre reinforced composite material. They also
occupy the largest volume fraction of the composite. Reinforcing fibres as such can take up only
its tensile load. But when they are used in fibre reinforced composites, the surrounding matrix
enables the fibre to contribute to the major part of the tensile, compressive, and flexural or shear
strength and stiffness of FRP composites.
1.2.1 Glass fibres[56]
The most common fibre used in polymeric fibre reinforced composites is the glass fibre. The
main advantage of the glass fibre is its low cost. Its other advantage are its high tensile strength,
low chemical resistance and excellent insulating properties. Among its disadvantages are its low
tensile modulus somewhat high specific gravity, high degree of hardness and reduction of tensile
strength due to abrasion during handling. Moisture decreases the glass fibre strength. Glass fibres
are susceptible to sustained loads, as they cannot withstand loads for long periods.
Two types of glass fibres are used in FRP industries. They are E-glass and S-glass . E-glass has
the lowest cost among all fibres.
S-glass has high tensile strength. Its typical composition is 65% SiO2 , 25% Al2O3 and 10%
MgO. The cost of s-Glass is 20-30 times that of E-glass. The tensile strength of S-glass is 33%
greater and the modulus of elasticity is 20% higher than that of E-glass. The principal advantages
of S-glass are its high strength-to-weight ratio, its superior strength relation at elevated
temperature and its high fatigue limit. In spite of its high cost, its main application area is in
aerospace components such as rocket mortars.
1.2.2 Carbon fibres[56]
Carbon fibres are charactyerised by a combination of high strength, high stiffness and light
weight. The advantages of carbon fibres are their very high tensile strength-to-weight ratio, high
tensile modulus-to-weight ratio, very low coefficient of thermal expansion and high fatigue
strength. The disadvantages are their low impact resistance and high electrical conductivity. Due
3
to the high cost the use of the carbon fibres is justified only in weight critical structures, that is
mostly applied to aerospace industry.
1.2.3 Aramid fibres[56]
Kevlar aramid is made of carbon , hydrogen, oxygen and nitrogen and is essentially an aromatic
organic compound. The advantages of aramid fibres are their low density, high tensile strength
and low cost.
Characteristics of Kevlar 49 are its high strength and stiffness, light weight, vibration damping,
resistance to damage, fatigue and stress ruptures. Another variety Kevlar 29 which is of low
density and high strength. Kevlar 29 is used in ropes, cables and coated fabrics for inflatables.
The principal disadvantages of aramid fibres are their low compressive strength and the
difficulty in cutting or machining. For structures or structural components where compression
and bending are predominant such as in a shell, aramid fibres can be used only when it is
hybridized with glass or carbon fibres.
A more advanced variety of Kevlar fibre is Kevlar 149. Of all commercially available aramid
fibres, it has the highest tensile modulus as it has 40% higher modulus than Kevlar 49. The
strain at failure for Kevlar 149 is; however, lower than that of Kevlar 49. Aramid fibres are
costlier than E-glass, but are cheaper than carbon fibres.
1.2.4 Boron fibres[56]
Boron fibres are characterized by their very high tensile modulus. Boron fibres have relatively
large diameters and due to this they are capable of withstanding large compressive stress and
providing excellent resistance to buckeling. Boron fibres are , however , costly and in fact are
costlier than most varities of carbon fibres. The application area of boron fibres at present is
restricted to aerospace industries only.
1.2.5 Ceramic fibres[56]
Ceramic fibres are mainly used in application areas dealing with elevated temperature. Examples
of ceramic fibres are silicon carbide and aluminium oxide. Ceramic fibres has an advantage in
4
that they have properties such as high strength, high elastic modulus with high temperature
capabilities and are free from environmental attack.
1.3 Polymeric matrix[56]
Polymers are divided into two broad categories: thermoplastic and thermoset. Thermoplastic
polymers are those which are heat softened ,melted and reshaped as many times as desired. But a
thermoset polymer cannot be melted or reshaped by the application of heat or pressure.
The advantages of thermoplastic matrices are their improved fracture toughness over the
thermoset matrix and their potential of much lower cost in the manufacturing of finished
composites.
Traditionally, thermoset polymers are widely used as a matrix material for fibre reinforced
composites in structural composite components. Thermoset polymers improve thermal stability
and chemical resistance.
For the purpose of a simple classification, we may divide the thermosets into five categories:(1) Polyster resin,
(2) Epoxy resin,
(3) Vinyl ester resin,
(4) Phenolic resin and
(5) High performance resin.
1.3.1 Polyster resins[56]
The most commonly used resin in glass reinforced plastic construction is the polyster resin and
they have exhibited good performance. The main advantages of polyester resins are their
reasonable cost and ease with which they can be used.
1.3.2 Epoxy resins[56]
Epoxy resins are mostly used in aerospace structures for high performance applications. It is also
used in marine structures, rarely though, as cheaper varieties of resins other than epoxy are
5
available. the extensive use of epoxy resins in industry is due to :- (1) the ease with which it can
be processed,(2) excellent mechanical properties and (3) high hot and wet strength properties .
1.3.3 Vinyl ester resins[56]
Vinyl ester resin is superior to polyester resin because it offers greater resistant to water. These
resins provide superior chemical résistance and superior retention properties of strength and
stiffness at elevated temperature. In construction and marine industries , vinyl ester resins have
been widely used in boat construction.
1.3.4 Phenolic resins[56]
The main characteristics of phenolic resins are their excellent fire resistance properties. As such
they are now introduced in high temperature application areas. The recently developed cold-cure
varieties of phenolic resins are used for contact moulding of structural laminates.
Phenolic resins have inferior mechanical properties to both polyester resins and epoxy resins, but
have higher maximum operating temperature, much better flame retardant and smoke and toxic
gas emission characteristics. Due to the above advantages , phenolic resins are the only matrix
used in aircraft interior.phenolic resins are increasingly used in internal bulkheads, decks and
furnishings in ships.
1.4 Application of composites
1. Marine field
2. Aircraft and Space
3. Automotive
4. Sporting goods
5. Medical Devices
6. Commercial applications.
6
1.5 The Timoshenko Beam Theory
It is well known the classical theory of Euler–Bernoulli beam assumes that(1) the cross-sectional plane perpendicular to the axis of the beam remains plane after
deformation (assumption of a rigid cross-sectional plane);
(2) The deformed cross-sectional plane is still perpendicular to the axis after deformation.
The classical theory of beam neglects transverse shearing deformation where the transverse shear
stress is determined by the equations of equilibrium. It is applicable to a thin beam. For a beam
with short effective length or composite beams, plates and shells, it is inapplicable to neglect the
transverse shear deformation. In 1921, Timoshenko presented a revised beam theory considering
shear deformation1 which retains the first assumption and satisfies the stress-strain relation of
shear.
Advantages:-
1. High resistance to fatigue and corrosion degradation.
2. High ‘strength or stiffness to weight’ ratio.
3. High resistance to impact damage.
4. Improved friction and wear properties.
5. Improved dent resistance is normally achieved. Composite panels do not sustain
damage as easily as thin gage sheet metals.
6. Due to greater reliability, there are fewer inspections and structural repairs.
7
1.6 PRESENT INVESTIGATION
In the current investigation the main objective is to find out the free vibration of generally
laminated composite beams based on first-order shear deformation theory and derived through
the use of Hamilton’s principle. The Poisson effect, rotary inertia, shear deformation and
material coupling among the bending, extensional and torsional deformations are embraced in
the formulation. A dynamic stiffness matrix is made to solve the free vibration of the generally
laminated composite beams [54]. The dynamic finite element method deals with the mass
distribution within a beam element exactly and thus it provides accurate dynamic characteristics
of a composite beam. Natural frequencies and mode shapes are obtained for the generally
laminated composite beams. The natural frequencies are investigated and comparisons of the
current results with the available solutions in literature are presented [54].
Also the current investigation is based on the higher order shear deformation theories, for the
dynamic analysis of the simply supported laminated composite beam. Numerical results have
been computed for various boundary conditions for the homogeneous and laminated composites
beams and the numerical results are compared with the results of other theories available in
literature [54, 55].
After the comparisons of results we noticed that the theories predict the natural frequencies of
the beams better than the other higher order shear deformation theories [55].
Apart from the presentation of analytical solutions to the vibration problems of the composite
laminated beams, two node finite elements of eight degrees of freedom per node are also
investigated in this present analysis to determine the natural frequencies of simply-supported and
clamped- free laminated composite beams for which analytical solutions cannot be obtained
using the higher order shear deformation theories [55]. Numerical results obtained for the above
problems compared to the analytical and finite element solutions available in the literature [54,
55].
The present results are compared with solutions available in the literature and obtained by the
help of MATLAB and ANSYS software.
8
Chapter 2
LITERATURE REVIEW
LITERATURE REVIEW
The fiber-reinforced composite materials are ideal for structural applications where high
strength-to-weight and stiffness-to-weight ratios are required. Composite materials can be
tailored to meet the particular requirements of stiffness and strength by altering lay-up and fiber
orientations. The ability to tailor a composite material to its job is one of the most significant
advantages of a composite material over an ordinary material. So the research and development
of composite materials in the design of aerospace, mechanical and civil structures has grown
tremendously in the past few decades. It is essential to know the vibration characteristics of these
structures, which may be subjected to dynamic loads in complex environmental conditions. If the
frequency of the loads variation matches one of the resonance frequencies of the structure, large
translation/torsion deflections and internal stresses can occur, which may lead to failure of
structure components. A variety of structural components made of composite materials such as
aircraft wing, helicopter blade, vehicle axles, and turbine blades can be approximated as
laminated composite beams, which requires a deeper understanding of the vibration
characteristics of the composite beams. The practical importance and potential benefits of the
composite beams have inspired continuing research interest. A number of researchers have been
developed numerous solution methods in recent 20 years.
Raciti and Kapania [2] collected a report of developments in the vibration analysis of laminated
composite beams.
Chandrashekhara et al. [3] found the accurate solutions based on first order shear deformation
theory including rotary inertia for symmetrically laminated beams.
The laminated beams by a systematic reduction of the constitutive relations of the threedimensional anisotropic body and found the basic equations of the beam theory based on the
parabolic shear deformation theory represented by Bhimaraddi and Chandrashekhara [4].
A third-order shear deformation theory for static and dynamic analysis of an orthotropic beam
incorporating the impact of transverse shear and transverse normal deformations developed by
Soldatos and Elishakoff [5].
9
The exact solutions for symmetrically laminated composite beams with 10 different boundary
conditions, where shear deformation and rotary inertia were considered in the analysis developed
by Abramovich [6].
Hamilton’s principle to calculate the dynamic equations governing the free vibration of
laminated composite beams. The impacts of transverse shear deformation and rotary inertia were
included, and analytical solutions for unsymmetrical laminated beams were obtained by applying
the Lagrange multipliers method developed by Krishnaswamy et al. [7].
The free vibration behavior of laminated composite beams by the conventional finite element
analysis using a higher-order shears deformation theory. The Poisson effect, coupled extensional
and bending deformations and rotary inertia are considered in the formulation studied by
Chandrashekhara and Bangera [8].
Abramovich and Livshits [9] presented the free vibration analysis of non-symmetric cross-ply
laminated beams based on first-order shear deformation theory.
Khdeir and Reddy [10] evolved the analytical solutions of various beam theories to study the free
vibration behavior of cross-ply rectangular beams with arbitrary boundary conditions.
Biaxial bending, axial and torsional vibrations using the finite element method and the first-order
shear deformation theory examined by Nabi and Ganesan [11].
The analytical solutions for laminated beams based on first-order shear deformation theory
including rotary inertia obtained by Eisenberger et al. [12].
Banerjee and Williams [13] evolved the exact dynamic stiffness matrix for a uniform, straight,
bending–torsion coupled, composite beam without the effects of shear deformation and rotary
inertia included.
10
Teboub and Hajela [14] approved the symbolic computation technique to analyze the free
vibration of generally layered composite beam on the basis of a first-order shear deformation
theory. The model used considering the effect of Poisson effect, coupled extensional, bending
and torsional deformations as well as rotary inertia.
An exact dynamic stiffness matrix for a composite beam with the impacts of shear deformation,
rotary inertia and coupling between the bending and torsional deformations included presented
by Banerjee and Williams [15].
An analytical method for the dynamic analysis of laminated beams using higher order refined
theory developed by Kant et al. [16] .
Shimpi and Ainapure [17] presented the free vibration of two-layered laminated cross-ply beams
using the variation ally consistent layer wise trigonometric shear deformation theory.
The in-plane and out-of-plane free vibration problem of symmetric cross-ply laminated
composite beams using the transfer matrix method analyzed by Yildirim et al. [18].
Yildirim et al. [19] examined the impacts of rotary inertia, axial and shear deformations on the
in-plane free vibration of symmetric cross-ply laminated beams.
The stiffness method for the solution of the purely in-plane free vibration problem of symmetric
cross-ply laminated beams with the rotary inertia, axial and transverse shear deformation effects
included by the first-order shear deformation theory developed by Yildirim [20].
Mahapatra et al. [21] presented a spectral element for Bernoulli–Euler composite beams.
Ghugal and Shimpi [22] preposed a review of displacement and stress-based refined theories for
isotropic and anisotropic laminated beams and discussed various equivalent single layer and
layer wise theories for laminated beams.
11
Higher-order mixed theory for determining the natural frequencies of a diversity of laminated
Simply-Supported beams presented by Rao et al. [23] .
A new refined locking free first-order shear deformable finite element and demonstrated its
utility in solving free vibration and wave propagation problems in laminated composite beam
structures with symmetric as well as asymmetric ply stacking proposed by Chakraborty et al.
[24].
A spectral finite element model for analysis of axial– flexural–shear coupled wave propagation
in thick laminated composite beams and derived an exact dynamic stiffness matrix proposed by
Mahapatra and Gopalakrishnan [25].
A new approach combining the state space method and the differential quadrature method for
freely vibrating laminated beams based on two-dimensional theory of elasticity proposed by
Chen et al. [26].
Chen et al. [27] reported a new method of state space-based differential quadrature for free
vibration of generally laminated beams.
Ruotolo [28] proposed a spectral element for anisotropic, laminated composite beams. The
axial-bending coupled equations of motion were derived under the assumptions of the first-order
shear deformation theory and the spectral element matrix was formulated.
A two-noded curved composite beam element with three degrees-of-freedom per node for the
analysis of laminated beam structures. The flexural and extensional deformations together with
transverse shear deformation based on first-order shear Deformation theories were incorporated
in the formulation. Also, the Poisson effect was incorporated in the formulation in the beam
constitution equation presented by Raveendranath et al. [29].
12
A complete set of equations governing the dynamic behavior of pre-twisted composite space
rods under isothermal conditions based on the Timoshenko beam theory. The anisotropy of the
rod material, the curvatures of the rod axis, and the effects of the rotary inertia, the shear, axial
deformations and Poisson effect were considered in the formulation reported by Yildirim [30] .
Banerjee [31,32] reported the exact expressions for the frequency equation and mode shapes of
composite Timoshenko beams with cantilever end conditions. The impacts of material coupling
between the bending and torsional modes of deformation together with the effects of shear
deformation and rotary inertia was taken into account when formulating the theory.
Bassiouni et al. [33] proposed a finite element model to investigate the natural frequencies and
mode shapes of the laminated composite beams. The model needed all lamina had the same
lateral displacement at a typical cross-section, but allowed each lamina to rotate a different
amount from the other. The transverse shear deformation was included.
A new variational consistent finite element formulation for the free vibration analysis of
composite beams based on the third-order beam theory proposed by Shi and Lam [34].
Chen et al. [35] presented a state space method combined with the differential quadrature method
to examined the free vibration of straight beams with rectangular cross-sections on the basis of
the two-dimensional elasticity equations with orthotropy.
The vibration analysis of cross-ply laminated beams with different sets of boundary conditions
based on a three degree-of-freedom shear deformable beam theory. The Ritz method was
adopted to determine the free vibration frequencies presented by Aydogdu [36].
A refined two-node, 4 DOF/node beam element based on higher-order shear deformation theory
for axial–flexural– shear coupled deformation in asymmetrically stacked laminated composite
beams. The shape function matrix used by the element satisfied the static governing equations of
motion developed by Murthy et al. [37].
13
The free vibration behavior of symmetrically laminated fiber reinforced composite beams with
different boundary conditions. The impacts of shear deformation and rotary inertia were
considered and the finite-difference method was used to solve the partial differential equations
describing the free vibration motion analyzed by Numayr et al. [38].
The free vibration analysis of laminated composite beams using two higher-order shear
deformation theories and finite elements based on the theories. Both theories considered a quintic
and quartic variation of in plane and transverse displacements in the thickness coordinates of the
beams, respectively, and satisfied the zero transverse shear strain/stress conditions at the top and
bottom surfaces of the beams developed by Subramanian [39] .
A new layer wise beam theory for generally laminated composite beam and contrasted the
analytical solutions for static bending and free vibration with the three-dimensional elasticity
solution of cross-ply laminates in cylindrical bending and with three-dimensional finite element
analysis for angle-ply laminates developed by Tahani [40] .
A 21 degree-of-freedom beam element, based on the FSDT, to study the static response, free
vibration and buckling of unsymmetrical laminated composite beams. They enlisted an accurate
model to obtain the transverse shear correction factor preposed by Goyal and Kapania [41].
Finite elements have also been developed based on Timoshenko beam theory [42]. Most of the
finite element models developed for Timoshenko beams possess a two node-two degree of
freedom structure based on the requirements of the variation principle for the Timoshenko’s
displacement field.
A Timoshenko beam element showing that the element converged to the exact solution of the
elasticity equations for a simply supported beam provided that the correct value of the shear
factor was used proposed by Davis et al. [43].
14
Thomas et al. [44] proposed a new element of two nodes having three degrees of freedom per
node, the nodal variables being transverse displacement, shear deformation and rotation of crosssection. The rates of convergence of a number of the elements were compared by calculating the
natural frequencies of two cantilever beams. Further this paper gave a brief summary of different
Timoshenko beam elements.
For the first time a finite element model with nodal degrees of freedom which could satisfy all
the forced and natural boundary conditions of Timoshenko beam. The element has degrees of
freedom as transverse deflection, total slope (slope due to bending and shear deformation),
bending slope and the first derivative of the bending slope presented by Thomas and Abbas [45].
A second-order beam theory requiring two coefficients, one for cross-sectional warping and the
other for transverse direct stress, was developed by Stephen and Levinson [46].
A beam theory for the analysis of the beams with narrow rectangular cross-section and showed
that his theory predicted better results when compared with elasticity solution than Timoshenko
beam theory. Though this required no shear correction factor, the approach followed by him to
derive the governing differential equations was variationally inconsistent developed by Levinson
[47].
Later Bickford [48] represented Levinson theory using a variational principle and also showed
how one could obtain the correct and variationally consistent equations using the vectorial
approach. Thus the resulting differential equation for consistent beam theory is of the sixth order,
whereas that for the inconsistent beams theory is of the fourth-order.
An improved theory in which the in-plane displacement was assumed to be cubic variation in the
thickness coordinate of the beam whereas the transverse displacement was assumed to be the
sum of two partial deflections, deflection due to bending and deflection due to transverse shear.
This theory does not impacts the effect of transverse normal strain and does not satisfy the zero
strain/stress conditions at the top and bottom surfaces of the beam reported by Krishna Murty
[49] .
15
A higher order beam finite element for bending and vibration problems of the beams. In this
formulation, the theory imagines a cubic variation of the in-plane displacement in thickness coordinate and a parabolic variation of the transverse shear stress across the thickness of the beam.
Further the theory satisfies the zero shear strain conditions at the top and bottom surfaces of the
beam and neglects the effect of the transverse normal strain developed by Heyliger and Reddy
[50].
A C0 finite element model based on higher order shear deformation theories including the effect
of the transverse shear and normal strain and the finite element fails to satisfy the zero shear
strain conditions at the top and bottom surfaces of the beam proposed by Kant and Gupta [51].
The free vibration analysis of the laminated composite beams using a set of three higher order
shear deformation theories and their corresponding finite elements. These theories also fail to
satisfy the zero-strain conditions at the top and bottom surfaces of the beams. Further the impacts
of the transverse normal strain were not included in the theories investigated by Marrur and Kant
[52].
An analytical solution to the dynamic analysis of the laminated composite beams using a higher
order refined theory. This model also fails to satisfy the traction- free surface conditions at the
top and bottom surfaces of the beam but has included the effect of transverse normal strain
preposed by Kant et al. [53].
16
2.1 OUTLINE OF THE PRESENT WORK:-
In the current investigation the main objective is to find out the free vibration of generally
laminated composite beams based on first-order shear deformation theory and derived through
the use of Hamilton’s principle. The Poisson effect, rotary inertia, shear deformation and
material coupling among the bending, extensional and torsional deformations are embraced in
the formulation. A dynamic stiffness matrix is made to solve the free vibration of the generally
laminated composite beams. The dynamic finite element method deals with the mass distribution
within a beam element exactly and thus it provides accurate dynamic characteristics of a
composite beam. Natural frequencies and mode shapes are obtained for the generally laminated
composite beams. The natural frequencies are investigated and comparisons of the current results
with the available solutions in literature are presented. For the results software package using for
the coding and programming is MATLAB and ANSYS 12.
This thesis contains seven chapters including this chapter.
A detailed survey of relevant literature is reported in chapter 2.
In chapter 3 dynamic analysis of laminated composite beam using first order shear deformation
theory and higher order theories including finite element method formulation is carried out
common boundary conditions, such as clamped-free, simply supported, clamped-clamped and
Clamped- simply supported has been analyzed.
In chapter 4 details of computational approach have been outlined. How to coding in MATLAB
software and ANSYS 12 have been outlined step – by- step procedure.
In chapter 5 important results (natural frequencies and mode shapes) drawn from the present
investigations reported in chapters 3 and 4, this chapter including results obtained by MATLAB
and ANSYS 12 for the various boundary condition of laminated composite beam.
Finally in chapter 6 important conclusions drawn from the present investigations reported in
chapters 3-5 along with suggestions for further work have been presented.
17
Chapter 3
THEORY AND FORMULATION
3.1 FIRST ORDER SHEAR DEFORMATION THEORY [54]
MATHEMATICAL FORMULATION
A generally laminated composite beam, is made of many piles of orthotropic materials, principal
material axis of a ply may be oriented at an angle with respect to the x axis. Consider the origin
of the beam is on mid-plane of the beam and x-axis coincident with the beam axis. As shown in
fig.1,
z
y
x
h
b
L
Fig. 1. Geometry of a laminated composite beam [54]
Where,
L= length of the beam,
b= breadth of the beam,
h= thickness of the beam.
Based on first- order shear deformation theory, assumed displacement field for the laminated
composite beam can be written as-
u  x , z , t   u 0  x , t   z  x , t  ,
v
x ,
z,t

 z
(1 i)
 x , t ,
(1j)
18
(1k)
w  x, z , t   w 0  x, t  ,
where, u 0=axial displacements of a point on the mid plane in the x-directions,
w 0= axial displacements of a point on the mid plane in the z-directions
 = rotation of the normal to the mid-plane about the y axis,
 = rotation of the normal to the mid-plane about the y axis,
t  time.
The strain-displacement relations are given by- (by theory of elasticity)—
 x  u 0 / x  z / x
(2i)

(2j)
xz 
w 0 / x
 xy   / x
(2k)
kx   / x
(2l)
k xy   / x
(2m)
By the classical lamination theory, the constitutive equations of the laminate can be obtained as-
 N x   A11
 N y   A 12

 
 N xy   A 16


M
x

  B 11
 M y   B 12

 
M
xy

  B 16
A 12
A16
B 11
B 12
A 22
A 26
A 26
A 66
B 12
B 16
B 22
B 26
B 12
B 16
D 11
D 12
B 22
B 26
B 26
B 66
D 12
D 16
D 22
D 26
Where,(I,j=1,2,6)
19
B 16    x0 


B 26    y0 
B 66    xy 


D 16    x 
D 26    y 


D 66    xy 
(3)
N x , N y and N xy are the in-plane forces,
M x , M y and M xy are the bending and twisting moments,
 x ,  y and  xy are the mid-plane strains,
 x ,  y and  xy are the bending and twisting curvatures,
Aij , Bij and Dij are the extensional stiffnesses, coupling stiffnesses and bending stiffnesses, respectively.
for the case of laminated composite beam,
N
y
and N
xy
, t h e i n -p la n e f o r c e s a n d th e b e n d in g m o m e n t M
 y0 ,  xy and the curvature  y assum ed to be non-zero
Then, now equation (3) can be rewritten as-
 N x   A11

 
M
 x    B11
M  
 xy   B16
B11
D11
D16
B16  u0 / x 


D16    / x 


D66   / x 
(4)
Now considering the effect of transverse shear deformation then,-
Qxz  A55 xz  A55  w0 / x    ,
(5)
20
y
 0.
Where,-
Qxz is the transverse shear force per unit length and
 A11 B11 B16  A B B   A A B  A A B 1  A A B T

  11 11 16   12 16 12   22 26 22   12 16 12 
B11 D11 D16   B11 D11 D16   B12 B16 D12 *A26 A66 B26  *B12 B16 D12 

 
 
 
 

B16 D16 D66  B16 D16 D66  B26 B66 D26  B22 B26 D22  B26 B66 D26 
(6)
The laminate stiffness coefficients A , B , D (I,j= 1,2,6) and the transverse shear stiffness
A₅₅ which are functions of laminate ply orientation , material properties and stack sequences,
are given ash /2
Aij 

Qij dz

Qij zdz

Qij z 2 dz
 h /2
h /2
Bij 
 h /2
h /2
Dij 
 h /2
(7)
h /2
A55  k

Q55 dz
(8)
 h /2
21
Where,k is the shear correction factor.
The transformed reduced stiffness constants Qij ( i,j=1,2,6) are given as-

Q11  Q11 cos4   2  Q12  2Q66  sin 2  cos2   Q22 cos 2 

Q12   Q11  Q22  4Q66  sin 2  cos 2   Q12 sin 4   cos 4 

Q22  Q11 sin 4   2  Q12  2Q66  sin 2  cos 2   Q22 cos 4 

(9i)

(9j)

(9k)
Q16   Q11  Q22  2Q66  sin  cos 3    Q12  Q22  2Q66  cos  sin 3 
(9l)
Q26   Q11  Q22  2Q66  sin 3  cos    Q12  Q22  2Q66  cos 3  sin 
(9m)

Q66   Q11  Q22  2Q66  2Q12  sin 2  cos 2   Q66 sin 4   cos 4 

Q55  G13 cos 2   G23 sin 2 
(9n)
(9o)
Where,∅ is the angle between the fiber direction and longitudinal axis of the beam.
The reduced stiffness constants Q11 , Q12 , Q22 and Q66 can be obtained in terms of the engineering
constants[57]Q11 
E1
,
1   12 21
Q12 
 12 E2
,
1  12 21
Q22 
E2
,
1  12 21
Q66  G12
(10)
22
The total strain energy V of the laminated composite beam given as—
L
1
V    N x x0  M x x  M xy xy  Qxz  xz b.dx
20
(11)
Substituting  x0 ,  x ,  xy and  xz values from equation (2) into equation (11) then –
L
1
V    N x u0 / x  M x  / x  M xy  / x  Qxz  w0 / x    b.dx
20
(12)
Total kinetic energy T of the laminated composite beam is given as –
L h /2
1
2
2
2
T      u / t    v / t    w / t   b.dx.dz

2 0  h/2 
(13)
Where,ρ is the mass density per unit volume.
Now substituting u, ν and ω from equation (1) into equation (13) and after integration with
respect to z we get—
L
1
2
2
2
2
T    I1  u0 / t   I 3   / t   2 I 2  u0 / t   / t   I 3   / t   I1  w0 / t   b.dx

20
(14)
Where,-
h /2
I1 

 dz

 zdz

 z 2 dz
 h /2
h /2
I2 
 h /2
h /2
I3 
 h/ 2
(15)
23
By the use of Hamilton’s principle, the governing equations of motion of the laminated
composite beam can be expressed in the formt2
  T   V dt  0
(16)
t1
At t  t1 and t 2 -
 u0   w0      0
After substitution the variational operations yields the following governing equation of motion-
  2u
 I1  20
 t

  2

I
 2 2

 t
  2u 0 

  2

A

B
 11  2  11  2

 x 
 x

  2

B
 16  2

 x

0

  2 w0 
  2 w0 
  
 I1  2   A55  2   A55 
0
 x 
 t 
 x 
  2
I3  2
 t
  2u0 
  2u0 

  2

I

B

D
 2  2  11  2 
11 
2

 t 
 x 
 x
  2
I3  2
 t
  2u0


B
 16  2

 x

  2

D

16 
2

 x
(17i)
(17j)

  2

D

16 
2

 x

  2

D

66 
2

 x

 w0 
  A55 
  A55  0
 x 


0

(17k)
(17l)
Note:-
If the Poisson effect is ignored, the coefficients in equation (17)[ A11 , B11 , B16 , D11 , D16 , D66 ]
should be replaced by the laminate stiffness coefficients [ A11 , B11 , B16 , D11 , D16 , D66 ].
24
3.1.1 Dynamic finite element formulation [54]
Equation (17) have solutions that are separable in time and space, and that the time dependence
is harmonic, like as-
u0  x, t   U  x  sin t ,
(18i)
w0  x, t   W  x  sin t ,
(18j)
  x, t     x  sin t ,
(18k)
  x, t     x  sin t ,
(18l)
Where,
ω is the angular frequency,
U  x  , W  x  ,   x  and   x  are the amplitudes of the sinusoidally varying longitudinal
displacement, bending displacement, normal rotation and torsional rotation respectively.
Now after the substitution of equation (18) values in equation (17), the following differential
eigenvalue problem is obtained:-
 2 I1U   2 I 2  A11U   B11  B16   0
(19i)
 2 I1W  A55W   A55   0
(19j)
 2 I3   2 I 2U  B11U   D11  D16   A55W   A55  0
(19k)
 2 I3   B16U   D16  D66    0
(19l)
Where the superscript primes denote the derivatives with respect to x.
The solution to equation (19) are given by-
U  x   Ae x
25
W  x   Be x
  x   Ce x
  x   De x
(20)
After the substitution of equation (20) into equation (19) the equivalent algebraic eigenvalue
equations are obtained and the equations have non-trivial solutions when the determinant of the
coefficient matrix of A , B , C and D vanishes. Now consider that determinant is zero , then the
characteristics equations, which is an eight-order polynomial equation in ĸ :-
 4 8  3 6   2 4  1 2  0  0
(21)
Where:-,


2
2
 4   A55 B16 D11  2 B11 B16 D16  B11 D66  A11 D16  D11 D66

(22i)


 A D 2I  B 2 D I  A D D I  A D 2  D D I  
55 11 66 1
11
16
11 66
1
 55 16 1 11 66 1


 2
3   2 A55 B11 D66 I 2  2 B11 D66 B11 I1  A55 I 2 


2
2
 A55 B11 I3  A11 A55 D11  D66 I 3  B16 D11 I1  A55 I 3








(22j)



 D 2 I 2  D D I 2  2 B D I I  2B D I I  A D I 2 
2
16 1
11 66 1
16 16 1 2
11 66 1 2
55 66 2
2 2



 2   A55 B16  A11 D66 I1 
2
2
I  A11 A55 I 3 


  B11  B16  A11  A55 D11  D66 I1  2 A55 B11I 2
 3




(22k)


2







 D11 I12  I 2 2 B11 I1  A55 I 2    
2


  2 
1    A55 I1 D66 I1  A11 I3  D66 I1  I 2  I1I 3  I 3 

 A11  A55 I1 I3
 


 



4







(22l)

0  I1 I3 6  A55 I1    I 2 2  I1 I3   2

(22m)
26
Now the fourth order polynomial equation for the roots χ must be solved. Where χ=κ² has
substituted into equation (21) to reduce into a fourth order polynomial equation. The solutions
can be found as follows-
 4  a1  3  a2  2  a3   a4  0
(23)
Where:-
a1  3 / 4
a2   2 /  4
a3  1 / 4
a4   0 /  4
(24)
The fourth- order equation(23) can be factorized as-

2


 p1   q1  2  p2   q2  0
(25)
Where:-
 p1  1 
2

    a1  a1  4a2  41 
 p2  2

 q1  1 
a11  2a3

 q    1 
a12  4a2  41 
 2  2 
(26)
And 1 is one of the roots of the following equation-
 3  a2 2   a1a3  4a4     4a2 a4  a3 2  a12 a4   0
(27)
Then the roots of equation (23) can be written as-
 1 
 
 2 
 3 
 
 4 
 p1
p12

 q1
2
4
 p2
p2 2

 q2
2
4
(28)
27
The roots of equation (27) can be written asa2
 2 Q cos  / 3
3
a
2  2  2 Q cos    2  / 3
3
a2
3   2 Q cos    4  / 3
3
1 
(29)
Where:3 

  cos 1  R / Q 


Q
R
1 2
a2  3a1a3  12a4
9


1
2a23  9a1a2 a3  27 a32  27 a12 a4  72a2 a4
54

3
DQ R

2
(30)
The general solutions to equations (19) are given by-
U  x   A1e1 x  A2e 1 x  A3e 2 x  A4e 2 x  A5e3 x  A6e 3 x  A7 e4 x  A8e  4 x
4

 jx
=  A2i 1e
j 1
 A2i e
 j x

(31i)
W  x   B1e1 x  B2 e 1 x  B3e 2 x  B4 e 2 x  B5e3 x  B6e 3 x  B7e 4 x  B8e 4 x
4

 jx
=  B2i 1e
j 1
 B2i e
 j x
(31j)

  x   C1e1 x  C2e 1 x  C3e 2 x  C4 e 2 x  C5e 3 x  C6e  3 x  C7 e 4 x  C8e  4 x
4

 jx
=  C2i 1e
j 1
 C 2i e
 j x
(31k)

  x   D1e1 x  D2e 1 x  D3e 2 x  D4e  2 x  D5e3 x  D6e 3 x  D7 e 4 x  D8e  4 x
4

 jx
=  D2i 1e
j 1
 D2i e
 j x
(31l)

28
Where-
1 
2 
3 
4 
1
2
3
4
The relationship among the constants is given byA2 j 1  t j C2 j 1
(32i)
A2 j  t j C2 j
B2 j 1  tj C2 j 1
B2 j  tj C2 j
(32j)
D2 j 1  tj C2 j 1
D2 j  tj C2 j
(32k)
Where:- (j=1-4)

t j   B16 D16 j 4  B11 j 2  I 2 2

tj   A55 j / A55 j 2  I1 2

 D

 j 2  I 3 2  /  j

66

(33j)
tj   j 2 B11 B16 j 2  A11 D16 j 2  D16 I1  B16 I 2  2 /  j


2

 j   B16  j 4  A11 j 2  I1 2
(33i)
 D
 j 2  I 3 2
66
 
(33k)

(33l)
29
Sign convention:-
+
+
Ny
Nx
Mx
+
Q xz
Mx
+
M xy
M xy
Fig. 2. Sign convention for positive normal force N x (x), shear force Qxz (x), bending moment M x
(x) and torque MQ
xy (x) [54]
xz
M xy1
1
W1
Qxz1
M x1
1
W2
Qxz 2
Nx1
U1
M xy 2
2
Nx2
U2
M x2
2
Fig.3. boundary conditions for displacements and forces of composite beam [54]
From fig. 2 the expression of normal force N x (x), shear force Qxz (x), bending moments M x
(x) and torque M xy (x) can be obtained from equations (5), (6) and (31) asN x  x   A11
dU
d
d
 B11
 B16
dx
dx
dx
30
4
=  A11 j t j  B11 j  B16 j tj * C2 j 1e


j 1
 jx
 C2 j e
 j x

(34i)
dW


Qxz  x     A55
 A55 
dx


4
=
 A

jx
 j t j  A55 * C2 j 1e
55
j 1
 C2 j e
 j x

(34j)
dU
d
d 

M x ( x )    B11
 D11
 D16

dx
dx
dx 

4
=   B11 j t j  D11 j  D16 j tj * C2 j 1e


j 1
 jx
 C2 j e
 j x

(34k)
dU
d
d 

M xy  x    B16
 D16
 D66

dx
dx
dx 

4
=  B16 j t j  D16 j  D66 j tj * C2 j 1e


j 1
 jx
 C2 j e
 j x

(34l)
From fig. 3. The boundary conditions for displacements and forces of the laminated composite
beam are given as—
At:-
x =0;
U  U1
W  W1
  1
N x   N x1
Qxz  Qxz1
M x  M x1
  1
M xy  M xy1
31
(35i)
At:-
x =L;
U  U2
W  W2
  2
  2
Nx  Nx2
Qxz  Qxz 2
M x  M x2
M xy  M xy 2
(35j)
Substituting equations (35) into equations (31), the nodal displacements defined by fig.(3) can be
expressed in terms of C as-
D0    R C
(36)
Where, D0  is the nodal degree of freedom vector.
D0   U1
C  C1
W1 1 1 U 2 W2  2  2 
(37i)
C3 C5 C7 C2 C4 C6 C8 
 t1

 t1
 1

 t1
 R  t e1L
1
t e1L
 1 L
e1
  1L
t1e
(37j)
t2
t3
t4
t1
t2
t3
t2
1
t
t3
1
t
t4
1
t
t1
1
t
t2
1
t
t3
1
t
t2e
t3e
t4e
2
2L
2L
3
3L
3L
4
4L
4L
1
1L
t1e
1L
t2e
t3e
t4e
t1e
e2L e3L e4L
e1L
t2e2L t3e3L t4e4L t1e1L
32
2
2L
t2e
2L
t2e
e2L
te2L
2
3
3L
t3e
3L
t3e
e3L
te3L
3
t4


t4 
1 

t4 

t4e4L 
t4e4L 

4L
e


t4e4L 
(37k)
Substituting equations (35j) into equations (31), the nodal forces defined in fig. 3 can be
expressed in terms of C as-
F0    H C
(38)
Where,  F0  is the nodal force vector.
F0    N x1
Qxz1 M x1 M xy1 N x 2 Qxz 2 M x 2 M xy 2 
 t1

 t1

 t
 1
 t1
 R     1L
 t1e

 t1e1L

 t1e1L

 t1e1L
(39i)
t2
t3
t4
t1
t2
t3
t2

t2

t2
te 2 L
t3

t3

t3
te3 L
t4

t3

t4
te 4 L
t1

t1
t
t1e 1L
t2

t2

t2
te 2 L
t3

t3

t3
te  3 L
t2 e 2 L

t2 e 2 L

t 2 e 2 L
t3e3 L

t3e3 L

t3e3 L
t4 e 4 L

t4 e 4 L

t 4 e 4 L
t1e 1L

t1e 1L
te 1L
t2 e  2 L

t2e  2 L

t2 e 2 L
t3e  3 L

t3e 3 L

t3e  3 L
2
3
4
1
1
2
3
t4


t4 



t4 

t4 

t4e  4 L 

t4e  4 L 


t4 e  4 L 

t4e  4 L 
(39j)
Where, (j=1-6)
tj  A11 j t j  B11 j  B16 j tj

t j   A55 j t j  A55


tj   B11 j t j  D11 j  D16 j tj


tj  B11 j t j  D16 j  D66 j tj



(40)
33
Now from the equations (36) and (38) relationship between the nodal force vector and nodal
degree of freedom vector can be written as-
1
F0    H  R  D0    K 0 D0 
(41)
Where,
1
 K 0    H  R 
And  K 0  is the frequency-dependent dynamic stiffness matrix.
34
3.2 HIGHER ORDER SHEAR DEFORMATION THEORY [55]
MATHAMATICAL FORMULATION
Z
Z
2h
X
Y
b
a
Fig.4. geometry and co-ordinate system of the beam [55]
.
u  x, z , t  and w  x, z , t  are the displacement in the x and z directions, respectively, at any point
(x,z).
h = half thickness of the beam,
i
= orientation angle of the i th layer with respect to the x-axis.
a = length of the beam,
b = width of the beam.
35
The displacement fields for the first theory are taken as-
u  u0  z
 wb n
z 2  w sh
z 5 w2


x
3h 2 x
5h 4 x
w  w bn  w sh
2
  z 4 
 z
   w 2  1     w 4
h
  h  
(1)
The displacement fields for the second theory are taken as-
 wbn
z 2  w sh
z 5 w2
u  u0  z


x
3h 2 x
5h 4 x
w  w bn  w sh
4
2

 z
 z 
   w 2  1     w 4
h
 h  

(2)
Where:
u0  x, t  Are the displacements due to extension,
wbn  x, t  Are the displacements due to bending,
wsh  x, t  Are displacements due to shear deformation.
The terms w2  x, t  and w4  x, t  are the higher terms to include sectional wrapping and all these
variables are measured at the mid surface of the beam z=0.
The displacement fields of the two theories are combined as follows:-
u  u0  z
 wbn
z 3  w sh
z 5  w2


x
3h 2 x
5h 4 x
z
w  wbn  wsh   
h
2n  2
4 2 n

n 
z
w2   1 1   
 w4
  h 

36
(3)
Where, n=0 for the first theory and n=1 for the second theory.
The strain – displacement relationship for these theories as-
 xx
u 0
 2 wbn
z 3  2 wsh
z 5  2 w2

z
 2
 4
x
x 2
3h  x 2
5h  x 2
(4)
z 2 n1
z 3 2 n
n
w


1
4

2
n
w4




2
h 2 n 2
h 4 2 n
(5)
 zz   2n  2 
2 n 2
4 2 n
 w4
 z 2  wsh  z 
z 4  w2
n 
z
 xz   1  2 
  
 4
  1 1   

h  x
  h 
 x
 h  x  h 
(6)
The strain fields are rewritten as-
 xx   10  z  11  z 3  13  z 5  15
(7)
 zz  z 2 n 1 31  z 3  2 n  33
(8)
 xz   40  z 2 41  z 2 n  2 42  z 4 43  z 4  2 n 44
(9)
Where,
u0
x2
 w bn
1
1  
x 2 2
1  w sh
 13   2
3 h 2 x 2
1  w2
 15  
5h 4 x 2
 10 
37
1
 31   2 n  2 
 33
 40
 41
 42
 43
 44
w2
h
1
n
    1  4  2 n  4  2 n w4
h
 w sh
n w4

   1
x
x
1  w sh
 2
h 2 n  2x
w2
1
 
x
h
1 w2
 4
h x
42n
  w4
n  1 
    1   

  h 
  x
2n2
(10)
The stress – strain relationship for the kth layer is given as-
k
 xx 
 Q11



 zz    Q13
 
 xz 
 0
Q13
Q33
0
0 
0 
Q 44 
k
  xx 
 
  zz 
 
  xz 
k
(11)
Where:-
Q11 , Q13 , Q33 and Q44 are the reduced material constants from the three dimensional orthotropic
elasticity matrix.
By the use of Hamilton’s principle, the equation of motion is given ast
  T
 U d t  0
(12)
0
Where, T is the kinetic energy and it is given asT 
1
2
Where,
   u
2
v
u
and

 w 2 dV
w
(13)
being time derivatives of u and w , and ρ is the density of the material.
U is the potential energy which is given as-
38
U 
1
 xx  xx   zz zz   xz  xz dV
2 v
(14)
Substituting equation (13) and equation (14) into equation (12) –

 u0
 2 wbn z 3  2 wsh z 5  2 w2
z


 xx xx   zz zz   xz xz    
xt 3h 2 xt 5h 4 xt
 t

  u0
 2 wbn z 3  2 wsh z 5  2 w2 

z
 2
 4

  t
t

x

t
3
h

x

t
5
h

x

t


4 2 n

0 v   wbn wsh  z 2n2 w2
n
 z   w4 

 
  1 1    

  

t  h 
t
  t
  h   t 
2 n2
4 2 n
  w
 wsh  z 
 w2
n 
 z    w4 
bn



 
  1 1    

t
t
 t
h
  h   t 






 dVdt





(15)
Integrating the equation (15) by parts, and collecting the coefficients  wbn ,  wsh ,  w2 and  w4 ,
the equation of motion in terms of stress resultant are given as-
N   2u 0
 3 wbn
I 3  3 wsh
I 5  3 w2 
  I0

I


0
1
x 
t 2
 x t 2 3h 2  x t 2 5 h 4  x t 2 
(16a)
2M 
 3u 0
 4 wbn
I 4  4 wsh
I 6  4 w2 
   I1
 I2 2 2  2 2 2  4 2 2 
x 2 
x t 2
x t
3h  x  t
5 h x t 
2
2
2
 wbn
 wsh I 2 n  2  w2
I 4  2 n   2 w4
n 
I0
 I0
 2n2
   1  I 0  4  2 n 
0
2
t 2
t 2
h
t 2
h

 t
(16b)
1  2 L  Q1 1 Q2    I 3  3u0
I 4  4 wbn
I 6  4 wsh
I 8  4 w2 








 
3h 2 x 2  x h 2 x   3h2 xt 2 3h 2 x 2t 2 9h 4 x 2 t 2 15h 6 x 2 t 2 
  2 wbn
 2 wsh I 2 n  2  2 w2
I 4 2 n   2 w4 
n 
  I0
 I0
 2n2
  1  I 0  4 2 n  2   0
t 2
t 2
h
t 2
h

 t 

(16c)
39
3
I 6  4 wbn
I8  4 wsh
I10  4 w2 
1  2 P 2n  2
 1 Q3 1 Q4    I 5  u0

V







1
 2n2
 
3h 2 x 2 h2 n  2
x h4 x   5h6 xt 2 5h 4 x 2 t 2 15h 6 x 2t 2 25h8 x 2 t 2 
h
I
 2 wbn I 2 n  2  2 wsh I 4 n  4  2 w2
I 6   2 w4 
n  I 2 n 2
  22nn22




1

   2 n 2 6  2   0
2
2 n 2
2
4n4
2
h

t
h

t
h

t
h  t 
h

(16d)
I 4  2 n   2 wbn
4  2n
1 Q5  
 Q1
V2  
 42 n
 I 0  42 n 

2
h 4 2 n
x  
h
 x h
 t
I 4  2 n   2 wsh  I 2 n  2 I 6   2 w2
2 I 4 2 n I 8 4 n   2 w4
n 

I





1
I



 0 h 4  2 n  t 2
 h 2 n  2 h 6  t 2
 0 h 4  2 n  h 8  4 n  t 2  0






(16e)
The laminate stiffness constants are given asNL zi 1
 A11 , A12 , A13 , A14    
i 1


Q11i 1, z , z 3 , z 5 dz
zi
NL zi 1

 A15 , A16    
i 1

Q13i z 2 n 1 , z 3  2 n dz
zi
NL zi 1
 B11 , B12 , B13    
i 1


Q11i z 2 , z 4 , z 6 dz
zi
NL zi 1
 B14 , B15    
i 1


Q13i z 2 n  2 , z 4  2 n dz
zi
N L z i 1
 C 11    
i 1
 
Q11i z 8 dz
zi
NL zi 1
 C12 , C13    
i 1


Q13i z 2 n  2 , z 6  2 n dz
zi
NL zi 1
 D11    
i 1
 
Q11i z 10 dz
zi
NL zi 1
 D12 , D13    
i 1


Q13i z 2 n  6 , z 8  2 n dz
zi
40
NL z i  1
 F11    
i 1


i
Q 33
z 6  4 n dz
zi
NL zi 1
 E11 , E12    
i 1


i
Q33
z 4 n  2 , z 4 dz
zi
NL zi 1
 G11 , G12 , G13 , G14 , G15    
i 1


i
Q44
1, z , z 2 n  2 , z 4 , z 4  2 n dz
zi
N L z i 1
 H 11 , H 12 , H 13    
i 1


i
Q 44
z 2 n  4 , z 6 , z 6  2 n dz
zi
N L zi 1
T1 1    
i 1


i
Q 44
z 8 4 n d z
zi
NL zi  1
 R11 , R12  
 
i 1


i
Q 44
z 4 n  4 , z 2 n  6 dz
zi
NL z i  1
 S11 , S12    
i 1


i
Q 44
z 8 , z 8  2 n dz
zi
NL is the number of layers. The stress resultant defined as –
NL zi  1
 N , M , L, P    
i 1
 xx 1, z , z 3 , z 5  dz
zi
NL z i  1
V1 , V2    
i 1
 zz  z 2 n 1 , z 3  2 n  dz
zi
NL zi 1
 Q1 , Q2 , Q3 , Q4 , Q5    
i 1
 xz 1, z , z 3 , z 5 , z 7  dz
zi
The mass moment of inertia given as-
 I 0 , I1 , I3 , I 4 , I5 , I 6 , I 2 n2 , I 42n , I8 , I10 , I 4n 4 , I84 n  
NL zi 1
   1, z, z
i 1
2

, z 3 , z 4 , z 5 , z 6 , z 2 n  2 , z 4 n  2 , z 8 , z10 , z 4 n  4 , z 8 4 n dz
zi
(17)
41
The equation of motion are expressed in terms of the displacement as follows-
 2u0
 3 wbn A13  3 wsh A14  3 w2 2n  2
w

A
 2
 4
 2 n 2 A15 2
12
2
3
3
3
x
x
3h x
5h x
h
x
w4   2u0
 3 wbn I 3  3 wsh
I 5  3 w2 
n  4  2n 
  1  4 2 n  A16
  I0
 I1


0
x  t 2
xt 2 3h 2 xt 2 5h 4 xt 2 
h

A11
(18a)
 3u0
 4 wbn B12  4 wsh B13  4 w2 2 n  2
 2 w2
 2 w4
n  4  2n 

B



B


1
B


11
14
 h 4 2 n  15 x 2
x3
x 4
3h 2 x 4
5h 4 x 4
h 2n 2
x 2


2
2
2
2
3
 wbn
 wsh  I 2 n  2   w2
I 4 2 n   w4
 u0
 4 wbn
I 4  3 wsh
I 6  4 w2
n
 I0

I



1
I


I

I


0


0
1
2
 0 h 4 2 n  t 2
 h 2 n  2  t 2
t 2
t 2
xt 2
x 2 t 2 3h 2 xt 2 5h 4 x 2 t 2




(18b)
A12
A13  3u0 B12  4 wbn B13  4 wsh C11  4 w2 
2G12 G14   2 wsh




G

 4 

11
3h 2 x 3 3h 2 x 4
9h 4 x 4 15h6 x 4 
h2
h  x 2
G13 G14 H11 H12   2 w2  I3  3u0
I 4  4 wbn
I 6  4 wsh
I8  4 w2 
 2n  2
C









 3h 2 n 2 12 h 2 n 2 h 4 h 2n  4 h 6  x 2  3h 2 xt 2 3h 2 x 2t 2 9h 4 x 2t 2 15h 6 x 2t 2 




2
2
2
  2 wbn

 wsh  I 2 n 2   w2
I
n
 w
  I0
 I0
  2 n 2  2   1  I 0  4422nn  2 4   0
2
2
t
t
h
h
 t

 t 

(18c)
A14  3u0 B13  4 wbn C11  4 wsh D11  4 w2 2n  2
u0 2n  2
 2 wbn




A

B

15
14
5h 4 x 3 5h 2 x 4 15h 6 x 4
25h82 x 4
h 2 n 2
x
h 2 n 2
x 2 2
G13 G14 H11 H12   wsh  4n  4
R11
2 R12 S11   w2
 2n  2
 3h 2 n 4 C12  h 2 n 2  h 4  h 2 n  4  h 6  x 2   5h 2 n 6 D12  h 4 n 4  h 2 n 6  h8  x 2




G13
H12 G14
S12   2 w4 2n  2
n  4  2n
n  2 n  2  4  2 n 
  1  82 n D13  2 n  2  6  4  8 2 n  2  2 n 2 E11w2   1
E12 w4
h
h
h
h
h
h6
 5h
 x
 I 5  3u0
I 6  4 wbn
I8  4 wsh
I10  4 w2   I 2 n  2  2 wbn I 2 n  2  2 w0  I 4 n 4   2 w2 
 6
 4 2 2

 2 n 2



2
5h x t 15h 6 x 2 t 2 25h8 x 2 t 2   h 2 n 2 t 2
h
t 2  h 4n  4  t 2 
 5h xt
I 6   2 w4
n  I 2 n 2
  1  2 n 2  6  2  0
h  t
h
(18d)
42
4  2n
u0 4  2n
 2 wbn  4  2n C13
G12 G15
H13   2 wsh
A16
 4 2 n B15
   4 2 n
 G11  2  4  2 n  6 2 n 

2
h 4 2n
x
h
x 2
3h 2
h
h
h
 h
 x
G13 G14 H12
S12   2 w2 
G15 
1 
T11    2 w4
n
n
 4  2n D13







1
G



1
G

   11 42n    42 n  15 42 n  2
 h 4  2 n 5h 4 h 2 n  2 h 4
h6 h8 2 n  x 2 
h
h
h




  x
2
2
2
 2n  2  4  2n 
I
I
n   4  2n  

  wsh 
  wbn

E12 w2   1  4 2 n  F11w4   I 0  4422nn 
  I 0  4422nn 

6
2
2
h
h
h

 t

 t
 h

2
2 I 4  2 n I8  4 n   2 w4
n
 I 2 n  2 I 6   w2



1
I

 h 2 n 2 h 6  x 2    0 h 4  2 n  h8 4 n  x 2  0




(18e)
3.2.1 Analytical solution of the equation of motion [55]
Closed form solution for the above equations can be obtained for a simply supported beam by
assuming –
u0 
 A  t  cos px ,  w
bn
, w sh , w 2 , w 4 
   B  t  , C  t  , D  t  , E  t   sin px
Where, p 
(19)
n
a
After substitution equation (19) into equation (18) , the following sets of equations is obtained-
 K  X    M  X  0
(20)
Now the elements of the stiffness matrix [K] are given asK11   p 2 A11
,
K12  p 3 A12
, K13  p3
A13
A
2n  2
, K14  p 3 144  p 2 n 2 A15
2
3h
5h
h
B
n  4  2n 
K15   1  4 2 n  A16 , K 22   p 4 B11 , K 23   p 4 122 ,
3h
h

K 24   p 4
B
2G
G 
n
 4  2n 

K 25   1 p 2  4  2 n  B15 , K 33   p 4 134  p 2  G11  212  144  ,
9h
h
h 
h


43
B13
2n  2
 p 2 2 n 2 A14
4
5h
h
K 34   p 4
C11
G
G
H
H
 2n  2
 p 2  2 n  2 C12  2 n13 2  144  2 n11 4  12
6
15h
h
h
h
h6
 3h

,

G
G
H 
n
 4  2n C
K 35    1 p 2   4  2 n 132  G11  122  4 152 n  6 132 n  ,
3h
h
h
h
 h

K 44   p 4
D11
R
2R
S  2n  2
 4n  4
 p 2  2 n 6 D12  4 n11 4  2 n12
 118   2 n  2 E11 ,
8
6
25h
h
h
h  h
 5h
G
H
G
S 
n  4  2n
n  2 n  2  4  2n 
K 45   p 2  1  82 n D13  2 n13 2  12
 144  8122 n    1
E12 ,
6
h
h
h
h
h6
 5h


G 
1
n
n
K 55   p  1  G11  4 152 n    1 4 2 n
h
h



2
2
T11  
n   4  2n  

 G15  4  2 n     1  4  2 n  F11 .
h


 h

And the mass matrix written as-

 I0

 pI
 1

 I
 M    p 3h32

 I5
p 4
 5h

 0

pI1

  I 0  p2 I 2
p

I3
3h2
I 

  I0  p2 42 
3h 

I 

  I 0  p 2 64 
9h 

I 

  I0  p2 42 
3h 

I 
I 
I
I
  22nn22  p2 64    22nn22  p 2 8 6 
5h 
15h 
h
h
I
I
n
n


  1  I0  4422nn    1  I0  4422nn 
h
h




And the displacement vector are expressed as-
X   A
B C D E
 X    A
 E
B C D

44
I5
5h4



I 4 2 n 
n
 I 2 n 2

2 I6 
  2 n 2  p
  1  I0  42n 
4 

5h 
h
h



I
I
n
 I 2 n 2




  2 n 2  p 2 8 6 
  1  I0  4422nn 

15h 
h
h




I
I
I
I
n
 I 4n  4



2
8
10
  1  22nn22  p2 10 8  
 h4n4  p 15h6 25h8 
25h  


h
I 
2I
I
n I
n

  1  22nn22  p 2 10 8    1  I0  4422nn  8844nn 
25h 
h
h 
h

p
0
It is assumed that to find the solution of equation (20)
A
B C D E   A0 B0 C0 D0 E0  e int
Where, A0 B0 C0 D0 E0 are the constants, and ωn is the natural frequency.
By substituting the values for A, B, C, D and E into equation (20) the natural frequencies and the
corresponding mode shapes can be obtained [54].
3.2.2 Finite element formulation[54]
For the present analysis we assumed, each element having two nodes and each node having eight
degrees of freedom. Linear polynomials are used for the nodal variables u0 and w4 and hermite
cubic polynomials are used for the other nodal variables of the elements.
The displacements fields given are rewritten in the matrix form as-
U d    Z d d 
(21)
T
U d   u w
(22)

z3
1
0
z
0


3h 2

Zd   
0
0 1 0 1

d   u0 wbn

0
z
 
h

z5
5h 4
2n  2
0



4 2n



z
n
 1 1    
  h
 
0
wbn
wsh
w2

wsh
w2
w4 
x
x
x

(23)
(24)
The strain field associated with equation (21) is given as-
    Z id  
(25)
45
T
    xx
 zz  xz 
1 z z 5

 Zid   0 0 0
0 0 0

z5
0
0
(26)
0
z
2 n 1
0
   10 11 13 15  31  33  40
0
0 0
z  2n 0 0
0
1 z2
0
0
3
z
2 n 2
0
0
z4
0 

0 
z 4 2 n 
(27)
(28)
T
 41  42  43  44 
From equation (18) we getT
 U d   v     dV
x h
T
T
 b      Z id    dzdA
0 h
x
(29)
h

T 
T
 b       Z id    dz dx
0
 h

x
T
=b    S d dx
0
Where the stress resultant Sd  given as-
S d    N
M L V1 V2 QL Q2 Q3 Q4 Q6 
h
T
S d     Z id    dz
h
NL
zi 1
 Zid T Q Z id   dz
i 1
  D0  
 
(30)
zi
Where,
NL
 D0    
i 1
zi1
zi
 Zid T Q  Z id  dz
Now substituting equation (30) into equation (29)x
T
 U d  b     D0 Sd dx
(31)
0
46
From equation (13)t
t
   TdVdt      u u  w  w dVdt
0 v
0 v
t
T
      U d 
0 v
t
T
      d 
U dVdt
d
 Z id T  Z id ddVdt
0 v
t x
 zi 1

T
   b    d     Z id   Z id  dz  d dxdt
 z

0 0
 i

t x
T
  b    d   I m oi  d dxdt
T
(32)


0 0
Here the mass moment of inertia matrix –
z i 1
 I m oi  

T
 Z  Z  d z
id
id
(33)
zi
Now substituting equations (31) and (32) into Hamilton’s principle, the expression for the
equation of motion is given ast x
t x
T
T
    d   I moi  d dxdt  b      D0   dxdt  0

0 0
0 0
47
(34)
3.2.3 Derivation of stiffness and consistent mass matrices [55]
The displacement vector within an element can be expressed in terms of the nodal degree of
freedom as-
d     N   
(35)
Where,
1 0
0 1



 0 x

0 0
    
0 0

0 0

0 0


0 0
0
0
0
0
0
0
1

x
0
0
0
0 

0

0

0

0

0


1
0
1

x
0
0
0
And the shape function are taken as-
 N11

0
N   0

0
0

0
N1
0
N2
0
0
0
0
0
0
0
0
0
0
N 21
0
0
N3
0
N4
0
0
0
0
0
0
0
0
0
0
N1
N2
0
0
0
0
0
0
N3
N4
0
0
0
0
0
0
N1
N2
0
0
0
0
0
0
N3
N4
0
0
0
0
0
0
N11
0
0
0
0
0
0
0
48
0

0
0

0
N 21 
And the nodal degree of freedom given by-


 wbn 
 wsh 
  w2 
w
w
.........
w
w
01 bn2 
214 
216 
 sh2  t 


t

t






15
3
5


   u
The strain curvature vector within an element is given as-
    B  
Where,
1

0


0


0


0


 B    0

0


0


0

0



 0
0
0
0
0
0
2

x2
0

1 2
3h 2 x 2
0
0
0
0
0
0
0
0

1 2
5h 4 x2
2n  2
h 2n2
0

x
1 
 2
h x
0
0
0
0
0
0
0
0
0

h
x
1 
 4
h x
1
2n2
0
0



0



0



0


0


n 1
  1   4  2 n  
h 42n
 N

n 

 1
x


0



0



0


n 1
 1


42n
h
x

49

(36)
3.2.4 Equations of motion [54]
To derive equation of motion using Hamilton’s principle, equation (36) and (35) are substituted
into equation (34)t x
 
T
  
T
T
  N   I moi 
0 0
t x
 


T
  N   dxdt  b      B T  D0  B    dxdt  0
0 0
And also we can write:el
M 

el
el
 K 
 
el
0
Where the element mass matrix and stiffness matrix are given as-
 M el
NL
T
 b      N 
 I moi T   N  dx
i 1
 K el
NL
 b
i 1
T
  B   D  B  dx
0


1/2
Non dimensional natural frequency =   n a 2   / 4 E1h 2  .
50
(37)
Chapter 4
COMPUTER PROGRAM
4.1 COMPUTER PROGRAM FOR THE LAMINATED COMPOSITE BEAM
A computer program is developed to perform all the necessary computations in MATLAB and
ANSYS (12) environment.
Advantages of MATLAB
– Ease of use
– Platform independence
– Predefined functions
– Plotting
• Disadvantages of MATLAB
– Can be slow
– Expensive
4.2 Using MATLAB to design and analyze composite laminate beam [59]
This work deals with the generation of MATLAB script files that assists the user in the design of
a composite laminate to operate within safe conditions. The inputs of the program are the
material properties, material limits. Equations based on first order shear deformation theory and
higher order beam theory given below. For three-dimensional composites were used to determine
the global and local stresses and strains on each layer. The free vibration analysis for these
theories by the coding on the MATLAB. The output of the program is the optimal number of
fibre layers required for the composite laminate, as well as the orientation of each layer.
The program promotes the user for the following inputs:
1. The material properties (E1, E2, G12 and ν12,ρ)
2. The number of layers (N)
3. The fibre angle (θ) of each layer
4. The thickness (t) of each layer
5. The length of the beam
51
6. Width of the beam.
7. The number of element.
8. Degree of freedom at each node.
9. Boundary conditions.
10. Relations between elastic constants for the calculation for stiffness and mass matrices.
11. Displacement vector.
These input parameters were used to determine the [Q], [T], and [Q] matrices for each layer.
All data concerning each layer (E1, E2, G12, ν12, θ, t, [Q], [T], and [Q ]) were stored in an array.
The data in this array was used to calculate the [A], [B], and [D] matrices. Using these matrices.
The following procedure illustrates the steps involved in this approach:
1) Calculate the [Q] matrix for each layer
2) Compute the [T] matrix for each layer.
3) Determine [Q ] matrix for each layer.
4) Find the location of the top and bottom surface of each layer, hk (k = 1 to n).
5) Calculate the [A], [B], and [D] matrices.
6) Calculate the laminate stiffness constants.
7) Now calculate the element stiffness matrix.
8) And then taken 5 number of elements and then calculate stiffness and mass matrices for
every one of them.
9) Now globalization of the stiffness matrix and mass matrices.
10) Reduced stiffness and mass matrices.
11) The built in MATLAB function “eig” is used to calculate, eigen values, eigen vectors,
and mode shape diagrams.
12) The non- dimensional natural frequencies are calculated.
Here in the problems we assumed 10 elements means 11 nodes and analyzed by two theories
1.first order shear deformation, 2. Second order shear deformation theory where the degree of
freedom at each node considered as 4 and for another mathematical approach considering the
52
degree of freedom at each node 8. After the calculation by the MATLAB code the results
compared with the references.
Flow chart given as-
Fig.5 Flow chart for MATLAB [59]
53
4.3 Using ANSYS (12) to design and analyze composite laminate beam [58]
The finite element simulation was done by finite element analysis package ANSYS (12). The
FEA software ANSYS includes time-tested, industrial leading applications for structural,
thermal, mechanical, computational fluid dynamics and electromagnetic analysis. ANSYS
software solves for the combined effects of multiple forces, accurately modeling combined
behaviors resulting from “metaphysics interaction”.
This is used to perform the modeling of the composite laminated beam and calculation of natural
frequencies with relevant mode shape. This is used to simulate both the linear and non-linear
effect of structural models in a static or dynamic environment. The purpose of the finite element
package was utilized to model the fibre reinforced polymer beam in three dimensions as
SHELL93, SHELL181, and SOLID 46. This package enables the user to investigate the physical
and mechanical behavior of the beam.
SHELL99
Linear Layered Structural Shell [58]
SHELL99 may be used for layered applications of a structural shell model. While SHELL99
does not have some of the nonlinear capabilities of SHELL91, it usually has a smaller element
formulation time. SHELL99 allows up to 250 layers. If more than 250 layers are required, a userinput constitutive matrix is available.
The element has six degrees of freedom at each node: translations in the nodal x, y, and z
directions and rotations about the nodal x, y, and z-axes.
Figure 6. SHELL99 Geometry [58]
54
xIJ = Element x-axis if ESYS is not supplied.
x = Element x-axis if ESYS is supplied.
LN = Layer Number
NL = Total Number of Layers
SHELL99 Input Data
Nodes
I, J, K, L, M, N, O, P
Degrees of Freedom
UX, UY, UZ, ROTX, ROTY, ROTZ
Real Constants
See Table 99.1: "SHELL99 Real Constants" for a description of the real constants.
Material Properties
If KEYOPT (2) = 0 or 1, supply the following 13*NM properties where NM is the
number of materials (maximum is NL):
EX, EY, EZ, ALPX, ALPY, ALPZ (or CTEX, CTEY, CTEZ or THSX, THSY, THSZ),
(PRXY, PRYZ, PRXZ or NUXY, NUYZ, NUXZ),
DENS, GXY, GYZ, GXZ, for each of the NM materials.
If KEYOPT (2) = 2, 3 or 4, supply none of the above.
Supply DAMP and REFT only once for the element (use MAT command to assign
material property set). See the discussion in "SHELL99 Input Data" for more details.
Surface Loads
Pressures --
55
face 1 (I-J-K-L) (bottom, in +Z direction), face 2 (I-J-K-L) (top, in -Z direction),
face 3 (J-I), face 4 (K-J), face 5 (L-K), face 6 (I-L)
Body Loads
Temperatures -T1, T2, T3, T4, T5, T6, T7, T8 if KEYOPT(2) = 0 or 1
None if KEYOPT(2) = 2, 3 or 4
Special Features
Stress stiffening
Large deflection
Table 2 SHELL99 Real Constants [58]
56
4.4 Procedure in modeling ANSYS [58]
There are major and sub important steps in ANSYS model,
1. Preprocessing
2. Solution stage
3. Post processing.
Flow chart given as-
Fig. 7 Flow chart for ANSYS
57
1. Requirement specification: Firstly it is required to give preference for what type
analysis you want to do, here we are analyzing for beam so given here structural part as
Shown in fig.8-
Fig.8 Preference menus on ANSYS
2. Preprocessor
Now next step is preprocessor, where the preprocessor menu basically used to inputs the
entire requirement thing for analysis such as- element type, real constraints, material
properties, modeling, meshing, and loads. Element menu contains – defined element type
and degree of freedom defined where we have to give the element structure type such as
BEAM, SOLID, SHELL, and the degree of freedom , here in this analysis selected part is
SHELL or SOLID fig . 9 shows this menu contained part –
Fig. 9 Defined element types
58
And degree of freedom menu shown in fig. 10. (Assumed 4 degree of freedom)
Fig. 10 Add degrees of freedom
After this the next step is to define the real constants set, which is contained the material
number, fiber orientation angle, and the thickness of each layer as shown in fig.11-
Fig. 11 Real constants set number
We assumed different thickness, different fiber orientation angle and different layer thickness
for different example which is given in chapter 5, Result and discussion.
The next step is to define material properties, by the help of material model menu in the
ANSYS12 as shown in fig 12-
59
Fig. 12 Define material model behaviors
For material model behavior considered material as orthotropic and for this given all the nine
input which is required (young modulus, modulus of rigidity and Poisson ratio).
Next step is to modeling the model given in numerical problems to analyze, means the
geometry shape of the structure, in this analysis we considered the shape of beam is
rectangular cross section. The key step is to model the model is- preprocessor-modelingcreate-key points-in the active cs plane, as shown in fig 13.-
Fig. 13 Create key points in active co-ordinate system
60
After the completion of this stage now meshing menu is introduced to mesh the problem, in
our problem we considered the element division is 10. How to mesh in the ANSYS is
showing in fig. 14. The step to follow the mesh is given as1. Meshing
2. Mesh tool
3. Define lines
4. Define global
5. Define layer
6. Mesh the volume
7. Shape should be hexagon
8. Then apply meshing
Fig. 14 Mesh tool
61
The next step is to define the boundary condition by the help of define load menu, here in
this analysis we defined the different boundary condition for different problems.
3. Solution stage:-Now in this menu solution stage we have to introduce the analysis type
means static analysis, harmonic analysis, modal analysis etc. in our problems we selected
modal analysis for frequency and mode shape analysis. As shown in fig. 15 and fig. 16-
Fig 15 New analyses
Fig. 16 Modal Analysis
62
After that the next step is to solve the analysis or problems by the help of solve menu. If there is
no error then it will be showing like that “solution is done” as shown in fig 17.
Fig 17. Solution status
4. Post processing:
This menu is helpful to find the output of the problems. Such as –
1. Result summery
2. Failure criteria
3. Plot results
4. List results
5. Result summery
6. Nodal calculation.
We found the output natural frequencies and mode shapes after the application of this
menu which is shown in chapter 5, result and discussion.
63
Chapter 5
RESULT AND DISCUSSION
RESULT AND DISCUSSION
In this chapter MATLAB code [59] and ANSYS 12 [58] software package are using for the
results given below, the procedure to obtained the results on ANSYS given on the chapter 4 and
MATLAB coding procedure for the numerical results are given in chapter 4.
The written MATLAB program must be first verified in order to ensure the subsequent analyses
are free of error. Therefore the result obtained from the analysis is compared with available
results of references. Natural frequencies obtained from this MATLAB coding and ANSYS are
listed in tables and those results comparing with the available results of references for the
composite laminated beam with different boundary conditions.
And mode shapes are presented by graphs for different boundary conditions.
Here in this chapter different numerical example taken for the analysis of natural frequencies and
mode shapes of the composite laminated beam, where the numerical example contained the
following data1. Material properties of the laminated composite beam.
2. Length of the laminated composite beam.
3. Width of the laminated composite beam.
4. Thickness of the laminated composite beam.
5. Lay-up of layers.(angles of fibre)
6. Density of the material.
7. Boundary conditions.
In this section the MATLAB code require only five material properties (E1,E2,G12,v12 and v21)
but ANSYS requires nine material property inputs. This is because plane stress assumption. In
this chapter the results obtained for the two theories 1. First order shears deformation theory and
2. Higher order shear deformation theory and comparing with the results of other theories which
is available in literature. In every example we considered the number of element is equal to 10.
SOLID 45 and SHELL 99 are considered for the analysis of composite laminated beam on the
ANSYS 12 [58].
64
First order shear deformation
The dynamic stiffness method formulated previously (first order shear deformation) is directly
applied to the illustrative examples of generally laminated composite beams to obtain the
numerical results. The influences of Poisson effect, material anisotropy, shear deformation and
boundary conditions on the natural frequencies of the laminated beams are investigated.
Example 1.
Consider a glass-polyester composite beam of rectangular cross – section with all fiber angles
arranged to 45 degree (45°/45/°45°/45°). The material properties of the beam are given as –
E1= 37.41 GPa
,
E2= 13.67 GPa ,
G12=5.478 GPa
,
G13= 6.03 GPa
G23=6.666 GPa
,
ν12= 0.3 , ρ= 1968.9 kg/m³,
,
L= length of the composite laminated beam =0.381 m,
b= width of the laminated composite beam= 25.4 mm,
h= thickness of the each ply = 25.4 mm.
Boundary conditions:For first order shear deformation theory, considered each node have four degree of freedom,
laminated composite beam divided into 10 elements, the assumed displacements are given
previously in chapter 3 .
Now the boundary conditions assumed areFor the clamped boundaryU = longitudinal displacements =0
W = bending displacements = 0
 = normal rotation =0
65
 = torsional rotation = 0.
For the simply supported boundary condition –
U = longitudinal displacements =0
W = bending displacements = 0
M x =bending moment per unit length = 0
 = torsional rotation = 0.
For the free boundary:-
N x = normal force per unit length = 0,
Qxz = transverse shear force per unit length =0,
M x = bending moment per unit length=0,
M xy = twisting moment per unit length = 0.
The dynamic stiffness matrix for the first order shear deformation theory which is formulated in
chapter 3 calculated by the help of MATLAB software [59] for 10 element division, each node
having 4 degree of freedom the dynamic stiffness matrix found (44 by 44).
After the consideration of different boundary conditions the global stiffness matrices reduced and
formed new reduced global stiffness matrices. For clamped – clamped boundary condition
reduced global stiffness matrix formed as [36×36] matrix. For clamped- simply supported
boundary condition reduced global stiffness matrix formed as [38×38]. For clamped- free
boundary condition reduced global stiffness matrix formed as [40×40]. For simply supportedsimply supported boundary condition reduced stiffness matrix formed as [38×38].
Now all the formulation for first order shear deformation theory coded by MATLAB [59] and
the results (natural frequencies and mode shapes) found by different boundary conditions are
listed in the tables and compared with available results of references.
66
The first five natural frequencies of the laminated composite beam with various boundary
conditions are calculated and the numerical results are introduced in table 3.
To validate the proposed formulation, the present results are compared with the theoretical
results of references.
Table 3
Natural frequencies (in Hz) of glass-polyester laminated composite beam (45°/45/°45°/45°)
Mode no.
Clamped-
Clamped-
Clamped-
Clamped-
Clamped-
Clamped-
Simply
Simply
clamped
clamped
simply
simply
free
free
supported-
supported-
present
Ref.[54]
supported
supported
present
Ref.[54]
simply
simply
present
Ref.[54]
supported
supported
present
Ref.[54]
1
760.1
763.2
502.4
529.9
86.4
120.5
307.2
348.1
2
2074.34
2087.9
1694.1
1700.7
635.2
752.4
1192.4
1346.2
3
4002.03
4051.5
2851.6
3514.3
1845.6
2092.9
2951.4
3017.6
4
6230.3
6612.9
5231.2
5937.5
3915.4
4062.2
4866.1
5285.3
5
8532.12
5627.33
5964.1
6213.4
5214.1
5632.4
7201.9
7402.4
It is observed from table 3 that the present results are in very good agreement with the theoretical
results of ref. [54].
Now the ANSYS 12 [58] results presented natural frequencies and mode shapes for example 1
given above is-(procedure on ANSYS for modal analysis given on chapter 4),
Natural frequencies obtained for clamped-clamped laminated composite beam are given as67
Fig. 18 Natural frequencies (Hz) for clamped-clamped (glass-polyester laminated composite
beam) (45°/45/°45°/45°)
In fig. 18 a column shown as sets it means the mode number and for different mode numbers
natural frequency obtained as shown in fig. 18. Now the natural frequencies obtained from the
ANSYS 12 is not good agreement with the present and theoretical results of references.
First mode shape of laminated composite beam (glass-polyester) with x – components of
displacement is shown in fig 19.
68
Fig. 19. First mode shape of glass-polyester laminated beam (45°/45/°45°/45°)
For the second mode-
Fig. 20 Second mode shape of glass-polyester laminated beam (45°/45/°45°/45°)
For the third, fourth and fifth mode ANSYS mode shape presented as-(fig. 21)
69
Fig 21 Third, fourth and fifth mode shapes
70
Natural frequencies obtained for clamped-free laminated composite beam are given as(In the case of glass-polyester laminated composite beam with (45°/45/°45°/45°) layup)
Fig. 22 Natural frequencies (Hz) for clamped-free (glass-polyester laminated composite beam )
(45°/45/°45°/45°)
First to fifth mode shapes of laminated composite beam (glass-polyester) with x – components of
displacement are shown in fig 23.
Fig. 23 (a) first mode shape
71
Fig. 23 (b) Second mode shape
Fig. 23 (c) third mode shape
Fig. 23 (d) fourth mode shape
Fig. 23 (e) fifth mode shape
Natural frequencies obtained for simply supported-simply supported laminated composite beam
are given as-(In the case of glass-polyester laminated composite beam with (45°/45/°45°/45°)
layup)
72
Fig. 24 Natural frequencies (Hz) for simply supported-simply supported (glass-polyester
laminated composite beam) (45°/45/°45°/45°)
First to fifth mode shapes of laminated composite beam (glass-polyester) with x – components of
displacement are shown in fig 25.
Fig. 25 (a) first mode shape
73
Fig. 25 (b) second mode shape
Fig. 25 (c) third mode shape
Fig. 25 (d) fourth mode shape
Fig. 25 (e) fifth mode shape
Natural frequencies obtained for clamped-simply supported laminated composite beam are given
as-(In the case of glass-polyester laminated composite beam with (45°/45/°45°/45°) layup)
Fig. 26 Natural frequencies (Hz) for simply clamped-simply supported (glass-polyester laminated
composite beam) (45°/45/°45°/45°)
74
First to fifth mode shapes of laminated composite beam (glass-polyester) with x – components of
displacement are shown in fig 27.
Fig. 27 (a) first mode shape
Fig. 27 (b) second mode shape
Fig. 27 (c) third mode shape
Fig. 27 (d) fourth mode shape
Fig. 27 (e) fifth mode shape
75
Example 2.
Consider a graphite-epoxy composite beam of rectangular cross – section with all fiber angles
arranged to (30°/50°/30°/50°). The material properties of the beam are given as –
E1= 144.80 GPa
,
E2= 9.65 GPa ,
G12=4.14 GPa
,
G13= 4.14 GPa
,
G23=3.45 GPa
,
ν12= 0.3 , ρ= 1389.2 kg/m³,
L= length of the composite laminated beam =0.381 m,
b= width of the laminated composite beam= 25.4 mm,
h= thickness of the each ply = 25.4 mm.
The first five natural frequencies of the laminated composite beam with various boundary
conditions are calculated and the numerical results are introduced in table 4. To validate the
proposed formulation, the present results are compared with the theoretical results of references.
Table 4
Natural frequencies (Hz) of graphite-epoxy laminated composite beam (30°/50°/30°/50°).
Mode no.
Clamped-
Clamped-
Clamped-
Clamped-
Clamped-
Clamped-
Simply
Simply
clamped
clamped
simply
simply
free
free
supported-
supported-
present
Ref.[54].
supported
supported
present
Ref.[54].
simply
simply
present
Ref.[54].
supported
supported
present
Ref.[54].
1
589.3
637.9
421.4
467.3
93
105.3
340
353.7
2
1420.8
1647.8
1218.4
1391.3
542.1
635
1054.2
1114.3
3
2912.43
3000
1328.7
1504.6
1451.7
1678.9
2154.4
2434.2
4
3167.54
3397.8
1824.4
2095.1
1642.4
1970.6
2965.1
3450.3
5
4116.45
4712.5
2954.4
3547.3
2187.9
2546.2
3567.4
4264.5
It is observed from table 4 that the present results are in very good agreement with the theoretical
results of ref.[54].
76
Now the ANSYS 12 [58] results presented natural frequencies and mode shapes for example 2
given above is-(procedure on ANSYS for modal analysis given on chapter 4), Natural
frequencies obtained for clamped-clamped laminated composite beam are given as-
Fig. 28 Natural frequencies (Hz) of clamped-clamped supported (graphite-epoxy laminated
composite beam) (30°/50°/30°/50°).
First to fifth mode shapes of laminated composite beam (graphite-epoxy) with x – components of
displacement are shown in fig 29.
Fig. 29 (a) first mode shape
77
Fig. 29 (b) second mode shape
Fig. 29 (c) third mode shape
Fig. 29 (d) fourth mode shape
Fig. 29 (e) fifth mode shape
78
Natural frequencies obtained for clamped-free laminated composite beam are given as-
Fig. 30 Natural frequencies (Hz) of clamped-free supported (graphite-epoxy laminated
composite beam) (30°/50°/30°/50°).
First to fifth mode shapes of laminated composite beam (graphite-epoxy) with x – components of
displacement are shown in fig 31.
Fig. 31 (a) first mode shape
79
Fig. 31 (b) second mode shape
Fig. 31 (c) third mode shape
Fig. 31 (d) fourth mode shape
Fig. 31 (e) fifth mode shape
For other boundary condition the results are same as example 1, and its mode shape also same
like as previous example 1.
Example 3.
Consider a graphite-epoxy composite beam of rectangular cross – section with all fiber angles
arranged to (30°/50°/30°/50°). The material properties of the beam are given as –
80
E1= 144.80 GPa
,
E2= 9.65 GPa ,
G12=4.14 GPa
,
G13= 4.14 GPa
,
G23=3.45 GPa
,
ν12= 0.3 , ρ= 1389.2 kg/m³,
L= length of the composite laminated beam =0.572 m,
b= width of the laminated composite beam= 25.4 mm,
h= thickness of the each ply = 25.4 mm.
The first five natural frequencies of the laminated composite beam with various boundary
conditions are calculated and the numerical results are introduced in table 5. To validate the
proposed formulation, the present results are compared with the theoretical results of references.
Table 5
Natural frequencies (Hz) of graphite-epoxy laminated composite beam (30°/50°/30°/50°).
Mode no.
Clamped-
Clamped-
Clamped-
Clamped-
Clamped-
Clamped-
Simply
Simply
clamped
clamped
simply
simply
free
free
supported-
supported-
present
Ref.[54].
supported
supported
present
Ref.[54].
simply
simply
present
Ref.[54].
supported
supported
present
Ref.[54].
1
201.65
382.1
421.4
211.3
44.1
46.9
140
158.5
2
754.1
1154.6
1218.4
648.4
186.4
288.9
454.2
512.5
3
1254.4
1201.4
1328.7
1296.3
1254.1
1504.9
1004.4
1151.0
4
2149.7
1746.5
1824.4
2097.7
2097.7
2391.7
1765.1
1862.6
5
2482.4
2614.3
2254.4
2576
2576.0
2619.4
2267.4
2567.9
It is observed from table 5 that the present results are in very good agreement with the theoretical
results of ref.[54].
81
Only displacement value has changed due to the change of length, on the ANSYS program. And
all other values frequencies and mode shapes same as example 2. So here it is not required to
introduced the again mode shapes pictures.
Table 5 lists the results for the natural frequencies of the composite beam with E1= 144.80 GPa
and L= 0.572 m, and Table 4 lists the results for the natural frequencies of the composite beam
with E1= 144.80 GPa
and L= 0.381 m, shows the effect of slender ratio on the natural
frequencies of the composite beam. It is clear that the natural frequency decreases with the
increase of beam length.
Example 4.
Consider a T300/976 graphite/epoxy composite beam of rectangular cross – section with all fiber
angles arranged to (30°/-60°/30°/-60°). The material properties of the beam are given as
E1= 221 GPa
,
G12=4.8 GPa
,
G23=3.45 GPa
E2= 6.9 GPa ,
G13= 4.14 GPa
,
,
ν12= 0.3 , ρ= 1550.1 kg/m³,
L= length of the composite laminated beam =0.381 m,
b= width of the laminated composite beam= 25.4 mm,
h= thickness of the each ply = 25.4 mm.
The first five natural frequencies of the laminated composite beam with various boundary
conditions are calculated and the numerical results are introduced in table 6. To validate the
proposed formulation, the present results are compared with the theoretical results of references.
82
Table 6
Natural frequencies (Hz) of T300/976 graphite-epoxy laminated composite beam (30°/-60°/30°/60°).
Mode no.
Clamped-
Clamped-
Clamped-
Clamped-
Clamped-
Clamped-
Simply
Simply
clamped
clamped
simply
simply
free
free
supported-
supported-
present
Ref.[54].
supported
supported
present
Ref.[54].
simply
simply
present
Ref.[54].
supported
supported
present
Ref.[54].
1
201.65
382.1
421.4
211.3
44.1
46.9
140
158.5
2
754.1
1154.6
1218.4
648.4
186.4
288.9
454.2
512.5
3
1254.4
1201.4
1328.7
1296.3
1254.1
1504.9
1004.4
1151.0
4
2149.7
1746.5
1824.4
2097.7
2097.7
2391.7
1765.1
1862.6
5
2482.4
2614.3
2254.4
2576
2576.0
2619.4
2267.4
2567.9
It is observed from table 6 that the present results are in very good agreement with the theoretical
results of ref.[54].
Now the ANSYS 12 [58] results presented natural frequencies and mode shapes for example 4
given above is-(procedure on ANSYS for modal analysis given on chapter 4),
Natural frequencies obtained for clamped-clamped laminated composite beam are given as-
Fig. 32 Natural frequencies (Hz) of clamped-free supported (T300/976 graphite-epoxy laminated
composite beam) (30°/-60°/30°/-60°).
83
First to fifth mode shapes of laminated composite beam (graphite-epoxy) with x – components of
displacement are shown in fig 33.
Fig. 33 (a) first mode shape
Fig. 33 (b) second mode shape
Fig. 33 (c) first mode shape
Fig. 33 (d) first mode shape
Fig. 33 (e) first mode shape
84
For other different boundary conditions natural frequencies found by ANSYS 12 [58] is
presented in table 6 and mode shapes as likely to be previously shown on example 1 and 2.
Example 5.
Consider a AS4/3501-6 graphite/epoxy composite beam of rectangular cross – section with all
fiber angles arranged to (0°/90°/0°/90°). The material properties of the beam are given as
E1= 144.80 GPa
,
E2= 9.65 GPa ,
G12=4.14 GPa
,
G13= 4.14 GPa
,
G23=3.45 GPa
,
ν12= 0.3 , ρ= 1550.1 kg/m³,
L= length of the composite laminated beam =0.381 m,
b= width of the laminated composite beam= 25.4 mm,
h= thickness of the each ply = 25.4 mm.
The first five natural frequencies of the laminated composite beam with various boundary
conditions are calculated and the numerical results are introduced in table 7. To validate the
proposed formulation, the present results are compared with the theoretical results of references.
Table 7
Natural frequencies (Hz) of AS4/3501-6 graphite-epoxy laminated composite beam
(0°/90°/0°/90°).
Mode no.
Clamped-
Clamped-
Clamped-
Clamped-
Clamped-
Clamped-
Simply
Simply
clamped
clamped
simply
simply
free
free
supported-
supported-
present
Ref.[54].
supported
supported
present
Ref.[54].
simply
simply
present
Ref.[54].
supported
supported
present
Ref.[54].
1
901.65
1054.4
721.4
784.9
151.1
181.8
540
557.8
2
1354.1
2509.2
2218.4
2223.9
986.4
1088.5
1754.2
1892.6
3
3954.4
4281.5
3828.7
4038.6
1054.1
1132.7
3604.4
3730.9
4
5849.7
6215.8
5824.4
6031.5
2097.7
2265.4
5865.1
5669.5
5
7982.4
8239.9
7954.4
8104.8
2476.0
2693.8
7467.4
7672.0
85
It is observed from table 7 that the present results are in very good agreement with the theoretical
results of ref.[34].
ANSYS 12 results for natural frequencies are same as MATLAB [59] results as shown in table 7,
and mode shapes figures for different boundary conditions are same as previous example 4.
Here table 7 listed the first five natural frequencies of the laminated composite beam with
various boundary conditions, which are obtained by use of the present dynamic stiffness
formulation.
Table 8
Natural frequencies (Hz) of T300/976 graphite-epoxy laminated composite beam 30°/-60°/30°/60°).considering shear deformation ignored.
Mode no.
Clamped-clamped
Clamped-simply
Clamped-free
Simply
Ref.[54]
supported
Ref.[54]
supported
Ref.[54]
supported-simply
Ref.[54]
1
674.1
496.2
106.2
396.4
2
1837.7
1515
659.5
1165
3
3541.8
3060.5
1819.2
2716.9
4
4837.3
4763.4
2419.1
4150.8
5
5714.9
5013
3035.9
4880.4
Comparison between table 6 and table 8. In order to analyses the effect of shear deformation on
the natural frequencies, table 8 shows the first five natural frequencies of the laminated
composite beam without the shear deformation embrace. The influence of shear deformation is to
decrease the natural frequencies. In general, the shear deformation has a more revealing on the
higher natural frequencies.
Mode shapes by MATLAB [59]:Now the next result is to present the five normal mode shapes using MATLAB for the example
2, considering clamped-free boundary condition, as shown in fig. 34All the data considering for this mode shapes taken from example 2.
86
Fig. 34 First five normal mode shapes of clamped-free laminated composite beam for example 2.
87
Now the next result is to present the five normal mode shapes using MATLAB [59] for the
example 2, considering simply supported-simply supported boundary condition, as shown in fig.
35-
88
Fig. 35 First five normal mode shapes of simply supported- simply supported laminated
composite beam for example 2.
Now the next result is to present the five normal mode shapes using MATLAB [59] for the
example 3, considering simply clamped-free supported boundary condition, as shown in fig. 36-
89
Fig. 36 First five normal mode shapes of clamped-free laminated composite beam for example 3.
For all other examples plotting of the mode shapes will be similar as fig. 34, 35 and 36 for
similar supported.
It is clear from fig. 34 to fig. 36 that the axial mode can be ignored for the first five modes; the
fourth and fifth modes in fig. 34 and the fourth mode in fig. 36 are imposing torsional modes, the
other modes are bending-torsion coupled modes. The material anisotropy has relatively
negligible effect on the mode shapes and the slender ratio has noticeable effect on all five modes
especially on the fifth node.
2. Higher order (second order) shear deformation theory
In this part, numerical results computed using the finite elements developed based on the higher
order shear deformation theories are compared with the theoretical results calculated for various
types of the laminated composite beams.
For this goal numerical results computed for simply supported and clamped-free laminated
composite beams in ref. [55] are utilized.
Further finite element results are also available for clamped-free and simply-supported beams in
ref. [55].
Numerical results computed for the finite elements HOBT4 and HOBT5 having four and five
degree of freedom per node, respectively is shown in table 9 and 10 for comparison purpose.
The present finite elements based theories which is given in chapter 3 (higher order beam theory)
is based on the first and second theories are represented as present theory 1(PT1) and present
theory 2(PT2) , respectively.
In all examples of the laminated composite beams the number of elements considered as 10 and
the results given in table 11-14 are non –dimensional frequencies.
MATLAB coding [59] procedure for the higher order shear deformation theory for PT1 and PT2
are given in chapter 4.
90
The element division for all problems considered as 10 and the degree of freedom for these two
theories are considered as 8 which are given in finite element formulation chapter 3. The input
data for MATLAB [59] same as previously theory first order shear deformation theory there is
only one change the degree of freedom.
Table 9 represents a comparison of non- dimensional natural frequencies calculated by various
finite elements based on higher order shear deformation theories. The numbers of layers are 6
and the lamination schemes are (0/0/90/90/0/0) respectively. The material properties are taken
from the example 4.
Another data considered asa= length of the laminated composite beam = 15 m
b= width of the laminated composite beam = 1 m
h= thickness of the laminated composite beam = 1 m
And the boundary considered assumed asSimply supported-simply supported beam.
Table 9
Comparison of non- dimensional natural frequencies of a simply – simply supported laminated
composite beam (0/0/90/90/0/0)
Mode no.
HOBT4[55]
HOBT5[55]
PT1(present)
PT1(ref.[55])
PT2(present)
PT2(ref.[55])
1
1.6561
1.6561
1.4254
1.4363
1.2346
1.4362
2
3.9232
3.9232
3.1654
3.3348
3.2465
3.3349
3
6.1913
6.1913
4.9654
5.4392
5.1354
5.4483
4
8.4704
8.4704
7.5482
7.8305
7.4658
7.8598
5
10.8035
10.8035
10.134
10.4952
10.4652
10.5196
91
It is observed from the table 9 that the prediction of the natural frequencies by PT1 and PT2 are
better than that of HOBT4 and HOBT5.
Table 10 represents a comparison of the laminated results of the natural frequencies for a
clamped- free laminated composite beam with the stacking sequence of layers are (0/90/90/0)
and number of layers are 4.
The material properties used for this purpose is taken from example 5 and the length, width and
thickness of the beam are as follows:a= length of the laminated composite beam = 15 m
b= width of the laminated composite beam = 1 m
h= thickness of the laminated composite beam = 1 m
Table 10
Comparison of non- dimensional natural frequencies of a clamped-free supported laminated
composite beam (0/90/90/0)
Mode no.
HOBT4[55]
HOBT5[55]
PT1(present)
PT1(ref.[55])
PT2(present)
PT2(ref.[55])
1
0.9241
0.9241
0.9185
0.9225
0.9134
0.9222
2
4.9852
4.9852
4.7658
4.9209
4.861
4.9212
3
11.8323
11.8323
11.234
11.5957
11.547
11.5963
4
19.5734
19.5734
17.224
17.3237
17.234
17.3237
5
27.7205
27.7205
18.924
19.0663
18.554
19.0693
It is observed from the table 10 that the prediction of the natural frequencies by PT1 and PT2 are
better than that of HOBT4 and HOBT5.
92
Chapter 6
CONCLUSION AND SCOPE FOR THE FUTURE
WORK
CONCLUSION
(1) The natural frequencies of different boundary conditions of laminated composite beam have been
reported. The program result shows in general a good agreement with the existing literature.
(2) Natural frequencies and mode shapes are obtained using first order shear deformation and higher
order beam theory for different types of laminated composites. It was found that natural
frequency increases with increase in mode of vibration.
(3) It is found that natural frequency is minimum for clamped –free supported beam and maximum
for clamped-clamped supported beam .In between these two, natural frequencies of simple-simple
and clamped-simple supported beam lies respectively.
(4) Mode shape was plotted for differently supported laminated beam with the help of MATLAB
[59] and ANSYS [58] to get exact idea of mode shape. Vibration analysis of laminated composite
beam was also done on ANSYS [58] to get natural frequency and same trend of natural frequency
was found to be repeated.
(5) The dynamic stiffness method defined previously is directly applied to the explained examples of
generally laminated composite beams to obtain the natural frequencies, the impact of Poisson
effect, slender ratio, material anisotropy, shear deformation and boundary conditions on the
natural frequencies of the laminated beams are analyzed. And it is found that the present results
are in very good agreement with the theoretical results of references.
(6) We assumed different examples and it is found that natural frequencies increase with the value of
E1 increases.
(7) It is found that natural frequencies decrease with the increase of beam length.
(8) The material anisotropy has relatively negligible effect on the mode shapes and the slender ratio
has considerable effect on all five modes especially on the fifth mode.
(9) In order to investigate the effect of shear deformation on the natural frequencies, it is found that
the shear deformation has a more relevance effect on the higher natural frequencies , means the
effect of shear deformation is to decrease the natural frequencies.
(10)
Two node higher order finite elements of eight degrees of freedom per node have also
been evolved for the vibration problems of the beam. Numerical results are calculated for various
lamination scheme and boundary conditions. The numerical results are compared with those of
the other higher order beam theories available in literature. The comparison study shows that the
present theories predict the natural frequencies of the laminated composite beams better than the
other higher order theories available in literature.
93
Scope for the future work:-
1. An analytical formulation can be derived for modelling the behaviour of laminated
composite beams with integrated piezoelectric sensor and actuator. Analytical solution
for active vibration control and suppression of smart laminated composite beams can be
found. The governing equation should be based on the first-order shear deformation
theory (Mindlin plate theory),
2. The dynamic response of an unsymmetrical orthotropic laminated composite beam,
subjected to moving loads, can be derived. The study should be including the effects of
transverse shear deformation, rotary and higher-order inertia. And also we can provide
more number of degree of freedom about 10 to 20 and then should be analyzed by higher
order shear deformation theory.
3. The free vibration characteristics of laminated composite cylindrical and spherical shells
can be analyzed by the first-order shear deformation theory and a meshless global
collocation method based on thin plate spline radial basis function.
4. An algorithm based on the finite element method (FEM) can be developed to study the
dynamic response of composite laminated beams subjected to the moving oscillator. The
first order shear deformation theory (FSDT) should be assumed for the beam model.
5. The damping behavior of laminated sandwich composite beam inserted with a visco elastic layer can be derived.
94
Chapter 7
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