“VIBRATION AND STABILITY OF LAMINATED COMPOSITE DOUBLY CURVED SHELLS BY HIGHER

“VIBRATION AND STABILITY OF LAMINATED COMPOSITE DOUBLY CURVED SHELLS BY HIGHER
“VIBRATION AND STABILITY OF LAMINATED
COMPOSITE DOUBLY CURVED SHELLS BY HIGHER
ORDER SHEAR DEFORMATION THEORY”
A THESIS SUBMITTED IN PARTIAL FULFILMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
Master of Technology
in
Structural Engineering
By
NEMI SHARAN
Roll No.-209CE2048
DEPARTMENT OF CIVIL ENGINEERING
NATIONAL INSTITUTE OF TECHNOLOGY, ROURKELA
ODISHA-769008
MAY 2011
“VIBRATION AND STABILITY OF LAMINATED
COMPOSITE DOUBLY CURVED SHELLS BY HIGHER
ORDER SHEAR DEFORMATION THEORY”
A THESIS SUBMITTED IN PARTIAL FULFILMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
Master of Technology
in
Structural Engineering
By
NEMI SHARAN
Roll no-209CE2048
Under the guidance of
Dr. A.V. Asha
DEPARTMENT OF CIVIL ENGINEERING
NATIONAL INSTITUTE OF TECHNOLOGY, ROURKELA
ODISHA-769008
MAY 2011
National Institute of Technology
Rourkela
CERTIFICATE
This is to certify that the thesis entitled “VIBRATION AND STABILITY OF
LAMINATED COMPOSITE DOUBLY CURVED SHELLS BY A HIGHER ORDER
SHEAR DEFORMATION THEORY” submitted by Mr. NEMI SHARAN in partial
fulfillment of the requirements for the award of Master of Technology Degree in Civil
Engineering with specialization in Structural Engineering at the National Institute of
Technology, Rourkela (Deemed University) 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.
Date:
Dr. A.V.Asha
Department of Civil Engineering
National Institute of Technology
Rourkela – 769008
ACKNOWLEDGEMENT
The satisfaction and euphoria on the successful completion of any task would be incomplete
without the mention of the people who made it possible whose constant guidance and
encouragement crowned my effort with success.
I am grateful to the Dept. of Civil Engineering, NIT ROURKELA, for giving me the
opportunity to execute this project, which is an integral part of the curriculum in M.Tech
programme at the National Institute of Technology, Rourkela.
I would also like to take this opportunity to express heartfelt gratitude for my project guide
Dr. A.V Asha, who provided me with valuable inputs at the critical stages of this project
execution. My special thanks are due to Prof. M. Panda, Head of the Civil Engineering
Department, for all the facilities provided to successfully complete this work
I am also very thankful to all the faculty members of the department, especially Structural
Engineering
specialization
for
their
constant
encouragement,
invaluable
advice,
encouragement, inspiration and blessings during the project.
Submitting this thesis would have been a Herculean job, without the constant help,
encouragement, support and suggestions from my friends, especially Alok, Pragyan, Javed,
Trishanu, for their timely help. I will relish your memories for years to come.
Last but not the least I would like to thank my parents, who taught me the value of hard
work by their own example. I would like to share this bite of happiness with my mother and
father. They rendered me enormous support during the whole tenure of my stay at NIT,
Rourkela.
Date:
Nemi Sharan
Roll No:-209CE2048
M.Tech (Structural Engineering)
Department of Civil Engineering
NIT, Rourkela,Odisha
CONTENTS
Page No
ABSTRACT
i
LIST OF TABLES
ii
LIST OF FIGURE
iv
NOTATIONS
v
CHAPTER 1 :
1.1
1.2
1.3
CHAPTER 2 :
INTRODUCTION
1-4
General
Application of shells
Vibration of composite Shells
1
3
4
LITERATURE REVIEW
5-10
2.1
Introduction
5
2.2
Objective and scope of present investigation
10
CHAPTER 3:
THEORY AND FORMULATION
11-36
3.1
Introduction
12
3.2
Basic assumption
13
3.3
Geometry of Shell
13
3.4
Strain Displacement Relation
16
3.4.1 First Order Shear Deformation Theory
16
3.4.2 Higher Order Shear Deformation Theory
18
3.5
Stress-Strain Relations
20
3.6
Governing Equation
21
3.6.1
Strain Energy
21
3.6.2
Kinetic Energy
23
3.6.3
Hamilton’s Principle
24
3.7
Stress Resultants and Stress Couples
26
3.8
Boundary Conditions
33
CHAPTER 4:
NUMERICAL RESULTS AND DISCUSSIONS
37-48
4.1
Introduction
38
4.2
The validation of the formulation and comparison of results
38
4.3
Numerical result
41
4.3.1 Vibration analysis
42
4.3.2 Stability analysis
46
CHAPTER 5:
5.1
CONCLUSION
5.2
Scope of future work
49-51
52
REFERENCES
53-57
APPENDIX
58-61
ABSTRACT
The present study deals with a higher order shear deformation theory of laminated shells as
suggested by Reddy and Liu. The theory is based on a displacement field in which the
displacements of the middle surface are expanded as cubic functions of the thickness
coordinate, and the transverse displacement is assumed to be constant through the thickness.
This displacement field leads to the parabolic distribution of the transverse shear stresses (and
zero transverse normal strain) and therefore no shear correction factors are used. The theory
is also based on the assumption that the thickness to radius ratio of shell is small compared to
unity and hence negligible.
The governing equations are derived in orthogonal curvilinear coordinates. These equations
are then reduced to those of doubly curved shell. All the quantities are suitably nondimensionalised. The Navier solution has been used which gives rise to a generalized
eigenvalue problem in matrix formulation. The natural frequencies for vibration and buckling
loads of laminated orthotropic doubly curved shells and panels with simply supported ends
are obtained.
The eigenvalues, and hence the frequency parameters are calculated by using a standard
computer program. To check the derivation and computer program, the frequencies in HZ for
different layer are compares with earlier results.
The lowest value of frequency parameter and buckling load are computed for the laminated
composite doubly curved shell. The effects of various parameters such as number of layers,
aspect ratio, modular ratio, etc on the above are studied.
Frequency also increases as number of layers of the shell increases for symmetric cross-ply
layout. But when there is unsymmetrical cross-ply layout, then frequency decreases. With the
increasing of modular ratio, non-dimensional frequency is also increasing.
i
LIST OF TABLES
Table Number
4.1
Description
Page
Comparison of lowest non dimensional Frequency
39
for a simply supported laminated composite
spherical shell (a/h=100)
4.2
Comparison of lowest non dimensional Frequency
40
for a simply supported laminated composite
spherical shell (a/h=10)
4.3
Comparison of non-dimensional buckling load for a
41
simply supported laminated composite doubly
curved shell (a/h=10)
4.4
4.5
Variations in non-dimensional frequency parameter
for laminated composite paraboloid shell by higher
order shear deformation theory
Variations in non-dimensional frequency parameter
42
43
for laminated composite elliptical shell by higher
order shear deformation theory.
4.6
Variations in non-dimensional frequency parameter
43
for laminated Composite doubly curved shell by
higher order shear deformation theory with variation
of modular ratio
4.7
Variations in non-dimensional frequency parameter
44
with different layers for laminated composite doubly
curved shells a/h=100
4.8
Variations
in
non-dimensional
buckling
load
46
parameter for laminated composite paraboloidal
shell by higher order shear deformation theory
4.9
Variations
in
non-dimensional
buckling
load
parameter for laminated composite elliptical shell by
higher order shear deformation theory
ii
47
4.10
Variations
in
non-dimensional
buckling
load
47
parameter for laminated composite doubly curved
shell by higher order shear deformation theory with
variation of modular ratio of [0/90/90/0] and
a/h=100
4.11
Variations
in
non-dimensional
buckling
load
parameter with different layers for laminated
composite doubly curved shells a/h=100
iii
48
LIST OF FIGURES
FIGURE NUMBER
DESCRIPTION
PAGE
1.2.1
Elliptical paraboloid shell
04
1.2.2
Hyperbolic paraboloid shell
04
3.1
Geometry of a laminated shell
10
3.2
Stress and moment resultants
22
4.1
Variation in non-dimensional frequency of
Hyper paraboloid shells with R2/a
4.2
Variation of non-dimensional frequency of
cross-ply elliptical shell with R1/a
4.3
45
45
Variation of non-dimensional frequency of
cross-ply spherical shell, hyperbolic and
elliptical paraboloid shell with modular ratio
iv
46
LIST OF NOTATIONS
The principal symbols used in this thesis are presented for easy reference. A symbol is used
for different meaning depending on the context and defined in the text as they occur.
English
Notation
Description
A1 , A2
Lames Parameter or Surface Metrics
Aij , Bij , Dij
Laminate Stiffnesses
Eij , Fij , H ij
,C
C
The matrix notation described in appendix
The distance between points  ,  , z  and   d ,   d , z  dz 
dS


dV
Volume of the shell element
dt
Time derivative
E1 , E2
Longitudinal and transverse elastic moduli respectively
G12 , G13 , G23
In plane and Transverse shear moduli
h
Total thickness of the shell
hk
Distance from the reference surface to the layer interface
k211, k2 22
The shear correction factors.
k1 , k2 , k6
Quantities whose expression are given in equation [11.b]
0
0
0
2
2
2
1
1
k1 , k 2 , k 6 , k1
k2 , k6 , k4 , k5
Quantities whose expressions are given in equation [11.b]
v
L1 , L2 , L3
lames coefficient
n1 and n2
axial load applied in α and β direction
a ,b
size of the spherical shell
N i , M i , Pi , Q1
Q2 , K 1 , K 2
Stress resultant and stress couples
M,M
Matrix notation described in appendix
m
Axial half wave number
n
Circumferential wave number
Q ij (k)
Reduced stiffness matrix of the constituent layer
R
Radius of the reference surface of spherical shell
R1 , R2
Principal radii of curvature
T
Kinetic energy
U, V, W, 1,  2
Amplitude of displacement
Us
Strain energy
U , V , W , 1 ,  2
Non dimensionalised amplitude of displacement
u, v, w
Displacement component at the reference surface
u, v, w
Non dimensionalised displacement component at any point in
the shell
Ii , I i , I i '
Inertia terms defined by equation
{X}
Column matrix of amplitude of vibration or eigenvector.
vi
{X}
Non dimensionalised form of column matrix
Greek
Notation
Description
,,z
Shell coordinates
I
Strain components
1 , 2 , 4
0
0
0
 50 , 60
Quantities whose expressions are given in equation[11.b]
 ,
Non dimensionalised axial and circumferential wave parameter
m
n
Material density
i
Stress components
1 ,  2
Rotation at z = 0 of normals to the mid-surface with respect to
 and  axes

Natural circular frequency parameter
vii
CHAPTER 1
INTRODUCTION
1
CHAPTER 1
INTRODUCTION
1.1 GENERAL:
An increasing number of structural designs, especially in the aerospace, automobile, and
petrochemical industries are extensively utilizing fiber composite laminated plates and shells
as structural elements. The laminated orthotropic shell belongs to the composite shell
category. One of the important factors in the analysis of the layered shells is its individual
layer properties, which may be anisotropic, orthotropic or isotropic.
A shell is a curved, thin walled structure. Two important classes of shells are plates
(shells which are flat when un-deformed) and membranes (shells whose walls offer no
resistance to bending). Shells may be made of a single homogeneous or anisotropic material
or may be made of layers of different materials.
The primary function of a shell may be to transfer loads from one of its edges to
another, to support a surface load, to provide a covering, to contain a fluid, to please the eye
or a combination of these.
A thin shell is defined as a shell with a thickness which is small compared to its other
dimensions and in which deformations are not large compared to thickness. A primary
difference between a shell structure and a plate structure is that, in the unstressed state, the
shell structure has curvature as opposed to plates. Membrane action in a shell is primarily
caused by in-plane forces (plane stress), though there may be secondary forces resulting from
flexural deformations. Where a flat plate acts similar to a beam with bending and shear
stresses, shells are analogous to a cable which resists loads through tensile stresses. The ideal
thin shell must be capable of developing both tension and compression.
At the present time, many of the existing methods of analysis for multilayered
laminated orthotropic plates and shells are direct extensions of those developed earlier for
homogeneous isotropic and orthotropic plates and shells. In the classical lamination theory
(CLT), the well known Kirchhoff-love hypothesis is assumed to be verified. The range of
applicability of the C.L.T solution has been well established for laminated flat plates by
Pagano (1969, 1970). These analyses have indicated that the hypothesis of no deformable
2
normal, while acceptable for isotropic plates and shells, is often quite unacceptable for
multilayered anisotropic plates and shells with very large ratio of Young’s modulus to shear
modulus, even if they are relatively thin.
A Mindlin-type first-order transverse shear theory deformation theory (S.D.T) has
been first developed by Dong and Tso (1972) for multilayered anisotropic shells. The present
study deals with a higher-order but simple shear deformation theory of laminated shells as
proposed by Reddy and Liu [24] for plates and shallow shells. The theory is based on a
displacement field in which the displacements of the middle surface are expanded as cubic
functions of the thickness coordinate, and the transverse displacement is assumed to be
constant through the thickness. The latter assumption is equivalent to neglecting the
stretching of the normal to the middle surface of the shell. This displacement field leads to
the parabolic distribution of the transverse shear stresses (and zero transverse normal strain),
and therefore no shear correction factors are used. The governing equations are derived in
curvilinear orthogonal coordinates. These equations are then reduced to doubly curved shell.
The free vibration and stability analysis of the doubly curved shell is carried out by
computing lowest value of frequencies and buckling loads for various shell parameters.
1.2 APPLICATION OF SHELLS
Thin shell structures are light weight constructions using shell elements. These elements are
typically curved and are assembled to large structures. Typical applications are fuselages of
aero planes, boat hulls and roof structures in some buildings.
Shell structures are mainly used in industrial applications such as automobile, civil,
aerospace and petrochemical engineering. Various types of shells are used in civil field such
as conoid, hyperbolic paraboloid and elliptical paraboloid shell. All are used for roofing to
cover large column-free areas.
Laminated composites are such type of material which has high strength to weight
and strength to stiffness ratios. The mechanical properties of the laminated composites
depend on the degree of orthotropy of the layers, ratio of the transverse shear modulus to the
in-plane shear modulus and stacking sequence of laminates. Many of the classical theories
developed for thin elastic shells are based on the Love-Kirchhoff assumptions in which the
3
normal to the mid-plane before deformation is considered to be normal and straight after the
deformation.
Figure:1.2.1 Elliptical Paraboloidal Shell
Figure:1.2.2 Hyperbolic Paraboloidal Shell
1.3 VIBRATION OF COMPOSITE SHELLS
Shell vibration was first studied by Germaine in 1821.Aron (1874) gave a set of five
equations which reduced to plate equations when curvatures were zero. Logical extensions of
the beam and plate equations for both transverse and in-plane motion were introduced by
Love (1863-1940) in 1888. Love’s equations brought the basic development of the theory of
vibration of continuous structures, which have a thickness that is much less than any length or
surface dimensions, to a satisfying end.
4
CHAPTER 2
LITERATURE REVIEW
5
CHAPTER 2
LITERATURE REVIEW
2.1 INTRODUCTION
The increasing use of high performance fibre reinforced laminated shell structures in
aerospace and other applications are well documented. A reliable prediction of the response
of these laminated shells or doubly-curved panels must account for transverse shear
deformation. Since the matrix material is of relatively low shearing stiffness as compared to
the fibers, polymeric composite shell type structures are highly prone to transverse shear
related fatigue failures. Additionally, a solution to the problem of deformation of laminated
shells of finite dimensions must satisfy the prescribed boundary conditions, which introduce
additional complexities into the analysis.
Most of the investigations use
(a) The classical lamination theory (CLT) or the first-order shear deformation theory
(FSDT). (A detailed review of CLT-based analyses is available in, e.g., Bert
and Reddy [6], and Chaudhuri et al. [8], while that relating to their FSDT
counterparts is available in, e.g., Abu Arja and Chaudhuri [1], Chaudhuri and
Abu Arja [9], and Librescu et al. [17].)
(b) Higher-order shear deformation theory(HSDT)
Leissa and Kadi (1971) worked on the effect of curvature upon the vibration frequencies of
shallow shells. For this purpose, the shell was chosen to have a rectangular boundary
supported by shear diaphragms and yielded exact, closed form solutions for convenient
comparison in the linear, small deflection regime and, at the same time, represented a
situation which could readily occur in practical application. He extended his analysis into non
linear, large deflection regime by assuming mode shapes satisfying the non-linear field
equations of motion and compatibility approximately by means of the Galerkin procedure.
Bhimaraddi (1983) worked on the free, undamped vibration of an isotropic circular
cylindrical shell with higher order displacement model, giving rise to a more realistic
6
parabolic variation of transverse shear strains. In this work in-plane inertia, rotatory inertia,
and shear deformation effects on the dynamic response of cylindrical shells was studied.
Bhimaraddi (1991) studied the free vibration analysis of homogeneous and laminated
doubly curved shells on rectangular planform and made of an orthotropic material using the
three- dimensional elasticity equations. A solution was obtained utilizing the asumption that
the ratio of the shell thickness to its middle surface radius is negligible as compared to unity.
However, it was shown that by dividing the shell thickness into layers of smaller thickness
and matching the interface displacement and stress continuity conditions, very accurate
results could be obtained even for very thick shells.
Touratier (1991) generalized the geometrically linear shear deformation theories for small
elastic strains for multilayered axisymmetric shells of general shape without any assumption
other than neglecting the transverse normal strain. He used a certain sine shear function and
no shear correction factor was used. He presented a new theory and compared with the
classical theories for a simply-supported thick laminated cylindrical shell under an internal
pressure.
Leissa and Chang (1996) derived a rigorous theory which governs the linearly elastic
deformation of shells made of laminated composite materials including shear deformation
and rotary inertia effects. The equations derived are applicable for static and dynamic
problems for shells of arbitrary curvature, but constant thickness.
Qatu (1999) derived accurate equations of deformation for laminated composite deep, thick
shells. These equations include shells with a pre- twist and accurate force and moment
resultants which are considerably different from plates. He also derived consistent set of
equations of motion, energy functionals and boundary conditions.
Tabiei and Tanov(1999) worked on the finite element formulation of a higher order shear
deformation shell element for non linear dynamic analysis with explicit time integration
scheme. They combined co rotational approach with velocity strain equations of a general
third order theory in the formulation of a four noded quadrilateral element with selectively
reduced integration.
7
Jayasuriya et al (2002) worked on development of isoparametric, finite element for doubly
curved anisotropic laminated shells of revolution. Variable-order Legendre polynomials were
used in the derivation of interpolation functions. They also developed a user -friendly finiteelement code based on the higher-order theory and isoperimetric hierarchical finite-element
formulation, and a numerical verification and application of the above model for the study of
static and dynamic analysis with mechanical and hydrothermal loadings.
Messina (2003) described the dynamics of freely vibrating multilayered doubly curved
shells. The relevant governing differential equations, associated boundary conditions and
constitutive equations were derived from one of Reissners mixed variational theorems. Both
the governing differential equations and the related boundary conditions were presented in
terms of resultant stresses and displacements. In spite of the multi-layer nature of the shell,
the theory was developed as if the shell were virtually made of a single layer.
Sudhakar et al (2003) formulated a degenerate shell element using higher order shear
deformation theory taking the piezoelectric effect in account. An eight-noded element was
used to derive global coupled electro elastic behavior of the overall structure.
Amabili (2005) investigated vibrations of doubly curved shallow shells with rectangular
base, simply supported at the four edges and subjected to harmonic excitation normal to the
surface in the spectral neighborhood of the fundamental mode . In-plane inertia and
geometric imperfections were taken into account. The solution was obtained by Lagrangian
approach.
Oktem and Chaudhuri (2006) worked on the Fourier series approach to solve a system of
five highly coupled linear partial differential equations, generated by the HSDT-based
laminated shell analysis, with the C4-type simply supported boundary condition prescribed
on two opposite edges, while the remaining two edges are subjected to the SS3-type
constraint. The numerical accuracy of the solution was ascertained by studying the
convergence characteristics of the deflection and moment of a cross-ply spherical panel, and
also by comparison with the available FSDT (first-order shear deformation theory) based
analytical solution.
8
Adam (2006) studied the nonlinear flexural vibrations of shallow shells composed of three
thick layers with different shear flexibility, which were symmetrically arranged with respect
to the middle surface. He considered shell structures of polygonal platform were hard hinged
simply supported (i.e. all in-plane rotations and the bending moment vanish) with the edges
fully restrained against displacements in any direction. He formulated kinematic field
equations layer wise by first order shear deformation theory. Numerical results of rectangular
shallow shells in nonlinear steady-state vibration were presented for various ratios of shell
rise to thickness, and non-dimensional load amplitude.
Kulikov and Carrera (2008) worked on higher shell models by new concept of I surfaces
inside the shell body. They introduced N (N≥3) I surfaces. The general 3N-parameter shell
model was proposed in the framework of the Lagrangian description.
Amabili and Reddy (2009) worked on the use of higher order shear deformation non linear
theory for shells of generic shape, taking geometric imperfections into account. They found
that results were obtained by keeping non-linear terms of the Von Karman type for
amplitudes of about two times the shell thickness.
Tripathy and Suryanarayan (2009) presented vibration and buckling of thin-walled tubular
beam shells typical of automotive structures, which are fabricated by joining sheet metal
stampings along the two longitudinal edges with periodic spot welds, adhesive bonding, or
combination of spot welds and bonding, known as weld bonding. Solutions were obtained for
such beam shells of rectangular cross section with two opposite ends simply supported. The
beam shell was modeled as an assembly of the constituent walls and Levy-type formulation
was used to obtain a series solution for the transverse displacement of each of the walls.
9
2.2 OBJECTIVE AND SCOPE OF PRESENT INVESTIGATION
In this project, analytical solution of frequency characteristics and buckling loads of
laminated composite doubly curved shells will be presented. Higher order shear deformation
theory as proposed by Reddy and Liu will be used for the formulation.
The governing equations have been developed. These equations are then reduced to the
equations of motion for doubly curved panels and the Navier solution has been obtained for
cross-ply laminated composite doubly curved panels. The resulting equations are suitably
non-dimensionalized. The Eigen value problem is then solved to obtain the free vibration
frequencies and buckling loads.
10
CHAPTER 3
THEORY AND FORMULATION
11
CHAPTER-3
THEORY AND FORMULATION
3.1 INTRODUCTION
In the present study, laminated composite doubly curved shells are considered. A number of
theories exist for layered composite shells. Many of these theories were developed originally
for thin shells, and are based on the Kirchhoff-Love kinematic hypothesis that straight lines
normal to the undeformed midsurface remain straight and normal to the middle surface after
deformation and undergo no thickness stretching.
The higher order theory is based on a displacement field as proposed by Reddy and
Liu in which the displacements of the middle surface are expanded as cubic functions of the
thickness coordinate and the transverse displacement is assumed to be constant through the
thickness. The additional dependent unknowns introduced with the higher order powers of
the thickness coordinate are evaluated in terms of the derivatives of the transverse
displacement and the rotation of the normal at the middle surface. This displacement field
leads to the parabolic distribution of the transverse shear strains and hence shear stresses and
zero transverse normal strain, and therefore no shear correction factors are used. In first order
shear deformation theory, shear correction factors are used.
The present study deals with the free vibration and buckling characteristics of thin
laminated composite cross-ply doubly curved panels. The displacement components u, v, and
w in the α, β, and z directions in a laminate element can be expressed in terms of the
corresponding mid-plane displacement components u°, v° , w°, and the rotations ɸ1 , ɸ2 of the
mid-plane normal along α and β axes. The higher order shear deformation theory is based on
a displacement field in which the displacements of the middle surface are expanded as cubic
functions of the thickness coordinate and the transverse displacement is assumed to be
constant through the thickness. This displacement field leads to parabolic distribution of the
transverse shear stresses and zero transverse normal strain and hence no shear correction
factors are used.
12
The governing equations including the effect of shear deformation are presented in
orthogonal curvilinear co-ordinates for laminated composite shells. These equations are then
reduced to the governing equations for free vibration of laminated composite cross-ply
doubly curved shells. The equations are suitably non-dimensionalised. The Navier solution
has been used and the generalized eigen value problem so obtained in matrix formulation is
solved to obtain the eigen values which are the natural frequencies and the buckling loads.
3.2 BASIC ASSUMPTIONS
A set of simplifying assumptions that provide a reasonable description of the behavior
of thin elastic shells is used to derive the equilibrium equations that are consistent with the
assumed displacement field.
1. No slippage takes place between the layers.
2. The effect of transverse normal stress on the gross response of the laminate is assumed
to be negligible.
3. The line elements of the shell normal to the reference surface do not change their
length after deformation.
4. The thickness coordinate of the shell is small compared to the principal radii of
curvature (z/R1, z/R2 <<<1).
5.
Normal to the reference surface of the shell before deformation remains straight, but
not necessarily normal, after deformation (a relaxed Kirchhoff -Love hypothesis).
3.3 GEOMETRY OF SHELL
Figure 1 shows an element of a doubly curved shell. Here (α, β, z) denote the orthogonal
curvilinear coordinates (shell coordinates) such that α and β curves are lines of curvature on
the mid surface, z = 0, and z-curves are straight lines perpendicular to the surface, z = 0. For
the doubly curved shells discussed here, the lines of principal curvature coincide with the coordinate lines. The values of the principal curvature of the middle surface are denoted by K 1
and K2.
13
Figure.3.1 Geometry of a laminated shell
The position vector of a point on the middle surface is denoted by r and the position of a
point at distance, z, from the middle surface is denoted by R. The distance, ds, between points
(α, β, z) and  α dα, β d β, z dzis determined by
(ds) ² dr dr
dr 
(1)
r
r
d 
d


(2)
The magnitude ds of d r is given in equation (2), the vectors
r
r
and
are tangent to the


and β coordinate lines. Then equation (1) can be modified as
ds 2 
r r
r r
r r
(d ) 2 
(d ) 2  2
d .d
 
 
 
(3)
The following derivation is limited to orthogonal curvilinear coordinates which coincide
with the lines of principal curvature of the neutral surface. The third term in equation (3) thus
becomes
14
2
r r
r r

d .d  2
.
cos d .d  0
 
 
2
(4)
Where we define
r r
r

  
r r
r

  
2
 A1
2
(5)
2
 A2
2
Now the equation (3) becomes
ds 2  A12 (d ) 2  A2 2 (d ) 2
(6)
This equation is called the fundamental form and A1 and A2 are the fundamental form
parameters, Lame parameters, or surface metrics. The distance, dS, between points (α, β, z)
and (α dα, β d β, z dzis given by
ds 2  d R.d R.  L12 (d ) 2  L2 2 (d ) 2  L3 2 (dz) 2
In which d R 
R
R
R
d 
d 
dz and L1, L2, and L3 are the Lame’s coefficients.


z


z 
z
L1  A1 1   ; L2  A2 1   ; L3 =1
 R2 
 R1 
The vectors
(7)
(8)
R
r
R
r
and
are parallel to the vectors
and
.




From the figure the elements of area of the cross section are

z
da1  L1 d .dz  A1 1   d .dz
 R1 
(9)
15

z 
da 2  L2 d .dz  A2 1   d .dz
 R2 
The strain displacement equations of a shell are an approximation, within the assumptions
made previously, of the strain displacement relations referred to orthogonal curvilinear
coordinates. In addition, it is assumed that the transverse displacement, w, does not vary with
z.
3.4
STRAIN DISPLACEMENT RELATIONS
3.4.1 FIRST ORDER SHEAR DEFORMATION THEORY
According to the first order shear deformation theory, the displacement field is given by:

z 
u  1  u  z1
R1 

(10)

z 
v  z 2
v  1 
 R2 
ww
Here (u,v,w) = the displacement of a point (α, β, z) along the (α, β, z) coordinates; and (u, v,
w) = the displacements of a point (α, β, 0). Now substituting equation [10] in the strain
displacement relations referred to an orthogonal curvilinear coordinate system we get
 1   1 0  zk1
 2   2 0  zk 2
 4   40
(11.a)
 5   50
 6   6 0  zk 6
Where,
16
 10 
1 u w
.
 ;
A1  R1
 20 
1 v w
.
 ;
A2  R2
 60 
v 1 u

;
A1  A2 
1
.
 4 0  2 
1 w u
 ;
A2  R1
 5 0  1 
1 w v
 ;
A1  R2
……………………
k1 
1 1
.
;
A1 
k2 
1  2
.
;
A2 
k6 
1 2 1 1 1  1
1  1 v 1 u 
;
.

   

A1  A2  2  R1 R2  A1  A2  
11.b
1 and  2 are the rotations of the normals to the reference surface, z = 0, about the α and β
coordinate axes, respectively. The displacement field in equation [10] can be used to derive
the general theory of laminated shells.
17
3.4.2 HIGHER ORDER SHEAR DEFORMATION THEORY
The higher order displacement field relations as proposed by Reddy and Liu [24] are

z 
1  w 
 4 
u  1  u  z1  z 3  2  1  

R1 
A1   
 3h 


z 
1  w 
 4 
v  z 2  z 3  2   2 


v  1 
R
A


3
h




2 
2


(12)
ww
Here (u,v,w) = the displacement of a point (α, β, z) along the (α, β, z) coordinates; and (u, v,
w) = the displacements of a point (α, β, 0). Now substituting equation [12] in strain
displacement relations referred to an orthogonal curvilinear coordinate system, we get

 z k
 1   1 0  z k 1 0  z 2 k1 2
 2   20
0
2
 z 2k2

2

 4   4 0  z 2 k 41
 5   5 0  z 2 k 51

 6   6 0  z k6 0  z 2 k6 2

……………………….
Where,
18
(13)
 10 
1 u w
.
 ;
A1  R1
 20 
1 v w
.

;
A2  R2
 60 
v 1 u

;
A1  A2 
1
.
 4 0  2 
1 w
;
A2 
 5 0  1 
1 w
;
A1 
k1 
1 1
.
;
A1 
k2 
1  2
.
;
A2 
0
0
1  2 1 1 1  1
1  1 v
1 u 

;
.

  

A1  A2  2  R1 R2  A1  A2  
w 
 4 

  2   2 
 
 h 
k6 
0
k4
1
w 
 4 
1
k 5   2  1 

 
 h 
2w 
 4  
2

k1   2  1 
2 
 3h    
k2
2
 4
  2
 3h
 4
2
k 6   2
 3h
…………….
(14)
2
  2  w 



2 
   
2w
  2 1



2


  



Where 1 and  2 are the rotations of the normals to the reference surface, z = 0, about α and
β coordinate axes, respectively. The stress-strain relations are as given in equation (15).
19
3.5 STRESS-STRAIN RELATIONS:
The stress-strain relation for the kth orthotropic layer takes the following form:
 1k 
 
k
k
k 
Q11
Q12
0
0 Q16
 k 
 k
k
k 
 2
0
0 Q26 
Q12 Q22
 k 
k
k

0 Q44 Q45
0 
 4   0


 
k
k
0 Q45
Q55
0 
 0
 5k 
Q k Q k
k 
 
0
0 Q66
26
 16

 k

 6 
 1k 
 
 k 
 2
 k 
 4 
 
 5k 
 
 k
 6 
(15)
For special orthotropic material, in which the principal axis direction coincides with the axis
of the material direction,
Q16K  Q26K  Q45K  0
Then,
 1k 
 
k
k
Q11
Q12
0
0
0 
 k 


k
k
 2
Q12 Q22
0
0
0 

 k 
k
0 Q44
0
0 
 4    0


 
k
0
0 Q55
0 
 0
 5k 
 0
k 
 
0
0
0 Q66


 k
 6 
 1k 
 
 k 
 2
 k 
 4 
 
 5k 
 
 k
 6 
(16)
For generalized plane stress condition, the above elastic moduli
are related to the
engineering constants as follows:
Q11 
E1
1  12 21
Q12 
E1 21
E 2 12

1  12 21 1  12 21
Q22 
E2
1  12 21
(17)
20
Q44 = G13
Q55 = G23
Q66 = G12
E1  12

E2  21
3.6 GOVERNING EQUATIONS
The governing differential equations, the strain energy due to loads, kinetic energy and
formulations of the general problem are derived on the basis of Hamilton’s principle.
The equation of motion is obtained by taking a differential element of the shell as
shown in Figure 3.1. The figure 3.1 shows an element with internal forces like membrane (N1,
N2, and N6), shearing forces (Q1, and Q2) and the moment resultants (M1, M2 and M6).
3.6.1 STRAIN ENERGY
The strain energy of a differential shell element can be written as,
U
1
2 
   
1 1
z
  2 2   6 6   4 4   5 5 .dV
(18)
dV = Volume of shell element

z  
z 
.d .d .dz.
dV  A1 1   A2 1 
 R1   R2 
(19)
The variation of strain energy U is given by
U 
1
2 
   
z
1
1
  2 2   6 6   4 4   5 5 .dV
(20)
The equation [20] is independent of the material property. Substituting the variation of strain
function and dV in equation (20),
21
U
1
2 
  [ N 
 M 1 k1  N 2 2  M 2 k 2  N 6 w1  M 6 1  N 6 w2  M 6 2 
0
0
1 1
z


z 
z 
 5 ] A1 A2 dd .
Q1 1   4  Q2 1 
 R1 
 R2 
(21)
The variation of strain energy is given as:
U 
1
2 
  [ N 
1
z
0
1
 M 1k1  N 2 2  M 2k 2  N 6w1  M 6 1  N 6w2  M 6 2 
0


z 
z 
 5 ] A1 A2 dd .
Q1 1   4  Q2 1 
 R1 
 R2 
(22)
Substituting for  10 ,  2 0 , k1 , k 2 .... .in equation (22) and integrating the resulting expression, the
equation becomes,
U    {
 
 N2

( N1 A2 )
A N A A
A (M 1 A2 )
A
u  N1v 1  1 1 2 w  M 1 2 1 
1  N 2u 2


R1



A1 A2
( N 2 A1 )
A (M 2 A1 )
A ( N 6 A2 )
A
w 
v  M 21 2 
 2  N 6u 1 
v  M 61 1
R2







( M 6 A2 )
A ( N 6 A1 )
A ( M 6 A1 )
u 
 2  N 6v 2 
u  M 6 2 2 
 1  Q1 A1 A2  1   





R1 


(Q1 A2 )
u  (Q2 A1 )
w  Q2 A1 A2  1   
w d .d

R


2

 N u  N v  Q w  M   M  A d
  N v  N u  Q w  M   M  A d

1
6
2
1
6
1
2
1
2
6
2
2
2
6
1
22
1

(23)
3.6.2 KINETIC ENERGY
If U be the displacement vector, the kinetic energy of the shell element is given by,
T 


1

U
.
U
dV
2 v
(24)

where  is the mass density and U represents differentiation with respect to time.
U  U1 n1  U 2 n2  W n1
 u  2  v  2  w  2 h 2    2    2  
h
1
T
   2   A1 A2 .d .d
         




2
 t   t   t  12  t   t   
(25)
The variation of kinetic energy is given as
T  h 

       h2     
 u  u  v  v w w 12 (1  1  2  2 )A1 A2 .d .d
(26)

This equation contains time derivatives of the variations, ie  u etc. To eliminate these terms,
integrating equation (26) by parts (the variations of limits t = t 1 and t = t2 must vanish) and
neglecting the terms h
3

12 1 and h
3

12 2 which represent rotatory inertia, the above
equation reduces to,
t2
t2

T .dt   h 
t1
t1

    
 u  u  v  v w


w A1 A2 .d .d .dt

(27)
If the shell is subjected to both body and surface forces and if q 1,q2 and q n are the
components of body and surface forces along the parametric lines, then the variation of work
done by the external loads are,
Ws  

 q u  q v  q wA .A .d .d
1
2
n
1
2


We1   N1u  N 6v  Q1w  M 11  M 6 2 A2 .d



We 2   N6u  N 2v  Q2w  M 61  M 2 2 A1.d

23
(28)
3.6.3 HAMILTON’S PRINCIPLE
The equations of equilibrium are derived by applying the dynamic version of the principle of
virtual work, which is the Hamilton’s Principle.
It states that among the set of all admissible configurations of system, the actual motion
t2
makes the quantity  L.dt stationary, provided the configuration is known at the limits t = t 1
t1
t2
and t = t2. Mathematically this means
  L.dt
t1
Here, L is called Lagrangian and is equal to
L= T - (U –V)
(29)
Where, T = Kinetic energy, U= Strain energy, V= potential of all applied loads,  =
Mathematical operation called variation. It is analogous to partial differentiation. It is clear
from equation [29] that the Lagrangian consists of kinetic, strain energy and potential of
applied loads.
By applying the dynamic version of the principle of virtual work (Hamilton’s
principle), integrating the displacement gradients by parts in the resulting equation and
setting the coefficients of δu, δv, δw, δφ 1 and δφ2 to zero separately, the following equations
of equilibrium are obtained:
u :
 

 ( A2 M 1 )
( A1M 6 ) 
( A2 N 2 ) ( A1 N 6 )
A
A
A
A
1 w


 N 2 2  N 6 1  k1 
 M6 1  M2 2 
 A1 A2
  q1 A1 A2   I 1u  I 21  I 3







A2  
 


(30.a)
v :
 
 '
' 
' 1 w
( A2 N 6 ) ( A1 N 2 )
 ( A2 M 6 )
( A2 M 6 ) 
A
A
A
A

 N1 1  N 6 2  k 2 
 M1 1  M 2 1 
 A1 A2
  q 2 A1 A2   I 1v  I 2  2  I 3












A




2


(30.b)
24
w :
 ( A2 Q1 )  ( A1Q2 )

 A1 A2 k1 N1  A1 A2 k 2 N 2 


4  

3h 2  

 1  ( A2 P1 )    1  ( A2 P2 )    1  ( A1P6 )    1  ( A2 P6 )    P1 A1    P2 A2    P6 A1    P6 A2 
















A


    A1     A2
    A2     A1     A1     A2  
 1     A2
4   ( A2 K1 )  ( A1K 2 ) 
2w
2w
 A1 A2 

 n2
 I1w
  q n A1 A2  n1
2  
2
 
h 

 2


  ( A u)
2  I  ( A21 )  I  ( A1v)  I  ( A1 2 )  16 I 7
5
5
I 3
3








9 h4

 

 




    A1 w
  
A2 w


 
A1     A2   

(30.c)
1 :
 ( M 1 A2 )  ( M 6 A1 )
A
A
A
A
A 
4   A2 P1    A1 P6 

 M2 2  M6 1  M6 1 

 P2 2  P6 1  
2  









 
3h
Q1 A1 A2 
4
h2
 

1 w
K 1 A1 A2   I 2 u  I 41  I 5
 A1 A2
A


1


(30d)
 2 :
 ( M 6 A2 )  ( M 2 A1 )
A
A
A
A 
4    A2 P6    A1 P2 

 M1 2  M 6 2 

 P1 1  P6 2  







 
3h 2  
Q1 A1 A2 
 

1 w
K 2 A1 A2   I 2 v  I 42  I 5
 A1 A2
A


2
h


4
2
(30e)
q1, q2, qn can be defined as the transverse loads
'
The inertias I 1 and I 1 (i=1, 2, 3, 4, 5) are defined by the equations,
I 1  I1  2k1I 2
'
I 1  I1  2k2 I 2
I 2  I 2  k1 I 3 
4
4k
I 4  12 I 5
2
3h
3h
25
'
I 2  I 2  k2 I 3 
I3 
'
I3
4
4k
I  12 I 5
2 4
3h
3h
4
4k
I  12 I 5
2 4
3h
3h
I 4  I3 
I5 
where
4
4k
I  22 I 5
2 4
3h
3h
(31)
8
16
I  2 I7
2 5
3h
9h
4
16k
I  41 I 7
2 5
3h
9h
I , I , I , I , I , I      1, , , , , d
n
1
1
1
1
1
1
hk 1
k
2
3
4
6
k 1
hk
Where 𝝆(k) is the density of the material of the kth layer.
3.7 STRESS RESULTANTS AND STRESS COUPLES
Figure 3.2: STRESS AND MOMENT RESULTANTS
26
(32)
Let N1 be the tensile force, measured per unit length along
β coordinate line, on a cross
section perpendicular to α coordinate line. Then the total tensile force on the differential
element in the α direction is
h 2
N1. A2 

1
da2 dz
h 2
(33)
In which h = the thickness of the shell (z = -h/2 and z = h/2 denote the bottom and top
surfaces of the shell) and da2 is the area of cross section. Using equation (9), the remaining
stress resultants per unit length are given by:
 
z  
 
 1 1 
R2  
 
 
 
 2 1  z  
R1  
 


z  
 
 N1 
 6 1  R  
N 
2 
2
 



 
 N12 
z  
 
 5 1 


R2  
 
 N 21 

h 2 

z  
Q1 

 




1



  4  R1  dz
Q2   h 2  

M 1 






 z 1  1  z  

R2  
M 2 


M 



z 
 12 



z

1


 2
M 21 

R1  





z 

 z 6 1 
R2 




 z 1  z  
6

R1  



……………………
34
In contrast to the plate theory (where 1/R1 =0,1/R2 = 0), the shear stress resultants, N12 and
N21, and the twisting moments, M12 and M21, are, in general, not equal. For shallow shells the
terms z/R1 and z/R2 can be neglected in comparison with unity. Hence N12 = N21 = N6 and M12
= M21 = M6.
27
.
The shell under consideration is composed of finite number of orthotropic layers of uniform
thickness, as shown in Figure 2. In view of assumption 1, the stress resultant in equation [34]
can be expressed as
N
hk
M i   k 1   i 1, z .dz...................i  1,2,6
n
i,
k
hk 1
hk
Qi  k 1   i .dz...................i  4,5
n
……………………
k
(35)
hk 1
In which n = the number of layers in the shell; hk and hk-1 is the top and bottom z coordinates
of the kth lamina.
Substitution of equation [11] and [15] into equation [35] leads to the following expression for
the stress resultants and stress couples in the first order shear deformation theory:
N i  Aij . j  Bij .K j ;
0
0
M i  Bij  j  Dij K j ;
0
0
………………..
Q1  A45 . 4  A55 5 ;
0
0
(36)
Q2  A44 . 4  A45 5 ;
0
0
Here Aij, Bij and Dij denote the extensional, flexural-extensional coupling, and flexural
stiffness. They may be defined as:
A
ij
, Bij , Dij  
hk
n
 Q
k 1 hk  i
ij
k
(1, z , z 2 ).dz.
(37)
For i, j = 1, 2, 4, 5, 6.
hk and hk-1 are the distances measured as shown in figure -1
According to the higher order theory, similarly,
N
hk


3
i , M i , Pi   k 1   i 1, z , z .dz...................i  1,2,6 
n
k
hk 1
Q1 , K1   k 1   4 k 1, z 2 .dz
n
hk
hk 1
……………………
28
(38)
Q2 , K 2   nk 1   5 k 1, z 2 .dz
hk
hk 1
In which n = the number of layers in the shell; hk and hk-1 is the top and bottom z coordinates
of the kth lamina.
Substituting of equation [13] and [15] into equation [38] leads to the following expression for
the stress resultants and stress couples
N i  Aij . j  Bij .k j  Eij k j ;
0
0
2
M i  Bij  j  Dij k j  Fij k j ;
0
0
2
Pi  Eij  j  Fij k j  H ij k j ;
0
0
2
i, j   1,2,6
Q1  A5 j . j  D5 j k j ;
0
1
Q2  A4 j . j  D4 j k j ;
0
1
K1  D4 j  j  F4 j k j ;
0
1
K 2  D5 j  j  F5 j k j ;
0
1
………………......
(39)
 j  4,5
where Aij, Bij , etc. are the laminate stiffnesses expressed as
A , B
ij
ij , Dij , Eij , Fij , H ij   
n
hk
Q
k 1 hk i
k
ij
(1, z , z 2 , z 3 , z 4 , z 5 ).dz .
(40)
The strain displacement relations (13) are substituted in the equations for the stress resultants
and stress couples given in equation (39). Since the solution for the equations of motion is
done by using the Navier solution, therefore such a solution exists only for specially
antisymmetric cross ply laminate for which the following laminate stiffnesses are zero.
29
The expression for the stress resultants and stress couples so obtained are then substituted
into the equation of motion (30). The equation of motion in terms of the displacements for
doubly curved shells hence reduces to
( A11  2 B11k1  D11k12 )
 2u
 2u
2

(
A
66  2 B66 k1  D 66k 1 )

 2
 2
A12  B12k 2  A66  B66k 2  B12k1  D12k 2 k1  k1 B66  D66k 2 k1 
 2v


w
4
3w
4
3w
4
3w
A11k1  A12k 2  B11k1  B12k 2 k1   2 E11 3  2 F11k1 3  2 E12  2 E66  F12k1  2 F66 
 3h

3h

3h
 2
2
2
2
4
4
4
4

  1 
  1
  B11  2 E11  D11k1  2 F11k1 
  B66  2 E66  D66 k1  2 F66 

2
2
3h
3h
3h
3h

 

 
2
4
4
4
4

  2
B

E

B

E

D
k

F
k

D
k

F
k

66
66
12 1
12 1
66 1
66 1 
 12 3h 2 12
3h 2
3h 2
3h 2
 

w
I 1u  I 21  I 3

A66  B66k1  B21k 2  D21k 2 k1  k 2 B66  D66k 2 k1  B21k1  A21 
 A66  2 B66 k 2  D66 k 22 
 2v
 2v
2



A

2
B
k

D
k

22
22 2
22 2
 2
 2
A21k1  A22k 2  k1k 2 B21  k

 2u

B22 
2
2
w
4
3w
4
3w
 2 E 22

F
22
 3h
 3 3h 2
 3
4
3w


2
E

E

F

F
66
21
66
21
3h 2
 2 
2
 21
 21
4
4

  1
  B66  2 E 66 
 B21
 2 E 21

 3h

3h

 
 21
 21
 21
 21
4
4
D66 k 2
 2 F66 k 2
 D21k 2
 2 F21k 2

 3h

 3h

2
2
4
4
4
4

  2 
  2
 B66  3h 2 E 66  3h 2 F66 k 2  D66 k 2   2   B22  3h 2 E 22  3h 2 F22 k 2  D22 k 2   2
' w

 I 1v  I 22  I 3

30
u
 k1  A11k1  A12 k 2   k 2  A21k1  A22 k 2 w 

v
 k1  A12  B12 k 2   k 2  A22  B22 k 2 


 k1  A11  B11k1   k 2  A21  B21k1 
 
4
4

 4
 
 
 k1  B11  2 E11   k 2  2 E 21k 2  B21    A55  2 D55  1
3h
h

 3h
 
 
 
 
4
4

 4
 
 
 k1  B12  2 E12   k 2   2 E 22  B22    A44  2 D44  2
3h
h

 3h
 
 
 
2
 4
4
4

  w
  2 E11k1  2 k 2 E 21   A55  2 D55 

2
3h
h

 
 3h
2
 4
4
4

  w
 3h 2 k1 E12  3h 2 k 2 E 22  k 2 E 21  k 2 E 22    A44  h 2 D44   2 



3
3
2


 u
 w 
4
 3v
  1



E

F
k

2
E

E11  F11k1  3  k1  k 2  2   F11  2 H 11 

12
12
2
66
3


3h
 2  

 



 3 2
4
4w
 3v



H

E

F
k

F
k

E

 F12

11
12
21 2
22 2
22
2
2
 4
 3
   3h

3


4 
4
4
  2


F

H

F

H



21
21
22
22
3
3h 2 
3h 2
3h 2

 


4
3
 u
 4 H  4 H   w  2 E


66
 3h 2 21 3h 2 22   4

 2


3
3
4
 v
4
 w

  2
k F

 F 
H
 H 66
2
2
 2 66  2  66 3h 2 66   2

 


2
 2w
  2 w
   ' v   1
2 
 4 
   2  I 7  2 
  I1 w

I

I

I
3
5
5



 
 2      
 3h 
 
31
2
4
4
4
4
4

 u 
 w
B

D
k

E

F
k
  B11k1  B12 k 2  2 E11k1  2 E12 k 2  A55  2 D55 
11 1
11
11 1 
2
2
2
 11
3h
3h
3h
3h
3h

 

 
2
4
4

  v
  B12  D12 k 2  B66  D66 k 2  2 E12  2 F12 k 2  E 66  F66 k 2 

3h
3h

 
2
8
16

  1
D

F

H
 11 3h 2 11 9h 4 11   2
2
4
4
4
16
4
16

  2
  D12  2 F12  D66  2 F66  2 F12  4 H 12  2 F66  4 H 66 
3h
3h
3h
9h
3h
9h

 
3
4
 3 w 16
3w  4
4
16
  w
 2 F11 3  4 H 11 3   2 F12  2 F66  4 H 12  2
3h

9h

3h
9h
 3h
  
2
4
4

 u
  B66  D66 k1  2 E 66  2 F66 k1  2 
3h
3h

 
2
8
16
2w
4

  1 16


D

F

H

H
 66 3h 2 66 9h 4 66   2 9h 4 66     A55  3h 2 D55 1




4 
4
4 
4

 w
 2  D55  2 F55 1  2  D55  2 F55 
h 
3h
h 
3h

 

w
 I 2 u  I 41  I 5

32
2
2
4
4
4
4

 v 
  u
 B66  D66 k 2  3h 2 E 66  3h 2 F66 k 2   2   B66  k1 D66  B21  k1 D21  3h 2 E 66  3h 2 F66 k1   




2
2
4
4
4
16
4
4
16

  1 
  2
 D66  3h 2 F66  D21  3h 2 F21  3h 2 F66  9h 4 H 66     D66  3h 2 F66  3h 2 F66  9h 4 H 66   2




3
2
4
4
4
 8
  w

 v


  2 F66  2 F21  2 H 66 

B

D
k

E

F
k

F
k

E

22
22 2
21
21 2
22 2
22 
2
2
3h
3h
3h 2
 3h
   
 
2
4
4
4 
4
4

 w 
   2


B
k

B
k

E
k

E
k

D

F

F

H

F

H



21 1
21 2
22 2 
22
22 
 22 3h 2 22 3h 2 21 3h 2 21
2
 22 2
3h 2
3h 2

  

 
3
16
16
4
4
 4
 w 

F

H

H

A

D

22
21
22
44
44

2
4
4
3
2
 3h
 
9h
9h
h
h2



4


 D44  2 F44  2 
h



4
4 
4
 w
 A44  h 2 D44  h 2  D44  h 2 F44  




w
 I 2 v  I 42  I 5

3.8 BOUNDARY CONDITIONS
Up to now, the analysis has been general without reference to the boundary conditions. For
reasons of simplicity, only simply supported boundary conditions are considered along all
edges for the shell. The boundary conditions for the simply supported doubly curved shell are
obtained as given below
N1 = 0, v = 0, w = 0,
Following the Navier solution procedure, the following solution form which satisfies the
boundary conditions in the above equation is assumed:
33
u  U cos m . sin n 
e i t
v  V sin m . cos n 
e it
  W sin m . sin n  e it
1  1 cos m . sin n  e
it
……………
(42)
2   2 sin m . cos n  e it
where , m 
m
n
and U, V, W,  1 and  2 are the maximum amplitudes, m and
, n 
a
b
n are known as the axial half wave number and circumferential wave number respectively.
Introducing the expressions [42] into the governing equations of motion in terms of
displacements [41], the following equation in matrix form is obtained, which is a general
eigen value problem.
C X    2 M X 
C X   BX 
(43)
(43.a)
Where,
2
is the eigenvalue

is the critical buckling load
{X}
is a column matrix of amplitude of vibration or eigenvector.
[C], [M] and [B] are 5 x 5 matrices.
For convenience, the elements of the above matrices are suitably non-dimensionalised as
follows
34
U  Uh
V  Vh
W  Wh
1  1
………………….
2  2
Aij  Aij Q2 h
(44)
Eij  E ij Q2 h 4
Bij  B ij Q2 h 2
Fij  F ij Q2 h 5
Dij  D ij Q2 h 3
H ij  H ij Q2 h 7
(- (bar) on top indicates non-dimensionalised quantities)
And

I1  I1  1h

I 2  I 2  1h 2

I 3  I 3  1h 3

I 4  I 4  1h 4

I 5  I 5  1h 5

I 7  I 7  1h 7
(45)
 I  , I  , I  ,.......... are the non-dimensionalised quantities
 1 2 3
Also
~
I 1  I 1 1 h  I 1  1 h
4 
~

I 2   I 2  I 4   1 h 2  I 2  1 h 2
3 

8
16 
~

I 3   I 3  I 5  I 7   1 h 3  I 3  1 h 3
3
9 

4 
~

I 5   I 5  I 7   1 h 5  I 5  1 h 5
3 

……………….
(46)
After non–dimensionalising the terms, the equation [43] in matrix form can be written as
given below
35
H X   X 
2
H X  X 
(47)
(47.a)
Where,
 
2
b 
2
2
Q
2
4
a
n
b h (1   )
1
2
(48)
2
12
21

A non-trivial solution for the column matrix X will give the required eigenvalues, which
are the values of the square of the frequency parameter  in the present case. The lowest
value of  is of particular interest.
 will give the critical buckling load.
36
CHAPTER 4
RESULTS&DISCUSSION
37
CHAPTER 4
NUMERICAL RESULTS AND DISCUSSIONS
4.1 INTRODUCTION
The frequency parameters and buckling loads are calculated by using a computer program for
laminated composite doubly curved shells. The results obtained using the present theory are
compared to earlier results and are tabulated. The numerical values of the lowest value of
frequency parameter and non-dimensional buckling load are presented for various shell
parameters in this chapter to study the effect of number of layers, the orientation of layers and
the ‘a/h’ ratio (side by thickness ratio).
SOLUTION OF EIGEN VALUE PROBLEM AND COMPUTER PROGRAM
A standard subroutine in the computer program to find the eigenvalue of matrices has been
used, which consists of root power method of iteration with Wielandt’s deflection technique.
The program is called RTPM, which is capable of finding the required number of roots in
descending order. The change of the sign of the determinant value is checked for values of
one percent on either side of the root to verify the convergence. The RTPM program gives
 
  H  then the highest
the highest value of eigenvalue first, so if the M matrix is taken as C
 1
value of  2

1
,

2
 is obtained, that is the lowest value of  and  which is of particular


interest. These programs are written in FORTRAN language.
4.2 The validation of the formulation and comparison of results
Using the formulation developed above, numerical studies are carried out. The lowest value
of the frequencies has been calculated at first for two layer, three layer and four layer
laminated composite spherical shells for various values of R/a and a/h by higher order shear
deformation theory. These results are compared with earlier available results. This also serves
to check on the validity of the present theory, as the results are mostly agreeable. Table 4. 1
and Table 4.2 shows the comparison of present results and those of J .N. Reddy [24] for the
38
a 2 2 of [0/90], [0/90/0] and [0/90/90/0]
non-dimensional frequency parameter  2  Q22
2 2
b h E2
simply supported spherical shell. The geometrical and material properties used are
E11  25E 22 ,
G12  G13  0.5E 22 ,
G 23  0.2E 22 ,
 12  0.25,
 1
For all the panels, a/b =1 and R1  R2  R .
Table 4.1: Comparison of lowest non-dimensional frequency for a simply supported
laminated composite spherical shell (a/h=100)
R/a
0 / 90 / 90 / 0
0 / 90 / 0
0 / 90
Theory
[24]
Present
Difference
[24]
value
Present
Difference
[24]
value
Present
Difference
value
5
HSDT
28.840
28.886
0.046
31.020
30.927
-0.093
31.100
31.026
-0.074
10
HSDT
16.710
16.811
0.101
20.350
20.245
-0.105
20.380
20.300
-0.08
20
HSDT
11.840
11.911
0.071
16.620
16.511
-0.109
16.63
16.540
-0.09
50
HSDT
10.060
10.230
0.17
15.420
15.291
-0.129
15.42
15.320
-0.1
100
HSDT
9.784
9.964
0.18
15.240
15.110
-0.13
15.23
15.140
-0.09
Plate
HSDT
9.688
9.687
-0.001
15.170
15.171
0.001
15.170
15.169
-0.001
39
Table 4.2: Comparison of lowest non-dimensional frequency for a simply supported laminated
composite spherical Shell (a/h=10)
R/a
0 / 90 / 90 / 0
0 / 90 / 0
0 / 90
Theory
[24]
Present
Difference
[24]
value
Present
Differen
value
ce
[24]
Present
Difference
value
5
HSDT
9.337
9.332
-0.005
12.060
12.110
0.05
12.040
12.649
0.609
10
HSDT
9.068
9.095
0.027
11.860
11.948
0.088
11.840
12.498
0.658
20
HSDT
8.999
9.035
0.036
11.810
11.907
0.097
11.790
12.460
0.67
50
HSDT
8.980
9.019
0.039
11.790
11.896
0.106
11.780
12.450
0.67
100
HSDT
8.977
9.017
0.04
11.790
11.894
0.104
11.780
12.448
0.668
Plate
HSDT
8.976
8.877
-0.099
11.790
11.890
0.100
11.780
12.445
0.665
As can be seen from the Table 4.1 and Table 4.2, the values are quite agreeable.
The validation of the formulation for buckling load is similarly done by comparing the results
with that of Librescu et al [17]. It can be seen from Table 4.3 that the results are in
agreement.
40
Table 4.3: Comparison of non-dimensional buckling load for a simply supported
laminated composite doubly curved shell (a/h=10)
E11  40E 22 ,
G12  G13  0.6E 22 ,
Curvature
 12  0.25,
G 23  0.5E 22 ,
Present work
 1
Theory
FSDT [17]
HSDT [17]
Rx/a=5, Ry/a=5
12.530
12.214
12.431
Rx/a=10, Ry/a=5
12.236
11.822
12.007
Rx/a=10, Ry/a=20
11.861
11.479
11.697
Rx/a=20, Ry/a=20
11.797
11.406
11.610
Plate
11.753
11.353
11.555
4.3 NUMERICAL RESULTS:
The governing equations are solved for doubly curved shells (spherical, parabolic, and
elliptical paraboloid shells). Analysis is carried out for simply supported boundary condition.
The material properties for the parametric study are assumed as:
E11  25  109 ,
G23  0.2 109
E22  01109 ,
G12  0.5 109 ,
G13  0.5 109
a
 100
h
 12  0.25
The geometry for various shells unless otherwise specified is
For spherical shell:
R1
R
 5, 2  5
a
a
For hyperbolic paraboloid shell:
R1
R
 5 , 2  5
a
a
For elliptical paraboloidal shell:
R
R1
 5 , 2  7.5
a
a
41
4.3.1 VIBRATION ANALYSIS
The first study is done to investigate the variation in R/a ratio on the non-dimensional
frequency parameter.
Table 4.4 shows the variation of R/a ratio on the non-dimensional frequency parameters of
the hyperbolic paraboloid shell.
Table 4.4: Variation in non-dimensional frequency parameter with change in curvature
ratio (R/a) for laminated composite hyperbolic paraboloidal shell
R1
a
R2
a
-1
0/90
0/90/90/0
a/h=200
a/h=100
a/h=10
a/h=200
a/h=100
a/h=10
1
45.590
32.826
12.582
12.352
12.330
10.166
-2
2
19.494
15.377
9.402
14.354
14.328
11.826
-3
3
13.533
11.798
9.028
14.764
14.737
12.167
-4
4
11.567
10.721
8.958
14.910
14.883
12.269
-5
5
10.774
10.308
8.945
14.979
14.952
12.346
-10
10
9.990
9.919
8.966
15.071
15.044
12.422
Here it is seen that as R/a ratio increases, then non-dimensional frequencies are decreasing.
42
The study is extended to elliptical paraboloidal shells by increasing the R/a ratio, keeping the
R1/R2 ratio constant, as given in Table 4.5.
Table 4.5: Variation in non-dimensional frequency parameter with change in curvature
ratio (R/a) for laminated composite elliptical paraboloidal shell
a/h =100
R1/a
a/h=10
R2/a 0/90
0/90/90/0 0/90/0/90/0 0/90
1
1.5
107.105
107.763
107.634
12.928
15.095
15.093
2
3.0
56.682
57.788
57.730
10.270
13.216
13.180
3
4.5
38.821
40.438
40.398
9.613
12.800
12.755
4
6.0
29.924
31.996
31.978
9.363
12.648
12.599
5
7.5
24.692
27.175
27.166
9.242
12.577
12.526
10
15
15.034
18.855
18.868
9.076
12.480
12.427
0/90/90/0 0/90/0/90/0
In case of elliptical paraboloidal shell, with the increase of R/a ratio, non-dimensional
frequency decreases. With the increase of thickness parameter (a/h), i.e., as thickness
decreases, then frequency parameter is also increasing.
Table 4.6 shows the variation of modular ratio on the non-dimensional frequency parameter.
Four layer cross-ply [0/90/90/0] and a/h = 100 value is chosen for the study.
Table 4.6: Variation in non-dimensional frequency parameter with modular ratio for
laminated composite doubly curved shells of [0/90/90/0] and a/h=100
Hyperbolic
Elliptical
paraboloidal shell
paraboloidal shell
R1/a= -5,R2/a=5
R1/a= 5,R2/a=7.5
R1/a=R2/a=5
25
14.952
27.175
31.026
40
18.432
29.465
33.131
60
22.220
32.101
35.538
E 11
E 22
Spherical shell
With the increasing modular ratio, it is seen that the non-dimensional frequency parameter is
also increasing. But hyperbolic paraboloidal shell has less non-dimensional frequency; hence
it is more stable than the other two doubly curved shells.
43
Table 4.7 shows the variation of the non-dimensional frequency with number of layers for all
shell geometries.
Table 4.7: Variation in non-dimensional frequency parameter with number of layers for
laminated composite doubly curved shells a/h=100
Shell
Curvature
0/90
0/90/0
0/90/90/0
0/90/0/90/0
Spherical
R1/a=R2/a=5
28.886
30.928
31.026
31.009
Elliptical
R1/a=5,R2/a=7.5
24.692
27.090
27.175
27.166
Hyper
R1/a= -5,R2/a=5
10.308
14.924
14.952
14.977
paraboloid
Above table shows that as number of layers changes, non-dimensional frequencies generally
increase.
Some of the above results are also presented in graphical form as shown in Figures 4.1-4.3.
Figure 4.1: Variation in non-dimensional frequency with R2/a of hyperbolic
paraboloid shells for various a/h ratios
44
Figure 4.2: Variation of non-dimensional frequency with R1/a of elliptical paraboloid shell
for various a/h ratios
Figure 4.3: Variation of non-dimensional frequency of cross-ply spherical shell, hyperbolic
and elliptical paraboloid shell with modular ratio.
45
4.3.2 STABILITY ANALYSIS
For determination of non-dimensional buckling load, the geometrical and material properties
used are
E11  25E 22 ,
G12  G13  0.5E 22 ,
G 23  0.2E 22 ,
 12  0.25,
 1
Table 4.8 shows the variation in non-dimensional buckling load with R/a ratio for a
hyperbolic paraboloidal shell. It is observed that here as R/a ratio increases, non-dimensional
buckling load decreases.
Table 4.8: Variation in non-dimensional buckling load with curvature ratio (R/a) for
laminated composite hyperbolic paraboloidal shell
R1
a
R2
a
-1
0/90
0/90/90/0
a/h=200
a/h=100
a/h=10
a/h=200
a/h=100
a/h=10
1
254.088
132.165
20.560
18.594
18.529
12.720
-2
2
40.545
25.285
9.925
21.935
21.859
15.040
-3
3
19.003
14.468
8.828
22.584
22.505
15.490
-4
4
13.745
11.822
8.563
22.813
22.734
15.650
-5
5
11.869
10.876
8.471
22.919
22.840
15.724
-10
10
10.138
10.001
8.398
23.061
22.982
15.822
46
Similar study is done next for the laminated composite elliptical paraboloid shell in Table 4.9
and the same variation is observed.
Table 4.9: Variation in non-dimensional buckling load with curvature ratio (R/a) for
laminated composite elliptical paraboloid shell
a/h=100
R1/a R2/a 0/90
a/h=10
0/90/90/0 0/90/0/90/0 0/90
0/90/90/0 0/90/0/90/0
1
1.5
1310.338 1327.928 1324.926
19.186 26.322
26.301
2
3.0
335.768
349.220
348.530
11.169 18.449
18.338
3
4.5
154.832
168.000
167.737
9.644
17.006
16.877
4
6.0
91.445
104.575
104.461
9.106
16.502
16.367
5
7.5
62.092
75.218
75.173
8.856
16.269
16.131
10
15
22.935
36.076
36.123
8.521
15.959
15.816
Table 4.10 shows the variation of modular ratio on the non-dimensional buckling load
for all shell geometries. Four layer cross-ply was taken with a/h ratio of 100. The nondimensional buckling load is found to increase with increase in modular ratio for all shell
geometries.
Table 4.10: Variation in non-dimensional buckling load with modular ratio for laminated
composite doubly curved shells [0/90/90/0] and a/h=100.
Hyperbolic
Elliptical
paraboloid shell
shell
R1/a= -5,R2/a=5
R1/a= 5,R2/a=7.5
R1/a=R2/a=5
25
98.235
75.218
22.840
40
112.059
88.452
34.710
60
128.961
105.006
50.442
E 11
E 22
47
paraboloid Spherical shell
Table 4.11 shows the variation of the non-dimensional frequency with number of layers for
all shell geometries.
Table 4.11: Variation in non-dimensional buckling load with number of layers for
laminated composite doubly curved shells a/h=100
Shell
Curvature
0/90
0/90/0
0/90/90/0
0/90/0/90/0
Spherical
R1/a=R2/a=5
85.161
97.606
98.235
98.126
Elliptical
R1/a=5,R2/a=7.5
62.092
74.748
75.218
75.173
Hyper
R1/a= -5,R2/a=5
10.876
22.756
22.840
22.918
paraboloid
Above table shows that with increase in number of layers, the non-dimensional buckling load
generally increases.
48
CHAPTER 5
CONCLUSION
49
CHAPTER 5
5.1 CONCLUSION:
The governing equations including the effect of shear deformations have been presented in
orthogonal curvilinear coordinates for laminated orthotropic doubly curved shells. The theory
is based on a displacement field as proposed by Reddy and Liu [24] in which the
displacements of the middle surface are expanded as cubic functions of the thickness
coordinate and transverse displacement is assumed to be constant through the thickness. This
displacement field leads to the parabolic distribution of the transverse shear strain and hence
shear stresses and therefore no shear correction factors are used.
The governing equations of motion are derived by integrating the displacement
gradients by parts and setting the coefficients of
u , v , w , 1 and  2 to zero separately.
In the present study, for first two equations the moment terms are also considered.
The governing equations are then specialized for a doubly curved shell. These
equations have been solved for simply supported doubly covered shells and the associated
eigenvalue problem has been solved by means of a computer program. Lowest frequencies
and buckling loads have been considered throughout.
The results of the present theory have been compared with the earlier results available
by Reddy and Liu [24]. Results are compared and found that it matches that of Reddy and Liu
for spherical shells. The study is extended to vibration of hyperbolic and elliptical paraboloid
shells and influence of various parameters like aspect ratio, number of layers, modular
ratio,etc on the same are studied. The non-dimensional buckling load for axial compression in
one direction is also studied for various shell geometries and variation in other parameters.
Following conclusions were observed and are summarized again below:
1) With the increasing modular ratio, non-dimensional frequency is also increasing. But
hyperbolic paraboloid shell has less non-dimensional frequency as compared to
elliptical paraboloid shell and spherical shell.
2) With the increase of curvature ratio R/a frequency decreases.
3) With the increase of a/h ratio, frequency also increases.
50
4) Increase of thickness parameter (a/h) ratio then non-dimensional frequency of the
doubly curved shell increases.
5) It is also observed that as number of layers of the shell increases, buckling load
increases.
6) In comparison among spherical, hyperbolic paraboloid shell and ellipitical paraboloid
shell, the hyperbolic paraboloidal shell has least non-dimensional buckling load.
7) In case of both shell geometries, with the increase of curvature ratio(R/a) buckling
load decreases.
8) As a/h ratio increases, the non-dimensional buckling load also increases.
9) As number of layers increases, in general the non-dimensional frequency and the nondimensional buckling load increases.
Thus by the above study it can be seen that by suitably changing the
orientation of the layers or the number of layers, the properties of the laminated
composite doubly curved shells can be tailored to suit the particular needs.
51
5.2 SCOPE FOR FUTURE WORK
1) HSDT can be used to study the vibration and buckling characteristics of thick
laminated composite shells.
2) HSDT is more important for laminated composite plates and shells because of greater
shear effect
3) Incorporation of finite element techniques to take care of higher shear deformation
theory instead of analytical method used in this work.
52
REFERENCES
53
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54
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55
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41–55
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Element For Metal Forming And Crashworthiness Analysis: Part 1. Formulation And
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56
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Engineering Mechanics Division, 99(EM1): pp.157-181.
57
APPENDIX
58
The non-dimensionalised coefficients of the [ Cij ] and [ Mij ] matrices of Higher order shear
deformation theory are given below,

C (1,1)  A11  2 KHB11  K 2 H 2 D11
S
2
m
2


 A66  2 KHB66  K 2 H 2 D66  2 n
 S

C (1,2)  A12  A66   H B12  B66   KH B12  B66   KH 2 D12  D66 
m
n

 J  4H
E12  2E66   KH F12  2F66 m  2 n 
C (1,3)   KHA11  HA12  K 2 H 2 B11  KH 2 B11 m  
SH  3SJ



4
  mH
 3 E11  KHF11  S 3 J
3
2

4
4
4
4
 m 
 
C (1,4)   B11  E11  KHD11  KHF11  2   B66  E66  KHD66  KHF66  2 n 
3
3
3
3
 S

 

4
4

 
C (1,5)  B12  B66   E12  E66   KH D12  D66   KH F12  F66  m n
3
3

 S
 S

C (2,1)  A66  A21   KH B21  B66   H B21  B66   KH 2 D12  D66 
m
n


 2m
C (2,2)   A66  2 HB66  H 2 D66 2  A22  2 HB22  H 2 D22  2 n 
S







n J 4
 2 m n H
2
2
 2 E66  E21  2 HF66  HF21 
 KHA21  HA22  KH B21  H B22
H
3
S 2J
C (2,3)   
 3
 4 n H E22  F22 
 3 J








4
4

 
C (2,4)  B21  B66   H D21  D66   E21  E66   H F21  F66  m n
3
3

 S
2

4
4
4
4
 m 
 
C (2,5)   B66  HD66  E66  HF66  2   B22  HD22  E22  HF22  2 n 
3
3
3
3
 S

 

2
 2

8
4
4

 m
2
K
H
A

K
A

K
A

A
J

A

8
D

KH
E

H
E

H
E

16
F
 11
11
12
21
22
55
11
12
21
55 


2
3
3
3

 S




4
8
4
16
16


C (3,3)    A44  8 D44  KHE21  HE22  KHE12  16 F44  2 n  4 2 H 11H 2  4 m  2 H 11H 2  4 n  
 

3
3
3
9S J
9J



 16 H 2

 2 2 H 12  H 21  4 H 66  2 m  2 n

 9S J



59
3

4
4
 m J 4 
  mH
A

8
D

16
F

KH
B

KH
E

H
B

H
E

F

H
 11
55
55
11
11
21
21 
11 
 55
3
3
3

 SH 3 
 S J

C (3,4)  
 4  F  4 H  2 F  8 H  m  2 n H
66
66
 3  21 3 21
3
SJ








4
4
4
8
 n J 4 

  F12  H 12  2 F66  H 66 
 A44  8D44  16 F44  KHB12  KHE12  HB22  HF22 
3
3
3
3
3

 H

C (3,5)  

2
  m  n H  4  F  4 H  3 n H

22
22
 S 2 J

3
3
J

2

4
4
4
4
J2 
  m 

C (4,4)   D11   2 F11  H 11   2   D66   2 F66  H 66   2 n  A55  8D55  16 F55  2 
3
3
3
3
H 
 S




4
16
  
C (4,5)   D12  D66  2 F12  2 F66   H 12  H 66  m n 
3
9
 S 

C (1,3)  C (3,1)
C (2,3)  C (3,2)
C (1,4)  C (4,1)
C (1,5)  C (5,1)
C (2,4)  C (4,2)
C (2,5)  C (5,2)
C (3,5)  C (5,3)
C (5,4)  C (4,5)
60
M (1,1)  ( I1'  2 K1hI 2' )
M (1.2)  M (2,1)  0
4
4
m
M (1,3)  M (3,1)  ( I 4'  K1hI 5' )
h
3
3
a
M (1,4)  M (4,1)  ( I 2'  K1hI 3' 
4 ' 4
I 4  K1hI 5' )
3
3
M (1,5)  M (5,1)  0
M (2,2)  ( I1'  2 K 2 hI 2' )
4
4
n
M (2,3)  M (3,2)  ( I 4'  K 2 hI 5' )
h
3
3
b
M (2,4)  M (4,2)  0
M (2,5)  ( I 2'  K 2 hI 3' 
M (3,3)  
4 ' 4
I 4  K 2 hI 5' )
3
3
2 h
16 ' 2 h 2
I 7 ( m ( )   n ( ) 2 )  I1'
9
a
b
4
16
h
M (3,4)  M (4,3)  ( I 5'  I 7' ) m
3
9
a
4
16
h
M (3,5)  M (5,3)  ( I 5'  I 7' ) n
3
9
b
8
16
M (4,4)  M (5,5)  ( I 3'  I 5'  I 7' )
3
9
B (3,3)  
2
61
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