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A STUDY ON THE DYNAMIC INSTABILITY OF

CYLINDRICAL SHELL DUE TO PARAMETRIC

EXCITATION

By

DEBABRATA PODDER ROLL NO.: 000810402005

REGN. NO.: 81964 of 2001 - 2002

EXAM. ROLL NO.: M4CIV 10-05

Under the Guidance of

DR. PARTHA BHATTACHARYA

A Thesis Paper to be submitted in Partial Fulfillment of the Requirements for the Degree of

Master of Engineering in Civil Engineering

(Specialization: Structural Engineering)

At the

Department of Civil Engineering

Faculty of Engineering and Technology

Jadavpur University

Kolkata – 700 032

(ii)

DEPARTMENT OF CIVIL ENGINEERING

FACULTY OF ENGINEERING AND TECHNOLOGY

JADAVPUR UNIVERSITY

KOLKATA – 700 032

CERTIFICATE OF RECOMMENDATION

This is to certify that the thesis titled, “A Study on the Dynamic Instability of Cylindrical

Shell due to Parametric Excitation”, that is being submitted by Debabrata Podder

(Roll no. 000810402005) to Jadavpur University for the partial fulfillment of the

requirements for awarding the degree of Master of Civil Engineering (Structural Engineering)

is a record of bona fide research work carried out by him under my direct supervision &

guidance.

The work contained in the thesis has not been submitted in part or full to any other university

or institution or professional body for the award of any degree or diploma.

Dr. Partha Bhattacharya

Reader

Department of Civil Engineering

Jadavpur University

Kolkata 700032

Countersigned by

_______________________ ________________________

Prof. S. Chakrabarti Prof. N. Chakraborti

Head of the Department Dean, FET

Department of Civil Engineering Jadavpur University

Jadavpur University Kolkata 700032

Kolkata 700032

(iii)

DEPARTMENT OF CIVIL ENGINEERING

FACULTY OF ENGINEERING AND TECHNOLOGY

JADAVPUR UNIVERSITY

KOLKATA – 700 032

CERTIFICATE OF APPROVAL

This thesis paper is hereby approved as a credible study of an engineering subject

carried out and presented in a manner satisfactorily to warrant its acceptance as a pre-

requisite for the degree for which it has been submitted. It is understood that, by this approval

the undersigned do not necessarily endorse or approve any statement made, opinion

expressed or conclusion drawn therein but approved the thesis paper only for the purpose for

which it is submitted.

Board of Thesis Paper Examiners:

1.

2.

(iv)

ACKNOWLEDGEMENT

I gratefully acknowledge the resourceful guidance, active supervision and constant

encouragement of my supervisor, Dr. Partha BhattacharyaDr. Partha BhattacharyaDr. Partha BhattacharyaDr. Partha Bhattacharya of the Department of Civil

Engineering, Jadavpur University, Kolkata, who despite his other commitments could find

time to help me in bringing this Thesis to its present shape. I do convey my sincere thanks

and gratitude to him.

I also acknowledge my gratefulness to all Professors and staffs of Civil Engineering

Department, Jadavpur University, Kolkata, for extending all facilities to carry out the

present study.

I also thankfully acknowledge the assistance and encouragement received from my

family members, friends and others during the preparation of this Thesis.

_______________________________

Debabrata Podder

Jadavpur University, Kolkata M.C.E. (Structural Engineering)

Date: Roll No.:- 000810402005

Regn. No.:- 81964 of 2001-’02

Exam. Roll No.:- M4CIV 10-05

(v)

ABSTRACT

Oscillatory instability of shell structures has been a major cause of concern in many

branches of engineering. The dynamic instabilities are the result of pulsating time varying

loads mainly inertial or thermal in origin. The greatest danger posed by such instabilities is

that the failure is very quick and abrupt. In parametric instability the rate of increase in

amplitude is generally exponential and thus potentially dangerous, while in typical resonance

due to external excitation the rate of increase in response is linear. More over damping

reduces the severity of typical resonance, but may only reduce the rate of increase during

parametric resonance. Parametric instability occurs over a region of parameter space and not

at discrete points. It may occur due to excitation at frequencies remote from the natural

frequencies. Researchers have worked in understanding the behavior of such instabilities.

With this particular concept in mind, a theoretical formulation has been developed in

the present work for the analysis of singly curved surfaces subjected to in plane periodic

loading and undergoing parametric excitation. A four noded iso-parametric shell element

having five mechanical degrees of freedom per node, using Mindlin and Reissener’s shallow

shell theory has been developed in MATLAB platform. The first order shear deformation and

effect of rotary inertia has been considered. The results obtained by the present FE code for

static, free vibration and buckling analysis are verified with the ANSYS finite element

software. Parametric instability studies have been carried out for cylindrical shells having

different fibre orientations, various geometric properties with different R/a ratio. A

generalized Rayleigh proportional damping has been considered for all the cases to study the

shift of the stability point with respect to frequency ratio in various cases. The obtained

results are discussed in detail and conclusions highlighting the important findings are made.

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CONTENTS

CERTIFICATE OF RECOMMENDATION II

CERTIFICATE OF APPROVAL III

ACKNOWLEDGEMENT IV

ABSTRACT V

SYMBOLS IX-X

LIST OF FIGURES XI-XII

LIST OF TABLES XIII

CHAPTER 1: INTRODUCTION 1-13

1.1 GENERAL INTRODUCTION 1

1.2 TYPES OF DYNAMIC INSTABILITY 2-3 1.2.1 Impulsive loading 3

1.2.2 Circulatory loads 3

1.2.3 Aero elastic problems 3

1.2.4 Buckling in the testing machine 3

1.3 PARAMETRIC EXCITATION 4-6

1.4 SHELLS 7

1.4.1 Shell as a Structural Form 7

1.4.2 Parametric Excitation on Shell Structure 7

1.5 LITERATURE REVIEW 8-11

1.5.1 Literature review on parametric excitation 8-9

1.5.2 Literature review on shell 9-10

1.5.3 Literature review on shell structures under parametric-

instability or dynamic instability 10-11

1.6 OBJECTIVE AND SCOPE OF THE PRESENT WORK 12

1.7 ORGANIZATION OF REPORT 12-13

(vii)

CHAPTER-2: CONSTITUTIVE EQUATIONS 14-25

2.1 INTRODUCTION 14

2.2 COMPOSITE MATERIALS 14

2.3 LAMINA AND LAMINATE 14-15

2.4 ASSUMPTIONS REGARDING THE BEHAVIOR OF A

LAMINATE 15

2.5 MACRO MECHANICAL BEHAVIOR OF COMPOSITE

LAMINATES 15-18

2.6 DISPLACEMENT MODELLING 18-20

2.7 STRESS -STRAIN RELATIONS FOR A LAMINATE 21-24

2.8 ENERGY FORMULATION 24-25

CHAPTER–3: FINITE ELEMENT FORMULATION 26-43

3.1 INTRODUCTION 26

� � 3.2 FORMULATION 27-43

3.2.1 Selection of element 27-28

3.2.2 Strain-Displacement relations 29-31

3.2.3 Structural stiffness matrix 31-32

3.2.4 Element mass matrix 32-33

3.2.5 Geometric stiffness matrix 33-38

3.2.6 Element load vector 39

3.4.7 Governing equations of motion 39-40

3.4.8 Stability equations 41-43

3.4.8.1 Calculation of damping for the present case (C) 42

CHAPTER-4: RESULTS AND DISCUSSION 44-66

4.1 INTRODUCTION 44

4.2 STATIC ANALYSIS 44-47

4.2.1 Isotropic cantilever shell 45-46

4.2.2 Composite cantilever shell 46-47

(viii)

4.3 FREE VIBRATION ANALYSIS 47-50



4.4 BUCKLING ANALYSIS 50-53



4.4 PARAMETRIC INSTABILITY STUDY 53-66



4.4.2.1 Effect of various fibre

orientations on stability 55-62

4.4.2.2 Effect of thickness on stability 62-64

4.4.2.3 Effect of various geometries on stability 64-66

CHAPTER-5: CONCLUSIONS 67-68

5.1 GENERAL CONCLUSIONS 67-68

5.2 SCOPE FOR FUTURE WORK 68

REFERENCES 69-72

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SYMBOLS

t,h Thickness of the laminate (m)

� Angle of orientation of fiber in a lamina (Degree)

E Young’s modulus of elasticity (N.m-2

)

ν Poisson’s ratio

G Modulus of rigidity (N.m-2

)

� Density of the material (kg.m-3

)

Qij Elastic modulli of an orthotropic material (N.m-2

)

{ }� Stress vector (N.m-2

)

{ }� Strain vector

u Displacement along X direction (m)

v Displacement along Y direction (m)

w Displacement along Z direction (m)

yθ Rotation about Y axis

xθ Rotation about X axis

{ }� Curvatures

{N} Force resultant vector (N.m-1

)

{M} Moment resultant vector (N.m.m-1

)

T Kinetic energy of the system (N.m)

U Potential energy of the system (N.m)

[B] Strain-displacement matrix

[K] Elastic stiffness matrix

[K�] Geometric stiffness matrix

[Z] Position matrix

[N] Shape function matrix

[J] Jacobian matrix

[M] Mass matrix

{p} Mechanical load vector (N)

{U} Displacement vector (m)

{ }U� Velocity vector (m.s-1

)

{ }U�� Acceleration vector (m.s-2

)

(x)

[T] Transformation matrix

[C] Damping matrix

L Length of the shell (m)

a Width of the shell

R Radius of curvature of shell (m)

� n Natural frequency (Rad.s-1

)

[�] Inertia matrix

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LIST OF FIGURES

Fig No. Name of Figures Page No.

1.1 Pendulum with a moving support 5

1.2 Stability Diagram 6

1.3 Solutions of Mathieu Equation 6

2.1 Laminate construction 15

2.2 Axis system of a unidirectional stressed Lamina 16

2.3� � Displacement field along respective coordinate axis. 19

2.4 Deformation of the laminate in X-Z and Y-Z plane 19

2.5 A general n-layered laminate 21

2.6 Stress resultants of a laminated shell element. 21

3.1 Four noded quadratic isoparametric element 27

4.1 Finite element model of a cantilever shell subjected to

unit mechanical transverse load at the free edge.

The mesh consists of (20 x 25) elements. 46

4.2 Mode shape for 1st natural frequency (19.93 rad/sec)

for fibre orientation 45/0/45 and R/a=50, L=2m, a=1m,

t=0.03m for E-Glass Epoxy composite. 49

4.3 Mode shape for 2nd

natural frequency

(105.96 rad/sec) for fibre orientation 45/0/45 and R/a=50,

L=2m, a=1m, t=0.03m for E-Glass Epoxy composite. 49

4.4 Mode shape for 3rd

natural frequency (223.83 rad/sec)

for fibre orientation 45/0/45 and R/a=50, L=2m,

a=1m, t=0.03m for E-Glass Epoxy composite. 50

4.5 Finite element model of a cantilever shell

subjected to unit mechanical compressive load at the

free edge. The mesh consists of (20 x 25) elements. 51

4.6 Mode shape for 1st buckling load factor (14116)

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4.7 Mode shape for 2nd

buckling load factor (128080)



4.8 Mode shape for 3rd

buckling load factor (358300)



4.9 Stability plot for cylindrical cantilever Aluminum shell with and without

damping (R/a = 25, L=2.0 m,a=1.0 m, thickness = 0.03 m,

E = 70 GPa, ν = 0.3) 54

4.10 Stability plots for different fibre orientations

0/�/0 with R/a ratio 15 and 100, L=2m, a=1m, t=0.03 m

for cantilever composite shell. 56

4.11 Variation of stability point for different

fibre orientations 0/�/0 with R/a ratio 15 and 100. 57

4.12 Stability plots for different fibre orientations 90/�/90 with R/a ratio 15 and

100, L=2m, a=1m, t=0.03 m for cantilever composite shell. 59

4.13 Variation of stability point for different fibre orientations

90/�/90 with R/a ratio 15 and 100. 59

4.14 Stability plots for different fibre orientations �/�/� with R/a ratio 15 and 100,

L=2m, a=1m, t=0.03 m for cantilever composite shell. 61

4.15 Variation of stability point for different fibre orientations

�/�/� with R/a ratio 15 and 100 61

4.16 Stability plots for different thickness variation with

R/a ratio 15 and 100, L=2m, a=1m and fibre orientation

0/0/0 for a cantilever composite shell. 63

4.17 Variation of stability point for different thickness

values with R/a ratio 15 and 100, fibre orientation – 0/0/0,

L = 2 m, a = 1 m of a cantilever composite cylindrical shell. 63

4.18 Stability plots for different geometries with R/a ratio 15 and 100, t = 0.03 m,

fibre orientation 30/30/30 for a cantilever composite shell. 65

4.19 Variation of stability point for different L/a ratios

with R/a ratio 15 and 100, fibre orientation 30/30/30,

t = 0.03 m of a cantilever composite cylindrical shell. 65

(xiii)

LIST OF TABLES

Table No. Name of Tables Page No.

4.1 Mesh convergence study of non-dimensional transverse

tip deflection (w/) for an Isotropic cantilever shell

(L=2m a=0.5m, t=0.02m of R/a = 30) 45

4.2 Non-dimensional transverse deflection at x = 0.5 m

and y = 0 m of a cantilever Aluminum shell

(L= 2m, a=1m, t=0.03 m). 45

4.3 Non-dimensional transverse deflection (w/) of at x = 0.5 m

and y = 0 m of a cantilever shell

(L= 2m, a=1m, t=0.03 m for R/a = 50). 47

4.4 Mesh convergence study of non-dimensional

first natural frequency of isotropic cantilever shell

(L= 2m, a=1m, t=0.03 m). 47

4.5 Non-dimensional first natural frequency of composite

cantilever shell (L= 2m, a=1m, t=0.03 m.) 48

4.6 Non-dimensional first buckling load factor of

isotropic cantilever shell (L= 2m, a=1m, t=0.03 m). 50

4.7 Non-dimensional first buckling load factor of

composite cantilever shell for R/a ratio 50

(L= 2m, a=1m, t=0.03 m). 51

4.8 Non-dimensional 1st natural frequency and

buckling load factor for 0/�/0 orientation for

cylindrical shell of L=2m, a=1m and t=0.03m. 57

1�

CHAPTER – 1

INTRODUCTION

1.1 GENERAL INTRODUCTION

Failures in many engineering structures fall into one of the two simple categories: (1)

material failure and (2) structural instability due to form failure. In material failure, the

stresses in the structure exceed the specified safe limit, resulting in the formation of cracks

which cause failure. Material failure can be adequately predicted by analyzing the structure

on the basis of equilibrium conditions or equations of motion that are written for the initial

undeformed configuration of the structure. By contrast, the prediction of failures due to

structural instability requires equations of equilibrium or motion to be formulated on the basis

of deformed configuration of the structure. In many instances, instability is not directly

associated with the failure of the overall structure. For example, if the skin of a plate or shell

like structure wrinkles or locally buckles, the entire structure does not fail. However, if a

portion of the structural element between two adjacent parts becomes unstable, the entire

structure fails catastrophically. Thus, stability plays a very important part in designing a

structure.

The load at which a structure becomes unstable can be, in the simplest approach,

regarded as independent of the material strength or yield limit; it depends on loading of the

structure, structural geometry and size, especially slenderness, and is governed primarily by

the stiffness of the material, characterized, for example, by the elastic modulus. Failures of

elastic structures due to structural instability have their primary cause in geometric effects:

the geometry of deformation introduces nonlinearities that amplify the stresses calculated on

the basis of the initial undeformed configuration of the structure. In the structural instability

due to form failure, though the stresses may not exceed the safe value, the structure may not

able to maintain its original form. Here, the structure does not fail physically but may deform

to some other shape due to intolerable external disturbance.

The instability problems, according to the type of loading can be divided into two

categories: (1) static instability and (2) dynamic instability. If the loading are static in nature

the instability problems encountered are called static instability and if the loading varies with

time it is called dynamic instability.

Buckling problems in membrane structures due to static loading is a kind of static

instability problems. Buckling occurs when the conditions of loading are such that

compressive stresses get introduced and when a member or a structure converts membrane

strain energy into strain energy of bending with no change in externally applied load. When

the magnitude of the load on a structure is such that the equilibrium changes from stable to

neutral, the load is called critical load. Buckling of bars, frames, plates and shells may occur

as a structural response to membrane forces. The membrane forces alter the bending stiffness

of a structure. Thus buckling occurs when compressive membrane forces are large enough to

2�

reduce the bending stiffness to zero for some physically possible deformation mode. If the

membrane forces are reversed – that is, made tensile rather than compressive – bending

stiffness is effectively increased. This is called stress stiffening effect. The effect of

membrane forces are accounted for by a matrix [k�] that augments the conventional stiffness

matrix [k]. Matrix [k�] has given various names, as follows: initial stress stiffness matrix,

differential stiffness matrix, geometric stiffness matrix, and stability coefficient matrix.

In certain cases time varying loads acts axially on the structures which may lead to

loss of dynamic equilibrium of the system resulting in instability of the system. Among the

problems of the dynamic stability of structures probably the best known subclass can be

constituted by the problems of parametric excitation, or parametric resonance. If the ordinary

resonance of forced vibrations occurs when the natural and exciting frequencies are equal

(primary resonance), then parametric resonance occurs when the exciting frequency is equal

to double the frequency of free vibrations (principal parametric resonance). Another essential

difference of parametric resonance lies in the possibility of exciting vibrations with

frequencies smaller than the frequency of the principal resonance. Finally, qualitatively new

in parametric resonance is the existence of continuous regions of excitation (regions of

dynamic instability). A typical example is the initially straight prismatic column whose two

ends are simply supported and upon which a periodic axial compressive load is acting. Such a

column is known to develop lateral oscillations if its straight-line equilibrium is disturbed.

Depending upon the magnitude and the frequency of the pulsating axial load, the linear Hill

or Mathieu equation defining the lateral displacements of the column may yield bounded or

unbounded values for these displacements.

As the work is going to be on the dynamic instability of cylindrical shell due to

parametric excitation, this subclass is discussed in some more detail in a separate section.

1.2 TYPES OF DYNAMIC INSTABILITY

In the introduction to the first English edition of their monumental textbook entitled

Engineering Dynamics, Biezeno and Grammel [1] explain that, following Kirchhoff’s

definition, dynamics is the science of motion and forces, and thus includes statics, which is

the study of equilibrium, and kinetics, which treats of the relationship between forces and

motion. Dynamics is generally accepted as the antonym of statics in everyday usage, and this

is the sense in which it is used in the title of International Conference on Dynamic Stability

of Structures.

A number of significantly different concepts can be included in the in the meaning of

the term dynamic stability of structures. One of them is the stability of an elastic system

subjected to forces that are functions of time. Another is the study of stability of a system

subjected to constant forces as long as the study is carried out with the aid of the dynamic

3�

equations of motion; such an investigation is designated by Ziegler as a stability analysis with

the aid of the kinetic criterion. Excluding parametric resonance the other types of dynamic

instability problems are listed as follows:

1.2.1 Impulsive loading:

In the second subclass of the dynamic stability of structures, buckling of column

under step loading and impulsive loading can be studied. It can be shown that a suddenly

applied load can cause collapse even it is smaller than the Euler load. At the same time, the

column need not be damaged by a suddenly applied load greater than the Euler load if the

load is removed after a sufficiently short time.

1.2.2 Circulatory loads:

The third subclass can be constituted by problems of buckling under stationary

circulatory loads, that is, under loads not derivable from a potential and not explicitly

dependent on time. A static linear analysis leads to the conclusion that a column, one of

whose ends is rigidly fixed while the other is subjected to a compressive load of constant

magnitude whose direction is always tangent to the deformed column axis, does not buckle,

whatever be the magnitude of the load.

1.2.3 Aero elastic problems:

Interaction between the non-conservative aerodynamic forces and the elastic structure

of airplanes and missiles can give rise to theoretically interesting and practically important

problems. They are dealt with, as a rule, by specialists known as aeroelasticians.

1.2.4 Buckling in the testing machine:

Of the many possible time-dependent loading conditions not yet mentioned, one,

buckling under the conditions prevailing in the ordinary testing machine, presents special

interest. In industry, most compressed structural elements are designed on the basis of Euler’s

theory of buckling, or with the aid of one of the modifications of Euler’s theory to account for

inelastic behavior. The practical suitability of these theories is judged, as a rule, on the basis

of a comparison with buckling loads obtained in conventional mechanical or hydraulic testing

machine. In 1949 Hoff J. Nicholas [2] drew attention to the fact that the behavior of dynamic

system consisting testing machine and test column does not necessarily agree with that of a

compressive element in an airplane hitting the ground or in a bridge subjected to dead and

live loads; nor do the initial and boundary conditions assumed in Euler’s theory agree with

those prevailing in the testing machine.

�

�

4�

1.3 PARAMETRIC EXCITATION

The problem of parametric resonance arises in many branches of physics and

engineering. One of the important problems is that of dynamic instability which is the

response of mechanical and elastic systems to time-varying loads, especially periodic loads.

There are cases in which the introduction of a small vibrational loading can stabilize a system

which is statically unstable or destabilize a system which is statically stable. In contrast with

the case of external excitations in which large response can’t be produced by a small

excitation unless the frequency of excitation is close to one of the natural frequencies of the

system (primary resonance), a large response can be produced by a small parametric

excitation when the frequency of the excitation is close to double the natural frequencies of

the system (principal parametric resonance). A physical system undergoes a parametric

forcing if one of its parameters is modulated periodically with time.

A common familiar example of parametric excitation of oscillations is given by the

playground swing on which most people have played in childhood. The swing can be treated

as a physical pendulum whose reduced length changes periodically as the child squats at the

extreme points, and straightens when the swing passes through the equilibrium position. It is

easy to illustrate this phenomenon in the classroom with the following simple experiment. Let

a thread with a bob hanging from its one end pass through a little ring fixed in a support.

Some small length of the other end of the thread that is held in the hand is pulled each time

the swinging bob passes through the middle position and the thread is released to its previous

length each time the bob reaches the maximum deflection. These periodic variations of the

pendulum length with the frequency twice the frequency of natural oscillation cause the

amplitude to increase progressively. In practice parametric excitation can occur in structural

systems subjected to vertical ground motion, aircraft structures subjected to turbulent flow,

and in machine components and mechanisms. Other examples are longitudinal excitation of

rocket tanks and their liquid propellant by the combustion chambers during powered flight,

helicopter blades in forward flight in a free-stream that varies periodically, and spinning

satellites in elliptic orbits passing through a periodically varying gravitational field. In

industrial machines and mechanisms, their components and instruments are frequently

subjected to periodic or random excitation transmitted through elastic coupling elements. A

few examples include those associated with electromagnetic and aeronautical instruments,

vibratory conveyers, saw blades, belt drives and robot manipulators etc.

The motion of a particle of mass m attached to one end of a mass-less rod of length l,

while the other end of the rod is attached to a point under the influence of a prescribed

acceleration as shown in Figure 1.1 is considered. Applying Newton’s second law of motion

in the direction perpendicular to the rod leads to

�� (1.1)

Hence, ��

� � �� (1.2)

5�

which is an equation with variable coefficients. For small oscillations about � � �, the above

equation (1.2) can be linearized to yield

��

� � � � �� (1.3)

Here it is assumed that F is a periodic, i.e. !� � "#� � !��$ Obviously the components of

F(t) are also periodic. The rest points of equation (1.3) are %�& �'( � �&�� )�* #& ��$ The

rest point (0, 0) corresponds to the down position of the pendulum and #& �� to the up

position. The Equation 1.3 is a Hill’s equation. The solution of these equations may be either

bounded or unbounded, i.e. the rest points are stable or unstable, depending on the parameters

in the equation. This leads to the remarkable result that the upward or downward rest points

may be either stable or unstable. Hence, it is of interest to characterize the stability/instability

by a stability diagram as a function of the parameters in the equations. There are many

stability results for this type of equation, which are based on Floqet theory. The theory

characterizes the problem of stability of solutions in terms of transition surfaces in parameter

space that separate regions of stability and instability. Crossing one of these surfaces leads to

a change of stability.

6�

Figure-1.2. Stability Diagram

Figure 1.3 Solutions of Mathieu Equation

7�

1.4 SHELLS

Shells are common structural elements in many engineering structures, including

pressure vessels, submarine hulls, ship hulls, wings and fuselages of airplanes, pipes,

exteriors of rockets, missiles, automobile tires, concrete roofs, containers of liquids, and

many other structures. The theory of laminated shells includes the theories of ordinary shells,

flat plates, and curved beams as special cases.

1.4.1 Shell as a Structural Form:

Thin shells are an example of strength through form as opposed to strength through

mass. The effort in design is to make the shell as thin as practical requirements will permit so

that the dead weight is reduced and the structure functions as a membrane free from large

bending stresses. By this means, a minimum of materials is used to the maximum structural

advantage. Shells of double curvature are among the most efficient of known structural

forms. Most shells occurring in nature are doubly curved. Shells of eggs, nuts, and the human

skull are commonplace examples. These naturally occurring shells are hard to crack or break.

1.4.2 Parametric Excitation on Shell Structure:

The problem of parametric resonance can occur in various shell type of structures

when the structural system is under time-varying loads, especially periodic loads. Some

common examples are given below:

� The shell of the rocket can be under parametric excitation or resonance when

the huge amount of energy releases from it’s backwards and as a result the

body of the rocket goes upwards. Here the reaction force due to the release of

that energy can be treated as time varying loads.

� If a dynamic system is mounted on a dome structure, the structure can be

under parametric excitation due to the periodic or time varying load

producing from that system.

� The ship-haul can be under parametric excitation in case of propeller-wave

force interaction.

� An aircraft body can be under parametric excitation in case of propeller-air

force interaction.

� A fluid filled structure subjected to base excitation can be under parametric

excitation or resonance.

8�

1.5 LITERATURE REVIEW

A literature review of work related to dynamic instability of orthotropic cylindrical

shell under parametric excitation has been carried out and presented in this chapter. For that

purpose at first an independent literature review on various structures under parametric

excitation is done. Then a literature review on general shell structures is carried out. Lastly, a

literature review on shell structures under parametric instability or dynamic instability has

been done in different sub-sections.

1.5.1 Literature review on parametric excitation:

The phenomenon of parametric resonance was first observed by Faraday [3]. He

noted that surface waves in a fluid-filled cylinder under vertical excitation exhibited twice the

period of the excitation itself.

Melde [4] tied a string between a rigid support and the extremity of the prong of a

massive tuning fork of low pitch. He observed that the string could be made to oscillate

laterally, although the exciting force is longitudinal, at one half the frequency of the fork

under a number of critical conditions of string mass and tension and fork frequency and

loudness.

Strutt [5] provided a theoretical basis for these observations and performed further

experiments with a string attached to one end of the prong of a tuning fork.

The results of Strutt [5] were amplified by Stephenson [6] and he observed the

possibility of exciting vibrations when the frequency of the applied axial excitation is a

rational multiple of the fundamental frequency of the lateral vibration of the string.

Stephenson [6] seems to be the first to point out that a column under the influence of a

periodic load may be stable even though the steady value of the load is twice that of the Euler

load.

A lengthy investigation was presented by Raman [7] which is beautifully and

profusely illustrated with photographs of vibrating springs.

Beliaev [8] analyzed the response of a straight elastic hinged-hinged column to an

axial periodic load of the form �� +, � +- ./� 0� . He obtained a Mathieu equation for the

dynamic response of the column and determined the principal parametric resonance

frequency of the column. The results show that a column can be made to oscillate with the

frequency -1 0 if it is close to one of the natural frequencies of the lateral motion even though

the axial load may be below the static buckling load of the column.

Beliaev’s investigation was completed by Andronov and Leontovich [9], and

Lubkin and Stoker [10] and Mettler [11] presented detailed analysis of this problem. These

results were verified experimentally by Gol’denblat [12], Bolotin [13], and Somerset and

Evan-Iwanowski [14].

9�

Krylov and Bogoliubov [15] used the Galerkin procedure to determine the dynamic

response of a column with arbitrary boundary conditions under the influence of

multiharmonic axial forces.

Chelomei [16] studied the parametric resonance of a column.

Kochin [17] examined the mathematically related problem of the vibrations of a

crankshaft, and Timoshenko [18] and Bonderenko [19] treated another mathematically

related problem in connection with the vibrations of the driving system of an electric

locomotive.

Asfar and Masoud [20] studied a single-degree freedom parametrically excited

system coupled with a lanchester damper, a mass-dash pot device.

Bolotin [13] gave an extensive treatment of dynamic stability of shallow, cylindrical,

and spherical shells, while Hsu [21] gave a review of the parametric excitation and snap-

through instability phenomenon of shells.

A number of physical systems contain pipes conveying fluid. The velocity often has

an unsteady component induced by the pumps. Thus parametric and combination instabilities

might occur in such pipes. These were studied theoretically by Chen [22], Ginsberg [23],

Paidoussis and Issid [24], Bohn and Herrmann [25], and Paidoussis and Issid [26].

P. Bhattacharya, S. Homann and M. Rose [27] studied the effects of piezo actuated

damping on parametrically excited laminated composite plates with feedback control

methodology. The study showed that piezo electric damping plays a positive role on the

stability behavior of laminated plates.

1.5.2 Literature review on shell:

A number of theories exist for layered anisotropic shells. Many of this theories were

developed originally for thin shells, and are based on the Kirchhoff-Love kinematic

hypothesis that straight lines normal to the undeformed mid-surface remain straight and

normal to the middle surface after deformation and undergo no thickness stretching. Surveys

of various shell theories can be found in the works of Naghdi [28] and Bert [29] and a

detailed study of thin ordinary (i.e., not laminated) shells can be found in the monographs by

Kraus [30], Ambartsumyan [31], and Vlasov [32].

The first analysis that incorporated the bending-stretching coupling (owing to

unsymmetric lamination in composites) is due to Ambartsumyan. In his analysis,

Ambartsumyan assumed that the individual orthotropic layers were oriented such that the

principal axis of material symmetry coincided with the principal coordinates of the shell

reference surface. Thus Ambartsumyan’s work dealt with what is now known as laminated

orthotropic shells, rather than laminated anisotropic shells; in laminated anisotropic shells,

the individual layers are, in general, anisotropic and the principal axes of material symmetry

of the individual layers coincide with only one of the principal coordinates of the shell (the

thickness normal coordinate).

10�

Dong, Pister, and Taylor [33] formulated a theory of thin shells laminated of

anisotropic material that is an extension of the theory developed by Stavsky [34] for

laminated anisotropic plates to Donnell’s shallow shell theory. Cheng and Ho [35] presented

an analysis of laminated anisotropic cylindrical shells using Flugge’s shell theory. A first

approximation theory for the un-symmetric deformation of non-homogeneous, anisotropic,

elastic cylindrical shells was derived by Widra and Chung [36] by means of the asymptotic

integration of the elasticity equations. For a homogeneous, isotropic material, the theory

reduces to Donnell’s equations.

All the theories listed above are based on Kirchhoff-Love’s hypotheses, in which the

transverse shear deformation is neglected. These theories, known as the Love’s first

approximation theories, are expected to yield sufficiently accurate results when

� the radius to thickness ratio is large,

� the excitations are within the low frequency range, and

� the material anisotropy is not served.

However, the application of such theories to layered anisotropic composite shells could lead

to 30% or more errors in deflections, stresses, and frequencies.

Whitney and Sun [37] developed a shear deformation theory for laminated

cylindrical shells that includes transverse shear deformation and transverse normal strain as

well as expansional strains. Recently, Reddy [38] presented a generalization of Sander’s shell

theory (1959) to laminated, doubly-curved anisotropic shells. The theory accounts for

transverse shear strains and the Von-Karman (nonlinear) strains.

1.5.3 Literature review on shell structures under parametric instability or dynamic

instability:

S.K. Sahu and P.K.Dutta [39] studied the dynamic stability of curved panels with

cutouts. In their literature the parametric instability behavior of curved panels with cutouts

subjected to in-plane static and periodic compressive edge loadings were studied using finite

element analysis. The first order shear deformation theory was used to model the curved

panels, considering the effects of transverse shear deformation and rotary inertia. The theory

used was the extension of dynamic, shear deformable theory according to the Sander's first

approximation for doubly curved shells, which can be reduced to Love's and Donnell's

theories by means of tracers. The effects of static load factor, aspect ratio, radius to thickness

ratio, shallowness ratio, boundary conditions and the load parameters on the principal

instability regions of curved panels with cutouts were studied in detail using Bolotin's

method.

Tension buckling and dynamic stability behavior of laminated composite doubly

curved panels subjected to partial edge loading was studied by L. Ravi Kumar, P.K. Datta

and D.L. Prabhakara [40]. This paper deals with the study of tensile buckling, vibration and

dynamic stability behavior of multi-laminated curved panels subjected to uniaxial in-plane

11�

point and patch tensile edge loadings by using the finite element method. The effect of first

order shear deformation theory was used to model the doubly curved panels.

Dynamic instability characteristics of laminated composite doubly curved panels

subjected to partially distributed follower edge loading was studied by L. Ravi Kumar, P.K.

Datta and D.L. Prabhakara [40]. This paper deals with the study of vibration and dynamic

instability characteristics of laminated composite doubly curved panels, subjected to non-

uniform follower load, using finite element approach. In their work they used first order shear

deformation theory to model the doubly curved panels. They also considered the effects of

shear deformation and rotary inertia. The formulation was based on the extension of dynamic,

shear deformable theory according to Sanders first approximation for doubly curved

laminated shells, which can be reduced to Love’s and Donnell’s theories by means of tracers.

The modal transformation technique was applied to reduce the number of equilibrium

equations for subsequent analysis. Structural damping was introduced into the system in

terms of viscous damping. Instability behavior of curved panels had been examined

considering the various parameters such as width of edge load, load direction control,

damping, influence of fiber orientation and lay up sequence etc.

Vibration, buckling and dynamic stability of cracked cylindrical shells was studied by

M. Javidruzi, A. Vafai, J.F. Chen and J.C. Chilton [41]. This paper presents a finite

element study on the vibration, buckling and dynamic stability behavior of a cracked

cylindrical shell with fixed supports and subject to an in plane compressive/tensile periodic

edge load. The effects of crack length and orientation were analyzed.

S.N. Patel, P.K. Datta and A.H. Sheikh [42] studied the Buckling and dynamic

instability analysis of stiffened shell panels. The static and dynamic instability characteristics

of stiffened shell panels subjected to uniform in-plane harmonic edge loading were

investigated in this paper. The eight-noded iso-parametric degenerated shell element and a

compatible three-noded curved beam element were used to model the shell panels and the

stiffeners, respectively.

Parametric instability of doubly curved panels subjected to non-uniform harmonic

loading was studied by S.K. Sahu and P.K. Dutta [39]. The parametric instability

characteristics of doubly curved panels subjected to various in-plane static and periodic

compressive edge loadings, including partial and concentrated edge loadings were studied

using finite element analysis. The first order shear deformation theory was used to model the

doubly curved panels, considering the effects of transverse shear deformation and rotary

inertia.

Parametric instability of thick, orthotropic, circular cylindrical shell was studied by C.

W. Bert and V. Birman [43]. The dynamic instability of simply supported, finite-length,

circular cylindrical shells subjected to parametric excitation by axial loading were

investigated analytically. The theory used was a general first-order shear deformable shell

theory.

12�

1.6 OBJECTIVE AND SCOPE OF THE PRESENT WORK

From the critical review of the existing literatures in the field of dynamic instability of

orthotropic shells under parametric excitation done in the previous section, it is found that a

very few literature is available in the area of parametric instability of orthotropic shell

structures and the effect of damping on instability behavior is almost non-existent.

The objectives of the present work are given as below:

� To develop a four noded iso-parametric shell element having five mechanical

d.o.f. per node.

� To determine the natural frequency and corresponding natural mode shape.

� To evaluate the buckling load factor and corresponding buckling mode shape.

� To find the instability regions of a cylindrical shell structure for different radius of

curvature to width ratio for isotropic as well as various fiber orientations under

periodic in-plane loading (parametric excitation).

� To estimate the effect of damping on the instability behavior of circular

cylindrical shells.

The present work has been developed in the MATLAB environment using four noded

iso-parametric element. The various numerical results are compared with ANSYS finite

element software. The different aspects of the present work are presented systematically in

various sections which have been briefly outlined in the next section.

1.7 ORGANIZATION OF REPORT

With the objectives stated above, the present work is presented in five chapters.

Chapter 1: Introduction

A general introduction to the present work, types of dynamic instability, a short note

on parametric excitation is given. A literature review is presented to give a broad

understanding of the previous works related to the present work. Finally the objective

and scope of the present work is presented.

Chapter 2: Constitutive Equations

In this chapter a general introduction to composite materials, lamina and laminate are

given. Constitutive equations of laminated composites and displacement modeling

have been shown. Finally the energy formulation for the present physical problem has

given.

13�

Chapter 3: Finite Element Formulation

In this chapter, a brief discussion on the element shape functions is given. The strain-

displacement equations are derived. The structural stiffness matrix, mass matrix,

geometric stiffness matrix are derived. Finally the stability equations are presented.

Chapter 4: Results and Discussion

This chapter presents the results for static, buckling and free vibration analysis using

developed FE code and compares with ANSYS finite element software. Parametric

instability of a cylindrical shell has been carried out both for isotropic and composite

cases.

Chapter 5: Conclusions

Major conclusions drawn from the present work along with a brief outline on the

scope of future work on the present investigation are highlighted in this chapter.

��

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�

�

�

�

�

�

�

14�

CHAPTER-2

CONSTITUTIVE EQUATIONS

2.1 INTRODUCTION

In this chapter a brief discussion is done on laminated composites. Subsequently after

making assumptions regarding the behavior of a laminate, the macro-mechanical behavior of

composite laminate is studied and the constitutive equations are presented. The

transformation of the elastic constants from on to off axis system has also been done to

correlate the stresses in the reference axis and that in the principal material axis. The

displacement modeling has shown and the expressions for the stress resultants in a laminate

are also derived. Finally, the energy formulation is given.

2.2 COMPOSITE MATERIALS

A composite material consists of two or more materials and offers a significant weight

saving in a structure in view of its high strength to weight and high stiffness to weight ratios.

Further, in a fibrous composite, the mechanical properties can be varied as required by

suitably orienting the fibres. In such a material the fibres are the main load bearing members,

and the matrix, which has low modulus and high elongation, provides the necessary

flexibility and keeps the fibres in position and protects them from environment.

Depending on the matrix material, composites can be further classified into the

following categories:

� Metal matrix composites (MMC)

� Polymer matrix composites (PMC)

� Ceramic matrix composites (CMC)

Depending on the reinforcement used, they can be classified into the following groups:

� Fibre reinforced composites

� Particle reinforced composites

� Flake/Plate reinforced composites

In the present work unidirectional fiber reinforced composites are considered.

2.3 LAMINA AND LAMINATE

A lamina, in general is a thin sheet with fibres oriented in some direction. Such a sheet

can be characterized as two dimensional, with orthogonal material properties. It is, however,

15�

not capable of carrying any load. Hence, for practical purposes, a structure consisting of

several laminae (laminate) is used.

Figure-2.1 Laminate construction

2.4 ASSUMPTIONS REGARDING THE BEHAVIOR OF A LAMINATE

� Laminate is made of perfectly bonded laminae.

� The bonds are infinitesimally thin and no lamina can slip relative to other. This

implies that displacements are continuous across the lamina boundaries. As a

result, the laminate behaves like a lamina with special properties.

� After displacement, a line originally straight and perpendicular to the middle

surface of the laminate remains straight and but is not necessarily perpendicular

to the middle surface.

� Constant normal strain is present.

2.5 MACRO MECHANICAL BEHAVIOR OF COMPOSITE LAMINATES

The stiffness of a composite laminate changes with the change in ply orientation. The

particular axis which is chosen for conveniently solving the problem, is known as the loading

axis or the reference axis and for the fibre reinforced composites, another axis system which

is parallel and perpendicular to the fibre orientation is convenient for the calculation of

material properties, is known as the principal material axis. The axis system as described

above is shown below in Figure 2.2

16�

Figure-2.2 Axis system of a unidirectional stressed Lamina

The stress-strain relations in principal material coordinates for a lamina of an orthotropic

material under plane stress conditions are

��

�

��

�

�

��

�

��

�

�

23

13

12

2

1

τ

τ

τ

σ

σ

=

��

��

�

55

44

66

2212

1211

0000

0000

0000

000

000

Q

Q

Q

QQ

QQ

��

�

��

�

�

��

�

��

�

�

23

13

12

2

1

γ

γ

γ

ε

ε

�� (2.1)��

�

where, [ ]Q i,j=1,2,6,4,5 are reduced stiffness for plane stress and are defined in terms of the

engineering constants as�

2112

111

1 υυ−=

EQ (2.2)

2112

222

1 υυ−=

EQ (2.3)

2112

121

2112

2122112

11 υυ

υ

υυ

υ

−=

−===

EEQQ (2.4)

1266 GQ = ; 2344 GQ = ; 3155 GQ = (2.5)

17�

The relationships between stresses of the principal material axis and the reference axis are

234-&1 � �5 2346&7 (2.6)

The relationships between strains of the principal material axis and the reference axis are

284-&1 � �5 2846&7 (2.7)

where, [T] is the transformation matrix and is given by the following equation

�5 ��

��

�

−−

−

−

)(0022

000

000

00

00

22

22

22

nmmnmn

mn

nm

mnmn

mnnm

(2.8)

where,

m = cosθ ; n = sinθ ; θ = orientation of the fiber with reference axis.

Now when a lamina is loaded in the reference axis X-Y, the relationship between stresses in

the reference X-Y axis and that in the principal material axis 1-2 is given by

2346&7 � �5 9-234-&1 (2.9)

i.e.

��

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��

�

�

��

�

��

�

�

yz

xz

xy

y

x

τ

τ

τ

σ

σ

��

��

��

�

−−

−

−

)(0022

000

000

00

00

22

22

22

nmmnmn

mn

nm

mnmn

mnnm

��

��

�

55

44

66

2212

1211

0000

0000

0000

000

000

Q

Q

Q

QQ

QQ

��

�

��

�

�

��

�

��

�

�

23

13

12

2

1

γ

γ

γ

ε

ε

(2.10)

Substituting strains in the 1-2 axis in terms of X-Y reference axis Equation 2.9 will take the form as

2346&7 � �5 9-�: ;&<�5 284=&> i , j = 1,2,6,5,4 (2.11)

18�

Equation- 2.11 will take the form as

��

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��

�

�

��

�

��

�

�

yz

xz

xy

y

x

τ

τ

τ

σ

σ

=

��

��

�

5545

5444

662616

262212

161211

''000

''000

00'''

00'''

00'''

QQ

QQ

QQQ

QQQ

QQQ

��

�

��

�

�

��

�

��

�

�

yz

xz

xy

y

x

γ

γ

γ

ε

ε

(2.12)

where [ ]′Q i,j = 1,2, 6,4,5 are the transformed reduced stiffnesses which are given in terms of

reduced stiffnesses, ijQ as

11'Q = 11Q m4 + 2( 12Q + 66Q ) n2m2 + 22Q n4 (2.13)

12'Q = ( 11Q + 22Q - 4 66Q ) n2m2 + 12Q (n4+m4) (2.14)

22'Q = 11Q n4 + 2( 12Q + 2 66Q ) n2m2 + 22Q m4 (2.15)

16'Q = ( 11Q - 12Q - 2 66Q ) nm3 + ( 12Q - 22Q + 2 66Q ) n3m (2.16)

26'Q = ( 11Q - 12Q - 2 66Q ) n3m + ( 12Q - 22Q +2 66Q ) nm3 (2.17)

66'Q = ( 11Q + 22Q - 2 12Q - 2 66Q ) n2m2 + 66Q (n4+m4) (2.18)

44'Q = 44Q m2+ 55Q n2 (2.19)

55'Q = 44Q n2+ 55Q m2 (2.20)

45'Q = 54'Q = ( 55Q - 44Q ) mn (2.21)

2.6 DISPLACEMENT MODELLING

The following assumptions are considered for the displacement model

� The material behavior is linear and elastic.

� The thickness of the laminate is small compared to other dimensions. � Displacement u, v, w are small compared to the laminate thickness.

19�

� Normal to the mid-surface before deformation remains straight but is not necessarily normal to the mid-surface after deformation.

� Constant normal strain is present.

Figure-2.3 Displacement field along respective coordinate axis.

Figure-2.4 Deformation of the laminate in X-Z and Y-Z plane

20�

Employing first order shear deformation theory the displacement u, v, w on the shell can be

expressed as,

yzuu θ+= 0 ; xzvv θ−= 0 ; 0ww = (2.22)

where, u, v, w are the translational displacement along X, Y and Z axes at a distance z from

the mid-plane while the notations with suffix ‘0’ denotes the same at mid-plane, and yx θθ ,

are rotation of shell element about X and Y axis respectively.

The strain-displacement relations for a shell element are as follows:

xz

R

W

x

uz

y

x

xxxx∂

∂+��

��

�−

∂

∂=+=

θκεε 000

yz

R

W

y

vz x

y

yyyy∂

∂−��

�

�

��

�

�−

∂

∂=+=

θκεε 000

�

��

∂

∂−

∂

∂+��

�

�

��

�

�−

∂

∂+

∂

∂=+=

xyz

R

w

y

u

x

vz xy

xy

xyxyxy

θθκγγ 0000 2

(2.23)

xyzyzyzy

w

z

v

y

wz θκγγ −

∂

∂=

∂

∂+

∂

∂=+= 0000

yzxzxzxx

w

z

u

x

wz θκγγ +

∂

∂=

∂

∂+

∂

∂=+= 0000

where, 866& 877 are the normal strains in X and Y directions respectively, and ?67& ?7@& ?@6 are

the shear strains in X-Y, Y-Z, and X-Z plane respectively.

The curvatures are expressed as,

A6 � B�7B= C A7 � � B�6B> C A67 � B�7B> � B�6B=

and the mid-plane strains are expressed in terms of the mid-plane displacements as,

��

��

�−

∂

∂=

x

xR

W

x

u 000ε

; ��

�

�

��

�

�−

∂

∂=

y

yR

W

y

v 000ε

; ��

�

�

��

�

�−

∂

∂+

∂

∂=

xy

xyR

w

y

u

x

v 0000 2γ

xyzy

w

z

v

y

wθγ −

∂

∂=

∂

∂+

∂

∂= 0000

; yzx

x

w

z

u

x

wθγ +

∂

∂=

∂

∂+

∂

∂= 0000

2.7 STRESS

The stresses in the k

laminate strains and curvatures as

��

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��

�

�

STRESS -STRAIN RELATIONS FOR A LAMINATE

The stresses in the k


��

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��

�

�

��

�

��

�

�

yz

xz

xy

y

x

τ

τ

τ

σ

σ

=

��

�

16

12

11

0

0

'

'

'

Q

Q

Q

Figure

STRAIN RELATIONS FOR A LAMINATE

Figure

The stresses in the kth layer of a laminated composite


2616

2212

1211

00

00

''

''

''

QQ

QQ

QQ

Figure- 2.6 Stress resultants of a laminated shell element.


Figure- 2.5 A general n

of a laminated composite


45

44

66

26

16

'0

'0

0'

0'

0'

QQ

QQ

2.6 Stress resultants of a laminated shell element.

21�


2.5 A general n-layered laminate

of a laminated composite

��

55

54

'

'

0

0

0

Q

Q

��

�

��

�

�

��

�

��

�

�

yzo

xzo

xyo

yo

xo

γ

γ

γ

ε

ε



layered laminate

of a laminated composite can be expressed in terms of the

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�

��

�

�

��

�

��

�

�

+

��

�

��

�

�

0

0xy

y

x

yz

xz

xy z κ

κ

κ


can be expressed in terms of the

��

�

��

�

�



(2.24)


(2.24)

22�

The resultant forces and moments acting on a laminate are obtained by integrating the

stresses in each layer or lamina through the laminate thickness, as given by

�−=2/

2/

t

tii dzN σ ; �−=

2/

2/

t

tii zdzM σ (2.25)

Ni is the force resultant (force per unit length) of the cross section of the laminate and Mi is

the moment resultant (moment per unit length) as shown in the Figure-2.6.

The total of force and moment resultants for an n-layered laminate can be defined as

dz

N

N

Nt

t

kxy

y

x

xy

y

x

�−��

��

�

��

��

�

=��

��

�

��

��

�2/

2/

σ

σ

σ

; zdz

M

M

Mt

t

kxy

y

x

xy

y

x

�−��

��

�

��

��

�

=��

��

�

��

��

�2/

2/

σ

σ

σ

(2.26)

The integration indicated in the above equations can be rearranged to take advantage of the

fact that the stiffness matrix for a lamina is constant within the lamina. Thus, the stiffness

matrix goes outside the integration over layer, but is within the summation of force and

moment resultants for each layer. When the lamina stress-strain relations are substituted,

��

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��

��

�

��

��

�

+��

��

�

��

��

�

��

��

�

=

��

��

�

��−−

=

Zk

Zk

xy

y

xZk

Zko

xy

o

y

o

xN

k

kkxy

y

x

zdzdz

QQQ

QQQ

QQQ

N

N

N

111

662616

262212

161211

'''

'''

'''

κ

κ

κ

γ

ε

ε

(2.27)

��

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��

�

��

��

�

��

��

�

+��

��

�

��

��

�

��

��

�

=

��

��

�

��−−

=

Zk

Zk

xy

y

xZk

Zko

xy

o

y

o

xN

k

kkxy

y

x

dzzzdz

QQQ

QQQ

QQQ

M

M

M

1

2

11

662616

262212

161211

'''

'''

'''

κ

κ

κ

γ

ε

ε

(2.28)

However, we should recall thato

xε , o

yε , o

xyγ , xκ , yκ and xyκ are not functions of z but are

mid-plane values so can be removed from under the summation signs. Thus the above equations can be written as

��

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�

��

��

�

��

��

�

+��

��

�

��

��

�

��

��

�

=

��

��

�

xy

y

x

o

xy

o

y

o

x

xy

y

x

BBB

BBB

BBB

AAA

AAA

AAA

N

N

N

κ

κ

κ

γ

ε

ε

662616

262212

161211

662616

262212

161211

(2.29)

23�

��

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�

��

��

�

��

��

�

+��

��

�

��

��

�

��

��

�

=

��

��

�

xy

y

x

o

xy

o

y

o

x

xy

y

x

DDD

DDD

DDD

BBB

BBB

BBB

M

M

M

κ

κ

κ

γ

ε

ε

662616

262212

161211

662616

262212

161211

(2.30)

The force and moment resultants together can be expressed as

��

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��

�

�

��

�

��

�

�

��

��

�

=

��

�

��

�

�

��

�

��

�

�

xy

y

x

xy

y

x

xy

y

x

xy

y

x

DDDBBB

DDDBBB

DDDBBB

BBBAAA

BBBAAA

BBBAAA

M

M

M

N

N

N

κ

κ

κ

γ

ε

ε

0

0

0

662616662616

262212262212

161211161211

662616662616

262212262212

161211161211

(2.31)

where,

[ ]bC

DDDBBB

DDDBBB

DDDBBB

BBBAAA

BBBAAA

BBBAAA

=

��

��

�

662616662616

262212262212

161211161211

662616662616

262212262212

161211161211

=constitutive matrix for bending (2.32)

Where, Aij are extensional stiffnesses, Bij are coupling stiffnesses, and Dij is the bending

stiffnesses and are given as below.

�=

−−=n

k

kkkijij ZZQA1

1 )()'( (2.33)

�=

−−=n

k

kkkijij ZZQB1

122 )()'(

2

1 (2.34)

�=

−−=n

k

kkkijij ZZQD1

133 )()'(

3

1 (2.35)

24�

The stress resultants Mxz, Myz for the laminates can be written in terms of the constitutive matrix for the shear [C]s and shear strain xzγ , yzγ as,

��

��

��

=�

��

yz

xz

yz

xz

AA

AA

M

M

γ

γ

5554

4544 (2.36)

Aij, approximated as

Aij =� −− )()( 1kkkoffij ZZQ (2.37)

where, Qoff, ij = off axis reduced stiffness (elastic constants) for plane stress.

2.8 ENERGY FORMULATION

In the present work, structural displacements occur due to external mechanical

loading. Fundamental equations of physical phenomenon can be deduced from time

dependent variational principle i.e. the Hamilton’s principle. It states that the motion of a system from time t0 to t1 is such that the time integral of the difference between the kinetic

and potential energies is stationary for the true path. This may be expressed mathematically as,

�� −==1

0

1

0

)(t

t

t

t

dtUTLdtI (2.38)

where, T and U are the kinetic and potential energies of the system. Now the associated Euler- Lagrange’s equation for n d.o.f system is,

( ) )39.2(,...2,1,.

==−∂

∂−��

�

�

��

�

�

∂

∂iQUT

qq

T

dt

dI

i

In the above equation, T is the kinetic energy and U is the potential energy of the system and qi’s are the generalized coordinates.

The strain energy of the system due to mechanical strain can be expressed as,

{ } { }��=

=n

kV

kTk

M dVU12

1σε (2.40)

25�

The work done due to mechanical loading is given by,

{ }�=V

T

M dvFuW )41.2(

where,

u = Displacement d.o.f in x-direction

F = Mechanical Force vector

� = Stress

� = Strain.

Now, the functional I can be expressed in the variational form for a time interval from t0 to t1 by the use of Equation-2.40 and 2.41 as,

{ } { }� ��

��

��

−==

1

012

1t

t

n

k v

kTk dvdtI σεδδ { } [ ] � ��

��

+�

′′−

1

0

}{}{}{2

1t

t A

m

T

v

TdAFudtdvuu δρ (2.42)

Now if the structural system is subjected to a pulsating axial compressive force P(t) = P

s + P

t

cos�t, acting along its undeformed axis where � is the excitation frequency of the dynamic

load component, Ps is the static and P

t is the amplitude of the time dependent component of

the load. In this case a residual strain will exist in the structural system. For this case, the

strain energy of the system can be expressed as follows: �

(2.43)

where, UL and UNL is the linear and nonlinear strain energy respectively. The expression for

UNL is given in chapter-3 and the expression for UL is given in Equation 2.40. UNL is the

nonlinear strain energy due to compressive residual forces.

�

NLLM UUU +=

26�

CHAPTER – 3

FINITE ELEMENT FORMULATION

3.1 INTRODUCTION

The finite element method is the most popular numerical tool available to the present

day engineer. It is quite versatile and is being used to solve a wide variety of engineering

problems. The following are the basic steps involved in the finite element analysis.

� The continuum is discretised into many sub-regions called finite elements.

They are of arbitrary size, shape and orientation.

� Each element is assumed to be connected to the neighbouring elements only

at a finite number of discrete points called nodes. Depending on the type of

the problems these nodes are located.

� The displacements at the nodes are assumed as the basic unknowns of the

problem. The total number of these nodal displacement components is called

the number of degrees of freedom of the finite element model. The larger this

number, the more accurate is the solution, although more expensive

computationally.

� After getting the element nodal displacement vector, the interior displacement

field within the element is obtained by interpolation functions or shape

functions.

� Once the displacement field within the element is known, the strain field can

be obtained by making use of the strain-displacement relations.

� By making use of the stress-strain relationships, the stress field in the element

can now be obtained.

� The principal of virtual work, Hamilton’s principal and governing equations

of motion are now used to arrive at the element stiffness matrix, element load

vector and element mass matrix.

� Numerical integration schemes are used to convert the resulting set of integral

equations into a set of algebraic equations. The solution of these algebraic

equations represents an approximate solution of the given

physical/mathematical problem.

� The element stiffness matrix, element mass matrix and the element load

vectors for various elements are added together to arrive at the global stiffness

matrix, global mass matrix and global load vector. Both of them are related

by the global displacement vector to form a system of linear algebraic

equations.

� The displacement boundary conditions are applied to make the global

stiffness matrix and global mass matrix non-singular.

27�

3.2 FORMULATION

The finite element formulation for the present case has been described in the

following subsections.

3.2.1 Selection of element

In the present work an attempt has been made to develop a four node isoparametric

2-D finite element with five mechanical degrees of freedom (u, v, w, xθ , yθ ) per node based

on Mindlin-Reissner shallow shell theory.

Figure- 3.1 Four noded quadratic isoparametric element

The mapping between the natural coordinates (D, E) and the physical coordinates

(X,Y) is related by certain functions Ni(D,E). Thus, if (D, E) coordinates of a point in the

natural coordinate system is known, then the coordinates of the corresponding point in the

physical coordinate system (X, Y) is given by

�=

=4

1i

ii xNX

�=

=4

1i

ii yNY ; where, xi and y

i are the nodal co-ordinates of the element.

Similarly, the components of displacement vector at any point of the element are given by

�=

=4

1i

iiuNu �� =

=4

1i

ii vNv �� =

=4

1i

ii wNw �� =

=4

1iixix N θθ ��

=

=4

1iiyiy N θθ �

where, u, v, w, �x, �

y having a subscript ‘i’ are the nodal displacements.

28�

The element displacements as given in equation can be written in the matrix form as follows:

��

�

��

�

�

��

�

��

�

�

��

��

�

=

��

�

��

�

�

��

�

��

�

�

yi

xi

i

i

i

i

i

i

i

i

y

x

w

v

u

N

N

N

N

N

w

v

u

θ

θ

θ

θ

0000

0000

0000

0000

0000

i.e , { } [ ]{ }edNu = (3.1)

The shape functions for an four noded quadrilateral element is given by,

N 1 = )1)(1(4

1ηξ −− ; N 2 = )1)(1(

4

1ηξ −+ ;

N 3 = )1)(1(4

1ηξ +− ; N 4 = )1)(1(

4

1ηξ ++ ;

The interpolation polynomials given above can be concisely written as

F; � -G H � DD;�H � EE;�& � � H �� I (3.2)

where, �i and �

i are the values of natural co-ordinates at node ‘i’.

The transformation from the local (�-�) co-ordinate to the global (x-y) co-ordinate is carried out as follows:

[ ] �

��

=�

��

−

η

ξ

i

i

iY

iX

N

NJ

N

N 1 ;

where, [ ] �

��

=

ηη

ξξ

ii

ii

YX

YXJ (3.3)

[J] is called the Jacobian matrix.

29�

3.2.2 Strain-Displacement Relations

The strains for the shell element can be calculated by using the constitutive equations

given below:

��

�

��

�

�

��

�

��

�

�

+

��

�

��

�

�

��

�

��

�

�

=

��

�

��

�

�

��

�

��

�

�

0

00

0

0

0

0

xy

y

x

xz

yz

xy

y

x

xz

yz

xy

y

x

k

k

k

Z

γ

γ

γ

ε

ε

γ

γ

γ

ε

ε

(3.4) where, �x, �y, �xy, �yz, �xz are the strains at a distance z from the mid-plane of the shell, �x0,

�y0

, �xy0

, �yz0

, �xz0 are the mid-plane strains and x, y, xy are the shell curvatures.

The above equation can also be written as in the following form:

{ } 155885 ][][ xxx

xz

yz

xy

y

x

dBZ=

��

�

��

�

�

��

�

��

�

�

γ

γ

γ

ε

ε

(3.5)

where, [Z] is the position matrix and {d} is the nodal d.o.f as given below:

��

��

�

=

10000000

01000000

0000100

0000010

0000001

][ z

z

z

Z ; { }

��

�

��

�

�

��

�

��

�

�

=

y

x

w

v

u

d

θ

θ0

0

0

(3.6a)

The strain-displacement matrix [B] can be written as:

�=

��

��

�

∂

∂

−∂

∂∂

∂

∂

∂−

∂

∂−

∂

∂

−∂

∂

∂

∂

−∂

∂

−∂

∂

=4

1

000

000

000

0000

0000

002

000

000

][i

ii

ii

ii

i

i

xy

iii

y

ii

x

ii

Nx

N

Ny

N

y

N

x

N

y

Nx

N

R

N

x

N

y

N

R

N

y

N

R

N

x

N

B (3.6b)

30�

The above [B] matrix can be divided into two parts, bending and shear strain-displacement

matrices.

Bending strain-displacement matrix consists of bending strains, shear strains and shell

curvatures.

�=

��

��

�

∂

∂

∂

∂−

∂

∂−

∂

∂

−∂

∂

∂

∂

−∂

∂

−∂

∂

=4

1

000

0000

0000

002

000

000

][i

ii

i

i

xy

iii

y

ii

x

ii

b

y

N

x

N

y

Nx

N

R

N

x

N

y

N

R

N

y

N

R

N

x

N

B (3.7)

Shear strain-displacement matrix consists of transverse shear strains.

�=

��

��

�

−∂

∂

−∂

∂

=4

1 000

000][

i

i

i

i

i

s

Nx

N

Ny

N

B (3.8)

The strain-displacement relationship for the shell in the matrix operator form is derived from

Equation 2.23 and is expressed as:

{ }{ }{ } �

��

�

��

�

�

��

�

��

�

�

��

�

��

�

�

��

�

��

�

�

��

��

�

∂

∂

∂

∂−

∂

∂−

∂

∂∂

∂

−∂

∂

−∂

∂

∂

∂

−∂

∂

−∂

∂

=

��

�

��

�

�

��

�

��

�

�

=

4

3

2

1

0

0

0

41

0

0

0

0

0

000

0000

0000

000

000

002

000

000

y

x

toi

ii

i

i

i

i

i

i

y

iii

y

ii

x

ii

xy

y

x

zx

yz

xy

y

x

w

v

u

y

N

x

N

y

Nx

N

Nx

N

Ny

N

R

N

x

N

y

N

R

N

y

N

R

N

x

N

θ

θ

κ

κ

κ

γ

γ

γ

ε

ε

(3.9)

31�

which can also be written as,

(3.10)

{ } { }edB][0 =ε

(3.11)

where [B] is given by equation (3.6b)

3.2.3 Structural stiffness matrix

The strain energy due to mechanical strain for shell as expressed in Equation 2.4 is

given as,

{ } { }��=

=n

kV

kTk

M dVU12

1σε

Now combining Equation 2.4 and Equation 2.12 the potential energy for the laminate due to

strain is given by,

{ } { }��=

=n

kV

kkTk

M dVQU1

][2

1εε (3.12)

Now, substituting Equation 3.11 in Equation 3.12 one gets,

(3.13)

where, [C] is the material matrix, given in Equation 2.32 and 2.36.

Equation 3.13 can also be written as,

(3.14)

Here, [KM] is the element stiffness (mechanical) matrix and is given by,

(3.15)

{ } { }edN ]][[0 ∂=ε

� �=a

e

b

TT

eM dxdydBCBdU0 0

}{]][[][}{2

1

}]{[}{2

1eM

T

eM dKdU =

� �=a b

T

M dxdyBCBK0 0

]][[][][

32�

To obtain the numerical value of the element stiffness, a numerical integration technique using Gauss-

Quadrature Method is adopted. A reduced order 2 x 2 integration and 1 x1 integration is carried out

for bending part and shear part respectively to avoid shear locking in thin shells.

The element stiffness matrix given in equation (3.15) can be written in the natural coordinate system

as,

. (3.16)

3.2.4 Element mass matrix

The equations of motion can be expanded and without considering the external force effect can be written as,

2

2

t

u

zyx

zxyxxx

∂

∂=

∂

∂+

∂

∂+

∂

∂ρ

ττσ (3.17a)

2

2

t

v

zyx

zyyyxy

∂

∂=

∂

∂+

∂

∂+

∂

∂ρ

τστ (3.17b)

2

2

t

w

zyx

zzyzxz

∂

∂=

∂

∂+

∂

∂+

∂

∂ρ

τττ (3.17c)

Substituting the expanded form of u, v and w according to Equation 2.22 one can express the equations of motion as follows,

��

��

∂

∂+

∂

∂=

∂

∂+

∂

∂+

∂

∂2

2

20

2

tz

t

u

zyx

yzxyxxxθ

ρττσ

(3.18a)

[ �

∂

∂−

∂

∂=

∂

∂+

∂

∂+

∂

∂2

2

2

02

tz

t

v

zyx

xzyyyxy θρ

τστ (3.18b)

20

2

t

w

zyx

zzyzxz

∂

∂=

∂

∂+

∂

∂+

∂

∂ρ

σττ (3.18c)

Integrating Equation 3.18a with respect to z, we get

dzt

zdzt

u

y

N

x

N

h

h

y

h

h

yxxx ��−−

∂

∂+

∂

∂=

∂

∂+

∂

∂ 2

2

2

22

2

2

02

..θ

ρρ

� �− −

=1

1

1

1

]][][[][][ ηξddJBCBKT

M

33�

2

2

22

02

1t

It

uI

y

∂

∂+

∂

∂=

θ (3.19)

Similarly, Equations 3.18b and 3.18c are integrated with respect to z and Equations 3.18a to

3.18c are integrated with respect to z2

. Finally, the inertia matrix [J] is expressed as, �

[ ]

��

��

�

−

−

=

32

32

1

21

21

000

000

0000

000

000

II

II

I

II

II

ρ ��(3.20)

with �−

=2

2

2321 ),,1).(.(,,

h

h

dzzzzIII ρ �

Kinetic energy equation can written as follows

{ } [ ] [ ][ ]{ } dydxdNNdT e

T

A

T

e .2

1�� ρ�=

Or,

{ } [ ]{ }e

T

e dMdT ��

2

1= (3.21)

where, M=mass matrix

= [ ] [ ]{ }dxdyNNT

a

ρ� (3.22)

3.2.5 Geometric stiffness matrix

Structural mechanics may include three types of non-linearity,

� Material nonlinearity � Contact nonlinearity

� Geometric nonlinearity

In the present problem, geometric nonlinearity is considered. In linear problems, it is assumed that the geometry of the element remains basically unchanged during the loading process so

that the first order, infinitesimal, linear strain approximation can be used. If accurate determination of stresses is needed, geometric nonlinearity may have to be considered.

34�

The nonlinear strains which are also known as the Green-Lagrangian strains are given as,

��

��

��

��

�

∂

∂+�

�

��

�

∂

∂+�

�

��

�

∂

∂+

∂

∂=

222

2

1

x

w

x

v

x

u

x

uxε �

��

��

��

��

�

∂

∂+��

�

��

�

∂

∂+��

�

��

�

∂

∂+

∂

∂=

222

2

1

y

w

y

v

y

u

y

vyε (3.23)

��

��

�

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂+

∂

∂=

y

w

x

w

y

v

x

v

y

u

x

u

x

v

y

uxyγ

Total strain energy for the shell element can be written as,

Strain energy = (Strain energy)linear + (Strain energy)non-linear

The nonlinear strain energy per unit volume for shell can be expressed as,

(Strain energy) NL= -1 3,8KL

where 3, is the initial stress developed due to the external loading and 8KL is the nonlinear strain

vector.

3, � �36M 37, 367, , ��

��

�

��

��

�

=

xyNL

yNL

xNL

NL

γ

ε

ε

ε ��(3.24)

The non-linear strain terms as expressed in Equation 3.24 are,

��

��

��

��

�

∂

∂+�

�

��

�

∂

∂+�

�

��

�

∂

∂=

222

2

1

x

w

x

v

x

uxNLε �

��

��

��

��

�

∂

∂+��

�

��

�

∂

∂+��

�

��

�

∂

∂=

222

2

1

y

w

y

v

y

uyNLε ��(3.25)

��

��

�

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂

∂

∂=

y

w

x

w

y

v

x

v

y

u

x

uxyNLγ �

�

35�

After expanding,

��

��

�

∂

∂

∂

∂+

∂

∂−

∂

∂−

∂

∂−�

�

��

�

∂

∂+�

�

��

�

∂

∂+�

�

��

�

∂

∂=

xx

uz

R

u

x

w

x

v

R

w

x

u

R

w

x

w

x

v

x

u y

xxyx

xNL

θε 0000000

2

0

2

0

2

0 22222

1

��

�

�

��

�

�

��

��

�+�

�

��

�

∂

∂+��

�

��

�

∂

∂+��

�

∂

∂−+

∂

∂+

∂

∂

∂

∂−

∂

∂−

222

200000

x

yxy

x

y

x

y

xxy

xx

x

y

Rxxz

x

w

RRR

u

R

w

xxx

v

R

w

x

θθθθθθθθ�

��

��

��

�+

��

�

�

��

�

�+��

�

��

�+

2

0

2

0

2

0

xxyx R

u

R

w

R

w��(3.26a)

��

��

�

∂

∂

∂

∂+

∂

∂−

∂

∂−

∂

∂−��

�

��

�

∂

∂+��

�

��

�

∂

∂+��

�

��

�

∂

∂=

yy

uz

R

v

y

w

y

v

R

w

y

u

R

w

y

w

y

v

y

u y

yyxy

yNL

θε 0000000

2

0

2

0

2

0 22222

1�

��

��

��

��

��

�

�

��

�

�+��

�

��

�

∂

∂+��

�

��

�

∂

∂+��

�

∂

∂+−

∂

∂+

∂

∂

∂

∂−

∂

∂−

222

200000

y

xxy

y

x

y

x

yy

xx

xy

y

Ryyz

y

w

RRR

v

R

w

yyy

v

R

w

y

θθθθθθθθ�

��

��

�

�

��

�

�+

��

�

�

��

�

�+

��

�

�

��

�

�+

2

0

2

0

2

0

yyxy R

v

R

w

R

w�� (3.26b)

��

∂

∂−

∂

∂−

∂

∂−

∂

∂−

∂

∂−

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂

∂

∂=

xyxyyxy

xyNLR

u

y

w

R

v

x

w

R

w

y

v

R

w

x

v

R

w

x

u

y

w

x

w

y

v

x

v

y

u

x

u 0000000000000000γ �

��

�

�

∂

∂+

∂

∂+

∂

∂

∂

∂−

∂

∂

∂

∂−

∂

∂−

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂−

xy

x

y

xxx

xy

yyy

x R

w

yR

w

xy

v

xyx

v

R

w

xy

u

xyx

uz

R

w

y

u 000000000 θθθθθθθ�

��

�

�

��

�

�−

∂

∂

∂

∂+

∂

∂

∂

∂+��

�

∂

∂−+

∂

∂−−

∂

∂+

y

x

x

yxxyyy

xyx

y

x

y

y

x

xy

x

RRyxyxz

yR

w

R

v

Ry

w

RRR

u

Rx

w θθθθθθθθθθθ 200000�

��

+++

xyxyxyxy R

w

R

w

R

v

R

u

R

w

R

w 000000��(3.26c)

The non-linear strains can be represented as,

{ } { }dRNL ][2

1=ε ��(3.27)

36�

where,

2*4 � NO,&6 O,&7 P,&6 P,&7 Q,&6 Q,&7 �6&6 �6&7 �7&7 �7&6 O,R6 P,R7 Q,R6 Q,R7 Q,R67 �6R7 �7R6S

and [R] is obvious from Equation 3.26. Now {d} can also be written as,

{d}=[G]{de}= [ ][ ] }{]........[.......... 41 edGG �

where,

[ ] �=

=

��

��

�

=4

1

,

,

,

,

,

,

,

,

,

,

41

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

i

x

i

y

i

xy

i

y

i

x

i

y

i

x

i

yi

xi

yi

xi

yi

xi

yi

xi

yi

xi

toii

R

N

R

N

R

N

R

N

R

N

R

N

R

N

N

N

N

N

N

N

N

N

N

N

G

�

Then the potential energy due to residual stresses for the element can be written as,

}{][][}{2

10σT

V

TT

eNL RGdU �= ��(3.28)

37�

Since, [R]T23,4 � 23,4�T 2*U4

UNL can be modified as,

� ==V

e

T

ee

TT

eNL dKddvdGGdU }]{[}{2

1}]{][[][}{

2

10 σσ �� (3.29)

in which [K�]=� �V

T dvGG ]][[][ 0σ is the geometric stiffness matrix due to residual stresses.

The geometric stiffness matrix can also be expressed in terms of residual stress resultants and the local coordinates of the element as,

� �− −

=1

1

1

1

]][[][][ ηξσσ ddJGSGKT

�� (3.30)

In the expression of the [S�] matrix Nx , Ny , Nxy, Mx, My, Mxy are the force and moment

resultants and were expressed using standard composite laminate modeling and the rest of the

terms like xyyx NNN ,, ��in the stress matrix are given by,

��

�

��

�

�

��

�

��

�

�

=��

��

�

��

��

�

xy

y

x

xy

y

x

xy

y

x

T

N

N

N

κ

κ

κ

γ

ε

ε

0

0

0

][ �� (3.31)

but, ][T can be written as,

��

��

�

=

662616662616

262212262212

161211161211

][

DDDDDD

DDDDDD

DDDDDD

T

where,

and �

=

−−=n

k

kkkijij ZZQD1

144 )()'(

4

1

The rest of the terms in [S�] matrix can be calculated as,

��

��

�

��

��

�

��

��

�

+��

��

�

��

��

�

��

��

�

=

��

��

�

xy

y

x

o

xy

o

y

o

x

xy

y

x

DDD

DDD

DDD

DDD

DDD

DDD

N

N

N

κ

κ

κ

γ

ε

ε

662616

262212

161211

662616

262212

161211

(3.32)

�=

−−=n

k

kkkijij ZZQD1

133 )()'(

3

1

�

�

Where, the stress matrix [S�] is given by,

�

�

�

�

�

�

�

�

�

�

�

�

�

�

��

��

�

−−−

−−−

+−−−−−−

−−

−−−−

−−−

−−−

−−

−−

−−

−−

−−−

−−−

−−−−

−−−−

−−

−−

=

xxyxyxxyx

xyyyxyyxy

yxxyxyxyyxyxxyxyxy

xyyyxyyxy

xyxxxyxyx

xyyyxyyxy

xxyxyxxyx

xyxxxyxyx

yxyxyyyxy

xyyyxyyxy

xxyxyxxyx

xyyyxyyxy

xxyxyxxyx

xyyyxyyxy

xxyxyxxyx

yxyxyyxxy

xyxxxyxyx

NNMMMM

NNMMMM

NNNNMMMMNNNN

NNMMNN

NNMMNN

MMNNNN

MMNNNN

MMNNMM

MMNNMM

MMNNMM

MMNNMM

MMNNNN

MMNNNN

NNMMNN

NNMMNN

NNMMNN

NNMMNN

S

00000000000

00000000000

00)(0000

00000000000

00000000000

00000000000

00000000000

00000000000

00000000000

00000000000

00000000000

00000000000

00000000000

00000000000

00000000000

00000000000

00000000000

][ σ

39��

3.2.6 Element load vector

The potential energy of the system due to external mechanical loading for each

element is written as,

{ } [ ] { }dApNdPT

A

T

e�= (3.33)

And { }p is the force vector specified on the surface at z = h n and is given by,

{ } [ ]xyxzyzzyx ppppppp =

3.4.7 Governing equations of motion

Now, combining all the terms for potential energy as well as the energy due to inertia

the total energy for each element is given as,

{ }[ ] [ ][ ]{ } { } [ ] { } { } [ ] [ ]{ }dAdNddAPNddAdBDBd e

T

A

T

e

TT

A

ee

T

A

ee�� ρ�� −−=∏

2

1

2

1 (3.34)

The Lagrangian ‘L’ is defined as

L=T-V

The Hamilton’s principle states that the variation of the Lagrangian during any time

interval t0 to t

1 must be equal to zero, i.e.

01

0

=� dtL

t

t

eδ (3.35)

Substituting from Equation 3.34 variational principle is applied to Equation 3.35 and on

integrating the last term by parts with respect to time one obtains,

{ } [ ] [ ][ ] { } { }[ { } [ ] [ ][ ] ] [ ] [ ][ ] { }{ }dtdddANNdANNdddtddANNd ee

t

t

T

AA

t

t

T

eee

A

T

t

t

T

e δρρδρδ ��

��

��

−= � ��1

0

1

0

1

0

(3.36)

The first term on the RHS in the Equation 3.36 vanishes at the limits because of the

agreement that {de} =0 at t=t

0 and t=t

1.

40��

The final form of Equation 3.37 can therefore be expressed as given in the following equation

in the element level,

[ ]{ } [ ]{ } { }e

e

e

e

ePdKdM =+�� (3.37)

The Equation 3.37 is derived for conservative system. In case of non conservative system

damping matrix introduce in Equilibrium Equations and expressed as

PKUUCUM =++ �� (3.38)

where,

M= mass matrix

C= damping matrix

K=stiffness matrices:

P = vector of externally applied loads,

U = displacement vectors

U� = velocity vectors

U�� = acceleration vectors

If residual stresses 3,� are present due to some axial external loading in the system or

structure, one have to consider the geometric stiffness part along with the normal stiffness

matrix. The effect of membrane forces are accounted for by geometric stiffness matrix. The compressive axial loading reduces the bending stiffness. If the membrane forces are reversed-

that is made tensile rather than compressive- bending stiffness is effectively increased. This effect is called stress stiffening. For compressive membrane forces Equation 3.37 will take

the form as below,

[ ]{ } [ ]{ } 0][][ =−+ e

e

e

edKKdM σ

�� (3.39)

If the axial external loading is dynamic and harmonic in nature, the loading function can be expressed as

+ � +, � +-�� (3.40)

where, P0 is the constant load part and P1 is the time varying part of that dynamic loading.

41��

3.4.8 Stability equations

In the present case the eigen bending vector (which is same as that of the buckling

shape for a cantilever structure) is used to uncouple the governing equation. . In the next step

the uncoupled equation is solved using the method of strained parameter and the detailed

procedure is shown. It must be noted that in the present case the constant part of the loading is not considered and the time dependent loading generating stiffness is only taken into

consideration.

As a first step the generalized displacement {d} are transformed into the model co-ordinate

using the transformation, {d (t)} =�V 2O��4 (3.41)

where [ψ ] is the modal state vector.

Hence Equation 3.39 can be written as considering the damping part:

W XYZX�Y � [ XZ

X� � \ � +\] ./��O�� (3.42)

The natural eigen frequency ^_1 � àb and

K

M

σσω = allows to rewrite Equation 3.42 in the

form

XYZ��X�Y � c

bXZX� � ^_1 � +^1] ./��O�� (3.43)

Let

�d � �" �

then,

∗

∗

∗==

dt

du

dt

dt

dt

du

dt

du

2

θ (3.44a)

4222

2

2

2

2

2 θθθθ∗∗∗

∗

∗∗=�

�

��

�=�

�

��

�=

dt

ud

dt

du

dt

d

dt

dt

dt

du

dt

d

dt

ud (3.44b)

Putting Equation 3.44a and 3.44b in Equation 3.43

( ) 0)()cos(24

222

2

2

=+++∗∗

tutPdt

du

M

C

dt

udn θωω

θθσ (3.45a)

0)()2cos(442 2

2

2

22

2

=��

��

�+++ ∗

∗∗tutP

dt

du

M

C

dt

udn σω

θω

θθ (3.45b)

42��

Equation 3.45b can be written as,

( ) 0)()2cos(222

=+++ ∗∗∗∗tutuu ψδµ �� (3.45c)

Equation 3.45c is Mathieu type of equation.

where,

2

2

4nω

θδ =∗ ,

θµ

M

C=∗ and

2

22

θ

ωψ σP

=∗

3.4.8.1 Calculation of the damping for the present case(C):

In the FE formulation, Rayleigh proportional damping has been considered, which has the

following form:

C = �� + K, where � and are constants. After application of the weighted modal matrix â

here results in

Λ+=+= βαβα IaKaaMaaCaTTT ��

(3.46)

where, I is a unit matrix and � is a diagonal matrix of the eigen-values.

Now,

θ

β

θ

α

θ

βα

θµ

M

K

M

KM

M

C+=

+==∗

(3.47)

for the condition of parametric resonance, � = 2�n , now putting in Equation 3.47 one gets,

22n

n

βω

ω

αµ +=∗

(3.48)

To evaluate � and 4 % damping for the first mode and 6 % for the second mode has been

considered and putting them in Equation 3.46 two simultaneous algebraic equation has been

solved. Now after getting the value of � and and putting them in Equation 3.48, �* has been

calculated to provide damping to the first mode only, i.e. for �n = �1 in the present case.

43��

The Floquet theory together with the strained parameter approach [44] is applied to Equation

3.45c. This results for small parameters ∗ψ in the expansions

( )

��

�

��

�

�

+��

��

�−±+

+−−±

+−

=

∗∗∗

∗∗∗

∗

∗

...1616

1

6

14

...8

141

...2

10

2

1

222

22

122

2

µψψ

ψµψ

ψ

δ (3.49)

In the derivation the unknown entities, ∗δ and u(t*) are expanded in a power series with

respect to ∗ψ under the additional assumption of a proportional Rayleigh damping

coefficient. This assumption is justified, because the stability is increased for higher damping.

The truncated series of Equation 3.49 are good approximation to the stability border.�This can be verified by a direct evaluation of the Floquet theory in combination with numerical

solution of the differential Equation 3.45c.�The expansion around zero is independent of the

damping coefficient. The expansion one and four correspond to excitation frequencies θ

around two times the natural frequency and around the natural frequency itself respectively.

In the present work, the second stability equation has been considered from Equation 3.49 for

stability study.

�

�

�

�

�

�

�

44��

CHAPTER-4

RESULTS AND DISCUSSION

4.1 INTRODUCTION

Based on the finite element formulation derived in the earlier chapter, a program has been developed in the MATLAB platform. Using the developed code, some problems have

been solved for static, free vibration and buckling analysis of cylindrical shell structures and the results are compared with those using ANSYS finite element software. Thereafter, studies

have been carried out to understand the instability behavior of isotropic as well as laminated

shell structures with different fibre orientations and with various radius of curvature to width

ratio. In case of stability studies a generalized Rayleigh proportional damping has been considered for all the cases.

4.2 STATIC ANALYSIS

In this section static analysis results are presented for cantilever shell. Mesh

convergence study of transverse tip deflection has been done for the isotropic case.

4.2.1 Isotropic cantilever shell

An isotropic cantilever shell for various radius of curvature to width ratio is subjected to transverse unit load distributed equally at the free edge. The geometric and material

properties used for the shell are given below.

Material properties:

Material: Aluminum

Young’s Modulus, E = 70 GPa

Poisson’s Ratio, � = 0.3

Geometric properties:

Length, L = 2 m

Width, a = 0.5 m

Thickness, t = 0.02 m

45��

Table 4.1 Mesh convergence study of non-dimensional transverse tip deflection (w/) for an Isotropic cantilever shell (L=2m a=0.5m, t=0.02m of R/a = 30)

Note: w/ = (w.t5.E)/(F.a4)

It can be seen from the above table that the result converges as we increase number of elements. But the computational time also increases as a result. So keeping two things in

mind i.e. computational time and accuracy level, an optimized mesh size distribution (20 x 20 for the present study) have been taken.

Table 4.2 Non-dimensional transverse deflection at x = 0.5 m and y = 0 m of a

cantilever Aluminum shell (L= 2m, a=1m, t=0.03 m)�

Note: w/ = (w.t5.E)/(F.a4)

It is observed from Table 4.2 that as the R/a ratio increases, w / at the mid node of the

free edge also increases. This is because, as R/a ratio increases the shell becomes shallower and as a result the stiffness gets reduced accordingly. In the case of plate, the maximum

deflection occurs as there is no curvature resulting in it being the least stiff among all the

cases. The finite element model of the cantilever shell is shown in Figure- 4.1. Here the maximum error is of the order 0.78 % for R/a ratio 15.

MESH PRESENT F.E

( w/)

ANSYS

( w/)

20 x 5

0.38277

0.39424

20 x 10

0.39105

20 x 12

0.39195

20 x 15

0.39273

20 x 20

0.39331

20 x 25

0.39359

20 x 30

0.39377

20 x 40

0.39392

R/a RATIO PRESENT F.E(w/) ANSYS(w/

)

15

0.0255 0.0257

50

0.0275 0.0275

100

0.0276 0.0276

Cantilever Plate (L=2 m, a=1 m, t=0.03 m )

0.0277 0.0277

46��

Figure- 4.1 Finite element model of a cantilever shell subjected to unit mechanical transverse

load at the free edge. The mesh consists of (20 x 25) elements.

4.2.2 Composite cantilever shell

In the present study, two typical two layered and eight layered laminates are

considered. The finite element model has been shown above in Figure-4.1. The material and geometrical properties used for the analysis are presented below.


Material: Carbon Fiber Composite

Young’s Modulus, E1 = 105 GPa; E2 = 6.13 GPa

Poisson’s Ratio, �12 = 0.317; Shear Modulus, G12 = 2.28 GPa; G13=G23=2.28 GPa


Length, L = 2 m

Width, a = 1 m


Fibre orientation:

2- Layered laminate = 450/-450; 8- layered laminate = (00/900/450/-450) s.

47��

The non-dimensional transverse displacements obtained at x = 0.5 m, y = 0 m from the present FE model and from ANSYS are given in Table 4.3. Here (20 x 25) element meshing has been considered in the FE modeling.

Table 4.3 Non-dimensional transverse deflection (w/) of at x = 0.5 m and y = 0 m of a

cantilever shell (L= 2m, a=1m, t=0.03 m for R/a = 50).

Note: w/ = (w.t5.E11)/ (F.a4)

It is observed from Table 4.3 that, as the number of layer is increasing w / at x = 0.5 m

and y = 0 m, also significantly getting reduced, which indicates the stiffness of the structure is

increasing. Here the maximum error is of the order 1.16 % for 450/-450 laminate.

4.3 FREE VIBRATION ANALYSIS

The free vibration analysis has been carried out both for isotropic as well as

composite shells. The finite element model and the boundary conditions are shown in Figure-

4.1.


The material properties are same as given in problem 4.2.1 and the geometric properties are same as given in problem 4.2.2. The results obtained by the present F.E modeling are compared with ANSYS and is given in Table 4.4. Here, the non-dimensional

frequency is taken as, ��

��

�=

211

2

tEL

ρωω

��

Table 4.4 Mesh convergence study of non-dimensional first natural frequency of isotropic cantilever shell (L= 2m, a=1m, t=0.03 m).

R/a RATIO

�

�

�

PRESENT FINITE ELEMENT ( )ω

�

�

ANSYS

( )ω �

20 x 10

20x15� 20x20

15

1.0862 1.0836 1.0831 1.0778�

50

1.0459 1.0448 1.0443 1.0429�

100

1.0427 1.0417 1.0414 1.0404�

FIBRE ORIENTATION PRESENT F.E ( w/

)

ANSYS ( w/)

450/-450 0.28706

0.29043

(00/900/450/-450)s 0.07886

0.07938

48��

It is observed from Table-4.4 that as R/a ratio reduce the non-dimensional first natural

frequency increases correspondingly. It is known that natural frequency, ^ � àb; where K is

the stiffness and M is the mass of the element. As the stiffness is only variable here, non-dimensional natural frequency will increase if the stiffness of the element also increases which describes the situation here, i.e. stiffness is increasing with the corresponding reduction of R/a ratio.


The geometric properties are same as given in problem 4.2.2. The results obtained by

the present FE modeling are compared with ANSYS and is given in Table 4.5. The material properties are as follows:

Material: E- Glass Epoxy Composite

Young’s Modulus, E1 = 39 GPa; E2 = 8 GPa

Poisson’s Ratio, �12 = 0.28; Shear Modulus, G12 = 3 GPa; G13=G23=3 GPa

Density, J � He�� \�f�g

Table 4.5 Non-dimensional first natural frequency of composite cantilever shell (L= 2m, a=1m, t=0.03 m.)

R/a RATIO FIBRE

ORIENTATION

PRESENT FINITE

ELEMENT

( )ω ANSYS ( )ω

50

90/0/90

1.0081 1.0045

45/0/45

0.5384 0.5372

0/90/45/-45/45/90/0

0.6849 0.6825

Here, the non-dimensional frequency is taken as, ��

��

�=

211

2

tEL

ρωω

��

It is observed from Table 4.5 that when most of the fibers are oriented along perpendicular to X- axis (i.e. 90/0/90) the natural frequency show higher value than it is oriented along other

angle to the X- axis (i.e. 45/0/45). This is because the stiffness is getting higher for the first case. Here it can be also observed that as the numbers of layers are increasing the first natural

frequency value is also increasing in the present case.

49��

Figure- 4.2 Mode shape for 1st natural frequency (19.93 rad/sec) for fibre orientation 45/0/45 and R/a=50, L=2m, a=1m, t=0.03m for E-Glass Epoxy composite.

Figure- 4.3 Mode shape for 2nd natural frequency (105.96 rad/sec) for fibre orientation

45/0/45 and R/a=50, L=2m, a=1m, t=0.03m for E-Glass Epoxy composite.

50��

Figure- 4.4 Mode shape for 3rd natural frequency (223.83 rad/sec) for fibre orientation

45/0/45 and R/a=50, L=2m, a=1m, t=0.03m for E-Glass Epoxy composite.

4.4 BUCKLING ANALYSIS

In this section, buckling analysis is carried out both for isotropic and composite cylindrical shallow shells. For this analysis axial compressive unit load has been evenly

distributed throughout the free edge of the shell at each node. The finite element modeling for buckling analysis has been shown in Figure-4.5.


The material and the geometric properties are same as taken in section-4.3.1. The results obtained by the present FE modeling are compared with ANSYS and is given in Table

4.6. Here, the non-dimensional buckling load is taken as, 3

11

2

tE

LN x=λ��

Table 4.6 Non-dimensional first buckling load factor of isotropic cantilever shell (L= 2m,

a=1m, t=0.03 m).

R/a RATIO

PRESENT FINITE

ELEMENT

( )λ

ANSYS ( )λ

15

0.2309 0.2292

50

0.2145 0.2144

100

0.2132 0.2133

51��

Figure- 4.5 Finite element model of a cantilever shell subjected to unit mechanical

compressive load at the free edge. The mesh consists of (20 x 25) elements.

It is observed from Table 4.6 that the first buckling load factors increases

correspondingly with the decrement of the R/a ratio, i.e. with the increment of the stiffness.

The maximum error is of the order of 0.74 % for R/a ratio 15.


The material and geometric properties are same as given in section 4.3.2. The results

obtained by the present F.E modeling are compared with ANSYS and is given in Table 4.8.

Table 4.7 Non-dimensional first buckling load factor of composite cantilever shell for R/a ratio 50 (L= 2m, a=1m, t=0.03 m).

FIBRE ORIENTATION

PRESENT FINITE

ELEMENT

( )λ ANSYS ( )λ

90/0/90

0.9839 0.9409

45/0/45

0.2614 0.2433

0/90/45/-45/45/90/0

0.4529 0.4360

52��

Figure- 4.6 Mode shape for 1st buckling load factor (14116) for fibre orientation 45/0/45 and R/a=50, L=2m, a=1m, t=0.03m for E-Glass Epoxy composite.

Figure- 4.7 Mode shape for 2nd buckling load factor (128080) for fibre orientation 45/0/45

and R/a=50, L=2m, a=1m, t=0.03m for E-Glass Epoxy composite.

53��

Figure- 4.8 Mode shape for 3rd buckling load factor (358300) for fibre orientation 45/0/45

and R/a=50, L=2m, a=1m, t=0.03m for E-Glass Epoxy composite.

The developed FE code seems to be working well for static, free vibration and

buckling analysis. With this background, stability studies due to parametric excitation have

been carried out in the subsequent sections.

4.4 PARAMETRIC INSTABILITY STUDY

Floquet theory together with the strained parameter approach gives the stability

equations which were shown in the previous chapter. Based on these equations the stability study has been carried out in this section. In this study one has to normalize global geometric

stiffness as well as the global structural stiffness matrix with a particular Eigen-vector. To

satisfy this condition the free vibration and the buckling mode-shape should be identical. In

the various sub-sections stability studies have been carried out for isotropic as well as orthotropic cantilever cylindrical shells. In the orthotropic case, shells with different fibre

orientations, different thickness and different geometries for different R/a ratio has been considered. For the damping effect, a generalized 4 % Rayleigh proportional damping for the

first mode and 6 % for the second mode have been considered for all the cases to study the

shifting of the point which cuts off that part of the regions of instability which borders on the

axis of the ordinate. In the figures presented in the next few pages xi = �* ; delta = * as

described in Equation-3.45c.

54��


In this subsection an aluminum shell having R/a ratio 25, of the following particular

material and geometric properties has been considered. The shell is subjected to an axial load and the stability curves with and without damping is plotted in Figure-4.9.


Material: Aluminium

Young’s Modulus, E = 70 GPa

Poisson’s Ratio, � = 0.3


Length, L = 2 m

Width, a = 1 m


Figure-4.9 Stability plot for cylindrical cantilever Aluminum shell with and without damping (R/a = 25, L=2.0 m, a=1.0 m, thickness = 0.03 m, E = 70 GPa, ν = 0.3)

55��

In Figure-4.9 the firm and dashed boundary line between stable and unstable region depicts the boundary lines without and with damping, respectively. It is clear from the figure

that the presence of damping cuts off that part of the regions of instability which borders on the axis of the ordinate, i.e. the damping decreases the unstable region by lifting it from delta

axis and narrowing its boundaries in the xi-delta plane.


In this section the study of the fibre orientation, effect of thickness, effect of various

geometric properties on the stability under separate subsections has been carried out for

different R/a ratios of a cylindrical cantilever shell. The laminated material properties are

given as follows:

Material: E- Glass Epoxy Composite

Young’s Modulus, E1 = 39 GPa ; E2 = 8 GPa

Poisson’s Ratio, �12 = 0.28; Shear Modulus, G12 = 3 GPa; G13=G23=3 GPa

Density, J � He�� \�f�g

4.4.2.1 Effect of various fibre orientations on stability

The different fibre orientations studied in this sub-section are as follows:

� 0/ � /0

a) 0/0/0 b) 0/30/0 c) 0/90/0

� 90/ �/ 90

a) 90/0/90 b) 90/30/90 c) 90/90/90

� � / �/ �

a) 30/30/30 b) 45/45/45 c) 90/90/90

CASE-1

Considering the first case i.e. 0/�/0, the stability charts are given both for R/a ratio 15 and 100 with L=2m, a=1m, t=0.03m in Figure-4.10. As mentioned above same amount of Rayleigh proportional damping i.e. 4 % for the first mode and 6 % for the second mode has been introduced for all the cases and after studying the shift of the stability point in various cases a plot has been shown in Figure-4.11.

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Figure-4.10 Stability plots for different fibre orientations 0/�/0 with R/a ratio 15 and 100, L=2m, a=1m, t=0.03 m for cantilever composite shell.

(a) 0/0/0, R/a = 15 (b) 0/0/0, R/a = 100

(c) 0/30/0, R/a = 15 (d) 0/30/0, R/a = 100

(e) 0/90/0, R/a = 15 (f) 0/90/0, R/a = 100

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Figure-4.11 Variation of stability point for different fibre orientations 0/�/0 with R/a ratio 15 and 100.

Table 4.8 Non-dimensional 1st natural frequency and buckling load factor for 0/�/0 orientation for cylindrical shell of L=2m, a=1m and t=0.03m.

R/a RATIO FIBRE

ORIENTATION

NON

DIMENSIONAL

NATURAL

FREQUENCY ( )ω

NON

DIMENSIONAL

BUCKLING LOAD

( )λ

15

0/0/0

0.4820 0.01844

0/30/0 0.4866

0.01966

0/90/0

0.5312 0.02410

100

0/0/0 0.4617 0.05420

0/30/0 0.4633 0.04540

0/90/0 0.4943 0.05840

Where, h � î1`j klmm�Yn as given in section 4.3.2 and o � KpLY

lYY�q as given in section 4.4.2.

It can be observed from (xi-value) does not change in Table 4.8dimensional natural fcase of R/a ratio. structural stiffness does not get altered significantly

The xiload to the eigen frequency. eigen frequency is similar, that is why ‘xi’(such a fibre

The lower xidamping is same for both the cases the portion of the instability zone in case of R/a ratio 100 gets much more reduced than the other casof R/a ratio 100. That is why the xibe said that, because the shell has higher stiffness in castability point

CASE-2

Now the fibre orientation 90/material properties are same as casebelow. Finally a plot has been given

t can be observed from alue) does not change in

4.8. From that table one observes that, dimensional natural fcase of R/a ratio. The natural frequency does not change significantly structural stiffness does not get altered significantly

The xi-value is nothing but a function load to the eigen frequency. eigen frequency is similar, that is why ‘xi’(such a fibre orientation does not show any change.

The lower xidamping is same for both the cases the portion of the instability zone in case of R/a ratio 100 gets much more reduced than the other casof R/a ratio 100. That is why the xibe said that, because the shell has higher stiffness in castability point gets reduced

Now the fibre orientation 90/material properties are same as case

. Finally a plot has been given

(a) 90/0/90, R/a = 15

(c) 90/45/90, R/a = 15

t can be observed from alue) does not change in each case of R/a ratio.

From that table one observes that, dimensional natural frequency and buckli

he natural frequency does not change significantly structural stiffness does not get altered significantly

value is nothing but a function load to the eigen frequency. Because the ratio of the change in the buckling load and the eigen frequency is similar, that is why ‘xi’(

orientation does not show any change.

The lower xi-value for R/a ratio 15 than R/a ratio 100 suggests that, though the damping is same for both the cases the portion of the instability zone in case of R/a ratio 100 gets much more reduced than the other casof R/a ratio 100. That is why the xibe said that, because the shell has higher stiffness in ca

gets reduced in this case.

Now the fibre orientation 90/material properties are same as case

. Finally a plot has been given

90/0/90, R/a = 15

(c) 90/45/90, R/a = 15

t can be observed from Figure-4.11 each case of R/a ratio.

From that table one observes that, requency and buckling load factor

he natural frequency does not change significantly structural stiffness does not get altered significantly

value is nothing but a function Because the ratio of the change in the buckling load and the

eigen frequency is similar, that is why ‘xi’(�orientation does not show any change.

value for R/a ratio 15 than R/a ratio 100 suggests that, though the damping is same for both the cases the portion of the instability zone in case of R/a ratio 100 gets much more reduced than the other case i.of R/a ratio 100. That is why the xi-value showsbe said that, because the shell has higher stiffness in ca

in this case.

Now the fibre orientation 90/�/90 is being considered here.material properties are same as case-1. The various cases of stability plot have been shown

. Finally a plot has been given to show the

90/0/90, R/a = 15

(c) 90/45/90, R/a = 15 58�

4.11 that, with the variation of each case of R/a ratio. This can be explained after

From that table one observes that, with the variation of ng load factor

he natural frequency does not change significantly structural stiffness does not get altered significantly and hence the results

value is nothing but a function (from EquationBecause the ratio of the change in the buckling load and the

eigen frequency is similar, that is why ‘xi’(�*) remains constant.orientation does not show any change.

value for R/a ratio 15 than R/a ratio 100 suggests that, though the damping is same for both the cases the portion of the instability zone in case of R/a ratio 100

e i.e. this damping is much more effective in case shows higher value in this case.

be said that, because the shell has higher stiffness in ca

/90 is being considered here.The various cases of stability plot have been shown

to show the shift of xi

, with the variation of This can be explained after

with the variation of ng load factor does not change significantly in each

he natural frequency does not change significantly and hence the results

quation-3.45c)Because the ratio of the change in the buckling load and the

*) remains constant. Hence the stability shi


damping is much more effective in case higher value in this case.

be said that, because the shell has higher stiffness in case of R/a ratio 15, the value of the

/90 is being considered here.The various cases of stability plot have been shown

shift of xi-value for different

(b)

(d) 90/45/90, R/a = 100

, with the variation of ‘�’ the stabilityThis can be explained after

with the variation of ‘�’ the first nondoes not change significantly in each

he natural frequency does not change significantly which signifies that and hence the results.

3.45c) of the ratio of Because the ratio of the change in the buckling load and the

Hence the stability shi


damping is much more effective in case higher value in this case. Alternately it can

se of R/a ratio 15, the value of the

/90 is being considered here. The geometric and The various cases of stability plot have been shown

value for different � values.

(b) 90/0/90, R/a = 100

(d) 90/45/90, R/a = 100

the stability point This can be explained after noticing

the first nondoes not change significantly in each

which signifies that the

of the ratio of buckling Because the ratio of the change in the buckling load and the

Hence the stability shift for


damping is much more effective in case lternately it can


The geometric and The various cases of stability plot have been shown

values.

90/0/90, R/a = 100

(d) 90/45/90, R/a = 100

int noticing

the first non-does not change significantly in each

the

buckling Because the ratio of the change in the buckling load and the

ft for


damping is much more effective in case lternately it can


The geometric and The various cases of stability plot have been shown

59��

Figure-4.12 Stability plots for different fibre orientations 90/�/90 with R/a ratio 15 and 100, L=2m, a=1m, t=0.03 m for cantilever composite shell.

Figure-4.13 Variation of stability point for different fibre orientations 90/�/90 with R/a ratio 15 and 100.

(e) 90/90/90, R/a = 15 (f) 90/90/90, R/a = 100

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Like Figure-4.11, it can also be seen in Figure-4.13 that the stability point does not vary at all with the variation of � i.e. the same explanation is also applicable here like as case-1. But after comparing Figure-4.11 and 4.13 one thing can be noted that, in case of Figure- 4.13 the xi-value are much lower than the values shown in Figure-4.11. Hence from this one thing can be concluded that, 90/�/90 fibre orientation is stiffer than 0/�/0 fibre orientation.

CASE-3

In this case �/�/� fibre orientation has been considered. The geometric and material properties are same as case-1 and 2. The various cases of stability plot have been shown below. Finally a plot has been given to show the shift of xi-value for different � values.

(d) 45/45/45, R/a = 100, � = 19.87 rad/sec

(a) 30/30/30, R/a = 15, � = 18.39 rad/sec (b) 30/30/30, R/a = 100, � = 17.57 rad/sec

(c) 45/45/45, R/a = 15, � = 20.83 rad/sec

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Figure-4.14 Stability plots for different fibre orientations �/�/� with R/a ratio 15 and 100, L=2m, a=1m, t=0.03 m for cantilever composite shell.

Figure-4.15 Variation of stability point for different fibre orientations �/�/� with R/a ratio 15 and 100

It can be observed from the result presented in Figure-4.15 that the fibre orientation affects the stability plot significantly for �/�/� fibre orientation. This can be attributed to the fact that with the change in the fibre orientation the stiffness of the structure changes. This

(f) 90/90/90, R/a = 100, � = 37.79 rad/sec (e) 90/90/90, R/a = 15, � = 38.97 rad/sec

62��

can be observed by the marked change of the eigen frequencies given in Figure-4.14. As it can be noted from Figure-4.15 that as the fibre orientation is going towards 900 the eigen frequency is also increasing correspondingly i.e. the stiffness of the structure is also increasing. That is why the xi-value is decreasing correspondingly or it can be said that, the structure is becoming more unstable with the application of same amount of damping.

4.4.2.2 Effect of thickness on stability

In this subsection the effect of thickness on stability has been carried out. The length and width of the shell is 2 m and 1 m correspondingly. Here R/a ratio of 15 and 100 have been considered and the fibre orientation 0/0/0 has been opted. The various cases of stability plot have been shown below. Finally a plot has been given to show the shift of xi-value for different thickness of the shell structure.

(a) R/a = 15, t = 0.002 m, � = 5.09 rad/sec (b) R/a = 100, t = 0.002 m, � = 1.36 rad/sec

(c) R/a = 15, t = 0.0075 m, � = 6.56 rad/sec (d) R/a = 100, t = 0.0075 m, � = 4.33 rad/sec

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Figure-4.16 Stability plots for different thickness variation with R/a ratio 15 and 100, L=2m, a=1m and fibre orientation 0/0/0 for a cantilever composite shell.

Figure-4.17 Variation of stability point for different thickness values with R/a ratio 15 and 100, fibre orientation – 0/0/0, L = 2 m, a = 1 m of a cantilever composite cylindrical shell.

(e) R/a = 15, t = 0.015 m, � = 9.88 rad/sec (f) R/a = 100, t = 0.015 m, � = 8.57 rad/sec

64��

The stability plot for thickness 0.03 m has been shown earlier in Figures-4.10 (a) and

(b). It can be observed from the result presented in Figure-4.17 that the thickness variation affects the stability plot significantly for 0/0/0 fibre orientation. This can be attributed to the fact that with the change in thickness the stiffness of the structure also changes. This can be observed by the marked change of the eigen frequencies in Figure-4.16 and Figure-4.10 (a) & (b). As it can be noted from Figure-4.16 that, as the thickness is increasing the eigen frequency is also increasing correspondingly i.e. the stiffness of the structure is also increasing. That is why the xi-value is decreasing correspondingly or it can be said that the same amount of damping is becoming ineffective with the increment of the thickness.

4.4.2.3 Effect of various geometries on stability

Lastly the effect of various geometries on the stability plot has been discussed. The material properties are same like other cases. A thickness of 0.03m and 30/30/30 fibre orientation has been taken into account. The various cases of stability plot have been shown below. Finally a plot has been given to show the shift of xi-value for different geometry of the shell structure.

(a) L = 0.5m, a =1m, � = 299.17 rad/sec (b) L= 0.5m, a =1m, � = 293.16 rad/sec

(c) L = 1m, a =1m, � = 74.45 rad/sec (d) L = 1m, a =1m, � = 71.98 rad/sec

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Figure-4.18 Stability plots for different geometries with R/a ratio 15 and 100, t = 0.03 m, fibre orientation 30/30/30 for a cantilever composite shell.

Figure-4.19 Variation of stability point for different L/a ratios with R/a ratio 15 and 100, fibre orientation 30/30/30, t = 0.03 m of a cantilever composite cylindrical shell.

(f) L = 2 m, a =1m, � = 17.57 rad/sec (e) L = 2 m, a =1m, � = 18.39 rad/sec

66��

It can be seen from Figure-4.19 that the stability point shifts considerably with the change in L/a ratio. This can be observed by the marked change of the eigen frequencies in

Figure-4.18. It is not clear to the present researchers what may be the exact reason for this particular behavior. The change of geometry from square to rectangular plate, other stress-

strain relations and Poisson-ratio influence likely causes these effects.

�

�

�

�

�

�

�

�

�

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CHAPTER-5

CONCLUSIONS

5.1 GENERAL CONCLUSIONS

In this present work, a finite element formulation is developed in the MATLAB

platform to study the parametric instability of isotropic and composite shells. A four noded isoparametric shallow shell element has been developed using Mindlin Reissener’s shallow

shell theory. The first order shear deformation effect and the effect of rotary inertia have been considered.

With the presently developed code, static, free-vibration and buckling analysis of

isotropic and composite shell structures have been carried out. By observing the results it is very clear that the presently developed code gives extremely comparable results with FE

software like ANSYS. So keeping this thing in mind, parametric stability studies were carried out for isotropic as well as composite cylindrical shell structures. Parametric instability

studies have been done with different fibre orientations, varying thickness and geometry and

with different R/a ratios in the present work.

Based on the analysis carried out in the earlier chapter the important conclusions

made are listed below:

� The results are close to ANSYS result with 20 x 20 mesh in the mesh convergence study for non-dimensional transverse tip deflection, in case of isotropic cantilever

shell. � Different R/a ratio and various fibre orientations play an important role in the static

transverse deflection of the shell. For lower R/a ratio, the transverse tip deflection is

lower, which signifies that the shell stiffness is higher in this case.

� If the fibre orientation is along the perpendicular direction with the plane of curvature,

the shell shows higher natural frequency, and higher buckling load which signifies

higher stiffness. � Region of instability reduces with application of damping.

� From Figure-4.11 and 4.13 it can be concluded that for 0/�/0 and 90/�/90 corresponding fibre orientations stability point does not change at all with the

variation of ‘�’, i.e. for these fibre orientations the natural frequency and buckling

load factor do not vary significantly, which signifies the structural stiffness does not

get altered significantly. But for 0/�/0 fibre orientation the stability point shows higher value than 90/�/90 fibre orientation, which signifies that because of the lower stiffness

the instability zone of 0/�/0 fibre oriented structure can be reduced much more than

90/�/90 fiber oriented structure with application of same amount of damping in both cases.

68��

� With the application of same amount of damping, lower R/a ratio shows lower stability point than the higher R/a ratio, which signifies that if the stiffness of the

structure increases the instability zone of the structure gets less reduced.�� From Figure-4.15 it can be concluded that as the fibre orientation is along the

perpendicular direction with the plane of curvature, xi-value decreases

correspondingly or it can be said that, the structure is becoming more unstable with

the application of same amount of damping.

� From Figure-4.17 it can be seen that as the ‘t/a’ ratio increases, the instability zone of

the structure increases with the application of same amount of damping. It is also interesting to note that as the ‘t/a’ ratio is going towards a higher value (0.03 in this

case) the R/a ratio does not play any role in the parametric instability of the structure, i.e. the xi-value converges to a same value as ‘t/a’ ratio increases.

� From Figure-4.19 it can be seen that, as the ‘L/a’ ratio increases the xi-value also

increases, which signifies that the same amount of damping becomes more effective

with the increment of the ‘L/a’ ratio. It can also be seen from the figure that, in lower ‘L/a’ ratio the ‘R/a’ ratio plays insignificant role.

5.2 SCOPE FOR FUTURE WORK

The work reported in this thesis is a limited part of a vast area of research on structural stability studies due to parametric excitation and it’s control. There are number of

complex problems that need to be solved in the present area of research. Some of the

important aspects pertaining to the present work that need attention are listed below:

� It can be extended to thermally excited load.

� The stability study can be carried out for higher modes. � The study can be extended to understand the time response behavior of the structure

due to parametric excitation.

� The stability study can be carried out for complex structures with variable boundary

conditions. � The numerical concept can be experimentally verified.

� The actual feedback control methodology can be implemented.

69��

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