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
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 Head of the Department Department of Civil Engineering Jadavpur University Kolkata 700032
Prof. N. Chakraborti Dean, FET Jadavpur University Kolkata 700032
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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 prerequisite 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.
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ACKNOWLEDGEMENT
I gratefully acknowledge the resourceful guidance, active supervision and constant encouragement of my supervisor, Dr. 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
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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 1.2 TYPES OF DYNAMIC INSTABILITY 1.2.1 Impulsive loading 1.2.2 Circulatory loads 1.2.3 Aero elastic problems 1.2.4 Buckling in the testing machine 1.3 PARAMETRIC EXCITATION 1.4 SHELLS 1.4.1 Shell as a Structural Form 1.4.2 Parametric Excitation on Shell Structure 1.5 LITERATURE REVIEW 1.5.1 Literature review on parametric excitation 1.5.2 Literature review on shell 1.5.3 Literature review on shell structures under parametricinstability or dynamic instability 1.6 OBJECTIVE AND SCOPE OF THE PRESENT WORK 1.7 ORGANIZATION OF REPORT
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1 2-3 3 3 3 3 4-6 7 7 7 8-11 8-9 9-10 10-11 12 12-13
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 3.2.1 Selection of element 3.2.2 Strain-Displacement relations 3.2.3 Structural stiffness matrix 3.2.4 Element mass matrix 3.2.5 Geometric stiffness matrix 3.2.6 Element load vector 3.4.7 Governing equations of motion 3.4.8 Stability equations 3.4.8.1 Calculation of damping for the present case (C) CHAPTER-4: RESULTS AND DISCUSSION 4.1 INTRODUCTION
27-43 27-28 29-31 31-32 32-33 33-38 39 39-40 41-43 42
44-66
44
4.2 STATIC ANALYSIS 4.2.1 Isotropic cantilever shell 4.2.2 Composite cantilever shell
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44-47 45-46 46-47
4.3 FREE VIBRATION ANALYSIS 4.3.1 Isotropic cantilever shell 4.3.2 Composite cantilever shell
47-50 47-48 48-50
4.4 BUCKLING ANALYSIS 4.4.1 Isotropic cantilever shell 4.4.2 Composite cantilever shell
50-53 50-51 51-53
4.4 PARAMETRIC INSTABILITY STUDY 4.4.1 Isotropic cantilever shell 4.4.2 Composite cantilever shell 4.4.2.1 Effect of various fibre orientations on stability 4.4.2.2 Effect of thickness on stability 4.4.2.3 Effect of various geometries on stability
53-66 54-55 55-66
CHAPTER-5: CONCLUSIONS
55-62 62-64 64-66 67-68
5.1 GENERAL CONCLUSIONS 5.2 SCOPE FOR FUTURE WORK REFERENCES
67-68 68 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)
θy
Rotation about Y axis
θx
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} {U}
Velocity vector
(m.s-1)
Acceleration vector
(m.s-2)
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[T]
Transformation matrix
[C]
Damping matrix
L
Length of the shell
a
Width of the shell
R
Radius of curvature of shell
(m)
Natural frequency
(Rad.s-1)
n
[ ]
(m)
Inertia matrix
(x)
LIST OF FIGURES Fig No.
Name of Figures
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
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
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
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
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
4.2
4.3
4.4
4.5
4.6
Page No.
Mode shape for 1st buckling load factor (14116)
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4.7
4.8
for fibre orientation 45/0/45 and R/a=50, L=2m, a=1m, t=0.03m for E-Glass Epoxy composite.
52
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.
52
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.
53
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
Variation of stability point for different fibre orientations 0/ /0 with R/a ratio 15 and 100.
57
4.11
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
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
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
4.16
4.17
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.
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65
LIST OF TABLES Table No. Name of Tables
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
Page No.
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
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
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
Mesh convergence study of non-dimensional first natural frequency of isotropic cantilever shell (L= 2m, a=1m, t=0.03 m).
47
Non-dimensional first natural frequency of composite cantilever shell (L= 2m, a=1m, t=0.03 m.)
48
Non-dimensional first buckling load factor of isotropic cantilever shell (L= 2m, a=1m, t=0.03 m).
50
Non-dimensional first buckling load factor of composite cantilever shell for R/a ratio 50 (L= 2m, a=1m, t=0.03 m).
51
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
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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
1
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
2
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.
3
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) (1.2)
Hence,
4
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.
5
Figure-1.2.
Stability Diagram
Figure 1.3 Solutions of Mathieu Equation
6
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.
7
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 0 if it is close to one of the natural frequencies of the lateral motion even though 1
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].
8
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 snapthrough 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).
9
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
10
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 nonuniform 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.
11
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.
12
Chapter 3: Finite Element Formulation In this chapter, a brief discussion on the element shape functions is given. The straindisplacement 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.
13
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,
14
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
15
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
Q11 Q12 0 0 0 σ1 Q12 Q22 0 0 0 σ2 0 Q66 0 0 τ 12 = 0 0 0 0 Q44 0 τ 13 τ 23 0 0 0 0 Q55
ε1 ε2 γ 12 γ 13 γ 23
(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
Q11 =
Q22 =
E1
(2.2)
1 − υ12υ 21
E2
(2.3)
1 − υ12υ 21
Q12 = Q21 ==
υ12 E 2 υ 21 E1 = 1 − υ12υ 21 1 − υ12υ 21
Q66 = G12 ; Q44 = G23 ; Q55 = G31
16
(2.4)
(2.5)
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 28 46&7
(2.7)
where, [T] is the transformation matrix and is given by the following equation
5
m2
n2
2
2
0 0 mn − mn n m 0 0 0 0 0 m −n 0 0 0 n m 2 − 2mn 2mn 0 0 (m − n 2 )
(2.8)
where, m = cos θ ; n = sin θ ; 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
σx σy i.e. τ xy τ xz τ yz
5
9- 234 -&1
(2.9)
m2
n2
Q11
Q12
0
0
0
2
2
Q12
Q22
0 0 0
0 0 0
0 Q66 0 0
0 0 Q44 0
0 0 0 Q55
0 0 − mn n m 0 0 mn 0 0 m n 0 0 0 −n m 0 2 2mn − 2mn 0 0 (m − n 2 )
ε1 ε2 γ 12 γ 13 γ 23
(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=&>
17
i , j = 1,2,6,5,4
(2.11)
Equation- 2.11 will take the form as
σx 0 0 Q'11 Q '12 Q'16 σy Q '12 Q' 22 Q' 26 0 0 τ xy = Q '16 Q' 26 Q' 66 0 0 τ xz 0 0 0 Q' 44 Q '54 τ yz 0 0 0 Q ' 45 Q' 55
εx εy γ xy γ xz γ yz
(2.12)
′ where [Q ] i,j = 1,2, 6,4,5 are the transformed reduced stiffnesses which are given in terms of
reduced stiffnesses, Qij as
Q'11 = Q11 m4 + 2( Q12 + Q66 ) n2m2 + Q22 n4
(2.13)
Q'12 = ( Q11 + Q22 - 4 Q66 ) n2m2 + Q12 (n4+m4)
(2.14)
Q' 22 = Q11 n4 + 2( Q12 + 2 Q66 ) n2m2 + Q22 m4
(2.15)
Q'16 = ( Q11 - Q12 - 2 Q66 ) nm3 + ( Q12 - Q22 + 2 Q66 ) n3m
(2.16)
Q' 26 = ( Q11 - Q12 - 2 Q66 ) n3m + ( Q12 - Q22 +2 Q66 ) nm3
(2.17)
Q' 66 = ( Q11 + Q22 - 2 Q12 - 2 Q66 ) n2m2 + Q66 (n4+m4)
(2.18)
Q' 44 = Q44 m2+ Q55 n2
(2.19)
Q'55 = Q44 n2+ Q55 m2
(2.20)
Q' 45 = Q'54 = ( Q55 - Q44 ) 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.
18
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
19
Employing first order shear deformation theory the displacement u, v, w on the shell can be expressed as,
u = u0 + zθ y ; v = v0 − zθ x ; w = w0
(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 θ x, θ y are rotation of shell element about X and Y axis respectively. The strain-displacement relations for a shell element are as follows:
ε xx = ε x0 + zκ x =
∂θ y ∂u 0 W0 − +z ∂x R x ∂x
ε yy = ε y0 + zκ y =
∂v 0 W0 ∂θ − −z x ∂y R y ∂y
∂θ y ∂θ x ∂v0 ∂u 0 2w0 + − +z − ∂x ∂y R xy ∂y ∂x
γ xy = γ xy0 + zκ xy =
γ yz = γ yz0 + zκ yz =
∂w0 ∂v0 ∂w0 + = −θx ∂y ∂z ∂y
γ zx = γ zx0 + zκ zx =
∂w0 ∂u 0 ∂w0 + = +θy ∂x ∂z ∂x
(2.23)
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 7 C A7 B=
B 6 C A67 B>
B 7 B>
B 6 B=
and the mid-plane strains are expressed in terms of the mid-plane displacements as,
ε x0 =
γ yz0 =
∂u 0 W0 − ∂x R x
ε y0 = ;
∂v 0 W0 − ∂y R y
γ xy0 = ;
∂v0 ∂u 0 2 w0 + − ∂x ∂y R xy
∂w ∂u ∂w ∂w0 ∂v 0 ∂w0 γ zx0 = 0 + 0 = 0 + θ y + = −θx ∂x ∂z ∂x ∂y ∂z ∂y ;
20
2.7 STRESS -STRAIN STRAIN RELATIONS FOR A LAMINATE
Figure 2.5 A general nn-layered Figurelayered laminate The stresses in the kth layer of a laminated composite can be expressed in terms of the laminate strains and curvatures as
σx σy τ xy τ xz τ yz
Q'11 Q'12 = Q'16 0 0
Q'12 Q ' 22 Q ' 26 0 0
Q'16 Q ' 26 Q ' 66 0 0
0 0 0 Q' 44 Q' 45
0 0 0 Q '54 Q '55
ε ox ε oy γ o xy + z γ o xz γ o yz
κx κy κ xy 0 0
Figure 2.6 Stress resultants of a laminated shell element. Figure-
21
(2.24)
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
Ni =
t/2 −t / 2
σ i dz ; M i =
t/2 −t / 2
(2.25)
σ i zdz
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
Nx
t/2
Ny = N xy
−t / 2
σx σy σ xy
Mx dz ; k
My = M xy
σx σy σ xy
t/2 −t / 2
(2.26)
zdz k
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,
Nx Ny N xy
N
= k =1 k
Mx My M xy
N
= k =1 k
Q'11 Q'12
Q'12 Q' 22
Q'16 Q' 26
Q'16
Q' 26
Q' 66
Q'11 Q'12
Q'12 Q' 22
Q'16 Q' 26
Q'16
Q' 26
Q' 66 o
Zk Zk −1 k
Zk Zk −1 k
o
ε xo ε y o dz + γ xy o
Zk Zk −1
ε xo ε y o zdz + γ xy o
κx κ y zdz κ xy
Zk Zk −1
κx κ y z 2 dz κ xy
(2.27)
(2.28)
o
However, we should recall that ε x , ε y , γ xy , κ x , κ y and κ xy 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
Nx A11 N y = A12 N xy A16
A12 A22 A26
A16 A26 A66
ε xo B11 o ε y + B12 γ xy o B16
22
B12 B22 B26
B16 B26 B66
κx κy κ xy
(2.29)
Mx
B11
B12
B16
M y = B12 M xy B16
B22 B26
B26 B66
ε xo D11 o ε y + D12 γ xy o D16
D12
D16
D22 D26
D26 D66
κx κy κ xy
(2.30)
The force and moment resultants together can be expressed as Nx Ny N xy Mx
=
My M xy
A11 A12 A16
A12 A22 A26
A16 A26 A66
B11 B12 B16
B12 B22 B26
B16 B26 B66
B11 B12 B16
B12 B22 B26
B16 B26 B66
D11 D12 D16
D12 D22 D26
D16 D26 D66
ε x0 ε y0 γ xy 0 κx κy κ xy
(2.31)
where, A11
A12
A16
B11
B12
B16
A12
A22
A26
B12
B22
B26
A16
A26
A66
B16
B26
B66
B11
B12
B16
D11
D12
D16
B12
B22
B26
D12
D22
D26
B16
B26
B66
D16
D26
D66
= [C ]b =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
(2.33)
(Q ' ij ) k ( Z k − Z k −1 )
Aij = k =1
Bij =
Dij =
1 2
n
(Q 'ij ) k ( Z 2 k − Z 2 k −1 )
(2.34)
1 n (Q 'ij ) k ( Z 3 k − Z 3 k −1 ) 3 k =1
(2.35)
k =1
23
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 γ xz , γ yz as,
M xz A = 44 M yz A54
A45 A55
γ xz γ yz
(2.36)
Aij, approximated as
(Qoffij ) k ( Z k − Z k −1 )
Aij =
(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, t1
t1
I = Ldt = (T − U )dt t0
(2.38)
t0
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,
d ∂T ∂ (T − U ) = QI , i = 1,2,... − . dt ∂ q ∂qi
(2.39)
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,
UM =
1 2
n k =1
{ε } {σ }dV k T
V
k
(2.40)
24
The work done due to mechanical loading is given by, WM = u T {F }dv
(2.41)
V
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, t1
δI = −δ dt t0
1 2
n
{ε } {σ }dv k T
k =1 v
k
t
−
1 1 {u ′}T [ρ ]{u ′}dv + dt 2v t0
{δu}T {Fm }dA
(2.42)
A
Now if the structural system is subjected to a pulsating axial compressive force P(t) = P + P s
t
cos t, acting along its undeformed axis where is the excitation frequency of the dynamic load component, P is the static and P is the amplitude of the time dependent component of s
t
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:
U M = U L + U NL
(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.
25
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.
26
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, θ x , θ y ) 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
X =
N i xi i =1
4
Y= i =1
N i yi ; where, xi and yi are the nodal co-ordinates of the element.
Similarly, the components of displacement vector at any point of the element are given by 4
4
u=
v=
N i ui
i =1
i =1
where, u, v, w,
x
,
y
4
N i vi
4
w=
N i wi i =1
θx =
4
N iθ x i i =1
having a subscript ‘i’ are the nodal displacements.
27
θy = i =1
N iθ y i
The element displacements as given in equation can be written in the matrix form as follows: u
Ni
0
0
0
0
ui
v
0
Ni
0
0
0
vi
w =
0
0
Ni
0
0
wi
θx θy
0 0
0 0
0 0
Ni 0
0 Ni
θ xi θ yi
i.e , {u} = [N ]{d e }
(3.1)
The shape functions for an four noded quadrilateral element is given by, 1 1 N 1 = (1 − ξ )(1 − η ) ; N 2 = (1 + ξ )(1 − η ) ; 4 4 1 1 N 3 = (1 − ξ )(1 + η ) ; N 4 = (1 + ξ )(1 + η ) ; 4 4
The interpolation polynomials given above can be concisely written as F;
G
H
where,
i
DD; H
and
i
EE; &
H
I
(3.2)
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:
N iX −1 N iξ = [J ] N iη N iY where, [J ] =
X iξ X iη
;
Yiξ Yiη
(3.3)
[J] is called the Jacobian matrix.
28
3.2.2 Strain-Displacement Relations
The strains for the shell element can be calculated by using the constitutive equations given below:
εx ε x0 εy ε y0 γ xy = γ xy0 + Z γ yz γ yz0 γ xz γ xz0 (3.4) where,
x,
0 0 0 y , xy , yz ,
kx ky k xy 0 0
are the strains at a distance z from the mid-plane of the shell, 0 xz are the mid-plane strains and x, y, xy are the shell curvatures. y, xy, yz,
xz
x
0
,
The above equation can also be written as in the following form:
εx εy γ xy = [ Z ]5 x 8 [ B ]8 x 5 {d }5 x1 γ yz γ xz
(3.5)
where, [Z] is the position matrix and {d} is the nodal d.o.f as given below:
1 0 0 0 [Z ] = 0 0 0
1 0 0 0
0 1 0 0
z
0 0 0 0
0 z 0 0 0 0 z 0 0 0 0 1 0 0 0 0
u0
0 0 0 1
;
v0 {d } = w0
(3.6a)
θx θy
The strain-displacement matrix [B] can be written as:
∂N i ∂x
0
0
0
0
∂N i ∂y
∂N i ∂y ∂N i ∂x
N i Rx N i − Ry N i − 2 R xy
0
0
4
0
0
0
0
∂N i ∂x
i =1
0
0
0
0
0
0
0
0
0
0
0
[B ] =
0
−
∂N i ∂y ∂N i ∂x
∂N i ∂y ∂N i − ∂x −
− N 0
29
i
0 ∂N i ∂y 0 N
i
(3.6b)
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. ∂N i ∂x
∂N i ∂y
[ B ]b = i =1
−
∂N i ∂y ∂N i ∂x
0 4
Ni Rx N − i Ry N −2 i R xy
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
∂N i ∂x
∂N i ∂y ∂N i − ∂x
(3.7)
0
−
∂N i ∂y
Shear strain-displacement matrix consists of transverse shear strains.
∂N i ∂y ∂N i ∂x
0 0
4
[ B]s = i =1
0 0
− Ni
0
0
− Ni
(3.8)
The strain-displacement relationship for the shell in the matrix operator form is derived from Equation 2.23 and is expressed as: ∂N i ∂x 0
ε x0 ∂N i ε y0 ∂y γ xy0 0 γ yz0 = 0 γ zx 0 κx κy 0 κ xy
0 ∂N i ∂y ∂N i ∂x 0 0
Ni Rx N − i Ry 2Ni − Ry ∂N i ∂y ∂N i ∂x −
0
0
0
0
0
0
0
0
0
0
0
0
0
0
− Ni
0
0
Ni
0
∂N i ∂x
∂N i ∂y ∂N i − ∂x −
u0 v0 w0
θx θy
{ }2 { }3 { }4
0 ∂N i ∂y
i =1to 4
30
1
(3.9)
which can also be written as,
{ε } = [∂ ][ N ]{d } {ε } = [ B]{d } 0
(3.10)
e
0
(3.11)
e
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,
UM
1 = 2
n k =1
{ε } {σ }dV k T
V
k
Now combining Equation 2.4 and Equation 2.12 the potential energy for the laminate due to strain is given by,
UM
1 = 2
n k =1
{ε } [Q ]{ε }dV k T
V
k
k
(3.12)
Now, substituting Equation 3.11 in Equation 3.12 one gets, a b
UM
1 {d e }T [ B ]T [C ][ B ]{d e }dxdy = 200
(3.13)
where, [C] is the material matrix, given in Equation 2.32 and 2.36. Equation 3.13 can also be written as,
UM =
(3.14)
1 {d e }T [ K M ]{ d e } 2
Here, [KM] is the element stiffness (mechanical) matrix and is given by, a b
[ B ]T [C ][ B ]dxdy
[K M ] =
(3.15)
0 0
31
To obtain the numerical value of the element stiffness, a numerical integration technique using GaussQuadrature 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,
1 1
[ B ]T [C ][ B ][ J ]dξdη
[K M ] =
.
(3.16)
−1 −1
3.2.4 Element mass matrix
The equations of motion can be expanded and without considering the external force effect can be written as, ∂σ xx ∂τ yx ∂τ zx ∂ 2u + =ρ 2 + ∂y ∂z ∂x ∂t ∂τ xy
∂σ yy
∂τ zy
(3.17a)
∂ 2v ∂t 2
(3.17b)
∂τ xz ∂τ yz ∂τ zz ∂2w + + =ρ 2 ∂x ∂y ∂z ∂t
(3.17c)
∂x
+
∂y
+
∂z
=ρ
Substituting the expanded form of u, v and w according to Equation 2.22 one can express the equations of motion as follows, ∂ 2θ y ∂σ xx ∂τ yx ∂τ zx ∂ 2u0 + + =ρ + z ∂x ∂y ∂z ∂t 2 ∂t 2
(3.18a)
∂ 2 v0 ∂ 2θ x + + = ρ[ −z 2 ∂x ∂y ∂z ∂t 2 ∂t 2 ∂τ xz ∂τ yz ∂σ zz ∂ w0 + + =ρ ∂x ∂y ∂z ∂t 2 ∂τ xy
∂σ yy
∂τ zy
(3.18b) (3.18c)
Integrating Equation 3.18a with respect to z, we get
∂N yx
∂N xx + = ∂x ∂y
h 2
−
2
h 2
2
∂ θy ∂ u ρ 20 dz + ρ .z. 2 dz ∂t ∂t h h 2
−
2
32
∂ 2θ y ∂ 2u0 = I1 2 + I 2 ∂t ∂t 2
(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 z . Finally, the inertia matrix [J] is expressed as, 2
0
0
0
I2
I1 0 − I2 0
0 I1 0 0
− I2 0 I3 0
0 0 0 I3
I1 0 [ρ ] = 0 0 I2
(3.20)
h 2
ρ .( z ).(1, z, z 2 )dz
with I 1 , I 2 , I 3 = −
h 2
Kinetic energy equation can written as follows T=
1 de 2A
{ } [N ] [ρ ][N ]{d }dx.dy T
T
e
Or, T=
1 de 2
{ } [M ]{d } T
(3.21)
e
where, M=mass matrix T
[N ] [ρ ]{N }dxdy
=
(3.22)
a
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.
33
The nonlinear strains which are also known as the Green-Lagrangian strains are given as,
∂u 1 + ∂x 2
∂u ∂x
∂v 1 εy = + ∂y 2
∂u ∂y
εx =
γ xy =
2
+
2
∂v ∂x
∂v + ∂y
2
∂w ∂x
+
2
∂w + ∂y
2
2
(3.23)
∂u ∂v ∂u ∂u ∂v ∂v ∂w ∂w + + + + ∂y ∂x ∂x ∂y ∂x ∂y ∂x ∂y
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= 3, 8KL 1
where 3, is the initial stress developed due to the external loading and 8KL is the nonlinear strain vector. 3,
36M
37,
367,
, ε NL
ε xNL = ε yNL γ xyNL
(3.24)
The non-linear strain terms as expressed in Equation 3.24 are, 1 2
∂u ∂x
1 = 2
∂u ∂y
ε xNL =
ε yNL
γ xyNL =
2
+
2
∂v ∂x
∂v + ∂y
2
+
2
∂w ∂x
∂w + ∂y
2
2
(3.25)
∂u ∂ u ∂ v ∂v ∂w ∂w + + ∂ x ∂ y ∂ x ∂ y ∂ x ∂y
34
After expanding,
ε xNL =
1 2
∂u 0 ∂x
2
+
∂v 0 ∂x
2
+
∂w 0 ∂x
2
−2
w 0 ∂u 0 w ∂v 0 ∂w u ∂u 0 ∂ θ y −2 0 − 2 0 0 + 2z R x ∂x R xy ∂x ∂x R x ∂x ∂x
∂θ y w0 ∂v 0 ∂θ x ∂θ x w0 u 0 θ y θ y ∂w0 − − + + − + z2 ∂x R x ∂x ∂x ∂x R xy R x R x R x ∂x w + 0 Rx
ε yNL
2
2
w0 + R xy
1 = 2
∂u 0 ∂y
u + 0 Rx
2
∂v0 + ∂y
2
γ xyNL =
2
w + 0 Ry
2
∂x
∂θ x + ∂x
2
+
θy
2
Rx
2
(3.26a)
2
∂w0 + ∂y
2
−2
w0 ∂u 0 w ∂v ∂w v ∂u 0 ∂θ y − 2 0 0 − 2 0 0 + 2z R xy ∂y R y ∂y ∂y R y ∂y ∂y
∂θ y w0 ∂v 0 ∂θ x ∂θ x w0 v 0 θ x θ x ∂w0 − − + − + + z2 ∂y R xy ∂y ∂y ∂y R y R y R y R y ∂y
w0 + R xy
∂θ y
∂θ y ∂y
2
∂θ x + ∂y
2
+
θx
2
Ry
2
v + 0 Ry
(3.26b)
∂ u 0 ∂u 0 ∂v 0 ∂v 0 ∂w 0 ∂w 0 ∂u 0 w 0 ∂v 0 w 0 ∂v 0 w 0 ∂w 0 v 0 ∂w 0 u 0 + + − − − − − ∂x ∂y ∂ x ∂y ∂x ∂y ∂x R xy ∂x R y ∂y R xy ∂x R y ∂y R x
−
∂u 0 w 0 ∂u 0 ∂θ y ∂θ y ∂u 0 ∂θ y w 0 ∂ v 0 ∂θ x ∂θ x ∂ v 0 ∂θ x w 0 ∂θ x w 0 +z + − − − + + ∂y R x ∂x ∂y ∂x ∂y ∂x R xy ∂x ∂y ∂x ∂y ∂x R y ∂y R xy
+
∂θ y ∂ θ y ∂θ x ∂ θ x θ y θ x ∂w 0 θ x u 0 θ x θ y ∂w 0 θ y v 0 w 0 ∂ θ y − − + − + z2 + − ∂x R y R x R y R x ∂y R x R y R x ∂y ∂x ∂y ∂ x ∂y Rx R y
+
w0 w0 u 0 v 0 w0 w0 + + R xy R y R x R y R x R xy
(3.26c)
The non-linear strains can be represented as,
{ε NL } = 1 [ R]{d }
(3.27)
2
35
where, 2*4
NO,&6 O,&7 P,&6 P,&7 Q,&6 Q,&7
6&6
6&7
7&7
7&6
O, P, Q, Q, Q, 6 7 S R6 R7 R6 R7 R67 R7 R6
and [R] is obvious from Equation 3.26. Now {d} can also be written as, {d}=[G]{de}= [[G1 ]..................[G4 ]]{d e } where,
[Gi ]i =1to 4 =
4 i =1
0 0
0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0 0 0
N i,x N i, y 0 0 0 0 0 0 0 0 Ni Rx
N i,x N i, y 0 0 0 0 0 0
N i,x N i, y 0 0 0 0
N i,x N i, y 0 0
N i,x N i, y
0
0
0
0
0
Ni Ry
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Ni Ry
0
0
0
0
0
Ni Rx
Ni Rx Ni Ry Ni R xy
Then the potential energy due to residual stresses for the element can be written as, U NL =
1 {d e }T [G ]T [ R ]T {σ 0 } 2V
(3.28)
36
23, 4 T 2*U 4
Since, [R]T23, 4
UNL can be modified as, U NL =
1 1 {d e }T [G ]T [σ 0 ][G ]{d e }dv = {d e }T [ K σ ]{d e } 2V 2
(3.29)
in which [K ]= [G ]T [σ 0 ][G ]dv is the geometric stiffness matrix due to residual stresses. V
The geometric stiffness matrix can also be expressed in terms of residual stress resultants and the local coordinates of the element as, 1 1
(3.30)
[G ]T [ S σ ][G ] J dξdη
[Kσ ] =
−1 −1
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 N x , N y , N xy in the stress matrix are given by,
ε x0 ε y0 γ xy0 = [T ] κx κy κ xy
Nx Ny N xy
(3.31)
but, [T ] can be written as, D11 [T ] = D12 D16
D12 D22 D26
D16 D 26 D66
D11 D12 D16
D12 D22 D26
D16 D26 D66
where, Dij =
1 3
n
(Q 'ij ) k ( Z 3 k − Z 3 k −1 )
k =1
and
Dij =
1 4
n
(Q 'ij ) k ( Z 4 k − Z 4 k −1 )
k =1
The rest of the terms in [S ] matrix can be calculated as, Nx D11 N y = D12 N xy D16
D12 D22 D26
D16 D26 D66
D11 ε xo o ε y + D12 γ xy o D16
D12 D22 D26
37
D16 D26 D66
κx κy κ xy
(3.32)
Where, the stress matrix [S ] is given by,
Nx
Nxy
0
0
0
0
0
0
Mxy
Mx
0
0
− Nx
0
− Nxy
0
0
Nxy 0
Nx 0
0 Nx
0 Nxy
0 0
0 0
0 − Mx
0 My − Mxy 0
Mxy 0
0 0
0 0
− Nxy 0
0 − Ny − Nxy − Nx
0 0
0 0
0
0
Nxy
Ny
0
0
− Mxy − M y
0
0
0
0
0
− Ny
− Nxy
0
0
0
0
0
0
Nx
Nxy
0
0
0
0
− Nx
− Nxy 0
0
0
Mxy
− Mx
0
0
0
0
Nxy
Ny
0
0
0
0
− Nxy
− Ny
0
0
0
My
− Mxy
0
0
− Mx
− Mxy 0
0
Nx
Nxy
0
0
0
0
0
Mxy
Mx
0
0
0 My
− Mxy − M y 0 0
0 0
0 0
Nxy 0
Ny 0
0 Ny
0 Nxy
0 0
0 0
0 My − Mxy 0
Mxy − My
0 0
0 0
Mxy
0
0
0
0
0
0
Nxy
Nx
0
0
− Mx
0
− Mxy
0
0
0
0
0
0
− Nx
− Nxy
0
0
0
0
Nx
Nxy
0
0
0
− Mxy Mx
0
0
0
0
− Nxy − Ny
0
0
0
0
Nxy
Ny
0
0
0
− My
Mxy
− Nx
− Nxy 0
0
0
0
0
0
− Mxy − Mx
0
0
Nx
0
Nxy
0
0
0
0
− Nxy
− Ny
0
0
Mxy
My
0
0
0
0
0
Ny
Nxy
0
0
− Nxy − Ny 0 0
− Nx 0
− Nxy 0
0 Mxy
0 My
Mx 0
Mxy 0
− My 0
0 Nxy − Mxy 0 0 − Mxy − M y 0
Nxy 0
(Nx + Ny ) 0 0 Ny
0 − Nxy
0
0
0
− Mx − Mxy 0
0
0
0
0
0
Nx
0 [Sσ ] = Mxy Mx
0
Mx
Mxy
0
− Nxy
3.2.6 Element load vector
The potential energy of the system due to external mechanical loading for each element is written as,
{d e }T [N ]T {p}dA
P=
(3.33)
A
{p} is the force vector specified on the surface at z = h n
And
{ p} = [ p x
py
pz
p yz
p xz
p xy
and is given by,
]
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, ∏e =
T 1 {d e }[B ]T [D ][B ]{d e }dA − {d e } [N ]T {P}dA − 1 d e 2A 2A A
{ } [N ] [ρ ]{d }dA T
T
(3.34)
e
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 t1 must be equal to zero, i.e. t1
(3.35)
δ Le dt = 0 t0
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, t1
{δd } [N ] [ρ ][N ]dA{d }dt = [{δd }{d } [N ] [ρ ][N ]dA ] T
T
e
t0
T
e
e
e
A
A
t1 t0
t1
T
[N ] [ρ ][N ]dA
− t0
{d }{δd }dt e
e
A
(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=t0 and t=t1.
39
The final form of Equation 3.37 can therefore be expressed as given in the following equation in the element level,
[M ]{d }+ [K ]{d } = {P } e
e
e
e
(3.37)
e
The Equation 3.37 is derived for conservative system. In case of non conservative system damping matrix introduce in Equilibrium Equations and expressed as (3.38)
MU + CU + KU = P
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 reversedthat 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,
[M ]{d }+ [[ K e
e
e
]
] − [ K σ ] {d e } = 0
(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.
40
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
XYZ X Y
[
XZ X
\
+\] ./
The natural eigen frequency ^_1 form XYZ
c XZ
X Y
bX
^_1
+^1 ] ./
O
(3.42)
a
` and ωσ = b
Kσ allows to rewrite Equation 3.42 in the M
O
(3.43)
Let d
"
then, du du dt ∗ θ du = = dt dt ∗ dt 2 dt ∗
(3.44a)
d 2u d du θ dt ∗ d du θ θ d 2 u θ 2 = = ∗ = dt 2 dt ∗ dt ∗ 2 dt dt dt ∗ 2 2 dt ∗2 4
(3.44b)
Putting Equation 3.44a and 3.44b in Equation 3.43 d 2 u θ 2 C θ du 2 2 + + ω n + Pωσ cos(θt ) u (t ) = 0 ∗2 ∗ dt 4 M 2 dt d 2 u 2C du 4 4 2 2 + + 2 ω n + P 2 ωσ cos(2t ∗ ) u (t ) = 0 ∗2 ∗ Mθ dt dt θ θ
(
)
41
(3.45a) (3.45b)
Equation 3.45b can be written as,
(
)
2
u + 2 µ ∗u + δ ∗ + 2ψ ∗ cos( 2t ∗ ) u (t ) = 0
(3.45c)
Equation 3.45c is Mathieu type of equation. where,
δ∗ =
4
θ
ωn 2 , µ ∗ = 2
2 Pω σ C and ψ ∗ = Mθ θ2
2
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 here results in
and
are constants. After application of the weighted modal matrix â
a T Ca = αa T Ma + βa T Ka = αI + βΛ where, I is a unit matrix and
(3.46)
is a diagonal matrix of the eigen-values.
Now,
µ∗ =
C αM + β K α β K = = + Mθ Mθ θ Mθ
(3.47)
for the condition of parametric resonance, = 2
µ∗ =
βω n α + 2ω n 2
n
, now putting in Equation 3.47 one gets,
(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.
42
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
δ∗ =
1 2 0 − ψ ∗ + ... 2 1 1 2 ∗2 ∗2 2 1 ± ψ − 4µ − ψ ∗ + ... 8
(
)
2 1 2 1 ∗2 4+ ψ∗ ± ψ − 16 µ ∗ 6 16
1 2
(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.
43
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
44
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) MESH
20 x 5 20 x 10 20 x 12 20 x 15 20 x 20 20 x 25 20 x 30 20 x 40 Note: w/ = (w.t5.E)/(F.a4)
PRESENT F.E ( w/)
ANSYS ( w/)
0.38277 0.39105 0.39195 0.39273
0.39424
0.39331 0.39359 0.39377 0.39392
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) R/a RATIO 15
50 100 Cantilever Plate (L=2 m, a=1 m, t=0.03 m ) Note: w/ = (w.t5.E)/(F.a4)
PRESENT F.E(w/)
ANSYS(w/)
0.0255
0.0257
0.0275
0.0275
0.0276
0.0276
0.0277
0.0277
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. 45
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 properties:
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
Geometric properties:
Length, L = 2 m Width, a = 1 m Thickness, t = 0.03 m Fibre orientation:
2- Layered laminate = 450/-450; 8- layered laminate = (00/900/450/-450) s.
46
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). PRESENT F.E ( w/)
FIBRE ORIENTATION
450/-450
0.28706
(00/900/450/-450)s
0.07886
ANSYS ( w/)
0.29043 0.07938
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 Figure4.1. 4.3.1 Isotropic cantilever shell
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, ω = ωL2
ρ E11t 2
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 )
20x15
20x20
1.0862
1.0836
1.0831
1.0778
1.0459
1.0448
1.0443
1.0429
1.0427
1.0417
1.0414
1.0404
20 x 10 15 50 100
47
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, ^
a
` ; where K is b
the stiffness and M is the mass of the element. As the stiffness is only variable here, nondimensional 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. 4.3.2 Composite cantilever shell
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, Density, J
12
He
= 0.28; Shear Modulus, G12 = 3 GPa; G13=G23=3 GPa \ 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
ANSYS (ω )
1.0081
1.0045
0.5384
0.5372
0.6849
0.6825
(ω )
90/0/90 45/0/45
50
0/90/45/-45/45/90/0
Here,
the
non-dimensional
frequency
is
taken
as,
ω = ωL2
ρ E11t 2
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.
48
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.
49
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. 4.4.1 Isotropic cantilever shell
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 N L2 4.6. Here, the non-dimensional buckling load is taken as, λ = x 3 E11t Table 4.6 Non-dimensional first buckling load factor of isotropic cantilever shell (L= 2m, a=1m, t=0.03 m).
R/a RATIO
15 50 100
PRESENT FINITE ELEMENT
ANSYS (λ )
0.2309
0.2292
0.2145
0.2144
0.2132
0.2133
(λ )
50
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. 4.4.2 Composite cantilever shell
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
90/0/90 45/0/45 0/90/45/-45/45/90/0
PRESENT FINITE ELEMENT
(λ )
ANSYS (λ )
0.9839
0.9409
0.2614
0.2433
0.4529
0.4360
51
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.
52
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.
53
4.4.1 Isotropic cantilever shell
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 properties:
Material: Aluminium Young’s Modulus, E = 70 GPa Poisson’s Ratio, = 0.3 Geometric properties:
Length, L = 2 m Width, a = 1 m Thickness, t = 0.03 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)
54
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.
4.4.2 Composite cantilever shell
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, Density, J
He
= 0.28; Shear Modulus, G12 = 3 GPa; G13=G23=3 GPa
12
\ 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/ a) b) c)
/0 0/0/0 0/30/0 0/90/0
90/ / 90 a) 90/0/90 b) 90/30/90 c) 90/90/90 / a) b) c)
/ 30/30/30 45/45/45 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.
55
(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
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.
56
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. FIBRE ORIENTATION
R/a RATIO
0/0/0 15
0/30/0
Where, ^ h
^i1 `j
k
lmm
NON DIMENSIONAL BUCKLING LOAD
0.4820
0.01844
0.4866
0/90/0
100
NON DIMENSIONAL NATURAL FREQUENCY (ω )
0.5312
0/0/0
0.4617
0/30/0
0.4633
0/90/0
0.4943
Y n as given in section 4.3.2 and o
57
Kp LY
lYY q
(λ )
0.01966 0.02410 0.05420 0.04540 0.05840 as given in section 4.4.2.
Itt can be observed from Figure-4.11 4.11 that,, with the variation of ‘ ’ the stability point int (xi-value) alue) does not change in each case of R/a ratio. This can be explained after noticing Table 4.8. 4.8 From that table one observes that, with the variation of ‘ ’ the first non nondimensional natural frequency frequency and buckli buckling ng load factor does not change significantly in each case of R/a ratio. The he natural frequency does not change significantly which signifies that the structural stiffness does not get altered significantly and hence the results results. The xi-value xi value is nothing but a function (from Equation quation-3.45c) 3.45c) of the ratio of buckling 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’( *) remains constant. Hence the stability shi shift ft for such a fibre orientation does not show any change. The lower xi xi-value 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 case i. i.e. this damping is much more effective in case of R/a ratio 100. That is why the xi xi-value shows higher value in this case. Alternately lternately it can be said that, because the shell has higher stiffness in case case of R/a ratio 15, the value of the stability point gets reduced in this case. CASE-2 Now the fibre orientation 90/ /90 is being considered here. The geometric and material properties are same as case-1. case The various cases of stability plot have been shown below.. Finally a plot has been given to show the shift of xi-value xi value for different values.
(a) 90/0/90, R/a = 15
(b) 90/0/90, R/a = 100
(c) 90/45/90, R/a = 15
(d) 90/45/90, R/a = 100 58
(e) 90/90/90, R/a = 15
(f) 90/90/90, R/a = 100
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.
59
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 case1. But after comparing Figure-4.11 and 4.13 one thing can be noted that, in case of Figure4.13 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.
(a) 30/30/30, R/a = 15,
(c) 45/45/45, R/a = 15,
= 18.39 rad/sec
= 20.83 rad/sec
60
(b) 30/30/30, R/a = 100,
= 17.57 rad/sec
(d) 45/45/45, R/a = 100,
= 19.87 rad/sec
(e) 90/90/90, R/a = 15,
= 38.97 rad/sec
(f) 90/90/90, R/a = 100,
= 37.79 rad/sec
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 61
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,
(c) R/a = 15, t = 0.0075 m,
= 5.09 rad/sec
= 6.56 rad/sec
(b) R/a = 100, t = 0.002 m,
(d) R/a = 100, t = 0.0075 m,
62
= 1.36 rad/sec
= 4.33 rad/sec
(e) R/a = 15, t = 0.015 m,
= 9.88 rad/sec
(f) R/a = 100, t = 0.015 m,
= 8.57 rad/sec
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.
63
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,
(c) L = 1m, a =1m,
= 299.17 rad/sec
(b) L= 0.5m, a =1m,
= 74.45 rad/sec
(d) L = 1m, a =1m,
64
= 293.16 rad/sec
= 71.98 rad/sec
(e) L = 2 m, a =1m,
= 18.39 rad/sec
(f) L = 2 m, a =1m,
= 17.57 rad/sec
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.
65
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 stressstrain relations and Poisson-ratio influence likely causes these effects.
66
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.
67
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.
68
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