رسالة دكتوراة بعنوان Parametric Resonance Characteristics of Laminated Composite Twisted Cantilever Panels
منتدى هندسة الإنتاج والتصميم الميكانيكى
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 رسالة دكتوراة بعنوان Parametric Resonance Characteristics of Laminated Composite Twisted Cantilever Panels

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عدد المساهمات : 16336
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تاريخ التسجيل : 01/07/2009
العمر : 32
الدولة : مصر
العمل : مدير منتدى هندسة الإنتاج والتصميم الميكانيكى
الجامعة : المنوفية

رسالة دكتوراة بعنوان Parametric Resonance Characteristics of Laminated Composite Twisted Cantilever Panels  Empty
مُساهمةموضوع: رسالة دكتوراة بعنوان Parametric Resonance Characteristics of Laminated Composite Twisted Cantilever Panels    رسالة دكتوراة بعنوان Parametric Resonance Characteristics of Laminated Composite Twisted Cantilever Panels  Emptyالسبت 01 أغسطس 2020, 1:05 am

أخوانى فى الله
أحضرت لكم
رسالة دكتوراة بعنوان
Parametric Resonance Characteristics of Laminated Composite Twisted Cantilever Panels
A thesis submitted to
National Institute of Technology, Rourkela
For the award of degree of
Doctor of Philosophy in Engineering
by
A.V. Asha
Under the supervision of
Prof. Shishir K. Sahu  

رسالة دكتوراة بعنوان Parametric Resonance Characteristics of Laminated Composite Twisted Cantilever Panels  P_r_c_10
و المحتوى كما يلي :


Contents
Abstract v
Contents vii
List of tables ix
List of figures xiv
Nomenclature xviii
List of Publications xxi
1. INTRODUCTION . 1
1.1: Introduction . 1
1.2: Importance of the present structural stability study 1
1.3: Outline of the present work . 2
2. REVIEW OF LITERATURE . 4
2.1: Introduction 4
2.2: Vibration and buckling of twisted panels 4
2.3: Dynamic stability of twisted panels . 19
2.4: Critical discussion . 23
2.5: Objectives and scope of the present study 25
3. THEORY AND FORMULATION 26
3.1: The Basic Problem . 26
3.2: Proposed Analysis . 27
3.2.1: Assumptions of the analysis . 28
3.3: Governing Equations 29
3.3.1: Governing Differential Equations 29
3.4: Dynamic stability studies 31
3.5: Energy Equations . 32
3.5.1: Formulation of Vibration and Static Stability problems . 35
3.6: Finite Element Formulation . 35
3.6.1: The shell element 36
3.6.2: Strain displacement relations 38
3.6.3: Constitutive Relations 39viii
3.6.4: Derivation of Element Matrices 44
3.6.5: Geometric stiffness matrix . 45
3.7: Computer program . 48
4. RESULTS AND DISCUSSIONS . 49
4.1: Introduction . . 49
4.2: Convergence study 50
4.3: Comparison with previous studies 51
4.4: Numerical results 55
4.5: Isotropic twisted panels 55
4.5.1: Non-dimensionalization of parameters 56
4.5.2: Boundary conditions . 56
4.5.3: Vibration and buckling studies . 56
4.5.4: Dynamic stability studies . 63
4.6: Cross ply twisted cantilever panels 66
4.6.1: Non-dimensionalization of parameters 66
4.6.2: Boundary conditions . 67
4.6.3: Vibration and buckling studies 67
4.6.4: Dynamic stability studies 81
4.7: Angle-ply twisted cantilever panels . 90
4.7.1: Non-dimensionalization of parameters 91
4.7.2: Boundary conditions 91
4.7.3: Vibration and buckling studies 91
4.7.4: Dynamic stability studies 103
5. CONCLUSIONS 112
5.1: Isotropic twisted panels . 113
5.2: Cross-ply twisted cantilever panels 115
5.3: Angle-ply twisted cantilever panels . 118
5.4: Scope for further work 123
REFERENCES 124
APPENDIX . 135ix
List of Tables
No. Title Page
4.1 Convergence of non-dimensional fundamental
frequencies of free vibration of isotropic twisted
plates . 50
4.2 Convergence of non-dimensional frequencies of
vibration of composite twisted cantilever plates with
45°/-45°/45° lamination . 51
4.3 Comparison of non-dimensional frequency
parameters (?) of the initially twisted isotropic
cantilever plate type blade 52
4.4 Comparison of non-dimensional fundamental
frequencies of vibration of graphite epoxy pretwisted
cantilever [?/-?/?] plates 53
4.5 Comparison of buckling loads for a thin untwisted (?
= 0°) angle-ply cylindrical panel with symmetric layup [0°/-?°/+ ?°/-90°]s 54
4.6 Variation of non-dimensional frequency parameter
with angle of twist for a square isotropic cantilever
plate 57
4.7 Variation of non-dimensional frequency parameter
with R
y/b ratio for a square isotropic cylindrical
cantilever panel 58
4.8 Variation of non-dimensional frequency parameter
with aspect ratio for an isotropic twisted cantilever
plate 58
4.9 Variation of frequency in Hz with b/h ratio for a
square isotropic twisted cantilever plate . 59x
4.10 Variation of non-dimensional frequency parameter
for different twisted cantilever curved panels 59
4.11 Variation of non-dimensional buckling load with
angle of twist for a square isotropic cantilever plate . 60
4.12 Variation of non-dimensional buckling load with
angle of twist for a square isotropic cylindrical
cantilever panel 61
4.13 Variation of non-dimensional buckling load with Ry/b
ratio for a square isotropic twisted cylindrical
cantilever panel 61
4.14 Variation of non-dimensional buckling load with
aspect ratio for an isotropic twisted cantilever plate 62
4.15 Variation of buckling load with b/h ratio for a square
isotropic twisted cantilever plate . 62
4.16 Variation of non-dimensional frequency parameter
with angle of twist for square cross-ply plates with
different ply lay-ups 67
4.17 Non-dimensional free vibration frequencies of square
cross-ply pretwisted cantilever plates with varying
angles of twist . 69
4.18 Non-dimensional free vibration frequencies of square
cross-ply pretwisted cantilever plates with varying
angles of twist (E-glass/epoxy) 70
4.19 Variation of non-dimensional frequency parameter
with R/a ratio for square cross-ply cylindrical and
spherical twisted cantilever shells 71
4.20 Comparison of non-dimensional frequency parameter
of square cross-ply twisted plates and square crossply twisted spherical shells (b/Ry = 0.25) . 72xi
4.21 Variation of non-dimensional frequency parameter
with aspect ratio for cross-ply twisted cantilever
plates with different ply lay-ups . 73
4.22 Variation of frequency in Hz with b/h ratio for square
cross-ply twisted cantilever plates with different ply
lay-ups 73
4.23 Variation of non-dimensional frequency parameter
with geometry for cross-ply twisted cantilever plates
with different ply lay-ups 74
4.24 Variation of non-dimensional frequency parameter
with degree of orthotropy of different square crossply twisted cantilever plates . 75
4.25 Variation of non-dimensional buckling load with
angle of twist for square cross-ply plates with
different ply lay-ups 76
4.26 Variation of non-dimensional buckling load with R/a
ratio for square cylindrical and spherical twisted
cross-ply shells . 77
4.27 Non-dimensional buckling load for square cross-ply
twisted plates and spherical twisted shells (b/Ry =
0.25) with different ply lay-ups 78
4.28 Variation of non-dimensional buckling load with
aspect ratio for cross-ply twisted cantilever plates
with different ply lay-ups 79
4.29 Variation of buckling load with b/h ratio for square
cross-ply twisted cantilever plates with different ply
lay-ups 80
4.30 Variation of non-dimensional buckling load with
geometry for square cross-ply twisted cantilever
panels with different ply lay-ups 80xii
4.31 Variation of non-dimensional buckling load with
degree of orthotropy (E1/E2) for different square
cross-ply twisted cantilever plates . 81
4.32 Variation of non-dimensional free vibration
frequencies with angle of twist and ply orientation of
angle-ply (?/-?/?) pretwisted cantilever plates 92
4.33 Variation of non-dimensional free vibration
frequencies with angle of twist and ply orientation of
angle-ply (?/-?/?) pretwisted cantilever panels . 94
4.34 Variation of non-dimensional free vibration
frequencies with Ry/b ratio of square angle-ply (?/-
?/?) pretwisted cantilever panels 95
4.35 Variation of non-dimensional frequency with aspect
ratio of laminated composite angle-ply (?/-?/?)
pretwisted cantilever plates 95
4.36 Variation of frequency in Hz with b/h ratio for square
laminated composite angle-ply (?/-?/?) pretwisted
cantilever plates . 96
4.37 Variation of non-dimensional frequency with degree
of orthotropy of square angle-ply (?/-?/?) pretwisted
cantilever plates . 97
4.38 Variation of non-dimensional buckling load with
angle of twist of square angle-ply(?/-?/?) pretwisted
cantilever plates . 98
4.39 Variation of non-dimensional buckling load with
angle of twist of square angle-ply(?/-?/?) pretwisted
cantilever plates with camber . 99
4.40 Variation of non-dimensional buckling load with
angle of twist of square laminated composite angleply (?/-?/?) pretwisted thick cantilever plates 99xiii
4.41 Variation of non-dimensional buckling load with
aspect ratio of laminated composite angle-ply (?/-?/?)
pretwisted cantilever plates . 100
4.42 Variation of non-dimensional buckling load with
angle of twist of rectangular angle-ply (?/-?/?)
pretwisted cantilever plates 101
4.43 Variation of non-dimensional buckling load with b/h
ratio of square angle-ply (?/-?/?) pretwisted cantilever
plates . 102
4.44 Variation of non-dimensional buckling load with
degree of orthotropy of angle-ply (?/-?/?) pretwisted
cantilever plates . 102xiv
List of Figures
No Title Page
3.1 Laminated composite twisted curved panel subjected
to in-plane harmonic loads . 27
3.2 Force and moment resultants of the twisted panel 30
3.3 Isoparametric quadratic shell element 36
3.4 Laminated shell element . 40
4.1 Comparison of results of instability regions of square
untwisted angle-ply panels(45°/-45°, 45°/-45°/45°/-
45°)of present formulation with Moorthy et al. 54
4.2 Variation of instability region with angle of twist of
the isotropic cantilever panel, a/b = 1, ? = 0°, 15° and
30°, ? = 0.2 63
4.3 Variation of instability region with static load factor
for a square isotropic twisted cantilever panel, a/b =
1, ? = 15°, ? = 0.0, 0.2, 0.4 and ? = 0.6 64
4.4 Variation of instability region with Ry/b ratio for a
square isotropic cylindrical twisted cantilever panel,
a/b = 1, ? = 15°, ? = 0.2 . 64
4.5 Variation of instability region with b/h ratio for a
square isotropic twisted cantilever plate, a/b = 1, ? =
15°, ? = 0.2 . 65
4.6 Variation of instability region with curvature for a
square isotropic twisted cantilever panel, a/b = 1, ? =
15°, ? = 0.2, b/Ry = 0.25 66
4.7 Variation of instability region with angle of twist of
the four layer cross-ply twisted plate [0°/90°/90°/0°],
a/b = 1, ? = 0°, 15° and 30°, ? = 0.2 . 82xv
4.8 Variation of instability region with number of layers
of the cross-ply twisted plate (2, 4, and 8 layers), a/b =
1, ? = 15°, ? = 0.2 . 83
4.9 Variation of instability region with static load factor
of a cross-ply twisted plate[0°/90°/90°/0°], a/b = 1, ? =
15°, ? = 0.0, 0.2, 0.4 and ? = 0.6 84
4.10 Variation of instability region with static load factor
of a cross-ply twisted plate
[0°/90°/0°/90°/0°/90°/0°/90°], a/b = 1, ? =15°, ? = 0.0,
0.2, 0.4 and ? = 0.6 . 84
4.11 Variation of instability region with aspect ratio of the
cross-ply twisted plate[0°/90°/90°/0°], ? = 15°, ? = 0.2,
a/b = 0.5, 1.0, 1.5 85
4.12 Variation of instability region with b/h ratio of the
four layer cross-ply twisted plate[0°/90°/90°/0°], a/b =
1 , ? = 15°, ? = 0.2, b/h = 200, 250 and 300 . 86
4.13 Variation of instability region with b/h ratio of the
cross-ply twisted plate[0°/90°/0°/90°/0°/90°/0°/90°],
a/b = 1 , ? = 15°, ? = 0.2, b/h = 200, 250 and 300 86
4.14 Variation of instability region with number of layers
of the cross-ply twisted cylindrical panel, a/b = 1, ? =
15°, ? = 0.2 and b/Ry = 0.25 . 87
4.15 Variation of instability region with number of layers
of the cross-ply twisted spherical panel, a/b = 1, ? =
15°, ? = 0.2 and b/Ry = 0.25, b/Rx = 0.25 87
4.16 Variation of instability region with number of layers
of the cross-ply twisted hyperbolic panel, a/b = 1, ? =
15°, ? = 0.2 and b/Ry = 0.25, b/Rx = ?0.25 . 88
4.17 Variation of instability region with curvature for a
cross-ply twisted cantilever panel [0°/90°], a/b = 1, ?
= 15°, ? = 0.2, b/Ry = 0.25 89xvi
4.18 Variation of instability region with curvature for a
cross-ply twisted cantilever panel [0°/90°/90°/0°], a/b
= 1, ? = 15°, ? = 0.2, b/Ry = 0.25 . 89
4.19 Variation of instability region with degree of
orthotropy of the cross-ply twisted cantilever panel
[0°/90°/90°/0°], a/b = 1, ? = 15°, ? = 0.2 . 90
4.20 Variation of instability region with angle of twist of
the angle-ply flat panel [30°/-30°/30°/-30°], a/b = 1, ?
= 0°, 15° and 30°, ? = 0.2 . 103
4.21 Variation of instability region with number of layers
of the angle-ply twisted panel [45°/-45°/45°/-45°], a/b
= 1, b/h = 250, ? = 15°, ? = 0.2 . 104
4.22 Variation of instability region with static load factor
of an angle-ply twisted panel [30°/-30°/30°/-30°], a/b =
1, ? = 15°, ? = 0.0, 0.2, 0.4 and ? = 0.6 . 105
4.23 Variation of instability region with ply orientation of
an angle-ply twisted panel [?/??/ ?/??], a/b = 1, ? =
15°, ? = 0.2, ? = 0° to 90° 106
4.24 Variation of instability region with aspect ratio of the
angle-ply twisted panel [30°/-30°/30°/-30°], a/b = 1, 2
and 4, ? = 15°, ? = 0.2 . 107
4.25 Variation of instability region with b/h ratio of the
angle-ply twisted panel [30°/-30°/30°/-30°], a/b = 1,
b/h =200, 250 and 300, ? = 15°, ? = 0.2 . 107
4.26 Variation of instability region with angle of twist of
the angle-ply cylindrical twisted panel [30°/-30°/30°/-
30°], a/b = 1, ? = 0°, 15° and 30°, ? = 0.2, b/Ry = 0.25 108
4.27 Variation of instability region with angle of twist of
the angle-ply spherical twisted panel [30°/-30°/30°/-
30°], a/b = 1, ? = 0°, 15° and 30°, ? = 0.2, b/Ry = 0.25,
b/Rx = 0.25 109xvii
4.28 Variation of instability region with angle of twist of
the angle-ply hyperbolic paraboloidal twisted panel
[30°/-30°/30°/-30°], a/b = 1, ? = 0°, 15° and 30°, ? =
0.2, b/Ry = 0.25, b/Rx = ?0.25 . 109
4.29 Variation of instability region with geometry for an
angle-ply twisted panel [30°/-30°/30°/-30°], a/b = 1, ?
= 15°, ? = 0.2, b/Ry = 0.25 110
4.30 Variation of instability region with degree of
orthotropy for an angle-ply twisted panel [30°/-
30°/30°/-30°], a/b = 1, ? = 15°, ? = 0.2, h = 2mm 111
6.1 Flow chart of computer programme . 137xviii
Nomenclature
The principal symbols used in this thesis are presented for easy reference. A
single symbol is used for different meanings depending on the context and
defined in the text as they occur.
English
a, b dimensions of the twisted panel
a/ b aspect ratio of the twisted panel
Aij, Bij, Dij and Sij extensional, bending-stretching coupling,
bending and transverse shear stiffnesses
b/ h width to thickness ratio of the twisted
panel
Strain-displacement matrix for the element
[D] stress-strain matrix
[Dp] stress-strain matrix for plane stress
dx, dy element length in x and y-direction
dV volume of the element
E11, E22 modulii of elasticity in longitudinal and
transverse directions
G12, G13, G23 shear modulii
h thickness of the plate
J Jacobian
k shear correction factor
[Ke] global elastic stiffness matrix
[ke] element bending stiffness matrix with
shear deformation of the panel
[Kg] global geometric stiffness matrix
[Kp] plane stiffness matrix
kx
, ky, kxy bending strains
[M] global consistent mass matrix
[me] element consistent mass matrixxix
Mx, My, Mxy moment resultants of the twisted panel
n number of layers of the laminated panel
[N] shape function matrix
Ni shape functions
N (t) in-plane harmonic load
Ns static portion of load N (t)
Nt amplitude of dynamic portion of load N (t)
Ncr critical load
Nx, Ny, Nxy in-plane stress resultants of the twisted
panel
Nx0, Ny0, Nxy0 external loading in the X and Y directions
respectively
[P] mass density parameters
q vector of degrees of freedom
Qx , Qy shearing forces
Rx, Ry, Rxy radii of curvature of shell in x and y
directions and radius of twist
T transformation matrix
u, v, w displacement components in the x, y, z
directions at any point
uo, vo, wo displacement components in the x, y, z
directions at the midsurface
U0 strain energy due to initial in-plane stresses
U1 strain energy associated with bending with
transverse shear
U2 work done by the initial in-plane stresses
and the nonlinear strain
V kinetic energy of the twisted panel
w out of plane displacement
xi, yi cartesian nodal coordinates
X, Y, Z global coordinate axis systemxx
Greek
? static load factor
? dynamic load factor
? shear strains
? x ,? y ,? xy strains at a point
?xnl, ?ynl, ?xynl non-linear strain components
?x, ?y rotations of the midsurface normal about
the x- and y- axes respectively
? non-dimensional buckling load
? Poisson’s ratio
?, ? local natural coordinates of the element
(?)k mass density of kth layer from mid-plane
? mass density of the material
? x ,? y ,? xy stresses at a point
?x
0
, ?y
0 and ?
xy
0 in-plane stresses due to external load
?
xy, ?xz, ?yz shear stresses in xy, xz and yz planes
respectively
frequencies of vibration
? non-dimensional frequency parameter
? frequency of excitation of the harmonic
load
? excitation frequency in radians/second
? angle of twist of the twisted panel
Mathematical Operators
? ?1 Inverse of the matrix
? T Transpose of the matrix
x y
, Partial derivatives with respect to x and yxxi
List of Publications out of this Work
Papers in International Journals
1. S. K. Sahu and A.V. Asha (2008): Parametric resonance characteristics of
angle- ply twisted curved panels, International Journal of Structural Stability and
Dynamics, Vol.8(1), pp.61-76
2. S. K. Sahu, A. V. Asha and R. N. Mishra (2005): Stability of Laminated
Composite Pretwisted Cantilever Panels, Journal of Reinforced Plastics and
Composites, Vol.24 (12), pp.1327-1334.
Papers Presented in Conferences
1. S. K. Sahu and A. V. Asha: Dynamic Stability of twisted laminated Composite
cross-ply panels, International Conference on Theoretical, Applied,
Computational and Experimental Mechanics (ICTACEM 2007), Dec 27-29,
2007 at IIT, Kharagpur
2. S. K .Sahu and A. V. Asha: Vibration and Stability of Cross-ply laminated
twisted cantilever plates, International conference on Vibration Problems, Feb 1-
3, 2007 at B.E College, Shibpur, Kolkata.
3. S. K.Sahu and A. V. Asha: Dynamic Stability of Laminated Composite twisted
curved Panels, IXth International conference on “Recent advances in Structural
Dynamics”, July 2006, Institute of Sound and Vibration Research, University of
Southampton, UK.


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