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 |  موضوع: رسالة دكتوراة بعنوان Parametric Resonance Characteristics of Laminated Composite Twisted Cantilever Panels السبت 01 أغسطس 2020, 1:05 am | |
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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
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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