Dynamic Instability of Laminated Composite Curved Panels in Hygrothermal Environment

Dynamic Instability of Laminated Composite Curved Panels in Hygrothermal Environment
Dynamic Instability of Laminated Composite
Curved Panels in Hygrothermal Environment
A thesis submitted to
National Institute of Technology, Rourkela
For the award of degree of
Doctor of Philosophy
in
Engineering
by
Manoj Kumar Rath
Under the supervision of
Prof. Shishir Kumar Sahu
Department of Civil Engineering
National Institute of Technology
Rourkela-769008, India
July 2012
Dedicated
To My Parents
i
CERTIFICATE
This is to certify that the thesis entitled “Dynamic Instability of Laminated
Composite curved panels in Hygrothermal Environment”, being submitted to the
National Institute of Technology, Rourkela, India by Manoj Kumar Rath for the
award of the degree of Doctor of Philosophy in Engineering is a record of bonafide
research work carried out by him under my supervision and guidance. This fulfills the
requirements of the regulations of the degree. The results embodied in this thesis have
not been submitted in part or full to any other university or institute for the award of
any degree or diploma.
NIT, Rourkela
Date:
Dr. Shishir Kumar Sahu
Professor, Department of Civil Engineering
National Institute of Technology
Rourkela-769008
Odisha, India
ii
Acknowledgement
I express my deep sense of gratitude and indebtedness to my thesis supervisor
Professor. Shishir Kumar Sahu, Department of Civil Engineering, National Institute
of Technology, Rourkela, for his invaluable encouragement, helpful suggestions and
supervision throughout the course of this work.
I express my sincere thanks to the Director, Prof. S.K. Sarangi, National Institute of
Technology, Rourkela for motivating me in this endeavor and providing me the
necessary facilities for this study.
I would like to thank Prof N. Roy, Head of the Civil Engineering Department and,
Prof. M. Panda, Professor, Civil Engineering department for their help and
cooperation during the progress of this work.
I would also like to thank Professor S.K. Sahoo of Mechanical Engineering
Department for their invaluable suggestions and help at various stages of the
experimental work.
I acknowledge with thanks the help rendered to me by Prof B. C. Roy, HOD, MM
Department. Prof B. B. Vernma and lab staff of MM, my friends and other staff of the
Civil Engineering Department for their continuous encouragement during the progress
of my work.
Last but not least, I am extremely grateful to my wife and children, Saswati and
Ayush, for their support and patience during this period.
The author is also thankful to Department of Science and Technology Govt of India
for their financial support through the R & D project No SR/S3/MERC/0009/2008 for
the material and facility during testing of Composites.
(Manoj Kumar Rath)
iii
ABSTRACT
Composite materials are increasingly used in aerospace, naval and high performance
civil engineering structures such as aerospace, submarines, automobiles. The
structural components, subjected to in-plane harmonic loads may undergo parametric
resonance or dynamic stability due to certain combinations of the applied in-plane
forcing parameters and natural frequency of transverse vibration. The parametric
instability itself requires investigation of vibration and buckling of structures. The
present study deals with free vibration, buckling and parametric resonance behavior of
laminated composite plates under in-plane periodic loading under varying temperature
and moisture. In this analysis, the effects of various parameters such as number of
layers, aspect ratios, side-to thickness ratios, ply orientations, static load factors,
lamination angle and the degree of orthotropic are studied.
A simple laminated plate model based on the first order shear deformation theory
(FSDT) is developed for the free vibration, buckling and parametric instability effects
of composite plates subjected to hygrothermal loading. The principal instability
regions are obtained using Bolotin’s approach employing finite element method
(FEM). An eight-node isoparametric quadratic element is employed in the present
analysis with five degree of freedom per node. The element is modified to
accommodate the laminated composite plates under hygrothermal environment,
considering the effects of transverse shear deformation and rotary inertia. The element
stiffness matrix, geometric stiffness matrix due to residual stresses, element mass
matrix, geometric stiffness matrix due to applied in-plane loads and nodal load vector
of the element are derived using the principle of minimum potential energy. They are
evaluated using the Gauss quadrature numerical integration technique. Reduced
integration technique is applied to avoid the possible shear locking. A computer
program based on FEM in MATLAB environment is developed to perform all
necessary computations.
The basic vibration and buckling experiments are performed on the industry driven
woven fiber Glass/Epoxy specimens subjected to hygrothermal environment. The
specimens were hygrothermally conditioned in a humidity cabinet where the
conditions were maintained at temperatures of 300K-425K and relative humidity
(RH) ranging from 0-1.0% for moisture concentrations.
iv
The numerical and experimental results show that there is reduction in natural
frequencies and buckling loads with increasing temperature and moisture
concentration for laminates both for simply supported and clamped boundary
conditions. The dynamic instability study using FEM revealed that, due to the static
component of load, the instability regions tend to shift to lower frequencies. The onset
of instability occurs earlier and the width of dynamic instability regions increases with
rise in temperature and moisture concentration for different parameters. With increase
in lamination angle, the width of the instability region becomes smaller. The onset of
instability occurs later for square plates than rectangular plates with wider instability
region with increase of aspect ratio. The ply orientation significantly affects the onset
of instability. It is observed that the excitation frequency increases with introduction
of curvatures from flat panel to doubly curved panel in hygrothermal environments.
Thus the instability behaviour of laminated composite panels is influenced by increase
in number of layers, aspect ratio, side to thickness ratio, increase in static and
dynamic load factor, geometry, material, ply lay-up and its orientation. This can be
utilized to tailor the design of laminated composite panels in hygrothermal
environment.
Keywords: parametric resonance, dynamic instability, woven fiber laminated
composite plates, excitation frequencies, hygrothermal environment.
v
Contents
1.
2.
Certificate
ii
Acknowledgement
iii
Abstract
iv
Contents
vi
List of tables
x
List of figures
xii
List of Publications
xix
Nomenclature
xxi
INTRODUCTION
1
1.1.
Introduction
1
1.2.
Importance of the present structural stability study
1
REVIEW OF LITERATURE
3
2.1.
Introduction
3
2.1.1.
Static behavior of composites in hygrothermal
3
environment
2.1.2.
Vibration of composite panels in hygrothermal
6
environment
7
2.1.2.1.
Plates
7
2.1.2.2.
Shells
9
2.1.3.
Buckling effects of composite panels in hygrothermal
10
environment
2.1.3.1.
Plates
11
2.1.3.2.
Shells
13
2.1.4.
Dynamic stability of composite panels in hygrothermal
13
environment
13
2.4.1.1.
Plates
2.4.1.2.
Shells
2.1.5.
Critical discussion
2.1.5.1.
Vibration of composite panels in hygrothermal
15
17
17
environment
vi
18
2.1.5.2.
Buckling effects of composite curved panels in
hygrothermal environment
2.1.4.3.
Dynamic stability of composite curved panels in
hygrothermal environment
2.1.4.4.
3.
4.
19
21
Aim and scope of the present study
MATHEMATICAL FORMULATION
22
3.1.
The Basic Problem
22
3.2.
Proposed Analysis
23
3.2.1.
Assumptions of the analysis
24
3.3.
Governing Equations
24
3.4.
Dynamic stability studies
25
3.5.
Constitutive Relations
27
3.6.
Strain Displacement Relations
29
3.7.
Finite Element Formulation
30
3.7.1.
Element elastic stiffness matrix
32
3.7.2.
Geometric stiffness matrix due to residual stresses
32
3.7.3.
Geometric stiffness matrix due to applied in-plane load
34
3.7.4.
The element mass matrix
35
3.8.
Solution process
36
3.9.
Computer program
36
EXPERIMENTAL PROGRAMME
38
4.1.
Introduction
38
4.2.
Materials required for fabrication of plates
38
4.3.
Fabrication procedure for static test specimens
38
4.4.
Fabrication procedure for vibration and buckling
40
4.5.
Hygrothermal treatment
41
4.6.
Static behavior experiment test
42
4.7.
Apparatus required for free vibration test
43
4.7.1.
Free vibration experiment test
43
4.8.
Buckling experiment test
45
vii
5.
RESULTS AND DISCUSSIONS
47
5.1.
Introduction
47
5.1.1.
Static behavior of woven fiber composites in
hygrothermal environment
47
5.2.
Flat panels
62
5.2.1.
vibration of woven fiber laminated composite flat
62
panels in hygrothermal environment
5.2.1.1.
Convergence study
62
5.2.1.2.
Comparison with previous studies
64
5.2.1.3
New results for free vibration
65
5.2.2.
Buckling effects of woven fiber laminated composite
72
flat panels in hygrothermal environment
5.2.2.1.
Convergence study
72
5.2.2.2.
Comparison with previous studies
73
5.2.2.3.
New results for buckling
74
5.2.3.
Dynamic stability of laminated Composite plates in
84
hygrothermal environment
5.2.4.
Non-dimensionalization of parameters
84
5.2.4.1.
Convergence study
84
5.2.4.2.
Comparison with previous studies
85
5.2.4.3.
New results for dynamic stability
86
5.3.
Curved panels
99
5.3.1.
Convergence study
99
5.3.2.
Comparison with previous studies
99
5.3.2.1.
Vibration of composite curved panels in hygrothermal
99
environment
5.3.2.2.
Buckling of composite curved panels in hygrothermal
100
Environment
5.3.2.3.
New results for dynamic stability
viii
101
6.
CONCLUSIONS
114
6.1.
115
Static behavior of composites in hygrothermal
environment
6.2.
6.3.
vibration of laminated composite flat panels in
hygrothermal environment
116
Buckling effects of laminated composite flat panels in
117
hygrothermal environment
6.4.
Dynamic stability of composite flat panels in
hygrothermal environment
6.5.
Dynamic stability of composite curved panels in
118
119
hygrothermal environment
6.6.
Further Scope of research
REFERENCES
120
121
ix
List of Tables
No.
Title
Page
5.1
Statistical variation of ILSS with loading speeds of Glass
47
Fiber: Epoxy composites for varying proportion
5.2
Statistical variation of ILSS with loading speeds of Glass
49
Fiber: Polyester composites for varying proportion
5.3
Statistical variation of ILSS with temperature of Glass Fiber:
51
Epoxy composites for varying proportion
5.4
Statistical variation of ILSS with temperature of Glass Fiber:
52
Polyester composites for varying proportion
5.5
Statistical variation of ILSS with moisture concetration of
54
Glass Fiber: Epoxy composites for varying proportion
5.6
Statistical variation of ILSS with moisture concetration of
54
Glass Fiber: Polyester composites for varying proportion
5.7
Statistical variation of ILSS with exposure time of Glass fiber:
56
Epoxy composites for varying proportion
5.8
Statistical variation of ILSS with exposure time of Glass fiber:
57
Polyester composites for varying proportion
5.9
Elastic moduli of glass fiber / epoxy lamina at different
61
temperatures
5.10
Elastic moduli of glass fiber / epoxy lamina at different
61
moisture
5.11
Convergence of non-dimensional frequencies of vibration for
63
SSSS four layered laminated composite plates for two
lamination sequences at 325K temperature
5.12
Convergence of non-dimensional frequencies of vibration for
63
SSSS four layered laminated composite plates for two
lamination sequences at 0.1% moisture concentration
5.13
Comparison of non-dimensional free vibration frequencies for
64
SSSS (0/90/90/0) plates at 325K Temperature
5.14
Comparison of non-dimensional free vibration frequencies for
x
65
SSSS (0/90/90/0) Plates at 0.1% moisture concentration
5.15
Convergence of non-dimensional critical load for SSSS four
72
layered Laminated composite plates at 325K temperature
5.16
Convergence of non-dimensional critical load for SSSS four
layered laminated composite plates
73
at 0.1% moisture
concentration
5.17
Comparison of non-dimensional critical load for SSSS
74
(0/90/90/0) four layered laminated composite plates at 325K
temperature and 0.1% moisture concentration
5.18
Convergence of non-dimensional excitation frequency for
85
SSSS four layered laminated composite plates for different ply
orientations at 325K temperature
5.19
Convergence of non-dimensional excitation frequency for
85
SSSS four Layered laminated composite plates for different
ply orientations at0.1% moisture concentration
5.20
Comparison of dimensional excitation frequencies SSSS four
86
layered laminated composite plates for cross-ply laminates
5.21
Comparison of non-dimensional free vibration frequencies for
100
SSSS (0/90/90/0) spherical shell at ambient temperature
5.22
Comparison of natural frequencies for SSSS (0/90/90/0) shell
100
at 1% moisture concentration
5.23
Comparison of Non-dimensional buckling loads of a square
simply supported symmetric cross-ply cylindrical laminated
curved panels with (0/90/0/90/0)
xi
101
List of Figures
No
3.1
Title
Page
Laminated composite curved panels under in-plane harmonic
22
loading under hygrothermal environment
3.2
Arbitrarily oriented laminated plate
23
3.3
Geometry of an n- layered laminate
23
3.4
Eight nodded isoparametric element
30
3.5
Flow chart of Program in MATLAB for instability of Composite
37
panels subjected to hygrothermal loads
4.1
Placing of woven glass fiber using on gel coat
41
4.2
Removal of air entrapment using steel roller
41
4.3
Fabrication of woven fiber composite plates
41
4.4
Instron 1195 UTM machine
41
4.5
Specimen in X direction for tensile testing
41
4.6
Specimen in 45o direction for tensile testing
41
4.7
Humidity Chamber
42
4.8
Temperature bath
42
4.9
Complete set up of Instron 1195 machine
43
4.10
Modal impact Hammer
44
4.11
FFT 3560-C analyzer
44
4.12
Display unit (laptop)
44
4.13
Iron frame for different boundary condition set-up
44
4.14
The test frame with specimen for four sides simply supported
44
boundary conditions
4.15
The test frame with specimen for four sides clamped boundary
44
conditions
4.16
Instron UTM machine with specimen
46
4.17
Composite test frame with specimen
46
4.18
Specimen before buckling
46
4.19
Specimen after buckling
46
5.1
Variation of ILSS with loading speeds of glass fibre/epoxy
51
composites for varying proportion
xii
5.2
Variation of ILSS with loading speeds of glass fibre/polyester
52
composites for varying proportion
5.3
Variation of ILSS with temperature of glass/epoxy composites for
53
varying proportion
5.4
Variation of ILSS with
temperature of glass fibre/polyester
54
composites for varying proportion
5.5
Variation of ILSS with moisture concetration of glass/epoxy
56
composites for varying proportion
5.6
Variation of ILSS with moisture concetration of glass/polyester
57
composites for varying proportion
5.7
Variation of ILSS with exposure time of glass/epoxy composites for
59
varying proportion
5.8
Variation of ILSS with exposure time of glass/Polyester composites
59
for varying proportion
5.9
Variation of Shear Modulus of elasticity with different temperature
60
of Glass/epoxy composites for 55:45 proportions
5.10
Variation of Modulus of rigidity with different
temperature of
60
glass/epoxy composites for 55:45 proportions
5.11
Variation of Modulus of elasticity with different
moisture
61
concentration of glass/epoxy composites for 55:45 proportions
5.12
Variation of Modulus of rigidity with different
moisture
61
concentration of glass/epoxy composites for 55:45 proportions
5.13
Variation of frequency in Hz with temperature for simply supported
67
16 layers [0/0]4S woven fiber composite plates
5.14
Variation of frequency in Hz with moisture concentration for simply
67
supported 16 layers [0/0]4S woven fiber composite plates
5.15
Variation of frequency in Hz with temperature for simply supported
68
of 16 layers [0/90] 4S, [45/-45]4S and [0/90]8, [45/-45]8 woven fiber
composite plates
5.16
Variation of frequency in Hz with moisture concentration for simply
supported of 16 layers [0/90]
4S,
[45/-45]4S and [0/90]8, [45/-45]8
woven fiber composite plates
xiii
68
5.17
Variation of frequency in Hz with temperature for simply supported
69
of 16 layers [0/0]4S, 12 layers [0/0]3S, 8 layers [0/0]2S woven fiber
composite plates
5.18
Variation of frequency in Hz with moisture concentration for simply
69
supported of 16 layers [0/0]4S, 12 layers [0/0]3S, 8 layers [0/0]2S
woven fiber composite plates
5.19
Variation of frequency in Hz with temperature for simply supported
70
of 16 layers [0/0]4S woven fiber composite plates
5.20
Variation of frequency in Hz with moisture concentration for simply
70
Supported of [0/0]4S woven fiber 16 layers composite plates
5.21
Variation of frequency in Hz with temperature for simply supported
71
of 16 layers [0/0]4S woven fiber composite plates
5.22
Variation of frequency in Hz with moisture concentration for simply
71
supported of 16 layers [0/0]4S woven fiber composite plates
5.23
Variation of frequency in Hz with temperature for clamped (c-c-c-c)
72
of 16 layers [0/0]4S woven fiber composite plates
5.24
Variation of frequency in Hz with moisture concentration for
72
clamped(c-c-c- c) of 16 layers [0/0]4S woven fiber composite plates
5.25
Variation of buckling load in KN with temperature of 16 layers
76
[0/90]4S woven fiber composite plates (C-F-C-F)
5.26
Variation of buckling load in KN with moisture concentration of 6
76
layers [0/90]4S woven fiber composite plates (C-F-C-F)
5.27
Variation of buckling load in KN with temperature of 16
layers[0/90]
4S,
77
[45/-45]4S and [0/90]8, [45/-45]8 woven fiber
composite plates(S-S-S-S)
5.28
Variation of buckling load in KN with moisture concentration of 16
layers [0/90]
4S,
78
[45/-45]4S and [0/90]8, [45/-45]8 woven fiber
composite plates (S-S-S-S)
5.29
Variation of buckling load in KN with temperature of 4
78
layers[0/90]1S, 8 layers [0/90]2S, 12 layers [0/90]3S woven fiber
composite plates (S-S-S-S)
5.30
Variation of buckling load in KN with moisture concentration of 4
layers [0/90]1S, 8 layers [0/90]2S, 12 layers [0/90]3S woven fiber
composite plates(S-S-S-S)
xiv
79
5.31
Variation of buckling load in KN with temperature of 16 layers
80
[0/90]4S woven fiber composite plates (S-S-S-S)
5.32
Variation of buckling load in KN with moisture concentration of 16
80
layers [0/90]4S woven fiber composite plates (S-S-S-S)
5.33
Variation of buckling load in KN with temperature of 16 layers
[0/90]4S
woven
fiber
composite
plates
81
(C-F-C-F)boundary
Condition
5.34
Variation of buckling load in9 KN with moisture concentration 16
81
layers [0/90]4S of woven fiber composite plates (C-F-C-F) boundary
condition
5.35
Variation of buckling load in KN with temperature of 16 layers
82
[0/90]4S woven fiber composite plates (S-S-S-S)
5.36
Variation of buckling load in KN with moisture concentration of 16
83
layers [0/90]4S woven fiber composite plates (S-S-S-S)
5.37
Variation of buckling load in KN with temperature of 16 layers
84
[0/90]4S woven fiber composite plates (C-F-C-F) boundary condition
5.38
Variation of buckling load in KN with moisture concentration of 16
84
layers [0/90]4S woven fiber composite plates (C-F-C-F)boundary
condition
5.39
Variation of instability regions with temperature at 325K for simply
89
supported of a/b=1, α= 0.2, woven fiber composite plates
5.40
Variation of instability regions with moisture concentration at 0.25%
89
for simply supported of a/b=1, α= 0.2, woven fiber composite plates
5.41
Variation of instability regions with temperature at 325K for simply
90
supported of a/b=1, α= 0.2, woven fiber composite plates
5.42
Variation of instability regions with moisture concentration at 0.25%
90
for simply supported of a/b=1, α= 0.2, woven fiber composite plates
5.43
Variation of instability regions with temperature at 325K for simply
91
supported of a/b=1, α= 0.2, woven fiber composite plates
5.44
Variation of instability regions with moisture concentration at 0.25%
91
for simply supported of a/b=1, α= 0.2, woven fiber composite plates
5.45
Variation of instability regions with temperature at 325K for simply
supported of [45/-45]4, woven fiber composite plates
xv
92
5.46
Variation of instability regions with moisture concentration at 0.25%
93
for simply supported of [45/-45]4, woven fiber composite plates
5.47
Variation of instability regions with different temperature for simply
93
supported of [45/-45]4, woven fiber composite plates
5.48
Variation of instability regions with different moisture concentration
93
for simply supported of [45/-45]4, woven fiber composite plates
5.49
Variation of instability regions with temperature at 325K for simply
94
Supported, α= 0.2, woven fiber composite plates
5.50
Variation of instability regions with moisture concentration at 0.25%
94
for simply supported, α= 0.2, woven fiber composite plates
5.51
Variation of instability regions with temperature at 325K for simply
95
supported of [45/-45]4, α= 0.2, woven fiber composite plates
5.52
Variation of instability regions with moisture concentration at 0.25%
96
for simply supported, α= 0.2, woven fiber composite plates
5.53
Variation of instability regions with temperature at 325K for simply
96
supported of [45/-45]4, α= 0.2, woven fiber composite plates
5.54
Variation of instability regions with moisture concentration at 0.25%
97
for simply supported of [45/-45]4, α= 0.2, (s-s-s-s) woven fiber
composite plates
5.55
Variation of instability regions with temperature at 325K for simply
98
supported of a/b=1, α= 0.2, woven fiber composite plates
5.56
Variation of instability regions with moisture concentration at 0.25%
98
for simply supported of a/b=1, α= 0.2, woven fiber composite plates
5.57
Variation of instability regions with different temperature for simply
99
supported of [45/-45]4, woven fiber composite plates
5.58
Variation of instability regions with different moisture concentration
99
for simply supported of [45/-45]4, woven fiber composite plates
5.59
Variations of instability region with static load factor of composite
103
symmetric laminated shell subjected to elevated temperature
(Temp=325K, a/b=1, Ry/b=Rx/b=5, b/t=100)
5.60
Variations of instability region with temperature of composite
laminated symmetric cross-ply (0/90/90/0) curved panel (a/b=1,
b/t=100, Ry/b=5)
xvi
103
5.61
Variations of instability region with temperature of composite
104
Laminated anti-symmetric angle-ply (45/-45/45/-45) curved panel
(a/b=1, b/t=100, Ry/b=5)
5.62
Variations of instability region with moisture of composite
104
laminated symmetric cross-ply (0/90/90/0) shell (Ry/b=5, a/b=1,
b/t=100)
5.63
Variations of instability region with moisture of composite
105
laminated anti-symmetric angle-ply (45/-45/45/-45) shell (Ry/b=5,
a/b=1, b/t=100)
5.64
Variations of curvature of composite laminated symmetric crossply(0/90/90/0)
curved
panel
with
elevated
105
temperature
(Temp=325K, a/b=1, b/t=100)
5.65
Effect of aspect ratio on instability region of (0/90/90/0) laminate
106
for elevated temperature (Ry/b=5, Temp=325K, b/t=100)
5.66
Effect of aspect ratio on instability region of (45/-45/45/-45)
106
laminate for elevated temperature (Ry/b=5, Temp=325K, b/t=100)
5.67
Effect of aspect ratio on instability region of (0/90/90/0) laminate
107
for elevated moisture (Ry/b=5, Mois=0.001, b/t=100)
5.68
Effect of aspect ratio on instability region of (45/-45/45/-45)
107
laminate for elevated moisture (Ry/b=5, Mois=0.001, b/t=100)
5.69
Effect of degree of orthotropy on instability region of (0/90/90/0)
108
laminate for elevated temperature (Ry/b=5, Temp=325K, b/t=100,
a/b=1)
5.70
Effect of degree of orthotropy on instability region of (45/-45/45/-
108
45) laminate for elevated temperature (Ry/b=5, Temp=325K,
b/t=100, a/b=1)
5.71
Effect of degree of orthotropy on instability region of (0/90/90/0)
108
laminate for elevated moisture (Ry/b=5, Mois=0.001, b/t=100,
a/b=1)
5.72
Effect of degree of orthotropy on instability region of (45/-45/45/45) laminate for elevated moisture (Ry/b=5, Mois=0.001, b/t=100,
a/b=1)
xvii
109
5.73
Effect of thickness on instability region of (0/90/90/0) laminate for
110
elevated temperature (a/b=1, Temp=325K, Rx/b= Ry/b = 5)
5.74
Effect of thickness on instability region of (45/-45/45/-45) laminate
110
for elevated temperature (a/b=1,b/t=100,Temp=325K, Rx/b=Ry/b=5)
5.75
Effect of thickness on instability region of (0/90/90/0) laminate for
110
elevated moisture (a/b=1, b/t=100, Mois=0.001, Rx/b= Ry/b = 5)
5.76
Effect of thickness on instability region of (45/-45/45/-45) laminate
111
for elevated moisture (a/b=1, b/t=100, Mois=0.001, Rx/b= Ry/b = 5)
effect of shallowness ratio
5.77
Effect of Ry/b on instability region of (0/90/90/0) laminate for
111
elevated temperature(a/b=1,b/t=100,Temp=325K,Ry=Rx = 1.5,2.5,5)
5.78
Effect of Ry/b on instability region of (0/90/90/0) laminate for
112
elevated moisture (a/b=1, b/t=100, Mois=0.1%, Ry=Rx = 1.5, 2.5, 5)
5.79
Effect of Ry/b on instability region of (45/-45/45/-45) laminate for
112
elevated temperature (a/b=1, b/t=100, Temp =325K, Ry=Rx=1.5,
2.5, 5)
5.80
Effect of Ry/b on instability region of (45/-45/45/-45) laminate for
113
elevated moisture (a/b=1, b/t=100, Mois=0.1%, Ry=Rx= 1.5, 2.5, 5)
5.81
Effect of different play orientation on instability region of anti-
113
symmetric angle-ply laminate for elevated temperature (a/b=1,
b/t=100, Temp=325K, Ry/b = 5)
5.82
Effect of different play orientation on instability region of antisymmetric angle-ply laminate for elevated temperature (a/b=1,
b/t=100, moist=0.1%, Ry/b = 5)
xviii
114
List of Publications out of this Research Work
Papers in International Journals
1.
M. K. Rath and S. K. Sahu (2011): Vibration of Woven Fiber Laminated
Composite Plates in Hygrothermal Environment, Journal of Vibration and
Control, Vol. 18 (13), pp. 1957-1970.
2.
M. K. Rath and S. K. Sahu (2011): Static Behavior of Woven Fiber Laminated
Composites in Hygrothermal Environment, Journal of Reinforced plastics and
Composites, Vol.30 (1), pp.61-76
3.
S. K. Sahu, M. K. Rath and R. Sahoo (2012): Parametric Instability of
Laminated Composite Doubly Curved Shell Panels Subjected to Hygrothermal
Environment, Advanced Materials Research Journal, Vol. 383-390, pp. 32123216.
4.
M. K. Rath and S. K. Sahu, Experimental and numerical study on buckling
effects of woven fiber laminated composite plates in hygrothermal
environment, Journal of Structural Engineering and Mechanics, under review
5.
S.K.Sahu, M.K.Rath and R.Sahoo,(2012) Parametric Resonance Characteristics
of Woven Fiber Composite Curved Panels in Hygrothermal Environment,
International Journal of Aeronautical and Space Sciences, 13(3), pp.332-348.
6.
M. K. Rath and S. K. Sahu, Parametric Instability of Woven Fiber Laminated
Composite Plates in Hygrothermal Environment, International Journal of
Mechanical Sciences, under review.
7.
M. K. Rath and S. K. Sahu, Parametric Instability of Composite Plates in
Hygrothermal Environment, International Jounal of Computational Methods in
Engineering Science and Mechanics, under review.
8.
M. K. Rath and S. K. Sahu, Dynamic Instability of Woven Fiber Laminated
Composite Plates in Hygrothermal Environment, International Journal of
Theoretical and applied Mechanics, under review.
xix
Papers Presented in International Conferences
1.
S. K. Sahu, M. K. Rath and R. Sahoo (2011): Parametric Instability of
Laminated Composite Doubly Curved Shell Panels Subjected to Hygrothermal
Environment, the International Conference on Manufacturing Science and
Technology (ICMST 2011), Sep 16-18, 2011 at Singapore.
2.
M. K. Rath and S. K. Sahu: Dynamic Response of Woven Fiber Laminated
Composite Plates in Hygrothermal Environment, 5th International Conference
on Theoretical, Applied, Computational and Experimental Mechanics
(ICTACEM 2010), Dec 27-29, 2010 at IIT, Kharagpur.
3.
M. K. Rath and S. K. Sahu: Buckling effects of composite plates in
hygrothermal Environment, 5th International Conference on Advances in
Mechanical Engineering (ICAME-2011), June 4-6, 2011 at SVNIT, Surat.
4.
M. K. Rath and S. K. Sahu: Dynamic Stability of Woven Fiber Composite
Plates
in
Hygrothermal
Environment,
International
Conference
on
Computational Methods in Manufacturing (ICCMM2011), Dec 15-16, 2011 at
IIT Guwahati.
5.
M. K. Rath and S. K. Sahu: Parametric Instability of Square Laminated Plates
in Hygrothermal Environment, Fourth International Conference on Structural
Stability and Dynamics (ICSSD2012), Jan 4-6, 2012 at MNIT, Jaipur.
xx
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
Plate dimensions along x and y axes, respectively
Aij , Bij , Dij Sij
Extensional, bending-stretching coupling, bending stiffnesses
and transverse shear stiffnesses
C, C0
Elevated and reference moisture concentrations
E11,E22
Young’s moduli of lamina in both 1 and 2 directions respectively
G12 , G13 , G23
Shear moduli of lamina with respect to 1, 2 and 3 axes
h
Thickness of plate
[K]
Elastic stiffness matrix
[Krg]
Geometric stiffness matrix due to hygrothermal load
[Kg]
Geometric stiffness matrix due to in-plane load
[M]
Mass matrix
Mx , My , Mxy
Internal moment resultants.
N
N
M x , M y , M xy
N
Non-mechanical moment resultants due to moisture and
temperature.
n
Number of layers of laminated composite plates
xxi
Nx , Ny , Nxy
N
N
N x , N y , N xy
In-plane internal stress resultants.
N
In-plane non-mechanical stress resultants due to moisture and
temperature.
Ni
Shape function at a node i
N a x , N a y , N a xy
Applied in-plane forces per unit length
N n x , N n y , N n xy
Applied in-plane forces per unit length
Ns
The static portion of load N (t)
Nt
The amplitude of the dynamic portion of N (t)
Qx , Qy
Transverse shear resultants.
Rx, Ry, Rxy
Radii of curvature of shell
T, TO
Elevated and reference moisture concentration
u, v, w
Displacement components in the x, y, z directions
Greek
α, β
Static and dynamic load factors
α1,α 2
Thermal coefficients along 1 and 2 axes of a lamina, respectively
β1, β2
Moisture coefficients along 1 and 2 axes of a lamina, respectively
x , y ,γxy
In-plane strains of the mid-plane.
ε xN ,ε yN ,ε xyN
Non-mechanical strains due to moisture and temperature
Kx , KY , Kxy
Curvature of the plate
θ
Fiber orientation in a lamina
xxii
κ
Shear correction factor
θx ,θy
Rotations of the plate about x and y axes
υ 12 , ν
21
Poisson’s ratios
ρ
Mass density
(ρ)k
Mass density of kth layer from mid-plane.
ξ ,η
Natural co-ordinates of an element
φ x , φy
Shear rotations in x-z and y-z planes, respectively
λ
Critical loads
ωn
Natural frequency
Ω
Excitation frequency
ε
Total strain
εl
Linear strain
εnl
Non-linear strain
Mathematical Operators
[ ]-1
Inverse of the matrix
[ ]T
Transpose of the matrix
∂ ∂
,
∂x ∂y
Partial derivatives with respect to x and y
Abbreviations
FSDT
First order shear deformation theory
DIR
Dynamic instability region
xxiii
CHAPTER 1
INTRODUCTION
1.1: Introduction
Composite materials are being increasingly used in aerospace, civil, naval and other
high-performance engineering applications due to their light weight, high specific
strength, high specific stiffness and low specific density, which reduce the overall
operational cost. Besides military aircraft like the B-2 bomber, Nighthawk F117-A
fighter, recent advancements in composites in the commercial aircraft sectors
including Boeing 787 and Airbus 350/380, all-composite empennages on the Boeing
7J7 and McDonnell Douglas MD-91X, is to limit sonic fatigue caused by the new fuel
efficient propfan or unducted fan (UDF) engines. Structures used in the above fields
are more often exposed to high temperature as well as moisture. The varying
environmental conditions due to moisture absorption and temperature seem to have an
adverse effect on the stiffness and strength of the structural composites. This wide
range of practical applications demands a fundamental understanding of their
vibrations, static and dynamic stability characteristics under hygrothermal conditions.
1.2: Importance of the present structural stability study
Structural elements under in-plane periodic forces may undergo unstable transverse
vibrations, leading to parametric resonance, due to certain combination of the values
of in-plane load parameters and natural frequency of transverse vibration. This
instability may occur below the critical load of the structure under compressive loads
over a range or ranges of excitation frequencies. Several means of combating
parametric resonance such as damping and vibration isolation may be inadequate and
sometimes dangerous with reverse results. In contrast to the principal resonance, the
parametric instability may arise not merely at a single excitation frequency but even
for small excitation amplitudes and combination of frequencies.
The distinction
between good and bad vibration regimes of a structure, subjected to in-plane periodic
loading can be distinguished through an analysis of dynamic instability region (DIR)
spectra. The calculation of these spectra is often provided in terms of natural
1
frequencies and the static buckling loads. So, the calculation of these parameters with
high precision is an integral part of dynamic instability analysis of composite plates
and shells in hygrothermal environment.
A comprehensive analysis of the vibration and buckling effects of plates and shells
has been studied exhaustively without considering hygrothermal effects. The vibration
and buckling aspects of laminated composite panels in hygrothermal environment,
which have increasing applications in recent years. The subject on dynamic instability
of laminated curved panels in hygrothermal environment has attracted the attention of
researchers since it is hitherto not well studied. It is clear from the above discussion
that the process of investigating the different aspects of vibration and stability studies
on woven fiber laminated composite plates structures in hygrothermal environment
are current problem of interest. All these advancements and design requirements place
a premium on an in-depth understanding of the response characteristics of such
structural components. A thorough review of earlier works done in this area becomes
essential to arrive at the objective and scope of the present investigation. The detailed
review of literature along with critical discussions is presented in the next chapter.
2
CHAPTER 2
REVIEW OF LITERATURE
2.1: Introduction
The vast uses of conventional metals, its alloys and the ever increasing demand of
composite materials in plates and shells are the subject of research for many years.
Though the investigations is mainly focused on dynamic instability analysis of flat
and curved panels in hygrothermal environment, some relevant research works on
free vibration, buckling and dynamic instability analysis of flat and curved panels
subjected to hygrothermal loadings are also considered for the sake of its relevance
with the present investigation. Some of the pertinent studies done recently are
reviewed elaborately and critically discussed to identify the lacunae in the existing
literature. The literature reviewed in this chapter is grouped into three major aspects
of dynamics as follows:
In each section, the various aspects of analysis covered are:
•
Vibration of laminated composite panels in hygrothermal environment
•
Buckling effects of laminated composite panels in hygrothermal environment
•
Dynamic instability of laminated composite panels in hygrothermal
environment
In each section, the various structural components covered are:
•
Plates
•
Curved panels
However, some studies on static/bending analysis of composites in hygrothermal
environment are studied before dynamic analysis for relevance and completeness.
2.1.1 Static behavior of composites in hygrothermal environment
High-Performance composite materials have received increasing consideration for
structural applications because of their low density, high strength, and high stiffness.
Their superior strength and stiffness properties, however, are often compromised by
3
the environment to which they are exposed. A general discussion of the affect of
environment on the structural behavior of composite materials was presented by many
researchers. Among the environmental factors, those which induce expansional strains
(volume change in the absence of surface tractions) are of particular concern. In the
case of advanced composite structures, such phenomena are primarily caused by an
increase in temperature, absorption by a polymeric matrix material of a swelling agent
such as water vapor, and by sudden expansion of absorbed gases in the matrix. In
addition to inducing residual stresses, expansional strains can affect the gross
response characteristics of a composite structure. In particular, bending deflections,
buckling loads and vibration frequencies can be considerably modified by the
presence of environmentally induced strains. Thus, if composite materials are to reach
their full potential, it will be necessary to consider such environmental factors as
temperature and humidity in structural analysis and design. Hence the changes in
static characteristics due to the hygrothermal effects seem to be an important
consideration in composite analysis and design, which are of practical interest.
An extensive review of earlier works on composite mechanics and its sophistication is
presented by Chamis [1989]. Shen and Springer [1976] performed tests on moisture
absorption and desorption of composite materials. A series of test using unidirectional
composites were performed in the temperature 300-425 K with the material
submerged both in moist air and in water. Ishai and Arnon [1977] studied the effects
of hygrothermal history on residual strength of glass fiber reinforced plastic
laminates. Structural glass fiber/epoxy laminates were exposed to various periods of
immersion in water at room and elevated temperatures and tested after oven drying.
Aditya and Sinha [1992] investigated on determination of diffusion coefficients for
the continuous hygrothermal exposure of laminates of different material combination.
Harding and Li [1992] predicted the interlaminar shear strength for glass/epoxy and
carbon/epoxy laminates at impact rate of strain by employing a new technique with
the help of a double-lap shear specimen, where failure occurs on predetermined
interfaces. Govindarajan et al. [1993] examined the reason of low interlaminar
strength and the consequent possibility of interlaminar failure in composite laminates.
Melvin et al. [1993] investigated the thermoplastics response to deformation of
carbon fiber/epoxy-resin composite laminates has been considered theoretically, and
compared with experimental results. It was found that the surface temperature is
4
strongly dependent on the near-surface lay-up. Harding and Dong [1994] studied the
effect of strain rate on the interlaminar shear strength of carbon-fiber-reinforced
laminates. Experimental results were obtained at quasi-static and an impact rate of
loading for the interlaminar shear strength parallel to the fibers in a unidirectional
laminates in which the fiber orientation is 0/90 and ±45. Selzer and Friedrich [1997]
examined the mechanical properties and failure behavior of carbon fibre-reinforced
polymer composites under the influence of moisture.
Shibasaki and Somiya [1999] investigated the time dependence of degradation
phenomena of plain woven AFRP (Aramid fiber reinforced plastic) in hot, wet
environmental exposure. Naik et al. [2002] presented interlaminar fracture
characterization for plain weave fabric E-glass/epoxy laminates. The double
cantilever beam test and the end notch flexure test has been used for loading the
specimen. Patel and Case [2002] examined the effects of hygrothermal ageing on the
durability of a graphite/epoxy woven composite material system. Experimental testing
showed that the initial and residual tensile properties of the aged material were
virtually unaffected by the imposed environmental aging. Baley et al. [2004]
investigated the relationship between Glass fiber/Polymer interfaces and interlaminar
properties of marine composites. Karbhari [2004] studied e-glass/vinyl ester
composites in aqueous environments. In this experiment aqueous immersion of eglass composites, fabricated by the resin infusion process (unidirectional and
bidirectional) were tested to assess the effect of temperature on the short beam shear
strength. Ray [2006] conducted mechanical tests on unidirectional carbon composite
laminates and woven fiber composite beam specimens at room temperature to assess
the environmentally induced damage in composites.
Botelho et al. [2006] studied the hygrothermal effects on the interlaminar shear
properties of carbon fiber/epoxy composites. The interlaminar shear strength was
measured by using short beam shear test and Iosiepescu shear strength. Lua et al.
[2006] investigated multi-scale dynamic failure prediction tool for marine composite
structures. A multi-scale computational framework was established to bridge the
response and failure prediction at constituent, ply, and laminated composite level.
Zenasni and Bachir [2006] examined the effect of hygro-thermo-mechanical aging on
the interlaminar fracture behavior of woven fabric fiber composite materials. Hygrothermo-mechanical ageing characterization was carried out on double cantilever beam
5
and notched flexural interlaminar fracture tests in order to determine the loss in crack
propagation resistance. Chan et al. [2007] proposed an inverse parameter
identification technique to determine the elastic interlaminar shear modulus of
composite laminates. The technique involved minimizing the difference between an
experimentally measured and a numerically determined material response by varying
the interlaminar shear modulus in the numerical model. Gigliotti et al. [2007]
presented the aspects of modeling and the simulation of hygrothermal deformation of
composite laminates. The investigation showed the ability of the model to handle
complex environmental loading, close to service condition. Sereir and Boualem
[2007] investigated the damage of hybrid composites under long term hygrothermal
loading and stacking sequence. Fu et al. [2008] studied analysis of inter-laminar
stresses for composite laminated plate with interfacial damage with a constitutive
model. Bergeret et al. [1989] examined the influence of the fiber/matrix interface on
ageing mechanism of glass fiber reinforced thermoplastic composites in a
hygrothermal environment. A study of the properties of short glass fiber reinforced
thermoplastic composites based on poly (ethylene terephthalate), poly (butylenes
terephthalate) and polyamide-6, 6 in an aggressive environment was reported. Pilli et
al. [2009] presented the influence of moisture on mechanical behavior and long- term
durability of composites. An accelerated humidity test technique was developed
where moisture ingression was obtained by increasing the pressure in the test
chamber. Tsai et al. [2009] investigated the absorption and diffusion of water in
carbon fiber/glass fiber hybrid composites. Water absorption experiments, mechanical
property test and dynamic mechanical analysis were performed after immersion in
water at different temperatures for up to 32 weeks. The behavior of structures
subjected to in-plane loads with hygrothermal load is less understood in comparison
with structures under ambient temperature. The above studies deal with static analysis
of composite plates subjected to hygrothermal loadings. The subsequent literature on
composite panels subjected to hygrothermal loadings are classified as vibration,
buckling and dynamic stability of composite panels in hygrothermal environment.
2.1.2 Vibration of panels in hygrothermal environment
Due to its significance in structural mechanics, large number of references in the
published literature deal with vibration behavior of panels subjected to in-plane
6
stresses. Exact solutions for panels are available only for free vibration under certain
uniform loading conditions under classical boundary conditions. An attempt to have
an numerical and experimental solution of the of the free vibration in hygrothermal
environment is an important task.
2.1.2.1 Plates
The previous studies on the bending, vibration and buckling of moderately thick
plates of different support conditions associated with elevated temperature is reviewed
by Tauchert [1991] through 1991. Whitney and Ashton [1971] studied the effect of
expansional strains on the elastic response of layered composite plates using a
generalized Duhamel-Newmann form of Hooke’s law. Numerical results indicate that
the expansional strains can substantially affect the gross response characteristics of a
composite material. Dhotarad and Ganesan [1978] examined the influence of thermal
gradient on natural frequency of rectangular plate vibration using finite difference
method and finite element method. Gandhi et al. [1988] investigated the nonlinear
vibration of moderately thick laminated composite plates in hygrothermal
environments. The shear deformable plate theory is modified to account for midplane
stretching due to large deflections and dimensional changes in hygrothermal
environment. Chen and Lee [1988] presented the thermally induced vibrations of a
simply supported orthotropic rectangular plate using differential equation. Chen and
Chen [1989] studied the free vibration of the laminated rectangular composite plate
exposed to steady state hygrothermal environment using finite element method.
Constantinos and Dimitri [1990] examined the effect of elevated temperatures,
absorbed moisture, and random external excitation on the dynamic response and
structure-borne noise transmission of discretely stiffened flat plates from laminated
composite material using analytical approach.
Sai Ram and Sinha [1992] investigated the effects of moisture and temperature on the
free vibration of laminated composite plates using finite element method. Noor and
Burton [1992] presented analytically the three –dimensional solutions for the free
vibrations and buckling of thermally stressed multilayered angle-ply composite plates.
Adams and Singh [1995] investigated on the dynamic properties of fiber-reinforced
epoxy composites by immersion in sea water. Liu and Huang [1995] studied the free
vibration analysis of laminated composite plates subjected to temperature changes
7
using finite element method to calculate the frequencies of vibration of symmetric
cross-ply plates. Eslami and Maerz [1995] investigated the vibration of a symmetric
cross-ply plate under unsteady temperature and moisture environment using finite
element method. Chen and Chou [1999] presented the free vibration analysis of
orthogonal-woven fabric composites analytically using one-dimensional elastodynamic analysis. Patel et al. [2003] studied static and dynamic characteristics of
thick composite laminates exposed to hygrothermal environment using a higher-order
finite element method. Rao and Sinha [2003] investigated the effects of temperature
and moisture on the free vibration and transient response of multidirectional
composites using three dimensional finite element analysis. Shen et al. [2004]
discussed in detail the effects of hygrothermal conditions on the dynamic response of
shear deformable laminated plates resting on elastic foundations using a micro-tomicromechanical analytical model. Huang et al. [2004] investigated the nonlinear
vibration and dynamic response of shear deformable laminated plates in hygrothermal
environments based on higher-order shear deformation plate theory and general Von
Karman-type equation of motion.
Young-Wann [2005] examined the vibration characteristics of initially stressed
functionally graded rectangular metal and ceramic plates in thermal environment
using Rayleigh Ritz method to obtain the frequency equation. Matsunaga [2007]
studied the free vibration and stability problems of angle-ply laminated composite and
sandwich plates subjected to thermal loading using the method of power series
expansion. Atas and Samna [2008] presented an overall view on impact response of
woven fabric composite plates. A number of tests were performed to examine the
damage process step by step from initiation of damage to final perforation.
Jeyaraj et al. [2009] described the vibration and acoustic response characteristics of a
fiber-reinforced composite plate in a thermal environment by considering the inherent
material damping property of the composite material using finite element method. Lal
and Singh [2010] investigated the stochastic free vibration of laminated composite
plates subjected to thermal loading with general boundary conditions, taking into
account the random material properties and thermal expansion coefficients using
finite element method. Gupta et al. [2010] studied the thermal gradient effect on
vibration of non-homogeneous rectangular plate having bi-direction thickness
variation using Rayleigh Ritz method to evaluate the fundamental frequencies.
8
Fakhari and Ohadi (2011) examined the large amplitude vibration of functional
graded material (FGM) plates under thermal gradient and transverse mechanical loads
using finite element method. Gupta and Sharma (2011) investigated the effect of
linear thermal gradient on vibrations of trapezoidal plates whose thickness varies
parabolically using the Raleigh Ritz Technique.
Most of the above studies deal with numerical analysis of vibration behavior of
unidirectional composite laminates subjected to hygrothermal conditions. But the
experimental studies on the subject are scarce in literature. Anderson and Nayfeh
(1996) determined the natural frequencies and mode shapes of laminated composite
plates using experimental modal analysis and finite element method. Strait et al.
[1992] reported experimentally the effect of seawater immersion on the impact
resistance of glass fiber reinforced composite materials. The results indicate that
moisture induced degradation can significantly reduce the impact resistance of glass
fiber reinforced epoxy composites. Naik et al. [2000] investigated the static behavior
of industry driven woven fabric laminated composite plates under transverse central
low-velocity point impact by using a modified Hertz law and a 3D transient finiteelement analysis. Chakraborty et al. [2000] presented a combined experimental and
numerical study of the free vibration of composite FRP plates to determine the
respective frequency response functions from which the modal parameters are
extracted using finite element method. Chaudhuri et al. [2005] presented a combined
theoretical and experimental investigation on free vibration of thin anisotropic fiber
reinforced plastic rectangular plates. Numerical results presented here pertain to the
resonant frequencies of five layer symmetric cross-ply plates with all edges clamped
and simply supported, which are, in turn compared with the corresponding
experimental results. Botelho et al. [2005] obtained experimentally the viscoelastic
properties, such as storage modulus and loss modulus of glass/epoxy composites
during hygrothermal conditioning. Zai et al. [2009] measured experimentally the
damping and dynamic stiffness of carbon/epoxy composite beam specimens with a
focus on the effect of moisture absorption.
2.1.2.2 Shells
The widespread use of shell structures in aerospace and hydrospace applications has
stimulated many researchers to study various aspects of their structural behavior. In
9
the present study an attempt is made to the reviews on shells in the context of the
present work and discussions are limited to vibration and stability.
Huang and Tauchert [1991] investigated the large deformation behavior of antisymmetric angle-ply laminates under non-uniform temperature loading. A finite
element procedure for the geometrically nonlinear analysis of linear viscoelastic
laminated composite systems subjected to mechanical and hygrothermal load was
presented by Marques and Creus [1994]. The formulation was implemented by
considering three-dimensional degenerated shell element. The vibration response of
flat and curved panels subjected to thermal and mechanical loads are presented by
Librescu and Lin [1996]. Gandhi et al. [1998] studied nonlinear vibration of
laminated composite plates in hygrothermal environments. In their analysis the
formulations were based on the first-order shear deformable plate theory (FSDPT)
and the numerical results were only for free vibration of a cantilevered laminated
composite beam. Parhi et al. [2001] investigated the effect of moisture and
temperature on the dynamic behavior of composite laminated plates and shells with or
without delaminations. The dynamic analysis of laminated cross-ply composite noncircular thick cylindrical shells subjected to thermal/mechanical load are carried out
based on higher-order theory was studied by Ganapathi et al. [2002]. The nonlinear
free vibration behavior of laminated composite shells subjected to hygrothermal
environment was investigated by Naidu and Sinha [2006]. The geometrically nonlinear vibrations of linear elastic composite laminated shallow shells under the
simultaneous action of thermal fields and mechanical excitations are analysed by
Ribeiro and Jansen [2008]. The vibration characteristics of pre- and post-buckled
hygro-thermo-elastic laminated composite doubly curved shells were investigated by
Kundu and Han [2009]. Panda and Singh [2011] studied the nonlinear free vibration
behaviour of single/doubly curved shell panel is addressed within the post-buckled
state where thermal post-buckling of shell panel is accounted for a uniform
temperature field.
2.1.3: Buckling effects of composite panels in hygrothermal
environment
The buckling of mechanical, civil engineering structures under compressive loading
has always been a important field of research with the introduction of steel a century
10
ago. More recently, there is a renewed interest with development of aviation and an
aerospace program during the 1960’s which is still expanding to offshore and nuclear
engineering.
2.1.3.1 Plates
Plenty of studies are available on buckling behavior of composite plates under
ambient temperature and moisture conditions and reviewed by Leissa [1987].
Tauchert [1991] reviewed the previous works on buckling and post buckling
characteristics associated with elevated temperatures of thin and moderately thick
plates having various plan forms and support conditions through 1991. Flags and
Vinson [1978] proposed the hygrothermal effects on the buckling of laminated
composite plates based on the Theorem of Minimum potential Energy. Chen and
Chen [1987] observed the thermal buckling of laminated composite plate subjected to
a temperature change using Galerkin’s method.
Thagaratnam et al. [1989] studied the buckling analysis of composite laminates for
critical temperature, based on linear theory and the finite element method using
semiloof elements. Chen et al. [1991] analysed the thermal buckling of antisymmetric cross-ply laminates having central circular holes subjected to uniform or
non-uniform temperature distribution using finite element method. Sai Ram and Sinha
[1992] investigated the effects of moisture and temperature on the buckling of
laminated composite plates using finite element method. Sai Ram and Sinha [1992]
studied the vibration and buckling of laminated plates with a cutout in hygrothermal
environment using the superposition method. Noor and Burton [1992] presented
analytically the three-dimensional solutions for the free vibrations and buckling of
thermally stressed multilayered angle-ply composite plates. Prabhu and Dhanraj
[1993] observed the thermal buckling analysis of symmetric cross-ply and symmetric
angle-ply laminates using the finite element method. Chao and Shyu [1996]
investigated the nonlinear buckling behavior of fiber-reinforced laminated composite
plates under hygrothermal environment using finite element method. Loughlan [1999]
studied the influence of bend-twist coupling on the shear buckling response of thin
laminated composite plates using finite strip method.
Thompson and Loughlan [2000] examined the control of the post buckling response
in thin composite plates using smart technology. Babu and Kant [2000] proposed with
11
a refined higher order finite element models for thermal buckling of laminated
composite and sandwich plates. Spallino and Thierauf [2000] presented the thermal
buckling optimization of laminated composite plates subjected to rise in temperature
using evolution strategies and a guided random-search method. Shen [2000] examined
the influence of hygrothermal effects on the postbuckling of shear deformable
laminated plates subjected to uniaxial compression using a micro-to-macromechanical analytical model of a laminate. Zenkour and EL-Sheikh [2001] presented
the buckling of anisotropic elastic plates analytically using simple and mixed shear
deformation theories for various boundary conditions.
Patel et al. [2003] studied the static and dynamic characteristics of thick composite
laminates exposed to hygrothermal environment using a higher-order finite element
method. Xiao and Chen [2005] investigated the dynamic and buckling analysis of a
thin elastic-plastic square plate in uniform temperature field using Hamilton’s
variational principle. Jones [2005] examined the thermal buckling of uniformly heated
unidirectional and symmetric cross-ply laminated fiber-reinforced composites
uniaxial in-plane restrained simply supported rectangular plates. Shariyat [2007]
examined the thermal buckling analysis of rectangular composite multilayered plates
under uniform temperature rise using a layerwise plate theory and von Karman straindisplacement equations. Matsunaga [2007] studied the free vibration and stability
problems of angle-ply laminated composite and sandwich plates subjected to thermal
loading using the method of power series expansion. Singh and Verma [2008]
investigated the hygrothermal effects on the buckling of laminated composite plates
with random geometric and material properties using finite element method. Kumar
and Singh [2008] presented the thermal buckling analysis of laminated composite
plates subjected to uniform temperature distribution using finite element method. Lal
et al. [2009] examined the effects of random system properties on thermal buckling
load of laminated composite plates under uniform temperature rise using finite
element method. Lal and Singh [2010] presented the effect uncertain system
properties thermo-elastic stability of laminated composite plates under nonuniform
temperature distribution using a C0 finite element method. Pandey et al. [2010]
observed the effects of moisture and temperature on the post buckling response of a
laminated composite plate subjected to hygro-thermo-mechanical loadings using finite
double Chebyshev series. Dash et al. [2011] presented an experimental study on the
12
effects of corrosion on elastic buckling and post buckling response of unidirectional
E-glass/epoxy composite rectangular plates subjected to compressive load and liquid
environment exposure.
2.1.3.2 Shells
The studies involving stability of composite shell under hygrothermal loads are much
less in literature. The finite element method was applied to study the problem of
moisture and temperature effects on the stability of a general orthotropic cylindrical
composite shell panels subjected to axial or in-plane shear loading by Lee and Yen
[1989]. Shen [1990] investigated the influence of hygrothermal effects on the
buckling and postbuckling of composite laminated cylindrical shells subjected to axial
compression using a micro-to-macro mechanical analytical model. The effect of
hygrothermal conditions on the post buckling of shear deformable laminated
cylindrical shells subjected to combined loading of axial compression and external
pressure was investigated using micro-to-macro mechanical analytical model by Shen
[2001]. The hygrothermoelastic buckling behaviour of laminated composite shells
were numerically simulated using geometrically nonlinear finite element method was
studiedby Kundu and Han [2009]. The effect of random system properties on the post
buckling load of geometrically nonlinear laminated composite cylindrical shell panel
subjected to hygro-thermo-mechanical loading is investigated by Lal et al. [2011].
2.1.4: Dynamic stability of composite panels in hygrothermal
Environment
The study of dynamic stability is an important class of problem in structural
mechanics and the first observation of parametric resonance or dynamic instability is
attributed to Faraday in 1831.
2.1.4.1 Plates
The behavior of composite panels subjected to in-plane periodic loads is much less
understood. Few literatures are available on dynamic instability of unidirectional
composite plates under ambient temperature and moisture subjected to periodic inplane loads. Bert and Birman [1987] studied the dynamic instability of shear
deformable anti-symmetric angle-ply plates by using finite element method.
Srinivasan and Chelapandi [1986] studied the dynamic stability of rectangular plates
13
due to periodic in-plane load by using finite strip method. Moorty et al. [1990]
presented the parametric instability of laminated composite shear deformable flat
panels subjected to in-plane edge loads using finite element method. Chen and Yang
[1990] investigated on the dynamic stability of laminated composite plates by
Galerkin finite element method. Zhou [1991] observed the theory of nonlinear
dynamic stability of composite laminated plates using Hamilton principle. Kwon
[1991] examined the dynamic instability of layered composite plates by finite element
method by using higher-order bending theory. Mond and Cederbaum [1992]
presented the dynamic stability of anti-symmetric laminated plates by using the
method of multiple scales. Liao and Cheng [1994] observed the dynamic instability of
stiffened laminated composite plates subjected to pulsating forces using finite element
equation of motion. Balmurugan et al. [1996] studied nonlinear dynamic instability of
laminated composite plates using finite element model.
Ganapathi [1998] proposed the dynamic instability of laminated composite plates
subjected to thermal loads using first order shear deformation theory and Lagrange’s
equation. Patel et al. [1999] investigated on the dynamic instability of laminated
composite plates supported on elastic foundations, subjected to periodic in-plane
loads, using C1 eight-nodded shear-flexible plate element. Chatopadhaya and Radu
[2000] studied the dynamic instability of composite laminates using a higher order
theory. The procedure implemented by using the finite element approach. Sahu and
Dutta [2000] investigated the dynamic instability of laminated composite rectangular
plates subjected to non-uniform harmonic in-plane edge loading using finite element
methods. Wang and Dawe [2002] proposed the dynamic instability of composite
laminated rectangular plates by using Lagrange’s formulation. Ravi Kumar et al.
[2003] examined the vibration and dynamic instability behavior of laminated
composite plates subjected to partially distribute non conservative follower forces.
Liew et al. [18] presented the dynamic analysis of laminated composite plates with
piezoelectric sensor/actuator patches using mesh free method.
Wu and Shih [2005] presented the dynamic stability of rectangular plates with an
edge crack by applying Galerkin’s method. Wu and Shih [2006] studied the dynamic
instability of arbitrarily laminated skew plates based on Von Karmans plate theory,
the large amplitude dynamic equation of thin laminated plates are derived by applying
double Fourier series. Chakrabarti and Sheikh [2006] studied the dynamic instability
14
of laminated sandwich plates subjected to in-plane partial edge loading by using finite
element method. Dey and Singh [2006] examined the dynamic stability characteristics
of simply supported laminated composite skew plates subjected to a periodic in-plane
load by using finite element approach. Saburcu and Evran [2006] observed the
dynamic stability of a rotating pre-twisted asymmetric cross-section Timoshenko
beam subjected to an axial periodic force. Patel et al. [2007] proposed the dynamic
instability analysis of stiffened shell panels subjected to partial edge loading along the
edges using Hill’s infinite determinant. Liew et al. [2007] investigated the dynamic
stability analysis of thin laminated cylindrical panels under static and periodic axial
forces by using the mesh-free Ritz method. Lanhe et al. [2007] studied the dynamic
stability analysis of FGM plates by the moving least squares differential quadrature
method. Udar and Datta [2007] investigated the combination of resonances in
parametrically excited simply supported laminated composite doubly curved panels
under uniform edge loading using finite element technique. Asha and Sahu [2008]
examined the dynamic stability of laminated composite pre-twisted cantilever panels
by using finite element method. Chen et al. [2009] studied the dynamic stability of
laminated hybrid composite plates subjected to periodic uniaxial and bending stress
and the instability region is marked by Bolotin’s method. Fazilati and Ovesy [2010]
presented the dynamic instability analysis of thin walled composite structures using
finite strip method. The effects of various parameters on the instability regions were
located using Bolotin’s approach. Biswas et al. [2011] studied the static and dynamic
instability characteristics of curved laminates with internal damage subjected to
follower loading using finite element approach
2.1.4.2 Shells
The parametric resonance characteristics of composite shells are studied by few
investigators without considering the hygrothermal effects. The dynamic stability or
phenomenon of parametric resonance in cylindrical shells under periodic loads has
attracted much attention due to its detrimental and de-stabilizing effects in many
engineering applications. This phenomenon in elastic systems was first studied by
Bolotin [1964], where the dynamic instability regions were determined. Yao [1965]
examined the non-linear elastic buckling and parametric excitation of a cylinder under
axial loads. The parametric instability of circular cylindrical shells was also discussed
by Vijayaraghavan and Evan-Iwanowskj [1967].
15
Based on the donnell’s shell equations, the dynamic stability of circular cylindrical
shells under both static and periodic compressive forces was examined by Nagai and
Yamaki [1988] using Hsu’s method. Bert and Birman [1990] extended Yao’s [1965]
approach to the parametric instability of thick orthotropic shells using higher-order
theory. Liao and Cheng [1994] proposed a finite element model with a 3-D
degenerated shell element and a 3-D degenerated curved beam element to investigate
the dynamic stability of stiffened isotropic and laminated composites plates and shells
subjected to in-plane periodic forces. Argento and Scott [1993] employed a
perturbation technique to study the dynamic stability of layered anisotropic circular
cylindrical shells under axial loading. Using the same method, Argento [1993] later
analyzed the dynamic stability of a composite circular cylindrical shell subjected to
combined axial and torsional loading. The study of the parametric instability behavior
of curved panels, the effects of curvature and aspect ratio on dynamic instability for a
uniformly loaded laminated composite thick cylindrical panel is studied by Ganapathi
et al. [1994] using finite element method.
The dynamic instability of laminated composite circular cylindrical shells is studied
by Ganapathi and Balamurugan [1998] using a C0 shear flexible two nodded
axisymmetric shell element. The effects of various parameters such as ply angle,
thickness, aspect ratio, axial and circumferential wave numbers on dynamic stability
are studied. The dynamic stability of thin cross-ply laminated composite cylindrical
shells under combined static and periodic axial force is investigated by NG et al.
[1998] using Love’s classical theory of thin shells. The effects of different lamination
scheme and the magnitude of the axial load on the instability regions are examined
using Bolotin’s method. Most of the above mentioned investigators studied the
dynamic stability of uniformly loaded closed cylindrical shells with a simply
supported boundary condition. The parametric instability of laminated composite
conical shells under periodic loads is studied of Ganapathi et al. [1999]. The
parametric resonance characteristics of laminated composite doubly curved panels
subjected to non-uniform loading was investigated by Sahu and Datta [2001]. The
dynamic stability behavior of laminated composite curved panels with cutouts
subjected to in-plane static and periodic compressive loads was studied by Sahu and
Dutta [2003]. A numerical technique is developed for the dynamic stability analysis
of composite laminated cylindrical shell under static and periodic axial forces by
16
mesh-free kp-Ritz method by Liew et al. [2006]. Quantitative results are presented to
show the effects of curvature, ply orientation, and degree of orthotropy, geometry and
number of layers of laminate on dynamic stability of composite plates for different
temperature and moisture concentrations.
The study of the dynamic instability of laminated composite plates and shells in
hygrothermal environment are relatively new and there were no references found in
this area.
2.1.5: Critical Discussion
On the whole, the focus of the research is changing from vibration to buckling effects
and then to dynamic stability. Recently more studies are conducted on woven fiber
composites than unidirectional composite materials. The structural configuration has
shifted from one dimensional beams to plates and shells. As regards to the
methodology, the focus is shifted from numerical method to experimental method for
free vibration and buckling effects in hygrothermal environment. More studies are
made for dynamic stability of composite plates and shells using finite element
methods in hygrothermal environment.
The study reveals that investigators are now concentrating on analysis of complicated
aspects of different parameters including hygrothermal conditions of plates and shells.
From the above review of literature, the inherent lacunae of earlier investigations
which need further attention of future researchers are summarized below.
2.1.5.1: Vibration of composite panels in hygrothermal environment
The literature on free vibration of laminated composite plates under ambient
temperature and moisture conditions is vast. However, the studies involving vibration
of woven fiber laminated plates subjected to hygrothermal conditions are less in
literature.
The previous studies on the bending, vibration and buckling of moderately thick
plates of different support conditions associated with elevated temperature is reviewed
by Tauchert [1991]. The free vibration of laminated composite plates in hygrothermal
environment by using finite element method has also been studied by (Chen and Chen
[1989], Sai Ram and Sinha [1992]) analysis lead to analytical method. Most of the
17
above studies deal with numerical analysis of vibration behavior of unidirectional
composite laminates subjected to hygrothermal conditions. However, experimental
studies on the free vibration are scarce in literature. Anderson and Nayfeh [1996]
determined the natural frequencies and mode shapes of laminated composite plates by
experimental modal analysis using finite element method. A combined experimental
and numerical study on the free vibration of composite plates to determine the
respective frequency response functions from which the modal parameters are
extracted using finite element method by (Chakraborty et al. [2000], Chaudhury et. al.
[2005]) without hygrothermal conditions.
The analysis of shell structures has a long history starting with the membrane theory
and then the bending theories. The nonlinear free vibration of composite shells in
hygrothermal environment has been studied by (Marques and Creus [1994], Librescu
and Lin [1996], Parhi et al. [2001], Naidu and Sinha [2006], Ribeiro and Jansen
[2008]). Nonlinear free vibration behaviour of single/doubly curved shell panel in
post-buckled state was investigated by (Kundu and Han [2009], Panda and Singh
[2011]). Most of the above studies deal with numerical analysis of free vibration
behaviour of unidirectional composite shells under ambient temperature and moisture
conditions. However, studies on the free vibration behavior of composite shells
subjected to hygrothermal conditions are scarce in literature for which this subject
research is for practical interest.
2.1.5.2: Buckling effects of composite panels in hygrothermal
environment
Plenty of studies are available on buckling behavior of composite plates under
ambient temperature and moisture conditions reviewed the previous works on
buckling and post buckling characteristics associated with elevated temperatures of
thin and moderately thick plates having various plan forms and support conditions by
(Leissa [1987], Tauchert [1991]). Many investigators worked on buckling analysis of
laminated composite plates in thermal environment using different methods by (Chen
and Chen [1987], Chen et al. [1991], Spallino and Thierauf [2000], Xiao and Chen
[2005], Jones [2005], Shariyat [2007]). Sai Ram and Sinha [1992] investigated the
effects of moisture and temperature on the buckling of laminated composite plates
using finite element method analytically. Studies were also conducted for composite
18
plates in thermal environment using finite element method by (Prabhu and Dhanraj
[1993], Singh and Verma [2008], Lal and Singh [2010]). Hygrothermal effects on the
buckling of laminated composite plates based on different methods have been
proposed by (Flags and Vinson [1978], Shen [2000], Pandey et al. [2010]). Most of
the above studies deal with numerical analysis of buckling effects of unidirectional
composite plates subjected to hygrothermal conditions. However, experimental
studies on the buckling effects are scarce in literature. So, the buckling behavior of
woven fiber laminated composite plates subjected to hygrothermal environments are
of tremendous technical importance for understanding the behavior of laminated
composite plates subjected in-plane loads.
Most of the above studies deal with numerical analysis of buckling effects of
unidirectional composite shells subjected to hygrothermal conditions. The recent
investigations are based on shear deformable cylindrical and doubly curved shells or
panels. The study on buckling effects of doubly curved shells subjected to in-plane
stresses in hygrothermal environment is new. Hygrothermal buckling and post
buckling of composite was investigated by (Shen [1990], Shen [2001]) using microto-macro mechanical analytical model. The hygrothermoelastic buckling behaviour of
laminated composite shells were numerically simulated using geometrically nonlinear
finite element method was studied by Kundu and Han [2009]. The effect of random
system properties on the post buckling load of geometrically nonlinear laminated
composite cylindrical shell panel subjected to hygro-thermo-mechanical loading is
investigated by Lal et al. [2011]. However, studies on the buckling effects of woven
fiber composite shells in hygrothermal environment are scarce in literature for which
this subject needs scope for peer attention.
2.1.5.3: Dynamic stability of composite panels in hygrothermal
Environment
The studies on dynamic stability of structures are much less in comparison to static
stability and got a boost after Bolotin’s [1964] contribution to the literature. The
dynamic stability of shear deformable plates was studied by Moorty et al. [1990]
using finite element method. Mond and Cederbaum [1992] presented the dynamic
stability of anti-symmetric laminated plates by using the method of multiple scales.
The instability results of thin simply supported plates (Dey and Singh [2006]) plates
19
are sparsely treated in literature. The dynamic stability of composite plates has been
studied using FEM by (Liao and Cheng [1994], Chen and Yang [1990], Chakrabarti
and Sheikh [2006]). Bert and Birman [1987] studied the dynamic instability of shear
deformable anti-symmetric angle-ply plates by using FEM. Nonlinear dynamic
stability of composite plates using different principle wsa proposed by (Cheng-ti
[1991], Balmurugan et al. [1996]). (Kwon [1991], Chatopadhaya and Radu [2000])
examined the dynamic instability of composite plates by finite element method using
higher-order theory. Studies were also conducted for the dynamic stability of
rectangular plates due to periodic in-plane load by using finite strip method by
(Srinivasan and Chelapandi [1986], Fazilati and Ovesy [2010]). Ganapathi [1998]
proposed the dynamic instability of composite plates subjected to thermal loads using
first order shear deformation theory and Lagrange’s equation. Few literatures are
available on dynamic instability of unidirectional composite plates under ambient
temperature and moisture subjected to periodic in-plane loads Sahu and Dutta [2000]
using finite element methods. No studies are available on parametric instability of
industry driven woven fiber laminated composite plates in hygrothermal environment
and thus it becomes the subject of this investigation.
The parametric resonance characteristics of composite shells are studied by few
investigators without considering the hygrothermal effects. The parametric instability
of circular cylindrical shells was studied by (Yao [1965], Vijayaraghavan and EvanIwanowskj [1967]. Based on the donnell’s shell equations, the dynamic stability of
circular cylindrical shells under both static and periodic compressive forces was
examined by Nagai and Yamaki [1988]. The dynamic stability of layered anisotropic
circular cylindrical shells under axial loading using perturbation technique was
investigated by (Argento and Scott [1993], Argento [1993]).
The dynamic instability of a uniformly loaded laminated composite thick cylindrical
panel is studied by Ganapathi et al. [1994] using finite element method. The dynamic
stability of thin cross-ply laminated composite cylindrical shells under combined
static and periodic axial force was investigated by NG et al. [1998] using Love’s
classical theory of thin shells. However, studies on the dynamic stability of woven
fiber composite shells in hygrothermal environment are scarce in literature for which
this subject interest is necessary for investigation.
20
2.1.6: Aim and scope of the present study
A review of the literature shows that a lot of work has been done on the vibration and
buckling of laminated composite plates in hygrothermal environment, but no
experimental work is reported in literature on vibration and buckling of industry
driven woven fiber composite plates subjected to hygrothermal conditions. Besides
this, the dynamic instability of composite plates in ambient environment is only
reported. No work is reported in literature on parametric instability of laminated
composite plates subjected to hygrothermal environment. The present study is mainly
aimed at filling some of the lacunae that exist in the proper understanding of the
dynamic stability of laminated composite plates in hygrothermal environment and inplane periodic loads. Based on the review of literature, the different problems
identified for the present investigation are presented as follows
• Experimental and numerical study on vibration of woven fiber laminated
composite plates in hygrothermal environment
• Experimental and numerical study on buckling effects of woven fiber
laminated composite plates in hygrothermal environment
•
Parametric instability of laminated composite plates in hygrothermal
environment
•
Parametric resonance characteristics of laminated composite shells in
hygrothermal environment
Due to its practical importance and uniqueness in the above fields, the influences of
various parameters such as aspect ratio, side to thickness ratio, static and dynamic
load factors, ply orientations, lamination angle, orthotropic on the parametric
resonance characteristics of laminated composite plates under higher temperature and
moisture environments are examined in detail. However, a complete parametric study
on the static behavior especially the variation of interlaminar shear strength on the
loading speed, different proportion of fiber to matrices, type of matrices, exposure
time of composite specimens subjected to hygrothermal loading is also is studied for
completeness.
21
CHAPTER 3
MATHEMATICAL FORMULATION
3.1: The Basic Problems
The basic configuration of the problem considered here is a doubly curved panel with
in-plane harmonic loadings under hygrothermal environment as shown in figure
3.1.The choice of the doubly curved panel geometry as a basic configuration has been
made so that depending on the value of curvature parameter, plate, cylindrical panel
and different doubly curved panels such as spherical configurations can be considered
as special cases.
The mathematical formulation for vibration, buckling and dynamic stability behavior
of laminated composite plates and shells subjected to moisture and temperature are
presented. Consider a laminated plate of uniform thickness ‘t’ consisting of a number
of thin lamiae, each of which may be arbitrarily oriented at an angle ‘θ’ with reference
to the X-axis of the co-ordinate system as shown in Figures 3.2 and 3.3.
Figure 3.1: Laminated composite curved panels under in-plane harmonic loading
under hygrothermal environment
22
Figure 3.2: Arbitrarily oriented laminated plate
Figure 3.3: Geometry of an n-layered laminate
3.2: Proposed Analysis
The governing equations for the dynamic stability of laminated composite doubly
curved panels subjected to hygrothermal loading are developed. The presence of
hygrothermal loading in the panel induces a in-plane stress field in the structures. This
necessitates the determination of stress field as a prerequisite to the solution of the
problem like vibration, buckling and dynamic stability behavior of plates and shells
with different temperature and moisture. As the thickness of the structure is relatively
smaller, the determination of stress field reduces to the solution of a plane stress
problem. Due to the in-plane harmonic load the equation of motion represents second
order Mathieu-Hill type. The development of the regions of instability arises from
Floquet’s theory and the solution is obtaining using Bolotin’s approach using finite
element method (FEM). The governing differential equations have been developed
using first order shear deformation theory (FSDT). The assumptions made in this
analysis are summarized as follows:
23
3.2.1: Assumptions of the Analysis
1. The analysis is linear with a few exceptions. This implies both linear constitutive
relations (generalized Hooke’s law for the material and linear kinematics) and small
displacement to accommodate small deformation theory.
2. The curved panels are of various shapes with no initial imperfections. The
considerations of imperfections are less important for dynamic loading.
3. This theory can be considered to be an extension of Sander’s theory to doubly
curved panels, considering transverse shear and rotary inertia. The straight line that is
perpendicular to the neutral surface before deformation remains straight but not
normal after deformation (FSDT). The thickness of the shell is small compared with
the principal radii of curvature. Normal stress in z direction is neglected.
4. The loading considered is axial with a simple harmonic fluctuation with respect to
time.
5. All damping effects are neglected.
3.3: Governing Equations
The governing differential equations for vibration of a shear deformable laminated
composite plates and shells in general are specified here. The behavior of laminated
composite plates in hygrothermal environment derived on the basis of first order shear
deformation theory subjected to in-plane loads are;
∂N x ∂N xy Q x Q y
∂ 2θ x
∂ 2u
+
+
+
= P1 2 + P2
∂x
∂y
R x R xy
∂t
∂t 2
∂ 2θ y
Qx
∂ 2v
+
+
+
= P1 2 + P2
∂x
∂y
R y R xy
∂t
∂t 2
∂ N xy
∂N y
Qy
2
N xy
∂Q x ∂Q y N x N y
∂2w
∂2w
a ∂ w
a
+
−
−
−2
+ N xa
+
N
+
N
xy
y
∂x
∂y
Rx
Ry
R xy
∂x∂y
∂x 2
∂y 2
2
∂2w
∂2w
∂2w
n ∂ w
n
+ N xn
+
N
+
N
=
P
xy
y
1
∂x∂y
∂x 2
∂y 2
∂t 2
∂ 2u
∂M x ∂M xy
∂ 2θ x
+
− Q x = P3
+ P2
∂x
∂y
∂t 2
∂t 2
24
(3.3.1)
(3.3.2)
(3.3.3)
(3.3.4)
∂M xy
∂x
+
∂M y
∂y
( P1 , P2 , P3 ) =
− Q y = P3
n
∂ 2θ y
∂t 2
+ P2
∂ 2v
∂t 2
(3.3.5)
zk
∑ ∫ (ρ )
k =1 z k − 1
k
(3.3.6)
(1, z , z 2 ) dz
3.4: Dynamic stability studies
The equation of motion for vibration of a laminated composite panel in hygrothermal
environment, subjected to generalized in-plane load, may be expressed in the matrix
form as:
[ ]
[M ]{q&&} + [[K ] + K r g − N(t)[K g ]]{q} = 0
(3.4.1)
‘q’ is the vector of degrees of freedoms (u, v, w, θx, θy). The in-plane load ‘N (t)’ may
be harmonic and can be expressed in the form:
N (t ) = N
s
+ N t Cos Ω t
(3.4.2)
Where Ns the static portion of load N (t), Nt the amplitude of the dynamic portion of
N (t) and Ω is the frequency of the excitation. The stress distribution in the panel may
be periodic. Considering the static and dynamic component of load as a function of
the critical load,
Ns = αNcr , Nt = βNcr
(3.4.3)
Where α and β are the static and dynamic load factors respectively. Using Eq. (5), the
equation of motion for panel in hygrothermal environment under periodic loads in
matrix form may be obtained as:
[ ]
[M ]{q&&} + [[K ] + K r g − αNcr [K g ] − βNcr [K g ]CosΩt]{q} = 0
(3.4.4)
The above Eq. (3.4.4) represents a system of differential equations with periodic
coefficients of the Mathieu-Hill type. The development of regions of instability arises
from Floquet’s theory which establishes the existence of periodic solutions of periods
T and 2T. The boundaries of the primary instability regions with period 2T, where
25
T=2 π/Ω are of practical importance and the solution can be achieved in the form of
the trigonometric series:
∞
q(t ) =
∑[{a }Sin(kΩt / 2) + {b }Cos(kΩt / 2)]
k =1, 3, 5,..
k
(3.4.5)
k
Putting this in Eq. (3.4.4) and if only first term of the series is considered, equating
coefficients of Sin Ωt/2 and Cos Ωt/2, the equation (3.4.4) reduces to
∞
[M ] ∑
K =1,3,5
 KΩ 
−

 2 
[ ]
2
[ ]

KΩt
 KΩt  
r
 {a K }Sin(
) + {bK }Cos
  + [[K ] + K g
2
 2 

[ ]

 kΩt 
 KΩt  
− αPcr K g − βPcr K g CosΩt ] a K Sin
 + {bK } + Cos
  = 0
 2 
 2 

{ }
(3.4.6)
Equating the coefficients of the sine and cosine terms leads to a series of algebraic
{ }
equations for the vectors a
K
and {b}K for determination of instability regions. For
non-trival solution, the infinite determinants of the coefficients of the groups of
homogeneous equation are equal to zero. Approximate solution can be obtained by
truncating the determinants. Principal instability regions, which is of practical interest
corresponds to K=1 and for this instability conditions leads to in line with the Bolotin.
[ [K ] + [K r g
] − αP [K ] ± 12 βP [K ] − Ω4 [M ] ]{q} = 0
2
cr
g
cr
g
(3.4.7)
Eq. (3.4.7) represents an eigenvalue problem for known values of α, β and Pcr . The
two conditions under the plus and minus sign correspond to two boundaries (upper
and lower) of the dynamic instability region. The above eigenvalue solution give of
Ω, which give the boundary frequencies of the instability regions for the given values
of α and β. In this analysis, the computed static buckling load of the panel is
considered as the reference load. Before solving the above equations, the stiffness
matrix [K] is modified through imposition of boundary conditions. The equation
reduces to other problems as follows:
Free Vibration: α=0, β=0 and Ω=2ω
[ [K] + [K r g ] ] − ωn 2
[M ]{q } =
26
0
(3.4.8)
Buckling: α=1, β=0 and Ω=0
[ [K ] + [K r g ] ] − λ [K g ]{q } = 0
(3.4.9)
3.5: Constitutive Relations
The constitutive relations for the Composite plate and shell subjected to moisture and
temperature is given by:
{F} = [D]{ε} − {F N }
Where
(3.5.1)
{F} = {N x , N y , N xy , M x , M y , M xy , Qx , Qy }T
{F } = {N
{ε } = {ε , ε
N
N
x
x
 A11
A
 12
 A16

B
[D] =  11
B12

 B16
 0

 0
N
N
N
N
}
, N y , N xy , M x , M y , M xy ,0,0
, γ xy , K x , K y , K xy ,ϕ x ,ϕ y }
T
T
y
A12
A16
B11
B12
B16
0
A22
A26
B12
B22
B26
0
A26
A66
B16
B26
B66
0
B12
B16
D11
D12
D16
0
B22
B26
D12
D22
D26
0
B26
B66
D16
D26
D66
0
0
0
0
0
0
S 44
0
0
0
0
0
S 45
0 
0 
0 

0 
0 

0 
S 45 

S 55 
The non-mechanical force and moment resultants due to moisture and temperature are
expressed as follows.
{N
{M
N
x
N
x
N
, N y , N xy
N
, M y , M xy
} = ∑ (Q ){ε } (z
N T
n
K =1
}
N T
=
ij
k
k
( ) {ε } (z
1 n
∑ Qij
2 K =1
k
− z k −1 )
2
k
− z k −1
For i, j = 1, 2, 6
2
)
For i, j = 1, 2, 6
(3.5.2)
(3.5.3)
k
Where
{ε }N = {ε xN , ε yN , ε xyN }T = [T ]{β1β 2 }k T (C − CO ) + [T ]{α1α 2 }k T (T − TO )
in which
27
(3.5.4)
[T ] = Transformation matrix due to moisture and temperature and is given as
cos 2 θ
[T ] =  sin 2 θ
 sin 2θ

sin 2 θ 

cos 2 θ 
cos 2θ 
(3.5.5)
The stiffness coefficient is defined as
(A , B , D ) = ∑ ∫ [Q ] (1, z, z ) dz
n
ij
ij
ij
k
2
k =1
n
S ij = κ ∑ ∫
k =1
(i, j =1.2.6)
ij k
z k −1
[Q ]
k
ij k
zk −1
dz
(3.5.6)
(i, j =4,5)
(3.5.7)
A shear correction factor of 5/6 is included in Sij for all numerical computations in
line with previous studies [Wang and Dawe 2002]
(Q )
ij
k
in equations 11 and 12 is defined as:
k
[Q ]
k
−1
ij k
Where
[ ] [T ]
[Q ij ]k = [T1 ] Qij
cos θ
− sin θ
[Q ]
ij k
[Q ]
ij k
sin 2 θ
cos 2 θ
2 sin θ cosθ
sin θ cosθ 

− sin θ cosθ 
cos 2 θ − sin 2 θ 
(3.5.9)
(3.5.10)
sin θ 
cos θ 
Q11 Q12 0 
= Q12 Q22 0 
 0
0 Q66 
Q
=  44
 0
(3.5.8)
( i, j = 4, 5)
2
 cos 2 θ
[T1 ] =  sin 2 θ
− 2 sin θ cosθ

[T2 ] = 
( i, j = 1, 2, 6)
[ ] [T ]
= [T2 ] Qij
−1
−T
1
For i, j = 1, 2, 6
(3.5.11)
0 
Q 55 
For i, j = 4, 5 (3.5.12)
28
In which the on axis stiffnesses are:
Q11 =
E11
(1 −ν 12ν 21 )
, Q12 =
E11ν 21
E22ν 12
E22
, Q21 =
, Q22 =
, Q66 = G12
(1 −ν 12ν 21 )
(1 −ν 12ν 21 )
(1 −ν 12ν 21 )
(3.5.13)
The off-axis stiffness values are:
Q11 = Q11 m 4 + 2(Q12 + 2Q66 ) m 2 n 2 + Q 22 n 4
Q12 = (Q11 + Q 22 − 4Q66 ) m 2 n 2 + Q12 ( m 4 + n 4 )
Q22 = Q11 n 4 + 2(Q12 + 2Q66 ) m 2 n 2 + Q22 m 4
(3.5.14)
Q16 = (Q11 − Q12 − 2Q66 ) m 3 n + (Q12 − Q22 + 2Q66 ) n 3 m
Q26 = (Q11 − Q12 − 2Q66 ) mn 3 + (Q12 − Q22 + 2Q66 ) m 3 n
Q66 = (Q11 + Q22 − 2Q12 − 2Q66 ) m 2 n 2 + Q66 ( m 4 + n 4 )
The stiffness corresponding to transverse deformations are:
Q44 = G13 m 2 + G 23 n 2
Q45 = (G13 − G 23 )mn
(3.5.15)
Q55 = G13 n 2 + G 23 m 2
Where m=cosθ and n=sinθ; and θ=angle between the arbitrary principal axis with the
material axis in a layer.
3.6: Strain Displacement Relations
Green-Lagrange’s strain displacement relations are presented in general throughout
the analysis. The linear part of the strain is used to derive the elastic stiffness matrix
and the non-linear part of the strain is used to derive the geometric stiffness matrix.
The total strain is given by
{ε } = {ε l } + {ε nl }
(3.6.1)
The linear generalized shear deformable strain displacement relations are
ε xl =
∂u w
+
+ zk x
∂x R x
29
(3.6.2)
ε yl =
γ xyl =
γ xzl =
∂v
w
+
+ zk y
∂y R y
(3.6.3)
∂u ∂v 2w
+ +
+ zkxy
∂y ∂x Rxy
(3.6.4)
∂w
u
v
+θx −
−
∂x
R x R xy
γ yzl =
(3.6.5)
∂w
v
u
+θy −
−
∂y
R y Rxy
(3.6.6)
The bending strains kj are expressed as,
kx =
∂θ y
∂θ x
, ky =
∂x
∂y
(3.6.7)
k xy =
∂θ y
∂θ x
1 1
1   ∂ v
∂u 


+
+ 
−
−
∂y
∂x
R x   ∂ x
2  R y
∂ y 
(3.6.8)
The non-linear strain components are as follows:
ε xnl
ε ynl
2
2
2
1  ∂u 
1  ∂v 
1  ∂w
v 
1   ∂θ
=   +   +
−
+ z 2  x
2  ∂y 
2  ∂y 
2  ∂y R y 
2   ∂y

γ xynl =
2
2
2
2
2
2
1  ∂u 
1  ∂v 
1  ∂w
u 
1 2  ∂θ x   ∂θ y  





−
+ z 
=   +   + 
 +
2  ∂x 
2  ∂x 
2  ∂x R x 
2  ∂x   ∂x  

 (3.6.9)
 ∂ θ x

  ∂ x
∂u  ∂u  ∂v  ∂v 
 +  +
∂x  ∂y  ∂x  ∂y 
 ∂ θ x
 
 ∂ y
  ∂θ y
 + 
  ∂x
 ∂w u

−
 ∂x R x
 ∂ θ y
 
 ∂ y



 ∂θ y

 + 

 ∂y
2




2

 (3.6.10)

 ∂w
v  + z.

−
 ∂y R y 



 
 
(3.6.11)
3.7: Finite element formulation
An eight nodded isoparametric element is used for static stability analysis of woven
fiber composite plates subjected to hygrothermal environment. Five degrees of
freedom u, v, w, θx and θy are considered at each node. The element is modified to
accommodate laminated materials and hygrothermal conditions of the panel, based on
30
first order shear deformation theory where u, v and w are the displacement
components in the x, y, z directions and
θ x and θ y are the rotations. The stiffness
matrix, geometric stiffness matrix due to residual stresses, geometric stiffness matrix
due to applied in-plane loads and nodal load vector of the element are derived using
the principle of minimum potential energy. The shape function of the element is
derived using the interpolation polynomial given below based on Pascal’s triangle for
convergence criteria.
The displacements are expressed in terms of their nodal values by using the element
shape functions and are given by.
u(ξ,η) = a1 + a2ξ + a3η + a4ξ 2 + a5ξη + a6η 2 + a7ξ 2η + a8ξη2
(3.7.1)
Figure 3.4: Eight nodded isoparametric element
The displacements are expressed in terms of their nodal values by using the element
shape functions and are given by.
8
8
8
i =1
i =1
i =1
u = ∑ N i u i , v = ∑ N i v i , w = ∑ N i wi
(3.7.2)
(3.7.3)
8
8
i =1
i =1
θ x = ∑ N iθ xi ,θ y = ∑ N iθ yi
The shape function Ni are defined as
Ni =
1
(1 + ξξ i )(1 + ηη 1 )(ξξ 1 + ηη 1 − 1)
4
Ni =
1
1 − ξ 2 (1 + ηη i )
2
For i =5, 7
(3.7.5)
Ni =
1
(1 − ξξ i ) 1 − η
2
(
For i =6, 8
(3.7.6)
(
)
2
)
31
For i =1, 2, 3 & 4 (3.7.4)
3.7.1: Element elastic stiffness matrix
The linear strain matrix {ε } is expressed as
Where
{ε} = [B]{δe }
(3.7.7)
{δ e } = {u1 , v1 , w,θ x1 ,θ y1.........., u8 , v8 , w8θ x8 ,θ y8 }T
(3.7.8)

 N i,x


0


 N
i, y

8
[B] = ∑ − 0
i =1

0

0

Ni

−

Rx

 − Ni

R xy

0
N i, y
N i,x
0
0
0
N
− i
R XY
N
− i
Ry
Ni
RX
NI
Ry
2N i
R xy
0
0
N i, x
0
0
N i, y
N i, x
Ni
N i, y
0
0
0
0

0 

0 


0 

0 

N i, y 

N i, x 

0 

Ni 


(3.7.9)
 1
1 
 is a term of Sander’s theory which accounts for conditions of
C 0 = 1 / 2
−
 R x RY 
zero strain for rigid body motion [Chandra sekhar 1989]
Element elastic stiffness matrix is given by
[K e ] = ∫−1 ∫−1 [B ] [D][B] J dξdη
+1 +1
T
(3.7.10)
[
3.7.2: Geometric stiffness matrix due to residual stresses K
r
ge
]
The non-linear strain equations are represented in matrix form:
ε nl = {ε xnl , ε ynl , γ xynl }T = [R]{d }/ 2
Where
{d} = {u x , u y , vx , v y , wx , wy ,θ x, x ,θ x, y ,θ y, x ,θ y, y ,θ x ,θ y }T
Equations {d }may be expressed as:
32
(3.7.11)
(3.7.12)
{d }= {G}{∂e }
Where
 N i, x
N
 i, y
 0

 0
 0

8
0
[G ] = ∑ 
0
i =1

 0
 0

 0

 0
 0
(3.7.13)
0
0
N i,x
N i, y
0
0
0
0
0
0
0
0
0
0
0
0
N i,x
N i, y
0
0
0
0
N i,x
N i, y
0
0
0
0
0
0
0
0
0
0
1
0






0 

0 
0 

0 
N i,x 

N i, y 

0 
1 
0
0
0
0
(3.7.14)
The geometric stiffness matrix due to residual stresses (hygrothermal loads) is given
by
[K ] = ∫ ∫ [G] [S ][G ] J dξdη
r
+1 +1
ge
T
−1 −1
(3.7.15)
Where
 S11
 S
 21
 0

 0
 0

0
[S ] = 0
0

 0
 S
 91
 S101

 0
 S121
S 22
0
0
0
0
0
0
S 92
S102
0
S122
S 33
S 43
0
0
S 73
S 83
0
0
S113
0
S 44
0
0
S 74
S 84
0
0
S114
0
S 55
S 65
0
0
0
0
0
0
S 66
0
0
0
0
0
0
S 77
S 87
0
0
0
0
S 88
0
0
0
S 99
S109
0
0
S110
0
0
















0 
0 0
(3.7.16)
In which
33
S22 = S44 = S66 = N r y
S11 = S33 = S55 = N r x
S21 = S43 = S65 = N r xy ,
S77 = S99 = Nx t 2 /12,
S88 = S1010 = N y t 2 / 12
S87 = S109 = N r xyt 2 /12,
− S73 = S91 = M r x ,
− S84 = S102 = M r y
r
r
− S74 = −S83 = S92 = S101 = M r xy , − S113 = S121 = Qr x , − S114 = S122 = Qr y (3.7.17)
[
3.7.3: Geometric stiffness matrix due to applied loads K
ge
]
The first three non-linear strain equations are represented in a matrix form:
{ε
, ε ynl , γ xynl } = [U ]{ f }/ 2
T
xnl
(3.7.18)
{ f } = {u x , u y , v x , v y , w x , w y ,θ x, x ,θ x, y ,θ y , x ,θ y , y
}
T
(3.7.19)
{ f } is expressed as:
{ f } = [H ]{δ e }
(3.7.20)
 N i,x
N
 i, y
 0

 0
8  0
[H ] = ∑ 
i =1  0
 0

 0
 0

 0
Where
0
0
N i, x
0
0
0
0
0
0
N i, y
0
0
N i,x
0
0
0
0
0
N i, y
0
0
0
N i,x
N i, y
0
0
0
0
0
0





0 
0 

0 
0 

0 
N i, x 

N i , y 
0
0
0
(3.7.21)
The geometric stiffness matrix due to applied in-plane loads is given by
[K ] = ∫ ∫ [H ] [P][H ] J dξdη
+1 +1
ge
T
−1 −1
34
(3.7.22)
 P11
P
 21
0

0
0
[P ] = 
0
0

0
0

 0
Where
P22
0
P33
0
0
0
0
0
0
P44
0
0
P55
P65
P66
0
0
0
0
0
0
0
0
0
0
P77
P87
P88
0
0
0
0
0
0
0
0
0
0
0
0
0
0
P99
P109







 (3.7.23)






P1010 
In which
P11 = P33 = P55 = N a x , P22 = P44 = P66 = N a y , P21 = P43 = P65 = N a xy ,
P77 = P99 = N a xt 2 /12’ P88 = P1010 = N a yt 2 /12, P87 = P109 = N a xyt 2 /12
(3.7.24)
3.7.4: The element mass matrix
[M e ] = ∫−1 ∫−1 [N ] [P][N ] J dξdη
T
+1 +1
(3.7.25)
Where the shape function matrix
Ni
0
8 
[N ] = ∑  0
i =1 
0
 0
0
0
0
Ni
0
0
Ni
0
0
0
0
Ni
0
0
0
 P1 0 0
0 P 0
1

[P1 ] =  0 0 P1

0 0 0
 0 0 0
In which, P1 =
n
0
0 
0

0
N i 
0 0
0 0
0 0

I 0
0 I 
(3.7.27)
ek
∑ ∫ ρ dz
(3.7.26)
And
I =
k =1 ek −1
n
ek
∑ ∫z
2
ρ dz
(3.7.28)
K =1 ek −1
The element load vector due to external transverse static load p per unit area is given
35
 p
{Pe } = ∫∫ N i  0 dxdy .
 0 
(3.7.29)
3.7.5: Load vector
The element load vector due to the hygrothermal forces and moments is given by
{P } = ∫ ∫ [N ] {q }J dξdη
T
+1 +1
a
a
e
j
−1 −1
(3.7.30)
3.8: Solution process
The element stiffness matrix, the initial stress stiffness matrix due to hygrothermal
load, the mass matrix, geometric stiffness matrix due to applied loads and the load
vectors of the element are evaluated by first expressing the integrals in local Natural
co-ordinates, ξ and η of the element and then performing numerical integration by
using
Gaussian
quadrature.
Then
the
initial
stress
resultants
i
i
N i x , N y , N i xy , M i x , M i y , M i xy , Qi x and Q y are obtained Then the element matrices
[ ], [K ]. The
are assembled to obtain the respective global matrices [K ] , [M ] , K
r
g
g
next part of the solution involves determination of natural frequencies, buckling load
and excitation frequencies from the eigenvalue solution of the equations Eq 3.4.7
through Eq 3.4.9.
3.9: Computer program
A computer program based on MATLAB environment developed to perform all
necessary computations. The composite panel is divided into a two-dimensional array
of rectangular elements. The element elastic stiffness and mass matrices are obtained
with 2x2 gauss points. The geometric stiffness matrix is essentially a function of the
in-plane stress distribution in the element due to applied load and residual stress
distribution due to hygrothermal load. The overall element stiffness and mass matrices
are obtained by assembling the corresponding element matrices using skyline
technique. Reduced integration technique is adopted in order to avoid possible shear
locking.
36
A flow chart shown in figure 3.5 is developed for the computational procedures for
dynamic instability of laminated composite panels under hygrothermal conditions and
in-plane harmonic loading.
Read Plate Geometry
Material Properties
Hygrothermal parameters
Generate Nodal connectivity
Identification of DOF
Constitutive Matrix
Derivation of Shape functions
Strain displacement matrix
Element Stiffness Matrix
Element Mass matrix
Element geometric stiffness matrix due to mechanical loads
Element geometric stiffness matrix due to hygrothermal conditions
Assembling
Overall Stiffness Matrix
Overall Mass matrix
Overall geometric stiffness matrix due to mechanical loads
Overall geometric stiffness matrix due to hygrothermal conditions
matrix
Boundary conditions
Eigenvalue Solver
Natural/Excitation
frequencies of vibration
Figure 3.5: Flow chart of Program in MATLAB for instability of Composite
panels subjected to hygrothermal loads
37
CHAPTER 4
EXPERIMENTAL PROGRAMME
4.1: Introduction
The experimental works carried out for the present free vibration and buckling test
under hygrothermal conditions are presented in this chapter. Some woven roving
Glass/Epoxy composite plates are fabricated for the present experimental work. The
percentage of fiber and matrix has taken as 55:45 in weight for fabrication of flat
panels.
4.2: Materials Required for Fabrication of Plates
The following constituent materials are used for fabricating the glass/epoxy fiber
plate:
•
Glass woven roving as reinforcement (FGP, RP-10)
•
Epoxy as resin
•
Hardener as catalyst (Ciba-Geigy, araldite LY556 and Hardener HY951)
•
Polyester resin (MEKP-Methyl Ethyl Ketone Peroxide)
•
Polyvinyl alcohol as a releasing agent
4.3: Fabrication procedure for static test specimen
In the present investigation for static behavior (ILSS), two different types of fiber:
matrix composite specimens were fabricated. These were (i) Glass: Epoxy (ii) Glass:
polyester, (E-glass fibers). Each type preparation of laminates was manufactured by
using three different types of weight fractions i.e. 55:45, 60:40 and 65:35. Woven
roving E-Glass fibers (FGP, RP-10) were cut into required shape and size according
to number of specimens required for testing. Each composite laminates consists of 16
plies of fiber in balanced form as per ASTM specification. For preparation of Epoxy
resin matrix, Hardener 8% (Ciba-Geigy, araldite LY556 and Hardener HY951) of the
weight of Epoxy were used and for polyester matrix 1% accelerator (cobalt octane
2%) was added first to the polyester resin then 1.5% of catalyst (MEKP-Methyl Ethyl
38
Ketone Peroxide) added to mixture and stirred thoroughly to get polyester matrix as
per ASTM D5687/D5687M-07 [2007]. Subsequent plies were placed one upon
another with matrix in each layer to obtain sixteen stacking plies as shown in figure
4.1. A hand roller was used to distribute resin uniformly, compact plies, and to
remove entrapped air to minimize void contents in the samples as shown in figure 4.2.
The mould and lay up were covered with a release film to prevent the lay up from
bonding to the mould surface. Then the resin impregnated fibers were placed in the
mould for curing. The laminates were cured at normal temperature (250C and 55 %
Relative Humidity) under a pressure of 0.2 mpa for 3 days as shown in figure 4.3.
After proper curing of laminates the release film were detached the specimens were
cut for inter laminar shear strength (ILSS) for three point bend test.
In addition to above the tensile test was conducted for four material constants i.e. E1,
E2, G12, and υ12 where the suffixes 1 and 2 indicate principal material directions. For
material characterization of composites, laminate having sixteen layers was fabricated
to evaluate the material constants. The constants are determined experimentally by
performing unidirectional tensile tests on specimens cut in longitudinal and transverse
directions, and at 450 to the longitudinal direction, as described in ASTM standard: D
3039/D 3039M-2008.The tensile tests specimens are having a constant rectangular
cross section in all cases. The dimension of the specimen was taken as below. The
specimens were cut from the plates themselves by diamond cutter or by hex. At least
four replicate sample specimens were kept in the hygrothermal chamber at different
temperature and moisture concentration for six hours and then tested and mean values
adopted. The tests specimens are shown in fig 4.5 and fig 4.6.
For measuring the Young’s modulus, the specimen was loaded in INSTRON 1195
universal testing machine (as shown in figure 4.9 monotonically to failure with a
recommended rate of extension (rate of loading) of 0.2mm/minute. Specimens were
fixed in the upper jaw first and then gripped in the movable jaw (lower jaw).Gripping
of the specimen should be as much as possible to prevent the slippage. Here, it was
taken as 50mm in each side for gripping. Initially strain was kept at zero. The load, as
well as the extension, was recorded digitally with the help of a load cell and an
extensometer respectively. From these data, engineering stress vs. strain curve was
plotted; the initial slope of which gives the Young’s modulus. The ratio of transverse
39
to longitudinal strain directly gives the Poisson’s ratio by using two strain gauges in
longitudinal and transverse direction. The shear modulus was determined using the
following formula from Jones
1
4
1
1 2
The values of material constants finally obtained experimentally for different
temperature and moisture concentrations are presented in Table 5.16 and 5.17.
4.4: Fabrication procedure for vibration and buckling
Contact moulding in an open mould by hand lay-up was used to combine plies of
woven roving in the prescribed sequence. The percentage of fiber and matrix has
taken as 55:45 in weight for fabrication of plates. A flat plywood rigid platform was
selected. A plastic sheet i.e. a mould releasing sheet was kept on the plywood
platform and a thin film of polyvinyl alcohol is applied as a releasing agent by use of
spray gun. Laminating starts with the application of a gel coat (epoxy and hardener)
deposited on the mould by brush, whose main purpose was to provide a smooth
external surface and to protect the fibers from direct exposure to the environment. Ply
was cut from roll of woven roving. Layers of reinforcement were placed on the mould
at top of the gel coat and gel coat was applied again by brush. Each composite
laminates consists of 16 plies of fiber in balanced form as per ASTM specification.
Any air which may be entrapped was removed using serrated steel rollers. The
process of hand lay-up was the continuation of the above process before the gel coat
had fully hardened. After completion of all layer, again a plastic sheet was covered
the top of last ply by applying polyvinyl alcohol inside the sheet as releasing agent.
Again one flat ply board and a heavy flat metal rigid platform were kept top of the
plate for compressing purpose. The plates were left for a minimum of 48 hours before
being transported and cut to exact shape for vibration and buckling testing. Figure 4.1
to 4.3 shows the fabrication process of laminated composite plates.
4.4.1: Details of test specimen
The size of specimen was taken as 235 mm×235 mm x 6mm size for plate, 200
mmx25 mm x 6mm size for tensile test and for 45 mm × 6 mm size for three point
bend test.
40
Figure 4.1: placing of woven glass fiber
using on gel coat
Figure 4.3: Composite plates
Figure 4.2: Removal of air entrapment
using steel roller
Figure 4.4: Instron 1195 UTM machine
Figure 4.6: Specimen in 450 directions
for tensile testing
Figure 4.5: Specimen in X direction
for tensile testing
4.5: Hygrothermal treatment
The specimens were hygrothermally conditioned in a humidity cabinet as shown in
figure 4.5, where the conditions were maintained at a temperature of 323K and
relative humidity (RH) ranging from 0-1% for moisture concentration as per ASTM
41
D5229/D5229M-04[2004]. The humidity cabinet had an inbuilt thermometer for
temperature and hygrometer for relative humidity measurements. The temperature
variation was maintained between 300K-425K whereas the RH was 0 in temperature
bath as shown in figure 4.6. The composite laminates were placed on perforated trays.
The hygrothermal conditioning was carried out for every six hours in a total period of
thirty six hours.
Figure 4.7 Humidity chamber
Figure 4.8 Temperature bath
4.6: Static Behaviour Experiment Test
Most commonly used test for inter laminar shear strength is the short beam under
three point bending. The specimen were tested for three point bend test on the
INSTON-1195 material testing machine as shown in figure 4.4 and 4.7 with different
cross head velocities to obtain inter laminar shear strength (ILSS) and to study the
effects of loading speed for different types of laminates. The test was conducted with
speed of 1mm / min, 10mm / min, 100mm / min, 200mm / min and 500mm / min with
constant span of 34mm. Then, yield load (max load) were obtained for each type of
laminates in which minimum five specimens were tested. Before testing the thickness
and width of the specimens were measured accurately at the midpoint. The test
specimen was placed in the test fixture and aligned so that its mid point was centered
and it’s long axis was perpendicular to the loading nose. The same procedure was
repeated for all the specimens. The specimens were tested at regular interval of time
at a constant cross head speed of 200mm / min. In all the above three cases there five
sample specimens were tested at each point of an experiment and then average value
was reported. The ILSS value was determined in accordance with ASTM
42
D2344/D2344M -06 [2006]. The interlaminar shear strength is calculated as
.
,
Where Pb is the breaking load in N, b is the width of specimen in mm and d is the
thickness of specimen in mm.
Figure 4.9: Complete set up of Instron 1195 machine
4.7: Apparatus required for free vibration test
•
Modal hammer. (2302-5)
•
Accelerometer. (B&K 4507)
•
FFT Analyzer. (Model B&K 3560-C)
•
Notebook with PULSE lab shop software.
•
Specimens to be tested (Composite plate size 0.235mX 0.235mx0.006m)
4.7.1: Free Vibration Experiment Test
The composite test specimens were fitted properly to the prefabricated iron frame as
shown in figure 4.11. The connections of FFT analyzer (Model B&K3560-C) as
shown in figure 4.9, laptop shown in figure 10, B&K4507 transducers, B&K2302-5
modal hammer as shown in figure 4.8, and cables to the system were done. The
PULSE Lab shop was used during the vibration measurement. The plate was excited
in selected points by means of impact hammer (Model 2302-5) and this resulting
vibration of the specimens was picked up by the accelerometer. The accelerometer
(B&K 4507) was mounted on the specimen as shown in figure 4.12 by means of bees
wax. The signal was then subsequently led to the analyzer, where its frequency
spectrum was also obtained using the pulse software. Various forms of Frequency
43
Response Functions (FRF) are directly
directly measured. The coherence is observed for
each set of measurement.
Figure 4.10: Modal Impact Hammer
Figures 4.11: The FFT 3560-C
3560 analyzer
Figure 4.12: Display unit
Figure 4.13: Iron Frame for different
B.C. Setup
Figure 4.14: The test frame with
specimen
(Simply supported boundary condition)
Figure 4.15: The test frame with
specimen
(Clamped boundary condition)
The output from the analyzer was displayed on the Lab shop screen. The modal
parameters obtained from experiments
experiments are natural frequencies and modal damping
factors as determined from the accelerance, by the “peak picking” method. The
44
four sides simply supported boundary conditions and four sides clamped position
of the composite plate specimens for testing is as shown in 4.12 and 4.13
respectively.
4.8: Buckling Experiment Test
In view of actual real time behavior of laminated composite plates, experimental
methods have become important in solving the buckling problem of laminated
composite plates. The specimen was clamped at two sides and kept free at two other
sides in an iron frame as shown in figure 4.15. The specimen was loaded in axial
compression by using an INSTRON 1195 Machine of 600 KN load capacity as shown
in figure 4.14. All specimens were loaded slowly unless buckling takes place as
shown in figure 4.16. Clamped boundary conditions were simulated along top and
bottom edges, restraining 2.5cm length. For axial loading, the test specimens were
placed between the two extremely stiff machine heads, of which the lower one was
fixed during the test; whereas the upper head was moved downwards by servo
hydraulic cylinder. All plates were loaded at constant cross-head speed of 0.5mm/min.
The shape of the specimen after buckling is as shown in figure 4.17. The load verses
end shortening curve was plotted. The displacement is plotted along x-axis and the
load was plotted on the y-axis. The load, which is the initial part of the curve deviated
linearity, is taken as the buckling load in KN in line with previous investigators.
45
Figure 4.16: Instron UTM machine with
specimen
Figure 4.18: Specimen before buckling
Figure 4.17: composite test frame with
specimen
Figure 4.19: Specimen after buckling
46
CHAPTER 5
RESULTS AND DISCUSSIONS
5.1: Introduction
The present chapter deals with the results of the analyses of vibration, buckling and
parametric resonance characteristics of woven fiber laminated composite plates in
hygrothermal environment using the formulation given in the previous chapter. Some
results on static behavior especially the variation of inerlaminar shear strength with
loading speed and exposure hours proportion of fiber to different matrices are
presented for completeness. The statistical integration of the test results is also
presented. As explained, the eight-node isoparametric quadratic shell element is used
to develop the finite element procedure. The first order shear deformation theory is
used to model the plate considering the effects of transverse shear deformation and
rotary inertia. The static behavior, vibration and stability characteristics of woven
fiber laminated composite plates and shells in hygrothermal environment are studied.
The parametric instability studies are carried out for woven fiber laminated composite
plates and shells subjected to in-plane periodic loads with static component of load to
consider the effect of various parameters in hygrothermal environment. The studies in
this chapter are presented separately for plates and shells as follows:
•
Convergence Study
•
Comparison with previous studies
•
New results
5.1.1: Static behaviour of woven fiber composites in hygrothermal
environment
The present study involves extensive experimental works to investigate the
hygrothermal effects on the mechanical behavior of Glass: Epoxy and Glass:
Polyester composites. Static tests involving three points bend tests and subsequent
interpolation of statistical parameters including interlaminar shear strength (ILSS)
47
appears to be of significant important for evaluation of static capacity of composite
structures in hygrothermal environment. The present investigation deals with the
failure behavior of woven fiber composites under different loading speeds and change
in material percentage constituents under hygrothermal conditions. The further
investigations were carried out on different types of composites with different
constituents of their weight fractions 55:45, 60:40, and 65:35 at a constant speed of
200mm/min in hygrothermal environment and different exposure time.
The statistical analysis of interlaminar shear strength (ILSS) with varying loading
speeds of 1,10,100,200 and 500 for different proportion of Glass fiber: Epoxy and
Glass fiber: Polyester such as 55:45, 60:40 and 65:35 of a glass/epoxy and a
glass/polyster are presented below in Table 5.1-5.2 respectively. The analysis is
carried out as per ASTM D-2344, section 12.2. The standard deviation and coefficient
of variation of ILSS is less in all higher loading speeds beyond 200mm/min in
comparison to lower loading speeds from 1 to100mm/min in glass fiber/epoxy
specimens in all three proportions. In glass fiber/polyster composites, the S.D and C.V
of ILSS is more for loading speeds beyond 200mm/ min but less from 1 to 100mm/
min in all three proportion. This shows the good degree of quality control of test
specimens preparation.
Table 5.1: Statistical variation of ILSS with loading speeds of Glass/Epoxy
composites for varying proportion
Loadingspeed(
mm/min)
(55:45)
ILSS for different proportion
( 60:40)
(65:35)
1
10
100
Mean
31
34.7
22.9
S.D
C.V
3.1
10
0.0395 0.98
1.5
6.5
Mean
36.7
31.6
21.5
S.D
1.93
0.98
0.92
C.V
5.25
3.123
4.279
Mean
39.4
28.4
19.1
S.D
1.46
2.74
0.181
C.V
9.4
9.64
0.94
200
28.4
0.0053 1.86
29
2.25
7.71
25.2
1.558
6.1
500
Average
27.1
28.8
0.0659 0.25
0.944 3.918
27.1
29.16
0.31
1.33
1.107
4.298
27
27.82
2.46
1.679
9.11
7.03
48
Table 5.2: Statistical variation of ILSS with loading speeds of Glass/Polyester
composites for varying proportion.
Loadingspeed
(mm/min)
(55:45)
ILSS for different proportion
( 60:40)
Mean
S.D
C.V
Mean
1
10
18.7
18.9
0.11
0.027
0.588 18.9
0.0075 18.7
100
14.3
1.113
0.544
200
22
2.97
500
19.7
Average
18.72
S.D
(65:35)
C.V
S.D
C.V
0.374
1.363
Mea
n
0.104 21.5
0.387 18.4
1.97
1.11
9.1
0.328
14.7
0.194
8.97
13.7
0.49
3.6
13.50
21.7
0.867
0.184 21.8
1.13
0.239
2.2
11.16
17.2
1.07
0.361
18
5.76
1.77
1.242
5.159
18.296 0.773
2.00
18.68 2.15
3.00
The variations of inter laminar shear strength (ILSS) with different cross head
velocity for glass:epoxy specimens are shown in fig.5.1. As shown, the ILSS of
woven roving glass fiber reinforced polymers showed a significant rate sensitivity. In
the three weight fraction of fiber-matrix of glass: epoxy, higher ILSS was obtained in
low loading speed. As the loading speed increased, the inter laminar shear strength
decreased upto 100mm / min for all the three weight fractions. The inter laminar shear
strength for glass: epoxy specimens increased with increase in loading speed upto
200mm/min, beyond that the inter laminar shear strength decrease marginlly or
remains constant for glass/epoxy with further increase in loading speed. The
glass/epoxy specimens with weight fraction 55:45 shows higher ILSS in lower
crosshead velocity and the specimens with weight fraction 65:35 shows lowest ILSS
among three fractions. However, glass/epoxy specimens with weight fractions 60:40
shows similar ILSS to that of 55:45 after crosshead velocity more than
200mm/minute. In general, the variation of inter laminar shear strengths of
glass/epoxy specimens are found significant with increasing loading speed.
49
Interlaminar shear strength
(mpa)
45
55:45"
40
60:40"
35
65:35"
30
25
20
15
10
0
100
200
300
400
500
600
Crosshead velocity(mm/min)
Figure 5.1: Variation of ILSS with different loading speeds of glass/epoxy
composites for varying proportion
In case of glass: polyster fibre matrix of three weight fractions, lower ILSS is
observed as shown in fig. 5.2 in comparison to glass/epoxy speciens at low loading
speed and then
decreased in loading speed upto 100 mm/min. The ILSS then
increased with increase in loading speed upto 200mm/ min. However, beyond that,
the ILSS decreased for all weight fraction of glass: polyester matrix resin. Woven
roving Glass Fiber Reinforced Polymer(GFRP) shows a significant rate sensitivity.
The variation of inter laminar shear strengths are found with increasing loading speed.
It is important note that a change in loading speed can result in a variation of failure
loads. When subjected to an increasingly higher impact velocity , a laminate behaves
like a more rigid beam or plate, which is less suceptible to bending. This shifts its
behavior from that of a flexible rigid beam with very low impact velocity and the
failure initiated from the rear surface to that occurs near the point of conact in the case
of much higher impact velocity. At intermediate velocities the complex behavior of
Interlaminar shear strength
(mpa)
mixed fracture modes are seen.
25
55:45"
60:40"
20
65:35"
15
10
0
100
200
300
400
500
600
Cross head velocity(mm/min)
Figure 5.2: Variation of ILSS with different loading speeds of glass/polyester
composites for varying proportion
50
It is observed from figure 5.1 and 5.2 that for woven roving glass fiber laminates with
Epoxy matrix shows higher inter laminar shear strengths than Polyster matrix for all
weight fractions, since epoxy resin is relatively low molecular weight than polymers
and capable of being processed under a variety of conditions. The major advantages
of epoxy resins is higher viscosity and exhibit low shrinkage during cure over polyster
resins.
The statistical analysis of ILSS with rise in temperature for different proportion of
Glass fiber: Epoxy and Glass:Polyester are given in Table 5.3 and Table 5.4
respectively. The glass/epoxy specimens shows higher value of mean ILSS of 32.86
for 60: 40 proportions. However, the glass/epoxy speimens with 55:45 proportion
shows least value of S.D. of 2.929 and CV of 8.336 respectively. In both the
proportions, the mean, S.D. and C.V. of ILSS decrease with increase of temperature
up to 520 C and then decrease with increase of temperature. The standard deviation
and coefficient of variation of ILSS is less with respect to rise in temperature from
270C to 2020C for glass/epoxy specimens for all proportion.
Table 5.3: Statistical variation of ILSS with temperature of Glass/Epoxy
composites for varying proportion.
ILSS for different proportion
Temperature
C
(55:45)
0
27
52
77
102
127
152
177
202
Average
Mean
18.6
30.4
20.5
32.8
30.7
31.2
41
35.9
30.13
S.D
3.15
3.236
1.862
2.86
3.46
4.20
3.07
1.60
2.929
( 60:40)
C.V
1.693
10.64
9.0
8.70
11.27
13.46
7.48
4.45
8.336
Mean
18.5
29.8
30.2
39.9
35
36.2
28.3
45
32.86
S.D
4.46
3.363
3.837
4.20
4.1
4.6
2.78
2.95
3.822
(65:35)
C.V
2.41
11.28
12.7
10.50
11.7
12.7
9.82
6.55
9.732
Mean
23
34.4
30.2
31.50
31.5
35.1
31.8
27.2
30.587
S.D
1.276
3.9
3.563
3.60
3.60
3.76
3.12
2.15
2.975
C.V
5.54
11.33
11.70
11.42
11.42
10.59
9.811
7.90
9.44
However, the glass/ polyester specimens shows much less mean value of ILSS in
comparison to glass/epoxy specimens. Within the glass/epoxy samples, the specimens
with proprtion 55:45 shows highest value of mean of 18.72. The specimens with
proprtion 60:40 shows least value of S.D. of 0.773 and C.V. of 2.00 showing better
quality control of specimens. The S.D. and C.V. of glass/polyster specimens are also
51
less in comparison to glass/epoxy specimens for increase in temperature from 270C to
2020C for all weight fractions.
Table 5.4: Statistical variation of ILSS with temperature of Glass/Polyester
composites for varying proportion.
ILSS for different proportion
Temperature
0
C
27
52
77
102
127
152
177
202
Average
(55:45)
( 60:40)
(65:35)
Mean
S.D
C.V
Mean
S.D
C.V
Mean
S.D
C.V
5.1
24.1
19.7
27
21.3
19.3
25.1
13
19.32
0.62
2.41
1.69
3.13
2.46
1.28
3.32
1.26
2.02
12.15
9.95
8.57
11.59
1.54
6.66
13.22
9.69
10.42
12
20
20.9
21.1
23.4
19.4
20.6
20.2
19.71
1.26
1.96
2.18
1.56
2.12
1.43
0.86
1.28
1.58
10.5
9.8
9.19
7.35
9.05
7.37
4.17
6.33
7.97
8.5
23.7
23.5
23.7
27.4
20.5
21.4
20.5
21.15
1.09
1.85
1.13
1.96
2.64
1.83
0.65
2.16
1.66
12.8
7.8
4.80
8.27
9.63
8.92
3.03
10.53
8.22
The variations of ILSS of glass/epoxy specimens with temperature is shown in shown
in fig 5.3. As shown, the values of ILSS for glass: epoxy specimen increase gradually
when the temperature is increased from 270C to 520C and then decrease upto 770C.
The ILSS for glass: epoxy specimen increased further with the increase of
temperature upto 1520C and there after decreased with increase of temperature for
glass: epoxy composite in 55:45 and 60:40 proportions.
Interlaminar shear strength
(mpa)
45
55:45"
40
60:40"
35
65:35"
30
25
20
15
10
0
25
50
75
100
125
150
175
200
225
250
Temperature (0C)
Figure 5.3: Variation of ILSS with different temperature of glass/epoxy
composites for varying proportion
However, the values of ILSS for glass: polyester specimen of 65:35 proportion
increase when the temperature is upto 520C then remains constant upto 1520C. The
52
values of ILSS then decrease with increase of temperature. For fiber: polyester
composites of 60:40 and 65:35 proportions, the values of ILSS increased gradually
when the temperature is increased upto 520C and remain constant upto 1020C.The
ILSS for glass: polyester specimens decreased when the temperature is increased upto
1520C and remains constant with increase in temperature except 55:45 proportion.
As shown in fig 5.4, the ILSS for glass: polyester specimen increase and then
decrease with temperature in every 250C upto 1770C and decreased thereafter at the
same constant speed of 200mm/ min. The exposure to elevated temperature can result
in degradation of mechanical properties, cracking and flaking of polymers. The first
form of damage in laminated composite is usually matrix microcracks. Matrix
microcracks cause degradation properties in composite laminates and also act as
precursors to other forms of damage leading to laminate failure. The moisture
absorption is more for sample specimen which is exposed at higher temperature. At
higher temperature, the thermal stresses are more which leads to higher mismatch in
the thermal strain.
Interlaminar shear strength
(mpa)
30
55:45"
25
60:40"
65:35"
20
15
10
5
0
25
50
75
100
125
150
175
200
225
250
Temperature (0C)
Figure 5.4. Variation of ILSS with different temperature of glass/polyester
composites for varying proportion
The statistical analysis of ILSS with rise in moisture concentration from 0.0% to 1.0%
at an interval of 0.25% for different proportion of Glass fiber: Epoxy and Glass fiber:
Polyester are given in Table 5.5 and Table 5.6 respectively. The mix 55:45 shows
mean value of ILSS of 23.8Mpa averaged over the different moisture concentrations
considered from 0% to 1.0%.
53
Table 5.5: Statistical variation of ILSS with moisture concetration of
Glass/Epoxy composites for varying proportion.
Moisture
concentration,C
(%)
0
0.25
0.50
0.75
1
Average
(55:45)
ILSS for different proportion
(60:40)
Mean
S.D
C.V
Mean
12.7
27.2
18.4
32.6
28.1
23.8
1.36
3.12
1.74
3.11
2.58
2.38
10.70
11.47
9.45
9.53
9.18
10.06
14.5
23
13.9
35.7
25.6
22.56
S.D
0.96
2.28
1.2
3.69
2.86
2.19
(65:35)
C.V
Mean
S.D
C.V
6.53
9.91
8.70
10.33
11.17
9.328
12
27.3
11.5
33.2
22.5
21.3
1.14
3.08
1.12
2.56
1.89
1.95
9.5
11.28
9.74
7.71
8.40
9.32
Table 5.6: Statistical variation of ILSS with moisture concetration of
Glass/Polyester composites for varying proportion.
Moisture
concentration,
C (%)
0
0.25
0.5
0.75
1
Average
(55:45)
Mean
9.4
26.6
8.9
24
18.9
17.56
S.D
1.07
2.13
1.12
2.27
1.67
1.65
ILSS for different proportion
(60:40)
C.V
11.38
8.0
12.58
9.45
8.83
10.04
Mean
15.41
16
10.6
24
19.3
17.06
S.D
1.79
1.56
1.20
2.56
1.45
1.71
C.V
11.62
9.75
11.32
10.66
7.51
10.17
Mean
18.7
25
10.4
26
21.9
20.4
(65:35)
S.D
2.21
2.69
1.27
1.78
1.58
1.90
C.V
11.81
10.76
12.21
6.84
7.21
9.76
The standard deviation and coefficient of variation of ILSS is less in all proportions
with rise in moisture concentration from 0 to 1 for all proportion in glass/epoxy and
glass/polyster specimens. Since moisture affects all components of the composite,
principally the matrix and the fibre matrix interface but also the fiber itself.
The plasticization process involves the interuption of the van der Waal’s bonds
between ethers, secondary amines and hydroxyl groups. Polymers with ketones and
imides are more resistant to hydrolysis, they have fewer polar groups and this reduces
their moisture sensitivity. Plasticization reduces residual stresses and increases
viscoelasticity. So the effect of increase in moisture concentration is very similar to an
increase in temperature as shown in fig. 5.3 through fig. 5.6. Mechanical damage from
moisture induced swelling, surface crazing and matrix microcracking. In general, the
glass/polyster shows less value of average ILSS in comparison to glass/epoxy
54
specimens.However, the value of SD is less in comparison to glass/epoxy specimens
showing better quality control.
The variation of ILSS with moisture concentrations of glass/epoxy specimens are
presented in fig.5.5. As shown in fig 5.5, the ILSS for glass:epoxy specimen increased
and decreased in every 0.25% moisture concentration gradually upto 1 for all
proportion of composites. In the glass:polyester composites, as shown in fig.5.6, all
the proportion behaves all most same as glaas :epoxy composites at the same constant
speed of 200mm/ min.When exposed to humid environment, the positive
concentration gradient between the environment and the specimen, the higher
pressure outside the specimen favours moisture absorption. When the same specimen
is exposed to a relatively dry environment, the conditions get reversed resulting in
moisture desorption. The relative rates of absorption and desorption controls the net
moisture pickup by the specimen. The ILSS values are seen to going downwards.
Amount of moisture absorbed by epoxy matrix is significantly greater than fibers
which absorb little or no moisture. This results in significant mismatch in moisture
induced volumetric expansion between matrix and ibers leading to evolution of
localized stress and strain fields in the composite at the interfacial region.
Inter laminar shear strength
(mpa)
40
35
30
25
55:45"
60:40"
65:35"
20
15
10
0
0.25
0.5
0.75
1
Moisture concentration (c%)
Figure 5.5: Variation of ILSS with different moisture concetration of glass/epoxy
composites for varying proportion.
55
Inter laminar shear strength
(mpa)
30
25
20
55:45"
60:40"
65:35"
15
10
5
0
0.25
0.5
0.75
1
Moisture concentration (c%)
Figure 5.6: Variation of ILSS with different moisture concetration of glass/
Polyester composites for varying proportion.
The statistical analysis of ILSS with rise in exposure time in hours for different
proportions of Glass fiber: Epoxy and Glass fiber: Polyester are given in Table 5.7
and Table 5.8 respectively. Unlike the previous cases, the glass/epoxy specimens at
proportion 65:35 shows highest value of mean value of ILSS out of all three mixes.
The standard deviation and coefficient of variation of ILSS decrease with increase in
exposure time for glass/epoxy specimens in all proportion.
Table 5.7: Statistical variation of ILSS with exposure time of Glass/Epoxy
composites for varying proportion.
Exposure
time
(hours)
5
10
15
20
25
Average
(55:45)
ILSS for different proportion
(60:40)
Mean
31.7
S.D
3.56
C.V Mean
11.23 33
S.D
3.67
31
30.3
30.2
16.3
27.9
2.42
1.56
1.69
0.78
2.0
7.80
5.14
5.59
4.78
6.90
2.23
1.79
1.23
1.36
2.05
26.6
24.8
24.4
16.3
25.02
56
C.V
11.12
8.38
7.21
5.04
8.34
8.01
(65:35)
Mean
33.3
S.D
3.26
C.V
9.78
33
32.6
32.4
26.1
31.48
2.78
2.25
2.56
2.19
2.60
8.42
6.90
7.90
8.39
8.27
Table 5.8: Statistical variation of ILSS with exposure time of Glass/Polyester
composites for varying proportion.
Exposure time
(hours)
5
10
15
20
25
Average
ILSS for different proportion
(55:45)
(60:40)
(65:35)
Mean S.D
C.V Mean S.D
C.V
Mean S.D
23.7
1.67
7.04 20.7
1.78 8.59
25.4 2.60
21.7
1.36
6.26 20.4
1.64 8.08
25
2.16
18.6
0.86
4.62 17
0.92 5.41
20.4 2.17
16.6
1.30
8.37 15.4
1.46 9.48
15.7 0.83
10.4 0.86
8.26 13.2
1.13 8.56
10.1 1.37
18.2 1.22
6.91 17.34 1.38 8.02
19.32 0.95
C.V
8.50
8.68
4.36
8.72
9.4
7.93
However, the mean value of ILSS drops suddenly toward the end of exposure time.
The low value of standard deviation and C. V. shows the good quality of sample
specimen preparation. Similarly, the glass/polyster specimens at 65: 35 proportions
show highest mean value of ILSS among the three mix proportions. The standard
deviation and coefficient of variation of ILSS decrease with increase in exposure time
glass/polyester specimens in all proportion with marginal increase towards the end of
exposure time like the glass/epoxy specimens.
The variations of ILSS with increase in exposure time of glass/epoxy and
glass/polyster specimens are shown in fig 5.7 and fig 5.8 respectively for all three
mixes. As shown in fig 5.7, it is observed that there is a significant continuous
reduction of ILSS with the increase in exposure time 20 hours onwards for three
proprtions. This may be due to more amount of absorved moisture. However, all the
glass/polyster specimens in fig. 5.8 shows considerable reduction of ILSS with
increase of exposure time from the beginning to end.
It was reported that the moisture absorption kinetic increases with more conditioning
time in glass:epoxy and glass:polyester composites for different proportion at the
same constant speed of 200mm/ min. The failure mechanism is due to environmental
exposure results in reduced interfacial stress transmissibility due to matrix
plasticization,chemical degradation. Matrix plasticization reduces matrix modulus.
Chemical degradation is the result of hydrolysis of interfacial bonds. Mechanical
degradation is the result in a function of matrix sweling strain.
57
Inter laminar shear strength
(mpa)
40
55:45"
60:40"
35
65:35"
30
25
20
15
5
10
15
20
25
30
Exposure time (hours)
Figure 5.7: Variation of ILSS with different exposure time of glass/epoxy
composites for varying proportion
Interlaminar shear strength
(mpa)
30
55:45"
60:40"
25
65:35"
20
15
10
5
10
15
20
25
30
Exposure time (hours)
Figure 5.8: Variation of ILSS with different exposure time of glass/Polyester
composites for varying proportion. Tensile testing of Composite under
hygrothermal loading
The effect of hygrothermal conditions on the elastic modulus of material is studied by
tensile testing of specimens at constant loading speed of 200mm/min. The variations
of Young’s modulii of materials in two orthogonal directions of woven fiber lamina is
shown in fig. 5.9. As a result, the tensile strength, ultimate compressive strength that
normally decreased with increase of temperature and moisture.
As shown in fig 5.9, both the modulii decrease with increase in temperature. In a
similar manner, study is also done to see the effects of hygrothermal conditions on the
shear modulus of specimens. The variations of shear modulus with temperature is
shown in fig. 5.10. As shown, the shear modulus also decrease with increase of
temperature.
58
Modulus of elasticity(Gpa)
10
E-11
9
E-22
8
7
6
5
25
50
75
100
125
150
Temperature 0C
Figure 5.9: Variation of Shear Modulus of elasticity with different temperature
of Glass/epoxy composites for 55:45 proportions
Modulus of rigidity(Gpa)
4
G-12
3
2
1
25
50
75
100
125
150
Temperature 0C
Figure 5.10: Variation of Modulus of rigidity with different temperature of
glass /epoxy composites for 55:45 proportions.
The study is then extended to examine the effect of moisture on modulus of elasticity
at the same loading speed of 200mm/min. As shown in figure 5.11, the modulus of
elasicity decreases with increase of moisture concentrations. Similarly the variations
of shear modulus with increase in moisture concentrations is shown in figure 5.12.
As observed, the shear modulus reduces with increase of moisture concentrations. As
shown in figure 5.9-5.12, that the modulus of elasticity and modulus of rigidity
decreased substantially with increase in temperature and moisture concentration
environment. Since temperature and moisture affects all component of composite,
principally the matrix and the fibre matrix interface also the fibre. Due to increase in
temperature and moisture, the longitudinal and transverse tensile strength varying
linearly. Also in the same linear variation appears in case of modulus of rigity in
hygrothermal environment. The different degradation rate for young’s modulus of
elasticity appers due to the difference in fibre-matrix adhesion of the coating on
59
fibres. It is noticed that along longitudinal fibre direction the modulus of elasticity
reduced more than transverse matrix direction because of transverse microcrack.
Modulus of elasticity(Gpa)
8
E-11
E-22
7
6
5
4
3
0
0.25
0.5
0.75
1
Mostureconcentration C%
Modulus of rigidity(Gpa)
Figure 5.11: Variation of Modulus of rigidity with moisture concentration
of glass/epoxy composites for 55:45 proportions.
3
G-12
2.5
2
1.5
1
0
0.25
0.5
0.75
1
Moisture concentration C%
Figure 5.12: Variation of Modulus of rigidity with moisture concentration
of glass/epoxy composites for 55:45 proportions.
The lamina materials properties at elevated moisture concentrations and temperatures
as per ASTM D3039/D3039M-[2008] are used in the present analysis are shown in
Tables 5.9 and 5.10 respectively.
60
Table 5.9: Elastic moduli of glass fiber / epoxy lamina at different temperatures
α1=0.3 X 10-6/OK, α2 =28.1 X 10- 6/OK, β1=0, β2=0.44
Temperature in (K)
Elastic moduli 300K
E1
7.9
325
350
7.6
375
7.1
400
425
6.7
6.5
6.3
E2
7.4
6.8
6.4
6.2
5.9
5.7
G12
2.9
2.6
2.3
2.1
1.8
1.6
ν12
0.4
0.43
0.41
0.35
0.36
0.35
Table 5.10: Elastic moduli of glass fiber/epoxy lamina at different moisture
concentrations α1=-0.3 X 10-6/OK, α2 =28.1 X 10-6/OK, β1=0, β2=0.44
Elastic moduli
0.25
0.5
0.75
1.0
7.9
7.6
7.5
7.3
7.2
E2
7.4
7.4
7.3
7.1
G12
2.9
2.9
2.8
2.7
2.6
ν12
0.4
0.4
0.4
0.39
0.39
E1
0.0
61
7.0
5.2: Flat panels
The studies in this section are grouped into three parts as follows:
•
Experimental and numerical study on vibration of woven fiber laminated
composite plates in Hygrothermal environment.
•
Experimental and numerical study on buckling effects of woven fiber
laminated composite plates in Hygrothermal environment.
•
Dynamic stability of woven fiber laminated composite plates in Hygrothermal
environment subjected to periodic loadings.
5.2.1: Vibration of woven fiber laminated composite flat panels in
Hygrothermal environment
The example considered here consists of a woven fiber laminated composite plate
0.235mx0.235mx0.006m subjected to hygrothermal loadings with four sides simply
supported and clamped boundary conditions as shown in figure 4.12 and 4.13
respectively.
5.2.1.1: Convergence Study
The convergence study is first done for non-dimensional frequencies of free vibration
of 4 layer symmetric cross-ply and symmetric angle-ply laminated composite plates at
a temperature of 325K and 0.1% moisture concentration for different mesh divisions
as shown in Table 5.9 and 5.10 respectively. As observed, a mesh of 10 ×10 shows
good convergence of the numerical solution for the free vibration of composite plates
in hygrothermal environment and this mesh is employed throughout for free vibration
analysis of woven fiber composite plates in hygrothermal environment.
62
Table 5.11: Convergence of non-dimensional frequencies of vibration for SSSS
four layered laminated composite plates for two lamination sequences
at 325K temperature
a/b =1, a/t= 100, At T=300K, E1=130Gpa, E2=9.5Gpa, G12=6Gpa, G13=G12,
G23=0.5G12, ν12=0.3, α1=-0.3 X 10-6/OK, α2 =28.1 X 10-6/OK
Non dimensional frequency, λ = ωna2
Mess Division
ρ / E2 t 2
Non- dimensional frequencies at 325K
Temperature
0/90/90/0
45/-45/-45/45
4x4
8.079
11.380
6x6
8.039
10.785
8x8
8.036
10.680
10x10
8.036
10.680
Table 5.12: Convergence of non-dimensional frequencies of vibration for SSSS
four layered laminated composite plates for two lamination sequences
at 0.1% moisture concentration
a/b =1, a/t= 100, At T=300K, E1=130Gpa, E2=9.5Gpa, G12=6Gpa, G13=G12,
G23=0.5G12, ν12=0.3 β1=0, β2=0.44
Non dimensional frequency, λ = ωna2
Mess Division
ρ / E2 t 2
Non- dimensional frequencies at 0.1% Moisture
concentration
0/90/90/0
45/-45/-45/45
4x4
9.422
12.383
6x6
9.387
11.858
8x8
9.384
11.765
10x10
9.384
11.765
63
5.2.1.2: Comparison with previous studies
The present formulation is validated for free vibration analysis of composite plates
subjected to temperature and moisture as shown in Table 5.11 & Table 5.12. The
square plate has four layers of Graphite / Epoxy composite. The four lowest nondimensional frequency parameters of the composite plate under hygrothermal
loadings obtained by the present finite element are compared with numerical solution
published by Sairam and Sinha [1992] and with those of Shen, Zheng and Huang
[2004] using a micro-to-macro mechanical analytical model. The present finite
element results show good agreement with the previous results in the literature.
Table 5.13: Comparison of non-dimensional free vibration frequencies for SSSS
(0/90/90/0) plates at 325K Temperature.
a/b=1, a/t=100, At T = 300K, E1= 130Gpa , E2= 9.5Gpa, G12= 6Gpa, G13=G12,
G23=0.5G12,
ν12=0.3, α1=-0.3 X 10-6/OK, α2 =28.1 X 10-6/OK
Non dimensional frequency, λ = ωna2
ρ / E2 t 2
Non- dimensional frequencies at 325K Temperature
Mode number
Shen, Zheng and
Sairam & Sinha
Huang [2004]
[1992]
Present FEM
1
7.702
8.088
8.079
2
17.658
19.196
19.100
3
38.312
39.324
39.335
4
44.038
45.431
45.350
64
Table 5.14: Comparison of non-dimensional free vibration frequencies for SSSS
(0/90/90/0) Plates at 0.1% moisture concentration.
a/b=1, a/t=100, At T = 300K, E1= 130Gpa , E2= 9.5Gpa, G12= 6Gpa, G13=G12,
G23=0.5G12, ν12=0.3, β1=0, β2=0.44
ρ / E t2
2
Non dimensional frequency, λ = ωna2
Non- dimensional frequencies at 0.1% Moisture concentration
Mode number
Shen, Zheng and
Sairam & Sinha
Huang [2004]
[1992]
Present FEM
1
9.413
9.429
9.422
2
19.867
20.679
20.597
3
39.277
40.068
40.084
4
45.518
46.752
46.708
5.2.1.3: New results for free vibration
New results are presented for modal analysis of woven fiber laminated composite
plates in hygrothermal environment. The FEM results based on the present formulation
are calculated for vibration frequencies in Hz with respect to rise in temperature and
moisture concentration is given below with the following parameter.
•
four different modes of frequencies
•
Ply orientation
•
Number of layers
•
Aspect ratios
•
Side to thickness ratios
The frequencies of vibration of woven fiber composite plates subjected to
hygrothermal environment are observed by using the experimental setup. The variation
of frequencies of vibration in Hz of laminated plates (both experimental and
numerical) for lowest four modes subjected to temperature is shown in fig 5.13. The
frequencies of vibration of composite plates decrease with increase of temperature due
to reduction of stiffness. The variation of frequencies in Hz of woven fiber laminated
plates, for lowest four modes subjected to moisture concentration is shown in fig 5.14.
The frequencies of vibration decrease with increase of percentage of moisture.
65
Frequency in Hz
1600
1400
1200
1000
800
600
400
200
0
Expmt-M1
FEM-M1
Expmt-M2
FEM-M2
Expmt-M3
FEM-M3
Expmt-M4
FEM-M4
300
325
350
375
400
425
Temperature in K
Figure 5.13: Variation of frequency in Hz with temperature for simply
supported of 16 layers [0/0]4S woven fiber composite plates
As increase in temperature beyond 400K and 1% moisture concentration for all four
lowest mode vibration frequencies are reduced. The experimentally and numerically
determined vibration frequencies of woven fiber composite plates with uniform rise in
temperature and moisture with respect to their mode shapes (lowest four modes) are
shown in figure 5.13 and 5.14 respectively.
The experimentally determined vibration frequencies of symmetric sixteen layers
cross-ply laminates with four edges simply supported, is 4%, 8%, 10%, 12% and 3%,
4%, 6%, 7% higher than its numerical counterparts with respect to lowest four modes
respectively in hygrothermal environment. The agreement between experimentally
and numerically determined resonant frequencies becomes less with higher modes.
The reason for variation of frequencies is due to interlaminar or transverse shear
deformation and rotary inertia become important even for thin laminate like present
one as the wavelength becomes smaller with the rise of the resonant frequency.
1200
Expmt-M1
FEM-M1
Expmt-M2
FEM-M2
Expmt-M3
FEM-M3
Expmt-M4
FEM-M4
Frequency in Hz
1000
800
600
400
200
0
0
0.25
0.5
0.75
Moisture Concentration in %
1
Figure 5.14: Variation of frequency in Hz with moisture concentration for simply
supported of 16 layers [0/0]4S woven fiber composite plates
66
Similar observation is also observed in the study of Vibration of composite plates
without hygrothermal loading by Chaudhury et al. (2005). The study is then further
extended to free vibration of woven fiber composite plates for different lamination
sequence. Sixteen layered symmetric and anti symmetric laminates with angle of fiber
orientations varying from 00 to 900 are analyzed. As shown in figure 5.15 and 5.16,
the frequencies of vibration in Hz decrease with increase in temperature and moisture
concentration for laminates with symmetric and anti-symmetric lay-up. It is observed
that the fundamental frequency of vibration for anti-symmetric laminates is more than
that for symmetric laminates. This indicates that the free vibration behavior is not
much affected by ply orientation for this lay-up. The study is further extended to
investigate the effects of other parameters numerically using FEM.
Frequency in Hz
250
200
symmetric cross-ply
symmetric angle-ply
150
anti symmetric cross-ply
100
anti symmetric angle-ply
50
0
300
325
350
375
400
425
Temperature in K
Figure 5.15: Variation of frequency in Hz with temperature for simply supported
of 16 layers [0/90] 4S, [45/-45]4S and [0/90]8, [45/-45]8 woven fiber composite plates
Frequency in Hz
250
symmetric cross-ply
symmetric angle-ply
anti symmetric cross-ply
anti symmetric angle-ply
200
150
100
50
0
0
0.25
0.5
0.75
Moisture Concetration in%
Figure 5.16: Variation of frequency in Hz with moisture concentration for simply
supported of 16 layers [0/90]4S, [45/-45]4S and [0/90]8, [45/-45]8 woven
fiber composite plates
The variation of frequencies in Hz of woven fiber composite plates subjected to rise in
temperature and moisture concentration with different number of layers is shown in
figure 5.17 and 5.18. The frequencies of vibration decrease with increase in
67
temperature and moisture for different number of layers. For laminates with
symmetric lay-up, the frequencies of vibration increases gradually with increase in the
number of layers. The severe hygrothermal environment shall reduce the stiffness of
the composite plates.
Frequency in Hz
250
Eight layers
Twelve layers
Sixteen layers
200
150
100
50
0
300
325
350
375
400
425
Temperature in K
Figure 5.17: Variation of frequency in Hz with temperature for simply supported
of 16 layers [0/0]4S, 12 layers [0/0]3S, 8 layers [0/0]2S woven fiber composite plates
In the present investigation, results are presented for laminates subjected to uniform
distribution of temperature and moisture concentration. Sixteen layered Glass
fiber/epoxy laminates with simply supported boundary condition have been analyzed
experimentally and all computations are made with FEM in MATLAB code.
250
Eight layers
Twelve layers
Sixteen layers
Frequency in Hz
200
150
100
50
0
0
0.25
0.5
0.75
1
Moisture Concentration in %
Figure 5.18: Variation of frequency in Hz with moisture concentration for simply
supported of 16 layers [0/0]4S, 12 layers [0/0]3S, 8 layers [0/0]2S woven fiber
composite plates
The vibration frequencies in Hz are reported. The aspect ratios considered are 0.5, 1.0
and 2 as shown in figure 5.19 and 5.20. As increase in aspect ratios the frequencies of
vibration decreases with increase in temperature and moisture concentration due to
reduction of stiffness of the plate. It is observed that for aspect ratio 1 and 2 beyond
temperature 400K and moisture concentration 0.75%, frequency of vibration is
68
decreased and approaches to zero with increase in temperature and moisture
Frequency in Hz
concentration due to Hygrothermal buckling starts beyond that point.
700
SSSS,FEM(a/b=0.5)
600
SSSS,FEM (a/b=1)
500
SSSS,FEM(a/b=2)
400
300
200
100
0
300
325
350
375
400
425
Temperature in K
Frequency in Hz
Figure 5.19: Variation of frequency in Hz with temperature for simply supported
of 16 layers [0/0]4S woven fiber composite plates
700
600
500
400
300
200
100
0
SSSS,FEM(a/b=0.5)
SSSS,FEM(a/b=1)
SSSS,FEM(a/b=2)
0
0.25
0.5
0.75
1
Moisture Concentration in %
Figure 5.20: Variation of frequency in Hz with moisture concentration for simply
supported of [0/0]4S woven fiber 16 layers composite plates
The side to thickness ratios is considered as 25, 40 and 50, as shown in figure 5.21
with increase in temperature and 5.22 with increase in moisture environment.
Changes in hygrothermal environment are function of thickness only. It is observed
that high temperature and moisture concentration will soften the composite plate. The
thicker plate has the stronger stiffness and naturally it has higher vibration frequency
as seen in the figure 5.21 and 5.22 in hygrothermal environment. The reason behind
the variation of vibration frequencies has rendered the plate more susceptible to
localized shear deformation.
69
Frequency in Hz
400
SSSS,FEM(b/t=25)
SSSS,FEM(b/t=40)
300
SSSS,FEM(b/t=50)
200
100
0
300
325
350
375
400
425
Temperature in K
Figure 5.21: Variation of frequency in Hz with temperature for simply supported
of 16 layers [0/0]4S woven fiber composite plates
Frequency in Hz
400
SSSS,FEM(b/t=25)
SSSS,FEM(b/t=40)
300
SSSS,FEM(b/t=50)
200
100
0
0
0.25
0.5
0.75
1
Moisture Concentration in %
Figure 5.22: Variation of frequency in Hz with moisture concentration for simply
supported of 16 layers [0/0]4S woven fiber composite plates
The experimental results for sixteen layered Glass fiber/epoxy with simply supported
boundary condition having side to thickness ratios are 40 and 50 are compared with
four edges clamped plates, as shown in figure 5.23 with increase in temperature and
figure 5.24 with increase in moisture environment. The first mode experimental
vibration frequency decrease with increase in side to thickness ratios for simply
supported and clamped boundary conditions. The experimental vibration frequency is
however higher than its numerical counterparts are due to elastic rigidities.
70
Frequency in hZ
CCCC,Expmt(b/t=40)
CCCC,Expmt(b/t=50)
SSSS,Expmt(b/t=40)
SSSS,Expmt(b/t=50)
700
600
500
400
300
200
100
0
300
325
350
375
400
425
Temperature in K
Figure 5.23: Variation of frequency in Hz with temperature for clamped of16
layers [0/0]4S woven fiber composite plates
The clamped plate is subjected to a more sever hygrothermal changes than simply
supported plates due to rigid boundaries. The hygrothermal environment shall reduce
the stiffness of the composite plates in both clamped and simply supported boundary
conditions. The zero frequency point means that the hygrothermal buckling will occur
at that point. The vibration frequencies for four sides clamped boundary condition has
higher vibration frequencies than simply supported one due to better elastic rigidities
and clamping effects at the edges.
CCCC,Expmt(b/t=40)
CCCC,Expmt(b/t=50)
SSSS,Expmt(b/t=40)
SSSS,Expmt(b/t=50)
700
Frequency in Hz
600
500
400
300
200
100
0
0
0.25
0.5
0.75
Moisture Concentration in %
Figure 5.24: Variation of frequency in Hz with moisture concentration for
clamped of 16 layers [0/0]4S woven fiber composite plates
71
1
5.2.2: Buckling effects of woven fiber laminated composite plates in
hygrothermal environment
The example considered here consists of a woven fiber laminated composite plate
0.235mx0.235mx0.006m under hygrothermal loadings with four sides simply
supported, two sides clamped and two sides free boundary conditions and subjected to
a uniaxial in- plane load as shown in figure 4.15
5.2.2.1: Convergence study
The convergence study is carried out for non-dimensional critical load at a
temperature of 325K and 0.1% moisture concentration for different mesh divisions is
shown in Table 5.13 and 5.14 respectively. As observed, a mesh of 10 ×10 shows
good convergence of the numerical solution for the free vibration and buckling
analysis of woven fiber composite plates in hygrothermal environment and this mesh
is employed throughout for free vibration and buckling analysis of woven fiber
composite plates in hygrothermal environment.
Table 5.15: Convergence of non-dimensional critical load for SSSS four layered
laminated composite plates at 325K temperature.
a/b =1, a/t= 100, At T=300K, E1=130Gpa, E2=9.5Gpa, G12=6Gpa, G13=G12,
G23=0.5G12, ν12=0.3, α1=-0.3 X 10-6/OK, α2 =28.1 X 10-6/OK
Critical load, λ = Nxcr / ( Nxcr)C=0% or T =300K
Mess Division
Non- dimensional critical load at 325K Temperature
0/90/90/0
45/-45/-45/45
4x4
0.4481
0.6120
6x6
0.4459
0.5818
8x8
0.4457
0.5764
10x10
0.4457
0.5745
72
Table 5.16: Convergence of non-dimensional critical load for SSSS four layered
laminated composite plates at 0.1% moisture concentration.
a/b=1, a/t=100, At T=300K, E1=130Gpa, E2=9.5Gpa, G12=6Gpa, G13=G12,
G23=0.5G12, ν12=0.3 β1=0, β2=0.44
Critical load, λ = Nxcr / ( Nxcr)C=0% or T =300K
Mess Division
Non- dimensional critical load at 0.1% Moisture
concentration
0/90/90/0
45/-45/-45/45
4x4
0.6095
0.7255
6x6
0.6079
0.7041
8x8
0.6078
0.7003
10x10
0.6078
0.7003
5.2.2.2: Comparison with previous studies
The present formulation is then validated for buckling analysis of composite plates for
temperature and moisture as shown in Table 5.15. The square plate has four layers of
Graphite / Epoxy composite. The non-dimensional critical load due to hygrothermal
loadings obtained by the present finite element is compared with analytical solution
published by Sairam and Sinha [1992] and Patel, Ganapathi and Makhecha [2002].
The present finite element results show good agreement with the previous analytical
results published in the literature for buckling of laminated composite plates.
73
Table 5.17: Comparison of non-dimensional critical load for SSSS (0/90/90/0)
layered laminated composite plates at 325K temperature and 0.1%
moisture concentration
a/b =1, a/t= 100, At T=300K, E1=130Gpa, E2=9.5Gpa, G12=6Gpa, G13=G12,
G23=0.5G12, ν12=0.3 α1=-0.3 X 10-6/OK, α2 =28.1 X 10-6/OK, β1=0, β2=0.44
Non-dimensional Critical load, λ = Nxcr / ( Nxcr)C=0% or T =300K
References
Non-dimensional critical load λ
At 325K
At 0.1%
Sairam & Sinha
[1992]
0.4488
0.6099
Patel,Ganapathi &
0.4466
0.6084
Makhecha [2002]
Present FEM
0.4481
0.6095
5.2.2.3. New results for buckling
Numerical results are presented for the static stability behavior of woven fiber
composite plates in hygrothermal environment for different parameter. The
geometrical and material properties of the laminated composite plates are:
a=b=0.235m, h=0.006m (unless otherwise stated). The material properties obtained
from tensile testing of glass/epoxy composite plates at different temperatures and
moisture as per ASTM D3039/D3039M- [2008] are shown in Table 5.9 and Table
5.10.
•
Ply orientation
•
Number of layers
•
Aspect ratios
•
Side to thickness ratios
The results for buckling loads in KN of both the numerical analysis and experimental
values with increase in temperature from 300K to 425K in every 25K rise in
temperature and 0 to 1% in every 0.25% rise in moisture concentration of sixteen
layered woven roving glass fiber/epoxy composites plates are presented for clampedfree-clamped-free boundary condition (CFCF). The variation of buckling loads with
increase in temperature and moisture concentrations of the plate is shown in figure
74
5.25 and 5.26. This shows that there is a good agreement between experimental and
numerical results within prescribed FEM formulation.
It is observed that due to the increase in temperature and moisture concentration, there
is a decrease in critical buckling loads due to reduction of stiffness and strength. The
effect of temperature generally causes a softening of the fibers and the effect of
moisture causes plasticization due to absorbed moisture.
Buckling Load in KN
45
Expmt
FEM
40
35
30
25
300
325
350
375
Temperature in K
400
425
Figure 5.25: Variation of buckling load in KN with temperature of 16 layers
[0/90]4S woven fiber composite plates (C-F-C-F)
The critical buckling load decreased severely with increase in temperature upto 425K
and moisture concentration beyond 1%, in which hygrothermal buckling appears. In
thermal buckling, the composite plate does not remain perfectly flat and suddenly
develops a large deformation due to critical temperature stress. The plates will begin
to deform as soon as thermal stresses are developed. The deformation will then
increase rapidly due to increase in temperature as shown in figure 5.25. The reduction
in buckling loads is more pronounced at lower temperature and higher moisture is due
to the lowered glass transition temperature at increased moisture concentration. The
buckling load is reduced by approximately 27% when the plate is subjected to 1%
moisture concentration and temperature increase to 400K.
The effects of reduction in the buckling loads are more prominent for increase in
temperature as compared to moisture concentration. It is clearly observed that the
detrimental effect of the increased moisture concentration and temperature on the
stability of the plate. It is also seen that the hygroscopic condition on the stability of
the plate becomes more significant in presence of the thermal loading.
75
Buckling Load in KN
45
Expmt
FEM
40
35
30
25
0
0.25
0.5
0.75
1
Moisture concentration in %
Figure 5.26: Variation of buckling load in KN with moisture concentration of 6
layers [0/90]4S woven fiber composite plates (C-F-C-F)
The variation of buckling load in KN for sixteen layer of composite plates with
simply supported boundary conditions are presented for a/b=1 and b/t=40, subjected
to uniform distribution of temperature in every increase in temperature of 25K from
300K to 425K and moisture concentration of every increase in 0.25% moisture from 0
to1% with different lamination sequence is analyzed by using FEM formulations in
figure 5.27 and 5.28 respectively. It is shown that the buckling loads for antisymmetric laminates are more than symmetric laminates with increase in uniform
temperature and moisture concentration environment. With increase in temperature
and moisture concentration the reduction in buckling loads is linear. Due to this
reason, the woven fiber laminated plates is marginally affected in thermal and
moisture environment. This reduction in buckling loads is due to the effect of
decrease in shear modulus.
The reduction of buckling loads with increase in temperature from 300K to 425K, for
anti- symmetric laminates are more than 18% and symmetric laminates are about to
15%. Similarly the reductions of buckling loads for increase in moisture concentration
from 0 to 1%, for anti-symmetric laminates are nearly 11% and symmetric laminates
are approximately 9%.
76
Buckling Load in KN
50
40
30
Symmetric cross-ply
20
Anti-symmetric cross-ply
10
Symmetric angle-ply
Anti-symmetric angle-ply
0
300
325
350
375
400
425
Temperature in K
Figure 5.27: Variation of buckling load in KN with temperature of 16 layers
[0/90] 4S, [45/-45]4S and [0/90]8, [45/-45]8 woven fiber composite plates (S-S-S-S)
Buckling Load in KN
50
40
30
Symmetric cross-ply
20
Anti-symmetric cross-ply
10
Symmetric angle-ply
Anti symmetric angle-ply
0
0
0.25
0.5
0.75
1
Mosture concentration in %
Figure 5.28: Variation of buckling load in KN with moisture concentration of 16
layers [0/90] 4S, [45/-45]4S and [0/90]8, [45/-45]8 woven fiber composite
plates (S-S-S-S)
The variation of buckling loads in KN are observed for a/b=1 for four, eight and
twelve layered simply supported cross-ply symmetric laminated plates subjected to
uniform distribution of temperature from 300K to 425K in every rise of 25K
temperature and moisture concentration from 0 to 1% in every rise of 0.25% moisture
are reported by using present FEM formulation in figure 5.29 and 5.30 respectively. It
is observed that the buckling loads increase with increase in number of layers up to
eight layers beyond that, increase in number of layers the buckling loads remain
constant at uniform increase in temperature and moisture concentration. It is also
noted that with increase in temperature hardening type non linear behavior of the
buckling loads for all layers of laminates. The buckling loads decrease with increase
in temperature and moisture concentration environment.
The reductions in buckling loads with increase in temperature from 300K to 425K are
22% and for increase in moisture concentration from 0 to 1% are approximately 9%.
This shows that the laminates are severely affected in higher thermal environment.
77
Buckling load in KN
50
45
40
35
30
25
20
15
10
Four layers
Eight layers
Twelve layers
300
325
350
375
Temperature in K
400
425
Figure 5.29: Variation of buckling load in KN with temperature of 4 layers
[0/90]1S, 8 layers [0/90]2S, 12 layers [0/90]3S woven fiber composite plates
(S-S-S-S)
The variation of buckling loads in KN for different aspect ratios such as a/b=0.5,
a/b=1 and a/b=2 for b/t=40, sixteen layered simply supported boundary conditions for
symmetric cross-ply laminated plates are subjected to uniform increase of temperature
from 300K to 425K in every 25K rise in temperature and moisture concentrations
from 0 to 1% in every rise of 0.25% moisture concentration by using FEM results are
as shown in figure 5.31 and 5.32 respectively. It is also observed that the buckling
loads increase with increase in aspect ratios of woven fiber laminated composite
plates in hygrothermal environment. Hygrothermal buckling also starts at room
temperature about 300K which is known as stress–free level. Similarly hygroscopic
buckling will start at 0.25% of moisture concentration which is also known as stressfree level. As the hygrothermal stress resultants are integrated quantities, their effect
Buckling load in KN
increases with the absorption of more moisture until equilibrium is reached.
50
45
40
35
30
25
20
15
10
Four layers
Eight layers
Twelve layers
0
0.25
0.5
0.75
1
Moisture concentration in %
Figure 5.30: Variation of buckling load in KN with moisture concentration of 4
layers [0/90]1S, 8 layers [0/90]2S, 12 layers [0/90]3S woven fiber
composite plates (S-S-S-S)
78
The reduction in buckling loads for increase in temperature from 300K to 425K are
12.61% for aspect ratio 0.5 and 22.42% for aspect ratios 1 and 9.23% for aspect ratio
2. Similarly the reduction in buckling loads for increase in moisture concentration
from 0 to 1% are 6% for aspect ratio 0.5 and 12% for aspect ratios 1 and 1% for
aspect ratio 2. The woven fiber laminated composite plates having aspect ratio more
Bucklimng Load KN
than 1 are stable and affected substantially in hygrothermal environment.
65
60
55
50
45
40
35
30
a/b=0.5
a/b=1
a/b=2
300
325
350
375
400
425
Temperature in K
Buckling Load in KN
Figure 5.31: Variation of buckling load in KN with temperature of 16 layers
[0/90]4S woven fiber composite plates (S-S-S-S)
65
60
55
50
45
40
35
30
a/b=0.5
a/b=1
a/b=2
0
0.25
0.5
0.75
1
Moisture concentration in %
Figure 5.32: Variation of buckling load in KN with moisture concentration of 16
layers [0/90]4S woven fiber composite plates (S-S-S-S)
The experimental results of buckling loads for different aspect ratios a/b=0.5, a/b=1
and a/b=2 for b/t=40, sixteen layered cross-ply symmetric woven fiber laminated
composite plates for clamped-free-clamped-free boundary conditions are subjected to
uniform distribution of temperature from 300Kto 425K in every rise in temperature of
25K and moisture concentration 0 to 1% in every rise of 0.25% moisture are
presented in figure 5.33 and 5.34 respectively. From the results it is clear that the
buckling loads are increase with increase in aspect ratios in hygrothermal
79
environment. The reduction of buckling loads with increase in temperature and
moisture concentration is linear. The buckling loads for CFCF boundary conditions
are more than four sides simply supported boundary conditions due to rigid
boundaries in hygrothermal environment.
The reduction in buckling loads from increase in temperature from 300K to 425K,
the experimental results for CFCF boundary conditions with aspect ratio for a/b=0.5 is
8.81%, a/b=1 is 11.04 and a/b=2 is 11.04% respectively. Similarly The reduction in
buckling loads from increase in moisture from 0 to 1%, the experimental results for
CFCF boundary conditions with aspect ratios for a/b=0.5 is 15.28%. a/b=1 is 19.7%
and a/b=2 is 22.92% respectively. The reduction in buckling loads for higher aspect
ratio is more than lower one in severe hygrothermal environment.
Buckling load in KN
65
CFCF, Expmt (a/b=0.5)
CFCF, Expmt (a/b=1)
CFCF, Expmt (a/b=2)
60
55
50
45
40
35
30
300
325
350
375
400
425
Temperature in K
Buckling load in KN
Figure 5.33: Variation of buckling load in KN with temperature of 16 layers
[0/90]4S woven fiber composite plates (C-F-C-F) boundary Condition
65
60
55
50
45
40
35
30
CFCF,Expmt (a/b=0.5)
CFCF, Expmt (a/b=1)
CFCF,Expmt (a/b=2)
0
0.25
0.5
0.75
1
Moisture concentration in %
Figure 5.34: Variation of buckling load in KN with moisture concentration 16
layers [0/90]4S of woven fiber composite plates (C-F-C-F) boundary condition
The variation of buckling loads for different side to thickness ratios b/t=25, b/t=40
and b/t=100 for a/b=1, sixteen layered cross-ply symmetric woven fiber laminated
80
plates for simply supported boundary conditions are subjected to uniform distribution
of temperature from 300Kto 425K in every rise in temperature of 25K and moisture
concentration 0 to 1% in every rise of 0.25% moisture are observed using FEM results
in figure 5.35 and 5.36. From the results it is clear that the buckling loads are more for
plates with lower side-to-thickness ratios than higher side-to-thickness ratios. The
reduction in buckling loads is linear with increase in temperature and moisture
concentration. This present method of analysis has been found efficient in order to
evaluate the buckling load of moderately thick composite laminated plate subjected to
hygrothermal loading. It is seen that the buckling load decreases with increase in
moisture concentration for different aspect and side-to-thickness ratios due to
degradation in material properties at higher temperature and moisture concentration.
It is also evident from the figure 35 that the moderately thick laminated composite
plates are more stable than thin laminated plates with increase in temperature and
moisture concentration environment. The random changes in thickness have more
impact on hygrothermal buckling loads scattering as compared to individual random
changes in material properties. The sensitivity of hygrothermal buckling loads is due
to variation in geometric and material properties which is dependent on thickness
only. The reduction in buckling loads with increase in temperature from 300K to
425K are 21.85% for b/t=25, 15.36% for b/t=50 and 24.82% for b/t=100. Similarly
the reduction in buckling loads with increase in moisture concentration from 0 to 1%
are 17.15% for b/t=25, 4.6% for b/t=50 and 4.32% for b/t=100. It is also observed that
the moderately thick woven fiber laminated composite plates are more susceptible to
hygrothermal environment and losses its stiffness and strength early as compared to
thin composite plates.
Buckling Load in KN
50
40
b/t=25
b/t=50
b/t=100
30
20
10
0
300
325
350
375
400
425
Temperature in K
Figure 5.35: Variation of buckling load in KN with temperature of 16 layers
[0/90]4S woven fiber composite plates (S-S-S-S)
81
Buckling Load in KN
50
40
30
20
b/t=25
b/t=50
b/t=100
10
0
0
0.25
0.5
0.75
1
Moisture concentration in %
Figure 5.36: Variation of buckling load in KN with moisture concentration of 16
layers [0/90]4S woven fiber composite plates (S-S-S-S)
The experimental results of buckling loads for different side to thickness ratios b/t=40
and b/t=50 for a/b=1, sixteen layered cross-ply symmetric woven fiber laminated
composite plates for clamped-free-clamped-free boundary conditions are subjected to
uniform distribution of temperature from 300Kto 425K in every rise in temperature of
25K and moisture concentration 0 to 1% in every rise of 0.25% moisture are
presented in figure 5.37 and 5.38 respectively.
From the results, it is clear that the buckling loads are decreases with increase in sideto-thickness ratio in hygrothermal environment. The reduction of buckling loads with
increase in temperature and moisture concentration is non linear, due to changes in
plate thickness. The critical buckling loads for CFCF boundary conditions are more
than four sides simply supported boundary conditions due to rigid boundaries in
hygrothermal environment.
The reduction in buckling loads from increase in temperature from 300K to 425K,
the experimental results for CFCF boundary conditions with side-to-thickness ratio
for b/t=40 is 20.54% and b/t=50 is 22.68% respectively. Similarly The reduction in
buckling loads from increase in moisture from 0 to 1%, the experimental results for
CFCF boundary conditions with side-to-thickness ratio for b/t=40 is 23.87% and
b/t=50 is 24.33% respectively. The reduction in buckling loads for higher side-tothickness ratio is more than lower values in hygrothermal environment due to lower
thickness of the plates.
82
50
Buckling load in KN
CFCF, Expmt (b/t=40)
45
CFCF, Expmt (b/t=50)
40
35
30
300
325
350
375
400
425
Temperature in K
Figure 5.37: Variation of buckling load in KN with temperature of 16 layers
[0/90]4S woven fiber composite plates (C-F-C-F) boundary condition.
Buckling load in KN
50
CFCF, Expmt (b/t=40)
CFCF, Expmt (b/t=50)
45
40
35
30
0
0.25
0.5
0.75
1
Moisture Concetration in %
Figure 5.38: Variation of buckling load in KN with moisture concentration of 16
layers [0/90]4S woven fiber composite plates (C-F-C-F) boundary condition
83
5.2.3: Dynamic instability of woven fiber laminated composite plates
in hygrothermal environment
The numerical results of the study of behavior of woven fiber composite plates
subjected to hygrothermal conditions under in-plane periodic loads are presented
using the formulations as follows.
5.2.4: Non-dimensionalization of parameters
Non-dimensional frequency λ = ωna2
ρ / E2 t 2
Non-dimensional Critical load, λ = Nxcr / ( Nxcr)
Non-dimensional excitation frequency Ω = Ωa
2
( ρ / E22 h2 )
(unless otherwise stated) is used throughout the dynamic instability studies, where
Ω is the excitation frequency in radians/second.
5.2.4.1: Convergence study
The convergence study is done for simply supported square 4 layer symmetric crossply and symmetric angle-ply laminated composite plates for elevated temperature and
moisture conditions for different mess divisions. Dynamic stability analysis of
laminated composite plates subjected to hygrothermal conditions under in-plane
periodic loads are presented in Table 5.18 and Table 5.19. As observed, a mesh of 10
×10 shows good convergence of the numerical solution of dynamic stability of woven
fiber composite plates and this mesh is employed throughout the dynamic stability
analysis of woven fiber composite plates in hygrothermal environment.
84
Table 5.18: Convergence of non-dimensional excitation frequency for SSSS four
layered laminated composite plates for different ply orientations
at 325K temperature
a/b =1, a/t= 100, At T=300K, E1=130Gpa, E2=9.5Gpa, G12=6Gpa, G13=G12,
G23=0.5G12, ν12=0.3, α1=-0.3 X 10-6/OK, α2 =28.1 X 10-6/OK , α = static load factor =
0.2, β = dynamic load factor = 0.2
Non-dimensional excitation frequency Ω = Ωa
Mess
Division
2
( ρ / E22 h2 )
Non-dimensional excitation frequency at 325K
Temperature
0/90/90/0
45/-45/-45/45
4x4
15.23
22.10
6x6
15.16
20.86
8x8
15.16
20.64
10x10
15.16
20.56
Table 5.19: Convergence of non-dimensional excitation frequency for SSSS four
layered laminated composite plates for different ply orientations at
0.1% moisture concentration.
a/b =1, a/t= 100, At T=300K,E1=130Gpa, E2=9.5Gpa, G12=6Gpa, G13=G12,
G23=0.5G12, ν12=0.3 β1=0, β2=0.44,α = static load factor = 0.2, β = dynamic load
factor = 0.2
Non-dimensional excitation frequency Ω = Ωa
Mess Division
2
( ρ / E22 h2 )
Non-dimensional excitation frequency at 0.1%
Moisture concentration
0/90/90/0
45/-45/-45/45
4x4
18.05
24.17
6x6
17.99
23.07
8x8
17.99
22.88
10x10
17.99
22.82
5.2.4.2: Comparison with previous studies
To validate the formulation, the frequencies of excitation frequency during parametric
resonance of laminated composite plates are computed and compared with previously
85
published results from literature without hygrothermal environment as shown in Table
5.20. The boundary frequencies of square plate have four layers of symmetric crossply laminates. The boundary frequencies without hygrothermal environment obtained
by the present finite element are compared with analytical solution obtained by Wang
and Dawe [2002]. The present finite element results show good agreement with the
previous analytical results published in the literature for boundary frequencies.
Table 5.20: Comparison of dimensional excitation frequency for SSSS four
layered laminated composite plates for symmetric cross-ply laminates
a/b =1, a/t= 100, At T=300K, G12/ E2= G13/ E2 =0.6, G23/ E2= 0.5, ν12= 0.25
Non-dimensional excitation frequency Ω = Ωa
α = static
β=
ωU
ωL
load factor
dynamic
Upper bound
Lower bound
load factor
frequency
frequency
0
0
144.57
0
0.3
0
2
( ρ / E22 h2 )
Present FEM
ωU
ωL
144.57
143.67
143.67
155.03
133.29
153.12
131.23
0.6
164.83
120.95
162.32
118.12
0
0.9
174.08
107.21
171.56
105.87
0
1.2
182.87
91.43
180.45
89.92
0
1.5
191.25
72.28
189.34
70.59
0
0
144.57
144.57
143.67
143.67
0.2
0.06
131.71
126.86
129.77
124.85
0.4
0.12
117.45
106.24
115.28
104.33
0.6
0.18
101.20
80.49
99.47
78.27
0.8
0.24
81.78
40.89
78.94
38.92
5.2.4.3: New results for dynamic stability
After validating the present formulation, the same is employed to study the effect of
different parameters on the dynamic instability effects of woven fiber composite
plates in hygrothermal environment The parameters are:
86
•
Number of layers
•
Aspect ratios
•
Side to thickness ratios
•
Static load factor
•
Increase in hygrothermal conditions
•
Ply orientation
•
Lamination sequence
•
Degree of orthotropy
The non-dimensional excitation frequency Ω = Ωa
2
( ρ / E22 h2 ) is used throughout
the dynamic instability studies, where Ω is the excitation frequency in radian/second.
The principal instability regions of woven fiber laminated composite flat panel
subjected to in-plane periodic loads is plotted with non-dimensional frequency Ω/ω
(ratio of excitation frequency to the free vibration frequency) versus the dynamic inplane load β. Here, a static load factor α=0.2 is taken for parametric study of
laminated composite plates in hygrothermal environment unless otherwise stated.
The structural instability may lead to large deflection or large amplitude vibrations of
structural elements leading to local or global failures. So the analysis is focused on the
determination of the primary instability region of laminated composite plates under
hygrothermal loads. The width of primary instability region frequencies is the
separation of the boundaries of the primary instability region for the given plate. This
can be used as an instability measure to study the influence of the other parameters.
This is the most dangerous zone and has the greatest practical importance. As can be
seen, the primary instability region that occurs in the vicinity of 2ω (α = β =0) and the
upper and lower excitation frequencies of the plates decrease with the increase of the
static load parameter. It is also observed that the primary instability region for each
plate increase with increasing static and dynamic load parameter, and the width of the
unstable zone is becoming more significant at the higher load parameter. The
spectrum of the values of parameters causing unstable motion is called the dynamic
instability region or DIR or parametric resonance. The industry driven woven fiber
composite plates subjected to hygrothermal environment are considered here to study
the effect of different parameters on the excitation frequency and width of instability
regions of composite plates. The geometrical and material properties of the laminated
87
composite plates are: a=b=0.235m, h=0.006m. The material properties obtained from
tensile testing of glass/epoxy composite plates at different temperatures and moisture
as per ASTM D3039/D3039M- [2008] are shown in Table 5.9 and Table 5.10.
The effect of increase in number of layers with thickness of the laminated composite
plates on the non-dimensional excitation frequency is illustrated in figure 5.39 at
temperature 325K. The 8, 12 and 16 layer laminated plates are having thickness of
6mm, 9mm and 12mm respectively. As observed, the onset of instability occurs later
with narrow instability regions with increase in number of layers due to higher
stiffness. As expected, woven fiber laminated plates is more stable with increase in
number of layers under periodic loads due to bending stretching coupling.
The
variation of excitation frequency with increase of dynamic load is studied for
composite plates with 0.25% moisture concentration for different increase in number
of layers as shown in figure 5.40. As observed, the onset of instability occurs earlier
and the width of instability regions becomes smaller with decrease in number of
layers.
Dynamic load factor
1
8Layers
12Layers
16Layers
0.8
0.6
0.4
0.2
0
6
7
8
9
10
11
12
Nondimensional excitation frequencies
Figure 5.39: Variation of instability regions with temperature at 325K for simply
supported of a/b=1, α= 0.2, woven fiber composite plates
The laminated composite plate affected severely and loses its strength and stiffness in
hygrothermal environment. All further parametric studies are done with an eight
layers laminate combinations. The effects of number of layers shift the instability
region to larger excitation frequencies.
88
Dynamic load factor
1
8Layers
12Layers
16Layers
0.8
0.6
0.4
0.2
0
6
7
8
9
10
11
12
Nondimensional excitation frequencies
Figure 5.40: Variation of instability regions with moisture concentration at
0.25% for simply supported of a/b=1, α= 0.2, woven fiber composite plates
The effect of increase in aspect ratio for a/b=0.5, 1 and 2 on the non-dimensional
excitation frequencies is analyzed for composite plates as shown in figure 5.41 at
temperature 325K. The onset of instability occurs earlier for composite plates with
increase in aspect ratios. With increase in aspect ratios, the excitation frequencies are
decreased, due to reduction of stiffness of the plates in hygrothermal environment.
Dynamic load factor
1
a/b=0.5
a/b=1
a/b=2
0.8
0.6
0.4
0.2
0
0
10
20
30
40
50
Nondimensional excitation frequency
Figure 5.41: Variation of instability regions with temperature at 325K for simply
supported of a/b=1, α= 0.2, woven fiber composite plates
The effect of increase in aspect ratios on the non-dimensional excitation frequencies is
studied for composite plates as shown in figure 5.42 at moisture concentration 0.25%.
The onset instability occurs later for square plates than rectangular plates with wider
instability region. The excitation frequencies are decreased with increase in aspect
ratios even for composite plates under moisture conditions.
89
Dynamic load factor
1
a/b=0.5
a/b=1
a/b=2
0.8
0.6
0.4
0.2
0
0
10
20
30
40
50
Nondimensional excitation frequency
Figure 5.42: Variation of instability regions with moisture concentration at
0.25% for simply supported of a/b=1, α= 0.2, woven fiber composite plates
The effect of increase in side to thickness ratio for b/t=25 and 50 on the nondimensional excitation frequencies is investigated for composite plates as shown in
figure 5.43 at temperature 325K. It is observed from the figure that the onset of
instability occurs earlier for increase in side to thickness ratio. The excitation
frequencies are decreased with increase in side to thickness ratio. The thick plates are
more dynamically stable than thin plates in temperature environment.
b/t=25
b/t=50
Dynamic load factor
1
0.8
0.6
0.4
0.2
0
0
5
10
15
Nondimensional excitation frequency
Figure 5.43: Variation of instability regions with temperature at 325K for simply
supported of a/b=1, α= 0.2, woven fiber composite plates
The effect of increase in side to thickness ratio for b/t=25 and 50 on the nondimensional excitation frequencies is investigated for composite plates as shown in
figure 5.44 at a moisture 0.25%. It is observed from the figure that the onset of
instability occurs earlier for increase in side to thickness ratio. The excitation
frequencies are decreased with increase in side to thickness ratio. The thick plates
having narrow instability region shows more stiffness and strength than thin plates in
hygrothermal environment. The width of instability regions increases due to increase
in dynamic loads.
90
Dynamic load factor
1
b/t=25
b/t=50
0.8
0.6
0.4
0.2
0
0
5
10
Nondimensional excitation frequency
15
Figure 5.44: Variation of instability regions with moisture concentration at
0.25% for simply supported of a/b=1, α= 0.2, woven fiber composite plates
The effect of increase in static load factor of eight layered anti-symmetric woven fiber
laminated composite plates on non-dimensional excitation frequency is analyzed in
figure 5.45 at temperature 325K. It is observed that with the increase of static load
factor from 0.4 to1, the onset of dynamic instability occurs earlier and the width of
dynamic instability region also increases. The reduction of excitation frequencies with
increase in static load factors from 0.4 to1 is more than 10% due to reduction of
stiffness and strength. All further studies are done with a static load factor α=0.2.
Dynamic load factor
1
α=0.4
α=0.6
α=0.8
α=1
0.8
0.6
0.4
0.2
0
9
10
11
12
Nondimensional excitation frequency
Figure 5.45: Variation of instability regions with temperature at 325K for simply
supported of [45/-45]4, woven fiber composite plates
The variations of dynamic instability regions with increase in static in-plane loads of
woven fiber composite plates at moisture concentrations 0.25% is observed as shown
in figure 5.46. The onset of instability occurs earlier with increase of compressive
static in-plane load, the instability region tend to shift to lower frequencies and wider.
The reduction of excitation frequencies with increase in static in-plane loads from 0.4
to 1 is more than 5%. The variations of dynamic instability of composite plates with
increase in temperature from 325K to 400K is shown in figure 5.47 with a static load
factor 0.2, it is observed that the onset of instability occurs earlier with increase in
91
temperature. The width of instability region decreased with increase in temperature
from 325K to 400K. The reduction of excitation frequencies with increase in
temperature from 325K to 400K is about 50%. Similarly with increase in moisture
concentration from0.25% to 1% is shown in figure 5.48 with static load factor 0.2, it
is observed that the onset of instability occurs earlier with increase in moisture
concentration. The width of instability region decreased with increase in moisture
concentration from 0.25% to 1%. The reduction of excitation frequencies with
increase in moisture concentration from 0.25% to 1% is more than 50%.
Dynamic load factor
1
α=0.4
α=0.6
α=0.8
α=1
0.8
0.6
0.4
0.2
0
9
9.5
10
10.5
11
Nondimensional excitation frequency
Figure 5.46: Variation of instability regions with moisture concentration at
0.25% for simply supported of [45/-45]4, woven fiber composite plates
325K
350K
375K
400K
Dynamic load factor
1
0.8
0.6
0.4
0.2
0
2
4
6
8
Non dimensional excitation frequencies
Figure 5.47: Variation of instability regions with different temperature for
simply supported of [45/-45]4, woven fiber composite plates
92
Dynamic load factor
1
0.25%
0.50%
0.75%
1%
0.8
0.6
0.4
0.2
0
2
4
6
8
Nondimensional excitation frequencies
10
Figure 5.48: Variation of instability regions with different moisture
concentration for simply supported of [45/-45]4, woven fiber composite plates
The study is further extended to study the effect of different laminated parameter on
composite plates under high temperature environment, here considered at temperature
325K. As shown in figure 5.49, it is observed that the onset of instability occurs for
symmetric laminates earlier than anti-symmetric laminates. The non-dimensional
excitation frequencies decreased 50% for laminates with symmetric lay-up than antisymmetric lay-up in temperature environment.
Dynamic load factor
1
[(0/90)2]s
[0/90]4
[(45/-45)2]s
[45/-45]4
0.8
0.6
0.4
0.2
0
4
6
8
10
12
Nondimensional excitation frequencies
Figure 5.49: Variation of instability regions with temperature at 325K for simply
supported α= 0.2, woven fiber composite plates
A study is also under taken to study the effect of different lamination sequence on
instability of composite plates under moisture environment. Similar symmetric and
anti-symmetric laminates having different lamination sequence with moisture
concentration 0.25% are considered in figure 5.50. The onset of instability occurs
earlier for symmetric laminates with wider instability regions, where as the narrow
instability regions for anti-symmetric laminates. Plates with 45o ply orientations seem
to be stiffer due to shifting of higher frequencies and narrow instability region. The
non-dimensional excitation frequencies decreased 50% for laminates with symmetric
lay-up than anti-symmetric lay-up in moisture environment.
93
[(0/90)2]s
[0/90]4
[(45/-45)2]s
[45/-45]4
Dynamic load factor
1
0.8
0.6
0.4
0.2
0
2
4
6
8
10
12
Nondimensional excitation frequencies
Figure 5.50: Variation of instability regions with moisture concentration at
0.25% for simply supported α= 0.2, woven fiber composite plates
The width of instability region increases with increase in both static and dynamic load
parameters. Further studies are done with anti-symmetric angle-ply laminated plates.
The effect of ply orientation 0o to 90o on the non-dimensional excitation frequencies
is considered here for anti-symmetric angle-ply laminated plates at temperature 325K
as shown in figure 5.51. The onset of instability occurs earlier with decrease of
lamination angle. The excitation frequencies beyond lamination angle 45o decreased
marginally. The excitation frequencies are decreased with increase in lamination
angle due to reduction of stiffness and strength of laminated plates. The reduction of
excitation frequencies with increase in lamination angle from 0o to 45o is about 25%.
The onset of instability and the width of instability region and its strength is highly
depends on lamination angle. The greater the lamination angle the smaller is the width
of instability region. The rectangular laminated plates show that the lamination angle
of 45o is symmetrical. For this the lamination angle of 45o seems to be the preferential
ply orientation for the lamination sequence which is due to the dominance effect of
bending-stretching coupling of laminates.
Dynamic load factor
1
0⁰
15⁰
30⁰
45⁰
60⁰
75⁰
90⁰
0.8
0.6
0.4
0.2
0
2
4
6
8
10
Nondimensional excitation frequencies
12
Figure 5.51: Variation of instability regions with temperature at 325K for
simply supported of [45/-45]4, α= 0.2, woven fiber composite plates
94
The effects of increase in lamination angle 0o to 90o on the non-dimensional
excitation frequencies is reported for composite plates at moisture concentration
0.25% is shown in figure 5.52. It is observed from the figure that the onset of
instability occurs for lower values of lamination angle. The reduction of excitation
frequencies with increase in lamination angle 0o to 45o is about 50%. Which shows
that the anti-symmetric angle-ply laminated plates is severely affected and reduced its
maximum strength and stiffness with increase in lamination angle in moisture
environment.
Dynamic load factor
1
0⁰
15⁰
0.8
30⁰
0.6
45⁰
0.4
60⁰
0.2
75⁰
0
90⁰
4
6
8
10
Nondimensional excitation frequencies
12
Figure 5.52: Variation of instability regions with moisture concentration at
0.25% for simply supported α= 0.2, woven fiber composite plates
The effect of degree of orthotropy is examined for the eight layered anti-symmetric
angle-ply laminated plates on the non-dimensional excitation frequency as shown in
figure 5.53 at temperature 325K. It is seen that with increase in the values of E1/E2
=10, 20 and 40, the onset of instability occurs earlier with decrease in degree of
orthotropy. The width of instability zones increases with increase of degree of
orthotropy in temperature environment.
Dynamic load factor
1
E1/E2=10
E1/E2=20
E1/E2=40
0.8
0.6
0.4
0.2
0
0
10
20
30
40
50
60
Nondimensional excitation frequencies
Figure 5.53: Variation of instability regions with temperature at 325K for simply
supported of [45/-45]4, α= 0.2, woven fiber composite plates
95
With the increase of degree orthotropy i.e. the increase in E1/E2 the instability occurs
at higher frequencies. The reduction of excitation frequency with decrease in E1/E2
=40 to 10 about 50%. This shows that there is worst variation of stiffness and strength
with increase in degree of orthotropy characteristics.
The variations of dynamic instability regions with increase in the values of degree of
orthotropy E1/E2 =10, 20 and 40 are presented as shown in figure 5.54 at a moisture
0.25%. The onset of instability occurs earlier with decrease in degree of orthotropy.
The width of instability region is narrow with increase in degree of orthotropy of
laminated plates, which shows more stiffness and strength in hygrothermal
environment.
Dynamic load factor
1
E1/E2=10
E1/E2=20
E1/E/2=40
0.8
0.6
0.4
0.2
0
0
10
20
30
40
50
60
Nondimensional excitation frequencies
Figure 5.54: Variation of instability regions with moisture concentration at
0.25% for simply supported of [45/-45]4, α= 0.2, woven fiber composite plates
The reduction of excitation frequency is about to 50% with decrease in degree of
orthotropy E1/E2 =40 to 10. The larger the ratio of E1/E2, for the high-modulus-fiber
material plates, the laminated plates is most stable and better stiffness in adverse
hygrothermal environment.
The effect of increase in dynamic load factor of eight layered anti-symmetric woven
fiber laminated composite plates on non-dimensional excitation frequency is analyzed
in figure 5.55 at temperature 325K. It is observed that with the increase of dynamic
load factor from 0.4 to 1, the onset of dynamic instability occurs earlier and the width
of dynamic instability region also increases. The reduction of excitation frequencies
with increase in dynamic load factors from 0.4 to 1 is more than 10% due to reduction
of stiffness and strength.
The variations of dynamic instability regions with increase in dynamic load factors of
woven fiber composite plates at moisture concentrations 0.25% is observed as shown
96
in figure 5.56. The onset of instability occurs earlier with increase of dynamic load,
the instability region tend to shift to lower frequencies and narrower. The reduction of
excitation frequencies with increase in dynamic loads from 0.4 to 1 is more than 10%.
This implies that the presence of the compressive static in-plane load reduces the
stiffness of the laminates, and thus shifts the resonance frequencies downwards in
hygrothermal environment.
1
β=0.4
β=0.6
β=0.8
β=1
Static load factor
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
12
14
Nondimensional excitation frequencies
Figure 5.55: Variation of instability regions with temperature at 325K for simply
supported of a/b=1, α= 0.2, woven fiber composite plates
1
β=0.4
β=0.6
β=0.8
β=1
Static load factor
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
Nondimensional excitation frequencies
12
Figure 5.56: Variation of instability regions with moisture concentration at
0.25% for simply supported of a/b=1, α= 0.2, woven fiber composite plates
It is observed from the above figure that the laminated plates become more
dynamically unstable with increase in dynamic load factor as compared to increase in
static load factor in hygrothermal environment.
The variations of dynamic instability of composite plates with increase in temperature
from 300K to 375K is shown in figure 5.57 with a dynamic load factor 0.2, it is
observed that the onset of instability occurs earlier with increase in temperature. The
width of instability region decreased with increase in temperature from 300K to
97
375K. The reduction of excitation frequencies with increase in temperature from
325K to 375K is about 20%.
Similarly with increase in moisture concentration from 0.25% to 1% is shown in
figure 5.58 with dynamic load factor 0.2, it is observed that the onset of instability
occurs earlier with increase in moisture concentration. The width of instability region
decreased with increase in moisture concentration from 0.25% to 1%. The reduction
of excitation frequencies with increase in moisture concentration from 0.25% to 1% is
about 50%.
Static load factor
1
300K
325K
350K
375K
0.8
0.6
0.4
0.2
0
8
9
10
11
Nondimensional excitation frequencies
12
Figure 5.57: Variation of instability regions with different temperature for
simply supported of [45/-45]4, woven fiber composite plates
Static load factor
1
0.25%
0.50%
0.75%
1%
0.8
0.6
0.4
0.2
0
1
2
3
4
5
6
Nondimensional excitation frequencies
Figure 5.58: Variation of instability regions with different moisture
concentration for simply supported of [45/-45]4, woven fiber composite plates
It is seen from the figure when static load factor remains constant α = 0.2, the
excitation frequency is less as compared to figure 5.57 and 5.58 but the plates is stable
in former than later with increase in temperature and moisture concentration
environment.
98
5.3: Curved panels
The study is further extended to Dynamic stability of woven fiber laminated
composite curved panels in hygrothermal environment subjected to periodic loadings.
5.3.1: Convergence study
The convergence study is done for simply supported square 4 layer symmetric crossply and symmetric angle-ply laminated composite plates and shells for elevated
temperature and moisture conditions for different mess divisions. Dynamic stability
analysis of laminated composite panels subjected to hygrothermal conditions are
presented in Table 5.18 and Table 5.19. As observed, a mesh of 10 ×10 shows good
convergence of the numerical solution of dynamic stability of woven fiber composite
panels and this mesh is employed throughout the dynamic stability analysis of woven
fiber composite plates and shells in hygrothermal environment.
5.3.2: Comparison with previous studies
To validate the formulation, the frequencies of vibration the buckling load and
excitation frequency during parametric resonance of laminated composite plates are
computed and compared with previously published results from literature.
5.3.2.1: Vibration of composite shells in hygrothermal environment
The present formulation is validated for free vibration analysis of composites for
shells with ambient temperature are compared with analytical solution published by
Reddy (1984), Chandrasekhar (1989) as shown in Table 5.20. The non-dimensional
frequency parameters for shell with hygrothermal condition is as shown in Table 5.21
is compared with results published by Parhi et al. (2001) and Naidu & Sinha (2006).
The present finite element results show good agreement with the previous numerical
results published in the literature for free vibration of laminated composite shells
subjected to hygrothermal conditions.
99
Table 5.21: Comparison of non-dimensional free vibration frequencies for SSSS
(0/90/90/0) spherical shell at ambient temperature
a/b=1, a/t=100, At T = 300K, E1= 125Gpa , E2= 5 Gpa, G12= 2.5Gpa, G13=G12,
G23=1G12, ν12=0.25
Non dimensional frequency, λ = ωna2
R/b
Reddy (1984)
1
2
3
4
5
10
126.33
68.294
47.415
37.082
31.079
20.38
Chandrasekhara
(1989)
126.7
68.294
47.553
37.184
31.159
20.417
ρ / E2 t 2
Present
126.460
68.364
47.459
37.110
31.097
20.376
Table 5.22: Comparison of natural frequencies for SSSS (0/90/90/0) shell at 1%
moisture concentration
a/b=1, a/t=100, At T = 300K, E1= 172.5Gpa, E2= 6.9Gpa (C=0), G12= 3.45Gpa,
E2= 6.17Gpa (C=1%), G13=G12, G23=1.38G12, ν12=0.25, β1=0, β2=0.44, ρ=1600
Fundamental natural frequency, λ = ωna2
Stacking sequence
R/a=5 (0/90)2
R/a=10 (0/90)2
References
Naidu & Sinha(2006)
Parhi et al. (2001)
Present
Naidu & Sinha(2006)
Parhi et al. (2001)
Present
C=0
201.82
202.02
201.93
129.13
129.20
129.08
ρ / E2 t 2 1/2 pi
*
C=1%
201.68
201.64
201.61
127.54
128.32
128.62
5.3.2.2: Buckling of composite shells in hygrothermal environment
The present formulation is then validated for buckling analysis of laminated
composite shells for normal temperatures and moistures. The non-dimensional critical
loads due to ambient temperature and moisture obtained by the present finite element
as shown in Table 5.22 is compared with analytical solution published by Scuiva &
Carrera (1990). The square plate has four layers of Graphite/Epoxy composite. The
present finite element results show good agreement with the previous numerical
results published in the literature for buckling of composite plates subjected in
hygrothermal environment.
100
Table 5.23: Comparison of Non-dimensional buckling loads of a square simply
supported symmetric cross-ply cylindrical laminated curved panels with
(0/0/90/0)
a/b=1,R/a=20.0, λ=Nxa2/E2 h3, E11=40E22, G23=0.6E2, G12=G13=0.5E22,
ν12=ν13=0.25
Theory
FSDT( Scuiva & Carrera)
(1990)
Present
a/h=50
a/h=100
35.42
35.843
35.235
36.803
5.3.2.3: New results for dynamic stability
After validating the present formulation, the same is employed to study the dynamic
stability effects of woven fiber composite shells in hygrothermal environment.
Numerical results are presented on the dynamic stability of cross-ply and angle-ply
laminated shell to study the effects of various parameters on instability regions. The
geometrical properties of the laminated shell are as follows a=500mm, b=500mm,
t=5mm, E1=130Gpa, E2=9.5Gpa, G12=6Gpa, G23/G12=0.5, υ12=0.3, G13=G12, α1=0.3*10^-6 /K, α2=28.1*10^-6 /K
The non-dimensional excitation frequency Ω= a2/ 22h2) is used throughout the
dynamic instability studies, where is the excitation frequency in radian/sec. The
principal instability regions of laminated composite curved panel subjected to in-plane
periodic loads is plotted with non-dimensional frequency Ω/ω (ratio of excitation
frequency to the free vibration frequency) versus the dynamic in-plane load β. The
analysis is focused on the determination of the primary instability regions of
laminated composite shells under hygrothermal loads.
The effect of static component of load for α= 0.0, 0.2, 0.4, 0.6 and 0.8 on the
instability region of laminated composite panel subjected to elevated temperature
325K is shown in fig.5.59. Due to increase of static component, the instability regions
tend to shift to lower frequencies and become wider. With increase in static load
factor from 0 to 0.8, the excitation frequency is reducing by 2.3%. All further studies
are made with a static load factor of 0.2 (unless otherwise mentioned).
101
Dynamic load factor
1
alpha=0
alpha=0.2
alpha=0.4
alpha=0.6
alpha=0.8
0.8
0.6
0.4
0.2
0
96
98
100
102
104
106
Non-dimensional excitation frequency
108
Figure 5.59: Variations of instability region with static load factor of composite
shell subjected to temperature (Temp=325K, a/b=1, Ry/b=Rx/b=5, b/t=100)
The variation of excitation frequency with dynamic load factor of composite
laminated simply-supported symmetric cross-ply square shells subjected to uniform
distribution of temperature from 300K and 325K is shown in fig.5.60. It is observed
that the onset of instability occurs earlier with wider DIR for symmetric cross-ply
laminated composite panels subjected to elevated temperature compared to composite
panel with normal temperature. With increase in temperature from 300K to 325K, the
excitation frequency is reducing by 31.6%. The width of instability region for
laminated panel with elevated temperature is increased by 46.67% from the plate with
Dynamic load factor
normal temperature for a dynamic load factor of 0.6.
1
temp=300K
0.8
temp=325K
0.6
0.4
0.2
0
20
30
40
50
60
Non-dimensional excitation frequency
Figure 5.60: Variations of instability region with temperature of composite
symmetric cross-ply (0/90/90/0) curved panel (a/b=1, b/t=100, Ry/b=5)
The variation of excitation frequency with dynamic load factor of composite
laminated simply-supported anti-symmetric angle-ply square shells subjected to
uniform distribution of temperature from 300K, 325K, 350K, 375K & 400K is shown
in fig.5.61. As shown, the onset of instability occurs earlier with wider DIR for antisymmetric angle-ply laminated composite shells subjected to elevated temperature
102
compared to composite shells with normal temperature. With increase in temperature
from 300K to 350K, the excitation frequency is reducing by 65.2%.
temp=0K
temp=325K
temp=350K
temp=375K
temp=400K
Dynamic load factor
1
0.8
0.6
0.4
0.2
0
20
40
60
80
100
Non-dimensional excitation frequency
Figure 5.61: Variations of instability region with temperature of composite antisymmetric angle-ply (45/-45/45/-45) curved panel (a/b=1, b/t=100, Ry/b=5)
The variation of excitation frequency with dynamic load factor of composite
laminated simply-supported symmetric cross-ply curved panel subjected to uniform
distribution of moisture concentration from 0% & 0.1% is shown in fig.5.62. It is
revealed that the onset of instability occurs earlier with wider DIR for symmetric
cross-ply laminated composite shells subjected to elevated moisture condition
compared to composite shells with normal moisture. When moisture concentration is
increased from 0% to 0.1% then excitation frequency reduces for about 21%. The
width of instability region for laminated shell with elevated temperature is increased
by 46.67% from the shell with normal temperature for a dynamic load factor of 0.6.
mois=0
mois=.001
Dynamic load factor
1
0.8
0.6
0.4
0.2
0
30
40
50
60
Non-dimensional excitation frequency
Figure 5.62: Variations of instability region with moisture of composite
symmetric cross-ply (0/90/90/0) shell (Ry/b=5, a/b=1, b/t=100)
The variation of excitation frequency with dynamic load factor of composite
laminated simply-supported anti-symmetric angle-ply shell subjected to uniform
distribution of moisture concentration from 0%, 0.1%, 0.25% & 0.5% is shown in fig.
5.63. It is revealed that the onset of instability occurs earlier with wider DIR for anti103
symmetric angle-ply laminated composite shells subjected to elevated moisture
condition compared to composite shells with normal moisture concentration. When
moisture concentration is increased from 0% to 0.25% then excitation frequency
Dynamic load factor
reduces for about 49.3%.
mois=0%
mois=0.1%
mois=0.25%
mois=0.5%
1.2
1
0.8
0.6
0.4
0.2
0
25
50
75
100
Non-dimensional excitation frequency
Figure 5.63: Variations of instability region with moisture of composite antisymmetric angle-ply (45/-45/45/-45) shell (Ry/b=5, a/b=1, b/t=100)
Studies have also been made in Fig.5.64 for comparison of instability regions for
different shell geometries. The effect of curvature on instability region of different
curved panels for a/b=1, flat panel (a/Rx = b/Ry = 0), Cylindrical (a/Rx = 0, b/Ry = 0.2)
& Spherical (a/Rx = b/Ry = 0.2) is investigated.
plate
cylindrical
spherical
Dynamic load factor
1
0.8
0.6
0.4
0.2
0
0
50
100
150
Non-dimensional excitation frequency
Figure 5.64: Variations of curvature of composite symmetric cross-ply
(0/90/90/0) curved panel with elevated temperature (Temp=325K,
a/b=1, b/t=100)
It is observed that the excitation frequency increases with introduction of curvatures
from flat panel to doubly curved panel in elevated temperature. The onset of dynamic
instability region occurs earlier with wider dynamic instability region (DIR) coming
from spherical laminated composite shell panel to laminated composite flat panel
subjected to uniform distribution of temperature.
104
The variation of instability region with dynamic load factor of composite laminated
simply supported symmetric cross-ply and anti-symmetric angle-ply shells with
excitation frequency subjected to uniform distribution of elevated temperature with
different aspect ratio (a/b =1, 2 & 3) are shown in figures 5.65 and 5.66.
From the fig5.65 and 5.66, it is observed that the onset of instability occurs earlier
with decrease of the aspect ratio with decreasing width of instability region & the
onset of instability occurs latter for rectangular anti-symmetric angle-ply laminated
composite shells than square shells subjected to elevated temperature but with wider
instability regions. The width of instability regions increased marginally for
rectangular plates than square plates with uniform rise in temperature and moisture
concentration. The increase in aspect ratio shifts the frequency of instability region to
higher values and reduces the dynamic stability strength. As a result the plates having
higher values of aspect ratios are dynamically unstable in elevated temperature and
Dynamic load factor
lose its stiffness.
a/b=1,b/t=100
a/b=2,b/t=100
a/b=3,b/t=100
1
0.8
0.6
0.4
0.2
0
0
50
100
Non-dimensional excitation frequency
Figure 5.65: Effect of aspect ratio on instability region of (0/90/90/0) laminate for
elevated temperature (Ry/b=5, Temp=325K, b/t=100)
a/b=1,b/t=100
a/b=2,b/t=100
a/b=3,b/t=100
Dynamic load factor
1
0.8
0.6
0.4
0.2
0
0
100
200
Non-dimensional excitation frequency
Figure 5.66: Effect of aspect ratio on instability region of (45/-45/45/-45) laminate
for elevated temperature (Ry/b=5, Temp=325K, b/t=100)
105
The variation of instability region with dynamic load factor of composite laminated
simply supported symmetric cross-ply and anti-symmetric angle-ply shells with
excitation frequency subjected to uniform distribution of elevated moisture with
different aspect ratio (a/b =1, 2 & 3) is shown in figures 5.67 for symmetric cross-ply
and 5.68 for anti-symmetric angle ply laminates.
From the figure 5.67 and 5.68, it is observed that the onset of instability occurs earlier
with decrease of the aspect ratio with decreasing width of instability region & the
onset of instability occurs latter for rectangular anti-symmetric angle-ply laminated
composite shells than square shells subjected to elevated moisture but with wider
instability regions. The width of instability regions increased marginally for
rectangular shells than square shells with uniform rise in moisture concentration. The
increase in aspect ratio shifts the frequency of instability region to higher values and
reduces the dynamic stability strength. As a result the shells having higher values of
aspect ratios are dynamically unstable in elevated moisture condition and lose its
stiffness.
a/b=1,Ry=2.5
a/b=2,Ry=2.5
a/b=3,Ry=2.5
Dynamic load factor
1
0.8
0.6
0.4
0.2
0
0
100
200
300
Non-dimensional excitation frequency
Dynamic load factor
Figure 5.67: Effect of aspect ratio on instability region of (0/90/90/0) laminate for
elevated moisture (Ry/b=5, Mois=0.001, b/t=100)
a/b=1,Ry=2.5
a/b=2,Ry=2.5
a/b=3,Ry=2.5
1
0.8
0.6
0.4
0.2
0
0
100
200
300
400
Non-dimensional excitation frequency
Figure 5.68: Effect of aspect ratio on instability region of (45/-45/45/-45) laminate
for elevated moisture (Ry/b=5, Mois=0.001, b/t=100)
106
The variation of excitation frequency with dynamic load factor of composite
laminated simply supported symmetric cross-ply and for anti-symmetric angle-ply
shells subjected to uniform distribution of temperature are shown in figures 5.69 and
5.70 The effect of degree of orthotropy is studied for E1/E2 = 10, 20, 40 keeping other
material properties constant.
E1/E2=10
E1/E2=20
E1/E2=40
Dynamic load factor
1
0.8
0.6
0.4
0.2
0
20
40
60
80
100
Non-dimensional excitation frequency
Figure 5.69: Effect of degree of orthotropy on instability region of (0/90/90/0)
laminate for elevated temperature (Ry/b=5, Temp=325K, b/t=100, a/b=1)
E1/E2=10
E1/E2=20
E1/E2=40
Dynamic load factor
1
0.8
0.6
0.4
0.2
0
30
60
90
120
150
Non-dimensional excitation frequency
Figure 5.70: Effect of degree of orthotropy on instability region of (45/-45/45/-45)
laminate for elevated temperature (Ry/b=5, Temp=325K, b/t=100, a/b=1)
It is observed that the onset of instability occurs latter with increase of degree of
orthotropy for symmetric cross-ply and anti-symmetric angle-ply laminated composite
shells subjected to elevated temperature but with wider instability regions. The
excitation frequency is reduced but the instability region is wider in cross-ply
symmetric laminate rather than anti-symmetric angle-ply laminate.
107
E1/E2=10
E1/E2=20
E1/E2=40
Dynamic load factor
1
0.8
0.6
0.4
0.2
0
20
40
60
80
100
Non-dimensional excitation frequency
Figure 5.71: Effect of degree of orthotropy on instability region of (0/90/90/0)
laminate for elevated moisture (Ry/b=5, Mois=0.001, b/t=100, a/b=1)
E1/E2=10
E1/E2=20
E1/E2=40
Dynamic load factor
1
0.8
0.6
0.4
0.2
0
30
60
90
120
150
Non-dimensional excitation frequency
Figure 5.72: Effect of degree of orthotropy on instability region of (45/-45/45/-45)
laminate for elevated moisture (Ry/b=5, Mois=0.001, b/t=100, a/b=1)
The variation of excitation frequency with dynamic load factor of composite
laminated simply supported symmetric cross-ply and for anti-symmetric angle-ply
shells subjected to uniform distribution of moisture are shown in figures 5.71 and
5.72. The effect of degree of orthotropy is studied for E1/E2 = 40, 20, 10 keeping other
material properties constant. It is observed that the onset of instability occurs latter
with increase of degree of orthotropy for symmetric cross-ply and anti-symmetric
angle-ply laminated composite shells subjected to uniform distribution of moisture but
with wider instability regions. The excitation frequency is reduced but the instability
region is wider in cross-ply symmetric laminate rather than anti-symmetric angle-ply
laminate.
The variation of excitation frequency with dynamic load factor of composite
laminated simply supported symmetric cross-ply and for anti-symmetric angle-ply
shells subjected to uniform distribution of temperature are shown in figures 5.73 and
5.74. The effect of radius to thickness ratio is studied for Rx/h = Ry/h = 625, 500, 375
keeping other geometries and material properties constant. It is observed from the fig.
that the onset of dynamic instability region occurs later with increase of Ry/h ratio but
108
with wider instability region. The excitation frequency is reduced but the instability
region is narrower in cross-ply symmetric laminate rather than anti-symmetric angleply laminate in elevated temperature.
Rx/h=Ry/h=375
Rx/h=Ry/h=500
Rx/h=Ry/h=625
Dynamic load factor
1
0.8
0.6
0.4
0.2
0
70
90
110
130
Non-dimensional Excitation frequency
Figure 5.73: Effect of thickness on instability region of (0/90/90/0) laminate for
elevated temperature (a/b=1, Temp=325K, Rx/b= Ry/b = 5)
Dynamic load factor
1
Rx/h=Ry/h=375
Rx/h=Ry/h=500
Rx/h=Ry/h=625
0.8
0.6
0.4
0.2
0
75
100
125
150
Non-dimensional excitation frequency
Figure 5.74: Effect of thickness on instability region of (45/-45/45/-45) laminate
for elevated temperature (a/b=1, b/t=100, Temp=325K, Rx/b= Ry/b = 5)
The variation of excitation frequency with dynamic load factor of composite
laminated simply supported symmetric cross-ply and for anti-symmetric angle-ply
shells subjected to uniform distribution of moisture are shown in figures 5.75 and
5.76. It is observed from the fig. that the onset of dynamic instability region occurs
earlier with decrease of Ry/h ratio but with narrower instability region. The excitation
frequency is reduced but the instability region is narrower in cross-ply symmetric
laminate rather than anti-symmetric angle-ply laminate in elevated moisture.
109
Dynamic load factor
Rx/h=Ry/h=375
Rx/h=Ry/h=500
Rx/h=Ry/h=625
1
0.8
0.6
0.4
0.2
0
60
90
120
150
Non-dimensional excitation frequency
Figure 5.75: Effect of thickness on instability region of (0/90/90/0) laminate for
elevated moisture (a/b=1, b/t=100, Mois=0.001, Rx/b= Ry/b = 5)
Dynamic load factor
1
Rx/h=Ry/h=375
Rx/h=Ry/h=500
Rx/h=Ry/h=625
0.8
0.6
0.4
0.2
0
60
90
120
150
Non-dimensional excitation frequency
Figure 5.76: Effect of thickness on instability region of (45/-45/45/-45) laminate
for elevated moisture (a/b=1, b/t=100, Mois=0.001, Rx/b= Ry/b = 5)
effect of shallowness ratio
The variation of excitation frequency with dynamic load factor of composite
laminated simply supported symmetric cross-ply and for anti-symmetric angle-ply
shells subjected to uniform distribution of temperature are shown in figures 5.77 and
5.78. The effect of shallowness ratio on instability regions is studied for Rx/a = Ry/b =
3, 5, 10 keeping other geometries and material properties constant. As seen from the
fig., the instability excitation frequency is higher for decrease of shallowness by
decreasing Rx and Ry. The onset of instability occurs earlier with increase of
shallowness ratio but with wide instability region. The excitation frequency is reduced
but the instability region is wider cross-ply symmetric laminate rather than antisymmetric angle-ply laminate in elevated temperature.
110
Dynamic load factor
1
Rx/a=Ry/b=3
Rx/a=Ry/b=5
Rx/a=Ry/b=10
0.8
0.6
0.4
0.2
0
25
75
125
175
Non-dimensional excitation frequency
Dynaic load factor
Figure 5.77: Effect of Ry/b on instability region of (0/90/90/0) laminate for
elevated temperature (a/b=1, b/t=100, Temp=325K, Ry=Rx = 1.5, 2.5, 5)
Rx/a=Ry/b=3
Rx/a=Ry/b=5
Rx/a=Ry/b=10
1
0.8
0.6
0.4
0.2
0
50
100
150
200
Non-dimensional excitation frequency
Figure 5.78: Effect of Ry/b on instability region of (0/90/90/0) laminate for
elevated moisture (a/b=1, b/t=100, Mois=0.001, Ry=Rx = 1.5, 2.5, 5)
The variation of excitation frequency with dynamic load factor of composite
laminated simply supported symmetric cross-ply and for anti-symmetric angle-ply
shells subjected to uniform distribution of moisture are shown in figures 5.79 and
5.80. The effect of shallowness ratio on instability regions is studied for Rx/a = Ry/b =
3, 5, 10 keeping other geometries and material properties constant.
As seen from the fig., the excitation frequency is higher for decrease of shallowness
by decreasing Rx and Ry. The onset of instability occurs earlier with increase of
shallowness ratio but with wide instability region. The excitation frequency is reduced
but the instability region is wider in cross-ply symmetric laminate rather than antisymmetric angle-ply laminate in elevated moisture.
111
Dynamic load factor
1
Rx/a=Ry/b=3
Rx/a=Ry/b=5
Rx/a=Ry/b=10
0.8
0.6
0.4
0.2
0
50
100
150
200
Non-dimensional excitation frequency
Figure 5.79: Effect of Ry/b on instability region of (45/-45/45/-45) laminate for
elevated temperature (a/b=1, b/t=100, Temp=325K, Ry=Rx = 1.5, 2.5, 5)
Dynamic loaod factor
1
Rx/a=Ry/b=3
Rx/a=Ry/b=5
Rx/a=Ry/b=10
0.8
0.6
0.4
0.2
0
50
100
150
200
Non-dimensional excitation frequency
Figure 5.80: Effect of Ry/b on instability region of (45/-45/45/-45) laminate for
elevated moisture (a/b=1, b/t=100, Mois=0.001, Ry=Rx= 1.5, 2.5, 5)
The variation of instability region with dynamic load factor of composite laminated
simply supported anti-symmetric angle-ply shells with excitation frequency subjected
to uniform distribution of temperature with different ply orientation is shown in figure
5.81. It is observed that the onset of instability occurs earlier for anti-symmetric
angle-ply laminated composite shells with 0 degree of ply orientation than the shells
with higher degree of ply orientation subjected to elevated moisture condition but
with narrow DIR. The value of ply orientation for which the instability region is
narrower is 45 and for the wider DIR the ply orientation value is 0. The instability
region is less wide for increase in lamination angle but the excitation frequencies are
decreased with decrease in uniform temperature distribution. The ply orientation for
00 seems to be the preferential ply orientation for the lamination sequence which is
due to dominance effect of bending-stretching coupling.
112
Dynamic load factor
1
theta=0
theta=15
theta=30
theta=45
0.8
0.6
0.4
0.2
0
25
50
75
100
125
150
Non-dimensional excitation frequency
Figure 5.81: Effect of different ply orientation on instability region of antisymmetric angle-ply laminate for elevated temperature (a/b=1, b/t=100,
Temp=325K, Ry/b=5)
Dynamic load factor
1
theta=0
theta=15
theta=30
theta=45
0.8
0.6
0.4
0.2
0
25
50
75
100
125
150
Non-dimensional excitation frequency
Figure 5.82: Effect of different ply orientation on instability region of antisymmetric angle-ply laminate for elevated moisture (a/b=1,
b/t =100, Mois=0.1%, Ry/b= 5)
The variation of instability region with dynamic load factor of composite laminated
simply supported anti-symmetric angle-ply shells with excitation frequency subjected
to uniform distribution of moisture with different ply orientation is shown in figure
5.82. It is observed that the onset of instability occurs earlier for anti-symmetric
angle-ply laminated composite shells with 0 degree of ply orientation than the shells
with higher degree of ply orientation subjected to elevated moisture condition but
with narrow DIR. The value of ply orientation for which the instability region is
narrower is 45 and for the wider DIR the ply orientation value is 0. The instability
region is less wide for increase in lamination angle but the excitation frequencies are
decreased with decrease in uniform moisture concentration.
113
CHAPTER 6
CONCLUSIONS
The present work deals with the experimental and numerical study on static, vibration
and buckling of woven fiber laminated composite plates in hygrothermal
environment. The parametric instability characteristics of composite flat and curved
panels are carried out using numerical approach by finite element approach. The
formulation is based on the first order shear deformation theory, taking into account
transverse shear and rotary inertia effects. The development of regions of instability
arises from Floquet’s theory developed by Bolotin and the boundaries of the primary
instability regions with period 2T, where T = 2 /Ω which are of practical importance
have been determined to study the effect of various parameters of the laminated
composite plates and shells on the dynamic instability regions.
The effects of various geometrical parameters like aspect ratio, side to thickness
ratio, static load factor, degree of orthotropy and lamination details on the vibration
and stability characteristics of woven fiber laminated composite plates and shells in
hygrothermal environment has been analysed.
A general formulation is presented for the vibration, buckling and dynamic stability of
laminated composite curved panels in hygrothermal environment subjected to inplane harmonic loads. The conclusions are presented separately for each cases.
114
6.1: Static behavior of composites in hygrothermal environment
Experimental investigations on static strength of composite specimens under
hygrothermal loading are carried out. The conclusions are summarized as:
In the same weight fraction of fiber matrix of glass: epoxy composites gives
higher inter laminar shear strength in all loading speed than glass: polyester
specimens.
The variations of inter laminar shear strength of woven fiber laminated
composites is significant for low loading speed and is not so prominent for high
speed.
Matrix resins such as polyester and epoxy are known to be highly rate sensitive.
Woven roving E-Glass fibers are found to be rate sensitive
The S.D and C.V for all specimens are low showing good quality control of
preparation of test specimens.
The variations of ILSS with respect to temperature are higher for all percentage
constituent in glass fiber: epoxy than glass fiber: polyester specimens.
The variations of ILSS with respect to moisture concentration are higher for glass
fiber: polyester than glass fiber: epoxy for all percentage constituents.
The interfacial adhesion for glass/ epoxy and glass/ polyester composites are more
affected by hygrothermal ageing at higher conditioning temperature and for more
exposure time.
The glass: epoxy composites are most affected with E22, and least affected in G12
in hygrothermal environment.
115
6.2: Vibration of composite flat panels in hygrothermal environment
The present study deals with the parametric study on free vibration behavior of
woven fiber composite plates subjected to uniform temperature and moisture
experimentally and comparing them using finite element method. From the
discussion, the following observation can be made:
There is a good agreement between natural frequencies of laminated
composite plates under hygrothermal environment.
The natural frequencies of vibration of fiber composite plates decrease with
increase of temperature due to reduction of stiffness for all laminates.
The frequencies of vibration of laminated plates also decrease substantially
with increase of moisture concentration for all laminates.
The frequencies of vibration for anti-symmetric laminates are higher than
symmetric laminates. However, the frequency decreases with increase of
hygrothermal conditions.
The frequencies of vibration of composite plates under hygrothermal loads are
more with increase in number of layers. However, the
frequency decreases
with increase of hygrothermal conditions.
The fundamental frequencies of vibration decrease with increase in aspect
ratios of plates in hygrothermal environment.
The frequencies of vibration increase with increase in thickness of composite
plates in hygrothermal environment.
116
6.3: Buckling effects of composite flat panels in hygrothermal
environment
The present study deals with the parametric study on buckling behavior of woven
fiber laminated composite plates subjected to uniform temperature and moisture
experimentally and comparing them using finite element method. From the
discussion, the following observation can be made:
There is a good agreement between the experimental and numerical results for
buckling of laminated composite plates at different temperature and moisture
concentration.
A general formulation on buckling of industry driven woven fiber composite
plates which accounts for hygrothermal effects due to moisture diffusion,
temperature and mechanical loads in addition to transverse shear, deformation and
bending-stretching coupling is presented.
The buckling loads decreases with increase in temperature and moisture
concentration due to reduction of stiffness and strength of laminated plates.
The buckling loads of laminated composite plates with hygrothermal loads
increase with increase in number of layers.
The reduction of buckling loads for anti-symmetric laminates is more than
symmetric laminates with increase in temperature and moisture concentration
environment.
The buckling loads are decreased with increase in side-to-thickness ratios in
hygrothermal environment.
The thick and moderately thick laminated composite plates are more stable than
thin plates in hygrothermal environment.
The buckling loads may reduce significantly depending upon the temperature,
moisture concentration, lamination sequence, side-to-thickness ratios and aspect
ratios.
The clamped-free-clamped-free boundary condition shows better buckling loads
compared to four sides simply supported edges in hygrothermal conditions.
The buckling loads are more for clamp-free-clamp-free boundary conditions as
compared to four sides simply supported edges in extreme hygrothermal
environment.
117
6.4: Dynamic instability of composite flat panels in hygrothermal
environment
The parametric instability study of woven fiber laminated composite plates in
hygrothermal environment subjected to periodic in- plane loads is examined. A
general formulation for dynamic instability of laminated composite plates subjected to
hygrothermal under in-plane periodic load is presented for the first time. From the
detailed study, the following observation can be made:
The excitation frequencies of laminated fiber composite plates decrease with
increase of temperature and moisture concentration due to reduction of stiffness
for all laminates. The greater is the lamination angle the smaller is the instability
region of composite plates in hygrothermal environment.
With increase in aspect ratios the excitation frequencies are decreased, due to
reduction of effective stiffness of the plates in hygrothermal environment.
Woven fiber laminated plates is more stable with increase in number of layers in
hygrothermal environment
The thick plates having narrow instability region shows more stiffness and
strength than thin plates in hygrothermal environment.
The width of instability zones decreases with increase of degree of orthotropy in
hygrothermal environment.
The high-modulus-fiber material plate is most stable in hygrothermal
environment.
The width of the instability regions increases with increase in both static and
dynamic load parameters.
Instability occurs earlier with an increase of the static compressive in-plane load,
the instability regions tend to shift to lower frequencies, showing a destabilizing
effect on the dynamic stability behavior of the plates in hygrothermal
environment.
118
6.5: Dynamic instability of composite curved panels in hygrothermal
environment
A general formulation for vibration, buckling and parametric resonance characteristics
of laminated composite curved panels subjected to hygrothermal loads is presented.
The excitation frequencies of laminated composite curved panels decrease with
increase of temperature due to reduction of stiffness for all laminates.
The excitation frequencies of laminated composite curved panels also decrease
substantially with increase of moisture concentration for all laminates.
Due to static component of load, the onset of instability shifts to lower
frequencies with wide instability regions of the laminated composite curved
panels.
The instability region is observed to be influenced by the lamination angle.
Increasing the thickness of the panels results in better dynamic stability strength.
Increasing the aspect ratio, shifts the frequencies of instability region to higher
values and reduces the dynamic stability strength.
The width of dynamic instability region is smaller for square panels than
rectangular panels.
The excitation frequency increases with introduction of curvatures from flat
panel to doubly curved panel in hygrothermal environment.
From the present studies, it is concluded that the instability behavior of woven fiber
laminated composite plates and shells is greatly influenced by the geometry,
lamination parameter and hygrothermal conditions. The figures dealing with variation
of the frequencies, buckling loads and dynamic instability regions are recommended
as design aids for flat and curved panels in hygrothermal environment. The above
recommendations for design of composite panels are valid within the range of
geometry and material considered in this study. So the designer has to be cautious
while dealing with structures subjected to hygrothermal loading. This can be utilized
to the advantage of tailoring during design of laminated composite structures in
hygrothermal environment.
119
6.6: Future scope of research
Possible extensions to the present work are:
forced vibration of laminated plates in hygrothermal environment.
dynamic stability of stiffened plates and shells subjected to hygrothermal
condition.
dynamic stability of delaminated plates and shells in hygrothermal environment.
dynamic stability of plates and shells involving arbitrary shaped openings in
hygrothermal environment.
The effects of damping on instability regions of plates and shells can be studied.
non-conservative forces like follower loading in hygrothermal environment.
large deflection and large amplitude vibration analyses of dynamic stability of
plates and shells subjected to hygrothermal condition.
Material nonlinearity may be taken into account in the formulation
can be plates and shells of varying thickness.
Besides all these, there is a large scope experimental investigation on dynamic
stability of plates and shells in hygrothermal environment.
120
REFERENCES
ASTM
Standard:
D5687/D5687M-07.
(2007):
Standard
Guide
for
Preparation of Flat Composite Panels With Processing Guidelines For
Specimen Preparation.
ASTM Standard: D5229/D5229M-04. (2004): Stanndard Test Method For
Moisture Absorption Properties And Equilibrium Conditioning of Polymer
Matrix Composite Materials.
ASTM Standard: D2344/D2344M-06. (2006): Standard Test Method for
Short-Beam Strength of Polymer Matrix Composite materials and Their
Laminates.
ASTM Standard: D3039/D3039M-08. (2008): Standard Test Method for
Tensile Properties of Polymer Matrix Composite Materials.
Adams, R.D and Singh, M.M. (1995): The effect of immersion in sea
water on the dynamic properties of fiber- reinforced flexibilised epoxy
composites, Journal of Composite Structures, Vol.31, pp. 119-127.
Anderson, T.J and Nayfeh, A.H. (1996): Natural frequencies and mode
shapes of laminated composite plates: Experiment and FEA, Journal of
Vibration and Control, Vol.2 (4), pp. 381-414.
Argento, A and Scott, R.A. (1993): Dyamic instability of laminated
anisotropiccircularcylindrical shells part II numerical results, Journal of
Sound and Vibration, Vol.162 (2), pp. 323-332.
Argento, A. (1993): Dynamic stability of of a composite circular cylindrical
shell subjected to combined axial and torsional loading. Journal of composite
materials, Vol.27(18).
Atas C and Sayman O. (2008): An overall view on impact response of
woven fabric composite plates, Journal of Composite Structures, Vol.82, pp.
336-345.
Aditya, P.K. and Sinha, P.K. (1992): Diffusion coefficient of polymeric
composites subjected to periodic hygrothermal exposure, Journal of
Reinforced Plastics and composites, Vol.9(1): 1035-1047.
121
Babu, C.S. and Kant, T. (2000): Refined higher order finite element models
for thermal buckling of laminated Composite and sandwich plates. Journal of
Thermal Stresses, Vol.23, pp.111-130.
Baley, Christophe., Grohens, Yves.Busnel, Frederic. and Devies, Peter.
(2004: Application of interlaminar shear test to marine composites, Journal
of Applied Composite Materials, 11, pp. 77-98.
Balamurugan, V. Ganapathi, M. and Varadan, T.K. (1996): Nonlinear
dynamic instability of laminated composite plates using finite element
method, Computers and Structures, Vol.60 (1), pp.125-130.
Berger, S., Moshonv, A. and Kenig, S. (1989): The effect of thermal and
hygrothermal ageing on the failure mechanism of graphite-fabric epoxy
composites subjected to flexural loading, Journal of composite, Vol.20 (4),
pp. 341-348.
Bert, C.W and Birman, V. (1987): Dynamic instability of shear deformable
anti- symmetric angle-ply plates, International Journal Solids Structures,
Vol.23, pp. 1053-1067.
Bert, C.W, and Birman, V. (1988): Parametric instability of thick,
orthotropic, circular cylindrical shells, Acta Mechanica, Vol.71, pp. 61-76.
Bolotin, V.V. (1964). The Dynamic stability of elastic systems, Holden-Day,
San Francisco, 19. Nagai, K and Yamaki, N. (1988). Dynamic stability of
circular cylindrical shells, Acta. Mech. 71 :61-76.
Biswas, S., Datta, P.K. and Kong, C.K. (2011): Static and dynamic
instability characteristics of curved laminates with internal damage subjected
to follower loading, Journal of Mechanical Engineering Science, Vol.225,
pp.1589-1600.
Botelho, E.C., Pardini,L.C. and Rezende, M.C. (2006): Hygrothermal
effects on the shear properties of carbon fiber/epoxy composites, Journal of
Material Science, Vol.41, pp.7111-7118.
Botelho, E.C., Pardini, L.C and Rezende, M.C. (2005): Hygrothemal
effects on damping behavior of metal/glass fiber/epoxy hybrid composites,
Journal of Material Science and Engineering A, Vol.399, pp. 190-198.
122
Chakraborty, S., Mkhopadhyay, M and Mohanty, A. R. (2000): Free
vibration responses of FRP composite plates, Experimental and Numerical
studies. Journal of Reinforced plastics and Composites, Vol.19 (7), pp.535551.
Chakrabarti, A. and Sheikh, A. H. (2006): Dynamic instability of laminated
sandwich plates subjected to in-plane partial edge loading, Journal of Ocean
Engineering, Vol.33, pp. 2287-2309.
Chan, A.,Chiu, W.K. and Liu, X.L.(2007): Determining the elastic
interlaminar shear modulus of composite laminates, Journal of Composite
Structures, Vol.80, pp. 396-408.
Chandrasekhar, K. Free vibration of anisotropic laminated doubly curved
shells, Computers and Structures, Vol. 33 (2), pp.435-440.
Chamis, C.C. (1989): Mechanics of composite materials: past,present,and
future, Journal of composites technology and research, Vol.11(1), pp. 3-14.
Chen, B and Chou, T-W. (1999): Free vibration analysis of orthogonalwoven fabric Composites, Journal of Composites part A, Vol.30, pp. 285297.
Chao, L-P. and Shyu, S-L. (1996): Nonlinear buckling of fiber-reinforced
composite plates under hygrothermal effects, Journal of the Chinese Institute
of Engineers, Vol.19 (6), pp.657-667.
Chatopadhyay, A. and Radu, A.G. (2000): Dynamic instability of
composite laminates, Journal of Computers and Structures, Vol.77, pp.453460.
Chaudhuri, R.A, Balaraman, K and Kunuakkasseril, V.X. (2005): A
combined theoretical and experimental Investigation on free vibration of thin
symmetrically laminated anisotropic plates. Composite Structures, Vol.67:
85-97.
Chen, L. W. and Chen, Y.M. (1989): Vibration of hygrothermal elastic
composite Plates, Journal of Engineering Fracture Mechanics, Vol.31 (2),
pp. 209-220.
Chen L. W. and Lee J. H. (1988): Vibration of thermal elastic orthotropic
plates, Journal of Applied Acoustics, Vol. 27 (4), pp.287-304.
123
Chen, W. J. L, P. D. and Chen, L. W. (1991): Thermal buckling behavior of
composite
laminated plates with a circular hole, Composite Structures,
Vol.18, pp. 379-397.
Chen L-W. and Yang J-Y. (1990): Dynamic stability of laminated
composite plates by the finite element method, Computers and Structures,
Vol.36 (5), pp. 845-851.
Chen, L-W and Chen, L-Y. (1987): Thermal buckling of laminated
composite plates, Composite Structures, Vol.8, pp.189-205.
Chun-Sheng, Chen. Wei-Ren, Chen. and Rean-Der, Chien. (2009):
Stability parametric vibrations of hybrid Composite plates, European Journal
of Mechanics
A/Solids, Vol.28, pp. 329-337.
Constantinos, S.L and Dimitri, A. B. (1990): Hygrothermal effects on
structure-borne Noise transmission of stiffened laminated composite plates,
Journal of Aircraft, Vol. 27, pp. 722-730.
Dash, P.K., Sathisbabu, R. and Ganesan, C. (2011): Effect of corrosive
environment on elasto-buckling strength of GFRC Plate, Asian Journal of
Material Science, Vol.3 (1), pp. 5-19.
Dey, P. and Singha, M. K. (2006): Dynamic stability analysis of composite
skew plates subjected to periodic in-plane load, Journal of Thin-Walled
Structures, Vol.44, pp. 937-942.
Dhotarad, M.S and Ganesan, N. (1978): Influence on thermal gradient on
natural frequency of rectangular plate vibration, Journal of Nuclear
Engineering and Design, Vol.52 (1), pp. 71-81.
Eslami H and Maerz, S. (1995): Thermally induced vibration of asymmetric
cross-ply plate with hygrothermal effects. American Institute of Aeronautics
of Astronautics Journal, Vol.33 (10), pp. 1986-1988.
Fakhari, V and Ohadi, A. (2011): Nonlinear vibration control of
functionally graded plate with piezoelectric layers in thermal environment,
Journal of Vibration and Control, Vol.17 (3), pp. 449- 469.
Fazilati, J. and Ovesy, H.R. (2010): Dynamic instability analysis of
composite laminated thin walled structures using two versions of FSM,
Journal of composite Structures, Vol.92, pp. 2060-2065.
124
Flaggs, D.L and Vinson, J.K. (1978): Hygrothermal effects on the buckling
of laminated composite plates, Journal of fiber Science and Technology,
Vol.11, pp. 353- 365.
Fu, Yiming. Li, Sheng. and Jiang, Yejie. (2008): Analysis of interlaminar
stresses for composite laminated plate with interfacial damage, Journal of
Acta Mechanica Solida Sinia, Vol.21(2), pp. 127-140.
Gandhi, M.V., Usman, M and Chao, L. (1988): Nonlinear vibration of
laminated composite plates in hygrothermal environments, Journal of
Engineering Material Technology, Vol.110 (2), pp. 140-146.
Ganapathi, M. (1998): Dynamic instability of laminates subjected to
temperature field, Journal of Engineering Mechanics, Vol.124, pp.11661168.
Ganapathi, M. Patel, B.P. and Pawargi, D.S. (2002): Dynamic analysis of
laminated cross-ply composite non-circular thick cylindrical shells using
higher-order theory, International Journal of Solid and Structures, Vol.39,
pp. 5945-5962.
Ganapati, M. Patel, B.P. Boisse, P and Touratier, M. (2000): Nonlinear
dynamic Stability characteristics of elastic plates subjected to periodic inplane load, International Journal Non- linear Mechanics, Vol.35, pp. 467480.
Ganapati, M., Patel, B.P., Boisse, P. and Touratier. M. (2000): Nonlinear
dynamic stability characteristics of elastic plates subjected to periodic inplane load, International journal Non-linear Mechanics, Vol.35, pp. 467-480.
Gigliotti, Marco., Jacquemin, Frederic., Molimard, Jerome. and
Vautrin,Alain. (2007): Modelling and Experimental characterisation of
hygrothermoelastic stress in polymer matrix composites, Journal of polymer
composites, Vol.247 (1), pp. 199-210.
Govindarajan, R., Krishna Murty, A.V., Vijaykumar, K. and Raghuram,
P. V. (1993): Finite element estimation of elastic interlaminar stresses in
laminates, Journal of Composite E ngineering, Vol.3 (5): pp. 451-466.
Gupta A.K, Panwar V and Vats RP. (2010): Vibrations of nonhomogeneous rectangular plate of variable thickness in both directions with
thermal gradient effect, Journal of Applied Mathematics, Vol.1, pp. 456- 463.
125
Gupta, A.K and Sharma, P. (2011): Effect of linear thermal gradient on
vibrations of trapezoidal plates whose thickness varies parabolically, Journal
of Vibration and Control, Vol.17 (8).
Harding, J. and Li, Y.L. (1992): Determination of interlaminar shear
strength for glass/epoxy and carbon/epoxy laminates at impact rates of strain,
Journal of Composite Science and Technology, Vol.45, pp.161-171.
Harding, J. and Dong, L. (1994): Effect of strain rate on the interlaminar
shear strength of carbon-fiber-reinforced laminates, Journal of Composite
Science and Technology, Vol.51, pp. 347-358.
Huang, X. L., Shen, Hui-Shen and Zheng, Jian-Jun. (2004): Nonlinear
vibration and dynamic response of shear deformable laminated plates in
hygrothermal environments, Journal of composites science and Technology,
Vol.64, pp. 1419-1435.
Huang, N.N. and Tauchert, T.R. (1991): Large deformations of laminated
cylindrical and doubly curved panels under thermal loading, Computers
Structures, Vol.41 (2), pp. 303-12.
Ishai, O. and Arnon, U. (1977): The effect of hygrothermal on residual
strength of glass fiber reinforced plastic laminates, Journal of testing and
evaluation,Vol.5(4),pp. 7.
Jeyaraj, P., Ganesan, N and Padmanabhan, C. (2009): Vibration and
acoustics response of a composite plate with inherent material damping in a
thermal environment, Journal of Sound and Vibration, Vol.320, pp. 322-338.
Jones, R. M. (2005): Thermal buckling of uniformly heated unidirectional
and symmetric cross-ply laminated fiber-reinforced composite uniaxial inplane restrained simply supported rectangular plates, Composites Part A,
Vol.36, pp.1355-1367.
Karbhari, V.M. (2004): E-Glass composites in aqueous environments,
Journal of composites for construction, Vol.8(2), pp. 148-156.
Kumar, S.K. and Singh, B .N. (2008): Thermal buckling analysis of SMA
fiber reinforced composite plates using layerwise Model, Journal Aerospace
Engineering, and Vol.53 (1), pp. 1-7.
Kundu, C.K and Han, Jae-Hung. (2009): Nonlinear buckling analysis of
hygrothermoelastic composite shell panels using finite element method,
Composite Part B, Vol.40, pp. 313-328.
126
Kundu, C.K. and Han. J.H. (2009): Vibration characteristics and snapping
behavior of Hygro-thermo-elastic composite doubly curved shells, Composite
Structures, Vol.91, pp. 306-317.
Kwon, Y.W. (1991): Finite element analysis of dynamic instability of
layered composite plates using a high-order bending theory, Computers and
Structures, Vol.38 (1), pp. 57-62.
Lal, A. and Singh, B. N. (2010): Stochastic free vibration of laminated
composite plates in thermal environments, Journal of Thermoplastic
Composite Materials, Vol.23, pp. 57-77.
Lal, A., Singh, B. N. and Kumar, R. (2009): Effects of random system
properties on the thermal buckling analysis of laminated composite plates.
Computers and Structures, Vol.87, pp.1119-1128.
Lal, A. and Singh, B. N. (2010): Effect of uncertain system properties on
thermo elastic stability of laminated Composite Plates under nonuniform
temperature distribution, International Journal of Applied Mechanics, Vol.2
(2), pp. 399-420.
Lal, A. Singh, B.N and Kale, S. (2011): Stochastic post buckling analysis of
laminated composite cylindrical shell panel subjected to hygrothermomechanical loading, Computers Structures, Vol. 93, pp. 1187-1200.
Lanhe, Wu. and Hangjur, W. Daobin. (2007): Dynamic stability analysis of
functionally graded material plates by the moving square differential
quadrature method, Journal of Composite Structures, Vol.77, pp. 383-394.
Laughlan, J. (1999): The influence of bend-twist coupling on the shear
buckling response of thin laminated composite plates, Journal of thin-walled
strut, Vol.34 (2), pp. 97-114.
Lee, S.Y and Yen, W.J. (1989): Hygrothermal effects on the stability of
cylindrical composite shell panel, Computers and Structures, Vol.33 (2), pp.
551-559.
Leissa, A.W. (1987): A review of laminated composite plate buckling,
journal Applied Mechanics Review, Vol.40 (5), pp. 575-591.
Liao, C.L and Cheng, C.R. (1994): Dynamic stability of stiffened laminated
composite plates and shells subjected to in-plane pulsating forces, Journal of
Sound and Vibration, Vol.174 (3), pp. 335-351.
127
Liao, C-Li. and Cheng, C-R. (1994): Dynamic stability of stiffened
laminated composite plates and shells subjected to in-plane pulsating forces,
International Journal for Numerical Methods in Engineering, Vol.37,
pp.4167-4183.
Librescu, L and Lin,W. (1996): Vibration of geometrically imperfect panels
subjected to thermal and mechanical loads, Journal of Spacecraft Rockets,
33(2), pp.285-91.
Liew, K.M., He, X.Q., Tan, M.J. and Lim, H.K. (2004): Dynamic analysis
of laminated composite plates with piezoelectric sensor/actuator patches
using the FSDT mesh free method, International Journal of Mechanical
Sciences, Vol.46, pp. 411-431.
Liew, K.M., Lee, Y.Y., Ng, T.Y. and Zhao, X. (2007): Dynamic stability
analysis of composite cylindrical panels, International Journal of Mechanical
Sciences, Vol.49, pp. 1156-1165.
Liew, K.M, Hu, Y.G. Zhao, X and Ng, T.Y. (2006): Dynamic stability
analysis of Composite laminated Cylindrical Shells via the mesh-free kp-Ritz
method, Journal of Composite Methods in Applied Mechanics and
Engineering, Vol.196, pp.147-160.
Liu, C-F and Huang, C-H. (1995): Free vibration of composite laminated
plates Subjected to temperature changes, Journal of Computers and
Structures, Vol.60, (1), pp. 95-101.
Lua, J., Gregory, W. and Sankar, J. (2006): Multi-scale dynamic failure
prediction tool for marine composite structures, Journal of material science,
Vol. 41(20), pp. 6673-6692.
Marques, S.P.C. and Creus, G.J. (1994).Geometrically nonlinear finite
element analysis of viscoelastic composite materials under mechanical and
hygrothermal loads, Computers and Structures, Vol. 53 (2), pp. 449-56.
Melvin, A.D., Lucia, A.C.and Solomos, G.P. (1993): The thermal response
to deformation to fracture of a carbon/epoxy composite laminate, Journal of
Composite Science and Technology, Vol.46, pp. 345-351.
Mond, M. and Cederbaum, G. (1992): Dynamic instability of antisymmetric laminated Plates, Journal of Sound and Vibration, Vol.154, pp.
271-279.
128
Moorty, J., Reddy, J. N. and Plaut R.H. (1990): Parametric instability of
laminated composite plates with transverse shear deformation, International
Journal Solids Structures, Vol.26, pp. 801-811.
Mutsunaga, H. (2007): Free vibration and stability of angle-ply laminated
composite and Sandwich plates under thermal loading, Journal Composite
Structures, Vol. 77(2), pp. 249-262.
Mutsunaga, H. (2007): Free vibration and stability of angle-ply laminated
composite and sandwich plates under thermal Loading, Journal Composite
Structures, Vol.77 (2), pp. 249-262.
Naik, N.K., Chandra sekhar, Y and Meduri, S. (2000): Damage in wovenfabric composites subjected to low- velocity impact. Journal of Composite
Science and Technology, Vol.60, pp. 731-744.
Nagai, K. and Yamaki, N. (1988): Dynamic stability of circular cylindrical
shells, Acta Mechanica, Vol.71, pp.61-76.
Naik, N.K., Reddy, K.S., Meduri, S.,Raju, N.B. and Prasad,P.P.(2002):
Interlaminar fracture characterization for plain weave fabric composites,
Journal of Material Science, Vol.37, pp. 2983-2987.
Naidu, N.V.S and Sinha, P.K. (2006): Nonlinear free vibration of laminated
Composite Shells in Hygrothermal condition, Journal of Composite
Structures, Vol.77, pp. 475-483.
NG, T.Y., Lam, K.Y and Reddy, J.N. (1998): Dynamic stability of cross-ply
laminated composite cylindrical shells, International Journal of Mechanical
Sciences, Vol. 8, pp. 805-823.
Noor, A. K and Burton, W. S. (1992): Three-dimensional solutions for the
free Vibrations and buckling of thermally stressed multilayered angle-ply
composite Plates, Journal of Applied Mechanics, Vol.59 (4), pp. 868-877.
Patel, B.P. Ganapathi, M. Prasad, K.R. and Balamurugan, V. (1999):
Dynamic instability of layered anisotropic composite plates on elastic
foundations Engineering Structures, Vol.21, pp. 988-995.
129
Patel, S. and Case, S.W. (2002): Durability of hygrothermally aged epoxy
woven composite under combined hygrothermal conditions, International
Journal of Fatigue, Vol.24, pp. 1295-1301.
Patel, B.P, Ganapathi, M and Makhecha, D.P. (2002): Hygrothermal effect
on the structural behavior of thick composite laminates using higher-order
theory, Journal of Composite structures, Vol.56, pp. 25-34.
Patel, B.P., Ganapathi, M. and Makhecha, D. P. (2003): Hygrothermal
effects on the structural behavior of thick Composites using higher-order
theory, Composite Structures, Vol.56, pp.25-34.
Parhi, P.K. Bhattacharyya, S.K and Sinha, P.K. (2001): Hygrothermal
effect on the dynamic behaviour of multiple delaminated composite plates
and shells, Journal of Sound and Vibration, Vol.248 (2), pp.195-214.
Pandey, R., Upadhaya, A.K. and Shukla, K.K. (2010): Hygro- thermo
elastic post buckling response of laminated composite Plates, Journal of
Aerospace Engineering, Vol.23, pp.1-13.
Patel, S.N., Datta, P.K and Sheikh, A.H. (2007): Dynamic instability
analysis of stiffened shell panels subjected to partial edge loading along the
edges, International Journal of Mechanical Sciences, Vol. 49, pp.1309-1324.
Pilli, Sonment., P, Simmons. and Kevin, L. (2009): A novel accelerated
moisture absorption test and characterisation, Journal of composites, partA,
Vol.(40), pp.1501-1505.
Prabhu, M. R. and Dhanaraj, R. (1993): Thermal buckling of laminated
composite plates, Cmputers and Structures, Vol.53 (1), pp.1-7.
Panda.S.K. and Singh, B.N. (2011): Large amplitude free vibration analysis
of thermallypost-buckled composite doubly curved panel using non-linear
FEM, Finite Element in Anaalysis and Design, Vol.47, pp. 378-386.
Rao, V.V.S and Sinha, P.K. (2003): Dynamic response of multidirectional
composites in hygrothermal environments, Journal of Composite structures,
Vol.64, pp. 329-338.
Ravi Kumar, L., Datta, P.K. and Prabhakara, D.L. (2003): Dynamic
instability characteristics of laminated composite plates subjected to partial
follower edge load with damping, International Journal of Mechanical
Sciences, Vol.45, pp.1429-1448.
130
Ray, B.C. (2006): Temperature effect durring humid ageing on interfaces of
glass and carbon fibres reinforced epoxy composites, Journal of Collid and
Interface science, Vol.298, pp.111-117.
Ribeiro, P. and Jansen, E. (2008): Non-linear vibrations of laminated
cylindrical shallow shells under thermomechanical loading, Journal of Sound
and Vibration,Vol. 315, pp.626-640.
Saburcu, M. and Evran, K. (2006): Dynamic stability of a rotating pretwisted asymmetric cross-section Timoshenko beam subjected to an axial
periodic force. International Journal of Mechanical Sciences, Vol.48, pp.
579-590.
Sahu, S.K. and Date, P.K. (2000): Dynamic instability of laminated
composite rectangular plates subjected to non-uniform harmonic in-plane
edge loading, Journal of Aerospace Engineering, Vol.214, pp. 295-312.
Sahu, S.K. and Asha, A.V. (2008): Parametric resonance characteristics of
angle-ply twisted curved Panels, International Journal of Structural Stability
and Dynamics, Vol. 8, pp. 61-76.
Sahu, S.K and Datta, P.K. (2001): Parametric resonance characteristics of
Laminated composite doubly curved shells subjected to non uniform loading,
Journal of Reinforced Plastics and Composite, Vol.20 (18), pp.1556-1576.
Sahu, S.K and Datta, P.K. (2003): Dynamic stability behaviour of laminated
composite curved panel with cutout, Journal of Engineering Mechanics
(ASCE), Vol.129, pp.11, 1245.
Sai Ram, K.S and Sinha, P.K. (1992): Hygrothermal effects on the free
vibration of laminated composite plates, Journal of Sound and Vibration,
Vol.158 (1), pp.133-148.
Sai Ram, K. S. and Sinha, P.K. (1992): Vibration and buckling of laminated
composite plates with a cutout in Hygrothermal Environment, ASME
Institution of Aerospace of Astrological Journal, Vol.30(9), pp. 2353-2355.
Sai Ram, K.S. and Sinha, P.K. (1992): Hygrothermal effects on the buckling
of laminated composite plates, Composite Structures, Vol. 21, pp. 233-247.
Selzer, R. and Friedrich, K. (1997): Mechanical properties and failure
behaviouof carbon fibre-reinfoeced polymer composites under the influence
of moisture, Journal of Composites part A, Vol.28A, pp. 595-604.
131
Sereir, Z. and Boualem, N. (2007): Damage of hybrid composites under
long term hygrothermal loading and stacking sequence, Journal of
Theoretical and applied fracture mechanics, Vol.47, pp. 145-163.
Shen, C.H. and Springer, G.S. (1976): Moisture absorption and desorption
of composite materials, Journal of composite materials, Vol.10, pp. 272-280.
Shibasaki, M. and Somiya, M. (1999): Time dependence of degadation
phenomena of plain woven FRP in hot, wet environmental exposure, Jounal
of mechanics of time-dependent materials, Vol.2, pp. 351-369.
Shen, Hui-Shen., Zheng, Jian-Jun and Hung, Xio-Lin. (2004): The effects
of hygrothermal conditions on the dynamic response of shear deformable
laminated plates resting on elastic foundations, Journal of Reinforced plastics
and Composites, 23(10): 1095-1113.
Shen, H-S. (1990): Elasto-plastic analysis for the buckling and postbuckling
of rectangular plates under uniaxial compression, Journal of Applied
Mathematics and Mechanics, Vol.11 (10), pp. 931-939.
Shen-H-S. (2000): Hygrothermal effects on the postbuckling of shear
deformable laminated plates, Computers Structures, Vol.53 (5), pp.193-1204.
Shen, S.H. (2001): The effects of hygrothermal conditions on the postbuckling of shear deformable laminated cylindrical shells, International
Journal of Solids and Structures, Vol.38, pp. 6357-6380.
Shariyat, M. (2007): Thermal buckling analysis of rectangular composite
plates with temperature-dependent properties based on a layer wise theory,
Journal of Thin-walled Structures, Vol.45, pp. 439-452.
Singh, B.N. and Verma, V.K. (2008): Hygrothermal effects on the buckling
of laminated composite plates with random geometric and material
properties, Journal of Reinforced Plastics and Composites, Vol. 28: pp. 409427.
Srinivasan, R.S. and Chellapandi, P. (1986): Dynamic stability of
rectangular laminated composite plates, Computers and Structures. Vol.24
(2), pp. 233-238.
Spallino, R. and Thierauf, G. (2000): Thermal buckling optimization of
composite laminates by evolution strategies, Computers and Structures,
Vol.78, pp.691-697.
132
Striat, L.H., Karasek, M.L. and Amateau, M.F. (1992): Effects of sea
water immersion on the impact resistance of glass fiber reinforced epoxy
composites, Journal of Composite Materials, Vol.26 (14), pp. 2118-2133.
Tauchert, T.R. (1991): Thermally induced flexure, buckling, and vibration
of plates, Journal of Applied Mechanics Review, Vol.44 (8), pp. 347-360.
Thangaratnam, K.R., Palaninathan and Ramachanran, J. (1989):
Thermal buckling of composite laminated plates, Computer and Structures,
Vol.32, pp.1117-1124.
Thompson, S.P. and Laughlan, J. (2000): The control of post buckling
response in thin composite plates using smart technology, Journal of thinwalled structures, Vol.36, pp. 231-263
Tsai, Y.I., Bosze, E.J., Barjastech, E. and Nutt, S.R. (2009): Inluence of
hygrothermal environment on thermal and mechanical properties of carbon
fiber/fiber glass hybrid composites, Jounal of composite science and
technology, Vol.69, pp. 432-437.
Udar, R.S. and Datta, P.K. (2007): Dynamic analysis of parametrically
excited laminated composite panels under non-uniform edge loading with
damping, Journal of Composite Structures, Vol.79, 356-368.
Vijayraghavan, A. and Evan-Iwanowski, R.M. (1967): Parametric
instability of circular cylindrical shell TASME, Journal of Applied
Mechanics, Vol.31, pp.985-990.
Wang, S. and Dawe, D.J. (2002): Dynamic instability of composite
laminated rectangular plates and prismatic plate structures, Journal of
Computer Methods in Applied Mechanics and Engineering, Vol.191,
pp.1791-1826.
Whitney, J.M and Ashton, J .E. (1971): Effect of environment on the elastic
response of layered composite plates, American Institute of Aeronautics of
Astronautics Journal, Vol.9 (9), pp. 1708-1713.
Wu, G-Y. and Shih, Y-S. (2005): Dynamic stability of rectangular plate with
an edge Crack, Journal of Computers and Structures, Vol.84, pp. 1-10.
Wu, G.Y. and Shih, Y.S. (2006): Analysis of dynamic instability for
arbitrarily laminated skew plates, Journal of Sound and Vibration, Vol.292,
pp. 315-340.
133
Xiao, S. and Chen, B. (2005): Dynamic and buckling analysis of a thin
elastic-plastic square plate in a uniform temperature field, Acta Mechanica
Sinica, Vol.21, pp.181-186.
Yao, J.C. (1965): Non-linear elastic buckling and parametric excitation of a
cylinder under axial loads, Journal of Applied Mechanics, Vol.29.pp.109115.
Young-Wann, K. (2005): Temperature dependent vibration analysis of
functionally graded rectangular plates, Journal of Sound and Vibration, Vol.
284(3-5), pp.531-549.
Zai, B.A., Park, M.K., Choi, H.S, Mehboob, H and Ali, R. (2009): Effect
of Moisture absorption on dynamic stiffness of carbon fiber/epoxy
composites, Journal of Mechanical Science and Technology, Vol. 23, pp.
2298-3004.
Zenkour, A.M. and EL-Sheikh, K. (2001): Buckling and free vibration of
elastic plates using simple and mixed shear deformation theories, Acta
Mechanica, Vol.46, pp.183-197.
Zhou, C. (1991): Theory of nonlinear dynamic stability for composite
laminated plates, Journal of Applied Mathematics and Mechanics, Vol.12, pp.
113-120.
Znasni, R. and Bachir, A.S. (2006): Effect of hygrothermomechanical aging
on the interlaminar fracture behaviour of woven fiber composite materials,
Journal of thermoplastic composite materials, Vol.19 (4), pp. 385-398.
134
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