Dynamic Stability of Delaminated Cross ply Composite Plates and Shells

Dynamic Stability of Delaminated Cross ply Composite Plates and Shells
Dynamic Stability of Delaminated
Cross ply Composite Plates and Shells
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
Master of Technology
In
Structural Engineering
By
JAVED ATHER
Roll No. 209CE2036
DEPARTMENT OF CIVIL ENGINEERING
NATIONAL INSTITUTE OF TECHNOLOGY
ROURKELA-769008, ORISSA
May 2011
Dynamic Stability of Delaminated
Cross ply Composite Plates and Shells
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
Master of Technology
In
Structural Engineering
By
JAVED ATHER
Roll No. 209CE2036
Under the guidance of
Prof. S. K. Sahu
DEPARTMENT OF CIVIL ENGINEERING
NATIONAL INSTITUTE OF TECHNOLOGY
ROURKELA-769008, ORISSA
May 2011
ACKNOWLEDGEMENT
It is with a feeling of great pleasure that I would like to express my most sincere heartfelt
gratitude to Prof. S.K Sahu, Professor, Dept. of Civil Engineering, NIT, Rourkela for
suggesting the topic for my thesis report and for his ready and able guidance throughout the
course of my preparing the report. I thank you Sir, for your help, inspiration and blessings.
I express my sincere thanks to Director of the institute, Prof. M. Panda, Professor and HOD,
Dept. of Civil Engineering NIT, Rourkela for providing me the necessary facilities in the
department. I would also take this opportunity to express my gratitude and sincere thanks to
my honourable teachers Prof. K.C Biswal, Prof. M.R. Barik, Prof. A.V. Asha and all other
faculty members for their invaluable advice, encouragement, inspiration and blessings.
Submitting this thesis would have been a Herculean job, without the constant help,
encouragement, support and suggestions from my friends, especially Itishree Mishra for
their time to help. It will be difficult to record my appreciation to each and every one of them
in this small space. I will relish your memories for years to come. I would also express my
sincere thanks to laboratory Members of Department of Civil Engineering, NIT, Rourkela.
I must like to thank my parents and other family members, for their support for choices in
all my life and their love, which has been a constant source of strength for everything I do.
JAVED ATHER
ROLL No.- 209CE 2036
National Institute of Technology
Rourkela
CERTIFICATE
This is to certify that the thesis entitled “DYNAMIC STABILITY OF DELAMINATED
CROSS PLY COMPOSITE PLATES AND SHELLS” submitted by Mr. JAVED
ATHER in partial fulfillment of the requirements for the award of Master of Technology
Degree in Civil Engineering with specialization in Structural Engineering at the National
Institute of Technology, Rourkela (Deemed University) is an authentic work carried out by
him under my supervision and guidance.
To the best of my knowledge, the matter embodied in the thesis has not been submitted to any
other University/ Institute for the award of any degree or diploma.
Date:
Prof. S.K. Sahu
Department of Civil Engineering
National Institute of Technology
Rourkela – 769008
CONTENT
Page No.
ABSTRACT…………………………………………………………………………………………1
INTRODUCTION…………………………………………………………………………… 11-13
LITERATURE REVIEW ON
VIBRATION………………………………………….15-17
STATIC ………………………………………………..17-20
DYNAMIC STABILITY……………………………20-21
AIM & SCOPE OF PRESENT STUDY…………21
THEORETICAL FORMULATION…………………………………………………………22-37
RESULTS AND DISCUSSIONS
VIBRATION ANALYSIS…………………………………………. 44-48
STABILITY ANALYSIS……………………………………………. 48-59
DYNAMIC STABILITY……………………………59-66
CONCLUSIONS……………………………………………………………………………… 67-69
REFERENCES……………………………………………………………………………………70-73
Abstract:
Fibre reinforced composite plates and shells are increasingly replacing traditional metallic
ones. The manufacturing process and service of the composite laminates frequently lead to
delamination. Delamination reduces the stiffness and strength of composite laminates
because they allow out of plane displacement of plies to occur
more easily. Dynamic
stability analysis is an integral part of most engineering structures. The present work deals
with the study of the effects of free vibration, buckling and dynamic stability of delaminated
cross ply composite plates and shells. A first order shear deformation theory based on finite
element model is developed for studying the instability region of mid plane delaminated
composite plate and shell.
The basic understanding of the influence of the delamination on the natural frequencies, nondimensional buckling load and non-dimensional excitation frequency of composite plates and
shells is presented. In addition, other factors affecting the vibration, buckling and dynamic
instability region of delaminated composite plates and shells are discussed.
The numerical results for the free vibration, buckling and dynamic stability of laminated
cross-ply plates and shells with delamination are presented. As expected, the natural
frequencies and the critical buckling load of the plates and shells decrease with increase in
delamination. Increase in delamination also causes dynamic instability regions to be shifted
to lower excitation frequencies.
1|Page
LIST OF SYMBOL
a, b = Plate dimensions along x- and y-axes, respectively.
[K] = bending stiffness matrix
[Kg ] = geometric stiffness matrix
= natural frequency
[M] = mass matrix
{∅} = eigenvectors i.e. mode shape
P = buckling load
α0 = static load factor
α1 = dynamic load factor
Pcr = critical buckling load.
= Fibre orientation in a lamina
2|Page
x ,y ,z = System of coordinate axes
t = thickness of plate
u, v, w = Displacement along x, y and z direction.
Nx, Ny, Nxy = In-plane internal force resultants per unit length
Zk, Zk- I = Top and bottom distances of a lamina from the mid-plane
Ni = Shape function at a node i
Kx, Ky, Kxy = Curvatures of a plate
Mx, My, Mxy = Internal moment resultants per unit length
Gl2, G13, G23 = Shear moduli of a lamina with respect to 1, 2, and 3 axes
E1, E2 = Young's moduli of a lamina along and across fibres, respectively
Qx, Qy = Transverse shear resultants per unit length
3|Page
LISTS OF TABLES
Table 1.1: Non-dimensional parameters of composite plates/shells
…………………39
Table 1.2: Natural frequencies (Hz) for mid-plane delaminated simply supported composite
spherical, cylindrical shells and plates with different % of delamination. ………………40
Table 1.3: Natural frequencies (Hz) for mid-plane delaminated simply supported composite
spherical and cylindrical shells with different % of delamination.
…………………..41
Table 2.1: Comparison of non-dimensional buckling loads of a square simply supported
doubly curved panel with (0/90) lamination.
…………………41
Table 2.2: Comparison of non-dimensional buckling loads of a square simply supported
symmetric cross-ply cylindrical shell panels with [0/90/0/90/0] lamination for different
length-to-thickness ratio (a/h).
…………………...42
Table 2.3: Comparison of critical buckling load with Radu et al (2002) for different mid
plane delamination length of the rectangular plates using cantilever boundary condition. .43
Table 3.1: Natural frequencies (Hz) for 25% delaminated cross ply-(0/90)n simply supported
composite spherical and cylindrical shells with different no. of layers. ……………….. 45
Table 3.2: Natural frequencies (Hz) for 25% delaminated cross ply-(0/90)2 simply supported
composite spherical and cylindrical shells with different aspect ratio.
…………………46
Table 3.3: Natural frequencies (Hz) for delaminated cross ply-(0/90)2 simply supported
composite spherical and cylindrical shells with different b/h ratio for R/a=5. ……………47
4|Page
Table 3.4: Natural frequencies (Hz) for delaminated cross ply-(0/90)2 simply supported
composite spherical and cylindrical shells with different orthotropic ratio for R/a=5. ……48
Table 4.1: Variation of non-dimensional buckling load with different no. of layers for 0%
………..49
delaminated composite shell.
Table 4.2: Variation of non-dimensional buckling load with different no. of layers for 0%
……….50
delaminated composite shell. Rx/a=10, Ry/a=10
Table 4.3: Variation of non-dimensional buckling load with different no. of layers for 0%
delaminated composite shell.
…………..51
Rx/a=20, Ry/a=20
Table 4.4: Variation of non-dimensional buckling load with different b/h ratio for 0%
delaminated composite shell.
…………52
Rx/a=5, Ry/a=5, E1=10E2,
Table 4.5: Variation of non-dimensional buckling load with different b/h ratio for 0%
delaminated composite shell.
…………..42
Rx/a=10, Ry/a=10, E1=10E2,
Table 4.6: Variation of non-dimensional buckling load with different degree of orthotropic
for 0% delaminated composite shell.
Rx/a=5, Ry/a=5, a/h=10,
………….53
Table 4.7: Variation of non-dimensional buckling load with different b/h ratio for 0%
delaminated composite shell.
Rx/a=10, Ry/a=10, a/h=10,
cross-ply-(0/90/0/90/0)…55
Table 4.8: Variation of non-dimensional critical buckling load with different no. of layers for
different percentage of delaminated composite shell.
Ry/a=10.
…………56
Table 4.9: Variation of non-dimensional critical buckling load with different no. of layers for
different percentage of delaminated composite shell.
5|Page
Rx/a=5, Ry/a=5
……….57
Table 5.1: Variation of non-dimensional critical buckling load with different no. of layers for
different percentage of delaminated composite plate.
……..58
Table 5.2: Variation of non-dimensional critical buckling load with different b/h ratio for
different percentage of delaminated composite plate
6|Page
………60
LISTS OF FIGURE
Figure 1: Laminated doubly curved composite shell axes
.......25
Figure 2: Layer details of shell panel.
……26
Figure 3: Multiple delamination model.
..….34
Figure 4: First natural frequency vs. no. of layer for simply supported composite shell with a
…..45
single mid-plane delamination.
Figure 5: First natural frequency vs. aspect ratio for simply supported composite shell with a
single mid-plane delamination.
Figure 6:
..….46
First natural frequency vs. delamination % at different b/h ratio for simply
supported composite shell with a single mid-plane delamination.
……47
Figure 7: First natural frequency vs. delamination % at different E1/E2 ratio for simply
supported composite shell with a single mid-plane delamination.
……..48
Figure 8: Variation of non-dimensional buckling load vs. no. of layers for simply supported
composite shell. Rx/a=5, Ry/a=5,
cross-ply-(0/90)n
………50
Figure 9: Variation of non-dimensional buckling load vs. no. of layers for simply supported
composite shell. Rx/a=10 Ry/a=10, cross-ply-(0/90)n
…….51
Figure 10: Variation of non-dimensional buckling load vs. no. of layers for simply supported
composite shell. Rx/a=20, Ry/a=20,
cross-ply-(0/90)n.
……52
Figure 11: Variation of non-dimensional buckling load vs. b/h ratio for simply supported
composite shell. Rx/a=5, Ry/a=5,
7|Page
cross-ply-(0/90).
…….53
Figure 12: Variation of non-dimensional buckling load vs. b/h ratio for simply supported
composite shell. Rx/a=5, Ry/a=5,
……54
cross-ply-(0/90)
Figure 13: Variation of non-dimensional buckling load vs. E1/E2 ratio for simply supported
composite shell. Rx/a=5, Ry/a=5,
…….55
cross-ply-(0/90).
Figure 14: Variation of non-dimensional buckling load vs. E1/E2 ratio for simply supported
composite shell. Rx/a=5, Ry/a=5,
……56
cross-ply-(0/90).
Figure 15: Variation of non-dimensional buckling load vs. no. of layers for simply supported
composite shell with different % of delamination. Ry =2,
cross-ply-(0/90)n
…….57
Figure 16: Variation of non-dimensional buckling load vs. no. of layers for simply supported
composite shell with different % of delamination. Rx/a=5, Ry/a=5,
cross-ply-(0/90)n ....58
Figure 17: Variation of non-dimensional buckling load vs. no. of layers for simply supported
composite plate with different % of delamination.
Cross-ply-(0/90)n
……59
Figure 18: Variation of non-dimensional buckling load vs. b/h ratio for simply supported
composite plate with different % of delamination.
Figure 19:
=125
Figure 20:
=25
Cross-ply-(0/90)
……60
Effect of delamination on instability region of [(0/90)2]s cross- ply plate for L/t
…….61
Effect of delamination on instability region of [(0/90)2]s cross- ply plate for L/t
…….62
Figure 21: Effect of % of delamination on instability region of delaminated 2-layers crossply plate.
……..62
Figure 22: Effect of % of delamination on instability region of delaminated 4-layers crossply plate.
8|Page
…….63
Figure 23: Effect of % of delamination on instability region for cross ply delaminated plate
…….63
for degree of orthotropy, E11/E22 =40.
Figure 24: Effect of % of delamination on instability region for cross ply delaminated plate
……64
for degree of orthotropy, E11/E22 =20.
Figure 25: Effect of aspect ratio on instability region for simply supported cross ply
delaminated plate. L/t=10, E1/E2 = 25, 𝞪=0.2.
......64
Figure 26: Effect of aspect ratio on instability region for simply supported cross ply
delaminated plate. L/t=10, E1/E2 = 25, 𝞪=0.2.
….65
Figure 27: Effect of static load factor on instability region of rectangular plate
(127*12.7*1.016)mm.
a=
127mm, b=12.7mm, t=1.016mm, stacking sequence= (0/90/0/90/90/0/90/0).
……65
Figure 28: Effect of % of delamination on the instability region of simply supported cross
ply (0/90) spherical shell: a/Rx = b/Ry =0 .25, 𝞪=0.2, a/b=1, a/h=10, E11=40E22,
G12=G13=0.6E22, G23=0.5E22,
=
=0.25.
………..66
Figure 29: Effect of % of delamination on the instability region of simply supported cross
ply (0/90) cylindrical shell: b/Ry =0 .25, 𝞪=0.2, a/b=1, a/h=10, E11=40E22, G12=G13=0.6E22,
G23=0.5E22,
=
……….66
=0.25.
Figure 30: Effect of curvature on instability region with 6.25% of delamination of simply
supported cross ply (0/90): a/Rx = b/Ry =0 .25, 𝞪=0.2, a/b=1, a/h=10, E11=40E22,
G12=G13=0.6E22, G23=0.5E22,
9|Page
=
=0.25.
………67
Figure 31: Effect of curvature on instability region with 25% of delamination of simply
supported cross ply (0/90): a/Rx = b/Ry =0 .25, 𝞪=0.2, a/b=1, a/h=10, E11=40E22,
G12=G13=0.6E22,G23=0.5E22,
=
=0.25.
……..67
Figure 32: 6.25% central delamination.
………..75
Figure 33: 25% central delamination.
………...75
Figure 34: 56.25% central delamination
……….76
Figure 35: Eight layered laminate without delamination.
……….76
Figure 36: Eight layered laminate with mid-plane delamination.
……….77
Figure 37: Eight layered laminate with three delamination.
10 | P a g e
……...77
CHAPTER 1
INTRODUCTION
Introduction:Composite laminates are widely used in engineering structures due to their excellent
properties, such as high strength-to-weight ratio, high stiffness-to-weight ratio and design
versatility etc. Delamination can cause serious structural degradation. Which is a debonding
or separation between individual plies of the laminate, frequently occurs in composite
laminates. Delamination may arise during manufacturing (e.g., incomplete wetting, air
entrapment) or during service (e.g., low velocity impact, bird strikes). They may not be
visible or barely visible on the surface, since they are embedded within the composite
structures. However, the presence of delamination may significantly reduce the stiffness and
strength of the structures and may affect some design parameters such as the vibration
characteristic of structure of structure. (e.g., natural frequency and mode shape).
Delaminations reduce the natural frequency, as a direct result of the reduction of stiffness,
which may cause resonance if the reduced frequency is close to the working frequency. It is
therefore important to understand the performance of delaminated composites in a dynamic
environment. The subject of predicting the dynamic and mechanical behaviour of
delaminated structures has thus attracted considerable attention.
Plate and Shell members have often been used in modern structural systems, because the
desired performance can be achieved by controlling the shape of those structures. In
particular, high-performance applications of laminated composites to plate and shell members
are advantageous because of their light weight, high specific stiffness and high specific
strength. However, laminated plates and shells subjected to in plane periodic forces may lead
11 | P a g e
to dynamic instability for certain combinations of load amplitude and disturbing frequency.
Furthermore, plates and shell members with delamination may result in significant changes to
their dynamic characteristics. Therefore, it is essential to study the effect of delaminations
simultaneously on the dynamic stability of layered shells under periodic loads. Composite
plates and shells are widely used in aerospace structures. These are often subjected to defects
and damage from both in-service and manufacturing events. Delamination is the most
important of these defects
Importance of the stability studies of delaminated composite shell
Structural elements subjected to in-plane periodic forces may lead to parametric resonance,
due to certain combinations of the values of load parameters. The instability may occur below
the critical load of the structure under compressive loads over wide ranges of excitation
frequencies. Several means of combating resonance such as damping and vibration isolation
may be inadequate and sometimes dangerous with reverse results. Structural elements
subjected to in-plane periodic forces may induce transverse vibrations, which may be
resonant for certain combinations of natural frequency of transverse vibration, the frequency
of the in-plane forcing function and magnitude of the in-plane load. The spectrum of values
of parameters causing unstable motion is referred to as the regions of dynamic instability or
parametric resonance. Thus the parametric resonance characteristics are of great importance
for understanding the dynamic systems under periodic loads.
Delamination between plies is one of the most common defects encountered in composite
laminates. Delamination can cause serious structural degradation. Which is a debonding or
separation between individual plies of the laminate, frequently occurs in composite laminates.
Delamination may arise during manufacturing (e.g., incomplete wetting, air entrapment) or
12 | P a g e
during service (e.g., low velocity impact, bird strikes). They may not be visible or barely
visible on the surface, since they are embedded within the composite structures. However, the
presence of delaminations may significantly reduce the stiffness and strength of the structures
and may affect some design parameters such as the vibration characteristic of structure of
structure. (e.g., natural frequency and mode shape). Delaminations reduce the natural
frequency, as a direct result of the reduction of stiffness, which may cause resonance if the
reduced frequency is close to the working frequency. It is therefore important to understand
the performance of delaminated composites in a dynamic environment. The subject of
predicting the dynamic and mechanical behaviour of delaminated structures has thus attracted
considerable attention.
13 | P a g e
CHAPTER 2
LITERATURE REVIEW
Introduction:-Thus the dynamic stability characteristics are of great technical importance
for understanding the dynamic systems under periodic loads. In structural mechanics,
dynamic stability has received considerable attention over the years and encompasses many
classes of problems. The distinction between “good” and “bad” vibration regimes of a
structure subjected to in-plane periodic loading can be distinguished through a simple
analysis of dynamic instability region. In modelling delamination, both, analytical as well as
numerical methods have been used in studying the dynamic and buckling behaviour of
composite laminates. Bolotin (1964) in his text on dynamic stability gives a thorough review
of the problems involving parametric excitation of structural elements. The dynamic
instability of composite plates and shells without delamination has been studied previously by
a host of investigators. The studies in this chapter are grouped into three major parts as
follows:
 Vibration
 Buckling
 Dynamic stability
The available literature on dynamic stability of delamination composite plates and shells is
very limited.
14 | P a g e
Free Vibration:
Tenek (1993) et al. studied vibration of delaminated composite plates and some applications
to non-destructive testing; he studied the impact of delamination on the natural frequencies of
composite plates, as well as delamination dynamics over a broad frequency range, using the
finite element method based on the three-dimensional theory of linear elasticity. For the case
of cantilever laminated plates, the method is successfully compared with experimental
observations.
Lee and Lee (1995) examined the free vibration of composite plates with delamination
around cut-outs. They presented a finite element approach is to analyze the free vibration of
square and circular composite plates with delamination around internal cut-outs. They
presented numerical examples including composite plates with delaminations around circular
holes or square cut-outs. And they discussed the effects of the cut-outs and the delamination
around the cut-outs on the natural frequencies and mode shapes.
Ju et al (1995) presented finite element analysis of free vibration of delaminated composite
plates. His study was based on Mindlin plate theory. He presented a finite element
formulation for the analysis of the free vibration of composite plates with multiple
delamination.
Chang (1998) investigated the vibration analysis of a delaminated composite plate subjected
to the axial load. The concept of continuous analysis was used to model the delaminated
composite plate as a plate on a discontinuous elastic foundation. The elastic adhesive layer
between the buckled composite plate and the undeformed substructure working as a
foundation to the plate is represented by linear parallel springs.
Williams and Addessio (1998) studied a dynamic model for laminated plates with
delaminations. They presented a dynamic, higher-order theory for laminated plates based on a
15 | P a g e
discrete layer analysis. The formulation includes the effects of delaminations between the
layers of the plate. The model implements a generalized displacement formulation at the
lamina level. The governing equations for the lamina are derived using vibrational principles.
Geubelle and Baylor (1998) observed Impact-induced delamination of composites using a 2D
simulation. The delamination process in thin composite plates subjected to low-velocity
impact is simulated using a specially developed 2D cohesive/volumetric finite element
scheme. Cohesive elements are introduced along the boundaries of the inner layers and inside
the transverse plies to simulate the spontaneous initiation and propagation of transverse
matrix cracks and delamination fronts.
Hou and Jeronimidis (1999) presented vibration of delaminated thin composite plates. He
studied experimental (employed free-free vibration system) and finite element modelling of
vibration of GFRP laminated circular plates. He shows that the resonant frequencies of low
velocity impacted plates are functions of matrix cracking and the local thickening coupling
with interlaminar delamination.
Ostachowicz and Kaczmarezyk (2001) studied the vibration of composite plates with SMA
fibres in a gas stream with defects of the type of delamination. They analyzed the dynamics
of a multi-layer composite plate with delaminations subjected to an aerodynamic load. They
proposed finite element model to predict the dynamic response of the system with embedded
shape memory alloys (SMA) fibres. They studied the effect of delamination on the natural
frequencies of delamination subjected to supersonic flow and determined the flutter
instability boundaries.
Hu et al (2002) investigated vibration analysis of delaminated composite beams and plates
using a higher-order finite element. In order to analyze the vibration response of delaminated
composite plates of moderate thickness, he proposed a FEM Model based on a simple higher-
16 | P a g e
order plate theory, which can satisfy the zero transverse shear strain condition on the top and
bottom surfaces of plates.
Chen et al (2004) examined the dynamic behaviour of delaminated plates considering
progressive failure process. A formula of element stiffness and mass matrices for the
composite laminates is deduced by using the first-order shear deformation theory combined
with the selecting numerical integration scheme.
Lee and Chung (2010) observed the finite element delamination model for vibrating
composite spherical shell panels with central cut-outs. They developed finite element model
of vibrating laminated spherical shell panels with delamination around a central cut-out is
based on the third-order shear deformation theory of Sanders. In the finite element
formulation for the delamination around cut-out, the seven degrees of freedom per each node
are used with transformations in order to fit the displacement continuity conditions at the
delamination region.
Static stability:
Hwang and Mao (2001) investigated failure of delaminated carbon/epoxy composite plates
under compression. Their work is to study the buckling loads, buckling modes, postbuckling
behaviour, and critical loads of delamination growth for delaminated unidirectional
carbon/epoxy composites. Nonlinear buckling analysis, which is a finite-element method
including contact elements to prevent the overlapping situation, was applied to predict the
delamination buckling loads.
Parhi et al. (2001) presented hygrothermal effects on the dynamic behaviour of multiple
delaminated composite plates and shells. They presented a quadratic isoparametric finite
element formulation based on the first order shear deformation theory for the free vibration
17 | P a g e
and transient response analysis of multiple delaminated doubly curved composite shells
subjected to a hygrothermal environment. For the transient analysis Newmark's direct
integration scheme is used to solve the dynamic equation of equilibrium at every time step.
Liu and Yu (2002) studied the Finite element modelling of delamination by layerwise shell
element allowing for interlaminar displacements. In his paper, he gave a brief review of the
layerwise shell element. He considered geometrically non-linear analysis involving finite
rotation and finite strain. According to his results he suggested that central delamination is
more damaging than the edge and corner delamination for it causes a greater reduction in the
strength and integrity of the plate, in particular in the early stage of shear mode delamination
Shan and Pelegri (2003) investigated the approximate analysis of the Buckling Behaviour of
Composites with Delamination. In this paper, they provided an insight into the governing
mechanism of the uni-axial compressive buckling of a delaminated composite. They
proposed an approximate method to analyse the buckling behaviour as the first step to further
investigate the effect of contact zone at the ends of a delamination. The agreement of the
analytical results with experiments in critical values of relative axial displacement verifies
with his model and approach.
Wang et al. (2003) observed Non-Linear Thermal Buckling for Local Delamination near the
Surface of Laminated Plates. An investigation is carried out to understand the non-linear
thermal buckling behaviour of local delamination near the surface of fibre reinforced
laminated plates. The shape of the delaminated region considered is rectangular, elliptic or
triangular..
Ju-fen et al. (2003) developed reference surface element modelling of composite plate/shell
delamination buckling and post buckling. The technique can be easily incorporated into any
finite element analysis programme for which the beam, plate and shell elements etc. satisfy
the Reissner–Mindlin assumption. They presented the reference surface element formulation
18 | P a g e
of a four-node Co quadrilateral membrane-shear-bending element and performed numerical
investigations for composite plates and shells with various delamination shapes.
Mümin and KüÇük (2004) investigated the buckling behaviour of simply supported woven
steel reinforced thermoplastic laminated plates with lateral strip delamination. He established
three dimensional finite element models of the square laminated plates. Each of these models
possesses four layers and a different delamination width between the second and third layers.
The orientation angle of the fibres is chosen as 0, 15, 30 and 45. He determined the buckling
loads for each model having different lateral delamination width.
Tafreshi (2006) examined Delamination buckling and postbuckling in composite cylindrical
shells under combined axial compression and external pressure. He investigated the
characteristics of the buckling and postbuckling behaviour of delaminated composite
cylindrical shells .The combined double-layer and single-layer of shell elements are
employed which in comparison with the three dimensional finite elements requires less
computing time and space for the same level of accuracy.
Yang and Fu (2006) investigated the Delamination growth of laminated composite cylindrical
shells. They derived the post buckling governing equations for the laminated cylindrical
shells, based on the variational principle of moving boundary, and the corresponding
boundary and matching conditions are given. At the same time, according to the Griffith
criterion, the formulas of energy release rate along the delamination front were obtained and
the delamination growth was studied.
Oh et al. (2008) observed buckling analysis of a composite shell with multiple delaminations
based on a higher order zig-zag theory. They developed a new three-node triangular shell
element based on a higher order zig-zag theory for laminated composite shells with multiple
delaminations. The baseline higher order zig-zag composite shell theory for multiple
19 | P a g e
delaminations has been developed in a general curvilinear coordinate system and in general
tensor notation.
Chirica et al. (2010) investigated the buckling analysis of the composite plates with
delaminations. In this paper they studied of the influence of elliptical delamination on the
changes in the buckling behaviour of ship deck plates made of composite materials is treated.
A delamination model has been developed by using the surface-to-surface contact option, in
licensed FEM code COSMOS/M. So, the damaged part of the structures and the undamaged
part were represented by well-known finite elements (layered shell elements).
Ovesy and Kharazi (2011) observed the stability analysis of composite plates with throughthe-width delamination. The analytical method is based on the first-order shear deformation
theory, and its formulation is developed on the basis of the Rayleigh-Ritz approximation
technique by the implementation of the polynomial series, which has been used for the first
time in the case of the mixed mode of buckling.
Dynamic Stability:
Radu and Chattopadhyay (2000) analyzed the dynamic stability of composite plates including
delamination using higher order theory and transformation matrix. He analyzed composite
plates with various thickness and delamination length and placement. Delamination affects
the instability regions by shifting them to lower, parametric resonance frequencies and by
modifying.
Jinhua and Yiming (2007) studied the analysis of dynamic stability for composite laminated
cylindrical shells with delaminations. they derived a set of dynamic governing equations for
the delaminated cylindrical shells By introducing the Heaviside step function into the
assumed displacement components and using the Rayleigh–Ritz method for minimizing the
20 | P a g e
total potential energy, Then, the dynamic governing equations are written as the Mathieu-type
equations to describe the parametric vibrating behaviour of the shells, and these equations are
solved by employing the Bolotin method. They presented numerical results for the dynamic
stability of laminated cylindrical shells with delaminations.
Park and Lee (2009) examined parametric instability of delaminated composite spherical
shells subjected to in-plane pulsating forces. The dynamic stability analysis of delaminated
spherical shell structures subjected to in-plane pulsating forces is carried out based on the
higher-order shell theory of Sanders. In the finite element (FE) formulation, the seven
degrees of freedom per each node are used with transformations in order to fit the
displacement continuity conditions at the delamination region. The boundaries of the
instability regions are determined using the method proposed by Bolotin.
Aim and Scope of present study:
The present study deals with the effect of delamination on dynamic characteristic of
composites.
The various modules are:

Vibration of delaminated composite shells

Buckling of delaminated composite shells

Dynamic stability of delaminated plates and shells
21 | P a g e
CHAPTER 3
MATHEMATICAL FORMULATION
The Basic Problems
This chapter involves the finite element mathematical formulation for vibration, static and
dynamic stability analysis of the plate and shell structures of various geometry with/without
delamination. The basic configuration of the problem considered here is a doubly curved
panel with mid plane single delamination subjected to in plane periodic load. The boundary
conditions are incorporated in the most general manner.
1.1. Proposed Analysis
The governing equations for the dynamic stability of the delaminated composite plates and
shell are derived on the basis of first order shear deformation theory .The element stiffness,
geometric stiffness and mass matrices are derived on the basis of principle of minimum
potential energy and Lagrange‟s equation.
Governing equation for analysis:-
Equation for free vibration is,
Where
frequency and
are the global stiffness and global mass matrices,
is the corresponding eigenvectors i.e. mode shape.
Equation for buckling analysis is,
22 | P a g e
is the natural
Where
and
are the bending stiffness matrix and geometric stiffness matrix. The
Eigen values of the above equation gives the buckling loads for different modes. The lowest
value of buckling load (P) is termed as critical buckling load
)of the structure.
Equation for dynamic stability analysis in line with [Radu and Chattopadhyay (2000) ],
Where M, K and
KG are the mass, the stiffness and the geometric matrices.
and
Represent the frequency,
are static and dynamic parameters taking values from 0 to 1.
FINITE ELEMENT ANALYSIS
A laminated doubly curved shell panel of length a, width b and thickness h consisting of n
arbitrary number of anisotropic layers is considered as shown in Figure 1. The layer details of
the laminate are shown in Figure 2. The displacement field is related to mid-plane
displacements and rotations as
u(x,y,z,t) = u0(x,y,z,t) + z
x
(x,y,t),
v(x,y,z,t) = v0(x,y,z,t) + z
y
(x,y,t),
w(x,y,z,t) =w0(x,y,z,t) ,
Where u, v and w are displacements in x, y and z directions, respectively, and the superscript
23 | P a g e
(0) corresponds to the mid-plane values. Here,
x
and
y
denote the rotations of the cross-
sections perpendicular to the y- and x-axis respectively. Using Sander‟s first approximation
theory for thin shells, the generalized strains in terms of mid-plane strains and curvatures are
expressed as
{
xy
xz
yz
}T =
}T + { kxx kyy kxy kxz kyz}T ,
Where,
=
And
=
And Rx , Ry and Rxy are the three radii of curvature of the shell element. The entire dynamic
24 | P a g e
Equations of equilibrium reduce to that for the plate, when the values of Rx, Ry, and Rxy
become
. The five degrees of freedom considered at each node of the element are u0, v0, w,
x and y. By using the eight-nodded element shape functions, the element displacements are
expressed in terms of their nodal values given by
u0 =
, v0 =
x=
=
y=
.
,
0
„s are the shape functions used to interpolate the generalized displacements
Where
,
,
and
at node i within an element.
Figure 1. laminated doubly curved composite shell axes
25 | P a g e
f igure 2. Layer details of shell panel.
The stress resultants are related to the mid-plane strains and curvatures for a general
laminated shell element as
=
Where
and
,
,
and
are in-plane stress resultants,
and
are moment resultants
are transverse shear stress resultants. The extensional, bending-stretching and
bending stiffness‟s of the laminate are expressed in the usual form as
(Aij, Bij, Dij) =
26 | P a g e
k
(1, z, z2 ) dz,
i , j = 1, 2, 6.
Similarly, the shear stiffness is expressed as
(Aij) =
k
dz,
i , j = 4, 5.
is the shear correction factor which is derived from the Timoshenko beam concept by
applying the energy principle is assumed as 5/6. It accounts for the non-uniform distribution
of transverse shear strain across the thickness of the laminate.
in equations (6) and (7) are the off-axis stiffness values defined as
k
k
=
=
,
k
k
,
i, j = 1, 2, 6.
i, j = 4, 5.
Where, m=cos , n=sin
k
are calculated in the conventional manner from the values of elastic and shear moduli
and the Poisson ratio values.
27 | P a g e
Q11 = E1/(1-υ12υ21) ,
Q12 = υ12E2/(1-υ12υ21), Q21 = υ12 E2/(1-υ12υ21) , Q22 = E2/(1-υ12υ21)
Q66= G12, Q44= G13, Q55= G23.
E1, E2 = Young‟s moduli of a lamina along and across the fibres, respectively
G 12, G13, G23 = Shear moduli of a lamina with respect to 1,2 and 3 axes.
υ12 ,υ21 = Poisson‟s ratios.
The element stiffness matrix, [Ke] is given by
Where [B] is stain-displacement matrix
[D] is the elasticity matrix
J is the Jacobean
Full integration (3x3) is used for bending stiffness, whereas reduced integration (2x2) is
employed to evaluate transverse stiffness of the elements. A 2 point integration eliminates
the shear locking in case of thin plates.
Similarly the consistent element mass matrix [Me] is expressed as
With [N], the shape function matrix and [𝜌], the inertia matrix.
The element load vector,{Fe} is presented by
28 | P a g e
Where {q} is the intensity of external transverse uniform loading.
The shape functions Ni are defined as
Ni= (1+ξξi)(1+𝜂𝜂i)( ξξi+ 𝜂𝜂i-1)/4
Ni= (1-ξ2)(1+𝜂𝜂i)/2
Ni= (1+ ξξi)(1-𝜂2)/2
i=1to4
i=5,7
i=6,8
Where ξ and 𝜂 are the local natural coordinates of the element and ξi and 𝜂i are the values
at ith mode. The derivatives of the shape function Ni with respect to x,y are expressed in term
of their derivatives with respect to ξ and 𝜂 by the following relationship
Where [J] =
Strain displacement Relations
Green-Lagrange‟s strain displacement is used throughout the structural analysis. The linear
part of the strain is used to derive the elastic stiffness matrix and non-linear part of the strain
is used to derive the geometrical stiffness matrix.
{ε}={ εl}+{ εnl}
The linear strains are defined as
29 | P a g e
Where the bending strains kj are expressed as
,
And C1 and C2 are tracers by which the analysis can be reduced to that of shear deformable
love‟s first approximations and Donnell‟s theories.
Assuming that w does not vary with z, the non-linear strains of the shell are expressed as
= [ (u,x+ w/Rx)2 +
+ (w,x
u/Rx)2]/2,
= [ (v,y+ w/Ry)2 +
+ (w,y
v/Ry)2]/2,
= [ (u,x+ w/Rx)u,y + v,x(v,y+ w/Ry) + (w,x
30 | P a g e
u/Rx) (w,y
v/Ry)],
= [ (u,x+ w/Rx)u,z + v,x v,z+ (w,x
u/Rx) w,z],
= [ (u,y+ w/Ry)u,z + v,y v,z+ (w,y
u/Ry) w,z],
The linear strain can be described in term of displacements as
=
Where
Generalised element mass matrix or consistent mass matrix:-
Where the shape function matrix
31 | P a g e
T
[N] =
[P] =
Where,
k
(1, z, z2 ) dz,
The element mass matrix can be expressed in local natural co-ordinate of the element as.
Geometric Stiffness Matrix:
The element geometric stiffness matrix is derived using the non-linear in-plane Green‟s
strains. The strain energy due to initial stresses is
Using non-linear strains, the strain energy can be written in matrix form as
U2=
32 | P a g e
=
[S] =
=
=
The in-plane stress resultants Nx
,
Ny, Nxy at each gauss point are obtained by applying
uniaxial stress in x-direction and the geometric stiffness matrix is formed for these stress
resultants.
=
The strain energy becomes
U2 =
=
Where element geometric stiffness matrix
[G] =
33 | P a g e
Multiple delamination modelling
Considering a typical composite laminate having p number of delamination is considered.
,
(1)
h
h/2
Figure 3: Multiple delamination model
Where ut0, vt0 are the mid-plane displacement of the tth sub laminate and zt0, the distance
between the mid-plane of the original laminate and the mid-plane of the tth sub laminate. The
strain component of any layer of a sub laminate are found from eqn (2) in the form of
(2)
34 | P a g e
In the order to satisfy the compatibility condition at the delamination‟s boundary, it is
assumed that transverse displacements and rotations at a common node for all the three sub
laminates including the original one are identical. Applying this multipoint constraint
condition,the midpoint displacements of any sub laminate t can be generalized as
,
(3)
The mid plane strain components of the tth sub laminate are derived from it as
(4)
Substituted eqn(4) into eqn(2),then we get strain components at any layer within a sub
laminate. For any lamina of the tth sub laminate, the in-plane and shear stresses are fined from
the relations
Integrating these stresses over the thickness of the sub laminate, the stress and moment
resultants of the sub laminate are desired which lend to the elasticity matrix of the tth sub
laminate is the form
35 | P a g e
Where
, i,j=1,2,6
, i,j=4,5
Here ht is the thickness of the tth sub laminate.
Analysis Method:The finite element formulation is developed for the dynamic analysis of laminated composite
shells with delamination using the first order shear deformation theory. An eight-nodded
continuous doubly curved isoparametric element is employed in the present analysis with five
degrees of freedom viz. u, v, w, x, and y at each node.
The eigenvalue equation for the free vibration analysis of laminated composite plate and shell
can be expressed as
Where
mass matrices are,
is the natural frequency and
the global stiffness and global
is the corresponding eigenvectors i.e.
mode shape.
The eigenvalue equation for the stability analysis of laminated composite plate and shell can
be expressed as
36 | P a g e
Where
and
are the bending stiffness matrix and geometric stiffness matrix. The
Eigen values of the above equation gives the buckling loads for different modes. The lowest
value of buckling load (P) is termed as critical buckling load
) of the structure.
The eigenvalue equation for the dynamic stability analysis of laminated composite plate and
shell can be expressed as
Where M, K
and KG are the mass, the stiffness and the geometric matrices.
and
Represent the frequency,
are static and dynamic parameters taking values from 0 to 1.
Computer Program:
A computer program based on finite element formulation developed in MATLAB 7.8.0. The
element bending stiffness, geometric stiffness and mass matrices are derived by using
functions. The program consists of sub function called inside the program. The sub function
performs specific work at different levels of analysis. In the beginning of the program,
dimensions of the plates and shells, and material properties of plates and shells defined by
appropriate symbol. In the program several function used such as assmbl for assembling the
stiffness, geometric stiffness and mass matrices. Compo function is used for finding the
constituent matrices. Gestiff is used for finding geometric stiffness, stiff is used for finding
bending stiffness, massfsdt is used for mass matrices, and gener is used for generalization of
bending stiffness, geometric stiffness and mass matrices. For delamination, delamcompo
function is used and also used two functions top and bottom for finding mid plane
delamination effect. Shaped is used for finding out the shape function and ndarr is for
boundary condition.
37 | P a g e
CHAPTER 4
RESULTS AND DISCUSSIONS
Introduction:
Recently composite plates and shells as a structural element are widely using in aerospace,
civil, mechanical and other engineering structures and playing a important role in such kind
of industries.
In this chapter, the result of vibration, buckling and dynamic stability of plate and shell
structures with and without delamination is presented by using above FEM formulation. The
instability regions are determined for composite plates and shell with and without
delamination and the results are presented for different delamination lengths and percentage.
The study is aimed upon the following studies.
 Comparison with previous studies
 Numerical Results
Boundary conditions:
Numerical results are presented for delaminated composite plates/shells with different
combination of boundary conditions. Shells of various geometry such as spherical (R y/Rx=1)
and cylindrical (Ry/Rx=0) are studied.
Further descriptions of boundary conditions are as follows:
I.
Simply supported boundary
v=w=
38 | P a g e
=0 at x=0, a and u=w=
=0 at y=0, b
II.
Clamped boundary
u=v=w=
III.
=
= 0 at x=0, a and y=0, b
Free edges
No restraint
Non-dimensionalisation of Parameters:
Following are the non-dimensional parameter using for vibration, buckling and excitation
frequency of dynamic stability analysis with the reference Bert, C.W. and Birman, V. (1988).
Table 1.1 Non-dimensional parameters of composite plates/shells
No
parameter
1
Frequency of vibration ( )
2
Buckling load ( )
3
Frequency of excitation ( )
Where
and
Composite plates/ shells
a2
a2
are in radian.
Comparison with previous studies:
The vibration, buckling of plates/shells and dynamic stability results of plates based on
present formulation are compared with that of existing literature.
Vibration of composite plates and shells
The result on free vibration of cross-ply delaminated plates and shell are compared with
result by parhi et al (2002) using first order shear deformation theory. The program of the
39 | P a g e
finite element formulation developed in MATLAB 7.8.0 and validated by comparing the
author‟s results with those available in the existing literature. For this free vibration analysis,
the boundary conditions used in present study is simply supported and composite (0/90/0/90)
spherical and cylindrical shell and plates with delamination is carried out with the following
geometric and material properties:
Present result of natural frequencies (Hz) for mid-plane delaminated plates, cylindrical and
spherical shells are shown in table 1 along with the result of Parhi (2002) et al.
.Geometry and material properties: a=b=0.5m,
a/h=100, R/a=5 and R/a=10,
=0.25,
Table1.2. Natural frequencies (Hz) for mid-plane delaminated simply supported composite
spherical, cylindrical shells and plates with different delamination.
% delamination
Stacking
Spherical shell
Cylindrical shell
sequence
(R/a=
(R/a=
)
present
)
Plate
(R/a= )
Parhi et present
Parhi et present
Parhi et
al(2002)
al(2002)
al(2002)
0
(0/90)2
129.1353
129.20
103.0197
103.03
92.7162
92.72
25
(0/90)2
104.5625
104.59
69.5945
69.60
52.9273
52.93
56.25
(0/90)2
98.3438
98.36
59.9258
59.88
39.5013
39.50
40 | P a g e
Table1.3. Natural frequencies (Hz) for mid-plane delaminated simply supported composite
spherical and cylindrical shells with different delamination.
% delamination
Stacking
Spherical shell
Cylindrical shell
sequence
(R/a=5)
(R/a=5)
present
Parhi et al present
Parhi et al
(2002)
(2002)
0
(0/90)2
201.8568
202.02
128.9892
129.04
25
(0/90)2
187.4469
187.51
104.5625
104.56
56.25
(0/90)2
183.9246
183.96
98.2757
98.24
Buckling of composite plates:
The present formulation is validated for buckling of cross-ply composite plates with and
without delamination with result by Librescu et al (1989), Sciuva and Carrera (1990), Sahu
(2001) and Radu and chattopadhyay (2002).
Table 2.1: Comparison of non-dimensional buckling loads of a square simply supported
doubly curved panel with (0/90) lamination.
a/b=1, a/h=10, E11=40E22, G12=G13=0.6E22, G23=0.5E22,
=
=0.25
Taken value, a=0.5, b=0.5, t=0.05, E11=160GPa, E22=4GPa, G12=G13=2.4GPa, G23=2GPa.
Spherical shell (0/90)
41 | P a g e
Curvature
Present Work
Sahu and Dutta(2001)
Librescu et al (1989)
Rx/a=5, Ry/a=5
12.018
11.920
12.214
Rx/a=10, Ry/a=5
11.649
11.515
11.822
Rx/a=10, Ry/a=20
11.245
11.250
11.479
Rx/a=20, Ry/a=20
11.187
11.164
11.406
11.114
11.115
11.353
Plate
Numerical computation is carried out to determine the capability of the present delaminated
doubly curved element to predict the buckling of laminated composite plates. The results
obtained by this present formulation are compared with the analytical and finite element
displacement approach results of Sciuva and Carrera (1990) and FEM result by Sahu and
Dutta for 0% of delamination.
Table 2.2: Comparison of non-dimensional buckling loads of a square simply supported
symmetric cross-ply cylindrical shell panels with [0/90/0/90/0] lamination for different
length-to-thickness ratio (a/h). a/b=1, R/a=20, E11=40E22, G12=G13=0.6E22, G23=0.5E22,
=
=0.25,
Taken value, a=0.5, b=0.5, t=0.05, E11=160GPa, E22=4GPa, G12=G13=2.4GPa, G23=2GPa.
Cylindrical Shell (0/90/0/90/0), Ry=10.
42 | P a g e
a/h
Present Work
Sahu
and
(2001)
Dutta Sciuva
and
Carrera(1990)
10
24.331
23.962
24.19
20
32.169
31.790
31.91
30
34.353
33.980
34.04
50
35.786
35.393
35.42
100
37.399
36.843
36.843
The results on buckling with different delamination length of cross-ply composite plates due
to dynamic load is compared with results by Radu and Chattopadhyay (2002) using higher
order shear deformation theory.
Table 2.3 Comparison of critical buckling load for different mid plane delamination length of
the rectangular plates using cantilever boundary condition.
a=
127mm, b=12.7mm, h=1.016mm, stacking sequence= (0/90/0/90/90/0/90/0).
43 | P a g e
Delamination length (mm)
Critical buckling load (N)
Present work
Radu
and
Chattopadhyay
(2002)
0
16.3296
16.336
25.4
15.8292
16.068
50.8
14.9085
15.054
Numerical Results:
Numerical validation of the governing equation of the composite plate and shell for vibration
and buckling is performed by solving the corresponding free vibration equation and buckling
equation eigenvalue problems. Finally, the dynamic stability phenomenon is investigated for
effect of layers of ply, different degree of orthotropic, different static load factor, different
length-thickness ratio and aspect ratio.
Numerical results of vibration are presented for simply supported square plate and shell
having following Geometry and material properties: a=b=0.5m, a/h=100, R/a=5 and R/a=10,
=0.25,
Numerical results of natural frequency of composite spherical and cylindrical shell having
different curvatures with 25% delamination for different no. Of layers with four sides simply
supported boundary condition are presented in Table 3.1. Variation of natural frequency with
no of layers of composite shell with 25% delamination is graphically presented in figure 4. It
observed that, as number of layers increase for 2 to 16 the natural frequency also increases.
44 | P a g e
Table3.1. Natural frequencies (Hz) for 25% delaminated cross ply-(0/90)n simply supported
composite spherical and cylindrical shells with different no. of layers.
No. of layers
Spherical shell
Cylindrical shell
(R/a=5)
(R/a=10)
(R/a=5)
(R/a=10)
2
188.8401
107.0587
107.0618
73.3193
4
187.4469
104.5625
104.5625
69.5945
8
190.7261
110.4770
110.4207
78.2414
16
191.5230
111.8759
111.8008
80.2093
Figure 4: First natural frequency vs. no. of layer for simply supported composite shell with a
single mid-plane delamination.
Natural frequency of composite spherical and cylindrical shell for different aspect ratio at
25% delamination is investigated in Table 3.2. The variation of natural frequency with
different aspect ratio is graphically presented in fig 5. It observed that as aspect ratio
increases the natural frequency decreases.
45 | P a g e
Table3.2. Natural frequencies (Hz) for 25% delaminated cross ply-(0/90)2 simply supported
composite spherical and cylindrical shells with different aspect ratio.
a/b
Spherical shell
Cylindrical shell
Rx/a=5, Ry/b=5
Rx/a= , Ry/b=5
0.5
293.8187
220.6784
1
187.4469
104.5625
1.5
153.3563
71.5290
2
134.7433
56.4072
Figure5. First natural frequency vs. aspect ratio for simply supported composite shell with a
single mid-plane delamination.
Natural frequency of delaminated composite cross-ply (0/90)2 spherical and cylindrical shell
for different side to thickness ratio is presented in table3.3. Variations of natural frequency
with different % of delamination with different b/h ratio are presented in fig 6. It represent
that as b/h ratio is increases, the natural frequency is also increases at 25 % of delaminated
case.
46 | P a g e
Table3.3. Natural frequencies (Hz) for delaminated cross ply-(0/90)2 simply supported
composite spherical and cylindrical shells with different b/h ratio for R/a=5.
% delamination
Spherical shell
b/h=100
b/h=50
Cylindrical shell
b/h=25
b/h=100
b/h=50
b/h=25
0
201.8568 256.4682 400.9718 128.9892 204.7105 371.4012
25
187.4469 208.1226 273.6824 104.5625 138.5553 226.2074
56.25
183.9246 195.9246 237.0790 98.2757
Figure6.
119.4906 179.8898
First natural frequency vs. delamination % at different b/h ratio for simply
supported composite shell with a single mid-plane delamination.
Table3.4 and fig 7 shows that the numerical result of natural frequency of spherical and
cylindrical shell with different degree of orthotropic with different % of delamination. Fig 7
represent the natural frequency decreases with % delamination increases at each different
degree of orthotropy.
47 | P a g e
Table3.4. Natural frequencies (Hz) for delaminated cross ply-(0/90)2 simply supported
composite spherical and cylindrical shells with different orthotropic ratio for R/a=5.
E1/E2
0
Spherical shell
Cylindrical shell
% delamination
% delamination
25
56.25
0
25
56.25
10
308.9156 238.2747 216.4470 271.7676 186.6011 157.7208
25
201.8568 187.4469 183.9246 128.9892 104.5625 98.2757
40
470.1998 300.8696 252.0983 445.2868 257.5297 197.9235
Figure7. First natural frequency vs. delamination % at different E1/E2 ratio for simply
supported composite shell with a single mid-plane delamination.
Numerical results of buckling are presented for simply supported square plate and shell.
The program of the finite element formulation developed in MATLAB 7.8.0 and validated by
48 | P a g e
comparing the author‟s results with those available in the existing literature. For this stability
analysis, the boundary conditions used in present study is simply supported. And composite
(0/90)n spherical and cylindrical shell and plates with and without delamination is carried out
with the following geometric and material properties: a/b=1, a/h=10, E11=40E22,
G12=G13=0.6E22, G23=0.5E22,
=
=0.25.
Numerical results of non-dimensional buckling load with different no of layers for 0%
delaminated composite plate and shell with different curvature are presented in table 4.1
,Table 4.2 , and Table 4.3 . The variation of non-dimensional buckling load with different no
of layers is presented in fig 8, fig 9, and fig 10. It observed that as no of layers increases the
non-dimensional buckling load increases.
Table 4.1: Variation of non-dimensional buckling load with different no. of layers for 0%
delaminated composite shell.
Rx/a=5, Ry/a=5
cross-ply-(0/90)n
No. of layers
Spherical shell
Cylindrical shell
Plate
3
22.0426
21.3819
21.2575
5
24.6990
23.9196
23.8427
7
25.7988
24.8905
24.8387
9
26.5486
25.4525
25.4205
49 | P a g e
Figure 8: Variation of non-dimensional buckling load vs. no. of layers for simply supported
composite shell. Rx/a=5, Ry/a=5,
cross-ply-(0/90)n
Table 4.2: Variation of non-dimensional buckling load with different no. of layers for 0%
delaminated composite shell. Rx/a=10, Ry/a=10
cross-ply-(0/90)n
No. of layers
Spherical shell
Cylindrical shell
Plate
3
21.4528
21.2887
21.2575
5
24.0540
23.8620
23.8427
7
25.0715
24.8516
24.8387
9
25.6833
25.4284
25.4205
50 | P a g e
Figure 9: Variation of non-dimensional buckling load vs. no. of layers for simply supported
composite shell. Rx/a=10 Ry/a=10, cross-ply-(0/90)n
Table 4.3: Variation of non-dimensional buckling load with different no. of layers for 0%
delaminated composite shell.
Rx/a=20, Ry/a=20
cross-ply-(0/90)n
No. of layers
Spherical shell
Cylindrical shell
Plate
3
21.3063
21.2633
21.2575
5
23.8954
23.8475
23.8427
7
24.8965
24.8419
24.8387
9
25.4854
25.4225
25.4205
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Figure 10: Variation of non-dimensional buckling load vs. no. of layers for simply supported
composite shell. Rx/a=20, Ry/a=20,
cross-ply-(0/90)n.
Table 4.4 and Table 4.5 represent the numerical result of non-dimensional buckling load with
different b/h ratio for 0% delamination of composite shell. The variation of non-dimensional
buckling load with different b/h ratio is presented in fig 11 and fig 12. It investigated that as
b/h ratio increases the non-dimensional buckling load increases.
Table 4.4: Variation of non-dimensional buckling load with different b/h ratio for 0%
delaminated composite shell.
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Rx/a=5, Ry/a=5, E1=10E2,
cross-ply-(0/90/0/90/0)
b/h
Spherical shell
Cylindrical shell
10
6.8671
6.6333
25
10.0591
9.0030
50
14.8080
10.5171
Figure 11: Variation of non-dimensional buckling load vs. b/h ratio for simply supported
composite shell. Rx/a=5, Ry/a=5, cross-ply-(0/90)n
.
Table 4.5: Variation of non-dimensional buckling load with different b/h ratio for 0%
delaminated composite shell. Rx/a=10, Ry/a=10, E1=10E2,
53 | P a g e
cross-ply-(0/90/0/90/0)
b/h
Spherical shell
Cylindrical shell
10
6.6183
6.6249
25
9.0448
8.7832
50
10.5710
9.5276
Figure 12: Variation of non-dimensional buckling load vs. b/h ratio for simply supported
composite shell. Rx/a=5, Ry/a=5,
cross-ply-(0/90)n
.
Numerical result of non-dimensional buckling load with different degree of orthotropic for
0% delamination composite shell is presented in table 4.6 and table 4.7. Variations of nondimensional buckling load with different degree of orthotropy shown in fig13 and fig14. It
observed that as degree of orthotropic increases the non-dimensional buckling load increases.
Table 4.6: Variation of non-dimensional buckling load with different degree of orthotropic
for 0% delaminated composite shell.
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Rx/a=5, Ry/a=5, a/h=10,
cross-ply-(0/90/0/90/0)
E1/E2
Spherical shell
Cylindrical shell
10
4.5802
4.2659
25
8.2576
7.7389
40
11.9450
11.1458
Figure 13: Variation of non-dimensional buckling load vs. E1/E2 ratio for simply supported
composite shell. Rx/a=5, Ry/a=5,
Table 4.7: Variation of non-dimensional buckling load with different E1/E2 ratio for 0%
delaminated composite shell. Rx/a=10, Ry/a=10, a/h=10,
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cross-ply-(0/90/0/90/0).
E1/E2
Spherical shell
Cylindrical shell
10
4.2730
4.2149
25
7.3335
7.6197
40
11.1338
10.9569
Figure 14: Variation of non-dimensional buckling load vs. E1/E2 ratio for simply supported
composite shell. Rx/a=5, Ry/a=5,
Numerical results of non-dimensional buckling load with different no of layers for 6.25%,
25% and 56.25% delaminated composite plate and shell with different curvature are
presented in Table 4.8, Table 4.9, and Table 5.1. The variation of non-dimensional buckling
load with different no of layers with different % of delamination is presented in fig 15, fig 16,
and fig 17. It observed that as no of layers increases the non-dimensional buckling load
increases for each delamination case and it also investigated that as % of delamination
increases the non-dimensional buckling load decreases.
Table4.8: Variation of non-dimensional critical buckling load with different no. of layers for
different percentage of delaminated composite shell. Ry/a=10. Cross-ply-(0/90)n
No of layers
Delamination %
6.25%
25%
56.25%
2
11.0223
10.1426
9.3741
4
14.7349
7.8369
4.7022
8
18.5797
11.7883
8.5228
56 | P a g e
Figure 15: Variation of non-dimensional buckling load vs. no. of layers for simply supported
composite shell with different % of delamination. Ry =2, cross-ply-(0/90)n
Table4.9: Variation of non-dimensional critical buckling load with different no. of layers for
different percentage of delaminated composite shell.
No of layers
Rx/a=5, Ry/a=5 cross-ply-(0/90)n
Delamination %
6.25%
25%
56.25%
2
11.5480
10.6895
9.3741
4
15.2722
8.3798
4.7022
8
9.9338
5.2532
9.0993
57 | P a g e
Figure 16: Variation of non-dimensional buckling load vs. no. of layers for simply supported
composite shell with different % of delamination. Rx/a=5, Ry/a=5,
cross-ply-(0/90)n
Table5.1: Variation of non-dimensional critical buckling load with different no. of layers for
different percentage of delaminated composite plate. Cross-ply (0/90)n
No of layers
Delamination %
6.25%
25%
56.25%
2
12.6494
9.8011
9.0518
4
14.9647
7.5238
8.6562
8
18.3863
11.5318
8.2304
58 | P a g e
Figure17: Variation of non-dimensional buckling load vs. no. of layers for simply supported
composite plate with different % of delamination.
Cross-ply-(0/90)n
Numerical results of non-dimensional buckling load with different side to thickness ratio for
6.25%, 25% and 56.25% delaminated composite plate and shell with different curvature are
presented in Table5.2. The variation of non-dimensional buckling load with different b/h
ratios with different % of delamination is presented in fig 18. It observed that as b/h ratio
increases the non-dimensional buckling load increases for each delamination case and it also
investigated that as % of delamination increases the non-dimensional buckling load
decreases.
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Table5.2: Variation of non-dimensional critical buckling load with different b/h ratio for
different percentage of simply supported delaminated composite plate. Cross-ply-(0/90)
b/h
Delamination %
6.25%
25%
56.25%
10
10.6494
9.8011
9.0518
20
11.8513
10.7830
9.8596
50
12.3561
11.1885
10.2022
Figure 18
Effect of % of delamination on non-dimensional critical buckling with varying
b/h ratio.
Numerical result for dynamic stability
Dynamic instability region are plotted for rectangular plates of stacking sequence [(0/90)2]s
for different delamination length. The laminates are made out of eight identical plied with
material properties:
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a= 127mm, b=12.7mm, t=1.016mm, stacking sequence= (0/90/0/90/90/0/90/0).
In this study the boundary conditions is one of the short edges fixed and opposite edge loaded
with dynamic buckling force. The non-dimensional excitation frequency
used throughout the dynamic instability studies. Where
= a2
is the excitation frequency in
radian /second.
The effect of delamination on instability regions for length to thickness ratio (L/t =125 and
25) is shown in fig19 and fig 20. The onset of dynamic instability regions occurs later for 0%
delamination.
Fig19. Effect of delamination on instability region of [(0/90)2]s cross- ply plate for L/t =125
61 | P a g e
Fig20.
Effect of delamination on instability region of [(0/90)2]s cross- ply plate for L/t =25
The effect of delamination for different no of layers is shown in fig 21 and fig 22. Dynamic
instability regions are plotted for 2 and 4 layer cross-ply rectangular plate. For both the cases
instability regions occurs later for 0% delamination.
Fig21. Effects of % of delamination on instability region of 2-layer cross-ply delaminated
plates.
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Fig22. Effect of % of delamination on instability region for 4-layer cross-ply delaminated
plate.
The effect of delamination is studied for degree the orthotropic E11/E22 =40 and 20, and keep
the other material parameters constants is shown in fig 23 and fig 24. Both the cases
instability regions occur later at 0% delamination.
Fig23. Effect of % of delamination on instability region for cross ply delaminated plate for
degree of orthotropy, E11/E22 =40.
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Fig 24: Effect of % of delamination on instability region for cross ply delaminated plate for
degree of orthotropy, E11/E22 =20.
Fig 25 and fig 26 Shows the effect of aspect ratio on instability region at 0% and 25%
delamination. It is observed that the onset of dynamic instability occurs much later with
increase of the aspect ratio and increasing width of instability regions.
Fig 25: Effect of aspect ratio on instability region for simply supported cross ply delaminated
plate. L/t=10, E1/E2 = 25, 𝞪=0.2.
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Fig 26: Effect of aspect ratio on instability region for simply supported cross ply delaminated
plate. L/t=10, E1/E2 = 25, 𝞪=0.2.
The effect of static component of load for α = 0.0, 0.2 and 0.4 on instability region is shown
in fig 27. A clamped-free-clamped-free boundary condition has taken into account in this
study with 6.25% delamination. Due to increase of static component the instability regions
tends to shift to lower frequencies and become wider.
Fig27. Effect of static load factor on instability region of rectangular plate
(127*12.7*1.016)mm.
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a= 127mm, b=12.7mm, t=1.016mm, stacking sequence= (0/90/0/90/90/0/90/0).
Fig 28 and fig 29 shows the dynamic instability regions for delaminated spherical shell at
different % of delamination. It is observed that the onset of dynamic instability occurs later
with decrease of delamination.
Fig28. Effect of % of delamination on the instability region of simply supported cross ply
(0/90) spherical shell: a/Rx = b/Ry =0 .25, 𝞪=0.2, a/b=1, a/h=10, E11=40E22, G12=G13=0.6E22,
G23=0.5E22,
=
=0.25.
Fig29. Effect of % of delamination on the instability region of simply supported cross ply
(0/90) cylindrical shell: b/Ry =0 .25, 𝞪=0.2, a/b=1, a/h=10, E11=40E22, G12=G13=0.6E22,
G23=0.5E22,
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=
=0.25.
The effect of curvature on instability region with 6.25% and 25% of delamination of simply
supported cross ply shown in fig 30 and fig 31. It shows in fig that the dynamic instability
occurs earlier in case of plate with increase of delamination.
Fig30 Effect of curvature on instability region with 6.25% of delamination of simply
supported
cross
ply
(0/90):
G12=G13=0.6E22, G23=0.5E22,
a/Rx=b/Ry=0.25,
=
𝞪=0.2,
a/b=1,
a/h=10,
E11=40E22,
=0.25.
Fig31 Effect of curvature on instability region with 25% of delamination of simply supported
cross ply (0/90): a/Rx = b/Ry =0 .25, 𝞪=0.2, a/b=1, a/h=10, E11=40E22, G12=G13=0.6E22,
G23=0.5E22,
67 | P a g e
=
=0.25
CHAPTER 5
CONCLUSION
A first order shear deformation theory based on finite element model has been developed for
studying the instability region of mid plane delaminated composite plate and shell.
The following observations are made from this study:Vibration study

The effects of dynamic behaviour on delaminated composite plates and shells under
free vibration conclude that for particular % of delamination, the natural frequencies
increase with increase of number of layers due to effect of bending-stretching
coupling.

With increase of aspect ratio, the natural frequency decreases.

With increase of % delamination, the natural frequency decreases and it also observed
the frequency of vibration increase with decrease of b/h ratio of cross ply panels with
delamination. This is due to reduction in stiffness caused by delamination.

With increase of % delamination the natural frequency decreases with different
degrees of orthotropic.

The frequency of vibration increases with increase of degree of orthotropy due to
increase of stiffness.
Buckling study

If no. of layers increases the non-dimensional buckling load increases and it also
investigated that as % of delamination increases the non-dimensional buckling load
decreases.
68 | P a g e

If b/h ratio increases the non-dimensional buckling load increases for each
delamination case and it also investigated that as % of delamination increases the nondimensional buckling load decreases.

The natural frequencies and the non-dimensional buckling load decrease with increase
in delamination length. This is due to the reduction in stiffness caused by
delamination.

With increase of degree of orthotropic the buckling load increases and it also
observed that buckling load increase with decrease of delamination.
Dynamic stability study

The onset of instability occurs earlier with increase in percentage of delamination.

It also observed that with increases of number of layers the excitation frequency
increases. This is due to increase of stiffness caused by bending-stretching coupling
with increase of layers.

The dynamic instability region occurs earlier with decrease of degree of orthotropy
and due to decrease of delamination the onset instability region shifted to lower
frequency to higher frequency and also the width of instability regions increased with
decrease of delamination.

The dynamic instability occurs much later with increase of aspect ratio and width of
instability region increase with increase of aspect ratio for delaminated cross ply
panel.

With increase of static load factor the instability region tends to shift to lower
frequencies and become wider showing destabilizing effect on the dynamic stability
behaviour of delaminated composite plate.
69 | P a g e

It observed that the onset of instability occurs at higher load frequencies with the
introduction of curvatures when compared with the flat member. This is probably due
to the influence of increased stiffness of shells. The onset of load frequency is heavily
dependent on the radius to length ratio.
Further scope of study:
In the Present study, natural frequency, buckling load and dynamic stability of
delaminated cross-ply composite plate and shell was determined numerically. The effect of
various parameters like percentage of delamination area, number of layers, aspect ratio,
degree of orthotropy and different side to thickness ratio was studied. The future scope of the
present study can be extending as follows:

Dynamic stability of multiple delaminated plate and shell can be studied.

The present study is based on linear range of analysis. It can also extend for nonlinear
analysis.

Dynamic stability of composite plates with circular, elliptical, triangular shaped
delamination can be studied.

The effect of damping on instability regions of delaminated composite plates and
shells can be studied.
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CHAPTER 6
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74 | P a g e
APPENDIX
Delamination Modelling:
For Single mid-plane delamination with different sizes like 0, 6.25%, 25%, 56.25% of the
total plate area is considered. The delamination sizes are assumed to increase from the centre
of the laminate and can be located anywhere along the thickness of the laminate. The
composite plates with different percentage of delamination are shown in figure 8.1 to 8.4.
Figure 32: 6.25% central delamination
Figure 33: 25% central delamination
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Figure 34: 56.25% central delamination
h
h/2
Figure 35: Eight layered laminate without delamination
76 | P a g e
h
h/2
Figure 36: Eight layered laminate with mid-plane delamination
h
h/2
Figure 37: Eight layered laminate with three delaminations
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