Finite element modelling for composite shell.pptx

1. Introduction
2. Literature Review
3. Finite Element Formulation
4. Experimental Procedure
5. Results and Discussion
6. Conclusion
7. Scope of Future Research
-: CONTENTS :-
6.11.2019
Contents

₪ Laminated Composites Plate and Shell Structures
₪ Used material: Laminated glass-FRP
₪ Motivation of the present study
₪ Major contributions of this thesis-
► Experimental investigation
► Theoretical investigation
► Parametric study
Chapter I
-: INTRODUCTION :-
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min min
1
max ,
10
h h
R L
 

 
 
1
2
L
R

1
2
L
R

min min
1 1
max ,
20 10
h h
R L
 
 
 
 
Thin shell (Tornabene and Fantuzzi, 2017):
 Moderately thick shell (Tornabene and Fantuzzi, 2017):
 Thick shell condition:
 Shallow shell (Raamachandran, 2013):
 Deep shell:
min min
1
max ,
20
h h
R L
 

 
 
Chapter I
(Contd.)
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Chapter I
(Contd.)
 Application of FRP laminates:
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Review of literature is presented in the following sub heads:
 Free Vibration Analysis of composite plates and shells with and
without cut-outs
 Free vibration analysis of skew composite plates and shells
 Free vibration analysis of skew composite plates and shells with
cut-outs
-: LITERATURE REVIEW :-
Chapter II
6.11.2019

 Free vibration analysis of composite laminates with and without cut-outs:
Reference Type of Laminate Approach Target Response
Olson and
Lindberg
[1971]
Shallow isotropic
shells
Shallow shell theory of
Novozhilo and
Experimental Investigations
Natural frequencies and mode shapes
Jenq et al.
[1993]
GRP cross-ply plates
Shear deformation theory
and Experimental
Investigations
Effects of the size of cut-outs, the
location of defectson the vibration
frequencies
Singh and Kumar
[1996]
Doubly curved
laminated shallow
shells
First order composite
shell theory
Free vibrational characteristics of shells
on quadrangular planforms
Qatu
[1999]
Laminated composite
deep/thick shells
First order shear
deformation theory
Accurate stress resultants
Ganeson and
Kadoli
[2004]
Isotropic
hemispherical shells
First order shear
deformation theory
Thermo-elastic buckling analysis and
free vibration analysis of shells with
cut-out at apex due to uniform
temperature rise
Shi et al.
[2004]
Arbitrarily laminated
plate
Galerkin method Free vibration analysis
Hota and Padhi
[2007]
Plates with cut-outs
First order shear
deformation
Free vibration of plates with arbitrary
shapes of cut-outs
Ulz and
Semercigil [2008]
Plates
ANSYS and Experimental
Investigations
Use of cut-out as a
dynamic vibration absorber
Asadi et al.
[2012]
Thick deep laminated
cylindrical shells
3D and various shear
deformation theories
Static and vibration analysis
Thakur and Ray
[2015]
Deep doubly curved
laminated shell
HSDT
Free vibration analysis
Chapter II
(Contd.)
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Critical Remarks:
Chapter II
(Contd.)
Free vibration analysis of composite plates and shells with and
without cut-outs:
 A good number of numerical investigations on dynamic analysis of
laminated plate and shells have been done using various kind of
approach and formulation to get more accurate result
 Very few studies on laminated composite plates or shells with
multiple cut-outs have been undertaken by the researchers
 A very less amount of experimental work has been conducted on
shells
6.11.2019

Reference Type of Laminate Approach Target Response
Kandasamy
and Singh
[2006]
Isotropic skew open
circular cylindrical
shells
Modified version of the
Rayleigh-Ritz method
Free vibration
analysis
Liew et al.
[2003]
Symmetrically
laminated plates
Moving least squares
differential quadrature
method based on FSDT
Free vibration
analysis
Vimal et al.
[2014]
Moderately thick
functionally graded
skew plates
FSDT
Free vibration
analysis
Park et al.
[2008]
Skew sandwich plates
with laminated
composite faces
HSDT
Dynamic
response
Kumar et al.
[2013]
Laminated composite
skew hypar shells
HSDT
Free vibration
analysis
Dey et al.
[2016]
Composite plates HSDT
Free vibration
analysis
 Free vibration analysis of skew composite plates and shells:
Chapter II
(Contd.)
6.11.2019

 Numerical analysis of laminated skew laminates is limited
 Cylindrical shell was considered for the numerical analysis of
laminated skew shells by majority of the researchers
 Experimental studies on skew composite plates and shells are quite a
few
Chapter II
(Contd.) Critical Remarks:
Free vibration analysis of skew composite plates and shells:
6.11.2019

Reference Type of Laminate Approach Target response
Lee
[2010]
Plate HSDT
Dynamic stability
analysis
Murthy et al.
[2013]
Skew plate with
circular cut-out
ANSYS
Free vibration
analysis
Chapter II
(Contd.)  Free vibration analysis of skew composite plates and shells
with cut-out:
6.11.2019

 A very few research works on the skew laminates with cut-out are
reported so far
 The lack of experimental as well as numerical study on laminated
skew plates with cut-out is also prominent
 Experimental dynamic analysis on laminated skew shell with cut-out
is not reported so far
Chapter II
(Contd.)
Critical Remarks:
Free vibration analysis of skew composite plates and shells
with cut-out:
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 The increasing demand of practical applications necessitates the detailed
analysis of laminated structures to achieve enhanced strength and
improved performance of laminated structures
 To figure out the influence of the physical features like shape, size,
position and number of cut-outs present in regular or skew plates and
shells
 Experimental investigation especially on skew shells with cut-out has
remained unexplored
 Fill the remarkable gap between requirement and very few existing
response experimental output data
 Carried out experiments and a numerical analysis with a simplified but
efficient finite element based model to obtain a realistic dynamic
characteristics of regular and skew laminated structures
Chapter II
(Contd.) Research Goal:
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Assumptions:
 Middle surface of the shell is considered as reference plane
 Effect of shear deformation is incorporated following the
Mindlin’s hypothesis
 The shallow shell theory is used in the present formulation
Chapter III
-: FINITE ELEMENT FORMULATION :-
1 2 3
4
5
6
7
8 9
Fig. 3.1: Plan view of nine
nodded element
9 9
1 1
r r r r
r r
x N x and y N y
 
 
 
where,
Xr , Yr = co-ordinates of the r-th nodal point
Nr = Lagrangian interpolation function
[1]
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Fig. 3.2: Deformation of shell panel in xz - plane
Chapter III
(Contd.) Effect of shear deformation:
x
x
y
y
w
x
w
y





 

 
  
 

   

   


 
 
øx & øy = Average shear rotation
over the entire plate or
shell thickness
θx & θy = Total rotations in bending
[2]
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9 9 9
1 1 1
9 9
1 1
, , ,
r r r r r r
r r r
x r r y r y
r r
u N u v N v w N w
N and N
   
  
 
  
 
  
 
[3]
[4]
[5]
Chapter III
(Contd.) The displacement fields at a point within the element:
The stress resultant-strain relationship:
{F} = [D] {ε}
where,
{F}T = [ Nx N y Nxy Mx My Mxy Qx Qy]
6.11.2019

X
Y
Z Nx
Ny
Nxy
Nyx
Fig. 3.3: In plane membrane normal stresses (Nx, Ny) and
In plane shear stress (Nxy, Nyx)
Chapter III
(Contd.)
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Fig. 3.4: Different forces acting on a rectangular plate during bending
Chapter III
(Contd.)
6.11.2019

 
























































y
x
y
x
y
x
x
y
w
x
w
x
y
y
x
x
v
y
u
R
w
y
v
R
w
x
u
y







/
/
/
/
/
/
/
/
/
/
/
/
and
and,
[6]
[7]
11 12 16 11 12 16
21 22 26 21 22 26
61 62 66 61 62 66
11 12 16 11 12 16
21 22 26 21 22 26
61 62 66 61 62 66
55 54
45 44
0 0
0 0
0 0
0 0
0 0
0 0
0 0 0 0 0 0
0 0 0 0 0 0
A A A B B B
A A A B B B
A A A B B B
B B B D D D
D
B B B D D D
B B B D D D
A A
A A
 
 
 
 
 
 
   

 
   
 
 
 
 
 
 
Chapter III
(Contd.)
6.11.2019

where,
Middle
Surface
and,
Chapter III
(Contd.)
=
=
=
=
Extensional stiffness:
Coupling stiffness:
Bending stiffness:
Shear stiffness:
6.11.2019

or, in short
 
























































































 

yr
xr
r
r
r
r
r
r
r
r
r
r
r
r
r
r
Y
r
x
r
w
v
u
N
y
N
N
x
N
x
N
y
N
y
N
x
N
x
N
y
N
R
y
N
R
x
N



9
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
0
     



9
1
r
e
r
r
B 

    

 B

[9]
[10]
[8]
Chapter III
(Contd.)
6.11.2019

Mass matrix:
      
1 1
1 1
T
e
K B D B J d d
 
 
 
  
The elemental stiffness matrix:
[12]
[11]
MLWORI (mass lumping without rotary inertia)-
(i=1,2,3,6,7,8,11,12,13,16,17,18,21,22,23,
26,27,28,31,32,33,36,37,38,41,42,43)
=
MLWRI (mass lumping with rotary inertia)-
(i=1,2,3,6,7,8,11,12,13,16,17,18,21,
22,23,26,27,28,31,32,33,36,37,38,41,42,43)
=
(i=4,9,14,19,24,29,34,39,44)
=
( i=5,10,15,20,25,30,35,40,45)
=
Chapter III
(Contd.)
[M] = ρh ∫∫ [N]T [N] dx dy
[13]
[14]
[15]
[16]
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The stiffness matrix and mass matrix having an order of 45×45 are
evaluated for all the elements and they are assembled together to
form the overall stiffness matrix [K0] and mass matrix [M0]. Once
[K0] and [M0] are obtained. The equation of motion may be
expressed as-
[K0]{δ} = 2[M0] {δ}
After incorporating the boundary conditions in the above equation it
has been solved to get frequency  for first four modes.
[17]
Chapter III
(Contd.)
6.11.2019

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