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Development of Curved Thermoplastic Composite Tubes Theory and Experiment
Prepared by:
Hamidreza Yazdani
Supervised by:
Dr. Suong. V. Hoa and Dr. Mehdi Hojjati
Outline
Problem statement
Literature review
Toroidal Elasticity Theory
Conclusions
Future works
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Problem Statement
• Aluminum landing gears have some technical issues• Fabrication is difficult and expensive • Failure due to the corrosion and Fatigue• The objective of this research project is to investigate
how a composite landing gear can be designed and manufactured using thermoplastic composite materials and automated fiber placement technology.
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Literature Review Finite element approach
(Redekop and Tan 1987) Analytical approach
– Shell bending theory (Boyle and Spence 1992)– Membrane theory (Bushnell 1981)– Toroidal Elasticity (TE)
Stress approach (Lang 1989)Displacement approach (Redekop 1991, 1993)
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Toroidal Elasticity for Orthotropic Materials
Displacement field for composite toroidal structures based on successive approximation method
The governing equations in three toroidal coordinate system
Method of successive approximation The governing equations in different orders Different order displacement components
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Governing Equations R is bend radius. Angular coordinates are θ and φ, and radial coordinate is
r.
1 1 1cos sin 0
1 2 1cos sin 0
1 1 12 cos 2 sin 0
r rrr r r
rr r
rr r
r r r
r r r
r r r
cosR r
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Governing Equations Kinematics relations Constitutive equations Governing Navier equations in toroidal coordinate
0 1 2
0 1 2
0 1 2
1 1 1 ˆ 0
1 1 1 ˆ 0
1 1 1 ˆ 0
U U U
V V V
W W W
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Method of Successive Approximation The solution is a series in terms of a small parameter
ε=1/R.
2 30 1 2 3
2 30 1 2 3
2 30 1 2 3
...
...
...
u u u u u
v v v v v
w w w w w
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Governing Equations for the Different Orders
The zeroth order:
The first order:
The second order:
The third order:
0
0
0
0
0
0
U
V
W
1 0
1 0
1 0
U U
V V
W W
2 1 0 0
2 1 0 0
2 1 0 0
ˆcos
ˆcos
ˆcos
U U U r U
V V V r V
W W W r W
2 23 2 1 0 1 0
2 23 2 1 0 1 0
2 23 2 1 0 1 0
ˆ ˆcos cos 2 cos
ˆ ˆcos cos 2 cos
ˆ ˆcos cos 2 cos
U U U r U r U U r
V V V r V r V V r
W W W r W r W W r
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General Solution Complementary solution
General form solution
0
0
0
sin
cos
sin
m
m
m
u ar n F
v br n F
w cr n F
1, 1, 1, 1,
2, 2, 2, 2,
1
266
55
44
1 1 1 1sin
1 1 1 1cos
sin 1
1
2
n n n n
n n n n
n n n n
n n n n
n n
m m m mn n m n m n m n m
m m m mn n m n m n m n m
m mn n n
n
u a B r b B r c B r d B r n Fn n n n
v a A r b A r c A r d A r n Fn n n n
w e r f r n F
Cm n
C
C C
4 2
4 2 2 2 2 211 11 22 11 44 44 22 12 44 12 22 44
1 1 2 10
2 2 2n n
n nm n C C C C C C n C C n C m C C
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Boundary Conditions The boundary conditions on the curved surfaces are
satisfied by each order
0rr r r
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The Zeroth Order Solution
1 1
2 222 22
11 11
0 0 0
1 10 0 0
0 0 0 0
10 0
0
0
0
0
Complementary solution
ln 0
Particular solution 0
0
0
0
C C
C Cu a r b r F
v c r d r F
w e f r F u
v d r F
w
u
v
w
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The First Order Solution Governing equations
Displacement components
1 1
1 1
1 1
1 1
1 1
20 23 131 1 1,0 1 1, 1 1,
4422
20 231 1 2,0 1 2, 1 2,
4422
1 1 1
2sin
22
cos2
2
sin
m mm m
m mm m
m m
d C Cu a B b B r c B r r F
CC
d Cv a A b A r c A r r F
CC
w e r f r F
1 0 1 13 23 0
1 0 1 23 0
1 0 1
2 sin
cos
0
U U U C C d F
V V V C d F
W W W
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The Second Order Solution Governing equations
Displacement components
1
2 2
1 1
2 2 2 2
1
2 2
12
1 12
112 66 55 1
sin 2
cos 2
1sin
2
mu u
m mv v v v
mw w
U a r b r F
V a r c r F b r d r F
W e r C C a r f r F
2 2 2 2 1
22 22
2 2 2 2 1
22 22
1
22 22
2 2 1
22 22
132 22 22 22 22
132 22 22 22 22
13
13 552 2 2 1
66
sin 2
cos 2
1 sin
m m m m mu u
m m m m mv v
mv v
m m mw w
u a r b r c r d r a r b r F
v a r b r c r d r b r d r F
a r c r F
Cw e r f r e r f r a r
C
F
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The Third Order Solution Governing equations
Displacement components
23 1 2 3 4 5
23 6 7 8 9 10
3 20
sin 2 cos sin sin cos sin cos 2 cos 2 cos
cos cos 2 cos sin cos sin sin 2 sin
sin 2
U R R R R R F
V R R R R R F
W R F
3 3 3 3
3 3 3 3
3 3 3 3
3 3 3 3
3 3 1, 3 1, 3 1, 3 1,
211 12 13 14 15
3 3 2, 3 2, 3 2, 3 2,
1 1 1 1sin 3
3 3 3 3
sin 2 cos sin sin cos sin cos 2 cos 2 cos
1 1 1 1
3 3 3 3
m m m mm m m m
m m m mm m m m
u a B r b B r c B r d B r F
a a a a a F
v a A r b A r c A r d A r
3 3
216 17 18 19 110
3 3 3 20
cos 3
cos cos 2 cos sin cos sin sin 2 sin
sin 2m m
F
a a a a a F
w e r f r a F
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Conclusions Toroidal elasticity is a developing three-dimensional
theory which can be used for the elastostatic analysis of thick-walled curved tubes.
A displacement based toroidal elasticity has been used. The governing equations in the toroidal coordinates are
much more complicated than those used in Cartesian coordinates.
The successive approximation method has been employed to simplify the governing equations.
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Previous and Future works Literature review Finding the appropriate theory Obtain the displacement field for isotropic materials Obtain the displacement field for a layer of orthotropic
materials Achieve the displacement field for a laminate of
orthotropic materials Analysis of the curved tube dynamically Comparison of the theoretical result with the
experimental one
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