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 1 cos sin 0

1 2 1 cos sin 0

1 1 1 2 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

000

UVW

1 0

1 0

1 0

U U

V V

W W

2 1 0 0

2 1 0 0

2 1 0 0

ˆcosˆcos

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 cosˆ 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,

12

66

55

44

1 1 1 1 sin

1 1 1 1 cos

sin 1

12

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 1 02 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 12 222 22

11 110 0 0

1 10 0 0

0 0 0 0

10 0

0

0

0

0

Complementary solution

ln 0

Particular solution 0000

C CC Cu a r b r F

v c r d r F

w e f r F u

v d r F

wuvw

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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

2sin

m mm m

m mm m

m m

d C Cu a B b B r c B r r FCC

d Cv a A b A r c A r r FCC

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

1 sin2

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 rC

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 1 sin 33 3 3 3

sin 2 cos sin sin cos sin cos 2 cos 2 cos

1 1 1 13 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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