FEM Analysis of Stress Distribution in Force-Balanced Coils

H. Tsutsui, S. Kajita, Y. Ohata, S. Nomura,

S. Tsuji-Iio, R. Shimada

Research Laboratory for Nuclear Reactors, Tokyo Institute of Technology

18th International Conference on Magnet TechnologyOctober 20-24, 2003, Morioka, Japan

Introduction

• The virial theorem is the relation between the kinetic and the potential energies. The theorem, which is derived only form the equilibrium, shows that the tension is required to hold the magnetic energy.

• Using the virial theorem, we extended and generalized Force-Balanced Coil which is a helical type hybrid coil of the toroidal field (TF) coil and the solenoidal coil, and showed the condition to minimize the stress working in the coil (virial-limit condition).

• In this work, we constructed a small device to prove our concept, and obtained stress distribution experimentally. The results are compared with those of numerical calculations with a shell model.

Principle of Force-Balanced Coil

FcI

Fca R

R FhFh

Fc

I

Fc

aR

aR Fh

I

Fh

Hoop Force by Toroidal Current

Centering Force by Poloidal Current

Force-Balanced Coil

Centering force is much reduced,

butstress distribution

is not investigated.

Virial Theorem

tensorstress

field magnetic :

densitycurrent :

0

0

0

:S

S

B

j

BjB

Bj

=⋅∇=×∇

=⋅∇+×µ

( )

tensorstress Maxwell

2

02

0

:T

IT

ST

−≡

=+⋅∇

BBBµ１

( )

stress principal :

0d2

d

0d Tr

M0

23

1

i

ii UVBV

V

σµ

σ >≡=

=+

∫∫∑

∫

=

ST Ω

Ω

∫≡

≡

VV

UV

d

~M

σσ

σσ

1~3

1

=∑=i

iσ

• Positive stress (tension) is required to hold the field.• Uniform tension is favorable.• Theoretical limit is determined.

Equilibrium Eq.

31~~~

321 === σσσ

Application to Thin Toroidal Shell

φ θ Sumation

Stress

Integral

Energy

ρπµ θ

∆−

RIa

2

220

16 ρπµ θ

∆RIa

2

220

8

RIa

4

220 θµ

RIa

4

220 θµ

RIa

2

220 θµ

RIa

4

220 θµ−

TF Coil (Major radius: R, minor radius: a, thickness: ∆ρ)

• We consider the toroidal coil with so large aspect ratio that toroidal effect is negligible.

• The current distribution is adopted which makes toroidal surface correspond to both current and magnetic surfaces.

• When torus is axisymmetric, the direction of principal stresses are φand θ.

Application to Helical Coil

ioAspect Rat :

Coil of Pitch:

1~~

28log2

28log

~

28log2

~

222

222

222

22

AIIN

AAAN

NAAA

AAANAN

φ

θ

φθ

φ

θ

σσ

σ

σ

≡

=+

−+

−−=

−+

−=

N

<σ>

σφ

σθ A=2A=5

A=10

~

~

~

0 5 10 15 20-2

-1

0

1

2

energy.in optimal is 21~~ == φθ σσ

Virial-Limit Condition

Shape of Coils

<σφ>=0N=9FBC

<σθ>=0N=4

<σφ>= <σθ>N=6VLC

A

N

σθ=0

σφ=0 σθ=σφ

0 5 10 15

10

20

30

Aspect Ratio A =4

Relations of pitch number and aspect ratio of Virial-Limit Coil (VLC) etc.

Comparison of Maximum Averaged Stress

• Neglecting the distribution of stress by thetoroidal effect, the maximum stress is reduced to 25% compared with that of traditional TF coil.

A

<σ>

TFC

PFC

FBC

~max

VLC10 200

1

2

3

0.5

N=∞

N=0

Toroidal Effect

• Distribution of stress in the toroidal shell with circular cross section is derived analytically by use of magnetic pressure.

z

rR

aρ

θφ

o

0

22

2µθφ BB

p−

≡

∫≡r

Rrrpraru 'd)'(')(

2)(

)(

Rru

RrarpT

rRruT

−−

−=

−=

φ

θ

Equilibrium of Magnetic Pressure and Stress

Distribution of Stress(large aspect ratio)

• When A=100, distribution of stress is flat.• There is no advantage of helical winding.

A=100

0

22

M

T

2ˆ

µθφ BB

UVp

−≡

θ

pσ

^^

TFC

<σφ>=0

<σφ>=<σθ>σφ

σθ

p

0 1 2 3

0.8

0.9

1

-1

0

1

2

θ

pσ

^^

TFC<σφ>=0<σφ>=<σθ>

σφ

σθ

p

0 1 2 3

0.6

0.8

1

1.2

-1

0

1

2

A=10

Distribution of Stress(low aspect ratio)

• When A<10, distribution of stress is important.

• Assumption of large aspect ratio is not held.

• Optimal distribution is achieved to minimize the stressat θ=π.

A=2

θ

pσ

^^

TFC<σφ>=0

<σφ>=<σθ>

σφ

σθ

p

σφ(π)=σθ(π)

0 1 2 3-2

0

2

4

-2

0

2

4

N=3

Frame of Device to demonstrate VLC

• It is made of Al with 2mm-14mm thickness so that the strain is 10-5

-10-4 when the maximum field is 1T.

288 mm400 mm512 mm

for Hall sensors

φ12

A

A

BB

C

C

25 mmA-A

B-B

C-C

84 mm100 mm112 mm

14 mm

Frame of aluminum alloyInner Coil

Outer Coil

Device to demonstrate Virial-Limit Condition

+I

-I

Frame

Outer CoilInner Coil

IOUTIIN

Positions of Strain Gauges

• In this work, data of triaxial strain gauges on the surface are used.

• surf3abc-0-1• surf3abc-45-1• surf3abc-90-1• surf3abc-135-1• surf3abc-180-1

7

10

13

O

a

b

8

9

11

12

bottom3a,b,c-0

bottom3a,b,c-45

bottom3a,b,c-90

bottom3a,b,c-135bottom3a,b,c-180

surf3a,b,c-0-1

surf3a,b,c-45-1

surf3a,b,c-90-1 surf3a,b,c-135-1*surf3a,b,c-180-1

surf3a,b,c-0-2

surf3a,b,c-45-2

surf3a,b,c-90-2

surf3a,b,c-135-2*surf3a,b,c-180-2

top1-0

top1-45top1-90

top1-135top1-180

Sensors of Toroidal Field

Sensors of Poloidal Field

1

2

3Current Lead

Shell model with periodic symmetry

Equilibirum Equation

α: amplitude of modem,n: poloidal /todoidal mode numbers

Expanding by einφ

Distribution of Stress

• Comparing the results of the experiments and the numerical calculations, a qualitative agreement of stress distribution between the calculation and the experiment is obtained in the toroidal direction, while discrepancies of stress in thepoloidal direction are not negligible.

Lines: shell model with A=4, α=0.1, m=6, n=18

0 ππ/2θ [rad]

σφφ~

PFCVLCTFC

PFC

VLC

TFC

-5

0

5

0 ππ/2

θ [rad]

σθθ~

PFCVLCTFC

PFC

TFC

VLC

-2

0

2

4

experiments experiments

+I

-I

Frame

Outer CoilInner Coil

IOUTIIN

Model in FEM Analysis

Elements in ConductorConductor Region

XY

Z

5.01+01

4.69+01

4.36+01

4.04+01

3.71+01

3.39+01

3.07+01

2.74+01

2.42+01

2.09+01

1.77+01

1.45+01

1.12+01

7.97+00

4.73+00

1.49+00 default_Fringe :Max 5.01+01 @Nd 317Min 1.49+00 @Nd 1058

XY

Z

• Coils and their winding form are replaced with a monolithic shell.

Only the outercoil is excited.

Electromagnetic force is applied to conductor region.

Stress Distribution

0 45 90 135 180-2

0

2

4

Poloidal angle (deg.)

Stre

ss (

MPa

)

Coil Coil

0 45 90 135 180-2

0

2

4

Poloidal angle (deg.)

Stre

ss (

MPa

)

Coil Coil

Stress distribution on the coil-shell system

（φ=0°） Shearing Stress

Normal stresses ( σθθ, σθφ , σφφ ) are dominant compared with shearing stresses.

Normal Stress

σｒｒ

σθθ

σφφ

σｒθ

σθφ

σφｒ

Comparison between FEM and Experiment

Toroidal direction Poloidal direction

FEM analysis can reconstruct experimental results.

0 45 90 135 180-1

0

1

2 Experiment FEM Shell Model

Poloidal angle (deg.)

σφ~

0 45 90 135 180-1

0

1

2 Experiment FEM Shell Model

Poloidal angle (deg.)

σθ~

Principal stress on upper surface

Tensile stress is large around the conductor, while compressive stress is large on the shell.

• Compressive stress is large on the shell where strain is measured.• Stress distribution strongly depends on the location of conductors.

Minimum principal stress

Compressive stress is large.

[MPa]

-2.8

-2.4

-2.0

-1.6

-1.2

-0.8

-0.4

0

compressive

tensiletensile

Tensile stress is large

[MPa]

0

0.5

1

1.5

2

2.5

3

3.5

Maximum principal stress

shell

shell

conductor

Coil

Coil

Coil

θ

Principal stress on poloidal cross section

Maximum principal stress Minimum principal stress

Coil

Coil

Coil

θ

（ φ＝ 0°）

Tensile stress is large in the conductor, while compressive stress is large in the form. Both stresses are concentrated around φ=π radian where magnetic field takes maximum.

[MPa]

-2.8

-2.4

-2.0

-1.6

-1.2

-0.8

-0.4

0

compres

tensileφ

z[MPa]

0

0.5

1

1.5

2

2.5

3

3.5

tensileφz

Summary I

• In order to verify the concept of the optimal coil to store magnetic energy based on the virial theorem, the device composed of a toroidal winding-form and two sets of helical coils wound on two layers was constructed.

• The experiments with the device show that our optimal coil (VLC) achieves the minimum and flat stress distribution, which is also obtained by the numerical calculations with the shell model.

• In order to complete the VLC concept, stress distribution is evaluated by FEM with a monolithic model, and good agreement with the experiment is obtained.

Summary II

• Compressive stress is produced mainly on the surface of the winding form where strains are measured, while tensile stress is produced mainly on the surface of the coil conductors.

• Stress distributions are quite different on inner and outer surfaces. Compressive stress is mainly produced in the shell region, and large compressive stress is produced around inner region where magnetic field takes the maximum value.

• A dense winding of coil conductors is required to reduce the compressive stress which is unfavorable for SMES.

FEM analysis shows that;