Approximate Calculation of U-shaped Bellows

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TSINGHUA SCIENCE AND TECHNOLOGY

ISSN 1007-0214 21/22 pp3 05~3 09

Volume 1, Number 3, September 1996

Approximate Calculation of U-shaped Bellows

Shen Zupei Л Й ^ )

Department of Engineering Physics , Tsinghua University, Beijing 100084

Abstrac t: U-shaped bellows are widely used for sealed connections that require some flexibility. Since the struc

ture of U-shaped bellows is complex , numerical methods are often used to calculate mechanical parameters such as

stiffness, d isplacem ent, etc. In this paper app roximate formulas are d erived for calculating the stiffness and the

stresses of a U-shaped bellows with a slender ring shell. These formulas can be used for designing bellows and se

lecting corrugation parame ters. Comparison betw een the results of the approximate caculation and a finite element

calculation showed that the approximate formulas are applicable for μ < 0 . 5.

Key words : bello ws; approximate calculation? stiffness; toroidal shell

л

2

л

2

Introduction

A U-shaped bel lows i s a th in , e las t ic ro ta t ional

shel l , consis t ing of semi toroidal shel l s and annular

p la tes . F ig . 1 show s one corrugat ion of a U -shape d

bel lows which may have many corru gat ion s. S ince

the bel lows are symmetric , only a hal f corrugat ion

AD is analyse d . In par t AD, AB is an exter nal

toro idal shel l , CD is an internal toroidal shell and В С

i s an annular p la te .

z

v

П

n

В

A

С

pt

Fig. 1 U-shaped bellows profile

In the bel lows analysis , a

3

is the radius of the

ci rcular cross sect ion , r

3

is the radius of revolution

for the external toro idal shel l , a

x

and r

x

are the radii

for the in ternal toro idal shel l , Ь = г

3

r

x

is the width

of the annu la r p l a t e , r

0

= — ri+r

3

) is the averag e ra

d i u s ,

and h is the shel l th ickness . The character i s t ic

parameters of the in ternal and the external toro idal

shells are

Manuscript received: 1995-10-04

w h e r e , h

x

= — , v is Poss ion 's ra t io .

/ 1 2 ( 1 - v

2

)

Analysis of the deformations and the stresses of

a U-shaped bel lows uses the theory of toro idal

shel l s . References [ l~ ~ 5] s tudied various approaches

for solving the toroidal shell equations and for

analysing bel lows deformat ions. These methods used

for calculat ing the st iffness are complex and the re

sul t s have no obvious re la t ion to the bel lows parame

t e rs .

Cla rk

[ 6 ]

discussed approximate formulas for a

U-s haped bel lows wi th a rough r ing shel l . T his pa

per d iscussed approximate formulas for a U-shaped

bel lows wi th a s lender r ing shel l . T he approx imate

analytic solution for toroidal shell under axial load

has been derived for / /<0 . Approximate formulas for

calculat ing the stresses and the st iffness of a U-

shaped bel lows were then derived using the approxi

mate so lu t ion .

1 Approximate Solution of the

Toroidal Shell Equation.

The complex variable equation for the toroidal

shell shown in Fig. 2 under axial load has been de

rived

1

-

23

. In the theory of thin sh el ls, for

a/r

0

<^l,

the

toroidal shell equation can be reduced to

d

2

X I—

j ^

+ i μ sin?> X = μ

Q

0

r

0

V r

0

cosp (1)

w h e r e ,



306 Tsinghua cience and Technology, September 1996, 1 (3) : 305 ^30 9

+ Q, 0 V 0

c o s ?

s i n ?

2)

Fig. 2 Toroidal shell coordinates

Q

0

i s th e axia l force pe r uni t l en gt h a t r„ . E is

t he e l a s t i c m odu l us . I f t o t a l a x i a l f o r c e i s F, t h e n Q

0

= F/2nr

0

.

T h e s i g n a n d t h e d i r e c t i o n o f t h e s t r e s s r e s u l

t a n t s a n d t h e d i s p l a c e m e n t s o f t h e t o r o i d a l s h e l l a r e

t he s a m e a s i n Z ha o e t a l

C 2 ]

. Q ^ i s t he s he a r f o r c e , Ν

φ

a n d Ν

θ

a r e t he s t r e s s r e s u l t a n t s i n t he ? a nd Θ d i r e c

t i ons , Μ

φ

a n d Μ

θ

a r e t h e b e n d i n g m o m e n t s i n t h e ?

a n d

Θ

d r i e c t i o n s ,

Ur

a n d

Uz

a r e t h e d i s p l a c e m e n t s i n

t h e r a n d Z d i r e c t i o n s , β is t he a ng u l a r r o t a t i on i n

t h e c o u n t e r - c l o c k w i s e d i r e c ti o n . T h e h o m o g e n e o u s

f o r m o f E q . ( 1 ) c a n be s o l ve d u s i ng a pow e r s e r i e s s o

l u t i o n , a s i n S h e n e t a l

L 3 ]

.

L e t

χ = Σ ^

S u b s t i t u t i n g i n t o E q . ( 1 ) a n d u s i n g t h e p o w e r s e r i e s

e x p r e s s i o n o f t h e t r i g o n o m e t r i c f u n c t i o n g i v e r e c u r

s i ve r e l a t i ons f o r t he c oe f f i c i e n t s a s f o l l ow s

1

n-l-m

C

n

=— \μ — 77 Y\ C

m

n(n — 1) f-4

. n — 2 — m

s in π

n > 2

(n — 2 — m)\

w h e r e , C

0

a n d C\ ar e u n d e t e r m i n e d c o m p l e x c o n

s t a n t s . N e g l e c t i n g t h e μ

2

t e r m s , t h e s o l u t i o n b e c o m e s

X =

[Co

. n - 2

s in — - — π

. n - 3

s in — - — π

C

}

-Λ Ψ

(n -

2 ) ' ^

1

(n -

3 )

S i m p l i f y i n g g i v e s h o m o g e n e o u s s o l u t i o n o f E q . ( 1 )

X = C

0

[ l - Ί μ ( φ - s i n ? ) ] +

C\{.4>— i / " ( 2 — 2 c o s ? — ? s i n ? ) ] ( 3 )

A d o p t i n g t h e c h a n g e o f c o e f f i c e n t s g i v e s o n e n o n - h o

m o g e n e o u s p a r t i c u l a r s o l u t i o n o f E q . ( 1 )

X

F

= — i μ Q

0

r

0

V r

0

[ c o s ? + Ί μ (— φ + — s i n 2 ? ) ]

N e g l e c t i n g t h e μ

2

t e r m s g i v e s t h e p a r t i c u l a r s o l u t i o n

X

F

=— Ί μ

Q

0

r

0

V r

0

c o s ? ( 4 )

F o r C

0

= A, + i Β

λ

a n d C

x

= A

2

+ i B

2

,

t h e n t h e g e n e r a l s o l u t i o n o f E q . ( 1 ) t a k e s t h e f o l l o w

i ng f o r m

X =(A

X

+ ÌB0Ì Ì - i / / ( ? - s i n ? ) ] +

(A

2

+ Ί Β

2

)[_φ — ϊμ (2 — 2 c o s ? — ? s i n ? ) ] —

Ψ Qor

0

V r

0

c o s ? ( 5 )

T h e s t r e s s r e s u l t a n t s i n t h e t o r o i d a l s h e l l m a y b e

e x p r e s s e d in t e r m s o f t h e g e n e r a l s o l u t i o n , N e g l e c t

i ng t he μ

2

t e r m s g i v e s

SmCP

- [ Λ + Β

ιμ

(φ — s i n ? ) + Л

2

? +

9

=

Β

2

μ (2 — 2 c os ? — ? s i n ? ) ] + - — c o s ?

2л г

0

6)

EhL·

\_Β

λ

Α

ι

μ ( φ —

s i n ? ) +Β

2

φ — Α

2

μ

( 2 - 2 c o s ? - ? s i n ? ) ] + ^ 7 7 ^ - c o s ? ( 7 )

V^Q

H = -

1

\_A

X

+ Β

λ

μ { ψ — s i n ? ) + A

2

? +

r

0

V r

0

Β

2

μ (2 — 2 c o s ?— ? s i n ? ) ]

Ν

φ

=Hcos<p+ Q

0

s i n ?

Ν

θ

=— η =\_Β

λ

μ {\ - c o s ? ) +A

2

+

a V r

0

Β

2

μ (8ΐη φ — ? c o s ? ) ]

A

1

M ,

[ - A ^ Q - co s ?) + Б

2

<2 V Г

0

— A

2

/ ^ ( si n ? — ? c o s ? ) ] + a Q

0

s i n ?

Μ

θ

^ ν Μ ,

( 8 )

( 9 )

( 1 0 )

( 1 1 )

( 1 2 )

( 1 3 )

u

r = ΐ ϊ Γ - [ β ι Μ 1 - c os ? ) + A

2

Eha

+ £>

2

/ *( si n? — ? c o s ? ) ]

a

( 1 4 )

C7

Z

= I tf/?cos?d? + I

-£τ(Ν

φ

— v Ng )sm cp a< p ( 1 5 )

J 0 J 0 -C,Ai

w h e r e , У a nd H a r e t he ve r t i c a l a nd ho r i z on t a l s t r e s s

r e s u l t a n t s . E i s t he e l a s t i c m od u l u s of t he m a t e r i a l .

T h e c o n s t a n t s A

x

,B

}

,A

2

a n d B

2

c a n b e d e t e r m i n e d u s

i n g t h e t o r o i d a l sh e l l b o u n d a r y c o n d i t i o n s . T h e s t r e s s

r e s u l t a n t s a n d t h e d i s p l a c e m e n t s o f t h e t o r o i d a l s h e l l

c a n t h e n b e c a l c u l a t e d .

2 Stresses in a U-shaped Bellows

under Axial Load

A s s u m e t h a t t h e i n t e r n a l a n d t h e e x t e r n a l

t o r o i d a l s h e l l s o f t h e U - s h a p e d b e l l o w s s h o w n i n

F i g . 1 s a t i s f y t h e c ond i t i on s μ ι < 0 an d μ

3

<^ 1 , t h a t b is

o f t he s a m e o r de r a s α

λ

, a

2

a n d a

3

a n d t h a t b<^r

0

, r ^

r

2

^r

0

. I n p a r t AD o f t h e U - s h a p e d b e l l o w s , F i g . 3 ,



Shen Zupei (ut *&*! ): pproximate Calculation of

U-shaped

Bellows

307

p a r t AB is the e x t e r n a l t o r o i d a l s h e l l and the s o l u t i o n

c o n s t a n t s are de f i ne d as A

3

, B

3

, A

4

a nd B

4

. P a r t

В С

i s the a n n u l a r p l a t e . P a r t

CD

is the i n t e r n a l t o r o i d a l

s h e l l w h i c h is o p p o s i t e to the i n t e r n a l t o r o i d a l s h e l l

w i t h —— ^ p ^ 0 , so t h a t Q<p, Μ

φ 9

β s h o u l d t a k e

n e g a t i v e s i g n

in the

a b o v e f o r m u l a s

and the

s o l u t i o n

c o n s t a n t s

ar e

de f i ne d

as Α

λ

, Β

λ

, A

2

and B

2

. The

a n g u

l a r r o t a t i o n s

and the

s h e a r f o r c e s v a n i s h

at

b o t h e d g e s

of

the

U - s h a p e d b e l lo w s

due to

s y m m e t r y .

At

po i n t

D <p

=

- ) ,

H

0

r

ΐΑ

1

-Β

1

μ ί± -Ό -Α

χ

-±

Β

2

μ (2 - - ) ] = 0

( 1 6 )

point

H

3

A

Ehh

x

V r

0

Β

2

·

Α (φ

=

= =

_ J _

г V г

0

Β <μ (2

Ehhi

Α

4

μ (2

f - Α

2

μ (2 -

-

Ι Α ,+Β ,μ ^

- | ) ] = ο

η ,

- f ]

=

- - ) ]

2

J

ϊ - » -

= 0

- i + ^ « f +

- 1

+ ß

4

— —

и

17)

18)

~2

~

19)

Fig. 3 Stress resultants and displacements

T h e v a l u e of H, M

r

,U

r

and ß for the t o r o i d a l

s h e l l and the v a l u e of t h e m for the a n n u a r p l a t e

s h o u l d be e q u a l at the j o i n t В and the j o i n t С , t h e r e

f o r e , t h e y

ar e

d e f i n e d w i t h

the

s a m e s y m b o l s .

For

p a r t CD , at f = 0

Ai

- M , = zB,

20 )

Н г

A,

/ ? , = -

Ehh,

~a

x

Eh

At

B

Ehh

2

Qo

F o r p a r t

AB

, at φ = 0

A,

M ,

B

4

я

2

= -

= i4

s

A = -

£ M

t

ß

3

+

_ « 3 _

EhhY

1

Ά

a3Eh

( 2 1 )

( 2 2 )

( 2 3 )

( 2 4 )

( 2 5 )

( 2 6 )

( 2 7 )

F o r the a n n u l a r p l a t e , the ge ne r a l s o l u t i on

1

-

53

of th e

e q u a t i o n s for an a n n u l a r p l a t e , u n d e r the c o n d i t io n of

b<^r

0

,

can be

a p p r o x i m a t e d

as

A

~ Eh

3 W o

r

2

M

г

- M,

6

-HO

)

28 )

( 2 9 )

N e g l e c t i n g the t e r m s of μ Α

{

(ί = 1 — 4) in Eq s. (1 7)

a n d (19) g i v e s the a p p r o x i m a t e e q u a t i o n s

π

Y

\ — ~^B

2

( 3 0 )

B

3

---B

A

( 3 1 )

T h e n , u s i n g Eqs. (2 0) , ( 22 ), (2 4) , (26) and (28)

g i v e s the s o l u t i o n s

:

Β

λ

=

— (α,α

3

+ ir ò -— —

в ,= -

1

π

h

α

3

α

λ

α

3

a\)Q

0

( 3 2 )

α

λ

α

3

^b ^

l

2 \ -

a i Q o 33)

-a\)Q

0

34)

2

α

λ

2 α

λ

α

3

α

}

α

3

α -χ

в > = е

α

λ

α

3

j b - ^ - - a \ ) Q

0

35)

2 α

1

α

3

A, ^

1

3

T h e c o n s t a n t s

А, (г = 1 — 4) can be

o b t a i n e d

f r o m

the

o t h e r e q u a t i o n s .

Д ar e of th e

o r d e r

of μ Β

{

(ί

—

1 ~ 4) , so th e

n e g l e c te d t e r m s

in Eqs. (1 7) and

( 1 9 ) a r e C K / /

2

)

.

T h e t e r m s of

μ Α

ί

(ί

= 1 — 4) can a l s o be n e g l e c t

e d w h e n c a l c u l a t i n g the a n g u l a r r o t a t i o n and the

b e n d i n g m o m e n t . S u b s t i t u t i n g B

{

(i = 1 — 4) i n t o

E q s .

(7) and (1 2)

g i v e s

the

a n g u l a r r o t a t i o n d i s t r i b u

t i o n a l o n g the U - s h a p e d b e l l o w s .

F o r the i n t e r n a l t o r o i d a l s h e l l ,

- _ <

y

+ ,»(«,«,

+ -

α

λ

+ а

ъ



308

Tsinghua Science a n d Technology, September

1 9 9 6 , 1

( 3 ) :

30 5 309

al)Q

0

Ehh\

T i Q o C O s ?

36)

F o r

th e

e x t e r n a l t o r o i d a l s h e l l

,

2

,π

vzEhh

2v

2

<p)(a

x

a

3

2 <2i + α

3

< 3 ^ o

T ~

h l

VQ

0

+

Q

0

c o s ^

37)

F o r t h e a n n u l a r p l a t e ,

1

Ehh\

[_ a

x

a

3

+ — b

1

( r

2

(2i

+ a

3

)

2

]Q<

r

i +

г

з

ч 2

38)

8

2

ч

2

T h e b e n d i n g m o m e n t d i s t r i b u t i o n

is

t h e n

:

f o r t h e i n t e r n a l t o r o i d a l s h e l l ,

M

2

i

π

/

а

а

ъ

φ

=—

— (α ^ + — b

— — —

π α

χ

2 α

χ

+ α

3

a ? ) Q o + a j Q o s i n ^

fo r

t h e

e x t e r n a l t o r o i d a l s h e l l ,

9

π α

3

2 a

x

+ a

3

al)Qo + a

3

Q

0

s m<p

T h e m a x i m u m b e n d i n g s tr e s s e s a r e

at

e d g e s

A

a n d

D

π

( 3 9 )

40)

*

=

7

У м

— ί— (α

2

— α

λ

) + b ι*

π

α

χ

-+- α

3

+ *.]§

2

<41

^ = Τ

σ

Μ

=

[_— {α

χ

— α

3

) + b —

π α

χ

,

n

6 Q

0

42)

A l l o t h e r s t r e s s e s are s m a l l e r a n d are of th e o r d e r

1

< ?Ν Θ

σ

Ν φ

ah

0t

Q<p

Ώ

aQo

Β μ

μ <*Μ

a

c

~ a

Q μ <*Μ

0

h

3 U-shaped Bellows Stiffness Form u

las

3 ·

1

A x i a l s t i f f n e s s f or m u l a

A s s u m i n g t h a t th e a x i a l d i s p l a c e m e n t of a U-

s h a p e d b e l l o w s is u

z

un de r t he a x i a l f o r c e F , t he a x i a l

s t i f f ne s s is de f i ne d as F/u

z

. The a x i a l d i s p l a c e m e n t

of t h e U - s h a p e d b e l lo w s can be c a l c u l a t e d f r o m

3

4

E q . ( 1 5 ) . B e c a u s e

aß

— I f T l Q o*

ψ ΐ^^ ~ ^1 3 Q<

£A V

0

^°

- aß Und-^j-Ν φ ^ ~^T-

2

QO - ( - ) W ) , t he t er ms

r

0

En

Eh

A

ro

r

0

of 0(μ

2

) c a n be n e g l e c t e d w h e n i n t e g r a t i n g E q . ( 1 5 ) .

T h e a x i a l d i s p l a c e m e n t s of th e U - s h a p e d b e l lo w s are

o b t a i n e d

as

f o l l o w s

:

fo r

t h e

i n t e r n a l t o r o i d a l s h e l l ,

W z l

=

a\Qo г 2

(

£ M ?

L

7 r

U s

fli +b

a

3

ч

^ a

x

+

a

fo r th e e x t e r n a l t o r o i d a l s h e l l ,

.

T-3

43)

alQo

Ehh\

[— a

x

— a

3

)

+

b

ö l

# i +

<2

3

4

fo r

the

a n n u l a r p l a t e ,

W o / ¥ 3

π

Α Ι , 1 .2 4

Ehh\

a

x

-\ -

a

3

2 12

44)

45)

T h e t o t a l a x i a l d i s p l a c e m e n t

of a

U - s h a p e d b e l l o w s

w i t h o n e c o r r u g a t i o n

is

d= 2(u

zX

+u

z2

+ u

z3

) (46)

T h e a x i a l s t i f f ne s s f o r m u l a

is

t h e n

Cd

= ^τ =

2rcr

0

Ehh

2

x

/l— a

x

a

3

(a

x

+ a

3

) + —

о т с Ь

+

( π

2

— 8 )

/

3 . 3Λ I л~ ~ L I

π

« 1 « 3 ^

2

2π (α? + a D + ± α , α φ + α

χ

+ α ,

] 47)

3 . 2

B e n d i n g s t i f f n e s s f or m u l a

W h e n

a

U - s h a p e d b e l l o w s

is

l o a d e d w i t h

a

b e n d

i n g m o m e n t

M

0

,

F i g .

4, the

be l l o w s w i l l ha ve

an an

n u l a r r o t a t i o n

a . Th e

b e n d i n g s t i f f n e s s

is

d e f i n e d

as

M

0

/a

.

T h e b e n d i n g m o m e n t m a y

be

g e n e r a t e d

by an

e qu i va l e n t ve r t i c a l f o r c e a c t i ng

on the

b e l l o w s .

The

ve r t i c a l f o r c e

per

u n i t l e n g t h h a s

a

c o s i n e d i s t r i b u t i o n

o n

th e

b e l l o w s e d g e ,

V =

F

0

c

o

s

# » w h e r e

V

0

is th e

ve r t i c a l f o r c e per u n i t l e n g t h at Θ = 0 . T h e r e l a t i o n

s h i p b e t w e e n

V

0

a n d

M

0

is

M

0

=

У

0

с о 8

2

в г Ш = TZAV,

T he ve r t i c a l f o r c e

is

m a x i m u m

at

Θ

=

0 w h e r e t h e a x i

a l d i s p l a c e m e n t

is £

0

.

A s s u m i n g t h a t

δ

0

is

a p p r o x i

m a t e l y e q u a l

to the

a x i a l d i s p l a c e m e n t

of th e U-

s h a p e d b e l l o w s u n d e r

th e

un i f o r m a x i a l f o r c e

Q

0

(Q

0

=

V

0

) , t h e n E q s . ( 4 3 ) , ( 4 4 ) and ( 4 5 ) c a n be u s e d to

c a l c u l a t e t h e d i s p l a c e m e n t £

0

. S i n c e t h e a n g u l a r r o t a

t i o n of t h e b e l l o w s # i s e q u a l to

S

0

/r

0

, t h e n th e b e n d

i ng s t i f f ne s s f o r m u l a is

a

0^ 0

O n

So/ г о ~ 2

b a

48)

Fig.

4

Bending loads



Shen Zupei (v£

*&*! )

: pproximate Calculation of

U-shaped

Bellows

3 0 9

4 Ь

ъ

С в = nrlE hh\/\_— a

l

a

z

a

l

+ α

3

) + — + 4αι<2

3

6

π 6

+

2π (# -f- α

3

;

(49)

5 Examples

The s t i ffness and the s t resses in a U-shaped bel

lows were ca lcula ted using the approximate formulas

derived above. The resul t s were compared to ca lcula

t ions made using the fini te element method

[7 ]

for the

geometries given in Table 1. The bellows material

has E = 196 GPa and v = 0. 3 .

The axial st iffness, the bending st iffness and the

max imum bend ing s t re ss a

M

a t the sym metric poin t D

of the internal toroidal shell were calculated for a U-

shaped bel lows wi th one corru gat ion , F ig . 1 , under

an axial force F = 100 N , T able 2 .

Table 1 Geom etr ic parameters for be l lows examples

r o / m m

< 2 i / m m

й з / m m

b/mm

h / m m

μ \

μ

e x a m p l e 1

100

1.0

1.0

1.0

0 . 2 5

0 . 1 3 2

0 . 1 3 2

e x a m p l e 2

100

1.6

2 . 0

2 . 0

0 . 4

0 . 2 1 1

0 . 3 3 0

e x a m p l e 3

100

2 . 5

3 . 0

3 . 0

0 . 5

0 . 4 1 3

0 . 5 9 5

The resul t s show that the approximate formulas

are accurate for μ

<C

0. 3 and th at t he dev iat ion is less

than 15% for μ — 0. 5 . Th e app roxim ate formulas

a re t he refo r app li cab l e for / / < 0 . 5 .

Table 2 Comp arison of ca lculated resul ts

e x a m p l e 1

e x a m p l e 2

e x a m p l e 3

W M N - m "

C 0 / k N - m

< W M P a

a p p r o x i m a t e f o r m u l a

f in i te e lement


f in i te e lement


f in i te e lement

1 9 . 8 5

1 9 . 8 7

9 9 . 2

1 0 0 . 6

2 2 . 9

2 3 . 6

1 2 . 8 0

1 2 . 8 8

6 4 . 0

6 7 . 8

1 7 . 7

1 7 . 6

7 . 1 1

7 . 3 8

3 5 . 6

4 2 . 1

1 7 . 0

1 6 . 5

6 Conclusions

Th e approxim ate formulas for a U-sh aped bel

lows with a thin shell derived in this paper are very

simple and can be used for calculat ing the st iffness

fo r μ <C 0. 5 . T he form ulas provide explici t relat io n

ship between the geometrica l parameters and the

st iffness. Th ese form ulas are very convenien t for de

signing bellows and for determining the bellows cor

ruga t ion pa ramete rs .

References

λ ^ Α , 1 9 7 9 , 1 9 ( 1 ) : 8 4 - 9 9

Q i a n W e i c h a n g . C a l c u l a t i o n s f o r s e m i

—

c i rcu lar a rc type

c o r r u g a t e d t u b e - A p p l i c a t i o n s o f t h e t h e o r y o f s l e n d e r r i n g

s h e l l s . J o u r n a l of T s i n g h u a U n i v e r s i t y , 1 9 7 9 , 1 9 ( 1 ) :

8 4 —

99

2 &»£, n«?, гш*. т ^ш т т № М Ш №

ж ^ ш - ^ ш т ш т ш ш - т * *

т , ΐ 9 8 ΐ , 2 Κ 2

:

ι -

14

Z h a o H u n g b i n , W u Z h e n h u i , S h e n Z u p e i . E q u a t i o n s in

c o m p l e x v a r i a b l e s u n d e r a x i a l s y m m e t r i c l o a d s a n d t h e i r

a s y m p t o t i c s o l u t i o n s . J o u r n a l o f T s i n g h u a U n i v e r s i t y ,

1 9 8 1

2 1 ( 2 ) : 1 - 1 4

i j ^ ^ J g , 1 9 8 1 , 2 1 ( 3 ) : 9 3 - 1 1 0

S h e n Z u p e i , W u Z h e n h u i , Z h a o H o n g b i n . S t r e s s e s a n d

d i s p l a c e m e n t s i n r o t a t i n g t o r o i d a l s h e l l s . J o u r n a l o f T s

i n g h u a U n i v e r s i t y , 1 9 8 1 , 2 1 ( 3 ) : 9 3 — 1 1 0

^ f t , 1 9 82 , 3 : 4 9 - 8 5

C h e n S h a n l i n . S y m m e t r i c a l s t r e s s a n d d e f l e c t i o n o f

S-type

c o r r u g a t e d t u b e s . J o u r n a l o f C h e n g d u U n i v e r s i t y o f S c i

e n c e a n d T e c h n o l o g y , 1 9 8 2 , 3 : 4 9 — 8 5

S Ä Ä f c . ^Ψ ^Ψ Ψ », 1 9 8 9 , 2 9 ( 6 ) : 5 8 - 6 6

S h e n Z u p e i , Z h a o H o n g b i n , W u Z h e n h u i . C o m p l e x

v a r i a b l e e q u a t i o n a n d a p p r o x i m a t e a s y m p t o t i c s o l u t i o n o f

t o r o i d a l s h e l l s u n d e r " W i n d - t y p e " l o a d s . J o u r n a l o f T s

i n g h u a U n i v e r s i t y , 1 9 8 9 , 2 9 ( 6 ) : 5 8 — 6 6

C l a r k R A , A n e x p a n s i o n b e l l o w s p r o b l e m . J o u r n a l of A p

p l ie d M e c h a n i c s , 1 9 7 0 , 5 :

6 1 —

69

А Ш 8 . 1 9 8 7, 4 ( 1 ) : 5 5- 61

S h e n Z u p e i , W u Z h e n b u i . F i n i t e e l e m e n t a n a l y s i s o f t h i n

s h e l l s o f r e v o l u t i o n . C o m p u t a t i o n a l S t r u c t u r a l M e c h a n i c s

a n d A p p l i c a ti o n s , 1 9 8 7 , 4 ( 1 ) :

5 5 —

61

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Approximate Calculation of U-shaped Bellows