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13th International Conference on Nuclear Engineering
Beijing, China May 16-20, 2005
ICONE 13-50470

TRANSVERSE VIBRATIONS OF MULTIPLY BELLOWS
EXPANSION JOINT ELASTICALLY RESTRAINED AGAINST
ROTATION
KAMESWARA RAO.C
Professor, Mechanical Engineering
Department, Muffakham Jah College
of Engineering & Technology
Hyderabad, India
Phone: 91 040 23811623
email: ckrao_52@yahoo.com

RADHAKRISHNA .M
Scientist, Design Engineering Division
Indian Institute of Chemical Technology
Hyderabad, India
Phone: 91 040 27193229
email: madabhushirk@yahoo.com

Keywords: Multiply bellows, elastic restraint, transverse and vibration
Abstract

layer acts as a nearly neutral fiber that offers

This paper presents the study of dynamic aspects of

significantly lower resistance to movement in the

the multiply bellows with elastically restrained ends

piping system than a single wall bellow. Therefore,

and under rotatory inertia. The effect of rotational

less force is required for actuating multiply bellows.

restraint and internal pressure load on response of

The stresses induced on the individual layers of

such bellows configuration is attempted. The coupled

bellows are a fraction of the stresses induced in the

response obtained is compared with single ply

single wall bellows of equal thickness. This results in

configuration. Two cases are considered –bellows

highest possible service life of bellows.

having same material and different materials for
The multiply design results in lower spring

plies.

rates for a given pressure capacity. Also the effective
cross sectional areas of multiply bellows is less

1. Introduction

compared to conventional bellows to accommodate a

Multiply bellows are generally used in
applications

where

lower forces and moments on the anchors, equipment

required than a single-ply construction. The multiply

and guides supporting the piping system. The anchors

construction increases the overall stiffness of the

and guides of the system can be dimensioned

system and reduces deflection. These bellows are

significantly smaller and more economically. This

manufactured

increases the life of the connected equipment and

multi-layered

capacity

given movement. These two reductions result in

is

as

higher-pressure

tubes

of

plies

generally ranging in number from 2 to 20 depending

raises the reliability of the overall piping system.

on diameter and operating pressure. Each of the thin-

1

Copyright © 2005 by CNS

This type of construction is well suited for

considers the ends of bellows as elastically restrained

applications with vibration or rapid cyclic motion

against rotation. The inter-ply interaction is not

because of the inherent damping provided by the

considered here and the plies are considered closely

relative inter-ply movement. Fig. 1, shows number of

fit.

plies and geometrical dimensions of multiply
It is also seen that no provision has been

bellows.

made to this effect in the expansion bellows [EJMA
1] code.
The equations for the transverse vibrations
of multiply bellows are derived. Case I considers a
multiply bellow that have same

material of

construction for all the three plies and Case II
considers

a

multiply

bellow

having

different

materials of construction. Results are obtained to
observe the effect of varying the rotational spring
stiffness parameter T from 10 to 1010 on natural
frequencies.
2. Theoretical Development
An attempt has been made here to derive the

Bellows having same dimensions as in thesis

transcendental frequency equation for transverse

[6], and having 3 numbers of plies of equal thickness

vibrations

multiple

are considered here for comparison purpose. The

materials of construction for plies. The innermost ply

maximum critical pressure and spring rate are

that is in contact with the fluid is of Alloy 400 and

calculated using the formulae given [3].

of

multiply

bellows

with

compatible to the fluid to resist corrosion and other
subsequent two plies are of austenitic stainless steel -

3. The Differential Equation

type 321 to take care of the pressure capacity.
The fundamental equation of motion of the
equivalent Timoshenko beam [5] is given by-

A literature search shows relatively little
work was carried out in predicting the dynamic

w
w
w
EI ∂
+ PπR ∂ ρA − J ∂
∂x
∂x
∂x ∂t
w
+m ∂
=0
∂t
4

response of multiply bellows. The previous works

2

m

4

contributed by Li Ting-Xin and Jakubauskas. V.F et
al. [2] studied the effect of classical fixed-fixed type

4

2

2

2

2

2

of boundary conditions for single ply bellows and

tot

obtained axial and transverse frequencies there on
[3].
Therefore, the paper is an attempt to obtain

2

(1)

Where EI is bending stiffness, P- internal pressure in

the flexural vibration characteristics of multiply

bellows, J – rotatory inertia per unit length, w –lateral

bellows subjected to an internal pressure and

deflection x-axial coordinate, Rm –mean radius of
2

Copyright © 2005 by CNS

bellows, mtot is the total mass of bellows per unit

dX ( x) 2
= Aα 2 sinh αx + Bα 2 cosh αx −
2
dX
Cβ 2 sin βx − Dβ 2 cos βx

length includes mass of bellows and fluid mass and ttime [4].

(9)
Using the technique of Separation of variables, the
Let the roots of the equation be α & β

lateral deflection of the bellows axis ‘w’ is expressed
asw (x, t) = X (x) .T (t)

α = − c2 + c4 + λ

(2)

β = c 2 + c 4 + λ4

Where w (t) is a parameter of ‘x’ and T (t) is the time

(10)

And

harmonic function T (t) = X (x) e iωt

4

(3)

λ =
4

‘ω’ is natural frequency

4

mtot .ω 2
EI

(11)

Differentiating the above equation (Eq.2) and

At the ends ‘A and B’, the bellows are connected to

substituting in the differential equation (Eq.1), we

short pipe nipples and so are considered to have a

get-

rotational stiffness of R1 and R2 respectively.

(

)

At x=0, the boundary conditions are

PπRm + Jω 2 ∂ 2 X
m
∂4 X
− w 2 tot X = 0
+
4
2
EI
EI
∂x
∂x
2

X (0) = 0

(4)
If c =

(PπR

+ Jω 2
2 EI
2

m

EI

)

(12)
At x=L, the boundary conditions are
(5)

X (L) = 0

Then equation (Eq.4) can be written as4

∂ 2 X (0)
∂X (0)
= −T1
2
∂x
∂x

EI

2

d X
d X
+ 2c 2 . 2 − λ4 X = 0
4
dx
dx

∂ 2 X ( L)
∂X ( L)
= −T 2
2
∂x
∂x
(13)

(6)
Where T1 =

The general solution of equation (6) is given by
X(x) = A sinh αx + B cosh αx + C sinβx + D cosβx
(7)

R1 L
R L
and T2 = 2
EI
EI

Applying the boundary conditions we get-

Where A, B, C &D are arbitrary constants

B+D=0

respectively. The first two derivatives of equation

[

]

(14)

EI B(αL) 2 − D( βL) 2 = T1 [A(αL) + C ( βL)]

(Eq.7) are as follows-

(15)

A sinh αL + B cosh αL + C sin βL
+ D cos βL = 0

dX ( x)
= Aα cosh αx + B sinh αx + Cβ cos βx
dx
+ Dβ sin βx

(16)

(8)
3

Copyright © 2005 by CNS

A{(αL )(αL sinh αL + T2 cosh αL )} +

B{(αL )(αL cosh αL + T2 sinh αL )} +

Mean diameter (Dm) –69.3mm, convolution height,

C {(βL )(T2 cosh βL − βL sin βL )} −

h=5.71mm; convolution pitch, q=5mm; bellows

D(βL )(T2 sin βL + βL cos βL ) = 0

thickness after forming, tp=0.28mm: number of
convolutions, N=9: pitch diameter of bellows,

(17)

Db=62.8mm: E400 at 400 deg F =2.03x1011N/m2:

The above set of equations can be written in matrix

E321=1.82x1011N/m2:

form.

6

operating

pressure,

2

P=0.862x10 N/m and bellows length, L = 0.045 m

1
1
1  A
 0
 T αL
2
βL2   B 
− αL
T1 βL
 1
 =0
sinh αL cosh αL sin βL cos βL  C 


c2
c3
− c 4   D 
 c1

respectively.
Case I: bellows have SS-321 as same material of
construction for all the three plies.

(18)

Case II: bellows have first inner ply as Alloy 400 and

c1 = (αL )(αL sinh αL + T2 cosh αL )

subsequent two plies as SS-321.

c 2 = (αL )(αL cosh αL + T2 sinh αL )
c3 = (βL )(T2 cos βL + βL sin βL )

The axial stiffness of bellows is computed by using

c 4 = (βL )(T2 sin βL + βL cos βL )

the expression given below3

t 
k1 = 4 Rm E321  p  .n321
h

(19)
Expanding the matrix and substituting for c1, c2, c3 &

(21)

t 
k2 = 4 Rm [E400 .n400 + E321.n321 ]. p 
h

c4 and on simplification, we get the final frequency
equation as follows-

3

(22)

Where, n321 and n400 represent number of plies of

(

)



1
2
2
(αL )2 + (βL )2 
 (αL ) − (βL ) +
T1 .T2


sinh αL sin βL + 2αL.βL(1 − cosh αL cos βL ) +

[

respective material of construction. According to
equations (21) and (22), k1=8.92x106N/m and k2=
9.26x106N/m.
The

]

1 1
 +  (αL )2 + (βL )2
 T1 T2 
[(αL )cosh αL sin βL − (βL )sinh βl cos βL] = 0

maximum

critical

pressure

bellows

can

withstand and bending stiffness are given by the
expressions -

Pmax =

(20)

π .k1, 2 .q
6.66 L2

EI = ¼.k.q.Rm2

The frequency equation is same for Case I and II
respectively.

(23)
(24)

Substituting the values for k1,2, q, L and Rm, we get

4. Exact Analysis

the values for maximum critical pressure, bending

The various dimensional parameters of U-shaped

stiffness and mtot, total mass of bellows per unit

bellows are same as those that have been considered

length. mtot for both the cases is same as the thickness

by Jakubauskas [3].

of ply assumed is same.

4

Copyright © 2005 by CNS

presents frequencies obtained for n321 and n400
2

EI (1) = 26.77Nm :

respectively.

2

EI (2) = 27.8Nm :
m tot =3.84kg/m

Table 2 presents the first mode frequencies by
varying the rotational restraint parameter T.

Also the rotatory inertia (J) of bellows per unit length
Table2: First Mode Frequency for various values
of T and at P=0.0Mpa

are found out as J = 0.00826354kgm
Substituting the above values in equation (Eq.5) and
(Eq.11) we get the values of c and λ for Case I & II
respectively.

T, Rotational

ω, Radian/s,

ω, Radian/s,

Restraint

n321=3

Ñ=1,
n400=3

2

(1) =

732 + 0.000154.ω

(25)

0.01

3004.5

3063.08

2

(1) =

0.3788.ω

(26)

1.0

3690.9

3762.9

(27)

10

4407.9

4494.0

2

5002.5

5100.3

3

5093.8

5193.5

The first lateral mode fundamental frequency is

4

10

5103.3

5203.0

obtained by using bisection method. The rotational

105

7364.6

5204.0

restraint T is varied from a minimum value of 0.1 to a

6

7364.8

5204.15

maximum of 10 at maximum and minimum internal

7

10

7364.9

5204.16

pressure as calculated. Equal rotations are assumed on

108

7364.9

5204.17

9

7364.9

5204.17

10

7364.9

5204.17

c

λ

2

c2 (2) = 720 + 0.000148.ω2
λ

2

(2) =

0.5127.ω

10

(28)

10

10

10

either ends and hence T=T1=T2.

10
10

5. Results and Discussion
Table 1 gives a comparison of the frequencies obtained

Fig 2: Transverse Vibrations of Multiply bellows
at P=0.0MPa

using the bisection method for multiply bellows vis-à-vis

Transverse Vibrations of Multiply Bellows
P=0.0MPa, number of plies n321 & n400=3,
T=T1=T2 and
Mode Number =1

to the single ply bellows having elastically restrained
ends presented by Rao.C.K et al. [5].
Table 1 Comparison of Frequencies at T1=T2=T= ∝
Pressure

Rao.C. K. [5]

n321=3

n400=3

Ñ

P, MPa

n=1, ω,

ω,

ω,

Radian/s

Radian/s

Radian/s

1

0.0

3684.1

7364.8

5204.1

1

10.0

3400.16

7036.9

4726.8

8.E+03
6.E+03
4.E+03
2.E+03

1.E-02

1.E+01

1.E+04

1.E+07

Frequency, radians/s

Mode

0.E+00
1.E+10

Rotational Restraint, T

It is seen that the frequencies obtained by considering

n321=3
n400=3

the ends as elastically restrained are higher than that
were found out for single ply bellows. Table 1

5

Copyright © 2005 by CNS

Table 3 presents the first mode frequencies by

T ransverse Vibrations of Multiply Bellows
P=10.0MPa, number of plies n 321 & n 400 =3,

varying the rotational restraint parameter T.
Table 3: First Mode Frequency for various values
of T and at P=10MPa
T, Rotational
ω, Radian/s,
ω, Radian/s,
Restraint

n321=3

8.E+03

Ñ=1

6.E+03

n400=3

4.E+03
2.E+03

Frequency, radians/s

T =T 1 =T 2 and
Mode Number =1

0.01

2098.8

2158.9

1.0

2999.7

3071.7

10

3847.6

3933.2

102

4515.5

4612.7

103

4616.0

4715.6

104

4626.3

4725.6

105

7036.8

4726.7

106

7036.9

4726.8

6. Conclusions

107

7036.9

4726.8

The exact method developed would determine the

108

7036.9

4726.8

dynamic characteristics of multiply bellows for its

109

7036.9

4726.8

lateral vibrations. The theoretical formulations are

1010

7036.9

4726.8

based on the equations of an equivalent Timoshenko

1.E-02

1.E+02

1.E+06

0.E+00
1.E+10

Rotational Restraint, T

n321=3
n400=3

beam. The influence of elastically rotational restraint
at either end is considered and the first fundamental

It is seen that as rotational stiffness, T increases and

mode frequency is quite significant.

approaches infinity, the lateral frequencies increases
in both the cases. It is interesting to note that the

References

transverse frequencies of multiply bellows having

1.

different materials of construction for plies lie

EJMA – “The Standards of the Expansion joint
Manufacturers Association Inc”. New York,

midway between single-ply and multiply bellows

ASME, 1984, p221.

with same materials of construction for plies.
2.

Fig 3: Transverse Vibrations of Multiply Bellows
at P=10MPa

Jakubauskas V.F, “Practical Predictions of
Natural Frequencies of Transverse Vibrations of
Bellows Expansion Joints”, Mechanika- Kaunas,
Technologija, 1998, No.3 (14), p47-52.

3.

Jakubauskas.V.F, “Transverse Vibrations of
bellows Expansion Joints”- PhD. Thesis –
Hamilton,

Ontario,

Canada:

McMaster

University, 1995, PP145-150.
4.

6

Kameswara

Rao.C

and

“Transverse

Vibrations

of

Radhakrishna.M,
Single

Bellows

Copyright © 2005 by CNS

Expansion Joint Restrained against Rotation”,
Proceedings of Tenth International Conference
on Nuclear Engineering, USA, Paper # 22090,
April 14-18,2002.
5.

Morishita.M. et al., “Dynamic Analysis Methods
of

Bellows

Including

Fluid

Structure

Interaction", ASME Pressure Vessels and Piping
Conference, Hawaii, July 23-27, 1989, P149157.
6.

Radhakrishna.M, “Vibrations and Stability of
Elastically Restrained Expansion Bellows in
Pipelines”- PhD. Thesis –Hyderabad, India,
Osmania University, 2003.

7

Copyright © 2005 by CNS


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