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Chellapilla 4 ICONE 11 Vibrations of Bellows 2003 .pdf



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ICONE 11
Eleventh International Conference on Nuclear Engineering
APRIL 20-24, 2003
Japan

“DRAFT”

ICONE 11-36400
SEISMIC RESPONSE OF ELASTICALLY RESTRAINED SINGLE BELLOWS
EXPANSION JOINT IN LATERAL MODE
KAMESWARA RAO.C
Bharat Heavy Electricals Limited
Hyderabad, India

RADHAKRISHNA . M
Indian Institute of Chemical Technology
Hyderabad, India

Keywords
Single bellows, elastic restraint, seismic, lateral mode
type of ends, which are infinitely stiff and have

Introduction

used the approximate method to determine the
Bellows expansion joints are used for absorbing
thermal movements of piping that results in a
substantial shortening of pipe length and thereby
reducing

the

plant

construction

costs

natural frequencies. Hence, it is observed that
this assumption will lead to an over estimation of
natural frequencies and incorrect determination
of seismic response.

considerably.
The paper attempts to derive an exact solution
It is seen that most of the nuclear and thermal
power plants around the world are located in the
seismic zones. The piping in these plants consists
of expansion bellows and needs to be analyzed
from seismic design point of view.

for the seismic response of U type of single
bellows that are considered elastically restrained
against rotation on either end. The fundamental
equation

of

motion

of

the

equivalent

Timoshenko beam is used to calculate the
spectral response of the bellows to a seismic

Hence, while designing an expansion bellow
from structural point of view, the seismic design
aspect is also to be given utmost importance.

excitation in the lateral direction. The shear
terms are neglected. The first and second lateral
mode frequencies are obtained keeping the
rotational spring stiffness values equal.

The formulae presented in EJMA and those
derived by Morishita et al, [1,3] cover classical
boundary conditions only like the fixed-fixed

The Differential Equation

[M] ÿ + [k]x = βb ÿ0(T)

The fundamental equation of motion of the

βb is the participation factor which for the

equivalent Timoshenko beam [5] is given by-

present study becomes equal to 1.0.

EI ∂

4

w

+ PπRm 2 ∂

2

w

∂x
∂x
w
+m ∂
=0
∂t
4

ρA − J
2

∂w
∂x ∂t

Differentiating the above equation (5) we get-

4

2

2

2

tot

(5)

2

(1)

y(x, T) = y(x).e(iωT)

(6)

ý (x, T) = y(x) (iω) e(iωT)

(7)

ÿ (x, T) = y(x) (i ω)2 e(iωT)

(8)

∴ÿ (x, T) =- ω2 y(x, T)

(9)

Where EI is bending stiffness, P- internal

Where ‘y’ is generalized coordinate in modal

pressure in bellows, J – rotatory inertia per unit

space. Assuming damping factor is equal to zero

length, w –lateral deflection x-axial coordinate,

i.e.[c] =0

R

m

–mean radius of bellows, mtot is the total

mass of bellows per unit length includes mass of

Let the solution of equation (9) be given as-

bellows and fluid mass and t-time [5].

Y= A sinh αx + B cosh αx + C sinβx + D cosβx
(10)

Using the technique of Separation of variables,

Where A, B, C&D are arbitrary constants

the lateral deflection of the bellows axis ‘w’ is

Boundary conditions are as follows-

expressed asw (x, t) = X (x) .T (t)

(2)

i.) y=0 at x=0
∂2y / ∂x2 = + T1 ∂y/∂x

(11)

Where w (t) is a parameter of x only and T (t) is
the time harmonic function such as T (t) = X (x) e

iωt

ii.) And y = 0 at x = L or x =1
(3)

∂2y / ∂x2 = - T2 ∂y/∂x

(12)

Where‘ω’ is the natural frequency
Now applying B.C (i) & (ii) we getThe modal expansion technique is used to obtain

B+D = 0 or B = -D

the seismic response of the beam. Applying the

A+C =0 or A = -C

modal expansion to equation (1) –we obtain the
equation of motion in modal space[M] ÿ + [c] ý + [k] y = βb ÿ0 (T)

Substituting in equation (10) and rewriting(4)

y= C (sinβx-sinhαx) + D (cosβx -coshαx)

(13)

y = D{[T2(sinhαx +sinβx) + ϕ (coshαx + cos βx)

/ T2 (cosβx -coshαx) - ϕ(sinhαx + sinβx)]
(sinβx-sinhαx) + (cosβx - coshαx)}

(14)

Exact Analysis –Seismic Response
The various dimensional parameters of U-shaped

Applying the boundary condition (ii) we get –

bellows are taken to be same as those that have
been considered by Morishita et al. [5]-

y= D{(cosβx - coshαx) + δ(sinβx - sinhαx)} (15)

Bellows

Where

convolution height, h=30mm; convolution pitch,

T2(sinhαx + sinβx) + φ(coshαx + cosβx)
δ=
T2(cosβx − coshαx) − φ(sinhαx + sinβx)
(16)

p=25mm;

inner

diameter

bellows

(Di)

thickness,

–545.8mm,
tb=0.563mm:

Bellows mean diameter, Dm=550.8mm: number
of convolutions, N=20: pitch diameter of
bellows, Dp=575.8mm and Bellows length L =
0.5 m respectively.

At x=1
The theoretical stiffness of bellows is computed
by using the expression given in EJMA,

sin β + sinh α
δ=
cos β − cosh α

k=387.02N/m.
(17)

in bellows and bending stiffness are given by the

Also equation (17) can be written as -

δ = −{

sin β − sinh α
cos β − cosh α

According to [3] the maximum internal pressure
expressions -

}
(18)

P max = π. kp / (6.666L2)

(22)

EI =¼.kp. Rm2

(23)

Substituting the values for k, p, L and Rm, we
get-

We know

Pmax =18.23Pa
√ (Pπ Rm2 + J ω2)
c = -----------------------√ 2 EI

EI = 0.1834Nm2.
(19)
Also the total mass (m tot) and rotatory inertia (J)

Let the roots of the equation (10) be α & β are
given as in reference [3]-

of bellows per unit length are found out asm tot = 440.76kg/m.
J = 2.3894kgm

4

α = √ -c + √ (c + ϕ )

(20)

β = √ c2 + √ (c4 + ϕ4)

(21)

2

4

Substituting the above numerical values in
equation (19), (20) and (21) we get –
c = √ (0.00001184 +6.5141ω2 )

(24)

λ = 7.0016√ω

(25)

Results and Discussion
Table 1 gives a comparison of the exact frequencies

Using the frequency expression derived and given

obtained using the bisection method for a single

in reference [4] the first and second lateral mode

bellows that are elastically restrained at both ends

frequencies are obtained using bisection method.

vis-à-vis to the fixed-fixed end results presented by
Morishita et al. [5].

The rotational restraint T is varied from a
10

minimum value of 0.01 to a maximum of 10 at
maximum internal pressure of P=18.233Pa. It is
assumed that there is equal rotations on either

Table 1 Comparison of Frequencies at T= ∝
Mode

Morishita [5]

Exact

Error, %

#

f, Hz

f, Hz

1

27.7

23.6

15

2

54.2

36.3

53

ends i.e.T=T1=T2.
It is seen that the frequencies obtained by
Now the spectral response of the bellows to a
seismic excitation in the lateral direction is
calculated

by

using

the

modal

expansion

technique. Applying this technique to equation
(1), the equation of motion in modal space is

considering the ends as elastically restrained are
lower than that was found out by Morishita et al.
Therefore, it can be concluded that the seismic
response by the present method is exact and not
an over prediction.

given by equation (5). From equation (5) the
maximum response lateral displacement of any
arbitrary section of bellow to a response spectrum
is given by-

y max = βb

Table 2 presents the first two mode frequencies
at T1=T2, the time period for the fundamental
mode frequency and the spectral acceleration.

Sa

(2πf )

2

.

Dp
y( x )
2.N.p
(26)

βb=1.0,
Where Sa is the spectral gravitational acceleration
at frequency ‘f’ and has units of mm/s2.

Table2: Fundamental Frequency for various
values of restraint parameter T=0.01 to ∝
T

t (sec)

Sa (mm/s2)

0.01

14.8635

0.0672

206.9

0.1

14.9301

0.0669

206.9

1.0

15.501

0.065

413.8

10

17.6262

0.0567

551.7

10

α
y( x ) = {(ψ(n + 1) − ψ(n )}
y

f (Hz)

2

20.502

0.0487

655.2

103

23.1246

0.0432

603.5

4

23.5662

0.0424

569.0

(27)

105

23.614

0.0423

567.3

Where ‘n’ is convolution number and is equal to

106

23.619

0.0423

567.3

N-1. Here in this case it is 19 as N=20

107

23.619

0.0423

567.3

8

23.619

0.0423

567.3

109

23.619

0.0423

567.3

10

23.619

0.0423

567.3

convolutions.

α
=1
y

10

10
10

109

0.0148

0.038

10

0.0148

0.038

10

The seismic gravitational acceleration obtained
at T=∝ and time period of t=0.042sec is
620.68mm/s2. Substituting this value in equation
(26), the maximum lateral displacement is
obtained as ymax = 0.016mm compared to
0.84mm by Morishita et al [5].
Fig 1:Time History Curve of Seismic Response
It is observed that for N=1 and at T=0.01 the
time period found out is 0.0672sec, while at
T=∝, the time period is 0.0423sec. It is seen that
as rotational stiffness increases and approaches
infinity, the time period of vibration decreases. A
standard time history graph of seismic wave and
its acceleration response is used for obtaining the
seismic accelerations (Sa) at various frequencies
as shown in Fig 1.

Conclusions
The exact method is developed in present study
to calculate the dynamic characteristics and
seismic response of bellows for its lateral
vibrations. The theoretical formulations are
based on the equations of an equivalent
Timoshenko beam, and the bending stress due to
these vibrations is estimated. In case of lateral
vibrations it is found that the influence of
elastically rotational restraint at either ends to its

Table 3 gives the maximum displacement, ymax
and bending stress σb values for varying
rotational restraint parameter T.

natural frequencies is significant.
References
1.

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

Table 3: Response of bellows to lateral seismic
excitation

York, ASME, 1984, p221.

T

ymax(mm)

σb(kg/mm2)

0.01

0.0136

0.035

0.1

0.0135

0.034

1.0

0.0251

0.065

10

0.0259

0.067

102

0.0227

0.058

103

0.0165

0.043

104

0.015

0.039

105

0.0148

0.038

106

0.0148

0.038

bellows Expansion Joints”- PhD. Thesis –

10

7

0.0148

0.038

Hamilton,

10

8

0.0148

0.038

University, 1995, PP145-150.

2.

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
Ontario,

Canada:

McMaster

4.

Kameswara Rao.C and Radhakrishna.M,
“Transverse Vibrations of Single Bellows
Expansion

Joint

Rotation”,

Proceedings

of

Tenth

Conference

on

Nuclear

International

Restrained

against

Engineering, USA, Paper # 22090, April 1418,2002.
5.

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

of

Bellows

Structure

Interaction",

Including
ASME

Fluid

Pressure

Vessels and Piping Conference, Hawaii,
July 23-27, 1989, P149-157.


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