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National Symposium on Advances in Structural Dynamics and Design
Structural Engineering Research Centre, Madras
January, 9 to 11, 2001

Finite Element Analysis of Vibrations of Rotationally
Restrained Fluid Conveying Pipes Resting on Soil
Medium
Simha, H.+ and Kameswara Rao, C.++
+

Design & Engineering Division, Indian Institute of Chemical Technology,
Hyderabad 500 007, INDIA. simha@iict.ap.nic.in; ++Patents & Standards
Translation Group, BHEL Corporate R&D Division, Hyderabad 500 093, INDIA

Abstract
This paper deals with the vibrations of rotationally restrained straight pipes conveying fluid
and resting on soil medium. An attempt has been made to analyse the dynamic behaviour of
straight Bernoulli-Euler pipes rotationally restrained at either ends including the effects of
flow velocity and foundation stiffness using the finite element approach. For the first time,
results are presented for the first two natural frequencies of rotationally restrained fluid
conveying pipes for varying values of the rotational restraint parameter, fluid flow velocity
parameter and foundation stiffness parameter. For extreme values of the rotational stiffness
parameter ( 0 and infinity), which represent the ideal boundary conditions, the results are
found to be in good agreement with those available in published literature.

INTRODUCTION
It is a well known fact that the velocity of the fluid has considerable effect on the
natural frequencies of the straight pipe conveying the fluid. There have been many
investigations on the effect of flow velocity on the natural frequencies of straight
pipes conveying fluid with simple boundary conditions such as hinged-hinged,
hinged-fixed and fixed-fixed. These boundary conditions are idealised cases and are
difficult to realise practically. A more realistic boundary condition is the rotational
restraint at both ends of the pipe. However, results are not available for rotationally
+

Scientist, ++Sr. DGM

National Symposium on Advances in Structural Dynamics and Design
Structural Engineering Research Centre, Madras
January, 9 to 11, 2001

restrained straight Bernoulli-Euler fluid conveying pipes resting on soil medium in
the published literature.
Since 1947, many investigations [1 - 7] have been carried out for studying the
vibration behaviour of fluid conveying pipes utilising various methods such as
Galerkin, Rayleigh-Ritz and Fourier series solutions. Stein and Tobriner [8],
Dermendjian-Ivanova [9] and Raghava Chary, Kameswara Rao and Iyengar [10],
have devoted their studies on fluid conveying pipes resting on elastic foundation.
As can be seen from the above discussion, many methods have been utilised by
investigators in solving this problem, but the finite element method was utilised by
only a few investigators such as Kohli and Nakra [11] and Pramila, Laukkanen
and Liukkonen [12]. However, the effect of foundation modulus is not included in
both these studies. In an earlier paper [13], the authors have presented numerical
results for the first three natural frequencies obtained from the mass and stiffness
matrices developed for straight Bernoulli-Euler fluid conveying pipes resting on soil
medium. These results were however for the three classical boundary conditions of
hinged-hinged, hinged-fixed and fixed-fixed.
The present paper deals with development of a finite element program for
rotationally restrained long pipes with internal flow and resting on Winkler
foundation. Different values of pipe rotational restraint parameter are considered in
generating results for the natural frequency of the piping system with variations in
the non-dimensional parameters defining the flow velocity and foundation stiffness.
For extreme values of the rotational restraint parameter, the results are in good
agreement with those available in published literature. The effects of soil inertia
along with shear deformation are not considered in the analysis.
EQUATIONS OF MOTION
The system under consideration consists of a pipe of length L with cross-sectional
area A, flexural rigidity EI, mass of pipe per unit length mp, constant axial flow
velocity vf, mass of fluid per unit length mf and foundation stiffness kf. Here the
extended Hamilton’s principle [6] is used and states that
t2

δ

t2

∫ (T − V) dt − ∫ m
t1

t1

f

& + vf w′ )δ w
vf ( w

dt = 0 ,

x =L

(1)

where the possible free end is assumed to be the right end, x = L. Here, T and V
denote the kinetic energy of the pipe with fluid and the potential energy respectively.
δ is the variation symbol. The dot indicates differentiation with respect to time, t,
and a prime indicates differentiation with respect to the spatial coordinate x. The
equation for kinetic energy is

National Symposium on Advances in Structural Dynamics and Design
Structural Engineering Research Centre, Madras
January, 9 to 11, 2001

T=

1
2

∫ [m
L

&
pw

]

& + v f w ′ ) 2 dx ,
+ mf v f 2 ( w ′ )2 + m f ( w

2

(2)

0

Potential energy is given by

V=

1
2

∫ [ E I w ′′

]

L

2

+ k f w 2 dx ,

(3)

0

The displacements and rotations of the nodes are interpolated independently, i.e.

w = N w a = [ N1

0 L] a ,

(4)

0 N 2 L] a ,

(5)

0 N2

and

w ′ = Nθ a = [ 0 N 1

Here, the vector of nodal parameters is

a =  w1


w1′

w2 ′ L ,

T

w2

(6)

and N1, N2, etc. are cubic functions. The first time-integral of equation (1) yields a
system of ordinary differential equations :

M &a& + G a& + K a = 0 ,
(7)
The mass matrix is
L



M = ( m p + m f ) N w T N w dx ,

(8)

0

The gyroscopic matrix is given by
L

T


G = m f v f  N w T N w ′ - N w ′ N w  dx ,


0



(9)

National Symposium on Advances in Structural Dynamics and Design
Structural Engineering Research Centre, Madras
January, 9 to 11, 2001

The stiffness matrix is composed of three parts :
L

Kb =

∫E

I N θ ′ N θ ′ dx ,
T

(10)

0

L

K g = − m f v f 2 N w ′ N w ′ dx ,
T



(11)

0

L



K f = k f N w T N w dx ,

(12)

0

Equation (10) is due to bending, equation (11) is due to the centripetal acceleration
of fluid particles, and equation (12) is due to continuous elastic foundation. For free
vibration analysis, we assume

a = Ae iω nt , where A is a constant and ω n is the circular natural frequency
As the present investigation is limited to the study of free vibration characteristics
only, the damping matrix [G], which effects the dynamic stability of the pipe is
neglected in the computations carried out. The final non-dimensional eigenvalue
problem solved here is

Ka − λMa = 0 , where λ =

m p L3ω n 2

is the frequency parameter

EI (m p + m f )

Contributions from a typical element to the stiffness and mass matrices respectively
are explicitly given below :
Stiffness Matrix

{
}
{
= { 4 − (2 / 15)v / n + (1 / 105)γ / n };
= { − 12 + 1.2v / n + (9 / 70)γ / n }n
= { − 6 + v / 10n + (13 / 420)γ / n }n
= {12 − 1.2v / n + (39 / 105)γ / n }n ; k = −k
= { 2 + v / 30n − (1 / 140)γ / n }; k = −k ; k

}

k11 = 12 − 1.2v f / n 2 + (39 / 105)γ / n 4 n 2 ; k 21 = 6 − v f / 10n 2 + (11 / 210)γ / n 4 n
k 22
k 31
k 32
k 33
k 42

2

4

f

2

4

2

f

2

4

2

4

f

2

41

f

2

f

32 ;

4

43

21

44

= k 22

National Symposium on Advances in Structural Dynamics and Design
Structural Engineering Research Centre, Madras
January, 9 to 11, 2001

Mass Matrix
m11 = 156/(420n2)
m31 = 54/(420n2)
m41 = - m32
m44 = m22

; m21 = 22/(420n3)
; m32 = 13/(420n3)
; m42 = - 3/(420n4)

; m22 = 4/(420n4)
; m33 = m11
; m43 = - m21

where n is the number of elements and

γ =

k f L4
EI

, the foundation stiffness parameter.

NUMERICAL RESULTS
Figure 1 shows the Bernoulli-Euler pipe element conveying fluid and resting on soil
medium. SR1 and SR2 are the rotational restraint stiffness parameters at either ends
of the pipe. For the purpose of this study, both the ends are assumed to have linear
restraint stiffness parameters ST1 and ST2 equal to infinity. Each node of the element
has two degrees of freedom as shown.
Numerical results are obtained from the computer program developed using the
above matrices with suitable modifications in the stiffness matrix to include the
effects of the rotational restraint stiffness parameters. The values of the first two
natural frequency parameter λ for various values of flow parameter vf and
foundation stiffness parameter γ for the case of equal rotational restraint at either
end (SR1 = SR2 ) are presented in Table 1. Table 2 gives the values of the first two
natural frequency parameter for various values of vf and γ for the case of
SR2 = SR1 / 100.
Figures 2 and 3 show the variation of the first two natural frequency parameters λ1
and λ2 respectively for various values of SR2, Vf and γ keeping the rotational
restraint parameter SR1 = 0.1. Similar results for SR1 = 10.0 and SR1 = 1000.0 are
presented in Figures 4 and 5, Figures 6 and 7 respectively.
The tables and figures presented in this paper give a wide range of results for the
vibrations of rotationally restrained Bernoulli-Euler pipes conveying fluid and
resting on elastic foundation. The authors believe that these results will be of
considerable use in the design of such piping systems.

National Symposium on Advances in Structural Dynamics and Design
Structural Engineering Research Centre, Madras
January, 9 to 11, 2001

CONCLUSIONS
The following conclusions can be drawn from the above study :


The results for the cases when the rotational restraint SR1 = SR2 = 0.0 ( hingedhinged ); SR1 = SR2 = infinity ( fixed-fixed ) and SR1 = 0.0, SR2 = infinity
(hinged-fixed ) are found to exactly tally with the results for classical boundary
conditions reported in a previous paper by the authors [13].



In all the cases studied here, the natural frequency of the piping system starts to
increase appreciably for values of SR2 in the range 0.01 to 10.0, for given values
of Vf, γ and SR1. Values of the natural frequency parameter remain essentially
constant for the values of SR2 < 0.01 and SR2 > 10. However, in all cases, the
natural frequency parameter decreases with increasing flow velocity parameter
and increase consistently with increasing foundation stiffness parameter.



The effect of foundation stiffness parameter γ on the first natural frequency
parameter λ1 is most profound. The frequency parameter increases appreciably
with increasing values of γ. This effect is not as appreciable for λ2.



The results are presented in a non-dimensional form and hence a designer can
obtain values of natural frequencies for any fluid conveying pipe by proper
interpolation.

ACKNOWLEDGEMENTS
Both authors are grateful to the managements of their respective organisations for
granting permission to carry out the study and publish the results.
REFERENCES
1.

I. I. GOLDENBLATT 1947 Stroezdat. Modern Problems of Vibrations and
Resistance in Engineering Construction.

2.

H. ASHLEY and G. HAVILAND 1950 Journal of Applied Mechanics Vol.
17, Trans. ASME, Vol. 72, 229-232. Bending Vibrations of a Pipeline
Containing Flowing Fluid.

3.

V. P. FEODOSYEV 1951 Inzhenernyisbornik Vol. 10, 169-170. Vibrations
and Stability of a Pipeline when a Liquid Flows Through it.

4.

G. W. HOUSNER 1952 Journal of Applied Mechanics Vol. 19, 205-208.
Bending Vibrations of a Pipe when a Liquid Flows Through it.

National Symposium on Advances in Structural Dynamics and Design
Structural Engineering Research Centre, Madras
January, 9 to 11, 2001

5.

G. H. HANDELMANN 1955 Quarterly of Applied Mathematics Vol. 13, No.
3. A Note on the Transverse Vibration of a Tube Containing Flowing Fluid.

6.

D. B. MCIVER 1973 Journal of Engineering Mathematics Vol. 7, 243-261.
Hamilton’s Principle for Systems of Changing Mass.

7.

S. NAGULESWARAN AND C. J. H. WILLIAMS 1968 Journal of
Mechanical Engineering Science Vol. 10, No. 3, 228-238. Lateral Vibration of
a Pipe Conveying Fluid.

8.

R. A. STEIN AND M. W. TOBRINER 1970 Journal of Applied Mechanics
Vol. , 906-916. Vibration of Pipes Containing Flowing Fluids.

9.

D. S. DERMENDJIAN-IVANOVA 1992 Journal of Sound and Vibration Vol.
157, No. 2, 370-374. Critical Flow Velocities of a Simply-Supported Pipeline
on an Elastic Foundation.

10. S. RAGHAVA CHARY, C. KAMESWARA RAO AND R. N. IYENGAR
1993 Proceedings of 8th National Convention of Aerospace Engineers, 266287, Institution of Engineers (India). Vibrations of Fluid Conveying Pipes on
Winkler Foundation.
11. A. K. KOHLI AND B. C. NAKRA 1984 Journal of Sound and Vibration Vol.
93, No. 2, 307-311. Vibration Analysis of Straight and Curved Tubes
Conveying Fluids by means of Straight Beam Finite Elements.
12. A. PRAMILA, J. LAUKKANEN AND S. LIUKKONEN 1991 Journal of
Sound and Vibration Vol. 144, No. 3, 421-425. Dynamics and Stability of
Short Fluid Conveying Timoshenko Element Pipes.
13. H. SIMHA AND C KAMESWARA RAO 1998 Paper communicated to the
Journal of Sound and Vibration. Finite Element Analysis of Vibrations of Fluid
Conveying Pipes Resting on Soil Medium.

National Symposium on Advances in Structural Dynamics and Design
Structural Engineering Research Centre, Madras
January, 9 to 11, 2001

Table 1 Values of first two natural frequency parameters λ for various values of flow velocity
parameter vf and foundation stiffness parameter γ with rotational stiffness parameters SR1 = SR2
SR1 = SR2

vf

1.00E-01
1.00E-01
1.00E-01
1.00E-01
1.00E-01
1.00E+01
1.00E+01
1.00E+01
1.00E+01
1.00E+01
1.00E+03
1.00E+03
1.00E+03
1.00E+03
1.00E+03

0
1
3
6
9
0
1
3
6
9
0
1
3
6
9

γ = 0.0
λ1
λ2
11.40749
41.14778
10.96567
40.66497
10.02374
39.68173
8.4153
38.15938
6.41534
36.57369
21.45771
59.2499
21.17999
58.87406
20.61224
58.11459
19.72708
56.95516
18.79645
55.77005
22.36453
61.66984
22.08772
61.29542
21.52237
60.53914
20.64247
59.3854
19.7196
58.20722

γ = 500.0
λ1
λ2
25.10241
46.83097
24.90474
46.40733
24.5046
45.54821
23.89178
44.22825
23.26277
42.86764
30.99086
63.32891
30.79922
62.97741
30.41158
62.26802
29.81875
61.18733
29.21141
60.08576
31.6255
65.59855
31.43036
65.24667
31.03566
64.53671
30.43208
63.4557
29.8138
62.35448

γ = 1000.0
λ1
λ2
33.61742
51.89547
33.47007
51.51349
33.17341
50.74091
32.72335
49.55944
32.26696
48.34909
38.21562
67.16064
38.06037
66.8293
37.74737
66.16121
37.27141
65.14514
36.78732
64.11161
38.73206
69.3049
38.57288
68.97194
38.25196
68.30071
37.76389
67.2802
37.26745
66.24259

National Symposium on Advances in Structural Dynamics and Design
Structural Engineering Research Centre, Madras
January, 9 to 11, 2001

Table 2 Values of first two natural frequency parameters λ for various values of flow velocity parameter vf and
foundation stiffness parameter γ with rotational stiffness parameters SR2 = SR1 / 100.0
SR1

SR2

vf

1.00E-01
1.00E-01
1.00E-01
1.00E-01
1.00E-01
1.00E+01
1.00E+01
1.00E+01
1.00E+01
1.00E+01
1.00E+03
1.00E+03
1.00E+03
1.00E+03
1.00E+03

1.00E-03
1.00E-03
1.00E-03
1.00E-03
1.00E-03
1.00E-01
1.00E-01
1.00E-01
1.00E-01
1.00E-01
1.00E+01
1.00E+01
1.00E+01
1.00E+01
1.00E+01

0
1
3
6
9
0
1
3
6
9
0
1
3
6
9

γ = 0.0
λ1
λ2
10.65084
40.32901
10.17603
39.83647
9.15272
38.83266
7.35513
37.27635
4.94124
35.65222
15.92336
49.79661
15.56861
49.37198
14.83202
48.51135
13.648
47.1905
12.34398
45.83105
21.90432
60.44271
21.62704
60.06756
21.06048
59.30966
20.17794
58.15305
19.25119
56.97138

γ = 500.0
λ1
λ2
24.76773
46.11322
24.56729
45.68308
24.16138
44.81044
23.53928
43.46868
22.90013
42.08421
27.45093
54.58665
27.24668
54.19956
26.83261
53.41677
26.19671
52.22014
25.54161
50.99496
31.30174
64.44627
31.10834
64.09456
30.71716
63.38483
30.11892
62.30391
29.50607
61.20243

γ = 1000.0
λ1
λ2
33.36825
51.2487
33.21975
50.86201
32.9207
50.07969
32.46687
48.88278
32.0065
47.65586
35.40556
58.98901
35.24744
58.63098
34.92834
57.90812
34.44224
56.80619
33.94663
55.682
38.46816
68.21526
38.31095
67.88308
37.994
67.21336
37.51199
66.19499
37.02173
65.15933

National Symposium on Advances in Structural Dynamics and Design
Structural Engineering Research Centre, Madras
January, 9 to 11, 2001

Fig. 1 Rotationally Restrained Fluid Conveying Pipe Element

36
34
32
30
28
26

1

24
Frequency

22
20
18
Vf = 0

16

Vf = 3

14

Vf = 6

12
γ = 0.0

10

Vf = 9

γ = 500.0
γ = 1000.0

8
6

4
1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04 1.0E+05 1.0E+06 1.0E+07
Stiffness Parameter SR2

Fig. 2 Variation of λ1 with SR2 for various values of vf & γ with SR1=0.1

National Symposium on Advances in Structural Dynamics and Design
Structural Engineering Research Centre, Madras
January, 9 to 11, 2001

60
58
56
54
52
Vf = 0
Frequency λ2

50

Vf = 3
Vf = 6

48

Vf = 9
46
44
γ = 0.0
γ = 500.0
γ = 1000.0

42
40
38
36
34
1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00 1.0E+01 1.0E+02

1.0E+03 1.0E+04 1.0E+05 1.0E+06

1.0E+07

Stiffness Parameter SR2

Fig. 3 Variation of λ2 with SR2 for various values of vf & γ with SR1=0.1

39
37
35
33
31

Frequency λ1

29
27
25
23
Vf = 0
21

Vf = 3
Vf = 6

19
17

γ = 0.0
γ = 500.0
γ = 1000.0

15

Vf = 9

13
11
1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04 1.0E+05 1.0E+06 1.0E+07
Stiffness Parameter SR2

Fig. 4 Variation of λ1 with SR2 for various values of vf & γ with SR1=10.0

National Symposium on Advances in Structural Dynamics and Design
Structural Engineering Research Centre, Madras
January, 9 to 11, 2001

70
68
66
64
62
Vf = 0
Frequency λ2

60

Vf = 3
Vf = 6

58

Vf = 9
56
54
γ = 0.0
γ = 500.0
γ = 1000.0

52
50
48
46
44
1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00 1.0E+01 1.0E+02

1.0E+03 1.0E+04 1.0E+05 1.0E+06

1.0E+07

Stiffness Parameter SR2

Fig. 5 Variation of λ2 with SR2 for various values of vf & γ with SR1=10.0

39
37
35
33
31

Frequency λ1

29
27
25
23

Vf = 0

21

Vf = 3
Vf = 6

19

Vf = 9

17

γ = 0.0
γ = 500.0
γ = 1000.0

15
13

11
1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04 1.0E+05 1.0E+06 1.0E+07
Stiffness Parameter SR2

Fig. 6 Variation of λ1 with SR2 for various values of vf & γ with SR1=1000.0

National Symposium on Advances in Structural Dynamics and Design
Structural Engineering Research Centre, Madras
January, 9 to 11, 2001

71
69
67
65
63
Vf = 0
Frequency λ2

61

Vf = 3
Vf = 6

59

Vf = 9
57
55
γ = 0.0
γ = 500.0
γ = 1000.0

53
51
49
47
45
1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00 1.0E+01 1.0E+02

1.0E+03 1.0E+04 1.0E+05 1.0E+06

1.0E+07

Stiffness Parameter SR2

Fig. 7 Variation of λ2 with SR2 for various values of vf & γ with SR1=1000.0


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