critical velocity of fluid conveying pipes resting on two ...

International Conference on Advances in Structural Dynamics and its Applications (ICASDA-2005), December 7-9, 2005, Visakhapatnam

CRITICAL VELOCITY OF FLUID CONVEYING PIPES RESTING ON TWO-PARAMETER FOUNDATION Kameswara Rao Chellapilla Professor, Mechanical Engineering Department Muffakham Jah College of Engineering & Technology Hyderabad, India Phone: 91-040-23811623 Email: [email protected]

H. Simha Scientist, Design & Engineering Division Indian Institute of Chemical Technology Hyderabad, India Phone: 91-040-27193229 Email: [email protected]

Abstract Pipelines are used extensively for transportation of fluids. The velocity of the fluid in the pipeline imparts energy to the pipeline making it to vibrate. It is well established from published literature that there exists a critical velocity of the fluid near which the natural frequency of the pipeline tends to zero. This is the required condition for buckling of the pipeline. Literature abounds with analyses, which give information on the influence of boundary conditions on the stability of fluid conveying pipes. However, much of these studies have been carried out for pipelines resting on Winkler type foundations. This has provided the motivation for studying the influence of non-Winkler type foundation on the critical velocities of a fluid-conveying pipe. The foundation considered in this study is a two-parameter foundation model , such as the Pasternak foundation. Expressions are derived for the critical flow velocity by utilizing Fourier series and Galerkin method for three simple boundary conditions, namely: Pinned-Pinned, Pinned-Fixed and Fixed-Fixed. Results are presented for varying values of the foundation stiffness parameter and interesting conclusions are drawn on the effect of the foundation parameters on the critical flow velocity of the pipeline. Introduction The technology of transporting fluids, especially petroleum liquids, through long pipelines which cover different types of terrain, has evolved over the years. Interest in studying the dynamic behaviour of such fluid conveying pipes was stimulated when excessive transverse vibrations were observed and subsequently analyzed first by Ashley and Haviland in 1950 [1] and later by Housner in 1952 [2]. Housner considered a simply supported beam model for the pipeline and analyzed it using a series solution approach and showed the existence of a critical flow velocity for a pipeline which could cause buckling. In 1955, Long [3] studied the influence of fixed-fixed and fixed-pinned boundary conditions on the critical velocity. In 1966, Gregory & Paidoussis [4] presented results on the dynamic behaviour of a cantilevered pipe conveying fluid. All the above studies did not consider elastic support conditions. When a pipeline rests on an elastic medium such as a soil, a model of the soil medium must be included in the governing differential equation. A very common structural model of the soil medium is the Winkler model, in which soil is represented by a series of constant stiffness, closely spaced linear springs. In 1970, Stein & Tobriner [5] studied the vibrations of a fluid conveying pipe resting on an elastic foundation. Lottati and Kornecki, in 1986 [6], studied the influence of the elastic foundation on the stability of the pipeline. Later, in 1992, Dermendjian-Ivanova [7] investigated the behaviour of a fluid conveying pipe resting on an elastic foundation and obtained the critical fluid velocity. In 1993, Chary [8] presented a detailed analysis of fluid conveying pipes resting on elastic foundation. He also considered the inertia of the foundation and alalyzed the seismic response of such pipelines. All these studies modeled the elastic foundation as a Winkler model. In this paper, the work of Chary has been extended to study the influence of a two-parameter foundation model on the critical flow velocity. Results are presented for various values of the foundation stiffness parameters.

1


Equation of motion The differential equation of motion for lateral displacement w(x,t) of a uniform fluid conveying pipe resting on a Winkler type elastic foundation is given by [8]:

EI

2 ∂4w ∂ 2w ∂2w 2 ∂ w + M + ρ Av + 2 ρ Av + k1 w = 0 ∂x 4 ∂t 2 ∂x2 ∂x ∂t

(1)

In the above equation, E denotes the modulus of elasticity, I is the moment of inertia of the the pipe section, M(=m+ρA) is the total mass of the pipe per unit length, m is the mass of the pipe alone per unit length, ρA is the mass of the fluid per unit length, v is the steady flow velocity and k 1 is the stiffness of the elastic medium per unit length of the pipe. In this equation, the elastic medium is modeled on the Winkler type foundation. Following the method given by Pantelides [9], The equation of motion for a fluid conveying pipe resting on a two-parameter foundation becomes:

∂4w ∂ 2w ∂ 2w ∂ 2w 2 EI + M 2 + ( ρ Av − k2 ) 2 + 2 ρ Av + k1 w = 0 ∂x 4 ∂t ∂x ∂ x∂ t

( 2)

In equation (2) above, k 2 represents the additional parameter defining the foundation, usually termed as the shear constant of the foundation. The model is shown in figure 1. Equation (2) is now solved for three simple boundary conditions. Pinned-pinned pipe The boundary conditions for a pinned-pinned pipe are

w( 0, t ) = w( L, t ) = 0 ∂ 2 w(0, t ) ∂ 2 w( L, t ) = =0 ∂x 2 ∂x 2

(3)

Taking the solution of equation (2) which satisfies the boundary conditions (3) as

w( x, t ) =

∑

an sin

n =1, 3,5,...

nπx sin ω j t + L

∑

a n sin

n = 2, 4, 6,...

nπx cos ω j t , j = 1,2,3,... L

(4)

Where ωj represents the natural frequency of the j th mode of vibration. Substitution of equation (4) in (2) and expanding in a Fourier series we have an equation of the form:

[K − ?

j

2

]

MI {a} = 0

(5)

where K is the stiffness matrix whose elements are enumerated in [8] and will not be repeated here, I is the identity matrix and aT ={a 1 ,a2 ,…..,a n }. Retaining the first two terms of the above equation, and setting the determinant equal to zero, we get

(

)

 256   Ω j4 −  β − 5π 2  V 2 − γ 2 + 17π 4 + 2γ 1 Ω j 2 +   9 

(

) (

)(

)(

)

2  4 2  4 π V − γ − V 2 − γ 2 5π 2γ1 + 20π 6 + 16π 8 + 17π 4γ1 + γ 12  = 0 2   

(6)

2


In equation (6), the following non-dimensional parameters have been used:

β=

k L4 k L2 ρA M ρA ; Ω j = ω j L2 , j = 1,2,3,... ; V = vL ;γ1 = 1 ;γ 2 = 2 M EI EI EI EI

When the fluid velocity reaches a certain value Vcr, the fundamental natural frequency becomes zero. Hence, setting Ω j = 0 in equation (8), we obtain:

(

) (

)(

)(

)

2  4 2 2 2 6 8 4 2  4 π V − γ − V − γ 2 5π γ1 + 20π + 16π + 17π γ1 + γ 1  = 0 2   

(7)

Solving equation (7) for V, we obtain the critical flow velocity for the pinned-pinned case.

Pinned-fixed and fixed-fixed pipe The boundary conditions for a pinned-fixed pipe are

w( 0, t ) = w( L, t ) = 0 2 ∂w(0, t ) ∂ w( L, t ) = =0 ∂x ∂x 2

(8)

And those for a fixed-fixed pipe are

w( 0, t ) = w( L, t ) = 0 ∂w(0, t ) ∂w( L, t ) = =0 ∂x ∂x

(9)

We assume the deflection of the pipe to be of the form

 x  w( x, t ) = ℜφ n  e iωt   L 

(10)  x  L

In equation (10), ℜ denotes the real part, φ n   is a series of beam eigen-functions ψ r (ξ ) given by:

x ψ r (ξ ) = cosh (λr ξ ) − cos (λ r ξ ) − σ r (sinh (λr ξ ) − sin (λr ξ )) , r = 1,2,3,...., n ; ξ =   (11) L cosh λr − cos λr σr = sinh λr − sin λ r In the above equation, λr is the frequency parameter of the pipe without fluid flow, which is considered as a beam, and it’s values are [10]

λ1 = 3.926602 and λ 2 =7.068583 for the pinned-fixed case and λ1 = 4.730041 and λ 2 =7.853205 for the fixed-fixed case 3


Substituting equation (10) in the equation of motion (2) gives

Ln = EI

∂ 4φ ∂x

4

(

+ ρAv − k 2 2

)

∂2w ∂x

2

+ 2 iωρ Av

(

)

∂φ + k 1 − Mω 2 φ = 0 ∂x

(12 )

Following the method given in [8], using Galerkin’s method and minimizing the mean square of the residual Ln over the length of the pipe, we have the following equations in V. For the pinned-fixed case:

(C11C22 − C12 C21 )(V 2 − γ 2 )

2

(

)[(

) ( ) ] [(λ14 + γ 1 )(λ2 4 + γ 1 )]= 0

(13)

) ] [(λ14 + γ 1 )(λ2 4 + γ 1 )] = 0

(14)

+ V 2 − γ 2 λ14 + γ 1 C 22 + λ2 4 + γ 1 C11 +

For the fixed-fixed case:

(C11C22 )(V 2 − γ 2 )

2

(

)[(

)

(

+ V 2 − γ 2 λ14 + γ 1 C 22 + λ 24 + γ 1 C11 +

In equations (13) and (14), the constants C11 etc. are integral values which are enumerated in [8].

Results Table 1 gives the numerical values of the critical velocity parameter. The two foundation parameters are varied from 1.0E-06 to 1.0E+4. The first value is equivalent to no foundation while the last value represents a very stiff foundation. The results are tabulated for all the three boundary conditions. In figures 2 3 & 4, the influence of γ2 on the critical velocity parameter of the pipe for a pinned-pinned boundary condition is shown for various values of γ1 . The results for the other two cases of pinnedfixed and fixed-fixed follow a similar trend. Figure 5 shows the variation of the critical velocity parameter with γ2 for various values of γ1 . Finally, a comparison of the individual effects of each of the two foundation parameters, when the other is equivalent to zero, on the critical velocity parameter is shown in figure 6.

Conclusions It is seen from figure 2 that there is not any perceptible change in the dynamic behaviour of the pipe until the shear constant of the two-parameter foundation γ2 takes a value of 10.0. The critical velocity increases slightly for the value of γ2 of 10.0. For a value of γ2 of 100.0, there is a sharp jump in the value of the critical velocity parameter and this trend continues for increasing values of γ2 , as shown in figures 3 and 4. Another observation from these plots is that, for lower values of γ2 , there is a sharp increase in the value of critical velocity for the Winkler foundation constant γ1 values greater than 10.0. The critical velocity does not seem to be effected by the value of the Winkler constant γ1 for higher values of γ2 . This is shown in another way in figure 5. It can be seen that the Winkler constant γ1 has little effect on the critical velocity especially for values of γ2 greater than about 100.0.

4


In figure 6, a comparison between the relative effects of each of the two foundation parameters is shown. The top curve shows that there is a sharp increase in the critical velocity when there is a progressive increase in the value of γ2 beyond 100.0. This curve represents the case where γ1 is near zero. The bottom curve shows the variation of critical velocity with γ1 when γ2 is near zero. It can be observed that the influence of the shear constant of the two-parameter foundation is more than that of the Winkler constant on the critical velocity.

References 1.

Ashley, H. and Haviland, G., (1950), “Bending Vibrations of a Pipe Line Containing Flowing Fluid”, J. Appl. Mech., Trans. ASME, September, pp. 229-232.

2.

Housner, G. W., (1952), “Bending Vibrations of a Pipe Line Containing Flowing Fluid”, J. Appl. Mech., Trans. ASME, June, pp. 205-208.

3.

Long, Jr., R. H., (1955), “Experimental and Theoretical Study of Transverse Vibration of a Tube Containing Flowing Fluid”, J. Appl. Mech., Trans. ASME, March, pp. 65-68.

4.

Gregory, R. W. and Paidoussis, M. P., (1966), “Unstable Oscillations of Tubular Cantilevers Conveying Fluid Proc. Roy. Soc (London), 293A, pp. 512-542.

5.

Stein, R. A. and Tobriner, M. W., (1970), “Vibration of Pipes Containing Flowing Fluids”, J. Appl. Mech., Trans. ASME, December, pp. 906-916.

6.

Lottati, I. And Kornecki, A., (1986), “The effect of an Elastic Foundation and of Dissipative Forces on the Stability of Fluid-Conveying Pipes”, J. Sound Vibration, 109(2), pp. 327-338.

7.

Dermendjian-Ivanova, D. S., (1992), “Critical Flow Velocities of a Simply Supported Pipeline on an Elastic Foundation”, J. Sound Vibration, 157(2), pp. 370-374.

8.

Chary, S. R., (1993), “Vibrations of Pipes Resting on Soil Medium”, M.S. Thesis, Indian Institute of Science, Bangalore.

9.

Pantelides, C. P., (1992), “Stability of Columns on Biparametric Foundations”, Comp. Str, 42(1), pp. 21-29.

10.

Rao, S. S., (1986), “Mechanical Vibrations, Addison Wesley Publishing Company, Massachusetts, p.386.

5


Table 1 – Values of the critical velocity parameter for various values of γ 1 and γ 2 for the three boundary conditions Pinned-pinned pipe VCr γ1 γ2

γ1

Pinned-fixed Pipe VCr γ2

γ1

Fixed-fixed pipe VCr γ2

1.00E-06 1.00E-05 1.00E+02 1.00E+03 1.00E+04

1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06

3.14159 1.00E-06 3.14159 1.00E-05 4.47233 1.00E+02 8.05039 1.00E+03 17.11086 1.00E+04

1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06

4.49975 1.00E-06 4.49975 1.00E-05 5.32942 1.00E+02 8.66386 1.00E+03 16.91955 1.00E+04

1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06

5.43433 5.43433 5.95246 9.40901 16.00158

1.00E-06 1.00E-05 1.00E+02 1.00E+03 1.00E+04

1.00E-05 1.00E-05 1.00E-05 1.00E-05 1.00E-05

3.14159 1.00E-06 3.14159 1.00E-05 4.47233 1.00E+02 8.05039 1.00E+03 17.11086 1.00E+04

1.00E-05 1.00E-05 1.00E-05 1.00E-05 1.00E-05

4.49975 1.00E-06 4.49975 1.00E-05 5.32942 1.00E+02 8.66386 1.00E+03 16.91955 1.00E+04

1.00E-05 1.00E-05 1.00E-05 1.00E-05 1.00E-05

5.43433 5.43433 5.95246 9.40901 16.00158

1.00E-06 1.00E-05 1.00E+02 1.00E+03 1.00E+04

1.00E+01 1.00E+01 1.00E+01 1.00E+01 1.00E+01

4.45753 1.00E-06 1.00E+01 4.45753 1.00E-05 1.00E+01 5.47738 1.00E+02 1.00E+01 8.6492 1.00E+03 1.00E+01 17.40061 1.00E+04 1.00E+01

5.4998 1.00E-06 1.00E+01 5.4998 1.00E-05 1.00E+01 6.19699 1.00E+02 1.00E+01 9.22293 1.00E+03 1.00E+01 17.21253 1.00E+04 1.00E+01

6.28745 6.28745 6.74031 9.9262 16.31105

1.00E-06 1.00E-05 1.00E+02 1.00E+03 1.00E+04

1.00E+02 1.00E+02 1.00E+02 1.00E+02 1.00E+02

10.48187 1.00E-06 1.00E+02 10.48187 1.00E-05 1.00E+02 10.95453 1.00E+02 1.00E+02 12.83778 1.00E+03 1.00E+02 19.81871 1.00E+04 1.00E+02

10.96575 1.00E-06 1.00E+02 10.96575 1.00E-05 1.00E+02 11.33149 1.00E+02 1.00E+02 13.23111 1.00E+03 1.00E+02 19.65379 1.00E+04 1.00E+02

11.38121 11.38121 11.63751 13.7306 18.8693

1.00E-06 1.00E-05 1.00E+02 1.00E+03 1.00E+04

1.00E+03 1.00E+03 1.00E+03 1.00E+03 1.00E+03

31.77845 1.00E-06 1.00E+03 31.77845 1.00E-05 1.00E+03 31.93747 1.00E+02 1.00E+03 32.63141 1.00E+03 1.00E+03 35.95527 1.00E+04 1.00E+03

31.94132 1.00E-06 1.00E+03 31.94132 1.00E-05 1.00E+03 32.06872 1.00E+02 1.00E+03 32.78814 1.00E+03 1.00E+03 35.86462 1.00E+04 1.00E+03

32.08632 32.08632 32.17812 32.99287 35.4408

1.00E-06 1.00E-05 1.00E+02 1.00E+03 1.00E+04

5.00E+03 5.00E+03 5.00E+03 5.00E+03 5.00E+03

70.78043 1.00E-06 5.00E+03 70.78043 1.00E-05 5.00E+03 70.85197 1.00E+02 5.00E+03 71.16747 1.00E+03 5.00E+03 72.7515 1.00E+04 5.00E+03

70.85371 1.00E-06 5.00E+03 70.85371 1.00E-05 5.00E+03 70.91123 1.00E+02 5.00E+03 71.23947 1.00E+03 5.00E+03 72.70675 1.00E+04 5.00E+03

70.91919 70.91919 70.96078 71.33393 72.49862

1.00E-06 1.00E-05 1.00E+02 1.00E+03 1.00E+04

8.00E+03 8.00E+03 8.00E+03 8.00E+03 8.00E+03

89.49787 1.00E-06 8.00E+03 89.49787 1.00E-05 8.00E+03 89.55446 1.00E+02 8.00E+03 89.80428 1.00E+03 8.00E+03 91.06471 1.00E+04 8.00E+03

89.55584 1.00E-06 8.00E+03 89.55584 1.00E-05 8.00E+03 89.60135 1.00E+02 8.00E+03 89.86135 1.00E+03 8.00E+03 91.02896 1.00E+04 8.00E+03

89.60766 89.60766 89.64057 89.93625 90.86281

1.00E-06 1.00E-05 1.00E+02 1.00E+03 1.00E+04

1.00E+04 1.00E+04 1.00E+04 1.00E+04 1.00E+04

100.0493 1.00E-06 1.00E+04 100.0493 1.00E-05 1.00E+04 100.1 1.00E+02 1.00E+04 100.3235 1.00E+03 1.00E+04 101.4534 1.00E+04 1.00E+04

100.1012 1.00E-06 1.00E+04 100.1012 1.00E-05 1.00E+04 100.1419 1.00E+02 1.00E+04 100.3746 1.00E+03 1.00E+04 101.4213 1.00E+04 1.00E+04

100.1476 100.1476 100.177 100.4417 101.2722

6


Figure 1 – Model of a pipe fluid conveying pipe resting on a two-parameter foundation

20

18

γ2=1.0Ε−06

16

γ2=1.0Ε−05 14

γ2=1.0Ε−04 γ2=1.0Ε−03

12

γ2=1.0Ε−02 Cr

V

10

γ2=1.0Ε−01 γ2=1.0Ε00

8

γ2=1.0Ε01 6

4

2

0 1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

1.00E+05

g1 Fig. 2 - Pinned-pinned pipe: Variation of V

Cr with

g1 for various values of

g2

Figure 2 – Pinned-pinned pipe: Variation of VCr with γ1 for various values of γ2

7


40

35

VCr

30

25

γ2=1.0Ε02 γ2=1.0Ε03

20

15

10 1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

1.00E+05

g1

Fig. 3 - Pinned-pinned pipe: Variation of V C r w i t h g 1 for various values of g 2 - Contd.

Figure 3 – Pinned-pinned pipe: Variation of VCr with γ 1 for various values of γ 2 – contd.

105

100

95

VCr

90

85

γ2=5.0Ε03 80

γ2=8.0Ε03 γ2=1.0Ε04

75

70 1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

1.00E+05

γ1 γ

γ

Fig. 4 - Pinned-pinned pipe: Variation of V w i t h 1 for various values of 2 - Contd. Figure 4 – Pinned-pinned pipe: Variation of VCr with γ 1 for various values of γ 2 – contd. Cr

8


120

100

γ1=1.0Ε−06 80

γ1=1.0Ε00 γ1=1.0Ε03

VCr

γ1=1.0Ε04 60

40

20

0 1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

1.00E+05

γ2

Figure 5 – Pinned-pinned Fig. 5 - pipe: Pinned-pinned Influence pipe: Influence of γ2 of γ 2on on the VCr V for for various various values ofvalues of γ 1 γ1 Cr

120

100

γ1=1.0Ε−06

VCr

80

γ2=1.0Ε−06

60

40

20

0 1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

1.00E+05

γ1, γ2 Fig. 6 - Pinned-pinned pipe: Comparison of the effect of γ 1 and the γ 2 on V C r

Figure 6 – Pinned-pinned pipe: Comparison of the effect of γ 1 and γ 2 on VCr

9