Choi 2000 Free Spanning Analysis of Offshore Pipelines

Ocean Engineering 28 (2001) 1325–1338

Free spanning analysis of offshore pipelines H.S. Choi Department of Naval Architecture and Ocean Engineering, Pusan National University, Pusan 609-735, South Korea

Received 18 May 2000; accepted 13 September 2000

Abstract

A rigorous procedure was established on the free span analysis of offshore pipelines. The closed clo sed for form m sol soluti utions ons of the bea beam–c m–colu olumn mn equ equati ation, on, con consid sideri ering ng ten tensio sion n and com compre pressi ssive ve force, were derived for the various possible boundary conditions. The solutions can be used to find the natural frequencies of the free spans using the energy balance concept. The results can be applied to improve the current design codes. The improved procedure will yield more realistic calculations of the allowable free span lengths of offshore pipelines. Some calculations are included to present the sensitivity of the axial forces on the allowable free spanning lengths. © 2001 Published by Elsevier Science Ltd. Keywords: Offshore pipeline; Vortex shedding; Allowable free span length

1. Introd Introduction uction

For a safe operation of offshore gas or oil pipeline during and after installation, the free span lengths should be maintained within the allowable lengths, which are determined deter mined during the desig design n stage stage.. Free spans may be caused by seabe seabed d uneve unevenness nness and change of seabed topology such as scouring or sand wave (Danish Hydraulic Institute, 1997). Once a free span longer than the allowable span length occurs, the free span may suffer the vortex-induced vibration (VIV) and consequently suffer the fatigue damages on the pipe due to the wave and current. The vortex shedding phenomenon results in two kinds of periodic forces on a free span of a pipe. Symmetrical vortices are shed when the flow velocity is low. A pipe will start to oscillate in-line with the flow when the vortex shedding frequency is about one-third of the natural frequency of a pipe span. Lock-in occurs when the vortex shedding frequency is half of the natural frequency. As the flow velocity increases further, the crossflow oscillation begins to occur and the vortex shedding frequency may approach 0029-8018/01/$ - see front matter © 2001 Published by Elsevier Science Ltd. PII: S 0 0 2 9 - 8 0 1 8 ( 0 0 ) 0 0 0 7 1 - 8

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H.S. Choi / Ocean Engineering 28 (2001) 1325–1338

the natural frequency of the pipe span. Amplified responses due to resonance between the vortex shedding frequency and natural frequency of the free span may cause fatigue damage. Thus, the determination of the critical allowable span lengths under the various environments becomes an important part of pipeline design. During the extensive study of vortex-induced vibration, it was found that the current design code is insuf ficient to include all the aspects of design, installation, and operation. The axial force (tension or compression) has a significant effect on determining the critical allowable span lengths. Therefore, this study suggests some improvement on the current design code regarding the vortex-induced vibration and free span analysis.

2. Parameters of vortex shedding analysis

The Strouhal number, S , is a non-dimensional number which relates the vortex shedding frequency, the diameter of the cylindrical pipe, and the velocity of the fl ow: t

f s D S  U

(1)

t

where f s is the vortex shedding frequency, U is the flow velocity normal to the pipe axis, and D is the diameter of the pipe. The Reynolds number, R , is used to determine the range of the vortex shedding: e

R

e

UD v

(2)



where v is the kinematic viscosity of fluid. Vortex shedding is well organized at sub-critical (300 R 3×105) and trans-critical ( R 3.5×106) ranges. At the critical range (3×105 R 3.5×106), vortex shedding is disorganized and vortex-induced motion is insignificant (Blevins, 1990). The stability parameter, K s, determines uniquely the maximum amplitude of vibrations (Sumer and Fredsoe, 1994) and is defined as: e

e

e

2med K s r D2

(3)

where d is the logarithmic decrement of damping, r is the mass density of the surrounding water, and me is the effective mass per unit length, including structural mass, added mass and the mass of any fluid contained within the pipe. Reduced velocity, V r, is used to determine the velocity ranges of the occurrence of VIV. V r is defined as: U V r f n D

(4)

where f n is the natural frequency of the pipe. Symmetric vortices are shed when V r falls between 1.0 and 2.2. When V r exceeds 2.2, vortices are shed alternately. Accord-

H.S. Choi / Ocean Engineering 28 (2001) 1325 – 1338

1327

ing to the Det Norske Veritas (DNV) codes (1981, 1991 and 1998), in-line vortex shedding may occur when 1.0V r3.5. As fl ow velocity increases further, the crossflow oscillation occurs and lock-in happens when V r is about 5.0.

3. Allowable span length by DNV 3.1. Allowable span lengths

Allowable span lengths are governed by the code limitations regarding maximum allowable stresses and by the onset of vortex shedding criteria. The allowable pipe span lengths can be chosen as the lesser lengths found from the span length criteria of the ANSI code: allowable stress (static), and the DNV code: onset of VIV (dynamic). Most of the allowable span lengths are governed by the onset of a VIV. Thus, this study is dealing with only the allowable span due to vortex shedding. 3.2. Calculation of allowable span lengths 3.2.1. In-line motion According to the DNV codes (1981, 1991 and 1998), in-line vortex shedding may occur when 1.0V r3.5 and K s1.8. Once the range of the in-line oscillations has been established, based on the upper and lower bound values of V r, the corresponding pipe spans can be determined. Fig. A.3 in the DNV code (1981) is used to obtain the lower bound value of V r based on K s for the onset of in-line motion. The upper bound value of V r is set at 3.5. The natural frequency of the pipe can then be calculated using the aforementioned relationship. The allowable pipe spans (lower bound values) are then computed by solving for the span length L, the following formula:

   

L

EI me

0.25

CV r D 2p U

0.5

(5)

where E is the modulus of elasticity, I is the moment of inertia, and C is the end boundary coef ficient. 3.2.2. Cross-flow motion For cross-flow oscillations, vortex shedding may occur when K s16 and V r falls within the range as determined in Fig. A.5 in the DNV code. The allowable pipe spans are then determined using the same procedure as Eq. (5) and cross-flow V r. 3.3. Maximum amplitude of vibration 3.3.1. In-line motion Once the stability parameter is determined, the maximum amplitude of motion can be directly obtained from Fig. A.4 in the DNV code (1981).

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3.3.2. Cross- fl ow motion The maximum amplitude of cross-flow motion can be determined by Fig. A.6 in the DNV code (1981). The mode shape parameters are required to get the maximum amplitude of cross-flow motion. 3.3.3. Mode shapes Free transverse vibration of the Bernoulli–Euler beam is governed by:

 

d2 d2 y EI 2 d x 2 d x

d2 y  r A 2 0 dt

(6)

where y is the displacement of the beam and r A is the mass per unit length of the beam. Assuming a harmonic motion given by y( x , t )Y ( x ) cos(w t a )

(7)

and substituting Eq. (7) into Eq. (6), one gets the eigenvalue equation: d4Y EI 4  r Aw 2Y 0 d x

(8)

On substituting

b

4

r Aw 2



(9)

EI

the fourth-order differential equation is obtained for a vibration of a uniform beam: d4Y 4  b Y 0 d x 4

(10)

The general solution of Eq. (10) may be written in the form Y ( x )C 1 sinh b x C 2 cosh b x C 3 sin b x C 4 cos b x

(11)

The four constants and b can be obtained using boundary conditions: Fixed end boundary condition: Y 0,

dY 0 d x

(12)

Pinned end boundary condition: d2Y Y 0, 2 0 d x

(13)

Free end boundary condition: d2Y d3Y 0, 0 d x 2 d x 3

(14)


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The mode shapes for each boundary condition are as follows: Fixed–free boundary condition: Y ( x )C {cosh b x cos b x K r(sinh b x sin b x )}

(15)

where (cosh b L+cos b L) K r (sinh b L+sin b L) Pinned–pinned boundary condition: Y ( x )C sin

r p x

L

(16)

Fixed–pinned boundary condition: Y ( x )C {cosh b x cos b x K r(sinh b x sin b x )}

(17)

where (cosh b L−cos b L) K r (sinh b L−sin b L) Fixed–fixed boundary condition: the equation of motion is the same as Eq. (17), but the different boundary conditions result in a different form of the mode shape.

3.3.4. Mode shape parameter, g The mode shape parameter, g , is defined as:



1/2

L



Y 2( x ) d x

g Y max L

 0

 



0

Y 4( x ) d x

(18)



where Y max is the maximum value of the mode shape. Once the mode shape parameter is obtained from Eq. (18), the maximum amplitude of cross-flow motion can be obtained from the DNV code (1981). 3.4. Results by DNV code approach

A pipe diameter of 324 mm (12 in.) and wall thickness of 16 mm (0.625 in.) was used for the calculations. Table 1 shows mode shape factors for various mode shapes and boundary conditions. Table 2 shows vortex shedding frequencies for in-line and cross-flow directions as the fl ow velocity increases. Figs. 1 and 2 show allowable span

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Table 1 Mode shape factors for various boundary conditions B.C.

Mode number

Mode shape factor

Fixed–free

1 2 3 4

1.3050 1.4987 1.5371 1.5634

Pinned–pinned

1 2 3 4

1.1547 1.1547 1.1547 1.1547

Fixed–pinned

1 2 3 4

1.1613 1.1934 1.2057 1.2124

Fixed–fixed

1 2 3 4

1.1670 1.1613 1.1824 1.1934

Table 2 Vortex shedding frequencies (Hz) Current speed (m/s)

0.3

0.6

0.9

1.2

1.5

In-line Cross-flow

0.480 0.190

0.961 0.390

1.441 0.600

1.922 0.820

2.402 1.046

lengths and Table 3 presents the amplitudes of motion for various boundary conditions. As mentioned in Section 2, the maximum amplitude of vortex-induced vibration is uniquely determined by the stability parameter. This fact indicates that the maximum amplitude is not controlled by the fluid speed. If fluid speed increases, the allowable span length will be reduced and the maximum amplitude remains the same.

4. Effect of axial force on span analysis 4.1. Effect of axial force

The load on the pipeline during the operation is not the same as the loading condition during installation. During installation, the pipeline may have residual tension


Fig. 1.

Fig. 2.

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Allowable span lengths by DNV (in-line).

Allowable span lengths by DNV (cross-flow).

due to the lay-barge method. During operation, the pipeline may have operational load due to the operational pressure and temperature. This operational loading may cause very high compressive force in part of the pipeline. Therefore, the effect of axial force is studied in this section and applied to the DNV code to calculate the modified allowable span lengths. 4.2. Governing equation and solutions

The axial force may greatly alter the shape of the elastic deflection and its influence on the equilibrium conditions cannot be neglected. When a pipeline is subjected to a transverse downward load of w( x ) per unit length, and an axial force, N x , the governing equation for a beam under an axial force is:

 

d2 d2 y EI 2 d x 2 d x

d2 y  N x 2w( x ) d x

(19)

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Table 3 Amplitude of response for various boundary conditions (m) B.C.

Fix–Fix

Pin–Pin

Fix–Pin

Fix–Free

In-line Cross-flow

0.025 0.429

0.025 0.378

0.025 0.381

0.025 0.384

The positive and negative N x respectively denote compression and tension. The general solutions of Eq. (19) are: w( x ) x 2 For N x 0: yC 1 sinh l x C 2 cosh l x C 3 x C 4 2 N x w( x ) x 4 For N x 0: yC 1 x C 2 x C 3 x C 4 24 EI 3

2

w( x ) x 2 For N x 0: yC 1 sin l x C 2 cos l x C 3 x C 4 2 N x

(20)

(21) (22)

where l=√| N x |/ EI The coef ficients C 1, C 2, C 3, and C 4 were determined from various boundary conditions and are presented in Table 4. The boundary conditions are provided by the end conditions: Fix end condition: Y 0,

dY 0 d x

(23)

Pinned end condition: d2Y Y 0, 2 0 d x

(24)

Free end condition: d2Y d3Y d y 0, EI N   0 x d x 2 d x 3 d x

(25)

When N x approaches zero, the deflections of Eqs. (20) and (22) converge to Eq. (21). If N x is positive and l approaches a critical value, then deflections of the beams increase indefinitely and Euler’s buckling load for the beams is yielded.

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2

4

C

2

C

0



1

3

C

2

C

C





1

C

C

l

l

0



2

0

L x w N − 2



I L E w 4 2

3

2

C

C





L x w N − 2

2

0

1

2

C

−

I L E w 4

2

) 1 − L L l l s n o i s c x L N l 2 ( l w −

x

N w 2 − l

x

0

N w 2 l

n o i t a u q e n m u l o c

L N w l x

1

C x

N

. C . B



e e r f – x i F

L I w E − 6

L N w l

0

+

x

) 1 L − l L h l n h i s s x o N c ( 2 w l



n i p – n i P

L I w E 2 − 1

0

) 1 L − L l n l i s s o x c N ( 2 w l −

+



1

2

C

0

1



1

C

C

C

C

l

l

l



) L 2 ) + L 2 ) l L h n l i ( · s L − l L h l · n L i s l + L h s l o c h ( n x i N s 2 2 l − 2 ( w

0

−

) 2 + ) L ) l L l h ( n · i L s l − L h s l · o L c + l L h l s o h ( c s x o N c 2 2 l − 2 ( w



n i p – x i F



) L l 2 ) − L 2 ) l L i n l s ( · − L L l l

I L E w 6 1

2

2

–

m a e b f o s n o i t u l o s n i s t n e 4 i c fi e f l b e a o T C

C



l

l

) 1 + L L l l h h s n o i s c x L N l 2 ( l w

2

C

L I w E 5 8 − 4

0

· n L i s l + s L o l c ( n x i N s 2 2 l − 2 ( w

−

) 2 ) − L 2 ) l L n l i ( · s − L L l · s l o L c l + s L o l c x s ( o N c 2 l 2 ( 2 w

+



0

) L )

l L + l

L h s o L c l 2 h − s L o l c · + L L l l h h i n n s i s + 2 2 ( − x ( N L l w 2

I L E w 4 2

) ) L L + s L o c l · L 2 l + L s l o · c L + l L n l i s n + i s 2 − 2 ( − x ( N L l w 2

L I w E 2 − 1

) L ) l 2 − s L o c l · L 2 l + L n l i · s L + L l n l i s s o + c 2 − 2 ( ( x L N w l 2

l l

l ·

−

) ) 2 L + l L h s l · o L c l 2 h − n L i s l · + L L l l h h n s i o s c + 2 2 ( − x ( N L l w 2



x

fi –

x i F



2

0

+

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4.3. Natural frequencies by energy method

To obtain the natural frequency of a beam under an axial load, the Rayleigh method can be applied. This method assumes that the maximum potential energy of the system is equal to its maximum kinetic energy. The fundamental frequency can be obtained by equating the potential and kinetic energy (Choi and Haun, 1994). n

gmi yi 1

i



w 2n

(26)

n

m y i

2 i

1

i



where yi are solutions obtained from Eqs. (20)–(22). 4.4. Lloyd ’s approximate formula

w n

  

C EI

L2

N x

0.5

1 PE

me

(27)

Eqs. (26) and (27) give the same natural frequency of beams if axial load becomes zero. The natural frequency equation of beams without axial force is as follows:



w nC

EI

me L4

(28)

Fig. 3 shows the axial load effect on the natural frequencies of the pipe for various boundary conditions. The natural frequencies significantly change with the axial load. The natural frequencies given by Lloyd are compared with exact solutions and

Fig. 3.

Effect of the axial force on the natural frequency.


Fig. 4.

1335

Natural frequencies for pinned–pinned boundary conditions (B.C).

are presented in Figs. 4–7. The approximate formula given by Lloyd is in good agreement with the exact solution, except for the free–fixed condition. 4.5. Allowable span lengths

The exact solutions of the beam equation under the axial load were used to calculate the natural frequency with the Rayleigh method. The natural frequencies obtained by Eq. (26) can be used with the current DNV code instead of Eq. (28). Thus, more realistic allowable span lengths can be obtained. For the improved method to calculate the allowable span lengths under the axial load, an iterative numerical method can be applied. Figs. 8 and 9 show allowable span lengths for various load factors

Fig. 5.

Natural frequencies for fixed–pinned B.C.

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Fig. 6.

Natural frequencies for fixed–fixed B.C.

Fig. 7.

Natural frequencies for fixed–free B.C.

Fig. 8.

Allowable span lengths with axial force (in-line).


Fig. 9.

1337

Allowable span lengths with axial force (cross-flow).

and boundary conditions. The allowable span lengths increase with an increment in tension, and decrease with an increment in compression.

5. Conclusions

1. If during the operation and installation of offshore pipelines high axial force is included, its effect could not be neglected. The results of the study show axial load effect on the natural frequencies and allowable span lengths of the pipeline for various boundary conditions. The natural frequencies significantly change with respect to the axial load. 2. The exact solutions of the beam–column equation are derived for various boundary conditions. The solutions are used to find natural frequencies with the energy balance method. This method was compared with Lloyd’s approximate method. The results of the two methods are in good agreement. 3. An improvement was made to the vortex shedding analysis of offshore pipelines. It can be applied to the current design code and will result in more accurate calculation of allowable free span lengths of offshore pipelines.

References Blevins, R.D., 1990. Flow-Induced Vibration. Van Nostrand Reinhold Co, New York. Choi, H.S., Haun, R.D., 1994. The effect of residual tension and free span-induced moments on vortex shedding of deep water pipelines. In: Fourth International Offshore and Polar Engineering Conference, Osaka, Japan, vol. 2, pp. 102 –109. Danish Hydraulic Institute (DHI), 1997. Pipeline Free Span Design. Design Guideline, vol. 1 (project PR-170-9522). Det Norske Veritas (DNV), 1981. Rules for Submarine Pipeline Systems (Appendix A).

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Det Norske Veritas (DNV), 1991. Environmental Condition and Environmental Loads. Classification Notes No. 30.5. Det Norske Veritas (DNV), 1998. Free Spanning Pipelines. Guidelines No. 14. Sumer, B.M., Fredsoe, J., 1994. A review on vibrations of marine pipelines. In: Fourth International Offshore and Polar Engineering Conference, Osaka, Japan, vol. 2, pp. 62 –71.

Choi 2000 Free Spanning Analysis of Offshore Pipelines

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