6. Pipeline vibration limiting

Pipeline vibration limiting regulations can be divided into the following categories:


The requirements [design documentation] state that "the maximum allowable amplitude of vibrations of the process pipelines is 0.2 mm at vibration frequency of max 40 Hz" [2].

Offshore pipeline specifications do not provide either limitations for the pressure pulsations or vibration limitations.

Low-frequency vibrations of the pipelines under principal modes, when such vibrations are close to be harmonic, can be easily evaluated on the basis of the amplitude of vibration displacement since in this case they are proportional to the stresses induced in the pipelines and can be regarded as a strength factor of the pipelines.

We get the following expression for k-form of the vibrations using formula [5], for the root mean square value of vibrations:

$$\sqrt{\sigma\_k^2(z)} = \left[ \int\_0^\infty \mathbb{C}\_k(\omega) \frac{\mathfrak{d}\_{\mathbb{Q}\mathbb{Q}}(\omega) d\omega}{\sqrt{\left(\omega\_0^2 - \omega^2\right)^2 + \left(2\beta\alpha\omega\_0\right)^2}} \right] \frac{ED}{2}$$

In the event of random vibrations, the combined stress in the pipeline is the following:

$$
\overline{\sigma^2}(z) = \sum\_{k=1}^N \overline{\sigma\_k^2}(z).
$$

However, in the regulatory documents for offshore pipelines, there are not only restrictions on pressure pulsations but also restrictions on vibrations. Low-frequency oscillations of pipelines along lower forms, when these oscillations are close to harmonic, can be conveniently estimated from the amplitude values of the vibrational displacement because in this case they are proportional to the stresses arising in the pipelines and are indicators of the strength of the pipelines.

Vibration velocity can be written as:

$$V\_l = \int\_0^T a(t)dt = \int\_0^T B\_0 \sin\omega t dt = B\_0 n / 2\pi f\_K \tag{20}$$

Development of the standardized vibration limits is complicated due to a wide variety of the

Vibration Strength of Pipelines

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However, for determination of the pipeline system reliability exposed to vibrations, it is required to consider the effect of vibrations on failures and malfunctions on the basis of certain assumptions regarding damage accumulation in the structures and failure occurrence. If stresses in the elastic elements of pipelines (supports) are considered failure criteria, the vibration strength of the supports shall be evaluated using calculation models of support structures. The private oilextracting companies apply burying of pipelines. It is just concealment of a problem. In the

During the analysis of the first fundamental forms of vibrations which lie within 20 Hz,

The following stages are included into the obligatory vibration tests of the pipeline: study of the operating conditions of the system and analysis of the dynamic loads acting on the pipeline; determination of the potential failure patterns; and selection of failure occurrence criteria due

According to the regulations of the Ministry of Gas Industry, emergency vibration level [2] is measured using vibration velocity Ve = 18 mm/s, and the warning vibration level is estimated

During the pipeline vibration analysis, it is necessary to know rigidity characteristics of the system components. The rigidity of pipe of permanent round section is characterized by the

Up to the present moment, frequency ranges have not been yet defined for the offshore pipelines where one or another vibration parameter shall be used for vibration limitation purposes.

Vmax ¼ vbasekη Vmax:

Amplitude of vibration stresses at different frequencies is determined during calculation of the

The main criterion of the pipeline vibration strength is detuning of natural frequencies fj from

Let us determine safe vibration velocity for the pipeline kinematically excited on the movable base plate. Amplitudes of vibration voltage at different frequencies are determined by the results of calculation of forced oscillations of the pipeline. The way to ensure the vibration

Vibrations of marine pipelines with a protective coating to decrease the interaction of the pipeline with the coating result in cracking in the coating of corrosion and damage to the coating.

discrete frequencies of the excitatory loads fip, defined as described in 2.2 [6].

resistance of the pipeline is to detune the natural frequencies fj of the structure [7].

.

requirements to the characteristics of the vibration capacity of various equipments.

seismic phenomena, vibrations arise in the thickness of the Earth.

vibration displacement is frequently used as a test parameter.

to vibrations.

by the exceedance Ve = 41 mm/s [9, 10].

forced vibrations of the pipeline [8].

following parameters: EI is bending rigidity, N�m2

The allowable amplitude of vibrations is defined as follows:

where V<sup>l</sup> is vibration velocity (mm/s); a is vibration acceleration (mm/s<sup>2</sup> ) with amplitude В0; fk is frequency of multiplicity factor k.

During the evaluation of the vibration strength, the maximum amplitude of equivalent vibration stresses shall be determined for each representative section of the pipeline. This amplitude is obtained as a result of various modal superpositions. Dimension of the transfer function depends on the type of disturbance and response against which the transfer function is determined.

When exposed to random vibrations, root-mean-square value of the maximum stress in the pipeline can be found using the transfer function for the maximum stress with respect to acceleration of the pipeline supports:

$$|H\_{\sigma}(\omega, z)| = \frac{\mathsf{C}\_{k}(\omega)\frac{\mathsf{ED}}{2}y\_{k}''(z)}{\sqrt{\left(\omega\_{0}^{2} - \omega^{2}\right)^{2} + \left(2\beta\omega\omega\_{0}\right)^{2}}}\tag{21}$$

where D is outside diameter of pipe.

Pipeline response to broadband random vibrations can be defined as a combined effect of several narrowband random vibrations. The narrowband vibrations of the pipelines occur as a response to the broadband excitation under low damping. The mean frequency of the narrowband vibrations can be calculated from the Rice's formula:

$$
\omega\_0^2 = \frac{\int\_{-\infty}^{\infty} \omega^2 \Phi\_{YY}(\omega) d\omega}{\int\_{-\infty}^{\infty} \Phi\_{YY}(\omega) d\omega} = \frac{R\_y^{\cdot}(0)}{R\_y(0)} = \frac{\sigma\_{\dot{y}}^2}{\sigma\_y^2} \tag{22}
$$

Root mean square value of the pipeline movement to be subject to vibrations can be calculated from the following formula:

$$
\sigma = \left[ \int\_{f\_1}^{f\_2} \eta\_f^2 \Phi(f) df \right]^{1/2}, \quad \text{here } \sigma = \sqrt{\sum\_{i=1}^N \eta\_{f\_i}^2 \Phi(f\_i) \Delta f\_i}. \tag{23}
$$

where η<sup>f</sup> is a dynamic response factor-relation between displacement amplitude of the anchor points for the pipeline and relative displacement amplitude of the pipeline sections at specified frequency; Ф(f) is a spectral density of the disturbed random vibration in the frequency band f<sup>1</sup> and f2; Δfi is an interval of the frequency band segmentation f1, f2; N is a number of intervals of the frequency band segmentation.

Development of the standardized vibration limits is complicated due to a wide variety of the requirements to the characteristics of the vibration capacity of various equipments.

However, for determination of the pipeline system reliability exposed to vibrations, it is required to consider the effect of vibrations on failures and malfunctions on the basis of certain assumptions regarding damage accumulation in the structures and failure occurrence. If stresses in the elastic elements of pipelines (supports) are considered failure criteria, the vibration strength of the supports shall be evaluated using calculation models of support structures. The private oilextracting companies apply burying of pipelines. It is just concealment of a problem. In the seismic phenomena, vibrations arise in the thickness of the Earth.

During the analysis of the first fundamental forms of vibrations which lie within 20 Hz, vibration displacement is frequently used as a test parameter.

The following stages are included into the obligatory vibration tests of the pipeline: study of the operating conditions of the system and analysis of the dynamic loads acting on the pipeline; determination of the potential failure patterns; and selection of failure occurrence criteria due to vibrations.

According to the regulations of the Ministry of Gas Industry, emergency vibration level [2] is measured using vibration velocity Ve = 18 mm/s, and the warning vibration level is estimated by the exceedance Ve = 41 mm/s [9, 10].

During the pipeline vibration analysis, it is necessary to know rigidity characteristics of the system components. The rigidity of pipe of permanent round section is characterized by the following parameters: EI is bending rigidity, N�m2 .

Up to the present moment, frequency ranges have not been yet defined for the offshore pipelines where one or another vibration parameter shall be used for vibration limitation purposes.

The allowable amplitude of vibrations is defined as follows:

Vibration velocity can be written as:

is frequency of multiplicity factor k.

acceleration of the pipeline supports:

where D is outside diameter of pipe.

from the following formula:

σ ¼

of the frequency band segmentation.

ð<sup>f</sup> <sup>2</sup> f 1 η2 <sup>f</sup> Φð Þf df " #<sup>1</sup>=<sup>2</sup>

band vibrations can be calculated from the Rice's formula:

ω2 <sup>0</sup> ¼ Ð ∞

determined.

30 System of System Failures

Vl ¼ ðT 0

a tð Þdt ¼

where V<sup>l</sup> is vibration velocity (mm/s); a is vibration acceleration (mm/s<sup>2</sup>

ðT 0

During the evaluation of the vibration strength, the maximum amplitude of equivalent vibration stresses shall be determined for each representative section of the pipeline. This amplitude is obtained as a result of various modal superpositions. Dimension of the transfer function depends on the type of disturbance and response against which the transfer function is

When exposed to random vibrations, root-mean-square value of the maximum stress in the pipeline can be found using the transfer function for the maximum stress with respect to

> <sup>2</sup> y<sup>00</sup> <sup>k</sup> ð Þ<sup>z</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

> > <sup>y</sup>ð Þ0 Ryð Þ<sup>0</sup> <sup>¼</sup>

> > > X N

i¼1 η2 f i Φ f <sup>i</sup> � �Δ<sup>f</sup> <sup>i</sup> : vuut (23)

σ2 y\_ σ2 y

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>0</sup> � <sup>ω</sup><sup>2</sup> � �<sup>2</sup> <sup>þ</sup> <sup>2</sup>βωω<sup>0</sup>

j j <sup>H</sup>σð Þ <sup>ω</sup>; <sup>z</sup> <sup>¼</sup> Ckð Þ <sup>ω</sup> ED

ω2

Pipeline response to broadband random vibrations can be defined as a combined effect of several narrowband random vibrations. The narrowband vibrations of the pipelines occur as a response to the broadband excitation under low damping. The mean frequency of the narrow-

�<sup>∞</sup> <sup>ω</sup><sup>2</sup>ΦYYð Þ <sup>ω</sup> <sup>d</sup><sup>ω</sup>

�<sup>∞</sup> <sup>Φ</sup>YYð Þ <sup>ω</sup> <sup>d</sup><sup>ω</sup> <sup>¼</sup> <sup>R</sup>}

Root mean square value of the pipeline movement to be subject to vibrations can be calculated

, here σ ¼

where η<sup>f</sup> is a dynamic response factor-relation between displacement amplitude of the anchor points for the pipeline and relative displacement amplitude of the pipeline sections at specified frequency; Ф(f) is a spectral density of the disturbed random vibration in the frequency band f<sup>1</sup> and f2; Δfi is an interval of the frequency band segmentation f1, f2; N is a number of intervals

Ð ∞

B<sup>0</sup> sin ωtdt ¼ B0n=2πf <sup>K</sup> (20)

� �<sup>2</sup> <sup>q</sup> (21)

) with amplitude В0; fk

(22)

$$\mathbf{V}\_{\text{max}} = \upsilon\_{\text{base}} k \eta \text{ V}\_{\text{max}}$$

Amplitude of vibration stresses at different frequencies is determined during calculation of the forced vibrations of the pipeline [8].

The main criterion of the pipeline vibration strength is detuning of natural frequencies fj from discrete frequencies of the excitatory loads fip, defined as described in 2.2 [6].

Let us determine safe vibration velocity for the pipeline kinematically excited on the movable base plate. Amplitudes of vibration voltage at different frequencies are determined by the results of calculation of forced oscillations of the pipeline. The way to ensure the vibration resistance of the pipeline is to detune the natural frequencies fj of the structure [7].

Vibrations of marine pipelines with a protective coating to decrease the interaction of the pipeline with the coating result in cracking in the coating of corrosion and damage to the coating.

#### Example.

#### Vibration strength of pipelines.

Low-frequency vibrations of the pipelines under principal modes, when such vibrations are close to be harmonic, can be easily evaluated on the basis of the amplitude of vibration displacement because in this case they are proportional to the stresses induced in the pipelines and can be regarded as a strength factor of the pipelines.

σ�<sup>1</sup> value can be defined either using reference data or Manson formula [4]:

<sup>σ</sup>�<sup>1</sup> <sup>¼</sup> <sup>1</sup>, <sup>75</sup>σB=N<sup>0</sup>:12, here <sup>N</sup> is a number of loading cycles.

Let us consider the following example of calculating the allowable stress amplitudes in the pipeline wall.

Researches were made for vibration speed (response characteristic: root mean square value of Vmax = 0.0103 cm/s) and pressure pulsation amplitude of ΔP = 0.5 MPa for the landfall section of the offshore pipeline with rated pressure of 17.5 MPa. Outside diameter of the pipeline is D = 406 mm, wall thickness is 17.5 mm. Material grade is X52 (σ<sup>в</sup> = 455 MPa, σ�<sup>1</sup> ≈ 916 MPa, E = 0.20457�106MPa, <sup>σ</sup><sup>e</sup> = 358 MPa). Pipe section modulus is W = 0.00199 m3 .

strength of the pipeline steel (see Table 1), stress amplitude σ<sup>Δ</sup><sup>t</sup> ¼ 38.322 MPa in the pipeline

Let us review the seismogram of seismic intensity 6 to MSK-64 scale for the vibration strength analysis of the landfall section of the offshore pipeline. The seismic impact is characterized by the following parameters: maximum acceleration amax = 0.94485 cm/s<sup>2</sup> at t = 0.12 s (Figure 2).

> <sup>¼</sup> <sup>2</sup><sup>π</sup> N∙Δt

Periodogram results can be interpreted as dispersed data at frequencies given in Table 2 and

Ser.no. Frequency Period Periodogram Density Spectral density in hamming window

ð Þ rad=s

Vibration Strength of Pipelines

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wall. Fatigue strength factor of the pipeline shall be at least п = 2.0 [3].

Figure 2. Accelerogram recording used during calculations.

We get the line spectrum of the signal consisting of the harmonics:

 0.00000 0.0 1488.55 0.035714 0.02174 46.0 2836.21 1828.18 0.241071 0.04348 23.0 1695.297 1839.68 0. 446429 0.06522 15.3333 1456.8 1528.67 0.241071 0.08696 11.5 1341.92 1355.41 0.035714

 0.1087 9.2 1255.43 1258.18 0.13044 7.66667 1174.63 1174.32 0.15217 6.67143 1092.56 1091.35 0.17391 5.75 1007.15 1005.84 0.19585 5.11111 918.245 917.192 0.21739 4.6 826.556 825. 941

where Тmin is a minimum period.

Figure 3.

<sup>Δ</sup><sup>ω</sup> <sup>¼</sup> <sup>2</sup><sup>π</sup> Tmin

Kt≔1

σBB�β�k=806.08

$$\text{norm1} \cdot \left(1 + \frac{\sigma \text{BB}}{\sigma \text{B}} \cdot \frac{1 + \text{rq}}{1 - \text{rq}}\right) = 64.214$$

If length l of the pipeline section is 30 m, displacement y0 is determined by the formula (7) and is equal to 0.026 m, considering the formula (9) vmax = 0.035 m

If pressure pulsation ΔР is 0.5 MPa and work pressure is 15.7 MPa (β = 0.8), stress variation on the pipeline wall is 6.386 MPa. Stress ratio (17) qr is 0.938. Stress amplitude in the pipeline wall is equal to σ<sup>Δ</sup><sup>q</sup> ¼ 12.553 MPa considering steel tensile strength and fatigue strength (refer to Table 1).

Temperature fluctuations Δt = 5� result in stress variation of 12.36 MPa in the pipeline wall and are equal to rt = 0.817 as shown in the formula (19). Allowing for tensile strength and fatigue


Table 1. The main physical characteristics of steel pipe [2].

Figure 2. Accelerogram recording used during calculations.

Example.

32 System of System Failures

pipeline wall.

σBB�β�k=806.08

Table 1).

Vibration strength of pipelines.

and can be regarded as a strength factor of the pipelines.

<sup>σ</sup>�<sup>1</sup> <sup>¼</sup> <sup>1</sup>, <sup>75</sup>σB=N<sup>0</sup>:12, here <sup>N</sup> is a number of loading cycles.

σ�<sup>1</sup> value can be defined either using reference data or Manson formula [4]:

0.20457�106MPa, <sup>σ</sup><sup>e</sup> = 358 MPa). Pipe section modulus is W = 0.00199 m3

nm1 � 1 þ

is equal to 0.026 m, considering the formula (9) vmax = 0.035 m

Name of the characteristics Steel

Table 1. The main physical characteristics of steel pipe [2].

Low-frequency vibrations of the pipelines under principal modes, when such vibrations are close to be harmonic, can be easily evaluated on the basis of the amplitude of vibration displacement because in this case they are proportional to the stresses induced in the pipelines

Let us consider the following example of calculating the allowable stress amplitudes in the

Researches were made for vibration speed (response characteristic: root mean square value of Vmax = 0.0103 cm/s) and pressure pulsation amplitude of ΔP = 0.5 MPa for the landfall section of the offshore pipeline with rated pressure of 17.5 MPa. Outside diameter of the pipeline is D = 406 mm, wall thickness is 17.5 mm. Material grade is X52 (σ<sup>в</sup> = 455 MPa, σ�<sup>1</sup> ≈ 916 MPa, E =

Kt≔1

1 þ rq 1 � rq

¼ 64:214

σBB σB �

If length l of the pipeline section is 30 m, displacement y0 is determined by the formula (7) and

If pressure pulsation ΔР is 0.5 MPa and work pressure is 15.7 MPa (β = 0.8), stress variation on the pipeline wall is 6.386 MPa. Stress ratio (17) qr is 0.938. Stress amplitude in the pipeline wall is equal to σ<sup>Δ</sup><sup>q</sup> ¼ 12.553 MPa considering steel tensile strength and fatigue strength (refer to

Temperature fluctuations Δt = 5� result in stress variation of 12.36 MPa in the pipeline wall and are equal to rt = 0.817 as shown in the formula (19). Allowing for tensile strength and fatigue

Yield strength, МPа 358 448 Strength limit Rm, МPа 455 530 Permissible stresses in metal pipes σ, МPа 255.6 292.8

.

Х52∅406 Х65∅711

strength of the pipeline steel (see Table 1), stress amplitude σ<sup>Δ</sup><sup>t</sup> ¼ 38.322 MPa in the pipeline wall. Fatigue strength factor of the pipeline shall be at least п = 2.0 [3].

Let us review the seismogram of seismic intensity 6 to MSK-64 scale for the vibration strength analysis of the landfall section of the offshore pipeline. The seismic impact is characterized by the following parameters: maximum acceleration amax = 0.94485 cm/s<sup>2</sup> at t = 0.12 s (Figure 2).

We get the line spectrum of the signal consisting of the harmonics:

$$
\Delta w = \frac{2\pi}{T\_{min}} = \frac{2\pi}{N \cdot \Delta t} \left(\text{rad/s}\right),
$$

where Тmin is a minimum period.

Periodogram results can be interpreted as dispersed data at frequencies given in Table 2 and Figure 3.



Mean mathematic amplitude Вк over a period of observation Т shall be written as:

<sup>B</sup><sup>0</sup> <sup>¼</sup> <sup>1</sup> T ðT 0

where BKð Þt is a behavior of k-х components of vibration spectrum.

a tð Þdt ¼

where V<sup>l</sup> is vibration velocity (mm/s); a is vibration acceleration (mm/s<sup>2</sup>

ðT 0

According to the standards of the Ministry of Gas Industry [6], an emergency vibration level is equal to the vibration speed Ve = 18 mm/s, and an alarm vibration level exceeds Ve = 41 mm/s. For pipeline sections more than 0.5 m, the vibration displacement span is restricted to 0.5 mm (refer to the standards of the National Compressor Engineering Association "Souzcompre-

Based on the allowable strength from [2], the allowable amplitude of vibrations is calculated for the pipeline of 408 mm diameter and 17.5 mm wall thickness, and allowable stress is 255.6 MPa. The effective span length of the pipeline section under review is 30 m. Amplitude of forced vibrations Y0 is 0.061 m. Safe vibration speed of the pipeline, which is excited kine-

Offshore pipelines can be regarded as new constructions in Russia. Industry-Specific Construc-

None of the private oil companies is interested in studying vibration strength of the pipelines; they make references to the existing standards. Offshore pipeline vibration analysis is very important and requires government involvement. Such researches should be performed by the

Private companies use a pipe burial method. The pipelines can be buried; however, it is not a

Vibrations of marine pipelines with a protective coating to decrease the interaction of the

way to solve the problem. Seismic impacts come from under the ground.

Vl ¼ ðT 0

Vibration velocity can be written as:

is frequency of multiplicity factor k.

matically on the moving platform, is 0.035 m/s.

tion Standard specifies dynamic vibrations of the pipeline.

pipeline with the coating result in cracking in the coating.

ssmach") [2].

7. Conclusion

large scientific institutes.

BKð Þt dt, (24)

B0sinωtdt ¼ B0n=2πf <sup>K</sup> (25)

) with amplitude В0; fk

Vibration Strength of Pipelines

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Table 2. Vibration impact periodogram.

Figure 3. Vibration impact periodogram.

Mean mathematic amplitude Вк over a period of observation Т shall be written as:

$$B\_0 = \frac{1}{T} \int\_0^T B\_K(t)dt,\tag{24}$$

where BKð Þt is a behavior of k-х components of vibration spectrum.

Vibration velocity can be written as:

Ser.no. Frequency Period Periodogram Density Spectral density in hamming window

 0.23913 4.18182 733. 242 733.16 0.26087 3.83333 639.703 640.204 0.28261 3.53846 547.464 548.566 0.30435 3. 28571 458.096 459.796 0.32609 3.06667 373.16 375.439 0.34783 2.875 294.168 296.99 0.36957 2.70588 222.536 225.855 0.3913 2.55556 159.563 163.32 0.41304 2.42105 106.391 110.52 0.43478 2.3 63.992 68.417 0.45652 2.19048 33.139 37.781 0.47826 2.09091 14.398 19.172

Table 2. Vibration impact periodogram.

34 System of System Failures

Figure 3. Vibration impact periodogram.

$$V\_{l} = \int\_{0}^{T} a(t)dt = \int\_{0}^{T} B\_{0} \sin \omega t dt = B\_{0} n / 2\pi f\_{K} \tag{25}$$

where V<sup>l</sup> is vibration velocity (mm/s); a is vibration acceleration (mm/s<sup>2</sup> ) with amplitude В0; fk is frequency of multiplicity factor k.

According to the standards of the Ministry of Gas Industry [6], an emergency vibration level is equal to the vibration speed Ve = 18 mm/s, and an alarm vibration level exceeds Ve = 41 mm/s. For pipeline sections more than 0.5 m, the vibration displacement span is restricted to 0.5 mm (refer to the standards of the National Compressor Engineering Association "Souzcompressmach") [2].

Based on the allowable strength from [2], the allowable amplitude of vibrations is calculated for the pipeline of 408 mm diameter and 17.5 mm wall thickness, and allowable stress is 255.6 MPa. The effective span length of the pipeline section under review is 30 m. Amplitude of forced vibrations Y0 is 0.061 m. Safe vibration speed of the pipeline, which is excited kinematically on the moving platform, is 0.035 m/s.

### 7. Conclusion

Offshore pipelines can be regarded as new constructions in Russia. Industry-Specific Construction Standard specifies dynamic vibrations of the pipeline.

None of the private oil companies is interested in studying vibration strength of the pipelines; they make references to the existing standards. Offshore pipeline vibration analysis is very important and requires government involvement. Such researches should be performed by the large scientific institutes.

Private companies use a pipe burial method. The pipelines can be buried; however, it is not a way to solve the problem. Seismic impacts come from under the ground.

Vibrations of marine pipelines with a protective coating to decrease the interaction of the pipeline with the coating result in cracking in the coating.
