3. Bending vibrations of the pipelines

Under bending vibrations of the pipelines, when distribution of stresses and vibration velocities is significantly different for various fixing conditions, factor с shall be determined individually for each case [3].

$$\frac{d^2y}{dx^2} + \frac{m\omega^2}{T}y = 0\tag{3}$$

Form of the elastic curve of the pipeline is expressed by a sine wave.

$$y(z) = Y\_0 \sin \frac{i\pi z}{L}, y(z) = Y\_0 \overline{y}(z) \tag{4}$$

where Y0 is an amplitude of vibrations.

Bending moment in the random location on the pipeline is equal to:

Figure 1. Force excitation of vibrations. (а) Excitation through an elastic support; (b) kinematic excitation of vibrations.

$$M(z) = EIy''(z) = -EI\frac{\text{i}^2 \pi^2}{L^2} Y\_0 \sin\frac{i\pi z}{L} \tag{5}$$

where I is a moment of inertia.

When calculations are made in regard of hazard assessment of the pipeline vibrations, criteria shall be used to determine the vibration behavior of the pipelines. Vibrations of the pipelines caused by external effects such as impact and earthquake can be described by a general

For example, in case of vibrations of the base plate to which a pipeline support is fixed (Figure 1а and b), the latter has a dynamic impact by which degree is determined by the

It is highly important to have an opportunity to determine stressed condition of the pipelines subject to vibrations in the form of elastic curves, which occur during vibrations caused by

It is used as a reliability factor of the structure under vibrations [1, 3]. Under harmonic

The harmonic vibrations are characterized by two parameters: frequency of vibrations and

<sup>v</sup><sup>0</sup> <sup>¼</sup> <sup>ω</sup>Y0; g<sup>0</sup> <sup>¼</sup> <sup>ω</sup><sup>2</sup>

Under bending vibrations of the pipelines, when distribution of stresses and vibration velocities is significantly different for various fixing conditions, factor с shall be determined individ-

mω<sup>2</sup>

d2 y dx<sup>2</sup> <sup>þ</sup>

y zð Þ¼ <sup>Y</sup><sup>0</sup> sin <sup>i</sup>π<sup>z</sup>

Form of the elastic curve of the pipeline is expressed by a sine wave.

Bending moment in the random location on the pipeline is equal to:

y ¼ Y<sup>0</sup> sin ωt (1)

Y<sup>0</sup> (2)

<sup>T</sup> <sup>y</sup> <sup>¼</sup> <sup>0</sup> (3)

<sup>L</sup> ,y zð Þ¼ <sup>Y</sup>0y zð Þ (4)

vibrations, the vibration velocity can serve as a reliability criterion for the pipeline.

Vibration velocity and vibration acceleration are expressed as follows:

3. Bending vibrations of the pipelines

where Y0 is an amplitude of vibrations.

integral of the forced vibration equation.

Stresses in the pipeline wall can be regarded as criteria.

support yield δA, Р = Y0δ<sup>A</sup> cos ωt.

external excitations.

24 System of System Failures

2. Vibration structure

displacement amplitude:

ually for each case [3].

Maximum stresses in the pipeline:

$$
\sigma\_{\text{makc}} = EI \frac{\text{i}^2 \pi^2}{\text{W} \text{L}^2} \text{Y}\_{0\text{v}} \tag{6}
$$

then W is a moment of resistance in the pipeline.

At initial approximation, certain typical ideal forms of vibrations are used for vibration analysis. Then, the natural vibration frequency of the pipeline is expressed by the following formula:

$$
\omega\_i = \frac{\dot{\mathbf{r}}^2 \pi^2}{L^2} \sqrt{\frac{\text{Elg}}{\rho F}} \tag{7}
$$

where F is a square section of the pipeline.

According to [3, 5], the maximum vibration velocity is determined from the following equation:

$$\mathbf{V}\_{\text{max}} = \mathbf{v}\_{\text{base}} \mathbf{K} \mathbf{\eta} \tag{8}$$

where vbase is a vibration velocity of the base plate; η is a dynamic magnification factor; k is a form engagement factor: <sup>k</sup> <sup>¼</sup> <sup>Ð</sup> <sup>1</sup> <sup>0</sup> ydz= Ð 1 <sup>0</sup> <sup>y</sup><sup>2</sup> dz, y, z are dimensionless forms of pipeline vibrations and current coordinate.

Expansion bends are regarded as vibration damping elements for the pipelines to ensure their vibration resistance. The expansion bends prevent the transfer of vibrations along the pipeline.

#### 4. Analysis of pipeline vibrations

Stresses across the cross sections of the pipeline under natural vibrations can be determined from the following equation:

$$
\sigma\_k = \frac{ED}{2} \mathcal{C}\_k(\omega) \frac{\partial^2 y\_k(z)}{\partial z^2} . \tag{9}
$$

p1 ¼ þ <sup>δ</sup> � <sup>4</sup>Et

p1 <sup>¼</sup> <sup>4</sup><sup>E</sup>

Combined stress in the axial direction due to pressure pulsations and vibration:

Or

5. Spectral transforms

STð Þ¼ f

are interrelated by a complex Fourier series.

At small values of <sup>ω</sup>0: <sup>S</sup>ð Þ <sup>ω</sup> <sup>≈</sup> <sup>1</sup>

ð∞ �∞

¼ 1 T ∙ X∞ m¼�∞

coefficients of the function (15) into a Fourier series as follows:

F tðÞ¼ sin ω0t at 0 < t <

ω.

Then amplitude spectrum is determined by the following expression:

sTð Þt e

<sup>S</sup> <sup>m</sup> T � �∙<sup>δ</sup> <sup>f</sup> � <sup>m</sup>

The function containing n vibrations is described by the following equations,

the time function

D2 ω2

<sup>D</sup><sup>2</sup> � <sup>ω</sup><sup>2</sup>

Spectral density corresponds to the spectral form of the internal pressure sinusoidal vibrations u(t) = sin ω0t. Periodic function spectrum ST(f) is determined by a direct Fourier transform of

> 1 T X∞ m¼�∞

> > m¼�∞

and F tðÞ¼ 0 at 0 > t >

�j2πftdt <sup>¼</sup> S fð Þ<sup>∙</sup>

T � � <sup>¼</sup> <sup>X</sup><sup>∞</sup>

> <sup>T</sup> <sup>∙</sup><sup>S</sup> <sup>m</sup> T � �

2ω<sup>0</sup> <sup>n</sup>πω ω0

� � � �

ω2 <sup>0</sup> � ω<sup>2</sup>

� � � �

Complex weights of S-functions at frequencies multiple of 1/Т. They represent expansion

Time function sТ(t) and spectral function С[т] of the impact are the main characteristics. They

2πn ω0

Sð Þ¼ ω

C m½ �¼ <sup>1</sup>

� �ta (12)

Vibration Strength of Pipelines

27

http://dx.doi.org/10.5772/intechopen.72794

� �td (13)

σ<sup>t</sup> ¼ σ<sup>Δ</sup>pt þ σ<sup>V</sup> (14)

<sup>δ</sup> <sup>f</sup> � <sup>m</sup> T � �

C m½ �∙<sup>δ</sup> <sup>f</sup> � <sup>m</sup>

T � �

> 2πn ω0 :

(15)

(16)

Stresses acting on the pipeline can be expressed as follows:

$$
\sigma = Y\_0 \frac{EI\overline{y''}}{W(z)'} ,
$$

where EI is bending rigidity of pipe, N�m<sup>2</sup> .

Allowable amplitude of vibration equals to:

$$[Y\_0] = \frac{[\sigma] \mathcal{W}(z)}{E I \overline{y''}},\tag{10}$$

where [σ] is a permissible stress in the pipe metal.

The analysis of pipeline vibrations is performed using root mean square of instantaneous vibration parameters over a period determined by the following formula [8]:

$$\overline{y} = \frac{1}{T} \int\_0^T y^2 dt. \tag{11}$$

Measurement results of real pipeline vibrations show that such vibrations are of complex, and in some cases, they are of random nature. For the determination of stresses in the pipeline walls, the process loads Рр due to operating pressure of the product being transported, the effect of hydrostatic water head pressure shall be taken into consideration as well as timevariable loading on the pipeline such as pressure pulsations and seismic forces. It is effective to use spectral method during the analysis of the pipeline random vibrations [1, 3].

The internal pressure of the gas line generates random vibrations.

Elastic stress in the pipe walls can be expressed as follows:

$$\delta = \frac{F}{A} = \frac{\overline{p}\_2 D}{2t}, \text{ when } \overline{p}\_2 = 2Et\frac{\Delta D}{D^2} = \frac{4Et}{D^2}d,\tag{11a}$$

where δ is a dynamic stres, N/m<sup>2</sup> . Pressure changes in this manner p sin ωt ¼ p2 þ p3, in this case the balance between the elastic and internal force in the pipe wall shall be equal to:

$$
\overline{\mathbf{p}}\_1 = \overline{\mathbf{p}}\_2 + \overline{\mathbf{p}}\_3 = \frac{4Et}{D^2}\mathbf{d} + \mathbf{t}\delta\mathbf{a}
$$

As displacement and acceleration, at a given frequency, are related by �ω<sup>2</sup> , the expression can be changed to:

#### Vibration Strength of Pipelines http://dx.doi.org/10.5772/intechopen.72794 27

$$\overline{\mathbf{p}}\_1 = + \left[ \delta - \frac{4Et}{D^2 \alpha^2} \right] \mathbf{t} \tag{12}$$

Or

<sup>σ</sup><sup>k</sup> <sup>¼</sup> ED

Stresses acting on the pipeline can be expressed as follows:

where EI is bending rigidity of pipe, N�m<sup>2</sup>

26 System of System Failures

Allowable amplitude of vibration equals to:

where [σ] is a permissible stress in the pipe metal.

<sup>2</sup> Ckð Þ <sup>ω</sup>

σ ¼ Y<sup>0</sup>

.

½ �¼ Y<sup>0</sup>

<sup>y</sup> <sup>¼</sup> <sup>1</sup> T ðT 0 y2

use spectral method during the analysis of the pipeline random vibrations [1, 3].

The internal pressure of the gas line generates random vibrations.

<sup>A</sup> <sup>¼</sup> <sup>p</sup>2<sup>D</sup>

As displacement and acceleration, at a given frequency, are related by �ω<sup>2</sup>

Elastic stress in the pipe walls can be expressed as follows:

<sup>δ</sup> <sup>¼</sup> <sup>F</sup>

where δ is a dynamic stres, N/m<sup>2</sup>

be changed to:

vibration parameters over a period determined by the following formula [8]:

∂2 ykð Þz

EIy} W zð Þ,

½ � σ W zð Þ

The analysis of pipeline vibrations is performed using root mean square of instantaneous

Measurement results of real pipeline vibrations show that such vibrations are of complex, and in some cases, they are of random nature. For the determination of stresses in the pipeline walls, the process loads Рр due to operating pressure of the product being transported, the effect of hydrostatic water head pressure shall be taken into consideration as well as timevariable loading on the pipeline such as pressure pulsations and seismic forces. It is effective to

<sup>2</sup><sup>t</sup> , when <sup>p</sup><sup>2</sup> <sup>¼</sup> <sup>2</sup>EtΔ<sup>D</sup>

case the balance between the elastic and internal force in the pipe wall shall be equal to:

p1 <sup>¼</sup> p2 <sup>þ</sup> p3 <sup>¼</sup> <sup>4</sup>E<sup>t</sup>

<sup>D</sup><sup>2</sup> <sup>¼</sup> <sup>4</sup>Et

<sup>D</sup><sup>2</sup> <sup>d</sup> <sup>þ</sup> <sup>t</sup>δ<sup>a</sup>

. Pressure changes in this manner p sin ωt ¼ p2 þ p3, in this

<sup>∂</sup>z<sup>2</sup> : (9)

EIy} , (10)

dt: (11)

<sup>D</sup><sup>2</sup> d, (11a)

, the expression can

$$
\overline{\mathbf{p}}\_1 = \left[\frac{4E}{D^2} - \omega^2\right]td\tag{13}
$$

Combined stress in the axial direction due to pressure pulsations and vibration:

$$
\sigma\_t = \sigma\_{\Delta pt} + \sigma\_V \tag{14}
$$

#### 5. Spectral transforms

Spectral density corresponds to the spectral form of the internal pressure sinusoidal vibrations u(t) = sin ω0t. Periodic function spectrum ST(f) is determined by a direct Fourier transform of the time function

$$\begin{split} S\_T(f) &= \int\_{-\infty}^{\infty} s\_T(t) e^{-j2\pi \theta} dt = S(f) \cdot \frac{1}{T} \sum\_{m = -\infty}^{\infty} \delta \left( f - \frac{m}{T} \right) \\ &= \frac{1}{T} \cdot \sum\_{m = -\infty}^{\infty} S\left( \frac{m}{T} \right) \cdot \delta \left( f - \frac{m}{T} \right) = \sum\_{m = -\infty}^{\infty} \mathbb{C}[m] \cdot \delta \left( f - \frac{m}{T} \right) \end{split} \tag{15}$$

Complex weights of S-functions at frequencies multiple of 1/Т. They represent expansion coefficients of the function (15) into a Fourier series as follows:

$$\mathbf{C}[m] = \frac{1}{T} \cdot \mathbf{S}\left(\frac{m}{T}\right).$$

Time function sТ(t) and spectral function С[т] of the impact are the main characteristics. They are interrelated by a complex Fourier series.

The function containing n vibrations is described by the following equations,

$$F(t) = \sin\omega\_0 t \text{ at } 0 < t < \frac{2\pi n}{\omega\_0} \text{ and } F(t) = 0 \text{ at } 0 > t > \frac{2\pi n}{\omega\_0}.$$

Then amplitude spectrum is determined by the following expression:

$$S(\omega) = \left| \frac{2\omega\_0 \frac{n\pi\omega}{a\_0}}{\omega\_0^2 - \omega^2} \right| \tag{16}$$

At small values of <sup>ω</sup>0: <sup>S</sup>ð Þ <sup>ω</sup> <sup>≈</sup> <sup>1</sup> ω. Upon the completion of the vibration analysis according to the scheme of the single-degree-offreedom system (which includes the reduced weight of the pipeline and its components, and elastic support action), stresses and deformations in the support elements shall be calculated.

Transfer function of the maximum stress relative to acceleration of the pipeline supports can be written as

$$|H\_{\sigma}(\omega, z)| = \frac{\mathbb{C}\_{k}(\omega)\frac{\mathrm{ED}}{2}y\_{k}''(z)}{\sqrt{\left(\omega\_{0}^{2} - \omega^{2}\right)^{2} + \left(2\beta\alpha\omega\_{0}\right)^{2}}}.\tag{17}$$

duration of the time period shall be long enough to consider all potential delays. Pipe laying

<sup>σ</sup>�<sup>1</sup> <sup>¼</sup> <sup>1</sup>:75σB=N<sup>0</sup>:<sup>12</sup>

The requirements [design documentation] state that "the maximum allowable amplitude of

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

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

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

ω2

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

<sup>σ</sup><sup>2</sup> ð Þ¼ <sup>z</sup> <sup>X</sup> N

k¼1 σ2 <sup>k</sup> ð Þz :

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.

<sup>Φ</sup>QQð Þ <sup>ω</sup> <sup>d</sup><sup>ω</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

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29

http://dx.doi.org/10.5772/intechopen.72794

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

vibrations of the process pipelines is 0.2 mm at vibration frequency of max 40 Hz" [2].

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

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

• for pipeline vibration resistance under exposure to external vibrations.

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

ffiffiffiffiffiffiffiffiffiffiffi σ2 <sup>k</sup> ð Þz q

¼

ð∞ 0 Ckð Þ ω

2 6 4

period shall not exceed this time interval.

here N is a number of loading cycles.

6. Pipeline vibration limiting

• for pipeline soundness and quality

vibration limitations.

mean square value of vibrations:

Spectral density of the pipeline response to random excitations will be equal to

$$\boldsymbol{\Phi}\_{\rm YY}(\omega) = \frac{\boldsymbol{\Phi}\_{\rm QQ}(\omega)}{\left(\omega\_0^2 - \omega^2\right)^2 + \left(2\beta\omega\omega\_0\right)^4} \tag{18}$$

Considering the tensile strength and fatigue strength, the allowable amplitudes of stresses in the pipeline wall can be calculated from the following formulas:

$$\sigma\_{\Lambda m} = \frac{\sigma\_{-1}\beta k}{n\_m \left(1 + \frac{\sigma\_{-1}}{\sigma\_\beta} \frac{1 + q\_r}{1 - q\_r}\right)}, \quad \sigma\_{\Lambda t} = \frac{\sigma\_{-1}\beta k}{n\_t \left(1 + \frac{\sigma\_{-1}}{\sigma\_\beta} \frac{1 + q\_t}{1 - q\_r}\right)}\tag{19}$$

where σ<sup>В</sup> is a tensile strength; σ�<sup>1</sup> is a fatigue strength under symmetrical loading cycle; β is a coefficient that takes into consideration the effect of the pipeline surface finish on the fatigue strength: for new pipelines β = 0.80–0.85, for corrosion susceptible pipelines, this coefficient is reduced to β = 0.5; k is a stress concentration factor [1, 3, 4].

Stress ratio is shown as follows:

$$q\_r = \frac{P\_p - \Delta P}{P\_p + \Delta P}, \quad q\_t = \frac{P\_p D / (2\delta) - \sigma\_t}{P\_p D / (2\delta) + \sigma\_t}.$$

During the installation of the pipeline system, the rated natural load shall be assumed as maximum\* at the most probable sea condition for the time period under review, which is determined using (Нз, Тр), and applicable stream and wind conditions. Rated load is assumed as maximum at the most probable parameters of the natural environment (in other words, waves, stream, and wind- LE), and equals to.

$$\mathcal{R}(L\_E) = 1 - \frac{1}{N}$$

where R(LЕ) is a probability distribution function LЕ.

N is a number of loading cycles of minimum 3 h in length at a certain sea condition.

Note that the specific sea condition for the time period under review can be interpreted as a sea condition for an applicable location and period of pipe laying. Ordinary requirement is that the duration of the time period shall be long enough to consider all potential delays. Pipe laying period shall not exceed this time interval.

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

$$\sigma\_{-1} = 1.75 \sigma\_{\rm B} / N^{0.12}$$

here N is a number of loading cycles.

Upon the completion of the vibration analysis according to the scheme of the single-degree-offreedom system (which includes the reduced weight of the pipeline and its components, and elastic support action), stresses and deformations in the support elements shall be calculated. Transfer function of the maximum stress relative to acceleration of the pipeline supports can be

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

� �<sup>2</sup> <sup>q</sup> : (17)

� �<sup>4</sup> (18)

� � (19)

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

ΦQQð Þ ω

<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

Spectral density of the pipeline response to random excitations will be equal to

ΦYYð Þ¼ ω

the pipeline wall can be calculated from the following formulas:

σΛ<sup>m</sup> <sup>¼</sup> <sup>σ</sup>�<sup>1</sup>β<sup>k</sup> nm <sup>1</sup> <sup>þ</sup> <sup>σ</sup>�<sup>1</sup> σB 1þqr 1�qr

reduced to β = 0.5; k is a stress concentration factor [1, 3, 4].

qr <sup>¼</sup> Pp � <sup>Δ</sup><sup>P</sup>

Stress ratio is shown as follows:

waves, stream, and wind- LE), and equals to.

where R(LЕ) is a probability distribution function LЕ.

ω2

ω<sup>2</sup>

Considering the tensile strength and fatigue strength, the allowable amplitudes of stresses in

where σ<sup>В</sup> is a tensile strength; σ�<sup>1</sup> is a fatigue strength under symmetrical loading cycle; β is a coefficient that takes into consideration the effect of the pipeline surface finish on the fatigue strength: for new pipelines β = 0.80–0.85, for corrosion susceptible pipelines, this coefficient is

Pp <sup>þ</sup> <sup>Δ</sup><sup>P</sup> , qt <sup>¼</sup> PpD=ð Þ� <sup>2</sup><sup>δ</sup> <sup>σ</sup><sup>t</sup>

During the installation of the pipeline system, the rated natural load shall be assumed as maximum\* at the most probable sea condition for the time period under review, which is determined using (Нз, Тр), and applicable stream and wind conditions. Rated load is assumed as maximum at the most probable parameters of the natural environment (in other words,

R Lð Þ¼ <sup>E</sup> <sup>1</sup> � <sup>1</sup>

Note that the specific sea condition for the time period under review can be interpreted as a sea condition for an applicable location and period of pipe laying. Ordinary requirement is that the

N is a number of loading cycles of minimum 3 h in length at a certain sea condition.

N

PpD=ð Þþ 2δ σ<sup>t</sup>

,

� � , <sup>σ</sup><sup>Δ</sup><sup>t</sup> <sup>¼</sup> <sup>σ</sup>�<sup>1</sup>β<sup>k</sup>

nt <sup>1</sup> <sup>þ</sup> <sup>σ</sup>�<sup>1</sup> σB 1þqt 1�qrt

written as

28 System of System Failures
