Vibration Strength of Pipelines

Damage of offshore oil-and-gas pipelines results in significant pollution of the marine environment. Extent of pipeline damage during an earthquake depends on a number of factors: the seismic force and the seismic waves’ propagation direction, the geological and groundwater conditions, the operation and process duties, the pipeline design and the joints, the pipeline material’s characteristics, and the extent of pipeline wear. Requirements to the structural reliability of subsea pipelines are much stricter than those set to underground and overhead pipelines. An offshore pipeline is in the combined stressed condition. It is characterized by tension of the pipeline walls caused by pressure pulsations and cyclic bending due to the vibrations. This chapter tackles the vibration analysis issues. The main goal is to present to the industry specialists the principles of vibration assessment as applied to the offshore pipelines in seismic regions and to outline solutions of the vibration problems. The dynamic calculations of the offshore pipelines based on the natural frequency analysis are represented in the industrial construction standards only.


Introduction
During operation of the offshore pipelines, the vibration processes occur as a result of pump plant running, activation of shutoff valves, emergency shutdowns, and external effects. Vibration protection issues are highlighted in the manual (design documentation), but during the design stage of the offshore pipeline system, these factors are not taken into consideration. Sea pipelines are new type of constructions in Russia. Only in VSN R 42-81 [8] are considered dynamic problem definition laying sea pipelines. All private oil-extracting companies are not interested in researches. A research of vibrations of pipelines demands the state researches.
The provisions [2][3][4] take into account the following stresses acting on the offshore pipeline: pressure of the product being transported, temperature exposures, and weight coating; however, dynamic nature of loading on the pipeline wall is not considered during the operation.
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 integral of the forced vibration equation.
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 support yield δ A , Р =Y 0 δ A cos ωt.
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 external excitations.
Stresses in the pipeline wall can be regarded as criteria.

Vibration structure
It is used as a reliability factor of the structure under vibrations [1,3]. Under harmonic vibrations, the vibration velocity can serve as a reliability criterion for the pipeline.
The harmonic vibrations are characterized by two parameters: frequency of vibrations and displacement amplitude: Vibration velocity and vibration acceleration are expressed as follows:

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].
Form of the elastic curve of the pipeline is expressed by a sine wave.
where Y 0 is an amplitude of vibrations.
Bending moment in the random location on the pipeline is equal to:

System of System Failures 24
Mz ðÞ¼EIy 00 z ðÞ¼À EI where I is a moment of inertia.
Maximum stresses in the pipeline: 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: where F is a square section of the pipeline.
According to [3,5], the maximum vibration velocity is determined from the following equation: where v base is a vibration velocity of the base plate; η is a dynamic magnification factor; k is a form engagement factor: k ¼ Ð 1 0 ydz= Ð 1 0 y 2 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.

Analysis of pipeline vibrations
Stresses across the cross sections of the pipeline under natural vibrations can be determined from the following equation: Stresses acting on the pipeline can be expressed as follows: where EI is bending rigidity of pipe, NÁm 2 .
Allowable amplitude of vibration equals to: 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]: 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 Р р 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: where δ is a dynamic stres, N/m 2 . Pressure changes in this manner p sin ωt ¼ p 2 þ p 3 , in this case the balance between the elastic and internal force in the pipe wall shall be equal to: As displacement and acceleration, at a given frequency, are related by Àω 2 , the expression can be changed to: Combined stress in the axial direction due to pressure pulsations and vibration:

Spectral transforms
Spectral density corresponds to the spectral form of the internal pressure sinusoidal vibrations u(t) = sin ω 0 t. Periodic function spectrum S T (f) is determined by a direct Fourier transform of the time function 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: Time function s Т (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, Then amplitude spectrum is determined by the following expression: At small values of ω 0 : S ω ðÞ ≈ 1 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 Spectral density of the pipeline response to random excitations will be equal to Considering the tensile strength and fatigue strength, the allowable amplitudes of stresses in the pipeline wall can be calculated from the following formulas: where σ В is a tensile strength; σ À1 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: 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 (Н з , Т р ), 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-L E ), and equals to.
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 À1 value can be defined either using reference data or Manson formula [4]: here N is a number of loading cycles.

Pipeline vibration limiting
Pipeline vibration limiting regulations can be divided into the following categories: • for pipeline soundness and quality • for pipeline vibration resistance under exposure to external vibrations.
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: In the event of random vibrations, the combined stress in the pipeline is the following: 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: where V l is vibration velocity (mm/s); a is vibration acceleration (mm/s 2 ) with amplitude В 0 ; f k 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: 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: Root mean square value of the pipeline movement to be subject to vibrations can be calculated from the following formula: σ ¼ where η f 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 1 and f 2 ; Δf i is an interval of the frequency band segmentation f 1 , f 2 ; 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 V e = 18 mm/s, and the warning vibration level is estimated by the exceedance V e = 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, NÁm 2 .
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: 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 f j from discrete frequencies of the excitatory loads f ip , 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 À1 value can be defined either using reference data or Manson formula [4]: Kt≔1 σBBÁβÁk=806.08 If length l of the pipeline section is 30 m, displacement y 0 is determined by the formula (7) and is equal to 0.026 m, considering the formula (9) [2].

System of System Failures
strength of the pipeline steel (see Table 1), stress amplitude σ ∆t 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 a max = 0.94485 cm/s 2 at t = 0.12 s ( Figure 2).
We get the line spectrum of the signal consisting of the harmonics: where Т min is a minimum period.
Periodogram results can be interpreted as dispersed data at frequencies given in Table 2 and Figure 3.

System of System Failures 34
Mean mathematic amplitude В к over a period of observation Т shall be written as: where B K t ðÞis a behavior of k-х components of vibration spectrum.
Vibration velocity can be written as: where V l is vibration velocity (mm/s); a is vibration acceleration (mm/s 2 ) with amplitude В 0 ; f k 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 V e = 18 mm/s, and an alarm vibration level exceeds V e = 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 Y 0 is 0.061 m. Safe vibration speed of the pipeline, which is excited kinematically on the moving platform, is 0.035 m/s.

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.