**2.1 Basic deterministic calculations**

The system of domestic and foreign trunk oil pipelines that took shape in the second half of the twentieth and the beginning of the twenty-first century is characterized by multistage creation and development of integrated approaches to justifying their strength [1–4]. These approaches were initially formed on the basis of the fundamental theories of thin-walled shells, classical theories of strength; they made it possible to form the main computational methods for the selection of

*Combined Calculated, Experimental and Determinated and Probable Justifications… DOI: http://dx.doi.org/10.5772/intechopen.89036*

computational schemes and computational cases and the assessment of static strength, taking into account the types of stress and limit states.

The basic strength condition was then recorded in the simplest form:

$$
\sigma\_{\text{new}}^{\circ} \le \sigma\_{\text{ow}},
\tag{1}
$$

where is maximum operating voltage stress and is hazardous stress. For a thin-walled pipe with a diameter *D* with wall thickness , ring stresses are maximal:

(**Figure 3**).

The operating pressures in the main oil pipelines range from 2 to 10 MPa

with ground, underground, and underwater laying.

The trunk oil pipelines are operated in a very wide range of climatic conditions (from 70°С to +60°С) and natural hazards (seismic, landslides, geological faults),

Despite the large, more than century-old experience of research, testing, construction, and operation of oil trunk pipelines in the world and in Russia, there were large-scale accidents and disasters. These accidents were accompanied by the release of large amounts of oil (up to 100–600 thousand barrels) into the environment (land, water) with great environmental damage, fires, death and injury to people, and pollution of hundreds of hectares of land. Economic damages from such accidents are estimated at \$ 10–100 million. The total number of accidents on oil pipelines in the world over the past 20 years is more than 2000, and the number of large oil leaks is more than 4500. For every million tons of pumped oil, 3–5 tons fall

In general, the accident rate on the trunk oil pipelines is reduced. However, at

These data indicate the need for further research and practical development to

In recent years, four basic approaches to determining the strength, resource, and

**2. Solving the problems of strength by basic and calibration methods**

The system of domestic and foreign trunk oil pipelines that took shape in the second half of the twentieth and the beginning of the twenty-first century is characterized by multistage creation and development of integrated approaches to justifying their strength [1–4]. These approaches were initially formed on the basis of the fundamental theories of thin-walled shells, classical theories of strength; they made it possible to form the main computational methods for the selection of

present it is at the level of 0.1–0.3 per 1000 km per year (**Figure 2**).

reduce accidents and improve the safety of trunk pipelines.

safety of oil pipelines have emerged:

**2.1 Basic deterministic calculations**

• Deterministic

• Statistical

• Probabilistic

• Combined

**144**

(**Figure 1**).

*Oil trunk pipelines (Russia).*

*Probability, Combinatorics and Control*

**Figure 1.**

into leaks.

$$
\sigma\_{\text{max}}^{\circ} = \frac{P\_{\text{max}}^{\circ} \cdot D}{2\mathcal{S}} \,, \tag{2}
$$

where is maximum operating pressure (**Figure 3**).

**Figure 2.** *Accidents on oil pipelines (Venezuela, China, Russia).*

**Figure 3.** *The main design scheme.*

Since in engineering calculations of static strength according to (1) and (2), a whole set of design, technological, and operational methods remained unclear, permissible stresses with corresponding safety margins were entered into the calculation:

$$\mathbb{E}\left[\sigma\right] = \frac{\sigma\_{o\bullet}}{n\_{\bullet}} = \min\left\{\frac{\sigma\_{\rm r}}{n\_{\rm r}}, \frac{\sigma\_{\rm s}}{n\_{\rm s}}\right\}.\tag{3}$$

mechanical properties of pipe steels (200 ≤ ≤ 800, 350 ≤ ≤ 950) and a

Expressions (1)–(4) were and remain central to foreign strength standards [2–4]. In Russian practice [1, 4, 5], expressions (1) and (2) were retained, but the

where *n*, *K*1, and *K*<sup>н</sup> are the reliability factors for load, material, and purpose

At the stage of preliminary and detailed design, the main calculation is reduced

max � *D*

max and *D*, taking into account economically and techno-

max at each of the calculated sections of the pipeline depend on

In view of **Figure 4** and expressions (1)–(5) in the feasibility study of the

to the calculated determination of the minimum wall thickness of the pipeline

logically reasonable choice of pipe steel with characteristics (according to

*<sup>δ</sup>*min <sup>≥</sup> *<sup>p</sup>*<sup>э</sup>

their height position, which determines the hydrostatic part of the pressure, the

tion calculations with an assessment of the maximum hydraulic tests are carried out in accordance with (2), their comparison with permissible values in accordance with (3) and (5) and confirmation of the absence of destruction or the formation of

> <sup>≤</sup> *<sup>σ</sup>оп n*г *σ*

At the stage of construction and pre-hydraulic pressure tests *p*<sup>г</sup> ≥ *p*<sup>э</sup>

max <sup>¼</sup> *<sup>p</sup>*<sup>г</sup> � *<sup>D</sup>* 2*δ*min

*σ*г

ð5Þ

max and *D*, ensuring the

max, calibra-

<sup>2</sup>½ � *<sup>σ</sup> :* (6)

*:* (7)

decrease in strength margins (2.8 ≥ ≥ 1.5, 4.0 ≥ ≥ 1.8).

*Generalized scheme of research and rating of strength.*

*DOI: http://dx.doi.org/10.5772/intechopen.89036*

(*n,K*1,*K*<sup>н</sup> ≥ 1) and *m* is the operating condition ratio (*m* ≤ 1).

project, two main parameters are defined and assigned, *p*<sup>э</sup>

specified pipeline performance.

technical conditions or standards):

unacceptable plastic deformations

calculated values are variable in length.

according to the given *p*<sup>э</sup>

**Figure 4.**

Since the values *p*<sup>э</sup>

**147**

strength margins in (3) were presented in a differentiated form:

*Combined Calculated, Experimental and Determinated and Probable Justifications…*

Dangerous stresses were applicable ones corresponding to:


These methods are divided into two main groups:


The formation of methods of basic and calibration calculations is currently associated with the stages of the life cycle. At the same time, an important role is always played by scientific studies to substantiate the strength criteria, the choice of design schemes and design cases, followed by the introduction of safety margins. This is a scientific basis for solving applied problems of strength—the development of strength standards with their design characteristics (**Figure 4**).

For all pipe steels and to fulfill condition (3), the margins must satisfy.

$$1 \le n\_{\tau} \le n\_{\pi}.\tag{4}$$

The development of pipeline transport during the decades of the twentieth to twenty-first centuries [5, 6] was accompanied by a gradual increase in the

*Combined Calculated, Experimental and Determinated and Probable Justifications… DOI: http://dx.doi.org/10.5772/intechopen.89036*

#### **Figure 4.**

Since in engineering calculations of static strength according to (1) and (2), a whole set of design, technological, and operational methods remained unclear, permissible stresses with corresponding safety margins were entered into the

• Ultimate strength , which excluded the occurrence of fracture (the first

• Yield strength (or conditional yield strength ), which excluded the formation of unacceptable plastic deformations (the second most significant limiting state). For modern pipeline systems transporting petroleum and petroleum products, a number of main life cycle stages, as measured up to

• Operation of pipeline systems with diagnostic and repair and rehabilitation

• The withdrawal of sections of pipelines or pipeline systems from operation

• Verification calculations of strength for the used construction material

of strength standards with their design characteristics (**Figure 4**).

The formation of methods of basic and calibration calculations is currently associated with the stages of the life cycle. At the same time, an important role is always played by scientific studies to substantiate the strength criteria, the choice of design schemes and design cases, followed by the introduction of safety margins. This is a scientific basis for solving applied problems of strength—the development

For all pipe steels and to fulfill condition (3), the margins must satisfy.

The development of pipeline transport during the decades of the twentieth to

twenty-first centuries [5, 6] was accompanied by a gradual increase in the

• For each of these stages and for the entire life cycle, to date, in our country and abroad, certain approaches and methods to substantiate strength have been

Dangerous stresses were applicable ones corresponding to:

30–60 years, are introduced into the strength analysis:

• Construction and testing of pipeline systems

These methods are divided into two main groups:

significant limiting state).

*Probability, Combinatorics and Control*

• Feasibility study of the project

• Outline and detailed design

• Basic strength calculations

works

formed.

**146**

ð3Þ

ð4Þ

calculation:

*Generalized scheme of research and rating of strength.*

mechanical properties of pipe steels (200 ≤ ≤ 800, 350 ≤ ≤ 950) and a decrease in strength margins (2.8 ≥ ≥ 1.5, 4.0 ≥ ≥ 1.8).

Expressions (1)–(4) were and remain central to foreign strength standards [2–4]. In Russian practice [1, 4, 5], expressions (1) and (2) were retained, but the strength margins in (3) were presented in a differentiated form:

$$
\boldsymbol{m}\_o = \{\boldsymbol{\mu}\_r, \boldsymbol{\mu}\_u\} = \frac{\boldsymbol{n} \cdot \boldsymbol{K}\_1 \cdot \boldsymbol{K}\_n}{m},\tag{5}
$$

where *n*, *K*1, and *K*<sup>н</sup> are the reliability factors for load, material, and purpose (*n,K*1,*K*<sup>н</sup> ≥ 1) and *m* is the operating condition ratio (*m* ≤ 1).

In view of **Figure 4** and expressions (1)–(5) in the feasibility study of the project, two main parameters are defined and assigned, *p*<sup>э</sup> max and *D*, ensuring the specified pipeline performance.

At the stage of preliminary and detailed design, the main calculation is reduced to the calculated determination of the minimum wall thickness of the pipeline according to the given *p*<sup>э</sup> max and *D*, taking into account economically and technologically reasonable choice of pipe steel with characteristics (according to technical conditions or standards):

$$
\delta\_{\rm min} \ge \frac{p\_{\rm max}^{\circ} \cdot D}{2[\sigma]}.\tag{6}
$$

Since the values *p*<sup>э</sup> max at each of the calculated sections of the pipeline depend on their height position, which determines the hydrostatic part of the pressure, the calculated values are variable in length.

At the stage of construction and pre-hydraulic pressure tests *p*<sup>г</sup> ≥ *p*<sup>э</sup> max, calibration calculations with an assessment of the maximum hydraulic tests are carried out in accordance with (2), their comparison with permissible values in accordance with (3) and (5) and confirmation of the absence of destruction or the formation of unacceptable plastic deformations

$$
\sigma\_{\text{max}}^r = \frac{p^r \cdot D}{2\delta\_{\text{min}}} \le \frac{\sigma\_{\text{on}}}{n\_{\sigma}^r} \,. \tag{7}
$$

At the stage of operation for the time there is a possibility of accumulation of damage, and a decrease in wall thickness due to corrosion, erosion, as well as a change in mechanical properties

These parameters are determined according to periodic in-line inspection, as well as according to mechanical testing of samples from damaged sections of pipelines. The verification calculation of the strength for this stage is reduced to the assessment of the strength margin

$$m\_{\boldsymbol{\sigma}}(\boldsymbol{\tau}) = \frac{\sigma\_{\boldsymbol{\alpha}\boldsymbol{\eta}}(\boldsymbol{\tau})}{\sigma^{\boldsymbol{\sigma}}(\boldsymbol{\tau})};\tag{8}$$

*σ*э

possible to reduce the operating pressure to level.

*DOI: http://dx.doi.org/10.5772/intechopen.89036*

• Maximum values of operating pressure *р*<sup>э</sup>

expression (12) is detected.

*σ*тmin, *σ*<sup>в</sup>min*:*

This calculation should include:

• Minimum wall thickness *δ*min

In this case, you can write

*σ*э

<sup>с</sup> <sup>¼</sup> *<sup>F</sup><sup>σ</sup> <sup>p</sup>*<sup>э</sup>

• With average values of all parameters *р*<sup>э</sup>

<sup>с</sup> <sup>¼</sup> *<sup>F</sup><sup>σ</sup> <sup>p</sup>*<sup>э</sup>

*σ*э

• At extreme (extreme) values *р*<sup>э</sup>

max <sup>¼</sup> *<sup>F</sup><sup>σ</sup> <sup>p</sup>*<sup>э</sup>

*σ*э

operational factors.

pipeline due to

**149**

*2.2.1 Operating pressure*

values of *р*<sup>э</sup>

*р*э <sup>с</sup> < *р*<sup>э</sup>

max c <sup>≤</sup>½ � *<sup>σ</sup>* ;*σ*<sup>э</sup>

*Combined Calculated, Experimental and Determinated and Probable Justifications…*

For the scientific substantiation of the need and possibility of extending the operation of pipelines in cases of failure to meet the strength conditions, it is

, when *σ*<sup>э</sup>

Simultaneous fulfillment of conditions (1)–(3) requires the mandatory calculation of the strength of the pipeline—its pipes and sections, where the realization of

• Maximum values of the diameter *D*max of the pipe or section of the pipeline

max, *<sup>D</sup>*max, *<sup>δ</sup>*min <sup>≤</sup> ½ �¼ *<sup>σ</sup>* min *<sup>σ</sup>*<sup>в</sup>min

At the same time, both for condition (12) and for conditions (13) and (14), it is

<sup>≤</sup>½ �¼ *<sup>σ</sup>* min *<sup>σ</sup>*<sup>в</sup> *<sup>m</sup>*

Thus, according to (12)–(16), the calculated (average, minimum, maximum)

, *D, δ*, and *σ*т, *σ*<sup>в</sup> are due to a whole range of design, technological, and

*<sup>m</sup>* <sup>≤</sup> *<sup>р</sup>*<sup>э</sup>

due to changes in the actual pressures in a given pipe or in a given section of the

max, *<sup>D</sup>*min, *<sup>δ</sup>*max <sup>≤</sup>½ �¼ *<sup>σ</sup>* min *<sup>σ</sup>*<sup>в</sup> max

The statistical nature of operating pressures *р*<sup>э</sup> satisfies inequalities.

*р*э min <sup>≤</sup>*р*<sup>э</sup>

The strength condition according to expression (14) should be checked according to the statistical analysis and when conditions (12) are fulfilled.

advisable to give an assessment of the strength according to (1):

*<sup>m</sup>*, *Dm*, *δ<sup>m</sup>*

• The minimum values of the characteristics of mechanical properties

max

max c>½ � *σ :* (12)

<sup>с</sup> <sup>&</sup>lt; *<sup>σ</sup>*<sup>э</sup> <sup>≤</sup>½ � *<sup>σ</sup>* (13)

*n*в , *σ*<sup>т</sup>min *n*т (14)

*<sup>m</sup>*, *Dm*, *δm*, and *σ*<sup>т</sup> *<sup>m</sup>*, *σ*<sup>в</sup> *<sup>m</sup>*

(15)

max (17)

*n*в , *σ*<sup>т</sup> *<sup>m</sup> n*т

max, *δ*max, *D*min, and *σ*<sup>т</sup> max, *σ*<sup>в</sup> max

*n*в , *σ*<sup>т</sup> max *n*т *:* (16)

$$\sigma\_{\rhd}(\mathfrak{r}) = \frac{p\_{\max}^{\mathfrak{p}} \cdot D}{\delta\_{\min}(\mathfrak{r})};\tag{9}$$

$$
\sigma\_{os}(\tau) = \langle \sigma\_r(\tau), \sigma\_\*(\tau) \rangle. \tag{10}
$$

If for the analyzed stage margin по (8) is not less than in in (3) and (5), then the operation of the pipeline can be continued.

To calculate the estimated time of the next in-line inspection, it is necessary to have data on monitoring and on the basis of previous inline inspections operations.

If such initial information is absent, then the construction of calculated curves is possible:

$$\{\sigma\_{\cdot}(\boldsymbol{\tau}), \sigma\_{\upharpoonright}(\boldsymbol{\tau})\} = F^{\dagger}(\sigma\_{\cdot}, \sigma\_{\cdot}) \cdot \boldsymbol{\tau}^{\prime \ast\_{\boldsymbol{\tau}}}\big],\tag{11}$$

where —If such initial information is absent, then the construction of calculated curves is possible [6, 7] (0≤ ≤ 0.03 for in hours).

#### **2.2 Statistical strength analysis**

In actual practice, in the manufacture and testing of pipes, the construction of pipeline sections and the operation of pipeline systems, all specified parameters of expressions (1)–(11), are statistically variable, despite the determination of the main calculations in the design of pipelines throughout the system of design expressions.

The statistical analysis of the calculated parameters in the framework of the basic calculations of the strength of pipelines is aimed at establishing:


On this basis, two decisions are made about the possibility or impossibility of further operations of pipelines.

In the first case, the main requirement for the strength of pipelines must be met; in the second case, the strength is considered not ensured if the maximum operating stresses exceed the allowable.

*Combined Calculated, Experimental and Determinated and Probable Justifications… DOI: http://dx.doi.org/10.5772/intechopen.89036*

$$
\sigma\_{\text{max c}}^{\flat} \le [\sigma]; \sigma\_{\text{max c}}^{\flat} > [\sigma]. \tag{12}
$$

For the scientific substantiation of the need and possibility of extending the operation of pipelines in cases of failure to meet the strength conditions, it is possible to reduce the operating pressure to level.

$$p\_{\mathfrak{c}}^{\mathfrak{o}} < p^{\mathfrak{o}}, \quad \text{when } \sigma\_{\mathfrak{c}}^{\mathfrak{o}} < \sigma^{\mathfrak{o}} \le [\sigma] \tag{13}$$

Simultaneous fulfillment of conditions (1)–(3) requires the mandatory calculation of the strength of the pipeline—its pipes and sections, where the realization of expression (12) is detected.

This calculation should include:

At the stage of operation for the time there is a possibility of accumulation of damage, and a decrease in wall thickness due to corrosion, erosion, as well

These parameters are determined according to periodic in-line inspection, as well as according to mechanical testing of samples from damaged sections of pipelines. The verification calculation of the strength for this stage is reduced to the

*<sup>σ</sup>*эð Þ¼ *<sup>τ</sup> <sup>р</sup>*<sup>э</sup>

max � *D*

If for the analyzed stage margin по (8) is not less than in in (3) and (5), then

To calculate the estimated time of the next in-line inspection, it is necessary to

If such initial information is absent, then the construction of calculated curves is

where —If such initial information is absent, then the construction of calcu-

In actual practice, in the manufacture and testing of pipes, the construction of pipeline sections and the operation of pipeline systems, all specified parameters of expressions (1)–(11), are statistically variable, despite the determination of the main calculations in the design of pipelines throughout the system of design

The statistical analysis of the calculated parameters in the framework of the

• Minimum (min), average (m), and maximum (max) parameter values

• Deviations of the calculated parameters to the dangerous and safe side in

On this basis, two decisions are made about the possibility or impossibility of

In the first case, the main requirement for the strength of pipelines must be met; in the second case, the strength is considered not ensured if the maximum operating

• Comparability with the values adopted in the project documentation

basic calculations of the strength of pipelines is aimed at establishing:

comparison with the statistically determined.

have data on monitoring and on the basis of previous in-

*<sup>δ</sup>*minð Þ*<sup>τ</sup>* ; (9)

ð8Þ

ð10Þ

ð11Þ

as a change in mechanical properties

*Probability, Combinatorics and Control*

assessment of the strength margin

the operation of the pipeline can be continued.

lated curves is possible [6, 7] (0≤ ≤ 0.03 for in hours).

line inspections operations.

**2.2 Statistical strength analysis**

further operations of pipelines.

stresses exceed the allowable.

possible:

expressions.

**148**


In this case, you can write

$$
\sigma\_{\mathbf{c}}^{\circ} = F\_{\sigma} \left\{ p\_{\max}^{\circ}, D\_{\max}, \delta\_{\min} \right\} \le [\sigma] = \min \left\{ \frac{\sigma\_{\min}}{n\_{\text{a}}}, \frac{\sigma\_{\text{rmin}}}{n\_{\text{r}}} \right\} \tag{14}
$$

The strength condition according to expression (14) should be checked according to the statistical analysis and when conditions (12) are fulfilled.

At the same time, both for condition (12) and for conditions (13) and (14), it is advisable to give an assessment of the strength according to (1):

• With average values of all parameters *р*<sup>э</sup> *<sup>m</sup>*, *Dm*, *δm*, and *σ*<sup>т</sup> *<sup>m</sup>*, *σ*<sup>в</sup> *<sup>m</sup>*

$$\sigma\_{\mathfrak{c}}^{\mathfrak{o}} = F\_{\sigma} \left\{ p\_{\mathfrak{m}}^{\mathfrak{o}}, D\_{\mathfrak{m}}, \delta\_{\mathfrak{m}} \right\} \le [\sigma] = \min \left\{ \frac{\sigma\_{\mathfrak{a}, \mathfrak{m}}}{n\_{\mathfrak{a}}}, \frac{\sigma\_{\mathfrak{T}, \mathfrak{m}}}{n\_{\mathfrak{r}}} \right\} \tag{15}$$

• At extreme (extreme) values *р*<sup>э</sup> max, *δ*max, *D*min, and *σ*<sup>т</sup> max, *σ*<sup>в</sup> max

$$
\sigma\_{\text{max}}^{\circ} = F\_{\sigma} \left\{ p\_{\text{max}}^{\circ}, D\_{\text{min}}, \delta\_{\text{max}} \right\} \le [\sigma] = \min \left\{ \frac{\sigma\_{\text{b max}}}{n\_{\text{b}}}, \frac{\sigma\_{\text{r \, max}}}{n\_{\text{r}}} \right\}. \tag{16}
$$

Thus, according to (12)–(16), the calculated (average, minimum, maximum) values of *р*<sup>э</sup> , *D, δ*, and *σ*т, *σ*<sup>в</sup> are due to a whole range of design, technological, and operational factors.

#### *2.2.1 Operating pressure*

The statistical nature of operating pressures *р*<sup>э</sup> satisfies inequalities.

$$p\_{\min}^{\circ} \le p\_m^{\circ} \le p\_{\max}^{\circ} \tag{17}$$

due to changes in the actual pressures in a given pipe or in a given section of the pipeline due to

• Actuation systems to maintain the specified working pressure at pumping stations

$$p\_{\text{и min}}^{\circ} \le p\_{\text{и }m}^{\circ} \le p\_{\text{и max}}^{\circ} \tag{18}$$

Values Δ*D*<sup>э</sup>

where *D*<sup>э</sup>

ovalization.

min, *D*<sup>э</sup>

should take into account their increase.

*DOI: http://dx.doi.org/10.5772/intechopen.89036*

• At maximum operating stress *σ*<sup>э</sup>

*2.2.3 Pipeline wall thickness in operation*

account the main bending deformations.

possible to change the wall thickness downwards:

diameter *D*<sup>э</sup> of the pipeline, defined by (25).

• Due to corrosion and erosion damage

*δ*э

wall thickness should be used:

**151**

and strength conditions.

, as a rule, have a positive value due to the possible deformation

min, *<sup>D</sup>*<sup>э</sup>

min are minimum and maximum diameter in the zone of

For ovalized sections, the calculated determination of stresses according to (1)

Pipeline wall thickness *δ* has the most significant effect on operating stresses *σ*<sup>э</sup>

where *δ*<sup>п</sup> is design wall thickness and Δ*δ*<sup>т</sup> is technological tolerance for thickness. The change in wall thickness during pipe rolling can be neglected, taking into

When testing pipes at the manufacturing stage and during construction, it is

where *μ* is Poisson's ratio (0.3 ≤ *μ* ≤ 0.5) and Δ*D*<sup>э</sup> is a possible increase in the

<sup>Δ</sup>*δ*кэ <sup>¼</sup> *<sup>F</sup><sup>δ</sup> <sup>р</sup>*<sup>э</sup>

where *τ*<sup>э</sup> is operation time and *с*кэ is the rate of corrosion erosion damage. In the basic calculations of static strength according to the basic calculations of static strength according to the expressions (1)–(28), the minimum value of the

max <sup>¼</sup> *<sup>δ</sup>*<sup>п</sup> � <sup>Δ</sup>*δ*т, (26)

<sup>Δ</sup>*δ*<sup>и</sup> <sup>¼</sup> *<sup>F</sup><sup>δ</sup> <sup>р</sup>*и, <sup>Δ</sup>*D*<sup>э</sup> f g , *<sup>μ</sup>* , (27)

min ¼ *δ*<sup>п</sup> � Δ*δ*<sup>т</sup> � Δ*δ*<sup>и</sup> � Δ*δ*кэ*:* (29)

, *<sup>τ</sup>*<sup>э</sup> f g , *<sup>с</sup>*кэ , (28)

The calculated justification of static strength in the framework of the basic

max

The statistical variation of values, as well as diameters D, is due to:

max , (25)

The second factor of change in diameters *D*<sup>э</sup> can be ovalization of the cross-

under the action of test or operating modes with increased pressure.

maintaining the length of the perimeter of the pipeline):

calculations according to (1)–(21) should be mainly oriented:

• To the maximum values in *D*max in (23) and (25)

• Rolling sheet technology, which is a blank for pipes:

*δ*т min, *<sup>δ</sup>*<sup>т</sup>

• Due to plastic deformations from high-pressure tests *р*<sup>и</sup>

section during transportation, construction, and operation (usually while

*Combined Calculated, Experimental and Determinated and Probable Justifications…*

*<sup>D</sup>*<sup>э</sup> <sup>¼</sup> *FD <sup>D</sup>*<sup>э</sup>

• Deterministic design and actual operational differences of hydrostatic pressures Δ*р*<sup>г</sup> from changes in the profile of the heights of laying pipelines

$$
\Delta p\_{\rm r\ min}^{\rm 3} \le \Delta p\_{\rm r\ m}^{\rm 3} \le \Delta p\_{\rm r\ max}^{\rm 3} \tag{19}
$$


$$
\Delta p\_{\text{rc \ min}} \le \Delta p\_{\text{rc \ m}} \le \Delta p\_{\text{rc \ max}} \tag{20}
$$

• Deterministic design and actual operating pressure changes due to external effects on the pipeline (seismic, temperature, vibration, aero-hydrodynamic)

$$
\Delta p\_{\text{в min}} \le \Delta p\_{\text{в }m} \le \Delta p\_{\text{в max}} \tag{21}
$$

In the basic calculations using expressions (1)–(11), for deterministic and statistical estimates of the static strength of pipelines, pressure components should be included when the pipelines are operating at maximum design conditions:

$$p\_{\mathbf{p}}^{\flat} = p^{\flat} + \sum \Delta p^{\flat}.\tag{22}$$

Deeper in scope, cyclic pressure changes due to software changes in pipeline operation modes (start-up, shutdowns, performance change—throughput) are subject to accounting for cyclic strength and durability calibration calculations. Statistical information on the change in pressure is obtained from the registration data at pumping stations.

### *2.2.2 Diameter of pipelines in operation*

The diameter D, which is included in expressions (1), (4)–(6), and the pipeline, is characterized by the scattering of its actual values. It is due to pipe manufacturing technology and is reflected in the maximum and minimum technological tolerances on the diameter Δ*D*т:

$$\left\{D\_{\text{max}}^{\text{r}}, D\_{\text{min}}^{\text{r}}\right\} = D\_{\text{n}} \pm \Delta D\_{\text{r}},\tag{23}$$

where *D*п is design diameter.

Values Δ*D*<sup>т</sup> in either direction may be the same or different.

The diameters of *D*<sup>э</sup> pipes in various parts of pipelines that are fixed during operation during inspections and diagnostics of pipelines may differ from the diameters *D*<sup>т</sup> after the manufacture of pipes:

$$\left\{D\_{\text{max}}^{\circ}, D\_{\text{min}}^{\circ}\right\} = \left\{D\_{\text{max}}^{\circ}, D\_{\text{min}}^{\circ}\right\} \pm \Delta D^{\circ}.\tag{24}$$

*Combined Calculated, Experimental and Determinated and Probable Justifications… DOI: http://dx.doi.org/10.5772/intechopen.89036*

Values Δ*D*<sup>э</sup> , as a rule, have a positive value due to the possible deformation under the action of test or operating modes with increased pressure.

The second factor of change in diameters *D*<sup>э</sup> can be ovalization of the crosssection during transportation, construction, and operation (usually while maintaining the length of the perimeter of the pipeline):

$$D^{\circ} = F\_D \{ D^{\circ}\_{\text{min}}, D^{\circ}\_{\text{max}} \}, \tag{25}$$

where *D*<sup>э</sup> min, *D*<sup>э</sup> min are minimum and maximum diameter in the zone of ovalization.

For ovalized sections, the calculated determination of stresses according to (1) should take into account their increase.

The calculated justification of static strength in the framework of the basic calculations according to (1)–(21) should be mainly oriented:


#### *2.2.3 Pipeline wall thickness in operation*

• Actuation systems to maintain the specified working pressure at pumping

• Deterministic design and actual operational differences of hydrostatic pressures Δ*р*<sup>г</sup> from changes in the profile of the heights of laying pipelines

<sup>г</sup> min <sup>≤</sup> <sup>Δ</sup>*р*<sup>э</sup>

<sup>н</sup> *<sup>m</sup>* <sup>≤</sup> *<sup>р</sup>*<sup>э</sup>

<sup>г</sup> *<sup>m</sup>* <sup>≤</sup> <sup>Δ</sup>*р*<sup>э</sup>

Δ*р*гс min ≤ Δ*р*гс *<sup>m</sup>* ≤ Δ*р*гс max (20)

Δ*р*<sup>в</sup> min ≤ Δ*р*<sup>в</sup> *<sup>m</sup>* ≤ Δ*р*<sup>в</sup> max (21)

� � <sup>¼</sup> *<sup>D</sup>*<sup>п</sup> � <sup>Δ</sup>*D*т, (23)

• Deterministic design pressure changes due to changes in hydraulic resistance to the movement of oil and oil products (due to changes in flow areas, viscosity,

• Deterministic design and actual operating pressure changes due to external effects on the pipeline (seismic, temperature, vibration, aero-hydrodynamic)

In the basic calculations using expressions (1)–(11), for deterministic and statistical estimates of the static strength of pipelines, pressure components should be included when the pipelines are operating at maximum design conditions:

<sup>р</sup> <sup>¼</sup> *<sup>р</sup>*<sup>э</sup> <sup>þ</sup>XΔ*р*<sup>э</sup>

Deeper in scope, cyclic pressure changes due to software changes in pipeline operation modes (start-up, shutdowns, performance change—throughput) are subject to accounting for cyclic strength and durability calibration calculations. Statistical information on the change in pressure is obtained from the registration

The diameter D, which is included in expressions (1), (4)–(6), and the pipeline, is characterized by the scattering of its actual values. It is due to pipe manufacturing technology and is reflected in the maximum and minimum technological tolerances

The diameters of *D*<sup>э</sup> pipes in various parts of pipelines that are fixed during operation during inspections and diagnostics of pipelines may differ from the

> max, *<sup>D</sup>*<sup>т</sup> min � � � <sup>Δ</sup>*D*<sup>э</sup>

<sup>н</sup> max (18)

<sup>г</sup> max (19)

*:* (22)

*:* (24)

*р*э <sup>н</sup> min <sup>≤</sup>*р*<sup>э</sup>

Δ*р*<sup>э</sup>

and temperature of the transported working fluid)

*р*э

*D*т max, *<sup>D</sup>*<sup>т</sup> min

Values Δ*D*<sup>т</sup> in either direction may be the same or different.

• Deterministic design pressure changes Δ*р*гс

stations

*Probability, Combinatorics and Control*

data at pumping stations.

on the diameter Δ*D*т:

**150**

*2.2.2 Diameter of pipelines in operation*

where *D*п is design diameter.

diameters *D*<sup>т</sup> after the manufacture of pipes:

*D*э max, *<sup>D</sup>*<sup>э</sup> min � � <sup>¼</sup> *<sup>D</sup>*<sup>т</sup>

Pipeline wall thickness *δ* has the most significant effect on operating stresses *σ*<sup>э</sup> and strength conditions.

The statistical variation of values, as well as diameters D, is due to:

• Rolling sheet technology, which is a blank for pipes:

$$\left\{\delta^{\mathrm{r}}\_{\mathrm{min}}, \delta^{\mathrm{r}}\_{\mathrm{max}}\right\} = \delta\_{\mathrm{n}} \pm \Delta \delta\_{\mathrm{r}},\tag{26}$$

where *δ*<sup>п</sup> is design wall thickness and Δ*δ*<sup>т</sup> is technological tolerance for thickness. The change in wall thickness during pipe rolling can be neglected, taking into account the main bending deformations.

When testing pipes at the manufacturing stage and during construction, it is possible to change the wall thickness downwards:

• Due to plastic deformations from high-pressure tests *р*<sup>и</sup>

$$
\Delta \delta\_{\mathfrak{u}} = F\_{\delta} \{ p\_{\mathfrak{u}}, \Delta D^{\mathfrak{u}}, \mu \},
\tag{27}
$$

where *μ* is Poisson's ratio (0.3 ≤ *μ* ≤ 0.5) and Δ*D*<sup>э</sup> is a possible increase in the diameter *D*<sup>э</sup> of the pipeline, defined by (25).

• Due to corrosion and erosion damage

$$
\Delta \delta\_{\mathsf{k}\mathfrak{a}} = F\_{\delta} \{ p^{\flat}, \tau^{\flat}, c\_{\mathsf{k}\mathfrak{a}} \}, \tag{28}
$$

where *τ*<sup>э</sup> is operation time and *с*кэ is the rate of corrosion erosion damage.

In the basic calculations of static strength according to the basic calculations of static strength according to the expressions (1)–(28), the minimum value of the wall thickness should be used:

$$
\delta\_{\rm min}^{\rm 0} = \delta\_{\rm 0} - \Delta \delta\_{\rm \rm } - \Delta \delta\_{\rm \rm 0} - \Delta \delta\_{\rm \rm \rm \rm 0}. \tag{29}
$$

Expression (29) under condition (14) will mean the maximum increase in operating stresses *σ*<sup>э</sup> .

### *2.2.4 Characteristics of mechanical properties*

Mechanical properties with strength characteristics (*σ*в, *σ*т), (*R*1, *R*2), as well as *р*<sup>э</sup> , *D*, *δ* are stochastic. In order to ensure and justify the static strength of pipelines ,their minimum values should be entered into the calculation. The statistical variation in the characteristics of the mechanical properties of pipe steels is determined by a set of technological factors:


$$\{\sigma\_\mathfrak{u}^\flat, \sigma\_\mathfrak{r}^\flat\} = F\_\sigma \{d\_3, t\_\mathfrak{n}, e\_\mathfrak{n}, t^\flat\}. \tag{30}$$

The statistical information about the values *σ*<sup>э</sup> is obtained on the basis of the

*Combined Calculated, Experimental and Determinated and Probable Justifications…*

In combination with the statistical data on the hazard values of the criterial characteristics *σ*оп in the form of tensile strengths *σ*<sup>в</sup> and yield strengths *σ*<sup>т</sup> (or

;*n<sup>σ</sup>* max <sup>¼</sup> *<sup>σ</sup>*<sup>в</sup> max

*<sup>n</sup><sup>σ</sup>* ср <sup>¼</sup> *<sup>σ</sup>*<sup>в</sup> *<sup>m</sup>*

tical interpretation, can be considered as secured if the normatively specified

To make decisions about the admissibility of safety margins, *n<sup>σ</sup>* should be esti-

*σ*э *m* , *σ*<sup>т</sup> *<sup>m</sup> σ*э *m*

The strength of the pipeline, determined by the allowable stresses in the statis-

According to these statistics, it is possible to quantify statistical variations of the coefficients *m*, *n*, *K*1, and *K*н. On this basis, you can make a conclusion about the

Failure to comply with conditions (36) and (37) requires making decisions about conducting refined basic and calibration calculations by deterministic and statistical

The accumulation of statistical information in the form of histograms of the main design parameters of strength makes it possible to proceed to a probabilistic analysis in the form of a distribution of strength. They are reflected in regulatory calculations for limiting states and limiting resistances [5] through the safety factors for the material, load, working conditions and purpose, and load in

• to obtain the probability density functions *р* of external and internal effects

design resistances (yield strength *σ<sup>Т</sup>* and strength *σв*) with the subsequent determination of the probability of failure *Рр* in areas, where areas with

);

• to construct probability functions *<sup>Р</sup> <sup>σ</sup>*<sup>э</sup> ð Þ and *<sup>Р</sup>*{*σ<sup>Т</sup>* , *<sup>σ</sup>в*} with the definition of the relationship between strength margins {*n<sup>Т</sup>* , *<sup>n</sup>в*} and given probabilities *<sup>Р</sup> <sup>σ</sup>*<sup>э</sup> ð Þ, *Р*{*σ<sup>Т</sup>* , *σв*}, corresponding to the volume of the initial statistical and probabilistic

*σ*э min , *σ*<sup>т</sup> max *σ*э min *:* (34)

*n<sup>σ</sup>* max ≥ *n<sup>σ</sup> <sup>m</sup>* ≥ *n<sup>σ</sup>* min ≥ *nσ*<sup>н</sup>*:* (36)

) and the corresponding design stresses *σ*<sup>э</sup> and

f g *m*min, *K*IImax, *K*<sup>н</sup> max, *n*min >f g *m*,*K*1,*K*н, *n* <sup>н</sup>*:* (37)

*:* (35)

, *D*, *δ* on the basis of the

analysis of the stress–strain state by statistical parameters *р*<sup>э</sup>

design resistances *R*1, *R*2), a scatter can be obtained *nσ*ð Þ *n*в, *n*<sup>т</sup> :

entire system of expressions (1)–(33).

*DOI: http://dx.doi.org/10.5772/intechopen.89036*

mated, and their average values

methods.

expression (25):

information.

**153**

*<sup>n</sup><sup>σ</sup>*min <sup>¼</sup> *<sup>σ</sup>*<sup>в</sup> min *σ*э max , *σ*<sup>т</sup> min *σ*э max

margin *nσ*<sup>н</sup>ð Þ *n*вн, *n*тн satisfies the inequality

strength of the pipeline, if combinations are performed:

**2.3 Basics of probabilistic strength calculations**

The essence of this analysis [1, 6, 8] is:

extremely low probabilities (*Рр* ≤10�<sup>7</sup>

(the number of pressure *р*<sup>э</sup>

For widely used pipe steels, the increase in strength characteristics *σ*т, *σ*<sup>в</sup> is usually combined with a decrease in ductility.

In the main calculations of the strength of pipelines, it is recommended to use statistical data on the reduction of strength characteristics:

$$\left\{\sigma\_{\mathfrak{u}}^{\flat}, \sigma\_{\mathfrak{r}}^{\flat}\right\} = \min\{\sigma\_{\mathfrak{u}}, \sigma\_{\mathfrak{r}}\}.\tag{31}$$

#### *2.2.5 Reflection of statistical factors of strength in margin*

Use in domestic and foreign basic regulatory calculations of the strength of the system of strength margins *n<sup>σ</sup>* (when calculating the permissible stresses ½ � *σ* ) and reliability coefficients *K*1, *K*2, *m*, and *n* (when calculating the limiting states and resistances) makes it possible to obtain a connection between them in the form of expression (5).

All coefficients of expression (5) in a deterministic form, taking into account the statistics of design parameters for expressions (12)–(31), reflect the general variation of design, technological, and operational strength factors.

The strength margins *n<sup>σ</sup>* of (5) in the deterministic basic and calibration statistical calculations take into account the level of the necessary reduction of operating stress *σ*<sup>э</sup> compared to dangerous stresses *σ*<sup>э</sup> max <sup>&</sup>lt;*σ*оп :

$$m\_{\sigma} = \frac{\sigma\_{\text{on}}}{\sigma\_{\text{max}}^{\text{3}}}.\tag{32}$$

At the same time, dangerous stresses *σ*оп are understood not only as deterministic but also as statistical limits of strength *σ*<sup>в</sup> (to exclude one-time static damage) and plasticity *σ*<sup>т</sup> (to exclude one-time static damage) and plasticity (to prevent the formation of unacceptable plastic deformations):

$$
\sigma\_{\rm on} = \min\{\sigma\_{\rm b}, \sigma\_{\rm r}\}.\tag{33}
$$

*Combined Calculated, Experimental and Determinated and Probable Justifications… DOI: http://dx.doi.org/10.5772/intechopen.89036*

The statistical information about the values *σ*<sup>э</sup> is obtained on the basis of the analysis of the stress–strain state by statistical parameters *р*<sup>э</sup> , *D*, *δ* on the basis of the entire system of expressions (1)–(33).

In combination with the statistical data on the hazard values of the criterial characteristics *σ*оп in the form of tensile strengths *σ*<sup>в</sup> and yield strengths *σ*<sup>т</sup> (or design resistances *R*1, *R*2), a scatter can be obtained *nσ*ð Þ *n*в, *n*<sup>т</sup> :

$$
\mathfrak{n}\_{\sigma\text{min}} = \left\{ \frac{\sigma\_{\text{a min}}}{\sigma\_{\text{max}}^{\text{2}}}, \frac{\sigma\_{\text{r min}}}{\sigma\_{\text{max}}^{\text{2}}} \right\}; \mathfrak{n}\_{\sigma\text{ max}} = \left\{ \frac{\sigma\_{\text{a max}}}{\sigma\_{\text{min}}^{\text{2}}}, \frac{\sigma\_{\text{r max}}}{\sigma\_{\text{min}}^{\text{2}}} \right\}. \tag{34}
$$

To make decisions about the admissibility of safety margins, *n<sup>σ</sup>* should be estimated, and their average values

$$m\_{\sigma \text{ \text{cp}}} = \left\{ \frac{\sigma\_{\text{b \text{ }m}}}{\sigma\_m^{\text{\text{3}}}}, \frac{\sigma\_{\text{r \text{ }m}}}{\sigma\_m^{\text{\text{3}}}} \right\}. \tag{35}$$

The strength of the pipeline, determined by the allowable stresses in the statistical interpretation, can be considered as secured if the normatively specified margin *nσ*<sup>н</sup>ð Þ *n*вн, *n*тн satisfies the inequality

$$n\_{\sigma \text{ max}} \ge n\_{\sigma \text{ } m} \ge n\_{\sigma \text{ min}} \ge n\_{\sigma \text{n}}.\tag{36}$$

According to these statistics, it is possible to quantify statistical variations of the coefficients *m*, *n*, *K*1, and *K*н. On this basis, you can make a conclusion about the strength of the pipeline, if combinations are performed:

$$\{m\_{\min}, K\_{\text{IImax}}, K\_{\text{H max}}, n\_{\min}\} > \{m, K\_1, K\_\text{u}, n\}\_\text{u}.\tag{37}$$

Failure to comply with conditions (36) and (37) requires making decisions about conducting refined basic and calibration calculations by deterministic and statistical methods.

#### **2.3 Basics of probabilistic strength calculations**

The accumulation of statistical information in the form of histograms of the main design parameters of strength makes it possible to proceed to a probabilistic analysis in the form of a distribution of strength. They are reflected in regulatory calculations for limiting states and limiting resistances [5] through the safety factors for the material, load, working conditions and purpose, and load in expression (25):

The essence of this analysis [1, 6, 8] is:


Expression (29) under condition (14) will mean the maximum increase in oper-

Mechanical properties with strength characteristics (*σ*в, *σ*т), (*R*1, *R*2), as well

• Chemical composition and structural structure (grain size *d*)

• Modes of thermal and thermomechanical (*t*т) processing

• Temporary factors of aging and degradation in time *τ*

*σ*э <sup>в</sup>, *σ*<sup>э</sup> т

statistical data on the reduction of strength characteristics:

*2.2.5 Reflection of statistical factors of strength in margin*

*σ*э <sup>в</sup>, *σ*<sup>э</sup> т

tion of design, technological, and operational strength factors.

stress *σ*<sup>э</sup> compared to dangerous stresses *σ*<sup>э</sup>

formation of unacceptable plastic deformations):

, *D*, *δ* are stochastic. In order to ensure and justify the static strength of pipelines ,their minimum values should be entered into the calculation. The statistical variation in the characteristics of the mechanical properties of pipe steels is

• The level of preliminary plastic deformations *е<sup>n</sup>* during sheet rolling, rolling of

tube blanks, and testing of pipes, sections, and sections of pipelines

<sup>¼</sup> *<sup>F</sup><sup>σ</sup> <sup>d</sup>*3, *<sup>t</sup>*п, *<sup>е</sup>n*, *<sup>t</sup>*

For widely used pipe steels, the increase in strength characteristics *σ*т, *σ*<sup>в</sup> is

In the main calculations of the strength of pipelines, it is recommended to use

Use in domestic and foreign basic regulatory calculations of the strength of the system of strength margins *n<sup>σ</sup>* (when calculating the permissible stresses ½ � *σ* ) and reliability coefficients *K*1, *K*2, *m*, and *n* (when calculating the limiting states and resistances) makes it possible to obtain a connection between them in the form of

All coefficients of expression (5) in a deterministic form, taking into account the statistics of design parameters for expressions (12)–(31), reflect the general varia-

The strength margins *n<sup>σ</sup>* of (5) in the deterministic basic and calibration statistical calculations take into account the level of the necessary reduction of operating

At the same time, dangerous stresses *σ*оп are understood not only as deterministic but also as statistical limits of strength *σ*<sup>в</sup> (to exclude one-time static damage) and plasticity *σ*<sup>т</sup> (to exclude one-time static damage) and plasticity (to prevent the

*<sup>n</sup><sup>σ</sup>* <sup>¼</sup> *<sup>σ</sup>*оп *σ*э max

max <*σ*оп :

<sup>э</sup> f g*:* (30)

*:* (32)

*σ*оп ¼ minf g *σ*в, *σ*<sup>т</sup> *:* (33)

<sup>¼</sup> minf g *<sup>σ</sup>*в, *<sup>σ</sup>*<sup>т</sup> *:* (31)

ating stresses *σ*<sup>э</sup>

expression (5).

**152**

as *р*<sup>э</sup>

.

*Probability, Combinatorics and Control*

*2.2.4 Characteristics of mechanical properties*

determined by a set of technological factors:

usually combined with a decrease in ductility.

There is a simple relationship between probability *Р* and the amount of initial statistical information:

$$m = \frac{i - 0, 5}{P},\tag{38}$$

from (40) using the specified probability parameters *Р* in the range of 10�<sup>4</sup> to 10�<sup>5</sup>

*Combined Calculated, Experimental and Determinated and Probable Justifications…*

On the basis of (5) and (41), it is possible to analyze changes in the regulatory strength margin *nв* taking into account the probabilistic characteristics of the oper-

1 � *Zp* � *Vσ<sup>в</sup>*

where *np* is margin of strength for a given probability *Р* and *np* is margin reduction ratio*nв*. The relationship between *np* and *Р* in (42) with *Vσ*<sup>в</sup> ¼ 0, 05 and *<sup>V</sup>σ*<sup>э</sup> <sup>¼</sup> 0, 08 is shown in **Figure 6**. From the data in **Figures 5** and **<sup>6</sup>**, it can be seen that the greatest influence on the allowable change in the strength margins *nв* is

• At their intersections with other transport systems (rail, high-voltage, pipeline), with non-compliance with the allowable distances from other

This approach becomes significant and necessary for those cases when the assigned service lives and estimated durability are developed, and the in-line

The probabilistic approach acquires its practical relevance in the critical sections

<sup>1</sup> <sup>þ</sup> *Zp* � *<sup>V</sup>σ*<sup>э</sup> <sup>¼</sup> *<sup>n</sup><sup>в</sup>* � *np*, (42)

. Refinement of probabilistic calcula-

and above.

of trunk pipelines:

**Figure 5.**

**155**

ational loading *σ*<sup>э</sup> and the limits of strength *σ*в:

*DOI: http://dx.doi.org/10.5772/intechopen.89036*

observed when *Р* decreases from 0.5 to 10�<sup>3</sup>

facilities and infrastructures

inspections show increased defectiveness.

*Scheme for assessing the impact of probability P on the strength margin.*

• On water transitions

*np* ¼ *n<sup>в</sup>*

tions of strength at lower *Р* does not make much practical sense.

• In zones of geological faults, landslides, and seismic effects

where *i* is the sequence number of the measured value and *n* is the total number of measurements.

With a commonly used sample of 20 measurements, the value is *<sup>Р</sup>* = 2.5�10�<sup>2</sup> (or 2.5%).

To estimate the values of *Р* at the level of 10�<sup>2</sup> (or 1%) it is necessary to make already 50 measurements, and for the probability of 10�<sup>4</sup> – 5000.

In statistical and probabilistic studies of the mechanical properties of structural steels, the volume of samples n is in the range of 20–22.000 [6, 10]. According to the histogram of the limit distribution functions *σ*т, *σ*в, the functional *F* is obtained for the strength margins *n*т, *n*в:

$$\{n\_T, n\_\sigma\} = F\{P(\sigma^\circ), P(\sigma\_T, \sigma\_\theta)\}\tag{39}$$

The number of laboratory samples of steel 17G1S, cut from pipes in the initial state and after 40 years of operation is 28.

To solve probabilistic problems of strength in terms of expression (39) in the zone of small probabilities of destruction *Рр*, a large amount of statistical information is needed with samples measured in the hundreds and thousands, which is practically impossible in many real cases. In this connection, it is more promising to use expressions (38) and (39), which allow f g *nT*, *n<sup>в</sup>* estimating reserves for a given probability *Р* of calculated characteristics, corresponding to the availability of experiments on the distribution functions, *<sup>Р</sup> <sup>σ</sup>*<sup>э</sup> ð Þ and *<sup>Р</sup>*{*σ<sup>Т</sup>* , *<sup>σ</sup>в*}, with the choices of tens and hundreds.

**Figure 5** shows the scheme for the implementation of a probabilistic analysis of reserves: along the ordinate axis, the probabilities *<sup>Р</sup> <sup>σ</sup>*<sup>э</sup> ð Þ and *<sup>Р</sup>*ð Þ *<sup>σ</sup><sup>в</sup>* on a scale corresponding to the normal distribution law. Then by the median values *σ*<sup>э</sup> *<sup>m</sup>* и *σ<sup>в</sup><sup>m</sup>* for the probability *Р* = 50% and for other values of *Р* (*Р* < 50%).

$$(n\_{\sigma\circ})\_m = \frac{(\sigma\_\circ)\_m}{(\sigma^\circ)\_m}; (n\_{\sigma\circ})\_p = \frac{(\sigma\_\circ)\_p}{(\sigma^\circ)\_p}.\tag{40}$$

If, according to the results of statistical processing of values *σ*<sup>э</sup> and *σ<sup>в</sup>* the parameters of their probability distributions are obtained—(the coefficients of variation *Vσ*<sup>э</sup> and *Vσ<sup>в</sup>* and their average values *σ*<sup>э</sup> *m* � � *<sup>m</sup>*and ð Þ *σ<sup>в</sup> <sup>m</sup>*, then the calculated values ð Þ *<sup>σ</sup><sup>в</sup> <sup>р</sup>* and *<sup>σ</sup>*<sup>э</sup> ð Þ*<sup>р</sup>* for a given probability *<sup>P</sup>* are obtained from the expressions.

$$\left\{ (\sigma^{\flat}) p, (\sigma\_{\sigma})\_p \right\} = \left\{ (\sigma^{\flat})\_m, (\sigma\_{\sigma})\_m \right\} \left( \mathbf{1} - Z\_p \{ V \sigma^{\flat}, V \sigma\_{\sigma} \} \right), \tag{41}$$

where *Zp* is distribution quantile depending on *Р.*

For coefficients of variation in the range of *Vσ*<sup>э</sup> и *Vσ<sup>в</sup>* in the range of 0.03–0.1 the calculated probabilities *Рр* are obtained when the margin factors *nσ<sup>в</sup>*>1, 8 are in the range of 10�<sup>15</sup> to 10�<sup>5</sup> .

With the currently existing banks of data on operational load *σ*<sup>э</sup> and mechanical properties of pipe steels (*σ*в, *σ*Т), it is more reasonable to consider not determining the values of *Рр* in the area of their low values, but determining the strength margins *Combined Calculated, Experimental and Determinated and Probable Justifications… DOI: http://dx.doi.org/10.5772/intechopen.89036*

from (40) using the specified probability parameters *Р* in the range of 10�<sup>4</sup> to 10�<sup>5</sup> and above.

On the basis of (5) and (41), it is possible to analyze changes in the regulatory strength margin *nв* taking into account the probabilistic characteristics of the operational loading *σ*<sup>э</sup> and the limits of strength *σ*в:

$$n\_p = n\_a \frac{\mathbf{1} - Z\_p \cdot V \sigma\_o}{\mathbf{1} + Z\_p \cdot V \sigma^\circ} = n\_a \cdot \overline{n}\_p,\tag{42}$$

where *np* is margin of strength for a given probability *Р* and *np* is margin reduction ratio*nв*. The relationship between *np* and *Р* in (42) with *Vσ*<sup>в</sup> ¼ 0, 05 and *<sup>V</sup>σ*<sup>э</sup> <sup>¼</sup> 0, 08 is shown in **Figure 6**. From the data in **Figures 5** and **<sup>6</sup>**, it can be seen that the greatest influence on the allowable change in the strength margins *nв* is observed when *Р* decreases from 0.5 to 10�<sup>3</sup> . Refinement of probabilistic calculations of strength at lower *Р* does not make much practical sense.

The probabilistic approach acquires its practical relevance in the critical sections of trunk pipelines:


There is a simple relationship between probability *Р* and the amount of initial

*<sup>n</sup>* <sup>¼</sup> *<sup>i</sup>* � 0, 5

where *i* is the sequence number of the measured value and *n* is the total number

With a commonly used sample of 20 measurements, the value is *<sup>Р</sup>* = 2.5�10�<sup>2</sup>

To estimate the values of *Р* at the level of 10�<sup>2</sup> (or 1%) it is necessary to make

In statistical and probabilistic studies of the mechanical properties of structural steels, the volume of samples n is in the range of 20–22.000 [6, 10]. According to the histogram of the limit distribution functions *σ*т, *σ*в, the functional *F* is obtained

The number of laboratory samples of steel 17G1S, cut from pipes in the initial

To solve probabilistic problems of strength in terms of expression (39) in the zone of small probabilities of destruction *Рр*, a large amount of statistical information is needed with samples measured in the hundreds and thousands, which is practically impossible in many real cases. In this connection, it is more promising to use expressions (38) and (39), which allow f g *nT*, *n<sup>в</sup>* estimating reserves for a given probability *Р* of calculated characteristics, corresponding to the availability of experiments on the distribution functions, *<sup>Р</sup> <sup>σ</sup>*<sup>э</sup> ð Þ and *<sup>Р</sup>*{*σ<sup>Т</sup>* , *<sup>σ</sup>в*}, with the choices of

**Figure 5** shows the scheme for the implementation of a probabilistic analysis of

;ð Þ *<sup>n</sup>σ<sup>в</sup> <sup>р</sup>* <sup>¼</sup> ð Þ *<sup>σ</sup><sup>в</sup> <sup>р</sup>*

*m* � �

� � <sup>1</sup> � *Zp <sup>V</sup>σ*<sup>э</sup> f g ,*Vσ<sup>в</sup>*

*σ*<sup>э</sup> ð Þ*<sup>р</sup>*

reserves: along the ordinate axis, the probabilities *<sup>Р</sup> <sup>σ</sup>*<sup>э</sup> ð Þ and *<sup>Р</sup>*ð Þ *<sup>σ</sup><sup>в</sup>* on a scale corresponding to the normal distribution law. Then by the median values *σ*<sup>э</sup>

*σ*<sup>э</sup> ð Þ*<sup>m</sup>*

If, according to the results of statistical processing of values *σ*<sup>э</sup> and *σ<sup>в</sup>* the parameters of their probability distributions are obtained—(the coefficients of var-

values ð Þ *<sup>σ</sup><sup>в</sup> <sup>р</sup>* and *<sup>σ</sup>*<sup>э</sup> ð Þ*<sup>р</sup>* for a given probability *<sup>P</sup>* are obtained from the expressions.

For coefficients of variation in the range of *Vσ*<sup>э</sup> и *Vσ<sup>в</sup>* in the range of 0.03–0.1 the calculated probabilities *Рр* are obtained when the margin factors *nσ<sup>в</sup>*>1, 8 are in

With the currently existing banks of data on operational load *σ*<sup>э</sup> and mechanical properties of pipe steels (*σ*в, *σ*Т), it is more reasonable to consider not determining the values of *Рр* in the area of their low values, but determining the strength margins

<sup>¼</sup> *<sup>σ</sup>*<sup>э</sup> ð Þ*m*,ð Þ *<sup>σ</sup><sup>в</sup> <sup>m</sup>*

for the probability *Р* = 50% and for other values of *Р* (*Р* < 50%).

iation *Vσ*<sup>э</sup> and *Vσ<sup>в</sup>* and their average values *σ*<sup>э</sup>

*<sup>σ</sup>*<sup>э</sup> ð Þ*р*,ð Þ *<sup>σ</sup><sup>в</sup> <sup>р</sup>* n o

the range of 10�<sup>15</sup> to 10�<sup>5</sup>

**154**

where *Zp* is distribution quantile depending on *Р.*

.

ð Þ *<sup>n</sup>σ<sup>в</sup> <sup>m</sup>* <sup>¼</sup> ð Þ *<sup>σ</sup><sup>в</sup> <sup>m</sup>*

f g *nT*, *<sup>n</sup><sup>в</sup>* <sup>¼</sup> *F P <sup>σ</sup>*<sup>э</sup> <sup>f</sup> ð Þ, *<sup>Р</sup>*ð*σ<sup>Т</sup>* , *<sup>σ</sup>в*Þg (39)

already 50 measurements, and for the probability of 10�<sup>4</sup> – 5000.

*<sup>P</sup>* , (38)

*<sup>m</sup>* и *σ<sup>в</sup><sup>m</sup>*

*:* (40)

*<sup>m</sup>*and ð Þ *σ<sup>в</sup> <sup>m</sup>*, then the calculated

� �, (41)

statistical information:

*Probability, Combinatorics and Control*

of measurements.

tens and hundreds.

for the strength margins *n*т, *n*в:

state and after 40 years of operation is 28.

(or 2.5%).

• In zones of geological faults, landslides, and seismic effects

This approach becomes significant and necessary for those cases when the assigned service lives and estimated durability are developed, and the in-line inspections show increased defectiveness.

#### **Figure 6.**

*Relative decrease in strength margins n with changing probabilistic characteristics of loading and mechanical properties.*

The scientific basis of these calculations is the entire system of calculation

the strength characteristics *σ*т, *σ*<sup>в</sup> (200 ≤ *σ<sup>Т</sup>* ≤ 800; 420 ≤ *σ<sup>в</sup>* ≤ 920 MPa).

*Combined Calculated, Experimental and Determinated and Probable Justifications…*

main stages of the life cycle—design, construction, operation, and

*<sup>δ</sup>* <sup>≥</sup> *<sup>р</sup>D<sup>в</sup>*

decommissioning. Currently two tasks are being solved:

and assigned margin *nσ*,ð Þ *nT*, *n<sup>в</sup>* :

calculated by expression (6) (**Figure 8**).

This system has been and remains basic in all international practice [1–4] to the present time with the development of methods for the design, construction, and operation of trunk pipelines to ensure their strength and deformability expressed in a gradual decrease in margins *n* (1.8 ≥ *nT* ≥ 1.2; 2.5 ≥ *n<sup>в</sup>* ≥ 1.7) и and an increase in

All uncertainty factors included in the calculations and reflecting the operating conditions, design, and construction technologies were taken into account by the coefficients (*nσ*, *nT*, *nв*) and the standard purpose of guaranteed mechanical prop-

A generalized analysis of trends and parameters of the development of pipeline transport of oil and oil pipelines and methods for calculating the strength is made in

Expressions (1) and (2) are initial in assessing the strength of pipelines at all the

• The direct task of a deterministic basic calculation of the wall thickness *δ* of the pipeline at the design stage with a preliminary feasibility study of the diameter *D<sup>в</sup>* and pressure *р* as well as with the selected structural material *σоп*,ð*σв*, *σ<sup>Т</sup>* )

> <sup>2</sup>½ � *<sup>σ</sup>* <sup>¼</sup> *<sup>р</sup>D<sup>в</sup>* � *<sup>n</sup><sup>σ</sup>* 2*σоп*

Under these conditions, the wall thickness *δ* cannot be less than the value

(43), deterministic calibration calculations are performed with the following

½ � *р* ≤ 2

*σоп nσ*

• Check of permissible operating pressure [*р*] at specified

At the stages of construction, operation, and decommissioning on the basis of

*:* (43)

*:* (44)

expressions (1)–(42) (**Figure 7**).

*Block diagram of regulatory foreign and domestic calculations.*

*DOI: http://dx.doi.org/10.5772/intechopen.89036*

erties (*σ*в, *σ*Т).

[6, 7].

**Figure 7.**

objectives:

**157**
