**2. Combined researches of strength and life time**

For installations of a nuclear energy in our country and abroad in the second half of twentieth century, the whole complex of fundamental and application developments [1–7, 11–14, 20–23] on the creation of normative strength calculations of the equipment and pipelines for nuclear power plants has been executed. Thus in our country special meaning had the solution of policy-making bodies that the scientific adviser of research developments on a justification of norms had been defined the Academy of Sciences of the USSR (The A.A. Blagonravov Institute for Machine Sciences—the IMASH), and the head development engineer of norms—the Ministry of medium machine building of the USSR (The N.A. Dollezhal Research and Development Institute of Power Engineering—NIKIET).

The same organizations making all prototype models of reactors for the NPP established the total statement about the strength before starting a reactor in service. Such norms developed both in the USSR [1, 12] and in the USA [14] subsequently were developed according to international standards set by the International Atomic Energy Agency—IAEA [13]. Compared to home norms of an NPP design [1, 12], basic sections on calculations, monitoring, probability safety assessment, and a justification of life time extension have been included.

*Probability Modeling Taking into Account Nonlinear Processes of a Deformation and Fracture… DOI: http://dx.doi.org/10.5772/intechopen.88233*

**Figure 5.** *The structure of the main task solution at making and service of the NPP equipment.*

Long-term experience of home nuclear branch organizations and the academic institutes has allowed to form (**Figure 5**) the schematic diagram of the combined solution of tasks in view:


In considered norms, there are two cores sections: calculation of principal dimensions predominantly by criteria of a static strength and the verification calculations on a different combination of limiting states at low-cycle and high-cycle, long-term, vibration, seismic loads with initiation of static, cyclic, brittle, corrosion fracture, and also cyclic forming and radiation damage.

gate valves, and legs arose due to the lack of suitable technical diagnostics of these situations [11, 19], when faults in the form of cracks because of technological or operational fault attained of the limiting, intolerable sizes (10<sup>2</sup> to 1.5 <sup>10</sup><sup>3</sup> mm), affecting 50–70% of carrying cross-section and creating sharp magnification of runner chattering. Thus, the analysis of such situations was not envisioned by

For installations of a nuclear energy in our country and abroad in the second half of twentieth century, the whole complex of fundamental and application developments [1–7, 11–14, 20–23] on the creation of normative strength calculations of the equipment and pipelines for nuclear power plants has been executed. Thus in our country special meaning had the solution of policy-making bodies that the scientific adviser of research developments on a justification of norms had been defined the Academy of Sciences of the USSR (The A.A. Blagonravov Institute for Machine Sciences—the IMASH), and the head development engineer of norms—the Ministry of medium machine building of the USSR (The N.A. Dollezhal Research and

The same organizations making all prototype models of reactors for the NPP established the total statement about the strength before starting a reactor in service. Such norms developed both in the USSR [1, 12] and in the USA [14] subse-

International Atomic Energy Agency—IAEA [13]. Compared to home norms of an NPP design [1, 12], basic sections on calculations, monitoring, probability safety

quently were developed according to international standards set by the

assessment, and a justification of life time extension have been included.

**2. Combined researches of strength and life time**

Development Institute of Power Engineering—NIKIET).

normative calculations.

*The accident on the "Fukushima-1" NPP.*

**Figure 3.**

**Figure 4.**

**194**

*The accident on the Chernobyl NPP (CNPP).*

*Probability, Combinatorics and Control*

#### **Figure 6.**

*The flow chart of strain-gauging of power equipment: 1—steam generator, 2—reactor, 3—pressure compensator, 4—bubble tank, 5—a main coolant pump, 6—commutators, and 7—registering apparatuses.*

In the capacity of the most responsible and dangerous NPP components, nuclear reactor vessels, pipelines, pumps, steam generators, reactors, and machine halls have been accepted (**Figure 6**).

In an NPP with water-moderated power reactors (VVER) in the capacity of the major critical parts, it is possible to consider also the basic attachment fittings of reactor covers such as studs. Thus, the computational-experimental analysis of stress-strain states, strengths, and life times of a connection joint of reactor covers is conducted by improved methods in more detail (**Figure 7**).

Modeling and full-scale researches have allowed to define detailed stress distributions on threads (**Figure 9**) and in a cover (**Figure 10**). These facts have given the chance to obtain real history of service impacts and nominal and local stresses

*Probability Modeling Taking into Account Nonlinear Processes of a Deformation and Fracture…*

on all parts of a reactor main joint.

*DOI: http://dx.doi.org/10.5772/intechopen.88233*

*Zones and points of placing of measuring gauges on the NPP equipment.*

**Figure 8.**

**Figure 9.**

**197**

*A stress loading of a stud attachment fitting of a reactor cover.*

For reactor installations of home production, such analysis was fulfilled [2–4, 11, 15, 16] jointly by the academic institutes, head branch research, and designer organizations on all prototype models of reactors in our country and abroad (Bulgaria, Finland, Hungary, Czech, and China) with application of the foremost methods: model researches of covers, studs, pressing rings on models from stress-optical and metallic materials, full-scale researches on reactors at preoperational tests on all regimes (including emergency), and also at an initial stage (till 1–3 years) of service.

In particular, the fifth unit of the Kozloduy NPP (Bulgaria) has been developed and implemented [15] after a most complicated program of full-scale researches by methods of a strain measurement, a thermometry, a vibrometry for all components of a primary loop with 1000 measuring points of local stresses, pressure pulsations, and temperatures (**Figure 8**).

**Figure 7.** *The scheme of strain-gauging of a threaded connection of an attachment fitting of a cover.*

*Probability Modeling Taking into Account Nonlinear Processes of a Deformation and Fracture… DOI: http://dx.doi.org/10.5772/intechopen.88233*

**Figure 8.** *Zones and points of placing of measuring gauges on the NPP equipment.*

Modeling and full-scale researches have allowed to define detailed stress distributions on threads (**Figure 9**) and in a cover (**Figure 10**). These facts have given the chance to obtain real history of service impacts and nominal and local stresses on all parts of a reactor main joint.

**Figure 9.** *A stress loading of a stud attachment fitting of a reactor cover.*

In the capacity of the most responsible and dangerous NPP components, nuclear reactor vessels, pipelines, pumps, steam generators, reactors, and machine halls

*The flow chart of strain-gauging of power equipment: 1—steam generator, 2—reactor, 3—pressure compensator, 4—bubble tank, 5—a main coolant pump, 6—commutators, and 7—registering apparatuses.*

In an NPP with water-moderated power reactors (VVER) in the capacity of the major critical parts, it is possible to consider also the basic attachment fittings of reactor covers such as studs. Thus, the computational-experimental analysis of stress-strain states, strengths, and life times of a connection joint of reactor covers is

For reactor installations of home production, such analysis was fulfilled [2–4, 11, 15, 16] jointly by the academic institutes, head branch research, and designer organizations on all prototype models of reactors in our country and abroad (Bulgaria, Finland, Hungary, Czech, and China) with application of the foremost methods: model researches of covers, studs, pressing rings on models from stress-optical and metallic materials, full-scale researches on reactors at preoperational tests on all regimes (including emergency), and also at an initial stage (till 1–3 years) of service. In particular, the fifth unit of the Kozloduy NPP (Bulgaria) has been developed and implemented [15] after a most complicated program of full-scale researches by methods of a strain measurement, a thermometry, a vibrometry for all components of a primary loop with 1000 measuring points of local stresses, pressure pulsations,

conducted by improved methods in more detail (**Figure 7**).

*The scheme of strain-gauging of a threaded connection of an attachment fitting of a cover.*

have been accepted (**Figure 6**).

*Probability, Combinatorics and Control*

**Figure 6.**

and temperatures (**Figure 8**).

**Figure 7.**

**196**

the Fukushima-1 in Japan) added additional information baseline for such

• From the position of strengths (in its multicriteria expression)

• From the position of life time (in time and cyclic statement)

• From the position of inadmissibility of large plastic strains

To the traditional solution of a problem of service safety [2, 6–10, 20–25], three

*Probability Modeling Taking into Account Nonlinear Processes of a Deformation and Fracture…*

Traditional methods of strength justification were founded on a complex of determined characteristics of mechanical properties of materials and fracture criteria (yield point—σ*y*, ultimate strength—σ*u*, fatigue limit—σ1, and long-term strength—σ*lt*). On the basis of these parameters of strength and fracture (present in standard and technical specifications for reactor structural materials), the status of safety and life time margins (*n*σ, *nN*, *nτ*) has been generated. These margins are included in the reference, educational, and standard literature [1, 2, 12, 20–26]. Today, a common system of criteria and strength margins guaranteeing a fracture of nonadmission for equipment components at observance of the given service

Mathematical modeling at the determined normative requirements to strength

• To modeling parts of rods, plates, and thin shell types on the basis of analytical solutions of the theory of a strength of materials and theory of elasticity

• To modeling real objects on the basis of numerical solutions by finite-element

Research of seismic impacts was the most complicated at computational and

• By finite-element method (FEM) for all parts of the first circuit (**Figure 12**)

It has thus appeared that most high stresses and damages from seismic loads

In addition to normative calculations of reactors on [1] at the complicated regimes (**Figure 14**) of an assembly, test and service loading (assembly, a

On the basis of such modeling, nominal σ*<sup>n</sup>* and maximum local σmax stresses in concentration zones were defined. However, in these traditional approaches, normative materials often did not contain the direct data quantitatively instituting strength and life time of considered objects taking into account a statistical property of parameters σ*y*, σ*u*, σ1, and σ*lt*. Occurring actually dissipation of parameters for strength calculation and life time of a NPP environment is caused by instability of manufacturing procedures at production of structural materials and NPP bearing parts (reactor vessels, pipelines, pumps, and heat exchangers). In the last decades, this deficiency has been eliminated, and the sphere of the traditional analysis of serviceability of the NPP equipment includes the theory and criteria of life time and

method, finite difference method, and integral equations method

• By methods of physical modeling of a reactor with reactor internals

occur at the zone of attaching of pipelines to a reactor vessel.

development.

conditions is developed.

experimental modeling:

(**Figure 13**)

reliability [2, 20–27].

**199**

and life time came down to two approaches:

groups of approaches had a direct ratio:

*DOI: http://dx.doi.org/10.5772/intechopen.88233*

### **Figure 10.**

*Stress distribution diagrams in a cover, flanges, and studs.*

Computational and special experimental test bench researches of a dynamic stress loading and cyclical damages from seismic loads had a particular actuality.

On metallic modeling studs with a diameter from М12 to М110, data about life time on the basis of 10<sup>4</sup> –10<sup>5</sup> cycles have been obtained. These data have allowed to justify improved margins on strength and life time of analyzed studs.

The principal great value in results these researches had that facts that the maximum accumulated damages (to 70%) arose in regimes multiple tightening and seal failure of caps (**Figure 11**). This fact has demanded work on special activities to decrease the indicated damages [15, 16].

Formation of development trends at the standardization instituting serviceability and safety of a nuclear (power-generating equipment went in a direction of specification and complicating of applied methods and criteria [1–3, 11, 20–23]. Thus, accidents and disasters (the TMI in the USA, the CNPP in the USSR, and

*Probability Modeling Taking into Account Nonlinear Processes of a Deformation and Fracture… DOI: http://dx.doi.org/10.5772/intechopen.88233*

the Fukushima-1 in Japan) added additional information baseline for such development.

To the traditional solution of a problem of service safety [2, 6–10, 20–25], three groups of approaches had a direct ratio:


Traditional methods of strength justification were founded on a complex of determined characteristics of mechanical properties of materials and fracture criteria (yield point—σ*y*, ultimate strength—σ*u*, fatigue limit—σ1, and long-term strength—σ*lt*). On the basis of these parameters of strength and fracture (present in standard and technical specifications for reactor structural materials), the status of safety and life time margins (*n*σ, *nN*, *nτ*) has been generated. These margins are included in the reference, educational, and standard literature [1, 2, 12, 20–26]. Today, a common system of criteria and strength margins guaranteeing a fracture of nonadmission for equipment components at observance of the given service conditions is developed.

Mathematical modeling at the determined normative requirements to strength and life time came down to two approaches:


Research of seismic impacts was the most complicated at computational and experimental modeling:


It has thus appeared that most high stresses and damages from seismic loads occur at the zone of attaching of pipelines to a reactor vessel.

On the basis of such modeling, nominal σ*<sup>n</sup>* and maximum local σmax stresses in concentration zones were defined. However, in these traditional approaches, normative materials often did not contain the direct data quantitatively instituting strength and life time of considered objects taking into account a statistical property of parameters σ*y*, σ*u*, σ1, and σ*lt*. Occurring actually dissipation of parameters for strength calculation and life time of a NPP environment is caused by instability of manufacturing procedures at production of structural materials and NPP bearing parts (reactor vessels, pipelines, pumps, and heat exchangers). In the last decades, this deficiency has been eliminated, and the sphere of the traditional analysis of serviceability of the NPP equipment includes the theory and criteria of life time and reliability [2, 20–27].

In addition to normative calculations of reactors on [1] at the complicated regimes (**Figure 14**) of an assembly, test and service loading (assembly, a

Computational and special experimental test bench researches of a dynamic stress loading and cyclical damages from seismic loads had a particular actuality. On metallic modeling studs with a diameter from М12 to М110, data about life

The principal great value in results these researches had that facts that the maximum accumulated damages (to 70%) arose in regimes multiple tightening and seal failure of caps (**Figure 11**). This fact has demanded work on special activities to

Formation of development trends at the standardization instituting serviceability and safety of a nuclear (power-generating equipment went in a direction of specification and complicating of applied methods and criteria [1–3, 11, 20–23]. Thus, accidents and disasters (the TMI in the USA, the CNPP in the USSR, and

justify improved margins on strength and life time of analyzed studs.

*The diagram of stresses change in studs at sealing of the main joint of the VVER-1000 reactor.*

–10<sup>5</sup> cycles have been obtained. These data have allowed to

time on the basis of 10<sup>4</sup>

**Figure 10.**

**Figure 11.**

**198**

decrease the indicated damages [15, 16].

*Stress distribution diagrams in a cover, flanges, and studs.*

*Probability, Combinatorics and Control*

#### **Figure 12.**

*Loads and stresses in a connecting pipes zone of a reactor vessel at seismic impacts for YOZ plain—the computational scheme (a), response stresses (MPa) on outside (b) and interior (c) surfaces; for XOZ plain the calculation scheme (d) and stresses (e) on an interior surface.*

Calculation on Eq. (1) with the use of deformation criteria can be brought

*Probability Modeling Taking into Account Nonlinear Processes of a Deformation and Fracture…*

regimes of a stress loading are established in Eq. (1) with introduction of two margins *nσ<sup>a</sup>* and *nN*. Then, the computational curve of permissible values [*ea*] (or

imental data, life time decrease from the number of cycles of basic loading *N*<sup>0</sup> (cycle) to two-frequency life time *N*<sup>2</sup> (cycle) is considered [1, 28] in equation

*<sup>N</sup>*<sup>2</sup> <sup>¼</sup> *<sup>N</sup>*0*=χ; <sup>χ</sup>* <sup>¼</sup> *<sup>f</sup> <sup>h</sup>*

*<sup>a</sup>* ]) and [*N*] is accepted as lower enveloping curves on each of these margins. For the complicated regimes of a two-frequency loading (low-frequency with

Equation (1) is true for a wide band of life times (10<sup>0</sup> ≤ *N* ≤ 1012). Permissible

*al*=*σ* <sup>∗</sup>

*ah* (MPa), accordingly) on the basis of generalization of exper-

*f* 0 � �*<sup>η</sup> σ* ∗ *a*0 *σ* ∗ *ah*

where χ and η are dimensionless characteristics of a material and parameters of a

The same approach is used to calculate life time taking into account the presence

The presence of initial or service defects of cracks type with depth *l* is reflected in calculations of survivability on the basis of the equations of linear and a nonlinear fracture mechanics by change of stresses *KI* (MPa�m1/2) and strains *KIe* intensity

> *KI<sup>с</sup> nK<sup>σ</sup>*

where *KIc* and *KIec* are the critical (fracture) stresses and strains intensity factors,

*; KIe* ≤

*KIe<sup>с</sup> nKe*

*<sup>a</sup>* ¼ *ea* � *E* (*Е*—a

*<sup>a</sup>*<sup>0</sup> (MPa), and high-frequency

*,* (2)

*,* (3)

together to calculate by force criteria (оn stresses) to accept *σ* <sup>∗</sup>

frequency *fl* = *f*<sup>0</sup> (hertz) and amplitude of stress *σ* <sup>∗</sup>

*The diagram of change of service loading parameters.*

*DOI: http://dx.doi.org/10.5772/intechopen.88233*

of contact (wear resistance) and seismic impacts.

intensity factors, accordingly (*nK<sup>σ</sup>* ≤ *nKe*).

factors [2, 20, 29]*.* For one-time brittle or a ductile fracture,

*KI* <sup>¼</sup> *<sup>σ</sup>* ffiffiffiffi

*<sup>π</sup><sup>l</sup>* <sup>p</sup> � *<sup>f</sup>* <sup>к</sup> <sup>≤</sup>

accordingly; *nK<sup>σ</sup>* and *nKe* are the dimensionless margins on stresses and strains

Reliability of equipment *PQR*(*τ*) along with the account of the probabilistic approach to estimations of mechanical properties of a structural material is defined

modulus of elasticity).

with *fh* (hertz) and *σ* <sup>∗</sup>

two-frequency regime.

[*σ* <sup>∗</sup>

**201**

**Figure 14.**

**Figure 13.** *A research of a dynamic state of a reactor simulator at seismic excitation.*

tightening of studs, a hydroshaping testing, launch, capacity change, emergency operations, and shut-down) for events of occurrence of high levels of stresses improved strength, and life time calculations were carried out on the equations type

$$\varepsilon\_{d} = \frac{1}{(4N)^{m\_{\ell}} + \frac{1+r\_{\varepsilon}}{1-r\_{\varepsilon}}} \cdot \ln \frac{1}{1-\boldsymbol{\mu}\_{\varepsilon}} + 0,\\ \text{43} \frac{\sigma\_{b} \left(1+\boldsymbol{\mu}\_{\varepsilon}\right)}{E \cdot N^{m\_{\sigma}} \left(1 + \frac{1+r\_{\sigma}}{1-r\_{\sigma}}\right)},\tag{1}$$

where *ea* is the amplitude of strain at a design regime; *N* is the life time at a crack initiation stage, in cycles; σ*<sup>b</sup>* is the ultimate strength of a material (400 ≤ σ*<sup>b</sup>* ≤ 950 MPa); ψ*<sup>c</sup>* is the reduction of area in a neck of a specimen at singlepass rupture (0.3 ≤ ψ*<sup>c</sup>* ≤ 0.7); *re*, *r*<sup>σ</sup> are the cycle asymmetry parameters on strains and stresses, accordingly; and *me*, *m*<sup>σ</sup> are the characteristics of a real material (0.5 ≤ *me* ≤ 0.6), (0.08 ≤ *m*<sup>σ</sup> ≤ 0.12). Values of parameters in Eq. (1) *ea,* ψ*c, re*, *r*σ*, me*, and *m*<sup>σ</sup> are relative and dimensionless.

*Probability Modeling Taking into Account Nonlinear Processes of a Deformation and Fracture… DOI: http://dx.doi.org/10.5772/intechopen.88233*

**Figure 14.** *The diagram of change of service loading parameters.*

Calculation on Eq. (1) with the use of deformation criteria can be brought together to calculate by force criteria (оn stresses) to accept *σ* <sup>∗</sup> *<sup>a</sup>* ¼ *ea* � *E* (*Е*—a modulus of elasticity).

Equation (1) is true for a wide band of life times (10<sup>0</sup> ≤ *N* ≤ 1012). Permissible regimes of a stress loading are established in Eq. (1) with introduction of two margins *nσ<sup>a</sup>* and *nN*. Then, the computational curve of permissible values [*ea*] (or [*σ* <sup>∗</sup> *<sup>a</sup>* ]) and [*N*] is accepted as lower enveloping curves on each of these margins.

For the complicated regimes of a two-frequency loading (low-frequency with frequency *fl* = *f*<sup>0</sup> (hertz) and amplitude of stress *σ* <sup>∗</sup> *al*=*σ* <sup>∗</sup> *<sup>a</sup>*<sup>0</sup> (MPa), and high-frequency with *fh* (hertz) and *σ* <sup>∗</sup> *ah* (MPa), accordingly) on the basis of generalization of experimental data, life time decrease from the number of cycles of basic loading *N*<sup>0</sup> (cycle) to two-frequency life time *N*<sup>2</sup> (cycle) is considered [1, 28] in equation

$$N\_2 = N\_0/\chi; \chi = \left(\frac{f\_h}{f\_0}\right)^{\frac{\sigma\_{ab}^\*}{\sigma\_{ab}^\*}},\tag{2}$$

where χ and η are dimensionless characteristics of a material and parameters of a two-frequency regime.

The same approach is used to calculate life time taking into account the presence of contact (wear resistance) and seismic impacts.

The presence of initial or service defects of cracks type with depth *l* is reflected in calculations of survivability on the basis of the equations of linear and a nonlinear fracture mechanics by change of stresses *KI* (MPa�m1/2) and strains *KIe* intensity factors [2, 20, 29]*.* For one-time brittle or a ductile fracture,

$$K\_I = \sigma \sqrt{\pi l} \cdot f\_\kappa \le \frac{K\_{Ic}}{n\_{K\_\sigma}} ; K\_{Ic} \le \frac{K\_{Ic}}{n\_{K\_\sigma}} , \tag{3}$$

where *KIc* and *KIec* are the critical (fracture) stresses and strains intensity factors, accordingly; *nK<sup>σ</sup>* and *nKe* are the dimensionless margins on stresses and strains intensity factors, accordingly (*nK<sup>σ</sup>* ≤ *nKe*).

Reliability of equipment *PQR*(*τ*) along with the account of the probabilistic approach to estimations of mechanical properties of a structural material is defined

tightening of studs, a hydroshaping testing, launch, capacity change, emergency operations, and shut-down) for events of occurrence of high levels of stresses improved strength, and life time calculations were carried out on the equations type

*Loads and stresses in a connecting pipes zone of a reactor vessel at seismic impacts for YOZ plain—the computational scheme (a), response stresses (MPa) on outside (b) and interior (c) surfaces; for XOZ plain—*

where *ea* is the amplitude of strain at a design regime; *N* is the life time at a crack

(400 ≤ σ*<sup>b</sup>* ≤ 950 MPa); ψ*<sup>c</sup>* is the reduction of area in a neck of a specimen at singlepass rupture (0.3 ≤ ψ*<sup>c</sup>* ≤ 0.7); *re*, *r*<sup>σ</sup> are the cycle asymmetry parameters on strains and stresses, accordingly; and *me*, *m*<sup>σ</sup> are the characteristics of a real material (0.5 ≤ *me* ≤ 0.6), (0.08 ≤ *m*<sup>σ</sup> ≤ 0.12). Values of parameters in Eq. (1) *ea,* ψ*c, re*, *r*σ*,*

<sup>þ</sup> <sup>0</sup>*,* <sup>43</sup> *<sup>σ</sup><sup>b</sup>* <sup>1</sup> <sup>þ</sup> *<sup>ψ</sup><sup>c</sup>* ð Þ

*<sup>E</sup>* � *<sup>N</sup><sup>m</sup><sup>σ</sup>* <sup>1</sup> <sup>þ</sup> <sup>1</sup>þ*r<sup>σ</sup>*

1�*r<sup>σ</sup>*

*,* (1)

� ln <sup>1</sup> 1 � *ψ<sup>c</sup>*

*ea* <sup>¼</sup> <sup>1</sup>

**Figure 13.**

**200**

**Figure 12.**

*me*, and *m*<sup>σ</sup> are relative and dimensionless.

ð Þ <sup>4</sup>*<sup>N</sup> me* <sup>þ</sup> <sup>1</sup>þ*re*

*A research of a dynamic state of a reactor simulator at seismic excitation.*

*the calculation scheme (d) and stresses (e) on an interior surface.*

*Probability, Combinatorics and Control*

1�*re*

initiation stage, in cycles; σ*<sup>b</sup>* is the ultimate strength of a material

also (**Figure 15**) on probabilistic characteristics of service stress loading *Q*(*τ*) and life time *RNτ*(*τ*) on the basis of distribution functions *f* of service impacts *Q<sup>s</sup>* (*τ*) and of ultimate loads *Q <sup>c</sup>*(*τ*) for the given life times *Nc*, *τc*. Thus, usually "trees of events" and "trees of failures" on experience of previous service of analogous technosphere objects are used. In such statement, the risk can be defined as

$$R(\boldsymbol{\pi}) = \mathbf{1} - P\_{\mathcal{QR}}(\boldsymbol{\pi}).\tag{4}$$

At the present time, the first approach was found to be the largest application. However, subsequently, the second approach appears to be deciding owing to its higher precision at estimations of the remaining strength, life time, and safety.

*A structure and evolution of standardization methods on the determined justification of strength and life time*

*Probability Modeling Taking into Account Nonlinear Processes of a Deformation and Fracture…*

*with the use of physical modeling of materials behavior at a static, cyclic, and long-term loading.*

On the basis of the normative documents developed and accepted to present safety of power engineering as a whole, and NPPs in particular, the level of individual risks and risks of a possibility of accidents and disaster initiation should be estimated. In the process of perfecting NPPs and their nuclear reactors, these risks were reduced and will be reduced from 10<sup>4</sup> to 10<sup>8</sup> 1/year and less. For example, the reactor of natural safety with plumbeous heat-transfer agent will have a probability of fracture considerably below 10<sup>8</sup> 1/year [8, 11]. Individual risks of

The great importance for the analysis, support, and improvement of safety of the considered equipment within the limits of dominating and active concepts,

*Comparative data about a radiation-ecological risk for different directions of the electric power manufacture.*

–10<sup>7</sup> 1/year (**Table 1**).

**3. An estimation of risks and service safety**

*DOI: http://dx.doi.org/10.5772/intechopen.88233*

**Figure 16.**

**Table 1.**

**203**

nonnuclear power engineering lay within the limits 10<sup>4</sup>

More oriented on the quantitative solution of a safety problem for complicated NPP installations, capable to cause severe accidents and disasters, are new methods and criteria of the following groups [2, 6–8, 11, 18–21, 24–26, 29–33]:


From the above-stated, the up-to-date justification of strength, life time, reliability, survivability, safety, and risks (**Figure 16**) should be based on results of corresponding calculations and tests with observance of the special and new requirements established by corresponding normative-legal documents.

For long-term operated high-risk installations of a nuclear energetic to which the NPPs with reactors of the VVER concern, the BN and the RBMK types' rate, initial parameters of strength, life time, risk, and safety were defined in an explicit and implicit kinds on stages of their design and commissioning on acting then norms and rules which place at the different displayed in **Figure 16** footsteps (on time and analysis level).

Thereupon, during estimations of their state, two scientific and application approaches are possible:


**Figure 15.** *The scheme of determination of reliability, failures, accidents, and disaster probability РQR*ð Þ*τ .*

*Probability Modeling Taking into Account Nonlinear Processes of a Deformation and Fracture… DOI: http://dx.doi.org/10.5772/intechopen.88233*

#### **Figure 16.**

also (**Figure 15**) on probabilistic characteristics of service stress loading *Q*(*τ*) and life time *RNτ*(*τ*) on the basis of distribution functions *f* of service impacts *Q<sup>s</sup>*

objects are used. In such statement, the risk can be defined as

*Probability, Combinatorics and Control*

different stages of accidents and disaster evolution)

• Risk (in probability-economic statement)

disasters)

analysis level).

life time

**Figure 15.**

**202**

approaches are possible:

and criteria of the following groups [2, 6–8, 11, 18–21, 24–26, 29–33]:

of ultimate loads *Q <sup>c</sup>*(*τ*) for the given life times *Nc*, *τc*. Thus, usually "trees of events" and "trees of failures" on experience of previous service of analogous technosphere

More oriented on the quantitative solution of a safety problem for complicated NPP installations, capable to cause severe accidents and disasters, are new methods

• Survivability (ability and steadiness of operation at occurrence of damages at

• Safety (taking into account the risk criteria and characteristics of accidents and

For long-term operated high-risk installations of a nuclear energetic to which the NPPs with reactors of the VVER concern, the BN and the RBMK types' rate, initial parameters of strength, life time, risk, and safety were defined in an explicit and implicit kinds on stages of their design and commissioning on acting then norms and rules which place at the different displayed in **Figure 16** footsteps (on time and

Thereupon, during estimations of their state, two scientific and application

• To realize stage by stage an estimation of the initial, exhausted, and remaining

• To estimate current life time, as initial for the given level of the service damage

that has been accumulated in the previous operating period

*The scheme of determination of reliability, failures, accidents, and disaster probability РQR*ð Þ*τ .*

From the above-stated, the up-to-date justification of strength, life time, reliability, survivability, safety, and risks (**Figure 16**) should be based on results of corresponding calculations and tests with observance of the special and new requirements established by corresponding normative-legal documents.

*R*ð Þ¼ *τ* 1 � *РQR*ð Þ*τ :* (4)

(*τ*) and

*A structure and evolution of standardization methods on the determined justification of strength and life time with the use of physical modeling of materials behavior at a static, cyclic, and long-term loading.*

At the present time, the first approach was found to be the largest application. However, subsequently, the second approach appears to be deciding owing to its higher precision at estimations of the remaining strength, life time, and safety.
