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

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 nonnuclear power engineering lay within the limits 10<sup>4</sup> –10<sup>7</sup> 1/year (**Table 1**).

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


#### **Table 1.**

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


*FQ* f g *σ;e; t; τ* ¼ *FQ f* <sup>1</sup>

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

strains; *f*<sup>1</sup> is the functional dependence, which includes *σ<sup>τ</sup>*

*FL*f g *σ;e; N; τ* ¼ *f* <sup>1</sup>

elastic *me* components of cyclic strains *ea* [2, 7, 20, 21].

*FK σ;e;KI;KIe* f g *; τ; t* ¼ *FK*

For a crack resistance estimation,

corresponding margins [2, 7, 20, 21]. For a survivability estimation,

intensity factors [2, 7, 20, 21].

*FS*f g *R*ð Þ*τ ; nR* ¼ *S*ð Þ*τ* ≤

**205**

For a risk and safety estimation,

1 *nR*

through margin factor on risk *nR* defined in advance.

accordingly, *e<sup>τ</sup>*

*<sup>c</sup>*, *e<sup>τ</sup>*

*στ y ny ; στ u nu ; στ lt nσ ; eτ c ne ; τc nτ ; <sup>f</sup>* <sup>2</sup>ð Þ *<sup>m</sup>*

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

where *FQ* is the functional characterizing dependence of stresses from actual force impacts *Q*; σ, *е* are the operating in time *τ* at temperature *t* stresses and

*nσ*, *nе*, *n<sup>τ</sup>* are the margins accordingly on yield and strength stress points, on stresses,

where *FL* is the functional characterizing dependence of life time from ampli-

corresponding to them, and from plasticity of material *ψ<sup>c</sup>* (the relative cross throat at fracture) and exponents for an equation of a fatigue curve for plastic *mp* and

> *σ nσ ; e ne ; KIc nK ; KIec nKe ; τc nτ ; tc nt ,* (7)

where *FK* is the functional characterizing dependence of stresses *KI* and strains *KIe* intensity factors, from their critical values *KIc* and *KIec*, from stresses *σ* and strains *е* levels, from critical time to fracture *τ<sup>c</sup>* and critical temperature *tc* with

where *FLld* is the functional characterizing dependence of survivability parameter from values of service stresses and strains, causing material damage *d*, from sizes of faults (cracks) *l*, from crack growth rates on number of cycles *dl=dN*, and time *dl=dτ* parameters, and also from values of ranges of stresses *KI* and strains *KIe*

where *F*<sup>R</sup> is the functional, characterizing risk *R*ð Þ*τ* as analytical dependence of probability *P*ð Þ*τ* of occurrence on installation of an emergency situation of this or that type and probable loss *U*ð Þ*τ* in case of its implementation; *FS* is the functional characterizing parameter of safety *S*ð Þ*τ* , which bundles parameters of really occurring risk with its critical *Rc*ð Þ*τ* (limiting) and admitted ½ � *R*ð Þ*τ* (acceptable) values

*FLld <sup>σ</sup>;e; <sup>l</sup>; <sup>N</sup>; <sup>τ</sup>;KI* f g *;KIe* <sup>¼</sup> *FLld* <sup>Δ</sup>*KI* ð Þ *;* <sup>Δ</sup>*KIe ; dl*

yield, strength, and long-term stress points of a material for deformation time *τ,*

strains, and time; *f* <sup>2</sup>ð Þ *m* is the functional dependence (in most cases, power) for a hardening parameter *m* in elastoplastic field of a deformation [2, 20, 21]. Для оценки ресурса по параметрам числа *N* циклов и времени *τ*

> *σa nσ ; ea ne ; Nf nN*

tudes of stresses *σa*, strains *ea*, number of fracture cycles *Nf* , and margins

*,* (5)

*<sup>y</sup>*, *σ<sup>τ</sup>*

*<sup>c</sup>* is the critical values (at fracture) of strains at this time; *ny*, *nu*, and

*<sup>f</sup>* <sup>2</sup> *<sup>σ</sup>y; <sup>ψ</sup>c; mp; me*

*,* (6)

*dN ; dl dτ ,* (8)

*FR*f g *P*ð Þ*τ ; U*ð Þ*τ* ¼ *R*ð Þ*τ ;* (9)

*Rc*ð Þ¼ *τ* ½ �¼ *R*ð Þ*τ FM*f g *Rc*ð Þ*τ ; nR; M*ð Þ*τ ; mM ,* (10)

*<sup>u</sup>*, and *σ<sup>τ</sup>*

*lt* that are the

### **Table 2.**

*Types extreme (emergency and catastrophic) situations and level of protectability from them of high-risk installations.*

strategies, norms, orders, and margins has the level of a scientific-practical justification of the predictable and acceptable risks characterizing generally regular and limiting states of these installations.

For all spectrum of technosphere installation types of emergency and catastrophic situations, the level of their protectability and types of accompanying risks at transition from standard conditions operation in regular states to emergency and catastrophic at service can be described (**Table 2**) as:


The complex calculation-experimental analysis of the initial and remaining service life of an NPP is founded first of all on an estimation of service damages accumulation conditions at different service regimes taking into account corresponding state equations, and also on the study of conditions of transition in limiting states taking into account service kinetics of mechanical properties of materials, criteria of strength, crack resistance, and survivability.

Generally termed procedures are implemented with the use of a complex criteria equations, computational equations, and design parameters applied to the analysis and definition of regular and limiting states of engineering objects. The complex criteria include the following equations:

For an estimation of static and long-term strength,

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

$$F\_Q\{\sigma, e, t, \tau\} = F\_Q\left\{ f\_1\left(\frac{\sigma\_y^\tau}{n\_\mathcal{y}}, \frac{\sigma\_u^\tau}{n\_u}, \frac{\sigma\_{lt}^\tau}{n\_\sigma}, \frac{\sigma\_c^\tau}{n\_\epsilon}, \frac{\tau\_c}{n\_\tau}\right), f\_2(m) \right\},\tag{5}$$

where *FQ* is the functional characterizing dependence of stresses from actual force impacts *Q*; σ, *е* are the operating in time *τ* at temperature *t* stresses and strains; *f*<sup>1</sup> is the functional dependence, which includes *σ<sup>τ</sup> <sup>y</sup>*, *σ<sup>τ</sup> <sup>u</sup>*, and *σ<sup>τ</sup> lt* that are the yield, strength, and long-term stress points of a material for deformation time *τ,* accordingly, *e<sup>τ</sup> <sup>c</sup>*, *e<sup>τ</sup> <sup>c</sup>* is the critical values (at fracture) of strains at this time; *ny*, *nu*, and *nσ*, *nе*, *n<sup>τ</sup>* are the margins accordingly on yield and strength stress points, on stresses, strains, and time; *f* <sup>2</sup>ð Þ *m* is the functional dependence (in most cases, power) for a hardening parameter *m* in elastoplastic field of a deformation [2, 20, 21].

Для оценки ресурса по параметрам числа *N* циклов и времени *τ*

$$F\_L\{\ \sigma, e, N, \tau\} = \left\{ f\_1\left(\frac{\sigma\_a}{n\_\sigma}, \frac{e\_a}{n\_\epsilon}, \frac{N\_f}{n\_N}\right) f\_2\left(\sigma\_\mathcal{V}, \mathcal{W}\_\varepsilon, m\_p, m\_\epsilon\right) \right\},\tag{6}$$

where *FL* is the functional characterizing dependence of life time from amplitudes of stresses *σa*, strains *ea*, number of fracture cycles *Nf* , and margins corresponding to them, and from plasticity of material *ψ<sup>c</sup>* (the relative cross throat at fracture) and exponents for an equation of a fatigue curve for plastic *mp* and elastic *me* components of cyclic strains *ea* [2, 7, 20, 21].

For a crack resistance estimation,

$$F\_K\{\left.\sigma,e,K\_I,K\_{I\varepsilon},\tau,t\right\} = F\_K\left\{\frac{\sigma}{n\_\sigma},\frac{e}{n\_\epsilon},\frac{K\_{I\varepsilon}}{n\_K},\frac{K\_{I\varepsilon}}{n\_{K\varepsilon}},\frac{\tau\_\varepsilon}{n\_\tau},\frac{t\_\varepsilon}{n\_t}\right\},\tag{7}$$

where *FK* is the functional characterizing dependence of stresses *KI* and strains *KIe* intensity factors, from their critical values *KIc* and *KIec*, from stresses *σ* and strains *е* levels, from critical time to fracture *τ<sup>c</sup>* and critical temperature *tc* with corresponding margins [2, 7, 20, 21].

For a survivability estimation,

$${}\_{L}F\_{L\_{\rm id}}\{\sigma,e,l,N,\tau,K\_{I},K\_{I\varepsilon}\} = F\_{L\_{\rm id}}\left\{ (\Delta K\_{I},\Delta K\_{I\varepsilon}), \left(\frac{dl}{dN},\frac{dl}{d\varepsilon}\right) \right\},\tag{8}$$

where *FLld* is the functional characterizing dependence of survivability parameter from values of service stresses and strains, causing material damage *d*, from sizes of faults (cracks) *l*, from crack growth rates on number of cycles *dl=dN*, and time *dl=dτ* parameters, and also from values of ranges of stresses *KI* and strains *KIe* intensity factors [2, 7, 20, 21].

For a risk and safety estimation,

$$F\_{\mathcal{R}}\{P(\boldsymbol{\tau}), U(\boldsymbol{\tau})\} = R(\boldsymbol{\tau});\tag{9}$$

$$F\_{\mathcal{S}}\{R(\boldsymbol{\tau}), n\_{R}\} = \mathcal{S}(\boldsymbol{\tau}) \le \frac{1}{n\_{R}} R\_{\boldsymbol{\epsilon}}(\boldsymbol{\tau}) = [R(\boldsymbol{\tau})] = F\_{\mathcal{M}}\{R\_{\boldsymbol{\epsilon}}(\boldsymbol{\tau}), n\_{R}, M(\boldsymbol{\tau}), m\_{M}\},\tag{10}$$

where *F*<sup>R</sup> is the functional, characterizing risk *R*ð Þ*τ* as analytical dependence of probability *P*ð Þ*τ* of occurrence on installation of an emergency situation of this or that type and probable loss *U*ð Þ*τ* in case of its implementation; *FS* is the functional characterizing parameter of safety *S*ð Þ*τ* , which bundles parameters of really occurring risk with its critical *Rc*ð Þ*τ* (limiting) and admitted ½ � *R*ð Þ*τ* (acceptable) values through margin factor on risk *nR* defined in advance.

strategies, norms, orders, and margins has the level of a scientific-practical justification of the predictable and acceptable risks characterizing generally regular and

*Types extreme (emergency and catastrophic) situations and level of protectability from them of high-risk*

For all spectrum of technosphere installation types of emergency and catastrophic situations, the level of their protectability and types of accompanying risks at transition from standard conditions operation in regular states to emergency and

• Regular situations—occurring at installations operation in the breaking points established by norms and rules; risks for them controlled; and protectability

aftereffects from them predicted, risks for them controlled; and protectability

• Design emergency situations—arise at a runout of installation out of breaking points of regular regimes with predicted and acceptable aftereffects; risks for

important parts of installation with high losses and human sacrifices and with necessity of carrying out a recovery work; risks for them heightened; and the

• Hypothetical emergency situations—can arise at the not forecast in advance scenarios of evolution with the greatest possible losses and sacrifices; are characterized by high risks; protectability from them low; and restoration of

The complex calculation-experimental analysis of the initial and remaining ser-

Generally termed procedures are implemented with the use of a complex criteria equations, computational equations, and design parameters applied to the analysis and definition of regular and limiting states of engineering objects. The complex

vice life of an NPP is founded first of all on an estimation of service damages accumulation conditions at different service regimes taking into account

materials, criteria of strength, crack resistance, and survivability.

For an estimation of static and long-term strength,

corresponding state equations, and also on the study of conditions of transition in limiting states taking into account service kinetics of mechanical properties of

• Out-of-design emergency situations—arise at nonreversible damages of

• Regime emergency situations—occurring at a shift from service standard conditions at regular operation of potentially dangerous installations;

limiting states of these installations.

*Probability, Combinatorics and Control*

**Table 2.**

*installations.*

from them increased

from them sufficient

installations is impossible

criteria include the following equations:

**204**

catastrophic at service can be described (**Table 2**) as:

them analyzed; and protectability from them partial

level of protectability from them insufficient

Thus, the level of installation safety functionally (*FM*) depends on values of critical risk, from margin on a risk *nR*, and also from costs *M*ð Þ*τ* of carrying out steps to decrease danger (risk) of installation and from effectiveness factor of these costs *mM* [8, 18, 24].

• Use of joint guard from severe accidents by new organization of working

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

• Introduction in practice of making and service of reactors with an in-depth analysis of risks of occurrence and propagation of the emergency and catastrophic situations, considering both probabilities of these situations and

• Inclusion in the analysis of heightened life time, risks and safety of reactors of such base criteria as strength, life time, reliability, survivability, physical

• Orientation to escalating requirements to safety of the NPP formed by national

• Elimination of unreasonable conservatism in already accepted normative and technical documents and introduction in the safety analysis of new threats and

• Statement as the corner-stone fundamental and applied researches of safety of nuclear reactors of problems of forming of unified methodical baseline on integrated study of external and interior impacts of a wide spectrum, responses to these impacts of critical important bearing elements of the NPP in linear and

• Setting, justification, control, and monitoring of the major parameters of life time and safety of the NPP operation at regular and emergency situations for confinement of margins on strength, life time, and risks in safety breaking

Problems of safety maintenance on the basis of the concept of risks generally should to be decided with the use of the determined, statistical, probability, and combined methods of fracture mechanics and mechanics of disasters. Probabilities *PS* of realization in an NPP of system threats can be presented with the use of

where *PN* is the probability of occurrence of the unfavorable event, stipulated by the human factor; *PT* is the probability of such event stipulated by a state of an NPP components; and *PО* is the probability of its occurrence stipulated by an environ-

The type of functional Eq. (11) remains the same and for probabilities of risks realization included in the analysis at design, making, and service of the NPP. The great importance thus has that facts that the role of the human factor in appraisal *PS* at change *PN* is defined not only human controllers and the personnel, their professional qualities and a physiological state, but the experts, making solutions on all

Probabilities *PT* essentially depend on the level of protectability of the NPP from accidents and disasters. This protectability is defined by quality of their initial and current state, extent of degradation of installations at the given stage of service, and diagnosing and monitoring level. Such position indicates direct interacting of

*PS* ¼ *FPS*f g *PN; PT; PO ,* (11)

nonlinear fields of a deformation, damages, and fractures

their aftereffects

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

points

mental exposure.

**207**

protectability, and economic justification

and international laws, norms, and rules

risks (including risks of terrorism)

functional *FPS* [2, 6, 8, 18, 24–26, 29, 32, 33]

level of the hierarchy by safety of the NPP.

master schedules both in regular and in the emergency situations promoting to decrease of negative and dangerous aftereffects of accident propagation

The mentioned complex functional criteria in Eqs. (1)–(10) allow to implement the full sequence of installation calculation for the purpose of providing for its service safety, beginning from strength parameters and completing at protectability parameters with acceptable values of risk both on a design stage, and at concrete stages of service, including a decision made about life time extension.

At an estimation of the remaining life time on resistance to cyclic fracture, levels of cyclical stresses, cycle asymmetry parameters, a stress concentration, cyclical properties of a material, service temperatures, special conditions of loading, and residual stresses and strains are subject to analysis. Under these data calculation processes and parameters of impacts, fracture stresses and life time are defined. On the basis of such definition are the functionals that resulted above in Eqs. (4)–(10), which include calculation dependences (state equations, curve of deformations and fractures, and strain and force criteria). In improved calculation zones of welded joints, a plastic deformation in the most loaded zones, variety of operating conditions and impacts, and dispersion of characteristics of mechanical properties [2, 10, 20–29, 31, 34–36] are considered.

As appears from Eqs. (1)–(10) the computational-experimental justification of static, long-term, and cyclic strength, life time, and risks included in comprehensive analysis of conditions of safety service of the NPP equipment at regular and unnominal situations, sampling of types of limiting states, calculation schemes and calculation cases, methods of the analysis of stress-strain states, methods of preliminary diagnostics of technical state, assignment of margins on strength and on life times, study of probabilities of limiting states reaching, an estimation of risks of accidents and disasters [2, 9–11, 20–36].

The built-up calculation of curve (permissible amplitudes of stresses and life time at a cyclic loading, and also of the maximum stresses and time before fracture in the long term) is carried out for an estimation of initial and remaining life time on the basis of a schematization of history of loading, sampling of computational schemes, and computational cases. The calculation of initial and remaining life time is carried out in two alternatives: an approximate calculation and improved calculation.

The concept of an estimation, a diagnosis, and a prediction of service life of the NPP is correlated with the sampling of state variables of the equipment on the level of wearing and life time exhaustion. To define the factors and parameters influencing on life time, it is necessary to attribute maximum deviations of wall width and errors in measurement, a staging of prediction of life time, results of resource and strength researches, levels of diagnosing of installations, and influence of engineering preliminary diagnostics efficiency on the level of a fracture risk.

On the basis of summarizing of results of a life time design justification of reactors, it is possible to establish a dependence of life time on commissioning terms, for example, an NPP with VVER type reactor of all generations (**Figure 17**). To a twenty-first century kickoff in our country and abroad, the design life time (expected life) has increased to 40–60 years; by 2025, the design life time can increase to 100 years [1, 3, 7, 11, 24].

Thus, the key problems of design, manufacture, service, upgrading, and a leading-out from service of nuclear units of the following (the fourth and the fifth) generations with heightened characteristics of life time and safety are:

• Transition to new principles of reactor core build-up, sharply reducing severe accident possibility with its melting

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


Problems of safety maintenance on the basis of the concept of risks generally should to be decided with the use of the determined, statistical, probability, and combined methods of fracture mechanics and mechanics of disasters. Probabilities *PS* of realization in an NPP of system threats can be presented with the use of functional *FPS* [2, 6, 8, 18, 24–26, 29, 32, 33]

$$P\_S = F\_{PS} \{ P\_N, P\_T, P\_O \}, \tag{11}$$

where *PN* is the probability of occurrence of the unfavorable event, stipulated by the human factor; *PT* is the probability of such event stipulated by a state of an NPP components; and *PО* is the probability of its occurrence stipulated by an environmental exposure.

The type of functional Eq. (11) remains the same and for probabilities of risks realization included in the analysis at design, making, and service of the NPP. The great importance thus has that facts that the role of the human factor in appraisal *PS* at change *PN* is defined not only human controllers and the personnel, their professional qualities and a physiological state, but the experts, making solutions on all level of the hierarchy by safety of the NPP.

Probabilities *PT* essentially depend on the level of protectability of the NPP from accidents and disasters. This protectability is defined by quality of their initial and current state, extent of degradation of installations at the given stage of service, and diagnosing and monitoring level. Such position indicates direct interacting of

Thus, the level of installation safety functionally (*FM*) depends on values of critical risk, from margin on a risk *nR*, and also from costs *M*ð Þ*τ* of carrying out steps to decrease danger (risk) of installation and from effectiveness factor of these costs

The mentioned complex functional criteria in Eqs. (1)–(10) allow to implement

At an estimation of the remaining life time on resistance to cyclic fracture, levels of cyclical stresses, cycle asymmetry parameters, a stress concentration, cyclical properties of a material, service temperatures, special conditions of loading, and residual stresses and strains are subject to analysis. Under these data calculation processes and parameters of impacts, fracture stresses and life time are defined. On the basis of such definition are the functionals that resulted above in Eqs. (4)–(10), which include calculation dependences (state equations, curve of deformations and fractures, and strain and force criteria). In improved calculation zones of welded joints, a plastic deformation in the most loaded zones, variety of operating conditions and impacts, and dispersion of characteristics of mechanical properties [2, 10,

As appears from Eqs. (1)–(10) the computational-experimental justification of static, long-term, and cyclic strength, life time, and risks included in comprehensive

The built-up calculation of curve (permissible amplitudes of stresses and life time at a cyclic loading, and also of the maximum stresses and time before fracture in the long term) is carried out for an estimation of initial and remaining life time on the basis of a schematization of history of loading, sampling of computational schemes, and computational cases. The calculation of initial and remaining life time is carried out in two alternatives: an approximate calculation and improved calculation.

The concept of an estimation, a diagnosis, and a prediction of service life of the NPP is correlated with the sampling of state variables of the equipment on the level of wearing and life time exhaustion. To define the factors and parameters influencing on life time, it is necessary to attribute maximum deviations of wall width and errors in measurement, a staging of prediction of life time, results of resource and strength researches, levels of diagnosing of installations, and influence of engineer-

On the basis of summarizing of results of a life time design justification of reactors, it is possible to establish a dependence of life time on commissioning terms, for example, an NPP with VVER type reactor of all generations (**Figure 17**). To a twenty-first century kickoff in our country and abroad, the design life time (expected life) has increased to 40–60 years; by 2025, the design life time can

Thus, the key problems of design, manufacture, service, upgrading, and a leading-out from service of nuclear units of the following (the fourth and the fifth)

• Transition to new principles of reactor core build-up, sharply reducing severe

ing preliminary diagnostics efficiency on the level of a fracture risk.

generations with heightened characteristics of life time and safety are:

analysis of conditions of safety service of the NPP equipment at regular and unnominal situations, sampling of types of limiting states, calculation schemes and calculation cases, methods of the analysis of stress-strain states, methods of preliminary diagnostics of technical state, assignment of margins on strength and on life times, study of probabilities of limiting states reaching, an estimation of risks of

the full sequence of installation calculation for the purpose of providing for its service safety, beginning from strength parameters and completing at protectability parameters with acceptable values of risk both on a design stage, and at concrete

stages of service, including a decision made about life time extension.

*mM* [8, 18, 24].

20–29, 31, 34–36] are considered.

*Probability, Combinatorics and Control*

accidents and disasters [2, 9–11, 20–36].

increase to 100 years [1, 3, 7, 11, 24].

**206**

accident possibility with its melting

#### **Figure 17.**

*Characteristics of initial design (full line) and the prolonged expected life (lives times) of the NPPs with type reactors VVER of the first–the fifth generations.*

parameters *PT* and *PN* taking into account base parameters of reliability and quality of technosphere installations.

Probabilities *PS*, as it is known, depend on occurrence of dangerous natural processes (earthquakes, floods, hurricanes, tsunami, landslides, etc.) and also from a state of the NPP installations and, hence, from *PT*. Adoption unreasonable (from the point of view of risks) *R* (*τ*) solutions on arrangement of technosphere installations and zones of population residing does parameter *PS* dependent and from *PN*.

Losses *US* from realization of system threats generally can be recorded through the functional *FUS*

$$U\_S = F\_{US}\{U\_N, U\_T, U\_O\},\tag{12}$$

*S*ð Þ¼ *τ R*ð Þ*τ =Z*ð Þ*τ* ≤ 1*:* (14)

Such conditions occurred at the moment of Chernobyl disaster (1986), last years the twentieth centuries at damages of collecting channels of steam generators PGV-1000 type, on boundary line of centuries at damages of welds to a weld zone of the

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

In **Figure 18** the major role of improving of all service parameters of the NPP, and first of all life time and safety which promote decrease of probabilities of accidents and disasters occurrence *P*(*τ*) and accompanying them losses *U*(*τ*) is

When for the equipment of the concrete NPP, the relative system risks *RS* (for population *RN*, for technosphere installations *RT*, and for environment *RO*) are defined, the surface of limiting states on values of these system risks *RS* varying on

> *R*2 *<sup>N</sup>* <sup>þ</sup> *<sup>R</sup>*<sup>2</sup>

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*<sup>T</sup>* <sup>þ</sup> *<sup>R</sup>*<sup>2</sup> *O:*

(15)

principal circuital pipeline to the steam generator [4, 11].

*The time history of the relative risk levels and protectability.*

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

some random paths *V*(*R*) can be plotted (**Figure 19**).

*RS* ¼

visible.

**Figure 19.**

**209**

*The surfaces dangerous and safe states on values of risks.*

**Figure 18.**

where *UN* is the losses caused to the population at interacting of primary and secondary knocking factors at realization of strategic system threats; *UT* is the losses caused to technosphere installations; and *UO* is the losses caused to an environment.

Values *UN*, *UT*, and *UO* can be measured both in natural units (for example, a death-roll of people, number of the blasted installations, and the square of injured territories) and in equivalents (for example, in economic, monetary parameters).

As a whole, in Russia, taking into account social and economic transformations, global processes to power supply and experience and prospects of nuclear energetics development based characteristics of risks *R* of accidents and disasters of the natural-technogenic character, defined by their losses *U* (or severity) and probability *P* (or quantity), have rather complicated character of a time history *τ* with a common trend to increment (**Figure 18**).

Accepting that the relative risks *R*ð Þ*τ* increase eventually owing to natural aging processes, degradation, accumulation of damages, and level of safety *S*ð Þ*τ* depends on the relative protectability *Z*ð Þ*τ* .

$$
\overline{R}(\tau) = F\_R\{\overline{U}(\tau), \overline{P}(\tau)\}; \overline{S}(\tau) = F\_S\{\overline{R}(\tau), \overline{Z}(\tau)\}, \tag{13}
$$

where the fact of accident and disaster occurrence will correspond to the condition

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

**Figure 18.** *The time history of the relative risk levels and protectability.*

$$
\overline{S}(\tau) = \overline{R}(\tau) / \overline{Z}(\tau) \le 1. \tag{14}
$$

Such conditions occurred at the moment of Chernobyl disaster (1986), last years the twentieth centuries at damages of collecting channels of steam generators PGV-1000 type, on boundary line of centuries at damages of welds to a weld zone of the principal circuital pipeline to the steam generator [4, 11].

In **Figure 18** the major role of improving of all service parameters of the NPP, and first of all life time and safety which promote decrease of probabilities of accidents and disasters occurrence *P*(*τ*) and accompanying them losses *U*(*τ*) is visible.

When for the equipment of the concrete NPP, the relative system risks *RS* (for population *RN*, for technosphere installations *RT*, and for environment *RO*) are defined, the surface of limiting states on values of these system risks *RS* varying on some random paths *V*(*R*) can be plotted (**Figure 19**).

$$
\overline{R}\_S = \sqrt{\overline{R}\_N^2 + \overline{R}\_T^2 + \overline{R}\_O^2}.\tag{15}
$$

**Figure 19.** *The surfaces dangerous and safe states on values of risks.*

parameters *PT* and *PN* taking into account base parameters of reliability and quality

*Characteristics of initial design (full line) and the prolonged expected life (lives times) of the NPPs with type*

Probabilities *PS*, as it is known, depend on occurrence of dangerous natural processes (earthquakes, floods, hurricanes, tsunami, landslides, etc.) and also from a state of the NPP installations and, hence, from *PT*. Adoption unreasonable (from the point of view of risks) *R* (*τ*) solutions on arrangement of technosphere installations and zones of population residing does parameter *PS* dependent and from *PN*. Losses *US* from realization of system threats generally can be recorded through

where *UN* is the losses caused to the population at interacting of primary and secondary knocking factors at realization of strategic system threats; *UT* is the losses caused to technosphere installations; and *UO* is the losses caused to an environment. Values *UN*, *UT*, and *UO* can be measured both in natural units (for example, a death-roll of people, number of the blasted installations, and the square of injured territories) and in equivalents (for example, in economic, monetary parameters). As a whole, in Russia, taking into account social and economic transformations, global processes to power supply and experience and prospects of nuclear energetics development based characteristics of risks *R* of accidents and disasters of the natural-technogenic character, defined by their losses *U* (or severity) and probability *P* (or quantity), have rather complicated character of a time history *τ* with a

Accepting that the relative risks *R*ð Þ*τ* increase eventually owing to natural aging processes, degradation, accumulation of damages, and level of safety *S*ð Þ*τ* depends

where the fact of accident and disaster occurrence will correspond to the

*<sup>R</sup>*ð Þ¼ *<sup>τ</sup> FR <sup>U</sup>*ð Þ*<sup>τ</sup> ; <sup>P</sup>*ð Þ*<sup>τ</sup> ; <sup>S</sup>*ð Þ¼ *<sup>τ</sup> FS <sup>R</sup>*ð Þ*<sup>τ</sup> ; <sup>Z</sup>*ð Þ*<sup>τ</sup> ,* (13)

*US* ¼ *FUS*f g *UN; UT; UO ,* (12)

of technosphere installations.

*reactors VVER of the first–the fifth generations.*

*Probability, Combinatorics and Control*

common trend to increment (**Figure 18**).

on the relative protectability *Z*ð Þ*τ* .

condition

**208**

the functional *FUS*

**Figure 17.**

To reach the acceptable protectability of the NPP equipment, implementation of complex steps on the decrease of system risks *RS* is necessary.

If on axes *RT*, *RN*, and *RO* to put aside classes from 1 to 7 for accidents and disasters on extent of increment of their severity (1—local, 2—object, 3—district, 4—regional, 5—national, 6—global, and 7—planetary), then the quantitative assessment of extent of the NPP safety and any of its components by criteria of risks is represented possible. Such estimation is given by the radius vector in threedimensional space "*RT*-*RN*-*RO*". The strength and life time improvement on all stages of installation design, making, and service should promote decrease in danger of these installations.

For an NPP transfer in safe states with the use of risk criteria *RN*, *RT*, and *RO* (**Figure 19**), it is necessary to reduce the possibility (risk *RS*) of uncontrollable emission of potentially dangerous substances *W* and energies *E* and also a loss of control (disruption of data flows *I*),

$$
\overline{R}\_{\mathcal{S}} = \sqrt{\overline{R}\_{W}^{2} + \overline{R}\_{E}^{2} + \overline{R}\_{I^{\*}}^{2}} \tag{16}
$$

survivability) appear to be essentially more effective, than liquidating of aftereffects of catastrophic situations (type of the TMI, the CNPP, and the Fukushima-1). Values *M* at a suitable justification of activities on the decrease of risks can be considerable (in *mM* time) less losses *US* caused to economy by vulnerability of the

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

As it was already mentioned, safety of nuclear energy installations *S*(*τ*), as well

behaviors, sources, and scenarios of unfavorable events both for each of considered installations and for the given set of installations (group, batch, and series) at occurrence and propagation of unfavorable events and also the information on aftereffects for installations, persons, and an environment at occurrence,

In nuclear energetics with reactors of all types and all generations (from the first to the fourth) prior to the beginning of the twenty-first century, at failure analysis, the basic attention was given to parameter *P*(*τ*) that defined reliability of safety operation of the NPP. Special meaning was added thus to the forestalling and prevention of the heaviest on the aftereffects of catastrophic situations with the peak damages—melting of the core and a radioactivity runout for breaking points of all guard barriers—casings of the fuel element, cartridge, reactor vessel, reactor hall, and containment. In this case, reactor vessel fracture is extremely dangerous.

Significant aftereffects arise also at fracture of the basic elements of the first circuit of a reactor vessel and collecting channels of steam generators, pumps, volume compensators, bubbler tanks, and also housings and runners of turbines in the second circuit. These fractures amount the sixth group of the limiting states

If while in service of the NPP because of occurrence of damages of parts of the first circuit has arisen a radioactivity outside breaking points of the NPP and there were thus threats of bombarding radiation for the population, then it is necessary to

The leakages caused by partial damages (faults of crack type or depressurizations of connectors) and creating threats for human controllers and the personnel in

The third group of limiting states should be bundled to the considerable damages of the above-termed parts of the first and the second circuit without a radioactivity runout for breaking points of an NPP, which are not demanding their mandatory

The second group of limiting states concern occurrence in bearing structures of the NPP of partial damages without a radioactivity runout for breaking points of the first circuit, not demanding their substitution, but demanding carrying out of

The first group of limiting states is amounted by those of them which are bundled to damages and the faults that have fallen outside the limits admissible under inspection norms and calculation, but not demanding mandatory carrying

as all other complicated engineering systems, on the given interval of time *τ* is defined in Eq. (13) by two basic quantities: probability *P*(*τ*) of unfavorable event occurrence (an unfavorable situation) and probable loss *U*(*τ*) from this event. Values *P*(*τ*) and *U*(*τ*) are generally statistically uncertain, demanding for their quantitative assessment of great volumes of the information on the nature,

equipment for all types of NPPs.

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

propagation, and liquidation of unfavorable events.

This event concerns the seventh group of limiting states.

the NPP concern the fourth group of limiting states.

substitution.

**211**

repair-and-renewal operations.

creating threats to the population, the NPP, and the environment.

attribute these events to the fifth group of dangerous limiting states.

**4. The analysis of limiting states**

or to reduce the relative risks of accidents and disasters *RN*, *RT*, and *RO* as in Eq. (15) and *RW*, *RE*, and *RI* as in Eq. (16).

This result can be attained by the creation of monitoring systems for diagnostics and monitoring of risk parameters *RN*, *RT*, *RO*, *RW*, *RE*, and *RI* and guard *Z*(*τ*), and also by the introduction in the analysis of safety *S*(*τ*) scenarios of occurrence and propagation of emergency and catastrophic situations.

The state, regional and object control, regulating and providing of safety *S*ð Þ*τ* by system risks criteria *RS*ð Þ*τ* comes to the qualitative both quantitative statistical and determined analysis on the given interval of time Δ*τ* of all service parameters and to implementation of complex activities on decrease of system risks from actual unacceptable *RS* to acceptable (admissible) levels [*RS*]:

$$\overline{R}\_{\mathcal{S}} = F\_{\mathcal{R}}\{\overline{P}\_{\mathcal{S}}, \overline{U}\_{\mathcal{S}}\} \le \left[\overline{R}\_{\mathcal{S}}\right] = \left(\mathbf{1}/n\_{\mathcal{S}}\right) \cdot \overline{R}\_{\mathcal{S}c} = F\_{\mathcal{R}}\{\left[\overline{P}\_{\mathcal{S}}\right], \left[\overline{U}\_{\mathcal{S}}\right] \} = F\_{\mathcal{M}}\{m\_{\mathcal{M}}^{-1}[\overline{M}]\},\tag{17}$$

where *nS* is the safety factor on system risks; *RSc* is the unacceptable (critical) risk; [*PS*] and *US* � � are the acceptable (permissible) probabilities and losses; [*M*] is the necessary acceptable expenditures for decrease of risks; and *mM* is the costeffectiveness ratio 1ð Þ ≤ *mM* ≤10 .

Safety of the NPP by criteria of risks can be considered ensured if the inequality *nS* ≥ 1 is attained.

The interval of time Δ*τ* for which risks *RS* are defined usually is accepted to equal 1 year (Δ*τ* = 1 year).

According to Eqs. (15) and (16), control and planning with the use of the criteria baseline grounded on risks come to following primal tasks:

To the development of scientifically well-founded methods of the analysis of risks *RS* and their basic quantities *PS* and *US*

To decision making about the level of allowable values [*RS*], [*PS*], and [*US*] with an estimation of margin values *ns*

To scientifically well-founded level of definition of necessary expenditures [*M*] on decreasing risks with sampling and improving of efficiency of these expenditures *mM*

Thus, predicting, monitoring, and forestalling of accidents and disasters for an NPP (including by improving of all parameters of strength, life time and

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

survivability) appear to be essentially more effective, than liquidating of aftereffects of catastrophic situations (type of the TMI, the CNPP, and the Fukushima-1). Values *M* at a suitable justification of activities on the decrease of risks can be considerable (in *mM* time) less losses *US* caused to economy by vulnerability of the equipment for all types of NPPs.

As it was already mentioned, safety of nuclear energy installations *S*(*τ*), as well as all other complicated engineering systems, on the given interval of time *τ* is defined in Eq. (13) by two basic quantities: probability *P*(*τ*) of unfavorable event occurrence (an unfavorable situation) and probable loss *U*(*τ*) from this event. Values *P*(*τ*) and *U*(*τ*) are generally statistically uncertain, demanding for their quantitative assessment of great volumes of the information on the nature, behaviors, sources, and scenarios of unfavorable events both for each of considered installations and for the given set of installations (group, batch, and series) at occurrence and propagation of unfavorable events and also the information on aftereffects for installations, persons, and an environment at occurrence, propagation, and liquidation of unfavorable events.
