2. Theoretical foundation of probabilistic modelling for engineering systems

#### 2.1. Statement for probabilistic modelling problems of engineering systems

The peculiarity of the above multi-level concept (Figure 1) ensuring operability of engineering systems in the form (1) is that each of the stages I–VIII considers its own, specific, calculating situations (Figure 2). At each stage, special fundamental problems of the mechanics of solids are solved:

• boundary problems for stress determining in the most loaded elements, in cross sections A and local volumes V(x, y, z)

$$\{\sigma\_{i\dot{\jmath}}, e\_{\dot{\imath}\jmath}\} = F\_{\sigma}\{Q, A, V(x, y, z)\}$$

• experimental problems of obtaining metal deformation diagrams (equations of state)

$$\{\sigma\_{\max}, e\_{\max}\} = F\_{\sigma, e} \{\sigma\_{i\circ}, e\_{i\circ}\}.$$

• experimental problems of estimating the criterial values of stresses and deformations corresponding to the achievement of the conditions for breaking strength (fracture)

$$\{\sigma\_c, e\_c\} = F\_c \{\sigma\_b, e\_f\}.$$

The nominal stresses σ<sup>n</sup> and deformations en, local stresses σ<sup>l</sup> and strains el, fracture stresses σ<sup>f</sup> and deformations ef, as well as actual and critical dimensions of technological and operational defects l and lf are used as the determining parameters. The values of fracture stresses and deformations characterize the limiting states and are determined taking into account the loading regime in terms of the number of cycles N and time τ, the fatigue diagrams (σ<sup>f</sup> � Nf) and (ef � Nf); long-term strength (σ<sup>f</sup> � τf) and (ef � τf); fracture toughness (σ<sup>f</sup> � lf) and (ef � lf).

The basic characteristics of strength Rσ, stiffness Rδ, steadiness Rλ, and durability RN,<sup>τ</sup> are considered for design situations when the values of all parameters are in the deterministic limits established by the project:

$$\{R\_{\sigma}, R\_{\delta}, R\_{\lambda}\} = F\_{\mathbb{R}}\{ (\sigma\_n, \mathfrak{e}\_n); (\sigma\_l, \mathfrak{e}\_l); (\mathfrak{e}\_f, \mathfrak{e}\_f) \}\tag{3}$$

$$R\_{\mathcal{N},\boldsymbol{\tau}} = F\_{\mathcal{N},\boldsymbol{\tau}}\{ (\mathcal{N},\boldsymbol{\tau}); (R\_{\mathcal{o}}, R\_{\mathcal{\delta}}, R\_{\lambda}) \}\tag{4}$$

U, given by margin factors for nominal and local stresses n<sup>σ</sup> and nσl, nominal and local strains ne

As probabilistic modelling methods evolved, problems (3) and (4) were considered in a probabilistic formulation, when the basic parameters are given by probability distribution functions. In this case, by the methods theory of probability and reliability theory, one can obtain diagrams of limiting states in the coordinates (σf, ef) � (N, τ, δ, λ, l) for different probabilities P

The reliability of engineering systems is determined in the presence of probability distribution functions of the basic parameters of operability. In a general case, reliability is estimated by the given probabilistic properties P of the characteristics of strength, stiffness, steadiness, durability:

The survivability of engineering systems is considered for design situations and beyond design situations with considering damage accumulation processes D. In engineering, practice damage is characterized by the sizes of the technological and operational defects L or scalar

Survivability can be estimated in deterministic and probabilistic formulation by using expression:

f g <sup>R</sup>σ;Rδ; <sup>R</sup><sup>λ</sup> <sup>U</sup> <sup>¼</sup> FR ð Þ <sup>σ</sup>n=nσ;en=ne ; <sup>σ</sup>l=nσ<sup>l</sup> ð Þ ;el=nel ; <sup>σ</sup><sup>f</sup> <sup>=</sup>nσ<sup>f</sup> ;ef <sup>=</sup>nef (5)

f g RN,<sup>τ</sup> <sup>U</sup> ¼ FN,τf g ð Þ N=nN; τ=n<sup>τ</sup> ;ð Þ Rσ;Rδ;R<sup>λ</sup> (6)

Probabilistic Modelling in Solving Analytical Problems of System Engineering

http://dx.doi.org/10.5772/intechopen.75686

7

PP,N, <sup>τ</sup> ¼ FPf g Pj ð Þ Q; N; τ ;ð Þ Rσ;Rδ;Rλ; RN,<sup>τ</sup> (7)

; l <sup>¼</sup> Fl <sup>l</sup> f g <sup>i</sup>ð Þ <sup>N</sup>; <sup>τ</sup> ; <sup>i</sup> <sup>¼</sup> <sup>1</sup>; <sup>m</sup> (8)

and nel, destroying stresses nσf, and deformations of nef:

of their realization (1%, 50%, 99%) (Figure 3).

measure of accumulated damage d:

D ¼ f g d; l ; d ¼ Fd

Figure 3. Probabilistic diagrams of limiting states.

N Nf ; τ τf ; l lf ; δ δf ; λ λf ; σ σf ; e ef

If statistical properties are taken into account for combinations ð Þ σn;en ; σ<sup>l</sup> ð Þ ;el ; σ<sup>f</sup> ;ef , the characteristics of strength, stiffness and resource can be determined with use quantile of probability

Figure 2. State diagram of engineering systems.

U, given by margin factors for nominal and local stresses n<sup>σ</sup> and nσl, nominal and local strains ne and nel, destroying stresses nσf, and deformations of nef:

$$\{R\_{\boldsymbol{\sigma}}, R\_{\boldsymbol{\delta}}, R\_{\boldsymbol{\lambda}}\}\_{\mathcal{U}} = F\_{\mathcal{R}}\left\{ (\sigma\_n/\mathbf{n}\_{\boldsymbol{\sigma}}, \mathbf{e}\_n/\mathbf{n}\_{\boldsymbol{\varepsilon}}); (\sigma\_l/\mathbf{n}\_{\boldsymbol{\sigma}l}, \mathbf{e}\_l/\mathbf{n}\_{\boldsymbol{\varepsilon}l}); \{\sigma\_f/\mathbf{n}\_{\boldsymbol{\sigma}f}, \mathbf{e}\_f/\mathbf{n}\_{\boldsymbol{\varepsilon}f}\} \right\} \tag{5}$$

$$\{R\_{\mathcal{N},\tau}\}\_{\mathcal{U}} = F\_{\mathcal{N},\tau}\{ (\mathcal{N}/\mathfrak{n}\_{\mathcal{N}}, \tau/\mathfrak{n}\_{\tau}); (R\_{\sigma}, R\_{\delta}, R\_{\lambda}) \}\tag{6}$$

As probabilistic modelling methods evolved, problems (3) and (4) were considered in a probabilistic formulation, when the basic parameters are given by probability distribution functions. In this case, by the methods theory of probability and reliability theory, one can obtain diagrams of limiting states in the coordinates (σf, ef) � (N, τ, δ, λ, l) for different probabilities P of their realization (1%, 50%, 99%) (Figure 3).

The reliability of engineering systems is determined in the presence of probability distribution functions of the basic parameters of operability. In a general case, reliability is estimated by the given probabilistic properties P of the characteristics of strength, stiffness, steadiness, durability:

$$P p\_{\cdot, \mathcal{N}, \mathsf{T}} = F p\{P | \, (Q, \mathcal{N}, \mathsf{T}); \left(R\_{\mathcal{O}}, R\_{\mathcal{S}}, R\_{\lambda}, R\_{\mathcal{N}, \mathsf{T}}\right)\}\tag{7}$$

The survivability of engineering systems is considered for design situations and beyond design situations with considering damage accumulation processes D. In engineering, practice damage is characterized by the sizes of the technological and operational defects L or scalar measure of accumulated damage d:

$$D = \{d, l\}; d = F\_d\{\frac{N}{N\_f}, \frac{\pi}{\tau\_f}, \frac{l}{l\_f}, \frac{\delta}{\delta\_f}, \frac{\lambda}{\lambda\_f}, \frac{\sigma}{\sigma\_f}, \frac{e}{e\_f}\}; l = F\_l\{l\_l(N, \tau), i = 1, m\} \tag{8}$$

Survivability can be estimated in deterministic and probabilistic formulation by using expression:

Figure 3. Probabilistic diagrams of limiting states.

f g σmax;emax ¼ Fσ,e σij;eij

• experimental problems of estimating the criterial values of stresses and deformations corresponding to the achievement of the conditions for breaking strength (fracture)

σ<sup>c</sup> f g ;ec ¼ Fc σb;ef

The nominal stresses σ<sup>n</sup> and deformations en, local stresses σ<sup>l</sup> and strains el, fracture stresses σ<sup>f</sup> and deformations ef, as well as actual and critical dimensions of technological and operational defects l and lf are used as the determining parameters. The values of fracture stresses and deformations characterize the limiting states and are determined taking into account the loading regime in terms of the number of cycles N and time τ, the fatigue diagrams (σ<sup>f</sup> � Nf) and (ef � Nf); long-term strength (σ<sup>f</sup> � τf) and (ef � τf); fracture toughness (σ<sup>f</sup> � lf) and (ef � lf). The basic characteristics of strength Rσ, stiffness Rδ, steadiness Rλ, and durability RN,<sup>τ</sup> are considered for design situations when the values of all parameters are in the deterministic

f g Rσ;Rδ;R<sup>λ</sup> ¼ FR ð Þ σn;en ; σ<sup>l</sup> ð Þ ;el ; σ<sup>f</sup> ;ef

acteristics of strength, stiffness and resource can be determined with use quantile of probability

If statistical properties are taken into account for combinations ð Þ σn;en ; σ<sup>l</sup> ð Þ ;el ; σ<sup>f</sup> ;ef

(3)

, the char-

RN, <sup>τ</sup> ¼ FN,τf g ð Þ N; τ ;ð Þ Rσ;Rδ;R<sup>λ</sup> (4)

limits established by the project:

6 Probabilistic Modeling in System Engineering

Figure 2. State diagram of engineering systems.

:

$$L(\pi) = \{l\_i(N, \pi), i = 1, m\} \tag{9}$$

Theoretical basis for calculating the reliability of structures in form (7) was formulated for case

In the 1960s and 1970s, Polovko et al. developed methods for probabilistic calculation of fatigue life RNτ of and reliability of machine parts PP,N,<sup>τ</sup> according to expressions (5) and (6). These methods have been used to calculate the probability diagrams of fatigue characteristics

According to F. Freudenthal (1956) and M. Shinozuki (1983), the reliability problem was formulated for a finite number of random variables X = {xi, i = 1, n} (geometrical parameters, material properties, loads, environmental factors, etc.). In this case, it was assumed that probabilistic properties of random variables are determined by joint probability distribution function f(X). Reliability is determined by computing a multi-fold integral on given security

Ω<sup>S</sup>

Difficulty in computing this probability has led to the development of various approximation methods: the first-order reliability method (FORM) (Hasofer, Lind, 1974) and the second reliability method (SORM) (Tvedt L., 1988). These methods are widely used in engineering practice [4, 5]. The main drawback of these methods is that they relate to cases when changing random parameters in time does not exceed the limits of their statistical variability character-

The further development of probabilistic methods was obtained in the 1970–1980s of the twentieth century on the basis of three conceptual ideas [6]. The first idea was that the external conditions of operation structure and its reaction to these conditions are random processes. Therefore, probabilistic methods for calculating structures should be based on methods of the theory of random functions. The second idea was that the failure of the structures in most cases is a consequence of the accumulation of damage (d, l). These damages, reaching a certain value, begin to interfere with the normal operation of structures. The third idea was that the main indicator of reliability should be probability staying of system parameters in a certain permissible region. Violation of normal operation (accident)

f Xð ÞdX, Ω<sup>S</sup> ¼ f g XjG Xð Þ > 0 (13)

Probabilistic Modelling in Solving Analytical Problems of System Engineering

http://dx.doi.org/10.5772/intechopen.75686

9

of two random variables: the load q and the strength r (Figure 4a).

P Xð Þ¼ P Xf g¼ jG Xð Þ > 0 ∭

of aircraft, transport, and other equipment.

region ΩS:

ized by function f(X).

Figure 4. Failure models for calculating reliability.

In the probabilistic formulation, solution of problem (8) consists of obtaining probability distribution function of the survivability F(Ld,l) for given probabilistic properties of accumulated damage. Here it should be noted that at stresses above the yield point, the calculation of damages in the stress values d = σ/σ<sup>f</sup> is significantly more complicated. Therefore, the damage is calculated in terms of relative deformations, d = e/ef.

From these positions, in the analysis of survivability, new calculation cases are considered such as deviations from design situations, beyond design situations, and hypothetical situations that characterize the transition from failures to accidents and disasters (Figure 2).

Currently, national and international programs to ensure the safety of engineering systems, engineering infrastructures, and natural environment (Rio-1992, Johannesburg-2002, Kobe-2005, Hyogo-2015) focus attention on security characteristics S. Quantitative assessments of safety characteristics are based on complex analysis of reliability and survivability of engineering systems [2, 3]:

$$S(\pi) = F\_S\{P\_{P,N,\pi}, L\_{d,l}\} \tag{10}$$

The safety is the ability of the engineering systems to remain operative in damaged states and fracture states. In engineering practice, quantitative safety characteristics have become associated with the risks of accidents and disasters. Risk in quantitative form is defined as a function of the probabilities Pf of accidents and catastrophes and the associated losses Uf:

$$R(\pi) = F\_{\mathbb{R}}\left\{P\_{\mathcal{P},\mathcal{N},\pi}, L\_{d,l}; \mathcal{U}\_f\right\} = F\_{\mathbb{C}}\left\{P\_f, \mathcal{U}\_f\right\}, \\ P\_f = F\_f\{P\_{\mathcal{P},\mathcal{N},\pi}, L\_{d,l}\} \tag{11}$$

It is important to note that the probabilities Pf are estimated for beyond design and hypothetical situations, with the extreme values of Qextr operation parameters and extreme strength and resource characteristics, not envisaged by the project:

$$P\_f = F\_P\{P|\left(Q^{\text{extr}}, N, \tau\right); \left(R\_o, R\_\delta, R\_\lambda, R\_{N,\tau}\right)^{\text{extr}}\}\tag{12}$$

The presented analysis shows that probabilistic models and probabilistic methods acquire an increasingly important role in ensuring the operability of engineering systems.

#### 2.2. The development of traditional probabilistic methods

The development of theoretical foundations' probabilistic approaches to the analysis of the operability of engineering systems covers a significant historical period (stages I–VI, Figure 1). The first studies in this direction were carried out by M. Mayer (1926), N.F. Hotsialov (1929), and W. Weibull (1939). In these studies, the significant variation of strength characteristics for structural materials was shown, and the idea of introducing safety factors was proposed. Essential development of these studies was the work of N.С. Streletsky (1935). In his studies, the strength characteristics of materials (σf, ef) and load parameters (σn, en) were considered as random variables. The further development of this approach was made by A.R. Rzhanicyn (1947). In his works, the relationship between safety factor n<sup>σ</sup> and reliability P was established. Theoretical basis for calculating the reliability of structures in form (7) was formulated for case of two random variables: the load q and the strength r (Figure 4a).

In the 1960s and 1970s, Polovko et al. developed methods for probabilistic calculation of fatigue life RNτ of and reliability of machine parts PP,N,<sup>τ</sup> according to expressions (5) and (6). These methods have been used to calculate the probability diagrams of fatigue characteristics of aircraft, transport, and other equipment.

According to F. Freudenthal (1956) and M. Shinozuki (1983), the reliability problem was formulated for a finite number of random variables X = {xi, i = 1, n} (geometrical parameters, material properties, loads, environmental factors, etc.). In this case, it was assumed that probabilistic properties of random variables are determined by joint probability distribution function f(X). Reliability is determined by computing a multi-fold integral on given security region ΩS:

$$P(X) = P\{X|G(X) > 0\} = \iint\_{\Omega} f(X)dX, \Omega\_S = \{X|G(X) > 0\} \tag{13}$$

Difficulty in computing this probability has led to the development of various approximation methods: the first-order reliability method (FORM) (Hasofer, Lind, 1974) and the second reliability method (SORM) (Tvedt L., 1988). These methods are widely used in engineering practice [4, 5]. The main drawback of these methods is that they relate to cases when changing random parameters in time does not exceed the limits of their statistical variability characterized by function f(X).

The further development of probabilistic methods was obtained in the 1970–1980s of the twentieth century on the basis of three conceptual ideas [6]. The first idea was that the external conditions of operation structure and its reaction to these conditions are random processes. Therefore, probabilistic methods for calculating structures should be based on methods of the theory of random functions. The second idea was that the failure of the structures in most cases is a consequence of the accumulation of damage (d, l). These damages, reaching a certain value, begin to interfere with the normal operation of structures. The third idea was that the main indicator of reliability should be probability staying of system parameters in a certain permissible region. Violation of normal operation (accident)

Figure 4. Failure models for calculating reliability.

Lð Þ¼ τ l f g <sup>i</sup>ð Þ N; τ ; i ¼ 1; m (9)

Sð Þ¼ τ FSf g PP,N, <sup>τ</sup>; Ld,l (10)

Pf <sup>¼</sup> FP <sup>P</sup><sup>j</sup> Qextr; <sup>N</sup>; <sup>τ</sup> ;ð Þ <sup>R</sup>σ;Rδ;Rλ; RN,<sup>τ</sup> extr (12)

, Pf <sup>¼</sup> Ff f g PP,N,τ; Ld,l (11)

In the probabilistic formulation, solution of problem (8) consists of obtaining probability distribution function of the survivability F(Ld,l) for given probabilistic properties of accumulated damage. Here it should be noted that at stresses above the yield point, the calculation of damages in the stress values d = σ/σ<sup>f</sup> is significantly more complicated. Therefore, the damage

From these positions, in the analysis of survivability, new calculation cases are considered such as deviations from design situations, beyond design situations, and hypothetical situations

Currently, national and international programs to ensure the safety of engineering systems, engineering infrastructures, and natural environment (Rio-1992, Johannesburg-2002, Kobe-2005, Hyogo-2015) focus attention on security characteristics S. Quantitative assessments of safety characteristics are based on complex analysis of reliability and survivability of engineer-

The safety is the ability of the engineering systems to remain operative in damaged states and fracture states. In engineering practice, quantitative safety characteristics have become associated with the risks of accidents and disasters. Risk in quantitative form is defined as a function

It is important to note that the probabilities Pf are estimated for beyond design and hypothetical situations, with the extreme values of Qextr operation parameters and extreme strength and

The presented analysis shows that probabilistic models and probabilistic methods acquire an

The development of theoretical foundations' probabilistic approaches to the analysis of the operability of engineering systems covers a significant historical period (stages I–VI, Figure 1). The first studies in this direction were carried out by M. Mayer (1926), N.F. Hotsialov (1929), and W. Weibull (1939). In these studies, the significant variation of strength characteristics for structural materials was shown, and the idea of introducing safety factors was proposed. Essential development of these studies was the work of N.С. Streletsky (1935). In his studies, the strength characteristics of materials (σf, ef) and load parameters (σn, en) were considered as random variables. The further development of this approach was made by A.R. Rzhanicyn (1947). In his works, the relationship between safety factor n<sup>σ</sup> and reliability P was established.

that characterize the transition from failures to accidents and disasters (Figure 2).

of the probabilities Pf of accidents and catastrophes and the associated losses Uf:

<sup>¼</sup> Fc Pf ; Uf

increasingly important role in ensuring the operability of engineering systems.

Rð Þ¼ τ FR PP,N,τ; Ld,l; Uf

2.2. The development of traditional probabilistic methods

resource characteristics, not envisaged by the project:

is calculated in terms of relative deformations, d = e/ef.

8 Probabilistic Modeling in System Engineering

ing systems [2, 3]:

of the system was interpreted as an output of parameters from this area (Figure 4b). Taking these ideas into account, reliability was determined in the form:

$$P\{X(\tau)\} = P\{V(X,\tau) \in \Omega\_S, \tau \in [0,T]\}\tag{14}$$

The final level in this expansion is criterial risk, which allows the connection of systemic risk

Probabilistic Modelling in Solving Analytical Problems of System Engineering

http://dx.doi.org/10.5772/intechopen.75686

11

When constructing a system of models, implementing decomposition (16), it is necessary to take into account the following problem features of the engineering systems as the objects of risk analysis. First, in most cases, we have to analyse situations that have not been seen before, since the coincidence of all circumstances of disasters is an almost impossible event. Second, the analysis is carried out under conditions of high uncertainty associated with both the random nature of external influences and processes in the elements of systems, and with the ambiguity of objectives and safety criteria, as well as alternatives to decisions and their consequences. Third, the analysis is performed with time limit. At the stage of analysis of design decisions, these restrictions are determined by the design time, at the stage of opera-

These features make specific requirements for model representations, the computer, and the information base for risk analysis. The development of model representations and a computational technology is connected with the solution of a number of specific problems. The first task is to describe the engineering systems from the standpoint of integrity and hierarchy. The creation of a substantial and compact model with a large number of significant parameters belongs to the number of difficult tasks, even with the use of modern mathematical and

The second task is to formulate information support for risk analysis. This task has two aspects. The first aspect is related to the task of processing information. Information in the hierarchical system comes in the language of the level that is being analysed. For conclusions at a higher hierarchical level, generalization is required, and at a lower level, detailing this information is needed. In both cases, this translation is ambiguous. The second aspect is related to the need to construct hypotheses about the states of elements on base the available information. The reliability of such hypotheses depends on the level of completeness of infor-

The third task is connected with the choice of the risk criterion. It can be solved on the basis of an analysis or development of special indicators that have the necessary properties of indicators of limit states of engineering systems. This choice can also be ambiguous or multicriteria. Finally, the fourth task is to create theory and methods for risk analysis at given parameters. This apparatus can be considered as a set of mathematical models that reflect the mechanisms of catastrophes in a given sequence of the process of risk analysis. Here it is necessary to take into account the accidental nature of the catastrophe event of the system and the possibility of

a formalized description and measurement of the random parameters of the systems.

A separate and difficult task is modelling the processes of accumulation of damage. In the general case, it is necessary to consider multicriterial damage (MCD) for each element and

To take into account multifocal character of damages and their structural hierarchy, we use the principle of selective scale and select the hierarchy of scales M = {∪Mi, i = 1, n} on which damages develop. Each scale M<sup>i</sup> is considered as internal for a given level and is analysed by appropriate methods. For example, for the scale level of structural elements can be used by the

tion—by the time of response to an emergency or emergency situation.

with mechanisms of catastrophes.

computational technologies.

mation and its reliability.

multi-structural damage for system (MSD).

Thus, the interpretation of reliability as the probability of the fulfilment of some inequality connecting random variables gave way to a more adequate and in-depth interpretation in form emissions of random functions from an admissible region [6–8].

In subsequent years, based on the achievements of reliability theory, extensive studies were carried out to substantiate and improve the normative design calculations using probabilistic methods and reliability theory. Research is carried out in nuclear engineering construction [9], aerospace technology [10], and other industries. An important role in this was played by the achievements of fracture mechanics, taking into account the presence of technological and operational defects and structural damage [11, 12]. The development of probabilistic models of fracture mechanics made it possible to create a concept and methods for probabilistic risk analysis of engineering systems [13, 14].

At the present time, probabilistic modelling and probabilistic methods of calculation have become an integral part of a wide class of problems of statics and dynamics of engineering systems, in which randomness plays an essential role and is introduced by the variations of their geometric and physical properties. To this class belong the problems of strength of microinhomogeneous materials, composite materials, and structures, including nanomaterials and microstructures. Significant progress in this direction is associated with the development of numerical methods of analysis and computational technologies [15, 16].

### 2.3. New directions for solving engineering problems of security and safety by risk criteria

Engineering systems, with rare exceptions, are complex structures of elements of different nature. The problems of probability modelling of such structures turn out to be multivariate and lead to ambiguous solutions. In conditions of complexity and statistical diversity of states of the engineering systems, the diversity of elements, the multiplicity of the mechanisms of catastrophes, it seems unlikely that an integrated comprehensive risk model will be constructed in the near future. A more promising direction can be development individual models of risk, based on the representation of the engineering systems in the form of a structure Σ, consisting of subsystems σ and elements e [13].

$$
\Sigma = \underbrace{\cup}\_{i} \sigma\_{i} \left( \bigvee\_{j} e\_{ij} \right), \ i = 1, n, \ j = 1, m. \tag{15}
$$

These models must realize the decomposition of R-characteristics (risk-decomposition) of structure (13) in the next form [14].

$$R\_{\Sigma} \longrightarrow \{R\_i\} \longrightarrow \{R\_{\vec{\eta}}\} \longrightarrow \{R\_{\vec{\eta}k}\} \tag{16}$$

where R<sup>Σ</sup> is integral (system) risk, Ri is complex (subsystem) risk, Rij is elemental risk, and Rijk is criterial risk.

The final level in this expansion is criterial risk, which allows the connection of systemic risk with mechanisms of catastrophes.

of the system was interpreted as an output of parameters from this area (Figure 4b). Taking

Thus, the interpretation of reliability as the probability of the fulfilment of some inequality connecting random variables gave way to a more adequate and in-depth interpretation in form

In subsequent years, based on the achievements of reliability theory, extensive studies were carried out to substantiate and improve the normative design calculations using probabilistic methods and reliability theory. Research is carried out in nuclear engineering construction [9], aerospace technology [10], and other industries. An important role in this was played by the achievements of fracture mechanics, taking into account the presence of technological and operational defects and structural damage [11, 12]. The development of probabilistic models of fracture mechanics made it possible to create a concept and methods for probabilistic risk

At the present time, probabilistic modelling and probabilistic methods of calculation have become an integral part of a wide class of problems of statics and dynamics of engineering systems, in which randomness plays an essential role and is introduced by the variations of their geometric and physical properties. To this class belong the problems of strength of microinhomogeneous materials, composite materials, and structures, including nanomaterials and microstructures. Significant progress in this direction is associated with the development of

P Xf g ð Þτ ¼ PVX f g ð Þ ; τ ∈ ΩS; τ∈½ � 0; T (14)

these ideas into account, reliability was determined in the form:

emissions of random functions from an admissible region [6–8].

numerical methods of analysis and computational technologies [15, 16].

Σ ¼ ∪ i σ<sup>i</sup> ∪ j eij 

2.3. New directions for solving engineering problems of security and safety by risk

Engineering systems, with rare exceptions, are complex structures of elements of different nature. The problems of probability modelling of such structures turn out to be multivariate and lead to ambiguous solutions. In conditions of complexity and statistical diversity of states of the engineering systems, the diversity of elements, the multiplicity of the mechanisms of catastrophes, it seems unlikely that an integrated comprehensive risk model will be constructed in the near future. A more promising direction can be development individual models of risk, based on the representation of the engineering systems in the form of a structure Σ, consisting of sub-

These models must realize the decomposition of R-characteristics (risk-decomposition) of

where R<sup>Σ</sup> is integral (system) risk, Ri is complex (subsystem) risk, Rij is elemental risk, and Rijk

! Rijk

R<sup>Σ</sup> ! f g Ri ! Rij

, i ¼ 1, n, j ¼ 1, m: (15)

(16)

analysis of engineering systems [13, 14].

10 Probabilistic Modeling in System Engineering

criteria

systems σ and elements e [13].

structure (13) in the next form [14].

is criterial risk.

When constructing a system of models, implementing decomposition (16), it is necessary to take into account the following problem features of the engineering systems as the objects of risk analysis. First, in most cases, we have to analyse situations that have not been seen before, since the coincidence of all circumstances of disasters is an almost impossible event. Second, the analysis is carried out under conditions of high uncertainty associated with both the random nature of external influences and processes in the elements of systems, and with the ambiguity of objectives and safety criteria, as well as alternatives to decisions and their consequences. Third, the analysis is performed with time limit. At the stage of analysis of design decisions, these restrictions are determined by the design time, at the stage of operation—by the time of response to an emergency or emergency situation.

These features make specific requirements for model representations, the computer, and the information base for risk analysis. The development of model representations and a computational technology is connected with the solution of a number of specific problems. The first task is to describe the engineering systems from the standpoint of integrity and hierarchy. The creation of a substantial and compact model with a large number of significant parameters belongs to the number of difficult tasks, even with the use of modern mathematical and computational technologies.

The second task is to formulate information support for risk analysis. This task has two aspects. The first aspect is related to the task of processing information. Information in the hierarchical system comes in the language of the level that is being analysed. For conclusions at a higher hierarchical level, generalization is required, and at a lower level, detailing this information is needed. In both cases, this translation is ambiguous. The second aspect is related to the need to construct hypotheses about the states of elements on base the available information. The reliability of such hypotheses depends on the level of completeness of information and its reliability.

The third task is connected with the choice of the risk criterion. It can be solved on the basis of an analysis or development of special indicators that have the necessary properties of indicators of limit states of engineering systems. This choice can also be ambiguous or multicriteria.

Finally, the fourth task is to create theory and methods for risk analysis at given parameters. This apparatus can be considered as a set of mathematical models that reflect the mechanisms of catastrophes in a given sequence of the process of risk analysis. Here it is necessary to take into account the accidental nature of the catastrophe event of the system and the possibility of a formalized description and measurement of the random parameters of the systems.

A separate and difficult task is modelling the processes of accumulation of damage. In the general case, it is necessary to consider multicriterial damage (MCD) for each element and multi-structural damage for system (MSD).

To take into account multifocal character of damages and their structural hierarchy, we use the principle of selective scale and select the hierarchy of scales M = {∪Mi, i = 1, n} on which damages develop. Each scale M<sup>i</sup> is considered as internal for a given level and is analysed by appropriate methods. For example, for the scale level of structural elements can be used by the methods of fracture mechanics, and for the level of construction, by the methods of structural mechanics. It should be noted that if fracture of individual elements, caused by MCD, can be considered as independent events, then at structure level there is an agreed redistribution of loads, and formation of the focus of MSD should be considered as a cooperated process.
