1. Introduction

Sustainable development of social systems and the natural environment is determined by the state and prospects of the development of engineering and technology. Modern engineering and technology are created on the basis of the achievements of fundamental scientific research. Particular importance is the development of fundamental foundations of mechanics, which is the basis for the design and produce of engineering systems. New machines and structures are creating, based on achievements of construction mechanics, theories of elasticity, plasticity and strength of materials. Multivariance of design and engineering solutions to engineering

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

problems and increase of uncertainties associated with the manifestation of complex combinations of dangerous natural, technical and social factors in the creation and operation of technical objects require the application of new approaches. These approaches will increasingly be based on a combination of traditional deterministic and developing statistical and prospective probabilistic methods of modelling, calculation and testing. Of particular importance is the development of methods of statistical mechanics, probabilistic fracture mechanics, reliability theory and safety theory of engineering systems [1, 2].

f g R; S; Ll,d; PP,R; RN,τ; Rδ;Rλ;R<sup>σ</sup> ¼ Ψ φQð Þ Q; N; t; τ ;φσ σy; σb; E; ν; m;ψ;K1<sup>c</sup>

defects, and stress concentrators ασ.

tional of the following form:

engineering systems is necessary.

and local volumes V(x, y, z)

safety factors n.

systems

are solved:

where Ψ is a generalized function of technical state; φQ(.) is loading functional that takes into account load parameters Q, number of cycles N, temperature t, time τ of loading; φσ(.) is functional of physical and mechanical properties of structural materials, taking into account the yield strength σy, ultimate strength σb, fatigue limit σr, modulus of elasticity E, Poisson's ratio ν, hardening ratio m, ultimate deformation ψ, critical stress intensity factor K1c; φA(.) is a functional of constructive forms, taking into account cross sections area A, lengths l of the

Expression (1) can be considered for limiting states, under which the engineering system ceases to meet the requirements of operation, and for admissible states, determined by the system of

The modern stage of research of engineering systems takes into account the largest man-made accidents and disasters of nuclear, hydraulic and thermal power engineering objects, transport systems and in chemistry objects from twentieth to twenty-first centuries. Taking this into account, stages VII–VIII consider the protection of technical objects based on according risk criteria. The defining equation of this new direction of the engineering methodology of probabilistic modelling, calculation and experimental justification for security Z becomes the func-

The probabilistic characteristics play a decisive role in the structure of the functional (1) and (2). Therefore, for their analysis, further development of probabilistic modelling methods of

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

• boundary problems for stress determining in the most loaded elements, in cross sections A

<sup>σ</sup>ij;eij � � <sup>¼</sup> <sup>F</sup>σf g <sup>Q</sup>; <sup>A</sup>; V xð Þ ; <sup>y</sup>; <sup>z</sup>

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

2. Theoretical foundation of probabilistic modelling for engineering

2.1. Statement for probabilistic modelling problems of engineering systems

Zð Þ¼ τ FZf g R; S; Ll,d; PP,R;RN, <sup>τ</sup>;Rδ;Rλ;R<sup>σ</sup> (2)

� �; <sup>φ</sup>Að Þ ασ; <sup>l</sup>; <sup>A</sup>

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

5

n o (1)

Probabilistic Modelling in Solving Analytical Problems of System Engineering

At the present time, a multi-level concept has been developed to ensure the safe operation of engineering systems (Figure 1). This concept includes specific stages, requirements, criteria, calculated parameters and directions of development. Each higher level is created and developed on the achievements of the lower levels. At the first stages, the methodology of modelling engineering systems and the calculation and experimental validation of operability were based on deterministic methods, with elements of statistical analysis (stages I–III). Understanding the role of random factors in the disruption of operability led to the use of probabilistic methods of modelling and analysis (stages IV, V). At the end of the twentieth century, operability analysis of complex engineering systems began to use parameters of safety S and risk R of disasters. These parameters take into account natural, technical and social hazards (stages VI, VII). On this basis, by the end of the twentieth century, a complex of interconnected multi-level deterministic and probabilistic requirements to engineering systems and their parameters was formed: "strength R<sup>σ</sup> ! stiffness R<sup>δ</sup> ! steadiness R<sup>λ</sup> ! durability RN, τ ! reliability PP, R ! survivability Ll, d ! safety S". Each stage in the development of fundamental research and requirements in this structure corresponds to a certain practical result in the design, creation and operation of engineering systems: "indestructibility - preservation of size and shape durability - fault tolerance - survivability - risk of disasters". Risk is considered as a quantitative probabilistic measure of safety.

The basic equation for determining these characteristics of engineering systems can be written in the following form [1, 2]:


Figure 1. Structure of system for ensuring operability of engineering systems.

$$\Psi\left\{R,S,L\_{l,d},P\_{\mathcal{P},\mathcal{R}},R\_{\mathcal{N},\tau},R\_{\delta},R\_{\lambda},R\_{\sigma}\right\} = \Psi\left\{\varphi\_{\mathcal{Q}}(Q,N,t,\tau);\varphi\_{\sigma}\left(\sigma\_{\mathcal{Y}},\sigma\_{\delta},\mathbb{E},\nu,m,\psi,\mathcal{K}\_{1c}\right);\varphi\_{A}(\alpha\_{\sigma},l,A)\right\} \tag{1}$$

problems and increase of uncertainties associated with the manifestation of complex combinations of dangerous natural, technical and social factors in the creation and operation of technical objects require the application of new approaches. These approaches will increasingly be based on a combination of traditional deterministic and developing statistical and prospective probabilistic methods of modelling, calculation and testing. Of particular importance is the development of methods of statistical mechanics, probabilistic fracture mechanics, reliability

At the present time, a multi-level concept has been developed to ensure the safe operation of engineering systems (Figure 1). This concept includes specific stages, requirements, criteria, calculated parameters and directions of development. Each higher level is created and developed on the achievements of the lower levels. At the first stages, the methodology of modelling engineering systems and the calculation and experimental validation of operability were based on deterministic methods, with elements of statistical analysis (stages I–III). Understanding the role of random factors in the disruption of operability led to the use of probabilistic methods of modelling and analysis (stages IV, V). At the end of the twentieth century, operability analysis of complex engineering systems began to use parameters of safety S and risk R of disasters. These parameters take into account natural, technical and social hazards (stages VI, VII). On this basis, by the end of the twentieth century, a complex of interconnected multi-level deterministic and probabilistic requirements to engineering systems and their parameters was formed: "strength R<sup>σ</sup> ! stiffness R<sup>δ</sup> ! steadiness R<sup>λ</sup> ! durability RN, τ ! reliability PP, R ! survivability Ll, d ! safety S". Each stage in the development of fundamental research and requirements in this structure corresponds to a certain practical result in the design, creation and operation of engineering systems: "indestructibility - preservation of size and shape durability - fault tolerance - survivability - risk of disasters". Risk is considered as a quantita-

The basic equation for determining these characteristics of engineering systems can be written

theory and safety theory of engineering systems [1, 2].

4 Probabilistic Modeling in System Engineering

tive probabilistic measure of safety.

Figure 1. Structure of system for ensuring operability of engineering systems.

in the following form [1, 2]:

where Ψ is a generalized function of technical state; φQ(.) is loading functional that takes into account load parameters Q, number of cycles N, temperature t, time τ of loading; φσ(.) is functional of physical and mechanical properties of structural materials, taking into account the yield strength σy, ultimate strength σb, fatigue limit σr, modulus of elasticity E, Poisson's ratio ν, hardening ratio m, ultimate deformation ψ, critical stress intensity factor K1c; φA(.) is a functional of constructive forms, taking into account cross sections area A, lengths l of the defects, and stress concentrators ασ.

Expression (1) can be considered for limiting states, under which the engineering system ceases to meet the requirements of operation, and for admissible states, determined by the system of safety factors n.

The modern stage of research of engineering systems takes into account the largest man-made accidents and disasters of nuclear, hydraulic and thermal power engineering objects, transport systems and in chemistry objects from twentieth to twenty-first centuries. Taking this into account, stages VII–VIII consider the protection of technical objects based on according risk criteria. The defining equation of this new direction of the engineering methodology of probabilistic modelling, calculation and experimental justification for security Z becomes the functional of the following form:

$$Z(\tau) = F\_Z\{R, S, L\_{l,d}, P\_{P,R}, R\_{N,\tau}, R\_\delta, R\_\lambda, R\_\sigma\} \tag{2}$$

The probabilistic characteristics play a decisive role in the structure of the functional (1) and (2). Therefore, for their analysis, further development of probabilistic modelling methods of engineering systems is necessary.
