3.2. Reliability and risk assessment of metal-liner composite overwrapped pressure vessels

Metal-composite pressure vessels (MCOPVs) have found a wide application in aerospace and aeronautical industries. Such vessels should combine the impermeability and high weight efficiency with enhanced long-term safety and durability. To meet these requirements, theoretical

Figure 7. A complete probabilistic integrity diagrams: (a) probability of fatigue, (b) durability distributions, and (c) functions of equal safety factors.

and experimental studies on the mechanics of deformation and failure of MCOPVs are required [20].

The Phoenix approach based on the Weibull reliability model was used to determine the term R(τ, σ). To account the influence of structural-mechanical heterogeneity of MCOPV, a reference measure M<sup>0</sup> was introduced. It is assumed that within M0, the deformation of material is

M<sup>0</sup>

where M is the total "scale" of MCOPV; τ is the time; τ<sup>c</sup> is the characteristic (reference) time, which can be considered as time of failure during static test; σ<sup>p</sup> is stress corresponding to

Risk assessment was performed on the basis of an analysis of possible mechanisms of MCOPV destruction. The event of fracture of the MCOPV decays into two events: the destruction event of the liner and the event of destruction of the power composite shell. In the first case, the main parameter controlling the state of the liner is deformation, and in the second case, the stresses

f g Risk ! f g Leakage ! Pfð Þ¼ <sup>τ</sup>; <sup>ε</sup> <sup>P</sup> <sup>τ</sup>jε<sup>p</sup> <sup>≥</sup> <sup>ε</sup><sup>f</sup>

Reliability functions R(t,σ) for MCOPV in the orbit are shown in Figure 9. As can be seen from the figure, while ensuring the homogeneity of the properties of the composite sheath and

f g Fracture ! Pfð Þ¼ τ; σ P τjσ<sup>p</sup> ≥ σ<sup>f</sup> � � (

τ τc σP σf

� � � �<sup>α</sup> <sup>β</sup> ( ) (28)

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Probabilistic Modelling in Solving Analytical Problems of System Engineering

� �

(29)

19

uniform. The probability of failure-free operation R(τ, σ) can be expressed as follows:

<sup>R</sup>ð Þ¼ <sup>τ</sup>; <sup>σ</sup> exp � <sup>M</sup>

operating pressure Р; σ<sup>f</sup> is stress at burst pressure; and α, β are statistic parameters.

in the composite shell are the determining parameter:

Figure 9. Functions of reliability depending on the time and the stress level σр/σс.

Investigate reliability and risk fractures were based on results of numerical stress analysis and experimental tests' full-scale samples of MCOPV. The construction of MCOPV was having an axisymmetric ellipsoid-like shell of revolution with the minor to major diameter ration of about 0.6 (Figure 8a). The thin-welded liner was made of VT1–0 titanium alloy. The composite shell was formed by helical winding of IMS-60 carbon fibres impregnated with a polymer matrix. The stress analysis of MCOPV under internal pressure was performed using the finite element method. The calculations were carried out with finite element models developed to reflect all significant geometric and deformation characteristics of the composites vessel (Figure 8b).

Actual MCOPVs structures will exhibit a non-uniform distribution of stresses and deformation owing to a number of factors. These include the nuances of liner geometry and its interaction with the overwrap winding pattern, the relative stiffness of the liner to the overwrap, the lineroverwrap interface slips characteristics, and the presence of incompatible curvature changes. Load equilibrium in the bimaterial vessels requires that the total applied pressure be equal to the sum of the pressure carried by the individual components.

Taking this into account, the calculation of reliability function R(P, τ) included the evaluation of two components: the reliability R(DM) at the beginning of service and the reliability R(τ, σ) during operation—R(P, τ) = R(DM)�R(t, σ). The component R(DM) was estimated by means of a conventional "load-strength" model, assuming the Gaussian law for load and strength values of MCOPV [20]:

$$R(DM) = \text{Prob}\{DM > 1\} = \Phi\left\{\frac{\mu\_f - \mu\_p}{\sqrt{s\_f^2 + s\_p^2}}\right\} = \Phi\left\{\frac{DM - 1}{\sqrt{V\_f^2 DM^2 + V\_p^2}}\right\} \tag{27}$$

where Φ is the standard normal distribution function; μf, μ<sup>p</sup> are median values for P and Pf; Vf, Vp are coefficients of variations for P and Pf, DM = Pf/P is design safety margins.

Figure 8. A metal-lined composite pressure vessel: calculation scheme (a), finite element model (b).

The Phoenix approach based on the Weibull reliability model was used to determine the term R(τ, σ). To account the influence of structural-mechanical heterogeneity of MCOPV, a reference measure M<sup>0</sup> was introduced. It is assumed that within M0, the deformation of material is uniform. The probability of failure-free operation R(τ, σ) can be expressed as follows:

and experimental studies on the mechanics of deformation and failure of MCOPVs are requi-

Investigate reliability and risk fractures were based on results of numerical stress analysis and experimental tests' full-scale samples of MCOPV. The construction of MCOPV was having an axisymmetric ellipsoid-like shell of revolution with the minor to major diameter ration of about 0.6 (Figure 8a). The thin-welded liner was made of VT1–0 titanium alloy. The composite shell was formed by helical winding of IMS-60 carbon fibres impregnated with a polymer matrix. The stress analysis of MCOPV under internal pressure was performed using the finite element method. The calculations were carried out with finite element models developed to reflect all significant geometric and deformation characteristics of the composites vessel

Actual MCOPVs structures will exhibit a non-uniform distribution of stresses and deformation owing to a number of factors. These include the nuances of liner geometry and its interaction with the overwrap winding pattern, the relative stiffness of the liner to the overwrap, the lineroverwrap interface slips characteristics, and the presence of incompatible curvature changes. Load equilibrium in the bimaterial vessels requires that the total applied pressure be equal to

Taking this into account, the calculation of reliability function R(P, τ) included the evaluation of two components: the reliability R(DM) at the beginning of service and the reliability R(τ, σ) during operation—R(P, τ) = R(DM)�R(t, σ). The component R(DM) was estimated by means of a conventional "load-strength" model, assuming the Gaussian law for load and strength

> s2 <sup>f</sup> þ s<sup>2</sup> p

where Φ is the standard normal distribution function; μf, μ<sup>p</sup> are median values for P and Pf; Vf,

9 >=

>; <sup>¼</sup> <sup>Φ</sup> DM � <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V2

q

8 ><

>:

<sup>f</sup> DM<sup>2</sup> <sup>þ</sup> <sup>V</sup><sup>2</sup>

p

9 >=

>;

(27)

q

8 ><

>:

Vp are coefficients of variations for P and Pf, DM = Pf/P is design safety margins.

Figure 8. A metal-lined composite pressure vessel: calculation scheme (a), finite element model (b).

the sum of the pressure carried by the individual components.

R DM ð Þ¼ Prob DM f g <sup>&</sup>gt; <sup>1</sup> <sup>¼</sup> <sup>Φ</sup> <sup>μ</sup><sup>f</sup> � <sup>μ</sup><sup>p</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

red [20].

18 Probabilistic Modeling in System Engineering

(Figure 8b).

values of MCOPV [20]:

$$R(\tau, \sigma) = \exp\left\{-\frac{M}{M\_0} \left[\frac{\tau}{\tau\_c} \left(\frac{\sigma\_P}{\sigma\_f}\right)^{\alpha}\right]^{\beta}\right\} \tag{28}$$

where M is the total "scale" of MCOPV; τ is the time; τ<sup>c</sup> is the characteristic (reference) time, which can be considered as time of failure during static test; σ<sup>p</sup> is stress corresponding to operating pressure Р; σ<sup>f</sup> is stress at burst pressure; and α, β are statistic parameters.

Risk assessment was performed on the basis of an analysis of possible mechanisms of MCOPV destruction. The event of fracture of the MCOPV decays into two events: the destruction event of the liner and the event of destruction of the power composite shell. In the first case, the main parameter controlling the state of the liner is deformation, and in the second case, the stresses in the composite shell are the determining parameter:

$$\{\text{Risk}\} \rightarrow \begin{cases} \{\text{Leakage}\} \rightarrow P\_f(\tau, \varepsilon) = P\{\tau | \varepsilon\_p \ge \varepsilon\_f\} \\ \{\text{Fracture}\} \rightarrow P\_f(\tau, \sigma) = P\{\tau | \sigma\_p \ge \sigma\_f\} \end{cases} \tag{29}$$

Reliability functions R(t,σ) for MCOPV in the orbit are shown in Figure 9. As can be seen from the figure, while ensuring the homogeneity of the properties of the composite sheath and

Figure 9. Functions of reliability depending on the time and the stress level σр/σс.


probabilistic approaches address more general and more complicated situations, in which behaviour of the engineering system cannot be determined with certainty in each particular experiment or a situation. Probabilistic models enable one to establish the scope and limits of the application of deterministic theories and provide a solid basis for substantiated and goaloriented accumulation and the effective use of empirical data. Realizing the fact that probability of failure of engineering system is never zero, probabilistic methods enable one to quantitative assess the degree of uncertainty in various factors, which determine the safety of system

Probabilistic Modelling in Solving Analytical Problems of System Engineering

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21

Analysis of the largest man-caused and natural-technogenic catastrophes of recent years indicates the need to improve methods and means of ensuring the safety of the engineering systems. One of the main ways in this direction is to improve the historically established system of forming the scientific basis for engineering calculations of the characteristics of strength, stability, durability, reliability, survivability, and safety. Of decisive importance is the need to switch to a new methodological framework and principles for ensuring the safety of engineering systems by the criteria for risks of accidents and disasters. A special role in this direction is that security defines all the main groups of requirements for engineering systems:

\*, Vladimir Moskvichev<sup>1</sup> and Nikolay Machutov<sup>2</sup>

[1] Makhutov NA. Strength and Safety. Fundamental and Applied Research. Novosibirsk:

[2] Makhutov NA, editor. Strength, Resource, Survivability and Safety of Machines.

[3] Makhutov NA. Safety and Risks: System Research and Development. Novosibirsk:

[4] Kapur KC, Lamberson LR. Reliability in Engineering Design. New York: John Wiley &

[5] Shuhir E. Applied Probability for Engineering and Scientists. New York: McGraw-Hill;

1 Institute of Computational Technologies SB RAS, Krasnoyarsk, Russia

2 Mechanical Engineering Research Institute RAS, Moscow, Russia

and design on this basis a system with a low probability of failure.

strength, rigidity, stability, reliability, survivability, and risk.

\*Address all correspondence to: aml@ict.nsc.ru

Author details

Anatoly Lepikhin<sup>1</sup>

References

Nauka; 2008. 528 p

Nauka; 2017. 724 p

Sons; 1979. 586 p

1997. 533 p

Moskow: Librocom; 2008. p. 576

Table 1. Calculated estimates of risk fracture of MCOPV.

operating stresses not exceeding 0.5 of the strength level of the composite material, the reliability of MCOPV is ensured at a level of at least 0.999 at the end of operation time in orbit. When the relative stresses level is increased to 0.6, high reliability is ensured only in the first 500 h of operation. The load level of more than 0.6 is unacceptable for the MCOPVs.

Quantitative risk assessments were performed for the most dangerous scenarios. The risk calculation was performed for the time moments 1000 h (the beginning work on the orbit) and 15,000 h (at the end work on the orbit). The calculation of the risk fracture of MCOPV was carried out according to the Phoenix approach, replacing the standard fracture stress of the composite σ<sup>f</sup> by the numerical or experimental estimate of the actual value of the composite strength σc.

The results of the calculation are presented in the Table 1. The obtained risk assessments can be regarded as tentative, since they do not take into account possible processes of creep of liner and composite under load during the MCOPV operation. It should be noted that obtained values of the probabilities of destruction of the MCOPV belong to the class of extremely unlikely events. Therefore, the risk of destruction of the MCOPV in the orbit can be considered acceptable.
