Reliability Analysis, Safety Assessment and Optimization. Enrico Zio

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to be a function of the range of the stress intensity ΔK experienced in a loading cycle (shown in Figure 1.7). The Paris’ equation describing this is

      Figure 1.7 Illustration of Paris Law.

      StartFraction normal d a Over normal d upper N EndFraction equals upper C left-parenthesis upper Delta upper K right-parenthesis Superscript m Baseline comma(1.43)

      upper Delta upper K equals upper K Subscript m a x Baseline minus upper K Subscript m i n Baseline period(1.44)

      1.2.3.2 Archard’s Law for Wear

      The Archard’s wear equation is a simple model used to describe sliding wear, which is based on the theory of asperity contact [4]. The volume of the removed debris due to wear is proportional to the work done by friction forces. The Archard’s wear equation is given by

      where Q is the total volume of the wear debris produced, K is a dimensionless constant, W is the total normal load, L is the sliding distance, and H is the hardness of the softest contacting surfaces. It is noted that WL is proportional to the friction forces. K is obtained from experimental results and it depends on several parameters, among which are surface quality, chemical affinity between the material of two surfaces, surface hardness process, etc.

      1.3 System Reliability Modeling

      The methods to model and estimate the reliability of a single component were introduced in Section 1.2. Compared with the single component case, the system reliability modeling and assessment is more complicated. The term ‘system’ is used to indicate a collection of components working together to perform a specific function. The reliability of a system depends not only on the reliability of each component but also on the structure of the system, the interdependence of its components, and the role of each component within the system, etc. To compute the reliability of the system, it is essential to construct the model of the system, representing the above characteristics.

      The conventional approaches typically assume that the components and the system have two states: perfect working and complete failure [5]. Below, we introduce the reliability models of a binary state system with specific structures. Details about the multi-state system can be found in Chapter 3.

      1.3.1 Series System

      In a series system, all components must operate successfully for the system to function or operate successfully. It implies that the failure of any component will cause the entire system to fail. The reliability block diagram of a series system is shown in Figure 1.8.

      Let Ri(t) be the reliability of the ith component, i=1,2,…,n,, and Rs(t) be the reliability of the system. Let xi be the event that the ith component is operational and let x be the event that indicates system is operational. The reliability of the series system can be calculated by

      upper R Subscript s Baseline left-parenthesis t right-parenthesis equals upper P left-parenthesis x right-parenthesis equals upper P left-parenthesis x 1 comma x 2 comma ellipsis comma x Subscript n Baseline right-parenthesis equals product Underscript i equals 1 Overscript n Endscripts upper P left-parenthesis x Subscript i Baseline right-parenthesis period(1.46)

      Assume all the components in the series system are independent; if so, the reliability of the system can be expressed as

      upper R Subscript s Baseline left-parenthesis t right-parenthesis equals product Underscript i equals 1 Overscript n Endscripts upper R Subscript i Baseline left-parenthesis t right-parenthesis period(1.47)

      Considering that the component reliability is a number between 0 and 1, we have the following relationship

      upper R Subscript s Baseline left-parenthesis t right-parenthesis less-than m i n left-brace upper R 1 left-parenthesis t right-parenthesis comma upper R 2 left-parenthesis t right-parenthesis comma ellipsis comma upper R Subscript n Baseline left-parenthesis t right-parenthesis right-brace period(1.48)

      1.3.2 Parallel System

      In a parallel system, the system functions or operates successfully when at least one component function is working. It implies that the failure of all components will cause the entire system to fail. The reliability block diagram of a parallel system is shown in Figure 1.9.

      Denote Fs(t) as the probability of failure of the system. Denote Fi(t) as the probability of failure of component i. The system reliability can be expressed as

      upper R Subscript s Baseline left-parenthesis t right-parenthesis equals 1 minus upper F Subscript s Baseline left-parenthesis t right-parenthesis equals 1 minus product Underscript i equals 1 Overscript n Endscripts upper F Subscript i Baseline left-parenthesis t right-parenthesis equals 1 minus product Underscript i equals 1 Overscript n Endscripts left-bracket 1 minus upper R Subscript i Baseline left-parenthesis t right-parenthesis right-bracket period(1.49)

      It follows that

      upper R Subscript s Baseline left-parenthesis t right-parenthesis greater-than-or-equal-to m a x left-brace upper R 1 left-parenthesis t right-parenthesis comma upper R 2 left-parenthesis t right-parenthesis comma ellipsis comma upper R Subscript n Baseline left-parenthesis t right-parenthesis right-brace period(1.50)

      1.3.3 Series-parallel System

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