Reliability Analysis, Safety Assessment and Optimization. Enrico Zio

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.

Figure 1.8 Reliability block diagram of a series system.

      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

      (1.46)

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

      (1.47)

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

      (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.

Figure 1.9 Reliability block diagram of a parallel system.

      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

      (1.49)

      It follows that

      (1.50)

      1.3.3 Series-parallel System

A series-parallel system consists of m subsystems that are connected in series, with ni units connected in parallel in each subsystem, i=1,…,m. The reliability block diagram of a series-parallel system is shown in Figure 1.10.

      Figure 1.10 Reliability block diagram of a series-parallel system.

      Denote pij as the reliability of component j in subsystem i, 1≤i≤m, 1≤j≤ni. Let Pi be the reliability of the subsystem i, 1≤i≤m. First, the reliability of each subsystem is derived as for the parallel system, that is,

      (1.51)

      The reliability of the series-parallel system is, then,

      (1.52)

      1.3.4 K-out-of-n System

      For a system composed of n components, the system is operational if and only if at least k of the n components are operational. We call this type of system as k-out-of-n: G system, where G is short for Good. For a system composed of n components, the system fails if and only if at least k of the n components are failed. We call this type of system a k-out-of-n: F system. According to the definition, the series system is a 1-out-of-n: F system, where F is short for Failed. The parallel system is a 1-out-of-n: G system. We will mainly present the reliability of the k-out-of-n: G system here.

      Assume that the n components are identical and independent. Denote R as the reliability of each component, F as the unreliability of each component, q=1−p. Let Pi be the probability so that exactly i components are functional. In a k-out-of-n: G system, the number of functional components follows the binomial distribution with parameter n and R. The probability that exactly i components are functional, Pi, is