Reliability Analysis, Safety Assessment and Optimization. Enrico Zio

alt="upper F left-parenthesis t right-parenthesis equals StartFraction lamda Superscript beta Baseline Over upper Gamma left-parenthesis beta right-parenthesis EndFraction integral Subscript 0 Superscript t Baseline x Superscript beta minus 1 Baseline e Superscript minus lamda x Baseline d x comma t greater-than-or-equal-to 0 period"/>(1.34)

      If T denotes the failure time of an item with gamma distribution, the reliability function will be

      (1.35)

      The hazard rate function is

      (1.36)

      The mean, μ, and variance, σ2, are

      (1.37)

      1.2.2.4 Lognormal Distribution

      A random variable T follows the lognormal distribution if and only if the pdf (shown in Figure 1.6) of T is

Figure 1.6 The pdf of the lognormal distribution with μ=0, σ=0.954_.

      (1.38)

      where σ>0 is the shape parameter and μ>0 is the scale parameter of the distribution. Note that the lognormal variable is developed from the normal distribution. The random variable X=lnT is a normal random variable with parameters μ and σ. The cdf of the lognormal distribution is

      (1.39)

      where Φ(x) is the cdf of a standard normal random variable. If T denotes the failure time of an item with lognormal distribution, the reliability function of T will be

      (1.40)

      The hazard rate function is

      (1.41)

      The mean, μ, and variance, σ2, are

      (1.42)

       Example 1.3

      The random variable of the time to failure of an item, T, follows the following pdf:

      where t is in days and t≥0.

      1 What is the probability of failure of the item in the first 100 days?

      2 Find the MTTF of the item.

       Solution

      1 The cdf of the random variable isThe probability of failure in the first 100 days is

      2 The MTTF of the item is

      1.2.3 Physics-of-Failure Equations

      Different from the traditional reliability assessment approach, the Physics-of-Failure (P-o-F) represents an approach to reliability assessment based on modeling and simulation of the physical processes leading to the occurrence of failures in an item [2]. The P-o-F approach begins within the first stages of the design of the item. A model is constructed based on the customer’s requirements, service environment, and stress analysis [1]. Once the models are established, a reliability assessment can be conducted on the item.

      1.2.3.1 Paris’ Law for Crack Propagation

Paris’ law is a crack growth equation that gives the rate of growth of a fatigue crack [3]. The stress intensity factor K characterizes the load around a crack tip and the rate of crack growth is experimentally shown 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.

      (1.43)

where a is the crack length and dadN is the fatigue crack growth for a load cycle N. The material coefficients C and m are obtained experimentally and their values depend on environment, temperature, and stress ratio. The stress intensity factor range has been found to correlate with the rate of crack growth in a variety of different conditions, which is the difference between the maximum and minimum stress intensity factors in a load cycle, defined as

      (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

(1.45)

      where Q is the total volume of the