## 2.0 Reliability Index, β, and Probability of Failure, P_{f}

A structural component or the entire structure becomes more reliable as the probability of failure decreases. Thus, reliability may be expressed in terms of the probability of failure, e.g., 1 in 100 (i.e., 1 failure in 100 events), 1 in 1,000, etc. In statistical terms, reliability may also be thought of as the inverse of the coefficient of variation (COV). The COV is defined as the standard deviation (σ) divided by the mean value (μ). Using this definition the reliability increases as the standard deviation decreases, i.e., as the COV decreases. This simple concept is generally useful only when there is one normally distributed variable. In practice, there is a need for a more general definition of reliability that can express the probability of failure in terms of the coefficients of variation of various parameters that may or may not be normally distributed. One option is to express the reliability in terms of a reliability index, β, which expresses the probability of failure, P_{f}, as a function of the statistics of the loads, Q, resistances, R, and a limit state function, g. Failure may be defined as g < 0 for the case of g = φR – γQ where φ is the resistance factor and γ is the load factor. In other words, the probability of failure, P_{f}, represents the probability for the condition of failure at which the factored resistance, φR, will be less than factored loads, γQ. Once failure is defined, the rate of failure per numbers of simulation (physical or numerical) is determined and expressed in terms of reliability index, β.

The First-Order Second-Moment (FOSM) method based on the first two moments, i.e., mean (μ) and variance (V=σ^{2}), of the data for a random variable can be used to develop analytical closed form solutions for relationships between β and P_{f}. Such solutions are widely published in the open literature, e.g., Whitman (1984), Withiam, *et al.* (1998), Haldar and Mahadevan (2000). Figure 2.1 shows the relationship between β and P_{f} based on the FOSM approach for a normally distributed load and resistance and a log-normally distributed load and resistance. In some publications (e.g., Paikowsky *et al.*, 2004) the â€œnormalâ€ distribution relationship in Figure 2.1 is referred to as an â€œexactâ€ relationship. The log-normal relationship in Figure 2.1 is approximate and valid for 2 < β < 6 (Rosenblueth and Esteva, 1972).

**Figure 2.1. Relationships between ****β and P _{f} for the case of a single load and single resistance.**

Both normal and log-normal relationships yield approximately the same β (≈ 5 to 5.7) at the practical upper limit of the probability of failure between 1 in 5 million to 1 in 100 million; such values of β may be applicable to extremely critical works such as nuclear facilities. The two relationships continue to be reasonably similar for β > 2.5 while they diverge significantly for β < 2.5. As discussed in Section 4.0, the target reliability index, β_{T}, for substructure design is usually between 2.3 and 3.5. For this range of β-values, the two relationships are practically similar, see Figure 2.2.

**2.1 Probability of Survival, P**_{s}

_{s}

Instead of the reliability index one may choose to express P_{f} as the probability of survival, P_{s}. The value of P_{s} in percent is obtained from (100)(1- P_{f}) where P_{f} is expressed as a decimal. In addition to P_{f}, the value of P_{s} is also shown in Figure 2.2. Since the values of P_{s} are well above 90% in the zone of interest for subsurface design as shown in Figure 2.2, the probability of survival can be used just like the reliability index to avoid the negative connotation of the word â€œfailure.â€

**Figure 2.2. Relationships between ****β, P _{f} and P_{s} in the range of interest for substructures for the case of a single load and single resistance. **

### 2.2 Cases of Multiple Loads and Resistances with Non-normal Distributions

For cases where multiple loads and multiple resistances with varying non-normal distributions are evaluated, relationships such as those shown in Figure 2.1 may not be possible without significant simplifying assumptions in the FOSM approach. In such cases, numerical simulation techniques may be used to determine the probability of failure and the corresponding reliability index. The problem may then be evaluated without simplifying assumptions thus leading to more realistic results. Many numerical simulation techniques to perform reliability analysis are available and published, e.g., see Law and Kelton (1991), Haldar and Mahadevan (2000). The Monte-Carlo simulation technique works well with the overall LRFD approach since it has the capability to evaluate random events; this technique is briefly discussed next.