Non Deterministic Approaches

   In the reliability theory, variabilities of an engineering design can be characterized by the variations of a random system parameter set. These random variables are modeled with probability distribution functions and are representative of the uncertain model parameters. Thus, given a set of random variables $x=[x_{i}]^{T}$ , with i=1…n, the probability distribution of each random variable $x_{i}$ is described either by its cumulative distribution function (CDF*, $F_{x_{i}}(x_{i})$) or by its probability density function (PDF*, $f_{x_{i}}(x_{i})$) and is often bounded by tolerance limits on the system parameter values. Because of these limits, unbounded probability density functions, like the Gaussian distribution, are generally not suggested, as they allow events with non-zero probability density even at values that are far off the nominal value.

   For each realization of the set of input random variables, the performance of the system has to be evaluated. This performance is generally described by Performance Functions (PF, $G_{j}(x)$ with j=1..m) that, for a structural design, are usually the selected failure criteria. Thus, if $G_{j}(x)$ is one of the m system PFs, the system is considered to fail if $G_{j}(x)<0$ for at least one index j. The probability of failure of the system for every jth performance function is then the multi dimensional integral of the joint PDF* function of the set of random variables over the failure domain $\Omega_{j}$. This is expressed by the integral:

$F_{G_{j}}(0)=P[G_{j}(x)<0]=int_{\Omega_{j}}...int f_{X}(x)dx_{1}...dx_{n}$ where $\Omega_{j}:{x in RR^{n}:G_{j}(x)<0}$

(1)

   The event space $\Omega_{j}$ is the region of the stochastic space where only failure events occur. Thus, the integral of the joint probability density function $f_{X}(x)$ over $\Omega_{j}$ yields the probability of failure $P_{f,j}$ of the structure for the j^th failure criterion.

$P_{f,j}=F_{G_{j}}(0)=P(G_{j}(x)<0)$ with j=1...m

   In general, given a fixed performance index (or measure) $g_{j}$ of the system, Eq (1) can be generalized by computing the probability of exceedance of the selected $g_{j}$ threshold.

$F_{G_{j}}(g_{j})=P[G_{j}(x)<g_{j}]=int_{\Omega_{G_{j}}}... int f_{X}(x)dx_{1}...dx_{n}$ where $\Omega_{G_{j}}:{x in RR^{n}:G_{j}<g_{j}}$

(2)

   In this case, the event space $\Omega_{G_{j}}$ represents the region of the stochastic space where the performance parameter of the structure is below the prescribed quantity $g_{j}$.

   The main aim of the reliability analysis is to estimate the probability of failure of a structure, given a set of input random variables and a set of failure criteria defining the safe and fail regions $\Omega_{j}$ or $\Omega_{G_{j}}$. This estimation is usually carried on by approximating the multi dimensional integral with appropriate techniques. These include sampling methods and limit state approximations, usually computed by transforming the problem definition from the parameter space X into the standard normal space Y.

   The total probability of failure of the structure is given by the integral of the joint PDF* $f_{X}(x)$ over the union of all the failure domains. These failure domains are defined by the corresponding Performance Functions $G_{j}(x)$ and their threshold values $g_{j}$ (performance indexes or measures).