General Reliability problem

   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).

Limit State Approximations

 In most industrial cases, the integrals of Eq.(1) and of Eq.(2) (ref. this section) cannot be evaluated in closed form. Only for simple academic cases $f_{X}$ can be defined analytically. In such a case, the performance function $G_{j}(x)$ is available for each specific failure mode of the structure so that $\Omega_{j}$ can be defined. The boundary of the domain $\Omega_{j}$ is also called limit-state surface and the function $G_{j}(x)$ is called the limit-state function (LSF).

 In the most widely used reliability methods, approximations are made in the space of standard uncorrelated normal variates, Y, obtained from a transformation of the basic variables

$Y=T(X) $

(3)

 where the transformation T is expressed in terms of the distributions of X.

 In the space Y, denoted as the standard normal space, approximations of the probability of failure $P_{f,j}$ of Eq.(1) (ref. this section) are obtained by replacing the limit state surface with first or second order approximating surfaces. These surfaces are fitted to the limit state surface at points with minimal distance to the origin (design points).

The Reliability-Based Design Optimization (RBDO) Model

In engineering design, the traditional deterministic design optimization model has been successfully applied to systematically improve the system design process, yielding a reduction of the costs and an improvement of the final quality of the products. However, uncertainties in either engineering simulations and/or manufacturing processes exist. This calls for different optimization models that can yield not only an improvement in the design, but also a higher level of confidence. Thus, a reliability-based design optimization (RBDO) model for robust and cost-effective designs can be defined using mean values of the random system variables as design parameters and optimizing the cost subject to prescribed probabilistic constraints (like probabilities of failure) by solving a mathematically nonlinear programming problem.

The general RBDO model can be defined as:

${(min_{d}{Cost[d(\mu_{X})]}),("subject to " P_{f,j}=P(G_{j}(x)<0)lt= bar P_{f,j} " with " j=1...m ):}$

(4)

where:

• the cost function can be any function of the mean values $mu_{X}$ of the input parameters and
  
• $bar P_{f,j}$ is the probabilistic constraint that can be defined for each failure mode j and needs to be satisfied.

For this optimization problem, the constraint definition is expressed in terms of probability distributions and thus needs to be evaluated, for each optimization step, within the probability framework. Thus, for each iteration of the optimization loop, an estimation of the probabilistic constraint in terms of its multidimensional integral (see Eq. (1) and (2) in this section) has to be computed. For this purpose, different methods exist. Most of them apply a transformation of the input parameter space X to the standard normal space Y. In this space, if the limit state functions are linear, each probability of failure can be represented in terms of the reliability index $beta_{j}$ as:

$P_{f,j}=P(G_{j}(x)<0) rArr P_{f,j}=P(G_{j}(y)<0)= Phi(-beta_{j}) rArr beta_{j}=-Phi^{-1}(P_{f,j})$

(5)

Where $Phi(*)$ is the standard normal CDF* (zero mean and standard deviation 1).

This equation can be generalized as:

$F_{G_{j}}(g_{j})=P(G_{j}(x)<g_{j})=Phi(-beta_{G_{j}})$

(6)

Where $F_{G_{j}}(*)$ is the CDF* of the $j^{th}$ system response.

The same approach can be used also to express the probabilistic constraint of Eq.(4) in a different notation. In this case, the probability of failure $P_{f,j}$ will be the target probability of failure $bar P_{f,j}$ and the reliability index $beta_{j}$ the target reliability index $beta_{t,j}$.

$bar P_{f,j}=Phi(-beta_{t,j}) rArr beta_{t,j}=-Phi^{-1}(bar P_{f,j})$

(7)

Using Eq.(6), the second condition of Eq.(4) can be rewritten as:

$P_{f,j}=F_{G_{j}}(0)=Phi(-beta_{j})lt=Phi(-beta_{t,j})=bar P_{f,j} rArr beta_{j}gt=beta_{t,j}$

(8)

The relation expressed in Eq.(8) in terms of the reliability index, can also be expressed in terms of the performance measure through inverse transformation. In fact, using Eq.(7) and Eq.(8), the target probability of failure can be expressed in terms of the target performance measure as

$bar g_{j} = F_{G_{j}}^{-1}(bar P_{f,j})=F_{G_{j}}^{-1}[Phi(-beta_{t,j})]$

(9)

where $bar g_{j}$ is named "target probabilistic performance measure". It represents the value of the performance function "equivalent" to the target reliability index $beta_{t,j}$.

The expression of the probabilistic constraint of Eq.(8) and (9) can be used in the optimization problem of Eq.(4) to equivalently replace the original definition.