What is a fully probabilistic design approach?

In structural reliability, in a fully probabilistic design approach, the system design is selected such that it directly meets a target reliability requirement. The target reliability requirement is usually specified either as a target probability of failure or in terms of a target reliability index. 

The model and the parameters in a probabilistic design

The behavior of the system of interest is approximated by a model. However, the response of the system cannot be predicted exactly, as any model is subject to uncertainties. These uncertainties include model errors and uncertainties about the values of the input parameters of the model. In a fully probabilistic design approach, all (relevant) uncertainties are quantified explicitly in terms of a vector $\mathbf{X}$ of uncertain model parameters. The values of the elements of $\mathbf{X}$ are expressed probabilistically in terms of their joint distribution function. Let vector $\mathbf{z}$ denote the design parameters of the system that can be controlled during system design.

In structural reliability analysis, the response of the system is expressed in terms of the limit state function $g(\mathbf{X},\mathbf{z})$. The sign of this function quantifies whether the system's state is considered acceptable or undesired (where positive values are associated with an acceptable behavior; negative values indicate an undesired system state). For the basic reliability problem, the limit state function is expressed as
$$
g(\mathbf{X},\mathbf{z}) = R(\mathbf{X},\mathbf{z})-S(\mathbf{X})\,,
$$ where $R$ represents the capacity of the system and $S$ is the demand. Note that in the equation above only the capacity (and not the demand) is expressed as a function of the design parameters $\mathbf{z}$, as the demand (e.g., the load acting on the system) can usually not be influenced during system design. The reliability of the system depends on the design parameters $\mathbf{z}$ and is linked to the probability of failure $p_f(\mathbf{z}) = \Pr\left[g(\mathbf{X}\le 0,\mathbf{z})\right]$.

The fully probabilistic design approach

In a fully probabilistic design approach, the design parameters $\mathbf{z}$ are selected such that $p_f(\mathbf{z})$ meets the target reliability requirement. For example, the values of $\mathbf{z}$ can be determined using stochastic optimization.

Challenges of a fully probabilistic design approach

Even though a fully probabilistic design approach allows to select the design parameters such that a target reliability requirement is explicitly maintained, it is rarely used in practice. This is mainly due to the following reasons:

Numerically/computationally: Evaluating the probability of failure in structural reliability analysis is computationally challenging. This holds in particular, if evaluating the limit-state function involves solving a numerical model. For probabilistic design, the probability of failure $p_f(\mathbf{z})$ needs to be evaluated several times, as it is a function of the design parameters $\mathbf{z}$. Even if tailored methods for structural reliability are employed, the computational costs of a fully probabilistic design are often considerable.

Conceptually: A technical standard on quantifying the uncertainties about the model parameters $\mathbf{X}$ does not exist. Therefore, selecting an appropriate probabilistic model for $\mathbf{X}$ requires a high level of expertise in probabilistic modeling.

Amount of work for the engineer: Setting up the probabilistic model and explicitly quantifying all uncertainties requires a considerable amount of time. Contrary to that, a design with classical design approaches that are based on (partial) safety factors can usually be obtained much faster.

Nevertheless, some design problems cannot be solved with classical approaches. In such cases, a fully probabilistic design is a valuable tool to obtain a reliable design.