How to derive partial safety factors based on FORM?

A powerful method to approximate the probability of failure is FORM. As a by-product of FORM, the design-point $\mathbf{x}^\star$ is obtained, which is the most probable point for failure. The design-point acts as a reference point for deriving partial safety factors based on FORM.

What is the motivation for deriving partial safety factors? 

Typically, partial safety factors used in a semi-probabilistic design are specified in technical standards and were calibrated based on a target reliability requirement. For certain types of structures or problems, the technical standards might not provide suitable safety factors. In other problems, one might want to identify partial safety factors that are tailored to a specific type of structure (for a clearly defined scope). 

How to derive safety factors from a probabilistic design?

1. In order to identify suitable partial safety factors for a certain type of system, one first needs to conduct a fully probabilistic design of that system — such that the system's design meets the specified target reliability.
2. Next, the design point $\mathbf{x}^\star$ associated with that system's design is identified.
3. Based on the probabilistic model, characteristic values $x_{i,\mathrm{k}}$ for all uncertain model parameters $X_i$ are evaluated (with $i\in\{1,\ldots,M\}$, where $M$ denotes the number of uncertain model parameters).
4. A set of safety factors that is linked to the reliability of the structure's design can be evaluated as 
   $$
   \gamma_i = \frac{x^\star_i}{x_{i,\mathrm{k}}}\,,
   $$ where $x^\star_i$ denotes the $i$th element of the vector of the design point $\mathbf{x}^\star$, and $\gamma_i$ is a safety factor linked to the $i$ uncertain model parameter.

Note: The safety factors above are defined such that the design value $x_{i,\mathrm{d}}$ is obtained by multiplying the characteristic value $x_{i,\mathrm{k}}$ with the safety factor $\gamma_i$; i.e., $x_{i,\mathrm{d}} = \gamma_i \cdot x_{i,\mathrm{k}}$. This corresponds to the definition of safety factors if the associated model parameter acts as a load variable. For model parameters that act as resistance variables, the safety factor is usually defined as the reciprocal value of $\gamma_i$. For the discussion here, we do not distinguish between capacity and load parameters and define the safety factors of all parameter as . However, based on the design point, the model parameters can be classified as capacity and demand and the definition of safety factors can be adopted accordingly.

Note: For the definition above, the design values $x_{i,\mathrm{d}}$ are equal to the coordinates of the design point $x^\star_i$; i.e. $x_{i,\mathrm{d}}==x^\star_i$. That is why the most likely failure point is typically referred to as "design point".

Is there a unique set of safety factors for a specific target reliability level?

Above we defined the safety factors such that they are directly linked to the design point. Remember that at the design point, the limit-state function equals zero. However, also remember that the design point is just the most probable point within the failure domain — there exists actually an infinite number of points that all are associated with a zero value of the limit-state function. Any of these points could be used to derive safety factors that would result in an equally reliable design —  actually any of these points would return the same design if applied within a semi-probabilistic design. 

Thus, the design point is a natural choice as reference point for deriving partial safety factors — but it is not a unique choice. Many other combinations (infinitely many) of safety factors will result in an equally reliable design.

A challenge when calibrating safety factors is to select the safety factors such that the choice is robust with respect to the variations of the structures in the portfolio (the structures that will be designed using the partial safety factors).