## 3 Strength Reduction Assessment Methodology
### Reliability Framework  
A reliability framework shall consider the random nature of _resistance_ and _load effect_ functions, respectively \(R\) and \(U\), and the minimum required safety margins. In the practical field of structural concrete, engineers shall apply design code provisions based on a quasi-probabilistic approach, using for instance the concept of load and resistance factors [22]. Under-strength material factors (e.g., \(\phi_{s}\) for steel reinforcement and \(\phi_{c}\) for concrete) and overload factors (e.g., \(\lambda_{P}\) factors for each load category \(P\)) are derived from probabilistic data and are compatible with a maximum allowable probability of failure \(p_{f}\) or a minimum reliability index \(\beta_{min}\) ([13; 23]). This approach, also called the _partial safety factor approach_, can be extended to the field of ANFEA by using characteristic values of material inputs parameters (e.g., \(\phi_{c}f_{c}^{\prime}\) for concrete or \(\phi_{s}f_{y}\) for steel reinforcement) in the non-linear analyses. Different load combinations using different load factors can be considered for the search of the critical load path. This approach is not really suitable to all applications even if it is widely used in engineering practice due to its simplicity and consistency with current design code provisions [13; 24]. For example, in non-linear analyses involving thermal effects, the use of characteristic or design values of material properties would result in a weaker material which may lead to unrealistic redistribution of internal forces. A reliability framework has been suggested by second author in a previous work, for the context of concrete design using results of ANFEA [13]. It is based on the _global resistance factor approach_, and is well-suited for structural engineers who have access to nonlinear  
Figure 3: Strength reduction assessment methodology: (**a**) SRAM and original SRDM; (**b**) Schematic representation of assessment analyses in the principal loads space; (**c**) Reliability problem and global resistance factor; (**d**) Schematic representation of [PERSON]’s assessment analyses in the principal loads space. deterministic finite element packages with concrete models. This formwork suggests a reliability format that focuses _on the variability on the resistance side_ while using the concept of critical load paths on the load side which is compatible with the philosophy of existing design codes (Figure 3c). A global resistance factor \(\gamma_{G}\) is computed for each loading path and then used to deduce the design resistance value \(R_{d}\) from a mean resistance value \(R_{m}\) obtained by ANFEA, using the following equation:  
\[R_{d}=\frac{R_{m}}{\gamma_{G}} \tag{4}\]  
The global resistance factor is computed from the following expression:  
\[\gamma_{G}=e^{\beta_{min}\cdot\alpha\cdot V_{R}} \tag{5}\]  
where: \(\alpha=0.80\) and \(\beta_{min}\) is the minimum reliability index required by code provisions (typical values: 3.5 for a ductile failure and 4.25 for a brittle failure). \(V_{R}\) is the coefficient of variation or c.o.v of the resistance function \(R\) estimated through the so-called [PERSON]'s point estimate method [25]. Point estimate methods are simple and robust statistical methods used to evaluate statistical moments of a random function \(R\) (e.g., mean and variance) by using its values at particular points in the space of the input variables, called sampling points. These points are typically located around the mean value point plus or minus a standard deviation. [PERSON]'s point estimate method [25] was specifically used and validated in a previous study [13], to assess \(V_{R}\), using a limited number of ANFE analyses. Typically, \(2^{N}\) analyses are required to assess the statistical moments of \(R\), using the following equation:  
\[E(R^{m})\approx\sum\
olimits_{2^{N}\ points}p_{i}R_{i}^{m} \tag{6}\]  
where:  
* \(m\) is the moment order (e.g., \(m=1\) for the mean value).