Reliability
Eurocodes were developed with a reliability target \(\beta=3.8\) with the reference period of 50 years, which corresponds to the yearly probability of failure approximately \(10^{-6}\). The same reliability should be ensured by numerical design calculations, which is almost never done and the guidelines on how to assess reliability are not well-known and publicly available. This thesis presents a step-by-step procedure based on several documents – Eurocode background \cite{EC_Reliability_background}, SAFEBRICTILE project Final report \cite{SAFEBRICTILE_Final_Report}, and most importantly SAFEBRICTILE deliverable D1.1 \cite{SAFEBRICTILE_D1.1}.
This chapter provides step-by-step guidance for the reliability assessment of numerical design calculations according to Eurocode standards and is mostly built upon SAFEBRICTILE deliverable D1.1 \cite{SAFEBRICTILE_D1.1}, which was not published.
First, it is necessary to split the reliability of actions and resistances, which greatly simplifies the task. Eurocode assumes factors for resistance \(\alpha_R=0.8\) and for loads \(\alpha_E=0.7\). The probability that the actual resistance R is smaller than the design resistance \(R_d\) is then:
\[ P\left(R\le R_d\right)=\Phi\left(-\alpha_R\cdot\beta\right)=\Phi\left(-0.8\cdot3.8\right)=\Phi\left(-3.04\right) \]
In other words, 1183 specimens should fail out of a million. Then a set of experiments should be specified, and experimental resistance \(r_{e,i}\) determined. This may seem like an easy task, but the load resistance is often an ambiguous term that needs to be clarified. For the example of tensile rupture, the load resistance is assumed to be at maximum load.
Then numerical design model is created for each experiment and the numerical resistances \(r_{t,i}\) are obtained. The geometrical and material properties are set as measured in the experiment. The numerical model should be subjected to a validation and verification process as described in FprEN 1993-1-14:2024 -- Chapter 7 \cite{FprEN1993-1-14}.
The next step is to determine the basic variables that affect the results. These variables are for example plate thickness or material yield strength. The mean values and coefficient of variation for the basis for calibration of partial factors are now codified in FprEN 1993-1-1:2017 – Table E.1 [6]. Interestingly, the variability in dimensional properties does not match the manufacturing tolerances, e.g. in EN 10029:2010 [7], but it outlines the assumptions, upon which the reliability of Eurocodes is built.
Next, the mean value of correction factor b should be calculated:
\[ b=\sum_{i=1}^{n}{r_{e,i}r_{t,i}}/\sum_{i=1}^{n}r_{t,i}^2\]
Where \(n\) is the number of specimens. FprEN 1993-1-14 -- Annex A \cite{FprEN1993-1-14} states that \(b\) should be in the range \(0.8 < b < 1.25\). SAFEBRICTILE \cite{SAFEBRICTILE_Final_Report} is even more strict -- range \(0.85 < b <1.15\). However, for numerical design calculation models with bilinear material model and small plastic strain limit, it is expected to reach high values of \(b\), e.g. 1.3. Such higher values are normal when neglecting strain-hardening and should not be a surprise or reason to terminate the reliability assessment due to inaccuracy.
Then calculate coefficient of variation of the error \(V_\delta\) by following equations:
\[ \delta_i=\frac{r_{e,i}}{b\cdot r_{t,i}} \]
\[ \Delta_i=\ln{\left(\delta_i\right)} \]
\[ s_\Delta^2=\frac{1}{n-1}\sum_{i=1}^{n}\left(\Delta_i-\bar{\Delta}\right)^2 \]
\[ V_\delta=\sqrt{\exp{\left(s_\Delta^2\right)-1}}\]
Where \(\bar{\Delta}\) is the average of \(\Delta_i\). The next step is a calculation of coefficient of variation \(V_{rt}\) related to the basic input variables. For that, the geometrical and material properties should be changed in these numerical models to mean values according to Table E.1 \cite{FprEN1993-1-1}. The load resistances of these numerical models are \(r_{t,i}(\underline{X_m})\). Furthermore, the numerical models with slightly changed mean values should be calculated. For example, if the basic variables are plate thickness and yield strength, a set of numerical models with changed plate thickness (load resistances \(r_{t,i}(\Delta t))\) and another set of numerical models with changed yield strength should be calculated (load resistances \(r_{t,i}({\Delta f}_y)\)). The change should be small but meaningful, and it should not completely change the failure mode. The authors suggest using values close to the standard deviation of the basic variable \(\sigma\).
\[ V_{r,t,i,\Delta t}^2=\left[\frac{\left(r_{t,i}\left(\underline{X_m}\right)-r_{t,i}\left(\Delta t\right)\right)}{\Delta t}\cdot\sigma\left(t\right)\right]^2\]
\[ V_{r,t,i,\Delta f_y}^2=\left[\frac{\left(r_{t,i}\left(\underline{X_m}\right)-r_{t,i}\left(\Delta f_y\right)\right)}{\Delta f_y}\cdot\sigma\left(f_y\right)\right]^2\]
\[ V_{rt,i}^2=\left(V_{rt,i,\Delta t}^2+V_{rt,i,\Delta f_y}^2\right)/r_{t,i}^2(\underline{X_m})\]
\[ V_{rt}^ =\sum_{i=1}^{n}\sqrt{V_{rt,i}^2}\]
It is probable that many numerical models will have more basic variables. In that case, expand the formulas accordingly. The values of \(V_{rt}\) depend mainly on the relationship of the result to the basic variables. If the relationship is linear, such as for tensile rupture, the expected values of \(V_{rt}\) are about 5\%. For phenomenon, where relationship is quadratic or even on a power of three, such as plate bending or buckling, the values of \(V_{rt}\) will be higher. It is difficult to establish \(V_{rt}\) specifically for every experimental set, and \(V_{rt}\) may be set based on previous experience; e.g. first generation of Eurocode was built mainly with \(V_{rt}=7\%\).
Finally, the two errors, \(V_\delta\) and \(V_{rt}\) are added together to coefficient of variation:
\[ V_r=\sqrt{V_\delta^2+V_{rt}^2}\]
Now variation coefficients \(Q\) can be calculated either for individual \(V_{rt,i}\) for each specimen, or global \(V_{rt}\):
\[ Q_{rt,i}=\sqrt{\ln{\left(V_{rt,i}^2+1\right)}}, \;\; \textrm{or for global } V_{rt}: \;\; Q_{rt}=\sqrt{\ln{\left(V_{rt}^2+1\right)}}\]
\[Q_\delta=\sqrt{\ln{\left(V_\delta^2+1\right)}}\]
\[ Q_i=\sqrt{\ln{\left(V_\delta^2+V_{rt,i}^2+1\right)}}, \;\; \textrm{or for global } V_{rt}: \;\; Q=\sqrt{\ln(V_r+1)} \]
The design resistance is calculated according to the number of specimens \(n\) (for global \(V_{rt}\), replace \(Q_{rt,i}\) for \(Q_{rt}\) and \(Q_i\) for \(Q\)):
\[ r_{d,i} =
\begin{cases}
b \cdot r_{t,i}\left(\underline{X_m}\right) \exp\left(-k_{d,\infty} \cdot \left(\dfrac{Q_{rt,i}^2}{Q_i}\right) - k_{d,n} \cdot \left(\dfrac{Q_\delta^2}{Q_i}\right) - 0.5 \cdot Q_i^2\right), & \text{for } n \leq 100 \\
b \cdot r_{t,i}\left(\underline{X_m}\right) \exp\left(-k_{d,\infty} \cdot Q_i - 0.5 \cdot Q_i^2\right), & \text{for } n > 100
\end{cases}
\]
Where \(k_{d,n}\) and \(k_{d,\infty} \) are from EN 1990 – Table D2 \cite{EN1990}, and FprEN 1993-1-14 -- A3(4) \cite{FprEN1993-1-14} suggests that \(V_x\) unknown should be used. Often, the number of specimens is higher than 30 and interpolation between 30 and infinity should be made. Authors suggest treating 100 specimens as infinity, i.e., \(k_{d,100}=k_{d,\infty}=3.04\) and interpolate between 30 and 100.
Finally, the partial safety factor can be calculated for each model and then the average:
\[ \gamma_{M,i}=r_{nom,i}/r_{d,i}\]
\[ \gamma_M=\frac{1}{n}\sum_{i=1}^{n}\gamma_{M,i} \]
This partial safety factor should be below acceptable limit, otherwise, it should be applied to each numerical design calculation using this model settings.
SAFEBRICTILE \cite{SAFEBRICTILE_D1.1} recommends acceptable levels of \(\gamma_M\) according to current Eurocode practice based on coefficient of variation \(V_r\); see Table \ref{tab:acceptance_gammaM}. These acceptance limits are within the target reliability according to Equation (1). In case \(\gamma_M\) is below acceptance limit, it does not need to be applied.
\begin{table}[ht]
\centering
\begin{tabular}{|c|c|}
\hline
\textbf{Range of $V_r$} & \textbf{Acceptance limit for $\gamma_M$} \\
\hline
$0.00 < V_r < 0.04$ & $1.03$ \\
$0.04 \leq V_r < 0.20$ & $1.03 + 0.75 \cdot (V_r - 0.04)$ \\
$V_r \geq 0.20$ & $1.15$ \\
\hline
\end{tabular}
\caption{Recommended acceptance limits of \(\gamma_M\) \cite{SAFEBRICTILE_D1.1} }
\label{tab:acceptance_gammaM}
\end{table}
Furthermore, coefficient of variation \(V_\delta\) may be reduced by two options:
\begin{itemize}
\item Sample subdivision: The set of specimens may be divided into groups, for which one model or model settings is more appropriate than the other. For example, some phenomena may be well captured only by GMNIA. Another appropriate subdivision may be into mild steel and high-strength steel. Note that with decreasing number of specimens in the set, \(k_{d,n}\) increases.
\item Tail approximation. This is useful for cases where many results of numerical model are very safe compared to experiments, which leads to high coefficient of variation \(V_\delta\). Such very safe cases may be disregarded in the reliability assessment, which leads to decrease in \(b\) value but also decrease in \(V_\delta\). Basically, the samples may be disregarded from the safest (smallest \(r_{t,i}/r_{e,i}\)) until partial safety factor \(\gamma_M\) decreases. This takes into account the fact that the distribution is not normal, but rather log-normal and the tail-end may be ignored. The value of \(k_{d,n}\) remains unchanged.
\end{itemize}
Experimental database
Experimental database is shown here as a list containing:
Year – First author name – DOI – Experiment type – Number of specimens
ideaCon project is always attached for the readers to review. The whole database is accessible through GitHub.
2025 – Francavilla – https://doi.org/10.1016/j.engstruct.2025.120118 – T-stub – 5 specimens
2024 – Xu – https://doi.org/10.1002/eng2.12869 – T-stub – 10 specimens
2024 – Luo – https://doi.org/10.1016/j.istruc.2024.106936 – Double split T-joint – 4 specimens
2022 – Song – https://doi.org/10.12989/scs.2022.42.4.569 – End plate (stainless steel) – 6 specimens
2022 – Lin – https://doi.org/10.1016/j.istruc.2022.06.039 – T-stub – 8 specimens
2022 – Gao – https://doi.org/10.1016/j.tws.2022.110119 – Double split T-joint – 4 specimens
2022 – Bezerra – https://doi.org/10.1016/j.jcsr.2022.107242 – T-stub – 4 specimens
2021 – Ozkılıç – https://doi.org/10.1016/j.jcsr.2021.106908 – T-stub – 35 specimens
2021 – Zhao – https://doi.org/10.1016/j.jcsr.2021.106919 – T-stub – 19 specimens
Results
