# Development of Stochastic Fatigue Model of Reinforcement for Reliability of Concrete Structures
## 2 Materials and Methods
### Statistical Analysis of Fatigue Data of Steel Reinforcing Bars  
For steel reinforcement bars used in concrete S-N, curves are recommended by various international codes (such as Model code 2010, Model code 1990, DNV OS C 502, EN 1992-1) [28; 29; 30; 31] and are generally written as:  
\[n_{i}=K\;\Delta s_{i}^{-m}, \tag{1}\]  
or  
\[\log(n_{i})=\log(K)-m\;\log(\Delta s_{i}), \tag{2}\]  
where \(n_{i}\) is the number of cycles to failure with stress range \(\Delta s_{i}\) in test number, \(i\). \(K\) and \(m\) are fatigue parameters to be fitted by MLM here using test data [31]. To account for uncertainties in fatigue life, Equation (2) can be rewritten [22]:  
\[\log(n_{i})=\log(K)-m\;\log(\Delta s_{i})+\varepsilon, \tag{3}\]  
where \(\varepsilon\) represents the uncertainty of the fatigue life model and is modelled by a stochastic variable with mean value equal to zero and standard deviation, \(\sigma_{\varepsilon}\). \(\varepsilon\) is often assumed to have a Normal distributed [31]. The Likelihood function to be used to estimate the optimal values of the parameters \(K\), \(m\), and \(\sigma_{\varepsilon}\) from test data is written [22]:  
\[L(K,\;m,\sigma_{\varepsilon})=\prod_{i=1}^{n_{i}}P[\log(K)-m\;\log(\Delta s_{ i})+\varepsilon=\log(n_{i})]\times\prod_{i=n_{i}+1}^{n_{i}+n_{\varepsilon}}P[\log(K)-m\; \log(\Delta s_{i})+\varepsilon>\log(n_{i})]. \tag{4}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Data Number (Index)** & **Number of Cycles to Failure** & **Stress Range [MPa]** & **Run-Out** \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 1: Data [1]. Here, \(n_{i}\) is the number of stress cycles to failure or to run-out with stress range \(\Delta s_{i}\) in test number \(i\). \(n_{F}\) is the number of tests where failure occurs, and \(n_{R}\) is the number of tests where failure did not occur after \(n_{i}\) stress cycles (run-outs). The total number of tests is \(n=n_{F}+n_{R}\). \(K\), \(m\), and \(\sigma_{\varepsilon}\) are obtained from the optimization problem \(max_{K,m,\sigma_{\varepsilon}}L(K,m,\sigma_{\varepsilon})\), which can be solved using a non-linear optimization algorithm [31]. Run-outs contain information which from a statistical point of view has to be included in the statistical modelling in order to be consistent with all tests performed. This paper describes how run-outs can be included using the MLM. The number of cycles where the tests are stopped are often chosen in order to limit the costs and time used for the test campaign. The terms in Equation (4) can be obtained from Equation (5) [22]:  
\[ \begin{split} P[\log(K)-\text{mlog}(\Delta s_{i})+\varepsilon= \log(n_{i})]&=\frac{1}{\sqrt{2}\pi\sigma_{\varepsilon}}\exp \left(-\frac{1}{2}\left(\frac{\log(K)-mlog(\Delta s_{i})-\log(n_{i})}{\sigma_{ \varepsilon}}\right)^{2}\right),\\ P[\log(K)-\text{mlog}(\Delta s_{i})+\varepsilon>\log(n_{i})]& =\Phi\left(\frac{\log(K)-\text{mlog}(\Delta s_{i})-\log(n_{i})}{\sigma_{ \varepsilon}}\right).\end{split}  \tag{5}\]  
The parameters \(K\), \(m\), and \(\sigma_{\varepsilon}\) are generally determined using a limited number of data. Consequently, the estimates are subject to statistical/parameter uncertainty. Since the parameters are estimated by the MLM, they become asymptotically (number of data should be \(>\)25-30). Normal distributed stochastic variables with expected values equal to the maximum-likelihood estimator and a covariance matrix equal to [32]:  
\[C_{K,m,\sigma_{\varepsilon}}=[-H_{K,m,\sigma_{\varepsilon}}]^{-1}=\left[ \begin{array}{ccc}\sigma_{K}^{2}&\rho_{K,m}\sigma_{K}\sigma_{m}&\rho_{K, \sigma_{\varepsilon}}\sigma_{K}\sigma_{\sigma_{\varepsilon}}\\ \rho_{K,m}\sigma_{K}\sigma_{m}&\sigma_{m}^{2}&\rho_{m,\sigma_{\varepsilon}} \sigma_{m}\sigma_{\sigma_{\varepsilon}}\\ \rho_{K,\sigma_{\varepsilon}}\sigma_{K}\sigma_{\sigma_{\varepsilon}}&\rho_{m, \sigma_{\varepsilon}}\sigma_{m}\sigma_{\sigma_{\varepsilon}}&\sigma_{\sigma_{ \varepsilon}}^{2}\end{array} \right].