# Working State of ECC Link Slabs Used in Continuous Bridge Decks
## 2 The Structural Working State Theory
### Detection of the Structural Working State Characteristics  
By directly reading the \(E_{j,\mathrm{norm}}\)-\(L_{j}\) curve, some of the working state leaps might be difficult to be recognized. In order to address this problem, the Mann-Kendall (M-K) criterion widely used in climatological, meteorological, and hydrological field is introduced to detect the trend of GSED sum over loading steps [31; 32; 33]. This non-parametric statistical analysis method is insensitive to outliers and free from data distributions [34; 35; 36]. By examining the sign of all pairwise differences of observed values, the mutations are distinguished from the temporal series. The sequence of \(E_{j,\mathrm{norm}}\) (the loading step \(j=1\), \(2\),..., \(n\)) is assumed to be statistically independent. Actually, the relevant and independent elements coexist in the struc-tural working status at different loading steps to a certain ex-tent. According to the [PERSON]'s principle, structural components which are located far away from each other have little spatial relevance or mutual effects, leading to the significant independence of the structural responses (strains, deflections, etc.) at different loca-tions. Additionally, the inherent randomness in experimen-tal models and material properties brings about many independent elements at different loading steps. Also, this analysis method could be feasible from the result-oriented perspective, which will be verified later. With these considerations, the leap characteristics of structural working state can be detected using the M-K criterion. In the application of the M-K criterion, a new stochastic variable \(d_{k}\) at the \(k\)th loading step can be defined by:  
\[d_{k}=\sum_{i=1}^{k}m_{i}(2\leq k\leq n),m_{i}=\left\{ \begin{array}{ll}+1,& \mbox{$E_{\mathrm{norm}}(i)>E_{\mathrm{norm}}(j)(1\leq j\leq i)$}\\ 0,&\mbox{otherwise}\end{array} \right., \tag{3}\]where \(m_{i}\) is the cumulative number of samples; \"+1\" means adding one more to the existing value if the inequality on the right side is satisfied for the \(j\)th comparison. The mean value E(\(d_{k}\)) and variance Var(\(d_{k}\)) of \(d_{k}\) can be calculated by:  
\[ \begin{array}{ll}\mbox{E}(d_{k})=k(k-1)/4,&(2\leq k\leq n)\\ \mbox{Var}(d_{k})=k(k-1)(2k+5)/72,&(2\leq k\leq n)\end{array}  \tag{4}\]  
Assuming that the \(\{E_{\rm norm}\ (i)\}\) sequence is statistically independent, a new statistic \(U\!F_{k}\) is defined by:  
\[U\!F_{k}=\left\{ \begin{array}{ll}0,&k=1\\ (d_{k}-\mbox{E}(d_{k}))/\sqrt{\mbox{Var}(d_{k})},&2\leq k\leq n\end{array}  \right.. \tag{5}\]  
Thus, an \(U\!F_{k}\)-\(L_{j}\) curve can be formed using the \(U\!F_{k}\) data. A similar procedure proceeds the inverse \(\{E_{\rm norm}\ (i)\}\) sequence, which is denoted by \(\{E^{\prime}_{\rm norm}\ (i)\}\) as following:  
\[E_{\rm norm}(i)=E^{\prime}_{\rm norm}(n-i+1), \tag{6}\]  
where \(n\) is the sample capacity. Similarly, the stochastic variable \(d^{\prime}{}_{k}\) at the \(k\)th loading step is defined by:  
\[d^{\prime}_{k}=\sum_{i=1}^{k}m_{i}(2\leq k\leq n),m_{i}=\left\{ \begin{array}[] {ll}+1,&E^{\prime}_{\rm norm}(i)>E^{\prime}_{\rm norm}(j)(1\leq j\leq i)\\ 0,&\mbox{otherwise}\end{array} \right., \tag{7}\]  
where \(m_{i}\) is still the cumulative number of samples; \"+1\" means adding one more to the existing value if the inequality on the right side is satisfied for the \(j\)th comparison. The mean value E(\(d^{\prime}{}_{k}\)) and the variance Var(\(d^{\prime}{}_{k}\)) of \(d^{\prime}{}_{k}\) can be calculated as follows:  
\[ \begin{array}{ll}\mbox{E}(d^{\prime}_{k})=k(k-1)/4,&(2\leq k\leq n)\\ \mbox{Var}(d^{\prime}_{k})=k(k-1)(2k+5)/72,&(2\leq k\leq n)\end{array} , \tag{8}\]  
where \(d^{\prime}{}_{k}\) represents the degree of the ascending trend of \(\{E^{\prime}_{\rm\ j,norm}\ (i)\}\) sequence. It is important to note that the inversed sequence has a developing trend contrary to the original one.