## 4 Damage of Mortise-Tenon Joint
### Damage in the End of the Tenon
When the damage length is greater than 100 mm, the two curves are almost linear, and the slopes of the two curves are \(-0.00404\) mm\({}^{-1}\) and \(-0.00806\) mm\({}^{-1}\). The former value is nearly half of the latter one. This is because the slope can be regarded as the reduction of the stiffness of the damaged model. If a given damage length \(l\) causes a reduction \(d_{\rm S}(l)\) on the averaged normalized stiffness of the single damaged model, then the averaged normalized stiffness \(\overline{NK_{\rm S}}(l)\) of this model can be derived by  
Figure 4.0  
When both of the tenons are damaged with the same damage length \(l\), and the damages at the two tenons are not supposed to affect each other, thus the reduction on the averaged normalized stiffness will be \(2d_{\rm S}(l)\), and the averaged normalized stiffness \(\overline{NK_{\rm B}}(l)\) will be  
Figure 4.1  
The right part of Equation (4) \(2\overline{NK_{\rm S}}(l)-1\) is calculated and listed in the last row of Table 3 and is also plotted in Figure 29 labelled by 2 Single-1. It can be seen that the \(2\overline{NK_{\rm S}}(l)-1\) is almost identical with \(\overline{NK_{\rm B}}(l)\). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Damage Length (mm)** & **0** & **40** & **60** & **80** & **100** & **120** & **140** & **160** & **180** & **200** \\ \hline \multirow{2}{*}{ANS} & Single & 1 & 0.997 & 0.988 & 0.959 & 0.911 & 0.840 & 0.756 & 0.667 & 0.584 & 0.507 \\  & Both & 1 & 0.994 & 0.975 & 0.917 & 0.822 & 0.684 & 0.519 & 0.353 & 0.174 & 0.016 \\ \hline \multicolumn{1}{c}{2 Single-1 [FOOTNOTE:]Footnote : Single-1 means twice the values in the row of ‘Single’ and then subtract 1.[ENDFOOTNOTE] } & 1 & 0.994 & 0.976 & 0.917 & 0.823 & 0.680 & 0.512 & 0.334 & 0.167 & 0.014 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Averaged normalized stiffness (ANS) of damaged models. Figure 28: Normalized stiffness of damaged models: (**a**) Single tenon damaged; (**b**) both tenons damaged. Figure 30 presents the normalized dissipated energy of the models, and these curves share the similar characteristics with the normalized stiffness. The normalized dissipated energy has a slight rising trend, especially for the models with large damage length, although the slopes of these curves are negligible. Therefore, the averaged normalized dissipated energy is achieved through the same way as the averaged normalized stiffness and is given in Table 4 and Figure 31.  
Just as the averaged normalized stiffness, the changing trends of the averaged normalized dissipated energy with the damage length smaller and bigger than 100 mm are significantly different. The averaged normalized dissipated energy can be regarded as a bilinear curve with the two straight lines intersecting at the damage length of 100 mm, which is remarked by the dash line in Figure 31. The slopes of the left part of the curves are prominently smaller than those of the right part. When the damage length is 100 mm, the averaged normalized energy of both models are 0.975 and 0.946, respectively, which are greater than the averaged normalized stiffnesses with the same damage length. When the damage length is smaller than 100 mm, the reductions on the dissipated energy are all greater than 0.95, which means the damage length smaller than 100 mm has little effect on the frame. When the damage length reaches 180 mm, the reductions on dissipated energy of single and both damaged \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Damage Length (mm)** & **0** & **40** & **60** & **80** & **100** & **120** & **140** & **160** & **180** & **200** \\ \hline \multirow{2}{*}{ANDE} & Single & 1 & 0.995 & 0.985 & 0.979 & 0.975 & 0.930 & 0.850 & 0.764 & 0.682 & 0.612 \\  & Both & 1 & 0.991 & 0.975 & 0.960 & 0.946 & 0.866 & 0.714 & 0.549 & 0.406 & 0.268 \\ \hline \multirow{2}{*}{2 Single-1} & 1 & 0.990 & 0.970 & 0.958 & 0.950 & 0.861 & 0.699 & 0.528 & 0.364 & 0.225 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Averaged normalized dissipated energy (ANDE) of damaged models.