## 4 Damage of Mortise-Tenon Joint
### Damage in the End of the Tenon  
In this section, the tenon is simplified to be cut off perpendicular to the longitudinal direction of the beam, as is seen in Figure 25. This cutoff damage is designated to happen in the left tenon or in both tenons. The damage length of the tenon is from 40 mm to 200 mm with intervals of 20 mm. Therefore, there are 18 damaged models generated in this section. Here, the damage length along with the letter S (for single tenon damaged) or B (for both tenons damaged) are used to denote the damaged models. For example, 40 mm-S model means that the damage length of the tenon is 40 mm and only the left tenon of the model is damaged. It should be noted that the length of the tenon exceeding the column is 100 mm, as seen in Figure 4. Therefore, when the damage length is bigger than 100 mm, the remaining part of the tenon is all inside the column. The hysteresis curves of the last loading cycle of the damaged models as well as the undamaged model are given in Figure 26. It should be noted that not all curves derived from the damaged models are given in the figure since they share the similar varying trend. Figure 26a shows the results of the models that only have their left tenon damaged (labelled with S), while Figure 26b are related to the models with both tenons damaged (labelled with B). It can be seen from both figures that the 40 mm-S  
Figure 24: Normalized dissipated energy. Figure 25: Damage in the end of the tenon. model and the 40 mm-B model have identical hysteresis curves with the undamaged model, which means this cutoff part contributes little to the strength and stiffness of the frame. This is consistent with the stress diagram in Figure 10 since the stress in that cutoff part is almost zero. Besides, the strength and the dissipated energy of the last cycle both reduce with the increase of the damage length of the tenon. From the comparison between these two figures, when the damage lengths are the same, the strength and the dissipated energy of the single tenon damaged model are greater than the model with two damaged tenons. Figure 27 demonstrates the varying trend of the stiffness of the damaged and undamaged models. It can be seen from both figures that, at the same displacement amplitude, the stiffness of the damaged models gets smaller with the enlargement of the cutoff part. For the 200 mm-B model, negative stiffness is observed when the displacement amplitude reaches 140 mm, which is also seen in Figure 26b. To better understand the influence of this cutoff damage on the stiffness of the model, the normalized stiffnesses of the damaged models are given in Figure 28. Although the normalized stiffness of different damaged models changes with respect to the displacement amplitude, the changing range of each curve is very limited. Therefore, the normalized stiffness of each model is regarded as a constant value, which is derived by averaging the normalized stiffnesses of the model at different displacement amplitudes, and is labelled as Averaged normalized stiffness (ANS) in Table 3.  
Figure 27: Stiffness of the damaged models: (**a**) Single tenon damaged; (**b**) both tenons damaged. Figure 26: Hysteresis curves of last cycle: (**a**) Single tenon damaged; (**b**) both tenons damaged. When the damage length is smaller than 100 mm, there is still some remaining part of the tenon outside the column. Therefore, the tenon of the damaged model contacts with the mortise at the abovementioned intersection points as the undamaged model. However, when the damage length exceeds 100 mm, the remaining part of the tenon is all surrounded by the column, which leads to the contact points between the tenon and the mortise moving to the inner of the column. Therefore, the varying trends of the averaged normalized stiffness with the damage length before and after 100 mm are quite different as plotted in Figure 29. When the damage length is 100 mm, the averaged normalized stiffnesses of the single tenon damaged model and the both tenons damaged model are 0.91 and 0.82, respectively.