## 4 Damage of Mortise-Tenon Joint
### Damage in the Top of the Tenon  
As can be seen in Figure 3b, the tenon part is often damaged and the cross-section of this part decreases a lot, especially in the upper place where the tenon and the mortise are in contact. In this section, this type of damage is simplified by reducing the height of the tenon of the left joint, with the length and width of the tenon remain unchanged, while the tenon of the right joint is not damaged, as shown in Figure 20. The original height of the tenon is 180 mm, and the reduction of the height is 5 mm, 10 mm, 15 mm and 20 mm. The hysteresis curves of the damaged models as well as the undamaged one are plotted in Figure 21a, and the results of the last loading cycle are also given in Figure 21b for better comparison. It can be seen from both figures that the reaction forces at different displacement amplitudes of the damaged models are far smaller than those of the undamaged model, which indicates the smaller stiffness of the damaged models. Besides, the areas enveloped by the curves, which are known as dissipated energy, decrease with the increase of the reduction of the tenon. Figure 19: Normalized dissipated energy: (**a**) Single joint damaged models; (**b**) both joints damaged models. Figure 20: Damage in the top of the tenon. The stiffnesses of each models are demonstrated in Figure 22a. For clear comparison between the stiffnesses of the undamaged model and the damaged models, the normalized stiffnesses are plotted in Figure 22b by dividing the stiffness of the damaged models by that of the undamaged model at the same displacement amplitude. From these two figures, it can be concluded that the height reduction of the tenon can significantly decrease the stiffness of the damaged models, especially when the displacement amplitude is small. When the tenon is intact, it is initially in contact with the mortise, so that the interaction between the two parts works from the beginning. However, in the damaged models with a height reduction of the tenon, there is an empty space between the tenon and the mortise of the left joint, which indicates that the two parts are not in contact with each other at the very beginning. Therefore, the interaction between them takes effect when the columns moves laterally to different displacements for different height reductions of the tenon. Once the tenon and the mortise begin to squeeze each other, there will be a sudden increase in both the hysteresis curves and the stiffness curves, which can be seen in Figures 21 and 22. It is clear in Figure 22b that the normalized stiffnesses of the damaged models with height reductions of 5 mm, 10 mm, 15 mm and 20 mm get larger prominently at the displacements of 40 mm, 100 mm, 140 mm and 140 mm, respectively. When these displacements are not reached by the corresponding models, the normalized stiffnesses of the damaged models are very close. For example, the normalized stiffnesses of the four damaged models at the displacement of 20 mm and 40 mm are around 0.2 and 0.3 respectively. Then, with the displacement increasing from 40 mm to  
Figure 21: Hysteresis curves: (**a**) All cycles; (**b**) last cycle. Figure 22: Stiffness and normalized stiffness (**a**) Stiffness; (**b**) normalized stiffness. 100 mm, the normalized stiffness of the 5 mm-reduction model is much larger than those of other three models, while these three models have the similar normalized stiffness. This phenomenon can be seen more directly from the vertical compressive stresses S33 plotted in Figure 23. Points 1-4 are the intersection points of the left mortise-tenon joint, while points 5-8 are related to the right joint, as can be seen in Figure 12. Since only the left joint is damaged, the envelope curves of S33 at points 1-4 of the damaged models are quite different from the curves of the undamaged model. When the displacement is small, the vertical compressive stresses S33 of the damaged models at these four points are nearly 0, which are far smaller than those of the undamaged model.