## 5 Conclusions  
In order to establish a formula that characterizes joint roughness, considering the mechanical contribution of each asperity order, shear tests were carried out on 39 groups of red sandstone joint specimens with seven different roughness degrees (5\({}^{\rm\#}\)-10\({}^{\rm\#}\) standard JRC profile and flat) and two asperity orders under three different normal stresses. The mechanical contribution of each asperity order was analyzed, and the relationship between the mechanical contribution of each asperity order and its statistical parameters was explored. The conclusions are as follows:  
1. A new decomposition method of joint surface roughness is proposed. The ensemble empirical mode decomposition (EEMD) in this method does not require an a priori basis function, and the critical decomposition level criterion in this method has a clear mathematical definition. The standard JRC profile can be decomposed into waviness and unevenness by the method combining the EEMD and the critical decomposition level.
2. The mechanical contribution of waviness and unevenness is related to normal stress. The ratio between them is less affected by normal stress. The mechanical contribution of the waviness and the unevenness decreases with the increase in normal stress, which means that as normal stress increases, the influence of the joint surface morphology on the shear strength gradually decreases. At the same time, the mechanical contribution ratio of waviness and unevenness hardly changes with the change in normal stress.
3. The mechanical contribution of waviness and unevenness is related to their statistical parameter contribution. The relationship between the mechanical contribution ratio of waviness and unevenness and its statistical parameter ratio can be describe by the power function. The mechanical contribution ratio of waviness and unevenness decreases with the increase in their statistical parameter ratio, which means that it is difficult to accurately characterize the roughness of the joint surface by considering only the contribution of the statistical parameter. In order to characterize the roughness of the joint surface accurately, we should consider not only the contribution of the statistical parameter, but also the mechanical contribution.
4. A formula for characterizing the roughness of the joint surface is established that considers the morphology contribution of waviness and unevenness and their mechanical contributions at the same time. Compared with the formula that only considers the statistical parameters of the joint surface, this formula fully considers the mechanical contribution of each asperity order. This formula is of great significance for accurately characterizing the roughness of the joint surface and evaluating the shear strength of the joint.  
Conceptualization, Z.Y. and Y.Y.; methodology, Z.Y.; software, Z.Y.; writing--original draft preparation, Z.Y.; writing--review and editing, B.L. and Y.L.; supervision, Y.Y.; funding acquisition, Y.Y. All authors have read and agreed to the published version of the manuscript. This research was funded by the scientific research project funded by Department of Education of Hubei Province grant number Q20201109 and the Natural Science Foundation of HubeiProvince grant number 2020 CFB123 and Hubei Province Key Field R&D Program Project grant number 2020 BCA082.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Data is contained within the article.  
**Conflicts of Interest:** The authors declare no conflict of interest.