## 7 Conclusions  
In this paper, the hysteretic behaviors of X-shaped metal dampers with an oblique angle were investigated experimentally through a series of cyclic loading tests. Additionally, a theoretical method and finite element analysis model were also proposed to predict the elastic stiffness and hysteretic behavior of dampers, respectively. The main conclusions are drawn as follows:  
1. In the initial loading stage, parallelogram hysteretic loops were observed in specimens W30-O15\({}^{\circ}\) and W30-O30\({}^{\circ}\) before reaching the critical buckling load of X-shaped metal plates. After that, obvious buckling behavior was found as specimens were mainly in compression and shear in the negative direction. On the contrary, the tensile behavior of X-shaped metal plates would eliminate the out-of-plane deformation. 2. The stiffening ribs and cover plates can effectively constrain the out-of-plane deformation of X-shaped steel plates. However, due to the tensile and shear behavior of specimens in the positive direction, asymmetric hysteretic loops were also observed in specimens with oblique angles and buckling-restrained devices. 3. The oblique angle could effectively improve the stiffness, strength, and energy dissipation of the XMD. When the oblique angle of the XMD with stiffening ribs increased from 0\({}^{\circ}\) to 30\({}^{\circ}\) at the applied displacement of 8.4 mm, the mean strengths and cumulative energy dissipation of specimens increased by about 80.77% and 80.57%, respectively. In addition, stiffening ribs with simple design details and higher strength were more suitable for application in engineering structures. 4. A theoretical calculation method was proposed to predict the elastic stiffness of specimens with an oblique angle, and the theoretical elastic stiffness matched well with the tests results. Additionally, the relative deviation for all specimens was approximately 6%, indicating that the proposed method can effectively predict the elastic stiffness of the XMD with an oblique angle. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Specimen** & \(\Delta_{\text{p}}\)**/mm** & \(F_{\text{e}}\)**/kN** & \(F_{\text{n}}\)**/kN** & \(F_{\text{n}}\)**/\(F_{\text{e}}\)** \\ \hline W30-O15\({}^{\circ}\) & 12.00 & 21.18 & 19.66 & 0.93 \\ W30-O30\({}^{\circ}\) & 12.00 & 20.27 & 23.85 & 1.18 \\ W20-O0-LS & 12.00 & 25.23 & 26.00 & 1.03 \\ W30-O0\({}^{\circ}\)-LS & 12.00 & 27.26 & 28.40 & 1.04 \\ W30-O15\({}^{\circ}\)-LS & 12.00 & 34.90 & 34.28 & 0.98 \\ W30-O30\({}^{\circ}\)-LS & 10.20 & 45.89 & 52.18 & 1.14 \\ W30-O0\({}^{\circ}\)-LP & 12.00 & 26.26 & 29.57 & 1.13 \\ W30-O15\({}^{\circ}\)-LP & 12.00 & 31.98 & 33.90 & 1.06 \\ W30-O30\({}^{\circ}\)-LP & 12.00 & 42.82 & 41.24 & 0.96 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The comparison of experimental and numerical results. 5. The hysteretic behaviors of specimens with an oblique angle were predicted well using the proposed FEA model. For specimens with cover plates, the hysteretic loops of the proposed FEA model were fatter than those of test specimens due to the gap between X-shaped metal plates and cover plates. Conceptualization, X.Z. and L.D. ; methodology, S.L. ; software, M.Z. ; validation, L.D., M.Z., and J.Y. ; formal analysis, M.Z and L.D. ; investigation, X.Z. ; resources, S.L. ; data curation, M.Z. ; writing--original draft preparation, X.Z. and L.D. ; writing--review and editing, J.Y. and X.D. ; visualization, L.D. and X.D. ; supervision, X.Z. and S.L. ; project administration, X.Z. ; funding acquisition, X.Z., S.L., and L.D. All authors have read and agreed to the published version of the manuscript. This research was funded by the National Natural Science Foundation of China (Grant No. 52308167), Open Research Subject of Architectural Design and Research Institute Ltd., Southeast University (2020-12), China Postdoctoral Science Foundation (Grant No. 2022M723003), Science and Technology Research Project of Henan Province (Grant Nos.