## 5 Concluding Remarks  
In this paper, the combined finite-discrete element method (FDEM) was employed to study the first-order seismic capacity of masonry arch structures subjected to both gravity and a constant horizontal acceleration. The simulation results are validated with data from the existing literature.  
In Sections 3.1 and 3.2, the seismic capacity factors for circular and basket-handle arch structures were predicted by the FDEM, and the results were compared with those from the limit analysis and the UDEC. The FDEM results show that the failures are dominated by a four-hinge failure mechanism should sliding be avoided. Both the seismic capacity factor \(\lambda^{\star}\) and the hinge locations were well verified. Supporting the high buttresses, the buttressed arches are much more vulnerable to seismic motions than the corresponding unbuttressed arches. In Section 3.3, the seismic capacity of a multi-span buttressed pointed masonry arch structure was studied with the FDEM, and the obtained results were almost identical to that from the limit analysis, illustrating the reliability and robustness of the FDEM in analysing arch structures with sophisticated shapes.  
A parametric investigation on \(H/h\), \(B/R_{c}\) and the friction coefficient \(\mu\) was conducted in Section 4 to evaluate their effects on \(\lambda^{\star}\). It was assumed that sliding was avoided when \(H/h\) and \(B/R_{c}\) varied, and thus the failure was dominated by pure hinging. It was found that \(\lambda^{\star}\) decreases with the increase of \(H/h\), while it increases along with an increase in \(B/R_{c}\). Moreover, a small \(\mu\) can result in pure sliding or combined sliding-hinging failure, leading to a lower \(\lambda^{\star}\). If sliding occurs, \(\lambda^{\star}\) increases along with an increase in \(\mu\). When \(\mu\) reaches some threshold, pure hinging failure dominates and \(\lambda^{\star}\) becomes almost constant, indicating that sliding is unlikely to happen any longer. In reality, the load multiplier \(\lambda\) is usually no higher than one. Though only simple investigations have been performed over limited parameters, a more comprehensive parametric study and general conclusions regarding the friction, shape of the arches etc., can be expected in future work.  
In general, the FDEM has been proven as a useful and reliable tool in predicting the first-order seismic capacity of masonry arch structures subjected to both gravity and constant horizontal acceleration. Thus, this method can be applied to evaluate the mechanical behaviour of masonry arch structures subjected to more sophisticated loads [48], or with steel-ties between buttresses [49], in future.  
Conceptualisation, X.C.; methodology, X.C.; software, X.C.; validation, W.L.; formal analysis, X.C., W.L. and Y.C.; investigation, W.L. and H.W.; resources, X.C. and Y.C.; writing--original draft preparation, X.C. and W.L.; writing--review and editing, X.C., H.W. and A.H.C.C.; supervision, A.H.C.C.; project administration, X.C.; funding acquisition, X.C. and W.L. All authors have read and agreed to the published version of the manuscript.  
Figure 20: Sliding failures: (**a**) pure sliding, buttressed circular arch, \(\beta=120^{\circ}\), \(\mu=0.2\), \(\lambda=0.16\); (**b**) combined hinging and sliding, buttressed basket-handle arch, \(\beta=180^{\circ}\), \(\mu=0.6\), \(\lambda=0.16\).  
**Funding:** This research was funded by the National Natural Science Foundation of China (grant number 51808368), Qinglan Project of Jiangsu Province of China, and the Postgraduate Research & Practice Innovation Program of Jiangsu Province.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Not applicable.  
**Acknowledgments:** The authors highly appreciate the support of the FDEM program 'Y' from [PERSON].  
**Conflicts of Interest:** The authors declare no conflict of interest.