Abstract: This paper proposes a novel procedure for the limit analysis of 2D masonry arch bridges under arbitrary loading conditions, based on the framework of a free discontinuity method. The method characterizes failure mechanisms through free crack-induced discontinuities, utilizing grid node discretization to model the masonry structure and constructing discontinuities via internode connections. Its core mathematical representation lies in a system of constraint equations derived from the compatibility conditions of discrete points. The ultimate bearing capacity and associated failure mechanisms are obtained by solving a mathematical programming problem, which minimizes a kinematic functional with the collapse crack pattern as a variable. To validate the accuracy and effectiveness of the method, the comparisons with benchmark cases from literatures are conducted. Furthermore, for both bare arches (without backfill) and backfilled arch bridges, the calculations of ultimate bearing capacity and sensitivity analyses of key parameters are performed. These results are systematically compared with those from the rigid block discrete element method. The findings demonstrate that the free discontinuity failure analysis method proposed accurately predicts the actual failure modes of masonry structures under loadings, providing a robust theoretical framework for assessing the limit states of masonry arch bridges.