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This thesis presents analysis methods designed to rapidly estimate the load-carrying capacity of backfilled masonry arches. To help engineers to analyse backfilled masonry arches, various methods exist. However, some of these are over-simplified; others are in contrast very complicated and time consuming for engineers to master. In industry simple, accurate, and rapid methods are potentially very useful. Thus the development of such methods has been the main focus of this thesis. One available numerical strategy is computational limit analysis. Limit analysis is a method that can obtain the failure load without the need to carry out an iterative elastic-plastic analysis. Limit analysis theorems are employed to obtain upper- and lower-bounds on the collapse load. Discontinuity layout optimization (DLO) is a limit analysis procedure that can be used to determine the critical layout of discontinuities and associated upper-bound limit load for plane plasticity problems. However, the computational cost of DLO can be quite high. In order to address this, a fast running DLO procedure is proposed. For lower-bound analysis, a soil-fill stress field model is proposed. In this model the Mohr-Coulomb criterion is strictly obeyed in the stress field in the backfilled masonry arches. Both literature benchmark problems and backfilled masonry arch bridge problems are solved to show the efficacy of the upper- and lower-bound methods. Another strategy is limit equilibrium. The limit equilibrium method has been used to obtain approximate solutions for stability problems. The method is usually considered as an approximate means of constructing a slip-line field. Differential planar soil elements are proposed here to simulate concave or convex arching effects in the retained soil backfill around arch barrels. An anisotropic stress distribution is applied to model the backfill material that in reality is often heterogeneous in nature. In this method, equilibrium considerations can be used to solve the masonry arch bridge problems by simple static methods. A number of examples from the literature and industry are analysed to demonstrate the efficacy of the limit equilibrium method. The choice of method is dependent on the principal goal of the assessment. To balance degree of accuracy and computational cost, the proposed rapid analysis strategies have been designed to provide a simple and fast means of assessing bridge load-carrying capacity. The upper- and lower-bound strategies provide rigorous solutions, but sometimes underestimate the bridge load-carrying capacity especially of the weak backfill strength. The limit equilibrium strategy provides approximate solutions which could address this issue.