Yes there are many ways to assess arches, of varying theoretical rigour.
In the UK Highways situation, you could use the methods set out in the highways standard (technically an advice note) BA16/97, which is freely available here.
The fastest method is the ‘modified MEXE’, which is modified from a military method intended for use in the field (i.e., to answer the question, can I move this artillery piece over that bridge now). Chapter 3 in the standard tells you how to do it. It’s not so useful except for vehicle loads, however.
Annex E discusses various assessment methods, however, and references a number of elements of software. They are all getting somewhat long-in-the-tooth now.
The software I use most often is either Archie (which is referenced in BA16/97, though it has evolved considerably) or Ring. I have a natural affinity for Archie (having been using it intermittently since the early 90s), but in some respects (notably treatment of barrel separation) Ring is superior.
The analyses are limit analyses, and not about assessing moments, shears etc. but rather solve for a collapse mechanism recognising the particulate nature of masonry. In the case of Archie (which is the most clearly expressed method), it solves for a zone of thrust solution in which there is a discrete zone of thrust within the thickness of the barrel that satisfies all the necessary constraints (geometrical, masonry strength etc.). In the screenshot on the Obvis front page you can see this – the orange lines are a zone of thrust that satisfies the constraints and are in equilibrium with factored loading.
Likewise, Ring analyses the arch as a system of interactions between rigid blocks meeting constraints and determines that an equilibrium solution exists, determining the additional factor that could be applied to all loads and still determine an equilibrium solution for every position of load.
It would theoretically be possible to analysis arches as a non-linear (compression-only) material within an FEA code. Indeed, CIRIA C656 (‘Masonry arch bridges: condition appraisal and remedial treatment’, a publication that is unfortunately not freely available) discusses this as the third class of analysis (after empirical, e.g. MEXE, and collapse mechanisms e.g. Archie and Ring). I’ve never done it in practice – if I need an analysis of an arch I’d use one of the two specialist analysis programs noted.