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The present paper proposes a new methodology for the load-bearing capacity analysis of 2D masonry constructions by extending and reformulating the Continuous Airy-based for Stress Singularities (CASS) method, with a particular focus on the incorporation of volume forces, crucial to solve mechanical problems involving masonry constructions accurately. The masonry material is modelled using a Normal, Rigid, No-Tension (NRNT) model, largely adopted by the scientific community as material parameters, often unknowable, are not needed. The load-bearing capacity problem is derived from an energy-based formulation and framed as a classic limit analysis approach, allowing for the direct determination of the maximum incremental load that a masonry structure can withstand, along with the corresponding internal stress pattern. The structural domain is discretised with a simple finite element mesh, and the Airy stress potential is adopted to enforce the internal equilibrium directly. The boundary value problem is then solved as second-order cone programming (SOCP). The CASS method is shown to offer modelling and computational advantages. Indeed, it accurately describes the mechanical response, including the ability to capture singular stress fields typically exhibited by masonry structures, particularly where cracks appear. The adoption of the Airy potential allows for a reduction in the number of explicit constraints from the problem. Moreover, the formulation of the boundary value problem as a SOCP ensures the existence of a unique load multiplier while offering computationally fast solutions even for large problems. Several numerical examples are presented to demonstrate the CASS potential. Specifically, the ability to capture singular stress patterns diagonally crossing the finite elements showcases the CASS mesh independence, providing a straightforward approach to modelling complex geometries and loading conditions. • Mesh-independent formulation for 2D seismic assessment of masonry façades. • Incorporation of volume forces into the CASS method. • Validation against experimental, analytical, and numerical solutions. • Singular stresses captured as concentrated on narrow bands. • The CASS method reveals the resisting compressive structure activated at the onset of collapse. [ABSTRACT FROM AUTHOR]