# Seismic Assessment of Historical Masonry Buildings at Different Scale Levels: A Review
## 3 Small-Scale Approaches
### Limit Analysis Methods  
Limit analysis (LA) methods represent a powerful tool to assess the failure load associated with different collapse mechanisms of structures that can be disaggregated into blocks. The presence of joints determining natural predefined weaknesses planes makes this kind of discrete modelling and analysis particularly attractive, especially for structures such as arches, piers and abutments of bridges [122]. For similar reasons, it can be successfully used for buildings not characterised by box behaviour due to the absence of rigid floors or the presence of potential partial collapses, as frequently happens in the case of historical constructions [3; 122].  
LA generally refers to the lower-bound (static) or upper-bound (kinematic) method for the evaluation of the collapse multiplier and the assessment of the damage; it gives results only for the ultimate condition [123], neglecting intermediate steps, ultimate displacement and post-peak response. LA methods cannot simulate brittle failure modes (e.g., limited shear or compression strength).  
They are hardly able to account for all the possible mechanisms detectable by plastic hinges in a complex structural system [124]; therefore, they are usually less accurate compared to other methods.  
Thanks to the reduced computational effort, LA methods were largely used in the past, even for historical masonry constructions, since they required a limited number of input parameters concerning materials, information usually missing in such structures [125]. Limit analysis for masonry structures was introduced by [PERSON], exploiting the three well-known hypotheses of (i) masonry infinite compressive strength, (ii) no sliding between parts, and (iii) zero tensile strength [126; 127]. Over the years, several studies, based on [PERSON] formulations, addressed the analysis of arches and bridges modelled as a discrete system of rigid blocks [128; 129; 130]. [PERSON] and [PERSON] [131] approached the issue of introducing block sliding through a non-linear mathematical problem, appealing to avoid assumptions concerning the arrangements of blocks and interfaces. The problem was difficult to solve compared to that resulting from the classical theory and generally based on the associated flow rule. Minimizing the load factor to achieve the safest solution was proposed but further invalidated by [PERSON] and [PERSON] [132], [PERSON] and [PERSON] [133] as being unrealistic in the presence of Coulomb friction. [PERSON] and [PERSON] [125] and [PERSON] and [PERSON] [134] proposed further developments accounting for limited compressive stress, and [PERSON] and-Loi [135] adopted a mathematical programme with equilibrium constraints. [PERSON], [PERSON] [136] proposed an approach requiring iterative application of linear formulations, later extended to three-dimensional structures, including torsional effects, by [PERSON], [PERSON] [137].  
Full three-dimensional limit analysis of masonry buildings was proposed by [PERSON], [PERSON] [138], including mortar joint cohesion and masonry crushing issues. The upper bound limit analyses and in- and out-of-plane loaded walls were used by adopting the Mohr-Coulomb failure criterion for bricks, implementing interfaces with tension cut-off and cap in compression for the mortar joints [139]. Particularly interesting in the field of ancient buildings were the contributions by [PERSON] [140], [PERSON] and [PERSON] [141], [PERSON] [142], [PERSON] [143], which allowed identifying recurrent failure modes by observing real damages in traditional constructions. Kinematic LA was applied to rigid blocks, providing useful results even for buildings without box behaviour. In Lagomarsino [121], the procedure has been further implemented by means of a combination with the capacity spectrum method.  
Other examples concerning limit analysis methods are provided in Roca Cervera [3] and in [PERSON], [PERSON] [117].