# Seismic Assessment of Historical Masonry Buildings at Different Scale Levels: A Review
## 3 Small-Scale Approaches
### Finite Element Methods
#### 3.2.1 FEM Modelling: Strategies and Issues Applied to Cultural Heritage
Modelling of Materials  
Nonlinear constitutive laws have been more widely used for the seismic assessment of 3D structures since they are able to represent the behaviour of large and complex buildings by setting a reduced number of parameters. Thanks to the diffusion of commercial FE codes, the current scientific contributions generally use laws based on fracture mechanics (smeared crack models) [166; 167; 168], plasticity theory [169; 170; 171; 172; 173], and damage mechanics [174; 175]. Several models currently adopted for masonry structures were initially developed for reinforced concrete ones, e.g., [PERSON], [PERSON] [176], [PERSON], [PERSON] [177] and [PERSON] [178]. Despite their limitations in the multi-level anisotropy and the heterogeneity of masonry compared to concrete, their use spread thanks to the availability of commercial FE software and the relative ease of their use. For historical buildings, isotropic models have been generally preferred to the (correct) anisotropic ones since a complete characterisation of the mechanic behaviour along all the geometrical axes would require expensive experimental campaigns; besides, masonry could be highly irregular within the same building (and even component), therefore, requiring the determination of multiple constitutive laws. Some attempts to develop a proper anisotropic model for masonry have been made, e.g., [PERSON], [PERSON] [179], [PERSON], [PERSON] [180; 181], but the consistent computational effort required and the numerous parameters to be set by these approaches make them more suitable for research aims than for real-world case study applications. For a more detailed review of constitutive masonry models, the reader can refer to [PERSON] et al. [117]. Modelling of Components  
Once identified the geometry, the FEM model can be realized through different element typologies depending on the analysis goals. For vertical structures, solid elements are used when great accuracy is needed, such as when structural and architectural features are strongly related to each other (e.g., complex pillars with capitals). In this way, elements/components are simulated realistically, but a high computational effort is required, therefore limiting their applicability to buildings with reduced dimensions or easy geometry [182; 183; 184; 185; 167; 186; 187], or portions of them [174]. In Orlando, [PERSON] [185], the problem of simulating multi-layer walls with solid elements is faced with reference to the analysis of the Rocca Strozzi (Campi Bisenzio, Florence, Italy). Double-leaf walls, composed of two external layers of good quality masonry and a heterogeneous inner infill, were modelled as an equivalent solid continuum characterised by the same thickness as the existing walls and a homogenised Young's modulus. A similar procedure was also present in [PERSON], [PERSON] [161]. The use of shell elements entails a relevant simplification, reducing the numerical demand and therefore being recommendable, even for more complex structures [186; 187; 156; 168; 166; 168]. Another example of the use of bidimensional elements for masonry walls can be found in Lourenco, Trujillo [187], which also introduces the possibility of accounting for wall damage: the damage pattern surveyed in the church of St. George of the Latines Church (Famagusta, Cyprus) was included within the model by reducing the thicknesses of elements located within the damaged areas. Modelling the horizontal structures has always represented a significant issue since, unlike recent buildings, the floors of historical-masonry buildings are mainly composed of vaults and timber floors, which are hardly schematised through infinitely rigid planes. When dealing with timber frame structures, besides modelling frame elements [188], the horizontal structures may be simulated through concentrated mass [185; 172; 182; 156], therefore neglecting their stiffness contribution. This assumption implies that the floors do not guarantee any load transfer from heavily damaged walls to still-efficient structural elements.