while ten-node, three-side isoparametric solid pyramid elements have been adopted for the more complicated ones. Interfaces between blocks have been modelled with triangular twelve-node quadratic interface elements.

The resulting model comprises 5081 elements and 9188 nodes. Figure 12 shows the mesh of the pillar, and the details of the masonry inserts acquired by the NDT analysis. The displacements at the base of the pillar were fixed, while the top of the pillar was elastically constraint, in order to account for the interaction with the surrounding structure of the Cathedral. Two main load were considered: the dead load (of the pillar and of the surrounding structure), and an horizontal seismic load provided as an horizontal ground acceleration.

Figure 13a shows a detail of the interface elements between each block of the pillar, while Figure 13b shows the deformed mesh under the effect of the dead load. The elastic modulus of the stone block, as deduced from the experimental test, was 2000 MPa, while the Poisson ratio was equal to 0.12. The mean compressive strength was equal to 9 MPa. For the present sandstone, an average tensile strength of 3 MPa, and fracture energy of 50 Nm were adopted.

Figure 12. Finite element mesh of the pillar (a). Details of the masonry inserts (b).

It is worth noting that, under the effect of the sole gravity load, the maximum compressive stress do not exceed the value of 2 MPa, which is low compared with the strength of the sandstone. On the other hand, due to the heterogeneity of the pillar, tensile stresses may arise that approach the tensile strength.

Figure 14. Principal compressive stress (a). Deformed mesh under the seismic acceleration (b).

Figure 12. Finite element mesh of the pillar (a). Details of the masonry inserts (b).

Figure 14. Principal compressive stress (a). Deformed mesh under the seismic acceleration (b).

A seismic load has also been considered. If a return period of 140 years takes place, an horizontal acceleration of 1.288 m/s2 is applied (30% probability occurrence in a period of 50 years). In this case, the compressive and tensile stresses increases, as shown in Figure 14a. The pillar deforms mainly according to his first modal shape, as shown in Figure 14b.

Due to the horizontal acceleration, cracking can take place in the pillar. A detail of crack nucleation is shown in Figure 15. The crack occurrence provided by the analysis agrees quite well with the crack localization provided by the AE recording. Cracking correspond both to diffuse cracking in the continuum elements in the sandstone blocks and opening or sliding of the discrete interface between blocks.

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