Issue 51

E. Mousavian et alii, Frattura ed Integrità Strutturale, 51 (2020) 336-355; DOI: 10.3221/IGF-ESIS.51.25 339 superscript is used for them. It is worth noting that, in the following, indices i and j are respectively used to enumerate blocks and interfaces between the blocks from the apex (point A) to the support (point B), i.e. interface j is the lower interface of block i . (a) (b) (c) Figure 2 . Thrust-lines for the a) half dome, b) lune with  <  , and c) base thrust-line. (a) (b) Figure 3 : Coordinates of the centroids of blocks on the a) central-arc C and of the control points on the b) base thrust-line, with the enumeration of blocks and their interfaces. The z coordinates of the control points constructing the base thrust-line are the variables of the optimization objective function f ( z i ). For a set of control points with coordinates x i b and z i modelling a base thrust-line, half of the thickness for the dome is obtained as max     2 2 b i i x z R         . As a result, the objective function of the optimization is: f ( z i ): t /2 = Min max     2 2 b i i x z R         (1) where t is the minimum thickness of the masonry dome.