Issue 51

G. Cocchetti et alii, Frattura ed Integrità Strutturale, 51 (2020) 356-375; DOI: 10.3221/IGF-ESIS.51.26 358 friction, as in the present setting. Moreover, the critical value of friction coefficient up to induce possible sliding may increase with the opening angle of the arch (specifically for over-complete horseshoe arches, see later discussion in Section 2.3), thus possibly approaching the ranges of friction coefficients that may be encountered in practice and maybe leading to the potential appearance of sliding within the failure mode. Figure 1 : Sketches of a symmetric semi-circular masonry arch under self-weight: (a) characteristic parameters, primarily: half-angle of embrace  , inner rupture joint angular position  , thickness to radius ratio  , non-dimensional horizontal thrust h ; (b) purely- r otational collapse mode; (c) m ixed sliding-rotational collapse mode; (d) purely- s liding collapse mode. The present analytical and numerical results appear to be consistent to ones numerically developed in the existing literature, specifically concerning the effects of friction in masonry arches. For instance, Gilbert et al. [25] have numerically investigated the role of friction, by estimating for a semi-circular arch a minimum thickness to radius ratio  = 0.1068, in the presence of a purely-rotational collapse mode, when  is greater than 0.396 (friction angle larger than  = 21.60°). This marks, at decreasing friction, the transition from the purely-rotational mode to the mixed sliding- rotational mode. Furthermore, a value of  = 0.31 (  = 17.22°) was found, which then located the shift from the mixed sliding-rotational mode to the purely-sliding mode. In Gilbert et al. [25] a characteristic, sort of L-shaped diagram depicts the critical value of thickness to radius ratio  as a function of friction coefficient  , with a double kink at these two critical values of  , a constant value of  for  > 0.396 and rapidly-growing values of  right on  = 0.31. These outcomes appear in good agreement with numerically developed earlier results by Sinopoli et al. [26-28], which individuated the above-mentioned kinks respectively at  = 0.395 (  = 21.55°) and  = 0.309 (  = 17.17°). Here, the following “exact” analytical results for the complete semi-circular arch are going to be consistently derived:  rm = 0.395832 (  rm = 21.5952°), for the transition from purely-rotational to mixed sliding-rotational modes;  ms = 0.309215 (  ms = 17.1824°), for the transition between mixed and purely-sliding modes. ( a ) ( b ) ( c ) ( d )