Issue 51

G. Cocchetti et alii, Frattura ed Integrità Strutturale, 51 (2020) 356-375; DOI: 10.3221/IGF-ESIS.51.26 370 hinge at the haunches moves further down, at decreasing  , from the location at purely-rotational collapse. This is due to the non-linear increasing trends of  and  at lowering h (fixed here by friction) in the purely-rotational solution (Fig. 3). Figure 7 : Least thickness to radius ratio  at variable friction coefficient  , obtained by the analytical and numerical analyses. The mixed sliding-rotational mode range,  ms <  <  rm , is limited by  ms = 0.309215 (  ms = 17.1824°) and  rm = 0.395832 (  rm = 21.5952°). The shaded region represents the possible arch equilibrium states of couples (  ,  ). Reducing friction  sets a required increase of least thickness to radius ratio  . At  =  ms = 0.309215 (  ms = 17.1824°) an additional sliding joint opens up at  ms = 0.499796 rad = 28.6362°, when  =  ms = 0.200637, with h = h ms = 0.485714. At  =  ms , this sliding joint coexists with a hinge at the haunches at  m (  ms ) = 1.05616 rad = 60.5134°, and the corresponding 2-dof mixed collapse mode would be any linear combination of 1-dof collapse modes in Figs. 1c and 1d. Any larger value of  >  ms would instead represent limit equilibrium conditions at  =  ms for which purely-sliding collapse would develop (Fig. 1d). In practice, at  =  ms any value of  >  ms would correspond to limit states associated to purely-sliding modes in Fig. 1d and the arch would no longer be able to withstand under self-weight. Nothing should be said, by the present static approach, for values of  <  ms , since then equilibrium is no-longer possible at any value of  . Thus, it may be concluded that the shaded region in Fig. 7 represents the equilibrium states of couples (  ,  ) allowing for arch equilibrium under self-weight. The inferior boundary of this domain is set by constant line  =  r at  ≥  rm , then by curve  =  m (  ) at  ms ≤  ≤  rm , finally by vertical line  ≥  ms at  =  ms (Fig. 7). Tab. 3 below further resumes all the main solution characteristics obtained by the analytical results, for the various traced collapse modes of the complete semi-circular arch (2  =  ), at reducing friction, according to the above detailed discussion and the corresponding illustration of the collapse modes. Notice that the collapse mechanisms were already drawn, on scale, in Figs. 1b,c,d, as indeed corresponding to the main cases on the 1 st , 3 rd and 5 th rows of Tab. 3, now reported. Remark that these constitute the three basic collapse cases that are revealed by the study. The other two intermediate rows in the table represent transition cases, among couples of them, where any possible combination of the two underlying mechanisms meeting at such a transition instance becomes possible. C ONCLUSIONS he role that friction coefficient  at the theoretical (radial) joints of a continuous circular masonry arch plays in the definition of geometrical collapse of continuous arches has been fully investigated, with specific reference illustration to the classical case of the complete semi-circular arch (half-angle of embrace  =  /2 = 90°). The analytical treatment presented in [5,6,8] for purely-rotational collapse has been complemented and extended, by releasing Heyman hypothesis 3 of no sliding failure and looking at the induced changes of the collapse mode through the role of a reducing friction, leading to sliding onset. T