Issue 51

G. Cocchetti et alii, Frattura ed Integrità Strutturale, 51 (2020) 356-375; DOI: 10.3221/IGF-ESIS.51.26 366   h  [rad] [deg]   h  [rad] [deg] 0.7 0.107426 0.621772 0.951141 54.4963 0.35 0.152920 0.549779 1.01227 57.9986 0.3959 0.107426 0.621772 0.951141 54.4963 0.34 0.163977 0.534071 1.02392 58.6661 0.395832 0.107426 0.621772 0.951141 54.4963 0.33 0.175448 0.518363 1.03499 59.3003 0.3958 0.107455 0.621721 0.951188 54.4991 0.32 0.187338 0.502655 1.04548 59.9016 0.39 0.112750 0.612611 0.959654 54.9841 0.31 0.199653 0.486947 1.05540 60.4702 0.38 0.122192 0.596903 0.973737 55.7910 0.3093 0.200531 0.485847 1.05608 60.5088 0.37 0.132031 0.581195 0.987190 56.5618 0.309215 0.200637 0.485714 1.05616 60.5134 0.36 0.142273 0.565487 1.00003 57.2974 0.3092 No equilibrium solution Table 1 : “Exact” critical solution values of triplet  , h ,  obtained for the complete semi-circular arch by the analytical analysis at variable friction coefficient  . Sliding joints appear at  =  /2 rad = 90° for 0.309215 =  ms ≤  ≤  rm = 0.395832 and at  =  s = 0.499796 rad = 28.6362° for  =  ms = 0.309215. Comment on present mixed mode at variable opening angle of the arch The previous relations, generally set for any half-angle of embrace  , and illustrated in detail for the considered reference case of the complete semi-circular arch (  =  /2), could be used to further explore the output of the discovered mixed mode, at reducing friction, for arches with variable opening angles. It can be shown that the analytical solution for the representation of the mixed mode here derived holds true for a range of opening angles up to the following limit one (0 <  ≤  lm ). Indeed, if one seeks the condition leading to the common satisfaction of systems of three Eqns. (1) and three Eqns. (13), namely those that mark the rotational mode and the sliding mode, one achieves the following numerical solution of the six equations in six unknowns  lm (limit opening angle for present mixed sliding-rotational mode),  lm (friction coefficient),  lm (thickness to radius ratio), h lm (non-dimensional horizontal thrust),  R lm , (angular position of inner rotational joint),  S lm (angular position of inner sliding joint): ( ) /2 2.48716 142.504 ( cot 0.844185), 1.41527 ( atan =0.955669 54.7558 ), 0.679605, 0.0978058, 1.03749 59.4435, 0.297052 = 17.0198° lm lm lm lm lm lm lm lm lm R S lm lm rad A rad h rad rad          = =  = = = = =  = = = = = (21) Thus, the documented solution of mixed mode is achieved until for an angle of embrace of about  lm = 142.5°, anyway requiring considerable friction (and thickness) for the arch to withstand under self-weight, in the limit condition of least thickness. At that value of  (marking a rather open, thick, horseshoe arch), the two solutions for rotational collapse and sliding collapse, together with that for mixed mode collapse, unify all together and are co-present at the same time for the values of the characteristic coefficients reported in Eqn. (21). Further numerical results and representations of the collapse characteristics of the circular masonry arch at variable angle of embrace, implicitly described by the present analytical treatment, and fully consistent with such analytical solutions, are additionally presented in [9,10], by a separate numerical treatment, based on a comprehensive non-linear Mathematical Programming formulation and implementation. A simpler, straightforward numerical strategy is instead proposed next, for independent and complete validation of the achieved analytical results.