Issue 51

G. Cocchetti et alii, Frattura ed Integrità Strutturale, 51 (2020) 356-375; DOI: 10.3221/IGF-ESIS.51.26 368 Within the optimisation process, toward the determination of least-thickness condition t = t min still making the whole self-standing equilibrium possible, cell constraints are then represented by resistance conditions (23) and by the enforcement that, at crown section A, Y = H > 0 and | X | = | M A | ≤ Y t /2. Equilibrium values of N (  ), T (  ), M (  ) from Eqns. (8) and (22) are then determined at discretised angular positions, with angle-step  = 0.001 rad . The optimum search of t min is iteratively performed, by varying the initial values of X = M A , Y = H and t , in order to satisfy all given constraints (with tolerances in the order of 10 -6 ). Rotational and sliding joints are respectively detected, and their angular position  recorded, when limit conditions (23) a and (23) b , with numerical equalities, are reached (Fig. 5). A high value of friction coefficient is first set, namely  = 0.7 (   35°), which should allow for predicting a purely-rotational collapse mode, due to Heyman hypothesis 3 (see analytical results in Tab. 1). The analysis is then repeated at lowering values of friction coefficient  , which has been reduced by steps up to  = 0.0001, until the numerical algorithm becomes no-longer able to find equilibrium solutions. Transition conditions at  =  rm and  =  ms have been numerically found, by looking at activated hinge and/or sliding joints (Figs. 5 and 6). Results have been recorded in terms of characteristic arch parameters  ,  , h at variable  . Salient numerical outcomes are reported in Tab. 2 below (to be consistently compared with analytical results in earlier Tab. 1). It may be noted from Tab. 2 that for  = 0.3959 (  = 21.5986°) the algorithm numerically detects the simultaneous presence of a rotational and a sliding joint at  =  = 90°. This provides a consistent numerical estimate of “exact” friction coefficient  rm = 0.395832 (  rm = 21.5952°), as earlier analytically derived. Further, the lack of equilibrium solutions for  < 0.3093 (  < 17.1868°) provides a consistent numerical approximation of “exact” bound  ms = 0.309215 (  ms = 17.1824°). These numerical results are also consistent with earlier independent numerical outcomes in [25-28], as mentioned in the Introduction. Also, the approximate numerical values of characteristic parameters  ,  , h fit quite well with the “exact” predictions from the analytical approach (Tab. 1), as analysed, represented and resumed next.   h Hinge joints  [deg] Sliding joints  [deg]   h Hinge joints  [deg] Sliding joints  [deg] 0.7 0.107426 0.621772 0-54.4883-90 - 0.35 0.152920 0.549779 0-58.0120 90 0.396 0.107426 0.621772 0-54.4883-90 - 0.34 0.163977 0.534071 0-58.5563 90 0.3959 0.107663 0.621878 0-54.4883-90 90 0.33 0.175448 0.518363 0-59.3011 90 0.3958 0.107456 0.621721 0-54.4883 90 0.32 0.187338 0.502655 0-59.9027 90 0.39 0.112750 0.612611 0-54.9753 90 0.31 0.199653 0.486947 0-60.5043 90 0.38 0.122191 0.596903 0-55.7774 90 0.3094 0.200406 0.486004 0-60.5043 90 0.37 0.132031 0.581195 0-56.6082 90 0.3093 0.200531 0.485847 0-60.5043 28.6479-90 0.36 0.142273 0.565487 0-57.2958 90 0.3092 No equilibrium solution Table 2 : Approximate critical solution values of triplet  , h ,  obtained for the complete semi-circular arch by the numerical analysis at variable friction coefficient  . S UMMARY OF ANALYTICAL AND NUMERICAL OUTCOMES igs. 7-8 below all together resume the present analytical and numerical results, by showing classical collapse characteristics of the masonry arch  ,  , h at variable (reducing) friction coefficient  . Numerical data out of the analysis in Section 3 are over-scored with cross markings on continuous analytical trends (parametric plots) out of the “exact” analysis in Section 2, with very good matching among them. Specifically, Fig. 7 first reports a main outcome of the derivation, as the limit curve in the (  ,  ) plane, in the sort of typical representation pointed out by Gilbert et al. [25] and Sinopoli et al. [26-28], and recently by Aita et al. [23], there generalised to masonry arches of various typologies and shapes. F