Issue 51

G. Cocchetti et alii, Frattura ed Integrità Strutturale, 51 (2020) 356-375; DOI: 10.3221/IGF-ESIS.51.26 361  [rad] A =  cot(  /2) Heyman CCR  2 −  2 /3  [rad]  Heyman CCR  4 /12  [rad] h Heyman CCR 1 −  2 /3 in Eqns. (5): f 2 – 2 gS + S 2 = 0. This corresponds to a stationary (maximum) point of the curves  (  ) or  ( A ) at variable arch opening [6,8]. The characteristic solution for  = A =  /2 keeps in the pre-peak branch ( A + ,  – , h + ) of solution (5). Solutions (5) a , (5) b and (5) c can be analytically plotted as a function of hinge position  , as depicted in Fig. 2, with comparison between correct CCR solution [6] and classical (say “approximated”) Heyman solution. Similar representations occur as well for Milankovitch solution accounting for the true self-weight distribution along the circular arch, though leading to the following more involved “cubic problem” : ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) 2 3 2 2 3 3 2 2 3 2 2 3 2 3 3 2 0 3 12 12 0 6 3 3 2 3 2 2 0 S g S A g f g S A f g A g S f g g S g f S h S f g S h g f f S h g f     =     − + − − + + + − − + − = − − − − + − =  − − (7) with rather similar results and minor differences, to those recovered for the above “quadratic problem” [6]. Thus, the value of Milankovitch solution is not further inspected here, since focus is now going to be made on the effect of reducing friction, on correct CCR solution, vs. “approximated” Heyman one, in the kept hypothesis of self-weight distribution along geometrical centreline of the circular arch. Moreover, CCR solutions (5) b and (5) c can be analytically plotted by parametric plots (  (  ), h (  )) and (  , h (  )), for 0 ≤  ≤  s  , showing (with important implications in the present forthcoming analysis on the role of reducing friction) non-linear dependencies of  ( h ) and  ( h ) at variable non-dimensional horizontal thrust h (Figs. 3a,b), in the critical condition of purely-rotational collapse (Fig. 1b). Figure 2 : Analytical parametric plot of solution triplet A (  ),  (  ), h (  ) as a function of inner hinge angular position  for CCR and Heyman solutions, with common trends for small  ( A =  cot(  /2),  : half-angle of embrace;  : thickness to radius ratio; h : non- dimensional horizontal thrust; refer to Fig. 1).