Issue 51

G. Cocchetti et alii, Frattura ed Integrità Strutturale, 51 (2020) 356-375; DOI: 10.3221/IGF-ESIS.51.26 359 In the present paper, two sorts of analyses on the role of reducing friction in circular masonry arches are carried-out, as respectively presented in Sections 2 and 3, with mutually consistent results, as eventually outlined in resuming Section 4: • First, a complete analytical approach in the wake of previous original analytical solutions for purely-rotational collapse [6,8] is developed, towards the characterisation of the mixed sliding-rotational collapse mode and of the relevant friction bounds. This analytical analysis starts from the presumed purely-rotational collapse mode due to Heyman [1-4] and, by decreasing friction, derives, through a static approach, the modes that subsequently arise, with explicit “exact” solutions. • Second, a comprehensive, original self-implemented numerical approach is developed, which sets an optimisation analysis in the sense of the lower-bound (static) theorem of Limit Analysis , within a classical commercial spreadsheet wherein an optimisation function is made available. There, the constitutive behaviour of the circular masonry arch is stated and equilibrium conditions are varied towards the determination of the critical least-thickness condition and of the attached collapse mode. Thus, the collapse mode, whether purely-rotational, mixed sliding-rotational or purely-sliding, is not there a priori hypothesised, while correctly numerically recovered, through an optimisation process, with outcomes that turn out fully coherent with those “exact” ones from the previous analytical analysis, and also to the above-quoted findings from the preceding literature [25-28]. A NALYTICAL APPROACH tarting from the classical analysis of purely-rotational collapse of circular masonry arches in the so-called Couplet-Heyman problem [1-4], the determination of collapse characteristics  ,  , h for a symmetric circular arch of general half-angle of embrace  (Figs. 1a,b) may be stated by the solution of the following system of three characteristic equations, in terms of unknown non-dimensional horizontal thrust variable h [5,6,8]: 2 2 2 (2 ) sin 2(1 cos )(1 /12) 2 (2 )cos 2 (1 /12), cot 2 2 (2 ) (sin cos ) 2sin (1 /12) cot (1 / 6) (2 )sin 2 M M M e CCR M H h h h h A A h h h                                 − − − + = =  + − −   = = − + =  +   − + − +  = = = − + − −   1 2 (1) Eqns. (1) a and (1) b represent two equilibrium relations, respectively the rotational equilibrium of any upper portion AB of the half-arch (symmetry conditions apply) with respect to inner intrados hinge at haunch B, and of whole half-arch AC with respect to extrados hinge at shoulder C (Fig. 1b); the latter introduces the dependence on opening angle  , through variable A =  cot(  /2). Importantly, Eqn. (1) c truly corresponds to the correct tangency condition of the line of thrust (locus of pressure points) at haunch intrados B and may be derived from the following stationary condition: num[ ] ( ) ( ) 0 den[ ] h dh h h h d h      = =  = =  1 1 1 e 1 (2) where symbol ( )′ denotes a differentiation with respect to  [6] and advantage has been taken from the quotient rule of differentiation. Notice that the correct statement of this tangency condition actually sets a main difference to classical Heyman solution, which stated the tangency condition on the thrust force itself ( h = h H in Eqn. (1) c ), and may be considered as leading to a sort of approximation of the true solution , at least until when  keeps small. Moreover, in Eqns. (1)  CCR and  M are two control flags allowing to shift from classical Heyman solution (  CCR = 0,  M = 0), to the correct solution of Heyman problem (  CCR = 1,  M = 0), and to Milankovitch solution considering the true S