Issue 51

G. Cocchetti et alii, Frattura ed Integrità Strutturale, 51 (2020) 356-375; DOI: 10.3221/IGF-ESIS.51.26 372 thrust h decreases with it, as set by friction, linearly for 2  =  , by ranging from h r to about 0.78 h r . At the same time,  non-linearly increases from  r to about 1.87  r (then, critical thickness almost doubles) and  also increases from near  r ≃ 54.5° to about 1.11  r ≃ 60.5°. An inner sliding joint finally appears at  =  ms at a different  s location near 30° from the crown. Table 3 : Summary of analytical collapse characteristics at variable friction coefficient  for the complete semi-circular arch with 2  =  (refer also to Fig. 1). This analytical and numerical investigation has enquired the role of friction in the framework of classical Heyman masonry arch analysis. Focus has been explicitly made on the case of a complete semi-circular arch (2  =  ), to report detailed results and understand crucial features, specifically for the present mixed sliding-rotational mode. The analysis could be further generalised to cases of general half-opening angles  , as implicitly analytically here defined and independently numerically outlined in [9,10] by an innovative non-linear mathematical programming procedure, accounting for both static and kinematic admissibilities, all together. This considers a general formulation apt to address possible issues of non-normality in the prediction of the Limit Analysis formulation and potential instances of non-uniqueness in the determination of the limit thickness condition. Indeed, these aspects have been previously discussed in the literature [36-39], particularly in the context of accounting for friction effects in masonry constructions [40-45]. Specifically, Gilbert et al. [25] have mentioned that "Casapulla and Lauro (2000) have identified a special class of non-associative friction problems for which provably unique solutions exist. The class comprises arches with symmetrical loading and geometry." , as handled in the present case. These outcomes have also been confirmed by the recent analysis by Aita et al. [23]. Despite, further, general analytical and numerical formulations shall investigate the subject, in inspecting if friction reduction effect may induce a resulting non-uniqueness in the prediction of the least-thickness condition, as instead still recorded in the present setting devoted to the analysis of symmetric masonry arches under self-weight, for more unspecific configurations and loading conditions. Collapse mode Friction coefficient  Collapse characteristics  ,  , h (  =  / 2) Purely- rotational ( r )  >  rm 1 2 0, / 2 90 ( , ) 0.951141 54.4963 ( , ( )) ( , ) 0.107426 0.621772 R R r r R r e r r rad h h rad h h A h h h             = = =   = = =   + =    = =    =  Rotational/ mixed shift ( rm )  =  rm = 0.395832 (  rm = 21.5952°) 0 0.951141 54.4963 , / 2 90 R rm R rm R S rm rm rad rad      = = =  = = =  0.107426, rm  = 0.621772 rm h = Mixed sliding- rotational ( m )  ms <  <  rm 1 0, / 2 90 ( , ) ( ) ( , ) ( , ) ( ) ( ) / 2 R S m m R m e m m rad h h h h h h h h                     = = =   = =  + =    = =    = =  Mixed/ sliding shift ( ms )  =  ms = 0.309215 (  ms = 17.1824°) 0, / 2 90 1.05616 60.5134 , 0.499796 28.6362 R S ms ms R ms S ms rad rad rad      = = =  = =  = =  0.200637, ms  = 0.485714 ms h = Purely- sliding ( s )  =  ms 0.499796 28.6362 / 2 90 S s S s rad rad    = =  = =  , 0.200637, s ms    = 0.485714 s ms h h = = –  <  ms No equilibrium solution