Issue 51

G. Cocchetti et alii, Frattura ed Integrità Strutturale, 51 (2020) 356-375; DOI: 10.3221/IGF-ESIS.51.26 362 This remarkably shows that  ( h ) becomes a non-linear increasing function at decreasing h and that  ( h ), in the pre-peak branch, is also a non-linear increasing function at lowering h . Thus, if h is set by external conditions (e.g., here, through friction reduction), in the least-thickness limit equilibrium state,  and  vary accordingly to the trends represented in Figs. 3a,b (non-linearly and with opposite concavities). These considerations display a crucial implication in the forthcoming investigation analysis at reducing friction. ( a ) ( b ) Figure 3 : Analytical parametric plot as a function of non-dimensional horizontal thrust h (  ): (a) least thickness to radius ratio  (  ); (b) inner hinge angular position  . Reducing friction The whole above solution holds true in Heyman sense for high values (infinite, in the limit) of friction coefficient  = tan  , apt to prevent sliding failure within the arch. By imaging now to decrease friction coefficient  (not present in system (1)) from infinity or from such high values, one seeks when, and where in the arch, a first sliding joint may arise, for a critical value of  =  rm marking the transition between purely-rotational and mixed sliding-rotational modes. This should occur when limit sliding activation condition T / N =   is reached for the first time, where T (  ) and N (  ) are the shear (clock-wise positive) and normal (compression positive) components of the internal thrust force at each theoretical (radial) joint of the continuous arch. From the translational equilibrium of any upper portion AB of the half-arch of general half-opening  , one gets, in non-dimensional terms: ( ) ( ) sin cos ( ) ( ) cos sin T t h w r N n h w r            = = −     = = +  (8) It may be noticed that non-dimensional internal force components t (  ) and n (  ) are just functions of geometrical angular position variable  , at a given value of horizontal thrust h (which is constant along the arch). Specifically, thickness to radius ratio parameter  does not intervene in Eqns. (8). At the shoulders of the arch (  =  ), internal action component ratio t / n becomes: sin cos ( ) cos sin t h n h         − = = + (9) For the complete semi-circular arch, at the half-arch extremes (namely crown A and shoulder C) one has, respectively:  = 0, t = 0, n = h , thus t / n = 0;  =  =  /2, t = h sin  –  cos  = h , n = h cos  +  sin  =  /2, thus t / n = 2 h /  .