Issue 51

G. Cocchetti et alii, Frattura ed Integrità Strutturale, 51 (2020) 356-375; DOI: 10.3221/IGF-ESIS.51.26 369 Figure 5 : Numerical optimisation results for the trends of shear T (  ) and moment M (  ) at friction coefficient  = 0.3959. Figure 6 : Numerical optimisation results for the trends of shear T (  ) and moment M (  ) at friction coefficient  = 0.3093. Moreover, Fig. 8 accordingly as well shows the non-linear trend of inner angular position  variation at reducing friction  in the mixed collapse mode and the associated (linear) decreasing trend of non-dimensional horizontal thrust h , as limited by reducing friction  at the sliding joint forming at the springing of the arch at  =  =  /2. This constitutes a key feature for the appearance and interpretation of the arising mixed mode, as revealed by the present analytical and numerical investigation on the role of reducing friction. On the hierarchy of the collapse modes at variable (decreasing) friction coefficient  , it may be resumed that for  >  rm the collapse mode in the least-thickness condition is purely-rotational (Fig. 1b). Collapse characteristics  ,  , h remain unvaried at changing  : since purely-rotational collapse is uniquely determined by arch geometry, they are not dependent on the values of friction coefficient if greater than  rm . At transition  =  rm = 0.395832 (  rm = 21.5952°), a first sliding joint appears at the shoulder of the arch (  =  =  /2), for a non-dimensional horizontal thrust h = h r matching h = h  (  ) =  /2  . This marks the transition from purely-rotational to mixed sliding-rotational collapse modes. The simultaneous presence of a hinge and a sliding joint at the shoulders is detected in both analytical and numerical analyses. The 2-dof collapse mode in such a transition state could be represented by any linear combination of 1-dof modes in Figs. 1b and 1c. When friction coefficient  is then further decreased from  rm , non-dimensional horizontal thrust h keeps fixed by friction as h = h  (  ). The least thickness required for equilibrium is forced to increase. Indeed, since a lower friction coefficient is related to a lower resistance to sliding, a larger section (thickness) is needed to prevent collapse. Also, the purely-rotational collapse mode cannot be further triggered, since a thickness larger than value  r required by purely-rotational collapse (independent on  ) is needed to avoid sliding. Hence, the collapse mode that appears first, when thickness is decreased from a super-critical value to the critical one, is the mixed sliding-rotational mode in Fig. 1c. At the same time, the inner