Issue 51

G. Cocchetti et alii, Frattura ed Integrità Strutturale, 51 (2020) 356-375; DOI: 10.3221/IGF-ESIS.51.26 365 Notice that, once friction mark  ms is known, thus h ms =  /2  ms , stationary condition (13) a , i.e.  2 = h (1– h ), gives the following explicit expression of the angular position of the inner sliding joint for the purely-sliding collapse mode: (1 ) 1 2 2 s ms ms ms ms h h        = − = −    (18) Since h ms is near 0.5,  s is also near 0.5 (see circle insert in Fig. 4a). Mixed sliding-rotational mode At this stage, the friction boundaries for the appearance of the mixed sliding-rotational collapse mode have been located. Notice that: • at  =  rm , any 2-dof linear combination of 1-dof modes in Figs. 1b and 1c is possible; • at  =  ms , any 2-dof linear combination of 1-dof modes in Figs. 1c and 1d is feasible; • in range  ms <  <  rm , the mixed-mode mode in Fig. 1c is found, with variable position  m (  ) of inner hinge B, thickness to radius ratio  m (  ) and horizontal thrust h m (  ), at variable friction coefficient  . This last occurrence is ruled by a new system of governing equations, in place of those in system (1), in which second equilibrium Eqn. (1) b is replaced by sliding equation h = h  =  /2  , namely: 1 ( , ) ( , ) ( , ) e h h h h h h         =  =  = (19) The solution of this system actually brings back to the previous analysis for the purely-rotational mode. Indeed, equations h = h 1 and h = h e are still the same, with same solutions (5) b and (5) c for  (  ) and h (  ), as previously explained. By setting h = h  =  /2  in the expression of h (  ) in Eqn. (5) c and solving for  (  ) = h (  ) 2/  , or by eliminating h =  /2  in two Eqns. (19) a and (19) c , this leads to trends  m (  ) and  m (  ). These can be analytically plotted, by parametric plots at variable  rm ≤  ≤  ms (Figs. 7-8, Section 4), where  ms can be found from  (  ), Eqn. (5) b , at  =  ms . Similarly,  ms is found as  (  ms ), at h ms =  /2  ms , so that: 1.05616 60.5134 , 0.200637, 0.485714 ms ms ms rad h   = =  = = (20) Since this leads to an increase of  (  ) at decreasing  , constant trace  =  r of the purely-rotational mode is abandoned, since  (  ) is higher and thus provides a new least thickness condition (Fig. 7). Accordingly, the hinge at the shoulder, co-present with the sliding joint there at  =  rm , closes down. The inner haunch hinge B keeps instead on, and moves further down at decreasing  . Basically, the trends of  (  ) and  (  ) are read in Figs. 7 and 8a, as they were in Figs. 3a and 3b at decreasing h . Indeed, h is limited by friction to h  =  /2  , with the linear decreasing trend at lowering  represented in Fig. 8b. Such a trend is linear for the exposed case of  =  /2. Tab. 1 reports analytically-evaluated (“exact”) mixed-mode collapse characteristics  ,  , h at variable friction coefficient  . At new transition  =  ms , trends  m (  ),  m (  ), h m (  ) stop. Limit equilibrium states associated to modes in Fig. 1d do not depend on thickness parameter  , thus they would require any value of  >  ms in the least thickness condition. Thus, the inner hinge at the haunch also closes down and from the two modes in Figs. 1c and 1d, co-present at  =  ms , only the purely-sliding mode in Fig. 1d survives for  >  ms . However, all these states at  =  ms are right-away limit equilibrium states, thus equilibrium is no-longer possible in practice, at any value  >  ms . Notice also that, at  =  ms , inner hinge and sliding joints are differently located, respectively at  =  ms = 1.05616 rad = 60.5134° (hinge joint) and  =  s = 0.499796 rad = 28.6362° (sliding joint), thus interestingly at nearly 60° and 30°. The present analytical outcomes are going to be further commented in Section 4, with comparison as well to independent, matching, numerical results by a self-made spreadsheet implementation, as derived in the next section.