Issue 46

L.U. Argiento et alii, Frattura ed Integrità Strutturale, 46 (2018) 226-239; DOI: 10.3221/IGF-ESIS.46.21 233 External actions Horizontal lever arm ( x j ) Vertical lever arm ( y j ) Ai i i W n b hv   2 v 1 2 N i j j i n n h             ( ) ( ) Bi i ci i W n n L v hb      2 L v  1 2 N i ci j j i n n n h              * 2 1 tan N Ci c ci i j j i W n h b n        * 1 tan 2 N c j j i h n v              1 2 N ci j j i n n h               2 * tan 2 c ci Di i n h W b     * * 1 tan tan 3 N c ci c j j i n h h n v        1 2 3 N ci j j i n n h             c pi i i Q Lq     tan N c pi i j c i j i Q n h q        2 c pi L    tan 2 N j c c pi j i n h       N j j i n h   Internal actions Horizontal lever arm ( x j ) Vertical lever arm ( y j )   1 2 ci ci gi i n n F vhb f    - 1 3 N ci j j i n n h             1 1 1 ( ) i i qi i ci i j j j ci j j F n n b h q n b h n vf                    - 1 2 N ci j j i n n h             Table 2 : Analytical expressions of the external and internal actions together with their lever arms. . Moreover, from the condition of translational equilibrium it is possible to define the load factor  s related to the mechanism of pure sliding; it is: 1 1 1 1 1 1 1 N N gi qi i i s N N N N N Ai Bi Ci Di i i i i i i F F W W W W Q                      (12) As already discussed in [22] it is always  s ≥ f , resulting in particular  s = f when tan  c = tan  b . This implies that the pure sliding failure rarely occurs, although the following parameters can increase the vulnerability to this kind of mechanism: the slenderness of the unit blocks together with the absence of overloading, a low number of rows or a low value of the friction coefficient [19, 28]. In these cases the entire wall is involved in the mechanism and the load factor tends to its maximum value coincident with the friction coefficient. This means that the friction coefficient represents an upper bound of the load factor and the condition  s = f marks the transition from the mixed rocking-sliding to pure sliding mechanism. Lastly, Tab. 3 reports the expressions of the external actions for a three-storey masonry wall, when α p 2 <  c <  p3 . It can be derived from Fig. 4 that the forces W C1 and W D1 at storey 1 and W B 3 and W C 3 at storey 3 are null. Finally, being  c greater both than  p 1 and  p 2 , Q 1 and Q 2 assume the maximum value depending on the length L of the wall; Q 3 , instead, depends on the inclination of the crack line  c , being  c ≤  p 3 .