PSI - Issue 44

Alessandra Maione et al. / Procedia Structural Integrity 44 (2023) 1372–1379 Alessandra Maione et al. / Structural Integrity Procedia 00 (2022) 000 – 000

1375

4

does not contribute to strength as well as the rows above it; these are represented as continuous yellow lines in Fig. 1a. In the XZ reference system with origin O r , the coordinates of points A j and the related crack inclination angles are: (a) (b)

Fig. 2. (a) Forces involved in the limit equilibrium of the macro-block related to a crack inclination angle  c j , r ; (b) dimensions of masonry units.

A j − ) z A j ,r ⁄ ]

z A j,r = ( k r + 1) ;

–1 [( x

 c j ,r = tan

(3)

x A j =L a – L 0d ;

being t f the thickness of the OOP-loaded wall. Hence, for each  c j,r ; being r fixed, the equality condition between the overturning and stabilizing moments, due to all the forces acting on the macro-block, provides the load multipliers  j,r as follows: , ( , ) = , ∙ , + , ∙ , + , ∙ , + , ∙ , + ∙ , ∙ , + ∙ , ∙ , + = − , ∙ , + , ∙ , + , ∙ , + , ∙ , (4) where M kr is the resultant moment provided by the k r active anchors, as better addressed in the next section. The forces acting on the macro-block and their lever arms are sketched in Fig. 2a for a generic  c j,r . In Eq. (4), the maximum frictional forces due to self-weight and overload F g and F q , respectively, are reduced by the coefficient  that accounts for the reduced contact surfaces in a mixed rocking-sliding motion, according to Casapulla et al. (2018). It depends on the inclination angle  b = tan – 1 v / h of a stair-stepped crack along the overlapped joints of the masonry units (Fig. 2b). Finally, adopting the kinematic method of limit analysis, the horizontal capacity of a strengthened masonry system coincides with the minimum value  * among all the load multipliers depending on both parameters  c j,r and r . Actually, it can be calculated in two steps as follows: ∗ = , ( , ) ( = 1, … , ), ∀ = 0, … , − 1 (5a) ∗ = ∗ ( = 0, … , − 1) (5b) Among the angles  c j,r , the angle  c * providing  * defines the crack inclination of the expected failure mechanism, whereas the related r gives the vertical position of the hinge in the strengthened condition. 2.2. Assessment of the maximum resultant moment provided by the anchors Herein, two approaches to assess the maximum resultant moment provided by the active grouted anchors are