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

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The simplified approach proposed in this paper is based on a closed-form formulation for the resultant moment provided by the anchors aimed at overcoming the need for a time-consuming non-linear analysis at the early stages of a design process. For this purpose, a single parameter, i.e., the pull-out strength T * (Fig. 1b), is required to characterize the anchor behavior. Indeed, it is assumed that, for small displacements, as those characterizing the onset of the failure mechanism, the active anchors work in the elastic field, and their tensile forces are linearly distributed along the height of the macro-block (Fig. 3b). Thus, the contribution of the generic active row T j+i,r (1 ≤ i ≤ k r ) can be expressed as a function of that provided by the first active one T j+ 1, r , which is assumed to reach the pull-out strength T * ( T j+ 1, r = T *). Based on Eq. (8), the force T j+i , r provided by the generic active row j + i is: T j+i,r ( , ) = ( +1− ) ∙ T* (11) With the lever arms z j+i,r given by Eq. (8), the resultant moment provided by the k r active rows is: = ∑ [ ∙( +1− ) 2 ∙ ∗ ] =1 = ∙( +1)∙(2 +1) 6 ∙ ∗ (12) The described approaches are compared in Fig. 4a-b in terms of the load multipliers  r  related to two strengthening layouts, which differ for the number m of the rows of anchors (8 and 16 rows). The anchors have a length L a = 1.5 m and are equally spaced along the height H c,0 – d t = 4 m of a T-shaped connection of masonry walls. For the masonry walls, the specific weight of 16 kN/m 3 and the friction coefficient f = 0.6 are considered, whereas the other parameters sketched in Fig. 1a and Fig. 2b are: L f = 3 m, t s = 0.5 m, t f = 0.5 m, h = 0.1 m, tan  b = 0.5 , q s q f = 0 kN/m. Fig. 4a-b shows that the two approaches provide very similar results in terms of the load multiplier  * r for both strengthening layouts and whatever r . Furthermore, both approaches provide the horizontal capacity of the strengthened system  * (  * = min  r * ) for r = 0, i.e., when the hinge O r is at the base (see Eq. (2)), confirming that the variability of the hinge position can reasonably be neglected in the strengthened condition, as in Maione et al. (2021). (a) (b)

Fig. 4. Comparison between the NLK and the simplified approaches in evaluating  r

 , for (a) m = 8 and (b) m = 16.

Table 1 summarizes all the results related to the case r = 0; it highlights a good agreement concerning the crack inclination angle  c * , the number of active rows k r and the anchors resultant moment M kr . Although the value of M kr becomes more than double when m increases from 8 to 16, the minimum load multiplier   , which also considers the masonry contribution, increases by 20% only. The issue suggests the relevance of identifying the minimum strengthening solution to achieve a given horizontal capacity, as addressed in the following section.

Table 1. Results provided by the compared approaches for r = 0. r [-]  c* [°]  * [-]

r [-]

M kr [kNm]

k 1 1 3 3

Simplified approach NLK approach Simplified approach NLK approach

0 0 0 0

37 37 37 37

0.352 0.352 0.423 0.430

20.25 20.34 47.25 50.27

m = 8

m = 16