X

Figure 5. Reduction of resistance vs. slendernesses Xc and X in EC6'03 (a) and NTC'07 (b).

order to reduce the vulnerability of the walls to the out-of-plane collapses.

In this paragraph, the out-of-plane collapse mechanisms of a wall with different restraint conditions at the edges are initially examined and the collapse multipliers are evaluated; then, the ancient rules of the art and the European (EC8'03) and Italian (NTC'07) seismic provisions for the unreinforced masonry buildings, which mainly consist of geometrical limitations, are analyzed; finally the resistance of masonry walls to horizontal loads, designed according to the above geometrical limitations (ancient rules of the art and seismic codes) is evaluated.

3.1 Collapse multipliers of horizontal loads

In the case of uniform distribution of horizontal loads, the application of the principle of virtual works to the wall 1L of Figure 2a, allows to explicit the relationship between the geometrical h/s ratio and the out-of-plane collapse multiplier a1 (defined as maximum horizontal load to self weight load ratio):

Figure 6. Comparison among the multipliers a of the walls 1L, 2L, 3L and 4L.

When the wall is sufficiently retrained at the top and bottom by the floor diaphragms (wall 2L) the multiplier a becomes (Hendry et al. 1997):

Instead, if the wall is stiffened on one or on two vertical edges (wall 3L and 4L, respectively), in the hypothesis that the yield lines are slanted at 45°, the collapse multipliers a are, respectively:

The Eqs. (12) and (13) show that the multipliers a3 and a4 depend on the geometrical slenderness h/s and on the aspect ratio h/b of the wall.

In the case of h/s = 10, the curves (1)^(4) associated to Eqs. (10)^(13) are plotted in Figure 6. It can be observed the influence of both edge restraints conditions and aspect ratio h/b on the out-of-plane resistance of the wall. In fact, the diagram of Figure 6 shows that: (i) the curves (1) and (2) (corresponding to wall 1L and 2L, respectively) do not depend on the aspect ratio h/b; (ii) in presence of floor diaphragms at the top of the wall (wall 2L), the multiplier a increases of 8 times with respect to the case of wall 1L; (iii) in presence of wall 3L and 4L with 1 or 2 stiffened vertical edges, the collapse multiplier a shows a further increase when the h/b ratio increases. For example, in Figure 6 a comparison among the multipliers a1, a2, a3 and a4 is provided when h/b = 1.

3.2 Ancient rules of art

Regarding the stability of unreinforced masonry walls under horizontal loads, J. B. Rondelet (1802), in his

Figure 8. Collapse multipliers for the rules of Rondelet.

historical treatise asserts that a wall will have strong stability if h/s=8, medium stability if h/s = 10, low stability if h/s = 12.

From the well-known study on the collapse mechanisms for unreinforced masonry panels characterized by different restraint conditions, Rondelet realizes that, for the same degree of stability, the thickness s can be reduced when the distance b between the transversal wall decreases. Particularly, in the case of a wall of one floor and unique room building, when the beams of the roof are well fixed to the wall and there are no roof thrust, the minimum value of the thickness s which guarantees an adequate degree of stability for the wall, can be calculated using the following empirical rule by Rondelet:

In term of h/s ratio, the Eq. (14) can be written as follow:

The geometrical Rondelet's rule (15) is plotted in Figure 7 (curve (d)) together with the limits that Rondelet gives for wall 1L: h/s = 8, strong stability (curve

(a)); h/s = 10, medium stability (curve (b)); h/s = 12, low stability (curve (c)).

Particularly, the curve (d) shows that the required h/s ratio to guarantee the stability of the wall increases with the aspect ratio h/b of the masonry panel; for example, if h = 3.5 m, for h/b = 1 it is necessary to have a minimum value of h/s = 17, i.e. a wall thickness s = 21 cm; instead, for h/b = 1.5 it is necessary to have a minimum value of h/s = 21, i.e. a wall thickness s = 17 cm.

The above Rondelet's rules, in terms of collapse multiplier a (Eqs. (10) and (13)), lead to the curves (a)^(d) reported in Figure 8. The comparison among these curves shows the great increase of resistance to out-of-plane loads of wall 4L (curve (d)) with respect to wall 1L (curves (a) ^ (c)). Further, the trend of curve (d) shows that the multiplier a4 increases with h/b ratio; indeed, the curve (d) is asymptotic to the vertical line h/b = 1.5. In such meaning, Rondelet writes that the collapse of wall is impossible when the distance b between the transversal stiffening walls is low.

3.3 Seismic codes for unreinforced masonry buildings

In order to reduce the attitude to out-of-plane collapse of unreinforced masonry panels, also the modern European seismic codes (EC8'03 and NTC'07 (seismic part)) limit the h/s ratio through the conventional slenderness Xc (Eq. (2)). Particularly, in presence of masonry wall with natural stone units, the Eurocode 8 prescribes the following limit:

while, the new Italian code NTC'07 recommends the following limits:

seismic /one I (^O.JSg) & 2 <a,«0.25g): X,( = —- <10 (17) seismic zone 3 lSg) & 4 (at-0.0Jg>: = —— < ! 2 (18)

The above limitations on conventional slenderness Xc, and consequently on h/s ratio, are more restrictive with respect to the case of buildings in non-seismic zone. In Figure 9 the comparison among the analyzed seismic limitations are provided in term of h/s ratio. In the cases of wall 1L and 4L, it can be observed a substantial coincidence between the limitations of EC8'03 and NTC'07, especially when h/b < 1 (forwall 4L); instead, for h/b > 1 the limitations prescribed by EC8'03 are more conservative than the Italian provisions ones. Finally, also for this diagram it is possible to note the effect of edge restraints on the h/s limits.

Vii.—^-^(1): EC8'03 (pa=0.75); -NTC (seismic zones 3 and 4) =$^-«-(2): EC8'03 (p2=1)

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