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geometry, all the walls have height h = 350 cm; the width b varies between 233 and 1400 cm in order to have walls with aspect ratio between 1.5 and 0.25. The thickness sis set equal to 30,35,45 cm, in order to have walls with h/s ratio equal to 12, 10 and 8, respectively.

Four-nodes shell elements (S4R5 elements) are used to model the masonry tuff walls; reduced integration is used for the shell elements; the number of integration points through the thickness of shell element is equal to five.

All the examined walls have been subjected to non linear analyses using a smeared cracking approach as implemented in the computer code Abaqus.

Figure 13. Specimens 350-350-35: (a) comparison among the push-over curves and the collapse multipliers; (b) deformed shapes with stress tensor vectors of the F.E.M. models; (c) collapse mechanisms considered for limit analyses.

In order to correctly calibrate the model parameters, reference has been made to the curve fitting procedure was made by Giordano A. (2002), which utilizes the results of the experimental tests on masonry tuff walls.

The applied loads are the self weight and the horizontal load, which increases with monotonic low up to the end of the analysis.

4.2 Results and comparisons

In Figure 13a, a summary of the non linear analyses for the walls 350-350-35 is reported; particularly, the pushover curves are depicted. In this diagram, the collapse multipliers calculated through the limits analyses and already provided in Figure 6 are also reported. The comparison among the nonlinear static and limit analyses shows that the F.E.M. results provide collapse multipliers higher than limit analyses, because the tensile strength of masonry is considered in the Abaqus models.

Indeed, in Figure 13b and c the visualization of the F.E.M. deformed shapes, with stress tensor vectors, and the hypothesized collapse mechanisms are reported. The comparison shows a good agreement between the Abaqus and limit analyses.

In Figure 14 the comparison among the collapse multipliers computed by means of F.E.M. analysis and of limit analyses (Eqs. (10) ^(13)) is provided. Particularly, the diagrams refers to walls characterized by h/s ratio equal to 10. In almost all analyzed specimens (Table 1) the diagrams confirm the previous

Figure 14. Comparison among the collapse multipliers of walls with h/s = 10 evaluated through F.E.M. and limit analyses.

Figure 14. Comparison among the collapse multipliers of walls with h/s = 10 evaluated through F.E.M. and limit analyses.

Figure 15. Comparison among the collapse multipliers evaluated through nonlinear static and limit analyses on walls designed according to the geometrical requirements of EC8'03 (a) and NTC'07 (in seismic zones) (b).

observation regarding the multipliers a; in fact, the values of a computed through pushover analyses are generally higher than the values calculated with the limit analyses. Finally, also the curves associated to the results of nonlinear static analyses carried out on walls 3L and 4L show an increasing trend with the aspect ratio h/b; this confirms the beneficial effect of the vertical edge restraints on the resistance of the walls to out-of-plane collapses.

The Figure 15 shows the comparison among the F.E.M. results and the curves relative to collapse multipliers a of the walls 3L and 4L associated to EC8'03

(Fig. 15a) and NTC'07 (Fig. 15b). The thicknesses s of the analyzed specimens have been obtained by Figure 9 curves (4) (wall 3L) and (6) (wall 4L) for EC8'03, and by Figure 9 curves (8) and (9) for NTC'07. In the case of EC8'03, the results of F.E.M. analyses show an increasing trend of the curves (Fig. 15b), while in the case NTC'07 it possible to note a sub-horizontal trend of the F.E.M. curves.

Moreover, when h/b ratio is larger than one, the NTC'07 curves show a slight decrease of collapse multipliers; this seems to underline that the limitations of NTC'07 give very large values of maximum h/s ratio when the wall is characterized by high values of aspect ratio h/b.

Finally, the comparison among the F.E.M. results reported in the curves (a) of Figure 15 shows that the EC8'03 is on the safe side with respect to NTC'07; in fact, it can be observed that the curve "Abaqus 4L" of Figure 15a is characterized by higher values of collapse multipliers a than the curves "Abaqus 4L" plotted in Figure 15b.

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