Figure 24. Maximum thrust surface's eccentricity obtained by the limit state analysis with finite friction for different friction coefficients |X.

show that two plastic hinges might form: the first one located at the extrados, for x equal to 1.5m(9y = 30°), and the second one at the intrados, for x equal to 2.78 m (0j = 65°). Figure 24 shows the likely collapse mechanism.

In addition, the vault could fail by way of sliding. Using the limit state analysis with finite friction it is possible to evaluate the minimum thickness which satisfies at the same time equilibrium and compatibility. The application of this new theory on pavilion vaults shows that, for a friction coefficient higher than 0.48, the shear force T are everywhere smaller than N. | , hence the results are unrelated to the friction coefficient and sliding mechanism are prevented.

On the other hand, in the 0.3-0.48 friction coefficient range, the vault is able to find a new equilibrium system, in which the stress resultants are unvaried, but the eccentricity considerably increases (Fig. 24). This means that the thickness necessary for the vault's stability would increase and consequently the safety factor decrease. It is necessary to specify that the maximum eccentricity is located near the support, where typically the spandrel ensures a big thickness.

For friction coefficient lower than 0.3 the Excel's solver cannot find a feasible solution, i.e. it cannot satisfy all constraints.

Moreover, the structural behaviour of masonry vaults could vary with the rise-span ratio f/l. As pointed out by Palladio (1570) and Guarini (1737), the most frequent geometry in XVI-XVII century palaces would typically results in f/l ratios equal to 1/3, 1/4 and

Figure 27. Comparison between the eccentricities obtained by the limit state analysis for the rise-span ratio equal to 1/3 (rise = 2 m), 1,5 (rise = 1,5 m) and 1,2 (rise = 1,2 m) respectively (central slice).

Figure 25. Comparison between the meridian forces S obtained by the limit state analysis after meridian cracking and before meridian cracking for the rise-span ratio equal to 1/3 (rise = 2 m), 1,5 (rise = 1,5 m) and 1,2 (rise = 1,2 m) respectively (central slice).

Figure 27. Comparison between the eccentricities obtained by the limit state analysis for the rise-span ratio equal to 1/3 (rise = 2 m), 1,5 (rise = 1,5 m) and 1,2 (rise = 1,2 m) respectively (central slice).

Figure 25. Comparison between the meridian forces S obtained by the limit state analysis after meridian cracking and before meridian cracking for the rise-span ratio equal to 1/3 (rise = 2 m), 1,5 (rise = 1,5 m) and 1,2 (rise = 1,2 m) respectively (central slice).

Figure 26. Comparison between the meridian forces S obtained by the limit state analysis after meridian cracking and before meridian cracking for the rise-span ratio equal to 1/3 (rise = 2 m), 1,5 (rise = 1,5 m) and 1,2 (rise = 1,2 m) respectively (slice near to the diagonal).

1/5. Figures 25 and 26 show the comparison between the meridian forces S obtained with the limit state analysis after cracking and before cracking for pavilion vaults over a plane square area, of semicircular curvature and of rise equal to 2 m (f/l = 1/3), to 1,5 m (f/l = 1/4) and to 1,2 m (f/l = 1/5) respectively. As it can be observed, the force S increase with decreasing rise-span ratio, nevertheless, decreasing the rise, the classical membrane solution, that represents the stress field before meridian cracking, converges on the optimum solution. In fact, reducing the f/l ratio, also the eccentricity's trend is towards zero (Fig. 27) and the tensile area near to the spring decreases (Fig. 27); this means that, in shallow vaults, the geometry is closer to the thrust surface generated by the gravity load distribution, and compression hoop stresses Hp are present for a grater portion of the surface (Fig. 28). This means that the spread of the cracks that often develops along the diagonal and in the centre of the web depends on the vault's rise and hence, for f/l ratio smaller than about

Comparison between the hoop stresses obtained by the limit state analysis for pavilion vaults with a rise-span ratio equal to 1/3 (rise = 2 m), 1,5 (rise = 1,5 m) and 1,2 (rise = 1,2m).

Comparison between the hoop stresses obtained by the limit state analysis for pavilion vaults with a rise-span ratio equal to 1/3 (rise = 2 m), 1,5 (rise = 1,5 m) and 1,2 (rise = 1,2m).

1/5, the classical membrane theory could be used with good approximation.

As it is well known, the horizontal thrust increase if the vault's rise reduces; furthermore it is possible to observe that its trend along the perimeter walls tends to become constant for shallower vaults (Fig. 29). This means that, reducing the rise/span ratio, the natural

Figure 30. Qualitative developments of natural arches for pavilion vaults with a rise-span ratio equal to 1/3 (rise = 2 m), 1/5 (rise = 1.5 m) and 1/2 (rise = 1,2 m).

The results obtained by the limit state analysis with finite friction show that this schematisation, especially for complex vaults, do not provide information about the actual stress field. In fact, in the case of pavilion vaults, it is emerged the importance of the interaction between the arches for the evaluation of the crack pattern and the horizontal thrust.

Therefore, the present paper provide a new analysis tool for a computer procedure able to value both the rotational and sliding mechanisms, the three-dimensional effects in the vaults, very important especially for complex vaults and the vaults' safety factor, given by the minimum thickness over the actual thickness ratio.

Figure 30. Qualitative developments of natural arches for pavilion vaults with a rise-span ratio equal to 1/3 (rise = 2 m), 1/5 (rise = 1.5 m) and 1/2 (rise = 1,2 m).

arches, that develop in the pavilion vaults and that cause the transfer of meridian stresses from the centre of the web to the diagonal, affect a smaller area near to the diagonal, as it can been see from the qualitative pictures in Figure 30.

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