## Comparison With Finite Element Model

The validation of the proposed approach based on the limit state analysis is carried out by comparing the results in terms of stress resultants with the output obtained by advanced non linear FE modelling.

The Finite Element Model has been constructed using the Algor V19 FE program: shell elements are used for all uncracked masonry portions and contact elements are inserted at the interface between shell elements where tensile stress develops.

The following material properties have been assumed: specific weight y = 1850 kg/m3; Poisson's ratio v = 0.15; Elastic Modulus E = 5000 MPa.

For an easier comparison between the results of the limit state analysis and the FE analysis, the FE mesh has half the size of the limit state analysis along the parallel but the same number of subdivisions along the meridian, (Fig. 17).

To simulate the presence of perimeter walls high 3 m and thick 0.50 m, Shell elements are been arranged

Figure 17. Axonometrie showingthe Finite Elements model of the pavilion vault.

Figure 18. Comparison between the meridian forces S obtained by the limit state analysis and the Finite Elements Analysis along the central slice.

Figure 20. Comparison between the eccentricities obtained by the limit state analysis and the Finite Elements Analysis along the slice near the diagonal.

Figure 18. Comparison between the meridian forces S obtained by the limit state analysis and the Finite Elements Analysis along the central slice.

along the supports and they are constrained by fixed nodal boundary conditions at the abutments.

Figures 18 and 19 show the comparison between the meridian forces S obtained by the optimised limit state analysis and the Finite Elements Models along the central slice and along the slice near to diagonal respectively. Both curves present the same trend, characterized by a change of curvature in the cracking area, this means that the new theory based on limit state analysis with finite friction accurately predicts the behaviour of pavilion vaults, identifying the actual crack pattern and the meridian resultant S field. As it can be seen, in agreement with the safety theorem of limit state analysis associated with lower bound approaches, the values of S as calculated by the procedure are always slightly greater than the actual state of stress identified by the FE analysis.

Also the normal force N and shear force T have the same trend in the two models.

Furthermore, the eccentricity and hence the position of the thrust surface is accurately calculated with

Figure 20. Comparison between the eccentricities obtained by the limit state analysis and the Finite Elements Analysis along the slice near the diagonal.

the limit state analysis. As it can be seen, again in favour of safety, the values computed by the procedure are slightly greater than the values obtained with the FE, however the location of relative minimum and maximum and absolute values identify with great accuracy the actual position of plastic hinges along the slices.

The identification of the position of the extrados hinge is particularly critical, as this also define the position along the arch of the maximum horizontal thrust and hence is essential for the correct positioning of ties or for the construction of abutments.

Values of horizontal thrust have been compared for the two analyses for and angle 0 = 65 degrees and for the condition of full lateral restraint at the support for the F.E. model.

As it can be observed in Figure 21 where the thrust is plotted along a half parallel between midspan (y = 0) and the diagonal, the trend is again similar in the two cases with the values estimated by the procedure being clearly equal to the F.E at midspan and slightly overestimated as progresses toward the diagonal with a maximum difference of 20% .

Hence it is possible to affirm that the new theory is able to provide the actual crack pattern and the stresses field in the masonry vaults.

Figure 21. Comparison between the horizontal thrust obtained by the limit state analysis and the Finite Elements Analysis.
Figure 22. Horizontal thrust along the supports obtained by the simplified arch model (a) and the actual horizontal thrust obtained by the limit state analysis (b).