Lower Bound Approach With Finite Friction

Masonry pavilion vaults, despite their common use in past centuries, in particular in XVI-XVII century palaces, have not been studied in any greater depth yet, principally because due to their singularity of shape and presence of cuspide form along the diagonal, the complex 3-dimensinal state of stress that develops in presence of uniform gravity loading does not lend itself easily to simplified approaches.

These vaults have continuous support along walls; have finite curvature in the direction of the meridians, but infinite curvature along horizontal plane. This means that there can be no contribution to recentre the line of thrust from hoop stresses and hence membrane theory is inherently non applicable. This means that such type of vaults is often affected by cracks along the diagonals, where the geometry of the vault is farthest from the geometry of the thrust surface generated from the gravity load distribution. Cracks can also appear toward the centre of the web of each portion so, in the past, they are been often modeled as independent arches.

However the simplified arch model, does not account for the capacity of the vault to transfer load associated with shear and for the arch effect that can be develop within the horizontal strips due to their non negligible thickness, similarly to the arch effect accounted for in walls or slabs.

On the basis of this considerations and suitably simulating the boundary conditions and the condition of equilibrium that arise along the diagonal due to the singularity of the surfaces along the diagonals, limit state analysis can be applied to this family of vaults to determine their state of stress at failure.

The main objective of the present section is to understand, through the limit state analysis approach, the structural behaviour of pavilion vaults, in order to evaluate with more accuracy the actual stress field that can cause cracks.

Moreover this method allows finding the thrust line position and the admissible thrust surfaces, leading to the calculation of the minimum thickness which satisfies at the same time equilibrium and compatibility. This is of great importance with regard to strengthening interventions because it allows determining the vaults' safety factor, which can be expressed, as proposed by Heyman as the ratio between the geometric thickness over the minimum required thickness, but also the accurate position of the hinges at failure and hence the appropriate positioning of ties or other thrust contrasting devices.

The simplest case to analyse is a pavilion vault over a plane square area and of semicircular curvature, which is also the most frequent occurrence according to the technical literature (Scamozzi, 1615), subject to self weight loading.

Figure 3 shows the geometric characteristic of the vault's generatrix, where Ris its radius, f is the vault's rise, lis the span. In the numerical procedure the slices that make up half of the web between to successive ribs are considered, these being present only along diagonal. The ribs according to historic the technical literature, as shown in figure 3, can be described by

Figure 3. Geometric characteristics of the vault's generatrix.

Figure 3. Geometric characteristics of the vault's generatrix.

Figure 4. Geometric characteristic of the diagonal and of the vault's generatrix, used for the centring work (Levi, 1932).

the equation of an ellipsis (Curioni, 1870; Breymann, 1885; Levi, 1932).

Assuming the n slices are made up of m blocks, each block is identified, in the global system, by the coordinates of its centre of mass:

for i = 1 to m and k = 1 to n were ak is the horizontal angle between the x axis in the global system and the horizontal projection of x-axis in the local system as shown in Figure 5 and 0i

Figure 5. Generic block with the coordinates of its centre of mass and the angles ak, 9i and 9i.

is the angle between the vertical and the perpendicular to the generatrix. The relation between 0i and 0i is:

Hence, the eccentricity of the thrust surface at each point is:

and the minimum constant thickness t required is:

This can be the objective function to minimise.

The meridian, hoop and shear stresses are calculated considering the coordinates of the generic surface of thrust at the interface between blocks. Subsequently, by limiting the maximum values of the internal shear force Tj, component of Sj parallel to the interface, which can not be greater than the frictional strength at each block interface, the thrust surface's zi coordinates that satisfy the material constraints are determined.

The coordinates of the generic thrust surface at the interface are:

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