## The Vault Modelling As An Equivalent Membrane

The role of floor slab in the 3D model of a building was previously introduced. The response depends on the whole system and both connection and load transfer effect due to the floors are very significant.

In this context, the floor modelling as isotropic or orthotropic membrane elements seems to be suitable. As proposed in Galasco (2004), the elastic element describing the floor is identified by a principal direction, with Young's modulus Ex , while Ey is the Young's modulus along the perpendicular direction, v is the Poisson ratio and Gx,y the shear modulus.

The moduli of elasticity Ex and Ey represent the normal stiffness of the membrane and each of them structurally describes the connection degree between the floor and the vertical wall parallel to its reference direction, both in linear and non-linear phases. In a few words, the normal stiffness of the floor along X-axis provide a link between the piers of a wall parallel to X-axis, influencing the axial force the spandrels, but it does not control the loads transferred to a wall parallel to Y-axis.

The term Gx y represents instead the shear stiffness of the floor and the horizontal force transfer among the walls, both in linear and non-linear phases.

The evaluation of these quantities may be rather simply identified in case of recurrent floor typologies, ascribing it to the structural role shown by some specific elements. For example, the shear stiffness of an r.c. floor with beams and slab is mainly ascribable to the slab. In this example, the beam axial stiffness leads to the definition of the Young's modulus of the equivalent plane element.

On the contrary, in case of vaults, beside thickness and material properties, the stiffening contribution strongly depends on shape and geometrical proportion (e.g., rise-to-span ratio). So, the definition of the elasticity moduli to be attributed to the equivalent plane element may be highly arbitrary.

The proposed modelling strategy starts from the elastic numerical simulation of vault response, in case of pre-defined load configurations, aiming to schematize the axial-only and pure shear behaviour by imposing special force or displacement conditions (Fig. 2).

The scope is to establish a functional relationship among the elastic parameters of the equivalent plane element and the significant quantities of the vaults. This obviously excludes the rigorous theoretical investigation of the vault behaviour as a membrane and flexural shell.

In this way, an ideal equivalence is set between the typology of vault examined and the plane element, having the same planimetric dimension (squared plan

Figure 2. Configurations of boundary conditions in order to schematize the axial-only and pure shear behaviour.

L x L), thickness (s) and material (characterized by the elastic moduli Ex, Ey and Gxy). An isotropic plane element is accounted for if the vault typology is symmetrical (e.g. cloister and cross vault) and an orthotropic one if unsymmetrical (e.g. barrel vault). In the first case, aiming to evaluate the vault equivalent Young's modulus Ey, a uniform displacement state along x-axis (Au) is imposed to one of the orthogonal sides. The opposite one is fixed, as in Figure 2.

From the isotropic linear elastic relationships, the numerical FEM results may be interpreted as follows. The Young's modulus Ey is given by eq. (1).

where: E is the Young's modulus of the plane element, a is the normal stress, e the axial strain, Au is the applied displacement to one side, n is the number of nodes on this side, rk is the reaction nodal force in the k-th node along the analysed direction.

In order to evaluate the vault equivalent shear modulus Gy, an auto-equilibrated system of forces (having a total value F on each side) is applied. The horizontal displacements of one edge of the model are adequately constrained. By means of the isotropic linear elastic relationships, eq. (2) provides the ratio between Gy and G (the shear modulus of the plane element).

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