Structural modelling

Figure 2.15 shows the extruded models of both each structural unit and whole complex in 3D view (SAP 2000, v. 9.1.6). Due to the high level of variability and irregularity, the geometrical modelling is affected by some approximation, always on the safe side. The assumptions for the main sub-structures, such as the vault, the floor beam, the truss, are following specified (Mazzolani et al., 2004).

V: Vault (Fig. 2.15a, d)

Ribs are modelled as single elements with a rectangular cross-section, 4×15 cm², and a broken axis. They are restrained at the extremities by means of hinges, whereas the connection between ribs and splines is modelled as a fixed joint. Considering the high variability of the rib spans a band criterion has been applied, in order to define the ribs inter-axis, by plotting the envelope of each rib, superimposing the four quadrants, where the plan can be divided by the symmetry axes, and using the axis of each band. In this way, the average position of the rib axes is defined, obtaining a symmetric scheme as respect to the symmetry axes. Splines are modelled as elements with rectangular

cross-section, 3×5 cm², spanning between the ribs, with almost the same inter-axis as the ribs. They are fixed at the extremities to the ribs. The horizontal central grid at the vault crown is modelled by considering elements with circular cross-section: along the X direction, 8 cm diameter, spanning 35 cm; along the Y direction 10 cm diameter, spanning 1.00 m; at the perimeter 10 cm diameter. The grid elements are restrained at the extremities to the perimeter elements by means of hinges, whereas the connection between orthogonal grid elements is modelled as a fixed joint. The vault-floor beam wooden links are modelled with square cross-section 4×4 cm².

F: Floor beam (Fig. 2.15b, d)

Primary beams are modelled by a single element with a circular cross-section of a constant diameter (25 cm, mean value of the actual ones), assuming that the nail connection between the two timber stocks, which compose the beams, provides a fixed joint. The single large shaped beam is modelled with the real sizes. Beams are arranged at the same spacing, 0.95 m, it corresponding to the mean value of the actual distance. The inclined struts and the secondary beams are modelled as elements with a circular cross-section. 12 cm diameter. The system of five secondary floor beams located near the façade wall has been neglected, being limited to a small area.

Concerning the restrain conditions, the following assumptions are made: for each floor beam, such as primary and secondary beams and inclined struts, the restrain at the extremities is considered as a simple hinge; the connection between secondary longitudinal beams and primary ones, realized by the system of metallic stirrups, which also embrace the tie beam of the truss, is modelled by a constraint, which allows equal displacements along the X, Y, Z directions and equal rotation around the axes; the connection between secondary transversal beams and primary ones is modelled by a constraint, which allows equal displacements along the X, Y, Z directions and equal rotation around the X axis (considering that the bending stiffness of the floor beam is larger than the torsional stiffness of the transverse secondary beam).

T: Truss (Fig. 2.15c, d)

All structural elements of the truss are modelled with the actual detected cross-section dimensions (Fig. 2.13a).

Th e a n c i en t t i mb er st ru c t u re s o f t h e R o ya l Pa la c e of Na p l es :

d i a gn osi s, a n a l ysi s a n d rest o ra t i on 8 5

Figure 2.15. Diplomatic Hall (II) − Structural modelling: a) Vault; b) Floor beam; c) XZ and YZ planes; d) 3D model.

XZ plane YZ plane

X Y

a)

b)

d)

Y X

Z

c)

Aiming at analysing the behaviour of the structures during their service life, from the beginning until today, the following phases and the corresponding structural models have been identified:

Erection phases (new timber)

- Phase 1: Erection of the beam floor structure (Model F1)

The structure consists of floor beam and related stiffening elements, such as primary beams, secondary longitudinal and transversal beams, longitudinal and transversal struts. Only dead load is considered (G_{1, floor});

- Phase 2: Erection of the truss (Model F2)

The structure is integrated by the truss above. Only dead load is considered (G1, floor + truss);

- Phase 3: Erection of the vault structure (Model V1)

The vault structure is independent by the existing floor-truss system. Only dead load are applied (G_{1, vault});

- Phase 4: Beam floor completion (Model F3)

The geometrical model is equal to Model F2. The beam floor is completed by non-structural elements. Therefore, on the floor primary beams all permanent loads are applied (G1, floor + truss + G_{2, floor});

- Phase 5: Vault-Floor connection and vault completion (Model VF1)

The geometrical model consists of the beam floor, the truss and the vault.

The vault is completed by lathing and stucco; therefore, all permanent load are applied on the vault grid too (G_{1, vault} + G_{2, vault});

- Phase 6: Complete structure in service conditions (Model VF2)

The structure is completed. Both permanent and variable actions are considered for the beam floor (G1, floor + truss + G2, floor + Qfloor).

Phases at the present state (ancient timber) - Phase 7: Complete structure (Model VF3)

The geometrical and loading model is equal to Model VF1, but the material is modelled as ancient timber, taking into account of the degradation effects during the time;

- Phase 8: Complete structure in service conditions (Model VF4)

The geometrical and loading model is equal to Model VF2, but the material is modelled as ancient timber;

Th e a n c i en t t i mb er st ru c t u re s o f t h e R o ya l Pa la c e of Na p l es :

d i a gn osi s, a n a l ysi s a n d rest o ra t i on 8 7

- Phase 9: Damaged structure (Model VF5)

The complete geometrical model lacks of the rotten beam 1, adjacent to the main façade, and it is characterized by the truss partially supported by the floor beams for considering that the truss strut at the main façade side is completely inefficient.

- Phase 10: Damaged structure in service conditions (Model VF6)

On the damaged structure, both permanent (G) and variable actions (Q) are considered. serviceability load (Model VF2) considering new timber properties, has been reconstructed. Then the effect of the degradation due to ancient timber, creep and moisture has been considered (Models VF3 and VF4), in order to evaluate the deformation of the vault at the present state, which determined the crack distribution requiring the restoration of the fresco.

In Figure 2.16 the deformed configurations of the vault are depicted, with reference to one of the structure quadrant at a single alignment of ribs where the wooden links, which connect the vault to the floor beams, are located. The reference points are marked with a circle on the vault model plan section.

Displacements have been scaled by an amplification factor equal to 5, for evidencing the deformed shape. As it appears, the vaulted ceiling undergoes a generalized flattening, which consists of sagging at the centre with a maximum displacement equal to about 9 cm and elevation near the masonry supports, with a maximum displacement equal to about 2 cm.

In Figure 2.17, together with the crack distribution at the intrados of the vault, the bending moment distribution and the corresponding deformed configuration are drawn with reference to a central YZ structural section.

The results of the stress state analysis and the safety checks emphasize that some ribs of the vault carpentry do not accomplish the strength requirements, due to the fact that the links between vault and floor beams located near the