Hi

Diaphragm arch differential rocking remarkable crack pattern surveyed in A, D, E, G, H Collapse surveyed in C

Single leave vault differential bending

Single leave vault differential bending

C = 1.0% C = 3.5% C = 5.0% C = 5.0% C = 5.2% (F) C = 3.3% (G) C = 7.8% (H) C = 10.7% (I)

remarkable crack pattern surveyed in A, B, D, E, G, H

Collapse surveyed in C

remarkable crack pattern surveyed in A, D, E, G, H Collapse surveyed in C

SS. Faustino e Giovita (Montemaderno, Brescia)

SS. Faustino e Giovita (Montemaderno, Brescia)

SS. Pietro e Paolo (Preseglie, Brescia)

SS. Pietro e Paolo (Preseglie, Brescia)

Madonna della Rocca, Sabbio Chiese, Brescia.

Madonna della Rocca, Sabbio Chiese, Brescia.

San Lorenzo Church, Clibbio, Brescia.

San Lorenzo Church, Clibbio, Brescia.

San Pietro Church (Roè Volciano, Brescia)

San Pietro Church (Roè Volciano, Brescia)

Sant'Antonio, (Roè Volciano, Brescia)

Sant'Antonio, (Roè Volciano, Brescia)

San Rocco, (Roè Volciano, Brescia)

San Rocco, (Roè Volciano, Brescia)

Santissima Trinità, (Roè Volciano, Brescia)

Santissima Trinità, (Roè Volciano, Brescia)

Santa Maria Assunta (Bione, Brescia)

Santa Maria Assunta (Bione, Brescia)

where: FS is the seismic force triggering the mechanism; C is the load collapse multiplier (also regarded as vulnerability index); W TOT is the structure total weight._

of full rocking, when the crack pattern is fully developed, no abutment buttress action can be accounted for, because the resisting ideal struts become parallel to each other. As a result, the arch thrust has to be entirely confined by the existing ties.

Figure 1. Overturning of the façade induced by the roof seismic thrust.

Figure 2. Transverse arch resisting mechanisms under a) vertical loads (rest condition) and b) horizontal loads (case of full rocking, Giuriani et al. 2008).

Figure 3. Madonna della Rocca Church, Sabbio Chiese, Italy. Transverse arch crack pattern induced by the seismic action.

Figure 2. Transverse arch resisting mechanisms under a) vertical loads (rest condition) and b) horizontal loads (case of full rocking, Giuriani et al. 2008).

Furthermore, following the cracking triggered by the rocking motion, the span of ideal arch significantly increases and so does the arch lateral thrust (Figure 2b). The span of the ideal arch in rocking condition can be conservatively assumed to be equal to the span of the ideal arch in rest condition (L*) plus the thickness of the abutment (d) (Giuriani et al. 2007, 2008). For typical case studies the ratio d/L* was observed to range

Figure 4. Detail of the horizontal crack pattern at the transverse arch abutment base induced by the seismic action.

between 0,15 ^ 0, 2, thus when shifting from rest condition to rocking the arch thrust could be 30 ^ 40% larger than the thrust at rest condition.

In the case of full rocking, the decrease of the abutment buttress action and the increase in the ideal arch span cause the existing tie over-tension. When the tie resistance is barely sufficient to withstand the traction force in rest condition, no extra resources are available in case of earthquake, and the collapse of the tie can be expected.

Figure 5. Detail of the welded tie rod in San Antonio Church (Manerba, Brescia, Italy). The tie rod collapsed during the 2004 earthquake.

Given this foreword, the failure of the existing ties in San Antonio Church (Roe Volciano, Figure 5) was interpreted as a consequence of the onset of the transverse arch full rocking mechanism. In some other churches, where weak existing ties were also surveyed, the over-tension caused either the tie unthreading from the masonry walls from exceeding the anchorage resistance (Madonna della Rocca church, Sabbio Chiese), or the tie yielding (San Lorenzo Church in Clibbio).

As for the vulnerability assessment, the evaluation of the safety factor with respect to the onset and developement of the mechanism can be obtained by reference to the traditional approach of the principle of virtual work (Abruzzese and Lanni 1999). The same approach can be also addressed to define the structure capacity curve (Housner 1963, Curti et al. 2006), to study the abutment out-of-phase rocking (Como et al. 1991), and to evaluate the tie tension (Lagomarsino etal. 2004).

A simplified method was proposed in Giuriani et al. (2007 and 2008) for the evaluation of the collapse multiplier and the tie tension in the case of either over resistant or weak ties. The seismic collapse multiplier was evaluated by introducing an ideal additional horizontal constraint at the tie level, and by enforcing that its reaction force R was nil (Figure 2b).

2.3 Differential rocking of the diaphragm arches and differential deflection of the thin groin vault

The different stiffness of the transverse arches along the nave can be regarded as another source of seismic vulnerability of the churches (Table 1). The maximum top displacement experienced by the transverse arches during the rocking motion depends on the geometry and stiffness of the structure, therefore

Figure 6. Crack pattern induced by the differential transverse arch rocking. Differential rocking, which is maximum in the first and last bay of the nave, is the result of the different stiffness of the transverse arches.

differential displacements can be expected between adjacent transverse arches (Figure 6). Noticeable differential displacement can also occur in the case of out-of-phase rocking of neighboring diaphragm arches.

The largest differential displacements are usually experienced by the first and last bay transverse arches. As a matter of fact, the façade rarely undergo rocking, unlike the nearby transverse arch; and the triumphal arch is typically much stiffer than the neighboring transverse arch. This way, the vaults covering the first and last bays are subjected to shear distortion. When differential rocking is severe, shear distortions can be remarkable and diagonal cracks may form in the vault ring (Figure 6). The damage is more severe for thinner vaults, like the single leave vaults. In the case of very strong earthquake differential rocking may even cause the collapse of the vault.

When subjected to earthquake loads, the nave vaults also experience differential deflection of the vault groins (Figure 7). Traditional masonry groins are connected to the perimeter walls at the imposts, whereas the vault rings are built adjacently to the perimeter walls. This result in the structural discontinuities shown in Figure 7.

contact contact

vault support at the springing structural discontinuity

Figure 7. Differential flexural mechanism of the groins of the single leave vaults.

vault support at the springing structural discontinuity

Figure 7. Differential flexural mechanism of the groins of the single leave vaults.

When subjected to the seismic actions Fo illustrated in Figure 7, groins a and b rotates counterclockwise around points A and B. the vault groin a pushes against the perimeter wall, whereas groin b detaches from the perimeter wall and the discontinuity width increases for increasing seismic actions. As a result, the central portion of the vault is subjected to concentrated differential deflection. The mechanism is amplified by the rocking of the diaphragm arches.

In the case of very strong seismic action, cracks induced by the differential deflection may extend through the vault ring thickness causing its collapse.

Severe shear distortions induced by differential rocking and differential deflection of the vault were recognized as the major causes of total and partial collapse of the single leave barrel vaults of the first and second bay of the main nave of San Pietro Church in Roe Volciano Brescia (Figure 8).

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