1 Introduction

Response spectrum linear dynamic analyses can represent an appropriate approach for interpreting the effects of both gravity and lateral loads on churches. Indeed, the assumption that modern software are able to reproduce even highly non linear behaviour of such a type of buildings is often a false illusion. In fact, such an assumption clashes with the scarce knowledge that pervades the complex modelling process of both complex mechanical features of the base material and constructional details.

On the other hand, the structural performance of masonry buildings is significantly influenced by the developing of cracks, which produce the detachment of some parts with a variation of the global structural behaviour. Linear analyses are particular apt for determining the zones of the building that are prone to experience stresses concentrations, even though such analyses are not appropriate to represent the structural response in case of extensive damage. To this purpose, it is compulsory to support the obtained results with engineeristic judgements or the outcomes obtained from more sophisticated analyses.

An example is given by the “two-step” procedure described and used in Brandonisio et al. (2013), which has been adopted for interpreting the state of damage occurred on a sample of four churches stricken by the 2009 L’Aquila Earthquake. In the ‘‘first step’’ the linear behaviour of the churches has been analysed by 3D finite element models, in order to determine the static and dynamic properties, as well as the distribution of horizontal forces acting on each macro-element. In the ‘‘second step’’ the whole structure has been dissected in the constituting macro-elements. They have been separately analysed in the non-linear range up to collapse, for determining the horizontal strength of each macro-element. The procedure has been finally completed by an additional ‘‘sub-step’’, concerning the evaluation of out-of-plane potential failures.

The same type of methodology has been recently used by Castellazzi et al. (2013), where the results obtained from limit analysis and nonlinear finite element analysis have been conducted on some macro-elements of the Madre Santa Maria del Borgo church in San Nicandro Garganico (Italy). The FEM numerical models allowed the Authors to focus their attention on those mechanisms involving the façade, as these seemed the most dangerous for the church stability.

Previously, De Matteis and Mazzolani (2010) used a linear FE model for analysing the Fossanova gothic cathedral in the inner Lazio (Italy). The Authors showed that when the response spectrum analysis is implemented on reliable FEM model calibrated on the basis of ambient vibration tests (De Matteis et al. 2008) and the related results are properly integrated by the outcomes obtained by linear kinematic analyses of the main macro-elements, the response of the whole church may be correctly interpreted up to the collapse. Such outcomes were confirmed by shacking table tests carried out on a scaled model of the church (Krstevska et al. 2009).

In line with the above methodology, in the present study the calibration process of a FEM numerical model of the Santa Gemma church in Goriano Sicoli is shown. This is used for performing a response spectrum linear analysis able to give indication on the stress distribution experienced by the church during the 2009 L’Aquila earthquake. The results, which integrate the ones presented in the companion paper dealing with both linear and non-linear kinematic analyses, allow to give a comprehensive evaluation of the level of damage attained by the church during the seismic event. In addition, based on the obtained results, useful indications on the structural fragilities characterizing similar churches belonging to the same regional area might be provided.

2 Ambient vibration test

The church under investigation is an asset upheld by the Ministry Cultural Supervisor. Therefore, destructive tests have been not carried out and a different strategy for getting an adequate knowledge of the building material properties has been actuated. In particular in order to identify the dynamic behavior of the church in the present damaged state (vibration frequencies and mode shapes), in the framing of a cooperation between the University of Chieti-Pescara and the IZIIS Institute of Skopje (Macedonia), within the European Research Project COST C26 (Tashkov et al. 2010; Krstevska et al. 2010, 2011), non destructive ambient vibration tests (AVTs) have been carried out (Aras et al. 2011). The outcomes have been accounted for the calibration of the mechanical features of the material to be used in the numerical model.

The AVTs have been carried out by means of three seismometers (Ranger type) which have been positioned, for each measurement, on each window of the building (Fig. 1a, b). The measured signal, amplified by four channel signal conditioner (Fig. 1c) and properly filtered, has been collected by a high- speed data acquisition system, which transformed the analogical signal to a digital one.

Fig. 1
figure 1

Measurement instrument for AVTs

Full size image

The ARTEMIS software (Fig. 2a) has been used to define the spectra density curves (Fig. 2b) and mode shapes of the structures, as well as to detect the relative main frequencies (f), period (T) and damping factors (ξ).

Fig. 2
figure 2

a Processing data and b obtained spectra density curves (data elaborated by the IZIIS Institute, Skopje, Macedonia)

Full size image

In Fig. 3 the position of seismometers is shown. It should be noted that, due to heavy damage state, measurement instruments were not placed inside the church.

Fig. 3
figure 3

Instruments arrangement for AVTs

Full size image

The main three natural frequencies f with the associated damping values ξ and mode shapes are shown in Table 1. It is apparent that the first mode (f 1 = 2.402 Hz, ξ 1 = 1.42 %) is mainly translational in the transversal direction, with larger deformation developing in the central parts of the navies and in the transept zone, with a torsional rotation due to the in-plane rigidity of the main façade; the second mode (f 2 = 3.027 Hz, ξ 2 = 1.34 %), is mainly torsional; the third mode (f 3 = 3.562 Hz, ξ 3 = 1.02 %) in addition to transversal deformation of the navies, evidences longitudinal displacements, especially of the main façade.

Table 1 Ambient vibration test results
Full size table

It should be noted that generally speaking the revealed mode shapes evidence mainly the development of global deformation, as the local modes involving single parts of the structure are limited. This result states the effectiveness of the provisional structures which were applied on the structure, apart from the case of the façade that in the third mode seems to have an independent vibration mode (Campitiello et al. 2011).

3 The numerical model

3.1 Set-up of the model

The proposed model has been assembled by referring to the current conditions of the building, therefore reproducing the geometrical features and the revealed crack patterns of the structure in the damaged state (Fig. 4a) (De Matteis et al. 2011). The ABAQUS 6.7 (SIMULIA 2007) software has been used to this purpose. Tetrahedral solid C3D4 finite elements of sizes ranging between 0.3 and 0.4 m have been adopted mesh. The dimension of the elements have been selected in relation to the structural macro-elements, so to allow a proper distribution of the mesh through the walls thickness. Then the adopted mesh has been validated through a sensitivity analysis. Thus, 584,617 finite elements joined by 127,001 nodes have been used.

Fig. 4
figure 4

The proposed model with the adopted mesh configuration for the damaged building: a whole building, b longitudinal section

Full size image

The effect of the existing reinforced concrete roof has been considered in terms of load on the RC beams and diaphragm action, which has been simulated by applying a coupling-kinematic constraint between reference points. In order to account for the effect provided by iron rod ties, a wire type connector has been defined and a preload force of 50 % of its yielding strength has been applied.

The most significant existing cracks have been reproduced by deleting the surrounding finite elements (Fig. 4b), thus eluding the bilateral contact between the related parts.

A specific weight of 19 and 18 kN/m3 has been assumed for the irregular and the brick masonry elements respectively, while a value of 25 kN/m3 has been considered for the concrete parts. A poison ratio of 0.2 has been assigned to all materials.

A specific procedure has been followed to fix the material elastic moduli. In a first stage, the bottom limit value (870 MPa) of the range proposed by the Italian code (NTC 2008) for old stone masonry has been assumed for all the parts composed by monolithic masonries made of irregular stones (walls and columns). Similarly, elastic modules of 2400 and 20000 MPa have been considered for the parts made of brick masonry (arches, vaults and dome) and reinforced concrete, respectively. Moreover, a very low value (almost zero) has been fixed for the filling material on the top of the vaults.

For masonry element, the above values have been amplified in order to account for the possible presence of good mortar (50 %, as indicated by the code) and then reduced by a proper “damage reduction factor” I d in order to account for the diffused cracks revealed during inspections. This factor has been calibrated by using an iterative procedure based on a sequence of modal analyses (Lanczos method), each one run with a different I d value, which has been progressively adjusted in order to obtain the same modal frequencies and shapes retrieved by the AVTs measurements. The obtained values of I d , together with the equivalent elastic modulus considered for each part of the model, are listed in Table 2. In Table 3 the data referred to the first six modes are given, while the corresponding modal shape are shown in Fig. 5.

Table 2 Mechanical features of materials—“damaged model”
Full size table
Table 3 Modal analysis results on the “damaged model”
Full size table
Fig. 5
figure 5

Main mode shapes and related frequencies (numerical analysis)

Full size image

It is possible to observe that the first two frequencies involve almost the 5.0 % of the total mass of the building for each of the two principal directions. This is due to the fact that they mainly refer to the tower vibration modes, which have not been measured during the AVTs. Therefore, for the model calibration, such modes have been neglected. On the contrary, the third mode (2.3911 Hz), which is mainly transversal, with a torsional rotation respect to the main façade, the fourth mode (3.105 Hz), which is mainly longitudinal with an evident deformation of the main façade and the fifth mode (3.827 Hz), which is mainly torsional, fit well the values of the main frequencies obtained by experimental measurements. Also, the sixth mode, which has not been identified during the AVT tests, is completely torsional and is characterized by a modal frequency of 4.27 Hz.

It is interesting to observe that the fourth mode obtained by the numerical analysis is characterized by the vibration in the longitudinal directions of the main façade, with a participating mass of 7.96 % of the total, which is evidently related to a local vibration mode as it was also identified by experimental tests. Anyway, it should be recognized that is not very easy to compare the shape of the numerical modes with the ones identified by experimental measurements, as the latter are characterized by a rough graphical representation due to the limited number of measured points.

3.2 Damage interpretation by fem analysis

A response spectrum linear dynamic analysis has been implemented, with the main scope to identify the parts of the church affected by higher stress concentrations in order to confirm the localization of the cracks. It has to be pointed out that such a goal has been successfully pursued also in previous studies (De Matteis et al. 2007, 2008).

To this purpose, the model previously presented has been adjusted, by eliminating the disconnections simulating the cracks and by restabilising the elastic modulus of masonry elements by setting equal to 1.0 the aforementioned damage reduction factors I d , as shown in Table 4. In such a table the maximum compression–tension (fCmax − fTmax) strength of the masonry materials, which will be useful for the interpretation of the analysis results, are also listed. In absence of in situ direct tests, these values have been conventionally assumed as proportional to the elastic modulus. In particular, the maximum compression strength has been assumed as fCmax = E/1000, while the maximum tensile strength fTmax has been posed as fCmax/10.

Table 4 Mechanical features of the materials—“undamaged model”
Full size table

One hundred modes have been considered obtaining a participating mass larger than the 85 % of the total mass in the three principal directions of the building. In Fig. 6 the participating masses in the three principal directions expressed as percentage of the global mass of the church are provided together with the related vibration frequencies.

Fig. 6
figure 6

Participating mass percentages and relative frequencies—“undamaged model”

Full size image

As first step, a static analysis under permanent loads only has been carried out in order to establish the state of the church before the mainshock. The revealed stress distribution highlighted a satisfying safety condition for the structure. However at the main arcade located between the transept and the naves the tensile stress exceeded the conventional strength of 0.24 MPa (Fig. 7). This means that before the earthquake, some cracks were probably present on these elements. It is worthy of being noticed that this situation could be ascribed to the heavy reinforced concrete roofs surmounting the aisles. In addition it could be observed that the columns are subjected to a compressive stress of almost 1.0 MPa, this meaning that before the earthquake they were quite close to a situation of incipient squashing.

Fig. 7
figure 7

Static analysis results

Full size image

In the second step the response spectrum analysis has been carried out. Due to the absence of time-history in situ registrations, both the horizontal and the vertical seismic components (see Fig. 8) have been referred to the data retrieved by the Ortucchio station (Pacor et al. 2011) (download of the records: http://itaca.mi.ingv.it, last access 23/09/2013. Station Code: ORC—Lat. 42.33; Long. 13.33; Site Classification EC8: A*; Morphologic Classification: R; Epicentral Distance: 49.3 km; horizontal PGA: 64.07 cm/s2; vertical PGA: 30.50 cm/s2), which was the one closest to the location of the building under investigation.

Fig. 8
figure 8

Scaled response spectra of the ORC (Ortucchio) station: horizontal (WE-comp) and vertical (UP-comp) components

Full size image

The horizontal spectrum has been scaled to the PGA equal to 0.15 g, according to the result obtained by attenuation laws provided by Sabetta and Pugliese (1987). Moreover, the vertical PGA has been scaled to the value of 0.06 g, which has been estimated according to a “back analysis”, based on the development of some local mechanisms, such as squashing of the columns.

The seismic input has been applied in the three principal directions of the building. The 100 % of the maximum spectral accelerations have been considered in the longitudinal direction, whereas the 30 % have been applied transversally, according to the combination rule given by many codes. Such assumption resulted particularly effective as the obtained cracks, at the end of the analysis, well matched the ones actually provoked by the earthquake. In the vertical direction the spectral accelerations depicted in Fig. 8 have been considered with their entire value. The modes of modal analysis have been combined by applying the SRSS (Square-Root-of-Sum-of-Squares) summation method. The obtained outcomes have been therefore superimposed to the effects due to gravity loads.

The obtained results have been interpreted in terms of damage distribution by identifying the parts characterized by higher tensile stresses. It can be observed that the model reproduces the revealed state of damage, with peak stresses in the parts of the church actually affected by cracks. For the sake of example, in Fig. 9, a comparison of the results related to the main façade is shown.

Fig. 9
figure 9

Comparison between FEM model and real church: response of the main façade

Full size image

A higher tensile stress (equal to about 0.33 MPa) was found at the base of the bell tower (Fig. 9, view 1). Moreover, a stress concentration is recognizable at the spandrels upon the architraves of the doors of both the nave and the left aisle (Fig. 9, views 2–6), with principal tensile stress equal to 0.35 MPa. The consistency of the proposed FE models can be observed also at the opening edges (Fig. 9, view 7), where the principal directions (Fig. 9, views A and B) allow to identify the same crack propagations observed for the real church.

The analysis results concerning the back walls of the church allows to notice principal tensile stresses of about 0.31 MPa, in particular at the apse window, where many cracks have been surveyed on the real church (Fig. 10, view 3). Moreover, at the circular external wall (Fig. 10, view 1), a significant stress is evident, consistently with the incipient activation of the apse overturning. Both the principal stress values (0.43 MPa) and directions on the drum panels (Fig. 10, view A) justify the cracks shown in Fig. 10b, while the principal tensile stresses at the basis of the bell tower highlight failures due to overturning bending moments (Fig. 10, view B).

Fig. 10
figure 10

Comparison between FEM model and real church: response of the back walls

Full size image

It is also important to observe that the FEM model reproduces the main failures occurred at the connection points of the parts of the church characterized by geometric irregularities, as shown in Fig. 11. Moreover, a significant stress concentration arose in the parts that were more involved, during the earthquake, by kinematic mechanisms, namely the façade and the transept, as it is still depicted in Fig. 11. This means that the model is able to give information on the possible activation of the above mechanisms, even though it is not able to reproduce their development.

Fig. 11
figure 11

Effects of geometric irregularities

Full size image

Figure 12 evidences the peak stresses at the connection between the nave and the transept walls (view 1) and on the left aisle wall (view 3), where many diagonal cracks have been observed.

Fig. 12
figure 12

Comparison between FEM model and real church: response of the left aisle external walls

Full size image

Similarly, stress concentrations are evident on the external walls of the right aisle (see Fig. 13, view 4), evidently due to the forces transmitted by the reinforced concrete roof. Moreover, the stress value of about 0.20 MPa revealed in the middle of the wall is consistent with the punching phenomena of one of the tie anchor plate observed in the immediate aftermath of the earthquake. Also, a tensile values of 0.32 MPa has been detected at the spandrel over the sacristy architrave (view 2), consistently with the cracks shown in Fig. 13.

Fig. 13
figure 13

Comparison between FEM model and real church: response of the right aisle external walls

Full size image

In Fig. 14 the stress distribution in the transversal and longitudinal arches, as well as in the vaults are highlighted. Maximum tensile stresses resulted equal to 0.57 MPa at the nave (view 5) and 0.44–0.55 MPa at the transept (view 3) under the dome. They are consistent with the diagonal cracks in the transept vaults and the transversal crack of the vault of the nave at the connection with the main arches. With reference to the aisle triumphal arches, tensile values raging between 0.42 and 0.78 MPa (views 1 and 2) have been detected, which are compatible with the deep cracks at the keystone location. Similarly, tensile values of about 0.40 MPa (view 6) have been noticed at the barrel vaults in correspondence with the main façade, consistently with the surveyed damage.

Fig. 14
figure 14

Comparison between FEM model and real church: response of arches and vaults

Full size image

4 Conclusions

In this paper a FEM numerical model of the St. Gemma Church in Goriano Sicoli has been used with the purpose of interpreting the main fragilities that this church, which is representative of a wide population of buildings belonging to the same territorial area, evidenced during the 2009 L’Aquila earthquake.

The model adopted, which has been calibrated by means of ambient vibration tests, has been used in order to carry out a linear dynamic analysis. Such an analysis allowed to detect the main parts of the church characterized by stress concentrations, namely the arch-vault systems, the edges of the openings, the points of geometric irregularities, the bell tower base and the columns. Such results appear to be in a very good agreement with the damage experienced by the real church under earthquake loads. The results of the developed analysis allowed to confirm that the vertical component of the earthquake played a significant role in damage occurred, especially in relation to the columns and vaults. Also, it has been pointed out that some vulnerabilities of the church have been even sharpened by the erroneous application of some interventions after past earthquakes, such as the reinforced concrete roofs placed on the top of the lateral naves. In addition, it has been highlighted that the revealed lateral wall mechanism has been favoured by the ineffectiveness at local level of some iron ties.

The obtained results, together with the ones presented in the companion paper (Criber et al. 2013), have to be intended as a useful guidance for future studies at regional scale that the author are planning to implement, which are directed to assess the seismic vulnerability of a population of churches having the same typology and belonging to the same territorial area of the one considered in the current paper.