Structural Analysis of the Dome of San Cerbone Cathedral in Massa Marittima (Italy)

Home , Clerestory

SAHC2014 – 9th International Conference on Structural Analysis of Historical Constructions F. Peña & M. Chávez (eds.) Mexico City, Mexico, 14–17 October 2014

STRUCTURAL ANALYSIS OF THE DOME OF SAN CERBONE CATHEDRAL IN MASSA MARITTIMA (ITALY)

G. Angelini 1, A. De Falco 2 and D. Pellegrini 3

1 University Centre "Logistics Systems", University of Pisa, Italy Villa Letizia, via dei Pensieri, 60 - Livorno [email protected]

2 Department of Energy, Systems, Territory and Constructions Engineering, University of Pisa largo Lucio Lazzarino, 1 - 56126 Pisa, Italy [email protected]

3 Institute of Information Science and Technologies "A. Faedo" ISTI-CNR, Pisa via G. Moruzzi 1, 56124 Pisa, Italy [email protected]

Keywords: Masonry dome, non-linear elasticity, masonry-like material, numerical methods.

Abstract. In this paper the assessment of the dome of the medieval cathedral of Massa Marit- tima (Italy) is carried out in order to evaluate the causes of its heavy damage. The deformation and the crack patterns on the vault and on the surface of the masonry drum are particularly complex and seem at first glance incomprehensible. In the context of the safety assessment of ancient masonry constructions, structural modeling can provide an important contribution to diagnose the damage while representing at the same time a valuable instrument to predict the effectiveness of consolidation interventions. This paper shows how the finite element numerical approach can be successfully employed as an assessment tool, provided that it is driven by the empirical intuitive method based on kinematic evaluations. The analysis has been conducted via the NOSA-ITACA code, developed in the framework of the project “NOSA-ITACA - Tools for modelling and assessing the structural behavior of ancient constructions” (2011-2013) promoted by the Tuscany Region (Italy). The code models masonry as a nonlinear elastic material, known as masonry-like (or no-tension) material, and is a code suitable to assess the structural behavior of masonry constructions. Initially, the different structural elements composing the dome were investigated to analyze their specific role and their typical pathology. The structure safety factor was successfully evaluated thanks to a detailed three-dimensional model of the dome with its substructure. Once validated the model in the light of the surveyed crack patterns, was also employed to design adequate strengthening interventions. For the sake of comparison, a commercial code was also used to perform the same analysis with a different constitutive equation for material thus offering the opportunity to highlight the peculiarity of NOSA-ITACA code. G. Angelini, A. De Falco and D. Pellegrini

1 INTRODUCTION Conservation and valorization of the historic architectural heritage in Italy constitute a top- ic of great relevance. Many historic masonry buildings are in dire need of restoration and their preservation often requires the study of their behavior under static and seismic loads. The safety assessment of ancient masonry buildings, however, is a particularly complex task. On one hand, there is limited knowledge on the internal morphology of the constructive elements, the material characteristics and assembly techniques. On the other hand, the significance of calculation schemes which must necessarily be very simplified is quite low. For this reason it is essential for the structural modeling to be driven by the engineer's intuition of what the static behavior of the construction might be, as much as by actual experience. In this context, the structural modeling may represent a valuable diagnostic tool. In particular, numerical codes play a crucial role by providing important information on the stress field, the distribution of cracked regions and their possible evolution. They may also constitute a valid support in choosing and designing both the seismic retrofitting and the strengthening of masonry buildings. This paper presents the structural analysis of the dome of Massa Marittima medieval cathedral in order to investigate the cause of its complex crack pattern. The Latin cross shaped cathedral, built in the XIII century on a travertine rock layer, is considered one of the major monuments in the South of Tuscany (Figure 1a). The global height is approximately 25 m, 15 m up to the summit of the clerestory walls. The vertical structures are made of stone masonry, as well as the cross vaults of the secondary aisles, the chapels and the apse, whereas the vaults of the principal nave and those of the presbitery are made of clay brick and lime mortar, as well as the dome. The dome, located at the cross of the church, at the end of the main nave, was once attributed to the XV th century by architecture historians. However recent archaeological investigations show that the dome actually belongs to the church implant of the XIII th century. A serious concern for the dome safety is a direct consequence of the crack pattern which is a rather diffused cracking along meridian lines, much more pronounced in the drum. The structural analysis was conducted via a special finite element code which models the masonry as nonlinear elastic material, known as "masonry-like" (or no-tension) material. The structure safety factor assessment was performed thanks to a detailed three-dimensional model that, once calibrated according to the surveyed crack patterns, was also employed for the design of appropriate reinforcement interventions. The additional study carried out by modeling the masonry as Drucker-Prager material via a commercial code confirms the validity of the analysis performed assuming the "masonry-like" model.

2 THE CRACK PATTERN IN THE DOME The dome consists of a brick masonry octagonal shaped cloister vault of about 7 m in height, surrounded by blind arcades and set at 16 m above the ground on a masonry drum in- scribed in a rectangle of 10 x 11.20 m (Figure 1b). Four trumpet squinches provide the con- nection with the substructure which is composed of the longitudinal arcades of the main nave and two transverse masonry arches (Figure 2a). The arch located at the front of the church is made of plastered brick masonry and is equipped by a chain, whereas the other, located above the presbitery, is made of travertine blocks with no plaster. The crack pattern mainly affects the supporting arches, the trumpet squinches and the drum. Inside the church, four dense bundles of hairline cracks, which spring from the base of the squinches (Figure 2b), go up on the oblique faces of the dome following the direction of some

2 Structural analysis of the dome of San Cerbone Cathedral in Massa Marittima (Italy) of the vault edges (Figure 3). These fissures also appear on the outside of the church on the external drum surface as few but important cracks and they continue on the tiburium within the blind arcades (Figure 4a). On the inner drum face which is above the transversal brick masonry arch, a series of cracks runs horizontally and enters the arch ring at the haunch on the left side of the church by bending toward the ends (Figure 3a). Figure 3a also shows the opposite face of the drum, where a heavy crack pattern with a parabolic trend is clearly visible.

a b

Figure 1 a) The Cathedral of Massa Marittima. b) The dome.

a b Figure 2. a) Cross section of the church in the axis of the dome and pictures of the two transversal arches. b) A trumpet squinch.

The travertine arch instead shows clear separations between the blocks at the haunches on the extrados, as shown in Figure 3, while a crisis in compression is visible at the arch intrados on the right side of the church (Figure 4b).

3 G. Angelini, A. De Falco and D. Pellegrini

The central issue is now the origin of the cracks and whether the crack pattern affects the safety of the structure, which requires the causes of cracking to be identified or, at least, to be hypothesized. Material degradation, due to environmental actions, is another phenomenon which also requires the assessment of the dome to be performed.

Figure 3. a) Cross section in the dome axis (view towards the front of the church) with the opposite face of the drum. b) Crack pattern on the dome intrados. c) Cross section in the dome axis (view towards the apse) with the opposite face of the drum.

What follows firstly presents the safety assessment of the dome and drum system through an historical survey and the observation of structural deformations. Secondly, the numerical modeling performed via a finite element code is carried out to confirm the hypotheses empiri- cally deduced.

3 THE SAFETY ASSESSMENT OF THE DOME

3.1 Historical survey Sporadic archival documents denonunce that the dome has always shown several weak- nesses over the centuries. According to Petrocchi [1], the dome suffered instability since 1463. It is not clear what kind of problems had occurred, whether they involved the roof structure or that of the masonry dome. Hypothesis on the roofing system are made using a sequence of old photographs (Figure 5). It is clear that before 1898 the dome was onion-shaped with different covering systems, whereas from 1930 it was covered by planar faces. In 1930 a masonry wall was erected over the blind arcades cornice, in order to support the new pitched roof with plane faces and to correct the difference in level of the existing construction (more than 10 cm). In 1938 the dome intrados damage was partially repaired, but already in 1947 new local reconstructions and cement injections were needed. The structural consolidation, aimed at stopping the slow but steady increase of the cracking, was approved in 1991, within a larger restoration project planned for the whole Cathedral monumental complex. However, never there was a proper intervention to eliminate the causes of the dome instability which, therefore, have never been clearly identified. From the historical evidence, supported by various archaeological and geometrical surveys, it clearly appears that the dome has not experienced important collapses, while it has surely suffered damage to its coverage wooden structure. However, even excluding the case of suc-

4 Structural analysis of the dome of San Cerbone Cathedral in Massa Marittima (Italy) cessive reconstructions of the dome with a different shape from the original one, further at- tempts to intervene on the cracking may have occurred, as demonstrated by the rectangular niche on the tiburium North-West side which is depicted in the photograph of Figure 5, dated back to the end of the XIX th century. Indeed, the rectilinear lintel cannot have had any function other than that of simply repairing a local damage.

a b Figure 4. a) Crack on the extrados of the drum oblique West side, close to the edge. b) Crack pattern on the travertine arch: backside and front side at the arch right haunch.

Figure 5. The dome before 1898 to date. Left: the first onion-shaped dome (1898photo). Centre: the successive onion-shaped dome (1914photo). Right: the actual dome after the restoration of the ’30 (2013photo).

3.2 Survey of the structural deformations The shape of the dome surface and its substructure were carefully reconstructed on the basis of a laser scan survey. Figure 6 shows the geometric development of the external surfaces in the North-West area, obtained by photogrammetric techniques. In particular, it can be noted that the tops of both the tiburium and the drum cornices are significantly different in height on the different sides. The bed joints exhibit the lowest altitude on the central face, which is located above the brick masonry arch, while the same phenomenon is repeated also on the opposite side of the dome. It can also be observed that the rows affected by the cracks present a sharp slope increase. This is well demonstrated by the trapezoidal shape of the masonry wall that was built in 1930 to sustain the roof (Figure 6). The measurement of the pillars height and of their overhangs, however, excludes the presence of soil settlements. The intrados shape of the brick masonry arch deviates from that ideally circular of about 15 cm, found in correspondence of the keystone. Moreover, the lines of the bed joints of the drum, furrowed by deep inclined cracks with convergent trend, present a notable depression that reaches 10 cm in the middle of the rows. The lowering of the arch profile determined an evident general slope of the dome towards the aisle and the consequent cracks in the drum

5 G. Angelini, A. De Falco and D. Pellegrini oblique sides. The walls of the clerestory also show deformation outwards in correspondence of the arch. This situation is currently not yet stabilized as clearly shown by the fall of little glass spies installed on the cracks of the drum during the 80's and by the opening of new small cracks on the lime mortar layer used to glue them (Figure 7). Finally, a serious material failure has been highlighted by removing the plaster from a small portion of the arch ring, where sub vertical cracks were present, as shown in Figure 8.

Figure 6. Geometric development of the dome outer surface in the North-West area: rectified and geo-referenced images.

On the basis of the above considerations it is now possible to formulate hypotheses about the causes of the damage. Excluding that the dome was rebuilt in ancient times, but accepting the possibility of its partial collapse, it can be argued that the crack pattern is very old and it is due to the brick masonry arch. The latter in fact is not restrained laterally and made up of low resistant and very deformable materials. The subvertical cracks on the trumpet squinches and on the oblique sides of the drum are a consequence of the dome transversal supports deformability which is higher than the one of the longitudinal supports. The travertine arch, however, is less deformable than the other and more effectively restrained by the presence of the bell tower on one side and the rectory building on the other side. For these reasons the travertine arch does not exhibit the same damage as the brick masonry arch, although showing local crushing at the haunches on the right side. The trend of the crack pattern on the masonry arch ring and of the overlying drum side suggests the stress distribution depicted in Figure 8 which actually determined the local material crushing. The damaged material is currently no longer capable of sustaining the loads in a safe way, while the only device left to prevent the collapse of the whole structure is the iron tie which was applied over a century ago. The tie rod is subjected to a considerable effort, judging by the response to percussions, and it cannot be inspected at the ends within the masonry, where corrosion may have already compromised the bearing capacity of the device. The above mentioned phenomenon, originally caused by improper proportioning of the arch, was probably exacerbated with time due to materials ageing and is currently evolving, despite the presence of the iron tie. This consequently arises the issue of material degradation. These assumptions have been validated through the finite element modeling.

6 Structural analysis of the dome of San Cerbone Cathedral in Massa Marittima (Italy)

Figure 7. Scheme of the brick masonry arch on the side towards the church front with the overlying damaged drum. The zoom shows trace of a spy glass with a crack on the mounting material.

Figure 8. Scheme of the brick masonry arch on the side towards the apse with an indication of the possible stress pattern. The zoom shows a part of the arch ring with numerous cracks.

4 THE STRUCTURAL MODELING

4.1 The numerical modeling A masonry peculiar characteristic is the low tensile strength, so, masonry can be assumed to be a no-tensile-resistant material. The equilibrium is required to be satisfied by admissible stress fields, which imply pure compression everywhere and compatibility of the strain field is assured by superposing an additional fracture field to the elastic strain field. The relation between the stress state and the possible active fracture strain can be set using a fracture-law which is analogous to the associated flow-law in plasticity. In this case the "masonry-like" model was adopted, therefore assuming a little tensile strength and a linear elastic behavior under compression. The analysis was performed via the finite element code NOSA-ITACA [2]. It is a freeware software for nonlinear analyses created by the MMS Lab of the Italian National Research Council of Pisa (Italy) and equipped with the open source graphics platform SALOME [3], for the pre- and post-processing. The code was also developed within the framework of the project “NOSA-ITACA - Tools for modeling and assessing the structural behavior of ancient constructions” funded by the Re- gion of Tuscany (2011-2013). It has been successfully applied to a number of studies on existing buildings [4] and [5].

7 G. Angelini, A. De Falco and D. Pellegrini

Through the analyses performed it was possible to calculate the stress and the fracture fields and therefore assess the structure’s safety. The results are also reported in terms of the line of thrust, which allows the graphic evaluation of the static safety [6]. The material characteristics which are assumed in the following are listed in table 1. Table 1: mechanical properties for masonry

Elastic modulus, Unit weight, Poisson Masonry type E [MPa] γ [kN/m 3] ratio, ν Brickwork (dome) 1500 18 0.2 Three leaf stone masonry (columns) 2000 22 0.2 Travertine blocks (posterior arch) 10000 24 0.2

Static problems are solved by using quadrilateral four-node shell elements based on the Love-Kirchhoff hypothesis. Moreover, in every model only the material self-weight is considered, being by far the most relevant load in masonry structures and also the essential factor to secure stability.

4.2 The submodels In order to assess the static behavior both of the dome and of its supporting structures, simple calculation models with different complexity were developed. They allowed assessing, in the first place, the function of the structural elements and, thereafter, the causes of the damage. Analyses were carried out initially by modeling the architectural elements above the drum base in three turns: i) the vault alone, ii) the vault with the drum and iii) the vault with the drum and the tiburium. In each case, the tensile strength value required to achieve the convergence is null, the maximum compressive stress does not overcome 0.6 MPa and the horizontal thrust is lower than 15 kN. Figure 9 shows the dome complete model, restrained by hinges at its base, with the fracture map and the distribution of the minimum principal stresses on the extrados surface.

Figure 9. Left: FEM model of the dome simply supported at the base. Centre: fracture map. Right: distribution of the minimum principal stresses on the extrados [daN/m 2].

The result obtained is an admissible stress state merely using compressions, which certain- ly does not violate the limit condition for material, whatever its actual tensile strength. More- over analyses highlighted that the tiburium blind arcades play the buttress role for the dome. It is noteworthy the excellent proportioning of the structure. The implemented models do not justify the damage detected when the base supports are rigid. In this case, the cracks presence at the edges (Figure 9) has no pathological significance.

8 Structural analysis of the dome of San Cerbone Cathedral in Massa Marittima (Italy)

In order to evaluate the load bearing capacity of the supporting structure, the transversal arches were studied separately. Two plane models were loaded with the overlying dome, thus neglecting its stiffness and strength contribution. For the sole posterior arch, the presence of adjacent structures was simulated via horizontal restraints on the whole height (Figure 10). This simplified analysis, which does not take into account the interaction with the overlying structure, shows that only the travertine arch is capable to survive, thanks to the effective con- trast of the adjacent structures, and its material good quality. On the contrary, the equilibrium of the brick masonry arch requires a minimum tensile strength value of 0.5 MPa, which is too high for any masonry type. This therefore demonstrates that the damage surveyed is due to the bad proportioning of the brick masonry arch that cannot survive without the structural contribution of the dome.

Figure 10. FEM models of the transversal substructures. Left: posterior travertine arch. Centre: brick masonry arch. Right: brick masonry arch deformed.

4.3 The global model The dome with its substructure was thus modeled using a three-dimensional mesh composed with 31682 elements (Figure 11a). Displacements of the base nodes were restrained in all directions, whereas only the horizontal displacements were restrained for the lateral nodes, in order to take into account the presence of the adjacent structural elements. The model, which does not include the iron tie of the brick masonry arch, reached the convergence with zero tensile strength of the dome material and with 0.01 MPa tensile strength of the substructure material. The results are reported in Figure 11. It is noteworthy that the cracks distribution obtained from the FEM model matches the actual crack pattern detected on the edges of the dome, in the wall above the brick masonry arch and close to the edge of the oblique sides of the drum (Figure 11b). In particular, the circumferential fracture strain on the extrados of the brick masonry arch ring (Figure 11d) is actually present (Figure 11e). Regarding the stress level, the model highlights stress concentrations on the intrados at the haunches of both arch rings. It also shows heavy local gradients and average compression values over 1.5 MPa for the uncracked section of the brick masonry arch and about 14 MPa for the uncracked section of the posterior arch (Figure 11c and 11f). The reported values widely justify the stress peaks which are able to produce the local material failure. The original representation of the line of thrust on the two arch rings (red lines in Figure 12) suggests the idea of the stress pattern assumed in Figure 8. Indeed, on the base of the obtained thrust values, it can be observed that the greatest load affects the ring haunches. The phenomenon is more pronounced for the brick masonry arch, where the internal load starting from about 250 kN proceeds upwards, following the red dashed lines. This can be clearly deduced by observing the variation of normal force that vanishes in the middle of the arch ring (Figure 12a). This also confirms the previous considerations on the substructure failure to find an equilibrium without the contribution of the overlying structures.

9 G. Angelini, A. De Falco and D. Pellegrini

a b

c d

f g

Figure 11. FEM global model. a) The mesh. b) Fracture strain pattern on the dome extrados. c) and d) Minimum principal stress and fracture strain, respectively, on the brick masonry arch side. e) Circumferential fracture on the extrados of the brick masonry arch ring. f) and g) Minimum principal stress and fracture strain, respectively, on the travertine arch side.

It must be noticed that the model is able to reproduce the actual structure behavior, so, once calibrated in sight of the actual damage, it was also employed in order to predict the effectiveness of strengthening interventions. An example of this is the stabilizing intervention

10 Structural analysis of the dome of San Cerbone Cathedral in Massa Marittima (Italy) with prestressed tie rods on both arches, aimed at mitigating the compression level at the intrados and at recentering the thrust line at the arch haunches. In particular, once the original global model has reached the convergence under its own weight, horizontal self-equilibrated loads were applied at the arch haunches. The most favorable situation for both arches may be achieved by applying a prestress load of about 180 kN at a distance of about 1.60 m from the keystone intrados. Figure 12 highlights the advantages of the prestressed tie rods, especially with regard to the brick masonry arch, as it can be deduced by observing the blu dashed line and the blu values.

a b Figure 12. The thrust lines on the arch rings: a) brick masonry arch; b) travertine arch. The red line is the current thrust line, the blue dashed line is the one in presence of prestressed tie-rods.

Finally, the model equipped by the iron tie to the brick masonry arch was also studied. One can conclude that maybe if a tie rod had been applied from the beginning of the construction, the damage could have been avoided. In order to emphasize the good result obtained from the assumption of no-tension material, the structure was modeled also as Drucker-Prager material. Friction and cohesion values for which the convergence of the model was obtained correspond to the material strength values listed in table 2. The results are shown in Figure 13 in the form of distribution of minimum principal stress and plastic strain. Table 2: mechanical properties assumed for Drucker-Prager model

Compressive Tensile Cohesion, c Friction coef- Masonry type strength, σc strength, σt [MPa] ficient, ϕ [MPa] [MPa] [°] Brickwork (dome) 2.5 0.08 0.180 73° Three leaf stone masonry (columns) 3.0 0.08 0.205 75° Travertine blocks (posterior arch) 10.0 0.08 0.360 81°

The comparison with the previously obtained results shows that the stress distribution on the arch rings has a certain correspondence with what presented in Figure 11, with regard to both i) stress concentrations at the arch haunches and ii) tensile stress on the drum over the keystone (Figure 13a). Nevertheless, the plastic strain distribution does not reproduce at all the detected crack pattern (Figure 13b).

5 CONCLUSIONS The mechanical response of masonry domes is a challenging issue also for modern engi- neers due to several reasons. In the first place, the modeling results can be very sensitive to the accuracy of the three-dimensional geometry that must be carefully measured. Moreover, the masonry exhibits a non-isotropic and non-linear response that strictly depends also on the units arrangement which essentially remains unknown despite the archaeological surveys. In

11 G. Angelini, A. De Falco and D. Pellegrini addition, constitutive models are still research issues and the mechanical characterization of the materials is in general almost impossible in monumental structures. On the other hand, given that one of the masonry peculiar characteristic is the low tensile strength, then the "masonry-like" material, or no-tension material, which is implemented in the finite element code NOSA-ITACA, is able to reproduce many typical aspects of masonry structures particularly the vaulted ones.

a b Figure 13. Drucker-Prager model - view of the brick masonry arch side. a) Distribution of minimum principal stresses. b) Distribution of plastic strains.

In this case study, the numerical tool was able to provide significant information, although only considering global quantities. In particular, it allowed to: i) recognize the origin of the crack pattern, due to deformability of the arches in which the drum is set; ii) appreciate the order of magnitude of the internal loads and iii) estimate the residual safety of the structure. Finally last but not least outcome of this paper is the quantitative esteem of the stabilizing role of new iron chains placed in transversal arches in view of the strengthening intervention.

ACKNOWLEDGEMENTS The Region of Tuscany is gratefully acknowledged for support activities.

REFERENCES [1] L. Petrocchi, Massa Marittima: arte e storia , Venturi, Firenze, 1900. [2] http://www.nosaitaca.it/en [3] http://www.salome-platform.org/ [4] M. Lucchesi, C. Padovani, G. Pasquinelli, N. Zani, Masonry constructions: mechanical models and numerical applications - Lecture Notes in Applied and Computational Me- chanics, Vol. 39 . Springer-Verlag, Berlin Heidelberg, 2008. [5] V. Binante, S. Briccoli Bati, M. Girardi, M. Lucchesi, C. Padovani, D. Pellegrini, A case study for the NOSA-ITACA Project: the "Voltone" in Livorno. B.H.V. Topping, P. Iványi eds. 14 th Intern. Conf. on Civil, Structural and Environmental Engineering Com- puting, Civil-Comp Press, Stirlingshire, UK, Paper 79, 2013. [6] J. Heyman, The masonry arch , J. Wiley &Sons, 1982.