Stress Analysis of San Vitale's Basilica in Ravenna: Current State and Mid

Structural Analysis of Historical Constructions, New Delhi 2006 P.B. Lourenço, P. Roca, C. Modena, S. Agrawal (Eds.) Stress Analysis of San Vitale’s Basilica in Ravenna: Current State and Mid-term Predictions Alberto Taliercio and Luigia Binda Politecnico di Milano, Department of Structural Engineering, Milan, Italy ABSTRACT: A finite element model was developed to analyze the Basilica of San Vitale in Ravenna, a Byzantine building which suffers diffused cracking and excessive deformation. In the structural analyses account was taken of permanent loads and ground settlements increasing in time. The tensile stresses predicted by a linear elastic stress analysis agree with most of the observed cracks. Assuming the ground settlements to increase at the currently estimated rate, the stresses in several parts of the Basilica might seriously endanger the stability of the building during the present century. 1 PREFACE The Byzantine Basilica of San Vitale in Ravenna (Italy) dates back to the VI century AD; it is one of the most important early Christian monuments in Italy, especially renowned for the beauty of its mosaics. A plan of the present-day building is shown in Fig. 1. Most of the walls, the vaults and the pillars consist of brick masonry with thick mortar joints. The walls are solid, except for a dome lantern, which was found to consist of three leaves. The dome is made of spindle-shaped fictile tubes embedded in mortar (see Mirabella-Roberti et al. 1995 for details). The columns are made of marble. Most of the building is decked by timber roofs, topped by ei- ther tiles or lead slabs. The church as it appears today (see Fig. 2) is the outcome of a number of extensions, demoli- tions and restorations: for a detailed description of the changes in the layout of the building during the centuries and the strengthening works carried out in the past century, readers are referred to Binda et al. (1995). Between the XIX and the XX century, massive interventions were done with the aim of restoring "the original lines" of the building, including the demolition of several chapels. These works, along with the settlements of the ground, moist and soft, where Ravenna stands, are likely to have strongly affected the statics of the building, which was first found to be in danger at the beginning of the XX century. Accordingly, steel rods were inserted between several pairs of pillars, and some parts of the Basilica were restored or reconstructed. Although these restoration works brought some relief to the building, the Basilica still has to be constantly monitored to detect its movements and prevent possible new faults. In 1998 a monitoring system was installed to survey the differential displacements at a number of benchmarks located both inside and outside the Basilica: further details on the monitoring network can be found in Mirabella-Roberti and Guzzetti (2001). Comparing the measurements in No- vember 2003 and November 1998, the ground settlements below the Basilica turn out to be un- even: in particular, the differential displacements of the benchmarks in the apsidal zone were found to range between −0.05 and −0.3 mm, whereas they increased up to −0.9 mm on the op- posite side. This indicates that, on the whole, the building is rotating north-westwards. The present tilt of the Basilica confirms this trend, as a difference in height exists at the ground level 2006 Structural Analysis of Historical Constructions between the two parts of the church of about 12 cm, rising toward the apse. The average subsi- dence of the building is of the order of 0.1 mm per year. In this paper, the results are shown of structural analyses, based on a finite element model of the Basilica, with the aim of assessing the short- and mid-term safety of the building. Account was taken of permanent loads (self-weight) and boundary displacements increasing in time. TS PE4 PE3 S3 S2 PE2 S1 S4 P3 PE5 P4 P2 P5 P1 P6 P8 S5 P7 S7 S6 TC PE6 PE7 Figure 1 : Present plan of the Basilica; left: ground floor; right: women’s gallery floor. 2 NUMERICAL MODEL Thanks to available drawings (Deichmann 1969-1976), to previous topographical surveys of part of the building (Binda et al. 1995), and to chemical and mechanical investigations (Binda & Baronio 1996, Binda et al. 1999), the geometry of the Basilica and the main physical properties of the materials are reasonably well defined. The finite element model developed does not virtu- ally neglect any structural element and accounts for the lack of symmetries in the building. The model is shown in Fig. 3: it consists of 277411 ten-noded, bilinear strain tetrahedra, with a total of 4923037 nodes. The finite elements employed were preferred to constant stress, four-noded tetrahedra, to allow for the possibility that pillars and walls are subjected to both in-plane com- pression and bending, which would make the vertical stresses vary linearly across the wall thickness. They also allow curved surfaces to be modelled with more accuracy than with flat- sided finite elements. The size of the elements was chosen so as to have two elements across the wall thickness to accommodate the niches existing in several walls: thus, the average size of the elements is about 60 cm. Because of the complexity of the geometric model, a simplified (linearly elastic, isotropic) constitutive law had to be assumed to keep the computing time within reasonable limits. In the present study, all the materials forming the Basilica were supposed to be linearly elastic and isotropic. This is a rough approximation for brick masonry and fictile tubes, which are macroscopi- cally orthotropic, but taking anisotropy into account would make the numerical model even more cumbersome. The presence of the few multi-leaf walls was disregarded, so that all the walls are supposed to be homogeneous. The numerical model of the Basilica was created using a commercial finite element code (ABAQUS®, vers. 6.4). The timber roofs were not modelled, but rather indirectly taken into account as dead load acting on the model. The same applies to the upper part of the bell-tower (TC in Fig. 1), about 20 m high, which enters the model as a uniform load acting upon the lower, discretized part. The brickwork filling a number of arches in the Basilica was disregarded and only the bearing contribution of the arches was taken into account. According to (Deichmann 1969-1976), an infill was inserted between drum and dome up to 30 cm above the arches; in the Alberto Taliercio and Luigia Binda 2007 analyses, the loose infill was given an elastic modulus equal to 1/10 of that of the surrounding brick masonry and a slightly lower unit weight. Figure 2 : View of the Basilica of San Vitale Figure 3 : View of the finite element model of the Basilica. (courtesy of Mrs Anna Pasta). The macroscopic elastic modulus of brick masonry was estimated through a double flat-jack tests performed in the outside wall. The unit weight was assessed from tests on samples of masonry taken from different locations. The elastic modulus of the dome masonry along the paral- lels was computed as a weighted average of the moduli of mortar (500 to 5000 MPa) and fictile tubes (10000 and 20000 MPa). The remaining thermomechanical properties of the different materials, including marble, were taken from the literature (Lenczner 1972, Stagg and Zienkiewicz 1968). In particular, the materials, excluding marble, were given the same coefficient of thermal expansion, which is an average of the values found in the literature for masonry in different directions (Lenczner 1972). The thermomechanical properties employed in the numerical analyses are summarized in Ta- ble 1. The numerical model allows also for the steel rods inserted between four pairs of pillars (P2- PE2, P3-PE3, P5-PE5 and P7-PE7 in Fig. 2). The rods were given a diameter of 2 cm and an elastic modulus of 200 GPa. Table 1 : Material properties employed in the finite element analyses. brick ma- dome masonry Infill sonry marble elastic modulus, MPa 1800 180 2600 70000 Poisson’s ratio 0.1 0.2 0.2 0.3 density, kg/m3 1600 1400 1450 2750 coefficient of thermal expansion, °K−1 6e-6 4e-6 3 NUMERICAL RESULTS 3.1 Study of the effects of the dead loads The first finite element analysis performed allows only for the permanent loads affecting the structure. The self-weight of the discretized structural elements was directly taken into account as a body force (see Table I for the values of the densities employed). The weights of the non- discretized parts of the building were converted into uniform pressures acting upon the elements underneath. In the present analysis, the finite element model is perfectly constrained to the ground and no kinematic constraint is imposed between the finite element model and the surrounding build- ings. 2008 Structural Analysis of Historical Constructions The main results of the f.e. analysis are summarized in Figs. 4 and 5, which show contour plots of the maximum (tensile) principal stress in the building compared with the surveyed crack pattern in the Basilica. Tensile stresses are mainly localized in the vaults, the highest values (about 0.25 − 0.30 N/mm2) being found at the keystones of the exedras. At the ground floor, the most important cracks are found at the keystone in sectors S2, S3, S4 and S7 (refer to Fig. 2); minor cracks exist in the lunettes. At the level of the women’s gallery (Fig. 5), the crack pattern is more widespread than at the ground floor (Fig.

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