Structural Analysis of Historical Constructions, New Delhi 2006P.B. Lourenço, P. Roca, C. Modena, S. Agrawal (Eds.)

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 dur-ing 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 bench-marks 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 pre-sent tilt of the Basilica confirms this trend, as a difference in height exists at the ground level

Stress Analysis of San Vitale’s Basilica in Ravenna: CurrentState 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.

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.

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 iso-tropic. 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 ac-count 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

P1 P2 P3 P4

P5 P6 P7 P8

PE2

PE3 PE4

PE5

PE6 PE7

TS

TC

S1

S2 S3

S4

S5

S6 S7

2006 Structural Analysis of Historical Constructions

Alberto Taliercio and Luigia Binda

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 (courtesy of Mrs Anna Pasta).

Figure 3 : View of the finite element model of the Basilica.

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 ma-sonry 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 mate-rials, excluding marble, were given the same coefficient of thermal expansion, which is an aver-age 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-sonry Infill

dome ma-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.

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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. 4). The zones subjected to tension are wide-spread along the entire keystones, the maximum principal stress attaining values of the order of 0.03 to 0.07 N/mm2 in most of the vaults. As the tensile strength of masonry can be approxi-mately assumed to be 0.20 N/mm2 (Lourenço 2002), the intensity and the diffusion of the tensile stresses induced by the self-weight alone suffice to explain the existing cracks.

The compressive stress attains values around of −0.4 MPa in the lower part of the building, except the columns; in the bell-tower, higher values are encountered (−0.65 MPa at the base). The walls are found to be subjected to in-plane compression and out-of-plane bending: the stresses at the base are of the order of −0.2 N/mm2 at the inner side and of −0.3 / −0.4 N/mm2 at the outer side, raising to −0.45 / −0.55 N/mm2 in the outer pillars and the pilasters. The com-puted values are far below the compressive strength of brick masonry, which can be estimated to be about 4 N/mm2. The same applies to the inner columns: here, compressions of the order of −6.8 / −7.0 N/mm2 are encountered at the ground floor and of −3.5 N/mm2 at the level of the women’s gallery. These values are much lower than the compressive strength of marble, which is around 60 / 100 N/mm2 according to the literature (Stagg and Zienkiewicz 1968). Also the columns are subjected to compression and bending, so that tensile stresses are found in some pedestals. According to the computations, the force exerted by the steel rods, inserted at the be-ginning of the XX century, is extremely low.

Figure 4 : FE analysis under the dead loads. Left: contour plots of the maximum (tensile) principal stress at the ground floor (view from below, values in kN/m2). Right: crack pattern in the vaults of the ambula-

tory at the ground floor.

3.2 Study of the effects of the ground settlements It has been pointed out in the Preface that San Vitale’s Basilica, as well as the entire region where Ravenna stands, is prone to subsidence and that this phenomenon is not uniform across the building. To investigate the effects of the differential settlements of the ground below the Basilica, a finite element analysis was performed by subdividing the building into six zones where settlements were found to be nearly uniform (see Fig. 6), and prescribing to the nodes at the base of each zone the average settlement surveyed by the benchmarks over five years (No-vember 1998 − November 2003). Zone no. 1 (including the apse, the north sacristy, sector S1, part of sector S2 and one of the buttresses) experiences the smaller settlements (around 0.10 mm); thus, it was taken as reference for the remaining zones and it was prescribed no displace-

Alberto Taliercio and Luigia Binda

ment in the numerical analysis. The values of the displacements δv prescribed to the remaining zones (positive downward) are listed in Fig. 6.

Figure 5 : FE analysis under the dead loads. Left: contour plots of the maximum (tensile) principal stress at the level of the women’s gallery (view from below, values in kN/m2). Right: crack pattern in the vaults

of the women’s gallery.

The results of the finite element analysis, taking only the effects of the differential settle-ments into account, are summarized in Fig. 7, where the contour plots of the maximum (tensile) principal stress in the vault of the ambulatory at the ground floor are shown. The tension peaks corresponding to the black areas may not be all meaningful, as some of them are located at the boundaries between zones undergoing uniform displacements and are strongly affected by the assumed discontinuity in the ground settlements. In the rest of the building, the highest increase in tension induced by the differential displacements occurs at the outer side of the intrados of the vault in sectors S4 and S5, and is of the order of 0.025N/mm2. Tensile stresses up to 0.018 N/mm2 are attained near the sacristy in sector S7. The same stress pattern is found at the women’s gallery level, with slightly lower tension peaks. In any case, the increase in stress in-duced by the differential displacements is quite low, so that the settlements cumulated over five years do not seem to have any detrimental effect upon the statics of the Basilica.

According to the continuously increasing subsidence of the ground below the Basilica, a nu-merical analysis was carried out with the aim of estimating the increase in stress over the next 100 years, assuming the rate of the ground settlements to be constant and equal to that moni-tored over the last five years. As the analysis is linearly elastic, the results obtained are purely an indication of the actual situation that will occur by the end of the current century. The values of the ground settlements prescribed to the F.E. model are piecewise constant, and equal to the values employed in the previous analysis (see Fig. 6) multiplied by twenty.

As pointed out in the Preface, the non-uniformity of the ground settlements gives rise to an overall tilt of the Basilica north-westwards. To take the rotation of the building into account, in the “mid-term analysis” the acceleration of gravity was also given horizontal components corre-sponding to a rotation of 0.27 deg in an ideal vertical plane passing through sectors S1 and S5 (Fig. 2), the latter moving downward respect to the former. This rotation was computed assum-ing a rise of 17 cm between the opposite sides of the Basilica, 36 m apart, somewhat higher than the current difference in level (i.e., 12 cm).

The results of the analysis are summarized in Figs. 8 and 9; they will be compared with the results presented in Par. 3.1, where only the permanent loads were taken into account.

In terms of compressive stresses (see Fig. 9), no significant change is obtained in the bell-tower, the narthex and the sacristies, whereas the situation in the main body of the Basilica is aggravated by the settlements. Especially, in one of the pilasters of sector S6 values around −0.90 / −1.20 N/mm2 are found. As a general remark, if settlements are disregarded the inner and outer pillars turn out to be more or less evenly loaded (with stresses around −0.25 / −0.40

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N/mm2 in the outer pillars, and around −0.30 / −0.45 N/mm2 in the inner ones); the differential settlements and the rotation of the building make the stresses in the pillars different, as they range between −0.30 / −0.50 N/mm2 in PE2 and PE3 to −0.60 / −0.75 N/mm2 in PE7, etc.). The obtained values are usually much lower than the estimated compressive strength of masonry (about 4 N/mm2), but are of an intensity which might activate a process of microcracking in some parts of the building. The compressive stress in the columns is also increased by the set-tlements (in some columns, of 50-60 % respect to the analysis with perfect restraints), and changes from column to column: the highest compressions are found in the columns close to P4 and P6 at the ground floor (17 N/mm2). These values are far below the compressive strength of marble (see Sec. 3.1). At present, however, cracks exist in some of the columns in the exedrae (Binda et al. 1995), whose origin is not clear according to the strength properties available for marble and the computed stress: this point deserves further investigations.

zone 1: δv = 0 mm zone 2: δv = 0.10 mm zone 3: δv = 0.30 mm

zone 4: δv = 0.35 mm zone 5: δv = 0.60 mm

Figure 6 : Subdivision of the boundary nodes at the base of the finite element model into five zones, un-dergoing uniform settlements.

For the tensile stresses, at the ground floor they are still localized at the key of the covering

vault of the ambulatory (see Fig. 8); the peak values of these stresses (0.08 to 0.16 N/mm2 at the key) are noticeably higher than the values obtained in Par. 3.1 where settlements were ne-glected, and are dangerously close to the estimated tensile strength of the material (0.15 / 0.20 N/mm2). Again, the even higher peaks in tension corresponding to the darkest areas in Fig. 9 are not meaningful, as they correspond to the discontinuities in settlements prescribed at the bound-ary of the model. At the level of the women’s gallery, the stress distribution matches that ob-tained at the ground floor, with tensions localized at the key of the vault (up to 0.12 / 0.15 N/mm2). These results match the crack pattern observed in the Basilica (see Par. 3.1). The ten-sile stresses at the keystones of the exedras, which attain dangerous values under the self-weight alone (0.25 to 0.30 N/mm2 − see Par. 3.1) are not significantly affected by the settlements. Fi-nally, note that also the tensile stresses at the key of the barrel vault in the narthex attain re-markable values (0.16 to 0.20 N/mm2) because of the settlements.

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Figure 7 : Contour plots of the maximum (tensile) principal stress (in kN/mm2) at the ground floor

induced by the differential displacements surveyed over five years (view from below).

Figure 8 : Contour plots of the maximum (tensile) principal stress (in kN/m2) induced by the self-weight and the estimated ground settlements in 100 years in

the vault of the ground floor (view from below).

4 CONCLUDING REMARKS

According to the finite element model developed, stress concentrations were detected in San Vitale Basilica, with peak values near the tensile strength of masonry, accounting for the dead loads only (Par. 3.1). These peaks are located in the neighbourhood of the measured cracks, mainly at the key of the vaults covering the deambulatory and the women’s gallery, indicating that the self-weight itself can explain most of the damage of the building.

The “mid-term” numerical analysis pointed out that, should the ground settlements increase at the presently estimated rate, the severity of the state of stress in several parts of the Basilica might seriously endanger the statics of the building during the current century (Par. 3.2).

According to the linear analyses presented, the steel rods inserted in the Basilica at the begin-ning of XX century turn out to be stressed only to a moderate extent. This is due to the fact that non-linearities (including cracking and material damage) are neglected in the analyses per-formed so far. The brittleness of the material and the crack propagation during the centuries, in-duced both by the sustained loads and the increasing settlements, justify the insertion of the iron chains.

The performed analyses constitute only a first step toward the understanding of the structural behaviour of the Basilica because of the adopted simplified constitutive law. In the prosecution of the research, single substructures will be individually modelled by prescribing boundary con-ditions allowing for the real interactions between adjacent parts of the Basilica using the find-ings of the present work. The reduced geometric complexity of the future models will allow an appropriate, more realistic, constitutive law for the materials to be taken into account, consider-ing the material brittleness and anisotropy, to help predicting past and future crack propagation.

ACKNOWLEDGMENTS

The authors are indebted to Mrs Anna Pasta, who developed the finite element model of the Ba-silica and performed the numerical analyses presented in the paper, and to Prof. Francesco Guzzetti who provided the measurements of the monitoring network. The assistance of the Monuments and Fine Arts Office of Ravenna in the research is also acknowledged. Most of the historical data were taken from the PhD Thesis of Dr Nora Lombardini, whose invaluable con-tribution is gratefully acknowledged by the authors.

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Figure 9 : Contour plots of the minimum (compressive) principal stress (in kN/m2) induced by the self-

weight and the estimated ground settlements in 100 years. Upper picture: south side; lower picture: north side

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Binda, L. and Baronio, G. 1996. Byzantine concretes: the role of thick masonry joints containing crushed bricks. Proc. RILEM Conf. on ‘Concrete: from material to structure’, Arles (F), p. 289-309.

Binda, L., Tedeschi, C. and Baronio, G. 1999. Mechanical behaviour at different ages, of masonry prisms with thick mortar joints reproducing a Byzantine masonry. Proc. 8th North American Masonry Con-ference, University of Austin, TX (USA), CD-ROM.

Binda, L., Lombardini, N. and Guzzetti, F. 1995. San Vitale in Ravenna: a survey on materials and struc-tures. In Historische Bauwerke - Konstruktiv sichern, behutsam konservieren schonend nutzen; Internationale Tagung des SFB 315, Karlsruhe (D), p. 113-124.

Deichmann, F.W. 1969-1976. Ravenna, Hauptstadt des spätantiken Abendlande (in German), 3 volumes. Wiesbaden (D).

Lenczner, D. 1972. Elements of loadbearing brickwork. Oxford: Pergamon press. Lourenço, P.B. 2002. Computations on historic masonry structures. Progress in Structural Engineering

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RILEM TC 177-MDT Workshop ‘On-site control and non-destructive evaluation of masonry struc-tures’, Mantua (I), November 12-14, 2001, p. 311-321.

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Stagg, K.G. and Zienkiewicz, O.C. (eds.) 1968. Rock mechanics in engineering practice. London: John Wiley & Sons.

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