DEVELOPING MASONRY VAULT MODELS FOR GLOBAL ASSESSMENT

DEVELOPING MASONRY VAULT MODELS FOR GLOBAL ASSESSMENT Thomas E. Boothby (1), Paola Condoleo (2), Alberto Taliercio (2) and Luigia Binda (2) (1) The Pennsylvania State University, University Park, PA, USA (2) Politecnico di Milano, Milano, Italy Abstract The development of a model for a masonry vault involves a complex series of interrelated decisions on geometry, modeling, material selection, meshing, and boundary conditions. It is necessary from the outset to decide the level of simplification that will be applied to the actual geometry of the structure. This decision takes account of the information desired from the model, the quantity and accuracy of the geometric information available. The use of shell or solid elements dictates both the constraints on meshing the model and the subsequent performance of the model. The boundary conditions are both critical to the results of the model and extremely difficult to assess. In this article, we present the process of working from a large amount of survey information to develop first a viable geometrical model, then the process of meshing used to make this into a working finite element model, and finally the process of using field-acquired vibration data to update the model as necessary. Key words Masonry, gothic church, domical cross vault, geometrical survey, finite elements 1. INTRODUCTION The first of the great Florentine basilicas to be completely covered with cross vaults, the Dominican church of Santa Maria Novella (1279-1355) has long been recognized as one of the outstanding examples of Gothic architecture in Italy. It served as a structural model for the nave of the Cathedral of Florence, among others, and continued to be much admired throughout the Renaissance. One of the major factors that accounts for its success as a design is the unified interior space of the nave with its soaring domical cross vaults. Like their counterparts in other regions of Europe, Italian builders of the Gothic era 551

experimented with new types and methods of construction, such as the domical vaults of Santa Maria Novella raised high on slender shafts, so much more daring than the domical vaulted churches of Romanesque and Gothic Lombardy, and achieved without the aid of the iron tie rods so often seen in Italian Gothic buildings. In so doing, the builders of Santa Maria Novella created that airy interpenetration of space that would come to characterize Tuscan Gothic architecture. The research carried out at Santa Maria Novella focuses primarily on the nave (Fig. 1), with the aim of understanding why and how it came to have this distinctive design. Ultimately, the authors would like to reconstruct the process of design and construction, identifying specific situations where builders made critical decisions regarding design and structure. In so doing, it is possible to gain a better appreciation of the achievements of the builders of Santa Maria Novella as they created what is essentially a Florentine Gothic Structural System. Mathematical models will also be used to better support the results of the investigation. The particular distinctive features that draw our attention are the use of domical rib vaults, that is, vaults whose crowns change level along the longitudinal and transverse axis, resulting in a sense of a high canopy over each bay, the use of high side aisles with crypto buttressing, concealed above the aisle vaults. (See Figure 1) This multi-disciplinary investigation has been carried out by means of collaboration between engineers and architectural historians, and collaborations between the Pennsylvania State University and Politecnico di Milano. The necessity of collaborations between architectural historians and engineers has resulted from the structural character of many of the questions surrounding the construction of the nave and the ability of certain engineering techniques to provide information on construction non-destructively. The general study has been described in other articles and conference papers [1, 2, 3, 4]. The particular study described in this article is the steps in the development of an analytical model of the vaults, which will eventually be used in predicting the vault behavior, with particular emphasis on the crack pattern (see Fig. 2), and in understanding the influence of the vaults on the remainder of the structure. Figure 1: General view of nave, looking northwards towards altar Figure 2: Surveyed crack pattern at the extrados of Bay 5 552

2. GEOMETRICAL SURVEY The main objective of the geometrical survey was to characterize, on a quantitative level, the shape of the square and rectangular vaults of the church. During the campaign it became necessary to modify the measuring procedure, in order to get precise geometrical information on certain irregularities of the structure. The survey was carried out using a reflectorless total station: therefore, no targets were placed on the surveyed elements. The work consisted of three phases: the first two concerned the extradoses of Bays 2 and 5, while the third regarded the pillars, the wall and transverse arches of the nave and the aisle, and, more in detail, the intradoses of Bays 2 and 5. Concerning the first phase (extrados of Bay 2), advantage was taken of a closed polygonal line, consisting of 6 station points, which allowed a detailed location of the points of the perimeter walls, the intersection of the vault with the filling, the principal generatrices and directrices as well as the diagonals. It is worth remarking that it was impossible to record the points of the keystone: for the highest points this impossibility was due to the interference with the wooden structure of the roof, whereas for the lower points the problem consisted in the low value of the angle between the axis of the total station and the pitch of the vault. During the second phase (extrados of Bay 5) a closed polygonal of 5 points was created: the choice of these points was crucial, because it was used to locate the same elements as in the case of Bay 2, by means of a more refined grid of points and thus allowed a precise description of the parallels and the meridians. The location of redundant points, which is the main difference with the survey of Bay 2, was finalized at the control of the geometric irregularities, such as the possible lack of symmetry and the sag of the quadrant boundaries. In the third phase a closed polygonal of 7 points was used. The principal and diagonal arches were measured, as well as several refinement points on the groins. All the points were then reported on sketches. The main finding of this phase was a hole in Bay 2 (rectangular vault), which was not detected in the first phase because of interferences between the total station and the structural elements. All the data collected were finally elaborated numerically. All the points that were not considered significant were deleted; dedicated software was then used to connect the remaining points with triangles, which made the interpolation with level contours possible. The contours are shown in detail in Figure 3, with a plan view shown in and an isometric view in. Figure 3: Results of geometrical survey: Level curves of Bay 5 vault 10 cm contour interval plan isometric 553

3. DEVELOPMENT OF THE GEOMETRICAL MODEL The structural finite element model is based on the geometric modeling of the structure. The shape of the vaults, bounding arches, ribs, and fill needs to be determined in order to define the final form of the model used for the structure. The development of the geometrical model requires many decisions on the level of smoothing and simplification to be used: excessive smoothing may result in an unrealistic structural model that behaves noticeably differently from the actual structure. Because the data on the shape of the vaults were acquired at discrete points, and have a certain measurement error, insufficient averaging and smoothing result in an unmanageable model, having too many irregular surfaces to mesh, and may lead to an incorrect representation of the vault, with significant errors, such as excessive bending, introduced due to the irregularities of the surface. The process of developing the model of the square vault that is described in this paper is partly based on the observation that the shape of the vault follows straight contour lines between the octant lines of the vault (see Figure 3 ). In fact, the lines along the axis of the nave, and at the vault midpoint perpendicular to the nave are also straight, rising towards the center at a uniform slope. The only curved lines are the diagonal ribs of the vault, which are segments of a circular arch (fifth point quinto acuto in the rectangular bays). This is doubtless a construction expedient, with the possibility of building the vault by laying straight, horizontal lagging between the quadrant lines and the diagonal ribs. For the initial vault model, which represents an averaged effect of the vault, symmetry is supposed about the quadrant lines. These lines are laid out by a best-fit line to all the observed points within 10 cm of a quadrant line. That is, the points to the east and west of the axis of the church were merged into one data set, and a best-fit line was constructed for the top of the vault perpendicular to the axis at mid-bay. The line along the axis of the vault was handled similarly. The arc of the diagonal was constructed by combining data from above the vault with data taken on the inside face of the vault webbing from below the vault. Again, data from all four quadrants were merged and a best-fit circular arc was found. With the octant lines set up, straight horizontal level curves were drawn, and a surface fit to the frame of octant lines and level lines. 4. FINITE ELEMENT MODEL The ribs are modeled using solid tetrahedral elements. The SOLID92 element in ANSYS was chosen. This is a 10-node tetrahedral element, having a mid-side none on each edge. It supports large deformations, inelasticity, plasticity, geometric non-linearities, and creep. The ribs are meshed with an approximate element edge length of 20 cm, which results generally in three or four elements through the thickness of the rib. The vault webbing is modeled with a SOLID91 layered shell element. There are two reasons for the choice of this element. First, this element supports the identification of the shell element surface with the front or back of the shell, which permits the geometry of the shell to be coordinated easily with the geometry of the ribs. Working from survey data, it was expedient, as described above, to construct the back surface of the entire vault, and to develop the geometry of the ribs on the basis of this previously defined geometry. This procedure does not easily permit offsets of the surface of the vault web. The extreme lowest corners of the vaults are not meshed, as the acute triangular surface in this location gives rise to pathological element shapes. In the following phase of the analysis, the fill in this area will dictate the behavior of this portion of the vault to a much 554

greater extent than the vault webbing. A vault thickness of 35 cm was measured through several holes in the vault, and a relatively uniform thickness of vault was confirmed by radar testing and impact-echo analysis. The portions of vault webbing between the ribs are kept separate from the ribs, with coincident but distinct edges, and are meshed separately. This allows more latitude in choosing mesh configurations in the vault web and in the ribs. This also gives considerably more control over what type of continuity conditions are enforced between the vault ribs and the vault shell. By meshing the ribs with target elements, and the vault web edges with contact elements, the bonded contact option can be selected. This is similar to a contact/target pair, in that the coincident but separate rib and shell cannot interpenetrate. However, because bonded contact has been specified, they cannot separate either. Using a shell/solid type of bonded contact, the local element degrees of freedom that are constrained can be specified. One of the variables investigated in the following section is the selection of fully restrained (translations and rotations restrained) and simply supported vault webs (translations only constrained). Only the self-weight of the material is considered as external load in the FE analyses shown in the next section. The material density is taken equal to 2000 kg/m 3, according to measurements on bricks found at the site. 5. STRUCTURAL MODELING DECISIONS AND RESULTS The establishment of a model vault involves a considerable amount of smoothing of the survey data. In order to establish the effect of this smoothing, we investigate the difference between an averaged vault, made of a composite of the survey data from the four quadrants of the vault, as described in Section 4 above, and a model in which various irregularities are allowed to remain. Although the lines along the quadrant boundaries of the vaults are nominally straight, as described in Section 3, there is, in some of the vaults, an apparent sag of up to 10 cm along this line in some of the vaults,. This sag was modeled by introducing a parabolic error with maximum amplitude of 10 cm along this line before proceeding with the modeling of the vault surface. The structural effects on the resulting model were compared to the initial model, with no apparent differences noted, at least for the linearly elastic model used in this portion of this study. In the measurements of the Bay 5 vault, a curious difference in height (of approximately 20 cm) between the east and west wall arches was noted. Although the model used in most of this study reflects symmetry about two axes, an additional model was constructed, in which the difference in wall arch height was incorporated. The displacement results for this model (with the shell/rib connection pinned), are shown in Figure 4. Comparing these results with those pertinent to the symmetric model, where the difference in height of the arches is disregarded, with pinned vault/rib connections (Fig. 4 ), it can be observed that qualitatively there is no significant difference in terms of contour plots. The maximum deflection is computed near the center of the south cell in the asymmetric model, and is 0.913 mm. In the symmetric model, the maximum deflection is obtained at a similar location of both the south and the north cell (adjacent to the wall arches), and is 0.893 mm. According to these remarks, only the symmetric case is dealt with in the continuation of the paper. The vaults typically exhibit cracking along the diagonals (see Fig. 2). These cracks are 555

generally present along part of the length of the diagonal in two or three of the four quadrants. Assuming the vaults to be simply pinned to the ribs is an attempt to match the existing boundary conditions for the vault cells. Under this assumption, a maximum vertical deflection of 0.893 mm is obtained (Fig. 4). In order to study the conditions that may have caused the cracks to develop, it is instructive to introduce rotational fixity between the ribs and vault cells. Fixity was introduced by constraining rotational degrees of freedom in the shell-solid bonded contact/target pair available in ANSYS. The immediate effect of this fixity was to reduce overall immediate elastic deflections of the vault subjected to self-weight by approximately 30%: the maximum vertical displacement decreases to 0.667 mm in the fixed model (Fig. 4(c)). Figure 4: Contour plots of the vertical displacement of the vault (in m units) according to: the asymmetric/pinned model, the symmetric/pinned model, and (c) the symmetric/fixed model (c) Comparing the contour plots of the maximum (tensile) principal stress in the vault, computed according to the two models, it can be observed that at the top side the overall distribution is similar (Fig. 5). Conforming to intuition, tensile stresses are somewhat higher by the mid-span of the cells (namely, the north and south) in the pinned model (Fig. 5), whereas they increase at the vault/rib connections in the fixed model (Fig. 5). Also, there is some stress localization in the cells at the level of the crown of the wall arches and the transverse arches. The highest stresses are of the order of 0.2/0.25 MPa. Note that 0.2 MPa is a reasonable value for the tensile strength of masonry [5]. 556

The distribution of the principal tensile stress at the bottom side of the vaults is totally different when computed according to the pinned model (Fig. 6) or the fixed model (Fig. 6). The assumption of fixity between vaults and ribs definitely alleviates the stress state at the intrados. Whereas tensile stresses exceeding 0.15 MPa are found in wide regions of the vault (mainly in the north-south cells) using the pinned model, the maximum principal stress is far below this value (except for some regions at the border of the intrados) using the fixed model. Figure 5: Contour plots of the maximum principal stress (in Pa units) at the extrados (topside) of the vault according to the pinned model and the fixed model Figure 6: Contour plots of the maximum principal stress (in Pa units) at the intrados (bottomside) of the vault according to the pinned model and the fixed model In terms of minimum (compressive) principal stress, the comparison between the two models leads essentially to opposite observations. Whereas there is no substantial difference at the bottom-side of the cell (see Fig. 8), compressive stresses definitely decrease (in absolute value) at the top-side when moving from the pinned model (Fig. 7) to the fixed model (Fig. 7). In both cases, the highest stresses are found by the rib springers, and exceed 0.5 MPa, 557

but whereas compressions exceed 0.30 MPa in wide parts of the cells using the pinned model, this level is seldom attained using the fixed model. Note that in both cases the distribution of the compressive stresses is more uniform than that of the tensile stresses, which would indicate that the structural behaviour of the vault is similar to that of a dome. Figure 7: Contour plots of the minimum principal stress (in Pa units) at the extrados (top-side) of the vault according to the pinned model and the fixed model Figure 8: Contour plots of the minimum principal stress (in Pa units) at the intrados (bottomside) of the vault according to the pinned model and the fixed model Finally, an analysis was carried out to investigate the effect of a detachment between the vault and the side walls. This detachment, with variable severity, is frequently encountered in medieval vaulting, and may be explained by a rotation of the side walls, or by cracking due to the extensive tensile stress concentration clearly visible in Fig. 6. In these analyses, the borders of the vaults parallel to the nave were supposed to be completely free. The vaults are supposed to be pinned to the ribs. The results obtained are shown in Figs. 9 and 10. Comparing e.g. Fig. 6 to Fig. 10, showing the contours of the maximum principal stress at the intrados computed with and without constraints at two of the vault sides, respectively, it 558

is evident that tensile stresses are dramatically increased by the constraint release, especially at the restrained boundaries of the model where it amply exceeds 0.25 MPa. Also, comparing Fig. 7 to Fig. 9, showing the contours of the minimum (compressive) principal stress at the extrados computed with two of the vault sides constrained or free, respectively, it can be noted that there is an increase in the highest compressions due to the constraint release, mostly by the rib springers. In the central part of the vault, the boundary conditions have a reduced effect on the stress state. Figure 9: Contour plots of the maximum and minimum principal stress (in Pa units) at the extrados (top-side) of the vault, assuming the vault to be detached from the side walls Figure 10: Contour plots of the maximum and minimum principal stress (in Pa units) at the intrados (bottom-side) of the vault, assuming the vault to be detached from the side walls 6. CONCLUSIONS AND FURTHER STEPS According to the remarks made in Sec. 5, it can be stated that the assumption of fixity of the vaults to the ribs has crucial effects in the evaluation of the stress state in the vault. When applied to this and other case studies, its reliability should be thoroughly assessed case by 559

case, as it definitely underestimates the extreme stresses in the vault compared to the most conservative assumption of vaults simply supported by the ribs. The exact nature of this connection requires further investigation. Small geometric variations in the shape of the vault, including asymmetry of the wall arches, do not appear to have a significant effect on the stress state and the deflections in the vaults. Taking the detachment of the vault from the side walls into account increases the peak values of the tensile and the compressive stresses over the stresses in the model with the constrained sides. Unrealistically high tensile stresses are computed, which cannot be carried by masonry without cracking. This point should be further investigated, taking a nonlinear constitutive law for masonry into account and modeling detachment in a less severe, more accurate way. The model initially developed for the analysis of the vaults of Santa Maria Novella is linearly elastic. Refinements will be added to this model, including the modeling of the vault fill as a solid mass, connected to the vaults through a contact surface, the representation of the two-wythe brick vault shell through layered elements, and the addition of plasticity, cracking, creep, and other material non-linearities to all or part of the vault, as appropriate. At an advanced stage of the development of the linearly elastic model, comparisons will be made to the results of modal testing of the vaults carried out in July 2006. This analysis will be refined considerably, by adjusting the model, adjusting material properties, and adjusting boundary conditions in view of the experimental results, before proceeding with the introduction of non-linearities to the model. Further investigations will also take account of the mass of fill at the four corners of the vault and its role in increasing or mitigating stresses in the vault. ACKNOWLEDGEMENTS This project was funded by a grant from the Kress Foundation European Preservation Program, administered by the World Monuments Fund. The authors are also grateful to the Comune of Florence and the Dominican Convent of Santa Maria Novella for granting access to this important building. REFERENCES [1] Binda, L., Boothby, T., Condoleo, P., Cardani, G, Cantini, L., and Smith, E., Santa Maria Novella and the Development of a Florentine Gothic Structural System, Proc. Int. Symp. on Studies on Historical Heritage (SHH-07), Antalya, Turkey, September 17-21, 2007. [2] Boothby, T., Binda, L., and Smith, E., Structural and Historical Assessment of Santa Maria Novella, Florence, Italy, Proc. 10th North American Masonry Conference, St. Louis (MO), USA, June 3-6, 2007. [3] Smith, E.B., Santa Maria Novella e lo sviluppo di un sistema gotico italiano, In Arnolfo di Cambio e la sua epoca: Costruire, scolpire, dipingere, decorare, Atti del Convegno Internazionale di Studi, Firenze-Colle di Val d'elsa, 7-10 marzo 2006, Vittorio Franchetti Pardo (Ed.), Rome : Viella, 2006, 289-298. [4] Erdogmus, E., Boothby, T.E., and Smith, E.B. Structural appraisal of the Florentine gothic construction system, Journal of Architectural Engineering, 13(1) (2007) 9-17. [5] Lourenço, P.B., Computations on historic masonry structures, Progress in Structural Engineering and Materials, 4(3) (2002) 301-309. 560