Structural Modeling of the Vaults of St. Julien in Couleuvre, France

Thomas E. Boothby¹, Huriye Sezer Atamturktur², John Ochsendorf³, Andrew Tallon⁴, and Stephen Murray⁵

¹ Professor of Architectural Engineering, The Pennsylvania State University – University Park, PA, USA 16802
² Graduate Student, The Pennsylvania State University – University Park, PA USA 16802
³ Assistant Professor, Department of Building Technology, Massachusetts Institute of Technology, Cambridge, MA, USA 02139.
⁴ Ph.D. Candidate, Department of Art History and Archeology, Columbia University, New York, NY.
⁵ Professor, Department of Art History and Archaeology, Columbia University, New York, NY.

Abstract

The twelfth-century church of St. Julien in Couleuvre, France is characteristic of Bourbonnais Romanesque construction practices, with relatively heavy barrel vaults with transverse arches surmounting a nave of modest height. The thrust of the nave vaulting is resisted by vaulted aisles and external buttresses. In the present study, the static behavior of the vaults of St. Julien is studied by the development of a finite element model. The predicted dynamic behavior of this model is validated by experimental modal analysis, by comparison of experimentally and analytically determined frequencies and mode shapes. Boundary conditions and material properties are established by obtaining reasonable match between the experimental and analytical results. The validated analytical model is studied for stress distribution, magnitudes of horizontal thrust, horizontal/vertical reaction ratios, and for the effectiveness of the buttressing scheme.

Introduction

The region in France known in the Middle Ages as the Bourbonnais, which corresponds roughly with the modern département of the Allier, lies at the heart of France; it flourished in the eleventh and twelfth centuries immediately prior to the dramatic turn of history when France became France. The rural landscape has not suffered from the damaging effects of industrialization: population has actually declined sharply over the last century, leaving intact both the relationship between village and surrounding countryside—and, most importantly, an unprecedented collection of some two-hundred medieval churches, which have suffered remarkably little damage over the years. Traditional assessments of these buildings (arrangement by chronological priority to tell the story of “progress,” or grouping in regional “schools”) typically portray the Bourbonnais as an architectural vacuum into which “influences” penetrated from the Auvergne, Burgundy and Berry. For the last five years, the Columbia University Field School in Romanesque Architecture has sought a more positive way to represent this extraordinary group of edifices by taking a macroscopic rather than microscopic view, to attempt to discern patterns of appropriation, both structural and stylistic, that defy traditional modes of representation (www.learn.columbia.ed/bourbonnais). What emerges is a group of heterogeneous buildings, each a unique synthesis of the great architecture of surrounding regions with a strong local building tradition, which nonetheless explore a limited range of structural and stylistic ideas (Figure 1).

Most churches are laid out in a cross plan, with three or four nave bays, a crossing with transept arms that may or may not project, and a small, single-bay choir. Often the nave is flanked by side aisles, but only in rare cases are windows pierced in the upper wall over the side aisles. The churches are mostly barrel-vaulted and divided into bays by rectangular transverse ribs which spring from often ornate high capitals. Vaults are constructed of rubble and sometimes brick, bathed in mortar and set in compression during drying by a series of stone wedges driven in at the crown. Compound piers are formed of a square, rectangular, or cruciform core with semicircular or rectangular responds extending from each face. Bell towers are a common feature, and more often than not are located directly above the crossing.

Because the churches in the Bourbonnais are fairly straightforward, the structural problems from which they suffer are correspondingly less difficult to assess. One of the most interesting phenomena is that of “retrofitting”: at some point, generally in the twelfth century, an older wooden-roofed nave is vaulted in stone. In most cases, the outer walls, despite the addition of heavy buttresses, have yielded and the vault is deformed; in some cases, the vault has failed altogether.

Description of St. Julien, Couleuvre

The parish church of Saint-Julien in Couleuvre is one of the most spectacular churches of the area for reasons of size and quality of construction. A four-bay nave with pointed barrel vault is flanked by rather narrow aisles with groin vaults and intersected by a projecting transept with pointed barrel vault. The nave supports are unusually substantial—the deeply-projecting dosseret and attached colonnette toward the central vessel provides internal buttressing. Rectangular chapels are set to the east of the transept arms. Nave and transept belong to the second part of the twelfth century; the main vault is said to be later. The flat chevet with its rib vault belongs to the mid-thirteenth century.

There is a gallery with upstairs chapel at the west end of the nave and a projecting staircase turret to serve it. The staircase also gives access to the vaults, a rare privilege in a region where most nave roofs are accessed through trap doors or are posed directly on top of the vaults. During the summer of 2006, a team of computer scientists from Columbia University made a complete three-dimensional high-definition surveying scan of interior and exterior volumes, including the space above the nave and side aisle vaults. This scan, of over six million data points, is of unprecedented accuracy and provided the raw material from which the sections were extracted for structural analysis. The plan of the church is depicted in Figure 2. Figure 3 provides a general view of the nave (a) and a detailed view of the nave vaulting, as seen from below (b). The building is more easily studied also because the easternmost nave vault was replaced with a plaster and lath vault; the cross-section of the adjacent vault with its materials and style of construction are thus available. This is illustrated in Figure 4.

Thrust line analysis

To analyze the structural performance of Couleuvre and other churches in the Bourbonnais region, a team at MIT has been developing new interactive analysis tools (Block et al, 2006). These two-dimensional tools are freely available on the internet (http://web.mit.edu/masonry). The goal of the analysis is to determine the relative safety of various church geometries and to provide a comparison of the structural behavior across a large body of buildings. The principle of the analysis is based primarily on static equilibrium. Assuming that the masonry structure can carry compressive forces, but no tension force, thrust line analysis is used to approximate the forces which hold the structure in equilibrium.

Heyman (1995) has formalized the limit analysis of masonry structures over the last 40 years. By assuming that masonry is composed of a series of rigid blocks acting in compression, the stability of masonry structures can be determined through the use of thrust lines, or internal compressive lines of force. At any one location, the thrust line can be considered as the resultant of all of the forces acting above that particular point. To demonstrate the safety of historic masonry, the thrust line must be shown to lie within the masonry everywhere. The primary value of thrust line analysis is that it can determine the limits of collapse for masonry, and therefore can be used to illustrate the relative safety of the building.

Figure 5 illustrates the result of a thrust line analysis for a section through the pier at Couleuvre. The left side of the image shows the data points from the laser scan of the section, which were used to create the geometry. The right side shows the idealized geometry, the resulting thrust line, and the support reactions to maintain equilibrium. Due to the relative symmetry of the section, the thrust line analysis can be assumed to be symmetrical about the centerline of the church. This assumption neglects small differences in the height and surcharge depth between the north and south aisle vaults. The analysis is further based on the following assumptions:

1. The section is constant in thickness of one meter and the longitudinal dimensions of the church (into the page) are neglected;
2. All masonry in the section has a density of 2000 kg/m³.
3. The weight of the timber roof is neglected;
4. Rubble fill on top of the vaults is assumed to act at the horizontal lines drawn above the arches;
5. The central arch acts at the minimum thrust state.
6. The side aisle arch acts with the same thrust value as the larger central arch so that the resultant force in the internal pier is purely vertical.

Historic masonry structures are often statically indeterminate, that is to say, that the internal forces are unknown. A rigid block arch can contain a wide range of horizontal thrust values, though very small support movements can lead to the formation of hinges giving a more precise knowledge of the internal forces. For example, any outward movement of the supports will cause an arch to tend towards the minimum horizontal thrust state (Heyman 1995). The laser scan of the church illustrates that the aisle wall and the exterior buttresses have leaned slightly outwards over the centuries, and therefore the central arch is likely acting at a minimum thrust state. The horizontal thrust of the central arch is found to be approximately 12 kN or 1.3 tons. The horizontal thrust is remarkably small, which is due to the relatively short span, tall height of the pointed arch, and large thickness of the vault and overlying fill. This thrust value is resisted by the combined buttressing effect of the exterior walls as well as the internal piers. Assuming a constant thickness into the page of one meter, the support reaction on the internal pier R_main is 436 kN or 48 tons, and the reaction on the exterior wall R_side is 288 kN or 32 tons.

For safe structural performance, the thrust line should fall within in the central third of the masonry at the base of the pier to ensure that the entire section is in compression. Here it is assumed that the side aisle arch provides an equal and opposite thrust to balance the horizontal force from the central arch, so that the resulting force R_main falls in the center of the internal pier. This assumption gives the most unfavorable conditions in the exterior buttress. (In reality, it is likely that the side aisle arch will thrust with a slightly lower value, so that the pier reaction R_main moves outward and the exterior buttress reaction R_side moves inward.) Even with the unfavorable assumption that the exterior buttress provides the entire horizontal force reaction, it is seen that the exterior wall reaction R_side falls within the middle third of the buttress, demonstrating the inherent stability of this church geometry. This confirms the satisfactory structural performance of this small church over many centuries. In summary, thrust line analysis can be used to explore the range of possible equilibrium conditions for a masonry structure. Interactive thrust line analysis tools, together with tutorials, are freely available at the web pages listed in the references for this paper.

3-D Finite Element Analysis of Nave Bay

A three-dimensional finite element analysis of the main vessel was conducted using the commercial finite element code ANSYS, Version 7.0 (ANSYS, Inc. 2006). Three bays of the nave are modeled, in order to be able to interpret the results of dynamic testing on a bay, including the effect of the two adjacent bays. The cruciform piers were modeled using solid elements. The transverse arches that occur in line with the piers are modeled using 2 × 2 SOLID95 elements in cross section, following the approximately circular curvature of the vault. These are rectangular solid elements provided with mid-side nodes. The vault webbing is modeled using SHELL181 elements, which are suitable for moderately thick shells, and admit both membrane and bending stresses.

Efficient modeling of the cruciform piers necessitated the use of tetrahedral solid elements, SOLID92. Although tetrahedral elements in general are less accurate than rectangular elements in capturing stresses within the body of a solid, the general interest of this finite element analysis is an accurate representation of the vaults. The piers are meant to simulate the appropriate boundary conditions, and the test of their accuracy is the quality of the validation described in the following section.

The masonry is modeled as a linearly elastic isotropic material, using a density of 2000 kg/m³. Within the rubble vault webbing, a modulus of elasticity of 2.0 GPa, as determined by impact-echo experiments, and a Poisson's ratio of 0.20 are used. In the piers and transverse arches, the modulus of elasticity is modified to 8.0 GPa, to reflect the superior quality of the masonry work. The boundary conditions consist solely of full vertical and horizontal restraint at the base of the piers. Symmetry-type boundary conditions are applied at the limits of the three-bay model, that is, displacements are restrained in the longitudinal direction. No horizontal restraint is applied to the piers to represent the effect of the aisle vaulting. The model is illustrated in Figure 6.

Results of the finite element analysis are divided into two types: static and dynamic. The dynamic results consist of natural frequencies and mode shapes used for comparison to the experimental results for validation of the analytical model, while the static results are obtained by gravity loading on the validated analytical model. The static results are eventually used for comparison to the two-dimensional thrust line results described in the preceding section.

The dynamic results include the sequence of natural frequencies listed in Table 1 The list includes a rough description of the mode shape for each of the natural frequencies found in the analytical model. A vault mode primarily involves the vault, without significant contribution of the piers. In a symmetric vault mode, the crown moves up and down, while in an antisymmetric vault mode, the crown is a node. Shaded modes have insignificant vertical displacement in the bay where the vibrations were measured, and are unlikely to be detected in the experimental set-up used in this program.

The mode shapes of modes 1, 3, 5, and 7, which were detected experimentally, are illustrated in Figure 7.

Mode	Fequency (Hz)		Description
	analytical	exp.
1	4.8	4.9	antisymmetric vault
2	10.9		sway
3	11.0	10.4	symmetric vault
4	14.3	14.6	antisymmetric vault, transverse pier
5	18.4	18.8	antisymmetric vault
6	18.9		sway
7	19.6	19.6	symmetric (2 nodes) vault, transverse pier
13	29.6	29.4	symmetric (2 nodes) vault, longitudinal pier

Experimental modal analysis of nave bay

A single nave bay was subjected to an investigation by experimental modal analysis. Accelerometers with a frequency response range of 1-100 Hz and a sensitivity of 1g/v were placed in an array in the bay (Figure 8), and accelerometer locations were excited using a 2.7 kg instrumented hammer. Hammer impacts were applied at each of the accelerometer locations in turn. A multi-channel data acquisition card with dynamic signal analysis capabilities was used to record the signals from the hammer and the accelerometers. A bandwidth of 0-125 Hz was used for the acquisition of accelerometer data. An average of three sets of responses for each excitation location was collected, and the coherence of the data was recorded.

Of the greatest interest is the Frequency Response Function, the frequency domain response of an accelerometer divided by the frequency domain signature of the impact hammer. The frequency response function is used to determine resonant frequencies, while the comparison of the imaginary part of the time domain response allows the qualitative determination of mode shapes by comparing the phase response of different accelerometer locations. In general, based on the similarity of reciprocal data—that is, the comparison of the response at point A due to an impact at point B to the reciprocal response at point B due to an impact at point A, the response of the vaults can be characterized as linear.

The data are also subjected to a more thorough computer-based investigation of mode shapes, based on the recorded points. The resulting mode shapes are used for comparison to the analytically determined mode shapes. Using the combined analytical and experimental results, analytical parameters, such as elastic constants of the piers, arches, and vault and boundary conditions can be adjusted so that correlation between the frequencies shown in Table 1 and the mode shapes shown in Figure 7 are found between the analytical and experimental data. In this case, boundary conditions were unchanged. The principal adjustment was the reduction of the modulus of elasticity of the vault material to correspond to the value determined in the impact-echo testing program described below. This method requires only matching of the frequencies and mode shapes between experimental and analytical data—although it is certainly possible to claim that a good match has been achieved to the fundamental frequency and mode shape of the analytical model, the low coherence of the experimental data below this frequency makes it impossible to claim that this is the lowest natural frequency of the prototype. The resulting model is considered to be validated. All further static analysis is completed on the validated model. General information and other specific examples of the application of this method are available in Atamturktur (2006).

Results Of Nave Bay Analysis

The horizontal thrust at the base of the arch, found by the finite element analysis, is 18 kN, compared to 12 kN for the minimum thrust analysis conducted in the "Thrust Line Analysis" section. This is consistent with the expectation that the thrust will not be minimum for an elastic solution. Figure 9 shows third principal stress on a cross section taken through the buttress and the transverse arch. An incipient hinge at the crown is evident, due to the presence of tensile stresses on the intrados at that location. At the haunches and at the springing, the absence of tensile stresses shows that the thrust line remains closer to the center of the arch, producing larger horizontal thrust than the minimum thrust state shown in Figure 5. The maximum compressive stress is found to be approximately 0.59 MPa, at the outside of the base of the pier. The maximum tensile stress, found by plotting the first principal stress, is also approximately 0.50 MPa, occurring on the inside face of the arch at the crown. All other tensile stresses in the main system are significantly lower. A maximum tensile stress of 0.09 MPa and a maximum compressive stress of 0.16 MPa were found in the vault webbing. Also noteworthy is the maximum horizontal displacement at the top of the pier of 0.5 mm, significantly below the observed displacement.

Conclusions

A coordinated approach, involving historical investigations, archaeological investigations, detailed building surveys, limit states analysis, finite element analysis, and model validation by experimental modal analysis has resulted in a more thorough understanding of a structure representative of a regional style of architecture. This building has had fewer modifications than most of the other churches in the region due to its better structural conception. Neither the limit analysis nor the finite element analysis reveals serious defects in the structural design of this building.

The simplified limit states analysis, which considers only the conditions at the transverse arches, is appropriate for the assessment of this building, as the bending stresses in the vaults are significantly lower. This implies that the vaulting system is effective in concentrating gravity loading at the location of the piers, and justifies the use of very crude vault construction. The observed leaning of the interior piers and exterior buttresses is two orders of magnitude greater than the instantaneous elastic displacements calculated on the basis of a validated analytical model. As a result, the explanation for these deformations must be sought in long-term or inelastic behavior, such as inelastic deformations, creep, accumulating cracking, or foundation settlement.

Acknowledgments

The authors would like to thank the Andrew W. Mellon Foundation for providing financial support for the study of Romanesque churches in the Bourbonnais region of France. Charles-Henri de Lobkowicz generously provided accommodation and other support during fieldwork in France. Claire Lackner and Paul Blaer provided the laser scan of the church, and Philippe Block carried out the thrust line analysis.

References

This reference list is furnished to provide additional detail on the methods used in this paper, and is by no means intended to provide a complete bibliography of any of the topics of the paper. Readers interested in such information should refer to the reference lists in the first two referenced papers.

1. Atamturktur, H.S. Structural Assessment of Guastavino Domes, M.S. Thesis, The Pennsylvania State University, May 2006.
2. Block, P., Ciblac, T., and Ochsendorf, J.A., “Real-time Limit Analysis of Vaulted Masonry Buildings,” Computers and Structures, Vol. 84, No 29-30, pp. 1841-1852, November 2006.
3. Cabri Geometry II Plus, Cabrilog sas, France. Available from: http://www.cabri.com/.
4. Génermont, M. and P. Pradel, Eglises de la France: l’Allier, Paris, 1938
5. Heyman, J., The Stone Skeleton: Structural Engineering of Masonry Architecture, Cambridge University Press, 1995.
6. http://www.learn.columbia.edu/bourbonnais. Romanesque Churches of the Bourbonnais, accessed September 13, 2006.
7. http://web.mit.edu/masonry/interactiveThrust/. Thrust line analysis tools, accessed September 10, 2006.