Design and analysis of a strengthening system based on steel tie rods to stabilize a Gothic Cimborio
Abstract
The dissertation focuses on the structural condition of the Sant Cugat monastery, a medieval Gothic complex where the central Cimborio plays a decisive role in the global equilibrium of the church. Due to its geometry and significant mass, the Cimborio generates substantial horizontal thrusts that are transmitted to the supporting arches and ultimately to the bell tower. This phenomenon has led to a progressive outward inclination of the tower, confirmed by monitoring data, indicating a long-term risk of structural failure.
The work develops a structural assessment based on graphic statics to understand the internal force flow and to evaluate the safety of the structure. Based on this analysis, strengthening strategies are proposed, primarily involving prestressed steel tie rods and, if necessary, foundation reinforcement using micro-piles. The study aims not only to stabilize the structure but also to provide a rational methodology applicable to similar historical constructions.
1. Introduction
1.1 History
The Saint Cugat monastery is a historically significant medieval complex located near Barcelona, representing an important example of Gothic religious architecture. Over centuries, the structure has undergone multiple phases of construction, modification, and repair, each contributing to its current structural configuration. These historical interventions, while preserving the monument, have also introduced uncertainties regarding material properties, geometry, and load paths. Understanding this historical evolution is essential because the current structural behavior cannot be dissociated from past construction techniques and interventions.
Modifications of the monastery between the 14th and 15th centuries.
1.2 Architecture and fittings
Architecturally, the church is composed of three naves intersected by a transept, above which rises the Cimborio, an octagonal dome-like structure. This Cimborio is supported by a system of main arches and ribs that channel loads downwards to the supporting pillars. The vaults further distribute loads across the structure, creating a complex three-dimensional load transfer mechanism.
However, unlike modern structures, Gothic constructions rely heavily on compressive behavior, making them particularly sensitive to horizontal forces. The bell tower, positioned adjacent to this system, is directly influenced by these forces, making it a critical element in the structural assessment.
1.3 Previous interventions and strengthening
Previous interventions have attempted to address structural issues, though often without a complete understanding of the global behavior of the system. Monitoring systems installed in recent years have revealed that the deformation of the bell tower is ongoing and accelerating. These findings highlight that earlier strengthening measures were either insufficient or not properly targeted, reinforcing the need for a comprehensive structural reassessment.
Existing steel ties from previous interventions.
2. General Arrangement
General Layout
2.1 Bell tower
Tower height is stated as ~41.2 m; structure is mainly limestone walls and vaults; the ground level includes the Chapel of Mercy, subdivided in the 16th century to accommodate a Renaissance organ; the tower is located between grids 4 and 5 and supports adjacent vault portions and loads from arches 3 and 4 (p. 23).
The Church’s bell tower
2.2 Cimborio/Dome
The Cimborio is an octagonal dome with 8 ribs, each supporting one-eighth of the vault, transferring load to the main supporting arches (p. 25).
This is the dominant structural element in terms of load generation. Its large mass produces vertical loads that are transformed into horizontal thrusts due to its geometry. These thrusts are transferred through the ribs and arches, creating a system that is stable only if all forces remain within the compressive capacity of the masonry. The analysis demonstrates that the Cimborio is the primary source of the structural imbalance observed in the building.
Cimborio of the church
2.3 Main arches
Four main arches support the dome and are supported by four major corner pillars; their heavy dome loads generate significant horizontal thrust, identified as a key driver of tower loading and assessed later via graphic statics (p. 25).
Main supporting arches
2.4 Vaults
The vaults contribute to the redistribution of loads across the structure. While they are not the primary source of instability, their interaction with the arches and ribs influences the global equilibrium. The vaults also introduce additional loads that must be considered in the analysis, particularly in the context of thrust distribution.
(1) Vault 1 between grids C, D and 4,5; (2) Vault 2 between grids C, D and 3,4; (3) Vault 3 between grids B, C and 3,4; (4) Layout of vaults.
3. Critical Damage and Previous Monitoring
The monitoring data clearly indicates that the bell tower is experiencing progressive tilting, which is consistent with the theoretical understanding of the load transfer mechanism. Cracks and deformations observed in the structure further confirm that the thrust lines are approaching or exceeding the geometric boundaries of the masonry elements.
This condition is critical because, according to limit analysis principles, the stability of masonry structures depends on the thrust line remaining within the section. Therefore, the observed damage is not merely superficial but indicative of a deeper structural imbalance.
4. Graphic Statics Analysis
Horizontal thrust on the bell tower is computed via graphic statics, with the load path framed from Cimborio weight → main arches → adjacent/secondary arches → bell tower, with additional vault thrust contributions (p. 29).
Working assumptions include: rubble-masonry fill under the Cimborio with density 22 kN/m³; the same density for ribs, vaults, pillars, and arches; vault thickness 20 cm; Cimborio rubble masonry treated as structural so thrust lines may lie outside ribs but within fill; other vault fills treated as weak with near-null compressive strength; and symmetry enabling analysis of one side then doubling final loads (p. 29).
4.1 Methodology
The methodology is based on graphic statics, a classical approach that allows the visualization of force flow through geometric constructions such as force polygons and funicular polygons. This method is particularly suitable for masonry structures because it aligns with the assumption of compressive-only behavior.
In addition, limit analysis principles are applied, particularly the lower bound theorem, which states that if a valid thrust line can be found within the structure, equilibrium is ensured. However, due to uncertainties in geometry and material properties, the results are interpreted qualitatively rather than as exact numerical predictions.
Resultant force
Arches are divided into virtual voussoirs including overburden; individual load vectors are summed polygonally to obtain the resultant magnitude and direction (p. 30).
Funicular polygon for a symmetric arch structure.
Funicular polygon method
The funicular polygon method is used to determine the line of action of the resultant force. By constructing geometric relationships between forces, the method identifies where the resultant force acts within the structure, providing insight into load paths.
Thrust lines
Thrust lines represent the path of compressive forces within the structure. Since multiple thrust lines are possible, the objective is to identify those that remain within the masonry boundaries. The position of these lines is crucial in determining whether the structure is stable or at risk of forming collapse mechanisms.
Thrust lines for a symmetric arch structure.
4.2 Dome/Cimborio
Slicing and loads distribution
The Cimborio is analyzed by dividing it into slices, allowing the calculation of loads acting on each segment. This approach simplifies the complex geometry into manageable components while preserving the essential structural behavior.
4.3 Main arches supporting the Cimborio
Cimborio loads are transferred through ribs to main arches; edge small-arch loads are also carried by the main arches; graphic statics is applied after load discretization (p. 35).
Loads distribution
The arches receive concentrated loads from the Cimborio and distribute them to the supports. The analysis shows that these loads are not symmetrically distributed, leading to uneven stress conditions.
Thrust lines
The thrust lines in the arches approach the edges of the section, indicating a reduced safety margin. This behavior explains the development of cracks and deformation observed in the structure.
4.4 Vaults
Each vault was analyzed individually to understand its contribution to the overall load distribution. Although their influence is secondary compared to the Cimborio, they still affect the equilibrium of the supporting elements and must be included in the global analysis (more detailed information in the dissertation).
4.5 Arches 3 and 4
Secondary arches 3/4 transfer thrust from Cimborio, main arches 1/2, and vaults to the bell tower and are treated as struts; equilibrium of pillars C4/C5 and thrust lines in arches 3/4 are analyzed to estimate transferred force (p. 40).
Equilibrium of the pillars
Pillar C4 and arch 3 are modeled in 2D, then the resulting tower transfer load is doubled to account for arch 4 and pillar C5 (p. 40). Forces from the Cimborio, main arches, and vaults are applied at their locations at pillar top; the arch 3-to-pillar force is treated as unknown and explored as a range; for each case, the resultant and its eccentricity are obtained graphically; acceptability requires both (i) at least one thrust line inside arch 3 and (ii) a pillar resultant within the pillar base footprint (p. 40).
Summary of thrust forces.
Thrust lines
Because arch 3/4 force cannot be predicted solely by graphic statics, 13 solutions are generated for a range of transferred forces; horizontal thrust varies from 780 kN to 150 kN (p. 43). The selection criteria are: resultant inside pillar base, thrust line within arches 3 and 4, and moment equilibrium for arches 3 and 4 (p. 43).
4.6 Bell tower
The bell tower is analyzed as a structure subjected to combined vertical and horizontal loads. The results confirm that the horizontal thrust from the Cimborio is the primary cause of its tilting, and is unstable without intervention.
4.7 Summary of results
This section presents a summary of all the results obtained from the graphic static analysis, the equilibrium of the tower and the pillar before applying any strengthening interventions.
The larger the eccentricity of the pillar the smaller the forces transferred to the tower and hence the eccentricity of the belltower is reduced. The smaller the eccentricity of the pillar, the larger the force transferred to the belltower and the bigger the eccentricity of the tower is.
Variation of belltower eccentricity with the eccentricity of the pillars
Final thrust lines solution.
5. Strengthening Solutions
Two stabilization routes are developed: prestressed tie rods on main arches as the primary system; foundation strengthening via micropiles as a secondary/contingency system (pp. 47, 54).
5.1 Applications of prestressed tie rods
Prestressed tie rods are selected to counteract large thrust from main arches; passive ties are rejected because activation requires prior structural deformation (p. 47). Tie placement is constrained: an “optimum” tie is stated at the end of arch curvature, but it is lowered due to side vault constraints and anchor plate placement limitations (p. 47). The graphic-statics procedure is repeated for the same 13 pre-intervention thrust-line solutions to quantify reductions in transferred forces through arches 3/4 and changes to bell-tower eccentricity for two tie-force cases: 400 kN and 800 kN (p. 47).
Case 1: stressing the ties to a force of 400 kN
In this case, the applied prestress partially compensates for the horizontal thrust, improving stability but not fully eliminating the problem.
Case 2: stressing the ties to a force of 800 kN
A higher prestress level results in a more significant improvement, bringing thrust lines further zone and reducing the risk of instability.
- Anchor plate design
The anchor plates are designed to ensure proper انتقال of forces from the tie rods to the masonry without causing local damage. - Steel Relaxation of the ties
Long-term effects such as relaxation and loss of prestress are considered, as they may reduce the effectiveness of the strengthening over time. - Summary of results
The tie rod solution is shown to be effective, particularly at higher prestress levels, although careful monitoring is required to ensure long-term performance.
5.2 Foundation strengthening with micro-piles
This solution is considered as a complementary or alternative measure, particularly if the tie rods alone are insufficient.
Loads taken by the micro-piles
Micro-piles redistribute loads to deeper, more competent soil layers, reducing stress on the existing foundation.
Micro piles design
The design is based on assumed soil properties, highlighting the need for further geotechnical investigation.
Shear check at interface between concrete and roman foundation
This step ensures compatibility between new and existing materials.
Verification of prestressing reinforcement
The reinforcement is checked to ensure it can safely carry the applied loads.
Concrete beams design
New beams are designed to integrate the micro-piles into the structural system.
6. Conclusions and recommendations
Large forces from Cimborio and vaults are identified as drivers of bell-tower tilt, interacting with complex geotechnical behavior (p. 63). Graphic statics is used to estimate loads before and after adding tie forces; ties are found theoretically capable of reducing tower load eccentricity, but not guaranteed to eliminate the problem, motivating a secondary micropile scenario (p. 63). The micropile option is quantified as 21 micropiles Ø225 mm (4 north, the rest south) (p. 63).
Implementation guidance for tie rods includes: slow stressing under engineer supervision per drawing SA8-D-DRW-300-REV0; on-site inspection for new cracks or increased crack widths; use of stainless steel for durability; drilled holes sized for ties and left ungrouted for reversibility; maintaining monitoring after stressing to 400 kN; monitoring tie tension because deformation may increase force; keeping tension below 800 kN despite a stated breaking load of 1308 kN; and only considering further stressing based on monitoring and tie force limits (p. 63).
For micropiles and foundation strengthening: adoption is conditioned on tie inadequacy; the presented design is tied to multiple geotechnical assumptions requiring detailed investigation; pillar eccentricity assumed ~0.5 m may differ and should be refined using monitoring; micropile lengths rely on limited geotechnical data and may change; Roman foundation depth (1.6 m) and compressive strength (10 MPa) must be verified via sampling and lab testing; foundation injection with grout is recommended to fill voids and raise compressive strength to at least 10 MPa; prestressing reinforcement should follow SA8-D-DRW-301-REV0 with corrugated ducts and post-stressing grouting; anchor protection is required for corrosion resistance; monitoring should continue even after micropile strengthening (p. 64)
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