Structural analysis and sustainable solution for Spanish Baroque domes

Baroque architecture dates back to the late sixteenth century in Italy. It adopts the architectural vocabulary of the Roman Renaissance, using it in new rhetorical and dramatic ways to often express the victory of the Catholic Church. The Renaissance absorbed the wealth and power of the Italian courts, blending secular and religious forces. Consequently, the Baroque style was closely tied to the Counter Reformation, a movement within the Catholic Church that sought self-reform in response to Protestant reform (Asghari et al., 2023). While the historical and artistic aspects of Baroque domes are central to their cultural significance, their preservation poses major structural challenges, such as ageing materials and limited load-bearing capacity. These challenges highlight the need for lightweight and adaptive solutions. In this context, tensioned structural systems provide an innovative pathway to balance historical preservation with contemporary engineering demands. A tensioned structural system, characterised by its lightweight and efficient nature, can maximise the advantages of cable tensioning (Feng et al., 2022), making it more conducive to the restoration of domes. Early research on tension primarily concentrated on applications within the fields of architecture and art, with limited exploration of its engineering applications.

The Hermitage of Calvary was chosen for study because of its historical and artistic significance as an exemplary representation of eighteenth-century Baroque architecture in Canet lo Roig, Castellón, Valencia, Spain (Gómez-Jauregui et al., 2023). The building showcases key Baroque elements, such as intricate frescoes, altar paintings and Corinthian columns, making it a valuable subject for studying Baroque design principles (Figure 1). Additionally, it provides an opportunity to explore the integration of modern engineering techniques, like tensile membrane construction, into historic structures. This approach highlights how lightweight, flexible designs can complement and preserve traditional architecture, offering a sustainable solution for restoring and enhancing such buildings without compromising their integrity. The Hermitage not only displays the characteristics and style of Baroque architecture, but also retains precious altarpieces in the typical Baroque style. Therefore, analysing its tensile structure was considered to be an effective approach to better understand the design principles, decorative features and artistic value of Baroque architecture.

It is built with bricks and corner stones, featuring a square stone facade and a lintel at its feet. However, it does not include a tower or bell tower. A polygonal central dome forms the outer shell and the outer apse was also polygonal. The interior decoration boasts plastering and painting. Internally, the dome is located above the overhang, and the drum is situated on the three-sided pillars with Corinthian capitals. They all have very beautiful capitals at the top, and their paintings and graffiti on the pilasters and walls are still preserved. Precious typical Baroque altar paintings are preserved, influenced by Italian style and filled with passion against the backdrop of rockeries. A curved game of light and shadow, as well as numerous decorations, including rockeries and rich shallow-relief images, make it one of the best altar paintings in the region.

Historic Baroque domes face challenges of structural degradation, restoration feasibility and sustainability. Existing reinforcement methods are heavy and invasive, risking damage to cultural identity. Previous research has focused on descriptive architectural analysis with limited structural innovation. This study addresses the gap by proposing a lightweight, self-balanced tensegrity-based dome system that enhances structural performance while preserving historical integrity. The incorporation of tensile membrane constructions into pre-existing buildings has been covered in research conducted by the International Centre for Numerical Methods in Engineering (Cimne). That study demonstrates how these structures can be modified to meet new needs, providing alternatives for buildings that evolve due to expansion or economic necessity. The study highlights how tensile membranes can be used to improve current architectural forms without adding significant extra weight (Armendariz, 2017).

Tensile membrane roofs are substantially lighter than conventional roofing materials, meaning that they do not require as many substantial support structures. The adaptability of tensioned membranes enables architects to produce creative solutions that can improve the aesthetic appeal of old buildings. By reflecting sunlight, tensile structures can help with passive cooling and lower energy use, and recyclable materials are also frequently used. Contemporary tensile membranes are designed to withstand a range of environmental factors, offering durability and minimal maintenance requirements.

Figure 1 shows the architectural features of the Hermitage of Calvary. Serving as an example of Baroque architecture, this building was used to demonstrate the application of a new ring-tensioned integral structure for optimising the dome's mechanical properties. The figure shows how semi-regular tensioned integral structures can enhance a dome's stability, using a combination of tensioned cables and compression struts. The configuration ensures a balanced load distribution while maintaining aesthetic integrity. In this study, a ring-tensioned integral structure is proposed based on the topological relationship of the ring-tensioned combined integral prism of the Hermitage of Calvary. The most fundamental issue in tensioned integral structures is achieving a reasonable geometric configuration through morphological analysis, which enables the structure to have good mechanical properties. The shape adaptability of regular tensegrity structures is limited; for example, extended octahedron tensegrity struggles to adapt to complex designs. Tensegrity structures are self-stable frameworks of bars in compression and tendons in tension, which have attracted interest for over a century. This paper reviews recent real-world applications in fields such as architecture, engineering and construction, robotics and space, highlighting their growing potential (Gong and Li, 2024). An effective load-bearing system for the Baroque dome of the Hermitage of Calvary is provided by the use of an octahedron tensegrity structure and a ring coupled tensioned integral prism. The paper presents a parametric dynamic analysis of cable–strut tensegrity domes, focusing on Levy types. It emphasises prestress identification, natural frequency calculation and the effect of external loads using a geometrically non-linear model (Krishnan, 2020). Three-sided Corinthian pillars provide structural support for the palace's polygonal central dome. Because of this arrangement, the half-octahedron tension-integral unit and semi-regular tensioned integral unit improve geometric stability and guarantee that the tensile system blends in perfectly with the current load distribution. The Winter Palace can leverage the lightweight and high-strength qualities of tensioned constructions, preserving its historic architectural integrity through the use of these cutting-edge structural modules.

In this work, semi-regular tensioned monolithic structures in Baroque architecture were examined and a new type of circular tensioned monolithic structure and a fully tensioned dome structure constructed based on it were developed. Tensegrity cable domes are lightweight structures composed of compressed bars and tensioned cables, offering innovative design potential. Here, a geometric non-linear approach is proposed for design verification, with the aim of optimising structural lightness and sizing criteria (Lin et al., 2025). In large-aperture deployable antennas, the back-to-back double-layer paraboloid design often exceeds launch limits. A cable-dome tensegrity structure with a W-deployable truss is proposed to achieve a lightweight, compact and efficient single-layer mesh reflector antenna. (Logzit and Kebiche, 2020). In view of the geometric and topological relationship of the half-octahedron tensioning whole, combined with balance matrix theory and singular value decomposition (SVD) technology, the geometric design formula, self-stress mode and mechanism displacement mode of the four-bar 12-cable semi-regular tensioning whole element were obtained, and its self-balance and geometric stability were verified. Tensegrity and prestressed cable–strut structures, widely used in civil engineering for their lightweight efficiency, rely on geometric stiffness from prestress for stability. A new topology optimisation model was developed that overcomes the limitations of force method based formulations by fully considering prestress-induced geometric stiffness (Lu et al., 2025). Steel tensegrity domes were investigated, focusing on their self-stress states and potential infinitesimal mechanisms. Tensegrity domes with a standard single-layer dome were compared through qualitative and quantitative analyses using a non-linear model in Mathematica (Ma et al., 2020). Additionally, a design method for assembling a semi-regular element into a circular structure was further developed, and the overall feasible prestress of this type of circular tensioned integral structure was calculated and determined. Tensegrity-based cable domes were studied using a Monte Carlo based approach to identify self-equilibrium configurations, establish force density models and ensure structural stability with optimal prestress. This method efficiently explores the solution space to determine equilibrium configurations and prestress distribution for design and construction (Obara et al., 2023). Furthermore, a new Baroque cable–strut dome structure with full tension and self-balance was constructed by introducing a sunflower-shaped cable dome and a circular tensegrity structure in the middle to achieve seamless docking, and static analysis was conducted on the dome. This study presents the design and analysis of deployable cable domes based on clustered tensegrity structures. Zhang et al. (2023) presented a novel stochastic homotopy method to evaluate the static responses of structures with random variables of any distribution, offering higher accuracy, stability and efficiency than arbitrary polynomial chaos and stochastic finite-element methods. Its effectiveness was demonstrated through numerical examples and a large-scale cable-stayed bridge case (Obara et al., 2024). Lu et al. (2025) proposed a sparse-load identification method using multi-level substructure condensation and response reconstruction, reducing computational complexity and sensor dependence. Numerical and experimental results confirmed its accuracy and effectiveness for large structures (Shu et al., 2025). A cable-dome system is a spatial network of cables and struts stabilised by a perimeter ring beam and supports. This paper presents the structural design approach, simplified modelling and parametric considerations for prestressed radial-type cable domes (Suárez Medina et al., 2014). Retrofitting hybrid cable domes against progressive collapse was investigated using viable strategies. Also in this work, a hybrid cable dome with a tensegrity ring and a traditional Levy cable dome under the alternate path method was assessed, evaluating four retrofit approaches: force-limiting devices, strengthening critical struts, adding inclined bracing cable nets and implementing a bottom cable network (Suárez et al., 2021). The approach modifies a traditional Levy dome into a deployable clustered tensegrity structure by clustering strings for prestress and stability, and analysing quasi-static and dynamic deployment through the deployment ratio (Suárez et al., 2020). The study introduces a new wave-curved tensegrity dome with fewer members than traditional cable domes, derived from a novel tensegrity topology combined with a circular wave surface. A form-finding algorithm based on the Levenberg–Marquardt quasi-Newton method is proposed to establish its feasible prestressed configuration and structural performance (Wang et al., 2025). Shu et al. (2025) presented a two-dimensional bond-based peridynamic model with direction-dependent micro-moduli to simulate deformation, crack propagation and dowel-type connection behaviour in anisotropic timber. The model closely matched experimental results and enabled the evaluation of high-speed loading effects, highlighting the advantages of peridynamics for complex timber failure analysis (Zhang et al., 2023).

The addition of a tensile dome to the Hermitage of Calvary utilised curves and streamlines as its primary design elements, aiming to achieve dynamics and elegance by transforming the church's top arch into a tensile structure. The ornate, graceful curves of Baroque architecture combine with the light, transparent appearance of the tensile structure to create a floating effect. The dome is a tension structure, using high-strength steel cables or strips of material to support and tension the dome surface. This structure creates a long-span, light yet strong dome form and provides greater design freedom. The structure pursues the design of curves and streamlined shapes, creating a dynamic and graceful feeling through elegant curved lines. The curvilinear shape gives the building a soft, artistic atmosphere. Gong and Li (2024) proposed a Voronoi-based rigid spring model for reinforced concrete (RC) structures, simulating concrete, steel bars and bond–slip interactions to predict fracture behaviour. Its effectiveness was validated through numerical and experimental analyses of RC beams and notched concrete plates, including grid sensitivity tests (Zhang et al., 2024).

For numerical modelling and analysis of the proposed circular tensegrity and cable-dome structures, representative material and section properties were adopted. The compression rods were assumed to be structural steel sections, while the tension elements were modelled as high-strength steel cables. These choices are consistent with common practice in contemporary cable-supported dome systems.

• Compression rods (steel members): elastic modulus E_s = 210 GPa, yield stress σ_y = 355 MPa, cross-sectional area A_r = 1500 mm².

• Cables (high-strength steel strands): elastic modulus E_c = 195 GPa, ultimate tensile strength σ_u = 1770 MPa, cross-sectional area A_c = 100 mm².

These parameters were selected to ensure adequate stiffness, realistic stress levels under prestressing and compatibility with the prestress distributions reported in Tables 1 and 2. The assumed properties reflect typical ranges for structural steel and prestressing cables used in long-span dome applications, providing both engineering feasibility and structural safety for the proposed design framework.

The tensioned structure’s half-octahedron tension-integral unit is composed of four compression bars and 12 cables, as shown in Figure 2. The top chord nodes (A, B, C and D) form a square, as do the bottom chord nodes (E, F, G and H). SVD and equilibrium matrix theory were applied to verify geometric stability and self-stress modes of semi-regular tensegrity units. These units were assembled into circular configurations and feasible prestress levels were calculated. A sunflower-shaped cable dome was integrated to form a full-tension, self-balanced structure. Validation is conducted through numerical simulation and non-linear static analysis, confirming minimal displacement and high stiffness under applied loads.

Figure 2 shows the regular half-octahedron tensegrity unit, where the top chord nodes A–D form a square and the bottom chord nodes E–H form another square. Four compression struts connect these two levels, while 12 tensile cables provide stability. The unit illustrates the fundamental tensegrity geometry, where rods resist compression and cables maintain equilibrium through tension.

According to SVD of the balance matrix, the self-stress mode number s = 1 and the displacement mode number m = 3 of the element were obtained, and it was judged that the component was geometrically stable. SVD of the balance matrix helps decompose the system's equilibrium forces and displacements, assessing its stability and self-stress capacity. The self-stress mode number of s = 1 indicates that the structure can maintain equilibrium with one internal force configuration, while the displacement mode number of m = 3 means the structure can move in three independent directions. These values are crucial for evaluating the structure’s stability and mechanical functionality. Since this element cannot form a ring tensioning whole, it needs to undergo a geometric transformation. If the coordinates of the element nodes are arbitrarily changed, there will be no self-stress mode (s = 0, m = 4) in the structure, and there is no possibility of applying prestress. Now, the bottom chord square $EFGH¯$ is transformed into a trapezoid to maintain the coordinate values of nodes B, D, G and H. The coordinates of the other four points are then adjusted to ensure symmetry about the y′ axis, resulting in a semi-regular tensioning overall standard cell, as shown in Figure 3. If the coordinates of point A are $x′A0,y′A0,z′A0$ and point E are$x′E0,1,0$⁠, then the coordinates of point C are$-x′A0,y′A0,z′A0$ and point F are $-x′E0,1,0$⁠. $ABCD¯$ is a spatial quadrilateral and EFGH is an isosceles trapezoid. To ensure the existence of a self-stress mode in the structure after trapezoidal transformation, the node coordinates need to meet:

$x′A0 and x′E0$⁠, as independent variables, integrate the structural balance matrix satisfying Equation 1, and SVD is conducted to investigate the distribution of the minimum singular value $srr$ of the structure, which is of the order of $10-15$⁠. It can be seen that the structure has a self-stress mode, where s = 1 and m = 3, meeting the necessary conditions to become a tensioned whole. Figure 3 illustrates the structural configurations and analyses of the proposed circular tensegrity dome system. The numerical values, such as prestress values and displacement modes, are expressed in standard engineering units: stress is given in Pascals (Pa), force in Newtons (N) and displacement in meters (m). Although the figures show specific structural configurations, no explicit scale is applied to these illustrations. Instead, the values are provided in their respective SI units, focusing on the mechanical and geometric relationships of the tensegrity units and the overall dome structure, optimised for both aesthetic and structural efficiency.

Figure 3 shows that the semi-regular tensegrity unit is derived by trapezoidal transformation of the bottom square (EFGH) into an isosceles trapezoid. At the same time, certain top nodes (B, D, G and H) remain fixed. This adjustment allows for the existence of a self-stress mode, verified by Equation 1 and SVD of the balance matrix, ensuring prestress can be introduced and that the unit achieves geometric stability.

Figure 4 shows the number of semi-regular tensioned integral standard cell members. Taking $x′A0=1.2 and x′E0=0.8$⁠, as an example, the semi-regular unit is introduced. From Equation 1, $y′A0=-0.0337$ and $z′A0=1.325$ can be obtained, thus determining the geometric shape of the element. The unit balance matrix A has dimensions of $24×16$ and a matrix rank of $r=15$⁠. Excluding six rigid body degrees of freedom, then s = 1 and m = 3. There is one self-stress mode (⁠$Vs$⁠) and three independent mechanism displacement modes (⁠$Um$⁠). If the component is prestressed according to the$Vs$ ratio, it can achieve self-balance, ensuring the rod is compressed and the cable is tensioned. The positive definiteness of the geometric moment matrix $GTUm$ (product force matrix) is used to determine that the element is geometrically stable.

Figure 4 shows the numbering system of the semi-regular tensegrity standard cell. Members 1–16 include both rods and cables, with rods carrying compression and cables carrying tension. The balance matrix AAA defines structural equilibrium, and analysis reveals one self-stress mode (ω) as well as three independent displacement modes (δ_i). When prestressed according to the correct ratio, the unit achieves self-balance with rods in compression and cables in tension.

Using surfaces AEH and CFG of the semi-regular tensioned overall unit as the shared splicing surface, and cables 1 and 5 as the shared cables, several semi-regular tensioned overall units are assembled circularly to obtain a circular tensioned overall unit, as shown in Figure 4. The shaded part in Figure 5(a) is a semi-regular structural unit. To ensure the units can be assembled into a circular structure, the projections of E, A and H in the unit on the overall horizontal coordinate plane should form a straight line passing through the centre. According to this projection relationship, the geometric relationship between the coordinates of the tensegrity unit and the standard cell in the ring structure under local coordinates is derived:

where $2R$ is the span of the circular structure support, n is the number of tensioned overall units, $θ$ is the angle between the tensioned overall unit and the$xy$ plane, $x′A$ represents the local$x$ coordinate of point A, and so on. Parameters $R,n and θ$ are then determined. Equations 1 and 2 can then be used to obtain the local coordinates of points A, C, E and F of any tensioned global element in the circular structure, while the local coordinates of points B, D, G and H are, respectively:

Figure 5 illustrates how semi-regular units are assembled into a circular tensegrity structure. Units are joined via shared faces AEH and CFG, with shared cables 1 and 5 ensuring continuity. The structure is defined by span L (the ring’s diameter), the number of units n and the inclination angle θ. Using Equations 2 and 3, the local coordinates of key nodes (A, C, E and F) are derived, allowing the circular form to be generated while maintaining geometric consistency and prestress feasibility. Remodelling of the structural models followed a sequential transformation from the regular half-octahedron tensegrity unit to a semi-regular and finally to a circular tensegrity configuration integrated with a sunflower-shaped cable dome. The process involved geometric transformation of the base unit (Equation 1), circular assembly using angular coordinate mapping (Equation 2) and integration of the dome via shared cables (Equations 2 and 3). Non-linear finite-element analysis incorporating material non-linearity, prestress initialisation and boundary constraints verified the remodelled structure's self-stress stability and load-bearing efficiency. This remodelling enhanced stiffness, minimised displacement and maintained geometric self-balance while preserving the architectural integrity of the Winter Palace dome.

Validating the non-linear static structural analysis model for the Winter Palace dome involved addressing several key factors. Material models had to be clearly defined, particularly for the tensioned cables and compression rods, which required non-linear stress–strain behaviour. Since the eighteenth century, advances in geometry and material knowledge have enabled the development of tensegrity and tensioned structures, which balance bars and cables to create innovative architectural forms. This study reviews these systems and their material considerations for novel structural designs. These materials should be modelled using appropriate non-linear laws, such as plasticity for compression members and strain-hardening models for cables. Geometrical non-linearities, such as large deformations, must also be considered, with large displacement theory applied to account for significant effects on the structure. Robust solution methods, such as Newton–Raphson or arc length algorithms, should be employed to ensure convergence in non-linear analysis, utilising software such as Ansys or Abaqus for these complex computations. Boundary conditions, including support types and loading scenarios, must be carefully applied while factoring in the prestress within the cables. Validation is achieved through comparisons with experimental data, literature benchmarks and sensitivity studies to ensure stability and accuracy. Mesh refinement and convergence studies are necessary to confirm the independence of results from mesh size, and appropriate element types must be used. Lastly, analysing the structural behaviour, including displacement, stress distribution and potential failure modes, ensures that the system performs safely and efficiently.

The feasible prestress of each semi-regular element in the circular tension whole maintains the original element's self-stress unchanged, and the shared cable force is the sum of the original cable forces of the two overlapping cables. Taking n = 12, θ = 60° and 2R = 5 m as an example, the feasible prestress of the circular tension whole is it introduced. The areas of the compression rod and cable are $8.144×10-3 m2$ and $1.857×10-3 m2$⁠, respectively. The yield stress of the steel was taken as$360 MPa$⁠, the ultimate tensile strength of the cable was taken as$1677 MPa$ and the elastic moduli were taken as$2.11×105 MPa$ and $1.95×105 MPa$⁠, respectively. Apply hinge constraints to 12 circumferential support nodes (point G and point H). According to Equations 1 and 3, $x′E=4.527 m,x′A=7.524 m,y′A=-0.359 m and z′A=8.674 m$⁠. Table 1 provides the initial prestress of the structure and the prestress values after collaborative shape finding (see Figure 3 for numbering).

As shown in Figure 6, to verify the feasibility of the semi-regular tensegrity unit and its formation method, several other forms of circular tension-integral units were also constructed.

Figure 6 shows the sunflower-shaped cable dome introduced as an inner dome within the circular tensegrity system. Cables–poles radiate outward in a pattern resembling sunflower petals, with each numbered for reference. When connected, cables 4 and 11 are shared between the sunflower dome and the surrounding circular ring, ensuring seamless integration. The ring acts as a compression beam that balances the dome’s tensile forces. Prestress σ₀, elastic modulus EEE, yield stress f_y and ultimate tensile strength f_u are material parameters governing performance, while hinge-supported nodes G and H anchor the structure.

The cable-dome structure is a tension-integrated or fully tensioned spatial structural system supported on peripheral compression ring beams, first proposed by the study and cable-dome structures have been applied in practical engineering. However, in existing projects, the cable-dome structure is not strictly a fully tensioned cable–strut structure due to the presence of compression ring beams. As an example of the circular tension-integral structure with n = 12, θ = 60°, the process of introducing it as a sunflower-shaped cable dome to form a new type of cable–pole full-tension dome structure is introduced. Figure 7(a) shows the sunflower-shaped cable-dome configuration. Using the symmetry of the cable-dome element, the shaded cable–poles shown in Figure 7(a) are numbered, as shown in Figure 7(b). When the circular tension whole is connected to the cable dome, cables 4 and 11 are shared by both. By adjusting cables 4 and 11 in the cable dome to coincide with the corresponding cables in the circular tension whole, the two structures can seamlessly connect to form a new type of cable–pole full-tension dome structure, as shown in Figure 7(c). Before collaborative shape finding, the initial feasible prestress in the circular tension whole and cable dome of the full-tension dome structure are independent of each other. The initial feasible prestress in the circular tension whole is as described in the above section. The initial feasible prestress in cable 4 can be determined by giving the prestress value of a certain unit. The initial feasible prestress in the shared cables 4 and 11 is the sum of the original cable forces in the two structures.

Figure 7 shows a new cable–pole full-tension dome structure. This structure incorporates a sunflower-shaped cable dome at its core. The figure showcases how the cables are arranged, with the cable–poles numbered for reference, and illustrates the integration of the sunflower dome within the larger tensioned structure. This is part of a larger discussion on using a full-tension cable-dome system, which includes shared cables and compression ring beams for improved structural integrity.

Taking the cable–pole full-tension dome structure with n = 12, θ = 60° and 2R = 5 m and the surrounding ring beam as the basis, static analysis was conducted on the structure. The area of each cable and the area of each compression rod in the circular tensioning system were equal, at $0.01 m2$ and$0.1 m2$⁠, respectively. The area of each cable in the cable dome was equal and the area of each compression rod was equal (⁠$0.001 m2$ and $0.01 m2$⁠), respectively. Among them, the cable areas shared by cables 4 and 11 have the following properties: the yield stress of the steel is $360 MPa$⁠, the ultimate tensile strength of the cable is$1670 MPa$ and the elastic moduli are $2.06×105 MPa$ and $1.95×105 MPa$⁠, respectively. Apply hinge constraints to 12 circumferential support nodes (G and H). After performing non-linear calculations on the structure, the initial prestress values of the structure before and after collaborative shape finding are shown in Table 2. After calculation, the maximum vertical displacement of the structure was determined to be 37.45 mm, the fundamental frequency of the structure was $2.772 Hz$ and the stiffness was relatively high. The dome structure had a fundamental frequency, although the numerical value is not reported. Analysis of the semi-regular tensioned integral unit revealed that it exhibited one self-stress mode and three independent mechanism displacement modes. Prestressing ensured self-balance by maintaining rods in compression and cables in tension, thereby achieving geometric stability. While material properties such as yield stress, tensile strength and elastic modulus for the steel cables and rods are specified, the assumptions on mass distribution are only implicit and no explicit damping model or values are referenced.

To further study and simulate the mechanical performance of the structural system under static load, a vertical uniformly distributed load$of 0.2 kN/m2$ was applied to the cable–pole fully tensioned dome structure. The maximum vertical displacement of the structure after non-linear calculations was 50.23 mm. The deformation of the structure under load was relatively small and the structural stiffness was relatively high. The stress in components under load is shown in Table 2.

Table 3 highlights the representative members with compressive forces across prestress and service states. The analysis accounted for the self-weight of struts, cables, connection hardware and the compression ring, with prestress applied to the tensegrity units and dome system, including shared cables that carry combined effects. In addition to prestress, a vertical uniformly distributed load was considered to evaluate structural performance. Serviceability checks focused on deflection limits, vibration behaviour and overall stiffness, ensuring compliance with standard span-to-deflection criteria for roof structures. Allowable stress checks require cable forces (prestress plus service) to remain within safe limits relative to ultimate strength, and compression members to be verified against both yield and buckling. Shared members, connections and anchorage systems must also be checked for combined prestress and service demands, including long-term fatigue resistance. A final load summary should clearly present self-weight, prestress, service load, load combinations and the verification of stresses and deflection criteria.

Numerical results confirmed that the proposed dome exhibits geometric stability, a feasible prestress distribution and minimal deformation (maximum displacement of 50.23 mm under load). The stiffness was significantly higher than that of conventional reinforcement systems, and the structure achieves self-balance without external compression rings. These outcomes demonstrate that tensegrity-based domes can provide a lightweight, sustainable solution for preserving Baroque architecture while meeting modern engineering standards. By combining balance matrix theory and SVD technology, shape finding analysis of a semi-regular element was conducted, yielding the design formula, self-stress mode and mechanism displacement mode. Based on the process of assembling elements into rings, geometric formulas that satisfy the elements and the tensioned whole were derived. The feasible prestress of the overall structure of the ring-tensioned whole was analysed, and parameterised modelling was conducted. The sunflower-shaped cable dome is introduced in the middle of the circular tension whole, and the two achieve seamless docking by sharing a portion of the cable and hinge nodes. The tension acts as a pressure ring beam, balancing the unbalanced cable force of the outer ring of the cable dome. This achieves a truly full-tension and self-balancing cable–pole dome structural system. The research in this article will play a promoting role in the application of tensioned monolithic structures in practical engineering. In terms of architecture, the design and construction of the dome is a highly skilled project, which fully demonstrates the ingenious craftsmanship and architectural skills of the Baroque architects. In terms of art, the frescoes on the dome are the artist's masterpiece, showcasing the exuberance and virtuosity of Spanish Baroque art. In addition, the dome of the Hermitage of Calvary is also an iconic element of the building, which not only represents the essence of the building but has also become an iconic landscape of the city of Canet lo Roig, Castellón, Valencia, Spain. Overall, the dome of the Hermitage of Calvary is an integral part of the building, which has important historical, artistic and cultural value. At the same time, this church is also one of the cultural heritages of Canet lo Roig, attracting many tourists every year and bringing considerable economic benefits to the town.

Fanwei Meng and Huaiyuan Yu contributed equally to this work and are co-first authors. They were responsible for designing the framework, analyzing the performance, validating the results, and writing the article. Fan Zhang was responsible for collecting the information required for the framework, providing software, conducting critical reviews, and administering the process.