Content area
Tensegrity structures have developed greatly in recent years due to their unique mechanical, structural, and mathematical properties. This study presents the design and fabrication of a tensegrity structure prototype. A pretensioning device is designed, and it is directly integrated into the tension element. This component enables precise application and regulation of cable pretension. Another instrumentation device was designed to enable internal force monitoring during structural testing. A physical prototype of the second member of the Octahedron family, known as the expanded octahedron, was constructed using 1 m long steel struts with a rigid auxiliary support frame specifically designed for this purpose. This frame allows the geometry of the tensegrity structure to be controlled at any stage of the fabrication process, and it proved highly effective—maximum nodal displacements were restricted to ±0.4 mm, and the final prestress state in all 24 cables was achieved within a tight tolerance of ±5% (i.e., 600 ± 30 N). This paper provides an essential methodological reference for the structure’s fabrication and assembly, supporting future experimental analysis of its mechanical response.
Full text
1. Introduction
Tensegrity structures are defined by Pough [1] as “a set of discontinuous compression components interacting with a set of continuous tensile components to define a stable volume in space”. A more precise definition is given in [2] based on their fundamental characteristics: The structure is free-standing, without any external support; The structural members are straight; There are only two different types of structural members: struts carrying compression and cables carrying tension; The struts do not contact with each other at their ends.
Their structural efficiency (low weight combined with high load-bearing capacity), deployability, and visually distinctive shapes have attracted growing interest in both engineering and architectural fields. These characteristics make them attractive for applications ranging from space structures [3,4,5] and adaptive buildings [6,7,8,9] to robotics [10,11,12] and morphing systems [13,14,15].
Numerous studies have investigated the theoretical modeling of tensegrity systems. As identified in [16], the main areas of research related to tensegrity structures include form-finding methods [17,18,19], optimization algorithms [20,21,22], shape control methods [23,24,25], and dynamic stability analysis [26,27,28]. While extensive research on the theoretical modeling of tensegrity structures exists, fewer studies have focused on their actual construction and assembly procedures. Despite this, the construction of tensegrity structures remains a challenging task, often representing a major obstacle to their widespread application in engineering practice [29]. This is because the construction of a tensegrity structure involves controlling its geometry during the prestress introduction [30,31] while also considering element-length tolerances and actual material behavior [32,33].
The implementation of the desired self-stress state is a crucial aspect in the physical realization of the tensegrity concept. The self-stress state stabilizes existing internal infinitesimal mechanisms and, consequently, provides stiffness to the structure. In [29], a physical model of the simplest three-dimensional tensegrity unit (namely, the simplex) was built. The initial self-stress state was achieved by adjusting the measured forces in the struts through variations in strut length. A similar strategy to obtain the desired self-stress state was adopted in [34], where a new fabrication method for a quadruplex module tensegrity was proposed. However, in this case the prestresses of the cables were measured by a tension meter. In [35], a tensegrity based on the simplex with and additional bar was constructed. As the proposed tensile unit has a plane of symmetry, tightening the turnbuckle of two cables induced the required tension of all the other cables which had a specified length. The prestresses of the cables were determined by measuring the frequency of the sound generated by plucking them (similar to a guitar string). Conversely, in [36], the required state of self-stress of a simplex tensegrity was achieved by tightening the turnbuckle located on each cable of the structure. The same approach was followed in [37], where a triplex tensegrity was built and the tension of each cable was introduced by adjusting the turnbuckles.
Another fundamental aspect in the physical realization of tensegrity systems concerns the connection systems between structural elements (i.e., nodes). Nodes play a central role in the mechanical behavior and constructability of tensegrity systems. In [37,38], laser-cut devices made from 20 mm thick stainless-steel sheets were designed and manufactured to form the nodes. In [34], a custom-fabricated hollow, lightweight pin connection system was developed for quadruplex module tensegrities, enabling adjustable cable orientations. Another purpose-built solution is presented in [29], where joints were designed and fabricated by welding gusset plates onto truncated cylinders and using mortise-and-tenon connections to ensure the concurrency of all elements while allowing their orientation according to the theoretical geometry.
The present study introduces a pretensioning device, an instrumentation device and a rigid auxiliary frame to address the challenges of prestress introduction, internal forces monitoring, and geometry control, respectively. The pretensioning device enables the controlled application and regulation of the initial internal force in each cable, thereby establishing and controlling the self-stress state of the structure. Additionally, a separate instrumentation device is incorporated into the cables to allow subsequent monitoring of internal forces, providing a means to evaluate the structural behavior and performance of the tensegrity system over time and the control of the initial prestress. Finally, the rigid auxiliary frame ensures that the compression members remain in their final geometric configuration while the cables are installed and tensioned.
A physical prototype of the expanded octahedron tensegrity (that belongs to the Octahedron family [39,40]) was designed and fabricated to validate the proposed concepts. The prototype was built using steel cables and bars and equipped with the pretension and instrumentation devices, which enable controlled tensioning and internal force monitoring. Nodes designed in this work were realized using standard, low-cost, off-the-shelf components readily available in most hardware or construction supply stores. This choice not only simplifies manufacturing and assembly but also demonstrates that tensegrity structures can be efficiently built using accessible and affordable materials. The constructed prototype represents a single, representative module of the expanded octahedron. While a complete, large-scale application would involve the assembly of multiple such modules, the proposed node solution and the assembly methodology are fully scalable to larger dimensions and complex assemblies.
Despite the theoretical development of tensegrity structures in recent years, there still are limitations to their widespread adoption in structural applications due to the lack of comprehensive, verified, and accessible methodologies for their construction and precise geometric control.
Addressing this gap, the novelty of this work lies in the development of a fully documented and reproducible assembly protocol, complemented by low-cost integrated devices (pretensioning and instrumentation) that achieve a degree of control rigorously acceptable for structural engineering (±0.4 mm displacement and ±5% pretension tolerance).
The ultimate purpose of this work is to meticulously document the entire fabrication and assembly methodology, establishing a comprehensive reference for the subsequent experimental study and analysis of the structure’s mechanical behavior under external loading. By outlining key design principles, materials, and construction method, it aims to support researchers and practitioners in developing reliable and efficient tensegrity systems for scalable applications in structural engineering. This document is intended to guide both theoretical understanding and practical implementation.
The work is organized as follows: Section 2 describes the procedure used to compute the equilibrium configuration and theoretical self-stress state of the prototype. Section 3 presents the physical model of the tensegrity, the rigid auxiliary frame, the pretensioning and instrumentation devices, and the nodes, specifying the materials used in their fabrication. Section 4 outlines the experimental tests conducted for the material characterization, determination of the rigidity of the auxiliary frame, calibration of the instrumentation device, and strength testing of the nodes. Section 5 details the proposed fabrication procedure, emphasizing the assembly sequence, connection details, and geometric control strategy underlying the construction of a relatively complex tensegrity structure, namely the expanded octahedron. Finally, Section 6 summarizes the conclusions reached in the work.
2. Form-Finding and Geometry Control
Determining a self-equilibrated configuration (commonly referred to as the form-finding process) is a critical step in the design of tensegrity structures. Several approaches have been proposed in the literature to address this problem [41,42,43,44,45,46,47], among which the Force Density Method (FDM) [48,49] is one of the most widely used. The inputs of the procedure are the force:length ratio or force density q for each of the m elements, and the connectivity matrix C, which describes how the n nodes in the mesh are linked to one another. The connectivity matrix C () is constructed so that if a member (cable or strut) k connects nodes i and j (with i < j), the ith and jth elements of the kth row are set to 1 and −1, respectively, and 0 otherwise. The m force:length ratios are collected in the vector q (). The FDM formulates the equilibrium equations of the structure as:
(1)
with D = CT·Q·C (), known as the force density matrix, and Q the diagonal square matrix of vector q. Equation (1) correspond to a set of homogeneous equations since no external loads or external supports are considered in the form-finding of a tensegrity structure. Solutions of Equation (1) for x, y, and z different from the trivial solution imply that D cannot have full rank, a condition that is fulfilled given the definition of D:(2)
An additional condition must be imposed on matrix D. To obtain a tensegrity structure in space, that is, 3D, it is necessary to fulfill the non-degeneracy condition [2], according to which the rank deficiency of matrix D must be at least 4. Therefore, certain relationships among the q values of the members must be set so that the first three coefficients of the characteristic polynomial of D vanish. Further insights into the application of the FDM in the form-finding of tensegrity structures can be found in [18]. If only two different force:length ratios are considered, one for struts, qs, and another for cables, qc, then [39] provides the relationship that needs to be fulfilled to obtain the pth super-stable [2,50] member of the Octahedron family:
(3)
with p ≠ 1. Therefore, for the expanded octahedron (p = 2) qs = −3/2 qc.The final three-dimensional configuration of a tensegrity structure is determined by the linear combination of vectors that form a basis for the null space of matrix D. In [51], an analytical approach was introduced that allows designers to control the geometry of tensegrities belonging to the Octahedron family. This method is grounded on the observation that the struts in these structures can be classified into three distinct groups, each characterized by identical lengths and spatial orientations. Depending on the specific configuration within the Octahedron family, the formulation specifies the parameters governing the linear combination of the vectors. These parameters are expressed in terms of the lengths of the strut groups and the angles between them, thus providing full control over the geometry of the structure. In the prototype developed in this study, the strut groups are arranged perpendicularly to one another, and each strut has a length of 1.0 m.
The members of the Octahedron family are composed by rhombic cells. A rhombic cell is formed by four nodes connected through four cables and one strut (see the rhombic cells of the expanded octahedron in Figure 1a). The resulting geometry of the expanded octahedron (with struts of 1.0 m length) is shown in Figure 1b.
3. Description of the Materials
As previously stated, the tensegrity structure considered in this study is an expanded octahedron, as shown in Figure 1. The struts have a nominal length of 1.000 m, while the cables present an ideal length of 0.612 m. It should be emphasized that these values correspond to the ideal geometry and differ from the actual dimensions of the physical prototype. This deviation originates from the practical realization of the nodes. In the theoretical model, nodes are dimensionless points where struts and cables intersect. In contrast, within the physical prototype, each node is materialized through a set of auxiliary components possessing finite dimensions. These components modify the effective connection points between struts and cables, thereby increasing the relative distance between nodes. As a result, the real lengths of both struts and cables must be adjusted to compensate for this additional separation.
A description of the components employed in the fabrication of the specimen is provided in the following sections.
3.1. Struts
The 1 m long struts are made of welded, cold-drawn steel tubes in accordance with UNE-EN 10305-3 [52]. The tubes have an outer diameter of 20 mm and a wall thickness of 1.5 mm. According to the manufacturer, the steel exhibits yield and ultimate tensile strengths of 383 MPa and 408 MPa, respectively. Since the elastic modulus was not provided by the manufacturer, tensile tests were carried out to determine its value. The tests are described in Section 4.
3.2. Tension Elements
The tensile elements in the structure are no longer referred to as simple “cables”, since, in the physical prototype, they are assembled from multiple components. Figure 2 shows a scheme of an actual tensile element.
The tensile elements are anchored to the nodes by means of shackles. Each element incorporates, at one end, a pretensioning device consisting of two threaded eyebolts (one right-hand and one left-hand) connected through a central cylindrical body, which enables controlled shortening or elongation of the assembly to introduce the desired pretension.
The cable itself is a 2 mm diameter 6 × 7 Fiber-Core (FC) steel wire rope, manufactured in accordance with UNE-EN 12385-4 [53]. It has an ultimate tensile strength of 1770 MPa, which yields an ultimate load of 2.35 kN. The cable is directly attached to one of the eyebolts of the pretensioning device through an eye termination, in which DIN 6899/A [54] galvanized steel thimbles (12 mm internal diameter) are inserted and secured by crimping aluminum ferrules according to UNE-EN 13411-3:2023 [55]. The thimbles prevent both wire wear and crushing at the cable eyes. At the opposite end, the cable is connected to the instrumentation device using a similar termination method as the pretensioning device. Finally, the instrumentation device is connected to the corresponding node by means of a steel snap hook and a shackle. The combination of the snap hook and the shackle avoids the effect of any secondary moment from being registered by the instrumentation device.
3.3. Pretensioning and Instrumentation Devices
The geometric design of the pretensioning device responds to the need to introduce the initial prestress state of the tensegrity structure. It must act as a conventional turnbuckle, shortening or lengthening the steel cable to adjust the level of prestress. Traditional commercial turnbuckles have a relatively low load-to-size ratio. Therefore, a specifically designed device was developed in this work.
The central body of the pretensioning device is a 60 mm long, 8 mm diameter cylindrical component made of free-machining stainless steel SS303. It includes a 4.2 mm through hole and two M5 threaded ends (20 mm each, one right-hand and one left-hand) where the eyebolts are inserted.
In addition to the pretensioning device, a dedicated instrumentation device was incorporated into selected tensile elements to enable accurate monitoring of internal forces during both pretensioning and subsequent structural testing. This device provides a direct measurement of the axial force in the cable, allowing the evaluation of the self-stress state, the structural response under external actions, and the long-term performance of the tensegrity system.
The central body of the instrumentation device is a 60 mm long, 8 mm diameter cylindrical component made of aluminum alloy 2011. It includes a 4.2 mm through hole and two M5 threaded ends (20 mm each) where the eyebolts are inserted. Aluminum 2011 was selected over steel to reduce stiffness and enhance axial strain sensitivity for sensing purposes. Two 3 mm strain gauges (one longitudinal and one transverse) are bonded onto the cylinder and connected to a data acquisition system (HBM Quantum MX1615B) in a half-bridge configuration, which amplifies the signal and improves measurement sensitivity. The instrumentation device and the arrangement of the strain gauges are shown in Figure 3.
3.4. Nodes
The nodes in the structure are based on steel M12 eyebolts anchored at the ends of the steel tubes by means of expansion anchors. Each anchor consists of an external steel expanding sleeve with an initial diameter of 16 mm and a length of 80 mm (see Figure 4a). When the eyebolt is tightened, the sleeve expands radially against the inner wall of the tube, ensuring a secure mechanical connection between the eyebolt and the strut. The eyebolt facilitates the connection of the shackles of the tensile elements (see Figure 4b).
3.5. Rigid Auxiliary Frame
Given that the number of elements to be connected in the tensegrity structure is relatively high, an auxiliary frame was built to assist and facilitate the fabrication process. Moreover, since preserving the geometry of the structure during construction is particularly important, this frame also serves to maintain the desired configuration throughout the assembly by fixing the position of the struts.
The rigid auxiliary frame is built using 30 × 30 mm Bosch Rexroth anodized aluminum profiles. These profiles combine high bending stiffness with low weight and feature a wide range of standardized accessories that allow quick and easy connection between bars. This modular system also facilitates the assembly process and enables straightforward adjustments during the construction of the structure.
As the struts in the expanded octahedron to be built generate three orthogonal planes (see Figure 1), the frame itself consists of three square frames aligned with these planes (see Figure 5a). To provide additional stiffness, two extra bars are added to each square frame to support the struts, called secondary bars. Near the ends of these additional bars, rods are provided with brass clamps to secure the tensegrity struts (see Figure 5b). These rods enable the precise setting of the angles between the struts in the completed tensegrity structure.
To ensure the accuracy of the auxiliary frame, its fabrication was carried out under rigorous quality control, verifying the orthogonality of its corners and the parallelism and distance between the axes of the supports employing a laser alignment station and measuring devices with a geometric tolerance of 0.01 mm.
4. Experimental Campaign
To accurately determine the mechanical behavior of the materials and devices employed in the prototype, a series of experimental tests were conducted. These tests were specifically designed to characterize relevant mechanical parameters, such as stiffness, deformation response, and sensitivity. The results obtained provided quantitative data essential for the subsequent calibration and validation of the components, ensuring their reliable performance within the overall structural system.
4.1. Struts
Two different experiments were carried out on the steel tubes corresponding to the struts. On one hand, tensile tests were performed to determine the elastic modulus of the tubes. On the other hand, compression tests were conducted to determine their buckling capacity, as these members are primarily subjected to compression in the tensegrity structure.
4.1.1. Tensile Tests
To determine the elastic modulus of the steel tubes used as struts, tensile tests were performed on three specimens, each 330 mm in length. The strut specimens were manufactured by introducing expansion anchors at both ends to reinforce these zones in order to prevent local deformation or crushing caused by the pressure of the testing machine jaws. Each reinforced zone was 90 mm long, leaving 150 mm of hollow section in the middle of the specimen. Special care was taken to align the reinforced zones with the gripping zones, ensuring that the free length between the jaws was not affected by the reinforced sections. An extensometer was placed at the central portion of each specimen to record its elongation in that region (see Figure 6a). The average elastic modulus remains 149.056 GPa with a coefficient of variation of 7.1%. Table 1 summarizes the values obtained from the three tensile tests on the tubes, and Figure 6b,c show the typical load–displacement and stress–strain responses of a tube specimen.
4.1.2. Buckling Tests
The buckling test specimens were 1000 mm long bar segments, identical to the struts of the structure to be built. To achieve a simply supported configuration (pinned–pinned), the anchors of the eyebolts (Figure 4a) used in the actual joints were attached to both ends. A threaded rod was inserted into each anchor, with a cap nut at its tip, ensuring contact at a single point while allowing free rotation during loading.
A custom gripping fixture (Figure 7a) was fabricated to allow proper clamping by the testing machine jaws while accommodating the cap nut (Figure 7b). The fixture was machined from a solid steel circular bar of 25 mm in diameter; a 50 mm long segment was cut using a 26 mm drill bit and used to machine a conical profile at the end of the piece on the lathe, forming a socket for the cap nut. Each specimen was then placed between the two gripping fixtures and loaded in compression until failure by buckling occurred after reaching its maximum compressive capacity (Figure 7d).
Given the strong influence of initial imperfections on the buckling strength of structural members, nine specimens were tested to ensure a representative dataset. The maximum load at buckling for each tested specimen is reported in Table 2. A noticeable scatter was observed in the results, with a coefficient of variation of 15.86%, attributed to the aforementioned initial imperfections. To account for this variability, the first decile was adopted instead of the average value (8.835 kN) as the representative buckling load of the struts, resulting in a value of 7.203 kN.
4.2. Tensile Elements
As described previously, the tensile elements are assembled from different components. To assess their mechanical performance, two sets of tests were conducted. The first focused on characterizing the cable itself, determining its ultimate load and elastic modulus. The second evaluated the complete tensile element, quantifying the overall deformability of the system and confirming that failure occurred in the cable.
4.2.1. Cable Behavior
Three cable specimens were tested under uniaxial tension. All specimens were fabricated following the same procedure used for the cables of the tensegrity structure: each end was terminated with an eye loop reinforced with galvanized steel thimbles. The total length between the outer ends of the thimbles was 500 mm.
Preliminary tests were carried out to determine the number of aluminum ferrules required to prevent slippage at the eye terminations. It was found that using two aluminum ferrules provided the best performance. The ferrules were installed in accordance with the UNE-EN 13411-3:2023 standard [55], ensuring that the applied crimping pressure did not damage the wire rope.
For the tensile tests, the specimens were mounted on a universal testing machine, replacing the conventional clamping jaws with a pin-type fixture, see Figure 8a. Each pin passed through the thimbles, reproducing the real boundary conditions of the connection with the shackles in the tensegrity structure.
The tests were performed under monotonic loading up to failure, controlling the displacement between pins at a rate of 1.5 mm/min. An extensometer was placed at the central region of the specimen to record its elongation and determine the elastic modulus of the cable, see Figure 8a.
Table 3 summarizes the results obtained from the tensile tests on the three cable specimens. The average ultimate load was 3.17 kN, Figure 8b, showing very limited dispersion among samples (coefficient of variation below 3%). The cables exhibited a linear elastic response up to approximately 80% of their ultimate load, at which point the extensometer was removed to prevent damage, Figure 8c. Beyond this stage, the tests continued up to failure, which occurred in the wire strands without any evidence of slippage at the eye terminations. The mean elastic modulus was 146.75 GPa, which is consistent with typical values reported for 6 × 7 FC steel wire ropes with a nominal tensile strength of 1770 MPa.
4.2.2. Tensile Element Behavior
Three specimens reproducing the configuration of the complete tensile element (Figure 2) were also tested under uniaxial tension. The setup of these tests was similar to that used in the previously described cable experiments. The average maximum load reached was 3.14 kN, comparable to that sustained by the isolated wire rope. The average elongation at break was 14.684 mm, which is considered sufficient to accommodate deformations without compromising the global stability of the tensegrity structure. Table 4 summarizes the results obtained from the three specimens. Figure 9 shows the comparison of the force–displacement behavior between a cable specimen and an assembled tensile element specimen. It can be observed that the complete tensile element exhibits a stiffer response than the cable alone. An additional observation from these tests is that failure consistently occurred in the cable.
4.3. Instrumentation Device
Prior to their use in the assembly of the tensegrity structure, the 24 instrumentation devices were calibrated to establish the relationship between the half-bridge strain gauge readings and the resulting tensile force. This calibration enables accurate control and monitoring of the target prestress during assembly, as well as subsequent assessment of the structural response under external loading. As mentioned in Section 3.3, two 3 mm strain gauges were bonded at the midspan of the instrumentation device cylinder (one oriented longitudinally and the other transversely). When connected in a half-bridge configuration to the data acquisition module, the strain output is amplified by a factor of approximately 1.33, given that the Poisson’s ratio of aluminum is around 0.33. This configuration also provides thermal compensation, minimizing temperature-induced measurement drift.
Before the calibration test was performed, it was essential to ensure that the measurement conditions matched the service conditions within the tensegrity structure. Since the measuring devices are cylinders connected to the cables via threaded eyebolts (Figure 3), any movement in these connections could affect the measurement accuracy. Therefore, before beginning the calibration process, the eyebolts were securely fixed into the cylinders using a locknut and thread locking adhesive to prevent any subsequent rotation or shifting. This rigid fixation had an additional benefit: it captures the secondary bending moment introduced by the axial load due to possible thread misalignment at both ends of the cylinder. This ensures the secondary moment effect remains identical during both the calibration process and in-service monitoring, guaranteeing that the strain measurement accurately reflects the true axial force state.
To calibrate the pretensioning devices, each unit was subjected to a tensile test in a universal testing machine up to 1.5 kN, utilizing a constant crosshead speed of 1.5 mm/min to ensure quasi-static testing conditions. The load introduced by the machine and the strain measured by the strain gauges were registered by an HBM Quantum MX1615B data acquisition system at a sampling rate of 10 Hz. This load is approximately 50% of the cable’s ultimate load and reasonably covers the pretensioning load level to be introduced. The load provided by the machine was recorded along with the strain measurement from the half-bridge. The procedure was repeated three times to ensure the representativeness of the measurements. All collected data were grouped, and the linear relationship between the load and strain magnitudes was obtained through linear regression. The 95% confidence intervals were computed for the slope and the intercept of the regression. The maximum measurement uncertainty, ΔF95%, was calculated by applying the principles of uncertainty propagation (as defined in the Guide to the Expression of Uncertainty in Measurement [56]) to the 95% confidence intervals of the linear regression parameters (slope and intercept). This calculation provides a conservative estimate of the maximum force error across the calibration range, ensuring the reported precision is robust. Table 5 shows the obtained results.
4.4. Nodes
As mentioned in Section 3.4, the structural nodes are formed by expansion eyebolts inserted into the ends of the hollow struts (Figure 4). The manufacturer specifies a tightening torque of 50 Nm for these components; however, the employed tubes cannot safely withstand torques above 40 Nm. This torque value becomes relevant only if the node is subjected to a resultant force from the tensile elements that induces tension in the strut (a situation that is unlikely to occur). Nevertheless, since the expansion eyebolt must be tightened to ensure its correct placement within the node, it is necessary to determine the maximum torque that can be applied without causing excessive deformation at the tube end during the expansion of the eyebolt sleeve. Once this limit torque is established, the corresponding maximum tensile force that the tube could safely resist is subsequently evaluated.
4.4.1. Tensile Element Behavior
For this test, three tube specimens, each 300 mm in length, were prepared. One end of the specimen was clamped in a bench vise, and an expansion eyebolt was inserted into the opposite end. The eyebolt was then tightened in 2 Nm increments using a calibrated torque wrench. Since the expansion of the eyebolt sleeve occurs approximately 80 mm from the tube end, ten measurements of the outer diameter were taken at that location with a caliper accurate to 0.01 mm after each torque step. The average diameter and its standard deviation were subsequently calculated. The maximum allowable torque was considered to have been reached when the variation from the initial outer diameter exceeded 0.5% (a deformation at which the diameter increase is visually detectable). Table 6 summarizes the results of the tests performed. Finally, a tightening torque of 20 Nm was adopted.
4.4.2. Tension Test on the Node
Three additional tube specimens, each 300 mm in length, were prepared for the tensile tests. At one end of the specimen, an anchor was introduced to reinforce it (following the same procedure used for the strut tensile tests), while an expansion eyebolt was installed at the opposite end and fastened using the torque determined in the previous test (20 Nm). The assembled specimens were then tested under uniaxial tension in a universal testing machine, applying the load through the eyebolt until failure occurred. The failure mode was consistently the extraction of the eyebolt. Table 7 presents the results. Although a high dispersion was observed in the maximum sustained load, this value was always greater than the maximum load sustained by both the isolated cable and the entire tensile element (see Table 3 and Table 4, respectively).
4.5. Rigid Auxiliary Frame
The control of the geometry of a tensegrity structure during the prestress introduction is a complex task because the equilibrium shape of the structure is directly related to the prestress state. For this reason, an auxiliary frame was employed as a geometric reference throughout the assembly and prestressing process. Its main function is to ensure the correct positioning of the nodes of the structure, allowing the bars to be fixed precisely and the cables to be tensioned progressively until the equilibrium shape is reached. In addition, the frame must provide sufficient stiffness to withstand the forces induced by prestressing without appreciable deformation.
An experimental test was carried out to verify that the stiffness of the auxiliary frame was adequate and that the displacements occurring during the prestressing phase remained negligible. The test consisted of assembling the tensegrity structure while applying a prestress level of 600 N to the tensile elements (corresponding to 1470.6 N of compression in the struts, according to Equation (3) and considering the lengths of cables and struts). This value was selected because it represents approximately 20% of the ultimate load capacity of the tensile element (3.17 kN), ensuring both structural rigidity and a sufficient safety margin. Simultaneously, six HBM 1-WA/100 mm LVDTs were used to measure the relative displacement between the two parallel struts of each pair of compressed members. In the expanded octahedron, three such pairs of parallel struts are present.
The LVDTs were placed to detect two critical geometric deviations: changes in the distance between parallel struts and loss of parallelism between them. Although the ideal position was near the strut-frame connection points (brass clamps), the actual spacing between the LVDT axes was constrained to 740 mm due to the clearance required by the auxiliary supports and fixtures used for reliable sensor attachment.
Figure 10a shows the general experimental setup and the arrangement of the LVDTs with their nomenclature, while Figure 10b provides a close-up view of one sensor and the auxiliary components used to attach it to a strut and measure the displacement in contact with the opposite member of the pair.
Figure 11 shows the displacement records from the six LVDTs, while Figure 12 presents the variation in the angle between each pair of parallel bars during the pretensioning process. This angle was computed by considering that its tangent equals the ratio between the differential readings of the two LVDTs in the same group and the distance separating them. As observed in the figures, the maximum displacement recorded during the procedure was 0.7008 mm, and the maximum angular variation between bars of the same group was 0.0418°. The final displacements measured by all LVDTs remained below ±0.4 mm, and the variation of angles between initially parallel bars was always below ±0.5°. These results confirm that the deformations of the auxiliary frame during the prestressing process are negligible, demonstrating that its stiffness is sufficient to preserve the designed geometry of the tensegrity structure throughout assembly.
5. Fabrication Method
The assembly of all the elements that compose the tensegrity structure follows a logical protocol. Firstly, the auxiliary frame is mounted, ensuring the struts are placed on the secondary bars of the frame. The placement must ensure that parallel struts face each other, and that no secondary bar with the same orientation (i.e., supporting the struts) remains in the span between them. Before proceeding, the nodes presented in Section 3.4 must be installed and tightened at the ends of the struts using the tightening torque determined in the tests (20 Nm). It is crucial that the pair of eyebolts of the nodes on each strut is aligned such that the circle (the loop) lies in the same plane.
Once the complete frame is assembled, the struts are provisionally placed on the secondary bars using metallic clamps. Subsequently, the final position of the struts is achieved by leveling the frame and centering the struts themselves. This process ensures the correct distances between struts, the parallelism among struts within the same group, and the perpendicularity with the other groups.
At this stage, the tensile elements are provisionally connected in a loose state between the struts’ eyebolts using shackles, which greatly facilitates the assembly process, following the connection scheme of the expanded octahedron. Since the tensile elements are asymmetric (having a pretensioning device at one end and an instrumentation device at the other), and four tensile elements converge at each eyebolt, they must be strategically placed. The arrangement used ensures that each eyebolt anchorage accommodates two pretensioning devices and two instrumentation devices.
Following the provisional placement, the structure was subjected to a pretensioning process. The target initial tension for each tensile element was set at 600 N. The pretensioning was applied in a staged manner, aiming to achieve 50% of the final pretensioning load in each tensile element during each stage. The pretensioning process was considered complete when the tensile force indicated by every load-measuring device fell within the tolerance range of ±5% of the final pretensioning force (±30 N).
The successful execution of the staged pretensioning process is illustrated in Figure 13, which presents the load history recorded by all 24 load-measuring devices. To ensure clarity, the data are presented across six subfigures, so that each one represents the four cables that compose the six basic cells of the expanded octahedron (Figure 1a). Each subplot demonstrates the stepwise application of the load and the final convergence of all elements into the tolerance band defined by the target load of 600 ± 30 N.
The tensioning process was performed node by node, starting at Node 1 and proceeding sequentially to tighten the cables converging at each node. This interconnected structural behavior is evidenced in Figure 13 by the fact that all curves show an increase in tension simultaneously with the application of tension to any single cable, demonstrating a high degree of correlation between the tensile elements. The stepped shape of the curves confirms this sequential and methodical application of the load. The final configuration of the structure was achieved after 4100 s of work, that is, 1 h, 8 min, and 20 s. Considering the complexity of the geometry and the number of elements involved, this execution time is deemed highly efficient.
The fabrication of the tensegrity structure is finalized with its extraction from the auxiliary frame once the target prestressing state has been achieved. The selection of Bosch Rexroth aluminum profiles and their specific accessories allows us to benefit from both the system’s high rigidity during assembly and its inherent ease and speed of mounting/disassembly. The frame is removed by detaching the necessary connecting bars in a controlled sequence, ensuring the tensegrity structure is released without being disturbed.
6. Conclusions
In this work, the complete fabrication and prestressing procedure of an expanded-octahedron tensegrity structure is presented, highlighting the necessary steps and control measures for ensuring that its final geometry corresponds to the theoretical equilibrium configuration. The results obtained throughout the assembly process allow the following conclusions to be drawn: The tensile element defined in this work exhibited highly consistent behavior, with low dispersion in ultimate load and a well-defined elastic response. Failure consistently occurred in the cable, confirming that the termination system with thimbles and ferrules performed adequately and did not compromise structural integrity. The introduction of prestress and the monitoring of the internal force of each tensile element have been achieved by incorporating a pretensioning and an instrumentation device, respectively, into the tensile elements. The pretensioning device defined in this work enables controlled application and regulation of the internal force of each tensile element. On the other hand, the instrumentation device has proven to be effective for monitoring the internal force. They showed a stable and repeatable linear response, allowing reliable calibration through strain measurements obtained from a half-bridge configuration. In tensegrity structures, nodes act as pin-jointed connections. The physical realization of this type of connection plays a central role in the mechanical behavior and constructability of tensegrity structures. The nodes defined in this work are based on steel M12 eyebolts anchored at the ends of the steel tubes by means of expansion anchors. Several experimental tests have been carried out to determine the tightening torque and the maximum tensile force that the node could safely resist in relation to the tensile elements connected. The node proposed allows for the accommodation of four tensile elements without disturbing each other (two pretensioning devices and two instrumentation devices). The introduction of prestress in a tensegrity structure requires strict geometric control, as the final equilibrium shape is highly sensitive to deviations in nodal positioning. For this reason, the use of a rigid auxiliary frame proved essential. Its stiffness ensured that the imposed prestressing forces did not induce significant deformations, as confirmed experimentally by LVDT measurements showing maximum displacements within ±0.4 mm and angular deviations always within ±0.5°. The assembly protocol developed in this work provides a clear and reproducible methodology for positioning the struts, aligning the eyebolts, and connecting the tensile elements. The use of provisional supports, combined with a systematic leveling and centering process, guarantees the parallelism and orthogonality conditions required by the expanded-octahedron geometry. The staged pretensioning strategy was successful in achieving a uniform and con-trolled prestress state in all tensile elements. Applying the tension in increments of approximately 50% of the target load allowed for progressive equilibration of the structure, avoiding sudden instabilities. The final forces measured by the 24 sensors fell within the prescribed tolerance of 600 ± 30 N, demonstrating both the precision of the procedure and the capacity of the system to reach a stable self-equilibrated state.
The proposed methodology constitutes a reliable framework for the laboratory construction of tensegrity structures, offering a validated sequence of operations (geometric control, auxiliary frame verification, assembly of struts and cables, and staged pretensioning) that ensures the reproduction of the intended tensegrity form. This framework provides a robust basis for future experimental research and for the practical implementation of tensegrity systems in structural applications. This foundation is being actively used in ongoing work, which will focus on critical areas such as subjecting the assembled prototype to external mechanical loading to characterize its mechanical behavior; developing and validating a detailed numerical model using the measured parameters; and executing a long-term study to quantify the effects of creep, stress relaxation, and temperature changes on the pretension force loss.
Conceptualization M.A.F.-R. and J.F.C.-M.; Methodology, M.A.F.-R. and J.F.C.-M.; Validation J.F.C.-M. and P.L.-J.; Formal Analysis, J.F.C.-M.; Investigation, M.A.F.-R., J.F.C.-M., M.H.-G.-M. and P.L.-J.; Resources, P.L.-J. and M.H.-G.-M.; Data Curation, J.F.C.-M. and M.H.-G.-M.; Writing—Original Draft Preparation, J.F.C.-M.; Writing—Review and Editing, M.A.F.-R. and M.H.-G.-M.; Project Administration, M.A.F.-R.; Funding Acquisition, M.A.F.-R. All authors have read and agreed to the published version of the manuscript.
The data that support the findings of this study are available from the corresponding author, J.F. Carbonell-Márquez, upon reasonable request.
The authors wish to thank Felipe García-Sánchez and Germán Castillo-López from the University of Málaga for their invaluable suggestions. The authors also wish to acknowledge the assistance provided by University of Málaga laboratory technician José Bernardo Beltrán Ramos in performing the presented tests.
The authors declare no conflicts of interest.
Footnotes
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.
Figure 1 Expanded octahedron (colored lines represent the struts; each color refers to a group): (a) schematic view of the six rhombic cells composing the expanded octahedron, showing the numbering of the nodes (black) and the detailed numeration of the 24 tensile elements (gray, italicized) by cell; (b) perspective view; (c) front view; (d) side view; (e) plan view.
Figure 2 Tensile element for the physical prototype of the expanded octahedron. All the components are identified.
Figure 3 Instrumentation device: (a) instrumentation device with threaded eyebolts; (b) detail of the longitudinal and (c) transverse strain gauges bonded to the instrumentation device.
Figure 4 Expansion anchor eyebolt for node assembly: (a) main dimensions; (b) detail of the tensile elements connected to the eyebolt via shackles.
Figure 5 Rigid auxiliary frame for tensegrity fabrication: (a) three orthogonal square frames; (b) complete frame with additional bars equipped with metal clamps for the struts.
Figure 6 Tensile test for a tube specimen: (a) experimental setup; (b) load–displacement curve; (c) stress–strain response for the same tube.
Figure 7 Buckling test for tube specimens: (a) detail of one of the gripping fixture devices; (b) cap nut at the socket of the gripping fixture; (c) general setup; (d) force-displacement response for one of the tube samples subjected to the buckling test.
Figure 8 Cable tensile test: (a) experimental setup; (b) load–displacement curve; (c) stress–strain response for the same cable.
Figure 9 Force–displacement curves of the isolated cable and the assembled tensile element.
Figure 10 General configuration of the rigid auxiliary frame test with the nomenclature given to each LVDT (a) and detail of the 3-A LVDT attachment to the struts (b).
Figure 11 LVDT displacement measurements used to monitor frame deformations during the prestressing process of the tensegrity structure.
Figure 12 Evolution of the angle between parallel struts, computed from LVDT measurements during the prestressing process of the tensegrity structure.
Figure 13 History of the measured load versus time for the 24 tensile elements during the staged pretensioning process. The insets show a magnified view of the final convergence into the tolerance band, defined by the dashed lines at 630 N and 570 N (i.e., ±5% of the target load of 600 N). Data are grouped into six subfigures corresponding to the cables in each basic cell of the expanded octahedron (
Results of tensile tests on the steel tubes.
| Specimen | Elastic Modulus (GPa) |
|---|---|
| 1 | 160.988 |
| 2 | 139.781 |
| 3 | 146.399 |
| Average | 149.056 |
Results of buckling tests on the steel tubes.
| Specimen | Buckling Load (kN) |
|---|---|
| 1 | 11.039 |
| 2 | 9.485 |
| 3 | 7.628 |
| 4 | 6.459 |
| 5 | 7.389 |
| 6 | 8.395 |
| 7 | 10.065 |
| 8 | 10.040 |
| 9 | 9.012 |
| Average | 8.835 |
Results of tensile tests on 6 × 7 FC steel wire rope specimens.
| Specimen | Maximum Load (kN) | Elastic Modulus (GPa) |
|---|---|---|
| 1 | 3.288 | 145.101 |
| 2 | 3.172 | 151.971 |
| 3 | 3.113 | 143.185 |
| Average | 3.171 | 146.752 |
Results of tensile tests on the tensile element specimens.
| Specimen | Maximum Load (kN) | Elongation at Break (mm) |
|---|---|---|
| 1 | 3.129 | 14.560 |
| 2 | 3.129 | 15.647 |
| 3 | 3.162 | 13.846 |
| Average | 3.140 | 14.684 |
Linear regression parameters for the calibration of the instrumentation devices.
| Device | Slope (N/µε) | Intercept (N) | R2 | ΔF95% (N) | ||
|---|---|---|---|---|---|---|
| Value | 95% CI Half-Width (10−4) | Value | 95% CI Half-Width | |||
| 1 | 3.048 | 8.274 | −5.925 | 0.195 | 0.99976 | 0.593 |
| 2 | 3.345 | 7.302 | −4.563 | 0.158 | 0.99984 | 0.479 |
| 3 | 3.705 | 4.286 | −2.458 | 0.086 | 0.99996 | 0.257 |
| 4 | 3.330 | 7.111 | −1.933 | 0.152 | 0.99985 | 0.467 |
| 6 | 3.297 | 5.030 | −6.187 | 0.106 | 0.99992 | 0.332 |
| 7 | 3.344 | 3.981 | −1.101 | 0.094 | 0.99995 | 0.269 |
| 8 | 3.771 | 4.087 | 3.765 | 0.077 | 0.99996 | 0.237 |
| 9 | 3.541 | 9.589 | −2.238 | 0.187 | 0.99974 | 0.591 |
| 11 | 3.319 | 2.207 | −0.471 | 0.049 | 0.99999 | 0.147 |
| 12 | 3.430 | 7.447 | −7.584 | 0.157 | 0.99985 | 0.478 |
| 13 | 3.853 | 2.279 | 0.951 | 0.040 | 0.99999 | 0.127 |
| 14 | 3.111 | 12.756 | −8.711 | 0.290 | 0.99944 | 0.892 |
| 16 | 3.919 | 8.736 | 11.620 | 0.159 | 0.99984 | 0.487 |
| 17 | 3.058 | 9.532 | −7.130 | 0.211 | 0.99965 | 0.668 |
| 18 | 2.791 | 8.733 | −8.908 | 0.226 | 0.99968 | 0.684 |
| 19 | 4.117 | 9.237 | 0.572 | 0.151 | 0.99981 | 0.486 |
| 21 | 3.902 | 11.614 | 14.159 | 0.207 | 0.99971 | 0.645 |
| 22 | 3.401 | 4.434 | 1.081 | 0.092 | 0.99994 | 0.284 |
| 23 | 3.937 | 10.866 | 3.106 | 0.188 | 0.99973 | 0.600 |
| 24 | 4.127 | 18.542 | 11.653 | 0.310 | 0.99932 | 0.981 |
| 26 | 3.970 | 13.232 | 5.616 | 0.249 | 0.99967 | 0.744 |
| 27 | 3.537 | 4.572 | −0.429 | 0.089 | 0.99994 | 0.281 |
| 28 | 3.513 | 5.586 | −0.552 | 0.123 | 0.99993 | 0.358 |
| 29 | 5.178 | 31.197 | 14.904 | 0.412 | 0.99876 | 1.316 |
Results of the diameter change versus tightening torque.
| Tightening Torque (Nm) | Diameter (mm) | Relative Change (%) | |
|---|---|---|---|
| Average | σ | ||
| 0 | 19.94 | 0.02 | |
| 10 | 19.97 | 0.02 | 0.12 |
| 12 | 20.00 | 0.01 | 0.26 |
| 14 | 20.01 | 0.01 | 0.33 |
| 16 | 20.03 | 0.02 | 0.42 |
| 18 | 20.05 | 0.02 | 0.51 |
| 20 | 20.06 | 0.02 | 0.58 |
| 22 | 20.08 | 0.02 | 0.70 |
| 24 | 20.11 | 0.03 | 0.84 |
Results of tensile tests on the nodes.
| Specimen | Maximum Load (kN) |
|---|---|
| 1 | 3.527 |
| 2 | 5.343 |
| 3 | 9.326 |
| Average | 6.065 |
1. Pugh, A. An Introduction to Tensegrity; University of California Press: Berkeley, CA, USA, Los Angeles, CA, USA, 1976; ISBN 0520030559
2. Zhang, J.Y.; Ohsaki, M. Tensegrity Structures: Form, Stability, and Symmetry; Springer: New York, NY, USA, 2015; 6.
3. Chen, M.; Goyal, R.; Majji, M.; Skelton, R.E. Deployable Tensegrity Lunar Tower. Earth and Space 2021: Space Exploration, Utilization, Engineering, and Construction in Extreme Environments, Proceedings of the 17th Biennial International Conference on Engineering, Science, Construction, and Operations in Challenging Environments, Virtually, 19–23 April 2021; American Society of Civil Engineers (ASCE): Reston, VA, USA, 2021; pp. 1079-1092. [DOI: https://dx.doi.org/10.1061/9780784483374.100]
4. Zhang, J.; Shi, C.; Geng, K.; Chen, Y.; Guo, H.; Liu, R.; Kou, Z. Design and Cushioning Performance Analysis of Spherical Tensegrity Structures. Aerospace; 2025; 12, 453. [DOI: https://dx.doi.org/10.3390/aerospace12060453]
5. Knap, L.; Świercz, A.; Graczykowski, C.; Holnicki-Szulc, J. Self-Deployable Tensegrity Structures for Adaptive Morphing of Helium-Filled Aerostats. Arch. Civ. Mech. Eng.; 2021; 21, 159. [DOI: https://dx.doi.org/10.1007/s43452-021-00292-6]
6. Wang, Y.; Senatore, G. Design of Adaptive Structures through Energy Minimization: Extension to Tensegrity. Struct. Multidiscip. Optim.; 2021; 64, pp. 1079-1110. [DOI: https://dx.doi.org/10.1007/s00158-021-02899-y]
7. Renner, M.; Spyridonos, E.; Dahy, H. Tensegrity FlaxSeat: Exploring the Application of Unidirectional Natural Fiber Biocomposite Profiles in a Tensegrity Configuration as a Concept for Architectural Applications. Buildings; 2024; 14, 2490. [DOI: https://dx.doi.org/10.3390/buildings14082490]
8. Angelucci, G.; Mollaioli, F.; Tardocchi, R. A New Modular Structural System for Tall Buildings Based on Tetrahedral Configuration. Buildings; 2020; 10, 240. [DOI: https://dx.doi.org/10.3390/buildings10120240]
9. Froli, M.; Laccone, F.; Maesano, D. The TVT Glass Pavilion: Theoretical Study on a Highly Transparent Building Made with Long-Spanned TVT Portals Braced with Hybrid Glass-Steel Panels. Buildings; 2017; 7, 50. [DOI: https://dx.doi.org/10.3390/buildings7020050]
10. Tang, G.; Yang, Q.; Lian, B. Design and Experimentation of Tensegrity Jumping Robots. Appl. Sci.; 2024; 14, 3947. [DOI: https://dx.doi.org/10.3390/app14093947]
11. Hao, S.; Liu, R.; Lin, X.; Li, C.; Guo, H.; Ye, Z.; Wang, C. Configuration Design and Gait Planning of a Six-Bar Tensegrity Robot. Appl. Sci.; 2022; 12, 11845. [DOI: https://dx.doi.org/10.3390/app122211845]
12. Yue, X.-H.; Yin, X.; Sun, Z.-Y.; Liu, L.-Y.; Wang, Y.; Xu, G.-K.; Cao, C.; Zhang, L.-Y. Flexible, Lightweight, Tunable Robotic Arms Enabled by X-Tensegrity Inspired Structures. Compos. Struct.; 2024; 344, 118331. [DOI: https://dx.doi.org/10.1016/j.compstruct.2024.118331]
13. Chen, M.; Liu, J.; Skelton, R.E. Design and Control of Tensegrity Morphing Airfoils. Mech. Res. Commun.; 2020; 103, 103480. [DOI: https://dx.doi.org/10.1016/j.mechrescom.2020.103480]
14. Shen, Y.; Chen, M.; Skelton, R.E. Markov Data-Based Reference Tracking Control to Tensegrity Morphing Airfoils. Eng. Struct.; 2023; 291, 116430. [DOI: https://dx.doi.org/10.1016/j.engstruct.2023.116430]
15. Sun, J.; Li, X.; Xu, Y.; Pu, T.; Yao, J.; Zhao, Y. Morphing Wing Based on Trigonal Bipyramidal Tensegrity Structure and Parallel Mechanism. Machines; 2022; 10, 930. [DOI: https://dx.doi.org/10.3390/machines10100930]
16. Obara, P. Assessment of Tensegrity Structures: Static, Dynamic, and Dynamic Stability Analysis; 1st ed. Taylor and Francis: London, UK, 2025; ISBN 978-1-032-87468-5
17. Xu, Z.; Lu, J.; Chen, L.; Zhang, Y. A Form-Finding Method of Hyperelastic Tensegrity Considering Geometric and Material Nonlinearity. Eng. Struct.; 2025; 343, 121096. [DOI: https://dx.doi.org/10.1016/j.engstruct.2025.121096]
18. Hernández-Montes, E.; Fernández-Ruiz, M.A.; Gil-Martín, L.M.; Merino, L.; Jara, P. Full and Folded Forms: A Compact Review of the Formulation of Tensegrity Structures. Math. Mech. Solids; 2018; 23, pp. 944-949. [DOI: https://dx.doi.org/10.1177/1081286517697372]
19. Fan, L.; Xu, R.; Shi, P.; Feng, X.; Chen, Y. Simplified Form-Finding for Tensegrity Structures through Reference Joints of Symmetry Orbits. Structures; 2023; 49, pp. 1157-1167. [DOI: https://dx.doi.org/10.1016/j.istruc.2023.02.006]
20. Dong, Y.; Yuan, X.; Li, S.; Wang, Y.; Samy, A.; Dong, S. An Optimization-Based Method for Reshaping Tensegrity Structures towards Target Configurations. Structures; 2025; 75, 108733. [DOI: https://dx.doi.org/10.1016/j.istruc.2025.108733]
21. Wang, Y.; Xu, X.; Luo, Y. Topology Optimization of Tensegrity and Prestressed Cable-Strut Structures Considering Geometric Stiffness. J. Struct. Eng.; 2025; 151, 4025055. [DOI: https://dx.doi.org/10.1061/JSENDH.STENG-14127]
22. Feng, X.; Shen, J.; Zhao, W.; Lyu, H.; Su, Y. A Novel Method for Optimizing Energy Absorption of Tensegrities with Multi- Self-Stress Modes. Structures; 2024; 64, 106500. [DOI: https://dx.doi.org/10.1016/j.istruc.2024.106500]
23. Yao, J.; He, Z.; Zhang, L.; Li, S. Machine Learning-Based Optimization of the Shape Control Strategy for a Sunflower-like Tensegrity-Membrane Structure. Eng. Struct.; 2025; 343, 121011. [DOI: https://dx.doi.org/10.1016/j.engstruct.2025.121011]
24. Dong, Y.; Yuan, X.; Ma, S.; Li, S.; Samy, A.; Dong, S. Research on a Novel Tensegrity Torus with Superior Shape Adaptability. Structures; 2024; 63, 106283. [DOI: https://dx.doi.org/10.1016/j.istruc.2024.106283]
25. Liu, H.; Song, J.; Luo, A. Analysis of a Tensegrity Camber Morphing Airfoil. Mech. Res. Commun.; 2024; 137, 104272. [DOI: https://dx.doi.org/10.1016/j.mechrescom.2024.104272]
26. Huang, K.; Zhao, Z.; Lin, C.; Shi, M.; Ye, D.; Luo, J.; Wu, Z.; Jiang, J.; Wang, X. Towards Validated Dynamic Models for Tensegrity Structures: Parametric Modelling, Vibration Testing, and Model Updating of a Hexagonal Prism. J. Sound Vib.; 2025; 618, 119350. [DOI: https://dx.doi.org/10.1016/j.jsv.2025.119350]
27. Song, N.; Wang, M.; Wang, X.; Peng, H. A Novel Machine Learning Method for Real-Time Dynamic Analysis of Tensegrity Flexible Multibody Systems. Nonlinear Dyn.; 2025; 113, pp. 19047-19074. [DOI: https://dx.doi.org/10.1007/s11071-025-11152-w]
28. Obara, P.; Solovei, M. Influence of the Initial Prestress Level on the Distribution of Regions of Dynamic Instability of Geiger Domes. Appl. Sci.; 2024; 14, 7512. [DOI: https://dx.doi.org/10.3390/app14177512]
29. Feron, J.; Rhode-Barbarigos, L.; Latteur, P. Experimental Testing of a Tensegrity Simplex: Self-Stress Implementation and Static Loading. J. Struct. Eng.; 2023; 149, 4023073. [DOI: https://dx.doi.org/10.1061/JSENDH.STENG-11517]
30. Yuan, X.; Liang, X.; Li, A. Shape and Force Control of Prestressed Cable-Strut Structures Based on Nonlinear Force Method. Adv. Struct. Eng.; 2016; 19, pp. 1917-1926. [DOI: https://dx.doi.org/10.1177/1369433216652411]
31. Xu, X.; Luo, Y. Non-Linear Displacement Control of Prestressed Cable Structures. Proc. Inst. Mech. Eng. G J. Aerosp. Eng.; 2009; 223, pp. 1001-1007. [DOI: https://dx.doi.org/10.1243/09544100JAERO455]
32. Chen, L.; Jiang, Z.; Gao, W.; Zhou, Y. Identification and Adjustment of the Pretension Deviation in Cable-Strut Tensile Structures. KSCE J. Civ. Eng.; 2020; 24, pp. 143-152. [DOI: https://dx.doi.org/10.1007/s12205-020-1473-4]
33. Luo, B.; Sun, Y.; Guo, Z.X.; Pan, H. Multiple Random-Error Effect Analysis of Cable Length and Tension of Cable-Strut Tensile Structure. Adv. Struct. Eng.; 2016; 19, pp. 1289-1301. [DOI: https://dx.doi.org/10.1177/1369433216634534]
34. Lee, S.; Jeong, J.; Ahn, S.; Lieu, Q.X.; Lee, J. Performance of Quadruplex Module Tensegrities Using New Pin-Jointed Connections. J. Constr. Steel Res.; 2020; 172, 105763. [DOI: https://dx.doi.org/10.1016/j.jcsr.2019.105763]
35. Hagiwara, N.; Fujishiro, Y. Experimental and Numerical Analysis of a Novel Structure Obtained by Joining Two Tensegrity-Based Units Using Simple Construction Methods. Int. J. Space Struct.; 2024; 39, pp. 157-171. [DOI: https://dx.doi.org/10.1177/09560599241267780]
36. Botis, M.F.; Imre, L.; Cerbu, C. Computer-Aided Design of a Tensegrity Structure. Structures; 2022; 38, pp. 340-360. [DOI: https://dx.doi.org/10.1016/j.istruc.2022.01.084]
37. Rutkiewicz, A.; Małyszko, L. Experimental and Numerical Static Tests of Tensegrity Triplex Modules. Bull. Pol. Acad. Sci. Tech. Sci.; 2024; 72, 151674. [DOI: https://dx.doi.org/10.24425/bpasts.2024.151674]
38. Małyszko, L.; Rutkiewicz, A. Response of a Tensegrity Simplex in Experimental Tests of a Modal Hammer at Different Self-Stress Levels. Appl. Sci.; 2020; 10, 8733. [DOI: https://dx.doi.org/10.3390/app10238733]
39. Fernández-Ruiz, M.A.; Hernández-Montes, E.; Gil-Martín, L.M. Topological Design of the Octahedron Tensegrity Family. Eng. Struct.; 2022; 259, 114211. [DOI: https://dx.doi.org/10.1016/j.engstruct.2022.114211]
40. Fernández-Ruiz, M.A.; Hernández-Montes, E.; Carbonell-Márquez, J.F.; Gil-Martín, L.M. Octahedron Family: The Double-Expanded Octahedron Tensegrity. Int. J. Solids Struct.; 2019; 165, pp. 1-13. [DOI: https://dx.doi.org/10.1016/j.ijsolstr.2019.01.017]
41. Tibert, A.G.; Pellegrino, S. Review of Form-Finding Methods for Tensegrity Structures. Int. J. Space Struct.; 2003; 18, pp. 209-223. [DOI: https://dx.doi.org/10.1260/026635103322987940]
42. Lee, S.; Lieu, Q.X.; Vo, T.P.; Lee, J. Deep Neural Networks for Form-Finding of Tensegrity Structures. Mathematics; 2022; 10, 1822. [DOI: https://dx.doi.org/10.3390/math10111822]
43. Liu, H.; Sanaullah,; Vumiliya, A.; Luo, A. A Resource-Efficient Form-Finding Approach to Tensegrity Structures. Eng. Comput.; 2024; 41, pp. 1-17. [DOI: https://dx.doi.org/10.1108/EC-07-2023-0354]
44. Zhang, P.; Zhou, J.; Chen, J. Form-Finding of Complex Tensegrity Structures Using Constrained Optimization Method. Compos. Struct.; 2021; 268, 113971. [DOI: https://dx.doi.org/10.1016/j.compstruct.2021.113971]
45. Gan, B.S.; Zhang, J.; Nguyen, D.-K.; Nouchi, E. Node-Based Genetic Form-Finding of Irregular Tensegrity Structures. Comput. Struct.; 2015; 159, pp. 61-73. [DOI: https://dx.doi.org/10.1016/j.compstruc.2015.07.003]
46. Zhou, Y.; Wang, Y.; Li, S.; Yang, C.; Luo, Y. An Improved Dog-Leg Method for Form-Finding of Tensegrity Structures. Comput. Struct.; 2024; 292, 107237. [DOI: https://dx.doi.org/10.1016/j.compstruc.2023.107237]
47. Zhou, Y.; Ma, M.; Luo, Y. Innovative Form-Finding Method and Modeling Platform for Cable Dome Design. Proceedings of the IASS 2024 Symposium; Zurich, Switzerland, 26–30 August 2024; Block, P.; Boller, G.; Dewolf, C.; Pauli, J.; Kaufmann, W. International Association for Shell and Spatial Structures (IASS): Madrid, Spain, 2024.
48. Schek, H.-J. The Force Density Method for Form Finding and Computation of General Networks. Comput. Methods Appl. Mech. Eng.; 1974; 3, pp. 115-134. [DOI: https://dx.doi.org/10.1016/0045-7825(74)90045-0]
49. Linkwitz, K.; Schek, H.-J. Einige Bemerkungen Zur Berechnung von Vorgespannten Seilnetzkonstruktionen. Ing. Arch.; 1971; 40, pp. 145-158. [DOI: https://dx.doi.org/10.1007/BF00532146]
50. Zhang, J.Y.; Ohsaki, M. Stability Conditions for Tensegrity Structures. Int. J. Solids Struct.; 2007; 44, pp. 3875-3886. [DOI: https://dx.doi.org/10.1016/j.ijsolstr.2006.10.027]
51. Carbonell-Márquez, J.F.; Fernández-Ruiz, M.A.; Hernández-Montes, E.; Gil-Martín, L.M. Control of Geometry and Stability of Tensegrities in the Octahedron and X-Octahedron Families. Comput. Struct.; 2024; 305, 107547. [DOI: https://dx.doi.org/10.1016/j.compstruc.2024.107547]
52.
53.
54.
55.
56.
© 2025 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.