Parametric study of tensegrity structures under seismic loading

Tensegrity structures are spatial structures made up of an assembly of pin-jointed, prestressed continuous tension members and discontinuous compressive members, arranged in different complex shapes. Limited research has been carried out to analyse the dynamic response of these structures under seismic loading. This paper presents a parametric study to identify the suitability of different forms of these structures under combined vertical gravity and lateral seismic loadings. Different tensegrity shapes were analysed, as simplex and complex structures, under wind and earthquake loadings. These responses of the structures were also compared to that of conventional steel structures of the same shapes. Different parameters such as the effect of varying bar diameter, bar thickness, cable diameter, and tower height, were studied, and behavioural trends were identified. It was found that if a suitable tensegrity shape is chosen and optimized, its behavioural trends can be very similar to that of the conventional steel structures, while saving materials and therefore cost. Modal frequencies of the structure were also identified and found to be low, which is beneficial under earthquake loading. Thus, these systems can prove to be a valuable replacement to conventional systems under adverse dynamic loading.


Introduction
"Tensegrity systems are free-standing pin-jointed cable networks in which a connected system of cables is stressed against a disconnected system of struts and extensively, any freestanding pin-jointed cable networks composed of building units that satisfy aforesaid definition." [1]. Hence, a tensegrity structure experiences the compression and tension forces completely separated. However, they can only exist if there is sufficient pre-tension in the tensile element to reach a state of self-stress which forms the initial mechanical state of the system, before any loading. The optimum self-stress is achieved by several form-finding methods. A "simplex" module is one of the elemental Tensegrity modules that can be joined to create grids/masts or more complex structures made of the same/different patterns [2]. Many researchers have identified simplex tensegrity shapes that can be used for different applications while highlighting several beneficial features of tensegrity systems in comparison to conventional systems, e.g., tensegrity systems have no points of local weakness, economic use of material due to its efficiency, cables in tension can make full use of high-strength materials as large cross sections due to member buckling are not necessary, no experienced torque or torsion, large tensegrity constructions would be easy since the structure is self-scaffolding, compressive members lose stiffness, while tension members gain stiffness as the structures are loaded, a large stiffness-to-mass ratio can be achieved by increasing the use of tensile members (cables), the compressive members are connected with joints, large displacement and deployability in a compact volume is possible which allows for saving in transportation and assembly costs [3], and the structure is tuneable due to its deployability, small adjustments can easily be made to the loaded or damaged structure.
Nevertheless, Tensegrity structures do have their limitations, e.g., there is no clear guide for the design or analysis of these structures, strict topological constraints, bar overcrowding might occur as the design becomes larger, and high pre-stress forces are required for critical load support that could cause difficulty in large scale construction.
Jáuregui [2] presented a comprehensive study of tensegrity structures and their applications. It was found that double-layer tensegrity grids have possible applications as walls, roofing and covering structures. Applications such as earthquake-resistant buildings, bridges, and shelters were proposed since these structures are extremely resilient, have rapid load transfer, and have the potential to be shock absorbent. Another application is towers, and the Needle Tower, designed by Snelson [4] is a well-known example. Snelson [4] had suggested that for large scale towers, the passageway formed by the crossed compression members could serve as a central shaft for a lift passage. Jáuregui [2] proposed other various applications such as lightning conductors and turbines installed on top of tensegrity towers. Furthermore, the ability of tensegrity arches and spline beams to support large-span membranes shows that their properties are suitable for accommodation of asymmetric loads and avoidance of stress concentration, due to torsional freedom.
Burkhardt [5] focused on highly-triangulated tensegrity design methods, which are useful for dome design. Study of Tensegrity domes showed that they allow for fabrication of largescale structures. These domes could encompass very large areas with only minimal support at their perimeters. Suspending structures above the earth on such minimal foundations would allow the suspended structures to escape terrestrial confines in areas where this is useful, such as congested or dangerous areas as well as urban areas.
Gilewski et al. [6] carried out research using the decomposition matrix method to check if some structures are true tensegrities. The research has shown that towers composed of interconnected tensegrity modules/simplexes are the best-known tensegrity structures. An example is the Warnow Tower which, at 62.3 meters, is the tallest tensegrity tower ever built. It consists of 6 simplex modules, each 8.3 m in height, stacked on top of each other.
A study of the structural efficiency of Tensegrity grids was undertaken by Wang [1]. The general finding was that Tensegrity grids are not very efficient in structures, due to their unconventional load-transfer patterns. The low structural efficiency is affected by three factors; isolation of struts, long bars subject to buckling and bar inclination that causes the reduction of the resistant lever arms for struts and cables. Hence, Wang [1] suggests that tensegrity grids are suitable in small spans and when aesthetics are required. However, Wang [1] performed some tests and identified what can be done to improve the properties of long span tensegrity structures. These include, adding diagonal cables to each simplex to evenly distribute internal forces, using struts with larger outer diameter and smaller wall thickness for buckling resistance, and increasing of the modular length and designing less modules reducing the structural self-weight. Having less modules makes simplex construction and connection easier.
Feng [7] studied the structural behaviour and types of tensegrity domes by analysing the first tensegrity dome, Georgia Dome, using the numerical analysis method as well as using non-linear software. It was found that structural behaviour of a tensegrity dome differs from the conventional due to its failure mode. It was found that the failure mode of a tensegrity dome is characterized by the slackening of the ridge and diagonal cables in the central section of the dome. Hence, Feng [7] showed through analysis that the best way to design an elliptical tensegrity dome and increase its bearing capacity is to increase the prestressed forces in the ridge and diagonal cables on the inner and upper layer of the central section of the dome.
Barbarigos et al. [8] analysed a "hollow rope" tensegrity pedestrian bridge through a study of three potential module configurations: square, pentagon and hexagon. The bridge was composed of four identical modules which are symmetrical about the centre. It was shown that the most suitable module is the pentagon ring for the bridge application in terms of torsion, self-weight, serviceability, rigidity, economical configuration as well having lowest differential displacements, while the least efficient is the hexagon.
Roshna and Anuragi [9] conducted a study to design and analyse the behaviour of a tensegrity BTS (Base Transceiver Station) tower, which was compared to an existing telecommunication tower, for a given load condition. It was found that the deflection of the tensegrity tower was almost twice the deflection of the existing tower, however, there was a significant reduction of the structure's total weight and number of members.
Seifollahi and Sadeghi [10] analysed the seismic behaviour of tensegrity barrel vaults. ANSYS software was used to perform linear and nonlinear analysis. Tensegrity barrel vaults with different rise to span ratios and variable initial strains were produced. When the tensegrity barrel vaults undergo horizontal accelerograms, a failure of kind of tension rupture or compression buckling occurs in the members. But under vertical accelerograms none of the rupture or buckling of the members were observed. Under horizontal earthquake actions, for low initial strains, the governing failure mode is rupture of tensile elements, but for intermediate and large initial strain, the failure mode is buckling of bars.
The literature reviewed indicates that tensegrities can be extremely efficient if designed correctly. However, if these structures are to be used in application, either as components in large structures, or on their own, studying the behaviour of these systems is critical. Recent research in tensegrities mostly cover aspects such as form-finding, investigation on the stability, and self-stress of tensegrity structures. However, a gap exists when it comes to the seismic analysis of tensegrity systems and their frequency modes. Hence, the understanding of their structural behaviour is required, before they can be applied in practice. Under seismic loading, how will they deform? How are their modal frequencies? Can they be designed/modified to reduce deformation? Are tensegrity structures efficient under earthquake loading? To identify the quantified level of the actual or anticipated seismic performance associated with the direct damage of a tensegrity structure to a specified ground shaking, analytical performance was conducted in this research using SAP2000. The overall tensegrity structural performance was compared with a conventional steel structure of the same shape undergoing the same parametric changes and subjected to the same loading, to help identify any benefits or drawbacks of using pure tension and compression members.

Methodology
Initially, two different simplex tensegrity modules were modelled and analysed so that complex tensegrity structures could be generated from them. These structures were tested under varying parameters. Seismic loading, using the 1940 El-Centro Earthquake data, was applied to the models with the help of SAP2000 analysis from which results were extracted and analysed. To ensure that the results obtained are validated, control models were analysed throughout, ensuring the same parametric changes as the tensegrity structures. The control models have the exact same geometry, but they are made from compression members (bars) and have no cables. Finally, results from all models were gathered and compared, for identification of behavioural patterns.

Analyses types
The main analyses types that have been performed in this research are, combined loading (dead + wind), response-spectrum, and nonlinear time history analyses.
Modal analysis is used to determine the frequencies and mode shapes of a structure. These modes are useful to understand the behaviour of the structure. This analysis is also the basis for modal superposition in response-spectrum and modal time-history. In SAP2000, Eigenvector Modal analysis was used to determine the undamped free-vibration mode shapes and frequencies of the system. The critical outputs from this analysis are the cyclic and natural frequency, as they were used for comparison with the critical earthquake frequency, to identify for resonance, as well as the deformation under the various dynamic analysis.
The El Centro earthquake, which occurred on May 18, 1940, had a magnitude of 7.1 on the Richter scale and perceived intensity X (extreme) on the Mercalli scale. From the El-Centro Time-history Accelerogram, it is observed that the maximum acceleration value g is about 0.35 m/s 2 , which were used for the response spectrum. From the Amplitude Spectrum Accelerogram, three major peaks are observed at 5, 3.5, and 2.5 Hz. These values were used to check against the modal frequency of the structures for resonance.
Response spectrum is a linear-dynamic statistical analysis method which measures the contribution from each natural mode of vibration to indicate the likely maximum seismic response of a structure. Hence, the maximum acceleration value for El-Centro Earthquake, 0.35 m/s 2 , were used in SAP2000 to define the response spectrum function. Note that the function is defined using Eurocode 8-2004, and all parameters are set by default. For any response quantity such as displacement, stress, force, etc., the total response should be computed using the SRSS method, as specified in Eurocode 8. This analysis offers a standardized solution to evaluate structures and it is an initial step to understanding behaviour before performing a time-history analysis Time-history analysis is used to determine the dynamic response of a structure under time-varying loading. It is the behavioural study of a structure under a past earthquake. It provides information on time-wise fluctuations of structural parameters, such as forces and deflections, and it captures duration effects such as changes in stiffness and strength. In this research, nonlinear modal time-history analysis is performed. Modal superposition is used in this case as it provides a highly efficient and accurate procedure for performing time-history analysis. Moreover, the analysis is transient, since El-Centro earthquake is a one-time event.
It is important to note that time-history analysis is performed at discrete time steps, which are dependent on the El-Centro data file. Only the horizontal acceleration components are considered in this research. Response spectrum analysis is less conservative than static analysis and is most representative of actual structural behaviour.

Parametric study
For two different tensegrity modules/shapes, the effect of varying the bar diameter, bar thickness, cable diameter, and tower height is studied. Usually, optimization procedures are used to find the optimum sections of tensegrity systems. However, in this study, the sections have been chosen based on their behavioural performance, in terms of deformation and stress. For the first three studied parameters, the parameter being investigated will vary while the other 2 will remain constant. This effect is studied only on the simplex modules and their respective control structures. Based on the results of these parameters, a suitable bar diameter, thickness, and a cable diameter were chosen for the final studied parameter, which is the tower height.

Simplex tensegrities
The two simplex tensegrity modules used in this research are shown in Table 1. The modules (adopted from previous research) are stable and in state of equilibrium. One of the advantages of a tensegrity structure over other kinds of structural systems is its ability to permit creation of complex structures by joining simplex tensegrities together, since they are self-stable. This means that any number of extension of the assembled pieces could be created. Hence, using this principle, two towers, using the previously shown modules, were created. For these two modules, 5 tower structures with varying heights, i.e., 5m, 25m, 50m, 75m, and 100m, have been modelled, while all other properties remain constant. To create a tower, each of the modules were replicated, rotated 180 degrees about the y-axis, and then placed on top of the original modules. This creates a 5m tall tower ( Figure  1). Hence, the 5 m tower was replicated as many times as necessary to create towers of different heights. The Module 2 tower is similar to the Warnow tower discussed in Gilewski et al. [6].

Material properties
Steel of grade S470 N/mm 2 was used for all the bars/struts, while tendons of grade 1770 N/mm 2 have been used for the cable sections.

Bar section (CHS) properties
Circular hollow sections (CHS) have been used for the models. 10 simplex modules with varying CHS in total have been modelled, 5 of which, i.e. CHS 85 6, CHS 105 6, CHS 125 6, CHS 145 6 and CHS 165 6, the diameter was varied, and 5 of which, i.e., CHS 125 4, CHS 125 5, CHS 125 6, CHS 125 8 and CHS 125 10, the thickness was varied. The cable diameter was kept as 10mm throughout. Based on the results of these parametric changes, a suitable value for the diameter and thickness were be used for the grid and tower models.

Cable prestress
Since optimization procedures is outside the scope of this research, a set prestressing force value has been used. Hence, the prestress was used as a control measure in the models. The prestress was applied on the cables in the models on SAP2000 by limiting the tension at one

Loading details
The models presented above were subjected to the following loading conditions, using SAP2000 v.20,  Self-weight (Dead load),  Static wind loading,  Modal Loading which includes response spectrum using maximum acceleration value from El-Centro earthquake and time-history loading using El-Centro earthquake. The response of the structure to the above-mentioned loading conditions were observed and the net deflection of the structure was noted for comparison. Combined loading were conducted with the help of EN 1990:2002.

Analysis output
The following were extracted from each model after the analysis:  Maximum stress,  Peak nodal displacement (Lateral and vertical) at the highest point of every structure,  Critical Modal Frequency.

Findings and discussion
This section presents the main results of the research, highlighting the key findings and their significance. After that, analysis and comparison of the two different tensegrity modules versus the control structure is presented. The discussion is carried out section by section following the order: 1. Effects of bar (CHS) diameter, bar (CHS) thickness, and cable diameter, 2. Effect of tower height, 3. Critical frequency modes, 4. Module 1 vs module 2.

Effect of bar (CHS) diameter, bar (CHS) thickness and cable thickness
There is a need to clarify that when the combined loading (dead + wind) is being considered, the effects are mostly due to self-weight. It was found that the effects of wind are negligible. This may be attributed to the shape of the structure, since it is an open structure with no shells/faces for the wind to exert pressure on. The bar (CHS) diameter, bar (CHS) thickness and cable diameter have negligible effects on response spectrum and time-history analyses, and thus not included. This will be discussed further on. Typical plots of the above mentioned parameters versus displacements and stresses are presented in Fig. 2 through Fig. 7. It is observed that the larger the CHS diameter, the lower the deflection and stresses under selfweight, however, this must be limited as to not affect the slenderness of the bars. No clear behavioural pattern can be discerned in varying the bar (CHS) thickness. The effect of varying CHS thickness has varying effects on module 1, whilst having almost no effect on module 2. A thickness of 6mm is chosen to be used for the CHS, as anything less than that would increase the chance of failure due to slenderness, as well as yielding. Effect of the varying cable diameter shows stationary effects which is attributed to the large value of prestress force applied on the cables. Using optimization algorithms would give better results for choosing the suitable sections of the structure with regards to the parameters considered in this section. CHS of diameter 125mm and thickness of 6mm are used in the following models alongside cables of diameter 10mm.

Effect of tower height
Typical plots of displacements versus tower height under combined loading, response spectrum analysis and time history analysis are shown in Fig. 8 through Fig. 10. Under the combined loading, it was observed that Tower 1 is more critical under the vertical displacement, as it displays a nonlinear increase in deformation up to 1m at a tower height of 100m. Whereas Tower 2 experiences similar increasing trends for both lateral and vertical displacements of 40mm and 250mm at a 100m tower height, respectively. As height increases, the stiffness decreases with the same composition of the structural elements, which results in the increase in overall lateral deflections. Both control towers experience almost negligible deformations at a constant rate at all tower heights.
For the response spectrum and time history analyses, the vertical displacement is negligible, as expected. The response spectrum analysis indicated that tower 2 is more critical, displaying an initial nonlinear increase which becomes linear from 50m to 100m, to reach a deformation of 1.2m. However, this behaviour is quite different from its control tower which displays a linear increase in deformation up to 160mm at 100m height. Tower 1 displays a linear increase in deformation up to 300mm at 100m height, and its control tower displays similar behaviour. From the time-history analysis, Towers 1 and 2 behave similarly, displaying a logarithmic growth to 300mm at 100m height. The stresses experienced by tower 1 display an exponential growth, and tower 2 shows a stationary trend, however, the final stress values at 100m height are similar (170 and 150 N/mm 2 respectively). Like the deformation, the stresses experienced by tower 1 and its control are closer in values than tower 2.

Critical frequency modes
The modal frequencies of the studied structures were checked, and those that coincided with the critical frequencies were noted. It was observed that tensegrity structures experience higher number of critical modes in comparison to the conventional systems. This could be due to several reasons. Since it is known that mass and stiffness affect the frequency, these are the factors that should be considered. It was already known that tensegrity structures have less self-weight in comparison to conventional structures, and that leaves one possible factor, which is the stiffness. The stiffness of the tensegrity structures, in comparison to the conventional system, is low. Hence, they are more likely to experience resonance. If the stiffness is altered, then the frequency can be varied as required. The stiffness of tensegrities depends on the level of prestress, as well as the geometry, hence very stiff or flexible tensegrity systems can be created. In this research, under the El-Centro loading, increasing the prestress would increase the stiffness and yield better results in terms of critical frequencies. Nevertheless, low stiffness can be beneficial, as it allows the structure to vibrate with the earthquake, rather than try to resist it. Hence, collapse can be avoided. It was also observed, that tower 2 has slightly less critical modes, and this can be attributed to its lower mass in comparison to tower 1, whilst having the same prestress. It should be noted that the modal frequencies of the individual modules were not discussed, since the structures are small and stiff, and their initial frequency modes are high, so they do not experience resonance.

Concluding remarks and recommendations
All in all, the benefits of tensegrity structural system were proven through several previous studies, which were mostly focused on form-finding and static loading. Thus, this research focused on exploring the behaviour patterns of tensegrity systems with different shapes, under various static and dynamic loading conditions. A parametric study was conducted to understand the effect of varying bar and cable sections, as well as to put the theory of creating complex structures from simplex tensegrity modules to the test. The modal analysis study of these structures, mainly in the forms of towers and grids, is a first of its kind, which creates opportunities for further research to follow. Even though this research was limited in several areas and has room for improvement, it helps in setting base trends of tensegrity structure behaviour to be used in future work. Combining the economic benefits and the promising structural behaviour that could be further improved through research, tensegrity structures This research can be expanded by exploring more simplex tensegrity shapes, as the geometry of the tensegrity can have significant effect on its behaviour. The effects of symmetry, irregularity, and number of supports can be studied. Using these shapes, a variety of tensegrity systems should be created, for various applications and under different loadings. These should also be tested under dynamic loading. Coordinate exchanges, a computer programming utility, can be used to generate these different structures. For instance, a tensegrity arch can be created with coordinate transformation from a vertical tower.
As prestress is a very important factor for tensegrity structures, the optimization of the stress will increase the accuracy of the results. Moreover, a range of prestress values above and below the optimized prestress can be studied to understand the effect of varying prestress on the structure and its parameters.
The authors would like to acknowledge the support of the University of Nottingham Malaysia.