Form-finding of free-form tensegrity structures by genetic algorithm–based total potential energy minimization

Abstract

Free-form tensegrities are composed of randomly connected cable and strut elements. The complexity of these structures causes determination of their self-equilibrium form to be a formidable task. There can be an infinite number of solutions with different forms, but it is difficult to identify the best form in terms of stability. Based on the fact that stability of structures is inversely proportional to potential energy, a genetic algorithm minimization process is developed to determine the self-equilibrium form of free-form regular tensegrity structures. The capability of the form-finding process on determination of the most stable form with minimum potential energy is investigated using two main parameters of free-form regular tensegrities which are cable–strut length ratio at rest and number of strut elements. The computational performance of the proposed method is also tested using free-form tensegrities with different number of structural elements.

Keywords

form-finding free-form tensegrity genetic algorithm stability total potential energy

Introduction

The idea of tensile integrity structures was introduced in the beginning of 20^th century (Fuller, 1962). Four years after the invention, new configurations such as continuous tension and discontinuous compression structures were practiced by Kenneth Snelson (1965). The first non-regular tensegrity structure was invented by Buckminster (1975) and further studies were focused on form-finding of tensegrity structures (Tibert and Pellegrino, 2003). Different methods for the design of tensegrity structures were developed such as dynamic relaxation (Belkacem, 1987; Day and Bunce, 1969; Motro and Nooshin, 1984), force density (Linkwitz, 1999; Linkwitz and Schek, 1971; Schek, 1974), matrix iteration (Lu et al., 2007, 2015) and reduced coordinates (Sultan et al., 1999). Computational methods were also developed by various authors (Chen et al., 2012; Estrada et al., 2006; Li et al., 2010; Masic et al., 2005; Pagitz and Tur, 2009; Zhang et al., 2006a; Zhang and Ohsaki, 2006) for the design of complex tensegrity structures. Self-stress state and effect of pre-stress on the balanced form of tensegrities were investigated and analysed in recent studies (Chen et al., 2014, 2015; Chen and Feng, 2012; Koohestani, 2015). Further information about tensegrity frameworks was presented by Mirats Tur and Juan (2009).

Connelly used and developed energy methods (Connelly, 1982, 1993; Connelly and Back, 1998; Connelly and Terrell, 1995). These studies showed that energy methods can be used to perform form-finding of tensegrity structures. Pellegrino (1986) proposed nonlinear programming method that performs form-finding of tensegrity structures using a constrained energy minimization approach. Achievements in computation power supported this process and search methods have been developed for the design of tensegrity structures (Li et al., 2010). Paul et al. (2005) introduced evolutionary algorithms for determination of tensegrity structures with non-regular forms. Xu and Luo (2010a) converted the form-finding problem of non-regular tensegrity structures into a constrained optimization problem and also developed a simulated annealing algorithm (Xu and Luo, 2010b) to perform pre-stress optimization and form-finding. Koohestani and Guest (2013) proposed a genetic algorithm form-finding method based on minimization of eigenvalues of the force density matrix and developed analytical and numerical form-finding methods. Faroughi et al. (2014) used a genetic algorithm based on the rank deficiencies on the geometry, the pre-stress coefficients and the semi-positive definite condition of the stiffness matrix. A node-based method was developed by Gan et al (2015). Another study based on optimization of force density is performed by Cai and Feng (2015). Ohsaki and Zhang (2015) used a nonlinear programming approach to investigate stability of the self-equilibrium shape. Studies show that different numerical methods were developed to determine the self-equilibrium state of the tensegrity structures. Metaheuristic techniques and energy methods were used for the purpose of form-finding; however, minimization of total potential energy (TPE) has never been done using metaheuristic techniques in order to perform form-finding of free-form regular tensegrity structures.

Zhang et al. (2006b) classified the tensegrity structures into three types, regular, semi-regular and non-regular. According to this classification, all cable and strut elements of the regular tensegrity structures have the same lengths at rest. Several studies were performed (Motro, 2011; Tachi, 2012; Tran and Lee, 2011) for the purpose of finding the free-form tensegrity structures with regular and irregular elements; however, conventional methods are efficient on tensegrities with specific scale (Paul et al., 2005; Rieffel et al., 2009). It is necessary to develop new effective methods in order to deal with the form-finding problem of free-form regular tensegrities.

The aim of this study is to demonstrate a new approach based on TPE minimization for determination of the self-equilibrium form of free-form tensegrity structures. Based on the fact that structures tend to have a form with minimum TPE, form-finding of free-form tensegrity structures is performed by TPE minimization using genetic algorithm. This study meets evolutionary algorithms with the principle of minimum potential energy for the first time in order to determine the stable form of free-form tensegrities. Self-equilibrium states are determined, through an evolution process, using only the basic information about these complex tensegrity structures. The TPE of self-equilibrium forms of randomly generated regular tensegrities are investigated with two main design parameters which are stress-free cable–strut length ratio and the number of strut elements.

Design of free-form tensegrity structures

Tensegrity structures are classified as regular, semi-regular and non-regular based on element lengths and the ones with constant element length are called regular tensegrity structures. In this study, a single unit is used for the design of free-form regular tensegrity structures which is composed of one strut and four cable elements as is illustrated in Figure 1. According to the developed design rule, each end of a strut should be connected to four cable elements and length ratio of cable and strut elements at rest is constant. Properties of the design rule are summarized in Table 1.

Figure 1.

Illustration of a single unit with one strut and four cable elements.

Table 1.

Properties of the design rule.

Number of units	n
Number of strut elements	n
Number of cable elements	4n
Assembly of units	Random
Length of strut elements	Constant
Length of cable elements	Constant
Ratio of initial cable and strut length	Constant

A free-form regular tensegrity structure is composed of n number of units. Each strut element is connected to eight different cable elements, four cables at each end. The cable elements are connected to two different strut elements and six different cable elements. After the initiation of n number of units, free end of the cables are connected to free end of the struts randomly according to the rule that a cable element cannot be merged with two ends of a single strut element. The number of strut elements cannot be less than 5.

Form-finding of free-form tensegrities

Ziegler (1968) stated that the TPE of a conservative system takes a stationary value in an equilibrium state; moreover, equilibrium configuration is stable if this stationary state of the system demonstrates a minimum. This principle can be interpreted in a way to determine the equilibrium configuration of a structure. Examples of the principle can be found in some simple and classical applications (Timoshenko and Gere, 1961), but it is widely used in making derivations for many other energy methods or finite element method. The development of metaheuristic algorithms enabled application of the minimum energy principle to real-life problems without recourse to differentiation of the energy function or using other incremental techniques. A search can be carried out to find the stable form which minimizes the TPE of the structure. In the case of no external load, this method can be used to determine the self-equilibrium form of tensegrity structures.

The proposed form-finding process minimizes TPE using genetic algorithms to determine the self-equilibrium form of tensegrity structures. The idea behind this method is that the stability of the structures is inversely proportional to potential energy (Alfutov et al., 2013; Gambhir, 2004; Godoy, 1999). This nature of the proposed method allows determination of the best form in terms of stability. Starting from a random form, evolution process increases the stability and decreases the TPE by modifying the structure. The form with minimum possible TPE is obtained at the end of the process which is the most stable form of the selected configuration.

In this study, form-finding of free-form tensegrities is performed based on a set of assumptions (Feng and Guo, 2015; Ohsaki and Zhang, 2015). The elements of the structure are assumed to be straight, self-weight of the structure is neglected, element failure related to yielding and buckling is not considered and no external loads are acted on the system. The process requires simple information about the structure which are element type (cable or strut), nodal joints (connectivity patterns), material properties (modulus of elasticity), stress-free element lengths and cross-sectional area of the elements. The initial coordinates of nodal joints can be determined randomly.

Genetic algorithm for TPE minimization

The basic and well-known steps of the simple genetic algorithm are used for the purpose of TPE minimization of free-form tensegrities. Starting from an initial form, the most stable form is determined after a series of evolution steps. Flowchart of the metaheuristic algorithm used for this process is illustrated in Figure 2 (Toklu and Uzun, 2016).

Figure 2.

General procedure for TPE optimization using metaheuristic optimization techniques (Toklu and Uzun, 2016).

Grefenstette (1986) previously studied the parameters of genetic algorithm process. In this study, the parameters of the evolution process are determined according to the results of that study. Genetic algorithm operators, elitism, selection, crossover and mutation, are applied to base configurations $(C_{B})$ , after evaluation of the fitness and during the creation of new configurations $(C_{N})$ . A dynamic mutation range with a convergence limit is used.

Genetic algorithm process finds the optimum solution using a set of solutions called population which is composed of members. Each member in this study corresponds to a solution which defines the coordinates of nodal joints in three-dimensional space. The nodal joints correspond to genes of chromosome of each member. i^th member of a population $(M_{i})$ with n number of nodal joints is defined with a vector where genes are nodal joints $(N_{j})$ which are composed of axial coordinates, $x_{j}$ , $y_{j}$ and $z_{j}$ . In this study, a population is composed of 30 members

M_{i} = [N_{j = 1}^{i}, N_{j = 2}^{i}, …, N_{j = n}^{i}]

(1)

N_{j}^{i} = [x_{j}^{i}, y_{j}^{i}, z_{j}^{i}]

(2)

The fitness of all members are evaluated and 10% of the fittest members in the population are determined to be elite members. Elite members are mated to create new child members. The selection process of the mated members is performed according to the fitness of each member using roulette wheel selection method. After the selection process, one-point crossover is performed at each mate with a probability of 0.95.

The non-elite child members are mutated according to a predetermined mutation rate of 0.04. A gene is mutated by changing axial coordinates $(x_{j}, y_{j}, z_{j})$ of the node within a dynamic spherical mutation range. In this study, initial mutation range is determined to be 0.01% of the stress-free length of the element with maximum element length (MEL). The mutation range is expanded after an improvement in the fitness of the best member of the population by a factor of 1.1 and decreased if the fitness is not changed by multiplying with 0.9. The minimum mutation range in this study is 0.1% of the initial mutation range.

Fitness function

The fitness of the genetic algorithm energy minimization process is the total of potential energies of all cable and strut elements. The sum of potential energies of m number of elements is calculated using equation (3) where s is the strain energy density, A is the cross-sectional area and $L'$ is the stress-free length of l^th element. In this equation, $A_{l} L'_{l}$ determines the volume of an element. The volume of l^th element is determined as $V_{l}$ and the modified fitness function is given in equation (4)

F = \sum_{l = 1}^{m} s_{l} A_{l} L'_{l}

(3)

F = \sum_{l = 1}^{m} s_{l} V_{l}

(4)

Strain energy density of an element is calculated using equation (5) where $σ$ is the stress and $ε$ is the strain. Stress is defined in terms of modulus of elasticity $(E)$ and strain. The tensegrity structures have some restrictions in the case of structural analysis. Cable elements cannot be compressed, and accordingly deformation of them cannot be negative. Strain energy density of a cable element is determined to be 0 if its strain is less than 0

s = {\begin{matrix} 0 & ε < 0 \\ \int_{0}^{ε} σ d ε = \int_{0}^{ε} E ε d ε = E ε^{2} / 2 & ε ⩾ 0 \end{matrix}

(5)

The strain of an element is defined in equation (6) where $L'$ is the stress-free length and L is the length of the l^th element. The final form of the TPE as a function of strain is given in equation (7)

ε_{l} = \frac{L_{l} - L'_{l}}{L'}

(6)

F (ε) = \sum_{l = 1}^{m} V_{l} E {ε_{l}}^{2} / 2

(7)

Analysis of free-form tensegrities

The presented method for form-finding of free-form tensegrity structures is analysed using different number of strut elements, cable–strut length ratios and cable–strut cross-sectional area ratios. Initially, 8 strut tensegrity is used to show the ability of the proposed method to find self-equilibrium form. Investigations are performed from a wider perspective using structures with different cable–strut length ratios. The effect of cable–strut length ratio is tested with different number of strut elements and a combined analysis is performed using two design parameters. In all, 17 different strut numbers (8:1:24) are investigated at 35 different cable–strut length ratios (0.70:0.02:0.04) using 10 different generations which means 5950 tests are performed. Finally, the effect of the percent of cross-sectional area between cable and strut elements is investigated to demonstrate its effect on the self-equilibrium form of free-form regular tensegrities. For this purpose, 72 tests are performed using four different strut numbers (8, 12, 16 and 20) at 18 different cable–strut cross-sectional area ratios (0.02:0.04:0.70).

TPE is a term related to strain energy of the structural elements and lower strain in a structure is the preferred case. However, tensegrity structures should balance compressive strain in struts with tensional strain in cables in order to have a stable form. Based on this fact, a tensegrity structure should have a positive TPE to sustain stability, but magnitude of the TPE should not be high that can cause high magnitude of stain in the elements. It can be stated that cable–strut cross-sectional area ratio should be determined in a way that provides minimum necessary TPE for a stable form.

Form-finding of 8 strut free-form tensegrity

Form-finding of 8 strut free-form tensegrity is performed through an evolution process of genetic algorithm starting from an initial random form. In the beginning, structural form of all members in the population are determined to be a form compressed into a single point. The cross-sectional area of cable and strut elements of 8 strut tensegrity is determined to be 10 and 100 mm², respectively. In this part of the study, cable–strut length ratio is set to 0.5 and the length of strut elements at rest is determined to be 100 mm. Initially, a sample investigation is performed to illustrate the stages of genetic algorithm form-finding process. The nodal connections after random connection of the elements are given in Table 2. The ratio between stress-free length of cable and strut elements is also investigated with separate form-finding processes.

Table 2.

Properties of the free-form tensegrity and final element lengths after TPE minimization process.

No.	ET	NC	FL	No.	ET	NC	FL	No.	ET	NC	FL	No.	ET	NC	FL
1	S	2-1	97.62	11	S	6-5	97.34	21	S	10-9	99.42	31	S	14-13	97.35
2	C	2-7	61.12	12	C	6-1	52.46	22	C	10-1	52.40	32	C	14-5	52.62
3	C	2-13	53.58	13	C	6-13	51.37	23	C	10-11	50.20	33	C	14-15	58.85
4	C	2-3	53.67	14	C	6-11	56.35	24	C	10-15	53.59	34	C	14-11	57.50
5	C	2-9	36.57	15	C	6-3	61.60	25	C	10-7	50.00	35	C	14-9	53.77
6	S	4-3	97.30	16	S	8-7	97.55	26	S	12-11	98.54	36	S	16-15	97.43
7	C	4-1	56.19	17	C	8-5	58.05	27	C	12-9	50.00	37	C	16-5	60.91
8	C	4-15	53.51	18	C	8-13	58.40	28	C	12-15	56.27	38	C	16-1	58.24
9	C	4-13	61.64	19	C	8-9	52.50	29	C	12-5	52.54	39	C	16-7	54.06
10	C	4-7	51.45	20	C	8-3	50.45	30	C	12-3	55.08	40	C	16-11	50.00

S: strut; C: cable; ET: element type; NC: nodal joints; FL: final length (mm).

Genetic algorithm evolution process is ended after 24,936 generations when TPE of the best member of the population is decreased to 26,451,301.956 N mm. TPE in logarithmic scale with respect to the number of generations in logarithmic scale and mutation range through the generations in logarithmic scale are given in Figure 3. TPE seems to be stabilized after 9000 generations and the evolution process is ended when the termination criterion is satisfied after about 15,936 generations. This stage of the process can be determined as fine-tuning period. Evolution process starts with initial mutation range of 0.1% of MEL. The mutation range has two peaks that reach to 73.235824 and 67.907003 mm in the generations of 106 and 141, respectively. This fluctuation shows the capability of dynamic mutation range that prevents the algorithm to stick to a local optimum. The algorithm is ended when the mutation range is decreased below 0.01% of the initial mutation range.

Figure 3.

Variation of mutation range and TPE through generations.

The nodal joint coordinates of the final form of the structure, the best member of the population, are given in Table 3. The first nodal joint is fixed to its initial coordinates while other nodes are free to move in the three-dimensional space. The final length of the elements are given in Table 2. The results show that all strut elements are in a compressed form as expected. Most of the cable elements are in tension, but two of them sustain their stress-free initial element length. The nature of the free-form tensegrities allows this type of forms. The stability of the structure is sustained by the cable elements in tension without the requirement of the intensity occured in these two cable elements.

Table 3.

Coordinates of nodal joints at equilibrium in terms of distance from the origin (mm).

Joint no.	Initial (x, y, z)	Final (x, y, z)	Joint no.	Initial (x, y, z)	Final (x, y, z)
1	(0, 0, 0)	(0.000, 0.000, 0.000)	9	(0, 0, 0)	(74.950, −32.188, 69.234)
2	(0, 0, 0)	(56.788, −59.331, 52.778)	10	(0, 0, 0)	(23.766, −44.177, −15.151)
3	(0, 0, 0)	(18.764, −43.809, 87.322)	11	(0, 0, 0)	(21.715, −1.195, 10.707)
4	(0, 0, 0)	(16.142, −52.975, −9.512)	12	(0, 0, 0)	(60.051, −73.292, 65.874)
5	(0, 0, 0)	(74.750, −23.033, 70.221)	13	(0, 0, 0)	(6.081, −75.621, 46.929)
6	(0, 0, 0)	(−17.171, −30.489, 39.089)	14	(0, 0, 0)	(76.940, −15.361, 18.206)
7	(0, 0, 0)	(51.718, −19.516, 6.678)	15	(0, 0, 0)	(62.432, −71.756, 9.675)
8	(0, 0, 0)	(40.641, −64.311, 92.625)	16	(0, 0, 0)	(34.268, 14.672, 44.743)

The minimization process seems to be stabilized after 200 generations as illustrated in Figure 4, but this is specious. Without the use of logarithmic scaling, fluctuations of the TPE from generation to generation cannot be observed. The form of the structure at 100^th and 200^th generations, after a sharp decrease in TPE, with initial and final forms are illustrated in Figure 4. During the evolution process, mutation range is always lower than the length of strut elements. This allows the formation of the final form through an expansion process starting from its initial compressed shape.

Figure 4.

Evolution of a random configuration of 8 strut free-form tensegrity through generations.

Form-finding of 8 strut tensegrity structures with different cable–strut length ratios is performed using the same process. The length of strut elements in all tests is fixed to 100 mm and the cable lengths are determined based on the cable–strut length ratio. In all, 34 different cable–strut length ratios from 0.04 to 0.70 with a step size of 0.02 are investigated at 10 different configurations. These 10 configurations are generated randomly according to the design rule. The only difference between these configurations is the nodal joints of the elements.

TPE of the self-equilibrium form of 340 different tensegrity structures (34 cable–strut length ratio at 10 different configurations) is given in Figure 5. In the illustration, cable–strut length ratios are given in the form of 1 − (L_c/L_s), where L_c represents the cable length and L_s represents the strut length. The increase in TPE is observed with decreasing cable–strut length ratio. In order to differentiate variations, TPE is also given in logarithmic scale. Deviation of TPE is higher when the length of cable and strut elements are similar. Different configurations have similar TPE at their self-equilibrium form when the length of the cable elements are decreased.

Figure 5.

TPE of different configurations of 8 strut free-form tensegrity at various cable–strut length ratios.

Skelton et al. (2002) stated that tensegrities have shape-changing flexibility without effecting stiffness and they are able to change stiffness without making any shape change. In this study, cable–strut length ratio is expected to affect stiffness according to the fact that compressive elements loss stiffness and tensile members gain stiffness when they are loaded. Decreasing the cable–strut length ratio, decreasing the cable length, increases the strain energy stored in both cable and strut elements which means compression in strut elements and tension in cable elements are increased. In the balanced form, strut elements are exposed to negative load and cable elements are affected by positive load. According to the increase in TPE, it can be stated that stiffness of the tensegrity structures is affected by cable–strut length ratio.

Combined effect of cable–strut length ratio and number of elements

Form-finding of the tensegrity structures with 12, 16, 20 and 24 strut elements at various cable–strut element length ratios are performed in order to investigate the combined effect of these two design parameters. Investigations are performed using a constant cable–strut cross-sectional area ratio of 0.1 when the cross-sectional area of the strut elements is fixed to 100 mm². The length of all strut elements in all tests is determined to be 100 mm and 34 different cable–strut length ratios from 0.04 to 0.70 with a step size of 0.02 are investigated at 10 randomly generated configurations. The total potential energies after form-finding of 340 different tensegrity structures are plotted in Figure 6. An increase in TPE is observed with decreasing cable–strut length ratio in all cases. The results in logarithmic scale show that deviation of TPE increases in all cases with increasing cable–strut length ratio.

Figure 6.

TPE of 10 configurations of 12, 16, 20 and 24 strut free-form tensegrities at different cable–strut length ratios.

In order to investigate the differentiation of TPE related to the number of strut elements, the average of 10 configurations of each cable–strut length ratio is determined for each number of strut separately and the results are plotted in Figure 7. The logarithmic scale for TPE makes it clear that this differentiation according to the number of struts is clear at higher and lower cable–strut length ratios; however, differentiation is not clear at the cable–strut length ratios around 0.5. There occurs an intersection point when cable–strut length ratio is between 0.62 and 0.64. These investigations aroused the requirement of separate investigations related to the number of strut elements at each cable–strut length ratio. For this purpose, a new set of analyses are performed using configurations with 17 different strut numbers from 8 to 24 at eight different cable–strut length ratios from 0.46 (1 − 0.46 = 0.54 in the figure) to 0.32 (1 − 0.32 = 0.68 in the figure).

Figure 7.

Average TPE of 10 configurations of 12, 16, 20 and 24 strut free-form tensegrities at various cable–strut length ratios.

The results of investigations on TPE related to the number of strut elements are illustrated separately in Figure 8. TPE at each point represents the average of tests with 10 different configurations. Information about cable–strut length ratio in this figure is given in the form of 1 − (L_c/L_s) to ensure consistency with previous plots. The results show that TPE decreases with increasing number of strut elements when the cable–strut length ratio is 0.46; however, this trend changes with decreasing cable–strut length ratio. TPE seems to increase clearly with increasing number of strut elements when the ratio is 0.32 (1 − 0.32 = 0.68).

Figure 8.

TPE related to number of strut elements at cable–strut length ratios of 0.46 (1 − 0.46 = 0.54) to 0.32 (1 − 0.32 = 0.68).

The trends of TPE distribution related to the number of strut elements are analysed in terms of power function which is given in equation (8). In this equation, $E_{p}$ represents TPE while a and b are constants. The number of strut elements is defined with $n_{s}$ . Constant b is the term which is related to positive or negative trend of the function. The positive value represents the increase in TPE with increasing number of strut elements. At a cable–strut length ratio which provides a power function with constant b of 0, TPE remains the same with increasing number of strut elements

E_{p} = a (n_{s})^{b}

(8)

Constant b is almost 0 when the cable–strut length ratio is a value between 0.36 and 0.38 (1 − 0.36 = 0.64 and 1 − 0.38 = 0.62). All structures with different number of strut elements are expected to have the same TPE at this ratio. However, this investigation is done using the average of 10 different configurations. Theoretical estimation is not expected to fit perfectly with the results of different random configurations.

After careful investigations on the behaviour of free-form tensegrity structures, random configurations with 12, 16, 20 and 24 strut elements are created other than previous investigations and their final forms are illustrated in Figure 9. The nodal connections of these free-form tensegrities are also given in the illustration. All solutions including these final tests are performed using a computer with Intel Core i7-3610 CPU that works at a clock speed of 2.3 GHz.

Figure 9.

Sample illustrations of self-equilibrium form of free-form tensegrity structures with 12, 16, 20 and 24 struts.

The results of the tests with random configurations are given in Table 4. TPE seems to be stable within a narrow range between 126 and 225 kN m. The number of generations created during the genetic algorithm evolution process and solution time also seem to be affected by the number of strut elements. The increase in this design parameter causes an increase in the solution time and the number of generations.

Table 4.

Results of investigations of random configurations.

Number of struts	Number of joints	TPE (N mm)	Solution time (s)	Number of generations
12	24	157,755,725.008	388.19	38,511
16	32	197,131,632.186	826.95	62,956
20	40	225,194,660.898	3300.53	191,820
24	48	126,878,824.268	5643.35	275,257

TPE: total potential energy.

Conclusion

The design of tensegrity structures is a challenging task that requires complex calculations. In order to overcome difficulties of the analytical methods, computational methods are widely used for design purposes. In this study, an evolutionary search technique and the minimum energy principle are combined, and a search method is developed to perform form-finding of free-form regular tensegrity structures. This form-finding technique is also used to investigate the design parameters of the free-form regular tensegrity structures.

The free-form tensegrities have a complex nature because of random connection of cable and strut elements. Investigations using a random configuration of 8 strut free-form tensegrity showed that the proposed method of form-finding is able to determine the self-equilibrium form of tensegrities. In order to propose the capabilities of the method, genetic algorithm process is analysed through generations. Evolution of the structure at different evolution steps is visualized. Recorded data of fitness and mutation range are presented using plots in logarithmic scales.

In all, 6022 different form-finding tests were performed using the genetic algorithm–based TPE minimization approach. The combined effect of the design parameters is investigated with the data collected during these tests. The results provide information not only about the investigated conditions but also show the trends that can allow analysis of various other configurations of regular but not free-form tensegrities with different design parameters.

TPE is a measure of strain energy stored in the elements of a structure. Analyses showed that at some stages TPE is about to be 0. This is not an expected condition because strain energy should be stored in the elements of a structure in order to satisfy the balance of tension with compression in cable and strut elements. The design parameters of a free-form tensegrity structure directly affect TPE, and accordingly these parameters should be determined in a way that provide a balanced TPE.

The presented method uses evolutionary algorithms for energy minimization in order to determine self-equilibrium form of the tensegrity structures. When compared to other form-finding methods, this computationally feasible method showed that it is possible to perform form-finding of tensegrities using simple calculations without requirement of complex formulations. In addition, this method is based on the inverse relation between stability and potential energy. Minimization of potential energy allows determination of the most stable form. However, evolution process has a possibility to stuck into a local minimum in spite of global minimum. In order to verify the results, it is necessary to perform several solutions of a problem using different parameter sets. This is an unfavourable property of the presented method as it increases the computation cost. Another property that affects the computation cost is the number of cable and strut elements. The results showed that the structures with higher number of elements required more generations and solution time.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship and/or publication of this article.

References

Alfutov

Balmont

Evseev

(2013) Stability of Elastic Structures. Berlin, Heidelberg: Springer.

Belkacem

(1987) Recherche de forme par relaxation dynamique des structures réticulées spatiales autocontraintes. Thése de Docteur Ingénieur, INSA Toulouse, Toulouse.

Buckminster

(1975) Non-symmetrical tension-integrity structures. Available at: https://www.google.com/patents/US3866366

Cai

Feng

(2015) Form-finding of tensegrity structures using an optimization method. Engineering Structures 104: 126–132.

Chen

Feng

(2012) Generalized eigenvalue analysis of symmetric prestressed structures using group theory. Journal of Computing in Civil Engineering 26(4): 488–497.

Chen

Feng

(2012) Novel form-finding of tensegrity structures using ant colony systems. Journal of Mechanisms and Robotics 4(3): 031001.

Chen

Feng

Zhang

(2014) A necessary condition for stability of kinematically indeterminate pin-jointed structures with symmetry. Mechanics Research Communications 60: 64–73.

Chen

Feng

. (2015) Efficient symmetry method for calculating integral prestress modes of statically indeterminate cable-strut structures. Journal of Structural Engineering 141(10): 04014240.

Connelly

(1982) Rigidity and energy. Inventiones Mathematicae 66(1): 11–33.

10.

Connelly

(1993) Rigidity. In: Gruber

Wills

(eds) Handbook of Convex Geometry. Amsterdam: Elsevier Publishers Ltd, pp. 223–271.

11.

Connelly

Terrell

(1995) Globally rigid symmetric tensegrities. Structural Topology 21: 59–78. http://upcommons.upc.edu/handle/2099/1099

12.

Connelly

Back

(1998) Mathematics and tensegrity: group and representation theory make it possible to form a complete catalogue of ‘strut-cable’ constructions with prescribed symmetries. American Scientist 86(2): 142–151.

13.

Connelly

Terrell

(1995) Globally rigid symmetric tensegrities. Structural Topology 21. Available at: http://www.math.cornell.edu/~connelly/Connelly.Terrell.pdf

14.

Day

Bunce

(1969) The analysis of hanging roofs. Arup Journal 3: 30–31.

15.

Estrada

Bungartz

H-J

Mohrdieck

(2006) Numerical form-finding of tensegrity structures. International Journal of Solids and Structures 43(22): 6855–6868.

16.

Faroughi

Kamran

Lee

(2014) A genetic algorithm approach for 2-D tensegrity form finding. Advances in Structural Engineering 17(11): 1669–1680.

17.

Feng

Guo

(2015) A novel method of determining the sole configuration of tensegrity structures. Mechanics Research Communications 69: 66–78.

18.

Fuller

(1962) Tensile-integrity structures. Available at: https://www.google.com/patents/US3063521

19.

Gambhir

(2004) Stability Analysis and Design of Structures. New York: Springer.

20.

Gan

Zhang

Nguyen

D-K

. (2015) Node-based genetic form-finding of irregular tensegrity structures. Computers & Structures 159: 61–73.

21.

Godoy

(1999) Theory of Elastic Stability: Analysis and Sensitivity. Philadelphia, PA: Taylor & Francis.

22.

Grefenstette

(1986) Optimization of control parameters for genetic algorithms. IEEE Transactions on Systems, Man, and Cybernetics 16(1): 122–128.

23.

Koohestani

(2015) Automated element grouping and self-stress identification of tensegrities. Engineering Computations 32(6): 1643–1660.

24.

Koohestani

Guest

(2013) A new approach to the analytical and numerical form-finding of tensegrity structures. International Journal of Solids and Structures 50(19): 2995–3007.

25.

Feng

X-Q

Cao

Y-P

. (2010) A Monte Carlo form-finding method for large scale regular and irregular tensegrity structures. International Journal of Solids and Structures 47(14): 1888–1898.

26.

Linkwitz

Schek

H-J

(1971) Einige bemerkungen zur berechnung von vorgespannten seilnetzkonstruktionen. Ingenieur-Archiv 40(3): 145–158.

27.

Linkwitz

(1999) Formfinding by the direct approach and pertinent strategies for the conceptual design of prestressed and hanging structures. International Journal of Space Structures 14(2): 73–87.

28.

Shu

(2015) Form-finding analysis of irregular tensegrity structures by matrix iteration. Advanced Steel Construction 11(4): 507–516.

29.

Luo

(2007) Mobility and equilibrium stability analysis of pin-jointed mechanisms with equilibrium matrix SVD. Journal of Zhejiang University: Science A 8(7): 1091–1100.

30.

Masic

Skelton

Gill

(2005) Algebraic tensegrity form-finding. International Journal of Solids and Structures 42(16): 4833–4858.

31.

Mirats Tur

Juan

(2009) Tensegrity frameworks: dynamic analysis review and open problems. Mechanism and Machine Theory 44(1): 1–18.

32.

Motro

(2011) Structural morphology of tensegrity systems. Meccanica 46(1): 27–40.

33.

Motro

Nooshin

(1984) Forms and forces in tensegrity systems. In: Nooshin

(ed.) Proceedings of the Third International Conference on Space Structures. Amsterdam: Elsevier, pp. 180–185.

34.

Ohsaki

Zhang

(2015) Nonlinear programming approach to form-finding and folding analysis of tensegrity structures using fictitious material properties. International Journal of Solids and Structures 69–70: 1–10.

35.

Pagitz

Tur

(2009) Finite element based form-finding algorithm for tensegrity structures. International Journal of Solids and Structures 46(17): 3235–3240.

36.

Paul

Lipson

Cuevas

FJV

(2005) Evolutionary form-finding of tensegrity structures. In: Proceedings of the 2005 conference on genetic and evolutionary computation, Washington, DC, 25–29 June, pp. 3–10. New York: ACM.

37.

Pellegrino

(1986) Mechanics of kinematically indeterminate structures. PhD Thesis, University of Cambridge, Cambridge.

38.

Rieffel

Valero-Cuevas

Lipson

(2009) Automated discovery and optimization of large irregular tensegrity structures. Computers & Structures 87(5): 368–379.

39.

Schek

H-J

(1974) The force density method for form finding and computation of general networks. Computer Methods in Applied Mechanics and Engineering 3(1): 115–134.

40.

Snelson

(1965) Continuous tension, discontinuous compression structures. Available at: http://www.google.co.in/patents/US3169611

41.

Skelton

Adhikari

Pinaud

Chan

Helton

(2001) An introduction to the mechanics of tensegrity structures. In: Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, FL, December, Vol. 5, pp. 4254–4259. IEEE.

42.

Sultan

Corless

Skelton

(1999) Reduced prestressability conditions for tensegrity structures. In: Proceedings of 40th ASME structures, structural dynamics and materials conference, St. Louis, MO, April, pp. 2300–2308. DOI: dx-doi-org.web.bisu.edu.cn/10.2514/6.1999-1478

43.

Tachi

(2012) Interactive freeform design of tensegrity. In: Hesselgren

Sharma

Wallner

al.

(eds) Advances in Architectural Geometry 2012. Vienna: Springer, pp. 259–268.

44.

Tibert

Pellegrino

(2003) Review of form-finding methods for tensegrity structures. International Journal of Space Structures 18(4): 209–223.

45.

Timoshenko

Gere

(1961) Theory of Elasticity Stability. New York: McGraw-Hill.

46.

Toklu

Uzun

(2016) Analysis of tensegric structures by total potential optimization using metaheuristic algorithms. Journal of Aerospace Engineering. Epub ahead of print 14 March. DOI: 10.1061/(ASCE)AS.1943-5525.0000571.

47.

Tran

Lee

(2011) Determination of a unique configuration of free-form tensegrity structures. Acta Mechanica 220(1–4): 331–348.

48.

Luo

(2010b) Force finding of tensegrity systems using simulated annealing algorithm. Journal of Structural Engineering 136(8): 1027–1031.

49.

Luo

(2010a) Form-finding of nonregular tensegrities using a genetic algorithm. Mechanics Research Communications 37(1): 85–91.

50.

Zhang

Ohsaki

Kanno

(2006a) A direct approach to design of geometry and forces of tensegrity systems. International Journal of Solids and Structures 43(7): 2260–2278.

51.

Zhang

Ohsaki

(2006) Adaptive force density method for form-finding problem of tensegrity structures. International Journal of Solids and Structures 43(18): 5658–5673.

52.

Zhang

Maurin

Motro

(2006b) Form-finding of nonregular tensegrity systems. Journal of Structural Engineering 132(9): 1435–1440.

53.

Ziegler

(1968) Principles of Structural Stability. Waltham, MA: Blaisdell Publishing Company.