Research on optimum design and construction process of tensegrity tower structures

Abstract

Tensegrity tower structures are innovative lightweight structures that feature a system of tension members arranged such that compressive members do not directly interact. In this article, a 7-stage tensegrity tower structure was designed and a modified genetic algorithm was used to optimize the prestress distribution within and shape of the tower. The optimization result shows the structure’s lateral stiffness improves considerably compared with the initial model. Moreover, in order to determine the construction process, a construction scheme utilizing initial prestress and compensating prestress was proposed and a computational program named compensating prestress design program was developed. This program can help identify target tension and joint coordinates at each construction step such that the cable tension distribution in the completed structure meets designer-specified target values. Sensitivity analysis was also introduced in this program and Key Cable Theory was proposed as a result in order to improve the precision of compensating prestress design program. An illustrative example showed after applying compensating prestress design program, tension errors were all controlled within a small range.

Keywords

construction process genetic algorithm optimization tensegrity tower structure

Introduction

Tensegrity structures are prestressable truss-like systems involving tensile cable elements and compressive struts. In recent years, the concept of tensegrity has been widely used to build long-span roof structures, such as cable domes, but its application in vertical direction is seldom reported, except for some sculptures and non-functional structures, for example, Needle Tower (Snelson, 1990) and Warnow Tower in Rostock (Schlaich, 2004). This structural form has received much attention in aerospace engineering because of its lightweight, deployable, and controllable characteristics (Sultan et al., 1999, 2001, 2002a, 2002b; Sultan and Skelton, 1997, 1998a, 1998b). Despite these potential advantages, however, tensegrity towers seldom see practical use. In this article, a 7-stage tensegrity tower structure is proposed to function as TV tower, while also providing a tourist platform on the top storey.

Tensegrity structures are usually designed by trial and error and engineering intuition. Yet, structural efficiency of tensegrity structures is highly dependent on its shape and distribution of prestress, and thus, optimization can offer advantages in systematically exploring this design variable space. Several optimization attempts have been made on cable domes (Kawaguchi et al., 1999; Yuan and Dong, 2002), where design variables included prestress distribution, shape, and cross-section area. Masic et al. (2006) used Sequential Quadratic Programming (SQP) to optimize the topology and geometry of some tensegrity structures, including a singular valve decomposition (SVD) tensegrity tower. All these attempts proved optimization played a critical role in improving structural behavior.

Since tensegrity tower structures are seldom built, and thus, there is not much literature or experience regarding their construction. The Warnow Tower, 62.3 m in height, is an exception to this. Its tensegrity units were all pre-fabricated off-site and then hoisted for connection to the base structure. Its struts of adjacent stages are linked on ends, which technically violates the definition of tensegrity that compressive members should be discrete (Motro, 1992), but this simplification made construction much easier. As for other prestress structures like cable domes and suspen-domes, in-depth studies on constructions have been made (Guo et al., 2009; Wang et al., 2012; Yuan and Dong, 2001) and many practical structures have been built. It should be mentioned that prestress compensation methods refer to situations when prestressing cannot be performed only once, but rather tension change requires subsequent tensioning in an iterative manner. Numerically, this is often performed by iteratively applying compensating temperature loads to the cables, often used in space grid structures (Zhuo and Koichiro, 2004) and suspen-domes (Wang et al., 2012).

This article is organized as follows. In section “Introduction to the model,” an introduction of the model is made, including topology, geometry, nomination, and so on. In section “Optimum design of tensegrity tower structure,” prestress and shape optimization are conducted. In section “Illustrative example,” a methodology for construction process design is proposed along with a computational program compensating prestress design program (CPDP). An example of this methodology is given in section “Illustrative example” and we finish with some conclusions in section “Conclusion.”

Introduction to the model

This proposed structure’s topology is largely inspired by Snelson’s Needle Tower. It consists of seven 3-bar tensegrity units, which are connected with draw and sling cables. As the struts are discrete, it is a true tensegrity structure. There is a 30° rotation angle between the upper and lower triangle of the unit, and the rotation directions of two adjacent units are opposite in order to offset the twisting behavior of the entire structure. A tube for the elevator with a diameter of 3.5 m was added inside the structure and attached to the outer structure with tube draw and sling cables. The configuration of the structure is shown in Figures 1 –4. Each cable name represents a group, including three cables, which share the same design tension. Both draw and sling cables include two sets of length: the shorter is named draw/sling cable A and the longer is named draw/sling cable B. A prefix denoting position is added to each group of cables, for example, “1-stage upper horizontal cable” represents the upper horizontal cables of the first stage unit, “1-2-stage draw cable A” represents the draw cables A connecting the first and second stage units, and the same goes for other cables.

Figure 1.

3-bar tensegrity unit.

Figure 2.

Connection between units.

Figure 3.

Cables for the elevator tube.

Figure 4.

Combination of two stages.

The total height of this structure is 86.28 m, and an antenna will potentially be added on the top making it to reach 120 m high. The elevation view and 3D view are shown in Figures 5 and 6, respectively.

Figure 5.

Elevation view.

Figure 6.

3D view.

The compressive bars are made of steel tubes with a Young’s modulus of 206 GPa, and the cables are made of steel wires with a Young’s modulus of 195 GPa and yield stress of 1870 MPa. The parameters denoting the shape are shown in Table 1. The assignment of sections is shown in Table 2.

Table 1.

Parameters denoting the shape.

Stage	Length of diagonal cables (m)	Unit height (m)	Unit overlap height (m)	Inclination angle of bars
1	22	21.25	–	45°
2	20	19.32	5.5	45°
3	18	17.39	5.0	45°
4	16	15.46	4.5	45°
5	14	13.52	4.0	45°
6	12	11.59	3.5	45°
7	10	9.66	3.0	45°

Table 2.

Assignment of sections.

Stage	Strut section (mm)	Cable diameter (mm)
1	1000 × 20	77.64
2	900 × 20	77.64
3	800 × 20	68.39
4	750 × 20	68.39
5	650 × 20	68.39
6	530 × 20	56.36
7	425 × 16	56.36

A special joint is specifically designed to facilitate compensating prestress. The joint, referred to as the Dragon Claw Joint, is shown in Figure 7 and consists of a hollow sphere joint that is attached to the end of struts, with some antenna tubes on it. A set of threaded tubes is welded to the antenna tube and the cable goes through the threaded tubes. The tension of the cables is then adjusted by screwing the threaded tubes.

Figure 7.

Dragon Claw Joint.

A common finite element (FE) software, SAP2000, is used in this article for nonlinear solving. Cables and struts are simulated with two-node frame element.

Optimum design of tensegrity tower structure

In a typical genetic algorithm (Goldberg, 1989), the evolution starts with a population of randomly generated designs that are encoded into binary strings (genes). The population in each iteration is called a generation. In each generation, the fitness of every design in the population is evaluated and the designs with larger fitness have a higher possibility of being selected from the current population to serve as “parents.” New designs are then creating by mixing components of two parent designs through operations known as crossover and mutation. In this work, we use two-point crossover with mutation rates of 1%. The resulting new designs represent the next generation, and this evolution continues until convergence, typically defined as a large number of iterations that the best, or elite, design is unchanged.

The prestressed tensegrity tower problem is challenging in that it is very easy for the system force distribution to lead to cables slack. To help overcome this challenge, we have modified the typical genetic algorithm slightly. If an elite design has slack cables and remains elite for 50 generations while some cables are still slack, the mutation rate of genes of the slack cables is changed from 1% to 50%, which significantly increases the versatility of the population and helps avoid early convergence of infeasible designs with slack cables. If the elite design then changes, the mutation rate is returned to 1%.

Design variables

In most FE software, a common approach to applying prestress in cable models is to apply an internal temperature load to the cables. There are 58 groups of cables in the proposed structure, the temperature loads of these 58 groups were defined as design variables, denoted as T_n , with n = 1, 2,…, 58.

The structure’s height was prescribed, and thus, only four design variables denoting the structure’s shape are used as follows:

D ₁: the length of diagonal cables in the first stage, equal to the diameter of circumscribed circle of three upper or lower horizontal cables;

d: the length difference of diagonal cables in two adjacent units. That is to say, the length of the second stage diagonal cables is $D_{1} - d$ , the third $D_{1} - 2 d$ , and so on;

$ζ$ : ratio between the overlap height and the height of lower stage. As shown in Figure 8, taking the first and second stages for example, $ζ = s_{1} / H_{1}$ . This value is uniform for all stages, that is, $ζ = s_{1} / H_{1} = s_{2} / H_{2} = s_{3} / H_{3} = s_{4} / H_{4} = s_{5} / H_{5} = s_{6} / H_{6}$ ;

$μ$ : the parameter denoting the joint of tube draw and sling cables on the elevator tube. This value is also uniform for all stages. As shown in Figure 8, $μ = a_{1} / b_{1}$ .

Figure 8.

Definition of $ζ$ and $μ$ .

We introduce some limits to these variables to prevent significant shape changes. $D_{1}$ , $ζ$ , and $μ$ are independent variables, their constraints are as follows

\begin{matrix} D_{1} \in [21.5, 22.5] \\ ζ \in [0.2, 0.3] \\ μ \in [0.25, 0.35] \end{matrix}

Since the total height is fixed to 83.183 m, the following constraint is needed

\sum_{i = 1}^{7} H_{i} - \sum_{i = 1}^{6} s_{i} = 83.183

(1)

where $H_{i} = 0.966 D_{i}, i = 1, 2, 3, \dots, 7$ ; $s_{i} = ζ \cdot H_{i}, i = 1, 2, 3, \dots, 6$ ; $D_{i} = D_{1} - (i - 1) \cdot d, i = 1, 2, 3, \dots, 7$ .

Equation (1) can be written as

(7 - 6 ζ) D_{1} - (21 - 15 ζ) d = 83.183

(2)

Considering the constraints of D ₁ and $ζ$ , we get

d \in [\frac{5.8 D_{1} - 86.11}{18}, \frac{5.2 D_{1} - 86.11}{16.5}]

In order to prevent some unreasonable values of d which may make the structure out of shape, we introduce some limits to d

d \in [min \frac{5.8 D_{1} - 86.11}{18, 2.5}, max \frac{5.2 D_{1} - 86.11}{16.5, 1.5}]

Objective function

The governing load in tensegrity towers is typically lateral wind loads and, with generally low lateral stiffness, lateral displacements and vibrations often become the governing design constraints. We therefore look to minimize the lateral deflection $δ$ at the top of the tower, or conversely maximize its reciprocal

max . K = \frac{1}{δ}

(3)

A nonlinear FE analysis is performed to calculate deformations under self-weight and wind load according to Chinese Structural Load Code 2012.

The stress in the members is constrained as follows

0.08 < \frac{σ}{[σ]} < 1

(4)

where $[σ]$ represents allowable stress; 0.08 is calculated according to Chinese Technical Specification for Cable Structures, representing the lower bound of stress level to prevent the existence of slack cables, whose deflection in mid-span exceed 1/250.

As typical in genetic algorithms, a penalty function is used to penalize infeasibility. This is computed as

\begin{matrix} δ_{1} = \sum (max (\frac{σ_{d}}{[σ]}, 1) - 1) \\ + \sum (max (\frac{σ_{w}}{[σ]}, 1) - 1) \end{matrix}

(5a)

\begin{matrix} δ_{2} = 3 \times [\sum (0.08 - min (\frac{σ_{d}}{[σ]}, 0.08)) \\ + \sum (0.08 - min (\frac{σ_{w}}{[σ]}, 0.08))] \end{matrix}

(5b)

where $σ_{d}$ stands for stress under dead load case and $σ_{w}$ stands for stress under wind load case. The penalty function is calculated in this way because we want $δ_{1}$ , $δ_{2}$ , and $δ$ to have approximately the same order of magnitude, making the penalty strict enough to ensure the feasibility of optimization result.

Then our objective function can be written as

max . K = \frac{1}{δ + δ_{1} + δ_{2}}

(6)

The optimization problem is expressed as follows

{\begin{matrix} max . 1 / (δ + δ_{1} + δ_{2}) \\ \begin{matrix} subject to : \\ T_{n} \in [- 327.68, 0] \\ D_{1} \in [21.5, 22.5] \\ ζ \in [0.2, 0.3] \\ μ \in [0.25, 0.35] \end{matrix} \end{matrix}

Optimization result

Before running the optimization, we attempted to design the prestress distribution by trial and error, as is often done in practice. This proved to be quite challenging due to the complexity and scale of the tower. Modifications to individual cable prestressing frequently led to slack being introduced somewhere in the structure, and satisfying the constraints conditions was quite time-consuming and challenging. After a significant number of iterations, we arrived at a feasible solution having a top node displacement of 0.474 m (K = 2.108). This solution had the least displacement in all feasible solutions we got in this trial-and-error process, and no cable was slack.

The optimization problem was then solved with the genetic algorithm using a convergence criterion of 100 generations without a change in the elite design. The optimized solution is feasible and features a top point displacement of 0.268 m on the top (K = 3.731), an improvement of 77% over the ad hoc-designed case. No cable experiences slack or exceeds the stress constraint under both dead and wind load case. The optimized shape parameters are D ₁ = 22.453 m, d = 1.884 m, ζ = 0.2959, µ = 0.3324. Although these shape parameters provide only minor shape change over the original design, they have significant impact on the structural behavior. The evolution of fitness function is shown in Figure 9. The optimized design tension of cables is shown in Table 3.

Figure 9.

Evolution of fitting function.

Table 3.

Optimized design tension.

Cable group	Tension (kN)	Cable group	Tension (kN)	Cable group	Tension (kN)
1-stage upper horizontal	693.452	3-4-stage draw B	354.922	5-6-stage sling A	527.854
1-stage diagonal	2713.612	3-4-stage sling A	351.323	5-6-stage sling B	820.956
1-2-stage draw A	3041.565	3-4-stage sling B	691.394	4-stage elevator sling	865.023
1-2-stage draw B	1895.671	2-stage elevator sling	546.236	5-stage elevator draw	295.555
1-2-stage sling A	732.157	3-stage elevator draw	835.905	6-stage lower horizontal	172.733
1-2-stage sling B	2184.342	4-stage lower horizontal	800.993	6-stage upper horizontal	129.701
1-stage elevator draw	1231.839	4-stage upper horizontal	582.765	6-stage diagonal	1145.162
2-stage lower horizontal	241.624	4-stage diagonal	2394.974	6-7-stage draw A	698.168
2-stage upper horizontal	1001.262	4-5-stage draw A	557.596	6-7-stage draw B	984.542
2-stage diagonal	2731.814	4-5-stage draw B	458.972	6-7-stage sling A	677.361
2-3-stage draw A	2110.915	4-5-stage sling A	403.185	6-7-stage sling B	356.334
2-3-stage draw B	1378.535	4-5-stage sling B	406.052	5-stage elevator sling	1161.096
2-3-stage sling A	684.585	3-stage elevator sling	919.79	6-stage elevator draw	244.232
2-3-stage sling B	1534.734	4-stage elevator draw	41.785	7-stage upper horizontal	177.509
2-stage elevator draw	382.548	5-stage lower horizontal	1049.997	7-stage lower horizontal	298.256
1-stage elevator sling	983.889	5-stage upper horizontal	303.405	7-stage diagonal	1594.387
3-stage lower horizontal	301.627	5-stage diagonal	2344.391	7-stage elevator draw	557.838
3-stage upper horizontal	565.571	5-6-stage draw A	1185.66	6-stage elevator sling	510.085
3-stage diagonal	2531.539	5-6-stage draw B	866.522	7-stage elevator sling	759.136
3-4-stage draw A	660.35

CPDP

The construction process of tensegrity structures is extremely critical to the final stress state and may be quite challenging to optimize since prestressing of new cables may lead to significant changes in already installed cables. For towers, this problem becomes more severe as gravity loads accumulate in vertical direction and any twisting or deformation will be enlarged. Unlike roof structures, tensegrity towers should be built stage by stage rather than prestressing as a whole. Our analyses show that after initial prestress, cables usually cannot achieve the design tension and some cables even get slack, so compensating prestress is required. Thus, the whole construction process can be divided into two parts: initial prestress and compensating prestress. In order to have all cable tension achieve our target tension, a compensating prestress design method is proposed, along with a computational program named CPDP.

Basic assumptions

During construction, cables are prestressed in groups. As defined above, each group includes three cables with the same design tension. However, the tension of cables in the same group may vary during construction because of inevitable twisting behavior of the structure. Considering the number of prestressing devices, feasibility, and precision, it is assumed that not more than four groups of cables are prestressed at the same time. Since construction errors are not considered here, each cable will undergo compensating prestress for only once, rather than time and time again.

Birth and death elements technique is adopted for construction process analysis in this section. Elements get “killed” when their stiffness multiply by a severe reduction factor, like 10⁻⁶. In the same manner, elements become “alive” when their stiffness return to their full original values. We first “kill” all elements and make them alive stage by stage; thus, construction process is simulated (Computers and Structures, Inc., 2015).

The parameters related to the prestressing are defined as follows:

In the kth iteration of CPDP, $T_{i}^{k}$ is the cable tension after initial prestress, $P_{i}^{k}$ is the cable tension after compensating prestress, where i is the serial number of cables, i = 1, 2,…, 174;

$P_{i}^{d}$ is the design (target) tension of each cable, in other words, our target tension;

${err}_{i}^{k}$ is the tension error after the k iteration, ${err}_{i}^{k} = (P_{i}^{k} - P_{i}^{d}) / P_{i}^{d}$ .

Key Cable Theory

Sensitivity analysis is innovatively introduced in this methodology to figure out the cables that have the most influence on the system, herein termed Key Cables. These Key Cables are sensitive to the tension change of other cables or are more likely to influence other cables tension. These cables were identified by computing Spearman’s rank correlation coefficient for Monte Carlo simulation as described below.

Since there are many cables in the model, we divide all cables into three categories to conduct sensitivity analysis separately, selecting the most influential cables in each category, and then put these cables together to conduct sensitivity analysis to find out the Key Cables for the whole structure. The three categories are as follows: unit cables (including all diagonal cables, upper and lower horizontal cables), draw and sling cables (including all draw and sling cables A/B), and elevator tube cables (including all elevator tube draw and sling cables). The variation from the target prestress magnitude (temperature load in the FE model) of each cable was assumed to follow a normal distribution with unit mean and a standard deviation of 0.04. 20,000 simulations was performed and Spearman’s rank correlation coefficient was calculated according to the displacement of the top point of the tower. The results are shown in Table 4 where cables with small correlation coefficient are omitted.

Table 4.

Sensitivity analysis result for each category.

Rank	Unit cables	Correlation coefficient	Draw and sling cables	Correlation coefficient	Elevator tube cables	Correlation coefficient
1	1-stage diagonal	0.76	1-2-stage sling B	−0.78	1-stage elevator tube draw	−0.92
2	2-stage diagonal	−0.44	1-2-stage draw A	−0.58	1-stage elevator tube sling	0.15
3	2-stage lower horizontal	0.29	4-5-stage draw B	−0.50
4	4-stage upper horizontal	0.11	5-6-stage draw B	−0.45
5			2-3-stage draw A	−0.43

The 11 cables listed in Table 4 are selected to conduct the overall sensitivity analysis. The result are listed in Table 5, cables with small correlation coefficient are omitted. We can see that the tension change of 1-stage elevator tube draw cable, 1-2-stage sling cable B, and 1-2-stage draw cable A exhibits the most influence on the structure. Also the influence of 1-stage diagonal cable, 2-stage diagonal cable, and 2-stage lower horizontal cable is not negligible. These cables can be defined as Key Cables.

Table 5.

Overall sensitivity analysis result.

Rank	Cable	Correlation coefficient
1	1-stage elevator tube draw	−0.59
2	1-2-stage sling B	−0.57
3	1-2-stage draw A	−0.44
4	1-stage diagonal	0.20
5	2-stage diagonal	−0.11
6	2-stage lower horizontal	0.09
7	4-5-stage draw B	0.03

Computational procedure of CPDP

Temperature compensation is the fundamental numerical method in CPDP, which means applying a compensating temperature load according to the difference between design tension and actual tension. We introduce a parameter $β$ , named as acceleration coefficient to control temperature compensation. The initial compensating temperature load can be calculated as follows

Δ t_{i}^{0} = β_{i}^{0} \cdot \frac{P_{i}^{d} - T_{i}^{k}}{α E A_{i}}

(7)

where $Δ t_{i}^{0}$ represents the initial value of compensating temperature load, $P_{i}^{d}$ represents the design tension of cables, $T_{i}^{k}$ represents the actual cable tension after initial prestress, $α$ represents coefficient of thermal expansion of cables, E represents Young’s Modulus of cables, $A_{i}$ represents the cross-section area of cables, $β_{i}^{0}$ represents the initial value of $β$ , it can be a positive or negative value, also can be 0, which means corresponding cables will not be adjusted. The larger the absolute value of $β$ , the more aggressive the compensation.

Simply speaking, the ultimate goal of CPDP is to find $β_{i}^{k}$ , the precise value of $β$ , such that the compensating temperature load can be calculated. The whole process of CPDP can be divided into two parts: determining the rough value of $β$ and determining the precise value of $β$ . In the first part, the tension error after compensating prestress should be as small as possible, better not more than 10% for most cables. The procedure for the first part is as follows:

Step 1. Build the FE model.

Step 2. Customize $P_{i}^{d}$ and $β_{i}^{0}$ .

Step 3. Run initial prestress load case using birth and death elements.

Step 4. Calculate $Δ t_{i}^{0}$ by equation (7).

Step 5. Run compensating prestress load case, using $Δ t_{i}^{0}$ as compensating temperature.

Step 6. Extract cable tension $P_{i}^{1}$ when compensating prestress load case is completed, and calculate ${err}_{i}^{1} = (P_{i}^{1} - P_{i}^{d}) / P_{i}^{d}$ . According to ${err}_{i}^{1}$ , adjust $β_{i}^{0}$ and iterate steps 2–6 until most cables’ tension errors decrease to an acceptable range, like 10%.

Although $β_{i}^{0}$ is determined via trial and error, it is not significant effort to have most cables’ tension error meet our requirements, leaving only several groups with larger errors. As for selecting $β_{i}^{0}$ such that other cables do not deviate from their designed tension, we typically must iterate within CPDP. At this time, we can only adjust some Key Cables rather than all. It is the so-called Key Cable Theory. Following steps 1–6, Key Cable Theory is realized as follows:

Step 7. Identify Key Cables; customize maximum tension error err_max and maximum iteration number k_max .

Step 8. For Key Cables, $β_{i}^{k} = β_{i}^{k - 1} \cdot (1 - {err}_{i}^{k})$ ; for other cables, $β_{i}^{k} = β_{i}^{0}$ . Replace $β_{i}^{k - 1}$ with $β_{i}^{k}$ and run compensating prestress load case again.

Step 9. Extract cable tension $P_{i}^{k + 1}$ when compensating prestress load case is completed, and calculate ${err}_{i}^{k + 1} = (P_{i}^{k + 1} - P_{i}^{d}) / P_{i}^{d}$ .

Step 10. For Key Cables, if ${err}_{i}^{k} \cdot {err}_{i}^{k + 1} > 0$ , then $β_{i}^{k + 1} = β_{i}^{k} \cdot (1 - {err}_{i}^{k + 1})$ ; if ${err}_{i}^{k} \cdot {err}_{i}^{k + 1} < 0$ , then adopt linear interpolation to $β_{i}^{k - 1}$ and $β_{i}^{k}$ , $β_{i}^{k + 1} = ({err}_{i}^{k} β_{i}^{k} - {err}_{i}^{k + 1} β_{i}^{k - 1}) / ({err}_{i}^{k} - {err}_{i}^{k + 1})$ . For other cables, $β_{i}^{k + 1} = β_{i}^{0}$ . Replace $β_{i}^{k}$ with $β_{i}^{k + 1}$ and run compensating prestress load case again.

Step 11. If all $| {err}_{i}^{k} |$ are less than err_max , iterations terminate. Otherwise, iterate steps 8 and 9 until all $| {err}_{i}^{k} |$ are less than err_max , or maximum iteration number k_max is reached.

Step 12. Output $β_{i}^{k}$ and ${err}_{i}^{k}$ , CPDP ends.

The flow chart of CPDP is shown in Figure 10.

Figure 10.

Flow chart of CPDP.

Illustrative example

Considering feasibility and constructability, an initial prestress process is proposed as follows:

Step 1. Prestress the 1-stage unit and fix the lower ends of struts to the ground with hinges.

Step 2. Hoist the 1-stage elevator tube to position and prestress the elevator tube draw and sling cables at the same time to connect the tube to the 1-stage unit.

Step 3. Prestress the 2-stage unit on the ground, then hoist it to position with some measures to prevent it from wavering. Three hoisting strings are used to hoist the unit, shown in Figure 11.

Step 4. Prestress 1-2-stage draw and sling cables at the same time to designated tension.

Step 5. Remove the hoisting strings of the 2-stage unit.

Step 6. Hoist the 2-stage elevator tube to position and prestress the elevator tube draw and sling cables at the same time to connect the tube to the 2-stage unit. Then, weld the 2-stage tube with 1-stage tube.

Step 7. Iterate steps 3–6 until all seven stages are finished, 26 steps in total.

Figure 11.

Hoisting of a unit.

In this example, we assign design tension $P_{i}^{d}$ with the cable tension in Table 3 and assign temperature load value in initial prestress load case with those in Table 3. By conducting steps 1–3 of CPDP, we can get tension errors after initial prestress, summarized in Table 6.

Table 6.

Summary of cable tension errors after initial prestress load case.

Tension error	−100% to −90%	−90% to −50%	−50% to −10%	−10% to 10%	10% to 50%	Above 50%
Quantity	3	2	56	53	45	15
Proportion (%)	1.72	1.15	32.18	30.46	25.86	8.62

Detailed statistics show that all three 1-2-stage sling cable B are slack, with a tension error of −96.55%. If we require all tension error less than 10%, then only 53 cables (30.46%) meet our requirement. We also find that the cable tension of cables in the same group varies as a result of twisting behavior.

We then conduct the compensating prestress. Since the initial prestress is from bottom to top, the compensating prestress is from top to bottom; thus, it can start right after initial prestress is completed. The compensating procedure is as follows:

Step 1. Compensate 7-stage elevator tube draw and sling cables.

Step 2. Compensate 7-stage unit cables, including diagonal and horizontal cables.

Step 3. Compensate 6-7-stage draw and sling cables.

Step 4. Iterate steps 1–3 for lower modules until all 1-stage cables are compensated, 20 steps in total.

Next, we conduct steps 4–6 of CPDP to determine $β_{i}^{0}$ . It is recommended to start from unit magnitude and then adjust each $β_{i}^{0}$ according to the feedbacks of ${err}_{i}^{1}$ . The final $β_{i}^{0}$ mostly ranges from 0.5 to 1.5. We use this set of $β_{i}^{0}$ to calculate temperature load and run compensating prestress load case. The tension error after compensating prestress is shown in Table 7.

Table 7.

Summary of tension errors after compensated by $β_{i}^{0}$ .

Tension error	Below −10%	−10% to −3%	−3% to 3%	3% to 10%	Above 10%
Quantity	1	10	149	12	2
Proportion (%)	0.57	5.75	85.63	6.90	1.15

Detailed data show only three cables have tension errors that are larger than 10%, including a 1-2-stage sling cable A (−14.5%), a 1-stage upper horizontal cable (10.4%), and a 3-stage elevator tube draw cable (12.0%). We then progress to the second part of CPDP, starting with assigning Key Cables. It is a critical step for the processing of CPDP because proper assignment of Key Cables improves the precision dramatically. The assignment and reasons are shown as follows:

According to the sensitivity analysis mentioned above, 1-stage elevator tube draw cable, 1-2-stage sling cable B, and 1-2-stage draw cable A are assigned as Key Cables.

Since 1-2-stage sling cable A becomes slack after initial prestress, as tension is relatively small and is easily influenced by other cables but hardly influences others, we assign this group as Key Cables.

The tension error of 1-2-stage sling cable A and 3-stage elevator tube draw cable are larger than 10%; moreover, their tension is relatively large, which means they are difficult to be influenced by other cables, so we put these two groups as Key Cables.

If we only assigned cables mentioned above as Key Cables, after running CPDP we find that the 3-stage and 5-stage elevator tube draw cables deviate from design tension considerably, so we also assign these two groups as Key Cables.

We customize err_max = 5%, k_max = 5 and continue conducting steps 8–12. When k = 2, CPDP ends and the summary of final tension errors is shown in Table 8.

Table 8.

Summary of tension errors after CPDP ends.

Tension error	−5% to −1%	−1% to 1%	1% to 5%
Quantity	23	124	27
Proportion (%)	13.22	71.26	15.52

The maximum tension error is 4.4%, and most tension errors lie within the range of −1% to 1%, which is quite satisfactory. Then, we can extract the target tension and joint coordinates of each construction step to guide our actual construction process.

Conclusion

In this article, a 7-stage tensegrity tower structure was designed and its prestress distribution and shape were optimized. After optimization, all cables satisfied stress constraints and the lateral stiffness was improved dramatically compared with those designed manually by trial and error.

A methodology to design compensating prestress procedure was proposed and a computational program CPDP was coded. This program is based on temperature compensation method, seeking compensating temperature load at each construction step to simulate the prestressing magnitude. Taking the optimized structure as example, this methodology successfully helped us to find a feasible construction process and tension errors were controlled within a small range. Since the outcomes of CPDP heavily rely on the assignment of Key Cables, we should fully consider all factors to make the best assignments. It is worth mentioning that the methodology and CPDP may also be applied to other prestressed structures.

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

Computers and Structures, Inc. (2015) SAP2000 Help Documentation. Berkeley, CA: Computers and Structures, Inc.

Goldberg

(1989) Genetic Algorithms in Search, Optimization and Machine Learning. Reading, MA: Addison-Wesley.

Guo

Yuan

Dong

. (2009) Global-process pretensioning analysis of suspen-dome. Engineering Mechanics 26(1): 198–203.

Kawaguchi

Tatemichi

Chen

(1999) Optimum shapes of a cable dome structure. Engineering Structures 21: 719–725.

Masic

Skelton

Gill

(2006) Optimization of tensegrity structures. International Journal of Solids and Structures 43: 4687–4703.

Motro

(1992) Tensegrity systems: the state of the art. International Journal of Space Structures 7(2): 75–83.

Schlaich

(2004) The Messeturm in Rostock—a tensegrity tower. Journal of International Association for Shell and Spatial Structures 45: 93–98.

Snelson

(1990) Correspondance avec R. Motro.

Sultan

Skelton

(1997) Integrated design of controllable tensegrity structures. In: Proceedings of the ASME International Congress and Exposition, vol. 54, pp. 27–37.

10.

Sultan

Skelton

(1998a) Force and torque smart tensegrity sensor. In: Proceedings of SPIE 5th symposium on smart structures and materials, San Diego, CA, 24 July, vol. 3323, pp. 357–368. Bellingham, WA: SPIE.

11.

Sultan

Skelton

(1998b) Tendon control deployment of tensegrity structures. In: Proceedings of SPIE 5th symposium on smart structures and materials, San Diego, CA, 24 July, vol. 3323, pp. 455–466. Bellingham, WA: SPIE.

12.

Sultan

Corless

Skelton

(1999) Peak to peak control of an adaptive tensegrity space telescope. In: Proceedings of SPIE 6th symposium on smart structures and materials, Newport Beach, CA, 4 June, vol. 3667, pp. 190–201. Bellingham, WA: SPIE.

13.

Sultan

Corless

Skelton

(2001) The prestressability problem of tensegrity structures: some analytical solutions. International Journal of Solids and Structures 38(30–31): 5223–5252.

14.

Sultan

Corless

Skelton

(2002a) Linear dynamics of tensegrity structures. Engineering Structures 24(6): 671–685.

15.

Sultan

Corless

Skelton

(2002b) Symmetrical reconfiguration of tensegrity structures. International Journal of Solids and Structures 39(8): 2215–2234.

16.

Wang

Luo

Guo

(2012a) Algorithm for compensating tension revision of cable force based on temperature compensation. Journal of Southeast University (Natural Science Edition) 42(1): 125–130.

17.

Wang

Cheng

You

. (2012b) Research on construction techniques of cable dome roof. Journal of Building Structures 33(4): 67–76.

18.

Yuan

Dong

(2001) Inverse analysis of construction process of cable dome. Journal of Building Structures 22(2): 75–79.

19.

Yuan

Dong

(2002) Nonlinear analysis and optimum design of cable domes. Engineering Structures 24(7): 965–977.

20.

Zhuo

Koichiro

(2004) Tensile force compensation analysis method and application in prestressed space grid structures tension construction. China Civil Engineering Journal 37(4): 38–41.