Int. J. Structural Engineering, Vol. 1, Nos. 3/4, 2010

241

Overall stiffness evaluation and shape optimisation ofa tensegric structure
Pei-Shan Chen
Department of Civil Engineering and Architecture andGraduate School,
Hachinohe Institute of Technology,
Oobiraki 88-1, Myo, Hachinohe City, Aomori 013-8501, Japan
Fax: +81 178 25 5018
E-mail: chen@hi-tech.ac.jp
Abstract: The present paper promotes a shape optimisation method for
maximum overall stiffness of a tensegric structure, and reports a study on theoverall stiffness of a tensile structure. General reduced gradient method isadopted for the shape optimisation, and a tensegric dome is taken as anexample for the numerical demonstration. The results of numerical analysis
show that the nodal displacements of the example decreases from 20%–30%
compared to that of the initial shape.
Keywords: shape optimisation; tensegrity; tensile structure; cable dome;
minimum displacement; maximum stiffness; overall stiffness.
Reference to this paper should be made as follows: Chen, P-S. (2010)‘Overall stiffness evaluation and shape optimisation of a tensegric structure’,Int. J. Structural Engineering, Vol. 1, Nos. 3/4, pp.241–254.
Biographical notes: Pei-Shan Chen received his DrEng at the House
University. He is a Professor at the Department of Civil Engineering andArchitecture as well as Graduate School in the Hachinohe Institute ofTechnology (HIT). Before teaching in the HIT, he had worked as an Engineerin Chief for more than ten years and designed many big span spatial structuresand high-rise buildings. Now, his research interests are related to shape
optimisation, creating new structure system, seismic design, and historicalstructures. He is an author of a great deal of research studies published atnational and international journals, conference proceedings.

1Introduction

A tensile structure is known as a flexible structure, especially when it is deformed against
asymmetrical loads as well as wind loads. To achieve high stiffness in a tensile structure,
therefore, high prestressed forces in members are necessary, which usually negatively
affects the cost and method of construction. However, the structural performances of atensile structure are heavily dependent on its shape and prestresses introduced in thestructure. Such a structural feature makes it very important to maximise the overall
stiffness of a tensile structure by optimising its shape and distribution of prestressed
forces through a mathematical programming, and that has been the research subjects ofseveral former studies of the author (Chen et al., 1993; Chen and Kawaguchi, 1993).

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Overall stiffness evaluation and shape optimisation of a tensegric structure245

squares of all vector elements. Considering the equilibrium equation [equation (6)], theauthor promotes a function below as the stiffness evaluation function.

(F = DT D = QT K −1)K Q
T−1(8)= QT KQ(where K = K −1) K . Equation (8) is a sum of positive values D ≥ 0, which is very
T−12idifferent from that presented in the last section.
In order to discuss if such an evaluation function results in a similar conclusion asequation (7), let us again assume that the first element of the load vector is zero, Q1 = 0.
Consequently, the equation (8) can be rewritten as the equation below.
⎛ 0 ⎞⎡ K11K1n ⎤ ⎜ ⎟⎢⎥ ⎜ Q2 ⎟F = ( 0, Q2 ,… , Qn ) ⎢⎥⎜ ⎟⎢ K n1 … K nn ⎥ ⎜ ⎟⎣⎦ Q⎝ n⎠n=(9)n∑∑ Q K Qiij

ji=2 j =2Then, we get the similar conclusion as equation (7). However, the present methods canalso be used to evaluate the vertical stiffness of a tensegric structure. In common, theshape of a tensegric structure is determined by radii and rises, and such geometricdimensions affect the nodal displacements including the components where the load
components are zeroes. So, the present method is efficient for the shape optimisation for
maximum overall stiffness.
If equation (8) is regarded as a virtue work and weighting factor method presented inSubsection 2.1, by using equation (1) and assuming, ψi = Di, an important result can befound that a big displacement component acts as a heavy weighting factor while a smallcomponent acts as a light one, such that evaluation function (8) provides heavy weighting
factors acting on the big displacements, F = ∑ψiDi = ∑DiDi. Then, we can again infer thatthe present method is efficient for the evaluation of overall structural stiffness.

3Optimisation theory for a tensile structure

3.1 The mathematical analysis model
In this study, the author promotes an optimisation method for maximum overall stiffnessof a tensegric structure. Based on the studies in former sections, the author chooses theminimum displacement method, demonstrated in Subsection 2.3, to evaluate the overall
stiffness of the structure.
For a tensegric structure with n degrees of freedom, the objective function F1 forminimum displacements can be expressed as the following equations.

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P-S. ChenFigure 3 Member symbols

A distributed load of 300 N/m2 was assumed to be the dead load applying downwards atthe top of the roof. For the wind loads, a design velocity load of 2.52 kN/m2 and a windforce coefficient (for entire roof surface, upward) of C = 0.8 were assumed.
Table 1Symbols

Materials of major members
Material

Symbols

Material

R12 – φ88 + 2 – φ53 + 2 – φ50

D12 – φ67 + 1 – φ60

R22 – φ53 + 4 – φ50

D22 – φ88

R32 – φ53 + 2 – φ50

D32 – φ50

R44 – φ50

D44 – φ50

H15 – φ100

P1

φ – 609.6 × 22.0

H22 – φ88 + 6 – φ100

P2

φ – 711.2 × 22.0

H34 – φ88P3

φ – 508.0 × 22.0

Note: Member symbols are referred in Figure 3.
Figure 4 Prestresses in diagonal cables

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P-S. Chen

1to evaluate the virtual work done by a virtual load set

2to evaluate the magnitude of the displacement vector.

Both of the promoted methods are efficient for the optimisation for maximum overall
stiffness of a tensegric structure.
The present paper demonstrated an optimisation method for maximum overall
stiffness of a tensegric structure. By the numerical analysis, the author demonstrated thatthe present optimisation is an effective method to find the optimum shape with maximum
stiffness. Consequently, the numerical analysis results show that the verticaldisplacements decreased 20–30% comparing to that of its initial shape. The present
optimisation can determine the member sections automatically and output more useful
information for design decisions.
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