Abstract
In this chapter, we present formulations of the stiffness matrices, including the tangent, linear, and geometrical stiffnesses for a prestressed pin-jointed structure. Moreover, three different stability criteria—stability, prestress-stability, and super-stability—are presented for its stability investigation. It is demonstrated that super-stability is the most robust criterion, and therefore, it is usually preferable in the design of tensegrity structures. Furthermore, to guarantee a super-stable structure, we present the necessary conditions and sufficient conditions, which will be extensively used in the coming chapters.
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Notes
- 1.
In this book, stability is investigated up to the second-order term of increment of the total potential energy. Some ‘unstable’ structures in this book may be actually stable if higher-order terms are included, see, for example, Example 4.4.
- 2.
For steel material, for instance, its Young’s modulus is appropriately 205 GPa; i.e., \(E = 205\times 10^9\) N/m\(^2\).
- 3.
The structure is indeed stable, if higher-order terms of increment of the total potential energy are taken into consideration. However, we investigate stability of a structure by considering only up to the second-order term in this book, which is sufficient for most cases.
- 4.
Signs of the prestresses should not be changed, while the magnitude of the prestresses can be arbitrarily scaled in proportion satisfying the self-equilibrium equations.
- 5.
A two-dimensional free-standing pin-jointed structure has at least three members (having the triangular appearance), and a three-dimensional free-standing structure has six (having the tetrahedral appearance). However, these simplest structures cannot carry any prestress in their members. To ensure the possibility of carrying prestresses, the number \(m\) of member is always larger than \((d^2+d)/2\) for a \(d\)-dimensional free-standing prestressed pin-jointed structure.
References
Connelly, R. (1999). Tensegrity structures: why are they stable? In M. F. Thorpe & P. M. Duxbury (Eds.), Rigidity theory and applications (pp. 47–54). New York: Kluwer Academic/Plenum Publishers.
Connelly, R., & Whiteley, W. (1996). Second-order rigidity and prestress stability for tensegrity frameworks. SIAM Journal on Discrete Mathematics, 9(3), 453–491.
Golub, G. H., & Van Loan, C. F. (1996). Matrix computations (3rd ed.). Baltimore: Johns Hopkins University Press.
Gray, A. (1997). Modern differential geometry of curves and surfaces with mathematica (2nd ed.). Boca Raton: CRC Press.
Lanczos, C. (1986). The variational principles of mechanics (4th ed.). New York: Dover Publications.
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Zhang, J.Y., Ohsaki, M. (2015). Stability. In: Tensegrity Structures. Mathematics for Industry, vol 6. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54813-3_4
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DOI: https://doi.org/10.1007/978-4-431-54813-3_4
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