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ISSN 1759-3158
Edited by: B.H.V. Topping, G. Montero, R. Montenegro
Chapter 22

A Combined Topological, Geometric, Statical and Stiffness Method to Compute the Stability of Pin-Jointed Structural Assemblies

H. Deng* and A.S.K. Kwan+

*Department of Civil Engineering, Zhe Jiang University, Hang Zhou, P.R. China
+Cardiff School of Engineering, Cardiff University, United Kingdom

Full Bibliographic Reference for this chapter
H. Deng, A.S.K. Kwan, "A Combined Topological, Geometric, Statical and Stiffness Method to Compute the Stability of Pin-Jointed Structural Assemblies", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Innovation in Computational Structures Technology", Saxe-Coburg Publications, Stirlingshire, UK, Chapter 22, pp 475-499, 2006. doi:10.4203/csets.14.22
Keywords: structural stability, static-kinematic analysis, geometrical stability, pre-stressed mechanism, infinitesimal mechanism.

Structural stability is the capability of a system to maintain its current equilibrium state. The stability of pin-jointed assemblies (including those with infinitesimal mechanisms) is dependent on a number of different factors, and has traditionally been studied with the focus on only one, or a limited number, of these factors. We maintain that structural stability is dependent on factors including topology, geometry, statics and member stiffness, and consequently, we cannot fully examine structural stability and leave out any one of these aspects. This paper broadens conventional concepts of structural stability by combining all the contributing factors together using the energy criterion and geometric nonlinearity theory, and further explains some non-generic stability phenomena of pin-jointed bar assemblies in a unified and coherent way. A classification for stability conditions is thus presented using analysis of the components of the tangential stiffness matrix.

The energy criterion for structural stability and the properties of a stiffness matrix are briefly examined with view to introduce a new analytical expression in vectorial mathematics of the tangential stiffness matrix of pin-jointed bar assembly. Special attention is given to the geometric stiffness matrix, to enable consideration of the geometric nonlinearity, and also the stiffening effect from internal force. The information on mechanisms and self-stress found in the compatibility matrix is also briefly summed up.

The different factors that affect structural stability are discussed by examining the properties of the constituents of the tangential stiffness matrix under different conditions, and a unified classification for stability conditions of pin-jointed bar assembly is thereby presented. Structural stability is then examined for structures without internal force, mechanisms and infinitesimal mechanisms. The static-kinematic analysis of pin-jointed bar assemblies is re-investigated from the viewpoint of structural stability. Necessary and sufficient conditions of "Maxwell's rule" for stability of a pin-jointed assembly, as well as the criterion based on rank analysis of equilibrium matrix, are provided. The effect of member stiffness on intrinsic stability (that is, structural stability independent of internal forces) is also considered, and a technique for determining "necessary bars" (bars necessary for stability) is presented with proof.

The stability of mechanisms stiffened by self-stress or external loading is also examined. Such mechanisms and those of tensegrities have been investigated in the literature, and a stability determination criterion based on physical explanations for these systems was developed by Calladine and Pellegrino [1,2]. This paper provides the theoretical proof for Calladine and Pellegrino's criterion, and further irregularities are also demonstrated. Mechanisms stiffened by loads are also discussed, and a determination criterion for such systems similar to that for prestressed mechanisms is also presented.

Although infinitesimal mechanisms, and especially their order, are still a matter of debate in the literature, we show that the approach in this paper has a contribution to make on this topic. We show that it is necessary to carry out analysis of high-order variations of potential energy to properly show the characteristics of these kinds of assemblies.

Calladine, C.R., Pellegrino, S., "First-order infinitesimal mechanisms", Int. J. Solids and Struct. 27, 505-515, 1991. doi:10.1016/0020-7683(91)90137-5
Pellegrino, S., Calladine, C.R., "Matrix analysis of statically and kinematically indeterminate frameworks", Int. J. Solids and Struct. 22, 409-428, 1986. doi:10.1016/0020-7683(86)90014-4

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