Abstract
In recent years, group theory has been gradually adopted for computational problems of solid and structural mechanics. This paper reviews the advances made in the application of group theory in areas such as stability, form-finding, natural vibration and bifurcation of novel prestressed structures. As initial prestress plays an important role in prestressed structures, its contribution to structural stiffness has been considered. General group-theoretic approaches for several problems are presented, where certain stiffness matrices and equilibrium matrices are expressed in symmetry-adapted coordinate system and block-diagonalized neatly. Illustrative examples on structural stability analysis, force-finding analysis, and generalized eigenvalue analysis on cable domes and cable-strut structures are drawn from recent studies by the authors. It shows how group theory, through symmetry spaces for irreducible representations and matrix decompositions, enables remarkable simplifications and reductions in the computational effort to be achieved. More importantly, before any numerical computations are performed, group theory allows valuable and effective insights on the behavior or intrinsic properties of a prestressed structure to be gained.
Highlights
Symmetry is one of the most common and important features in nature
Through different areas of structural mechanics and illustrative examples drawn from recent work of the authors, this study will describe important developments and applications of group theory for novel prestressed structures
General group-theoretic approaches for the involved problems are presented, where certain stiffness matrices and equilibrium matrices are expressed in symmetry-adapted coordinate system and block-diagonalized
Summary
Symmetry is one of the most common and important features in nature. Different forms of symmetry can be observed from a microscopic view to macroscopic view, from atoms and crystals to large-scale space structures. As an important branch of mathematics and vector algebra, group theory is a powerful tool for systematic analysis on symmetric systems Zlokoviâc [15] and Zingoni [16] showed how to evaluate irreducible representations and symmetry subspaces for symmetric structures They summarized the main advantage of group theory through its applications in structural stability, vibration and control. Kangwai et al [3] briefly described how to utilize group theory to establish a symmetry-adapted coordinate system and perform static analysis on symmetric structures. Through different areas of structural mechanics and illustrative examples drawn from recent work of the authors, this study will describe important developments and applications of group theory for novel prestressed structures. General group-theoretic approaches for the involved problems are presented, where certain stiffness matrices and equilibrium matrices are expressed in symmetry-adapted coordinate system and block-diagonalized
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