Influence of symmetry on the stability behaviour of plane and space frames

Kaluba, Chisanga

Influence of symmetry on the stability behaviour of plane and space frames

Thesis / Dissertation

2024

Publisher

University of Cape Town

Department

Department of Civil Engineering

Faculty

Faculty of Engineering and the Built Environment

Abstract

Symmetry refers to a property of an object or a system that remains unchanged or invariant under a transformation, such as reflection, rotation, or scaling. This study sought to investigate how symmetry influences the buckling behaviour of open frames as well as closed polygonal and prismatic frames. This was achieved by employing analytical methods developed in this study and the finite element method to study plane and space frames. The analytical results were used to validate the finite element models developed in Abaqus. The finite element models were then used to carry out more detailed studies on the influence of symmetry on the stability behaviour of plane frames. In addition, the study also proposed an alternative method for tracing the post buckling paths of lattice domes. Two group theoretic approaches to the buckling analysis of plane frames were developed based on the matrix stiffness method, and slope deflection method respectively. This was achieved by using group theory to decompose an n-dimensional problem into smaller independent problems of much smaller dimensions. The buckling loads and mode shapes are then computed with much less computational effort and numerical problems of ill conditioning are avoided. This was very useful for the group theoretic slope deflection approach developed in this study. The group theoretic slope deflection approach was used to obtain analytical results for the buckling loads and modes of plane frames. Another advantage of the group theoretic approach developed was that insights on the symmetry of the buckling modes were obtained even before detailed computations for eigenvalues are carried out. The analytical results obtained from the group theoretic slope deflection method were used to validate Finite element models created in Abaqus of the Cnv plane and space frames, and Dnh space frames. These finite element models were then used to conduct more detailed investigations of the influence of symmetry on the stability behaviour of plane and space frames. This study found that for plane frames symmetric to a Cnv group in terms of stiffness and loading, the lowest buckling load had an eigenmode with the symmetry of a subgroup of Cnv with the highest order of elements. The study found that the order of emergence of symmetries for eigenmodes of higher eigenvalues was from the subgroups of Cnv with the highest order of elements to those with the lowest order of elements and this order of emergence repeats every 2n eigenvalue. The study also found that when eigenvalues are categorised by the symmetry group of their eigenmodes, eigenvalues whose eigenmodes have the symmetries of subgroup of Cnv with the highest order of elements, formed the lower bound eigenvalues; and the eigenvalues whose eigenmodes have the Cn symmetries formed the higher bound eigenvalues. Further, this study found that when the group index of an eigenmode is greater than two, the respective eigenvalue is a repeating eigenvalue; and that when the group index is equal to two, then the respective eigenvalue is non-repeating. This study also found that space frames that are Cnv symmetric in terms of stiffness and subjected to a loading arrangement symmetric to a subgroup of Cnv, display the buckling behaviour of a frame that is symmetric to that particular subgroup in terms of stiffness and loading. Lastly this study found that, plane frames that are Cnv symmetric in terms of loading and subjected to Cnv/2 loading display the buckling behaviour of a plane frame symmetric to Cnv with respect to loading and stiffness for the case where n/2 is even. Some of the findings of these two approaches (matrix formulation and slope-deflection) have already been reported in a publication of the American Society of Civil Engineers, and presented at an international conference. For laterally unrestrained Cnv symmetric space frames, this study found that for n>2: the buckling mode for the lowest buckling value is C1v symmetric and the frame sways along one line of symmetry; and the lowest buckling value is a repeating buckling value. For Dnh symmetric space frames in terms of stiffness and loading, the post-buckling behaviour was investigated in addition to the buckling behaviour. Such space frames were found to display the following stability behaviour: the symmetries of the buckling modes are subgroups of Dnh; the order of emergence of symmetries of buckling modes repeats every 4n buckling value; the buckling values whose buckling modes have the Cn symmetries form the higher bound buckling values when all buckling values are arranged by buckling mode symmetry. This study also found that space frames that are Dnh symmetric in terms of stiffness and subjected to a loading arrangement symmetric to a subgroup of Dnh, display the buckling behaviour of a frame that is symmetric to that particular subgroup in terms of stiffness and loading. Lastly, this study found that the post-buckling behaviour of space frames is stable and is independent of the beam to column stiffness. This study also demonstrated how the post buckling equilibrium paths of lattice space domes can be traced using a load excitation method, without having to use special path switching algorithms. In this method, the pattern of the applied load arrangement to excite a particular bifurcation path is based on the symmetry of the deformation pattern of the dome on that particular bifurcation path. Thus, the post buckling equilibrium paths can be traced by simply using geometric non-linear analysis in a Finite element software such as Abaqus.

Keywords

Engineering

Reference:

Collections

PhD / Doctoral

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