Exploiting the Equilibrium Matrix to Ensure the Geometric Stability of Planar Trusses
Publication: Practice Periodical on Structural Design and Construction
Volume 27, Issue 1
Abstract
Equilibrium can be used to determine the geometric stability (kinematic determinacy) of a structure. For initial (conceptual) design, analyzing a structure’s equilibrium matrix first eliminates the added computational expense associated with performing a complete structural analysis if the design is determined to be unstable. This paper describes an economical procedure for ensuring the geometric stability of complex planar trusses using the equilibrium matrix and its transpose, the compatibility matrix, to support initial design. Equilibrium and compatibility matrices were derived for an individual truss member, and then the assembly process to produce the global equilibrium matrix (and global compatibility matrix) was demonstrated. Several examples illustrated the concepts, beginning with considering only the equilibrium of key joints (nodes) and progressing to formal analyses of the structure’s equilibrium and compatibility matrices using linear algebra. It was shown that structures can be simultaneously geometrically unstable (possessing one or more mechanisms) and statically indeterminate (possessing one or more states of self-stress). Analysis of a complex planar truss fully illustrated the utility of the procedure, and, by using the analysis results and associated visualization tools, strategies for rendering an unstable truss stable were presented.
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Data Availability Statement
All data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.
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Published In
Practice Periodical on Structural Design and Construction
Volume 27 • Issue 1 • February 2022
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© 2021 American Society of Civil Engineers.
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Received: Mar 5, 2021
Accepted: Jul 23, 2021
Published online: Sep 27, 2021
Published in print: Feb 1, 2022
Discussion open until: Feb 27, 2022
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