Exploiting the Equilibrium Matrix to Ensure the Geometric Stability of Planar Trusses

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.