Abstract
This paper presents an algebraic approach that unifies both the elastic and limit theories of static structural analysis. This approach reveals several previously unpublished improvements, which are based on several dualities in the mathematical description. Firstly, we show a novel duality between the solutions to two different problems: the elastic solution to the internal forces of an externally loaded structure, and the nodal displacements induced by prestressing in one or several elements of the same structure. This duality is proven and discussed. The application of this solution to the limit state analysis is very productive, and includes the determination of the ductility requirements necessary to achieve full plastic behaviour, and the assessment of the prestress needed to limit or eliminate such requirements. The unified framework also allows to obtain the elasto-plastic deformed state at the beginning of the plastic structural collapse. We have also detailed the theoretical duality between two classes of structures—hyperstatic and hypostatic—which was derived from the linear algebra principles that define these solutions. Finally, we studied and exposed the dimensionality reduction of structural problems given by the singular value decomposition and the eigenvalues problem. An illustrative example that clearly illustrates all these points is provided.
Highlights
Different theoretical approaches to structural analysis allow engineers to model and interpret the behaviour of structures in different contexts and fields of application
We demonstrate the ability of eigenvalue and single value decomposition (SVD) problems to solve the elasto-plastic state reached by the structures at the beginning of a full plastic collapse when applied to the compatibility conditions
N 1⁄4 N, and where no internal mechanisms are exhibited. It does possess r = n − N redundant internal forces that cannot be solved via the N equilibrium equations F 1⁄4 BTf, which is the set of equations for which the solution differs in fd, depending on whether plastic or elastic material conditions are applied
Summary
Different theoretical approaches to structural analysis allow engineers to model and interpret the behaviour of structures in different contexts and fields of application. This paper aims to present a unified algebraic review of static analysis in order to show, prove and use several dualities that exist within the structural theory. This methodology provides new insights into the established theories. In the process of proving this relationship, the self-stress state spaces are deduced and subsequently analysed. These results can be used to compute the ductility requirements necessary to achieve a full plastic structural response. We present several illustrative examples of the previous theoretical concepts and the conclusions (§§5 and 6)
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