Static Modal Analysis: A Review of Static Structural Analysis Methods Through a New Modal Paradigm

• Published in: Archives of computational methods in engineering Vol. 31; no. 6; pp. 3409 - 3440
• Main Authors: Feron, Jonas, Latteur, Pierre, Pacheco de Almeida, João
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.08.2024; Springer Nature B.V
• Subjects: Compatibility; Eigenvalues; Eigenvectors; Engineering; Equilibrium; Kinematics; Mathematical and Computational Engineering; Mathematical models; Modal analysis; Paradigms; Resonant frequencies; Review Article; Singular value decomposition; State-of-the-art reviews; Static equilibrium; Static structural analysis; Trusses; Vibration analysis
• ISSN: 1134-3060; 1886-1784
• DOI: 10.1007/s11831-024-10082-x

This article is a state-of-art review on static structural computations for pin-jointed structures, revising the last forty years of scientific research on the subject matter through the introduction of static modal analysis . This novel paradigm is inspired by the so-called singular value decomposition (SVD) of the equilibrium matrix and by dynamic modal analysis. In dynamics, modal analysis requires the solution of an eigenvalue problem, which returns the natural frequencies of the structure and the corresponding mode shapes of vibration, the eigenvectors. The application of the static modal analysis to the four types of linear trusses—determinate or indeterminate from the static and kinematic viewpoints—allows re-interpreting the well-known force method and displacement method of structural analysis. Central to this proposal is the solution of static equilibrium and compatibility equations in a modal space where the relations between the extensional, inextensional, and self-stress modes are unequivocally identified. Their physical interpretation, also at the equilibrium and compatibility levels, is discussed and illustrated by key accompanying examples of structures subjected to external loads. Several original diagrammatic representations of the static modal analysis contribute to the overall understanding and implementation of the mathematical relations. This approach brings out new aspects of the interrelationship between the force and displacement methods, which strengthen their complementarity.