Geometrically nonlinear Hessian eigenmode decomposition for local stability analysis of thin-walled structures
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ORIGINAL ARTICLE
Geometrically nonlinear Hessian eigenmode decomposition for local stability analysis of thin‑walled structures Aref K. L. Kzam1 · Humberto B. Coda2 Received: 1 June 2018 / Accepted: 21 May 2019 © Springer-Verlag London Ltd., part of Springer Nature 2019
Abstract A nonlinear geometric formulation based on position and unconstrained vector is originally proposed to evaluate the local stability of thin-walled members. The Hessian is decomposed into linear and geometric equivalent installments. As the Hessian matrix exhibits a definite positive quadratic form, stability condition for large displacements requires that the Hessian matrix positivity be verified at each load small steps. This condition must be established by evaluating the smallest eigenvalue signal at the critical point imminence. The positional finite element method is formulated from a total Lagrangian reference. The mechanical system equilibrium is guaranteed by the total potential energy stationary principle. The constitutive law of Saint–Venant–Kirchhoff is obtained from the linear relationship between the second Piola–Kirchhoff stress tensor and Green–Lagrange deformation tensor. Some examples demonstrate the applicability of the methods. Keywords Eigenvalue · Eigenvector · Geometrical nonlinearity · Hessian decomposition · Thin-walled members
1 Introduction An equilibrium configuration is no longer stable when a succession of stable configurations is terminated by a critical point. The theory of stability is crucial for structural, aerospace, nuclear, and offshore engineering. According to Elishakoff [1], Euler (1707–1783) and Koiter (1914–1997) established the paradigms of the classical and modern theories of structural instability. The classical approach was originally investigated for [2] that proposed the critical force of elastic buckling. Koiter [3] presented a structural sensitive under the influence of small imperfections. Both theories were a milestone in the study of structural instability models. These models reflect the linear and nonlinear viewpoints of the phenomenon. Despite significant theoretical advances in these areas, new research is required, particularly in the field of thin-walled * Aref K. L. Kzam [email protected] 1
Federal University for Latin-American Integration, Tancredo Neves Av., 6731, POBox: 2102, Foz do Iguaçu, Paraná 85867‑970, Brazil
São Carlos School of Engineering, University of São Paulo, Trabalhador São Carlense Av., 400, São Carlos, São Paulo 13566‑960, Brazil
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slender structures [4]. State-of-the-art approaches on the mechanical behavior of these structures can be found in the works of [5–10]. These works describe the geometric hypotheses and influence of the section shape and dimensions on the instability phenomenon of slender pieces. For a review on the instability of steel structures with slender profiles, the reader is referred to Erkmen and Mohareb [11]. In this paper, the authors highlight the classical studies of Michell [12], Prandtl [13], and Reissner [14] in the
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