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# Graduate Thesis of David Mora ProgramPhD in Applied Sciences with mention in Mathematical Engineering, Universidad de Concepción
Enrollment Year2006
Senior Year2010
Thesis TitleFinite element methods for stability problems of thin structures

#### Thesis Summary:

The main objective of this thesis is to analyze numerical methods for the approximation of the coefficients and modes of buckling of thin structures. Specifically, the finite element approximation of the problem of plate and beam buckling is studied. In the first work, a formulation in terms of the moments for the buckling and vibration problems of a non-convex elastic polygonal plate modeled by the Kirchhoff-Love equations is studied. For the discretization, finite elements are considered linear and continuous pieces for all variables. Using the spectral theory for compact operators, optimal convergence results are obtained for the autofunctions (transverse displacement) and a double order for the eigenvalues ​​(buckling coefficients). In the second work, we study the problem of buckling of an elastic plate modeled by the Reissner-Mindlin equations. This problem leads to the spectral study of a non-compact operator. It is shown that the essential spectrum of the same is well separated from the relevant eigenvalues ​​(buckling coefficients) to be calculated. For the numerical approximation, finite elements of low order (DL3) are used. Adapting the spectral theory for non-compact operators, we show optimal convergence for the auto-functions and a double order for the eigenvalues, with error estimates independent of the thickness of the plate, which shows that the proposed method is "locking-Free"). In the third work, a low order finite element method is studied for the problem of buckling of a nonhomogeneous beam modeled by the Timoshenko equations. A spectral characterization of the continuous problem is given and using spectral theory for non-compact operators, optimal convergence orders are shown for the autofunctions (transverse displacement, rotations and shear stresses) and a double order for eigenvalues ​​(buckling coefficients) Also with constants independent of the thickness of the beam. In all cases, numerical tests are presented confirming the theoretical results obtained.

Thesis Director(s) Carlos Lovadina, Rodolfo Rodríguez
Thesis Project Approval Date2008, January 25
Thesis Defense Date2010, March 31
Professional MonitoringAs of March 2010, Assistant Professor of the Departamento de Matemáticas of the Universidad del Bío Bío, Concepción.
PDF ThesisDownload Thesis PDF #### ISI Publications from the Thesis

Carlo LOVADINA, David MORA, Rodolfo RODRíGUEZ: A locking-free finite element method for the buckling problem of a non-homogeneous Timoshenko beam. ESAIM: Mathematical Modelling and Numerical Analysis, vol. 45, 4, pp. 603-626, (2011).

Carlo LOVADINA, David MORA, Rodolfo RODRíGUEZ: Approximation of the buckling problem for Reissner-Mindlin plates. SIAM Journal on Numerical Analysis, vol. 48, 2, pp. 603-632, (2010)

David MORA, Rodolfo RODRíGUEZ: A piecewise linear finite element method for the buckling and the vibration problems of thin plates . Mathematics of Computation, vol. 78, 268, pp. 1891-1917, (2009)

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