Sumários
T8 - Finite element programming for truss structures
11 outubro 2017, 09:30 • Miguel Matos Neves
Convergence. Dirichelet and Newmann type of problems. Potential energy, elastic strain energy and work done by external forces.
Solution of a two bar truss (Beer&Johnston, Mechanics of Materials, Fig 1.1). Analytical solution: Axial efforts (method of joints -> sine law), displacements by energy conservation. Development of a simple finite element code (using array programming features). 2 node linear bar finite element and coordinate transformation. Preprocessing, solution and postprocessing (The Matlab/Octave draft code developed during the lesson is available at Fenix web page of this course). Comparison of results and superconvergence.
AP4L - 1D FEs with quadratic approximation
9 outubro 2017, 10:30 • Miguel Matos Neves
Solution of Prob. 7 using 1D FE with quadratic approximation. Deduction of the shape functions Fi1, Fi2 and Fi3. Obtaining Ke and Fe for the prob. 7. Assembling with 2 FEs. Imposing boundary conditions. Solution of the linear system of equations. Comparison with solution obtained
using 1D FE with linear approximation. Octave script for
shape functions Fi1, Fi2 and Fi3, available at PDF link (main page).
AP4 - 1D FEs with quadratic approximation
9 outubro 2017, 09:30 • Miguel Matos Neves
Solution of Prob. 7 using 1D FE with quadratic approximation. Deduction of the shape functions Fi1, Fi2 and Fi3. Obtaining Ke and Fe for the prob. 7. Assembling with 2 FEs. Imposing boundary conditions. Solution of the linear system of equations. Comparison with solution obtained
using 1D FE with linear approximation.
T7 - Quadratic Finite Element and Finite element programming
9 outubro 2017, 08:00 • Miguel Matos Neves
Finishing the Prob. 1.9
from collection of exercises. Superconvergence in this case and exact solution. How FE treat discontinuities in proprieties (e.g. change of material). How to code a FE program in Octave/Matlab, using
array programming? Development of a short script for the solution of Prob. 7. Comparison with exact solution and non-superconvergence of the FEM solution in this case. Introduction to the 1D quadratic finite element, requirements to the shape functions, deduction of the shape functions and Lagrange polinomial interpolation.
T6 - Comparing solution without superconvergence and notes on hyperstaticity
4 outubro 2017, 09:30 • Miguel Matos Neves
Solution of a problem of type -u''+u=1 and comparison between analytical and FEM (linear element) solutions. Finite element of Cables, bars, shafts, thermal conduction + convection, ... presents superconvergence property. Absence of super-convergence in 2nd order differential equations with the term b*u (example: prob 1.7 from collection of exercises). External static indeterminacy of structures and its difficulty with classical methods. Reasons why static indeterminacy is not an issue with FEM (hyperstatic example: prob. 1.9 from collection of exercises).