Finite Elements

Module aims

This course presents the fundamentals of the finite element method together with its practical use in solving structural problems

Learning outcomes

By the end of this course the student should be able to:

  • Comprehend the link between the displacement based finite element method and the basic structural methods of virtual work and minimum total potential energy.
  • Demonstrate the underlying concepts of discretization via shapes functions, piecewise integration, etc.
  • Perform basic element development.
  • Describe the requirements for convergence of the method.
  • Value how the method can be used to solve a range of problems and to be aware of potential pitfalls.
  • Characterise the consequences on the solution of the assumed interpolation functions.
  • Specify techniques for choosing good meshes.
  • Outline how the results, especially stresses, can be interpreted to give the most accurate estimates.
  • Model and analyze simple structures using commercial finite-element packages (in particular, Nastran and ABAQUS).
     

Module syllabus

Review of Matrix Algebra: Basic matrix operations, including finding the transpose of a matrix, addition of matrices, differentiation etc, notation, Gauss elimination and Cholesky factorisation.  
Displacement method for structural analysis – rod element: development of a stiffness matrix for a two-node rode element using the displacement method, assembly of element stiffness matrix, consideration of boundary conditions, solving for displacements and element stresses.
General Formulation of Finite Element Theory: re-introduction of basic 2-D elasticity, the principle of minimum potential and equivalence to the principle of virtual work.
Rod elements: development of a simple structure using rod elements, steps required for a finite element analysis, transforming local variable to global axes in 2-D and 3-D
Beam Element (Kirchhoff and Timoshenko): Kirchhoff element formulation, hermitian shape functions, bending moment and shear force in beam, accuracy. Timoshenko element formulation and advantages over Kirchhoff element, shear locking.
Numerical Integration: Gauss quadrature, required order of integration, element instabilities.
Triangular Membrane Elements: constant strain triangle, higher-order triangular element, usage in auto-meshing.
Quadratic Membrane Elements: four-node quadrilateral element, eight-node quadrilateral element.
Modelling Strategies and potential pitfalls: mesh generation, mesh density, element distortion, incorrect element connection, mixing of element types.
Introduction to the use of finite-element software: Use of a selected commercial package to be used for the tutorial/lab sessions, common features found in most commercial packages will be emphasised, results interpretation, stress averaging.
 

Pre-requisites

AERO50008 Structures 2
 

Teaching methods

Lecture, Tutorials.

Assessments

Examined Assessment
2 hour written examination in January (90%)
Coursework assignment (10%) based on the use of commercial finite element software (ABAQUS and Nastran).
Non-Examined Assessment
2 x Progress tests (peer marked)
 

Reading list

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Core

Supplementary