# Aeronautical Engineering (MEng)

## 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.
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)

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### Core

• #### The finite element method : its basis and fundamentals / [electronic resource]

Zienkiewicz, O. C.,

7th ed., Butterworth-Heinemann

• #### The finite element method : its basis and fundamentals

Zienkiewicz, O. C.,

7th edition., Butterworth-Heinemann

Fish, J.

Wiley

### Supplementary

Hitchings, D

NAFEMS

• #### Fundamentals of finite element analysis

Hutton, David V

McGraw Hill

• #### An introduction to the finite element method

Reddy, Junuthula Narasimha

3rd, McGraw Hill

• #### Concepts and applications of finite element analysis

Cook, Robert D

4th, Wiley