Finite Elements

Module aims

This module offers an in-depth understanding of the theory and practice of finite element methods, with application to solving real-life structural problems. Starting with the general formulation of finite element theory, the module proceeds with the formulation of 2D and 3D elements. Rods, beams and plate elements are developed rigorously. Modelling Strategies and potential pitfalls are discussed. You will be introduced to the use of finite-element software, emphasising common features found in most commercial packages and focusing on results interpretation.

Learning outcomes

On successfully completing this module, you should be able to: 1. Relate the displacement based finite element method to the basic structural methods of virtual work and minimum total potential energy.  2. Apply discretization via shape functions and piecewise integration to develop basic elements for structural analysis.  3. Discuss the requirements for convergence of the finite element method. 4. Apply FEA to solve a range of problems and demonstrate awareness of potential pitfalls.  5. Characterise the consequences of the assumed interpolation functions on the quality of the solution.  6. Choose appropriate high-quality meshes for a wide range of structural problems.  7. Interpret the results of FE computations, especially stresses, considering levels of accuracy required.  8. Model and analyse 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, 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


 

Teaching methods

The module will be delivered primarily through large-class lectures introducing the key concepts and methods, supported by a variety of delivery methods combining the traditional and the technological. The content is presented via a combination of slides, whiteboard and visualizer.Learning will be reinforced through tutorial question sheets and laboratory exercises, featuring analytical, computational and experimental tasks representative of those carried out by practising engineers.

Assessments

Exam: 2 hour written examination in January (90%)
Coursework:  Commercial finite element software-based assignment (10%)

 

Reading list

Core

Supplementary

Module leaders

Dr Ajit Panesar