## MECH3900 Finite Element Methods of Analysis

### 20 creditsClass Size: 290

Module manager: Dr A Jones
Email: A.C.Jones@leeds.ac.uk

Taught: Semesters 1 & 2 View Timetable

Year running 2019/20

### Pre-requisites

 MECH1230 Solid Mechanics MECH1520 Engineering Mathematics MECH2610 Engineering Mechanics

Module replaces

MECH 3105MECH 3230MECH 3825

This module is not approved as a discovery module

### Objectives

On completion of this module, students should:
1) have an understanding of the basic principles of structural analysis using finite element methods.
2) understand the principle of the minimum structural potential method and be able to apply this principle to derive equations for:
b) the analysis of linear buckling of structures
c) the analysis of dynamic systems
3) Understand the concept of the shape function and be able to derive stiffness matrices for 1D and 2D elements under different modes of loading.
4) Be able to analyse 2 and 3 dimensional structures under static loading by the construction of global stiffness matrices, application of boundary and symmetry conditions and extraction of unknown simultaneous equations.
5) Be able to determine the critical forces for the buckling of simple 2D structures
6) Be able to describe the phenomenon of vibration with reference to the exchange of energy that takes place within the mass and stiffness matrices used to describe a flexible body.
7) Be able to determine the undamped natural frequencies and modes of vibration of simple 2D truss and beam structures. Understand the link between the experimental free vibration behaviour of a structure and the validation of the equivalent linear structural FEM.
8) Understand the computational processes used to solve such problems using Gaussian elimination and Choleski factorisation.
9) Be familiar with a finite element software package, be able to use the software to construct models of structures under static and dynamic loading and evaluate the results.
10) Be able to identify and choose the correct loads and boundary conditions to represent symmetry and real-life problems.
11) Understand the limitations of finite element modelling, be able to evaluate the accuracy of the model results, assess the most suitable methods of analysis for different types of structure and understand how to interpret and validate the results.
12) Understand the processes and assumptions that underpin the definition and development of a FEM to answer a specific engineering question
13) Understand the concepts of composite material behaviour and develop models of single and multiple-directional laminates

Learning outcomes
1. Knowledge and Understanding.
On completion of this module students should be able to explain and illustrate:
· The basic principles of structural analysis using finite element methods, including shape functions and stiffness matrices (in 1D and 2D).
· The principle of the minimum structural potential and be able to apply this principle to derive equations for key static structural problems.
· The phenomenon of vibration with reference to the exchange of energy that takes place within the mass and stiffness matrices used to describe a flexible body.
· The link between the experimental free vibration behaviour of a structure and the validation of the equivalent linear structural FEM.
· The concepts of composite material behaviour and develop models of single and multiple-directional laminates.

2. Skills and Attributes:
(i) Intellectual
· Understand the processes and assumptions that underpin the definition and development of a FEM to answer a specific engineering question.
· Understand the limitations of finite element modelling, be able to evaluate the accuracy of the model results, assess the most suitable methods of analysis for different types of structure and understand how to interpret and validate the results.
(ii) Applied / software based
· Be familiar with a finite element software package and be able to use the software to construct models of structures under loading and evaluate the results.
· Be able to identify and choose the correct loads and boundary conditions to represent symmetry and real-life problems.
· Be able to analyse structures under static loading using the finite element formulation and appropriate solution techniques.
· Be able to determine the critical forces for the buckling of simple structures.
· Be able to determine the undamped natural frequencies and modes of vibration of simple structures.
· Be able to develop models of single and multiple-directional laminates.

Upon successful completion of this module the following UK-SPEC learning outcome descriptors are satisfied:

A comprehensive knowledge and understanding of the scientific principles and methodology necessary to underpin their education in their engineering discipline, and an understanding and know-how of the scientific principles of related disciplines, to enable appreciation of the scientific and engineering context, and to support their understanding of relevant historical, current and future developments and technologies (SM1m)
Knowledge and understanding of mathematical and statistical methods necessary to underpin their education in their engineering discipline and to enable them to apply a range of mathematical and statistical methods, tools and notations proficiently and critically in the analysis and solution of engineering problems (SM2m)
A comprehensive knowledge and understanding of mathematical and computational models relevant to the engineering discipline, and an appreciation of their limitations (SM5m)
Understanding of engineering principles and the ability to apply them to undertake critical analysis of key engineering processes (EA1m)
Ability to identify, classify and describe the performance of systems and components through the use of analytical methods and modelling techniques (EA2)
Ability to apply quantitative and computational methods, using alternative approaches and understanding their limitations, in order to solve engineering problems and implement appropriate action (EA3m)
Apply their skills in problem solving, communication, information retrieval, working with others, and the effective use of general IT facilities (G1)

Skills outcomes
Use of software for finite element model development, solution and post-processing.

### Syllabus

- Matrix algebra
Revision of matrix operations
Methods for solving simultaneous equations using Gaussian elimination and Choleski factorisation.

- Fundamentals of the FE method for structural analysis
Principle of minimum structural potential
Derivation of potential equations
Introduction to shape functions
Finite element formulation
Derivation of stiffness matrix
Relationship between nodal displacements and stress/strain.

- Truss elements
Derivation of shape function and stiffness matrix for truss element
Derivation of body force vector
Transformation in 2 and 3D of truss element
Assembly of global stiffness matrix and load vectors
Application of boundary conditions
Solution of equations.

- Beam elements
Derivation of shape function and stiffness matrix for beam element
Derivation of distributed load force vector.

- 2D elasticity finite elements
Plane stress, plane strain and axisymmetric stress states
Constant strain triangle for plane stress: derivation of stiffness matrix
Force vectors in 2D
Constant strain triangle for plane strain: derivation of stiffness matrix
Constant strain triangle for axisymmetric stress state: derivation of stiffness matrix.

- Buckling
Derivation of structural potential for buckling analysis
Derivation of geometric matrices for truss and beam elements
Solution for simple structures to determine critical load.

- The process of finite element analysis
The creation of a mathematical idealisation of the real structure.
The development of the finite element model: assumptions, choice of element type, constraints and convergence of mesh.
Model verification: the need for verification; alternative theoretical approaches; experimental.

- Structural dynamics
Derivation of structural potential for natural frequency analysis
Derivation of consistent mass matrices for truss and beam elements
Solution for simple structures to determine the natural frequency and mode shapes
The importance of free vibration behaviour in validating a FEM.

- Composites
Elastic behaviour and strength of unidirectional lamina
Elastic behaviour of multidirectional laminates
Implementation into finite element models

### Teaching methods

 Delivery type Number Length hours Student hours Class tests, exams and assessment 1 2.00 2.00 Group learning 1 32.00 32.00 Lecture 44 1.00 44.00 Practical 14 1.00 14.00 Independent online learning hours 48.00 Private study hours 60.00 Total Contact hours 92.00 Total hours (100hr per 10 credits) 200.00

### Private study

Independent learning - Online desktop recordings outlining introductory model build tasks are provided for initial self-study of the software techniques (in preparation for lab sessions).

Example theoretical problems and worked solutions are provided in semester 1 and 2 to provide active learning of theoretical aspects.

Private study - writing reports for laboratory assessments and revision for exam.

### Opportunities for Formative Feedback

- Laboratory project on FE submitted and feedback given on first part during first semester.
- Full report submitted at the start of semester 2 and feedback given.
- Laboratory project on modal analysis and feedback given during second semester.

### Methods of assessment

Coursework
 Assessment type Notes % of formal assessment Group Project 1 project with initial tasks to learn software 20.00 Group Project I project with initial tasks to learn software 20.00 Total percentage (Assessment Coursework) 40.00

Coursework marks carried forward and 60% resit exam OR 100% exam

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 2 hr 60.00 Total percentage (Assessment Exams) 60.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated