2017/18 Undergraduate Module Catalogue

MECH3900 Finite Element Methods of Analysis

20 creditsClass Size: 250

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

Taught: Semesters 1 & 2 (Sep to Jun) View Timetable

Year running 2017/18

Pre-requisites

MECH1230	Solid Mechanics
MECH1520	Engineering Mathematics
MECH2610	Engineering Mechanics

This module is mutually exclusive with

MECH3730

Structural 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:
a) structures under point and distributed static loading
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 stress wave propagation and dynamic material models as they affect the analysis of impact problems
14) Understand the concepts of composite material behaviour and develop models of single and multiple-directional laminates

Overall Grading/Criteria for marking
70-100 - Outstanding
Demonstrates a high level of understanding of the modeling process. The reports are clearly laid out and the content has a high level of accuracy. Extensive and relevant checks are undertaken on all aspects of the modeling process and these are carefully taken into account in the presentation of the results and in the conclusions.
Demonstrates a comprehensive understanding of the underpinning principles that can be synthesised and applied to new problems. Very few errors made in calculations and assumptions used are justified.
60-69 - Very Good
Demonstrates clear understanding of the modeling process with all of the key aspects reported. Most of the necessary checks have been undertaken and there is evidence of these having been taken into account in the results and conclusions. Demonstrates a clear understanding of the underpinning principles with correct use of equations in most cases. Some minor errors in calculations or assumptions made.
50-59 - Good
Shows a reasonable understanding of the modeling process and reports cover most of the basic issues. Content is generally accurate but not always clearly presented. Several checks are undertaken and some are taken into account in the conclusions. Demonstrates an understanding of the basic principles but with frequent errors and use of incorrect equations or false assumptions.
40-49 - Satisfactory
Shows evidence of having understood some of the key aspects of the finite element modeling process but reports contain some errors or omissions. Some limited evidence of chec ks undertaken is presented but these are not taken into account in the analysis of the results and conclusions. Demonstrates some understanding of the underpinning principles but approach to questions is unclear and unsystematic.

0-39 Fail
Shows little understanding with limited or unclear evidence of the modeling process undertaken. Very limited if any report of checks undertaken on the models. Conclusions drawn are unclear or do not follow from results. Evidence of failure to understand basic finite or apply basic finite element principles. Calculations contain significant errors or the wrong use of equations and are frequently inaccurate.

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
On-line Learning	10	2.00	20.00
Class tests, exams and assessment	1	2.00	2.00
Group learning	1	15.00	15.00
Lecture	33	1.00	33.00
Practical	20	2.00	40.00
Private study hours			90.00
Total Contact hours			110.00
Total hours (100hr per 10 credits)			200.00

Private study

Independent learning - online self-assessment questions including some multiple choice questions with automated feedback and some with full worked solutions to provide active learning of theoretical aspects during Semester 1.

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	40.00
Total percentage (Assessment Coursework)		40.00

In first semester, students learn software package and undertake series of tasks (10%). A project is then set with a full report submitted by the end of semester 1 (30%).

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

Reading list

The reading list is available from the Library website

Last updated: 05/09/2018

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