Comments and corrections to Dr Llewellyn Morse: l.morse@ucl.ac.uk
1. Introduction
Structural failure in plates forming e.g. the exterior of automobiles or fuselages of aircraft, is usually
initiated at the most highly stressed points of the structure, typically near a sharp corner or a hole. For
example, an aircraft window, as seen in Figure 1, is one of the most highly stressed areas in an aircraft.
For this reason, estimating the stress concentration introduced by geometrical discontinuities is of practical
importance in the field of structural integrity monitoring. In this assignment, the finite element method is
used to examine the stress field in the vicinity of the hole that is present in a plate. The results are compared
with the predictions from the Theory of Elasticity; Topic 1.
2. Assignment Aim
The aims of this assignment are the following:
• To show how the finite element method can be used to solve stress analysis problems.
• To obtain practical experience in using the commercial finite element package ABAQUS
• To demonstrate critical analysis of the output obtained from the finite element analysis calculations,
commenting on mesh convergence, element behaviour, and justifying technical arguments with
experimental and theoretical results from literature.
Figure 1. Aircraft fuselage window.
2
3. Stress Solution in the Vicinity of a Hole in an Infinite Elastic Plate
The stress field around the hole in an infinite elastic plate (Figure 2) has been derived in lectures. In the (r,
θ) coordinate system, these components are [1]:
This solution indicates that the maximum stress concentration factor (SCF) S, in this plate is in the vicinity
of the hole:
(2)
The plate, of course, does not have to be loaded under uniaxial tension. Cases of biaxial loading and shear
are common – e.g. pure shear is obtained by imposing 2 = −1. For these cases, the solution of (1) and
superposition can be used to obtain exact solutions.
Figure 2. Plate with a hole, subjected to a biaxial load.
 2cos341
2
1
1
2
1
4
4
2
2
2
2






+−+





−=
r
a
r
a
r
a
rr
 2cos31
2
1
1
2
1
4
4
2
2






+−





+=
r
a
r
a
  2sin321
2
1
4
4
2
2






−+−=
r
a
r
a
r
(1)
=
=
( = , = 2⁄ 3 2⁄ )
= 3.0
3
4. Coursework Tasks and Report
For your FE model setup, assume that the plate is made of an aluminium alloy, with material properties
E = 70 GPa, and ν = 0.33. The stress biaxiality ratio is given in the table below, according to the first letter
of your first name.
Table 1. Stress biaxiality ratios according to the first letter of your first name.
First Letter of
First Name
Stress biaxiality ratio
⁄
First Letter of
First Name
Stress biaxiality ratio
⁄
A-B 5.0 M-N 0.9
C-D 4.5 O-P 0.8
E-F 4.0 Q-R 0.7
G-H 3.0 S-T 0.5
I-J 2.0 U-W 0.4
K-L 1.5 X-Z 0.3
Your report should include the following sections:
1. Description of the finite element model setup:
• Geometry of the plate
• Boundary conditions
• Element type and justification of choice
• Mesh configuration used and mesh convergence, including the physical quantity used for monitoring
convergence, and the convergence criterion and convergence threshold used. NB: Typically, the
quantity used for judging convergence is either the quantity of interest, i.e. something that you need
to investigate, or a quantity that is most sensitive to mesh density. The convergence criterion is a
numerical measurement that shows you how fast you are approaching mesh-independence of results.
The convergence threshold is a target to reach mesh independence of results.
For the above items, include diagrams to support your descriptions.
2. Post-Processing and Examination of Results
• Theoretical calculation of the stress field around the hole.
• Distribution (field output) of the three components of the stress field in the plate from the FE analysis.
• Plots of:
∞ vs. ,
∞ vs. ,
∞ vs. for = 0,
4
,
2
from the FE analysis and theory
(superimposed), where
∞ is the maximum applied stress (this will be either 1 or 2 depending
on your stress biaxiality ratio) and is the radial distance from the centre of the hole.
3. A discussion on:
• The location and magnitude of the maximum stress concentration factor (SCF), defined as =
ℎ
∞
where
ℎ is the maximum stress component in the vicinity of the hole. Which stress component
( or or ) is largest, and where? How do your FE analysis calculations compare with the
theoretical predictions for the location and magnitude of S? If there are discrepancies, what are the
reasons?
• The effect of the plate dimensions on the stress concentration factor (SCF) for the biaxiality ratio
assigned to you. How do you expect the finite dimensions of the plate to affect the stress
concentration observed at the hole, and for what reason?
For both discussion items, provide quantitative arguments, and support them with references and your
own calculations/analyses.
4
5. Due Date and Length of Report
The report is individual. It shall be submitted on Moodle by Thursday 30 November 2023 at 14:00.
The main body of the report (excluding title page, table of contents, references) shall not exceed 1500
words. There is no need to use appendices in this report.
6. Marking Scheme
See “Elasticity Coursework Grading Rubric 23-24.pdf” on Moodle.
7. AI Guidance
AI Guidance Category for this coursework: Category 2 (AI tools can be used in an assistive role) (Link
to UCL Web Page)
Examples of where AI might be used in an assistive role in this coursework include:
• Drafting the structure of the report
• Supporting the writing process in a limited manner e.g. proofreading
Examples of where AI cannot be used in this coursework:
• Drafting content
• Researching
The use of generative AI must be acknowledged in an ‘Acknowledgements’ section of any piece of
academic work where it has been used as a functional tool to assist in the process of creating academic
work (Link to UCL Web Page). Minimum requirement to include in acknowledgement:
• Name and version of the generative AI system used; e.g. ChatGPT-3.5
• Publisher (company that made the AI system); e.g. OpenAI
• URL of the AI system.
• Brief description (single sentence) of context in which the tool was used.
For example:
“I acknowledge the use of ChatGPT 3.5 (Open AI, https://chat.openai.com) to proofread my final draft.”
References
[1] S.P. Timoshenko and J.N. Goodier, Theory of Elasticity, McGraw- Hill, 3rd ed, 1970