MEC2008 – Final Course assignment
1

Task A

Stress Invariants [ /35]
The stress state in a steel structure (E=210GPa, ν=0.3) is given as components (σx, σy, σz, τxy, τyz, τzx)
in standard units (Pascals; Pa) in the table on Pages 3-7. Write a MatLab script m-file to calculate the
following:
A1. the three invariants I1, I2 and I3: [ /5.5]

I
1
= s
x
+ s
y
+ s
z
=
I
2
= s
x
s
y
+ s
y
s
z
+ s
z
s
x
- t
xy
2
- t
yz
2
- t
zx
2
=
I
3
= s
x
s
y
s
z
+ 2t
xy
t
yz
t
zx
- s
x
t
yz
2
- s
y
t
zx
2
- s
z
t
xy
2
=

A2. the hydrostatic stress, σh: : [ /3.5]
s
h
=
I
1
3
=
A3. the second deviatoric invariant, J2, and thus the von Mises equivalent stress, σv: [ /4.5]

J
2
= I
2
-
I
1
2
3
=
s
v
= -3J
2
=

A4. by finding the three roots of the cubic polynomial: [ /5.5]
l
3
- l
2
I
1
+ lI
2
- I
3
= 0
the principal stresses σ1, σ2 and σ3:

s
1
=
s
2
=
s
3
=

A5. the maximum shear stress, τmax: [ /4.5]
t
max
= 1
2
max s
1
- s
2
, s
2
- s
3
, s
3
- s
1{ } =
A6. find the three eigenvalues (i.e., the three principal stresses again) and their corresponding
eigenvectors v1, v2 and v3: [ /7.5

s
1
= v
1
=



æ
è
ç
ç
ö
ø
÷
÷
s
2
= v
2
=



æ
è
ç
ç
ö
ø
÷
÷
s
3
= v
3
=



æ
è
ç
ç
ö
ø
÷
÷

A7. the strain energy density, U/δV: [ /4]

U
dV
=
1
2E
I
1
2
- 2(1+ n)I
2{ } =
MEC2008 – Final Course assignment
2

Task B
Trajectory Animation [ /30]

The motion of two particles is described by the vectors r1(t) and r2(t):


,


Create an m-file called ‘TrajAnim.m’ for showing the animation of these trajectories:
B1. Define r1(t) and r2(t) and/or the individual coordinate functions x1(t), y1(t), etc. in your m-file.
[ /15]
B2. Plot the particles on the same animated plot in order to show the particle trajectories. Particle
1 should be a red circle and Particle 2 should be a green circle. Connect both to the origin with
dotted, black lines, and have a solid blue line running from one particle to the other. [ /15]

Task C
Pendulum / Hemisphere [ /35]

The motion of a pendulum is identical to sliding in a frictionless hemispherical basin:

r(z) = R 1- 1- z / R( )
2


The radius (and height) of the basin is R. The differential equations describing the motion are:


d2x
dt 2
= x = -
gx
R
3 1- x / R( )
2
- 2 1- h / R( )( )
d2z
dt 2
= z = g 2
h
R
- 4 + 2
h
R
æ
èç
ö
ø÷
z
R
+ 3
z
R
æ
èç
ö
ø÷
2æ
è
ç
ö
ø
÷
where g = 9.81ms-2.

The student has to write a function m-file, ‘Hemisphere.m’, which takes R, h and the desired duration
(Tf) as inputs.
C1. Add a help section at the start (after the function declaration) explaining what the function
does and what the inputs are. [ /7]
C2. Check that the user has supplied valid inputs. The basin radius (R), starting height (h) and
duration (T) should all be greater than zero. Also, the starting height should be less than the
basin height (i.e., the basin radius). [ /7]
C3. Define the basin shape function, r(z). Use this to determine the starting position. Display this
neatly to the user, e.g., for z=1 and R=5, then x=r(1)=3, as shown below:
Initial position is (3,1). [ /5]
C4. The solution for the pendulum when the angle of swing is small has period 2p R / g .
Display this value neatly to the user. [ /6]
C5. Solve the differential equation for vertical motion, letting the figure appear for the user to see
the solution. [ /10]
____________________________________________________________________________________________________________________________________________________________________________________
Submission. The deadline for submission of this assignment is Friday 12th May 2023 at 4pm. In
order to obtain full credit, you should submit your m-file(s).
Please add in the m-file comments for each question to report the answer(s) and to explain
commands and rationale used for filing the code. Use appropriate engineering units & notation to
state the answers. Late submission will be penalized (-60%). No Plagiarisms!
MEC2008 – Final Course assignment
3


Student
σx
(Pa)
σy
(Pa)
σz
(Pa)
τxy
(Pa)
τyz
(Pa)
τzx
(Pa)

MEC2008 – Final Course assignment
7
Student
σx
(Pa)
σy
(Pa)
σz
(Pa)
τxy
(Pa)
τyz
(Pa)
τzx
(Pa)