EML 4507
Fall 2010
Exam 11
Write only on the front side. If you want, write your
name in the back. All problems carry equal points
Solution
Problem 1
:
The state of stress at a point is given by
MPa
2
0
0
[ ]
0
4
0
0
0
1
If the yield strength
Y
of the material is equal to 6 MPa,
calculate the Safety Factor
N
using: (a) Maximum principal
stress theory; (b) Maximum shear stress theory; and (c) von
Mises’s Theory
.
Solution
:
The stress matrix is a diagonal matrix and hence the normal
stresses themselves are the principal stresses.
1
2
3
4;
2;
1.
(a)
1
6
1.5
4
Y
SF
(b)
1
3
max
max
4
( 1)
2.5
2
2
6
1.2
2
5
Y
SF
(c)
2
2
2
1
2
2
3
3
1
2
19
4.36
6
1.38
4.36
VM
Y
VM
SF
Problem
2
: The state of stress at a point is given by:
13
0
0
[ ]
0
13
0
0
0
13
Calculate the normal stress
n
and shear stress
n
on a plane
whose normal is given by (3/13, 4/13, 12/13).
Solution
:
The traction on an arbitrary plane is given by
T
n
Since
is a diagonal matrix, the matrix multiplication is
simple:
13
0
0
3
3
1
0
13
0
4
4
13
0
0
13
12
12
T
Magnitude of this traction is:
9
16
144
13
T
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 Fall '11
 Kim
 Geometry, Force, Shear Stress, Finite Element Analysis, Stress, Diagonal matrix

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