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32
Finite Element Analysis and Design
10. With respect to the coordinate system
xyz
, the state of stress at a point
P
in a solid is
20
0
0
[ ]
0
50
0 MPa
0
0
50
(a)
m
1
,
m
2
and
m
3
are three mutually perpendicular vectors such that
m
1
makes 45º
with both
x
 and
y
axes and
m
3
is aligned with the
z
axis. Compute the normal
stresses on planes normal to
m
1
,
m
2
, and
m
3
.
(b)
Compute two components of shear stress on the plane normal to
m
1
in the
directions
m
2
and
m
3
.
(c)
Is the vector
n
= {0, 1, 1}
T
a principal direction of stress?
Explain.
What is the
normal stress in the direction
n
?
(d)
Draw an infinitesimal cube with faces normal to
m
1
,
m
2
and
m
3
and display the
stresses on the positive faces of this cube.
(e)
Express the state of stress at the point
P
with respect to the
x
′
y
′
z
′
coordinates
system that is aligned with the vectors
m
1
,
m
2
and
m
3
?
(f) What are the principal stress and principal directions of stress at the point
P
with
respect to the
x
′
y
′
z
′ coordinate
s system?
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 Fall '08
 Staff
 Finite Element Analysis, Stress

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