1
AE 321 – Solution of Homework
#3
1.
The components of the traction at a given plane is given by
j
ji
j
ij
n
i
n
n
T
σ
σ
=
=
)
(
or
T
n
( )
[
]
=
σ
[ ]
n
[ ]
,
(1)
where
ij
σ
represent the symmetric stress tensor and
j
n
represents the j component of the
outward normal unit vector of the plane under consideration.
(a)
State of stress:
σ
ij
[ ]
=
σ
0
0
0
σ
0
0
0
σ
with unit vector normal to the plane:
n
j
[ ]
=
1
2
−
1
2
0
After substituting into Equation (1), we obtain the traction on the plane under consideration
T
n
( )
[
]
=
σ
0
0
0
σ
0
0
0
σ
1
2
−
1
2
0
=
2
σ
2
−
2
σ
2
0
or
T
n
( )
=
2
2
σ
e
1
−
e
2
(
)
(b)
State of stress:
σ
ij
[ ]
=
σ
σ
0
σ
−
σ
0
0
0
0
the plane under consideration is the same as in
(a).
Applying Equation (1)
T
n
( )
[
]
=
σ
σ
0
σ
−
σ
0
0
0
0
1
2
−
1
2
0
=
0
2
σ
0
or
T
n
( )
=
2
σ
e
2
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2
2.
a. Compute the components of traction on the plane passing through P whose outward
normal vector
n
makes equal angles with the coordinate axes.
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 Fall '07
 waller
 Force, 1 Pa, 5 Pa, 0 PA, 6 Pa, 6kPa

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