Solution of sample problems first exam
1.
The nonzero components of the stress tensor are
xy
=
xz
=4Mpa
. What is
the normal component of the stress on a plane perpendicular to the vector
(1,2,2).
First normalize the vector to unit length
=(1/3,2/3,2/3).
Now
0 4 4
1/3
16/3
400 2
/
3
4
/
3
/
3
4
/
3
t
So
t
=(4/3)(4
i
+
j
+
k
)
Mpa. The normal component of
t
is
t
=32/9Mpa
2.
For the stress state defined in Problem 1, verify that the direction of the
vector (0,1,1) is a principal direction. What is the principal stress
associated with that direction?
A principal direction is an eigenvector, and when you multiply it by the matrix
it gets multiplied by the eigenvalue (which is the principal stress) without
changing direction.
044 0
0
0
400 1
0 01
0
1
so indeed the vector is an eigenvector, and the eigenvalue, equal to the
principal stress, is zero.
3.
The displacement field in meters is given by v=0.01xyz, with the other
components being zero. What are the components of the engineering strain
at x=y=z=1m?
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 Fall '09
 Kim
 Shear Stress

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