ME 3201B Transformation of Stress

1
ME 3201B Transformation of Stress
2/21/2005
Recall the transformation of stress is given in
general by For Plane Stress, all stresses
are confined to act in one plane and at least
three components of stress are zero. Thus, all
transformations occur about a single axis
y
y
sy
q
sxy
x
sxy
sxy
sx
sx
x
sx
sxy
sxy
sy
sy
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The Transformation of Stress for Plane Stress can
be simplified by the following Notice
This is called the first invariant of
stress.
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Plotting these functions -stress
(normal and shear) vary continuously with
rotation -reach maximum and minimum values at
90? intervals - maximum and minimum normal
stresses occur where shear 0.
Principal Stresses
sx
sxy
q
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Find Direction of Principal Stresses
sxy
2qp
Plug expressions into equation for sx.
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We can use the first invariant of stress to solve
for s2 Combining with previous equation, we
have For plane stress, remember sz 0 and
is the third principal stress by default. All
principal stress directions are separated by
90?. Maximum shear stress
Planes of max shear also contain normal stresses
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All of these transformations of stress for 2-D
can be represented Graphically MOHRS CIRCLE
sy ,sxy
sxy
sx, sy,
2qp
-sx, -sy,
sx ,sxy
sxy
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All of these transformations of stress for 2-D
can be represented Graphically MOHRS CIRCLE
sy ,sxy
sxy
sx, sy,
2qp
-sx, -sy,
sx ,sxy
sxy