Civil Engineering Reference
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This gives
w
u
x
H
x
w
x
w
u
y
H
y
w
y
(4.34)
H
0
z
w
u
w
u
w
u
w
u
y
x
z
z
J
,
J
,
J
xy
yz
xz
w
y
w
x
w
y
w
x
Complete plane strain can be split into the plane strain case already discussed and an
antiplane strain or St Venant torsion component for which H
x
= H
y
= H
z
= J
xy
= 0 and
w
u
z
J
zy
w
y
(4.35)
w
u
z
J
zx
w
x
In complete plane strain it is possible to have shear strains and stresses acting in the
z
-direction.
y
V
V
V
x
V
x
x
W
xy
W
xy
W
xy
y
V
y
V
y
D
x
Figure 4.7
Transformation of stresses in two dimensions
Sometimes it is necessary to compute the magnitudes of stress or strain in directions
which do not coincide with the global axes. In this c
as
e a
transformation
of stress or
strain is necessary. The transformation of local stresses
ı
acting on planes in the material
parallel with the
x yz
axes to global stresses V
acting on cuts parallel with the
x, y, z
axes can be written as
,,
(4.36)
ı T ı
V
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