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13.2.2
Material Law
The material law of Chapter 1, Eq. (1.32) for a three-dimensional continuum appears as
−
ν
−
ν .
x
y
z
......
σ
x
σ
y
σ
z
......
1
−
ν .
−
ν
1
0
−
ν
−
ν
1
1
E
...
... ...
......
......
......
=
(13.24)
.
2
(
1
+
ν)
γ
xy
γ
τ
xy
τ
.
0
2
(
1
+
ν)
xz
xz
.
2
(
1
+
ν)
γ
τ
yz
yz
E
−
1
=
σ
z
is smaller than other stress components
and can be neglected. This approximation may be questionable in the neighborhood of
concentrated transverse loading. With
Recall from Eq. (13.22) that
=
0
.
The stress
σ
z
σ
=
0
,
inversion of Eq. (13.24) gives
z
0
.
σ
x
σ
y
τ
xy
···
ν
x
y
γ
xy
···
1
0
.
ν
1
0
00
1
−
ν
2
.
··· ···
E
=
(13.25)
1
−
ν
2
··· ······ ······
.
τ
xz
τ
yz
1
−
ν
2
γ
xz
γ
yz
0
.
1
−
ν
2
σ
=
E
Frequently, the two expressions for
τ
xz
and
τ
yz
are written as
τ
=
G
γ
xz
,
τ
=
G
γ
(13.26)
xz
yz
yz
with
E
=
G
2
(
1
+
ν)
Figure 13.4 shows the stress components defined in Eq. (13.25). The stresses
σ
x
,
σ
y
,
τ
xy
,
τ
xz
,
and
τ
yz
are defined similarly to those in the engineering beam theory. There, it was conve-
nient to replace cross-sectional stresses by their resultant forces. For plates, we choose to
utilize forces and moments per unit length. These are the
stress resultants
(Fig. 13.5).
m
x
m
y
m
xy
=
σ
x
σ
y
τ
xy
=
τ
yx
t
2
=
zdz
Bending and Twisting Moments
t
2
m
yx
−
(13.27)
q
x
q
y
τ
dz
t
2
xz
=
Shear Forces
τ
t
2
yz
−
The signs (directions) of the moments and forces in Fig. 13.5 correspond to those of the
stress components in Fig. 13.4.
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