Civil Engineering Reference
In-Depth Information
9.2 PLANE STRESS
A commonly occurring situation in solid mechanics, known as
plane stress,
is
defined by the following assumptions in conjunction with Figure 9.1:
1.
The body is small in one coordinate direction (the
z
direction by
convention) in comparison to the other dimensions; the dimension in the
z
direction (hereafter, the thickness) is either uniform or symmetric about
the
xy
plane; thickness
t
, if in general, is less than one-tenth of the smallest
dimension in the
xy
plane, would qualify for “small.”
2.
The body is subjected to loading only in the
xy
plane.
3.
The material of the body is linearly elastic, isotropic, and homogeneous.
The last assumption is not required for plane stress but is utilized in this text as
we consider only elastic deformations.
Given a situation that satisfies the plane stress assumptions, the only nonzero
stress components are
x
,
y
,
and
xy
.
Note that the nominal stresses perpendic-
ular to the
xy
plane
(
yz
)
are zero as a result of the plane stress assump-
tions. Therefore, the equilibrium equations (Appendix B) for plane stress are
∂
x
∂
z
,
xz
,
x
+
∂
xy
=
0
∂
x
(9.1)
∂
y
∂
y
+
∂
xy
=
0
∂
y
where we implicitly assume that
Utilizing the elastic stress-strain
relations from Appendix B, Equation B.12 with
z
=
xz
=
yz
=
xy
=
yx
.
0,
the nonzero
stress components can be expressed as (Problem 9.1)
E
x
=
ε
x
+
ε
y
)
2
(
1
−
E
y
=
2
(
ε
y
+
ε
x
)
(9.2)
−
1
E
xy
=
)
xy
=
G
xy
2(1
+
where
E
is the modulus of elasticity and
is Poisson's ratio for the material.
In the shear stress-strain relation, the shear modulus
G
=
E
/
2(1
+
)
has been
introduced.
The stress-strain relations given by Equation 9.2 can be conveniently written
in matrix form:
=
1
0
x
y
xy
ε
x
ε
y
xy
E
1
0
(9.3)
00
1
−
2
1
−
2
