Civil Engineering Reference
In-Depth Information
1
2
x
1
1
2
(2
x
=
tF
BY
2
f
(
b
)
2
y
2
x
2
d
x
=
tF
BY
−
y
)d
y
d
x
0
0
0
2
3
=
tF
BY
2
=
tF
BY
3
=−
0
.
0189 lb
1
2
x
1
=
tF
BY
2
=
tF
BY
2
=
tF
BY
3
f
(
b
)
3
y
2
x
2
d
x
y
d
x
d
y
=−
0
.
0189 lb
0
0
0
showing that the body force is equally distributed to the element nodes.
If we now combine the concepts just developed for the CST element in plane
stress, we have a general element equation that includes directly applied nodal
forces, nodal force equivalents for distributed edge loadings, and nodal equiva-
lents for body forces as
f
(
p
)
f
(
b
)
[
k
]
{}={
f
}+
+
(9.48)
where the stiffness matrix is given by Equation 9.24 and the load vectors are as
just described. Equation 9.48 is generally applicable to finite elements used in
elastic analysis. As will be learned in studying advanced finite element analysis,
Equation 9.48 can be supplemented by addition of force vectors arising from plas-
tic deformation, thermal gradients or temperature-dependent material properties,
thermal swelling from radiation effects, and the dynamic effects of acceleration.
9.3 PLANE STRAIN: RECTANGULAR ELEMENT
A solid body is said to be in a state of
plane strain
if it satisfies all the assump-
tions of plane stress theory
except
that the body's thickness (length in the
z
direction) is
large
in comparison to the dimension in the
xy
plane. Mathemati-
cally,
plane strain
is defined as a state of loading and geometry such that
∂
w
∂
∂
u
z
+
∂
w
∂
v
z
+
∂
w
ε
z
=
z
=
0
xz
=
x
=
0
yz
=
y
=
0
(9.49)
∂
∂
∂
∂
(See Appendix B for a discussion of the general stress-strain relations.)
Physically, the interpretation is that the body is so long in the
z
direction that
the normal strain, induced by only the Poisson effect, is so small as to be negli-
gible and, as we assume only
xy
-plane loadings are applied, shearing strains
are also small and neglected. (One might think of plane strain as in the example
of a hydroelectric dam—a large, long structure subjected to transverse loading
only, not unlike a beam.) Under the prescribed conditions for plane strain, the
