Civil Engineering Reference
In-Depth Information
x
u
(
x
,
t
)
d
x
1
2
u
2
(
x
2
,
t
)
x
u
1
(
x
1
,
t
)
d
x
(a)
(b)
Figure 10.7
(a) Bar element exhibiting time-dependent displacement. (b) Free-body diagram of a
differential element.
where
is density of the bar material. Note the use of partial derivative operators,
since displacement is now considered to depend on both position and time. Sub-
stituting the stress-strain relation
=
E
ε
=
E
(
∂
u
/∂
x
)
, Equation 10.51 becomes
2
u
2
u
∂
E
∂
=
∂
(10.52)
∂
x
2
t
2
Equation 10.52 is the
one-dimensional wave equation,
the governing equation
for propagation of elastic displacement waves in the axial bar.
In the dynamic case, the axial displacement is discretized as
u
(
x
,
t
)
N
2
(
x
)
u
2
(
t
)
(10.53)
where the nodal displacements are now expressed explicitly as time dependent,
but the interpolation functions remain dependent only on the spatial variable.
Consequently, the interpolation functions are identical to those used previously
for equilibrium situations involving the bar element:
N
1
(
x
)
=
1
−
(
x
/
L
)
and
N
2
(
x
)
=
N
1
(
x
)
u
1
(
t
)
+
L
. Application of Galerkin's method to Equation 10.52 in analogy to
Equation 5.29 yields the residual equations as
L
=
x
/
N
i
(
x
)
E
∂
A
d
x
2
u
2
u
∂
−
∂
=
0
i
=
1, 2
(10.54)
∂
x
2
t
2
0
Assuming constant material properties, Equation 10.54 can be written as
L
L
2
u
2
u
N
i
(
x
)
∂
N
i
(
x
)
∂
A
d
x
=
AE
d
x
i
=
1, 2
(10.55)
∂
t
2
∂
x
2
0
0
Mathematical treatment of the right-hand side of Equation 10.55 is identical to
that presented in Chapter 5 and is not repeated here, other than to recall that the
result of the integration and combination of the two residual equations in matrix
form is
1
u
1
u
2
f
1
f
2
AE
L
−
1
=
⇒
[
k
]
{
u
}={
f
}
(10.56)
−
11
