Biomedical Engineering Reference
In-Depth Information
Since
σ
=
Edu
/
dx
, the natural boundary condition at
x
=
L
reads
EA
du
dx
=
F
.
(6.24)
Because the equilibrium equation Eq. (
6.20
) is a second-order differential equa-
tion,
two
boundary conditions must be specified, one must be an essential bound-
ary condition (the displacement must be specified at least at one point to avoid
rigid body displacement) and the other may either be an essential or natural bound-
ary condition. The combination of the equilibrium equation with appropriate
boundary conditions is called a (determinate)
boundary value
problem.
Example 6.2
x
F
L
x
=
0
As a first example the solution of a well-defined boundary value problem for a
homogeneous bar without a distributed load is analysed. Consider a bar of length
L
that has a uniform cross section, hence
A
is constant, and with constant Young's
modulus
E
. There is no volume load present, hence
q
=
0. At
x
=
0 the displace-
ment is suppressed, hence
u
0, while at the other end of the bar a force
F
is applied. Then, the boundary value problem is fully described by the following
set of equations:
=
0at
x
=
EA
du
dx
d
dx
=
0f r 0
<
x
<
L
u
=
0 t
x
=
0
EA
du
dx
=
F
at
x
=
L
.
Integrating the equilibrium equation once yields
EA
du
dx
=
c
,
where
c
denotes an integration constant. This may also be written as
du
dx
=
c
EA
.
Because both the Young's modulus
E
and the cross section area
A
are constant,
integration of this relation gives
c
EA
x
+
d
,
with
d
yet another integration constant. So, the solution
u
(
x
) is known provided
that the integration constants
c
and
d
can be determined. For this purpose the
u
=
