Biomedical Engineering Reference
In-Depth Information
boundary conditions at both ends of the bar are used. First, since at
x
=
0 the
displacement
u
=
0, the integration constant
d
must be zero, hence
c
EA
x
.
u
=
Second, at
x
=
L
the force is known, such that
x
=
L
=
EA
du
dx
EA
c
F
=
EA
=
c
.
So, the (unique) solution to the boundary value problem reads
F
EA
x
.
u
=
The strain
ε
is directly obtained via
du
dx
=
F
EA
,
ε
=
while the stress
σ
follows from
F
A
σ
=
E
ε
=
as expected.
Example 6.3
Consider, as before, a bar of length
L
, clamped at one end and loaded by a force
F
at the other end of the bar. The Young's modulus
E
is constant throughout the bar,
but the cross section varies along the axis of the bar. Let the cross section
A
(
x
)
be given by
A
0
1
,
x
3
L
A
=
+
with
A
0
a constant, clearly representing the cross section area at
x
=
0.
x
F
L
x
=
0
The boundary value problem is defined by the same set of equations as in the
previous example:
EA
du
dx
d
dx
=
0f r 0
<
x
<
L
u
=
0 t
x
=
0
EA
du
dx
=
F
at
x
=
L
.
