Biomedical Engineering Reference
In-Depth Information
x
F
L
1
L
2
(a)
F
N
1
N
1
N
2
N
2
L
1
L
2
(b)
Figure 6.6
A two bar system and associated free body diagram.
Clearly, based on the previous examples, for the first bar the displacement field is
given by
N
1
E
1
A
1
x
.
For the second bar the following boundary value problem holds:
d
dx
u
1
=
E
2
A
2
du
2
dx
=
0f r
L
1
<
x
<
L
1
+
L
2
u
2
=
0 t
x
=
L
1
+
L
2
E
2
A
2
du
2
dx
=
N
2
at
x
=
L
1
.
The solution of this system is given by
N
2
E
2
A
2
(
L
1
+
L
2
).
We should realize that neither
N
1
nor
N
2
is known so far. However, there are two
additional equations that have to be satisfied. Force equilibrium of the slice (see
Fig.
6.6
) requires that
N
2
E
2
A
2
x
−
u
2
=
−
N
1
+
N
2
+
F
=
0,
while the displacement field must be continuous at
x
=
L
1
: the two bars must
remain fitting together, hence
u
1
(
L
1
)
=
u
2
(
L
1
) .
(6.25)
Based upon the solution for
u
1
and
u
2
it follows that
N
1
E
1
A
1
L
1
,
u
1
(
L
1
)
=
