Biomedical Engineering Reference
In-Depth Information
q
N
(
x
+
Δ
x
)
N
(
x
)
F
x
x
L
Δ
x
(b) Free body diagram of a slice at
position
x
x
+
(a) Representation of a bar
Figure 6.1
Bar and free body diagram of a slice of the bar.
slice has a length
x
. The net force on the left side of the slice equals
N
(
x
), while
on the right side of the slice a force
N
(
x
+
x
) is present. The net force on the
right side of the bar may be different from the net force on the left side of the slice
due to the presence of a so-called distributed volume force. A volume force
Q
is a
force per unit of volume, and may be due to, for instance, gravity. Integration over
the cross section area of the slice yields a load per unit length, called
q
.
If the distributed load
q
is assumed constant within the slice of thickness
x
,
force equilibrium of the slice implies that
N
(
x
)
=
N
(
x
+
x
)
+
q
x
.
(6.1)
This may also be written as
N
(
x
+
x
)
−
N
(
x
)
+
q
=
0.
(6.2)
x
If the length of the slice
x
approaches zero, we can write
N
(
x
+
x
)
−
N
(
x
)
dN
dx
,
lim
x
→
=
(6.3)
x
0
where
dN
/
dx
denotes the derivative of
N
(
x
) with respect to x. The transition
expressed in Eq. (
6.3
) is illustrated in Fig.
6.2
. In this graph a function
N
(
x
)is
sketched. The function
N
(
x
) is evaluated at
x
and
x
+
x
, while
x
is small.
When moving from
x
to
x
+
x
the function
N
(
x
) changes a small amount: from
N
(
x
)to
N
(
x
+
x
). If
N
/
x
defines the tan-
gent line to the function
N
(
x
) at point
x
, and hence equals the derivative of
N
(
x
)
with respect to
x
. Notice that this implies that for sufficiently small
x
is sufficiently small the ratio
x
:
