Biomedical Engineering Reference
In-Depth Information
F
y
e
y
Δ
A
Δ
F
Δ
e
y
Δ
F
x
e
x
F
z
e
z
Δ
e
x
e
z
Figure 8.1
Force
F
on an infinitesimal surface element having area
A
.
8.2
From stress to force
Suppose, that a free body diagram is created by means of an imaginary cutting
plane through a body. The cutting plane is chosen in such a way that it coincides
with the
xy
-plane (Fig.
8.2
). On the imaginary cutting plane a stress vector
s
is
given as a function of
x
and
y
. How can the total force vector acting on that plane
be computed based on this stress vector? The complete answer to this question is
somewhat beyond the scope of this course because it requires the integration of a
multi-variable function. However, the more simple case where the stress vector is
a function
x
(or
y
) only, while it acts on a rectangular plane at constant
z
is more
easy to answer and sufficiently general to be useful in the remainder of this chap-
ter. Therefore, suppose that the stress vector is a function of
x
such that it may
be written as
s
=
s
x
(
x
)
e
x
+
s
y
(
x
)
e
y
+
s
z
(
x
)
e
z
.
(8.5)
Let this stress vector act on a plane
z
=
0, that spans the range 0
<
x
<
L
and that
has a width
h
in the
y
-direction. The resulting force vector on the plane considered
due to this stress vector is denoted by
F
=
F
x
e
x
+
F
y
e
y
+
F
z
e
z
.If
s
x
,
s
y
and
s
z
are
constant
, the net force is simply computed by multiplication of the stress vector
components with the surface area
hL
in this case:
F
=
s
x
hL
e
x
+
s
y
hL
e
y
+
s
y
hL
e
z
.
(8.6)
For
non-constant
stress vector components (e.g. as visualized in Fig.
8.3
),
the force components in the
x
-,
y
- and
z
-direction due to the stress vector
s
are
obtained via integration of these components over the domain in the
x
-direction
and multiplication with the width
h
of the plane (which is allowed because the
stress components are constant in the
y
-direction):
