Biomedical Engineering Reference
In-Depth Information
resistance to bending. Equations for calculating the area moment of inertia for few
commonly utilized cross-sections are summarized in Table 5.5.
EXAMPLE 5.8
A company is investigating a new plate for use in a bone fracture. They have a plate that
is 1 cm thick and 3 cm wide. They want to know whether Case A or Case B orientation is
better for resisting bending. Could you help them?
Solution: For a rectangular cross-section,
3
/12
I
=
h
3
3*1
For Case A,
4
I
0.25 cm
=
=
12
3
1* 3
For Case B,
4
I
2.25 cm
=
=
12
Since the moment of inertia for Case B is significantly higher, the developed stresses
and deflection are significantly less. Hence, orientation according to Case B is preferred.
Substituting for (
E
/
r
) in (5.33) in terms of
σ
x
gives the expression
σ
MI
=
x
b
zz
Rearranging, the variation in axial stress throughout the cross-section is cal-
culated as
Table 5.5
Formula for Moment of Inertia of Common Shapes
Area Moments
of Inertia (I)
Polar Moments
of Inertia (J)
Cross-Sectional Shape
1
12
a
1
6
a
Square of side length
a
4
4
Circle with radius
r
π
π
4
r
4
2
r
4
Circular tube with inner radius
r
i
and outer
radius
r
o
π
π
(
)
(
)
4
4
4
4
r
−
r
r
−
r
o
i
o
i
4
2
Ellipse with a minor axis of 2
a
and major
axis 2
b
π
π
(
)
4
ab
3
4
ab a
2
+
b
2
3
96
a
3
80
a
Equilateral triangle with side length
a
4
4
1
12
bh
0.1406
bh
3
Rectangle of
b
breadth and
h
height
3
