Biomedical Engineering Reference
In-Depth Information
10
7
Pa) and is approximately the same of that of granite
the strength of steel (50
10
7
Pa).
An important consideration to be made in the case of bones is that their
mechanical response depends on the rate at which the force is applied. Bones are
more resistant under a given load when it is applied quickly than if the same load is
applied slowly.
Example 7.2
Consider the femur of an adult male. Its dimensions are: length
¼
50 cm, radius
¼
1.5 cm and the radius of the internal part that contains the
bone marrow
¼
0.4 cm. Consider that one of the femurs supports a body weight
force of 700 N of a person who is walking. Find: (a) the compressive stress applied
to this femur and (b) the amount of femur shortening caused by this load. Consult
Table
7.3
.
(20
(a) Let us begin by calculating the effective area of the femur that supports the
weight: it is in the form of a ring and the area can be obtained by subtracting
from the total area the area containing the marrow:
A
effetive
¼ π
(
r
femur
)
2
(
r
marrow
)
2
(1.5
2
- 0.4
2
cm
2
)
6.56 cm
2
π
¼ π
¼
¼
6.56
10
4
m
2
σ ¼
10
4
m
2
).
T
/
A
¼
(700 N)/(6.56
10
6
Pa
Comparing this result with atmospheric pressure, we verify that this stress is
equivalent to 10.56 atm, which is very large.
Hence,
σ ¼
1.07
10
6
Pa)/(0.94
10
10
Pa)
10
4
(b)
ε ¼ σ
/
Y
¼
(1.07
¼
1.138
10
4
)
10
4
m
Δ
L
¼
L
i
ε ¼
(0.50 m)(1.138
¼
0.57
0.06 mm
Example 7.3
Studies with human bones demonstrate that they behave elastically,
for deformations smaller than 0.5 %. Find the tension and the compression at the
elastic limit for an adult humerus which is 0.20 m in length with a cross-sectional
area of 3.0 cm
2
. Suppose that the elastic properties of the humerus are the same as
for the femur. Consult Tables
7.2
and
7.3
.
Tension
10
10
¼
T
t
¼ σ
t
A
¼
Y
t
ε
A
¼
Y
t
(
Δ
L
/
L
i
)
A
¼
(1.6
Pa)(0.005)
10
4
m
2
)
(3.0
T
t
¼
24 kN
Compression
10
10
Pa)(0.005)
¼
T
c
¼ σ
c
A
¼
Y
c
ε
A
¼
Y
c
(
Δ
L
/
L
i
)
A
¼
(0.94
10
4
m
2
)
T
c
¼
(3.0
14 kN
Note that the humerus remains in the elastic phase for loads (masses) as large as
2.4 tons, when tensioned, and 1.4 tons, when compressed.
Exercise 7.3
In Example 1.1, the force applied to a leg of an adult in the traction
apparatus was found to be 76.6 N. Supposing that the tibia has the same elastic
properties as the femur, find its elongation in percentage when this force is applied.
Consider the cross-sectional area of the tibia to be 3.3 cm
2
.
R.W. McCalden, J.A. McGeough, M.B. Barker, and C.M. Court-Brown
published in 1993 a paper with the title
Age-Related changes in the Tensile
Properties of Cortical Bone
in the J. Bone and Joint Surg. Am. 75 (A-8)
