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人類誌, J. Anthrop. Soc. Nippon79(4):323-336 (1971)
Cross-Section of Human Lower Leg Bones
Viewed from Strength of Materials
Tasuku KIMURA
Department of Anthropology, Faculty of Science The University of Tokyo
Abstract The bones of the lower leg were examined from the viewpoint of strength of materials. The area, the moment of inertia of area and the polar
moment of inertia of area of the cross-section at the middle of the lower leg bones were calculated. The resistance of the bone against the normal force, against
the bending moment and against the torsion can be shown by these properties of the cross-section. The properties of the shape of the bones do not correlate with the age of the specimen. The sexual dimorphism is clear. The fibula is
very much weaker than the tibia. The index of cross-section has no direct cor- relation with the strength of bones nor with the curvature of tibia shaft.
INTRODUCTION
The mechanical strength of the long
bone has already been discussed from the
viewpoint of the strength of materials.
The long bone can be regarded as the
beam on which external forces are being
applied. The shape of the transverse cross-
section of the beam is related with the
strength of the beam. The forces acting
on the beam are mainly the normal force,
the bending moment and the torsion. The
area of the cross-section shows the resis-
tance against the normal force. The mo-
ment of inertia of the area shows the re-
sistance against the bending moment. The
polar moment of the inertia of the area
shows the resistance against the torsion.
In this study the cross-section of the
middle of the human lower leg bones was
examined. The area of the all kinds of
human long bones was studied by AOJI
et al. (1959). The report on the polar mo-
ment of inertia on the cross-section of the
bone has not been appeared except in the
paper by FRANKEL and BURSTEIN (1965).
The strongest working force on the long
bone is the bending moment as stated by
PAUWELS (1948), especially so on the tibia
(KIMURA, 1966). The moment of inertia
of area must be examined to know the
strength of the long bones. The reports
on the moment of inertia have been not
many. KNESE et al. (1954) reported on
many sections of all the long bones, but
their number of samples is small. JERN-
BERGER (1970) examined the minimum mo-
ment of inertia on many sections of five
tibiae. FRANKEL and BURSTEIN (1965)
calculated the moment of inertia on the
tibia at the mid-shaft and at the fractured
sites. I read the unpublished data on the
324 T. KIMURA
long bones and compared them with mine
through the courtesy of Dr. Hiromi SUZU-
KI.
MATERIALS
Materials in this study consisted of
pairs of tibia and fibula of seventeen Jo-mon males, twenty-two recent Japanese
males and twenty recent Japanese females
(Table 1). All the bones, so far as is known, appeared to be normal. All recent
samples were left side. The right and left
bone were mixed in the Jomon samples.
The recent Japanese skeletons were ex-
amined through the courtesy of the De-
partment of Anatomy, Faculty of Medi-
cine, The University of Tokyo.
Table 1, Recent specimens.
The skeletons of the Jomon man (prehi-
storic food gatherer in Japan more than
two thousands years ago) were excavated
in the Honshu Island. All of them were
nearly in a perfect state of preservation.
They were stored in the Section of An-
thropology, University Museum, The Uni-
versity of Tokyo. The Jomon samples
were not taken at random. The flat tibiae,
which are rare in the recent ones, were
picked up purposely. The Jomon samples
were used as supplementary in this study.
The ages of the recent specimens were
from 40 to 82 years in the male and from
53 to 84 years in the female. It means
that these specimens were of relatively
old age.
METHODS
The bi-articular length in this study is
MARTIN'S condylo-talar length of the tibia.
This length can roughly be regarded as
the physiological length of the lower leg
and the tibia. The physiological axis pas-
ses through the centers of the upper and
lower articular surfaces of the tibia. The
frontal plane of the lower leg in this stu-
dy consists of the center of the upper
medial articular surface, the center of the
upper lateral articular surface and the
center of the lower articular surface.
The middle of the bi-articular length of
the lower leg is cut horizontally to show
the cross-section. A glass plate with a 1
mm mesh is put on this section and a
photograph is taken. The area and the
moment of inertia of the cross-section are
examined on the photograph by the nu-
merical method. Only the compact subs-
tance is regarded as the bone material in
this study. The spongy substance was
included in the marrow. The mechanical
strength of the spongy substance is much
weaker than that of the compact subs-
tance. The area of the spongy substance
is very small in the middle of the body.
It may be possible to exclude this subs-
tance in this part of the bone. The area,
A, is the area of the compact substance
in this study. The total area, Atot, means
the sum of A and the area of the marrow,
Am.
The axis parallel to the maximum an-
Cross-Section of Human Lower Leg Bones 325
Fig. 1, Cross-section of the lower leg bones showing the axes. Sag : Sagittal axis of the lower leg. Front : Frontal axis. X : X-axis of the tibia. Y : Y-axis. Max : Principal axis of the fibula in the direction of the maximum mo-ment of inertia. Min : Principal axis in the direction of the minimum moment of inertia.
O: Centroid. N: Physiological axis of the low-er leg. a: anterior side, p: posterior side. m: medial side. 1: lateral side.
tero-posterior diameter (Y-axis) is used
as the principal axis of the tibia for the
simplification of integration (Fig. 1). The
transverse axis, X-axis, forms a right an-
gle with the Y-axis. The nearly maximum moment of inertia with respect to the X-
axis, Ix, is in the direction of Y-axis. The
nearly minimum moment of inertia with
respect to the Y-axis, Iy, is in the direction
of X-axis. This simplification can be al-
lowed as shown by the results in this
study. The maximum and the minimum
moment of inertia of the fibula, Imax and
Imin with respect to the principal axis of
the cross-section are determined by the
numerical integration. The polar moment
of inertia, Ip, is the sum of Ix and Iy in the
tibia or of Imax and Imin in the fibula.
When the normal force (P) works on
the beam, the stress (*) will be
*=P/A A : area of the cross-section
The area shows the resistance against the
normal force.
The moment of inertia of an area from
the neutral axis z (Iz) is given by
Iz =* Ay2dA y : distance from z
when the pure bending is applied on a
beam, the maximum stress (*x)max on a
cross-section of the beam in the axial (x)
direction appears on the most outside part
of the cross-section from z ; that is,
(*x)max=Mh/I2=M/ Z
M : bending moment
h : maximum height on the cross-section
from z
Z : section modulus for z
The larger the moment of inertia or the section modulus is, the greater is the re-
sistance of the beam against the bending
moment.
The maximum shearing stress (*max) of
the circular shaft which is produced by the torsion is
Mt : torsional moment
*max=Mtd/2Ip d: diameter Ip : polar moment of inertia
The polar moment of inertia shows the
resistance of the circular shaft against the
torsion. The long bone can be considered
as the circular hollow shaft. The lower
leg bones, however, are not exactly the
circular shaft. The problem of the torsion
of the non-circular shaft is complicated,
due to the warping of the cross-section.
The polar moment of inertia of the lower
leg bones will show a rough standard of
326 T. KIMURA
the strength against the torsion. Further
details of the strength of the beam can be
found in the textbook dealing with the
strength of materials.
RESULTS
The personal records of the specimen
and the data on the tibia and fibula at
the middle of the bi-articular length of
the lower leg are shown in Tables 1 to 3.
The correlation coefficients between two
properties of them are shown in Table 4
and Fig. 2. The data on each the recent
specimen are shown in Appendices 1 to 3.
At first, the influence of the sampling
must be commented on. There is no sig-
nificant correlation between the index of
the cross-section and the area of the tibia.
Table 2. Tibia.
Cross-Section of Human Lower Leg Bones 327
Table 3, Fibula.
The moment of inertia and the polar mo-
ment of inertia have almost no correlation
with the index (Fig. 2-1). For these a-
bove reasons, discussions will be made also
on the Jomon specimen. The recent speci-
mens in this study were of old age. The
age of the specimen, however, is correlated
significantly neither with the area, with
the moment of inertia (Fig. 2-2) nor with
the polar moment of inertia.
The bones of the female have a small
area, moment of inertia and the polar mo-
ment of inertia compared with the bones
of the male (Fig. 2). The difference of the
mean area between the male and the
female is greater than the standard devi-
ation of each in case of both the tibia and
the fibula. The differences of the moment
of inertia and of the polar moment of in-
ertia between the male and the female are
also larger than the standard deviations
of each.
The fibula has a very small area, mo-
ment of inertia and polar moment of in-
ertia compared with the tibia. The mean
of the area of the fibula is less than thirty
percent of that of the tibia. The mean
moment of inertia and the mean polar
moment of inertia are less than ten per-
cent. The standard deviations of the mo-
ment of inertia and polar moment of in-
ertia of the fibula are relatively larger
than those of the tibia.
The bi-articular length of the tibia cor-
relates with the maximum moment of in-
ertia and the polar moment of inertia.
The length correlates with the area of the
tibia when the sexes are not considered.
The stature and the body weight correlate
with the area, the moment of inertia and
the polar moment of inertia of the tibia
and the fibula in case of not considering
the sexes. When considering by sexes,
they do not correlate well especially in
328 T. KIMURA
Table 4. Correlation coefficients.
case of the female.
The moment of inertia of area correlates
with the diameters in both the tibia and
fibula. The index of the cross-section at
the middle of the tibia correlates with Iy/
Ix. The index of the cross-section of the
fibula correlates with Imin/Imax.
The tibia with a wide area has also a
large moment of inertia or polar moment
of inertia (Fig. 2-3). The ratio of the ef-
fective area, A divided by Atot, becomes
large when the area become wide (Fig. 2-
4). The area of the marrow shows almost
no correlation with the total area.
The area of the tibia significantly cor-
relates with that of the fibula. But the
moment of inertia of the tibia and that of
the fibula do not show a good correlation.
The polar moment of inertia of the tibia
correlates with that of the fibula.
The index of the curvature of the tibia
using the centroid, the index being the
Cross-Section of Human Lower Leg Bones 329
Fig. 2. Scatter-diagramm showing the correlations.
(2-1) Between Ix and the index at mid-shaft of the tibia.
(2-2) Between the age of specimens and A of the tibia.
(2-3) Between Ix and A of the tibia.
(2-4) Between A and the ratio of the effective area of the tibia.
distance from the centroid of the cross-
section to the physiological axis divided
by the bi-articular length, shows no sig-
nificant correlation with the index of the
cross-section. The centroid of the tibia
in this section is placed forward of the
330 T. KIMURA
physiological axis and about 1mm back- ward of the mid-point of the maximum
antero-posterior diameter.
The principal axis on the cross-section
of the tibia diverges from the Y-axis only
from eight degrees laterally to ten degree
medially at the maximum. The difference
between Ix and Imax of the tibia is less
than two percent and between Iy and Imin
is less than six percent in this study.
The principal axis diverges from the sa-
gittal axis from 18 to 42 degrees laterally.
DISCUSSION
The area of the tibia and the fibula re-
ported by AOJI et al. (1959) is the average
of the right and left bones. The area of
the tibia reported by them is slightly
smaller than that of the present study.
The moment of inertia of the European
bones (KNESE et al., 1954; JERNBERGER,
1970) is rather large than that of the Ja-
panese in this study.
AOJI et al. (1959) believed that the
sectional area diminishes after sixty years
in the female and eighty years in the male.
The age of the specimens does not corre-
late with the area, with the moment of
inertia nor with the polar moment of in-
ertia in the present study, though the
specimens are rather old ones. It is im-
possible to find the correlation between the area of the bones and the age in the
data by Aoji et al. which include the
specimens of the Japanese in the twenties
and thirties. YAMADA (1970, p. 20, 255)
reported that the mechanical properties of
bones decrease in the old age group. The
present study is concerned only with the
mass and shape of the bone and the me-
chanical properties are not being con-
sidered.
Compared with the data reported by the
Nutrition Section, Ministry of Health and
Welfare in 1967, the mean stature of the
present recent samples is slightly lower
within the same range of age, and the
body weight is much lower. Since none
of the recent specimens died accidentally,
they may have suffered from the disease
for some period of time. This may be one
of the reasons why the body size does not
show a good correlation with the area nor
with the moment of inertia. On the other
hand, the length of the tibia is not affec-
ted by the disease and may show to some
extent the body mass of the specimen.
The long bone shows a large resistance
especially against the bending. The bone
which is strong against the normal force
is also strong against bending and torsion.
The area of the compact bone becomes
very wide when the total area becomes
wide. The bone with a wide area has the
thick compact substance. In other words,
the shape of the cross-section is not simi-
lar in the bones with a wide and a small
area. The marrow of the bone with a
wide total area is not necessarily large.
The sexual dimorphism in the cross-
section of the long bones is very clear.
The female has very weak bones compared
with the male, though considering her
small body size. The male bone is not
only large in size externally, but also has
a thick compact substance.
The centroid of the cross-section at the
mid-shaft of the tibia is situated forward
Cross-Section of Human Lower Leg Bones 331
of the physiological axis and backward of
the mid-point of the maximum antero-
posterior diameter. When the compressive
normal force is applied to the physio-
logical axis, the load on the cross-section
at the mid-shaft becomes eccentric. The
entire body of the tibia can be seen as a
curved column as discussed by KIMURA
(1966). If the posterior part of the bone stretches out and forms a buttress, the
curvature of the shaft can become small.
The index of the cross-section, however,
has no correlation with the curvature.
The flatness of the tibia does not increase
the strength against the normal force.
The significance of the tibia shaft has
been discussed by many investigators. As
shown by the correlation coefficient in
Table 4 and in Fig. 2-1, the flat tibia is not
absolutely strong. The index of the cross-
section of the tibia shows the ratio of the
strength in the antero-posterior direction
to that in the transverse direction. The
index of the cross-section of the fibula also
shows the ratio of the strength between
the principal axes. The bending force
acting on the tibia is mainly in the antero-
posterior direction because of the move-
ment of the muscles and articulation. The
strength in the antero-posterior direction
will be more important than that in the
transverse direction. On this point, the
flatness would be regarded as a suitable
form of the tibia.
The strength of the tibia correlates with
that of the fibula. The strength of fibula
is not related closely with the body size.
The variation of the properties of the fi-
bula is relatively large as seen by the
standard deviation. One of the reasons for
this variation will be that the fibula does
not bear a large portion of the strength
of the lower leg. The area and the mo-
ment of inertia of the fibula are much
smaller than those of the tibia. The tibio-
fibular connections are not rigid ones.
The forces acting on the lower leg may be
sustained mainly by the tibia. It is not
possible to consider the size and shape of
the cross-section of the fibula only from
the mechanical viewpoint of the fibula
alone.
The strength against the pure bending
is shown by the section modulus. The
maximum section modulus of the mid-
shaft of the tibia appears at the posterior
side which is in the direction of the
maximum moment of inertia. It is because
the anterior side is pointed and has a large
height from the neutral axis compared
with the posterior side. The minimum
section modulus is at the lateral side
where it is high because of the existance
of the crista interossea. Usually the lar-
ger the moment of inertia is, the greater
is the section modulus on both sides of
the axis. For simplification, in this study
the moment of inertia is used to show the
resistance against the bending.
The bone is heterogeneous and aniso-
tropic. An experimental study is necce-
sary to clarify the mechanical properties
of the bone. The strength of the bone,
however, could be shown to a certain deg-
ree by the shape of the cross-section in
this study.
332 T. KIMURA
SUMMARY
The area, the moment of inertia of area
and the polar moment of inertia of area
of the cross-section at the middle of the
human lower leg bones were examined to
learn about the strength of the bone from
the viewpoint of the strength of materials.
The specimens were obtained from twenty-
two recent Japanese males, twenty recent
Japanese females and seventeen Jomon ma-les.
The results are shown in Tables 1 to 4
and Fig. 2 and summarized as follows :
1) The age of the specimens does not cor-
relate with the area, with the moment of
inertia nor with the polar moment of in-
ertia ; 2) The bones of the female have
distinctly a small area, moment of inertia
and polar moment of inertia compared
with the bones of the male; 3) There will
be a tendency that a large body mass is
associated with a large strength of the
bone ; 4) The shape of the cross-section
with a wide area is not similar to that
with a small area ; 5) The strength of the
fibula is much smaller than that of the
tibia and the deviation of the properties
of the fibula is greater than that of the
tibia ; 6) The index of the cross-section
at the middle has no direct correlation
with the strength of the bones nor with
the curvature of the tibia, and the index
of the tibia shows a ratio of strength be-
tween the antero-posterior and transverse
directions ; and 7) The principal axis at
the mid-shaft of the tibia is parallel to
the maximum antero-posterior diameter.
ACKNOWLEDGMENT
This study is indebted to Professor Ta-
dahiro OOE, Associate Professor Ichiyoh
ASAMI and Dr. Toshiro KAMIYA of the
Department of Anatomy, Faculty of Me-
dicine, the University of Tokyo for allow-
ing and assisting me to examine the recent
specimens. Thanks are also expressed to
Mr. Hisao BABA and Mr. Yasuo FUKUSHIMA
of the Department of Anthropology, Facul-
ty of Science, the University of Tokyo for
their assistance of this study.
REFERENCES
AOJI, O., MOTOJIMA, T, and BANDO, T., 1959:
On the effective sectional areas and maximum compressive loads of diaphysis of human long
bones. J. Kyoto Pref. Med. Univ., 65: 979-983
(Japanese with English summary). FRANKEL, V. H, and BURSTEIN, A.H., 1965;
Load capacity of tubular bone. Biomechanics
and Related Bio-Engineering Topics (Ed. R.
M. KEN E DI), pp. 381-396. Oxford.
JERNBERGER, A., 1970: Mesurement of stability of tibial fractures. Acta Orthop. Scand., Sup-
pl. No.135. KIMURA, T., 1966: An experimental study of
the form of the human tibia from the bio-
mechanical point of view. J. Anthrop. Soc.
Nippon., 74: 119-227. KNESE. K-H., HAHNE, 0. H, and BIERMANN, H.,
1954: Festigkeitsuntersuchungen an mensch-
lichen Extremitatenknochen. Gegenbauers
Morph., 96: 141-209.
厚生省公衆衛 生局栄養課(編),1969:国 民栄養の現
状,昭 和42年 度 国民栄養調査成績.東 京.
(Nutrition Section, Public Health Bureau, Ministry of Health and Welfare, The report
of the national nutrition survey in Japan,
1967. Tokyo. In Japanese)
PAUWELS, F., 1948: Die Bedeutung der Baup-
rinzipien des Stutz- and Bewegungsapparates
fur die Beanspruchung der Rohrenknochen.
Cross-Section of Human Lower Leg Bones 333
Z. Anat. Entwickl. Gesch., 114: 129-166.
*, 1950: Die Bedutung der Muskelkrafte
fur die .Regelung der Beanspruchung des
Rohrenknochens wahrend der Bewegung der
Glieder. Z. Anat. Entwickl. Gesch., 115: 327-
351.
YAMADA, H., 1970: Strength of biological ma-
terials (Ed. F. G. EVANS). Baltimore.
(Received June 3, 1971)
材 料 力 学 的 に 見 た ヒ トの 下 腿 骨 横 断 面
木 村 賛
東京大学理学部人類学教室
長骨の強 さは断面形 において材料力学的にお しはか ることができる.長 骨 に加 わる外力は主 として軸力,曲
げモー メ ン ト,〓 りの三つで ある.断 面の面積の大 きさは軸力 に対 す る抵抗 の大 きさを示す.断 面二次 モーメ
ントは曲げに対す る抵抗 の大 きさを ほぼ示 している.断 面二次極モーメ ン トは丸軸 の〓 りに対す る抵抗 の大 き
さを示す.
本研究では現代 日本人男性22側,女 性20側,縄 文時代人 男性17側 の下腿骨中央 断面において面積,断 面二次
モー メン ト,断 面二次極モー メン トの三種の数値が調べ られた.骨 の持つ機械的性質の違いについて は考慮せ
ず,緻 密質部分の形 状によってわか る強さのみが論 じられている.骨 の力学的性質の解明には実験 的研 究が不
可 欠で ある.
断面 の形状に関 して下記の結果が得 られ た.1)資 料の年令 は下腿骨の断面 積,断 面 二次モーメ ン ト,断 面
二次極 モー メン トの大 きさのいずれ とも相関がない.2)女 性骨 は男性骨 と比較 し三種 の 数値がすべて非 常に
小 さい. 3)体 の大きさ と骨 の丈夫 さとには関係があ るよ うで ある.4)断 面積 の大 きさが異な る骨 の断面 の
形状 は相似形でない.大 きな骨の緻密質は厚 くなる.5)腓 骨 は脛骨 と比べて三種の数値が著 しく小 さ く,そ
のば らつきの程度が比較 的大であ る.6)断 面係数 は骨 の強 さと直接の相関はない.脛骨 の弯 曲と も相関がない.
断面係 数は前後方向及び横方 向の 断面二次モーメ ン トの比 と相関 している.7)脛 骨中央断面 の 主軸は最大前
後径 と平行で ある.
334 T. KIMURA
Appendix 1. Recent Japanese specimens.
Cross-Section of Human Lower Leg Bones 335
Appendix 2. Tibia of recent Japanese.
336 T. KIMURA
Appendix 3, Fibula of recent Japanese.