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人類誌, J. Anthrop. Soc. Nippon 79(4):323-336 (1971)

Cross-Section of Human Lower

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 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 is very much weaker than the . The index of cross-section has no direct cor- relation with the strength of bones nor with the curvature of tibia shaft.

human long bones was studied by AOJI INTRODUCTION et al. (1959). The report on the polar mo- The mechanical strength of the long ment of inertia on the cross-section of the bone has already been discussed from the bone has not been appeared except in the viewpoint of the strength of materials. paper by FRANKEL and BURSTEIN (1965). The can be regarded as the The strongest working force on the long beam on which external forces are being bone is the bending moment as stated by applied. The shape of the transverse cross- PAUWELS (1948), especially so on the tibia section of the beam is related with the (KIMURA, 1966). The moment of inertia strength of the beam. The forces acting of area must be examined to know the on the beam are mainly the normal force, strength of the long bones. The reports the bending moment and the torsion. The on the moment of inertia have been not area of the cross-section shows the resis- many. KNESE et al. (1954) reported on tance against the normal force. The mo- many sections of all the long bones, but ment of inertia of the area shows the re- their number of samples is small. JERN- sistance against the bending moment. The BERGER (1970) examined the minimum mo- polar moment of the inertia of the area ment of inertia on many sections of five shows the resistance against the torsion. tibiae. FRANKEL and BURSTEIN (1965) In this study the cross-section of the calculated the moment of inertia on the middle of the human lower leg bones was tibia at the mid-shaft and at the fractured examined. The area of the all kinds of sites. I read the unpublished data on the 324 T. KIMURA long bones and compared them with mine 53 to 84 years in the female. It means through the courtesy of Dr. Hiromi SUZU- that these specimens were of relatively KI. old age.

MATERIALS METHODS Materials in this study consisted of The bi-articular length in this study is pairs of tibia and fibula of seventeen Jo- MARTIN'S condylo-talar length of the tibia. mon males, twenty-two recent Japanese This length can roughly be regarded as males and twenty recent Japanese females the physiological length of the lower leg (Table 1). All the bones, so far as is and the tibia. The physiological axis pas- known, appeared to be normal. All recent ses through the centers of the upper and samples were left side. The right and left lower articular surfaces of the tibia. The bone were mixed in the Jomon samples. frontal plane of the lower leg in this stu- The recent Japanese skeletons were ex- dy consists of the center of the upper amined through the courtesy of the De- medial articular surface, the center of the partment of , Faculty of Medi- upper lateral articular surface and the cine, The University of Tokyo. center of the lower articular surface.

Table 1, Recent specimens. 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- The skeletons of the Jomon man (prehi- tance is regarded as the bone material in storic food gatherer in Japan more than this study. The spongy substance was two thousands years ago) were excavated included in the marrow. The mechanical in the Honshu Island. All of them were strength of the spongy substance is much nearly in a perfect state of preservation. weaker than that of the compact subs- They were stored in the Section of An- tance. The area of the spongy substance

thropology, University Museum, The Uni- is very small in the middle of the body. versity of Tokyo. The Jomon samples It may be possible to exclude this subs- were not taken at random. The flat tibiae, tance in this part of the bone. The area, which are rare in the recent ones, were A, is the area of the compact substance

picked up purposely. The Jomon samples in this study. The total area, Atot, means

were used as supplementary in this study. the sum of A and the area of the marrow, The ages of the recent specimens were Am. from 40 to 82 years in the male and from The axis parallel to the maximum an- Cross-Section of Human Lower Leg Bones 325

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) Fig. 1, Cross-section of the lower leg bones direction appears on the most outside part showing the axes. Sag : Sagittal axis of the of the cross-section from z ; that is, lower leg. Front : Frontal axis. X : X-axis of the tibia. Y : Y-axis. Max : Principal axis of (*x)max=Mh/I2=M/Z the fibula in the direction of the maximum mo- ment of inertia. Min : Principal axis in the M : bending moment direction of the minimum moment of inertia. h : maximum height on the cross-section : Centroid. N: Physiological axis of theO low- from z er leg. a: anterior side, p: posterior side. Z : section modulus for z m: medial side. 1: lateral side. The larger the moment of inertia or the tero-posterior diameter (Y-axis) is used section modulus is, the greater is the re- as the principal axis of the tibia for the sistance of the beam against the bending simplification of integration (Fig. 1). The moment. transverse axis, X-axis, forms a right an- The maximum shearing stress (*max) of gle with the Y-axis. The nearly maximum the circular shaft which is produced by moment of inertia with respect to the X- the torsion is axis, Ix, is in the direction of Y-axis. The Mt : torsional moment nearly minimum moment of inertia with max=Mtd/2Ip d: diameter * respect to the Y-axis, Iy, is in the direction Ip : polar moment of inertia of X-axis. This simplification can be al- The polar moment of inertia shows the lowed as shown by the results in this resistance of the circular shaft against the study. The maximum and the minimum torsion. The long bone can be considered moment of inertia of the fibula, Imax and as the circular hollow shaft. The lower Imin with respect to the principal axis of leg bones, however, are not exactly the the cross-section are determined by the circular shaft. The problem of the torsion numerical integration. The polar moment of the non-circular shaft is complicated, of inertia, Ip, is the sum of Ix and Iy in the due to the warping of the cross-section. tibia or of Imax and Imin in the fibula. The polar moment of inertia of the lower When the normal force (P) works on leg bones will show a rough standard of 326 T. KIMURA the strength against the torsion. Further the lower leg are shown in Tables 1 to 3. details of the strength of the beam can be The correlation coefficients between two found in the textbook dealing with the properties of them are shown in Table 4 strength of materials. and Fig. 2. The data on each the recent specimen are shown in Appendices 1 to 3. RESULTS At first, the influence of the sampling The personal records of the specimen must be commented on. There is no sig- and the data on the tibia and fibula at nificant correlation between the index of the middle of the bi-articular length 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- The fibula has a very small area, mo- ment of inertia have almost no correlation ment of inertia and polar moment of in- with the index (Fig. 2-1). For these a- ertia compared with the tibia. The mean bove reasons, discussions will be made also of the area of the fibula is less than thirty on the Jomon specimen. The recent speci- percent of that of the tibia. The mean mens in this study were of old age. The moment of inertia and the mean polar age of the specimen, however, is correlated moment of inertia are less than ten per- significantly neither with the area, with cent. The standard deviations of the mo- the moment of inertia (Fig. 2-2) nor with ment of inertia and polar moment of in- the polar moment of inertia. ertia of the fibula are relatively larger The bones of the female have a small than those of the tibia. area, moment of inertia and the polar mo- The bi-articular length of the tibia cor- ment of inertia compared with the bones relates with the maximum moment of in- of the male (Fig. 2). The difference of the ertia and the polar moment of inertia. mean area between the male and the The length correlates with the area of the female is greater than the standard devi- tibia when the sexes are not considered. ation of each in case of both the tibia and The stature and the body weight correlate the fibula. The differences of the moment with the area, the moment of inertia and of inertia and of the polar moment of in- the polar moment of inertia of the tibia ertia between the male and the female are and the fibula in case of not considering also larger than the standard deviations the sexes. When considering by sexes, of each. they do not correlate well especially in 328 T. KIMURA

Table 4. Correlation coefficients.

case of the female. large when the area become wide (Fig. 2- The moment of inertia of area correlates 4). The area of the marrow shows almost with the diameters in both the tibia and no correlation with the total area. fibula. The index of the cross-section at The area of the tibia significantly cor- the middle of the tibia correlates with Iy/ relates with that of the fibula. But the Ix. The index of the cross-section of the moment of inertia of the tibia and that of fibula correlates with Imin/Imax. the fibula do not show a good correlation. The tibia with a wide area has also a The polar moment of inertia of the tibia large moment of inertia or polar moment correlates with that of the fibula. of inertia (Fig. 2-3). The ratio of the ef- The index of the curvature of the tibia fective area, A divided by Atot, becomes 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- nificant correlation with the index of the section to the physiological axis divided cross-section. The centroid of the tibia by the bi-articular length, shows no sig- in this section is placed forward of the 330 T. KIMURA

physiological axis and about 1mm back- mass and shape of the bone and the me- ward of the mid-point of the maximum chanical properties are not being con- antero-posterior diameter. sidered.

The principal axis on the cross-section Compared with the data reported by the of the tibia diverges from the Y-axis only Nutrition Section, Ministry of Health and from eight degrees laterally to ten degree Welfare in 1967, the mean stature of the

medially at the maximum. The difference present recent samples is slightly lower between Ix and Imax of the tibia is less within the same range of age, and the than two percent and between Iy and Imin body weight is much lower. Since none is less than six percent in this study. of the recent specimens died accidentally, The principal axis diverges from the sa- they may have suffered from the disease gittal axis from 18 to 42 degrees laterally. for some period of time. This may be one of the reasons why the body size does not DISCUSSION show a good correlation with the area nor The area of the tibia and the fibula re- with the moment of inertia. On the other ported by AOJI et al. (1959) is the average hand, the length of the tibia is not affec- of the right and left bones. The area of ted by the disease and may show to some the tibia reported by them is slightly extent the body mass of the specimen. smaller than that of the present study. The long bone shows a large resistance The moment of inertia of the European especially against the bending. The bone bones (KNESE et al., 1954; JERNBERGER, which is strong against the normal force 1970) is rather large than that of the Ja- is also strong against bending and torsion. panese in this study. The area of the compact bone becomes AOJI et al. (1959) believed that the very wide when the total area becomes sectional area diminishes after sixty years wide. The bone with a wide area has the in the female and eighty years in the male. thick compact substance. In other words, The age of the specimens does not corre- the shape of the cross-section is not simi- late with the area, with the moment of lar in the bones with a wide and a small inertia nor with the polar moment of in- area. The marrow of the bone with a ertia in the present study, though the wide total area is not necessarily large. specimens are rather old ones. It is im- The sexual dimorphism in the cross- possible to find the correlation between section of the long bones is very clear. the area of the bones and the age in the The female has very weak bones compared data by Aoji et al. which include the with the male, though considering her specimens of the Japanese in the twenties small body size. The male bone is not and thirties. YAMADA (1970, p. 20, 255) only large in size externally, but also has reported that the mechanical properties of a thick compact substance. bones decrease in the old age group. The The centroid of the cross-section at the present study is concerned only with the mid-shaft of the tibia is situated forward Cross-Section of Human Lower Leg Bones 331 of the physiological axis and backward of standard deviation. One of the reasons for the mid-point of the maximum antero- this variation will be that the fibula does posterior diameter. When the compressive not bear a large portion of the strength normal force is applied to the physio- of the lower leg. The area and the mo- logical axis, the load on the cross-section ment of inertia of the fibula are much at the mid-shaft becomes eccentric. The smaller than those of the tibia. The tibio- entire body of the tibia can be seen as a fibular connections are not rigid ones. curved column as discussed by KIMURA The forces acting on the lower leg may be (1966). If the posterior part of the bone sustained mainly by the tibia. It is not stretches out and forms a buttress, the possible to consider the size and shape of curvature of the shaft can become small. the cross-section of the fibula only from The index of the cross-section, however, the mechanical viewpoint of the fibula has no correlation with the curvature. alone. The flatness of the tibia does not increase The strength against the pure bending the strength against the normal force. is shown by the section modulus. The The significance of the tibia shaft has maximum section modulus of the mid- been discussed by many investigators. As shaft of the tibia appears at the posterior shown by the correlation coefficient in side which is in the direction of the Table 4 and in Fig. 2-1, the flat tibia is not maximum moment of inertia. It is because absolutely strong. The index of the cross- the anterior side is pointed and has a large section of the tibia shows the ratio of the height from the neutral axis compared strength in the antero-posterior direction with the posterior side. The minimum to that in the transverse direction. The section modulus is at the lateral side index of the cross-section of the fibula also where it is high because of the existance shows the ratio of the strength between of the crista interossea. Usually the lar- the principal axes. The bending force ger the moment of inertia is, the greater acting on the tibia is mainly in the antero- is the section modulus on both sides of posterior direction because of the move- the axis. For simplification, in this study ment of the muscles and articulation. The the moment of inertia is used to show the strength in the antero-posterior direction resistance against the bending. will be more important than that in the The bone is heterogeneous and aniso- transverse direction. On this point, the tropic. An experimental study is necce- flatness would be regarded as a suitable sary to clarify the mechanical properties form of the tibia. of the bone. The strength of the bone, The strength of the tibia correlates with however, could be shown to a certain deg- that of the fibula. The strength of fibula ree by the shape of the cross-section in is not related closely with the body size. this study. The variation of the properties of the fi- bula is relatively large as seen by the 332 T. KIMURA

SUMMARY ACKNOWLEDGMENT

The area, the moment of inertia of area This study is indebted to Professor Ta- and the polar moment of inertia of area dahiro OOE, Associate Professor Ichiyoh of the cross-section at the middle of the ASAMI and Dr. Toshiro KAMIYA of the human lower leg bones were examined to Department of Anatomy, Faculty of Me- learn about the strength of the bone from dicine, the University of Tokyo for allow- the viewpoint of the strength of materials. ing and assisting me to examine the recent The specimens were obtained from twenty- specimens. Thanks are also expressed to two recent Japanese males, twenty recent Mr. Hisao BABA and Mr. Yasuo FUKUSHIMA Japanese females and seventeen Jomon ma- of the Department of Anthropology, Facul- les. ty of Science, the University of Tokyo for The results are shown in Tables 1 to 4 their assistance of this study. and Fig. 2 and summarized as follows : REFERENCES 1) The age of the specimens does not cor- relate with the area, with the moment of AOJI, O., MOTOJIMA, T, and BANDO, T., 1959: On the effective sectional areas and maximum inertia nor with the polar moment of in- compressive loads of of human long ertia ; 2) The bones of the female have bones. J. Kyoto Pref. Med. Univ., 65: 979-983 distinctly a small area, moment of inertia (Japanese with English summary). and polar moment of inertia compared FRANKEL, V. H, and BURSTEIN, A.H., 1965; Load capacity of tubular bone. Biomechanics with the bones of the male; 3) There will and Related Bio-Engineering Topics (Ed. R. be a tendency that a large body mass is M. KEN E DI), pp. 381-396. Oxford. associated with a large strength of the JERNBERGER, A., 1970: Mesurement of stability bone ; 4) The shape of the cross-section of tibial fractures. Acta Orthop. Scand., Sup- with a wide area is not similar to that pl. No.135. KIMURA, T., 1966: An experimental study of with a small area ; 5) The strength of the the form of the human tibia from the bio- fibula is much smaller than that of the mechanical point of view. J. Anthrop. Soc. tibia and the deviation of the properties Nippon., 74: 119-227. of the fibula is greater than that of the KNESE. K-H., HAHNE, 0. H, and BIERMANN, H., tibia ; 6) The index of the cross-section 1954: Festigkeitsuntersuchungen an mensch- lichen Extremitatenknochen. Gegenbauers at the middle has no direct correlation Morph., 96: 141-209. with the strength of the bones nor with 厚 生 省 公 衆衛 生局 栄 養 課(編),1969:国 民 栄 養 の現 the curvature of the tibia, and the index 状,昭 和42年 度 国民 栄 養 調 査成 績.東 京. of the tibia shows a ratio of strength be- (Nutrition Section, Public Health Bureau, tween the antero-posterior and transverse Ministry of Health and Welfare, The report of the national nutrition survey in Japan, directions ; and 7) The principal axis at 1967. Tokyo. In Japanese) the mid-shaft of the tibia is parallel to PAUWELS, F., 1948: Die Bedeutung der Baup- the maximum antero-posterior diameter. 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. 351. 1950: Die Bedutung der *, Muskelkrafte YAMADA, H., 1970: Strength of biological ma- fur die .Regelung der Beanspruchung des terials (Ed. F. G. EVANS). Baltimore. Rohrenknochens wahrend der Bewegung der (Received June 3, 1971) Glieder. Z. Anat. Entwickl. Gesch., 115: 327-

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.