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Various Methods for Measuring the Geometrical Properties of the Long 人 類 誌.J. Anthrop. Soc. Nippon 90 (1):1-16 (1982) Various Methods for Measuring the Geometrical Properties of the Long Bone Cross Section with Respect to Mechanics Banri ENDO and Hideo TAKAHASHI Department of Anthropology Faculty of Science The University of Tokyo Abstract Owing to the recent development of various kinds of equipment for the measurement of irregular figures, papers dealing with the long bone cross section related to mechanics are increasing in the field of anthropology. However, there have been few explanations of the actual methods, equipments and equations for measuring these properties. This paper explains briefly the geometrical properties of the cross section in relation to mechanics as well as their actual use in functional morphology related to anthropology and shows various methods and equations for the actual measurement of these geometrical properties. Some of the equations presented in this paper have never or rarely been published before. for example, by KOCH (1917), PAUWELS INTRODUCTION (1950, 1954, 1968) and KIMURA (1971, 1974). Use has been made of circumference, Manual measurement, however, is very sagittal and transverse diameters, maxi- time-consuming work. Owing to the recent mum and minimum diameters etc. as the development of various kinds of equip- geometrical properties of the cross section ment for measuring irregularly shaped of long bones in historical morphology. planes, many researchers, such as MINNS These properties, however, are not availa- ct al (1976), LoVEJOY et al (1976), KIMURA ble for the analysis in biomechanics or (1980), MILLER and PURKEY (1980), BURR et functional morphology. The geometrical al (1981) and TAKAHASHI (in press), have properties of the cross section, such as begun to measure them. Nevertheless, few area, moment of inertia, position of cen- explanations of the methods and equations troid, principal axes, radius of gyration for measuring them with various equip- and section modulus are useful in asse- ments have been done, except those by ssing the complex relationship between PAUWELS (1968), by NAGURKA and HAYES structure and function of bone. (1980) and by KIMURA (1980). Moreover, The measurement of these geometrical they are rather specialized in using certain properties of the long bone cross section specific equipment. has sometimes been manually carried out, This report aims to explain briefly the Article No. 8141 geometrical properties related to mecha- 2 B. ENDO and H.TAKAHASHI nics as well as their application to bone The closed plane has a centroid, which morphology and to show various methods is designated as C in Fig. 1. Let the Car- and equations for using different equip- tesian coordinates have their origin at the ments in order to obtain the geometrical centroid, as shown in Fig. 1. The moment properties of the long bone cross section of inertia about the x axis is defined as related to mechanics in a more generaliz- Ix = *y2dA, ed form. Some of the equations have the same moment about the y axis as never or only rarely been described in ly=*x2dA published papers and even in textbooks on strength of materials. The methods desc- and the product of inertia as ribed in this report are limited to those Ixy =*xydA. based on the Cartesian coordinate system If the coordinates rotate, these moments which is easy to use with the various become larger or smaller. At a certain devices. angle of rotation the former two become GEOMETRICAL PROPERTIES OF simultaneously maximum and minimum THE CROSS SECTION RELATED respectively. Let them at this angle be TO MECHANICS designated as II and III, then The cross section of long bones, as shown in Fig. 1, is a kind of the closed plane. The area of the closed plane is defined as They are called principal moments of A = *dA. inertia. According to Fig. 1, they are also defined as II=*2dA III= *2dA The angle thus rotated is The rotated axes (I and II in Fig. 1) are called principal axes. The polar moment of inertia is defined as Ip =*r2dA. The r of the polar coordinate system can Fig.1. Cross section of a femoral shaft with be obtained from the Cartesian coordinate its centroid (C), coordinates through cent- system as x2*y2. Therefore, roid (x and y) and principal axes (I and II). Hatched area is integrated. Measurement of geometrical properties of bone 3 and also is fairly similar in shape. =II+IIIIp . When the external forces act on a beam, The radii of gyration about the x and reactions to cope with them occur in it. y axes are If this beam is virtually cut normally to its longitudinal axis to make a couple of The principal radii of gyration are cross sections facing one another, forces appear on each of the cross sections, also Let the maximum distance from the coupling with one another. These forces margin of the closed plane normal to the may be called the internal forces or the axis through the centroid be e. This exists cross-sectional forces. The internal forces on both sides of the axis. They are usual- are usually classified into three kinds: ly designated as e1 and e2 (see Fig. 3). The axial force (normal force), shearing force section modulus (also called the modulus and bending moment. The internal forces of resistance) is defined on each side of are the resultant of the forces continuous- the axis as ly distributed over the cross section. The force per unit area of the distributed force is called stress. The stress is either the It should be noted that the moment of normal stress or the shearing stress. inertia of the cross section may also be When a beam is acted on by the exter- defined even about any arbitrary axis which does not pass through the centroid. However, the moment of inertia takes a certain value which is proper to the cross section, when the axis passes through the centroid. Therefore, the term "moment of inertia" is generally used with respect to the axis through the centroid. MECHANICAL SIGNIFICANCE OF THE GEOMETRICAL PROPERTIES OF THE CROSS SECTION The relationship between geometrical properties of the cross section and their Fig. 2. A beam acted on by three external mechanical significance is based on the forces (P1, P2. and P3) in equilibrium as a beam theory. The beam means a slender simplified model of long bone acted on by but not too long bar, the cross section of joint forces and muscular force (top) and intensity diagrams of internal forces dist ri- which is always the same or gradually buted in the beam (bottom). The equilibrium changes along its longitudinal axis. equations are P1 cos *-P2 cos *- P3cos *=0, Therefore, the beam theory is approxi- P1 sin *-P2 sin *+P3 sin *=0 and l1P2 sin * mately applicable to the long bone which P3sin *=0. l 4 B. ENDO and H. TAKAHASHI nal forces, the internal forces are distri- buted over the whole length of the beam in a certain manner which is related to the condition of the external forces. Fig. 2 shows an example of the internal force distribution.. As concerned with a cross section at an arbitrary position on the beam, the axial force produces the normal stresses uni- formly distributed over the cross section. Designating the axial force as P and the normal stress as *, the intensity of the normal stresses is Fig.3. Cross section of a tibia for explanation of normal stress at dA in bending about the The bending moment also produces normal principal axis (I). stresses, but their intensity is not uniform. When the axis of the bending moment (M) and the principal axis of the cross section are the same, the stress at dA is in the case of Fig . 3. Intensity of stress depends on the distance between the axis and the dA. The maximum intensity of stress appears on each extremity of the axis as If the axis of the bending moment is in- clined counterclockwise from the princi- pal axis at an angle of * as shown in Fig. 4, the intensity of the stress at the dA Fig.4. Cross section of a tibia for explanation is of normal stress at dA in bending about an axis inclined from the principal axis. I and II, principal axes; *, inclination angle of the and the inclination of the neutral axis of axis of bending moment; *,inclination angle bending is of the neutral axis of bending; * and *,dis- tances of dA from the I and II axes respec- tively. Measurement of geometrical properties of bone 5 to the torsional force which also produces the shearing stresses, but this relation is limited to the approximately circular cross section (solid or hollow). GEOMETRICAL PROPERTIES OF BONE CROSS SECTION IN RELATION TO FUNCTIONAL MORPHOLOGY Bone is a part of the locomotor system. Therefore, a bone is acted on externally by joint forces and muscular forces during body movement or body support. These forces produce internal forces as already mentioned. Although the axial force is strong in the bone, the bending moment produces more intensive stresses which tend to concentrate to certain localities of bone. Thus the bending moment has usu- Fig. 5. Cross section of a tibia for explana- ally far larger influence on the strength tion of shearing stress at dA. I and II, of bone. Consequently, the moment of principal axes; b,breadth; *,height; bi and breadth and height at the level of dA; *,*i, inertia of the bone cross section plays a distance between dA and II axis; *, angle very important role in terms of the resis- between the II axis and the line from dA tance to bending.
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