Chapter VI: Calculation of the State of Stress in the Femoral Neck

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1 Acta Radiologica SSN: (Print) (Online) Journal homepage: Chapter V: Calculation of the State of Stress in the Femoral Neck To cite this article: (1957) Chapter V: Calculation of the State of Stress in the Femoral Neck, Acta Radiologica, 47:sup146, 66-94, DO: / To link to this article: Published online: 14 Dec Submit your article to this journal Article views: 339 Full Terms & Conditions of access and use can be found at

2 CHAPTER V CALCULATON OF THE STATE OF STRESS N THE FEMORAL NECK When studying mathematically the strength of the femoral neck, was obliged to idealize as follows. The calculations were limited to the stresses in one cross-section of the femoral neck midway between points 0 and H (see fig. 1). This cross-section was approximated to an elliptoid ring in accordance with fig. 24. The resultant of forces that act upon the femoral head always passes through its center. According to HRSCH and BRODETT (1956) the trabecular system plays a relatively subordinate role in the strength of the femoral neck as a whole.-n the present calculations the influence of the trabecular system was taken into account in such a way that the thickness of the cortical shell was increased somewhat (about 20 per cent). The material was assumed to be isotropic, and Hooke s law was regarded as valid. Having regard to the above assumptions, the results of the calculations will also be approximate. Hence the ultimate strength of the femoral neck is difficult to evaluate, not least in that the calculations were concerned only with a single cross-section. These calculations nevertheless brought out entirely new aspects of problems which had not previously been possible to analyze or explain satisfactorily. n my own opinion, this was the foremost result of the analyses reported here. This view is founded upon, inter alia, the conclusions that are summarized on page 95. Two elementary cases were analyzed: (a) Forces acting upon the femoral head in a plane through the cervical axis and perpendicular to the principal plane. (b) Forces acting upon the femoral head in the principal plane. n addition, a more general case is discussed: (c) A fall on the hip. Forces in a Plane Through the Cervical Axis and Perpendicular to the Principal Plane General Analysis The femoral head is assumed to be subjected to a force R, acting in a plane at right angles to the principal plane, in accordance with fig. 25. n a cross-section through the middle of the femoral neck, the following forces must then act:

3 P = RH cos 5 T = R, sin 6 L Mx = - R, sin 5 2 Mu = ~R,cos~.* (7) 67 The following normal stresses can be calculated for e.g. point 0: P cos 5 0 =----R P A- HA evidently From the above equations can be computed as a function of angle 5, if L = 40 mm and other data are inserted according to page 65. The normal stress can be similarly calculated for points As regards the shearing stress due to shear force T, only a very rough estimate can be made. Disregarding the torsional effect of the shear force (see page 63) and assuming the coefficient v to be 2.5 (cf. page 65) we obtain, according to equation (6), for sin 5 sin 5 -=v-- z RH A A The normal and shearing stresses calculated in the above way are plotted diagrammatically in fig. 26. From this diagram it emerges that the femoral neck, as is natural, will be under the greatest load, with regard to both tensile and compressive and shearing stresses, if angle E &z 90 degrees; i.e., if force RH acts at right angles to the cervical axis. The use of the diagram in fig. 26 is illustrated e.g. on page 76.

4 Qo r. J Fig ' Femoral Neck Fractures Due to Muscle Force of the External Rotators Smith (1953) considers that if medial fractures of the femoral neck can be produced intravitally by external forces, then similar fractures should be duplicable on specimens.-n experiments on 115 specimens he sought to duplicate an external trauma by applying static and dynamic forces either in the vertical or in the

5 lateral direction-in the latter case, that is, against the trochanter from the side. n no case did he obtain a medial fracture. Smith drew the corollary of this total failure and thought that medial fractures had a muscular etiology. n his view, the external rotators in the hip may give rise to fractures, provided that normal external rotation is prevented. This may occur if the foot is fixed against the ground by the weight of the body. Here, Smith considers that muscle contraction will give rise to a retroflexion fracture. He demonstrates his theory with an experimental series having the arrangement shown in fig. 27.-Specimens were fixed in a U-shaped device; static and dynamic forces were then applied in the direction of Posterior the arrow. Thirty-three experiments were conducted, each of which resulted in medial fractures-a finding which Smith contrasted to the aforementioned results. n point of fact, an experiment as shown in fig. 27 does not demonstrate the hypothesized course of events; moreover, it is most unlikely that any other type of fracture than medial could have been produced with this experimental arrangement. Smith s experiments are nevertheless interest- Fig. 27. Principle of Smith s experiing in that they shed some light on the bending mental determinations of the strength of the femoral neck. strength of the femoral neck under static and dynamic forces. The static force producing fractures thus ranged from 500 to 2,400 lbs (227--1,090 kp) and Smith calculated a mean of 900 lbs (410 kp). The standard deviation was very large and the mean was based on twenty observations. Knowing the bending strength of the femoral neck, it is worthwhile returning to Smith s line of thought and attempting to calculate the muscle force required in order to overcome the bending strength. For if this muscle force lies within reasonable limits, then some consideration will have to be given to the relevant mechanism of origin. Smith s hypothesis is illustrated by fig. 28, where the force exerted by the rotator muscles is represented by force a, which under normal conditions rotates the femur backward about axis b. n so doing, the lateral part of the trochanter describes a movement along arc b,. On the assumption that normal rotation is possible, component c will have a torsional effect and component d will act as a compressive force in the direction of the cervical axis, pressing the femoral head into the acetabulum. n Smith s opinion, no fracture-promoting force exists under these conditions. f, however, the foot is fixed against the ground by the body s weight, normal rotation cannot occur. Here muscle contraction, according to Smith, will give rise to a bending moment tending to bend the femoral neck forward, i.e., to produce 69

6 70,Transection of Axis of femoral shatt External rotary muscle spine ~v lschial tuberosity Fig. 28. Effect of the muscle force on the femoral neck according to Smith. a retroflexion fracture. n this case, Smith says it is component d that causes fracture by a tendency to rotate about axis e and form a forward-bending moment that he indicated as arc el. Smith s reasoning is, in essential respects, ob- scure and incorrect.-for this reason found it advisable to study the significance of the external rotator muscles in the mechanism of femoral neck fractures. therefore analyzed a few different cases of loading and viewed a relevant example in relation to Smith s experimental findings. This enabled me to check the plausibility of the assumption that muscle forces in normal subjects are capable of producing femoral neck fractures. n the following considerations have ventured to idealize the state of affairs as shown in fig. 29. Muscle force Q is acting on the intertrochanteric crest, and is assumed to have its line of action in plane c-c, b-b. Axis b-b is perpendicular to diaphysial axis a-a and to cervical axis c-c, which in Fig. 29. turn forms angle u with the diaphysial axis (see also fig. 1). n the following the conditions will be studied for bending of the cervical neck in plane b-b, c-c. 1. Loading According to Smith s Theories for Fractures Smith claims that muscle force Q will bend the femoral neck forward if the foot is fixed against the ground, thus preventing the leg from rotating. This theory, interpreted consistently, might be visualized as in fig. 30. The fixation

7 of the foot against the ground can be replaced by a moment along axis a-a (fig. 29), the component of which along axis d-d in fig. 30 is denoted by M, (d-d is perpendicular to plane b-b, c-c). Equilibrium will now be possible only if the femoral head is affected by a force RH, having components R, and R2. From fig. 30 we accordingly obtain: Ri = Qi R, = Q29 i.e., -t=p and for a section through the middle of the femoral neck: L M = R,* / -E - -C t! L Fig L 2 Since bending moment M has the direction indicated in fig. 30, the femoral neck in this example will be subject to backward bending and not forward bending as Smith arbitrarily assumed. As regards Smith s account of the action of the external rotators, would also call attention to the following considerations: Smith claims that component d (see fig. 28) has no external rotational effect. This is erroneous; d has a torsional moment with respect both to point b and to point e in Smith s diagram. f this were not so, then theoretically the situation might arise in which the femur, in an internally rotated position where force a is parallel with the cervical axis, could not be rotated back into place. A detailed analysis of Smith s load example thus shows that backward bending of the femoral neck is obtained which, in medical terminology, implies that the femoral head will be in anteflexion. We know empirically that such displacements are uncommon, and this militates against the mechanism of origin described by Smith.

8 72 t is remarkable that a number of authors have accepted Smith s hypotheses uncritically, both as to the direction of bending of the femoral neck and as to other demonstrable errors. 2. Loading with Friction in the Hip Joint t has been assumed here that friction in the hip joint will give rise to a friction force which may counterbalance muscle force Q (see external fig. 31). rotators This implies it would that be possible by tensing to lock the the femoral head in the acetabulum. This assumption has been considered quite implausible. &r 0 Fig Loading with nfluence of nertial Forces from the Femur A further hypothetical case of loading is shown in fig. 32. The muscle force and the reaction from the pelvis are denoted, as before, by Q and RH respectively. Under the influence of these forces alone, the femur would promptly rotate about the center of the femoral head, even if Q were quite insignificant. However, the femur and soft parts surrounding it offer, by virtue of the mass they represent, a certain amount of resistance to any movement-in this case to a rotary movement in the acetabulum. This resistance is represented by force F. Since F is by nature an inertial force, its magnitude is dependent partly upon the mass to be accelerated and partly upon the magnitude of the acceleration. The greater the mass to be accelerated, the slower will be the rotary movement. t is unnecessary in this connection to introduce any special assumptions regarding Fig. 32.

9 the magnitude of the mass; the only essential postulate is that the muscle force acts so instantaneously as to leave no time for major rotation of the femur or pelvis before the femoral neck has fractured. Moment equilibrium with respect to point m in fig. 32 makes it necessary for the line of action of RH to pass through m; 73 h where tg E = - L With regard to point 0, we obtain the equation that is to say, for a section through the middle of the femoral neck the following equation is evidently valid: The action of the bending moment, would emphasize, is now such that it seeks to bend the femoral neck forward, in contrast to the example in fig. 30, where the action is opposite (the direction of M is reversed). 4. Combination of 1 and 3 The bending moment, as pointed out, has directly opposite directions in cases 1 and 3. Hence it is worthwhile investigating a case where we have both a moment M, and an inertial force F. For this combination, reference should be made to fig. 33, from which we obtain Fig. 33. ' - L -i 2

10 74 and n this connection, interest attaches only to the equation for bending moment M in the femoral neck, which can evidently be written alternatively as follows: Q2 *h By introducing Md = 0 into the first member of this equation, we get M = 2 i.e., a return to case 3. By introducing F = 0 instead into the second member, QiL we return to case 1, i.e., M = --. Every conceivable combination of M and F 2 can evidently be represented by a straight line in the manner shown in fig. 34. Fig. 34. t seems plausible to assume that in practice we are here concerned neither with the clear-cut case 1 nor with case 3; rather, we should reckon with a combination of the two. How this combination must be made in order to match the actual conditions cannot be decided; ie., it is not possible to specify exactly the final point on straight line A-B. Yet it should be observed that whatever com-

11 bination is postulated, the bending moment will always be numerically less than that in the respective clear-cut cases. n particular, if it be assumed that F = Q1, i.e., M, = Qz - h, then the bending moment in the femoral neck M = 0; that is to say, there will be no tendency for the femoral neck to bend either backward or forward. ntroduction of the following assumed values further yields Qz. h M=-- 2 M=--- h = 25 mm L = 40 mm /3 = 30" cos (2- = 11 - Q kp-mm (load example 3). 2 Qi. L 2 --Q sin * Q kp-mm (load example 1) that is to say, the two clear-cut bending moments are of approximately the same magnitude but have opposite directions. 5. Smith's Experimental nvestigations of Femoral Neck Fractures Smith, as mentioned in the foregoing, produced experimental fractures of the femoral neck as shown in fig. 27, and which may thus be visualized as in fig. 35, from which we obtain: K R -- H - 2 (C = goo) 75 K T,pJ - ' L ' 2 Fig. 35.

12 76 Possibility of Fracture Due to Muscle Force of the External Rotators From the considerations enumerated under (4) above, it is evident that the clear-cut case 3 (fig. 32) is the most unfavorable one with respect to a bending moment M. This load also has a tendency to bend the femoral neck in the common direction, i.e., forward, and for these reasons it is taken as a basis for the following analyses. For case 3 we have, according to the aforegoing: RH cos E = Q cos h tgt=- L nsertion of h = 25 mm, L = 40 mm, and /3 = 30 degrees gives = 32" RH = 1.02 Q From the diagram in fig. 26 it emerges that the normal stress is greatest at point 0; and here the following is valid: i.e., kp/mm RH kp ' U ~- - u = * 0.02 Q (T = Q kp/mm2 n Smith's experimental studies of fractures 5 = 90" K R -- H- 2 t will be seen from the diagram in fig. 26 that the normal stress at point 0 is in this case a tensile stress numerically equal to the compressive stress at point 0. Since the tensile strength of the material is lower than its compressive strength, point 0 would be the most critical one, with regard to fracture, if normal stresses alone were taken into account. However, am disregarding this factual state of affairs and assuming that the normal stresses at points 0 and 0 are about equally critical for fracture. For point 0 we then obtain, according to the diagram: i.e., 0. kp /mm kp -- - RH (T = -- - K = K kp/mm2 2

13 Assuming that the maximum normal stress is critical for fracture, the condition required for muscle force Q to be equivalent, in the fracture sense, to force K is evidently: (T = Q = K kp/mm2 77 i.e., Q = 0.57 K ~- With K = 410 kp (see page 69) the muscle force required will be 9 = 235 kp n the foregoing calculations it has been shown that and 0 are those where the normal stress is greatest. t is evident from the diagram in fig. 26, however, that even points 0 are subjected to (fairly) substantial shearing stresses, and point 0 also to a substantial normal stress. A more accurate evaluation of the ultimate strength here would necessitate combining the stress with the aid of one of the hypotheses mentioned on page 64. n fact it can be shown that a more exact calculation, based on the combined stress, will give about the same factor 0.57 between Q and K. Experimental Determination of the Muscle Force of the External Rotators t is impossible to say without further ado if the muscle force calculated in this way is plausible, i.e., whether a human being is capable of producing a force of this magnitude. have been unable to find in the literature any data concerning the force of the external rotators with the femur in different degrees of rotation. n order to gain some idea of this force, an investigation was conducted with the experimental arrangements illustrated in fig. 36. The first experimental series consisted of 11 wrestlers between the ages of 20 and 30 years. They were tested both while recumbent and while sitting; in each of these postures the femur was studied without rotation and in two different degrees of rotation, namely: internal rotation of 30 degrees external rotation of 30 degrees. The degree of rotation was measured as the angle formed by the lower leg with the vertical line, the lower leg being kept at right angles to the femur. The other experimental series consisted of 19 women between the ages of 21 and 47 years. They too were examined both recumbent and sitting, but here the femur was not rotated. During the examination the other leg was fixed in the same position as the tested leg in order to prevent compensatory movements of the

14 78 Fig. 36. Experimental determination of the muscle force of the external rotators. body. Careful checks were made to ensure that the line which had been stretched between the ankle and the scale was always perpendicular to the lower leg. The scale, which had been placed at my disposal by Lindellvdgar i Stockholm AB, allowed, on maximum loading, a change only of a degree or so in the rotation position of the femur. For calculation of the moment, the lever was measured in each individual case. t was defined as the distance between the lower border of the patella and the line s attachment above the ankle. The mean values of the moments computed in these experiments are tabulated below. The figures in the second column denote the rotation position of the femur. When the foot is moved medially and the angle formed by the lower leg with the vertical line is measured on the medial side, the femur will thus be in external rotation.

15 79 Subjects Position of Femur Sitting Moment M, Recumbent Wrestlers 30" internal rotation 5,716 kp-mm 6,882 kp-mm 0" 5,368 kp-mm 5,674 kp-mm 30" external rotation 3,658 kp-mm 3,856 kp-mm Women 0" 3,101 kp-mm 3,045 kp-mm From these experimentally determined values the corresponding muscle force can be computed (see figs. 29 and 30). Muscle force Q is first resolved into its components Q1 and Qz, both of which lie in plane b-b, c-c. Here, Q1 has the following moment with respect to axis g-g: (for angle s, see page 38). Ml = Q1 - L sin s Component Qz has, with respect to the same axis, moment but Mz = Qz - h sin s Q1 = Q sin B Qz = Q cos B Mv= Q sin s (L sin B + h cos b) ntroduction of the following values: gives h=25 mm L=40 mm B = 30" s = 48" Q = * M v kp. The M, values set forth in the above tabulation thus correspond to the following values for muscle force Q:

16 80 Subjects Position of Femur Muscle force Q Sitting Recumbent Wrestlers 30" internal rotation 183 kp 221 kp 0" 172 kp 182 kp 30" external rotation 117 kp 124 kp Women 0" 100 kp 98 kp The calculations of the strength of the femoral neck indicated that the muscle force required to produce a cervical fracture amounted to about 235 kp. Comparison with the values tabulated above yields the following results: The possibility of femoral neck fracture due to a muscle force Q seems to be extremely slight in normal cases, for not even the muscular force that a wrestler can accomplish is sufficient to produce a fracture. f the comparison is based on the muscular force of which a normal person is capable and which may be estimated at about 125 kp, the factor of safety will be 2. n my opinion, however, the factor of safety will in practice be at least 3 or 4, for the following fundamental reasons, among others: (a) The comparative calculations were based on case 3, which in all respects was the least favorable. All other equally conceivable loads are tantamount to a requisite muscular force exceeding that computed for case 3. (b) f Q = 125 kp be introduced into equation (ll), page 76, we obtain: cr = * 125 = kp/mm2 which constitutes very slight stress by comparison with the compressive strength of the bone tissue, which has a minimum of 12 kp/mm2 (see page 57). (c) With the procedure employed in Smith's experimental investigations, a minute flaw was obtained at the point where force K acts. n reality no such flaw occurs, of course, when muscle force Q acts, and Smith's experimental results may accordingly be directly misleading. (d) Experimental investigations as described below showed that muscle force Q should amount to about 700 kp if a possibility of fracture is to exist. According to case 3 (page 76), RH = 1.02 Q i.e., RH and Q are approximately equal. n these experimental investigations specimens of the upper end of the femur were fixed at the shaft, after which a static force was applied against the femoral head in the same direction as force R,. Angle E was in no case less than 32 degrees; that is, the line of action of RH passed through or somewhat dorsal to the inter-

17 trochanteric crest. The static load was gradually increased, but the ultimate stress was reached only in one of ten specimens tested, due to difficulties of fixation. See the tabulation below Woman Man Man Woman Woman Man Woman Woman Woman Man 78 years kp 80 H kp 64 )) kp 72 )) kp Shaft fracture 48 H kp 56 H kp 71 o kp 70 o kp 75 o kp 61 o kp Mean 696 kp A fracture was obtained only in case 4 and was localized to the shaft. n no case was any femoral neck fracture produced by fhe loads applied. On several occasions angle & increased, under increasing loads, due to inadequate fixation, and in a few cases it was estimated at about 40 degrees. The values tabulated above are therefore to be regarded as minimum ones. nsertion of R, = Q = 700 kp in equation (ll), page 76, gives CJ = = kp/mm2 This constitutes a stress which, having regard to the compressive strength of the bone tissue, is sufficient to produce a fracture (normal values, according to page 57: kp/mm2). f these experimental findings are taken as a basis for evaluation of the possibility of fracture due to the muscle force of the external rotators, the factor of safety is found to be 5-6. The experimental determination of the muscle force as described on page 77 relates to the voluntary muscle force. The involuntary muscle force probably can reach higher values, but even if it is estimated at twice the voluntary muscle force there will still be an ample margin of safety. Forces in the Principal Plane General Analysis n this case the conditions can be defined as in fig. 37, where the force acting on the femoral head is R, and the forces in the section are P, T and M,, for which the following are valid:

18 82 4x Fig. 37. P = Rv cos p T = R, sin y The normal stress, at an arbitrary point, due to P and M, will evidently be: P cos y - Rv *- up=--= A A f we here insert L = 40 mm and x = - 6 for points 0 and 8, x = - (a, + 6) for point 0, x = a2-6 for point (3, and other data according to page 65, we can calculate at points 0, 0, 8 and (3 as a function of angle y. Shearing stress z due to T is calculated from T t = -y- A where v = 0 at points 0 and v = 2.5 at points 0 and 0; i.e., at these points we have: t _- sin p RV A The stresses computed in this may are introduced diagrammatically in fig. 38, where negative values for p are also included (later on, angle 5 is used for defining

19 83 \ "\,

20 84 the direction of R,, instead of p. Angle is therefore included in the diagram in accordance with equation = 90" + y). The use of the diagram is illustrated by examples later on in this paper. Conditions Associated with Walking on Level Ground When a person stands still on both legs, the weight of the body, G, may be assumed to be equally distributed, on the whole, between the two legs. About two-thirds of the body weight is referable to the trunk, head, arms, etc.; and if normal muscle Fig. 39. Load on the femoral head when standing on the right leg. Fig. 40.

21 tone be disregarded, each femoral head will evidently bear about one-third of the total body weight. Entirely different conditions arise when a person stands on one leg or walks on level ground. n standing on one leg, the pelvis and trunk are prevented from descending-i.e., rotating in the hip joint-by muscle force N, which has attachments on the upper end of the femur and on the pelvis; see fig. 39. The non-weight- G bearing leg is assumed to be hanging free, and can then be represented by force - 6 in accordance with the figure. The conditions for equilibrium of the pelvis will be: N sin u = R, sin / G + - G + N cos u = R, cos / N *c=-g.b +-G*2b 3 6 PAUWELS (1935) reported the following mean values for a, c and b: a = 21", c = 40 mm and b = 85 mm nsertion of the values gives N = 2.13 G R, = 2.92 G /3 = 15.1" n walking, a load arises which has not only the above-mentioned static component but also a dynamic component. Pauwels analyzed this state of affairs in detail and showed that the dynamic component served very substantially to increase load R, on the femoral head. He stated that R, in this instance amounted to about four and a half times the body weight (i.e., an approximately 50 per cent increase as compared with the above), and, moreover, he estimated angle j3 at about 16 degrees. t seems quite clear that a further increase of R, might arise under certain unfavorable conditions, which nevertheless are to be regarded as fully normal (for example, ascending and descending stairs, carrying heavy objects, sudden movements of the body, etc.). The assumption is by no means implausible that under such conditions R, might amount to six or possibly eight times the body weight. t is now worthwhile calculating that angle y which R,, directed against the femoral head, forms with the cervical axis. For the femoral position shown in fig. 40, y is obtained from the equation: $9=u-p-v-p 85

22 86 From Pauwels' data concerning the femoral position in subphase 16 of locomotion, d and Hg can be determined at d = 50 mm and Hg = 760 mm. (These values are also approximately valid for subphase 14, when the load on the femoral head reaches its maximum.) We thus find: d 50 tg 91 = - = - = Hg 760 i.e., q= 4" By further inserting u = 54" (see page 29), p = 6" (see page 36), = 16" (see above) we obtain: y = 28" f this value for y is also taken to be valid for subphase 14 where Rvismaximum, and we disregard the fact that R, does not lie exactly in the principal plane (the deviation in this respect is of very little importance), then fig. 38 can be directly used for determining the stresses in a section through the femoral neck. The following interesting findings emerge: CT (a) At point 0, i.e., on the superior aspect of the femoral neck, --roand RV hence (T 0, regardless of the magnitude of R,. This explains why the compact bone on the upper aspect of the neck is usually very thin. CT (b) At points 0 and 0 we obtain a compressive stress - of relatively small RV t magnitude. Nor is shearing stress - appreciable at these points. RV (c) At point 0 we find a compressive stress of the same magnitude as the shearing stress at points 0 and 0. Of considerable interest is to insert a mean value for R, that will be valid for walking or the like. Let us assume that G = 60 kp and R, = 5 G, i.e., R, = 300kp. We then obtain the following: Point 0 Points 0 and 0 (T = = kp/mm2 (T = e300 = kp/mm2 t = = kp/mm2 Having regard to the ultimate strength of the bone tissue (see page 57), it may be concluded that the factor of safety for this type of load is fairly great, perhaps 3 to 5. Purely Vertical Loads There have been a large number of experimental investigations with the femoral head subjected to loads directed more or less parallel with the diaphysial axis.

23 Still disregarding the fact that this load does not lie exactly in the principal plane, the state of stress can be read from fig. 38, with angle y = u = 54". The following interesting conditions are found: At point 0, i.e., on the superior surface of the femoral neck, there is a substantial tensile stress of a similar order of magnitude to the compressive stress at on the inferior surface. Points 0 and 0 are free of tensile and compressive stresses, but on the other hand are subject to considerable shearing stresses. HRSCH and BRODETT (1956) conducted investigations with strain gauges attached round the femoral neck in a section which probably approximated that in my experiments. The femoral head was subjected to vertical load (y = 54") up to about 100 kp. Judging by the results that were reported, the authors obtained elongation of the upper surface of the femoral neck, coinciding with an approximately similar compression of the inferior surface. The strains, moreover, were proportional to the magnitude of the load; that is to say, Hooke's law can be assumed to have been valid.-the strain is accordingly a direct measure of the stresses at the different points, and the authors' results with regard to the stress distribution fully accord in principle with my interpretation of the stress diagram in fig. 38. n this connection the following considerations are, in my view, worth observing. According to the aforegoing, a shearing stress z alone acts at points 0 and 0. Yet according to theoretical stress calculations, there must simultaneously be a shearing stress of equal magnitude that forms an angle of 90 degrees with the former one and which therefore has the same direction as the cervical axis. These two shearing stresses give rise to tensile and compressive stresses a, and cr2 respectively (principal stresses), acting upon a small element as shown in fig. 41. Since the tensile strength of the bone tissue is lower than its compressive strength, there is some indication, at least at point 0, that the fracture direction will be perpendicular to tensile stress cr, i.e., almost vertical. t is well known that in experimental investigations with vertical loads on the femoral head, vertical fractures almost invariably result, and the explanation of this conceivably lies, to some extent, in the aforegoing considerations. Fig

24 88 From the magnitudes of the breaking loads reported in the literature, we obtain an approximate mean value of 800 kp for R,, and according to fig. 38 the corresponding tensile stress at point 0 will be (r = = 10.4 kp/mm2 This stress is of the same order of magnitude as the tensile strength of the material, and the agreement is therefore satisfactory. General Considerations Falls on the Hip By a fall on the hip is usually meant that the body falls in such a way that the trochanter takes the impact. This view will be a premise in the following discussion. When, in a person who falls, the trochanter strikes the ground and the fall is arrested, it will be subject to a reaction force F that is directed from the ground against the trochanter. With respect to the femur this reaction force corresponds partly to a force R which is directed from the pelvis toward the femoral head, and partly to a force which depends on the weight and momentum of that part of the leg which is located distal to the trochanter. The direction of these two last-named forces is opposite to that of F, and the sum of them is approximately F. n addition to these vertical forces it is necessary, in a fall on the hip, to take into account those horizontal forces which result from friction with the underlying surface. These forces will be especially appreciable in oblique falls. n the following it will be assumed for the time being that the direction of fall is vertical and that the horizontal forces may be disregarded. Detailed study and analysis of these forces and their effect on the femur, requires knowledge of the latter s position in relation to the underlying surface. A fairly good idea of this can be obtained by studying different ways of falling and analyzing the positions that may be assumed by the femur. Position of Femur in Relation to the Underlying Surface The possibilities of falling in such a way that the trochanteric region will strike the ground are determined and limited partly by the range of movement in the hip joint and partly by the topography in relation to surrounding soft tissues and the pelvis. t will here be assumed, for the time being, that antetorsion angle t = 0, and hence that the antetorsion plane coincides with the frontal plane of the femur. When any part of the trochanter strikes the underlying surface, the femur s position in relation to that surface can be defined by angles e and r in the manner shown by the determinations described in Chapter 11, page 36. When r = 90

25 degrees, the lateral surface of the trochanter will constitute the point of support. When r exceeds 90 degrees the point of support will be anterior to, and when it is less than 90 degrees, posterior to, the lateralmost point on the trochanter. When e = 0 the femoral shaft will be horizontal. Angle e can never be greater than angle u. The various positions which the femur may assume in falls on the hip have been studied both in man and in human pelvis-femur-tibia specimens. n discussing various types of falls, importance attaches e.g. to the position of the femur with respect to angle r. Even a cursory study of the possible types of falls shows that r cannot be very much greater than 90 degrees. Values of up to degrees may conceivably occur in exceptional cases, but higher values are inconceivable. The reasons for this lie partly in the limited range of extension in the hip joint, and partly in the powerful muscles that invest the anterior aspect of the trochanteric region. With flexion in the hip joint, r may exceed 90 degrees only if there is coincident pronounced external rotation. Even in this position, however, r will be only slightly more than 90 degrees. n falls on the hip there is generally a history of slipping or tripping, the leg sliding away from or folding beneath the patient, who thus falls more or less directly upon the hip. Probably there is often coincident flexion of the knee, and this is important because any tendency of the femur to rotate will thus be reduced. f e = 0 and the knee is flexed at right angles, then r = 90 degrees and the lateralmost part of the trochanter will be the point of support. f e from this initial position is assumed to increase when the foot remains on the ground, then a position will gradually arise where r = 0. All those positions which the femur assumes when r decreases from 90 degrees toward zero, and the foot still rests on the ground, can readily be explained in connection with a fall on the hip. The lesser the flexion of the knee joint, the lower will be the e values required when r decreases from 90 degrees to zero. t may be objected that there is little possibility of falling in such a way that r = 0; for this presupposes that the pelvis too will come into contact with the ground-sometimes, perhaps, before the trochanter. When e is maximum, i.e., equal to u, the tip of the trochanter will be the point of support. n this position the femoral head will be somewhat above the underlying surface, and the possibility is not precluded that a femoral neck fracture may result, although it seems most likely that in this position there will be a fracture through the tip of the trochanter. have considered that under the given assumptions the femur's position in relation to the underlying surface may be determined by the following values: r=o+90" e = 0 +the value for angle u 89

26 90 Fig. 42. Plane TAB lies in, and plane TAC perpendicular to, the principal plane. The two angles can be combined in different ways, and hence the variations in position will be great. n the above schematic presentation the principal plane has not been taken into account. n discussing the stresses that arise in the femoral head and neck when a person falls, it is advisable to take a different starting point than the frontal plane of the femur; the natural approach is the principal plane, which constitutes the plane of symmetry of the femoral head and neck. n the experimental investigations, the femoral position was determined as in fig. 42. Here angle 5 between the cervical axis and the underlying surface is measured instead of elevation angle e. This constitutes an advantage in that the magnitude of u need not be taken into account. Angle 6 is measured instead of r, since the former is a criterion of the inclination of the principal plane relative to the underlying surface. Forces Acting on the Femoral Head When, in a fall on the hip, the trochanteric region strikes the underlying surface and the fall is arrested, forces arise which in various ways produce stresses in the upper end of -the femur. A force R, produced by the weight of the body and the

27 kinetic energy acquired by the body in falling, will always be directed toward the femoral head. t is thus a case of a dynamic force and, since the falling movement is arrested abruptly, R will be an impact force. The magnitude of this impact force cannot be calculated in the individual case, since a number of factors that should be taken into account are unknown. On page 108 the magnitude of the impact force is roughly estimated, subject to certain conditions and assumptions. Force R can be resolved into components, as shown in fig. 42: which falls along the cervical axis, i.e., corresponds to R, cos in fig. 25 ~_ and to R, cos y~ in fig. 37; which lies in the principal plane and is perpendicular to the cervical axis, i.e., corresponds to R, sin y~ in fig. 37; which is perpendicular to both of the above and corresponds to R, sin 5 in fig. 25. When a fracture occurs, force RTB, in view of its direction, will tend to give rise to varus-valgus displacement, and RT, to anteflexion or retroflexion displacement. The following definitions of displacement positions are used: Varus-valgus: Displacement in the principal plane, i.e., in plane TAB. Ante- and retroflexion: Displacements in a plane perpendicular to the principal plane and coinciding with the cervical axis; i.e., in plane TAC. Rotational displacements: Displacements in a plane perpendicular to the cervical axis; i.e., in plane TBC. n a fall on the hip the position of the femur, defined by angle e, may, according to an earlier analysis, vary between e = 0 and e = u. t is manifest that angle 5 at the same time will vary between 5 = u and [ = 0. t can never be appreciably greater than u. With the direction that R has in a fall on the hip, this means that no tendency to displacement in varus can arise through the influence of the fracturing force. - BOHLER (1938) introduced the designations abduction and adduction fractures. These terms, it is true, had existed previously, but since 1938 they have been generally accepted and widely employed. The phonetic similarity and the risk of confusion led NYSTROM (1938) to introduce the terms uazgus and uarus. Nystriim s nomenclature has been generally adopted in the Scandinavian literature, which in my view is quite correct for more reasons than the risk of confusion. The terms abduction and adduction fractures refer not only to the displacement position; they also indicate the mode of origin, and in this respect must be regarded as misnomers. Adduction fractures cannot, in any case, result from falls on the hip, and in my opinion these designations should be abandoned insofar as they refer to medial fractures of the femoral neck, at all events for the displacement positions. 91

28 92 t is theoretically possible with the aid of earlier equations to calculate the state of stress that may arise in a cross-section of the femoral neck with falls on the hip in different positions of the femur. Since calculations of this kind are exceedingly complicated, no attempt has been made at a more general analysis. Special cases with forces in and perpendicular to the principal plane, on the other hand, can be readily analyzed with the aid of figs. 26 and 38. f the fall is such that the load lies in the principal plane (i.e., 0 < 5 < 60 in fig. 38), the femoral neck will be much weaker than it is for the load directions corresponding to "walking on level ground" or "purely vertical load". f we seek to determine the magnitude of the breaking load on the basis of the stress at point 0, with e.g. [ = 30 degrees, we find that 0 kp/mm ~ RV kp which with 0 = - 13 kp/mm2 gives R, = 400 kp. That this force may very easily occur in a fall on the hip is evident e.g. from the calculations of the magnitude of the impact force on page 108. t emerges, moreover, from fig. 38 that the magnitude of the breaking load is dependent only to a slight degree on that of angle 5, and that the stresses in other parts of the cross-section may be quite substantial. f, however, the fall is such that the load lies in a plane perpendicular to the principal plane (which can be shown to correspond to 6O"<t<9O0) then, according to fig. 26, considerable stresses evidently may still arise in the abovementioned cross-section. f the force acting on the femoral head is approximately 400 kp, there will be a stress at point 0 in the interval 60" < < go", for which the following will be approximately valid (T = = - 10 kp/mm2 This stress, as will be seen, is close to the compressive strength of the bone tissue. There is no plausible reason to assume that the strength of the femoral neck will be appreciably greater in respect of other load directions associated with falls on the hip. From the aforegoing considerations it must not be concluded, however, that fractures will always arise in the cross-section defined; for it is not unlikely that the strength can be less in other parts of the upper end of the femur. - This was confirmed also by the experimental investigations, where fractures were obtained in the trochanteric and subtrochanteric regions. t seems quite reasonable, would emphasize here, that under some conditions torsional fractures of the femoral

29 shaft will occur instead of cervical fractures. This, too, was verified by the present experiments. The significance of twisting moments along the cervical axis has not yet been discussed in this paper. That such moments may be produced by the force directed against the femoral head is evident from the considerations adduced on pages 63 and 105. Further twisting moments may result from friction in the hip joint, which in turn depends on the major impact forces that may develop with a fall on the hip. f in a fall on the hip the trochanteric region takes the impact, then it follows that no part of the pelvis must reach the underlying surface before or coincident with the trochanteric region. At the impact (at all events in the initial phase) the pelvis will thus rest only against the femoral head. Theoretically it is conceivable that the pelvis will be balanced in an unchanged position on the femoral head, provided the center of gravity is directly above the latter. This state of balance is very unstable and, in practice, can scarcely occur. The deformation of the body leads to shifts, as a result of which the pelvis sinks toward one side in order to assume a more stable condition of equilibrium. (See fig. 43.) This movement of the pelvis means that the latter must rotate about the femoral head. f the center of gravity is located, from the outset, at the side of a vertical line through the femoral head, the pelvis will have a more unstable position and the tendency to rotate will be greater, since the force from the center of gravity will have a longer lever. 93 a b Fig. 43 a. n the fall on the hip the trochanter has struck the ground. Fig. 43 b. The final phase, in which the pelvis and distal part of the femur have sunk to the ground.

30 94 n a fall on the hip the center of gravity will usually be located fairly close to the vertical line through the femoral head. f the friction in the hip joint is sufficiently great, the femoral head will be subjected to a torsional stress that may conceivably be of some significance for the fracture mechanism. Experimental investigations into the friction associated with increased compression of the hip joint (see page 105) show that in rotation about the cervical axis the friction increases with the degree of compression; at first gradually, then at an increasing rate, and finally, at compressions between 500 and 700 kp, reaching such values that the rotary movement of the pelvis may cause substantial torsional stresses in the femoral head and neck. The direction of torsion in a fall on the hip appears to be such that it tends, in the main, to twist the anterior aspect of the femoral head upward. Force R is not the only significant factor in the fracture mechanism. f, at the moment of impact with the underlying surface, the femur is elevated-i.e., angle e exceeds O-the part of the femur distal to the trochanter will be affected by a force, perpendicular to the underlying surface, that will be dependent on the weight and kinetic energy of the femur. The point of application of the force is the femur s center of gravity. This force seeks to abolish the elevation, and since its lever is very long in comparison with that of force R, it is not impossible that the femur will sink towards the underlying surface. n so doing, the trochanter will serve as a fulcrum. The femoral neck and head will be forced upward against the acetabulum, and coincidently the head will describe a slight rotary movement. The compressive and friction forces in the hip joint will increase, and as a result the anterior surface of the femoral head will tend to be rotated upward (see fig. 43).