On the mathematical analysis of stress in the human femur

J. Biomechunicr,

1972. Vol. 5. pp. 203-215.

ON THE

Pergamoa

Press.

Printed

in Great

Britain

MATHEMATICAL ANALYSIS IN THE HUMAN FEMUR*?

OF STRESS

E. F. RYBICKI, F. A. SIMONEN and E. 9. WEIS, Jr.+ Applied Mathematics and IMechanics Division. Battelle’s Columbus Laboratories, Ohio43201, U.S.A.

Columbus.

Abstract-A mathematical model for the behavior of the human femur is used to examine the effect of muscle forces on the resulting stresses and strain energy during a one-legged stance. The tesuits show that tension in the tensor faciae latae can very effectively serve to counteract bending stresses in the femur while also reducing the strain energy. It was found that the choice of mathematical model to represent the behavior of the femur is important. Stresses in the femur are calculated using both beam theory and a continuum theory in the form of a finite element computer program. The analyses include joint and muscle loadings. The results of the study show that beam theory is appropriate for the calculation of stresses in the shaft of the femur, but gives an inaccurate picture of the stress distribution in the regions of the greater trochanter and the femoral head as well as the areas of muscle attachment. In these regions a continuum theory is required. INTRODUCTION

of mechanical loads and resulting stresses in the femur has been the subject of a number of investigations. The classical work of Koch (19 17) encompassed both a detailed geometrical description of a femur and the calculation of stresses induced by force loadings assumed to occur during certain normal activities. Koch compared the principal stress lines with the natural trabecu: lar pattern in the femoral head, in connection with an evaluation of Wolff’s ( 1892) theories of bone architecture. In the photoelastic works of Pauwels (195 l), the femur was represented experimentally as a continuum and certain muscle loadings were simulated. Because a planar representation was inherent in the photoelastic model, it did not simulate many of the aspects of the cross-sectional shape of the femur included in the mathematical beam representation of Koch. It did, however, demanstrate that tension in a muscle from the greater trochanter to the lateral femoral condyle could reduce the bending stress in the femur. Other experimental stress

THE

PREDICTION

analysis investigations include the use of a colophonium coating by Pauwels (1950) and Kiintscher (1935) and a brittle stress coating by Evans and Lissner (1948) and Frankel ( 1960). The experimental results of Evans and Lissner (1948) and the mathematical analysis of Koch (19 17) are in agreement about the location of the maximum tensile stress for loading at the femoral head and femoral condyle. The work of Frankel focuses on the femoral neck for a wide range of load conditions applied to the femoral head. The work of Toridis (1%9) followed another direction of refinement. Retaining the approximation of beam theory, the femur was modeled as a three-dimensional space curve. This analysis included torsional as well as bending effects. Mathematical models of portions of the femur such as the condylar structures have been used by Burstein e? al. (1970) to calculate stresses and displacements for variations in geometry and elastic properties. While the importance of muscle loadings as sources of stress in the skeletal system has long been recognized, efforts to define such

*Received6 July 197 I. +This resetich was supported by Battelle Institute’s Engineering Science Program. *Division of Orthopaedics, The Ohio State University.

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forces in a quantitative manner and relate them to stresses in bones have been quite limited. In relation to the femur, the forces produced by the hip abductor muscles have received the greatest attention. Treatments of this subject are given by lnman (1947), Williams and Lissner (1962) and McLeish (1970). The influence of the hip abductor muscles on fractures of the femoral neck has been demonstrated by Hirsch (1965). The tests showed that .laboratory fractures of embalmed femurs resembled clinical fractures only if the loading of the abductor muscles was simulated in the tests. Experimental investigations to determine joint forces and moments at the hip and knee during walking have been conducted, for example, by Bresler and Frankel(1950). Paul (1967), Morrison (1967) and Rydell (1966). Muscle forces have been predicted by Paul (197 1), Williams ( 1968) and Blount ( 1956). The work described here is part of a larger program to define and investigate the mechanical factors influencing the structural characteristics and regeneration of bone. Accomplishing the objectives of this program requires a knowledge of the range of mechanical loadings acting on a bone and a capability to predict the stresses, strains, and strain energy produced by these loadings. Because the muscle forces are not generally known, but can be said to lie within a certain range, one aspect of such a study is an investigation of the effect of various muscle loadings on the states of stress, strain, and strain energy of a bone. Another aspect concerns the selection of an appropriate mathematical model to represent the behavior of a bone. This paper concerns these two aspects of the larger research program as they apply to the human femur. The first part of the paper is an analysis of the effect of certain muscle loadings on the stresses and strain energy of the human femur. The second part concerns the selection of appropriate mathematical models to predict the stress behavior of bones. The loading for the analysis presented here

and E. B. WEIS, Jr.

consists of the body weight. Although the specific magnitude of this loading pertains to a static or low inertial force condition, the dynamic cases of running and brisk walking can be estimated from the case considered here. One approximate method for treating dynamic loading is in a quasi-static fashion by adding inertial forces to the body weight. Doing this increases the body weight by a multiplicative factor and also increases the stresses by the same factor. The muscles considered are the hip abductors and the tensor fasciae latae. These muscles were chosen because equilibrium requirements provide a range of values for their muscle forces. Other muscle groups such as the ilio-psoas, the short hip rotators, the hip abductors, and the gluteus maximus are active, but it is more difficult to assess values for these muscle forces. While the tensor fasciae latae serves well to demonstrate how the bending stress and the strain energy can be influenced by muscle forces, it is very likely that further reduction of maximum stresses and the strain energy can be obtained by including other muscle groups. STRESSES AND STRAIN ENERGY PRODUCED CERTAIN JOINT LOADS AiiD IMUSCLES

BY

The mechanical loadings on bones can be conveniently classified as joint loads and muscle loads. In the following calculations, the stresses and internal strain energy for a human femur were calculated for various combinations of such loadings. In this study, attention was focused on the one-legged stance as one representative type of activity. For this portion of the study, the stress calculations were based on the beam model of the human femur developed by Koch (1917). Adoption of this model was a matter of convenience, since the geometric and crosssectional data as given were sufficiently complete to perform a realistic analysis. It is inherent to this model that the axis of the femur has no curvature except in the frontal plane. This simplification is reasonable when

ANALYSIS

OF STRESS

IN THE HUMAN

the interest is the critical stresses in the femur which are produced by bending within this plane. Rapid calculation of stress for many loadings was facilitated by a small program for the digital computer. The femur was described mathematicalliy in terms of the cross-sections shown in Fig. 1. At each cross-section the coordinates of the neutral axis for bending HIP abductor

mLlscle

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205

were prescribed. along with the cross-sectional area and moment of inertia. Externally applied axial loads, shear forces, or bending moments can be placed anywhere on the model. The computer program then calculates the resultant forces at each cross-section and the stress distribution based on standard equations for stresses in beams. For this part of the study, the interest was primarily in the stresses in the shaft of the femur, hence beam theory was adequate for this portion of the bone.

.Jo~nt force 2 521 lb /

F=

te

-+-t-l32

Fig. I. Cross-section description of femur showing joint and muscle loadings.

Remits

of stress analysis

The first case considered was a simple axial load applied to the head of the femur. This type of loading was treated by Koch (19 17) and Toridis (1969). and provided useful stress results for comparisons with muscle induced stresses. Figure 2 shows the distribution of tensile stress on the lateral side of the femur. This stress is produced by an axial load of 169 lb applied to the head of the femur, and reacted by a force of equal magnitude and opposite direction at the distal end. The 169 lb force corresponds to the weight of a 200 lb man minus the weight of one leg. Figure 2 shows a maximum tensile stress in the shaft of the femur of 1680 psi. The distribution of stress is the same as that given by Koch as the stress in the femur due to walking. The one-legged stance considered here cannot be maintained without the action of the hip abductor muscles. The force due to these muscles is indicated on Fig. 1. The resultant acts on the greater trochanter and creates a moment to prevent rotation of the body about the hip joint. This force was estimated to have a value of 358 lb. The addition of this force increases the axial compression in the femoral neck, giving a joint load at the head of the femur of 521 lb. These forces were determined from data given in an account by Williams and Lissner ( 1962) of an investigation by Inman (1947). When the abductor muscle force (assumed to be produced by only the gluteus medius and gluteus minimus muscles) is included in the

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169 lb

Joint

1400

I oodinq

psi/

1680

Fig. 2. Stress in femur due to simple axial loading.

calculation, the maximum tension in the femoral shaft becomes 6640 psi as shown in Fig. 3. This is an increase of about four times the stress obtained neglecting the abductor muscle force. The gluteus medius and gluteus minimus muscles, however, only partially account for

and E. B. WEIS, Jr.

the muscle loadings. In the final phase of the study, the force in the tensor faciae latae was considered. This muscle has its origin at the anterior crest of the ilium and its insertion on the iliotibial tubercle located on the proximal lateral portion of the tibia. The forces due to the tensor faciae latae are indicated in Fig. 1. This muscle force is not necessarily required to maintain force equilibrium of the body. However this muscle produces a pair of self equilibrating forces that may act to reduce stresses in the femur. Calculations were performed to determine if such an effect can be produced, and a magnitude of muscle tension that would be desirable. Strictly from an equilibrium requirement, the allowable values of tension in the tensor faciae latae (for the case described here) range from O-3 58 lb. To study the effect of the force in this muscle on the stresses in the femur, the force in the tensor faciae latae was varied from O-358 lb. The forces in the gluteus medius and gluteus minimus muscles were adjusted so that the sum of the forces in these two muscles and the tensor faciae latae was 358 lb. The forces acting on the femoral head were not changed. The resulting range of stress distributions on both the lateral and medial surfaces of the femur are shown in Fig. 4. In the proximal part of the femur, extending to the greater trochanter, all the stress curves are coincident because the force acting on the head of the femur remains constant. At low values of force in the tensor faciae latae. the femur is stressed in tension on the lateral surface and in compression on the medial surface. Increasing the force in the tensor faciae latae reverses this pattern. At very high values, high tensile stresses are in fact produced on the medial side where compression had previously existed. Figure 5 shows the calculated values of the maximum tensile and compressive stress in the femur as a function of the assumed tension in the tensor faciae latae. It is seen that both the tensile and compressive stresses in the shaft of the femur have minimum values for a muscle tension of

ANALYSIS

OF STRESS

IN THE

HUMAN

FEMUR

$07

521 lb Joint loadmq

.................. .................. .................. ..................

.................. .................. .................. .................. ............... ............. ............................. ............................. .............................

Fig. 3. Stress in femur due to combined loadings of hip joint and hip abductor muscles.

about 300 lb. Also shown is the elastic strain energy stored in the femur, and this too is a minimum for a muscle tension of the same general magnitude. While a muscle tension of about 300 lb minimizes the stress in the shaft of the femur, the maximum tensile stress in the

neck of the femur (about 3000 psi) was unchanged by the variation in muscle tension. Figure 5 shows that a tensic-t in the tensor faciae latae of about 200 lb is needed to reduce the maximum tensile stress in the shaft of the femur to the 3000 psi value corresponding to

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E. F. RYBICKI, F. A. SIMONEN and E. B. WEIS, Jr.

Forte in the te!nso f ascioe lotoe

STRESS

ALONG

LATERAL

SIDE

Fig. 4(a)

the neck. This suggests that a tension of about 200 lb is a desirable value. Although the tensor faciae latae is capable of exerting a tension of 200 lb, it is not known whether the body functions in such a manner that this value of tension actually develops in the onelegged stance position. Paul (197 1) estimates a tension of about 100 lb for a 150-lb person,

which if increased proportionately, corresponds to 135 lb for a 200 lb man. Although less than the desired value of 200 lb suggested by the present analysis, it nevertheless results in an appreciable reduction of stress (45 per cent) as shown in Fig. 5. In the absence of any muscle tension, the tensile stress in the shaft of the femur is 6640 psi, which is unreasonably

ANALYSIS

[71::l’il

Tensile

m

Compressive

OF STRESS

IN THE

HUMAN

‘09

FEMUR

stress

stress Force

f ascioe

STRESS

ALONG

MEDIAL

in tensor I otae

SIDE

Fig. 4(b) Fig. 4. Stress distribution in femur as a function of the force in the tensor faciae latae.

high. Although the tensile strength of the femur is greater than this (lO,OOO- 12,000 psi), it is not reasonable to expect stresses of such magnitude under a condition of mild activity as the one-legged stance, and under conditions of repeated loading where the fatigue strength of bone is relevant. Also if

B.M. Vol. S No. 2-F

the human body is to be considered an efficient mechanical structure, as is suggested by for example Wolff (1892) and Frost (1964). then the tensile stresses in the femur should be much lower than the compressive stresses, since bone is about twice as strong in compression than tension. This condition is met

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F. A. SIMONEN

Maximum

compressive

Maximum

Force

in tensor

and E. B. WEIS, Jr.

tensile

fasciae

stress

stress

lotae, lb x

IO2

Fig. 5. Maximum stress and strain energy in femur vs. muscle tension in tensor faciae latae.

.

with the 2004b tension in the tensor faciae latae, but not in its absence or with excessive tension in this muscle.

CONTINUUM ANALYSIS

The stresses in the shaft of the femur may appropriately be analyzed using slender beam theory. However in the regions of the femoral head, the greater trochanter, and areas of muscle attachments, the assumptions inherent in beam theory become questionable. In such regions of rapidly changing geometry, it is not reasonable to assume that plane sections remain plane during deformation or that shear deformations may be neglected. Such complex geometries as posed by the femur can be accurately represented by using recent advances in computer techniques for the analysis of stress in structural components. This section describes the application of one such

technique to the calculation of the stress in the proximal third of the femur. Analytical method In this study application was made of the finite element method. In recent years this method has become widely used for the analysis of stresses in engineering structures and the reader is referred for example to Zienkiewicz (1967) for a detailed treatment of the subject. The essential aspects of the finite element method can best be discussed in terms of the mathematical model of the proximal third of the femur shown in Fig. 6. As indicated the proximal third of the femur is subdivided into a grid of elemental areas, hence the terminology *‘finite element?. The stress is assumed in the theory to be the same at all points within each element, but may vary arbitrarily from element to element.

ANALYSIS

OF STRESS

IN THE

HUMAN

FEMUR

Fig. 6. Finite element model for the proximal third of the femur.

In this study. a computer program due’to Wilson (I 965) was used. This program treated the mathematical model of Fig. 6 as one of plane stress, which involves an idealization insofar as the geometry of the femur is concerned. However the fact that the bone is essentially circular in cross-sectional shape was approximately accounted for in the foilowing manner. The computer program is general in that the material properties may vary continuously from element to element. It was, however, recognized that such variations are analogous to the variations in thickness of a planar cross-section. Thus the bone was represented as a plate of variable thickness, with the thickness variation being treated mathematically within the program as a variation in material stiffness (i.e. elastic modulus). To illustrate the difference between the beam theory results and the continuum representation, homogeneous isotropic properties were assumed. However, the computer program has the capability to account for such deviations in bone properties as occur in the femoral head.

Comparison of stress distributions from finite element and beam analyses

Calculations were performed for two of the cases of loading considered in the above section. These were the simple axial joint load and the joint load combined with the loading due to the hip abductor muscle group. All loadings were of the same magnitude as those used in the above calculations using beam theory. Both the joint and muscle loadings were treated as distributed surface loadings as indicated in Figs. 7 and 8. At the left end of the mathematical model in Fig. 6 a displacement constraint condition was imposed. The resulting stresses calculated on this crosssection then represented the required reaction forces developed in the shaft of the femur in response to the applied loads. Stress distributions. Shown in Figs. 7 and 8 are the calculated distribution of stress along the lateral and medial surfaces of the femur. The magnitude of the stress is indicated as the distance between the stress curve and surface of the bone. The stress component plotted acts tangent to the surface of the bone, and the

E. F. RYBICKI,

212

Finite -Q

Q--

element

F. A. SIMONEN

and E. B. WEIS, Jr.

analysis

Beam analysis

Stress

scale,psi

Fig. 7. Calculated stress distributions for the proximal third of the femur due to joint loading.

&-

-

4

Finite

element

Beam

analysis

analysis

Fig. 8. Calculated stress distributions

02ooo4ooo Stress

scale,psi

for the proximal third of the femur due to joint loadirig and hip abductor muscles.

ANALYSIS

OF STRESS

directions of the arrows indicates tension or compression. Tensile stresses act on the lateral surface and compressive stresses on the medial surface. Results of the beam theory analysis are shown by the dotted curves. The difference between the beam theory and the finite element analysis is significant. While both methods are in agreement in their prediction of smaller stresses in the head region of the femur than in the shaft, it is seen that the beam theory gives higher stresses than the finite element results. In the beam theory the bone is forced to support the applied loads by deforming only in a beam bending mode. In contrast the finite element analysis allows the bone to deform as an elastic continuum. Thus the same applied loads can be supported with stresses of lower magnitude. It is interesting to note that the beam calculation predicts bending stress in the greater trochanter. where the continuum calculation shows that stresses are small in

IN THE

HUMAN

213

FEMUR

this region. The finite element calculations confirm that the deformations in the femoral shaft are primarily that of beam bending. The stress distributions on the lateral and medial sides of the femur for a force of 300 lb in the tensor faciae latae are shown in Fig. 9. A comparison of Figs. 8 and 9 shows how the stress distributions are changed due to this activity of the tensor faciae latae. The magnitude of the stresses in the femoral head remained relatively unchanged, but the stresses in the shaft changed by a factor of about three. A comparison of the continuum solution and the beam bending solution is presented in Fig. 9. This comparison also shows that the beam solution produces much larger stresses than the continuum solution. Stress trajectories. The results of both the beam and finite element calculations allowed stress trajectories to be plotted and these are shown in Fig. 10. The pattern determined by beam theory is in basic agreement with that

muscle

Finite (3- ---_O

element

analysis

Beom onolysis ” 0-

2000 4000 Stress stole, psi

Fig. 9. Calculated stress distributions in the proximal femur including forces in the tensor faciae latae.

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E. F. RYBICKI,

(0)

(cl

F. A. SIMONEN

4not loml lood,“g, 0e2m anatysls

A~IOI fmte

pnt

loading,

element

analysts

and E. B. WEIS. Jr.

(bl

Jomt loadmg wth htp abductor muscles, beam anatyos

(d)

Joant loadmq wth htp abductor muscles, tlmte element analysts

Fig. 10. Calculated stress trajectories for the proximal third of the femur.

determined by Koch (19 17). It is seen that for the two assumed loading states there is little difference within each theory between the patterns. The lines of principal stress which were calculated for the 300 lb force in the tensor faciae latae were very similar to those shown in Fig. 10 for zero tensor faciae latae force. However, the patterns are somewhat different depending on the analytical method. In fact with beam theory, the stress trajectory patterns are in large measure predetermined by the basic assumptions of beam theory. Therefore the finite element calculation is the more valid approach since no preconditions are imposed on how the stresses may vary within the bone cross-section.

Koch (1917) and other investigators, have postulated that the natural trabecular pattern in the head of the femur can be correlated with calculated stress line patterns. This contention has been severely criticized. It is. of course, difficult to prove or disprove whether the trabecular pattern is associated with imposed stresses or due to other physiological reasons such as nourishment requirements. It is seen that the stress line pattern predicted by the continuum theory differs somewhat from that predicted by the beam theory of Koch. Further refinements of the continuum calculation to include details of inhomogeneity and anisotropy present in the internal bone structure would probably produce further changes in

ANALYSIS

OF STRESS

the predicted stress line pattern. The stress trajectory theory may also be questioned because correlations have generally been made with the appearance of the trabecular pattern as seen in a frontal section. Viewed in other sections, a correlation may not be evident. Further clarification of the stress trajectory theory requires a stress analysis that accounts for more detail in the complexities of the internal structure of the femoral head. SUJIMARY

Using a mathematical model to represent the behavior of the human femur, the effect of muscle forces on the stresses and strain energy was investigated for a one-legged stance. It was found that the tensor faciae latae can have a pronounced effect on both the maximum stress and the total strain energy of the femur. A suitable tension in this muscle can reduce the maximum tensile and compressive stresses by a factor of 3 and the strain energy by a factor of 10. It is interesting to note that because of the loadings and geometry of the femur. the value of force in the tensor faciae latae that minimized the maximum stress was not the same as that which minimizes the strain energy for the femur. (See Fig. 5.) It was found that the choice of mathematical model to represent the behavior of a bone is important. A continuum model should be used in regions where loads are applied or where the shape of the femur is unlike that of a slender beam. In this study. the proximal third of the femur was modeled as a continuous elastic solid, and this provided an improved picture of the stresses in the femoral head over that given by beam theory. It was concluded, however, that beam theory can give acceptable results for stresses in the shaft of the femur. REFERENCES Bresler, B. and Frankel. J. P. (1950) The forces and moments in the leg during walking. Trans. Am. Sot.

IN THE mech.

HUMAN Engrs

215

FEMUR

72.27.

Blount. W. P. (1956) Don’t throw away the cane. Jr. Swg.

J. Bone

38‘4,695.

But-stein. A. H.. Shaffer. 8. W. and Frankel, V. H. (1970) Elastic analysis of condylar structures. Presented at ASME Winter Annual Meeting, November 29 to December 3. 1970. New York. Paper No. 70-WA/ BHF-I. Evans. F. G. and Lissner. H. R. (1948) Stresscoat. deformation studies of the femur under static vertical loading. Anat. Rec. 100. 159- 190. Frankel. V. (1960) The Femoral Neck. Almquist and Wiksells. Uppsala. Frost. H. M. (1964) The Laws of Bone Structure. Charles Thomas. Springfield. Illinois. Hirsch. C. (1965) Forces in the hip-joint. Biomechanics and Related Bio-Enaineerina Totks. (Proceedines of a symposium held-in Glasgow. September, ib64) (Edited by R. M. Kenedi) pp. 341-350. Pergamon Press. Oxford. Inman. V. T. (1947) Functional aspects of the abductor muscles of the hip. J. Bone Jr. Slug. (Am.) 29.607-619. Koch. J. C. (1917) The laws of bone architecture. Am. J. Anat.

21. 177-298.

Kiintscher. G. (1935) Die Bedeutung der Darstellung des Kraftflusses im Knochen fir die Chirumie. Arch. klin. Chir. 182.489-55 I. McLeish. R. D. and Charnley. J. (1970) Abduction forces in the one-legged stance. J. Biomechnnics 3. 19 I-209. Morrison. J. B. (1967) The forces transmitted by the human knee joint during activity. Ph.D. Thesis. University of Strathclyde. Glasgow. Paul. J. P. (1967) Forces transmitted by joints in the human body. Proc. lnstn mech. Engrs 181.8. Paul. J. P. (1971) Load actions on the human femur during walking and some stress resultants. Exp. Mech. 121-175.

Pauwels. F. (1950) Ubeer die mechanische Bedeutung der groberen Kortikalisstruktur beim normalen und pathologischen verbogenen Rohrenknochen. Anat. iVachr. 1.53-57. Pauwels. F. (195 I) Uber die Bedeutung der Bauprinzipien des Stutz-und Bewegungsapparates fur die Beanspruchung der Rohrenknochen. Actn Anat. 11. 207727. Rydell. N. W. ( 1966) Forces acting on the femoral head prosthesis. Acta. orthop. stand.. Suppi. 88. Toridis. T. G. (1969) Stress analysis of the femur. J. Biomechanics. 2. 163-174. Williams. M. and Lissner. H. R. (1962) Biomechnnics of Human Motion. W. B. Saunders Company. Williams. J. F. (1968) A force analysis of the hip joint. Biomed.

Engng

3.365-370.

Wilson. E. L. (1965) Structural analysis of axisymmetric solids. AMA J. 3. 2269. Wolff. J. (1892) Das Geset: der Transformation der Knochen. Guano. Berlen. Ziekiewicz, 0. C. and Cheung. Y. K. (1967) The Finite Element Method in Strrtcturnl nnics. McGraw-Hill, London.

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Continuum

Mech-