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Mechanics of Skeleton and Tendons

R. McNeill Alexander

10.1002/cphy.cp010202

Source: Supplement 2: Handbook of Physiology, The Nervous System, Motor Control

Originally published: 1981

Published online: January 2011

Full Article on Wiley Online Library



Abstract

The sections in this article are:

1 Basic Kinematics
2 Kinematics of Joints
3 Number of Muscles Required to Work Joints
4 Muscle Attachments
5 Moment Arms and Pennation Patterns
6 Properties of Skeletal Materials
7 Statics of Skeleton
8 Dynamics of Skeleton
9 Dynamics of Body
10 Stresses and Strains in Tendons
11 Stresses and Strains in Bones
12 Engineering Design of Bones
13 Conclusion
Figure 1. Figure 1.

A: two positions of a body moving in a plane, showing how instantaneous center (x0, y0) can be located. B‐D: mechanisms consisting of rigid bars joined by hinges with axes perpendicular to paper. Each of these mechanisms has 1 degree of freedom.
Figure 2. Figure 2.

A‐C: examples of joints that allow 1 degree of freedom; x, y, and z are distances; θ is an angle. D, E: 2 degrees of freedom. F: 3 degrees of freedom of relative movement.
Figure 3. Figure 3.

Diagrams of bones and some of the ligaments of a human elbow and forearm: h, humerus; r, radius; u, ulna. A: section. B: medial view.

From Alexander 2
Figure 4. Figure 4.

Outlines of skull drawn from X‐radiographs of rats (Rattus norvegicus) feeding. Top outlines show sequence of jaw positions involved in biting with incisors. Bottom outlines show sequence involved in chewing. Arrows A, B, C indicate sequence of movement in each case.

From Hiiemae 39 with permission from Zool. J. Linn. Soc., vol 50, © 1971 Linnean Society of London
Figure 5. Figure 5.

A: outlines of a human tibia and fibula. Dots show successive positions of the instantaneous center ofa normal knee during bending from 20° flexion. B: mechanism representing a knee joint. Further explanation is given in the text.

A data from Soudan et al. 66
Figure 6. Figure 6.

Diagrams of human knee joint; acl, pcl, anterior and posterior cruciate ligaments, respectively.

From Barnett et al. 14
Figure 7. Figure 7.

Diagrammatic sections through bones of the wrist. A: primate. B: ungulate. Forearm is on left and hand on right in each case. Wrists are shown extended (A1, B1) and flexed (A2, B2). C: diagram of bones of a mammalian wrist. Wrist is extended (C1) and shows radial deviation (C2). sf, Stop facet.

From Yalden 74
Figure 8. Figure 8.

Anterior views of skeleton of thorax. A: monkey (Macaco). B: man.

From Schultz 59
Figure 9. Figure 9.

Skeleton and some of the muscles of human arm. Labels include bi., biceps; br. rad., brachioradialis; h., humerus; pr. quad., pronator quadratus; pr. ter., pronator teres; rad., radius; sup., supinator; uln., ulna.

From Young 77
Figure 10. Figure 10.

Diagram showing some of the bones and muscles of a hind leg of a typical mammal. Broken lines separate epiphyses from diaphyses. Dark areas show tendons and aponeuroses.
Figure 11. Figure 11.

A: section through the plantaris muscle of a wildebeest (Connochaetes). B: section through the interosseous muscle of forelimb of a sheep (Ovis). Dark areas show aponeuroses.
Figure 12. Figure 12.

Diagrams of a hinge joint with 3 alternative but equivalent extensor muscles. These diagrams are explained further in text. r, Moment arm of muscle about joint.
Figure 13. Figure 13.

A: diagram of a human arm with weight hanging from thumb. B: free‐body diagram of distal phalanx of thumb in A. F1, weight of load (67 N); F2, weight of phalanx; F3, F4, forces exerted on phalanx by the 2 muscles attached to it, the flexor pollicis longus and extensor pollicis longus; F5, reaction at joint. Lines of action of these forces are distant (e.g., x1) from instantaneous center of joint where F5, is assumed to act. F1 and F2 are vertical; F3‐F5 act at angles θ3‐θ5 to vertical.

A from Borelli 17
Figure 14. Figure 14.

A: diagram of skull of a mammal, showing temporalis and masseter muscles. B‐E: lower jaws of Martes, showing the forces (P, P′, T, M) that act on it in various circumstances described in text. Force P acts on lower carnassial tooth. R, reaction at jaw articulation.

From Smith and Savage 63
Figure 15. Figure 15.

A: diagram of a knee showing forces acting on tibia. Forces are F, external; T, exerted by quadriceps; CN and CF, normal and tangential, exerted by condyles of femur; L, exerted by ligaments. B: mechanism equivalent to the joint. At point O, PQ (rigid bar) intersects perpendicular from articulating surfaces. Hence O must be instantaneous center of joint.

From Alexander 8
Figure 16. Figure 16.

Free‐body diagrams of a soccer player's leg just before the foot kicks a ball. A: lower leg and foot. F, force exerted by quadriceps muscles extending knee; R, reaction at joint acting at angle γ; −m‘x’, inertia force; −I‘θ’, inertial torque; mg, leg segment weight. Angle α gives acceleration direction. B: similar diagram of whole leg.
Figure 17. Figure 17.

A: force‐platform record of a 36‐kg dog taking off for a running long jump. Only the vertical (FY) and longitudinal (FX) components of force are shown. An upward displacement of FX record represents a backward force exerted by dog on the platform (i.e., a forward force exerted by platform on the dog). B: outline traced from film of same jump, showing force exerted by 1 foot at the instant when it was greatest. C: forces (in Newtons) acting on 1 foot at the instant shown in B.

A from Alexander 3; B and C from Alexander 4
Figure 18. Figure 18.

Two outlines traced from films of the same 68‐kg man. In A: running across force platform. B: taking off for a standing jump from the force platform. Force acting on right foot is represented by an arrow. The instant illustrated in each case is the one at which the moment of this force about the knee was greatest.
Figure 19. Figure 19.

Outlines traced from films of a 68‐kg man and a 35‐kg dog on a force platform, showing the magnitude (in Newtons) and direction of the force exerted by 1 foot. A: man walking. B: man running. C: dog trotting.

From Alexander and Goldspink 7
Figure 20. Figure 20.

Schematic graphs based on force‐platform records, showing vertical and longitudinal components of force on the ground plotted against time. A: man walking. B: man running. Continuous lines show forces exerted by individual feet. Dotted lines during double‐support phases in A show total force exerted by the 2 feet. Bars under graphs show when feet are on ground.
Figure 21. Figure 21.

Diagram showing how right leg and shoulder move in human walking.

From Carlsöo 20
Figure 22. Figure 22.

Schematic graphs of mechanical energy against time. A: 70‐kg man walking at 1.4 m/s. B: 10.5‐kg wallaby (Protemnodon) hopping at 2.4 m/s. PE, potential energy; KE, kinetic energy; EE, elastic strain energy in Achilles tendons. Bars under graphs show when feet are on ground.

Data for A from Cavagna and Margaria 22; data for B from Alexander and Vernon 12
Figure 23. Figure 23.

Cross sections drawn to same scale. Rod and 3 tubes would all be equally strong in bending if made of same material. Numbers indicate k, ratio of internal diameter to external diameter.
Figure 24. Figure 24.

Graphs of weight/unit length against k, (internal diameter): (external diameter), for cylindrical tubes of equal strength in bending. Bottom line shows the weight of empty tube. Top line shows weight of the same tube filled with a substance that has one‐half the density of the tube wall.
Figure 25. Figure 25.

Transverse sections of a fore (A) and a hind (B) cannon bone of an ox. Sections were cut halfway along the bones.
Figure 26. Figure 26.

Top, diagrams of 2 bones on which vertical forces act. F, force exerted by a muscle; P, external force; R, reaction at a joint. Bottom, graphs showing distribution of bending moments along the same bones. Bending moment is shown as positive if it tends to rotate distal part of bone counterclockwise.
Figure 27. Figure 27.

Graph of section modulus against distance from distal end of the bone (tibia of a 24‐kg dog). Overall length of bone was 17 cm. Values refer to bending moments in sagittal plane. Two points are given for each cross section because due to asymmetry the maximum tensile stress was not numerically equal to maximum compressive stress at opposite face of bone. Cross sections are also shown, with anterior edges uppermost.

From Alexander 5
Figure 28. Figure 28.

A, B: diagrams illustrating principles of stress reduction by balanced loads and by bracing, respectively. C: pelvic girdle and part of skeleton of supporting leg of a man who has the other leg off the ground. D: arm skeleton of man holding a heavy object on his palm. Arrows represent forces. Further explanation is given in the text.

From Alexander 2
Figure 29. Figure 29.

A vertical longitudinal section of a human calcaneum, with lines showing directions in which trabeculae run.
Figure 30. Figure 30.

Diagram of relationship between stresses. When direct stresses act vertically and horizontally, shear stresses act at 45°, and vice versa. A: block enclosing broken outline, which is square but tilted at 45°. B: square deformed to rhombus is sheared. C: square stretched to rectangle.

From Alexander 2
Figure 31. Figure 31.

Pattern of stresses acting in heel of a person standing on his toes, or running with toes on ground and heel off the ground. Diagram is based on data obtained by photoelastic analysis of a model. Continuous lines show directions of principal compressive stresses. Dotted lines show directions of principal tensile stresses. Black areas represent Achilles tendon and plantar aponeurosis. Stippled band represents epiphysial plate.

From Smith 64


Figure 1.

A: two positions of a body moving in a plane, showing how instantaneous center (x0, y0) can be located. B‐D: mechanisms consisting of rigid bars joined by hinges with axes perpendicular to paper. Each of these mechanisms has 1 degree of freedom.



Figure 2.

A‐C: examples of joints that allow 1 degree of freedom; x, y, and z are distances; θ is an angle. D, E: 2 degrees of freedom. F: 3 degrees of freedom of relative movement.



Figure 3.

Diagrams of bones and some of the ligaments of a human elbow and forearm: h, humerus; r, radius; u, ulna. A: section. B: medial view.

From Alexander 2


Figure 4.

Outlines of skull drawn from X‐radiographs of rats (Rattus norvegicus) feeding. Top outlines show sequence of jaw positions involved in biting with incisors. Bottom outlines show sequence involved in chewing. Arrows A, B, C indicate sequence of movement in each case.

From Hiiemae 39 with permission from Zool. J. Linn. Soc., vol 50, © 1971 Linnean Society of London


Figure 5.

A: outlines of a human tibia and fibula. Dots show successive positions of the instantaneous center ofa normal knee during bending from 20° flexion. B: mechanism representing a knee joint. Further explanation is given in the text.

A data from Soudan et al. 66


Figure 6.

Diagrams of human knee joint; acl, pcl, anterior and posterior cruciate ligaments, respectively.

From Barnett et al. 14


Figure 7.

Diagrammatic sections through bones of the wrist. A: primate. B: ungulate. Forearm is on left and hand on right in each case. Wrists are shown extended (A1, B1) and flexed (A2, B2). C: diagram of bones of a mammalian wrist. Wrist is extended (C1) and shows radial deviation (C2). sf, Stop facet.

From Yalden 74


Figure 8.

Anterior views of skeleton of thorax. A: monkey (Macaco). B: man.

From Schultz 59


Figure 9.

Skeleton and some of the muscles of human arm. Labels include bi., biceps; br. rad., brachioradialis; h., humerus; pr. quad., pronator quadratus; pr. ter., pronator teres; rad., radius; sup., supinator; uln., ulna.

From Young 77


Figure 10.

Diagram showing some of the bones and muscles of a hind leg of a typical mammal. Broken lines separate epiphyses from diaphyses. Dark areas show tendons and aponeuroses.



Figure 11.

A: section through the plantaris muscle of a wildebeest (Connochaetes). B: section through the interosseous muscle of forelimb of a sheep (Ovis). Dark areas show aponeuroses.



Figure 12.

Diagrams of a hinge joint with 3 alternative but equivalent extensor muscles. These diagrams are explained further in text. r, Moment arm of muscle about joint.



Figure 13.

A: diagram of a human arm with weight hanging from thumb. B: free‐body diagram of distal phalanx of thumb in A. F1, weight of load (67 N); F2, weight of phalanx; F3, F4, forces exerted on phalanx by the 2 muscles attached to it, the flexor pollicis longus and extensor pollicis longus; F5, reaction at joint. Lines of action of these forces are distant (e.g., x1) from instantaneous center of joint where F5, is assumed to act. F1 and F2 are vertical; F3‐F5 act at angles θ3‐θ5 to vertical.

A from Borelli 17


Figure 14.

A: diagram of skull of a mammal, showing temporalis and masseter muscles. B‐E: lower jaws of Martes, showing the forces (P, P′, T, M) that act on it in various circumstances described in text. Force P acts on lower carnassial tooth. R, reaction at jaw articulation.

From Smith and Savage 63


Figure 15.

A: diagram of a knee showing forces acting on tibia. Forces are F, external; T, exerted by quadriceps; CN and CF, normal and tangential, exerted by condyles of femur; L, exerted by ligaments. B: mechanism equivalent to the joint. At point O, PQ (rigid bar) intersects perpendicular from articulating surfaces. Hence O must be instantaneous center of joint.

From Alexander 8


Figure 16.

Free‐body diagrams of a soccer player's leg just before the foot kicks a ball. A: lower leg and foot. F, force exerted by quadriceps muscles extending knee; R, reaction at joint acting at angle γ; −m‘x’, inertia force; −I‘θ’, inertial torque; mg, leg segment weight. Angle α gives acceleration direction. B: similar diagram of whole leg.



Figure 17.

A: force‐platform record of a 36‐kg dog taking off for a running long jump. Only the vertical (FY) and longitudinal (FX) components of force are shown. An upward displacement of FX record represents a backward force exerted by dog on the platform (i.e., a forward force exerted by platform on the dog). B: outline traced from film of same jump, showing force exerted by 1 foot at the instant when it was greatest. C: forces (in Newtons) acting on 1 foot at the instant shown in B.

A from Alexander 3; B and C from Alexander 4


Figure 18.

Two outlines traced from films of the same 68‐kg man. In A: running across force platform. B: taking off for a standing jump from the force platform. Force acting on right foot is represented by an arrow. The instant illustrated in each case is the one at which the moment of this force about the knee was greatest.



Figure 19.

Outlines traced from films of a 68‐kg man and a 35‐kg dog on a force platform, showing the magnitude (in Newtons) and direction of the force exerted by 1 foot. A: man walking. B: man running. C: dog trotting.

From Alexander and Goldspink 7


Figure 20.

Schematic graphs based on force‐platform records, showing vertical and longitudinal components of force on the ground plotted against time. A: man walking. B: man running. Continuous lines show forces exerted by individual feet. Dotted lines during double‐support phases in A show total force exerted by the 2 feet. Bars under graphs show when feet are on ground.



Figure 21.

Diagram showing how right leg and shoulder move in human walking.

From Carlsöo 20


Figure 22.

Schematic graphs of mechanical energy against time. A: 70‐kg man walking at 1.4 m/s. B: 10.5‐kg wallaby (Protemnodon) hopping at 2.4 m/s. PE, potential energy; KE, kinetic energy; EE, elastic strain energy in Achilles tendons. Bars under graphs show when feet are on ground.

Data for A from Cavagna and Margaria 22; data for B from Alexander and Vernon 12


Figure 23.

Cross sections drawn to same scale. Rod and 3 tubes would all be equally strong in bending if made of same material. Numbers indicate k, ratio of internal diameter to external diameter.



Figure 24.

Graphs of weight/unit length against k, (internal diameter): (external diameter), for cylindrical tubes of equal strength in bending. Bottom line shows the weight of empty tube. Top line shows weight of the same tube filled with a substance that has one‐half the density of the tube wall.



Figure 25.

Transverse sections of a fore (A) and a hind (B) cannon bone of an ox. Sections were cut halfway along the bones.



Figure 26.

Top, diagrams of 2 bones on which vertical forces act. F, force exerted by a muscle; P, external force; R, reaction at a joint. Bottom, graphs showing distribution of bending moments along the same bones. Bending moment is shown as positive if it tends to rotate distal part of bone counterclockwise.



Figure 27.

Graph of section modulus against distance from distal end of the bone (tibia of a 24‐kg dog). Overall length of bone was 17 cm. Values refer to bending moments in sagittal plane. Two points are given for each cross section because due to asymmetry the maximum tensile stress was not numerically equal to maximum compressive stress at opposite face of bone. Cross sections are also shown, with anterior edges uppermost.

From Alexander 5


Figure 28.

A, B: diagrams illustrating principles of stress reduction by balanced loads and by bracing, respectively. C: pelvic girdle and part of skeleton of supporting leg of a man who has the other leg off the ground. D: arm skeleton of man holding a heavy object on his palm. Arrows represent forces. Further explanation is given in the text.

From Alexander 2


Figure 29.

A vertical longitudinal section of a human calcaneum, with lines showing directions in which trabeculae run.



Figure 30.

Diagram of relationship between stresses. When direct stresses act vertically and horizontally, shear stresses act at 45°, and vice versa. A: block enclosing broken outline, which is square but tilted at 45°. B: square deformed to rhombus is sheared. C: square stretched to rectangle.

From Alexander 2


Figure 31.

Pattern of stresses acting in heel of a person standing on his toes, or running with toes on ground and heel off the ground. Diagram is based on data obtained by photoelastic analysis of a model. Continuous lines show directions of principal compressive stresses. Dotted lines show directions of principal tensile stresses. Black areas represent Achilles tendon and plantar aponeurosis. Stippled band represents epiphysial plate.

From Smith 64
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Article Title: Mechanics of Skeleton and Tendons
 

How to Cite

R. McNeill Alexander. Mechanics of Skeleton and Tendons. Compr Physiol 2011, Supplement 2: Handbook of Physiology, The Nervous System, Motor Control: 17-42. First published in print 1981. doi: 10.1002/cphy.cp010202