Bones

Bones are the stiffest of all musculoskeletal tissue. Long bones are irregular hollow cylinders filled with a loose cellular tissue (the marrow) containing blood and other matter. The compact bone is a composite of organic and inorganic material, the organic phase being nearly all collagen. The inorganic phase consists of water and a mineral salt called hydrox-yapatitie. Electron microscopy has shown that the hydroxyapatite is in the form of very fine needles only 15 nm wide and up to 10 times as long. These needles are brittle but strong. The bone matrix in which they are embedded lowers the stiffness and protects the needles from breaking. The average value of Young's modulus (stiffness coefficient) is 20,000 N/mm2, about one-tenth of that for steel. If the bones taken from a cadaver are dried and then tested, they fail at a tensile stress of about 100

N/mm2 at a strain of 0.004. Bones, as is fitting for their irregular shapes, carry complex stress systems. Let us illustrate this with the following example.

Example 6.9. Bone Fracture. A man whose humerus had fractured earlier believed that he was now healed but that he had a stiff elbow. He lay on a bed and asked a friend to gently help him flex his elbow (Fig. 6.13a). His friend placed one hand on the forearm just below the elbow and the other just above the wrist and pushed in opposite directions with a force of 15 N. The distance between his hands was 19 cm. Apparently, he had pushed too strongly; the humerus failed at its weakest point, the original fracture site, shown as BB' in the figure. Determine the maximum tensile stress that occurred at the fracture site during bending. Assume that the normal stress varied linearly along the cross section of the humerus. The diameter of the humerus was 7 cm and the thickness of the compact bone was 1.5 cm at the site of the fracture.

Figure 6.13a-c. Fracturing of the humerus bone of a person with stiff elbow, resulting from incorrect manipulation at the forearm (a). The free-body diagram of part of the arm is shown in (b). The stress distribution on a cross section of the humerus is illustrated in (c). The original fraction was at BB'.

Figure 6.13a-c. Fracturing of the humerus bone of a person with stiff elbow, resulting from incorrect manipulation at the forearm (a). The free-body diagram of part of the arm is shown in (b). The stress distribution on a cross section of the humerus is illustrated in (c). The original fraction was at BB'.

Solution: The free-body diagrams of the arm is shown in Fig. 6.13b. For simplicity in analysis we neglected the weight of the arm. Because the elbow was stiff, it did not bend during the manipulation of the forearm. The force couple exerted on the forearm was transmitted to the upper arm. According to the free-body diagram, the magnitude of the net moment acting on a cross section of the humerus is given as follows:

What kind of stresses does a bending moment cause on a cross section normal to the long axis of the bone? We had seen earlier (in Chapter 5) that bending moment caused axial stress in a cantilever beam. If the humerus could be considered as a linearly elastic solid, the stress distribution would be linear (Fig. 6.13c). The stress is minimum at the inner edge and maximum at the outer edge. This maximum stress (o-o) is related to the moment M acting on the cross section by the formula:

in which h denotes the maximum dimension of the cross section in the y direction, and Jx, axial moment of inertia, is a geometric parameter that depends only on the shape of the cross section.

for a rectangle with width b and height h, and

for an annulus with inner radius r and outer radius R.

The cross section of the humerus occupied by compact bone could be represented as an annulus with outer radius equal to 3.5 cm and thickness equal to 1.5 cm. Then, h = 7.0 cm and Jx = 105 cm4. Under these conditions the maximum normal stress ao corresponding to the cross-sectional moment of 2.85 N-m would be equal to ao = 285 N-cm (3.5 cm)/(105 cm4) = 9.5 N/cm2

Although this value is less than the yield stress for failure of normal young adult bone, the yield stress in the crack region would be expected to be much less because the fracture had not completely healed at the time.

Was this article helpful?

0 0
Getting Started With Dumbbells

Getting Started With Dumbbells

The use of dumbbells gives you a much more comprehensive strengthening effect because the workout engages your stabilizer muscles, in addition to the muscle you may be pin-pointing. Without all of the belts and artificial stabilizers of a machine, you also engage your core muscles, which are your body's natural stabilizers.

Get My Free Ebook


Post a comment