A long bone shaft is modelled as a circular tube of outer radius and inner radius

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Biophysics & Medical Physics · Bioelectricity, the senses and the physics of the body

Forces, Machines and Signals, Ch Bones, blood and body size

Problem

A long bone shaft is modelled as a circular tube of outer radius

r_o

and inner radius

r_i

, carrying a bending moment

M_b = 35\,\mathrm{N\,m}

(the model femur value in single-leg stance). Three tubes are compared, all built from the same amount of bone material, so all have the same cross-sectional area

A = 377\,\mathrm{mm^2}

: ((a)) a solid rod,

r_i = 0

; ((b)) the model femur,

r_i = 7\,\mathrm{mm}

; ((c)) a more hollow tube,

r_i = 10\,\mathrm{mm}

. For each case find

r_o

from the fixed area, then the second moment of area

I = \pi(r_o^4 - r_i^4)/4

, and the peak surface bending stress

\sigma_{\max} = M_b r_o/I

. State the trend in stress as the bone is made more hollow at constant mass, and explain it in terms of where the material sits relative to the neutral axis.

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