5.9 Geometric properties

Structural Properties

Depends on shape and material!

We have seen that there is a geometric component to stiffness
We will examine three properties which are critical
- Cross sectional area (\(A\))
- Area moment of inertia (\(I\))
- Polar (torsional) moment of inertia (\(J\))

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The area is critical in axial load because the stress is inversely proportional to the area

\[\sigma = \frac{P}{A} = \frac{P}{b \cdot h}\]

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The area moment of inertia (\(I\)) goes as the third power of the thickness (about the axis of bending) \[I = \frac{b h^3}{12}\]
The bending stiffness (\(EI\)) is directly proportional to \(I\)

Additionally, \(h\) also critical in bending because of its relation the maximum stress

\[\sigma_{\mathrm{max}} = \frac{M h}{2 I} = \frac{6 M}{b h^2}\]

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Similarly, the area moment of inertia for a hollow bone is: \[I = \frac{\pi (R^4-r^4)}{4}\]

Consider an application: IM Nails

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Equal values of mean width, equal thickness

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\[GJ_{\mathrm{open}} = G \frac{b t^3}{3} = G \frac{\pi d t^3}{3}\] \[GJ_{\mathrm{ef}}^{\mathrm{cir}}=G \frac{\pi d^{3} t}{4}\]

Equal values of mean width, equal thickness

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Assume: \[t=3 \mathrm{mm}\] \[d=25 \mathrm{mm}\]

The ratio is: \[\frac{J_{\mathrm{ef}}^{\mathrm{cir}}}{J_{\mathrm{open}}}=52\]

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Bone is viscoelastic: its stress-strain characteristics are dependent upon the rate of loading
Example: trabecular bone becomes stiffer in compression the faster it is loaded.