Deformation - Orthopaedic Biomaterials in Research and Practice - page 38

Area
6.16 cm 2

r o
r i

1.4 cm 1.6

0.77

1.8 2.0

- 1.13 1.43

I (rel)
σ OF (rel)

1.0 1.8 2.3 3.1

1.0 0.63 0.55 0.46

(a) (b) (c) (d)

FIGUre 2.13 structural efficiency of circular cross-sections.

This example may explain why most bones have medullary spaces;
their presence is a sign of relative structural efficiency. It also helps to
understand why endosteal absorption and periosteal accretion occur in
osteoporosis: since stiffness is proportional to the product E * I , when E
decreases, as in osteoporosis, a response that increases I by remodeling
will help restore stiffness without increasing the amount of bone present.

Buckling There is, however, a limit to which this principle may be extended. A

bone is generally considered a thick-walled tube, since its cortical thick-
ness ( t ) is rarely less than one-eighth its radius ( r ). Tubes with ratio ( t / r )
less than one-eighth tend to behave as curved sheets rather than as tubes;
thus, different formulas are required to calculate moments of inertia, as
seen in Table 2.4. What is different about thin-walled tubes is that they
are subject to buckling : local, concentrated deformation.
Buckling in bending is difficult to discuss in general because it depends
on many factors, including the shape and location of the supports. However,
buckling may occur under any type of loading and is usually considered in
compressive (end) loading of a cylinder, as in Figure 2.14.
There is a critical force, F C , that will produce buckling:

2 EI

L

⋅

F =

C

2

e

where L e is an effective length, dependent on the degree of constraint at
the ends of the column. If the column ends are free, then L e = L ; if they
are fixed, then L e = L /2.
It is useful to restate this relationship as follows:

2 EA

Lk

⋅

F =

C

2

(

e /

)