Biomedical Engineering Reference
In-Depth Information
F
/
A
σ
y
Buckling
Buckle
No buckling
F
F
F
>
F
c
100
200
Small
Intermediate
L
e
/
K
FIGUre 2.14
Conditions for buckling.
where
A
is the cross-sectional area and
k
is called the radius of gyration
(= (
I
/
A
)
1/2
) (=
r
/2 for a column).
It is then possible to make a plot of critical stress (critical force/
A
)
versus the ratio
L
e
/
k
. For low values of this ratio (below approximately
50), the critical stress for buckling exceeds the yield stress, so that no
buckling occurs. There is a small intermediate range (50
<
L
e
/
k
<
100) in
which buckling also does not occur but in which the yield stress appears
to be modestly reduced. Finally, there is a range in which buckling always
occurs (
L
e
/
k
>
100). This region is called the region of elastic buckling,
since structural deformation, although resulting in plastic deformation
as loads redistribute, begins as an elastic instability.
Buckling may occur easily in a lamellar structure, such as bone, if
delamination produces individual portions of material with very high
L
e
/
k
ratios. This phenomenon is probably responsible in part for the
so-called buckle fracture that is seen in poorly mineralized immature
bone. Trabecular buckling may also occur in osteoporotic vertebral bod-
ies as loss of horizontal spicules radically increases
L
e
in the remaining
vertical spicules.
Torsion
In torsion (Figure 2.6), the major stress is a shear stress that increases
with radial distance from the center of a rod. The shear stress, τ, at any
point is given by
τ
=
M
∙
r
/
K
where
M
is the torsional moment at that point,
r
is the radial distance
from the rod axis (the neutral axis in torsion), and
K
is the polar moment
of inertia, an expression of the distribution of material across the cross
section of the rod. Values of
K
for various cross-sectional shapes are
given in Table 2.5. As before, the torsional stiffness (resistance to tor-
sional deformation under load) depends on the value of
K
.
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