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View Full Version : Bone Deformation (theoretical) Summary (again!)

randerson75
11-29-2001, 06:42 PM
In my previous summary (slightly premature I guess) I stated that - This is
assuming that the force applied to the beam is at the very end of the beam
and causes a moment (M) of F x L (see below). In fact the following
statement holds true.

If the deflection is due to a point load applied to the end of a beam, x=(F
L^3)/(3 E I). If it is due to a point
moment: x=(M L^2)/(2 E I). The equations are different because the point
force results in a linearly decreasing moment in the beam from end to end,
while the point moment is a constant moment along the entire length.

Regards Ross

Original Summary

Firstly thanks you to all who replied to my message (see original message
below)

Firstly the major thing was that the formula below is slightly wrong (if a
formula can ever be 'slightly' wrong!) and should read as follows -

X=(M.L^2)/(2.E.I)

This is assuming that the force applied to the beam is at the very end of
the beam and causes a moment (M) of F x L.

The other thing is that to calculate I for a bone (which we all know is a
hollow cylinder) you need to use a slightly different equation to calculate
I, this is as follows (I had this one covered but wasn't going to 'reveal'
it to my students mathematically just the theory, but now I think I'll fully
cover the difference).

I=(pi.(ro^4 - ri^4))/4 where ro is the outer bone radius and ri is the inner
bone radius.

Hope this clears te answer up for everyone out there.

Regards Ross

-----Original Message-----
From: Ross.Anderson [mailto:ross.anderson@ul.ie]
Sent: 29 November 2001 12:04
To: BIOMCH-L@NIC.SURFNET.NL
Subject: [BIOMCH-L] Bone Deformation (theoretical)

I am about to discuss the ability of us to look at bone (from a simple point
of view) as a cantilever with a circular cross sectional area with my 3rd yr
sport and exercise students. From my old physics notes (circa 1991!) I have
the following equation for estimating the linear deviation that occurs at
the unsupported end of the beam.

X=(4.M.L^2)/(3.pi.E.r^4)

Where M is the bending moment, L is the length of the beam, E is the
stiffness of the material (Young's Modulus), and r is the radius of the
beam.

In another, more general format, this can be expressed as

X=(M.L^2)/(3.E.I)

Where I is the second moment of area calculated around the neutral axis, and
for a beam with a circular cross section is

I=(pi.r^4)/4

I have been getting results, theoretical, from these equations which do not
make sense to me. I have been using realistic values for all unknowns in
trying to calculate x (the linear deviation) but when the calculation is
done using SI units the deviation is massive (longer than the beam!). I
have tried converting units to mm and this gives very small values, I can
see mathematically what is happening but the values are highly unrealistic.

Can anyone tell me whether

1 - the formula I am using is correct, if not what is recommended.
2 - what would be a typical linear deviation for a bone?

3 - Any other comments on the methodology etc I am using here.

Regards Ross

Ross Anderson__________________________________________ _______

Dept of Sport and Exercise Sciences and
Centre for Biomedical Electronics
University of Limerick
IRELAND
Tel - +353 (0) 86 6090866 or +353 (0) 61 202810
Fax - +353 (0) 61 330431
e-Mail - ross.anderson@ul.ie
WWW - www.ul.ie/~pess/staff/ross/

Ross Anderson__________________________________________ _______

Dept of Sport and Exercise Sciences and
Centre for Biomedical Electronics
University of Limerick
IRELAND
Tel - +353 (0) 86 6090866 or +353 (0) 61 202810
Fax - +353 (0) 61 330431
e-Mail - ross.anderson@ul.ie
WWW - www.ul.ie/~pess/staff/ross/

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