Your logic is correct, but I think Jim's assertion was that if you
allowed the top board to deflect "normally" under our assumed 1/2 load
that the bottom of the top board would enforce a deflection profile on
the top of the bottom board. The resulting curvature imposed on the
bottom board should result in an identical moment by your spring
analogy, but the larger radii would result in a lower moment, if every
so slightly  hence the apparent paradox. And the number of angels
depends in the size of the pin head, no?
Jordan
Robert Kazanjy wrote:
I think we're debating "the number of angels that can
dance on the head of a pin"
but
how about this........
if both beams have the same deflection & both have the same
section properties
then are they not unlike two springs in parallel? they share the load
equally
equal load, equal deflection, equal stress state?
for all practical purpose, variations in sections, friction,
etc......the two members see the same moment
cheers
Bob
btw the deflections HAVE to be the same, how could they not be?
The moment distribution & stress state is such that deflection
compatibility is maintained.
On 3/27/07, Jordan Truesdell, PE <seaint1(nospamat)truesdellengineering.com>
wrote:
I'm not convinced, no
direspect intended. The circle case is
nonsensical, as the condition can not exist  the curve is
noncircular (M varies for the beam in question) and the method of
analysis for deflections used in common engineering does not allow for
such deflections within the equation assumptions. I'll offer that the
curvature may not be exact in real life, but they will be the same, to
the precision afforded by the design assumptions required for the
analysis of slender beams which are, as we all know, imprecise, but
well within the accuracy of our knowledge of the materials for common
efficient spans.
Well, you forced me to do the @#$^@ math (caffeine, late night, too
easily distracted). I got a difference between the bottom and top of a
circle (L/360 deflection in the nonsensical circular segment, 10'
span, 3.5" deep members) of 1.00005. (someone can check me  I got a
radius of 565.5 x span; no idea if this is right, I may have flipped a
sin/tan somewhere) .
Actually, I suppose the whether the top and bottom of the member bent
in the shape of a quartic curve "nests" would tell us whether the
moments are actually identical. But that wouldn't work either as that
ignores shear deflection, varying stresses triaxial compression states,
and deformations we routinely ignore (St. Venant's principle).
Rather than admit I've talked myself into a corner, I'm going to claim
 at this point  that the apparent paradox is a function of our
simplification of the slender beam assumptions. :)
Jordan
Daryl Richardson wrote:
Jordan,
Actually, I believe Jim Lutz is
correct.
If you take this curvature to the extreme the two beams will form a
wheel with one beam inside of the other, hence, they can not have
exactly the same radius of curvature. I don't want to be bothered
doing the arithemetic; but I would estimate that the ratio
R(bot)/R(top) is probably about 1.001
(just a guess) for beams deflecting about
L/360,
therefore, the stresses and deflections of the two beams are equal for
most practical purposes.
Respectfully presented,
H. Daryl Richardson

Original Message 
Sent:
Tuesday, March 27, 2007 10:09 AM
Subject:
Re: Stacked Floor Joists
You're considering that the ends don't move in your paradoxical case.
By allowing the ends freedom of motion (rotation and lateral
deflection), the curvature radii will be the same.
Jordan
Jim Lutz wrote:
I have been on vacation and just read this
thread and offer the following
paradox.
Stacked members may share the same
deflection, but does this
mean they share the same bending moment? If
the members stay in contact, then the member on the concave side
necessarily has a smaller radius of curvature
and should have slightly higher bending moment than the lower member.
(Remember
1/R = M/EI from basic mechanics?) So if the moments aren
't
the same, how can the deflections be the same?
That's the paradox.
Jim Lutz, P.E., S.E
.
720 3rd
Avenue, Suite 1200
Seattle,
WA
981041820
206 505
3400
Ext 126
206 505
3406
(Fax)
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