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 I'm not convinced, no direspect intended.  The circle case is non-sensical, as the condition can not exist - the curve is non-circular (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 non-sensical 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 ----- From: Jordan Truesdell, PE 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. 720 3rd Avenue, Suite 1200 Seattle, WA 98104-1820 206 505 3400 Ext 126 206 505 3406 (Fax) ******* ****** ******* ******** ******* ******* ******* *** * Read list FAQ at: http://www.seaint.org/list_FAQ.asp * * This email was sent to you via Structural Engineers * Association of Southern California (SEAOSC) server. To * subscribe (no fee) or UnSubscribe, please go to: * * http://www.seaint.org/sealist1.asp * * Questions to seaint-ad(--nospam--at)seaint.org. Remember, any email you * send to the list is public domain and may be re-posted * without your permission. Make sure you visit our web * site at: http://www.seaint.org ******* ****** ****** ****** ******* ****** ****** ******** ******* ****** ******* ******** ******* ******* ******* *** * Read list FAQ at: http://www.seaint.org/list_FAQ.asp * * This email was sent to you via Structural Engineers * Association of Southern California (SEAOSC) server. To * subscribe (no fee) or UnSubscribe, please go to: * * http://www.seaint.org/sealist1.asp * * Questions to seaint-ad(--nospam--at)seaint.org. Remember, any email you * send to the list is public domain and may be re-posted * without your permission. Make sure you visit our web * site at: http://www.seaint.org ******* ****** ****** ****** ******* ****** ****** ********