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The following images are taken from Mechanics of Materials, Ferdinard Beer et al. 6th ed (2012) Just wondering that why the effective length of the column is taken 0.7 L (highlighted in the sample problem) ? Souldn't it be 0.5 L since in the xy plane we have two plates supporting the column with a reaction force in the y direction and a moment (which is the same as if the column were fixed at both ends). Any thoughts would be appreciated.

enter image description here enter image description here

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The text says... "smooth and rounded fixed plates". So although the plates are fixed, they are not rigidly attached to the column. They will restrain translation in the y and that is all. So for all intents and purposes you can consider the top support to be "pinned". And the effective length can be determined from The first figure, condition c).

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  • $\begingroup$ Coreect me if i'm wrong... if the plates have finite thickness then wouldn't the reactions at the top and bottom of the plates ensure a moment even if they are not rigidly attached to the column ? $\endgroup$ May 7 '19 at 21:49
  • $\begingroup$ From a purely applied standpoint, you may be correct that if you sit a plate column tightly between two rigid plates then there would be some moment resistance. However from the wording of this problem, it specifically states what movement is restrained at the top of the column. I.E. "Two smooth and rounded fixed plates restrain end A from moving in one of the vertical planes...." Because of this wording you are safe to assume pinned at the top. Also physically, you would need rather large plates, and they would have to be tight with the column to really restrain rotation at the top. $\endgroup$
    – ShadowMan
    May 7 '19 at 22:17
  • $\begingroup$ Ok, well I agree, all doubts cleared... also from the wording as you said "smooth and rounded" we don't know to what extent ? Is it rounded enough at the contact to have a point (or line) contact or not (surface contact) which dictates the existing of moment resistance. Additionally, also depends upon the clearance as if the clearance is large enough we will rather have a free end for some angular displacement (depends on the reader). Anyways, thanks for the help. No more doubts $\endgroup$ May 7 '19 at 23:40
  • $\begingroup$ Your analysis of the details is commendable, that type of thinking will help you alot in the future in general for these types of problems, we have to try to infer some of the idealizations they are making. Typically they will be pretty straight forward. But here just the fact that they say smooth and rounded basically tells us that it is smooth and rounded enough to allow rotation. These problems are trying to teach fundamentals and will usually not be trick questions. We need to look for the keywords in the problem description then continue with the solution. $\endgroup$
    – ShadowMan
    May 8 '19 at 0:18
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    $\begingroup$ @YousufKhan The plates have "rounded" ends, so the contact between each plate and the column is only a line, not a finite area. Imagine the column was held between two circular cylinders. $\endgroup$
    – alephzero
    May 8 '19 at 10:16
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In real life cases these diagrams are all highly exaggerated to convey the idea that effective length has to do with the inflection point and or, it starts at a pin support.

Otherwise for all intents and purposes the smooth rounded edge of the two plates are close enough to a pin support at that joint.

Now if we want to be too fussy, even if the two plates are too tight and fit to the column, under the load they yield just a smidge to make it a pin connection. And we would see the yield line as shiny horizontal score lines after.

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