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Author: Autar Kaw
Autar Kaw (http://autarkaw.com) is a Professor of Mechanical Engineering at the University of South Florida. He has been at USF since 1987, the same year in which he received his Ph. D. in Engineering Mechanics from Clemson University. He is a recipient of the 2012 U.S. Professor of the Year Award. With major funding from NSF, he is the principal and managing contributor in developing the multiple award-winning online open courseware for an undergraduate course in Numerical Methods. The OpenCourseWare (nm.MathForCollege.com) annually receives 1,000,000+ page views, 1,000,000+ views of the YouTube audiovisual lectures, and 150,000+ page views at the NumericalMethodsGuy blog. His current research interests include engineering education research methods, adaptive learning, open courseware, massive open online courses, flipped classrooms, and learning strategies. He has written four textbooks and 80 refereed technical papers, and his opinion editorials have appeared in the St. Petersburg Times and Tampa Tribune.
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The Numerical Methods Guy 
Autar – Coincidentaly I have just been looking at column buckling on my blog, using a Runge-Kutta approach:
http://newtonexcelbach.wordpress.com/2010/06/19/elegant-solutions-column-buckling-and-the-hole-through-the-middle-of-the-earth/
For a point load at the top of the column I get exactly the same result as the Euler buckling equation (within machine accuracy), but applying it to a uniform column under self weight I get a slightly different answer to the equation you posted (about 7.2% less). Do you know of a freely available derivation of the Timoshenko equation?
Doug Jenkins
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The self-weight and applying the point load to the top of column are two different problems. This is because the load on the top problem results in a reaction equal to the applied load at any cross-section. The self-weight problem reaction at any cross-section increases from zero at the top to the full weight at the bottom.
I am not aware of a freely available derivation of the Timoshenko equation.
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[quote] The self-weight problem reaction at any cross-section increases from zero at the top to the full weight at the bottom.
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Yes, I allowed for that. I’ll look into it further when I have time and let you know if it is me or Timoshnko that has got it wrong!
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Ever spend hours working on something and end up with something so obvious you can’t understand why you didn’t do it that way in the first place?
Anyway, you will be glad to know that the Runge-Kutta solution to the self-buckling problem gives exactly the same answer as the Timoshenko equation (when you do it properly). I’ll post details on my blog when I have time.
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