robinho

Hi studiot, thanks so much for answering. Actually I already got my answer to this particular question. By the way, is there Michell problems in the book you scanned?

I wonder if there're are tables of these in any book?

Hi studiot, thanks so much for answering. I want a physical explanation. I don't think you answered directly why dRx/dx=-dRy/dy. If we take a differential element on the boundary and analyze force balance, can we get that?

By the way what is the best branch on this website to post elasticity questions?

Airy stress function in boundary conditions in 2D elasticity has a requirement that its x and y derivatives be related to the x and y boundary tractions Rx and Ry, i.e.: dAiry/dy=Rx, dAiry/dx=-Ry. Then dRx/dx=-dRy/dy? How to understand this identity?

Thank you very much, though I haven't found any warping or bending stress functions for bars with such simple cross sections as circles or rectangles yet. I'll keep scanning. Maybe I should've called them stress functions from the beginning, stress functions satisfying harmonic PDEs or biharmonic PDEs.

Thank you very much for your reply😛. I look for analytic solutions. I'm aware that analysis is only valid for a range of width to length or depth to length ratios. But analytic solutions for torsion isn't aspect ratio sensitive. Yes, I refer to torsion. When you twist a bar and if the bar is not of round cross section, there is longitudinal (z) displacement and this is where the warping function psi(x,y) comes from. When the cross section is round, psi=0. I don't have the book Formulas for Stress, Strain, and Structural Matrices. Don't know if I can find it online.

thank you so much! I checked the books you suggested. For example, I couldn't find warping function formulae of common cross sections of a beam. Imagine having a straight bar of known cross section, then twist it from both ends, and compute the longitudinal displacement, then we need a warping function satisfying a poisson equation. The solution depends only on the cross section, not anything else. Is there a table listing such warping functions?

Where can one find warping and bending function formulae of common cross sections of a beam in St Venant problems in elasticity?