resultant 9residual0 stresses in a steel beam

huangyhg

resultant (residual) stresses in a steel beam
i was wondering if someone could help me determine the resultant stresses in a steel beam after it has been loaded past yield and then released.
i've worked through this some with bending of plates.  it's kind of a pain and the problem is in the details.
one method is to assume that the entire cross section has been loaded past yield.  then as the bending moment is reduced, stresses drop the quickest at the outer faces until stress is reversed there.  for example, the tension flange will then have residual compression, which must be balanced by tension back in the web to give zero moment, and vice versa on the compression flange.
alternate method is to assume linear amount of permanent strain starting from some point away from the neutral axis and calculate stresses required to give zero moment on the cross section.
doing this with plate, i assumed that the stress-strain curve was a straight line to yield, then constant stress thereafter.  of course, in reality, the whole plot is a curve, and the actual yield stress will be higher than the minimum specified yield stress, so the whole procedure is pretty approximate.  the normal mc/i is also assuming a straight beam, so as you roll it, that assumption gradually becomes worse.
you didn't say why you needed this.  but i might point out that residual stresses are not normally considered in design.  so if you roll a beam to a radius, whatever residual stress is there is just ignored, and similarly with plates in pipe and pressure vessel design.  and of course, a beam may have all kinds of residual stresses just from the manufacture in the first place, which are likewise ignored.
is this an indeterminate beam going through plastic hinging or just partial yielding of a cross section?
just partial yielding of a cross section
do you know your material hardening behaviour?  this will affect the resulting stress distribution and depth of yield.
might pay to start with the elastic-perfectly plastic model, then move to simple linear hardening.
googling those might get you underway.