orsional-flexural buckling of columns

huangyhg

torsional-flexural buckling of columns
does anyone know the formula for the critical buckling axial load (or stress) of a singly symmetric wide flange  about a restrained axis?
a little background. lrfd, second edition, p. 6-110, equation a-e3-6 gives the type of buckling i'm looking for except that there is no restrained axis.  dr. joseph yura and dr. todd helwig give the equation for a doubly symmetric wide flange about a restrained axis in "bracing for stability" (notes from on seminar put on by aisc that i highly recommend).
an example of a restrained axis would be one flange is not allowed to move laterally but it can rotate about the long axis of the column.
oh, by the way, for you guys that are going to say just do an fea.  i already have, but i need a closed form solution that i can include in a program that i am writing.
for you math wizards, i have the derivation of the formula i mentioned that's in lrfd.  if i wasn't so rusty at diff. eq. i could probably change some of the boundary conditions and derive it.  if anybody's a glutton for punishment i can point you in the right direction.
thanks
for this comment i use as guide
design of steel structures 3d. ed.
gaylord, gaylord and stallmeyer
section 4-21 deals with the buckling of columns with one axis of symmetry...says
column will fail in one of 2 buckling modes, i.e., bend buckling about the x axis or a combination of twisting and bending about yhe y axis (rx and rtb to deliver the slenderness).
rt for pure torsional failure and ry intervene in the evaluation of rtb (eq 4-56 in text).
yet you are preventing on weak y axis buckling.
hence may view is that you need check the slenderness for rt an rx, simply. a program can be written for this.
i assume bending on y is prevented in the whole shape, not just at one flange. were the restraint delivered by sheet metal, my understanding of what i am saying is corresponding to the sheet meeting the shear center at the web this able to fully restrain on weak y axis bending.
if you don't prevent the lateral buckling of one flange, the shape could still have lateral torsional buckling (ltb), in the non restrained flange, hence flexural torsional in the shape (since free to rotate).
yet it is not the case you quote.
if you don't have it a good reference (but may still fail to have what you exactly seek) is
flexural-torsional buckling of structures
n.s. trahair
crc, boca raton, fl, 1993
see 'theory of elastic stability' by timoshenko and gere.
chapter on torsional buckling, sub section, buckling by torsion and flexure.
if you are thinking only of diaphragm braced columns (braced in one face) and ready to settle for one empirical equation, p.502, section 13.5.2 of
guide to stability design criteria for metal structures
5th ed.
theodor v. galambos
john wiley & sons 1998
has one empirical equation. it merely adds to the axial compressive strength for weak axis buckling pyy one term
ad x geff
ad is cross sectional area of the diaphragm normal to the column axis and contributing to the support of the member and
geff is the effective shear modulus of the diaphragm.