Including

lastic pergeo ical m tee of Japan, one may find Kato’s approximate formulas for the buckling loads of heavy columns with various end conditions. Using the Rayleigh–Ritz method and the appropriate choice of Problem Definition and Formulation Fig. 1 shows a column of length L, flexural rigidity D, subjected to both its own weight per unit length and an end concentrated load P. The x-y axes are also shown in the figure. The ends of the column may be clamped, pinned, free, or have a sliding restraint. The problem at hand is to determine the buckling capacity of such a loaded column. The governing differential equation for the buckling of such a loaded column is given by Chai and Wang 2006 d4w dx4 + + 1 − x d2w dx2 dw dx = 0 1 wher