Consider the cross-section of a beam made up of thin uniform elements having thickness t(t << a) shown in the figure. The (x, y) coordinates of the points along the center-line of the cross-section are given in the figure.

The coordinates of the shear center of this cross-section are :

Two steel columns $$P$$ (Length $$L$$ and yield strength $${f_y} = 250\,\,MPa)$$ and $$Q$$ (length $$2L$$ and yield strength $${f_y} = 500\,\,MPa$$) have the same cross-sections and end-conditions. The ratio of buckling load of column $$P$$ to that of column $$Q$$ is

The ratio of the theoretical critical buckling load for a column with fixed ends to that of another column with the same dimensions and material, but with pinned ends, is equal to

The effective length of a column of length $$L$$ fixed against rotation and translation at one end is