A beam of square cross section (a = length of each side) is bent in the plane of a diagonal (see figure). By removing a small amount of material at the top and bottom corners, as shown by the shaded triangles in the figure, we can increase the section modulus and obtain a stronger beam, even though the area of the cross section is reduced.
(a) Determine the ratio β defining the areas that should be removed in order to obtain the strongest cross section in bending.
(b) By what percent is the section modulus increased when the areas are removed?

/XA Ba