The section modulus of a beam of square section of each side ‘b’ placed with a diagonal horizontally is 

Concept:

Section modulus of a square of side s (here s = a/2), when square beam is placed with two sides horizontal:

\(\begin{array}{l} {Z_1} = \frac{{{I_1}}}{{{y_1}}} = \frac{{\frac{{{s^4}}}{{12}}}}{{\frac{s}{2}}} = \frac{{{s^3}}}{6}\\ {Z_1} = \frac{{{s^3}}}{6} = \frac{{{{\left( {\frac{a}{2}} \right)}^3}}}{6} = \frac{{{a^3}}}{{48}} \end{array}\)

Section modulus of a square of side s (here s = a/2), when square beam is placed with two sides horizontal:

\(\begin{array}{l} {Z_1} = \frac{{{I_1}}}{{{y_1}}} = \frac{{\frac{{{s^4}}}{{12}}}}{{\frac{s}{2}}} = \frac{{{s^3}}}{6}\\ {Z_1} = \frac{{{s^3}}}{6} = \frac{{{{\left( {\frac{a}{2}} \right)}^3}}}{6} = \frac{{{a^3}}}{{48}} \end{array}\)

Section modulus of a square of side a, when square section is placed with a diagonal horizontal:

\({Z_2} = \frac{{{a^3}}}{{6\sqrt 2 }}\)

Additional Information

Ratio of the section modulii of a square beam when it is placed with a diagonal horizontal to the square  when it is placed with two sides horizontal is:

\(\frac{{{Z_2}}}{{{Z_1}}} = \frac{{48}}{{6\sqrt 2 }} = 5.65\)