The correct answer is $\frac{{3{\text{P}}}}{{{{\text{a}}^2}}}$.
The bending stress is calculated by the following formula:
$$\sigma = \frac{M}{I}$$
where:
- $\sigma$ is the bending stress
- $M$ is the bending moment
- $I$ is the moment of inertia
The bending moment is calculated by the following formula:
$$M = P\cdot d$$
where:
- $M$ is the bending moment
- $P$ is the load
- $d$ is the distance from the load to the centroid of the beam
The moment of inertia of a square beam is calculated by the following formula:
$$I = \frac{1}{36}{\text{a}^4}$$
where:
- $I$ is the moment of inertia
- $a$ is the side of the square
Substituting the expressions for $M$ and $I$ into the expression for $\sigma$, we get:
$$\sigma = \frac{{P\cdot d}}{{\frac{1}{36}{\text{a}^4}}}$$
$$\sigma = \frac{{36{\text{P}}}}{{{{\text{a}}^4}}}\cdot d$$
In this case, the load is applied at the centroid of one of the quarters of the square. The distance from the load to the centroid of the beam is $\frac{a}{2}$. Therefore, the bending stress is:
$$\sigma = \frac{{36{\text{P}}}}{{{{\text{a}}^4}}}\cdot \frac{a}{2}$$
$$\sigma = \frac{{3{\text{P}}}}{{{{\text{a}}^2}}}$$