A simply supported beam carries varying load from zero at one end and w at the other end. If the length of the beam is a, the maximum bending moment will be A. $$\frac{{{\text{wa}}}}{{27}}$$ B. $$\frac{{{\text{w}}{{\text{a}}^2}}}{{27}}$$ C. $$\frac{{{{\text{w}}^2}{\text{a}}}}{{\sqrt {27} }}$$ D. $$\frac{{{\text{w}}{{\text{a}}^2}}}{{9\sqrt 3 }}$$

The correct answer is $\frac{{{\text{w}}{{\text{a}}^2}}}{{27}}$.

The maximum bending moment in a simply supported beam occurs at the point of contraflexure, which is the point where the bending moment is zero. In this case, the point of contraflexure is located at $\frac{a}{2}$ from the left end of the beam.

The bending moment at any point along the beam can be calculated using the following equation:

$$M = \frac{w}{2} x \left( \frac{a}{2} – x \right)$$

where $w$ is the load per unit length, $x$ is the distance from the left end of the beam, and $a$ is the length of the beam.

Substituting $x = \frac{a}{2}$ into the equation for $M$, we get the following expression for the maximum bending moment:

$$M_{\text{max}} = \frac{w}{2} \left( \frac{a}{2} \right)^2 = \frac{{{\text{w}}{{\text{a}}^2}}}{{27}}$$

Therefore, the maximum bending moment in a simply supported beam carrying a varying load from zero at one end and $w$ at the other end is $\frac{{{\text{w}}{{\text{a}}^2}}}{{27}}$.

The other options are incorrect because they do not take into account the fact that the maximum bending moment occurs at the point of contraflexure, which is located at $\frac{a}{2}$ from the left end of the beam.