The maximum deflection due to a load W at the free end of a cantilever of length L and having flexural rigidity E$$I$$, is A. $$\frac{{{\text{W}}{{\text{L}}^2}}}{{2{\text{E}}I}}$$ B. $$\frac{{{\text{W}}{{\text{L}}^2}}}{{3{\text{E}}I}}$$ C. $$\frac{{{\text{W}}{{\text{L}}^3}}}{{2{\text{E}}I}}$$ D. $$\frac{{{\text{W}}{{\text{L}}^3}}}{{3{\text{E}}I}}$$

The correct answer is $\frac{{{\text{W}}{{\text{L}}^3}}}{{3{\text{E}}I}}$.

The maximum deflection due to a load $W$ at the free end of a cantilever of length $L$ and having flexural rigidity $EI$ is given by the following equation:

$$\delta = \frac{{{\text{W}}{{\text{L}}^3}}}{{3{\text{E}}I}}$$

where:

• $\delta$ is the maximum deflection,
• $W$ is the load,
• $L$ is the length of the cantilever,
• $E$ is the Young’s modulus, and
• $I$ is the moment of inertia.

The equation can be derived using the following steps:

1. Assume that the cantilever is a uniform beam with a constant cross-section.
2. Apply the principle of superposition to find the deflection of the beam due to the load $W$.
3. The deflection of the beam due to the load $W$ is given by the following equation:

$$\delta = \frac{{M}}{{EI}}$$

where:

• $M$ is the bending moment at the free end of the beam,
• $E$ is the Young’s modulus, and

• $I$ is the moment of inertia.

• The bending moment at the free end of the beam is given by the following equation:

$$M = WL$$

where:

• $W$ is the load, and

• $L$ is the length of the cantilever.

• Substituting equation (2) into equation (3) gives the following equation:

$$\delta = \frac{{WL}}{{EI}}$$

1. The maximum deflection occurs at the free end of the beam, where the bending moment is maximum.
2. The maximum deflection is given by the following equation:

$$\delta = \frac{{{\text{W}}{{\text{L}}^3}}}{{3{\text{E}}I}}$$