The effective stiffness of a cantilever beam of length L and flexural rigidity EI subjected to a transverse tip load W is

The correct option is 3EI/L³.

Step-by-Step Explanation:

1. Understanding the System from the Image:
As shown in the provided diagram, we have a horizontal cantilever beam of length L, which is fixed at the left support and free at the right end. A transverse tip load W acts downwards at the free end. The flexural rigidity of the beam is EI.

2. Deflection of the Beam Tip:
From standard structural mechanics (using methods such as double integration, Macaulay's method, or moment-area theorems), the maximum downward deflection _$\delta$ at the free tip of a cantilever beam of length L under a concentrated point load W at the tip is given by:

$$\delta =\frac{W{L}^3}{3EI}$$

3. Calculating the Effective Stiffness:
Stiffness (k) is defined as the force required per unit displacement. Mathematically, it is the ratio of the applied load (W) to the corresponding deflection (_$\delta$):

$$k=\frac{W}{\delta }$$

Substituting the expression for _$\delta$ into this equation:

$$k=\frac{W}{\left(\frac{W{L}^3}{3EI}\right)}$$

Simplifying the fraction, the load term W cancels out from the numerator and denominator:

$$k=\frac{3EI}{L^3}$$

Therefore, the effective stiffness of the cantilever beam under a transverse tip load is 3EI/L³.