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 Derivation:

1. Deflection at the Free End:
As shown in the image, we have a cantilever beam of length L fixed at the left end and subjected to a transverse concentrated load W at the free tip (right end). According to Euler-Bernoulli beam theory, the maximum downward deflection at the tip of the beam, denoted as $\delta$, under a tip load W is given by the standard formula:

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

where:
• W is the transverse tip load,
• L is the length of the cantilever beam,
• E is the Young's modulus of the beam material, and
• I is the area moment of inertia of the beam's cross-section (with EI representing the flexural rigidity).

2. Definition of Effective Stiffness:
The effective stiffness (denoted by k) of a structural element is defined as the ratio of the applied load to the resulting deflection along the line of action of the load:

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

3. Calculating Effective Stiffness:
Substituting the expression for deflection $\delta$ into the stiffness equation:

$$k=\frac{W}{\left[\frac{W{L}^3}{3EI}\right]}$$

Simplifying this expression by canceling out the load term W:

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

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