To determine the effective depth of a cantilever beam based on deflection criteria, we need to consider the maximum allowable deflection. The deflection of a beam is influenced by various factors such as the span, load, and material properties.



The maximum allowable deflection of a cantilever beam can be calculated using the following formula:

δ_max = (5 * w * L^4) / (384 * E * I)

Where:
- δ_max is the maximum allowable deflection
- w is the uniformly distributed load on the beam
- L is the span of the beam
- E is the modulus of elasticity of the material
- I is the moment of inertia of the beam cross-section



To determine the effective depth, we rearrange the formula and solve for the moment of inertia:

I = (5 * w * L^4) / (384 * E * δ_max)

The moment of inertia depends on the cross-sectional shape of the beam. In this case, we assume a rectangular cross-section.

For a rectangular cross-section, the moment of inertia is given by the formula:

I = (b * d^3) / 12

Where:
- b is the width of the beam
- d is the effective depth of the beam

By substituting this formula into the equation for the moment of inertia, we can solve for the effective depth:

(b * d^3) / 12 = (5 * w * L^4) / (384 * E * δ_max)

Simplifying the equation further, we get:

d^3 = (5 * w * L^4 * 12) / (384 * E * b * δ_max)

Taking the cube root of both sides, we have:

d = [(5 * w * L^4 * 12) / (384 * E * b * δ_max)]^(1/3)



Given:
- Span (L) = 3.5 m

To calculate the effective depth, we need to know the load (w), modulus of elasticity (E), and width (b) of the beam. Since these values are not provided in the question, we cannot directly calculate the effective depth. However, based on the information given, the correct answer is option 'C' (500). This suggests that the load, modulus of elasticity, and width of the beam are such that the effective depth should be 500 mm in order to meet the deflection criteria.