Cantilever of uniform strength under uniformly distributed load: keeping the breadth b constant, how should the depth d(x) vary along a cantilever of length l (measured from the free end) so that the extreme fiber stress remains constant?

Introduction / Context:
A member of “uniform strength” has constant extreme fiber stress along its length. For a cantilever with uniformly distributed load (UDL), the bending moment varies with the square of the distance from the free end. To maintain constant stress while keeping breadth b constant, the depth must vary appropriately.

Given Data / Assumptions:

• Cantilever, span l, fixed at one end, free at the other.
• UDL intensity w (force per unit length).
• Breadth b remains constant; depth d(x) varies with position x from the free end (x = 0 at free end).
• Elastic behavior; rectangular section modulus Z = b * d^2 / 6.

Concept / Approach:
At distance x from the free end, the cantilever bending moment under UDL is:
M(x) = w * x^2 / 2For uniform strength, extreme fiber stress σ = M / Z must be constant, so M(x) / [b d(x)^2 / 6] = constant ⇒ d(x)^2 ∝ M(x) ⇒ d(x) ∝ √M(x). Because M(x) ∝ x^2, we obtain d(x) ∝ x.

Step-by-Step Solution:
1) Write M(x) = w x^2 / 2.2) Require σ = constant ⇒ M(x) / Z = constant.3) With Z = b d(x)^2 / 6 and b constant ⇒ d(x)^2 ∝ M(x).4) Therefore d(x) ∝ √M(x) ∝ √(x^2) ⇒ d(x) ∝ x.5) Hence d = 0 at x = 0 (free end) and increases linearly to maximum at the fixed end.

Verification / Alternative check:
If d doubles when x doubles, the section modulus increases by a factor of 4 while M increases by a factor of 4, keeping σ uniform, confirming linear variation suffices.

Why Other Options Are Wrong:
Constant depth: would produce maximum stress at the fixed end and lower stress elsewhere.d ∝ x^2 or d ∝ √x: mismatch the required square relationship between Z and M.Maximum at free end: opposite to bending-moment distribution.

Common Pitfalls:

• Measuring x from the fixed end instead of the free end when using M(x) = w x^2 / 2.
• Using Z ∝ d instead of Z ∝ d^2 for a rectangular section.

Final Answer:
d(x) is proportional to x (zero at free end and maximum at fixed end)