Strength of Materials — Cantilever beam with end load: effect of doubling the width on tip deflection A cantilever beam of rectangular cross-section carries a single concentrated load at the free end. If the breadth (width) b of the section is doubled while the depth h and all other parameters (load, span, material) remain unchanged, by what ratio does the free-end deflection change?

Introduction / Context:For a cantilever beam with a point load at the free end, the tip deflection depends inversely on the flexural rigidity EI. For a rectangular section, the second moment of area I is proportional to the width b and to the cube of the depth h. This question checks understanding of how geometric changes alter stiffness and deflection.

Given Data / Assumptions:

• Cantilever length L; end load P.
• Rectangular section with width b and depth h.
• Material modulus E is unchanged; only b is doubled.
• Small deflection Euler–Bernoulli beam theory applies.

Concept / Approach:

The tip deflection of a cantilever with an end load is given by δ = PL^3 / (3EI). For a rectangle, I = bh^3/12 about the strong axis. If b doubles, I doubles; hence δ becomes half. No other parameter changes, so the ratio of new to old deflection is 1/2.

Step-by-Step Solution:

Verification / Alternative check:

Doubling any parameter that increases I linearly will halve the deflection, consistent with δ ∝ 1/I. A quick dimensional check confirms consistency.

Why Other Options Are Wrong:

• 2 or 8 imply the deflection increases, which contradicts increased stiffness.
• 1/8 would require I to increase by a factor of 8, which does not occur by doubling b.
• 3 has no basis in the formula.

Common Pitfalls:

• Confusing the roles of width and depth; deflection is far more sensitive to h since I ∝ h^3.
• Applying the cantilever formula for a different loading or support case.

Final Answer:

1/2.