Beam deflection trends: For a simply supported rectangular beam, which proportionality best characterizes how midspan deflection varies with section dimensions and span?

Introduction / Context:Qualitative proportionalities are frequently examined in structural analysis to anticipate stiffness changes without full calculations. For a simply supported beam under standard loads, deflection depends on E, span L, and the second moment of area I of the cross-section. For a rectangular section, I is highly sensitive to depth.

Given Data / Assumptions:

• Simply supported beam, rectangular section.
• Linear elasticity (E constant).
• Comparisons concern geometric influence, not changes in load magnitude.

Concept / Approach:Take the classic case of a central point load where δ ∝ L^3/(EI). For a rectangle, I = bd^3/12. Holding width b constant, δ ∝ 1/d^3. This sharp dependence makes depth the dominant stiffness driver in beams.

Step-by-Step Solution:Use δ ∝ 1/I and I ∝ d^3 ⇒ δ ∝ 1/d^3.Thus, doubling depth reduces deflection by 2^3 = 8 times.Other proportionalities (to width or length) exist, but the question seeks the most characteristic statement highlighting depth sensitivity.

Verification / Alternative check:For a uniformly distributed load, δ_max ∝ L^4/(E*I); the inverse cube dependence on depth remains because I still includes d^3.

Why Other Options Are Wrong:'Directly proportional to its weight': weight is not a defining variable unless using self-weight loading specifically.'Inversely proportional to its width': true if depth fixed, but the depth-cube dependence is more defining; the standard single best choice emphasizes depth.'Directly proportional to the cube of its length': only for a specific loading; generalized wording can mislead.

Common Pitfalls:Assuming linear dependence on depth; ignoring that I scales with d^3 for rectangles; mixing load type effects.

Final Answer:Inversely proportional to the cube of its depth