Base plate design for a circular column on a square base: If d is the column diameter, D is the side of the square base plate, p is the allowable bearing pressure on concrete, and a = (D − d) / 2 is the cantilever projection on each side, what plate thickness t is commonly used from cantilever bending theory?

Introduction / Context:
Steel column bases are often idealized as plates cantilevering beyond the column footprint under uniform concrete bearing. Plate thickness is sized so bending stress does not exceed the permissible value.

Given Data / Assumptions:

• Circular column diameter d on a square plate side D.
• Cantilever projection on each side: a = (D − d) / 2.
• Allowable concrete bearing pressure p and allowable plate bending stress f_b.
• Uniform bearing under the plate for sizing.

Concept / Approach:
For a unit-width cantilever of length a under uniform pressure p, the maximum moment at the fixed edge is M = p * a^2 / 2 per unit width. The extreme fiber bending stress for a rectangular plate strip of thickness t is sigma = 6 * M / t^2 (for unit width), leading to a closed-form sizing.

Step-by-Step Solution:

Verification / Alternative check:
Compare with tabulated values in design handbooks for typical p and f_b; results align within standard tolerances for plate design.

Why Other Options Are Wrong:
Different constants (2, 6, 1/3) do not follow from the cantilever derivation; option (e) uses a nonphysical D/d factor not implied by bending theory.

Common Pitfalls:
Using gross projection D/2 instead of a; neglecting local bearing beneath the column stub; ignoring biaxial effects when loads are eccentric.

Final Answer:
t = a * sqrt(3 * p / f_b)