Two simply supported beams B1 and B2 carry equal central point loads W. Beam B1 has span l with a 1 × 1 (arbitrary units) rectangular section; beam B2 has span 2l with a 2 × 2 rectangular section. What is the ratio of the maximum flexural stress σ_B1 : σ_B2?

Introduction / Context:
Flexural (bending) stress for a simply supported beam with a central point load depends on the maximum bending moment and the cross-section's section modulus. Comparing two beams with different spans and section sizes illustrates how geometry dominates stress levels for the same load.

Given Data / Assumptions:

• B1: span = l, section = 1 × 1; B2: span = 2l, section = 2 × 2.
• Equal central load W on each beam.
• Both sections oriented with depth equal to the second dimension listed.
• Linear elastic bending; small deflection theory.

Concept / Approach:
For a central load, M_max = W * L / 4. Bending stress at extreme fibre: σ = M * c / I, with c = h/2 and I = b * h^3 / 12 (rectangular section). Compute σ for each beam and form the ratio.

Step-by-Step Solution:
For B1: L = l, b = 1, h = 1 → I_1 = 1*1^3/12 = 1/12, c_1 = 1/2.M_1 = W*l/4, so σ_1 = (W*l/4) * (c_1 / I_1) = (W*l/4) * ((1/2) / (1/12)) = (W*l/4) * 6 = 1.5 * W * l.For B2: L = 2l, b = 2, h = 2 → I_2 = 2*2^3/12 = 16/12 = 4/3, c_2 = 1.M_2 = W*(2l)/4 = W*l/2, so σ_2 = (W*l/2) * (1 / (4/3)) = (W*l/2) * (3/4) = 0.375 * W * l.Ratio σ_1 : σ_2 = (1.5)/(0.375) = 4.

Verification / Alternative check:
Section modulus Z = I/c; Z_1 = (1/12)/(1/2) = 1/6, Z_2 = (4/3)/1 = 4/3. Using σ = M/Z gives the same result.

Why Other Options Are Wrong:

• 2, 1, 1/2, 1/4 do not satisfy the computed ratio given the strong increase in section modulus for B2.

Common Pitfalls:
Using I proportional to b*h^2 instead of b*h^3; forgetting that span doubles the moment for the same W but section size has a larger effect here.

Final Answer:
4