Reinforced concrete columns (elastic theory): A column has longitudinal steel equal to 1% of the net concrete area. With modular ratio m = 10, if Ps and Pc are the loads carried by steel and concrete respectively, what is the ratio Ps/Pc expressed as percent?

Introduction / Context:Under the elastic (transformed section) theory, steel and concrete in a short column share load in proportion to their moduli and areas because they undergo the same strain. The modular ratio m = Es/Ec scales the steel area to an equivalent concrete area, letting us apportion the load between materials.

Given Data / Assumptions:

• Percentage of longitudinal steel As = 1% of net concrete area Ac.
• Modular ratio m = 10.
• Short column behavior; same strain in steel and concrete.

Concept / Approach:

Because strain is common, stresses relate as σs = m * σc. Load carried: Ps = As * σs, Pc = Ac * σc. Therefore, Ps/Pc = (As * m * σc) / (Ac * σc) = m * (As/Ac). With As = 0.01 Ac, Ps/Pc = 10 * 0.01 = 0.10 = 10%.

Step-by-Step Solution:

Verification / Alternative check:

Transformed section: Aeq = Ac + m As = Ac + 10(0.01 Ac) = 1.10 Ac. Steel carries 0.10 Ac worth of equivalent area; steel share vs. concrete share is 0.10:1.00 → 10%.

Why Other Options Are Wrong:

• 0.1 or 1 or 1.1 represent 0.1%, 1%, or 1.1%—not consistent with m = 10 and 1% steel.
• 11 overstates due to rounding; exact is 10% with given data.

Common Pitfalls:

• Treating 1% as of gross area including steel instead of net concrete area; here definition explicitly uses concrete area, giving a clean 10% result.

Final Answer:

10