Why is sigma_s = m * sigma_c in the RC column?

Steel and concrete shorten by the same amount, so their strains are equal

Steel and concrete carry equal loads

Steel and concrete have equal areas

Steel and concrete have the same modulus

Which pair of stresses is obtained?

Concrete 5.56 N/mm^2, steel 103.73 N/mm^2

Concrete 103.73 N/mm^2, steel 5.56 N/mm^2

Concrete 5.56 N/mm^2, steel 5.56 N/mm^2

2080 Bhadra · 6 marks

Reinforced concrete column stresses in steel and concrete

Short answer

Using modular ratio m=18.67, equal strain gives σs = m σc. Equilibrium of total load P = σc Ac + σs As yields σc = 5.56 N/mm² and σs = 103.73 N/mm² (both compressive).

A reinforced concrete column of size $230\ \text{mm}\times400\ \text{mm}$ has 8 steel bars of 12 mm diameter and subjected to an axial compression of 600 kN, find the stresses developed in steel and concrete, take modular ratio as 18.67.

The steel and concrete shorten by the same amount, so the strain is equal and $\sigma_s = m\,\sigma_c$ with the modular ratio $m = E_s/E_c = 18.67$ .

Areas

\begin{aligned} A_s &= 8\times\frac{\pi}{4}(12)^2 = 904.78\ \text{mm}^2\\ A_g &= 230\times400 = 92000\ \text{mm}^2\\ A_c &= A_g - A_s = 91095.22\ \text{mm}^2 \end{aligned}

Equilibrium

P = \sigma_c A_c + \sigma_s A_s = \sigma_c\,(A_c + mA_s)

600\times10^3 = \sigma_c\,(91095.22 + 18.67\times904.78) = \sigma_c\,(107987.44)

\sigma_c = 5.556\ \text{N/mm}^2,\qquad \sigma_s = m\sigma_c = 18.67\times5.556 = 103.73\ \text{N/mm}^2

Check: load carried by concrete 506.14 kN + steel 93.86 kN = 600.00 kN.

Answer: stress in concrete = 5.56 N/mm² and stress in steel = 103.73 N/mm² (both compressive).

✓ Answer checked by Claude Opus 5.5 (high effort)

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Reinforced concrete column stresses in steel and concrete

Areas

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