Neutral Axis of a Cracked Section

Hardcracked-sectionneutral-axistransformed-section

A singly reinforced rectangular beam has b = 300 mm, effective depth d = 450 mm, As = 1,000 mm², and modular ratio n = Es/Ec = 8. For a cracked elastic section, ignore all tensile concrete and use the transformed steel area nAs. Find the neutral-axis depth x below the top compression face.

A123.4 mm
B130.5 mm
C225 mm
D51.5 mm

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Answer: B. 130.5 mm

Equation: first moment of the transformed area about the neutral axis is zero: bx²/2 = nAs(d − x). Inputs: nAs = 8(1,000) = 8,000 mm². Substitution: 150x² + 8,000x − 3,600,000 = 0, so x = [−8,000 + √(8,000² + 4·150·3,600,000)]/300 = 130.5 mm. Wrong: 123.4 mm uses (n − 1)As = 7,000 mm², which applies to an uncracked section where the steel displaces concrete; 225 mm is d/2, the uncracked gross-section centroid location, ignoring cracking and steel; 51.5 mm uses As without the modular ratio n.

\frac{b x ^{2}}{2}

= n A_{s} (d - x)

\Rightarrow 150 x^{2} + 8000 x - 3.6 \times 1 0^{6}

= 0

\Rightarrow x

= \frac{- 8000 + 800 0 ^{2} + 4 ( 150 ) ( 3.6 \times 1 0 ^{6} )}{300}

= 130.5

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