Cracked section properties

Transformed to concrete

\[\begin{align} n & = \frac{E_{s}}{E_{c}} \\ p & = \frac{b}{n A_{s}} \\ q & = \frac{(n - 1) A'_{s}}{n A_{s}} \\ x & = \frac {\sqrt{(q + 1)^{2} + 2 p (q d' + d)} - q - 1}{p} \\ I_{c} & = \frac{b x^{3}}{3} + (n - 1) A'_{s} (x - d')^{2} + n A_{s} (x - d)^{2} \end{align}\]

Symbol Description
\(A_{s}\) area of bottom steel
\(A'_{s}\) area of top steel
\(b\) width of section
\(d\) depth to bottom steel (effective depth)
\(d'\) depth to top steel
\(E_{c}\) Young’s modulus of concrete
\(E_{s}\) Young’s modulus of rebar
\(I_{c}\) second moment of area of section transformed to concrete
\(n\) modular ratio
\(x\) depth to neutral axis

For cracked in hogging, flip upside down

Transformed to steel

\[ I_{s} = \frac{I_{c}}{n} \]

Derivation

Assumptions for cracked section properties:

Before we can calculate the second moment of area, we need an expression for \(x\), the depth to the neutral axis. To find an expression for \(x\), we note that the first moment of area about the neutral axis equals zero.

\[ \sum A \overline{x} = 0 \]

\[\begin{align} b x \frac{x}{2} + (n - 1)A'_{s} (x - d') + n A_{s} (x - d) & = 0 \\ \frac{b}{2n A_{s}} x^{2} + \frac{(n - 1) A'_{s}}{n A_{s}} (x - d') + (x - d) & = 0 \end{align}\]

Simplify using the following substitutions

\[\begin{align} p & = \frac{b}{n A_{s}} \\ q & = \frac{(n - 1) A'_{s}}{n A_{s}} \end{align}\]

\[\begin{align} \frac{p}{2} x^{2} + q (x - d') + (x - d) & = 0 \\ \frac{p}{2} x^{2} + (q + 1) x - q d' - d & = 0 \end{align}\]

Solve for \(x\) using the quadratic equation

\[\begin{align} x & = \frac {-(q + 1) + \sqrt{(q + 1)^{2} + 2 p (q d' + d)}}{p} \\ x & = \frac {\sqrt{(q + 1)^{2} + 2 p (q d' + d)} - q - 1}{p} \end{align}\]

Write expression for second moment of area

\[\begin{align} I &= \frac{b x^{3}}{12} + b x ( \frac{x}{2} )^{2} + (n - 1) A_{s} (x - d')^{2} + n A'_{s} (x - d)^{2} \\ I &= \frac{b x^{3}}{3} + (n - 1) A'_{s} (x - d')^{2} + n A_{s} (x - d)^{2} \end{align}\]