# 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:

• Reinforcement is transformed to equivalent concrete section
• Concrete in tension is ignored

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}