# Moment Of Inertia Of A Cracked Reinforced Concrete Section

Most reinforced concrete beams have cracks, leading to a reduced stiffness. In structural verifications of reinforced concrete elements like beams, we use the moment of inertia as the representation of the stiffness.

Cracks happen, when the tensile stress resistance of concrete is reached. Then the concrete canโt take up any tensile forces any more and the reinforcement gets activated and takes up all tensile forces.

**Here’s the formula of the moment of inertia of a cracked reinforced concrete section (you can find a step-by-step calculation later in the article ๐๐):**

$$I_a = \frac{w \cdot x^3}{12} + w \cdot x \cdot (x/2)^2 + \alpha \cdot A_{s.t} \cdot (d_{eff.b} – x)^2$$

In some situations, reinforced concrete sections are **not** cracked. In that case, the moment of inertia is a lot bigger. **Check out this guide to learn how to calculate it for uncracked sections.**

Alright, let’s get started. ๐๐

## Process Of Calculating The Moment Of Inertia Of A Cracked Reinforced Concrete Section

Here are the steps, we need to follow to calculate the moment of inertia of a cracked rectangular reinforced concrete section (weโll calculate and define all parameters in the next section):

- Define the geometrical and material properties of the reinforced concrete section (Longitudinal reinforcement at bottom d
_{s.b}and top d_{s.t}, effective height of the reinforcement d_{eff}, E-moduli of steel E_{s}and concrete E_{c.eff}, cross-sectional dimensions h and b, etc.) - Ratio between moduli:

$$\alpha = \frac{E_s}{E_{c.eff}}$$ - Height of the compression area:

$$x = \frac{\alpha \cdot A_{s.b}}{w} \cdot (-1 + \sqrt{1 + \frac{2 \cdot w \cdot d_{eff.b}}{\alpha \cdot A_{s.b}}})$$ - Moment of inertia (cracked section)

$$I_a = \frac{w \cdot x^3}{3} + \alpha \cdot A_{s.t} \cdot (d_{eff.b} – x)^2$$

In the last section of this post, we’ll run through an example calculation of a cracked reinforced concrete cross-section. ๐๐

## Example Calculation Of The Moment Of Inertia

Now, letโs apply this to an example with the following parameters.

#### Geometrical parameters

Cross-sectional height | $h = 400 mm$ |

Cross-sectional width | $w = 300 mm$ |

Concrete cover | $c = 25 mm$ |

Stirrup diameter | $d_{s.s} = 8 mm$ |

Bottom reinforcement – diameter | $d_{s.b} = 16 mm$ |

Bottom reinforcement – number of bars | $n_{s.b} = 4$ |

Bottom reinforcement – effective height | $d_{eff.b} = h – c – d_{s.s} – d_{s.b}/2 = 41 mm$ |

Bottom reinforcement – cross-sectional area | $A_{s.b} = n_{s.b} \cdot (d_{s.b}/2)^2 \cdot \pi = 804.2 mm^2$ |

#### Material parameters

E-modulus of reinforcement | $E_s = 200 GPa$ |

E-modulus of concrete (C25) EN 1992-1-1 Table 3.1 | $E_{cm} = 31 GPa$ |

Creep factor (assumption*) | $\varphi_{28} = 2.0$ |

Effective e-modulus of concrete | $E_{c.eff} = \frac{E_{cm}}{1 + \varphi_{28}} = 10.3 GPa$ |

1. First, weโll calculate the ratio between the reinforcement and concrete moduli:

$$\alpha = \frac{E_s}{E_{c.eff}} = 19.4$$

2. Next, weโll calculate the neutral axis of the cracked section:

We’ll find the depth of the neutral axis by equilibrium of the first moment of areas of the tension (steel) and compression (triangle). But don’t get confused by the triangle here. When calculating the static moment of the compression zone, we only need the area of it and the lever arm is x/2.

Equilibrium of the 1. moment of areas:

$$w \cdot x \cdot x/2 = \alpha \cdot A_{s.b} \cdot (d_{eff.b} – x)$$

Now, let’s solve that equation for x to get the height of the compression area:

$$x = \frac{\alpha \cdot A_{s.b}}{w} \cdot (-1 + \sqrt{1 + \frac{2 \cdot w \cdot d_{eff.b}}{\alpha \cdot A_{s.b}}}) = 148 mm$$

3. Then, as the last step, the moment of inertia of the cracked section:

$$I_a = \frac{w \cdot x^3}{12} + w \cdot x \cdot (x/2)^2 + \alpha \cdot A_{s.t} \cdot (d_{eff.b} – x)^2$$

$$I_a = \frac{w \cdot x^3}{3} + \alpha \cdot A_{s.t} \cdot (d_{eff.b} – x)^2 = 1.072 \cdot 10^9 mm^4$$

## Conclusion

The cracked moment of inertia is used for example in the design of reinforced concrete beams. You can find a step-by-step tutorial here:

Or in when verifying the crack width of flat slabs. ๐๐