Building rehabilitation sometimes requires to use **beams that are not found in standard shapes,** unique pieces that we usually create by means of **adding steel plates**. Before developing a complete structural model, we should err on the side of caution and check that specific shape.

One of the most important features we need to know is the **Section Modulus of the beam’s cross section** (and more specifically: the elastic section modulus – S, or Wel in Eurocodes-). The main purpose of getting S (or W) is that it is a very simple way of calculating the flexural resistance of that beam.

And as the Section Modulus is simply defined as S = I / y, being I the area moment of inertia (or the second moment of the area) and y the distance from the centroid or neutral axis to the furthest point of that section, we just need to discover the value of I.

I_x = ʃ y^2 dA \text{ and } Iy = ʃ x^2 dA

**Steiner’s theorem** allows us to get it working with smaller regular pieces. The area moment of inertia of a shape related to a certain Z’ axis is equal to the area moment of inertia of that shape, related to its own Z axis (the one passing through the body’s center of mass) plus a product of the area and the distance between the two axis:

I_x = ∑ I_x,i+ ∑(Ai \ x \ di^2)

So if we get the centroid of the complete shape, we can have its **area moment of inertia** by adding every piece’s I related to this new axis :

I_x = ∑ I_x,i+ ∑(Ai \ x \ di^2)

Once we get it, we just need to divide it between the distance to the furthest point to know S.

## Example: Creation of a T section adding two plates of 150 x 10 mm

I_x = ∑ I_x,i+ ∑(Ai \ x \ di^2)

I_a = bh^3/12 = 12500 mm^4 \text{ } Aa \ x \ da^2 = 2400000 mm^4

I_b = bh^3/12 = 2812500 mm^4 \text{ } Ab \ x \ db^2 = 2400000 mm^4

\text{So }I_x = 7625000 mm^4

\text{And then: }W_{sup} = 169444 mm^3 \text{ } W_{sup} = 66304 mm^3