# Transforming the Area of a Hollow Square Section filled with concrete into the equivalent area of just steel

I need to complete the 3D stiffness matrix of steel concrete composite columns under all kinds of stresses (axial and rotational as shown in the image below). So because the section is a Square Hollow Section 150 x 150 x 16mm, filled with concrete, I need to transform the concrete into equivalent steel. How can I go about this to obtain the equivalent, Young's Modulus E, Shear Modulus G, Area to use and Section Moment of area I and Polar Moment of area J. Any advice on the procedure to solve this? Images of the section and the matrix are below.

• May 12 at 10:59

You can simply calculate "equivalent" cross-sectional stiffness values.

Note that all non-zero terms of the stiffness matrix depend on one of these values: $$EA$$ or $$EI$$. These are the cross-sectional axial and rotational stiffnesses, respectively.

Obviously, when looking at entire beams, another important parameter is the beam's length $$L$$ (to multiple exponents), but here we're just looking at the cross-section, so we only care about $$EA$$ and $$EI$$.

Given that the stiffness matrix only cares about $$EA$$ and $$EI$$, we can actually replace them entirely with $$K_a$$ and $$K_r$$ (axial and rotational stiffness).

The wonderful thing about doing this is that it exposes an important aspect of the stiffness matrix: it doesn't care about the beam's elastic modulus, area or moment of inertia. It only cares about its stiffness.

We can therefore do whatever we want with $$E$$, $$A$$ and $$I$$ as we see fit, so long as we get a reasonable stiffness at the end.

In this case, we're dealing with a composite beam made up of a square concrete section and a hollow square steel section. How do we find the stiffness of this beam? Well, a good first guess is to simply add the stiffness of each of component:

\begin{align} K_a &= E_c A_c + E_s A_s \\ K_r &= E_c I_c + E_s I_s \end{align}

(the math is a bit simple in this case since both concrete and steel have the same centroid; if that weren't the case, then their moments of inertia would have to be calculated relative to the composite centroid, see here an example of how to do so)

This is a pretty conservative method: you aren't taking into consideration any interaction between the components. For example, the concrete's true axial stiffness will likely be larger than $$E_c A_c$$ due to the lateral constraint imposed by the steel section (as described here).