How to find the Stiffness of a Composite Section?

In order to calculate the bending stiffness (i.e. second moment of area) of a composite section made up of 2 or more materials, you need to transform all of them into a common material based on the relative Youngs Modulus. 

There's two ways to do this:

• convert the weaker materials into the stronger (stiffer) materials
• or vice versa.

For example, if you have a steel material and timber material connected together, you can convert the timber into an "effective steel section" based on the ratios of the Youngs Moduli.

For the procedure and examples described below, we will be converting the weaker material into the stronger material.

General Procedure

Step 1: Find the modular ratio which describes the relative stiffness between the two materials. The weaker material should be E₁

Step 2: The geometry of the weaker material needs to be adjusted based on the transformed area. In this case, we are also adjusting the width, not the height about Y-Y.  The new width of the weaker material is:

Transformed width,

Step 3: Calculate the new neutral axis of the transformed section. 

Step 4: Now that we have the transformed geometry of the composite section, we can calculate the equivalent second moment of area using the parallel axis theorem...

Now that we have the section stiffness, we can check the stresses and deflections of our composite section. Let's see an example of the above steps applied to some real world examples...

Flitch Beam Example

Consider the flitch beam above which consists of a 12 mm thick steel plate (200 mm deep) sandwiched by two sections of 50x200 mm sawn timber connected using thru-bolts. Lets consider the Youngs modulus to be E₁ = 12,000 MPa for the timber sections, and E₂ = 205,000 MPa for the steel plate.

Step 1: Modular ratio,
Step 2: Transformed width of timber sections,

The new transformed section now looks like this:

Notice that because the neutral axis of the timber is the same as the steel, there is no need to calculate the neutral axis position (skip step 3). The new second moment of area can be simply calculated as:

Step 4: Second moment of area calculations...





Therefore 

With the benefit of the steel plate, the composite section is about 5 times more stiff against bending!

Steel Beam with Concrete Slab Example

Let's consider another example of a steel beam with a concrete slab above. In this case, we shall use:

• UB-533x210x82 steel beam
  A = 105 cm2, I = 47,500 cm4, E_s = 205,000 MPa
  
  
• Flat slab above (t = 120 mm), effective width of slab to be 1.8 m
  E_concrete = 30,000 MPa
  
  

Step 1: Calculate modular ratio,
Step 2: Transformed width,

Step 3: New neutral axis,   

The new transformed section looks as follows:

Step 4: Transformed section second moment of area, it is a lot easier to calculate without making any mistakes by taking our units in cm instead of mm...



If we try to consider how much the stiffness of this system has increased compared to considering just the steel beam along, the "stiffness modifier" can be calculated as:

Stiffness modifier,

Steel Beam Composite Comparison

Comparison of composite action between steel beam and (i) a concrete slab, and (ii) a CLT slab.

Coming soon and on request...