# Composite beam calculation

I'm building a shelf out of an Oriented Strand Board (OSB-3), reinforced with steel strip attached to it's bottom side. Please check my calculations for deflection.

The OSB thickness is $h_{OSB}=18 \text{ mm}$, it's width $b_{OSB}=500 \text{ mm}$, I googled OSB's modulus of elasticity to be $E_{OSB}=3.5 \text{ GPa}$. For steel strip it's $h_{st}=4 \text{ mm}$, $b_{st}=60 \text{ mm}$, $E_{st}=200 \text{ GPa}$. The shelf span $L=2000 \text{ mm}$. Load is uniformly distributed $q=500 \text{ N/m}$.

I approach the task with utmost utilitarianism, using online tools and off-the shelf formulas:

1. I transform area of OSB into steel: $$b_{eq}=\dfrac{E_{OSB}}{E_{st}} \cdot b_{OSB}=\dfrac{3.5}{200}\cdot 500=8.75 \text{ mm}$$

2. I find moment of inertia of uniform inverted T-beam from step 1, using an on-line calculator: $$I=16079 \text{ mm}^4$$

3. I obtain max deflection at center $$d=\dfrac{5qL^4}{384E_{st}I}=\dfrac{5\cdot500\text{ N/m}\cdot(2000\text{ mm})^4}{384\cdot200\text{ GPa}\cdot16079\text{ mm}^4}=32 \text{ mm}$$

My questions:

1. Does OSB modulus of elasticity look legit?
2. Did I miss something, are the calculators and formulas appropriate here?

The modulus of elasticity depends on the grade of OSB you're adopting. The code BS EN 300:2006 has a table of OSB properties, including the modulus of elasticity. This site describes OSB and presents all of these properties for each grade.

OSB has four grades, each with a different modulus of elasticity:

• OSB/1 - General purpose boards and boards for interior fitments (including furniture) for use in dry conditions ($E = 2.5\text{ GPa}$)
• OSB/2 - Load-bearing boards for use in dry conditions ($E = 3.5\text{ GPa}$)
• OSB/3 - Load-bearing boards for use in humid conditions ($E = 3.5\text{ GPa}$)
• OSB/4 - Heavy-duty load-bearing boards for use in humid conditions ($E = 4.8\text{ GPa}$)

So your adopted modulus is adequate if you're using OSB/2 or 3, but if you're making furniture, you're probably using OSB/1, in which case you should adopt $E = 2.5\text{ GPa}$. This would increase the deflection to 41.25 mm.

One possible issue is certifying that there is in fact deformation compatibility between the OSB and the steel strip. Simple glue may not be sufficent to guarantee this. If the steel and the OSB are allowed to deform independently, even just a little, that will dramatically reduce the stiffness and increase the deflection of the section.

• The issue of deformation compatibility is perfectly answered in WoodWorking SE. I'm going to use countersunk screw head to mitigate any slippage. – Gleb May 6 '16 at 19:51
• OSB is mostly used in construction here in Ukraine, so OSB-3 is much more widely available, that's what I'm using. – Gleb May 6 '16 at 19:56