Composite Design - Modular Ratios

[DougP100]

Hi:

I am in the process of analyzing an existing floor to determine the maximum load it can take. It was constructed in the 1930's, and is a composite floor featuring a 4" lightly-reinforced slab that sits on a W6x12 beam. The beam is entirely encased in concrete, and the slab and beam appear to have been cast at the same time.

I am analyzing this as a T-Beam with the slab (or actually the upper portion of the slab) as the compression element and the W6x12 as the tension element. My question regards the modular ratio.

I have computed the modular ratio, n, as 8.6, using E = 29,000,000 for the steel and ACI 8.5.1 for the concrete.

I have approached the analysis from 2 points of view using the Theory of Transformed Section - Converting everything to steel, and converting everything to concrete.

I have calculated the location of the centroid from both perspectives. From All-steel P.O.V. (Point of View) I divided n into the concrete values and solved by completing the square. From All-Concrete P.O.V., I multiplied the steel values by n and solved by completing the square. Both ways yielded the same centroid location. My problem arises when I compute the Moment of Inertia, I, of the transformed section.

When I approach things from the all-steel perspective, I have divided n into the I of the concrete before adding A x d squared. When using allowable f values for steel and concrete, and using the standard f = Mc/I equation, I get one set of allowable moment values.

When I approach things from the concrete perspective, by multiplying the steel areas by n, and leaving the I values alone, and using f=Mc/I for the same allowable f values, I get significantly lower moment values. I had believed that they should be the same.

I suspect that this may be a Math error on my part and I deparately need to brush up on my algebra. Or perhaps I goofed up and one way is better than the other. Maybe I am 100% wrong in my analysis methods. Any way, could someone shed some light on where I am screwing up? I am a geotechnical engineer, and I don't do this too often....only when I have to.

Thanks!!

 

[DrAmmar]

Hi
You can refer to any book on steel design to see exactly where you had done a mistake! I recommend the book by Jack C. McCormac titled "Structural Steel Design". However note that you should be careful about the concrete section that you should include in the transformed section; only that part in compression should be included. I guess that your mistake could be that you included the whole concrete section in the transformed section; which , of course, is not correct.

 

[DougP100]

Thanks for your reply.

I actually own the 3rd edition of that book and neglected to refer to it. I never realized that there was composite design examples in it. I was busy refering to McCormac's Concrete design book, plus a few other references I have (Gaylord and Gaylord, Wang and Salmon). You have been very helpful and I appreciate it.

My problem seems to lie in my calculations of the composite moment of inertia along with where I use the modular ratio in the stress calculations. Fortunately, McCormac gives some very good information and examples for this, more in-depth than the other references I was using.

For what it's worth, I was only using the compression area of the concrete in my calculations. I was just misusing the modular ratio.

Thanks again!