Re: how much can concrete bend before it breaks

G’Day, Roger!



This analysis is a little messy. It has to be done in two parts.  First, 
how much can an unreinforced concrete slab support? Second, how much will 
it deflect under that load?



Concrete can support tension load, but it is not very good at it. The 
value of allowable tension strength in concrete is approximately 1/10 its 
strength in compression. Plus, when it begins to fail, it will go crack-
pow, all of a sudden—a catastrophic failure.

I started by calculating how much load your one meter by 15 cm beam 
spanning five meters could support. The formula for allowable stress is f 
= M/S, where f is the allowable stress in the material, M is the bending 
moment in the beam generated by the load applied, and S is the section 
modulus, which is a property of the shape being used. 



S for a rectangle is width times thickness squared, divided by 6. Moment M 
for a uniform load is M = w x l^2/8. By reanalyzing these two equations 
and solving first for M using a known stress, then for w, I got a 
ridiculously small value of 16 kg per square meter. But this neglects the 
weight of the concrete.  This beam would weigh 360 kg per square meter, 
meaning that it couldn’t support its own weight!

Now what? The carrying capacity of a beam increased by the square of its 
thickness, while the weight increases linearly. So by increasing the 
thickness, we eventually get to the point where the beam will work. It 
looks like a beam a meter thick could span 5 meters. It would weigh 2400 
kg per meter. And I had to assume a high strength concrete was used.

Now part two: Deflection in a beam is calculated using the formula: 
deflection = w x l^4 / (384 E I). W is the weight per foot, l is the span 
length, raised to the fourth power. I is the Moment of Inertia, a property 
of the section. For a rectangle, I = width times thickness cubed divided 
by 12. (Divide moment of inertia by the distance from the mid-point to the 
bottom edge, and you get the section modulus which we used earlier) 

E is Young’s Modulus, or the elasticity of the material:



https://en.wikipedia.org/wiki/Properties_of_concrete 





For concrete, the value of E is empirical. The ACI code allows E to be 
4700 (f’c)^.5 MPa. Combining all the factors into the equation for 
deflection I get that our one meter thick beam would deflect 1.6 cm under 
its own weight before failing.

That’s why concrete is reinforced with steel bars. Steel is almost 100 
times stronger than concrete in tension, plus when it fails it yields 
slowly, giving notice as it were that the structure is overloaded. 

Your 15 cm thick slab with a nominal amount of steel in it should be able 
to support 1000 kg per square meter.