Minimum and Maximum Reinforcement for an Experimental Beam

1. Minimum Reinforcement

In an earlier lesson (here) we calculated the strength of our experimental beam. We already have a couple links at our disposal for minimum reinforcement ... here (Item 7) ... and ... here (Item 5). Also, Table 13.3 of your Ambrose text gives minimum reinforcement values (ratios) for some sets of f_y and f '_c, as well as Code equations on the preceding page.

Minimum reinforcement in equation form ...

... ρ _min = 3 √f '_c / f_y ... but not less than ... 200 / f_y ...

For our example ...

... ρ _min = 3 (√4000) / 60,000 ... but not less than ... 200 / 60,000 ...

Gives ..

... 0.0032 ... but not less than ... 0.0033 ...

So ...

... ρ_min = 0.0033.

Our beam ... (1 - # 6, blah, blah ... d = 7.5 in., and so on ...)

... ρ = A_s/(bd) = 0.44 / (5.5 x 7.5) = 0.01067 ...

GOOD! ... our ρ = 0.01067 ... which is not less than the minimum ... ρ _min = 0.0033.

2. Maximum Reinforcement

For members loaded primarily in flexure (bending) the amount of tension reinforcement is limited by Code to an amount that will strain 0.004 by the time the concrete crushes. This will result in a reinforcement ratio limitation of (not showing the derivation) ...

ρ _{max, 0.004} = 0.364 β₁ f 'c / f_y ,

where,

β₁ = established by the Code and is 0.85 for f '_c of 4000 psi and less ... and decreased by 0.05 per thousand f '_c but in no case less than 0.65.

For our example,

ρ_{max, 0.004}= 0.364 β₁f '_c / f_y = 0.364 (0.85) 4000/ 60,000 = 0.0206.

Our beam ...

... ρ = 0.01067 ...

So, ... GOOD! ... the amount of reinforcement in our beam is less than the maximum amount ... way less.

Way good!

This `says' that the steel will yield before, actually `way before', the concrete crushes.

Hmmmm ...

... this seems contradictory to what we found in terms of the load to `first yield the steel' versus the load to `first crush concrete' ... which were really pretty close to each other (P_y = 9600 lb and P_c = 10,100 lb). And it is (contradictory). Actually, when we calculated the load that will first bring the concrete to f '_c ... we assumed that concrete is `linear' in stress and strain up to it's crushing point. And it is not (linear). Beyond about ½ of f '<sub>c</sub> the concrete takes on stress `slower' with increasing strain (bent-ness) ... and, in fact, will reach f '_c and still not break (crush) until strained (bent) even more. At higher strains (bent-ness) the stress actually drops off a bit, with the concrete crushing at a strain of ... 0.003. It is this 0.003 value that is embedded in the equation for ρ _max. And, it is this non-linearity of stress and strain that will cause us (all other things equal) to way underestimate P_c (depending on how we define it) using a linear stress distribution or f_c = M y_c / I type calculation. What actually happens as the beam is more bent is that more of the concrete takes on high stress; the stresses don't exceed f '_c, but more concrete is stressed at a value nearer f'_c, instead of a linear (straight line) drop from f'c to zero.

Maybe I could say it like this: with 1 - #6 bar, according to theory, the extreme fibers of concrete are getting near f '_c as the steel reaches yield. If we double the amount of steel, a whole lot more of the concrete will be stressed at a higher level, indeed, near crushing.

3. Summary

Our first two Code checks are good. We have greater than the minimum amount of reinforcement, which `says' that we have enough steel to `catch' the beam as it first cracks in flexural tension. Also, we have less than the maximum amount, which means that the steel will be forced to yield before (way before, in our case) the concrete crushes. And even though the load to first yield the steel is not much less than the load to first bring concrete to f '_c, we know that at these higher stresses concrete is not linear ... not only will it take more bending to get the extreme fibers to f '_c, it will take even more to actually get the concrete to crush.

References

Strength and Deflection Calculations of an Experimental Beam, Jeff Filler, Associated Content.

Rebar, Jeff Filler, Associated Content.

Flexure of Reinforced concrete Beams, Jeff Filler, Associated Content.

Simplified Engineering for Architects and Builders, Ambrose, J. and P. Tripeny, 10^th edition, John Wiley & Sons, Hoboken, New Jersey.

Building Code Requirements for Structural Concrete, ACI 318, American Concrete Institute, P.O. Box 9094, Farmington hills, Michigan, 48333.

Design of Concrete Structures, Christian Meyer, Prentice Hall, 1996.