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Rho = 0.0033, "Effective Beam Width"

Rho = 0.0033, "Effective Beam Width"

Rho = 0.0033, "Effective Beam Width"


I am designing a concrete beam which, due to other constraints, has a width dimension quite a bit larger than may be required due to the applied loading.

Normally in a beam, the minimum amount of steel is limited by the rho = 0.0033 requirement.

I noticed that if I reduced the width of the beam (compression side) and replaced it with, say marshmallow, I meet the rho = 0.0033 and achieve the necessary design strength.

I maintain that by replacing the marshmallow with concrete again, the beam can only get stronger (extra dead weight already allowed for), and my beam is sufficient.

Would anyone out there have a problem with this design, using the effective width of beam?

Regards to all,


RE: Rho = 0.0033, "Effective Beam Width"

rho is not intended to "achieve the necessary design strength", but rather, to achieve a level of ductility to avoid sudden failure should the beam be overloaded.

Read the commentary in the ACI 318 for section 10.5.1.  Yes, replacing with marshmallow is "weaker", but with a lot of concrete, and not a minimum level of steel, the non-marshmallow beam (i.e. brittle concrete) can become a safety hazard in that it would fail suddenly, with no deflection warning given.

RE: Rho = 0.0033, "Effective Beam Width"

I agree with JAE and would like to add to it.  Minimum rho values are given so that the cracked concrete section is not stronger than the uncracked section.  As the beam is loaded from zero to its maximum, it begins as uncracked and the concrete takes tension.  

When the section cracks the tension in the concrete is suddenly transferred to the steel reinforcement.  Supplying a minimum rho is designed to ensure that when this happens the steel will not fail suddenly and in a brittle manner.  I think you should follow the code.  

Your other option is the note that says if the steel supplied is so much over what is required then rho minimum may be waved.  I don't have my code with me at the moment to point the chapter and section to you.  I believe it's in the same chapter as the one that gives you minimum rho.

RE: Rho = 0.0033, "Effective Beam Width"

Yep - there is that provision in ACI where you can set your As(min) equal to the SMALLER of either

1.  The rho that we discuss above.
2.  1.33 x As(req'd), where As(req'd) is the steel calculated as necessary from the applied moment Mu.

RE: Rho = 0.0033, "Effective Beam Width"


I must say, the code clearly states that we want to insure that fracture is ductile, as you pointed out JAE and UsfSE.

But now I have a new problem.

The old rho is based upon the equ 200/fy which yields 0.0033 for 60 ksi yield rebar.  

The code as of 02 requires compliance with a second equation also of (3 SQRT fc)/fy.

This equation yields 0.0033 for fc at 4400 psi. Unfortunately (?), my breaks came in somewhat higher. Now I no longer meet the minimum steel requirement and the failure will no longer be ductile.

I do not see how this problem can be avoided because 95% of the cylinder tests must be higher than the specified strength of concrete.

Now that's a poser isn't it?

Regards to all,


RE: Rho = 0.0033, "Effective Beam Width"

"This equation yields 0.0033 for fc at 4400 psi. Unfortunately (?), my breaks came in somewhat higher. Now I no longer meet the minimum steel requirement and the failure will no longer be ductile."

Concrete test cylinder breaks are used to check the consistancy of the concrete and are only a relative indication of in-place concrete strength.  The standard test cylinders are usually field cured for 24 hours then either steam cured or water bath cured, neither of which is done to the in-place concrete, so their strength is NEARLY always higher than the inplace concrete of the same age.

Following your logic, we would have to redesign every concrete structure after recieving the lab's test report on the cylinder breaks.  You must be consistant in your design calculations and in reality, they have little to do with the cylinder break result.

RE: Rho = 0.0033, "Effective Beam Width"

I agree - and one other point - the ACI Code says that you must be at least 3 x sqrt(f'c)/fy.  Note that the value is f'c, not fc.  There is a distinct difference.

f'c is the specified compressive strength of concrete at 28 days, not the actual compressive strength.  So technically, you don't have a problem here.

RE: Rho = 0.0033, "Effective Beam Width"

Just as a side note I want to add that minimum rho for negative reinforcing is higher than that for positive reinforcing.  

RE: Rho = 0.0033, "Effective Beam Width"


First of all, I'd like to make the comment that I appreciate bouncing these ideas around amongst my colleagues.

The commentary in the ACI 318 for section 10.5.1 has answered my first question so thanks for the response JAE.

My second question is more of a reality check than anything else. I agree that it is impossible to redesign every structure based upon the actual compressive strength of concrete, which is used in a structure.

I sort of disagree with the statement that the strength of the tested concrete cylinders is "NEARLY always higher than the inplace concrete of the same age". I count on getting at least the specified concrete strength in the field. Also, as I understand the hydration process, concrete continues to get stronger with age.

I suppose that in truth, we have to design a concrete beam to be ductile in failure based upon all the code requirements.

In reality the concrete may very well have a higher compressive strength than specified, therefore a non-ductile failure zone may be encountered not much different from the "effective width" method previously discussed.

When it comes to design though, I suppose I will stay away from using an "effective width" method.

Regards to all,


RE: Rho = 0.0033, "Effective Beam Width"

I'll relate one story from a number of years ago that relates to the redesign comment I made earlier in this thread.  

A structural engineer designs the precast columns, beams, double tee roof plank and wall panels for a large food warehouse.  During construction, the general contractor discovers that there is excessive deflection in the beams supporting the roof double tees so he calls a halt to construction, shores the beams and calls the design engineer.  After a few weeks of delay, the engineer finds that his calculations were in error and there is no excess capacity at all in his design.  The building is not in danger of colapse, but there is more than the allowable deflection in the precast beams.  

However, he finds that the concrete cylinder breaks are about 1,000 psi higher than his design f'c.  He revises his calculations using the higher concrete strength to "prove" that the building is OK.  To the best of my knowledge there was no field testing done on the precast beams in question.

The question for the "good of the order" is: Is this acceptable practice?

RE: Rho = 0.0033, "Effective Beam Width"

Hmmm...good story and interesting question.

I would think that I would also get field cores to double check my cylinder breaks, but even with that, the cylinder breaks are at 28 days and fc would go up over time slightly more anyway.

Also, our designs include a phi factor, of which part of it is the variation of f'c.  Since the concrete was placed and the tests performed, the statistical uncertainty of f'c would then be lower - but how to quantify the portion of f'c variation within phi is not easily determined...if at all.

I suppose that the building in this story is still standing?

RE: Rho = 0.0033, "Effective Beam Width"


I agree with you, at a minimum in a case like this the concrete in the beams should have been tested in the field to verify the in place strength, not just rely on the cylinder tests.  IF the inplace strength were higher than the design f'c, then I would use the higher strength.  However, the higher than allowable deflection would still have to be justified in some way and in this case I don't think it was, it was just accepted.  

Yes, the building is still standing and is now approximatly 30+ years old and has survived some pretty heavy snow falls. A testament to continued concrete strength gain? conserative phi factor? or just lucky?

VirtualEngineer:  The reason for my statement regarding test cylinder strengths "NEARLY always higher than the inplace concrete of the same age" is because of the difference in curing that the cylinders get.  Concrete strength is a function not only of the materials but of the curing it receives and test cylinders usually receive better treatment than the in-place concrete.  That is why low cylinder breaks get everybody's attention so quickly.  The cylinder tests check the mix design and consistancy of the concrete, not the in-place strength, unless they are field cured.

RE: Rho = 0.0033, "Effective Beam Width"


I think it is quite common for the in place concrete to be significantly higher than the specified strength. The difference between the compressive strength of in place concrete and lab cured concrete is something of a side issue in that regard.

As soon as this occurs a beam element designed to fail in a ductile manner could easily fall in the region of beam design where a brittle failure is expected.

Is there anything we can do about it? I doubt it.

The beam will always be stonger than the originally designed beam. The situation is similar to a beam with less than the minimum steel requred to prevent brittle failure.

I don't want to kid myself about the situation, and I think this is just an example where the code is biting its own tail a little bit.



RE: Rho = 0.0033, "Effective Beam Width"

Just curious, but a rho min of (3*sqrt f'c)/fy has always been in the code, just above the (200/fy)at equation (10-3).  Have I been missing something?

Also, I thought because of the large deviation in concrete mixing, your cylinder breaks should almost always be higher than the compressive strength used for design.  Say your mix design has a reliability of 95%, shouldn't 95 out of 100 breaks be greater than or equal to design f'c?  

Been a few years since I was in school, just trying to refresh myself as well.  So let me know if I've lost it.

RE: Rho = 0.0033, "Effective Beam Width"


I just looked in my ACI 318-95 and that section is shown as revised.  Perhaps the older codes made a recommendation of (2.67 sqrt f'c)/fy as a minimum (I found this formula in an old textbook) whereas the 95 code makes it a requirement.

For using the effective "beam width method" I described above, I have set up my mathcad worksheet to automatically increase the design moment by a ductility factor of 1.33.

As JAE says above, technically the design is based upon the specified compressive strength. From a legal point of view the engineer is covered.

If the actual strength goes up to some higher strength (which as blake says is 95% of the time) a brittle failure may occur but at least the engineer followed the letter of the ACI code.

As I stated above, I think the code is biting its own tail a little bit on this point.



RE: Rho = 0.0033, "Effective Beam Width"

Keep in mind that if your f'c goes up, the load at crushing goes way up as well so a high fc is not the end of the world...the critical, abrupt failure would get less critical at higher and higher loads as the potential of the higher overload would reduce.  But I can't visualize the statistics here so I'm sure its still not a desired outcome to have higher than expected concrete strengths.

The minimum limit, at least through the late 70's, into the 80's and 90's was always simply set at 200/fy.  Just in recent editions of ACI 318 have they used the 3sqrt(f'c) with 200/fy as another check limit.

RE: Rho = 0.0033, "Effective Beam Width"

You beat me to the next post, I've been thinking more since that last post and I think the reliability is actually 90%.  In a flexural member, that is what you desire for a ductile failure.  The concrete should be stronger than the steel.  If the steel is the weak link, but not too weak, the steel will yield slowly.  

RE: Rho = 0.0033, "Effective Beam Width"

I don't want to steer you guys wrong, so I broke out the text book,  ACI has established the allowable risk as 10%, so the 90% thing was correct.  But, it can't be too high, there is a formula including the standard deviation, which I guess is arbitrarily chosen, that limits how high above f'c the concrete can be, called fcr.  Wow, its been a while since I thought about mix design.  Thanks for the engineering therapy guys.

RE: Rho = 0.0033, "Effective Beam Width"

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