R.C. Beam question

[DWHA]

This problem came out of a PE study guide. The question is asking what is the amount of reinforcing steel required for the R.C. beam. If you look at the solution to the question it states “The depth of the neutral axis is ct = 0.375dt” Where did they get the 0.375 from? according to my calcs, the answer is not even correct. I am coming up with the As,rqd = 2.67 sq. in. however the book says 2.93 sq. in. is required. Has anyone seen this before?

 

[frv]

It's important to note that the code does not allow a strain smaller than .004 (at the strength limit) for steel in non-prestressed flexural members.

 

[miecz]

I used 3 ksi for the concrete stress. Maybe I should have assumed 0.85*3 ksi, in which case, the concrete strain is 0.00081, and the steel strain is 0.00155. I was trying to correct the error in my previous post, where I said the minimum steel strain was 0.005. See the attached worksheet.

 

[Lion06]

That assumes a linear stress-strain relationship. That might be more appropriate for service level loading analysis, but that's not valid for strength level analysis, is it? That assumes a linear stress-strain relationship, which breaks down for concrete at failure. That's the whole reason for using the rectangular stress block. If it were triangular, that would be easy, the rectangle is to approximate the parabolic (or whatever shape it takes, - we studied the Hognestaad model in a grad class) stress-strain relationship at failure level loading.

I agree that the concrete doesn't reach a strain of 0.003 at it's maximum stress, but it won't fail (in compression)until (theoretically) it reaches that 0.003 strain. This what we're concerned with.

Am I missing something?

 

[miecz]

EIT-

I agree that my approach assumes a linear stress-strain relationship and that doesn't seem appropiate for strength level analysis. I'll have to take another look at this, but I can't today. Got work to get out.

 

[NoName1]

To answer the "Most Nearly" question, I would say 4 since the steel required is for a beam with capacity to carry the moment. Three (3) does not carry the moment, so it does not matter if that is closest to the calculted number. It just doesn't work.

But that most nearly part always throws me off with these tests.

 

[miecz]

EIT-

Do you have PCA's Notes on ACI 318? Section 6 explains this pretty well. My worksheet is based on Example 6.1.

 

[Lion06]

I have the '05 version. It shows the concrete strain at 0.003, and the steel strain at 0.00523. Is this the one you're referring to, or do you using a different version?

 

[BAretired]

As I recall, the strain is deemed to be linear but the stress-strain curve is not linear. The actual stress curve is curved but is approximated by the Whitney stress block which is rectangular. I believe that a curved block is used in Europe. The Canadian code is similar to ACI although there could be minor differences in the phi values.

BA

 

[Lion06]

BA-
Right, plane sections remain plane, so the strain diagram is linear, but the stress-strain relationship of the concrete isn't. Either way, we always assume a concrete strain of 0.003 when determining steel strains right?

 

[miecz]

EIT-

I have the '02 version, but example 6.1 has [?]_s=.00523. Interesting, there is a article in the section called "Tension-Controlled Sections and Transition" that shows the area of steel required at the transition point between tension controlled and compression controlled. The area required at this point for the problem above is 2.93in[²]/ft. I believe that [ø] is 0.9 for any area of steel less than that amount.

 

[BAretired]

SEIT,

Our code used to assume a maximum concrete strain of 0.003 but the 1994 version bumped it up to 0.0035. I'm not sure why and I don't know what the current code says about that.

What I normally do is guess at the depth of compression block, then calculate As required based on my guess. I then calculate the depth of compression block required to match T = As*Fy. If it is greater than my first guess, which it almost never is, I recalculate As. The reason I do it that way is because I can't be bothered to solve the quadratic equation.

Ordinarily, I don't worry about the strain in the concrete, because I choose under-reinforced beams whenever I can. The maximum concrete strain, whether 0.003 or 0.0035 combined with the strain at first yield of the steel i.e. Fy/E determines the location of the neutral axis based on a straight line strain diagram.

To consider a strain of 0.005 in the steel seems to be a strange thing to do. I am wondering why the examiner put that in the exam question.

BA

 

[Lion06]

miecz-
Are you referencing the problem in the OP? If so, I agree. What I was noting is that you can't select an adequate number of #8's and be under 2.93, therefore phi will go down just by selecting the appropriate number of #8 bars. If you keep it as a req'd area of steel then phi is 0.9, because it's less than 2.93.

 

[miecz]

What I was noting is that you can't select an adequate number of #8's and be under 2.93, therefore phi will go down just by selecting the appropriate number of #8 bars

Somehow, I missed that. One would think that if 2.93 in[²] is adequate, then 3.14 [²] would surely suffice, but you never know. I once had a case with a bridge stringer that was composite in the negative moment region and worked with a certain amount of deck reinforcing. But, if we added reinforcing to the deck, it moved the plastic neutral axis away from the compression flange, which made the section non-compact, and effectively dropped the strength of the section by 30%, according to the (AASHTO) code.

 

[TERIO]

I don't have anything to add about RC beam calcs, but the link below gives some guidance on the "most nearly" questions.

http://www.ppi2pass.com/ppi/PPIInfo_pg_myppi-faqs-pefaq.html#mc

 

[BAretired]

This link might help in understanding differences between ACI and CSA codes.

http://structsource.com/analysis/types/concrete.htm

BA