Conc. Rect. Beam Derivation - McCormac 2

[Howlyn2]

Hello, I am having a little trouble deriving the percentage of steel reinforcement for a beam using rho and Rn (Flexural Resistance Factor). The results are given in Design of Reinforced Concrete by McCormac & Brown - Equation (3-3), attached for reference. I however have not been able to make the correlation. Does anyone have a reference illustrating the steps to get here? I believe the Whitney Stress block --> a = (As/ 0.85 f'c)*(As/b*d)* d can be helpful but I cannot find the result.

Similarly are the any other helpful "cheat sheets" / sources showing derivations of useful concrete equations like this? Thank you in advance.

 

[271828]

Did you try the following?

1. Eq. 3-2: divide both sides by phi*b*d^2
2. Eq. 3-2: make the left side Rn
3. Quadratic formula to get rho.
4. simplify

 

[Howlyn2]

Hello 271828 and thank you so far. I am with you there to Step #: 4. The simplification and cancelling out of variables is what I was missing. Perhaps I should have been clearer, but that's what I intend on looking for, a proof if you will.

 

[dik]

For reasonable amounts of reinforcing... assume moment arm is 0.9 d and calculate the As, calculate the depth of the compression block, and then use this value to determing the new compression block moment arm. It converges very quickly to the 'exact' value. ;[gen rev]

-----*****-----
So strange to see the singularity approaching while the entire planet is rapidly turning into a hellscape. -John Coates

-Dik

 

[TLHS]

See attached. Then discard the second solution because it requires negative steel, which is apparently still not available in the industry.

 

[Ingenuity]

...requires negative steel, which is apparently still not available in the industry.

That was funny nerd humor

 

[IDS]

TLHS beat me to it, but here's mine anyway:

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[dik]

What are your beam dimensions, concrete strength, concrete cover, main bar and stirrup size, and moment? Steel grade assumed to be 60ksi stuff.

-----*****-----
So strange to see the singularity approaching while the entire planet is rapidly turning into a hellscape. -John Coates

-Dik

 

[JLNJ]

I have always solved the quadratic for the steel required rather than express it as a percentage using rho.

 

[dik]

It's generally faster, with a calculator, to iterate the solution as I noted.

-----*****-----
So strange to see the singularity approaching while the entire planet is rapidly turning into a hellscape. -John Coates

-Dik

 

[TLHS]

I either just make a really conservative compression block estimate to do it dik's way by hand, but often without iterating, or I use the calc sheet I have set up that just tells me actual steel amounts. I mostly stopped doing reinforcement percentages as a thing for bending members when I moved away from using the design aid tables.

 

[dik]

Reiterating it is fast to do on a calculator and provides nearly an 'exact' answer. It converges very quickly.

-----*****-----
So strange to see the singularity approaching while the entire planet is rapidly turning into a hellscape. -John Coates

-Dik

 

[IDS]

dik - how is iterating faster than using a single equation?

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[TLHS]

I was saying I use an approximate check if I just need rough numbers and if it's close enough I don't bother iterating. dik was saying I'm being lazy because it takes thirty seconds!

 

[Howlyn2]

Hello All, Thank you so much TLHS & IDS. This was not for a specific problem unlike DIK suggested. Instead this was an exercise for me to better understand the equations presented in the text and to also be more comfortable using tables also in the text's appendices based upon these iterations. Appreciate your time all.

 

[dik]

Have you ever used a hand calculator? It can be a lot faster.

-----*****-----
So strange to see the singularity approaching while the entire planet is rapidly turning into a hellscape. -John Coates

-Dik

 

[IDS]

Have you ever used a hand calculator? It can be a lot faster.

I used pocket calculators from 1972 until probably mid-90's when I realised that everything could be way faster, better documented and more easily checked if you used a spreadsheet.

But we are probably getting a little off-topic


Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[JoelTXCive]

See attached........

I was always getting confused by the various tabular methods to solve for concrete beam design, so several years ago I ground through the math to get a better understanding.

The tables are similar but different in the various popular resources (SERM, Wight & MacGregor Textbook, & PCA Circular Tank Manual)

I believe some of the tabular data tables incorporate a Phi factor of 0.9 and some tables do not.

Attached is the Wight & MacGregor equation worked out; and then comments on how the other tabular methods differ.

Enjoy!

 

[TLHS]

We're all a bunch of crazy people.