Reinforced Concrete Design (with Flowchart) - Concrete Beam with Tension Reinforcement Only

Have you heard of M_u/4d?

That’s something I learned at work from one of my bosses who has been a licensed SE since 1984! It’s basically a quick way to check your numbers for concrete flexure (which I will show you later in the post).

Apparently that’s how engineers used to do quick checks – there’s no reasons why we can’t still use the same technique today.

In this post, you will learn how to design a reinforced concrete beam step-by-step, with my “simple-to-follow” flowchart. By creating and using the flowchart, I was able to recall the info needed without having to memorize anything which I hope I can help you do the same.

The Goals

There a few things that I want to help you achieve by the end of the post:

To be able to come up with the required reinforcement without having to “re-read” anything.
Recall how to design without memorizing.
A step-by-step procedure that can be easily followed so that you don’t miss all the little “fine-prints” such as minimum/maximum reinforcement requirement…etc.

Note that the flowchart mainly addresses the design of a rectangular beam (not T-beams) with tension reinforcing only. T-beam design will come in a later post.

Assumptions

I want to point out that since you are taking the SE exam, you most likely have some idea about all of the concrete properties (brittle) and the design theories (compression zone…etc.) so I am not going to elaborate too much about them.

If you do need more info though, please let me know – I am more than happy to help.

Another thing I want to mention is that there are many many ways to do the same design and this is just the one that I am most familiar with.

OK let’s get to this!

Flowchart

The flowchart itself is pretty self-explanatory (maybe because I created it?) but I am going to give you a quick rundown anyway. Please do not hesitate to let me know if you got any questions.

Click Here to get the Flowchart

What's Given?

Usually, you should already have something to get started with on your design:

M_u: Factored design moment based on the worst case from your load combinations.
f'_c: Specified compressive strength. This is typically 3000, 4000, or 5000 psi.
f_y: Specified yield strength of reinforcement. Usually 60,000 psi for new buildings and 40,000 psi for older buildings.
b: Width of the beam.
d: Usually the total beam depth – cover – 1/2 of the rebar diameter. I am going to assume you know what this is.

What Are We Trying to Determine?

Ultimate goal is of course to find the reinforcements you need that makes the beam work.

Quick Check

$\frac{{ M }_{ u }}{4d}=?\:in^2$$

Before you do anything, it’s a good idea to do this first. Like I mentioned in the beginning, it’s a very quick way to get a rough number for the reinforcement you need.

I know the units don’t make sense but just go with it and test it out. You want to make sure the unit for your “M_u” is [kip-ft] and “d” is [in]. The result will be in [in²].

I’ll show you an example at the end.

Step-by-Step Explanations

#	Equation	Action	Notes/Explanation
1	$\frac { { M }_{ u } }{ { f }_{ c }^{ ' }b{ d }^{ 2 } } =?$	Calculate	You are going to calculate this number a lot. It's used for determining the $\rho$ factor which you can see in the next item.
2	Use Appendix A Table (of the SE Reference Manual)	Determine $\omega$	This basically tabulates the equation $\frac { { M }_{ u } }{ { f }_{ c }^{ ' }b{ d }^{ 2 } } =\omega (0.9-0.5294\omega)$ so that you don't have to plug in the numbers. For example, if you had calculated $\frac { { M }_{ u } }{ { f }_{ c }^{ ' }b{ d }^{ 2 } }=0.1976$ , using the table, you get $\omega$ =0.259. Pretty handy.Let me know if you don't have the table, I can create one and post it when I get a chance. You can also solve the equation using quadratic formula if needed.
3	$\rho =\omega \frac { { f }_{ c }^{ ' } }{ { f }_{ y } }$	Calculate	This is basically the reinforcement ratio you need. You still need to verify min/max and $\phi$ …etc.
4	${ \rho }_{ min }=MAX\left( \frac { 3\sqrt { { f }_{ c }^{ ' } } }{ { f }_{ y } } ,\frac { 200 }{ { f }_{ y } } \right)$	Calculate (make sure your units are in psi)	This is the minimum reinforcement ratio required. I have tabulated the number for the most common case: if ${ f }_{ c }^{ ' }=3000\:psi$ and ${ f }_{ y }=60000\:psi$ , then ${ \rho }_{ min }=0.0033$ .
5	$\beta _{ 1 }=\begin{cases} 0.85,\quad if\quad { f }_{ c }^{ ' }\le 4000\quad \\ 0.85-\frac { { f }_{ c }^{ ' }-4000 }{ 20000 } ,\quad if\quad 4000\le { f }_{ c }^{ ' }\le 8000 \\ 0.65,\quad if\quad { f }_{ c }^{ ' }>8000 \end{cases}$	Calculate	This will be used in a couple of equations later.
6	$\rho _{ max }=0.364{ \beta _{ 1 } }\frac { f_{ c }^{ ' } }{ f_{ y } }$	Calculate	This is the maximum reinforcement ratio which was derived from the requirement of minimum net tensile strain at nominal strength.
7	${ \rho }_{ req }=MIN\left[ MAX\left( \rho ,{ \rho }_{ min } \right) ,{ \rho }_{ max } \right]$	Calculate	After comparing these three numbers, we should now know how much reinforcement is needed that meets both the minimum and maximum requirements.PS: I wrote it in this format because I am used to writing like this in Excel functions which I assume you might be as well.
8	$\rho _{ t }=0.319{ \beta _{ 1 } }\frac { f_{ c }^{ ' } }{ f_{ y } }$	Calculate	This is for checking to see if the beam is “tension controlled” or “compression controlled”. See below.
9	${ \rho }_{ req }<\rho _{ t }?$	Compare	If the statement is true, then we know that the beam is “tension controlled”.
10	${ A }_{ s.req }={ \rho }_{ req }A$	Calculate	This simply converts the ratio to an actual number so that you can decide on the number of bars and the size of bars.Note that this number is based on a tension controlled section and has already accounted for $\phi = 0.9$ .
11	$a=\frac { A_{ s }{ f }_{ y } }{ 0.85f_{ c }^{ ' }b }$	Calculate	This is the size of the compression block (see diagram on the very top). We need this to calculate the location of neutral axis and the corresponding $\phi$ factor if the section is compression controlled.
12	$c=\frac { a }{ { \beta }_{ 1 } }$	Calculate	Location of neutral axis from the top fiber.
13	$\phi =MAX\left( \left( 0.23+\frac { 0.25 }{ \frac { c }{ d } } \right) ,0.65 \right)$	Calculate	Corresponding $\phi$ factor as mentioned earlier.
14	${ A }_{ s.req(c) }={ A }_{ s.req }\left( \frac { 0.9 }{ \phi } \right)$	Calculate	Revised (increased) reinforcement required since the section is “compression controlled” which has lower $\phi$ .

Concrete Beam Design Example

Now let's run through this flowchart with an actual example.

Given

${ M }_{ u }=225\: kip\cdot ft$
${f}_{c}^{'}=3000 \: psi$
${f}_{y}=60,000 \: psi$
$b = 12\:in$
$beam\:depth = 22\:in$
$d = 20\:in$ (say cover is 1-1/2″ and we are using #8 bar: $22" - 1.5" - 0.5" = 20"$ )

Quick Check

$\frac{{ M }_{ u }}{4d}=\frac{225}{4\times20 }=2.8125\:in^2$$

This is the quick and dirty way to check the required reinforcement. We'll come back to verify when we have an actual solution.

Flowchart

#	Equation	Results	Notes/Explanation
1	$\frac { { M }_{ u } }{ { f }_{ c }^{ ' }b{ d }^{ 2 } } =?$	0.1875
2	Use Appendix A Table (of the SE Reference Manual)	0.243	Use the table, the number listed that is closest to 0.1875 is 0.1874 – which corresponds to $\omega=0.243$ .
3	$\rho =\omega \frac { { f }_{ c }^{ ' } }{ { f }_{ y } }$	0.0122
4	${ \rho }_{ min }=MAX\left( \frac { 3\sqrt { { f }_{ c }^{ ' } } }{ { f }_{ y } } ,\frac { 200 }{ { f }_{ y } } \right)$	0.0033
5	$\beta _{ 1 }=\begin{cases} 0.85,\quad if\quad { f }_{ c }^{ ' }\le 4000\quad \\ 0.85-\frac { { f }_{ c }^{ ' }-4000 }{ 20000 } ,\quad if\quad 4000\le { f }_{ c }^{ ' }\le 8000 \\ 0.65,\quad if\quad { f }_{ c }^{ ' }>8000 \end{cases}$	0.85
6	$\rho _{ max }=0.364{ \beta _{ 1 } }\frac { f_{ c }^{ ' } }{ f_{ y } }$	0.0155
7	${ \rho }_{ req }=MIN\left[ MAX\left( \rho ,{ \rho }_{ min } \right) ,{ \rho }_{ max } \right]$	0.0122
8	$\rho _{ t }=0.319{ \beta _{ 1 } }\frac { f_{ c }^{ ' } }{ f_{ y } }$	0.0136
9	${ \rho }_{ req }<\rho _{ t }?$	Yes; therefore tension governs.
10	${ A }_{ s.req }={ \rho }_{ req }A$	2.9171 in²	Compare to the quick calc we did above (2.8125 in²), we are fairly close! Therefore, we know that we didn't make any computational errors.Based on this, we will need 4-No.8 bars (0.79 in² x 4 = 3.16 in²).Note that if the A_s you use is significantly larger than A_s.req, you should repeat steps 7 and 9 to ensure that you didn't exceed the maximum reinforcing and to ensure that tension still governs.For example: $\rho=\frac{3.16}{12\times {20}}=0.0132$ $\rho_{max}=0.0155\:>\:0.0132$ It is still less than the max so we are OK. $\rho_{t}=0.0136\:>\:0.0132$ Tension still governs – we are OK here as well.
11	$a=\frac { A_{ s }{ f }_{ y } }{ 0.85f_{ c }^{ ' }b }$	5.720 in	Not really necessary if tension governs but it is useful later if you want to calculate out the actual capacity. I can demonstrate how this works in a future post.
12	$c=\frac { a }{ { \beta }_{ 1 } }$	6.729 in
13	$\phi =MAX\left( \left( 0.23+\frac { 0.25 }{ \frac { c }{ d } } \right) ,0.65 \right)$	–	Not applicable since tension governs.
14	${ A }_{ s.req(c) }={ A }_{ s.req }\left( \frac { 0.9 }{ \phi } \right)$	–	Not applicable since tension governs.

And done!

Now if you change M_u to 300 kip-ft, you can see how it works if compression governs (I won't demonstrate here).

Final Thoughts

Of course, as I mentioned earlier, just like any other engineering topics, this is not the only way to design a rectangular concrete beam but it's a fairly straightforward way to do it.

Note that I didn't cover shear design as it will come in a later post.

Is this helpful? Let me know what you think in the comments below!

Reinforced Concrete Design (with Flowchart) – Concrete Beam with Tension Reinforcement Only

The Goals

Assumptions

Flowchart

What's Given?

What Are We Trying to Determine?

Quick Check

Step-by-Step Explanations

Concrete Beam Design Example

Given

Quick Check

Flowchart

Final Thoughts

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The Goals

Assumptions

Flowchart

What's Given?

What Are We Trying to Determine?

Quick Check

Step-by-Step Explanations

Concrete Beam Design Example

Given

Quick Check

Flowchart

Final Thoughts

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