Happy Thursday! This is Back of the Envelope – the structural engineering newsletter that helps you become slightly smarter each week.

Today, I am going to share a quick tip about flexural reinforcing for concrete beams. I learned this a decade ago from an old book, and I still find it helpful to this day.

Let’s dive in.

(Estimated read time = 1 minute 36 seconds)

Back-of-the-envelope Equation

Alright, the tip is a super short equation to help you estimate the required flexural reinforcing.

It goes like this:

As = Mu/(4d), where,

• As = required area of reinforcing

• Mu = design moment

• d = usually [beam depth] – [cover] – [1/2 rebar diameter]. (Definition from ACI is “distance from extreme compression fiber to centroid of longitudinal tension reinforcement”)

For units, As is [inch^2], d is [inch], and Mu is [kip-ft]. (I know, it doesn’t add up but bear with me here for a second.)

Here is an example of how this would work:

Say you have an 18” beam with Mu = 100 kip-ft.

• If we assume we’ll use #8 bars, “d” in this case would be 18-1.5-0.5 = 16”.

• The equation would then get you As = 100 / (4 x 16) = 1.56 in^2.

• Which means, you’ll probably need about two #8s (2 x 0.79 = 1.58 in^2).

You see how quickly that was? You could even do this on your phone!

When would you use this?

Now, remember that the equation is meant to be used only as a quick back-of-the-envelope check.

It’s not precisely accurate since it’s missing other parameters such as fy, f’c, beam width, and whether tension/compression controls.

Regardless, it’s useful in the following scenarios:

• You are checking someone else’s work

• You are double-checking your own work

• You want to quickly verify a computer software output

• You want to impress your date

Using the equation, you could quickly check whether the design is in the ballpark without pulling out your fancy TI-83 or TI-84s (could get awkward during a dinner date?)

How was it derived?

You might be wondering, how was the equation derived?

It’s actually just an extrapolation of this equation:

If you set Mu = phi x Mn, and then move the variables around: As = Mu / (phi x fy x j x d)

If we assume phi = 0.9, fy = 60, j = 0.85, and account for the unit, then As = Mu / (0.9 x 60 x 0.85 / 12 x d) -> As = Mu / (3.8d)

So 4d is roughly in the ballpark.

And that’s it for now – thanks for reading and I hope you find this helpful.

Andy Lin
Back of the Envelope/Structural Engineer HQ