# Envelope #18: Super-Quick Concrete Beam Check

Andy Lin

August 25, 2022

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.

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