How do I EXACTLY calculate the k-factor for metal bending?

Introduction

I've recently started to work with metals and the process of bending them. There is a lot more math involved than I initially thought. All the break press operators just have a standardized list of k-factors that they either use or don't. But these are just estimates and this chart can be found here. I decided against using the sheet as I wanted to find the answer myself, and not just have it handed to me.

Note: A lot of these formulas in this post may look different depending on what source you go to, but they're either simplified or re-written

How do you exactly calculate a k-factor in metals?

Now what is the k-factor?

The K-factor is the ratio of the neutral axis to material thickness (as shown in the equation below).

$$K=\frac{t}{Mt}$$

t = neutral axis, Mt= Material Thickness

The Material thickness is pretty self explanatory but what is the neutral axis? The neutral axis is a zone in a metal where the metal is neither in tension nor compression and that the only force is the sheer force (which is at its maximum). When Bending a metal, the inside of the curve compresses while the outside expands.

Calculating the Neutral Axis can be found Here if you are interested. I'm not even sure that is needed, but I went through the formulas there.

The K-factor is basically just a percentage of the relocation of the neutral axis and how much that axis shifts when a metal is bent.

the value of the k-factor will enable you to predict the total amount of elongation that will occur within a given bend. The k-factor allows you to calculate the bend allowance, the outside setback, the bend deduction, and the flat layout of the precision part you’re forming. The neutral axis does not suffer any change [of] length during a bending operation, but does move toward the inside surface by a percentage, that percentage being the k-factor. This relocating or shifting of the neutral axis—from 50 percent of the material thickness to a new location equal to or less than 50 percent of the material thickness—is the reason why the part elongates during forming.

The typical percentage ranges from 0.25 to 0.5 depending on the metal and the bends.

Note: There are a bunch of variables that play a roll on how much the neutral axis shifts in a bend metal. When a metal has more than one bend and/or bends in opposite directions, it changes the neutral axis location.

Bend Allowance

The Bend allowance is the dimensional amount added to a part through elongation during the bending process.

$$BA=\frac{A}{360}\times 2\pi \left ( R+K\times Mt \right )$$

BA = Bend Allowance, A = Bend Angle, R = Internal Radius, K = K-factor

K-factor as one of the variables? I thought we are looking for the k-factor? So the search continues!

Bend Reduction

For Bend Reduction, I think the best explanation I got for this was from Wikipedia.

The bend deduction BD is defined as the difference between the sum of the flange lengths. The outside set back (OSSB) is the length from the tangent point of the radius to the apex of the outside of the bend. The bend deduction (BD) is twice the outside setback minus the bend allowance.

And yes, there is a formula for this as well!

$$BD=2\left ( R+Mt \right )tan\frac{A}{2} - BA$$

When Bending a Metal, the inside compresses and the outside expands. This forces the metal to expand at the ends and changes the length of your metal.

To Work out what the length of the flat piece of metal needs to be, we need to calculate the Bend Allowance or Bend Deduction that tells us how much we need to add or subtract to our leg lengths to get exactly what we want.

Calculating K-Factor

At this point, We have defined the variables that we need to start working at the k-factor formula that has been provided! (This formula stays relatively the same between each website or book that I checked.)

$$K=\frac{-R+\frac{BA}{\frac{\pi A}{180}}}{Mt}$$

And now here is the first problem I encountered. When you sub in $$\frac{A}{360}\times 2\pi \left ( R+K\times Mt \right )$$ for BA, it uses the k-factor in the formula. At the end of the simplification $$k=k$$.

That is where I got stuck for a week and I recently found this:

$$k=\frac{log,min(100,\frac{max(20R,Mt)}{Mt})}{2log(100)}$$

Note: there is not supposed to be a comma after the log in the numerator of the first fraction, I just don't know how else to write it

For the following example, I plugged in $$R = 1/8 = 0.375$$ and $$Mt = 1/2 = 0.500$$. I think I am doing the math correctly?

$$k=\frac{log,min(100,\frac{max(20\times 0.375,0.500)}{0.500})}{2log(100)}$$

Then I decided to multiply $$20$$ by both $$0.375$$ and $$0.5$$. If this is incorrect and I only multiply it by 0.375, I will fix it in an edit.

$$k=\frac{log,min(100,\frac{max(7.5,10)}{0.500})}{2log(100)}$$

Since $$7.5 < 10$$ then use $$10$$ as it is the max and divide by $$0.5$$ and simplify.

$$k=\frac{log,min(100,20)}{2log(100)}$$

Now since we are looking for the minimum, I used $$20$$

$$k=\frac{log(20)}{2log(100)}$$

Evaluate

$$k=0.325$$

This seems correct as the range for k-factors is $$0.25$$ through $$0.5$$ but I am not entirely sure. Do I need to use the actual neutral axis formula and use that when calculating the k-factor?

Another thing that is bothering me is that IF that is the correct answer, how do you calculate it for different metals? Carbon Steel, Aluminum, Stainless Steel, etc. all have different metallurgical properties and sometimes vary in composition depending on what "tolerance" they are made. That is why I'm not using a chart because these results may vary greatly. This second question I will ask in a separate thread, but if anyone also knows where to start on that would be much appreciated.

• Having just watched several Youtube videos showing metal bending then a range is used as it cannot be calculated precisely. There are many things in engineering where variable constants need to be used, like the Coefficient of discharge for orifice plates... or friction factors... Commented Jul 21, 2021 at 15:16
• @SolarMike yes, in a fabrication factory, you do go by conventions and such charts. It's not practical to sample every single metal sheet in order to find out it's composition. Those charts are very useful and I stand by them. The reason I first asked this question is because different metals have different metallurgical properties that affect the neutral axis positioning. Being able to theoretically find the precise answer (like even to three decimal places) has been my goal. The engineers I work with agreed that you can find the k-factor accurately, although they never did the calculations Commented Jul 21, 2021 at 15:48
• You are right that the metallurgical properties of different metals can affect the k-factor. The k-factor is usually determined experimentally or through trial and error, as it can vary based on the specific material, thickness, bend angle, and other factors.
– user13416
Commented Mar 19, 2023 at 1:24