Cut length, bend gain, K-Factor, ...

Warren Hall

February 08, 2025 05:24

Cut length, bend gain, K-factor, bend allowance, bend compensation, set back, and so on. What does it all mean? How does it all work?

Why?

But first, why. Why cut length? Why bend gain? Why K-factor, bend allowance, bend compensation, set back, and so on in the first place?

There are two primary approaches to produce cut and bent bar - cut then bend or bend then cut. Off-coil machines follow the bend then cut approach, straighten coiled reinforcing steel and feeding it through , bending where needed, finally cutting the piece from the coil once the piece is formed. In contrast, following the cut and bend approach, a length is first cut from straight bar or straightened coil, then, sometime later and possibly on a different machine, the straight length is bent as needed.

The length to cut is not so important for the bend then cut approach as the piece is cut from coil once formed. Following the cut then bend approach however, the length to cut is of considerable importance. It is important to know just how long to cut the bar so that when the bar is bent, the produced piece is neither too short nor too long. A long piece could be cut or trimmed to achieve the shape specified however this requires additional work and this takes time and costs. It would be better if the length to cut was just right length in the first place so no additional work was necessary.

Say for example, we want to produce the shape below. We want to produce the shape from 40mm bar and have leg lengths 1000mm and 1500mm.

It is desirable to know that if we cut a 2404.7mm length of straight bar (in practice 2405mm or to an lesser tolerance), when it is bent 90°, it will form the shape with leg lengths 1000mm and 1500mm. The shape will neither be too short nor will the shape be too long - it will be just right.

Understanding cut length, bend gain, K-Factor, and so on

To understanding cut length, bend gain, K-Factor, and so on, let's take the following shape as our example.

The shape has leg lengths 1000mm and 1500mm comes with the assumption that the legs meet at 90°.

In reality, the shape is not two lines meeting at a sharp point but is instead a single piece of bar, with thickness, bent around a pin.

In this example, let's use 40mm bar is used and bend around a 200mm pin (five times the diameter of the bar).

Consider the shape in three parts - the straight sections of each leg and the curved section formed by bending 90° around the pin.

The lengths of straight sections of the legs are readily calculated by subtracting a bar thickness (40mm in this example) and the pin radius (200mm / 2 = 100mm in this example) from leg length. This reduction is called the set back.

The set back in this example is 40mm + 100mm = 140mm

So the straight section of the 1000mm leg is 1000mm - 140mm = 860mm

And the straight section of the 1500mm leg is 1500mm - 140mm = 1360mm

Easy enough. Now what about the bend? How much steel is in the bend? Do we follow arc on the inner radius? Or use the arc on the outer radius? Or somewhere in between?

When steel is bent, the inner surface undergoes compression and the outer surface undergoes tension. There is a arc between the inner surface and outer surface where the steel is neither under compression nor under tension. This path is called the neutral axis. The arc length of neutral axis corresponds to the length material used when forming the bend and is termed bend allowance.

The neutral axis varies between materials, bending radius, and so on.

The bend allowance formula is

The formula might look a little scary but after some time with the formula, it's simply the arc length formula. The arc length formula allows the calculation of arc length by simply multiplying the the bend angle in radians by the radius. We are interested in the arc length of the neutral axis (at the neutral axis radius) as the bar is bent to bend angle

The bend angle can be calculated from degrees rather than from radians

The radius of the neutral axis is the inner bend radius plus some fraction (K-factor) of the thickness

Substituting in to the arc length formula, we arrive at the bend allowance formula

Let's take a look at what K-factor is doing

A K-factor of 0 would place the neutral axis on the inner surface.

A K-factor of 1 would place the neutral axis on the outer surface.

A K-factor of 0.5 would place the neutral axis half way between the inner surface and the outer surface.

So the K-factor is essentially placing the neutral axis between the inner surface and outer surface.

In this example, the material has a K-factor of 0.44.

This places the neutral axis off the middle of the material, slightly towards the inner surface.

With all of that, bend allowance should hopefully be less scary. Here it is again.

It is essentially calculating the arc length along the neutral axis through the bend. That's it.

So in this example

What does this say? What does this mean? It says that the length of bar that will be used or needed to produce a 90° bend in bar 40mm thick with a K-factor of 0.44 around a pin of radius of 100mm will be 184.7mm. That's it.

And now we have all three components, we can work out the length of the material required

The straight section of 1000mm leg = 860mm

The Bend Allowance = 184.7mm

The straight section of 1500mm leg = 1360mm

The sum of the parts is 860 + 184.7 + 1360 = 2404.7

What does this say? It says if we have a straight section of bar 2404.7mm long (in practice 2405mm or to an lesser tolerance), the bar can be bent 90° in the right place and achieve a shape with leg lengths 1000mm and 1500mm.

But that's not how the shape is specified. Instead, the shape is specified from the ends of the bar to the intersection point of the outside surfaces of the legs.

Here's a reminder.

But we've worked out that the straight length should be 2404.7mm. Here's where 1000mm and 1500mm end on on the straight length.

We can see that the two leg lengths (1000mm and 1500mm) overlap - the dimension for 1000mm ends somewhere in the dimension for 1500mm and vise versa. This means the straight length must be less than 1000mm + 1500mm = 2500mm and indeed we have found it to be 2404.7mm. Finally we can see where a negative bend gain is coming in - it's an effect of specifying the shape as leg lengths but forming the shape around a pin.

So, how to we work this out?

Well, we've already done it.

Putting in terms of full leg length, we get

Rearranging a bit to put the entire leg lengths together, we get

To formalize things a little, bar thickness + pin radius for this shape (90°-angle bend) is called the set back

So with set back defined, we get

What does this say? It says to find the cut length, we can take the leg lengths as they are, subtract twice the set back and add the bend allowance

To formalize a little further

Substituting back in

And here is our bend gain or bend compensation

Now we have everything we need to work through our example

First, from earlier, bend allowance

Then set back

And from bend allowance and set back, bend compensation (bend gain)

And finally substituting to find length

Finally, we can see our bend gain or bend compensation. To form this shape with leg lengths 1000mm and 1500mm from 40mm bar having a K-factor of 0.44, bending around pin of radius 100mm, the sum of the leg lengths must be increased (bar gain) by -95.3mm (a reduction in length) to arrive at a cut length of 2404.7mm.

Single bends

We worked through the 90°-angle bend example above to understand cut length, bend gain, K-Factor, and so on. Now we need to look at the possible (single) bend cases.

Bar is bent from straight typically between 0° and 180°. The scheduling handbook details obtuse-angle bends, 90°-angle bends and acute-angle bends. Obtuse-angle bends are formed when bar is bent from straight by greater than 0° and less than 90°. 90°-angle bends are formed when bar is bent from straight by 90°. Acute-angle bends (including 180° hook) are formed when bar is bent from straight by greater than 90° and less than or equal to 180°.

For each bend, the bend allowance formula is identical. After all, the bend allowance formula determines the length of the neutral axis. To illustrate this, for an identical pin and identical bar (identical thickness and K-factor), all that is changing is the bend angle - bend radius, K-factor, and thickness all remain the same

What differs between the bends however is the set back. This difference is owing to the manner in which the shape is dimensioned. For 90°-angle bends and acute-angle bends, the dimension is taken, parallel to the bar, to the tangent with the outer surface of the bar through the bend. For obtuse-angle bends, the dimension is taken, parallel to the bar, to the intersection point made by the outer surfaces of the straight bar either side of the bend.

As we look at the bends, they are general in nature (between 0° and 90° for example instead of say specifically 53°) so we need to work with symbols rather than values and so we need to move from a wordy-formula to something more concise.

Particularly for the obtuse-angle bend case, we will use the following that the bend angle as a deflection from straight is the same as the angle around the pin.

Let's call the bend angle theta.

Angles on a straight line add up to 180. So the adjacent angle is 180 - theta

The straight sections of the bar are tangent to the pin circumference and so normal to the radius (90° to the radius)

There are 360° on the inside of a 4-sided shape

So the angle at the pin center is the same as the bending angle

90°-angle bends

90°-angle bends are formed when bar is bent from straight by 90°. The dimension is taken from a point on the outer surface of the bend normal to the outer surface of the bar.

To calculate the cut length, we need to add up the components

Putting the components in terms of leg lengths

Separating out leg lengths

The set back for 90°-angle bends

Bend compensation

Worked example

We'll reuse the example from above where we worked to understand

We want to work out cut length

But to work out cut length, we bend compensation

And to work out bend compensation, we need bend allowance and we need set back

First, set back

Then bend allowance

Now we can work out bend compensation

And finally, cut length

Acute-angle bends

Acute-angle bends are formed when bar is bent from straight by greater than 90° and less than 180°. The dimension is taken from a point on the outer surface of the bend tangent to the outer surface of the bar.

To calculate the cut length, we need to add up the components

Putting the components in terms of leg lengths

Separating out leg lengths

The set back for acute-angle bends

Bend compensation

Worked example

We want to work out cut length

But to work out cut length, we bend compensation

And to work out bend compensation, we need bend allowance and we need set back

First, set back

Then bend allowance

Now we can work out bend compensation

And finally, cut length

180° hook

The 180° hook is an acute-angle bend formed when bar is bent from straight by 180°. A bar dimension is taken from a point on the outer surface of the bend normal to the outer surface of the bar.

To calculate the cut length, we need to add up the components

Putting the components in terms of leg lengths

Separating out leg lengths

The set back for 180° hooks

Bend compensation

Worked example

We want to work out cut length

But to work out cut length, we bend compensation

And to work out bend compensation, we need bend allowance and we need set back

First, set back

Then bend allowance

Now we can work out bend compensation

And finally, cut length

And here you can see the straight length. To form a 180° hook with leg lengths 1000 and 1500, an additional 89.45mm must be added to the sum of the leg lengths (1000+1500=2500) to get 2589.45.

Obtuse-angle bends

Obtuse-angle bends are formed when bar is bent from straight by greater than 0° and less than 90°. The dimension is taken from the intersection point of the outer surfaces of the bars.

The obtuse-angle bend (bar bent from straight by less than 90° ) is a separate case. Owing to the manner in which the shape is dimensioned in the bar schedule (to an intersection point made by the outer surfaces of the bars), unlike the 90° bend case, the 180° hook case and the acute-angle case, the set back for this case is dependent on the bend angle

To calculate the cut length, we need to add up the components

Putting the components in terms of leg lengths

Separating out leg lengths

But how do we calculate the set back for obtuse-angle bends?

We need to pause for a bit and work through determining the set back for obtuse-angle bends.

Using our result from earlier

We first bisect angle theta

We now have a right-angle triangle to apply some trigonometry

Throw in some terms

Returning to our cut length formula

We now have set back

Bend compensation

Worked example

We want to work out cut length

But to work out cut length, we bend compensation

And to work out bend compensation, we need bend allowance and we need set back

First, set back

Then bend allowance

Now we can work out bend compensation

And finally, cut length

Multiple bends

We've worked to understand how bend gain works and seen the various single bend cases. What happens when there are multiple bends?

To calculate the cut length, we need to add up the components

Putting the components in terms of leg lengths

Separating out leg lengths

From earlier work, the set back for acute-angle bend (AB)

From earlier work, the set back for 90°-angle bend (BC)

From earlier work, the set back for obtuse-angle bend (CD)

From earlier work, the set back for hook bend (DE)

Substituting back in

Bend compensation for acute-angle bend (AB)

Bend compensation for 90°-angle bend (BC)

Bend compensation for obtuse-angle bend (CD)

Bend compensation for hook bend (DE)

And this is the clever thing about show shapes are specified, set back is used, bend allowance and bend deduction work - you can simply add up all of the lengths then add bend compensation for each of the bends. Neat. Very neat.