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Analyzing the k-factor in sheet metal bending: Part II

k-factor is the most significant among the math constants in sheet metal fabs. It seems the most basic value while calculating bending allowance (BA) and bending deductions (BD). If you gain the correct k-factor, your precise bending will be ensured. 

A Slight Overview

The neutral axle is a theoretic space that lies at 50% of material thicknesses (Mt) when it is not stressed yet and is plain. In the bending process, the axle is shifted into the bend. K-factor determines the shifting extent of the neutral axle. To be more specific, k-factor is the location of the neutral axle in the aftermath of bending, denoted as ‘’t’’, divided by material thicknesses (k-factor equals t/Mt).

A variety of ratios influence the value of the k-factor. One of the ratios is the bending radii relating to the material thicknesses (in accordance with the suppliers` specifications) as well as to the border amidst acute and the least bending in air formation when the pressures have higher significance compared with the piercing pressures, finally producing creases on the central-line of bends.

Directions of grains matter for the k-factor. So do the material thicknesses and rigidity. 

Bend Methods  

Formation methods include air, bottom and coin formations. It should be noted that bottom and coin formations are quite different.

In the coin formation materials get fully contacted with the sidewalls of punches and dies (Fig. 1) From this time on, excessive strength is imposed to the metal. Punching tips penetrate the neutral axle, next punches and dies approach within an area, which appears no more than the material thickness.

This makes the materials thinner at the base of strokes. This tonnage load is sufficiently great to force metallurgical structures to rebuild, making it possible for you to produce the least radii required. Acute, clear inner radii are the final aim of coin bending.

Bottom formation unlike coining, demands a gap among punch and die angles. The downward punching point causes the sheet to roll up about the punch, when it goes on applying strength, the sheet opens to fit the die angles. (Fig.2)

The real bottom formation happens when the inner bending radii get contracted under the forces of the punching nozzle, then slimming the sheet while bending.

Fig. 1

Air formation is the most prevailing in the progressive precision bend formation. (Fig. 3). It implies trial pointed bending, this means that tooling contacts bend at 3 spots- at the punching nozzle and 2 radii that lead to the die aperture. How the material expands and compresses in the formation depends on its characteristics. Not like bottom and coin formations, air bending produces floating radii according to the die aperture, and angles depending on how deep the punch penetrates in the die area. Considerably smaller tonnage is required in comparison with bottom and coin formations. A precise press brake and tools are necessary for this formation. An old-version press brake won`t be suitable for air formation.

 What is the influence of these bend formations on the k-factor? Air formation serves as a base to define the k-factor, neutral axle as well as bending allowance. In comparison with air formation, bottom forming provides relatively high k-factor valuation. Some studies have provided evidence that while bottom forming, k-factor values increase up to 15 % for the same materials and tools as at the air formation. The reason of this is the significant amounts of deformity occurring at the radii.

Coin formation excludes any stress in materials. This is achieved using pressure the greatness of which is enough to bring the entire metal on the radii and in the adjacent smooth area is brought to yielding strength. Stress relief is an important fact explaining why the coin formation prevents back spring. This removal of inner stresses forces the neutral axle to return to the bending inner surfaces contrary to the positioning of the neutral axle at the bottom formation.

Die Breadth

As soon as the thicknesses of materials are increased, the coefficient k becomes small, as long as the suitable die hole is used on tap. Still, while increasing the thicknesses and keeping the same combinations of die and punches, the result is not quite the same. In this case increased thicknesses bring about frictions as well as the reduction of the material capability to glide upon the die radii. As a consequence, more material deformity happens at bends causing the coefficient k to rise.

On the other hand, while keeping material thicknesses and decreasing die breadth, coefficient k will increase. Research has proved that the small die hole enlarges the coefficient k. In case of invariable thicknesses, dies of small sizes require relatively higher strength to gain the same bending angles.

Friction Factor

This is the ratio of the frictional strength amongst 2 subjects, as each of them moves against the other. The kinetic frictional factor resists the movement. It is the traction strength of 2 objects, as 1 object passes the next one. 

The frictional factor is dependent upon the object(s), that cause frictions-here, the sheets gliding upon the die radius. Values range from zero (meaning that there is not any friction) to one.

What can this indicate? When the sheet thickness increases, the coefficient k reduces. The frictional factor, stresses as well as pressures, occurring at the formation, return. 

Overview of Components

The increase of the coefficient k indicates that the neutral axle winds up at the center of the material thicknesses. The decrease of the coefficient k indicates that the neutral axle moves on into the inner sides of bending.

Fig. 2

Then, let us overview the components of the coefficient k, and let us begin with the bending radii.  Imagine decreasing the inner bending radii in relation with the sheet thicknesses. During the bend of smaller radii with grains, cracks might be caused on the outer sides of bending. While piercing the bending lines at the inner bending radii with excessively acute punching tips, grains can extend upon the outer side of bending, causing the neutral axle to go inwards and also reducing the coefficient k.

While retreating from air formation to bottom formation, the coefficient K is increased as a result of deformity and considerably narrow bending radii. While replacing bottom formation with coin formation, the coefficient k gets smaller because of the stress relief, and the neutral axle goes towards the inner sides of bending.

As soon as materials thicken and harden, the coefficient k gets smaller. As long as the sheet thicknesses get changed, but the tools remain the same, the bend strength will change. This is how it can be reasoned that the coefficient k increases together with sheet thicknesses if the formation is carried out through the same combinations of punches and dies. Keeping the constancy of thicknesses and using narrow dies will cause the increase of the coefficient k.

Precision Level 

This is the essence of the interaction of components. Prior to passing to the equation, look over Fig.5, where each term being further discussed is shown.

Mostly the average value of the coefficient k is accepted as 0.4468. For the formula of bending allowance, the average value of the coefficient k is as the following:  

BA = [(0.017453 × Ir) + (0.0078 × Mt)] × outer bending angles

 “0.0078” is the outcome of π/180 × 0.446—and 0.446 is the average coefficient k.

Some workshop experts offer other fast ways to calculate the coefficient k, such as the ratio between radii and sheet thicknesses. In case the radii are 2 times as little as the sheet thicknesses, the k value will be 0.33. In case the radii appear 2 times as much as the sheet thicknesses, the value will be 0.5. This will work well in case you form a box for dump trucks.

For higher accurate figures, decide on your coefficient k value out of the table depicted in Fig.6.

Fig. 3

Testing Measurement

For higher precision, it is recommended to estimate the coefficient k from the very scratches founded on bending tests. Note that even a little change will cause the value of the coefficient k to shift. Mostly, defining the exact coefficient k requires at least 3 testing specimens with the same grades and thicknesses of materials, finely formed in unchanged circumstances and grain directions. For the calculation of the k value, one must gain some essential data particularly on bending allowances and inner radius. All the testing pieces must be measured, then, the average values of the coefficient k must be inserted into the formula.

For the beginning, note the accurate measuring of testing samples. To define the inner radius the bent pieces must be measured through pin or radii gauges or more accurately, through optic comparators.

You may face some difficulties to measure the bending allowance. It is the arch span of the neutral axle, which has moved on inwards at the bend. Flat dimensions must be measured prior to bending, next, the bend allowance can be found.

Calculating BA for Formation of 90 Degrees

For 90-degree bending the entire outer dimensions of bent parts should be measured, the Mt and measured Ir should be subtracted off the measurement of outer flanges. Thus, you will gain the inner leg size. Fold together the 2 inner leg sizes and take the flat size. Thus, bend allowance (BA) will be got: 

Inner leg size for bends of 90-degree angularity is equal outer dimensions – Mt – Ir

Estimated inner leg dimension – estimated flat = BA

The given equation is suitable for a bend of 90 degrees. It can be explained by the relation of the radii and leg dimension in case of angularity of 90 degrees. The reason of this is that the flat-legged lengths correspond to the contact points.

BA For More or Less Than 90-degree Angularity  

The situation can be somewhat challenging while calculating BA for more or less than 90-degree angles. Begin with each estimated point of testing sample and identify the inner leg sizes through right-angular trigonometry.

Fig. 4

It should be noted that the trigonometric equation is not the single option. You are free to make use of other references whether on-line or at your workshop to decide on other options offering solutions to varied sides and angularities of right-angled triangles.

Now let us get into less than 90-degree outer angles. Take the outer bending angularity of 60 degrees as shown in Fig. 6. Each step beneath relates to the same ones given in Fig. 6. Each step should be repeated for the 2nd inner leg.

Step 1: Estimate the size of A on the testing sample.

Step 2: Gain the size of B by adding the size of A to the Mt.

Step 3: Take the measurement of the inner radius through pin and radii gauges and optic comparators.

Step 4: Count on the external setbacks (OSSB), which equals [tangent (outer bending angle/2) × (Mt + Ir). The OSSB provides side a in the green triangle. As the outer bending angle is a 60-degree one, angle C in the green triangle is of 30 and B is of 60 degrees. Thus, you can gain side b: b = a × sine B. Side b equals C, which is the measurement to the contact points on the material outer side. It should be noted that At this bending angularity, C corresponds to or becomes close to the material thickness; however, size C will vary with bending angles, that is why OSSB should be used to estimate the suitable position of size C.

Step 5: The size of D equals c in the red right-angled triangle. Line a is the material thickness. B in the violet triangle is the outer bending angularity of 60 degrees. Thus, C in the violet triangle is of 30-degree angularity (60 + 30 + 90 = 180). In case of 90-degree edged material B in the red one is of 60-degree angularity (30 + 90 + 60 = 180). At this point line c in the red one of the triangles can be solved: c = a × cosine B.

Step 6: After the estimation of B, C, D the size of E can be calculated: E = B – (C + D).

Step 7: E provides the dimension of b in the violet triangle. As soon as the violet triangle angularities become apparent, a could be solved. This will give the size of F, the inner leg span: a= b/cosine C.

What about workpieces possessing more bending angles than 90-degree ones? Fig. 7 shows that the same steps should be followed again taking any estimated dimension as a start upon the testing sample, then passing through each correct triangle, identify the inner leg size. The process should be repeated to identify the next leg.

Fig. 5


  1. Take the measurement of A over the testing sample.
  2. Take the measurement of the inner radius through pin and radii gauges and optic comparators.
  3. B is equal to the c in the red triangle. Line a is the material thickness. With cohesive angles of 30 and 90 degrees, B appears of 60-degree angularity (30 + 90 + 60 = 180). c will be c = a × cosine B
  4. As soon as B is identified, C can be found: C = A – B
  5. After measuring the inner radius, you can pass to the measuring the dimension of a in the blue triangle for inner setbacks (ISSB): ISSB = [tangent (outer bending angle/2) × Inner radius.
  6. Line a is the inner setback. C must be of 30-degree angularity (60 + 90 +30 = 180). At this point, b in the blue triangle is possible to calculate. This can provide the size of D: b = a × sine B.
  7. After finding D, it is possible to figure out E: E = C – D. Thus you will gain b in the violet triangle.
  8. This provides the solution of a in the violet triangle, then the size of F, the inner leg span: a = b/cosine C.

At last, you have gained the inner leg sizes. In the same way as you formed a bending of 90 degrees, fold the inner leg sizes and subtracting the flat dimensions you will be able to find the bending allowance. 

Estimated inner leg sizes – flat dimension=Bending Allowance

Fig. 6

Calculations for k

Having the inner radius and bending allowance for testing samples, these can be included in the formula given below: k-factor = [(180 × BA) /(π × Outer bending angle × Mt)] – (Ir / Mt)

The following should be repeated unless you gain up to 3 testing samples. Only then it is proper to strike the average of the coefficient k. Thus, you can gain the most correct k for your applications.

The Y Factor

Still, it is actually possible to gain levels of higher accuracy. Having figured out the k factor, it might be used to define the Y factor. The latter considers stress of materials.

How can the y factor be defined and what is its relation to the k factor? They are closely related. They directly influence the elongation of bends in the formation. The base for the calculation of Y is the k factor.

The CAD software can use the Y factor rather than the k while determining BA and BD, which allows creating more accurate flat templates for sheet-metal parts. You might as well look over the figure on Y factor. Or else, Y factor could be calculated through this equation:

Y-factor = (k-factor × π) / 2

In case of using Y factor, certain adjustment should be made in accordance with your bending. Quite a different equation is used to determine the bending allowance: BA = [(π/2) × inner radius ] + (Y-factor × material thickness )×(outer bending angle / 90)

Finally your bending calculation can include custom-made k factor as well as Y factor. Now we can look into the process together with bending formulas:

Fig. 7
  1. Form up to 3 testing samples.
  2. Figure out the inner radius and bending allowance by measuring samples.
  3. Determine the k factor: k-factor = [(180 × bending allowance) (π×outer bending angle × material thickness)]-(inner radius /material thickness).
  4. For more precision determine the Y factor: Y-factor = (k-factor × π) / 2.

While making the necessary preparations to produce parts, include the coefficient k (if necessary the coefficient Y)in the equation of the bending allowance. The entire bend precision will be formed through BD, sizes of flat layouts:

Bending allowance with k factor = {[(π/180) × Inner radius] + [(π/180 × k-factor) × material thickness)] × outer bending angle

Bending allowance with Y-factor = BA = [(π/2) × Inner radius] + (Y-factor × Material thickness) × (outer bending angle / 90)

OSSB = [Tangent (Bending angle/2) × (Material thickness + Inner radius)

BD = (2 × OSSB) – BA

The calculation of the coefficient k will provide such essential facts as die widths, formation methods and factor of frictions. 

Surely, not every bending needs this combination. Actually, mostly applied coefficient k 0.4468 is suitable on daily basis. Still, some cases, particularly those requiring customized k and Y factors, might lack these features.

k or K factor?

What do they differ? k factor (not capital) serves as a base for calculation of the neutral axle transfer at the bend. K factor (with capital letter) calculates the outer setbacks (OSSB), which must be known prior to bending, as it is used to define bending deduction (BD) and the tangent locations together with bending radii.

Calculation of K factor is easier than that of k (for neutral axle shifts). K is just the tangent to half bending angles. In case of bending of 90 degrees, it is: K = tan (90/2) = 1. A K-factor for a 60-degree bend is K = tan (60/2) = 0.5773. Actually, it is included in the determination of outer setback:  

OSSB = [Tangent (Bending angle/2) × (Material thickness + Inner radius)

Did you notice K in the first part of the formula? Tangent (Bending angle/2). 

In addition, using the outer or inner bending angles to calculate the outside setback is dependent upon the methods of flat layouts.

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