By Ronald W. Leigh, Ph.D.

December 5, 2008
Copyright © 1994, 2006 Ronald W. Leigh

This paper explains the formulas used by the Bend Allowance Calculator.

Drawings of sheet metal parts usually show the part in its finished (bent) state.  However, the designer should supply information about the layout of the flat sheet stock before it is bent, which requires the calculation of the amount of material used at each bend.  This amount of material at each bend is referred to as the bend "allowance."  The individual bend allowances are shown in a developed (flat) view of the part which can thus include overall dimensions and the locations of key features prior to bending.

A.  The Formulas

The bend allowance for any bend is the length of the neutral arc, as shown below.  This neutral arc can be calculated from the material thickness (T), inside radius (R), and the angle through which the material is bent (A).

Bend with neutral arc and variables

The neutral arc, in a side view of the bend, represents the theoretical cylindrical plane of material that is neither stretched nor compressed.  The material on the outside of this plane is stretched, while the material on the inside of this plane is compressed.  But the length of this arc is the same before and after bending.

This plane occurs a certain fraction of the material thickness into the material measured from the inside surface.  This fraction ranges between 1/4 and 1/2, but is never greater than 1/2.  For rough calculations, some designers use a fixed fraction for all bends (such as .3, or .333, or .4).  However, the fraction actually varies from one bend to the next because the fraction depends on the ratio between the size of the inside radius and the thickness of the material.  The fraction is closer to 1/4 when the the bend is "profound" or "tight," that is, the inside radius is relatively small and the material is relatively thick.  The fraction moves from 1/4 toward 1/2 as the radius gets larger or the material gets thinner.

The first formula below provides a fraction (K) that is based on experimental results with soft steel as reported in various reference manuals.

FIRST FORUMLA (applies to soft steel)
 K =  ((R/T)/16) + .25    (K never greater than .5) 

The above formula applies to "medium hard" materials such as soft steel and aluminum.  For softer materials such as soft copper and soft brass replace the .25 with .21.  For harder materials such as hard copper, bronze, CRS, and spring steel, replace the .25 with .28.

After K is determined, a second formula is used to calculate the bend allowance.

 Bend allowance for any angle   =  A · π · (R + K·T) / 180 
Thus, the bend allowance for a 90° angle  =  π · (R + K·T) / 2

These formulas can be used whether you are working in inches or millimeters.  The derivation of both formulas is explained at the end of this article.

B.  Print Reading Pitfalls

Four cautions are necessary when working with sheet metal drawings and calculating bend allowances.

Misleading bend dimensions

C.  Inaccurate Tables in Machinery's Handbook

The three bend allowance tables in the Machinery's Handbook (Industrial Press) are based on three different formulas which use a constant fraction K for each different material.
For soft materials such as soft copper and soft brass, K is always .350.
For medium-hard materials such as soft steel, K is always .408.
For hard materials such as bronze, CRS, and spring steel, K is always .452.
The fact that these tables do not recognize variations in K (as the radius and the thickness change) makes them unacceptably inaccurate.  The use of a different K for different radius bends (determined by the first formula above) yields much more accurate results.

D.  Other Sources of Inaccuracy

Even when using the more accurate fraction K from the first formula above, we must keep in mind that any formula based on experience is still only a theoretical approximation.  In the case of sheet metal bends, the above formulas do not take into account several facts.

Even so, the error introduced by these factors is far less than the error introduced by the assumption of a constant K.

However, there are many production factors which can affect the actual amount of material used in a given bend.

Since the designer usually cannot control these factors, he/she must offer the best general estimate possible and expect those who fabricate the part to know how to compensate for these factors as needed.  See the following articles:

S. M. Adams, Bend Allowance Overview
Olaf Diegel, Bend Works: The fine-art of Sheet Metal Bending

E.  Derivation of the Formulas

First formula

The first formula is based on experience rather than purely on mathematics.  By plotting experimental data we can construct a suitable formula.  Here is the data reported by Pollack and Ostergaard.

Herman Pollack,
Tool Design, 2nd ed.,
Prentice Hall, 1988,
page 465
If  R < T         then K = 1/4
If  T <= R <= 2T  then K = 1/3
If  R > 2T        then K = 1/2

D. Eugene Ostergaard,
Basic Die Making,
McGraw-Hill, 1963,
page 26

If  R < 2T         then K = .33
If  2T <= R <= 4T  then K = .4 
If  R > 4T         then K = .5 

Ostergaard also indicates that for a zero-radius, 90° bend the allowance is .5T, which is the same result obtained from the second formula above when K = .3183.  These data are graphed below.

True K would certainly increase gradually (without the stair-steps), thus it is assumed that the green line is a fairly realistic representation of K (up to R/T = 4).  This line's slope is 1/16 and its y-intercept is .25, so the formula is:

K = ((R/T) / 16) + .25    (K never greater than .5)

Second formula

The second formula is easily derived from the formula for the circumference of a circle.

Full circumference of any circle  =  π · diameter  =  π · 2 · radius

Consider the neutral arc as part of a full circle which has a radius that is larger than the inside radius.  The amount larger is merely K·T, so we substitute as follows.

Full circumference of neutral "circle"  =  π · 2 · (R + K·T)

Since the neutral arc is only a portion of a full circle, we place the fraction A/360 in the formula as follows.

Length of neutral arc  =  A · π · 2 · (R + K·T) / 360  =  A · π · (R + K·T) / 180