# Minimum Bend Radius for Plate Bending

Although Vulcan was best known for products based on sand castings, many of Vulcan products–especially those produced by the Special Products Division–were fabricated. An important operation in fabrication is plate bending. Although plate bending is usually associated with sheet metal formation like Vulcan’s vibratory power packs, Vulcan also bent plate which is thicker than normal sheet metal. An example of a bent plate part is the motor guard for the 2300/2300L/2300A vibratory hammer, shown in the photo above.

One question which is hard to get an answer for is “how small does the bending radius have to be before the plate breaks?” This piece attempts to furnish an answer for that. An interesting study of this was done by Datsko and Yang (1959), and that forms the basis for this piece, although the solution presented needs to be used with care.

Let’s start by defining some nomenclature.

As is the case with any bending, the process of bending induces a moment in the plate around a bending mandrel. The concept is to bend a plate of thickness $t$ to an inner radius $R$. Implicit in the diagram above is that the neutral axis–the axis at which the tensile stresses above it change into the compressive stresses below it–is halfway between the upper surface and the lower surface of the plate. For plates that aren’t too thick and those where the elastic limit of the material isn’t exceeded, that’s fine: however, the whole point of plate bending is to induce plasticity and thus make the bend permanent. So the neutral axis will shift off of centre.

Datsko and Yang (1959) discovered that the minimum bend radius could be determined using one parameter: the reduction in area $A_r$ in percent of the material, a common property determined during tension tests. Without going into the details of their theory, they discovered the following:

First, for values of $A_r <20%$, the minimum ratio of the bend radius to the plate thickness $\frac{R}{t}$ can be estimated from theory as follows: $\frac{R}{t}=50\,{A_{{r}}}^{-1}-1$ (1)

Second, for values of $A_r >20%$ the minimum ratio of the bend radius to the plate thickness $\frac{R}{t}$ can be estimated from theory as follows: $\frac{R}{t}={\frac {\left (100-A_{{r}}\right )^{2}}{200\,A_{{r}}-{A_{{r}}}^{2}}}$ (2)

The “trout in the milk” for the equations is that the values aren’t the same when $A_r =20%$; Equation (1) yields $\frac{R}{t} =1.5$ and Equation (2) yields $\frac{R}{t} =1.78$.

Third, Datsko and Yang (1959) also recommend “from laboratory experience” that this equation $\frac{R}{t}=60\,{A_{{r}}}^{-1}-1$ (3)

“…is a very satisfactory relationship for the ductile materials that do have a shift in the position of the neutral axis.”

The three relationships are plotted below. The bend radius results from the three equations of Datsko and Yang (1959) The red line is Equation (1), the blue line is Equation (2), and the green line is Equation (3).

Let’s do an example using Equation (3). Consider an ASTM A514 Gr. B (a “T1” type steel) with a yield strength of 100 ksi (689 MPa) and a reduction of area of 40%. Direct substitution in to Equation (3) yields $\frac{R}{t}=60\times{40}^{-1}-1 = 0.5$. This simply means that the bending radius (again on the inside of the bend) can be one half of the thickness. This will strike many designers as low. Square and rectangular tubing, for example, have corner radii where the outer radius is 2-3 times the plate thickness, which translates into an $\frac{R}{t}$ value of 1-2 times the plate thickness. Leaders at the Special Products Division with square tubing for the main rails of the leaders. Although it’s not a close-up, you can see the corner bending in the square tubing.

The key problem here is whether the steel has the reduction in area the “typical” values indicate. In some cases it is difficult to obtain the reduction in area. In others the steel is not properly tempered and thus does not have the ductility one should expect. Defensive design in these cases is crucial. Although Datsko and Yang (1959) considered non-ferrous materials, Vulcan did not widely use these in its products, especially in its fabricated ones. The actual strength of these bent corners is also an open question.

The method described here give a “theoretical minimum” to the bend radius, but should be used with a generous “factor of safety” to insure that bent plate areas retain their material integrity after bending.

## References

Datsko, J., and Yang, C.T. (1959) “Correlation of Bendability of Materials With Their Tensile Properties.” The University of Michigan Industry Program of the College of Engineering. Report IP-401, November.