The bending line construction

Thus, a is both the height of the point above the line of application of the force A and the appropriate distance along the scaled moment curvature diagram. It should be noted that as the units of Mare [force][length][length] and of P [force][length], the unit of MP is [length]. This construction permits the curvature at any point to be obtained directly. For example, at point B on the curved sheet, the curvature is given by the corresponding point B in the diagram on the right. The angle of the normal to the bent sheet, α, can also be found from this construction. As shown on the left-hand side of Figure 6.25, the change in the angle of the normal over a small distance along the sheet ds is dα. The corresponding change in height is da, where da = ds cos α As ds = ρdα, we obtain cos α dα = 1 ρ da 6.49 Integrating between the points A and B on the sheet, we obtain α B α A cos α dα = sin α B − sin α A = a B o 1 ρ da 6.50 The term on the right is the area between the curve OEB and the vertical axis. This can be found by a graphical method from an experimental moment curvature diagram or calculated from a theoretically determined characteristic. Thus, knowing the direction of the normal at A, the direction of the normal at B can be determined.

6.8.3 Examples of deflected shapes

It is often useful to know the region in a bent sheet where the deformation will be elastic. If the elasticplastic transition is known in the moment curvature diagram, e.g. at point E on the curve in Figure 6.25, the sheet will be elastically deformed between A and the height corresponding to E. On release of the force, this portion of sheet will spring back to a straight line. The effect of material properties on the deformed shape of sheets bent by line forces is illustrated in Figure 6.26. In a, the material is rigid, perfectly plastic. The area between the moment curve and the axis C between A and B is zero and therefore, from Equation 6.50, the normals to the sheet at these points are parallel and the sheet is straight. There is a plastic hinge at B. The value of the force P is uniquely determined as M p P = a B In Figure 6.26b the moment curve for a strain-hardening sheet is shown on the right. The differences between the sines of the angles of the normals is given by the area between the curve OB and the axis OC. In Figure 6.26c, a linear moment curve is shown and the 102 Mechanics of Sheet Metal Forming a b c a A O 1r M p P P B a B a A a O 1r P B B a A a B a O 1r P B a A a B Figure 6.26 Bending line construction for a sheet bent by a horizontal line force for a a rigid, perfectly plastic sheet, b a strain-hardening sheet, and c a sheet having a linear stress–strain relation. curvature of the sheet will be proportional to the height above the line of application of the force. In Figure 6.27, a sheet is bent between two smooth, parallel plates. This is similar to the operation of hemming where the edge of a sheet is bent over itself or another sheet. If we assume that the force is applied at the tangent point and is normal to the plates, then the construction is shown in the diagram. The curvature will vary from zero at the tangent point to a maximum at the nose. In the manufacture of articles such as hose clips, a strip may be bent over a form roll as shown in Figure 6.28. The curvature of the strip should match that of the form roll at the tangent point, i.e. 1ρ B = 1R. If the force to bend the strip is applied by a small roller at A and at the instant shown the strip is horizontal, then the angle of the normal at Bending of sheet 103