shown in the preceding section, for example in Figure 6.17. This shows a characteristic determined by analysis, but techniques also exist to obtain such a diagram experimentally.
In studying practical problems it is highly desirable to use a moment diagram determined experimentally as inaccuracies may exist in curves calculated from tensile test data due,
among other things, to:
• inaccuracies in data from tensile test at small strains in the region of the elastic plastic transition;
• the yield criterion adopted being only an approximation; • variation of properties through the thickness of the sheet; and
• anisotropy in the sheet that is not well characterized. Assuming that a reasonable moment curvature characteristic is available, the determination
of the bent shape of the sheet is often time-consuming. In this section, a construction is described called the bending line that will give quickly some of the information needed.
6.8.2 The bending line construction
In Figure 6.25, a sheet bent by a horizontal force per unit width of sheet P is shown on the left. On the right, a scaled version of the moment curvature characteristic is drawn
such that the curvature axis is collinear with the line of action of the force. The ordinate of the curve is obtained from the moment curvature diagram by dividing the moment by the
force P . At some point a distance s along the sheet at a height a the moment is M
= P a, and hence
a =
M P
6.48
MP B
E A
P a
A
O a
a da
ds
r S
B
1r
Figure 6.25 Diagram of the deflected shape of a sheet bent by a line force P per unit length on the
left, and the scaled moment curvature characteristic on the right.
Bending of sheet 101
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
