Worked example curving a strain-hardening sheet

The change in curvature is given by Equation 10.11, i.e. 1ρ = − dσ 1 E ′ dε 1 1ρ Given that ρ = a 2 + h 2 2h ≈ a 2 2h for small values of h a, we obtain h = a 2 2 1ρ Hence h = 0.8 2 2 0.22 × 800 × 10 6 220 × 10 9 1 0.0015 + ε 1 0.78 1 5 m Evaluating, for 0.3 strain, the springback is 3.5 mm and for 1.5 strain it is 1.3 mm.

10.4 Bending a rigid, perfectly plastic sheet under tension

If the bend radius is in the range of about 3 to 10 times the sheet thickness, it is reasonable in an approximate analysis to consider the material as rigid, perfectly plastic, as discussed in Section 6.5.2; also we neglect through-thickness stress and assume plane strain bending. For very small radius bends, these assumptions may not be justified. If a rigid, perfectly plastic sheet is subjected to a tension less than the yield tension and then to a moment sufficient to generate some curvature, the strain and stress distributions will be as shown in Figure 10.6. The neutral axis will be at some distance e from the mid-surface. For a positive tension the neutral surface is below the mid-surface and if the sheet is in compression, i.e. T is negative, it is above the mid-surface. The strain in Figure 10.6b, from Equation 6.3 and 6.4, is ε 1 = ε a + ε b = e ρ + y ρ 10.13 S e T T M a b c T M s 1 S e 1 − S Figure 10.6 a Stress distribution in a sheet prior to bending. b Strain distribution in the sheet after bending. c Stress distribution after bending. 144 Mechanics of Sheet Metal Forming Applying the equilibrium equation, Equation 6.7, we obtain T = −e −t2 −S dy + t 2 −e S dy = 2Se or e = T 2S = t 2 T T y 10.14 where the tension to yield the sheet in the absence of tension is T y = St. The moment equilibrium equation, Equation 6.8, gives M = −e −t2 −Sy dy + t 2 −e Sy dy = S t 2 2 − e 2 and substituting the plastic moment M p in the absence of tension as given by Equation 6.21, we obtain for combined tension and moment, that the moment is M = St 2 4 1 − T T y 2 = M p 1 − T T y 2 10.15 It is seen that the presence of an axial force on the sheet will significantly reduce the moment required to bend the sheet and this will be true for both tensile and compressive forces as the sign of the tension T in Equation 10.15 will be immaterial, although, as indicated, the position of the neutral axis in Figure 10.6 does depend on the sign of the applied force.

10.5 Bending and unbending under tension

A frequent operation in sheet forming is dragging sheet over a radius as illustrated in Figure 10.7. The sheet moves to the right and is suddenly bent at A. It then slides against friction over the radius and is unbent at B. An example of such a process is shown in draw die forming in Figure 4.3. At the region DC the sheet slides against friction over the die radius. This tension is sufficient to yield the sheet and the process can only be performed with strain-hardening material. Referring to Figure 4.3, there are three important effects: T A B T f v t f r t Figure 10.7 Dragging a sheet over a tool radius under tension. Combined bending and tension of sheet 145