Worked example moments Bending without tension

From Equation 6.10, the elastic modulus in plane strain is E ′ = E 1 − υ 2 = 76.9 GPa From Equation 6.18, the limiting elastic bending moment is M e = St 2 6 = 138.6 × 10 6 2 × 10 −3 2 6 = 92.3 Nmm The radius of curvature, from Equation 6.19, is ρ e = E ′ t 2S = 76.9 × 10 9 × 2 × 10 −3 2 × 138.6 × 10 6 = 0.555 m or 555 mm The fully plastic moment from Equation 6.21 is M p = St 2 4 = 3 2 M e = 138 Nmm

6.6 Elastic unloading and springback

If a sheet is bent by a moment to a particular curvature, as shown in Figure 6.15, and the moment then released, there will be a change in curvature and bend angle. The length of the mid-surface is l = ρθ This will remain unchanged during unloading as the stress and strain at the middle surface are zero. From this, we obtain θ = l 1 ρ 6.28 Differentiating Equation 6.28, in which l = constant, we obtain θ θ = 1ρ 1ρ 6.29 M M q q + ∆q r + ∆r r Figure 6.15 Unloading a sheet that has been bent by a moment without tension. 92 Mechanics of Sheet Metal Forming

6.6.1 Springback in an elastic, perfectly plastic material

The assumed stress–strain curve for an elastic, perfectly plastic material that undergoes reverse loading is shown in Figure 6.16. This neglects any Bauschinger effect; this is the phenomenon of softening on reverse loading that is observed in many materials. From Figure 6.16, a change in stress of σ 1 = −2S can occur without the material becoming plastic. If we assume that the unloading of the sheet will be an elastic process, then the elastic bending equations, Equations 6.17, can be written in difference form, i.e. M I = σ 1 y = σ 1 max t 2 = E ′ 1 ρ 6.30 ∆s 1 = −2 S e 1 s 1 S − S Figure 6.16 Elastic, perfectly plastic material model with reverse loading. For a sheet that has been bent to the fully plastic moment, the unloading curve will be parallel to the elastic loading line as shown in Figure 6.17. Noting the similar triangles, we see that for a change in moment of −M p , 1ρ 1ρ e = M M e = −M p M e M Moment Curvature 6 St 2 4 St 2 M p = 1r e ∆1r 1r 1r M e = Figure 6.17 Moment, curvature diagram for an elastic, perfectly plastic sheet showing unloading from a fully plastic moment. Bending of sheet 93