Elastic bending Bending without tension

where I = t 3 12 is the second moment of area for a unit width of sheet and 1ρ is the curvature. The limit of elastic bending is when the outer fibre at y = t2 reaches the plane strain yield stress S. The limiting elastic moment is given by M e = St 2 6 6.18 and the curvature at this moment is 1 ρ e = 2S E ′ t 6.19 Within this elastic range, the moment, curvature diagram is linear as shown in Figure 6.8, i.e. M = E ′ t 3 12 1 ρ 6.20 The bending stiffness of unit width of the sheet is E ′ t 3 12.

6.5.2 Rigid, perfectly plastic bending

If the curvature is greater than about five times the limiting elastic curvature, a rigid, perfectly plastic model, Equation 6.12, as shown in Figure 6.6c, may be appropriate, although this will not give information on springback. The stress distribution will be as shown in Figure 6.9. In Equation 6.8, the stress is constant and integrating, we obtain the so-called fully plastic moment M p as M p = St 2 4 6.21 The moment will remain constant during bending and is illustrated in Figure 6.10. S M p s 1 Figure 6.9 Stress distribution for a rigid, perfectly plastic material bent without tension. 88 Mechanics of Sheet Metal Forming 4 St 2 M r 1 Curvature Figure 6.10 The moment curvature diagram for a rigid, perfectly plastic sheet bent without tension. t 2 t 2 S − S y e M s 1 Figure 6.11 Stress distribution for an elastic, perfectly plastic sheet bent without tension.

6.5.3 Elastic, perfectly plastic bending

For curvatures beyond the limiting elastic curvature 1ρ e and below that where the moment reaches the fully plastic moment M p , an elastic, perfectly plastic model, as in Section 6.4.1, is often used. The model is illustrated in Figure 6.6b; the flow stress is constant and for plane strain, σ 1 = 2 √ 3 σ f = S. The stress distribution is illustrated in Figure 6.11; for y y e , the material is plastic with a flow stress S. As the curvature increases, y e decreases and at any instant is given by ε b y =y e = y ρ = S E ′ i.e. y e = S E ′ 1 1ρ = m t 2 6.22 From Equation 6.19, m = 1ρe 1ρ and 1 ≥ m ≥ 0. The equilibrium equation, from Equation 6.16, is M = 2 y e E ′ y ρ y dy + t 2 y e Sy dy = St 2 12 3 − m 2 6.23 The moment, curvature characteristic is shown in Figure 6.12 and it may be seen that this is tangent to the elastic curve at the one end and to the fully plastic curve at the other. Bending of sheet 89