Unbending at B Worked example drawing over a radius

The increase in tension as the sheet straightens is T = 300 × 10 6 1.91 × 10 −3 2 4 × 0.8 × 9 × 10 −3 1 + 339 573 2 = 51 kN m and the exit tension is 299 + 51 = 350 kNm. The change in thickness at the unbend is t t = − 1.91 2 × 9 339 573 = −0.0628 and the thickness increment is −0.0628 × 1.91 = −0.12, and the final thickness is 1.91 − 0.12 = 1.79 mm.

10.6 Draw-beads

In draw die forming as in Chapter 4, draw-beads are used to generate tension in the sheet. A draw-bead is illustrated schematically in Figure 10.10. The sheet enters at the left with a small tension T , and then undergoes a series of bending or unbending processes marked by the broken lines in the diagram. At each bending or unbending point, following Equation 10.20, the tension will increase, by T = 1 4η T y ρt 1 + T T y 2 and the thickness decrease, as indicated in Equation 10.21, by t t = − 1 2ρt T T y Between the points where the curvature changes, there will be an increase in tension, T f , due to friction, of T + T f = T exp μθ T T f h t n q r Figure 10.10 A draw-bead used to increase the tension in a sheet from T to T f as it passes through from left to right in a draw die. 150 Mechanics of Sheet Metal Forming where the angle of wrap θ is related to the depth of engagement h as shown in Figure 10.10. The tension generated by a bead can be increased by reducing the bend ratio ρt, and increasing the depth of the bead h, which will increase the angle θ . For a given bead, the tension will increase with the flow stress of the sheet S, with the incoming thickness t and with the friction coefficient μ. It may be seen from Chapter 4 that in order to maintain a constant strain distribution in a part, the tension applied should increase if the above variables increase. Therefore a draw-bead is, to some extent, a self- compensating device in a draw die that will adjust the tension as material properties and friction change.

10.7 Exercises

Ex. 10.1 A sheet of aluminium, 1.85 mm, thick and having a constant plane strain flow stress of 180 MPa is curved over a form block having a radius of 600 mm. If the plane strain elastic modulus is 78 GPa, determine the tension required to a initiate plastic deformation, and b to make the sheet fully plastic. [Ans: 111, 333 kNm] Ex. 10.2 For the operation in Exercise 10.1, determine the tension and the moment when the plastic deformation zone has penetrated to the mid-surface. What is the final radius of curvature after unloading? Sketch the approximate form of the residual stress distribution in the sheet. [Ans: 278 kNm, 34 Nmm, 1.19 m] Ex. 10.3 Steel sheet is curved by stretching over a frictionless form block as in Figure 10.1. The radius of curvature of the block is 2.5 m. The final mean plastic strain in the sheet is 0.012 1.2. Aluminium sheet is substituted and the same final strain preserved. The plane strain elastic modulus and stress strain curves and the thickness are given below. Compare the final radius of curvature of each sheet. [Ans: 2.56 v. 2.59 m] Material E ′ GPa Stress, strain, MPa Thickness, mm Steel 220 σ 1 = 700ε 0.2 1 0.8 Aluminium 78 σ 1 = 400ε 0.2 1 1.2 Ex. 10.4 A steel sheet is drawn over a radius ρ = bt, as in Figure 10.7. If the back tension is 60 of the yield tension in plane strain, show how the stress σ 1 S and the thickness reduction factor −tt vary with the bend ratio b = ρt in the range 3 b 10 for the first bend. Assume the efficiency factor is 1. Combined bending and tension of sheet 151