Stress distribution in small radius bends

R R dq d r s 3 s 1 s 3 + ds 3 r Figure 6.22 Stresses on an element at a radius r in a small radius bend. To investigate the stresses, we assume a Tresca yield criterion as in Section 2.4.4. In the region where the bending stress is tensile, the greatest stress is σ 1 and the least is σ 3 ; therefore, σ 1 − σ 3 = S = σ f and Equation 6.46 becomes dσ 3 dr − S r = 0 If the stress on the outer surface is zero, integrating between the limits r = R and r, we obtain σ 3 = −S ln R r 6.47a In the region where the bending stress is compressive, the yield criterion gives that σ 1 − σ 3 = −S = −σ f Integrating Equation 6.46 between the limits r = R and r, and assuming that the stress on the inner surface is zero, we obtain σ 3 = −S ln r R 6.47b The stress distributions are shown schematically in Figure 6.23. It should be noted that these are for a non-strain-hardening material in which S is constant. The through thickness stresses given by Equations 6.67a and b are equal at a radius of r b = RR o As noted earlier, this is the radius at which the bending strain reverses. Bending of sheet 99 R R r b = RR o 12 s 2 s 1 s 1 s s 2 s 3 Figure 6.23 Stress distributions in bending a rigid, perfectly plastic sheet to a small bend ratio. In determining the stress distributions for a strain-hardening material it is necessary to determine the strain ratios in each element and from these evaluate the effective strain as outlined in the preceding section. The result for a material obeying a strain-hardening stress strain law and bent to a bend ratio of 2.5 ρ = 2.5t is shown in Figure 6.24. − 800 − 400 400 800 s 3 s 2 s 1 s 1 s f s MPa s 2 −0.5 0.5 2 yt Figure 6.24 Stress distributions for a rigid, strain-hardening sheet bent to a bend ratio of 2.5.

6.8 The bending line

6.8.1 The moment curvature characteristic

The determination of the shape of a sheet that is bent under the action of a force or moments depends on knowing the moment curvature characteristic. Examples of this are 100 Mechanics of Sheet Metal Forming 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