Worked example tensions Principal tensions or tractions

The effective stress is σ = 530 × 0.246 0.246 = 375 MPa. The principal stresses are σ 1 = σ 2 = σ and the principal tensions are T 1 = T 2 = 375 × 10 6 × 0.626 × 10 −3 = 236 kNm. For plane strain, α = 0.5, β = 0 and the strain at maximum tension is ε ∗ 1 = n = 0.246 and ε = 2 √ 3ε 1 and σ = √ 32σ 1 . The thickness at maximum tension is, t = t exp − 1 + β ε ∗ 1 = 0.8 exp −0.246 = 0.626 mm and following the above, T 1 = 281 kNm, T 2 = αT 1 = 140 kNm Thus for plane strain the maximum tension is greater, but the thickness at maximum tension is the same.

3.8 Summary

In the analysis of a sheet metal forming operation, different regions of the sheet will have a particular forming path that can often be considered as proportional and represented by a line in the stress and strain diagrams. These diagrams are the ‘charts’ in which the course of the process can be plotted. In subsequent chapters these diagrams will be used extensively.

3.9 Exercises

Ex. 3.1 A small circle 5.0 mm in diameter is printed on the surface of an undeformed low carbon steel sheet with thickness 0.8 mm. Then the sheet is deformed in a plane stress proportional process. It is noted after unloading that the circle has been distorted into an ellipse with major and minor diameters of 6.1 mm and 4.8 mm respectively. The effective stress strain relation is: σ = 600ε 0.22 MPa a Assuming that the loading is monotonic, what is the ratio of stresses α? b Determine the tension T 1 and T 2 . c Calculate the effective strain. [Ans: 0.328; 329.2 kNm, 108.2 kNm; 0.21] Ex. 3.2 Consider the case of a constant thickness, b uniaxial tension, and c plane strain, where σ 1 σ 2 and σ 3 = 0. For each case, compare the ratio of σ , the effective stress, and τ max , the maximum shear stress. [Ans: √ 3 ; 2 ; √ 3 .] Ex. 3.3 In a deep-drawn cup as shown in Figure 3.2, the strains in the centre of the base, a, half-way up the cup wall, b, and in the middle of the flange,c, are as follows: a 0.015, 0.015, b 0.050, 0.000, and c 0.150, −0.100. Strain-hardening in the material is negligible so that the effective stress is constant at 300 MPa. The initial thickness is Deformation of sheet in plane stress 43 0.50 mm. Determine at each point, the thickness and the major tension acting along the line shown in the sectioning plane. [Ans: 0.485, 0.476, 0.476 mm; 146, 165, 125 kNm] Ex. 3.4 The average effective strain in a steel stamping is 0.03. If friction is neglected and all the plastic work done is converted into heat, what would be the average temperature rise? The effective stress strain law is σ = 600 0.01 + ε 0.22 MPa; for steel the specific heat is 454 Jkg ◦ C and the density is 7850 kgm 3 . [Ans : ≈ 2 ◦ C] 44 Mechanics of Sheet Metal Forming