_______________________________________________________________________ PROBLEM (9.103) A plate of thickness t = 5 mm is made of a cast-aluminum alloy for which σ u = 70 MPa and σ u ' = 175 MPa. Calculate the force P required to punch a d = 15 mm-diameter hole through the plate, as depicted in Fig. P9.103 and discussed in Sec. 2.5. Apply: (a) The maximum normal stress theory of failure. (b) Colulomb-Mohr theory of failure. Assumption: The shear force is taken to be uniformly distributed through the plate thickness. SOLUTION Uniform shear stress τ acts on a typical element shown in Fig. (a). (a) σ 1 = σ u or σ3 = σu P = σu π td P = π tdσ u = π (0.005)(0.015)(70 ×106 ) or P = 16.49 kN (b) σ1 σ 3 − = 1; σu σu ' τ (1 + τ = P π dt σ 1 = −σ 3 = τ σu ) = σu σu ' From which P= tdσ u 1+ or σu ' σu = (0.005)(0.015)(70 ×106 ) 175 1+ 70 P = 1.5 kN Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other reproduction or translation of this work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful.