Part 2: Limit Collapse of a Cracked Beam a) A long solid rectangular section beam, length L=1.4m, width b=40mm and depth 2h=100mm, is subject to uniform distributed loading W per unit length, as shown. The beam is made from alloy steel with yield strength oy = 665 MPa. W po x Wy L/2 16bh²oy 3L² L/2 Show that first yield will occur simultaneously on the top and bottom surfaces at the middle of the beam when the load W = Wy, where: 2h [2 marks]/n[2 Marks] b) The beam material is elastic perfectly plastic, with yield stress dy. Sketch the following stress distributions through the depth of the beam -h ≤ y ≤ h) at the mid-span: • Elastic • Partially plastic • Limit state From equilibrium (i.e. lower bound limit load approach), show that the limit load W₁ is: WL 8bh²oy L² Calculate the numerical value of W₁./nc) Visual inspection of the beam identifies a crack of undefined length a across the full section width b located mid-span at the bottom surface of the beam, as shown. a 9m 2h

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