Consider the following biaxially loaded thin plate with a central circular cutout: Txy ot dy The tangential stress () at any location (0) along the edge of the cutout can be expressed as a function of the nominal plate stresses x, y, Txy and the angle 0: o = [1-2cos20] + [1 +2cos 20]oy - [4sin20]txy i. x > 0,0y = +0xTxy = 0 ii. o, > 0,0,= -0x, Txy = 0 iii. dx > 0, y = 0x, Txy = 0x a) Derive expressions for the tangential stress () for the following conditions and indicate at what angle (0) o is maximum: b) What condition from (a) will produce the lowest fatigue life along the hole edge for fully-reversed constant amplitude cycling of ax? 0 Ox da Aa da Omax omin time c) Assume the plate is constructed from 2024-T4 aluminum alloy and has the following properties": Elastic Properties Monotonic Material Strengths a=of (2N,)b E % Tu dB a', b (MPa) (MPa) (MPa) (MPa) 70,300 0.345 303 476 631 (MPa) 900 -0.102 V What is the minimum predicted fatigue life along the hole edge if o, is cycled between +60MPa and - 60MPa? Assume condition (a.ii) above applies. d) What is the minimum predicted fatigue life along the hole edge if the amplitude of o, is 100 MPa and the stress ratio is R=+0.10? Assume condition (a.iii) above applies and report the results using both the Morrow and SWT mean stress effects relations. e) What stress ratio (R) will produce a minimum fatigue life of 1 x 10cycles if the amplitude of o, is 120 MPa. Assume condition (a.i) above applies and use the SWT mean stress relation.