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1. A steel component (E = 200 GPa, ν = 0.29) is subjected to a state of plane stress (σz = τxz = τyz = 0) when the design loads are applied. At the critical point in the member, the stress components are given in Figure 1. The material has a yield stress, σyield = 450 MPa. For (a) – (d), calculate the factor of safety based on the following yield criteria: (i) Maximum principal stress theory (Rankine’s theory); (ii) Maximum shear stress theory (Tresca’s theory); (iii) Maximum distortion energy theory (von Mises theory). (a) (b) (c) (d) Figure 1 x y 130 MPa 20 MPa 100 MPa x y 100 MPa 50 MPa 120 MPa x y 100 MPa 30 MPa 150 MPa x y 100 MPa 40 MPa 150 MPa 2 2. At a point on the surface of a component the stresses are σx = −130 MPa, σy = 50 MPa, and τxy = 25 MPa as shown in Figure 2. Using Mohr’s circle, determine: (a) the stresses acting on an element inclined at an angle, θ, of 60° clockwise, (b) the principal stresses, and (c) the maximum in-plane shear stresses. Figure 2 x y 130 MPa 25 MPa 50 MPa 3 3. A shaft containing two fillets, as shown in Figure 3, is machined from alloy steel heat-treated to 300 BHN and rotates at 3000 rpm, whilst the imposed load, F, remains static. The shaft is designed to have infinite lifetime with respect to fatigue failure. It is found when the imposed load is increased, failure will occur at the two fillets at the same time. Calculate: (a) the maximum allowable load given a factor of safety of 2; (b) the distance L4; (c) the fatigue life in the number of cycles if the shaft is subjected to the loading pattern as shown in Table 1. Given: D = 40 mm; d = 30 mm; L1 = 50 mm; L2 = 100 mm; L3 + L4 = 200 mm; R1 = 2 mm; R2 = 1 mm. Figure 3: A shaft containing two fillets Table 1: Loading sequence of varying amplitudes Load Number of cycles (n) 3Fallow 30 7Fallow 20 Fallow 50 6Fallow 40 5Fallow 25 where Fallow is the maximum allowable load found in (a). L1 R1 F R2 L4 L3 L2 d D d 4 4. A shaft containing two fillets, as shown in Figure 4, is machined from alloy steel and rotates at 3000 rpm, whilst the imposed load, F, remains static. The shaft is designed to have infinite lifetime with respect to fatigue failure. The ultimate tensile strength is 1000 MPa. (a) If the shaft is not transmitting power, given a factor of safety of 2, calculate the maximum allowable load; (b) At the maximum allowable load from (a), if the shaft is transmitting power of 40 kW, calculate the factor of safety. Given: D = 40 mm; d = 30 mm; L1 = 50 mm; L2 = 150 mm; L3 = 90 mm; R1 = R2 = 3 mm. Figure 4: A shaft containing two fillets

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