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A = 25B + 3.2 x 108 = (—2.88 + 3.2) 10s =0.32 x 108 substituting in the given expression for hoop stress,

115x106

aH = 0.32 x 108 --—---1.6 x \06T + 1.6 x 106 x 247

rAt r = 0.2, aH = (0.32 - 2.88 - 3.2 + 3.96)10x = -180 MN/m2 At r = 0.3, aH = (0.32 - 1.28 - 1.6 + 3.96)108 = +140 MN/m2

The maximum tensile circumferential stress therefore occurs at the outside radius and has a value of 140 MN/m2. The maximum compressive stress is 180 MN/m2 at the inside radius.

Problems

Unless otherwise stated take the following material properties for steel:

4.1 (B). Determine equations for the hoop and radial stresses set up in a solid rotating disc of radius r commencing with the following relationships:

B f.xv2 r2

Hence determine the maximum stress and the stress at the outside of a 250 mm diameter disc which rotates at 12000 rev/min. [76, 32.3 MN/m2.]

4.2 (B). Determine from first principles the hoop stress at the inside and outside radius of a thin steel disc of 300 mm diameter having a central hole of 100 mm diameter, if the disc is made to rotate at 5000 rev/min. What will be the position and magnitude of the maximum radial stress?

[38.9. 12.3 MN/m2; 87 mm rad; 8.4 MN/m2.] 43 (B). Show that the tensile hoop stress set up in a thin rotating ring or cylinder is given by nH = fxo'r'

Hence determine the maximum angular velocity at which the disc can be rotated if the hoop stress is limited to 20 MN/m2. The ring has a mean diameter of 260 mm. |3800 rev/min.]

4.4 (B). A solid steel disc 300 mm diameter and of small constant thickness has a steel ring of outer diameter 450 mm and the same thickness shrunk onto it. If the interference pressure is reduced to zero at a rotational speed of 3000 rev/min. calculate

(a) the radial pressure at the interface when stationary:

(b) the difference in diameters of the mating surfaces of the disc and ring before assembly.

The radial and circumferential stresses at radius r in a ring or disc rotating at w rad/s are obtained from the following relationships:

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