25 mm

332 (B). A solid shaft 40 mm diameter is made of a steel the yield point of which in shear is 150 MN/m2. After yielding, the stress remains constant for a very considerable increase in strain. Up to the yield point the modulus of rigidity C = 80 GN/m2. If the length of the shaft is 600 mm calculate:

(a) the angle of twist and the twisting moment when the shaft material first yields;

(b) the twisting moment when the angle of twist is increased to twice that at yield. [3.32°; 1888, 2435 Nm.]

333 (B). A solid steel shaft, 76 mm diameter and 1.53 m long, is subjected to pure torsion. Calculate the applied torque necessary to cause initial yielding if the material has a yield stress in pure tension of 310 MN/m2. Adopt the Tresca criterion of elastic failure.

(b) If the torque is increased to 10% above that at first yield, determine the radial depth of plastic penetration. Also calculate the angle of twist of the shaft at this increased torque. Up to the yield point in shear, G = 83 GN/m2.

(c) Calculate the torque to be applied to cause the cross-section to become fully plastic.

334 (B). A hollow steel shaft having outside and inside diameters of 32 mm and 18 mm respectively is subjected to a gradually increasing axial torque. The yield stress in shear is reached at the surface when the torque is I kNm, the angle of twist per metre length then being 7.3°. Find the magnitude of the yield shear stress.

If the torque is increased to 1.1 kN m. calculate (a) the depth to which yielding will have penetrated, and (b) the angle of twist per metre length.

State any assumptions made and prove any special formulae used.

335 (B). A hollow shaft, 50 mm diameter and 25 mm bore, is made of steel with a yield stress in shear of 150 MN/m2 and a modulus of rigidity of 83 GN/m2. Calculate the torque and the angle of twist when the material first yields, if the shaft has a length of 2 m.

On the assumption that the yield stress, after initial yield, then remains constant for a considerable increase in strain, calculate the depth of penetration of plastic yield for an increase in torque of 10% above that at initial yield. Determine also the angle of twist of the shaft at the increased torque.

336 (C). A steel shaft of length 1.25 m has internal and external diameters of 25 mm and 50 mm respectively. The shear stress at yield of the steel is 125 MN/m2. The shear modulus of the steel is 80 GN/m2. Determine the torque and overall twist when (a) yield first occurs, (b) the material has yielded outside a circle of diameter 40 mm, and (c) the whole section has just yielded. What will be the residual stresses after unloading from (b) and (c)?

(2.88, 3.33, 3.58 kN m; 0.0781, 0.0975, 0.1562 rad, (a) 19.7, -9.2, -5.5 MN/m2, (b) 30.6, -46.75 MN/m2.]

337 (B). A shaft having a diameter of 90 mm is turned down to 87 mm for part of its length. If a torque is applied to the shaft of sufficient magnitude just to produce yielding at the surface of the shaft in the unturned part, determine the depth of yielding which would occur in the turned part. Find also the angle of twist per unit length in the turned part to that in the unturned part of the shaft. [U.L.] [5.3 mm; 1.18.]

338 (B). A steel shaft, 90 mm diameter, is solid for a certain distance from one end but hollow for the remainder of its length with an inside diameter of 38 mm. If a pure torque is transmitted from one end of the shaft to the other of such a magnitude that yielding just occurs at the surface of the solid part of the shaft, find the depth of yielding in the hollow part of the shaft and the ratio of the angles of twist per unit length for the two parts of the shaft. [U.L.] [1.5 mm; 1.0345:1.]

339 (B). A steel shaft of solid circular cross-section is subjected to a gradually increasing torque. The diameter of the shaft is 76 mm and it is 1.22 m long. Determine for initial yield conditions in the outside surface of the shaft (a) the angle of twist of one end relative to the other, (b) the applied torque, and (c) the total resilience stored.

Assume a yield in shear of 155 MN/m2 and a shear modulus of 85 GN/m2. If the torque is increased to a value 10% greater than that at initial yield, estimate (d) the depth of penetration of plastic yielding and (e) the new angle of twist. [B.P.] [3.35°; 13.4 kN m; 391 J; 4.3 mm; 3.8°.]

3.40 (B). A solid steel shaft, 50 mm diameter and 1.22 m long, is transmitting power at 10 rev/s.

(a) Determine the power to be transmitted at this speed to cause yielding of the outer fibres of the shaft if the yield stress in shear is 170 MN/m2.

(b) Determine the increase in power required to cause plastic penetration to a radial depth of 6.5 mm, the speed of rotation remaining at 10 rev/s. What would be the angle of twist of the shaft in this case? g for the steel is 82 GN/m2. [BP.] [262 kW. 52 kW, 7.83°.]

3.41 (B). A marine propulsion shaft of length 6 m and external diameter 300 mm is initially constructed from solid steel bar with a shear stress at yield of 150 MN/m2.

In order to increase its power/weight ratio the shaft is machined to convert it into a hollow shaft with internal diameter 260 mm, the outer diameter remaining unchanged.

Compare the torques which may be transmitted by the shaft in both its initial and machined states:

(a) when yielding first occurs,

(b) when the complete cross-section has yielded.

If, in service, the hollow shaft is subjected to an unexpected overload during which condition (b) is achieved, what will be the distribution of the residual stresses remaining in the shaft after torque has been removed?

[795 kN m, 346 kN m, 1060 kN m, 370 kN m; -10.2, +11.2 MN/m2.]

3.42 (C). A solid circular shaft 100 mm diameter is in an elastic-plastic condition under the action of a pure torque of 24 kN m. If the shaft is of steel with a yield stress in shear of 120 MN/m2 determine the depth of the plastic zone in the shaft and the angle of twist over a 3 m length. Sketch the residual shear stress distribution on unloading. G = 85 GN/m2. [0.95 mm; 4.95°.]

3.43 (C). A column is constructed from elastic - perfectly plastic material and has a cross-section 60 mm square. It is subjected to a compressive load of 0.8 MN parallel to the central longitudinal axis of the beam but eccentric from it. Determine the value of the eccentricity which will produce a fully plastic section if the yield stress of the column material is 280 MN/m2.

What will be the values of the residual stresses at the outer surfaces of the column after unloading from this condition? [7 mm; 213,-97 MN/m2.]

3.44 (C). A beam of rectangular cross-section with depth d is constructed from a material having a stress-strain diagram consisting of two linear portions producing moduli of elasticity e\ in tension and e2 in compression.

Assuming that the beam is subjected to a positive bending moment M and that cross-sections remain plane, show that the strain on the outer surfaces of the beam can be written in the form

where R is the radius of curvature.

Hence derive an expression for the bending moment M in terms of the elastic moduli, the second moment of area I of the beam section and R the radius of curvature.

3A5 (C). Explain what is meant by the term "autofrettage" as applied to thick cylinder design. What benefits are obtained from autofrettage and what precautions should be taken in its application?

(b) A thick cylinder, inside radius 62.5 mm and outside radius 190 mm, forms the pressure vessel of an isostatic compacting press used in the manufacture of sparking plug components. Determine, using the Tresca theory of elastic failure, the safety factor on initial yield of the cylinder when an internal working pressure Pw of 240 MN/m2 is applied.

(c) In view of the relatively low value of safety factor which is achieved at this working pressure, the cylinder is now subjected to an autofrettage pressure of Pa = 580 MN/m2.

Determine the residual stresses produced at the bore of the cylinder when the autofrettage pressure is removed and hence determine the new value of the safety factor at the bore when the working pressure P„ is applied.

The yield stress of the cylinder material o\, = 850 MN/m2 and axial stresses may be ignored.

3.46 (C). A thick cylinder of outer radius 190 mm and radius ratio K = 3.04 is constructed from material with a yield stress of 850 MN/m2 and tensile strength 1 GN/m2. In order to prepare it for operation at a working pressure of 248 MN/m2 it is subjected to an initial autofrettage pressure of 584 MN/m2.

Ignoring axial stresses, compare the safety factors against initial yielding of the bore of the cylinder obtained with and without the autofrettage process. [1.53, 8.95.]

3.47 (C). What is the maximum autofrettage pressure which should be applied to a thick cylinder of the dimensions given in problem 3.46 in order to achieve yielding to the geometric mean radius?

Determine the maximum hoop and radial residual stresses produced by the application and release of this pressure and plot the distributions of hoop and radial residual stress across the cylinder wall.

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