Page 269 - Mechanical Behavior of Materials

Problems and Questions 271

6.25 A solid shaft of diameter 50 mm is subjected to an axial load P = 200 kN and a torque
T = 1.5kN·m. Determine the maximum normal stress and the maximum shear stress.
6.26 A solid shaft of diameter d is subjected to an axial load P and a torque T.
(a) Derive an expression for the maximum shear stress as a function of d, P, and T.
(b) If P = 200 kN and T = 1.5kN·m, what is the smallest diameter such that the
maximum shear stress does not exceed 100 MPa?
6.27 A simply supported beam 0.50 m long has a rectangular cross section of depth 2c = 60 mm
and thickness t = 40 mm. A vertical force P = 40 kN is applied at midspan, as in Fig. A.4(a),
and also an axial force F = 100 kN is applied along its length. Determine the maximum
normal stress and the maximum shear stress.
6.28 Consider an internally pressurized thick-walled spherical vessel, as in Fig. A.6(b).
(a) Develop an equation for the maximum shear stress at any radial position in the vessel,
expressing this as a function of the radii r 1 , r 2 , and R, and the pressure p.Alsoshow
that the overall maximum shear stress in the vessel occurs at the inner wall.
(b) Assume that the vessel has an inner diameter of 100 mm and an outer diameter of
150 mm and contains an internal pressure of 300 MPa. Then determine the principal
normal and shear stresses at the inner wall.
(c) For the same case as in (b), plot the variations of σ r , σ t , and τ max versus R.
6.29 Consider an internally pressurized thick-walled tube, as in Fig. A.6(a). Assume that the tube
has closed ends, but neglect the localized effects of the end closure.
(a) Develop an equation for the maximum shear stress at any radial position in the vessel,
expressing this as a function of the radii r 1 , r 2 , and R, and the pressure p.Alsoshow
that the overall maximum shear stress in the vessel occurs at the inner wall.
(b) Assume that the vessel has an inner diameter of 80 mm, an outer diameter of 100 mm,
and contains an internal pressure of 100 MPa. Then determine the principal normal and
shear stresses at the inner wall.
(c) For the same case as in (b), plot the variations of σ r , σ t , σ x , and τ max versus R.
6.30 A thick-walled tube has closed ends and is loaded with an internal pressure of 75 MPa and a
torque of 30 kN·m. The inner and outer diameters are 80 and 120 mm, respectively.
(a) Determine the maximum shear stress in the tube. Neglect the localized effects of the
end closure. (Suggestion: Using Fig. A.6(a), calculate τ max at the inner and outer walls
and at several intermediate radial positions.)
(b) Plot the variations of σ r , σ t , and σ x due to the pressure, τ tx due to the torsion, and τ max ,
all versus R.
6.31 Proceed as in Prob. 6.30(a) and (b), except change the numerical values as follows:
internal pressure 90 MPa, torque 15 kN·m, and inner and outer diameters 48 and 78 mm,
respectively.
6.32 A rotating annular disc as in Fig. A.9 has inner radius r 1 = 90, outer radius r 2 = 300, and
thickness t = 50 mm. It is made of an alloy steel and rotates at a frequency of f = 120
revolutions/second.
(a) Calculate values of the radial and tangential stresses, σ r and σ t , for a number of values
of the variable radius R, and then plot these stresses as a function of R.
(b) Determine the values and locations of the maximum normal stress and the maximum
shear stress in the disc.
(Problem continues)