6.16 Exercises
219
10. A curved bar bent into a arc of a circle having internal radius “a” and external
radius “b” is subjected to a bending couple M at its end. Determine the stresses
σ r , σ θ and τ r θ .
11. For the stress function, ϕ = Ar
2 log r , where A is a constant, compute the stress
components σ r , σ θ and τ r θ .
12. A thick cylinder of inner radius 150 mm and outer radius 200 mm is subjected
to an internal pressure of 15 MN/m
2 . Determine the radial and hoop stresses in
the cylinder at inner and outer surfaces.
13. The internal and external diameters of a thick hollow cylinder are 80 mm and
120 mm respectively. It is subjected to an external pressure of 40 MN/m
2 , when
the internal pressure is 120 MN/m
2 . Calculate the circumferential stresses at
the external and internal surfaces and determine the radial and circumferential
stresses at the mean radius.
14. A thick-wall cylinder is made of steel (E = 200 GPa and ν = 0.29), has an
inside diameter of 20 mm, and an outside diameter of 100 mm. The cylinder is
subjected to an internal pressure of 300 MPa. Determine the stress components
σ r and σ θ at r = a = 10 mm, r = 25 mm and r = b = 50 mm.
15. A long closed cylinder has an internal radius of 100 mm and an external radius
of 250 mm. It is subjected to an internal pressure of 80 MPa. Determine the
maximum radial, circumferential and axial stresses in the cylinder.
16. A solid disc of radius 200 mm is rotating at a speed of 3000 rpm. Determine
the radial and hoop stresses in the disc if ν = 0.3 and ρ = 8000 kg/m
3 . Also
determine the stresses in the disc if a hole of 30 mm is bored at the centre of
the disc.
17. A disc of 250 mm diameter has a central hole of 50 mm diameter and runs at
4000 rpm. Calculate the hoop stresses. Take ν = 0.25 and ρ = 7800 kg/m
3 .
18. A turbine rotor 400 mm external diameter and 200 mm internal diameter
revolves at 1000 rpm. Find the maximum hoop and radial stresses assuming the
rotor to be thin disc. Take the weight of the rotor as 7700 kg/ m
3 and Poisson’s
ratio as 0.3.
19. Check Whether the Following is a Stress Function.
φ =
Ar
2
+ Br
2
+
C
r 2 + D
cos 2θ
20. Show that φ =
Ae
α y
+ Be
−α y
+ Cye
α y
+ Dye
−α y
sin α x represents
stress function.
21. The curved beam shown in Fig. 6.23 has a circular cross-section 50 mm in
diameter. The inside diameter of the curved beam is 40 mm. Determine the
stress at B when P = 20 kN.
22. A crane hook carries a load W = 20 kN as shown in Fig. 6.24. The cross-section
mn of the hook is trapezoidal as shown in the figure. Find the total stresses at
points m and n. Use the data as given b 1 = 40 mm, b 2 = 10 mm, a =
30 mm and c = 120 mm
219
10. A curved bar bent into a arc of a circle having internal radius “a” and external
radius “b” is subjected to a bending couple M at its end. Determine the stresses
σ r , σ θ and τ r θ .
11. For the stress function, ϕ = Ar
2 log r , where A is a constant, compute the stress
components σ r , σ θ and τ r θ .
12. A thick cylinder of inner radius 150 mm and outer radius 200 mm is subjected
to an internal pressure of 15 MN/m
2 . Determine the radial and hoop stresses in
the cylinder at inner and outer surfaces.
13. The internal and external diameters of a thick hollow cylinder are 80 mm and
120 mm respectively. It is subjected to an external pressure of 40 MN/m
2 , when
the internal pressure is 120 MN/m
2 . Calculate the circumferential stresses at
the external and internal surfaces and determine the radial and circumferential
stresses at the mean radius.
14. A thick-wall cylinder is made of steel (E = 200 GPa and ν = 0.29), has an
inside diameter of 20 mm, and an outside diameter of 100 mm. The cylinder is
subjected to an internal pressure of 300 MPa. Determine the stress components
σ r and σ θ at r = a = 10 mm, r = 25 mm and r = b = 50 mm.
15. A long closed cylinder has an internal radius of 100 mm and an external radius
of 250 mm. It is subjected to an internal pressure of 80 MPa. Determine the
maximum radial, circumferential and axial stresses in the cylinder.
16. A solid disc of radius 200 mm is rotating at a speed of 3000 rpm. Determine
the radial and hoop stresses in the disc if ν = 0.3 and ρ = 8000 kg/m
3 . Also
determine the stresses in the disc if a hole of 30 mm is bored at the centre of
the disc.
17. A disc of 250 mm diameter has a central hole of 50 mm diameter and runs at
4000 rpm. Calculate the hoop stresses. Take ν = 0.25 and ρ = 7800 kg/m
3 .
18. A turbine rotor 400 mm external diameter and 200 mm internal diameter
revolves at 1000 rpm. Find the maximum hoop and radial stresses assuming the
rotor to be thin disc. Take the weight of the rotor as 7700 kg/ m
3 and Poisson’s
ratio as 0.3.
19. Check Whether the Following is a Stress Function.
φ =
Ar
2
+ Br
2
+
C
r 2 + D
cos 2θ
20. Show that φ =
Ae
α y
+ Be
−α y
+ Cye
α y
+ Dye
−α y
sin α x represents
stress function.
21. The curved beam shown in Fig. 6.23 has a circular cross-section 50 mm in
diameter. The inside diameter of the curved beam is 40 mm. Determine the
stress at B when P = 20 kN.
22. A crane hook carries a load W = 20 kN as shown in Fig. 6.24. The cross-section
mn of the hook is trapezoidal as shown in the figure. Find the total stresses at
points m and n. Use the data as given b 1 = 40 mm, b 2 = 10 mm, a =
30 mm and c = 120 mm