II B. Tech. (Mett. Engg.)

Strength of Materials

Assignment – 1

1. Find the minimum diameter of a steel wire with which a load of 1000 * (No.) N can

be raised so that the stress in the wire may not exceed 130 N/mm2. For the diameter

chosen, find the extension of the wire, if it is 4 m long. Take E = 2 x 105 N/mm

2.

2. A rod circular in section tapers from 20 mm diameter at one end to 10 mm diameter at

the other end and is 200 mm long. On applying an axial pull of 1000 * (No.) N it was

found to extend by 0.068 mm. Find the Young’s modulus of the material of the rod.

3. Prove that the total extension of a uniformly tapering rectangular bar of length L,

widths a and b and thickness t, when the bar is subjected to an axial load P is given by

b

alog

)ba(Et

PLdL e

where E = Young’s Modulus.

4. The following data given in Table -3, were obtained during a tension test of an

aluminum alloy. The initial diameter of the test

specimen was 12.8 mm. and the gauge length was 50

mm. Plot the stress-strain diagram and determine the

following mechanical properties: (a) proportional limit;

(b) modulus of elasticity; (c) yield point; (d) yield

strength at 0.2 % offset; (e) ultimate strength and (f)

rupture strength.

5. A tie bar has enlarged ends of square section 60 cm x

60 cm. If the middle portion of the bar is also of square

section. Find the size and length of the middle portion,

if the stress there is 140 N/ mm2 and the total extension

of the bar is 0.14 mm. Take E = 2 x 105 N/mm

2.

6. A metallic bar 250 mm x 80 mm x 30 mm is subjected to a force of 20 kN (tensile),

30 kN (tensile) and (No.) kN ( compression) along x, y, and z directions respectively.

Determine the change in the volume of the block. Take E = 2 x 105 N/mm

2 and

Poisson’s ratio = 0.25.

7. Compare the strain energy absorbed by the bars A and B under an axial force P. Bar A

is a tapered circular bar of length l, diameter d at one end is uniformly increasing to

diameter 2dat the other end. Bar B is a stepped bar of length l, with diameter d upto

half of its length and 2d for the remaining half.

8. An unknown weight falls by (No.) mm on to a collar rigidly attached to the lower end

of a vertical bar 3.25 m long and 600 mm2 in section. If the maximum instantaneous

extension is known to be 2.5 mm, find the corresponding stress and the magnitude of

the falling weight. Take E = 2 x 105 N/mm

2.

Table -3 Test data

Load

(kN)

Elongation

(mm)

0.00 0.00

10.50 0.06

21.09 0.11

31.59 0.17

42.23 0.22

52.73 0.28

57.27 0.38

63.64 0.51

65.45 0.64

65.91 1.52

66.36 2.03

67.27 2.54

66.36 3.05

61.82 Fracture

87.5 kN

300 mm

Fig. 4

87.5 kN

60 mmx 60 mm 60 mmx 60 mm

9. A small light piston 125 mm2 in area compresses oil in a

rigid container of 15 litre capacity. When a weight of 45

N is gradually applied to the piston its movement is

observed to be 15 mm. Find the bulk modulus of the oil.

If a weight of 18 N falls from a height of 72 mm on to the

45 N load as shown in Fig. 9, determine the maximum

pressure developed in the oil container, neglecting the

effect of friction and the loss of energy at the impact.

10. A cantilever PQRS, 7 m long is fixed at P such that PQ=QR=2m and RS=3m. It

carries loads of 5, 3, (No.) kN at Q, R, S respectively in addition to UDL of 1 kN/m

run between P and Q and 2 kN/m run between R and S. Draw the shear force and

bending moment diagram.

11. For the given beam conditions, draw the shear force and bending moment diagram.

12. Draw the shear force and bending moment

diagrams for abeam shown in Fig. 12

13. Shear force diagram for the loaded beam is

shown in Fig. 13. Determine the loading

and the nature of the beam and sketch it

neatly. Hence, determine B.M. diagram

indicating important ordinates and the

points of contraflexure, if any.

14. Shear force diagram for the loaded

beam is shown in Fig. 14.

Determine the loading and the

nature of the beam and sketch it

neatly. Hence, determine B.M.

diagram indicating important

ordinates and the points of

contraflexure, if any.

Fig. 9

a=125 mm

δ1

W1=45N

V=15 Litres

B A

1 m 3 m 1 m Fig. 11

(No.) kN/m 80 kN/m

B A

4 m 2 m Fig. 12

2 kN/m

2 m

C D

(No.) kN 5 kN

1 m

40kN

36

kN 16

kN

24kN

2 m 1 m 1.6m 2.4 m

Fig. 13

20kN

2 m

10

V kN

2nd

degree curve

2

8

_

+

3 2 1 2

x (m)

Fig. 14

15. The horizontal beam of channel section shown in Fig. 15 is 4 m long and is simply

supported at the ends. Calculate the

maximum uniformly distributed load it

can carry if the tensile and compressive

stresses must not exceed 25 MN/m2 and

45 MN/m2 respectively.

16. A simply supported beam and its cross-

section are shown in Fig. 16. The beam

carries a load W = (No.) kN as shown in

Fig. 16 Its self weight is 7 kN/m.

Calculate the maximum normal stress at

1-1.

17. A steel Stanchion is built of a rolled steel joist of I-section 45 cm x 20 cm united by

1.5 cm thick and 30 cm wide plates fastened on each flange. The length of the

stanchion is 5 m and is freely supported at both ends. For the I-section: Ixx = 35060

cm4. Find (i) Moment of inertia of the enlarged section

(ii) Greatest central point load the beam will carry if the bending stress is not to

exceed 120 MN/m2.

(iii) Minimum length of the 30 cm x 1.5 cm plates. (Fig. 17)

18. A vertical flag staff standing 10 metres above the ground is of square section

throughout, the dimensions being 100 mm x 100 mm at the top tapering uniformly to

200 mm x200 mm at the ground. A horizontal pull of 10* (No.) N is applied at the top,

the direction of the loading being along a diagonal of the section. Find the maximum

stress due to bending and the position from the top.

19. A beam of uniform strength and varying rectangular section is simply supported over a

span of 4 m. It carries a uniformly distributed load of 15 kN/m run. The uniform

strength is 100 MN/m2. Determine: (i) The depth at a distance of 1.5 m from one end if

20 mm

20 mm 120 mm

100 m

m

Fig. 15

20 mm

A

1.2 m 1.2 m

Fig. 16

B

W = (No.) kN

1.2 m

7 kN/m

10cm 15cm

30cm

20cm

30 cm

30 cm

20 cm

45 c

m

48 c

m

N A

I-section

Plate

Plate

Fig. 17

the breadth is same throughout and equal to 200 mm. (ii) The breadth at the centre of

the span if the depth is constant throughout the length of the beam and is equal to 150

mm.

20. A beam section shown in Fig. 20 is subjected

to a shear force of 10 kN. Plot a graph

showing the variation of shear stress along the

depth of the section. Determine also the ratio

of maximum shear stress and mean shear

stress.

21. The section of a beam is rectangular 40 mm

wide x 80 mm deep with two circular holes as shown in Fig. 21. The vertical shear

force on the section is 5 kN. Determine the magnitude of distance d if the shear stress

along the plane AA is not to exceed 0.62 N/mm2.

d

d

80

mm

R=10 mm

40 mm

Fig. 21

A A

30

mm

20 mm 20 mm

60 m

m

Fig. 20

20 mm

II B. Tech. (Mett. Engg.)

Strength of Materials

Assignment – 2

1. The diameter of a cylindrical shell made of steel is 3 m. The sheet is subjected to

an internal pressure of 1 MN/m2 gauge. Find out the thickness of the shell plate if

ultimate tensile stress of the mild steel is 480 MN/m2. Longitudinal joint

efficiency of the shell is 80% and factor of safety is 6.

2. The gauge pressure in a boiler of 1.5 m diameter and 12.5 mm thickness is

2 MN/m2, find the longitudinal and circumferential stresses in the boiler plate and

circumferential, longitudinal and volumetric strains. Take E = 200 GN/m2 and

Poisson’s ratio = 0.25

3. A cylindrical shell 3m long and 50 cm in diameter and 1.25 cm thick is at

atmospheric pressure. What would be its dimensions when it is subjected to an

internal pressure of 2 MN/m2? Take E = 200 GN/m

2 and Poisson’s ratio = 0.25.

4. A vertical steam boiler is 2 m internal diameter and 4 m high. It is constructed

with 20 mm thick plates for a working pressure of 1 MN/m2 . If the plates are flat,

Calculate (i) The stress in the circumferential plates due to resisting the bursting

effect. (ii) The stress in the circumferential plate due to the pressure on the end

plates. (iii) The increase in length, diameter and volume. Assume the Poisson’s

ratio as 0.3 and E= Take E = 200 GN/m2

5. A thin spherical shell 1.5 m in diameter, with its wall of 1.25 cm thickness is filled

with the fluid at atmospheric pressure. What intensity of pressure will be

developed in it if 160 cm3 more of fluid is pumped into it? Also calculate the hoop

stress at that pressure and increase in diameter. Take E = 200 GN/m2 and

Poisson’s ratio = 10/3.

6. Find the ratio of thickness to internal diameter for a tube subjected to an internal

pressure when the pressure is 5/8 of the value of the maximum permissible

circumferential stress. Find the increase in internal diameter of such a tube 100

mm internal diameter when the internal pressure is 100 MN/m2. Also find the

change in the wall thickness of the tube. Take E = 200 GN/m2 and Poisson’s

ratio = 0.25.

7. To measure the longitudinal and circumferential strains, strain gauges were fixed

on the outer surface of a closed thick cylinder of diameter ratio 2.5. At an internal

pressure of 230 MN/m2 these strains were recorded as 9.18 x 10-5 and 36.9 x 10-5

respectively. Determine the values of (i) Poisson’s ratio (ii) Young’s Modulus (iii)

Modulus of Rigidity.

8. A compound cylinder is formed by shrinking one cylinder on to another, the final

dimensions being internal diameter 120 mm, external diameter 240 mm and

diameter at junction 200 mm. After shrinking on the radial pressure at the

common surface is 10 MN/m2. (i) Calculate the necessary difference in diameters

of the two cylinders at the common surface. (ii) What is the minimum

temperature through which outer cylinder should be heated before it can be

slipped on? Take E = 200 GN/m2 and α = 0.000011per °C

9. The external diameter of steel collar is 200 mm and the internal diameter

decreases by 0.125 mm when shrunk on to a solid steel shaft of 125 mm diameter.

If E = 205 GN/m2 and Poisson’s ratio is 0.304. Find (i) Radial pressure between

the collar and the shaft (ii) Circumferential stress at the inner surface of the tube

(iii) Reduction in diameter of the shaft

10. A compound cylinder is to be made by shrinking an outer tube of 150 mm external

radius on to an inner tube of 75 mm internal diameter. Determine the common

radius at the junction if the greatest circumferential stress in the inner tube is to be

two-third of the greatest circumferential stress in the outer tube.

Use Double Integration Method for the following problems

11. A steel sleeve is pressed on to a steel shaft of 50 mm diameter. The radial pressure

between the steel shaft and sleeve is 20 MN/m2 and the hoop stress at the inner

radius of the sleeve is 56 MN/m2. An axial compressive force of 50 kN is applied

to the shaft. Determine the change in radial pressure. Take E = 200 GN/m2 and

Poisson’s ratio = 0.3

12. A 250 mm long cantilever of rectangular section 48 mm wide and 36 mm deep

carries a uniformly distributed load. Calculate the value of w if the maximum

deflection in the cantilever is not to exceed 1 mm. Take E = 70 x 109 GN/m2.

13. A 2 m long cantilever of rectangular section 100 mm wide and 200 mm deep

carries a uniformly distributed load of 2 kN per meter run for a length of 1 from

the free end and a point load of 1 kN at the free end. Calculate the deflection at the

free end. Take E = 105 GN/m2.

14. A simply supported 6 m long rolled steel joist carries a uniformly distributed load

of 8 kN/m length. Determine slope and deflection at a distance of 2 m from one

end of the beam.

Use Macaulay’s Method for the following problems

15. A steel girder of uniform section, 14 m long is simply supported at its ends. It

carries point loads of 120 kN and 80 kN at two points 3 m and 4.5 m from the two

ends respectively. (i) Calculate the deflection of the girder at the two points under

the two loads (ii) The maximum deflection. Take I = 16 x 10-4

m4 and E = 210 x

106 kN/m

2.

16. A beam AB of 8 m span is simply supported at the ends. It carries a point load of

10 kN at a distance of 1m from the end A and a uniformly distributed load of 5

kN/m for alength of 2 m from the end B. If I = 16 x 10-4

m4, determine

(i) deflection at the mid span (ii) maximum deflection (iii) slope at the end A.

17. A beam AB of span 6 m and of flexural rigidity EI = 8 x 104 kNm

2 is subjected to

a clockwise couple of 600 kNm at a distance of 4 m from the left end, Find (i) the

deflection at the point of application of the couple (ii) the maximum deflection

18. A beam ABC 13 m long is supported at A and B, such that AB = 10 m and

overhang BC = 3 m. It carries a point load of 4 kN from the end A and a uniformly

distributed load of 0.4 kN/m over the entire overhang. Determine (i) slope at the

end A (ii) deflection at the free end C (iii) Maximum deflection.

Use Moment Area Method for the following problems

19. A beam 3 m long, simply supported at its ends, is carrying a point load (W) at its

centre. If the slope at the ends of the beam is not to exceed 1°, find the deflection

at the centre of the beam.

20. A simply supported beam of circular cross section is 5 m long and is of 150 mm

diameter. What will be the maximum value of the central load if the deflection of

the beam does not exceed 12.45 mm. Also calculate the slope at the supports. Take

E = 2 x108 kN/m

2.

21. A simply supported beam of 4 m span carries a U.D.L. of 20 kN/m on the whole

span and in addition carries a point load of 40 kN at the centre of span. Calculate

the slope at the ends and maximum deflection of the beam. Take E = 200 GN/m2

and I = 5000cm4

22. A beam 4 m long is freely supported at the ends. It carries concentrated loads of

20 kN each at points 1 m from the ends. Calculate the maximum slope and

deflection of the beam and slope and deflection under each load.

EI = 13000 kN/m2.