Set 6

NRI INSTITUTE OF INFORMATION SCIENCE & TECHNOLOGY BHOPAL

COMPILED BY AMIT SINGH 9827740442 Page 1

PROBLEM 1:- A steel rod 20mm diameter passes centrally through a steel tube

25mm internal diameter and 40mm external diameter. The tube is 750mm long

and is closed by rigid washer of negligible thickness which is fastened by nuts

threaded on the rod. The nuts are tightened until the compressive load on the

tube is 20kN. Calculate the stresses in the tube and the rod.

Find the increase in these stresses when one nut is tightened by one quarter of a

turn relative to the other. There are 0.4 threads per mm length. Take

E=200GN/m2.

(Answer: -

𝐶𝑎𝑠𝑒 1 − : 𝜎𝑡 = 26.1𝑀𝑃𝑎 𝑐 ,𝜎𝑟 = 63.6𝑀𝑃𝑎 (𝑡)

𝐶𝑎𝑠𝑒 2 − : 𝜎𝑡 = 48.47𝑀𝑃𝑎 𝑐 , 𝜎𝑟 = 118.19𝑀𝑃𝑎 (𝑡))

PROBLEM 2:- A steel rod 20mm in diameter passes centrally through a steel tube

of 25mm internal diameter and 30mm external diameter. The tube is 800mm long

and is closed by rigid washers of negligible thickness which are fastened by nuts

threaded on the rod. The nuts are tightened until the compressive load on the

tube is 20kN. Calculate the stresses in the tube and the rod.

Find the increase in these stresses when one nut is tightened by one-quarter of a

turn relative to the other. There are four threads per 10mm. Take 𝐸 = 2 ×

105𝑁/𝑚𝑚2.



(Answer: -

Case 1: When the compressive load of 20kN

𝜎𝑡 = 92.599 𝑁/𝑚𝑚2 (𝑐) ,

𝜎𝑟 = 63.66 𝑁/𝑚𝑚2 (𝑡)

Case 2: When one nut is tightened by one quarter of a turn

𝜎𝑡 = 92.59 𝑁/𝑚𝑚2 ,

𝜎𝑟 = 63.65 𝑁/𝑚𝑚2 )

PROBLEM 3:- A copper sleeve, 21mm internal and 27mm external diameter,

surrounds a 20mm steel bolt, one end of the sleeve being in contact with the

shoulder of the bolt. The sleeve is 60mm long. After putting a rigid washer on the

other end of the sleeve, a nut is screwed on the bolt through 10 degrees. If the

pitch of the threads is 2.5mm, find the stresses induced in the copper sleeve and

steel bolt. ES=200GN/m2, Ec=90GN/m2

(Answer: -𝜎𝑠 = 56.6 𝑀𝑁/𝑚2 (𝑡) , 𝜎𝑐 = 78.75 𝑀𝑁/𝑚2 (𝑐))



PROBLEM 4:- A steel bolt of 500mm length and 18mm diameter passes coaxially

through a steel tube of same length and 20mm internal diameter and 30mm

external diameter. The assembly is rigidly fixed at its both ends by washers. If one

of the nuts is tightened through 450, find the stresses developed in the steel rod

and steel tube. Take pitch of the threads as 2.4mm and E for the steel as 200GPa.

(Answer: - 𝜎𝑏 = 72.8𝑀𝑃𝑎 (𝑡),𝜎𝑡 = 47.2𝑀𝑃𝑎 (𝑐))

PROBLEM 5:- A steel rod 20mm diameter passes centrally through a copper tube

of 25mm internal diameter and 35mm external diameter. Copper tube is 800mm

long and is closed by rigid washer of negligible thickness, which is fastened by nut

threaded on the rod.

The nuts are tightened till the load on the assembly is 20kN. Calculate the initial

stresses in the copper tube and steel rod. Also calculate increase in the stresses,

when one nut is tightened by one-quarter of a turn relative to the other. Take

pitch of the thread as 1.6mm. Take E for steel and copper as 200GPa and 100GPa

respectively.

(Answer: - 𝜎𝑐 = 21.2𝑀𝑃𝑎, 𝜎𝑠 = 31.8𝑀𝑃𝑎, 𝜎𝑐1 = 28.6𝑀𝑃𝑎 𝑐 , 𝜎𝑠1 =

42.9𝑀𝑃𝑎 (𝑡)

PROBLEM 6:- A solid copper rod 300mm long and 40mm diameter passes axially

inside a steel tube of 50mm internal diameter and 60mm external diameter. The

composite bar is tightened by using rigid washers of negligible thickness.

Determine the stresses in copper rod and steel tube, when the nut is tightened so

as to produce a tensile load of 100kN in the copper rod.



(Answer: - 𝜎𝑐 = 79.5𝑀𝑃𝑎 (𝑡), 𝜎𝑠 = 115.75𝑀𝑃𝑎 (𝑐))

PROBLEM 7:- A steel bolt and sleeve assembly is shown. The nut is tightened up

on the tube through the rigid end blocks until the tensile force in the bolt is 40kN.

If an external load 30kN is then applied on the end blocks, tending to pull them

apart, estimate the resulting force in the bolt and sleeve.

(Answer: -

Stress due to tightening the nut 𝜎𝑠𝑏 = 81.5𝑀𝑃𝑎(𝑡), 𝜎𝑠𝑠 = 36.23𝑀𝑃𝑎(𝑐),

Stresses due to tensile force, 𝜎𝑠𝑏 = 16𝑀𝑃𝑎(𝑡), 𝜎𝑠𝑠 = 20𝑀𝑃𝑎(𝑡),

𝑅𝑒𝑠𝑢𝑙𝑡𝑎𝑛𝑡 𝑠𝑡𝑟𝑒𝑠𝑠, 𝑅𝑠𝑏 = 0.0478𝑀𝑃𝑎(𝑡), 𝜎𝑅𝑠𝑠 = 0.0179𝑀𝑃𝑎(𝑐))



PROBLEM 8:- A steel rod of 10mm diameter passes centrally through a copper

tube of external diameter 40mm and internal diameter 30mm and length 2m. The

tube is closed at each end by 20mm thick steel plates which are screwed by the

nuts. The nuts are tightened until the copper tube is reduced to length 1.9996m.

Find the stresses in the rod and the tube.

If the whole assembly is heated through 600C, what are then the stresses in the

rod and the tube? Assuming that the thickness of the plates remains unchanged?

Es=210GN/m2 𝛼𝑠 = 12 × 10−6per0C

Ec=100GN/m2 𝛼𝑐 = 17.5 × 10−6per0C

(Answer: -

𝐶𝑎𝑠𝑒 1 − : 𝜎𝑐 = 20𝑀𝑃𝑎 𝑐 ,𝜎𝑠 = 140𝑀𝑃𝑎 (𝑡)

𝐶𝑎𝑠𝑒 2 − : 𝜎𝑐 = 27.17𝑀𝑃𝑎 𝑐 , 𝜎𝑠 = 190𝑀𝑃𝑎 (𝑡))

PROBLEM 9:- A steel rod of 16mm diameter and 3m length passes through a

copper tube of 50mm external and 40mm internal diameter and of the same

length. The tube is closed at each end with the help of 30mm thick steel plates

which are tightened by nuts till the length of the copper tube is reduced by

0.6mm. The temperature of the whole assembly is raised by 560C. Determine the

stresses in the steel and copper before and after the rise of temperature. Assume



that the thickness of the steel plates at the ends do not change during tightening

of the nuts.

𝐸𝑠 = 210𝐺𝑃𝑎 𝛼𝑠 = 12 × 10−6/0C 𝐸𝑐 = 100𝐺𝑃𝑎 𝛼𝑐 = 17 × 10−6/0C

(Answer: - Stress in copper=29.87MPa (c), Stress in Steel=104.9MPa (t))

PROBLEM 10:- A steel rod, one of 75mm diameter and other of 55mm diameter,

are joined end to end by means of a turn buckle. The other end of each rod is

rigidly fixed and there is initially a small tension in the rods. If the effective length

of each rod is 4.5m, find the increase in this tension when the turn buckle is

turned by one-quarter of a turn. On one end of the bigger diameter rod there are

0.15 threads per mm length, while there are 0.2 threads per mm length on the

other rod.

Neglect the extension of the turn buckle. Find also what rise in temperature

would nullify the increase in tension.

𝐸𝑠 = 200𝐺𝑃𝑎, 𝛼𝑠 = 12 × 10−6/0C



(Answer: - 𝑃 = 200241𝑁 (t), 540C)

PROBLEM 11: - A steel rod 40mm diameter is fixed concentrically in a brass tube

having outside and inside diameters of 60mm and 50mm respectively. Both the

rod and the tube are 0.5m long and their ends are level. The compound rod is

held between two stops which are exactly 0.5m apart and the temperature of the

bar is then raised by 500C.

(i) Find the stresses in the rod and the tube if the distance between the

stops

(a) Remains constant

(b) Is increased by 0.2mm

(ii) Find the increase in the distance between the stops if the forces exerted

between them is 60kN.

𝐸𝑠 = 200𝐺𝑃𝑎 𝛼𝑠 = 1.2 × 10−5/0C

𝐸𝑏 = 90𝐺𝑃𝑎 𝛼𝑏 = 2.1 × 10−5/0C

(Answer: - σs = 120MN/m2, σb = 94.5MN/m2, σs =40MN/m2, σb =58.5MN/m2(c),

0.262mm)

PROBLEM 12: - A steel tube 2.4cm external diameter and 1.8cm internal diameter

encloses a copper rod 1.5cm diameter to which it is rigidly connected at the two

ends. If at a temperature of 100C there is no longitudinal stress, calculate the

stresses in the rod and the tube when the temperature is raised to 2000C.



𝐸𝑠 = 2.1 × 105𝑁/𝑚𝑚2 𝛼𝑠 = 11 × 10−6 𝑝𝑒𝑟0𝐶 𝐸𝑐 = 1.0 × 105𝑁/𝑚𝑚2 𝛼𝑐 = 18 × 10−6 𝑝𝑒𝑟0𝐶

(Answer: - σs = 83.13MN/m2, σc = 93.41MN/m2)

PROBLEM 13: - A copper bar 50mm in diameter is placed within a steel tube

75mm external diameter and 50mm internal diameter of exactly the same length.

The two pieces are rigidly fixed together by two pins 18mm in diameter, one at

each end passing through the bar and tube. Calculate the stresses induced in the

copper bar, steel tube and pins if the temperature of the combination is raised by

500C.

𝐸𝑠 = 210𝐺𝑃𝑎 𝛼𝑠 = 11.5 × 10−6 𝑝𝑒𝑟0𝐶 𝐸𝑐 = 105𝐺𝑃𝑎 𝛼𝑐 = 17 × 10−6 𝑝𝑒𝑟0𝐶

(Answer: - σs = 16.5MN/m2, σc = 20.63MN/m2, τpin = 79.57MN/m2)

PROBLEM 14: - A brass rod of 6mm diameter and 1m long is joined at one end to

a rod of steel 6mm diameter and 1.3m long. The compound rod is placed in a

vertical position with the steel rod at the top and connected top and bottom are

connected to rigid fixing in such a way that it is carrying a tensile load of 3.5kN.

An attachment is fixed at the junction of the two rods and to this a vertical load of

1.3kN is applied downwards. Calculate the stresses in steel and brass. The

temperature is then raised through 300C. What are the final stresses in steel and

brass?



𝐸𝑠 = 200𝐺𝑃𝑎 𝛼𝑠 = 12 × 10−6 𝑝𝑒𝑟0𝐶 𝐸𝑏 = 85𝐺𝑃𝑎 𝛼𝑏 = 19 × 10−6 𝑝𝑒𝑟0𝐶

(Answer: - σs = 96.39MN/m2 (t), σb = 50.41MN/m2 (t))

PROBLEM 15: - A steel rod of 30mm diameter is enclosed in a brass tube of 42mm

external diameter and 32mm internal diameter. Each is 360mm long and the

assembly is rigidly held between two stops 360mm apart. The temperature of the

assembly is then raised by 500C. Determine

(i) Stress in the tube and the rod

(ii) Stresses in the tube and the rod if the stops yields by 0.15mm

(iii) Yield of the stops if the force at the stops is limited to 60kN.

𝐸𝑠 = 205𝐺𝑃𝑎 𝛼𝑠 = 11 × 10−6 𝑝𝑒𝑟0𝐶 𝐸𝑏 = 90𝐺𝑃𝑎 𝛼𝑏 = 19 × 10−6 𝑝𝑒𝑟0𝐶



(Answer: - σb = 85.5MPa (c), σs = 112.75MPa(c), σb = 48MPa (c), σs = 27.33MPa(c),

0.127mm)

PROBLEM 16: - Figure shows a frame work, the rigid horizontal bar ABC is

supported by a steel rod and a copper bar. The mechanism is stress free at 200C.

Find the stresses in the bars when the temperature rises to 500C. Take area of

steel and copper as 800mm2 and 600mm2 respectively.

𝐸𝑠 = 200𝐺𝑃𝑎 𝛼𝑠 = 12 × 10−6 𝑝𝑒𝑟0𝐶 𝐸𝑐 = 120𝐺𝑃𝑎 𝛼𝑐 = 18 × 10−6 𝑝𝑒𝑟0𝐶



(Answer: -𝜎𝑠 = 35.48𝑀𝑃𝑎, 𝜎𝑐 = 135𝑀𝑃𝑎)

PROBLEM 17: - Three bars, of aluminum, steel, and copper, support a rigid bar

which is centrally placed. All the bars have the same area. The aluminum bar is

1m long. Find the length of the steel and the copper bar and the stresses in the

bars if the rigid bar remains horizontal when the temperature rises by 400C.

𝐸𝑎𝑙 = 70𝐺𝑁/𝑚2 𝛼𝑎𝑙 = 23 × 10−6 𝑝𝑒𝑟0𝐶 𝐸𝑠 = 200𝐺𝑁/𝑚2 𝛼𝑠 = 12 × 10−6 𝑝𝑒𝑟0𝐶 𝐸𝑐 = 120𝐺𝑁/𝑚2 𝛼𝑐 = 16 × 10−6 𝑝𝑒𝑟0𝐶



(Answer: - Ls = 1.917m, Lc = 1.438m, σs= 96MPa, σc = 76.8MPa)

PROBLEM 18: - A bar of length L and cross-sectional area A is fixed to a rigid

support at one end and to a spring of stiffness k at the other end as shown in the

figure. Derive a formula for stress in the bar caused by an increase in temperature

∆𝑡 if the coefficient of thermal expansion is α for the bar.

(Answer: -𝜎 =𝐸𝛼∆𝑡

1+𝐸𝐴

𝑘𝐿

)



PROBLEM 19: - A steel bar 1.5m long and 12mm diameter is supported by a rigid

support at one end and a spring of stiffness 10kN/mm at the other. If the

temperature of the bar is raised by 200C, find the stress in the bar.

𝛼 = 11.6 × 10−6 𝑝𝑒𝑟0𝐶 𝐸 = 200𝐺𝑃𝑎

(Answer: - 𝜎 = 18.4𝑁/𝑚𝑚2)

PROBLEM 20: - A copper rod and a steel rod are joined together. There is a gap of

0.1mm between the rigid support and the end of the bar at 270C. Determine the

stresses in the bars when the temperature becomes 500C.

𝐸𝑠 = 200𝐺𝑁/𝑚2 𝛼𝑠 = 12 × 10−6 𝑝𝑒𝑟0𝐶 𝐸𝑐 = 120𝐺𝑁/𝑚2 𝛼𝑐 = 16 × 10−6 𝑝𝑒𝑟0𝐶

(Answer: - −0.062𝑚𝑚)

PROBLEM 21: -A bar, 20mm diameter and fixed at A, is stretched with a force of

10kN to bring to support B and fix it. Length AB = 2m. Temperature is 270C.



Determine at what temperature will the stress become zero? What will be the

stress in the bar if the temperature rises to (i) 400C (ii) 500C.?

𝐸𝑠 = 200𝐺𝑁/𝑚2 𝛼𝑠 = 12 × 10−6 𝑝𝑒𝑟0𝐶

(Answer: - −23.4𝑁/𝑚𝑚2(𝑐))

PROBLEM 22: - The steel rod shown is in two parts. It has a diameter of 10mm for

a length of 1m and of 20mm for the remaining length of 1.5m. If it is constrained

between two supports A and B and is stress free at 200C, find the stress in the

material when it is heated to 700C.

If the part CB was of brass, what would be the stress in the two materials?

𝐸𝑠 = 200𝐺𝑁/𝑚2 𝛼𝑠 = 12 × 10−6 𝑝𝑒𝑟0𝐶 𝐸𝑏 = 105𝐺𝑁/𝑚2 𝛼𝑏 = 19 × 10−6 𝑝𝑒𝑟0𝐶

(Answer: - 𝜎𝐴𝐶 = 218.18𝑁/𝑚𝑚2, 𝜎𝐵𝐶 = 54.54𝑁/𝑚𝑚2)



PROBLEM 23: - A steel bolt of cross-sectional area 600mm2 is fitted snugly in a

900mm2 aluminum collar. The pitch of the bolt is 2mm. Determine

(i) The stress in steel bolt and aluminum collar if the bolt is given 1/8th turn.

Take moduli of steel and aluminum as 210GPa and 70GPa.

(ii) If the temperature also increased by 500C. Take the coefficient of

thermal expansion for steel and aluminum to be 𝛼𝑠 = 18 ×

10−6𝑚𝑚/𝑚𝑚/𝑜𝐶 and 𝛼𝑎 = 23 × 10−6𝑚𝑚/𝑚𝑚/𝑜𝐶 respectively.

(iii) If an external load of 60kN is applied on the same bolt and aluminum

collar assembly. Calculate the stresses in steel bolt and aluminum collar.

Answer: -

CASE 1 𝜎𝑠 = 175𝑀𝑃𝑎 𝑡 𝜎𝑎 = 116.7 𝑀𝑃𝑎 𝑐

CASE 2 𝜎𝑠 = 192.5𝑀𝑃𝑎 𝑡 𝜎𝑎 = 128.34 𝑀𝑃𝑎 𝑐

CASE 3 𝜎𝑠 = 125.83𝑀𝑃𝑎 𝑡 𝜎𝑎 = 150.56 𝑀𝑃𝑎 𝑐

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