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Unit V
Q.1 Derive the following relation for torsion:
T/J = τ/R = Gθ/l
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Q.2. A steel shaft 50 mm diameter and 500 mm long is subjected to a twisting moment of 1100
N-m, thetotal angle of twist being 0.6°. Find the maximum shearing stress developed in the
shzaft and modulusof rigidity. [Ans. 44.8 MPa; 85.6 kN/m2]
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Q.3. A shaft is transmitting 100 kW at 180 r.p.m. If the allowable stress in the material is 60
MPa, find the suitable diameter for the shaft. The shaft is not to twist more than 1° in a length
of 13 metres. Take C = 80 GPa. [Ans. 105 mm]
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Q.4. Design a suitable diameter for a circular shaft required to transmit 90 kW at 180 r.p.m.
The shear stress in the shaft is not to exceed 70 MPa and the maximum torque exceeds the
mean by 40%. Also find the angle of twist in a length of 2 metres. Take C = 90 GPa. [Ans. 80
mm; 2.116°]
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Q.5 Design a hollow shaft required to transmit 11.2 MW at a speed of 300 r.p.m. The
maximum shear stress allowed in the shaft is 80 MPa and the ratio of the inner diameter to
outer diameter is 3/4.[Ans. 240 mm; 320 mm]
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Q.6. Compare the weights of equal lengths of hollow shaft and solid shaft to transmit a given
torque for the same maximum shear stress. The material for both the shafts is same and inside
diameter is 2/3 of outside diameter in case of hollow shaft. [Ans. 0.56]
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Q.7. A spindle as shown in Fig. 5.1is a part of an industrial brake and is loaded as shown. Each
load P is equal to 4 kN and is applied at the mid point of its bearing. Find the diameter of the
spindle, if the maximum bending stress is 120 MPa.
[Ans. 22 mm]
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Q.8.A shaft is supported in bearings, the distance between their centres being 1 metre. It
carries a pulley in the centre and it weighs 1 kN. Find the diameter of the shaft, if the
permissible bending stress for the shaft material is 40 MPa.
[Ans. 40 mm]
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Q.9. A hollow shaft with diameter ratio 3/5 is required to transmit 450 kW at 120 rpm with a
uniform twisting moment. The shearing stress in the shaft must not exceed 60 N/mm 2 and the
twist in a length of 2.5m must not exceed 1o. Calculate the minimum external diameter of the
shaft satisfying these conditions. Take the modulus of rigidity C= 8* 104 N/mm2
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Q.10. A stepped solid circular shaft is built in at its ends and subjected to an externally applied
torque T0 at the shoulder as shown in the figure. Determine the angle of rotation ɵ0 of
the shouldersection where T0 is applied ?
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Q.11. A hollow cast iron column whose outside diameter is 200mm has a thickness of 20mm.
It is 4.5m long and is fixed at both the ends. Calculate the safe load by Rankin’s formula using
a factor of safety of 4. Calculate the slenderness ratio and the ration of Euler’s and Rankin’s
critical loads.
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Q12. A composite shaft ABC is fixed at ends A and C as shown in Fig. It is subjected to 500
Nm torque at B. Determine (i) Relative torque at A and B, (ii) Maximum shear stress in each
material, (iii) Angle of twist at B.
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Q.13. A solid shaft of 160mm diameter has the same cross-sectional area as that of hollow
shaft of the same material of inside diameter 110mm.
(i) Find the ratio of power transmitted by the two shafts of same angular velocity,
(ii) Compare angle of twist in equal lengths of these shafts when stressed equally.
Take-
GBrass = 40 GPa , GSteel = 79 GPa
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Q.14 Compare the weights of equal lengths of hollow and solid shaft to resist same torsional
moment for same maximum shear stress. Assume internal diameter 0.8 times the external
diameter for hollow shaft.
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Q.15. Derive Rankin’s formula for buckling of columns with hinged ends. Also state the
limitations of Euler’s formula.
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Q.16. From the first principle, derive equation for Euler load PE for a column with lower end
fixed and upper end hinged. Hence show that effective length of this column is 0.707L where
‘L’ is the length of equivalent pin ended i.e. column carrying same load.
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Q.17. A solid circular bar 25m long and 120mm diameter was found to extend 1.2mm under
tensile load of 52kN. The bar is used as strut. Determine crippling load and also safe load
taking F.O.S. 3 with the following end condition:
(i) Both ends are fixed
(ii) One end fixed and other end hinged
(iii) Both ends are pin Jointed.
Use Euler’s equation.
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Q.18. A bar of length 4m is used as simply supported beam and subjected to UDL of 30kN/m
over the whole span, deflects 15mm at centre. Determine buckling loads when it is used as
column with the following end conditions:
(i) Both ends are pin jointed;
(ii) One end fixed and other end hinged;
(iii) Both ends fixed.
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Q.19.Compare the ratio of strength of a solid steel column to that of a hollow column of
internal diameter equal to 3/4th of its external diameter. Both the columns have the same crosssectional areas, lengths and end conditions. [Ans. 25/7]
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Q.20.. Find the Euler’s crippling load for a hollow cylindrical steel column of 38 mm external
diameter and 35 mm thick. The length of the column is 2.3 m and hinged at its both ends. Take
E = 200 GN/m2. Also determine the crippling load by Rankine’s formula, using σc = 320 MPa
; and a =1/7500 [Ans. 17.25 kN ; 17.4 kN]
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Q.21.. Determine the diameter of the pistion rod of the hydraulic cylinder of 100 mm bore
when the maximum hydraulic pressure in the cylinder is limited to 14 N/mm2. The length of
the piston rod is 1.2 m. The factor of safety may be taken as 5 and the end fixity coefficient as
2. [Ans. 45 mm]
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Q.22. Find the diameter of a piston rod for an engine of 200 mm diameter. The length of the
piston rod is 0.9m and the stroke is 0.5 m. The pressure of steam is 1 N/mm2. Assume factor
of safety as 5.
[Ans. 31 mm]
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Q.23.Determine the diameter of the push rod made of mild steel of an I.C. engine if the
maximum force exerted by the push rod is 1500 N. The length of the push rod is 0.5 m. Take
the factor of safety as 2.5 and the end fixity coefficient as 2.
[Ans. 10 mm]
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Q.24. The eccentric rod to drive the D-slide valve mechanism of a steam engine carries a
maximum compressive load of 10 kN. The length of the rod is 1.5 m.
Assuming the eccentric rod hinged at both the ends,
Find
(a) Diameter of the rod, and
(b) Dimensions of the cross-section of the rod if it is of rectangular section. The depth of the
section is twice its thickness.
Take factor of safety = 40 and E = 210 kN/mm2.
[Ans. 60 mm ; 30 × 60 mm]
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Q.25. The connecting rod of a four stroke cycle Diesel engine is of circular section and of
length 550 mm. The diameter and stroke of the cylinder are 150 mm and 240 mm respectively.
The maximum combustion pressure is 4.7 N/mm2. Determine the diameter of the rod to be
used, for a factor of safety of 3 with a material having a yield point of 330 MPa.
Find also the maximum bending stress in the connecting rod due to whipping action if the
engine runs at 1000 r.p.m. The specific weight of the material is 7800 kg/m3.
[Ans. 33.2 mm ; 48 MPa]
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Q.26.Determine the crippling load for a solid round bar of dia.5cm and length 3m when it is
used as a column. Take Young’s modulus =2.1 x 105N/mm2. when,
1.
When one end is fixed and the other end is free.
2.
When two end are fixed .
3.
When one end is fixed and the other end is hinged.
ANS-[16.822KN,269.152KN,134.576]
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Q.27. Determine the dimensions of an I-section connecting rod for an internal combustion
engine having the following specifications :Diameter of the piston = 120 mm , Mass of
reciprocating parts = 350 kg/m2 of piston area ,Length of connecting rod = 350 mm, Engine
revolutions per minute = 1800, Maximum explosion pressure = 3 N/mm2,Stroke length = 180
mm
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Q.28. Determine the crippling load for a Timber column having cross section 15cm x20cm and
6m long when both ends are fixed. Take Young’s modulus =17.5KN/mm2 and
Also safe load of column if factor of safety is 3.
[1079.48KN,360KN]
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Q.29. Determine Euler’s Crippling Load for I-section joist 40cm x 20cm x 1cm and 5m long
which is used as column with both ends fixed. Take Young’s modulus =2.1 x 105N/mm2
er end hinged;
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Q.30. A T-section 150 mm × 120 mm × 20 mm is used as a strut of 4 m long hinged at both
ends. Calculate the crippling load, if Young’s modulus for the material of the section is 200
kN/mm2.
[703 kN]
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Q.31. Calculate the diameter of a piston rod for a cylinder of 1.5 m diameter in which the
greatest difference of steam pressure on the two sides of the piston may be assumed to be 0.2
N/mm2.The rod is made of mild steel and is secured to the piston by a tapered rod and nut and
to the crosshead by a cotter. Assume modulus of elasticity as 200 kN/mm2 and factor of safety
as 8. The length of rod may be assumed as 3 metres.
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Q.32. The maximum load on a petrol engine push rod 300 mm long is 1400 N. It is hollow
having the outer diameter 1.25 times the inner diameter. Spherical seated bearings are used for
the push rod. The modulus of elasticity for the material of the push rod is 210 kN/mm2. Find a
suitable size for the push rod, taking a factor of safety of 2.5.
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Q.33. A connecting rod of length l may be considered as a strut with the ends free toturn on the
crank pin and the gudgeon pin. In the directions of the axes of these pins, however, it may
be considered as having fixed ends. Assuming that Euler’s formula is applicable, determine the
ratio of the sides of the rectangular cross-section so that the connecting rod is equally strong in
both planes of buckling.
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34. What do you understand by a column or strut ? Explain the various end conditions of a
column or strut.
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Q.35. Define ‘slenderness ratio’. How it is used to define long and short columns?
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Q.36 State the assumptions used in Euler’s column theory.
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Q.37. What is equivalent length of a column? Write the relations between equivalent length
and actual length of a column for various end conditions.
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Q.38. Explain Johnson’s formula for columns. Describe the use of Johnson’s formula and
Euler’s formula.