CE6306 STRENGTH OF MATERIALS
UNIT I STRESS, STRAIN AND DEFORMATION OF SOLIDS
Part-A (2 Marks)
1. A rod of diameter 30 mm and length 400 mm was found to elongate 0.35mm when it was
subjected to a load of 65 KN. Compute the modulus of elasticity of the material of this rod.
2. What is strain energy and write its unit in S.I. system?
3. Define ‘volumetric strain’.
4. What is proof resilience?
5. State the principle of super position.
6. Define “Shear stress”.
7. State Hooke’s Law
8. Define bulk modulus.
9. The young’s modulus and the shear modulus of material are 120G pa and 45 G pa
respectively. What is its bulk modulus?
10.calculate the instantaneous stress produced in a bar of cross sectional area 1000mm2 and 3 m
long by the sudden application of a tensile load of unknown magnitude, if the corresponding load
take E=200Gpa.
11. Define poison’s ratio.
12. What is thermal stress?
13. The strain induced in an ms bar of rectangular section having width equal to twice the depth
is 2.5x10-5.the bar is subjected to a tensile load of 4KN .find the section dimensions of the bar.
Take E=0.2x106N/mm2
14. Define modulus of rigidity.
15. Define shear stress and shear strain.
16. What are the uses of a Mohr’s circle?
Part-B (16Marks)
1. Two vertical rods one of steel and other of copper are each rigidly fixed at the top and 600 mm
apart. Diameters and lengths of the rods are 25 mm and 5 m respectively. A cross bar fixed to the
rods at the lower end carries a load of 7 k N such that the cross bar remains horizontal even after
loading. Find the steps in each rod and the position of the load on the cross bar. Assume the
modulus of elasticity for steel and copper as 200 KN/mm2 and 100 KN/mm2 respectively.
2. A cast iron flat 300 mm long and 30 mm (thickness) x 60 mm (width) uniform cross section, is
acted upon by the following forces:
30 KN tensile in the direction of the length
360 KN compressions in the direction of the width
240 KN tensile in the direction of the thickness.
Calculate the direct strain, net strain in each direction and change in volume of the flat. Assume
the modulus of elasticity and Poisson’s ratio for cast iron as 140 KN/ mm2 and 0.25 respectively.
3. A steel wire 6mm diameter is used for lifting a load of 1.5 KN at its lowest end, the length of
the wire hanging vertically being 160 meters. Taking the unit weight of steel=78KN/m3 and
E=2x105N/mm2, calculate the elongation of the wire.
4. A steel bar 4cmx4cm in section, 3 meters long is subjected to an axial pull of 128 KN. Taking
E=20x1010 N/m2 calculate the alteration in the length of the bar. Calculate also the amount of
energy stored in the bar during extension.
5. A Mild steel rod of 20 mm diameter and 300 mm long is enclosed centrally inside a hollow
copper tube of external diameter 30 mm and internal diameter 25 mm. The ends of the rod and
tube are brazed together, and the composite bar is subjected to an axial pull of 40 KN. If E for
steel and copper is 200 GN/m2 and 100 GN/m2 respectively, find the stresses developed in the
rod and the tube also find the extension of the rod.
6. A bar of 30 mm diameter is subjected to a pull of 60 KN. The measured extension on gauge
length of 200 mm is 0.09 mm and the change in diameter is 0.0039 mm. calculate the Poisson’s
ratio and the values of the three modulii.
7. (i) An alloy circular bar ABCD 3 m long is subjected to a tensile force of 50 KN as shown in
figure. If the stress in the middle portion BC is not to exceed 150 M Pa, then what should be its
diameter? Also find the length of the middle portion, if the total extension of the bar should not
exceed by 3 mm. Take E = 100 G Pa.
(12)
(ii) A circular bar rigidly fixed at its both ends uniformly tapers from 75 mm at one end to 50
mm at the other end. If its temperature is raised through 26 K, what will be the maximum stress
developed in the bar. Take E as 200 G Pa and α as 12 x10-6 /K for the bar material .
(4)
8. (i) In an experiment, a bar of 30 mm diameter is subjected to a pull of 60 KN. The measured
extension on gauge length of 200 mm is 0.09 mm and the change in diameter is 0.0039 mm.
Calculate the Poisson’s ratio and the values of the three modulii.
(12)
(ii) An alloy specimen has modulus of elasticity of 120 G Pa and modulus of rigidity of 45 G Pa.
Determine the Poisson’s ratio of the material.
(4)
9. The following data relate to a bar subjected to a tensile test;
Diameter of the bar =30mm
Tensile load P=54KN
Gauge length l=300mm
Extension of the bar=0.112mm
Change in diameter=0.00366mm
Calculate (i)Poisson’s ratio (ii)The values of three modulii
10. Three bars made of copper, zinc and aluminium of equal length are rigidly connect at their
ends as shown in Fig.(i)
They have cross-sectional areas of 250mm2, 375 mm2 and 500mm2 respectively. If the
compound member is subjected to a longitudinal pull of 125KN, estimate the proportion of load
carried on each rod and the induced stresses. Take Ecu=130Gn/m2,EZn=100GN/m2,Eal=80GN/m2
UNIT II TRANSVERSE LOADING ON BEAMS AND STRESSES IN BEAM
Part-A (2 Marks)
1.
2.
3.
4.
5.
6.
List out the types of beam.
What is point of contra flexure?
Mention the various types of beam with respect to support conditions.
What is meant by shear centre?
Mention and sketch any two types of supports and cording for the beams.
Sketch the bending stress as well as shear stress distribution for a beam of rectangular
cross section.
7. What is the value of bending moment corresponding to a point having a zero shear force?
8. Define shear force and bending moment.
9. Sketch the bending and shear stress distribution for a ’T’ section.
10. A cantilever beam of 3m long carries a point load of 20 kN at its free end .calculate SF
and BM diagram
11. Draw the shear force diagram for a cantilever beam of span 4m and carrying a point load
of 50KN at midspan.
12. Mention any two assumptions made in the theory of simple bending.
13. Sketch the bending moment diagram of a cantilever beam subjected to udl over the entire
span.
14. Draw the shear force diagram for a cantilever beam of span 4m and carrying a udl of
2KN/m over the entire span.
15. A rectangular beam 150mm wide and 200mm deep is subjected to a shear force of 40
KN. Determine the average shear stress and maximum shear stress.
Part-B (16Marks)
1. A cantilever 1.5 m long is loaded with a uniformly distributed load of 2 kN/m run over
a length of 1.25 m from the free end. It also carries a point load of 3 k N at a distance of
0.25 m from free end. Draw the shear force and bending moment diagrams of cantilever.
2. A timber beam of rectangular section is to support a load of 20 kN uniformly distributed
over a span of 3.6 m when beam is simply supported. If the depth of the section is to be
twice the breadth, and the stress in the timber is not to exceed 7 N/mm2, find the
dimensions of the cross-section.
3. A simply supported beam of span 6 meters carries a point load of 1 kN at 1 metre from
the left end and another point load of 4 kN at 1 metre from the right end. It carries a
uniformly distributed load of 2 kN/m run over a distance of 2 meters at mid span of the
beam. Find the SF and BM values. Draw SFD and BMD
4. A timber beam 150 mm 
its ends and
has a span of 3.5 m. The maximum safe allowable stress in bending is 7500 kN/m2. Find
the maximum safe UDL which the beam can carry. What is the maximum shear stress in
the beam for the UDL calculated?
5. A cast iron pipe 300 mm internal diameter, metal thickness 15 mm, is supported at two
points 6 m apart. Find the maximum bending stress in the metal of the pipe when it is
running full of water. Assume the specific weight of cast iron and water as 72 kN/m3
and 10 kN/m3 respectively.
6. Draw the shear force and bending moment diagrams for the beam shown in Fig.Q 12(a).
Also determine the maximum bending moment and location of point of contra flexure.
7. A Cantilever 1.5 m long carries a load of 2 tons at its free end, and another load 1 ton at
a distance of 0.5 m from the free end. Draw shear force and bending moment diagrams
for the cantilever.
8. A beam of triangular cross section having base width of 100 mm and height of 150 mm
is subjected to a shear force of 15 KN. Find the value of maximum shear stress, and
sketch the shear stress distribution along the depth of beam.
9. A simply supported beam of 4 m span is carrying loads as shown in figure. Draw the
shear force and bending moment diagrams for the beam.
10. A horizontal beam 10 m long is carrying a uniformly distributed load of 1 kN/m. The
beam is supported on two supports 6 m apart. Find the position of the supports, so that
bending moment on the beam is as small as possible. Also draw the shear force and
bending moment diagrams.
UNIT III TORSION
Part-A (2 Marks)
1. List the loads normally acting on a shaft.
2. Write the equation of torsion acting in a circular shaft.
3. Write down the simple torsion formula with the meaning of each symbol for circular
cross section.
4. Define stiffness of spring and mention its unit in SI system.
5. Write the assumption for finding out the shear stress of a circular shaft, subjected to
torsion.
6. What is meant by stiffness of spring?
7. What is a laminated spring?
8. Write the expression for torsional rigidity of solid circular shaft.
9. Find the torque which a shaft of 50mm diameter can transmit safely , if the allowable
shear stress is 75 N/mm2.
10. Differentiate open coil helical spring from the closed coil helical spring and state the
type of stress induced in each spring due to an axial load
11. Find the minimum diameter of shaft required to transmit a torque of 29820 Nm if the
maximum shear stress is not to exceed 45 N/mm2.
12. Write down the equation of torsion showing the various terms invovled in it.
13. A closelycoiled helical spring is to carry a load of 500N. its mean coil diameter is to be
10 times that of the wire diameter.calculate these diameters if the maximum shear stress
in the material of the spring is to be 80 MN/m2 .
14. Define polar modulus of a section .what is the polarmodulus value for a hollow circular
section of 100 mm external diameter and 40 mm internal diameter.
15. What is the maximum shear stress produced in a bolt of diameter 20 mm when it is
tightened by a spanner which exerts a force of 50 N with a radius of action of 150mm?
16. A closelycoiled helical spring of 10 mm in diameter having 10 complete turns, with mean
diameter 120 mm is subjected to an axial load of 200N. determine the maximum shear
stress and stiffness of the spring. Take G = 9x104 N/mm2 .
17. What is the power transmitted by circular shaft subject to a torque of 700KN-m at 110
rpm.
18.calculate the maximum torque that a shaft of 125 mm diameter a can transmit, if the
maximum angle of twist is 1° in a length of 1.5m.take C = 70x103 N/mm2.
Part-B (16Marks)
1.(a) Calculate the power that can be transmitted at a 300 rpm by a hollow steel shaft of 75
mm external diameter and 50 mm internal diameter when the permissible shear stress for the
steel is 70 N/mm2 and the maximum torque is 1.3 times the mean. Compare the strength of
this hollow shaft with that of an solid shaft. The same material, weight and length of both the
shafts are the same.
(b) A helical spring of circular cross section wire 18 mm in diameter is loaded by a force of
500 N. The mean coil diameter of the spring is 125 mm. The modulus of rigidity is 80
kN/mm2. Determine the maximum shear stress in the material of the spring. What number of
coils must the spring have for its deflection to be 6 mm?
2. (a) A hollow steel shaft of outside diameter 75 mm is transmitting a power of 300 kW at
2000 rpm. Find the thickness of the shaft if the maximum shear stress is not to exceed 40
N/mm2.
(b) A close coiled helical spring is to have a stiffness of 1.5 N/mm of compression under a
maximum load of 60 N. The maximum shearing stress produced in the wire of the spring is
125 N/mm2. The solid length of the spring is 50 mm. Find the diameter of coil, diameter of
wire and number of coils C=4.5*104 N/mm2.
3. (a) A solid cylindrical shaft is to transmit 300 kN power at 100 rpm. If the shear stress is
not to exceed 60 N/mm2, find its diameter. What percent saving in weight would be obtained
if this shaft is replaced by a hollow one whose internal diameter equals to 0.6 of the external
diameter, the length, the material and maximum shear stress being the same.
(b) A closely coiled helical spring of round steel wire 10 mm in diameter having 10 complete
turns with a mean diameter of 12 cm is subjected to an axial load of 250 N. Determine
i.
The deflection of the spring
ii. Maximum shear stress in the wire and
iii. Stiffness of the spring and
iv.
Frequency if vibration.
4. (a) A solid shaft is subjected to a torque of 1.6 kN-m. Find the necessary diameter of the
shaft, if the allowable shear stress is 50 MPa. The allowable twist is 1o for every 20 diameter
length of the shaft. Take C=80 GPa.
(b) A closely coiled helical spring of round steel wire 5 mm in diameter having 12 complete
coils of 50 mm mean diameter is subjected to an axial load of 150 N. Find the deflection of
the spring and the maximum shearing stress in the material. Modulus of rigidity (C) = 80
GPa. Also, find stiffness of the spring.
5. (a) Determine the dimensions of a hollow circular shaft with a diameter ratio of 3:4 which
is to transmit 60 kW at 200 rpm. The maximum shear stress in the shaft is limited to 70 GPa
and the angle of twist to 3.8o in a length of 4 m. For the shaft material the modulus of rigidity
is 80 GPa.
(b) A close coiled helical spring is required to absorb 2250 joules of energy. Determine the
diameter of the wire, the mean coil diameter of the spring and the number of coils necessary
if
(i) The maximum stress is not to exceed 400 MPa,
(ii) The maximum compression of the spring is limited to 250 mm and
(iii) The mean diameter of the spring is eight times the wire diameter.
For the spring material, rigidity modulus is 70 GPa.
6. (a) (i) What do you mean by the strength of the shaft? Compare the strength of solid and
hollow circular shafts.
(ii) Find the diameter of a solid shaft to transit 90 kW at 160 rpm, such that the shear stress is
limited to 60 N/mm2. The maximum torque is likely to exceed the mean torque by 20%. Also
find the permissible length of the shaft, if the twist is not to exceed 1 degree over the entire
length. Take rigidity modulus as 0.8 * 105 N/mm2.
(b) A close-coiled helical spring is to have a stiffness of 1 kN/m of compression under a
maximum load of 45 N and maximum shearing stress of 126 MPa. The solid length of the
spring (i.e., when the coils are touching) is to be 45 mm. Find the diameter of the wire and
mean diameter of the coil required. Take G = 42 * 103 N/mm2.
7. (a) A solid shaft is subjected to a torque of 45 kNm. If the angle of twist is 0.5 degree per
meter length of the shaft and shear stress is not to exceed 90 MN/m2, find
(i) Suitable diameter of the shaft
(ii) Final maximum shear stress and the angle of twist per meter length. Modulus
of rigidity= 80GN/m2.
(b)A closely coiled helical spring having 12 coils of wire diameter 16 mm and made with coil
diameter 250 mm is subjected to an axial load of 300 N. Find axial deflection, strain energy
stored and torsional shear stress. Modulus of rigidity = 80 GN/m2.
UNIT IV DEFLECTION OF BEAMS
Part-A (2 Marks)
1. What are the advantages of Macaulay method over the double integration method, for finding
the slope and deflections of beams?
2. State the limitations of Euler’s formula.
3. Define crippling load.
4. State Mohr’s theorem.
5. State any three assumption made in Euler’s column theory.
6. What are the different modes of failures of a column?
7. Write down the Rankine formula for columns.
8. What is effective or equivalent length of column?
9. Define Slenderness Ratio.
10. Define the terms column and strut.
Part-B (16Marks)
1. A simply supported beam of 10 m span carries a uniformly distributed load of 1 kN/m over the
entire span. Using Castigliano’s theorem, find the slope at the ends. EI = 30,000 kN/m2. (16)
2. A 2m long cantilever made of steel tube of section 150 mm external diameter and10mm thick
is loaded. If E=200 GN/m2 calculate (1) The value of W so that the maximum bending stress is
150 MN/m (2) The maximum deflection for the loading.
(16)
3. A beam of length of 10 m is simply supported at its ends and carries two point loads of 100
KN and 60 KN at a distance of 2 m and 5 m respectively from the left support. Calculate the
deflections under each load. Find also the maximum deflection. Take I = 18 X 108 mm4 and
E = 2 X 105.
(16)
4. i) A column of solid circular section, 12 cm diameter, 3.6 m long is hinged at both ends.
Rankine’s constant is 1 / 1600 and σc = 54 KN/cm2. Find the buckling load.
ii) If another column of the same length, end conditions and rankine constant but of
12 cm X 12 cm square cross-section, and different material, has the same buckling load,
find the value of _c of its material.
(16)
5. A beam of length of 6 m is simply supported at its ends. It carries a uniformly distributed
load of 10 KN/m as shown in figure. Determine the deflection of the beam at its mid-point
and also the position and the maximum deflection. Take EI=4.5 X 108 N/mm2.
(16)
6. An overhanging beam ABC is loaded as shown is figure. Determine the deflection of the
beam at point C. Take I = 5 X 108 mm4 and E = 2 X 105 N/mm2. (16)
7. A cantilever of length 2m carries a uniformly distributed load 2 KN/m over a length of 1m
from the free end, and a point load of 1 KN at the free end. Find the slope and deflection at the
free end if E = 2.1 X 105 N/mm2 and I = 6.667 X 107 mm4.
(16)
8. A cantilever of length 2 m carries a uniformly distributed load of 2.5 KN/m run for a length of
1.25 m from the fixed end and a point load of 1 KN at the free end. Find the deflection at the
free end if the section is rectangular 12 cm wide and 24 cm deep and E=1X 104N/mm2. (16)
9. Determine the section of a hollow C.I. cylindrical column 5 m long with ends firmly built in.
The column has to carry an axial compressive load of 588.6 KN. The internal diameter of
the column is 0.75 times the external diameter. Use Rankine’s constants
a = 1 / 1600, σc = 57.58 KN/cm2 and F.O.S = 6.
(16)
UNIT V THIN CYLINDERS, SPHERES AND THICK CYLINDERS
Part-A (2 Marks)
1. List out the stresses induced in thin cylindrical shell due to internal pressure.
2. What are principal planes and stresses?
3. What is the stress developed in thin cylinders when they are subjected to internal fluid
pressures?
4. What are assumptions involved in the analysis of thin cylindrical shells.
5. Define principle planes.
6. List out the modes of failure in thin cylindrical shell due to an internal pressure.
7. What do you mean by principal plane?
8. Draw conjugate beam for a cantilever carrying uniformly distributed load over the entire
span.
9. Give conjugate beams for the cantilever beam and simply supported beam.
10. Write down the two area moment theorems for the evaluation of slope and deflection.
11. Write the expression for strain energy due to torsion.
12. State Castiglione’s first theorem.
13. The principal stress at a point is 100 N/ mm2 (tensile) and 50 N/ mm2 (compressive)
respectively. Calculate the maximum shear at this point.
14. A spherical shell of 1 m diameter is subjected to an internal pressure 0.5 N/ mm2. Find
the thickness if the allowable stress in the material of the shell is 75 N/ mm2.
Part-B (16Marks)
1. (a) A steel cylinder shell 3 m long which is closed at its ends, had an internal diameter of
1.5 m and a wall thickness of 20 mm. Calculate the circumference and longitudinal stress
induced and also the change in dimensions of the shell if it is subjected to an internal
pressure of 1.0 N/mm2. Assume the modulus of elasticity and poisons ratio for steel as
200 kN/mm2 and 0.3 respectively.
(b) The state of stress at a certain point in a strained material is show in fig. Calculate (i)
Principal stress, (ii) Inclination of the principal planes, (iii) Normal, shear and resultant
stress on the plane MN.
2. (a) A cylindrical thin drum 80 cm in diameter and 3 m long has a shell thickness of 1 cm.
If the drum is subjected to an internal pressure of 2.5 N/mm2, determine
(i) change in diameter
(ii) change in length
(iii) change in volume Take E = 2*105 N/mm2 and poisson’s ratio = 0.25
(b) At a certain point in a strained material, the intensities of stresses on two planes at
right angles to each other are 20 N/mm2 and10N/mm2 both tensile. They are accompanied
by a shear stress of Magnitude 10 N/mm2. Find graphically or otherwise, the orientation
of principal planes and evaluate the principal stresses.
3. (a) A cylindrical shell 3 meter long which is closed at the ends has an internal diameter of
1 m and a wall thickness of 15 mm. Calculate the circumference and longitudinal stress
induced and also change in the dimensions of the shell if it is subjected to an internal
pressure of 1.5 * 106 N/m2. Take E = 20* 1010 N/m2 and Poisson’s ratio = 0.3.
(b) A short metallic column of 500 mm2 cross sectional area carries an axial load 100 kN.
For a plane inclined at 60o with the direction of load, calculate normal stress, tangential
stress, resultant stress, maximum shear stress and obliquity of the resultant stress.
4. (a) A cylindrical shell 800 mm in diameter, 3 m long is having 10 mm metal thickness. If
the shell is subjected to an internal pressure of 2.5 N/ mm2
(i) the change in diameter
(ii) the change in length
(iii) the change in volume
Take E = 200kN/mm2 and 1/m =0.25.
(b) The state of stress (in N/mm2) acting at a certain point of the strained material is
shown in fig Q.15 (b). Compute
(i) The magnitude and nature of principal stresses and
(ii) The orientation of principal planes
75
45
45
75
5. (a) A thin cylinder shell 1.5 m long, internal diameter 300 mm and wall thickness 10 mm
is filled up with a fluid at atmospheric pressure. If the additional fluid of 300 * 103 mm3
is pumped in the shell, find the pressure exerted by the fluid on the shell. Take E = 2 *
105 N/mm2 and 1/m =0.3. Also find the hoop stress induced.
(b) Stresses at a point are px = 80 N/ mm2, py = -35 N/ mm2, q = 11.5 N/ mm2. Determine
principal planes, principal stresses and maximum shear stress.
6. (a) The normal stresses in two mutually perpendicular directions are 110 N/ mm2 and 47
N/ mm2 both tensile. The complementary shear stresses in these directions are of intensity
63 N/ mm2. Find the principal stresses and its planes.
(b) A cylindrical vessel 2 m long and 500 mm in diameter with 10 mm thick plates is
subjected to an internal pressure of 2 MPa. Calculate the change in volume of the vessel.
Take E = 200 GPa and poisson’s ratio = 0.3 for the vessel material.