Sm 5

C.K.GOPALAKRISHNAN, AP/MECH, MAHALAKSHMI ENGINEERING COLLEGE, TRICHY
UNIT-5
PART-A
1. Define principal stress and principal plane.
The magnitude of normal stress, acting on a principal plane is know n as principal
stresses.
The planes w hich have no shear stress are know n as principal planes.
2. What are the uses of Mohr’s circle?
It is used to find out the normal, resultant and principal stresses and their planes.
3. List the methods to find the stresses in oblique plane?
Analytical method , Graphical method.
4. What is Mohr’s circle method?
It is a graphical method to determine normal, tangential and resultant stresses on any
oblique planes and position and magnitude of principal stresses.
5. What is the radius of Mohr’s circle?
Radius of Mohr’s circle is equal to the maximum shear stress.
6. What are the planes along which the greatest shear stresses occur?
Greatest shear stress occurs at the planes w hich is inclined at 45o
to its normal.
7. Differentiate thick and thin shells.
S.No Thin cylinder Thick cylinder
1 The ratio of w allthickness to the
diameter of the cylinder is less than
1/20
The ratio of w allthickness to the
diameter of the cylinder is more than
1/20
2 Circumferential stress is assumed to
be constant throughout w allthickness.
Circumferential stress varies from inner
to outer w allthickness.
8. Define circumferential and Hoop stress.
The stress acting along the circumference of the cylinder is called circumferential stress
whereas the stress acting along the length of the cylinder is know n as longitudinal
stress.
9. Define thin shell.
If the thickness of the w all of the cylinder vessel is less than 1/20 of its internal diameter,
the cylinder vessel is know n as thin cylinder.
10. Name the stresses induced in a thin walled cylinder subjected to internal fluid
pressure.
Circumferential stress and longitudinal stress.
11. Explain the failure of a thin cylinder due to internal pressure.
If the stress induced in the cylinders exceed the permissible limit, the cylinder is likely to
fail in any one of the follow ing tw o ways. (i) It may split into tw o troughs and (ii) It may
split up into tw o cylinders.
12. List the assumptions made in the analysis of thin cylinders.
The assumptions are the stresses are uniformly distributed throughout the w all
thickness.

13. How will you find major principal stress and minor principal stress? Alsomention
how to locate the directionof principal planes.
14. Write the formulae to calculate stresses when a point in a member is subjected to
direct stress in two mutually perpendicular directions accompanied by a simple
shear stress.
15. Write the formulae to calculate stresses w hen a point in a member subjected in
direct stress in one direction.
16. Give the expression for maximum shear stress in a two dimensional stress
system.
17. Define theories of failure.
Failure theory is the science of predicting the conditions under w hich solid materials fail
under the action of external loads. The failure of a material is usually classified into brittle
failure (fracture) or ductile failure (yield). Depending on the conditions most materials
can fail in a brittle or ductile manner or both.

PART-B
1. A short metallic column of 500 mm2
cross sectional area carries an axial load
compressive load of 100 kN. For a plane inclined at 60o
with the direction of load,
calculate (i) normal stress (ii) Tangential stress (iii) Resultant stress (iv)Maximum
shear stress (v) obliquity of resultant stress.
Solution:

2. The Principal stresses at a point in a bar are 200 N/mm 2
(tensile) and 100 N/mm2
(compressive). Determine the resultant stress in magnitude and direction on a
plane inclined at 60o to the axis of the major principal stress. Also determine the
maximum intensity of shear stress in the material at that point.

3. The normal stresses at a point on two mutually perpendicular planes are 140 MPa
(tensile) and 100 MPa (compressive). Determine the shear stress on these planes
if the maximum principal stress is limited to 150 MPa (tensile). Determine also the
following: (i) Minimum principal stress (ii) Maximum shear stress and its plane
(iii) Normal, shear and resultant stresses on a plane which is inclined at 30o anti-
clockwise to X-Xplane.

4. An elemental cube is subjected to tensile stresses of 30 N/mm2
and 10 N/mm2
acting on two mutually perpendicular planes and a shear stress of 10 N/mm 2
on
these planes. Draw the Mohr’s circle of stresses and determine the magnitude
and direction of principal stresses and also the greater shear stress.

5. 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
and 10 N/mm2
both tensile. They are
accompanied by a shear stress of magnitude 10 N/mm 2
. Find graphically or
otherwise, the orientation of principal planes and evaluate the principal stresses.

6. At a point in a stressed material, the stresses on the vertical plane are 50 MPa and
40 MPa anticlockwise in effect. On the horizontal plane, the normal stress is 30
MPa compressive stress. Determine (i) Principal stresses and their planes (ii)
Maximum shear plane and the stresses on it. (iii) The stresses on an inclined
plane at 30o
anticlockwise from VP and the resultant stress on it (iv) Stresses on
an inclined plane at 40o
anticlockwise from horizontal plane and the resultant
stress on it.

7. An element in a strained material has tensile stress of 500 N/mm 2
and a
compressive stress of 350 N/mm2
acting on two mutually perpendicular planes
and equal shear stress of 100 N/mm 2
on these planes. Find the principal stresses
and their planes. Find also maximum shear stress and normal stress on the plane
of maximum shear stress.

8. A material is subjected to two mutually perpendicular tensile direct stresses of 40
MPa and 30 MPa together with a shear stress of 20 MPa, shear stress being clock -
wise on the face carrying the 40 MPa tensile stress. Determine (i) The stresses on
a plane making an angle of 40o counter-clockwise to the plane of the 40 MPa
stress; (ii) The principal stresses and their planes; (iii) The maximum shear
stress and its plane.

9. A point in a strained material is subjected to mutually stress of 600 N/mm2
(tensile) and 400 N/mm2
(compressive). It is also subjected to a shear stress of
100 N/mm2
. Draw Mohr’s circle and find the principal stresses and maximum
shear.

10. At a point in a strained material, the principal stresses are 100 MPa (tensile) and
60 MPa (compressive). Determine the normal stress, shear stress, resultant
stress on a plane inclined at 50 degree to the axis of major principal stress. Also
determine the maximum shear stress at the point.

11. The state of stresses at a point on two mutually perpendicular planes is as below
σx = 160 MPa (Tension); αy = 80 MPa (compression) and Ʈ = 50 MPa (acting in the
positive direction on positive X plane). Determine the following: (i) Principal
stresses and their planes (ii) Maximum shear stress (iii) Normal and shear stress
on a plane which is inclined 30o
and clockwise to positive x plane.

12. At a point within a body there are two mutually perpendicular stresses of 80
N/mm2
and 40 N/mm2
of tensile in nature. Each stress is accompanied by a shear
stress of 60 N/mm2
. Determine the normal, shear and resultant stress on an
oblique plane at an angle of 45 degree withthe axis of the major principal stress.
Solution:

13. At a point in a strained material, there is a horizontal tensile stress of 100 N/mm 2
and an unknown vertical stress. There is also a shear stress of 30 N/mm 2
on these
planes. On a plane inclined at 30o
to the vertical, the normal stress is found to be
90 N/mm2
tensile. Find the unknown vertical stress and also the principle stresses
and maximum shear stress.

14. A cylindrical shell 3 m long which is closed at the ends has an internal diameter 1
m and wall thickness of 15 mm. Calculate the circumferential and longitudinal
stresses induced and also change in dimensions if the internal pressure is 1.5
N/mm2
. E = 2x 105
N/mm2
, µ =0.3.

15. A cylindrical shell 100 cm long, and 25 cm in internal diameter having thickness of
metal as 8 mm, is filled with a fluid at atmospheric pressure. If the additional fluid
of 30 cm3
is pumped in the shell. Take E = 200 GPa and µ = 0.3. Also find the
hoop stress induced.

16. A thin cylinder is 3.5 m long, 90 cm in diameter, and the thickness of the metal is
12 mm. It is subjected to an internal pressure of 2.8 N/mm2. Calculate the change
in dimensions of the cylinder and the maximum intensity of shear stress induced.
E = 200 GPa and Poisson’s ratio =0.3.

17. Derive the expression for the change in diameter and for the change in volume of
volume of a thin spherical shell when it is subjectedto an internal pressure.

18. 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/mm 2
, determine (i)
change in diameter, (ii) change in length and (iii) change in volume. Take E = 2 x
105
N/mm2
and Poisson’s ratio =0.25.

19. Derive relations for change in length, thickness and volume of a thin cylinder
subjectedto an internal pressure. Also explain the failure of thin cylinders.

20. A boiler is to be made of 20 mm thick plate having a limiting tensile stress of 120
MPa. If the efficiencies of the longitudinal and circumferential joints are 70% and
30% respectively, determine the maximum permissible diameter of the shell for an
internal pressure of 2 MPa. When the shell diameter is 1.5 m, find the permissible
intensity of internal pressure.

21. A thin cylindrical shell has an internal diameter of 250 mm, has walls 5 mm thick
and is 1 m long. It is found to change in internal volume by 19200 mm3 when
filled with a liquid at a pressure ‘p’. If E = 200 GPa and Poisson’s ratio = 0.25 and
assuming rigid end plates, determine: (i) the values of hoop and longitudinal
stresses (ii) the change in internal diameter of the cylinder (iii) the change in
length and (iv) the necessary change in pressure p to produce a further increase
in internal volume of 10 %. The liquid may be assumed incompressible.

22. A shell 4.5 m long, 900 mm in diameter is subjected to an internal pressure of 1.1
N/mm2
. If the thickness of the shell is 8.5 mm, find the circumferential and
longitudinal stresses. Find also the maximum shear stress and changes in
dimensions of shell. E= 2.1 x 105
N/mm2
, µ = 0.33.
Solution:

23. Derive the relation for strain energy in terms of principal stresses.
Solution:
24. A cylindrical drum 600 mm in diameter has to withstand an internal pressure of 1.8
N/mm2. Calculate the necessary wall thickness for a factor of safety of 3 if the
criterion of failure is the maximum strain energy and the elastic limit in pure
tension is 237 N/mm2. Take µ= 0.3.

25. Derive the expressionfor equivalent bendingmomentand equivalenttorque.

Sm 5

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