CE-1259, Strength of Materials
UNIT I
STRESS, STRAIN DEFORMATION OF SOLIDS
Part -A
1. Define strain energy density.
2. State Maxwell’s reciprocal theorem.
3. Define proof resilience.
4. State Castigliano’s theorem.
5. Define modulus of resilience.
6. Explain strain energy in pure shearing.
7. Give some points that are to be remembered while use Castligliano’s theorem.
8. Give strain energy and deflection under axial load.
9. Give stress (instantaneous) due to suddenly applied load.
10. Give strain energy and deflection under bending.
Part - B
1. (i) Derive a relation for strain energy due to shear force.
(ii) Derive a relation for maximum deflection of a simply supported beam with
uniformly distributed load over entire span. Use strain energy method.
2. Determine the deflection at C of the beam given in fig below. Use principal of
virtual work.
W
L/2
A
B
L
C
3. The external diameter of a hollow shaft is twice the internal diameter. It is subjected
to pure torque and it attains a maximum shear stress ‘τ’. Show that the strain energy
stored per unit volume of the shaft is (5τ2 / 16C). Such a shaft is required to transmit
5400kw at 110 r.p.m with uniform torque the maximum stress not exceeding 84
MN/m2 Determine
(i)
the shaft diameters
(ii)
The strain energy stored per m3. Take C = 90 GN/m2.
4. Using Castigliano’s theorem, determine the deflection of the free end of the
cantilever beam shown in fig. A is fixed and B is free end. Take EI = 1.9MNm2.
16 kN
30 kN
20kN/m
A
B
1m
1m
1m
5. Using Castigliano’s theorem, obtain the deflection under a single concentrated load
applied to a simply supported beam shown in fig. Take EI = 2.2 MNm2.
6. Fig shows a cantilever, 8m long, carrying a point load 5kN at the centre and a
uniformly distributed load of 2kN/m for a length 4m from the end B. If EI is the
flexural rigidity of the cantilever find the reaction at the prop.
7. A simply supported beam of span l is carrying a concentrated load ‘W’ at the centre
and a uniformly distributed load of intensity of w per unit length. Show that
Maxwell’s reciprocal theorem holds good at the centre of the beam.
8. Compare the strain energies of the following two shafts subjected to the same
maximum shear stress in torsion.
(i)
A hollow shaft having outer diameter n times the linear dia.
(ii)
A solid shaft.
Masses, lengths and materials of the two shafts are same.
9. A beam of length ‘l’ simply supported at the ends is loaded with a point load ‘W’ at
a distance ‘a’ from one end. Assuming that the beam has constant cross-section with
moments of inertia as I and Young’s modulus of elasticity for the material of the
beam as E, find the strain energy of the beam and hence find the deflection under
the load. Strain energy due to shearing may be neglected.
10. A 1m long beam rectangular in section 30mm wide x 40mm deep is supported on
rigid supported at its ends. If it is struck at the centre by a 12kg mass falling through
a height of 60mm find:
(i)
The instantaneous stress developed.
(ii)
The instantaneous strain energy stored in the beam.
UNIT II
BEAMS - LOADS AND STRESSES
Part -A
11. A fixed beam AB of length 3m is having moment of inertia I = 3 x 10 6 mm4 the
support B sinks down by 3 mm. If E= 2 x 105 N/mm2, find the fixing moments.
12. What will be the fixed end moment for a beam subjected to uniformly varying load,
which is maximum at the centre and minimum at supports?
13. Define Fixed beam and continuous beam.
14. Give the advantages of continuous beam over simply supported beam.
15. Give the advantages of fixed beam over simply supported beam.
16. A cantilever of length 6m carries a point load of 48kN at its centre. The cantilever is
propped rigidly at the free end. Determine the reaction at the rigid prop.
17. Give fixed end moments for any 4 different types of loading on a beam.
18. Give theorem of three moment equation, when all supports remain at the same level.
19. Give theorem of three moment equation, when flexural rigidity for spans is same
and supports are at same level.
20. Give theorem of three moment equation, when continuous beam has a fixed end.
Part - B
11. A fixed beam AB of length 6m carries point loads of 160 kN and 120 kN at a
distance of 2 m and 4 m from the left end A. Find the fixed end moments and the
reactions at the supports. Draw BM and SF diagrams.
12. A fixed beam of 6m span is loaded with point loads of 150 kN at distance 2m from
each support. Draw the B.M.D and S.F.D. Find also the maximum deflection. Take
E=2x108 kN/m2, and I = 8 x 108 mm4.
13. A fixed beam of 6m span is subjected to a concentrated couple of 150 kNm
applied at a section 4m from the left end. Find the end moments from the first
principles. Draw B.M and S.F diagrams also.
14. A simply supported beam of span 10m carries a uniformly distributed load of
1152N per unit length. The beam is propped at the middle of the span. Find the
amount by which the prop should yield, in order to make all the three reactions
equal. Take E = 2 x 105 N/mm2 and I = 108 mm4.
15. A cantilever AB of span 6m is fixed at the end ‘A’ and propped at the end B. It
carries a point load of 50kN at the mid span. Level of the prop is the same as that of
the fixed end.
(i)
Determine reaction at the prop.
(ii)
Draw the S.F and B.M diagrams
16. A fixed beam of 8m span carries a uniformly distributed load of 40 kN/m run over
4m length starting from left end and a concentrated load of 80kN at a distance of 6m
from the left end. Take EI = 15000 kNm2. Find
(i)
Moments at the supports.
(ii)
Deflection at centre of the beam.
17. A fixed beam of 6m span carries a uniformly distributed load of 2kN/m run. If
E=2x108 kN/m2 and I = 0.48x10-4m4, find
(i) Bending moment at the centre;
(ii) Maximum deflection.
18. A fixed beam of 6m span carries a uniformly distributed load of 40kN/m run over
4m length starting from left hand end and a concentrated load of 80kN at a distance
of 6m from the left hand end. Find:
(i) Moment at the supports;
(ii) Deflection at centre of the beam. Take EI = 15000kNm2.
19. Fig shows a beam simply supported at the supports A and C and is continuous over
the supports B. Assuming EI is constant draw the bending moment and shear force
diagrams.
2 kN
A
D
4 kN
B
E
1.8m
C
1.8m
3.6m
2.4m
20. Fig shows a continuous beam ABCD having three equal spans of length ‘l’ each. It
carries a uniformly distributed load w/unit length over its entire length. It is freely
supported on all supports, which are at the same level. Draw SFD and BMD for this
beam.
21. A continuous beam ABCD of uniform cross-section is loaded as shown in fig. Find
(i) Bending moments at the supports B and C;
(ii) Reactions at the supports.
Draw B.M and S.F diagrams also.
22. For the continuous loaded beam ABCD shown in fig find
(i)
Moments at the supports;
ii) Reactions at the supports.
Draw the B.M and S.F diagrams also.
23. Fig shows a continuous beam ABCD (with loads) of uniform cross section. If the
support B sinks by 10mm, Find the following:
(i)
Moments at the supports;
(ii)
Reactions at the supports;
(iii) Points of contra flexure,
Take E = 2 x 108 kN/m2 and I = 8.5 x 10-5 m4.
UNIT III
TORSION
UNIT IV
BEAM DEFLECTION
UNIT V
ANALYSIS OF STRESSES IN TWO DIMENSIONS