STM Objective

STRENGTH OF MATERIAL

4 THEORY OF SIMPLE BENDING

1. Two beams of equal cross - section area are subjected to equal bending moment. If one beam has

square cross-section and the other has circular section, then. [IES-99]

(a) both beams will be equally strong.

(b) circular section beam will be stronger.

(c) square section beam will be stronger.

(d) the strength of the beam will depend on the nature of loading.

2. Which one of the following portions of the loaded beam shown in the given fig is subjected to pure

bending ? [IES-99]

l W 2l W
l

AB CDE

l

(a) AB (b) DE (c) AE (d) BD

3. Which one of the following statements is correct? Abeam is said to be of uniform strength if

(a) The bending moment is the same throughout the beam. [IES-2007]

(b) The shear stress is the same throughout the beam.

(c) The deflection is the same throughout the beam.

(d) The bending stress is the same at every section along its longitudinal axis

4. Abeam cross-section is used in two different orientations as shown in the fig. given below.[IES-96]
b/2

b b
b/2

(A)
(B)

Bending moments applied to the beam in both case are same. The maximum bending stresses induced

in cases (A) and (B) are related as

(a) σA = σB (b) σA = 2σB (c) σA = σB / 2 (d) σA = σB / 4

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STRENGTH OF MATERIAL

5. A homogenous, simply supported prismatic beam of width B, depth D and span L is subjected to a

concentrated load of magnitude P. The load can be placed anywhere along the span of the beam. The

maximum flexural stress developed in beam is (GATE-2003) (CE)

2 PL 3 PL 4 PL 3 PL

a) b) c) d)

3 BD2 4 BD2 3 BD2 2 BD2

6. A rectangular section beam subjected to a bending moment M varying along its length is required to

develop same maximum bending stress at any cross-section. If the depth of the section is constant,

then its width will vary as. [IES-95]
(a) M
(b) √ M

(c) M2 (d) 1/M

7. Consider a simply supported beam with a uniformly distributed load having a neutral axis (NA) as

shown. For points P (on the neutral axis) and Q (at the bottom of the beam) the state of stress is best

represented by which of the following pairs ? (GATE-2011) (CE)

→→ Q→ b) P→→ → Q→
a) P → → →

→ →

→→ → Q→ d) → P → →
c) P →
Q
→
→

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STRENGTH OF MATERIAL

8. For the configuration of loading shown in the given fig., the stress in fibreAB is given by.

5 [IES-95]
P
P
e A 10
B
A

(a) P (tensile) ((b) P )P.e.5 (compressive)
A A Ixx

((c) P )P.e.5 (compressive) (d) P (compressive)
A Ixx A

9. A short column of symmetric cross-section made of a brittle material is subjected to an eccentric

vertical load P at an eccentricity ‘e’. To avoid tensile stress in the short column , the eccentricity ‘e’

should be less than or equal to. →→.............................eeP→ [IES-2001]

(a) h / 12 ......... →→ ↓
(b) h / 6 →
(c) h / 3 L
(d) h / 2 → ↓

↓
↓

.

10. BeamAis simply supported at its ends and carries udl of intensity w over its entire length. it is made of

steel havingYoung’s modulus E.Beam B is a cantilever and carries a udl of intensity w/4 over its entire

length. It is made of brass having Young’s modulus E/2. The two beams are of same length and have

some cross-sectional area. if σA and σB denote the maximum bending stresses developed in beams

A and B, respectively then which one of the following is correct? [IES-2005]

(a) σA/σB = 1.0
(b) σA/σB < 1.0

(c) σA/σB >1.0
(d) σA/σB depends on the shape of cross-section

11. The maximum bending stress induced in a steel wire of modulus of elasticity 200 kN/mm2 and diam

eter 1 mm when wound on a drum of diameter 1 m is aproximately equal to (GATE-1992) (CE)

a) 50 N/mm2 b) 100 N/mm2

c) 200 N/mm2 d) 400 N/mm2

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STRENGTH OF MATERIAL

12. A beam with the cross-section given below is subjected to a positive bending moment (causing

compression at the top) of 16k N-m acting around the horizontal axis. The tensile force acting on the

hatched area of the cross-section is (GATE-2006) (CE)

a) zero 75 mm
b) 5.9 kN
c) 8.9 kN 11111112222222333333344444445555555666666677777778888888999999900000001111111 25 mm
d) 17.8 kN 50 mm

50 mm 50 mm

13. For the component loaded with a force F as shown in the figure, the axial stress at the corner point

P is 111222333444555666777 (GATE-2008) (ME)

<> →
< P

a) F(3L - b) b) F(3L + b) L

4b3 4b3 F

< L-b >

c) F(3L - 4b) d) F(3L - 2b) <L >

4b3 4b3 2b
2b

14. A frame of square cross-section of (a x a) is shown in the figure. The stress near the fixed end on the

upper sided of the frame is 111111111111222222222222< → (GATE-2007) (PI)

a) 58 P b) 59 P 10 a
>
a2 a2
61 P 100 a
c) 60 P
d)
a2
a2

15. A cantilever beam XY is made of stepped circular shaft of diameters 100 mm and 50 mm, as shown

in the following figure. The cantilever is subjected to two concentrated bending moments, one of

100 Nm at point Y and another of 200 Nm at point Z. The maximum bending stress (in MPa)

experienced by the cantilever is (GATE2010) (PI)

a) 1.02 X 1111111111122222222222X Z Y
b) 3.06
c) 8.15 <
d) 16.30
<

< << << <

1m 1m 1m

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STRENGTH OF MATERIAL

16. A cantilever beam has the square cross section of 10 mm x 10 mm. It carries a transverse load of

10 N. Considering only the bottom fibres of the beam, the correct representation of the longitudinal

vaiation of the bending stress is (GATE-2005) (ME)

111111111222222222333333333 10 N→ 11111111222222223333333344444444555555556666666677777777→
→ 10 mm
< 1m >< 1m >

a) 60 MPa b) 60 MPa

c) 400 MPa d) 400 MPa

17. Two bars one made of aluminum and the other of steel are glued together along its length as shown in
the fig. The combined structure is subjected bending loads. Strain gauges S1 and S 2 are placed on
the top and 30 mm from the top, respectively, of the aluminium layer. The gauge s3 is placed at the
bottom of the steel layer, The tensile steel strains measured in gauges S1 and S2 are 4 x 10-06 and
1 x 10-06 respectively. If the beam depth is 240 mm, the maginitude of the strain gauge reading in
gauge S3 at the base of the steel layer is.

(a) 4 x 10-6 S1 30 mm
(b) 1 x 10-6 S2 240 mm
(c) 10 x 10-6
(d) 20 x 10-6

S3

18. A steel plate is bent into a circular arc of radius 10 m. If the plate section be 120 mm wide and 20mm
thick, then the maximum bending stress is equal to (Take E = 2 x 105 N/mm2)

(a) 100 N/mm2 (b) 150 N/mm2 (c) 200 N/mm2 (d) 300 N/mm2

19. In the above problem the maximum bending moments due to that stress is equal to.

(a) 800 Nm (b) 1600 Nm (c) 2000 Nm (d) 2400 Nm.

20. For a square sectional beam bent as shown in the figure, the exaggerated view of the deformed cross

section is.

Beam

(a) (b)

(c) (d)

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STRENGTH OF MATERIAL

21. AT-Section is used in inverted position shown in fig. over a span of 400 mm. Due to application of
forces shown in fig. the longitudinal strain gauge at ‘F’ registers a compressive strain of 1.5 mm /mm
E = 200 GPa, INA = 2176 mm4. The maximum value of ‘P’ in Newton is.

4 mm P 2P 3P
6 mm
F
NA

↔100 100100
100
50 mm

(a) 0.188 MN (b) 0.288 MN (c) 0.348 MN (d) 0.678 MN

22. A wooden beam of width B and depth D is strengthened by two steel plates of thickness ‘t’ and depth
‘D’ on the both sides of the beam. The allowable stress in wood is σ and modular ratio of steel to

wood is ‘m’. The allowable bending moment is. IES-93(CE)

(a) σ D2 (B + mt) t → B →t ↓

6

(b) σ D2 (B +2 mt)

6 D

(c) σ D2 (2B + mt)

6

(d) σ D2 (2B + 2mt) ↓

6

23. Atapered cantilever beam of constant thickness is loaded as shown in the sketch below. The bending

stress will be 111111111111111222222222222222333333333333333444444444444444 (GATE-1988) (ME)

6PL <> x P
fd2
1111122222< L >

a) maximum near the fixed end 1

2 b) maximum at x = L

c) maximum at x = L 2

3 d) uniform through the length

24. A horizontal beam with square cross-section is simply supported with sides of the square horizontal
and vertical and carries a distributed loading that produces maximum bending stress σ in the beam.

When the beam is placed with one of the diagonals horizontal themaximum bending stress.

(a) (1/ 2) σ. (b) σ

(c)√2σ. (d) 2 σ.

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STRENGTH OF MATERIAL

25. The ratio of curvature of the 3 loose beams (b x d) placed one over the other to an integral beam

(b x 3d) is,

(a) 9 (b) 1 / 9 (c) 1.0 (d) 27 (e) 1 / 27

26. A 6 metre long supported wooden beam of rectangular section 10 cm x 20 cm deep is strengthened

by mild steel plates 0.5 cm x 10 cm wide at the top and bottom fibre over the entire length. Find the

minimum supportable uniformly distributed load considering failures in steel and wood due to flexure.

Weakening of wood due to screws and weakening of the steel plates due to drilled holes may be

ignored. (GATE-1987) (ME)

→ 100mm → →→ 5mm

Premissible tensile stress for steel = 156.8 N/mm2 11112222333344445555666677778888999900001111222233334444555566667777888899990000

Permissible tensile stress for wood = 14.89 N/mm2 200mm
Young’s modulus of mild steel = 1.96 x 105 N/mm2

Young’s modulus of wood = 0.117 x 105 N/mm2 111222333444555666777888999000111222333444555666777888999000→→
Statement For Linked Data Questions 27 and 28
5mm

A simply supported beam of span length 6m and 75 mm diameter carries a udl of 1.5 kN/m

27. What is the maximum value of bending moment ? (GATE-2006) (ME)

a) 9 KN-m b) 13.5 KN-m c) 81 KN-m d) 125 KN-m

28. What is the maximum value of bending stress ?

a) 162.98 MPa b) 325.95 MPa c) 625.95 MP d) 651.90 MPa

29. A cantilever beam of length L is subjected to a moment M at the free end. The moment of inertia of the

beam cross section about the neutral axis is l and the Young’s modulus is E. The magnitude of the

maximum deflection is (GATE-12)

a) ML2 b) ML2
2EI EI

c) 2ML2 d) 4ML2
EI EI

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STRENGTH OF MATERIAL

30. Consider a cantilever beam, having negligible mass and uniform flexural rigidity, with length 0.01m.

The frequency of vibration of the beam, with a 0.5 kg mass attached at the free tip, is 100Hz. The

flexural lrigidity (in N.m2) of the beam is ___________ (GATE-14-Set 1)

31. The flexural rigidity (EI) of a cantillever beam is assumed to be constant over the length of the beam

shown in figure. If a load P and bending moment PL/2 are applied at the free end of the beam then the

value of the slope at the free end is (GATE-14-Set 2)

a) 1PL2 PL2
2EI b) EI

c) 3PL2 5PL2
2EI d)

2EI

32. Acantilever beam of length,L, with uniform cross-section and flexural rigidity, EI, is loaded uniformly
by a vertical load, w per unit length. The maximum vertical deffection of the beam is given by
(GATE-14-Set 2)

wL4 wL4 wL4 d) wL4
a) 8EI b) c) 4EI 24EI

16EI

33. Consider a simply supported beam of length, 50h, with a rectangular cross-section of depth,h, and

width,2h, The beam carries a vertical point toad, P, at its mid -point. The ratio of the maximum shear

stress to the maximum bending stresss in the beam is (GATE-14-Set 3)

(a) 0.02 (b) 0.10 (c) 0.05 (d) 0.01

34. A force P is applied at a distance x from the end of the beam as shown in the figure. Whta would be the

value of x so that the displacement at ‘A’is equal to zero? (GATE-14-Set 3)

a) 0.5 L

b) 0.25 L

c) 0.33 L

d) 0.66 L

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STRENGTH OF MATERIAL

35. A frame is subjected to a load P as shown in the figure. The frame has a constant flexural rigidity EI.
The effect of axial load is neglected. The deflection at point A due to the applied laod P is
(GATE-14-Set 2)

a) 1PL3
3EI
2PL3

b) 3EI
PL4

c) EI
4PL3

d) 3EI

36. A cantilever beam with flexural rigidity of 200 Nm2 is loaded as shown in the figure. The deflection

(in mm) at the tip of the beam is _____________ (GATE-15-Set 1)

37. A cantilever beam with square with square cross-section of 6 mm side is subjected to a load of 2 kN

normal to the top surface as shown in the figure. The young’s modulus of elasticity of the material of

the beam is 210 GPa. The magnitude of slope (in radian) at Q (20 mm from the fixed end)

is __________ (GATE-15-Set 2)

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STRENGTH OF MATERIAL

38. A simply-supported beam of length 3L is subjected to the loading shown in the figure.

It is given that P = 1 N, L = 1 m and Young’s modulus E = 200 Gpa. The cross section is a square

with dimensions 100 mm. The bending stress (in Pa) at the point Alocated at the top surface of the

beam at a distance of 1.5 L from the left end is (Indicate compressive stress by a negative sign and

tensile stress by a positive sign) (GATE-16-SET-1)

39. A 1 m x 10 mm x 10 mm cantilever beam is subjected to a uniformly distributed load per unit length

of 100 N/m as shown in the figure below. The normal stress (in Mpa) due to bending at point P is

_____________ (GATE-PI-16)

**********************

THEORY OF SIMPLE BENDING (ANS.)

1-c, 2-d, 3-d, 4-b, 5-d, 6-a, 7-a, 8-b, 9-b, 10-d, 11-c, 12-c, 13-d, 14-d, 15-c, 16-a, 17-d, 18-c, 19-b,
20-a, 21-b, 22-b, 23-d, 24-c, 25-a, 26-sol., 27-b, 28-b., 29 - a, 30 - 0.06573Nm2, 31 - b, 32 - a,
33 -d, 34 - c, 35 - d, 36 - 0.264mm, 37 - 0.1587 radian, 38-0, 39-300

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STRENGTH OF MATERIAL

IV UNIT : THEORY OF SIMPLE BENDING

PRACTICE QUESTIONS :

1. The beams, one having square cross section and another circular cross-section, are subjected to the same amount

of bending moment. If the cross sectional area as well as the material of both the beam are the same then

(GATE-2003) (ME)

a) Maximum bending stress developed in both the beams is the same

b) The circular beam experience more bending stress than the square one

c) The square beam experience more bending stress than the circular one

d) As the material is same both beams will experience same deformation

2. If the area of cross section of a circular section beam is made four times, keeping the loads , length support

conditions and material of the beam unchanged , then the quantities (List - I) will change through different factors

(List-II). Match the list - I with the List - II and select the correct answer using the code given below the lists:

[IES-2005]

List-I List-II

A Maximum bending moment 1. 8

B. Deflection 2. 1

C. Bending stress 3. 1/8

D. Section Modulus 4. 1/16

Codes:-

ABCD

(a) 3 1 2 4

(b) 2 4 3 1

(c) 3 4 2 1

(d) 2 1 3 4

3. A steel wire of diameter 2 mm is would on a rigid drum of diameter 2 m. If the Young’s modulus of the steel is

200 GPa, the maximum stress (in MPa) in the steel wier is (GATE-2007) (PI)
d) 400
a) 50 b) 100 c) 200

Statement for linked answer questoins 4 and 5

A mass less beam has a loading pattern as shown in figure. The beam is of rectangular cross section with a width

of 30 mm and height of 100 mm. (GATE-2005) (ME)

→ 3kN/m

→
→
RA 2m 2m

4. The maximum bending moment occurs at

a) Location B b) 2675 mm to the right of A

c) 2500 mm to the right of A d) 3225 mm to the right of A

5. The maximum magnitude of bending stress (in MPa) is given by

a) 60.0 b) 67.5 c) 200.0 d) 225.0

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STRENGTH OF MATERIAL

Statement For Linked Data Questions 6 and 7

A simply supported beam of span length 6m and 75 mm diameter carries a udl of 1.5 kN/m

6. What is the maximum value of bending moment ? (GATE-2006) (ME)

a) 9 KN-m

b) 13.5 KN-m

c) 81 KN-m

d) 125 KN-m

7. What is the maximum value of bending stress ?

a) 162.98 MPa

b) 325.95 MPa

c) 625.95 MP

d) 651.90 MPa

8. Consider the following statements :

For each component in a flitched beam under the action of a transverse load,

1. The radius of curvature will be difference.

2. The radius of curvature will be same.

3. The maximum bending stress will be same.

4. The maximum bending stress will dependent upon the modulus elasticity of the material of component.

Which of these statements are correct. ICS-00 (CE)

(a) 1 and 3 (b) 1 and 4

(c) 2 and 3 (d) 2 and 4

9. A square beam 4 cm x 4cm is cut into 4 equal parts as shown in fig. The ratio of strength of integral beam to that of

loose beam is.

(a) 1 (b) 2

(c) 4 (d) 16

10. A timber beam of rectangular section 100mm x 50mm is simply supported at the ends, has 30mm x 10mm steel strip

securely fixed to the top surface as shown in the given fig. The centroid of the “Equivalent timber beam” in this case

from the top surface. 11111112222222333333344444445555555 30 x 10 Steel strip
(a) is 5 mm

(b) is 30mm 30mm
(c) is 15mm Timber 100 mm
(d) can not be predicted.

50 mm

11. Aflat ribbon of steel 3 mm wide and 0.5 mm thick is wound round a cylinder 500 mm in diameter. The maximum stress

in the steel ribbon is ‘N/mm2’ is.

(a) 100 (b) 200

(c) 400 (d) None.

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STRENGTH OF MATERIAL

12. The modular ratio of the materials used in the flitched beam is 10 and the ratio of the allowable stresses is also 10.
Four different sections of the beam are shown in the fig. The material shown hatched has larger modulus of
elasticity and allowable stress than the rest.

(a) 111111111111222222222222333333333333 d 111111111111222222222222333333333333 d
t
t (b) 111222333444555666777888999000111222333

d d

1111222233334444555566667777888899990000111122223333

d d

(c) (d)

111122223333444455556666777788889999000011112222 d+2t d+2t

111222333444555666777888999000111222

Which one of the following statements is true for the beam under consideration ?

(a) All the given sections would support the same magnitude of load.

(b) Sections II,III and IV would support equal loads which is more than what section I would support.

(c) Sections I and II would support equal loads which is more than what section III and IV would support.

(d) Section II would support greatest load.

13. A freely supported beam of length 6m is subjected to a U.D.L of 3 kN/m over the entire span.the size of the beam

is 50 cm x 100 cm. The maximum bending stress developed at the top fibre at the support is.

(a) 0.6 N/m2 (b) 0.16 N/m2

(c) 0 (d) 1.6 N/m2

14. A cantilever beam of length 2m having dimensions of cross section is 40mm x 60mm. This beam failed by applying

a force of 5kN at the free end. The bending stress at the failure is given by.

(a) 420 N/mm2 (b) 110 N/mm2

(c) 200 N/mm2 (d) 100 N/mm2

15. The cross-section of a beam is shown in fig. I Its Ixx is equal to 3 x 106 mm4, it is subjected to a load as shown in

fig.-II. The maximum tensile stress in the beam would be.

y1 = 70 mm↓ 0.3kN ↓↓ 0.3kN
↓
3m ↓ 3 m↓3 m 4m 3m ↓

y2 = 70 mm (II)

↓ (b) 21 MN/m2
(d) 21 N/m2
(I)

(a) indeterminable as data is insufficient.

(c) 21 kN/m2

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STRENGTH OF MATERIAL

16. Consider the following statements about flitched beams : (ICS-98(CE))

1. A flitched beam has a composite section made of two or more materials joined together in such a manner that they

behave as a unit piece and each material bends to the same radius of curvature.

2. The total moment of resistance of a flitched beam is equal to the sum of the moments of resistances of individual

sections.

3. Flitched beams are used when a beam of one material, if used alone, would require quite a large cross-sectional

area.

Of these statements :

(a) 1,2 & 3 correct (b) 1 & 2 correct

(c) 1 & 3 correct (d) 2 & 3 correct

17. A beam is said to be uniform strength when.

(a) Maximum shear stress is same every where. (b) Maximum bending stress is same every where.

(c) Both of the above (d) None.

18. A rectangular section beam subjected to a bending moment M varying along its length is required to develop same

maximum bending stress at any cross - section. if the depth of the section is constant, then its width will vary as.

(a) M (b) √M (c) M2 (d) 1 / M

19. The ratio of strength of ‘n’ loose beams (b x d) placed one over the other to the strength of one integral beam

(b x nd) is, (b) n (c) 1 / n (d) 1 / n2
(a) n2

20. A cantilever of span ‘l’ and of uniform depth is of uniform strength. It is subjected to a point load ‘w’ at free end. If

the max. fibre stress is not to exceed ‘f’, the width of the cantilever at the fixed end is.

3wl 3wl 6wl 6wl
(a) 2fd2 (b) √2fd2 (c) √fd2 (d) fd2

21. A beam made of steel is subjected to pure bending Yielding of the material in the beam will take place when the

maximum bending stress is equal to. (ICS-99 (CE))

(a) two times the yield stress of steel.
(b) √2 times the yield stress of steel.

(c) half the yield stress of steel

(d) the yield stress of steel

22. A portion of a beam between two sections is said to be in pure bending when there is .

(a) Constant B.M and S.F

(b) Constant S.F and zero B.M

(c) Constant B.M and zero S.F

(d) None of the above.

23. Curvature of beam is equal.

(a) EI./M (b) M / EI (c) ME / I (d) MI / E

24. Which of the following statements is correct.

1 : A square section is more economical in bending than the circular section ofor same strength.

2 : The modulus of the square section is less than that of circular section of same area of cross-section.

(a) 1 and 2 (b) 1 only (c) 2 only (d) none.

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STRENGTH OF MATERIAL

25. A T- section is used as a simply supported beam with uniform loading. The maximum bending stresses for a given

load will occur at the . ↓ (IES-93 (CE))

(a) top of the section y2 A
(b) C.G of the section ↓
(c) mid - point of the depth of section.
(d) bottom of the section y1
↓

26. A beam cross - section is used in two different orientations as shown in the fig. given below :

→→ b/2→ ↓
b →↓

b/2 b

↓↓

Bending moments applied to the beam in both cases are same.The maximum bending stresses induced. in cases (A)

and (B) are related as. IES -97 (ME)

(a) σA = σB (b) σA = 2σB.
(c) σA = (σB / 2) (d) σA = (σB / 4)

27. A beam has rectangular section 100 mm x 200 mm. If it is subjected to a maximum B.M. of 4 x 107 N.mm, then the

maximum bending stress developed would be. ICS-00 (CE)

(a) 30 N/mm2 (b) 60 N/mm2

(c) 90 N/mm2 (d)120 N/mm2

28. The flexural stresses at top and bottom of a ‘T-section’ of 30 cm depth are 50 and 150 ‘N/mm2’ is. Distance of N.A.

from top is

(a) 7.5 (b) 15 (c) 22.5 (d) none.

29.. A cantilever beam has the square cross section of 10mm x 10mm.It carries a transverse load of 10N. Considering

only the bottom of the beam, the correct representation of the longitudinal of the bending stress is . (G-ME 2005)

10 kN
↓

1m 1m

(a) (b) (c) (d)

60 MPa 400 MPa

400 MPa 60 MPa

30. Section Modulus for a hollow rectangular section of outside dimensions is B.D and inside dimensions are b,d.

Then the section modulus is given by. (c) (BD3 - bd3) / 6 D (d) (BD3 - bd3) / 12 D
(a) (BD2 - bd3) / 32 D (b) (BD3 - bd3) / 16 D

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STRENGTH OF MATERIAL

31. Three smooth wooden battens are placed one above the other and are subjected to bending moments M as shown
in the given fig. The bending stress distribution is given by.

M M↓
D

↓ ↓↓

↓↓ (c) ↓ ↓
(a) (b)
(d)

DD DD

↓↓ ↓↓

32. Consider the following statements for a beam based on theory of bending:

1. Strain developed in any fibre is directly proportional to the distance of fibre from neutral surface.

2. For flexural loading and linearly elastic action the neutral axis pass through the centroid of cross - section.

3. The assumption of the plane cross - sections remaining plane will not hold good during inelastic action.

4. Instances in which the neutral axis does not pass through the centroid of a cross - section include a homogenous

symmetrical beam (with respect to neutral axis) and subjected to inelastic action)

Which of the statements given above are correct?

(a) 1 , 2 , 3 and 4 (b) 1 , 2 and 4

(c) 3 and 4 (d) 1 and 2

33. For a beam of uniform strength keeping its depth constant, the width will vary in proportion to (M is bending

moment). (b) √M (c) M2 (d) none of these
(a) M

34. A rectangular cross - sectioned beam is made of one steel plate sandwiched between two aluminium plates of

double the thickness of the steel plate. The ratio of the normal stress in the fibres of the steel to that in the

aluminium plates at the same distance from the CG is (Es > Eat)
(a) equal to one

(b) more than one

(c) less than one

(d) uncertain to define with the data given

35. The ratio of the maximum bending stress in the flange to that in the web of an I - section at a section on a beam is

always.

(a) less than one (b) equal to one

(c) more than one (d) no exact relation as above

36. he maximum bending moment due to a moving load on a fixed ended beam occurs.

(a) at a support (b) always at the midspan

(c) under the load only (d) none of the above

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STRENGTH OF MATERIAL

37 The bending stress on a beam section is zero at

(a) centroid of the section (b) top fibre

(c) bottom fibre (d) depends on the moment of inertia.

33 An increase in temperature on the top fibre of a simply supported beam will cause.

(a) upward direction (b) downward direction

(c) no deflection (d) upward or downward deflection

39. The shear flow in a section can be defined as

(a) total shear force (b) total shear stress at point

(c) direction of the shear stress (d) none of the above

40. A cantilever of constant depth carries a udl on the whole span. To make the maximum stress at all section the same,

the width of the section at a distance x from the free end should be proportional to .

(a) x (b) √x (c) x2 (d) x3

41. The width of a beam of uniform strength having a constant depth d length L, simply supported at the ends with a

central load W is.

a) 2WL 3WL 2f L 3 fL2
3fd2 b) 2fd2 c) 3Wd4 d) 3Wd

*****************************

“We must accept finite disappointment,
but we must never lose infinite hope.”

IV UNIT : THEORY OF SIMPLE BENDING PRACTICE QUESTIONS :
1-b, 2-b, 3-c, 4-c, 5-b, 6-b, 7-b, 8-b, 9-b, 10-d, 11-b, 12-d, 13-c, 14-a, 15-b, 16-d, 17-b, 18-a, 19-c, 20-d, 21-d, 22-c, 23-b, 24-b,
25-d, 26-b, 27-b, 28-a, 29-a, 30-c, 31-c, 32-d, 33-a, 34-b, 35-c, 36-a, 37-a, 38-a, 39-b, 40-c, 41-b.

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STRENGTH OF MATERIAL

5 SHEAR STRESS DISTRIBUTION IN BEAMS

1. The transverse shear stress acting in a beam of rectangular cross-section, subjected to a transverse

shear load,is. (G-08)

(a) variable with maximum at the bottom of the beam.

(b) variable with maximum at the top of the beam.

(c) uniform.

(d) variable with maximum at the neutral axis.

2. Which of the following diagrams correct the distribution of transverse shear stress across the depth

of a rectangular beam subjected to varying bending moment along its length ? (GATE-1990) (ME)

a) b) v) d)

111111111111111111222222222222222222333333333333333333444444444444444444555555555555555555666666666666666666777777777777777777888888888888888888999999999999999999000000000000000000111111111111111111 11111111111111111112222222222222222222333333333333333333344444444444444444445555555555555555555666666666666666666677777777777777777778888888888888888888999999999999999999900000000000000000001111111111111111111222222222222222222233333333333333333334444444444444444444555555555555555555566666666666666666667777777777777777777888888888888888888899999999999999999990000000000000000000 111111111111111111122222222222222222223333333333333333333444444444444444444455555555555555555556666666666666666666777777777777777777788888888888888888889999999999999999999 111111111111111111222222222222222222333333333333333333444444444444444444555555555555555555666666666666666666777777777777777777888888888888888888999999999999999999

3. The ratio of maximum shear stress developed in beam of rectangular section to that of the average
shear stress is 1.5
(G-94)

4. For a given shear force across a symmertrical I - section, the intensity of shear is maximum at
(GATE-1994) (CE)

a) Extreme fibres
b) Centroid of the section
c) At the junction of the flange and the web on the web
d) At the junction of the flange and the web out on the flange

5. I - Section of a beam is formed by gluing wooden planks as shown in the figure below. If this beam

transmits a constant vertical shear force of 3000 N, the glue at any of the four joints will be subjected

to a shear force (in kN) permeter length) of (GATE-2006) (CE)

a) 3.0 111111111122222222223333333333444444444455555555556666666666777777777788888888881111111111111111999999999922222222222222220000000000111111111133333333333333332222222222333333333344444444445555555555666666666677777777778888888888 50 mm
b) 4.0 200 mm
c) 8.0 50 mm
d) 10.7

50 mm 75 mm

200 mm

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STRENGTH OF MATERIAL

6. If a beam of rectangular cross-section is subjected to a vertical shear force V, the shear force carried

by the upper one-third of the cross-section is (GATE-2005) (CE)

a) zero b) 7V
c) 8V
27
27 V

d)

3

7. The shear stress at the neutral axis in a beam of traingular section with a base of 40 mm and height

20 mm, subjected to a shear force of 3 kN is (GATE-2007) (CE)

a) 3 MPa b) 6 MPa c) 10 MPa d) 20 MPa

----------------- xxxxxxxxxxxx ----------------

“Misery is almost always the result of thinking”

SHEAR STRESS DISTRIBUTION IN BEAMS (ANS)
1-d, 2-b, 3-1.5, 4-b, 5-b, 6-c, 7-c.

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STRENGTH OF MATERIAL

UNIT - V : SHEAR STRESS DISTRIBUTION IN BEAMS

PRACRICE QUESTIONS

1. The ratio of maximum shear stress developed in beam of rectangular section to that of the average

shear stress is.......... (G-94)

2. The ratio of average shear stress to the maximum shear stress in a beam with a square cross-section is: (G-98)

(a) 1 (b) 2 / 3 (c) 3 / 2 (d) 2

3. A sudden change in shear stress distribution diagram at the cross-section shows.

(a) Sudden change in depth of the section (b) Sudden change in width of the beam

(c) Both of the above (d) none.

4. The shape of the shear stress distribution diagram for a rectangular beam is.

(a) Parabola (b) Rectangle

(c) Hyperbola (d) none.

5. The ratio of maximum shear stress developed in a beam of rectangular section to that of the average shear stress

is .

(a) 1 (b) 2 / 3 (c) 3 / 2 (d) 2.

6. The shear stress at any section at a distance ‘y’ from N.A is given by.

(a) q = F Ay / Ib (b) q = F A/ y Ib (c) Ib/F Ay (d) none

7. A rectangular beam of 100mm wide is subjected to maximum shear force of 50kN, the corresponding maximum shear

stress being 3N/mm2. The depth of the beam is equal to.

(a) 200 mm (b) 250 mm (c) 300 mm (d) none.

8. The ratio of average shear stress to the maximum shear stress in a beam with a square cross-section is.

(a) 1 (b) 2 / 3 (c) 3 / 2 (d) 2

9. For a rectangular cross-section, the maximum shear stress is % more than average shearstress.

(a) 25% (b) 50% (c) 75% (d) 90%

10. In a triangular section of depth ‘h’ meters, the maximum shear stress occurs at.

(a) 2 h/3 from base (b) h/2 from base (c) at the N.A of the triangle (d) 2 h/4 from base

11. For a circular section, qmax is equal to times that of average shear stress.

(a) 3 / 2 (b) 4 / 3 (c) 9 / 8 (d) 2 / 3

12. The shear stress distribution shown in fig.-I represents a beam with cross-section.

(a) (b)

(c) (d)

UNIT - V : SHEAR STRESS DISTRIBUTION IN BEAMS [ PRACRICE QUESTIONS ]ANS. :
1- 1.5, 2-b, 3-b, 4-a, 5-c, 6-a, 7-b, 8-b, 9-b, 10-b, 11-b, 12-sol.

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STRENGTH OF MATERIAL

6 TORSION

1. Maximum shear stress developed on the surface of a solid circular shaft under pure torsion is

240 MPa. If the shaft diameter is doubled than the maximum shear stress developed corresponding to

the same torque will be (GATE 2003 ME)

a) 120 MPa b) 60 MPa c) 30 MPa d) 15 MPa

2. A steel shaftA‘ of diameter ”d‘ and length ‘l’is subjected to a torque ”T‘. Another shaft B‘ made of

aluminium of the same diameter ”d‘ and length 0.5 l is also subjected to the same torque ”T‘.

The shear modulus of steel is 2.5 times the shear modulus of aluminium. The shear stress in the steel

shaft is 100 MPa. The shear stress in the aluminium shaft,n MPa, is. (G-2K)

(a) 40 (b) 50 (c) 100 (d) 250

3. A shaft is simultaneously subject to a torque ‘T’ and a bending moment ‘M’, the ratio of maximum

shear stress to bending stress is.

(a) M / T (b) T / M (c) 2M / T (d) T / 2M.

4. A circular shaft of length ‘L’, a uniform cross - sectional area ‘A’and modulus of rigidity ‘G’is

subjected to a twisting moment that produces maximum shear stress ‘’ in the shaft. Strain energy in

the shaft is given by the expression τ2A L/k G where k is equal to. (ICS-CE-02)

(a) 2 (b) 4 (c) 8 (d) 16

5. A steel shaft ‘A’of diameter ‘d’and length ‘l’is subjected to a torque ‘T’. Another shaft ‘B’made of

aluminium of the same diameter ‘d’ and length 0.5 l is also subjected to the same torque ‘T’. The shear

modulus of steel is 2.5times the shear modulus of aluminium. The shear stress in the steel shaft is

100 MPa. The shear stress in the aluminium shaft, in MPa, is. (G-ME-2000)

(a) 40 (b) 50 (c) 100 (d) 250

6. Two hollow shafts of the same material have the same length and outside diameter, shaft 1 has inner

diameter equal to one third of the outer diameter and shaft 2 has internal diameter equal to half of the

outer diameter if both the shafts are subjected to the same torque, the ratio of their twists θ1/ θ2 will

be equal to. [IES-98]

(a) 16/81 (b) 8/27 (c) 19/27 (d) 243/256

7. A solid shaft of diameter 100mm, length 1000mm is subjected to a twisting moment ‘T’, the maximum

shear stress developed in the shaft is 60N/mm2.Ahole of 50mm diameter is now drilled throughout the

length of the shaft. To develop a maximum shear stress of 60 N/mm2 in the hollow shaft, the torque

‘T’ must be reduced by [IES-98]

a) T/4 b) T/8 c) T/12 d) T/16

8. A round shaft of diameter ‘d’and length ‘l’fixed at both ends ‘A’and ‘B’, is subjected to a twisting

momen ‘T’at ‘C’at a distance of l/4 from A (see fig.) The torsional stresses in the parts AC and CB

will be. A B [IES-97]
(a) equal C
(b) in the ratio of 1 : 3

(c) in the ratio of 3 : 1 l /4
(d) indeterminate. l

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STRENGTH OF MATERIAL

9. A circular rod of diameter d and length 3d is subjected to a compressive force F acting at the top
point as shown below. Calculate the stress at the bottom-most support poiont A
(GATE 1993 ME)
3d F
-----------------------

>>

-----------------------
- - - - - - - - - - - - - - - - - - - - - - d- - - - - - - - - - - <- - - - - - - -

A

10. Asolid shaft of diameter, d and length L is fixed at both the ends.Atorque, T0 is applied at a distance,

L/4 from the left end as shown in the figure given below. (GATE 2009 ME)

11111111111111112222222222222222-3333333333333333- - - T 0----------- - - - - - - - - - - - - - - - - - - - - - - - - - - - 11111111111111112222222222222222
--
-

L/4 3L/4

The maximum shear stress in the shaft is

16T b) 12T 8T d) 4T
a) πd3 πd3 c) πd3 πd3

11. The compound shaft shown is built-in at the two ends. It is subjected to a twisting moment T at the
middle. What is the ratio of the reaction torques T1 and T2 at the ends ? (GATE 1993 ME)

T 1111111111111111111111222222222222222222222233333333333333333333334444444444444444444444 T←← 2d 111111111111111111111122222222222222222222223333333333333333333333
1 d ←←

T
2

ll

1 1 1 1
a) b) c) d) 2

16 8 4

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STRENGTH OF MATERIAL

12. A torque to 10 N-m is transmitted through a stepped shaft as shown in figure. The torisonal stiffnesses

of individual sections of lengths MN, NO and OP are 20 N-m/rad. 30 N-m/rad and 60N-m/rad

respectively. The angular deflection between the ends M and P of the shaft is (GATE 2004 ME)

OP

>T = 10 N -m M N

>T

-----------------------------------------

a) 0.5 rad b) 1.0 rad c) 5.0 rad d) 10.0 rad

13. Two shaftsAand B are made of the same material. The diameter of shaft B is twice that of shaftA. the

ratio of power can be transmitted by shaft A to that of shaft B is (GATE 1994 ME)

a) 1 b) 1 c) 1 d) 1
16 8 42

14. A shaft subjected to torsion experiences a pure shear stress on the surface. The maximum principal

stress on the surface which is at 45o to the axis will have a value. (G-2003)

(a) τ cos 450 (b) 2 τ cos 450
(c) τ cos2 450 (d) 2τ sin 450 cos 450.

15. A torque T is applied at the free end of a stepped rod of circular cross-sections as shown in the

figure. The shear modulus of the material of the rod is G. The expression for d to produce an

angular twist θ at the free and is (GATE 2011 ME)

1 1

a) 32TL 4 b) 18TL 4
πθG πθG

16TL 1 1
4
c) πθG d) 2TL 4
πθG

1111111111111111111-2222222222222222222-<3333333333333333333--------L--2-d----->--<---d-L--/2----->-<>
<>

16. A hollow circular shaft has an outer diameter of 100 mm and a wall thickness of 25 mm. The allowable
shear stress in the shaft is 125 MPa. The maximum torque the shaft can transmit is (GATE 2009 CE)
a) 46 kN-m
b) 24.5 kN-m
c) 23 kN-m
d) 11.5 kN-m

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STRENGTH OF MATERIAL

17. The maximum and minimum shear stresses is a hollow circular of outer diameter 20 mm and thickness

2 mm, subjected to a torque of 92.7 N-m will be (GATE 2007 CE)

a) 59 MPa and 47.2 MPa b) 100 MPa and 80 MPa

c) 118 MPa and 160 MPa d) 200 MPa and 160 Mpa

18. A circular solid shaft of span L = 5 m fixed at one end and free at the other end. A twisting moment

T = 100 kN-m is applied at the free end. The torsional rigidity GJ per unit angular twist is 50000 kN-

m2/rad. Following statements are made for this shaft. (GATE 2004 CE)

1. The maximum rotation is 0.01 rad 2. The torional strain energy is 1kN-m

With reference to the above statements, which of the following applies ?

a) Both statements are true b) Statement 1 is ture but 2 is false

c) Statement 2 is true but 1 is fasle d) Both the statements are false

Questions 19, 20 are based on the following data :

19. The shaft shown in fig. rotates at 200 rpm with 30 kW and 15 kW power is taken off at A and B

respectively. The

actual power applied at C is 45 kW. The rigidity modulus of the material is 8.5 x 104 MPa.

20. The maximum shear stress developed in the shaft in MPa is.

(a) 58.36 (b) 25.94 (c) 15.14 (d) none.

21. The angle of twist in degrees of the gear ‘A’relative to C is.

(a) 2.36 ← 4m ← 2m
(b) 5.87

(c) 7.14 50mm dia 75mm

(d) 1.56

AB C

22. The maximum shear stress in a solid shaft of circular-section having diameter d subjected to a torque
T is τ. If the torque is increased by four times and the diameter of the shaft is increased by two times,

the maximum shear stress in the shaft will be (GATE 2008 CE)
d) τ/4
a) 2τ b) τ c) τ/2

23. Two solid circular shafts of radit R1 and R2 are subjected to same torque. The maximum shear stresses
developed in the two shafts are τ1and τ 2. If R1/R2 = 2, then τ1/τ2. is__________ (GATE-14)

24. Consider a stepped shaft subjected to a twisting moment applied at B as shown in the figure.Assume

shear modulus, G = 77 GPa. The angle of twist at C (in degree) is________________ .

(GATE-15-Set 1)

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STRENGTH OF MATERIAL

25. A hollow shaft (d0=2di where d and d are the quter and inner diameters respectively) needs to
0 i

transmit 20kW power at 3000 RPM. If the maximum permissible shear stress is 30 MPa, d is
0

(a) 11.29mm (b) 22.58mm (c) 33.87mm (d) 45.16mm

(GATE-15-Set 2)

26. A hollow shaft of 1 m length is designed to transmit a power of 30 kW at 700 rpm. The maximum
permissible angle of twist in the shaft is 10. The inner diameter of the shaft is 0.7 imes the quter
diameter. The modulus of rigidity is 80 GPa. The outside diameter (in mm) okf the shaft is______

(GATE-15-Set 2)

*******************

TORSION (ANS)
1-c, 2-c, 3-d, 4-b, 5-c, 6-d, 7-d, 8-c, 9-sol, 10-b, 11-a, 12-b, 13-, 14-d, 15-b, 16-c, 17-b, 18-b,
19-sol, 20-a, 21-c, 22-c, 23 sol, 24 - 0.2368 degree, 25 - b, 26 - 44.5212 mm.

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STRENGTH OF MATERIAL

UNIT - VI (A)
TORSION

PRACTICE QUESTIONS :
1. For a circular shaft of diameter d subjected to torque T, the maximum value of the shear stress is

(GATE 2006 ME)

64T 32T c) 16T 8T

a) πd3 b) πd3 πd3 d) πd3

2. Torsional rigidity is. (1984)

(a) also known as flexural rigidity.

(b) the product of modulus of elasticity and moment of inertia.

(c) the torque which develops unit twist per unit length.

(d) the product of shear modulus and moment of inertia.

3. A 3-meter long steel cylindrical shaft is rigidly held at its two ends. A pulley is mounted on the shaft at 1 meter

from one end; the shaft is twisted by applying torque on the pulley. The maximum shearing stresses developed

in 1 m and 2 m lengths are respectively _______________ τ1 and τ2. The ratio τ2:τ1 is . [IES-97]

(a) 1/ 2 (b) 1 (c) 2 (d) 4

4. Maximum shear stress in a solid shaft of diameter D and length L twisted through an angle θ is τ . A hollow haft

of same material and length having outside and inside diameters of D and D/2 respectively is also twisted through

the same angle of twist θ . The value of maximum shear in the hollow shaft will be. [IES-97]

(a) 16 / 15τ (b) 8 / 7 τ (c) 4 / 3 τ (d) τ

5. For a power transmission shaft transmitting power P at N rpm, its diameter is proportional to: [IES-05]

( )P 1/3 ( )(b) P 1/2
N
(a)

N

( )P 2/3 ( )P

(c) (d)

N N

6. The two shafta AB and BC, of equal length and diameters d and 2d , are made of the same material. They are joined

at B through a shaft coupling, while the ends A and C are built-in (cantilevered). A twisting moment T is applied to

the coupling. If TA and TC represent the twisting moments at the ends A and C, respectively, then (GATE 2005 ME)
2d 111111111222222222333333333 C
A 1111111222222233333334444444 ←←d
←←B

LL

a) TC = TA b) TC = 8TA
c) TC = 16TA d) TA = 16TC

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STRENGTH OF MATERIAL

7. A solid circular shaft of 60 mm diameter transmits a torque of 1600 N-m. The value of maximum shear developed is

(GATE 2004 ME)

a) 37.72 MPa b) 47.22 MPa

c) 57.72 MPa d) 67.72 MPa

8. A solid circular shaft of diameter d and length L is fixedatone end and free at the other end. A torque T is applied

at the free end. The shear modulus of the material is G. The angle of twist at the free end is (GATE 2010 CE)

a) 16TL b) 32TL c) 64TL d) 128TL

πd4G πd4G πd4G πd4G

9. For the cantilever brracket, PQRS, loaded as shown in the adjoining figure (PQ = RS = L, and QR = 2 L), which of the

following statements is FALSE (GATE 2011CE)

Fixed→ S R
>
<
2L
P

Q
L
W
a) The portion RS has a constant twisting moment with a volue of 2WL.
>
b) The portion QR has a varying twisting moment with a maximum value of WL.<

c) The portion PQ has varying bending moment with a maximum value of WL.↓
↓
(d) The portion PQ has no twisting moment.

10. A shaft diameter 10 mm transmits 100 W of power at an angular speed of 800 rad / s. The maximum shear

stress (in MPa) developed in the shaft is π (GATE 2008 PI)

a) 2 b) 4 c) 8 d) 16

11.. The diameter of shaft A is twice the diameter of shaft B and both are made of the same material. Assuming both the

shafts to rotate at the same speed. the maximum power transmitted by B is. [IES-01]

(a) the same as that of A (b) half of A

(c) 1/8 th of A (d) 1/4 th of A

12. The outside diameter of a hollow shaft is twice that of its inside diameter. The torque carrying capacity of this shaft

is Mt1. A solid shaft of the same material has the diameter equal of the hollow shaft. the solid shaft can carry a

torque of Mt2. The ratio Mt1 / Mt2 is. [IES-01]

(a) 15 / 16 (b) 3 / 4

(c) 1 / 2 (d) 1 / 16

13 A shaft is subjected to torsion as shown. [IES-02]

TT
Which of the following figures represents the shear stress on the element LMNOPQRS?

↓ ↓ ↓

(c)
↓
↓
↓
(a) ↓ (b) ↓ ↓ (d) ↓

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STRENGTH OF MATERIAL

14. A shaft is subjected to simultaneous action of a torque T, bending moment M and an axial thrust F. Which one of

the following statements is correct for this situation? [IES-04]

(a) One extreme end of the vertical diameter fibre is subjected to maximum compressive stress only.

(b) The opposite extreme end of the vertical diametral fibre is subjected to tensile/compressive stress only

(c) Every point of the surface of the shaft is subjected to maximum shear stress only.

(d) Axial longitudinal fibre of the shaft is subjected to compressive stress only.

15. Consider the following statements: [IES-08]

Maximum shear stress induced in a power transmitting shaft is.

1.directly proportional to torque being transmitted. 2.inversely proportional to the cube of its diameter.

3.directly proportional to its polar moment of inertia.

Which of the statements given above are correct?

(a) 1 , 2 and 3 (b) 1 and 3 only. (c) 2 and 3 only (d) 1 and 2 only.

16. The diameter of a solid shaft is D. The inside and outside diameters of a hollow shaft of same material and length are
D /√3 and 2D /√3respectively. What is the ratio of the weight of the hollow shaft to that of the solid shaft?

(a) 1 : 1 (b) 1 : √3 (c) 1 : 2 [IES-07]
(d) 1 : 3

17. What is the maximum torque transmitted by a hollow shaft of external radius R and internal radius r ? [IES-06]

(a) π (R3 - r3)fs (b) π (R4 - r4)fs (c) π (R4 - r4)fs (d) π (R4 - r4)fs

16 2R 8R 32

18. A hollow shaft of the same cross-section area and material as that of a solid shaft,transmits : [IES-05]

(a) Same torque (b) Lesser torque

(c) More torque (d) Cannot be predicted without more data.

19. A solid circular rod AB of diameter D and length L is fixed at both ends. A torque T is applied at a section X such that

AX = 1/4 and BX = 3L/4. What is the maximum shear stress developed in the rod? [IES-04]

(a) 16T/πD3 (b) 12T/πD3 (c) 8T/πD3 (d) 4T/πD3

20. A solid shaft transmits a torque T. The allowable shearing stress is τ . What is the diameter of the shaft? [IES-08]

16T 32T (c) 3 16T (d) 3 T
τ τ
(a) 3 πτ (b) 3 πτ

21. The ratio of torque carrying capacity solid shaft to that of a hollow shaft is given by ; [IES-08]

(a) (1 - K4) (b) (1 - K4 )-1

(c) K4 (d) 1 / K4

22. While transmitting the same power by a shaft, it its speed is doubled, what should be its new diameter if the

maximum shear stress induced in the shaft remains same? [IES-06]

(a) 1/2 of the original diameter. (b) 1/√2 of the original diameter

(c) √2 of the original diameter (d) 1 of the original diameter.
(2)1/3

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STRENGTH OF MATERIAL

23. A chalk stick is twisted to failed to failure by applying opposite torques T at the two ends. take a square element

ABCD with two sides parallel to the longitudinal axis of the stick. (G-2002)

(a) Show the free body diagram with principal stresses.

(b) Find out the principal stresses 1 and 2 and the principal planes.

(c) Show the plane on which failure / fracture will take place.

24. Twisting moment is a moment applied in the plane of cross section acting.

(a) along longitudinal axis. (b) about longitudinal axis (c) about neutral axis. (d) none of the above.

25. Plor moment of inertia is . (1984)

(a) the M.I. about an axis in the plane of cross-section (b) the product of inertia

(c) about the axis of member (d) none of the above.

26. In a shaft subjected to pure twist, the shear stress at anysection is maximum at . (1987)

(a) centre of section (b) mid radius (c) surface (d) 3/4 radius from centre.

27. When a shaft is subjected to pure twisting, the type of stress developed in the shaft is. (1987)

(a) bending stress (b)axial stress (c) shear stress (d) normal stress.

28. For a rectangular shaft subjected to torsion, the maximum shear stress occurs at

(a) A B C
(b) B A
(c) C D

(d) D

29. For transmitting same power, all properties remaining same, between a solid and hollow shaft.

(a) solid shaft is economical (b) hollow shaft is economical

(c) both cost same (d) none .

30. A circular shaft is subjected to torsion. The shear stress in the cross section . (1984)

(a) varies parabolically with the maximum stress occuring at the centre.

(b) uniform over the section.

(c) varies linearly with the radius with the maximum at the circumference and zero at the center.

(d) None of the above.

31. A shaft subjected to torsion experience a pure shear stress t on the surface. The maximum principal stress on the

surface which is at 450 to the axis will have a value. (G- ME-03)

(a) t cos 450 (b) 2 t cos 450

(c) t cos2 450 (d) 2t sin 450 cos 450

32. A chalk piece firmly fixed at one end and applied with a clock-wise torque at the other end. The crack formed will

be.

(a) clockwise with axis 450 (b) anti clockwise with axis 450

(c) vertical (d) not able to assess.

33. A long shaft of diameter d is subjected to twisting moment T at its ends. The maximum normal stress acting at its

cross-section is equal to. (G-CE-06)
(d) 64 T /πd3
(a) zero (b) 16 T /πd3 (c) 32 T /πd3

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STRENGTH OF MATERIAL

34. A fixed beam AB is carrying two concentrated loads 2 kNand 1 KN at distances of 0.5 m on either side of center

of span in a direction normal to the axis of the beam as shown below.

1 kN

0.5

A 0.5 B
2 kN

The twisting moment on the beam ‘AB’ in kN.m is.

(a) 0.5 (b) 1 (c) 1.5 (d) zero.

35. Two shafts are of equal length and same material. The external dia of both the shafts are same and internal dia. of

hollow shaft is half of that of external dia. The ratio of the strength of solid to hollow shaft is.

(a) 8 / 7 (b) 17 / 16 (c) 16 / 15 (d) 15 / 16

36. A shaft 10 cm dia. and 2 m long is subjected to a torque of 800 Kg.m. The maximum shear stress in Kg/sq.cm.,

eveloped is.

(a) 308.6 (b) 407.4 (c) 128.6 (d) 216.2

37. The two shafts AB and BC, of equal length and diameters and 2d, are made of the same material. They are joined

at B through a shaft coupling, while the ends A and C are built-in (cantilevered), A twisting moment T is applied

to the coupling. If TA and TC represent the twisting moments at the ends A and C respectively, then.

CA B D

→ → →

L L L
L →→ →

(a) TA = TC (b) TA = 8TC (c) TA = 16TC (d) TC = 16TA

38. A shaft turning 150 rpm is subject to a torque of 150 Kg.m. The H.P transmitted by the shaft is .

(a) 100 π (b) 10 π (c) 10 (d) 5 π

39. Two shats ‘A’ and ‘B’ are transmitting same power shaft ‘A’ is subjected to a torque of 10 kN-m and 100 rpm . If

shaft ‘B’ is subjected to 150 rpm, the torque to which it should be subjected in ‘kN-m’ is .

(a) 6.67 (b) 7.5 (c) 15 (d) None.

40. A composite bar of circular cross section I subjected to an axial force P and a torque T, as shown in fig. At the

interface ‘AA’ A
(a) A both normal and shear stress are continuous

(b) either normal of shear stress is continuous ←
(c) shear strain is continous . ← P
(d) normal strain is continous . A ← T

************ E1,π1 E1,π1
UNIT- VI : TORSION [PRACTICE QUESTIONS]ANS. :

1-c, 2-c, 3-a, 4-d, 5-a, 6-c, 7-a, 8-b, 9-b, 10-a, 11-c, 12-a, 13-d, 14-d, 15-d, 16-a, 17-b, 18-d, 19-b, 20-a, 21-a, 22-d, 23-sol, 24-

b, 25-c, 26-c, 27-c, 28-d, 29-b, 30-c, 31-d, 32-a, 33-a, 34-a 35-c, 36-b, 37-c, 38-b, 39-a, 40-a.

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STRENGTH OF MATERIAL

UNIT - VI (B)
SPRINGS

1. The figure shows arrangements of springs. They have stiff nesses K1 and K2 as marked. Which of the following

2K K (GATE 1987 ME)
arragements offers a stiffness = 1 2

K1 + 2K2

(a) (b) (c) (d)

K1 K1 K1 K2 K2 K2 K1 K2
K1

K1

K2 K1
K2

2. The deflection of a spring with 20 active turns under a load of 1000 N is 10 mm. the spring is made into two pieces each

of 10 active coils and placed in parallel under the same load. The deflection of this system is (GATE 1995 ME)

a) 20 mm b) 10 mm c) 5 mm d) 2.5 mm

3. A weighing machine consists of a 2 kg pan resting on a spring in this condition, with the pan resting on the spring,

the length of the spring is 200 mm. When a mass of 20 kg is placed on the pan, the length of the spring becomes 100

mm. For the spring, the undeformed length L and the spring constant k (stiffness) are (GATE 2005 ME)

a) L = 220 mm, K = 1862 N/m b) L = 210 mm, K = 1960 N/m

c) L = 200 mm, K = 19860 N/m d) L = 200 mm, K = 2156 N/m

4. A compression spring is made of music wire of 20 mm diameter having a shear strength and shear modulus of 800 MPa

and 80 GPa respectively. The mean coil diameter is 20 mm, free length is 40 mm and the number of active coils is 10. If

the mean coil diameter is reduced to 10 mm. the stiffness of the spring is approximately (GATE 2008 ME)

a) Decreased by 8 times b) Decreased by 2 times

c) Increased by 2 times d) Increased by 8 times

********

SPRINGS (Ans.) : 1- b,2- d, 3 - b, 4 - d

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STRENGTH OF MATERIAL

UNIT - VII

DEFLECTIONS AND SLOPE

1.. A caantilever beam is shown in the figure. The moment to be applied at free end for zero vertical

deflection at that point is 9KN (GATE 1998 CE)
a) 9 kN.m clock wise
b) 9 kN.m anti-clock wise 111111111122222222223333333333
c) 12 kN.m clock wise

d) 12 kN.m anti-clock wise < 2m >

2. A cantilever type gate hinged at Q is shown in the figure. P and R are the centres of gravity of the

cantilever part and the counterweight respectively. The mass of the cantilever part is 75 kg. The mass

of the counterweight, for static balance is (GATE 2008 ME)

a) 75 kg . . .R Q P
b) 150 kg
c) 225 kg
d) 300 kg

< >< >

0.5 m 2.0 m

3. The ratio of maximum deflection of a beam simplysupported at its ends with .
(i) a central load of ‘W’ and
(ii) a u.d.l over entire length of total ‘W’ is.
(a) 8 / 5
(b) 1 / 4
(c) 3 / 2
(d) 5 / 8

4. Consider the beamAB shown in the figure below. Part AC of the beam is rigid while Part CB has the

flexural rigidity EI. Identify the correct conbination of deflection at end B and bending moment at end

A. respectively. (GATE 2006 CE)

a) P L3 , 2PL 111111222222A C P
B
3EI
LL
b) P L3 , PL

3EI

c) 8 P L3 , 2PL

3EI

d) 8 P L3 , PL

3EI

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STRENGTH OF MATERIAL

5. For the structure shown below, t11111111h22222222e33333333<verticalE3dILeflection> at point A is given by (GATE 2000 CE)
a)
PL3 A
81 EI
<>
b) 2PL3 EI 3L

81 EI

c) Zero P EI >
3L
d) PL3 L

72 EI <

6. A frame of two arms of equal length L is shown in the adjacent figure. The flexural rigidity of each arm

of the fram is EI. The vertical deflection at the point of application of load P is (GATE 2009 ME)

a) PL3 b) 2PL3 111112222233333 P

3EI 3EI L
PL3
d) 4PL3 111222333444555
c)
3EI
EI

L

7. A cantilever beam of span l subjected to a uniformly distributed load ‘w’ per unit length resting on a
rigid prop at the tip of cantilever. The magnitude of the reaction of at the prop is : (GATE 1994 CE)

1 2 3 4

a) W1 b) W1 c) W1 d) W1

8 8 8 8

8. The bending moment (in kNm units) at the mid-span location X in the beam with overhangs shown

below is equal (GATE 2001 CE)

20kN 10 kN

X spring
suppport

< 1m >< 1m >< 1m >< 1m >

a) 0 b) 10 c) - 15 d) - 20

9. The stepped cantilever is subjected to moments, M as shown in the figure below. The vertical

deflection at the free end (neglecting the self weight) is (GATE 2008 CE)

a) ML2 ML2 111111111111222222222222333333333333 2EI EI
L/2
8EL b) M
L/2
c) ML2 4EL

2EL d) zero

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STRENGTH OF MATERIAL

10. Two identical cantilevers are loaded as shown in the respective fig. If slope at the free end of the
cantilever in fig. E is θ,the slope at free end of the cantilever in fig. F will be

(a) 1θ (b) 1 θ L LP
3 2

2 E F
3 M = PL/2
(c) θ (d) θ

11. A cantilever beam of cross section (b x h) 20 x40 mm and of length 233 mm is supporting a load 1 kN

at the free end. A simply supported beam made of same material and having a cross section (b x h)

15 x 30 mm with indentical load and deflection at centre will have a span of (GATE 2005 PI)

a) 100 b) 220 c) 400 d) 530

12. A “H” Shaped frame of uniform flexural rigidity EI is loaded as shown in the figure. The relative

outward displacement between points K and O is R R (GATE 2003 CE)
M
a) R L h2 I
h
EI

b) R L2 h JN

EI

c) R L h2 h

3EI

R L2 h KO
L
d)

3EI

Statement for Linked Answer Ques 13 & 14
Atriangular-shaped is shown in the figure. Thee young’s modulus of the material of the beam is E.A
concentrated load P applied at the free and of the beam (GATE-ME-11)

13. The area moment of inertia of inertia about the neutral axis of a cross-section at a distance x measured

from the free end is (GATE 2011 ME)

bxt3 bxt3 bxt3 xt3

a) 61 b) 121 c) 241 d) 12

14. The maximum deflection of the beam is (GATE 2011 ME)

24Pl3 12Pl3 8Pl3 6Pl3

a) Ebt3 b) Ebt3 c) Ebt3 d) Ebt3

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STRENGTH OF MATERIAL

Statement for Linked Answer Questions 15 and 16 (GATE 2007 CE) (2 x2 = 4)
A two span continuous beam having equal spans each of length L is subjected to a uniformly
distributed load w per unit length. The beam has constant flexural rigidity.
15. The reaction at the middle support is

a) w L 5wL c) 5wL 5wL

b) 4 d) 8

2

16. The bending moment at the middle support is

a) wL2 b) wL2 c) wL2 d) wL2

4 8 12 16

17 The two cantileversAand B shown in the given fig. have the same uniform cross-section and the same

material. Free end deflection of cantilever ‘A’is d . The value of mid-span deflection of the cantilever

‘B’ is P

A L
L

P

B

L L

(a) 1 δ (b) 2δ 53PLP2L2
2 3 2EIEI

(c) δ. (d) 2δ

18. The flexural rigidity (EI) of a cantilever beam is assumed to be constant over the length of beam as

shown in the figure. If a load P and bending moment PL/2 are applied at the free end of the beam then

the value of the slope at the free end is (GATE-ME-14-SET-2)

1 PL2 b)
(d)
(a)

2 EI

(c)

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STRENGTH OF MATERIAL

19. A cantilever beam of length L, with uniform cross-section and flexural rigidity EI, is loaded uniformly
by a vertical load w per unit length. The maximum vertical deflection of the beam is given by
(GATE-ME-14-SET-2)

wL4

(a) (b) (c) (d)

8EI

20. A force P is applied at a distance x from the end of the beam as shown in the figure. What would

be the value of x so that the displacement at ‘A’is equal to zero ? (GATE-ME-14-SET-3)

(a) 0.5 L

(b) 0.25 L

(c) 0.33 L

(d) 0.66 L

21. Aframe is subjected to a load P as shown in the figure. The frame has a constant flexural rigidity EI.
The effect of axial load is neglected. The deflection at pointA due to the applied load P is
(GATE-ME-14-Set-4)

(a) 1 PL3 24PwwLPLL3L4443
13224E64EIEEEIIII
3 EI

(b)

(c)

(d)

22. A cantilever beam with flexural rigidity of 200 N.m2 is loaded as shown in the figure. The deflection

(in mm) at the tip of the beam is ________. (GATE-15-Set-1)

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STRENGTH OF MATERIAL

23. A cantilever beam with square cross section of 6 mm side is subjected to a load of 2 kN normal to
the top surface as show in figure. The Young’s modulus of elasticity of the material of the beam is
210 Gpa. The magnitude of slope (in radian) at Q (20 mm from the fixed end) is _______
(GATE-15-SET-2)

24. A simply supported beam of length 2L is subjected to a moment M at the mid-point x = 0 as shown

in the figure. The deflection in the domain 0<x<L is given by

W = −Mx (L − x)(x + c)
12EIL

Where E is the Young’s modulus, I is the area moment of inertia and c is a constant (to be

determined) (GATE-16-SET-2)

The slope at the center x = 0 is

(a) ML/(2EI)

(b) ML/(3EI)

(c) ML/(6EI)

(d) ML/(12EI)

25. Abeam of length L is carrying a uniformly distributed load w per unit length. The flexural rigidity of the

beam is EI. The reaction at the simple support at the right end is (GATE-16-SET-3)

wL 3wL (c) wL wL
(a) 2 (b) 4 (d)

8 8

**************************

7. DEFLECTIONS AND SLOPE (Ans.) CLASS WORK
1-c, 2-d, 3-a, 4-a, 5-c, 6-b, 7-c, 8-c, 9-c, 10-d, 11-c, 12-a, 13-b, 14-d, 15-c, 16-a, 17-c, 18-b, 19-a,
20-c, 21-d, 22-0.26mm, 23-0.158 Radians, 24-c, 25-b.

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STRENGTH OF MATERIAL

UNIT - VII

DEFLECTION AND SLOPE

PRACTICE QUESTIONS

1. .In a real beam, at an end, the boundary condition of zero slope and zero vertical displacement exists. In the

corresponding conjugate beam, the boundary conditions as this and will be : (GATE 1992 CE)

a) Shear forces = 0 and bending momenet = 0 b) Slope = 0 and vertical displacement = 0

c) Slope = 0 and bending moment = 0 d) Shear force = 0 and vertical displacement = 0

Questions 2 And 3 Are Based On Common Data

A steel beam of breath 120 mm and height 750 mm is loaded as shown in figur. Assume modulus of elasticity as

200 GPa. 120 kN/m (GATE 2004 ME) (2x2 = 4M)

↓↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓

15 m

2. The beam is subjected to a maximum bending moment of

a) 3375 KN-m b) 4750 KN-m c) 6750 KN-m d) 8750 KN-m

3. The value of maximum deflection of the beam is

a) 93.75 mm b) 83.75 mm c) 73.75 mm d) 63.75 mm

4. A cantilever beam of span, ‘L’ in subjected to a downward load of 800 kN uniformly distributed over its length and

a concentrated upward load P at its free end. For vertical displacement to be zero at the free end, the value of P is :

(GATE 1992 CE)

a) 300 kN b) 500 kN c) 800 kN d) 1,000 kN

5. A propped cantilever beam of span L, is loaded with uniformly distributed load of intensity w/unit length, all

through the span, Bending moment at the fixed end is (GATE 1995 CE)

a) WL2 WL2 WL2 d) WL2
8 b) c) 24

2 12

6. A two span beam with an internal hinge is shown below

1111111122222222 a Hinge d

c 1122334455
b

111222333444555

Conjugate beam corresponding to this beam is

(GATE 2000 CE11)2233445566

a b c d b) a b c d

a) 11112222d11111111333322222222444455556666 11112222333344445555

111122223333444455556666 111222333444555666 d

b a bc 11112222333344445555

c) a c d) 1111222233334444111222333444555

1122334455

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STRENGTH OF MATERIAL

7. A cantilever beam of span ‘L’ is loaded with a concentrated load ‘P’at the free end. Deflection of the beam at the free

end is (GATE 1997 CE)

PL3 5PL3 PL3 PL3
a) b) c) d)

48 EI 384 EI 3 EI 6 EI

8. A cantilever beam XY of length 2 m and cross-sectional dimentions 25 mmx25mm is fixed at x and is subjected to a

moment of 100 N-m and an unknown force P at the free end Y as shown in the figure. The young’s modulus of the

material of the beam is 200 GP . If the deflection of the free end Y is zero, then the value of P (in N) is
a
(GATE 2008 PI)

a) 67 111111122222223333333 P
b) 75 Y
c) 133
d) 150

9. A beam having uniform cross-section carries as uniformly distributed load of intensity q per unit length over its
entire span, and its mid-span deflection is δ .The value of mid-span deflection of the same beam when the same load

is distributed with intensity varying from 2q unit length at one end to zero at the other end is

(a) 1/3 δ (b) 1/2 δ. (c) 2/3 δ (d) δ.

10. A cantilever beam carries a load W uniformly distributed over its entire length. If the same load is placed at the free

end of the same cantilever, then the ratio of maximum deflection in the first case to that in the second case will be.

(a) 3/8 (b) 8/3 (c) 5/8 (d) 8/5

11. The given fig shows a cantilever of span ‘L’ subjected to a concentrated load ‘P’ and a moment ‘M’ at the free end.

Deflection at the free end is given by. P

M

(a) PL2 + ML2 (b) ML2 + PL3 (c) ML2 + PL3 (d) ML2 + PL3
2EI 2EI 3EI 3EI 2EI
3EI 2EI 48EI

12. For a cantilever beam of length ‘L’ flexural rigidity EI and loaded at its free end by a concentrated load W, match

List-I with List - II and select the correct answer.

List - I List - II

A. Maximum bending moment- 1. WL
B. Strain energy 2. WL2/ 2EI
C. Maximum slope 3. WL3/ 3EI.
D. Maximum deflection 4. W2L3/ 6EI

Codes :

AB CD

(a) 1 4 3 2

(b) 1 4 2 3

(c) 4 2 1 3

(d) 4 3 1 2

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STRENGTH OF MATERIAL

13. A simply supported beam with width ‘b’ and depth ‘d’ carries a central load W and undergoes deflection δ at the

centre. If the width and depth and interchanged, the deflection at the centre of the beam would attain the value.

(a) d δ (b) d δ 3 3/2
b b
(c) d δ (d) d
b b

14. A simply supported beam of constant flexural rigidity and length 2L carries a concentrated load ‘P’ at its mid-span
and the deflection under the load is . If a cantilever beam of the same flexural rigidity and length ‘L’ is subjected to
a load ‘P’ at its free end.

(a) d δ (b) δ (c) 12δ (d) 4δ
b

15. A cantilever beam of rectangular cross-section is subjected to a load W at its free end. If the depth of the beam

is doubled and the load is halved, the deflection of the free end as compared to original deflection will be.

(a) half (b) one - eighth (c) one - sixteenth (d) double

16. A cantilever of length L, moment of inertia l, Young’s modulus E carries a concentrated load W at the middle its

length. The slope of cantilever at the free end is. (b) WL2/4EI
(a) WL2/2EI (d) WL2/16EI
(c) WL2/8EI

17. A cantilever beam of length l is subjected to a concentrated load P at a distanceof 1/3 from the free end. What is

the deflection of the free end of the beam ? (EI is the flexural rigidity)

2PI3 3PI3 14PI3 15PI3
(a) (b) (c) (d) 81EI

81EI 81EI 81EI

18 Match list - I with list - II and select the correct answer using the code given below the lists :

List - I List - II

(Long colum) (Critical load)
1. π2 EI /4l2.
A. Both ends hinged - 2. 4π2 EI /4l2.
3. 2π2 EI /4l2.
B. One end fixed and other end free - 4. π2 EI /l2.

C. Both ends fixed -

D. One end fixed and other end hinged-

Codes : AB C D
21 4 3
(a) 41 2 3
(b) 23 4 1
(c) 43 2 1
(d)

19. Maximum deflection of a cantilever beam of length / carrying uniformly distributed load w per unit length will be.

(a) wI4/ (EI) (b) wI4/ (4EI)

(c) wI4/ (8EI) (d) wI4/ (384EI)

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STRENGTH OF MATERIAL

20. Match list - I with list - II and select the correct answer using the code given below the lists:

List - I List - II

(Formula/ theorem/ method) (Deals with topic)

A. Clapeyron’s theorem - 1. Deflection of beam

B. Maculay’s method - 2. Eccentrically loaded column

C. Perry’s formula - 3. Riveted joints

4. Continuous beam.

Codes :

ABC

(a) 3 2 1

(b) 4 1 2

(c) 4 1 3

(d) 2 4 3

21. If the deflection at the free end of a uniformly loaded cantilever beam is 18 mm and the slope of the deflection curve

at the free end is 0.02 radius then the length of the beam is.

(a) 0.8 m (b) 1.0 m (c) 1.2 m (d) 1.5 m

22. A cantilever carries a total u.d.l of ‘W’ over its entire length and a force ‘W’ acts at its free end upwards. The net

deflection a the free end is.

(a) zero (b) 5WL3 / 24EI upward

(c) 5WL3 / 24EI downward (d) none

23. A S.S.beam of width ‘b’ and depth ‘d’ is subject to a point load ‘w’ at its centre causing deflection ‘y’ at that

point. If the beam be turned such that width ‘d’ and depth ‘b’ and be subjected to same load at same point, the

central deflection would be. (1987)

(a) (d / b)y (b) (dy / b)2

(c) (d / b)2 y (d) (b / d)y

24. A S.S.beam of circular cross section with dia ‘d’ adn length ‘I’ carries concentrated load ‘W’ at centre of beam.

The strength of beam is proportional to.

(a) d2 / l (b) d3 / l (c) l / d2 (d) l / d3

25. A uniform beam of length ‘L’ is simply supported and symmetrically supported on a span ‘l’ The ratio ‘L’/ l so that

the upward deflection at each end equals the downward deflection at mid span due to central point load of ‘W’ is.

(a) 5 / 3 (b) 3 / 5 (c) 2 / 5 (d) 5 / 2

26. A cantilever beam of span ‘2M’ is subjected to a u.d.l of 10 kN/m. If EI = 2 x 1010 kN.mm2.The maximum deflection in

mm is.

(a) 1000 (b) 1 (c) 0.5 (d) none.

27. Consider the beam AB shown in the fig. below. Part AC of the beam is rigid while part CB has the flexural rigidity EL.

Identify the correct combination of deflection at end B and bending moment at end A, respectively. (G-CE 06)

(a) PL3 / 3 EI,2PL P
(b) PL3 / 3 EI, PL

(c) 8PL3 / 3 EI,2PL A C B
(d) PL3 / 3 EI,2PL L L

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STRENGTH OF MATERIAL

28. A S.S.beam carries a point load at mid span. The relation between maximum deflection ‘y’ and maximum bending

stress ‘f’ is given by,

(a) y = f l2 / 6 Ed (b) y = f l / 6 Ed (c) y = f l2 / 3 E (d) y = 6 Ed / f l2

29. A simply support laterally loaded beam was found to deflect more than a specified value. Which of the following

measures will reduce the deflection ? (G-03)

(a) Increase the area moment of inertia

(b) Increase the span of the beam.

(c) Select a different material having lesser modulus of elasticity.

(d) Magnitude of the load to be increased.

30. A cantilever beam of cross-section (b x h) 20 x 40 mm and of length 233 mm is supporting a load of 1 kN at the free

end. A simply supported beam made of same material and having a cross section (b x h) 15 x 30 mm with identical

load and deflection at the centre will have a span of. (G-prod-05)

(a) 100 (b) 220 (c) 400 (d) 530

31. A lean elastic beam of given flexural rigidity, EI, is loaded by a single force F as shown in fig. How many boundary

conditions are necessary to determine the deflected centre line of the beam?

(a) 5 (c) 3 Undeflected ↓F
(b) 4 (d) 2 position

32. Two identical cantilever beams are supported as shown, with their ree ends in contact through a rigid roll . After

the load P is applied, the free ends will have . (G-05)

↓P

(a) equal deflections but not equal slopes (b) equal slopes but not equal deflections

(c) equal slopes as well as equal deflections (d) neither equal slopes nor equal deflections .

33. A conceentrated load P acts at the middle of a simply supported beam of span 1 and flexural rigidity EI. Another

simply supported beam of identical material, geometry and span is being acted upon by an equivalent distributed

load (w =p/l ) spread over the entire span. The central deflections in both the beams are identical.............(T/F).

(G-94)
34. For the composite beam shown in fig, flexural rigidities EI of AB and DC are equal to105N- cm2, add EI of BD is

2 x 105N-cm2. Using moment - area theorem, determine the location and magnitude of maximum deflection

between B and C. (G-98)

A 100N/cm111111111222222222333333333444444444555555555666666666777777777888888888999999999000000000111111111222222222333333333444444444555555555↑666666666777777777D600N C

B

→ 6 cm →→ 7cm 3 cm

→→ →

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STRENGTH OF MATERIAL

35. A simply supported laterally loaded beam was found to delfect more than a specified value. Which of the following

measures will reduce deflection ? (GATE 2003 ME)

a) Increase the area moment of inertia

b) Increase the span of the beam

c) Select a different material having lesser modulus of elasticity

d) Magnitude of the load to be increased

Linked Answer Questions (GATE 2010 CE)

Statement of Linked Answer Questions 36 and 37 :

In the cantilever beam PQR shown in figure below, the segment PQ has flexural EI and the segment QR has infinite

flexural rigidity W
R
P 111111222222333333 EI Q Rigid
B

LL

36. The deflection and slope of the beam at ‘Q’ are resepctively

5WL2 3WL2 WL3 WL2
a) and b) and
6EI 2EI 6EI 2EI

WL3 WL2 WL3 3WL2
c) and d) and
2EI EI 3EI 2EI

37. The deflection of the beam at ‘R’ is

8WL3 5WL3 c) 7WL3 8WL3
a) b) 3EI d)

EI 6EI 6EI

38. A cantiliever beam AB of length L rigidly fixed at end A, is uniformly loaded with intensity q (downwards) over two-
third of its length from the free end B as shown in the figure. The modulus of elasticity is E and the moment of inertia
about the horizontal axis is I. The angle of rotation at the free and end under the applied load is (GATE 2011 PI)

a) 7qL3 13qL3 A 1111111111111111122222222222222222333333333333333334444444444444444455555555555555555 1111222233334444555566667777888899990000q11112222333344445555666677778888999900001111
48 EI b) L/3 B

11qL3 72 EI 2L/3
c)
d) qL3
60 EI 24 EI

39. In the propped cantilever beam carrying a uniformly distribued load of ‘w’ N/m shown in the following figure, the

reaction at the support ‘B’ is (GATE 2002 CE)

5 3 111111111111122222222222223333333333333 w kN/m
a) wL b) wL →→→→→→→→→→

8 8

1 3 < L B
c) wL d) wL
>
2 4

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STRENGTH OF MATERIAL

40. A simply supported beam carrying a concentrated load W at mid-span deflects d1 under the load. If the same beam

carries the load W such that it is distributed uniformly over entire length and undergoes a deflection at the mid

span.The ratio d1 : d2 is

(a) 3 / 8 (b) 8 / 5 (c) 5 / 8 (d) 8 / 3

41. I = 375 x 10-6 m4 ll
P
l = 0.5 m
E = 200 GPa

2l l

Determine the stiffness of the beam shown in the above fig.

(a) 12 x 1010N/m (b) 10 x 1010 N/m (c) 4 x 1010 N/m (d) 8 x 1010 N/m

42. A cantilever beam of rectangular cross-section carries a point load ‘W’ at its free end. If the depth of the beam is

doubled, and the load halved.The deflection at the free end, as compared to its original value, will be (1987)

(a) 1/2 (b) 1/4 (c) 1/8 (d) none.

43. A cantilever beam ‘AB’ fixed at ‘A’ and carrying a load ‘W’ at the free end ‘B’ is found to deflect by ‘y’ at the mid

point of ‘AB’. The deflection of ‘B’ due to a load W/2 at the mid point will be.

(a) 2 y (b) y (c) y / 2 (d) y / 4

*********************

“People may doubt what you say,
bit they will believe what you do”

UNIT - VII : DEFLECTIONS AND SLOPE & SLOPES AND DEFLECTIONS
PRACTICE QUESTIONS
1-a, 2-a, 3-a, 4-a, 5-a, 6-a, 7-c, 8-b, 9-d, 10-a, 11-b, 12-b, 13-b, 14-c, 15-c, 16-c, 17-c, 18-b, 19-c, 20-b, 21-c, 22-b, 23-c, 24-b,
25-a, 26-b, 27-a, 28-a, 29-a, 30-c, 31-c, 32-c, 33-F, 34-sol., 35-a, 36-a, 37-c, 38-b, 39-b,40-b, 41-c, 42-d, 43-c.

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STRENGTH OF MATERIAL

CHAPTER - VIII
COLUMNS AND STRUTS

One Mark Questions : (GATE 1998 ME)
1. If the length of a column is doubled, the critical load becomes

a) 1 of the original value 1

2 b) of the original value

c) 1 of the original value 4
1
8
d) of the original value

16

2. For the case of a slender column of length 1, and flexural rigidity EI built-in at its base and free at the top,

the Euler’s critical buckling load is (GATE 1994 ME)

a) 4π2EI 2π2EI π2EI π2EI

l2 b) c) l2 d) 4l2

l2

3. A pin-ended column of length L, modulus of elasticity E and second moment of the cross-sectional area

I is loaded centrally by a compressive load P. The critical bucking load (Pcr) is given by
(GATE 2006 ME)

EI π2 EI

a) Pcr = π2L2 b) Pcr = 3L2

c) P= πEI π2 EI
cr
L2 d) Pcr = L2

4. The kern area (core) of a solid circular section column of diameter, D, is a concentric circle of diameter,

d, equal to (GATE 1992 CE)

a) D/8 b) D/6

c) D/4 d) D/2

5. The axial load carrying capacity of a long column of given material. Cross-sectional area,Aand length

L, is governed by to (GATE 1992 CE)

a) Strength of its material only

b) Its flexural figidity only

c) Its slenderness ratio only

d) Both flexural rigidity and slenderness ratio

6. When a column is fixed at both ends corresponding Euler’s critical load is (GATE 1994 CE)

a) n2EI b) 2n2EI

L2 L2

c) 3n2EI d) 4n2EI

L2 L2

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STRENGTH OF MATERIAL

7. Four column of the same material and having identical geometric properties are supported in different

ways as shown below (GATE 2000 CE) 11112222 111222333444555 11112222

112233445566 11112222 1122334455 11223344
I II III IV

It is required to order these four beams in the increasing order of their respective first bucking loads. The

correct order is given by

a) I, II, II ,IV b) III, IV, II, I

c) II, I, IV, III d) I, II, IV, III

8. A long structural column (length = L) with both ends hingedis acted upon by an axial compressive load,

P. The differential equation governing the bending of column is given by :

Where y is the structural lateral deflection and EI is the flexural rigidity. The first critical load on column

responsible for its blucking is given by (GATE 2003 CE)

d2y
= - Py
EI dx2

a) π2EI √2π2 EI
b)
L2
L2
c) 2π2EI 4π2 EI

L2 d)

L2

9. The effective length of a column of length L fixed against rotation and translation at one end is
(GATE 2010 CE)

a) 0.5 L

b) 0.7 L

c) 1.414 L

d) 2 L

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STRENGTH OF MATERIAL

10. For a long slender column of uniform cross section, the ratio of critical buckling load for the case with

both ends clamped to the case with both ends hinged is (GATE - 12)

a) 1
b) 2
c) 4
d) 8

11. Consider a steel (Young’s modulus E = 200 GPa) column hinged on both sides. Its heights is 1.0 m
and corss-section is 10 mm x 20 mm. The lowest Euler critical bucking load (in N) is_____
(GATE - 15)

Two Marks Questions :

1. The rod PQ of length L and with flexural rigidity EI is hinged at both ends. Four what minimum force F

is expected to buckle ? 1111111111111111122222222222222222334455P66774885-990-00(--11-Q22334455F (GATE 2008 CE)

a) π2EI b) √2π2EI c) π2 EI d) π2 EI
2L2
L2 L2 √2L2

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STRENGTH OF MATERIAL

2. A column has a rectangular cross-section of 10 mm x 20 mm and a length of 1m. The slenderness r ratio

of the column is close to (GATE 2011 ME)

a) 200 b) 346 c) 477 d) 1000

3. Arigid rodAB of length Lis hinged atAand is mintained in its vertical position by two springs constants
K attached at end B. The system is under stable equilibrium under the action of laod P when P < Pcr The
system will be in unstable equilibrium when Pattains a value greater than : (GATE 1990 CE)

11111C P k 11111D22222

L RIGID
ROD
---------------
A11223344

a) kL b) k/L c) 2kL d) 4 kL

4. The maximum tensile stress at the section X-X shown in the figure below is (GATE 2005) (CE)

L/3 L/3 L/3

b
X

d/2

-------------------- P d

d/2

X
L/2 L/2

8P 6P 4P 2P

a) bd b) bd c) bd d) bd

5. The buckling load P = Pcr for the column AB in figure, as KT approaches infinity, becomes α π2EI

L2

111111111222222222333333333 P 111111111222222222
A

L Flexural rigidity. EI

Torsional spring
of stiffness KT

11223344B

Where α is equal to (GATE 2006 CE)

a) 0.25 b) 1.00 c) 2.05 d) 4.00

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STRENGTH OF MATERIAL

6. A steel column, pinned at both ends, has a bucking load of 200 kN. If the column is restrained against

lateral movement at its mid-height, its bucking load will be

(GATE 2007 CE)

a) 200 kN b) 283 kN c) 400 kN d) 800 kN

7. A rigid bar GH of length L is supported by a hinge and a spring of stiffness K as shown in the figure

below. The bucking load, Pcr for the bar will be (GATE 2008 CE)

P K 111
H

L

G
1122334455

a) 0.5 KL b) 0.8 KL c) 1.0 KL d) 1.2 KL

8. Cross-section of a column counsisting of two steel strips, each of thickness t and width b is shown in the

figure below. The crtical loads of the column with perfect bond and without bond between the strips are

P and P0 respectively. The ratio P/P0 is (GATE 2008 CE)

1111111222222233333334444444555555566666667777777888888899999990000000111111122222223333333 t
t

b

a) 2 b) 4 c) 6 d) 8

9. A short column of length L having cross-sectional area of 50 mm by 100 mm is pinned at the ends. The

proportional limit of the column is 250 MPa and modulus of elasticity is 200 GPa. The minimum length

of the column (in m) at which it will buckle elastically is (GATE 2011 PI)

a) 5.25 b) 2.25 c) 1.65 d) 1.15

**********

“What seems impossible one minute becomes, through faith,
possible the next”

8. COLUMNS AND STRUTS (CLASS WORK)
One Mark Questions : 1 - b, 2 - d, 3 - d, 4 - a, 5 - d, 6 - d, 7 - b, 8 - a, 9 - d, 10 - c, 11 - 3289.868.
Two Marks Questions : 1- b, 2- b, 3 - c, 4 - a, 5 - c, 6 - d, 7 - c, 8 - b, 9 - d

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STRENGTH OF MATERIAL

CHAPTER - 9
THIN CYLINDERS

One Mark Questions :

1. A cylindrical tank of radius r and wall thickness t has flat end closurs. The tank is subjected to an internal
pressure P. The longuitudinal (σx) and the circumferential (σθ) stress respectively are given by :
(GATE 1990 CE)

2. A thin cylindrical vessel of mean diameter C and of length ‘L’closed at both the ends is subjected to a

water pressure ‘P’. The value of hoop stress and longitudinal stress in the shell shall be respectively

(GATE 1991 CE)

a) PD . PD b) PD . PD c) PD . PD d) PD . PD
2t 4t 8t 8t 4t 8t t 2t

3. A thin walled cylindrical pressure vessel having a radius of 0.5 m and wall thickness of 25 mm is

subjected to an internal pressure of 700 kPa. The hoop stress developed is (GATE 2008 CE)

a) 14 MPa b) 1.4 MPa c) 0.14 MPa d) 0.014 MPa

4. A thin walled spherical shell is subject to an internal pressure, if the radius of the shell is increased by 1%

and the thickness is reduced by 1%, with the internal pressure remaining the same, the percentage

change in the circumferential (hoop) stress is (GATE-12)

a) 0 b) 1 c) 1.08 d) 2.02

5. A long thin walled cyclindrical shell, closed at both the ends, is subjected to an internal pressure. The

ratio of the hoop stress (circumferential stress) to longitudinal stress develped in the shell is

(GATE-2013)

(a) 0.5 (b) 1.0 (c) 20. (d) 4.0

6. A thin gas cycllinder with an internal radius of 100mm is subjet to an inernal pressure of 10 MPa.The

maxmim permissible working stress is restricted to 100 MPa. The minimum cyclinder wall thickness

(in mm) for safe design must be__________ (GATE-2013- Set 4)

7. A gas is stored in a cyclindrical tank of inner radius 7 m and wall thickness 50mm. The gauge pressure

of the gas is 2 MPa. The maxmim shear stress (in MPa) in the wall is (GATE-2015- Set 2)

(a) 35 (b) 70 (c) 140 (d) 280

8. A cyclinerical tank with closed ends is filled with compressed air at a pressure of 500 kPa. The inner

radius of the tank is 2 m, and it has wall rhickness of 10 mm.the magnitude of maximum in-plane shear

stress (in MPa) is___________ (GATE-2015- Set 3)

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