STM objective

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

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. For the piping system shown in fig, (EI)AB = 1011 N-cm2 and (EI)BC = 8 x 1011 N-cm2 . The
axial rigidity of DC, (EF)DC = 107 N. Determine the load on DC.
(G-98).

5375N CC

AB 

100 cm 200 cm 50cm



D

5. Consider the beam AB shown in the figure below. PartAC 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)

P

a) P L3 , 2PL b) P L3 , PL AC B

3EI 3EI LL

c) 8 P L3 , 2PL d) 8 P L3 , PL

3EI 3EI

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MECHANICAL ENGINEERING

6. For the structure shown below, the vertical deflection at point Ais given by (GATE 2000 CE)

a) PL3 < 3L >A <>

81 EI EI EI 3L

b) 2PL3 P >
EI
81 EI
L
c) Zero
< 3L
d) PL3

72 EI

7. A frame of two arms of equal length L is shown in the adjacent figure. The flexural rigidityof 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 L
P
3EI 3EI
PL3 L
d) 4PL3
c)
3EI
EI

8. 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

9. 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

10. 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 EI
2EI
8EL b)
M
c) ML2 4EL L/2 L/2

2EL d) zero

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STRENGTH OF MATERIAL

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

(a) 1 (b) 1 L LP
3 2

2 E F
3 M = PL/2
(c)  (d) 

12. 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

13. 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

14. 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) b) c) d) 12

61 121 241

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

24Pl3 12Pl3 8Pl3 6Pl3

a) b) c) d) Ebt3

Ebt3 Ebt3 Ebt3

Statement for Linked Answer Questions 16 and 17 (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.
16. The reaction at the middle support is

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

2 4 d)

8

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MECHANICAL ENGINEERING

17. The bending moment at the middle support is

a) wL2 wL2 c) wL2 d) wL2

4 b) 12 16

8

18 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) 1  (b) 2 A L
2 3 L

(c) . (d)  B P
L L

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

7. DEFLECTIONS AND SLOPE (Ans.) CLASS WORK
1-c, 2-d, 3-a, 4-sol, 5-a, 6-c, 7-b, 8-c, 9-c, 10-c, 11-d, 12-c, 13-a, 14-b, 15-d, 16-c, 17-a, 18-c.

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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

Hinge

c d

ab

Conjugate beam corresponding to this beam is

(GATE 2000 CE)

a b c d a b c d

a) b)

c) a b d a b c d

c d)

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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 P
b) 75

c) 133 Y
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  .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  (b) 1/2 . (c) 2/3  (d).

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 3EI 2EI 3EI 3EI 2EI 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  at the

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

(a) d (b) d  3 3/2
b b
(c) d  (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  (b)  (c)  (d) 4
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
(d) 81EI
(a) (b) (c)

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. 2 EI /4l2.
A. Both ends hinged - 2. 42 EI /4l2.
3. 22 EI /4l2.
B. One end fixed and other end free - 4. 2 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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MECHANICAL ENGINEERING

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 AC B
(d) PL3 / 3 EI,2PL

LL

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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)

100N/cm 600N C

A D

B

6 cm  7cm 3 cm



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MECHANICAL ENGINEERING

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

EI Q Rigid
PR

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 q
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 w kN/m
a) wL b) wL

8 8

1 3 LB
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-d, 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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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) 42EI 22EI 2EI 2EI

l2 b) c) d) 4l2

l2 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 P= 2 EI
cr
a) Pcr = 2L2 b) 3L2

c) P= EI  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 figidityonly

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)

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) 2EI 2 EI
b)
L2
L2
c) 2EI 2 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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MECHANICAL ENGINEERING

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 ? (GATE 2008 CE)

P

4 5-0-(- - Q F
-

a) 2EI b) 2EI c) 2 EI d) 2 EI
L2
L2 L2 L2

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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 bytwo 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 P attains a value greater than : (GATE 1990 CE)

C P D
k

L RIGID
ROD
---------------
A

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) b) c) d)

bd bd bd bd

5. The buckling load P = P for the column AB in figure, as K approaches infinity, becomes  2EI
cr T L2

P

A

L Flexural rigidity. EI

Torsional spring
of stiffness K

T

B

Where  is equal to b) 1.00 c) 2.05 (GATE 2006 CE)
a) 0.25 d) 4.00

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MECHANICAL ENGINEERING

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)

PK

H

L

G

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)

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 MP and modulus of elasticity is 200 GP . The minimum length
aa

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 (x) and the circumferential () stress respectively are given by :
(GATE 1990 CE)

2. Athin 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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MECHANICAL ENGINEERING

Two Marks Questions :
1. A thin cylinder of 100 mm internal diameter and 5 mm thickness is subjected to an internal pressure of
10 MPa and a torque of 200 Nm. Calculate the magnitudes of the principal stress. (GATE 1996
ME)
Statement For Linked Questions 02 And 03
A cylindrical contaner of radius R - 1m, wall thickness 1 mm is filled with water upto a depth of 2m and
suspended along with its upper rim. The density of water is 1000 kg/m3 and acceleration due to gravity is 10
mps. The self weight of the cylinder is negligible. The formula for hoop stress in a thin walled cylinder can be
used at all points along the height of the cylindrical container. (GATE 2008 ME) (2x2 = 4)

.................................... 2m

....................................

1 m ....................................

....................................

2. The axial and circumferential stress (a, 0) experienced by the cylinder wall at middepth (1m as
shown ) are

a) ( 10, 10) MPa b) (5, 10) MPa c) (10, 5) MPa d) (5, 5) MPa

3. If the Youngs’modulus and Poission’s ratio of the container material are 100 GPa and 0.3, respectively.

The axial strain in the cylinder wall at mid height is

a) 2 x10-5 b) 6 x 10-5 c) 7 x 10 -5 d) 1.2 x 10-5

4. A thin-walled long cylindrical tank of inside radius r is subjected simultaneouly to internal gas pressure p

and axial compresive force F at its ends. In order to produce ‘pure shear’state of stress in the wall of

the

cylinder, F should be equal to (GATE 2006 CE)
d) 4Pr2
a) Pr2 b) 2Pr2 c) 3Pr2

******

“What the mind of man can conceive and believe,
it can achieve”

9 . THIN CYLINDERS CLASS WORK

One Mark Questions : 1 - b, 2 - a, 3 -a , 4 - d, 5 - c, 6 - 10 mm, 7 - c, 8 - 25 MPa

Two Marks Questions : 1 - 102 MPa, 2 - b, 3 - c, 4 - c

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STRENGTH OF MATERIAL

TEST PAPERS ANSWER KEYS

TEST PAPER : UNIT - I (ANS.)
1-a, 2-b, 3-a, 4-c, 5-a, 6-b, 7-a, 8-a, 9-b, 10-b, 11-b, 12-b, 13-d, 14-a, 15-b, 16-d, 17-d, 18-d,
19-b, 20-b, 21-b, 22-a, 23-b, 24-b, 25-c, 26-c, 27-d, 28-d, 29d, 30-a.

TEST PAPER - STM : UNIT - II (Ans.)

1-d, 2-b, 3-c, 4-a, 5-b, 6-a, 7-b, 8-d, 9-d, 10-b, 11-b, 12-b, 13-d, 14-c, 15-d, 16-c, 17-d, 18-b, 19-a, 20-c, 21-d, 22-a, 23-b, 24-a,
25-d, 26-c, 27-d, 28-c, 29-a, 30-b.

UNIT - 3 : TEST QUESTIONS (ANS.) : (COMPLEX STRESS AND STRAINS)
1-a, 2-b, 3-a, 4-d, 5-c, 6-a, 7-a, 8-b, 9-d, 10-d, 11-c, 12-b, 13-a, 14-a, 15-d, 16-c, 17-a, 18-a, 19-c,
20-d, 21-c, 22-d, 23-b, 24-d, 25-d, 26-a, 27-a, 28-b, 29-b, 30-d.

UNIT : IV, V & VI = TEST QUESTIONS [ANS.]
1-c, 2-c, 3-a, 4-b, 5-b, 6-a, 7-sol, 8-a, 9-b, 10-a, 11-b, 12-b, 13-a, 14-d, 15-d, 16-a, 17-a, 18-sol,
19-c, 20-sol, 21-b, 22-a, 23-b, 24-b, 25-a, 26-c, 27-b, 28-d, 29-c, 30-c.

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