STM objective

STRENGTH OF MATERIAL

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

stress on the surface which is at 450 to the axis will have a value (GATE 2003) (ME)

a) cos 450 b) 2 cos 450

c) cos2 450 d) 2 sin 450 cos 450

7. The maximum and minium principal stresses respectively from the Mohr’s circle are

(GATE 2003) (ME)

Mohar circle

a) + 175 MPa, -175 MPa

b) + 175 MPa, +175 MPa <>
c) 0, -175 MPa
175 MPa,

d) 0, 0

8. The directions of maximum and mimimum principal stresses at the point “P” from the Mohr’s circle are

(GATE 2003) (ME)

a) 0, 900 b) 900 , 0 c) 450, 1350 d)All directions

9. The three-dimensional state of stress at a point is given by (GATE 1990) (ME)

30 10 -10
[] = 10 0 20 MN / m2
20 0
-10

The shear stress in the x-y plane at the same point is then equal to

a) zero MN/m2 b) -10 MN/m2

c) 10 MN/m2 d) 20 MN/m2

10. Consider the following statements :

If two planes at right angles carry only shear stress of magnitude ‘q’ then the.

(i) Dia of Mohr’s circle could equal ‘2q’

(ii) Centre of Mohr’s circle would lie at origin.

(iii) Principal stresses are unlike and have magnitude ‘q’

(iv)Angle betweem principal plane and plane of maximum shear would be equal to 450

Of these statements :

(a) 1 & 2 are correct

(b) 2 & 4 are correct

(c) 3 & 4 are correct

(d) 1,2,3 & 4 are correct.

11. A bar of square section is subjected to a pull of 10,000kg. If the maximum allowable shear stress

on any section is 500 kg/cm2, then the side of the square section will be.
(a) 5cm
(b) 10 cm
(c) 15 cm
(d) 20 cm

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MECHANICAL ENGINEERING

12. Match List -I (State of stress) with List-II (kind of loading) and select the correct answer using the

codes given below the lists: (IES-96)

List - I

A. B. CD

List - II
1 Combined bending and torsion of circular shaft.
2 Torsion of circular shaft.
3 Thin cylinder subjected to internal pressure.
4 Tie bar subjected to tensile force.

Codes:
ABCD

(a) 1 2 3 4
(b) 2 3 4 1
(c) 2 4 3 1
(d) 3 4 1 2

13. State of stress at a point in a strained body is shown in Fig.A. Which one of the figure .given below
represents correctly the Mohr’s circle for the state of stress?

y y
(a) (b) x

x

(c) y y
x (d)

x

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STRENGTH OF MATERIAL

14. In the given fig.  and  are the maximum and minimum principal stresses. In order that the
12
resultant stress on the plane AB is .

{[( 1+ 2, ) / 2 ]2 + [( 1- 2) / 2]}1/2 2 C
The value of ‘0’ should be
^ A 

(a) 300 1
(b) 450

(c) 600

(d) 750 B

15. If an element subjected to pure shearing stress ‘q’, then the maximum principal stress is equal to.

(a) 2q (b) q / 2 (c) q (d) 1 + q2

16. The state of stress at a point in a stressed element is shown in fig. The maximum tensile stress in the

element will be . 

(a) 20N/mm2 

(b) 10 2 N/mm2 
(c) 10 N/mm2 10 N/mm2
(d) zero

17. A cast iron block of 5 cm2 cross section carries an axial load of 10t. The maximum shear stress in the

block is given by.

(a) 2000 kg/cm2 (b) 1000 kg/cm2

(c) 500 kg/cm2 (d) 250 kg/cm2.

18. Pick the incorrect statement from the following four statements :

(a) On the plan which carries maximum normal stress, the shear stress is zero.

(b) Principle planes are mutually orthogonal.

(c) On the plane which carries maximum shear stress, the normal stress is zero.

(d) The principle stress axes and principle strain axes coincide for an isotropic material.

19. On a plane, resultant stress is inclined at an angle of 450 to the plane. If the normal stress is

100 N/mm2, the shear stress on the plane is .

(a) 71.5 N/mm2 (b) 100N/mm2

(c) 86.6N/mm2 (d) 120.8N/mm2

20. At a point in a stressed body the state of stress on two planes 450 apart is as shown below. Determine

the two principal stresses (GATE 1993) (ME)

3 MPa 8 MPa

v 45 >

2 MPa v

3 MPa

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MECHANICAL ENGINEERING

21. A solid circular shaft of diameter 100 mm is subjected to an axial stress of 50 MPa. It is furhter

subjected to a torque of 10 KN-m. The maximum principal stress experineced on the shaft is closest

to (GATE 2008) (ME)

a) 41 MPa b) 82 MPa

c) 164 MPa d) 204 MPa

22. When the two principal stresses are equal and like, the resultant stress on any plane is .

(a) equal to the principal stress (b) zero

(c) one half principal stress (d) one third of the principal stress.

23. Which of the following Mohr’s circles shown below qualitatively and correctly represents the state

of plane stress at a point in a beam above the neutral axis, where it is subjected to combine shear

and bending compressive stresses (GATE 1993) (CE)



 (tensile)  (tensile)

a) b)

   (tensile)
d)
 (tensile)

c)

24. In a two dimensional analysis, the state of stress at a point is shown below. If  = 120 MPa and

 = 70 MPa,  and y, are respectively (GATE 2004) (CE)
x
^y
a) 26.7 MPa and 172.5 MPa
AB = 4

BC = 3 A 
 ^ 
b) 54 MPa and 128 MPa AC = 5

 

x

c) 67.5 MPa and 213.3 MPa B C x
d) 16 MPa and 138 MPa y

25. Match the items in List-I with those in List-II(with reference to the given figure) and select the correct

answer using the codes given below the lists :

Units shown are in MPa.

List - I List-II

(A) Maximum principle stress (1) 9.0

(B) Minimum principle stress (2) 8.0

(C) Maximum shear stress (3) 5.0

(D) Normal stress on the plane of maximum shear stress. (4) 18.0

(5) 13.0

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STRENGTH OF MATERIAL

Codes : 

ABCD 
(a) 4 3 5 2

(b) 4 2 5 3

(c) 4 5 1 2 8 18
(d) 2 5 1 3

26. Atwo dimensional fluid element rotates like a rigid body.At a point within the element, the pressure

is 1 unit. Radius of the Mohr’s circle, characterizing the state at the point is (GATE 2008) (ME)

a) 0.5 unit b) 0 unit

c) 1 unit d) 2 unit

27. A strain rosette, as shown in the figure, has three strain gauges PL, Q and R,. If the values of strain

indicated in the three strain gauges are (GATE 2008) (PI)

 = 100 x 10-6,
P

Q = 150 x 10-6 and

 = 200 x 10-6,
R

The largest principal strain is

a) 200 x 10-6 b) 250 x 10-6

c) 300 x 10-6 d) 350 x 10-6

Questions 28,29 and 30 are based on the following data:
A mild steel plate is stressed as shown in the fig.Before stressing, a circle of 300 mm diameter is
drawn on the plate. Poisson’s ratio = 0.3; modulus of elasticity = 200GN/m2.

28. The principal stresses developed in MPa

respectively.

(a) 65,13 (b) 78,0

(c) 69.82,-17.82 (d) 57.2,-5,2.

29. After stressing the plate the inscribed circle becomes on ellipse, the angle of major axis of the ellipse

with the plane of P1 = 65 MPa is (b) 41.630
(a) 20.80 (d) 30.320

(c) 13.50

30. The lengths of the major and minor axes, in mm, of the ellipse into which the circle deforms after
stressing. q = 20N/mm2

(a) 300.101,299.98  P2=13N/mm2

(b) 300.094,289.76

(a) 318.34,299.56 P1=65N/mm2
(d) none





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31. The state of stress at a point under plane stress condition is (GATE-12)

 = 40 MPa
xx

xy = 100 MPa and

 = 40 MPa
xy

The radius of Mohr’s circle representing the given state of stress in MPa is

a) 40 b) 50 c) 60 d) 100

32. In a structure subjected to fatigue loading. the minimum and maximum stresses developed in a cycle

are 200 MPa and 400 MPa respectively. The value of stress amplitude (in MPa) is _______

(GATE-14-Set 2)

33. In a plane stress condition, the components of stress at a point are  =20 MPa,  =80 MPa and
x y

 = 40 MPa. The maxmim shear stress (in MPa) at the point is (GATE-15-Set 2)
xy

(a) 20 b) 25 (c) 50 (d) 100

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

Complex Stress and Strains (Ans.)
1-d, 2-a, 3-c, 4-c, 5-c, 6-d, 7-b, 8-d, 9-c, 10-d, 11-b, 12-c, 13-c, 14-b, 15-c, 16-c, 17-b, 18-c, 19-b,
20-sol, 21-b, 22-a, 23-c, 24-c, 25-b, 26-b, 27-a, 28-c, 29-c, 30-a, 31 -b, 32 - 100MPa, 33 - c

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STRENGTH OF MATERIAL

PRACTICE QUESTIONS : (COMPLEX STRESS AND STRAINS)

1. If the two principal strains at a point are 1000 x 10-6 and -600 x 10-6, then the maximum shear strain is

(GATE 1996) (ME)

a) 800 x 10-6 b) 500 x 10-6 c) 1600 x 10-6 d) 200 x 10-6

2. At a point in two-dimensional stress system sx = 100 N/mm2,sx = txy= 40 N/mm2 .What is the radius

of the Mohr circle for stress drawn with a scale of : 1 cm = 10N/mm2. (IES-05)

(a) 3cm (b) 4cm (c) 5 cm (d) 6 cm

3. The Mohr’s circle of plane stress for a point in a body is shown. The design to be done on the basis of the maximum

shear stress theory for yielding. Then yielding will just begin if the designer chooses a ductile material whose yield

strength is. (MPa) (GATE 2004) (ME)
a) 45 MPa

b) 50 MPa

c) 90 MPa  (MPa)
d) 100 MPa.
-100 -10

4. The state of two dimensional stresses acting a concrete lamina consists of a direct tensile stress,  = 1.5 N/mm2,
x

and shear stress,  = 1.20 N/mm2, which cause cracking of concrete. Then the tensile strength of the concrete in

N/mm2 is (GATE 1994) (CE)

a) 1.50 b) 2.08

c) 2.17 d) 2.29

5. The magnitude of the only shear stresses acting at a point in plane stress situation is 7.5 N/mm2. The magnitudes

of the principle stresses will be (GATE 1990) (ME)

a) + 15.0 N/mm2 and -7.5 N/mm2 b) + 7.5 N/mm2 and -15.0 N/mm2

c) + 7.5 N/mm2 and -7.5 N/mm2 d) + 10.0 N/mm2 and -7.5 N/mm2

6. If the two principal strains at a point are 1000 x 10-6 and -600 x 10-6 , then the maximum shear strain is.

(a) 800 x 10-6  
(b) 500 x 10-6
(c) 1600 x 10-6 1000 x 10-6
(d) 200 x 10-6
E

600 x 10-6

E

7. An elastic body is subjected to a tensile stress X in a particular direction and a compressive stress Y in its
perpendicular direction. X and Y are unequal in magnitude. On the plane of maximum shear stress in the body there

will be (GATE 1989) (ME)

a) no normal stress b) also the maximum normal stress

c) the minimum normal stress d) both normal stress and shear stress

8. If the principal stresses in a plane stress problem are  = 100 MPa,  = 40 MPa, the magnitude of the shear stress
1 2

(in MPa) will be (GATE 2009) (ME)

a) 60 b) 50 c) 30 d) 20

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MECHANICAL ENGINEERING

9. The symmetry of stress tensor at a point in the body under equilibrium is obtained from (GATE 2005) (CE)

a) Conservation of mass

b) Force equilibrium equations

c) Moment equilibrium equations

d) Conservation of energy

10. The components of strain tensor at a point in the plane strain case can be obtained by measuring longitudinal strain

in following directions (GATE 2005) (CE)

a) Along any two arbitrary directions

b) Along any three arbitrary directions

c) Along two mutually orthogonal directions

d) Along my arbitrary direction

11. The major and minor principal stresses at a point are 3 MPa and -3 MPa respectively. The maximum shear stresses

at the point is (GATE 1997) (CE)

a) zero b) 3 MPa c) 6 MPa d) 9 MPa

12. Which oneof the following Mohr’s circles represents the state of pure shear ? (IES-2K)
(a) (b)


o o


 
(c) o  (d) o 

13. In a two dimensional problem, the state of pure shear at a point is characterized by . (IES-01)

(a) x = xy and xy = 0 (b) x = xy and xy  0
(c) x = y and xy  0 (d) x = 0.5y and xy  0

14. In a strained material, normal stresses on two mutually perpendicular planes are x and y (both alike)
accompanied by a shear stress xy. One of the principal stresses will be zero, only if.

(a) xy = x x y (b) xy = x x y
2

(c) xy = x x y (d) xy = 2x x 2y

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STRENGTH OF MATERIAL

15. In a Mohr’s circle, the radius of the circle is taken as.

( )(a) 2 2
x + y + xy) x - y +
( )2 xy)2

2 (b) 2

x - y 2
( )(c) + xy) 2 ( )(d) x - y 2 - xy)2

2

where, sx and sy are normal stresses along x and y directions respectively,and txy is the shear stress.
16. The radius of Mohr’s circle of stress of a strained element is 20 N/mm2 and minor principal tensile stress

is 10 N/mm2. The major principal stress is .

(a) 30 N/mm2 (b) 50 N/mm2 (c) 60 N/mm2 (d) 100 N/mm2

17. For a general two dimensional stress system, what are the coordinates of the centre of Mohr’s circle?

(a) x - y) , 0 (b) 0 , x + y)
2 2

( )(c) x + y , 0 (d) 0 , x - y)
2 2

18. Maximum shear stress in a Mohr’s circle .

(a) is equal to radius of Mohr’s circle. (b) is greater than radius of Mohr’s circle.

(c) is less than radius of Mohr’s circle (d) could be any the above.

19. The principal stresses at a point in two dimensional stress system are 1 and 2 and corresponding principal strains
are 1 and 2. If E and denote Young’s modulus and Poisson’s ratio, respectively then which one of the following
is correct?

(a)  = E 1 (b)  = 1 E 1+ 2]
- 2

(c) = E 1 - 2] (d)  = E 1- 2)
1 - 2

20. The principal strains at a point in a body, under biaxial state of stress, are 1000 x 10-6 and -600 x 10-6 what is the

maximum shear strain at that point? (b) 800 x 10-6.
(a) 200 x 10-6. (d) 1600 x 10-6.
(c) 1000 x 10-6

21. The plane of maximum shear stress has normal stress that is.

(a) maximum (b) minimum

(c) zero (d) None of the above

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MECHANICAL ENGINEERING

22. An element is subjected is to two mutually perpendicular unlike, but stresses of ‘p’. The radius of Mohr’s

circle will be.

(a) zero (b) p / 2 (c) p (d) 2p

23. If x, y and  xy are the rectangular stress components at a point. The radius of Mohr’s circle is .

(a) 2 x - 2 y + 2xy (b) (x + y)2 + 2xy

(c) (2y + 2x) + 2xy (d)  [(x - y) / 2]2 +2xy

24. If principal stresses in a two-dimensional case are (-) 10 MPa and 20 MPa respectively, then maximum shear stress

at the point is (GATE 2004) (CE)

a) 10 MPa b) 15 MPa

c) 20 MPa d) 30 MPa

25. Match correct pairs between list I and list II for the questions. (G-94)

List- I List-II

(a) Strain rosette - 1. Critical Speed

(b) Beams - 2. Mohr’s circle

(c) Section modulus - 3. Coil springs

(d) Wahl’s stress factor - 4. Flexural rigidity

(e) Fatigue - 5. Endurance limit

(f) Somer field number - 6. Core Section.

26. A small element at the critical section of a component is in a bi-axial state of stress with the two principal stresses

being 360 MPa and 140 MPa. The maximum working stress according to Distortion Energy Theory is . (G-97)

(a) 220 MPa (b) 110 MPa

(c) 314 MPa (d) 330 MPa.

27. The normal stresses at a point are x 10 MPa and, y2 MPa;the shear stress at this point is 4 MPa. The maximum

principal stress at this point is: (G-98)

(a) 16 MPa (b) 14 MPa

(c) 11MPa (d) 10 MPa.

28. The number of components in a stress at a point in three dimensions is. (G-2002)

(a) 3 (b) 4

(c) 6 (d) 9

29. The fig. shows the state of stress at a certain point in a stressed body. The magnitudes of normal stresses in the x

and y direction are 100 MPa and 20 MPa respectively. The radius of Mohr’s stress circle representing this state of

stress is. y (G-2004)
(a) 120

(b) 80 x x
(c) 60 y
(d) 40

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STRENGTH OF MATERIAL

30. If an element of a stressed body is in a state of pure shear with a magnitude of 80 N/mm2, the magnitude of maxi

mum principle stress at the location is

(a) 80 N/mm2 (b) 113.4 N/mm2 (c)120 N/mm2 (d) 56.57 N/mm2

31. If two principle strains at a point are 1000 x 10-6 and -600 x 10-6 then the maximum shear strain is.

(a) 800 x 10-6 (b) 500 x 10-6

(c) 1600 x 10-6 (d) 200 x 10-6

32. If ‘e1’ and ‘e2’ are major and minor princple strains the greatest shear strains equal to.

(a) (e1 - e2) /2 (b) (e1 - e2) (c) 2 (e1 - e2) (d) none.

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

PRACTICE QUESTIONS : (COMPLEX STRESS AND STRAINS)
1-c, 2-c, 3-c, 4-c, 5-c, 6-c, 7-d, 9-c, 9-c, 10-b, 11-b, 12-d, 13-b, 14-c. 15-b, 16-b, 17-c, 18-a, 19-b,
20-d, 21-d, 22-c, 23-d, 24-b, 25=a-2, c-4, d-3, e-5, 26-c, 27-c, 28-c, 29-c, 30-a, 31-c, 32-b.

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TEST PAPER

STM : UNIT - III

TIME : 90 MINUTES ] [ M.M. : 50 MARKS

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

.

1. The state of plane-stress at a point is given by  = 200 MPa,  = 100 MPa and   = 100 MPa. The maximum
x y 1 xy

shear stress (in MPa) is (GATE 2010) (ME)

a) 111.8 b) 150.1 c) 180.3 d) 223.6

2. An axially loaded bar is subjected to a normal stress of 173 MPa. The shear stress in the bar is

(GATE 2007) (CE)

a) 75 MPa b) 86.5 MPa c) 100 MPa d) 122.3 MPa

3. The state of plane stress in a plate of 100 mm thickness is given as.=100 N/mm2,= 200 N/mm2 Young’s modulus =

300 N/mm2, Poisson’s ratio = 0.3 The stress developed in the direction of thickness is. (IES-2K)

(a) 0 (b) 90 N/mm2 (c) 100 N/mm2 (d) 200 N/mm2.

4. Consider a two dimensional state of stress given for an element as shown in the diagram given below:

y 100MPa

x 200MPa 200MPa

100MPa

what are the co-ordinates of the centre of Mohr’s circle ?

(a) (0,0) (b) (100,200) (c) (200,100) (d) (50,0)

5. A point in a two dimensional state of strain is subjected to pure shearing strain of magnitude xy radians. which

one of the following is the maximum principal strain?

(a) xy (b) xy  2 (c) xy / 2 (d) 2 xy

6. Principal strains at a point are + 100 x 10-6and -200x10-6. What is the maximum shear strain at the point?

(a) 300 x 10-6. (b) 200 x 10-6 (c) 150 x 10-6 (d) 100 x 10-6

7. Disruptive strength is the maximum strength of a metal, when.

(a) subjected to 3 principal tensile stresses at right angles to one another and of equal magnitude.

(b) loaded in tension (c) loaded in compression (d) loaded in shear.

8. Consider the Mohr’s circle shown above: what is the state of stress represented by this circle?

(a) x = y  0, xy = 0  ............................
(b) x = y  0, xy  0 ................................ 
(c) x = 0, y  xy  0
(d) x  0, y  xy  0 0

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STRENGTH OF MATERIAL

9. Consider the following statements:

1.Two- dimensional stresses applied to a thin plate in its own plane represent the plane stress condition.

2. Under plane stress condition, the strain in the direction perpendicular to the plane is zero.

3. Normal and shear stresses may occur simultaneously on a plane.

Which of the above statements is / are correct?

(a) 1 only (b) 1 and 2

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

10. In a bi-axial stress problem, the stresses in x and y directions are x 200 MPa and y 100 MPa. the

maximum principal stress in MPa, is. (G-2K)

(a) 50 (b) 100

(c) 150 (d) 200.

11. A body is subjected to a pure tensile stress of 100 units. What is the maximum shear produced in the
body at some oblique plane due to the above?

(a) 100 units (b) 75 units (c) 50 units (d) 0 units

12. In a bar subjected to axial tension, if the normal stress on section 1-1, perpendicular to its axis is  ,

then the normal stress ‘n ‘ on an inclined section 2-2, making an angle ‘0’ with the section 1-1 will be.

(a) n = cos  (b) n = cos2

(c) n =  /cos (d) n - sin2

13. The cross section of a bar subjected to a uniaxial tensile stress’p’, the tangential stress on a plane

inclined at ‘o’ to the cross section of the bar would be .

(a) p. (sin 2 q )/2 (b) p. sin 2q

(c) p. (cos 2q )/2 (d) p. cos 2q

14. The maximum principle stress for the stresss state shown in the fig. is

(a)  
(b) 2  
(c) 3 
(d) 1.5   
 

15. Match list-I with list-II for plane stress system. (The units are in MPa)

List- I List- II

(normal stress) (Shear stress)

(A) [+50, + 50] (1) 25

(B) [+150, + 50] (2) zero

(C) [0, + 50] (3) , 50

(D) [-50, -200] (4) 75

Codes :

ABCD

(a) 2 3 4 1

(b) 3 2 4 1

(c) 4 1 3 2

(d) 2 3 1 4

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16. The magnitude of the only shear stresses acting at a point in a plane stress situation is 7.5 Nmm2. The
magnitude of principle stress will be.

(a) + 15.0 N/mm2 and -7.5 N/mm2 (b) + 7.5 N/mm2 and 15.0 N/mm2

(c) + 75 N/mm2 and -7.5 N/mm2 (d) + 10.0 N/mm2 and 7.5 N/mm2

17. On an element shown in fig. the stresses are (in MPa) y = 30 x= 11 xy = 30 = xy

The radius of Mohr’s circle and the principle stresses 1 , 2 (in MPa).

Radius(r) 1, 2 y 

(a) 50 - 20,120

(b) 55 - 11,030

(c) 60 - 20,140  x
(d) 70 - 0,140 

 
18. The principle strains in ‘x’ and ‘y’ directions are 0.0006 and 0.0002 respectively.The major principle stress in

‘N/mm2 is (Poisson’s ratio 0.2, E = 1.92 x 105 MPa).

(a) 128 (b) 64 (c) 112 (d) none.

19. Consider the following statements :

i. In a member subjected to uniaxial tensile force the maximum normal stress is the external load divided by the

maximum cross sectional area.

ii. When the structural member is subjected to uniaxial loading, the shear stress is zero on a plane where the normal

stress in maximum.

iii. In a member subjected to uniaxial loading, the normal stress on the planes of maximum shear stress is less than

the maximum.

Of the these statements.

(a) i & ii are correct (b) i & iii are correct

(c) ii & iii are correct (d) i, ii & iii arecorrect.

20. The rectangular block shown in the given fig. is subjected to pure shear of intensity q. if represents the principal

plane and the principal stresses are s1, s2 then the values of 0 s1 and s2 will be respectively.

(a) 00, 900 + q and q D A
(b) 300, 1200 + q and q Eq
(c) 450, 1350 + q/2 and q/2 q
(d) 450, 1350 + q and - q q 

B

21. Consider the following statement :

In a uni-dimensional stress system, the principal plane is defined as on.

(i) Shear stress is zero (ii) Shear stress is maximum

(iii) Normal stress is zero. (iv) Normal stress is maximum.

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

(c) 1 & 4 correct (d) 3 & 4 correct

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STRENGTH OF MATERIAL

22. Consider the following statements: (IES-98)

State of stress in two dimensions at a point in a loaded component can be completely specified by indicating the

normal and shear stresses on.

1. a plane containing the point.

2. any two planes passing through the point.

3. two mutually perpendicular planes passing through the point.

of these statements

(a) 1 and 3 are correct. (b) 2 alone is correct.

(c) 1 alone is correct. (d) 3 alone is correct

23. Plane stress at a point in a body is defined by principal stresses 3 and . The ratio of the normal stress to the

maximum shear stress on the plane of maximum shear stress is . (IES-2K)

(a) 1 (b) 2 (c) 3 (d) 4

24. Normal stresses of equal magnitude , of opposite signs, act at a point of a strained material in perpendicular

direction. What is the magnitude of the stress on a plane inclined at 450 to the applied stresses ?

(a) 2 (b) / 2 (c) / 4 (d) 0

25. Assertion (A) : Mohr’s construction is possible for stresses, strains and area moment of inertia.

Reason (R) : Mohr’s circle represents the transformation of second - order tensor.

(a) Both A and R are individually true and R is the correct explanation of A.

(b) Both A and R are individually true but R is not the correct explanation of A.

(c) A is true but R is false.

(d) A is false but R is true.

Questions 26,27 and 28 are based on the following data:

At a given point in a machine element, following stresses measured : 150 MPa. (T) and zero shear

horizontal plane and 75 MPa (C) on a plane perpendicular to this plane.

26. The normal and shear stresses, in MPa, at this point on a plane inclined at 600 to the horizontal .

(a) 93.75,97.42 150 MPa  600
(b) 93.75,0  
(c) 0, 97.42  
(d) 150
75 MPa






27. The maximum shear stress developed on MPa is

(a) 112.5 (b) 37.5

(c) 150 (d) 75

28. The normal stresses on the plane of maximum shear stress in MPa is .

(a) 112.5 (b) 37.5

(c) 150 (d) 75

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Questions 29 and 30 are based on the following data:

The state of stress at a point P in a two dimesional loading is such that the Mohr’s circle is a point located at

175 MPa on the positive normal stress axis.

29 Determine the maximum and minimum principal stresses respectively from the Mohr’s circle.

(a) + 175 MPa, -175MPa (b) + 175 MPa, +175MPa

(c) 0, -175MPa. (d) 0,0

30. Determine the directions of maximum and minmum principal stresses at the point “P” from the Mohr’s circle.

(a) 0,900 (b) 900, 0

(c) 450,1350 (d) all directions.

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

“It is not important what you believe, only that you believe”

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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) A = B (b) A = B (c) A =B / 2 (d) A =B / 4

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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 

  
c) P d)  P
Q 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  L
(b) h / 6  
(c) h / 3
(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 A and B denote the maximum bending stresses developed in beams

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

(a) A/B = 1.0

(b) A/B < 1.0
(c) A/B >1.0
(d) A/B 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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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 25 mm
c) 8.9 kN 50 mm
d) 17.8 kN
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 (GATE-2008) (ME)

<>
<
a) F(3L - b) b) F(3L + b) P

4b3 4b3 L
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  (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 XX 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)

10 N

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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MECHANICAL ENGINEERING

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 and modular ratio of steel to

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

(a) D2 (B + mt) t  B t

6

(b) D2 (B +2 mt)

6 D

(c) D2 (2B + mt)

6

(d) D2 (2B + 2mt) 

6

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

stress will be (GATE-1988) (ME)

P

6PL x <>
fd2 < L >

a) maximum near the fixed end b) maximum at x = 1 L

2 2

c) maximum at x = L d) uniform through the length

3

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  in the beam.

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

(a) (1/ 2). (b) 

(c)2. (d) 2 .

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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 uniformlydistributed 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  5 mm

Premissible tensile stress for steel = 156.8 N/mm2 

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

Young’s modulus of wood = 0.117 x 105 N/mm2

5 mm

Statement For Linked Data Questions 27 and 28

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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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 Adue to the applied laod P is

(GATE-

14-Set 2)
1PL3

a) 3EI
2PL3

3EI
b) PL4

EI
c) 4PL3

3EI
d)

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)

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

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.

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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 GP , the maximum stress (in MP ) in the steel wier is (GATE-2007) (PI)
aa 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

R 
A 
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. 30 x 10 Steel strip

(a) is 5 mm 30mm
(b) is 30mm Timber 100 mm
(c) is 15mm
(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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MECHANICAL ENGINEERING

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.

d d

(a) (b)

t d d
t

d d

(c) (d)

d+2t d+2t

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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MECHANICAL ENGINEERING

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) A = B (b) A = B.

(c) A = B / 2) (d) A = B / 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.

MM

D


(a) (b) (c) (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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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)

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)

50 mm

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

50 mm 75 mm

200 mm

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MECHANICAL ENGINEERING

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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MECHANICAL ENGINEERING

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 2A L/k G where k is equal to. (ICS-CE-02)

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

5. Asteel 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 1/ 2 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 partsAC 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 Lis 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)

T0

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

L/4 3L/4

The maximum shear stress in the shaft is 8T
c) d3
16T b) 12T d) 4T
a) d3 d3 d3

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)

T1 T

d 2d

T
2

ll

1 1 1 1
a) 16 b) 8 c) 4 d) 2

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MECHANICAL ENGINEERING

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)  cos 450 (b) 2  cos 450

(c)  cos2 450 (d) 2 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  at the free and is (GATE 2011 ME)

1 1

a) 32TL 4 b) 18TL 4
G G

16TL 1 1
4
c) G d) 2TL 4
G

< L >< L/2 >
------------2-d----------d---------
<>
<>

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 . 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 c) /2 (GATE 2008 CE)
a) 2 b)  d) /4

23. Two solid circular shafts of radit R1 and R2 are subjected to same torque. The maximum shear stresses
1and  1/2.
developed in the two shafts are 2. If R1/R2 = 2, then 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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MECHANICAL ENGINEERING

25. A hollow shaft (d0=2di where d0 and di are the quter and inner diameters respectively) needs to
transmit 20kW power at 3000 RPM. If the maximum permissible shear stress is 30 MPa, d0 is
(a) 11.29mm (b) 22.58mm (c) 33.87mm (d) 45.16mm

(GATE-15-Set 2)
26. Ahollow 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) d3 b) d3 d3 d) d3

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  1 and 2. The ratio 2:1 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 is  . 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  . The value of maximum shear in the hollow shaft will be. [IES-97]

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

 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)




Ad 2d C

B

LL

a) T = T b) T = 8T
CA CA

c) T = 16T d) T = 16T
CA AC

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MECHANICAL ENGINEERING

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

d4G d4G d4G d4G

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
WL
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  (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?

(a)  (b)  (c)  (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)  (R3 - r3)fs (b)  (R4 - r4)fs (c)  (R4 - r4)fs (d)  (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/D3 (b) 12T/D3 (c) 8T/D3 (d) 4T/D3

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

16T 32T (c) 3 16T (d) 3 T

(a) 3  (b) 3   

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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MECHANICAL ENGINEERING

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 BC
(b) B
(c) C A
(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 /d3
(a) zero (b) 16 T /d3 (c) 32 T /d3

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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) T = T (b) T = 8T (c) T = 16T (d) T = 16T
AC AC AC CA

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  (b) 10  (c) 10 (d) 5 

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 .  T

************ E1,1 A
UNIT - VI : TORSION [PRACTICE QUESTIONS]ANS. : E ,

11

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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MECHANICAL ENGINEERING

UNIT - VI (B)
SPRINGS

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

arragements offers a stiffness = 2K1K2 (GATE 1987 ME)
K + 2K

12

(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

TEST PAPER

STM : UNIT - IV, V & VI

TIME : 90 MINUTES ] [ M.M. : 50 MARKS

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

1. A rectangular beam is to be cut from a circular log of wood of diameter D. The ratio of the two sides of the rectangle

for strongest section in bending should be. (IES-92 (CE))
(d) 3 / 4
(a) 2 (b) 3 / 2 (c) 1 / 2

2. Two beams A and B are simply supported subjected to identical loads. The two beams have same width, but beam

A has double the depth of beam B. The ratio of section Modulus of beam A and B is.

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

3. The ratio of flexural strength of a square section with its two sides horizontal to its diagonal horizontal. is.

(a) 2 (b) 2 / 5 (c) 22 (d) 2

4. Two 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 beams are the same then.

(G-03-ME)

(a) maximum bending stress developed in both the beams is the same.

(b) the circular beam experiences more bending stress than the square one.

(c) the square beam experiences more bending stress than the circular one.

(d) as the material is same both the beams will experiencesame deformation.

5. The assumption that transverse sections which are plane and normal before bending remain plane and normal after

bending results along a cross section.

(a) Linear variation of stress (b) Linear variation of strain, (c) Constant strain. (d) Non linear variation of strain.

6. A beam of square section is placed horizontally with one diagonal placed horizontally, the maximum shear stress

occurs at.

(a) 3 / 8d from top (b) At the N.A (c) 3 / 8 d from N.A (d) none.

7. In a beam of circular cross-section, the shear stress variation across a cross-section is.

(a) (b)

(c) (d)

8. A solid circular shaft is subjected to pure torsion. the ratio of maximum shear stress to maximum normal stress at

any point would be. [IES-99]

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

9 A circular shaft subjected to twisting moment results in maximum shear stress of 60 MPa. Then the maximum

compressive stress in the material is. [IES-02]

(a) 30MPa (b) 60MPa (c) 90MPa (d) 120MPa

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MECHANICAL ENGINEERING

10. A circular section rod ABC is fixed at ends Aand C.It is subjected to torque T at B. AB=BC=L and the polar moment

of inertia of portions AB and BC are 2 J and J respectively. If G is the modulus of rigidity. what is the angle of twist

at point B? [IES-05]

TL TL TL 2TL
(a) (b) (c) GJ (d)

3GJ 2GJ GJ

11. A beam of square section 200mm x 200mm is fixed as a cantilever of span 3m. what u.d.l can be applied through out
the length of the beam if maximum stress is not to exceed 5 N/mm2

(a) 14.8 kN/m (b) 1.48 kN/m (c) 148 kN/m (d) 7.2 kN/m

12. If the maximum flexural stress in timber joist of the flitched beam shown in fig. is 7N/mm2, the maximum stress in steel

is N/mm2 is Es / Et = 20. tbt
(a) 7

(b) 140

(c) 240

(d) 280

13. The ratio of moment of resistance of solid circular shaft of diameter ‘D’ and a hollow shaft of external diameter D and

internal diameter ‘d’ is. (b) (D4 - d4) / D4 (c) (D2 - d2) / D2 (d) (D3 - d3) / D3
(a) D4 / (D4 - d4)

14. A simply supported beam is made of two wooden planks of same width resting one upon the other without friction

and with out connection. The upper plank is of half the thickness as compared to lower plank. The assembly is

loaded by a uniformly distributed load on the entire span. The ratio of the maximum stress developed between top

and bottom planks will be.

(a) 1 : 16 (b) 1 : 8 (c) 1 : 4 (d) 1 : 2

15. 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

16. Match List - I with List - II and select the correct answer by using codes given below the lists:

List - I 1. bh3 List - II
A. Moment of inertia of a triangle about its base - 36

B. Moment of inertia of a rectangle about its centro - bh3
idal axis parallel to the base. 2. 12

C. Moment of inertia of a rectangle about the base - bh3
of the rectangle. 3. 3

D. Moment of inertia of a triangle about centroidal - bh3
axis parallel to the base. 4. 12

Codes:-

ABCD

(a) 4 2 3 1

(b) 1 2 4 3

(c) 3 2 1 4

(d) 1 4 2 3

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STRENGTH OF MATERIAL

17. The cross-sections of the beams of equal length are a circle and a square whose permissible bending stress are

same under same maximum bending. the ratio of their weights is.

(a) 1.118 (b) 1.338

(c) 1.228 (d) 1.108

18. A beam has a triangular cross-section having base 40 mm and attitude 60 mm. If this section is subjected to a shear

force of 36000N, the maximum shear stress in the cross-section would be.

(a) 60 N/mm2 (b) 36 N/mm2 (c) 45 N/mm2 (d) 30 N/mm2

19. Consider the following statements :

When a beam of square cross-section is used with a diagonal in a vertical position.

1. the shear stress distribution across the section of the beam will be maximum at the neutral axis.

2. the shear stress distribution across the section of the beam will be zero both at top and bottom.

3. the maximum stress does not occur at neutral axis.

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

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

20. Shear stress distribution in different types of cross-section of a beam is stated as follows .

I.For rectangular cross-section, the shear stress is maximum at the neutral axis.

II.For circular cross-section, the shear stress is maximum at the neutral axis.

III.For I-section, the shear stress is maximum at the junction of web and flange.

Choose the correct combination

(A) I, II and III are correct

(B) I, and II are correct

(C) II and III are correct

(D) I, and III are correct

21. A stepped shaft of uniform material with shear modulus of 105 MPa is shown in fig. The lengths and polar oment

of inertia are indicated in fig.. A torque of 5000 N-m is applied at the free end. The twist in radians at the free end

is.

Ip = 2.5 cm4
Ip = 10 cm4



100 cm

100 cm

(a) 0.3 (b) 2.5 (c) 0.2 (d) 0.7

22. The maximum shear stress produced in a shaft is 5N / mm2. The shaft is of 40 mm dia. The value of twisting

moment is .

(a) 62.8 N-m (b) 628 N-m

(c) 125.6 N-m (d) None.

23. A solid circular shaft of 6m length is built in at its ends and subjected to an externally applied torque 9 t.m at a

distance of 2m from left end. The respective torque in t.m at the left end and the right end are respectively.

(a) 3,6 (b) 6,3

(c) 4,5 , 4.5 (d) 5,4

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MECHANICAL ENGINEERING

24. A circular shaft shown in the fig is subjected to a torsion T at two points A and B .the torsional rigidity of the of

pertions CA and BD is GJ1 and that of the portion AB is GJ2. The rotations of the shaft at points A and B are 

and  . The rotation  is . (Gate-CE-05)

CA B D

  

L L L
L  

(a) TL/(GJ1 + GJ2) (b) TL/GJ1 (c) TL/GJ2 (d) TL/(GJ1 - GJ2)

25. A solid circular shaft of 10 cm dia and 1 m length is subjected to a torque of 8kN.m.If modulus of rigidity 8 x 104

N/mm2 , the maximum angle of twist is.

(a) 32 / 1000  radius (b) 1000 / 32  radius (c) 32 / 1000  degrees. (d) None.

 A steel shaft 5 cm dia. and 50 cm long, is subject to a twisting couple of 11,000 Kg.cm. If the angle of twist be

0.6 degree, the modulus of rigidity of material of shaft in ‘Kg/cm2’ is .

(a) 0.762 x 106 (b) 0.936 x 106 (c) 0.856 x 106 (d) 0.892 x 106

27. A shaft has to transmit 105kW at 160 rpm. If the shear stress is not to exceed 65 MPa and the twist in a length of

3.5 m must not exceed 1 degree, the rigidity modulus of material is 8 x 104 MPa. The diameter of the shaft in ‘mm’

is ?

(a) 78.9 (b) 112.5 (c) 234.76 (d) none.

28. A shaft transmits 80 H.P at 60 rpm .If the maximum torque is 30 % greater than the mean, the maximum torque in

Kg.m.is. (b) 3200 /  (c) 1920 /  (d) 3900 / 
(a) 1800 / 

Questions 29 and 30 are based on the following data :

A solid shaft is to transmit 452.8 HP at 2 Hz. If the shear stress of the material must not exceed 80 MPa .

29. The diameter of the shaft required in ‘mm’ is.

(a) 50.38 (b) 2.3 (c) 119.5 (d) 250.67

30. The percentage savings in weight if the above solid shaft is replaced by a hollow shaft whose internal diameter

is equal to 0.6 times the external diameter, If the maximum shear stress remain un-changed ?

(a) 5 (b) 10 (c) 30 d) 20

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

“What a person believes is not as important
as how a person believes”

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