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GATE SOLVED PAPERCivil EngineeringStrength of Materials

Copyright © By NODIA & COMPANY

Information contained in this book has been obtained by authors, from sources believes to be reliable. However, neither Nodia nor its authors guarantee the accuracy or completeness of any information herein, and Nodia nor its authors shall be responsible for any error, omissions, or damages arising out of use of this information. This book is published with the understanding that Nodia and its authors are supplying information but are not attempting to render engineering or other professional services.

NODIA AND COMPANYB-8, Dhanshree Tower Ist, Central Spine, Vidyadhar Nagar, Jaipur 302039Ph : +91 - 141 - 2101150www.nodia.co.inemail : [email protected]

GATE SOLVED PAPER - CESTRENGTH OF MATERIALS

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YEAR 2013 ONE MARK

Q. 1 Creep strain is(A) caused due to dead load only (B) caused due to live load only

(C) caused due to cyclic load only (D) independent of load

Q. 2 The “Plane section remain plane” assumption in bending theory implies(A) strain profile is linear (B) stress profile is linear

(C) both profiles are linear (D) shear deformation is neglected

Q. 3 A symmetric I-section (with width of each flange mm50= , thickness of web mm10= ) of steel is subjected to a shear force of 100 kN. Find the magnitude of

the shear stress (in N/mm2) in the web at its junction with the top flange.

Q. 4 Two steel column P (length L and yield strength MPaf 250y = ) have the same cross-section and end condition the ratio of bucking load of column P to that of column Q is(A) 0.5 (B) 1.0

(C) 2.0 (D) 4.0

YEAR 2013 TWO MARKS

Q. 5 2D stress at a point is given by a matrix

xx

yx

xy

yy

st

ts> H MPa

10030

3020= > H

The maximum shear stress in MPa is(A) 50 (B) 75

(C) 100 (D) 110

YEAR 2012 ONE MARK

Q. 6 The Poisson’s ratio is defined as

(A) lateral stressaxial stress (B) axial strain

lateral strain

(C) axial stresslateral stress (D) laterial strain

axial strian

Q. 7 The following statement are related to bending of beamsI. The slope of the bending moment diagram is equal to the shear force.

II. The slope of the shear force diagram is equal to the load intensity.

III. The slope of the curvature is equal to the flexural rotation

IV. The second derivative of the deflection is equal to the curvature.

The only FALSE statements is

GATE SOLVED PAPER - CE STRENGTH OF MATERIALS

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(A) I (B) II

(C) III (D) IV

Q. 8 The ratio of the theoretical critical buckling load for a column with fixed ends to that of another column with the same dimensions and material, but with pinned ends, is equal to(A) 0.5 (B) 1.0

(C) 2.0 (D) 4.0

YEAR 2012 TWO MARKS

Q. 9 The simply supported beam is subjected to a uniformly distributed load of intensity w per unit length, on half of the span from one end. The length of the span and the flexural stiffness are denoted as l and El respectively. The deflection at mid-span of the beam is

(A) Elwl

61445 4 (B) El

wl7685 4

(C) Elwl

3845 4 (D) El

wl1925 4

Q. 10 This sketch shows a column with a pin at the base and rollers at the top. It is subjected to an axial force P and a moment M at mid-height. The reaction(s) at R is/are.

(A) a vertical force equal to P

(B) a vertical force equal to /P 2

(C) a vertical force equal to P and a horizontal force equal to /M h

(D) a vertical force equal to /P 2 and a horizontal force equal to /M h

Q. 11 A symmetric frame PQR consists of two inclined members PQ and QR connected at ‘Q ’ with a rigid joint and hinged at ‘P ’ and ‘R ’. The horizontal length PR is l . If a weight W is suspended at Q , the bending moment at Q is


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(A) Wl2 (B) Wl4

(C) Wl8 (D) zero

YEAR 2011 ONE MARK

Q. 12 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 ?

(A) (B)

(C) (D)

YEAR 2011 TWO MARKS

Q. 13 For the cantilever bracket, PQRS , loaded as shown in the adjoining figure (PQ RS L= = , and, QR L2= ), which of the following statements is FALSE ?

(A) The portion RS has a constant twisting moment with a value of WL2 .

(B) The portion QR has a varying twisting moment with a maximum value of WL .

(C) The portion PQ has a varying bending moment with a maximum value of WL .

(D) The portion PQ has no twisting moment.

Statement for Linked Q. 14 and 15 :A rigid beam is hinged at one end and supported on linear elastic springs (both having a stiffness of ‘k ’) at point ‘1’ and ‘2’, and an inclined load acts at ‘2’, as shown.


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Q. 14 Which of the following options represents the deflections 1d and 2d at points ‘1’ and ‘2’ ?

(A) kP

52 2

1d = b l and kP

54 2

2d = b l (B) kP

52

1d = b l and kP

54

2d = b l

(C) k

P52

21d = c m and k

P54

22d = c m (D) k

P52 2

1d = c m and kP

54 2

2d = c m

Q. 15 If the load P equals 100 kN, which of the following options represents forces R1 and R2 in the springs at point ‘1’ and ‘2’?(A) 20 kNR1 = and 40 kNR2 = (B) 50 kNR1 = and 50 kNR2 =(C) 30 kNR1 = and 60 kNR2 = (D) 40 kNR1 = and 80 kNR2 =

YEAR 2010 ONE MARK

Q. 16 Two people weighing W each are sitting on a park of length L floating on water at /L 4 from either end. Neglecting the wright of the plank, the bending moment at the centre of the plank is

(A) WL8 (B) WL16

(C) WL32 (D) zero

Q. 17 The major and minor principal stresses at a point are 3 MPa and 3- MPa respectively. The maximum shear stress at the point is(A) zero (B) 3 MPa

(C) 6 MPa (D) 9 MPa

Q. 18 The number of independent elastic constants for a linear elastic isotropic and homogeneous material is(A) 4 (B) 3

(C) 2 (D)1

Q. 19 The effective length of a column of length L fixed against rotation and translation at one end is(A) 0.5 L (B) 0.7 L

(C) 1.414 L (D) 2 L

Q. 20 A solid circular shaft of diameter d and length L is fixed at one 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

(A) d GTL164p

(B) d GTL324p

(C) d GTL644p

(D) d G

TL1284p


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YEAR 2010 TWO MARKS

Q. 21 For the simply supported beam of length L , subjected to a uniformly distributed moment M kN-m per unt length as shown in the figure, the bending moment (in kN-m) at the mid-span of beam is

(A) zero (B) M

(C) ML (D) /M L

Q. 22 A disc of radius r has a hole of radius /r 2 cut-out as shown. The centroid of the remaining disc (shaded portion) at a radial distance from the centre “O” is

(A) /r 2 (B) /r 3

(C) /r 6 (D) /r 8

YEAR 2009 ONE MARK

Q. 23 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(A) 14 MPa (B) 1.4 MPa

(C) 0.14 MPa (D) 0.014 MPa

Q. 24 The point within the cross sectional plane of a beam through which the resultant of the external loading on the beam has to pass through to ensure pure bending without twisting of the cross-section of the beam is called(A) moment centre (B) centroid

(C) shear centre (D) elastic center

YEAR 2009 TWO MARKS

Q. 25 Consider the following statements :1. On a principal plane, only normal stress acts.

2. On a principal plane, both normal and shear stresses act.

3. On a principal plane, only shear stress acts

4. Isothermal state of stress is independent of frame of reference.

Which of the above statements is/are correct ?(A) 1 and 4 (B) 2 only

(C) 2 and 4 (D) 2 and 3


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Q. 26 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(A) 46 kNm (B) 24.5 kNm

(C) 23 kNm (D) 11.5 kNm

Q. 27 Match List-I (Shear Force Diagrams) beams with List-II (Diagram of beams with supports and loading) and select the correct answer by using the codes given below the lists :


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Codes : a b c d(A) 3 1 2 4(B) 3 4 2 1(C) 2 1 4 3(D) 2 4 3 1

Statement for Linked Q. 28 & 29 :In the cantilever beam PQR shown in figure below, the segment PQ has flexural rigidity EI and the segment QR has infinite flexural rigidity

Q. 28 The deflection and slope of the beam at Q are respectively

(A) 65

23

EIWL and EI

WL3 2 (B) 3 EI 2WL and EI

WL3 2

(C) 2 EIWL and EI

WL3 2 (D) 3 EI 23WL and EI

WL3 2

Q. 29 The deflection of the beam at R is

(A) EI8 WL3

(B) 6 EI5 WL3

(C) 3 EI7 WL3 (D) 6 EI

8 WL3

YEAR 2008 ONE MARK

Q. 30 A mild steel specimen is under uniaxial tensile stress. Young’s modulus and yield stress for mild steel are 2 105# MPa and 250 MPa respectively. The maximum amount of strain energy per unit volume that can be stored in this specimen without permanent set is(A) 156 Nmm/mm3 (B) 15.6 Nmm/mm3

(C) 1.56 Nmm/mm3 (D) 0.156 Nmm/mm3

YEAR 2008 TWO MARKS

Q. 31 Cross-section of a column consisting to two steel strips, each of thickness t and width b is shown in the figure below. The critical loads of the column with perfect bond and without bond between the strips are P and P0 respectively. The ratio

/P P0 is


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(A) 2 (B) 4

(C) 6 (D) 8

Q. 32 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 buckling load, Pcr , for the bar will be

(A) 0.5 KL (B) 0.8 KL

(C) 1.0 KL (D) 1.2 KL

Q. 33 The maximum shear stress in a solid shaft of circular cross-section having diameter d subjected to a subject to a torque T is t. 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(A) 2t (B) t(C) /2t (D) /4t

Q. 34 A vertical rod PQ of length L is fixed at its top end P and has a flange fixed to the bottom end Q . A weight W is dropped vertically from a height ( )h L< on to the flange. The axial stress in the rod can be reduced by(A) increasing the length of the rod

(B) decreasing the length of the rod

(C) decreasing the area of cross-section of the rod

(D) increasing the modulus of elasticity of the material

Q. 35 The maximum tensile stress at the section X -X shown in the figure below is

(A) bdP8

(B) bdP6

(C) bdP4

(D) bdP2


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Q. 36 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

(A) 8 EIML2

(B) 4 EIML2

(C) 4 EIML2

(D) Zero

Statement for Linked Q. 37 and 38 :Beam GHI is supported by three pontoons as shown in the figure below. The horizontal cross-sectional area of each pontoon is 8 m2, the flexural rigidity of the beam is 10000 kNm2 and the unit weight of water is 10 kNm3

Q. 37 When the middle pontoon is removed, the deflection at H will be(A) 0.2 m (B) 0.4 m

(C) 0.6 m (D) 0.8 m

Q. 38 When the middle pontoon is brought back to its position as shown in the figure above, the reaction at H will be(A) 8.6 kN (B) 15.7 kN

(C) 19.2 kN (D) 24.2 kN

YEAR 2007 ONE MARK

Q. 39 An axially loaded bar is subjected to a normal stress os 173 MPa. The stress in the bar is(A) 75 MPa (B) 86.5 MPa

(C) 100 MPa (D) 122.3 MPa

Q. 40 A steel column, pinned at both end, has a buckling load of 200 kN. If the column is restrained against lateral movement at its mid-height, it buckling load will be(A) 200 kN (B) 283 kN

(C) 400 kN (D) 800 kN


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Q. 41 For an isotropic material, the relationship between the Young’s modulus (E), shear modulus (G) and Poisson’s ratio (m) is given by

(A) ( )

G E2 1 m= + (B) ( )E

E2 1 m= +

(C) ( )

G E1 2m= + (D) ( )G

E2 1 2m= -

YEAR 2007 TWO MARKS

Q. 42 A metal bar of length 100 mm is inserted between two rigid supports and its temperature is increased by 10 Cc . If the coefficient of thermal expansion is 12 10 per C6# c- and the Young’s modulus is 2 105# MPa, the stress in the bar is(A) zero (B) 12 MPa

(C) 24 MPa (D) 2400 MPa

Q. 43 A rigid bar is suspended by three rods made of the same material as shown in the figure. The area and length of the central rod are 3 A and L , respectively while that of the two outer rods are A2 and L2 , respectively. If a downward force of 50 kN is applied to the rigid bar, the forces in the central and each of the outer rods will be(A) 16.67 kN each (B) 30 kN and 15 kN

(C) 30 kN and 10 kN (D) 21.4 kN and 14.3 kN

Q. 44 The maximum and minimum shear stresses in a hollow circular shaft of outer diameter 20 mm and thickness 2 mm, subjected to a torque of 92.7 Nm will be

(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

Q. 45 The shear stress at the neutral axis in a beam of triangular section with a base of 40 mm and height 20 mm, subjected to a shear force of 3 kN is(A) 3 MPa (B) 6 MPa

(C) 10 MPa (D) 20 MPa

Q. 46 U1 and U2 are the strain energies stored in a prismatic bar due to axial tensile forces P1 and P2, respectively. The strain energy U stored in the same bar due to combined action of P1 and P2 will be(A) U U U1 2= + (B) U U U1 2=(C) U U U< 1 2+ (D) U U U> 1 2+


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YEAR 2006 ONE MARK

Q. 47 Mohr’s circle for the state of stress defined by 300

030> H MPa is a circle with

(A) centre at (0, 0) and radius 30 MPa (B) centre at (0, 0) and radius 60 MPa

(C) centre at (30, 0) and radius 30 MPa (D) centre at (30, 0) and zero radius

Q. 48 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

(A) zero (B) dT163p

(C) dT323p (D

dT643p

Q. 49 The buckling load P Pcr= for the column AB in figure, as KT approaches infinity,

becomes LEI2

2

ap

Where a is equal to(A) 0.25 (B) 1.00

(C) 2.05 (D) 4.00

YEAR 2006 TWO MARKS

Q. 50 A thin-walled long cylindrical tank is inside radius r is subjected simultaneously to internal gas pressure p and axial compressive 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(A) p r2p (B) p r2 2p(C) p r3 2p (D) p r4 2p

Q. 51 Consider the beam AB shown in the figure below. Part AC of the beam is rigid while Part CB has the flexural rigidity EI . Identify the correct combination of deflection at end B and bending moment at end A, respectively

(A) ,EIPL PL3 2

3

(B) ,EIPL PL3

3

(C) ,2EIPL PL3

8 3 (D) ,EI

PL PL38 3


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Q. 52 A simply supported beam AB has the bending moment diagram as shown in the following figure.

The beam is possibly under the action of following loads(A) Couples of M at C and 2M at D

(B) Couples of 2M at C and M at D

(C) Concentrated loads of M/L at C and 2M/L at D

(D) Concentrated load of M/L at C and couple of 2M at D

Q. 53 A beam with the cross-section given is subjected to a positive bending moment (causing compression at the top) of 16 kNm acting around the horizontal axis. The tensile force acting on the hatched area of the cross-section is

(A) zero (B) 5.9 kN

(C) 8.9 kN (D) 17.8 kN

Q. 54 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

(A) zero (B) V277

(C) V278

(D) V3

Q. 55 For the section shown below, second moment of the area about an axis /d 4 distance above the bottom of the area is

(A) bd483

(B) bd123

(C) bd487 3

(D) bd33


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Q. 56 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 joint will be subjected to a shear force (in kN per meter length) of

(A) 3.0 (B) 4.0

(C) 8.0 (D) 10.7

Common Data For Q. 57 and 58 :Consider a propped cantilever beam ABC under two leads of magnitude P each as shown in the figure below. Flexural rigidity of the beam is EI .

Q. 57 The reaction at C is

(A) ( )LPa

169 upwards (B) ( )L

Pa169 downwards

(C) ( )LPa89 upwards (D) ( )L

Pa89 downwards

Q. 58 The rotation at B is

(A) ( )EIPLa

165 clockwise (B) ( )EI

PLa165 anticlockwise

(C) ( )EIPLa

1659 clockwise (D) ( )EI

PLa1659 anticlockwise

YEAR 2005 ONE MARK

Q. 59 The symmetry of stress tensor at a point in the body under equilibrium is obtained from(A) conserved of mass

(B) force equilibrium equations

(C) moment equilibrium equations

(D) conservation of energy


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Q. 60 The components of strain tensor at a point in the plane strain case can be obtained by measuring longitudinal strain in following directions(A) along any two arbitrary directions

(B) along any three arbitrary directions

(C) along two mutually orthogonal directions

(D) along any arbitrary direction

YEAR 2005 TWO MARKS

Q. 61 If principal stresses in a two-dimensional case are 10- MPa and 20 MPa respectively, then maximum shear stress at the point is(A) 10 MPa (B) 15 MPa

(C) 20 MPa (D) 30 MPa

Q. 62 The bending moment diagram for a beam is given below :

The shear force at sections aal and bbl respectively are of the magnitude(A) 100 kN, 150 kN (B) zero, 100 kN

(C) zero, 50 kN (D) 100 kN, 100 kN

Q. 63 A circular shaft shown in the figure is subjected to torsion T at two point A and B. The torsional rigidity of portions CA and BD is GJ1 and that of portion AB is GJ2. The rotations of shaft at points A and B are 1q and 2q . The rotation 1q is

(A) GJ GJTL1 2+

(B) GJTL

1

(C) GJTL

2 (D) GJ GJ

TL1 2-

YEAR 2004 ONE MARK

Q. 64 For linear elastic systems, the type of displacement function for the strain energy is(A) linear (B) quadratic

(C) cubic (D) quartic


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YEAR 2004 TWO MARKS

Q. 65 In a two dimensional stress analysis, the state of stress at a point is shown below. If 120 MPas = and 70 MPat = , andx ys s , are respectively,

(A) 26.7 MPa and 172.5 MPa (B) 54 MPa and 128 MPa

(C) 67.5 MPa and 213.3 MPa (D) 16 MPa and 138 MPa

Q. 66 For the linear elastic beam shown in the figure, the flexural rigidity, EI is 781250 kNm2. When 10w kN/m= , the vertical reaction RA at A is 50 kN. The value of RA for 100w kN/m= is

(A) 500 kN (B) 425 kN

(C) 250 kN (D) 75 kN

Q. 67 A homogeneous, 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

(A) BDPL

32

2 (B) BDPL

43

2

(C) BDPL

34

2 (D) BDPL

23

2

Q. 68 A circular solid shaft of span 5L m= is fixed at one end and free at other end. A twisting moment 100T kNm= is applied at the free end. The torsional rigidity GH is 50000 kNm /rad2 .Following statements are made for this shaft :1. The maximum rotation is 0.01 rad

2. The torsional strain energy is 1 kNm

With reference to the above statements, which of the following applies ?(A) Both statements are true

(B) Statement 2 is true but 2 is false

(C) Statement 2 is true but 1 is false

(D) Both the statements are false


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Common Data For Q. 69 and 70 :A three-span continuous beam has an internal hinge at B. Section B is at the mid-span of AC. Section E is at the mid-span of CG. The 20 kN load is applied at section B whereas 10 kN loads are applied at sections D and F as shown in the figure. Span GH is subjected to uniformly distributed load of magnitude 5 kN/m. For the loading shown, shear force immediate to the right of section E is 9.84 kN upwards and the hogging moment at section E is 10.31 kNm

Q. 69 The magnitude of the shear force immediate to the left and immediate to the right of section B are respectively(A) 0 and 20 kN

(B) 10 kN and 10 kN

(C) 20 kN and 0

(D) 9.84 kN and 10.16 kN

Q. 70 The vertical reaction at support H is(A) 15 kN upward

(B) 9.84 kN upward

(C) 15 kN downward

(D) 9.84 kN downward

YEAR 2003 ONE MARK

Q. 71 A bar of varying square cross-section is loaded symmetrically as shown in the figure. Loads shown are placed on one of the axes of symmetry of cross-section. Ignoring self weight, the maximum tensile stress in N/mm2 anywhere is

(A) 16.0 (B) 20.0

(C) 25.0 (D) 30.0


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Q. 72 A curved member with a straight vertical leg is carrying a vertical load at Z , as shown in the figure. The stress resultants in the XY segment are

(A) bending moment, shear force and axial force

(B) bending moment and axial force only

(C) bending moment and shear force only

(D) axial force only

YEAR 2003 TWO MARKS

Q. 73 The state of two dimensional stresses acting on a concrete lamina consists of a direct tensile stress, 1.5 N/mmx 2s = , and shear stress, 1.20 N/mm2t = , which cause cracking of concrete. Then the tensile strength of the concrete in N/mm2 is(A) 1.50

(B) 2.08

(C) 2.17

(D) 2.29

Q. 74 A " "H shaped frame of uniform flexural rigidity EI is located as shown in the figure. The relative outward displacement between points K and O is

(A) EIRLh2

(B) EIRL h2

(C) EIRLh3

2

(D) EIRL h3

2


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Q. 75 A simply supported beam of uniform rectangular cross-section of width b and depth h is subjected to linear temperature gradient 0c at the top and Tc at the bottom, as shown in the figure. The coefficient of linear expansion of the beam material is a . The resulting vertical deflection at the mid-span of the beam is

(A) LTh8 upward

2a

(B) hTL8 upward

2a

(C) LTh8 downward

2a

(D) hTL8 downward

2a

Q. 76 List I shows different loads acting on a beam and List II shows different bending moment distributions. Match the load with the corresponding bending moment diagram.

Codes : A B C D(A) 4 2 1 3(B) 5 4 1 3(C) 2 5 3 1(D) 2 4 1 3


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Q. 77 A long structural column (length L= ) with both ends hinged is acted upon by an axial compressive load, P . The differential equation governing the bending of column is given by :

EIdxd y

2

2

Py=-

Where y is the structural lateral deflection and EI is the flexural rigidity. The first critical load on column responsible for its buckling is given by

(A) LEI2

2p (B)

LEI2

2

2p

(C) LEI22

2p (D) L

EI42

2p

YEAR 2002 ONE MARK

Q. 78 The shear modulus Ĝh, modulus of elasticity E^ h and the Poisson’s ratio v^ h of a material are related as

(A) Gv

E2 1

=+^ h6 @

(B) Ev

G2 1

=+^ h6 @

(C) Gv

E2 1

=-^ h6 @

(D) GvE

2 1=

-^ h6 @

Q. 79 For the loading given in the figure below, two statements (I and II) are made

I. Memeber AB carries shear force and bending moment

II. Member BC carries axial load and shear force

Which of these statement is true?

(A) Statement I True but II is False (B) Statement I is False but II is True

(C) Both statement I and II are True (D) Both statement I and II are False

YEAR 2002 TWO MARKS

Q. 80 In the propped cantilever beam carrying a uniformly distributed load of /N mw , shown in the following figure, the reaction at the support B is

(A) wL85 (B) wL8

3

(C) wL21 (D) wL4

3


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YEAR 2001 ONE MARK

Q. 81 The bending moment (in kNm units) at the mid span location X in the beam with overhangs shown below is equal to

z

(A) 0 (B) –10

(C) –15 (D) –20

**********


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ANSWER KEY

STRENGTH OF MATERIALS

1 2 3 4 5 6 7 8 9 10

(A) (A) 71.12 (D) (A) (B) (C) (D) (B) (C)

11 12 13 14 15 16 17 18 19 20

(D) (A) (B) (B) (D) (D) (B) (C) (D) (B)

21 22 23 24 25 26 27 28 29 30

(A) (C) (A) (C) (A) (C) (A) (A) (A) (D)

31 32 33 34 35 36 37 38 39 40

(B) (C) (C) (A) (A) (C) (B) (C) (B) (D)

41 42 43 44 45 46 47 48 49 50

(A) (C) (C) (B) (C) (D) (D) (A) (D) (C)

51 52 53 54 55 56 57 58 59 60

(A) (A) (C) (B) (C) (B) (C) (A) (C) (B)

61 62 63 64 65 66 67 68 69 70

(B) (C) (B) (B) (C) (B) (D) (B) (A) (B)

71 72 73 74 75 76 77 78 79 80

(C) (D) (C) (A) (D) (D) (A) (A) (A) (B)

81

(C)

nodia and company - deewanbittal.files.wordpress.com · intensity w per unit length, on half of the...

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