Strength of Material for GATE.pdf

CONTENTS VOL 1

ENGINEERING MECHANICS

AM 1 Equilibrium of Forces AM 3

AM 2 Structure AM 40

AM 3 Friction AM 81

AM 4 Virtual Work AM 117

AM 5 Kinematics of Particle AM 128

AM 6 Kinetics of Particles AM 157

AM 7 Plane Kinematics of Rigid body AM 190

AM 8 Plane Kinetics of Rigid body AM 206

STRENGTH OF MATERIALS

SM 1 Stress and Strain SM 3

SM 2 Axial Loading SM 41

SM 3 Torsion SM 86

SM 4 Shear Force and Bending Moment SM 118

SM 5 Transformation of Stress and Strain SM 179

SM 6 Design of Beams and Shafts SM 226

SM 7 Deflection of Beams and Shafts SM 270

SM 8 Column SM 315

SM 9 Energy Methods SM 354

THEORY OF MACHINES

TM 1 Analysis of Plane Mechanism TM 3

TM 2 Velocity and Acceleration TM 20

TM 3 Dynamic Analysis of Slider - Crank and Cam TM 38

TM 4 Gear - Trains TM 59

TM 5 Fly Wheel TM 91

TM 6 Vibration TM 109

MACHINES DESIGN

MD 1 Static and Dynamic Loading MD 3

MD 2 Joints MD 22

MD 3 Shaft and Shaft Components MD 54

MD 4 Spur Gears MD 71

MD 5 Bearings MD 88

MD 6 Clutch and Brakes MD 105

CONTENTS VOL 2

FLUID MECHANICS

FM 1 Basic Concepts and Properties of Fluids FM 3

FM 2 Pressure and Fluid Statics FM 33

FM 3 Fluid Kinematics & Bernouli Equation FM 80

FM 4 Flow Analysis Using Control Volumes FM 124

FM 5 Flow Analysis Using Differential Method FM 172

FM 6 Internal Flow FM 211

FM 7 External Flow FM 253

FM 8 Open Channel Flow FM 289

FM 9 Turbo Machinery FM 328

HEAT TRANSFER

HT 1 Basic Concepts & Modes of Heat-Transfer HT 3

HT 2 Fundamentals of Conduction HT 34

HT 3 Steady Heat Conduction HT 63

HT 4 Transient Heat Conduction HT 94

HT 5 Fundamentals of Convection HT 114

HT 6 Free and Force Convection HT 129

HT 7 Radiation Heat Transfer HT 155

HT 8 Heat Exchangers HT 181

THERMODYNAMICS

TD 1 Basic Concepts and Energy Analysis TD 3

TD 2 Properties of Pure Substances TD 28

TD 3 Energy Analysis of Closed System TD 52

TD 4 Mass and Energy Analysis of Control Volume TD 76

TD 5 Second Law of Thermodynamics TD 106

TD 6 Entropy TD 136

TD 7 Gas Power Cycles TD 166

TD 8 Vapor and Combined Power Cycles TD 199

TD 9 Refrigeration and Air Conditioning TD 226

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CONTENTS VOL 3

MANUFACTURING PROCESS

INDUSTRIAL ENGINEERING

OPERATION RESEARCH

CONTENTS VOL 4

ENGINEERING MATHEMATICS

EM 1 Linear Algebra EM 1

EM 2 Differential Calculus EM 24

EM 3 Integral Calculus EM 46

EM 4 Directional Derivatives EM 67

EM 5 Differential Equation EM 79

EM 6 Complex Variable EM 103

EM 7 Probability and Statistics EM 123

EM 8 Numerical Methods EM 142

VERBAL ANALYSIS

VA 1 Synonyms VA 1

VA 2 Antonyms VA 16

VA 3 Agreement VA 26

VA 4 Sentence Structure VA 37

VA 5 Spellings VA 58

VA 6 Sentence Completion VA 87

VA 7 Word Analogy VA 111

VA 8 Reading Comprehension VA 135

VA 9 Verbal Classification VA 148

VA 10 Critical Reasoning VA 153

VA 11 Verbal Deduction VA 168

QUANTITATIVE ANALYSIS

QA 1 Number System QA 1

QA 2 Surds, Indices and Logarithm QA 14

QA 3 Sequences and Series QA 28

QA 4 Average, Mixture and Alligation QA 44

QA 5 Ratio, Proportion and Variation QA 59

QA 6 Percentage QA 75

QA 7 Interest QA 89

QA 8 Time, Speed & Distance QA 99

QA 9 Time, Work & Wages QA 112

QA 10 Data Interpretation QA 126

QA 11 Number Series QA 145

SOLVED PAPER

SP 1 Engineering Mathematics SP 3

SP 2 Engineering Mechanics SP 65

SP 3 Strength of Materials SP 90

SP 4 Theory of Machines SP 138

SP 5 Machine Design SP 189

SP 6 Fluid Mechanics SP 218

SP 7 Heat Transfer SP 265

SP 8 Thermodynamics SP 303

SP 9 Refrigeration and Air-Conditioning SP 358

SP 10 Manufacturing Engineering SP 375

SP 11 Industrial Engineering SP 448

SP 12 General Aptitude SP 496

SM 1STRESS AND STRAIN

Common Data For Q. 1 and 2A long wire of tungsten ( 190 /kN mT 3g = ) hangs vertically from a high-altitude balloon, is shown in figure.

SM 1.1 If the ultimate strength (or breaking strength) is 1500 MPa, the greatest length that it can have without breaking, is(A) 3950 m (B) 7900 m

(C) 1975 m (D) 790 m

SM 1.2 If the same wire hangs from a ship at sea ( 10 / )kN msea water 3g = , the greatest length is(A) 8300 m (B) 2075 m

(C) 7500 m (D) 3750 m

Common Data For Q. 3 and 4The 650 N load is applied along the centroidal axis of the member as shown in figure. Take 60cq = .

SM 1.3 The resultant internal normal and shear forces in the member at section a a- , which passes through point A, is(A) N 0= , V 0= (B) 50 NN = , 650 NV =(C) N 0= , 650 NV = (D) 650 NN = , V 0=


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SM 1.4 The resultant internal normal and shear forces in the member at section b b- , which passes through point A, is(A) 325 NN = , 563 NV = (B) 650 NN = , 563 NV =(C) 563 NN = , 325 NV = (D) 325 NN = , 1126 NV =

SM 1.5 In the figure shown, link BC of 6 mm thickness is made of a steel with a 450 MPa ultimate strength in tension. If the structure is being designed to support a 20 kN load P with a factor of safety of 3, its width w should be

(A) 13.9 mm (B) 55.6 mm

(C) 27.8 mm (D) 41.7 mm

SM 1.6 In figure shown, the two-member frame is subjected to the distributed loading. Member CB has a square cross section of 35 mm on each side and take 8 /kN mw =. The average normal stress and average shear stress acting at section b-b , are

(A) 4.41MPas = , 5.88 MPat = (B) 11.76 MPas = , 4.41MPat =(C) 8.82 MPas = , 5.88 MPat = (D) 5.88 MPas = , 4.41MPat =

Common Data For Q. 8 and 9A solid bar of circular cross section has a hole of diameter /d 4 drilled laterally through the center of the bar as shown in figure below. The allowable average tensile stress on the net cross section of the bar is allows .

SM 1.7 The formula for the allowable load Pallow that the bar can carrying in tension, is(A) . d0 27 allow2# s (B) . d0 54 allow2# s(C) . d0 675 allow2# s (D) . d0 54 allow# s


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SM 1.8 If the bar is made of brass with diameter 40 mmd = and 80 MPaallows = , the value of Pallow is(A) 86.5 kN (B) 70 kN

(C) 172 kN (D) 35 kN

SM 1.9 An axial load P is supported by a short 250 0.67W # column of cross-sectional area 8580 mmA 2= and is distributed to a concrete foundation by a square plate as shown in figure. If the average normal stress in the column must not exceed 150 MPa and the bearing stress on the concrete foundation must not exceed 12.5 MPa, the side a of the plate which will provide the most economical and safe design is

(A) 103 mm (B) 321 mm

(C) 8.6 mm (D) 160 mm

SM 1.10 The column shown in figure, is subjected to an axial force of 8 kN at its top. What is the average normal stress acting at section a -a ?

(A) 1.82 MPa (B) 3.64 MPa

(C) 0.91MPa (D) 2.73 MPa

SM 1.11 A round bar of 10 mm diameter is made of aluminum alloy, as shown in figure. When the bar is stretched by axial forces P , its diameter decreases by 0.016 mm. The magnitude of the load P is (Take 72 GPaE = , 0.33n = , 480 MPaYs = )

(A) 27.4 kN (B) 54.8 kN

(C) 13.7 kN (D) . kN37 7

SM 1.12 A steel bar of length 2.5 m with a square cross section 100 mm on each side is subjected to an axial tensile force of 1300 kN as shown in figure. The increase in





volume of the bar is (Take 250 MPaYs = , 200 GPaE = , .0 3n = )

(A) 8112 mm3 (B) 4868 mm3

(C) 3245 mm3 (D) 6490 mm3

Common Data For Q.14 and 15Three steel plates, each 16 mm thick, are joined by two 20 mm diameter rivets as shown in the figure.

SM 1.13 If the load 50 kNP = , the largest bearing stress acting on the rivets is(A) 39 MPa

(B) 156 MPa

(C) 78 MPa

(D) 117 MPa

SM 1.14 If the ultimate shear stress for the rivets is 180 ,MPa what force Pu is required to cause the rivets to fail in shear ? (Disregard friction between the plates.)(A) 170 kN (B) 57 kN

(C) 226 kN (D) 113 kN

SM 1.15 The small block of 5 mm thickness is shown in figure. If the stress distribution at the support developed by the load varies as shown, the force F applied to the block and the distance d to where it is applied, respectively, are

(A) 220 mm (B) 110 mm

(C) 165 mm (D) 55 mm

SM 1.16 The bar has a cross-sectional area of (400 10 )m6 2# - . If it is subjected to a uniform axial distributed loading along its length and to two concentrated loads as shown in figure, the average normal stress in the bar as a function of x for





0 0.5 mx< # , is

(A) . . MPax47 5 20 0-^ h (B) 67.5 MPax(C) . . MPax47 5 20 0+^ h (D) 27.5 MPax

Common Data For Q. 18 and 19In the figure shown, a hollow box beam ABC of length L is supported at end A by a 20 mm diameter pin that passes through the beam and its supporting pedestals. The roller support at B is located at distance /L 3 from end A.

SM 1.17 If load P is equal to 10 kN, the average shear stress in the pin is(A) 15.9 MPa (B) 31.8 MPa

(C) 63.6 MPa (D) 7.95 MPa

SM 1.18 If the wall thickness of the beam is equal to 12 mm, the average bearing stress between the pin and the box beam will be(A) 41.7 MPa (B) 125.1MPa

(C) 83.4 MPa (D) 20.85 MPa

SM 1.19 Rods AB and BC shown in figure, have diameters of 4 mm and 6 mm, respectively. The vertical load of 8 kN is applied to the ring at B . If the average normal stress in each rod is equivalent then this stress will be





(A) 237 MPa (B) 316 MPa

(C) 474 MPa (D) 158 MPa

Common Data For Q. 20 amd 21A steel plate ( /kN m77 3g = ) of dimensions 2.5 1.2 0.1 m# # is hoisted by a cable sling that has a clevis at each end as shown in figure. The pins through the clevises are 18 mm in diameter and are located 2.0 m apart. Each half of the cable is at an angle of 32c to the vertical.

SM 1.20 For above conditions, the average shear stress avert in the pins will be(A) 8.9 MPa (B) 6.7 MPa

(C) 13.4 MPa (D) 26.8 MPa

SM 1.21 The average bearing stress bs between the steel plate and the pins is(A) 22.7 MPa (B) . MPa15 2

(C) 7.57 MPa (D) 30.3 MPa

SM 1.22 Two solid cylindrical rods AB and BC are welded together at B and loaded as shown in figure. If the average normal stress must not exceed 150 MPa in either rod, the smallest allowable values of the diameters d1 and d2 are





(A) 45.2 mmd1 = , 20.1 mmd2 = (B) 22.6 mmd1 = , 40.2 mmd2 =(C) 20.1 mmd1 = , 45.2 mmd2 = (D) 40.2 mmd1 = , 22.6 mmd2 =

SM 1.23 Members AB and AC of the truss as shown, consist of bars of square cross section made of the same alloy. It is known that a 20 mm square bar of the same alloy was tested to failure and that an ultimate load of 120 kN was recorded. If a factor of safety of 3.2 is to be achieved for both bars, the required dimension of the cross section of the bar AB is

(A) 27 mma = (B) 12 mma =(C) 13.5 mma = (D) 6 mma =

SM 1.24 The two steel members are joined together using a 60c scarf weld as shown in figure. The average normal and average shear stress resisted in the plane of the weld are

(A) 8 MPaavgs = , 4.62 MPaavgt = (B) 4.62 MPaavgs = , 8 MPaavgt =(C) 4.62 MPaavgs = , 16 MPaavgt = (D) 16 MPaavgs = , 4.62 MPaavgt =

SM 1.25 A steel pipe of 300 mm outer diameter is fabricated from 6 mm thick plate by welding along a helix which forms an angle of 25c with a plane perpendicular to the axis of the pipe. If a 250 kN axial force P is applied to the pipe, the normal and shearing stresses in directions respectively normal and tangential to the weld are





(A) 18.5 MPa- , 17.28 MPa (B) 37.1MPa- , 34.56 MPa(C) 18.5 MPa- , 34.56 MPa (D) 37.1MPa- , 17.28 MPa

SM 1.26 A 6 kN load is supported by two wooden members of 75 125mm mm# uniform rectangular cross section which are joined by the simple glued scarf splice as shown in figure. The normal and shearing stresses in the glued splice respectively, are

(A) 565 kPa, 206 kPa (B) 282 kPa, 206 kPa

(C) 565 kPa, 103 kPa (D) 282 kPa, 103 kPa

SM 1.27 In the figure shown, the wooden members A and B are to be joined by plywood splice plates which will be fully glued on the surface in contact. If the clearance between the ends of the members is to be 8 mm and the average shearing stress in the glue is not to exceed 800 kPa, the smallest allowable length L will be

(A) 308 mm (B) 150 mm

(C) 300 mm (D) 292 mm





SM 1.28 In the figure shown, the frame is subjected to the distributed loading of 2 /kN m. What is the required diameter of the pins at A and B if the allowable shear stress for the material is 100 MPaallowt = ? Both pins are subjected to double shear.

(A) 2.6 mmd = (B) 7.8 mmd =(C) 5.2 mmd = (D) 10.4 mmd =

SM 1.29 A specially designed wrench is used to twist a circular shaft by means of a square key that fits into slots (or keyways) in the shaft and wrench as shown in the figure. The shaft has diameter ,d the key has a square cross section of dimensions b b# and the length of the key is c . The key fits half into the wrench and half into the shaft (i.e., the keyways have a depth equal to /b 2). When a load P is applied at distance L from the center of the shaft, the formula for the average shear stress avert in the key is( Hints : Disregard the effects of friction, assume that the bearing pressure between the key and the wrench is uniformly distributed)

(A) bc d b

PL2 +^ h (B) bc d bPL22 +^ h

(C) bc d b

PL23+^ h (D) bc d bPL24 +^ h

SM 1.30 The two wooden members shown in figure supports a 20 kN load, are joined by plywood splices fully glued on the surface in contact. The ultimate shearing stress in the glue is 2.8 MPa and the clearance between the members is 8 mm. If





a factor of safety of 3.5 is to be achieved, the required length L of each splice is

(A) 216 mm (B) 208 mm

(C) 200 mm (D) 104 mm

SM 1.31 A torque T0 is transmitted between two flanged shafts by means of four 20 mm bolts as shown in figure. The diameter of the bolt circle is 150 mmd = . If the allowable shear stress in the bolts is 90 MPa, the maximum permissible torque will be

(A) 16.96 kN m- (B) 8.48 kN m-

(C) 12.72 kN m- (D) 4.24 kN m-

SM 1.32 The cross section of an aluminium tube serving as a compression brace in the fuselage of a small airplane is shown in the figure. The outer diameter of the tube is 25 mmd = and the wall thickness is 2.5 mmt = . If the factors of safety with respect to the yield stress and the ultimate stress are 4 and 5 respectively, the allowable compressive force Pallow is (Take 270 MPaYs = , 310 MPaus = )

(A) 9.5 kN (B) 12.0 kN

(C) 11.0 kN (D) 13.7 kN

SM 1.33 In the figure shown, a long steel wire ( 77.0 /kN m3g = ) hanging from a balloon carries a weight W at its lower end. The 4 mm diameter wire is 25 m long. The tensile yield stress for the wire is 350 MPaYs = and a margin of safety against yielding of 1.5 is desired. The maximum weight Wmax that can safety be carried is (Include the weight of the wire in the calculations.)





(A) 1783 N (B) 1711 N

(C) 1759 N (D) 1735 N

SM 1.34 What is the smallest dimensions of the circular shaft and circular end cap if the load it is required to support is 150 kNP = ? The allowable tensile stress, bearing stress and shear stress is ( ) 175 MPat allows = , ( ) 275 MPab allows = and

115 MPaallowt = .

(A) 15.8 mmd1 = , 26.4 mmd3 = , 44.6 mmt =(B) 26.4 mmd1 = , 44.6 mmd3 = , 15.8 mmt =(C) 44.6 mmd1 = , 26.4 mmd3 = , 15.8 mmt =(D) 44.6 mmd1 = , 15.8 mmd3 = , 26.4 mmt =

SM 1.35 The assembly shown in figure, consists of three disks A, B and C are used to support the load of 140 kN. The allowable bearing stress for the material is ( ) 350 MPab allows = and allowable shear stress is 125 MPaallowt = . The smallest diameter d1 of the top disk, the diameter d2 within the support space and the diameter d3 of the hole in the bottom disk are





(A) 27.6 mmd1 = , 22.6 mmd2 = , 35.7 mmd3 =(B) 22.6 mmd1 = , 35.7 mmd2 = , 27.6 mmd3 =(C) 22.6 mmd1 = , 27.6 mmd2 = , 35.7 mmd3 =(D) 35.7 mmd1 = , 22.6 mmd2 = , 27.6 mmd3 =

SM 1.36 In the structure shown, an 8 mm diameter pin is used at A and 12 mm diameter pins are used at B and D . The ultimate shearing stress is 100 MPa at all connections and the ultimate normal stress is 250 MPa in each of the two links joining B and D . If an overall factor of safety of 3.0 is desired, the allowable load P is

(A) 7.7 kN (B) 14.04 kN

(C) 3.97 kN (D) 3.72 kN

SM 1.37 The bar shown in figure, is held in equilibrium by the pin supports at A and B. The support at A has a single leaf and therefore it involves single shear in the pin and the support at B has a double leaf and therefore it involves double shear. The allowable shear stress for both the pins is 125 MPaallowt = . If 1 mx = and

12 /kN mw = , the smallest required diameter of pins A and B are (Neglect any axial force in the bar.)

(A) 957 mmdA = , 20.6 mmdB = (B) 10.3 mmdA = , 9.57 mmdB =(C) 19.14 mmdA = , 10.3 mmdB = (D) 9.57 mmdA = , 10.3 mmdB =

SM 1.38 Two plates, each 3 mm thick, are used to splice a plastic strip as shown below. If the ultimate shearing stress of the bonding between the surface is 900 kPa and

1500 NP = , the factor of safety with respect to shear will be

(A) 2 (B) 3.6

(C) 5.4 (D) 1.8





SM 1.39 The cable shown in figure has a specific weight g (weight/volume) and cross-sectional area A. If the sag s is small, so that its length is approximately L and its weight can be distributed uniformly along the horizontal axis, the average normal stress in the cable at its lowest point C is

(A) sL2

2

s g= (B) sL

8sg=

(C) sL8

2

s g= (D) sL4

2

s g=

SM 1.40 An elastomeric bearing pad consisting of two steel plates bonded to a chloroprene elastomer, is subjected to a shear force V during a static loading test as shown in figure. The pad has dimensions 150 mma = , 250 mmb = and the elastomer has thickness 50 mmt = . When the force V equals 12 kN, the top plate is found to have displaced laterally 8.0 mm with respect to the bottom plate.

The shear modulus of elasticity G of the chloroprene is(A) 0.5 MPa (B) 1MPa

(C) 4 MPa (D) 2 MPa

SM 1.41 A metal bar AB of weight W is suspended by a system of steel wires arranged as shown in the figure. The diameter of the wire is 2 mm and the yield stress of the steel is 450 MPa. The maximum permissible weight Wmax for a factor of safety of 1.9 with respect to yielding, is





(A) N685 (B) 10 N28

(C) 1370 N (D) 2740 N

Common Data For Q. 42 and 43In figure shown below, link BD consists of a single bar 30 mm wide and 12 mm thick. Each pin has a 10 mm diameter.

SM 1.42 If 0cq = , the maximum value of the average normal stress in link BD is (A) zero (B) 72 MPa

(C) 24 MPa (D) 48 MPa

SM 1.43 If 90cq = , the maximum value of the average normal stress in link BD is (A) 83 MPa (B) 125 MPa

(C) 42 MPa (D) 44.5 MPa

SM 1.44 The rigid beam AC shown in figure, is supported by a pin at A and wires BD and CE . If the load P on the beam causes the end C to be displaced 10 mm downward, the normal strain developed in wires CE and BD are

(A) .0 00025CEe = , 0.0107BDe = (B) .0 0025CEe = , .0 00107BDe =(C) .0 025CEe = , 0.0107BDe = (D) .0 00107CEe = , .0 0025BDe =

SM 1.45 The rigid beam shown in figure, is supported by a pin at A and wires BD and CE . If the load P on the beam is displaced 10 mm downward, the normal strain developed in wires CE and BD are





(A) .1 43 10CE 3#e = - , .3 58 10BD 3#e = -(B) .1 43 10CE 3#e = - , .1 79 10BD 3#e = -(C) .1 79 10CE 3#e = - , .1 43 10BD 3#e = -(D) .2 86 10CE 3#e = - , .1 79 10BD 3#e = -

SM 1.46 A steel pipe is to carry an axial compressive load 1200 kNP = as shown in figure. A factor of safety of 1.8 against yielding is to be used. If the thickness t of the pipe is to be one-eighth of its outer diameter, the minimum required outer diameter dmin is (Take 270 MPaYs = )

.

(A) 153 mm (B) 76.5 mm

(C) 114.75 mm (D) 38.25 mm

Common Data For Q. 47 and 48A circular aluminum tube of length 400 mmL = is loaded in compression by forces P as shown in figure. The out-side and inside diameters are 60 mm and 50 mm, respectively. A strain gage is placed on the outside of the bar to measure normal strains in the longitudinal direction.

.

SM 1.47 If the measured strain is 550 10 6#e = - , the shortening d of the bar is(A) 0.220 mm (B) 2.20 mm

(C) 0.022 mm (D) 1.10 mm

SM 1.48 If the compressive stress in the bar is intended to be 40 MPa, the load P should be(A) 17.35 kN (B) 34.6 kN

(C) 69.4 kN (D) 52.0 kN





Common Data For Q. 49 and 50A suspender on a suspension bridge consists of a cable that passes over the main cable is shown in figure and supports the bridge deck, which is far below. The suspender is held in position by a metal tie that is prevented from sliding downward by clamps around the suspender cable. Let P represent the load in each part of the suspender cable and q represent the angle of the suspender cable just above the tie. Also, let allows represent the allowable tensile stress in the metal tie.

SM 1.49 The minimum required cross-section area of the tie is

(A) tanA Pminallows

q= (B) /tanA P2min allowq s=(C) /cotA P2min allowq s= (D) cotA Pmin

allowsq=

SM 1.50 If 130 kNP = , 75cq = and 80 MPaallows = , the minimum area will be(A) 6064 mm2 (B) 12130 mm2

(C) 435 mm2 (D) 870 mm2

SM 1.51 An elastomeric bearing pad consisting of two steel plates bonded to a chloroprene elastomer, is subjected to a shear force V during a static loading test as shown in figure. The pad has dimensions 150 mma = , 250 mmb = and the elastomer has thickness 50 mmt = . When the force V equals 12 kN, the top plate is found to have displaced laterally 8.0 mm with respect to the bottom plate.

The shear modulus of elasticity G of the chloroprene is(A) 0.5 MPa (B) 1MPa

(C) 4 MPa (D) 2 MPa

SM 1.52 Part of a control linkage for an airplane consists of a rigid member CBD and a





flexible cable AB . Originally the cable is unstretched. If a force is applied to the end D of the member and causes a normal strain in the cable of 0.0035 /mm mm, the displacement of point D is

(A) 21.9 mm (B) 4.38 mm

(C) 43.8 mm (D) 8.76 mm

Common Data For Q. 53 and 54The material distorts into the dashed position is shown in figure.

SM 1.53 The average normal strains xe , ye and the shear strain xyg at A are(A) 0x ye e= = , 0.0798 radxyg =(B) 0xe = , .0 00319ye = , 0.0798 radxyg =(C) .0 00319xe = , 0ye = , 0.0798 radxyg =(D) .0 00319x ye e= = , 0.0798 radxyg =

SM 1.54 The average normal strain along line BE is(A) 0.179 /mm mm- (B) 0.0179 /mm mm(C) 0.0179 /mm mm- (D) 0.179 /mm mm

SM 1.55 In the figure shown, the bar is originally 300 mm long when it is flat. It is subjected to a shear strain defined by . x0 02xyg = , where x is in millimeters. It is distorted into the shape shown, where no elongation of the bar occurs in the x direction. The displacement yD at the end of its bottom edge will be





(A) 2.03 mm (B) 4.06 mm

(C) 1.015 mm (D) 3.045 mm

Common Data For Q. 56 and 57The steel wires AB and AC support the 200 kg mass. The allowable axial stress for the wires is 130 MPaallows = . Take the unstretched length of AB to be 750 mm and 200 GPaEst = .

SM 1.56 The required diameter of each wire is(A) 3.23 mmdAC = , 7.08 mmdAB = (B) 3.54 mmdAC = , 3.23 mmdAB =(C) 3.23 mmdAC = , 3.54 mmdAB = (D) 6.46 mmdAC = , 3.54 mmdAB =

SM 1.57 What is the new length of wire AB after the load is applied ? (A) 749.51 mm (B) 750.49 mm

(C) 751 mm (D) 749.00 mm

SM 1.58 The plug shown in figure has a diameter of 30 mm and fits within a rigid sleeve having an inner diameter of 32 mm. Both the plug and the sleeve are 50 mm long. What is the axial pressure p , that must be applied to the top of the plug to cause it to contact the sides of the sleeve ? (Take 5 MPaE = , .0 45n = )

(A) 926.25 kPa (B) 370 kPa

(C) 741 kPa (D) 555.75 kPa

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SM 2AXIAL LOADING

Common Data For Linked Q. 1 and 2A 15 mm diameter rod is subjected to a 3.5 kN tensile force as shown in figure. An elongation of 11 mm and a decrease in diameter of 0.62 mm are observed in a 120 mm gage length.

SM 2.1 What will be the modulus of elasticity of the material ?(A) 216 MPa (B) 270 MPa

(C) 162 MPa (D) 108 MPa

SM 2.2 The Poissons ratio and Modulus of rigidity of the material respectively, are (A) .0 15, 93.125 MPa (B) .0 90, 36.75 MPa

(C) .0 60, 55.8 MPa (D) .0 45, 74.5 MPa

SM 2.3 An elastomeric bearing with 220 mmb = and 30 mma = is shown in figure. For a maximum lateral load 19 kNP = and a maximum displacement 12 mmd = , the shearing modulus G and the shear stress t respectively, are

(A) 1.08 MPa, 43 kPa2 (B) 0.54 MPa, 431 kPa

(C) 1.08 MPa, 862 kPa (D) 1.08 MPa, 6 kPa21





SM 2.4 A 200 mm length rod of steel is shown in figure. The dilatation e and the change in volume of the rod respectively, are (Take 200 GPaE = , .0 30n = )

(A) e 121 10 6#= - , 23 mmv 3D = (B) e 121 10 6#= - , 18.4 mmv 3D =(C) e 242 10 6#= - , 18.4 mmv 3D = (D) e 242 10 6#= - , 13.8 mmv 3D =

SM 2.5 For the axial loading shown in figure, the loading is hydrostatic with 70 MPax y zs s s= = =- . The change in height and the change in volume of the

brass cylinder respectively, are

(A) 0.031 mm3, 521 mm (B) 0.031 mm- , 521 mm3-(C) 0.031 mm, 521 mm3 (D) 0.031 mm3- , 521 mm-

Common Data For Q. 6 and 7A fabric is subjected to a biaxial loading that results in normal stresses

120 MPaxs = and 160 MPazs = as shown in figure. The properties of the fabric can be approximated as 87 GPaE = and .0 34n = .





SM 2.6 The change in length of sides AB and BC , are(A) 0.103 mmABd = , 0.075 mmBCd = (B) 0.075 mmABd = , 0.103 mmBCd =(C) 0.075 mmABd = , 0.206 mmBCd = (D) 0.150 mmABd = , 0.103 mmBCd =

SM 2.7 The change in length of diagonal AC , is(A) 0.61 mm (B) 1.22 mm

(C) 0.122 mm (D) 0.244 mm

SM 2.8 An aluminium plate is subjected to a centric axial load which causes a normal stress s as shown in figure. Before the loading, a line of slope :2 1 is scribed on the plate. When 125 MPas = , the slope of the line is (Take 74 GPaE = , .0 33n = )

(A) .1 995 (B) .19 95

(C) .0 1995 (D) .3 99

SM 2.9 In the figure shown, a bar AB having length L and axial rigidity EA is fixed at end A. At the other end a small gap of dimension s exists between the end of the bar and a rigid surface. A load P acts on the bar at point C , which is two-third of the length from the fixed end. If the support reactions produced by the load P are to be equal in magnitude, what should be the size s of the gap ?

(A) s EAPL4= (B) s EA

PL3=

(C) s EAPL6= (D) s EA

PL9=

SM 2.10 The T -shaped arm ABC shown in the figure lies in a vertical plane and pivots about a horizontal pin at A. The arm has constant cross-sectional area and total weight W . A vertical spring of stiffness k supports the arm at point B . What will be the elongation d of the spring due to the weight of the arm ?

(A) kW2d = (B) k

W43d =





(C) kW34d = (D) k

W4d =

SM 2.11 In the figure shown, a cable with a restrainer at the bottom hangs vertically from its upper end. The cable has an effective cross-sectional area 40 mmA 2= and an effective modulus of elasticity 130 GPaE = . A slider of mass 35 kgM = drops from a height 1.0 mh = onto the restrainer. If the allowable stress in the cable under an impact load is 500 MPa, what is the minimum permissible length L of the cable ?

(A) 7.4 mm (B) 9.25 mm

(C) 11.1 mm (D) 5.55 mm

SM 2.12 In the figure shown, a uniform bar AB of weight 25 NW = is supported by two springs. The distance between the springs is 350 mmL = and the spring on the right is suspended from a support that is distance 80 mmh = below the point of support for the spring on the left. At what distance x from the left-hand spring should a load 18 NP = be placed in order to bring the bar to a horizontal position ?

(A) 67.5 mm (B) 135 mm

(C) 270 mm (D) 202.5 mm

SM 2.13 A rigid bar ABCD is pinned at point B and supported by springs at A and D . A load P acts at point C is shown in figure. If the angle of rotation of the bar due to the action of the load P is limited to 3c, what will be the maximum permissible load (Pmax) ?





(A) 2.25 kN (B) 1.8 kN

(C) 1.35 kN (D) 0.9 kN

SM 2.14 The rigid bar shown in figure is supported by the two short white spruce wooden posts ( 9.65 GPaE = ) and a spring. The spring has a stiffness of 2 /MN mk = and an unstretched length of 1.02 m. If each of the posts has an unloaded length of 1 m and a cross-sectional area of 600 mm2, what will be the vertical displacement of A and B after the load is applied to the bar ?

(A) 4.42 mmA Bd d= = (B) 4.42 mAd = , 8.84 mmBd =(C) 2.21 mmAd = , 4.42 mmBd = (D) 4.42 mAd = , 2.21 mmBd =

SM 2.15 The aluminium strap as shown in figure, is subjected to an axial force of 30 kN. What is the elongation of the strap ? (Take 70 GPaEal = )

(A) 1.19 mm (B) 1.58 mm

(C) 0.79 mm (D) 2.37 mm

SM 2.16 The A-36 steel pipe ( 200 GPaE = ) has a 6061-T6 aluminum core ( 68.9 GPaE =). It is subjected to a tensile force of 200 kN as shown in figure. The pipe has an outer diameter of 80 mm and an inner diameter of 70 mm. What will be the average normal stress in the aluminium and the steel due to this loading ?





(A) 27.5 MPaals = , 39.9 MPasts =(B) 79.9 MPaals = , 27.5 MPasts =(C) 27.5 MPaals = , 79.9 MPasts =(D) 55 MPaals = , 79.9 MPasts =

SM 2.17 The column shown in figure, is constructed from high-strength concrete with 25 GPaEc = and four A-36 steel reinforcing rods with 200 GPaEst = . It is

subjected to an axial force of 800 kN. If one-fourth of the load is carried by the steel and three-fourth by the concrete, the required diameter of each rod will be

(A) 16.9 mm (B) 33.9 mm

(C) 11.3 mm (D) 28.2 mm

SM 2.18 In the figure shown, a round brass bar of diameter 20 mmd1 = has upset ends of diameter 26 mmd2 = . The lengths of the segments of the bar are 0.3 mL1 = and

0.1 mL2 = . Quarter-circular fillets ( .K 1 6. ) are used at the shoulders of the bar and the modulus of elasticity of the brass is 100 GPaE = . If the bar lengthens by 0.12 mm under a tensile load P , what is the maximum stress maxs in the bar ?

(A) 23 MPa (B) 69 MPa

(C) 46 MPa (D) 34.5 MPa

SM 2.19 The 10 mm diameter steel bolt is surrounded by a bronze sleeve as shown in figure. The outer diameter of this sleeve is 20 mm and its inner diameter is 10 mm. If the yield stress for the steel is ( ) 6 0 MPa4Y sts = and for the bronze ( ) 520 MPaY brs = , what will be the magnitude of the largest elastic load P that can be applied to the assembly ? (Take 200 GPaEst = and 100 GPaEbr = )





(A) 122 kN (B) 82 kN

(C) 126 kN (D) 204 kN

SM 2.20 In the figure shown, the joint is made from three A-36 steel plates that are bonded together at their seams. Each plates has a thickness of 5 mm and 200 GPaE = . What will be the displacement of end A with respect to end B when the joint is subjected to the axial loads as shown ?

(A) 2.45 mm (B) 4.91 mm

(C) 0.0491 mm (D) 0.491 mm

Common Data For Q. 21 to 23Two boards are joined by gluing along a scarf joint, as shown in figure. For purpose of cutting and gluing, the angle a between the plane of the joint and the faces of the boards must be between 10c and 40c. Under a tensile load P , the normal stress in the boards is 4.9 MPa.

SM 2.21 What are the normal and shear stresses acting on the glued joint if 20ca = ?(A) 0.57 MPas =q , 0.79 MPat =q (B) 0.57 MPas =q , 1.58 MPat =q(C) 1.58 MPas =q , 0.57 MPat =q (D) 1.14 MPas =q , 1.58 MPat =q

SM 2.22 If the allowable shear stress on the joint is 2.25 MPa, what is the largest permissible value of the angle a ?(A) .22 2c (B) .44 4c

(C) .33 3c (D) .27 8c





SM 2.23 For what angle a will the shear stress on the glued joint be numerically equal to twice the normal stress on the joint ?(A) 19.9c (B) 16.6c

(C) 33.2c (D) 26.6c

Common Data For Q. 24 and 25.In the shown figure, a 15 mm diameter steel shaft AC is supported by a rigid collar, which is fixed to the shaft at B . It is subjected to an axial load of 80 kN at its end.

SM 2.24 The uniform pressure distribution p on the collar required for the equilibrium, is(A) 10.9 MPa (B) 21.8 MPa

(C) 14.54 MPa (D) 32.7 MPa

SM 2.25 What is the elongation on segment BC and segment BA ?(A) 1.13 mmBC BAd d= = (B) 0BCd = , 1.13 mmBAd =(C) 1.13 mmBCd = , 0BAd = (D) 0BC BAd d= =

SM 2.26 A plastic bar ABC ( 4.0 GPaE = ) of length L consists of two parts of equal lengths but different diameters is shown in figure. Segment AB and BC have diameters 100 mmd1 = and 60 mmd2 = , respectively. Both segments have length /2 0.6 mL = . A longitudinal hole of diameter d is drilled through segment AB for

one-half of its length (distance /4 0.3 mL = ). Compressive loads 110 kNP = act at the ends of the bar. If the shortening of the bar is limited to 8.0 mm, what is the maximum allowable diameter dmax of the hole ?

(A) 11.9 mm (B) 7.96 mm

(C) 23.9 mm (D) 2.39 mm





Common Data For Linked Q. 27 and 28The elastoplastic rods AB and BC are made of mild steel with 200 GPaE = and

345 MPaYs = is shown in figure. The rods are stretched until end has moved down 9 mm. Neglect the stress concentrations.

SM 2.27 The maximum value of the force P is(A) 332 kN (B) 415 kN

(C) 166 kN (D) 249 kN

SM 2.28 What will be the permanent set measured at point A and B after the force has been removed ?(A) 0Ad = , 6.37 mmBd = (B) 6.37 mmA Bd d= =(C) 0Ad = , 0Bd = (D) 6.37 mmAd = , 0Bd =

SM 2.29 A post AB is tapered uniformly throughout its height H as shown in figure. The cross sections of the post are square with dimensions b b# at the top and . .b b1 5 1 5# at the base. Assume that the angle of taper is small and disregard

the weight of the post itself. The shortening d of the post due to the compressive load P acting at the top is

(A) EbPH

32

2d = (B) EbPH

23

2d =

(C) EbPH

43

2d = (D) EbPH

52

2d =





SM 2.30 A rectangular bar of length L has a slot in the middle half of its length as shown in figure. The bar has width b , thickness t and modulus of elasticity E . The slot has width of /b 4. What will be the expression for the elongation d of the bar due to the axial load P ?

(A) /L E8 7d s= (B) 7 /4L Ed s=(C) /L E7 6d s= (D) 7 /8L Ed s=

SM 2.31 A long slender bar in the shape of a right circular cone with length L and base diameter d hangs vertically under the action of its own weight is shown in figure. The weight of the cone is W and the modulus of elasticity of the material is E . Assume that the angle of taper of the cone is small. What will be the expression for the increase in the length of the bar due to its own weight ?

(A) /WL dE2d p= (B) /WL d E2 2d p=(C) /WL d E2 2d p= (D) /WL d E4 2d p=

Common Data For Linked Q. 32 and 33In the figure shown, rod AB consists of two cylindrical portions AC and BC , each with a cross-sectional area of 1750 mm2. Portion AC is made of mild steel with 200 GPaE = , 250 MPaYs = and portion BC is made of high-strength steel with 200 GPaE = , 345 MPaYs = . A load P is applied at point C . Assume both steels to be elastoplastic.





SM 2.32 If P is gradually increased from zero to 975 kN and then reduced back to zero, the maximum deflection of C is(A) 1.46 mm (B) 0.146 mm

(C) 0.292 mm (D) 2.92 mm

SM 2.33 The permanent deflection of C is(A) 0.027 mm (B) 0.27 mm

(C) 2.7 mm (D) 0.135 mm

Common Data For Q. 34 and 35In the figure shown, a plastic rod AB of length 0.5 mL = has a diameter 30 mmd1 =. A plastic sleeve CD of length 0.3 mc = and outer diameter 45 mmd2 = is securely bonded to the rod so that no slippage can occur between the rod and the sleeve. The rod is made of an acrylic with modulus of elasticity 3.1GPaE1 = and the sleeve is made of a polyamide with 2.5 GPaE2 = .

SM 2.34 When the rod is pulled by axial forces 12 kNP = , the elongation of the rod is(A) 1.09 mm (B) 1.91 mm

(C) 0.547 mm (D) 0.818 mm

SM 2.35 What will be the elongations respectively, when (a) the sleeve is extended for the fully length of the rod and (b) the sleeve is removed ?(A) 0.68 mm, 0.274 mm (B) 0.136 mm, 1.37 mm

(C) 1.36 mm, 2.74 mm (D) 2.72 mm, 0.137 mm

SM 2.36 The specimen as shown in figure has a allowable stress of 140 MPa and .K 2 10=. If the raised portions are removed ( .K 1 0= ) at the ends of the specimen, the percentage change in the maximum allowable magnitude of the centric load P , will be

(A) %90 (B) %110

(C) %10 (D) %11

SM 2.37 The fixed-end bar ABCD consists of three prismatic segments as shown in the figure. The end segments have cross-sectional area 840 mmA1 2= and length





200 mmL1 = . The middle segment has cross-sectional area 1260 mmA2 2= and length 250 mmL2 = . Loads PB and PC are equal to 25.5 kN and 17.0 kN, respectively. What will be the compressive axial force FBC in the middle segment of the bar ?

(A) 22.5 kN (B) 30 kN

(C) 15 kN (D) 7.5 kN

SM 2.38 The rod ABC consist of two portions is shown in figure. Both portions are made of aluminium for which 70 GPaE = and the magnitude of P is 4 kN. For zero deflection at A, the value of Q is

(A) 16.67 kN (B) 65.6 kN

(C) 21.87 kN (D) 32.8 kN

SM 2.39 A composite bar of square cross section with dimensions b b2 2# is constructed of two different metals having moduli of elasticity E1 and E2 as shown in figure The two parts of the bar have the same cross-sectional dimensions. The bar is compressed by forces P acting through rigid end plates. The line of action of the loads has an eccentricity e of such magnitude that each part of the bar is stressed uniformly in compression. What will be the eccentricity e of the loads ?

(A) eE E

b E E2 2 1

2 1= +-^^ hh (B) e E Eb E E2 2 12 1= -

+^^ hh(C) e

E Eb E E2 1 2

1 2= -+^^ hh (D) e E Eb E E2 1 21 2= +

-^^ hh





SM 2.40 A rigid bar of weight 800 NW = hangs from three equally spaced vertical wires, two of steel and one of aluminum is shown in figure. The wires also support a load P acting at the midpoint of the bar. The diameter of the steel wires is 2 mm and the diameter of the aluminium wire is 4 mm. If the allowable stress in the steel wires is 220 MPa and that is 80 MPa in the aluminum wire, what load Pallow can be supported ? (Take 210 GPaEst = , 70 GPaEal = )

(A) 3.0 kN (B) 1.5 kN

(C) 2.25 kN (D) 4.5 kN

SM 2.41 A plastic cylinder is held snugly between a rigid plate and a foundation by two steel bolts as shown in figure. Data for the assembly are as follows :Length 200 mmL = , pitch of the bolt threads 1.0 mmp = , modulus of elasticity for steel 200 GPaEs = , modulus of elasticity for the plastic 7.5 GPaEp = , cross-sectional area of one bolt 36.0 mmAs 2= , 1n = and cross-sectional area of the plastic cylinder 960 mmAp 2= .What will be the compressive stress ps in the plastic when the nuts on the steel bolts are tightened by one complete turn ?

(A) 31.25 MPa (B) 18.75 MPa

(C) 25 MPa (D) 15.63 MPa

SM 2.42 The uniform wire ABC of unstretched length l2 , is attached to the supports as shown in figure and a vertical load P is applied at the midpoint B . The cross-sectional area of the wire is denoted by A and the modulus of elasticity is by E . For l





(A) /l P AE2 /1 3d = ^ h (B) l AEP /1 2d = b l(C) l AE

P /1 3d = b l (D) l AEP/2 3d = b l

SM 2.43 A nylon thread is subjected to a 8.5 N tensile force. If 3.3 GPaE = and the length of the thread increases by . %1 1 , the diameter of the thread and the stress in the thread respectively, are(A) 5.46 mm, 36 MPa (B) 0.36 mm, 54.6 MPa

(C) 0.546 mm, 36 MPa (D) 3.6 mm, 546 MPa

SM 2.44 In the figure shown, the 4 mm diameter cable BC is made of steel with 200 GPaE = . If the maximum stress in the cable and the elongation of the cable

must not exceed 190 MPa and 6 mm respectively, the maximum load P can be

(A) 9.94 kN (B) 1.988 kN

(C) 0.994 kN (D) 19.88 kN

SM 2.45 The rigid bar is supported by the pin-connected rod CB of cross-sectional area of 14 mm2 as shown in figure and is made from 6061-T6 aluminium ( 68.9 GPaE =). What will be the vertical deflection of the bar at D when the distributed load is applied ?

(A) 12.97 mm (B) 21.63 mm

(C) 17.3 mm (D) 6.48 mm

SM 2.46 In the figure shown, the distributed loading is supported by the three suspender bars AB , EF and CD . The bars AB and EF are made from aluminium (

70 GPaEal = ) and CD is made from steel ( 200 GPaFst = ). Each bar has a cross-sectional area of 450 mm2. If allowable stress of ( ) 180 MPaallow sts = in the steel and ( ) 94 MPaallow als = in the aluminium is not exceeded, the maximum intensity w of the distributed loading will be





(A) 28.35 /kN m (B) 45.9 /kN m

(C) 120.8 /kN m (D) 68.5 /kN m

SM 2.47 The links AB and CD are made of aluminium ( 75 )GPaE = and has a cross-sectional area of 125 mm2 each. If they support the rigid member BC as shown in figure, the deflection of point E , is

(A) 0.055 mm (B) 1.10 mm

(C) 0.110 mm (D) 0.55 mm

SM 2.48 An axial force of 60 kN is applied to the assembly as shown by means of rigid end plates. The normal stress in the brass shell and the corresponding deformation of the assembly respectively, are

(A) 47.5 MPa, 0.113 mm (B) 47.5 MPa, 0.226 mm

(C) 22.6 MPa, 0.475 mm (D) 11.3 MPa, 0.475 mm

SM 2.49 The rigid beam as shown in figure, is supported by the three posts A, B and C . Posts A and C have a diameter of 60 mm and are made of aluminium, for which 70 GPaEal = and ( ) 20 MPaY als = . Post B is made of brass, for which

100 GPaEbr = and ( ) 590 MPaY brs = . If 130 kNP = , the largest diameter of post B so that all the post yield at the same time, is





(A) 3.56 mm (B) 1.78 mm

(C) 8.9 mm (D) 17.8 mm

Common Data For Q. 50 and 51.Three steel rods ( 200 )GPaE = supports a 36 kN load P as shown in figure. Each of the rod AB and CD has a 200 mm2 cross-sectional area and rod EF has a 625 mm2 cross-sectional area.

SM 2.50 The change in length of rod EF is(A) 0.762 mm (B) 0.0762 mm

(C) 0.1524 mm (D) 7.62 mm

SM 2.51 What will be the stresses in rods AB and EF ?(A) 30.5 MPaABs = , 38.1MPaEFs = (B) 15.25 MPaABs = , 38.1MPaEFs =(C) 30.5 MPaABs = , 19.05 MPaEFs = (D) 38.1MPaABs = , 30.5 MPaEFs =

Common Data For Linked Q. 52 and 53Consider the figure shown.

SM 2.52 What will be the compressive force in the bars shown, after a temperature rise





of 96 Cc ?(A) 162.75 kN (B) 217.3 kN6

(C) 271.25 kN (D) 108.5 kN

SM 2.53 The corresponding change in length of the bronze bar is(A) 0.1823 mm (B) 0.1215 mm

(C) 0.24 mm25 (D) 0.081 mm

SM 2.54 In the figure shown, a bar AB of length L is held between rigid supports and heated non-uniformly in such a manner that the temperature increase TD at distance x from end A. The temperature increase is given by the expression

/T T x LB 3 3D D= , where TBD is the increase in temperature at end B of the bar. If the material has modulus of elasticity E and coefficient of thermal expansion a, What is the expression for the compressive stress cs in the bar ?

(A) E T

8cBs a D= ^ h (B) E T2c Bs a D= ^ h

(C) E T

6cBs a D= ^ h (D) E T4c Bs a D= ^ h

SM 2.55 Five bars, each having a diameter of 10 mm support a load P is shown in figure. If the material is elastoplastic with yield stress 250 MPaYs = , the plastic load Pp is

(A) 55 kN (B) 82.5 kN

(C) 110 kN (D) 41.25 kN

***********

SM 3TORSION

SM 3.1 The torque which may be applied to a solid shaft of 90 mm outer diameter without exceeding an allowable shearing stress of 75 MPa, is(A) 21.6 kN m- (B) 10.8 kN m-

(C) 16.3 kN m- (D) 5.4 kN m-

SM 3.2 In the figure shown, the link acts as part of the elevator control for a airplane. The attached aluminum tube has an inner diameter of 25 mm and a wall thickness of 5 mm. What will be the maximum shear stress in the tube when the cable force of 600 N is applied to the cables ?

(A) 7.25 MPa (B) 20.6 MPa

(C) 10.3 MPa (D) . MPa14 5

SM 3.3 The solid rod BC has a diameter of 30 mm and is made of aluminum for which the allowable shearing stress is 25 MPa. Rod AB is hollow and has an outer diameter of 25 mm. It is made of brass for which the allowable shearing stress is 50 MPa. Which of the following is the largest inner diameter of rod AB for which the factor of safety is the same for each rod ?

(A) 11.39 mm (B) 7.59 mm

(C) 5.7 mm (D) 15.18 mm

Common Data For Q. 4 and 5.The steel shaft of a socket wrench has a diameter of 8.0 mm and a length of 200 mm is shown in figure. The allowable stress in shear is 60 MPa and 78 GPaG = .

SM 38 Torsion SM 3




SM 3.4 What is the maximum permissible torque Tmax that may be exerted with the wrench?(A) 6.03 N m- (B) 7.53 N m-

(C) 4.53 N m- (D) 3.76 N m-

SM 3.5 Through what angle f will the shaft twist under the action of the maximum torque? (Disregard any bending of the shaft.)(A) 2.75c (B) 1.65c

(C) 2.20c (D) 1.37c

SM 3.6 The solid brass rod ( 39 )GPaAB G = is bonded to the solid aluminum rod ( 27 )GPaBC G = as shown in figure. The angle of twist at A and B are

(A) 0.741 , 1.573A Bc cf f= = (B) 0.831 , 0.741A Bc cf f= =(C) 0.741 , 0.831A Bc cf f= = (D) 1.573 , 0.741A Bc cf f= =

SM 3.7 A plastic bar of diameter 50 mm is to be twisted by torques T as shown in figure, until the angle of rotation between the ends of the bar is .5 0c. If the allowable shear strain in the plastic is 0.012 rad, what is the minimum permissible length of the bar?

(A) 113.75 mm (B) 136.5 mm

(C) 182 mm (D) 227.5 mm

Common Data For Q. 9 and 10An aluminum bar of solid circular cross section is twisted by torques T acting at

SM 3 Torsion SM 39




the ends as shown in figure. The dimensions and shear modulus of elasticity are as follows: 1.2 mL = , 30 mmd = and 28 GPaG = .

SM 3.8 The torsional stiffness of the bar is(A) 1860 N m- (B) 1395 N m-

(C) 2325 N m- (D) 1163 N m-

SM 3.9 The design specifications of a 2 m long solid circular transmission shaft require that the angle of twist of the shaft not exceed 3 when a torque of 9 kN-m is applied. Which of the following is the required diameter of the shaft if the shaft is made of a steel with an allowable shearing stress of 90 MPa and a modulus of rigidity of 77 GPa ?(A) 41.06 mm (B) 79.9 mm

(C) 39.9 mm (D) 82.1 mm

Common Data For Q. 10 and 11A circular tube of outer diameter 70 mmd3 = and inner diameter 60 mmd2 = is welded at the right-hand end to a fixed plate and at the left-hand end to a rigid end plates as shown in figure. A solid circular bar of diameter 40 mmd1 = is inside of and concentric with the tube. The bar passes through a hole in the fixed plate and is welded to the rigid end plate.The bar is 1.0 m long and the tube is half as long as the bar. A torque 1000 N mT -= acts at end A of the bar. Also, both the bar and tube are made of an aluminum alloy with shear modulus of elasticity 27 GPaG = .

SM 3.10 The maximum shear stresses in both the bar and tube are(A) 79.6 , 64.6MPa MPabar tubet t= = (B) 32.3 , 79.6MPa MPabar tubet t= =(C) 79.6 , 32.3MPa MPabar tubet t= =(D) 39.8 , 32.3MPa MPabar tubet t= =

SM 3.11 The angle of twist at end A of the bar is(A) 7.07c (B) 9.43c

SM 40 Torsion SM 3




(C) 11.79c (D) 5.89c

SM 3.12 The composite shaft as shown is to be twisted by applying a torque T at end A. The modulus of rigidity is 77 GPa for the steel and 27 GPa for the aluminum. If the allowable stresses are not to be exceeded 60 MPasteelt = and 45 MPaaluminumt =,the largest angle through which end A may be rotated, is

(A) 4.13c (B) 1.65c

(C) 2.65c (D) 6.63c

SM 3.13 The torques TA and TB as shown, are exerted on pulleys A and B which are attached to solid circular shafts AB and BC. In order to reduce the total mass of the assembly, which of the following is the smallest diameter of shaft BC for which the largest shearing stress in the assembly is not increased ?

(A) 29.9 mm (B) 39.8 mm

(C) 49.7 mm (D) 24.9 mm

SM 3.14 A uniformly tapered tube AB of hollow circular cross section is shown in the figure. The tube has constant wall thickness t and length L. The average diameters at the ends are dA and d d2B A= . The polar moment of inertia is represented by the approximate formula /4J d t3. p . What will be the angle of twist f of the tube when it is subjected to torques T acting at the ends ?

.

(A) GtdTL

23

A4f p= (B) Gtd

TL43

A3f p=

SM 3 Torsion SM 41




(C) GtdTL

43

A4f p= (D) Gtd

TL2

3A3f p=

SM 3.15 For a given allowable stress, which of the following is the ratio T/w of the maximum allowable torque T and the weight per unit length w for the hollow shaft shown ?

(A) ccc

2 1all2

2212

gt -c m (B) 2c cc1all2 221

2

gt +c m

(C) 2c

cc1all1

2212

gt +c m (D) 2c cc1all1 221

2

gt -c m

SM 3.16 A prismatic bar AB of length L and solid circular cross section of diameter d is loaded by a distributed torque of constant intensity t per unit distance as shown in figure. What will be the angle of twist f between the ends of the bar ?

(A) GdtL4

4

2f p= (B) GdtL16

4

2f p=

(C) GdtL12

4

2f p= (D) GdtL8

4

2f p=

SM 3.17 A solid circular bar of diameter 50 mmd = shown in figure, is twisted by a torque 500 N mT -= . At this value of torque, a strain gage oriented at 45c to the axis

of the bar gives a reading 339 10 6e #= - . What is the shear modulus G of the material?

(A) 22.5 GPa (B) GPa30

(C) . GPa37 5 (D) GPa45

SM 3.18 The drive shaft for a truck of outer diameter 60d mm2 = and inner diameter 40d mm1 = is running at 2500 rpm as shown in figure. If the shaft transmits 150

kW, what is the maximum shear stress in the shaft?

SM 42 Torsion SM 3




(A) 10.5 MPa (B) . MPa12 6

(C) . MPa16 8 (D) MPa21

Common Data For Q. 19 and 20.The solid aluminum shaft has a diameter of 50 mm and an allowable shear stress of 6 MPaallowt = . The largest torque T1 is applied to the shaft and it is also subjected to the other torsional loadings. It is required that T1 act in the direction shown.

SM 3.19 What will be the largest torque T1 ?(A) 215 N m- (B) 2 N m58 -

(C) N m172 - (D) N m129 -

SM 3.20 The maximum shear stress within regions CD and DE are(A) 2 MPaCDt = , 2. MPa58DEt = (B) 2.58 MPaCDt = , 2 MPaDEt =(C) MPa4CDt = , 2. MPa58DEt = (D) . MPa5 16CDt = , MPa4DEt =

SM 3.21 The 60 mm diameter solid shaft is subjected to the distributed and concentrated torsional loadings as shown in figure. The shear stress at points A and B are

(A) . MPa18 86At = , . MPa14 1Bt = (B) . MPa9 43At = , . MPa14 1Bt =(C) . MPa9 43At = , . MPa7 05Bt = (D) . MPa18 86At = , . MPa7 05Bt =

SM 3.22 The steel shafts are connected together using a fillet weld as shown in figure. If the torque applied to the shafts is 60 N mT -= , the average shear stress in the weld along the critical section a-a is

SM 3 Torsion SM 43




(A) 1.17 MPa (B) . MPa0 87

(C) 1. 7 MPa4 (D) . MPa0 735

SM 3.23 The propellers of a ship are connected to a solid steel ( 75GPaG = ) shaft that is 60 m long and has an outer diameter of 340 mm and inner diameter of 260 mm. If the power output is 4.5 MW when the shaft rotates at 20 rad/s, the maximum torsional stress in the shaft and its angle of twist respectively, are(A) 22.15 , .MPa 11 9c (B) 22.15 , .9MPa 5 6c

(C) . , .9MPa44 3 5 6c (D) . , 11.9MPa44 3 c

SM 3.24 The 8 mm diameter bolt with 75 GPaG = is screwed tightly into a block at A as shown in figure. What will be the couple force F that should be applied to the rigid wrench and the corresponding displacement of each force F , needed to cause 18 MPa of maximum shear stress in the bolt ?

(A) 12.06 , 0.720N mm (B) 6.03 , 0.720N mm

(C) 12.06 , 0.0720N mm (D) 6.03 , 0.0720N mm

SM 3.25 The steel jacket CD has been attached to the 40 mm diameter steel shaft AE by means of rigid flanges welded to the jacket and to the rod. The outer diameter of the jacket is 80 mm and its wall thickness is 4 mm. If 500 N-m torques are applied as shown in figure, the maximum shearing stress in the jacket is

SM 44 Torsion SM 3




(A) 12.24 MPa (B) 7.65 MPa

(C) 15.3 MPa (D) 9.18 MPa

SM 3.26 A hollow shaft is to transmit 250 kW at a frequency of 30 Hz. The shearing stress must not exceed 50 MPa. What will be the outer diameter of the shaft for which the ratio of the inner diameter to the outer diameter is 0.75 ?(A) 29.12 mm (B) 72.75 mm

(C) 43.65 mm (D) 58.2 mm

SM 3.27 The assembly is made of A-36 steel ( 75GPaG = ) and consists of a solid rod of 15 mm diameter connected to the inside of a tube using a rigid disk at B as shown in figure. If the tube has an outer diameter of 30 mm and wall thickness of 3 mm, the angle of twist at A will be

(A) 0.90c (B) .2 70c

(C) .3 60c (D) . 01 8 c

SM 3.28 A propeller shaft of solid circular cross section and diameter d, is spliced by a collar of the same material as shown in figure. The collar is securely bonded to both parts of the shaft. What should be the minimum outer diameter d1 of the collar in order that the splice can transmit the same power as the solid shaft ?

(A) . d1 49 (B) . d0 819

(C) 1. d221 (D) . d0 794

SM 3.29 A solid circular bar ABCD with fixed supports, is acted upon by torques T0 and T2 0 at the locations as shown in the figure. The maximum angle of twist maxf of

the bar is

(A) GJT L3max

0f = (B) GJT L

43

max0f =

(C) GJT L

53

max0f = (D) GJ

T L56

max0f =

SM 3.30 A steel pipe of 60 mm outer diameter is to be used to transmit a torque of 350 N-m. A series of 60 mm outer-diameter pipes is available for use. The wall

SM 3 Torsion SM 45




thickness of the available pipes varies from 4 mm to 10 mm in 2 mm increments. If the allowable shearing stress is not to be exceeded 12 MPa , the thickness of the pipe will be

(A) 7 mm (B) 8 mm

(C) 6 mm (D) 10 mm

SM 3.31 A steel shaft transmits 150 kW at a speed of 360 rpm and for steel 77 GPaG = . If the allowable maximum stress and the angle of twist in a 2.5 m length will not exceed 50 MPa and 3c respectively, the diameter of the shaft will be(A) 35.4 mm (B) 37 mm

(C) 74 mm (D) 70.8 mm

Common Data For Q. 36 and 37.In given figure, the shaft is made of red brass with GPaG 37= and has an elliptical cross section. It is subjected to the torsional loading.

.

SM 3.32 The maximum shear stress within regions AC and BC are(A) 0.955 MPaACt = , 1.59 MPaBCt = (B) 1.59 MPaACt = , 1.91MPaBCt =(C) 0.795 MPaACt = , 0.955 MPaBCt =(D) 1.59 MPaACt = , 0.955 MPaBCt =

SM 3.33 The angle of twist of end B relative to end A is(A) .0 207c (B) .2 07c

(C) .0 414c (D) .1 04c

SM 3.34 In the figure shown, a solid circular shaft AB of diameter d is fixed against rotation at both ends and a circular disk is attached to the shaft at the location shown. If the allowt is the allowable shear stress in the shaft and assume that a b>, what is the largest permissible angle of rotation maxf of the disk ?

SM 46 Torsion SM 3




(A) /b Gd2max allow 3f t= ^ h (B) /b Gd2max allow 2f t= ^ h(C) /b Gd2max allow 4f t= ^ h (D) /b Gd2max allowf t= ^ hCommon Data For Q. 35 and 36A solid shaft of 54 mm diameter is made of mild steel which is assumed to be elastoplastic with 145 MPaYt = .

SM 3.35 What will be the maximum shearing stress and the radius of the elastic core respectively, caused by the application of a torque of magnitude 4 kN-m ? (A) 129.4 , 27MPa mm (B) 129.4 , 54MPa mm

(C) 64.7 , 27MPa mm (D) 64.7 , 54MPa mm

SM 3.36 What will be the maximum shearing stress and the radius of the elastic core respectively, if a torque of 5 kN-m is applied ?(A) 145 , 46.8MPa mm (B) 145 , 23.4MPa mm

(C) 290 , 23.4MPa mm (D) 290 , 46.8MPa mm

SM 3.37 A hollow steel shaft ACB of outside diameter 50 mm and inside diameter 40 mm is held against rotation at ends A and B as shown in figure. Horizontal forces P are applied at the ends of a vertical arm that is welded to the shaft at point C. If the maximum permissible shear stress in the shaft is 45 MPa, the allowable value of the forces P will be

(A) 1693.75 N (B) .5 N2032

(C) N2710 (D) . N3387 5

SM 3.38 A high-strength steel symmetric tube, having the mean dimensions is shown in figure and a thickness of 5 mm. If it is subjected to a toque of 40 N mT -= , what will be the average shear stress developed at points A and B ?

SM 3 Torsion SM 47




(A) 357 PakA Bt t= = (B) 357 PakAt = , 178.5 PakBt =(C) 178.5 PakA Bt t= = (D) 178.5 PakAt = , 357 PakBt =

SM 3.39 In the figure shown, the inner circle of the tube is eccentric with respect to the outer circle, due to a fabrication error. By what percentage is the torsional strength reduced when the eccentricity e is one-fourth of the difference in the radii ?

(A) %50 (B) %75

(C) %25 (D) No change

Common Data For Q. 40 and 41A solid steel bar of diameter 25.0 mmd1 = is enclosed by a steel tube of outer diameter 37.5 mmd3 = and inner diameter 30.0 mmd2 = as shown in figure. Both bar and tube are held rigidly by a support at end A and joined securely to a rigid plate at end B. The composite bar which has a length 550 mmL = , is twisted by a torque 400 N mT -= acting on the end plate.

SM 3.40 The maximum shear stresses 1t and 2t in the bar and tube respectively, are(A) 32.7 MPa1t = , MPa492t = (B) MPa491t = , . MPa32 72t =(C) . MPa64 71t = , MPa492t = (D) 32.7 MPa1t = , . MPa24 52t =

SM 3.41 If the shear modulus of the steel is 80 GPaG = , the torsional stiffness kT of the composite bar is(A) . kN m16 7 - (B) . kN m13 95 -

SM 48 Torsion SM 3




(C) . kN m27 9 - (D) . kN m22 3 -

SM 3.42 The solid circular drill rod AB is made of a steel which is assumed to be elastoplastic with 160 MPaYt = and 77 GPaG = as shown in figure. If a torque 5 kN mT -= is applied to the rod and then removed, the maximum residual shearing stress in the rod is

(A) 44.9 MPa (B) 204 MPa

(C) 43.7 MPa (D) 115 MPa

SM 3.43 A 1.25 m long steel angle has an 127 76 6.4L # # cross section is shown in figure. The thickness of the section is 6.4 mm and its area is 1252 mm2. Neglect the effect of stress concentrations. What will be the largest torque T which may be applied and the corresponding angle of twist respectively ? (Take 60 MPaallt = , 77 GPaG = )

(A) 157 N m- , 8.72c (B) 196.25 N m- , 5.45c

(C) 117.75 N m- , 10.9c (D) 98.125 N m- , 6.54c

Common Data For Q. 44 and 45A solid shaft has a diameter of 40 mm, length of 1 m and 80 GPaG = . It is made from an elastic-plastic material having a yield stress of 100 MPaYt = .

SM 3.44 What will be the maximum elastic torque TY and the corresponding angle of twist respectively ?(A) 2.52 , .kN m 3 58c- (B) . , .kN m1 26 1 79c-

(C) . , 3.58kN m1 26 c- (D) 2.52 , .kN m 1 79c-

SM 3.45 What is the angle of twist if the torque is increased to .T T1 2 Y= ?(A) .3 64c (B) .4 86c

(C) .6 08c (D) 3.04c

SM 3 Torsion SM 49




Common Data For Q. 46 and 47.

Equal torques are applied to thin-walled tubes of the same length L, same thickness t and same radius c. One of the tubes has been slit lengthwise as shown in figure.

SM 3.46 What is the ratio /b at t of the maximum shearing stresses in the tubes ?(A)

tc32

2

(B) tc3

(C) tc3 2 (D)

tc32

SM 3.47 The ratio /b af f of the angles of twist of the shafts is(A) t

c3 (B) tc32

2

(C) tc3 2 (D)

tc32

SM 3.48 A steel tube having the cross section as shown in the figure. The tube has length 1.5 mL = and is subjected to a torque 10 kN mT -= . What will be the shear

stress t and the angle of twist f respectively ? (Take 76 GPaG = )

(A) 1 . , 0.MPa7 5 285c (B) , 0.MPa35 285c

(C) 1 . , 0.MPa7 5 570c (D) , 0.MPa35 570c

***********

SM 4SHEAR FORCE AND BENDING MOMENT

SM 4.1 For the beam with overhangs as shown in figure, if one load acts downward and the other upward, the shear force V and bending moment M at the midpoint of the beam will be

(A) ,V LbP M2 0= = (B) ,V L

bP M Lb P2 2 2= =

(C) , 0V LbP M2= = (D) ,V L

bP M Lb P

2 22= =

SM 4.2 Two metric rolled-steel channels are to be welded along their edges and are used to support the loading as shown in figure. If the allowable normal stress for the steel is 150 MPa, the section modulus of the beam is

(A) mm45 103 3#^ h (B) mm90 103 3#^ h(C) mm180 103 3#^ h (D) mm135 103 3#^ h

SM 4.3 In figure shown, the beam is subjected to the load P at its center. What will be the placement a of the supports for maximum moment and the absolute maximum bending stress in the beam ?

(A) 0, bdPL

23

2 (B) L4 , bd

PL32

2

(C) L2 , bdPL

32

3 (D) L4

3 , bdPL

23

3

SM 4.4 In figure shown, the beam ABC is simply supported at A and B , and has an overhang from B to C . The loads consist of a horizontal force 4.0 kNP1 = acting





at the end of a vertical arm and a vertical force 8.0 kNP2 = acting at the end of the overhang. If the widths of the beam and vertical arm are neglected, the shear force V and bending moment M , at a cross section located 3.0 m from the left-hand support using centerline dimensions, will be

(A) . , .kN kN mV M1 0 7 0 -=- =- (B) . , .kN kN mV M1 0 7 0 -= =(C) . , .kN kN mV M1 0 7 0 -= =- (D) . , .kN kN mV M1 0 7 0 -=- =

SM 4.5 Consider the beam and loading of the 1 0 1 .S 0 1 5# rolled steel section as shown in figure. If the section modulus is . 10 mS 49 6 6 3#= - , the maximum normal stress due to bending will be

(A) . MPa19 35 (B) . MPa9 65

(C) . MPa38 7 (D) . MPa29 03

SM 4.6 For the beam and loading shown in figure, if the grade of timber used has an allowable normal stress of 12 MPa, the width b of the beam is

(A) mm96 (B) mm24

(C) mm32 (D) mm48

SM 4.7 In figure shown, the beam AB supports a uniformly distributed load of 2 /kN m and two concentrated loads P and Q . The normal stress due to bending on the bottom edge of the beam is 56.9 MPa- at A and 29.9 MPa- at C . What will be the magnitudes of the loads P and Q ?





(A) 250 , 500N NP Q= = (B) 50 , 50N NP Q0 2= =(C) , 500N NP Q1000= = (D) 50 , 00N NP Q0 10= =

SM 4.8 What will be the equations of the shear force and bending-moment curves for the beam and loading shown in figure ?

(A) ,cos sinV w L Lx M w L L

x0 22

02

pp

pp= =

(B) ,cos sinV w L Lx M w L L

x02

02

pp

pp= =

(C) ,sin cosV w L Lx M w L L

x0 0 2pp

pp= =

(D) ,sin cosV w L Lx M w L L

x0 2 0 2 2pp

pp= =

Common Data For Q. 9 and 10Beams AB, BC and CD have the cross sections as shown in the figure and are pin-connected at B and C. The allowable normal stress is 110 MPa in tension and

150 MPa- in compression.

SM 4.9 If beam BC is not to be over stressed, what will be the largest permissible value of P ?(A) . kN4 01 (B) . kN5 02

(C) .01 kN3 (D) . kN8 02

SM 4.10 What will be the corresponding maximum distance a for which the cantilever beams AB and CD are not to be over stressed ?(A) 2.45 m (B) . m6 54

(C) . m4 09 (D) . m3 27

SM 4.11 Consider a timber beam of length 16 mL = and width 75 mmb = as shown in figure. The dead load carried by each beam, including the estimated weight of the beam, is a uniformly distributed load 350 /N mwD = . The live loads can be represented by a uniformly distributed load 600 /N mwL = . If a 6 kN concentrated load P applied at the midpoint C of each beam, what will be the minimum allowable depth h of the beams using LRFD (Load and Resistance Factor Design) ? : 50 , 1.2, 1.6 .Use Data MPa and 0 9U D Ls g g f= = = =





(A) 312.72 mm

(B) 383. mm0

(C) . mm541 64

(D) . mm270 82

SM 4.12 In figure shown, the aluminium machine part is subjected to a moment of 75 N mM -= . What will be the bending stress created at points B and C on the

cross section ?

(A) 3.61 , .MPa MPa1 55B Cs s= =(B) 3.61 , 1.55MPa MPaB Cs s=- =(C) 3.61 , 1.55MPa MPaB Cs s= =-(D) 1.55 , 3.61MPa MPaB Cs s= =

SM 4.13 The aluminium strut has a cross-sectional area as shown in figure. If it is subjected to a moment 8 kN mM -= , the bending stress acting at points A and B are

(A) . , .MPa MPa4 49 49 4A Bs s= =(B) 4.49 , 49.4MPa MPaA Bs s=- =(C) 4 .4 , 4. 9MPa MPa9 4A Bs s= =-(D) 4 .4 , 4. 9MPa MPa9 4A Bs s= =

SM 4.14 The member shown in the figure is subjected to an internal bending moment of 40 kN mM -= . What will be the largest bending stress developed in it ?





(A) 161.25 MPa (B) . MPa96 75

(C) . MPa129 0 (D) . MPa80 63

SM 4.15 The smooth pin of 20 mm diameter is supported by two leaves A and B and subjected to a compressive load of 0.4 kN as shown in figure. What will be the absolute maximum bending stress in the pin ?

(A) 248.25 kPa (B) 331.0 kPa

(C) . kPa413 75 (D) . kPa206 85

Common Data For Q. 16 and 17A shaft made of a polymer having an elliptical cross-section is shown in figure. It resists an internal moment of 50 N mM -= .

SM 4.16 What will be the maximum bending stress developed in the material using the flexure formula, where (0.08 )(0.04 )m mIz 4

1 3p= ?(A) 372.75 kPa (B) . kPa310 65

(C) . kPa621 25 (D) . kPa497 0

SM 4.17 What will be the maximum bending stress developed in the material using integration ?(A) . kPa621 25 (B) . kPa497 0

(C) . kPa310 65 (D) 372.75 kPa





Common Data For Q. 18 and 19For the beam and loading of the 310 52S # rolled steel shape as shown in figure, the section modulus is 10 mS 625 6 3#= - .

SM 4.18 What will be the magnitude and location of the maximum bending moment, respectively using singularity functions ?(A) 18 ,1.94kN m m-- (B) . ,1.94kN m m28 28 -

(C) ,1.94kN m m27 -- (D) 18 ,1.94kN m m-

SM 4.19 The maximum normal stress due to bending is(A) 56.5 MPa (B) . MPa33 9

(C) . MPa45 2 (D) . MPa28 25

Common Data For Q. 20 and 21A timber beam is supported and loaded as shown in figure. The available stock consists of beams with a 12 MPa allowable stress and a rectangular cross section of 30 mm width and depth h varying from 80 to 160 mm in 10 mm increments.

SM 4.20 What will be the magnitude and location of the maximum bending moment, respectively for the beam and loading using singularity functions ?(A) . ,4.kN m m0 872 0-

(B) 0.872 , .kN m m1 5-

(C) 0.872 , .0kN m m2 94-

(D) 0.872 ,3.kN m m0-

SM 4.21 What will be the most suitable depth h that can used ?(A) 140 mm (B) 1 0 mm3

(C) 1 0 mm1 (D) 1 0 mm2

Common Data For Q. 22 and 23In the figure shown, a beam is made of a material that has a modulus of elasticity Ec in compression and Et in tension.





SM 4.22 The location c of the neutral axis, is

(A) E Eh Et c

c

- (B) E Eh Et c

c

+

(C) E Eh Et c

t

- (D) E Eh Et c

t

+SM 4.23 If it is subjected to the bending moment M , an expression for the maximum

tensile stress in the beam having the dimensions shown, will be

(A) bhM

EE E3

c

t c2

-e o (B) bhM

EE E3

t

t c2

+e o(C)

bhM

EE E3

t

t c2

-e o (D) bhM

EE E3

c

t c2

+e oSM 4.24 The 65 mm diameter steel shaft is subjected to the two loads that act in the

directions as shown in the figure. If the journal bearings at A and B do not exert an axial force on the shaft, the absolute maximum bending stress developed in the shaft will be

(A) 203.75 MPa (B) . MPa163 0

(C) . MPa101 85 (D) . MPa122 25

Common Data For Q. 25 and 26The composite beam as shown in figure, is made of aluminium ( )A and red brass ( )B . Take 68.9 GPaEal = and 101GPaEbr = .

SM 4.25 What will be the dimension h of the brass strip so that the neutral axis of the beam is located at the seam of the two metals ?(A) 82.6 mm (B) mm413

(C) . mm41 3 (D) . mm4 13





SM 4.26 If the allowable bending stress for the aluminium is ( ) 128 MPaallow als = and for the brass is ( ) 35 MPaallow brs = , what maximum moment will this beam support ?(A) . kN m6 60 - (B) . kN m36 5 -

(C) 29.2 kN m- (D) . kN m43 8 -

Common Data For Q. 27 and 28A beam AB consists of a cast-aluminium plate of uniform thickness b and length L, is support the load as shown in figure.

SM 4.27 If the beam is to be of constant strength, the expression for h in terms of x , L and h0 for portion AC of the beam will be

(A) h h Lx2

0= (B) h h Lx

0=

(C) h h Lx20= (D) h h L

x32

0=SM 4.28 If 800 mmL = , 200 mmh0 = , 25 mmb = and 72 MPaallows = , the maximum

allowable load P will be(A) 30 kN (B) 0 kN12

(C) 0 kN6 (D) kN45

SM 4.29 Consider a beam reinforced with A-36 steel straps ( 200 GPaEst = ) at its sides as shown in figure. If the beam is subjected to a bending moment of 4 kN mMz -= , the maximum stress developed in the wood ( 13.1GPaEw = ) and steel are

(A) . , .MPa MPa4 55 0 30st ws s= = (B) 4.55 , .MPa MPa2 28st ws s= =(C) . , 0.MPa MPa2 28 60st ws s= = (D) . , 0.MPa MPa0 60 30st ws s= =

SM 4.30 The cantilever beam AB consisting of a cast-iron plate of uniform thickness b and length L, is to support the distributed load ( )w x as shown in figure. If the beam is of constant strength, the expression for h in terms of x , L and h0 will be





(A) h h Lx /

0

2 3= b l (B) h h Lx /0 5 2= b l(C) h h L

x /0

1 2= b l (D) h h Lx /0 3 2= b lSM 4.31 For the section shown in figure, the allowable bending stress is 50 MPacs = in

compression and 120 MPats = is in tension. What will be the greatest magnitude of the applied forces P ?

(A) 79.75 kN

(B) . kN110 4

(C) . kN159 5

(D) . kN55 2

SM 4.32 A cantilevered machine element of cast aluminium and in the shape of a solid of revolution of variable diameter d, is being designed to support a horizontal distributed load 20 /kN mw = as shown in figure. If the machine element is to be of constant strength with 72 MPaallows = and 300 mmL = , the smallest allowable value of d0 will be

(A) 503 mm (B) 50.3 mm

(C) 5.03 mm (D) . mm100 6

SM 4.33 A curved bar of rectangular cross section is subjected to a couple as shown in figure. What will be the maximum tensile and compressive stress acting at section a a- ?





(A) 1.02 , 792MPa kPat cs s= = (B) 792 , 1.02kPa MPat cs s= =(C) 2.04 , 792MPa kPat cs s= = (D) 1.02 , 396MPa kPat cs s= =

SM 4.34 For the tapered beam as shown in the figure, if 150 kNP = , what will be the distance x of the transverse section in which the maximum normal stress occurs and the corresponding value of the normal stress ?

(A) 0.4 , 78.15m MPaxm ms= = (B) 0. , .m MPax 2 156 3m ms= =(C) 0.4 , .m MPax 156 3m ms= = (D) 0. , 78.15m MPax 2m ms= =

SM 4.35 Bar as shown in figure has a thickness of 10 mm. If the maximum bending stress at ,A B and C is same and 1.5K = , what will be the length L of the center portion of the bar ?

(A) 95 mm (B) 95 mm0

(C) 9.5 mm (D) . mm19 0

SM 4.36 Consider a machine element of cast aluminium and in the shape of a solid of revolution of variable diameter d to support a distributed load w as shown in figure. If the machine element is to be of constant strength, the expression for d in terms of ,x L and d0 is





(A) d d Lx

Lx4 1

/

0

1 3= +a k& 0 (B) d d Lx Lx2 1 /0 1 3= -a k& 0(C) d d L

xLx4 1

/

0

1 3= -a k& 0 (D) d d Lx Lx2 1 /0 1 3= +a k& 0SM 4.37 In the figure shown, the box beam is made of an elastically perfectly plastic

material for which 250 MPaYs = . If the plastic moment Mp is applied and then released, what will be the residual stress in the top and bottom of the beam ?

(A) . MPa317 14 (B) . MPa67 1

(C) . MPa158 6 (D) . MPa134 2

SM 4.38 The beam shown in figure, is made of elastic perfectly plastic material. If 50 mma = and 230 MPaYs = , what will be the maximum elastic moment and

the plastic moment that can applied to the cross section ?

(A) 50.7 , 86.25kN m kN mM MY p- -= =(B) 86.25 , 50.7kN m kN mM MY p- -= =(C) . , .kN m kN mM M101 4 86 25Y p- -= =(D) . , .kN m kN mM M50 7 43 15Y p- -= =

SM 4.39 Consider the beam and loading as shown in figure, the shear force and bending moment diagrams for the beam will be





SM 4.40 For a simple beam AB supporting two equal concentrated loads P as shown in figure, the shear-force and bending-moment diagrams are





SM 4.41 The shear force and bending-moment diagrams for the beam and loading shown in the figure, are




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