Gate Mechanical Sm By Nodia

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

***********

CONTENTS MANUFACTURING PROCESS

INDUSTRIAL ENGINEERING

OPERATION RESEARCH

VOL 3

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 1 STRESS AND STRAIN

Common Data For Q. 1 and 2 A long wire of tungsten ( gT = 190 kN/m3 ) 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 ( gsea water = 10 kN/m3), the greatest length is (A) 8300 m (B) 2075 m (C) 7500 m (D) 3750 m

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

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) N = 50 N , V = 650 N (C) N = 0 , V = 650 N (D) N = 650 N , V = 0

GATE Mechanical Engineering in 4 Volume NODIA SM 4

Demo Ebook

Page 4

Stress and Strain

SM 1

SM 1.4

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

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 (C) 27.8 mm SM 1.6

(B) 55.6 mm (D) 41.7 mm

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 w = 8 kN/m . The average normal stress and average shear stress acting at section b-b, are

(A) s = 4.41 MPa , t = 5.88 MPa (C) s = 8.82 MPa , t = 5.88 MPa

(B) s = 11.76 MPa , t = 4.41 MPa (D) s = 5.88 MPa , t = 4.41 MPa

Common Data For Q. 8 and 9 A 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 sallow .

SM 1.7

The formula for the allowable load Pallow that the bar can carrying in tension, is (A) 0.27d 2 # sallow (B) 0.54d 2 # sallow (C) 0.675d 2 # sallow (D) 0.54d # sallow

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Page 5

Stress and Strain

SM 5

SM 1.8

If the bar is made of brass with diameter d = 40 mm and sallow = 80 MPa , 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 W 250 # 0.67 column of cross-sectional area A = 8580 mm2 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 (C) 8.6 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 (C) 0.91 MPa SM 1.11

(B) 3.64 MPa (D) 2.73 MPa

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 E = 72 GPa , n = 0.33 , sY = 480 MPa )

(A) 27.4 kN (C) 13.7 kN SM 1.12

(B) 321 mm (D) 160 mm

(B) 54.8 kN (D) 37.7 kN

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

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GATE Mechanical Engineering in 4 Volume NODIA SM 6

Demo Ebook

Page 6

Stress and Strain

SM 1

volume of the bar is (Take sY = 250 MPa , E = 200 GPa , n = 0.3 )

(A) 8112 mm3 (C) 3245 mm3

(B) 4868 mm3 (D) 6490 mm3

Common Data For Q.14 and 15 Three 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 P = 50 kN , 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 (C) 165 mm SM 1.16

(B) 110 mm (D) 55 mm

The bar has a cross-sectional area of (400 # 10-6) m2 . 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

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Page 7

Stress and Strain

SM 7

0 < x # 0.5 m , is

(A) ^47.5 - 20.0x h MPa (C) ^47.5 + 20.0x h MPa

(B) 67.5x MPa (D) 27.5x MPa

Common Data For Q. 18 and 19 In 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) 125.1 MPa (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

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GATE Mechanical Engineering in 4 Volume NODIA SM 8

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Page 8

Stress and Strain

(A) 237 MPa (C) 474 MPa

SM 1

(B) 316 MPa (D) 158 MPa

Common Data For Q. 20 amd 21 A steel plate ( g = 77 kN/m3 ) 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 taver 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 sb between the steel plate and the pins is (A) 22.7 MPa (B) 15.2 MPa (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

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Demo Ebook Stress and Strain

(A) d1 = 45.2 mm , d2 = 20.1 mm (C) d1 = 20.1 mm , d2 = 45.2 mm SM 1.23

(B) d1 = 22.6 mm , d2 = 40.2 mm (D) d1 = 40.2 mm , d2 = 22.6 mm

(B) a = 12 mm (D) a = 6 mm

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) savg = 8 MPa , tavg = 4.62 MPa (C) savg = 4.62 MPa , tavg = 16 MPa SM 1.25

SM 9

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) a = 27 mm (C) a = 13.5 mm SM 1.24

Page 9

(B) savg = 4.62 MPa , tavg = 8 MPa (D) savg = 16 MPa , tavg = 4.62 MPa

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

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GATE Mechanical Engineering in 4 Volume NODIA SM 10

Demo Ebook Stress and Strain

(A) - 18.5 MPa , 17.28 MPa (C) - 18.5 MPa , 34.56 MPa SM 1.26

SM 1

(B) - 37.1 MPa , 34.56 MPa (D) - 37.1 MPa , 17.28 MPa

A 6 kN load is supported by two wooden members of 75 mm # 125 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 (C) 565 kPa, 103 kPa SM 1.27

Page 10

(B) 282 kPa, 206 kPa (D) 282 kPa, 103 kPa

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 (C) 300 mm

(B) 150 mm (D) 292 mm

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SM 1.28

SM 11

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 tallow = 100 MPa ? Both pins are subjected to double shear.

(B) d = 7.8 mm (D) d = 10.4 mm

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 taver in the key is ( Hints : Disregard the effects of friction, assume that the bearing pressure between the key and the wrench is uniformly distributed)

PL bc ^2d + b h 3PL (C) bc ^2d + b h (A)

SM 1.30

Page 11

Stress and Strain

(A) d = 2.6 mm (C) d = 5.2 mm SM 1.29

Demo Ebook

2PL bc ^2d + b h 4PL (D) bc ^2d + b h (B)

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

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GATE Mechanical Engineering in 4 Volume NODIA SM 12

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Page 12

Stress and Strain

SM 1

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

(A) 216 mm (C) 200 mm SM 1.31

(B) 208 mm (D) 104 mm

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 d = 150 mm . If the allowable shear stress in the bolts is 90 MPa, the maximum permissible torque will be

(A) 16.96 kN- m (C) 12.72 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 d = 25 mm and the wall thickness is t = 2.5 mm . 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 sY = 270 MPa , su = 310 MPa )

(A) 9.5 kN (C) 11.0 kN SM 1.33

(B) 8.48 kN- m (D) 4.24 kN- m

(B) 12.0 kN (D) 13.7 kN

In the figure shown, a long steel wire ( g = 77.0 kN/m3 ) 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 sY = 350 MPa 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.)

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Demo Ebook Stress and Strain

(A) 1783 N (C) 1759 N SM 1.34

SM 13

(B) 1711 N (D) 1735 N

What is the smallest dimensions of the circular shaft and circular end cap if the load it is required to support is P = 150 kN ? The allowable tensile stress, bearing stress and shear stress is (st) allow = 175 MPa , (sb) allow = 275 MPa and tallow = 115 MPa .

(A) d1 = 15.8 mm , (B) d1 = 26.4 mm , (C) d1 = 44.6 mm , (D) d1 = 44.6 mm , SM 1.35

Page 13

d 3 = 26.4 mm , d 3 = 44.6 mm , d 3 = 26.4 mm , d 3 = 15.8 mm ,

t = 44.6 mm t = 15.8 mm t = 15.8 mm t = 26.4 mm

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 (sb) allow = 350 MPa and allowable shear stress is tallow = 125 MPa . The smallest diameter d1 of the top disk, the diameter d2 within the support space and the diameter d 3 of the hole in the bottom disk are

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GATE Mechanical Engineering in 4 Volume NODIA SM 14

Demo Ebook

Page 14

Stress and Strain

SM 1

(A) d1 = 27.6 mm , d2 = 22.6 mm , d 3 = 35.7 mm (B) d1 = 22.6 mm , d2 = 35.7 mm , d 3 = 27.6 mm (C) d1 = 22.6 mm , d2 = 27.6 mm , d 3 = 35.7 mm (D) d1 = 35.7 mm , d2 = 22.6 mm , d 3 = 27.6 mm 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 (C) 3.97 kN SM 1.37

(B) 14.04 kN (D) 3.72 kN

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 tallow = 125 MPa . If x = 1 m and w = 12 kN/m , the smallest required diameter of pins A and B are (Neglect any axial force in the bar.)

(A) dA = 957 mm , dB = 20.6 mm (C) dA = 19.14 mm , dB = 10.3 mm SM 1.38

(B) dA = 10.3 mm , dB = 9.57 mm (D) dA = 9.57 mm , dB = 10.3 mm

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

(A) 2 (C) 5.4

(B) 3.6 (D) 1.8

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SM 1.39

SM 1.40

Demo Ebook

Page 15

Stress and Strain

SM 15

The cable shown in figure has a specific weight g (weight/volume) and crosssectional 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) s =

gL2 2s

(B) s =

gL 8s

(C) s =

gL2 8s

(D) s =

gL2 4s

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 a = 150 mm , b = 250 mm and the elastomer has thickness t = 50 mm . 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) 1 MPa (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

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GATE Mechanical Engineering in 4 Volume NODIA SM 16

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Page 16

Stress and Strain

(A) 685 N (C) 1370 N

SM 1

(B) 1028 N (D) 2740 N

Common Data For Q. 42 and 43 In 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 q = 0c, 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 q = 90c, 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) eCE = 0.00025 , eBD = 0.0107 (C) eCE = 0.025 , eBD = 0.0107 SM 1.45

(B) eCE = 0.0025 , eBD = 0.00107 (D) eCE = 0.00107 , eBD = 0.0025

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

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Page 17

Stress and Strain

(A) (B) (C) (D) SM 1.46

Demo Ebook

eCE eCE eCE eCE

= 1.43 # 10-3 , = 1.43 # 10-3 , = 1.79 # 10-3 , = 2.86 # 10-3 ,

SM 17

eBD = 3.58 # 10-3 eBD = 1.79 # 10-3 eBD = 1.43 # 10-3 eBD = 1.79 # 10-3

A steel pipe is to carry an axial compressive load P = 1200 kN 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 (Take sY = 270 MPa ) d min is

. (A) 153 mm (C) 114.75 mm

(B) 76.5 mm (D) 38.25 mm

Common Data For Q. 47 and 48 A circular aluminum tube of length L = 400 mm 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 e = 550 # 10-6 , 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

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GATE Mechanical Engineering in 4 Volume NODIA SM 18

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Page 18

Stress and Strain

SM 1

Common Data For Q. 49 and 50 A 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 sallow represent the allowable tensile stress in the metal tie.

SM 1.49

The minimum required cross-section area of the tie is (B) A min = 2P tan q/sallow (A) A min = P tan q sallow (C) A min = 2P cot q/sallow (D) A min = P cot q sallow

SM 1.50

If P = 130 kN , q = 75c and sallow = 80 MPa , 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 a = 150 mm , b = 250 mm and the elastomer has thickness t = 50 mm . 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) 1 MPa (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

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Page 19

Stress and Strain

SM 19

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 (C) 43.8 mm

(B) 4.38 mm (D) 8.76 mm

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

SM 1.53

The average normal strains ex , ey and the shear strain gxy at A are (A) ex = ey = 0 , gxy = 0.0798 rad (B) ex = 0 , ey = 0.00319 , gxy = 0.0798 rad (C) ex = 0.00319 , ey = 0 , gxy = 0.0798 rad (D) ex = ey = 0.00319 , gxy = 0.0798 rad

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 gxy = 0.02x , 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 Dy at the end of its bottom edge will be

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GATE Mechanical Engineering in 4 Volume NODIA SM 20

Demo Ebook

Page 20

Stress and Strain

(A) 2.03 mm (C) 1.015 mm

SM 1

(B) 4.06 mm (D) 3.045 mm

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

SM 1.56

The required diameter of each wire is (A) dAC = 3.23 mm , dAB = 7.08 mm (C) dAC = 3.23 mm , dAB = 3.54 mm

(B) dAC = 3.54 mm , dAB = 3.23 mm (D) dAC = 6.46 mm , dAB = 3.54 mm

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 E = 5 MPa , n = 0.45 )

(A) 926.25 kPa (C) 741 kPa

(B) 370 kPa (D) 555.75 kPa ***********

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SM 2 AXIAL LOADING

Common Data For Linked Q. 1 and 2 A 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 Poisson’s 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 b = 220 mm and a = 30 mm is shown in figure. For a maximum lateral load P = 19 kN and a maximum displacement d = 12 mm , the shearing modulus G and the shear stress t respectively, are

(A) 1.08 MPa, 432 kPa (C) 1.08 MPa, 862 kPa

(B) 0.54 MPa, 431 kPa (D) 1.08 MPa, 216 kPa

GATE Mechanical Engineering in 4 Volume NODIA

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SM 2

Axial Loading

SM 21

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 E = 200 GPa , n = 0.30 )

(A) e = 121 # 10-6 , Dv = 23 mm3 (C) e = 242 # 10-6 , Dv = 18.4 mm3 SM 2.5

(B) e = 121 # 10-6 , Dv = 18.4 mm3 (D) e = 242 # 10-6 , Dv = 13.8 mm3

For the axial loading shown in figure, the loading is hydrostatic with sx = sy = sz =- 70 MPa . The change in height and the change in volume of the brass cylinder respectively, are

(A) 0.031 mm3 , 521 mm (C) 0.031 mm, 521 mm3

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

Common Data For Q. 6 and 7 A fabric is subjected to a biaxial loading that results in normal stresses sx = 120 MPa and sz = 160 MPa as shown in figure. The properties of the fabric can be approximated as E = 87 GPa and n = 0.34 .

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GATE Mechanical Engineering in 4 Volume NODIA SM 22

Demo Ebook

Page 22

Axial Loading

SM 2

SM 2.6

The change in length of sides AB and BC , are (B) dAB = 0.075 mm , dBC = 0.103 mm (A) dAB = 0.103 mm , dBC = 0.075 mm (C) dAB = 0.075 mm , dBC = 0.206 mm (D) dAB = 0.150 mm , dBC = 0.103 mm

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 s = 125 MPa , the slope of the line is (Take E = 74 GPa , n = 0.33 )

(A) 1.995 (C) 0.1995 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 = PL 4EA (C) s = PL 6EA SM 2.10

(B) 19.95 (D) 3.99

(B) s = PL 3EA (D) s = PL 9EA

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) d = W 2k

(B) d = 3W 4k

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SM 2

Axial Loading

SM 23

(C) d = 4W 3k 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 A = 40 mm2 and an effective modulus of elasticity E = 130 GPa . A slider of mass M = 35 kg drops from a height h = 1.0 m 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 (C) 11.1 mm SM 2.12

(B) 9.25 mm (D) 5.55 mm

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

(A) 67.5 mm (C) 270 mm SM 2.13

(D) d = W 4k

(B) 135 mm (D) 202.5 mm

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 ) ?

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GATE Mechanical Engineering in 4 Volume NODIA SM 24

Demo Ebook Axial Loading

(A) 2.25 kN (C) 1.35 kN SM 2.14

The rigid bar shown in figure is supported by the two short white spruce wooden posts ( E = 9.65 GPa ) and a spring. The spring has a stiffness of k = 2 MN/m 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 ?

(B) dA = 4.42 m , dB = 8.84 mm (D) dA = 4.42 m , dB = 2.21 mm

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

(A) 1.19 mm (C) 0.79 mm SM 2.16

SM 2

(B) 1.8 kN (D) 0.9 kN

(A) dA = dB = 4.42 mm (C) dA = 2.21 mm , dB = 4.42 mm SM 2.15

Page 24

(B) 1.58 mm (D) 2.37 mm

The A-36 steel pipe ( E = 200 GPa ) has a 6061-T6 aluminum core ( E = 68.9 GPa ). 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 ?

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SM 2

Axial Loading

SM 25

(A) (B) (C) (D) SM 2.17

sal sal sal sal

= 27.5 MPa , sst = 39.9 MPa = 79.9 MPa , sst = 27.5 MPa = 27.5 MPa , sst = 79.9 MPa = 55 MPa , sst = 79.9 MPa

The column shown in figure, is constructed from high-strength concrete with Ec = 25 GPa and four A-36 steel reinforcing rods with Est = 200 GPa . 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 (C) 11.3 mm SM 2.18

In the figure shown, a round brass bar of diameter d1 = 20 mm has upset ends of diameter d2 = 26 mm . The lengths of the segments of the bar are L1 = 0.3 m and L2 = 0.1 m . Quarter-circular fillets ( K . 1.6 ) are used at the shoulders of the bar and the modulus of elasticity of the brass is E = 100 GPa . If the bar lengthens by 0.12 mm under a tensile load P , what is the maximum stress smax in the bar ?

(A) 23 MPa (C) 46 MPa SM 2.19

(B) 33.9 mm (D) 28.2 mm

(B) 69 MPa (D) 34.5 MPa

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 (sY ) st = 640 MPa and for the bronze (sY ) br = 520 MPa , what will be the magnitude of the largest elastic load P that can be applied to the assembly ? (Take Est = 200 GPa and Ebr = 100 GPa )

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GATE Mechanical Engineering in 4 Volume NODIA SM 26

Demo Ebook Axial Loading

(A) 122 kN (C) 126 kN SM 2.20

Page 26 SM 2

(B) 82 kN (D) 204 kN

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 E = 200 GPa . 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 (C) 0.0491 mm

(B) 4.91 mm (D) 0.491 mm

Common Data For Q. 21 to 23 Two 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 a = 20c ? (A) sq = 0.57 MPa , tq = 0.79 MPa (B) sq = 0.57 MPa , tq = 1.58 MPa (C) sq = 1.58 MPa , tq = 0.57 MPa (D) sq = 1.14 MPa , tq = 1.58 MPa

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

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SM 2

Axial Loading

SM 27

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) dBC = dBA = 1.13 mm (B) dBC = 0 , dBA = 1.13 mm (C) dBC = 1.13 mm , dBA = 0 (D) dBC = dBA = 0

SM 2.26

A plastic bar ABC ( E = 4.0 GPa ) of length L consists of two parts of equal lengths but different diameters is shown in figure. Segment AB and BC have diameters d1 = 100 mm and d2 = 60 mm , respectively. Both segments have length L/2 = 0.6 m . A longitudinal hole of diameter d is drilled through segment AB for one-half of its length (distance L/4 = 0.3 m ). Compressive loads P = 110 kN 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 d max of the hole ?

(A) 11.9 mm (C) 23.9 mm

(B) 7.96 mm (D) 2.39 mm

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GATE Mechanical Engineering in 4 Volume NODIA SM 28

Demo Ebook

Page 28

Axial Loading

SM 2

Common Data For Linked Q. 27 and 28 The elastoplastic rods AB and BC are made of mild steel with E = 200 GPa and sY = 345 MPa 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 (C) 166 kN

(B) 415 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) dA = 0 , dB = 6.37 mm (B) dA = dB = 6.37 mm (C) dA = 0 , dB = 0 (D) dA = 6.37 mm , dB = 0

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 1.5b # 1.5b 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) d = 2PH2 3Eb (C) d = 3PH2 4Eb

(B) d = 3PH2 2Eb (D) d = 2PH2 5Eb

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SM 2

Axial Loading

SM 29

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) d = 8sL/7E (C) d = 7sL/6E SM 2.31

(B) d = 7sL/4E (D) d = 7sL/8E

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) d = 2WL/pdE (C) d = WL/2pd 2 E

(B) d = 2WL/pd 2 E (D) d = 4WL/pd 2 E

Common Data For Linked Q. 32 and 33 In 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 E = 200 GPa , sY = 250 MPa and portion BC is made of high-strength steel with E = 200 GPa , sY = 345 MPa . A load P is applied at point C . Assume both steels to be elastoplastic.

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GATE Mechanical Engineering in 4 Volume NODIA SM 30

Demo Ebook

Page 30

Axial Loading

SM 2

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 (C) 2.7 mm

(B) 0.27 mm (D) 0.135 mm

Common Data For Q. 34 and 35 In the figure shown, a plastic rod AB of length L = 0.5 m has a diameter d1 = 30 mm . A plastic sleeve CD of length c = 0.3 m and outer diameter d2 = 45 mm 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 E1 = 3.1 GPa and the sleeve is made of a polyamide with E2 = 2.5 GPa .

SM 2.34

When the rod is pulled by axial forces P = 12 kN , 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% (C) 10% SM 2.37

(B) 110% (D) 11%

The fixed-end bar ABCD consists of three prismatic segments as shown in the figure. The end segments have cross-sectional area A1 = 840 mm2 and length

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SM 2

Axial Loading

SM 31

L1 = 200 mm . The middle segment has cross-sectional area A2 = 1260 mm2 and length L2 = 250 mm . 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 (C) 15 kN SM 2.38

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

(A) 16.67 kN (C) 21.87 kN SM 2.39

(B) 30 kN (D) 7.5 kN

(B) 65.6 kN (D) 32.8 kN

A composite bar of square cross section with dimensions 2b # 2b 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 ?

b ^E2 - E1h 2 ^E2 + E1h b ^E1 + E2h (C) e = 2 ^E1 - E2h (A) e =

b ^E2 + E1h 2 ^E2 - E1h b ^E1 - E2h (D) e = 2 ^E1 + E2h (B) e =

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SM 2.40

SM 2

A rigid bar of weight W = 800 N 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 Est = 210 GPa , Eal = 70 GPa )

(B) 1.5 kN (D) 4.5 kN

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 L = 200 mm , pitch of the bolt threads p = 1.0 mm , modulus of elasticity for steel Es = 200 GPa , modulus of elasticity for the plastic E p = 7.5 GPa , crosssectional area of one bolt As = 36.0 mm2 , n = 1 and cross-sectional area of the plastic cylinder A p = 960 mm2 . What will be the compressive stress sp in the plastic when the nuts on the steel bolts are tightened by one complete turn ?

(A) 31.25 MPa (C) 25 MPa SM 2.42

Page 32

Axial Loading

(A) 3.0 kN (C) 2.25 kN SM 2.41

Demo Ebook

(B) 18.75 MPa (D) 15.63 MPa

The uniform wire ABC of unstretched length 2l , is attached to the supports as shown in figure and a vertical load P is applied at the midpoint B . The crosssectional area of the wire is denoted by A and the modulus of elasticity is by E . For d << l , the deflection at the midpoint B , is

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SM 2

Axial Loading

SM 33

(A) d = 2l ^P/AE h1/3 1/3 (C) d = l b P l AE

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

SM 2.43

A nylon thread is subjected to a 8.5 N tensile force. If E = 3.3 GPa 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 E = 200 GPa . 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 (C) 0.994 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 ( E = 68.9 GPa ). What will be the vertical deflection of the bar at D when the distributed load is applied ?

(A) 12.97 mm (C) 17.3 mm SM 2.46

(B) 1.988 kN (D) 19.88 kN

(B) 21.63 mm (D) 6.48 mm

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 ( Eal = 70 GPa ) and CD is made from steel ( Fst = 200 GPa ). Each bar has a crosssectional area of 450 mm2 . If allowable stress of (sallow) st = 180 MPa in the steel and (sallow) al = 94 MPa in the aluminium is not exceeded, the maximum intensity w of the distributed loading will be

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GATE Mechanical Engineering in 4 Volume NODIA SM 34

(B) 45.9 kN/m (D) 68.5 kN/m

(B) 1.10 mm (D) 0.55 mm

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 (C) 22.6 MPa, 0.475 mm SM 2.49

SM 2

The links AB and CD are made of aluminium (E = 75 GPa) and has a crosssectional 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 (C) 0.110 mm SM 2.48

Page 34

Axial Loading

(A) 28.35 kN/m (C) 120.8 kN/m SM 2.47

Demo Ebook

(B) 47.5 MPa, 0.226 mm (D) 11.3 MPa, 0.475 mm

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 Eal = 70 GPa and (sY ) al = 20 MPa . Post B is made of brass, for which Ebr = 100 GPa and (sY ) br = 590 MPa . If P = 130 kN , the largest diameter of post B so that all the post yield at the same time, is

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SM 2

Axial Loading

SM 35

(A) 3.56 mm (C) 8.9 mm

(B) 1.78 mm (D) 17.8 mm

Common Data For Q. 50 and 51. Three steel rods (E = 200 GPa) 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

SM 2.51

The change in length of rod EF is (A) 0.762 mm (C) 0.1524 mm

(B) 0.0762 mm (D) 7.62 mm

What will be the stresses in rods AB and EF ? (A) sAB = 30.5 MPa , sEF = 38.1 MPa (B) sAB = 15.25 MPa , sEF = 38.1 MPa (C) sAB = 30.5 MPa , sEF = 19.05 MPa (D) sAB = 38.1 MPa , sEF = 30.5 MPa

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

SM 2.52

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

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GATE Mechanical Engineering in 4 Volume NODIA SM 36

Demo Ebook

Page 36

Axial Loading

of 96cC ? (A) 162.75 kN (C) 271.25 kN

SM 2

(B) 217.36 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) 0.2425 mm (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 DT at distance x from end A. The temperature increase is given by the expression DT = DTB x3 /L3 , where DTB 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 sc in the bar ?

(A) sc = (C) sc = SM 2.55

Ea ^DTB h 8 Ea ^DTB h 6

(B) sc = (D) sc =

Ea ^DTB h 2

Ea ^DTB h 4

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 sY = 250 MPa , the plastic load Pp is

(A) 55 kN (C) 110 kN

(B) 82.5 kN (D) 41.25 kN ***********

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SM 3 TORSION

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 (C) 10.3 MPa SM 3.3

(B) 20.6 MPa (D) 14.5 MPa

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 (C) 5.7 mm

(B) 7.59 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 G = 78 GPa .

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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 AB (G = 39 GPa) is bonded to the solid aluminum rod BC (G = 27 GPa) as shown in figure. The angle of twist at A and B are

(A) fA = 0.741c, fB = 1.573c (C) fA = 0.741c, fB = 0.831c SM 3.7

(B) fA = 0.831c, fB = 0.741c (D) fA = 1.573c, fB = 0.741c

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 (C) 182 mm

(B) 136.5 mm (D) 227.5 mm

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

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SM 39

the ends as shown in figure. The dimensions and shear modulus of elasticity are as follows: L = 1.2 m , d = 30 mm and G = 28 GPa .

SM 3.8

SM 3.9

The torsional stiffness of the bar is (A) 1860 N-m (C) 2325 N-m

(B) 1395 N-m (D) 1163 N-m

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 11 A circular tube of outer diameter d 3 = 70 mm and inner diameter d2 = 60 mm 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 d1 = 40 mm 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 T = 1000 N- m acts at end A of the bar. Also, both the bar and tube are made of an aluminum alloy with shear modulus of elasticity G = 27 GPa .

SM 3.10

The maximum shear stresses in both the bar and tube are (A) tbar = 79.6 MPa, ttube = 64.6 MPa (B) tbar = 32.3 MPa, ttube = 79.6 MPa (C) tbar = 79.6 MPa, ttube = 32.3 MPa (D) tbar = 39.8 MPa, ttube = 32.3 MPa

SM 3.11

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

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GATE Mechanical Engineering in 4 Volume NODIA SM 40

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 tsteel = 60 MPa and taluminum = 45 MPa ,the largest angle through which end A may be rotated, is

(B) 1.65c (D) 6.63c

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 (C) 49.7 mm SM 3.14

SM 3

(D) 5.89c

(A) 4.13c (C) 2.65c SM 3.13

Page 40

Torsion

(C) 11.79c SM 3.12

Demo Ebook

(B) 39.8 mm (D) 24.9 mm

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 dB = 2dA . The polar moment of inertia is represented by the approximate formula J . pd3 t/4 . What will be the angle of twist f of the tube when it is subjected to torques T acting at the ends ?

. (A) f = 3TL 4 2pGtd A

(B) f =

3TL 4pGtd A3

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SM 41

3TL (D) f = 3TL 3 2pGtd A 4pGtd A4 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 ? (C) f =

SM 3.15

SM 3.16

2 (A) c2 tall c1 - c 12 m 2g c2

2 (B) c2 tall c1 + c 12 m 2g c2

2 (C) c1 tall c1 + c 12 m 2g c2

2 (D) c1 tall c1 - c 12 m 2g c2

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 ?

2 (A) f = 4tL 4 pGd 2 (C) f = 12tL4 pGd

SM 3.17

A solid circular bar of diameter d = 50 mm shown in figure, is twisted by a torque T = 500 N- m . At this value of torque, a strain gage oriented at 45c to the axis of the bar gives a reading e = 339 # 10-6 . What is the shear modulus G of the material?

(A) 22.5 GPa (C) 37.5 GPa SM 3.18

2 (B) f = 16tL4 pGd 2 (D) f = 8tL 4 pGd

(B) 30 GPa (D) 45 GPa

The drive shaft for a truck of outer diameter d2 = 60 mm and inner diameter d1 = 40 mm is running at 2500 rpm as shown in figure. If the shaft transmits 150 kW, what is the maximum shear stress in the shaft?

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Torsion

(A) 10.5 MPa (C) 16.8 MPa

SM 3

(B) 12.6 MPa (D) 21 MPa

Common Data For Q. 19 and 20. The solid aluminum shaft has a diameter of 50 mm and an allowable shear stress of tallow = 6 MPa . 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 (C) 172 N-m

(B) 258 N-m (D) 129 N-m

SM 3.20

The maximum shear stress within regions CD and DE are (A) tCD = 2 MPa , tDE = 2.58 MPa (B) tCD = 2.58 MPa , tDE = 2 MPa (C) tCD = 4 MPa , tDE = 2.58 MPa (D) tCD = 5.16 MPa , tDE = 4 MPa

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) tA = 18.86 MPa , tB = 14.1 MPa (C) tA = 9.43 MPa , tB = 7.05 MPa SM 3.22

(B) tA = 9.43 MPa , tB = 14.1 MPa (D) tA = 18.86 MPa , tB = 7.05 MPa

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

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(A) 1.17 MPa (C) 1.47 MPa

SM 43

(B) 0.87 MPa (D) 0.735 MPa

SM 3.23

The propellers of a ship are connected to a solid steel (G = 75GPa ) 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 MPa, 5.96c (C) 44.3 MPa, 5.96c (D) 44.3 MPa, 11.9c

SM 3.24

The 8 mm diameter bolt with G = 75 GPa 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 N, 0.720 mm (C) 12.06 N, 0.0720 mm SM 3.25

(B) 6.03 N, 0.720 mm (D) 6.03 N, 0.0720 mm

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

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Page 44

Torsion

(A) 12.24 MPa (C) 15.3 MPa

SM 3

(B) 7.65 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 (G = 75GPa ) 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 (C) 3.60c SM 3.28

(B) 2.70c (D) 1.80c

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) 1.49 d (C) 1.221 d SM 3.29

(B) 0.819 d (D) 0.794 d

A solid circular bar ABCD with fixed supports, is acted upon by torques T0 and 2T0 at the locations as shown in the figure. The maximum angle of twist fmax of the bar is

(A) f max = T0 L 3GJ (C) f max = 3T0 L 5GJ SM 3.30

(B) f max = 3T0 L 4GJ (D) f max = 6T0 L 5GJ

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

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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 (C) 6 mm SM 3.31

(B) 8 mm (D) 10 mm

A steel shaft transmits 150 kW at a speed of 360 rpm and for steel G = 77 GPa . 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 G = 37GPa 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) tAC = 0.955 MPa , tBC = 1.59 MPa (B) tAC = 1.59 MPa , tBC = 1.91 MPa (C) tAC = 0.795 MPa , tBC = 0.955 MPa (D) tAC = 1.59 MPa , tBC = 0.955 MPa

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 tallow is the allowable shear stress in the shaft and assume that a > b , what is the largest permissible angle of rotation fmax of the disk ?

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GATE Mechanical Engineering in 4 Volume NODIA SM 46

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Page 46

Torsion

(A) f max = ^2btallow h /Gd 3 (C) f max = ^2btallow h /Gd 4

SM 3

(B) f max = ^2btallow h /Gd 2 (D) f max = ^2btallow h /Gd

Common Data For Q. 35 and 36 A solid shaft of 54 mm diameter is made of mild steel which is assumed to be elastoplastic with tY = 145 MPa . 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 MPa, 27 mm (B) 129.4 MPa, 54 mm (C) 64.7 MPa, 27 mm (D) 64.7 MPa, 54 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 MPa, 46.8 mm (B) 145 MPa, 23.4 mm (C) 290 MPa, 23.4 mm (D) 290 MPa, 46.8 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 (C) 2710 N SM 3.38

(B) 2032.5 N (D) 3387.5 N

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 T = 40 N- m , what will be the average shear stress developed at points A and B ?

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(A) tA = tB = 357 kPa (C) tA = tB = 178.5 kPa SM 3.39

Page 47 SM 47

(B) tA = 357 kPa , tB = 178.5 kPa (D) tA = 178.5 kPa , tB = 357 kPa

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% (C) 25%

(B) 75% (D) No change

Common Data For Q. 40 and 41 A solid steel bar of diameter d1 = 25.0 mm is enclosed by a steel tube of outer diameter d 3 = 37.5 mm and inner diameter d2 = 30.0 mm 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 L = 550 mm , is twisted by a torque T = 400 N- m acting on the end plate.

SM 3.40

The maximum shear stresses t1 and t2 in the bar and tube respectively, are (A) t1 = 32.7 MPa , t2 = 49 MPa (B) t1 = 49 MPa , t2 = 32.7 MPa (C) t1 = 64.7 MPa , t2 = 49 MPa (D) t1 = 32.7 MPa , t2 = 24.5 MPa

SM 3.41

If the shear modulus of the steel is G = 80 GPa , the torsional stiffness kT of the composite bar is (A) 16.7 kN-m (B) 13.95 kN-m

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Demo Ebook Torsion

(C) 27.9 kN-m SM 3.42

SM 3

(D) 22.3 kN-m

The solid circular drill rod AB is made of a steel which is assumed to be elastoplastic with tY = 160 MPa and G = 77 GPa as shown in figure. If a torque T = 5 kN- m is applied to the rod and then removed, the maximum residual shearing stress in the rod is

(A) 44.9 MPa (C) 43.7 MPa SM 3.43

Page 48

(B) 204 MPa (D) 115 MPa

A 1.25 m long steel angle has an L 127 # 76 # 6.4 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 tall = 60 MPa , G = 77 GPa )

(A) 157 N-m , 8.72c (C) 117.75 N-m , 10.9c

(B) 196.25 N-m , 5.45c (D) 98.125 N-m , 6.54c

Common Data For Q. 44 and 45 A solid shaft has a diameter of 40 mm, length of 1 m and G = 80 GPa . It is made from an elastic-plastic material having a yield stress of tY = 100 MPa . 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) 1.26 kN-m, 1.79c (C) 1.26 kN-m, 3.58c (D) 2.52 kN-m, 1.79c

SM 3.45

What is the angle of twist if the torque is increased to T = 1.2TY ? (A) 3.64c (B) 4.86c (C) 6.08c (D) 3.04c

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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

SM 3.47

SM 3.48

What is the ratio tb /ta of the maximum shearing stresses in the tubes ? 2 (B) 3c (A) 3c2 t t 2 (C) 3c (D) 3c2 t t The ratio fb /fa of the angles of twist of the shafts is 2 (B) 3c2 (A) 3c t t 2 (C) 3c (D) 3c2 t t A steel tube having the cross section as shown in the figure. The tube has length L = 1.5 m and is subjected to a torque T = 10 kN- m . What will be the shear stress t and the angle of twist f respectively ? (Take G = 76 GPa )

(A) 17.5 MPa, 0.285c (C) 17.5 MPa, 0.570c

(B) 35 MPa, 0.285c (D) 35 MPa, 0.570c ***********

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SM 4 SHEAR FORCE AND BENDING MOMENT

SM 4.1

SM 4.2

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 = 2bP , M = 0 L

2 (B) V = 2bP , M = 2b P L L

(C) V = bP , M = 0 2L

2 (D) V = bP , M = b P 2L 2L

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) ^45 # 103h mm3 (C) ^180 # 103h mm3

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, 3PL2 2bd (C) L , 2PL3 2 3bd SM 4.4

(B) ^90 # 103h mm3 (D) ^135 # 103h mm3

(B) L , 2PL2 4 3bd (D) 3L , 3PL3 4 2bd

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 P1 = 4.0 kN acting

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Shear Force and Bending Moment

SM 4

at the end of a vertical arm and a vertical force P2 = 8.0 kN 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 lefthand support using centerline dimensions, will be

(A) V =- 1.0 kN, M =- 7.0 kN-m (C) V = 1.0 kN, M =- 7.0 kN-m SM 4.5

Consider the beam and loading of the S100 # 11.5 rolled steel section as shown in figure. If the section modulus is S = 49.6 # 10-6 m3 , the maximum normal stress due to bending will be

(A) 19.35 MPa (C) 38.7 MPa SM 4.6

(B) 9.65 MPa (D) 29.03 MPa

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) 96 mm (C) 32 mm SM 4.7

(B) V = 1.0 kN, M = 7.0 kN-m (D) V =- 1.0 kN, M = 7.0 kN-m

(B) 24 mm (D) 48 mm

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 ?

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(A) P = 250 N, Q = 500 N (C) P = 1000 N, Q = 500 N SM 4.8

Page 53 SM 53

(B) P = 500 N, Q = 250 N (D) P = 500 N, Q = 1000 N

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

2 px (A) V = w 0 L cos2 px , M = w 0 L 2 sin L p L p 2 (B) V = w 0 L cos px , M = w 0 L sin px 2 p L L p 2 (C) V = w 0 L sin px , M = w 0 L cos px p p L L 2 (D) V = w 0 L sin2 px , M = w 0 L cos2 px p p L L

Common Data For Q. 9 and 10 Beams 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) 4.01 kN (B) 5.02 kN (C) 3.01 kN (D) 8.02 kN

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 ? (B) 6.54 m (A) 2.45 m (C) 4.09 m (D) 3.27 m

SM 4.11

Consider a timber beam of length L = 16 m and width b = 75 mm as shown in figure. The dead load carried by each beam, including the estimated weight of the beam, is a uniformly distributed load wD = 350 N/m . The live loads can be represented by a uniformly distributed load wL = 600 N/m . 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) ? Use Data: sU = 50 MPa, gD = 1.2, gL = 1.6 and f = 0.9

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GATE Mechanical Engineering in 4 Volume NODIA SM 54

312.72 mm 383.0 mm 541.64 mm 270.82 mm

sB = 3.61 MPa, sC = 1.55 MPa sB =- 3.61 MPa, sC = 1.55 MPa sB = 3.61 MPa, sC =- 1.55 MPa sB = 1.55 MPa, sC = 3.61 MPa

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

(A) (B) (C) (D) SM 4.14

SM 4

In figure shown, the aluminium machine part is subjected to a moment of M = 75 N- m . What will be the bending stress created at points B and C on the cross section ?

(A) (B) (C) (D) SM 4.13

Page 54

Shear Force and Bending Moment

(A) (B) (C) (D) SM 4.12

Demo Ebook

sA = 4.49 MPa, sB = 49.4 MPa sA =- 4.49 MPa, sB = 49.4 MPa sA = 49.4 MPa, sB =- 4.49 MPa sA = 49.4 MPa, sB = 4.49 MPa

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

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(A) 161.25 MPa (C) 129.0 MPa SM 4.15

Page 55 SM 55

(B) 96.75 MPa (D) 80.63 MPa

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 (C) 413.75 kPa

(B) 331.0 kPa (D) 206.85 kPa

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

SM 4.16

What will be the maximum bending stress developed in the material using the flexure formula, where Iz = 14 p (0.08 m) (0.04 m) 3 ? (A) 372.75 kPa (C) 621.25 kPa

SM 4.17

(B) 310.65 kPa (D) 497.0 kPa

What will be the maximum bending stress developed in the material using integration ? (A) 621.25 kPa (B) 497.0 kPa (C) 310.65 kPa (D) 372.75 kPa

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Page 56

Shear Force and Bending Moment

SM 4

Common Data For Q. 18 and 19 For the beam and loading of the S 310 # 52 rolled steel shape as shown in figure, the section modulus is S = 625 # 10-6 m3 .

SM 4.18

What will be the magnitude and location of the maximum bending moment, respectively using singularity functions ? (A) - 18 kN-m, 1.94 m (B) 28.28 kN-m, 1.94 m (C) - 27 kN-m, 1.94 m (D) 18 kN-m, 1.94 m

SM 4.19

The maximum normal stress due to bending is (A) 56.5 MPa (B) 33.9 MPa (C) 45.2 MPa (D) 28.25 MPa

Common Data For Q. 20 and 21 A 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) 0.872 kN-m, 4.0 m (B) 0.872 kN-m, 1.5 m (C) 0.872 kN-m, 2.094 m (D) 0.872 kN-m, 3.0 m

SM 4.21

What will be the most suitable depth h that can used ? (A) 140 mm (B) 130 mm (C) 110 mm (D) 120 mm

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

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Page 57

Shear Force and Bending Moment

The location c of the neutral axis, is h Ec (A) Et - Ec (C)

h Et Et - Ec

SM 57

(B)

h Ec Et + Ec

(D)

h Et Et + Ec

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) 3M2 e Et - Ec o (B) 3M2 e Et + Ec o bh bh Ec Et (C) 3M2 e Et - Ec o (D) 3M2 e Et + Ec o bh bh Et Ec

SM 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 (C) 101.85 MPa

(B) 163.0 MPa (D) 122.25 MPa

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

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) 413 mm (C) 41.3 mm (D) 4.13 mm

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SM 4.26

Demo Ebook

Page 58

Shear Force and Bending Moment

SM 4

If the allowable bending stress for the aluminium is (sallow) al = 128 MPa and for the brass is (sallow) br = 35 MPa , what maximum moment will this beam support ? (A) 6.60 kN-m (B) 36.5 kN-m (C) 29.2 kN-m (D) 43.8 kN-m

Common Data For Q. 27 and 28 A 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 h 0 for portion AC of the beam will be (B) h = h 0 x (A) h = h 0 2x L L x (C) h = h 0 (D) h = h 0 2x 2L 3L

SM 4.28

If L = 800 mm , h 0 = 200 mm , b = 25 mm and sallow = 72 MPa , the maximum allowable load P will be (A) 30 kN (B) 120 kN (C) 60 kN (D) 45 kN

SM 4.29

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

(A) (B) (C) (D) SM 4.30

sst = 4.55 MPa, sw = 0.30 MPa sst = 4.55 MPa, sw = 2.28 MPa sst = 2.28 MPa, sw = 0.60 MPa sst = 0.60 MPa, sw = 0.30 MPa

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 h 0 will be

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Page 59

Shear Force and Bending Moment

x 2/3 (A) h = h 0 b L l

x 5/2 (B) h = h 0 b L l

x 1/2 (C) h = h 0 b L l

x 3/2 (D) h = h 0 b L l

SM 59

For the section shown in figure, the allowable bending stress is sc = 50 MPa in compression and st = 120 MPa is in tension. What will be the greatest magnitude of the applied forces P ?

(A) 79.75 kN (B) 110.4 kN (C) 159.5 kN (D) 55.2 kN 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 w = 20 kN/m as shown in figure. If the machine element is to be of constant strength with sallow = 72 MPa and L = 300 mm , the smallest allowable value of d 0 will be

(A) 503 mm (C) 5.03 mm SM 4.33

(B) 50.3 mm (D) 100.6 mm

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 ?

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Shear Force and Bending Moment

(A) st = 1.02 MPa, sc = 792 kPa (C) st = 2.04 MPa, sc = 792 kPa SM 4.34

(B) st = 792 kPa, sc = 1.02 MPa (D) st = 1.02 MPa, sc = 396 kPa

(B) xm = 0.2 m, sm = 156.3 MPa (D) xm = 0.2 m, sm = 78.15 MPa

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

(A) 95 mm (C) 9.5 mm SM 4.36

SM 4

For the tapered beam as shown in the figure, if P = 150 kN , 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) xm = 0.4 m, sm = 78.15 MPa (C) xm = 0.4 m, sm = 156.3 MPa SM 4.35

Page 60

(B) 950 mm (D) 19.0 mm

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 d 0 is

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Shear Force and Bending Moment

1/3 (A) d = d 0 & 4x a1 + x k0 L L 1/3 (C) d = d 0 & 4x a1 - x k0 L L

SM 4.37

1/3 (B) d = d 0 & 2x a1 - x k0 L L 1/3 (D) d = d 0 & 2x a1 + x k0 L L

(B) 67.1 MPa (D) 134.2 MPa

The beam shown in figure, is made of elastic perfectly plastic material. If a = 50 mm and sY = 230 MPa , what will be the maximum elastic moment and the plastic moment that can applied to the cross section ?

(A) (B) (C) (D) SM 4.39

SM 61

In the figure shown, the box beam is made of an elastically perfectly plastic material for which sY = 250 MPa . If the plastic moment M p is applied and then released, what will be the residual stress in the top and bottom of the beam ?

(A) 317.14 MPa (C) 158.6 MPa SM 4.38

Page 61

MY MY MY MY

= 50.7 kN-m, M p = 86.25 kN-m = 86.25 kN-m, M p = 50.7 kN-m = 101.4 kN-m, M p = 86.25 kN-m = 50.7 kN-m, M p = 43.15 kN-m

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

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GATE Mechanical Engineering in 4 Volume NODIA SM 62

SM 4.40

Demo Ebook

Page 62

Shear Force and Bending Moment

SM 4

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

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SM 4.41

Demo Ebook

Shear Force and Bending Moment

Page 63 SM 63

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

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GATE Mechanical Engineering in 4 Volume NODIA SM 64

SM 4.42

Demo Ebook

Page 64

Shear Force and Bending Moment

SM 4

Consider the compound beam as shown in figure. It is supported by a smooth plate at A which slides within the groove and so it cannot support a vertical force, although it can support a moment and axial load. Which of the followings are the shear force and bending moment diagrams for this beam ?

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Shear Force and Bending Moment

Page 65 SM 65

Common Data For Q. 43 and 44 Consider the beam and loading as shown in figure.

SM 4.43

The shear force and bending-moment diagrams are

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GATE Mechanical Engineering in 4 Volume NODIA SM 66

Demo Ebook

Page 66

Shear Force and Bending Moment

SM 4

SM 4.44

The maximum normal stress due to bending is (A) 12.89 MPa (B) 10.75 MPa (C) 21.49 MPa (D) 17.19 MPa

SM 4.45

In the figure shown, a simply supported beam ABC is loaded by a vertical load P which is acting at the end of a bracket BDE . The shear-force and bendingmoment diagrams for beam ABC , are

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Shear Force and Bending Moment

Page 67 SM 67

SM 4.46

Consider the figure as shown. Which of the followings are the required shear force and bending-moment diagrams for the beam and loading ?

SM 4.47

Consider a beam ABC , which is simply supported at A and B , and has an overhang BC as shown in figure. The beam is loaded by two same forces P and a clockwise couple of moment Pa that act through the arrangement as shown. Which of the following options show the required shear-force and bending-moment diagrams for beam ABC ?

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GATE Mechanical Engineering in 4 Volume NODIA SM 68

SM 4.48

Demo Ebook

Page 68

Shear Force and Bending Moment

SM 4

Which of the followings are the required shear force and bending-moment diagrams for the beam and loading shown in the figure ?

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SM 4.49

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Shear Force and Bending Moment

Page 69 SM 69

Consider the beam and loading as shown in figure. The shear force and bendingmoment diagrams for the beam, are

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GATE Mechanical Engineering in 4 Volume NODIA SM 70

Demo Ebook

Page 70

Shear Force and Bending Moment

SM 4

SM 4.50

Which of the followings are the required shear force and bending-moment diagrams for the beam and loading shown in the figure ?

SM 4.51

For the beam and loading as shown in figure, the shear force and bending-moment diagrams are

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SM 4.52

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Shear Force and Bending Moment

Page 71 SM 71

For the beam shown in figure, correct representation of the shear force and bending moment diagrams are

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GATE Mechanical Engineering in 4 Volume NODIA SM 72

SM 4.53

Demo Ebook

Page 72

Shear Force and Bending Moment

SM 4

The shear force and bending moment diagrams for the beam shown in the figure, are

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Page 73 SM 73

SM 4.54

A cantilever beam AB carries a uniform load of intensity w over one-half of its length as shown in figure. What will be the shear-force and bending-moment diagrams for the beam ?

SM 4.55

In figure shown, the beam ABCD has overhangs that extend in both directions for a distance of 4.2 m from the supports at B and C , which are 1.2 m apart. The shear-force and bending-moment diagrams for this overhanging beam are

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GATE Mechanical Engineering in 4 Volume NODIA SM 74

SM 4.56

Demo Ebook

Page 74

Shear Force and Bending Moment

SM 4

Two segments of uniform load is supported by a simple beam AB and two horizontal forces acting at the ends of a vertical arm as shown in figure. Which of the followings are the shear-force and bending-moment diagrams for this beam ?

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Shear Force and Bending Moment

Page 75 SM 75

In figure shown, a simple beam AB is subjected to a counterclockwise couple of moment M 0 acting at distance a from the left-hand support. Which of the followings are the shear-force and bending-moment diagrams for this beam ?

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GATE Mechanical Engineering in 4 Volume NODIA SM 76

SM 4.58

Demo Ebook

Page 76

Shear Force and Bending Moment

SM 4

The shear force and bending moment diagrams for the beam as shown in figure, are

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

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SM 5 TRANSFORMATION OF STRESS AND STRAIN

SM 5.1

Consider the given state of stress as shown in figure. By using a method of analysis based on the equilibrium of that element, the normal and shearing stresses exerted on the oblique face of the shaded triangular element, are

(A) s =- 0.521 MPa , t =- 56.4 MPa (B) s = 0.521 MPa , t = 56.4 MPa (C) s =- 0.521 MPa , t = 56.4 MPa (D) s = 0.521 MPa , t =- 56.4 MPa SM 5.2

In figure shown, the stresses acting on element A in the web of a train rail are found to be 42 MPa tension in the horizontal direction and 140 MPa compression in the vertical direction. The shear stresses of magnitude 60 MPa act in the directions shown. What will be the stresses acting on an element oriented at a counterclockwise angle of 48c from the horizontal ?

(A) (B) (C) (D) SM 5.3

sx sx sx sx

1

1

1

1

=- 118.2 MPa , sy = 20.2 MPa , tx y =- 84.2 MPa = 118.2 MPa , sy = 20.2 MPa , tx y =- 84.2 MPa =- 118.2 MPa , sy =- 20.2 MPa , tx y =- 84.2 MPa =- 118.2 MPa , sy = 20.2 MPa , tx y = 84.2 MPa 1

1

1 1

1 1

1

1

1 1

1 1

Two members of uniform cross section 50 # 80 mm are glued together along plane a -a , which forms an angle of 25c with the horizontal. If the allowable stresses for the glued joint are s = 800 kPa and t = 600 kPa , what will be the largest axial load P ?

GATE Mechanical Engineering in 4 Volume NODIA SM 78

Demo Ebook

Transformation of Stress and Strain

(A) 6.27 kN (C) 7.8 kN SM 5.4

SM 5

(B) 7.04 kN (D) 3.9 kN

Consider the state of stress as shown in figure. What will be the principal planes and the principal stresses ?

(A) (B) (C) (D) SM 5.5

Page 78

qp =- 30.96c, 59.04c, smax =- 52 MPa , smin =- 84 MPa qp =- 30.96c, 59.04c, smax = 52 MPa , smin =- 84 MPa qp =- 30.96c, 59.04c, smax = 84 MPa , smin = 52 MPa qp = 30.96c, - 59.04c, smax = 52 MPa , smin =- 84 MPa

A rectangular plate of dimensions 100 mm # 250 mm is formed by welding two triangular plates and subjected to a compressive stress of 2.5 MPa in the long direction and a tensile stress of 12.0 MPa in the short direction as shown in figure. What will be the normal stress sw acting perpendicular to the line of the weld and the shear stress tw acting parallel to the weld ?

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

sw =- 10 MPa , tw =- 5 MPa sw =- 10.0 MPa , tw = 5.0 MPa sw = 10.0 MPa , tw = 5.0 MPa sw = 10.0 MPa , tw =- 5.0 MPa

Common Data For Q. 6 and 7 Two steel plates of uniform cross section 10 # 80 mm are welded together as shown in figure. The centric forces of 100 kN are applied to the welded plates and the in-plane shearing stress parallel to the weld is 30 MPa.

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SM 5.6

Demo Ebook

Page 79

Transformation of Stress and Strain

The angle b is (A) 61.32c (C) 14.34c

SM 79

(B) 28.68c (D) 75.66c

SM 5.7

What will be the corresponding normal stress perpendicular to the weld ? (A) 87.9 MPa (B) 117.3 MPa (C) 146.7 MPa (D) 73.35 MPa

SM 5.8

Consider the state of plane stress as shown in figure. The value of txy for which the in-plane shearing stress parallel to the weld is zero and the corresponding principal stresses using Mohr’s circle, are

(A) (B) (C) (D) SM 5.9

txy = 2.89 MPa , sA = 12.77 MPa , sB = 1.23 MPa txy =- 2.89 MPa , sA =- 12.77 MPa , sB = 1.23 MPa txy =- 2.89 MPa , sA = 12.77 MPa , sB =- 1.23 MPa txy =- 2.89 MPa , sA = 12.77 MPa , sB = 1.23 MPa

In figure shown, the grain of a wooden member forms an angle of 15c with the vertical. For the state of stress shown, the in-plane shearing stress parallel to the grain and the normal stress perpendicular to the grain using Mohr’s circle, are

(A) txlyl = 0.3 MPa , sxl = 2.92 MPa (C) txlyl =- 0.3 MPa , sxl =- 2.92 MPa

(B) txlyl = 0.3 MPa , sxl =- 2.92 MPa (D) txlyl =- 0.3 MPa , sxl = 2.92 MPa

Common Data For Q. 10 and 11 In figure shown, a propeller shaft subjected to combined torsion and axial thrust is designed to resist a shear stress of 63 MPa and a compressive stress of 90 MPa.

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SM 5.10

SM 5.11

Demo Ebook

Page 80

Transformation of Stress and Strain

What will be the principal stresses ? (A) s1 = 32.4 MPa , s2 = 122.4 MPa (C) s1 =- 32.4 MPa , s2 = 122.4 MPa

SM 5

(B) s1 = 32.4 MPa , s2 =- 122.4 MPa (D) s1 =- 32.4 MPa s2 =- 122.4 MPa

The maximum shear stresses and associated normal stresses are (A) save =- 77.4 MPa , t = ! 45 MPa (B) save = 45 MPa , t = ! 77.4 MPa (C) save =- 45 MPa t = ! 77.4 MPa (D) save = 77.4 MPa , t = ! 45 MPa

Common Data For Q. 12 and 13 An element in uniaxial stress is subjected to tensile stresses sx = 55 MPa as shown in the figure. Use Mohr’s circle for further calculation.

SM 5.12

SM 5.13

What will be the stresses acting on an element oriented at an angle q =- 30c from the x axis (minus means clockwise) ? (A) sx = 41.2 MPa , sy = 27.5 MPa , tx y = 23.8 MPa (B) sx = 41.2 MPa , sy = 13.8 MPa , tx y =- 23.8 MPa (C) sx = 27.5 MPa , sy = 13.8 MPa , tx y = 23.8 MPa (D) sx = 13.8 MPa , sy = 41.2 MPa , tx y = 23.8 MPa 1

1

1 1

1

1

1 1

1

1

1 1

1

1

1 1

The maximum shear stresses and associated normal stresses are (A) tmax = 27.5 MPa , savg = 13.75 MPa (B) tmax = tavg =- 27.5 MPa (C) tmax = 27.5 MPa , savg = 27.5 MPa (D) tmax = tavg = 13.75 MPa

Common Data For Q. and 15 An element in biaxial stress is subjected to stresses sx =- 60 MPa and sy = 20 MPa as shown in the figure.

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SM 5.14

Demo Ebook

Page 81

Transformation of Stress and Strain

SM 81

Using Mohr’s circle, the stresses acting on an element oriented at a counterclockwise angle q = 22.5c from the x axis, are (A) sx =- 48.28 MPa , sy = 8.28 MPa , tx y = 28.28 MPa (B) sx =- 20 MPa , sy = 8.28 MPa , tx y = 28.28 MPa (C) sx = 8.28 MPa , sy = 48.28 MPa , tx y = 28.28 MPa (D) sx = 48.28 MPa , sy =- 20 MPa , tx y = 28.28 MPa 1

1

1

1

1

1

1

1 1

1 1

1 1

1

1 1

SM 5.15

What will be the maximum shear stresses and associated normal stresses using Mohr’s circle ? (A) tmax =- 40 MPa , savg = 20 MPa (B) tmax = tavg = 20 MPa (C) tmax = tavg = 40 MPa (D) tmax = 40 MPa , savg =- 20 MPa

SM 5.16

For the state of stress shown in figure, what will be the range of values of q for which the normal stress sx is equal to or less than 50 MPa ?

(A) 110.085c # q # 290.085c (C) 110.085c # q # 196.524c

(B) 16.524c # q # 110.085c (D) 16.524c # q # 196.524c

Common Data For Q. 17 and 18 For the state of plane stress shown in figure, consider both in-plane and out-ofplane shearing stresses.

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Demo Ebook

Page 82

Transformation of Stress and Strain

SM 5

SM 5.17

The maximum shearing stress when sy = 20 MPa , is (A) 80 MPa (B) 160 MPa (C) 180 MPa (D) 100 MPa

SM 5.18

The maximum shearing stress when sy = 140 MPa , is (A) 110 MPa (B) 80 MPa (C) 60 MPa (D) 220 MPa

SM 5.19

An element in plane stress is subjected to stresses sx = 21 MPa , sy = 11 MPa and txy = 8 MPa as shown in figure. The angle q is positive when counterclockwise and negative when clockwise. Using Mohr’s circle, the stresses acting on an element oriented at an angle q = 50c from the x axis, is

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

sx sx sx sx

1

1

1

1

= 23.01 MPa , sy =- 8.99 MPa , tx y = 6.31 MPa = 23.01 MPa , sy = 8.99 MPa , tx y = 6.31 MPa =- 8.99 MPa , sy = 23.01 MPa , tx y = 6.31 MPa = 8.99 MPa , sy = 23.01 MPa , tx y = 6.31 MPa 1

1 1

1

1 1

1

1

1 1

1 1

Common Data For Q. 20 and 21 In figure shown, an element in plane stress is subjected to stresses sx = 3100 kPa , sy = 8700 kPa and txy =- 4500 kPa . Use the Mohr’s circle.

SM 5.20

SM 5.21

What will be the principal stresses ? (A) s1 = 5300 kPa , s2 = 5900 kPa (C) s1 = 11200 kPa , s2 = 600 kPa

(B) s1 = 600 kPa , s2 = 11200 kPa (D) s1 = 11200 kPa , s2 = 5300 kPa

The maximum shear stresses and associated normal stresses are (A) tmax =- 5300 kPa , savg = 5900 kPa (B) tmax = 5300 kPa , savg = 5900 kPa (C) tmax = avg = 5300 kPa (D) tmax = avg = 5900 kPa

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SM 5.22

Page 83

Transformation of Stress and Strain

SM 83

A rectangular steel plate with thickness t = 10 mm is subjected to uniform normal stresses sx and sy as shown in figure. Strain gauges A and B oriented in the x and y directions, respectively, are attached to the plate. The gauge readings are ex = 480 # 10-6 (elongation) and ey = 130 # 10-6 (elongation). What will be the stresses sx and sy and the change in the thickness Dt of the plate ? (Take E = 200 GPa , n = 0.30 )

(A) (B) (C) (D) SM 5.23

Demo Ebook

sx = 60.2 MPa , sy = 114.1 MPa , sx = 114.1 MPa , sy = 60.2 MPa , sx = 60.2 MPa , sy = 114.1 MPa , sx = 114.1 MPa , sy = 60.2 MPa ,

Dt = 2610 # 10-6 mm Dt = 2610 # 10-6 mm Dt = 2610 # 10-6 mm Dt =- 2610 # 10-6 mm

A brass cube 50 mm on each edge is compressed in two perpendicular directions by forces P = 175 kN as shown in figure. The change Dv in the volume of the cube and the strain energy U stored in the cube respectively, are (Take E = 100 GPa , n = 0.34 )

(A) - 56 mm3 , 4.04 J (C) - 56 mm3 , 8.08J

(B) - 56 mm3 , 2.02 J (D) 56 mm3 , 4.04 J

Common Data For Q. 24 and 25 A circle of diameter d = 200 mm is etched on a brass plate as shown in figure. The plate has dimensions 400 # 400 # 20 mm . Forces are applied to the plate, producing uniformly distributed normal stresses sx = 42 MPa and sy = 14 MPa . (Take E = 100 GPa and n = 0.34 )

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GATE Mechanical Engineering in 4 Volume NODIA SM 84

SM 5.24

Page 84

Transformation of Stress and Strain

SM 5

The change in length Dac of diameter ac and Dbd of diameter bd , are (A) (B) (C) (D)

SM 5.25

Demo Ebook

Dac = 0.0745 mm , Dbd = 560 # 10-6 mm Dac =- 0.0745 mm , Dbd = 560 # 10-6 mm Dac = 0.0745 mm , Dbd =- 560 # 10-6 mm Dac =- 0.0745 mm , Dbd =- 560 # 10-6 mm

The change Dt in the thickness of the plate, the change Dv in the volume of the plate and the strain energy U stored in the plate respectively, are (A) 0.00381 mm, - 573 mm3 , 25 J (B) - 0.00381 mm , 573 mm3 , 25 J (C) - 0.00381 mm , 573 mm3 , 50 J (D) 0.00381 mm, - 573 mm3 , 50 J

Common Data For Q. 26 and 27 In figure shown, a cube of granite ( E = 60 GPa , n = 0.25 ) with sides of length a = 75 mm is subjected under triaxial stress. The compressive strains in the material are ex =- 720 # 10-6 and ey = ez =- 270 # 10-6 .

SM 5.26

The normal stresses sx , sy and sz acting on the x , y and z faces of the cube, are (A) sx = sy =- 64.8 MPa , sz =- 43.2 MPa (B) sx =- 64.8 MPa , sy = sz =- 43.2 MPa (C) sx = sz =- 64.8 MPa , sy =- 43.2 MPa (D) sx = 64.8 MPa , sy = sz =- 43.2 MPa

SM 5.27

The maximum shear stress tmax in the material is (A) 8.1 MPa (B) 10.8 MPa (C) 13.5 MPa (D) 6.75 MPa

SM 5.28

The change Dv in the volume of the cube and the strain energy U stored in the cube respectively, are (A) - 532 mm3 , 14.8 J (B) 532 mm3 , 14.8 J (C) - 532 mm3 , 28.7 J (D) 532 mm3, 7.4 J

SM 5.29

A shaft AB of 38 mm diameter is made of steel for which the yield strength is sY = 250 MPa . The magnitude of the torque T for which yield occurs when P = 240 kN , is (Hint: Use the maximum shearing stress criterion)

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Page 85

Transformation of Stress and Strain

(A) 717 N-m (C) 538 N-m

SM 85

(B) 896 N-m (D) 448 N-m

SM 5.30

A spherical steel pressure vessel has a diameter of 250 mm and a wall thickness of 6 mm. The maximum gauge pressure is 8 MPa in the vessel. If the ultimate stress in the steel used is sU = 400 MPa , the factor of safety with respect to tensile failure is (A) 7.56 (B) 5.04 (C) 10.08 (D) 2.52

SM 5.31

In the figure shown, a cylindrical portion of the compressed air tank is fabricated of 6 mm thick plate, welded along a helix forming an angle b = 30c with the horizontal. If the shearing stress parallel to the weld is 30 MPa, what will be the gage pressure ?

(A) 3.41 MPa (C) 2.56 MPa SM 5.32

(B) 4.26 MPa (D) 2.31 MPa

In the figure shown, the tank has a 180 mm inside diameter and a 12 mm wall thickness. If the tank contains compressed air under a pressure of 8 MPa, what will be the magnitude T of the applied torque for which the maximum normal stress in the tank is 75 MPa ?

(A) 21.36 kN-m (C) 17.06 kN-m

(B) 12.76 kN-m (D) 10.68 kN-m

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GATE Mechanical Engineering in 4 Volume NODIA SM 86

Demo Ebook

Page 86

Transformation of Stress and Strain

SM 5

Common Data For Q. 33 and 34. A brass ring of 160 mm outside diameter fits exactly inside a steel ring of 160 mm inside diameter as shown in figure, when the temperature of both rings is 5cC . The temperature of the rings is then raised to 55cC .

SM 5.33

The pressure exerted by the brass ring on the steel ring is (A) 1.75 MPa (B) 1.4 MPa (C) 2.8 MPa (D) 1.05 MPa

SM 5.34

The tensile stress in the steel ring is (A) 28 MPa (C) 35 MPa

(B) 21 MPa (D) 17.5 MPa

SM 5.35

A solid steel sphere of diameter d = 150 mm is subjected to hydrostatic pressure p such that its volume is reduced by 0.4%. What will be the pressure p, the volume modulus of elasticity K for the steel and the strain energy U stored in the sphere ? (Take E = 210 GPa , n = 0.3 ) (A) p = 350 MPa , K = 175 GPa , U = 2470 J (B) p = 700 MPa , K = 350 GPa , U = 2470 J (C) p = 700 MPa , K = 175 GPa , U = 2470 J (D) p = 700 MPa , K = 175 GPa , U = 1235 J

SM 5.36

In the figure shown, an element of material subjected to plane strain, has strains as follows: ex = 220 # 10-6 , ey = 480 # 10-6 and gxy = 180 # 10-6 . What will be the strains for an element oriented at an angle q = 50c ?

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

ex ex ex ex

1

1

1

1

= 461 # 10-6 , ey = 239 # 10-6 , gx y =- 225 # 10-6 =- 461 # 10-6 , ey = 239 # 10-6 , gx y = 225 # 10-6 = 461 # 10-6 , ey = 239 # 10-6 , gx y = 225 # 10-6 =- 461 # 10-6 , ey =- 239 # 10-6 , gx y = 225 # 10-6 1

1 1

1

1

1 1

1 1

1

1 1

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SM 5.37

Demo Ebook

Transformation of Stress and Strain

Page 87 SM 87

An aluminum plate is loaded in biaxial stress by normal stresses sx = 86.4 MPa and sy as shown in figure. A strain gauge is bonded to the plate at an angle f = 21c and strain measured by the gauge is e = 946 # 10-6 . What are the shear strains (gmax) xy , (gmax) xz and (gmax) yz ? (Take E = 72 GPa and n = 1/3 )

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

-6 -6 -6 ^gmaxhxy =- 1200 # 10 , ^gmaxhxz = 1600 # 10 , ^gmaxhyz = 399 # 10 -6 -6 -6 ^gmaxhxy = 1200 # 10 , ^gmaxhxz =- 1600 # 10 , ^gmaxhyz = 399 # 10 -6 -6 -6 ^gmaxhxy = 1200 # 10 , ^gmaxhxz = 1600 # 10 , ^gmaxhyz =- 399 # 10 -6 -6 -6 ^gmaxhxy = 1200 # 10 , ^gmaxhxz = 1600 # 10 , ^gmaxhyz = 399 # 10

Common Data For Q. 38 and 39 The rosette shown in figure, has been used to determine the following strains at a point on the surface of a crane hook : e1 =+ 420 m , e2 =- 45 m , e4 =+ 165 m

SM 5.38

What should be the reading of gauge 3 ? (A) - 300 m (B) 300 m (C) 255 m (D) 210 m

SM 5.39

The principal strains and the maximum in-plane shearing strain are (A) ea = 435 m , eb =- 315 m , gmax = 750 m (B) ea = 435 m , eb = 315 m , gmax = 750 m (C) ea =- 435 m , eb =- 315 m , gmax = 750 m (D) ea = 435 m , eb =- 315 m , gmax =- 750 m

SM 5.40

Consider the figure shown. What will be the sum of the three strain measurements ( e1 + e2 + e3 ) made with a 60c rosette which is independent to the orientation of the rosette ? Here eave is the abscissa of the centre of the corresponding Mohr’s circle for strain.

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GATE Mechanical Engineering in 4 Volume NODIA SM 88

Demo Ebook

Transformation of Stress and Strain

(A) eave (C) 3eave SM 5.41

A single strain gauge is cemented to a solid 96 mm diameter aluminium shaft at an angle b = 20c with a line parallel to the axis of the shaft as shown in figure. If G = 27 GPa , the torque T corresponding to a gauge reading of 400 m, is

(B) 5.84 kN-m (D) 4.38 kN-m

In figure shown, the beam is subjected to a shear of V = 15 kN and has width of w = 125 mm . What will be the web’s shear stress at A and B ?

(A) tA = 1.99 MPa , tB = 3.3 MPa (C) tA = 1.99 MPa , tB = 1.65 MPa SM 5.43

SM 5

(B) 6eave (D) 2eave

(A) 7.3 kN-m (C) 3.65 kN-m SM 5.42

Page 88

(B) tA = 1.65 MPa tB = 1.99 MPa (D) tA = 3.98 MPa , tB = 1.65 MPa

The strut shown in figure, is subjected to a vertical shear of V = 130 kN . Which of the following is the resultant shear force developed in the vertical segment AB ?

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Transformation of Stress and Strain

(A) 50.3 kN (C) 37.7 kN SM 5.44

(B) 62.9 kN (D) 25.15 kN

(B) 18.37 MPa (D) 14.7 MPa

What will be the smallest allowable diameter of the solid shaft ABCD of figure shown when tall = 60 MPa and the radius of disk B is r = 120 mm ?

(A) 19.6 mm (C) 196 mm SM 5.46

SM 89

The T-beam is subjected to the loading as shown in figure. What will be the maximum transverse shear stress in the beam at the critical section ?

(A) 29.4 MPa (C) 7.35 MPa SM 5.45

Page 89

(B) 39.2 mm (D) 3.92 mm

Member AB has a uniform rectangular cross section of 10 # 24 mm . For the loading shown in figure, the normal and shearing stresses at point H , are

(A) sx = 32.5 MPa , txy = 14.06 MPa (C) sx = 32.5 MPa , txy =- 14.06 MPa

(B) sx =- 32.5 MPa , txy = 14.06 MPa (D) sx =- 32.5 MPa , txy =- 14.06 MPa

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GATE Mechanical Engineering in 4 Volume NODIA SM 90

SM 5.47

SM 5

The beam is made from three polystyrene strips that are glued together as shown in figure. If the glue has a shear strength of 80 kPa, the maximum load P that can be applied without causing the glue to lose its bond, is

(B) 178.5 N (D) 297.5 N

In the figure shown, a 13 kN force is applied to the 60 mm diameter cast-iron post ABD . At point H , the principal stresses and the maximum shearing stresses are

(A) (B) (C) (D) SM 5.49

Page 90

Transformation of Stress and Strain

(A) 357 N (C) 238 N SM 5.48

Demo Ebook

sA = 4.3 MPa , sB =- 93.4 MPa , t =- 48.9 MPa sA =- 4.3 MPa , sB =- 93.4 MPa , t = 48.9 MPa sA = 4.3 MPa , sB = 93.4 MPa , t = 48.9 MPa sA = 4.3 MPa , sB =- 93.4 MPa , t = 48.9 MPa

The aluminium strut is 10 mm thick and has the cross section as shown in figure. If it is subjected to a shear of V = 150 N , the shear flow at points A and B are

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Transformation of Stress and Strain

(A) qA = 1.39 kN/m , qB = 2.50 kN/m (C) qA = 1.39 kN/m , qB = 1.25 kN/m SM 5.50

Page 91 SM 91

(B) qA = 2.79 kN/m , qA = 2.50 kN/m (D) qA = 2.78 kN/m , qB = 1.25 kN/m

A beam is constructed from five boards bolted together as shown in figure. If the bolts are spaced s = 250 mm apart and the applied shear is V = 35 kN , the maximum shear force developed in each bolt is

(A) 6.64 kN (C) 10.62 kN

(B) 5.31 kN (D) 3.98 kN

Common Data For Q. 51 and 52 A force P is applied to a cantilever beam by means of a cable attached to a bolt, located at the center of the free end of the beam as shown in figure. A load P acts in a direction perpendicular to the longitudinal axis of the beam.

SM 5.51

The normal stress at point a in terms of P , b, h , l and b is cos b sin b cos b sin b (A) 6Pl : (B) 6Pl : + bh h b D bh h b D cos b sin b cos b sin b (C) 3Pl : (D) 3Pl : + bh h b D bh h b D

SM 5.52

What will be the values of b for which the normal stress at a is zero ? (B) b = tan-1 b b l (A) b = tan-1 b h l b 2h (C) b = tan-1 b b l (D) b = tan-1 b 2h l h b

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GATE Mechanical Engineering in 4 Volume NODIA SM 92

SM 5.53

Demo Ebook

Transformation of Stress and Strain

(B) 70 mm (D) 43.75 mm

Consider an angle of thickness 3 mm and a height of h = 100 mm as shown in the figure. If it is subjected to a shear of V = 50 N , the shear flow at point A and the maximum shear flow in the angle are

(A) qA = 0 , q max = 375 N/m (C) qA = 187.5 N/m , q max = 375 N/m SM 5.55

(B) qA = 0 , q max = 187.5 N/m (D) qA = 187.5 N/m , q max = 187.5 N/m

The stepped support is subjected to the bearing load of 50 kN as shown in figure. The maximum and minimum compressive stress in the material are

(A) smax = 11 MPa , smin = 5.5 MPa (C) smax = 11 MPa , smin = 0 SM 5.56

SM 5

What will be the location e of the shear center (point O ), for the thin-walled member having a slit along its side ?

(A) 52.5 mm (C) 87.5 mm SM 5.54

Page 92

(B) smax = 5.5 MPa , smin = 0 (D) smax = 16.5 MPa , smin = 0

Several forces are applied to the pipe assembly as shown in figure. Each section of pipe has inner and outer diameter respectively equal to 36 mm and 42 mm. What will be the normal and shearing stresses at point H , located at the top of the outer surface of the pipe ?

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Transformation of Stress and Strain

(A) sH =- 20.2 MPa , tH = 2.82 MPa (C) sH = 20.2 MPa , tH =- 2.82 MPa SM 5.57

(B) sH = 20.2 MPa , tH = 2.82 MPa (D) sH =- 20.2 MPa , tH =- 2.82 MPa

(B) 13.2 kN (D) 8.25 kN

Consider the assembly shown in figure. Each piston has a radius of 45 mm and the cylinder has a wall thickness of 2 mm. The maximum force P that can be exerted on each of the two pistons so that the circumferential stress component in the cylinder does not exceed 3 MPa, is

(A) 1060 N (C) 848 N SM 5.59

SM 93

What will be the magnitude of the load P as shown in figure that will cause a maximum normal stress of smax = 200 MPa in the link at section a -a ?

(A) 9.9 kN (C) 16.5 kN SM 5.58

Page 93

(B) 636 N (D) 530 N

The cap on the cylindrical tank is bolted to the tank along the flanges as shown in figure. The tank has an inner diameter of 1.5 m and a wall thickness of 18 mm. If the pressure in the tank is p = 120 MPa , what will be the force in the 16 bolts that are used to attach the cap to the tank and also, specify the state of stress in the wall of the tank ?

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GATE Mechanical Engineering in 4 Volume NODIA SM 94

Demo Ebook

Page 94

Transformation of Stress and Strain

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

s1 = 25 MPa , s1 = 50 MPa , s1 = 75 MPa , s1 = 50 MPa ,

s2 = 50 MPa , s2 = 25 MPa , s2 = 50 MPa , s2 = 75 MPa ,

Fb = 133 kN Fb = 133 kN Fb = 133 kN Fb = 133 kN

***********

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SM 5

SM 6 DESIGN OF BEAMS AND SHAFTS

SM 6.1

A 1.5 m long cantilever beam AB is loaded by a couple M 0 at its free end as shown in figure. The distance from the top surface of the beam to the neutral surface is 75 mm and the longitudinal normal strain at the top surface is 0.001 . What will be the curvature k and the vertical deflection d at the end of the beam ?

(A) k = 0.1333 m-1 , d = 15.0 mm (C) k = 0.01333 m-1 , d = 1.5 mm SM 6.2

Wide-flange beam as shown in figure, is made of a high-strength, low-alloy steel for which sY = 345 MPa and sU = 450 MPa . If a factor of safety of 3.0 is used and the effect of fillets is neglected, what will be the largest couple that can be applied to the beam when it is bent about the y -axis ?

(A) 42.3 kN-m (C) 70.3 kN-m SM 6.3

(B) k = 0.1333 m-1 , d = 1.5 mm (D) k = 0.01333 m-1 , d = 15.0 mm

(B) 35.15 kN-m (D) 56.3 kN-m

A nylon spacing bar has the cross section as shown in figure. If the allowable stress for the grade of nylon used is 24 MPa, the largest couple Mz that can be applied to the bar, is

(A) 2.38 kN-m (C) 2.975 kN-m

(B) 1.785 kN-m (D) 1.487 kN-m

GATE Mechanical Engineering in 4 Volume NODIA SM 96

SM 6.4

sA = 116.4 MPa , sB =- 87.3 MPa sA =- 116.4 MPa , sB =- 87.3 MPa sA =- 116.4 MPa , sB = 87.3 MPa sA = 116.4 MPa , sB = 87.3 MPa

Each girder of a highway bridge has a cantilever length of 46 m and an I -shaped cross section with dimensions as shown in the figure. The load on each girder is 11.0 kN/m, which includes the weight of the girder. What will be the maximum bending stress in a girder due to this load?

(B) 129 MPa (D) 193.5 MPa

For the extruded beam shown in figure, the allowable stress is 120 MPa in tension and 150 MPa in compression. What will be the largest couple M that can be applied ?

(A) 1.89 kN-m (C) 7.58 kN-m SM 6.7

SM 6

In the figure shown, if the couple acts in a vertical plane, the stress at point A and point B are

(A) 96.75 MPa (C) 161.25 MPa SM 6.6

Page 96

Design of Beams and Shafts

(A) (B) (C) (D) SM 6.5

Demo Ebook

(B) 5.65 kN-m (D) 3.79 kN-m

The box beam as shown in figure has an allowable bending stress of sallow = 10 MPa and an allowable shear stress of tallow = 775 kPa . What will be the maximum

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Page 97

Design of Beams and Shafts

SM 97

intensity w of the distributed loading that it can safely support ?

(A) 1.51 kN-m (C) 2.01 kN-m

(B) 6.04 kN-m (D) 3.02 kN-m

Common Data For Q. 8 and 9 Consider the beam as shown in figure which has an allowable stress of 120 MPa.

SM 6.8

What will be the largest couple M that can be applied to the beam ? (A) 18.34 kN-m (B) 9.17 kN-m (C) 4.585 kN-m (D) 6.875 kN-m

SM 6.9

If the cross section of the beam is a square of 80 mm side, what will be the largest couple M ? (A) 7.68 kN-m (B) 10.24 kN-m (C) 15.36 kN-m (D) 5.12 kN-m

SM 6.10

Vertical wood beams AB of thickness t = 120 mm is used to construct a small dam of height h = 2.0 m as shown in the figure. If the beams are simply supported at the top and bottom and the weight density of water is g = 9.81 kN/m3 , the maximum bending stress smax in the beams will be

(A) 1.4 MPa (C) 2.8 MPa SM 6.11

(B) 3.15 MPa (D) 2.10 MPa

In figure shown, the tapered beam supports a concentrated force P at its center. If it is made from a plate that has a constant width b, what will be the absolute

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GATE Mechanical Engineering in 4 Volume NODIA SM 98

Demo Ebook

Page 98

Design of Beams and Shafts

SM 6

maximum bending stress in the beam ?

2 (A) smax = 3PL2 8bh 0 (C) smax = 3PL2 8bh 0

(B) smax = 3PL3 8bh 0 3 (D) smax = 3PL2 4bh 0

Common Data For Q. 12 and 13 A beam of rectangular cross section is to be sawed from a log of circular cross section as shown in figure. A couple M is applied to the rectangular beam.

SM 6.12

For the smallest possible maximum stress sm , the ratio d/b will be 2 (B) (A) 2 3 (C) 2

SM 6.13

3

What will be the ratio d/b, for which the radius of curvature of the beam is maximum ? 3 (B) (A) 2 2 (C)

SM 6.14

(D)

(D) 2

3

For the simple beam AB and loading shown in figure, consider the following data: P = 5.4 kN , L = 3.0 m , d = 1.2 m , b = 75 mm t = 25 mm , h = 100 mm and h1 = 75 mm . What will be the maximum tensile stress st and maximum compressive stress sc due to the load P ?

(A) st = 73.2 MPa , sc = 43.9 MPa (C) st = 36.6 MPa , sc = 43.9 MPa

(B) st = 43.9 MPa , sc = 73.2 MPa (D) st = 36.6 MPa , sc = 87.8 MPa

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GATE Mechanical Engineering in 4 Volume NODIA SM 6

SM 6.15

Demo Ebook

Design of Beams and Shafts

Page 99 SM 99

The reinforced concrete beam as shown in figure, is subjected to a positive bending moment of 175 kN- m . The modulus of elasticity is 25 GPa for the concrete and 200 GPa for the steel. What will be the stress in the steel and the maximum stress in the concrete ?

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

ss = 247.5 MPa , sc =- 26.0 MPa ss = 330 MPa , sc =- 26.0 MPa ss = 247.5 MPa , sc = 32.5 MPa ss = 330 MPa , sc = 32.5 MPa

Common Data For Q. 16 to 18 A cantilever beam AB of triangular cross section has length L = 0.8 m , width b = 80 mm and height h = 120 mm as shown in figure. The beam is made of brass weighing 85 kN/m3 .

SM 6.16

What will be the maximum tensile stress st and maximum compressive stress sc due to the beam’s own weight ? (A) st = 1.36 MPa , sc = 2.72 MPa (B) st = 5.44 MPa , sc = 2.72 MPa (C) st = 2.72 MPa , sc = 5.44 MPa (D) st = 2.72 MPa , sc = 1.36 MPa

SM 6.17

If the width b is doubled, the stresses will be (A) half (B) double (C) remain same (D) one-Fourth

SM 6.18

If the height h is doubled, the stresses will be (A) half (B) double (C) remain same (D) one-Fourth

SM 6.19

Consider the design of a reinforced concrete beam as shown in figure. The design is said to be balanced when the maximum stresses in the steel ss and concrete sc are equal to the allowable stresses. If Ec and Es are the moduli of elasticity of concrete and steel, respectively and d is the distance from the top of the beam to the reinforcing steel, the distance x from the top of the beam to the neutral axis to achieve a balanced design, must be

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GATE Mechanical Engineering in 4 Volume NODIA SM 100

Demo Ebook

Page 100

Design of Beams and Shafts

d 1 + Ec sc Es ss d (C) x = 1 - Ec sc Es ss (A) x =

SM 6

d 1 - Ec ss Es sc d (D) x = 1 + Ec ss Es sc (B) x =

Common Data For Q. 20 and 21 Consider the two bars as shown in figure (a) and (b). The allowable stress used in the design of a steel bar is 80 MPa.

SM 6.20

If the bar is designed with grooves having semicircular portions of radius r = 15 mm and K = 1.92 as shown in figure (a), what will be the largest couple M that can be applied to the bar ? (A) 2.5 kN-m (B) 3.125 kN-m (C) 0.625 kN-m (D) 1.25 kN-m

SM 6.21

If the bar is redesigned by removing of material above the grooves ( K = 1.57 ) as shown in figure (b), the largest couple M that applied to the bar can be (A) 1.53 kN-m (B) 3.825 kN-m (C) 2.04 kN-m (D) 3.06 kN-m

SM 6.22

A 450 mm long cantilever beam AB of circular cross section supports a load P = 400 N acting at the free end as shown in figure. The beam is made of steel with weight density of 77 kN/m3 and an allowable bending stress of 60 MPa. If the effect of the beam’s own weight is considered, the required diameter d min of the beam is

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Page 101

Design of Beams and Shafts

(A) 39.51 mm (C) 31.61 mm

SM 101

(B) 23.71 mm (D) 19.75 mm

Common Data For Q. 23 and 24 A tapered cantilever beam AB of length L has square cross sections and supports a concentrated load P at the free end as shown in figure. The width and height of the beam vary linearly from hA at the free end to hB at the fixed end.

SM 6.23

If hB = 3hA , what will be the distance x from the free end A to the cross section of maximum bending stress ? (B) 2L (A) L 2 (C) L (D) L 4

SM 6.24

What will be the ratio of the maximum stress smax to the largest stress sB at the support ? (A) 1 (B) 4 (C) 2 (D) 6

SM 6.25

In figure shown, a bar of rectangular cross section is made of steel which is assumed to be elastoplastic with sY = 320 MPa and subjected to a couple M parallel to the y -axis. What will be the moments Ma of the couple for which yield first occurs and Mb for the 5 mm thick plastic zones at the top and bottom of the bar ?

(A) Ma = 115.2 N-m , Mb = 171.2 N-m (C) Ma = 143.75 N-m , Mb = 171.2 N-m

(B) Ma = 115.2 N-m , Mb = 214 N-m (D) Ma = 143.75 N-m , Mb = 214 N-m

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GATE Mechanical Engineering in 4 Volume NODIA SM 102

SM 6.26

Demo Ebook

Design of Beams and Shafts

SM 6

Consider a steel beam of the cross section as shown in figure. If the steel is elastoplastic with a yield strength of 240 MPa, the plastic moment MP of the beam is

(A) 14.26 kN-m (C) 23.76 kN-m SM 6.27

Page 102

(B) 11.88 kN-m (D) 19.01 kN-m

A beam of width t has a depth that varies as shown in figure. If it supports a concentrated force P at its end, the absolute maximum bending stress in the beam and its location x , are

b1 L , s = 3PL max b 2 - b1 2tb2 ^b2 - b1h 3PL (C) x = b2 L , smax = b 2 - b1 2tb1 ^b2 - b1h (A) x =

b1 L , s = 3PL max b 2 - b1 2tb1 ^b2 - b1h 3PL (D) x = b1 L , smax = b 2 + b1 2tb1 ^b2 - b1h

(B) x =

Common Data For Q. 30 and 31 Consider a solid bar of elastoplastic material of rectangular cross section. The bending moment and radius of curvature at the onset of yield are, MY and rY , respectively. SM 6.28

The radius of curvature when a couple of moment M = 1.25MY is applied to the bar, is (A) r = 0.70711 rY (B) r = 1.414 rY (C) r = 0.50 rY (D) r = 2 rY

SM 6.29

What will be the radius of curvature after the couple is removed ? (A) rR = 0.70711 rY (B) rR = 6.09 rY (C) rR = 1.732 rY (D) rR = 2.236 rY

SM 6.30

A tall signboard supported by two vertical beams consisting of thin walled, tapered circular tubes is shown in figure. Each beam is represented as a cantilever AB of length L = 8.0 m subjected to a lateral load P = 2.4 kN at the free end. The tubes have constant thickness t = 10.0 mm and average diameters dA = 90 mm and dB = 270 mm at ends A and B , respectively. The moment of inertia at any cross section is I = pd3 t/8 and the section modulus is S = pd 2 t/4 . At what distance x from the free end does the maximum bending stress occur and what is

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Design of Beams and Shafts

SM 103

the magnitude smax of the maximum bending stress ?

(A) 5.0 m, 47.13 MPa (C) 2.5 m , 23.56 MPa SM 6.31

A shaft is supported on journal bearings as shown in figure that do not offer resistance to axial load. The allowable normal stress for the shaft is sallow = 80 MPa . Using the maximum distortion energy theory of failure, the smallest diameter of the shaft that support the loading will be

(A) 15 mm (C) 10 mm SM 6.32

(B) 4.0 m , 37.73 MPa (D) 3.0 m , 28.33 MPa

(B) 20 mm (D) 25 mm

Consider a cantilever beam AB having rectangular cross sections with constant width b and varying height hx . The beam is subjected to a uniform load of intensity w as shown in figure. If the beam is fully stressed, what will be the height hx ?

(A) hx = 2hB x L (C) hx = hB x L

2

(B) hx = hB x 2L (D) hx = hB x L

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GATE Mechanical Engineering in 4 Volume NODIA SM 104

SM 6.33

Demo Ebook

Page 104

Design of Beams and Shafts

SM 6

For the beam shown in figure, if the magnitude of the horizontal force P is 8 kN , the stresses at point A and B will be

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

sA =- 102.8 MPa , sB = 80.6 MPa sA = 102.8 MPa , sB =- 80.6 MPa sA = 80.6 MPa , sB = 102.8 MPa sA =- 80.6 MPa , sB = 80.6 MPa

Common Data For Q. 35 and 36 A laminated plastic beam of square cross section is built up by gluing together three strips of 10 mm # 30 mm in cross section each as shown in figure. The beam has a total weight of 3.2 N and is simply supported with span length of L = 320 mm . Consider the weight of the beam.

SM 6.34

If the allowable shear stress in the glued joints is 0.3 MPa, the maximum permissible load P that placed at the midpoint, will be (A) 402 N (B) 488 N (C) 480 N (D) 405 N

SM 6.35

If the allowable bending stress in the plastic is 8 MPa, the maximum permissible load P that placed at the midpoint, will be (A) 450 N (B) 488 N (C) 448 N (D) 402 N

SM 6.36

What will be the stress at point B , (a) for the loading shown in figure, (b) if the 60 kN loads applied at points 2 and 3 are removed ?

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SM 105

^sB ha = 8.33 MPa , ^sB hb =- 7.64 MPa ^sB ha =- 8.33 MPa , ^sB hb = 7.64 MPa ^sB ha = 6.25 MPa , ^sB hb = 8.23 MPa ^sB ha = 7.64 MPa , ^sB hb = 6.25 MPa

The machine element shown in figure is formed by welding a steel rod to a steel plate. Centroid of the cross section is at C and Iz = 4195 mm 4 . If the allowable stress is 135 MPa, what will be the largest force P that can be applied to the element ?

(A) 3.175 kN (C) 3.81 kN SM 6.38

Page 105

Design of Beams and Shafts

(A) (B) (C) (D) SM 6.37

Demo Ebook

(B) 1.9 kN (D) 2.54 kN

A C-shaped steel ( E = 200 GPa ) bar of square cross section of side 40 mm is shown in figure. The strain on the inner edge of the bar is found to be 450 m. What will be the magnitude of the forces P ?

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GATE Mechanical Engineering in 4 Volume NODIA SM 106

Demo Ebook

Design of Beams and Shafts

(A) 9 kN (C) 6.25 kN SM 6.39

A 60 kN eccentric axial force P acts at point D , which is located 25 mm below the top surface of the steel bar as shown in figure. What will be the depth d of the bar for which the compressive stress at point B is maximum and the corresponding stress at point B ?

(B) d = 75 mm , sB =- 20 MPa (D) d = 150 mm , sB =- 10 MPa

An aluminium pole for a street light of weight 4600 N which supports an arm of weight 660 N is shown in figure. The outside diameter of the pole at its base is 225 mm and its thickness is 18 mm. If the center of gravity of the arm is 1.2 m from the axis of the pole, what will be the maximum tensile and compressive stresses st and sc , respectively, in the pole at its base due to the weights ?

(A) (B) (C) (D) SM 6.41

SM 6

(B) 11.25 kN (D) 13.5 kN

(A) d = 75 mm , sB = 10 MPa (C) d = 75 mm , sB = 20 MPa SM 6.40

Page 106

st = 961 kPa , sc =- 1860 kPa st = 1860 kPa , sc =- 961 kPa st = 1410 kPa , sc = 1860 kPa st = 450 kPa , sc = 1410 kPa

A couple M = 300 N- m is applied to a beam of rectangular cross section as shown in figure, in a plane forming an angle b = 60c with the vertical. What will be the stresses at point A, B and D ?

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SM 107

sA =- 3.37 MPa , sB =- 18.6 MPa , sD = 3.37 MPa sA = 3.37 MPa , sB = 18.6 MPa , sD = 3.37 MPa sA = 3.37 MPa , sB = 18.6 MPa , sD =- 3.37 MPa sA =- 3.37 MPa , sB = 18.6 MPa , sD = 3.37 MPa

A couple M acts in a vertical plane and is applied to a beam oriented as shown in figure. What will be the angle that the neutral axis forms with the horizontal plane and the maximum tensile stress in the beam, respectively ?

(A) 19.52c, 190 MPa (C) 5.475c, 190 MPa SM 6.43

Page 107

Design of Beams and Shafts

(A) (B) (C) (D) SM 6.42

Demo Ebook

(B) 19.52c, 95 MPa (D) 5.475c, 95 MPa

An axial load P = 30 kN is applied to a short section of a C 150 # 12.2 rolled steel channel as shown in figure. Sectional properties for C 150 # 12.2 rolled steel channel are as below:

What will be the largest distance a for which the maximum compressive stress is 60 MPa? (A) 46.65 mm (B) 93.3 mm (C) 77.75 mm (D) 62.2 mm SM 6.44

The Z section shown in figure is subjected to a couple M 0 acting in a horizontal plane. The I max = 2.28 # 106 mm 4 , I min = 0.23 # 106 mm 4 and the principal axes are 25.7c and 64.3c. If the maximum stress is not to exceed 80 MPa, the largest permissible value of the moment M 0 of the couple is

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GATE Mechanical Engineering in 4 Volume NODIA SM 108

(B) 3.969 kN-m (D) 2.646 kN-m

(B) 21.4 kN (D) 5.35 kN

In figure shown, a is the distance from the line of action of the 3 kN force to the vertical plane containing the center of curvature of the bar. If the allowable stress in the bar is 150 MPa, the largest permissible value of a is

(A) 60.9 mm (C) 93.37 mm SM 6.47

SM 6

In figure shown, a box beam of wood is constructed of two 260 mm # 50 mm and two 260 mm # 25 mm boards. The boards are nailed at a longitudinal spacing of s = 100 mm . If each nail has an allowable shear force F = 1200 N , the maximum allowable shear force Vmax will be

(A) 16.05 kN (C) 10.7 kN SM 6.46

Page 108

Design of Beams and Shafts

(A) 7.938 kN-m (C) 1.323 kN-m SM 6.45

Demo Ebook

(B) 32.5 mm (D) 125.87 mm

For the curved beam and loading shown in figure, the stresses at point A and point B are

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GATE Mechanical Engineering in 4 Volume NODIA SM 6

A beam of the cross section shown in figure is extruded from an aluminium alloy for which E = 72 GPa . If the couple shown acts in a vertical plane, what will be the maximum stress in the beam and the corresponding radius of curvature ?

(B) smax = 56.25 MPa , r = 6.4 m (D) smax = 56.25 MPa , r = 12.8 m

Consider the built-up wooden beam as shown in figure. It is subjected to a vertical shear of 8 kN and the nails are spaced longitudinally every 60 mm at A and every 25 mm at B and Ix = 1.504 # 109 mm 4 . What will be the shearing force in the nails at B ?

(A) 549 N (C) 411.75 N SM 6.50

SM 109

sA = 63.9 MPa , sB =- 52.6 MPa sA =- 63.9 MPa , sB =- 52.6 MPa sA = 63.9 MPa , sB = 52.6 MPa sA =- 63.9 MPa , sB = 52.6 MPa

(A) smax = 112.5 MPa , r = 12.80 m (C) smax = 112.5 MPa , r = 6.4 m SM 6.49

Page 109

Design of Beams and Shafts

(A) (B) (C) (D) SM 6.48

Demo Ebook

(B) 823 N (D) 686.25 N

A vertical shear V causes a maximum shearing stress of 75 MPa in the hatshaped extrusion as shown in figure. The corresponding shearing stress at point a and point b, are

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GATE Mechanical Engineering in 4 Volume NODIA SM 110

Demo Ebook

Design of Beams and Shafts

(A) ta = tb = 20.7 MPa (C) ta = 20.7 MPa , tb = 41.4 MPa SM 6.51

(B) ta = 41.4 MPa , tb = 20.7 MPa (D) ta = tb = 41.4 MPa

(B) 332.5 kN-m (D) 166.25 kN-m

A thin-walled beam of uniform thickness has the cross section as shown in figure. What will the dimension h for which the shear center O of the cross section is located at a distance e = 25 mm from the center of the flange AB ?

(A) 92.75 mm (C) 74.2 mm SM 6.53

SM 6

The section shown in figure is formed by welding three plates together of 12 mm thickness each. For a vertical shear of 100 kN, the shear flow through the welded surfaces is

(A) 199.5 kN-m (C) 266 kN-m SM 6.52

Page 110

(B) 55.65 mm (D) 46.37 mm

An extruded beam of 3 mm wall thickness has the cross section as shown in figure and subjected to a 10 kN vertical shear. What will be the shearing stress at point A and the maximum shearing stress in the beam ?

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

Page 111 SM 111

tA = 31.3 MPa , tmax = 50.7 MPa tA = 25.35 MPa , tmax = 62.61 MPa tA = 25.35 MPa , tmax = 31.3 MPa tA = 50.7 MPa , tmax = 62.6 MPa ***********

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SM 7 DEFLECTION OF BEAMS AND SHAFTS

Common Data For Q. 1 and 2 Consider the loading of cantilever beam AB shown in figure.

SM 7.1

The equation of the elastic curve for the cantilever beam AB , is 2

(B) y =- Px (3L - x) 6EI

2

(D) y =- Px (3L - x) 3EI

(A) y =- Px (3L + x) 3EI (C) y =- Px (3L + x) 6EI SM 7.2

2

2

The deflection and slope at the free end respectively, are 3 2 3 2 (B) PL , PL (A) PL , PL 2EI 6EI 6EI 3EI 3 2 (C) PL , PL 3EI 2EI

3 2 (D) PL , PL 4EI 4EI

Common Data For Q. 3 to 5 The deflection curve for a simple beam AB shown in figure, is given by the following equation: 4 y =- w40 L sin px L p EI

SM 7.3

The load acting on the beam is expressed as (B) w = w 0 sin px (A) w = w 0 cos pL x L (C) w = w 0 cos px (D) w = w 0 sin pL x L

SM 7.4

The reactions RA and RB at the supports, are (A) RA = RB = w 0 L p

(B) RA =- w 0 L , RB = w 0 L p p

(C) RA = w 0 L , RB =- w 0 L p p

(D) RA = RB =- w 0 L p

GATE Mechanical Engineering in 4 Volume NODIA SM 7

SM 7.5

Demo Ebook

Page 113

Deflection of Beams and Shafts

SM 113

The maximum bending moment M max , is 2 2 2 (B) w 0 L (A) w 0 L p p2 2 w 02 L (C) w 0 L (D) p2 p2

Common Data For Q. 6 and 7 Consider the cantilever beam and loading shown in figure below.

SM 7.6

SM 7.7

SM 7.8

What will be the deflection at B ? 4 (A) 11wL 192EI

4 (B) 11wL 384EI

3 (C) 11wL 384EI

2 (D) 11wL 384EI

What is the slope at B ? 4 (A) 5wL 48EI

3 (B) 5wL 48EI

2 (C) 5wL 48EI

3 (D) 5wL 16EI

A uniformly loaded steel wide-flange beam with simple supports is shown figure. It has a downward deflection of 10 mm at the midpoint and angles rotation equal to 0.01 radians at the ends. If the maximum bending stress 90 MPa and the modulus of elasticity is 200 GPa, what will be the height h the beam.

(A) 96 mm (C) 9.6 mm SM 7.9

in of is of

(B) 19.2 mm (D) 48 mm

A gold-alloy microbeam attached to a silicon wafer behaves like a cantilever beam, subjected to a uniform load is shown in figure. The beam has length L = 27.5 mm and rectangular cross section of width b = 4.0 mm and thickness t = 0.88 mm . The total load on the beam is 17.2 mN . If the deflection at the end of the beam is 2.46 mm , what is the modulus of elasticity Eg of the gold alloy?

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GATE Mechanical Engineering in 4 Volume NODIA SM 114

Demo Ebook

Deflection of Beams and Shafts

(A) 75 GPa (C) 60 GPa SM 7.10

In the figure shown, uniformly distributed loads are applied to beam AE . If the beam is a W200 # 35.9 rolled shape and that L = 3 m , w = 5 kN/m , I = 34.4 # 106 mm 4 and E = 200 GPa , the distance of the center of the beam from the x axis is

(B) 3.97 mm (D) 39.7 mm

What will be the reaction at the roller support for the beam and loading shown in figure ?

(A) 11w 0 20L (C) 11w 0 30L SM 7.12

(B) 11 w 02 L 40 (D) 11 w 0 L 40

For the beam and loading shown in figure, the reaction and bending moment at A, are

(A) RA = 1 w 0 L , MA =- 5 w 0 L2 96 4 (C) RA =- 1 w 0 L , MA =- 5 w 0 L2 96 4 SM 7.13

SM 7

(B) 80 GPa (D) 46 GPa

(A) 7.94 mm (C) 0.397 mm SM 7.11

Page 114

(B) RA = 1 w 0 L , MA = 5 w 0 L2 96 4 (D) RA =- 1 w 0 L , MA = 5 w 0 L2 96 4

A simple beam AB is subjected to a distributed load of intensity w = w 0 sin px/L as shown, where w 0 is the maximum intensity of the load. What will be the

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Page 115

Deflection of Beams and Shafts

SM 115

deflection y max at the midpoint of the beam using the fourth-order differential equation of the deflection curve (the load equation) ?

4 (A) w 04 L 2p EI 4 (C) w40 L p EI

SM 7.14

SM 7.15

In the figure shown, a simple beam with a uniform load is pin supported at one end and spring supported at the other. The spring has stiffness k = 48EI/L3 . What will be the angle of rotation qA at support A ? (Hint: Use third-order differential equation (the shear-force equation)).

3 (A) 5wL 48EI

2 (B) 5wL 96EI

3 (C) 5wL 16EI

3 (D) 5wL 96EI

In the figure shown, the two wooden meter sticks are separated at their centers by a smooth rigid cylinder having a diameter of 50 mm. Each stick has a width of 20 mm, a thickness of 5 mm and Ew = 11 GPa . What will be the force F that must be applied at each end in order to just make their ends touch ?

(A) 0.6875 N (C) 1.375 N SM 7.16

4 (B) w30 L p EI 4 (D) w20 L p EI

(B) 2.75 N (D) 2.06 N

Beam ACB hangs from two springs is shown in figure. The springs have stiffnesses k1 and k2 and the beam has flexural rigidity EI . What is the downward displacement of the midpoint C of the beam when the load P is applied ? (Use EI = 216 kN-m2 )

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GATE Mechanical Engineering in 4 Volume NODIA SM 116

Demo Ebook

Page 116

Deflection of Beams and Shafts

(A) 25 mm (C) 20.5 mm

SM 7

(B) 2.5 mm (D) 4.5 mm

Common Data For Q. 17 and 18 Consider the beam and loading shown in figure.

SM 7.17

SM 7.18

SM 7.19

The slope at end A is (A) M 0 (3b2 + L2) 6EIL (C) M 0 (3b2 - L2) 6EIL

(B) M 0 (3b2 - L2) 3EIL (D) M 0 (3b2 + L2) 3EIL

The deflection of point C is (A) M 0 ab (b + a) 6EIL (C) M 0 ab (b - a) 3EIL

(B) M 0 ab (b - a) 6EIL (D) M 0 ab (b - a) 4EIL

For the beam and loading shown in figure, the reaction at A and the slope at C are

(A) RA = 3 M 0 , qC = 1 M 0 L 16 EI 2 L (C) RA = 3 M 0 , qC = 1 M 0 L 8 EI 2 L SM 7.20

(B) RA = 3 M 0 , qC = 1 M 0 L 16 EI 4 L (D) RA = 3 M 0 , qC = 1 M 0 L 8 EI 4 L

A cantilever beam AB supporting a parabolic load defined by the equation w = w 0 x2 /L2 is shown in figure. What will be the angle of rotation qB and deflection yB at the free end of the beam ?

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SM 7.21

Demo Ebook

Page 117

Deflection of Beams and Shafts

3 4 (A) qB = w 0 L , yB = 13w 0 L 10EI 90EI

3 4 (B) qB = w 0 L , yB = 13w 0 L 10EI 180EI

3 4 (C) qB = w 0 L , yB = 13w 0 L 20EI 90EI

3 4 (D) qB = w 0 L , yB = 13w 0 L 20EI 180EI

SM 117

The beam is subjected to the load P as shown in figure. If the deflection at C is zero and EI is constant, the magnitude of force F that must be applied at the end of the overhang C , is

(A) P 2 (C) 3P 4

(B) P 4 (D) 2P

Common Data For Q. 22 and 23 Rigid bars are welded to the steel rod AD as shown in figure. (Take E = 200 GPa ).

SM 7.22

For the loading shown, what will be the deflection at point B ? (A) 0.30 mm (B) 15 mm (C) 1.5 mm (D) 0.15 mm

SM 7.23

What will be the slope at end A ? (A) 5.65 # 10-3 rad (C) 9.35 # 10-3 rad

(B) 4.67 # 10-3 rad (D) 7.50 # 10-3 rad

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GATE Mechanical Engineering in 4 Volume NODIA SM 118

SM 7.24

Page 118

Deflection of Beams and Shafts

SM 7

Consider a W360 # 39 beam and loading is shown in figure. If E = 200 GPa and I = 102 # 106 mm 4 , the slope at end A and the deflection at point C , are

(A) (B) (C) (D) SM 7.25

Demo Ebook

qA = 1.65 # 10-3 rad , yC = 2.7 mm qA = 2.58 # 10-3 rad , yC = 2.7 mm qA = 2.58 # 10-3 rad , yC = 27.0 mm qA = 1.65 # 10-3 rad , yC = 27.0 mm

For the cantilever beam shown in figure, the slope and deflection at point B respectively, are (Take E = 200 GPa and I = 2.53 # 106 mm 4 )

(A) 5.56 # 10-3 rad , 25 mm (C) 5.56 # 10-3 rad , 5 mm

(B) 11.12 # 10-3 rad , 5 mm (D) 5.56 # 10-3 rad , 2.5 mm

Common Data For Q. 26 and 27 In figure shown, a 50 mm diameter A-36 steel shaft is supported by the bearings at A and B . The bearings exert only vertical reactions on the shaft and take E = 200 GPa for the steel shaft.

SM 7.26

What will be the slope of the shaft at A and B , respectively ? (A) qA = 0.00968 rad , qB = 0.0105 rad (B) qA = 0.00968 rad , qB = 0.00105 rad (C) qA = 0.0968 rad , qB = 0.00105 rad (D) qA = 0.0105 rad , qB = 0.00968 rad

SM 7.27

What will be the maximum deflection of the shaft? (A) 8.16 mm (B) 16.32 mm (C) 0.816 mm (D) 81.6 mm

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SM 7.28

SM 7.29

SM 119

The frame ABCD shown in figure is squeezed by two collinear forces P , acting at points A and D . The flexural rigidity EI is constant throughout the frame and disregard the effects of axial deformations. Considering the effects of bending due the loads P , what is the decrease y in the distance between points A and D ?

2 (A) PL (2L - 3a) 3EI

2 (B) PL (2L + 3a) 6EI

2 (C) PL (2L - 3a) 6EI

2 (D) PL (2L + 3a) 3EI

A simple beam AB supports two concentrated loads P at the positions shown in the figure. A support C at the midpoint of the beam is positioned at distance d below the beam before the loads are applied. The magnitude of the loads P so that the beam just touches the support at C , is (Data: d = 10 mm , L = 6 m , E = 200 GPa and I = 198 # 106 mm 4 )

(B) 80 kN (D) 32 kN

A central beam BD is joined at hinges to two cantilever beams AB and DE . All beams have the cross section shown in figure. For the loading shown, if the deflection at C is not to exceed 3 mm and E = 200 GPa , the largest allowable value of w is

(A) 151.25 N-m (C) 75.95 N-m SM 7.31

Page 119

Deflection of Beams and Shafts

(A) 64 kN (C) 48 kN SM 7.30

Demo Ebook

(B) 121.5 N-m (D) 91.15 N-m

In figure shown, beam AC rests on the cantilever beam DE . A W410 # 38.8 rolledsteel shape is used for each beam for which I = 127 # 106 mm 4 and E = 200 GPa . For the loading shown in figure, the deflection at point B and at point D , are

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GATE Mechanical Engineering in 4 Volume NODIA SM 120

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Deflection of Beams and Shafts

(A) yB = 5.76 mm , yD = 23.1 mm (C) yB = 4.61 mm , yD = 23.1 mm SM 7.32

Page 120 SM 7

(B) yB = 10.38 mm , yD = 23.1 mm (D) yB = 10.38 mm , yD = 9.2 mm

A steel shaft is used to support a rotor that exerts a uniform load of 5 kN/m within the region CD of the shaft as shown in figure. The bearings exert only vertical reactions on the shaft. What will be the slope of the shaft at the bearings A and B ? (Take E = 200 GPa )

(A) qA = 0c, qB = 0.175c (C) qA = 0.175c, qB = 0.350c

(B) qA = 0.175c, qB = 0c (D) qA = qB = 0.175c

Common Data For Linked Answer Q. 33 and 34 The cantilever beam ACB shown in figure, has moments of inertia I2 and I1 in parts AC and CB , respectively.

SM 7.33

Using the method of superposition, the deflection yB at the free end due to the load P , is 3 3 (B) PL b1 + 7I2 l (A) PL b1 + 7I1 l 24EI2 I2 24EI1 I1 3 (C) PL b1 - 7I1 l 24EI2 I2

3 (D) PL b1 + 7I1 l 24EI1 I2

SM 7.34

What will be the ratio r of the deflection yB to the deflection y1 at the free end of a prismatic cantilever with moment of inertia I1 carrying the same load ? (A) 1 c1 - 7I2 m (B) 1 c1 + 7I1 m 8 4 I1 I2 (C) 1 c1 + 7I1 m (D) 1 c1 + 7I2 m 8 8 I2 I1

SM 7.35

A simple beam AB of length L is loaded at the left-hand end by a couple of moment M 0 as shown in figure. What will be the angle of rotation qA at support A by the strain energy of the beam and using Castigliano’s theorem?

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Deflection of Beams and Shafts

(A) M 0 L 2EI (C) M 0 L 3EI SM 7.36

SM 7.37

Page 121 SM 121

(B) M 0 L 6EI (D) M 0 L 4EI

For the cantilever beam and loading shown in figure, what will be the deflection and slope at end D caused by the couple M 0 ?

2 (A) qD = 11 M 0 a , yD = 25 M 0 a 6 EI 12 EI

2 (B) qD = 11 M 0 a , yD = 25 M 0 a 12 EI 6 EI

2 (C) qD =- 11 M 0 a , yD =- 25 M 0 a 6 EI 12 EI

2 (D) qD =- 11 M 0 a , yD =- 25 M 0 a 12 EI 6 EI

For the beam and loading shown in figure, what will be the slope at end A and the deflection at midpoint C ? (Use data P = 8 kN , I = 16.6 # 106 mm 4 and E = 200 GPa )

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

qA = 0.3125 # 10-3 rad , yC = 1.016 mm qA = 31.25 # 10-3 rad , yC = 10.16 mm qA = 3.125 # 10-3 rad , yC = 1.016 mm qA = 3.125 # 10-3 rad , yC = 10.16 mm

Common Data For Q. 43 and 44 Consider the beam and loading as shown in figure with I = 556 # 106 mm 4 and E = 200 GPa .

SM 7.38

SM 7.39

SM 7.40

The slope at point A is (A) - 2.35 # 10-3 rad (C) 3.92 # 10-3 rad

(B) 3.14 # 10-3 rad (D) - 1.96 # 10-3 rad

The deflection at point D , is (A) 4.15 mm (C) 8.3 mm

(B) 41.5 mm (D) 20.75 mm

The cantilever beam shown in figure, supports a triangularly distributed load of

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Page 122

Deflection of Beams and Shafts

SM 7

maximum intensity w 0 . The deflection dB at the free end B by using Castigliano’s theorem, is

2 (A) w 0 L 30EI

4 (B) w 0 L 30EI

3 (C) w 0 L 15EI

3 (D) w 0 L 30EI

Common Data For Q. 46 and 47 A W310 # 74 beam is loaded as shown in figure. It has I = 165 # 106 mm 4 and E = 200 GPa .

SM 7.41

SM 7.42

SM 7.43

SM 7.44

The slope at point C , is (A) 0.521 # 10-3 rad (C) 1.042 # 10-3 rad

(B) - 2.084 # 10-3 rad (D) - 0.65 # 10-3 rad

The deflection at point D , is (A) 0.271 mm (C) 5.42 mm

(B) 27.1 mm (D) 2.71 mm

The cantilever beam ACB shown in figure, is subjected to a uniform load of intensity w acting between points A and C . The angle of rotation qA at the free end A by using the modified form of Castigliano’s theorem, is

3 (A) 7wL 32EI

3 (B) 7wL 16EI

3 (C) 7wL 96EI

3 (D) 7wL 48EI

The rigid bars BF and DH are welded to the rolled-steel beam AE for which I = 4.77 # 106 mm 4 and E = 200 GPa as shown in figure. If c = 0.4 m , the deflection at point B and at the midpoint C of the beam, are

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Deflection of Beams and Shafts

(A) yB = 0.175 mm , yC = 0.882 mm (C) yB = 0.882 mm , yC = 0.175 mm SM 7.45

SM 123

(B) yB = 1.75 mm , yC = 0.882 mm (D) yB = 0.175 mm , yC = 8.82 mm

In figure shown, an overhanging beam ABC is subjected to a couple MA at the free end. The lengths of the overhang and the main span are a and L, respectively. What will be the deflection yA at end A by using the modified form of Castigliano’s theorem ?

(A) MA a ^2L - 3a h 6EI (C) MA a ^3a - 2L h 6EI SM 7.46

Page 123

(B) MA a ^2L + 3a h 6EI (D) MA a ^3L + 2a h 6EI

The steel strip AB shown in figure, is 5 mm wide and 10 mm high. What will be the deflection at the end B of the clamped steel strip ? (Take E = 200 GPa )

(A) 0.015 mm (C) 0.15 mm

(B) 15.0 mm (D) 1.50 mm

Common Data For Q. 46 and 47 Beam DE rests on the cantilever beam AC is shown in figure. A square rod of side 10 mm is used for each beam with E = 200 GPa .

SM 7.47

What will be the deflection at end C , if the 25 N-m couple is applied to end E

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Page 124

Deflection of Beams and Shafts

of beam DE ? (A) 59.4 mm (C) 29.7 mm

SM 7

(B) 11.88 mm (D) 5.94 mm

SM 7.48

What will be the deflection at end C , if the 25 N- m couple is applied to end C of beam AC ? (A) 0.675 mm (B) 67.5 mm (C) 6.75 mm (D) 1.35 mm

SM 7.49

A weight W = 20 kN falls through a height h = 1.0 mm onto the midpoint of a simple beam of length L = 3 m as shown in figure. The beam is made of wood with square cross section (dimension d on each side) and E = 12 GPa . If the allowable bending stress in the wood is sallow = 10 MPa , what will be the minimum required dimension d ?

(A) 140 mm

(B) 2.81 mm

(C) 28.1 mm

(D) 281 mm

***********

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SM 8 COLUMN

SM 8.1

A rigid bar AB is shown in figure. If the spring at A has a stiffness of k , the critical load Pcr will be

(A) kL/2 (B) kL (C) kL2 /2 (D) kL2 SM 8.2

What will be the critical buckling load for the column shown in figure when the material assumed to be rigid ?

(A) kL/2 (B) kL/4 (C) kL/6 SM 8.3

(D) kL/3

A rigid bar AD is attached to two springs of stiffness k each and is in equilibrium in the position as shown in figure. If the equal and opposite loads P and Pl remain horizontal, the magnitude of the critical load for the system, will be

GATE Mechanical Engineering in 4 Volume NODIA SM 8

SM 127

(B) ka/2l (D) ka2 /l

In the figure given, each spring can act in either tension or compression. If m = 125 kg , h = 700 mm and the stiffness of each spring is k = 2.8 kN/m , which of the following is the value of distance d for which the equilibrium of the rigid rod AB is stable in the position shown ?

(A) 245 mm (C) 392 mm SM 8.5

Page 127

Column

(A) ka2 /2l (C) ka2 /4l SM 8.4

Demo Ebook

(B) 294 mm (D) 490 mm

In figure shown, the steel pipe is fixed supported at its ends for which Est = 200 GPa and sY = 250 MPa . If it is 4 m long with outer diameter of 50 mm and support an axial load of P = 100 kN without buckling, what will be its required thickness ?

(A) 4.44 mm (C) 7.4 mm

(B) 5.92 mm (D) 3.7 mm

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GATE Mechanical Engineering in 4 Volume NODIA SM 128

SM 8.6

SM 8

A horizontal beam AB is pin-supported at end A and carries a load Q at end B as shown in the figure. The beam is supported at C by a pinned-end column. The column is a solid steel bar of square cross section and have a length L = 1.8 m , side dimension b = 60 mm and E = 200 GPa . Based upon the critical load of the column, if the factor of safety with respect to buckling is 2.0, the allowable load Q is

(B) 657.97 kN (D) 548.27 kN

A rectangular column with cross-sectional dimensions b and h is pin-supported at ends A and C as shown in figure. At mid-height, the column is restrained in the plane of the figure but is free to deflect perpendicular to the plane of the figure. What will be the ratio h/b such that the critical load is the same for buckling in the two principal planes of the column ?

(A) 1 (C) 3 SM 8.8

Page 128

Column

(A) 767.7 kN (C) 109.7 kN SM 8.7

Demo Ebook

(B) 4 (D) 2

An A-36 steel column has a length of 4 m and is pinned at both ends as shown in figure. If the cross sectional area has the dimensions shown with E = 200 GPa and sY = 250 MPa , the critical load is

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GATE Mechanical Engineering in 4 Volume NODIA SM 8

(B) 244 kN (D) 366 kN

(B) 4 kN (D) 5.2 kN

In figure shown, the steel bar AB of the frame is pin-connected at its ends. It has Est = 200 GPa and sY = 360 MPa . For buckling about the y-y axis due to the applied load of P = 18 kN , the factor of safety is

(A) 4.76 (C) 6 SM 8.12

(B) 17.02 kN (D) 22.7 kN

A load P can be applied to the structure shown in figure. Consider only bucking in the plane of the structure and E = 200 GPa . If a factor of safety of 2.6 is required, what will be the largest value of P ?

(A) 6.4 kN (C) 2.4 kN SM 8.11

SM 129

A steel tube ( E = 200 GPa ) shown in figure is 5.0 m long. It has 100 mm outer diameter and 16 mm wall thickness. The critical load of tube is

(A) 305 kN (C) 183 kN SM 8.10

Page 129

Column

(A) 28.38 kN (C) 14.19 kN SM 8.9

Demo Ebook

(B) 2.38 (D) 3

A strut AB is compressed by the action of tensile forces F in a cable that makes an angle a = 75c with the strut as shown in figure. The strut is a circular tube of aluminium with E = 72 GPa It has outer diameter d2 = 50 mm and inner

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GATE Mechanical Engineering in 4 Volume NODIA SM 130

Demo Ebook

Page 130

Column

SM 8

diameter d1 = 40 mm . The strut is 1.0 m long and is pin-connected at both ends. If factor of safety is 2.5 with respect to the critical load, the allowable force F in the cable will be

(A) 128.7 kN (C) 29.2 kN SM 8.13

(B) 99.5 kN (D) 51.49 kN

A compression member of 7 m effective length is made by welding together two L152 # 102 # 12.7 angles as shown in figure. Data for angles are y = 50.3 mm , A = 3060 mm2 , Ix = 7.20 # 106 mm 4 , Iy = 2.64 # 106 mm 4 , x = 25.3 mm .

If E = 200 GPa and a factor of safety is 2.2, the allowable centric load for the member will be (A) 370.5 kN (B) 168.4 kN (C) 185.25 kN (D) 336.8 kN SM 8.14

A distributed loading is supported by two pin-connected columns, each having a solid circular cross section as shown in figure. If AB is made of aluminium with Eal = 70 GPa , (sY ) al = 100 MPa and CD is made of steel with Est = 200 GPa , (sY ) st = 250 MPa , what will be the required diameter of each column so that both are on the point of buckling at the same time ?

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GATE Mechanical Engineering in 4 Volume NODIA SM 8

Demo Ebook Column

Page 131 SM 131

(A) dAB = 40.2 mm , dCD = 68.8 mm (B) dAB = 34.4 mm , dCD = 80.4 mm (C) dAB = 34.4 mm , dCD = 40.2 mm (D) dAB = 68.8 mm , dCD = 40.2 mm SM 8.15

The truss ABC shown in the figure supports a vertical load W at joint B . The distance between supports is 7.0 m and joint B is restrained against displacement perpendicular to the plane of the truss. If each member is a slender circular steel pipe ( E = 200 GPa ) with outside diameter 100 mm and wall thickness 6.0 mm, what will be the critical value Wcr of the load ?

(A) 203 kN (B) 117.1 kN (C) 190 kN (D) 247 kN SM 8.16

In figure shown, member AB is made of steel. It is pinned at its ends for x - x axis buckling and fixed at its ends for y -y axis buckling. The factor of safety with respect to buckling is 3. What will be the maximum allowable load P that can be applied to member BC without causing member AB to buckle ?

(A) 26.3 kN (B) 13.15 kN (C) 29.6 kN (D) 14.8 kN SM 8.17

In the assembly shown in figure, force P is applied to the handle so that the steel control rod BC does not buckle. The rod has a diameter of 25 mm. Take Est = 200 GPa and sY = 250 MPa . What will be the maximum value of force P that can be applied ?

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GATE Mechanical Engineering in 4 Volume NODIA SM 132

SM 8

(B) 14.95 kN (D) 37.38 kN

Consider only buckling in the plane of the structure shown in figure. If the load P = 5.2 kN and E = 200 GPa , the factor of safety for the structure is

(A) 1.135 (C) 2.27 SM 8.19

Page 132

Column

(A) 29.9 kN (C) 59.8 kN SM 8.18

Demo Ebook

(B) 3.13 (D) 6.2

A vertical post AB is embedded in a concrete foundation and held at the top by two cables. The cables are tightened equally by turnbuckles as shown in figure. The post is a hollow steel tube with modulus of elasticity of 200 GPa. It has a outer diameter of 40 mm and thickness of 5 mm. If factor of safety against Euler buckling in the plane of the figure is 3.0, what will be the maximum allowable tensile force Tallow in the cables ?

(A) 39.34 kN (C) 18.1 kN

(B) 26.22 kN (D) 78.67 kN

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SM 8.20

Demo Ebook Column

SM 133

Column AB carries a centric load P of magnitude 72 kN. Cables BC and BD are taut and prevent motion of point B in the xz plane as shown in figure. Moments of inertia for column are Ix = 48.9 # 106 mm 4 and Iy = 4.73 # 106 mm 4 . Neglect the tension in the cables. If factor of safety is 2.3 and E = 200 GPa , what will be the maximum allowable length L using Euler’s formula ?

(A) 9.06 m (C) 7.55 m SM 8.21

Page 133

(B) 15.1 m (D) 12.08 m

The brace shown in figure is an A-36 steel rod of 50 mm diameter. For the brace CD is not to be buckled, the maximum distributed loading that can be applied to the wide-flange beam will be (Use: Est = 200 GPa and sY = 360 MPa )

(A) 11.82 kN/m (C) 5.91 kN/m

(B) 9.46 kN/m (D) 7.1 kN/m

Common Data For Q. 22 and 23 The roof beams of a warehouse are supported by pipe columns as shown in figure. Each column has outer diameter d2 = 100 mm , inner diameter d1 = 90 mm and length L = 4.0 m . The columns have fixed supports at the base with modulus E = 210 GPa .

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GATE Mechanical Engineering in 4 Volume NODIA SM 134

Demo Ebook

Page 134

Column

SM 8

SM 8.22

If we assume: (1) the upper end is pinned and the beam prevents horizontal displacement, (2) the upper end is fixed against rotation and the beam prevents horizontal displacement. What will be the critical load Pcr of one of the columns ? (A) ^Pcr h1 = 219 kN , ^Pcr h2 = 54.7 kN (B) ^Pcr h1 = 447 kN , ^Pcr h2 = 875 kN (C) ^Pcr h1 = 875 kN , ^Pcr h2 = 447 kN (D) ^Pcr h1 = 54.7 kN , ^Pcr h2 = 219 kN

SM 8.23

If we assume: (1) the upper end is pinned but the beam is free to move horizontally, (2) the upper end is fixed against rotation but the beam is free to move horizontally. What will be the critical load Pcr of one of the columns ? (A) ^Pcr h1 = 447 kN , ^Pcr h2 = 875 kN (B) ^Pcr h1 = 219 kN , ^Pcr h2 = 54.7 kN (C) ^Pcr h1 = 54.7 kN , ^Pcr h2 = 219 kN (D) ^Pcr h1 = 875 kN , ^Pcr h2 = 447 kN

SM 8.24

An aluminium ( E = 72 GPa ) tube AB of circular cross section is fixed at the base and pinned at the top to a horizontal beam as shown in figure. The horizontal beam supports a load Q = 200 kN . The tube has outside diameter d = 100 mm . If the desired factor of safety with respect to Euler buckling is 3.0, the required thickness t of the tube will be

(A) 9.15 mm (C) 15.25 mm

(B) 12.2 mm (D) 1.22 mm

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SM 8.25

SM 135

The 50 mm diameter bronze rod ( E = 103 GPa ) is fixed supported at A and has a gap of 2 mm from the wall at B as shown in figure. If the contact at B acts as a pin, what will be the increase in temperature DT to buckle the rod ?

(B) 101cC (D) 353.5cC

In figure shown, the line of action of the axial load P of magnitude 270 kN is parallel to the geometric axis of the column AB and intersects the x axis at e = 14 mm . What will be the deflection of the midpoint C of the column and the maximum stress in the column, respectively ?

(A) 23.14 mm, 40 MPa (C) 11.57 mm, 80 MPa SM 8.27

Page 135

Column

(A) 303cC (C) 50.5cC SM 8.26

Demo Ebook

(B) 23.14 mm, 80 MPa (D) 11.57 mm, 40 MPa

An aluminium ( E = 73 GPa ) box column of square cross section is fixed at the base and free at the top as shown in figure. The outside dimension b of each side is 100 mm and the thickness t of the wall is 8 mm. A compressive load P = 50 kN is acting on the top outer edge of the column at the midpoint of one side. If the deflection at the top is not to exceed 30 mm, what will be the longest permissible length L max of the column ?

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GATE Mechanical Engineering in 4 Volume NODIA SM 136

Demo Ebook Column

(A) 1.66 m (C) 2.76 m SM 8.28

SM 8

(B) 1.38 m (D) 2.21 m

A brass ( E = 120 GPa ) pipe has the cross section as shown in figure. It has an axial load P applied 10 mm from its geometric axis. What will be the load P for which the horizontal deflection at the midpoint C is 5 mm and the corresponding maximum stress in the column, respectively ?

(A) 528.8 kN, 109.5 MPa (C) 528.8 kN, 218 MPa SM 8.29

Page 136

(B) 151.6 kN, 109.5 MPa (D) 151.6 kN, 218 MPa

The steel column with Est = 200 GPa and sY = 360 MPa supports the two eccentric loadings as shown in figure. It is pinned at its top and fixed at the bottom. If it is assumed to be fully braced against buckling about the y -y axis, the maximum deflection of the column and the maximum stress in the column are

(A) y max = 12.3 mm , (B) y max = 24.3 mm , (C) y max = 24.3 mm , (D) y max = 12.3 mm ,

smax = 199 MPa smax = 199 MPa smax = 398 MPa smax = 398 MPa

Common Data For Q. 30 and 31 A column of 6.5 m effective length has sY = 250 MPa and E = 200 GPa . Assume allowable stress design is used.

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Page 137

Column

SM 137

SM 8.30

If it is made from the W250 # 49.1 rolled-steel shape for which A = 6250 # 10-6 m2 and rmin = 49.2 # 10-3 m , the allowable centric load will be (A) 230 kN (B) 276 kN (C) 368 kN (D) 460 kN

SM 8.31

If it is made from the W250 # 80 rolled-steel shape for which A = 10200 # 10-6 m2 and rmin = 65 # 10-3 m , the allowable centric load will be (A) 1145 kN (B) 575.5 kN (C) 687 kN (D) 916 kN

SM 8.32

A steel ( E = 250 GPa ) post AB of hollow circular cross section is fixed at the base and free at the top as shown in figure. A cable CBD passes through a fitting that is welded to the side of the post. The cable is pretensioned by tightening the turnbuckles. The inner and outer diameters are d1 = 96 mm and d2 = 110 mm , respectively and the length L = 4.0 m . The distance between the plane of the cable (plane CBD ) and the axis of the post is e = 100 mm and the angles between the cable and the ground are a = 53.13c. If the deflection at the top of the post is limited to d = 20 mm , what will be the maximum allowable tensile force T in the cable ?

(A) 8.3 kN (C) 16.6 kN

(B) 13.26 kN (D) 6.63 kN

Common Data For Q. 33 and 34 A pinned-end steel ( E = 210 GPa ) column of length L = 2.1 m is shown in figure. It has inside diameter d1 = 60 mm and outside diameter d2 = 68 mm . A compressive load P = 10 kN acts with eccentricity e = 30 mm .

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Page 138

SM 138

Column

SM 8

SM 8.33

What will be the maximum compressive stress smax in the column ? (A) 29.1 MPa (B) 24.25 MPa (C) 58.2 MPa (D) 38.8 MPa

SM 8.34

If the allowable stress in the steel is 50 MPa, what will be the maximum permissible length L max of the column ? (A) 5.03 m (B) 7.54 m (C) 2.52 m (D) 10.06 m

SM 8.35

A column made of sawn lumber (C = 0.8 , KCE = 0.3 and E = 10 GPa ) has 114 mm # 140 mm cross section. Its effective length is 3.5 m. For the grade of wood used, the adjusted allowable stress for compression parallel to the grain is sC = 7.6 MPa . What will be the maximum allowable centric load for the column ? (A) 45.4 kN (B) 34.5 kN (C) 56.75 kN (D) 28.37 kN

SM 8.36

A compression member has the cross section as shown in figure and an effective length of 1.55 m. If it is made from the aluminium alloy 6061-T6 for which sallow = 139 - 0.868 ( Lr ), the allowable centric load will be e

(A) 638 kN (C) 213 kN

(B) 319 kN (D) 425 kN

Common Data For Q. 37 and 38 A steel pipe column with pinned ends (K = 1) has an outside diameter of 220 mm and wall thickness of 12 mm. The design formulas for structural-steel columns are as follows: (Take E = 200 GPa and sY = 250 MPa ) V 3 ^KL/r h ^KL/r h3 W 5 F.S. 1 = + 3 8 ^KL/r hc 8 ^KL/r hc3 W W for ^KL/r h # ^KL/r hc W 2 W KL r / h sallow = 1 1 - ^ W p sY F.S. 1 f W 2 ^KL/r hc2 X V F.S. 2 - 1.92 W W for ^KL/r h $ ^KL/r hc W 2 ^KL/r hc W sallow = sY 2F.S. 2 ^KL/r h2 W X

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Demo Ebook

Page 139

Column

SM 139

SM 8.37

The allowable axial load Pallow for L = 5 m is (A) 1086 kN (B) 905 kN (C) 543 kN (D) 724 kN

SM 8.38

The allowable axial load Pallow for L = 10 m , is (A) 584 kN (B) 292 kN (C) 438 kN (D) 365 kN

SM 8.39

An aluminium structural tube is reinforced by riveting two plates to it as shown in figure. This structure is used as a column of 1.7 m effective length. If sallow = (372 # 103) / (L/r) 2 for 2014 T6-aluminium alloy, what will be the maximum allowable centric load ?

(A) 240.4 kN (C) 156.2 kN SM 8.40

A centric load P is supported by the steel ( sY = 250 MPa and E = 200 GPa ) bar AB as shown in figure. Taking allowable stress design in consideration, what will be the smallest dimension d of the cross section that can be used when P = 108 kN ?

(A) 30.1 mm (C) 40.1 mm SM 8.41

(B) 120.2 kN (D) 198.2 kN

(B) 20.1 mm (D) 25.1 mm

A rectangular column shown in figure is made of sawn lumber with C = 0.8 and KCE = 0.30 . It has an adjusted allowable stress for compression parallel to the grain sC = 8.3 MPa and a modulus of elasticity E = 11.1 GPa . Using the allowable-stress method, the largest allowable effective length L will be

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GATE Mechanical Engineering in 4 Volume NODIA SM 140

SM 8

(B) 6.01 m (D) 4.81 m

A compression member AB is made of a steel for which sY = 250 MPa and E = 200 GPa . It is free at its top A and fixed at its base B as shown in figure. The allowable bending stress is equal to 120 MPa and the eccentricities ex and ey are equal. What will be the largest allowable common value for e using the interaction method ?

(A) 77.5 mm (C) 7.75 mm SM 8.43

Page 140

Column

(A) 3.61 m (C) 5.41 m SM 8.42

Demo Ebook

(B) 0.775 mm (D) 0.0775 mm

A eccentric load P = 85 kN is applied at a point located at a distance e = 30 mm from the geometric axis of the rod AB as shown in figure. The allowable bending stress is 140 MPa. What will be the smallest diameter d using interaction method. Assume the rod is made of the aluminium alloy 6016-T6 for which 3 sallow, centric = 315 #L10 2 MPa . ( r)

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Page 141

Column

(A) 70.0 mm (C) 7.0 mm SM 8.44

Demo Ebook

SM 141

(B) 35.0 mm (D) 14.0 mm

A steel tube of 80 mm outer diameter is to carry a load P = 93 kN with an eccentricity of 20 mm as shown in figure. The tubes available for use are made with wall thicknesses in increments of 3 mm from 6 mm to 15 mm. Taking allowable-stress method in consideration, what will be the thickness of lightest tube that can be used ? (Take E = 200 GPa and sY = 250 MPa )

(A) t = 6 mm (C) t = 12 mm

(B) t = 9 mm (D) t = 15 mm

***********

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SM 9 ENERGY METHODS

Common Data For Q. 1 and 2. The load deformation diagram shown in figure, has been drawn from data obtained during the tensile test of a specimen of structural steel. The cross-sectional area of the specimen is 250 mm2 and the deformation was measured using a 500 mm gage length.

SM 9.1

The modulus of resilience of the steel is (A) 187.5 kJ/m3 (B) 112.5 kJ/m3 (C) 75 kJ/m3 (D) 150 kJ/m3

SM 9.2

The modulus of toughness of the steel is (A) 63 MJ/m3 (B) 42 MJ/m3 (C) 84 MJ/m3 (D) 28 MJ/m3

SM 9.3

The rod ABC is made of a steel and the loading is shown in figure. The strain energy of the steel when P = 25 kN and the corresponding strain-energy density of portions AB and BC of the rod are (Take E = 200 GPa )

(A) U = 12.18 J , uAB = 15.83 kJ/m3 , uBC = 19.3 kJ/m3 (B) U = 9.0 J , uAB = 15.83 kJ/m3 , uBC = 38.6 kJ/m3 (C) U = 12.18 J , uAB = 15.83 kJ/m3 , uBC = 38.6 kJ/m3 (D) U = 12.18 J , uAB = 19.8 kJ/m3 , uBC = 38.6 kJ/m3

GATE Mechanical Engineering in 4 Volume NODIA SM 9

SM 9.4

(B) 3.72 J (D) 0.744 J

In the figure shown, rods AB and BC are made of a steel for which the yield strength sY = 300 MPa and the modulus of elasticity E = 200 GPa . What will be the maximum strain energy that can be acquired by the assembly without causing any permanent deformation when the length a of rod AB is 2 m ?

(B) 40 J (D) 15 J

A-36 steel shaft shown in figure, has a radius of 40 mm and G = 75 GPa . What will be the torsional strain energy in the shaft ?

(A) 14.9 J (C) 1.49 J SM 9.7

SM 143

Consider a rod assembly shown in figure. Portion AB is steel, BC is brass and CD is aluminum. For steel, brass and aluminum Est = 200 GPa , Ebr = 101 GPa and Eal = 73.1 GPa , respectively. What will be the strain energy in the assembly ?

(A) 30 J (C) 10 J SM 9.6

Page 143

Energy Methods

(A) 1.86 J (C) 0.372 J SM 9.5

Demo Ebook

(B) 149 J (D) 74.5 J

A solid circular cross section stepped shaft of steel (G = 80 GPa ) is shown in figure. It has length L = 0.80 m , diameter d2 = 40 mm and diameter d1 = 30 mm . If the angle of twist is 1.0c, the strain energy U of the shaft is

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GATE Mechanical Engineering in 4 Volume NODIA SM 144

Consider the tapered rod AB shown in figure. What will be the expression for the strain energy of the tapered rod if the cross-sectional area at end B is A min ?

(C)

(B)

P2L 2EA min

P2L 4EA min

2 2 (D) P L 4EA min

Consider a A-36 steel beam with A = 2300 mm2 , I = (9.5 # 106) mm 4 and E = 200 GPa as shown in figure. The total axial and bending strain energy in the beam is

(A) 2.5 J (C) 496 J SM 9.10

SM 9

(B) 1.38 J (D) 2.3 J

2 2 (A) P L 2EA min

SM 9.9

Page 144

Energy Methods

(A) 1.15 J (C) 1.84 J SM 9.8

Demo Ebook

(B) 49.6 J (D) 493 J

In figure shown, a thin-walled hollow tube AB of conical shape has constant thickness t and average diameters dA and dB at the ends. What will be the strain energy U of the tube when it is subjected to pure torsion by torques T ?

2 (A) U = T L # c dA 2+ d2 B m 2pGt dAdB 2 2 2 (C) U = T L # c d A + d B m dA dB 2pGt

2 (B) U = T L # c dA 2+ d2 B m pGt dAdB 2 2 2 (D) U = T L # c d A + d B m dA dB pGt

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GATE Mechanical Engineering in 4 Volume NODIA SM 9

SM 9.11

2

(B) 2.37 P l EA

2 2

(D) 2.37 Pl EA

2

3

All the members of a truss shown in figure, are made of aluminium ( E = 72 GPa ) and have the uniform cross-section area. What will be the strain energy of the truss for the loading shown ?

(A) 1015 J (C) 1269 J

SM 9.14

SM 145

In the truss shown in figure, all members are made of the same material and have the uniform cross-sectional area as indicated. The strain energy of the truss when the load P applied, is

(C) 2.37 P l EA

SM 9.13

Page 145

Energy Methods

(A) 2.37 Pl EA

SM 9.12

Demo Ebook

(B) 761 J (D) 635 J

In the figure shown, the beam has a rectangular cross section of area A and the shear modulus is G . What is the shear strain energy in the beam ?

2 3 (A) Ui = w L 20GA

3 (B) Ui = wL 20GA

2 2 (C) Ui = w L 24GA

2 3 (D) Ui = w L 24GA

In the assembly shown in figure, the beam is made of aluminum ( E = 73.1 GPa ) and has a square cross section 50 mm by 50 mm. The rods are made of steel (

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GATE Mechanical Engineering in 4 Volume NODIA SM 146

Demo Ebook

Page 146

Energy Methods

SM 9

E = 200 GPa ) and have a circular cross section with a 20 mm diameter. What will be the bending strain energy in the beam and the axial strain energy in each of the two rods ?

(A) (B) (C) (D) SM 9.15

^Ui ha = 2.24 kJ , ^Ui hb = 1.02 kJ ^Ui ha = 1.02 J , ^Ui hb = 2.24 kJ ^Ui ha = 1.02 kJ , ^Ui hb = 2.24 J ^Ui ha = 2.24 J , ^Ui hb = 1.02 kJ

In the figure shown, the steel beam is supported on two springs, each having a stiffness of k = 8 MN/m . If Est = 200 GPa and I = (5 # 106) mm 4 , what will be the strain energy in each of the springs and the bending strain energy in the beam ?

(A) ^Ui hsp = 4.0 J , ^Ui hb = 0.1 J (C) ^Ui hsp = 1.0 J , ^Ui hb = 0.4 J SM 9.16

Consider the steel beam ( E = 200 GPa and I = 165 # 106 mm 4 ) and loading shown in figure. The strain energy due to bending is

(A) 596 J (C) 1192 J SM 9.17

(B) ^Ui hsp = 0.1 J , ^Ui hb = 4.0 J (D) ^Ui hsp = 0.4 J , ^Ui hb = 1.0 J

(B) 397 J (D) 894 J

In the beam shown in figure, the shear modulus is G . The deflection of the beam at its center caused by shear, is

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GATE Mechanical Engineering in 4 Volume NODIA SM 9

SM 9.18

Demo Ebook

Page 147

Energy Methods

SM 147

2 (A) 3P L 12bhG

(B) 3PL 10bhG

(C) 3PL 20bhG

2 (D) 3PL 10bhG

A heavy flywheel rotating at N revolutions per minute, is rigidly attached to the end of a shaft of diameter d as shown in figure. If the bearing at A suddenly freezes, what will be the maximum angle of twist f of the shaft?

(Let L = length of the shaft, G = shear modulus of elasticity and Im = mass moment of inertia of the flywheel about the axis of the shaft. Also, disregard friction in the bearings at B and C and disregard the mass of the shaft.) (A) f = 2N 2 # 15d (C) f = N 2 # 15d SM 9.19

(B) f = 2N2 # 5d (D) f = 2N # 15d

2p I m L G 2pIm L G

Rod AC is made of aluminium (G = 73 GPa) and is subjected to a torque T applied at end C as shown in figure. If the portion BC of the rod is hollow and has an inside diameter of 16 mm, the strain energy of the rod for a maximum shearing stress of 120 MPa, is

(A) 9.2 J (C) 14.7 J SM 9.20

2p I m L G 2p I m L G

(B) 11 J (D) 18.4 J

The A-36 steel bars with E = 200 GPa are pin connected at B and C as shown in figure. If they each have a diameter of 30 mm, the slope at E will be

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GATE Mechanical Engineering in 4 Volume NODIA SM 148

Demo Ebook Energy Methods

(A) 3.15c (C) 1.57c SM 9.21

Page 148 SM 9

(B) 0.315c (D) 15.7c

Consider a uniformly loaded simple beam AB of span length L and rectangular cross section as shown in figure. If the beam has a maximum bending stress smax due to the uniform load, the strain energy U stored in the beam is

2 (A) 4bhLsmax 45E

2 (B) 4bhLsmax 45EI

2 (C) 4b hLsmax 45E

2 2 (D) 4b hLsmax 45EI

Common Data For Q. 22 and 23 A bar is 4 m long and has a diameter of 30 mm. It is to be used to absorb energy in tension from an impact loading. SM 9.22

If it is made of steel for which E = 200 GPa and sY = 800 MPa , the total amount of elastic energy that it can absorb, is (A) 3.39 kJ (B) 2.83 kJ (C) 5.65 kJ (D) 4.52 kJ

SM 9.23

If it is made from an aluminium alloy for which E = 70 GPa and sY = 405 MPa , the total amount of elastic energy that it can absorb, is (A) 3.31 kJ (B) 4.14 kJ (C) 2.48 kJ (D) 2.07 kJ

SM 9.24

A simple beam AB of length L supports a concentrated load P at the midpoint is shown in figure. What will be the strain energy of the beam from the equation of the deflection curve and the deflection d under the load P ?

SM 9.25

2 3 3 (A) P L , PL 8EI 16EI

2 3 3 (B) P L , PL 96EI 48EI

2 3 3 (C) P L , PL 48EI 24EI

2 3 3 (D) P L , PL 16EI 32EI

The A-36 steel bolt ( E = 200 GPa ) is required to absorb the energy of a 2 kg mass that falls from h = 30 mm . If the bolt has a diameter of 4 mm, its required length L so that the stress in the bolt does not exceed 150 MPa, is

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Demo Ebook Energy Methods

(A) 8.5 mm (C) 850 mm SM 9.26

(B) 85 mm (D) 0.850 mm

(B) 395 MPa , 2.37 mm (D) 395 MPa , 0.237 mm

The state of stress shown in figure, occurs in a machine component made of brass for which sY = 160 MPa . If sz = 45 MPa , the machine component will be (Hint: Use the maximum-distortion energy criterion)

(A)Yield (C)Not depend on sz SM 9.28

SM 149

The steel beam AB ( Est = 200 GPa and sY = 250 MPa ) acts to stop the oncoming railroad car, which has a mass of 10 Mg and is coasting towards it at V = 0.5 m/s . The beam is simply supported and only horizontal forces occur at A and B . The railroad car and the supporting framework for the beam remains rigid. If beam is struck at its center by the car, the maximum stress developed in the beam and the maximum deflection of the beam respectively, are

(A) 237 MPa, 3.95 mm (C) 237 MPa, 3.95 # 10-3 mm SM 9.27

Page 149

(B)Not Yield (D)None of these

In the figure shown, a uniform rod AB is made of a brass for which sY = 125 MPa and E = 105 GPa . Collar D moves along the rod and has a speed V0 = 3 m/s as it strikes a small plate attached to end B of the rod. If factor of safety is 4 and the rod is not to be permanently deformed, the largest allowable mass of the collar is

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GATE Mechanical Engineering in 4 Volume NODIA SM 150

SM 9

(B) 2.992 kg (D) 1.995 kg

A single 6 mm diameter steel pin B is used to connect the steel strip DE to two aluminum strips, each of 20 mm width and 5 mm thickness as shown in figure. If the allowable shearing stress for the pin at B is tallow = 85 MPa , what will be the maximum strain energy that can be acquired by the assembled strips for the loading shown ? (Take Esteel = 200 GPa and Ealu. = 70 GPa )

(A) 8.46 J (C) 84.6 J SM 9.30

Page 150

Energy Methods

(A) 0.998 kg (C) 3.99 kg SM 9.29

Demo Ebook

(B) 42.3 J (D) 0.846 J

Collar D is released from rest in the position shown in figure and is stopped by a small plate attached at end C of the vertical rod ABC . What will be the mass of the collar for which the maximum normal stress in portion BC is 125 MPa ?

(A) 4.76 kg (C) 3.57 kg

(B) 1.19 kg (D) 5.95 kg

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Demo Ebook

Page 151

Energy Methods

SM 151

Common Data For Q. 31 and 32 The 2 kg block D is dropped from the position shown in figure onto the end of a 16 mm diameter rod. The modulus of elasticity of the material is E = 200 GPa .

SM 9.31

What will be the maximum deflection of end A ? (A) 17.7 mm (B) 23.6 mm (C) 29.5 mm (D) 14.75 mm

SM 9.32

The maximum bending moment and the maximum normal stress in the rod respectively, are (A) - 157.6 N-m , 64.4 MPa (B) 64.4 N-m , 157.6 MPa (C) - 64.4 N-m , 157.6 MPa (D) 157.6 N-m , 64.4 MPa

SM 9.33

In the figure shown below, a block of weight W is dropped from a height h onto the horizontal beam AB and hits it at point D . The maximum deflection ym at point D can be expressed as

SM 9.34

(A) yst c1 -

1 + 2h m yst

(B) yst c

1 + 2h - 1m yst

(C) yst c1 +

1 - 2h m yst

(D) yst c1 +

1 + 2h m yst

In the figure shown, the beam is made of oak, for which E = 11 GPa . What will be the slope and displacement at A respectively ?

(A) 5.75 # 10-3 rad , 2.74 mm (C) 5.75 # 10-3 rad , 27.4 mm

(B) 57.5 # 10-3 rad , 27.4 mm (D) 5.75 # 10-3 rad , 0.274 mm

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GATE Mechanical Engineering in 4 Volume NODIA SM 152

SM 9.35

(B) 15 M 0 L 16 EI (D) 16 M 0 L 5 EI

In the beam and loading shown in figure, the deflection at point B is (Take E = 200 GPa and I = 28.9 # 106 mm 4 )

(B) 0.508 mm (D) 50.8 mm

The A-36 steel shaft has a diameter of 30 mm as shown in figure. The displacement at pulley B is

(A) 0.0478 mm (C) 0.478 mm SM 9.38

SM 9

Consider the figure shown below. What will be the slope at point B caused by the couple M 0 using the method of work and energy ?

(A) 5.08 mm (C) 1.016 mm SM 9.37

Page 152

Energy Methods

(A) 16 M 0 L 15 EI (C) 5 M 0 L 16 EI SM 9.36

Demo Ebook

(B) 47.8 mm (D) 4.78 mm

Consider a A-36 structural steel beam with E = 200 GPa and moment of inertia I = (125 # 106) mm 4 as shown in figure. What is the slope of the beam at B ?

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GATE Mechanical Engineering in 4 Volume NODIA SM 9

(B) 1.24c (D) 2.48c

(B) 17.9 mm (D) 0.359 mm

A block of weight W is placed in contact with a beam at some given point D and released as shown in figure. What will be the ratio ym /dst of resulting maximum deflection ym at point D and the deflection dst due to a static weight W applied at D ?

(A) 1 (C) 4 SM 9.41

SM 153

Bar ABC has a rectangular cross section of 300 mm by 100 mm. Attached rod DB has a diameter of 20 mm as shown in figure. Consider only the effect of bending in ABC and axial force in DB . If both members are made of steel with E = 200 GPa , the vertical displacement of point C due to the loading, is

(A) 0.017947 mm (C) 1.79 mm SM 9.40

Page 153

Energy Methods

(A) 0.124c (C) 0.0124c SM 9.39

Demo Ebook

(B) 2 (D) 3

The 12 mm diameter steel rod ABC has been bent into the shape as shown in figure. If E = 200 GPa and G = 77.2 GPa , the deflection of end C caused by the 150 N force, will be

(A) 11.57 mm (C) 14.47 mm

(B) 8.67 mm (D) 7.23 mm

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GATE Mechanical Engineering in 4 Volume NODIA SM 154

SM 9.42

SM 9.43

SM 9

A disk of radius a has been welded to end B of the solid steel shaft AB . A cable is then wrapped around the disk and a vertical force P is applied to end C of the cable as shown in figure. The radius of the shaft is r . If neglecting the deformations of the disk and that of the cable, the deflection of point C caused by the application of P , is

3 2 (A) dC = PL c1 - 2 Ea 2 m 3 GL 3EI

3 2 (B) dC = PL c1 - 3 Ea 2 m 2 GL 3EI

3 2 (C) dC = PL c1 + 3 Ea 2 m 2 GL 3EI

3 2 (D) dC = PL c1 + 2 Ea 2 m 3 GL 3EI

Each A-36 steel member shown in figure, has a moment of inertia I = (125 # 106) mm 4 and E = 200 GPa . What will be the slope at A using Castigliano’s theorem ?

(B) 2.89c (D) 1.445c

In the figure shown, each A-36 steel member has a cross-sectional area of 400 mm2 . What will be the vertical displacement of point D ?

(A) 1.16 # 10-3 mm (C) 1.16 # 10-2 mm SM 9.45

Page 154

Energy Methods

(A) 0.578c (C) 0.289c SM 9.44

Demo Ebook

(B) 0.116 mm (D) 1.16 mm

Consider the truss shown in figure, each A-36 steel member has a cross-sectional area of 400 mm2 and E = 200 GPa . By using Castigliano’s theorem , what will be

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GATE Mechanical Engineering in 4 Volume NODIA SM 9

Demo Ebook

Page 155

Energy Methods

SM 155

the vertical displacement of point A ?

(A) 33.1 mm (C) 11.03 mm SM 9.46

In the figure shown, each member of truss has a cross-sectional area of 400 mm2 and it is made of steel with E = 200 GPa . By using Castigliano’s theorem, the vertical displacement of joint C , is

(A) 3.75 mm (C) 0.750 mm SM 9.47

(B) 0.0375 mm (D) 0.375 mm

Consider the figure shown below, each member of the truss is made of steel. The cross-sectional area of member BC is 800 mm2 and for all other members the cross sectional area is 400 mm2 . For the loading shown, which one of the following is the deflection of point D caused by the 60 kN load ?

(A) 103 mm (C) 1.03 mm SM 9.48

(B) 0.0993 mm (D) 0.0331 mm

(B) 10.3 mm (D) 0.103 mm

Each bar as shown in figure, is made of A-36 steel and has a cross-sectional area of 600 mm2 . What will be the vertical displacement of joint A using the conservation of energy ?

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GATE Mechanical Engineering in 4 Volume NODIA SM 156

(B) 0.536 # 10-3 mm (D) 0.536 mm

(B) 3.38 mm (D) 33.8 mm

Each member of the truss as shown, is made of steel with E = 200 GPa and has a cross-sectional area of 500 mm2 . The horizontal deflection of joint B , is

(A) 0.468 mm (C) 4.68 mm SM 9.51

SM 9

For each member as shown in figure, A = 400 mm2 and E = 200 GPa . Using Castigliano’s theorem, the vertical displacement of joint B is

(A) 0.338 mm (C) 6.76 mm SM 9.50

Page 156

Energy Methods

(A) 10.72 mm (C) 5.36 mm SM 9.49

Demo Ebook

(B) 46.8 mm (D) 0.0468 mm

In the given figure, a uniform rod of flexural rigidity EI is bent and loaded as shown. What will be the vertical and horizontal deflection of point A ?

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GATE Mechanical Engineering in 4 Volume NODIA SM 9

SM 9.52

Page 157

Energy Methods

SM 157

3 3 (A) dV = 1 PL , dH = 0.1443 PL 6 EI EI

3 3 (B) dV = 0.1443 PL , dH = 1 PL 6 EI EI

(C) dV = 1 PL , dH = 0.1443 PL 6 EI EI

2 2 (D) dV = 1 PL , dH = 0.1443 PL 6 EI EI

The steel bars BE and AD have each a 5 # 15 mm cross section as shown in figure. Assuming that lever ABC is rigid and E = 200 GPa , the deflection of point C is

(A) 0.164 mm (C) 16.4 mm SM 9.53

Demo Ebook

(B) 1.64 mm (D) 1.64 # 10-3 mm

In the figure shown, the steel rod BC has a 24 mm diameter and the steel cable ABDCA has a 12 mm diameter. If E = 200 GPa , the deflection of point D caused by the 12 kN load, is

(A) 1.111 mm (C) 0.111 mm

(B) 11.1 mm (D) 1.11 # 10-3 mm ***********

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