gate strength of materials book

8/10/2019 GATE Strength of Materials Book

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Strength of Materials

For

Mechanical Engineering

By

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Syllabus SOM

THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th

Cross, 10th

Main, Jayanagar 4th

Block, Bangalore-11

080 65700750 i f @th t d C i ht d W b th t d

Syllabus for

Strength of Materials

Stress and strain, stress-strain relationship and elastic constants, Mohrs circle for plane stress and plane

strain, shear force and bending moment diagrams, bending and shear stresses, deflection of beams,

torsion of circular shafts, thin and thick cylinders, Eulers theory of columns, strain energy methods,

thermal stresses.

Analysis of GATE Papers

(Strength of Materials)

Year Percentage of marks Overall Percentage

2013 5.00

8.29

2012 11.00

2011 9.00

2010 5.00

2009 6.00

2008 13.33

2007 8.00

2006 8.00

2005 9.33


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Content SOM

THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th

Cross, 10th

Main, Jayanagar 4th

Block, Bangalore-11

080 65700750 i f @th t d C i ht d W b th t d P i

C O N T E N T S

Chapters Page no.

1. Simple Stress and Strain 1 33

Introduction 1

Simple Stress & Strain 1

Hookes Law 1 2

Stress Strain Diagram 3 4 Possions Ratio 4 5

Cylindrical Pressure Vehicle 5 6

Spherical Pressure Vehicle 7

Solved Examples 8 21

Assignment 1 22 24

Assignment 2 24 27

Answer keys 28

Explanations 28 33

2. Shear Force and Bending Moment 34 - 58

Introduction 34

Shear and Moment 34 35

Shear Force and Bending Moment Relationships 35 37

Propped Means & Fixed Beams 37 40

Singularity Function 40 41


Assignment 1 49 52

Assignment 2 52 54

Answer keys 55 Explanations 55 58

3. Stresses in Beams 59- 76

Introduction 59

Bending Stress 59 60

Important Points 60

Shear Stress 61 62

Composite Beams 62


Assignment 1 68 70

Assignment 2 70 71


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Content SOM


Cross, 10th

Main, Jayanagar 4th

Block, Bangalore-11

080 65700750 i f @th t d C i ht d W b th t d P ii

Answer keys 72

Explanations 72 76

4. Deflection of Beams 77 - 105

Introduction 77

Double Integration Method 77 78

Area Moment Method 78 79

Maxwells Law of Reciprocal Deflections 79 85


Assignment 1 95 97

Assignment 2 98 99

Answer keys 100 Explanations 100 105

5. Torsion 106 - 133

Introduction 106

Torsion 106 - 107

Torsion of Shafts 108 109

Combined Bending and Torsion 109 110


Assignment 1 119 122 Assignment 2 123 124

Answer keys 125

Explanations 125 - 133

#6. Mohrs ircle

134 - 150

Introduction 134

Mohrs Circle 134 135

Applications : Thin Walled Pressure Vessel 135 136

Solved Example 137 141

Assignment 1 142 144


Answer key 148

Explanation 148 150

7. Strain Energy Method 151 - 163

Introduction 151

Elastic strain Energy in Uniaxial Loading 151

Elastic strain Energy in Flexural Loading 151 152

Elastic strain Energy in Torsional Loading 152 Castiglianos Theorem 152


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Content SOM


Cross, 10th

Main, Jayanagar 4th

Block, Bangalore-11

080 65700750 i f @th t d C i ht d W b th t d P iii

Impact or Dynamic Loading 152 - 153




Answer keys 161

Explanations 161 163

8. Columns and Struts 164

180

Introduction 164

Columns & Struts 164 165

Eulers Theory of Buckling 165 166

Effective Length of Columns 166 167 Limitations of Eulers Formula 167

Rankines Formula 168




Answer keys 178

Explanations 178180

Module Test 181 205

Module Test 181 191

Answer keys 192

Explanations 192 205

Reference 206


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Chapter 1 SOM


Cross, 10th

Main, Jayanagar 4th

Block, Bangalore-11

080 65700750 i f @th t d C i ht d W b th t d P 1

CHAPTER 1

Simple Stress and Strain

Introduction

Strength of materials deals with the elastic behavior of materials and the stability of members.The concept of strength of materials is used to determine the stress and deformation of axially

loaded members, connections, torsion members, thin-walled pressure vessels, beams,eccentrically loaded members and columns. In this chapter we will study the stress and strain

due to axial loading, temperature change and thin walled pressure vessels.

Simple Stress Strain

Stress is the internal resistance offered by the body per unit area. Stress is represented as forceper unit area. Typical units of stress are N/m2, ksi and MPa. There are two primary types of

stresses: normal stress and shear stress. Normal stress,is calculated when the force is normal

to the surface area whereas the shear stress is calculated when the force is parallel to the

surface area.

normal to areaP

=

A

parallel to areaP=

A

Linear strain (normal strain, longitudinal strain, axial strain is a change in length per unit

length. Linear strain has no units. Shear strainis an angular deformation resulting from shearstress. Shear strain may be presented in units of radians, percent or no units at all.

=

L

parallel

= = tan Height

[in radians]

Hookes Law: xial and Shearing Deformations

Hookes law is a simple mathematical relationship between elastic stress and strain: Stress is

proportional to strain. For normal stress, the constant of proportionality is the modulus ofelasticity (Youngs Modulus) E.

E


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Chapter 1 SOM


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The deformation of an axially loaded member of original length L can be derived from Hookes

law. Tension loading is considered to be positive, compressive loading is negative. The sign of

the deformation will be the same as the sign of the loading.

AE

PL

ELL

This expression for axial deformation assumes that the linear strain is proportional to the

normal stressE

and that the cross-sectional area is constant.

When an axial member has distinct sections differing in cross-sectional area or composition,

superposition is used to calculate the total deformation as the sum of individual deformations.

AE

LP

AE

PL

When one of the variables (e.g., A), varies continuously along the length,

AE

dLP

AE

PdL

The new length of the member including the deformation is given by

LLf

The algebraic deformation must be observed.

Hookes law may also be applied to a plane element in pure shear. For such an element, the shear

stress is linearly related to the shear strain, by the shear modulus (also known as the modulus ofrigidity), G.

G

The relationship between shearing deformation, s and applied shearing force, V is then

expressed by

AG

VLs


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Chapter 1 SOM


Cross, 10th

Main, Jayanagar 4th

Block, Bangalore-11


Stress-Strain Diagram

Proportional Limit:

It is the point on the stress strain curve up to which stress is proportional to

strain.Elastic Limit:

It is the point on the stress strain curve up to which material will return to itsoriginal shape when unloaded.

Yield Point:

It is the point on the stress strain curve at which there is an appreciable elongationor yielding of the material without any corresponding increase of load indeed the load actually

may decrease while the yielding occurs.

Ultimate Strength:

It is the highest ordinate on the stress strain curve.

Rupture Strength:It is the stress at failure

Modulus of Resilience

The work done on a unit volume of material, as a simple tensile force is gradually increased fromzero to such a value that the proportional limit of the material is reached, is defined as the

modulus of resilience. This may be calculated as the area under the stress-strain curve from the

origin up to the proportional limit

Modulus of Toughness

The work done on a unit volume of material as a simple tensile force is gradually increased fromzero to the value causing rupture is defined as the modulus of toughness. This may be calculated

as the entire area under the stress-strain curve from the origin to rupture. Toughness of a

material is its ability to absorb energy in the plastic range of the material.

Stress

P

A

Yield point

Elastic limitProportional limit

Rupture

strength

Ultimate strength

Actual rupture

strength

0Strain

L


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Chapter 1 SOM


Cross, 10th

Main, Jayanagar 4th

Block, Bangalore-11


Percentage Elongation

The increase in length of a bar after fracture divided by the initial length and multiplied by 100 isthe percentage elongation. Both the percentage reduction in area and the percentage elongationare considered to be measures of the ductility of a material.

Strain Hardening

If a ductile material can be stressed considerably beyond the yield point without failure, it is said

to strain-harden. This is true of many structural metals.

Poissons Ratio: Biaxial and Triaxial Deformations

Poissons ratio , is a constant that relates the lateral strain to the axial strain for axially loaded

members.

axial

lateral

Theoretically Poissons ratio could vary from zero to 0.5, but typical values are 0.33 for

aluminum and 0.3 for steel and maximum value of 0.5 for rubber.

Poissons ratio permits us to extend Hookes law of uniaxial stress to the case of biaxial stress.Thus if an element is subjected simultaneously to tensile stresses in x and y direction, the strain

in the x direction due to tensile stress x is x/E. Simultaneously the tensile stress y will

produce lateral contraction in the x direction of the amount y/E, so the resultant unitdeformation or strain in the x direction will be

EE

yxx

Similarly, the total strain in the y direction is

EE

xy

y

Hookes law can be further extended for three-dimensional stress-strain relationships andwritten in terms of the three elastic constants, E, G and. The following equations can be used to

find the strains caused due to simultaneous action of triaxial tensile stresses:

zyxxE

1

xzyyE

1

yxzz E1


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Chapter 1 SOM


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Block, Bangalore-11


G

xy

xy

G

yz

yz

G

zxzx

For an elastic isotropic material, the modulus of elasticity E, shear modulus G and Poissons ratio

are related by

12

EG

12GE

Thebulk modulus (K) describes volumetric elasticity or the tendency of an object's volume to

deform when under pressure; it is defined as volumetric stress over volumetric strain, is theinverse of compressibility. The bulk modulus is an extension of Young's modulus to three

dimensions.

For an elastic, isotropic material, the modulus of elasticity E, bulk modulus K, Poissons ratio

are related by

21K3E

Thermal stresses

Temperature causes bodies to expand or contract. Change in length due to increase intemperature can be expressed as

L L. .t

Where, L is the length, (/oC) is the coefficient of linear expansion, t (oC) is the temperature

change.

From the above equation thermal strain can be expressed as:

If a temperature deformation is permitted to occur freely no load or the stress will be induced in

the structure. But in some cases it is not possible to permit these temperature deformations,which results in creation of internal forces that resist them. The stresses caused by theseinternal forces are known as thermal stresses.

When the temperature deformation is prevented, thermal stress developed due to temperature

change can be given as:

E..t

Cylindrical Pressure Vehicle

Hoop Stress :

A pressure vessel is a container that holds a fluid (liquid or gas) under pressure.

Examples include carbonated beverage bottles, propane tanks and water supply pipes. In a smallpressure vessel such as a horizontal pipe, we can ignore the effects of gravity on the fluid. If the
http://en.wikipedia.org/wiki/Bulk_modulushttp://en.wikipedia.org/wiki/Stress_%28physics%29#Stress_deviator_tensorhttp://en.wikipedia.org/wiki/Compressibilityhttp://en.wikipedia.org/wiki/Compressibilityhttp://en.wikipedia.org/wiki/Stress_%28physics%29#Stress_deviator_tensorhttp://en.wikipedia.org/wiki/Bulk_modulus


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gate strength of materials book

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