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8/10/2019 GATE Strength of Materials Book
1/12
8/10/2019 GATE Strength of Materials Book
2/12
Strength of Materials
For
Mechanical Engineering
By
wwwthegateacademy com
http://www.thegateacademy.com/http://www.thegateacademy.com/http://www.thegateacademy.com/http://www.thegateacademy.com/http://www.thegateacademy.com/http://www.thegateacademy.com/http://www.thegateacademy.com/8/10/2019 GATE Strength of Materials Book
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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
8/10/2019 GATE Strength of Materials Book
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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
Solved Examples 42 48
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
Solved Examples 63 67
Assignment 1 68 70
Assignment 2 70 71
8/10/2019 GATE Strength of Materials Book
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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 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
Solved Examples 86 94
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
Solved Examples 111 118
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
Assignment 2 145 147
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
8/10/2019 GATE Strength of Materials Book
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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 iii
Impact or Dynamic Loading 152 - 153
Solved Examples 154 157
Assignment 1 158 159
Assignment 2 159 160
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
Solved Examples 169 174
Assignment 1 175 176
Assignment 2 176 177
Answer keys 178
Explanations 178180
Module Test 181 205
Module Test 181 191
Answer keys 192
Explanations 192 205
Reference 206
8/10/2019 GATE Strength of Materials Book
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Chapter 1 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 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
8/10/2019 GATE Strength of Materials Book
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Chapter 1 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 2
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
8/10/2019 GATE Strength of Materials Book
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Chapter 1 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 3
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
8/10/2019 GATE Strength of Materials Book
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Chapter 1 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 4
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
8/10/2019 GATE Strength of Materials Book
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Chapter 1 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 5
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_modulus8/10/2019 GATE Strength of Materials Book
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