Simple stresses and strains

SIMPLE STRESS & STRAIN
JISHNU V
ENGINEER
BHEL

1.1 LOAD
 Load is defined as the set of external forces

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acting on a mechanism or engineering structure
which arise from service conditions in which the
components work
Common loads in engineering applications are
tension and compression
Tension:- Direct pull. Eg:Force present in lifting
hoist
Compression:- Direct push. Eg:- Force acting on
the pillar of a building
Sign convention followed: Tensile forces are
positive and compressive negative

1.1.1TYPES OF LOAD
 There are a number of different ways in which

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load can be applied to a member. Typical loading
types are:
A) Dead/ Static load- Non fluctuating forces
generally caused by gravity
B) Live load- Load due to dynamic effect. Load
exerted by a lorry on a bridge
C) Impact load or shock load- Due to sudden
blows
D) Fatigue or fluctuating or alternating loads:
Magnitude and sign of the forces changing with
time

1.2 STRESS
 When a material is subjected to an external

force, a resisting force is set up within the
component, this internal resistance force per unit
area is called stress. SI unit is N/m²(Pa).
1kPa=1000Pa, 1MPa=10^6 Pa, 1 Gpa=10^9Pa, 1
Terra Pascal=10^12 Pa
 In engineering applications, we use the
the original cross section area of the specimen
and it is known as conventional stress or
Engineering stress

1.3 STRAIN
 When a body is subjected to some external

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force, there is some change of dimension of the body.
The ratio of change of dimension of the body to its
original dimension is known as strain
Strain is a dimensionless quantity
Strain may be:- a) Tensile strain b) Compressive
strain c) Volumetric strain d) Shear strain
Tensile strain- Ratio of increase in length to original
length of the body when it is subjected to a pull force
Compressive strain- Ratio of decrease in length to
original length of the body when it is subjected to a
push force
Volumetric strain- Ratio of change of volume of the
body to the original volume
Shear strain-Strain due to shear stress

1.4.1TYPES OF DIRECT
STRESS
 Direct stress may be normal stress or shear

stress
 Normal stress (σ) is the stress which acts in
direction perpendicular to the area. Normal stress
is further classified into tensile stress
 Tensile stress is the stress induced in a
body, when it is subjected to two equal and
opposite pulls (tensile forces) as a result of which
there is a tendency in increase in length
 It acts normal to the area and pulls on the area

1.4.1 TYPES OF DIRECT STRESS
(Tensile stress)
 Consider a bar subjected to a tensile force P at its

ends. Let
A= Cross sectional area of the body
L=Original length of the body
dL= Increase in length of the body due to its pull
P
ς= Stress induced in the body
e= Tensile strain
Consider a section X-X which divides the body into
two halves

(Tensile stress)
 The left part of the section x-x, will be in

equilibrium if P=R (Resisting force). Similarly the
right part of the section x-x will be in equilibrium if
P=R (Resisting force)

(Tensile stress)
 Tensile stress (ς)= Resisting force/ Cross sectional

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area= Applied force/Cross sectional area=P/A
Tensile strain= Increase in length/Original length= dL/L
Compressive stress:- Stress induced in a body, when
subjected to two equal and opposite pushes as a
result of which there is a tendency of decrease in
length of the body
It acts normal to the area and it pushes on the area
In some cases the loading situation is such that the
stress will vary across any given section. In such
cases the stress at any given point is given by
ς= Lt ΔA 0 ΔP/ ΔA= dP/dA= derivative of force w.r.t
area

(Compressive stress)

 Compressive stress=Resisting force/ cross sectional

area= Applied force/ cross sectional area
 Compressive strain= Decrease in length/ Original length= dL/L
 Sign convention for direct stress and strain:- Tensile
stresses and strains are considered positive in sense
producing an increase in length. Compressive stresses
and strains are considered negative in sense producing
decrease in length

(Shear stress)
 Shear stress :- Stress Induced in a body, when

subjected to two equal and opposite forces which
are acting tangentially across the resisting
section as a result of which the body tends to
shear off across that section
 Consider a rectangular block of height h, length L
and width unity. Let the bottom face AB of the
block be fixed to the surface as shown. Let P be
the tangential force applied along top face CD of
the block. For the equilibrium of the block, the
surface AB will offer a tangential reaction force R
which is equal in magnitude and opposite in
direction to the applied tangential force P

(Shear stress)
 Consider a section X-X cut parallel to the applied force

which splits rectangle into two parts

 For the upper part to be in equilibrium; Applied force

P=Resisting force R
 For the lower part to be in equilibrium; Applied force
P=Resisting force R
 Hence, shear stress τ= Resisting force/Resisting area=P/L
x 1=P/L
 Shear stress is tangential to the area on which it acts

(Shear stress)
 As the face AB is fixed, the rectangular section ABCD

will be distorted to ABC1D1, such that new vertical
face AD1 makes an angle φ with the initial face AD

 Angle φ is called shear strain. As φ is very small,
 φ=tan φ=DD1/AD=dl/h

 Hence shear strain=dl/h

1.5 ELASTICITY & ELASTIC
LIMIT
 The property of a body by virtue of which it undergoes

deformation when subjected to an external force and
regains its original configuration (size and shape)
upon the removal of the deforming external force is
called elasticity.
 The stress corresponding to the limiting value of
external force upto and within which the deformation
disappears completely upon the removal of external
force is called elastic limit
 A material is said to be elastic if it returns to its
original, unloaded dimensions when load is removed.
 If the external force is so large that the stress exceeds
the elastic limit, the material loses to some extent its
property of elasticity. If now the force is removed, the
material will not return to its original shape and size
and there will be a residual deformation in the
material

1.6 HOOKE’S LAW & ELASTIC
MODULI
 Hooke’s law states that: “ When a body is loaded

within elastic limit, the stress is proportional to
strain developed” or “Within the elastic limit the
ratio of stress applied to strain developed is a
constant”
 The constant is known as Modulus of elasticity or
Elastic modulus or Young’s modulus
 Mathematically within elastic limit
Stress/Strain=ς/e=E
ς= P/A; e =ΔL/L
E=PL/A Δ L

1.7 HOOKE’S LAW & ELASTIC
MODULI
 Young's modulus (E) is generally assumed to be

the same in tension or compression and for most
of engineering applications has a high numerical
value. Typically, E=210 x 10^9 N/m² (=210 GPa)
for steel
 Modulus of rigidity, G= τ/φ= Shear stress/ shear
strain
 Factor of safety= Ultimate stress/Permissible
stress
 In most engineering applications strains donot
often exceed 0.003 so that the assumption that
deformations are small in relation to orinal
dimensions is generally valid

1.8 STRESS-STRAIN CURVE
(TENSILE TEST)
 Standard tensile test involves subjecting a

circular bar of uniform cross section to a gradually
increasing tensile load until the failure occurs
 Tensile test is carried out to compare the
strengths of various materials
 Change in length of a selected gauge length of
bar is recorded by extensometers
 A graph is plotted with load vs extension or stress
vs strain

(TENSILE TEST)

(TENSILE TEST DIAGRAM)
 A Limit of proportionality; It is the point where the
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linear nature of the stress strain graph ceases
B Elastic limit; It is the limiting point for the condition
that material behaves elastically, but hooke's law does
not apply . For most practical purposes it can be often
assumed that limit of proportionality and elastic limits
are the same
Beyond the elastic limits, there will be some
permanent deformation or permanent set when the
load is removed
C (Upper Yield point), D (Lower yield point) Points
after which strain increases without correspondingly
high increase in load or stress
E Ultimate or maximum tensile stress; Point where
the necking starts
F Fracture point

1.9 CONSTITUTIVE
RELATIONSHIPS BETWEEN
STRESS & STRAIN
 A) 1-Dimensional case (due to pull or push or shear

force)
ς=Ee
 B) 2-Dimensional case
 Consider a body of length L, width B and height H. Let
the body be subjected to an axial load. Due to this
axial load, there is a deformation along the length of
the body. This strain corresponding to this
deformation is called longitudinal strain.
 Similarly there are deformations along directions
perpendicular to line of application of fore. The strains
corresponding to these deformations are called lateral
strains

1.9 CONSTITUTIVE
STRESS & STRAIN

1.9 CONSTITUTIVE
STRESS & STRAIN
 Longitudinal strain is always of opposite sign of

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that of lateral strain. Ie if the longitudinal strain is
tensile, lateral strains are compressive and vice
versa
Every longitudinal strain is accompanied by
lateral strains in orthogonal directions
Ratio of lateral strain to longitudinal strain is
called Poisson’s ratio (μ); Mathematically,
μ=-Lateral strain/Longitudinal strain
Consider a rectangular figure ABCD subjected a
stress in σx direction and in σ y direction

1.9 CONSTITUTIVE
STRESS & STRAIN
 Strain along x direction due to ςx= ς x/E

Strain along x direction due to ς y=-μ x ςy/E
Total strain in x direction ex= ς x/E - μ x ςy/E
Similarly total strain in y direction, ey= ς y/E - μ x
ςx/E
 In the above equation tensile stresses are
considered as positive and compressive stresses
as negative
 C) 3 Dimensional case:Consider a 3 D body subjected to 3 orthogonal
normal stresses in x,y and z directions
respectively

1.9 CONSTITUTIVE
STRESS & STRAIN
 Strain along x direction due to ςx= ς x/E

Strain along x direction due to ς y=-μ x ςy/E
Strain along x direction due to ς z=-μ x ςz/E
Total strain in x direction ex= ς x/E - μ x (ςy/E +
ςz/E )
Similarly total strain in y direction, ey= ς y/E - μ x
(ςxE + ςz/E )
Similarly total strain in z direction, ez= ς z/E - μ x
(ςxE + ςy/E )

1.10 ANALYSIS OF BARS OF
VARYING CROSS SECTION
 Consider a bar of different lengths and of different

diameters (and hence of different cross sectional
areas) as shown below. Let this bar be subjected to
an axial load P.

 The total change in length will be obtained by adding

the changes in length of individual sections
 Total stress in section 1: ς1=E1 x ΔL1/L1
ς1 x L1/E1=ΔL1
ς1=P/A1; Hence ΔL1=PL1/A1E1
 Similarly, ΔL2=PL2/A2E2; ΔL3=PL3/A3E3

VARYING CROSS SECTION
 Hence total elongation ΔL=Px (L1/A1E1+L2/A2E2 +

L3/A3E3)
 If the Young’s modulus of different sections are the
same, E1=E2=E3=E; Hence ΔL=P/Ex (L1/A1+L2/A2
+ L3/A3)
 When a number of loads are acting on a body, the
resulting strain, according to principle of
superposition, will be the algebraic sum of strains
caused by individual loads
 While using this principle for an elastic body which is
subjected to a number of direct forces (tensile or
compressive) at different sections along the length of
the body, first the free body diagram of individual
section is drawn. Then the deformation of each
section is calculated and the total deformation is
equal to the algebraic sum of deformations of
individual sections

1.11 ANALYSIS OF UNIFORMLY
TAPERING CIRCULAR ROD
 Consider a bar uniformly tapering from a diameter
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D1 at one end to a diameter D2 at the other end
Let
P Axial load acting on the bar
L Length of bar
E Young’s modulus of the material

1. 11 ANALYSIS OF UNIFORMLY
TAPERING CIRCULAR ROD
 Consider an infinitesimal element of thickness dx, diameter Dx at

a distance x from face with diameter D1.
Deformation of the element d(Δx)= P x dx/ (Ax E)
Ax=π/4 x Dx²; Dx= D1 - (D1 – D2)/L x x
Let (D1-D2)/L=k; Then Dx= D1-kx
d(ΔLx)= 4 x P x dx/(π x (D1-kx)² x E)
Integrating from x=0 to x=L 4PL/(πED1D2)
Let D1-kx=λ; then dx= -(d λ/k)
When x=0, λ=D1; When x=L, λ=D2

1.12 ANALYSIS OF UNIFORMLY
TAPERING RECTANGULAR BAR

COMPOSITE SECTIONS
 A bar, made up of two or more bars of equal

lengths but of different materials rigidly fixed with
each other and behaving as one unit for
elongation and shortening when subjected to
axial loads is called composite bar.
 Consider a composite bar as shown below
 Let
P Applied load
L Length of bar
A1 Area of cross section of Inner member
A2 Cross sectional area of Outer member

COMPOSITE SECTIONS
 Strain developed in the outer member= Strain

developed in the inner member
ς1/E1 = ς2/E2
 Total load (P)= Load in the inner member (P1) +
Load in the outer member (P2)
 ς1 x A1 + ς2 x A2= P
 Solving above two equations, we get the values
of ς1, ς2 & e1 and e2

1.14 STRESS & ELONGN.
PRODUCED IN A BAR DUE TO ITS
SELF WEIGHT
 Consider a bar of length L, area of cross section A

rigidly fixed at one end. Let ρ be the density of the
material. Consider an infinitesimal element of
thickness dy at a distance y from the bottom of the
bar.

 The force acting on the element considered= weight

of the portion below it=ρAgy

1.14 STRESS & ELONGN.
PRODUCED IN A BAR DUE TO ITS
SELF WEIGHT
 Tensile stress developed= Force acting on the

element/Area of cross section= ρgy.
 From the above equation, it is clear that the

maximum stress at the section where y=L, ie at
the fixed end (ρgL) and minimum stress is at the
free end(=0)
 Elongation due to self weight

1.15 STRESS IN BAR DUE TO
ROTATION

1.16 THERMAL STRESS
 Thermal stresses are the stresses induced in a body due

to change in temperature. Thermal stresses are set up in a
body, when the temperature of the body is raised or
lowered and the body is restricted from expanding or
contracting
 Consider a body which is heated to a certain temperature
Let
L= Original length of the body
Δ T=Rise in temp
E=Young's modulus
α=Coefficient of linear expansion
dL= Extension of rod due to rise of temp
 If the rod is free to expand, Thermal strain developed
et= Δ L/L=α x Δ T

1.16 THERMAL STRESS
 The extension of the rod, Δ L= L x α x Δ T
 If the body is restricted from expanding

freely, Thermal stress developed is ςt/et=E
 ςt= E x α x Δ T
 Stress and strain when the support yields:If the supports yield by an amount equal to
δ, then the actual expansion is given by the
difference between the thermal strain and δ
Actual strain, e= (L x α x Δ T – δ)/L
Actual stress= Actual strain x E= (L x α x Δ T –
δ)/L x E

Simple stresses and strains

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Simple stresses and strains