Related to Stress and Strain.

1. JAHANGIRABAD INSTIUTE OF TECHNOLOGY
BARABANKI
Department of Mechanical Engineering
Strength of Materials
Or
Mechanics of Solid
June 10, 2017 RAVI VISHWAKARMA

2. CONTENTS
1. Introduction
2. Normal Stress & Strain
3. Shear Stress & Strain
4. Strain Energy
5. Impact Loads & Stresses
6. Principle Stress & Strain, Maximum Shear Stress
7. Mohr’s stress circle
8. Equilibrium equation
9. Generalized Hooks Law
10. Theories of failure
June 10, 2017 RAVI VISHWAKARMA

3. Introduction
Engineering study the mechanics of materials mainly in order to
have a means of analyzing and designing various machines and
load bearing structure.
I should emphasize that the engineer’s role is not limited to
analyzing subjected to given loading conditions, it is of even
greater importance to design new structure and machines that is,
to select the appropriate structural components to perform a
given task, some models examples will help the reader to gain a
deeper understanding of the problems explained here.
June 10, 2017 RAVI VISHWAKARMA

4. Normal Stress & Strain
If the applying force is normal to the surface then the developed stress
is known direct stress. Direct stress mainly depends upon the
direction of force. Basically its of two types-
Tensile Stress
Compressive stress
The normal stress (σ) in a material is defined as σ ≡ F/A
where F is the force (either tension or compression) acting
perpendicular to an imaginary plane surface passing through a
piece of material and A is the cross section area.
June 10, 2017 RAVI VISHWAKARMA

5. Tensile Stress
When force is applied in such a way, due to this the length of the
body increases, this force is called tensile force and stress
developed by this force is known as tensile stress.
June 10, 2017 RAVI VISHWAKARMA
FF

6. Compressive Stress
June 10, 2017 RAVI VISHWAKARMA
The force is applied in such a way, due to this length of the body
decreases. This type of force is known as compressive force & the
stress which is developed by using compressive force is known as
compressive stress.
FF

7. Strain
After applying load on body, the body gets deformed. This deformation
is measured in terms of a dimensionless quantity which is known as
unit strain or called strain. This is of three types-
1. Tensile strain
2. Compressive strain
3. Shear Strain
June 10, 2017 RAVI VISHWAKARMA

8. Shear Stress
A shear stress, often denoted τ (Greek: tau), is defined as the
component of stress coplanar with a material cross section.
Shear stress arises from the force vector component parallel to
the cross section.
June 10, 2017 RAVI VISHWAKARMA

9. As we know that the shear stresses acts along the surface. The
action of the stresses is to produce or being about the deformation
in the body consider the distortion produced b shear sheer stress
on an element or rectangular block
τ τ
This shear strain or slide is f and can be defined as the change in right angle. or
The angle of deformation g is then termed as the shear strain. Shear strain is
measured in radians & hence is non – dimensional i.e. it has no unit. So we have
two types of strain i.e. normal stress & shear stresses.
June 10, 2017 RAVI VISHWAKARMA
Shear Strain

10. Strain Energy
Strain energy is defined as the energy stored in a body due to
deformation. The strain energy per unit volume is known as strain
energy density and the area under the stress-strain curve towards
the point of deformation. When the applied force is released, the
whole system returns to its original shape. It is usually denoted by
U.
The strain energy formula is given as,
U = F δ / 2
Where,
δ = compression,
F = force applied.
Cont..
June 10, 2017 RAVI VISHWAKARMA

11. June 10, 2017 RAVI VISHWAKARMA
When stress σ is proportional to strain ,ϵ the strain energy
formula is given by,
U = 1 / 2 V σ ϵ
Where,
σ = stress,
=ϵ strain,
V = volume of body
Regarding young’s modulus E, the strain energy formula is
given as,
U = σ2
/ 2E × V.
Where,
σ = stress,
E = young’s modulus,
V = volume of body.

12. Impact Loads
An Impact load is one which having kinetic energy strikes
instantaneously on a body, the body is subjected to what is called
Impact load. That load may be produces in tension, Compression,
torsion or bending or combination of both.
Let us consider a circular body ‘AB’ of length L , cross section area
‘A’ , its assumed that the mass M falls freely without any loss of
energy through height ‘h’,
June 10, 2017 RAVI VISHWAKARMA
AL
E
×⇒
2
energyStrain
2
σ h
L

13. Principal Stress & Strain
The principal stresses are the components of the stress tensor
when the basis is changed in such a way that the shear stress
components become zero.
Maximum and minimum normal strain possible for a specific
point on a structural element. Shear strain is 0 at the orientation
where principal strain occurs.
June 10, 2017 RAVI VISHWAKARMA

14. Mohr’s Circle
June 10, 2017 RAVI VISHWAKARMA
xσ yσ

15. June 10, 2017 RAVI VISHWAKARMA
L
P
N Q
O M
τ
X Axis
Y Axis

16. Equilibrium equation
The equilibrium equations in three dimensional based on
Newton's first law are:
Cont…
June 10, 2017 RAVI VISHWAKARMA
z
y
x

17. June 10, 2017 RAVI VISHWAKARMA
0
0
0
=+
∂
∂
+
∂
∂
+
∂
∂
=+
∂
∂
+
∂
∂
+
∂
∂
=+
∂
∂
+
∂
∂
+
∂
∂
z
zzyzxz
y
yyxy
x
zxyxxx
f
zyx
f
z
y
yx
f
zyx
σσσ
σσσ
σσσ
The above equilibrium equations in Cartesian
coordinates system.

18. Generalized Hooks Law
The generalized Hooke's Law can be used to predict the
deformations caused in a given material by an arbitrary
combination of stresses.
The linear relationship between stress and strain—
June 10, 2017 RAVI VISHWAKARMA
EEE
EEE
EEE
zyx
z
zyx
y
zyx
x
σσ
υ
σ
υε
σ
υ
σσ
υε
σ
υ
σ
υ
σ
ε
+−−=
−+−=
−−=

19. Theories of failure
The interesting thing in the Failure theories is that, just by
looking at the name of the theory you will be able to formulate
condition of failure in an actual case. Just make sure that your
concept of STT and Principal stresses are clear. The theories along
with its usability is given below.
1.Maximum principal stress theory – (Good for brittle
materials)
According to this theory when the maximum principal stress
induced in a material under complex load condition exceeds the
maximum normal strength in a simple tension test the material
fails. So the failure condition can be expressed as
June 10, 2017 RAVI VISHWAKARMA
ultσσ ≥1

20. June 10, 2017 RAVI VISHWAKARMA
2.Maximum shear stress theory – (Good for ductile materials)
According to this theory when the maximum shear strength in actual case
exceeds maximum allowable shear stress in simple tension test the material
case. Maximum shear stress in actual case in represented as
Maximum shear stress in simple tension case occurs at angle 45 with load,
so maximum shear strength in a simple tension case can be represented as
Comparing these 2 quantities one can write the failure condition as
2
31
max,
σστ
−
=act
2
max,45
y
simp
σ
ττ ==
( ) yσσσ
2
1
2
1
31 ≥−

21. June 10, 2017 RAVI VISHWAKARMA
EEE
strain act
321
max,
σσ
υ
σ
−−=
3.Maximum normal strain theory – (Not recommended)
This theory states that, when the maximum normal strain in actual case is
more than maximum normal strain occurred in simple tension test case the
material fails. The maximum normal strain in actual case is given by
Maximum strain in simple tension test case is given by
So condition of failure according to this theory is
E
strain
yσ
=max
EEE
321 σ
υ
σ
υ
σ
−−

22. 4.Total strain energy theory - Good for ductile material
According to this theory when the total strain energy in actual case
exceeds the total strain energy in simple tension test at the time of
failure, the material fails. The total strain energy in actual case is given
by
The total strain energy in simple tension test at time of failure is given
by
June 10, 2017 RAVI VISHWAKARMA
([ )133221
2
1
2
1
2
1 2
2
1
.. σσσσσσυσσσ ++−++=actEST
E
EST
y
simp
2
..
2
σ
=

23. June 10, 2017 RAVI VISHWAKARMA
5.Shear strain energy theory - Highly recommended
According to this theory when the shear strain energy in the actual case
exceeds shear strain energy in simple tension test at the time of failure
the material fails.