Simple Strees and Stains

1. Unit-1
SIMPLE STRESSES & STRAINS
Presented By
Dr. Sivasankara Raju R
Associate Professor
Dept. of Mechanical Engg.
AITAM, Tekkali-532201
Andhra Pradesh

2. Pre-requisite of SOM
• Remember basic concepts of Forces and
Moments, Introduction to Equilibrium
• Types of loads and Free Body Diagram
• Classification of Materials and properties
• Types of Mechanical Testing
• Basic Engineering Applications

3. CONTENTS
• INTRODUCTION
• TYPES OF STRESS AND STRAIN
• St.VENANTS PRINCIPLE
• STRESS-STRAIN DIADRAM FOR MILD STEEL
• STRESS-STRAIN DIAGRAMS FOR BRITTLE MATERIALS
• WORKING STRESS,FACTOR OF SAFETY
• LINEAR STRAIN ,LATERAL STRAIN & POISSON’S RATIO
• BARS WITH VARYING CROSS SECTION
• COMPOSITE BARS
• THERMAL STRESSES
• ELASTIC CONSTANTS & RELATIONSHIP

4. Introduction
• When an external force acts on a body, the body tends to undergo
some deformation. Due to cohesion between the molecules, the body
resists deformation. This resistance by which material of the body
opposes the deformation is known as strength of material.
(Or)
• The strength of a material is the ability of that material to withstand
an applied stress without failure.
• Within a certain limit (i.e., in the elastic stage) the resistance offered
by the material is proportional to the deformation brought out on the
material by the external force. Also within this limit the resistance is
equal to the external force (or applied load).
• Beyond the elastic stage, the resistance offered by the material is less
than the applied load. In such a case, the deformation continues, until
failure takes place.

5. Types of loading

6. STRESS :
Stresses are expressed as the ratio
of the applied force divided by
the resisting area
• Mathematically:
σ = Force /Area
• Units:
• N/m2 or Pascal.
• 1kPa = 1000Pa,
• 1 Mpa= 106 Pa
TYPES OFSTRESS
There are two types of stress
1) Normal Stress
1.1) T
ensile stress
1.2) Compressive stress
2) Combine Stress
2.1) Shear stress
2.2)Tortional stress

7. 1)Normal Stress (σ : Sigma)
The resisting area is perpendicular to the
applied force
1) Tensile Stress:
•It is a stress induced in a body when it is
subjected to two equal and opposite pulls
(Tensile force) as a result of which there is
tendency in increase in length.
•It acts normal to the area and pulls on the
area.
2) 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.

8. 2) Combined Stress:
Acondition of stress that cannot be
represented by a single resultant stress.
2.1) Shear stress:
Forces parallel to the area resisting the
force cause shearing stress.
It differs to tensile and compressive
stresses, which are caused by forces
perpendicular to the area on which they
act.
Shearing stress is also known as
tangential stress.
2.2) Tortional stress:
The stresses and deformations
induced in a circular shaft by a
twisting moment.

9. Units of stress
• The unit of stress depends upon the unit of load (or force) and
unit of area.
• In M.K.S. units, stress unit is kgf/m². In the S.I. units, stress
unit is N/m².
• 1 N/m² = 1 N/(100 cm)2 = 1 N/104 cm² = 10-4 N/cm² or 10-6
N/mm²
• 1 N/mm² =106 N/m².
• 1 N/m² = 1 Pascal = 1 Pa.
• The large quantities are represented by kilo, mega, giga and
terra.
• The small quantities are represented by milli, micro, nana and
pica.

10. STRAIN (ε: Epsilon)
When a body is subjected to some external force, there
is some change in the dimension of the body. The ratio of
change in dimension of body to its original dimension is
called as strain.
Strain is a dimensionless quantity.
Longitudinal or Linear Strain
Strain that changes the length of a line
without changing its direction.
Can be either compressional or
tensional.
Compression
Longitudinal strain that shortens an
object.
T
ension
Longitudinal strain that lengthens an
object.

11. TYPES OF STRAIN
1. Tensile Strain
2. Compression Strain
3. Volumetric Strain
4. Shear Strain
1) T
ensile Strain:
Ratio of increase in length to the original
length of the body when it is subjected to
a pull force.
Tensile strain = Increase in length/ Original
Length
= dL/L
2) Compressive Strain:
Ratio of decrease in Length to the original
length of body when it is subjected to push
force.
Compressional Strain = Decrease in
length/Original Length
= dL/L
3) V
olumetric Strain:
Ratio of change of volume to the original
volume.
Volumetric Strain= dV/V
4) Shear Strain
Strain due to shear stresses
Sign convection for direct strain
Tensile strains are considered
positive in case of producing
increase in length.
Compressive strains are
considered negative in case of
producing
decrease in length.

12. St.Venant's Principle
It states that the stress measured at any point on an axially loaded
cross section is uniform if the measured location is far enough
away from the point of application of load or any discontinuity in
the members cross section.

13. Need for STRESS-STRAIN diagrams
• Stress-Strain Diagrams and their interrelationship is one of the
primary requirements in the selection of Engineering
materials for majority purposes.
• Utilization of any material in the manufacturing process
without having proper knowledge of its stress-strain behavior
is undesirable and must not be practiced.

14. STRESS-STRAIN RELATION
• The stress-strain relation of any
material is obtained by conducting
tension test in the laboratories on a
standard specimen.
• Different materials behave
differently and their behaviour in
tension and in compression differ
slightly. a typical tensile test
specimen of mild steel. Its ends
are gripped into universal testing
machine. Extensometer is fitted to
test specimen which measures
extension over the length L1
• The length over which extension is
measured is called gauge length.

15. Tensile
Properties
https://msestudent.com/stress-strain-and-the-
stress-strain-curve/

16. Stress-Strain Curves for Ductile Materials
• Portion OA: This portion is absolutely straight,
where the stress is proportional to strain and the
material obeys Hooke’s law (σ =E ϵ). The value of
stress at point A is called proportional limit.
• Portion AB: In this portion, Hook’s law is not
obeyed, although the material may still be elastic.
The point B indicates the elastic limit.
• Portion BC: In this portion, the metal shows a
strain even without increase in stress and the
strain is not fully return when load is removed.
• Portion CD: Yielding start in this portion and there
is a drop of stress at the point D directly after
yielding begins at C. The point D is termed as
lower yield point and C is called upper yield point.
• Portion DE: After yielding has taken place at D,
further straining takes place at this portion by
increasing the stress and the stress–strain curve
continues to rise up to the point E. Strain in this
portion is about 100 times that of portion O-A. At
the point E, the bar begins to form a local neck.
The point E is termed as ultimate tensile stress
point.
• Portion EF: In this portion, the load is falling off
from the maximum and fracture at F takes place.
The point F is termed as fracture or breaking
point and the identical stress is called breaking
stress.

21. SALIENT POINTS ON STRESS-STRAIN
DIAGRAM
Limit of Proportionality (A): It is the limiting value of the stress up to which stress is
proportional to strain.
Elastic Limit: This is the limiting value of stress up to which if the material is stressed
and then released (unloaded) strain disappears completely and the original length is
regained. This point is slightly beyond the limit of proportionality.
Upper Yield Point (B): This is the stress at which, the load starts reducing and the
extension increases. This phenomenon is called yielding of material. At this stage strain
is about 0.125 per cent and stress is about 250 N/mm2
Lower Yield Point (C): At this stage the stress remains same but strain increases for
some time.
Ultimate Stress (D): This is the maximum stress the material can resist. This stress is
about 370–400 N/mm2 . At this stage cross-sectional area at a particular section starts
reducing very fast. This is called neck formation. After this stage load resisted and hence
the stress developed starts reducing.
Breaking Point (E): The stress at which finally the specimen fails is called breaking
point. At this strain is 20 to 25 %.

22. Hooke’s Law
• Robert Hooke, an English mathematician conducted several ex
concluded that stress is proportional to strain up to elastic limit
Hooke’s law.
• Thus Hooke’s law is, up to elastic limit 𝜎 ∝ e where 𝜎 is stre
strain
hence, 𝜎 = Ee
where E is the constant of proportionality of the material, k
modulus of elasticity or Young’s modulus.
• Units for Young’s Modulus are N/mm2

23. NOMINAL STRESS-STRAIN
CURVE vs
TRUE STRESS-STRAIN CURVE

24. Measures of ductility
• Percentage elongation and percentage reduction in area are the
measure the ductility of material.
• Percentage Elongation: It is defined as the ratio of the final ext
original length expressed, as percentage.
• Thus, Percentage Elongation =
𝐿′−𝐿
𝐿
× 100 where L-original
rupture.
• The code specify that original length is to be five times the diam
considered.
• In case of ductile material percentage elongation is 20 to 25 %.

25. Measures of ductility
• Percentage Reduction in Area: It is defined as the ratio of m
in the crosssectional area to original cross-sectional are
percentage.
• Percentage Reduction in Area =
𝐴−𝐴′
𝐴
× 100 where A–orig
min c/s area.
• In case of ductile material, A′ is the diameter at the neck.
broken pieces of the specimen are to be kept joining each othe
• For steel, the percentage reduction in area is 60 to 70

26. FACTOR OF SAFETY
In practice it is not possible to design a mechanical component or
structural component permitting stressing up to ultimate stress for
the following reasons:
1.Reliability of material may not be 100 per cent. There may be
small spots of flaws.
2.The resulting deformation may obstruct the functional
performance of the component.
3.The loads taken by designer are only estimated loads.
Occasionally there can be overloading. Unexpected impact and
temperature loadings may act in the lifetime of the member.
4. There are certain ideal conditions assumed in the analysis (like
boundary conditions). Actually ideal conditions will not be
available and, therefore, the calculated stresses will not be 100
per cent real stresses.

27. FACTOR OF SAFETY
• Hence, the maximum stress to which any member is designed is
much less than the ultimate stress, and this stress is called
Working Stress.
• Thus Factor of Safety = Ultimate Stress / Working Stress
• In case of elastic materials, since excessive deformation create
problems in the performance of the member, working stress is
taken as a factor of yield stress or that of a 0.2 proof stress (if
yield point do not exist).
• Factor of safety for various materials depends up on their
reliability.
• The following values are commonly taken in practice:
1.For steel – 1.85 2. For concrete – 3 3. For timber – 4 to 6

28. EXTENSION / SHORTENING OF A
BAR
• Consider the bars
shown in figure From the definition of
Therefore Strain, e =
∆
𝐿
From Hooke’s Law we
E = Stress / Strain
E=
𝜎
𝑒
∆ =
𝑷𝑳
𝑨𝑬

29. Problems on elongation of a bar
Problem 1:A circular rod of diameter 16 mm an
long is subjected to a tensile force 40 kN. The m
elasticity for steel may be taken as 200 kN/mm
stress, strain and elongation of the bar due to ap
load.
Solution:Load P = 40 kN = 40 × 1000 N E =
kN/mm2 = 200 × 103 N/mm2
L = 500 mm d = 16 mm Theref
area A =
𝜋
4
𝑑2 = 201.06 mm2

30. A specimen of steel 20 mm diameter with a gauge length of 200
mm is tested to destruction. it has an extension of 0.25 mm under
a load of 80 kN and the load at elastic limit is 102 kN. the
maximum load is 130 kN. the total extension at fracture is 56
mm and diameter at neck is 15 mm. find (i) the stress at elastic
limit. (ii) young’s modulus. (iii) percentage elongation. (iv)
percentage reduction in area. (v) ultimate tensile stress

48. • Stresses in x,y and z directions
• Strains in x,y and z directions
• Volumetric strain = change in volume/original
volume

52. Data given
Linear strain
Lateral strain
Poisson’s ratio
Elongation/deformation ∆
Young’s modulus/modulus of elasticity
Modulus of rigidity
Bulk modulus

56. • Data given
• Relation between E and G
• Relation between E and K
• Longitudinal/linear stress
• Linear strain
• Volumetric strain ev = ex(1-2𝜇)
• Change in volume

58. Composite / Compound bars

59. Composite / Compound bars

61. • Let Ps and Pabe the loads shared by steel and Al strips
• Write the equation of equilibrium 1
• Write the equation of compatibility 2
• Solve 1 and 2 to get Ps and Pa
• Find stress in Steel (𝜎𝑠)and Aluminium(𝜎𝑎)
• Calculate extension in the compound bar

64. • Let Ps and Pc be the loads shared by steel and copper
• Write the equation of equilibrium 1
• Write the equation of compatibility 2
• Solve 1 and 2 to get Ps and Pc
• Find stress in Steel (𝜎𝑠)and Copper(𝜎𝑐)