Lecture Notes on Simple Stress and Strain

1. Strength of Material
KME - 502
Lecture – 2
STRESSES AND STRAINS
SASWAT KUMAR DAS
28/08/2020

2. Syllabus
 Unit -I
 Compound Stress and Strains
 3-D Stress, Theories of failure
 Unit -II
 Stresses in Beam
 Deflection of Beams
 Torsion
 Unit – III
 Helical and Leaf Springs
 Column and Struts

3. Syllabus
 Unit – IV
 Thin cylinders and spheres
 Thick cylinders
 Unit – V
 Curved Beams
 Unsymmetrical Bending

4. Stress-Strain Diagram
• A plot of Strain vs. Stress.
• The diagram gives us the behavior of the material and
material properties.
• Each material produces a different stress-strain diagram.

5. Stress - Strain Diagram
(Behaviour of mild-steel rod under tension)
Elastic region
slope =Young’s(elastic) modulus
yield strength
Plastic region
ultimate tensile strength
strain hardening
fracture
Point A : Limit of proportionality , Point B : Elastic limit
Point C : Upper yield point , Point D : Lower yield point
Point E : Ultimate or maximum strength point
Point F : Fracture / Rupture point

6. • Elastic Region (Point O – B)
- The material will return to its original shape after the
material is unloaded(like a rubber band).
- The stress is linearly proportional to the strain in
this region.
εEσ 
: Stress (N/m2 , N/mm2)
E : Elastic modulus (Young’s Modulus) (N/m2, N/mm2)
: Strain (m/m, mm/mm)
σ
ε
ε
σ
E or
Stress-Strain Diagram

7. Elasticity and Hooke’s Law
• All solid materials deform when they are stressed, and as
stress is increased, deformation also increases.
• If a material returns to its original size and shape on
removal of load causing deformation, it is said to be
elastic.
• If the stress is steadily increased, a point is reached when,
after the removal of load, not all the induced strain is
removed.
• This is called the elastic limit.

8. Hooke’s Law
• States that providing the limit of proportionality of a material
is not exceeded, the stress is directly proportional to the
strain produced.
• If a graph of stress and strain is plotted as load is gradually
applied, the first portion of the graph will be a straight line.
• The slope of this line is the constant of proportionality called
modulus of Elasticity, E or Young’s Modulus.
• It is a measure of the stiffness of a material.

9. - The strain, or elongation over a unit length, will behave
linearly (as in y=mx +b) and thus predictable.
-The material will return to its original shape (Point 1) once an
applied load is removed.
- The stress within the material is less than what is required to
create a plastic behavior (deform or stretch significantly
without increasing stress).
The ELASTIC Range Means:
Stress-Strain Diagram

10. Plastic Region (Point C – F)
- Point C : Yield Strength : a point at which permanent
deformation occurs.
- If the material is loaded beyond the yield strength,
the material will not return to its original shape
after unloading.
- It will have some permanent deformation.
Stress-Strain Diagram

11. Strain Hardening (Point D – E)
- When yielding has ended, a further load can be applied
to the specimen, resulting in a cure that rises
continuously but becomes flatter until it reaches a
maximum stress referred to as ultimate stress at point E.
- The rise in the curve is called Strain Hardening.
Stress-Strain Diagram

12. Tensile Strength (Point E)
- The largest value of stress on the diagram is called
Tensile Strength(TS) or Ultimate Tensile Strength
(UTS)
- It is the maximum stress which the material can
support without breaking.
Fracture (Point F)
- If the material is stretched beyond Point E, the stress
decreases as necking and non-uniform deformation
occurs.
- Fracture will finally occur at Point F.
Stress-Strain Diagram

13. Elasticity
 Shear Modulus/Modulus of rigidity – ratio of shear stress
to shear strain.
 Young’s modulus/Modulus of elasticity- ratio of tensile
or compressive stress to tensile or compressive
strain.
 Factor of safety=
Shear stress 
C / G  Shear strain 
Tensilestrain
Tensile stress Compressive stress 
E / Y   
Compressive strain e
Max stresses
Working stresses

14. Modulus of Elasticity
If the strain is "elastic" Hooke's law may be used to
define
Young's modulus is also called the modulus of elasticity or
stiffness and is a measure of how much strain occurs due to a
given stress. Because strain is dimensionless, Young's modulus
has the units of stress or pressure.
Strain x A
E =
Stress
=
W

L
Youngs Modulus

15. •Consider a circular bar of diameter d and length l, subjected
to a tensile force P.
•Due to tensile force, the length of the bar increases by an
amount δl and the diameter decreases by an amount δd.
•Every direct stress is accompanied by a strain in its own
direction is known as linear strain and an opposite kind of
strain in every direction, at right angles to it, is known as
lateral strain.
•Linear and Lateral Strain are always opposite in nature.
•Vice-versa in the case of circular bar of diameter d and
length l, subjected to a compressive force P.
Linear and LateralStrain
20

16. •When a body is stressed within elastic limit, the lateral
strain bears a constant ratio to the linear strain.
𝐿𝑎𝑡𝑒𝑟𝑎𝑙 𝑆𝑡𝑟𝑎𝑖𝑛
𝐿𝑖𝑛𝑒𝑎𝑟 𝑆𝑡𝑟𝑎𝑖𝑛 = 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡
•This constant is known as Poisson's ratio and is denoted
by (1/m) or μ.
Poisson’s Ratio

17. Volumetric Strain
• Hydrostatic stress refers to tensile or compressive stress in
all dimensions within or external to a body.
• Hydrostatic stress results in change in volume of the
material.
• Consider a cube with sides x, y, z. Let dx, dy, and dz
represent increase in length in all directions.
• i.e. new volume = (x + dx) (y + dy) (z + dz)

18. Volumetric Strain Contd.
x y z
= dx/x + dy/y + dz/z
Neglecting products of small quantities:
New volume = x y z + z y dx + x z dy + x y dz
Original volume = x y z
= z y dx + x z dy + x y dz
Volumetric strain, V/ V = z y dx + x z dy + x y dz
v  x   y  z (Volumetric strain of a
rectangular body subjected to three mutually perpendicular
forces)