Simple Stresses and strains, Stress strain Diagram, Loads, Numerical

1. UNIT 1
Stresses and Strains
by
Dr Priya Gajjal
Associate Professor
Department of Mechanical Engineering
AISSMS College of Engineering

2. UNIT-01.
SIMPLE STRESSES & STRAINS

3. Content
 Introduction
• Load
• Stress
• Strain
• Poisson’s Ratio

4. Outcomes
 DEFINE Various types of stresses and strain developed on determinate
and indeterminate member.

5. Introduction
 Solid Mechanics defined as ability, to resist its failure & behaviour,
under the action of external forces.
LOAD:
• Any engineering design which is built up of a number of members is
in equilibrium under the action of external forces and the reactions
at the point of support.
• Each individual member of the design is subjected to external forces
which constitute the load on the member.
Solid Mechanics

6. Types of Load
Load
Surface
Load
Linearly
Distributed
Load
Point
Load
Axial
Load
Eccentric
Load
UDL
UVL
Impact
Load
Tensile
Load
Compressive
Load

7. Types of Load
 Surface Load
Forces acting on a
surface
F (Surface Load)
 Linearly Distributed Load
F (Linearly Distributed Load)
Solid Mechanics

8. Types of Load
 Point Load/ Concentrated Load  Axial Load
Point Load/Concentrated Load
Axial Load
Solid Mechanics

9. Types of Load
 Eccentric Load  Uniformly Distributed Load
Eccentric Load
Point Load
Uniformly Distributed Load

10. Types of Load
 Uniform Varying Load  Impact Load
Uniform Varying Load
Solid Mechanics

11. Types of Load
 Compressive Load
 Tensile Load

12. Stress
 The force of resistance offered by a body against the deformation is
called the stress.
𝑹𝒆𝒔𝒊𝒔𝒕𝒊𝒗𝒆 𝑭𝒐𝒓𝒄𝒆
𝑺𝒕𝒓𝒆𝒔𝒔 =
𝑪𝒓𝒐𝒔𝒔 − 𝒔𝒆𝒄𝒕𝒊𝒐𝒏𝒂𝒍 𝑨𝒓𝒆𝒂
F 𝑷 𝑵
𝝈 =
𝑨
=
𝑨
𝑵
(
𝒎𝟐 𝒐𝒓
𝒎𝒎𝟐)

13. Types of Stress

14. Stress
 Direct Stresses
 Normal Stresses: Normal stress is the stress which acts in a direction
perpendicular to the area.
 Tensile Stress
∴ 𝑇𝑒𝑛𝑠𝑖𝑙𝑒 𝑆𝑡𝑟𝑒𝑠𝑠 𝜎𝑇
𝑓𝑜𝑟𝑐𝑒 (F)
=
𝐶𝑟𝑜𝑠𝑠 − 𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝐴𝑟𝑒𝑎
=
𝑇𝑒𝑛𝑠𝑖𝑙𝑒 𝐿𝑜𝑎𝑑 (𝑃)
𝐶𝑟𝑜𝑠𝑠 − 𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝐴𝑟𝑒𝑎
F 𝑃
∴ 𝜎𝑇 =
𝐴
=
𝐴
Solid Mechanics

15. Stress
 Direct Stresses
 Normal Stresses: Normal stress is the stress which acts in a direction
perpendicular to the area.
 Compressive Stress
∴ Compressive Stress
𝜎C
𝑅𝑒𝑠𝑖𝑠𝑡𝑖𝑛𝑔 𝑓𝑜𝑟𝑐𝑒 (𝑅)
=
𝐶𝑟𝑜𝑠𝑠 − 𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝐴𝑟𝑒𝑎
=
Compressive 𝐿𝑜𝑎𝑑 (𝑃)
𝐶𝑟𝑜𝑠𝑠 − 𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝐴𝑟𝑒𝑎
F 𝑃
∴ 𝜎𝑇 =
𝐴
=
𝐴

16. Stress
 Direct Stresses
 Shear Stresses:
𝑆ℎ𝑒𝑎𝑟 𝑆𝑡𝑟𝑒𝑠𝑠 =
𝑆ℎ𝑒𝑎𝑟 𝑅𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝑆ℎ𝑒𝑎𝑟 𝐴𝑟𝑒𝑎
𝑃
𝜏 =
𝐴

17. Stress
 Indirect Stresses
 Bending Stress:
𝑴𝒎𝒂𝒙 = 𝑷 × 𝒍 (𝑵. 𝒎)
Solid Mechanics

18. Stress
 Indirect Stresses
 Torsion:
𝝉 = 𝑭 × 𝒓 (𝑵. 𝒎)
Solid Mechanics

19. Stress
 Combined Stresses
 Bending + Torsion:

20. Strain
 Strain is a measure of the deformation produced in the member by the
load.
 The change in dimension divided by original dimension is defined as
strain.
∴ 𝑺𝒕𝒓𝒂𝒊𝒏 𝜀
𝑪𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒅𝒊𝒎𝒆𝒔𝒊𝒐𝒏
=
𝑶𝒓𝒊𝒈𝒊𝒏𝒂𝒍 𝒅𝒊𝒎𝒆𝒏𝒔𝒊𝒐𝒏

21. Types of Strain
Solid Mechanics

22. Types of Strain
 Linear Strain / Longitudinal Strain
 Deformation of body along the direction of applied force is known as
Linear strain or Longitudinal strain
 Tensile Strain
𝑻𝒆𝒏𝒔𝒊𝒍𝒆 𝑺𝒕𝒓𝒂𝒊𝒏 =
𝑶𝒓𝒊𝒈𝒊𝒏𝒂𝒍 𝑳𝒆𝒏𝒈𝒕𝒉
𝑪𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝑳𝒆𝒏𝒈𝒕𝒉 𝜹𝒍
=
𝒍
Solid Mechanics

23. Types of Strain
 Linear Strain / Longitudinal Strain
 Compressive Strain
𝑪𝒐𝒎𝒑𝒓𝒆𝒔𝒔𝒊𝒗𝒆 𝑺𝒕𝒓𝒂𝒊𝒏 =
𝑶𝒓𝒊𝒈𝒊𝒏𝒂𝒍 𝑳𝒆𝒏𝒈𝒕𝒉
𝑪𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝑳𝒆𝒏𝒈𝒕𝒉 𝜹𝒍
=
𝒍
Solid Mechanics

24. Types of Strain
 Lateral Strain
𝑻𝒆𝒏𝒔𝒊𝒍𝒆 𝑺𝒕𝒓𝒂𝒊𝒏 =
𝑶𝒓𝒊𝒈𝒊𝒏𝒂𝒍 𝑫𝒊𝒂𝒎𝒆𝒕𝒆𝒓
𝑪𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝑫𝒊𝒂𝒎𝒆𝒕𝒆𝒓 𝜹𝒅
=
𝒅
Solid Mechanics

25. Poisson’s Ratio
 It is a ration of lateral strain to linear strain is known as poisson’s ratio.
For most engineering materials the value of µ is between 0.25 and 0.33
• Poison's ratio (μ) = lateral strain / longitudinal strain

26. Hooke’s law
• Within elastic limit of material, Stress is directly proportional to strain.
𝑆𝑡𝑟𝑒𝑠𝑠 ∝ 𝑆𝑡𝑟𝑎𝑖𝑛
𝜎 ∝ 𝜀
𝜎 = E 𝜀
Where,
E=Constant of proportionality
(Modulus of Elasticity or Young’s Modulus)
E
Const
Strain
Stress



27. Hooke’s Law
∴ 𝐸 =
𝑆𝑡𝑟𝑒𝑠𝑠 ∝ 𝑆𝑡𝑟𝑎𝑖𝑛
𝜎 ∝ 𝜀
𝜎
𝜀
∴ 𝐸 = 𝑃/𝐴
𝛿𝑙/𝑙
∴ 𝐸 =
𝑃
𝑙
𝐴𝛿𝑙
∴ 𝛿𝑙 = 𝑃𝑙
𝐴𝐸

28. Stress Strain Diagram

29. Stress Strain dig (Cont)
• Proportional Limit (1 –2) :-
The stress is linearly proportional to the strain in this region i.e. Hookes law
obeyed.
ε
σ
E 
• Elastic Limit (2 –3) :-The material will return to its original shape after the
material is unloaded.
• Yield Limit (3-4) :- The point where permanent deformation occurs.
• Strain Hardening(5-6):- Raising the yield strength by permanently straining
the material is called Strain Hardening.

30. Stress Strain dig (Cont)
• Ultimate Tensile Strength (Point 6 ):-
It is the maximum stress which the material can support without breaking.
• Fracture (Point 7) :- If the material is stretched beyond Point 6, the
stress decreases as necking and non-uniform deformation occur.

32. Stress-Strain Diagram (Ductile)

33. Stress Strain Diagram for Ductile and Brittle
Material
• Ductile material:-Having yield point, after
ultimate point fracture occurs.
• Brittle material:- Brittle materials do not
have a yield point, ultimate strength and
breaking strength are the same.
• Ex- cast iron, glass, stone, concrete etc.

34. Factor of Safety
 The ratio of ultimate load to allowable load is known as factor of safety.
𝐹𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑆𝑎𝑓𝑒𝑡𝑦 =
𝑈𝑙𝑡𝑖𝑚𝑎𝑡𝑒 𝐿𝑜𝑎𝑑
𝐴𝑙𝑙𝑜𝑤𝑎𝑏𝑙𝑒 𝐿𝑜𝑎𝑑
 Factor of safety can also be defined using yield stress criterion. Factor of
safety also defined as the ratio of yield stress to allowable stress.
𝐹𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑆𝑎𝑓𝑒𝑡𝑦 =
𝑌𝑖𝑒𝑙𝑑 𝑆𝑡𝑟𝑒𝑠𝑠
𝐴𝑙𝑙𝑜𝑤𝑎𝑏𝑙𝑒 𝑆𝑡𝑟𝑒𝑠𝑠

35. Factor of Safety
 Factor of safety is also defined on the basis of ultimate stress as the ratio of
ultimate stress to allowable stress.
𝐹𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑆𝑎𝑓𝑒𝑡𝑦 =
𝑈𝑙𝑡𝑖𝑚𝑎𝑡𝑒 𝑆𝑡𝑟𝑒𝑠𝑠
𝐴𝑙𝑙𝑜𝑤𝑎𝑏𝑙𝑒 𝑆𝑡𝑟𝑒𝑠𝑠