Basic Concepts of Simple Stresses and Strains
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Concept of Stress and Strain
Stress
- In mechanics, stress is defined as the force per unit area on the body.
- Stress (σ) = Force applied / Area of application of force
- Its S. I. unit is N/m2.
Strain
- In mechanics, the strain is defined as the deformation occurred in the body in the direction of force applied divided by the initial dimensions of the body.
- Strain (ε) = Deformation occurred in the body in the direction of the force / Initial dimensions of the body
- It has no unit or it is dimensionless.
St. Venant's Principle
In a body, the stresses and strains produced in the points which are
sufficiently remote from the points of application of load depend only upon
the static resultant of the loads and not on the distribution of loads.
Suppose, there is a body and load is being applied to it, then the stress
and strain produced in the vicinity of the points of application of loads
are large. But, for the points, which are sufficiently remote from the
points of application of load (which are at a relatively short distance),
the stress and strain distribution for them becomes uniform.
Hooke's Law
Within the elastic limit of the material, the strain produced in the body is
directly proportional to the stress applied to it.
σ ∝ ε
σ /ε = constant
Therefore, E = σ /ε
where E is Young's modulus of elasticity
Mathematically, Hooke's law is expressed as:-
F = -kx
where F is Force
k is Spring constant
x is the Extension in length
Stress and Strain Diagram for Mild Steel
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Stress and Strain Diagram for Mild Steel |
Elements of Stress and Strain Diagram for Mild Steel
Proportional Limit
It is the region up to which body obeys Hooke's law.
OA represents the proportional limit.
In this limit, stress (σ) is directly proportional to strain (ε).
σ ∝ ε
σ /ε = constant
Therefore, E = σ /ε
where E is Young's modulus of elasticity
Elastic Limit or Yield Point
It is the region or the point up to which the body returns to its original shape or size when the load is removed completely.
Beyond this limit, the body ceases to return to its original shape or size when the load is removed completely.
Beyond this limit or after the yield point is passed, plastic deformation starts to appear in the body.
OB represents the elastic limit.
There are two types of yield points:-
Upper yield point- The point at which maximum load or stress is required to initiate the plastic deformation of the body.
Lower yield point- The point at which minimum load or stress is required to maintain the plastic behaviour of the material.
Ultimate Stress Point or Breaking Stress
The amount of stress or the point up to which the body can withhold the maximum stress before its failure. Beyond this point, the material starts losing its strength and offers less resistance and finally breaks or fails.
Fracture Point
The point at which failure of the body occurs or the body breaks.
For a brittle material, the ultimate stress point and fracture point are close to each other.
For a ductile material, the ultimate stress point and fracture point are far from each other.
Greater is the distance between the ultimate stress point and fracture point from each other more will be the ductility of the material.
Modulus of Resilience
The area under the curve which is covering the elastic limit. It is the energy absorbed per unit volume up to the elastic limit.
Modulus of Resilience = (1/2) * σ * ε
Modulus of Toughness
It is the area of the whole curve (from point O to E). It is the energy absorbed per unit volume up to the breaking point.
Elasticity and Plasticity
Elasticity
- Elasticity is the property by virtue of which a material returns to its original shape or size when the load or external forces are completely removed.
- In the elasticity, the atoms are displaced from their original lattice site and they return to their original lattice site when the load is completely removed.
- For elastic deformation, the force required is less.
Plasticity
- Plasticity is the property by virtue of which a material retains its deformed state when the load or external forces are completely removed.
- In the plasticity, the atoms are displaced from their original lattice site and they retain their new position when the load is completely removed.
- For plastic deformation, the force required is more.
Types of Stresses and Strains:-
Stress and Strain are of two types:-
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Normal Stress and Strain
When the direction of the deforming force is
perpendicular to the cross-sectional area of the body, then the stress on
the body is called normal stress and the corresponding strain produced is
the normal strain.
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Shear Stress and Strain
When the direction of the deforming force is parallel
to the cross-sectional area of the body, then the stress on the body is
called shear stress and the corresponding strain produced is the shear
strain. In the case of shear stress, the strain produced in the body is
measured by the angle through which the body distorts.
Normal Stress and Strain are of two types:-
- Longitudinal stress and strain- When the two cross-sectional areas of the body experience equal and opposite force, then the stress on the body is called longitudinal stress and the corresponding strain produced is the longitudinal stress.
- Bulk or volumetric stress and strain- When the deforming force acts from all the dimensions and cause a change in the volume of the body, then the stress on the body is called bulk stress or volumetric stress and the corresponding strain produced is the bulk strain or volumetric strain.
Bulk or volumetric strain = Change in volume (𝛿V) / Original volume (V)
Longitudinal stress and strain are of two types:-
- Tensile stress and strain- The type of longitudinal stress which results in an increase in the length of the body is called tensile stress and the corresponding strain produced is the tensile strain.
- Compressive stress and strain- The type of longitudinal stress which results in a decrease in the length of the body are called compressive stresses and the corresponding strain produced are the compressive strains.
Temperature Stresses and Strains
- If the temperature of the body changes, its dimensions will also change. The stresses produced in the body due to change in its temperature are called temperature stresses and the corresponding strain produced are called temperature strains.
- If the temperature of the body is lowered, then the stress produced in the body will be tensile in nature whereas if the temperature of the body is raised, then the stress produced in the body will be compressive in nature.
- The extension in the bar of uniform cross-section due to a rise in temperature = α(t2-t1)l
where α is the Coefficient of linear expansion
t2 is the Final temperature
t1 is the Initial temperature
l is the length of the bar
- If this extension in the bar is prevented by fixing its ends or by some external force, the temperature strain produced = (α(t2-t1)l / l) = α(t2-t1)
- Therefore, the corresponding temperature stress developed = α(t2-t1)E
Working Stress
Working stress is the actual stress in a material when a load is applied to
the body.
Allowable stress
The maximum safe stress (such as yield strength) that a material can
withhold is termed as allowable stress.
Factor of Safety
- FS = actual stress in the structure / maximum stress limit (such as yield strength)
- It indicates how far the actual stress is below the limiting stress.
- The value of FS must be greater than or equal to 1.
FS >= 1
- Values of FS range from 1.15 to as high as 10.
- Similarly, the margin of safety is calculated as:-
MS = FS- 1
Linear Strain and Lateral Strain
Linear Strain
- The strain produced when a body is subjected to load.
- Linear strain or primary strain = Change in length / Original length
Lateral Strain
- The strain produced in a body at right angles to the direction of the applied load is known as lateral strain.
- Lateral strain or secondary strain = change in length of the bar at right angles to the direction of the applied load / original length
Poisson's Ratio
- The ratio of lateral strain to linear strain is termed as Poisson's ratio.
- Poisson's ratio (μ) = lateral strain / linear strain = 1 / m
So, lateral strain = linear strain / m
where m is a constant and its value varies between 3 and 4 for different
materials
Elastic Moduli
Modulus of Elasticity
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The ratio between stress and strain within the elastic limit when the body follows Hooke's law.
- Modulus of elasticity (E)= Stress / Strain
Modulus of Rigidity
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The ratio between shear stress and shear strain.
- Modulus of rigidity (C, N or G) = Shear strain / Shear strain
Bulk or Volume Modulus of Elasticity
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The ratio between normal stress (on each face of a solid cube) to
volumetric strain.
- The bulk modulus of elasticity (K) = Normal stress (on each face of a solid cube) / Volumetric strain
Strain Energy
When a load is applied on the body within the elastic limit, the body undergoes deformation and returns to its original shape when the load is released. For the time being, loaded, strain energy is stored in the body due to its deformation.
Resilience
When loaded externally, the strain energy stored in the body within the
elastic limit is called 'Resilience'.
Modulus of Resilience
- Proof Resilience per unit volume of the body is called modulus of resilience.
- It is the mechanical property of the body.
- It indicates the capacity of the body to bear shocks.
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