Stress and Strain Most Important Notes and formula for Mechanical Engineering GATE ,ESE,PSU exams

By Akhil Gupta|Updated : December 29th, 2021

Stress and Strain is the first topic in Strength of Materials which consists of various types of stresses, strains, and different properties of materials which are important while working on them.

Stress and Strain is the first topic in Strength of Materials which consist of various types of stresses, strains, and different properties of materials which are important while working on them.

Stress: The force of resistance per unit area, offered by a body against deformation is known as stress.

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Fig.1: Stress

It is denoted by a symbol ‘σ’. And mathematically expressed as

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As particular stress generally holds true only at a point, therefore it is defined mathematically as

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TYPES OF STRESSES:

only two basic stresses exist.

(1) Normal stress and

(2) Shear stress.

(i) Normal stresses:

If the force applied are perpendicular or normal to areas concerned, then these are termed as normal stresses.

The normal stresses are generally denoted by a Greek letter (σ).

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Fig.2 Uniaxial Normal Stress

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Fig.3: Biaxial Normal Stress

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Fig.4: Triaxial Normal Stress

Tensile or compressive stresses:

The normal stresses can be either tensile or compressive depending upon the direction of the load.

Sign convention: The tensile forces are termed as (+ve ) while the compressive forces are termed as negative (-ve).

(ii)Shear Stress:

when cross-sectional area of a block of material is subject to a distribution of forces which are parallel to the area concerned. Such forces are associated with a shearing of the material, are known as shear forces. The stress produced by these forces are known as shear stresses.

 

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Fig.5: Shear stress

It is generally denoted by ‘τ’ and expressed as

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The complementary shear stresses are equal in magnitude.

 The same form of relationship can be obtained for the other two pair of shear stress components to arrive at the relations,

 τxyyx

τyz=τzy

Sign convections for shear stresses:

Shear stress tending to turn the element Clockwise is taken as Positive.

Shear stress tending to turn the element Anticlockwise is taken as Negative.

 

 

STRAIN

When a prismatic bar is subjected to axial load, it undergoes a change in length, as indicated in Figure. This change in length is usually called deformation.

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Fig.6: Deformation of bar under axial load

 If the axial force is tensile, the length of the bar is increased, while if the axial force is compressive, there is shortening of the length of the bar.

The deformation (i.e. elongation or shortening) per unit length of the bar is termed as strain and denoted by ε or e.

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Fig.7: Classification of strain

 

(i) Longitudinal strain:

The ratio of axial deformation along the length of the applied load to the original length of the body is known as longitudinal (or linear) strain.

L = Length of the body,

P = Tensile force acting on the body,

δL = Increase in the length of the body in the direction of P.

Then,

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(ii) Lateral strain:

The strain at right angles to the direction of applied load is known as lateral strain. Let a rectangular bar of length L, breadth b and depth d is subjected to an axial tensile load P. The length of the bar with increase while the breadth and depth will decrease.

δL = Increase in length,

δb = Decrease in breadth, and

δd = Decrease in depth.

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Fig.8: strain

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(iii)Shear Strain

Change in initial right angle between two-line elements which are parallel to x and y-axis respectively.

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Fig.9: Shear strain

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 STRESS - STRAIN DIAGRAM

The mechanical properties of a material are determined in the laboratory by performing tests on small specimens of the material, in the materials testing laboratory. The most common materials test is the tension test performed on a cylindrical specimen of the material.

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Fig.16: Stress strain Diagram

A = Proportional Limit          Oa = Linear Deformation

B = Elastic Limit                  Ob = Elastic Deformation

C = Yield Point                    bd = Perfect Plastic Yielding

C’ = Lower Yield Point          de = Strain hardening

E = Ultimate Strength         ef = Necking

F = Rupture Strength/ Fracture strength

It is customary to base all the stress calculations on the original cross-sectional area of the specimen, and since the latter is not constant, the stresses so calculated are known as Nominal stresses.

LINEAR ELASTICITY: HOOKE’S LAW

The slope of the stress-strain curve is called the young’s modulus of elasticity (E):

The slope of the stress-strain curve,

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This equation is known as Hooke’s law.

Type of Metal Behaviour

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Fig.17: Stress strain Diagram

 

 ELONGATION OF BAR UNDER DIFFERENT CONDITION

 UNIFORMLY TAPERING CIRCULAR BAR

Let us now consider a uniformly tapering circular bar, subjected to an axial force P, as shown in Figure. The bar of length L has a diameter d1 at one end and d2 at the other end (d2 > d1).

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Fig.18: Uniformly tapering circular bar

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 PRINCIPLE OF SUPERPOSITION

Bars In Series

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Fig.19: A prismatic bar subjected to multiple loads

 It stated that

"The resultant strain in the bar will be equal to the algebraic sum of the strains caused by the individual forces acting along the length of the member.

Thus, if a member of uniform section is subjected to a number of forces, the resulting deformation (ΔL) is given by

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Sample and Previous Year question of  asked in GATE exam

Q 1: The true stress (σ)(σ) - true strain (ε)(ε) diagram of a strain hardening material is shown in figure. First, there is loading up to point A, i.e., up to stress of 500 MPa and strain of 0.5. Then from point A, there is unloading up to point B, i.e., to stress of 100 MPa. Given that the Young’s modulus E = 200 GPa, the natural strain at point B (εB)(εB) is _________ (correct to three decimal places).

Q2 : A bar is compressed to half of its original length. The magnitude of true strain produced in the deformed bar is _________________ (correct to two decimal places).

 

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