Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

Mechanical Engineering SSC JE (Technical)

Mechanical Engineering : Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

The document Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev is a part of the Mechanical Engineering Course Mechanical Engineering SSC JE (Technical).
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Chapter 2

PRINCIPAL STRESSSTRAIN AND THEORIES OF FAILTURE

ANALYSIS OF PRINCIPAL STRESSES

  •  Principal stresses are direct normal stresses acting on mutually perpendicular planes  on which shear stresses are zero. The planes which carry zero shear stresses are  known as principal planes.
  •  Case-1 : If principal stresses acting on two mutually perpendicular planes are σ1 and σ2 then, normal and shear stresses on a plane n – n which is inclined at an angle  θ with the plane of σ1 are given by

Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  •  Case-2 : If σx and σare normal stresses and txy is shear stress acting on the  mutually perpendicular planes then the normal and shear stresses on any plane n-n  inclined at an angle θ with the plane of σare given by

 Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
Special case-1 : If θ becomes such that ζx'y' on this plane becomes zero then this plane will be known as principal plane and the angle of principal plane is given by
Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
The magnitude of principal stresses σ1 and σ2 are given by
Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
Special case-2 : The plane of maximum shear stress lies at 45° to the plane of principal stress and magnitude of ζmax is given by
Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
Note that planes of tmax carry equal and alike normal stresses. The normal stress on plane of tmax is given by
Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
Therefore resultant stress on the plane of ζmax is
Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
The angle of obliquity of sr with the direction of σn is given by
Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
Special case-3 : In case of pure shear element, the principal stresses act at 45° to the plane of pure shear stress.
Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
 Properties of Mohr’s Circle for Stress :

  •  Mohr’s circle is the locus on normal and shear stresses on an element with the  changing angle of plane in 2 dimensional case.

Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  •  The radius of Mohr’s circle is equal to maximum shear stress.

Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  •  The centre of circle always lies on s axis and its co-ordinates are (σn, 0)

Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
Note : Sum of normal stresses on two mutually perpendicular planes remain constant i.e. σ+ σ2 = σx + σy = constant

  •   In case of pure shear element σ1 = + ζand σ2 = – ζ therefore centre of the  circle coincides with the origin,

Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  •  In case of an element inside the static fluid, ζ = 0 because principal stresses are  equal and alike therefor Mohr’s circle reduces into a point

COMBINED BENDING & TORSION  

  •  Let a shaft of diameter ‘d’ be subjected to bending moment ‘M’ and a twisting  moment ‘T’ at a section. At any point in the section at radius ‘r’ and at a distance y  from the neutral axis, the bending stress is given by

Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
and shear stress is given by
Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
Where I = Moment of inertia about its NA and Ip = Polar moment of Inertia.

  •  The location of the principal planes through the point is given by tan

Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  • The principal stresses are given by

Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  •   The maximum shear stress is given by

Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  •  The position of principal planes is given by

Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
EQUIVALENT BENDING MOMENT & EQUIVALENT TORQUE

  •   Let ‘Me’ be the equivalent bending moment which acts alone producing the maximum  tensile stress equal to σ, as produced by M and T.

Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
Therefore
Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  •  Let ‘Te’ be the equivalent torque, which acts alone producing the same maximum  shearing stress ζmax as produced by M and T.

Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
ANALYSIS OF PRINCIPAL STRAINS  

  •  Case-1 : If ∈1 and ∈are principal strains in two mutual perpendicular directions in  plane stress problem then principal stresses are given by

Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  •   If ∈1 and ∈2 are principal strains in x and y directions respectively, then normal and  shear strain in any other direction x' are given by

Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  •  If ∈1,2and φxy are normal and shear strain in x – y plane the normal and shear  strain in x' – y' plane are given by

Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  • Special case : If φx'y' = 0 then magnitude of principal strains and their plane are given by

Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  •  Properties of Mohr’s circle for strain

Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
The radius of Mohr’s circle is half of maximum shear strain i.e.
Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
Therefore Diameter of Mohr’s circle,
Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
STATIC LOADING & DYNAMIC LOADING

  •  When load is increased gradually from zero to P, it is called static loading. Under  static loading the normal stress ’σ’ developed due to load P is given by

Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  •  When load is applied suddenly, then the normal stress ’σ’ due to load P is given by

Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
Hence, maximum stress intensity due to suddenly applied load is twice the stress intensity produced by the load of the same magnitude applied gradually.
Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

THEORIES OF ELASTIC FAILURE

  •   Function of theories of elastic failure is to predict the behaviour of materials in  simple tensile test when elastic failure will occur under any condition of applied  stress.
  •  Maximum principal stress theory (Rankine)

This assumes that max. principal stress in the complex system reaches the elastic limit stress in simple tension and failure occurs when
σ1 = σy ... for tension
Failure can occur in compression when least principal stress (σ3) reaches the elastic
limit stress in compression i.e.
σ3 = σy ... for compression
It is well suited for brittle materials. Failure envelope occurs when
(a) σ1 or σ2 = σyt or σyc
(ii) σ3 = 0
 
Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  • Max shear stress theory (Guest - Tresca)

This assumes that max shear stress in the complex stress system becomes equal to that at the yield point in simple tensile test.
Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
This theory holds good for ductile materials. For like stresses in Ist and IIIrd quadrant
σ1 = σy or σ2 = σy
For unlike stresses in IInd or IVth quadrant
 Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
Note : Aluminium alloys & certain steels are not governed by the Guest theory.

  • Max principal strain theory (Saint Venant)

This assumes that failure occurs when max. strain in the complex stress system equals that at the yield point in the tensile test
1 – μσ2 – μσ3) = σy
Failure should occur at higher load because the Poisson's ratio reduces the effect in perpendicular directions
Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  •  Maximum strain energy theory (Haigue)or Total strain Energy Theory : -  

This assumes that failure occur when total strain energy in the complex system is equal to that at the yield point in tensile test.
Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
It is fairly good for ductile materials.
Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
 Maximum shear strain energy theory or distortion energy theory (Mises-Henky Theory).
Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
The properties are similar in tension and compression
Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  •  Failure of most ductile materials is governed by the distortion energy criterion or  Von mises theory.

Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  •  Factor of safety = 
    Chapter 2 Principal Stress - Strain and Theories of Failure - Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
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