Mechanical Engineering  >  Mechanical Engineering SSC JE (Technical)  >  Principal Stress-Strain & Theories of Failure

Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

Document Description: Principal Stress-Strain & Theories of Failure for Mechanical Engineering 2022 is part of Mechanical Engineering SSC JE (Technical) preparation. The notes and questions for Principal Stress-Strain & Theories of Failure have been prepared according to the Mechanical Engineering exam syllabus. Information about Principal Stress-Strain & Theories of Failure covers topics like and Principal Stress-Strain & Theories of Failure Example, for Mechanical Engineering 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Principal Stress-Strain & Theories of Failure.

Introduction of Principal Stress-Strain & Theories of Failure in English is available as part of our Mechanical Engineering SSC JE (Technical) for Mechanical Engineering & Principal Stress-Strain & Theories of Failure in Hindi for Mechanical Engineering SSC JE (Technical) course. Download more important topics related with notes, lectures and mock test series for Mechanical Engineering Exam by signing up for free. Mechanical Engineering: Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
1 Crore+ students have signed up on EduRev. Have you?

PRINCIPAL STRESS-STRAIN 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

Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

  •  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

 Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
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
Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
The magnitude of principal stresses σ1 and σ2 are given by
Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
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
Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
Note that planes of tmax carry equal and alike normal stresses. The normal stress on plane of tmax is given by
Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
Therefore resultant stress on the plane of ζmax is
Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
The angle of obliquity of sr with the direction of σn is given by
Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
Special case-3 : In case of pure shear element, the principal stresses act at 45° to the plane of pure shear stress.
Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
 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.

Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

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

Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

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

Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
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,

Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

  •  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

Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
and shear stress is given by
Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
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

Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

  • The principal stresses are given by

Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

  •   The maximum shear stress is given by

Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

  •  The position of principal planes is given by

Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
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.

Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
Therefore
Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

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

Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
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

Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

  •   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

Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

  •  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

Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

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

Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

  •  Properties of Mohr’s circle for strain

Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
The radius of Mohr’s circle is half of maximum shear strain i.e.
Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
Therefore Diameter of Mohr’s circle,
Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
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

Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

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

Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
Hence, maximum stress intensity due to suddenly applied load is twice the stress intensity produced by the load of the same magnitude applied gradually.
Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

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
 
Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

  • 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.
Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
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
 Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
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
Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

  •  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.
Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
It is fairly good for ductile materials.
Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
 Maximum shear strain energy theory or distortion energy theory (Mises-Henky Theory).
Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
The properties are similar in tension and compression
Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

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

Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

  •  Factor of safety =
    Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
The document Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering is a part of the Mechanical Engineering Course Mechanical Engineering SSC JE (Technical).
All you need of Mechanical Engineering at this link: Mechanical Engineering

Related Searches

ppt

,

Exam

,

Summary

,

Viva Questions

,

pdf

,

Important questions

,

Previous Year Questions with Solutions

,

Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

,

mock tests for examination

,

Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

,

past year papers

,

practice quizzes

,

Free

,

Extra Questions

,

video lectures

,

Objective type Questions

,

Sample Paper

,

shortcuts and tricks

,

Semester Notes

,

MCQs

,

Principal Stress-Strain & Theories of Failure Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

,

study material

;