Principal Stress - Strain and Theories of Failure Civil Engineering (CE) Notes | EduRev

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Civil Engineering (CE) : Principal Stress - Strain and Theories of Failure Civil Engineering (CE) Notes | EduRev

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Principal Stress - Strain and Theories of Failure
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 and Theories of Failure Civil Engineering (CE) Notes | EduRev
Principal Stress - Strain and Theories of Failure Civil Engineering (CE) Notes | EduRev

Case-2 : If σx and σy are 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 σx are given by

 Principal Stress - Strain and Theories of Failure Civil Engineering (CE) Notes | EduRev
Principal Stress - Strain and Theories of Failure Civil Engineering (CE) 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

 Principal Stress - Strain and Theories of Failure Civil Engineering (CE) Notes | EduRev
The magnitude of principal stresses σ1 and σ2 are given by

 σor σ2 = (σxy)/2 ± √[(σxy/2)22]

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 and Theories of Failure Civil Engineering (CE) Notes | EduRev

Note that planes of ζmax carry equal and alike normal stresses. The normal stress on plane of  ζmax is given by

 Principal Stress - Strain and Theories of Failure Civil Engineering (CE) Notes | EduRev

Therefore resultant stress on the plane of Тmax is

 Principal Stress - Strain and Theories of Failure Civil Engineering (CE) Notes | EduRev

The angle of obliquity of σr with the direction of σn is given by

 Principal Stress - Strain and Theories of Failure Civil Engineering (CE) Notes | EduRev

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 and Theories of Failure Civil Engineering (CE) Notes | EduRev

σ1 = +  ζxy
σ= – ζxy ·

  • 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 and Theories of Failure Civil Engineering (CE) Notes | EduRev

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

 Principal Stress - Strain and Theories of Failure Civil Engineering (CE) Notes | EduRev

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

 Principal Stress - Strain and Theories of Failure Civil Engineering (CE) Notes | EduRev

Note : Sum of normal stresses on two mutually perpendicular planes remain constant i.e.σ1 + σ2 = σ+ σy = constant

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

 Principal Stress - Strain and Theories of Failure Civil Engineering (CE) 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

 Principal Stress - Strain and Theories of Failure Civil Engineering (CE) Notes | EduRev
and shear stress is given by

 Principal Stress - Strain and Theories of Failure Civil Engineering (CE) 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

 Principal Stress - Strain and Theories of Failure Civil Engineering (CE) Notes | EduRev

  •  The principal stress es are given by

 Principal Stress - Strain and Theories of Failure Civil Engineering (CE) Notes | EduRev

  • The maximum shear stress is given by

 Principal Stress - Strain and Theories of Failure Civil Engineering (CE) Notes | EduRev

  • The position of principal planes is given by

 Principal Stress - Strain and Theories of Failure Civil Engineering (CE) Notes | EduRev

EQUIVALENT BENDING MOMENT & EQUIVALENT TORQUE

  • Let ‘Me’ be the equivalent bending moment which acts alone producing the maximum tensi l e stress equal to σ1, as produced by M and T.

 Principal Stress - Strain and Theories of Failure Civil Engineering (CE) Notes | EduRev
Therefore
Principal Stress - Strain and Theories of Failure Civil Engineering (CE) Notes | EduRev

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

 Principal Stress - Strain and Theories of Failure Civil Engineering (CE) Notes | EduRev

ANALYSIS OF PRINCIPAL STRAINS

  •  Case-1 : If ε1 and ε2 are principal strains in two mutual perpendicular directions in plane stress problem then principal stresses are given by

 Principal Stress - Strain and Theories of Failure Civil Engineering (CE) Notes | EduRev
Principal Stress - Strain and Theories of Failure Civil Engineering (CE) Notes | EduRev

  • If ε1 and ε2 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 and Theories of Failure Civil Engineering (CE) Notes | EduRev
Principal Stress - Strain and Theories of Failure Civil Engineering (CE) Notes | EduRev

  • If εx, εy and 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 and Theories of Failure Civil Engineering (CE) Notes | EduRev
Principal Stress - Strain and Theories of Failure Civil Engineering (CE) Notes | EduRev

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

 Principal Stress - Strain and Theories of Failure Civil Engineering (CE) Notes | EduRev

  • · Properties of Mohr’s circle for strain

 Principal Stress - Strain and Theories of Failure Civil Engineering (CE) Notes | EduRev

The radius of Mohr’s circle is half of maximum shear strain i.e.

 Principal Stress - Strain and Theories of Failure Civil Engineering (CE) Notes | EduRev

Therefore Diameter of Mohr’s circle,

 Principal Stress - Strain and Theories of Failure Civil Engineering (CE) 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
σ = (P/A)
When load is applied suddenly, then the normal stress ‘σ’ due to load P is given by
  σ = (2P/A)

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 and Theories of Failure Civil Engineering (CE) 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

 Principal Stress - Strain and Theories of Failure Civil Engineering (CE) 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.

 Principal Stress - Strain and Theories of Failure Civil Engineering (CE) 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

 Principal Stress - Strain and Theories of Failure Civil Engineering (CE) Notes | EduRev
Principal Stress - Strain and Theories of Failure Civil Engineering (CE) 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 (σ1 – µσ2 – µσ3) = σy
    Failure should occur at higher load because the Poisson's ratio reduces the effect in perpendicular directions

 Principal Stress - Strain and Theories of Failure Civil Engineering (CE) Notes | EduRevRHOMBUS

  •  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 and Theories of Failure Civil Engineering (CE) Notes | EduRev

It is fairly good for ductile materials.

 Principal Stress - Strain and Theories of Failure Civil Engineering (CE) Notes | EduRev  ELLIPSE

  •  Maximum shear strain energy theory or distortion energy theory (Mises-Henky Theory).

 Principal Stress - Strain and Theories of Failure Civil Engineering (CE) Notes | EduRev

The properties are similar in tension and compression

 Principal Stress - Strain and Theories of Failure Civil Engineering (CE) Notes | EduRev  ELLIPSE
Principal Stress - Strain and Theories of Failure Civil Engineering (CE) Notes | EduRev

  •  Failure of most ductile materials is governed by the distortion energy criterion or Von mises theory.
    σ1 = σ2 = σ3; cylinder
    σ= σ2 and σ3 = 0; ellipse ·
     
  • Factor of safety =  Tensile strength / Allowable working stress
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