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
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
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
The magnitude of principal stresses σ1 and σ2 are given by
σ1 or σ2 = (σx+σy)/2 ± √[(σx-σy/2)2+Τ2]
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
Note that planes of ζmax carry equal and alike normal stresses. The normal stress on plane of ζmax is given by
Therefore resultant stress on the plane of Тmax is
The angle of obliquity of σr with the direction of σn is given by
Special case-3 : In case of pure shear element, the principal stresses act at 45° to the plane of pure shear stress.
σ1 = + ζxy
σ2 = – ζxy ·
The radius of Mohr’s circle is equal to maximum shear stress.
Note : Sum of normal stresses on two mutually perpendicular planes remain constant i.e.σ1 + σ2 = σx + σy = constant
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
and shear stress is given by
Where I = Moment of inertia about its NA and Ip = Polar moment of Inertia.
EQUIVALENT BENDING MOMENT & EQUIVALENT TORQUE
ANALYSIS OF PRINCIPAL STRAINS
Special case : If φx'y' = 0 then magnitude of principal strains and their plane are given by
The radius of Mohr’s circle is half of maximum shear strain i.e.
Therefore Diameter of Mohr’s circle,
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.
THEORIES OF ELASTIC FAILURE
Failure envelope occurs when
(a) σ1 or σ2 = σyt or σyc
(ii) σ3 = 0
Note : Aluminium alloys & certain steels are not governed by the Guest theory.
It is fairly good for ductile materials.
The properties are similar in tension and compression