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PRINCIPAL STRESSSTRAIN AND THEORIES OF FAILTURE
ANALYSIS OF PRINCIPAL STRESSES
Special case1 : 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 σ_{1} and σ_{2} are given by
Special case2 : The plane of maximum shear stress lies at 45° to the plane of principal stress and magnitude of ζ_{max} is given by
Note that planes of tmax carry equal and alike normal stresses. The normal stress on plane of tmax is given by
Therefore resultant stress on the plane of ζ_{max} is
The angle of obliquity of sr with the direction of σ_{n} is given by
Special case3 : In case of pure shear element, the principal stresses act at 45° to the plane of pure shear stress.
Properties of Mohr’s Circle for Stress :
Note : Sum of normal stresses on two mutually perpendicular planes remain constant i.e. σ_{1 }+ σ_{2} = σ_{x} + σ_{y} = constant
COMBINED BENDING & TORSION
and shear stress is given by
Where I = Moment of inertia about its NA and I_{p} = Polar moment of Inertia.
EQUIVALENT BENDING MOMENT & EQUIVALENT TORQUE
Therefore
ANALYSIS OF PRINCIPAL STRAINS
The radius of Mohr’s circle is half of maximum shear strain i.e.
Therefore Diameter of Mohr’s circle,
STATIC LOADING & DYNAMIC LOADING
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
This assumes that max. principal stress in the complex system reaches the elastic limit stress in simple tension and failure occurs when
σ_{1} = σ_{y} ... for tension
Failure can occur in compression when least principal stress (σ_{3}) reaches the elastic
limit stress in compression i.e.
σ_{3} = σ_{y} ... for compression
It is well suited for brittle materials. Failure envelope occurs when
(a) σ_{1} or σ_{2} = σ_{yt} or σ_{yc}
(ii) σ_{3} = 0
This assumes that max shear stress in the complex stress system becomes equal to that at the yield point in simple tensile test.
This theory holds good for ductile materials. For like stresses in I^{st} and III^{rd }quadrant
σ_{1} = σ_{y} or σ_{2} = σ_{y}
For unlike stresses in II^{nd} or IV^{th} quadrant
Note : Aluminium alloys & certain steels are not governed by the Guest theory.
This assumes that failure occurs when max. strain in the complex stress system equals that at the yield point in the tensile test
(σ_{1} – μσ_{2} – μσ_{3}) = σy
Failure should occur at higher load because the Poisson's ratio reduces the effect in perpendicular directions
This assumes that failure occur when total strain energy in the complex system is equal to that at the yield point in tensile test.
It is fairly good for ductile materials.
Maximum shear strain energy theory or distortion energy theory (MisesHenky Theory).
The properties are similar in tension and compression
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