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. ·
Case1 : If principal stresses acting on two mutually perpendicular planes are σ_{1} and σ_{2} then, normal and shear stresses on a plane n – n which is inclined at an angle θ with the plane of σ_{1} are given by
Case2 : If σ_{x} and σ_{y} are normal stresses and t_{xy} is shear stress acting on the mutually perpendicular planes then the normal and shear stresses on any plane nn inclined at an angle θ with the plane of σ_{x} are given by
Special case1 : If θ becomes such that ζ_{x'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
σ_{1 }or σ_{2} = (σ_{x}+σ_{y})/2 ± √[(σ_{x}σ_{y}/2)^{2}+Τ^{2}]
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 ζ_{max }carry equal and alike normal stresses. The normal stress on plane of ζ_{max} is given by
Therefore resultant stress on the plane of Т_{max} is
The angle of obliquity of σ_{r} 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.
σ_{1} = + ζ_{xy}
σ_{2 }= – ζ_{xy ·}
The radius of Mohr’s circle is equal to maximum shear stress.
Radius,
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
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 I_{p} = Polar moment of Inertia.
EQUIVALENT BENDING MOMENT & EQUIVALENT TORQUE
Therefore
ANALYSIS OF PRINCIPAL STRAINS
Special case : If φ_{x'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) σ_{1} or σ_{2} = σ_{yt } or σ_{yc}
(ii) σ_{3} = 0
Note : Aluminium alloys & certain steels are not governed by the Guest theory.
RHOMBUS
It is fairly good for ductile materials.
ELLIPSE
The properties are similar in tension and compression
ELLIPSE
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Test: Solid Mechanics Level  1 Test  25 ques 
Test: Solid Mechanics Level  2 Test  25 ques 
1. What is the difference between stress and strain? 
2. How are principal stresses calculated? 
3. What are the theories of failure in materials? 
4. How do the theories of failure help in material design? 
5. What are the limitations of the theories of failure? 
Test: Solid Mechanics Level  1 Test  25 ques 
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