State the theories of elastic failure. Explain maximum normal stress theory and maximum shear stress theory with equations.
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Explain Maximum shear stress theory. {Maximum shear stress theory is also called as?}
Theories of Failures
Maximum shear stress theory is explained below
The principal theories of failure for a member are as follows:
(i) Maximum principal or normal stress theory
(ii) Maximum shear stress theory
(iii) Maximum principal or normal strain theory
(iv) Maximum strain energy theory
(v) Maximum distortion energy theory
Maximum normal stress theory (Maximum principal stress theory or Rankines theory
According to this theory, the elastic failure occurs when the greatest principal stress reaches the elastic limit value in a simple tension test irrespective of the value of other two principal stresses. Taking factor of safety (F. S.) into consideration, the maximum principal or normal stress (σt) is given by, σt = σyt / F. S. (for ductile materials) σt = σu / F. S. (for brittle materials) where, σyt = Yield point stress in tension as determined from simple tension test σu = Ultimate stress This theory ignores the possibility of failure due to shear stress, therefore it is not used for ductile, However, for brittle materials which are relatively strong in shear but weak in tension and compression, this theory is generally used. This theory is also known as maximum principal stress theory or Rankine’s theory.
Maximum Shear Stress Theory (Guest’s theory or Tresca’s theory)
Maximum Shear Stress Theory According to this theory, the failure or yielding occurs at a point in a member when the maximum shear stress reaches a value equal to the shear stress at yield point in a simple tension test. Mathematically, τmax = τyt / F. S. where, τmax = Maximum shear stress τyt = Shear stress at yield point as determined from simple tension test F. S = Factor of safety. Since the shear stress at yield point in a simple tension test is equal to one half the yield stress in tension, therefore τmax = σyt / (2 x F. S.).This theory is mostly used for designing members of ductile materials. This theory is also known as Guest’s theory or Tresca’s theory.
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