[Machine Design] - Theories of failure under static load

1] Maximum Principal or normal Stress theory ( Rankine Theory) -->


According to this theory, the members fails when maximum principal or normal stress in system reaches to limiting value.
Limiting stresses
For ductile material is yield stress
For brittle material (do not have well definite yield point) is ultimate stress.
According to above theory, taking factor of safety (F.S.) in account , the principal or normal stress (δt1 ) in a bi-axial stress system is given by



δt1= δyt/F.S. For ductile materials
δt1= δut/ F.S. For brittle materials

Where,
δyt =Yield stress in tensile test
δut = Ultimate stress



This theory is generally used for brittle materials.[FONT=&quot]

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2] Maximum Shear Stress theory (Guest’s or Tresca’s theory ) -->
According to this theory, the failure is occurs at a point in member when maximum shear stress reaches to the value equal to shear stress at yield point in simple tension test.
Mathematically
τmax = τyt/F.S.
where,
τmax = Maximum shear stress in system
F.S.= factor of sefty
τyt = Shear stress at yield point determined from simple tension test.

3] Maximum principal (or normal) Strain theory (Also known as Saint Venant's Theory)
According to this theory the failure/ yielding occurs at a point when maximum principal strain reaches to limiting value of strain.
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where
δ[SUB]t1[/SUB]& δ[SUB]t2[/SUB] = Maximum & minimum principal stresses.
ε= strain at yield point.
1/m = Passions ratio
E= Young’s Modulus.
F.S.= Factor of safety

This theory is rarely used due to unreliable results in many cases.

{Reserved For 4] Maximum Strain energy theory (also known as Haigh’s theory)}

{Reserved For 5] Maximum distortion energy theory (also known as Hencky & Von Mises theory)}

Now below we can discuss above topic.

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How to find factor of safety using  (MSST) without given yield stress