Theory of failure - Rankine's theory

Rankine's Theory or Maximum Principal stress theory

     According to this theory, Failure of the structure takes place when the major principal tensile stress reaches the value beyond elastic limit in simple tension and minor principal stress(compressive in nature) reaches a value beyond elastic limit in simple compression.

Consider the three dimensional structure subjected to stresses in three mutually perpendicular directions, let σ1 , σ2 and σ3 be the normal stress in all 3 respective directions in which , σ1 and σ2(Major principal stress) be the normal stress in tensile nature and σ3 (Minor principal stress) normal stress  which is  compressive in  nature.Therefore, According to this theory, the failure of the material takes place as follows

• σ1 and σ2 ≥ σt in simple tension.
• σ3  ≥ σc in simple compression

Therefor if the material has to be safe against failure, then the maximum principle stress should not exceed permissible stress.

• σ1 and σ2 = σp
• Since,  σp = σt / FOS

where  σp = permissible stress , σt = Tensile stress after elastic limit
Limitation of Rankine's theory

• This theory is only applicable for structure made of brittle materials subjected to all loading conditions.Since brittle materials are weak in tension.
• This theory is not suited for structures made of ductile material, since the ductile material is weak in shear.But only in some conditions the theory holds good for ductile materials .Some of the conditions are as follows

1. Under uni axial state of stress when maximum shear stress is equal to half of maximum principal stress.
2. Under bi axial state of stress when there is occurrence of like forces and also when maximum shear stress is equal to half of maximum principal stress.
3. Under hydrostatic condition .i.e, when shear stress at all the boundaries is equal to zero.