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.

THEORY OF FAILURES - Introduction

THEORIES OF FAILURE 

     When a structural element has to be designed,then it is important to determine its strength and stability based on the elasticity,yielding and fracture.As a element alone , it is easy to determine its structural behavior in terms of simple stress and strain.But in reality, the structure cannot be analysed as a part since , load applied and its corresponding stress and strain behavior will be in complex form.Some of the theories were designed to predict failure pattern for simple loads were also declared to be feasible for complex loading also.But these theories used to predict failure pattern in complex loading , failed to provide proper results in practical conditions, hence they are termed as theories of failure.The following are the five theories of failure put forth by researchers.

1. Rankine's theory or Maximum Principal Stress theory.
2. Columb's theory or Maximum Shear stress theory.
3. St Venant's theory or Maximum Strain theory.
4. Beltrami and Haigh's theory or Maximum strain energy theory.
5. Von-Mises theory or Maximum Energy distortion theory.