Stress Concentration - 2 | Design of Machine Elements - Mechanical Engineering PDF Download

Theoretical basis of stress concentration 

Consider a plate with a hole acted upon by a stressσ . St. Verant’s principle states that if a system of forces is replaced by another statically equivalent system of forces then the stresses and displacements at points remote from the region concerned are unaffected. In figure-3.2.3.1 ‘a’ is the radius of the hole and at r=b, b>>a the stresses are not affected by the presence of the hole.

3.2.3.1F- A plate with a central hole subjected to a uni-axial stress

Here, σ_x = σ_y  , σ = 0, τ_xy = 0 For plane stress conditions:

This reduces to

such that 1^st component in σ_r and σ_θ is constant and the second component varies with θ . Similar argument holds for r_τθ if we write r_τθ =  The stress distribution within the ring with inner radius r_i a = and outer radius or b = due to 1^st component can be analyzed using the solutions of thick cylinders and the effect due to the 2^nd component can be analyzed following the Stress-function approach. Using a stress function of the form φ = θ R (r) cos2θ the stress distribution due to the 2^nd component can be found and it was noted that the dominant stress is the Hoop Stress, given by

This is maximum at θ= ±π 2 and the maximum value of  Therefore at points P and Q where r =a  σ_θ is maximum and is given by σ_{θ =} 3σ i.e. stress concentration factor is 3.