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-188.8.131.52 ‘a’ is the radius of the hole and at r=b, b>>a the stresses are not affected by the presence of the hole.
184.108.40.206F- A plate with a central hole subjected to a uni-axial stress
Here, σx = σy , σ = 0, τxy = 0 For plane stress conditions:
This reduces to
such that 1st component in σr and σθ is constant and the second component varies with θ . Similar argument holds for rτθ if we write rτθ = The stress distribution within the ring with inner radius ri a = and outer radius or b = due to 1st component can be analyzed using the solutions of thick cylinders and the effect due to the 2nd 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 2nd 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 σθ is maximum and is given by σθ = 3σ i.e. stress concentration factor is 3.