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**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.

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