holeinplate - Problem: Find the stress state in a plate...

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Problem: Find the stress state in a plate with a circular hole of radius a. The plate is subjected to a simple tensile stress of 0 σ . 0 σ y x Solution: The boundary conditions for the problem are On the edge of the hole at r = a, π θ = θ τ π θ = θ σ θ 2 0 , 0 ) , a ( 2 0 , 0 ) , a ( r rr (a) and on the edges of the infinite plate, that is ) y , x ( r At the right and left edges ±∞ = ±∞ = x y x x y x xy xx , 0 ) , ( , ) , ( 0 τ At the top and bottom edges ±∞ = ±∞ = y y x y y x xy yy , 0 ) , ( , 0 ) , ( (b) The solution to the problem of a plate with no hole is given by the stress function ) 2 cos 1 ( r 4 1 y 2 1 2 o 2 0 θ σ = φ σ = φ (1) Hence the stresses in the problem of a plate with no hole subjected to a uniaxial tensile stress is 1
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θσ τ θ σσ 2 sin 2 1 ), 2 cos 1 ( 2 1 ) 2 cos 1 ( 2 1 0 0 0 = = + = r r (2) The above will be the stress state far from the center of the hole. Hence, the stress function can be assumed to be of the form θ + = φ 2 Cos ) r ( f ) r ( f 2 1 Hence gives 0 4 = φ [ 0 2 Cos ) r ( f ) r ( f 2 1 4 = θ + ] (3) [] 0 2 Cos ) r ( f ) r ( f 2 4 1 4 = θ + Since the first term does not include ‘ θ ’ terms and the second term does, [ 0 2 Cos ) r ( f 0 ) r ( f 2 4 1 4 = θ = ] (4) respectively, give 0 4 1 4 1 0 1 1 2 2 2 2 2 2 2 2 2 1 2 1 2 2 2 = + + = + + r f dr df r dr f d r dr d r dr d dr df r dr f d dr d r dr d (5) The above two linear ordinary differential equations have the solution respectively as 8 2 7 4 6 2 5 2 4 3 2 2 2 1 1 c r c r c r c f c r ln c r c r ln r c f + + + = + + + = Using and the stress-Airy Stress Function relationships
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This note was uploaded on 11/22/2011 for the course EML 6653 taught by Professor Law during the Fall '09 term at University of South Florida.

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holeinplate - Problem: Find the stress state in a plate...

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