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EN0175-10 - EN0175 10 05 06 Maximum and minimum shear...

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EN0175 10 / 05 / 06 Maximum and minimum shear stresses in a solid An general stress state = 33 32 31 23 22 21 13 12 11 σ σ σ σ σ σ σ σ σ σ , when expressed in the principal directions, becomes diagonalized as = III II I σ σ σ σ 0 0 0 0 0 0 1 e v 2 e v 3 e v I II III t v n σ s σ n v The traction on an arbitrary plane with normal n v is 3 3 2 2 1 1 e n e n e n n t III II I v v v v v σ σ σ σ + + = = The magnitude of is therefore t v ( ) 2 1 2 3 2 2 2 2 2 1 2 n n n t III II I σ σ σ + + = v The normal stress on the plane is 2 3 2 2 2 1 n n n n n t n III II I n σ σ σ σ σ + + = = = v v v v Note that 2 2 2 t n s v = + σ σ , therefore ( ) 2 2 3 2 2 2 1 2 3 2 2 2 2 2 1 2 2 n n n n n n III II I III II I s σ σ σ σ σ σ σ + + + + = We wish to find the maximum/minimum values of s σ subject to constraint (because is a unit vector). 1 2 3 2 2 2 1 = + + n n n n v Introduce Lagrangian multiplier λ , 1
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