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Stress Transformation - Plane Stress &amp; Mohr's Circle

# Stress Transformation - Plane Stress &amp; Mohr's...

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CIVL 311 Stress Transformations – Plane Stress F n = 0 : σ n = σ x cos 2 θ + σ y sin 2 θ + 2 τ xy sin θ cos θ σ n = σ x + σ y ( ) 2 + σ x σ y ( ) 2 cos2 θ + τ xy sin2 θ F t = 0 : τ nt = − σ x σ y ( ) 2 sin2 θ + τ xy cos2 θ σ x = σ x + σ y ( ) 2 + σ x σ y ( ) 2 cos2 θ + τ xy sin2 θ σ y = σ x + σ y ( ) 2 σ x σ y ( ) 2 cos2 θ τ xy sin2 θ τ x y = − σ x σ y ( ) 2 sin2 θ + τ xy cos2 θ Principal Stress ( σ max and σ min ) and τ max : σ p = σ x + σ y ( ) 2 ± σ x σ y 2 2 + τ xy ( ) 2 at tan2 θ p = τ xy σ x σ y 2 τ max = ± σ x σ y 2 2 + τ xy ( ) 2 at tan2 θ s = − σ x σ y 2 τ xy Summary: There are two principal planes, 90 ° apart, located by θ p . On these planes are the principal stress values σ p = σ max and σ min . There are two other planes containing τ max , also 90 ° apart, located by θ

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