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Lecture5a - τ σ Triaxial Stress State ve sense shown 3D...

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Unformatted text preview: τ σ Triaxial Stress State (+ve sense shown) 3D Principal – Triaxial Stress max int min σ σ σ ≥ ≥ 1 2 3 σ σ σ ≥ ≥ 3D Stress – Principal Stresses 3 2 1 2 3 I I I σ σ σ- +- = The three principal stresses are obtained as the three real roots of the following equation: where 1 2 2 2 2 2 2 2 3 2 x y z x y x z y z xy xz yz x y z xy xz yz x yz y xz z xy I I I σ σ σ σ σ σ σ σ σ τ τ τ σ σ σ τ τ τ σ τ σ τ σ τ = + + = + +--- = +--- I 1 , I 2 , and I 3 are known as stress invariants as they do not change in value when the axes are rotated to new positions. Stress Invariants for Principal Stress 3 2 1 3 3 1 3 2 2 1 2 3 2 1 1 σ σ σ σ σ σ σ σ σ σ σ σ = + + = + + = I I I 1 2 2 2 2 2 2 2 3 2 x y z x y x z y z xy xz yz x y z xy xz yz x yz y xz z xy I I I σ σ σ σ σ σ σ σ σ τ τ τ σ σ σ τ τ τ σ τ σ τ σ τ = + + = + +--- = +--- Zero shear stress on principal planes Mohr’s Circle?...
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Lecture5a - τ σ Triaxial Stress State ve sense shown 3D...

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