lec16 - 2.001 - MECHANICS AND MATERIALS I Lecture #17...

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±± ± 2.001 - MECHANICS AND MATERIALS I Lecture # 11/8/2006 Prof. Carol Livermore Recall: Stress Transformations [ σ ]= ± σ xx σ xy ± . ± σ xy σ yy ± σ xz = σ yz = σ zz =0 ± [ σ ± σ σ x x x y σ σ x y ± . σ x ± x ± = σ xx + σ + σ xx + σ cos 2 θ + σ xy sin 2 θ 22 σ y ± y ± = σ xx + σ σ xx σ cos 2 θ σ xy sin 2 θ σ x ± y ± = σ xx σ sin 2 θ + σ xy cos 2 θ 2 1 17
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±² ³ Mohr’s Circle σ xx yy > 0 σ xy > 0 C =( σ, 0) = σ xx + σ , 0 2 2 σ xx σ R =+ σ 2 2 xy Principal stresses: σ 1 = σ + R σ 2 = σ R Mohr’s circle 2 θ corresponds to θ in physical space. 2
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±² 2 ( σ x ± x ± σ ) 2 = σ xx σ yy cos 2 θ + σ xy sin 2 θ 2 2 σ x 2 ± y ± = σ xx σ sin 2 θ + σ xy cos 2 θ 2 2 ±± σ ) 2 =c o s 2 2 θ + σ 2 sin 2 2 θ +( σ xx sin 2 θ ) cos 2 θ ( σ xy σ xx 2 σ xy σ )( σ xy 2 σ 2 = σ xx σ sin 2 2 θ + σ 2 cos sin 2 θ cos 2 θ x ± y ± 2 xy 2 2 θ ( σ xx σ ) σ 2 σ xx σ + σ 2 ( σ x ± x ± σ ) 2 + σ x 2 ± y ± = xy 2 These both describe the same circle. This serves as a proof of Mohr’s circles as
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This note was uploaded on 05/05/2010 for the course MSE 2.001 taught by Professor Carollivermore during the Fall '06 term at MIT.

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lec16 - 2.001 - MECHANICS AND MATERIALS I Lecture #17...

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