1-5 - CHAP 1 Stress-Strain Analysis 23 5. Find the...

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CHAP 1 Stress-Strain Analysis 23 5. Find the principal stresses and the corresponding principal stress directions for the following cases of plane stress. (a) σ xx = 40 MPa, σ yy = 0 MPa, τ xy = 80 MPa (b) σ xx = 140 MPa, σ yy = 20 MPa, τ xy = −60 MPa (c) σ xx = −120 MPa, σ yy = 50 MPa, τ xy = 100 MPa Solution: (a) The stress matrix becomes 40 80 MPa 80 0 xx xy xy yy   To find the principal stresses, the standard eigen value problem can be written as   0    In The above problem will have non-trivial solution when the determinant of the coefficient matrix becomes zero: 40 80 0 80 0 xx xy xy yy   The equation of the determinant becomes:       2 40 80 80 40 6400 0   The above quadratic equation yields two principal stresses, as 1 102.46MPa and 2 62.46MPa  .
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1-5 - CHAP 1 Stress-Strain Analysis 23 5. Find the...

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