# Show in two dimensions that the maximum shear stress t m = {[(s xx – s yy )/2] 2 + (t xy ) 2 } (1/2)

Show in two dimensions that the maximum shear stress τ_{m}

= {[(σ_{xx} − σ_{yy})/2] ^{2} + (τ_{xy})^{2}}^{(1/2)}

is invariant under a rotation of the reference axes and is equal to one-half

the difference of the major and minor principal stresses in the considered

plane.

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Show in two dimensions that the maximum shear stress t m = {[(s xx – s yy )/2] 2 + (t xy ) 2 } (1/2)

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Given the strains:

_{xx} = 2,000 _{yy}

= 3,000 _{zz} = 4,500

γ_{xy} = −200 γ_{yz} = 300

γ_{zx} = 225

where x = east, y = north, z = up, compression is positive,

the units are microinches per inch, Young’s modulus E = 5.0(10)6 psi, and shear

modulus G = 2.0(10)6 psi, find the corresponding stresses for a linear,

homogeneous, isotropic elastic response.