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 τm
= {[(σxx − σyy)/2] 2 + (τxy)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

γxy = −200 γyz = 300
γzx = 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.

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