What is the maximum shear stress in the stress element shown?

The stress state shows the normal stresses, σ_x and σ_y, to be positive (tension) but the shear stress arrows are in the negative direction. Thus, the stress state is

     σ_x = 10.1 ksi
     σ_y = 6.2 ksi
     τ_xy = -3.7 ksi

The maximum shear stress is,

This could be either positive or negative due to the square root. Angle that the max shear stress acts at is,

     

     2θ_τ-max = 27.79^o (or -152.21 )
     θ_p = 13.90^o

     τ_max = 4.182 ksi @ 13.90^o

It is interesting to note, the new normal stresses are,

          σ_x' = σ_y' = (σ_x+ σ_y)/2 = 8.15 ksi