Noting that the components of the stress vector S in terms of principal stresses

Question

Noting that the components of the stress vector S in terms of principal stresses can be written as S_x = n_x*_1, S_y = n_y*_2, and S_z = n_z*_3, show the equation below is valid.

[(S_x)^2]/(_1)^2 + [(S_y)^2]/(_2)^2 + [(S_z)^2]/(_3)^2 = 1

This equation is stress ellipsoid, the surface of which is shown in the first quaddrant. The stress ellipsoid graphically depicts that the principal stresses are the maximum and minimum stresses at a point. The surface of the ellipsoid describes all possible stress states at a point.

Explanation / Answer

From the info given, we have

nx = Sx/1     ny = Sy/2    nz = Sz/3   

Since n = nx i + ny j + nz k is a unit vector, we have

nx2 + ny2 + nz2 =1   =>

(Sx/1)2 + (Sy/2)2 + (Sz/3)2 =1     

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