Tabular solution of the stress-cubic equation
Edwin P. Russo, P.E.
Duane J. Jardine
Mechanical Engineering Dept.
University of New Orleans
New Orleans, La.
Principal stresses are defined as the maximum and minimum normal stresses in a plane. Principal stresses are perpendicular to each other and oriented such that shear stresses are zero. For 3D stress states, principal stresses equal the roots of the general stress-cubic equation:
where I1, I2, and I3 are known as the stress invariants and are given by:
Classical techniques involving the computation of inverse cosines, cosines of multiple angles, and so forth, can be used to find the roots of Eq 1. But the equation can be transformed into a much simpler form that has only one coefficient:
The transformation provides a convenient way to tabularize the roots of Eq 1. However, for the roots to all be real:
Otherwise, one real and two complex roots will result, which is meaningless for the stress-cubic equation. The Table lists the roots of Eq 5 for various values of the coefficient, Q. Use these roots and Eq 7 to obtain the roots of the stress-cubic equation. Here's how:
As an example, find the principle stresses given:
Substituting these numbers into the stress-invariant equations gives
The stress-cubic equation becomes:
Eq 8 gives:
From the Table: 1= 1.0880; 2= 0.20915; 3= 0.87889
Eq 7 then gives:
Similarly, σ2 = 25.6 MPa and σ3 = 66.0 MPa
Interested readers may want to expand the Table to include finer increments of Q. Besides principle stresses, the method may also help solve other important engineering problems including pump curves, eigenvalues, hydraulic jumps, control systems, spillway flow, and moving wave/bores.