Advanced Mechanics of Materials and Applied Elasticity, 6th Edition

Appendix B. Solution of the Stress Cubic Equation

B.1 Principal Stresses

Many methods may be used to solve a cubic equation. A simple approach for dealing with Eq. (1.33) is to find one root, say σ₁, by plotting it (σ as abscissa) or by trial and error. The cubic equation is then factored by dividing by (σ_p – σ₁) to arrive at a quadratic equation. The remaining roots can be obtained by applying the familiar general solution of a quadratic equation. This process requires considerable time and algebraic work, however.

What follows is a practical approach for determining the roots-of-stress cubic equation, Eq. (1.33):

σp3−I1σp2+I2σp−I3=0(a)

where

I1=σx+σy+σzI2=σxσy+σxσz+σyσz−τxy2−τyz2−τxz2I3=σxσyσz+2τxyτyzτxz−σxτyz2−σyτxz2−σzτxy2(B.1)

According ...