Advanced Strength and Applied Elasticity, Fourth Edition

Appendix B Solution of the Stress Cubic Equation

B.1. Principal Stresses

There are many methods in common usage for solving a cubic equation. A simple approach for dealing with Eq. (1.28) 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 (1.28):

(a)

where