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Advanced Mechanics of Materials and Applied Elasticity, Fifth Edition by Saul K. Fenster, Ansel C. Ugural

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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. (1.33) 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 (σpσ1) 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 (1.33):

Image

(a)

where

(B.1) ...

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