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Simplification of the Parametric Transformation Equations for Analysis of Plane Stress

For the analysis of plane stress, there are two methods, transformation of equation and Mohr’s circle. With the former method, the two principle angles are found by taking derivative to determine the extreme value of stresses. However, it is unknown which one of the two principle angles corresponding to the maximum stress. Moreover, the relationship between the two methods is inexplicit, as the angle of the rotation of the element is eliminated to derive the equation of Mohr’s circle. Using trigonometric relations, considering two systems of sign conventions for shear stresses, simplified parametric transformation equations are obtained. A unique principle angle is easy to be found corresponding to a principle stress. It is obvious that the graph of the simplified equations is a circle. The relationship of the angle of the rotation of the element and the angular parameter of the circle is explicit.