Beer & Johnston — Transformations of Stress and Strain: Problem 7.147

7.147 Using a 45° rosette, the strains \(\epsilon_1\), \(\epsilon_2\), and \(\epsilon_3\) have been determined at a given point. Using Mohr’s circle, show that the principal strains are: \[\epsilon_{max, min} = \frac{1}{2}(\epsilon_1 + \epsilon_3) \pm \frac{1}{\sqrt{2}}[(\epsilon_1 - \epsilon_2)^2 + (\epsilon_2 - \epsilon_3)^2]^{1/2}\] (Hint: The shaded triangles are congruent.)