Principal Stress for the Case of Plane Stress Principal Stress (2D) Formula Home Mechanics of Matl. Stress Plane Stress Principal Stress Mohr's Circle Mohr's Circle Usage Mohr's Circle Examples Strain Hooke's Law Applications Pressure Vessels Rosette Strain Gages Failure Criteria Calculators Stress Transform Strain Transform Principal Stress Principal Strain Elastic Constants Resources Bibliography  Login   Home Membership Magazines Forum Search Member Calculators  Materials  Design  Processes  Units  Formulas  Math Principal Directions, Principal Stress The normal stresses (sx' and sy') and the shear stress (tx'y') vary smoothly with respect to the rotation angle q, in accordance with the coordinate transformation equations. There exist a couple of particular angles where the stresses take on special values. First, there exists an angle qp where the shear stress tx'y' becomes zero. That angle is found by setting tx'y' to zero in the above shear transformation equation and solving for q (set equal to qp). The result is, The angle qp defines the principal directions where the only stresses are normal stresses. These stresses are called principal stresses and are found from the original stresses (expressed in the x,y,z directions) via, The transformation to the principal directions can be illustrated as: Maximum Shear Stress Direction Another important angle, qs, is where the maximum shear stress occurs. This is found by finding the maximum of the shear stress transformation equation, and solving for q. The result is, The maximum shear stress is equal to one-half the difference between the two principal stresses, The transformation to the maximum shear stress direction can be illustrated as: Home  Membership  About Us  Privacy  Disclaimer  Contact  Advertise Copyright © 2019 eFunda, Inc.