23.2: Principal Stresses
The graphical depiction of normal and shearing stress equations is represented by a circle, demonstrating the interplay between these stresses under different angular conditions. The center of this circle C, located on the vertical axis, represents the average normal stress, while its radius shows the range of stress variations. At points A and B, where the circle intersects the horizontal axis, the maximum and minimum normal stresses are observed, occurring without shearing stress. These pivotal points define the principal stress planes, where only normal stress, known as principal stress, exists.
To determine the values of the maximum and minimum normal stresses, one must adjust the mean stress by the circle's radius. Identifying the principal plane that carries the maximum or minimum normal stress requires inputting the angular parameter into the normal stress equation.
The points along the circle's vertical diameter indicate areas of maximum shearing stress, which arise when the normal stress equals the average stress. This condition leads to two orientations, each 90° apart, identifying the peak shearing stress locations.
An important observation is the 45° angular difference between the planes experiencing maximum shearing stress, and the principal stress planes. This geometrical relation highlights the essential connection between normal and shearing stresses, providing fundamental insight into stress distribution and interaction within materials under various loading scenarios.
