23.2:
Principal Stresses
The normal and shearing stress equations of the transformed plane, when graphed, form a circle that demonstrates their relationship for any given angular parameter.
The circle's center, relative to the vertical axis, represents the average normal stress, with its radius indicating the spread of these stress values.
The circle intersects the horizontal axis at two points, signifying the maximum and minimum normal stresses, occurring with zero shearing stress. These points define principal planes of stress where only normal stress, known as principal stress, exists.
The maximum and minimum normal stresses can be identified by adding or subtracting the average stress from the radius.
The principal plane that experiences maximum or minimum normal stress is identified by substituting the angular parameter into the normal stress equation.
The largest shearing stress is represented by points on the circle's vertical diameter, obtained when the normal stress equals the average stress, resulting in two 90° orientations that predict maximum shearing stress.
The planes of maximum shearing stress and the principal planes are 45° apart.
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.