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.