23.4 : Mohr's Circle for Plane Stress

Mohr's circle is a graphical method for identifying the state of stress at a point in a material, making it easier to analyze stress transformations under plane stress conditions. This two-dimensional technique visualizes both normal and shearing stresses on an element.

Consider a set of Cartesian coordinates. The horizontal and vertical axes correspond to normal stress (σ) and shearing stress (τ), respectively. Two points, points A and B, are defined by the normal and shear stresses on the element. The coordinates of point A are located on the plane based on shear and normal stresses on the element. The coordinates of point A are (σ_x, -τ_xy), and the coordinates of point B are (σ_x, τ_xy). Mohr's circle is created by drawing a line between A and B. Point O, which crosses the horizontal axis, is the center of Mohr's circle. O is exactly halfway between points A and B.

Points X and Y, where the circle intersects the horizontal (normal stress) axis, indicate the maximum and minimum principal stresses. The orientation of these principal planes, denoted by θ_p, is half the angle between a line from O to point X (the maximum principal stress) and the line connecting points A and B. θ_p is the angle between the principal plane and the original coordinate system. The radius from O to the highest point of the circle represents the maximum shearing stress.

Mohr's circle offers vital insights into the behavior of materials, highlighting the magnitudes and orientations of principal and shearing stresses, which are essential for structural design and analyzing material failure.