23.4:
Mohr's Circle for Plane Stress
Mohr's circle is a visual representation of stress transformation on an element. The stress components in this element are plotted on a graph to create Mohr's circle for a square element experiencing plane stress.
If the shearing stress is positive, point A is plotted below the horizontal axis, and point B is plotted above. If it is negative, their positions are reversed.
The midpoint, O, of the line connecting points A and B lies on the horizontal axis and serves as the circle's center. A circle, drawn with O as its center and the line AB as its diameter, is called Mohr's circle.
The abscissae of points X and Y, where the circle intersects the horizontal axis, represent the maximum and minimum principal stresses, respectively.
The angle AOX equals twice the angle θp. The orientation, θp, of the principal plane corresponding to point X, can be obtained by halving the angle AOX measured on Mohr's circle.
The radius of Mohr's circle in the vertical direction corresponds to the magnitude of the maximum shearing stress.
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 (σx, -τxy), and the coordinates of point B are (σx, τxy). 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 θp, is half the angle between a line from O to point X (the maximum principal stress) and the line connecting points A and B. θp 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.