Stress Transformation and Mohr's Circle: Visualizing 2D Plane Stress

Overview

In this video, the concept of a 2D stress element under plane stress is used to illustrate how rotating the element changes the normal and shear stresses. The talk introduces the stress transformation equations, explains the significance of the rotation angle theta, and shows how maximum and minimum normal stresses occur on principal planes where shear is zero. The Moore's circle is presented as a visual tool to determine stresses for any orientation, including the relationship between circle radius, principal stresses, and the doubled angle between Mohr's circle and the physical element. A brief discussion on extending these ideas to three dimensions closes the piece.

Introduction to Stress Transformation and Mohr's Circle

The video begins by describing the stress element as a snapshot of normal and shear stresses at a point within a body and specifies a 2D plane-stress state for simplicity. It emphasizes that the choice of orientation for the stress element affects the measured components, even though the actual state in the body remains unchanged. The stress transformation equations are introduced as the method to compute new normal and shear stresses when the element is rotated by an angle theta, with theta defined as a counterclockwise rotation.

From a simple axial-loading example, the speaker shows how the stress components on the X and Y faces evolve as the element rotates. The key takeaway is that at some angles the normal stress attains extreme values while the shear stress vanishes. Those angles correspond to the principal planes, on which the normal stresses are called principal stresses, denoted sigma1 and sigma2. The angle that yields the principal stresses is thetaP, and, because the rotation of axes on the element corresponds to a rotation on the Mohr circle, this angle is doubled on the circle.

The Geometry of Mohr's Circle

The Moore circle is introduced as a graphical method to read off stresses for any orientation without directly solving the stress transformation equations. The horizontal axis represents normal stress, the vertical axis shear stress, with positive shear defined as tending to rotate the element counterclockwise. Points corresponding to the stresses on the X and Y faces are plotted, and the diameter joining these points defines the circle. The radius of the Mohr circle equals the maximum shear stress, a valuable visual cue for failure analysis and design decisions.

Reading Principal Stresses and Angles on the Circle

By locating where the circle intersects the horizontal axis, one can read off the principal stresses: they are the center coordinate on the X-axis plus or minus the circle radius. The video notes that the maximum and minimum principal stresses occur at those intersection points, and the corresponding principal angles on the original stress element relate to twice the physical rotation angle. This doubling principle is a recurring theme in Mohr circle analysis.

Extending to Three Dimensions

Although the presentation centers on two-dimensional plane stress, it briefly extends the ideas to three dimensions. In 3D, there are three principal stresses, ordered from largest to smallest, and Mohr's circle generalizes to a series of circles that enclose all possible normal and shear stress combinations for the 3D element. The visual concept remains the same: principal stresses occur on planes where shear vanishes, and their geometry can be explored graphically or analytically.

Recap and Practical Implications

The video concludes by highlighting the importance of principal stresses in predicting failure and guiding design choices. It recaps the transformation equations, the principal-plane concept, and the Mohr circle as an intuitive, practical tool for engineers working with stress analysis in planar bodies and welds. The final notes point to the 3D extension as a natural progression for more complex stress states.