Mohr Circle

Mohr’s Circle is a graphical method for analyzing a 2D plane-stress state at a point, converting the normal and shear stress transformation equations into a circle in σ-τ space. For stresses σx, σy, τxy, the center is average normal stress and the radius is maximum in-plane shear stress; it directly gives principal stresses and their orientation.

Mohr’s Circle comes into play whenever a part, fixture, spindle interface, bond line, or fastener group sees combined axial stress and shear; it reveals the worst-case normal and shear stress on any rotated plane. In CNC workholding, it helps determine whether a clamp layout pushes a thin wall, extrusion, or cast housing toward yielding or cracking at a sharp internal corner. In millwork, it checks how screw patterns, dowel joints, and laminate interfaces handle combined pull-out and racking loads, since shear-dominated failure initiates along the plane of largest shear. In tolerance engineering, it clarifies how an oblique load path or misaligned locator redirects stress relative to grain or toolpath-induced residual stress, because peak stress rarely aligns with the applied load. The practical value is not drawing the circle but quickly identifying principal directions and maximum shear planes to guide clamp placement, rib orientation, fillet sizing, grain direction, and stress-relief decisions.