Mohr's Circle and Principal Stresses — Full Method Guide

The centre C of the circle sits at the average of the two normal stresses: C = (σx + σy)/2. The radius R is the straight-line distance from the centre to the plotted stress point A = (σx, τxy) on the σ–τ plane: R = √[((σx − σy)/2)² + τxy²]. The principal stresses σ1 and σ2 are the rightmost and leftmost points of the circle (where shear is zero), separated by the full diameter 2R. The maximum in-plane shear stress τmax equals R, acting on planes rotated 45° from the principal planes.

The principal angle θp — the angle from the original x-axis to the σ1 direction, measured in the physical element — is half the angle swept on the circle from point A to the σ1 axis. The atan2 form handles all quadrants correctly and is what the calculator uses.

Once σ1 and σ2 are known, two standard criteria assess the margin against yielding. The Von Mises (distortion-energy) criterion computes an equivalent stress σVM from the principal stresses and compares it to the uniaxial yield strength Sy. For plane stress (no out-of-plane load, so σ3 = 0), the formula below applies. The Tresca (maximum-shear-stress) criterion uses the span between the largest and smallest principal stress across all three planes — including the out-of-plane zero. Tresca is always equal to or more conservative than Von Mises; Von Mises fits experimental yield data for ductile metals slightly better and permits up to ≈15% more capacity along the shear diagonal.

The safety factor in both cases is the ratio of the material's yield strength to the equivalent stress (Von Mises) or to the Tresca stress (2·τmax_abs). A safety factor above 1.0 means the stress state is inside the yield surface; below 1.0 means yielding is predicted. For plane stress with σ3 = 0, the Tresca criterion evaluates the absolute maximum shear across all three principal-stress pairs — not just the in-plane pair — which is why it can be more conservative when σ1 and σ2 have opposite signs.

Mohr's Circle also gives the normal and shear stresses on any plane inclined at angle θ to the x-axis. Rotating by θ on the element corresponds to rotating by 2θ on the circle. The transformed stresses follow the standard stress-transformation equations below. These are useful when you need to verify the stress on a specific weld plane, slip plane, or joint interface — not just the worst-case principal orientation.

The Mohr's Circle calculator computes the transformed stresses σx', σy', τx'y' for any angle θ you specify, and overlays the result on the circle diagram so you can read off the geometry directly. This is particularly useful for checking weld throat stresses or verifying the stress state at a critical section defined by the geometry rather than the principal direction.

When a component carries an out-of-plane stress — a thick-walled pressure vessel, a triaxially loaded weld, or any section where the through-thickness stress σz is not negligible — you must consider all three principal stresses. The general Von Mises equivalent uses all three principals: σVM = √(½·[(σ1−σ2)²+(σ2−σ3)²+(σ3−σ1)²]). The absolute maximum shear stress is half the span of the largest-to-smallest principal: τmax_abs = (σ1 − σ3)/2 after sorting σ1 ≥ σ2 ≥ σ3. Neglecting the third principal when it is tensile can be non-conservative: it enlarges the principal-stress span and raises the absolute maximum shear stress above the in-plane value.