STRESS ANALYSIS & MOHR'S CIRCLE

Stress analysis converts a multi-axial state of stress (σx, σy, τxy) into a single scalar that can be compared to a material yield strength. Two principal directions exist where shear vanishes and normal stress is maximum or minimum; these are the principal stresses σ1 and σ2. The maximum in-plane shear stress sits at 45° to those axes. Mohr's circle plots normal stress on the x-axis and shear stress on the y-axis, parameterizing the rotation of the stress tensor as you change reference frame.

• σ_avg = (σx + σy)/2
• R = sqrt(((σx - σy)/2)² + τxy²)
• σ1, σ2 = σ_avg ± R
• τ_max = R
• σ_vM = sqrt(σ1² - σ1·σ2 + σ2²)
• FoS = Sy / σ_vM

References & StandardsShigley 9.6, Norton 5.4, ASME BPVC VIII-1 UG-23

Use this module any time a part sees combined loading: a pressure-vessel wall (hoop + longitudinal + radial), a shaft under torque + bending, a weld throat under shear + tension, or a bolted bracket where eccentric load creates simultaneous tension and shear at the same fastener. The von Mises criterion gives a single number to compare to yield for ductile materials; the maximum-shear (Tresca) criterion is more conservative. For brittle materials, use the maximum principal-stress criterion instead.