Mohr's Circle Calculator

How to Use This Calculator

1. Enter σx, σy (normal stresses) and τxy (shear stress) in MPa.
2. Results show principal stresses σ₁, σ₂, maximum shear τmax, and principal angle θp.
3. Use Failure Criteria tab with yield strength to check Tresca and von Mises safety.
4. Use Stress Transformation tab to find stresses on any rotated plane.
5. Professional mode adds stress invariants and safety factor.

Formula

Example

Frequently Asked Questions

• What is Mohr's Circle and what does it represent? ▼
  Mohr's Circle is a graphical and analytical tool developed by German engineer Otto Mohr in 1882 to visualize the stress state at a point in a material. When you have normal stresses σx and σy acting on two perpendicular faces, plus a shear stress τxy, the circle shows how these stresses transform as you rotate the plane of interest. The horizontal axis represents normal stress and the vertical axis represents shear stress. Every point on the circle corresponds to the stress state on a particular plane through the material. The center of the circle sits at the average normal stress ((σx + σy)/2), and the radius equals the square root of ((σx − σy)/2)² + τxy². This elegant geometry encodes all possible normal and shear stress combinations for any orientation at that point. Engineers use it to find the principal stresses (maximum and minimum normal stresses on planes with zero shear), the maximum shear stress, and the orientations of these critical planes.
• How are principal stresses different from normal stresses? ▼
  Normal stresses are the stresses acting perpendicular to a given face of a material element — their magnitude depends on which face (orientation) you choose to examine. Principal stresses σ₁ and σ₂ are special: they are the maximum and minimum normal stresses among all possible orientations, and they act on planes where the shear stress is exactly zero. The principal stresses are intrinsic to the stress state at a point — they do not change with coordinate system choice. Mathematically, σ₁ = ((σx + σy)/2) + R and σ₂ = ((σx + σy)/2) − R, where R is the Mohr's Circle radius. Principal stresses are critical for structural analysis because materials tend to fail along principal stress planes (brittle fracture in tension) or along maximum shear planes (ductile yielding). Knowing the principal stresses allows engineers to compare directly against material yield and ultimate strength values without worrying about orientation effects.
• Why is shear stress important in failure analysis? ▼
  Shear stress drives ductile failure — the slip of atomic planes past each other in metals, which is the physical mechanism of plastic yielding. The Tresca (maximum shear stress) criterion states that yielding begins when τmax = Sy/2, where Sy is the uniaxial yield strength. The von Mises criterion (also called distortion energy theory) states that yielding occurs when the von Mises stress σ_VM = √(σ₁² − σ₁σ₂ + σ₂²) equals Sy. Both criteria are derived from shear-based reasoning. In geology, shear stress drives fault slip: the Mohr-Coulomb failure criterion τ = c + σ·tan(φ) predicts when rock will fracture, where c is cohesion and φ is the internal friction angle. This is directly related to Mohr's Circle — fault slip occurs when the circle touches the Coulomb failure envelope. Understanding shear is also critical in geotechnical slope stability, soil bearing capacity, and pile design.
• What is the difference between plane stress and plane strain? ▼
  Plane stress assumes that all stresses act in two dimensions — specifically that σz = τxz = τyz = 0. This applies to thin plates, sheet metal, and surface coatings where the material is free to deform in the thickness direction. Plane strain assumes that strain in the z-direction is zero (εz = 0), which occurs in very thick or long structures where z-direction deformation is physically constrained — like a long dam, a buried pipe, or a long geological fault. The 2D Mohr's Circle analysis applies directly to both cases for the in-plane stresses, but the three-dimensional principal stresses differ: in plane stress the third principal stress σ₃ = 0; in plane strain σ₃ = ν(σ₁ + σ₂) where ν is Poisson's ratio. For fracture mechanics, the stress state (plane stress vs plane strain) determines fracture toughness: plane strain conditions give lower apparent toughness, which is why thick sections are more prone to brittle fracture — a key factor in pressure vessel and pipeline design codes.
• How is Mohr's Circle used in geology vs engineering? ▼
  In engineering, Mohr's Circle is used daily in structural design, machine component analysis, and materials testing. Engineers use it to determine whether applied loads will cause yielding (via Tresca or von Mises criteria), where fatigue cracks are likely to initiate (perpendicular to σ₁), and how to orient structural members for maximum efficiency. In geology and geomechanics, the same mathematics predicts fault orientation and slip conditions using the Mohr-Coulomb failure criterion. When tectonic stresses build until the Mohr Circle touches the Coulomb failure envelope (τ = c + σn·tan(φ)), a fault will slip — this is the physical basis of earthquake occurrence. Petroleum engineers use Mohr's Circle to predict borehole stability during drilling: knowing in-situ stress magnitudes and orientations allows wellbore orientation to be chosen to minimize breakout and collapse risk. Mining engineers apply it to pillar design and open-pit slope stability.

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