Mohr's Circle — Stress Analysis

Mohr's Circle — Stress Analysis and Transformation

Mohr's Circle is one of the most important graphical tools in mechanics of materials. Developed by Christian Otto Mohr in 1882, it provides a graphical method for determining principal stresses, maximum shear stress, and stress transformation at a point under plane stress. By plotting normal stress (σ) on the horizontal axis and shear stress (τ) on the vertical axis, engineers can instantly visualise how stresses transform as the orientation of the plane changes. This simulator lets you explore Mohr's Circle interactively — dragging Point X on the σ–τ plane, animating the rotation angle θ, and watching the stress element and the corresponding point on the circle update simultaneously.

Understanding Plane Stress and the Stress Element

In plane stress analysis, we consider a thin element where all stresses act in one plane. The state of stress at a point is defined by three components: the normal stress σx acting in the x-direction, the normal stress σy acting in the y-direction, and the shear stress τxy acting on the x- and y-faces. A positive normal stress indicates tension (pulling the element apart), while a negative value indicates compression. The stress element is a small square drawn at the material point showing all these stress components with arrows on each face. When the element is rotated by an angle θ, the stress components transform according to the stress transformation equations, and Mohr's Circle provides a graphical representation of these equations.

Constructing Mohr's Circle

To construct Mohr's Circle: (1) Plot point X = (σx, τxy) and point Y = (σy, −τxy). (2) Connect X and Y with a straight line — this is the diameter of the circle. (3) The centre of the circle is at (σavg, 0) where σavg = (σx + σy) / 2. (4) The radius of the circle is R = √(((σx − σy) / 2)² + τxy²). The rightmost point on the σ-axis gives the maximum principal stress σ1 = σavg + R, and the leftmost gives the minimum principal stress σ2 = σavg − R. The top and bottom of the circle give the maximum shear stress τmax = R.

Stress Transformation Equations

The transformed normal stress on an inclined plane at angle θ is given by σn = σavg + R · cos(2θ − 2θp), and the transformed shear stress is τn = R · sin(2θ − 2θp). As θ varies from 0° to 180°, the point on the circle travels a full 360°. This 2:1 relationship between the rotation on the physical element and the angle on Mohr's Circle is a fundamental property — press Play in the simulator to watch it animate.

Applications in Engineering Design

Mohr's Circle is widely used in structural, mechanical, and aerospace engineering for failure analysis and design. The Von Mises equivalent stress σv = √(σ1² − σ1·σ2 + σ2²) is derived from principal stresses obtained via Mohr's Circle and is compared against the material yield strength to determine whether yielding occurs. Engineers use Mohr's Circle in pressure vessels, shafts under combined loading, welded joints, and composite materials. The Show-Calculations modal in this simulator walks through each step in classical mathematical notation.

Key Formulas at a Glance

For quick reference, the essential Mohr's Circle formulas are: Centre: σavg = (σx + σy) / 2. Radius: R = √(((σx − σy) / 2)² + τxy²). Principal stresses: σ1 = σavg + R, σ2 = σavg − R. Maximum shear: τmax = R = (σ1 − σ2) / 2. Principal angle: θp = ½ · arctan(2τxy / (σx − σy)). Transformation: σn = σavg + R·cos(2θ − 2θp), τn = R·sin(2θ − 2θp). Von Mises: σv = √(σ1² − σ1·σ2 + σ2²).

Failure Theories and Mohr's Circle

Mohr's Circle provides the principal stresses needed for all major failure theories. The Maximum Normal Stress Theory (Rankine) predicts failure when σ1 exceeds the ultimate tensile strength — applicable to brittle materials. The Maximum Shear Stress Theory (Tresca) predicts yielding when τmax exceeds the shear yield strength — slightly conservative for ductile metals. The Distortion Energy Theory (Von Mises) is the most accurate for ductile materials. The Failure-theories learning panel below applies all three to the current stress state with a colour-coded safety summary.

Worked Example — A Plane Stress Element Solved End-to-End

Take an element with σ_x = 80 MPa, σ_y = −30 MPa, τ_xy = 40 MPa. Plug these into the simulator and walk the circle:

Read the simulator’s rotated-element overlay to verify visually: rotate the element by 18° and the shear traction disappears (only normal stresses remain); rotate it by 63° (= 18 + 45) and the shear traction reaches its maximum.

Reading the Circle — What Each Point Means Physically

The Mohr’s circle is more than a calculation aid; every point on it corresponds to a physical face of the same material element rotated by some angle:

• Centre C — the hydrostatic (volume-changing) component of the stress. Shifts the whole circle horizontally; does not affect failure of ductile materials, which respond to the deviatoric (distortion) component.
• Radius R — the maximum shear stress this state can produce on any rotated plane. R = 0 means the state is purely hydrostatic and no shear failure is possible (deep-water rock far below failure).
• Top of circle (above σ axis) — the plane on which the shear stress is maximum and positive (tending to rotate the element clockwise when viewed in standard convention).
• Rightmost point — the principal plane carrying σ₁; zero shear here, so this is the plane on which a brittle crack is most likely to open.
• Angle on the circle = 2× angle in physical space. If you rotate the physical element by 18°, the corresponding point moves 36° around the circle. This factor of two trips up almost every student; remembering it is the key insight.

When 2D Mohr’s Circle Is Not Enough — The 3D Extension

Plane stress assumes the out-of-plane direction carries zero stress. That works for thin sheets, beams in bending, and the surface of pressurised pipes — but breaks for thick pressure vessels, deeply embedded bolts, and any state where all three principal stresses are non-trivial. In 3D, three principal stresses σ₁ ≥ σ₂ ≥ σ₃ define three circles, one for each pair, plotted on the same σ−τ axes.

The true maximum shear stress is then:

τ_max,3D = (σ₁ − σ₃) / 2

Critical example: a thin-walled pressure vessel under internal pressure has in-plane principal stresses of hoop σ_h and axial σ_a = σ_h/2; both positive. The 2D Mohr’s circle gives τ_max,2D = (σ_h − σ_a)/2 = σ_h/4. But the third principal stress (radial through the wall) is σ₃ ≈ −p ≈ 0 (or slightly negative). The true τ_max,3D = (σ_h − 0)/2 = σ_h/2 — twice the 2D answer. A vessel designed on 2D Mohr’s circle alone would be undersized for shear-controlled (Tresca) failure.

A Real Case — Diagnosing a Fillet-Weld Crack with Mohr’s Circle

A fillet weld attaching a bracket to a column transfers a vertical load 80 mm out from the column face. Hand-calculate the stress state at the weld root for an applied load of 12 kN on a 6 mm-throat, 100 mm-long weld:

• Direct shear from the load: τ = F / (throat × length) = 12000/(6·100) = 20 MPa.
• Bending stress from the 80 mm eccentricity: M = 12·0.08 = 0.96 kN·m. Z (weld) ≈ 6·100²/6 = 10000 mm³. σ = M/Z = 0.96×10⁶/10000 = 96 MPa (tensile at the top of the weld).
• Plug σ_x = 96, σ_y = 0, τ_xy = 20 into the simulator: σ₁ = 100.1 MPa, σ₂ = −4.1 MPa, τ_max = 52.1 MPa, principal angle θ_p = 11.3° from the weld axis.
• Crack orientation: brittle weld cracks open perpendicular to σ₁. The simulator predicts a crack starting at the weld root and propagating at 11.3° to the column face — matching the typical 10−15° angle observed in failed bracket welds. A picture of the cracked weld would show this slant tellingly.

Selected References

• Hibbeler, R. C. — Mechanics of Materials, 10th ed., Pearson, Chapter 9 (Stress Transformation). The canonical undergraduate reference.
• Boresi, A. P., Schmidt, R. J. — Advanced Mechanics of Materials, 6th ed., Wiley, for the 3D extension.
• ASME BPVC Section VIII Division 2 — Alternative Rules for Construction of Pressure Vessels, which uses Mohr’s 3D principal stresses in its stress-intensity assessment.
• Eurocode 3 (EN 1993-1-1) — the European steel-structures code, which references Tresca and von Mises criteria computed from Mohr’s principal stresses for weld and bolt verification.

Mohr’s Circle Formulas — 2D Stress Transformation

Explore Related Simulators

If you found this Mohr’s Circle simulator helpful, explore our Thin-Walled Pressure Vessel simulator, Stress–Strain Curve simulator, Beam Bending simulator, and Shaft Torsion simulator for more hands-on practice.