Mohr's Circle Calculator

Compute principal stresses and max shear using Mohr's circle relationships. Educational use only.

This page documents the scope, inputs, outputs, and computational approach of the Mohr's Circle Calculator on steelcalculator.app. The interactive calculator is designed to run in your browser for speed, but this documentation is written so the page remains useful (and indexable) even if JavaScript is not executed.

What this tool is for

Fast screening and iteration while you are exploring a design space.
Creating a repeatable calculation workflow that a reviewer can audit.
Learning the terminology and the “shape” of a typical check for Mohr's circle / plane stress transformation.

What this tool is not for

It is not a complete design package and does not replace the governing standard, project specification, or an engineer's judgment.
It is not a substitute for system-level checks (global stability, constructability, fatigue/seismic detailing, etc.).
It does not guarantee compliance with any specific standard, because compliance depends on configuration, edition, and jurisdictional requirements.

Key concepts this page covers

principal stresses
max shear
orientation

Inputs and naming conventions (high-level)

The calculator UI may present different groupings depending on the selected standard or mode, but inputs generally fall into these categories:

1) Actions / demands
Values that represent the loading on the component you are checking (forces, moments, pressures). Ensure you understand whether the workflow expects factored actions (strength) or service actions (serviceability), and keep that consistent across your verification.

2) Geometry and detailing parameters
Dimensions that define the physical configuration (spacing, thickness, eccentricity, end conditions). Many “unexpected” results come from geometry assumptions that are implicitly different from the real detail.

3) Material properties
Strength values (yield/ultimate), stiffness values (E), and any standard-specific parameters that affect resistance models.

4) Standard / method selection
The same physical configuration can be checked using different methods, with different reduction factors and definitions. A tool can only be unambiguous when you lock down the standard and edition you are matching.

The most common inputs for this tool include: σx, σy, τxy, sign convention.

Outputs you should expect

A well-behaved calculator output should be both summary-friendly and auditable:

A small set of headline results (pass/fail indicators, utilization ratios, controlling mode).
Intermediate values that let you reproduce at least one limit state independently (areas, lever arms, coefficients).
Clear units on every numeric value and a statement of the method used.

If the output is not auditable, treat it as a black box and do not rely on it for anything beyond quick intuition.

Computation approach (what happens under the hood)

This calculator is intended to implement a deterministic sequence of steps:

Normalize inputs into a consistent internal unit system (for example, all lengths in meters, all forces in newtons), then convert back for display.
Derive secondary parameters that are not explicitly entered (for example, effective areas, lever arms, eccentricities, or effective lengths). These are often where standards differ.
Evaluate candidate limit states relevant to Mohr's circle / plane stress transformation. Each limit state produces a resistance (or allowable) that can be compared to the demand.
Compute utilization as a dimensionless ratio (demand divided by resistance, or resistance divided by demand depending on convention). The controlling utilization is the maximum across the evaluated checks.
Render the report with intermediate values and the controlling failure mode, so a user can trace “why” the governing mode controls.

The implementation should also apply predictable rounding rules: keep higher precision internally, and only round for display. This is essential for stable regression tests.

Verification workflow (recommended QA steps)

This section is not a design instruction; it is a quality-assurance pattern for checking any engineering calculator.

Unit sanity check: confirm that each input has the unit you think it has. A common failure mode is mixing MPa and Pa, or mm and m.
Independent replication: pick one limit state (or one equation) and replicate it with an independent method (hand check, spreadsheet, or trusted reference). You are validating the method, not chasing an exact rounded match.
Sensitivity test: change one input in a direction that should clearly increase or decrease the capacity (for example, increase thickness) and confirm the output changes logically.
Boundary test: test extreme-but-possible values to make sure the UI doesn't silently overflow, divide by zero, or return NaN/Infinity.
Documentation: record the standard/mode, inputs, and the controlling output in a calculation note format so the result can be reviewed later.

For a structured approach, see: How to verify calculator results.

Stress Transformation Formulas

The stress transformation equations convert a known plane-stress state (σx, σy, τxy) to the stresses on a plane at any angle θ measured counter-clockwise from the x-axis.

Normal and shear stress on a rotated plane

σ(θ) = (σx + σy)/2 + (σx - σy)/2 · cos(2θ) + τxy · sin(2θ)
τ(θ) = -(σx - σy)/2 · sin(2θ) + τxy · cos(2θ)

These two equations define every point on Mohr's circle. The circle centre is at σavg = (σx + σy)/2 on the σ-axis, and the radius is R = √[((σx − σy)/2)² + τxy²].

Principal stresses

σ₁ = σavg + R  (maximum principal stress)
σ₂ = σavg - R  (minimum principal stress)

Principal angle: θp = (1/2) arctan(2τxy / (σx − σy))

At the principal planes (θ = θp and θ = θp + 90°), the shear stress τ = 0.

Maximum shear stress

τmax = R = √[((σx − σy)/2)² + τxy²]

Occurs at θs = θp + 45° (45° from the principal planes)
Normal stress at max-shear planes: σavg = (σx + σy)/2

Mohr's circle properties

Property	Formula	Notes
Centre	C = (σx + σy)/2	On the σ-axis
Radius	R = √[((σx − σy)/2)² + τxy²]	Distance from centre to any point
σ₁ (max principal)	C + R	Rightmost point on circle
σ₂ (min principal)	C - R	Leftmost point on circle
τmax (max shear)	R	Top and bottom of circle
Principal angle θp	(1/2) arctan(2τxy / (σx − σy))	Physical rotation angle
Angle on circle	2θ (double angle)	Circle angle = 2 × physical angle

Yield Criteria for Structural Steel

Von Mises (distortion energy) criterion

The von Mises equivalent stress for plane stress (σz = 0):

σvm = √[σx² - σx·σy + σy² + 3τxy²]

Yield check: σvm ≤ Fy

This is the most commonly used yield criterion for structural steel. AISC 360 uses von Mises implicitly in combined-stress interaction formulas. For pure shear (σx = σy = 0): σvm = √3 · τ = 1.732τ, so shear yield stress τy = Fy/√3 ≈ 0.577Fy.

Tresca (maximum shear stress) criterion

τmax = (σ₁ - σ₂)/2 ≤ Fy/2

Yield check: σ₁ - σ₂ ≤ Fy

Tresca is more conservative than von Mises. For pure shear: τy = Fy/2 = 0.50Fy, which is lower than the von Mises value of 0.577Fy. AISC 360 uses a shear yield of 0.6Fy (approximating von Mises).

Comparison of yield criteria

Criterion	Shear Yield	Equivalent Stress	Use In
Von Mises	Fy/√3 = 0.577Fy	σvm = √(σ₁² − σ₁σ₂ + σ₂²)	AISC 360, EN 1993
Tresca	Fy/2 = 0.50Fy	σ₁ − σ₂	Conservative checks
AISC (nominal)	0.6Fy	Direct shear formula	AISC Chapter G, J

Worked Example — Beam Web Stress State

Problem: A W18x46 beam (A992 steel, Fy = 50 ksi) carries a factored moment Mu = 250 kip-ft and factored shear Vu = 80 kips at a critical section near a support. Evaluate the principal stresses and maximum shear in the beam web at mid-depth.

Step 1 — Stress components at mid-depth of web

W18x46 properties: d = 18.06 in, tw = 0.305 in, Ix = 712 in⁴

Bending stress at mid-depth (y = 0 from neutral axis):
  σx = M·y/I = 0 (neutral axis)
  → Bending stress is zero at the neutral axis

Shear stress at neutral axis (maximum):
  For W-shapes, VQ/(I·tw) at the NA ≈ Vu/(d·tw) for preliminary estimate
  τxy ≈ Vu/(d·tw) = 80,000/(18.06 × 0.305) = 14,525 psi ≈ 14.5 ksi

Step 2 — Principal stresses

At the neutral axis: σx = 0, σy = 0, τxy = 14.5 ksi
σavg = 0
R = τxy = 14.5 ksi

σ₁ = 0 + 14.5 = 14.5 ksi (tension)
σ₂ = 0 - 14.5 = -14.5 ksi (compression)
τmax = 14.5 ksi

θp = (1/2) arctan(2×14.5 / 0) = (1/2)(90°) = 45°

The principal stresses at the neutral axis are equal and opposite (pure shear state), oriented at 45° to the beam axis.

Step 3 — Von Mises check

σvm = √[0 - 0 + 0 + 3×(14.5)²] = √(630.75) = 25.1 ksi
Fy = 50 ksi
σvm / Fy = 25.1/50 = 0.50 → OK (well below yield)

Step 4 — AISC shear check comparison

AISC nominal shear capacity (Chapter G):
  Cv = 1.0 (web not slender for W18x46)
  Vn = 0.6Fy·Aw = 0.6 × 50 × (18.06 × 0.305) = 165.2 kips
  φVn = 1.0 × 165.2 = 165.2 kips

  Vu / φVn = 80/165.2 = 0.48 → OK

The AISC shear check (0.6Fy × Aw) implicitly uses a von Mises shear yield of approximately 0.6Fy, which is consistent with the principal stress analysis.

Common pitfalls and how to avoid confusion

Sign convention: Tension is positive on Mohr's circle. A common error is plotting compressive stresses (negative σ) on the wrong side of the origin, which flips the entire circle. Always confirm: positive σ is to the right.
Double angle: Mohr's circle uses 2θ, so a 15° physical rotation corresponds to 30° on the circle. Forgetting the factor of 2 is the most common mistake in graphical construction.
Axis confusion: major/minor axis properties, sign conventions, and local coordinate systems can flip a result. Confirm whether the input stress state uses x-y or some other local convention before computing.
Plane stress assumption: Mohr's circle as presented here applies to plane stress (σz = 0). Thick pressure vessels, tunnel linings, and triaxial stress states require 3D Mohr's circles with all three principal stresses.
Yield criterion mismatch: AISC uses approximately von Mises (0.6Fy shear yield). If you check principal stress against Tresca (0.5Fy), you will get a more conservative but potentially misleading result.

Data handling, privacy, and offline behavior

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Frequently Asked Questions

What are principal stresses and how do you find them from Mohr's circle? Principal stresses are the normal stresses acting on planes where shear stress is zero — they represent the maximum and minimum normal stress values for a given stress state. On Mohr's circle, the principal stresses σ₁ and σ₂ are the two points where the circle intersects the horizontal axis (τ = 0). Algebraically, σ₁,₂ = (σx + σy)/2 ± √[((σx − σy)/2)² + τxy²]. The principal planes are oriented at an angle θp = (1/2)arctan(2τxy / (σx − σy)) from the original x-face.

What is maximum shear stress and where does it occur on Mohr's circle? The maximum in-plane shear stress is the radius of Mohr's circle: τmax = √[((σx − σy)/2)² + τxy²]. It occurs on planes rotated 45° from the principal planes. At the planes of maximum shear stress, there is also a normal stress equal to the average stress (σx + σy)/2 — these planes are not stress-free. For structural steel yield criteria under combined loading, τmax is compared to the shear yield stress τy = Fy/√3 (von Mises) or Fy/2 (Tresca).

How do you transform stresses to a plane rotated by angle θ? The stress transformation equations give the normal stress σθ and shear stress τθ on a plane at angle θ from the x-axis: σθ = (σx + σy)/2 + (σx − σy)/2·cos(2θ) + τxy·sin(2θ), and τθ = −(σx − σy)/2·sin(2θ) + τxy·cos(2θ). On Mohr's circle this corresponds to rotating the point representing the x-face by 2θ around the circle centre. Note that the angle on the circle is twice the physical rotation angle of the plane.

What is the difference between plane stress and plane strain? Plane stress assumes that all stress components in the out-of-plane direction are zero (σz = τxz = τyz = 0) — this applies to thin plates and webs where the through-thickness stress is negligible. Plane strain assumes that out-of-plane strains are zero (εz = 0), which applies to thick sections or long prismatic members where the geometry constrains expansion. In plane strain, an out-of-plane stress σz = ν(σx + σy) is induced even though no force is applied in that direction. Mohr's circle and stress transformation apply directly to both states for the in-plane components.

When is principal stress analysis needed in structural steel design? Principal stress analysis is most relevant when a steel element is subject to combined normal and shear stresses simultaneously — for example, in beam webs near supports where high shear and bending moment coexist (the critical section for combined stress), in gusset plates with eccentric load paths, or in welded connections where throat stresses combine normal and shear components. AISC 360 checks for combined loading in webs (Section G) and weld throat (Section J2) implicitly use principal stress concepts; Mohr's circle makes these limits intuitive and helps identify the orientation of critical planes for detailing.

What are the principal stresses for σx = 10 ksi, σy = 4 ksi, τxy = 3 ksi? Using the principal stress formula: σavg = (10 + 4)/2 = 7 ksi; R = √[((10 − 4)/2)² + 3²] = √[9 + 9] = √18 = 4.24 ksi. Therefore σ₁ = 7 + 4.24 = 11.24 ksi ≈ 11.2 ksi, σ₂ = 7 − 4.24 = 2.76 ksi ≈ 2.8 ksi, and τmax = 4.24 ksi. The principal plane angle is θp = (1/2) arctan(2 × 3 / (10 − 4)) = (1/2) arctan(1.0) = 22.5°. To verify: on the principal planes, the shear stress is zero and the Mohr's circle diameter equals σ₁ − σ₂ = 8.48 ksi = 2R ✓.

Disclaimer (educational use only)

This page is provided for general technical information and educational use only. It does not constitute professional engineering advice, a design service, or a substitute for an independent review by a qualified structural engineer. Any calculations, outputs, examples, and workflows discussed here are simplified descriptions intended to support understanding and preliminary estimation.

All real-world structural design depends on project-specific factors (loads, combinations, stability, detailing, fabrication, erection, tolerances, site conditions, and the governing standard and project specification). You are responsible for verifying inputs, validating results with an independent method, checking constructability and code compliance, and obtaining professional sign-off where required.

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