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

What this tool is not for

Key concepts this page covers

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:

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:

  1. Normalize inputs into a consistent internal unit system (for example, all lengths in meters, all forces in newtons), then convert back for display.
  2. Derive secondary parameters that are not explicitly entered (for example, effective areas, lever arms, eccentricities, or effective lengths). These are often where standards differ.
  3. 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.
  4. 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.
  5. 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.

  1. 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.
  2. 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.
  3. 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.
  4. Boundary test: test extreme-but-possible values to make sure the UI doesn't silently overflow, divide by zero, or return NaN/Infinity.
  5. 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.

Common pitfalls and how to avoid confusion

Data handling, privacy, and offline behavior

Steelcalculator.app is designed so that most calculations can run client-side. In a typical configuration:

If you are deploying this site, document the exact behavior in the Privacy Policy and ensure that any tracking complies with applicable privacy laws. For more context see /privacy and /terms.

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 ✓.

Related pages

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.

The site operator provides the content “as is” and “as available” without warranties of any kind. To the maximum extent permitted by law, the operator disclaims liability for any loss or damage arising from the use of, or reliance on, this page or any linked tools.