Mohr's Circle Calculator: Complete Guide to Stress Analysis

✓ Verified Content: All equations, formulas, and reference data in this simulation have been verified by the Simulations4All engineering team against authoritative sources including NIST, peer-reviewed publications, and standard engineering references. See verification log

Any structural engineer will tell you: the stress you measure depends entirely on how you orient your measuring device. Apply tension in the x-direction, and a strain gauge at 45 degrees sees pure shear. This seemingly paradoxical result tripped up early engineers until Christian Otto Mohr, a German civil engineer working in the 1880s, devised an elegant graphical solution that transforms abstract trigonometry into visual intuition.

The load has to go somewhere, and as it travels through material, it creates stress states that look completely different depending on your viewing angle. Materials don't fail because of "x-direction stress" or "y-direction stress"; they fail because of the maximum stress, regardless of orientation. This is where structures fail: at the critical planes where principal stresses reach their peak values, often at unexpected angles that simple calculations miss.

This interactive Mohr's Circle calculator reveals those critical orientations instantly. Whether you're designing pressure vessels, analyzing shaft failures, or preparing for engineering exams, Mohr's Circle transforms the complex stress transformation equations into a single circle that shows you exactly where maximum stresses occur, and why structures fail where they do.

How to Use This Simulation

This interactive Mohr's Circle analyzer transforms stress states between arbitrary orientations and identifies principal stresses and maximum shear. The load has to go somewhere, and this tool shows you exactly how the same stress state looks completely different depending on the angle of your coordinate system.

Main Controls

Input Parameters

Results Display

The simulation presents Mohr's Circle with synchronized outputs:

• Mohr's Circle Diagram: The circle itself, with the current stress point highlighted and moving as you rotate. Any structural engineer will tell you that reading this circle quickly becomes second nature with practice.
• Principal Stresses σ1, σ2: Maximum and minimum normal stress values, found where the circle crosses the horizontal axis.
• Principal Angle θp: The rotation needed to reach principal orientation (where shear is zero).
• Maximum Shear τmax: The circle radius, representing maximum shear stress regardless of orientation.
• Transformed Stresses: Current σx', σy', and τx'y' values at the selected rotation angle.
• Stress Element: Animated 2D element showing stresses on each face as you rotate.

Tips for Exploration

1. Watch the 2θ relationship. Rotate the element by 15° and observe the stress point move 30° around the circle. This doubling is fundamental to Mohr's Circle geometry. Any structural engineer will tell you that this is why principal stresses occur at θp, but maximum shear occurs at θp ± 45°.

2. Start with pure tension. Set σx = 100 MPa, σy = 0, τxy = 0. The circle passes through the origin and (100, 0). Now add σy = 50 MPa and watch the circle shift right. The load has to go somewhere, and adding a second normal stress moves the center of the circle.

3. Add shear to normal stress. Start with biaxial tension (σx = σy = 100 MPa), creating a point-circle with zero radius. Now increase τxy and watch the circle grow. This is where structures fail: shear stress always creates a larger maximum stress than the applied normal stresses alone suggest.

4. Find pure shear's hidden tension. Set σx = σy = 0 and τxy = 100 MPa. The principal stresses are ±100 MPa at 45° orientations. Experienced designers know that this is why torsion failures often occur along helical lines: the principal tension from pure shear acts at 45° to the shaft axis.

5. Explore pressure vessel stresses. Click the pressure vessel preset (biaxial tension with σy = 0.5σx) and confirm that principal stresses are simply σx and σy when there's no shear. Then add a small shear and watch the principal directions rotate slightly, increasing the maximum stress beyond what simple hoop stress formulas predict.

Why Learn Mohr's Circle? Benefits for Engineers and Students

Understanding Mohr's Circle is not just an academic exercise - it is a fundamental skill that separates competent engineers from those who merely memorize formulas. Here is why this tool matters:

For Engineering Students

1. Visual Understanding of Abstract Concepts Stress transformation involves trigonometric equations that can feel disconnected from physical reality. Mohr's Circle provides a geometric interpretation that makes these equations intuitive. When you rotate a stress element by angle theta, you can literally see the stress point move around the circle by 2 theta. This visual connection helps cement understanding in ways that equations alone cannot.

2. Exam Performance Mohr's Circle problems appear consistently on FE (Fundamentals of Engineering) exams, PE (Professional Engineer) exams, and university finals in Mechanics of Materials, Solid Mechanics, and Structural Analysis courses. Students who master the graphical method solve problems faster and with fewer errors than those who rely solely on memorized formulas.

3. Foundation for Advanced Topics Mohr's Circle concepts extend directly to strain analysis, 3D stress states, yield criteria (von Mises, Tresca), and failure theories. Without solid grounding in 2D stress transformation, these advanced topics become significantly harder to grasp.

For Practicing Engineers

1. Quick Sanity Checks Even with FEA software doing the heavy lifting, engineers need to verify results make physical sense. A quick Mohr's Circle sketch can confirm whether computed principal stresses are reasonable and catch simulation errors before they become costly mistakes.

2. Design Intuition Experienced engineers develop intuition about how loads create stresses. Mohr's Circle builds this intuition by showing how combined loading states interact. When you understand that adding shear always increases the circle radius (and thus the maximum shear stress), you make better design decisions instinctively.

3. Communication with Colleagues Mohr's Circle provides a universal visual language for discussing stress states. Sketching a quick circle during design reviews communicates complex stress conditions more effectively than reciting numbers.

Practical Applications: Where Engineers Use Mohr's Circle Daily

Pressure Vessel Design

Chemical plants, refineries, and power stations rely on pressure vessels operating at high pressures and temperatures. Engineers use Mohr's Circle to:

• Determine weld stresses where longitudinal and circumferential stresses meet
• Calculate maximum shear stress for fatigue analysis at nozzle connections
• Verify that combined pressure and thermal stresses stay within allowable limits

The hoop stress in a cylindrical vessel is twice the axial stress - Mohr's Circle instantly shows the resulting principal stresses and maximum shear.

Rotating Machinery

Shafts in turbines, pumps, and motors experience simultaneous bending and torsion. At any point on the shaft surface:

• Bending creates normal stress varying from tension to compression
• Torsion creates shear stress (see our Torsion of Shafts calculator)
• Mohr's Circle combines these to find the critical principal stresses for fatigue analysis

Without this analysis, shafts would either fail prematurely or be overdesigned and wasteful.

Aircraft Structures

Modern aircraft experience complex loading: cabin pressurization, wing bending, fuselage torsion, and landing loads all create multiaxial stress states. Stress engineers use Mohr's Circle concepts to:

• Analyze skin panels under combined tension and shear
• Design riveted and bonded joints
• Predict fatigue crack initiation sites

Civil and Structural Engineering

Bridges, buildings, and infrastructure face combined loading from gravity, wind, seismic events, and thermal expansion:

• Beam webs experience combined bending and shear stresses
• Connection plates have complex stress fields
• Concrete structures need principal stress analysis for crack prediction

Geotechnical Engineering

The Mohr-Coulomb failure criterion - the foundation of soil mechanics - is literally a failure envelope drawn on Mohr's Circle. Geotechnical engineers use this daily for:

• Slope stability analysis
• Foundation bearing capacity
• Retaining wall design
• Tunnel stability assessment

How to Draw Mohr's Circle: Step-by-Step Guide

Follow these steps to construct Mohr's Circle for any plane stress state:

1. Step 1: Identify your stress state: σx, σy, and τxy from the given problem
2. Step 2: Calculate the center: C = (σx + σy) / 2
3. Step 3: Plot Point X at coordinates (σx, τxy) - representing the x-face
4. Step 4: Plot Point Y at coordinates (σy, -τxy) - representing the y-face
5. Step 5: Draw the circle with center C passing through points X and Y
6. Step 6: The radius R = √[((σx - σy)/2)² + τxy²]
7. Step 7: Principal stresses: σ₁ = C + R, σ₂ = C - R (where circle crosses σ-axis)

Understanding Principal Stresses with Mohr's Circle

Mohr's Circle principal stress analysis reveals the maximum and minimum normal stresses at a point. These principal stresses occur on planes where shear stress equals zero. For yield prediction using these values, see our Principal Stress & Von Mises Calculator.

Types of Stress States

Mohr's Circle Formulas: Quick Reference

Mohr's Circle Example Problems: Solved Step-by-Step

Example 1: Finding Principal Stresses

Problem: A stress element has σx = 80 MPa, σy = 20 MPa, τxy = 30 MPa. Find the principal stresses.

Solution using Mohr's Circle calculator:

1. Center C = (80 + 20) / 2 = 50 MPa
2. Radius R = √[(80-20)/2)² + 30²] = √[900 + 900] = 42.43 MPa
3. σ₁ = 50 + 42.43 = 92.43 MPa
4. σ₂ = 50 - 42.43 = 7.57 MPa
5. τmax = 42.43 MPa (occurs at 45° from principal planes)

Try this in the simulator above to verify!

Example 2: Pure Shear Analysis

Problem: A shaft under torsion has τxy = 60 MPa with no normal stresses. Find principal stresses.

Solution:

1. σx = 0, σy = 0, τxy = 60 MPa
2. Center C = 0 (circle centered at origin)
3. Radius R = 60 MPa
4. σ₁ = +60 MPa, σ₂ = -60 MPa (equal tension and compression at 45°)

Example 3: Pressure Vessel Stress

Problem: A thin-walled cylinder has hoop stress σh = 120 MPa and axial stress σa = 60 MPa. Find τmax.

Solution:

1. These ARE the principal stresses (no shear on these faces)
2. σ₁ = 120 MPa, σ₂ = 60 MPa
3. τmax = (120 - 60) / 2 = 30 MPa

3D Mohr's Circle vs 2D: When to Use Each Method

This calculator handles plane stress Mohr's Circle (2D). Here is when to use 2D vs 3D Mohr's Circle:

Important: For 3D stress states, you need three Mohr's Circles representing the three principal stress pairs (σ₁-σ₂, σ₂-σ₃, σ₁-σ₃). The absolute maximum shear stress is τabs = (σmax - σmin) / 2.

Learning Objectives

After mastering this Mohr's Circle stress analysis tool, you will be able to:

• Construct Mohr's Circle from any 2D stress state
• Calculate principal stresses σ₁ and σ₂ graphically and analytically
• Determine maximum shear stress and its orientation
• Understand why 2θ appears on the circle (vs θ in physical space)
• Apply stress transformation to real engineering problems
• Distinguish when to use 2D vs 3D analysis

Exploration Activities

Activity 1: Verify Example Problem 1

1. Enter σx = 80 MPa, σy = 20 MPa, τxy = 30 MPa in the calculator
2. Observe the Mohr's Circle and verify σ₁ ≈ 92.4 MPa, σ₂ ≈ 7.6 MPa
3. Rotate θ to find the principal angle (~22.5°)
4. Confirm shear stress approaches zero at principal orientation

Activity 2: Maximum Shear Stress Plane

1. Set σx = 100, σy = 0, τxy = 0 (uniaxial tension)
2. Find τmax on the circle (should be 50 MPa)
3. Rotate θ to 45° to confirm this is where τmax occurs
4. This explains why ductile materials fail at 45° in tension tests!

Activity 3: Hydrostatic Stress State

1. Set σx = 100, σy = 100, τxy = 0
2. The circle shrinks to a point - why?
3. All orientations have the same stress, no shear possible
4. This is why materials do not fail under hydrostatic compression

Activity 4: Sign Convention Practice

1. Set σx = 50, σy = -30, τxy = 40 (mixed tension/compression)
2. Note how the circle crosses both positive and negative σ regions
3. σ₁ is always the rightmost point, σ₂ the leftmost

Real-World Applications of Mohr's Circle

• Pressure Vessels: Cylindrical tanks have hoop stress = 2× axial stress; Mohr's Circle finds τmax for weld design
• Aircraft Structures: Combined pressurization and bending creates complex biaxial stress requiring transformation
• Bridge Design: Combined bending and shear at supports requires principal stress analysis
• Rotating Shafts: Combined torsion and bending creates 2D stress state at surface (use with Von Mises criterion for yield prediction)
• Geotechnical Engineering: Mohr-Coulomb failure criterion uses Mohr's Circle for soil stability
• Composite Materials: Stress transformation needed for off-axis ply analysis

Material Properties for Design

Challenge Questions

1. Conceptual: If σx = σy and τxy = 0, what does Mohr's Circle look like? What does this mean physically?
2. Calculation: Given σx = 100 MPa, σy = -50 MPa, τxy = 75 MPa, calculate σ₁, σ₂, and θp.
3. Analysis: A pressure vessel has σhoop = 150 MPa, σaxial = 75 MPa. What is the absolute maximum shear stress (considering 3D effects with σradial ≈ 0 at surface)?
4. Application: Why do brittle materials fracture perpendicular to the maximum principal stress, while ductile materials show 45° slip lines?
5. Design: You need τmax < 100 MPa for safety. If σx = 200 MPa and σy = 0, what is the maximum allowable τxy?

Common Mistakes to Avoid

Any structural engineer will tell you that Mohr's Circle seems simple until you make one of these common errors:

• 2θ Confusion: Remember that angles on Mohr's Circle are 2x the physical angle. This is where structures fail in exam problems, and occasionally in real designs.
• Sign Convention: Be consistent - some texts use CW shear as positive, others CCW. In the real world, nothing matters more than consistent conventions.
• Forgetting 3D Effects: Even in "2D" problems, σz = 0 creates a third principal stress affecting τabs
• Principal vs Maximum: Principal stresses can be negative (compression)
• Point vs Circle: Equal biaxial stress gives a point, not a circle - τmax = 0 in-plane. The load has to go somewhere, but when stresses are equal everywhere, shear disappears.

Frequently Asked Questions

What is the physical meaning of Mohr's Circle?

Mohr's Circle provides a geometric representation of stress transformation. Every point on the circle represents the stress state on a plane at some orientation. The x-coordinate gives normal stress, the y-coordinate gives shear stress. As you rotate the physical element, the stress point moves around the circle at twice the rotation rate [1].

Why are principal stresses important in design?

Principal stresses determine material failure. Ductile materials fail when shear stress exceeds yield (von Mises/Tresca criteria), while brittle materials fail when principal stress exceeds tensile strength. Knowing σ₁ and σ₂ allows engineers to predict if a component will fail under given loading [2].

How do I handle negative (compressive) stresses?

Negative stresses are plotted to the left of the origin on the σ-axis. The circle construction works the same way. For example, σx = -50 MPa, σy = -100 MPa gives a circle entirely in the compression region, but τmax calculation is unchanged [3].

When should I use 3D Mohr's Circle analysis?

Use 3D analysis when σz ≠ 0 (thick sections, triaxial loading). For plane stress (σz = 0), the absolute maximum shear stress τabs = max(|σ₁|, |σ₂|, |σ₁ - σ₂|)/2, which may be larger than the in-plane τmax when σ₁ and σ₂ have the same sign [4].

Can Mohr's Circle be used for strain analysis?

Yes, Mohr's Circle works for strain transformation by replacing σ with ε and τ with γ/2 (engineering shear strain divided by 2). Principal strains and maximum shear strain are found the same way [5].

References

1. MIT OpenCourseWare (Course 2.002): Comprehensive coverage of stress transformation and Mohr's Circle derivation. Available at: https://ocw.mit.edu/courses/2-002-mechanics-and-materials-ii-spring-2004/ (Creative Commons BY-NC-SA)

2. HyperPhysics: Stress and Strain: Clear explanations of stress concepts with interactive diagrams. Available at: http://hyperphysics.gsu.edu/hbase/permot3.html (Educational use permitted)

3. Engineering Toolbox: Stress, Strain and Young's Modulus: Reference data for material properties and stress analysis. Available at: https://www.engineeringtoolbox.com/stress-strain-d_950.html (Free educational use)

4. OpenStax University Physics: Chapter on stress, strain, and elastic deformation with worked examples. Available at: https://openstax.org/details/books/university-physics-volume-1 (Creative Commons BY)

5. NPTEL: Strength of Materials: Indian Institute of Technology video lectures on Mohr's Circle. Available at: https://nptel.ac.in/courses/112107146 (Free educational access)

6. eFunda: Mohr's Circle Calculator: Online reference for stress transformation equations. Available at: https://www.efunda.com/formulae/solid_mechanics/mat_mechanics/mohr_circle.cfm (Free access)

7. University of Cambridge DoITPoMS: Stress Analysis: Teaching and learning packages for stress concepts. Available at: https://www.doitpoms.ac.uk/tlplib/metal-forming-1/stress.php (Educational use)

8. MatWeb: Material Property Data: Database of material yield and ultimate strengths. Available at: https://www.matweb.com/ (Free basic access)

About the Data

The stress transformation equations used in this simulation are derived from equilibrium of an infinitesimal stress element, a fundamental result in continuum mechanics. The material property data in the reference table comes from MatWeb and Engineering Toolbox databases. Yield strengths are typical values for common alloys; actual values depend on heat treatment, processing, and manufacturer specifications. The calculator assumes linear elastic behavior and small deformations.

How to Cite

Simulations4All. (2025). Mohr's Circle Calculator: Complete Guide to Stress Analysis. Interactive Engineering Education Platform. Retrieved from https://simulations4all.com/simulations/mohrs-circle

For academic work, include the access date and note that this is an interactive educational simulation based on classical stress transformation theory from mechanics of materials.

Verification Log