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Mohr's Circle Calculator

Resolve center, radius, principal stresses, transformed stress, and maximum in-plane shear.

Mohr's Circle inputs

Result section

warningPlane-stress Mohr circle

Maximum principal stress

112.426 MPa

Secondary results

Circle center: 70 MPa
Circle radius: 42.4264 MPa
Minimum in-plane principal stress: 27.5736 MPa
Maximum in-plane shear: 42.4264 MPa
Principal angle: 22.5 deg
Transformed sigma x prime: 110.981 MPa
Transformed shear: -10.9808 MPa

Unit conversion summary

Unit-normalization summary: 7.00000e+7 Pa; 4.24264e+7 Pa

Warnings

Calculations are performed locally in your browser and are not submitted to a third-party calculation API.
Positive normal stress is tension. Negative normal stress is compression. Positive shear follows the displayed stress-element convention.

Assumptions

This is a two-dimensional plane-stress Mohr circle only.
The displayed shear-axis convention matches the stress-transformation formulas.

Calculation steps

Calculate center7.00000e+7 Pa
Calculate radius4.24264e+7 Pa
Transform stress1.10981e+8 Pa, -1.09808e+7 Pa

Mohr's circle for plane stress

C=(sx+sy)/2; R=sqrt(((sx-sy)/2)^2+txy^2)

C = (sigma_x + sigma_y)/2, R = sqrt(((sigma_x - sigma_y)/2)^2 + tau_xy^2).

C = 7.00000e+7 Pa

R = 4.24264e+7 Pa

theta_p = 22.5 deg

Technical visual

Engineering interpretation

Mohr's circle maps plane-stress transformation, principal stresses, maximum in-plane shear, and doubled angle relationship.

What is Mohr's circle and how does this Mohr's Circle Calculator work?

Mohr's circle is a graphical transformation of a two-dimensional stress state. Its centre and radius show the principal stresses, maximum in-plane shear stress, and the relationship between physical-plane rotation and angle on the stress circle.

Mohr's Circle Calculator applies this concept to its defined inputs and workflow. It is a 2D plane-stress visualizer and calculator. Its governing relationship or workflow is stated as follows: Mohr's circle uses the normal-stress average and in-plane shear radius.

The calculator evaluates Stress input mode, Stress unit, Material preset, and Display precision and returns Mohr's Circle, base-unit normalization, formula substitution, and calculation steps using the stated method. The selected units and sign or reference conventions must describe one consistent physical case. Users should interpret the output within the assumptions shown on the page and repeat the check with trusted reference values when the result affects engineering or research decisions. This is a plane-stress Mohr circle only; it is not a full 3D Mohr-circle envelope.

How to use the mohr's circle calculator

Select the mode that matches the stress, material, section, or thermal data you already have.
Enter values with units; the calculator normalizes inputs to SI internally.
Review the formulas, normalized values, assumptions, and warnings before transferring the result to another check.
Use the transformed stress and principal-angle outputs to check hand calculations or interpret local 2D stress states.

Input explanations

Mohr's circle mode is limited to plane stress.
The entered angle is the physical-plane rotation angle; Mohr's circle uses twice that angle internally.
Circle center is the average normal stress and circle radius is the maximum in-plane shear stress.

Notation

Formula notation
Symbol	Meaning	Unit
C	Mohr circle center	Pa
R	Mohr circle radius	Pa
theta	Physical plane angle	deg or rad
2 theta	Mohr circle angle	deg or rad

Worked examples

Plane stress: sigma x = 100 MPa, sigma y = 40 MPa, tau xy = 30 MPa gives center = 70 MPa and radius about 42.426 MPa.

A physical angle of 30 deg corresponds to a 60 deg rotation on Mohr's circle.

At the principal angle, transformed shear stress is zero within numerical tolerance.

Cross-tool workflow

The Mohr's Circle Calculator is a 2D plane-stress visualizer and calculator. It reports circle center, radius, principal angle, transformed stress, and maximum in-plane shear.

Use principal stresses in Factor of Safety Calculator

Common mistakes

Forgetting that Mohr's circle angle is twice the physical material-plane angle.
Mixing shear sign conventions between textbooks and FEA postprocessors.
Using a 2D Mohr circle to represent a full 3D stress state.
Treating the drawing as a mesh-independent FEA validation.

Limitations

This is a plane-stress Mohr circle only; it is not a full 3D Mohr-circle envelope.
No graphical export or solver-result import is included.
It does not check material failure, fatigue, or design-code compliance.

References

Roark's Formulas: Young, Budynas, and Sadegh, Roark's Formulas for Stress and Strain.
Shigley's Mechanical Engineering Design: Budynas and Nisbett, Shigley's Mechanical Engineering Design, McGraw-Hill.
Mechanics of Materials: Hibbeler, Mechanics of Materials, Pearson.
Theory of Elasticity: Timoshenko and Goodier, Theory of Elasticity, McGraw-Hill.

What does Mohr's circle show?

It shows how normal and shear stress on a plane vary with rotation for a 2D plane-stress state.

Why is the angle doubled?

Stress transformation equations contain sin 2 theta and cos 2 theta, so Mohr's circle rotates through twice the physical angle.

What is the circle radius?

The radius equals the maximum in-plane shear stress.

Can this draw full 3D Mohr circles?

No. This tool is limited to the 2D plane-stress circle.

Can I use the output in factor-of-safety checks?

Yes, transfer the principal stresses when the 2D stress-state assumptions are valid.

Does this calculator prove that a part is safe?

No. It reports an educational preliminary stress or factor comparison. Real design decisions need applicable standards, validated models, material data, and engineering review.

Can these results replace commercial FEA software?

No. ScholarTool provides transparent hand-calculation support and postprocessing checks, not a replacement for validated FEA solvers.

Professional support from ScholarEase Consultancy Services LLP

Professional FEA, simulation setup, solid-mechanics calculations, technical documentation, and research support are available through the connected services website. These calculators remain free to use without purchasing services.

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Engineering disclaimer

This tool is intended for educational, estimation, and preliminary solid-mechanics support. Always verify critical engineering calculations with applicable standards, validated software, material data, and qualified professional judgment.

Need Professional Support?

ScholarTool is powered by ScholarEase Consultancy Services LLP. Professional engineering, simulation, research, and technical documentation support is available through the connected services website.

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Mohr's Circle inputs

Result section

warningPlane-stress Mohr circle

Maximum principal stress

112.426 MPa

Secondary results

Circle center: 70 MPa
Circle radius: 42.4264 MPa
Minimum in-plane principal stress: 27.5736 MPa
Maximum in-plane shear: 42.4264 MPa
Principal angle: 22.5 deg
Transformed sigma x prime: 110.981 MPa
Transformed shear: -10.9808 MPa

Unit conversion summary

Unit-normalization summary: 7.00000e+7 Pa; 4.24264e+7 Pa

Warnings

Calculations are performed locally in your browser and are not submitted to a third-party calculation API.
Positive normal stress is tension. Negative normal stress is compression. Positive shear follows the displayed stress-element convention.

Assumptions

This is a two-dimensional plane-stress Mohr circle only.
The displayed shear-axis convention matches the stress-transformation formulas.

Calculation steps

Calculate center7.00000e+7 Pa
Calculate radius4.24264e+7 Pa
Transform stress1.10981e+8 Pa, -1.09808e+7 Pa

Mohr's circle for plane stress

C=(sx+sy)/2; R=sqrt(((sx-sy)/2)^2+txy^2)

C = (sigma_x + sigma_y)/2, R = sqrt(((sigma_x - sigma_y)/2)^2 + tau_xy^2).

C = 7.00000e+7 Pa

R = 4.24264e+7 Pa

theta_p = 22.5 deg

What is Mohr's circle and how does this Mohr's Circle Calculator work?

How to use the mohr's circle calculator

Select the mode that matches the stress, material, section, or thermal data you already have.
Enter values with units; the calculator normalizes inputs to SI internally.
Review the formulas, normalized values, assumptions, and warnings before transferring the result to another check.
Use the transformed stress and principal-angle outputs to check hand calculations or interpret local 2D stress states.

Symbol

Meaning

Unit

Mohr circle center

Mohr circle radius

theta

Physical plane angle

deg or rad

2 theta

Mohr circle angle

deg or rad

References

Roark's Formulas: Young, Budynas, and Sadegh, Roark's Formulas for Stress and Strain.

Shigley's Mechanical Engineering Design: Budynas and Nisbett, Shigley's Mechanical Engineering Design, McGraw-Hill.

Mechanics of Materials: Hibbeler, Mechanics of Materials, Pearson.

Theory of Elasticity: Timoshenko and Goodier, Theory of Elasticity, McGraw-Hill.

Mohr's Circle inputs

Stress state

Result section

Secondary results

Unit conversion summary

Warnings

Assumptions

Calculation steps

Mohr's circle for plane stress

Technical visual

Engineering interpretation

What is Mohr's circle and how does this Mohr's Circle Calculator work?

How to use the mohr's circle calculator

Input explanations

Notation

Worked examples

Cross-tool workflow

Common mistakes

Limitations

References

Related tools

Professional support from ScholarEase Consultancy Services LLP

Print summary

Engineering disclaimer

Need Professional Support?

Mohr's Circle inputs

Stress state

Result section

Secondary results

Unit conversion summary

Warnings

Assumptions

Calculation steps

Mohr's circle for plane stress

Technical visual

Engineering interpretation

What is Mohr's circle and how does this Mohr's Circle Calculator work?

How to use the mohr's circle calculator

Input explanations

Notation

Worked examples

Cross-tool workflow

Common mistakes

Limitations

References

Related tools

Professional support from ScholarEase Consultancy Services LLP

Print summary

Engineering disclaimer

Need Professional Support?