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Principal Stress Calculator

Resolve plane or 3D stress states into sorted principal stresses for FEA postprocessing checks.

Principal Stress inputs

Result section

warningPlane stress closed-form solution

Maximum principal stress

112.426 MPa

Secondary results

Principal stress sigma 1: 112.426 MPa
Principal stress sigma 2: 27.5736 MPa
Principal stress sigma 3: 2.98023e-14 MPa
Maximum shear stress: 56.2132 MPa
Mean stress: 46.6667 MPa
Principal angle: 22.5 deg

Unit conversion summary

Unit-normalization summary: 1.12426e+8 Pa; 2.75736e+7 Pa; 2.98023e-8 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

Stress tensor is symmetric.
Plane stress assumes sigma z, tau yz, and tau zx are zero.

Calculation steps

Resolve stress statePlane stress closed-form solution
Sort principal stresses1.12426e+8 Pa, 2.75736e+7 Pa, 2.98023e-8 Pa
Calculate maximum shear5.62132e+7 Pa

Plane stress principal-stress formula

sigma_1, sigma_2, sigma_3 = eigenvalues(sigma)

sigma_avg plus or minus radius, with sigma3 = 0 for plane stress.

sigma1 = 1.12426e+8 Pa

sigma2 = 2.75736e+7 Pa

sigma3 = 2.98023e-8 Pa

Technical visual

Engineering interpretation

Principal stresses are the normal stresses on planes where shear stress is zero. They are sorted for comparison workflows.

What is principal stress and how does this Principal Stress Calculator work?

Principal stresses are the normal stresses acting on orientations where shear stress vanishes. They are the eigenvalues of the symmetric stress tensor and are commonly ordered as maximum, intermediate, and minimum values for stress-state interpretation, failure criteria, and comparison with solver output.

Principal Stress Calculator applies this concept to its defined inputs and workflow. It supports plane stress, full symmetric 3D stress tensors, and already solved principal-stress inputs. Its governing relationship or workflow is stated as follows: Principal stresses are the eigenvalues of the symmetric stress tensor.

The calculator evaluates Stress input mode, Stress unit, Material preset, and Display precision and returns Principal Stress, 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. The tool does not read solver result files or evaluate mesh convergence.

How to use the principal stress 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 transfer links when the sorted principal stresses should feed Von Mises, Tresca, or factor-of-safety checks.

Input explanations

Plane stress uses sigma x, sigma y, and tau xy with sigma z, tau yz, and tau zx equal to zero.
Three-dimensional mode accepts a symmetric Cauchy stress tensor.
Principal-stress mode accepts already solved sigma 1, sigma 2, and sigma 3 values.
Positive normal stress is tension; negative normal stress is compression.

Notation

Formula notation
Symbol	Meaning	Unit
sigma x, sigma y, sigma z	Normal stress components	Pa
tau xy, tau yz, tau zx	Shear stress components	Pa
sigma 1, sigma 2, sigma 3	Sorted principal stresses	Pa
tau max	Maximum shear stress	Pa

Worked examples

Plane stress: sigma x = 100 MPa, sigma y = 40 MPa, tau xy = 30 MPa gives in-plane principal stresses about 112.426 MPa and 27.574 MPa.

Pure shear: tau xy = 50 MPa gives principal stresses +50 MPa and -50 MPa.

Hydrostatic 3D stress: sigma x = sigma y = sigma z gives equal principal stresses and zero deviatoric equivalent stress.

Cross-tool workflow

The Principal Stress Calculator supports plane stress, full symmetric 3D stress tensors, and already solved principal-stress inputs. It reports sigma 1, sigma 2, sigma 3, maximum shear stress, and normalized SI values.

Use in Von Mises Stress Calculator Use in Tresca Stress Calculator

Common mistakes

Mixing sign conventions from different FEA postprocessors without checking tensor orientation.
Using displayed nodal extrapolated stresses as if they were fully validated design stresses.
Treating a scalar equivalent stress as a replacement for standards-based design review.
Ignoring stress concentrations, fatigue, temperature effects, welds, bolts, or load combinations.

Limitations

The tool does not read solver result files or evaluate mesh convergence.
Plane stress assumes out-of-plane normal and shear stresses are zero.
Material failure, fatigue, plasticity, buckling, and contact are outside this calculator.

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 are principal stresses?

Principal stresses are normal stresses on planes where the shear stress is zero.

How are plane-stress principal values calculated?

They use the average normal stress plus or minus the Mohr-circle radius, with sigma 3 set to zero for plane stress.

Does three-dimensional mode solve a full tensor?

Yes. It solves the eigenvalues of a symmetric 3x3 stress tensor.

Why are the stresses sorted?

Most downstream checks expect sigma 1 >= sigma 2 >= sigma 3.

Is maximum shear the same as Tresca equivalent stress?

Tresca equivalent stress is two times the maximum shear stress.

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.

Visit ScholarEase

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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Principal Stress inputs

Result section

warningPlane stress closed-form solution

Maximum principal stress

112.426 MPa

Secondary results

Principal stress sigma 1: 112.426 MPa
Principal stress sigma 2: 27.5736 MPa
Principal stress sigma 3: 2.98023e-14 MPa
Maximum shear stress: 56.2132 MPa
Mean stress: 46.6667 MPa
Principal angle: 22.5 deg

Unit conversion summary

Unit-normalization summary: 1.12426e+8 Pa; 2.75736e+7 Pa; 2.98023e-8 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

Stress tensor is symmetric.
Plane stress assumes sigma z, tau yz, and tau zx are zero.

Calculation steps

Resolve stress statePlane stress closed-form solution
Sort principal stresses1.12426e+8 Pa, 2.75736e+7 Pa, 2.98023e-8 Pa
Calculate maximum shear5.62132e+7 Pa

Plane stress principal-stress formula

sigma_1, sigma_2, sigma_3 = eigenvalues(sigma)

sigma_avg plus or minus radius, with sigma3 = 0 for plane stress.

sigma1 = 1.12426e+8 Pa

sigma2 = 2.75736e+7 Pa

sigma3 = 2.98023e-8 Pa

What is principal stress and how does this Principal Stress Calculator work?

How to use the principal stress 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 transfer links when the sorted principal stresses should feed Von Mises, Tresca, or factor-of-safety checks.

Input explanations

Plane stress uses sigma x, sigma y, and tau xy with sigma z, tau yz, and tau zx equal to zero.
Three-dimensional mode accepts a symmetric Cauchy stress tensor.
Principal-stress mode accepts already solved sigma 1, sigma 2, and sigma 3 values.
Positive normal stress is tension; negative normal stress is compression.

Symbol

Meaning

Unit

sigma x, sigma y, sigma z

Normal stress components

tau xy, tau yz, tau zx

Shear stress components

sigma 1, sigma 2, sigma 3

Sorted principal stresses

tau max

Maximum shear stress

Worked examples

Plane stress: sigma x = 100 MPa, sigma y = 40 MPa, tau xy = 30 MPa gives in-plane principal stresses about 112.426 MPa and 27.574 MPa.

Pure shear: tau xy = 50 MPa gives principal stresses +50 MPa and -50 MPa.

Hydrostatic 3D stress: sigma x = sigma y = sigma z gives equal principal stresses and zero deviatoric equivalent stress.

Common mistakes

Mixing sign conventions from different FEA postprocessors without checking tensor orientation.

Using displayed nodal extrapolated stresses as if they were fully validated design stresses.

Treating a scalar equivalent stress as a replacement for standards-based design review.

Ignoring stress concentrations, fatigue, temperature effects, welds, bolts, or load combinations.

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.

Principal Stress inputs

Stress state

Result section

Secondary results

Unit conversion summary

Warnings

Assumptions

Calculation steps

Plane stress principal-stress formula

Technical visual

Engineering interpretation

What is principal stress and how does this Principal Stress Calculator work?

How to use the principal stress 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?

Principal Stress inputs

Stress state

Result section

Secondary results

Unit conversion summary

Warnings

Assumptions

Calculation steps

Plane stress principal-stress formula

Technical visual

Engineering interpretation

What is principal stress and how does this Principal Stress Calculator work?

How to use the principal stress 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?