Strength of Materials & Mechanics

Principal Stress Calculator — von Mises Stress & 2D/3D Yield Check

Calculate von Mises equivalent stress, three principal stresses and absolute maximum shear stress from plane stress, principal-stress inputs or a full 3D stress state, with optional yield utilization and safety-factor screening.

Unit-aware inputs Deterministic calculation Engineering interpretation
Calculation workspace

Enter the known values and review the calculated result

Deterministic calculation
01
Parameters

Input parameters

Use consistent values and select the intended engineering units.

Calculation mode

Plane stress components

Principal stresses

3D stress components

Material and yield screening [Optional]

02
Output

Results

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Calculation tools Save, export and manage your results

Save your calculation, export available results, or store a finalized engineering artifact in your account.

Engineering reference

Method, application and limitations

Review the calculation method, intended application and engineering assumptions before using the result in a design decision.

01
Method

Formula and calculation method

Stress tensor and principal stresses

For a three-dimensional stress state, the symmetric stress tensor is represented by:

σ = [[σx, τxy, τzx], [τxy, σy, τyz], [τzx, τyz, σz]]

The three principal stresses are obtained from this stress state and ordered as σ1 ≥ σ2 ≥ σ3.

von Mises equivalent stress from 3D components

σvm = √{0.5[(σx − σy)2 + (σy − σz)2 + (σz − σx)2] + 3(τxy2 + τyz2 + τzx2)}

where σvm is the von Mises equivalent stress, σx, σy and σz are normal stress components, and τxy, τyz and τzx are shear stress components. All quantities have units of stress.

Plane-stress mode

Plane stress sets σz = 0, τyz = 0 and τzx = 0. The von Mises equation therefore reduces to:

σvm = √(σx2 − σxσy + σy2 + 3τxy2)

The two in-plane principal-stress roots are:

σa,b = (σx + σy) / 2 ± √{[(σx − σy) / 2]2 + τxy2}

The two in-plane roots and the out-of-plane value of zero are then ordered to obtain the reported σ1, σ2 and σ3.

von Mises stress from principal stresses

When all three principal stresses are entered directly, the calculator first orders them from maximum to minimum and evaluates:

σvm = √{0.5[(σ1 − σ2)2 + (σ2 − σ3)2 + (σ3 − σ1)2]}

Absolute maximum shear stress

τmax = (σ1 − σ3) / 2

This is the absolute maximum shear stress associated with the complete principal-stress range, not only the in-plane shear stress.

Optional yield screening

If a yield strength Re is provided:

Yield utilization = 100 · σvm / Re [%]

n = Re / σvm

where n is the calculated yield safety factor. If a required safety factor nreq is also entered, the evaluated criterion is satisfied when n ≥ nreq.

When σvm is effectively zero, utilization is zero. The calculator treats an entered required-safety-factor criterion as satisfied, but does not report a finite numerical safety-factor value.

02
Application

When to use this calculator

When to use this calculator

Use this calculator when the stress state at a point is already known and you need to transform it into principal stresses, von Mises equivalent stress and absolute maximum shear stress.

Typical applications

  • Evaluate a plane-stress state from σx, σy and τxy.
  • Calculate von Mises stress and maximum shear stress when all three principal stresses are already known.
  • Evaluate a complete three-dimensional stress state from the six independent normal and shear stress components.
  • Post-process stress components obtained from analytical calculations, finite-element analysis or another engineering model.
  • Compare von Mises stress with an entered or material-derived yield strength.
  • Calculate yield utilization and yield safety factor for an initial distortion-energy yield screening.
  • Check an optional user-defined required safety factor.

Choose the input mode that matches the data you already have

  • Plane stress: enter σx, σy and τxy. Out-of-plane components are set to zero.
  • Principal stresses: enter all three principal stresses. The calculator orders them automatically before evaluating the results.
  • 3D stress components: enter the known components of the three-dimensional stress tensor.

When not to use this calculator

This calculator does not derive local stresses from forces, moments, torque, cross-section geometry or beam loading. Determine the relevant local stress components first, then evaluate that stress state here.

It also does not calculate principal directions, principal-plane angles, Mohr's circles, fatigue life, fracture, buckling, stress concentrations, nonlinear plastic response or compliance with a specific design standard.

03
Decision support

How to interpret the result

How to interpret the results

von Mises equivalent stress, σvm

σvm reduces the distortional part of a multiaxial stress state to one non-negative equivalent stress. When yield strength is supplied, this is the stress used by the calculator for the optional yield screening.

Larger differences between the normal stress components increase σvm. Increasing shear stress also increases σvm. Adding the same hydrostatic stress to all three normal components does not increase von Mises stress.

Principal stresses

  • σ1 — maximum principal stress.
  • σ2 — intermediate principal stress.
  • σ3 — minimum principal stress.

These are the normal stresses acting in principal directions where the corresponding shear stresses are zero. The calculator reports their values but does not calculate the principal directions themselves.

Absolute maximum shear stress, τmax

τmax = (σ1 − σ3) / 2 represents the largest shear-stress magnitude associated with the complete principal-stress range.

Yield utilization

When Re is provided, yield utilization is 100σvm / Re. A value of 100% corresponds to σvm = Re.

Utilization below 100% means the calculated von Mises stress is below the entered yield strength. This alone does not establish that a required design safety factor has been satisfied.

Yield safety factor

For nonzero von Mises stress, the reported safety factor is n = Re / σvm. A larger value indicates a larger ratio between the entered yield strength and the calculated equivalent stress.

Result status

  • Safe: a required safety factor was entered and the calculated criterion is satisfied. A zero von Mises stress also satisfies an entered required-factor criterion.
  • Unsafe: the calculated safety factor is below the entered required value, or — when no required factor is specified — the nonzero von Mises stress reaches or exceeds the entered yield strength.
  • Info: no yield strength was supplied, the stress state is hydrostatic with zero von Mises stress, or the von Mises stress is below yield strength but no required design safety factor was specified.
  • Invalid: the selected mode does not contain sufficient valid input data or the optional yield-screen inputs are inconsistent.

A satisfactory von Mises screening result confirms only the criterion evaluated here. It does not establish complete component safety or compliance with all relevant failure modes and design requirements.

04
Worked case

Calculation example

Example — plane stress with yield and required safety-factor check

Consider a point in a component with the following plane-stress state:

  • σx = 120 MPa
  • σy = 40 MPa
  • τxy = 30 MPa
  • Re = 250 MPa
  • nreq = 2.0

Plane-stress mode sets σz = τyz = τzx = 0.

1. Calculate the principal stresses

The average in-plane normal stress is:

σavg = (120 + 40) / 2 = 80 MPa

The Mohr-circle radius for the in-plane stress transformation is:

R = √{[(120 − 40) / 2]2 + 302} = 50 MPa

The two in-plane principal stresses are therefore:

80 + 50 = 130 MPa

80 − 50 = 30 MPa

Including the out-of-plane principal stress of 0 MPa and sorting the three values gives:

  • σ1 = 130 MPa
  • σ2 = 30 MPa
  • σ3 = 0 MPa

2. Calculate von Mises equivalent stress

σvm = √(1202 − 120 · 40 + 402 + 3 · 302)

σvm = 117.90 MPa

3. Calculate absolute maximum shear stress

τmax = (130 − 0) / 2 = 65.00 MPa

4. Calculate yield utilization

Utilization = 100 · 117.90 / 250 = 47.2%

5. Calculate yield safety factor

n = 250 / 117.90 = 2.12

6. Interpret the required safety-factor check

The calculated safety factor of 2.12 is greater than the entered required value of 2.0. The required safety-factor criterion is therefore satisfied for the von Mises yield screening performed by this calculator.

This result does not verify fatigue, fracture, buckling, stress concentrations or other possible failure modes.

05
Model boundaries

Assumptions and limitations

Assumptions and limitations

  • The calculation evaluates the stress state at a point. It does not determine the spatial stress distribution through an entire component.
  • The three-dimensional stress state is represented by a symmetric stress tensor using three normal stresses and three independent shear stresses.
  • Plane-stress mode assumes σz = 0, τyz = 0 and τzx = 0.
  • In plane-stress and 3D-component modes, stress components that are not entered are treated as zero. At least one applicable component must be provided.
  • Principal-stress mode requires all three principal stresses. The entered values are sorted internally before the results are evaluated.
  • Stress signs must be used consistently with the convention employed by the source calculation or analysis.
  • The calculator determines principal stress values but not principal directions, principal-plane angles or Mohr's-circle geometry.
  • The optional yield check is a von Mises distortion-energy screening based on the entered yield strength. It is primarily applicable to materials for which a von Mises yield criterion is appropriate.
  • A material-selected or manually entered yield strength must represent the actual material grade, condition, temperature and other relevant service conditions.
  • The calculator does not derive stress components from forces, moments, torque, pressure or cross-section geometry.
  • Local stress concentrations caused by holes, notches, fillets, keyways, weld details, contact or other geometric discontinuities are not calculated. Any such effects must already be represented in the entered stress state.
  • The calculation does not model plastic redistribution, strain hardening, nonlinear material behavior, anisotropic yielding or pressure-dependent failure criteria.
  • Fatigue, fracture mechanics, buckling, creep, wear, impact, instability and other failure mechanisms are outside this calculation.
  • The result does not demonstrate compliance with any particular design standard. Final design verification should follow the applicable code, material specification, loading requirements and required safety factors.
  • Safety-critical or complex components may require additional finite-element analysis, experimental validation or specialist engineering review.
06
Questions

Frequently asked questions

What does this principal stress calculator calculate?
It calculates the three ordered principal stresses, von Mises equivalent stress and absolute maximum shear stress from a plane-stress state, three entered principal stresses or a full 3D stress state. If yield strength is provided, it can also calculate yield utilization and yield safety factor.
How is von Mises stress calculated for plane stress?
For plane stress, the calculator sets σz, τyz and τzx to zero and uses σvm = √(σx² − σxσy + σy² + 3τxy²). The result is an equivalent stress used for distortion-energy yield screening.
Can I calculate von Mises stress directly from principal stresses?
Yes. Enter all three principal stresses. The calculator orders them as σ1 ≥ σ2 ≥ σ3 and calculates σvm = √{0.5[(σ1 − σ2)² + (σ2 − σ3)² + (σ3 − σ1)²]}.
Can this calculator evaluate a full 3D stress state?
Yes. The 3D mode accepts σx, σy, σz, τxy, τyz and τzx. The calculator determines the principal stresses, von Mises equivalent stress and absolute maximum shear stress from the resulting stress tensor. Components left blank in this mode are treated as zero, but at least one stress component must be entered.
What maximum shear stress does the calculator report?
It reports the absolute maximum shear stress based on the extreme principal stresses: τmax = (σ1 − σ3) / 2. For a plane-stress problem this can differ from the maximum shear stress calculated only within the xy plane because the zero out-of-plane principal stress is also included.
How is the yield safety factor calculated?
When yield strength Re is provided and von Mises stress is nonzero, the yield safety factor is n = Re / σvm. Yield utilization is 100σvm / Re. If a required safety factor is entered, the calculator compares the calculated value directly with that requirement.
What happens to von Mises stress under hydrostatic stress?
If all three principal stresses are equal, their differences are zero and the von Mises equivalent stress is zero. The principal stresses themselves can still be nonzero. This reflects the fact that the distortion-energy von Mises criterion is insensitive to a purely hydrostatic stress state.
Is the von Mises criterion appropriate for brittle materials?
The von Mises criterion is primarily used as an initial yield criterion for ductile materials. Brittle materials, anisotropic materials, pressure-sensitive materials and other special material systems may require different failure criteria and should not be accepted solely from a von Mises result.
Does this calculator determine stresses from beam loads, torque or section geometry?
No. This calculator requires a stress state as its input. It does not convert forces, bending moments, torque or cross-section dimensions into local stresses. Use the appropriate axial, bending, shear or torsion calculation first, then evaluate the resulting stress components here.
Does the calculator provide principal directions or a Mohr's circle?
No. It reports principal stress values, von Mises stress and absolute maximum shear stress, but it does not calculate principal directions, principal-plane angles or a graphical Mohr's circle.
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Engineering notice. Verify input data, assumptions, applicable standards and required safety factors before using results for design release or safety-critical decisions.