Tension/compression and bending stress add to a resultant normal stress, shear and torsion stress add to a resultant shear stress; only then are both combined into the equivalent stress via the GEH criterion.
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Tension/compression and bending stress add to a resultant normal stress, shear and torsion stress add to a resultant shear stress; only then are both combined into the equivalent stress via the GEH criterion.
σz,d=40 MPa, σb=60 MPa, τs=0, τt=30 MPa give σres=100 MPa, τres=30 MPa and σv=√(100²+3·30²)≈112.7 MPa.
Distortion-energy (von Mises) criterion for ductile materials under predominantly static load; fatigue and endurance strength require a separate check.
This calculator superimposes normal and shear stresses from tension/compression, bending, shear and torsion into resultant stresses, then forms the von Mises (distortion-energy, GEH) equivalent stress.
Tension/compression stress σz,d and bending stress σb are both normal stresses and add to σres = σz,d + σb. Shear stress τs and torsional stress τt are both shear stresses and add to τres = τs + τt. Only then are both combined via the GEH criterion into the equivalent stress σv = √(σres² + 3·τres²), which can be compared directly against the material's yield strength.
Picture two separate tills at a currency exchange: all normal-stress contributions are summed in one till, all shear-stress contributions in a second. Only the final balances of both tills are then combined via the GEH formula into a single risk value — normal and shear contributions must not be mixed beforehand.
σv = √[(σz,d+σb)² + 3·(τs+τt)²]
σv = √[(σz,d+σb)² + 3·(τs+τt)²]σz,d = √(σv²−3·τres²) − σbτt = √[(σv²−σres²)/3] − τs| Symbol / input | Meaning |
|---|---|
| Equivalent stress σv | Combined stress per GEH (von Mises); compare against the yield strength or the static safety factor calculator. |
| Tension/compression stress σz,d | Normal stress from a concentric tension or compression force, σz,d = F/A. |
| Bending stress σb | Normal stress from bending, σb = Mb/Wb. |
| Direct shear stress τs | Shear stress from a transverse (shear) force, τs = Fs/A. |
| Torsional shear stress τt | Shear stress from torsion, τt = T/Wt. |
σz,d is the normal stress from a tension or compression force (σz,d = F/A), σb the bending stress (σb = Mb/Wb), τs the shear stress from a transverse force (τs = Fs/A), and τt the torsional stress (τt = T/Wt). If a load type does not occur, set the corresponding value to 0.
First calculate each of the four component stresses individually using the known section formulas and enter them here to get σv. To solve backward (e.g. allowable σb for a given σv), enter σv and the other three quantities.
σz,d = 40 MPa, σb = 60 MPa, τs = 0 and τt = 30 MPa give σres = 100 MPa, τres = 30 MPa and σv = √(100² + 3·30²) ≈ 112.7 MPa.
σv ≈ 112.7 MPa is the stress that, under purely uniaxial tension, would produce the same material stress state as the actual multiaxial combination. Compare it against the yield strength Rp, e.g. via the static yield safety factor calculator.
All four inputs and the result are usually given in MPa (N/mm²); all four must share the same stress unit.
The distortion-energy (von Mises, GEH) criterion uses factor 3 in front of the shear-stress square (σv = √(σ²+3τ²)); the maximum shear stress criterion (Tresca) uses factor 4 (σv = √(σ²+4τ²)). GEH is considered more realistic for most ductile metals and is therefore preferred by Roloff/Matek and most modern design methods; Tresca always errs on the safe side (yields a somewhat higher equivalent stress) and is occasionally used for a more conservative estimate.
The GEH equivalent stress is used for ductile parts under simultaneous bending, torsion, tension/compression and shear, such as shafts, brackets, levers and arbitrarily shaped machine parts.
GEH applies to ductile materials under predominantly static load. For brittle materials, the maximum normal stress criterion is more appropriate instead. Under cyclic or fluctuating load, this calculator does not replace a fatigue-strength verification (e.g. per DIN 743 for shafts).
Common mistake: A common mistake is adding all four stresses directly under one shared square root instead of first summing normal and shear stresses separately and only then applying the GEH formula. The factor 3 in front of the shear term is also sometimes forgotten or confused with the Tresca criterion's factor 4.
Because σres and τres each act along the same physical direction (normal or tangential to the section); only the GEH formula then links both directions into a single equivalent value.
Set the corresponding value to 0; the formula then automatically reduces to the remaining terms.
For ductile materials (most steels), yes; for brittle materials like cast iron, the maximum normal stress criterion is often more appropriate.
Use the static yield safety factor calculator to check σv against the yield strength Rp and obtain the safety factor.
No, it only gives the static equivalent stress for one load state; cyclic loading requires a separate fatigue-strength calculation (e.g. per DIN 743).