Can I just use von Mises equivalent stress for fatigue?

For proportional (in-phase) loading where the principal directions stay fixed, a von Mises or Tresca equivalent amplitude is often adequate. For non-proportional loading, where the principal axes rotate during the cycle, equivalent-stress methods can be badly non-conservative — that is where critical-plane criteria are needed.

What is a critical-plane method?

It searches over candidate material planes for the one that maximises a damage parameter combining shear and normal action, on the physical basis that fatigue cracks nucleate and grow on specific planes. Findley, Fatemi–Socie and Smith–Watson–Topper are common variants, differing in whether they are shear- or tension-dominated.

Why is non-proportional loading more damaging?

Rotating principal axes activate more slip systems and cause additional cyclic hardening in many materials, so the same equivalent amplitude does more damage than under proportional loading. Critical-plane parameters that include the normal stress on the shear plane capture this; a scalar equivalent stress does not.

Fatigue · guide

Multiaxial fatigue assessment

Shafts, pressure parts and welded joints rarely see simple uniaxial cycling. Under combined bending, torsion and pressure the question becomes which plane fails and how to score the damage on it. Picking the right multiaxial method is the difference between safe and badly wrong.

Proportional vs non-proportional

The first question is whether the loading is proportional — all components rising and falling in phase, so the principal stress directions stay fixed — or non-proportional, where they rotate during the cycle. That distinction decides which family of methods is valid.

Equivalent-stress methods

For proportional loading, a signed von Mises (or Tresca) equivalent amplitude reduces the multiaxial state to a uniaxial one that can be put on an ordinary S–N curve:

σ_eq,a = √( σ_x² − σ_x·σ_y + σ_y² + 3·τ_xy² ) (von Mises form)

This is convenient but loses the directionality of damage and ignores the extra hardening of non-proportional paths — so its validity ends where the principal axes start rotating.

Critical-plane methods

Critical-plane criteria scan all material planes for the one maximising a damage parameter, reflecting the physical reality that cracks form on particular planes. Three are widespread:

Findley: max over planes of ( τ_a + k·σ_n,max )

Fatemi–Socie: τ_a · ( 1 + k·σ_n,max / σ_y ) (shear-dominated)

Smith–Watson–Topper: σ_n,max · ε_a (tension-dominated)

Shear-dominated parameters (Findley, Fatemi–Socie) suit ductile metals whose cracks grow by shear; the tensile SWT parameter suits materials and regimes where crack growth is normal-stress driven. The normal stress on the critical plane is what lets these capture non-proportional damage.

Open the calculatorMultiaxial fatigue calculator →Assess combined loading with von Mises, Findley, Fatemi–Socie and SWT, with the critical-plane search and the damage parameter on each plane.

Choosing a method

Match the criterion to the failure mechanism and the loading: proportional and ductile, an equivalent stress may suffice; non-proportional or life-critical, use a critical-plane parameter calibrated to the material. Stress-invariant approaches (Sines, Crossland) using the second deviatoric invariant and hydrostatic stress are a middle ground for high-cycle proportional design.

Frequently asked

Can I just use von Mises equivalent stress for fatigue?: For proportional (in-phase) loading where the principal directions stay fixed, a von Mises or Tresca equivalent amplitude is often adequate. For non-proportional loading, where the principal axes rotate during the cycle, equivalent-stress methods can be badly non-conservative — that is where critical-plane criteria are needed.
What is a critical-plane method?: It searches over candidate material planes for the one that maximises a damage parameter combining shear and normal action, on the physical basis that fatigue cracks nucleate and grow on specific planes. Findley, Fatemi–Socie and Smith–Watson–Topper are common variants, differing in whether they are shear- or tension-dominated.
Why is non-proportional loading more damaging?: Rotating principal axes activate more slip systems and cause additional cyclic hardening in many materials, so the same equivalent amplitude does more damage than under proportional loading. Critical-plane parameters that include the normal stress on the shear plane capture this; a scalar equivalent stress does not.

References

W.N. Findley, "A theory for the effect of mean stress on fatigue of metals under combined torsion and axial load or bending," J. Eng. Industry, 1959.
A. Fatemi, D.F. Socie, "A critical plane approach to multiaxial fatigue damage including out-of-phase loading," Fatigue Fract. Eng. Mater. Struct., 1988.
K.N. Smith, P. Watson, T.H. Topper, "A stress–strain function for the fatigue of metals," J. Materials, 1970.
D.F. Socie, G.B. Marquis, "Multiaxial Fatigue," SAE International.

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