Steinberg Formulation utilizes all three stress occurrences (1σ, 2σ, 3σ) and their rate of occurrence along with the Miner’s rule in order to compute the total fatigue damage of the system.

Where:

= actual number of cycles at or below the 1σ level (0.6831 ).

= actual number of cycles at or below the 2σ level (0.271 ).

= actual number of cycles at or below the 3σ level (0.0433 ).

= allowable number of cycles (from fatigue curve) at 1σ, 2σ, 3σ stress levels.

are obtained by using the S-N relation and the (1σ, 2σ, 3σ) stresses to find the corresponding number of cycles. If the Bilinear curve is used, the solver chooses the appropriate curve to interpolate on, based on the value of the stresses.

is defined as the statistical frequency which is obtained as follows:

Where:

= RMS Stress Velocity Result (stress based on velocity result)

= RMS Stress Result

Narrow Band formulation is a generalized method where the stress ranges are assumed to have a Rayleigh distribution. The formula to calculate Fatigue damage is given by,

Where:

= Statistical frequency

t: Exposure Duration

σ: Equivalent Alternating Stress

: Gamma function.

A, m: SN curve properties from the equation NSm = A, where S = Stress Amplitude.

If we are using the Bilinear form of the SN curve, this formula changes to:

where:

and:

is the stress point of intersection of the two SN Curves.

= lower incomplete gamma function.

The Wirsching Formulation can be described as a correction factor to the Narrow Band Formulation in order to account for Wideband scenarios. Instead of using a different, more complicated method for Wideband cases, we calculate Fatigue Damage using Narrow Band formulation and simply apply the Wirsching correction factor to it, as shown:

Where is the wide band correction factor.

Where:

= 0.926-0.033

= 1.587 - 2.323

= Bandwidth Factor

= Irregularity Factor

= Spectral Moments

is the fatigue strength exponent obtained from the Linear SN Curve.

Since Wirsching does not have a specialized formula for the Bilinear SN Curve, the average of the two fatigue strength exponents (m and r) is used in the Bilinear case.