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Spectral method

The main assumption underlying the spectral fatigue analysis is that the damage resulting from a given stress history can be computed through the stress spectrum, associated duration and SN curve.

The stress spectrum corresponding to a given wave spectrum will be computed through stress RAO that provides the stress cycle amplitude as a function of the wave elevation.

The first step in the spectral fatigue analysis consists in splitting the irregular wave scatter diagram (WSD) in blocks. The WSD is representative of environmental sea state conditions to be encountered over the whole service life. These blocks are also referred to as bins in the following.

Basically, from stress and wave elevation time series, stress RAOs are computed using several algorithms according to the type of the considered wave (regular or irregular): FFT, standard deviation, cross-spectrum. Several wave incidences are defined (named bin) and these stress RAOs are computed for each bin. Bins is composed of a set of wave spectrum (Jonswap, Ochi-Hubble, …).

Then, damage is calculated using the stress RAOs, a distribution law (Rayleigh or Dirlik) and a wave spectrum. Damage is computed for each wave spectrum of each bin and then a global damage is calculated by summing all the damages (according to the Miner-Palmgrens’s rule).

The method is illustrated in the sketch below :

Spectral approach diagram

Response spectrum and distribution law

The stress spectral response is derived either from:

• combination of stress RAOs and wave spectra:

\[ S_{Stress}(\omega)=RAO_{stress}(\omega)^2*S_{wave}(\omega) \]

• direct simulations in frequency domain.

From the stress response spectrum, the stress range distribution is calculated considering a Rayleigh distribution for a narrow banded vibrations or a Dirlik distribution for multi-peaks solution.

• Rayleigh distribution:

A significant stress range and its associated apparent period (zero up-crossing period) are computed: $$ \Delta \sigma_s=4\sqrt{m_{0,stress}} $$ $$ T_{z,stress}=2\pi\sqrt{\frac{m_{0,stress}}{m_{2,stress}}} $$

The stress range distribution is the deduced using the probability law : $$ P(\Delta \sigma>\Delta \sigma^.)= e^{-2\frac{\Delta \sigma^2 }{\Delta \sigma_s^2 }} $$

• Dirliks formula:

\[P (\Delta \sigma)=\frac{D_1e^{\frac{-Z}{Q}}+\frac{D_2 Z}{R^2}e^{\frac{-Z^2}{2R^2}}+D_3Ze^{\frac{-Z^2}{2}}}{2\sqrt{m_{0,stress}}} \]

with

\(D_1=\frac{2(x_m-\gamma^2)}{1+\gamma^2}\) ; \(D_2=\frac{1-\gamma-D_1+D_1^2}{1-R}\) ; \(D_3=1-D_1-D_2\) ; \(Z=\frac{\Delta \sigma}{2\sqrt{m_{0,stress}}}\)
and
\(Q=\frac{1.25(\gamma-D_3-D_2R)}{D_1}\);\(R=\frac{\gamma-x_m-D_1^2}{1-\gamma-D_1+D_1^2}\) ; \(\gamma=\frac{m_{2,stress}}{\sqrt{m_{0,stress} m_{4,stress}}}\);\(x_m=\frac{m_{1,stress}}{m_{0,stress}}\sqrt{\frac{m_{2,stress}}{m_{4,stress}}}\)

Damage assessment

Damage associated to one sea-state :
For a given sea-state, the damage assessment associates a distribution law with a fatigue curve. Indeed, the damage formula is:

\[ D=\int_0^\infty \frac{n_{SeaState}}{n_{Allowable}}p(\Delta\sigma)d\Delta\sigma \]

With :

\(n_{SeaState}\) : corresponds to the number of cycle of the sea-state during one year \((365*24*3600/T)\)

For a rayleigh distribution, \(T\) corresponds to the up-crossing period: \(T_z=2\pi\sqrt{\frac{m_0}{m_2}}\)

For a Dirlik distribution, \(T\) corresponds to the crest period: \(T_c=2\pi\sqrt{\frac{m_2}{m_4}}\)

\(n_{Allowable}\): corresponds to the allowable number of cycle (computed with fatigue SN or TN curve parameters: \(n_{Allowable}=\frac{a}{\Delta\sigma^m}\), or \(Log(N) = Log(a) – mLog(\Delta\sigma)\))

So it comes :

\[ D=n_{SeaState}\frac{1}{a}\int_0^{\infty} (\Delta\sigma)^m p(\Delta\sigma)d\Delta\sigma \]

If the fatigue curve has several slopes, the damage formula becomes:

\[ D=n_{SeaState}\sum_{i=1}^{i=p}\frac{1}{a}\int_0^{\infty} (\Delta\sigma)^{m_i} p(\Delta\sigma)d\Delta\sigma \]

Cumulative assumptions

The basic assumption Miner-Palmgrens’s rule is considered to deduce the total fatigue damage from the different sea-states. The damage relative to different sea-states may be added to one another:

\[ D_{swells}=\sum_{bin=1}^{numberOfBins}p(bin)D(bin) \]

Where: \(D(bin)\) is the contribution to damage of bin \(p(bin)\) is the overall percentage of occurrence of sea-states gathered in that bin.

Similarly, the yearly contribution of bin \(b\) to the total fatigue damage is a weighted sum of every swell contribution:

\[ D(b)=\sum_{S}p(S)D(S) \]

Where: \(D(S)\) is the contribution to damage of swell \(S\)
\(p(S)\) is the probability of occurrence of swell \(S\) inside bin \(b\)
For a given bin: \(\sum_Sp(S)=1\)
At the end, the resulting global Lifetime \(L\) is the inverse of the total damage D.