2.1.8. Ship Fatigue Damage Assessment (FDA) with Ramseries

2.1.8.1. Introduction

A description of the methodology followed in RamSeries regarding fatigue damage assessment analysis is provided in this document. The procedure is based upon those described in different codes and recommendations (API Recommended Practice (Ref. [API_2002]), Det Norske Veritas (Ref. [Det_Norske_2005]) and Lloyds Register (Ref. [Lloyds_Register_2009])).

Some of those codes define their highest level of assessment as an spectral direct calculation procedure, using hydro-dynamic loads with motion analysis, and finite element analysis (FEA). The procedure implemented in RamSeries goes further, and proposes a FEA dynamic analysis in the time domain. This analysis consists in a dynamic structural FEA which comprises the whole ship geometry -with mesh refinement in the required structural details subjected to FDA-, and includes dynamic wave loads coming from analyses performed with time-domain wave radiation/diffraction solver code SeaFEM (Ref. [Compassis_2015]). These wave loads will correspond either to monochromatic waves which, properly combined, produce the global wave statistical data representing the ships voyages throughout its service life (linear approach), or to a complete wave spectrum analysis in the time domain (non-linear approach).

Thanks to such analysis, RamSeries FDA is capable of obtaining the stress ranges and associated number of cycles (using the Rain-flow Counting Method), which, together with a selection of suitable S − N fatigue curves (from fatigue design codes) allows the calculation of Fatigue Damage on the assumption of the Palmgren-Miner linear cumulative fatigue damage law.

This document presents two different approaches for RamSeries FDA, one is based on the superposition of monochromatic wave loads in combined load cases, plus still water loads and ship loading condition representing the actual service conditions. This approach assumes the validity of the linear superposition of loads and therefore is only valid to analyze the linear response of the structure. The second is fully general and is based on the direct analysis of the different service conditions obtained from a voyage simulation.

2.1.8.1.1. Voyage simulation

The voyage simulation refers to the procedure for obtaining the service profile matrix, which contains the conditional probabilities of occurrence of each sea-state with respect to heading, ship loading condition, and speed. This matrix is the starting point for the fatigue damage assessment analysis.

2.1.8.2. Generation of load cases

2.1.8.2.1. Linear approach

This section presents the procedure to obtain the actual service conditions (combined load cases) for the FDA, based on the superposition of monochromatic wave loads in combined load cases, plus still water loads and ship loading condition.

Monochromatic wave loads of unitary height ( \(h_{\omega}\) = 1 m) will be calculated for “r” regular spaced frequency sampling points (at least 25, r ≥ 25), from 0.2 rad/s to 1/2 rad/s. These waves will be used to generate the loads due to actual wave spectra, by combining them using the procedure which will be described in the following lines.

The different wave spectra used to generate the service profile matrix will be given as a wave scatter. A typical wave scatter will consist on a table of p × q probabilities (for each sea state, identified by a significant wave height, H_s, and wave period, T_s):

(2.180)\[[T{z_{i}},H{s_{j}}], \quad i=1,\ldots,p, \quad j=1,\ldots,q\]

A standard wave spectrum S(\(\omega\)) will be applied for studying the mentioned number of given frequencies,

(2.181)\[\omega_{k}, \quad k=1,\ldots,r\]

combined with a given range of advancing velocities,

(2.182)\[V_{l}, \quad l=1,\ldots,s\]

Also, a given number of ship loading conditions (draft and trim), \(\delta_{m}\), and main wave directions, \(\theta_{n}\), must be considered:

(2.183)\[\delta_{m}, \quad m=1,\ldots,t\]

(2.184)\[\theta_{m}, \quad n=1,\ldots,u\]

So, each Wave Combined Load Case will correspond to a sea state, wave direction, advancing velocity, and ship condition, and will have some associated data [1], which is the following:

• Total time (\(t^{tot}\)): Total duration of exposure (device expected service life).

• Simulation time (\(t^{sim}\)): Total duration of simulation (actual time simulated for each Combined Load Case, or non-linear run).

• Initialization time (\(t^{ini}\)): Time interval to be discarded at the beginning of the stress time history, in order to avoid not yet stabilized stress values [2]

Each Combined Load Case (CLC) can be written as a linear combination of simple Wave Load Cases (W LC) corresponding to each monochromatic wave, plus the linear combination of simple Static Load Cases (SLC), which include self-weight loads, cargo loads, hydrostatic loads, and any other requested for being considered, each of them with their corresponding coefficient if needed. Thus, the stress response can be expressed as:

(2.185)\[\sigma(CLC_{h})= \sum_{f=1}^{v} \psi^{h}_{f} \cdot \sigma^{h}(SLC_{f})+\sum_{k=1}^{r} \xi^{h}_{k} \cdot \sigma^{h}(WLC_{k}), \quad ( h=1,\ldots, p \times q \times s \times t \times u)\]

Being:

• \(\xi^{h}_{k}\), the corresponding spectra coefficients for each sampling frequency

  (\(\xi^{h}_{k} = \sqrt{2S_{ij}(\omega^{h}_{k})\Delta\omega}\)).

• \(\psi^{h}_{f}\), the corresponding static load case coefficient (if needed).

In order to maintain the linearity, monochromatic wave analysis will be performed for each frequency \(\omega^{h}_{f}\) (modified by the different advancing velocities). The wave formation resistance force, corresponding to the advancing velocity, will have to be added to each combined load case. Also, a monochromatic wave simulation including currents could be performed (with Kelvin linearization and assuming that small non-linearities are being neglected).

2.1.8.2.2. Non-Linear approach

In this approach, structural FEM non-linear dynamic simulations will be performed in RamSeries. These simulations will include several time-history runs, each one of them corresponding to the combination of given data, as mentioned in the linear approach (\(\delta_{m}\), \(\theta_{m}\), \(V_{l}\)). Such data, together with the chosen wave spectrum (characterised also by its significant wave heights and periods [ \(T_{z_{i}}\); \(H_{s_{j}}\) ]), will define the time domain simulations in SeaFEM. The result of those simulations (time histories of pressures over the hull) together with the corresponding static loads, will be used to define the different load cases for the simulation in RamSeries. So, in this case, the stress response, will be expressed as:

(2.186)\[\sigma_{h} = \sigma^{h}([T{z_{i}},H{s_{j}}], \delta_{m} , \theta_{n} , V_{l} , SLC), \quad (h=1,\ldots, n)\]

2.1.8.2.3. Ocean wave spectra

The presented procedures for FDA are based on the generation of the wave load cases by characterizing the sea state by using standard wave spectrum. Two of the most commonly used spectrum are:

• ISSC spectrum. Also known as Bretschneider or modified Pierson-Moskowitz):

  (2.187)\[S( \omega ) = \frac{5}{16} \cdot H^{2}_{s} \cdot \omega^{4}_{m} \cdot exp\left(\frac{-5}{4} \cdot \frac{\omega}{\omega_{m}}^{-4}\right)\]

  Being:

  • \(\omega\): frequency.

  • \(\omega_{m}\): peak frequency

  • \(H_{s}\): significant wave height

• JONSWAP spectrum

  (2.188)\[S( \omega ) = \frac{\alpha \cdot g^{2}}{16 \cdot \pi^{4}}\omega^{-5} \cdot exp\left(\frac{-5}{4} \cdot \frac{\omega}{\omega_{m}}^{-4}\right) \cdot \gamma^{b}\]

  With:

  • g: gravitational constant

  • \(b = exp(-0.5 \cdot (\frac{\omega}{\omega_{m}}-1)^{2})\)

  • \(\sigma\) = \(\sigma_{1}\), for \(\omega\) ≤ \(\omega_{m}\), and \(\sigma\) = \(\sigma_{2}\) for \(\omega\) > \(\omega_{m}\)

  • \(\gamma\), \(\sigma\), \(\sigma_{1}\) and \(\sigma_{2}\) are custom data

2.1.8.3. FDA procedure

The procedure implemented in RamSeries for analysing the damage caused by cyclic stresses (fatigue), outputs a colour contour fill (over welded joints or critical shell zones) of the total lifetime accumulated fatigue damage DT over a specified service period, given as follows:

(2.189)\[D_{T} = \sum_{h=1}^{M} \sum_{m=1}^{R} \frac{n_{h,m}}{N_{h,m}} \cdot \left[ \frac{t^{tot}_{h}}{t^{sim}_{h}} \right] \cdot DFF \leq 1, ((M = p \times q \times s \times t \times u))\]

Being \(D = \sum_{m=1}^{R} \frac{n_{h,m}}{N_{h,m}}\) the accumulated short term fatigue in a hot-spot, for a given ship loading condition, ship speed, ship heading to waves and sea-state (for a combined load-case, \(CLC_{h}\)), or for a non-linear run in accordance with Palmgren-Miner rule.

With:

• DFF: design fatigue factor.

• \(n_{h,m}\): number of stress cycles in stress block (h), for stress range (m).

• \(N_{h,m}\): number of cycles to failure at constant stress range \(\Delta\sigma_{h,m}\).

The design fatigue factors, according to DNV-OS-C101, are:

DFF	Structural element
1	Internal structure, accessible and not welded directed to the submerged part.
1	External structure, accessible for regular inspection and repair in dry and clean conditions.
2	Internal structure, accessible and welded directly to the submerged part.
2	External structure not accessible for inspection and repair in dry and clean conditions.
3	Non-accessible areas, areas not planned to be accessible for inspection and repair during operations.

For obtaining the stress ranges (\(\Delta\sigma_{h,m}\)) and their corresponding cycle counts \(n_{h,m}\), the Rainflow Counting method (Ref. [Tom_Irvine_2011]) is applied to the stresses time history of each combined load case or non-linear run. The number of ranges (R) is defined by the user. The output of the Rain flow Counting method is a table like the following:

Ranges (MPa)	Cycle counts
\(\Delta\sigma_{h,1}\)	\(n_{h,1}\)
…	…
…	…
\(\Delta\sigma_{h,1}\)	\(n_{h,m}\)
…	…
…	…
\(\Delta\sigma_{h,R}\)	\(n_{h,R}\)

2.1.8.4. Fatigue Strength S-N Curves

This procedure adopts hot spot stress approach, together with Palmgren-Miner cumulative damage rule to calculate fatigue damage of structural details. Corresponding Classification Society code hot spot stress design S − N curves are to be used for the evaluation of fatigue damage. Where considered appropriate, additional stress concentration factors (SCF) may be applied to account for construction tolerances and plate thickness effects. For structural steel, the S-N curves used are:

(2.190)\[\log_{10}(N_{m}) = \log_{10}(A) - M \cdot \log_{10}\left(\Delta\sigma^{*}_{m}\left( \frac{t}{t_{ref}}\right)^{K}\right)\]

With

(2.191)\[\Delta\sigma^{*}_{m} = SCF \cdot \Delta\sigma_{m}\]

And:

• t: Thickness through which a crack will most likely grow. \(t = t_{ref}\) is used when \(t < t_{ref}\).

• \(t_{ref}\): Reference thickness: \(t_{ref} = 32\, \text{mm}\) for tubular joints, \(t_{ref} = 25\, \text{mm}\) for welded connections other than tubular joints.

• M, K, \(log_{10}(A)\): Given by the user, for the different structural details and environments.

• SCF: Stress Concentration Factor.

[1]

This associated data also applies for each run of the non-linear approach.

[2]

This applies specially for non-linear approach