# On the risk of fatigue failure of structural elements exposed to bottom wave slamming – Impulse response regime

Romain Hascoët, Nicolas Jacques

► To cite this version:

Romain Hascoët, Nicolas Jacques. On the risk of fatigue failure of structural elements exposed to bottom wave slamming – Impulse response regime. Applied Ocean Research, 2025, 154, pp.104411. 10.1016/j.apor.2024.104411. hal-05344936

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Research paper

# On the risk of fatigue failure of structural elements exposed to bottom wave slamming – Impulse response regime

Romain Hascoët ID*, Nicolas Jacques ID

ENSTA Bretagne, CNRS UMR 6027, IRDL, 2 rue François Verny, 29806 Brest Cedex 9, France

<figure><img src="image_2.png" /></figure>

## ARTICLE INFO

**Keywords:**

Water wave

Slamming

Fatigue

Risk of failure

Sea state

Marine structure

## ABSTRACT

This study aims to investigate whether fatigue damage induced by bottom wave slamming can be a failure mode, important to consider when sizing a marine structural element. The body exposed to wave impacts is assumed to have a shape and structural arrangement such that the duration of wave-impact loads is short relative to the structure's vibratory response time. In this dynamical regime, fatigue is found to be a potentially important failure mechanism: accounting for the risk of failure due to fatigue damage may result in design constraints that are significantly more conservative than those based on the risk of ultimate strength exceedance. The role of fatigue damage depends on the elevation of the body. It is predominant for low elevations, for which slamming events are frequent. Since this study aims to provide general insight, the specific details of the body, such as its shape and structural arrangement, are not specified. Instead, a general framework is used for the analysis. The way forward to address a specific case study, possibly including the effects of forward and seakeeping motions, is briefly explained.

## 1. Introduction

Fatigue damage can be a critical failure mechanism to consider in the design of ships and offshore structures. One of the primary environmental factors causing time-varying mechanical stress, which leads to fatigue damage, is exposure to water waves. Water-wave loads can induce time-varying stress through several mechanisms: (i) non-resonant response (possibly quasistatic) to the continuous wave loading (see, e.g., Jha and Winterstein, 2000; Li et al., 2013, 2014), (ii) resonant response to the continuous wave loading (springing mechanism, see, e.g., Naess, 1992; Winterstein et al., 1994; Storhaug, 2007), (iii) transient events related to high-order wave nonlinearities (see, e.g., Faltinsen et al., 1995; Tromans et al.; Bachynski and Moan, 2014), (iv) transient events induced by wave slamming (see, e.g., Wienke and Oumeraci, 2005; Barhoumi and Storhaug, 2014; Hulin et al., 2023). Much research has been devoted to the experimental investigation and the modeling of the fatigue of ships, driven by the global response of their hull girders to wave loads (see Temarel et al., 2016; Dong et al., 2022 for recent reviews). Ship hull girders are subject to wave-loading mechanisms (i-ii-iv). Wave slamming is usually regarded as a secondary source of fatigue damage (see, e.g., Mao et al., 2010; Andersen and Jensen, 2013), acting through the global stress-response of the hull girder (whipping mechanism). In contrast, to our knowledge, configurations where wave-slamming is the primary source of fatigue damage have not been considered in the literature. The latter situation may occur for structural elements, either on ships or offshore installations,

that respond locally to wave slamming.

The objective of this paper is to investigate whether fatigue damage could be a failure mode relevant to the design of a marine structural element, whose main source of stress variations is bottom wave slamming. The exposed body is assumed to be above the still water level, at a certain elevation. If the body is close to the mean sea level, it will frequently experience water entry and emersion cycles. If its elevation is increased, the body will remain mostly out of the water, with wave impacts occurring less frequently. In both cases, the exposed body will experience bottom-slamming loads due to the successive water entries. In this context, slamming events are defined as the moments when the body crosses the sea free-surface, into the water domain. It is important to note that even moderate water waves may generate substantial slamming loads (especially for blunt bodies), since the structural element is assumed to enter the water domain from a fully emerged position. The novelty of this study lies in the consideration that the local response of the structural element to bottom-wave slamming may constitute the dominant contribution to fatigue damage.

A priori, the modeling of the risk of failure of a structure subjected to wave impacts requires considering two potential failure modes:

* Failure due to stress locally exceeding a specific threshold. The considered threshold is usually the ultimate tensile strength (shortened hereinafter to "ultimate strength") or the yield strength of the material. The choice between the ultimate and

<footnote>* Corresponding author.
E-mail address: romain.hascoet@ensta-bretagne.fr (R. Hascoët).</footnote>
yield strengths depends on whether the occurrence of localized plastic deformation is acceptable. When the structural element is designed for a low probability of failure, this failure mode relates to extreme events (i.e., extreme wave impacts) that have a low probability of occurrence over the considered exposure time.

* Failure due to fatigue damage. Fatigue damage will accumulate through a (very) large number of impacts, experienced over the exposure time.

A key point in modeling the first failure mode is apprehending rare and extreme events. This presents challenges in two regards. Firstly, it requires modeling the probability of occurrence of extreme environmental conditions: here, extreme waves, which are themselves generated by extreme sea states. Second, the response of the system to these extreme conditions needs to be estimated, with the substantial challenge posed by the fact that the kinematics of extreme waves remains uncertain.

The second failure mode (fatigue damage) may be less challenging in terms of excitation and response modeling, as most of the damage may be caused by a class of events that are not particularly rare and extreme. For such a class of events, in situ or lab measurements may be “readily” accessible, while modeling is less complicated and less subject to uncertainties. However, to properly model this second failure mode, a key challenge lies in the fact that the long-term fatigue damage results from the accumulation of a large number of random events (here wave impacts), whose underlying probability distribution (here related to the properties of the encountered sea states) is also random. In this respect, this makes ‘risk-based’ analyses (i.e., the sizing of the marine structure for a given probability of failure) more involved than for the first failure mode.

This paper investigates the relative importance of these two failure modes, when designing a structural element with respect to the risk induced by bottom wave slamming. The configuration is assumed to be such that the structural element experiences a large number of wave impacts over its lifetime. The case where the risk of wave impact occurrence is small over the exposure time is not relevant to fatigue damage. The latter situation would typically arise when the elevation of the exposed body (vertical position of the body relative to the still water level) is significantly larger than the typical significant wave height in the geographical zone of interest (see, e.g., *Sweetman and Winterstein, 2003*; *Lim and Kim, 2019*). As the relative importance of the two failure modes depends on the total number of experienced wave impacts, this study explores how the results vary with the assumed exposure duration and body elevation. Additionally, it aims to identify the dominant classes of sea states and wave impacts, that contribute most to the risk. The study adopts a general framework, allowing the findings and conclusions to remain qualitatively applicable across various contexts. The structural element exposed to wave slamming is assumed to have a bottom shape and a structural arrangement such that the characteristic duration of slamming loads is short compared to the structure’s response timescale. The local vibratory stress response is assumed to be dominated by a single structural mode. This dominant vibratory mode is assumed to be responsible for most of the fatigue damage.

The framework considered in the present study is further elaborated in Section 2. It details the assumptions regarding the structural stress response following wave impact (Section 2.1) and the resulting fatigue damage (Section 2.2). Additionally, it outlines the assumptions related to the modeling of sea state occurrence (Section 2.3), and impact occurrence within a given sea state (Section 2.4). Section 3 introduces the case studies to be considered for investigation and discussion. Section 4.1 investigates the evolution of the failure probability as a function of the structural sizing, exposure time, and failure mode. The dominant classes of sea states and wave impacts responsible for the risk of fatigue failure are identified in Section 4.2. The effect of a change in the elevation of the exposed body is investigated in Section 4.3. The paper ends with a discussion in Section 5, followed by a conclusion in Section 6.

## 2. Framework and assumptions

A flow diagram summarizing the approach is presented in Fig. 1. Each component of the methodology is detailed below.

### 2.1. Structural response to wave impacts: impulse regime

During a hydrodynamic impact on an elastic body, transient loads will induce a vibratory response of the body structure, provided that the loads evolve sufficiently quickly relative to the structure’s response timescale (period of the dominant mode) – see Faltinsen (2006, 2000), Korobkin and Khabakhpasheva (1999), Scolan (2004), Sun and Faltinsen (2006), Maki et al. (2011), Piro and Maki (2013) for a sample of studies on slamming and the resulting hydroelastic response. In the context of bottom-wave slamming, the vertical component of the fluid at impact may be considered as the main variable relevant to slamming loads (see, e.g., Ochi and Motter, 1973; Rassinot and Mansour, 1995; Wang et al., 2002; Hermundstad and Moan, 2007; Wang and Guedes Soares, 2016). The impact timescale, $t_{imp}$, can be expressed as follows:

$$t_{imp} = \frac{h_{imp}}{V}, \qquad (1)$$

where $h_{imp}$ is the characteristic water-entry depth over which the transient loads develop, and $V$ is the vertical component of the characteristic water-entry velocity. It is important to note that $h_{imp}$ may be significantly smaller than the size of the exposed body, particularly for blunt-shaped bodies, such as a tubular element of circular section (see, e.g., Cointe and Armand, 1987), or a water foil (see, e.g., Hascoët et al., 2019). In this study, the response timescale of the body’s structure, $t_{vib}$, is assumed to be much longer than $t_{imp}^1$:

$$t_{vib} \gg t_{imp}. \qquad (2)$$

Then, as illustrated in Fig. 2, the slamming load may be treated as an impulse from the perspective of structural response. Hereafter, this dynamical regime is referred to as the *impulse regime*. (Faltinsen, 1999) explored this impulse regime and the transition to the non-impulse regime, through numerical simulations of wedge-shaped structures entering water at constant velocity.

Considering a structural detail within the body, the impact-induced vibratory response will translate into a series of stress cycles that may induce fatigue damage. For simplicity, a single structural mode is assumed to dominate the vibratory response, as well as the resulting fatigue damage: hereafter, $t_{vib}$ is used to refer to the eigen period of the dominant mode. Note that the dominant mode is not necessarily the mode having the lowest eigen frequency, but the mode that leads to the largest stress amplitude (for the considered loading). Hydroelastic coupling must be considered when evaluating $t_{vib}$, due to added-inertia effects.

Following the impulse excitation, if there are no further rapid changes in the hydrodynamic load, the mechanical vibrations will decay. The damping of the vibratory response is assumed to be of viscous type (see, e.g., Schmitz and Smith, 2012), which is typically valid for metal structures. For simplicity, the damping ratio of the dominant mode, $\zeta$, is assumed to be constant and significantly smaller than unity, $\zeta \ll 1$. The assumption $\zeta \ll 1$ is usually valid for metal structures in air (Schmitz and Smith, 2012). In the context of hull structures, measured values of damping ratios are typically around 1% (see, e.g., Storhaug et al., 2007; Storhaug, 2014; Orlowitz and Brandt, 2014; van Zijl et al., 2021). In the specific context of structural elements exposed to wave slamming, the main uncertainty may lie in the potential effects of hydroelastic coupling on the damping ratio. This subject has received limited attention in the literature (see Abrahamsen

<footnote n="1">¹ In practice, results obtained for a simple harmonic oscillator suggest that $t_{vib}/t_{imp} \ge 3$ may be sufficient.</footnote>
<figure><img src="image_3.png" /><figcaption>Fig. 1. Flow diagram of the approach. The input data of the problem are indicated by blue boxes with square corners. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 2. Illustration of the impulse stress-response regime considered in the present study. The stress response is assumed to be dominated by a single structural mode. In this dynamical regime, the stress amplitude of the first cycle, $s^{(1)}$, is approximately proportional to the load impulse (i.e., the time-integral of the load).</figcaption></figure>

et al., 2023 for a recent study addressing this matter). Several experimental studies (see, e.g., Campbell and Weynberg, 1980; Suja-Thauvin et al., 2017; Wang et al., 2020; Moalemi et al., 2023) show that water impacts do generate a vibratory structural response. Although the damping ratio is not evaluated in these studies, it is clearly smaller than unity. Moreover, in lab experiments, the measurement of impact forces often requires a special treatment to compensate for the perturbations induced by the vibrations of the mock-up (Antonini et al., 2021; Tassin

et al., 2024).

Based on the aforementioned assumptions regarding structural damping, the stress response of a structural detail, induced by a wave impact, can be modeled as a series of cycles with exponentially decaying amplitudes, which can be approximated as follows:

$$ \frac{s^{(k)}}{s^{(1)}} \approx \exp\{-2\pi\zeta(k-1)\}, \quad (3) $$

where $s^{(1)}$ and $s^{(k)}$ represent the amplitudes of the 1st and kth cycles, respectively (see Fig. 2 for an illustration). The structural vibratory response is assumed to have time to mostly dampen between successive wave impacts.

According to Eq. (3), the sequence of stress-cycle amplitudes, triggered by a wave impact, is entirely determined by the amplitude of the first cycle, $s^{(1)}$. In the present study, the magnitude of the stress response is assumed to depend solely on the vertical component of the fluid velocity.<fnref n="2" /> The slamming impulse, which is the time integral of the slamming force, responds linearly to the impact velocity. This linear relationship arises from two factors: (i) slamming loads are proportional to the square of the water-entry velocity (if the latter is assumed to be constant), and (ii) the characteristic water-entry duration, as defined

<footnote n="2">² Depending on the shape of the slamming-exposed body, and the desired degree of accuracy, it may be important to include additional kinematic variables – such as the free-surface slope, the horizontal fluid velocity, or the fluid acceleration – as inputs to the considered slamming model (see, e.g., Scolan and Korobkin, 2015; Helmers et al., 2012; Hascoët et al., 2019; Hascoët et al., 2020; Hascoët et al., 2021).</footnote>
by Eq. (1), is inversely proportional to the impact velocity. Then, if the stress responds linearly to the load impulse (which is expected if condition (2) is fulfilled and if the structure has a linear elastic behavior), the amplitude of the first cycle may be modeled as follows:

$$s^{(1)} = S_{*} \left( \frac{V}{1 \text{ m/s}} \right), \quad (4)$$

where $S_*$ is a stress normalization factor that depends on the sizing of the structural element under consideration. This study focuses on the relative importance of the two aforementioned failure modes as a function of $S_*$, without specifying the underlying structural element. The effective estimation of $S_*$ for a specific structural element is illustrated in Appendix C.

## 2.2. Fatigue damage and failure criteria

This subsection details the approach used to model fatigue damage. In Section 2.2.1 the assumed SN curve pattern is introduced. Section 2.2.2 then explains how the randomness of the SN curve is modeled. Finally, Section 2.2.3 briefly discusses the failure criteria considered in this study.

### 2.2.1. SN curve pattern

Various empirical laws have been proposed to model the fatigue stress-life relationships of metals (see Bannantine et al. (1990), Stephens et al. (2000) for reviews on the subject). These relationships are commonly known as “SN curves”, where N represents the number of cycles to failure at a given stress-cycle amplitude (or range), S. The metal considered in the present marine context is steel, as it is widely used in the naval and offshore industries. For fatigue damage occurring in the high cycle regime ($10^7 > N > 10^4$) or in the very high cycle regime ($N > 10^7$), classification societies provide guidelines regarding the SN curve patterns to be used for the fatigue assessment of ship structures (see, e.g., Bureau Veritas, 2020; Det Norske Veritas, 2021a; American Bureau of Shipping, 2017). These SN curve patterns are composed of two power-law branches:

* A high cycle power-law branch where $S \propto N^{-1/m}$, which is valid for $10^7 > N > 10^4$. Classification societies recommend an index m ranging from 3 to 4, depending on the geometry and surface finishing of the structural detail. The value m = 3 is used in the illustrative examples investigated below.

* A very high cycle power-law branch where $S \propto N^{-1/(m+\Delta m)}$, starting from $N = 10^7$, with $\Delta m = 2$ being the value recommended by classification societies. The two branches connect at $N = 10^7$. In order to limit the number of cycles that need to be considered, the very-high cycle branch is truncated above $N = 10^{12}$; i.e., an endurance limit is set to $S(N = 10^{12})$. This truncation is adopted for computational convenience and does not affect the qualitative findings of this study.

Through the illustrative examples provided in Section 4, it will be shown that this SN curve pattern – limited to $N > 10^4$ – is sufficient to estimate the risk of fatigue failure, provided the exposed body is close enough to the mean water level. This latter condition ensures a large number of wave impacts over the body’s life. However, as the body’s elevation above the mean water level increases, the frequency of wave impacts decreases, and at a certain point, low cycle fatigue becomes relevant in the reliability analysis (this point is specifically addressed in Section 4.3). Furthermore, in the following sections, the risk induced by fatigue is compared with the risk of failure due to the material’s ultimate strength being exceeded. For these two reasons, the risk analysis must account for two additional factors: (i) the damage contribution of stress cycles within the low cycle regime; (ii) the possibility that a single extreme event leads to failure through the exceedance of the ultimate strength. In order to cover these two aspects, the SN curve pattern proposed by classification societies has been extended into the

low cycle region ($N < 10^4$) by introducing a third power-law branch. To decide on how to parametrize this third branch, the ultimate strength of the material, $S_u$, has been considered to be a good proxy for the stress cycle amplitude at $N = 1$, i.e., $S(N = 1) \equiv S_0 \approx S_u$. On the other hand, the yield strength of the material, $S_c$, may be considered as a good proxy for the transition between the low cycle and high cycle regimes, i.e., $S(N = 10^4) \approx S_c$. Given that most steel types used in the marine industry typically have $S_u/S_c \approx 2$ (see, e.g., Bureau Veritas, 2021; Det Norske Veritas, 2021b; American Bureau of Shipping, 2022), these considerations have led to the choice $S_0 = 2 \times S(N = 10^4)$. These assumptions produce the SN curve pattern shown in the left panel of Fig. 3.

The mean stresses of cycles are expected to be close to zero (full stress reversal) in the impulse regime. Therefore, the effect of mean stress on fatigue life does not need to be considered in this study.

### 2.2.2. SN curve randomness

Fatigue tests show significant dispersion in the results obtained from experiments repeated in the same conditions. For a given fixed stress amplitude, the observed number of cycles to failure can vary considerably between trials. As stated by (Stephens et al., 2000) this variability may arise from “inherent material variability, variations in heat treatment and manufacturing, variations in specimen or component geometry, and variations from differences in the testing conditions”. In this study, the structural detail under study is linked to an SN curve, $n(S)$, which is modeled as a random curve. For a given structural detail specimen, the realized instance of the SN curve is assumed to hold during the whole life of the specimen. Then, the fatigue damage, $D_{cum}$, accumulated by the specimen is computed using Miner’s rule (Miner, 1945). Assuming that the stress loading history consists of a collection of M cycles of amplitudes $s_1, s_2, \dots, s_M$, Miner’s rule reads:

$$D_{\text{cum}} = \sum_{k=1}^{M} \frac{1}{n(s_k)}. \quad (5)$$

For a given stress amplitude, the statistical properties of $n(S)$, can be estimated by conducting fatigue tests at different constant stress amplitudes, using a different specimen for each test. However, for a realized SN curve (i.e., for a given specimen) the covariance structure between $n(S_1)$ and $n(S_2)$, the numbers of cycles to failure at two different stress levels, $S_1$ and $S_2$, is more difficult to probe, as a fatigue test cannot be repeated twice with the same specimen. In the literature, this matter has been investigated by modeling $n(S)$ as the realization of a stochastic process (see, e.g., Liu and Mahadevan, 2007; Paolino and Cavatorta, 2014). For the present study, two extreme assumptions have been tested:

1. The SN curve is randomized through a global shift factor applied to the median SN curve, $N(S)$. In other words, for a given specimen, the knowledge of $n(S)$ at a given stress level fully determines the realized SN curve. This approach corresponds to the one proposed by classification societies (Bureau Veritas, 2020; Det Norske Veritas, 2021a). In this study, the logarithm of the random shift factor is assumed to follow a centered normal distribution of standard deviation $\delta_{BV} = 0.25 \times \ln(10)$ (which is the value recommended by Bureau Veritas, Bureau Veritas, 2020).

2. The randomness of $\ln[n(S)]$ is modeled as a Gaussian white noise: i.e., for a given specimen, the values of $\ln[n(S_1)]$ and $\ln[n(S_2)]$, at two different stress levels $S_1$ and $S_2$, are independent random Gaussian variables. The standard deviation of the white noise is set to $\delta = \delta_{BV}$. This approach is used for example in Shen et al. (2000), Rathod et al. (2011).

<footnote n="3">³ An alternative option could have been to map $S_u$ to $S(N = 1/2)$ or $S(N = 1/4)$; $N = 1/2$ and $N = 1/4$ respectively corresponding to a half-cycle and a single monotonic tensile ramp: the precise choice is unimportant in the present study.</footnote>
<figure><img src="image_5.png" /><figcaption>Fig. 3. Left: SN curve pattern used in this study. The different constraints which define this SN curve are annotated: (i) the stress amplitude at N = 1, S₀ ≡ S(N = 1), is used as a normalization factor of the proposed SN curve pattern. S₀ is expected to be close to the ultimate strength of the material; (ii) the stress level at the end of the low cycle branch is set to S = S₀/2 at N = 10⁴. The transition stress level, S₀/2, can be considered as a proxy for the yield strength; (iii) the power law index of the high cycle branch is equal to -1/3; (iv) the very high cycle branch starts at N = 10⁷ and has a power law index equal to -1/5. Right: SN curve randomness. In the illustration, the median SN curve is shown as a dashed line. The noisy gray curve illustrates the case where the randomness is modeled through option 2 (see Section 2.2.2). The solid gray curve illustrates the case where the randomness is modeled through a random shift factor, αshift (option 1). The dotted lines show the ±2δ band. The “-2δ” curve (i.e., the dotted line on the left of the median SN curve) matches the curve pattern shown on the left panel.</figcaption></figure>

These two scenarios are illustrated in the right panel of Fig. 3. Computations have shown that option 1 yields the most conservative constraints in the regime of small failure probabilities (which is usually the regime relevant for the design of a marine system). Therefore, it is the modeling option which has been adopted for the case studies presented below.

The fact that option 1 leads to more conservative constraints compared to option 2 – in the regime of small failure probabilities – can be understood as follows. On the one hand, for a given sequence of stress cycles of various amplitudes, the mean fatigue damage does not depend on the covariance structure of the SN curve. On the other hand, the dispersion of the realized fatigue damage (above and below its mean) will be greater when the SN curve randomness is modeled through assumption 1. This is because, in option 1, the damages induced at different stress amplitudes are positively correlated, whereas in option 2, they are independent. As a result, option 1 leads to longer tails in the distribution of the realized fatigue damage, which translates into sizing constraints being more conservative in the regime of small failure probabilities. The same trend has been found and similarly explained in previous studies (see Liu and Mahadevan, 2007; Paolino and Cavatorta, 2014), where further details are provided.

### 2.2.3. Failure criteria and failure modes

The structure is considered to fail due to fatigue when the cumulated damage, D_cum Eq. (5), exceeds one. As the SN curve is randomized through a global shift of the SN curve pattern, the ultimate strength, S_u, can be formally identified as the stress-cycle amplitude at which the realized SN curve intersects N = 1 (thereby, S_u is a random variable). Then, two failure modes can be distinguished: (i) failure due to a cumulated fatigue damage exceeding one, D_cum > 1 (with no occurrence of ultimate strength exceedance); (ii) failure due to the occurrence of a stress cycle with an amplitude exceeding S_u.

### 2.3. Modeling of sea state occurrence

The long-term sea state history, to which the marine structure is exposed, is modeled as a sequence of short-term stationary sea states. The randomness of a given stationary sea state is parametrized through

its significant wave height, H_s (see Eq. (9) below), and its average wave period, T_z (see Eq. (11) below). The assumed sea state population is the one recommended by classification societies for the North Atlantic region (Bureau Veritas, 2019; Det Norske Veritas, 2019). The related joint density function of sea states in the plane (H_s, T_z), was derived by applying a conditional modeling approach:

$$P_{cs}(H_s, T_z) = f_{wbl}(H_s; \alpha_{cs}, \beta_{cs}, \gamma_{cs}) \cdot f_{logn}[T_z; \mu_{cs}(H_s), \sigma_{cs}^2(H_s)], \quad (6)$$

where f_wbl is the three-parameter Weibull density function of scale parameter αcs, shape parameter βcs, and position parameter γcs, and f_logn is the log-normal density function of parameters μcs and σcs². For the North Atlantic region, classification societies recommend the following parameter values: αcs = 3.041, βcs = 1.484, γcs = 0.661, μcs(Hs) = 0.70+1.27 Hs⁰.¹³¹ and σcs(Hs) = 0.1334+0.0264 exp(-0.1906 Hs), where Hs and Tz are in meters and seconds, respectively. This parametrization is used in this study.

Sea states are assumed to be stationary over a duration of 1 h. Successive sea states are modeled as independent events. In practice, an extreme (resp. moderate) sea state, is likely to be followed by another extreme (resp. moderate) sea state. Accounting for this effect would be important to accurately evaluate the risk of ultimate stress exceedance. Ignoring the serial dependence of sea states, as done in the present study, leads to conservative sizing limits with respect to the risk of extreme event occurrence (see, e.g., Mackay et al., 2021). Conversely, the serial dependence of sea states is unimportant to evaluate the risk of fatigue failure, as fatigue damage is built up over a large number of encountered sea states (see Section 4.2.1 for more details).

### 2.4. Modeling of impacts in a given sea state

The vertical fluid velocity at impact is the sole variable used to estimate hydrodynamic slamming loads, and the resulting structural vibratory response (see Section 2.1). The impacted body is assumed to be small enough in relation to water wave wavelengths, so that the body can be reduced to a single material point regarding the risk of slamming event. Given that ocean waves typically have wavelengths of a few hundred meters, the latter assumption translates into an upper limit of a few dozen meters on the horizontal extent of the body. In this context, a slamming event corresponds to the sea surface upcrossing
the material point. For the examples presented below, the exposed body is assumed to be fixed in the reference frame of the mean flow (i.e., the reference frame where the mean fluid velocity field of water waves is zero). The way to account for a forward motion, and/or a seakeeping motion is exposed in Appendix A. In the framework of linear wave theory, the free-surface elevation measured at the station of the material point, $\eta(t)$, can be modeled as a stationary Gaussian process, whose mean is zero (assuming $\eta(t)$ is measured from the mean water level) and whose power spectrum corresponds to the wave spectrum. The control material point is assumed to be at an altitude $z = a$, with respect to the mean water level. In the linear wave model, the vertical component of the fluid velocity, $w$, at the free surface, is equal to the time derivative of the free-surface elevation, $w = d\eta/dt$. Using level-crossing theory (see, e.g., Lindgren et al., 2013), it can be shown that $w|\eta(t) \uparrow a$, the vertical velocity given that $\eta(t)$ upcrosses the altitude $a$, follows a Rayleigh distribution, whose probability density function reads:

$$f_{w|\eta(t)\uparrow a}(w) = \frac{w}{m_2} \exp \left\{ -\frac{1}{2} \frac{w^2}{m_2} \right\}, \quad w \ge 0. \qquad (7)$$

The scale of the Rayleigh distribution is set by the second moment of the wave spectrum,

$$m_2 = \int_0^{+\infty} \omega^2 S(\omega) d\omega, \qquad (8)$$

with $S$ being the one-sided wave spectrum, and $\omega$ being the wave angular frequency. The significant wave height of the corresponding sea state is defined as:

$$H_s = 4\sqrt{m_0}, \qquad (9)$$

where $m_0$ is the zeroth moment of the wave spectrum:

$$m_0 = \int_0^{+\infty} S(\omega) d\omega. \qquad (10)$$

The average zero-upcrossing period (i.e., the average wave period), $T_z$, is related to $m_0$ and $m_2$ by:

$$T_z = 2\pi \sqrt{\frac{m_0}{m_2}}. \qquad (11)$$

By combining Eqs. (9)–(11), $m_2$ can be expressed as a function of the sea state characteristics:

$$m_2 = \left( \frac{\pi H_s}{2 T_z} \right)^2. \qquad (12)$$

In addition to the distribution of impact velocities (Eq. (7)), the frequency of slamming events (i.e., the average frequency of upcrossing events) is also required to evaluate the risk of failure due to wave slamming. Using Rice's formula (Rice, 1944, 1945) for a stationary Gaussian process, this frequency can be expressed as:

$$\mu_{\uparrow a} = \frac{1}{2\pi} \sqrt{\frac{m_2}{m_0}} \exp \left\{ -\frac{1}{2} \frac{a^2}{m_0} \right\}. \qquad (13)$$

Finally, by substituting Eqs. (9)–(11) into Eq. (13), the upcrossing frequency can be expressed as:

$$\mu_{\uparrow a} = \frac{1}{T_z} \exp\{-8(a/H_s)^2\}. \qquad (14)$$

Thus, from Eq. (7)–(12)–(14), it appears that both the impact frequency and the distribution of impact velocities, for a given sea state, are fully determined by the values of $H_s$ and $T_z$.

### 3. Case studies

Building on the framework introduced in the previous section, the question of whether slamming-induced fatigue can be an important failure mode is now addressed. To explore this question, a series of case studies have been investigated through numerical simulations. The considered case studies are described in Section 3.1. Section 3.2 provides a brief overview of the numerical setup implemented to conduct the numerical simulations.

#### 3.1. Simulated scenarios

The various configurations, considered in this study, have been defined by adopting simple assumptions and by varying key parameters, as follows:

* Long-term exposure duration: the body exposed to wave slamming, is assumed to operate continuously for a specific duration, $d_{lt}$, in the same geographical region. To investigate the effect of exposure duration, the following durations have been considered: $d_{lt} = 1$; 5; 20 yr.

* The random encounter of sea states is modeled by following the approach introduced in Section 2.3. Sea states are drawn from the North Atlantic population, assuming that sea states are sequentially independent. Each sea state is assumed to be stationary over a duration $d_{st} = 1$ hr.

* The detailed sizing of the structure is summarized by the stress-response normalization factor $S_*$, introduced in Eq. (4). The damping ratio of vibrations following wave impact (see Eq. (3)) is fixed at $\zeta = 10^{-2}$.

* It is assumed that the body is fixed in the reference frame of the mean flow, at a given altitude, $a$, relative to the mean water level (see Section 2.4). From Eq. (14), the mean number of impacts, for a single sea state exposure, is given by:

$$E[N_{ip}^{(st)}] = \frac{d_{st}}{T_z} \exp \left\{ -8 \left( \frac{a}{H_s} \right)^2 \right\}, \qquad (15)$$

which is used to determine the number of impacts generated by each encountered sea state. In principle, the number of impacts, generated by a given sea state, should be treated as a random variable. In this study, $E[N_{ip}^{(st)}]$ is used as a proxy for $N_{ip}^{(st)}$. This simplifying assumption is valid to evaluate the risk of fatigue-induced failure, since fatigue damage is built up by a very large number of wave impacts, themselves generated by a large number of encountered sea states.

* The randomness of the SN curve is modeled according to approach 1, as introduced in Section 2.2.2.

#### 3.2. Numerical methods

For each case study investigated, a total number of $N_{lt} = 10^4$ realizations of long-term exposure are simulated by using a Monte Carlo method. First, for each long-term realization, a sequence of encountered sea states, covering the long-term exposure duration $d_{lt}$, is drawn in the plane $(H_s, T_z)$. Then, for each sea state, the number of impacts is computed from Eq. (15), and the set of realized impact velocities is drawn from a Rayleigh distribution whose scale is determined by Eq. (12). This process yields a long-term collection of impacts, from which the long-term collection of stress cycles can be derived by using Eqs. (3)–(4), up to a factor $S_*$ (representative of the structure sizing). For numerical efficiency, the realized collections of stress-cycle amplitudes are reduced into histograms. Each long-term histogram of stress-cycle amplitudes is then paired with a realized SN curve (i.e., a material sample). Finally, for a given value of $S_*$, the long-term fatigue damage is calculated by applying Miner's rule (Eq. (5)).

### 4. Results

#### 4.1. Probability of failure

This section examines the probability of failure related to fatigue damage. Fig. 4 illustrates how the failure probability, $P_{tail}$, evolves as a function of the structural sizing factor, $S_*$. For a given configuration, $P_{tail}(S_*)$ is an increasing function, from 0 (for $S_* = 0$ there is no structural excitation, and thereby no damage) to 1 (for $S_* \to \infty$, failure occurs almost surely). As expected and shown in Fig. 4, the probability
<figure><img src="image_6.png" /><figcaption>Fig. 4. Risk of failure, $P_{\text{fail}}$, as a function of the structural sizing factor, $S_*$, Comparison of $P_{\text{fail}}$ estimates when the risk induced by fatigue is accounted for (curves labeled as ‘FAT’ in the legend), or when only the risk of ultimate strength exceedance is considered (curves labeled as ‘UTS’ in the legend). The elevation of the body is set to $a = 0$. Three different exposure durations are considered: $d_{\text{lt}} = 1$ yr (black), $d_{\text{lt}} = 5$ yr (dark gray), and $d_{\text{h}} = 20$ yr (light gray).</figcaption></figure>

of failure due to fatigue damage (solid lines in Fig. 4) rises when the exposure time is increased. For a fixed failure probability $P_{\text{fail}} = P_\epsilon$, within the range $P_\epsilon < 0.1$, increasing the exposure time from $d_{\text{lt}} = 1$ yr to $d_{\text{lt}} = 20$ yr results in a reduction of the structural sizing factor, $S_*$, by a factor $\approx 2.2$.

Fig. 4 also displays (as dashed lines) the probability of failure obtained when only the risk of ultimate strength exceedance is considered, while ignoring fatigue damage. Below, $P_{\text{fail}}^{(u)}$ is used to denote the probability of ultimate strength exceedance for a given sizing factor; and $S_*(u)$ is used to denote the sizing factor corresponding to a given probability of ultimate strength exceedance. Note that the estimate of $P_{\text{fail}}^{(u)}$ is rough, as the present approach has not been designed to specifically predict extreme events (i.e., extreme waves). As expected, for a given configuration and a given value of $S_*$, $P_{\text{fail}}^{(u)}$ is smaller than $P_{\text{fail}}$, since the risk induced by fatigue is ignored when computing $P_{\text{fail}}^{(u)}$. Furthermore, there is no overlap between the risk related to fatigue damage and the risk related to ultimate strength exceedance. Here, “no overlap” means that – as $S_*$ is decreased (for a given configuration) – $P_{\text{fail}}^{(u)}$ becomes vanishingly small before $P_{\text{fail}}$ starts to significantly deviate from unity. Consequently, in the region of the parameter space where $P_{\text{fail}} \lesssim 0.1$ the risk of failure is primarily due to fatigue damage, while the risk of ultimate strength exceedance brings no significant contribution. Also, for a given probability of failure, the structural sizing factor is noticeably larger (i.e., noticeably less conservative), when the risk due to fatigue damage is ignored. Among the considered configurations, for a small failure probability $P_\epsilon \lesssim 0.03$ (region of the parameter space usually relevant for practical applications), the ratio $S_*(P_\epsilon)/S_*(P_\epsilon)$ is in the range $\approx 3-5$, depending on the exposure duration.

## 4.2. Properties of realizations leading to failure

This section focuses on identifying the dominant classes of sea states and wave impacts that lead to failure. In the context of reliability analysis, a key question arises: is the damage leading to failure due to a large number of relatively frequent events, or is it driven by rare and extreme events? Additionally, when considering the modeling of wave impacts, an important related question concerns the degree of nonlinearity of risk-predominant sea states and waves. These two

questions are addressed in terms of sea states in Section 4.2.1, and in terms of individual waves in Section 4.2.2. Subsequently, Section 4.2.3 investigates the relative contributions of the different fatigue regimes to the risk of failure. These issues are analyzed for one specific exposure duration: $d_{\text{lt}} = 10$ yr. The structural sizing factor $S_*$ is set to a value such that the probability of failure is $P_{\text{fail}} = 2.3\%$, over the exposure duration. The elevation of the body is still set to $a = 0$. It has been checked that the specific choice of the considered configuration is not crucial for the qualitative features presented below.

### 4.2.1. Dominant class of damaging sea states

The left panel of Fig. 5 shows realized curves of cumulated damage as a function of the sea state wave steepness, $\kappa$, which is defined by:

$$ \kappa = \frac{2\pi H_s}{g T_z^2}, \qquad (16) $$

where $g$ is the acceleration due to gravity. The curves represented are restricted to long-term realizations that resulted in failure (recall that $P_{\text{fail}} = 2.3\%$ in the present case study); a total of 20 realizations resulting in failure are depicted. To construct each curve, encountered sea states – until failure occurrence – were first collected and sorted by their steepness in descending order. Therefore, Fig. 5 is suitable for identifying predominant classes of damaging sea states, in terms of wave steepness. The sample of realized curves $D_{\text{cum}}(\kappa)$ shows little dispersion. Approximately 80% of the cumulated damage is generated by sea states with wave steepness values within the range $\approx 0.030-0.072$, i.e., a range of relatively “common” steepness values.

### 4.2.2. Dominant class of damaging impacts

Continuing from the previous subsection, this one investigates which populations of water waves are predominant in terms of damage, with a focus on the degree of wave nonlinearity. To characterize the nonlinearity level of impacting waves, the slope of the free surface, at impact, is one quantity of particular interest. This quantity can be used as a proxy for the steepness of the impacting wave. Appendix B demonstrates that the conditional mean of the free-surface slope – given the impact velocity, $w$, and the sea state properties – can be used as a reasonable proxy for the realized value of the free-surface slope. Hereinafter, this conditional mean is denoted by $E[\eta_x|w]$; its expression is given in Appendix B, Eq. (B.3). According to Eq. (B.3), $E[\eta_x|w]$ is negative for $w > 0$, the latter being always satisfied for a free-surface upcrossing event (within the linear wave model). $E[\eta_x|w] < 0$ indicates that the conditional mean slope is downward in the direction of wave propagation.

The right panel of Fig. 5 shows, for a sample of realizations leading to failure, the cumulative damage, $D_{\text{cum}}$, as a function of the conditional mean slope, $E[\eta_x|w]$. To construct each curve, the encountered impacts – until failure – have been collected and sorted based on their conditional slope, in descending order of absolute value. Similarly to the situation observed regarding sea state steepness, in Section 4.2.1, the realized curves of cumulated damage as a function of $E[\eta_x|w]$ exhibit little scatter. Approximately 80% of the cumulated damage leading to failure is due to waves with $E[\eta_x|w]$ values within the range $\approx -16^\circ \rightarrow -6^\circ$. These slopes can be converted into steepness values of “equivalent” regular waves by using the following definition:

$$ k_{\text{reg}} A_{\text{reg}} \equiv |E[\eta_x|w]|, \qquad (17) $$

where $k_{\text{reg}}$ and $A_{\text{reg}}$ are the wave number and amplitude of the equivalent regular wave, respectively. The range of slopes $\approx -16^\circ \rightarrow -6^\circ$ corresponds to a range of equivalent-wave steepness $k_{\text{reg}}A_{\text{reg}} \approx (0.089 \rightarrow 0.033) \times \pi$. This range of steepness values corresponds to waves which are moderately nonlinear, and for which the linear wave model may represent a reasonable approximation.
<figure><img src="image_7.png" /><figcaption>**Fig. 5.** Left: Cumulative damage to failure, $D_{cum}$, as a function of sea state wave steepness, $κ$. Right: Cumulative damage as a function of the conditional mean of the free-surface slope, $η_{,x}$, given the impact vertical velocity, $w$, and the sea state properties (see Eq. (B.3)). In both panels the number of represented realizations is $N_{rea} = 20$. All curves are almost superimposed, which shows that the randomness of the realized sea-state sequence and wave sequence is unimportant in the present context.</figcaption></figure>

### 4.2.3. Dominant damage channels

This section examines the contribution of the different fatigue damage regimes. For all realizations leading to failure, the cumulated fatigue damage is comparably distributed between the high cycle and very high cycle regimes, with ≈ 39% of fatigue damage occurring in the high cycle regime and ≈ 61% in the very high cycle regime. There is virtually no wave impact contributing to fatigue damage in the low cycle regime: the probability of such an event occurring is extremely small when $P_{fail}$ is fixed to a low value ($P_{fail} = 0.023$ in the present section).

## 4.3. Effect of the elevation of the exposed body

In the examples presented in Sections 4.1–4.2, the control material point – representing the exposed body – has been assumed to lie at the mean water level. Consequently, all waves lead to water-entry events. In this configuration, it has been found that the risk related to fatigue damage leads to sizing constraints that are significantly more conservative compared to those obtained from the risk of ultimate strength exceedance. As the control point is moved away from the mean water level, this conclusion will not hold anymore, at some point. Indeed, as the material point elevation, $a$, increases, water-entry events become less frequent and tend to increasingly select high waves and large-$H_s$ sea states (see Eq. (15)), which gives the mean number of slamming events for a given sea state). If $a$ is increased to the point that the mean number of wave impacts become small over the exposure duration, fatigue is expected to become irrelevant to the risk of failure, since the number of cumulated stress cycles will necessarily be limited.

To apprehend the order of magnitude of $a$, for which there is a change in the nature of the risk, numerical simulations have been conducted for different values of $a$. Using the results obtained, Fig. 6 shows, as a function of $a$, the ratio $S_{*}^{(w)}(P_{fail})/S_{*}(P_{fail})$, for a fixed failure probability, $P_{fail} = 0.023$. As a reminder, $S_{*}(P_{fail})$ is the structural sizing factor which includes the risk related to fatigue damage, and $S_{*}^{(w)}(P_{fail})$ is the structural sizing factor obtained when only the risk of ultimate strength exceedance is accounted for. Fig. 6 shows a gradual decrease in $S_{*}^{(w)}/S_{*}$, as the elevation of the body increases. For an exposure duration $d_{lt} = 20$ yr (resp. $d_{lt} = 5$ yr), Fig. 6 shows that $S_{*}^{(w)}(P_{fail})/S_{*}(P_{fail})$ falls below 1.2 at $a \approx 6.9$ m (resp. $a \approx 5.8$ m). Note that, as $a$ increases, it may become important to account for the effect of wave nonlinearities both on the impact frequency and the distribution of the impact velocity. This point is discussed in Section 5.2, below.

<figure><img src="image_8.png" /><figcaption>**Fig. 6.** Effect of the elevation of the body. This figure shows the ratio $S_{*}^{(w)}/S_{*}$, as a function of the elevation of the body, for a failure probability fixed to $P_{fail} = 0.023$. Results are shown for two exposure durations: $d_{lt} = 5$ yr (dots); $d_{lt} = 20$ yr (empty circles). Further details can be found in the main body of the text.</figcaption></figure>

# 5. Discussion

Section 5.1 discusses the results presented above. Section 5.2 discusses the anticipated effect of wave nonlinearities, and how it may be included in the analysis. Section 5.3 briefly explains how the main parameters of the model may be inferred for a specific structural element. Finally, Section 5.5 highlights the key challenges that would need to be addressed, if the non-impulse regime were to be explored in future research.

## 5.1. Relevance of slamming-induced fatigue damage in the impulse regime

In the impulse regime, the risk of failure due to fatigue yields structural-sizing constraints which are noticeably more conservative than those derived solely from the risk of ultimate strength exceedance. In the examples considered in Section 4.1, for a given (small) failure
probability, $P_{\text{fail}}$, the structural sizing factor $S_*$ is 3 to 5 times smaller (depending on the exposure time) when fatigue is included in the risk analysis. The cumulated fatigue damage leading to failure mostly occurs in the high cycle and very high cycle regimes. The probability of a stress cycle reaching the low cycle branch is very small and can be considered negligible for practical purposes; this conclusion holds for $a \lesssim 6 \text{ m and } d_{\text{lt}} > 20 \text{ yr.}$

## 5.2. Accounting for wave nonlinearities

The cumulated fatigue damage is built up by a large number of encountered sea states and wave impacts, which are relatively “common”. The cumulated damage is mostly due to sea states which have characteristic wave steepness values in the range $\kappa \approx 0.030 - 0.072$. While these sea states are not particularly extreme in terms of probability of occurrence, the effect of wave nonlinearities on level-crossing statistics may be notable within this range of wave steepness. Nonlinearities will affect the predicted damage in two ways: (i) relative to Eq. (14), the frequency of wave impacts will be increased (resp. decreased) for positive (resp. negative) values of the elevation, $a$; (ii) the level-crossing conditioned distribution of wave kinematics (and thus Eq. (7)) will be modified. It is important to note that the extent of nonlinearity effects on level-crossing statistics depends on the elevation of the exposed body, $a$. To the second order, the effect of wave nonlinearities on level-crossing statistics remains modest as long as $a/H_s \ll 1$ (Hascoët, 2022). When $a$ is increased and becomes of the same order of $H_s$, the effect of wave nonlinearities becomes more notable: both the upcrossing frequency, and the scale of the upcrossing distribution of the vertical fluid velocity (at the free-surface), increase as $a$ increases.<fnref n="4" /> A preliminary investigation of the Derisk database (Pierella et al., 2021) – which is populated by numerical simulations based on a fully nonlinear potential code – has shown that these results hold beyond the second order.

Different approaches could be employed to include wave nonlinearities in the risk analysis. To the second-order, the effect of wave nonlinearities, both on the impact frequency and on the conditional distribution of wave kinematics, may be semi-analytically approached by using Edgeworth series (Hascoët, 2022). Edgeworth series are appropriate for modeling the bulk of level-crossing distributions, which is the relevant portion of distributions for fatigue analysis. Beyond the second order, another strategy could involve creating a library of upcrossing frequencies and conditional distributions, estimated for different sea state conditions. This library may be populated through experiments (in a lab or in the field), through simulations, or though a combination of both (using a multi-fidelity method). It should be noted that, for each sea state condition, the experiments (whether physical or numerical) would not need to be conducted over particularly long time frames, as the wave impacts that contribute to fatigue damage are not particularly rare events. Subsequently, a response model may be fitted to the library, and used as a replacement of the first-order wave model, in the long-term analysis. For instance, the recent database produced within the DeRisk project (Pierella et al., 2021) may be particularly well suited to produce such a library.

## 5.3. On the inference of the main model parameters for a specific structural element

The aim of this study has been to investigate whether fatigue damage should be considered as a relevant failure mode when sizing a structural element (e.g., an appendage on a ship, or a tubular element on a platform) that locally responds to bottom wave slamming. Here, “locally” applies both to the slamming loads and to the structural response. A simplified model has been introduced in Section 2.1, where

the body structure is characterized by its vibratory response time (eigen period of the dominant structural mode), $t_{\text{vib}}$, and the scale of the stress response to a wave impact, $S_*$ (introduced in Eq. (4)). The curves depicting $P_{\text{fail}}$ as a function of $S_*$, as shown in Fig. 4, may be used to estimate the probability of failure for a given structural element.

For this purpose, as a first step, the characteristic timescales $t_{\text{imp}}$ and $t_{\text{vib}}$ need to be estimated, for the structural element under study. The timescale $t_{\text{imp}}$ is related to the time evolution of the slamming loads during a wave impact. It should be evaluated for a range of impact velocities representative of the vertical fluid velocities encountered in water waves (i.e., a few m·s⁻¹), from water-entry experiments or numerical simulations. The vibratory response timescale (i.e., the eigen period of the dominant structural mode), $t_{\text{vib}}$, may be obtained through modal analysis, including the added-mass effect. Then, by comparing the characteristic times $t_{\text{imp}}$ and $t_{\text{vib}}$, the relevant stress-response regime can be identified. If $t_{\text{vib}} \gg t_{\text{imp}}$,<fnref n="5" /> the stress-response is expected to occur in the impulse regime, in which case the approach developed in this study may be applied to assess the risk of failure due to fatigue damage. In the other cases (i.e., $t_{\text{vib}} \lesssim t_{\text{imp}}$), a more advanced approach is needed to properly assess the risk of failure (see Section 5.5, below).

As a second step, the stress normalization factor, $S_*$, needs to be estimated before entering reliability curves as shown in Fig. 4. For a given detail of the considered structural element, $S_*$ may be estimated by measuring the hydroelastic response in lab experiments, full-scale (in situ) experiments, or numerical simulations. In the case of numerical simulations, high-fidelity modeling may be used, since the factor $S_*$ needs to be computed only once in the present stochastic approach. Using data from Faltinsen (1999), Appendix C illustrates, through a concrete example, how the parameter $S_*$ can be estimated from numerical simulations conducted at different impact velocities.

Whether $S_*$ should contain stress concentration factors, depends on the definition of the stress when entering the *SN* curve. The *SN* curve pattern used in this paper is schematic and does not assume a specific stress definition. The design *SN* curves proposed by classification societies depend both on the geometry of the structural detail, and on the definition of the stress used to enter the *SN* curve (see Bureau Veritas (2020), Det Norske Veritas (2021a), American Bureau of Shipping (2017) for details). It is important to ascertain which stress definition should be considered when using *SN* data.

Experiments or numerical simulations may also be valuable for other purposes: (i) checking that the amplitude of the response depends linearly on the impact velocity (impulse regime), as formulated in Section 2.1, (ii) confirming the identification of the dominant structural mode, and (iii) estimating the damping ratio, $\zeta$, of the dominant structural mode. It may happen that no single mode can be clearly identified as dominant, because multiple modes significantly contribute to fatigue damage. In such a case, the response may not be modeled through Eqs. (3)–(4), and a more advanced approach would be needed.

## 5.4. Applicability of the approach to the global response of a vessel

Several experimental studies have shown that slamming events (through the *whipping* mechanism) may constitute the dominant source of fatigue damage for the hull girder of slamming-prone vessels (see, e.g., Barhoumi and Storhaug, 2014; Storhaug et al., 2007; Storhaug, 2014; Pferdekämper and Bekker, 2024). In such a case, one might wonder whether the formalism introduced in Section 2 is applicable for the sizing of a structural detail whose design stress is directly related to the hull-girder response. In this perspective, the assumptions introduced in Section 2 would have to be checked. First, the proposed approach assumes that the impacted body is small enough relative to water wave wavelengths that it can be treated as a single material point regarding the risk of slamming event. For large vessels, this implies that the area

<footnote n="4">⁴ Both effects tend to make reliability predictions, based on the linear wave model, unconservative.</footnote>

<footnote n="5">⁵ See footnote 1.</footnote>
of the hull experiencing slamming loads (i) should be of limited extent, (ii) should be quite repeatable from one slamming event to another. Another key assumption is that slamming events can be considered as load impulses, from the perspective of the vibratory response. If these conditions are met, the proposed approach may be applicable. A critical experimental test of the applicability of the approach, would be to check whether the amplitude of the post-slam transient response is linearly correlated to the impact velocity (which should consider both wave motion and seakeeping motion), as formulated in Eq. (4). If so, the slope of the correlation between the impact velocity and the amplitude of the stress response of a structural detail would yield an estimate for $S_*$.

### 5.5. The non-impulse regime

In this study, the structural element has been assumed to respond to slamming loads in the impulse regime ($t_{vib} \gg t_{imp}$). In the other regimes (i.e., $t_{vib} \sim t_{imp}$ or $t_{vib} \ll t_{imp}$), a vibratory response may still be excited, even when the structural response timescale is much shorter than the impact timescale ($t_{vib} \ll t_{imp}$). This is because slamming loads may vary on timescales much shorter than the overall impact duration, $t_{imp}$. For instance, in the case of blunt bodies (such as a tubular element of circular section), the rise time of slamming loads may be much shorter than the decay time (see, e.g., Cointe and Armand, 1987). In the regime $t_{vib} \ll t_{imp}$ (regardless of whether a vibratory response is excited), the magnitude of the stress response is expected to be proportional to the slamming force, and thus to the square of the impact velocity (for an impact at constant velocity). This contrasts with the impulse regime, where the magnitude of stress response is proportional to the impact velocity (see Eq. (4)). This change in the velocity dependency exponent increases the weight of high-velocity impacts in the risk of failure. A preliminary investigation of this regime (results not shown) has been conducted using the same formalism as this study, but with Eq. (4) replaced by:

$$s^{(1)} = S_{*} \left( \frac{V}{1 \text{ m/s}} \right)^{2}. \qquad (18)$$

Compared to the impulse regime, this preliminary investigation reveals two key changes that significantly complicate the risk assessment:

1. The low cycle fatigue regime is found to play an important role in the risk of failure. In this study, fatigue damage is modeled using Miner's rule. A more advanced approach (considering time-history effects) would be necessary to accurately estimate fatigue damage in the low cycle regime.

2. Extreme wave impacts are found to play an important role in the fatigue-related risk of failure. The proper modeling of extreme wave impacts (probability of occurrence and wave kinematics) would require a specific approach, able to account for the nonlinearity of the most extreme waves.

## 6. Summary and conclusions

Within the context of the risk of failure induced by bottom wave slamming on marine structures, the main objective of this study has been to investigate whether fatigue damage can be a relevant failure mode (i.e., can lead to structural sizing being significantly more conservative) compared to the risk of ultimate strength exceedance. Without further specifications, the impacted body has been assumed to have a shape and a structural arrangement such that slamming loads can be considered as impulses from the structural response perspective (impulse regime). The sizing of the structure is modeled through a sizing factor, $S_*$, which sets the magnitude of the stress response, assuming the latter is dominated by a single vibratory mode. The SN curve pattern used in this study is an extended version of the fatigue curves recommended by classification societies for the high cycle and

very high cycle regimes. To combine the risks due to fatigue and due to the ultimate strength exceedance, the SN curve pattern has been extended into the low cycle regime, up to the material's ultimate strength.

In the impulse regime, fatigue damage is found to be a failure mode important to consider when designing a marine structure. For a given failure probability, it yields constraints on the sizing factor, $S_*$, which can be significantly more conservative (by up to a factor $\approx 5$, in the examples considered in this study) compared to the constraints derived solely from the risk of ultimate strength exceedance. This finding softens as the exposure time, $d_{lt}$, is decreased and/or the elevation of the body, $a$, is increased.

Investigating the non-impulse regime would require a more advanced approach that carefully addresses fatigue damage in the low cycle regime and properly models the occurrence of extreme waves. This could be the subject of a future study.

### CRediT authorship contribution statement

Romain Hascoët: Writing – review & editing, Writing – original draft, Validation, Software, Methodology, Formal analysis, Conceptualization. Nicolas Jacques: Writing – review & editing, Writing – original draft.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Appendix A. Accounting for seakeeping motions and/or forward motion

In the case studies considered above, the material point representing the body has been assumed to be fixed in the reference frame of the mean flow. This appendix explains how the approach can be adapted, when the structural element under study is attached to a moving platform. Two types of motion are considered: forward motion and/or seakeeping motions. Both types of motion are expected to affect the upcrossing frequency (see Eq. (13)) and the distribution of water-entry velocities (see Eq. (7)), but only through changes in the zeroth and second moments of the wave spectrum. With forward motion and seakeeping motion, the moments $m_0$ and $m_2$ appearing in Eqs. (13)–(7) should be replaced with (see Hascoët et al. (2020), Hascoët et al. (2021) for details):

$$\tilde{m}_0 = \int_{-\pi}^{\pi} d\theta \int_{-\infty}^{+\infty} d\omega |1 - H(\omega, \theta; V_s, \psi)|^2 G(\omega, \theta), \qquad (A.1)$$

and

$$\tilde{m}_2 = \int_{-\pi}^{\pi} d\theta \int_{-\infty}^{+\infty} d\omega [\omega - V_s \cos(\psi - \theta) k(\omega)]^2 |1 - H(\omega, \theta; V_s, \psi)|^2 G(\omega, \theta), \qquad (A.2)$$

where the quantities newly introduced are:

* $\theta$, the direction of wave propagation.

* $\psi$, the course of forward motion. Both $\theta$ and $\psi$ are measured relative to a reference direction attached to the floating platform.

* $k$, the wavenumber related to $\omega$ through the dispersion relation of water waves, $\omega^2 = g k \tanh(kh)$, with $g$ being the acceleration due to gravity, and $h$ the water depth.

* $V_s$, the speed of forward motion. The term $[\omega - V_s \cos(\psi - \theta) k(\omega)]$, appearing in Eq. (A.2), is the wave encounter frequency, induced by the forward motion.
• G, the bidimensional wave spectrum parametrized in terms of wave angular frequency, $\omega$, and direction of wave propagation, $\theta$.

• H, the transfer function accounting for the seakeeping vertical motion of the material point. The input of the transfer function is a regular wave of frequency $\omega$, and direction of propagation, $\theta$. The output is the response of the material point (representing the body) in terms of vertical motion. The term $1 - H$, appearing in Eqs. (A.1)–(A.2), may be viewed as the transfer function of the free surface elevation measured in the frame of the moving material point. Using the Response Amplitude Operators of the floating platform and the position of the structural element on the platform, the transfer function H can be readily expressed. The transfer function H may also account for the diffraction waves, as well as the waves generated by the steady (forward) and oscillatory motions of the platform, if the related transfer functions are known (see, e.g., Hermundstad and Moan, 2005). A priori, the transfer function H depends on the course and speed of forward motion, which is why $V_s$ and $\psi$ appear as parameters in Eqs. (A.1)–(A.2).

The elevation, $a$, to be used in Eq. (13), with $m_0$ and $m_2$ replaced by $\tilde{m}_0$ and $\tilde{m}_2$, is the mean elevation of the material point, relative to the mean water level. For more details about the effect of forward speed on the level-crossing distribution of wave kinematic variables, see Hascoët et al. (2021).

## Appendix B. Proxy for impacting-wave slopes

This appendix shows that the conditional mean of the local free-surface slope – given the impact velocity, $w$, and the sea state characteristics, $H_s$ and $T_z$ – can be used as a relevant proxy for the slopes of impacting waves. For this purpose, a two-dimensional frequency-direction wave spectrum is assumed:

$$G(\omega, \theta) = \mathcal{A}(\theta)S(\omega), \qquad (B.1)$$

where G is the two-dimensional wave spectrum, $\omega$ the wave angular frequency, and $\theta$ the wave direction of propagation. In Eq. (B.1), the frequency and direction dependencies have been assumed to be separable, with S being the one-sided wave frequency spectrum, and A the wave direction distribution; the function A satisfies:

$$\int_{-\pi}^{\pi} \mathcal{A}(\theta) \, d\theta = 1. \qquad (B.2)$$

The slope along the average wave direction, $\theta = \theta_0$, will be denoted as $\eta_{,x}$ and it will be used as a proxy for the steepness of the impacting waves. Within the linear wave model, $\eta_{,x}$ and $w$, non-conditioned to level-crossing, are jointly Gaussian, and are both independent of the free-surface elevation $\eta$ (see for instance Section 3.1 in Hascoët et al., 2021). Therefore, the conditional distribution of $\eta_{,x}$, given $w$, is also Gaussian. The conditional mean and conditional variance are given by:

$$E[\eta_{,x}|w] = \rho \sigma_{\eta_{,x}} \frac{w}{\sigma_w}, \qquad (B.3)$$

and

$$\mathrm{Var}[\eta_{,x}|w] = \sigma_{\eta_{,x}}^2 (1 - \rho^2), \qquad (B.4)$$

where $\rho$ is the nonconditional correlation coefficient between $\eta_{,x}$ and $w$, and $\sigma_w$ (resp. $\sigma_{\eta_{,x}}$) is the nonconditional standard deviation of $w$ (resp. $\eta_{,x}$). The nonconditional standard deviation of the vertical fluid velocity is given by $\sigma_w = \sqrt{m_2}$. Assuming an infinite water depth, the nonconditional standard deviation of $\eta_{,x}$ is given by (see Section 3.1 in Hascoët et al., 2021):

$$\sigma_{\eta_{,x}}^2 = \alpha_2 \frac{m_4}{g^2}, \qquad (B.5)$$

and the correlation coefficient is given by:

$$\rho \sigma_{\eta_{,x}} \sigma_w = E[\eta_{,x} w] = -\alpha_1 \frac{m_3}{g}. \qquad (B.6)$$

<figure><img src="image_9.png" /><figcaption>Fig. C.1. Cross section of the wedge-shaped structure considered in Faltinsen (1999). The maximum strain reported in Fig. C.2 is measured in the middle of the second longitudinal stiffener (the one pointed to by an arrow on the diagram). See Faltinsen (1999) for more details.</figcaption></figure>

In Eqs. (B.5)–(B.6), g is the acceleration due to gravity, the third- and fourth-order wave moments, $m_3$ and $m_4$, are defined by:

$$m_p = \int_{0}^{+\infty} \omega^p S(\omega) \, d\omega, \qquad (B.7)$$

and the coefficients $\alpha_1$ and $\alpha_2$ depend on the wave direction distribution as follows:

$$\alpha_p = \int_{-\pi}^{\pi} \mathcal{A}(\theta) [\cos(\theta - \theta_0)]^p \, d\theta. \qquad (B.8)$$

Note that $E[\eta_{,x}|w]$ and $\mathrm{Var}[\eta_{,x}|w]$ depend on the sea state spectrum through $\sigma_{\eta_{,x}}$, $\sigma_w$, and $\rho$. For simplicity, all sea states are assumed to have the same shape of two-dimensional wave spectrum. The wave direction distribution is assumed to be of “cosine-squared” type:

$$\mathcal{A}(\theta) = \begin{cases} (2/\pi) \cos^2 \theta & , \text{ for } |\theta| < \pi/2 \\ 0 & , \text{ for } |\theta| > \pi/2 \end{cases} \qquad (B.9)$$

with an average wave direction $\theta_0 = 0$. The wave frequency spectrum is assumed to have a JONSWAP shape (Hasselmann et al., 1973), with a peak-enhancement factor $\gamma = 3.3$. Besides, 1% of the wave energy is truncated at low and high frequency (in total 2% of wave energy is discarded). The high-frequency truncation ensures that the variance of the free-surface slope is finite. The spectrum is normalized following Eq. (9), after the truncation operation. Eqs. (B.5) to (B.9) can be used to numerically compute the correlation coefficient:

$$\rho \approx -0.922. \qquad (B.10)$$

The ratio of conditional mean and conditional standard deviation is given by:

$$\frac{E[\eta_{,x}|w]}{\sqrt{\mathrm{Var}[\eta_{,x}|w]}} = \frac{\rho}{\sqrt{1 - \rho^2}} \frac{w}{\sigma_w}. \qquad (B.11)$$

In the impulse stress-response regime, $\approx 95\%$ (resp. $\approx 50\%$) of the cumulated damage is built up by impacts whose velocities are such that $w/\sigma_w > 1.4$ (resp. $w/\sigma_w > 2.4$). Using Eqs. (B.10)–(B.11), $w/\sigma_w > 1.4$ corresponds to $E[\eta_{,x}|w]/\sqrt{\mathrm{Var}[\eta_{,x}|w]} < -3.3$, which means that the relative dispersion around the mean of $\eta_{,x}|w$ can be considered as moderate. Therefore, $E[\eta_{,x}|w]$ can be used as a relevant proxy for the realized slope $\eta_{,x}|w$.

## Appendix C. Estimation of the parameter $S_*$: illustrative example

This appendix shows how the parameter $S_*$ (introduced in Eq. (4)) can be estimated in a concrete situation. The considered case study is a wedge-shaped structure which was experimentally and numerically investigated by Faltinsen (1999). The study examines the importance of hydro-elastic effects as a function of wedge deadrise angle and impact velocity. Using a structural model based on hydroelastic orthotropic plate theory, coupled with a Wager-type water entry model, the author made numerical predictions of the maximum stress response in the
<figure><img src="image_10.png" /><figcaption>Fig. C.2. Nondimensional maximum strain, Q, as a function of a parameter P that is proportional to $t_{imp}/t_{vib}$ (see main text). Numerical data points are shown as circles. The data points with $P < 0.25$ are highlighted in red: they belong to the impulse regime, $t_{imp} \ll t_{vib}$. The red solid line is a least-squares fit of these data points with a linear function $Q = \Theta P$. [Data points were extracted from Fig. 17 in Faltinsen (1999).] (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

middle of a longitudinal stiffener (see Fig. C.1). The results are reported in Fig. C.2 (see Fig. 17 in Faltinsen (1999) for the original version). They are presented in terms of two nondimensional quantities:

$$P = \frac{\tan \beta}{V \sqrt{\rho L^3 / EI}}, \quad \text{on the x-axis,} \qquad (C.1)$$

and

$$Q = \frac{\epsilon_m EI \cdot \tan \beta}{z_a V^2 \rho L^2}, \quad \text{on the y-axis,} \qquad (C.2)$$

where $\epsilon_m$ is the maximum strain reached during the impact, $\beta$ is the deadhead angle of the wedge, $V$ is the impact velocity (assumed to be constant), $\rho$ is the mass density of the fluid, $L$ is the span of the wedge (i.e., its length along a generatrix), $E$ is the Young's modulus of the material, $I$ is the area moment of inertia of the longitudinal stiffener (and adjacent plate) about neutral axis, divided by the spacing between longitudinal stiffeners, and $z_a$ is the distance from neutral axis to maximum strain location. It can be shown that the parameter $P$ is proportional to the ratio $t_{imp}/t_{vib}$, $t_{imp}$ being the impact load timescale, and $t_{vib}$ being the wet natural period of the longitudinal stiffener (see Faltinsen (1999) for more details). In Fig. C.2, for $P \gtrsim 1.4$, $Q$ becomes approximately constant (close to 0.2), which implies $\epsilon_m \propto V^2$: this corresponds to the regime $t_{imp} \gg t_{vib}$, discussed in Section 5.5. For $P \lesssim 0.25$, $Q$ is approximately proportional to $P$, which implies $\epsilon_m \propto V$: this corresponds to the impulse regime, $t_{imp} \ll t_{vib}$, described in Section 2. The proportionality coefficient can be estimated by fitting a linear model $Q = \Theta P$ to the data points in the range $P < 0.25$: an ordinary least-squares regression (solid line in Fig. C.2) yields $\Theta \approx 0.71$. Since the amplitude of the first stress cycle is equal to the maximum stress in the impulse regime (see Fig. 2 for an illustration), an estimation of the parameter $S_*$, as defined in Eq. (4), is given by:

$$S_* = \frac{E\epsilon_m}{V} = \Theta z_a \sqrt{\rho EL/I}. \qquad (C.3)$$

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