review of ship slamming loads and responses

J. Marine Sci. Appl. (2017) 16: 427-445

DOI: https://doi.org/10.1007/s11804-017-1437-3

Review of Ship Slamming Loads and Responses

Shan Wang* and C. Guedes Soares

Centre for Marine Technology and Ocean Engineering (CENTEC), Instituto Superior Técnico, Universidade de Lisboa, Lisboa 1049-001, Portugal

Abstract: The paper presents an overview of studies of slamming on ship structures. This work focuses on the hull slamming, which is one of the most important types of slamming problems to be considered in the ship design process and the assessment of the ship safety. There are three main research aspects related to the hull slamming phenomenon, a) where and how often a slamming event occurs, b) slamming load prediction and c) structural response due to slamming loads. The approaches used in each aspect are reviewed and commented, together with the presentation of some typical results. The methodology, which combines the seakeeping analysis and slamming load prediction, is discussed for the global analysis of the hull slamming of a ship in waves. Some physical phenomena during the slamming event are discussed also. Recommendations for the future research and developments are made. Keywords: review, slamming probability, ship slamming, hydroelasticity Article ID: 1671-9433(2017)04-0427-19

1 Introduction1

When a ship sails in rough seas, she will impact the sea water because of the severe relative motions between the ship hull and the wave surface. The impulsive loads associated with the high pressure peaks occur when the ship bottom impact onto the wave surfaces with a high vertical velocity. This is often called ‘slamming’. The slamming event is mostly characterized by very large and impulsive pressures and forces that are considerably localized in time and space. The duration of the event is in the order of milliseconds.

Different types of sea water slamming exist and they are classified in two main categories: bottom slamming and breaking wave slamming. The pure bottom, straight or vertical slamming occurs when a ship structure impacts with the sea water surface, after having emerged in large amplitude motions. This type of slamming mostly occurs with smaller ships and increases with ship’s speed, occurring when they go out of the sea water and fall down again on the waves. For larger vessels, bottom slamming is more likely to happen in a ballast condition and in extreme sea states.

A similar type of slamming can occur at the underside of

Received date: 12-Jun-2017 Accepted date: 14-Oct-2017 Foundation item: Supported by Portuguese Foundation for Science and Technology (Fundação para a Ciência e Tecnologia-FCT) *Corresponding author Email: [email protected]

© Harbin Engineering University and Springer-Verlag GmbH Germany 2017

the deck of the offshore platforms, or on the connecting structure between the two hulls of a catamaran. Because the plate will have a very small deadrise angle, the wave impact loads can cause significant local structural damage, and produce transient girder loads in both longitudinal and transverse directions and induce a global whipping response.

Local structural damage because of the slamming loads has been reported by a lot of ships (Boesten, 2006; Kleefsman, 2005; Attfield, 1975; MAIB2008), especially for the ships sailing in the head waves with a high forward speed. For the Schiehallion floating production storage and offloading unit (FPSO) reported in Kleefsman (2005), a steep wave caused an indentation of the bow, which finally resulted in an evacuation of all personnel. As one of the deadliest marine disasters in 20th century, the tragedy of MV Estonia in 1994 was initialized by the failure of the bow door due to severe slamming. A last example of damage due to wave slamming was mentioned by Attfield (1975), who reported the failure of a horizontal structural member in the splash zone of the BP’s West Sole offshore platform in the southern North Sea in 1972. As reported in MAIB (2008), the accident of the container ship MSC Napoli in 2007, showed that slamming loads and associated whipping responses of a ship were not soundly incorporated in the rules of classification societies.

The examples mentioned above show that an accurate assessment of the slamming loads and the associated structural responses are of a great importance for the structural design and operation of a ship. An investigation of the slamming induced responses of a ship may require an accurate prediction of not only the slamming loads but also the conditions under which slamming occurs.

2 Slamming occurrence

2.1 Criterion of slamming occurrence The frequency of slamming occurrence is an important

reason for a ship master to reduce the speed or to change the heading. Since the ship motions and the free surface elevation in a sea state are both random in nature, the slamming occurrence which depends on these has been studied statistically.

The two initial studies on statistical characteristics of slamming are published in Tick (1958) and Ochi (1964a). The approach of Tick (1958) was to find the joint

Shan Wang, et al. Review of Ship Slamming Loads and Responses 428

probability density function for the occurrence of a slamming event and to integrate this function between the limits of the variables and then arrived at the expected number of slams per second. The general criteria he used are based on three parameters: 1) relative vertical motion between the structure and the sea surface, 2) the relative vertical velocity and 3) the angle between the cross-structure and free surface. It was assumed in Tick (1958) that the random processes of the three conditions were stationary, Gaussian and uncorrelated. Ochi (1964a) assumed that for all practical purposes the relative vertical motions and velocity between ship structure and the wave surfaces are Gaussian, random variables with a narrow-band frequency distribution. Only the first two criteria were used by Ochi, because he indicated that smallness of the keel line angle with respect to the sea surface was reflected in his concept of a threshold velocity. An experimental investigation has been performed by Guedes Soares et al. (2007) to determine the instants when a slamming event occurs for a specific bow shape. It has been concluded that the slamming at the bow is highly correlated with the local wave steepness.

Zarnick and Hong (1986) proposed a method to make an estimation of the number of slamming event per unit length for a SWATH cross-structure. The occurrence of a cross-structure slamming was also determined upon the three conditions mentioned above. Today the first two conditions are accepted in probabilistic slamming models as shown in the studies of Psaraftis (1978), Faltinsen (1993), Senjanovic and Parunov (2001), Wang and Guedes Soares (2015, 2016a), and Wang et al. (2016b).

Slamming occurrence of catamarans traveling in irregular waves with high speed has been studied in Thomas et al. (2011), Amin (2009), Matsubara (2011) and French et al. (2015). Slams were identified in Thomas et al. (2011) by defining two slam event criteria, namely, a threshold centre bow load and the change rate of the load. It has been shown that a slam occurs if the peak value is higher than 10 N (model scale) and the change rate of the load exceeds a threshold value with 5 000 N/s (model scale). Amin (2009) and Matsubara (2011) conducted a wavelet analysis, which is a time-frequency analysis methods, to strain gauge data, and found it to be an effective tool to identify both slam events and the subsequent whipping behavior. A more convenient technique to identify slam events was found by French et al. (2015) using the presence of the peak value in the pressure transducer data. The identification of the number and intensity of slam impacts of a high-speed vessel was used by Magoga et al. (2014) to establish the probabilistic models of slamming occurrences concerning the fatigue life. Full-scale experiments of hull girder stresses of an aluminium high-speed patrol boat were utilized to analyse the characteristics of slam events that were of importance in structural integrity assessment.

It was written in French et al. (2015) that a slam criterion was not necessary, instead only a peak detection was required to identify the peak pressures and thus slam events.

When a mono hull ship travels in seas at a low speed, sharp peak values could not be found in the recorded pressure. Model tests or full-scale measurements for the slamming loads are not practical for all of the ships in different sea states. It is more reasonable to estimate the hull slamming occurrence based on the relative motions and the velocity between the ship hull and the wave surface, namely the first two conditions mentioned initially. These two conditions are more relevant to the bottom slamming of a ship. The rational criteria should account for slamming loads and the associated structural responses used in the structural design.

2.2 Statistical methods The statistical approaches to slamming were discussed

in-depth by Henry and Bailey (1970) who mainly focused on the methods of Tick (1958) and Ochi (1964a). The statistical model has also been applied to determine the characteristics of the wave impact loads in Guedes Soares et al. (2017) and Wang et al. (2016a). Especially, the statistical method of Ochi (1964a, 1964b) to predict the probability of bottom slamming was reviewed by Kapsenberg (2011). The severe drawbacks of the method are that the vessel or the section is supposed first to fully emerge from the water and then enters into a flat horizontal surface of the fluid, and the definition of threshold velocity does not account for the shape of a ship. However, the method is easy to use and proved to be a valuable approach for predicting the probability of the bottom slamming of a ship advancing at a constant speed. This method was used in Wang and Guedes Soares (2015, 2016a) and Wang et al. (2016b) to estimate the probability of the bottom slamming of a chemical tanker with 170 m length advancing in extreme seas and analyze the effects of speed of the ship and the characteristics of the waves. The irregular waves can be described by a sea spectrum S(ω), with the assumption of a short-term stationary process. When the maxima (peak values) of the relative velocity between the ship hull and the sea waves follow the Rayleigh distribution, the probability of a slamming occurrence can be calculated as a joint probability between the bow emergence and exceedance of the threshold velocity during the re-entry. The expression of the joint probability was given by Jensen (2001):

2 2

2 2exp( ( ))

2 2cr

v r

V dP

(1)

where d is the draft, σv is the standard deviation of the relative vertical velocity VR and σr is the standard deviation of the relative vertical motion ζRM, which are expressed by:

2

2

0

( )( ) dR

va

VS

(2)

2

2

0

( )( ) dRM

ra

S

(3)

where 2v is the variance of the relative vertical velocity,

2r is the variance of the relative motion, and a is the

Journal of Marine Science and Application (2017) 16: 427-445 429

irregular wave amplitude. ( )R

a

V

and ( )RM

a

are the

transfer functions of the relative velocity and motion respectively. The threshold velocity is given by Ochi (1964a):

1/20.093( )crV gL (4) where g is the gravitational acceleration and L is the ship length.

The relative motions RM and velocities RV between

the ship hull and the sea surface were calculated using a time domain nonlinear seakeeping program which was proposed by Fonseca and Guedes Soares (1998a, 1998b) and extended in Rajendran et al. (2015) by accounting for the body nonlinearity in radiation/diffraction forces. Rajendran et al. (2016) developed the program by considering 2nd order Froude-Krylov force. The wave surfaces were modelled by a nonlinear Schrödinger equation as presented in Zhang et al. (2014a, 2015) and validated in Zhang et al. (2014b, 2016).

(a) TP=12 s, Hs=11.5 m

(b) TP=15.9 s, Hs=16.5 m

TP is the peak period, and Hs is the significant wave height

Fig. 1 Comparisons between the numerical probability distribution and Rayleigh distribition (Wang and Guedes Soares, 2016a)

(a) Fn=0

(b) Fn=0.04

(c) Fn=0.08

Fig. 2 Probability of slamming occurrence for a ship travelling in seas with different speeds (Wang et al., 2016b)

Fig 3 Distribution of the probability of slamming occurrence along the ship in the sea state with TP=12 s, Hs=11.5 m (Wang et al., 2016b)

Table 1 Probability of slamming occurrence for a ship section with Fn=0.08

Sea states Ship section Numerical Experiment

TP=12 s, Hs=11.5 m Section 3 0.050 7 0.033 5

TP=15.9 s, Hs=16.5 m Section 3 0.096 1 0.061 3

TP=12 s, Hs=9.7 m Section 3 0.027 6 0.016 8

Note: the section is located at the forward perpendicular for the cruising ship

Fig. 1 shows the probability distribution of the maximum values of the relative vertical velocity between a ship section and waves. It indicates that the probability distributions obtained from the numerical solution generally follow the Rayleigh distribution. The probability of a slamming occurrence for the chemical tanker is estimated statistically as a joint probability shown in Eq. (1). Fig. 2 shows the slamming probabilities for a section at the bow for the ship advancing at different speed and in various sea states. The probability distributions of slamming occurrence along the


longitudinal direction of the ship in the irregular waves with TP=12 s, Hs=11.5 m are plotted in Fig. 3. It shows that the slamming probability becomes higher for a ship traveling at a high speed in the waves with a higher significant wave height. The slamming event occurs frequently at the fore and aft parts, and the increased forward speed causes greatly increase on the slamming occurrence. As listed in Table 1, the statistical predictions of the slamming probability are higher than the results from experimental data. It is due to the slight differences of the peak values on the wave profile and ship motions between the numerical calculations and experimental data as shown in Wang and Guedes Soares (2016a).

This statistical method can be easily applied to estimate the probability of the bottom slamming ocurrence for a ship advancing in irregular waves. The results may provide insight to potential significant slamming events and help researchers to determine on which part the slamming loads should be assessed further. However, the statistical method does not account for the effects of ship’s shape or the bow-flare slamming.

3 Physical phenomena relevant to slamming

Different physical effects occur during slamming. To fully investigate the slamming loads and slamming induced structural responses, various phenomena like trapped air, compressibility effects, hydro-elastic interaction, and the free surface deformation etc., must be modelled properly (Xu and Duan, 2009).

The effects of viscosity and surface tension are negligible generally. An air cushion may be formed between the body and the sea waves when the local angle between the body surface and the sea wave surface is small at the position of the impact, which affects the interaction between air and water. Compressibility influences the flow of the air in the cushion (Faltinsen, 2005).

The experimental data of the water entry tests for flat rigid body obtained in Huera-Huarte et al. (2011) have showed that the asymptotic theory works well for assessing the loading on flat plates when the local impact angle is over 5° and the importance of the cushioning effect created by the trapped air in the initial stage of water slamming at 4° of deadrise has been evidenced. For large deadrise angles the impact velocity introduces practically no differences in the total force obtained.

Faltinsen (2005) stated that compressibility and the formation and collapse of an air cushion are important initially, and normally in a time scale that is smaller than the time scale for local maximum stresses to occur. So, their effects on the maximum local stress are generally small and it has been proved that the cushioning effect of the air pocket reduces the pressure on the structure. So, these effects are usually neglected for local slamming problems. However, one cannot exclude the possibility that the shape of the impacting free surface generates an air cushion of

sufficiently long time duration from a structural reaction point of view.

The continuity equation for the air cushion has been introduced by some researchers to study the effects on local structural response. A simplified model of water impact on the flat rigid plate was proposed by Verhagen (1967) to simulate the effect of the air cushion during the sea water entry. This model was improved by Yamamoto et al. (1983) so as to make the model more accurate. Iwanowski et al. (1993) improved the model by considering the fluid flows in both the air and water domains two-dimensional. Koehler and Kettleborough (1977) investigated numerically the problem for a rigid structure and showed that the cushioning effect of the air pocket reduces the pressure on the impacted structure.

The impulsive loads that may occur during water impact when the local angle between the body bottom and water surface is small, can cause important local dynamic hydroelastic effects. The hydroelastic response may lead to subsequent cavitation. The relative importance of hydroelasticity was studied by Faltinsen (1999), who showed that the significance of hydroelasticity for the local slamming-induced maximum stresses increased with the decreasing of the deadrise angle and the increasing of the impact velocity and natural period, and that the deadrise angle has to be small for the hydroelasticity to matter. The vibrations of the impacted body cause cavitation and ventilation. From a structural point of view, the important time scale is when the maximum stresses occur. This scale is defined by the highest wet natural period for the local structure. For the water entry of an elastic horizontal plate, it has been found by Faltinsen (1997) and Wang et al. (2016a) that the maximum strains have occurred before the cavitation has started. A Boundary Element Method (BEM) was developed by Wang and Faltinsen (2013) to investigate the air cavity formation during the high-speed water entry of wedges. Their study have showed that the cavity closure period is independent on the initial entry velocity of the wedge; the submergence depth of the wedge at pinch-off increases approximately linearly with respect to the initial entry velocity; and the cavity size is highly dependent on the mass with a larger mass causing a larger cavity.

In Wang et al. (2015a), the theoretical model of Wang et al. (2015b) and the numerical method of Wang and Faltinsen (2013) were used to investigate impact loads, motions and cavity dynamics for wedges with different deadrise angles vertically falling into the waves. It was found that gravity and acceleration of the wedge have negligible influence on the slamming pressures for a large Froude number or a small deadrise angle, before the flow separation from the body surface.

Experimental investigation of water impact on flexible bodies by Panciroli (2013) showed that large structural deformations may cause visible fluid and Structure Interaction (FSI) phenomena commonly neglected in analytical formulations or numerical solutions. There are


two possible fluid–structure interaction phenomena that never occur in rigid body water impacts: (i) the repetition of impacts and separation between the fluid and the structure in the region characterized by the fluid jet generated during the water entry and (ii) an under pressure region with a cylindrical wave front in the underwater fluid/structure interface.

4 Slamming loads

The initial approaches to estimate the slamming pressure and the associated responses of ship structure were commonly composed of two steps. By using some seakeeping program based on linear or non-linear ship motion theories (i.e. Belik et al., 1983; Guedes Soares, 1989; Rajendran et al., 2015), the relative motion between the ship and the waves was determined at the first step. When the entry velocity and location of a slamming event were determined, the slam induced loads on ship sections can be estimated by analytical models (i.e., Wagner, 1932; Howison et al., 1991; Korobkin, 2004; Mei et al., 1999), experimental studies (i.e., Aarsnes, 1996; Ramos et al., 2000; Engle and Lewis, 2003; Hermundstad and Moan, 2005; Tveitnes, 2008; De Backer et al., 2009; El Malki Alaoui et al., 2012; Tenzer et al., 2015; Wang et al., 2016c) and Computational Fluid Dynamics (CFD) approaches (i.e., Stenius et al., 2016, 2017; Alexandru et al., 2007; Veen et al., 2011, 2012; Facci et al., 2015; Facci and Ubertini , 2015).

4.1 Analytical formulations Simplified approaches, considering a rigid body for

calculating the hydrodynamic loading, which is then projected onto the flexible structure in quasi-static manner, have been applied to perform the structural analysis of a ship hull due to slamming loads.

To estimate the slamming force on a rigid body entering into water at a given speed, many methods have been used. The initial work was done by von Karman (1929) who estimated the water impact forces on the sea plane floats by using the momentum theory. The so-called Generalized von Karman Model (GvKM) was applied in Santos et al. (2013) for studying the water impact of axisymmetric bodies by using a desingularized variational numerical method. With the assumption of the potential flow and the flat-disc approximation, Wagner (1932) presented an asymptotic solution for the water impact of 2D rigid bodies with small deadrise. It was assumed that no air cavity formed during the impact. The wetted length on the section waswas determined by using the so called Wagner condition (Korobkin, 1996), which was applied as a transcendental equation in 2D (Howison et al., 1991), axisymmetric (Scolan and Korobkin, 2001; Korobkin and Scolan, 2006) and perforated (Molin and Korobkin, 2001) cases. The three-dimensional Wagner problem was derived by Scolan and Korobkin (2001) and Korobkin and Scolan (2006), and several exact solutions were found. The Wagner theory accounts for the pile-up effects and the estimations of the

wetting correction on the bottom of an entry body with the help of the Wagner condition are in good agreement with the experimental data. The theory can be easily used to impact problems and be developed further; however, the linearized boundary conditions and Bernoulli’s equation lead to overestimation of the slamming loads on an entering body.

The Wagner theory has been developed and improved. The nonlinearities of the jet flow in the intersection region between the body and free surface were taken into consideration by Armand and Cointe (1987) and Howison et al. (1991) using the asymptotic matching expansions. An analytical solution was derived in Dobrovol'skaya (1969) for a constant water entry by transferring the potential flow problem for a into a self-similar flow problem in a complex plane, with simplifications on the body shape. His solution is valid for an arbitrary deadrise angle when hydroelastic effects and compressibility are not important and air entrapment is not formed.

To improve the Wagner solution, some extra terms on the velocity potential distribution in the contact domain were included in the models (Logvinovich, 1973; Korobkin and Malenica, 2005; Zhao et al., 1996; Zhao and Faltinsen, 1993). The Original Logvinovich Model (OLM) (Logvinovich, 1973) assessed the slamming loads on a water entry body. The results agreed fairy well with the measured values even for the bodies with moderate deadrise angles. The Modified Logvinovich Model (MLM) approximately accounts for both the body shape and the nonlinear terms in the Bernoulli equation for predicting the slamming pressure on an entering body (Korobkin, 2004).

Zhao and Faltinsen (1993) generalized the work of Wagner (1932) by using linearized free-surface boundary conditions on the horizontal plane at the splash-up height and applying the actual body boundary conditions. Their results obtained from a boundary element agreed well with experimental data for the slamming force and the body pressure distribution of 2D sections. The method used in Zhao et al. (1996) is based on the so called generalized Wagner approach. Motivated by this method, Mei et al. (1999) proposed a semi-analytical solution for water impact problems of a wedge, circular cylinder and ship sections of Lewis form, while the instantaneous velocity of bodies was considered in Yettou et al. (2007). Another group of analytical models was proposed by Tulin (1957) and developed in Vorus (1996), Xu et al. (1998) and Savander et al. (2002) by neglecting the geometrical nonlinearity but accounting for the nonlinear terms in the boundary conditions of the impact problem. In addition, asymmetric impacts, which cause flow separation and an induced sway force, were also studied by these theories for large inclination angles. It also showed that even a small inclination angle may influence the water impact pressure. Khabakhpasheva et al. (2014) investigated the problem of symmetric rigid body entering water vertically at a given time-dependent speed based upon the so-called Generalised Wagner Model (GWM). The hydrodynamic pressure is


given by the non-linear Bernoulli equation and the conformal mapping of the flow region onto the lower half-plane was used.

The analytical models of water impact problem were discussed in-depth by Korobkin (2004), who was focused on the mathematical models that are based on the velocity potential calculated by the classical Wagner theory. The derivations of the OLM, MLM and GWM were provided and the models were verified by the numerical and experimental results. I was found that the OLM could be safely used for the bodies with a deadrise angle smaller than 30º, and the MLM should be perspective to be used in the practical calculations of water impact loads on a blunt body. Both of these two models improve the assessment of slamming loads on an water impacting body. The MLM was then developed in Qin et al. (2011) for considering the roll motions of the ship sections.

4.2 Model tests The experimental techniques of slamming problems were

well reviewed by Lewis et al. (2010) and Kapsenberg (2011). The biggest difficulty is the requirements for the measurement system and the data acquisition system. Very high samplings like 5–10 kHz are required to capture the peaks of the local pressures, and the pressure sensors are expensive and easily damaged by the slamming load.

For the symmetric cases, Stavovy and Chuang (1976) performed the experimental study on the water impact of rigid and elastic bodies with small deadrise, and correlated the maximum slamming pressure and the local deadrise angle by fitting the experimental data. Ochi and Motter (1973) investigated the slamming pressure, the pressure distribution and the total slamming force by analysing lots of experimental data. Zhao et al. (1996) conducted the water entry tests for a wedge with deadrise angle 30° and a bow-flare section. Their results showed that the maximum impact pressure occurred at the moment before flow separation. The model tests of free drop for two 2D sections, i.e., one wedge section and one bow-flare section were performed by Aarsnes (1996) to investigate the slamming loads on a 2D section with various roll angles. A high-speed Charge Coupled Device (CCD) camera were used by Lin and Shieh (1997) as the data acquisition system to simultaneously measure the pressure and flow field on a flat bottom body. Yettou et al. (2006) presented the experimental investigation of the pressure distribution on a free-falling wedge upon entering water. The pressure coefficients were reported for the cases with various drop heights, the deadrise angles and the masses. The existing Mei’s (1999) model that assumed a constant water entry velocity of the wedge were compared with experimental data, which showed that this method provided an adequate approximation of the maximum pressures measured.

Xu et al. (1999) investigated the acceleration of a 2D wedge section during asymmetric water entry. The model tests of a wedge of 20° deadrise angle with a heel angle ranging from 0 to 5° were performed. The wedges with

different weights were freely dropped from different heights. The wedges were free to roll after their entry into water, so the measurements cannot be directly compared with other experimental results because of different mass moments of inertia in roll motion. The initial asymmetric water entry of wedges was studied by Judge et al. (2004) considering both the horizontal and vertical impact speeds. A bow-flared section entering into calm water with different roll angle was studied by Sun and Faltinsen (2009) using numerical method and experimental analysis. Barjasteh et al. (2016) provided an experimental reference for investigating the asymmetric water entry of wedges with initial deadrise angle of 20° and 30°. The initial deadrise angle, inclination angle and impact speed were discussed. These wedges were freely falling from three different heights with inclination angles ranging from 0 to 15° with 5° increments. The effects of the inclined angles for the water entry of circular cylinders were experimentally studied in Wei and Hu (2015).

De Backe et al. (2009) experimentally investigated the water impact of 3D bodies, to predict the slamming loads on these buoys appropriate to the wave energy devices under consideration. An experimental and numerical analysis was performed for a circular cylinder either freely falling on or exiting the water in Colicchio et al. (2009). Schiffman and Spencer (1951) also give an explicit relationship of impact coefficient with f(β)=1.6 for a cone with deadrise angle 30°. E1 Malki et al. (2012) recently found the experimentally determined equivalent as f(β)=1.58 and the non-dimensional slamming coefficient f(β) depends only on the deadrise angle β. By means of high-speed shock machine, they studied the slamming coefficient on axisymmetric bodies, and found that Cs for hemisphere, unlike the cones, depends on the depth of immersions. A detailed investigation on the slamming pressure on a quasi-rigid cylinder during water entry was conducted by Van Nuffel et al. (2014).

An experimental method was described by Kapsenberg et al. (2002) to analyse the slamming loads on the aft body of a ship model. The total slamming force was calculated by integrating the measurements of the slamming pressures on the body. The bow flare slamming on a Ro-Ro vessel in regular oblique waves was studied numerically and experimentally by Hermundstad and Moan (2005). By using a nonlinear strip theory, the relative vertical motion between the ship and wave surface was predicted, and the slamming pressure was assessed by using a simplified 2D BEM method with a generalized 2D Wagner formulation. In the experimental study of Clauss et al. (2011), the ship motions in extreme waves, wave-induced load on the scaled model of a chemical tanker were measured and discussed. Based on the measured slamming pressures on a scaled model of planning craft advancing in regular waves with a high speed, the pressure distribution on the bottom plating was assessed by Santoro et al. (2014).

Recently, another group of model tests were conducted in Facci et al. (2015), Nila et al. (2013), Panciroli and Porfiri


(2013, 2015), Panciroli et al. (2015a, 2015b), Jalalisendi et al. (2015a, 2015b), Shams et al. (2015) and Shams and Porfiri (2015) by using the indirect reconstruction of the pressure from the Particle Image Velocimetry (PIV) data. Nila et al. (2013) and Panciroli and Porfiri (2013) proposed the PIV technique to reconstruct the slamming pressure from planar PIV measurements using 2D Poisson equations (Hosokawa et al., 2003) or incompressible Navier–Stokes equations (Murai et al., 2007; Jalalisendi et al., 2014). The PIV-based technique was validated by Nila et al. (2013) and Panciroli and Porfiri (2013) through experiments on wedges of 25° and 45° deadrise angle entering the water surface freely or at a constant speed. The PIV-based technique was then applied to experimentally investigate the water impact of curved rigid wedge-shaped sections (Panciroli et al., 2015a), three-dimensional bodies (Jalalisendi et al., 2015a, 2015b), asymmetric wedges (Shams et al., 2015) and elastic bodies (Shams and Porfiri, 2015; Panciroli et al., 2015b; Shams and Porfiri, 2015). The resulting velocity field is a spatially averaged approximation of the actual velocity field because the velocity vectors obtained from PIV system are based on cross-correlating the intensity distributions over small areas of the flow. Class IV lasers and high-resolution, are often used in the research with PIV method due to their low cost and high safety.

4.4 Numerical methods Slamming problems were investigated numerically by

using BEM (Alexandru, 2007; Zhao et al., 1996; Zhao and Faltinsen, 1993; Sun and Faltinsen, 1999), Smoothed Particle Hydrodynamics (SPH) (Alexandru, 2007; Veen and Gourlay, 2011, 2012; Panciroli et al., 2012; Ghadimi et al., 2013; Farsi and Ghadimi, 2016; Lind et al., 2015; Kai et al., 2009), Arbitrary Lagrangian-Eulerian (ALE) algorithm (Stenius et al., 2006, 2007; Das and Batra, 2011; Wang and Guedes Soares, 2012, 2013, 2014a, 2014d; Wang et al., 2014), and other CFD tools (Algarín et al., 2011; Swidan et al., 2013; Facci et al., 2016; Ghadimi et al., 2014; Mousaviraad et al., 2015). This section provides the presentation of the various numerical analyses of the water impact problems.

4.4.1 Boundary element method BEM was developed initially for two-dimensional water

entry problems in Greenhow and Lin (1985), and then it was developed and used to model the fluid domain during water entry of 2D and 3D bodies, including an arbitrary cross section with a jet flow approximation and rigid wedges (Zhao et al., 1996; Zhao and Faltinsen, 1993; Lin and Ho, 1994; Wu et al., 2004; Wu, 2012), curved bodies (Xiao and Batra, 2016.), horizontal circular cylinders and cylindrical shells (Sun and Faltinsen, 2006), asymmetric wedges (Xu et al., 2008), cones and spheres (Xu et al., 2011; Battistin and Iafrati, 2003), a Ro–Ro vessel (Hermundstad and Moan, 2005), and offshore platforms (Baarholm et al., 2001; Zhao et al., 2015).

As mentioned in Kapsenberg (2011), good solutions could

be obtained by using the BEMs at reasonable computational efforts with some assumptions. The compressibility effects of the fluid should be included as demonstrated by Ogilvie (1963). In the study of Zhao et al. (1996), the 2D Laplace equation was solved by using the BEM method with the approximation of the jet flows at the intersection between the body surface and the free surface. The method was validated by using the measurements from drop tests. This approach was used to vessels in 3D manner as presented by Hermundstad and Moan (2005) and Tuitman (2010). For the analysis of a whole ship slamming, the method is effective, however, it is restricted to head seas and the main problems are the existence of the bulbous bow and the choice of the angle for the 2D cuts.

Among numerical procedures, the BEM method can also account for a jet region. The moving boundaries are updated in each time step. The small contact angle between the body surface and the free surface on the jet may cause numerical errors near the intersection point during the calculations. One solution is to cut off the very thin jet near the intersection point. Zhao and Faltinsen (1993) introduced a small element normal to the body surface; however, the information of the jet flow was missing because of the cut off. Kihara (2004) introduced two models for the jet flow. One is the model to cut off the jet flow, and the other is the model to include convexity of the free surface shape. Kihara (2004) also indicated that gravity effects are important in the jet region. The procedure introduced by Sun (2007) controls the jet flow both when the jet is too thin and when the points on the jet cross to the inside of the body surface. The basic principle of these methods is to limit the contact angle to prevent numerical instabilities.

4.4.2 Smoothed particle hydrodynamics Overviews of the SPH method and the development were

given by Monaghan (1992, 1998) and Molteni et al. (2007). The SPH method is one of the meshless CFD approaches, which are suitable for the problems with large deformations. It was initially developed for compressible fluids (Gingold and Monaghan, 1997). Oger et al. (2006) indicated that a lack of stability near boundary pressure distribution in the SPH method might be due to some compressible effects generated by the sudden change of fluid flow characteristics at the very beginning of the impact. Incompressible smoothed particle hydrodynamics (Incom-SPH) were then used by Shao (2009), Aly et al. (2011) and Liu et al. (2013) to study the wave-structure interaction problems. As demonstrated in Shao (2009), the Incom-SPH method is a two-step semi-implicit hydrodynamic formulation of the SPH method and it can be used to capture the updated free surface and assessing the slamming pressuer during the fluid–structure interactions. For a free dropping object, its motion is followed by using an additional Lagrangian algorithm based on Newton's law to couple with the Incom-SPH program.

By using the SPH method, Veen and Gourlay (2012) predicted 2D slamming pressure which was then applied to


estimate the slamming force on a ship in head waves. Their results have showed that this method can be applied to investigate the slamming loads on 2D ship-sections and 3D bodies, when the proper related parameters are used.

4.4.3 Arbitrary Lagrangian-Eulerian method A general description of the ALE method and an

application of the algorithm were described in detail by Souli et al. (2000). The Euler–Lagrange coupling algorithm was developed by Aquelet et al. (2006) for the fluid and structure interaction problems. An ALE computation consists of two steps. The first step is a Lagrangian step. The second is called an advection step, which means that the state variables of the deformed material configuration are mapped back onto the reference mesh. The conservation laws of mass, momentum, and energy together with material constitutive equations are used to solve for the state variables.

The ALE method (Souli et al., 2000) and the coupling algorithm (Aquelet et al., 2006; Guo et al., 2013) are solved by using the explicit Finite Element Method (FEM) which is implemented in LS-DYNA as presented by Stenius et al. (2006, 2007), who studied the water entry problems for rigid wedge. Several parameters that influence the convergence of simulation, such as mesh density and contact stiffness were discussed. For a FSI problem, LS-DYNA searches the intersection between the Lagrangian parts and ALE parts. If a coupled Lagrangian node is detected inside an ALE element, LS-DYNA marks the Lagrangian-Eulerian coupling points at the time step. It then tracks the independent motion of the two materials over the time step based on the volume fraction in the element. Then the penetration distance of the ALE element across the Lagrangian element. Coupling force is calculated based on the penetration and then redistributed back onto the two materials. This method has been validated by comparing the experimental results with the numerical calculations by Luo et al. (2011b), Wang and Guedes Soares (2012, 2013, 2014a, 2014b, 2016a) and Wang et al. (2012, 2014, 2016a) for predicting the slamming loads on 2D and 3D ship structures. The ALE numerical algorithm can be used for predicting the slamming loads on an arbitrary simple ship component, though high computation effort is required as indicated in Wang and Guedes Soares (2014d). This method is also capable of predicting the slamming loads and associated responses of stiffened panels as presented in Luo et al. (2010, 2011a, 2012).

4.5 Comparisons of slamming loads on ship structures Alexandru et al. (2007) compared the simulations of 2D

slamming problems, by using Computational Fluid Dynamics (FLUENT and FLOW-3D codes), BEM method, and SPH and ALE algorithms. The slamming loads predicted from these methods agreed well generally. Some differences were observed on the peaks of slamming pressure for the wedge with a deadrise angle of 25°. More differences between the predictions of the slamming

pressures were found for a rigid bow section. Nevertheless, the results obtained were encouraging, but a more validation work and a parameter study still needs to be performed to obtain better numerical results. The application of these methods on slamming is still a big challenge for some issues involved. For instance, at the beginning of the water entry of a blunt body, fluid compressibility is significant but it is not usually taken into consideration in the numerical methods. The numerical convergence for ALE method is very sensitive to the mesh density and the contact stiffness of the model, and a very high computational effort is required.

Wang and Guedes Soares (2014c) reviewed the analytical models and numerical approaches for assessing the slamming pressure and total force on ship structures, including 2D wedge-shape sections, 2D circular cylinder section, 2D bow-flared sections and 3D cones. Compared with the available experimental data, the semi-analytical model of Mei et al. (1999), the simplified method of MLM and GWM and the numerical tool (BEM and ALE) have showed good agreement on the predictions of the pressure distribution and slamming force before the flow separates from the rigid bodies. Both the analytical formulations of Stavovy and Chuang (1976), Ochi and Motter (1973) were obtained by analysing lots of experimental data. The former formulation is only applicable for the wedge with small deadrise angle, while the latter can be applied to medium deadrise angle. For the non-dimensional coefficients of maximum slamming pressure on two-dimensional wedges with small deadrise angle, the results calculated by Mei et al. (1999) are about 9.8% lower, and the values predicted by using ALE algorithm are about 23% lower, compared with the calculations by using MLM. As to the slamming force, the predictions from Stavovy and Chuang (1976) are about 9.6% higher and the values from Ochi and Motter (1973) are about 58% lower, compared with the calculations using MLM. The predicted non-dimensional slamming force from ALE approach is about 19.6% lower than the values from MLM, considering the wedge with a deadrise angle of 20°. The semi-analytical model of Mei et al. (1999)’s could provide consistent predictions with the analytical models and numerical approaches, however, the solution can only be found numerically for a entering body with general shape. The results predicted from MLM indicates that the correction on the non-linear terms in Bernoulli equation together with the flat-disc approximation provides a satisfactory assessment of the slamming force on a water entry body with a small deadrise angle. The analysis of the results for the circular cylinder and the ship sections show that this model can also be applied to an arbitrary 2D section if the geometry of the body can be found. Furthermore, the non-dimensional coefficient of total slamming force on 3D cones with different deadrise angle from MLM also agrees will with the experimental and numerical results, though the predictions from ALE approach are about 3.8% higher than the calculations from MLM. Numerical tools provide satisfactory assessment of slamming pressure and total force


on 2D and 3D structures, though more careful selection of mesh size and contact parameters and high computational efforts are required for an accurate prediction. It is suggested that the MLM has much potential to be used in practical calculations of assessing slamming loads on marine structures.

Wang and Guedes Soares (2016a) investigated the slamming induced loads on the chemical tank model advancing in irregular waves. The ALE computations and MLM analytical solutions were compared with the measurements from the model tests of Clauss et al. (2011) in Figs. 4 and 5. For the slamming loads on the bow section as seen in Figure 4, the agreements between the numerical and experimental results are good, while the numerical and analytical predictions are much higher than the measurements for the stern section as shown in Fig. 5. The main differences are due to the three-dimensional effects. Due to the differences of the section’s geometry, the slamming loads curves in these two curves are totally different.

It has been proposed by Guedes Soares (1989) that there were two components for the slamming force on a ship hull. The first one is an impact component, which is caused by the initial impact between the ship bottom and wave surface and characterized by an impulse with a small duration of milliseconds. The second one is calculated by the rate of change of the hydrodynamic momentum as the hull enters into the water. In order to investigate the relationship between the geometry of a ship and the characteristics of slamming load, Fonseca et al. (2006) calculated the total slamming force for two ship sections with different geometries. Their results have showed that the slamming force component related with the hydrodynamic momentum is larger than the initial impulse for the section with a small deadrise angle at the keel and a big flare.

(a) Sensor 2

(b) Sensor 1

Fig. 4 Comparisons of the numerical results and experimental data for the slamming pressure on the bow section for a chemical tanker in extreme waves with TP=12 s, Hs=11.5 m (Wang and Guedes Soares, 2016a)

Fig. 5 Comparisons of the numerical results and

experimental data for the slamming pressure on the stern of a chemical tanker in extreme waves with TP=12 s, Hs=11.5 m (Wang and Guedes Soares, 2016b)

5 Stern slamming

More research has been performed to investigate the slamming on the frontal part of a ship. It has been proved that the probability of the slamming occurrence is the highest at the bow region, in the case of a high forward speed of the vessel sailing in extreme sea states, which usually lead to the most severe slamming events. In the case of a vessel with a zero or low forward speed and with a small draft at the stern, the aft part of the ship may impact with the wave surfaces frequently, particularly in following waves (Wang and Guedes Soares, 2016b).

When the stern of a ship enters into the water, the slamming loads with high amplitude and short-duration may occur as shown in Fig. 5, due to the flat bottom at the stern. These slamming loads may not only cause local damage of the hull structure, but also increase the global responses of the hull girder. An experimental method was presented by Kapsenberg et al. (2002) to predict slamming loads on the aft part of a ship model. The total slamming force was calculated by integrating the pressure distributions on the body. Kim et al. (2008) investigated the stern slamming on a Liquefied Natural Gas (LNG) carrier with twin skegs by using a three-dimensional CFD method. Oberhagemann et al. (2009, 2010) assessed the global design loads and responses of hull girder induced by slamming loads at the stern using a one-way coupling method.

6 Hydroelastic slamming

Hydroelasticity of ships consists of two main research aspects: global hydroelasticity and local hydroelasticity. Global hydroelasticity deals with the longitudinal bending and torsional twisting within a ship (Bishop and Price, 1979), while local hydroelasticity addresses the local deformation caused by a ship structure impacting a free surface (Faltinsen, 1999).

Bishop and Price (1979) did a pioneering work in formulating the hydroelastic theory of ship hull vibrations based on the modal analysis and considering both dry and wet mode shapes associated with a two-dimensional linear theory. Price and Wu (1985) have presented a


three-dimensional hydroelasticity theory, in which the structural modes are obtained by the finite element method and the fluid flow is solved by the three-dimensional potential theory. This theory has been applied to study different types of floating structures, i.e., Small Waterplane Area Twin-Hulled (SWATH) ships (Kean at al., 1991), and Very Large Floating Structures (VLFS) (Phan and Temarel, 2002). A review work of the hydroelasticity of ships has been presented by Hirdaris and Temarel (2009). In general it has been concluded that the global hydroelastic response of ship hulls is of importance for long and slender ships, including the recent ultra large containerships (Rajendran et al., 2015). For shorter ships this can only occur if they are made of composite materials, which are significantly more flexible than steel (Santos et al., 2009a, 2009b; Qin and Batra, 2009).

Local hydroelastic slamming, which is likely to be found more often, has been paid much attention since 1990s with the development of high speed marine vessels.

Kvalsvold and Faltinsen (1993, 1994) investigated theoretically the hydroelastic impact between the wet decks and the calm water. The theoretical calculations were compared with the experimental data by Kvalsvold et al. (1995). For a steel plate and an aluminium plate, more details on the comparison were presented in Kvalsvold and Faltinsen (1995), which included the numerical predictions, further experimental results and the calculations from a simplified approach which was proposed by Faltinsen (2000).

The review work of Faltinsen (2000) showed that the hydroelastic effect was related with the structural properties and geometry, the impact speed, and the local deadrise angle. Tenzer et al. (2015) experimentally studied the water impact for four wedge-shaped bodies, to investigate the slamming induced hydroelasticity effects on the slamming pressure peaks. Their study showed that the hydroelasticity effects on the hydrodynamic loads were found to be moderate.

6.1 Hydroelastic wave impact of wet-decks of a catamaran As multi-hull vessels tend to have better seakeeping and

wave-resistance characteristics, there is an increasing interest on multi-hull crafts both for naval and commercial applications. Local slamming is likely to happen on the wet-deck connecting two hulls of a catamaran (Davis and Whelan, 2007; Swidan et al., 2016; Swidan et al., 2017). A wet-deck has a wedge-shape cross section with a small deadrise angle, which can be zero or as large as it is for the wave-piercing catamaran.

Faltinsen et al. (1997) studied theoretically the hydroelastic slamming by representing the wet deck of a multihull as an Euler–Bernoulli beam and solving the fluid flow with a velocity potential. By using the classical Wagner (1932) theory and Euler–Bernoulli beam theory, the wave impact onto a simply supported elastic plate was analysed by Korobkin (1998) and Khabakhpasheva (2006). The slamming loads on flexible structures were calculated by using Wagner theory with linearized boundary conditions of the potential flow on the initial wave surface. This method was then developed by Korobkin et al. (2006) and

Khabakhpasheva and Korobkin (2013). Due to the flat-disk approximation of the Wagner theory, this method is only applicable to any elastic body with a small deadrise angle entering in to the sea water vertically at a moderate velocity.

The similar methods that couple the Euler–Bernoulli beam theory and Wagner (1932) theory were also used by Wang et al. (2015, 2016a), Wang and Guedes Soares (2016c) and Senjanovic et al. (2007, 2008). The asymptotic solution proposed by Faltinsen et al. (1997) was extended by Wang et al. (2016a) to include the effects of the forward speed of the ship and the compression force on the plate. For the water entry of an elastic horizontal plate, Fig. 6 compares the results of the deflections in the mid-point of the plate with different edge boundary conditions. The agreement between the different methods is satisfactory in the first half period of the structural vibration. During this period, the maximum deflection of the plate occurs. Fig. 7 presents the analytical results of the maximum deflections of the beam with various forward speeds and compressive force with L=0.5 m from the work of Wang et al. (2016a). It is found that the response of the beam increases as the forward speed and compressive force increase. The threshold values of them are dependent largely on the length of beam and the edge boundary conditions.

(a) Spring supported edge boundary condition

(b) Clamp-clamped edge boundary condition

Fig. 6 Deflection at the middle point of the plate with entry velocity of 2.5 m/s from Wang et al. (2016a)

The Euler–Bernoulli beam theory has been widely used to describe the vibration of a ship structure, neglecting the shear deformation and rotational inertia. Lv and Grenestedt (2015) investigate analytically the responses of ship hull bottom panels under slamming loads by using the linear elastic Euler–Bernoulli with the slamming loads modelled as a high-intensity peak followed by a lower constant pressure, traveling along the beam. The results showed that the lowest natural frequency of the bottom panel played a key role in


the structural response. When higher modes are important, the effect of shear deformation should be included.

Fig. 7 Maximum deflections on the plate with with respect

to forward speed and compressive force for the palte with L=0.5 m, V=2.5 m/s, and simply- supported edge boundary conditions from Wang et al. (2016a)

An alternative solution is the Timoshenko beam model, but it does not provide much difference on the results of bending moments. Kvalsvold and Faltinsen (1995) studied the hydroelastic water impact of a wet deck by representing the plate with Timoshenko beam with rotatory spring connections. It was indicated that the absolute maximum stress of the structure occurs in the free vibration phase, which was called added mass-restoring phase, and the local hydroelastic effects of the longitudinal stiffener were determined by the time scale associated with this impact phase. A theoretical hydroelastic analysis of an axially loaded uniform Timoshenko beam under slamming loads was performed in Datta and Siddiqui (2016) by considering intermediate end fixities. Lu et al. (2000) employed the coupled BEM/FEM to study the hydroelastic effects with the panel modelled as a Timoshenko beam. Hydroelastic responses of flat stiffened panels subjected to slamming loads were investigated by using the explicit finite element method in Luo et al. (2010, 2011a, and 2012) and Cheon et al. (2016). Less attention has been paid on the 3D fluid–structure interaction problems because of its complexity except in the axisymmetric case. By coupling the Wagner model with an elastic model for thin shells, Scolan (2004) investigated the hydroelastic impact of a cone. The three-dimensional Wagner problem for the deformable cone was studied numerically by Peseux et al. (2005) using a FEM code. They compared the experimental data of pressure distributions for the deformable cone models. Lakitosh and Ananthakrishnan (2012) considered the deformation of hull plates subjected to slamming loads. A suite of plate theories, ranging from small-amplitude liner undamped isotropic plate theory to damped nonlinear and stiffened plate theory, are applied to represent the plate vibrations due to the transient slamming force, which is determined based on the Wagner theory and empirical methods from Classification Society ABS Rules.

Wang (2016) has studied different types of wet-deck slamming induced responses for a catamaran. The first type is a rectangular plate hitting vertically an assumed wave

profile with a constant velocity, and the second one is a long wave hitting the frontal part of the wet-deck. For both of these two problems, the hydrodynamic loadings on the wet-deck are calculated by Wagner’s theory. The linear isotropic plate theory is used to represent the structural vibration of the first problem, while the Euler–Bernoulli beam theory is applied for the second one. Wagner’s model for two-dimensional body, which is derived from velocity potential, is used for prediction of the hydrodynamic pressure on a rigid body, with an assumption that the structural deformation doesn’t affect the loading on the wetted surface of the structure, and the structural responses induced by slammingload are calculated by mode superposition method. Fig. 8 plots the comparisons of the time history of structural deflection at the middle point of the plate from the analytical and FEM methods. The spectral analyses of structural deflections are conducted by using a Fast Fourier Transform (FFT). The results show that the first eigenmode dominates the structural oscillation of the plate, and the third eigenmode may affect the oscillation slightly. The maximum deflection increases as the impact velocity and wetted radial length increase, but decreases as the plate thickness becomes bigger. The differences between the results from the FEM and findings from the simplified model increase with the increasing impact velocity. The importance of hydroelasticity for the local slamming induced response on the plate is studied by the relation between the loading period and the natural period of vibration of the structure. It shows that a low impact velocity and a small wetted radial length of wave yield negligible effects of hydroelasticity.

(a) x=0, y=0

(b) t=0.01 s

Fig. 8 Comparison of time history of structural deflection at the middle point of the plate between theoretical results and FEM findings, bs=0.011 m, R=2 m, V=1 m/s


6.2 Water entry of an elastic wedge-shape plate As mentioned before, the wet-deck that connects the two

hull of a catamaran has also a wedge-shape cross section. The water entry of an elastic wedge-shape plate has been investigated by many researchers. Among them, Lu et al. (2000) studied the elastic wedges impact with water by using a fully coupled method. The BEM together with the fully nonlinear free surface condition was applied on the fluid flow, while the FEM was used to calculate the structural response based on the linear elastic theory. Korobkin et al. (2006) and Khabakhpasheva and Korobkin (2013) developed fully coupled methods that combine the beam finite element model with the Wagner theory. The fully coupled ALE/FEM has been used by Stenius et al. (2011, 2013), Wang and Guedes Soares (2014a, 2014b), Wang (2016) to investigate the hydroelastic effects during the water entry of an elastic wedge section. These studies showed that the largest hydroelastic effects appeared to be a time-lag effect, which did not affect the structural response dramatically. These observations correlate with trends identified in the quantifications of hydroelastic effects on slamming loaded panels in Stenius et al. (2013) based on a large number of systematic model experiments.

The structural analysis of ships and ship structural components subjected to slamming loads has also been performed by using simplified approaches for years, calculating the slamming loads on a rigid structure, and the loads are then applied on the elastic structure to estimate the response in a quasi-static manner (i.e., el Moctar et al., 2006; Lotfollahi-Yaghin et al., 2012; Schellin and el Moctar, 2007; Oberhagemann et al., 2009, 2010; Maki et al., 2011; Luo and Guedes Soares, 2012; Luo et al., 2014; Khabakhpasheva and Korobkin, 2013; Van Nuffel et al., 2014, Van Nuffel, 2014; Shams and Porfiri, 2015; Wang and Guedes Soares, 2016a; Wang, 2016). Maki et al. (2011) studied the water entry of an elastic wedge by using a loosely CFD/FEM coupled method. The CFD is applied to obtain the slamming pressure on a rigid body, and the pressure is projected onto a structural FEM model with the response calculated by using a modal analysis. The method proposed by Khabakhpasheva and Korobkin (2013) has been extended by Shams and Porfiri (2015) to address arbitrary boundary conditions and free fall impacts. The slamming loads and the associated responses of one 3D steel wedge with stiffened panels on both sides were calculated by Luo and Guedes Soares (2012). One uncoupled approach that combined the Wagner theory and a FEM was used to assess the stresses on the elastic structures. The predicted results agreed well with the experimental data from free falling tests, though the effects of structural responses on slamming pressure were not included in the simplified approach.

A simplified formulation that combines the Euler–Bernoulli beam theory for structural vibration and analytical models (GWM, OLM, MLM) derived from velocity potential for prediction of the hydrodynamic pressure on a rigid body, was proposed by Wang (2016) to

study the hydroelastic response of an elastic wedge plate impacting with calm water. The structural deflections of wedges with different deadrise angles from the simplified method were compared with published results of theoretical method and fully coupled ALE/FEM method (Wang et al., 2016a). The deflections of flexible wedges with various deadrise angles, impact velocities, and edge boundary conditions were computed and discussed. Fig. 9 shows the calculated structural deflections at the middle point of the wedge with different deadrise angles, together with the results from the fully coupled method of Lu et al. (2000). When the deadrise angle is 30°, the agreement between the simplified method and the ALE/FEM method is quite good, while the results from them are slightly larger than the calculations of Lu et al. (2000). When the deadrise angle is 45°, the results of the simplified method are higher than the calculations of the other two methods.

(a) β=30º, bs=8 mm

(b) β=45º, bs=8 mm

Fig. 9 Structural deflection at the middle point of the wedge

Combining the comparison of pressure distribution on rigid wedges, it was found that the structural deflection is not affected by the maximum pressure. Different impact velocities and edge boundary conditions were also applied in the calculation of the maximum structural deflections on a wedge with three plate thicknesses. The results showed that, as the plate thickness increased or the impact velocity decreased, the hydroelastic effect reduced enormously, thus a good agreement between the simplified method and the fully coupled ALE/FEM method was achieved. For the fully coupled ALE/FEM solver, it could be found that the in-plane fixation affected the deflection of a wedge with a small dearies angle, a thin plate thickness and a high impact velocity. The computational efficiency is improved by using the simplified method and the result is satisfying for thick


plate with a low impact velocity. Calculations have been implemented in one PC with 2.4 GHz processor and 8 Gigabytes of memory to obtain the structural responses of wedge subjected to slamming loads. The computational time is reduced to about 5 minutes by using the simplified method instead of about 24 hours by the ALE/FEM simulation for the wedge with β=30°, as an example.

6.3 Hull girder vibrations induced by slamming As demonstrated in Faltinsen (2005), the elastic

vibrations of a ship sailing in head sea are dominated by the two node longitudinal vertical bending. It is called whipping which also induces global shear force, bending moments and stresses. The natural period of the global two node bending is of the order of 1s when whipping matters. The structure can be considered to be locally rigid in the global structural analysis, given that local hydroelastic slamming has typically a time scale of the order of 10−2 s,

To assess the effects of the hull girder vibrations, transient coupled computations that combined the fluid and structural dynamics have been performed by el Moctar et al. (2006). The Reynolds-Averaged Navier–Stokes Equations (RANSE) solver COMET was applied to compute the fluid dynamic pressures, and the finite element software package ANSYS was used to perform the calculations of the structural response. The similar analysis was performed by Lotfollahi-Yaghin et al. (2012) for the dynamic response of the floating structure induced by slamming loads.

The hydroelastic ship slamming has been investigated using coupled CFD/FEM methods by el Moctar et al., (2006), Schellin and el Moctar (2007) and Oberhagemann et al., (2009). They have analysed the whipping and stern slamming on ships, and have compared their numerical results with experimental data. For this coupled CFD/FEM approach, the CFD-predicted rigid-body slamming forces are applied to the structural FEM model in a one-way manner. The added mass due to bending is taken into consideration in an approximate way using two-dimensional calculations.

Wave-induced slamming results in a transient vibratory response which is mainly in the fundamental two-node mode wet resonance. Significant stresses at the amidships will be induced. The slam-induced stresses in a containership have been studied experimentally by Ramos et al. (2000). It has been stated in Bishop and Price (1974) that modal analysis is likely to offer the best rewards for their study of ship strength. The structural dynamic behaviour of a fast patrol boat using both 2D and 3D modal analysis is studied in Santos et al. (2009a) based on non-uniform Timoshenko beam. One-way and two-way coupling approaches have been used by Paik et al. (2009) and Dhavalikar et al. (2015), to compute structural loads and ship motions of surface ships subjected to slamming. In el Moctar et al. (2006), Oberhagemann and el Moctar (2011) and el Moctar and Ley (2016), a numerical procedure based on the combined use of a BEM, a statistical analysis has been presented for determining the structural response of ships in extreme seas. OpenFOAM has been used by Southall et al. (2014), Seng et al. (2014) and Gadelho et

al. (2015) to simulate the FSI between ship structures and waves, by Jacobsen et al. (2012) and Hu et al. (2016) for wave generation and absorption.

When the transient vibrations of the ship hull are studied, the rigid body bending moment is usually ignored. The initial contributions to the combination of wave and whipping loads have been provided by Ferro and Mansour (1985). A probabilistic model based on the joint probability distribution of low-frequency wave-induced and high-frequency slamming-induced bending moments has been developed in Friis-Hansen (1994). The correlation analysis between wave and whipping bending moments has been performed and a practical method for calculation of the most probable load combination factor between considered bending moments has been presented by Ćorak et al. (2015).

7 Conclusions and recommendations

Slamming of ships has been studied extensively since the early 20th century, and it is still a very challenging subject. The paper provides a comprehensive review of the literature in the three areas: the slamming occurrence, the slamming loads and the slamming induced responses of ship structure. Ship slamming in waves can be divided into a local problem of one part of the ship’s structure impacting with the wave surface, and a global problem of the ship’s response induced by the slamming loads. This paper is mainly focused on the local slamming problem; however it is relevant to the global slamming problem.

For the bottom slamming occurrence of a ship, it has been investigated based on the relative motions and the velocity between the ship hull and the wave surface using the statistical method. When it comes to the bow-flare slamming, the relative angle between the ship hull and wave surfaces should be considered. The rational criteria should be related to slamming loads used in the structural design, or, ideally, to structural response due to slamming. To estimate accurately the slamming occurrence of a ship, the accurate prediction of ship motions in waves is important.

A large range of the methods are reviewed for the prediction of the slamming loads and slamming induced vibration of ship structures. Analytical solutions are available for some simple structures, and satisfactory results can be achieved. The numerical tools are better for the complex structures and some complicated factors (eg, compressibility effects and air cushion) are considered, however it is not easy to obtain accurate results and very high computation efforts are often required. It is recommend that the simplified approximation methods can be used in the preliminary design stage, and then the numerical tools can be applied for the analysis of specific structures.

For the global response of a ship due to slamming loads, the three aspects reviewed in this paper can be coupled. Firstly, the slamming occurrence on a ship can be assessed. Secondly, the slamming loads on the ship hull with high slamming occurrence are predicted. Thirdly, the ship hull is


approximated as a beam model and the global response of the ship is then calculated as the forced vibration induced by the computed slamming forces on different locations.

Acknowledgement

This work was performed within the Strategic Research Plan of the Centre for Marine Technology and Ocean Engineering, which is financed by Portuguese Foundation for Science and Technology (Fundação para a Ciência e Tecnologia-FCT).

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