Structural response analysis of slamming impact on free fall lifeboats

Marine Structures 54 (2017) 112e126

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Structural response analysis of slamming impact on free fall lifeboats* Jonas W. Ringsberg a, *, Svein Erling Heggelund b, Paul Lara c, Beom-Seon Jang d, Spyros E. Hirdaris e a

Department of Mechanics and Maritime Sciences, Chalmers University of Technology, Gothenburg, Sweden DNV GL, Høvik, Norway c Ship Structures Branch, Naval Surface Warfare Center, Carderock Division, West Bethesda, USA d Department of Naval Architecture and Ocean Engineering, Seoul National University, Seoul, South Korea e Lloyd's Register EMEA, Strategic Research Lab, Global Technology Centre, Southampton Bolderwood Innovation Campus, UK b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 5 September 2016 Received in revised form 8 February 2017 Accepted 28 March 2017 Available online 18 April 2017

The evaluation of impact induced slamming loads experienced by ships and offshore structures using advanced fluid structure interaction methods may be a challenging task involving complex and time consuming engineering solutions. This is the reason why to date the application of well understood and validated quasi-response approaches remains the most rational alternative used by experts for preliminary design assessment. Based on a benchmark study carried out by the International Ship and Offshore Structures Congress Technical Committee II.1 on Quasi-Static Response this paper demonstrates the practical use of “quasi-response” prediction methods for the assessment of impact loads on modern free fall lifeboats. The case study presented is considered relevant in terms of technical background and ship design for safety. Following a brief review rationalising the practical relevance of the engineering solutions examined, the influence of high speed impact is evaluated using linear-elastic and nonlinear beam models, as well as a nonlinear transient dynamic finite element analysis idealisation. Comparisons of the methods presented against experimental results led to the conclusion that the nonlinear quasi-dynamic beam approach accounts for the influence of the dynamic effects of strain by suitably idealising the effects of nonlinear geometric stiffness. It may therefore be more appropriate to employ this approach at preliminary stage, especially when conducting comparisons against simplistic linear methods used for rigid structures (e.g. stiffened steel and aluminium panels), or advanced nonlinear finite element analysis and other multi-physics methods that may be computationally complex and time consuming. © 2017 Elsevier Ltd. All rights reserved.

Keywords: Free fall lifeboat Geometric nonlinearities Preliminary ship design Quasi-response Slamming impact loads

1. Introduction Free-Fall Lifeboats (FFLBs) have shown rapid development since their commercial dawn in 1962. They are typically thin plated composite hull structures with many advantages over the traditional lifeboats lowered by wires. They can be deployed

* Paper prepared in conjunction with the ISSC committee work 2012-2015. Edited in cooperation with Professor Carlos Guedes Soares, chairman of ISSC2015 conference. * Corresponding author. E-mail address: [email protected] (J.W. Ringsberg).

http://dx.doi.org/10.1016/j.marstruc.2017.03.004 0951-8339/© 2017 Elsevier Ltd. All rights reserved.

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Nomenclature

List of abbreviations BEM Boundary Element Method CFD Computational Fluid Dynamics FE Finite Element FFLB(s) Free Fall Lifeboat(s) FSI Fluid Structure Interaction GNL Geometrically Nonlinear NL-FEA Nonlinear Finite Element Analysis OLF Norwegian Oil Industry Association (changed name to NOROG e Norsk Olje og Gass) S1 Plate strip (force transducer) in way of the stern of the FFLB S2 Plate strip (force transducer) in way of the bow of the FFLB List of symbols A Cross-section area of plate strip [m2] B Overall width of FFLB [m] Cp Pressure coefficient [
] D Dynamic factor [
] E Elastic modulus [MPa] G Shear modulus [MPa] Hs Significant wave height [m] I Moment of inertia [m4] L Characteristic length of plate strip [m] LOA Length overall [m] Lpp Overall length of FFLB [m] M Bending moment [Nm] madd Added mass [kg] N Normal force in plate strip [N] p Pressure load [Pa] P Point load acting on the plate strip [N] q(x) Distributed load on plate strip [N/m] R Characteristic radius of half circle in added mass calculations [m] T Wave period [s] t Plate strip thickness [m] t1 Duration of pressure pulse [s] Tn Natural period of the plate strip [s] V Velocity [m/s] w(x) Deflection of plate strip [m] x, y, z Coordinates [m] b Deadrise angle [degrees] ε Normal strain [ms, microstrain] r Density [kg/m3] y Poisson's ratio [
]

quickly, and due to their path through the water after impact, they can rapidly reach a safe position and velocity suited for an effective retreat from the oil platform or the ship. In 2005, the Norwegian Petroleum Safety Authority expressed concerns about the safety of the FFLBs used by the Norwegian offshore industry. This safety item followed test drops carried out at the Statoil platform Veslefrikk B, where damages were observed on FFLB superstructures. In response to this incident the “Norwegian Oil Industry Association (OLF) free fall lifeboat project” initiated by Statoil and lifeboat manufacturers played key role in instigating interest on technical issues associated with the design assessment and safety of FFLBs. Technical work carried out as part of this project formulated the basis of the existing state-of-the-art engineering approaches used for the assessment of impact loads on FFLBs and led to the development of the relevant offshore standard for design of free fall lifeboats DNV-OS-E406 [1]. In 2005 submissions by the Swedish and Norwegian Maritime Administrations to the International Maritime Organization's (IMO) Maritime Safety Committee (MSC) subcommittee on Design and Equipment (DE) also reviewed the impact of accidents and the role of model/ full scale testing in design development [2e4].

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Based on engineering experience, the design of a FFLB should ensure that the structure is not damaged in way of water entry. However, finding realistic design loads for a FFLB especially in way of water entry induced slamming remains a challenging task, especially because design assessment relies on the use of results from physical model tests, which encompass a large environmental test matrix, including wave headings, drop heights, and wind directions. Whereas by current standards this approach may be considered adequate to ensure safety and reliability under the most extreme environmental conditions [1], it is not so practical because of difficulties associated with: (i) replicating the worst point of slamming impact in a wave basin, and (ii) establishing a unified formulation for the reserve strength in drop conditions. In addition, the high pressure peaks during slamming can lead to very large deformations where the use of mathematically complex and computationally intensive tasks involving the coupling of Non-Linear Finite Element Analysis (NL-FEA) and fluid dynamics (e.g. boundary element methods - BEM or computational fluid dynamics - CFD) idealisations remains relevant [5e10]. This is why a well validated, practically rationalised, and simplified assessment tools and procedures are still considered of practical significance for preliminary design. This paper utilises practical tools and methods developed by the members of the International Ship and Offshore Structures Congress Technical Committee II.1 on Quasi-Static Response, and suggests a closed form conservative beam formula that accounts for the geometrically nonlinear effects (GNL) of large deformations occurring during impact. It is believed that the approach suggested could be used for the initial design or design assessments of slamming loads on FFLBs at preliminary design stage. In this sense, the work presented may be considered as an important milestone in: (a) Further rationalising the design assessment of this type of marine structures within the context of Classification Rules and design assessment procedures. (b) Assisting with design optimisation by suggesting a simple approach that could be used in the process of rationalising weight, cost, and manufacturing methods. (c) Assisting in the generation or assessment of safety cases or in type approval. (d) Providing better understanding of the physical processes involved in water entry of FFLBs.

2. Impact induced slamming for a FFLB The impact of a ship-like marine structure onto a wave fluid environment with a high relative velocity between the structure and the fluid is referred to as local slamming, a phenomenon directly applicable during the water entry phase of a launched FFLB. Slamming thus involves the interaction of fluid and structural components, and it encompasses a highly nonlinear and complex FSI domain. A rational engineering solution for the evaluation of impact loads used at preliminary design should idealise in a realistic fashion, or simplify with confidence, the influence of fluid actions on the structure's reserve strength. This section discusses some key items relevant to the dynamics of FFLB drop and the physics of impact induced slamming loads during water entry. Although the review is not exhaustive, it may be considered appropriate within the context of classifying the significance of the problem and providing sufficient background of relevance for preliminary design and design assessment. 2.1. Rigid body dynamics of a FFLB drop The launch of a FFLB (see Fig. 1a) includes an initial trajectory in air followed by a trajectory in water. The launching parameters for a skid-launched lifeboat in the stowed position are illustrated in Fig. 1b; Fig. 1c outlines the phases of a FFLB drop, which can be divided in five separate phases, namely: (1) release from launch system, rotation (note: only for skidlaunched lifeboats) and free-fall, (2) impact and water entry, (3) submergence, (4) ascent, and (5) resurfacing and sailing motion. The activation of the release mechanism initiates the launch event, sending the lifeboat down the skid (skidlaunched) or directly into free fall (drop-launched). During water impact maximum pressures occur. When the lifeboat first hits the water surface, high accelerations are experienced by the bow of the boat. This is bow impact. At that time, the fluid forces and the weight of the lifeboat cause the initial drop angle (and potential angular momentum induced during the rotation phase) to be reversed and the boat to returns to horizontal position leading to a stern impact. The quasi-response approaches presented in this paper focus explicitly in these local impact phenomena. 2.2. Physics and idealisation of impact loads on FFLB Most of the useful insights in the area of impact induced slamming (for the case of a FFLB or any other stiffened structure) can be realised by reviewing key experimental work (e.g. model scale or full scale drop tests), analytical solutions, and computational methods (e.g. BEM, CFD and NL-FEA approaches). A quasi-response approach should account for the relevant physics as practically as possible. This is the reason why in this section some of the key methods of relevance are briefly discussed; the keen reader may refer to e.g. Refs. [11e14].

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Fig. 1. The dynamics of a FFLB: (a) FFLB full scale test, (b) rigid body dynamics idealisation [1], and (c) launch test phases 1e5 [1].

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In way of water entry of the FFLB, assuming symmetric impact in two dimensional axisymmetric domain momentum slamming theory may be applicable. This implies that the force acting on the body can be evaluated by the rate of momentum transfer to the surrounding water, according to von Karman's [15] principle. This approach has been extended by Wagner [16], who proposed an asymptotic solution for water entry with small local dead-rise angles that considered water pile up effects. Although this approach has been widely used in area of marine hydrodynamics, it assumes that only small deformations that do not affect the fluid motion surrounding the structure are encountered. The hydrodynamic pressure is obtained using the Bernoulli equation, potential flow theory and linearized body boundary and free surface conditions. Recent analytical solutions introduced by Korobkin et al. [17] built up on this development, and focus on the influence of the wetted body area in time as another important characteristic of wave impacts that may influence the magnitude of the load applied onto the structure. However practical, none of these methods consider the influences of structurally induced GNL on dynamic response, which may be significant in way of a high impact dynamic response experienced by FFLBs researched in this paper. Computational methods, depending on their taxonomy [18] (e.g. potential flow, weakly nonlinear hydrodynamic impact methods in 2D and 3D, CFD etc.), may shed light on the influence of dynamic phenomena in way of the boundary domains (i.e. impact zone, free surface zone, etc.) of the solution. For example, the use of BEM for impact problems presented by Zhao et al. [19] is listed in Ref. [1] as one of several methods that can be used to calculate slamming loads. It is believed to suggest a feasible solution within the context of potential flow analysis, including linear or nonlinear free surfaces. In this method, the gravitational effect is usually neglected in a slamming problem due to the acceleration of the fluid during the event. However, gravity effects could be more influential in mild impacts, where the fluid acceleration cannot be assumed to be much larger than the gravitational acceleration. In this sense, they would not be considered to affect the idealisation of the high impact for the case of FFLB. On the other hand, based on later research findings [20], the weaknesses in the idealisation of the FSI phenomenon by linear or weakly nonlinear potential flow hydrodynamics may arise due to the limitation of such methods to idealise general flow separation, local jet flow with high velocities near the free surface intersection or the marked oscillations in way of the intersection of the structure and the fluid flow. Impacts may also introduce a singularity in the velocity potential causing the pressure to become infinite; an issue that may lead to limitations initiated from small submergence of the body unless the relative angle between the free surface and the surface of the body is small. However, the latter does not apply for the case of FFLB water entry. CFD methods may consider both the free surface and floating body conditions, they idealise air pockets and compressibility effects. However, the added CFD capability imposes computational uncertainties that may influence the accuracy of the solution and large demands for CPU time, where issues of unwanted numerical diffusion, slow convergence, and poor numerical stability are present [9]. In recent developments, the influence of fluid to structure coupling has been investigated by combining CFD (STAR-CCMþ) for flow, and finite element (Abaqus) for structural calculations [21,22]. The loads computed by CFD on a finite volume mesh are projected to a finite element model, carried out for a large number of snapshots separated by a fixed time interval. Alternatively, selective analyses can be performed for those cases that loads are believed to be the highest [23]. However, the large number of snapshots approach is expected to give a more complete picture of the structural capacity. The analyses are based on a quasi-static load approach, utilizing a one way coupling methodology, where hydrodynamic loads affect the structural deformations, but the structural response does not feed back to the fluid, hence lacking a two way coupling. In the case of slamming, both dynamic response and coupling effects can be important in the water entry phase. In addition, air compressibility can be important in the ventilation phase involving collapse of the air pocket created by the lifeboat. Recently, Tregde et al. [5,6] analysed these effects, and found that when air compressibility is included, the simulation time is significantly increased. However, the calculation was able to reproduce the unexpected and very high dynamic pressures. Thus, good agreement with full scale experimental data was obtained. The effect of air compressibility is relevant only for the aft section of the vessel, where the compressed air cavity makes a larger difference.

3. Free fall lifeboat experiments The numerical comparisons presented in this paper make use of model scale results derived for one of the 14 FFLBs, the Schat-Harding FF1000, studied by Marintek during the OLF joint industry project [23,24] (see Table 1). The tests were performed in a wave tank (length: 80.0 m; width: 10.5 m; depth: 10.0 m) where regular waves and irregular wave spectra were generated by a high capacity double flap wave generator. A wave probe for calibration of the waves was positioned in the middle of the wave tank, close to the water entry position.

Table 1 Main particulars of the case study lifeboat model. Parameter

Value

Overall length (Lpp) Overall width (B)  90 kg) Displacement (105%: light vessel þ 64 persons a Longitudinal centre of gravity (forward of stern) Radius of gyration in pitch (% of LOA)

12.57 m 3.34 m 16.8 tonnes 5.29 m 25%

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The model test results have been based on a 1:9 scale model, and a 50 drop angle. The pressure was measured using force transducers located on the bottom forward end (S2) and aft end (S1) of the lifeboat (see Table 2). These positions are thought to be adequately indicative of the impact loads the FFLB experiences in way of water entry and exit (see section 2.1 and Fig. 1c). Lifeboat performances were tested and assessed for different weather conditions ranging from still water to regular or irregular waves with the associated wind. The launch arrangement was placed on a working platform where the height of the platform was controlled by a winch system. The lifeboat model was launched at design angle and released from a magnetic slip hook. In order to control and regulate the water entry position in regular waves, the slip hook was released automatically by a signal from the wave probe when the wave surface passed a certain level. The hit point was checked by a second wave probe located approximately at the water entry position. The launch height was measured directly from the craft bow (the lowest point) at each launch condition. In some tests, pressure and accelerations were measured in a number of points of the lifeboat models. The instrumentation cables were then attached to the stern of the boat but they did not have any significant influence on the measured pressure. A high speed underwater video camera was used to record the underwater trajectory, and to estimate the relative velocity between the lifeboat bottom and the water surface. Fig. 3 shows an indicative example of the recording and postprocessing Table 2 Definition of locations of force transducers. Location

x (from stern)

y

z

S1 (Aft) S2 (Fore)

2.25 m 8.20 m

1.0 m 1.0 m

1.1 m 1.1 m

Fig. 2. Model test with the case study free fall lifeboat: (a) test setup and (b) body plan.

Fig. 3. Velocity at two cross section locations: for the aft (dotted line, 0.3 m from stern) and the fore (solid line, 12.3 m from stern) section. Zero velocity relative to water means that the section bottom is above sea surface.

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from a test performed in irregular waves (Hs ¼ 6.8 m, T ¼ 6.8 s). Zero velocity relative to water means that the section bottom is above sea surface, and marks the water entry moment of each sensor location. Whereas this is a representative and realistic test scenario it is acknowledged that it may not be included with ease in simplistic empirical formulae and for this reason introduces an uncertainty in the calculation of impact pressures (see section 5.1). 4. Numerical benchmark methods and analyses Benchmark comparisons focused on calculation results from different numerical methods namely: analytical plate strip models (linear and nonlinear idealisations), and FE models of different configuration and complexity (quasi-static linear, GNL linear-elastic and transient dynamic). 4.1. Linear plate strip model If the deformation of the FFLB is small (and hence linear), the lateral load can be assumed to be carried mainly by local plate in bending. The maximum strain in the middle of the plate field may then be calculated by modelling a strip of the composite plate as a simple beam (see Fig. 4). For the purposes of this study the strain was calculated using two linear plate strip methods. The first method is based on experiment results reported by Faltinsen [25]. Those were performed using stiffened aluminium and steel panels. The results of these experiments are summarised in Fig. 5, where a non-dimensional strain is plotted versus non-dimensional impact velocity. The stiffness of the composite plate strip considered in the study is within the range of the figure. The second method is based on beam theory. It takes into account the pure static response and a dynamic amplification factor. In this method, the static strain in the plate strip is expressed as ε ¼ (Mt)/(2EI) where t is the plate strip thickness, E is the elastic modulus, I is the moment of inertia, and M is the bending moment, M ¼ (pL2)/24, induced by a pressure load p over the length of a strip L. To evaluate the dynamic amplification factor the pressure pulse is assumed to be triangular; a standard loading pattern expressed by the dynamic amplification factor D ¼ pt1/Tn, where t1 is the duration of the pressure pulse and Tn is the natural period of the plate strip [27]. To simplistically account for hydrodynamic effects the influence of added mass

Fig. 4. Schematic of plate strip model of lifeboat bottom.

Fig. 5. Maximum non-dimensional strain amplitude in the middle of a longitudinal stiffener [25,26]. The solid line represents quasi-static orthotropic plate theory, and the dotted line represents hydroleastic beam theory with b ¼ 0.

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may be considered equivalent to that of a heaving semi-circle, i.e. madd ¼ 0.5rpR2, where r is the density of the water (1025 kg/m3), and R is a characteristic radius of the half circle (here, R ¼ L/2; see e.g. Fig. 6) [1,27]. 4.2. GNL plate strip model For large deformations, the load is carried mainly by membrane stresses induced across the plate strip. In this paper this is idealised using a GNL plate strip model with geometry same to that presented in Fig. 4 and load cases corresponding to a point load acting on the mid-span of the plate strip (see Fig. 6) and a uniformly distributed load. For the case of the point load the strain is derived by Eq. (3) after considering the principle of equivalence between equilibrium and kinematic conditions expressed by Eqs (1) and (2):

N¼ ε¼

P ¼ Eεt 2sinðqÞ

DL L

(1)

  ¼ cos
1 ðqÞ
1

(2)

  P ¼ sinðqÞ,ε ¼ sinðqÞ, cos
1 ðqÞ
1 2Et

(3)

After solving Eq. (3) for q, the strain can be calculated by Eq. (2). For the case of the uniformly distributed pressure load (see Fig. 7) the loading deflection w(x) is assumed to follow the standard sinusoidal shape:

wðxÞ ¼ wmax $sin

px

(4)

L

Assuming that static equilibrium in the vertical direction is fulfilled,

qL ¼ 2N$sinðqÞ ¼ 2N$sin

 w max p L

(5)

Therefore, the strain in the plate strip can be expressed as:

ε¼

DL L

¼

N EA

(6)

Using Taylor expansion, DL in Eq. (6) can be approximated as:

Fig. 6. Illustration of the point load case for the nonlinear plate strip model.

Fig. 7. Illustration of the GNL plate strip model with distributed load.

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0:25,ðwmax pÞ2 L

DLz

(7)

Combining Eqs (4)e(7) gives the expression of Eq. (8) which is solved numerically by comparing the left and right hand sides for different values of wmax.

     w   w EA DL EA wmax p2 max p max p $sin z2 q¼2 $ sin L L L L 2L L

(8)

4.3. Quasi-static and linear-elastic FE analysis models The geometry and FE model topology of Schat-Harding FF1000 FFLB were developed by DNV GL in CAD software 3DS Catia and meshed in Abaqus for CATIA [29]. A detailed presentation of the FE model and the composite material are presented in Ref. [21]. The design basis was taken from drawings and information provided by the lifeboat producer and owner. The FE model represents the major structural features including masses of significant size (passengers, engine, fuel tank, foam particulars, etc.). Fig. 8 presents FE mesh of the hull constructed of structural and buoyancy foam, and spray laminate with laminate properties as outlined in Table 3. The full model consists of approximately 600,000 elements, shell and beam elements representing the major structural components and solid elements representing the foam. The hull mesh size was nominally 50 mm. The FE model was unconstrained, and the inertia relief method was used to keep the FE model “balanced” under the applied external loads. The external loads were applied in the form of space average slamming pressures as defined in Ref. [28]; see section 5.1 for further description. A quasi-static linear-elastic FE analysis was performed, using Abaqus/CAE 6.13e3 [29] and MSC Nastran [30] FE analysis software programmes. As part of the benchmark study, the solvers and element formulations of these softwares were compared. Dynamic factors or material nonlinearities were not considered in the analysis. The elastic limit of the spray laminate is around 100 MPa, and the corresponding strain limit is around 13.3 ms The validity of the linear-elastic assumption of elastic material was ensured by comparing the strain limits derived from FE analysis (see section 5.4). Accordingly, equivalent static pressures were applied to the two plate strips of the FE model in way of the fore part (S2) and the aft part (S1) where the force transducers defined in Table 2 were positioned as depicted in Fig. 9. Since the hull profiles of the hull sections defining the strips are close to constant in the longitudinal direction, 3D geometric effects were neglected. However, the structural behaviour of this model is more like a plate while the strip model is more like a beam. This makes the structure stiffer and the deflection smaller compared to the strip models, even if the loads are applied in a similar fashion along the transverse strip.

Fig. 8. Hull FE mesh of the case study lifeboat model.

Table 3 Material properties of FE model. Property

Spray laminate

Structural foam

Buoyancy foam

Ex [MPa] Ey [MPa] Ez [MPa] yxy [
] yxz ¼ yyz [
] Gxy [MPa] Gxz ¼ Gyz [MPa] r [kg/m3]

7500 7500 5000 0.32 0.32 3000 1500 1460

42 42 42 0.3 0.3 e e 42

5 5 5 0.4 0.4 e e 100

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Fig. 9. Pressure loads applied on the FE model. Equivalent static pressure applied on S1 (red-left, upper-aft) and S2 (purple-right, lower-fore). The crosses indicate the locations of the force transducers S1 and S2. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

4.4. GNL, linear-elastic and transient dynamic FE analysis models In contrast to the models presented in sections 4.1e4.3, this analysis included as practically as possible GNL effects caused by large deformations and amplifications via quasi-static or dynamic impact loading patterns. The dynamic response is sensitive to load amplitude and duration since the pressure pulse duration Dt influences the dynamic response of the structure. Fig. 10 illustrates a long (quasi-static, Dt ¼ 11 s), and a short (dynamic, Dt ¼ 0.05 s) impulse duration, which were compared with regard strain response. Accordingly, GNL, linear-elastic and transient dynamic FE analysis were carried out for two plate strips of 100 mm width using the LS-DYNA explicit time integration FE solver [31]. Based on engineering experience it is believed that this type of analysis handled well nonlinearities including, for example, contact and material nonlinearities (note: not considered in this study) with relative ease as compared with implicit analysis. Figs. 11 and 12 illustrate the boundary conditions (B.C.) applied on selected strips in way of the aft (S1) and fore (S2) parts of the FFLB, respectively. The x-symmetric boundary condition is imposed on both the front and rear sides in the longitudinal direction, and the y-symmetric boundary conditions in way of the transverse centreline. Both strip models are fixed at knuckles and corners where a stiff longitudinal structural member crosses. Thus, the influence of strain rates was ignored and only the local plate response was considered in the analysis.

Fig. 10. The quasi-static and transient dynamic load applied to the strips of the FE model.

Fig. 11. Strip model of the aft (S1): (a) pressure-applied area (black), (b) y-symmetric B$C. (c) x-symmetric B$C. and (d) all-fixed B$C.

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Fig. 12. Strip model of the fore (S2): (a) pressure-applied area (black), (b) y-symmetric B$C. (c) x-symmetric B$C. and (d) all-fixed B$C.

5. Results and discussion Results presented in this section focus on benchmark comparisons based on maximum strain responses and deflections in way of the middle of the plate field because this represents the most onerous results. Throughout the benchmark comparisons pressures from the experimental data in way of S1 and S2 were used to load as applicable different numerical models. 5.1. Impact pressure calculation According to Faltisen [25] the space average pressure on the two plate strips was calculated as p ¼ 0.5rCpV2, where V is the velocity taken from the experimental data, r is the water density and Cp is a pressure coefficient calculated as Cp ¼ 2.5$(tan(b))
1.1 for a deadrise angle b. Note that this value represents the pressure over a broader area (i.e. several plate fields of a ship). Fig. 13 presents a comparison between calculated and measured pressures in way of strip locations S1 and S2. The calculated peak values deviate significantly from the measured values for two main reasons: (1) there is an influence of 3D geometric effects in the experiments that may be prominent especially in way of the bow area [28]; (2) the experiments were carried out for a specific wave environment (Hs ¼ 6.8 m; T ¼ 6.8 s) which is not included in the simplistic formulae properly and thereby an uncertainty is introduced in the calculation of the impact pressure. 5.2. Comparison of strip models Based on the linear methods presented in section 4.1 the strain in way of the middle of the plate field (L/2) was evaluated and results are summarised in Table 5. As shown in Table 4 the strain in S1 is twice as large as in S2. The second linear method is a simplified quasi-static method based on basic beam theory with a dynamic amplification factor. The natural period of the plate strip, Tn, is around 1.3 s and the duration of the pressure pulse T1 is 0.01 s (see Fig. 13). Thus, for a triangular pressure pulse, the dynamic amplification factor, D in section 4.1, was calculated as 0.0242. If the effect of hydrodynamic added mass is included, the natural period increases from 1.3 s to 4.7 s, whereas the dynamic amplification factor is reduced from 0.0242 to 0.007. Hydroelastic effects are important when tan(b) < 0.25V (rL3/(EI))0.5 [28]. For the study case under consideration the deadrise angle considered for the FFLB are 30 or 40 . Hence, tan(b) ¼ (for b ¼ 30 ) ¼ 0.58 or (for b ¼ 40 ) ¼ 0.83 and 0.25V

Fig. 13. Measured and calculated impact pressures in locations S1 and S2.

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Table 4 Linear method: results based on experimental data; S1 ¼ aft; S2 ¼ fore.

Velocity [m/s] Deadrise angle [deg.] Pressure [kPa] Pressure coefficient, Cp [
] Strain [ms]

S1

S2

Comment

17.6 30 570 4.2 1.07

12.8 40 140 1.7 0.54

From Fig. 3 From Fig. 13 Reference [28] Fig. 5

Table 5 Linear method: results from the simplified quasi-static method in way of plate strips S1 ¼ aft; S2 ¼ fore. Air (no added mass)

Velocity [m/s] Deadrise angle [deg.] Pressure coefficient, Cp [
] Pressure [kPa] Dynamic amplification factor [
] Maximum deflection [mm] Strain [ms]

Wet (added mass considered)

S1

S2

S1

S2

17.6 30 4.6 726 0.0242 32 3.57

12.8 40 3.0 255 0.0242 11 1.28

17.6 30 4.6 726 0.007 9 0.97

12.8 40 3.0 255 0.007 3 0.35

(rL3/(EI))0.5 z 0.1 and hydroelastic effects are not significant. By comparing Tables 4 and 5 it is shown that when the effect of hydrodynamic added mass is included, the results from the simplified quasi-static method are quite close to the results based on experimental data. Hence, the two methods show good agreement if the response is assumed linear. The GNL plate strip model presented in section 4.2 is used for two load cases namely (a) a point load, P, acting on the midspan of the plate and (b) a uniformly distributed load, q and summarised for plate strips in way of the fore and aft end of the structure in Table 5. For example, for a calculated pressure equal to 726 kPa in S1 (see Fig. 13), the magnitude of the point load, P, is equal to pL ¼ 675 kN/m. In way of the aft section, the corresponding deflection is 90 mm and the strain 19 ms For the load case with a uniformly distributed load q ¼ 726 kN/m, the maximum deflection and strain are calculated 74 mm and 14 ms, respectively. As summarised in Table 6 the point load model predicts about 35% higher strains in comparison with the uniform load model. This difference is considered to be small compared to the uncertainty in load estimation that may arise because of the actual spatial variation of the load magnitude locally.

5.3. FE analyses Quasi-static linear-elastic FE analysis was carried using Abaqus/CAE 6.13e3 [29] and MSC Nastran [30]. Generally, good agreement was achieved and the resultant strains excluding dynamic effects at S1 and S2 are presented in Table 7. The results are comparable with the results from the simplified quasi-static method presented in Table 5 when hydrodynamic added mass is not included. GNL, linear-elastic and transient dynamic FE analysis were also carried out for four cases using LS-Dyna [31]. The aim was to study the influence on the maximum transverse strains for impulse durations (Dt) of 11 s, 0.1 s, 0.05 s and 0.01 s. The three shortest impulse durations correspond to those from the experiments. The analysis with long impulse duration was carried out for comparison with the quasi-static methods. As shown in Fig. 14 maximum transverse strains occurred in the middle of two fixed points (i.e. in the “result point”). Note that in the fore strip, the S2 point exactly matches the middle point, but the S1 point is slightly off the middle point in the aft strip; the difference in strain in the result point and the exact location of S1 was found to be negligible. In the analytical strip models presented in sections 4.1 and 4.2, the strains were calculated at the midspan, and the exact positions of the S1 and S2 locations in the 3D FE (see section 4.3). the FE analysis with the longest pulse duration (Dt ¼ 11 s). Fig. 15 presents the transverse strains in the fore strip in way of three layers of the shell element (top, middle, and bottom layer) for the longest impulse duration, Dt ¼ 11 s, and the shorter impulse duration, Dt ¼ 0.05 s, respectively. The overall tendency of structural response at the fore strip (S2) is similar to the aft strip (S1), thus, only the results at S2 are explained here. It is seen that the difference between three layers, i.e. the bending strain, is small. The maximum strain is 5.45 ms, and the membrane strain is 5.0 ms This difference is even smaller when the impulse duration is shorter as identified in Fig. 15b. The results for all impulse durations are summarised in Table 8. It is clearly seen that the stiffness of the structure is so large that the dynamic response is almost the same as for the static analysis. That is, the dynamic amplification factor is close to 1 and the response is quasi-static. Compared with the FE analysis strain result for the entire boat (1.5 ms), the strain from the transient dynamic analysis using the strip FE model is much larger (5.45 ms). This is because the longitudinal stiffness of the plate between two neighbouring web sections is taken into account in the full FE model. The same difference was observed in the aft strip (S1): 3.9 ms (or 3.4 ms) versus 11.7 ms.

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J.W. Ringsberg et al. / Marine Structures 54 (2017) 112e126 Table 6 GNL strip method: S1 ¼ aft; S2 ¼ fore.

Pressure [kPa] Strain [ms] (point load, P) Strain [ms] (uniform load, q)

S1

S2

726 19.0 14.0

255 10.0 7.0

Table 7 Results from FE analysis using pressures from the experimental data. S1 Analysis code Velocity [m/s] Pressure [kPa] Strain [ms]

Abaqus/CAE 6.13e3 17.6 726 3.92

S2

S1

S2

12.8 255 1.49

Patran/Nastran 17.6 726 3.42

12.8 255 1.45

Fig. 14. Locations (result points) of maximum transverse strain from.

Fig. 15. Strain response in the fore strip (S2) from a FE analysis with (a)long pulse duration (Dt ¼ 11 s), and (b) short pulse duration (Dt ¼ 0.05 s).

5.4. Analysis summary The results from the eight different calculation methods are summarised in Table 9. The GNL transient dynamic FE model being the most realistic model (i.e. No. 8 in Table 9) has been used for final strength verification. However, a designer should

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Table 8 Maximum strain from transient analysis for S1 and S2. Pulse duration, Dt [s]

Strain [ms] in S1 (p ¼ 726 kPa)

Strain [ms] in S2 (p ¼ 255 kPa)

11 0.1 0.05 0.01

10.90 11.00 11.70 11.20

5.53 5.57 5.45 8.16

Table 9 Summary of results from calculated strains using eight methods. Method

Strain [ms] S1 (aft)

S2 (fore)

1. 2. 3. 4. 5. 6. 7. 8.

1.07 3.57 0.97 3.92 3.42 19.0 14.0 11.70

0.54 1.28 0.35 1.49 1.45 10.0 7.0 5.45

Experiments (steel and aluminium stiffened panels) Linear plate strip model, hydrodynamic added mass not included Linear plate strip model, hydrodynamic added mass included Quasi-static, linear-elastic FE analysis (Abaqus/CAE 6.13e3) Quasi-static, linear-elastic FE analysis (Patran/Nastran) GNL plate strip model, point load GNL plate strip model, uniformly distributed load GNL linear-elastic and transient dynamic FE analysis (Dt ¼ 0.05 s)

consider a large number of uncertainties at early design phase, and the conservative nature of the strain results, may be a limitation of an assessment utilizing a GNL plate strip model (see section 5.3). As shown in Table 9 the GNL transient dynamic analysis results (Method 8, Strain S2 ¼ 5.45 ms) are smaller than the nonlinear plate strip mode (Method 7, Strain S2 ¼ 7.0 ms). This is explained because of the larger initial deflection of the former method that may lead to amplification of membrane effects in way of the mid span of the plate and hence lead to lower loadinduced deflection. Similar results are confirmed for the aft plate strip section (S1). Another observation is that except the two GNL plate strip idealisations all models demonstrate strain limits below 13.3 ms This implies that the assumption of linearelastic material response adopted in the study is valid. Notwithstanding it could be expected that if material nonlinearities were taken into account the strain values of two GNL plate strip models (see Methods 6 and 7 in Table 9) would become marginally higher. 6. Conclusions This paper presented a benchmark study carried out by the International Ship and Offshore Structures Congress Technical Committee II.1 on Quasi-Static Response with the aim to demonstrate the practical use of quasi-response methods for the assessment of impact loads on modern FFLBs. In total, eight different calculation methods were compared based on analytical plate strip models (linear and nonlinear idealisations), and FE models of different configuration and complexity (quasi-static linear, GNL, linear-elastic and transient dynamic). The strains calculated by the GNL, linear-elastic and transient dynamic FE analysis showed that the strain calculated by the GNL may be large. Hence, the water entry impact loads are mainly carried by membrane stresses and the effect of plate bending may be considered to be small. The calculations also show that the response is close to quasi-static, i.e. the dynamic amplification factor is close to 1.0. Irrespective to modelling assumptions or solvers used strain responses are below the strain limit of plastic responses; a matter that implies the validity of linear-elastic material response. However, irrespective to loading assumption (point or uniformly distributed) the magnitude of strains from analytical models were larger than the results from the nonlinear FE analyses. One reason for this could be that the plate strip model disregards the bending stress in the plate. Comparisons of the methods presented against experimental results (i.e. Method 1) led to the conclusion that the GNL quasi-dynamic beam approach accounts for the influence of the dynamic effects of strain by suitably idealising the effects of GNL stiffness in case of high impact velocities. It may therefore be more appropriate to employ this approach at preliminary stage, especially when conducting comparisons against simplistic linear methods used for rigid structures (e.g. stiffened steel and aluminium panels), or advanced nonlinear finite element analysis, and other multi-physics methods that may be computationally complex and time consuming. Notwithstanding this conclusion does not replace the importance of using a GNL transient dynamic FE model in final strength verifications. Acknowledgments The results and opinions presented in this publication do not necessary express the commercial views of the corporate organisations represented by the authors. The authors wish to express their thanks of appreciation to the Norwegian Oil Industry Association (OLF), Schat-Harding, Statoil and Marintek for the permission to use the illustrations under Fig. 2 as well

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