Impact Response of a Double-Bottom Structure with High and Penetrated Girders and Floors

Abstract

This paper presents an experimental and numerical analysis of the response of a scaled double-bottom structure with high and penetrated girders and floors impacted vertically by a rock-shaped indenter. The specimen, scaled from the bottom structure of the power-battery cabin of a new energy ship, is struck by a spherical indenter. The special double-bottom structure is designed to protect the power batteries and to facilitate heat dissipation. The experimental overall impact response, vibration acceleration, and stress of the inner bottom plate are measured in order to evaluate the impact environment in the target cabin. The investigation provides valuable information to evaluate the safety of power-battery cabins in a ship grounding scenario. The experimental results show good agreement with the finite element analyses using the explicit LS-DYNA software. The numerical analysis outlines the influence of the structural openings on the impact response and also the effect of battery mass and striking velocity on the impact environment in the target cabin.

1. Introduction

New energy vessels, such as electric-powered ships, are developing rapidly, and the safety of power sources, such as power batteries, is of concern for developing ships of large sizes. The power batteries differ from other cargo and produce lots of heat during ship operations. In the ship design, the power batteries are often arranged on the inner bottom of ships. In the event of a ship grounding, the instantaneous impact on the ship bottom may cause the failure of internal equipment. To ensure the safety of power batteries, the design of double-bottom structures should consider not only their efficient heat dissipation but also their anti-grounding capacity. One of the good structural designs is double-bottom structures with high and penetrated girders and floors, i.e., multi-openings exist on the bottom girders and floors. The novel design complicates the impact behaviour of this type of double-hull structure. Therefore, it is of importance to assess accurately the impact force, deformation, vibration acceleration, and stress of the double-bottom structures during ship grounding.

In maritime navigation, grounding accidents can be categorised into two primary scenarios: drifting and powered grounding [1,2]. These scenarios are characterised by two types of impacts, i.e., vertical penetration, commonly known as “stranding”, and horizontal sliding, commonly known as “raking”, as depicted in Figure 1. In the case of stranding (Figure 1a), the ship’s bottom structures experience lateral penetration due to the combined effects of pitch and heave movements induced by wave motion. In the case of raking (Figure 1b), the ship’s hull slides over a seabed obstruction, and the bottom structures are subjected to severe plastic deformation and even rupture. In the safety assessment of the power-battery cabin, the stranding scenario should be more dangerous than the raking scenario. In the stranding scenario, the impact location may be underneath the target cabin, and the bottom structures experience very large dent deformation in a short period of time, causing high vibration acceleration in the ship cabin. In the raking scenario, the initial impact position is close to the bow, far from the target cabin in the midship, and the ship sliding time is relatively long; thus, the effect on the vibration in the ship cabin is less significant.

Liu et al. [3,4] reviewed the technical developments in ship collision analysis and the challenges to an accidental limit state (ALS) design method of ships based on collisions, which is emphasised to establish the reasonable acceptance criterion for the anti-collision structures in the ALS design of ships. Liu et al. [5] proposed a conceptual design framework for analysing the crashworthiness of double-hull structures in ship collisions and grounding, and the statistical seabed shapes and sizes are presented. In stranding scenarios, the topologies of seabed obstacles can be categorised into rock and shoal types [2]. The rock-type seabed obstruction is typically represented by a conical indenter characterised by its spreading angle [6,7,8,9].

Conducting a full-scale ship grounding experiment is widely recognised as the most valuable and reliable method for assessing the crashworthiness of ship structures. However, this approach is often constrained by many factors such as the prohibitive costs and the requirement for extensive testing facilities. As a practical alternative, large-scale impact testing offers a feasible method to examine and analyse the response of ship structures in accidental scenarios. A series of large-scale experiments on the stiffened panels and double-bottom structures were conducted to evaluate the structural crashworthiness [6,7,10,11,12,13,14]. Nevertheless, most of the experiments were carried out in a quasi-static manner to determine the relationship between contact force and structural deformation. The motivation of these experiments is to investigate the plastic deformation and fracture mechanisms of double-hull ship structures. They demonstrate the physical phenomenon to gain insight into the structural damage behaviour in collision and grounding accidents. In double-hull structures, the outer hull panel experiences membrane tension in the plastic deformation region and the rotation of plastic hinges [11,14], and the web girder experiences the plastically compressed folding deformation subjected to in-plane loading [15,16]. However, the investigation into the vibration acceleration of structures does not account for during and after the impact in the experimental tests. In general, a series of experiments are carried out in order to validate the analytical calculations and finite element simulations.

Finite element analysis offers a cost-effective approach to assess the response of double-bottom structures under impact. In such simulations, most numerical definitions have become standardised except the definition of material nonlinearities. The key material definitions are involved in plastic strain hardening, strain rate sensitivity, and dynamic fracture strain [3,17,18,19,20,21]. The material strain hardening is captured accurately by the true stress–strain curve, which is often described by a power–law relationship. The strain rate sensitivity is crucial in determining the response of structures under impact. For low-velocity impacts, the Cowper–Symonds model [22] is frequently employed by catering to low to moderate strain rate issues [23]. However, the numerical definition of dynamic fracture strain remains a challenge since the magnitude of fracture strain is influenced strongly by the size of finite elements. Ringsberg et al. [24] highlighted this challenge by presenting results from fifteen research groups who independently conducted numerical simulations of an indentation experiment on a double-hull structure. The investigation revealed variances in numerical and experimental absorbed energies ranging from −28% to 13% at maximum penetration, underscoring the difficulty of accurately predicting experimental outcomes, even among established researchers in the field. However, the numerical analysis of the vibration acceleration of structures under impact is still not carried out in the literature since special attention has been paid to this issue recently with the development of new energy ships.

The aim of this study is to investigate the impact response of ship hulls in stranding. A falling weight impact test and a series of numerical simulations are performed to investigate the crashworthiness of a scaled double-bottom structure with high and penetrated girders and floors impacted by a rigid rock-shaped indenter. The specimen is scaled from the bottom structure of the power-battery cabin of a new energy ship. The investigation analyses the experimental and numerical results, examining the structural force–displacement response, deformation pattern, and vibration acceleration response. The numerical analysis discusses the influence of the structural openings on the impact response and also the effect of battery mass and striking velocity on the impact environment in the target cabin.

2. Experimental Details

The aim of the experimental programme is to represent a ship-stranding event where a new-energy ship bottom structure impacts vertically on a rock (Figure 2). Compared to traditional double-bottom structures, the double-bottom structure is designed with high and penetrated girders and floors, i.e., there are multiple openings on both bottom girders and floors. This design is to satisfy not only the requirement of heat dissipation in the cabin of power batteries for serviceability safety but also the requirement of vibration reduction in the potential ship grounding accident for ALS design.

The ship stranding event is scaled and examined in the impact experiment. For reference, the main dimensions of the prototype structure and the scaled specimen are given in

Table 1. Main dimensions of the prototype structure and scaled specimen.

Main Dimensions	Unit	Prototype Structure	Scaled Specimen
Total length	mm	14,400	1440
Total width	mm	11,250	1125
Total height	mm	9800	980
Bottom height	mm	2800	280
Frame spacing	mm	800	160
Thickness of inner bottom plate	mm	20	4.0
Thickness of outer bottom plate	mm	20	4.0
Thickness of transverse floor	mm	15	3.0
Thickness of longitudinal girder	mm	15	3.0

. The scaling of length and thickness are 1:10 and 1:5, respectively. The three-dimensional configuration and typical cross-sections of the specimen are shown in Figure 3. The specimen includes all of the double-bottom structures and part of the cabin structures. The bottom structures of the specimen include 7 longitudinal girders and 10 transverse floors.

The mean value of the statistical geometry of rock in grounding accidents can be found in Refs. [5,25,26,27,28], and the medium shape can be expressed as conical with a tip radius of 4.0 m and a spreading angle of 27.5°. In the experiment, a scale of 1:10 is selected, and a spherical shape is designed to describe the rock tip. This simplification is reasonable since the predetermined indentation of the specimen is relatively small, so the contact area is the spherical shape.

The materials used to fabricate the specimen are high-strength steels, which have mechanical properties that are determined through quasi-static tension tests. Three tensile tests are conducted for each plate thickness (3.0 mm and 4.0 mm) in the specimen. The material mechanical properties are summarised in

Table 2. Mechanical properties of the steel.

Property	Unit	PL. 3 mm	PL. 4 mm
Yield stress	MPa	338	312
Ultimate tensile strength	MPa	462	445
Fracture strain	-	0.24	0.25

, and the corresponding tensile stress–strain curves are plotted in Figure 4.

The impact experiment is conducted using a falling-weight impact machine (see Figure 5a). An indenter attached to a large weight block (total mass of 1450 kg) is dropped from a known height (0.45 m) between guide rails onto the specimen fixed on the foundation support (see Figure 5b). Two accelerometers are installed on the indenter to record the acceleration–time data during the impact (Figure 6a). Multiplying these data by the total mass of the indenter assembly can obtain the contact force–time curve. With the prescribed height of the indenter (0.45 m), the initial impact velocity (3.0 m/s) can be calculated, and the incident kinetic energy is estimated at 6525 J. Using these known data, the force–displacement response and absorbed energy–displacement curve can be derived through numerical integration.

For capturing the structural response inside the cabin, two accelerometers are installed on the inner bottom plate of the specimen to record the acceleration response during and after the impact. Moreover, the three-direction strain gauges are arranged at the inner bottom plate. The layout of accelerometers and strain gauges is shown in Figure 6b. The numerical analysis was performed before the experimental test to search the critical regions in the specimen. It helps to obtain effective data using limited measurements in the experiment.

3. Finite Element Modelling

The numerical simulations are carried out using an explicit finite element analysis software LS-DYNA version 971. The numerical model comprises two components, specimen and indenter, as illustrated in Figure 7. The numerical definitions used in the impact simulations are in agreement with those established in prior numerical studies [29,30]; thus, they are described briefly in the present study. It mainly focuses on the practical methodology for defining the material properties of the specimen and indenter.

The specimen is modelled using Belytschko–Lin–Tsay shell elements, incorporating five integration points throughout its thickness. The selected mesh size is 20 mm, which is fine enough to create the model of the penetrated girders and floors in the double-bottom specimen (see Figure 3). The nodes at the specimen edges welded to the foundation support are fixed in all degrees of freedom.

The selected material model for the specimen is “Mat.024 Piecewise Linear Plasticity”, and this model facilitates the definition of an accurate stress–strain curve through an offset table. The material behaviour is depicted using a “combined material model”, where the flow stress curve is linked using two segments at the necking point [31]. In the first segment before necking, the true stress (σ_t) and true strain (ε_t) are expressed in terms of the engineering stress (σ_e) and the engineering strain (ε_e) as

$${\sigma }_{\mathrm{t}}={\sigma }_{\mathrm{e}}({\epsilon }_{\mathrm{e}}+1)$$

(1)

$${\epsilon }_{\mathrm{t}}=\ln ({\epsilon }_{\mathrm{e}}+1)$$

(2)

In the second segment beyond necking, the true stress and true strain relation is expressed by the power–law function:

$${\sigma }_{\mathrm{t}}=K{\epsilon }_{\mathrm{t}}^n$$

(3)

where the variables K and n are determined by extrapolating the true stress–strain data in the first segment. This combined material curve can well describe the true stress–strain relationship of steel materials. As no failure occurs in the impact test, the critical failure strain is not defined in the numerical model.

The strain rate sensitivity of the steel materials is evaluated using the Cowper–Symonds model [22]:

$${\sigma }_{\mathrm{d}}={\sigma }_0{\left(1+\frac{\stackrel{\cdot }{\epsilon }}{D}\right)}^{1/q}$$

(4)

where σ_d is the dynamic true stress corresponding to the plastic strain rate ε, σ₀ is the associated static true stress, and D and q are constants for a specific material. For the high-strength steels with a static yield stress of around 300 MPa, the material constants are often defined using D = 3200 s⁻¹ and q = 5 [29,32]. These values are also used in the current material definitions since they can describe approximately the strain rate effect of steel materials in low-velocity impacts.

The indenter assembly is modelled as a simplified rigid body (Figure 7), and the selected material model is “Mat.020 Rigid” in LS-DYNA. It is assigned an artificially large density to the rigid body, matching the same striking mass (1450 kg) in the experiment. The initial impact velocity (3.0 m/s) is defined in the free vertical translation of the rigid body. This simple modelling used to represent the actual experimental indenter assembly aligns with previous experimental and numerical investigations [29,30]. The contact between the specimen and indenter is set as “Automatic Surface to Surface” with a friction coefficient of 0.3 [30].

4. Experimental and Numerical Results

4.1. Impact Force and Deformation

The permanent deformations of the tested specimen obtained from the experiment and simulation are presented in Figure 8. The specimen is cut at the central cross-section after the experiment in order to observe the deformation inside the double-bottom structure (Figure 9). The experimental and numerical impact force–displacement curves are given in Figure 10. Generally, the plastic behaviours of specimens are well captured by the numerical simulation, particularly the permanent plastic deformation and the tendency of the force–displacement curve.

In the experiment, the force–displacement curve exhibits oscillatory behaviour throughout the impact, but this phenomenon is not observed in the numerical prediction. It is mainly caused by the slight vibration of the indenter assembly during the impact. Thus, this vibration is not captured in the present simulation.

To describe the characteristics of structural deformation in detail, the specimen’s deformation process is segmented into four stages, as shown in Figure 11. The end of stages (1) to (4) corresponds to points (1) to (4) in the force–time and force–displacement curves (see Figure 10).

Stage (1): The outer bottom plate experiences small plastic deformation underneath the spherical indenter, and its clamped condition is provided by the bottom floors and girders.

Stage (2): With the development of the local indentation of the outer bottom plate, the uppermost parts of the bottom girders mainly experience elastic deformation during this stage.

Stage (3): After the elastic limit, the bottom girders suffer substantial folding deformation caused by the large in-plane loading and the outer bottom plate suffers large plastic deformation until absorbing all incident kinetic energy.

Stage (4): It takes place during the rebound process, which is caused by the release of the elastic strain energy accumulated earlier. As a result, the specimen exhibits a permanent dent of about 23 mm.

4.2. Energy Absorption

The energy–displacement curve can be determined by integrating the corresponding force–displacement data. The behaviour of structural crashworthiness can be well illustrated by the contribution of each structural component to the entire energy absorption. This contribution can be separated by the post-processing in the numerical simulation, and the energy absorbed by the outer bottom plate, bottom floors, and girders is plotted in Figure 12.

Generally, most energy is absorbed by the outer bottom plate and the bottom floors. After the deformation of about 12 mm, the bottom floors absorb more energy than the outer bottom plate. It illustrates that the plastic deformation of the bottom floors dominates the structural resistance during the impact.

4.3. Acceleration Response

Acceleration responses are recorded at two measure points located on the inner bottom plate in the experiment (see Points 1 and 2 in Figure 6). Point 1 is located at the centre of the inner bottom plate, and Point 2 is at the cross-section of a bottom floor and a girder. The histories of the accelerations obtained from the experimental and numerical results are shown in Figure 13. The acceleration response measured at Point 1 is larger than the one at Point 2 due to the fact that the stiffness is smaller at the plate element. The magnitude of acceleration increases rapidly when the contact of the specimen and indenter starts. The experimental peak acceleration of Points 1 and 2 are 518 g and 456 g, respectively (g is the gravity acceleration). The acceleration shows a significant reduction in the vibration amplitude after reaching its peak due to the attenuation characteristics of high-frequency vibration.

The numerical results agree well with the experimental data, showing small discrepancies in peak values between the experimental and numerical results (9.7% and 6.5%). Thus, the finite element simulation is accurate enough to assess the vibration acceleration of the structure during and after the impact. In reality, the mass of equipment in the cabin affects the structural vibration behaviour; thus, this phenomenon will be discussed in Section 5.3.

4.4. Stress

The experimental and numerical stresses located at two representative points of the inner bottom plate (see Points 3 and 4 in Figure 6) are shown in Figure 14. The histories of experimental stresses during the impact are well predicted by the numerical simulation, particularly the tendency of the curves. Stress concentration occurs at the inner bottom plate underneath the indenter. Peak stresses occur at about 5 ms corresponding to the end of Stage (2) in Figure 10. Later, the reaction force remains almost constant so that the stresses change slightly. The experimental and numerical maximum stresses of the inner bottom plate are about 69 MPa during the impact within the elastic region. As the double bottom is high, the inner bottom plate does not suffer plastic deformation during the impact.

5. Discussion

Good agreements between the experiment and simulation show that the numerical definitions are reasonable to predict the response of a similar structure under low-velocity impact. As this response is affected by the structural openings on bottom floors and girders and the impact velocity of the indenter, these influences are discussed in the next subsections. Moreover, we discussed the effects of the mass of power batteries in the cabin on the acceleration response of the inner bottom plate during the impact.

5.1. Effect of Structural Openings

There are structural openings on the bottom floors and girders (see Figure 3), and the analysis results presented in Figure 10 and Figure 13 are used as the reference. Here, one more simulation is carried out for the double-bottom structure without these openings (Figure 15). The comparative analysis of the entire force–displacement responses and the plastic deformation of the specimen are shown in Figure 16, and the comparison of the acceleration response of the inner bottom plate is shown in Figure 17.

The structural openings slightly affect the entire structural plastic deformation and force–displacement response (see Figure 16). Compared to the reference structure, the structure without openings exhibits a similar deformation under the impact, with a decrease of only 1.2% in the maximum deformation. It indicates that the impact strength of the double-bottom structure is not significantly altered by the openings. The small discrepancy is mainly due to the fact that the openings are far from the large plastic deformation region and slightly affect the energy absorption of bottom floors and girders.

The structural openings strongly affect the acceleration response inside the cabin, and a comparison of the acceleration–time curves of Point 1 at the inner bottom plate is given in Figure 17. The structure without the openings experiences a much larger acceleration response during the impact. The peak value is 976 g, which is 2.1 times the reference structure. A wave is easier to propagate through an intact structure. The openings decrease the structural stiffness, avoiding excessive high-frequency vibration transmission. Therefore, the openings on the bottom floors and girders can reduce largely the vibrations in the ship cabin structure, minimising the impact on the internal equipment.

A topological optimisation of the structural openings is possible to provide some information for obtaining an optimal structural design. Nevertheless, this optimisation is often based on static analysis, and its combination with impact analysis is still challenging due to the different types of loads.

5.2. Effect of Impact Velocity

The incident energy of the indenter can remain the same when changing both impact mass and velocity. Here, the analysis results presented in Figure 10 and Figure 13 are used as the reference. Two more simulations are performed with the same impact energy using the impact velocities of 1.0 m/s and 2.0 m/s.

The resulting force–displacement curves of entire structures are plotted in Figure 18a. These responses are very close due to the same impact energy, and the small discrepancy is mainly caused by the strain rate effect of steel materials (see Equation (4)). The resulting force–time curves are shown in Figure 18b. With the same incident energy, a high impact velocity leads to a shorter contact time. The analysed acceleration responses at the inner bottom plate are plotted in Figure 19. The maximum accelerations at the inner bottom plate are relatively close, and they are 432 g, 457 g, and 472 g, corresponding to the impact velocities of 1.0 m/s, 2.0 m/s, and 3.0 m/s, respectively. The small discrepancy in peak acceleration is mainly due to the fact that the change in force–time curves is not significant until the first peak force (Figure 18b). It corresponds to the end of Stage (2) in Figure 10. However, the acceleration response time becomes shorter when the impact velocity increases since the peak force reaches earlier (see Figure 18b).

5.3. Effect of Battery Mass

The mass of power batteries inside the cabin of new energy ships is often very large, and the battery mass should strongly affect the acceleration response in the cabin. This mass is related to the distance of the maritime route; thus, the influence factor is discussed preliminarily in the present study. In view of the safety of power batteries, the vibration acceleration cannot be overlarge to avoid battery damage. Thus, the assessment of vibration acceleration in the target cabin is very important in the ship grounding event.

The analysis results presented in Figure 13 are used as the reference, and more simulations are performed to investigate the effect of battery mass. In the simulations, an element mass is defined to simulate the battery mass (Figure 20). A framed support is installed on the inner bottom plate to place the battery box, and they are fixed together using the multi-point constraint. In the present simulations, the defined masses are 300 kg, 600 kg, and 900 kg, respectively, and their corresponding peak accelerations are 35.6 g, 23.1 g, and 15.6 g. The resulting acceleration responses varying the battery mass are given in Figure 21. The acceleration data are taken from the mass point shown in Figure 20. Generally, a larger mass presents a smaller acceleration response, i.e., a larger battery mass can effectively mitigate the impact on the battery box. Here, the influence of mass is analysed simply in order to illustrate the influence factor, and plenty of work is still required in the preliminary design of the battery cabin.

6. Conclusions

Experimental and numerical analyses are carried out on the impact response of a double-bottom structure with high and perforated floors and girders, focusing on the acceleration response inside the cabin structure. The impact analysis of battery cabins during the grounding accident is beneficial to guide the structural design of new energy ships.

In the impacted double-bottom structure, the outer bottom plate undergoes dent deformation, and the bottom floors and girders suffer a folding deformation. The consistency between the experiment and simulation shows that the defined material properties well describe the strain, stress, and acceleration response states of the impacted double-bottom structures with high and penetrated girders and floors.

The design of structural openings on the bottom floors and girders does not reduce largely the impact strength of the double-bottom structure. Meanwhile, it effectively decreases the acceleration response of the internal structure, which is beneficial to the safety of power batteries.

The battery mass strongly affects the acceleration response inside the cabin, and a heavier battery box results in a smaller acceleration response. The assessment of vibration acceleration is very important to analyse the safety of batteries in ship grounding accidents.