Recent developments in ship collision analysis and challenges to an accidental limit state design method

1. Introduction

Ship collision analysis has developed rapidly over the last 10 years towards more accurate predictions of accidental consequences. However, there is a large gap when it comes to the Accidental Limit State (ALS) design based on collisions. This paper reviews the last 10 years of technical development in ship collision analysis and summarises the future challenges in both the accidental analysis and the ALS design. The development of ship collision analysis is not only to evaluate the consequence of accidental events but also to develop structural standards for improving the crashworthiness of ship structures.

Pedersen (2010) presented a review of prediction and analysis tools for collision and grounding analyses and outlined a probabilistic procedure to develop performance-based rules. To further review the analysis of ship collision and grounding damage, Ehlers (2011), Calle and Alves (2015) and Liu et al. (2018a)published review articles on the experimental, numerical, analytical and empirical methods for assessing the impact resistance of ship structures. The main focus of the present paper is on the advantages and disadvantages of the calculation methods, the recent developments and technical challenges in ship collision analysis, and the special technical needs for the ALS design method based on collisions.

The ship collision analysis procedure includes the assessment of external dynamics and internal mechanics. The external dynamics deals with the global rigid motion of ship collisions and the loss of collision energy, accounting for the main fluid forces interacting between the ship and the surrounding water; the internal mechanics reveals the response of ship structures caused by the released energy during collisions, which is related with impact force, damage and energy absorption of ship structures.

For the analysis of the external dynamics, Pedersen and Zhang (1998) derived closed-form analytical solutions, based on rigid body mechanics, to estimate the energy released for deforming ship structures impacted at arbitrary locations and angles. The accuracy of the analytical approach is related to the definition of the added mass coefficients for various motions, especially for the sway motion of the struck ship. In addition, Tabri et al. (2009) presented an analytical model to analyse ship collisions accounting for large forces raised from the sloshing in ship ballast tanks.

For the internal mechanics' analysis, the evaluation methods include experimental, numerical, analytical, empirical and semi-analytical methods (Liu et al., 2018a). For the experimental method, large-scale models have to be manufactured to avoid the scaling effect, which is cost-inefficiency in the financial and time aspects. With the development of computer hardware and software, it is possible to perform the finite element simulation of full-scale ship collisions. However, finite element simulation also requires human resources for numerical modelling and computational resources. The accuracy of the simulation is related to the experience of analysis experts together with experimental validations.

The recent research is mainly focused on the improvement of the accuracy of finite element simulations. The explicit finite element analysis is a well-established method to simulate structural impacts and quasi-static indentations, see for example the benchmark study reported by Ringsberg et al. (2018). The correct definitions of material parameters and boundary conditions are of considerable practical importance to assess the impact strength of structures. The boundary conditions should simulate accurately the physical constraints on the finite element model. The definition of material propertieshas to account for the material strain hardening, strain rate effect and dynamic failure. In the finite element model, standard inputs should be identified leading to the analysis procedure of ship collisions.

The analysis of external dynamics and internal mechanics are mostly treated independently. The decoupled analysis can well predict the ship's structural damage in right and large-angle collisions. Nevertheless, in an oblique collision the collision angle changes due to the sliding and rotation of the striking and struck ships. This phenomenon challenges the accuracy of the decoupled analysis since the penetration path is assumed to be a straight line. The coupled analysis of external dynamics and internal mechanics can well simulate the whole collision process together with time histories of the ship motions and the reaction forces (Brown, 2002; Pill and Tabri, 2011; Le Sourne et al., 2012; Liu et al., 2017b). It captures the realistic penetration path and predicts the global motion and local penetration of structures.

The most rapid tools for evaluating the crashworthiness of ship structures are analytical, empirical and semi-analytical methods. The analytical method is based on the upper bound theorem of plasticity, focusing on the evaluation of structural plastic deformation, reaction force and energy dissipation. Various analytical formulae have been proposed to evaluate the energy dissipation of relatively complex structures, such as stiffened plates, web girders and plate intersections (Zhang, 1999; Liu et al., 2021b). These methods can establish the global pattern of deformation by adding up all local contributions of the individual structural components.

According to the existing theories, no analytical method has been proposed to assess the fracture initiation and propagation of complex structures. The level of the analytical methods involving fracture is still in the stage of plate components, for example, plates subjected to a spherical indenter (Simonsen and Lauridsen, 2000; Liu et al., 2014). On the other hand, empirical formulae are too simplified to describe well the structural mechanics of ship collisions. Generally, it seems that the semi-analytical method is the preferred tool in the industry application keeping a good balance between rapidity and accuracy (Zhang et al., 2019b). This method is based on the math formulae from analytical analysis and the empirical parameters proposed by the experts.

It is necessary to assess the collision and crashworthiness in the design of ships that carry hazardous materials, such as oil, LNG and chemical tankers, whose accidents may lead to enormous environmental pollution. Also, the impact of ship fuel tanks should be involved for all ships. With the development of the LNG fuel tank used in modern green ships, the collision problem of LNG tanks becomes more critical. Based on this background, LR (2016) developed the Guidance Notes for Collision Assessment for the Location of Low-flashpoint Fuel Tanks. The development of the guidance notes shows clearly the advantage of moving towards ALS design, although it is not mandatory to carry out the collision analysis in the ship design stage.

2. Recent developments in finite element analysis

Nowadays ship collision analysis can predict with good approximation the accident consequences as a result of the development of the nonlinear finite element method over the last 10 years. Finite element modelling requires the definition of various parameters and most of them have become standard inputs in the numerical codes.

The ship structure is often modelled by four-node shell elements with five-integration points throughout the thickness, defining the Belytschko-Lin-Tsay shell element formulation. The recent developments in finite element analysis are mainly focused on the definitions of material nonlinearities and fracture including plastic strain hardening, strain rate effect and dynamic fracture. The investigations for the material characterisation of steels used in shell models are validated against the quasi-static punching and low-velocity impact experiments of plates and stiffened panels.

2.1. Material true stress-strain curve

Commonly, the material mechanical properties are determined by tensile tests. The material strain hardening to fracture can be well described by a true stress-strain curve, representing the basic plastic-flow characteristics of the material. The true stress-strain relationship is often converted from the recorded engineering stress-strain data until the onset of necking, using a simple or modified power-law form with a strength coefficient (K) and a strain-hardening exponent (n).

Investigations over 10 years ago include.

• Dieter (2000), • Johnson and Cook (1983), • Alsos et al. (2008), and • Ehlers (2010).

Investigations in recent 10 years include.

• Storheim and Amdahl (2017), and • Liu et al. (2017a).

Storheim and Amdahl (2017) discussed the sensitivity of material strain hardening in finite element analysis of ship collisions. Generally, the true stress-strain curves defined by various experts, based on the same engineering stress-strain curve, often show some discrepancy. However, it will not affect strongly the analysed results of impacted structures. In practice, the engineering stress-strain curve is not available at the design stage to determine the strain hardening of the material. The known material properties of structural steels from standards only include yield stress, ultimate tensile strength and engineering fracture strain. Based on the limited material data, Liu et al. (2017a)established analytical expressions to estimate the coefficients K and n for the description of the true stress-strain curve of steel materials used in the ship collision analysis.

2.2. Strain rate effect

Increasing the strain rate has the effect of increasing the yield stress and flow stress of steels. Nevertheless, there is not yet a standard procedure to define the dynamic material characteristics of steels in the low-velocity impact. Two approximate solutions have been adopted in recent investigations.

(1)
As the quasi-static indentation can represent a similar response as the low-velocity impact, the dynamic effects are omitted in the numerical simulations defining only the material true stress-strain curve obtained from quasi-static tensile tests. Although the strain rate effect is obvious for mild steels, the difference found between quasi-static and low-velocity impact tests reaches only magnitudes of 10%–20% in the reaction force and/or deflection (Yu and Jones, 1991, 1997; Corbett and Reid, 1993; Liu and Guedes Soares, 2015a, 2019; Gruben et al., 2016, 2017; Liu et al., 2018b).
(2)
The Cowper-Symonds model with the standard coefficients of D = 40.4 s−1and q = 5 is used to describe the material strain rate sensitivity in mild steel structures under the low-velocity impact (Cowper and Symonds, 1957). Nevertheless, the actual coefficients for mild steel vary significantly with the increase of plastic strain, for example, the values of D and q at the yield and the ultimate tensile stresses are different (Jones, 2013; Choung et al., 2013). The scaling of yield stress is often larger than the one of ultimate tensile stress in dynamic tensile tests.

Investigations over 10 years ago include.

• Cowper and Symonds (1957), • Johnson and Cook (1983), and • Tanimura et al., (2009).

Investigations in recent 10 years include.

• Tanimura et al. (2014), • Liu et al. (2018b), and • Liu and Guedes Soares (2019).

The dynamic flow stress changes with the strain magnitude as well as with the strain rate (Jones, 2006). Tanimura et al., (2009) proposed a constitutive model to simulate the strain rate sensitivity of steel materials over the wide strain rate range and the entire range of strain up to fracture strain. Its advantage was shown by comparing various rate-dependent constitutive models with experimental data (Tanimura et al., 2014).

Liu et al. (2018b) defined the mild steel dynamic material characteristics with true stress-strain curves at five strain rates obtained from quasi-static and dynamic tensile tests. The accuracy of this approach is illustrated by comparing the numerical prediction with the corresponding experimental result on the force-displacement response of a mild steel plate impacted by a hemispherical indenter. Unfortunately, most analysts do not have the dynamic true stress-strain curves as input into their numerical models. Thus, Liu and Guedes Soares (2019) proposed a simplified method to characterise the material strain rate effect at varied plastic strains in the simulations. It contributes to enhancing the industry practice in impact simulations of steel structures when only material quasi-static tensile test results are available.

2.3. Dynamic failure

Besides the material plastic strain hardening and strain rate effect, it is known that critical fracture strain is required for the prediction of the damage extent in structures. The material failure model is used with an element deletion strategy to simulate the fracture initiation and propagation through the finite element meshes. The evaluation of material fracture is sensitive to the mesh size, stress state and strain rate.

2.3.1. Mesh size sensitivity

In the numerical simulations of ship collisions, the shell elements are highly sensitive to the element size due to the discrete nature of the mesh. Some simplified failure criteria for shell elements have been proposed to obtain the material critical failure strain determined by the ratio of element size to plate thickness (Peschmann, 2001; Zhang et al., 2004; Liu et al., 2017a). The range of this ratio is between 5 and 20, which is suitable for coarse meshes.

Investigations over 10 years ago include.

• PES criterion proposed by Peschmann (2001), and • GL criterion proposed by Zhang et al. (2004).

Investigations in recent 10 years include.

• Liu criterion proposed by Liu et al. (2017a).

These criteria provide a rapid and simple tool to define the material failure strain. However, the failure criterion should have its scope of application. For example, the criterion proposed by Liu et al. (2017a) is particularly suitable for simulating penetrations with a rounded indenter, and it had been used in the Guidance Notes on Nonlinear Finite Element Analysis of Marine and Offshore Structures developed by ABS (2020). These criteria are calibrated against their selected experiments and it is difficult to find a universal failure criterion for coarsely meshed shell structures which can predict well the results of all existing experiments.

For fine meshes, the simulation of tensile failure initiation is inevitable to evaluate the structural response in accidental scenarios. Wiegard and Ehlers (2020) gave an overview of the analysis approaches to how the failure initiation can be simplified and simulated in finite element models of large thin-walled structures.

Generally, most of the proposed failure criteria are mesh dependent, since the failure strain is lower for the large elements. The mesh size sensitivity is involved in the actual size of the stress concentration region underneath the impactor, affected by the shape and size of the impactor. A sharper and/or smaller impactor causes a more concentrated plastic strain in the impacted structure, and thus in the numerical analysis the failure strain is more sensitive to the mesh size, i.e. the failure strain decreases rapidly with the increase of mesh size. To obtain rational analysis results, an expert is required to select reasonably a mesh size and a failure criterion.

2.3.2. Stress state sensitivity

The fracture initiation of ductile materials is sensitive to the stress triaxiality at the fracture area. A failure criterion should be more appropriate as it covers a wide range of stress triaxiality, evaluates different material failures such as compression, shear and tension, and considers the strain state sensitivity. The shape of the striker affects the history of stress triaxiality of the failing elements during the impact.

Investigations over 10 years ago include.

• Swift criterion (Swift, 1952), • RTCL criterion combined Rice and Tracey (RT) and Cockcroft and Latham (CL) criterion (Tørnqvist, 2003), • BWH criterion combined Bressan-Williams (BW) shear instability and Hill's criterion (H) for local necking (Alsos et al., 2008), and • Modified Mohr-Coulomb (MMC) criterion (Bai and Wierzbicki, 2010).

Investigations in recent 10 years include.

• Modified BWH criterion (Storheim et al., 2015a; Liu et al., 2022a), • Failure strain formula based on average stress triaxiality (Choung et al., 2012, 2014) • DSSE-HC criterion combined the concept of the Domain of Shell-to-Solid Equivalence (DSSE) and Hosford–Coulomb (HC) fracture criterion (Pack and Mohr, 2017), • Through-thickness damage regularisation scheme (Costas et al., 2019) • Failure criterion based on localised necking condition (Cerik et al., 2020), and • MSSRT criterion combined the Maximum Shear Stress and RT fracture criterion (Lu et al., 2021, 2022).

Recently, a large amount of investigations are carried out on the failure criteria of steel materials expanding their usage to a wide range of stress states. These new failure criteria are proposed often using the combination of various classical failure criteria to avoid the shortcoming of a single criterion. Cerik et al. (2019a, 2019b) evaluated the suitability of the DSSE-HC fracture model to simulate the rupture in large thin-walled structures due to biaxial stretching and confirmed the accuracy of the DSSE-HC model in the estimation of the displacement to fracture initiation in the stiffened panel penetration tests. Costas et al. (2019) proposed and validated a damage regularisation model for shell elements used in large-scale simulations. The model evaluated the ratio of bending to membrane loading in the elements based on the gradient of through-thickness plastic strain. Lu et al. (2021, 2022) developed the MSSRT criterion to analyse the fracture initiation and propagation in the tensile tests, plate punching tests and penetration tests of double-hull structures.

Generally, these criteria are calibrated against their selected experiments using fine meshes. Nevertheless, the stress triaxiality is averaged in the coarse meshes and thus the applicability of these criteria is still unclear for the coarse meshes. To achieve a universal criterion describing the stress state sensitivity, it seems sensible to carry out further work using fine meshes first and then expanding to the coarse meshes.

2.3.3. Strain rate sensitivity

The rupture of impacted structures is sensitive to the strain rate which is related to the impact velocity and stress concentration. This problem is not well described using mathematical equations.

Investigations over 10 years ago include.

• Inverse Cowper-Symonds model proposed by Paik (2007), and • Johnson-Cook fracture model (Johnson and Cook, 1985).

Investigations in recent 10 years include.

• An extension of the DSSE-HC model proposed by Cerik and Choung (2020, 2021).

As the dynamic failure of materials is affected by many factors, it is difficult to separate accurately the influence of strain rate on the dynamic failure. This is mainly because the effects of mesh size and stress state have also been understood comprehensively. Some mathematical descriptions were proposed many years ago (Johnson and Cook, 1985; Paik, 2007), but they are not accepted widely and applied in impact analysis.

2.3.4. Accuracy of fracture assessment

Calle and Alves (2015) presented a review of failure criteria used in finite element modelling of ship collision events. Ehlers et al. (2008), Storheim et al. (2015b), Marinatos and Samuelides (2015), Calle et al. (2017) and Kõrgesaar (2019) compared the accuracy of existing fracture criteria based on the simulations of experiments, and large discrepancies were found among them. Although much effort has been made by investigators, the scatter of the results between finite element analyses and experiments using various suggested fracture criteria is still very large. For example, in Storheim et al. (2015b), various approaches suggested by current design rules and proposed by researchers were used to predict the experimental results and the analysed errors range from about −60% to 100%; in Ringsberg et al. (2018), 15 experts carried out the numerical analyses with the known experimental results and the analysed errors are from −28% to 13%.

Many quasi-static punching and impact experiments were performed to calibrate the material failure criterion in the finite element analysis, for example, Villavicencio et al. (2013, 2014), Kõrgesaar et al. (2018b) and Chen et al. (2019, 2022). In the finite element analysis, the experimental boundary conditions have to be represented accurately to predict the experimental response.

2.4. Coupled analysis of external dynamics and internal mechanics

The advantage of the coupled analysis of external dynamics and internal mechanics is that it provides a more accurate description of the whole collision process together with time histories of the ship motions and the reaction forces. The coupled method can capture the realistic penetration path and predict the global motions and local penetration of structures. The coupled method is very suitable for the analysis of small-angle oblique collisions since the collision angle changes are caused by the sliding and rotation of the striking and struck ships.

Investigations over 10 years ago include.

• Brown (2002), • Pill and Tabri (2011), and • Le Sourne et al., (2012).

Investigations in recent 10 years include.

• Yu and Amdahl (2016), • Le Sourne et al., 2012, Yu et al., 2016a, Yu et al., 2016b, • Le Sourne et al., 2012, Yu et al., 2016a, Yu et al., 2016b, • Le Sourne et al., 2012, Yu et al., 2016a, Yu et al., 2016b,• Liu and Amdahl (2019), • Le Sourne et al., 2012, Yu et al., 2016a, Yu et al., 2016b, and • Le Sourne et al., 2012, Yu et al., 2016a, Yu et al., 2016b.

In the analysis of the external dynamics, the effect of the fluid is represented by non-dimensional coefficients of the added mass for each ship motion. The added mass for ship motion induced by the effect of the fluid still needs further investigation, especially the coefficient for the sway motion of the struck ship. To investigate the external dynamics of ship collisions in detail, Lee et al. (2017), Rudan et al. (2019) and Kim et al. (2021) applied the Fluid-Structure Interaction (FSI) analysis to couple structural dynamics and hydrodynamics. The embedded coupling algorithms are very complex and need to be used at once within a multi-physics analysis. It can well reveal the external mechanics of ship collisions, but the complex calculation is difficult to be applied in the ALS design.

3. Recent developments in analytical and semi-analytical methods

Analytical and semi-analytical methods for assessing structural damage and energy absorption have developed rapidly over the last 10 years. The analytical method can well predict the structural energy absorption at the plastic deformation stage; nevertheless, at the large penetration stage, the semi-analytical method has to be applied.

3.1. Analytical method

Simplified analytical methods have been derived to analyse the plastic collapse mechanisms of individual structural members of ship structures for energy absorption assessment. The response of complex marine structures is then estimated as a sum of the crushing loads of all structural components. For the pure plastic crushing modes without fracture, the simplified methods are based on the upper bound theorem of plasticity, where kinematically admissible collapse patterns are applied to find the collapse load. For these theoretical analyses, one of the most important steps is to accurately express the identified deformation modes of the structural components.

3.1.1. Stiffened panel

The analytical analysis of stiffened panel is represented by the combination of the deformation mechanisms of the individual plate and stiffener components. These works are based on the analytical analysis of plates (Wang et al., 1998; Simonsen and Lauridsen, 2000; Lee et al., 2004).

Investigations over 10 years ago include.

• Cho and Lee (2009).

Investigations in recent 10 years include.

• Liu et al. (2015), • Sun et al. (2015), • Yu et al. (2018), and • Zhang et al., 2019a, Zhang et al., 2020.

In the deformation pattern, both plate and stiffener dissipate the energy through membrane plastic tension of the structural elongation and rotation of the plastic hinges. For a stiffened panel under a specific indenter shape, the accuracy of analytical analysis is mainly dependent on the mathematical description of the energy absorption mechanism of the plate and stiffeners.

3.1.2. Web girder

In the analytical analysis of the web girder, the energy is dissipated through the rotation of the plastic hinges and the membrane tension over the plastically deformed regions. The analytical methods for unstiffened web girders can evaluate the crushing resistance of web girders longitudinally stiffened on the ship side, i.e. the stiffeners are perpendicular to the direction of the incoming striker.

Investigations over 10 years ago include.

• Wierzbicki and Driscoll (1995), • Simonsen and Ocakli (1999), • Zhang (1999), and • Hong and Amdahl (2008).

Investigations in recent 10 years include.

• Liu and Guedes Soares (2015b, 2016a), and • Liu et al. (2019).

The folding deformation of web girders can be divided into two parts, plastic deformation and elastic buckling zones (Liu and Guedes Soares, 2015b). Based on the failure mode, the crushing force can be well estimated. Moreover, as the transverse stiffeners crush axially and influence severely the resistance of stiffened web girders, Liu and Guedes Soares (2016a) presented an analytical method to evaluate the web girders transversely stiffened.

3.1.3. Double hulls

The impact resistance of double-side structures can be evaluated by adding up the crushing of stiffened plates and web girders. The accuracy of the analysis of double hulls is mainly determined by the analysis of the individual stiffened plates and web girders.

Investigations in recent 10 years include.

• Liu and Guedes Soares (2016b), and • Liu (2017).

Generally, the proposed analytical methods can only assess the impact resistance of double-hull structures in minor ship collisions with small structural damage. For assessing severe accidents with large penetration damage, semi-analytical methods have to be used.

3.2. Semi-analytical method

The analytical methods are not well suited for the evaluation of crack initiation and propagation. Besides the pure plastic deformation modes, researchers have identified major failure patterns for structural components in the ship's collision which include fractures, such as plate tension, folding/crushing, tearing/cutting and concertina tearing. As the semi-analytical method is the most preferred tool in the industry application, it is introduced in detail here.

Investigations over 10 years ago include.

• Minorsky (1959), • Pedersen and Zhang (2000), and • Ehlers and Tabri (2012).

Investigations in recent 10 years include.

• Zhang and Pedersen (2017), and • Liu et al. (2021b).

Zhang and Pedersen (2017) updated the semi-analytical formulae developed by Pedersen and Zhang (2000) for analysing the relationship between the absorbed energy and the damaged material volume for two damage modes, taking into account the structural arrangements, the material properties and the damage patterns.

Recently, this method was validated by Zhang and Pedersen (2017) and Zhang et al. (2019b) using a series of published experiments. Generally, this method can predict approximately the large penetration damage of double-hull structures. It, however, is possible to be improved with detailed investigations, especially revising the input parameters. In this semi-analytical method, a linear relationship is assumed between the energy absorbed by the plastic tension damage mode and the engineering strain at fracture. This assumption is valid in the uniaxial tension, but it is false to describe the relation in the plane-strain tension which failure often occurs at the ship side shells under impact. This assumption should overestimate the energy absorbed by side shells. Moreover, the analysis for the energy absorbed by the crushing and folding damage mode is based on web folding without fracture. The tearing of web girders, often occurring at the in-plane impact, dissipates much higher energy with structural fracture (Wierzbicki, 1995). The energy absorbed by web girders should be underestimated by the formula proposed by Zhang and Pedersen (2017).

A revised semi-analytical method was proposed by Liu et al. (2021b) to predict the resistance force of double-hull structures at the large penetration stage. The semi-analytical analysis is established by adding up the reaction forces induced by the plastic bending and crushing of components and the propagation of cracks, and the detailed analysis makes the results more credible.

4. ALS design based on collisions

Ship collision analysis procedures can be developed by Classification Societies to improve the crashworthiness of ship structures to reduce the high economic, environmental and human costs resulting from those sometimes-unavoidable accidents. A conceptual design framework for collision analysis was proposed to evaluate the crashworthiness of double-hull structures (Liu et al., 2021a); however, the statistical data of ship collision accidents are essential. For design purposes, rational definitions of input parameters are needed for potential striking ships, and also a set of acceptance criteria to quantify and classify the structural resistances.