Analysis of unsaturated soil columns with application to bulk cargo liquefaction in ships
Introduction
Approximately 3 billion tonnes of nominally dry, granular ore cargoes (iron ore, coal, grains, bauxite, phosphate) are transported annually by sea, and each year a small but significant number of large bulk carriers are lost at sea along with some of their crew. One of the reported and least understood causes of these ship losses is from the liquefaction and shifting of their cargo. This is often inferred from the rapid sinking of the vessels and is associated with many crew fatalities. For example, during the 10 years from 2008 liquefaction has been the reported cause of 9 bulk carrier losses, 19% of the total and for the loss of 101 lives, 54% of the total loss of life from bulk carriers (Intercargo, 2019). Liquefaction has also been reported in some cargoes without the loss of the ship (e.g. Lee, 2017).
Bulk cargoes are comprised of particles that can cover a wide range of sizes from clay sized (<2μm) to large gravel sized (60 mm) and many ore cargoes have particle size distributions that suggest they could be susceptible to liquefaction as they lie mostly within the range of gradings with the potential to liquefy (Tsuchida, 1970). Ore products can be variable, but typical iron ore fines and bauxite particle size distributions are shown in Fig. 1 together with the grading limits suggested by Tsuchida. Because ores are normally transported in a relatively dry state and water is required for liquefaction the problem is primarily confined to ores with significant fines (particles < 75 μm) fractions that are able to retain sufficient moisture during transport and loading into ships. During shipping, the moist, unsaturated cargo is subject to cyclic loading from ship vibrations and the rocking motions caused by sea waves which can result in large numbers of cycles of significant amplitude.
Many element tests, mostly using triaxial apparatus, have been used to study the cyclic response of partially saturated soils. These studies have shown that partial saturation significantly increases the resistance to liquefaction (e.g. Yoshimi et al., 1989, Unno et al., 2008, Tsukamoto et al., 2014) and it has been suggested that the initial degree of saturation needs to be greater than 0.7 combined with a low relative density for liquefaction to occur (Tsukamoto, 2019). Several model 1-g shaking table and centrifuge tests of partially saturated soils subject to cyclic loading have also been performed to investigate the potential for liquefaction, settlement and slope instability (e.g. GHAYOOMI, M., 2011, Higo et al., 2015, Mirshekari and Ghayoomi, 2017) but these have been focused on earthquake loading with small numbers of cycles and mostly sandy soils. Centrifuge tests investigating the rocking motion on partially saturated cargo materials (Atkinson and Taylor, 1994), have shown that saturated conditions can develop at the base of the cargo and cyclic loading can cause the generation of pore pressure that results in a loss of strength and a loss in resistance to the shifting of cargo. More recent rocking centrifuge tests have demonstrated another mechanism, termed dynamic separation, where water which rises to the surface, as a result of settlement, increasing saturation and elevated pore pressures, can progressively erode and redistribute the cargo towards one side of the ship (IMO, 2017). For these failure mechanisms to develop the cargo must have a low relative density, a tendency to compress and generate positive excess water pressures, and the initial moisture content must be reasonably close to full saturation on loading. In addition, the ship must experience sufficient cycles of wave loading of significant amplitude.
Current practice, as required by shipping codes, is that ore materials considered susceptible to liquefaction must be below a specified Transportable Moisture Limit (TML) when loaded onto a ship. The TML can be determined from a Proctor Fagerberg Test (PFT), a compaction test performed according to the ASTM Standard D-698, but with lower than standard compaction energies intended to reproduce the density of the loaded cargo. According to the International Maritime Solid Bulk Cargoes (IMSBC) code (IMO, 2013a), the TML should be taken as equal to the moisture content obtained at an 80% degree of saturation from the relevant PFT (IMO, 2013b). From a soil mechanics perspective, the reliance on a simple moisture content and assumed compaction energy is unlikely to capture the complexity of the response which varies with material type, hydraulic characteristics, drainage conditions, stress level, magnitude and number of cycles and thus there is a need for analytical and numerical models that can be adapted to the range of shipped products that can provide confidence in this simple TML procedure.
Experimental studies of ship cargo liquefaction, other than the centrifuge tests mentioned above, have been mostly limited to small (0.03 m3) 1 g tests (eg. Munro and Mohajerani, 2017) and many studies using cyclic triaxial and simple shear tests with varying degrees of saturation (eg. Wang et al., 2016, IMO, 2017, Kwa and Airey, 2019). The lack of well accepted numerical models for unsaturated soils subject to cyclic loading limits our ability to assess the ship cargo performance from these small-scale tests, which cannot capture the phenomena related to fluid flow. Previous numerical analyses of ship cargo liquefaction have treated the cargo as a viscous fluid (Zou et al., 2013, Zhang et al., 2020), modelled the cargo with a small number of discrete elements (Ju et al, 2018), modelled the cargo with an established dynamic saturated soil model using finite elements for a small number of cycles (Daoud et al, 2018) and used AI approaches that avoid the soil mechanics (Wu et al, 2020). These numerical analyses have either assumed the cargo is fully saturated or when unsaturated the hydraulic and mechanical behaviours have not been coupled. Numerical models that are able to obtain solutions to coupled unsaturated dynamic problems are limited to research codes (e.g. Bian et al., 2017, Matsumaru and Uzuoka, 2016, Zhang and Muraleetharan, 2018) and their ability to provide reliable predictions is hampered by the lack of well-established experimental data for calibration and validation. Additionally, the focus on sands and earthquake loading with small numbers of cycles in the soil mechanics community means that existing approaches, for example for estimating seismic compression (Ghayoomi et al., 2013, Zeybek and Madabhushi, 2019), cannot be simply applied and model parameters cannot be easily determined from the literature.
The ship cargo liquefaction problem is computationally demanding as capturing the essential physics of unsaturated cargos requires the use of a dynamic analysis that fully couples the mechanical and hydraulic behaviour of the cargo (e.g., Schrefler and Scotta, 2001, Ghorbani et al., 2020) together with an appropriate constitutive model that can consider plastic deformations associated with stress reversals. This gives rise to a high level of nonlinearity in the equations. It should also be noted that, unlike soils that experience high-frequency vibrations for a short period during earthquakes, cargoes are exposed to a high number of low-frequency cycles that can last for days during which consolidation can also occur. This unique type of loading, which results in the simultaneous presence of consolidation and dynamic loading, imposes an extra challenge in the simulation of the cargo response, particularly in selecting appropriate time step sizes to avoid numerical instability. It may also be noted that existing adaptive time marching solution schemes require extensive modification to be used for this type of analysis (Schrefler et al., 2006). This is because they are either developed for consolidation of saturated soils under static loads (e.g., Sloan and Abbo, 1999, Sheng et al., 2003), or uncoupled dynamic analysis (e.g. Wiberg and Li, 1994).
In this paper, a recently developed fully coupled unsaturated dynamic soil model (Ghorbani and Airey, 2021), which has been shown to successfully reproduce triaxial test behaviour, is used to perform some parametric studies to investigate the effects of initial density, degree of saturation and frequency of loading. To reduce the complexity of the problem, the analyses are limited to one-dimensional conditions with lateral acceleration applied to the base of a soil column. This is a considerable simplification of the conditions experienced by cargo in a ship hold subject to complex sea motions, but modelling the complete cargo hold and boundary conditions would lead to very large computational times, severely limiting the scope of any parametric study and increase the numerical stability challenge. This simplification has facilitated the parametric study and has allowed the numerical model performance to be demonstrated. The use of a soil column has also enabled us to capture the key processes occurring during cyclic loading including pore pressure changes and fluid flow (Lakeland et al., 2014). The applicability of the results to the problem of cargo instability are discussed.
Section snippets
Numerical Model
In the presented work, the fully coupled framework developed by (GHORBANI, J., 2016, Ghorbani et al., 2018a) is used which assumes that there are isothermal conditions, the compressibility of the solid particles can be ignored, and the relative accelerations of the non-solid phase compared to the solid are negligible. Finite element discretisation of the governing equations of motion of unsaturated soils are performed by using unknown displacements, pore water pressures and suctions,
Methodology
The numerical model is used to study the response of a one-dimensional column of unsaturated soil that is subjected to horizontal cyclic loading applied to its base. The column has an initial height of 10 m and a width of 1 m and is modelled under plane strain conditions using six-noded fully coupled elements. Nodes in the mesh at the same elevation have been constrained to move together in both horizontal and vertical directions to produce pseudo-one dimensional conditions in the column with a
Application to ship instability
The motivation for the analyses described above has been to provide insights into the processes occurring in ore cargos during shipping transport and hence to help explain observations of cargo liquefaction and the capsizing of large bulk carriers.
Fig. 11a shows a generic section for an ore on loading into a ship’s hold. There are a range of different sizes and shapes for both the holds and the cargo, but the width is typically in the range of 25 to 35 m and the height of the ore in the range
Conclusions
A fully coupled unsaturated dynamic numerical model has been used to study the response of a 10 m high soil column subjected to lateral shaking at its base. Although some numerical issues have prevented a full investigation of the problem, the results have nevertheless been able to demonstrate the interaction between initial void ratio, degree of saturation, permeability and frequency of loading. In particular, the analyses have demonstrated the important role of the initial hydraulic
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
This research was supported by the Australian Government through the Australian Research Council's Discovery Projects funding scheme (project DP19010364).
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Both authors David Airey and Javad Ghorbani have contributed equally to the paper.