Physical and theoretical modelling of embedded mooring line–seabed interaction in sands

$a$

cone penetrometer fitting parameter

$b$

passive resistance factor

D_r

relative density

$d$

fibre rope diameter

$dbar$

chain link bar diameter

$dc$

cone penetrometer diameter

$d50$

mean particle size

$En,Et$

chain geometric factors

|$E¯$|

mean error

$Eb¯$

bias

$F$

sliding (or frictional) resistance per unit length

$g$

Earth’s acceleration due to gravity

$Kp$

passive resistance factor

$llink$

link width

$m$

mass per unit length

$N$

normal resistance per unit length

$Nq$

chain bearing factor

$n$

cone penetrometer fitting parameter

$pa$

atmospheric pressure

$qc$

cone penetration resistance

$qc1N$

stress normalised cone penetration resistance

$qc,ssn$

steady-state stress normalised cone penetration resistance

$Ra$

chain link surface roughness

$s$

position along chain length

$su$

undrained shear strength

$T$

tension

$Ta$

tension at anchor padeye

$T0$

tension at sea floor

$wlink$

chain link length

$x$

horizontal distance

$x0$

horizontal displacement at sea floor

$z$

depth

$za$

depth to anchor padeye

$α$

chain interface friction ratio

$β$

dimensionless scaling factor for normal resistance

$βθ$

dimensionless scaling factor for normal resistance with dependency on local chain angle

$β0$

scaling factor on $βθ$

$θ$

local chain inclination

$θa$

chain inclination at anchor padeye

$μ$

friction coefficient

$ξ$

dimensionless scaling factor for sliding resistance

$σv′$

vertical effective stress

$ϕ′$

effective friction angle

Floating facilities are held in place by mooring lines that consist of chain and wire or synthetic rope connected to anchors. The mooring lines originate at the floating unit and terminate at an anchor located on the sea floor or within the seabed. The most common mooring configuration is a catenary shape in the water column, arriving at the sea floor horizontally, with a length of line lying on the sea floor. The final section is embedded in the seabed, forming an inverse catenary between the sea floor and the anchor load attachment point, termed the padeye (Fig. 1).

The embedded mooring line that forms an inverse catenary controls the fraction of the mooring load, $T0$⁠, that is resisted, and hence the load at the padeye, $Ta$⁠, and its inclination, $θa$⁠. Predicting the shape and load-carrying response of the embedded line is an integral part of the geotechnical and structural design of the anchor.

Also, the embedded mooring line’s inverse catenary shape changes under increased loading, releasing additional ‘slack’ into the mooring system. Excessive movement of the floating unit can occur if additional chain slack is released during operation (e.g. Neubecker & O’Neill, 2004).

The embedded mooring line’s inverse catenary shape and tension profile is modelled by way of the chain equilibrium equations (e.g. Vivatrat et al., 1982; Degenkamp & Dutta, 1989a, 1989b; Dutta & Degenkamp, 1989). The normal, $N$⁠, and sliding (or frictional) resistance, $F$⁠, as well as the weight, $mg$⁠, per unit length on the embedded chain, affect the tension, $T$ (Fig. 1). Integration of the chain equations gives the shape and the changing tension from the sea floor to the anchor. The governing equations are:

where $s$ is the position along the chain length and $θ$ is the local chain inclination (Vivatrat et al., 1982).

In fine-grained soils, $N$ and $F$ are linked to the undrained strength, $su$ (Degenkamp & Dutta, 1989a, 1989b; Dutta & Degenkamp, 1989). Corresponding parameters for the normal and sliding resistance in coarse-grained sediments are less well established (Neubecker & Randolph, 1995; Frankenmolen et al., 2016). Conventional bearing capacity theory suggests $N$ is proportional to depth, $z$⁠, and can be calculated by way of a conventional bearing factor, $Nq$⁠, combined with a friction coefficient, $μ$ (Neubecker & Randolph, 1995), such that:

where the bearing stress, $Nqγ′z$⁠, acts on an area per unit length that is scaled from the bar diameter, $dbar$⁠, by the factor, $En$⁠, which has been estimated as $En=2·5$ from model tests.

However, assessment of $Nq$ requires selection of an appropriate effective friction angle, $ϕ′$⁠, and an expression for $Nq=fϕ′$⁠, and does not allow for any inclination of the chain relative to the ground surface. There is therefore, uncertainty in the shape of the embedded mooring line inverse catenary and the resulting loading direction at the anchor padeye, with the consequence that mooring and anchor designs may be non-optimal. Any potential optimisations of anchor design could have significant benefit as floating wind energy is upscaled because projects require of the order of 300 anchors and mooring lines per gigawatt of installed capacity (based on 10 MW turbines anchored with three mooring lines/anchors (Cerfontaine et al., 2023a).

This uncertainty is addressed through a programme of chain–seabed interaction experiments conducted in a geotechnical centrifuge using carbonate and siliceous sand under drained conditions. The testing provides new data of the inverse catenary shape and load distribution in the embedded chain, which is used to calibrate and refine an improved model for the chain shape and tensioning response based on cone penetration test (CPT) tip resistance. The paper first describes the experiments and then validates the improved theoretical model against the test data. Finally, the implications of this improved model for anchor and embedded mooring line design are illustrated by a case study.

The experiments were conducted at a testing acceleration of 40g using the 3·6 m dia. 40 g-tonne beam centrifuge located at the National Geotechnical Centrifuge Facility in the University of Western Australia. The tests used carbonate and silica sands at different density states and used a studless chain with two different bar link diameters.

Model chains with bar diameters, $dbar$ = 4 and 6 mm were tested, equivalent to a prototype chain with $dbar$ = 160 and 240 mm, which is typical for mooring applications. The chain link dimensions, masses and surface roughness data are given in Fig. 2 and Table 1. The relative roughness is R_a/d₅₀ = 0·007–0·008 for the two sands, making the interface of intermediate roughness (Lings & Dietz, 2005).

As shown in Fig. 3, one end of the chain was attached to a fixed padeye located at a depth, $za$ = 125 mm. Two orthogonal load cells were pin-connected between the padeye and the container walls, giving the tensions, $TaH$ and $TaV$⁠, from which the anchor load, $Ta$⁠, and inclination, $θa$⁠, are found. The other end of the chain was on the sand surface and connected to a steel cable by way of a third load cell measuring $T0$⁠.

The experiments used a carbonate and a siliceous sand, with properties shown in Fig. 4 and Table 2. The $d50$ values correspond to a minimum $dbar/d50$ ratio of 18, which is sufficient to neglect particle size effects (Bolton et al., 1999), noting that particle breakage around the chain would increase this ratio further.

The sand samples were prepared by air pluviation with adjustments made to the fall height, hopper slot opening and travelling velocity to achieve dense and loose samples. During sample preparation the chain was connected to the anchor padeye and oriented vertically and draped over the edge of the box. The sand was filled flush to the top of the sample container and levelled with a straight edge so the sample height equalled the container depth of 225 mm. As drained behaviour was the focus in the experiments, the samples were dry.

Even though the prototype situation is offshore, the use of dry sand is convenient for modelling purposes, although it introduces an increase in the vertical effective stress level in the model compared to a saturated prototype, due to the sand having a dry effective weight rather than a saturated effective weight. A consequence is that the stiffness and strength response of the sand corresponds to behaviour over a wider range of stress level than would apply in the field over the depth range from the surface to the anchor padeye. However, any influence on the soil response will affect both the chain–soil behaviour as well as the CPT that is used for the sand characterisation in the accompanying prediction model, so this difference in effective weight is not a concern.

The use of dry sand in centrifuge studies is a common approach that has been adopted for various offshore foundation systems (White & Lehane, 2004; Cox et al., 2014; Nicolai et al., 2017; Cerfontaine et al., 2023b). When using these studies or the present work for further analysis or to validate prediction methods, it may be important to recognise that the geostatic stresses are higher than at an equivalent depth offshore (or onshore in saturated soil), with a resulting influence on strength and stiffness due to the stress-dependency of these properties.

Prior to the first stage of loading the chain was fixed at a point above the anchor padeye to ensure it remained vertical. CPTs used a cone with diameter, $dc$ = 7 mm and gave repeatable profiles of cone tip resistance, q_c, confirming the lateral homogeneity (Fig. 5). Fitted and adjusted profiles of q_c were generated to represent field-scale q_c profiles following Lehane et al. (2023). These adjustments take into account the effects of stress level and relative density on the measured cone tip resistance, and adjust for the shallow embedment effects that influence centrifuge CPT data due to the large penetrometer diameter relative to the sample depth. The Lehane et al. (2023) approach uses the stress-normalised cone tip resistance:

where $pa$ is atmospheric pressure; $σv′$ is the vertical effective stress; and $n$ is a constant equal to 0·5 in carbonate sands and 0·7 in silica sands (Lehane et al., 2023). In soil with depth-constant relative density, the shallow embedment effect causes $qc1N$ to vary with depth, $z$⁠, in the following form:

where $qc,ssn$ is a steady-state stress-normalised cone penetration resistance at large normalised penetration depths, $z/dc$⁠, and $a$ is an empirical curve-fitting parameter that fits the increase in $qc1N$ with increasing penetration depth. The CPT fitting parameters, $qc,ssn$ and $a$ that were adopted to provide the fitted q_c profiles in Fig. 5 are listed in Table 3. The values of $qc,ssn$ and $a$are within the range of published values for carbonate and silica sands included in Lehane et al. (2023). The adjustment is more significant for the dense sample, because the effect of the shallow cone embedment persists to a greater depth in denser soils. The adjusted $qc$ profiles were used in the subsequent $qc$-based chain–seabed interaction analysis, and are relevant to field-scale conditions. The presence of the load cells and chain within the samples meant that estimation of the sample density from sample mass and volume measurements was unreliable. As such, relative density was inferred for each sample using the approach outlined in Lehane et al. (2023). This gave relative densities of D_r = 0·93 and 0·25 for the dense and loose carbonate samples and D_r = 0·82 and 0·20 for the dense and loose silica samples, albeit that the relative densities inferred for the loose samples are lower than the range adopted to develop the Lehane et al. (2023) approach.

Six tests were conducted, each in a separate soil sample, with the test IDs given in Table 3. The free chain end was pulled by an electro-mechanical actuator at ∼0·5 mm/s, causing the embedded chain to cut through the soil and form the inverse catenary. Each test was continued up to $T0$ = 11·25 kN (⁠$T0$ = 18 MN at prototype scale), with unloading stages where the centrifuge was stopped and the chain profile was observed.

This section outlines typical results for a single test. The full set of results is set out later when compared to the predictive model. Fig. 6 shows a typical sea-floor load–displacement response (⁠$T0$ against $x0$⁠), and the loads at the padeye (⁠$TaH, TaV$ and $Ta$⁠). Unload–reload loops show when the centrifuge was stopped for observations. Fig. 6(b) shows the load transfer to the padeye as a proportion of the sea-floor load (that is $Ta/T0$⁠, $TaV/T0$ and $TaH/T0$⁠). Spikes arise from $T0$ reducing to zero during unloading, when the mobilised friction reverses to lock in load at the padeye.

Reloading after unloading led to subsequent increases in $Ta$ that were a continuation of the maximum $Ta$ profile measured in the previous loading stage. This suggests that any release in locked-in load or shear stress from the single unload–reload cycle had minimal effect on the load–displacement response.

The load–displacement responses are highly non-linear, with a rapid increase in tangent stiffness as the chain becomes straighter. The proportion of load transmitted to the padeye was typically $Ta/T0$ = 0·75 to 0·85 at high loads, with slightly higher ratios in the looser soils and for the larger chain. This ratio increases with increasing sea-floor load, showing that the embedded chain sustains a reducing proportion of the mooring load as it straightens.

A greater displacement, $x0$⁠, occurred in the loose samples under a given tension, confirming that the lower density allowed easier penetration by the chain, leading to a lower chain angle at the anchor padeye and a reduced vertical uplift, $TaV$⁠. This is evident in Fig. 7, which shows the load components at the anchor padeye, $TaH$ and$TaV$⁠, for each test. The axes of Fig. 7(a) are scaled equally, so the load vector (⁠$TaH$⁠, $TaV$⁠) is aligned in the true direction, which is colinear with the chain at the padeye.

The resultant padeye load is initially vertical, consistent with the chain shape, and progressively rotates as the inverse catenary is mobilised. Variation in the path between tests is consistent with the chain displacement being closer to vertical for the denser sample and the larger chain. The padeye chain angle, $θa$⁠, is also indicated in Fig. 7(a) and ends in the range, $θa$ = 27·5 – 64·5°. This same effect is evident in Fig. 7(b), which shows the variation of the load inclination at the anchor padeye, $θa$⁠, as the sea-floor load, $T0$⁠, is mobilised. In all tests, this inclination is initially close to 90°, but falls as the sea floor load increases towards the maximum value and the inverse catenary develops.

For coarse-grained soils under drained conditions, the normal and tangential chain–soil resistance can be estimated by scaling from the cone penetration resistance, $qc$ (Frankenmolen et al., 2016). The resulting expressions for $N$ and $F$ are:

where $β$ and $ξ$ are dimensionless scaling factors for normal and tangential resistance that replace the bearing and interface friction factors used for an undrained response.

Combining equations (7) and (8) leads to the friction coefficient between the chain and soil:

Rather than treating $ξ$and the ratio $Et/En$ as model inputs, it is more appropriate to use $μ$ since this has physical meaning as a friction coefficient. The basis for defining these two inputs, $β$ and $μ$⁠, is set out in the following sections.

A horizontal element of chain (i.e. $θ∼$0$°$⁠) is analogous to an embedded strip foundation, while a chain element oriented vertically (i.e. $θ∼90°$⁠) is analogous to a laterally loaded pile. Vertical penetration of a horizontal chain is therefore analogous to CPT tip penetration, with differences due to geometry differences. It can be assumed that the normal stress acting on the effective chain width scales directly with $qc$⁠.

However, as the chain inclination increases towards the padeye (⁠$θ→$90$°$⁠), the interaction changes towards the laterally loaded pile case, for which the ultimate lateral resistance is only a small fraction of $qc$ (e.g. Dyson & Randolph, 2001; Suryasentana & Lehane, 2014). A formulation commonly used for the limiting lateral bearing stress is $σh′=Kp2σv′$ (Barton, 1982; Fleming et al., 2009) where $Kp=(1+sin⁡ϕcs)/(1−sin⁡ϕcs)$⁠. To allow for this dependence of the normal resistance on chain inclination, the scaling factor on $qc$ becomes a function, $βθθ$⁠:

Combining equations (7) and (10) leads to:

For a horizontal chain (⁠$θ=0°)$⁠, equation (11) reduces to the original formulation in Frankenmolen et al. (2016) with a simple scaling factor, $β0$ on $qc$⁠; $N=Endbarβ0qc$⁠. For a vertical chain, $(θ$ =$π/2$ in radians or $90°$⁠), equation (11) reduces to $N=Endbar(bKp2σv′)$⁠. The variation in normal stress with chain inclination, $θ$⁠, follows an exponential trend, which is consistent with the change in mean stress through rotation of the principal stress direction in frictional materials (e.g. Bolton, 1991; Powrie, 2018).

Figure 8 plots the relationship between the vertical and horizontal components of the normalised pressure acting on the chain, $pN=N/Endbar/qc$ (that is $pNV$ = $pNsin⁡θ$ and $pNH=pNcos⁡θ$⁠) for different combinations of $β0$ and $b$⁠, together with the original Frankenmolen et al. (2016) formulation in equation (7) where $β$ = 0·625. The passive resistance factor, $b,$ controls the normal resistance as the chain rotates towards a vertical orientation. The selected combinations of $β0$ and $b$ provide a good fit to the load paths measured in the experiments, as discussed later in the paper.

The mobilised friction coefficient, $μ$⁠, in an embedded chain with an inverse catenary shape can be estimated from the tension and inclination at each end of the chain, ignoring the chain self-weight, using the ‘capstan’ (or Euler–Eytelwein) equation, which is (Neubecker & Randolph, 1995):

For a catenary mooring, for which $θ0=0°$⁠, equation (12) gives:

As equation (13) neglects the weight of the chain it is less accurate at lower chain loads when the self-weight is higher relative to the imposed tensions. This is evident in Fig. 9, which shows the evolution of $μ$ with increasing $T0$⁠. For $T0$ > 2 MN the friction coefficient stabilises at $μ ∼$ 0·25.

The inferred friction coefficients at high load in Fig. 9 are significantly lower than the range of friction coefficient, $μ$=  0·65 – 0·75, measured in chain surface drag tests (Frankenmolen et al., 2016). However, during the early loading stages, when more of the chain remains on the soil surface, higher values are inferred, which is consistent with the chain being pulled horizontally. The lower values of$μ$ in the embedded chain are linked to the higher level of mobilised normal resistance – that is full bearing failure, leading to penetration into the soil (Frankenmolen et al., 2016). This normal failure causes a reduction in the available frictional capacity, as found in other forms of combined multi-directional loading, such as the shape of vertical–horizontal combined loading envelopes in sand (e.g. Butterfield & Gottardi, 1994). For different seabed and chain or mooring-line combinations, selection of $μ$ could be based on model tests or tests on segments or elements of a line. It is interesting to note that in this study, $μ$ showed only small variation between dense and loose carbonate and silica sands, suggesting that $μ$ = 0·25 could be used more generally for inverse catenary chain–sand applications. However, further testing is needed to validate this.

Figure 10 shows calculated chain profiles for $dbar$ = 4 mm at $T0$ = 18 MN in loose carbonate sand with the adjusted $qc$ profiles shown in Fig. 5(b) and the same sets of $β0$ and $b$ shown in Fig. 8. The different profiles show that the $β0θ$ model causes the chain to cut more deeply because of the lower resistance, $pN$⁠, on the vertical portion, which allows the chain to rotate to a flatter angle close to the padeye (see inset Fig. 10, and the corresponding $pN$ profiles in Fig. 8). Also, the chain cuts deeper for lower values of $β0$⁠, which is due to the lower normal resistance on the near-horizontal portion of the chain (as also illustrated in Fig. 8).

The impact of varying the values of $β0$ and $b$ on the shape of the chain profile is small, but variations in $β0$ and $b$ have a greater effect on the load path responses at the anchor padeye and the chain tensioning responses. These are discussed in the following section.

In this section, the theoretical model is calibrated to the experimental results by fitting the measured load paths at the anchor padeye (Figs 11(a) and 11(b)). These load paths describe the chain–anchor interaction and influence the combined chain–anchor system capacity. The unload–reload loops in the experimental data were omitted during the calibration process.

The calculated load paths in the improved theoretical model were fitted by minimising the mean error, $|E¯|$⁠, by varying the model parameters $β0$ and $b$⁠. This mean error, $|E¯|$⁠, is defined as the sum of mean errors, $|E¯| =∑|Eı¯|$⁠, over all tests, where $|Eı¯|$ is the normalised area between the calculated and measured load paths (Fig. 12).

Values of mean error, $|E¯|,$across all tests are shown in Fig. 13(a) for different combinations of $β0$and $b$⁠. Optimal $β0$and $b$ sets give $|E¯|$= 5–7%, which is much better than $|E¯|$ = 25% using the original formulation (⁠$β$= 0·625). Other values of $β$ were also trialled, with the minimum $|E¯|$ = 16% obtained using $β$= 0·5 (see Fig. 13(a)), which is more than double the error from the modified $β0-b$ formulation.

The bias error for a single test, $Eb,ı¯$⁠, is defined as the difference between the over- and under-predicted regions of the load paths (Fig. 12). For the optimal β₀ – b pairs (that is with minimum $|E¯|$in Fig. 13(a)), the bias $Eb¯$ was within ±4% (represented by solid symbols in Fig. 13(b)). This compares to $Eb¯$ = 12% obtained with the original formulation and $β$= 0·5 (which gave the lowest $|E¯|$⁠).

The optimal parameter choice is $β0$ = 0·7 and $b$ = 1, such that $pN$ decreases to a minimum of $Kp2σv′$ as $θ→$90$°$ (consistent with lateral pile analysis, e.g. Fleming et al. (2009)). This combination of $β0$ and $b$gives minimum error and bias (Fig. 13).

The padeye load path data can also be compared with the embedded chain solution adopted by Neubecker & Randolph (1995), which features a linear variation in chain–soil resistance with depth, following a bearing factor approach (equation (4)). The bearing factor, $Nq$⁠, for chains embedded in sands is not well defined, but the solutions shown in Fig. 14 adopt values of $Nq$ which correspond to a pressure on the equivalent chain width (⁠$N/Edbar$⁠) that increases at the relevant rate, $kqc$⁠, as shown in Fig. 5 (i.e. equating equations (4) and (7)). This leads to values of $Nq$in the range of 40–292, which are consistent with bearing capacity theory (e.g. Hansen, 1970) for the relevant friction angles. This solution did not fit the measured load paths as well as the improved theoretical model, underpredicting the horizontal load, $TaH$ at low $Ta$ and overpredicting $TaH$ at high $Ta$ (Fig. 12).

Figure 15 compares the chain tensioning response in the experiments with this modified formulation (⁠$β0$ = 0·7 and $b$ = 1) and the original Frankenmolen et al. (2016) formulation with $β$ = 0·625. The resulting mean error for each test is $|Eı¯|$ ≤ 7% and ≤10% using the modified and original formulations, respectively, when $|Eı¯|$ is defined as in Fig. 13, but using tension and displacement rather than horizontal and vertical load components. Typically, the calculated chain tensions match the measurements well at lower displacements but tend to underestimate the displacement at high loads. This could be attributed to stretch of the chain, which is not accounted for in the inverse catenary model.

Figure 16 compares the calculated and measured loads at the padeye as a proportion of the sea-floor load (that is $Ta/T0$⁠, $TaV/T0$ and $TaH/T0$⁠). The modified formulation gives better agreement, particularly for smaller displacements and lower tensions. This is because in the modified formulation, the normal resistance on the chain is lower for higher chain inclinations than in the original formulation, causing more chain rotation at the padeye (as shown in Fig. 8). This results in higher $Ta/T0$⁠, consistent with the simplified analysis of equation (12).

Both the measurements and calculated responses show that $TaV$ develops initially, consistent with the initial vertical orientation of the chain, and significant $TaH$ is only mobilised at higher displacements (Fig. 16). Although some of the profiles show a reduction in $TaV/T0$ with increasing displacement, the absolute value of $TaV$ always increases.

The modified model matches the measured growth in $Ta/T0$ better than the original model. It also shows an earlier growth in $TaH/T0$⁠, with rotation of the chain at the padeye beginning at lower values of displacement, $x0$⁠. These two effects result from the lower normal resistance on the vertical chain elements, permitting the chain to rotate at the padeye under lower tensions. These improvements from the modified formulation are important, as the load inclination at the padeye influences the anchor capacity, as explored in the following section.

In this section, the embedded mooring line inverse catenary model is applied in an example application where the chain is connected to a suction caisson anchor with skirt length, $L$⁠, equal to the diameter, $D$ (i.e. $L/D=1$⁠) and the padeye is located at $z/L =2/3$⁠. The capacity of the caisson is assessed using the failure envelope formulations from Zhao et al. (2019), calculated for sands with the same properties as the tests in this paper (Tables 2 and 3). The only adjustment from the Zhao et al. (2019) approach is that a rough caisson surface is assumed, with a normalised interface friction angle, $δ/ϕ$ = 0·765, consistent with interface strength data from Potyondy (1961) and Westgate et al. (2021), whereas Zhao et al. (2019) modelled a smooth model caisson surface with $δ/ϕ$ = 0·5 to 0·7. The embedded mooring line–anchor system interaction at the anchor padeye is found by comparing the capacity envelope to the load paths created by the mooring line inverse catenary as it is tensioned. The ultimate capacity of the embedded mooring line–anchor system is governed by the intersection of the embedded mooring line load paths with the anchor failure envelope.

The simulated embedded mooring lines included chains with the same dimensions as the experiments (Table 4) and a synthetic fibre rope, since these are emerging as a mooring line option due to enhanced abrasion resistance using jackets or sheathes (Pillai et al., 2022). As the scaling factor for a rope is $En =1$⁠, the effective diameter of the rope is lower than that of the chains, which means that the rope has a higher potential for cutting into the seabed when tensioned. The synthetic rope properties (Table 4) were selected based on a Bridon-Bekaert MoorLine polyester rope (Bridon-Bekaert, 2021), which has a comparable minimum breaking strength (MBS) to the $dbar$ = 0·185 m chain adopted in the mooring of a reference 15 MW turbine catenary mooring system (Allen et al., 2020; Pillai et al., 2022).

Each embedded mooring line was tensioned to $T0$ = 18 MN, matching the limit in the experiments and below the lowest MBS (20·6 MN, see Table 4). The calculated load paths are provided in Fig. 17 for the two sand types and density states considered in the experiments, together with the caisson failure envelope based on Zhao et al. (2019). The embedded mooring line–anchor system capacity for each case is given in Table 5.

As the effective embedded mooring line diameter (⁠$Endbar$for the chains or $End$for the fibre rope) reduces, the load paths flatten and intersect the anchor envelope at higher $Ta$ and $T0$⁠, such that embedded mooring line–anchor system capacity is highest for the fibre rope, followed by the $dbar$= 0·16 m chain and the $dbar$= 0·24 m chain. This arises because a lower effective diameter allows the embedded mooring line to cut further into the seabed, reducing the load inclination at the padeye. The simulations show that this mechanism causes the fibre rope to provide a 23 – 65% improvement in system capacity compared to the larger chain and a 23 – 33% improvement compared to the smaller chain. Lower effective diameters also lead to higher displacements at the sea floor, $x0$ (Fig. 18).

The embedded mooring line–anchor interaction is also influenced by soil density. Mooring lines cut further into the looser sands, making the padeye load angle flatter than in the denser sands. In most cases, this causes the embedded mooring line–anchor system capacity in loose sand to exceed that in dense sand for the same anchor and mooring line combination, even although the failure envelope is larger for the anchor in the denser soil. This occurs because the reduction in anchor capacity due to the lower density is eclipsed by the increase in $Ta$ at failure due to the lower padeye load inclination. The higher $Ta$ at failure also leads to a higher tension at the sea floor, $T0$ (Table 5). This effect is not properly captured by other models for the embedded mooring line shape because the reduced lateral resistance on the vertical part of the mooring line is not captured.

These results highlight the potential for fibre rope or steel wire embedded mooring lines to form a flatter inverse catenary shape at lower mooring line tensions, allowing additional anchor capacity to be mobilised compared to an equivalent chain.

These tests involve only monotonic loading, but in offshore installations, chains are subjected to cyclic loads over their operational lifetime. Further cyclic testing and model development is required to characterise this response, which is beyond the scope of this study. However, the relative performance of different chains in different soil conditions under monotonic loading, which is the focus of the present study, might be expected to also apply to cyclic loading conditions.

This paper develops an improved CPT-based model for the response of embedded mooring lines, calibrated from model tests conducted in a geotechnical centrifuge, and demonstrates the resulting influence on embedded mooring line–anchor system design.

The test data provide new insights on the transmission of mooring line tension from the horizontal on-bottom position through the embedded line inverse catenary to the padeye of an anchor as the mooring line is progressively tensioned. The tests spanned two different sand types and density states, under drained conditions, and used two sizes of embedded mooring chain.

A new feature of the improved CPT-based model is the influence of line inclination on the limiting normal and sliding resistance through a scaling factor on $qc$ that varies with line inclination. This model was calibrated to the experimental data and was shown to give better agreement than models without inclination-dependence of the embedded line–soil interaction forces. In particular, the improved model better captures the onset of chain rotation at the padeye, by recognising that the normal resistance on a vertical chain element is lower than on a horizontal element. This matches the observed flattening of the padeye load observed in the experiments.

This improved model provides a more rigorous basis to assess mooring line–anchor interaction. Example simulations show that this improved theoretical model provides a simple basis to assess embedded mooring line–anchor system capacity when combined with combined loading failure envelopes for the attached anchor. These simulations demonstrate important interaction effects that could be harnessed for more efficient embedded mooring line and anchor designs.

A smaller diameter of embedded mooring line (or chain) leads to a flatter load inclination at the padeye, which is a more optimal loading direction for most anchors. For the example case of a suction caisson, a smaller diameter embedded mooring line led to higher embedded mooring line–anchor system capacity due to the change in padeye load inclination and the intersection point with the failure envelope. Emergent wire or rope-type mooring lines, which have a smaller effective diameter than chains of the same breaking strength, lead to flatter embedded catenaries and a lower load inclination at the padeye. As a result, these new mooring line types offer the potential to raise the holding capacity provided by a given size of anchor.

Soil density affects the anchor capacity and the embedded mooring line inverse catenary in ways that have opposite effects on the system capacity. The anchor capacity envelope is smaller in loose soil than in dense soil due to the lower soil strength, but the embedded mooring line inverse catenary cuts to a more favourable flatter orientation at the padeye, which mobilises higher horizontal anchor capacity. The simulations show that in many cases the latter effect eclipses the former effect, so the embedded mooring line–anchor system capacity is greater in loose sand than in dense sand.

Overall, the new embedded mooring line–sand interaction model developed in this paper provides a more rigorous basis to assess mooring and anchor interaction. This allows more optimal designs involving smaller and cost-effective mooring line and anchor systems to support the development of floating wind facilities, and other moored offshore facilities.

The experiments reported in this paper were partly funded by Shell Australia. This work was supported by the Royal Academy of Engineering under the Research Fellowship Programme, and by the UK EPSRC by way of the EPSRC Supergen Offshore Renewable Energy (ORE) Hub (grant EPSRC EP/Y016297/1). Katherine Kwa is supported by an RAEng Research Fellowship.