Applied Ocean Research 32 (2010) 284–297

Contents lists available at ScienceDirect

Applied Ocean Research

journal homepage: www.elsevier.com/locate/apor

Advancing pipe–soil interaction models in calcareous sandYinghui Tian ∗, Mark J. Cassidy 1, Christophe Gaudin 2Centre for Offshore Foundation Systems, The University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia

a r t i c l e i n f o

Article history:Received 19 August 2009Received in revised form18 January 2010Accepted 7 June 2010Available online 5 July 2010

Keywords:Pipe–soil interactionCentrifuge testCalcareous sandIntegration into FE

a b s t r a c t

The interpretation of geotechnical centrifuge experiments of a segment of pipe on calcareous sand isdescribed. Results of 33 individual tests are used to advance a theoretical plasticity model that describesthe load–displacement behaviour of a pipe when subjected to combined vertical and horizontal loading.A new model formulation is provided with all parameters calibrated from the experimental tests. It isshown that the modified model provides excellent agreement with the centrifuge data up to lateraldisplacement of two pipe diameters. This provides additional confidence in its use in the simulation oflong pipeline events. The principal purpose of the model is illustrated in a numerical example calculationof the response of a 1245m long pipeline. Integrating themodel into a structural analysis program allowsefficient modelling of the pipe–soil behaviour, as demonstrated for the partially buried pipe subjected to3 h of storm loads.

© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

As an integral part of the infrastructure of offshore oil andgas developments, pipelines play a significant role in transportinghydrocarbons between subsea wells, platforms and onshoreprocessing facilities. Pipelines laid directly on the seabed aresubjected to environmental hydrodynamic loads as well asinternal loading due to temperature changes. One of the majortechnical challenges to ensuring their on-bottom stability is toaccurately model pipe–soil interaction. Traditional geotechnicalapproaches have been empirically based on bearing capacityand frictional models ([1–3]), with break-out forces determinedthrough analytical closed-form solutions or empirically derivedand calibrated from experimental data [4,5].The use of a plasticity framework to encapsulate the behaviour

of a small section of a pipe and the underlying soil offers anattractive alternate framework [6]. By expressing the pipe–soilbehaviour purely in terms of the loads on the pipeline and thecorresponding displacements (see Fig. 1), a more fundamentalunderstanding of the pipe–soil interaction under combinedloading can be expressed in a terminology consistent with thepipeline structural analysis (see [7,8,6,9]). The pioneering work ofSchotman and Stork [10] has lead to further models developed

∗ Corresponding author. Tel.: +61 8 6488 7076; fax: +61 8 6488 1044.E-mail addresses: tian@civil.uwa.edu.au (Y. Tian), mcassidy@civil.uwa.edu.au

(M.J. Cassidy), gaudin@civil.uwa.edu.au (C. Gaudin).1 Tel.: +61 8 6488 3732; fax: +61 8 6488 1044.2 Tel.: +61 8 6488 7289; fax: +61 8 6488 1044.

0141-1187/$ – see front matter© 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.apor.2010.06.002

through combinations of geotechnical centrifuge tests [11–14],limit theory analysis [15] and finite element modelling [16].For predicting the behaviour of partially embedded pipes on

sand the three pipe–soil interaction models developed by Zhang[11] and Zhang et al. [13,12] are current state-of-the-art. Thesemodels were calibrated through geotechnical centrifuge tests ofmodel pipes on calcareous sand. Two pipes were used, one being160 mm long and 20 mm diameter and the other 350 mmlong and 70 mm diameter. With tests conducted at accelerationsof 50 and 15 times that of Earth’s gravity respectively, bothsimulated a 1 m diameter prototype pipe. The three models haveincreasing sophistication, from a single surface work hardeningelastoplastic model (EP), a bounding surface model (BS) to a two-yield surface kinematic hardening model colloquially named the‘‘bubble’’ model. Recently called the UWAPIPE models, details oftheir numerical implementation and recent modifications can befound in [9].Retrospective simulation of Zhang’s centrifuge tests using the

UWAPIPE models show good agreement, but only when thelateral displacement is less than half the pipe diameter [11].This severely limits the application of the model to prototypeconditions. Furthermore, a close fit is not surprising as mostof Zhang’s centrifuge experiments were deliberately aimed tocalibrate the model parameters (and not to represent realisticsubsea pipeline loads under hydrodynamic wave and currentaction). To investigate the limit of the UWAPIPE models to lateraldisplacement and to provide evidence of its capability undermore load paths, a suite of tests have been conducted in thebeam centrifuge at the University of Western Australia. Silty sandcollected in 169 m water depth from the North West Shelf ofAustralia has been used.

Y. Tian et al. / Applied Ocean Research 32 (2010) 284–297 285

Pipe segment

Pipe axis

Fig. 1. Combined loads and displacement on a pipe segment, and sign convention adopted.

2. Outline of the UWAPIPE model

The UWAPIPE EP model is based within the framework ofconventional work hardening plasticity theory and thereforeassumes purely elastic response inside a single expandable yieldsurface. It contains four components, (i) a combined loadingyield surface defining the allowable loading conditions, (ii) ahardening law that describes the expansion of this surfacewith embedment, (iii) an elasticity matrix defining the elasticresponse for increments inside the yield surface, and (iv) aflow rule describing incremental plastic displacements during anelastoplastic event. Thoughnot explicitly explored in this paper theBS model utilises a similar formulation, but allows for a nonlineartransitional elastoplastic domain inside the bounding surface byeffectively shrinking the EPmodel’s elastic region to a point [9,17].The BMaccounts for kinematic hardening through the introductionof a second smaller bubble surface.Some numerical problems surrounding the UWAPIPE models

[11,9] have also been addressed in this paper. For instance, singularpoints of the sharp vertexes on the yield surface and plasticpotential of the UWAPIPE model can cause numerical instabilities.The numerical robustness of the model has been increased in thispaper by proposing new forms of equations for the yield surface,hardening law and plastic potential. The centrifuge model testshave been used to calibrate and update the proposed UWAPIPE EPand BM models. Formulation details of the models are providedwhen discussing their experimental derivation, though a fullnumerical description of the more complex BM is also provided inthe Appendix.

3. Centrifuge tests

A series of geotechnical beam centrifuge tests were conductedat 50 times Earth’s gravity using a 20 mm diameter and 120 mmlong smooth model pipe. This represents a prototype pipe of 1 mdiameter and 6 m length, with all subsequent test results in thispaper to be presented in equivalent prototype dimensions (see [18]for scaling laws).Three individual centrifuge boxes of tests were conducted, with

a total of 42 pipe and 17 characterising CPT tests conducted. Onlyselected results are discussed here, as outlined in Table 1. However,the full set of experimental results and a detailed description of thetesting methodology are provided in [19].The silty sand used was collected in grab samples from 169

m water depth on the North West Shelf of Australia. It hasbeen utilised in previously centrifuge testing at COFS/UWA. Thesample’s particle size distribution is shown in Fig. 2. Also shownis the similar distribution of the calcareous sand used in the

Fig. 2. Particle size distribution.

qc (kPa)

w (

m)

Fig. 3. Cone penetration tests.

original centrifuge tests of Zhang [11]. Though the sandswere fromdifferent offshore locations the particle distribution is reasonablysimilar between the two soils.Miniature cone penetration (7 mm diameter) tests were

performed in every strongbox before and after the pipe–soilexperiments. Two cone penetration tests are plotted in comparisonwith those conducted by Zhang [11] in Fig. 3. The curves agreewell, especially when the penetration depth is less than 0.5 m(half a pipe diameter), which is the embedment depth of the

286 Y. Tian et al. / Applied Ocean Research 32 (2010) 284–297

Table 1Details of tests presented (Force: kN/m Displacement: m).

Test no. Test type Status before test Test control FigureV V0 w OLR 1w 1u 1V 1H

CW1 CW (Swipe) 66.25 66.25 0.24 1.00 0 0.27 Figs. 6, 7 and 15CW2 50.25 66.96 0.22 1.33 0 0.31 Figs. 6, 7 and 15CW3 33.60 66.78 0.22 1.99 0 0.2 Figs. 6, 7 and 15CW4 82.75 82.75 0.23 1.00 0 0.5 Figs. 6, 7 and 15CW5 117.14 117.14 0.32 1.00 0 0.5 Figs. 6, 7 and 15CW6 133.68 133.68 0.33 1.00 0 0.45 Figs. 6, 7 and 15CW7 32.69 32.69 0.13 1.00 0 0.5 Figs. 6, 7 and 15CW8 37.36 37.36 0.10 1.00 0 0.5 Figs. 6, 7 and 15CW9 66.52 66.52 0.26 1.00 0 0.4 Figs. 6, 7 and 15CW10 75.62 75.62 0.25 1.00 0 0.3 Figs. 6, 7 and 15

CV1 CV (Probe) 23.90 23.90 0.12 1.00 2 0 Figs. 10(a) and 12CV2 17.70 32.01 0.14 1.81 2 0 Figs. 10(a) and 12CV3 8.16 8.16 0.10 1.00 2.46 0 Figs. 10(a), 12 and 14CV4 3.94 15.36 0.13 3.90 2 0 Figs. 10(a), 12 and 15CV5 33.38 33.38 0.16 1.00 1 0 Figs. 10(a) and 12CV6 21.05 21.05 0.11 1.00 2.5 0 Figs. 10(a), (b) and 12CV7 4.27 41.55 0.15 9.73 4 0 Figs. 10(a), (b) and 12CV8 2.19 41.69 0.17 19.04 4 0 Figs. 10(a), (b) and 12CV9 10.87 10.87 0.06 1.00 5 0 Figs. 10(a), (b) and 12CV10 4.90 167.48 0.42 34.18 5 0 Figs. 10(a), (b) and 12

RD1 RD 41.19 41.19 0.18 1.00 0.15 1 Figs. 8, 10(a), 12 and 16RD2 41.22 41.22 0.20 1.00 0.15 0.5 Figs. 8, 10(a) and 12RD3 41.88 41.88 0.19 1.00 0.15 0.25 Figs. 10(a) and 12RD4 21.96 42.28 0.22 1.92 −0.15 1 Figs. 10(a) and 12RD5 21.81 41.90 0.22 1.92 −0.15 2 Figs. 10(a) and 12

RF1 RF 21.25 41.45 0.25 1.95 25 25 Figs. 10(a) and 12RF2 21.20 41.58 0.20 1.96 12.5 25 Figs. 10(a) and Fig. 12RF3 20.95 41.15 0.22 1.96 6.25 25 Figs. 10(a), 12 and 17RF4 21.16 41.78 0.23 1.97 −8.33 8.33 Figs. 10(a) and 12RF5 21.21 41.56 0.20 1.96 −4.17 12.5 Figs. 10(a) and 12RF6 21.65 41.76 0.17 1.93 −12.5 6.25 Figs. 10(a) and 12RF7 21.56 42.23 0.18 1.96 −16.67 4.17 Figs. 8 and 12

VC1 VC 80.62 80.62 0.16 1.00 −8.33− 0 Fig. 5

majority of the experiments. It is also the area of most engineeringsignificance. Figs. 2 and 3 suggest that the silty sand used inthis series of centrifuge tests has comparable properties with thecalcareous sand used in the original model development by Zhang[11].In some cases, external hydrodynamic loading and internal

thermal loading acting on the pipeline can change so fast that theexcess pore pressure accumulation in the silty sand does not haveenough time to dissipate. Consequently, the pipe–soil interactionbehaviour results in partially drained condition, which are morecomplex than the fully drained behaviour. However, a thoroughlyunderstanding of the basic drained behaviour is considered asfundamental and the basis for further partially drained experimentand models. All the centrifuge tests in this paper were designed tobe conducted under drained condition. A relative slow velocitywasadopted in the experiments (approximate 0.1mm/s inmost cases).Thenormalised velocityvD/cv was about 0.002,wherevrepresentsthe pipe velocity, D the pipe diameter and cv the consolidationcoefficient which was assumed to have a typical value for siltysand of 10−3m2/s. This normalised velocity is considerably smallerthan the value of 0.01 considered to be the transition frompartiallyto fully undrained behaviour [20,11,21]. Tap water was used asthe pore fluid during the tests ensuring that the model behavedunder fully drained condition. It was unnecessary to use the moreexpensive silicon oil in the drained tests although it might modelconsolidation degree more accurately.The sign convention adopted in representing the vertical and

horizontal forces (V ,H) and the corresponding displacements(w, u) of the centrifuge test results and in the numerical modelis illustrated in Fig. 1. Six different types of centrifuge tests wereconducted. These are named vertical monotonic loading (VM),

Fig. 4. Loading path illustration.

constantw (CW), constant V (CV), radial displacement (RD), radialforce (RF) tests and vertical cyclic loading (VC). All the loading(and/or displacement) paths of centrifuge tests are illustrated inFig. 4 and detailed in Table 1.

Y. Tian et al. / Applied Ocean Research 32 (2010) 284–297 287

Table 2Model parameters.

Constant Description Dimension Value Unit Note

khe Elastic stiffness (horizontal) F/L/L 8333 kN/m/m Consistent with [11] khe = kve is adopted. For spudcan modelling Cassidy andHoulsby [23] reported khe = 0.87kve , andDNV-RP-F105 [44] recommended khe = 0.75kve and khe = 0.67kve forsandy and clay soils respectively. Calculation of the dynamic stiffness inDNV-RP-F105 [44] could also be used to estimate vertical and horizontalstiffness at sites where limited soils data is available. Using this methodfor this example, kve = 11 860 kN/m/m and khe = 8890 kN/m/m iscalculated for a 1 m diameter and 0.02 m wall thickness pipeline on loosesand. Other methods include assuming a simple relationship betweenvertical elastic stiffness and plastic stiffness, e.g. kve = λ1kvp . This wasrecommended by Zhang [11] with λ1 taken as 20.

kve Elastic stiffness (vertical) F/L/L 8333 kN/m/m

kvp Plastic stiffness (vertical) F/L/L 420 kN/m/m Hardening law parameters provided are best fit of experimental vertical loadtests. For offshore scenarios, traditional bearing capacity methods, such asthose provided in DNV-RP-F105 [44], or directcalibration to CPT tests could be used.

ζ Hyperbolic parameter inhardening law

L 0.09 m Plastic embedment when hardening law slope is kvp/√2.

h0surface Dimension of yield surface(horizontal: no penetration)

– 0.15 – Values of h0 = 0.116, 0.122 and 0.154 suggested for circular footings on densesilica, very loose silica and loose carbonate sands under drained conditionrespectively [45,42,43]. Zhang [11] used an equivalent value of 0.112.

κ Increase of yield surface sizewith normalised depth

0.052 Depth normalised by diameter Zhang [11] used an equivalent value of 0.183.

η Tensile capacity (yield surfaceapex)

– 0 – Uplift capacity assumed to be 0 as centrifuge tests were carried out under drainedconditions. The value of η is a function of the drainage conditions with fasteruplift causing partially drained or undrained conditions. Zhang [11]recommended a value of 0.06, based on extension of the curve of the yield surface.

β1 Curvature factor for yieldsurface (low stress)

– 0.56 – Though slight variations in yield surface shape with depth were observed,constant β values are recommended for simplicity.

β2 Curvature factor for yieldsurface (high stress)

– 0.99 –

α Association factor (horizontal) – 3.5 – The curvature of the plastic potential is a best fit of the measured displacementand force increments. This plastic potential differs from Zhang [11] who assumeda constant transition from uplift to penetration at V = V0/10, a conclusion notconsistent with the experimental results presented here.

β3 Curvature factor for plasticpotential (low stress)

– 0.025 –

β4 Curvature factor for plasticpotential (high stress)

– 0.6 –

Kmax The maximum value of theplastic modulus

– 80000 – These three parameters are only used in the bubble model.They are based on best fit of experimental values in this suite of tests.As recommended by Rouainia and Muir Wood [46], r = 0.2 and ρ rangingbetween 2 and 5 are appropriate. These values are, however,as also recommended by Zhang [11].Kmax = 80 000 was adopted in this study. It is a large enough value toguarantee a very ‘‘stiff’’ status when the distance betweenthe force point and the conjugate point reaches a maximum value. ρcontrols the changing rate of the plastic modulus and r represents the bub-ble size. Both values provide predictions which are consistent withexperimental observations.

r Size ratio of bubble tobounding surface

– 0.2 –

ρ parameter controlling thechanging rate of plasticmodulus

– 3.0 –

The vertical monotonic loading tests were not carried outseparately but conducted prior to all the other tests as the initialpath O → 1 (Fig. 4). A vertical unloading 1 → 2 usuallyfollowed the initial path O → 1, which means the maximumhistorical vertical load was larger than the current load. Theterminology of ‘‘normal load’’ and ‘‘over load’’ are analogous to‘‘normal consolidation’’ and ‘‘over consolidation’’ in classical soilmechanics. They are utilised in this paper to distinguishwhether ornot a larger historical vertical load V0 was exerted [11]. The symbolOLR is defined likewise to describe the ratio of over-loading. Thatis, OLR = V0/V , where V is the current force and V0 the maximum‘‘equivalent’’ vertical force during the initial penetration.Each test consisted of this initial step, with testing conditions

outlined in Table 1. These include the current vertical load V ,the maximum initial load V0 and the embedment w. Load ordisplacement control was then utilised to manipulate the testprocess. These are shown in Fig. 4 with the arrows indicatingthe expected behaviour in load and displacement space. Fourcomponents V , H , w and u were sampled at discrete frequenciesduring the tests.

4. Use of Centrifuge tests to define the UWAPIPE model

The centrifuge testswere performed to define each of themodelcomponents and parameters, as now described. A list of parametervalues for the new model is provided in Table 2.

4.1. Elasticity

Elasticity describes the constitutive behaviour for incrementsinside the yield surface. The vertical and horizontal behavioursare regarded as an uncoupled linear relationship with thecorresponding elastic displacements

1F ={1V1H

}= De1Ue =

[kve 00 khe

]{1we

1ue

}(1)

where F, U are the force and displacement vectors; kve the verticalelastic stiffness and khe the horizontal elastic stiffness.1 representsan increment and the superscript e denotes an elastic component.

288 Y. Tian et al. / Applied Ocean Research 32 (2010) 284–297

V (kN/m)

w (

m)

kve

Fig. 5. Vertical loading–unloading curve.

The vertical elastic stiffness kve can be estimated as theloading–unloading–reloading gradient in the vertical tests, asshown in Fig. 5. The value of kve was derived as 8333 kN/m/m.This is an average of all the vertical unloading tests in the testingprogram (Table 1). Zhang [11] reported a similar value of 8000kN/m/m.The estimation of the horizontal elastic stiffness khe directly

from the centrifuge tests is more difficult. Based on elasticitytheory Gazetas and Tassoulas [22] proposed empirical equations torelate horizontal and vertical elastic stiffness for various embeddedfoundations (with the pipe–soil system assumed to be a stripfoundation here). An equivalent horizontal to vertical elasticstiffness, khe = kve, is adopted. This is consistent with Zhang’s [11]proposal, while Cassidy and Houlsby [23] reported khe = 0.87kvefor the modelling of inverted conical ‘‘spudcan’’ footings in sand.

4.2. Yield surface

The yield surface is the boundary separating the pure elasticand elastoplastic states. The size of the yield surface denoted byV0 and h0 is solely related to the vertical plastic displacement wp.The equation of the yield surface is proposed as:

F =(Hh0V0

)2− β2f

(VV0+ η

)2β1 (1−

VV0

)2β2= 0 (2)

where V and H are the vertical and horizontal forces; β1, β2control the curvature of the surface (including the location of thepeak horizontal force, Hmax) and follow the approach of Nova andMontrasio [24]; and η represents the tensile capacity. A value ofβf =

(β1+β2)β1+β2

ββ11 β

β22 (1+η)β1+β2

ensures that Hmax equals h0V0 irrespective

of the values assumed for β1 and β2.As verified by Tan [25], the constant vertical displacement tests

(often termed a ‘‘swipe’’ test) tracks the yield surface in combinedloading space, assuming that the vertical elastic stiffness kve ismuch larger than the plastic stiffness kvp. Both normal-loadedand over-loaded swipe tests were conducted (see Table 1). Priorto the swipes, the pipe was first pushed vertically to a certainvertical force V0 and a subsequent embedment w0 was reached.In the normal swipes, the vertical displacement w0 was thenkept as constant and the lateral displacement u was increaseduntil a residual resistance was obtained. In contrast, in the over-loaded swipes the vertical force was first unloaded before beingpushed laterally. The trajectory of the vertical and horizontal forces(V ,H) of 10 swipe tests are shown in Fig. 6. These have all beennormalised by their corresponding V0.

V/V0

H/V

0

Fig. 6. Ten swipe tests and the best fit yield surface.

V/V0

w/D

Fig. 7. Change of peak horizontal load in yield surface as a function of pipepenetration.

Because the centrifuge tests were carried out under drainedcondition using silty sand, the tensile capacity is assumed to benegligible (η = 0). In other cases, where a tensile or suctioncapacity could be reasonably assumed the value of η wouldbe greater than zero. It is therefore included in the numericalformulation provided. The values of β1 and β2 were best fit usingthe least squares method [26] to be 0.56 and 0.99 respectively. Thevalue of h0 varies from 0.15 to 0.17 when the pipe embedment isless than half a diameter (the detailed description of h0 is providedas part of the hardening law formulation below). As illustratedin Fig. 6 the best fit yield surface shows an asymmetry about theyield surface centre ( V02 , 0). The crest of the yield surface peaks at(VV0

)peak=

β1−β2ηβ1+β2

= 0.36, which is smaller than 0.47 reported

by Zhang [11]. The reason for this discrepancy is attributed tothe deeper embedments used in these swipe tests (Table 1). Therelationship between the (V/V0)peak and the relative embedmentw/D is illustrated in Fig. 7, where a decreasing trend of V/V0 withincreasing w/D can be observed. Although β1 and β2 could bevaried to account for these subtle changes in shape, they have, forsimplicity, been assumed to be constant with depth.

4.3. Hardening law

The hardening law defines the expansion and/or contractionof the size of the yield surface. In the UWAPIPE model there are

Y. Tian et al. / Applied Ocean Research 32 (2010) 284–297 289

V0

wp

Fig. 8. Comparison of hardening law and centrifuge experiments.

two parameters in the yield surface (Eq. (1)) that determine thesize of the surface. These values of V0 and h0 are contained inthe hardening law, though both are related directly to the verticalplastic displacementwp.In the vertical monotonic loading tests, the vertical force

measured is also V0 and the vertical plastic displacementwp can beevaluated by subtracting the elastic component we from the totalvertical displacementw that is measured.

wp = w − we = w −Vkve. (3)

By plotting V0 against wp in Fig. 8, a nonlinear relationship isinitially recorded before it is approaching an asymptote. Therefore,a hyperbolic relationship between V0 and wp was adopted as thehardening law.

V0 = kvp

(−ζ +

√ζ 2 + wp2

)(4)

where kvp is the slope of the asymptote and ζ denotes the plasticembedment when the tangential slope is kvp√

2. Best fit values of

kvp and ζ are 420 kN/m/m and 0.09 m respectively. Again a leastsquares fitting method was used from all the vertical monotonicloading tests. In the original UWAPIPE model, a simple linearrelationship between V0 and wp was used. This results in ahardening law with the same value of the plastic stiffness kvp =400 kN/m/m [11].A function between h0 andwp has also been derived, in this case

from the 10 swipe test results (Table 1). The dots in Fig. 9 show h0from each individual swipe tests (i.e. the maximum H/V0) againstthe vertical plastic penetration wp of the swipe (Table 1). A linearrelationship is adopted in this paper.

h0 = h0surface + κwp

D(5)

where h0surface is the h0 value assumed for a surface footings (orin this case a pipe with zero penetration) and κ is the increase ofh0 with plastic vertical penetration. These values were best fit tobe 0.15 and 0.052 respectively. The original UWAPIPE model usesa parameter of µ rather than h0 to represent the size of the yieldsurface in horizontal load [11]. However, Zhang’s test results havebeen converted to equivalent h0 and provided for comparison inFig. 9. A lower h0surface of 0.112 and a steeper increase κ = 0.183was used in the original UWAPIPE model.As just outlined, the hardening law parameters have been

derived through 4 VM and 10 CW tests. However, the 5 RD

wp /D

h0

Fig. 9. Normalised yield surface size h0 as a function ofwp .

and 7 RF tests also provide information on the validity of thecombined yield surface shape (Eq. (1)) and size (Eqs. (4) and(5)). For these latter tests, the value of the theoretical V0 at eachrecorded step of the test can be back-calculated by (i) estimatingwp from the total vertical displacement using Eq. (3) and thenh0 from Eq. (5), and then (ii) determining V0 for each incrementby solving the nonlinear equation of Eq. (1) using the recordedvertical and horizontal loads. The assumption of surface expansioncan be evaluated by comparing the experimentally derived valuesof V0 from the combined loading tests against the hardeninglaw V0 determined from the purely vertical load tests. Shown inFig. 10(a) is this comparison for the proportion of the tests whenthe horizontal displacement was less than 2 pipe diameters. Thisshows reasonable agreement, validating the hardening law andyield surface shape assumptions being proposed.However, some CV tests continued to horizontal displacements

up to 5 diameters (Table 1). These results are shown in Fig. 10(b),but only for horizontal displacements larger than 2 diameters.The deviation of the curves from the hardening law line indicatesthat the assumptions being used are no longer valid, with thebuilding of a berm in front of the pipe becoming too prominentto be accounted for with this simple model. It will be furthershown in the latter part of this paper that the modified modelperformswell when the horizontal displacement is less than 2 pipediameters. Further developments, including a more sophisticatedradial hardening law, are required to account for behaviour pastthis point.

4.4. Flow rule

During an elastoplastic step the relative magnitudes of thehorizontal and vertical plastic displacements are defined by theflow rule. Back-calculation of the plastic displacements from theCV, RD and RF tests showed that an assumption of associated flowwas not appropriate. Therefore, an alternative plastic potential hasbeen defined with a vector normal to it describing the plastic flowdirection. For simplicity a non-associated plastic potential with asimilar mathematical expression as the yield surface is used and iswritten as:

g =(H ′

αh0V0

)2− β2g

(VV0+ η

)2β3 (1−

VV0

)2β4= 0 (6)

where H ′ is the ordinate value of the corresponding point onthe plastic potential rather than the current load point. This isillustrated in Fig. 11. Theparametersα,β3 andβ4 control the plastic

290 Y. Tian et al. / Applied Ocean Research 32 (2010) 284–297

V0 (kN/m)

wp (

m)

V0 (kN/m)

wp (

m)

(a) u < 2D. (b) u > 2D.

Fig. 10. Comparison of experimentally derived V0 withwp from CV, RD and RF tests with the hardening law assumptions.

Fig. 11. Illustration of the flow rule.

potential shape, with βg =(β3+β4)

β3+β4

ββ33 β

β44 (1+η)β3+β4

again eliminating the

influence of β3 and β4 on the surface size (which is now definedby αh0V0). By adjusting the parameters of α, β3 and β4, the aboveequation can vary from associated (α = 1, β3 = β1 and β4 = β2)to a significantly non-associated flow rule.The experimental CV, RD and RF tests are used to calibrate

the parameters of Eq. (6), though this is not as straightforward asthe other three components of the model. In order to investigatethis, the incremental plastic displacement ratio denoted by theangle tan−1(dup/dwp) and the force ratio represented by the angletan−1(H/V ) are defined. The meaning of these angles is displayedin Fig. 11. The values can be simply evaluated from the measureddata at each increment of yield surface expansion. Results from CV,RD and RF tests are shown in Fig. 12, with each dot representingan experimental measurement increment. The tests results arethen compared to the theoretical paths assuming different plasticpotentials. The test results unequivocally indicate that associatedflow is an unreasonable assumption, with all tests having a lowerhorizontal to vertical plastic displacement ratio (i.e. they penetratemore than associated flow predicts). After several trail calculations

tan-1

(H

/V)

tan-1 (dup/dwp)

Fig. 12. Comparison of experimental displacement directions (points) andtheoretical predictions (lines).

the values of α = 3.5, β3 = 0.025 and β4 = 0.6 are regarded asthe best fit parameters to the dataset as a whole.The plastic potential of the original UWAPIPE used a different

mathematical form [11], and the relative ratios are also shownin Fig. 12. Although the flow rule of the original UWAPIPE agreesfairly well with most of the data points of this study, it gives poorpredictions when tan−1(H/V ) approaches zero (in the region ofthe surface apex around V = V0). This is due to the sharp tip atthe apex of the previous plastic potential (see [11]). On the otherhand, the rounded corner of the modified plastic potential of Eq.(6) facilitates a better fit to the experimental database, as well asrobustness in numerical implementation.

4.5. Kinematic hardening two-surfacemodel and numerical enhance-ments

The above four components of elasticity, yield surface, hard-ening law and flow rule comprise a complete framework of theconventional elastoplastic model. No plasticity will be predictedinside the yield surface and an abrupt change from elastic toelastoplastic occurs when an increment touches and expands theyield envelope. However, numerous experiments have showedthat an irreversible plastic strain (or deformation) occurs even forsmall increments inside the yield surface. Furthermore, a smooth

Y. Tian et al. / Applied Ocean Research 32 (2010) 284–297 291

D/2

w

Vheave /2

Vdisp

Vheave /2

Vheave

=0

Vheave

=Vdisp

Vheave

=0.5Vdisp

Barbosa-Cruz and Randolph (2005)

B

0.0 0.2 0.4 0.6 0.8 1.0w

(m

)B/D

0.5

0.4

0.3

0.2

0.1

0.0

Vertical resistance (kN/m)

Hardening law

qcB

0 40 80 120 160 200

w (

m)

0.5

0.4

0.3

0.2

0.1

0.0

(a) Contact width. (b) CPTs and hardening law.

Fig. 13. Vertical resistance and CPT.

transition of stiffness, rather than an abrupt reduction at theboundary surface is also frequently observed. The incapability ofthe conventional elastoplastic model to account for these featurescan be overcome by introducing a mechanism to also account forthe plasticity behaviour inside the yield surface [27–29]. Manynumerous constitutive models achieve this improvement throughinterpolating the hardening modulus [30,28,31,32]. Among thesemodels, the bounding surface and infinite nested surface frame-works are well known and widely used [33–35]. One model thatbelongs to the family of the boundary surface is the two-yield sur-face model with a kinematic hardening law regulating the man-ner of the inner surface [36]. It is colloquially named the ‘‘bubble’’model, as a smaller surface travels inside the outer boundary sur-face. The framework of the bubble model developed by Zhang [11]and Zhang et al. [12] is adopted in this paper. A simple rule to inter-polate the plastic modulus and a Mróz kinematic hardening law toregulate the motion of the bubble is used [11,37]. The detailed de-scription of the ‘‘bubble’’ model, with the new equations adoptedin this paper, is given in Appendix.

5. Discussion of model parameters and calibration with off-shore conditions

All of themodel parameters required for implementation of theUWAPIPE model are provided with a brief description in Table 2.The majority of these parameters can be used for similar sand

conditions in the field. However, the hardening law parametersmay need estimation from routine geotechnical measurements(such as a CPT). The hardening law parameters kvp and ζ effectivelyfit a function for the pure bearing capacity of a pipe in sand. Othermethods could include relating the vertical monotonic test to anoffshore CPT. The relationship between the vertical resistance forceV in the vertical monotonic test and the tip resistance qc can beestablished as:

V = qcB (7)

where B is the pipe and soil contact width. This value is, however,affected by the heaving of soil around the pipeline during thepenetration [5]. Following the method of Wantland et al. [2] andZhang [11] analytically predicted the relationships between thecontact area B and the embedment w assuming the heaved upvolume is equal to 0, 0.5 and 1 times the displaced soil volume,e.g. Vheave = 0 or 0.5Vdisp or Vdisp. Barbosa-Cruz and Randolph [38]also provided curves linking B andw from finite element study. As

shown in Fig. 13 (a), the analytical curve of Vheave = 0.5Vdisp agreeswell with the Barbosa-Cruz and Randolph result.If the vertical elastic stiffness ismuch larger than themonotonic

virgin loading stiffness, the vertical monotonic curve is close to thecurve of the hardening law, providing the possibility of using CPTresults to calibrate the hardening law. This is shown in Fig. 13(b)with the centrifuge CPT results, calibrated by assuming that theheave was half the displaced volume. It compares well to the pipederived hardening law, especially when the vertical penetrationis less than 0.35D. This provides a reasonable method for backcalculating parameter kvp and ζ from offshore CPT data.

6. Modelling the experiments

To evaluate the performance of the enhanced UWAPIPE model,retrospective numerical predictions using the bubble model arecompared with the centrifuge experiment results. A comparisonis also made using the original formulation of Zhang [11]. Theparameters of themodifiedmodel in this paper are listed in Table 2.

6.1. Constant V (CV) tests

A retrospective simulation of a normal constant V and an overconstant V test are shown in Figs. 14 and 15 respectively (Table 1CV3 and CV4). The load applied on the pipe was increased to 8.16kN/m for the normal constant V and to 15.36 kN/m then 3.94 kN/mfor the over constant V before the vertical load was held constantwhile the horizontal displacement was increased. The first stage iseffectively a check of the hardening law, with differences betweenthe simulations and experiments due to hardening law parametersaveraged over all of the tests being applied. The second stage seesexpansion of the yield surface. Figs. 14 and 15 show the UWAPIPEmodel predicting this expanding yield surfaces reasonably well,with the numerical simulation curves tracking both the horizontalload increases and vertical pipe penetration. The simulationsare particularly good when the horizontal displacement is lessthan 2 m (two pipe diameters), with the predictions increasinglydeviating from the test curves after that. This observation isexpected with the theoretical deviation away from the underlyingassumptions of the hardening law seen in Fig. 10(b).It is, however, an improvement on the previous formulation.

For comparison, the original UWAPIPE model formulation with 3different yield surface shape parameters (equivalent to h0surface =0.11, 0.14, 0.17) was also simulated, with results shown in

292 Y. Tian et al. / Applied Ocean Research 32 (2010) 284–297

Fig. 14. Retrospective simulation of normal CV test.

Fig. 15. Retrospective simulation of over constant V test.

Figs. 14 and 15. Only moderate agreement is seen and this islimited to small lateral displacement (consistent with the originalobservation of applicability only with half a pipe diameter lateraldisplacement, equivalent to 0.5 m in this case).

6.2. Constant displacement (RD) and constant force (RF) ratio tests

Experiment RD1 (Table 1) is a constant displacement ratio test,where the pipe is initially pushed vertically (to 41.19 kN/m) and

then displacement controlled in both degrees of freedom with theratio between vertical and horizontal increment kept constant. Acomparison of the numerical simulations and experimental resultsis given in Fig. 16. The UWAPIPE formulation provides a fairprediction of the horizontal and vertical load track (Fig. 16), againreaffirming the ability of the model to track an expanding surface.The constant load ratio test, where the ratio between vertical

and horizontal force was kept constant after an initial verticalloading–unloading process, is considered to most resemble

Y. Tian et al. / Applied Ocean Research 32 (2010) 284–297 293

Fig. 16. Retrospective simulation of a constant displacement ratio test.

practical load paths followed by an offshore pipeline. Shown inFig. 17 is an increasing horizontal to vertical load ratio of 4. Thepredictions of both the original and new formulation of UWAPIPEmodel are shown in Fig. 17. The new formulation model givesa considerably better agreement with the original model overpredicting the horizontal displacements whilst under-predictingthe vertical. The improved simulation is due to a more closelymatched flow rule (Fig. 11).

7. Application of UWAPIPE model in the stability analysis of anon-bottom pipe

Conventionally the pipe–soil interaction may bemodelled withlinear springs, with friction factors used to predict failure. Theadvantage of UWAPIPE is that it can be easily implemented withina conventional structural analysis program (refer to [17,37] fordetails about the implementation). Here the UWAPIPE attachedto each structural node calculates a nonlinear stiffness matrixunder the combined hydrodynamic loads. To show this and furthercompare the prediction results with the originalmodel an exampleanalysis of a long pipeline (1245 m) laid on the seabed is outlined.A 1245m long straight pipeline is assumed sitting on a perfectly

flat seabed with an initial embedment due to the pipe laying. Thestructural model representing the idealised pipeline is shown inFig. 18 and is modelled in ABAQUS. The 1245 m pipe is evenlydiscretized into 250 nodes and 249 beam elements (using ABAQUSB33H elements). The UWAPIPE model has been implemented as aone-nodded element as an ABAQUS UEL and numerically attachedto every node (no free span is considered). The pipeline ends isassumed to be free, i.e. no boundary restriction. Since the presentUWAPIPEmodel only accounts forV−H behaviour, axial behaviouris not considered in this example. The structural properties arelisted in Table 3.During pipe laying the vertical force pushing the pipe into

the soil is larger than the eventual submerged weight [9]. Ashighlighted by Cathie et al. [6], this load concentration factorcan be further increased due to vessel motion, welding operation

Table 3Computational parameters.

Parameter Value

Outer diameter (m) 1.0Wall thickness (m) 0.0312Pipe submerged weight (empty) (kN/m) 5.1Equivalent density (kg/m3) 5.69× 103

Young’s modulus of pipe (Pa) 2.1× 1011Poisson’s ratio of pipe 0.3

and wave and current scouring. In this example a low loadconcentration factor of 2 is used. Numerically for the UWAPIPEmodel, the soil underwent vertical loads of twice self-weight (2W )and then unloaded to W , which provides an initial size of thebounding surface of V0 = 2W . The bubble is considered to coincidewith the submerged weight (Fig. 19).Three hours of sea state of 10-year return period was

numerically generated from a spectrum. Therefore, the generatedthree-dimensional sea varied in space coordinates (i.e. differentpositions had different wave time series) (see [39,40] for details ofthe wave generation). The water particle velocity and accelerationnear the pipeline was calculated according to the surface wavetime series. The stormwas assumed perpendicular to the pipeline.Fig. 20(a) illustrates the horizontal and vertical force (Fh andFv) acting on just one node, in this case node 73 (a noderandomly chosen for illustrative purposes). Fig. 20(b) shows thehydrodynamic loading along the pipeline at time of 1000 s. Thenegative value of Fv denotes upward hydrodynamic uplift forceacting on the pipeline (see sign convention in Fig. 1 thoughremembering the self-weight of 5.1 kN/m is also applied).Fig. 21 shows the displacement history calculated for node

73. With the storm creating different loads on each node, thedisplacements changes along the pipe, with the final configurationat the end of the storm shown in Fig. 22. Both the maximumvertical penetration and lateral displacement are around Nodes70–80. These events coincide as the pipe capacity required duringthe large wave event is increasing due to the pipe penetration.

294 Y. Tian et al. / Applied Ocean Research 32 (2010) 284–297

Fig. 17. Retrospective simulation of a constant force ratio test.

Fig. 18. Computational model of a 1245 m pipeline.

Fig. 19. Initial condition of bubble model.

The new model formulation predicted less horizontal movementthan the original Zhang [11] model (8 times less for node 73). Oneof the major reasons was that the original formulation predicteduplift of the pipe under this storm and significant sliding of thepipe (at the position of the largest wave at ∼9700 s). This can beascribed to differences in the two flow rules. The ability to trace

the displacement paths along the pipeline is one of the strength ofthese models, and an important step towards displacement baseddesign.

8. Conclusion

In this paper, a state-of-the-art pipe–soil interactionmodel wasadvanced using new forms of yield surface, plastic potential andhardening laws. These were based on and calibrated to geotech-nical centrifuge observations. One of the major improvements isthat the modified model has been shown to be applicable withinhorizontal displacement of two pipe diameters. This providesmorescope for use in predicting themovement of long offshore pipeline,as was shown in this paper for a 10-year storm in typical Aus-tralian conditions. Centrifuge testswere also conducted under loadand displacement conditions that the pipemay be subjected to off-shore, with the radial load and displacement path tests instrumen-tal in calibrating the hardening law and flow rule proposed. Ex-tremely close agreement during retrospective simulations of thesetests provides confidence for the model application.New forms of the mathematical formulations were provided,

including a yield, bubble and plastic potential surface with thesharp points on the surface at V = −ηV0 and V = V0 roundedoff. This follows the recommendation of Nova and Montrasio[24] and significantly improves the numerical robustness. Fifteenmodel parameters are required for the bubble and twelve forthe elastoplastic model. Interpretation of the parameters werediscussed and recommended values given.

Y. Tian et al. / Applied Ocean Research 32 (2010) 284–297 295

Fh

(kN

/m)

Fv

(kN

/m)

(a) Hydrodynamic horizontal (Fh) and vertical (Fv) forces on node 73. (b) Hydrodynamic horizontal (Fh) and vertical (Fv) forces along the pipelineat 1000 s.

Fig. 20. Hydrodynamic loads.

Fig. 21. Displacement history of node 73.

Distance (m)

w (

m)

u (m

)

Node No.

Fig. 22. Final pipeline configuration at the end of the storm.

It should be noted that the accumulated bermwas not explicitlytaken into account in the model. The hardening law is still solely

dependent on the embedment. Therefore, further study is requiredto extend the model to consider the berm effect. As shown, thehardening law deviates from the centrifuge test results whenthe horizontal displacement is greater than two pipe diameters,most probably due to the increased effect of a berm being builtalongside the pipe. Incorporation of the accumulation of horizontalplastic displacements within the hardening law provides onemethod to explicitly account for this. Only drained centrifugetests were conducted in this paper’s study. However, fast loadssuch as hydrodynamics and thermal change can result in partiallydrained pipe–soil behaviour. Further study is required to conductcentrifuge tests to account for the displacement rate effect.

Acknowledgements

This research is being undertaken within the CSIRO Wealthfrom Oceans Flagship Cluster on Subsea Pipelines with fundingfrom the CSIRO Flagship Collaboration Fund. Dr. Jianguo Zhang’sdiscussion and explanation of the original UWAPIPE model ismuch appreciated. The first author thanks Prof. DaveWhite for hisconstructive suggestions.Mr. DonHerley is appreciated for his helpin carrying out the beam centrifuge tests. Mr. S. Shivanna Gowda isthanked for his help in the data processing.

Appendix. Kinematic hardening and the two-surface bubblemodel

The bounding surface, equivalent to the yield surface of theelastoplastic model, can be rewritten by introducing the boundingsurface centreM( (1+η)V02 , 0):

Fb =(H − HMh0V0

)2− β2f

(V − VMV0

+1+ η2

)2β1×

(1+ η2−V − VMV0

)2β2= 0 (8)

where VM = V0/2 and HM = 0 are the value of the coordinate ofthe bounding surface centreM .The bubble, r times the size of the bounding surface, centred at

N (VN ,HN ) can be written as:

fb =(H − HNh0rV0

)2− β2f

(V − VNrV0

+1+ η2

)2β1×

(1+ η2−V − VNrV0

)2β2= 0. (9)

296 Y. Tian et al. / Applied Ocean Research 32 (2010) 284–297

V

N

H

M

C

δ

maxδ

Plastic potential

Bounding surface

Bubble

Fig. 23. Bubble model illustration.

V

N'

N

M

C

H

13

2

Fig. 24. Kinematic hardening.

A slightly different form of the plastic potential from the EPmodel is applied by keeping the ordinate of the plastic potentialcentre flush with the bubble centre (Fig. 23). In doing this, themeaningless status of the normals of the bubble and plasticpotential in reverse directions can be avoided.

gb =(H − HNαh0V0

)2− β2g

(VV0+ η

)2β3 (1−

VV0

)2β4= 0. (10)

The kinematic hardening law regulating the movement of thebubble, in addition to the isotropic hardening law of the EP model,contains three parts in Eq. (1). They are (1) movement of thebounding surface; (2) expansion or contraction of the boundingsurface and (3) drag of the force point. The smooth translationof the bubble surface and a guarantee that it will never intersectwith the bounding surface is provided by the third part of thekinematic hardening law, which requires the motion of the bubble

to travel toward the bounding surface parallel the line connectingthe current load and the conjugate point. This is depicted in Fig. 24and follows, amongst others, [33,35,41].

FN = FM +(V0V0+h0h0

)1− rr

(F− FN)+ Λ (FC − F) (11)

where FC is the conjugate force point on the boundary face; Λ is apositive scalar; the superior dot denotes derivatives with respectto time.The plastic modulus is then interpolated according to the

distance between current force point and the conjugate pointvarying from 0 to δmax (Fig. 23).

K = KC + (Kmax − KC )(

δ

δmax

)ρ(12)

where Kmax is a large number and Kmax = 80 000 is adopted withgood agreement with the experimental results; KC is hardeningmodulus of the conjugate point; δmax = 2 (1− r)

∥∥∥−→MC∥∥∥ is themaximum possible distance to the conjugate point; ρ controllingthe variation rate of the plastic modulus and thus the decay of thestiffness is taken as 3.0 from fitting the numerical prediction withthe centrifuge test results.Summarising the bounding surface, bubble surface, flow rule,

hardening laws and elasticity, the final constitutive stiffnessmatrixDep can be straightforward obtained:

1F ={1V1H

}= Dep1U =

(De −

De ∂g∂F∂ fb∂FTDe

K + ∂ fb∂FTDe ∂g

∂F

){1w1u

}. (13)

References

[1] Lyons CG. Soil resistance to marine lateral sliding pipe lines. Dallas (TX): OTC;1973.

[2] Wantland GM, O’Neill MW, Reese LC, Kalajian EH. Lateral stability of pipelinesin clay; 1979.

[3] Lambrakos KF. Marine pipeline soil friction coefficients from in-situ testing.Ocean Engineering 1985;12(2):131–50.

[4] Brennodden H, Sveggen O, Wagner DA, Murff JD. Full-scale pipe–soilinteraction tests. Houston, (TX): OTC; 1986.

[5] Murff JD, Wagner DA, Randolph MF. Pipe penetration in cohesive soil.Geotechnique 1989;39(2):213–29.

[6] Cathie DN, Jaeck C, Ballard J-C, Wintgens J-F. Pipeline geotechnics — state-of-the-art. In: Proc. of the Int. Symp. on the Frontiers in Offshore Geotechnics:ISFOG 2005. Perth (Australia): Taylor and Francis Group; 2005. p. 95–114.

[7] Calvetti F, Di Prisco C, Nova R. Experimental and numerical analysis of soil-pipe interaction. Journal of Geotechnical and Geoenvironmental EngineeringASCE 2004;130(12):1292–9.

[8] Di Prisco C, Nova R, Corengia A. Amodel for landslide-pipe interaction analysis.Soils and Foundations 2004;44(3):1–12.

[9] Tian Y, Cassidy MJ. Modelling of pipe–soil interaction and its applicationin numerical simulation. International Journal of Geomechanics 2008;8(4):213–29.

[10] Schotman GJM, Stork FG. Pipe–soil interaction: a model for laterally loadedpipelines in clay. Houston (Texas): OTC; 1987.

[11] Zhang J. Geotechnical stability of offshore pipelines in calcareous sand, Ph.D.thesis, University of Western Australia; 2001.

[12] Zhang J, Stewart DP, RandolphMF. Modelling of shallowly embedded offshorepipelines in calcareous sand. Journal of Geotechnical and GeoenvironmentalEngineering ASCE 2002;128(5):363–71.

[13] Zhang J, Stewart DP, Randolph MF. Kinematic hardening model for pipeline-soil interaction under various loading conditions. The International Journal ofGeomechanics ASCE 2002;2(4):419–46.

[14] Hodder MS, Cassidy MJ, Barrett D. Undrained response of shallow pipelinessubjected to combined loading. In: BGA International Conference onFoundations 2008.

[15] RandolphMF,White DJ. Upper-bound yield envelopes for pipelines at shallowembedment in clay. Geotechnique 2008;58(4):297–301.

[16] Merifield R, White DJ, Randolph MF. The ultimate undrained resistance ofpartially embedded pipelines. Geotechnique 2008;58(6):461–70.

[17] Tian Y, Cassidy MJ. A practical approach to numerical modeling of pipe–soilinteraction. In: The 18th international offshore (ocean) and polar engineeringconference, 2008.

Y. Tian et al. / Applied Ocean Research 32 (2010) 284–297 297

[18] Garnier J, Gaudin C, Springman SM, Culligan PJ, Goodings D, Konig D, et al.Catalogue of scaling laws and similitude questions in geotechnical centrifugemodelling. International Journal of Physical Modelling in Geotechnics 2007;7(3):1–24.

[19] Tian Y, Cassidy M, Gaudin C. Centrifuge tests of shallowly embedded pipelineon silt sand. GEO:09475 report of centre for offshore foundation systems,Perth: the University of Western Australia, 2009.

[20] Finnie IMS. Performance of shallow foundations in calcareous soil, Ph.D. thesis,University of Western Australia; 1993.

[21] Chung SF, Randolph MF, Schneider JA. Effect of penetration rate on penetrom-eter resistance in clay. Journal of Geotechnical and Geoenvironmental Engi-neering ASCE 2006;132(9):1188–96.

[22] Gazetas G, Tassoulas JL. Horizontal stiffness of arbitrarily shaped embeddedfoundations. Journal of Geotechnical Engineering 1987;113(5):440–57.

[23] Cassidy MJ, Houlsby GT. On the modelling of foundations for jack-up units onsand. In: Proceedings of the Annual Offshore Technology Conference, 1999. p.783–95.

[24] Nova R, Montrasio L. Settlements of shallow foundations on sand. Geotech-nique 1991;41(2):243–56.

[25] Tan FSC. Centrifuge and theoreticalmodelling of conical footings on sand, Ph.D.thesis, University of Cambridge; 1990.

[26] Chapra SC, Canale RP. Numerical methods for engineers. Boston: McGraw-HillHigher Education; 2006.

[27] Krieg RD. A practical two surface plasticity theory. Transactions of the ASME.Series E, Journal of Applied Mechanics 1975;42(3):641–6.

[28] Dafalias YF, Popov EP. Cyclic loading for materials with a vanishing elasticregion. Nuclear Engineering and Design 1977;41(2):293–302.

[29] Hashiguchi K. Generalized plastic flow rule. International Journal of Plasticity2005;21(2):321–51.

[30] Dafalias YF, Popov EP. Amodel of nonlinearly hardeningmaterials for complexloading. Acta Mechanica 1975;21(3):173–92.

[31] Mróz Z, Norris VA, Zienkiewicz OC. An anisotropic, critical statemodel for soilssubject to cyclic loading. Geotechnique 1981;31(4):451–69.

[32] Dafalias YF. An anisotropic critical state soil plasticity model. MechanicsResearch Communications 1986;13(6):341–7.

[33] Mróz Z. On the description of anisotropic work-hardening. Journal of theMechanics and Physics of Solids 1967;15(1):163–76.

[34] Dafalias YF, Popov EP. Plastic internal variables formalism of cyclic plasticity.Transactions of the ASME. Series E, Journal of Applied Mechanics 1976;43(4):645–51.

[35] Mróz Z, Norris VA, Zienkiewicz OC. Application of an anisotropic hardeningmodel in the analysis of elasto-plastic deformation of soils. Geotechnique1979;29(1):1–34.

[36] Muir Wood D. Geotechnical modelling. London: Spon Press; 2004.[37] Tian Y, Cassidy MJ. The challenge of numerically implementing numerous

force-resultant models in the stability analysis of long on-bottom pipelines.Computers and Geotechnics 2010;37:216–312.

[38] Barbosa-Cruz ER, Randolph MF. Bearing capacity and large penetration of acylindrical object at shallow embedment. In: Gourvenec S, Cassidy M, editors.Proc. int. symposium on frontiers in offshore geotechnics — ISFOG 2005. Perth,Australia: Taylor & Francis Group; 2005. p. 615–21.

[39] Youssef B, Cassidy M, Tian Y. Implementing 3D hydrodynamic forces intointegrated pipeline analysis—UWAHYDRO. C2521, COFS, Perth: University ofWestern Australia, 2009.

[40] Youssef B, Cassidy M, Tian Y. Numerical modelling of hydrodynamic forces.C2520, COFS, Perth: University of Western Australia, 2009.

[41] Rouainia M, Muir Wood D. A Kinematic hardening constitutive model fornatural clays with loss of structure. Geotechnique 2000;50(2):153–64.

[42] Bienen B, Byrne BW, Houlsby GT, Cassidy MJ. Investigating six-degree-of-freedom loading of shallow foundations on sand. Geotechnique 2006;56(6):367–79.

[43] Byrne BW, Houlsby GT. Observations of footing behaviour on loose carbonatesands. Geotechnique 2001;51(5):463–6.

[44] DNV Recommended Practice RP-F105. Free spanning pipelines; 2002.[45] Houlsby GT, Cassidy MJ. A plasticity model for the behaviour of footings on

sand under combined loading. Geotechnique 2002;52(2):117–29.[46] Rouainia M, Muir Wood D. Implicit numerical integration for a kinematic

hardening soil plasticity model. International Journal for Numerical andAnalytical Methods in Geomechanics 2001;25(13):1305–25.