Post-preload undrained uniaxial capacities of skirted circular foundations in clay

Ocean Engineering 147 (2018) 355–369

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Post-preload undrained uniaxial capacities of skirted circular foundations in clay D. Fu *, C. Gaudin, Y. Tian, B. Bienen, M.J. Cassidy Centre for Offshore Foundation Systems and ARC Centre of Excellence for Geotechnical Science and Engineering, University of Western Australia M053, Perth, WA, 6009, Australia

A R T I C L E I N F O

A B S T R A C T

Keywords: Shallow footing Preloading Consolidation Bearing capacity Modified Cam-Clay

Improvement of the undrained capacities of foundation as a result of preloading, has received attention in offshore engineering only recently. It offers the benefit of optimising foundation design and reducing footprint and cost. This paper investigates the preloading performance of skirted circular foundations in clay using 3D coupled finite-element analyses. The increase in the ultimate undrained uniaxial vertical, horizontal and moment capacities is provided for non-dimensional groups of foundation skirt length, in situ undrained shear strength heterogeneity, magnitude of vertical preload and normalised consolidation time. An exponential relationship between preloading gain in capacity and normalised consolidation time is established, with the maximum preloading gain expressed as a function of the three other dimensionless groups. An approach to estimate the postpreload undrained ultimate uniaxial capacities of skirted circular foundations on normally consolidated clay is ultimately proposed. The influence of foundation geometry is discussed via comparison with the existing solutions.

1. Introduction Holding vertical load on a foundation in a preloading process improves its capacity due to consolidation and the associated dissipation of excess pore pressures in the soil. In offshore engineering, an ‘active’ preloading process may be used prior to operations to develop an acceptable margin of safety against environmental loading (Randolph and Gourvenec, 2011). Passive preloading, resulting from self-weight consolidation also increases the foundation capacity, though this is usually not considered in design. The majority of the research work undertaken to date focuses on the enhancement of the vertical undrained bearing capacity of shallow foundations in clay. Fig. 1 summarises the vertical capacity increase ηv,f after full consolidation for increasing levels of preloading P% (both normalised with the no-preload undrained capacity) from full-scale and reduced-scale experimental work, and from numerical analysis. Lehane and Jardine (2003) conducted field tests to investigate the effect of preloading on the undrained bearing capacity of an embedded solid square foundation in lightly over-consolidated soil (OCR approximately 2.5 at the foundation top level). A significant increase in the undrained capacity (ηv,f of 1.5) was measured after 11-year of consolidation under a sustained level of preloading of P% ¼ 65%. A finite-element model was

also implemented by Zdravkovic et al. (2003), providing further insight into the response of a preloaded strip foundation in clay with various ratios of over-consolidation (OCR ranging within 1, 2, 4, 9 and 25). A much larger gain in undrained capacity was calculated in normally consolidated clay than in over-consolidated clay. Lehane and Gaudin (2005) reported results of centrifuge testing on embedded solid square foundations. A considerable gain (ηv,f of 1.8) in undrained capacity was measured after full consolidation under a level of preloading of P % ¼ 65%, although the over-consolidation ratio (OCR) was approximately 8 at the foundation level. A series of centrifuge tests was conducted by Bienen et al. (2010) to investigate the effects of both preloading and consolidation on the undrained bearing capacity of a surface circular foundation on normally consolidated clay. The results showed an increase (ηv,f of 1.6) in undrained bearing capacity under a level of preloading of approximately P% ¼ 75%. As evident from Fig. 1, the improvement in vertical capacity can be significant (up to 80%), but varies greatly as a function of the foundation shape and embedment and the degree of over consolidation of the soil. More recently, further work has been undertaken to characterise more rigorously the gain in capacity due to preloading. Gourvenec et al. (2014) developed a framework to predict the gain in undrained vertical bearing capacity of a strip and circular foundation as a function of the

* Corresponding author. Current Address: State Key Lab of Hydraulic Engineering Simulation and Safety, Tianjin University, Weijin Road 92, Tianjin, 300072, China. E-mail address: [email protected] (D. Fu). https://doi.org/10.1016/j.oceaneng.2017.10.029 Received 26 May 2017; Received in revised form 27 August 2017; Accepted 16 October 2017 0029-8018/© 2017 Elsevier Ltd. All rights reserved.

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Fig. 1. Summary of published data of increase in undrained bearing capacity of shallow foundations with vertical preloading 10.

level of preloading and degree of consolidation. The changes in preload-driven elastic and plastic stress were assessed to evaluate the gain in average undrained shear strength. In parallel, Fu et al. (2015) conducted a series of centrifuge testing and numerical analyses to investigate the history of the response of the bearing capacity of a skirted circular foundation on soft clay. A time-dependent exponential increase in bearing capacity was proposed for any given level of preloading, based on the evolution in both elastic and plastic volume changes. The effect of the interface property was also investigated, which shows no influence on the gain in undrained bearing capacity, consistent with the observation of Gourvenec et al. (2014). A smaller body of research investigated the increase in combined vertical (V), horizontal (H) and Moment (M) capacity due to preloading. Bransby (2002) explored the preloaded vertical and horizontal load response of a strip foundation in normally consolidated soil. A higher increase in horizontal capacity was calculated compared with that in vertical capacity. More recently, Feng and Gourvenec (2015) and Vulpe et al. (2016) reported capacity for a surface rectangular (in combined VHM as well as the torsional T load direction) and for surface strip and circular foundation (in VHM) for different levels of preloading and duration of consolidation. This body of literature provides evidence of the increased foundation capacity due to preloading, and the importance of factors, such as foundation shape, interface property, loading direction, and initial stress state. However, the influence of the foundation skirt embedment has received less attention. This is of significant importance because the skirt transfers the load applied on the foundation to deeper soil, changing the region of primary consolidation. This particular aspect is addressed in this paper, which presents numerical results using coupled small-strain finite-element analyses to investigate the post-preload capacities of skirted circular foundations on normally consolidated clay. The skirt length, the initial undrained shear strength distribution in the soil, the magnitude of preloading and the duration of consolidation are considered. Based on the finite-element results, formulations to predict the postpreload undrained uniaxial capacities of skirted circular foundations are

Table 1 Soil characteristics for the kaolin clay used in numerical modelling. Parameters

Values

Slope of critical state line (CSL) in p'-qd space, M (critical friction angle in triaxial compression, φ0 tc) Void ratio at p' ¼ 1 kPa on CSL, ecs Virgin compression index, λ Swelling and recompression index, κ Shear Modulus, G0 Submerged unit weight, γ' (kN/m3) Permeability of soil, ks (m/s)

0.890 2.140 0.205 0.044 50p'0 6 1.0  109

proposed, as a function of:  the level of preloading P%, defined as the ratio of the applied vertical load Vp to the undrained vertical bearing capacity Vun,  the duration of application of the preload, defined by the time factor Tv ¼ cv0t/D2 (where cv0 is the initial in situ coefficient of consolidation at the skirt tip level, t is the elapsed time and D is the foundation diameter),  the foundation aspect ratios d/D (where d is the foundation skirt length), which is varied from 0 to 1,  and the soil strength heterogeneity ratio kD/su0 (where k is the soil strength gradient with soil depth z and su0 is the undrained shear strength at the skirt tip level d), which is varied from 0.5 to 5. Note that, a linear soil strength profile is adopted as su ¼ sum þ kz, where sum is the shear strength at the mudline. 2. Numerical model 2.1. Soil model and parameters Three-dimensional coupled small-strain finite element analyses were undertaken using the Modified Cam Clay soil model (Roscoe and Burland, 1968), as implemented in Abaqus (Dassault Systemes, 2010). The soil domain was modelled as a linear elastoplastic material. All 356

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Fig. 2. Mesh and geometry of skirted circular foundation with d/D of 0.5 (RP: reference point) 11.

Table 2 Summary of nomenclature. Vertical Load No-preload undrained ultimate load No-preload undrained capacity factor Post-preload undrained ultimate load full consolidation partial consolidation Absolute improvement in undrained ultimate load

Preloading gain after full consolidation

Preloading gain after partial consolidation

Normalised preloading gain

Q (general term) V Qun (general term) Vun Nc (general term) NcV Qpc (general term) Vpc,f Vpc,p ΔQ (general term) ΔV Vpc,f – Vun ηf (general term) ηV,f Vpc,f/Vun ηp (general term) ηV,p Vpc,p/Vun Uv (Vpc,p - Vun)/(Vpc,f - Vun) (ηV,p - 1)/(ηV,f - 1)

parameters adopted in this study are summarised in Table 1. The soil body was anisotropically consolidated with K0 ¼ 1 – sinϕ0 tc ¼ 0.612, where the critical friction angle ϕ0 tc was 22.8 in triaxial compression (Wroth, 1984). The submerged unit weight of soil was taken as γ' ¼ 6 kN/m3. The plane strain strength su varies with depth, following standard relationships (Wroth, 1984):

 λκ su su p' cs p' 0 Ms cos θL p' c0 λ 1 þ 2K0 p ffiffi ffi ¼ ¼ σ'v p' cs p' 0 σ'v 2p' 0 3 3

ks ks ð1 þ e0 Þp'0 ¼ mv γ w λγ w

Moment

H

M

Hun

Mun

NcH

NcM

Hpc,f Hpc,p

Mpc,f Mpc,p

ΔH Hpc,f – Hun

ΔM Mpc,f – Mun

ηH,f Hpc,f/Hun

ηM,f Mpc,f/Mun

ηH,p Hpc,p/Hun UH (Vpc,p - Vun)/(Vpc,f - Vun) (ηH,p - 1)/(ηH,f - 1)

ηM,p Mpc,p/Mun UM (Vpc,p - Vun)/(Vpc,f - Vun) (ηM,p - 1)/(ηM,f - 1)

e0 ¼ ecs þ ðλ  κÞln 2  κ lnp'0  ðλ  κÞlnp'c0

(3)

where ks is the permeability, p0 c0 is the initial pre-consolidation pressure, and p0 0 is the corresponding mean effective normal stress. Further details of the soil model and finite element modelling methods used in the present study can be obtained from Chatterjee et al. (2012), Fu et al. (2015 and 2017).

(1) 2.2. Geometry, foundation model and mesh

where σ0 v is the vertical effective stress, κ is the swelling and recompression index, λ is the virgin compression index, p0 0 is the initial mean normal effective stress, p0 c0 is the initial pre-consolidation pressure, p0 cs is the mean normal stress at failure, and θL is the Lode angle. The surcharge was applied on the top surface and across the foundation base to provide an initial soil strength heterogeneity ratio kD/su0 varying within the range [0.5, 1, 1.25, 1.4285, 2, 3.33 and 5]. Under these conditions, the coefficient of consolidation, cv0 (Eq. (2)), and initial void ratio, e0 (Eq. (3)) also vary with depth:

cv0 ¼

Horizontal

The skirted circular foundation was modelled with a diameter D of 14 m and a skirt tip pre-embedded in the soil to a depth d of 0–1 diameters (0, 0.2D, 0.3D, 0.5D, 0.7D, 0.8D and 1D). The embedded skirt tip and skirt base nodes were bounded to the adjacent soil elements, leading to a rough interface in shear with no detachment permitted between the soil and foundation. This is thought to be appropriate for representing offshore field conditions because the tensile resistance developed within the soil plug due to suction is confined by the periphery skirts. No permeability was allowed at the foundation interface. The reference point (RP) for loads and displacements is located along the foundation centreline at the skirt tip level.

(2)

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Fig. 3. Increase in undrained uniaxial capacities of skirted circular foundations (d/D ¼ 0, 0.2, 0.5, 1) with the level of preloading P% after full consolidation: (a) ηV,f; (b) ηH,f; (a) ηM,f 14.

The mesh shown in Fig. 2 (d/D ¼ 0.5) comprises 23,500 full integration stress-pore fluid continuum elements (C3D8P in the Abaqus/ Standard library). The minimum element length was approximately 0.002D, located beneath and around the skirt tip, to ensure the reliable simulation at the failure zone. A mesh sensitivity analysis was undertaken to examine the efficiency in calculation without compromising accuracy.

A typical three-dimensional finite element mesh used for the analyses of a skirted circular footing (d/D ¼ 0.5) is presented in Fig. 2, which represents a half-footing cut through the plane of symmetry. The mesh boundary extends a distance of 7D horizontally and 3.1D vertically from the tip level of the skirted circular foundation, with the horizontal displacement on the lateral boundaries and vertical displacement at the base boundary set to zero. A free boundary was allowed only at the top surface, which was defined as a permeable boundary.

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Fig. 4. Increase in undrained uniaxial capacities of surface circular foundation under P% of 10% and 90% after full consolidation: (a) ηV,f,P10,surf, ηH,f,P10,surf, ηM,f,P10,surf; (b) ηV,f,P90,surf, ηH,f,P90,surf, ηM,f,P90,surf 15.

2.3. Analysis procedure

3. Nomenclature

The numerical analyses included 4662 cases, involving the application of a vertical preload Vp on the foundation (yielding a level of preloading P% ¼ Vp/Vun  100%, where Vun is the ultimate vertical load), maintaining a period of time for consolidation. The foundation was then subjected to the following procedures: (1) further undrained penetration was imposed to establish the post-preload undrained ultimate vertical load, Vpc,p; (2) the horizontal or rotational side-swipe tests (Tan, 1990) were then carried out to derive the post-preload undrained horizontal load Hpc,p, or moment undrained load Mpc,p. Nine different levels of preloading P% from 10% to 90%, with intervals of 10%, and a series of consolidation times were included in the simulation for each foundation aspect ratio d/D and each soil strength heterogeneity kD/su0. Note that all no-preload undrained uniaxial capacities necessary to the analyses in this study are inferred from the solution described by Fu et al. (2017), with the same computation parameters.

The notation of loads involving both no-preload and post-preload analyses is summarised in Table 2. Note that more complicated subscripts involved in notations are not included, but the definition follows a simple rule: the combination of two general terms, undrained uniaxial capacity Q and preloading gain η, with several subscripts indicates different conditions. The subscripts comprised ‘V’ for the vertical direction; ‘H’ for the horizontal direction; ‘M’ for the moment direction; ‘un’ for the no-preload condition; ‘pc’ for the post-preload condition; ‘f’ for full consolidation; ‘p’ for partial consolidation; ‘P10’ for the P% of 10%; ‘P90’ for the P% of 90%; ‘skt’ for the skirted circular foundation; ‘surf’ for the surface circular foundation. For example, ηV,f,P10,surf denotes the preloading gain in the undrained vertical capacity under P% of 10% after the full consolidation for a surface circular foundation. For all analyses, an initial in situ coefficient of consolidation at the skirt tip level cv0 is adopted in this study to capture the normalised 359

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Fig. 5. Method to quantify the effect of embedment ratio on absolute improvement in undrained uniaxial capacities 16.

elapsed time Tv ¼ cv0t/D2.

involves a relatively small soil domain (see Fu et al., 2017). The latter moment is therefore only associated with enhancement in undrained shear strength of the soil above the skirt tip.

4. Results

4.1. Method to predict the increase in uniaxial capacity after full consolidation

When exploring the post-preload undrained uniaxial capacities of a skirted circular foundation in clay, an appropriate method is to investigate the ratio of the post-preload to the original undrained capacities (ηV,p ¼ Vpc,p/Vun, ηH,p ¼ Hpc,p/Hun, ηM,p ¼ Mpc,p/Mun). This is because a significant amount of studies exist to evaluate the original undrained capacities, considering various boundary conditions. For instance, Fu et al. (2017) outlined an approach to estimate the undrained uniaxial capacities of skirted circular foundations for any given d/D and kD/su0, which can be taken as a direct database for comparing the corresponding gains due to preloading studied in this paper. Increase in undrained uniaxial capacities after full consolidation (ηV,f ¼ Vpc,f/Vun, ηH,f ¼ Hpc,f/Hun, ηM,f ¼ Mpc,f/Mun). Fig. 3 shows the gains in the undrained uniaxial vertical, horizontal and moment capacities of skirted circular foundations of aspect ratios d/ D ¼ 0, 0.2, 0.5 and 1, under different levels of preloading P% (from 10% to 90% with intervals of 10%) after full consolidation. Five soil strength heterogeneity ratios (kD/su0 ¼ 0.5, 1, 2, 3.33 and 5), are selected for demonstration. A similar trend of increase for each of the undrained uniaxial capacities due to preloading and consolidation is observed. The trend exhibits a bilinear increase, with an inflexion point at about P% ¼ 10%, implying a different rate of increase in undrained shear strength between very low and high levels of preloading. This is because the increase in undrained shear strength resulting from consolidation arises from both elastic and plastic volume changes. The elastic compression associated with low P% gives a smaller change in volume than the plastic compression associated with large P% (Fu et al., 2015). For each d/D, the gain in all uniaxial capacities increases with increasing kD/su0, which reflects shallower failure mechanisms of higher strength heterogeneity and larger enhancement of the undrained shear strength due to preloading in the near surface soil. Although the trend is similar, the magnitude of increase in uniaxial capacity differs between the loading directions, with maximum ηV,f, ηH,f, ηM,f of 1.9, 3.2 and 3.4 respectively. These are obtained from calculations on the surface circular foundation (ηV,f, ηH,f) and skirted circular foundation with d/D of 0.2 (ηM,f) on the soil with kD/su0 of 5 and under the largest P% of 90%. This is attributed to the non-uniform increase in undrained shear strength (with the near surface soil experiencing the largest increase in strength) and the mobilised volume at failure under different directional loading. The vertical loading mobilises the largest soil domain, encompassing most of the consolidated material. The horizontal loading mobilises a smaller domain, limited to the sliding plane beneath the foundation and the passive and active side wedges along the skirts. Low foundation aspect ratios in soil with high strength heterogeneity lead to an internal scoop mechanism under moment loading, which

Considering the bilinear variation of gain in undrained capacities (ηV,f, ηH,f, ηM,f) with the level of preloading, the focus of this paper is on the assessment of ηV,f, ηH,f, and ηM,f under a transitional level of preloading P% of 10% and a relatively larger level of preloading P% of 90%. Based on this, the preloading gains in undrained capacities after full consolidation can be expressed as follows: For P%  10%

ηf ¼

ηf ;P10 P% 10%

(4)

For 10%  P%  90%

ηf ¼ ηf ;P10 þ

ηf ;P90  ηf ;P10 ðP%  10%Þ ð90%  10%Þ

(5)

where ηf is the preloading gain after full consolidation, representing ηV,f, ηH,f and ηM,f for each loading direction; subscripts P10 and P90 are associated with P% of 10% and 90%, respectively. The foundation aspect ratio d/D and the soil strength heterogeneity ratio kD/su0 are also considered in evaluating ηf,P10 and ηf,P90, considering their influence on the preloading gain as evident from Fig. 3. The influence of kD/su0 can be readily assessed from the response of the surface circular foundation (d/D ¼ 0). Numerical results under P% of 10% and 90% are shown in Fig. 4a and b, encompassing ηV,f, ηH,f, and ηM,f respectively, and the corresponding linear lines showing the best fit are also included, expressed as. For P% ¼ 10%

ηf;P10;surf

8 kD > 1:05 þ 0:007 > > > su0 > > > < kD ¼ 1:16 þ 0:02 > su0 > > > > > > : 1:10 þ 0:003 kD su0

vertical horizontal

(6)

moment

For P% ¼ 90%

ηf;P90;surf

360

8 kD > > > 1:75 þ 0:03 s > > u0 < kD ¼ > 2:52 þ 0:15 > > su0 > > : 2:29

vertical horizontal moment

(7)

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Fig. 6. Comparison the fitting method (Eq. (10)) with the numerical results of preloading gains in undrained uniaxial capacities of skirted circular foundation against kD/su0 at P% of 10% after full consolidation: (a) ηV,f,P10,skt, (b) ηH,f,P10,skt, (c) ηM,f,P10,skt 18.

Fig. 7. Comparison the fitting method (Eq. (10)) with the numerical results of preloading gains in undrained uniaxial capacities skirted circular foundations against kD/su0 at P% of 90% after full consolidation: (a) ηV,f,P90,skt, (b) ηH,f,P90,skt, (c) ηM,f,P90,skt 20.

where ηf,P10,surf and ηf,P90,surf are the preloading gains in the undrained capacities of the surface circular foundation after full consolidation at P% of 10% and 90%, respectively. The influence of d/D is evaluated by comparing the post-preload undrained capacity of a skirted circular and surface circular foundation. The differences essentially resides in the contribution of the soil above the skirt tip. Fig. 5 presents the method to quantify the effect of d/ D on the increase in undrained uniaxial capacities. The increase in

undrained capacities ΔQ (ΔV ¼ Vpc,f – Vun, ΔH ¼ Hpc,f – Hun and ΔM ¼ Mpc,f – Mun) of the skirted circular foundation is divided into two independent components: one (ΔQtop) is related to the soil above the skirt tip (see the region with the hatching shadow in Fig. 5), and the other (ΔQsurf) relies on the soil below the skirt tip (see the region with the square-dot shadow in Fig. 5), which can be evaluated by the increase in capacity for a surface circular footing with identical kD/su0 condition. ΔQ can therefore be expressed from the improvement evaluated for a 361

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Fig. 8. Variation of normalised preloading gains in undrained uniaxial capacities with normalised consolidation time Tv under different levels of preloading (d/D ¼ 0, kD/su0 ¼ 0.5) 21.

8   <1 d g ¼ 1 : D P10 1

surface circular foundation:

ΔQskt

  d ΔQsurf ¼g D

(8)

ΔQsurf ¼ ηf ;surf

  1 Qun;surf

g

Qun;skt þ ΔQskt g ¼ Qun;skt

 d   ηf ;surf  1 D Qun;skt Qun;surf

þ1

Hun;skt ¼ Hun;surf



(10)

     d sum d kd 1 þ 1:9ln 1 þ þ 1 þ 1:15ln 1 þ D D su0 su0 (11)



     d sum d kd 1 þ 8:15ln 1 þ þ 1 þ 5ln 1 þ D D su0 su0 (12)

" "  2  3 # Mun;skt d d d sum ¼ 1 þ 1:79 þ 1:02 þ 1:56 þ 0:65 D D D Mun;surf su0  2 # d d kd þ 0:05 þ 1:88 D D su0

8  0:5 d > > > 1 þ 0:18 > > D > > > >  0:33 < d ¼ 1 þ 0:41 > D > > > >  0:55 > > > d > : 1 þ 0:46 D

vertical horizontal

(15)

moment

Eqs. (14) and (15) give a value of g(d/D) higher than 1 within the range of P% of 10%–90%. This indicates that the foundation skirt length contributes positively to the absolute improvement in undrained capacities (ΔQ) under the same kD/su0 condition due to the added consolidation of soil above the skirt tip. At P% of 10%, g(d/D) is almost unity for any direction of loading with increasing d/D. This implies that at low P%, the soil above the skirt tip makes little contribution to the preloading gain of the skirted circular foundation. At P% of 90%, the variation of g(d/D) with d/D follows a power law increase with an increasing d/D along each loading direction. To determine the accuracy of this set of equations (Eq. (15)), both results of ηf,skt from finite element analyses and equations are shown in Figs. 6 and 7 for P% of 10% and 90%, respectively. Both sets of results are generally in close agreement, with the largest discrepancies within 5% observed at P% of 10% and kD/su0 of 5 under moment loading. Normalised preloading gain in undrained uniaxial capacities with elapsed time. It is not always possible to wait until full consolidation of the skirted footing, particularly in the costly offshore environment. Engineers, therefore, require knowledge of the increase in capacity with time during the preloading. Here we use the finite element analyses to define the normalised preloading gain U (a general term representing UV, UH, UM for each loading direction) with the elapsed time as a representative of the rate of increase in capacity with consolidation. We follow the strategy described by Fu et al. (2015) where:

where Qun,skt/Qun,surf is the ratio of the undrained capacities of the skirted circular foundation to that of the surface circular foundation, quantified by Fu et al. (2017), with the general term being replaced by V, H, and M for each loading direction for any d/D and kD/su0:

Vun;skt ¼ Vun;surf

  d D P90

(9)

where ηf,surf is calculated using Eqs. (4)–(7) for any given level of preloading; Qun,surf is the undrained uniaxial capacities of the surface circular foundation, representing Vun,surf, Hun,surf and Mun,surf. The preloading gains in undrained capacities for skirted circular foundations ηf,skt is obtained with a combination of Eqs. (8) and (9):

ηskt ¼

(14)

For P% ¼ 90%

where g(d/D) is a function of d/D, indicating the effect of foundation skirt length on the improvement in the undrained capacity due to preloading with consolidation; subscripts ‘skt’ and ‘surf’ distinguish the improvement between the skirted and surface foundations; and ΔQsurf is expressed as



vertical horizontal moment

(13)

V

To establish the expression of g(d/D), the numerical results of ηf,skt are used for back-calculation analyses. Note that the values of ηf,surf and Qun,skt/Qun,surf are also from numerical analyses rather than calculated from the equations above to ensure accuracy in the prediction of g(d/D). The corresponding fitting curves are: For P% ¼ 10%

UV ¼

pc;p  1 ηV;p  1 Vpc;p  Vun un ¼ ¼ VVpc;f Vpc;f  Vun  1 ηV;f  1 Vun

UH ¼

pc;p  1 ηH;p  1 Hpc;p  Hun un ¼ HHpc;f ¼ Hpc;f  Hun  1 ηH;f  1 Hun

(16a)

H

362

(16b)

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Fig. 9. Variation of normalised preloading gains in undrained uniaxial capacities against normalised consolidation time Tv with d/D of 0, 0.2, 0.5, 0.8, and 1 (P% ¼ 50%, kD/ su0 ¼ 0.5): (a) UV, (b) UH, (c) UM 23.

Fig. 10. Variation of normalised preloading gains in undrained uniaxial capacities against normalised consolidation time Tv with kD/su0 of 0.5, 1, 1.25, 2, 3.33, 5 (P% ¼ 50%, d/ D ¼ 0): (a) UV, (b) UH, (c) UM 25.

M

UM ¼

pc;p  1 ηM;p  1 Mpc;p  Mun un ¼ MMpc;f ¼ Mpc;f  Mun  1 ηM;f  1 Mun

partial consolidation. The variation of U, defined by the variation of time factor Tv ¼ cv0t/ D2, with different P%, d/D, and initial kD/su0 is discussed in the following sections.

(16c)

with ηV,p, ηH,p, and ηM,p being the preloading gains in the undrained vertical, horizontal and moment capacities under partial consolidation, and ηV,f, ηH,f, and ηM,f being the respective preloading gains corresponding to full consolidation, as evaluated by Eqs. (4) and (5). Similarly, Vpc,p, Hpc,p, and Mpc,p are the post-preload undrained capacities under

4.2. Effect of P% The magnitude of the vertical preloading changes the effective stress 363

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Fig. 9a, b and c, respectively. Results for P% ¼ 50% and kD/su0 ¼ 0.5 are selected for interpretation, ensuring that no kD/su0 effects are involved. The normalised preloading gain in each loading direction shows a slower consolidation response with increasing d/D. This reflects a larger volume of soil being mobilised during consolidation and longer drainage paths.

Table 3 Summary of T50 and n for Eq. (19). d/D

0

0.2

0.3

0.3 0.5

0.8

1

kD/su0

0.5 1 1.25 2 3.33 5 0.5 1 1.25 2 3.33 5 0.5 1 1.25 2 3.33 0.5 1 1.25 2 0.5 1 1.25 0.5 1

V

H

M

T50

n

T50

n

T50

n

0.018 0.016 0.015 0.013 0.011 0.009 0.040 0.034 0.031 0.030 0.028 0.025 0.048 0.044 0.043 0.041 0.035 0.065 0.065 0.065 0.065 0.080 0.075 0.070 0.090 0.080

0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85

0.011 0.010 0.010 0.010 0.009 0.007 0.038 0.040 0.042 0.044 0.046 0.046 0.045 0.048 0.049 0.050 0.055 0.065 0.065 0.065 0.065 0.080 0.075 0.070 0.090 0.080

0.60 0.62 0.64 0.65 0.66 0.68 0.78 0.80 0.82 0.83 0.84 0.85 0.80 0.85 0.90 0.92 0.95 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85

0.018 0.016 0.015 0.013 0.011 0.009 0.040 0.034 0.037 0.040 0.042 0.050 0.048 0.045 0.047 0.055 0.060 0.065 0.065 0.065 0.065 0.080 0.075 0.070 0.090 0.080

0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.82 0.83 0.84 0.85 0.85 0.85 0.90 0.92 0.95 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85

4.4. Effect of strength ratio kD/su0 Variations of UV, UH and UM with time factor Tv for different strength heterogeneity ratios are presented in Fig. 10a, b and c. Results are limited to P% ¼ 50% and d/D ¼ 0 for clarity. For each loading direction, a faster consolidation response is observed with increasing kD/su0 due to the corresponding distribution of the initial coefficient of consolidation. 4.5. Evaluation of U considering combined effects of P%, d/D and kD/su0 Based on the characteristics outlined previously, an exponential formulation is presented to depict the time responses of normalised preloading gain in undrained capacities:

U ¼ 1  0:5ðTv =T50 Þ

n

(17)

where T50 is the time at which the increase in uniaxial undrained capacity is half the maximum increase, and n is the factor controlling the shapes of the time response. Table 3 summarises the best fit values of T50 and n to UV, UH and UM for skirted circular foundations with different d/D. Note that the influence of kD/su0 is neglected here; instead, the consolidation responses with a relatively homogeneous soil strength profile (kD/su0 ¼ 0.5, with a corresponding relatively uniform profile of the coefficient of consolidation) is used to represent each d/D, which allows for a conservative and simpler formulation. The results are representative for all P%. The parameter T50 increases with d/D for UV, UH and UM, which implies longer time periods are required for consolidation. The parameter n maintains a constant value of 0.85 for UV and UM but for UH, shows an initial rise from approximately 0.60 at low d/D to a constant value of 0.85 at higher d/D. Such a change in shape of the consolidation response is presented in Fig. 9. Compared to UV, and UM, a major difference exists in the early stage of consolidation, with UH exhibiting a faster consolidation rate (Fig. 11a, d/D ¼ 0). However, this discrepancy becomes less pronounced with increasing d/D (Fig. 11b). This reflects the difference of the consolidation response across the soil domain. UV, UH and UM are associated with the evolution of undrained shear strength with consolidation within the soil domain mobilised by each direction of loading. Volumetric deformations under vertical preload develop from a region in the vicinity of the foundation to the far field with consolidation. Under

field due to consolidation, which also changes the coefficient of consolidation. The consolidation responses, therefore, vary with P%. The variations of UV, UH and UM with time factor Tv for different levels of preloading are presented in Fig. 8. Results for d/D ¼ 0, with a relatively homogeneous soil condition kD/su0 ¼ 0.5 are selected for presentation to avoid the combined effects of d/D and kD/su0. The normalised preloading gain in each loading direction shows a similar trend, with a slightly faster consolidation response being observed under higher P%. This implies that P% has a minimal influence on the normalised consolidation responses. The consolidation curves presented in this study are, therefore, mostly presented for P% of 50%, representing a good (and generally conservative) estimation of all results. 4.3. Effect of aspect ratio d/D With increasing d/D, the drainage paths within the soil lengthens and the consolidation response is affected. The variations of UV, UH and UM, with time factor Tv for different skirt embedment levels, are reported in

Fig. 11. Comparison of normalised preloading gains in undrained uniaxial capacities UV, UH, UM against consolidation time Tv (P% ¼ 50%, kD/su0 ¼ 0.5): (a) surface circular foundation; (b) skirted circular foundation with d/D of 0.2, 0.3, 0.5, 0.8 and 1 26. 364

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passive wedge region will be mobilised, reducing the difference in consolidation response between UH and UV and UM. 5. Summary of the method to calculate the post-preloaded uniaxial capacities Formulations are presented to calculate the preloaded undrained uniaxial capacities of skirted circular foundations:  as a function of the level of preloading P%  as a function of the duration of application of preload, defined by the time factor Tv  for embedment ratios d/D ranging from 0 to 1, and  for strength heterogeneity ratios kD/su0. The method incorporates the solutions to estimate the no-preload uniaxial undrained capacities presented in Fu et al. (2017) and the formulations for evaluating ηf and U presented in this paper. The calculation procedure comprises four steps: (1) Calculate the no-preload undrained uniaxial capacities Vun, Hun and Mun, using the expressions of the undrained uniaxial capacity factors Nc of the skirted circular foundation (Fu et al., 2017). (2) Calculate the preloading gains in undrained uniaxial capacities after full consolidation ηf using Eqs. (4) and (5). (3) Calculate the preloading gains in undrained uniaxial capacities for partial consolidation ηp, using ηf calculated from step 2 and Tv, following an expression derived from a combination of Eqs. (16) and (17):

  n ηp ¼ ηf  ηf  1 0:5ðTv =T50 Þ

(19)

Fig. 12 a, b and c show an example of a preloading gain in undrained vertical, horizontal and moment capacities η (ηV, ηH, ηM) with a normalised consolidation time Tv for a skirted circular foundation with d/ D ¼ 0.3. Four different kD/su0 (0.5, 1, 2, 3.33) and two different P% (of 20% and 90%) are included. (4) Calculate the post-preload undrained uniaxial capacities Qpc, which can be expressed as follows: For full consolidation

Qpc;f ¼ Qun ηf

(20a)

For partial consolidation

Qpc;p ¼ Qun ηp

(20b)

The overall procedure, with a summary of the formulation to calculate Nc, ηf and ηp is presented in Table 4. Following the method proposed above, an example is introduced here to illustrate the potential gains in the uniaxial capacities of a subsea foundation due to preloading with consolidation. A circular foundation, with a dimeter (D) of 10 m, rests on the normally consolidated clay with an undrained shear strength profile su ¼ 4.5 þ 1.5z (kD/su0 of 3.33) and a consolidation coefficient cv0 of 3 m2/year. A vertical preload V of 1160 kN is applied to the foundation, and durations of consolidation under the vertical preload of 3 and 6 months (within the range of typical time lag between installation and operation of pipelines) are considered. Following Fu et al. (2017), the no-preload undrained ultimate vertical capacity Vun, horizontal capacity Hun and moment capacity Mun are 2905.2 kN, 353.4 kN and 3852.4 kNmm. The preloading level P% corresponds to approximately 40%, which yields preloading gains ηf,V, ηf,H, and ηf,M under full consolidation of 1.33, 1.66 and 1.50. The 3-month (and 6-month) consolidation time, corresponds to Tv of 0.0075 (and 0.015), which will lead to the preloading gains ηp,V, ηp,H, and ηp,M being

Fig. 12. Example of preloading gains in undrained uniaxial capacities with normalised consolidation time, Tv (P% of 20%, 90%, d/D of 0.3, kD/su0 of 0.5, 1, 2, 3.33): (a) ηV; (b) ηH; (b) ηM 28.

vertical and moment load, a larger volume of soil is mobilised in consolidation (for low strength heterogeneities the failure mechanism under moment load is not confined to an internal scoop), hence UV and UM reflect the average consolidation response over a larger volume of soil. Under horizontal loading, at a lower d/D, soil is essentially mobilised along the sliding plane, and the corresponding UH is associated with a more localised consolidation beneath the foundation. At a higher d/D, a larger body of soil underneath the foundation and in the active and 365

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Table 4 Summary of equations required to predict the gain in capacity with preloading. V Non-preload capacity factors, NcV, NcH, NcMa

Equation Key parameters

Nc_surf

366 ηV,f, ηH,f, ηM,f (fully consolidated gain)

"

 2 #

NcV_surf ¼ 6:05 1 þ 0:14 skD  0:007 u0 Nc;r ¼

Nc,r

Preloading gains, ηV, ηH, ηM

H

Nc ¼ Nc;r Nc_surf

sum su0 Nc;r; homo

þ

kd su0 Nc;r; NC soil

Nc,r,homo

 NcV;r; homo ¼ 1 þ 1:9ln 1 þ

Nc,r,NC

 NcV;r; NC soil ¼ 1 þ 1:15ln 1 þ

soil

Equation (Exponential function)

ðTv =T50 Þn

a

 d D

"

Nc_surf ¼ 1

 NcH;r; homo ¼ 1 þ 8:15ln 1 þ  NcH;r; NC soil ¼ 1 þ 5ln 1 þ

 d D

 d D

 3 þ 1:56 Dd

 2

Key parameters

ηf,P10 ηf,P90

When P%  10%

ηf,surf   g Dd

ηV;f;P10;surf ¼ 1:05 þ 0:007 skD u0   ¼ 1 g Dd

ηH;f ;P10;surf ¼ 1:16 þ 0:02 skD u0   g Dd ¼ 1

ηM;f ;P10;surf ¼ 1:10 þ 0:003 skD u0   g Dd ¼ 1

When 10%  P%  90%,

ηf,surf   g Dd

ηV;f;P90;surf ¼ 1:75 þ 0:03 skD u0  0:5   ¼ 1 þ 0:18 Dd g Dd

ηH;f ;P90;surf ¼ 2:52 þ 0:15 skD u0    0:5 g Dd ¼ 1 þ 0:41 Dd

ηM;f ;P90;surf ¼ 2:29    0:5 g Dd ¼ 1 þ 0:46 Dd

Equation

d D

NcM;r; NC soil ¼ 0:65 þ 0:05 Dd þ 1:88

When P%  10%, ηf ¼ h   ηf ;P10ðP90Þ ¼ g Dd

d D

where, ηf for full consolidation and ηp for partial consolidation. ηH,f ηM,f

ηf ;P10 10%

P10ðP90Þ

kD su0

 2 NcM;r; homo ¼ 1 þ 1:79 Dd þ 1:02

ηp ¼ ηf  ðηf  1Þð0:5Þ ηV,f Tv ¼ cv0t/D2 Listed in Table 3

ηf Tv T50 n Equation (Bilinear piecewise function)

 2 #

 0:008 NcM_surf ¼ 0:67 1 þ 0:18 skD u0

Key parameters

η

η

f ;P90 f ;P10 P% When 10%  P%  90%, ηf ¼ ηf ;P10 þ ð90%10%Þ ðP%  10%Þ(3)  i ηf;surf;P10ðP90Þ 1 þ1 Nc;r

V;P10

V;P90

ηV,p, ηH,p, ηM,p (partially consolidated gain)



d D

kD su0

M

H;P10

H;P90

M;P10

M;P90

n

ηp ¼ ηf  ðηf  1Þð0:5ÞðTv =T50 Þ

Equations derived from Fu et al. (2017). Ocean Engineering 147 (2018) 355–369

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compared with existing solutions based on linear fits for surface strip (Vulpe et al., 2016) and surface rectangular foundations (Feng and Gourvenec, 2015) in Fig. 13 a, b, and c. The fitted expressions for preloading gains in undrained capacities are compared for P% ¼ 0%–70% after full consolidation for a surface circular foundation (d/D ¼ 0) in Fig. 10 a, b, and c. To ensure a direct comparison between the studies, a strength heterogeneity ratio kD/sum (kB/sum for the surface rectangular foundation) of 1.875 is considered. The surface strip foundation generally exhibits the largest increase due to preloading. This is because plane strain consolidation leads to a higher average increase in undrained shear strength. For the vertical loading direction, the surface circular foundation shows slightly lower preloading gains than the surface rectangular foundation. For the horizontal loading direction, the surface circular and rectangular foundations present similar (although slightly lower) preloading gains compared to the surface strip footing, implying a similar increase in the undrained shear strength of the near-surface soil under the same magnitude of vertical preload. The surface circular foundation has a higher preloading gain under a lower preloading level in horizontal loading direction compared to the surface strip foundation, which may be attributed to the bi-linear fit adopted in this study. In the moment direction the rectangular footing presents the lowest preloading gain along the loading direction; the rotation axis is the in the centre foundation base plane and parallel to the short edge, which is highly related to the largest volume of soil being mobilised by consolidation. The performance of other types of foundation in preloading gain in V, H and M are also compared. For the vertical loading direction, the surface circular foundation shows much higher preloading gains than the skirted circular foundation. This is because the periphery skirt allows consolidation to occur in stronger soil with lower compressibility and thus yields a lower increase in undrained shear strength. The development of consolidation in deep soil also results in a low preloading gain of the skirted circular foundation, and this tendency is more obvious with an increasing d/D. For the moment loading direction, a skirted circular foundation with a small d/D of 0.2 shows a large gain exceeding that of the surface foundations at large P%. This is because although the soil beneath the skirt tip has a lower increase in shear strength, the additional improvement in soil strength due to consolidation confined within the periphery skirt is mobilised under the moment loading at failure after consolidation. This tendency does not remain with increasing d/D. 7. Concluding remarks A set of comprehensive finite-element analyses were conducted to investigate the post-preload undrained uniaxial capacity of the skirted circular foundation on normally consolidated clay. The influences of the level of preloading, the foundation aspect ratio, the initial soil strength heterogeneity and the duration of consolidation are observed. The main conclusions are:

c

 Non-uniform increases in the uniaxial capacities in V, H and M for a given level of preload and consolidation time is attributed to the volume of consolidated soil encompassed by the failure mechanism.  The level of preloading reflects the magnitude of soil deformation in the consolidation soil region. Elastic compression dominates soil behaviour at low P% (lower than 10%) while the plastic compression dominates at higher P% (larger than 10%). The resultant preloading gains follow an approximately bilinear relationship against the magnitude of vertical preload.  Higher soil strength heterogeneity results in shallower soil being mobilised at failure, which shows larger increases in undrained shear strength due to consolidation.  Larger gains are associated with shorter skirt length because the periphery skirt transfers the vertical preload to deeper soil – already more competent soil.

Fig. 13. Comparison of preloading gains in undrained uniaxial capacities against level of preloading for different foundation geometries with published solutions (kD/sum or kB/sum of 1.875 and full consolidation): (a) ηV,f, (b) ηH,f, (c) ηM,f 30.

1.13 (1.20), 1.30 (1.41) and 1.20 (1.30) respectively. The vertical, horizontal and moment capacity are therefore predicted to increase to 3285.3 (3478.8) kN, 460.2 (497.9) kN and 4616.0 (5004.8) kNm, respectively. It is clear that the consolidation allows the foundation to resist greater environmental loads. If accounted in design, this could result in smaller and more economical foundations. 6. Comparison with available solutions The influence of the foundation geometry on the preloading gains in undrained uniaxial capacities is discussed in this section. Results are first 367

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Acknowledgements

A series of equations is proposed to evaluate the full consolidated gains in uniaxial undrained capacities under preloading. A further extension incorporates the entire consolidation history and a method to estimate the post-preload uniaxial capacities for any given P%, d/D and kD/su0 after a period of consolidation is proposed. Finally, this study has neglected a range of complexities, such as skirted foundation roughness, over-consolidation ratio and soil model sensitivity. These influences might be analysed independently to gauge the significance of foundation preloading issues. However, the general conclusion here will be useful for many practical foundation designs.

This work forms part of the activities of the Centre for Offshore Foundation Systems (COFS), which is supported by the Lloyd's Register Foundation as a Centre of Excellence and now forms one of the primary nodes of the Australian Research Council (ARC) Centre of Excellence for Geotechnical Science and Engineering. Lloyd's Register Foundation invests in science, engineering and technology for public benefit, worldwide.

notation

cv0 d d/D D e0 ecs H k kD/su0 kD/sum ks K0 mv M Ms n Nc Nc,r Nc,r,homo Nc,r,NC soil p0 0 p0 c0 p0 cs Q su su0 sum T50 Tv U V ΔH ΔM ΔV ΔQ η γw γ0 θL κ λ σ0 v ϕ0 tc

initial in situ coefficient of consolidation at the skirt tip level skirt length aspect ratio diameter of skirted circular foundation initial void ratio intercepts of the critical state line (CSL) with the compression plane (at p0 ¼ 1 kPa) general term for horizontal load shear strength gradient undrained shear strength heterogeneity non-dimensional undrained shear strength gradient coefficient of permeability coefficient of lateral earth pressure coefficient of volume compressibility general term for moment load gradient of critical state line (CSL) in the (p–qd) plane fitting parameter undrained uniaxial capacity factor ratio of the capacity factor for a skirted circular foundation divided by that for a circular foundation ratio of the capacity factor for a skirted circular foundation divided by that for a circular foundation on homogeneous soil ratio of the capacity factor for a skirted circular foundation divided by that for a circular foundation on normally consolidated soil initial mean normal effective stress pre-consolidation pressure (hardening parameter) mean normal effective stress at the failure state a general term for undrained ultimate vertical, horizontal and moment load undrained shear strength initial undrained shear strength at the skirt tip level initial undrained shear strength at the mudline fitting parameter normalised consolidation time factor normalised preloading gain general term for vertical load absolute preloading enhancement in undrained horizontal capacities absolute preloading enhancement in undrained moment capacities absolute preloading enhancement in undrained vertical capacities general term of absolute preloading enhancement in undrained vertical, horizontal and moment capacities general term of preloading gains water unit weight submerged unit weight of the soil Lode angle swelling and recompression index virgin compression index vertical effective stress critical friction angle in triaxial compression

Bransby, M.F., 2002. The undrained inclined load capacity of shallow foundations after consolidation under vertical loads. In: Proceedings of the 8th International Symposium on Numerical Models in Geomechanics (NUMOG VIII), pp. 431–437. Chatterjee, S., Yan, Y., Randolph, M.F., White, D.J., 2012. Elastoplastic consolidation beneath shallowly embedded offshore pipelines. Geotechnique Lett. 2, 73–79. Dassault Systemes, 2010. Abaqus Analysis Users' Manual. Simula Corp, Providence, RI, USA.

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