Content uploaded by Stephen Suryasentana
Author content
All content in this area was uploaded by Stephen Suryasentana on Sep 26, 2023
Content may be subject to copyright.
Small-strain, non-linear elastic Winkler model for uniaxial
1
loading of suction caisson foundations
2
3
4
Stephen K. Suryasentana1
5
Harvey J. Burd2
6
Byron W. Byrne3
7
Avi Shonberg4
8
9
Affiliations
10
1 Department of Civil and Environmental Engineering, University of Strathclyde,
11
Glasgow, UK;
12
stephen.suryasentana@strath.ac.uk (Orcid: 0000-0001-5460-5089)
13
2 Department of Engineering Science, University of Oxford, Oxford, UK;
14
harvey.burd@eng.ox.ac.uk (Orcid: 0000-0002-8328-0786)
15
3 Department of Engineering Science, University of Oxford, Oxford, UK;
16
byron.byrne@eng.ox.ac.uk (Orcid: 0000-0002-9704-0767)
17
4 Ørsted Wind Power, London, UK
18
avish@orsted.com
19
20
21
22
Corresponding author information
23
Stephen K. Suryasentana
24
stephen.suryasentana@strath.ac.uk
25
26
Number of words in the main text (excluding abstract and references)
27
2519
28
Number of figures
29
10
30
Number of tables
31
4
32
33
34
Date
35
1 May 2023
36
2
Abstract
37
Soils exhibit non-linear stress-strain behaviour, even at relatively low strain levels.
38
Existing Winkler models for suction caisson foundations cannot capture this small-strain,
39
non-linear soil behaviour. To address this issue, this paper describes a new non-linear
40
elastic Winkler model for the uniaxial loading of suction caissons. The soil reaction curves
41
employed in the model are formulated as scaled versions of the soil response as
42
observed in standard laboratory tests (e.g. triaxial or simple shear tests). The scaling
43
relationships needed to map the observed soil element behaviour onto the soil reaction
44
curves employed in the Winkler model are determined from an extensive numerical study
45
employing 3D finite element analysis. Key features of the proposed Winkler model
46
include: computational efficiency, wide applicability (it can be used for caisson design in
47
clay, silt or sand) and design convenience (the required soil reaction curves can be
48
determined straightforwardly from standard laboratory test results). The proposed model
49
is suitable for small and intermediate caisson displacements (corresponding to fatigue
50
and serviceability limit state conditions) but it is not applicable to ultimate limit state
51
analyses.
52
53
Keywords
54
Footings/foundations, soil/structure interaction, offshore engineering, numerical modelling
55
56
57
58
59
3
Notation
60
depth below ground level
61
suction caisson diameter
62
suction caisson embedded skirt length
63
suction caisson skirt thickness
64
horizontal displacement of foundation along x-axis
65
horizontal displacement of foundation along y-axis
66
vertical displacement of foundation
67
rotation of foundation about x-axis
68
rotation of foundation about y-axis
69
torsional rotation of foundation
70
secant shear stiffness of soil
71
small-strain shear modulus of soil
72
parameter governing the shape of the small-strain shear modulus profile
73
shear strain of soil
74
reference shear strain of soil
75
shape parameter governing the shape of the stiffness degradation curve for
76
deviatoric stress of a soil element
77
deviatoric strain of a soil element
78
reference deviatoric strain of a soil element
79
equivalent deviatoric strain due to normalised displacement of a soil reaction
80
lateral soil reaction along x-axis
81
lateral soil reaction along y-axis
82
axial soil reaction
83
moment soil reaction about x-axis
84
moment soil reaction about y-axis
85
torsion soil reaction
86
lateral displacement of the soil reaction along x-axis
87
lateral displacement of the soil reaction along y-axis
88
axial displacement of the soil reaction
89
rotational displacement of the soil reaction about x-axis
90
rotational displacement of the soil reaction about y-axis
91
torsional displacement of the soil reaction
92
secant stiffness of soil reaction
93
initial linear elastic stiffness of soil reaction
94
4
axial secant stiffness of the soil reaction
95
lateral secant stiffness of the soil reaction
96
rotational secant stiffness of the soil reaction
97
torsional secant stiffness of the soil reaction
98
displacement scaling factor for the vertical soil reaction stiffness
99
displacement scaling factor for the horizontal soil reaction stiffness
100
displacement scaling factor for the moment soil reaction stiffness
101
displacement scaling factor for the torsional soil reaction stiffness
102
103
5
Introduction
104
Suction caisson foundations are increasingly used to support offshore wind turbines in relatively
105
deep waters, where they may be more economical than monopile foundations. To facilitate the
106
efficient design of suction caissons for large-scale wind farm projects, design tools are needed
107
that are both rapid and reliable. Winkler-based design models have recently been proposed for
108
suction caissons (e.g. Suryasentana et al. 2022; Antoniou et al. 2022). Compared to detailed
109
computational simulations such as three-dimensional (3D) finite element analysis (FEA),
110
Winkler models offer advantages such as computational efficiency and ease of use.
111
OxCaisson (Suryasentana et al. 2022) is a Winkler model that provides accurate predictions of
112
the stiffness of suction caissons in linear elastic soil. However, its applicability is limited to low
113
load levels for which the soil response can be approximated as being linear elastic. To achieve
114
realistic predictions at higher load levels, the non-linear behaviour of soil must be considered.
115
However, existing non-linear Winkler models for caissons (e.g. Suryasentana et al. 2018;
116
Antoniou et al. 2022) do not consider the small-strain, non-linear behaviour of soil. The current
117
paper describes a new small-strain, non-linear elastic Winkler model to predict the caisson
118
response for uniaxial loading for small and intermediate soil strain levels, relevant to fatigue and
119
serviceability limit state conditions. A key feature of the proposed Winkler model is that the soil
120
reaction curves employed in the analysis are represented by scaled versions of the soil
121
response as observed in standard laboratory soil element tests.
122
Small-strain soil non-linearity
123
Soils exhibit non-linear stress-strain behaviour, even at small strains (Atkinson 2000).
124
Numerous studies have explored the non-linear variation of soil secant shear stiffness with
125
strain. For example, Hardin and Drnevich (1972) proposed a one-parameter () model to
126
describe the reduction of with increasing shear strain ,
127
(1)
where is the small-strain shear modulus and is a reference shear strain.
128
Darendeli (2001) proposed a two-parameter (, ) variant of Eq. 1 to better match laboratory
129
test data,
130
(2)
6
Darendeli (2001) determined best-fit values for ranging from 0.8 to 0.9 for sand and 0.9 to 1.0
131
for clay and silt. Vardanega and Bolton (2013) suggested values for ranging from 0.5 to 1.39
132
for clays, and 0.73 to 0.94 for silts.
133
Scaling relationships
134
Previous FEA studies on laterally loaded pile slices (e.g. Bransby 1999; Zhang and Andersen
135
2017) suggest that the - soil reaction behaviour can be approximately represented as a
136
scaled version of the stress-strain behaviour of a soil element as measured in laboratory tests.
137
This concept allows for the generation of site-specific - curves by scaling measured data from
138
laboratory tests. This scaling concept is employed in the current work.
139
Objectives of the paper
140
The paper has three principal objectives: (i) determine scaling relationships between soil
141
element stress-strain behaviour and soil reaction curves; (ii) develop a small-strain, non-linear
142
elastic Winkler model in which the soil reaction curves are formulated using these scaling
143
relationships; (iii) demonstrate that the proposed Winkler model provides an accurate
144
representation of the performance of a caisson for uniaxial loading as determined by 3D FEA.
145
146
Finite Element Calibration Analyses
147
To determine the scaling relationships, 3D FEA were conducted for a rigid caisson embedded in
148
a small-strain, non-linear elastic soil. Soil reaction curves extracted from the analysis are
149
compared with the stress-strain behaviour of the soil (as implemented in the finite element
150
model) to determine the required scaling factors. These analyses, which were conducted using
151
Abaqus v6.13 (Dassault Systemes 2014), employed a UMAT implementation of a non-linear
152
elastic soil constitutive model in which the secant shear stiffness is defined as,
153
(3)
The deviatoric strain is defined in the Appendix and is a model parameter. Note that
154
Eq. (3) is a generalised form of Eq. 2. The secant bulk modulus, , is determined via
155
(4)
where is a (constant) Poisson’s ratio. The soil is weightless. The depth variation of adopted
156
in the analyses is,
157
7
(5)
where is a reference stiffness, is a parameter (values were considered), is
158
the distance below ground level, and
is the caisson diameter. Four values of Poisson’s ratios
159
(), three values (0.0001, 0.0005, 0.002) and three values (0.4, 0.7,
160
1) were considered. The cases considered in this study are summarised in Table 2, which
161
encompasses a parameter range extensive enough to account for the majority of practical
162
applications. For example, the parameters in Table 2 include the approximate lower and
163
upper bound of the values observed by Vardanega and Bolton (2013) based on an extensive
164
database of soil laboratory tests.
165
The 3D finite element model was based on the procedures described in Suryasentana et al.
166
(2022). Five embedded skirt lengths () were analysed. The caisson was
167
modelled as being rigid via the use of rigid body constraints. A cylindrical mesh was employed –
168
as illustrated in Suryasentana et al. (2022) - with the diameter and depth both equal to .
169
Displacements were fixed at the bottom of the mesh domain and in the radial directions on the
170
vertical mesh boundaries. First-order, linear, brick elements C3D8 (C3D8H for ) were
171
employed for the soil and caisson. Tie constraints were used at the soil-caisson interface to
172
prevent contact breaking. Uniaxial displacements were applied at the caisson loading reference
173
point (LRP, see Fig. 1a) as indicated in Table 1.
174
175
The skirt soil reactions (i.e. the distributed force/moment applied by the soil to the caisson skirt)
176
and the base soil reactions (i.e. the concentrated force/moment applied by the soil to the
177
caisson base) are extracted from the FEA results following the procedures in Suryasentana et
178
al. (2022). There are six types of skirt and base soil reactions: , representing
179
lateral soil reaction along the x direction, lateral soil reaction along the y direction, axial soil
180
reaction, moment soil reaction about the x axis, moment soil reaction about the y axis and
181
torsional soil reaction, respectively.
The corresponding displacements are ,
182
representing lateral displacement of the soil reaction along the x direction, lateral displacement
183
of the soil reaction along the y direction, axial displacement of the soil reaction, rotational
184
displacement of the soil reaction about the x axis, rotational displacement of the soil reaction
185
about the y axis and torsional displacement of the soil reaction, respectively. Fig. 1b shows a
186
schematic diagram of the skirt and base soil reactions acting on a one-dimensional Winkler
187
model representation of the suction caisson structure. Based on the 3D FEA analysis in
188
Suryasentana et al. (2022), it was observed that the caisson lid has insignificant effects on the
189
distribution of the soil reactions on the skirt. Thus, the lid is not modelled in the Winkler model,
190
consistent with other researchers (e.g., Antoniou et al. 2022).
191
192
8
Results
193
The computed load-displacement behaviour of the caisson for an example case (
194
and ) for each of the four uniaxial displacement cases
195
specified in Table 1 is shown in Fig. 2. Fig. 3 shows the corresponding soil reactions extracted
196
from the FEA. As the skirt soil reactions are distributed soil reactions, the skirt soil reaction
197
curve in Fig. 3 should be integrated along the caisson skirt length and then added to the base
198
soil reaction curve in Fig. 3 in order to obtain the curve in Fig. 2. Fig. 4 presents the normalised
199
secant stiffness of the soil reactions, where and represent the secant stiffness of
200
the lateral, moment, axial and torsional soil reactions, respectively. It is evident that the
201
normalised secant stiffnesses in Fig. 4 decrease with increasing displacement in a manner
202
similar to the reduction of with increasing strain.
203
Scaling relationships
204
Fig. 4 suggests that the normalised displacements of the soil reaction curves can be scaled to
205
match the curves (Eq. 3). Suitable scaling factors () were determined using
206
least squares regression, with and representing the scaling factors for , ,
207
(and ), (and ), respectively. Different scaling factors were identified for the skirt and base
208
soil reactions. The scaled normalised displacement is referred to as the ‘equivalent deviatoric
209
strain’ , which is defined as the equivalent amount of deviatoric strain induced by the
210
normalised displacement. The four definitions corresponding to each displacement mode
211
are,
212
(6)
(7)
(8)
(9)
Fig. 5 illustrates the excellent agreement between the normalised secant stiffness of the soil
213
reactions and after applying the determined scaling factors.
214
Figs. 6 to 9 show the best fit values of the scaling factors for ; similar trends were
215
observed for and 1. Figs. 6 and 7 show the variations of the scaling factors for the skirt
216
soil reactions (
) and base soil reactions (
) with
217
respect to and , for . For both the skirt and base scaling factors, decrease
218
9
with , while tends to increase with . Figs. 8 and 9 show the variations of the scaling
219
factors with respect to and , for . For both the skirt and base scaling factors,
220
increase with .
221
The scaling factors were found to depend only on and , with having negligible
222
influence. Approximating functions were derived to represent these scaling factors, based on
223
the following functional forms:
224
(10)
(11)
Eqs. 10 and 11 are used to estimate the scaling factors () to transform the
225
normalised displacements into the ‘equivalent deviatoric strain’, as per Eqs. 6 to 9. Eq. 10 is
226
used for and Eq. 11 is used for
and
. Tables 3
227
and 4 provide the best-fit parameters for Eqs. 10 and 11 obtained through least squares
228
regression. The scaling factors determined by Eqs. 10 and 11 using these parameters are
229
shown as dashed lines in Figs. 6 to 9, for comparison with the FEA values. Good agreement
230
between the approximating functions and the FEA calculated scaling factors is clearly observed.
231
Soil reaction curves
232
On the basis of the scaling relationships described above, Eq. 3 can be used to represent the
233
soil reaction functions with a change of variables (
and ),
234
(12)
where is the secant stiffness of the soil reaction curves, and is the corresponding initial
235
stiffness (Suryasentana et al. 2022; Suryasentana and Mayne 2022). Using Eqs. 6 to 9, the
236
following equations describe how the secant stiffness of the soil reaction curves for each loading
237
mode change with displacements,
238
(13)
10
(14)
(15)
(16)
where Eqs. 13 to 16 apply to both the skirt and base soil reactions. Values of
,
,
239
and
are listed in Table 3 in Suryasentana et al. (2022), and
,
,
and
240
are listed in Table 5 in Suryasentana et al. (2022).
241
Proposed small-strain, non-linear elastic Winkler model
242
The proposed small-strain, non-linear elastic Winkler model employs the soil reaction curves
243
described in Equations 13 to 16, based on the soil-structure interaction framework described in
244
Suryasentana et al. (2022). It was observed that the -coupling soil reaction secant stiffness
245
do not follow the same ‘S-shaped’ stiffness degradation curve as per Fig. 4 and decreases
246
rapidly at increasing strain. As no scaling relationship was identified for and the artificial
247
rotational stiffness
defined in Suryasentana et al. (2022), a constant displacement
248
scaling factor was defined for both and
for simplicity. These components
249
therefore contribute to the initial caisson stiffness but degrade rapidly and have negligible
250
contributions at increasing displacements. Note that there is still coupling between the global
251
lateral load and moment for the caisson, even if there are no contributions from and
.
252
The proposed Winkler model’s calculations are shown in Fig. 2, which shows good agreement
253
with the corresponding 3D FEA results.
254
Equations 13 to 16 are calibrated for uniaxial displacements, whereas practical design cases
255
are more likely to involve uniaxial loads. Uniaxial displacement refers to the application of
256
displacement in a single direction while keeping displacements in all other directions at zero. On
257
the other hand, uniaxial loading refers to applying a load solely in one direction while
258
maintaining loads in all other directions at zero (with the caisson remaining free to displace in all
259
directions). To explore the performance of the Winkler model for uniaxial loads (rather than
260
displacements), additional 3D FEA were carried out for uniaxial applications of vertical,
261
horizontal, moment and torsional loading at the caisson LRP caisson. Fig. 10 compares the
262
resultant 3D FEA calculations with the corresponding Winkler model’s predictions for the
263
11
example case where and (i.e., the same
264
conditions as Fig. 2); the two sets of results agree well.
265
Discussion
266
The proposed Winkler model is suitable for predicting the performance of caisson foundations at
267
intermediate displacements to support the assessment of fatigue and serviceability limit state
268
design conditions. Site-specific calibrations are achieved by fitting the shear stiffness
269
degradation model in Eq. 2 to the available laboratory test data to determine appropriate value
270
of and . The value of is then transformed into (see Appendix). Site specific data
271
on small strain shear modulus and Poisson’s ratio are also required.
272
Compared to 3D FEA, the proposed Winkler model requires significantly less computational
273
time (e.g. 5 hours for 3D FEA versus 1 minute for the Winkler model in producing the
274
calculations for Fig. 2). Adopting the Winkler modelling framework offers some potential
275
advantages, such as the ability to use deformable elements (e.g. Timoshenko beam elements)
276
to represent caissons with flexible skirt. Previous studies (Suryasentana et al. 2022) have
277
shown that the soil reactions are largely decoupled from the caisson structural properties and
278
the same soil reactions can be used to estimate the stiffness of rigid and flexible caissons. For
279
users who prefer to use the caisson stiffness results directly, the caisson stiffness results from
280
the FEA study in this paper are also available at
281
https://github.com/autogeolab/caisson_smallstrain_fea. Caution is advised against extrapolating
282
beyond the parameter ranges in Table 2.
283
The proposed Winkler model is developed for uniaxial loading, whereas typical design cases
284
would likely involve the application of combined loads. The non-linear basis of the Winkler
285
model means that linear superposition does not strictly apply and so the model cannot be
286
guaranteed for combined loads. However, preliminary work has indicated that – when combined
287
loading cases typical of offshore wind applications (e.g., lateral-moment loading at some
288
constant ratio) are applied to the proposed Winkler model – a good agreement with
289
independent 3D FEA is typically achieved. It is uncertain if the Winkler model can be directly
290
applied to soil stiffness depth profiles that differ from those investigated in this paper, and if
291
approaches used for estimating caisson stiffness in layered soils (Suryasentana et al. 2023)
292
would be directly applicable to this model. Further work is needed to explore the applicability of
293
the model to general combined load cases, different soil profiles and identify any limitations with
294
the model. The proposed Winkler model is not suitable for the analysis of failure states. For
295
ultimate limit state assessments, alternative approaches employing elastoplastic Winkler models
296
(e.g. Suryasentana et al. 2018; Antoniou et al. 2022), macro-element models (e.g. Skau et al.
297
2018) or failure envelope models (e.g., Vulpe 2015; Suryasentana et al. 2020) would be
298
appropriate.
299
12
Previous studies (e.g., Bransby 1999; Zhang and Anderson 2017) have indicated that the lateral
300
pile soil reaction behaviour are scaled versions of soil element behaviour using their respective
301
elastoplastic soil constitutive models. This suggests the possibility of finding similar scaling
302
relationships for caisson soil reactions based on more advanced soil constitutive models.
303
However, advanced soil constitutive models often involve a large number of parameters, making
304
it more practical to develop non-scaling based formulations for the soil reactions based on a
305
smaller set of parameters, as demonstrated in the case of PISA soil reactions (Byrne et al.
306
2020).
307
Acknowledgments
308
Parts of the work described here were conducted during the DPhil studies of the first author at
309
the University of Oxford. The first author would like to acknowledge the generous support of
310
Ørsted Wind Power for funding his DPhil studentship at the University of Oxford. Byrne is
311
supported by the Royal Academy of Engineering under the Research Chairs and Senior
312
Research Fellowships scheme.
313
314
315
13
Appendix: Calculation of from triaxial and simple shear test results
316
The deviatoric strain is defined,
317
A1
where
and are displacements and coordinate directions respectively.
318
For simple shear tests, Eq. A1 simplifies to:
319
A2
where is the shear strain. For triaxial tests, Eq. A1 simplifies to:
320
A3
where and are the axial and radial strain, respectively.
321
14
References
322
Antoniou, M., Kourkoulis, R., Gelagoti, F., & Anastasopoulos, I. (2022). Simplified method for
323
performance-based seismic design of suction caissons supporting jacket offshore wind
324
turbines. Soil Dynamics and Earthquake Engineering, 155, 107169.
325
Atkinson, J. H. (2000). Non-linear soil stiffness in routine design. Géotechnique 50(5), 487–508.
326
Bransby, M. F. (1999). Selection of p-y curves for the design of single laterally loaded piles.
327
International Journal for Numerical and Analytical Methods in Geomechanics 23(15), 1909–
328
1926.
329
Byrne, B.W., Houlsby, G.T., Burd, H.J., Gavin, K.G., Igoe, D.J., Jardine, R.J., Martin, C.M.,
330
McAdam, R.A., Potts, D.M., Taborda, D.M. and Zdravković, L., (2020). PISA design model for
331
monopiles for offshore wind turbines: application to a stiff glacial clay till. Géotechnique,
332
70(11), pp.1030-1047.
333
Darendeli, M. B. (2001). Development of a new family of normalized modulus reduction and
334
material damping curves. PhD thesis, University of Texas at Austin.
335
Dassault Systemes (2014). Abaqus user manual. Simula Corp., Providence, RI.
336
Hardin, B. O. and Drnevich, V. P. (1972). Shear Modulus and Damping in Soils: Design Equations
337
and Curves. Journal of Soil Mechanics and Foundation Division (SM7), 667–692.
338
Skau K. S., Grimstad G., Page A. M., Eiksund G. R. and Jostad H. P. (2018). A macro-element
339
for integrated time domain analyses representing bucket foundations for offshore wind
340
turbines. Marine Structures, 59, 158–78.
341
Suryasentana, S. K., Byrne, B. W., Burd, H. J., & Shonberg, A. (2018). An elastoplastic 1D Winkler
342
model for suction caisson foundations under combined loading. Numerical Methods in
343
Geotechnical Engineering IX, 973-980. CRC Press.
344
Suryasentana, S. K., Dunne, H. P., Martin, C. M., Burd, H. J., Byrne, B. W., & Shonberg, A. (2020).
345
Assessment of numerical procedures for determining shallow foundation failure envelopes.
346
Géotechnique, 70(1), 60-70.
347
Suryasentana, S. K., Burd, H. J., Byrne, B. W., & Shonberg, A. (2022). A Winkler model for suction
348
caisson foundations in homogeneous and non-homogeneous linear elastic soil.
349
Géotechnique, 72(5), 407-423.
350
Suryasentana, S. K., & Mayne, P. W. (2022). Simplified method for the lateral, rotational, and
351
torsional static stiffness of circular footings on a nonhomogeneous elastic half-space based
352
on a work-equivalent framework. Journal of Geotechnical and Geoenvironmental Engineering,
353
148(2), 04021182.
354
Suryasentana, S. K., Burd, H. J., Byrne, B. W., & Shonberg, A. (2023). Modulus weighting method
355
for stiffness estimations of suction caissons in layered soils. Géotechnique Letters, 13(2), 1-8.
356
Vardanega, P. J. and Bolton, M. D. (2013). Stiffness of Clays and Silts: Normalizing Shear
357
Modulus and Shear Strain. Journal of Geotechnical and Geoenvironmental Engineering
358
9(September), 1575–1589.
359
15
Vulpe, C. (2015). Design method for the undrained capacity of skirted circular foundations under
360
combined loading: effect of deformable soil plug. Géotechnique, 65(8), 669-683.
361
Zhang, Y. and Andersen, K. H. (2017). Scaling of lateral pile p-y response in clay from laboratory
362
stress-strain curves. Marine Structures 53, 124–135.
363
364
16
Table 1. Displacement boundary conditions applied at LRP (see Fig. 1a) to determine the soil
365
reaction responses. For the analyses in this paper, was set to 0.1, which was sufficient to
366
observe the small-strain, non-linear behaviour of the soil reactions.
367
368
(rad)
(rad)
(rad)
Rigid Axial Translation
0
0
0
0
0
Rigid Lateral Translation
0
0
0
0
0
Rigid Rotation
0
0
0
0
0
Rigid Torsion
0
0
0
0
0
369
370
Table 2. Combinations of 3D FEA input parameters analysed for the calculation of displacement
371
scaling factors for the skirt and base soil reactions
372
Values analysed
0, 0.25, 0.5, 1, 2
0, 0.2, 0.4, 0.49
0, 0.5, 1
0.4, 0.7, 1
0.0001, 0.0005, 0.002
373
374
Table 3. Best-fit parameters of Eq. 10 for the displacement scaling factors for the skirt and base
375
soil reactions
376
0.064
0.044
0.062
0
0.167
0.316
0.799
1.732
0.427
0
0.293
0
0.63
0
0.33
0.885
0.067
0
0.07
0
0.144
0.325
0.68
1.074
0.101
-0.043
0.105
0
0.055
-0.046
0.324
1.85
0.383
0
0.251
0
0.229
0
0.375
1.399
0.094
0
0.257
0
0.019
0.228
0.609
1.505
377
7.684
-2.536
-4.855
-0.652
9.424
0
-4.637
-0.179
5.709
1.603
-3.676
-0.394
7.436
8.477
-5.341
-2.105
14.453
0
-5.951
-0.629
8.412
1.854
-4.276
-1.054
378
379
Table 4. Best-fit parameters of Eq. 11 for the displacement scaling factors for the skirt and base
380
soil reactions
381
0.36
0.172
0.426
0.354
0.171
0
0
0.193
0.305
-0.232
0.233
0.219
0.231
0.437
0.115
0.111
382
17
383
(a)
384
385
(b)
386
Figure 1. (a) Schematic diagram of a suction caisson foundation, where LRP represents the
387
loading reference point. (b) Distributed skirt soil reactions and the concentrated base soil
388
reactions acting on a one-dimensional Winkler model representation of the suction caisson
389
structure.
390
391
18
392
393
(a)
(b)
(c)
(d)
394
Figure 2. The 3D FEA results for the ground-level load-displacement behaviour of suction
395
caissons in small-strain, non-linear elastic soil, for and
396
, under the prescribed displacements detailed in Table 1. The corresponding
397
calculations by the proposed Winkler model are also included in these figures for comparison.
398
399
400
401
402
403
404
19
(a)
(b)
(c)
(d)
405
Figure 3. Load-displacement behaviour of the skirt and base soil reactions for
406
and , under the prescribed displacements detailed in Table
407
1.
408
409
20
(a)
(b)
*
(d)
410
411
Figure 4. Secant stiffness of the skirt and base soil reactions from Fig. 3, normalised by their
412
linear elastic counterparts, for and . The soil
413
constitutive relationship (Eq. 3) is also plotted here as grey dotted lines for comparison.
414
415
416
417
418
21
(a)
(b)
(c)
(d)
419
420
Figure 5. Secant stiffness of the skirt and base soil reactions, normalised by their linear elastic
421
counterparts, after applying the displacement scaling factors (), for
422
and . The soil constitutive relationship (Eq. 3) is also plotted
423
here as grey dotted lines for comparison.
424
425
426
22
427
(a)
(b)
*
(c)
(d)
428
Figure 6. Variations of the best-fit displacement scaling factors (
) for the
429
skirt soil reactions with respect to and for and . The dotted lines represent
430
the predictions of the approximating functions (Eqs. 9 and 10).
431
432
433
23
(a)
(b)
*
(c)
(d)
434
435
Figure 7. Variations of the best-fit displacement scaling factors (
) for the
436
base soil reactions with respect to and for and . The dotted lines represent
437
the predictions of the approximating functions (Eqs. 9 and 10).
438
439
440
441
24
442
(a)
(b)
*
(c)
(d)
443
Figure 8. Variations of the best-fit displacement scaling factors (
) for the
444
skirt soil reactions with respect to and for and . The dotted lines represent
445
the predictions of the approximating functions (Eqs. 9 and 10).
446
447
448
25
449
(a)
(b)
*
(c)
(d)
450
Figure 9. Variations of the best-fit displacement scaling factors (
) for the
451
base soil reactions with respect to and for and . The dotted lines represent
452
the predictions of the approximating functions (Eqs. 9 and 10).
453
454
455
456
457
458
26
459
(a)
(b)
(c)
(d)
460
Figure 10. The 3D FEA results for the ground-level load-displacement behaviour of suction
461
caissons in small-strain, non-linear elastic soil, for and
462
, for prescribed uniaxial loading. The corresponding Winkler model predictions are
463
included in these figures for comparison.
464
465