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1 INTRODUCTION

1.1 General Background

Winkler models are widely used to design deep foun-

dations such as piles. However, in recent work

(Gerolymos and Gazetas, 2006; Varun et al., 2009;

Suryasentana et al., 2017), Winkler models have also

been developed for shallow foundations such as cais-

son foundations. While these design methods may not

be as accurate as more rigorous approaches such as

the three-dimensional finite element (3DFE) method,

Winkler models have the advantages of being rela-

tively fast and easy to use.

Winkler models simplify the three-dimensional

(3D) foundation-soil interaction problem into a more

tractable one-dimensional (1D) problem, with the

foundation replaced by a beam and the soil continuum

by Winkler ‘springs’ (also termed as soil reactions in

this paper). To simulate the non-linear response of

soil, the Winkler models adopt non-linear elastic soil

reactions. Examples of such models include the p-y

and t-z methods (API, 2010; DNV, 2014) used to de-

sign laterally and axially loaded piles respectively.

Nevertheless, there are shortcomings with these

existing non-linear Winkler models. For example, the

non-linear elastic soil reactions used in the p-y or t-z

methods for piles cannot reproduce observed cyclic

loading phenomena such as permanent displacement

or hysteresis. Moreover, they cannot account for com-

bined loading effects on the failure state.

1.2 Proposed Model

Recently, Suryasentana et al. (2017) developed a

1D Winkler model, calibrated against 3DFE analyses,

to accurately predict suction caisson behaviour in lin-

ear elastic soil for six degrees of freedom (dof) load-

ing. However, this model can only be applied to load-

ing conditions where the soil response can be

approximated as linear elastic.

This paper extends the 1D Winkler model devel-

oped in Suryasentana et al. (2017) to allow predic-

tions of non-linear caisson behaviour in undrained

clay under combined planar vertical V, horizontal H

and moment M loading. The extension involves cou-

pling linear elastic soil reactions with local plastic

yield surfaces, which are calibrated against rigorous

3DFE failure state analyses.

The governing mechanics of the proposed 1D

model is based on the same elastoplasticity frame-

work used in 3DFE analyses. This allows straightfor-

ward reproduction of the 3DFE predictions, but with

higher efficiency due to dimensionality reduction.

Consequently, the proposed 1D model allows fast and

accurate solutions of caisson behaviour in elasto-

plastic soil for design assessment under fatigue, ser-

viceability and ultimate limit states (FLS, SLS, ULS).

This enables an efficient design process, with the 1D

model used to quickly shortlist potential designs from

a large candidate space, before further refinement is

conducted with 3DFE analyses.

An elastoplastic 1D Winkler model for suction caisson foundations under

combined loading

S.K. Suryasentana, B.W. Byrne & H.J. Burd

University of Oxford, UK

A. Shonberg

Ørsted Wind Power, London, UK

ABSTRACT: Most existing Winkler models use non-linear elastic soil reactions to capture the non-linear be-

haviour of foundations. These models cannot easily capture phenomena such as permanent displacement, hys-

teresis and the influence of combined loading on the failure states. To resolve these shortcomings, an elasto-

plastic Winkler model for suction caisson foundations under combined loading is presented. The proposed

model combines Winkler-type linear elastic soil reactions with local plastic yield surfaces to model the non-

linear soil response using standard plasticity theory, albeit in a simplified one-dimensional (1D) framework.

The results demonstrate that the model reproduces the appropriate foundation behaviour, comparing closely to

three-dimensional finite element (3DFE) analyses but with the advantage of rapid computation time.

2 METHODS

2.1 1D Model

The 1D model adopted in this paper is similar to that

detailed in Suryasentana et al. (2017) and it is briefly

described as follows. The 1D model is a simplified

representation of the original 3D caisson-soil interac-

tion problem, where the caisson structure and soil

continuum are replaced by a 1D rigid body and Win-

kler-type soil reactions respectively.

Figure 1 shows a schematic diagram of the original

3D caisson-soil problem and the 1D model represen-

tation. There are two types of soil reactions in this

model: distributed soil reactions that act along the

caisson skirt (referred to as the ‘skirt soil reactions’

and indicated as hskirt, mskirt and vskirt in Figure 1) and

concentrated soil reactions that act on the caisson

base, including the soil plug (referred to as the ‘base

soil reactions’ and indicated as hbase, mbase, vbase in

Figure 1). hskirt, vskirt and mskirt represent the distrib-

uted horizontal force, vertical force and rotational

moment along the skirt length, while hbase, vbase and

mbase represent the concentrated horizontal force, ver-

tical force and rotational moment at the base.

Figure 1. Schematic diagram of an embedded suction caisson

foundation (left) and its corresponding simplified 1D represen-

tation (right), where RP is the loading reference point. v, h and

m are the vertical, horizontal and rotational soil reactions.

There are, however, a few notable differences be-

tween the 1D model adopted in this paper and that de-

scribed in Suryasentana et al. (2017). First, as this pa-

per is only concerned with planar VHM loading, there

are only 3 components (v, h, m) for each soil reaction,

which correspond to the vertical w, horizontal u and

rotational ϴ degrees of freedom (dof). Second, unlike

the linear elastic soil assumed in Suryasentana et al.

(2017), the current paper assumes a linear elastic-per-

fectly plastic soil. This gives an ultimate limit to the

soil response, which the previous 1D model was not

able to capture. To simulate this behaviour, the 1D

model in this paper couples the linear elastic soil re-

actions with local plastic yield surfaces. These local

yield surfaces are a direct analogy of the elemental

yield surfaces in the soil reactions space (consisting

of v, h, m components). Just as the canonical yield

surfaces determine the set of allowable elemental

stress states, the local yield surfaces determine the set

of allowable soil reaction states.

The mechanics of the coupled soil reactions-yield

surfaces model can be explained by standard plastic-

ity theory. For soil reaction states lying inside the lo-

cal yield surface, the soil response is linear elastic

with the incremental response given by:

(1)

where p = soil reactions {v, h, m}, ke = elastic stiffness

matrix and u = local displacements {w, u, ϴ}. ke can

be obtained from Suryasentana et al. (2017) as the

caisson dimensions (L/D = 1) and elastic soil proper-

ties (ν = 0.49) adopted in this paper are identical.

However, for simplicity and faster numerical conver-

gence, the coupling terms between h and m in ke are

ignored (the exclusion of these coupling terms will

mainly impact the accuracy of the elastic horizontal

and rotational predictions). Thus, ke for the skirt and

base soil reactions are as follows:

(2)

(3)

where G = shear modulus of soil, D = caisson diame-

ter, z = depth below ground level (see Figure 1).

When the soil reaction states reach the local yield

surface, the soil response becomes elastoplastic, with

incremental behaviour given by:

(4)

where kep = elastoplastic stiffness matrix. By conven-

tion, the local yield surface f(p) is defined as follows:

f < 0 for states inside the yield surface, f = 0 for states

on the yield surface, and f > 0 for inadmissible states

outside the yield surface.

When elastoplastic yielding occurs, permanent

plastic displacements accumulate with the total dis-

placement increment u composed of elastic and

plastic parts:

(5)

The elastic displacement increment ue is determined

by the soil reaction increment through:

(6)

The plastic displacement increment up is determined

by the flow rule:

(7)

where g(p) is a plastic potential function and λ is a

non-negative, scalar plastic multiplier. When yielding

occurs, the incremental soil reaction p must remain

on the local yield surface. This is enforced by the con-

sistency condition:

= 0 (8)

Following the conventional approach for linear elas-

tic-perfectly plastic models, kep is obtained from:

(9)

For this paper, an associated flow rule is assumed i.e.

g(p) = f(p). The local yield surface f(p) is calibrated

using the limiting soil reactions extracted from the

3DFE analyses, which is described in Section 2.3.

The 1D model was implemented numerically using

the Galerkin finite element methodology, where two-

noded 1D soil elements (each with a linear shape

function and two Gauss points) representing the skirt

soil reactions are tied to two-noded 1D caisson rigid

bar elements. The base soil reaction is represented by

a lumped model tied to the bottom node of the deepest

caisson element. The explicit Runge-Kutta (4, 5) al-

gorithm (Dormand and Prince, 1980) was used for the

integration process during elastoplastic behaviour

and the full Newton-Raphson procedure was used to

obtain the system solution.

2.2 3DFE Model

The 3DFE analyses were carried out using the finite

element program ABAQUS v6.13 (Dassault Sys-

tèmes 2010). The 3DFE model consists of a suction

caisson foundation (of unit diameter D and unit skirt

length L = D) embedded in homogeneous soil, which

is similar to that used in Suryasentana et al. (2017).

The mesh domain is set as 8D for the diameter and

6D for the depth, which was verified to be large

enough to avoid boundary effects for load capacity

predictions. Mesh convergence analyses were also

carried out to determine the required mesh fineness.

Due to symmetry of the problem, only half of the cais-

son and soil domain was modelled. A typical mesh of

the 3DFE model is shown in Figure 2.

The soil was defined as weightless, homogeneous

and linear elastic-perfectly plastic. The soil is as-

sumed to obey the von Mises yield criterion with an

associated flow rule. The Young’s modulus E of the

soil is set as 10003su (where su is the undrained

shear strength) and the Poisson’s ratio is set as 0.49.

Figure 2. Mesh used for the 3DFE analyses. The diameter and

depth of the mesh domain is set as 8D and 6D respectively.

Fully-integrated, linear, brick elements C3D8H

were used to model the soil elements. The caisson

was modelled as being entirely rigid using rigid body

constraints. The caisson reference point was set at RP,

as shown in Figure 1. Contact breaking between the

caisson and soil was prevented using tie constraints at

the caisson-soil interface. Displacements were fixed

in all directions at the bottom of the mesh domain and

in the radial directions at the periphery.

2.3 Calibration of local yield surfaces

To calibrate the local yield surfaces in the proposed

1D model, a series of 3DFE analyses were carried out

to obtain the limiting soil reactions. These analyses

involved the determination of the global VHM failure

envelope of the caisson-soil interaction problem

(Bransby and Yun, 2009; Gourvenec and Barnett,

2011; Vulpe, 2015). This was done using mixed load

and displacement control, where load control was

used in the V load space while displacement control

was used in the HM load space. In total, four vertical

loads (V/V0 = 0, 0.25, 0.5, 0.75 where V is the vertical

load applied at RP and V0 is the uniaxial vertical load

capacity) were applied before displacement probes

were applied in the HM load space. This determines

the HM failure envelopes at fixed levels of V. It is hy-

pothesized that, just as there exists a failure envelope

that limits the global load space, there also exists a

limiting envelope in the soil reactions space (termed

as ‘local yield surface’ in this paper), which can be

identified using the limiting soil reactions extracted

from the 3DFE analyses.

The limiting soil reactions were extracted from the

3DFE results at the end of each displacement probe,

corresponding to a global failure state of the caisson-

soil interaction problem. For simplicity, the limiting

skirt soil reactions are assumed to be constant along

the skirt and are computed as the average of the soil

reactions along the skirt.

To represent the local yield surface, an ellipsoid

function f(p) is adopted:

(10)

where v0, h0 and m0 are the limiting uniaxial vertical,

horizontal and moment soil reactions (i.e. the uniaxial

capacities in the soil reactions space) and α is a pa-

rameter that governs the rotation of the ellipsoid in

the hm space. This ellipsoid function was adopted as

it has favourable theoretical properties such as global

convexity.

The unknown parameters v0, h0, m0 and α were

identified by running least squares regression against

the limiting soil reactions extracted from the 3DFE

results. The best-fit parameters for the skirt and base

local yield surfaces are shown in Table 1.

Table 1. Best-fit parameters for the skirt and base lo-

cal yield surfaces

_________________________________________________

Parameter Skirt Base

_________________________________________________

v0/su Askirt 9.1Abase

h0/su 2.07Askirt 1.34Abase

m0/su 0.19AskirtD 0.72AbaseD

α -1.23 -0.47

_________________________________________________

where D is the caisson diameter, Askirt (skirt surface area per

metre length basis) = πD and Abase = πD2/4

The local yield surface contours generated by

Equation 17 and the best-fit parameters in Table 1 are

compared against the 3DFE limiting soil reactions in

Figure 3. Although the global vertical load V is fixed

while the HM failure envelope is probed, the distribu-

tion of the vertical load between the skirt and base soil

reactions is not constant. Thus, each of the limiting

soil reactions is associated with a different v/v0 value.

To simplify the process, the average of these v/v0 val-

ues (for each dataset corresponding to a fixed V) are

used in Equation 17 to predict the hm contours for

each V/V0; their values are shown in the contour labels

in Figure 3. For V/V0 = {0, 0.25, 0.5, 0.75}, v/v0 = {0,

0.1, 0.35, 0.7} for the skirt soil reactions and {0, 0.32,

0.57, 0.77} for the base soil reactions.

It was observed that the predicted local yield sur-

face contours are good approximations to the limiting

base soil reactions at low vertical loads. However, at

higher vertical loads (V/V0 ≥ 0.5), there is less agree-

ment as the ellipsoid function cannot capture the

change in yield surface geometry. The fit is less than

ideal for the limiting skirt soil reactions as they do not

conform closely to an ellipsoidal shape.

However, although these simplified local yield

surfaces do not match well on a local level, they pro-

duce reasonably accurate global predictions (as will

be shown in Section 3.2).

Figure 3. Comparison of the 3DFE limiting skirt and base soil

reactions (as depicted by the markers in the figure) against the

local yield surface contours (as depicted by the grey dotted lines)

predicted by Equation 17 and Table 1. The average v/v0 values

(for each dataset corresponding to a fixed V) are shown in the

contour labels and they are used in Equation 17 to produce the

contours.

2.4 Evaluation of Models

To compare the predictions between the 1D model

and the 3DFE model, three types of evaluations were

implemented. First, the uniaxial load capacities

(which are the load capacities under the application

of V, H and M individually) were evaluated to assess

the accuracy of the models for simple loading cases.

Next, the influence of combined loading on failure

states was assessed by using the 1D and 3DFE models

to find the failure envelopes of the caisson in the VHM

load space. Finally, a single cyclic load test was sim-

ulated using both models to assess the capability of

capturing permanent displacement and hysteresis.

3 RESULTS

3.1 Uniaxial Load Capacities

Table 2 shows the uniaxial vertical V0, horizonal H0

and moment load capacities M0 predicted by the 1D

and 3DFE models. It is evident that the 1D model pre-

dictions agree very well with the 3DFE model predic-

tions, with the largest difference being only 1.51% for

the horizontal load capacity H0.

Table 2. Comparison of the uniaxial global load ca-

pacities predicted by the 1D and 3DFE models

_________________________________________________

Capacity 1D 3DFE Diff (%)

_________________________________________________

V0/Abasesu 13.12 13.12 0.00

H0/Abasesu 6.01 5.93 1.51

M0/AbaseDsu 3.7 3.7 -0.07

_________________________________________________

where D is the caisson diameter and Abase = πD2/4

Figure 4 compares the global load-displacement

predictions under uniaxial loading, where wRP, uRP

and ϴRP are the vertical, horizontal and rotational dis-

placements of the loading reference point RP. As ob-

served, the 1D and 3DFE model predictions tend to

reach the load capacity at different displacements.

Under pure vertical loading, the 1D model reaches

load capacity at a smaller displacement than the

3DFE model. Moreover, it can be seen from the close-

up inset that the 1D model load-displacement predic-

tion is bilinear. The first linear response is the elastic

response while the second linear response occurs

when the base soil reaction has reached its local yield

surface but the skirt soil reaction remains elastic.

Under pure horizontal loading, the 1D model pre-

dicts uRP/D of 0.1 and ϴRP of 0.129, which compares

well with the 3DFE model predictions of uRP/D of 0.1

and ϴRP of 0.139. Similarly, under pure moment load-

ing, the 1D model predicts ϴRP of 0.1 and uRP/D of

0.05, which compares well with the 3DFE model pre-

dictions of ϴRP of 0.1 and uRP/D of 0.0556.

Under pure horizontal or moment loading, the 1D

model load-displacement predictions are not bilinear

as both horizontal h and rotational m soil reactions

occur during these loadings. The influence of com-

bined h and m loading forces the soil reaction path to

track on the local yield surface during elastoplastic

yielding, until the global load capacity is reached.

Figure 4. Comparison of global load-displacement predictions.

The close-up insets focus on the results at small displacements.

3.2 Failure Envelopes

Figure 5 compares the predictions of the VH and VM

failure envelopes of the caisson in normalised forms,

where the loads are normalised by their respective

uniaxial capacities. The 1D model predictions of the

VH and VM failure envelopes match the 3DFE results

very well, albeit with a slight overprediction for the

VH failure envelope for some load cases.

Figure 5. Comparison of global failure envelope predictions in

the VH and VM load space.

Next, Figure 6 compares the predictions of the

VHM failure envelopes in normalised forms. Despite

the poor match of the local yield surfaces at the local

level (see Figure 3), the 1D model predictions of the

global HM envelope under fixed V loads match the

3DFE predictions reasonably well.

Figure 6. Comparison of global failure envelope predictions in

the VHM load space.

Nevertheless, given the mismatch (especially that of

the skirt local yield surface) in Figure 3, it is encour-

aging to see that the global failure envelope predic-

tions are not too sensitive to the accuracy of these lo-

cal yield surfaces. Furthermore, most loading

scenarios are in the quadrants where H and M have

the same sign. Thus, the mismatch in the quadrants

where H and M have different signs are of less prac-

tical concern.

3.3 Cyclic Loading

To assess whether the 1D model can simulate hyster-

etic behavior, a single cycle of positive and negative

vertical displacements (wRP/D = ±0.05) was pre-

scribed onto the caisson. Figure 7 shows the compar-

ison of the global load-displacement behavior under

this cyclic loading. It is clear that the 1D model is able

to simulate hysteresis, although the 1D model predic-

tions is a rather crude piecewise linear approximation

of the 3DFE model predictions.

By comparing the displacements at zero V load, it

is evident that the permanent displacement predic-

tions of the 1D and 3DFE models are in good agree-

ment. The 1D model comes with a built-in capability

for simulating effects such as permanent displace-

ment and hysteresis. This is not surprising as both the

1D and 3DFE models are based on fundamentally the

same elastoplasticity concepts, but with different

measures of ‘stress’ and ‘strain’.

Figure 7. Comparison of global load-displacement behavior un-

der cyclic vertical loading

4 DISCUSSION

The 1D model ignores much of the detail of the orig-

inal 3D continuum-based problem, with the aim of

appropriate simplification to provide a fast proxy to

the original problem. Despite the simplifying abstrac-

tions, the loss in accuracy is minimal, relative to the

large gains in computational efficiency. For example,

the 3DFE model took about 28 hours in total to run

the analyses presented in Section 3. By contrast, the

1D model took about 0.8 hours in total, yielding a

time saving of 97%.

This computational efficiency is very important

for design optimization involving multiple founda-

tions, such as that for an offshore wind farm. Whilst

3DFE is perhaps practical for design projects involv-

ing only a few foundations, it is clearly impractical

when there are hundreds of foundations. A tool such

as the trained 1D model offers the 3DFE accuracy but

with much higher efficiency, and therefore allows

more of the design space to be explored.

Furthermore, the proposed 1D model offers ad-

vantages over existing macro-element models for

shallow foundations (e.g. Cassidy 2004, Salciarini et

al. 2011). Given the localised nature of the soil reac-

tions and the yield surfaces, the 1D model may be

simply adapted to non-homogeneous or multi-layered

grounds with arbitrary yield strength profiles. This

contrasts with macro-element models, which can only

be adapted to ground profiles similar to that in the

original calibration. In other words, the 1D model is

a more generalised model by comparison with the

macro-element model.

The focus of this paper is a presentation of the map-

ping process from a 3DFE elastoplastic continuum

model to a 1D elastoplastic Winkler model, and a

demonstration of the accuracy of the approach. As

such, generalised formulations of the yield surfaces,

although established for caissons of L/D ≤ 2, are not

presented. They will be described in future publica-

tions.

There are, of course, some observed limitations

with the 1D model. For example, the load predictions

under purely vertical loading is bilinear. This could

be resolved by adding multiple or nested local yield

surfaces but this increase in accuracy comes at the ex-

pense of increased computational effort. Also, there

is room for improvement for the global failure enve-

lopes predicted by the 1D model and this can be

achieved by adopting a more expressive function with

more parameters to represent the local yield surface.

However, while there are ready solutions to these lim-

itations, it is advisable to consider whether the addi-

tional complexity balances the aim of providing a

rapid but approximate solution to the caisson-soil in-

teraction problem for preliminary designs, which can

then be refined using more advanced 3DFE analyses.

5 CONCLUSION

The main concern with Winkler models that use non-

linear elastic soil reactions to approximate the soil

continuum response is that they do not easily repro-

duce observed phenomena such as permanent dis-

placement, hysteresis and influence of combined

loading on failure states. This paper resolves this

shortcoming by proposing a 1D Winkler model that

couple linear elastic soil reactions with local plastic

yield surfaces that limits the allowable soil reaction

states. The results indicate that the proposed 1D

model compares favourably with the 3DFE model

predictions in terms of accuracy across a range of

loading states. The principal advantage, however, is

efficiency, as it takes only 3% of the computational

time required by the 3DFE model. Furthermore, un-

like macro-element models which can only be used

for ground profiles that are similar to the original cal-

ibration, the 1D model can be used for non-homoge-

neous or multi-layered grounds with arbitrary yield

strength profiles, making it a more general, and argu-

ably, useful model. Thus, the proposed 1D model of-

fers an efficient method to predict realistic, non-linear

behavior of caissons in elastoplastic soil.

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