DEVELOPMENT AND APPLICATION OF FORCE RESULTANT MODELS DESCRIBING

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					   DEVELOPMENT AND APPLICATION OF FORCE RESULTANT MODELS DESCRIBING
                    JACK-UP FOUNDATION BEHAVIOUR



                                                  M.J. Cassidy
                    Centre for Offshore Foundation Systems, the University of Western Australia

                                        C.M. Martin and G.T. Houlsby
                               Department of Engineering Science, Oxford University



ABSTRACT

As jack-ups have moved into deeper and harsher waters there has been an increased need to understand jack-up
behaviour and develop analysis techniques. One of the areas of significant development has been the modelling of
spudcan footing performance, where the load-displacement behaviour of the footings is required to be included in
any overall numerical model. Because they can be incorporated into conventional structural analysis programs, force
resultant models based on strain-hardening plasticity theory are appropriate replacements for the unrealistic
assumptions of pinned or linear spring footings. The development of these models for the analysis of spudcans on
both clay and sand is reviewed here. A formulation for a six-degrees-of-freedom model that describes the load-
displacement behaviour in the vertical, moment, horizontal, and torsion directions is also detailed. Using this model
any load or deformation path can be applied to the footing and the corresponding unknowns (deformations or loads)
calculated. This formulation allows the model to be implemented into three-dimensional structural analysis
programs, and examples of this are given. Some future challenges in this area are addressed, including the
development of models that account for cyclic loading behaviour.


KEY WORDS

Jack-up; spudcan foundations; footings/foundations; plasticity; model tests; soil-structure interaction; sand; clay


INTRODUCTION

Jack-ups play a vital role in the offshore industry, with proven flexibility and cost-effectiveness in field development
and operation. With a steadily increasing demand for their use in deeper waters and in harsher environments and
also a desire for longer commitment of a jack-up at a single location, especially in the role of a production unit, there
is an increased need to understand jack-up behaviour and to develop analysis techniques. The publication of the
‘Guidelines for the Site Specific Assessment of Mobile Jack-Up Units’ [1][2] and more recently the drafting of
ISO19905 ‘Site Specific Assessment of Mobile Offshore Units (MOUS)’ are examples of the offshore industry’s
desire both to standardise and to develop jack-up assessment procedures. One of the areas of significant
development has been the modelling of spudcan footing behaviour.

During installation, and in a perfectly calm sea, the vertical self-weight of a jack-up is the dominant loading on its
spudcan footings. During a storm, however, environmental wind and wave forces impose additional horizontal
loads, overturning moments and even torsion loads on the foundations, as well as altering the sharing of vertical load
among the footings. This combined loading of a footing is shown in Figure 1, and it results in a complex state of
stress and strain in the underlying soil. An understanding of spudcan performance under these conditions is essential
to the analysis of jack-up response, yet it is rarely possible (or indeed appropriate, unless a full geotechnical site
investigation has been undertaken) to conduct a finite element analysis in which the seabed soil is modelled in detail
using continuum elements. A more practicable option is to incorporate the foundation as a ‘macro element’
expressed purely in terms of the loads (force resultants) on the footing and the corresponding displacements. This
approach is directly analogous to the use of force resultants (axial force, bending moment and shear force) and nodal
displacements and rotations in the analysis of beams and columns. It has the major advantage that spudcans can be
incorporated directly into the structural analysis of a jack-up as ‘point’ elements at the bottom of each leg, without
any need for special transition or interface elements between the structure and the soil.

Under combined loading (Figure 1), the simplest approach is to model the spudcan with a pin joint or a set of linear
springs (the former being a special case of the latter). It is generally acknowledged that the pinned footing
assumption is conservative, because the inclusion of any rotational spudcan fixity – whether elastic or plastic – will
tend to reduce the critical member stresses at the leg/hull connection, and other responses such as the lateral hull
deflection. Note however that under dynamic loading, it has been shown that this is not always the case [3][4].
Historically, the focus has been on rotational spudcan behaviour because of its ‘obvious’ influence on bending
moments in the jack-up structure, yet accurate modelling of the other degrees of freedom is also essential,
particularly when assessing the ultimate capacity of a jack-up under extreme environmental loading [5][6][7]. Rig
failures involving, for example, shallow sliding of a windward spudcan and/or ‘plunging’ of a leeward spudcan
cannot be simulated in a realistic manner when the foundations are modelled as pin joints or linear springs.

During extreme loading events it is essential that any ‘general’ numerical spudcan model (force resultant or
otherwise) be able to account for nonlinear behaviour. One approach is to use a set of coupled nonlinear springs with
load-displacement curves described by algebraic expressions, perhaps obtained from curve-fitting to experimental
data or finite element results. Models of this type have been developed and used extensively (see e.g. [8][9][1]) but a
detailed review would not be appropriate here. The second major class of nonlinear spudcan models is based on
classical plasticity theory, utilizing concepts that are familiar from metal plasticity: elastic domain, yield surface,
flow rule and hardening law. This type of model has the major advantage that the complex interactions between the
various degrees of freedom are handled ‘automatically’ as part of the modeling framework. Although several
different models will be described, they all have a common theoretical basis that will be familiar to both Civil and
Mechanical Engineers.

In a plasticity-based numerical model of a spudcan, the macroscopic load-displacement behaviour is determined in
essentially the same way that a constitutive law for a metal (or a soil) relates stresses and strains. Loading is applied
incrementally, and the numerical plasticity model computes updated tangent stiffnesses for each step. The hardening
concept adopted is that at any given plastic penetration of the foundation into the soil, a yield surface of a certain
size is established in combined loading space. The size of the yield surface increases as the footing is pushed further
into the soil, though it has been found (in both experimental and numerical work) that its shape remains more or less
constant, as shown in Figure 2. The ‘backbone’ curve of vertical bearing capacity against plastic vertical penetration
can be determined either theoretically or empirically. As in standard plasticity theory, changes of load within the
current yield surface result only in elastic deformation. A loading path that intersects (and remains on) the yield
surface also gives rise to plastic deformation, with the components of incremental plastic displacement being
determined from the flow rule and hardening law.

The model just described is one of single-surface strain-hardening plasticity (the term strain-hardening is used
because the size of the yield surface is linked to displacement, which is analogous to strain in a constitutive model).
This paper reviews the development of strain-hardening plasticity models for use in the analysis of jack-ups, and
also outlines a six-degrees-of-freedom model. Future developments of multi-surface (and even infinite-surface)
plasticity models will be also be discussed.


DEVELOPMENT OF PLASTCITY MODELS FOR SPUDCANS

The use of interaction diagrams in solving soil-structure interaction problems was pioneered by Roscoe and
Schofield [10] in 1956, when they developed a method for calculating the fully plastic moment resistance of a short
pier foundation for a steel framework. It took two decades before Butterfield and co-workers further developed the
concept. In a lecture in December 1978, Butterfield pondered the idea of using interaction diagrams in load space,
without the numerous and cumbersome bearing capacity factors of traditional bearing capacity methods. The
invitation to this lecture is provided in Figure 3. Also shown are some of Butterfield’s notes highlighting the idea of
relating the shape of the interaction diagram and the (plastic) movements of the footing. Though these concepts
started to be used in other applications (see e.g. Butterfield and Ticof [11]), Schotman [12] was first to describe a
complete incremental plasticity model for a spudcan foundation in terms of force resultants. The model was framed
in planar (V:M:H) load space, though it still relied heavily on numerous assumptions. For instance, the yield surface
and hardening law were derived from Brinch Hansen’s semi-empirical bearing capacity formula, and the elasticity
constants and plastic potential were calibrated using finite element analyses of a plane strain ‘spudcan’. However,
Schotman did succeed in incorporating his spudcan model into a (linear elastic) jack-up structural analysis, and
some useful insights into the behaviour of the overall soil-structure system were obtained.

Since Schotman’s proposed model, there have been numerous experimental investigations aimed at providing the
data necessary to calibrate plasticity-based force resultant models of spudcans and other shallow foundations. Much
of this research has concentrated on silica sands [13][14][15][16][17][18][19][20] though there has been some work
involving soft clay [5][21][22]. However, the approach has also been investigated for shallow footings on
uncemented loose carbonate sand [23].
Swipe Tests Investigating Yield Surface Shape

Most of the above tests have concentrated on establishing the yield surface shape in combined loading space, and
‘swipe tests’ are an efficient means of investigation. In a swipe test, the footing is penetrated vertically to a
prescribed level, then subjected to a radial displacement excursion (radial here meaning horizontal, rotational or
torsional displacement, or a combination thereof). The load path followed can be assumed to be a track across the
yield surface appropriate to that penetration. This is shown in Figure 4 for the planar loading case. Tan [13] argued
in favour of this assumption when he made his detailed investigation of the (V:H) yield loci for various conical and
spudcan footings on saturated sand, though by using a ‘fixed’ loading arm only one track in combined load space
was investigated (also the case for Murff et al. [15]; Dean et al. [16]). To explore the full three-dimensional nature
of the planar-loading yield surface it is necessary to follow tracks along the surface at different M:H ratios. By using
the loading apparatus shown in Figure 5, a thorough investigation of the yield surface was achieved by Martin [5]
for spudcan footings on overconsolidated clay and by Gottardi et al. [19] and Byrne and Houlsby [23] for flat
circular footings on dense silica and loose carbonate sands respectively. These experiments were unique in that they
allowed fully automated and independent control in the vertical, horizontal and rotational directions. This paper
concentrates on the combined loading experiments performed using this apparatus at the University of Oxford.

In the testing of Martin [5], 75 individual swipe tracks were used to establish the yield surface shape for a spudcan
subjected to combined loading. As an example, load paths followed in V:M2/2R and V:H3 load space for a footing
with rotation and horizontal displacement applied simultaneously (and in a constant ratio) are shown in Figure 6. It
was observed in these tests that, while the moment and horizontal loads increase (and then slightly decrease towards
the end of the test) the vertical load continuously decreases, rapidly at first but slowing as a ‘critical state’ is
approached, the force point remaining almost stationary in V:M2/2R:H3 load space despite continued horizontal and
rotational displacement of the footing.

Also shown in Figure 4 is the technique, pioneered by Tan [13], for investigating the yield surface at low vertical
loads. The footing is driven to a particular vertical penetration, then unloaded to a low stress level before locking the
vertical displacement and making a swipe. An example of this is shown in Figure 7, where a 100mm diameter flat
footing has been loaded to 1600N on dense silica sand and then unloaded to 200N before being rotated at constant
vertical displacement. It is clear from test GG08 in Figure 7(b) that the moment load increases at an almost constant
vertical load (indicating elastic behaviour) until the yield surface is reached. The load path then meets up with the
same parabola as a swipe test directly from 1600N (GG04).

It has been found that, for shallow foundations, the size of the yield surface is primarily a function of the plastic
component of vertical displacement (see for example Martin and Houlsby [21][22]; Gottardi et al. [19]; Watson
[24]; Byrne [20]; Zhang [25]). The determination of appropriate vertical load-plastic penetration curves is discussed
in the numerical modelling section below.

Constant V and Radial Displacement Tests Investigating Load-Displacement Behaviour at Yield

With the yield surface shape established, an understanding of the load-displacement behaviour both before and
during yielding is required. For loads within the yield surface, an elastic stiffness matrix has usually been
determined through numerical finite methods [26][27][28]. For elastic-plastic load steps, experiments involving the
measurement of incremental plastic displacements at yield have been used to define a suitable (empirical) flow rule.
Typically constant V tests are used where, rather than holding the vertical penetration constant, the vertical load is
kept fixed while the footing is driven horizontally and/or rotated [5]. Radial displacement tests have also been used,
where various combinations of vertical, horizontal and rotational displacements are applied to the footing in a fixed
ratio [19][23]. Both sets of tests provide information on the load-displacement relationship (flow rule) at yield, as
well as secondary information about the expansion or contraction of the yield surface (which should be in
accordance with the strain-hardening relationship).

For both clay and sand, associated flow (normality to the yield surface) was observed in the moment-horizontal load
plane, but non-association in the vertical-horizontal and vertical-moment planes [21][22][29]. A discussion of
appropriate numerical formulations for capturing this non-association is given in the following section.


DETAILS OF THE STRAIN-HARDENING PLASTICITY MODEL

Having introduced the broad framework of the plasticity-based spudcan model, and having outlined some of the
experimental testing used to calibrate its features for various soil types, an overview of the numerical model for
planar loading (three degrees of freedom) will now be given. This represents a special case of the full six-degree of
freedom model (for M3, H2 and Q = 0), and an extension to the general case will be discussed. The sign convention
for both the planar case ( V , M 2 , H 3 ) and the six-degrees of freedom case ( V , H 2 , H 3 , Q , M 2 , M 3 ) are shown in
Figure 1. Further details of the development of the numerical models can be found, for clay, in Martin [5] and
Martin and Houlsby [22], and for sands, in Cassidy [30], Houlsby and Cassidy [31] and Cassidy et al. [29].

The model has four major components:

Yield Surface

For planar loading (V:M2/2R:H3) a yield surface of a similar form can be defined for both the clay and sand cases:

             H   M 2R  2aH3 M2 2R  (β1 + β2 )(β1+β2 ) 
                       2                    2                                    2          2β1        2β2
                                                                                     V           V
        f =  3  + 2
             h V   m V  − h m V 2 −  β β1β β2 
                                                                                     
                                                                                     V          1− 
                                                                                                   V       =0                             (1)
             0 0  0 0      0 0 0     1 2
                                                          
                                                                                     0            0



where V0 determines the size of the yield surface and indicates the bearing capacity of the foundation under purely
vertical loading (H3 = 0 and M 2 2 R = 0 ). Furthermore, V0 is governed by the vertical plastic penetration and is
determined from the strain-hardening law (Figure 2). The dimensions of the yield surface in the horizontal and
moment directions are determined by h0 and m0 respectively and a accounts for eccentricity (rotation of the
elliptical cross-section) in the M2/2R:H3 plane. The parameters β1 and β 2 round off the points of surface near
V V0 = 0 and V V0 = 1 . The yield surface shape for the clay case is shown in Figure 8. Appropriate yield surface
parameters are as follows, in which it should be noted that the parameters defining the shape of the surface do not
vary greatly for the different soil types:

                                          Clay                         Dense Silica Sand                          Loose Carbonate Sand
                           (Martin and Houlsby [22])                 (Houlsby and Cassidy [31])                     (Cassidy et al. [29])
          h0                              0.127                                 0.116                                      0.154
          m0                              0.083                                 0.086                                      0.094
          β1                              0.764                                  0.9                                       0.82
          β2                              0.882                                 0.99                                       0.82

In clay the eccentricity varies slightly with V V0 and takes the form

                     V  V 
        a = e1 + e2   − 1
                     V  V                                                                                                               (2)
                     0  0 

with e1 = 0.518 and e2 = 1.180 recommended by Martin and Houlsby [22]. For flat footings on sand the ellipse is
rotated in the other direction in M2/2R:H3 space, with a = −0.2 and a = −0.25 found to fit the experimental data for
dense silica sand and loose carbonate sand respectively. This fit of the experimental data by Equation 1 is shown in
Figures 10 and 11 for the M2/2R:H3 load plane, and a combined radial versus vertical load plane respectively.

For general loading Equation 1 can be extended to

                   2                  2                          2          2                                2
             H   M 2R  2aH3 M2 2R  H2   M3 2R  2aH2 M3 2R  Q 2R 
             h V   m V  − h m V 2 + h V  + m V  + h m V 2 + q V 
        f =  3  + 2                                             
             0 0  0 0      0 0 0    0 0  0 0       0 0 0    0 0 


          (β + β )(β1+β2 ) 
                                2          2β1        2β2
                                    V           V
        − 1 β 2 β                  
                                    V          1− 
                                                  V       =0                                                                              (3)
          β1 1β2 2 
                                   0            0



This extension of Equation 1 can be deduced because, due to symmetry, there can be no cross product terms of
H 2 H 3 , M 2 M 3 , H 2 M 2 , M 3 H 3 , nor any involving the torque Q . The size of the yield surface in the torsional
direction is determined by q0 and Figure 9 shows a typical shape of a normalised yield surface for purely vertical
and torsion loading based on the experimental data for a flat plate on sand. Further details of this vertical-torsion
behaviour and yield surface size can be found in Cheong [32].
Strain Hardening

The capacity of a spudcan usually increases when it is pushed further into the ground. The size of the yield surface is
not therefore fixed, but can be considered to increase with further plastic penetration. Though some experimental
evidence exists for linking the size of the yield surface not just with plastic vertical displacement, but with other
plastic displacement components as well [23][29], it is usual to assume that the size of the yield surface is defined
solely in terms of the pure vertical load capacity. The variation of V0 with plastic vertical displacement wp defines a
hardening law and the size of the yield surface and can be determined either by constructing curves based on bearing
capacity theory [5][21][22], experimental evidence (or prototype pre-loading) [31][29] or by a combination of both
[30][33].

An example of the latter is the hardening law describing the relationship between vertical load and plastic
penetration for a spudcan on sand:

                                                   2
                              kw         w 
                    (      )
                     1− fp  p  + f p p 
                             V          w 
                              0m         pm                                                              (4)
       V0 =                                                V
                                                          2 0m
                              kw pm  w p    w p 
            (       )
            1 − f p 1 −  2 −
                     
                                            +      
                                V0 m  w pm    w pm 
                                                 

Here k is the initial plastic stiffness, f p a dimensionless constant that describes the limiting magnitude of vertical
load, V0 m is the peak value of V0 , and w pm the value of plastic vertical penetration at this peak. A fit to
experimental data on dense silica data is shown in Figure 12. However, Cassidy and Houlsby [34] provide bearing
capacity values for V0 m for 360 combinations of cone angle, roughness and friction angle and Cassidy and Houlsby
[33] detail a method for calculating the response for geometric changes of the spudcan during partial penetrations.
As an example the values of N γ for a conical footing of apex angle 150° are shown in Figure 13. Martin [5] and
Houlsby and Martin [35] have tabulated bearing capacity factors for conical footings on clay with a linearly
increasing undrained strength profile; these factors are also included in the ‘Guidelines for the Site Specific
Assessment of Mobile Jack-Up Units’ [1].

Elastic Behaviour

The elastic response of the soil needs to be defined for any load increments within the yield surface. Bell [26] and
Ngo-Tran [27] showed using finite element methods that cross coupling exists between the horizontal and rotational
footing displacements. For three degrees of freedom the elastic behaviour can be expressed as

        dV              k1      0      0  dw e 
                                                      
                                          k 4  2 Rdθ2 
                                                        e
        dM 2 2 R  = 2GR  0      k2                                                                       (5)
        dH                              k 3  du3 
                                                      e
             3          0
                                  k4                   

where G is a representative shear modulus and k1 , k 2 , k 3 and k 4 are dimensionless stiffness factors. This is easily
extended into six-degrees of freedom with values of the additional torsion constant ( k5 ) given by Doherty and
Deeks [28] or Poulos and Davis [36]:

        dV            k1         0        0     0    0      0   dw e 
                      0                                                     
        dH 2                      k3       0     0    0     − k 4   du2 e 
                                                                   
        dH            0          0       k3    0     k4     0   du3 e 
            3
                 = 2GR                                                                                  (6)
        dQ 2 R        0          0       0     k5    0      0   2 Rdωe 
        dM 2 R        0          0       k4    0     k2     0  2 Rdθ2 e 
          2                                                                
        dM 2 R        0         − k4     0     0     0      k 2   2 Rdθ3 e 
          3                                                                

The shear modulus (G) linearly scales all of the stiffness coefficients in Equations 5 and 6 and for clay can be
determined by
       G = I r su                                                                                           (7)

where su is the undrained shear strength measured at 0.15 diameters below the reference point of the spudcan (taken
at the level at which the maximum diameter is reached). Ir is the rigidity index and can be calculated from:

              G   600
       Ir =     =                                                                                           (8)
              su OCR 0.25

In sands the shear modulus can be estimated by

                              0.5
        G      V         
           = g
               Ap        
                                                                                                           (9)
        pa     a         

where V is the spudcan vertical load, A the spudcan area and pa atmospheric pressure. The recommended value for
the dimensionless constant g for a relative density DR is

                    D 
       g = 230 0.9 + R                                                                                   (10)
                    500 

Equations 8 and 10 are based on results of the back-analysis of eight case records of jack-ups in the North Sea [37].
When used with the dimensionless stiffness factors of Equations 5 and 6, they represent higher stiffness levels than
are currently suggested in the ‘Guidelines for the Site Specific Assessment of Mobile Jack-Up Units’ [1]. Such
higher stiffness factors had been expected to be appropriate by some practitioners, but the case records provided a
firmer basis than had hitherto been available. However, as stated in Cassidy et al. [37] these case records were all
for relatively mild environmental conditions (significant wave heights between 4.1 and 9.85m) and a study using
harsher environmental conditions would be most valuable.

Flow Rule

When the load state touches and expands (or possibly contracts) the yield surface, plastic displacements occur.
Though the stiffness of the response is determined by this expansion (or contraction) through the hardening law, the
ratios of the plastic displacements are determined by the flow rule. The simplest form is associated flow, where the
yield surface also acts as the plastic potential, and the ratios of the plastic displacement components are determined
by

                    ∂f              ∂f               ∂d
       dw p = λ        , du3 p = λ      , dθ2 p = λ                                                        (11)
                    ∂V             ∂H 3             ∂M 2

where λ is a non-negative multiplier that can be determined from the requirement that an elastic-plastic load step
must remain on the yield surface.

Both the clay and sand experiments showed associated flow only in the M 2 2 R : H 3 plane, as indicated on Figure 8
for the clay experiments. To account for this non-association Martin [5] used a simple empirical modification of the
vertical component of Eqn 11:

                     ∂f
       dw p = ζλ                                                                                           (12)
                     ∂V

where a value of ζ = 0.6 is realistic. The situation in sands is somewhat different, with an associated flow rule
underpredicting the magnitude of vertical displacements at all load levels. Cassidy [30] introduced a plastic potential
function to allow accurate prediction of the plastic displacements. Further details of the expression can be found in
Cassidy [30], Houlsby and Cassidy [31] and Cassidy et al. [29].
VERIFICATION OF NUMERICAL MODEL

With the model outlined, the first stage is to ensure the model can retrospectively simulate the experiments used in
its calibration. The ability of the models for clay and sand to achieve this for a wide range of tests was established
by Martin and Houlsby [22] and Houlsby and Cassidy [31] respectively, and some example cases are discussed
below. In these simulations the values of the three control quantities (e.g. the displacements) used in the experiments
were taken as input, and the other three quantities (e.g. the loads) were calculated as output for comparison with the
measured experimental data. No idealisation of the experimental input data was carried out, so that the input values
contain all the minor fluctuations associated with experimental measurements.

Example of Simulation of Swipe Tests (for clay)

Figure 14 shows the observed and predicted load paths for the swipe test previously illustrated in Figure 6. The
simulation, like the test, involved simultaneous rotational and horizontal displacements, and in general there is good
agreement between the numerical simulation and the experimental results. However, this is fundamentally a test of
the yield surface shape and in the following example of a constant V test on sand a more thorough investigation of
all of the components is undertaken.

Example of Simulation of Constant V tests (for dense sand)

In the test shown in Figure 15, the numerical analysis was loaded to V ≈ 1600 N, before the vertical load was held
constant at around that value (with slight fluctuations according to the experimental data), whilst pure rotational
control models the subsequent loading. The constant V tests involve the expansion and then later contraction of the
yield surface and Figure 15 shows that the numerical formulation models expanding yield surfaces reasonably well,
reaching a similar peak for moment load as the experimental values. Once the peak value has been reached, the yield
surface then contracts back, as predicted by the post-peak performance of the hardening law. Figure 15 shows that
for the predominantly moment case this post-peak performance is adequately modelled, although the experimental
data did not continue until M/2R = 0. Further, Figure 15 shows that the flow rule satisfactorily predicts the vertical
displacements when compared with the rotational displacements, which were part of the input.


INCORPORATION INTO STRUCTURAL ANALYSIS PROGRAMS

Example Three-Dimensional Quasi-Static Analysis

The advantage of models based in a strain-hardening plasticity framework is that they can easily be implemented
within a conventional structural analysis program. To show this for the six-degree of freedom plasticity model
detailed in this paper an example analysis of a jack-up unit is outlined. The model has been implemented as a user
element in the commercially available finite element program ABAQUS [38]. It can be used as a one element
“macro model” to define the load-displacement behaviour of each spudcan foundation. The library of ABAQUS
structural elements has been used to model the three-dimensional jack-up structure with the user element attached to
the bottom node of each leg. The example jack-up structure and properties are shown in Figure 16. The truss legs are
assumed as equivalent sections and ABAQUS beam elements used to model them. The hull is simplified as an
equilateral triangle of beam elements interlinked by three inner beam members, as illustrated in Figure 17. Though
non-linearities in the leg/hull jack houses are recognised as significant [39][40], no attempt was made to include
these effects in this example, and a rigid leg/hull connection was assumed. For all of the analyses described here a
mean water level of 90m was assumed, and the rig size typical of a three-legged jack-up used in harsh North Sea
conditions.

The spudcan was assumed to have a shape as shown in Figure 18. A preload of 133MN per spudcan was applied
representing a multiple of 1.65 on the jack-up’s self-weight. Assuming sand with a friction angle of φ = 45o a
vertical plastic penetration of 3.0m (from the spudcan tip) was evaluated according to the bearing capacity
formulation outlined in Cassidy and Houlsby [33][34]. The example jack-up has been analysed in a static analysis
loaded at 35° from the “plane-frame” direction (Figure 17). The analysis consisted of the jack-up being loaded
vertically to the preload level, unloaded to its self-weight and then the environmental loads applied. A simplification
of the ocean environment has been used with uniform wave and current loading assumed to occur up to the mean
water level (90m) and wind loading applied at the top of the legs.

Figure 19 shows the reactions for all of the footings, with the spudcans underneath the respective corner of the hull
labeled S1, S2 and S3 in Figure 17. With preloading of the jack-up the initial yield surface size has been expanded
to V0 = 133 MN and all of the results in Figure 19 have been normalised by this initial V0 value. As the self-weight
of the rig is only 80.6 MN per spudcan, the environmental loads are applied at V / V0 ≈ 0.605 . This is shown in
Figure 19, where the distribution of forces between the legs and spudcans can also be observed.

After the preloading phase the environmental loads have been increased until failure of the footings occurred. In this
analysis the most leeward spudcan (S2) yielded first (observed by the change in slope and then non-linear
behaviour). As both other footings begin to yield, the horizontal load is increasingly carried by these spudcans and
the vertical load shedding from the windward footing is at a faster rate than for elastic behaviour. However, at the
same time the most windward (S1) footings moment carrying capacity is significantly reduced. By the end of the
analysis it is only carrying 65% of its peak. This shows the impact of the reduction in foundation fixity due to the
plasticity model. However, in the leeward footings the formulation is allowing an increase in moment carrying
capacity with increased yielding. This is due to considerable expansion of the yield surface. Significantly Figure 19
does show that the jack-up can sustain considerable load after the footings have initially yielded.

Example Dynamic Analysis with NewWave Loading

Conventionally, jack-up assessments have used the same quasi-static analysis methods employed for fixed
structures. However, the need to consider dynamic effects has long been acknowledged [41][42][43]. With use in
deeper water, the contribution of dynamic effects to the total response has become more important as the natural
period of the jack-up approaches the peak wave periods in the sea-state. The following example shows the
importance of the foundation model in any dynamic analysis with the plasticity model for sand compared with
conventional pinned and linear spring assumptions.

In this example the environmental wave loading is included using NewWave theory, a deterministic method
described by Tromans et al. [44] that accounts for the spectral composition of the sea, and can be used as an
alternative to both regular waves and full random time domain simulations of lengthy time periods. For this case
only in-plane loading is considered and Figure 20 shows the surface elevation of a NewWave in the time domain for
both the upwave and downwave legs. The wave is focused on the upwave leg at the reference time ( t = 0 s). The
sea-state can be described by the Pierson Moskowitz wave energy spectrum, with a significant wave height ( H s ) of
12m and a mean zero crossing period ( Tz ) of 10s.

The corresponding horizontal deck displacements due to this NewWave are shown in Figure 21 for three foundation
cases: pinned, plasticity model and linear springs. Pinned footings represent infinite horizontal and vertical stiffness,
but no rotational stiffness and the plasticity model as outlined in this paper. The linear springs uses finite stiffness
values as in the elastic region of the plasticity model (as in Eqn 4). After the NewWave passes, the rig can be seen to
be vibrating in its natural mode. With increased rotational fixity the natural periods decrease, with approximate
values of 9, 5, and 5 seconds for the pinned, plasticity model and linear springs respectively. In this example the
load combinations were contained entirely within the yield surface, thus giving a response identical to the linear
spring case. By increasing the NewWave crest amplitude to α = 15m or α = 18m, as shown in Figure 22, the
increased loading caused plastic displacements in the footings simulated with the plasticity model, shifting the entire
foundations and leaving a permanent offset in the displacement of the deck. This yielding of the footings occurred
during the peak of the NewWave. This direct indication of yielding is a major benefit in using elasto-plastic
formulations for the spud-can footings. The natural period after this event may also be modified by the plastic
behaviour


OTHER EXPERIMENTAL INVESTIGATIONS

Three-legged Jack-Up Pushover Experiments

With the models based on single footing experiments, investigation of their predictive capabilities of a three-legged
jack-up system is being undertaken at the University of Western Australia. A series of experiments conducted on a
1:250 scale jack-up has been performed on overconsolidated clay. The experimental set-up is shown in Figures 23
and 24 (see Vlahos et al. [7] for more details). By using a combination of weights and a horizontal actuator, the
model rig is subjected to a combination of vertical and horizontal loads at the hull level (simulating first a pre-
loading event and then a environmental loading situation). The resulting vertical, moment and lateral loads on all of
the individual legs, as well as the hull and footing displacements are measured. The aim of the experiments is to
investigate the load-displacement behaviour of the jack-up when subjected to monotonic and cyclic pushovers.
Typical experimental results are given in Vlahos et al. [7] and the ability of the force-resultant model outlined in this
paper to numerically simulate the experiments is also under investigation.
FUTURE CHALLENGES

Cyclic Loading

The strain-hardening plasticity model described in this paper has been developed from monotonic loading tests and
is inadequate for the modelling of cyclic behaviour. In the ocean environment, reversal of load paths and cyclic
behaviour can be expected and can cause both the reduction of the strength in the soil and hysteretic behaviour, and
(as yet) these are not properly accounted for. The current models incorporate one discrete yield surface, whereas it is
now increasingly recognised that yielding of a foundation is more gradual. When used for numerical prediction of
jack-up behaviour they predict a sudden reduction in stiffness where as in reality the process of spudcan yielding
entails a gradual reduction of stiffness.

The next major step forward is the development (or refinement) of models that provide realistic modelling of
behaviour during cycling, including a gradual degradation of stiffness with strain amplitude, is required. Several
approaches are possible, including boundary surface models or the use of multiple (or even infinite) numbers of
yield surfaces.

Results of a preliminary cyclic loading model are presented here. Its formulation uses a ‘continuous hyperplasticity’
approach, a method currently seeing considerable development in its application to soil mechanics and constitutive
modelling [45][46]. The advantage of continuous hyperplasticity is that the entire theoretical model can be specified
by defining just two scalar functionals and that the theory can describe models with an infinite number of plastic
strain components within a compact theoretical framework. The hyperplasticity approach has been found to simulate
well the load-displacement response of footings subject to combined cyclic load [20].

The example again uses NewWave loading in considering the dynamic response of the jack-up. Only in-plane
loading is considered and for this simple example there is only one upwave and one downwave leg. Figure 25 shows
the time history of the NewWave elevation of 15m and for a slightly different sea-spectrum (in this case the
JONSWAP). The foundations in this case are modelled by a preliminary version of a multiple yield surface model.
The important distinction is the nonlinearity exhibited at small displacements, as shown in the combined loading
response of the spudcans shown in Figure 26 and the moment rotation response of the upwave leg shown in Figure
27. This nonlinearity is occurring at very small moments and after the large NewWave passes a permanent
unrecoverable rotation is shown.

Full Scale Monitoring Programs

Already some use of monitored full-scale jack-up data has been incorporated into the force-resultant modelling
approach [37]. Numerical simulation of the platforms under storm loading has helped evaluate appropriate levels of
foundation stiffness and has lead to recommendations of increased stiffness values. However, this study was limited
to two-dimensional modelling with assumptions of wave and wind loading direction and subsequent 2D rig
orientation made. Furthermore, the horizontal deck displacements of the measured data and the numerical simulation
results could only be compared in the frequency domain and by the statistical magnitude of response (a random one
hour of sea-state was simulated and the magnitudes of response numerically evaluated). With the development of
three-dimensional models that will include cyclic loading the opportunity to simulate the exact loading and response
history (not just the response frequency and overall magnitude) would be beneficial. The ability of the foundation
models to simulate real jack-up prototype data, especially under cyclic wave loading, should be investigated. This
would produce a more thorough understanding of the level of fixity under spudcan footings and may-be further
elimination of conservative assumptions in industry guidelines.

Further monitoring and back analysis is therefore recommended. Any field monitoring should concentrate on
gathering high-quality site-investigation data so that the soils can be properly characterised. Other direct indications
of stiffness such as the measurement of soil rebound as the preload is dumped, would also be a valuable check


CONCLUSION

This paper has described the development of force resultant models describing spudcan behaviour for application in
the response analysis of jack-up platforms. A six-degree of freedom model based on the framework of single-surface
strain-hardening plasticity theory and capable of application in both sands and clay has been outlined. The ability of
these models to be incorporated within dynamic structural analysis programs has been demonstrated. Future
challenges do exist, including the modelling of realistic responses to cyclic loading and the development of models
for special applications, such as caissons. These challenges would be best addressed with an integrated approach of
experiments, numerical and theoretical developments, and importantly, monitoring of full scale jack-ups by
industry.


ACKNOWLEDGEMENTS

The research presented here has involved the contribution over many years of colleagues at Oxford University, The
University of Western Australia and elsewhere, and their efforts are gratefully acknowledged. The three-
dimensional jack-up example given here was computed in co-operation with Ms Britta Bienen and the experimental
three-legged jack-up shown is the doctoral research of Mr George Vlahos, both of The University of Western
Australia. Support from the Australian Research Council through the ARC Discovery grant scheme (DP0345424) is
gratefully acknowledged.


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[20] Byrne, B.W. (2000). Investigations of suction caissons in dense sand, DPhil Thesis, The University of Oxford.
[21] Martin, C.M. and Houlsby, G.T. (2000). Combined loading of spudcan foundations on clay: laboratory tests,
     Géotechnique 50, No 4, pp 325-338.
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     Géotechnique 51, No 5, pp463-466.
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     University of Oxford.
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     combined loading, Géotechnique 52, No 2, pp. 117-129.
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     Offshore Technology Conference, Houston, OTC 10995.
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     Géotechnique, 52, No. 9, pp. 687-692.
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[37] Cassidy, M.J., Houlsby, G.T., Hoyle, M. and Marcom, M. (2002). Determining appropriate stiffness levels for
     spudcan foundations using jack-up case records, Proc. 21st Int. Conf. on Offshore Mechanics and Arctic
     Engineering (OMAE), Oslo, Norway, OMAE2002-28085.
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     connections of jack-up rigs. M.Sc. Thesis, Delft University of Technology.
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     Conf. Jack-Up Platforms Design, City University, London.
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     of natural frequency and damping ratio of jackup rigs and some theoretical considerations. Proc. 14th Offshore
     Technology Conference, Houston, pp. 661-670, OTC 4287.
[42] Grenda, K.G. (1986). Wave dynamics of jackup rigs. Proc. 18th Offshore Technology Conference, Houston, pp.
     111-116, OTC 5304.
[43] Bradshaw, I.J. (1987). Jack-up structural behaviour and analysis methods. Mobile Offshore Structures,
     Elsevier, London, pp. 125-155.
[44] Tromans, P.S., Anaturk, A.R. and Hagemeijer, P. (1991). A new model for the kinematics of large ocean waves
     -applications as a design wave-. Proc. 1st Int. Offshore and Polar Engng Conf. , Edinburgh, 3, pp. 64-71.
[45] Puzrin, A.M. and Houlsby, G.T. (2001). Fundamentals of kinematic hardening hyperplasticity, International
     Journal of Solids and Structures, 38, No. 21, May, pp 3771-3794
[46] Puzrin, A.M. and Houlsby, G.T. (2001). A thermomechanical framework for rate-independent dissipative
     materials with internal functions, International Journal of Plasticity, 17, pp 1147-1165
[47] Cassidy, M.J. and Bienen, B. (2002). Three-dimensional numerical analysis of jack-up structures on sand.
     Proc. 12th Int. Offshore and Polar Engng Conf., Kitakyushu, Japan, 2, pp. 807-814.
                                                              H2 , u2       2
                                         2R


                                                                  M2, θ2


                                                                            3
                                                                  H3, u3
                                                        M3, θ3
                                  Q, ω
                                              V, w
                                         1


Figure 1 – Loads and corresponding displacements of a spudcan footing in six-degrees-of-freedom




                                                                           V0
                                                                           (describes apex of
                                                                           yield surface, Eqn. 4)


                   Vertical
                   plastic
                   displacement

                                                                     Vertical load
                          H3                                         penetration
                                                     Expansion
                                    M2/2R            of surface
                                                                     curve




                           Yield surface in
                           (V, M2/2R, H3) load space                 V


          Figure 2 - Expansion of yield surface with plastic vertical displacement




                Figure 3 – Invitation to and notes from Butterfield’s 1978 lecture
           Figure 4 – Expected load paths for experimental swipe tests (after Martin and Houlsby [21])




        Figure 5 – Combined vertical, moment and horizontal loading rig at the University of Oxford




Figure 6 – Example swipe test results for a spudcan on over-consolidated clay (after Martin and Houlsby [21])
                                                      200

                                                                                                                   GG04
                                                      150
                                                                        GG08




                                         M2/2R (N)
                                                      100


                                                       50


                                                         0
                                                             0      200        400      600         800     1000   1200     1400         1600
                                                                                                    V (N)

Figure 7 - Example swipe test with rotation for a flat 100mm circular footing on sand (after Gottardi et al. [19])



                                                     Associated flow vectors
                                                        shown dotted by
                                                      comparison (Eqn. 11)




 Figure 8 – Yield surface and flow rule for spudcan on over-consolidated clay (after Martin and Houlsby [22])


                              3000

                              2500
       Torsion Load,Q (Nmm)




                              2000

                              1500

                              1000

                              500

                                0

                              -500
                                     0           200              400          600            800           1000   -2   0   2    4   6    8   10 12 14 16 18
                                                                 Vertical Load, V (N)                                           Rotation, ω (deg)

  Figure 9 - Track of vertical torsion yield surface from swipe test of a 50mm flat plate on dense silica sand
                                                       hn
                                                 0.15
                                                                                                                Best fit
                                                                                                                GG03
                                                 0.10                                                           GG04
                                                                                                                GG05
                                                                                                                GG06
                                                 0.05
                                                                                                                GG28
                                                                                                                GG29

                                                 0.00                                              mn           GG07
                -0.15     -0.10         -0.05        0.00          0.05              0.10     0.15
                                                                                                                GG08
                                                                                                                GG10
                                                -0.05
                                                                                                                GG12



                                                -0.10



                                                -0.15



Figure 10 – Comparison of experimental and fitted numerical yield surface for sand in M2/2R:H3 load space (after
                                            Gottardi et al. [19])


                             1.20

                             q


                             1.00




                             0.80



                                                                          Best Fit
                             0.60
                                                                          GG03
                                                                          GG04
                                                                          GG05
                             0.40                                         GG06
                                                                          GG28
                                                                          GG29
                                                                          GG07
                             0.20
                                                                          GG08
                                                                          GG10
                                                                          GG12
                             0.00
                                 0.00           0.20        0.40              0.60          0.80         1.00
                                                                                                     v


      Figure 11 – Comparison of experimental and numerical yield surface for sand in combined moment-
                          horizontal:vertical load space (after Gottardi et al. [19])
                              2500
                                                                                                               Experiments
                              2000                                                                             Theory


                              1500




                      V (N)
                              1000

                               500

                                 0
                                     -1    0          1       2    3         4        5         6        7          8   9    10
                                                                         w p (mm)

            Figure 12 - Experimental data and fitted numerical vertical load-displacement curve


                                           10000


                                               1000


                                                100


                                                 10
                                          Nγ




                                                                                                     α=1
                                                                                                     α = 0.8
                                                  1
                                                                                                     α = 0.6
                                                                                                     α = 0.4
                                                0.1
                                                                                                     α = 0.2
                                                                                                     α=0
                                               0.01
                                                      0   5   10   15   20       25   30   35       40   45    50

                                                                             φ (°)


                     Figure 13 – Bearing capacity factors (Nγ) for a cone angle of 150°




                                Experimental results                                       Numerical simulations
Figure 14 – Retrospective simulation of combined loading swipe tests on clay (after Martin and Houlsby [22])
                                              200
                                                              Numerical
                                                              Experimental
                                              150




                                  M/ 2R (N)
                                              100



                                               50



                                                0



                                               -50
                                                     -1   0     1     2      3      4      5      6
                                                0

                                                1

                                                2
                                  w (mm)

                                                3

                                                4

                                                5             Numerical
                                                              Experimental
                                                6
                                                     -1   0     1     2      3      4      5      6

                                                                    2Rd θ (mm)

                  Figure 15 – Retrospective simulation of a constant V test on dense silica sand



                                              Values:
                                                                                                                   S1
  55 m
                                              For a single leg:
                                              E = 200GPa
                                              Iy = Iz = 10.843m4                        α=35°
                                              IT = 0.849m4
                                              A = 0.58m2                                                                       S2
                                              M = 2.85x106kg
115.2 m                   MWL
                                              G = 80GPa
                          90 m
                                              For hull:
                                                                                                                   S3
                                              M = 16.1x106kg




Figure 16– Idealised jack-up used in numerical analyses                             Figure 17 – Definition of loading direction α
           (after Cassidy and Bienen [47])
                                                                                               Load reference point (i.e.
                                                                                               numerical point of attachment
                                                                                               on jack-up legs)


                                                                                                                        0.8
                                                                                                                        1.8

                                                                                                                        2.9

                                   Radial measurements                                                          All units
                                   of conical underside                      2.25              7.75             are in m


                                                                                        10.0


                                 Figure 18 – Size and shape of the spudcan used
   0.10                                                                                                                                 0.10
H/V0                                                                                                                                 M/2RV0                                           First yield              S2
   0.09                                                                                                                                 0.09
                                    S1                                                                                                                                           S3

   0.08                                                                            S3                                                   0.08

   0.07                                                                                                                                                                                           Envelope for storm with
                                                    Envelope for storm with return period of 1,000,000 years
                                                                                                                                        0.07
                                                                                                                      S2                                                                          return period of
                                                                                                                                                                                                  1,000,000 years
   0.06                                                                                                                                 0.06
                                                                                                                                                                                               Envelope for storm with
                                                                                                                                                 S1                                            return period of
   0.05                                             Envelope for storm with return period of 10,000 years                               0.05                                                   10,000 years

                                                                                                                                                                                            Envelope for storm with
   0.04                                                                                                                                 0.04                                                return period of
                                                                                                                                                                                            100 years
   0.03                                               Envelope for storm with return period of 100 years
                                                                                                                                        0.03

   0.02                                                                                                                                 0.02
                                                                                                  First yield
   0.01                                                                                                                                 0.01

   0.00                                                                                                                                 0.00
      0.3                           0.4                0.5                 0.6                 0.7              0.8        0.9             0.3    0.4      0.5             0.6        0.7               0.8                 0.9
                                                                        V/V0                                                                                              V/V0



                                                       Figure 19 – Spudcan reactions for α = 35° (after Cassidy and Bienen [47])


                                     12
                                                    H s = 12 m                                                                                                              upwave legs
                                         8          T z = 10 s
                                                                                                                                                                            downwave leg
            surface elevation (m)




                                                    α = 12 m
                                         4

                                         0

                                     -4
                                                         upwave                         downwave
                                                           leg                             leg
                                     -8
                                                              x = 0m             x = 51.96m
                                    -12
                                             -60                            -40                                -20                  0                 20                  40                     60
                                                                                                                                 time (s)

                                                     Figure 20 - NewWave surface elevation at the upwave and downwave legs


                                          1
                                     0.8                                                                                                                         pinned
                                                                                                                                                                 Model C and linear springs
            deck displacement (m)




                                     0.6
                                     0.4
                                     0.2
                                          0
                                    -0.2
                                    -0.4
                                    -0.6
                                    -0.8
                                              -60                             -40                              -20                  0                 20                  40                      60
                                                                                                                                 time (s)

                                                         Figure 21 - Horizontal deck displacements due to NewWave loading
                          1
                        0.8
deck displacement (m)   0.6
                                                                                                                                      α = 18 m
                        0.4
                        0.2                                                                                                           α = 15 m
                          0
                        -0.2                                                                                                          α = 12 m
                        -0.4
                        -0.6
                        -0.8
                               -60             -40                  -20                          0                              20           40   60
                                                                                          time (s)

                                 Figure 22 - Horizontal deck displacements due to increasing amplitude NewWaves




                                                                                                     Vertical Actuator



                                                                                                              Hull
                                                                                                              Potentiometer
                                                                                                              & Tilt Sensor
                                                      Vertical Actuator Arm                                   Holder
                                                           Dead Weights                                         Horizontal Actuator



                                                             Footing
                                                             Potentiometer                                      Load Cell
                                                             & Tilt Sensor
                                                             Holder

                                                                                                               Reaction Frame


                                                             Testing Tank          Clay Sample



                                                                     Test Site
                                                                                                            Penetrometer
                                                                                                            Sites
                                 Figure 23 – Experimental set-up of the 1:250 scale jack-up (after Vlahos et al. [7])
                                                                                          Loading Direction




                                                                                           JUP-3
                                                                                           JUP-4



                                                          Figure 1 - Test Configuration and Site Plan




                                              Figure 24 – Model jack-up after push-over test is completed
                     20
                     15                                              Upwave leg
                     10                                              Downwave leg
Wave elevation (m)
                      5
                      0
                      -5

                     -10
                     -15
                           -80                                 -60             -40         -20            0           20      40            60         80
                                                                                                    Time (seconds)


                                           Figure 25 – Time history of the NewWave elevations on the jack-up legs



                                                              10

                                                              8                      Upwave leg

                                                              6
                                               Moment (MNm)




                                                              4

                                                              2

                                                              0

                                                              -2

                                                              -4
                                                               -0.005                 0               0.005            0.01         0.015
                                                                                               Rotation (degrees)


                                          Figure 26 – Load paths followed by footings using multiple surface model



                                          10

                                           8                                                       Upwave leg
                                                                                                   Downwave Leg
                                           6
                           Moment (MNm)




                                           4

                                           2

                                           0

                                          -2

                                          -4

                                          -6
                                                92                        94              96            98            100     102                104
                                                                                                 Vertical load (MN)


                     Figure 27 – Moment rotation response of the upwave spudcan using multi-surface model