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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. REFERENCES [1] SNAME T&R 5-5A (1997). Site specific assessment of mobile jack-up units. 1st Edition – Rev 1. Society of Naval Architects and Marine Engineers, New Jersey. [2] Noble Denton and Associates. (1987). Foundation fixity of jack-up units: a joint industry study. Noble Denton and Associates, London. [3] Williams M.S., Thompson R.S.G. and Houlsby G.T. (1998). Non-linear dynamic analysis of offshore jack-up units. Computers and Structures, 69(2), pp. 171-180. [4] Cassidy, M.J., Eatock Taylor, R. and Houlsby, G.T. (2001). Analysis of jack-up units using a Constrained NewWave methodology. Applied Ocean Research, 23, pp. 221-234. [5] Martin, C.M. (1994). Physical and numerical modelling of offshore foundations under combined loads. D.Phil. Thesis, University of Oxford. [6] Martin, C.M. and Houlsby, G.T. (1999). Jackup units on clay: structural analysis with realistic modelling of spudcan behaviour. Proc. 31st Offshore Technology Conf., Houston, OTC 10996. [7] Vlahos, G., Martin, C.M. and Cassidy, M.J. (2001). Experimental investigation of a model jack-up unit. Proc. 11th Int. Offshore and Polar Engng Conf., Stavanger, Norway, 2001-JSC-152, 1, pp. 97-105. [8] Arnesen, K., Dahlberg, R., Kjeøy, H. and Carlsen, C. A. (1988). Soil-structure interaction aspects for jack-up platforms. Proc. Conf. Behaviour Offshore Struct., Trondheim, 259-277. [9] Hambly, E. C., Imm, G. R., and Stahl, B. (1990). Jackup performance and foundation fixity under developing storm conditions. Proc. 22nd Offshore Technology Conf., Houston, OTC 6466. [10] Roscoe, K.H. and Schofield, A.N. (1956). The stability of short pier foundations in sand. British Welding Journal, August, pp. 343-354. [11] Butterfield, R. and Ticof, J. (1979). Design parameters for granular soils (discussion contribution). Proc. 7thEuropean Conf. Soil Mech. Fndn Engng, Brighton, 4, pp. 259-261. [12] Schotman, G.J.M. (1989). The effects of displacements on the stability of jackup spudcan foundations, Proc. 21st Offshore Technology Conf., Houston, OTC 6026. [13] Tan, F.S.C. (1990). Centrifuge and numerical modelling of conical footings on sand, PhD Thesis, Cambridge University. [14] Nova, R. and Montrasio, L. (1991). Settlements of shallow foundations on sand, Géotechnique 41, No 2, pp. 243-256. [15] Murff, J.D., Prins, M.D., Dean, E.T.R., James, R.G. and Schofield, A.N. (1992). Jackup rig foundation modelling. Proc. 24th Offshore Technology Conf., Houston, OTC 6807. [16] Dean, E.T.R., James, R.G., Schofield, A.N., Tan, F.S.C. and Tsukamoto, Y. (1993). The bearing capacity of conical footings on sand in relation to the behaviour of spudcan footings of jack-ups. Proc. Wroth Memorial Symp. “Predictive Soil Mechanics”, Oxford, pp. 230-253. [17] Gottardi, G. and Butterfield, R. (1993). On the bearing capacity of surface footings on sand under general planar loads, Soils and Foundations 33, No 3, pp. 68-79. [18] Gottardi, G. and Butterfield, R. (1995). The displacement of a model rigid surface footing on dense sand under general planar loading, Soils and Foundations 35, No 3, pp. 71-82. [19] Gottardi, G., Houlsby, G.T. and Butterfield, R. (1999). The plastic response of circular footings on sand under general planar loading, Géotechnique 49, No 4, pp 453-470. [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. [22] Martin, C.M. and Houlsby, G.T. (2001). Combined loading of spudcan foundations on clay: numerical modelling, Géotechnique 51, No 8, pp 687-700. [23] Byrne, B.W. and Houlsby, G.T. (2001). Observations of footing behaviour on loose carbonate sands, Géotechnique 51, No 5, pp463-466. [24] Watson, P.G. (1999). Performance of skirted foundations for offshore structures. PhD Thesis, University of Western Australia. [25] Zhang, J. (2001). Geotechnical stability of offshore pipelines in calcareous sand. PhD Thesis, University of Western Australia. [26] Bell, R.W. (1991). The analysis of offshore foundations subjected to combined loading, MSc Thesis, Oxford University. [27] Ngo-Tran, C.L. (1996). The analysis of offshore foundations subjected to combined loading. D.Phil. Thesis, University of Oxford. [28] Doherty, J.P. and Deeks, A.J. (2003). Elastic response of circular footings embedded in a non-homogeneous half-space, Dept. of Civil Engn. Report C:1687, University of Western Australia (accepted for publication in Géotechnique). [29] Cassidy, M.J., Byrne, B.W. and Houlsby, G.T. (2002). Modelling the behaviour of circular footings under combined loading on loose carbonate sand. Géotechnique, 52, No. 10, pp. 705-712. [30] Cassidy, M.J. (1999). Non-linear analysis of jack-up structures subjected to random waves, DPhil Thesis, The University of Oxford. [31] Houlsby, G.T. and Cassidy, M.J. (2002). A plasticity model for the behaviour of footings on sand under combined loading, Géotechnique 52, No 2, pp. 117-129. [32] Cheong, J. (2002). Physical testing of jack-up footings on sand subjected to torsion. Honours Thesis, University of Western Australia. [33] Cassidy, M.J. and Houlsby, G.T. (1999). On the modelling of foundations for jack-up units on sand, Proc. 31st Offshore Technology Conference, Houston, OTC 10995. [34] Cassidy, M.J. and Houlsby, G.T. (2002). Vertical bearing capacity factors for conical footings on sand. Géotechnique, 52, No. 9, pp. 687-692. [35] Houlsby, G.T. and Martin, C.M. (2003). Undrained bearing capacity factors for conical footings on clay, Géotechnique, 53, No. 5, pp 513-520 [36] Poulos, H.G. and Davis, E.H. (1974). Elastic solutions for soil and rock mechanics. John Wiley, New York. [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. [38] HKS (1998). ABAQUS Users’ Manual, Version 5.8. Hibbit, Karlsson and Sorenson, Inc. [39] Grundlehner, G.J. (1989). The development of a simple model for the deformation behaviour of leg to hull connections of jack-up rigs. M.Sc. Thesis, Delft University of Technology. [40] Spidsøe, N. and Karunakaran, D. (1993). Non-linear dynamic behaviour of jack-up platforms. Proc. of 4th Int. Conf. Jack-Up Platforms Design, City University, London. [41] Hattori, Y., Ishihama, T., Matsumoto, K., Arima, K., Sakata, N. and Ando, A. (1982). Full-scale Measurement 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

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