Elasto-plastic modelling of bucket foundations

Elasto-plastic modelling of bucket foundations M.F. Bransby, C.M. Martin Centre for Offshore Foundation Systems, The University of Western Australia

ABSTRACT: A work-hardening plasticity model is introduced for predicting the behaviour of bucket foundations under combinations of vertical, horizontal and moment loading. It is incorporated into a method of analysing multi-legged jacket structures with bucket foundations, and typical system behaviour is examined. Results from recent centrifuge model tests of a jacket structure with four bucket foundations are compared with the numerical predictions and good agreement is obtained. 1 INTRODUCTION Traditionally, offshore structures with multiple footings such as jacket structures used pile foundations. Increasingly, these foundations are being replaced by the suction caisson or bucket foundation. This foundation type is cheap, easy to install and has significant uplift capacity. It is particularly appropriate for the soft, low permeability sediments prevalent offshore. Environmental loading of offshore structures with multiple footings results in combined vertical (V), moment (M) and horizontal (H) loading of individual foundations. The distribution of foundation loads for structures with multiple footings depends on the form of the environmental loading, the stiffness and configuration of the structure and the stiffness of the footings at any instant. Hence, detailed models for footing response are required to allow accurate global analysis of offshore structures. Even for idealised, monotonic environmental loadings, methods for prediction of footing performance are not simple. Recently, both foundation performance and system behaviour have been analysed using plasticity methods. The undrained capacity of individual footings such as spudcans or buckets undergoing combined V-M-H loading is expressed in terms of a yield locus where f(V,M,H) = 0 at yield (e.g. Murff 1994, Tan 1990, Martin 1994, Bransby & Randolph 1998). With a suitable plastic potential and hardening rule, the footing behaviour can be modelled in a work-hardening plasticity framework (e.g. Tan 1990, Martin 1994). The behaviour of an entire structure-footing system is less well documented. Schotman (1989), Martin (1994) and Martin and Houlsby (1999) ana-

lysed jack-up rig behaviour using plastic footing models together with elastic analysis of the superstructure. Murff (1994) calculated the capacity of rigid structures with plastic foundations using bound plasticity theorems. As yet there is no work comparing the two approaches for the analysis of jacket structures with bucket foundations. This paper summarises findings on the capacity of individual bucket foundations and extends this work to elasto-plastic modelling. This is then used as part of the modelling of a rigid jacket structure subject to monotonic, short-term loadings. The results are examined and compared to a set of centrifuge model tests of a structure on overconsolidated kaolin. 2 ANALYSIS OF SINGLE BUCKET FOUNDATIONS 2.1 Geometry and soil conditions Bucket foundations will be considered with a diameter, D and skirt length, d (Figure 1). Although there is soil within the skirts, this is usually considered to be rigid as it deflects little during undrained loadings and remains in place during tension loadings because of suction (Tani & Craig 1995, Bransby & Randolph 1999). Loads and displacements are defined as shown in Figure 1 and both are measured at the centre of the footing at skirt tip level. The soil will be idealised with undrained shear strength, sus at the surface and a shear strength gradient, k (Figure 1). The shear strength at the skirt tip depth will be defined as suo (prior to plastic penetra-

Elasto-plastic modelling of bucket foundations Loads

V

2.3 Prediction of single load capacities, Vo, Mo and Ho

Undrained shear strength,

su

sus suo

M Skirt length, d

M.F.Bransby & C.M.Martin

H h

v

θ

Displacements

k Footing diameter, D

Depth, z

Figure 1. Geometry of a bucket foundation

tion or heave, suo = sus + kd). The side walls will have the capacity to mobilise a shear strength, τ = α su, where α is an adhesion factor for the soil-skirt roughness, but the base will be considered to be fully rough as it represents soil-soil shear. 2.2 Choice of yield locus Many analyses have been performed to investigate the detailed shape of the V-M-H yield locus for suction caissons (e.g. Ngo-Tran 1996, Ukritchon et al. 1997, Bransby & Randolph 1998a, b, 1999). Additionally, centrifuge model testing of caissons in clay, silt and sand has been carried out to determine the yield locus shape in V-H space (Watson, 1999). These studies suggest that the shape of the yield locus is very complex (though rather unaffected by embedment depth or foundation shape) and it is difficult to encapsulate the shape in a single mathematical expression. In addition, it has been confirmed that normality to the yield locus holds. This is as predicted from plasticity theory and follows because normality is obeyed at undrained element failure and because there is no detachment of the footing from the soil. Given the complexity of the yield locus, a simple curve fit similar to that used by Murff (1994) is employed here (Figure 2). Because the tension capacity of a bucket, Vt ≈ - Vo the equation simplifies to V f =   Vo

 M *   −1 +     M o   q

m

  H  +  H   o

   

n

   

1 p

(1)

where Vo, Mo and Ho are the foundation capacities under pure vertical, moment and horizontal loading and q, m, n and p are exponents chosen empirically. Exponents q = m = n = 2 and p = 1.6 were used in later analyses. H M /D

The undrained load capacities, Vo, Mo and Ho determine the size of the yield locus. These bearing capacities are calculated with bearing capacity factors Nc, Nm and Nh, which link the bearing pressure to the shear strength at skirt tip depth, suo e.g. Ho/A = Nh suo. A program has been written to calculate a lower bound value of Nc for any su profile using the method of characteristics. This method is appropriate for shallow foundations (d/D < 1) and the solutions have been verified with results from centrifuge model tests (Watson 1999). The horizontal capacity of the foundation is calculated using the upper bound mechanisms of Murff and Hamilton (1995). The moment capacity may also be predicted using the same set of mechanisms although results from alternative upper bound mechanisms (e.g. Bransby and Randolph 1998b) should be compared to check that the solution gives the lowest upper bound. 2.4 Plastic hardening rule Once yield of a footing has occurred, work hardening plasticity is employed using the yield locus of equation 1. The plastic hardening law is based on the fact that suo increases with plastic vertical penetration, vp of the footing. As vertical bearing capacity, Vo = Nc A suo. then

(

Vo = N c A s us + kd + kv p

)

(2)

The moment and horizontal capacities, Mo and Ho also work harden with su in an analagous manner. 2.5 Stiffness response It is assumed that foundation behaviour is linearly elastic inside the yield locus. The values of the stiffness constants were calculated for varying embedment ratios, but for uniform soil (Martin & Houlsby, 1999). An elasto-plastic model of the footing can be assembled using the above components in the manner of Schotman (1989) and Martin & Houlsby (1999). An elasto-plastic stiffness matix, Kep gives the instantaneous tangential stiffness of the footing once the load state and plastic penetration is known.

Vt

3 ANALYSIS OF JACKET STRUCTURES WITH ELASTO-PLASTIC FOOTINGS Vo V

Figure 2. V-M-H Yield locus for bucket foundation.

A jacket structure with bucket foundations may be idealised as shown in Figure 3. A dead load, W is 2

Elasto-plastic modelling of bucket foundations


Jacket structure

W

Environmental loading

H1

H2

H1 w

H2

F L

F

w H1 M1

H2 M2

V1

V2

Plan

Elevation

Figure 3. Parallel loading of jacket structure.

applied at the centre of the structure, representing its weight. The environmentally induced loading is idealised as a single horizontal load, F applied at a height, L above the skirt tips. One possible load direction is such that pairs of footings will be subjected to the same loading conditions due to symmetry (Figure 3). The subscript 1 is applied to the windward foundation pair; the subscript 2 refers to the leeward pair and the footings are at a spacing w from centre to centre. Because there is redundancy, many different load combinations fulfill equilibrium. Consider the displacements of the same jacket structure subject to the same environmental loading (Figure 4). The jacket is assumed to be rigid compared with the foundations and therefore will translate and rotate as a rigid body. Displacements are assumed to be small. Murff (1994) suggested that the movement of a rigid structure could be considered by calculating the equivalent centre of rotation x to the left and y below the centre of the structure at skirt tip level (Figure 4). The behaviour of the system is calculated incrementally (Martin & Houlsby, 1999). A small load increment is applied and the global displacements are calculated using the equilibrium and compatibility equations knowing the present loadings on the footings. The load increments on the individual footings are then found.

F Jacket structure w

θ

θ1 v1

h

h1

v

y

centre of rotation

θ2 v2

θ

Figure 4. Displacements of a jacket structure.

h2

Once a single increment is calculated, the stiffness matrices for the individual footings are recalculated using updated loads and vertical plastic displacements, and a new global load increment is applied. Typically, several thousand load increments were used to reach a failure state. Two programs were written to do this: the Fortran program of Martin and Houlsby (1999) was modified and a spreadsheet-based program was written specifically. Although the general solution method was similar, this allowed some cross-validation. Because many increments were required near failure, results from the faster Fortran program running on a Digital Personal Workstation are presented below. 4 LIMIT ANALYSIS METHODS FOR JACKET STRUCTURE ANALYSIS Murff (1994) presented limit analysis methods for predicting offshore structure capacity knowing the geometry of the structure and the yield locus equation of the footings. In the method it is assumed that there is no plastic penetration of any of the footings so that the yield loci do not change size. In the lower bound method, equilibrium is maintained together with the condition of f = 0 on each footing. The load distribution on each footing is varied to find the maximum F (Figure 3) which can be sustained by the system. A spreadsheet has been written to do this. 5 PROBLEM GEOMETRIES 5.1 Typical jacket loadings Consider a typical 4-leg jacket structure as shown previously in Figure 3 under the effects of a dead weight, W and a single equivalent monotonic horizontal load, F applied at a height, L. Load F may sometimes be applied in the direction shown in Figure 3. However, loading is likely from a range of directions. Consider the loading direction shown in Figure 5. Load is applied diagonally across the structure so that there is a single windward leg, two equally loaded middle legs and a single leeward leg. There are now three sets of unknown foundation loads, rather than two with the previous loading direction. If the load direction is between the angles shown in Figures 3 and 5, there will be four differently loaded legs. This is likely to require significantly more complex analysis. It is the loading direction most likely to cause foundation failure that should be analysed. This is often believed to be the condition of diagonal loading. Indeed, a set of centrifuge tests was recently conducted in the UWA beam centrifuge (Watson, 3



Jacket structure H2 H1

Environmental loading

W F

F H3

w

L

1.414 w H1 M1

H2

H2 M2

V1

H3 M3

V2

V3

Plan

Elevation Figure 5. Diagonal jacket structure loading.

1999) to investigate the behaviour of a square jacket structure with 4 caisson foundations under such a loading. 5.2 Centrifuge model testing The centrifuge model test configuration is shown in Figure 6. The caisson foundations were of prototype diameter 9 m and skirt length 3.6 m (d/D = 0.4) and were embedded in overconsolidated kaolin. The undrained shear strength was measured in flight using a cone penetrometer, a T-bar penetrometer and a shear vane. The average shear strength profile is shown on Figure 6. Vertical load, W was applied such that V/A = 45 kPa on each footing, which is calculated as V/Vo = 0.24. Before drainage could occur, a horizontal load, F was applied to the foundation with an effective loading height, L = 52 m. The maximum value of F observed was 8.4 MN (prototype), at which point the structure had rotated approximately 6o. 5.3 Elasto-plastic modelling The centrifuge model test was analysed numerically. The geometry was set exactly as the test. The soil strength profile was idealised with sus = 21.3 kPa and k = 1.217 kPa/m which was used in equation 2. The bearing capacity factor, Nc = 7.195 was calculated using stress characteristics. The bearing capacity factors, Nh =2.34 and Nm=0.7486 were calculated Plan view

Horizontal load, F Platform weight, W

using the Murff and Hamilton (1995) method with an approximate assumed linear strength profile in the upper 3.6 m. Elastic stiffness constants were chosen as presented by Martin and Houlsby (1999) and the rigidity index, G/su = 50, was chosen arbitrarily. To examine the effects of load direction, an analysis was also carried out for the load direction shown in Figure 3. For this analysis, the soil and foundation properties and loadings were unchanged, but the geometry was modified such that there were two rows of independently loaded footings at a spacing of 25.46 m = 36/√2. A third analysis was carried out to examine the effect of dead load, W. The geometry of the model test with diagonal loading was analysed, but an initial V/A = 120 kPa (V/Vo = 0.65) was applied. 6 RESULTS 6.1 Analysis of the centrifuge model test Figure 7 shows the load-displacement response from the elasto-plastic modelling of the diagonally loaded jacket structure for displacements measured at the base and centre of the foundation. Failure is observed at F ≈ 7.7 MN at which point significant rotation and lateral movement begins to occur. Examination of the load distribution on each footing shows the complexity of the system response (Figure 8). Initially there is an equal distribution of horizontal and moment loading between the footings, but as individual footings yield, loads become redistributed. The load states at which each footing yields as well as the failure load are annotated on Figures 7 and 8. Even though a majority of the foundations may yield well before system failure, no significant plastic hardening is possible for individual footings until all the foundations have yielded. This is because large plastic displacements consistent with plastic hardening would not be compatible with the small elastic displacements of other footings. Horizontal and moment loadings of the yielding leeward and windward foundations reduce so virtually all the horizontal loading at failure is taken by the middle foundations.

F 9 Lower bound

52 m 6

su,kPa

Assumed to be rigid

9m

0

10

20

30

40

F (M N)

36 m

50

0

3.6 m

Centrifuge tes t data Analy s es

3

Footing 3 yields Footing 1 yields

Rotation

Footing 2 yields Serviceability failure

Middle

Leeward

over-consolidated kaolin

Depth, m

3.6

Windward

Dis plac em ent

7.2

0

10.8

0

14.4

0.02

0.03

0.04

0.05

0.06

Rotation θ ( Rads ) , lateral dis placement h/D.

18

Figure 6. Centrifuge model geometry.

0.01

Figure 7. Jacket displacements from the model test and analysis

4


0.4 M/ADs uo

M/ADs uo

0.4

Footing 3

Footing 1 0.2

0

-8

x/w

0.6 Footing 2

0.3

0.2

0

0.4 Footing 1 0.2

0.2

Footing 3

0.4

0 V/As uo 4

0

8

1

2

Centrifuge analysis

H/As uo

H/As uo

2.4

Footing 1 yields Footing 3 yields Footing 2 yields Failure Lower bound

Footing 3 0.6 0 -4

Diagonal loading, large W

Footing 2

1.8 1.2

-8

0

0

-4

Footing 1

0.1

Footing 2

0

4

0.6

y/w

0.6


0.8 Footing 3 yields Footing 2 yields Footing 1 yields Serviceability failure

Parallel loading analysis

1 1.2

8

Figure 10. Rotation centre during jacket displacements.

V/As uo

Figure 8. Load distribution on diagonally loaded structure.

The footing displacement during loading shows that there is some change of displacement mechanism with increasing loading (Figure 9). This is clarified in the plot of system rotation centre shown on Figure 10. Initially, when all the footings are elastic, the mechanism is one of rotation about a point directly below the centre footing (x = 0, y = 0.117) whereas failure displacements cause a little settlement of the centre foundation (x = 0.134, y = 0.453). However, the failure mechanism is mainly one of rotation with the windward leg pulling out and the leeward leg penetrating. Interestingly, the mechanism changes are not gradual or uniform. When each footing yields for the first time there is a sudden jump in the rotation centre (Figure 10). Results are compared with those from the centrifuge model tests in Figure 7. The failure load is remarkably well predicted, but the initial stiffness of the system is not well replicated. This is not suprising as the footings are assumed to be totally linearly elastic when loads are within the yield locus. In reality, there will be significant plastic yielding of the footings within the yield locus due to progressive soil failure and soil softening around the footings. Failure of a real system is likely to be more gradual than predicted by the elasto-plastic method unless plasticity can be introduced within the footing yield loci. In Figure 7 the results are also compared to calculations using the lower bound method described in

section 4. The lower bound method gives F = 7.69 MN at failure. The individual footing loads calculated are shown on Figure 8. There is close agreement between these results and the ‘end-point’ loads obtained from the incremental analysis. 6.2 Analysis of alternative loading direction. Elasto-plastic analysis of the jacket geometry under parallel loading showed that the capacity was 7.9 MN, slightly higher than that for the diagonal loading condition with the same W (Figure 11). This was confirmed as the lower bound solution method gave a capacity of 7.94 MN. Again, loads redistribute as loading increases and the mechanisms change (Figure 10). 6.3 Jacket analysis with high vertical loading. The load displacement response of the diagonally loaded structure with higher W was noticeably more ductile after serviceability failure so that failure is more difficult to identify (Figure 11). The failure mechanism involved considerably more vertical penetration of the footings (Figure 10). 7 DISCUSSION Only simplified system analyses are presented above. There are many other factors that are not included such as structure flexibility, large displace-

0.03

0.03

0.02

0.02

Footing 3

0.01

9 7

0

v/D

0.05

0

0.02

F (M N)

0

0.04

h/D

0.04 Footing 2

0.03

h/D

Diagonal loading, large W

6

0

0.02 Footing 3

0.01 Footing 1

Footing 1 yields Footing 2 yields Footing 3 yields Failure

5 4 3

Rotation

2

Lateral dis plac em ent

1 0

0

v/D

0.05

Figure 9. Footing displacements for diagonally loaded structure

Foo tin g 3 yields Foo tin g 1 yields Foo tin g 2 yields Se rvice ab ility fa ilu re L ow er b o un d

0

0 -0.05

P arallel loading

8

0.01

Footing 1

-0.05

Rotation , θ

Rotation , θ

Footing 2

0.01

0.02 0.03 0.04 0.05 Rotation θ ( Rads ) , lateral dis placement h/D.

0.06

Figure 11. Displacement of parallel loaded jacket structure.

5

Elasto-plastic modelling of bucket foundations ments, the footing non-linearity within the yield locus and the detailed shape of the loci. Also, only three simple loading geometries have been considered. However, the above analyses have highlighted the complexity of a system with multiple footings and progressive failure. Despite individual footing ‘failure’ (in terms of loads), significant movement of the structure does not occur until all foundations have reached yield. Furthermore, load distribution among the footings is such that induced plastic displacements are kinematically admissible. The lower bound method might be expected to underestimate slightly the system failure load. However, because it does not allow for displacements, ductile systems may reach a serviceability failure before the lower bound load is attained. Generally, it appears that the larger the number of independently loaded footings, the more ductile the system. The failure modes observed in the analyses above mainly involve rotation of the structure at failure – the leeward leg embeds while the windward leg pulls out. For different soil conditions, dead loads and structure geometries, failure mechanisms can be very different, for example involving little rotation but vertical penetration and sliding of all the footings. Therefore, every prospective new structure should be analysed to examine the likely failure modes and load redistribution prior to reaching the overall ultimate capacity. 8 CONCLUSIONS Analyses have been carried out of rigid jacket structures with multiple bucket foundations modelled using work-hardening plasticity principles Through analysis of a jacket structure under three different loading conditions, the mechanics of the system have been examined. Significant load redistribution between footings allows much larger loads to be sustained than those which first cause yield of an individual foundation. Results were compared for loading of a jacket structure in different directions. The capacities were similar for diagonal and parallel loading, with the diagonal load capacity slightly lower. However, this is likely to be site-specific and depend on the geometry, loading and soil conditions. Results from the elasto-plastic and lower bound methods were very similar. Indeed, the lower bound method appears a simple, yet accurate method of predicting push-over capacity. However, for systems with significant ductility, it should be noted that serviceability failure may occur before these loads are reached so elasto-plastic analysis is required. The geometry of a jacket structure investigated in a centrifuge model test was analysed, and the results were compared with those from the centrifuge. Good

M.F.Bransby & C.M.Martin prediction was made of the ultimate capacity and the load-displacement response was also reasonably well predicted. The elasto-plastic footing models are shown to be a useful design tool for describing of footing behaviour within the context of an entire offshore structure. In addition, their use allows parametric studies leading to increased understanding of the behaviour of multi-footing systems. ACKNOWLEDGEMENTS The work forms part of the activities of the Special Research Centre for Offshore Foundation Systems, funded through the Australian Research Council’s Research Centres Program. Special thanks are due to Phil Watson who supplied his preliminary centrifuge data and Mark Randolph whose upper bound program was used. REFERENCES Bransby, M.F. &Randolph, M.F. 1998a. Combined loading of skirted foundations. Geotechnique 48, No 5. pp 637-655. Bransby, M.F.& Randolph, M.F. 1998b. The effects of skirted foundation shape on behaviour under combined V-M-H loading. Proc. ISOPE 98, Montreal,Canada. Vol 1, 543-548 Bransby, M.F.& Randolph, M.F. 1999. The effects of embedment on the undrained response of caisson foundations to combined loading. Soils and Foundations, In press. Martin, C.M. 1994. Physical and numerical modelling of offshore foundations under combined loads. D.Phil thesis, University of Oxford. Martin, C.M., Houlsby, G.T. 1999. Jackup units on clay: structural analysis with realistic modelling of spudcan behaviour, Proc. Offshore Technology Conf., OTC 10996. Murff, J.D. 1994. Limit analysis of multi-footing foundation systems. Proc of the 8th Int. Conf. On Comp. Methods and Advances in Geomechanics, Morgantown, 1, pp. 223-244. Murff, J.D. & Hamilton, J.M. 1995. P-ultimate for undrained analysis of laterally loaded piles, Jour. Of Geotech. Eng, 119, No.1, pp 91-107. Ngo-Tran, C.L. 1996. The analysis of offshore foundations subjected to combined loading. DPhil thesis, University of Oxford. Schotmann, G.J.M. 1989. The effects of displacement on the stability of jackup spud-can foundations. OTC 6026. Tan, F. 1990. Centrifuge and theoretical modelling of conical footings on sand. PhD Thesis, The University of Cambridge. Tani, K. & Craig, W.H. 1995. Bearing capacity of circular foundations on soft clay of strength increasing with depth, Soils and Foundations, Vol. 35, No. 4, pp 21-35. Ukritchon, B., Whittle, A.J., Sloan, S.W. 1997. Undrained limit analysis for combined loading of strip footings on clay. Jour. Of Geotech. And Geoenvironmental. Eng., Vol 124, No. 3, pp.265-276. Watson, P.G. 1999. PhD thesis, University of Western Australia, In preparation.

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