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Efficient finite element techniques for limit analysis of suction caissons under lateral loads
Computers and Geotechnics 24 (1999) 89±107
Ecient ®nite element techniques for limit analysis of suction caissons under lateral loads B. Sukumaran a,*, W.O. McCarron b, P. Jeanjean b, H. Abouseeda c a Rowan University, 201 Mullica Hill Road, Glassboro, NJ 08028, USA Amoco Worldwide Engineering and Construction, Houston, TX 77058, USA c Fugro-McClelland Marine Geosciences, Inc., Houston, TX 77274, USA
b
Received 10 August 1998; received in revised form 20 November 1998; accepted 4 December 1998
Abstract This paper documents the use of ®nite element analyses techniques to determine the capacity of suction caisson foundations founded in soft clays under undrained conditions. The stress±strain response of the soft clay is simulated using an elasto-plastic model. The constitutive model employed is the classical von Mises strength criterion with linear elasticity assumed within the yield/strength surface. Both two- and three-dimensional foundation con®gurations are analyzed. The three-dimensionality of the failure surface of the actual caisson requires that computationally intensive three-dimensional models be used. Suggestions are given on how to improve computational eciency by using quasi three-dimensional Fourier analyses with excellent results instead of true three-dimensional analyses. The ®nite element techniques employed are veri®ed against available classical limit solutions. Results indicate that both hybrid and displacement-based ®nite element formulations are adequate, with the restriction that reduced-integration techniques are often required for displacement-based formulations. # 1999 Elsevier Science Ltd. All rights reserved.
Nomenclature A B d D
bearing area width or diameter of footing displacement diameter
* Corresponding author. Tel.: +1-609-256-5324; fax: +1-609-256-5242; e-mail: sukumaran@rowan. edu 0266-352X/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S026 6-352X(98)0003 6-6
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DSS f L N P Su z
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direct simple shear (shear strength evaluation) coecient of friction length bearing factor load undrained shear strength of soil depth
1. Introduction As oshore exploration and development of oil ®elds reach water depths in the 1000 to 3000 m range, novel methods of anchoring production platforms become attractive due to cost savings associated with oshore installation activities. Surface production systems that are viable in these water depths include Tension Leg Platforms (TLP), Spar platforms, and laterally moored ship-shaped and semi-submersible vessels. TLPs are ¯oating structures anchored by vertical pretensioned tendons which exert considerable tensile forces on the foundation. In the Gulf of Mexico, vertical design loads for individual tendons range up to 27 MN (6000 kips), resulting in foundation lateral loads of 4 MN (900 kips). Laterally moored systems, on the other hand, have the dominant load in the horizontal direction; the horizontal load component being on the order of 9 MN (2000 kips), and the vertical component being less than half that of the horizontal component. At this time (1998), there are ®ve TLPs and two Spars installed and operating in the Gulf of Mexico. Several other TLPs and Spars are in various stages of design and fabrication. For Gulf of Mexico TLPs, driven piles have been the preferred method of anchoring tendons to the sea¯oor. This solution arises from a combination of engineering, geotechnical, fabrication, installation and cost constraints. Chief among these are the high accuracy (within 1 m) required for positioning the tendon anchoring points, and the normally consolidated cohesive soils without appreciable strength in the upper 20 m. These conditions combined with available oshore installation barges, underwater hammers capable of operating in 1300 m of water and simple fabrication of steel pile shapes lead to the preference of driven piles. The depth of penetration of TLP piles ranges up to 120 m. Spars are buoyant single-column hulls anchored by lateral moorings to foundation elements. In water depths less than approximately 1200 m (4000 ft), TLP and Spar systems are competitive economically, but TLPs have a more proven construction, technology and operating history. For greater water depths, the Spar platform oers some performance and economic advantages. Possible foundation systems for Spars include the traditional driven piles, drag anchors and suction caissons. Suction caissons have been used in shallow waters as foundations for single point moorings, jack-up drilling rigs, ®xed platforms and as anchors for ¯oating systems. Initial penetration of the suction caisson into the seabed occurs due to the self
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weight; subsequent penetration is by the `suction' created by pumping water out from the inside of the caisson. Suction caissons become attractive alternatives to driven piles in deepwater because of technical challenges and costs associated with the installation equipment. In addition, suction caissons also provide a greater resistance to lateral loads than driven piles because of the larger diameters typically used. The terminology for this foundation is sometimes misleading. `Suction' can refer to the method of installation or a component of foundation load resistance, or both. Fig. 1 shows a schematic view of a Spar platform anchored by mooring to suction caissons. The feasibility of suction caissons has been demonstrated in the North Sea foundations for the Snorre TLP [1], Europipe 16/11-E structure [2], and with centrifuge tests for Gulf of Mexico TLP conditions [3]. At present, the use of suction caissons are being extended to the Gulf of Mexico. Soil conditions in the North Sea (sti clays and sands) have so far lead to designs with penetration to diameter ratios typically less than 2. Because the deepwater shallow sediments in the Gulf of Mexico exhibit very low surface shear strength, it is necessary to increase the penetration to diameter ratio of the caisson to obtain satisfactory capacities. However, experience
Fig. 1. Schematic view of a spar platform anchored to suction caissons at the seabed.
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with the installation and behavior of caissons with large penetration to diameter ratios (up to 10) is limited. To investigate the installation, performance and capacity of such caissons, a series of ®eld tests, centrifuge tests, and numerical investigation have been commissioned by the industry. This paper presents the ®ndings of some numerical investigations aimed at validating numerical procedures to calculate the lateral capacity of caissons for deepwater foundations. 2. Some background on ®nite element limit analysis Available displacement based and hybrid (combined stress and displacement solution variables) based ®nite elements formulations are capable of accurately and eciently calculating limit loads for foundation systems. An important feature in the successful use of displacement based ®nite element formulations is the use of reduced integration techniques in many limit analysis investigations. The term `reduced' integration refers to the fact that a lower level (fewer sampling points) of numerical integration is being used than that theoretically required, to exactly integrate a polynomial of a certain order. Experience has indicated that the use of reduced integration techniques improves performance under conditions of nearly incompressible (entirely deviatoric plastic strains) response for von Mises and other pressure independent material strength models near the limit conditions. Historical discussions on the relative merits of full and reduced integration techniques are given by Zienkiewicz and Taylor [4] and Zienkiewicz et al. [5]. Nagtegaal et al. [6] discussed the success and failures of several fully integrated elements with respect to their ability to accurately predict limit loads in association with elastic-plastic material models. Sloan and Randolph [7] extended this work for plane strain (strip) and axisymmetric (circular) footing con®gurations, and proposed a triangular 15-noded element for use in axisymmetric problems. The performance of this element was later discussed by de Borst and Vermeer [8], and Whittle and Germaine [9]. Barlow [10,11] presented mathematical arguments that reduced integration in quadratic (eight-noded) elements enhances performance and solution convergence. Griths [12] presented the successful use of quadratic reduced-integration elements in plane-strain and axisymmetric conditions. Naylor [13] discussed the elements' performance for nearly incompressible conditions. Zienkiewicz and Taylor [4] demonstrate that reduced integration elements, of the type used here, satisfy the mathematical conditions of stability and convergence required in the `patch' test. While it is beyond the intended scope of this paper to review in detail the theoretical studies on reduced integration cited by these authors, two key ®ndings are summarized in the following for completeness of discussion. First, the minimum level of numerical integration (quadrature) necessary to insure stability and convergence of solution is that which correctly integrates the volume of the ®nite element. For the two-dimensional quadratic eight-noded element used here, this implies a four-point quadrature for reduced integration, as opposed to full quadrature which leads to a nine point requirement. Second, reduced integration leads to a weak singularity in the single element stiness and fewer internal constraints on the
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coupling between volumetric and deviatoric strains. This singularity does not appear in equilibrium equations when more than one element is involved. We therefore conclude that reduced integration techniques have a ®rm theoretical basis, supporting their application. However, as with any ®nite element, their robustness and accuracy in particular applications should be critically examined. Alternatives to the use of reduced integration exist, e.g. hybrid ®nite elements, or very high order displacement-based elements such as the 15-noded cubic strain triangle. Hybrid elements are available in commercial codes, such as ABAQUS [14], and are eective in the analysis of incompressible materials. The term hybrid stems from the use of both displacement and stress components as solution variables. In this case, the stress component included is the mean pressure. Detailed discussion of these elements are given by Zienkiewicz and Taylor [4] and HKS [14]. The performance of the hybrid elements are compared herein with results of displacement based elements. 3. Veri®cation of ®nite element modeling techniques The adequacy of the hybrid and reduced-integration elements are demonstrated in the following by virtue of their performance in accurately calculating the limit loads for three problems; some of which have typically proven to be problematic for a wide range of element formulations. The analyses were performed with the program ABAQUS [14]. The ®rst series of problems are plane strain (strip) and axisymmetric (circular) footings on the surface of purely cohesive soil. Secondly, deeply embedded footings in cohesive soil are considered. The ®nal problem relates to the ultimate lateral resistance of a circular pile cross section in a cohesive soil. The ®nite element analyses make use of isotropic elasticity combined with a von Mises or Tresca type strength surface. The implementation of the Tresca elastic±plastic constitutive model includes a non-associated ¯ow rule. Young's modulus is taken as approximately 1000 times greater than the undrained strength. Footing analyses presented here represent rigid footings and a weightless soil. The surface and deep footings are analyzed with smooth and perfectly rough interfaces, respectively. The footing limit load is conventionally denoted as Pult NSu A, where N is the bearing capacity factor, Su is the undrained strength, and A is the bearing area. Theoretical limit solutions presented here are based on the Tresca strength condition. The Tresca and Mohr±Coulomb criteria are identical when the Mohr±Coulomb internal friction parameter is zero, as is the case for a purely cohesive material. Finite element strength models used here include both the von Mises and Tresca criteria. Plane strain (strip) footings are analyzed with the von Mises condition matched to the Tresca strength using the procedure presented by Chen [15]. For circular footings, limit loads are presented for the Tresca criterion. Limit loads predicted with displacement and hybrid elements for strip and circular surface footings are shown in Fig. 2. The ®nite element mesh is shown in Fig. 2(a), and the load±displacement relationship in Fig. 2(b). Displacement (D) and hybrid (H) element formulations are indicated in Fig. 2(b), along with the order of integration, full/standard (S) and reduced (R). Thus, SD implies a standard integration
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combined with a displacement formulation. A small displacement formulation was used in the analyses to formulate the element stiness and equilibrium equations. Limit solutions are well de®ned for all conditions but the fully integrated displacement
Fig. 2. (a) Finite element mesh used in the analyses of the capacity of surface footings; (b) bearing capacity factor, N vs normalized displacement computed for surface strip and circular footings.
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formulation element in the circular footing case; as expected based on previous experience. Theoretical bearing capacity factors, N for smooth and rough circular footings are 5.69 and 6.05 [15], respectively. The theoretical bearing capacity factor for a strip footing is 5.14. The calculated bearing capacity factors for the strip footing range from 5.3 to 5.4. For circular footings, the calculated bearing capacity factors range from 5.77 upwards. The RH and RD analyses with the Tresca material model for the circular footing fall within a reasonable range of the theoretical solution. The bearing capacity factor of 5.77 from the RH analysis is essentially the exact solution for the smooth footing condition considered. The ®nite element mesh and results of limit analysis of embedded deep strip and circular footings are shown in Fig. 3. A large displacement formulation is required for this particular problem because of the relatively large stresses compared to the elastic moduli. Theoretical solutions for these conditions have been given by Chen [16], using upper-bound limit analysis, and Meyerhof [17], using limit equilibrium methods. The ratio of the depth of embedment to footing width is 4, and the shaft of the footing is smooth and allows no horizontal deformations. The theoretical solutions by Meyerhof [17] result in bearing capacity factors ranging from 8.85 to 9.74 for rough strip and circular footings, respectively. Chen [16] reported a bearing capacity factor of approximately 9 for deep strip footings. The results for the RD analyses for strip and circular footings indicate bearing capacity factors of 8.9 and 10.7, respectively. The RH analyses result in bearing capacity factors of 7.7 and 9.6 for strip and circular footings, respectively. In the present case, the fully integrated displacement elements (SD) perform poorly and do not reach a limit (results not presented). Accurate determinations of the bearing capacity of deep embedded footings are complicated by the fact that the plastically deforming region is con®ned within an elastic region, so that the plastically strained zone is not free to undergo the unlimited plastic deformation typically associated with limit conditions. In the present cases, the zone of plastic behavior is contained within about two footing diameters distance from the footing (results not shown). The ®nal illustrative example is that of determining the lateral resistance of a circular pile cross section. For this analysis, a plane strain idealization and von Mises strength criterion matched to the plane strain condition is adopted. The ®nite element mesh is shown in Fig. 4(a). The results are shown in Fig. 4(b) for three pile±soil interface conditions: rough with no separation, gapping (separation allowed, no tensile forces transmitted), and gapping with a frictional interface. The theoretical limit solutions have been presented by Randolph and Houlsby [18]. Those authors presented lower-bound solutions with bearing capacity factors, N of 9.14, 10.52, and 11.94 for friction coecients f of 0, 0.4, and 1, respectively. These correspond well with the results shown in Fig. 4(b) for the same conditions obtained for both reduced integration and hybrid elements. Hamilton et al. [19] reported experimental investigations showing N values generally between 10 and 12 for depths greater than four pile diameters. These results are further discussed later in the paper. Current practice in the oshore industry [20] assumes the lateral bearing capacity factor at large depths to be N 9. Based on the results presented here, limit solutions [18], and experimental investigation [19], the current API practice is conservative.
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Fig. 3. Bearing capacity factor, N vs normalized displacement for deeply embedded circular and strip footings.
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Fig. 4. (a) Finite element mesh used for determination of lateral pile capacity; (b) bearing capacity factor vs normalized displacement for circular pile.
4. Lateral load capacity analyses of suction caissons The ultimate holding capacity of a suction caisson anchored at a site with soil conditions similar to that found in the Gulf of Mexico were analyzed. The soil at the site where the suction caisson is expected to be anchored is a normally consolidated clay. The shear strengths are assumed to be zero at the seabed and increasing linearly with depth as given below:
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SDSS 1:41z kPA u
1
is the undrained static direct where z is the depth below seabed in meters and SDSS u simple shear strength. The submerged unit weight of the soil is 6.3 kN/m3. The ®nite element analyses were conducted using ABAQUS [14]. A von Mises shear strength idealization was used to model the clay. The von Mises model implies a purely cohesive (pressure independent) soil strength de®nition. The caisson is modeled as a weightless linear elastic material. 5. Two-dimensional load capacity analysis of suction caissons Two-dimensional analyses were performed to con®rm the mesh and boundary de®nitions selected for three-dimensional investigations are suitable for classical passive and active pressure problems. The caisson analyzed was 6.1 m in diameter with a penetration depth of 12.2 m below the mud line. The caisson length to diameter ratio is L/D=2. The caisson has a closed top during installation and operation. Initial horizontal stresses were de®ned with a coecient of lateral earth pressure Ko equal to 1. The ®nite element mesh shown in Fig. 5 was used for the analyses. The mesh model dimensions of 56 m width by 36.6 m depth was intended to minimize boundary eects on response. The mesh consists of eight-noded plane strain elements for the soil and two-noded linear beam elements for the caisson. The inclination of the load considered was assumed to be 28 with the horizontal, measured counterclockwise. Several points of attachment for the mooring line were
Fig. 5. Finite element mesh used for determining the capacity of suction caissons.
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considered to study the eect of the attachment point on the load capacity. The optimal load attachment point is that which produces maximum capacity. Fig. 6 shows a plot of load capacity vs point of attachment. The capacity generally increases with depth of attachment. This is in agreement with previously published results [21,22]. Fig. 7 shows the horizontal stress acting on the wall when it is constrained to translate horizontally with no rotation occurring. Both smooth and rough soil±wall interfaces are considered. The active and passive pressures computed at the limit conditions display the expected linear distribution with depth, and closely match the pressures calculated for classic active and passive pressure retaining wall response. The eect of the load attachment point on the failure mechanism produced was also studied. Fig. 8(a)±(c) shows the various failure mechanisms produced when the load is attached above, at and below the optimum point. Fig. 8(a) and (c) shows that when the load attachment point is above or below the optimal point, the caisson rotates. The failure mechanism is more rotational than translational. The shear zone mobilized is also less in area than if the load is attached at the optimal attachment point. It can be seen from Fig. 8(b) that when the load is attached at the optimal load attachment point, the failure mechanism is predominately translational. From Fig. 6, it can be seen that the load capacity only decreases slightly if the attachment point is below mid-height of the caisson.
Fig. 6. Load capacity of suction caisson (kN/m) vs depth to load attachment point obtained from the analyses of a plane strain model.
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Fig. 7. Horizontal stress acting on the wall on the active and passive side.
6. Comparison of load±deformation response of Fourier and three-dimensional analyses The three-dimensional response was analyzed using both three-dimensional and quasi three-dimensional Fourier analysis elements (CAXA) available within ABAQUS. CAXA elements are biquadratic, Fourier quadrilateral elements. These elements were used for the analyses of suction caissons because they allow non-linear, asymmetric deformations and loading. Two types of CAXA elements, namely the CAXA8R2 and CAXA8R4, were used in the analyses. These are eight-noded quadrilateral reduced integration elements that dier in the number of Fourier modes used for interpolation. CAXA8R2 elements use two Fourier modes for interpolation while the CAXA8R4 uses four. The number of elements and nodes in the mesh are 180 and 3616, respectively. A three-dimensional model having a similar mesh con®guration with 20-noded brick elements was also developed to compare the results from a quasi three-dimensional Fourier analysis and the actual three-dimensional response. The far boundaries of the model are modeled as perfectly rough (no translations allowed). The modeled caisson does not include a top plate, which results in free deformations of the top soil surface in the caisson soil plug. The plane of symmetry of the three-dimensional and axisymmetric models is identical to the plane-strain model shown in Fig. 5. The point of load application is at about mid-height of the caisson (5.97 m below the mud line). The inclination of the load was 32 with the horizontal, measured counterclockwise. The coecient of lateral earth pressure Ko is 0.8. The load-displacement curves for the three-dimensional model as well as the axisymmetric analyses using two and four Fourier modes
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Fig. 8. (a) Plot of displacement vectors indicating failure pattern when horizontal load is attached at the top of the caisson; (b) plot of displacement vectors indicating failure pattern when horizontal load is attached at the optimal attachment point; (c) plot of displacement vectors indicating failure pattern when horizontal load is attached below optimal attachment point.
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are shown in Fig. 9. The load indicated in Fig. 9 represents the load vector magnitude (resultant from horizontal and vertical components). The three-dimensional and Fourier analyses give the same limit loads, approximately 7700 kN. Finite element analyses with a purely horizontal applied load results in a limit load of approximately 5000 kN when the load is applied at mid-height of the caisson, and 2300 kN when the load is applied at the top of the caisson. Mur and Hamilton [23] present methods using upper-bound limit analyses, which give a capacity of 7000 kN for the case of pile translating horizontally in a soil mass with full adhesion and suction assumed on the back side of the pile. Those authors also compared their solutions with experimental centrifuge tests in kaolin clay previously presented by Hamilton et al. [19]. The present numerical results are compared with these limit analysis and experimental results in the following. Fig. 10 shows the non-dimensionalized normal (radial) stresses acting on the outside wall of the caisson from the axisymmetric±asymmetric analysis. The 0 plane is the plane along which the load is attached. The results in Fig. 10 were obtained by dividing the soil radial stresses adjacent to the caisson by the soil strength [Eq. (1)] at the respective depth. The left side of the plot in Fig. 10 represents results with an inclined (at 32 from the horizontal) load, while the right side represents results for a purely horizontal load. The results from the analysis with an inclined load generally show higher non-dimensionalized pressures than that for a horizontal load. This can be explained by the fact that the vertical load component applied on the face of the caisson tends to reduce rotation of the caisson. The non-dimensionalized stresses generally increase with depth when rotation of the caisson is limited, but when signi®cant
Fig. 9. Load capacity vs normalized displacement for the three-dimensional and axisymmetric±asymmetric analyses.
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Fig. 10. Normal stresses acting on the outside of the caisson at various depths below mudline.
rotation does occur, passive pressures are reduced at the leading edge of the base of the caisson. As shown in Fig. 11, the present numerical results compare reasonably with available limit solutions and experimental results. Fig. 11 includes ultimate lateral pressures computed from methods commonly used in the oshore industry developed by Matlock [24] for laterally loaded piles in soft clays. Centrifuge tests performed by Hamilton et al. [19,23] resulted in a mean bearing factor of 11 over a wide range of depths. The analytical limit solutions and experimental results in Fig. 11 are for a strength pro®le increasing linearly with depth at the same rate assumed for the present numerical results [Eq. (1)]. Mur and Hamilton [23] attributed the scatter shown in Fig. 11 at shallow depths to `both the low shear strength near the mud line and the inherent scatter in the soil-resistance derivation methodology.' The latter point refers to numerical procedures that infer bearing pressures on model piles from measured bending strains. The present numerically determined bearing factors are intermediate to those determined by Mur and Hamilton [23], and Matlock [24]. Limiting bearing factors N assumed by Mur and Hamilton, and Matlock are 12 and 9, respectively. The Matlock method results in lower lateral bearing factors than the experimental results and other numerical and analytical limit results shown in Fig. 11.
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Fig. 11. Predicted vs experimental soil resistance (reproduced with permission from ASCE [23]).
Randolph et al. [25] presented results for centrifuge tests of model caissons having L/D ratios of 2.3. Tests in a normally consolidated soil, having a linearly increasing shear strength with depth, resulted in an equivalent bearing factor of N Pult =Su DL 8:4, where Su is the shear strength at mid depth of the caisson. The point of load application was at 64% of the caisson depth, with a load inclination of approximately 11 from the horizontal. The equivalent bearing factor for the numerical results presented here for the inclined load at mid-depth is 10.1 (based on horizontal load component only). The equivalent N factors for horizontal loads at the mid-depth and top locations are 7.8 and 3.6, respectively. It is noted that the equivalent lateral bearing factors N are average values in the sense that shallow caissons include signi®cant resistance from shear along the bottom surface of the caisson soil plug that is included in the normalization with respect to the projected lateral bearing area of the caisson. Magnitudes of plastic strain are plotted in Fig. 12 on the deformed mesh for an inclined load analysis with Fourier elements. The zone of plastic action is contained within a distance of three caisson diameters of the caisson axis. This is also the zone of signi®cant soil deformation. The mobilized soil mass is roughly conical in shape and extends to a depth of one half diameter below the caisson base. The authors performed limit analyses with hybrid forms of the Fourier elements and found that these elements produced limit loads approximately 3% lower than the displacement based formulation. It can be concluded based on these results that CAXA8R2 or CAXA8R4 elements can be used for three-dimensional analyses of suction caissons without loss in accuracy compared to the full three-dimensional
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Fig. 12. Plastic strains developed in the soil surrounding the caisson.
formulations. Further, the de®nition of the ®nite element model is much less time consuming, and limit loads compare favorably with available experimental results. Investigations not presented here showed that the limit loads determined from analyses with and without the eect of soil self-weight are negligibly dierent. The reason for this can be seen in the form of the failure mechanisms shown in Fig. 8. Since the passive and active wedges are the same size, the work contribution due to the self-weights sum to zero. That is, the weight of material lifted in front of the caisson is the same as that pulled down on the opposite side, thus resulting in no net work being performed. Dierent results are expected if separation between the caisson and soil occurred on the active pressure side [21,25]. 7. Conclusions Few studies have been conducted to examine the response under loading of suction caissons in Gulf of Mexico clays. The present analyses used linear elasticity combined with the von Mises strength model to describe the deformation and strength properties typical of Gulf of Mexico deepwater clays. New insights were obtained on the extent of area in which displacements will occur due to loading of a suction caisson as well as the magnitude of horizontal stresses that are expected to develop on the caisson wall. The following conclusions can be drawn from the study: . The maximum anchor capacity is obtained when the load attachment point forces the caisson to have a translational mode of failure rather than a rotational mode of failure. This is in agreement with earlier ®ndings by Keaveny et al. [21], and Colliat et al. [22] but con¯icts with those of Mur and Hamilton
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. .
. . .
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[23] who concluded that translational and rotational mechanisms resulted in essentially the same limit loads. Inclined loads applied at the face of the caisson tend to reduce caisson rotation, resulting in greater lateral capacity. A pseudo three-dimensional model available in ABAQUS, utilizing the use of Fourier interpolation to de®ne approximate three-dimensional conditions can be used instead of an actual three-dimensional model for analyzing the capacity of a suction caisson with considerable computational time savings. While not pointed out elsewhere in this paper, the authors would like to note that the use of reduced integration procedures and the Fourier modeling technique oered large computational eciencies with no loss of accuracy. For the problems considered herein, using reduced integration procedures for three-dimensional problems resulted in a 40% reduction in solution time requirements compared to full integration procedures. Fourier solutions required approximately 20% of the computational time of full three-dimensional analyses. Limiting lateral bearing pressures on deep piles in cohesive soils are greater than those currently used in the design of oshore piles. Other limit analysis solutions and experimental observations support this conclusion. The lateral resistance of suction caissons is not aected by installation disturbance of the soil near the caisson wall since the source of resistance is the soil in the passive and active pressure zones. Accurate predictions of the capacity of suction caissons, footings and other embedded structures can be obtained from ®nite element analyses. Reduced integration elements were shown to produce well-de®ned limit conditions in both two- and three-dimensional conditions. Hybrid elements generally provided lower limit loads than displacement based formulations. The dierence in caisson capacities determined by the hybrid and displacement element formulations is relatively small.
These results have important practical implications for the estimation of the capacity and the design of suction caissons. References [1] Christophersen HP, Bysveen S, Stove OJ. Innovative foundation systems selected for the Snorre ®eld development. Proceedings of the Behavior of Oshore Structures Conference, 1992. p. 81±94. [2] Baerheim M, Heberg L, Tjelta TI. Development and structural design of the bucket foundations for the Europile jacket. Proceedings of the Oshore Technology Conference, 1995. p. 859±868. [3] Clukey EC, Morrison MJ. A centrifuge and analytical study to evaluate suction caissons for TLP applications in the Gulf of Mexico. Design and Performance of Deep Foundations: Piles and Piers in Soil and Soft Rocks, Dallas (TX), 1993. p. 141±156 [preprint]. [4] Zienkiewicz OC, Taylor RL. The ®nite element method, vol. 1, 4th ed., New York: McGraw±Hill, 1994. [5] Zienkiewicz OC, Taylor RL, Too JM. Reduced integration technique in general analysis of plates and shells. International Journal for Numerical and Methods in Geomechanics 1971;3:275±90. [6] Nagtegaal JC, Parks DM, Rice JR. On numerically accurate ®nite element solutions in the fully plastic range. Computer Methods in Applied Mechanics and Engineering 1974;4:153±77.
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[7] Sloan SW, Randolph MF. Numerical prediction of collapse loads using ®nite element methods. International Journal for Numerical and Analytical Methods in Geomechanics 1982;6:47±76. [8] De Borst R, Vermeer RA. Possibilities and limitations of ®nite elements for limit analysis. Goetechnique 1984;34:199±210. [9] Whittle AJ, Germaine JT. Behavior of suction caisson foundations. Annual technical report. Cambridge (MA): Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, 1993. [10] Barlow J. Optimal stress locations in ®nite element models. International Journal for Numerical Methods in Engineering 1976;10:243±51. [11] Barlow J. More on optimal stress points-reduced integration, element distortions and error estimation. International Journal for Numerical Methods in Engineering 1989;28:1487±504. [12] Griths DV. Elasto-plastic analyses of deep foundations in cohesive soil. International Journal for Numerical and Analytical Methods in Geomechanics 1982;6:211±8. [13] Naylor DJ. Stresses in nearly incompressible materials by ®nite elements with applications to the calculations of excess pore pressures. International Journal for Numerical Methods in Engineering 1974;8:443±60. [14] HKS, ABAQUS User's manualÐversion 5.7. Hibbitt, Karlsson and Sorensen, USA, 1997. [15] Chen WF. Constitutive equations for engineering materials. Plasticity and modeling, vol. 2. Elsevier, New York, NY, 1994. [16] Chen WF. Limit analysis and soil plasticity. Elsevier Scienti®c, New York, NY, 1975. [17] Meyerhof GG. The ultimate bearing capacity of foundations. Geotechnique 1951;2:301±32. [18] Randolph MF, Houlsby GT. The limiting pressure on a circular pile loaded laterally in cohesive soil. Geotechnique 1984;34:613±23. [19] Hamilton JM, Phillips R, Dunnavant TW, Mur JD. Centrifuge study of laterally loaded pile behavior in clay. Proceedings of the International Conference on Centrifuge, ISSMFE, 1991. [20] API, Recommended practice for planning, designing and constructing ®xed oshore platformsÐ working stress design. API RP 2A-WSD, 20th ed. Washington (DC): American Petroleum Institute, 1993. [21] Keaveny JM, Hansen SB, Madshus C, Dyvik R. Horizontal capacity of large scale model anchors. XIII International Conference on Soil Mechanics and Foundation Engineering, New Delhi, India (India), 1994. [22] Colliat JL, Boisard P, Andersen K, Schroeder K. Caisson foundations as alternative anchors for permanent mooring of a process barge oshore Congo. 27th Annual Oshore Technology Conference, 1995. [23] Mur JD, Hamilton JM. P-ultimate for undrained analysis of laterally loaded piles. ASCE Journal of Geotechnical Engineering 1993;119:91±107. [24] Matlock H. Correlations for design of laterally loaded piles in soft clay. Paper no. 1204. Houston (TX): Oshore Technology Conference, 1970. [25] Randolph MF, O'Neill MP, Stewart DP, Erbrich E. Performance of suction anchors in ®ne-grained calcareous soils. Proceedings of the Oshore Technology Conference. Paper no. 8831, 1998. p. 521± 529.
Ecient ®nite element techniques for limit analysis of suction caissons under lateral loads B. Sukumaran a,*, W.O. McCarron b, P. Jeanjean b, H. Abouseeda c a Rowan University, 201 Mullica Hill Road, Glassboro, NJ 08028, USA Amoco Worldwide Engineering and Construction, Houston, TX 77058, USA c Fugro-McClelland Marine Geosciences, Inc., Houston, TX 77274, USA
b
Received 10 August 1998; received in revised form 20 November 1998; accepted 4 December 1998
Abstract This paper documents the use of ®nite element analyses techniques to determine the capacity of suction caisson foundations founded in soft clays under undrained conditions. The stress±strain response of the soft clay is simulated using an elasto-plastic model. The constitutive model employed is the classical von Mises strength criterion with linear elasticity assumed within the yield/strength surface. Both two- and three-dimensional foundation con®gurations are analyzed. The three-dimensionality of the failure surface of the actual caisson requires that computationally intensive three-dimensional models be used. Suggestions are given on how to improve computational eciency by using quasi three-dimensional Fourier analyses with excellent results instead of true three-dimensional analyses. The ®nite element techniques employed are veri®ed against available classical limit solutions. Results indicate that both hybrid and displacement-based ®nite element formulations are adequate, with the restriction that reduced-integration techniques are often required for displacement-based formulations. # 1999 Elsevier Science Ltd. All rights reserved.
Nomenclature A B d D
bearing area width or diameter of footing displacement diameter
* Corresponding author. Tel.: +1-609-256-5324; fax: +1-609-256-5242; e-mail: sukumaran@rowan. edu 0266-352X/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S026 6-352X(98)0003 6-6
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DSS f L N P Su z
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direct simple shear (shear strength evaluation) coecient of friction length bearing factor load undrained shear strength of soil depth
1. Introduction As oshore exploration and development of oil ®elds reach water depths in the 1000 to 3000 m range, novel methods of anchoring production platforms become attractive due to cost savings associated with oshore installation activities. Surface production systems that are viable in these water depths include Tension Leg Platforms (TLP), Spar platforms, and laterally moored ship-shaped and semi-submersible vessels. TLPs are ¯oating structures anchored by vertical pretensioned tendons which exert considerable tensile forces on the foundation. In the Gulf of Mexico, vertical design loads for individual tendons range up to 27 MN (6000 kips), resulting in foundation lateral loads of 4 MN (900 kips). Laterally moored systems, on the other hand, have the dominant load in the horizontal direction; the horizontal load component being on the order of 9 MN (2000 kips), and the vertical component being less than half that of the horizontal component. At this time (1998), there are ®ve TLPs and two Spars installed and operating in the Gulf of Mexico. Several other TLPs and Spars are in various stages of design and fabrication. For Gulf of Mexico TLPs, driven piles have been the preferred method of anchoring tendons to the sea¯oor. This solution arises from a combination of engineering, geotechnical, fabrication, installation and cost constraints. Chief among these are the high accuracy (within 1 m) required for positioning the tendon anchoring points, and the normally consolidated cohesive soils without appreciable strength in the upper 20 m. These conditions combined with available oshore installation barges, underwater hammers capable of operating in 1300 m of water and simple fabrication of steel pile shapes lead to the preference of driven piles. The depth of penetration of TLP piles ranges up to 120 m. Spars are buoyant single-column hulls anchored by lateral moorings to foundation elements. In water depths less than approximately 1200 m (4000 ft), TLP and Spar systems are competitive economically, but TLPs have a more proven construction, technology and operating history. For greater water depths, the Spar platform oers some performance and economic advantages. Possible foundation systems for Spars include the traditional driven piles, drag anchors and suction caissons. Suction caissons have been used in shallow waters as foundations for single point moorings, jack-up drilling rigs, ®xed platforms and as anchors for ¯oating systems. Initial penetration of the suction caisson into the seabed occurs due to the self
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weight; subsequent penetration is by the `suction' created by pumping water out from the inside of the caisson. Suction caissons become attractive alternatives to driven piles in deepwater because of technical challenges and costs associated with the installation equipment. In addition, suction caissons also provide a greater resistance to lateral loads than driven piles because of the larger diameters typically used. The terminology for this foundation is sometimes misleading. `Suction' can refer to the method of installation or a component of foundation load resistance, or both. Fig. 1 shows a schematic view of a Spar platform anchored by mooring to suction caissons. The feasibility of suction caissons has been demonstrated in the North Sea foundations for the Snorre TLP [1], Europipe 16/11-E structure [2], and with centrifuge tests for Gulf of Mexico TLP conditions [3]. At present, the use of suction caissons are being extended to the Gulf of Mexico. Soil conditions in the North Sea (sti clays and sands) have so far lead to designs with penetration to diameter ratios typically less than 2. Because the deepwater shallow sediments in the Gulf of Mexico exhibit very low surface shear strength, it is necessary to increase the penetration to diameter ratio of the caisson to obtain satisfactory capacities. However, experience
Fig. 1. Schematic view of a spar platform anchored to suction caissons at the seabed.
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with the installation and behavior of caissons with large penetration to diameter ratios (up to 10) is limited. To investigate the installation, performance and capacity of such caissons, a series of ®eld tests, centrifuge tests, and numerical investigation have been commissioned by the industry. This paper presents the ®ndings of some numerical investigations aimed at validating numerical procedures to calculate the lateral capacity of caissons for deepwater foundations. 2. Some background on ®nite element limit analysis Available displacement based and hybrid (combined stress and displacement solution variables) based ®nite elements formulations are capable of accurately and eciently calculating limit loads for foundation systems. An important feature in the successful use of displacement based ®nite element formulations is the use of reduced integration techniques in many limit analysis investigations. The term `reduced' integration refers to the fact that a lower level (fewer sampling points) of numerical integration is being used than that theoretically required, to exactly integrate a polynomial of a certain order. Experience has indicated that the use of reduced integration techniques improves performance under conditions of nearly incompressible (entirely deviatoric plastic strains) response for von Mises and other pressure independent material strength models near the limit conditions. Historical discussions on the relative merits of full and reduced integration techniques are given by Zienkiewicz and Taylor [4] and Zienkiewicz et al. [5]. Nagtegaal et al. [6] discussed the success and failures of several fully integrated elements with respect to their ability to accurately predict limit loads in association with elastic-plastic material models. Sloan and Randolph [7] extended this work for plane strain (strip) and axisymmetric (circular) footing con®gurations, and proposed a triangular 15-noded element for use in axisymmetric problems. The performance of this element was later discussed by de Borst and Vermeer [8], and Whittle and Germaine [9]. Barlow [10,11] presented mathematical arguments that reduced integration in quadratic (eight-noded) elements enhances performance and solution convergence. Griths [12] presented the successful use of quadratic reduced-integration elements in plane-strain and axisymmetric conditions. Naylor [13] discussed the elements' performance for nearly incompressible conditions. Zienkiewicz and Taylor [4] demonstrate that reduced integration elements, of the type used here, satisfy the mathematical conditions of stability and convergence required in the `patch' test. While it is beyond the intended scope of this paper to review in detail the theoretical studies on reduced integration cited by these authors, two key ®ndings are summarized in the following for completeness of discussion. First, the minimum level of numerical integration (quadrature) necessary to insure stability and convergence of solution is that which correctly integrates the volume of the ®nite element. For the two-dimensional quadratic eight-noded element used here, this implies a four-point quadrature for reduced integration, as opposed to full quadrature which leads to a nine point requirement. Second, reduced integration leads to a weak singularity in the single element stiness and fewer internal constraints on the
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coupling between volumetric and deviatoric strains. This singularity does not appear in equilibrium equations when more than one element is involved. We therefore conclude that reduced integration techniques have a ®rm theoretical basis, supporting their application. However, as with any ®nite element, their robustness and accuracy in particular applications should be critically examined. Alternatives to the use of reduced integration exist, e.g. hybrid ®nite elements, or very high order displacement-based elements such as the 15-noded cubic strain triangle. Hybrid elements are available in commercial codes, such as ABAQUS [14], and are eective in the analysis of incompressible materials. The term hybrid stems from the use of both displacement and stress components as solution variables. In this case, the stress component included is the mean pressure. Detailed discussion of these elements are given by Zienkiewicz and Taylor [4] and HKS [14]. The performance of the hybrid elements are compared herein with results of displacement based elements. 3. Veri®cation of ®nite element modeling techniques The adequacy of the hybrid and reduced-integration elements are demonstrated in the following by virtue of their performance in accurately calculating the limit loads for three problems; some of which have typically proven to be problematic for a wide range of element formulations. The analyses were performed with the program ABAQUS [14]. The ®rst series of problems are plane strain (strip) and axisymmetric (circular) footings on the surface of purely cohesive soil. Secondly, deeply embedded footings in cohesive soil are considered. The ®nal problem relates to the ultimate lateral resistance of a circular pile cross section in a cohesive soil. The ®nite element analyses make use of isotropic elasticity combined with a von Mises or Tresca type strength surface. The implementation of the Tresca elastic±plastic constitutive model includes a non-associated ¯ow rule. Young's modulus is taken as approximately 1000 times greater than the undrained strength. Footing analyses presented here represent rigid footings and a weightless soil. The surface and deep footings are analyzed with smooth and perfectly rough interfaces, respectively. The footing limit load is conventionally denoted as Pult NSu A, where N is the bearing capacity factor, Su is the undrained strength, and A is the bearing area. Theoretical limit solutions presented here are based on the Tresca strength condition. The Tresca and Mohr±Coulomb criteria are identical when the Mohr±Coulomb internal friction parameter is zero, as is the case for a purely cohesive material. Finite element strength models used here include both the von Mises and Tresca criteria. Plane strain (strip) footings are analyzed with the von Mises condition matched to the Tresca strength using the procedure presented by Chen [15]. For circular footings, limit loads are presented for the Tresca criterion. Limit loads predicted with displacement and hybrid elements for strip and circular surface footings are shown in Fig. 2. The ®nite element mesh is shown in Fig. 2(a), and the load±displacement relationship in Fig. 2(b). Displacement (D) and hybrid (H) element formulations are indicated in Fig. 2(b), along with the order of integration, full/standard (S) and reduced (R). Thus, SD implies a standard integration
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combined with a displacement formulation. A small displacement formulation was used in the analyses to formulate the element stiness and equilibrium equations. Limit solutions are well de®ned for all conditions but the fully integrated displacement
Fig. 2. (a) Finite element mesh used in the analyses of the capacity of surface footings; (b) bearing capacity factor, N vs normalized displacement computed for surface strip and circular footings.
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formulation element in the circular footing case; as expected based on previous experience. Theoretical bearing capacity factors, N for smooth and rough circular footings are 5.69 and 6.05 [15], respectively. The theoretical bearing capacity factor for a strip footing is 5.14. The calculated bearing capacity factors for the strip footing range from 5.3 to 5.4. For circular footings, the calculated bearing capacity factors range from 5.77 upwards. The RH and RD analyses with the Tresca material model for the circular footing fall within a reasonable range of the theoretical solution. The bearing capacity factor of 5.77 from the RH analysis is essentially the exact solution for the smooth footing condition considered. The ®nite element mesh and results of limit analysis of embedded deep strip and circular footings are shown in Fig. 3. A large displacement formulation is required for this particular problem because of the relatively large stresses compared to the elastic moduli. Theoretical solutions for these conditions have been given by Chen [16], using upper-bound limit analysis, and Meyerhof [17], using limit equilibrium methods. The ratio of the depth of embedment to footing width is 4, and the shaft of the footing is smooth and allows no horizontal deformations. The theoretical solutions by Meyerhof [17] result in bearing capacity factors ranging from 8.85 to 9.74 for rough strip and circular footings, respectively. Chen [16] reported a bearing capacity factor of approximately 9 for deep strip footings. The results for the RD analyses for strip and circular footings indicate bearing capacity factors of 8.9 and 10.7, respectively. The RH analyses result in bearing capacity factors of 7.7 and 9.6 for strip and circular footings, respectively. In the present case, the fully integrated displacement elements (SD) perform poorly and do not reach a limit (results not presented). Accurate determinations of the bearing capacity of deep embedded footings are complicated by the fact that the plastically deforming region is con®ned within an elastic region, so that the plastically strained zone is not free to undergo the unlimited plastic deformation typically associated with limit conditions. In the present cases, the zone of plastic behavior is contained within about two footing diameters distance from the footing (results not shown). The ®nal illustrative example is that of determining the lateral resistance of a circular pile cross section. For this analysis, a plane strain idealization and von Mises strength criterion matched to the plane strain condition is adopted. The ®nite element mesh is shown in Fig. 4(a). The results are shown in Fig. 4(b) for three pile±soil interface conditions: rough with no separation, gapping (separation allowed, no tensile forces transmitted), and gapping with a frictional interface. The theoretical limit solutions have been presented by Randolph and Houlsby [18]. Those authors presented lower-bound solutions with bearing capacity factors, N of 9.14, 10.52, and 11.94 for friction coecients f of 0, 0.4, and 1, respectively. These correspond well with the results shown in Fig. 4(b) for the same conditions obtained for both reduced integration and hybrid elements. Hamilton et al. [19] reported experimental investigations showing N values generally between 10 and 12 for depths greater than four pile diameters. These results are further discussed later in the paper. Current practice in the oshore industry [20] assumes the lateral bearing capacity factor at large depths to be N 9. Based on the results presented here, limit solutions [18], and experimental investigation [19], the current API practice is conservative.
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Fig. 3. Bearing capacity factor, N vs normalized displacement for deeply embedded circular and strip footings.
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Fig. 4. (a) Finite element mesh used for determination of lateral pile capacity; (b) bearing capacity factor vs normalized displacement for circular pile.
4. Lateral load capacity analyses of suction caissons The ultimate holding capacity of a suction caisson anchored at a site with soil conditions similar to that found in the Gulf of Mexico were analyzed. The soil at the site where the suction caisson is expected to be anchored is a normally consolidated clay. The shear strengths are assumed to be zero at the seabed and increasing linearly with depth as given below:
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SDSS 1:41z kPA u
1
is the undrained static direct where z is the depth below seabed in meters and SDSS u simple shear strength. The submerged unit weight of the soil is 6.3 kN/m3. The ®nite element analyses were conducted using ABAQUS [14]. A von Mises shear strength idealization was used to model the clay. The von Mises model implies a purely cohesive (pressure independent) soil strength de®nition. The caisson is modeled as a weightless linear elastic material. 5. Two-dimensional load capacity analysis of suction caissons Two-dimensional analyses were performed to con®rm the mesh and boundary de®nitions selected for three-dimensional investigations are suitable for classical passive and active pressure problems. The caisson analyzed was 6.1 m in diameter with a penetration depth of 12.2 m below the mud line. The caisson length to diameter ratio is L/D=2. The caisson has a closed top during installation and operation. Initial horizontal stresses were de®ned with a coecient of lateral earth pressure Ko equal to 1. The ®nite element mesh shown in Fig. 5 was used for the analyses. The mesh model dimensions of 56 m width by 36.6 m depth was intended to minimize boundary eects on response. The mesh consists of eight-noded plane strain elements for the soil and two-noded linear beam elements for the caisson. The inclination of the load considered was assumed to be 28 with the horizontal, measured counterclockwise. Several points of attachment for the mooring line were
Fig. 5. Finite element mesh used for determining the capacity of suction caissons.
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considered to study the eect of the attachment point on the load capacity. The optimal load attachment point is that which produces maximum capacity. Fig. 6 shows a plot of load capacity vs point of attachment. The capacity generally increases with depth of attachment. This is in agreement with previously published results [21,22]. Fig. 7 shows the horizontal stress acting on the wall when it is constrained to translate horizontally with no rotation occurring. Both smooth and rough soil±wall interfaces are considered. The active and passive pressures computed at the limit conditions display the expected linear distribution with depth, and closely match the pressures calculated for classic active and passive pressure retaining wall response. The eect of the load attachment point on the failure mechanism produced was also studied. Fig. 8(a)±(c) shows the various failure mechanisms produced when the load is attached above, at and below the optimum point. Fig. 8(a) and (c) shows that when the load attachment point is above or below the optimal point, the caisson rotates. The failure mechanism is more rotational than translational. The shear zone mobilized is also less in area than if the load is attached at the optimal attachment point. It can be seen from Fig. 8(b) that when the load is attached at the optimal load attachment point, the failure mechanism is predominately translational. From Fig. 6, it can be seen that the load capacity only decreases slightly if the attachment point is below mid-height of the caisson.
Fig. 6. Load capacity of suction caisson (kN/m) vs depth to load attachment point obtained from the analyses of a plane strain model.
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Fig. 7. Horizontal stress acting on the wall on the active and passive side.
6. Comparison of load±deformation response of Fourier and three-dimensional analyses The three-dimensional response was analyzed using both three-dimensional and quasi three-dimensional Fourier analysis elements (CAXA) available within ABAQUS. CAXA elements are biquadratic, Fourier quadrilateral elements. These elements were used for the analyses of suction caissons because they allow non-linear, asymmetric deformations and loading. Two types of CAXA elements, namely the CAXA8R2 and CAXA8R4, were used in the analyses. These are eight-noded quadrilateral reduced integration elements that dier in the number of Fourier modes used for interpolation. CAXA8R2 elements use two Fourier modes for interpolation while the CAXA8R4 uses four. The number of elements and nodes in the mesh are 180 and 3616, respectively. A three-dimensional model having a similar mesh con®guration with 20-noded brick elements was also developed to compare the results from a quasi three-dimensional Fourier analysis and the actual three-dimensional response. The far boundaries of the model are modeled as perfectly rough (no translations allowed). The modeled caisson does not include a top plate, which results in free deformations of the top soil surface in the caisson soil plug. The plane of symmetry of the three-dimensional and axisymmetric models is identical to the plane-strain model shown in Fig. 5. The point of load application is at about mid-height of the caisson (5.97 m below the mud line). The inclination of the load was 32 with the horizontal, measured counterclockwise. The coecient of lateral earth pressure Ko is 0.8. The load-displacement curves for the three-dimensional model as well as the axisymmetric analyses using two and four Fourier modes
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Fig. 8. (a) Plot of displacement vectors indicating failure pattern when horizontal load is attached at the top of the caisson; (b) plot of displacement vectors indicating failure pattern when horizontal load is attached at the optimal attachment point; (c) plot of displacement vectors indicating failure pattern when horizontal load is attached below optimal attachment point.
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are shown in Fig. 9. The load indicated in Fig. 9 represents the load vector magnitude (resultant from horizontal and vertical components). The three-dimensional and Fourier analyses give the same limit loads, approximately 7700 kN. Finite element analyses with a purely horizontal applied load results in a limit load of approximately 5000 kN when the load is applied at mid-height of the caisson, and 2300 kN when the load is applied at the top of the caisson. Mur and Hamilton [23] present methods using upper-bound limit analyses, which give a capacity of 7000 kN for the case of pile translating horizontally in a soil mass with full adhesion and suction assumed on the back side of the pile. Those authors also compared their solutions with experimental centrifuge tests in kaolin clay previously presented by Hamilton et al. [19]. The present numerical results are compared with these limit analysis and experimental results in the following. Fig. 10 shows the non-dimensionalized normal (radial) stresses acting on the outside wall of the caisson from the axisymmetric±asymmetric analysis. The 0 plane is the plane along which the load is attached. The results in Fig. 10 were obtained by dividing the soil radial stresses adjacent to the caisson by the soil strength [Eq. (1)] at the respective depth. The left side of the plot in Fig. 10 represents results with an inclined (at 32 from the horizontal) load, while the right side represents results for a purely horizontal load. The results from the analysis with an inclined load generally show higher non-dimensionalized pressures than that for a horizontal load. This can be explained by the fact that the vertical load component applied on the face of the caisson tends to reduce rotation of the caisson. The non-dimensionalized stresses generally increase with depth when rotation of the caisson is limited, but when signi®cant
Fig. 9. Load capacity vs normalized displacement for the three-dimensional and axisymmetric±asymmetric analyses.
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Fig. 10. Normal stresses acting on the outside of the caisson at various depths below mudline.
rotation does occur, passive pressures are reduced at the leading edge of the base of the caisson. As shown in Fig. 11, the present numerical results compare reasonably with available limit solutions and experimental results. Fig. 11 includes ultimate lateral pressures computed from methods commonly used in the oshore industry developed by Matlock [24] for laterally loaded piles in soft clays. Centrifuge tests performed by Hamilton et al. [19,23] resulted in a mean bearing factor of 11 over a wide range of depths. The analytical limit solutions and experimental results in Fig. 11 are for a strength pro®le increasing linearly with depth at the same rate assumed for the present numerical results [Eq. (1)]. Mur and Hamilton [23] attributed the scatter shown in Fig. 11 at shallow depths to `both the low shear strength near the mud line and the inherent scatter in the soil-resistance derivation methodology.' The latter point refers to numerical procedures that infer bearing pressures on model piles from measured bending strains. The present numerically determined bearing factors are intermediate to those determined by Mur and Hamilton [23], and Matlock [24]. Limiting bearing factors N assumed by Mur and Hamilton, and Matlock are 12 and 9, respectively. The Matlock method results in lower lateral bearing factors than the experimental results and other numerical and analytical limit results shown in Fig. 11.
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Fig. 11. Predicted vs experimental soil resistance (reproduced with permission from ASCE [23]).
Randolph et al. [25] presented results for centrifuge tests of model caissons having L/D ratios of 2.3. Tests in a normally consolidated soil, having a linearly increasing shear strength with depth, resulted in an equivalent bearing factor of N Pult =Su DL 8:4, where Su is the shear strength at mid depth of the caisson. The point of load application was at 64% of the caisson depth, with a load inclination of approximately 11 from the horizontal. The equivalent bearing factor for the numerical results presented here for the inclined load at mid-depth is 10.1 (based on horizontal load component only). The equivalent N factors for horizontal loads at the mid-depth and top locations are 7.8 and 3.6, respectively. It is noted that the equivalent lateral bearing factors N are average values in the sense that shallow caissons include signi®cant resistance from shear along the bottom surface of the caisson soil plug that is included in the normalization with respect to the projected lateral bearing area of the caisson. Magnitudes of plastic strain are plotted in Fig. 12 on the deformed mesh for an inclined load analysis with Fourier elements. The zone of plastic action is contained within a distance of three caisson diameters of the caisson axis. This is also the zone of signi®cant soil deformation. The mobilized soil mass is roughly conical in shape and extends to a depth of one half diameter below the caisson base. The authors performed limit analyses with hybrid forms of the Fourier elements and found that these elements produced limit loads approximately 3% lower than the displacement based formulation. It can be concluded based on these results that CAXA8R2 or CAXA8R4 elements can be used for three-dimensional analyses of suction caissons without loss in accuracy compared to the full three-dimensional
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Fig. 12. Plastic strains developed in the soil surrounding the caisson.
formulations. Further, the de®nition of the ®nite element model is much less time consuming, and limit loads compare favorably with available experimental results. Investigations not presented here showed that the limit loads determined from analyses with and without the eect of soil self-weight are negligibly dierent. The reason for this can be seen in the form of the failure mechanisms shown in Fig. 8. Since the passive and active wedges are the same size, the work contribution due to the self-weights sum to zero. That is, the weight of material lifted in front of the caisson is the same as that pulled down on the opposite side, thus resulting in no net work being performed. Dierent results are expected if separation between the caisson and soil occurred on the active pressure side [21,25]. 7. Conclusions Few studies have been conducted to examine the response under loading of suction caissons in Gulf of Mexico clays. The present analyses used linear elasticity combined with the von Mises strength model to describe the deformation and strength properties typical of Gulf of Mexico deepwater clays. New insights were obtained on the extent of area in which displacements will occur due to loading of a suction caisson as well as the magnitude of horizontal stresses that are expected to develop on the caisson wall. The following conclusions can be drawn from the study: . The maximum anchor capacity is obtained when the load attachment point forces the caisson to have a translational mode of failure rather than a rotational mode of failure. This is in agreement with earlier ®ndings by Keaveny et al. [21], and Colliat et al. [22] but con¯icts with those of Mur and Hamilton
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. .
. . .
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[23] who concluded that translational and rotational mechanisms resulted in essentially the same limit loads. Inclined loads applied at the face of the caisson tend to reduce caisson rotation, resulting in greater lateral capacity. A pseudo three-dimensional model available in ABAQUS, utilizing the use of Fourier interpolation to de®ne approximate three-dimensional conditions can be used instead of an actual three-dimensional model for analyzing the capacity of a suction caisson with considerable computational time savings. While not pointed out elsewhere in this paper, the authors would like to note that the use of reduced integration procedures and the Fourier modeling technique oered large computational eciencies with no loss of accuracy. For the problems considered herein, using reduced integration procedures for three-dimensional problems resulted in a 40% reduction in solution time requirements compared to full integration procedures. Fourier solutions required approximately 20% of the computational time of full three-dimensional analyses. Limiting lateral bearing pressures on deep piles in cohesive soils are greater than those currently used in the design of oshore piles. Other limit analysis solutions and experimental observations support this conclusion. The lateral resistance of suction caissons is not aected by installation disturbance of the soil near the caisson wall since the source of resistance is the soil in the passive and active pressure zones. Accurate predictions of the capacity of suction caissons, footings and other embedded structures can be obtained from ®nite element analyses. Reduced integration elements were shown to produce well-de®ned limit conditions in both two- and three-dimensional conditions. Hybrid elements generally provided lower limit loads than displacement based formulations. The dierence in caisson capacities determined by the hybrid and displacement element formulations is relatively small.
These results have important practical implications for the estimation of the capacity and the design of suction caissons. References [1] Christophersen HP, Bysveen S, Stove OJ. Innovative foundation systems selected for the Snorre ®eld development. Proceedings of the Behavior of Oshore Structures Conference, 1992. p. 81±94. [2] Baerheim M, Heberg L, Tjelta TI. Development and structural design of the bucket foundations for the Europile jacket. Proceedings of the Oshore Technology Conference, 1995. p. 859±868. [3] Clukey EC, Morrison MJ. A centrifuge and analytical study to evaluate suction caissons for TLP applications in the Gulf of Mexico. Design and Performance of Deep Foundations: Piles and Piers in Soil and Soft Rocks, Dallas (TX), 1993. p. 141±156 [preprint]. [4] Zienkiewicz OC, Taylor RL. The ®nite element method, vol. 1, 4th ed., New York: McGraw±Hill, 1994. [5] Zienkiewicz OC, Taylor RL, Too JM. Reduced integration technique in general analysis of plates and shells. International Journal for Numerical and Methods in Geomechanics 1971;3:275±90. [6] Nagtegaal JC, Parks DM, Rice JR. On numerically accurate ®nite element solutions in the fully plastic range. Computer Methods in Applied Mechanics and Engineering 1974;4:153±77.
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