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Finite element modeling of suction anchors under combined loading
Marine Structures 22 (2009) 660–669
Contents lists available at ScienceDirect
Marine Structures journal homepage: www.elsevier.com/locate/ marstruc
Finite element modeling of suction anchors under combined loading H. Monajemi, H. Abdul Razak* Department of Civil Engineering, University of Malaya, 50603 Kuala Lumpur, Malaysia
a r t i c l e i n f o
a b s t r a c t
Article history: Received 24 July 2007 Accepted 15 February 2009
Suction anchors are subjected to inclined, quasi-horizontal or quasi-vertical loadings. The type of the structure and depth of water govern the inclination of the load. Under this load condition, suction anchor experiences a combination of horizontal and vertical translations, and rotation. Therefore the soil’s reaction to this load condition can be idealized as horizontal and vertical loadings, together with a moment. The magnitude and combination of the reactions depend on the load inclination, soil property and the point at which the load is applied. The behavior of the suction anchor subjected to the combined V–H-M loading is elaborated in this paper. This is to observe the effects of soil properties on the failure mechanism and ultimate capacity of the foundation. This was achieved by applying the pure horizontal and vertical displacements, and rotation and their combinations to the foundation, on V–H, V-M and H-M spaces and the yield-locus created for each space. The general purpose finite element program DIANA was used for this study. Non-linear analysis was conducted using a simplified elastic-perfectly-plastic model with Von-Mises yield criterion for saturated and consolidated clay. Ó 2009 Elsevier Ltd. All rights reserved.
Keywords: Suction anchor Combined loading Finite element
1. Introduction Suction foundations and anchors are used increasingly for fixed and floating offshore structures. They can be used as an alternative to piles and gravity foundations in fixed structures i.e. suction caisson or pile and drag anchors in floating structures i.e. suction anchor. The skirt length to the
* Corresponding author. E-mail address: [email protected] (H.A. Razak). 0951-8339/$ – see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.marstruc.2009.02.001
H. Monajemi, H.A. Razak / Marine Structures 22 (2009) 660–669
661
Fig. 1. Suction anchor and padeye on the skirt.
diameter ratio (L/D) is generally less than one in the suction caissons, which are used in jacket platforms or wind turbines and more than one for the suction anchors, which are used in tension leg platforms (TLPs), or floating structures [1]. Suction anchor is a hollow cylinder, open at the bottom and closed at the top as illustrated in Fig. 1. It is installed to the seabed initially with self-weight to provide a seal between the foundation’s skirt tip and the soil, and then later by pumping out the water from within the skirt. This creates a differential pressure at the top of the caisson and pushes it into the soil. Suction anchor, depending on the type of the structure, may be subjected to lateral, pull out and inclined loading through a cable which is connected to the load attachment point i.e. padeye on the skirt as shown in Fig. 2. Padeye depth and cable’s angle are important parameters that determine the ultimate capacity of the foundation. Cable angle generally depends on the system of the structure and water depth. For example, it is quasi-vertical in Snorre tension leg platform [2,3], quasi-horizontal in Nkossa process barge [4]. In general, for water depth more than 1000 m, inclined loading is provided by taut wire moorings [5]. The behavior of the suction anchors and piles under inclined and lateral loadings has been the subject of various studies. For the anchors subjected to lateral loading, the capacity of the foundation has been described using a dimensionless lateral unit resistance factor, NP which is:
DH ¼ NP Su DDZ where DH is the horizontal resistance force along the length, D is the pile’s diameter, Su is the undrained shear strength of material and NP is a dimensionless lateral unit resistance factor. Randolph [6] employed plasticity theory to estimate the lateral resistance and collapse mechanism of laterally loaded piles. The horizontal resistance of the pile was estimated by considering the pile as
662
H. Monajemi, H.A. Razak / Marine Structures 22 (2009) 660–669
D F Li φ
Lo Lf
P r O RH
RM RV
Fig. 2. Suction anchor subjected to an inclined loading.
Fig. 3. Finite element model of suction anchor.
a translating cylinder which does not have any rotation in combination with the horizontal displacement. It was deduced that the value of NP for smooth (a ¼ 0) and rough (a ¼ 1) surfaces were equal to 9.42 and 11.94, respectively. Murff [7] and Aubeny [8,9] improved previous studies using plastic limit analysis (PLA) theory by considering the pile as a translating and rotating cylinder and describe a velocity field to define the energy dissipation equations. The method has been able to include the moment combined with the horizontal loading, gap-forming behavior behind the pile, adhesion coefficient, soil weight and linear soil profile. Instead of investigating the moment capacity of the foundation individually, a reduction factor was applied to the value of NP to present the effect of rotation on the lateral capacity of the pile. Supachawarote [5,10] presented a finite element study of suction anchor subjected to inclined loading. For each load angle, there is a special load attachment depth for which the foundation only undergoes a displacement and no rotation occurs. The capacity of the foundation for the combined V–H
Table 1 Soil shear strength profiles. Soil profile
Region
Shear strength
Profile 1 Profile 2
Benchmark Gulf of Mexico
Su ¼ 1 kPa Su ¼ 1.25z kPa
H. Monajemi, H.A. Razak / Marine Structures 22 (2009) 660–669
663
Stress
Hardening Perfectly - Plastic Softening
Strain Fig. 4. Stress–strain behavior of stiffness.
loading for different L/D ratios and adhesion factors were evaluated in the study. Furthermore the effect of the soil separation on the tension side of the foundation has been considered in their study. In this study, in addition to H–V space, the behavior of suction anchor in H-M and V-M spaces is investigated using displacement and rotation control analysis for two different soil profiles. The effect of the soil shear strength profile on the position of center of stiffness and the ultimate horizontal, vertical and moment capacities of the foundation will be presented. 2. Problem description and finite element model Fig. 2 shows a suction anchor subjected to an inclined loading. The anchor’s diameter and skirt length are shown with D and Lf respectively. The load attachment point called padeye is at the depth Li from the top and the center of stiffness is at the depth Lo. The direction of loading is at an angle f to the
Fig. 5. Finding the center of stiffness.
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Fig. 6. Applied displacements and rotation at the center of stiffness and their reactions.
horizontal. Due to this load condition, suction anchor experiences horizontal and vertical displacement and rotates around the center of rotation at the depth Lc. This give rise to horizontal, vertical and moment reactions from the surrounding soil as shown in Fig. 2. The maximum value of each individual reaction depends on the soil behavior, anchor’s aspect ratio, adhesion factor, etc. Furthermore each reaction load can be affected by the other reactions and any combination of them reduces the maximum values of each reaction. Since the inclined loading, usually gives rise to a combination of horizontal and vertical displacements, and rotation, it is important to investigate the foundation’s behavior under combined loading. For this purpose, a 3-D finite element analysis was carried out using DIANA release 9. A suction anchor 5 m in diameter and 10 m in depth was modeled within a cylindrical soil block, 35 m in diameter and 20 m in depth as shown in Fig. 3. Two different soil shear strength profiles were selected as given in Table 1. The value of soil’s modulus ratio was adopted, with E/Su of 500, and Poisson’s ratio was taken as 0.49. The soil response was taken as elastic-perfectly plastic using a Von-Mises failure criterion (Fig. 4). The selected soil was assumed to be normally consolidated and therefore no gap occurs in the backside of the caisson. The soil-foundation interface was assumed rough and thus the adhesion factor a ¼ 1 was used. Foundation and surrounding soil were assumed weightless. 3. Center of stiffness Supachawarote [5,10] and Randolph [11] reported the depth of the center of stiffness of 0.7 Lf from their study. The results which have been found in this study showed the depth of the center of stiffness Lo is strictly dependant on the soil’s shear strength profile and Young’s modulus profile. To find the value of Lo in the finite element model, a horizontal load F was applied to the center point on top of the caisson (Fig. 5). It causes a moment reaction on the foundation. The value of this moment was obtained using a rotational fixed boundary on the center point. The position of center of stiffness can then be found simply by dividing the value of moment by the corresponding applied force. The applied load F was increased in 10 increments and values of Lo calculated for all of them. Taking the average gives value of 5.7 m (0.57 Lf) and 7.3 m (0.73 Lf) for soil profile 1 and 2 respectively. 4. Ultimate capacity of the anchor Generally two methods of analysis can be used to find the ultimate capacity of the anchor. Load control analysis and displacement control analysis. In load control analysis, an inclined load can be
-02 2.00 E
-03 1.0 0E -02 1.5 0E -02
7.00E+05 1.00E+07
4.00E+06
1.80E+06
6.00E+05
6.00E+06
2.00E+06
Su = 1 kPa 4.00E+06
1.50E+06 1.00E+06
2.00E+06
5.00E+05 0.00E+00 0
0.1
0.2
0.3
0.4
0.00E+00 0.6
0.5
Displaement (m) FH
FV
1.50E+06
5.00E+05
1.20E+06
4.00E+05
9.00E+05
3.00E+05 Su = 1.25 z kPa 2.00E+05
6.00E+05
1.00E+05
3.00E+05
0.00E+00 0
0.1
0.2
0.3
0.4
0.5
Moment (N.m)
2.50E+06
Moment (N.m)
8.00E+06
3.00E+06
Load F (N)
3.50E+06
Load F (N)
5.0
0.0 0E +
4.50E+06
0E
00
Rotation (rad)
5.0 0E -03 1.0 0E -02 1.5 0E -0 2 2.0 0E -0 2 2.5 0E -0 2
0 .0
0E +0 0
Rotation (rad)
665
2.50 E2-0 2
H. Monajemi, H.A. Razak / Marine Structures 22 (2009) 660–669
0.00E+00 0.6
Displaement (m) M
FH
FV
M
Fig. 7. Load-Displacement and Moment-Rotation relationships for both soil profiles.
applied to the skirt at different padeye depths and different load path angles [5]. The load can be increase step by step using non-linear analysis. By defining the failure criterion and soil strength, after a certain load the program will stop the analysis due to large deformations in the entire model. This method is much closer to reality but rather time consuming, especially for a model with large number of elements. This is due to the iterations which the program has to perform within each step to solve the stiffness matrix. In this study, displacement control analysis was used to find the ultimate horizontal, vertical and moment capacity of the suction anchor and their combinations. In this approach, instead of applying the load and finding the related displacement, a certain value of displacement/rotation is applied to the foundation in gradual steps and the related load/moment reaction calculated for each step. In this method no failure occurs in the model, but after a certain displacement, the reaction load does not show any significant increment and it presents the ultimate capacity of the foundation. Fig. 6 shows the applied displacements (dH, dV and dq) and their related reactions (RH, RV and RM) at the center of stiffness. Fig. 7 shows the load-displacement diagram and moment-rotation diagram for both soil profiles for pure horizontal and vertical displacements, and rotation. As can be seen, the plots resemble the elasticperfectly-plastic behavior of the soil as shown in Fig. 4. The horizontal part of the diagram shows a constant value of load/moment while the displacement/rotation increase, and the ultimate capacity of the anchor is given in Table 2. 5. Combined loading 5.1. Horizontal–Vertical (H–V) Fig. 8 shows the yield-locus diagram on H–V space. It demonstrates the behavior of the foundation when subjected to different combination of horizontal and vertical loadings. The concept of the yieldTable 2 Ultimate capacity of suction anchor. Shear Strength Profile
Su ¼ 1 kPa Su ¼ 1.25ZkPa
Ultimate Capacity Horizontal
Vertical
Moment
6.2 105 N 4.0 106 N
4.0 105 N 4.2 106 N
1.6 106 N.m 8.8 106 N.m
H. Monajemi, H.A. Razak / Marine Structures 22 (2009) 660–669
350E+05.
3.50E+06
3.00E+05
3.00E+06 2.50E+06
5.00E+05
0.00E+00
0.00E+00
FH (N) 30 15 H swipe V
60 45 75 V swipe H Yield - Locus
6 6 6 6 6 5 0 6 +0 +0 +0 +0 +0 +0 +0 +0 0E .00E .00E 0E 0E 50E 0E 0E 0 . 5 0 0 5 . 5 0 1 2 1. 2. 3. 3.
E+ 0
5 5 0 5 5 5 5 5 +0 +0 +0 +0 +0 E+0 +0 +0 0E .00E .00E .00E .00E 00E .00E 0 0 0 . . . 1 4 2 3 6 0 5 7
6
1.00E+06
5.00E+04
E+ 0
1.50E+06
1.00E+05
6
2.00E+06
1.50E+05
50
2.00E+05
00
2.50E+05
Su = 1.25Z kPa
4.00E+06
FV (N)
FV (N)
4.50E+06
Su = 1 kPa
4.00E+05
4.
4.50E+05
4.
666
FH (N)
30 15 H swipe V
60 45 75 V swipe H Yield - Locus
Fig. 8. Horizontal–vertical yield-locus.
locus is that the region within the locus defines the elastic state of the foundation, while the outside region demarcates the plastic state. Both side-swipe and constant displacement ratios have been used to obtain the shape of the yield-locus on H–V space. Side-swipe method was used by Tan [12] during his centrifuge tests. In this method, initially the displacement in one direction e.g. horizontal is prescribed to the foundation until ultimate load is reached. Subsequently, the second displacement direction i.e. vertical is applied during which the horizontal displacement increment is zero but the horizontal load reaction can still be monitored. Since elastic stiffness is much greater than the plastic stiffness, the stress path in the second stage almost follows the shape of the yield-locus. This test can be repeated by changing the applied displacements sequence with first vertical and then horizontal. In the constant displacement ratio, the inclined displacements are applied to the center of stiffness at different angles to find the ultimate capacity. Table 3 shows the information of the applied sideswipe and constant displacement ratio to the anchor. By connecting all the ultimate points together, the yield-locus as shown in Fig. 8 is obtained. 5.2. Horizontal-Moment (H-M) and Vertical-Moment (V-M) To find the yield-locus in the horizontal-moment and the vertical-moment spaces, series of displacements and rotations are applied to the center of rotation with different ratios as given in Table 3. The ultimate capacities are plotted with horizontal or vertical loading versus moment, thus Table 3 Various combinations of applied displacements and rotations. No
Displacement
Displacement – rotation (m/Rad)
1 2 3 4 5 6 7 8
Swipe: H then V Swipe: V then H dh/dv ¼ tan 15 dh/dv ¼ tan 30 dh/dv ¼ tan 45 dh/dv ¼ tan 60 dh/dv ¼ tan 75 –
dh/dq or dv/dq ¼ N dh/dq or dv/dq ¼ 50 dh/dq or dv/dq ¼ 25 dh/dq or dv/dq ¼ 15 dh/dq or dv/dq ¼ 5 dh/dq or dv/dq ¼ 2.5 dh/dq or dv/dq ¼ 1.5 dh/dq or dv/dq ¼ 0
H. Monajemi, H.A. Razak / Marine Structures 22 (2009) 660–669
Su = 1 kPa
1.80E+06 1.60E+06
9.00E+06
1.40E+06
8.00E+06 7.00E+06
M (N.m)
1.00E+06 8.00E+05 6.00E+05
6.00E+06 5.00E+06 4.00E+06 3.00E+06
50 E+ 4.0 06 0E + 4.5 06 0E +0 6
5 E+ 0
5
0 5 6 6 6 6 6 +0 +0 +0 +0 +0 +0 +0 0E .00E 00E 50E 0E 50E 00E 0 . . 0 0 5 2 1. 1. 3. 2.
3.
7.
00
5
E+ 0
6.
00
5
E+ 0
5.
00
E+ 0
4.
00
E+ 0
3.
00
E+ 0
E+ 0
00 2.
00
00
1.
5
0.00E+00
5
1.00E+06
0.00E+00
5
2.00E+06
2.00E+05 E+ 00
4.00E+05
0.
Su = 1.25Z kPa
1.00E+07
1.20E+06
M (N.m)
667
FH (N) 1
2
3
7
8
FH (N) 4
5
6
1
2
3 7
Yield - Locus
8
4 5 6 Yield - Locus
Fig. 9. Horizontal-moment yield-locus.
providing the shape of the yield-locus as shown in Figs. 9 and 10. The applied H–V, H-M and V-M displacements and rotation and their corresponding reactions are visualized in Fig. 11. 6. Discussion It was observed that for soil profile 1, the position of the center of stiffness was at 0.57 Lf. This was because the soil strength profile was uniform so that the distribution of the stiffness was almost in the middle of the anchor. For the soil profile 2 the strength profile was not uniform and the strength and the Young’s modulus increase with depth. The distribution of the stiffness was deeper and the position of the center of stiffness was lower than the middle at 0.73 Lf. Thus it was apparent that the optimum
1.80E+06
1.00E+07 Su = 1 kPa
1.60E+06
8.00E+06
1.40E+06
7.00E+06
M (N.m)
1.20E+06 1.00E+06 8.00E+05 6.00E+05
6.00E+06 5.00E+06 4.00E+06 3.00E+06 1.00E+06
0.00E+00
0.00E+00
5.
0.
5.
00 E+ 0 00 0 E+ 0 1. 00 5 E+ 0 1. 50 6 E+ 0 2. 00 6 E+ 0 2. 50 6 E+ 0 3. 00 6 E+ 0 3. 50 6 E+ 0 4. 00 6 E+ 0 4. 50 6 E+ 06
2.00E+06
2.00E+05
00 E+ 0 00 0 E+ 0 1. 00 4 E+ 0 1. 50 5 E+ 0 2. 00 5 E+ 0 2. 50 5 E+ 0 3. 00 5 E+ 0 3. 50 5 E+ 0 4. 00 5 E+ 0 4. 50 5 E+ 05
4.00E+05
0.
M (N.m)
Su= 1.25Z kPa
9.00E+06
FV (N)
FV (N) 1
2 7
3 8
4
5
6
Yield-Locus Fig. 10. Vertical-moment yield-locus.
1
2 7
3 8
4
5
Yield-Locus
6
668
H. Monajemi, H.A. Razak / Marine Structures 22 (2009) 660–669
D
D
Lf
Lf
δV
•
RH
D
•
δH
δV
Lf • • RM
RH
RV
δθ δ H
• •
δθ
RM
Rv
Fig. 11. Anchor’s displacement and rotation and corresponding reactions in H–V, H-M and V-M spaces.
position of the load-attached point was strictly dependant on the soil profile to achieve the maximum inclination or lateral capacity. Furthermore it was observed that the uniformity of the soil strength profile does not have a significant effect on the normalized horizontal loading and moment. On the other hand the horizontal and moment ultimate capacities are function of the average soil strength along the skirt length, and does not matter very much which strength profile provides this average value. Under vertical loading, the soil resistance was from the soil-skirt friction as well as at anchor’s tip. In this case in addition to the average shear strength of the soil, the vertical capacity was also a function of the soil strength at the tip. This strength was related to the soil strength profile so that for the soil profiles with increasing strength with depth, the vertical capacity increased significantly as shown in Fig. 12.
Su = 1 kPa Su = 1.25 Z kPa
14.00
FV / LDSu
12.00 10.00 8.00 6.00 4.00 2.00 0.00 0.00
2.00
4.00
6.00
8.00
10.00 12.00 14.00
FH / LDSu 35.00
25.00
25.00 20.00 15.00
20.00 15.00
10.00
10.00
5.00
5.00
0.00 0.00
Su = 1 kPa Su = 1.25 Z kPa
30.00
M / LDSu
M / LDSu
35.00
Su = 1 kPa Su = 1.25 Z kPa
30.00
2.00
4.00
6.00
8.00 10.00 12.00 14.00
0.00 0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00
FV / LDSu
FH / LDSu Fig. 12. Normalized yield-locus.
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7. Conclusions The 3-D finite element model of the suction anchor subjected to the inclined loading has been obtained for two different soil profiles. The position of the center of stiffness has been calculated and discussed for both soil profiles. The yield-locus diagram has been obtained for H–V, H-M and V-M spaces using displacement control analysis. The normalized yield-locus showed the effect of the soil strength uniformity on the ultimate capacity of the foundation. References [1] Byrne BW. Investigations of suction caissons in dense sand. PhD Thesis, University of Oxford, UK; 2000. [2] St4ve OJ, Bysveen S, Christophersen HP. New foundation systems for the Snorre development. In: Proceedings of the annual offshore technology conference, Houston, Paper 6882; 1992. [3] Christophersen HP, Bysveen S, Stove OJ. Innovative foundation systems selected for the snorre field development. In: Proceedings of the sixth international conference on the behavior of offshore structures (BOSS), vol. 1; 1992. p. 81–94. [4] Colliat JL, Boisand P, Gramet JC, Sparrevik P. Design and installation of suction anchor piles at a soft clay site in the Gulf of Guinea. In: Proceedings of the annual offshore technology conference, Houston, paper 8150; 1996. [5] Supachawarote C, Randolph MF, Gourvenec S. Inclined pull-out capacity of suction caissons. In: Proceedings of the 14th international offshore and polar engineering conference, Toulon, France; 2004. [6] Randolph MF, Houlsby GT. The limiting pressure on a circular pile loaded laterally in cohesive soil. Geotechnique 1984; 34(4):613–23. [7] Murff JD, Hamilton JM. P-Ultimate for undrained analysis of laterally loaded piles. ASCE Journal of Geotechnical Engineering 1993;119(1):91–107. [8] Aubeny CP, Moon SK, Murff JD. Lateral undrained resistance of suction caisson anchors. International Journal of Offshore and Polar Engineering 2001;11(2):95–103. [9] Aubeny CP, Murff JD. Simplified limit solutions for the capacity of suction anchors under undrained conditions. International Journal of Ocean Engineering 2005;32:864–77. [10] Supachawarote C, Randolph MF, Gourvenec S. The effect of crack formation on the inclined pull-out capacity of suction caissons. In: 11th international conference on computer methods and advances in geomechanics, Bologna, Italy. Patron Editor, CD; 2005. p. 577–84. [11] Randolph MF, Cassidy M, Gourvenec S. Challenges of offshore geotechnical engineering. In: 16th International conference on soil mechanics and geotechnical engineering Osaka; September, 2005. [12] Tan FS. Centrifuge and theoretical modeling of conical footings on sand. PhD Thesis, Cambridge University, UK; 1990.
Contents lists available at ScienceDirect
Marine Structures journal homepage: www.elsevier.com/locate/ marstruc
Finite element modeling of suction anchors under combined loading H. Monajemi, H. Abdul Razak* Department of Civil Engineering, University of Malaya, 50603 Kuala Lumpur, Malaysia
a r t i c l e i n f o
a b s t r a c t
Article history: Received 24 July 2007 Accepted 15 February 2009
Suction anchors are subjected to inclined, quasi-horizontal or quasi-vertical loadings. The type of the structure and depth of water govern the inclination of the load. Under this load condition, suction anchor experiences a combination of horizontal and vertical translations, and rotation. Therefore the soil’s reaction to this load condition can be idealized as horizontal and vertical loadings, together with a moment. The magnitude and combination of the reactions depend on the load inclination, soil property and the point at which the load is applied. The behavior of the suction anchor subjected to the combined V–H-M loading is elaborated in this paper. This is to observe the effects of soil properties on the failure mechanism and ultimate capacity of the foundation. This was achieved by applying the pure horizontal and vertical displacements, and rotation and their combinations to the foundation, on V–H, V-M and H-M spaces and the yield-locus created for each space. The general purpose finite element program DIANA was used for this study. Non-linear analysis was conducted using a simplified elastic-perfectly-plastic model with Von-Mises yield criterion for saturated and consolidated clay. Ó 2009 Elsevier Ltd. All rights reserved.
Keywords: Suction anchor Combined loading Finite element
1. Introduction Suction foundations and anchors are used increasingly for fixed and floating offshore structures. They can be used as an alternative to piles and gravity foundations in fixed structures i.e. suction caisson or pile and drag anchors in floating structures i.e. suction anchor. The skirt length to the
* Corresponding author. E-mail address: [email protected] (H.A. Razak). 0951-8339/$ – see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.marstruc.2009.02.001
H. Monajemi, H.A. Razak / Marine Structures 22 (2009) 660–669
661
Fig. 1. Suction anchor and padeye on the skirt.
diameter ratio (L/D) is generally less than one in the suction caissons, which are used in jacket platforms or wind turbines and more than one for the suction anchors, which are used in tension leg platforms (TLPs), or floating structures [1]. Suction anchor is a hollow cylinder, open at the bottom and closed at the top as illustrated in Fig. 1. It is installed to the seabed initially with self-weight to provide a seal between the foundation’s skirt tip and the soil, and then later by pumping out the water from within the skirt. This creates a differential pressure at the top of the caisson and pushes it into the soil. Suction anchor, depending on the type of the structure, may be subjected to lateral, pull out and inclined loading through a cable which is connected to the load attachment point i.e. padeye on the skirt as shown in Fig. 2. Padeye depth and cable’s angle are important parameters that determine the ultimate capacity of the foundation. Cable angle generally depends on the system of the structure and water depth. For example, it is quasi-vertical in Snorre tension leg platform [2,3], quasi-horizontal in Nkossa process barge [4]. In general, for water depth more than 1000 m, inclined loading is provided by taut wire moorings [5]. The behavior of the suction anchors and piles under inclined and lateral loadings has been the subject of various studies. For the anchors subjected to lateral loading, the capacity of the foundation has been described using a dimensionless lateral unit resistance factor, NP which is:
DH ¼ NP Su DDZ where DH is the horizontal resistance force along the length, D is the pile’s diameter, Su is the undrained shear strength of material and NP is a dimensionless lateral unit resistance factor. Randolph [6] employed plasticity theory to estimate the lateral resistance and collapse mechanism of laterally loaded piles. The horizontal resistance of the pile was estimated by considering the pile as
662
H. Monajemi, H.A. Razak / Marine Structures 22 (2009) 660–669
D F Li φ
Lo Lf
P r O RH
RM RV
Fig. 2. Suction anchor subjected to an inclined loading.
Fig. 3. Finite element model of suction anchor.
a translating cylinder which does not have any rotation in combination with the horizontal displacement. It was deduced that the value of NP for smooth (a ¼ 0) and rough (a ¼ 1) surfaces were equal to 9.42 and 11.94, respectively. Murff [7] and Aubeny [8,9] improved previous studies using plastic limit analysis (PLA) theory by considering the pile as a translating and rotating cylinder and describe a velocity field to define the energy dissipation equations. The method has been able to include the moment combined with the horizontal loading, gap-forming behavior behind the pile, adhesion coefficient, soil weight and linear soil profile. Instead of investigating the moment capacity of the foundation individually, a reduction factor was applied to the value of NP to present the effect of rotation on the lateral capacity of the pile. Supachawarote [5,10] presented a finite element study of suction anchor subjected to inclined loading. For each load angle, there is a special load attachment depth for which the foundation only undergoes a displacement and no rotation occurs. The capacity of the foundation for the combined V–H
Table 1 Soil shear strength profiles. Soil profile
Region
Shear strength
Profile 1 Profile 2
Benchmark Gulf of Mexico
Su ¼ 1 kPa Su ¼ 1.25z kPa
H. Monajemi, H.A. Razak / Marine Structures 22 (2009) 660–669
663
Stress
Hardening Perfectly - Plastic Softening
Strain Fig. 4. Stress–strain behavior of stiffness.
loading for different L/D ratios and adhesion factors were evaluated in the study. Furthermore the effect of the soil separation on the tension side of the foundation has been considered in their study. In this study, in addition to H–V space, the behavior of suction anchor in H-M and V-M spaces is investigated using displacement and rotation control analysis for two different soil profiles. The effect of the soil shear strength profile on the position of center of stiffness and the ultimate horizontal, vertical and moment capacities of the foundation will be presented. 2. Problem description and finite element model Fig. 2 shows a suction anchor subjected to an inclined loading. The anchor’s diameter and skirt length are shown with D and Lf respectively. The load attachment point called padeye is at the depth Li from the top and the center of stiffness is at the depth Lo. The direction of loading is at an angle f to the
Fig. 5. Finding the center of stiffness.
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H. Monajemi, H.A. Razak / Marine Structures 22 (2009) 660–669
Fig. 6. Applied displacements and rotation at the center of stiffness and their reactions.
horizontal. Due to this load condition, suction anchor experiences horizontal and vertical displacement and rotates around the center of rotation at the depth Lc. This give rise to horizontal, vertical and moment reactions from the surrounding soil as shown in Fig. 2. The maximum value of each individual reaction depends on the soil behavior, anchor’s aspect ratio, adhesion factor, etc. Furthermore each reaction load can be affected by the other reactions and any combination of them reduces the maximum values of each reaction. Since the inclined loading, usually gives rise to a combination of horizontal and vertical displacements, and rotation, it is important to investigate the foundation’s behavior under combined loading. For this purpose, a 3-D finite element analysis was carried out using DIANA release 9. A suction anchor 5 m in diameter and 10 m in depth was modeled within a cylindrical soil block, 35 m in diameter and 20 m in depth as shown in Fig. 3. Two different soil shear strength profiles were selected as given in Table 1. The value of soil’s modulus ratio was adopted, with E/Su of 500, and Poisson’s ratio was taken as 0.49. The soil response was taken as elastic-perfectly plastic using a Von-Mises failure criterion (Fig. 4). The selected soil was assumed to be normally consolidated and therefore no gap occurs in the backside of the caisson. The soil-foundation interface was assumed rough and thus the adhesion factor a ¼ 1 was used. Foundation and surrounding soil were assumed weightless. 3. Center of stiffness Supachawarote [5,10] and Randolph [11] reported the depth of the center of stiffness of 0.7 Lf from their study. The results which have been found in this study showed the depth of the center of stiffness Lo is strictly dependant on the soil’s shear strength profile and Young’s modulus profile. To find the value of Lo in the finite element model, a horizontal load F was applied to the center point on top of the caisson (Fig. 5). It causes a moment reaction on the foundation. The value of this moment was obtained using a rotational fixed boundary on the center point. The position of center of stiffness can then be found simply by dividing the value of moment by the corresponding applied force. The applied load F was increased in 10 increments and values of Lo calculated for all of them. Taking the average gives value of 5.7 m (0.57 Lf) and 7.3 m (0.73 Lf) for soil profile 1 and 2 respectively. 4. Ultimate capacity of the anchor Generally two methods of analysis can be used to find the ultimate capacity of the anchor. Load control analysis and displacement control analysis. In load control analysis, an inclined load can be
-02 2.00 E
-03 1.0 0E -02 1.5 0E -02
7.00E+05 1.00E+07
4.00E+06
1.80E+06
6.00E+05
6.00E+06
2.00E+06
Su = 1 kPa 4.00E+06
1.50E+06 1.00E+06
2.00E+06
5.00E+05 0.00E+00 0
0.1
0.2
0.3
0.4
0.00E+00 0.6
0.5
Displaement (m) FH
FV
1.50E+06
5.00E+05
1.20E+06
4.00E+05
9.00E+05
3.00E+05 Su = 1.25 z kPa 2.00E+05
6.00E+05
1.00E+05
3.00E+05
0.00E+00 0
0.1
0.2
0.3
0.4
0.5
Moment (N.m)
2.50E+06
Moment (N.m)
8.00E+06
3.00E+06
Load F (N)
3.50E+06
Load F (N)
5.0
0.0 0E +
4.50E+06
0E
00
Rotation (rad)
5.0 0E -03 1.0 0E -02 1.5 0E -0 2 2.0 0E -0 2 2.5 0E -0 2
0 .0
0E +0 0
Rotation (rad)
665
2.50 E2-0 2
H. Monajemi, H.A. Razak / Marine Structures 22 (2009) 660–669
0.00E+00 0.6
Displaement (m) M
FH
FV
M
Fig. 7. Load-Displacement and Moment-Rotation relationships for both soil profiles.
applied to the skirt at different padeye depths and different load path angles [5]. The load can be increase step by step using non-linear analysis. By defining the failure criterion and soil strength, after a certain load the program will stop the analysis due to large deformations in the entire model. This method is much closer to reality but rather time consuming, especially for a model with large number of elements. This is due to the iterations which the program has to perform within each step to solve the stiffness matrix. In this study, displacement control analysis was used to find the ultimate horizontal, vertical and moment capacity of the suction anchor and their combinations. In this approach, instead of applying the load and finding the related displacement, a certain value of displacement/rotation is applied to the foundation in gradual steps and the related load/moment reaction calculated for each step. In this method no failure occurs in the model, but after a certain displacement, the reaction load does not show any significant increment and it presents the ultimate capacity of the foundation. Fig. 6 shows the applied displacements (dH, dV and dq) and their related reactions (RH, RV and RM) at the center of stiffness. Fig. 7 shows the load-displacement diagram and moment-rotation diagram for both soil profiles for pure horizontal and vertical displacements, and rotation. As can be seen, the plots resemble the elasticperfectly-plastic behavior of the soil as shown in Fig. 4. The horizontal part of the diagram shows a constant value of load/moment while the displacement/rotation increase, and the ultimate capacity of the anchor is given in Table 2. 5. Combined loading 5.1. Horizontal–Vertical (H–V) Fig. 8 shows the yield-locus diagram on H–V space. It demonstrates the behavior of the foundation when subjected to different combination of horizontal and vertical loadings. The concept of the yieldTable 2 Ultimate capacity of suction anchor. Shear Strength Profile
Su ¼ 1 kPa Su ¼ 1.25ZkPa
Ultimate Capacity Horizontal
Vertical
Moment
6.2 105 N 4.0 106 N
4.0 105 N 4.2 106 N
1.6 106 N.m 8.8 106 N.m
H. Monajemi, H.A. Razak / Marine Structures 22 (2009) 660–669
350E+05.
3.50E+06
3.00E+05
3.00E+06 2.50E+06
5.00E+05
0.00E+00
0.00E+00
FH (N) 30 15 H swipe V
60 45 75 V swipe H Yield - Locus
6 6 6 6 6 5 0 6 +0 +0 +0 +0 +0 +0 +0 +0 0E .00E .00E 0E 0E 50E 0E 0E 0 . 5 0 0 5 . 5 0 1 2 1. 2. 3. 3.
E+ 0
5 5 0 5 5 5 5 5 +0 +0 +0 +0 +0 E+0 +0 +0 0E .00E .00E .00E .00E 00E .00E 0 0 0 . . . 1 4 2 3 6 0 5 7
6
1.00E+06
5.00E+04
E+ 0
1.50E+06
1.00E+05
6
2.00E+06
1.50E+05
50
2.00E+05
00
2.50E+05
Su = 1.25Z kPa
4.00E+06
FV (N)
FV (N)
4.50E+06
Su = 1 kPa
4.00E+05
4.
4.50E+05
4.
666
FH (N)
30 15 H swipe V
60 45 75 V swipe H Yield - Locus
Fig. 8. Horizontal–vertical yield-locus.
locus is that the region within the locus defines the elastic state of the foundation, while the outside region demarcates the plastic state. Both side-swipe and constant displacement ratios have been used to obtain the shape of the yield-locus on H–V space. Side-swipe method was used by Tan [12] during his centrifuge tests. In this method, initially the displacement in one direction e.g. horizontal is prescribed to the foundation until ultimate load is reached. Subsequently, the second displacement direction i.e. vertical is applied during which the horizontal displacement increment is zero but the horizontal load reaction can still be monitored. Since elastic stiffness is much greater than the plastic stiffness, the stress path in the second stage almost follows the shape of the yield-locus. This test can be repeated by changing the applied displacements sequence with first vertical and then horizontal. In the constant displacement ratio, the inclined displacements are applied to the center of stiffness at different angles to find the ultimate capacity. Table 3 shows the information of the applied sideswipe and constant displacement ratio to the anchor. By connecting all the ultimate points together, the yield-locus as shown in Fig. 8 is obtained. 5.2. Horizontal-Moment (H-M) and Vertical-Moment (V-M) To find the yield-locus in the horizontal-moment and the vertical-moment spaces, series of displacements and rotations are applied to the center of rotation with different ratios as given in Table 3. The ultimate capacities are plotted with horizontal or vertical loading versus moment, thus Table 3 Various combinations of applied displacements and rotations. No
Displacement
Displacement – rotation (m/Rad)
1 2 3 4 5 6 7 8
Swipe: H then V Swipe: V then H dh/dv ¼ tan 15 dh/dv ¼ tan 30 dh/dv ¼ tan 45 dh/dv ¼ tan 60 dh/dv ¼ tan 75 –
dh/dq or dv/dq ¼ N dh/dq or dv/dq ¼ 50 dh/dq or dv/dq ¼ 25 dh/dq or dv/dq ¼ 15 dh/dq or dv/dq ¼ 5 dh/dq or dv/dq ¼ 2.5 dh/dq or dv/dq ¼ 1.5 dh/dq or dv/dq ¼ 0
H. Monajemi, H.A. Razak / Marine Structures 22 (2009) 660–669
Su = 1 kPa
1.80E+06 1.60E+06
9.00E+06
1.40E+06
8.00E+06 7.00E+06
M (N.m)
1.00E+06 8.00E+05 6.00E+05
6.00E+06 5.00E+06 4.00E+06 3.00E+06
50 E+ 4.0 06 0E + 4.5 06 0E +0 6
5 E+ 0
5
0 5 6 6 6 6 6 +0 +0 +0 +0 +0 +0 +0 0E .00E 00E 50E 0E 50E 00E 0 . . 0 0 5 2 1. 1. 3. 2.
3.
7.
00
5
E+ 0
6.
00
5
E+ 0
5.
00
E+ 0
4.
00
E+ 0
3.
00
E+ 0
E+ 0
00 2.
00
00
1.
5
0.00E+00
5
1.00E+06
0.00E+00
5
2.00E+06
2.00E+05 E+ 00
4.00E+05
0.
Su = 1.25Z kPa
1.00E+07
1.20E+06
M (N.m)
667
FH (N) 1
2
3
7
8
FH (N) 4
5
6
1
2
3 7
Yield - Locus
8
4 5 6 Yield - Locus
Fig. 9. Horizontal-moment yield-locus.
providing the shape of the yield-locus as shown in Figs. 9 and 10. The applied H–V, H-M and V-M displacements and rotation and their corresponding reactions are visualized in Fig. 11. 6. Discussion It was observed that for soil profile 1, the position of the center of stiffness was at 0.57 Lf. This was because the soil strength profile was uniform so that the distribution of the stiffness was almost in the middle of the anchor. For the soil profile 2 the strength profile was not uniform and the strength and the Young’s modulus increase with depth. The distribution of the stiffness was deeper and the position of the center of stiffness was lower than the middle at 0.73 Lf. Thus it was apparent that the optimum
1.80E+06
1.00E+07 Su = 1 kPa
1.60E+06
8.00E+06
1.40E+06
7.00E+06
M (N.m)
1.20E+06 1.00E+06 8.00E+05 6.00E+05
6.00E+06 5.00E+06 4.00E+06 3.00E+06 1.00E+06
0.00E+00
0.00E+00
5.
0.
5.
00 E+ 0 00 0 E+ 0 1. 00 5 E+ 0 1. 50 6 E+ 0 2. 00 6 E+ 0 2. 50 6 E+ 0 3. 00 6 E+ 0 3. 50 6 E+ 0 4. 00 6 E+ 0 4. 50 6 E+ 06
2.00E+06
2.00E+05
00 E+ 0 00 0 E+ 0 1. 00 4 E+ 0 1. 50 5 E+ 0 2. 00 5 E+ 0 2. 50 5 E+ 0 3. 00 5 E+ 0 3. 50 5 E+ 0 4. 00 5 E+ 0 4. 50 5 E+ 05
4.00E+05
0.
M (N.m)
Su= 1.25Z kPa
9.00E+06
FV (N)
FV (N) 1
2 7
3 8
4
5
6
Yield-Locus Fig. 10. Vertical-moment yield-locus.
1
2 7
3 8
4
5
Yield-Locus
6
668
H. Monajemi, H.A. Razak / Marine Structures 22 (2009) 660–669
D
D
Lf
Lf
δV
•
RH
D
•
δH
δV
Lf • • RM
RH
RV
δθ δ H
• •
δθ
RM
Rv
Fig. 11. Anchor’s displacement and rotation and corresponding reactions in H–V, H-M and V-M spaces.
position of the load-attached point was strictly dependant on the soil profile to achieve the maximum inclination or lateral capacity. Furthermore it was observed that the uniformity of the soil strength profile does not have a significant effect on the normalized horizontal loading and moment. On the other hand the horizontal and moment ultimate capacities are function of the average soil strength along the skirt length, and does not matter very much which strength profile provides this average value. Under vertical loading, the soil resistance was from the soil-skirt friction as well as at anchor’s tip. In this case in addition to the average shear strength of the soil, the vertical capacity was also a function of the soil strength at the tip. This strength was related to the soil strength profile so that for the soil profiles with increasing strength with depth, the vertical capacity increased significantly as shown in Fig. 12.
Su = 1 kPa Su = 1.25 Z kPa
14.00
FV / LDSu
12.00 10.00 8.00 6.00 4.00 2.00 0.00 0.00
2.00
4.00
6.00
8.00
10.00 12.00 14.00
FH / LDSu 35.00
25.00
25.00 20.00 15.00
20.00 15.00
10.00
10.00
5.00
5.00
0.00 0.00
Su = 1 kPa Su = 1.25 Z kPa
30.00
M / LDSu
M / LDSu
35.00
Su = 1 kPa Su = 1.25 Z kPa
30.00
2.00
4.00
6.00
8.00 10.00 12.00 14.00
0.00 0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00
FV / LDSu
FH / LDSu Fig. 12. Normalized yield-locus.
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669
7. Conclusions The 3-D finite element model of the suction anchor subjected to the inclined loading has been obtained for two different soil profiles. The position of the center of stiffness has been calculated and discussed for both soil profiles. The yield-locus diagram has been obtained for H–V, H-M and V-M spaces using displacement control analysis. The normalized yield-locus showed the effect of the soil strength uniformity on the ultimate capacity of the foundation. References [1] Byrne BW. Investigations of suction caissons in dense sand. PhD Thesis, University of Oxford, UK; 2000. [2] St4ve OJ, Bysveen S, Christophersen HP. New foundation systems for the Snorre development. In: Proceedings of the annual offshore technology conference, Houston, Paper 6882; 1992. [3] Christophersen HP, Bysveen S, Stove OJ. Innovative foundation systems selected for the snorre field development. In: Proceedings of the sixth international conference on the behavior of offshore structures (BOSS), vol. 1; 1992. p. 81–94. [4] Colliat JL, Boisand P, Gramet JC, Sparrevik P. Design and installation of suction anchor piles at a soft clay site in the Gulf of Guinea. In: Proceedings of the annual offshore technology conference, Houston, paper 8150; 1996. [5] Supachawarote C, Randolph MF, Gourvenec S. Inclined pull-out capacity of suction caissons. In: Proceedings of the 14th international offshore and polar engineering conference, Toulon, France; 2004. [6] Randolph MF, Houlsby GT. The limiting pressure on a circular pile loaded laterally in cohesive soil. Geotechnique 1984; 34(4):613–23. [7] Murff JD, Hamilton JM. P-Ultimate for undrained analysis of laterally loaded piles. ASCE Journal of Geotechnical Engineering 1993;119(1):91–107. [8] Aubeny CP, Moon SK, Murff JD. Lateral undrained resistance of suction caisson anchors. International Journal of Offshore and Polar Engineering 2001;11(2):95–103. [9] Aubeny CP, Murff JD. Simplified limit solutions for the capacity of suction anchors under undrained conditions. International Journal of Ocean Engineering 2005;32:864–77. [10] Supachawarote C, Randolph MF, Gourvenec S. The effect of crack formation on the inclined pull-out capacity of suction caissons. In: 11th international conference on computer methods and advances in geomechanics, Bologna, Italy. Patron Editor, CD; 2005. p. 577–84. [11] Randolph MF, Cassidy M, Gourvenec S. Challenges of offshore geotechnical engineering. In: 16th International conference on soil mechanics and geotechnical engineering Osaka; September, 2005. [12] Tan FS. Centrifuge and theoretical modeling of conical footings on sand. PhD Thesis, Cambridge University, UK; 1990.