Two2Dimensional Large Deformation Finite Element ...

China Ocean Engineering , Vol. 20 , No. 2 , pp . 269 - 278

Ζ 2006 China Ocean Press , ISSN 089025487

Two2Dimensional Large Deformation Finite Element Analysis for the Pulling2up of Plate Anchor 3 WANG Dong ( 王 　栋) a , 1 , HU Yu2xia ( 胡玉霞) b and J IN Xia ( 金 　霞) c a

State Key Laboratory of Coastal and Offshore Engineering , Dalian University of Technology , Dalian 116024 , China

b c

Department of Civil Engineering , Curtin University of Technology , Perth 6102 , Australia

College of Computer Science and Technology , Dalian Maritime University , Dalian 116026 , China

( Received 14 October 2005 ; accepted 8 March 2006)

ABSTRACT Based on mesh regeneration and stress interpolation from an old mesh to a new one , a large deformation finite ele2 ment model is developed for the study of the behaviour of circular plate anchors subjected to uplift loading. For the deter2 mination of the distributions of stress components across a clay foundation , the Recovery by Equilibrium in Patches is ex2 tended to plastic analyses. ABAQUS , a commercial finite element package , is customized and linked into our program so as to keep automatic and efficient running of large deformation calculation. The quality of stress interpolation is testified by evaluations of Tresca stress and nodal reaction forces. The complete pulling2up processes of plate anchors buried in homogeneous clay are simulated , and typical pulling force2displacement responses of a deep anchor and a shallow anchor are compared. Different from the results of previous studies , large deformation analysis is of the capability of estimating the breakaway between the anchor bottom and soils. For deep anchors , the variation of mobilized uplift resistance with anchor settlement is composed of three stages , and the initial buried depths of anchors affect the separation embedment slightly. The uplift bearing capacity of deep anchors is usually higher than that of shallow anchors. Key words : finite elements ; plate anchor; clay ; large deformation ; uplift bearing capacity

1. Introduction Owing to the shift of offshore oil/ gas exploration to the deep water and ultra2deep water , several types of floating structures , such as tension leg platform and spar platform , have been developed. The foundations for large mooring systems include suction piles , suction caissons , drag anchors and plate anchors. In recent 10 years , the plate anchor has attracted increasing attention due to its economy , ac2 curate positioning and short installing time (Aubeny et al . , 2001 ; Wilde et al . , 2001) . The instal2 lation of a plate anchor involves three steps : ( 1) fixing a vertically oriented plate to a suction caisson , and inserting the caisson into the seabed by means of selfweight and suction force , ( 2) separating the caisson and plate , followed by retraction of the caisson , and ( 3) dragging the chain connected with the plate , and then rotating the anchor to an inclined or horizontal position. 3 This project financially supported by the National Natural Science Foundation of China ( Grant No. 50309001) , Dis2 covery Grant from Australian Research Council (DP0344019) and Young Teacher Foundation of Dalian University of Technology (2003) 1 Corresponding author. E2mail : wd. dlut @163. com

270

WANG Dong et al . / China Ocean Engineering , 20 ( 2) , 2006 , 269 - 278

The plate anchor is designed to offer uplift resistance ; thus , it is very different from common on2 shore footings in load2displacement response and failure mechanism. Moreover , the low permeability of clay seabed induces excessive pore pressure difference in the soils over and beneath the plate , which is referred to as‘uplift suction force’( Shin et al . , 1994) . The suction force tends to increase the resis2 tance of the anchor ; however , its development heavily depends on the soil property and pulling rate (Datta and Kumar , 1996) . Since it is difficult to determine the value of suction force in practice , the effect of suction force on the uplift resistance is neglected in previous studies , and the anchor behaviour in clay is simplified to two distinct conditions : immediate breakaway case and no breakaway case ( Rowe and Davis , 1982) . In the former it is assumed that soils separate from the back of anchors as soon as loading is applied , while there is no breakaway between soils and the anchor in the latter. In fact , for the soil beneath the anchor , the upward movement of the anchor leads to gradual unloading of overburden pressure , and then the soil will rebound and ’ adhere ’to the anchor in some moving dis2 tance until breakaway takes place. Not only is the mobilized pulling2up resistance of the anchor affect2 ed by anchor embedment , but also by the soil2anchor contact state. An axisymmetric large deformation finite element ( FE) model is developed in this study to simu2 late the pulling2up of circular plate anchors buried in homogeneous clay , and the mechanical be2 haviours of shallow anchors and deep anchors are compared. The purpose of developing the large defor2 mation model is to make up for the deficiencies of small2strain analyses from Total Lagrangian ( TL ) formulation or Updated Lagrangian ( UL ) formulation. Small2strain calculation based on the initial mesh ( TL ) can not describe reasonably the history of anti2uplift force , and some severely distorted ele2 ments will result in numerical divergence in UL.

2. Large Deformation FE Model The large deformation FE theories can be divided into two categories ( Susila and Hryciw , 2003) . The first is Arbitrary Lagrangian2Eulerian (ALE) formulation. This method permits material particles to move independently of the mesh evolution so that excessive mesh distortion is constrained , although defining boundary and loading conditions is very complex. The second one is an approach based on mesh regeneration and interpolation of field values from an old mesh to a new one , which accumulates the results from small strain algorithms in Lagrangian formulation. The large deformation simulation in this paper belongs to the latter , in which the pulling2out process of a plate anchor is composed of nu2 merous incremental steps. At the beginning of each step , the deformed geometry is re2meshed for elim2 ination of mesh distortion , and the filed values , such as stresses , are mapped from an old mesh to a new one. After that , the small2strain calculation is conducted and key data are extracted for the next step . From the above description , it is easy to conclude that we have to choose an accurate and effi2 cient interpolation technique to ensure the mapping precision of filed values. Here the Recovery by E2 quilibrium in Patches ( REP , Boroomand and Zienkiewicz , 1997) is employed to recover stress compo2 nents in the whole FE zone. Considering the difficulty in meshing irregular geometry , the quadratic tri2 angular axisymmetric element rather than the quadrangular element is adopted.

WANG Dong et al . / China Ocean Engineering , 20 ( 2) , 2006 , 269 - 278

271

2. 1 　Stress Recovery Based on REP

The Modified Unique Element Method (MUEM) has ever been used in large deformation analysis for skirted foundation ( Hu and Randolph , 2002) , but the efficiency of MUEM is low. Compared with MUEM , REP needs short time and gives high interpolation accuracy. Here the basic theory of REP is to be introduced briefly and our modification on REP will also be proposed.

Fig. 1. Patch of quadratic triangular elements.

The recovered stress components at any position in an element patch ( Fig. 1) are postulated to be expressed in an polynomial expansion P and a vector of unknown parameters a i

σi3 = Pa i .

( 1)

For the axisymmetric or plane elements 2 2 P = ( 1 　x 　y 　x 　y 　xy ) .

It is hoped that the nodal force obtained from recovered stress is equal to that from small strain analysis 3

i3 p

( 2)

i p

( 3)

T h

( 4)

∫B σ dΩ = ∑F ∫B σ dΩ = ∑F ∫B σ dΩ = ∫Β σ dΩ T

Ωp

i

T h

Ωp

i

3

T

Ωp

Ωp

where B = SN , N being the shape function. To satisfy Eq. ( 4) approximately , let

Πi =

∫

3

T B σi dΩ -

Ωp

∫

h

T B σi dΩ

T

3

∫B σ dΩ - ∫B σ dΩ T

Ωp

i

Ωp

5Πi 5 a i = 0 .

T h

Ωp

i

( 5) ( 6)

Substituting Eq. ( 5) into Eq. ( 6) results in ai = i

iT

[H

H =

i

H

∫B Ωp

]

-1

iT

I P dΩ

T i

i

H Fp i

( 7) ( 8)

where I represents a zero vector of unit value at the position of the i2th stress. Borromand and Zienkiewicz ( 1997) found that the minimization to the patch alone is not sufficient

272

WANG Dong et al . / China Ocean Engineering , 20 ( 2) , 2006 , 269 - 278

to keep the calculation stability. Thus an additional minimization to each element is attached to Eq. ( 8) ai =

Hi T Hi + α

∑H T H i e

patch

i e

-1

Hi T F ip + α

∑H T F i e

i e

( 9)

patch

where α is a weighted coefficient . Small strain calculations give stresses at element integration points , which are substituted into Eq. ( 9) to educe a i . And nodal stresses are recovered by Eq. ( 1) . Because of the stress discontinuity on the boundaries , boundary nodes can not be located as patch assembly nodes. To keep the interpolation accuracy , the following strategies are adopted in our appli2 cation. Firstly , in Fig. 1 , the patch assembly nodal stress and interior mid2side nodal stresses should be recovered in this patch. Secondly , if some vertex nodes and mid2side nodes are located on the boundaries of the domain , the stresses are also recovered in this patch. Thirdly , for special boundary nodes such as A1～A4 in Fig. 2 , there is no patch covering them. Zienkiewicz and Taylor ( 2000) ad2 vised to make an expansion from the interior patch near the special boundary nodes without any extra condition added. However , we have found that this solution is not rigorous enough to simulate the per2 formance of plate anchors. Our alternative is to avoid the coming into being of special boundary nodes when meshing soils.

Fig. 2. Special boundary nodes in an axisymmetric mesh.

In previous studies , the applications of REP are limited to linear elastic problems ( Borromand and Zienkiewicz , 1997) . For elasto2plastic analyses in this paper , recovered stress components at new integration points are possibly beyond the yield surface. Such stress components will be pulled back onto the yield envelop after the stress projection. 2. 2 　Procedure of Large Deformation Analysis

The commercial FE software package ABAQUS is involved in large deformation models developed. ABAQUS is employed for the small strain computation in the first incremental step , after that (1) Read stress components at every integration point and displacements of each element node from result file. ( 2) Recover stresses at element nodes using REP technique. (3) Regenerate deformed region by means of several key geometric points , and mesh the region with quadratic triangular elements. ( 4) Determine the old element containing a new interpolation point . The stress components at the

WANG Dong et al . / China Ocean Engineering , 20 ( 2) , 2006 , 269 - 278

273

new interpolation point are derived from the following equation

σ = Nσ3 ,

( 10)

3

where σ represents the recovered stress components at 6 old element nodes. If σ is out of the yield surface , the stress components should be corrected. ( 5) Input the recovered stresses at each new integration point into ABAQUS as the initial stress condition. This work is finished through ABAQUS user subroutine interface SIGINI. ( 6) Set the controlling parameters of small strain running in ABAQUS. The settlement of the an2 chor in each incremental step is determined in advance. ( 7) Repeat the above steps until the anchor reaches the prescribed embedment . A program in Fortran 90 format is developed for stress recovery , stress correction and the control of the running procedure. Since a pulling2out process is usually composed of several hundred incremen2 tal steps , the large deformation analysis has to be designed to run continuously and automatically. Our solution is to code special Python subroutines where the building of the small strain FE model and data extraction from the result file are set . Python is the build2in script language in ABAQUS ( HKS , 2002) , and the Python subroutines are called by the Fortran program mentioned above in every incre2 mental step . 2. 3 　Simulation of the Pulling up of the Plate Anchor

The large deformation model established is implemented to study the bearing capacity of the plate anchor. The stiffness of an anchor is much greater than that of the clay foundation , so the plate anchor does not appear directly in the simulation , but is replaced by some reasonable boundary condition. The friction between the anchor and soil is limited to completely rough or smooth. The pulling resistance of the anchor equals to the sum of reactive forces at corresponding boundary nodes. As for the boundary edges represented by the soil beneath the anchor , if the sum of reactive forces at 3 edge nodes is a tension force , this soil edge is estimated to be separated from the anchor. Once all such soil edges reach breakaway , the embedment of the plate anchor in respect to the original mudline is defined as separation depth hs .

3. Verification of Stress Interpolation In the following investigations , it is assumed that the circular anchor is buried horizontally in ho2 mogeneous clay. The clay is modeled as an elasto2perfectly plastic Tresca material with Young’s modu2 lus E = 500 S u and Possion’s ratio υ= 0. 49 , S u being the undrained strength of the clay. The unit weight and coefficient of earth pressure are selected as γ = 17 kN/ m3 and K0 = 1 , respectively. The suction force , due to the low permeability of clay , is not taken into consideration. The mobilized up2 lifting resistance is defined as : N mob = F/ ( AS u ) ,

( 11)

where F is the mobilized uplifting resistance at some embedment , and A is the area of plate. The maximum value of N mob in the whole pulling process is termed uplifting bearing capacity factor N c . The

WANG Dong et al . / China Ocean Engineering , 20 ( 2) , 2006 , 269 - 278

274

initial embedment of the anchor before loading , and the anchor embedment during the pulling process are referred to as hi and h respectively. In this section the quality of stress recovery and interpolation will be verified. Furthermore , the numerical results will be compared with those from limit analysis and laboratory testing. The influence of step displacement on load2displacement response will also be dis2 cussed.

Fig. 3. Comparison of Tresca stress before and after recovery.

The following are taken : S u = 10 kPa , plate diameter D = 2 m , plate thickness t = D / 20 , ini2 tial embedment hi = 2 . 5 D , incremental step settlement d s = D/ 200 , and the plate anchor is rough. In Fig. 3 , Tresca stresses before and after stress mapping are compared for h = 2 D . For clear demon2 stration , the mesh edges are hidden. It is observed that the REP technique in conjunction with correc2 tion of stress components beyond the yield surface makes a very good reproduction of the stress field. As shown in Eq. ( 4) , the basic thought of REP is to give nodal reactive forces from recovered stresses approximately equal to those from original stresses. The uplifting resistance F is further de2 rived from reactive forces at corresponding boundary nodes , so F is also a right parameter to reflect the reliability of stress interpolation. The load2displacement curve for d s = D/ 200 is presented in Fig. 4. Table 1 lists the fluctuation of mobilized force induced by remeshing. At different depths , the error is always smaller than 4 %. More importantly , there is no sign of error accumulation in the whole process of large deformation simulation. Error of pulling2up forces induced by stress recovery

Table 1 h/ D

2. 4

2. 2

2. 0

1. 8

1. 5

1. 2

0. 9

0. 6

Uplift force before interpolation ( kN)

425. 0

424. 7

424. 6

311. 6

295. 2

288. 5

249. 9

214. 0

Uplift force after interpolation ( kN)

413. 4

412. 7

413. 8

303. 2

286. 0

280. 6

244. 2

207. 8

Error ( %)

2. 73

2. 83

2. 54

2. 70

3. 12

2. 74

2. 28

2. 90

WANG Dong et al . / China Ocean Engineering , 20 ( 2) , 2006 , 269 - 278

275

When the rough circular anchor ( no thickness) is buried deeply , soils will always be attached to the anchor at the commencement of loading. The exact solution from limit analysis is N c = 13. 11 (Mar2 tin and Randolph , 2001) , compared with N c = 13. 24 from large deformation simulation as long as h i

≥2 . 5 D , the difference is only 0. 99 %. Fig. 5 shows the variation of uplift bearing capacity ( in weightless soil ) against h i under immediate breakaway condition. The numerical results agree well with small2scale model test data ( Shin et al . , 1994) . Based on the above comparisons for both immediate breakaway case and no breakaway case , it is believed that large deformation analyses can estimate the uplift bearing capacity accurately.

Fig. 4. The effect of step settlements on uplift force2 displacement curves.

Fig. 5. Variation of uplift bearing capacity factor with initial anchor embedment under immediate breakaway condition.

Like in the incremental method without iterations in traditional non2linear FE calculation , the an2 chor step settlement d s has to be small enough to keep numerical convergence and stability in large de2 formation studies. The effect of d s on load2displacement response is demonstrated in Fig. 4 for d s/ D = 1/ 25 , 1/ 50 , 1/ 100 and 1/ 200. The last two settlements lead to similar curves , suggesting that ds = D/ 100～ D / 200 should be chosen as incremental step settlement in the study of the behaviour of circular anchors. Fig. 6 compares the variations of N mob with normalized embedment h/ D from small strain FE and large deformation FE. The small strain analysis is implemented in ABAQUS/ CAE , where the friction2 less contact and/ or rough contact are set between soils and the anchor. It is observed that , in the rough case , the small strain computing stopped soon after the start of loading due to numerical non2 convergence. Although small strain FE can give the progress of a smooth anchor , the mesh distortion results in failure in predicting the breakaway between soil and the bottom of the plate. N mob keeps con2 stant after reaching the maximum value , and this is inconsistent with the phenomenon observed in small scale model testing ( Howes , 2003) . Fig. 7 demonstrates that the uplift bearing capacities of all 3 model anchors decrease gradually after breakaway takes place. The diameters of anchor models are 40 , 60 and 80 mm , respectively. In this testing , half the anchor model was embedded and pulled in trans2 parent soil to show the process of breakaway. From the comparison of Figs. 6 and 7 , it is concluded

WANG Dong et al . / China Ocean Engineering , 20 ( 2) , 2006 , 269 - 278

276

that large deformation analyses , rather than small2strain FE , succeed in describing the performance of plate anchors correctly. The bearing capacity of the rough anchor is about 6. 3 % higher than that of the smooth one.

Fig. 6. Comparison of results from small strain FE and

Fig. 7. Performance of plate anchor in model testing ( after Howes , 2003) .

large deformation FE.

4. Shallow Anchor and Deep Anchor The factors affecting the mobilized uplifting resistance include soil strength , overburden pressure and initial embedment ratio h i / D . The mechanical behaviours of deep anchors and shallow anchors will be studied through two typical examples. In the first example a rough anchor is supposed with D = 2 m and t = D / 20. The strength of clay is S u = 10 kPa. The initial embedment ratios are selected as hi / D = 3. 5 , 3. 0 , 2. 5 , 2. 0 and 1. 0 , respectively. The load2displacement responses from the large deformation model are shown in Fig. 8 (a) . When hi ≥2 D every load2displacement curve can be subdivided into three stages. During the initial stage , N mob reaches the maximum value rapidly , while it keeps constant in the next stage. The breakaway between soil and the anchor bottom indicates the commencement of the third stage where N mob decreases gradually. These kinds of responses can be named‘deep anchor’behaviour. It is fur2

ther found that the separation depth hs is almost independent of initial buried depth hi . When the deep anchor is pulled up , the overburden pressure imposed on the soil under the plate bottom is transferred to plate , resulting in the rebound of soil in this region. With the movement of the anchor , the stresses of the soil under the plate decrease gradually and finally reach balance with the surrounding field of gravity stress. The separation depth is mainly dependant on some value of initial gravity stress , the ini2 tial embedment of the anchor being insignificant . Fig. 8 ( a) shows that , if the anchor is pulled up from h i = D , the anchor bottom is separated from soil immediately. At this time , the uplifting resistance has not reached the maximum value yet . Such an anchor is named‘shallow anchor’, of which the bearing capacity factor N c = 7. 7 is far lower than N c = 13. 6 of the deep anchor. The working mechanism of shallow anchors is similar to that of the ones under immediate breakaway condition.

WANG Dong et al . / China Ocean Engineering , 20 ( 2) , 2006 , 269 - 278

277

In the second example , S u = 15 kPa , D = 5 m , t = D/ 20 , and the anchor is rough. The initial embedment ratios are 3. 0 , 2. 0 , 1. 5 and 0. 5 , respectively. The behaviours of deep anchors and shal2 low anchors are also observed in Fig. 8 ( b) .

Fig. 8. Load2displacement curves of deep anchors and shallow anchors.

5. Conclusions Based on mesh regeneration and stress interpolation , an axisymmetric large deformation model is developed and presented for the study of the pulling up process of circular plate anchors buried in uni2 form clay. The uplift bearing capacities from the large deformation model agree well with the results from limit analyses and small2scale model testing. The major conclusions are as follows. ( 1) Large deformation analysis has an advantage over small strain FE in predicting the separation between the soil and the bottom of the anchor , which is important in describing the variation of uplift2 ing resistance with anchor embedment . (2) Plate anchors can be divided into deep anchors and shallow anchors , the uplift bearing ca2 pacity of the former being far greater than that of the latter. The performance of deep anchors under pulling force is composed of three stages. ( 3) With regard to deep anchors , the effect of initial buried depth on separation embedment can usually be ignored. References Aubeny , C. P. , Murff , D. J . and Roesset , J . M. , 2001. Geotechnical issues in deep and ultra2deep waters , Pro2 ceedings of 10 th International Conference on Computer Methods and Advances in Geomechanics , Tucson , USA , 13～ 26. Boroomand , B. and Zienkiewicz , O. C. , 1997. An improved REP recovery and the effectivity robustness test , Interna2 tional Journal for Numerical Methods in Engineering , 40 , 3247～3277. Datta , M. and Kumar , P. , 1996. Suction beneath cylindrical anchors in soft clay , Proceedings of 6 th International Off 2 shore and Polar Engineering Conference , Los Angeles , USA , 544～548. Hibbit , Karlsson and Sorensen ( HKS) , Inc. , 2002. ABAQUS/ Scripting Manual , Version 6. 3 , Providence : HKS Inc. Howes , A. S. , 2003. An investigation into the behaviour and pullout capacity of plate anchors , Research Report to De2 partment of Civil Engineering , Curtin University of Technology.

278

WANG Dong et al . / China Ocean Engineering , 20 ( 2) , 2006 , 269 - 278

Hu , Y. and Randolph , M. F. , 2002. Bearing capacity of caisson foundations on normally consolidation clay , Soils and Foundations , 42 (5) : 71～77. Martin , C. M. and Randolph , M. F. , 2001. Application of the lower and upper bound theorems of plasticity to collapse of circular foundations , Proceedings of 10 th International Conference on Computer Methods and Advances in Geome2 chanics , Tucson , USA , 1417～1428. Rowe , R. K. and Davis , E. H. , 1982. The behaviour of anchor plates in clay , Gé otechnique , 32 (1) : 9～23. Shin , E. C. , Dass , R. N. , Omar , M. T. et al . , 1994. Mud suction force in the uplift of plate anchor in clay , Pro2 ceedings of 4 th International Offshore and Polar Engineering Conference , Osaka , Japan , 462～466. Susila , E. and Hryciw , R. D. , 2003. Large displacement FEM modeling of the cone penetration test ( CPT) in normal2 ly consolidated sand , International Journal for Numerical and Analytical Methods in Geomechanics , 27 , 585～602. Wilde , B. , Treu , H. and Fulton , T. , 2001. Field testing of suction embedded plate anchors , Proceedings of 11 th In2 ternational Offshore and Polar Engineering Conference , Stavanger , Norway , 544～551. Zienkiewicz , O. C. and Taylor , R. L. , 2000. The finite element method , 5th edition , Oxford : Butterworth2Heine2 mann.