(1.Maritime Research Centre,School of Civil and Environmental Engineering,Nanyang Technological University,Singapore 639798,Singapore;2.State Key Laboratory of Structural Analysis for Industrial Equipment,School of Naval Architecture Engineering,Dalian University of Technology,Dalian 116024,China)

0 Introduction

Fixed offshore platform is commonly deployed for offshore oil exploration.With the development of offshore industry,more platforms are designed at locations susceptible to rare and severe strong ductility level earthquake.In order to demonstrate that the offshore platform is stable and does not experience structure collapse,one may adopt the nonlinear soil-pile-structure interaction time history analysis as recommended by API RP 2A and ISO 19902.The analysis is used to evaluate whether the platform-foundation system meets structural reserve strength and energy dissipation requirements.In the case of a jacket offshore platform,the piles are typically founded at the bedrock deep below the water and soft soil.During an earthquake event,it is generally believed that the shock waves pass from the bedrock to the soil.The piles in the jacket structure are assumed being moved along with the bedrock are then moved through the soil and hydraulic,resulting in a complex Structure-Pile-Fluid-Soil interaction process.

To achieve lateral stability of an offshore jacket structure foundation,one has to establish safe attachment of the structure to the ground and in particular how the loads applied to the structure could be safely transferred to the surrounding soil.An established practice is the utilization of appropriate grouted or un-grouted piling system.In a grouted system the certain level of adhesion between the grout and steel surfaces may be achieved and translational movement of pile in the leg will be fixed-fixed ended condition.It was also reported in mechanical tests carried out that the presence of grout improved the strength and fatigue performance of the structural systems(Dedi,2009)[1].Un-grouted piling system is another method of piling system.The top of the pile in the method is fixed to top of the jacket by welding the both members,so that the leg and pile are allowed to have finite axial strain relative to each other but in normal direction they are bound to each other by the aid of wishbone elements.

Dynamic response of structures founded on soft soils is influenced by the soil properties,and the response is significantly different from that of the fixed base condition as a result of the interaction between the soil and the structure.Several studies(Gazetas,1991[2];Han and Cathrio,1997[3];Wu and Gan,1998[4];Inaba et al,2000[5];Hokmabadi et al,2011[6];Carbonari et al,2011[7])reported findings on seismic soil-pile-structure interactions and the effect of this phenomenon on the seismic response of the structures.There are three groups of analytical methods for studying the soil-pile-structure interaction,and they are:(1)Substructure Methods(or Winkler methods),in which a series of springs and dashpots are employed to represent the soil behaviour;(2)Elastic Continuum Methods;and(3)Numerical Methods based on a set of relevant governing equations.

Mardfekri et al[8]studied the behavior of laterally loaded monopole foundations using linear and nonlinear approach and assessed the accuracy of different pile-soil interaction model as compared to the results obtained using finite element model.Cyrus et al[9]conducted feasibility study of an un-grouted offshore jacket structure using the endurance time method.Tabeshpour[10]assessed the requirements for accurate modeling of pile-soil interaction of an offshore jacket structure.Wang et al[11]summarized the developments in grouted pile and its performance under different loadings.Various engineering methods and equivalent simplified models and methods have been applied in order to facilitate application and save computation time (Zhou et al,2014&2016)[12-13].The phenomenon of pile-fluid-soil interactions is frequently not considered in detailed during preliminary design of a jacket platform.The equivalent-pile method is often used to account for the pile-soil interaction due to the lack of soil data.The fluid-structure interaction is replaced by treating the fluid as added mass to account for the effects of fluid on the vibrating structure.

In the present study,finite element method using LS-DYNA software is employed to investigate the effects of piled foundation on the seismic response of offshore platform.For this purpose,the seismic behavior of the platform supported by two types of foundations including the Structure-Pile-Fluid-Soil interaction foundation is compared with the equivalent pile foundation with attached water.Secondly,the natural frequency of the jacket platform with these two foundation models is calculated respectively.By comparing the frequency result of the two models,the frequency of simplified equivalent pile model is modified,and then the modified coefficient relation curve is proposed in the paper.In association with the modified coefficient relation curve,the simplified equivalent pile model can be used to compute the vibration characteristic of an offshore platform effectively.In this way,the computing time will be saved and the precision of results will be improved.

1 Modeling techniques

The Structure-Pile-Fluid-Soil interaction is a complex phenomenon as it involves the nature of each element and the coupling relationship and interaction among the elements.In this study,the nonlinear finite element program LSDYNA was used to perform the numerical simulation.LS-DYNA is a fully functional explicit dynamic analysis software,which is used to solve all kinds nonlinear physics(geometry,material and interfacial contact)(Shi et al,2005[14],Bai,2005[15]).The software makes use of Arbitrary Lagrangian-Eulerian(ALE)algorithm which combines the advantages of the Lagrange algorithm and Euler algorithm,and is a real fluid-solid coupling algorithm.With regard to structural boundary motion,it has the characteristics of the Lagrange algorithm which can track the movement at the boundary of structure effectively.It also has the characteristics of Euler algorithm which can cause the inner grid to exist independently of the physical entity.Moreover,the location of grid can be modified in the process of solving according to the parameters defined,by which the grid will not suffer from severe distortion.This algorithm is very appropriate for handling the large deformation problem.The main characteristic of the fluid-solid coupling method in ALE algorithm is the model of structure and fluid which allows overlapping of the grid when building up and meshing the model(see Fig.1).The finite element meshes of the structure and fluid were constructed independently.ALE algorithm was used to perform the numerical simulation of fluid-solid coupling in this study①.

The interfacial contact between different moving objects was achieved by defining the possible contact surface,contact type and contact parameters.In the process of calculation,the contact interface was guaranteed not to be penetrated.As a result of soil-pile interaction,non-slip condition was not enforced and the friction induced due to relative movement of the objects moving on the contact surface was taken into account②.

Notes:①Fluid-solid coupling was achieved by using the key word*CONSTRAINED_LAGRANGE_N_SOLID in the LS-DYNA program;

Fig.1 ALE algorithm in LS-DYNA

②Soil-pile coupling was done by using the key word*CONTACT_ERODING_SURFACE_TO_SURFACE in the LS-DYNA program.

2 Analysis of three-dimensional(3D)finite element model

The 3D finite element model of a jacket platform contained the following parameters:the height of platform was 68 m,and the platform was located in a 30 m-deep sea.The mass of three decks from bottom to top was 93 tons,2 670 tons and 1 231 tons,respectively.The length of pile above soil surface was 30 m;the length of pile embedded in the soil was 25 m.The dimension of the section for the piles was Ф 1 333×20 mm;the dimension of the stay bars was Ф 800×10 mm.The living quarters,piles and stay bars were modeled using shell element.Two finite element models were used to analyze the velocity and acceleration at the top of the platform.Model 1 was a 3D model considering the actual physics of Structure-Pile-Fluid-Soil interactions(see Fig.2).Model 2 was a 3D model with attached-water and equivalent-pile effect(see Fig.3).

Fig.2 Model 1(Structure-Pile-Fluid-Soil interaction)

Fig.3 Model 2(attached water and equivalent pile)

In Model 1,the soil and fluid were modeled using solid element.Non-reflection domain boundary condition was applied to simulate infinite space.The bottom of piled foundation was simply supported.In Model 2,the equivalent-pile model was embedded by 6-times the pile diameter(according to the rule of China Classification Society).The mass of fluid was incorporated on the structure as added mass on the vibrating structure.

Fig.4 shows the time history of seismic wave.Node 2231 refers to the top of jacket structure as shown in Fig.2 and Fig.3.Figs.5-8 show the comparison results at node 2231 when L/D=4.5 and 8.5,respectively,where L is the pile spacing and D is the pile diameter.The results of response extremum are tabulated in Tab.1 to Tab.4.

Fig.4 Acceleration of seismic excitation

Fig.5 Comparison of velocity at node 2231when L/D=4.5

Fig.6 Comparison of acceleration at node 2231 when L/D=4.5

Tab.1 Comparison for extremum value of velocity at node 2231 when L/D=4.5

Tab.2 Comparison for extremum value of acceleration at node 2231 when L/D=4.5

Tab.3 Comparison for extremum value of velocity at node 2231 when L/D=8.5

Tab.4 Comparison for extremum value of acceleration at node 2231 when L/D=8.5

Fig.7 Comparison of velocity at node 2231 when L/D=8.5

Fig.8 Comparison of acceleration at node 2231 when L/D=8.5

Model 1 reflects the actual physics of the structure-pile-fluid-soil interactions with the dynamic response well damped and modulated.Model 2 produces large magnitude responses.These behaviors are shown clearly by comparing the responses of the two models.Model 1 with Structure-Pile-Fluid-Soil interaction tends to modify the time-history of velocity and acceleration.This is due to the presence of stiff pile elements in the soil damping the dynamic properties of the whole system.The maximum velocity and acceleration of Model 1 are lower than those of Model 2.It should be noted that although the velocity and acceleration are increased due to Structure-Pile-Fluid-Soil interaction,the largest velocity and acceleration are smaller.When L/D is increased from 4.5 to 8.5,the largest amplitude of velocity is reduced by nearly 30%and the largest amplitude of acceleration is reduced by more than 20%.It could be concluded that increasing L/D leads to reduction in the magnitude of the structural response.

One may conclude that the equivalent-pile model with attached water greatly simplifies the computational complexity and improves the computational efficiency at the expense of the accuracy of the dynamic response which may lead to unduly conservative engineering design of the platform.

3 Proposed frequency modification for equivalent pile model

With the wide-spread use of the finite element software,the emphasis has been set on full-physics 3D modeling of structures.Ideally the more comprehensive and more inclusive the models are,the more accurate are the simulation results.However,these comprehensive models are not readily adopted in engineering practice during the preliminary design stage.This is especially so for a jacket platform which poses a complex Structure-Pile-Fluid-Soil system.It is very complicated and time consuming to study its dynamic characteristics as a whole.A simpler and equivalent modeling approach with less computation resources is needed to establish the seismic response relatively quickly and economically.By comparing the frequency result of the 3D analysis of Model 1 and Model 2,one could note the frequency of the full-physics model(Model 1)and modify the equivalent model(Model 2)to achieve similar response through the use of a modified coefficient relation curve which is described herein.

The coefficient relation curve is typically established based on a simpler platform which could easily be visualized as a structure with the bulk of the mass lumped to the floor deck.In this study,the two models shown in Fig.2 and Fig.3 in Chap.2 are still adopted here.Fig.9 shows the arrangement of platform legs.The response frequencies were computed for various ratio of pile spacing to pile diameter(L/D)ranging from 4.5,5.5,6.5,7.5 to 8.5 where D=1.3 m,1.5 m and 1.7 m,respectively.The coefficient relationship is defined as φi=fi/fˉi,where fiis the frequency response from the 3D model which included the full physics of the Structure-Pile-Fluid-Soil interaction,and fˉiis the frequency response from the simplified equivalent pile model;i is the order of vibration.The authors found that the coefficient correlates best with L1.25/D2.The first order of modified coefficient φ1relation curve is shown in Fig.10.

Fig.9 Arrangement of piles

Fig.10 Modified coefficient relation curve

4 Conclusions

The following conclusions can be achieved by the numerical computation and comparison.

(1)The equivalent-pile model with attached water greatly simplifies the tedious calculation for dynamic response of offshore platforms and increases the calculation efficiency although the calculation results are a little bit rough.The equivalent calculation method is very convenient in engineering practice,especially in the preliminary design stage of offshore platforms.

(2)The natural period will become larger,and the vibration velocity and acceleration will change smaller after considering the Structure-Pile-Fluid-Soil interactions.In the dynamic response analysis of an offshore platform,the actual working state can be truly reflected so that the structural design can be more economical and reasonable by means of considering the Structure-Pile-Fluid-Soil interaction.

(3)Despite the advised models and computational method in large amount of literatures,the current engineering practice still prefers simplified structural calculation model and method in which a reasonable degree of precisions could be achieved.The modified coefficient relation curve proposed in the paper can be used to compute the vibration characteristic of an offshore platform effectively.Using the simplified equivalent pile model to compute the frequency of offshore platforms and amending the frequency result in association with the modified coefficient relation curve,the natural frequency after considering the Structure-Pile-Fluid-Soil interaction will be obtained.The computing time will be saved and the precision of results will be improved.

(4)In the analysis of various geometric nonlinear,material nonlinear and contact nonlinear problems,the size of grid has a certain impact on the computational time and result accuracy.Although the result will be closer to the real situation through refining mesh,it also increases the computing time significantly.Therefore,it is practical and economic to choose a reasonable mesh size by which the accuracy of result can be guaranteed and the computing time can be saved at same time.

Acknowledgements

This work is supported by the Fundamental Research Funds for the Central Universities.

[1]Dedi N.Analysis of grouted connection in monopile wind turbine foundations[M].LAP LAMBERT Academic publishing,2009.

[2]Gazetas G.Formulas and charts for impedances of surface and embedded foundations[J].Journal of Geotechnical Engineering,1991,117:1363-1381.

[3]Han Y C,Cathro D.Seismic behaviour of tall building supported on pile foundations.Seismic analysis and design for soilpile-structure interaction[J].Geotechnical Special publications,ASCE,1997(70):36-51.

[4]Wu S M,Gan G.Dynamic soil-structure interaction for high-rise buildings[J].Developments in Geotechnical Engineering,1998,83:203-216.

[5]Inaba T,Dohi H,Okuta K,Sato T,Akagi H.Nonlinear response of surface soil and NTT building due to soil-structure interaction during the 1995 Hyogo-ken Nanbu(Kobe)earthquake[J].Soil Dynamics and Earthquake Engineering,2000,20(5):289-300.

[6]Hokmabadi A S,Fakher A,Fatahi B.Seismic strain wedge model for analysis of single piles under lateral seismic loading[J].Australian Geomechanics,2011,46:31-41.

[7]Carbonari S,Dezi F,Leoni G.Linear soil-structure interaction of coupled wall-frame structures on pile foundations[J].Soil Dynamics and Earthquake Engineering,2011,31:1296-1309.

[8]Mardfekri M,Gardoni P,Roesset J M.Modeling laterally loaded single piles accounting for nonlinear soil-pile interactions[J].Journal of Engineering,2013(5):29-38.

[9]Cyrus M,Yazdi S R S,Javid A H.Utilization of endurance time method for seismic analysis of offshore structures[C]//The Fifth Iranian National Offshore Industry(OIC2013).Sharif University-Tehran,Iran,2012.

[10]Tabeshpour M R.Conceptual interpretation(Chapter 2).2800 seismic building design regulation(In Farsi)[M].Ebrahimian H,2006.

[11]Wang Z,Jiang S C,Zhang J.Structural performance of prestressed grouted pile-to-sleeve performance connection[J].Progress in Steel Building Structures,2010,14:304-311.

[12]Zhou B,Han X S,Tan S.A simplified computational method for random seismic responses of a jacket platform[J].Ocean Engineering,2014,82:85-90.

[13]Zhou B,Guo W,Han X S,Tan S.Random seismic response analysis of jacket structure with Timoshenko’s beam theory[J].Ships and Offshore Structures,2016,11(4):438-444.

[14]Shi D Y,Li Y C,Zhang S M.The analysis method and engineering examples of ANSYS/LS-DYNA(In Chinese)[M].Beijing:Qinghua University Press,2005.

[15]Bai J Z.Theoretical basis and example analysis of LS-DYNA3D(In Chinese)[M].Beijing:Science Press,2005.