1 Introduction

As the offshore developments of oil and gasmove into deeper water, the underst and ing of the fatigue performance of the steelcatenary riser(SCR)becomes critical to long-st and ing operation. The SCRs usuallytend to fatigue damage, especially in the region where the riser pipe reachesthe seabed, known as the ‘touchdown zone’(TDZ)(Hodder and Byrne, 2010). Oneof the key issues for SCR design is to assess the fatigue damage due torepetitive loading over the lifetime of the riser(Xu et al., 2006).Accurate evaluation of the fatigue life of an SCR remains a major challenge dueto uncertainty surrounding the interaction forces where the riser touches downon the seabed(Hodder et al., 2009). The results of the fatigue evaluationdepend significantly on the assumed riser-soil interaction model at the TDZ, which is still an area of uncertainty for designers.

The mechanisms that govern riser dynamicresponses in the TDZ are not easy to quantify. The riser moves cyclicallywithin the touchdown zone due to excitation from the vessel and wave loadings, softening and remolding the seabed soil. Vertically or in-plane, riser motionscreate transient variations in the riser-soil force, associated withlongitudinal translation of the touchdown point accompanied by changes in thetension. The cyclic expulsion and sucking-in of water between the riser and soil during vertical motions also tends to cause water entrainment into thesoil, thereby increasing the degradation of the shear strength. Horizontally orout-of-plane, riser motions reduce the vertical bearing capacity of the soil dueto the combined V-H loading imposed on the seabed. The embedment required tosupport a given vertical load need to be increased. Also, horizontal motionstend to sweep soil laterally away from the riser alignment, which leads to theformation of a trench around the riser.

The full riser-soil interaction responseswithin the TDZ of an SCR include interaction between the riser response and thesoil response. The actual conditions imposed on an element of a riser pipe areneither load controlled nor displacement controlled only(Hodder et al., 2009). Therefore, the riser-soil interaction response is dependent on a rangeof factors such as the seabed soil strength, loading conditions and riserdisplacement magnitude.

The riser-soil interaction forces within theTDZ strongly influence the SCR fatigue damage. Structural analyses of SCRsusually consider only vertical pipe-soil forces and incorporate the pipe-soilinteraction via linear springs(Mekha and Bhat, 2013). Fatigue life predictionsfor SCRs in the vicinity of the TDZ are heavily dependent on the assumedstiffness of the seabed. For accurate fatigue life predictions to be achieved, a reliable evaluation of the seabed stiffness is required. Nonlinear modelswhich incorporate tensile pipe-soil forces have been developed, including theCARISIMA model(Giertsen et al., 2004; Leira et al., 2004), STRIDEmodel(Thethi and Moros, 2001; Bridge et al., 2004), P-y curve model(Aubeny and Biscontin, 2008; 2009) and hysteretic seabed model(Randolph and Quiggin, 2009). The analysis results of the nonlinear models above indicatesignificantly decreased fatigue damage as compared to the linear idealization.

Plasticity theory, which has been applied toconstitutive modeling of both metals and soils, is gradually being used on amacroelement scale to model the combined loading behavior of shallowfoundations, as presented by Wood(2004), Cathie et al.(2005) and Tian and Cassidy(2008). The force-resultant models simulate the behavior of theentire foundation by combining the resultant forces directly with thecorresponding displacement(Cassidy et al., 2004) and supply analternate method to model the elements of riser or pipeline.

The force-resultant kinematic hardeningplasticity model in calcareous s and was originally presented by Zhang(2001) and Zhang et al.(2002a; 2002b). This novel approach is applied topipeline-soil interaction, which utilizes the bounding surface theory as theframework to describe the combined vertical and horizontal load-displacementbehavior of a pipeline in calcareous s and s. The method was calibrated usingcentrifuge test data of a prototype 1 m in diameter and an 8 m long pipeline oncalcareous s and s(Zhang, 2001). Tian and Cassidy(2008)provided a revision byformulating a different plastic potential function and the consistencycondition of the yield surface in derivation of the constitutive equation. Thenon-associated flow rule introduced by Zhang(2001)was modified so that theplastic potential surface could remain a similar shape and position with the yieldsurface. Tian et al.(2010)provided a new model formulation with allthe parameters calibrated from the experimental tests of a segment of pipe oncalcareous s and . The modified model has been shown to have excellent agreementwith the centrifuge data from the lateral displacement tests of the diametersof the two pipes. This provides additional confidence in the plasticity framework model’s use in the simulation of the pipeline and underlying soil.

Using a plasticity framework model to simulatethe behaviour of the pipeline and the underlying soil offers an alternate method.The more fundamental underst and ing of the pipe-soil interaction under thevertical and horizontal loading can be expressed by the parameters consistentwith the pipeline structural analysis, through expressing the pipe-soil interactionin terms of the loads and the corresponding displacements. Integrating theplasticity framework model into an FEM analysis program can describe the pipe-soilinteraction behavior efficiently, and the reasonable results can be achieved.

Similarly, a plasticity framework of theriser-soil interaction model in a clay soil seabed has been developed in this paper.The fatigue life of an SCR was analyzed by integrating the linear springs model and the plasticity framework model into a structural analysis program in thetime domain, respectively. According to the comparisons of the differentmodels, the fatigue life analysis result from the plasticity framework isreasonable and the horizontal effects of the riser-soil interaction can beincluded.

2 The plasticity framework model 2.1 Basic assumptions

The plasticity framework model is based on thetheory of kinematic hardening and critical state soil mechanics. A two-surfacemodel provided by Li and Meissner(2002)was developed for predicting theundrained behavior of saturated cohesive soils under cyclic loads. Thedevelopment of this model is based on the following assumptions similarly:

1)The riser-soil loading history is described with the bounding surface that is definitelydefined by the vertical settlement and represents the isotropic properties of soil.The bounding surface is a geometrical boundary, which can translate, contract and exp and in the V-H space, but the loading point cannot go outside of it.

2)The elastic domain is surrounded by theyield surface, which becomes to be a point and the plastic flow turns up for loadincrement within the bounding surface. Other than the classic yield surface, there is a loading surface within the domain surrounded by the bounding surfacethat represents a single loading event and reflects the anisotropic characteristicof the soil.

3)The loading surface could translate orexp and with the loading path within the bounding surface. The loading surface cantangentially contact with the bounding surface, but can not cross it.

4)At the time that the load point moves to thebounding surface, the plastic hardening modulus on the loading surface varies fromits local value to an appointed value on the bounding surface. The magnitude ofthe plastic hardening modulus lies on the relative condition of the twosurfaces.

5)The associated flow rule is utilized togovern the plastic flow for the loading surface. The positions of the loading and bounding surfaces in the V-H space are defined by the kinematic hardening rule.

2.2 Formulation ofthe model

The steel catenary riser in the touchdown zoneis assumed to be rigid and placed on the flat surface of homogeneous isotropicsoil. The riser pipe is embedded into the soil under the inner vertical loading and the horizontal soil loading. The resultant load contains the verticalloading and the horizontal loading, so the riser-soil interaction is defined inthe vertical and horizontal load space.

2.2.1 Bounding and loadingsurfaces

The bounding surface is assumed to be anelliptic form as:

where H and Vare the horizontal and vertical forces; isthe ratio of the two semi axes, $r = 1/\tan \theta $;a^m and b^m represent the coordinates of thecenter M of the bounding surface; a^m isthe semidiameter of the ellipse in the V direction; w^p is the vertical settlement of theriser pipe; m is the ordinal number of the loading eventsbut not the loading cycles number, m=0 means the initialloading during the loading process, m=1 represents thefirst loading or unloading event, and so on.

In order to keep a smooth transformationbetween the deformation processes inside the bounding surface, the loadingsurface is assumed to be similar to the bounding surface and alwaystangentially in contact with it, and their axes are parallel to each other, asshown in Fig. 1. Similar to Eq.(1), the loading surface is written as follows:

where a_f^m is the semidiameter of the loadingsurface in the major axis direction; and ξ^m and ζ^m are the coordinates of the center N^m of the loading surface.

2.2.2 Kinematic hardening rule

The kinematic hardening rule is described in detailby Li and Meissner(2002). The position of the memory center is defined by the hardeningrule firstly, which states that the memory center is located at the origin ofthe V-H space for virgin loading, and that it moves to the new load point forthe next loading where the loading path changes. The movement of the bounding and loading surfaces is controlled by the hardening rule else, which statesthat when the memory center gets its new position, the old bounding and loadingsurfaces in the last loading event are removed, and the new bounding and loading surfaces begin to play their roles in the new loading event. Thekinematic hardening rule is schematically represented in Fig. 2.

2.2.3 Flow rule and incremental relations

The associated flow rule for during the mth loading event iswritten as:

with where $L_f^m$ is the loading index and $\left\langle {L_f^m} \right\rangle $ is the Macauley operator, defined asfollows:$\left\langle {L_f^m} \right\rangle $=$L_f^m$, if $\left\langle {L_f^m} \right\rangle $ ≥0, otherwise $\left\langle {L_f^m} \right\rangle $ =0; K_m is the plastichardening modulus; n_V^m and n_H^m arethe unit vector components normal to f_m where

If the soil fluid and solid phases areincompressible, the loading index can be represented by:

and where

The differential equation of the loading pathis expressed as:

2.2.4 Evolvementrule

The size of the bounding surface is defined byspecifying the variation of the semidiameter w^p, which isthe only hardening parameter for the bounding surface:

with where φ and c arerespectively the internal friction angle in failure and the cohesion of clays, a⁰ is decided by the initial loading, w_o^p the vertical plastic settlement in the initial loading, and χ the soil state parameter.

The center coordinates of can be obtained from theproportionality relationship:

The center of f_m canbe derived from these two equations by replacing $a_c^m$ and $a_f^m$. The relationship between the point $R$ and ${f_m}$ can also be expressed in the proportionality equations:

The expression of R can be given from Eq.(1)by

with

where ${r_R}$ and ${r_M}$ expressthe value of at points R and M respectively. The semidiameter of ${f_m}$ can be obtained:

2.2.5 Plastic hardeningmodulus

The plastic hardening modulus is assumed to transformdepending on the relationship of the bounding and loading surfaces. Thehardening plastic modulus of ${f_m}$ can be representedby:

where $\delta = \left| {PR} \right|$, the distance of point P and R;${\delta _{\max }}$ is the maximum value of δ, ${\delta _{\max }} = \left| {MR} \right|$, shown in Fig. 1.γ is a material constant greater than 1.

3 Calculation example and results

To show the feasibility of the plasticityframework’s riser-soil interaction model, the finite element method has beenused to evaluate the fatigue life of a typical SCR example subjected to platformmotions and wave loadings in the method suggested by Dong et al.(2014).

The fatigue life of an SCR was analyzed by integratingthe plasticity framework model into a structural analysis program ABAQUS in thetime domain. The parameters of the example riser are shown in Table 1 and theplasticity framework model parameters are shown in Table 2.

The SCR nonlinear dynamic response analysis wascarried out under the wave and current forces coupled with the motions of thefloating in the time domain. The time histories of the combined stressesaccording to the dynamic analysis were employed to predict the riser fatiguelife by the method of the S-N curve and Rainflow counting technique. The SCRfatigue damage results of every seastate were added together with differentprobabilities of occurrence and the whole SCR fatigue damage and fatigue lifewere achieved. The motions of the floating were predicted in 6 degrees offreedom and the wave and current forces acting on the SCR were calculated usingMorison’s equation. The moment and tension responses of the SCR are shown inFig. 3.

The Von Mises combinedstress can be given as the expression:

where T is thetension force; A is the area of the riser pipe, $A = \frac{{\rm{\pi }}}{4}(D_o^2 - D_i^2);$Mis the moment force; I is the moment of inertia;${D_i}$ and ${D_0}$ are the inner and outer diameter ofthe riser pipe respectively.

The stress results of riser element number1241(3 087 m from the top end of SCR)in one seastate are shown in Table 3. The stress circulation number results ofriser element number 1241, direction 180°(The bottom of the riser pipe)are shown in Table 4.

The Doe-E S-N curve was used to estimate thefatigue damage of the riser. The expression can be written as:

where $S = {\rm{SCF}} \times {\rm{\Delta }}S$, SCF is the stress concentration factor, and arethe material constant.

So, the riser fatigue life can be predicted bythe method of the S-N curve and Rainflow counting technique. The fatigue damageof the example riser can be obtained in the SW direction consisting of 15seastates with different probabilities of occurrence. The SCR fatigue damageresults of every seastate were added together with the probability of occurrence.The whole SCR fatigue damage and fatigue life were achieved and the fatiguedamage results are shown in Fig. 4. It can be seen that the maximum fatiguedamage result is located at the TDZ.

4 Comparison of results and discussion

To illustrate that the result of the plasticityframework model is reasonable and reliable, the fatigue life of an SCR wasanalyzed by integrating the linear spring model and the plasticity frameworkmodel into a structural analysis program in the time domain, respectively.According to the comparisons of the different models, the fatigue life analysisresult of the plasticity framework is reasonable and the horizontal effects ofthe pipe-soil interaction can be included. The fatigue damage at the TDZ of thetwo models is shown in Fig. 5 and the fatigue damage at the TDZ of the twomodels in the vertical is shown in Fig. 6. It can be seen that the differenceof the results between the linear spring model and the plasticity frameworkmodel is more remarkable in the vertical.

To find out the reason why the fatigue resultof the plasticity framework model decreases evidently in the vertical, the comparisonanalysis between the combined stresses of the two models in the vertical and the horizontal directions is needed.

The maximum and minimum combined stresses atthe TDZ of the two models in the vertical are shown in Fig. 7 and Fig. 8 From the comparisonof the two figures, it can be judged that the margin between the maximum and minimum combined stresses of the spring element model is more remarkable, thatis, the cyclic amplitude is larger. So, the fatigue of the springelement model is more severe.

Also, the maximum and minimum combined stressesat the TDZ of the two models in the horizontal are shown in Fig. 9 and Fig. 10.Butjudged from the comparison of the two figures, the margin between the maximum and minimum combined stresses of the plasticity framework model is more obvious, the cyclic amplitude of the combined stress is larger, and the fatigue damageis higher than the other. It indicates that the effectof the horizontal reaction force between the riser and the soil can be summedup in the plasticity framework model.

5 Conclusions

This paper introduces a practical approach tointegrate the riser-soil interaction plasticity model into the finite element(FE)program. Attaching numerous force-resultant plasticity model elements to riser’sFinite Element nodes, the analysis of the riser-soil interaction becomescomputationally feasible. The 3D beam theory and FE displacement method areutilized to combined the model described in this paper into the FE program softwareABAQUS. The structural stiffness matrix was assembled with the plasticity modelby discretizing the riser pipe in the TDZ as beam elements. That is, thecontribution of the riser-soil plasticity model is incorporated into thestructural stiffness matrix using the FE displacement method. An SCRcalculation case demonstrates the feasibility of the suggested approach.

Although the plasticitymodel is only covering vertical and horizontal effects, combining the 3D beamtheory and the finite elementdisplacement method in implementing the model into the finite element method program would provide a more efficientapproach to simulate the riser-soil interaction with averaged sophistication ofthe structure and soil. The proposed approach facilitates the riser dynamicanalysis and can be used to evaluate the riser fatigue under complex loadingconditions. However, the axial friction force and rotation have not beenconsidered in the model or in this paper, which needs further development. Inconclusion, the proposed approach provides a new strategy for SCR fatigueanalysis and the research results should be helpful to the SCR design and analysis.