Yongqiang Dong and Liping Sun

College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China

1 Introduction1

As the offshore developments of oil and gas move into deeper water, the understanding of the fatigue performance of the steel catenary riser (SCR) becomes critical to long-standing operation. The SCRs usually tend to fatigue damage, especially in the region where the riser pipe reaches the seabed, known as the ‘touchdown zone’ (TDZ) (Hodder and Byrne, 2010). One of the key issues for SCR design is to assess the fatigue damage due to repetitive loading over the lifetime of the riser (Xuet al., 2006). Accurate evaluation of the fatigue life of an SCR remains a major challenge due to uncertainty surrounding the interaction forces where the riser touches down on the seabed (Hodderet al., 2009). The results of the fatigue evaluation depend 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 dynamic responses in the TDZ are not easy to quantify. The riser moves cyclically within the touchdown zone due to excitation from the vessel and wave loadings, softening and remolding the seabed soil.Vertically or in-plane, riser motions create transient variations in the riser-soil force, associated with longitudinal translation of the touchdown point accompanied by changes in the tension. The cyclic expulsion and sucking-in of water between the riser and soil during vertical motions also tends to cause water entrainment into the soil, thereby increasing the degradation of the shear strength. Horizontally or out-of-plane, riser motions reduce the vertical bearing capacity of the soil due to the combined V-H loading imposed on the seabed. The embedment required to support a given vertical load need to be increased. Also, horizontal motions tend to sweep soil laterally away from the riser alignment, which leads to the formation of a trench around the riser.

The full riser-soil interaction responses within the TDZ of an SCR include interaction between the riser response and the soil response. The actual conditions imposed on an element of a riser pipe are neither load controlled nor displacement controlled only (Hodderet al., 2009).Therefore, the riser-soil interaction response is dependent on a range of factors such as the seabed soil strength, loading conditions and riser displacement magnitude.

The riser-soil interaction forces within the TDZ strongly influence the SCR fatigue damage. Structural analyses of SCRs usually consider only vertical pipe-soil forces and incorporate the pipe-soil interaction via linear springs(Mekha and Bhat, 2013). Fatigue life predictions for SCRs in the vicinity of the TDZ are heavily dependent on the assumed stiffness of the seabed. For accurate fatigue life predictions to be achieved, a reliable evaluation of the seabed stiffness is required. Nonlinear models which incorporate tensile pipe-soil forces have been developed,including the CARISIMA model (Giertsenet al., 2004;Leiraet al., 2004), STRIDE model (Thethi and Moros, 2001;Bridgeet 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 indicate significantly decreased fatigue damage as compared to the linear idealization.

Plasticity theory, which has been applied to constitutive modeling of both metals and soils, is gradually being used on a macroelement scale to model the combined loading behavior of shallow foundations, as presented by Wood(2004), Cathieet al. (2005) and Tian and Cassidy (2008).The force-resultant models simulate the behavior of the entire foundation by combining the resultant forces directly with the corresponding displacement (Cassidyet al., 2004)and supply an alternate method to model the elements of riser or pipeline.

The force-resultant kinematic hardening plasticity model in calcareous sand was originally presented by Zhang (2001)and Zhanget al. (2002a; 2002b). This novel approach is applied to pipeline-soil interaction, which utilizes the bounding surface theory as the framework to describe the combined vertical and horizontal load-displacement behavior of a pipeline in calcareous sands. The method was calibrated using centrifuge test data of a prototype 1 m in diameter and an 8 m long pipeline on calcareous sands(Zhang, 2001). Tian and Cassidy (2008) provided a revision by formulating a different plastic potential function and the consistency condition of the yield surface in derivation of the constitutive equation. The non-associated flow rule introduced by Zhang (2001) was modified so that the plastic potential surface could remain a similar shape and position with the yield surface. Tianet al. (2010) provided a new model formulation with all the parameters calibrated from the experimental tests of a segment of pipe on calcareous sand. The modified model has been shown to have excellent agreement with the centrifuge data from the lateral displacement tests of the diameters of 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 simulate the behaviour of the pipeline and the underlying soil offers an alternate method. The more fundamental understanding of the pipe-soil interaction under the vertical and horizontal loading can be expressed by the parameters consistent with the pipeline structural analysis, through expressing the pipe-soil interaction in terms of the loads and the corresponding displacements. Integrating the plasticity framework model into an FEM analysis program can describe the pipe-soil interaction behavior efficiently, and the reasonable results can be achieved.

Similarly, a plasticity framework of the riser-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 the time domain, respectively. According to the comparisons of the different models, the fatigue life analysis result from the plasticity framework is reasonable and the horizontal effects of the riser-soil interaction can be included.

2 The plasticity framework model

2.1 Basic assumptions

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

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

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

3) The loading surface could translate or expand with the loading path within the bounding surface. The loading surface can tangentially contact with the bounding surface,but can not cross it.

4) At the time that the load point moves to the bounding surface, the plastic hardening modulus on the loading surface varies from its local value to an appointed value on the bounding surface. The magnitude of the plastic hardening modulus lies on the relative condition of the two surfaces.

5) The associated flow rule is utilized to govern 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 of the model

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

2.2.1 Bounding and loading surfaces

The bounding surface is assumed to be an elliptic form as:

whereHandVare the horizontal and vertical forces;ris the ratio of the two semi axes,r=1/tanq;andrepresent the coordinates of the centerMof the bounding surface;is the semidiameter of the ellipse in theVdirection;pwis the vertical settlement of the riser pipe;mis the ordinal number of the loading events but not the loading cycles number,m=0 means the initial loading during the loading process,m=1 represents the first loading or unloading event, and so on.

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

Fig. 1 Kinematic hardening model

2.2.2 Kinematic hardening rule

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

Fig. 2 The kinematic hardening rule

2.2.3 Flow rule and incremental relations

The associated flow rule forfmduring themth loading event is written as:

with

where

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

and

where

The differential equation of the loading path is expressed as:2.2.4 Evolvement rule

The size of the bounding surface is defined by specifying the variation of the semidiameterpw, which is the only hardening parameter for the bounding surface:

with

whereφandcare respectively the internal friction angle in failure and the cohesion of clays,0ais decided by the initial loading,the vertical plastic settlement in the initial loading, andcthe soil state parameter.

The center coordinates ofFmcan be obtained from the proportionality relationship:

The center offmcan be derived from these two equations by replacingand. The relationship between the pointRandfmcan also be expressed in the proportionality equations:

The expression ofRcan be given from Eq. (1) by

with

whereRrandMrexpress the value ofrat pointsRandMrespectively. The semidiameter offmcan be obtained:2.2.5 Plastic hardening modulus

The plastic hardening modulus is assumed to transform depending on the relationship of the bounding and loading surfaces. The hardening plastic modulus offmcan be represented by:

3 Calculation example and results

To show the feasibility of the plasticity framework’s riser-soil interaction model, the finite element method has been used to evaluate the fatigue life of a typical SCR example subjected to platform motions and wave loadings in the method suggested by Donget al. (2014).

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

Table 1 SCR data

Table 2 Model parameter

The SCR nonlinear dynamic response analysis was carried out under the wave and current forces coupled with the motions of the floating in the time domain. The time histories of the combined stresses according to the dynamic analysis were employed to predict the riser fatigue life by the method of the S-N curve and Rainflow counting technique. The SCR fatigue damage results of every seastate were added together with different probabilities of occurrence and the whole SCR fatigue damage and fatigue life were achieved. The motions of the floating were predicted in 6 degrees of freedom and the wave and current forces acting on the SCR were calculated using Morison’s equation. The moment and tension responses of the SCR are shown in Fig. 3.

Fig. 3 SCR tension and moment

The Von Mises combined stress can be given as the expression:

whereTis the tension force;Ais the area of the riser pipe,Mis the moment force;Iis the moment of inertia;iDandoDare the inner and outer diameter of the riser pipe respectively.

The stress results of riser element number 1241(3 087 m from the top end of SCR) in one seastate are shown in Table 3. The stress circulation number results of riser 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 the fatigue damage of the riser. The expression can be written as:

whereS=SCF×ΔS, SCF is the stress concentration factor,aandmare the material constant.

So, the riser fatigue life can be predicted by the method of the S-N curve and Rainflow counting technique. The fatigue damage of the example riser can be obtained in the SW direction consisting of 15 seastates with different probabilities of occurrence. The SCR fatigue damage results of every seastate were added together with the probability of occurrence. The whole SCR fatigue damage and fatigue life were achieved and the fatigue damage results are shown in

Fig. 4. It can be seen that the maximum fatigue damage result is located at the TDZ.

Table 3 Riser element stress results

Table 4 Stress circulation number results of riser element

Fig. 4 The fatigue damage of the SW direction

4 Comparison of results and discussion

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

Fig. 5 TDZ fatigue damage of the two models

Fig. 6 TDZ fatigue damage of the two models in the vertical

Fig. 7 The combined stress envelopes of the spring element model in the vertical

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

The maximum and minimum combined stresses at the TDZ of the two models in the vertical are shown in Figs. 7 and 8. From the comparison of 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, that is, the cyclic amplitude is larger. So, the fatigue of the spring element model is more severe.

Also, the maximum and minimum combined stresses at the TDZ of the two models in the horizontal are shown in

Figs. 9 and 10. But judged 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 damage is higher than the other. It indicates that the effect of the horizontal reaction force between the riser and the soil can be summed up in the plasticity framework model.

Fig. 8 The combined stress envelopes of the plasticity framework model in the vertical

Fig. 9 The combined stress envelopes of the spring element model in the horizontal

Fig. 10 The combined stress envelopes of the plasticity framework model in the horizontal

5 Conclusions

This paper introduces a practical approach to integrate the riser-soil interaction plasticity model into the finite element(FE) program. Attaching numerous force-resultant plasticity model elements to riser’s Finite Element nodes, the analysis of the riser-soil interaction becomes computationally feasible. The 3D beam theory and FE displacement method are utilized to combined the model described in this paper into the FE program software ABAQUS. The structural stiffness matrix was assembled with the plasticity model by discretizing the riser pipe in the TDZ as beam elements.That is, the contribution of the riser-soil plasticity model is incorporated into the structural stiffness matrix using the FE displacement method. An SCR calculation case demonstrates the feasibility of the suggested approach.

Although the plasticity model is only covering vertical and horizontal effects, combining the 3D beam theory and the finite element displacement method in implementing the model into the finite element method program would provide a more efficient approach to simulate the riser-soil interaction with averaged sophistication of the structure and soil. The proposed approach facilitates the riser dynamic analysis and can be used to evaluate the riser fatigue under complex loading conditions. However, the axial friction force and rotation have not been considered in the model or in this paper, which needs further development. In conclusion, the proposed approach provides a new strategy for SCR fatigue analysis and the research results should be helpful to the SCR design and analysis.

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