Bending behavior modeling of unbonded flexible pipes considering tangential compliance of interlayer contact interfaces and shear deformations
Introduction
While flexible pipe behavior under axisymmetric loads is well understood [34], [15], [20], [2], [26], [25], [29], determination of various bending characteristics is still a challenging problem. The difficulty is mainly due to the presence of helical layers and the freedom of helical components to slide against surrounding layers. So many authors put substantial effort into the post-slip behavior modeling of flexible pipes [5], [35], [24], [27], [11], [12], [19], [25], [29], [4], based on the assumption that the slip path of the armor wire follows either the strained helix (also known as the loxodromic path) or a geodesic curve.
Experimental results in terms of bending moment-curvature relationship generally show much smaller pre-slip bending stiffness and longer transition from the pre-slip to the full-slip stiffness than the analytical prediction [33], which implies that the pre-slip behavior of flexible pipes is not described adequately. In current stick-slip models, the pipe is simply modeled as a solid beam before gross slip initiates, i.e. plane sections remain plane during bending. Recently shear deformation in the multi-layer structure is identified as a potential source of the stiffness decrease in the stick regime [28]. However, it is believed that tangential compliance of interlayer contact interfaces is probably another mechanism which plays some part in the reduction of the pre-slip stiffness.
According to the theory of contact mechanics [10], micro-slip does occur between contacting bodies up to a certain limit value at which gross slip begins. The relative tangential displacement is accommodated in large part by elastic shear strain at the contact interface. This elastic behavior is characterized by tangential compliance. Analytical models considering the tangential compliance of contact interfaces can be found in the literature, mostly dealing with the analyses of spiral strands or cables [9], [8], [7], [21], [18]. The inclusion of tangential compliance is translated into a slowly decreasing cable flexural rigidity [7], [21], [18]. To our knowledge, the consideration of tangential compliance in flexible pipe analyses has not been performed.
The pre-slip behavior has a significant effect on the fatigue performance of dynamic applications of flexible pipes, such as a deep-water flexible riser. In global analyses the decrease of pre-slip stiffness can give larger curvatures compared with the current stick-slip models. This effect is even more pronounced if bending hysteresis is incorporated [30], [31]. In addition, shear deformations will delay the gross slip initiation, hence reducing the stress in the stick domain at a given curvature [28]. Considering the slope parameter in the S–N curve, a small reduction factor in stress can have a great impact on the resulting fatigue life.
In this paper, more rigorous analytical formulations are firstly given to describe the gross slip initiation and progression of the armor wire. The effects of tangential compliance and shear deformations on the layer bending behavior are then taken into account. Comparative studies with respect to the available experimental data are further carried out and encouraging correlations are found.
Section snippets
Assumptions and definitions
The main assumptions and definitions used throughout the analysis are:
- 1.
The slip path of the armor wire is assumed to follow either the strained helix or a geodesic curve. Both curves are considered as the limit curves for wire slip, and the actual path is probably somewhere in between. According to [23]; the transverse slip can be neglected when evaluating the dynamic bending stress. Hence, the strained helix assumption is employed in this study.
- 2.
The wire slippage in its axial direction is only
Case study
Witz [33] published a case study in the cross-section analysis of a 2.5-inch unbonded flexible pipe. Both construction details of the pipe and the experimental results were provided for axisymmetric and bending behavior. The bending response of the pipe was measured under two different internal pressures, 0 and 30 MPa. Berge et al. [1] also reported the full scale bending test results of a 4-inch flexible pipe under the internal pressures of 0.7, 10 and 25 MPa.
Fig. 6 shows the boundary angles
Discussion on the TC model
The TC model stands on the relationship between the relative displacement and the tangential force, i.e. Equation (46) and Equations (49), (50). In the derivation of Equation (46), the contact dimensions and the pressure distribution due to the normal force are determined according to the Hertz theory and these quantities are assumed to be unaffected by the subsequent application of the tangential force [10]. Each individual asperity is also assumed to follow the Hertz theory in the model
Concluding remarks
By evaluating the shear forces on the wire cross-section, this study firstly verifies the commonly made assumption that the wire slippage in its axial direction is only governed by the internal axial force and the available friction. Then more rigorous analytical formulations are given to describe the gross slip initiation and progression of the wire. Two mechanisms are further thought to contribute to the decrease of layer stiffness before gross slip begins. The first one introduces the
Acknowledgments
Financial support from the National Natural Science Foundation of China (Grant No. 51409044), China Postdoctoral Science Foundation (Grant No. 2014M551088) and the Fundamental Research Funds for the Central Universities (Grant No. DUT14RC(3)005) is gratefully acknowledged.
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