Dynamic curvature of a steel catenary riser on elastic seabed considering trench shoulder effects: an analytical model

Abstract

Prediction of the fatigue life of steel catenary risers (SCR) in the touchdown zone is a challenging engineering design aspect of these popular elements. It is publically accepted that the gradual trench formation underneath the SCR due to cyclic oscillations may affect the fatigue life of the riser. However, due to the complex nature of the several mechanisms involving three different domains of the riser, seabed soil, and seawater, there is still no strong agreement on the beneficial or detrimental effects of the trench on the riser fatigue. Seabed soil stiffness and trench geometry play crucial roles in the accumulation of fatigue damage in the touchdown zone. There are several studies about the effect of seabed soil stiffness on fatigue. However, recent studies have proven the significance of trench geometry and identified the touchdown point oscillation amplitude as a key factor. In this study, a boundary layer solution was adapted to obtain the dynamic curvature oscillation of the riser in the touchdown zone on different areas of seabed trenches with a range of seabed stiffness. The proposed analytical model was validated against advanced finite element analysis using a commercial software. A range of seabed stiffness was examined, and the corresponding fatigue responses were compared. It was observed that in the elastic seabed, the effect of soil stiffness is attributed to the curvature oscillation amplitude and to the minimum local dynamic curvature that SCR can take in the touchdown zone. The proposed analytical model was found to be a simple and reliable tool for riser configuration studies with trench effects, particularly at the early stages of riser engineering design practice.

Appendix. Dynamic equilibrium equations for the planar problem of a catenary riser

This Appendix brings a derivation that can be found in a more detailed analysis in Pesce [15], Chapter 4, Section 4.1. It is however essential for the understanding of the local analysis close to TDP, carried out through the boundary layer technique, in the main core of the text. Consider a planar problem of a riser suspended from a floating unity, whose static configuration is characterized by the functions $\theta \left( s \right)$, $T\left( s \right)$ and $Q\left( s \right)$, respectively, the angle of the line with respect to the horizontal, the effective tension and the shear force at a given section s. Let their dynamic counterparts be written as,

$$\begin{aligned} \Theta \left( {s,t} \right) & = \theta \left( s \right) + \gamma \left( {s,t} \right) \\ T\left( {s,t} \right) & = T\left( s \right) + \tau \left( {s,t} \right) \\ Q\left( {s,t} \right) & = Q\left( s \right) + \vartheta \left( {s,t} \right) \\ \end{aligned}$$

(36)

where $\gamma \left( {s,t} \right)$, $\tau \left( {s,t} \right)$ and $\vartheta \left( {s,t} \right)$ are the corresponding perturbed values, resulting from dynamic loads acting on the riser in the vertical plane. Let also $u\left( {s,t} \right)$ and $v\left( {s,t} \right)$ be small displacements around the static equilibrium configuration in their tangential and normal directions, respectively. To first order, the following well-known kinematic relation can be promptly derived (Fig. 14).

Fig. 14

Schematic view of planar problem

$$\gamma \left( {s,t} \right) = \frac{\partial v}{{\partial s}} + u\frac{{{\text{d}}\theta }}{{{\text{d}}s}}$$

(37)

Taking a small segment $\Delta s$, the resultant of effective tension and shear force projected onto the tangential and normal directions of the static configuration is readily obtained in the form:

$$\begin{aligned} \Delta F_{u} & = T\left( {s + \Delta s,t} \right)\cos \left( { \Delta \theta + \gamma \left( {s + \Delta s,t} \right)} \right) - T\left( {s,t} \right)\cos \gamma \left( {s,t} \right) \\ & \quad - \left( {Q\left( {s + \Delta s,t} \right)\sin \left( {\Delta \theta + \gamma \left( {s + \Delta s,t} \right)} \right) - Q\left( s \right)\sin \gamma \left( {s,t} \right)} \right) \\ \Delta F_{v} & = Q\left( {s + \Delta s,t} \right)\cos \left( { \Delta \theta + \gamma \left( {s + \Delta s,t} \right)} \right) - Q\left( s \right)\cos \gamma \left( {s,t} \right) \\ & \quad + \left( {T\left( {s + \Delta s,t} \right)\sin \left( { \Delta \theta + \gamma \left( {s + \Delta s,t} \right)} \right) - T\left( {s,t} \right)\sin \gamma \left( {s,t} \right)} \right) \\ \end{aligned}$$

(38)

If only first-order terms in $\Delta \theta$ and $\gamma$ are retained, Eq. (38) reduces to

$$\begin{aligned} \Delta F &_{u} \cong T\left( {s + \Delta s,t} \right) - T\left( {s,t} \right) - \left( {Q\left( {s + \Delta s,t} \right)\Delta \theta + Q\left( {s + \Delta s,t} \right)\gamma \left( {s + \Delta s,t} \right) - Q\left( s \right)\gamma \left( {s,t} \right)} \right) \\ \Delta F_{v} & \cong Q\left( {s + \Delta s,t} \right) - Q\left( s \right) + \left( {T\left( {s + \Delta s,t} \right)\Delta \theta + T\left( {s + \Delta s,t} \right)\gamma \left( {s + \Delta s,t} \right) - T\left( {s,t} \right)\gamma \left( {s,t} \right)} \right) \\ \end{aligned}$$

(39)

The dynamic equilibrium equation of the segment $\Delta s$ then reads

$$\begin{aligned} & \Delta F_{u} + h_{u} \Delta s - q\sin \theta \Delta s = m\frac{{\partial^{2} u}}{{\partial t^{2} }}\Delta s \\ & \Delta F_{v} + h_{v} \Delta s - q\cos \theta \Delta s = m\frac{{\partial^{2} v}}{{\partial t^{2} }}\Delta s \\ \end{aligned}$$

(40)

where

$$\begin{aligned} h_{u} \left( {s,t} \right) & = h_{u} \left( s \right) + \varpi_{u} \left( {s,t} \right) \\ h_{v} \left( {s,t} \right) & = h_{v} \left( s \right) + \varpi_{v} \left( {s,t} \right) \\ \end{aligned}$$

(41)

refer to the hydrodynamic forces, q is the immersed weight, and m is the mass of the structure, all per unit length. The terms $h_{u,v} \left( s \right)$ and $\varpi_{u,v} \left( {s,t} \right)$ are the components of the static and dynamic parcels of the hydrodynamic force in the tangential and normal direction, regarding the static configuration. The last ones are due to the relative external water flow with respect to the riser, at section s, usually modeled through the well-known Morison’s formula. Equations (40) transform into partial differential ones, by the usual process of taking the limit when $\Delta s \to 0$, in the following form:

$$\begin{aligned} & \frac{\partial T}{{\partial s}} - \left( {T\gamma + Q} \right)\frac{{{\text{d}}\theta }}{{{\text{d}}s}} - \frac{\partial }{\partial s}\left( {Q\gamma } \right) + h_{u} + \varpi_{u} - q\sin \theta = m\frac{{\partial^{2} u}}{{\partial t^{2} }} \\ & \frac{\partial Q}{{\partial s}} + \left( {T - Q\gamma } \right)\frac{{{\text{d}}\theta }}{{{\text{d}}s}} + \frac{\partial }{\partial s}\left( {T\gamma } \right) + h_{v} + \varpi_{v} - q\cos \theta = m\frac{{\partial^{2} v}}{{\partial t^{2} }} \\ \end{aligned}$$

(42)

Alternatively, Eq. (42) may be written with the use of the kinematic relation (37) as,

$$\begin{aligned} & \frac{\partial T}{{\partial s}} - \left[ {T\left( {\frac{\partial v}{{\partial s}} + u\frac{{{\text{d}}\theta }}{{{\text{d}}s}}} \right) + Q} \right]\frac{{{\text{d}}\theta }}{{{\text{d}}s}} - \frac{\partial }{\partial s}\left[ {Q\left( {\frac{\partial v}{{\partial s}} + u\frac{{{\text{d}}\theta }}{{{\text{d}}s}}} \right)} \right] + h_{u} + \varpi_{u} - q\sin \theta = m\frac{{\partial^{2} u}}{{\partial t^{2} }} \\ & \frac{\partial Q}{{\partial s}} + \left[ {T - Q\left( {\frac{\partial v}{{\partial s}} + u\frac{{{\text{d}}\theta }}{{{\text{d}}s}}} \right)} \right]\frac{{{\text{d}}\theta }}{{{\text{d}}s}} + \frac{\partial }{\partial s}\left[ {T\left( {\frac{\partial v}{{\partial s}} + u\frac{{{\text{d}}\theta }}{{{\text{d}}s}}} \right)} \right] + h_{v} + \varpi_{v} - q\cos \theta = m\frac{{\partial^{2} v}}{{\partial t^{2} }} \\ \end{aligned}$$

(43)

Notice that by using Eqs. (36b,c), (42) may be also rewritten as,

$$\begin{aligned} & \left\{ {\frac{\partial T}{{\partial s}} - Q\frac{{{\text{d}}\theta }}{{{\text{d}}s}} + h_{u} - q\sin \theta } \right\} + \left\{ {\frac{\partial \tau }{{\partial s}} - \left( {T\gamma + \vartheta } \right)\frac{{{\text{d}}\theta }}{{{\text{d}}s}} - \frac{\partial }{\partial s}\left( {Q\gamma } \right) + \varpi_{u} } \right\} = m\frac{{\partial^{2} u}}{{\partial t^{2} }} \\ & \left\{ {\frac{\partial Q}{{\partial s}} + T\frac{{{\text{d}}\theta }}{{{\text{d}}s}} + h_{v} - q\cos \theta } \right\} + \left\{ {\frac{\partial \vartheta }{{\partial s}} + \left( {\tau - Q\gamma } \right)\frac{{{\text{d}}\theta }}{{{\text{d}}s}} + \frac{\partial }{\partial s}\left( {T\gamma } \right) + \varpi_{v} } \right\} = m\frac{{\partial^{2} v}}{{\partial t^{2} }} \\ \end{aligned}$$

(44)

or, from (43),

$$\begin{aligned} & \left\{ {\frac{\partial T}{{\partial s}} - Q\frac{{{\text{d}}\theta }}{{{\text{d}}s}} + h_{u} - q\sin \theta } \right\} + \left\{ {\frac{\partial \tau }{{\partial s}} - \left( {T\left( {\frac{\partial v}{{\partial s}} + u\frac{{{\text{d}}\theta }}{{{\text{d}}s}}} \right) + \vartheta } \right)\frac{{{\text{d}}\theta }}{{{\text{d}}s}} - \frac{\partial }{\partial s}\left( {Q\left( {\frac{\partial v}{{\partial s}} + u\frac{{{\text{d}}\theta }}{{{\text{d}}s}}} \right)} \right) + \varpi_{u} } \right\} = m\frac{{\partial^{2} u}}{{\partial t^{2} }} \\ & \left\{ {\frac{\partial Q}{{\partial s}} + T\frac{{{\text{d}}\theta }}{{{\text{d}}s}} + h_{v} - q\cos \theta } \right\} + \left\{ {\frac{\partial \vartheta }{{\partial s}} + \left( {\tau - Q\left( {\frac{\partial v}{{\partial s}} + u\frac{{{\text{d}}\theta }}{{{\text{d}}s}}} \right)} \right)\frac{{{\text{d}}\theta }}{{{\text{d}}s}} + \frac{\partial }{\partial s}\left( {T\left( {\frac{\partial v}{{\partial s}} + u\frac{{{\text{d}}\theta }}{{{\text{d}}s}}} \right)} \right) + \varpi_{v} } \right\} = m\frac{{\partial^{2} v}}{{\partial t^{2} }} \\ \end{aligned}$$

(45)

The first terms in brackets, either in Eqs. (44) or (45) are, in fact, Love’s equations for the static equilibrium of curved bars on the plane. Therefore, they are identically null. The (perturbed) dynamic variables are, therefore, governed by the following coupled nonlinear partial differential equations:

$$\begin{aligned} & \frac{\partial \tau }{{\partial s}} - \left( {\left( {T + \tau } \right)\gamma + \vartheta } \right)\frac{{{\text{d}}\theta }}{{{\text{d}}s}} - \frac{\partial }{\partial s}\left( {\left( {Q + \vartheta } \right)\gamma } \right) + \varpi_{u} = m\frac{{\partial^{2} u}}{{\partial t^{2} }} \\ & \frac{\partial \vartheta }{{\partial s}} + \left( {\tau - \left( {Q + \vartheta } \right)\gamma } \right)\frac{{{\text{d}}\theta }}{{{\text{d}}s}} + \frac{\partial }{\partial s}\left( {\left( {T + \tau } \right)\gamma } \right) + \varpi_{v} = m\frac{{\partial^{2} v}}{{\partial t^{2} }} \\ \end{aligned}$$

(46)

or, given just in terms of the displacements $u\left( {s,t} \right)$ and $v\left( {s,t} \right)$,

$$\begin{aligned} & \frac{\partial \tau }{{\partial s}} - \left( {\left( {T + \tau } \right)\left( {\frac{\partial v}{{\partial s}} + u\frac{{{\text{d}}\theta }}{{{\text{d}}s}}} \right) + \vartheta } \right)\frac{{{\text{d}}\theta }}{{{\text{d}}s}} - \frac{\partial }{\partial s}\left( {\left( {Q + \vartheta } \right)\left( {\frac{\partial v}{{\partial s}} + u\frac{{{\text{d}}\theta }}{{{\text{d}}s}}} \right)} \right) + \varpi_{u} = m\frac{{\partial^{2} u}}{{\partial t^{2} }} \\ & \frac{\partial \vartheta }{{\partial s}} + \left( {\tau - \left( {Q + \vartheta } \right)\left( {\frac{\partial v}{{\partial s}} + u\frac{{{\text{d}}\theta }}{{{\text{d}}s}}} \right)} \right)\frac{{{\text{d}}\theta }}{{{\text{d}}s}} + \frac{\partial }{\partial s}\left( {\left( {T + \tau } \right)\left( {\frac{\partial v}{{\partial s}} + u\frac{{{\text{d}}\theta }}{{{\text{d}}s}}} \right)} \right) + \varpi_{v} = m\frac{{\partial^{2} v}}{{\partial t^{2} }} \\ \end{aligned}$$

(47)

On the other hand, the third planar static Love’s equation that relates bending moment and shear force may be written, in the absence of any external applied moment per unit length as,

$$\frac{\partial M}{{\partial s}} + Q = 0$$

(48)

Consistently with the kinematic relation (37), and considering that the slenderness of the structure makes the effect of the rotatory inertia negligible, the corresponding dynamic equation regarding the rotation would be written as:

$$\frac{\partial \mu }{{\partial s}} + \vartheta = 0$$

(49)

where $\mu \left( {s,t} \right)$ is the dynamic parcel of the bending moment. In fact, this is a quasi-static approximation.

Therefore, bending moment and shear may be said to be simply related by

$$\frac{\partial M}{{\partial s}} + Q = 0$$

(50)

On the other hand, from the three basic and usual hypotheses: (1) small strains; (2) linear relations between stresses and strains; (3) Kirchhoff’s ‘plane sections remain plane after deformation,’ the following constitutive equation may be assumed valid:

$$M\left( {s,t} \right) = EI\frac{\partial \Theta }{{\partial s}} = EI\chi \left( {s,t} \right)$$

(51)

where EI is the bending stiffness at section s, and $\chi \left( {s,t} \right)$ is the total curvature. The following relations, regarding the static and dynamic parcels, are then promptly derived:

$$\begin{aligned} & M\left( s \right) = EI\frac{{{\text{d}}\theta }}{{{\text{d}}s}} \\ & \mu \left( {s,t} \right) = EI\frac{\partial \gamma }{{\partial s}} \cong EI\left[ {\frac{{\partial^{2} v}}{{\partial s^{2} }} + \frac{\partial }{\partial s}\left( {u\frac{{{\text{d}}\theta }}{{{\text{d}}s}}} \right)} \right] \\ \end{aligned}$$

(52)

From (48) and (49), it follows that

$$\begin{aligned} & Q\left( s \right) = - \frac{\partial }{\partial s}\left( {EI\frac{{{\text{d}}\theta }}{{{\text{d}}s}}} \right) \\ & \vartheta \left( {s,t} \right) = - \frac{\partial }{\partial s}\left( {EI\frac{{\partial^{2} v}}{{\partial s^{2} }} + EI\frac{\partial }{\partial s}\left( {u\frac{{{\text{d}}\theta }}{{{\text{d}}s}}} \right)} \right) \\ \end{aligned}$$

(53)

Also, using (51) in (50) and then substituting the result in Eq. (42,b), it follows that

$$- \frac{{\partial^{2} }}{{\partial s^{2} }}\left( {EI\chi } \right) + \left( {{\text{T}} - \frac{\partial }{\partial s}\left( {EI\chi } \right)\gamma } \right)\frac{d\theta }{{ds}} + \frac{\partial }{\partial s}\left( {{\text{T}}\gamma } \right) + h_{v} + \varpi_{v} - q\cos \theta = m\frac{{\partial^{2} v}}{{\partial t^{2} }}$$

(54)

In the common case in which the bending stiffness is assumed constant along the line, Eq. (54) reduces to:

$$- EI\frac{{\partial^{2} \chi }}{{\partial s^{2} }} - \gamma EI\frac{{{\text{d}}\theta }}{{{\text{d}}s}}\frac{\partial \chi }{{\partial s}} + T\chi + \gamma \frac{\partial T}{{\partial s}} + h_{v} + \varpi_{v} - q\cos \theta = m\frac{{\partial^{2} v}}{{\partial t^{2} }}$$

(55)

Equation (55) is a fundamental result to be used in the local analysis, close to TDP, via the boundary layer technique. It can be used in the vicinity of the hang-off point as well, see [15].