To investigate the effects of breathing cracks on the dynamic characteristics, i.e., natural frequencies, vibration displacement, slope angle, and bending moment, of a drilling riser, a time-domain finite element method based on the fracture mechanics theory is developed. The correctness of the proposed method is verified, and the performance of the different dynamic characteristics for crack identification is evaluated. First, the influence of the breathing and open cracks on natural frequencies is explored. The maximum frequency shifts vs crack location and water depth are studied. The results of the small difference of natural frequencies between intact and cracked risers illustrate that the traditional frequency-based crack-detection method is not applicable for the risers. Second, the time-domain motions, orbit plots, Poincaré maps, and fast Fourier transform diagrams are extracted for discussing the effects of the breathing and open cracks and crack depth and location. Finally, the second and fourth derivatives of the root-of-mean-square (RMS) of the dynamic response characteristics are defined for crack identification. It is found that the moment RMS has a good performance in single- and multiple-crack identifications and that not only locations but also degrees of the breathing cracks can be well detected by the proposed indices. In short, several conclusions drawn are a benefit for the safety of a serviced drilling riser system.

## I. INTRODUCTION

Due to complex sea loads, especially the extreme wind and harsh waves and current, corrosion, high operation loads, low temperatures, wearing, fatigue, and other factors, cracks are unavoidably generated and propagated and significantly affect the safety of a drilling riser system. Because of its long-span and thin-wall characteristics, the drilling riser is one of the weakest parts of a drilling device. It is essential to investigate the effect of damage and damage detection on drilling risers. However, the traditional riser inspection techniques could prove inadequate for deepwater drilling risers because of more complex factor contributions^{1} and the requirement of the riser to be pulled to the surface and buoyancy blocks to be removed, causing difficult, dangerous, and time-consuming operations.^{2} As a consequence, the vibration-based damage-detection method, recently, has become a new trend to structural health monitoring and crack identification for marine structures, e.g., platforms, drilling risers, and pipelines.

Damage-detection methods using dynamic characteristics, e.g., natural frequencies, displacement, slope, strain, energy, and bending moment, have been widely reported in civil, aviation, petroleum, and ocean engineering in the literature. Tian *et al.*^{3} modeled a surface crack in a platform by the classical linear spring model and detected the crack using the structural intensity approach. Nguyen and Oterkus^{4} developed a modified peri-dynamic model to predict cracks and possible damages that may occur during the operation process in a marine platform. Zhou *et al.*^{5} studied the dynamic characteristics of a platform with a single-side open crack and concluded that crack sizes play an effective role in dynamic behaviors, where the maximum crack depth reaches half of the thickness of the beam. Mu *et al.*^{6} investigated a blind hole with 3 mm diameter and 2 mm depth in a carbon steel plate with 5 mm thickness using a modified time-reversal damage localization method. Min *et al.*^{7} considered a decrease in Young’s modulus with the structural damage and studied the natural frequencies and mode shapes of a cracked riser. The same damage assumptions using stiffness reduction are also available in the literature.^{8–11} Zhou *et al.*^{12} studied the effect of surface-open cracks on natural frequencies and mode shapes of a hang-off riser during deployment and retrieval. Dotti *et al.*^{13} modeled an elliptical hole as a crack and investigated the breathing effect using the classic bilinear stiffness model. Avramov and Raimberdiyev^{14} and Avramov and Malyshev^{15} studied the dynamic behaviors of beams with one-side and two-side breathing cracks, respectively. The bilinear stiffness modeling of breathing cracks was also used in studies.^{16–18} Another modeling of the breathing crack method accounts for the crack breathing effect as a time-varying stiffness, which is widely used in the traditional simple beams, i.e., the mostly simply supported and cantilever beams.^{19–21}

As evidenced in the literature, the studies on damage detection of marine risers are mainly based on the simple stiffness-reduction assumption and the open-crack assumption. Few pieces of research consider the opening and closing effect of breathing cracks even though the breathing effects and crack assessment have been widely studied in traditional structures. Two significant differences in geometric features between traditional beams and marine risers are the great slenderness ratio and small crack size. The first feature illustrates that the ratio of the riser length to the pipe diameter is much large and that the vibration difference caused by the breathing crack is probably a bit less sensitive.^{12} The other feature illustrates that cracks are not allowed to penetrate the thin wall in a drilling riser during normal operations to prevent liquid leakage.^{22} Therefore, the combination of the two factors may cause small crack identification in marine risers to be more difficult or even impossible. To investigate the influence of small cracks on the dynamic characteristics of a marine riser, a time-domain finite element method (FEM) based on the modeling of a breathing crack is developed in this study. To detect the small crack’s position and degree, several crack-identification indices are proposed and discussed with consideration of the reduction in flexural stiffness.

The remainder of this paper is structured as follows: A numerical model to obtain a riser’s vibration characteristics with consideration of the breathing crack effect, crack-detection theory, and numerical solution and comparison are presented in Sec. II. Natural frequency shifts due to breathing and open cracks vs crack locations and degrees are studied in Sec. III. The effects of breathing and open cracks and crack locations and degrees on the transverse-vibration characteristics are studied in Sec. IV. Single- and multiple-cracked drilling risers with different crack severities and positions are identified in detail using the proposed indices in Sec. V.

## II. TIME-DOMAIN DYNAMIC MODEL OF A RISER WITH BREATHING CRACK

### A. Transverse-vibration model by a time-domain FEM

As a long-span, thin-wall channel that connects the surface floater, e.g., vessel or platform, to the subsea wellhead, a drilling riser, usually suffering harsh waves and extreme loads, is one of the most crucial and weakest parts of marine structures. Figure 1 shows a drilling riser system. It mainly consists of a dynamic positioning (DP) vessel, an upper flex/ball joint (UFJ), a tensioner, a telescopic joint, riser joints, a lower flex/ball joint (LFJ), a lower marine riser package (LMRP)/blowout preventers (BOPs), and a wellhead.

For accurate mathematical modeling of the dynamics of a riser, a time-domain FEM is developed herein. The following assumptions are introduced to describe the riser’s motions:

The riser moves only in the plane, i.e.,

*yoz*plane in Fig. 1, where the coordinate origin*o*is at the UFJ, the*z*axis is along the water depth, and the*y*axis is parallel to the wave direction.The pipe is inextensible, and the tension is an axial variation, neglecting the longitudinal vibration.

Each riser joint is seen to be uniform, and the riser’s length

*L*is much greater than its maximum diameter*D*(i.e.,*L*/*D*≫ 1) so that the effects of shear on the riser dynamics can be ignored.A large deflection but small strain is considered, and the material is homogeneous and remains linearly elastic.

The rotatory inertia of the riser joint is negligible, the influence of the drilling rod is neglected, and the pipe is filled up with the drilling fluid.

Based on the assumptions, the riser is discretized into a series of beam elements. Each element has two nodes with four degrees of freedom according to FEM theory, as sketched in Fig. 2. The origin of local coordinates $z\u0302$ for each element is assumed to be located at the lower end. The deflection of the *e*th element *y* is a cubic polynomial function of local coordinates.

Thus, the node displacement *y* can be expressed by the product of the shape function matrix ** N**, polynomial vector

**, and state vector $\delta e=ui,\theta i,uj,\theta jT$,**

*H*where $H=1,z\u0302,z\u03022,z\u03023$ and

where *l*_{e} is the length of the *e*th beam element.

The first and second derivatives of the displacement at each cross section to the length is

Because the axial tension has a significant influence on the bending stiffness of the riser, the axial variable-tension due to the riser weight, pressure inside and outside the pipe, and heave motion of the telescopic joint is given as

where *T*_{b} is the tension at the bottom end of the riser; *T*_{d} is the dynamic component; *w*_{s} is the weight of riser per unit length; *A*_{o} and *A*_{i} are the cross-sectional area inside and outside the pipe; *P*_{o} and *P*_{i} are the pressures of external seawater and internal mud; *f*_{r} is the pretension factor; *a*_{v} and *ω*_{d} are the amplitude and frequency of the vessel’s heave motion; and *k*_{v} is the telescope joint’s equivalent stiffness, which is defined as

Therefore, the equivalent gravity of the riser per unit length is

where *γ*_{o} and *γ*_{i} are the weight of external seawater and internal mud.

Compared with the translational kinetic energy, the rotational energy of the riser is neglected because of the super slenderness-ratio assumption. Considering the riser mass and the additional seawater mass, the kinetic energy of an element is expressed as

The element’s potential energy arising from the bending moment, equivalent tension, and external loads is

where *E* is the elastic modulus, *I* is the moment of inertia of cross section, and $pz\u0302$is the lateral load.

Besides the force of the additional mass in Eq. (8), the lateral force includes two parts: one arising from the acceleration of seawater and the other arising from the velocities of seawater and the riser according to the modified Morison equation

where the wave force is thus expressed as

where

where *v*_{w} is the horizontal velocity of water participation.

To obtain the vibration characteristics conveniently, Airy wave theory is employed for the deepwater riser system. The free surface of the wave is a harmonic function of time,

The horizontal velocity and acceleration of water participation are

In reality, the free surface of the wave *η* is random, containing more than one frequency component. Thus, the JONSWAP spectrum is used, which is expressed as

where *H*_{s} is the significant wave height, *T*_{p} is the peak period, and *γ* is a peakedness factor, equal to 3.3.

The total kinetic energy and potential energy of the riser system are

Using the Lagrange equation,

the transverse-vibration differential equation of the drilling riser can be derived as the expression of the generalized displacement at nodes,

where

### B. A time-varying stiffness model for breathing cracks

When a drilling riser is cracked, the local flexibility and stiffness are changed. A breathing crack is assumed to be located in the middle of one cracked element, as shown in Fig. 3.

Neglecting the small amount of mass loss due to cracks, the flexible stiffness is affected by the crack depth and position. The local-flexibility change coefficient *c*_{ii} for the cracked riser can be obtained by referring to the work of Zheng and Fan^{24} and Zhou *et al.*^{12} as follows:

where *A*_{c} is the area of the crack, *γ* = *D*_{i}/*D*_{o}, *y*_{c} = *η*/*D*_{o}, and *K*_{I} is the stress intensity factor in the near region of a crack tip,

where $\xi \u2032=\xi +Do2/4\u2212\eta 2\u2212Do/2$ and $h\u2032=Do2\u22124\eta 2$, *x*_{c} = *ξ*′/*h*′, and $Fxc$ is a function of the local relative position *x*_{c},

The local-flexibility coefficient for the uncracked element *c*_{nocrack} is^{19,21}

Therefore, the total flexibility of the riser with an open crack is given by

The equivalent spring stiffness *k*_{c} for the riser with a closed crack is *k*_{c} = 1/*c*_{nocrack}, while the equivalent stiffness of the riser with an open crack is *k*_{o} = 1/*c*_{open},

To simulate the breathing effects of the crack, a time-varying stiffness model is herein utilized,

where *k*_{c} and *k*_{o} are the equivalent stiffness of the crack in fully closed and open states and *ω*_{b} is the frequency of the breathing crack.

The breathing crack model assumes that when

and when

The top and bottom boundary conditions are given by

### C. Numerical solution and comparison

The Newmark integration method, an unconditionally stable method with high integration precision, is utilized herein to solve the time-domain vibration equations [Eq. (19)]. Considering the nonlinearities of stiffness caused by the breathing crack, the Newton–Raphson iteration technique is employed at each time step during computation.

For verification of natural frequencies, the analytical solution of eigenequations for modal analysis was obtained in Ref. 11. The natural frequencies of the variable-tensioned riser without cracks are analytically solved by

where *J* is the order of the natural frequencies.

The parameters of the riser are as follows: length *L* = 152.4 m, outer and inner diameters *D*_{o} = 0.6096 m and *D*_{i} = 0.5778 m, elastic modulus *E* = 207 GPa, equivalent mass per unit length *m*_{e} = 995.92 kg/m, tension at LFJ *T*_{b} = 1.27 × 10^{3} kN, net weight of riser per unit length in seawater *w* = 3.12 × 10^{3} N/m, seawater density *ρ*_{w} = 1038.9 kg/m^{3}, and inner fluid density *ρ*_{m} = 1362.8 kg/m^{3}.

The results in the literature^{25,26} are used directly for comparison and verification of the modal results calculated in this study. As listed in Table I, the approximate values of the first five natural frequencies show that the modal results calculated by this study are consistent with those obtained by the analytical method and those in the literature.

. | ω_{1}
. | ω_{2}
. | ω_{3}
. | ω_{4}
. | ω_{5}
. |
---|---|---|---|---|---|

Dareing and Huang^{25} | 0.8150 | 1.8036 | 3.0876 | 4.7375 | 6.7890 |

Cheng et al.^{26} | 0.8150 | 1.8037 | 3.0878 | 4.7377 | 6.7896 |

Analytical solution^{11} | 0.8150 | 1.8037 | 3.0880 | 4.7380 | 6.7901 |

This study | 0.8150 | 1.8041 | 3.0887 | 4.7388 | 6.7900 |

For comparison with the time-domain response, the nonlinear coupled model of Lei *et al.*,^{23} with consideration of the axial and transverse effects under irregular waves, is employed herein. The basic computation parameters of the drilling-he riser system are listed in Table II.

Parameters . | Value . |
---|---|

Riser length, L | 500 m |

Outside diameter of riser, D_{o} | 0.5588 m |

Inside diameter of riser, D_{i} | 0.5080 m |

Steel tube material, ρ_{s} | 7850 kg/m^{3} |

Young’s modulus, E | 210 GPa |

Poisson’s ratio, μ | 0.3 |

Internal fluid density, ρ_{f} | 1200 kg/m^{3} |

Seawater density, ρ_{w} | 1025 kg/m^{3} |

Drag coefficient, C_{D} | 0.5 |

Inertia coefficient, C_{m} | 2.0 |

Pretension factor, f_{r} | 1.3 |

Significant wave height, H_{s} | 8.7 m |

Wave period, T_{p} | 12.3 s |

Parameters . | Value . |
---|---|

Riser length, L | 500 m |

Outside diameter of riser, D_{o} | 0.5588 m |

Inside diameter of riser, D_{i} | 0.5080 m |

Steel tube material, ρ_{s} | 7850 kg/m^{3} |

Young’s modulus, E | 210 GPa |

Poisson’s ratio, μ | 0.3 |

Internal fluid density, ρ_{f} | 1200 kg/m^{3} |

Seawater density, ρ_{w} | 1025 kg/m^{3} |

Drag coefficient, C_{D} | 0.5 |

Inertia coefficient, C_{m} | 2.0 |

Pretension factor, f_{r} | 1.3 |

Significant wave height, H_{s} | 8.7 m |

Wave period, T_{p} | 12.3 s |

The root-of-mean-square (RMS) of transverse displacement along the water depth, which is a useful index for describing the response energy, is investigated. Figure 4 shows the dynamic results calculated by the tension-coupled models developed in this study and by Lei *et al.*,^{23} respectively.

The displacements’ RMS illustrates that the results of this study agree well with those in the literature,^{12,23} with and without considering the heave motions of the vessel. Therefore, the numerical method in this study is proven to be valid and accurate enough to obtain the dynamic responses of the drilling riser when considering the influence of irregular waves^{27,28} and heave motions.

## III. NATURAL FREQUENCY SHIFT DUE TO CRACKS

### A. Frequency shift due to breathing and open cracks

The natural frequency of the riser is concerned with the time-varying stiffness and the breathing states of the crack. To study the effect of a breathing crack on natural frequencies, the intact and breathing- and open-cracked risers are calculated. As plotted in Fig. 5, the results of the riser with a breathing crack vary with time. The time-varying frequencies are less than the intact ones but larger than the open-cracked ones. However, the difference in the natural frequencies is little because of the weak influence of small cracks on such a huge riser system.

### B. Maximum shift of natural frequency vs crack depth and location

Because the natural frequencies are altered not only by the crack depth but also by the position, the effects of the crack depth and location on natural frequency shifts require detailed studies. As shown in Fig. 5, the riser with a fully open crack has the smallest natural frequencies. Therefore, the riser with an open crack is studied to explore the maximum effects of the crack depth and location on the natural frequency.

As shown in Fig. 6, the frequency decreases most obviously when the crack is located at about 100 m above the seabed. However, the variation of the natural frequencies of the open-cracked risers is so small that it may not be well recognized in practical engineering.

### C. Maximum difference of intact and cracked natural frequencies vs crack depth and location

To study the maximum shift of natural frequencies, the difference of natural frequencies *Δω* between the intact and racked risers is defined with the variation of the crack depth and location.

As shown in Fig. 7, the results illustrate that the natural frequencies are relatively significant when the crack is near 75 m above the seabed. The order of magnitude of the maximum difference is between −5 and −7. The small difference means that the frequency-based crack-detection method for traditional beam structures is no longer applicable for marine risers.

## IV. TRANSVERSE RESPONSE DIFFERENCE ANALYSIS

Under the irregular waves’ excitation, the node displacement *u*_{y}, slope angle around *x* axis *r*_{x}, and bending moment *m*_{x} are extracted to investigate the difference of the transverse dynamic responses of the intact riser and the riser with breathing cracks.

### A. Difference between breathing and open cracks

In Fig. 8, the time-domain responses of *u*_{y} and *r*_{x} between the breathing- and open-cracked risers are almost the same. The orbit plots and Poincaré maps also explain the small difference of *u*_{y} and *r*_{x} between the open- and breathing-cracked risers. The overlapped points mean the vibration of the risers is the standard periodic motion under irregular waves. However, considering the influence of the cracked stiffness *EI*, the values of bending moment responses are slightly different for the risers with open and breathing cracks.

### B. Time-domain response of the riser vs crack depth

For the risers with different crack depths, i.e., *a*_{c} = 5 mm, *a*_{c} = 15 mm, and *a*_{c} = 25 mm, respectively, the dynamic responses of nodal displacement *u*_{y}, slope angle *r*_{x}, and bending moment *m*_{x} of the cracked risers are shown in Fig. 9.

Figure 9 illustrates that the differences in node displacement *u*_{y} and slope angle *r*_{x} are small when the crack depth increases from 5 mm to 25 mm. Compared with *u*_{y} and *r*_{x}, the difference of bending moment *m*_{x} is relatively large with the increase in crack depth.

### C. Time-domain response of the riser vs crack location

As the maximum shift of the first natural frequency occurs when the crack is near −425 m, as shown in Figs. 6 and 7, and the riser is usually weakened near the top and bottom ends,^{12} the following cracked position, i.e., *L*_{c} = 1 m, 425 m, and 499 m, respectively, is typically assumed for discussion. Via calculation, the dynamic responses of *u*_{y}, *r*_{x}, and *m*_{x} of the cracked risers are shown in Fig. 10.

Figure 10 shows that the differences in node displacement *u*_{y} and slope angle *r*_{x} are still small when the crack location varies from 1 m, 425 m, to 499 m. Compared with *u*_{y} and *r*_{x}, the relatively large difference of bending moment *m*_{x} means that the cracks can be detected using the data of the bending moment.

## V. CRACK IDENTIFICATION USING RMS OF THE DYNAMIC CHARACTERISTICS

### A. By different dynamic characteristics

Because the effects of the breathing cracks on the dynamic characteristics are different, the crack detecting abilities of *u*_{y}, *r*_{x}, and *m*_{x} need to be investigated further. For convenient demonstration, the second derivative of the dynamic RMSs of *u*_{y}, *r*_{x}, and *m*_{x}, i.e., *RMS*_{uy}, *RMS*_{rx}, and *RMS*_{mx}, are defined as follows:

where *φ*^{RMS} = *RMS*_{uy}, *RMS*_{rx}, and *RMS*_{mx} and *i* is the *i*th node number.

Similarly, the fourth derivative of the dynamic *RMS*_{uy}, *RMS*_{rx}, and *RMS*_{mx} can be defined as

For a single cracked case, $c\phi i$and $s\phi i$ of RMSs of

*u*_{y},*r*_{x}, and*m*_{x}for the crack identification are plotted in Fig. 11.As shown in Fig. 11, the indices based on

*RMS*_{mx}of the bending moment provide more clarity of crack identification than those based on the node displacement and slope angle. In addition, the crack can be better identified based on the fourth derivative $s\phi i$ than the second derivative$c\phi i$.For the multiple cracked case, the crack identification using $c\phi i$ and $s\phi i$ of

*RMS*_{uy},*RMS*_{rx}, and*RMS*_{mx}is plotted in Fig. 12.As shown in Fig. 12, it is difficult to identify the cracks depending only on the data of

*RMS*_{rx}. Using*s*_{φ}and*c*_{φ}of*RMS*_{mx}, not only the positions but also the degrees of the breathing cracks can be clearly and accurately detected.

### B. By the second derivative of bending moment

Under the irregular waves, the performance of *s*_{my} and *c*_{my} of *RMS*_{mx} for crack identification needs to be investigated further considering the influence of time *t* and water depth *D*_{w}. Figure 13 shows the second derivative of *RMS*_{mx} vs time *t* and water depth *D*_{w}.

As shown in Fig. 13, the multiple cracked positions can be identified by the developed index$c\phi i$. The height in Fig. 13(a), the width in Fig. 13(b), and the color shade indicate the crack degrees. It is worth noting that the size and color of the plotted data near the top and bottom ends are the undesired interfering signals because of the boundary constraints.^{12}

### C. By the fourth derivative of bending moment

To improve the performance of crack identification, the fourth derivative *s*_{mx} of *RMS*_{mx} in Eq. (30) vs time *t* and water depth *D*_{w} is investigated and the results are plotted in Fig. 14.

Compared with Fig. 13, the multiple cracks of the drilling riser can be more accurately located and more precisely measured by using the index *s*_{mx} than *c*_{mx}. For instance, the detected locations are more significant, and the magnitudes are more distinguishable. In addition, the undesired signals are less along the whole riser length, and the interfering data near the top constraint are narrower.

## VI. CONCLUSIONS

A few new and original conclusions were drawn as follows:

The natural frequencies calculated in this study showed a close agreement with those obtained by the analytical method and the literature. Compared with the nonlinear combined axial and transverse coupled model in the literature, the developed model in this study is verified to be valid and accurate enough for the dynamic response analysis of a time-varying top-tensioned drilling riser.

The natural frequencies of the drilling riser in service were significantly affected by the cracks near 75 m above the seabed. Nevertheless, the magnitude order of the maximum difference between the intact and cracked frequencies is less than −5, which means that the frequency-based crack-detection method for traditional beams is no longer applicable for marine risers.

For the risers with breathing and open cracks or with different cracked locations and degrees, the time-domain response, orbit plots, Poincaré maps, and fast Fourier transform (FFT) diagram of the node displacement and slope angle were almost no different, while the bending moment’s response was slightly different due to the influence of the cracked

*EI*.The indices

*c*_{mx}and*s*_{mx}based on the RMS of the bending moment showed good performance in crack identification with the consideration of time*t*and water depth*D*_{w}influence. Compared with*c*_{mx}, the crack identification of*s*_{mx}was hardly disturbed by the undesired data along the riser and near the top boundary constraint.

In total, this study has investigated the effects of breathing cracks on the dynamic characteristics of a deepwater drilling riser and detected the single and multiple breathing cracks using different indices by a proposed time-domain FEM. However, the identification results are conservative because of the assumption of “zero” thickness of cracks. For more accurate research, a more special crack model with a certain thickness and a multi-factor coupling model is suggested to be considered in further work.

## ACKNOWLEDGMENTS

The authors acknowledge the financial support of the “National Natural Science Foundation of China” under Grant No. 52001044, “Fundamental Research Funds for Central Universities” under Grant No. 3132019041, “National Natural Science Foundation of China” under Grant No. 51779026, and “Liaoning Provincial Natural Science Foundation of China” under Grant No. 2020-HYLH-35. We also appreciate the kind comments and valuable suggestions from the reviewers and editors for improving the paper.

## DATA AVAILABILITY

The data that support the findings of this study are available within the article.

## NOMENCLATURE

### APPENDIX: MATRICES OF THE DISCRETIZED ELEMENTS

where $T\u0302oe$ is the equivalent tension of the *e*th beam element.