Very High Cycle Fatigue Life of Free-Spanning Subsea Pipeline Subjected to Vortex-Induced Vibrations

Abstract

Free-spanning subsea pipelines subjected to vortex-induced vibrations (VIVs) are particularly prone to fatigue failure. Existing flume observations indicated that the VIVs of a near-bed cylinder may be triggered effectively in moderate shear flows. This may imply that the vibration cycles of a spanned pipeline could be up to tens of millions. As such, very high cycle fatigue (VHCF) can occur during engineering service. The free span length is a key parameter for determining the structural natural frequency and the corresponding reduced velocity (Vr). On the basis of the dimensionless vibration amplitude A/D–Vr curve and the recommended S-N curves for high-strength steel pipelines with cathodic protection under seawater environments, a prediction method is proposed for the fatigue life of a free span undergoing VIVs. A parametric study is then performed to evaluate the fatigue life of the spanned pipelines with a focus on VHCF. It is indicated that the minimum fatigue life emerges at certain flow with a moderate velocity for a given span length. With a further decrease or increase in the flow velocity, the fatigue life would be enhanced correspondingly, which could be within the VHCF regime. Such nonlinear variation of the fatigue life with the span length and the flow velocity is attributed to being involved in various VIV branches of the A/D–Vr curve.

1. Introduction

Subsea pipelines have been widely employed for transporting oil and gas from underwater wells to offshore platforms and onshore processing facilities. The pipeline free-span scenario could be permanent when generated by seabed unevenness, artificial supports for pipeline crossing, or with short-to-long-term evolution by seabed mobility and local scour [1]. Under the action of ocean currents, a partially embedded pipeline may also be suspended because of tunnel erosion [2,3]. As such, free spans are frequently encountered or even inevitable, especially for a long laid pipeline for oil and gas exploitation in deep waters.

In subsea environments, a free-spanning pipeline may suffer from vortex-induced vibrations (VIVs), as illustrated in Figure 1. For an elastically mounted cylinder, the transverse VIVs can be effectively triggered when the reduced velocity (Vr) approaches a critical value $V{r}_{\mathrm{cr}}\approx 4.0$ under wall-free conditions [4]. Once the VIVs are initiated, the frequency of vortex shedding and that of the structural oscillation would collapse into an identical value, which is known as the “lock-in” or “synchronization” phenomenon. The maximum amplitude of the vibration principally depends on the combined mass damping parameter K_s [5]. In engineering practice, subsea pipelines are usually in the proximity of the seabed. For such a near-bed cylinder, the flow fields around the cylinder would be altered due to bed proximity effects, which would subsequently affect the VIV responses. Recently, Liu and Gao [6] investigated the bed proximity effects in triggering transverse vibration of a cylinder under low K_s conditions. It was found that as the cylinder became closer to the bed boundary (i.e., with a decrease in the gap-to-diameter ratio e/D from 2.0 to 0.1), the values of $V{r}_{\mathrm{cr}}$ significantly decreased (from 4.0 to 2.85), which was accompanied by a decrease in the vibration amplitudes. Previous flume observations by Gao et al. [7] indicated that once VIVs are triggered, the local scour around the vibrating pipe will be greatly enhanced due to the cyclic vibrations (e.g., the scour depth could be up to approximately 1.2D, which can be regarded as the wall-free condition for VIV responses).

In engineering practice, fatigue may take place under cyclic loading after a substantial period of service. The fatigue limit stress is generally defined as the highest stress at which specimens do not fail after 10⁷ loading cycles. However, the existing studies have extended this knowledge by warning that fatigue failure may occur at lifetimes greater than $N={10}^7$ at a stress which is lower than the conventional fatigue limit [8]. Generally speaking, the fatigue of metallic materials can be divided into three regions: low cycle fatigue (LCF), high cycle fatigue (HCF) and very high cycle fatigue (VHCF), in which the number of cycles to failure is beyond 10⁷. The fatigue behavior in the VHCF region is quite different compared with that in the conventional HCF region [9,10,11,12,13]. For instance, the fatigue behaviors of high-strength steels in the VHCF region show crack initiation at the inclusion site, while in the HCF region, the cracks are located preferentially at the surface [8]. The influence of hydrogen was found to be crucial for eliminating the fatigue limit in the extremely high cycle fatigue region [8,9].

For offshore structures subjected to typical wave and wind loading, significant fatigue failure may occur at N ≥ 10⁷ cycles (i.e., in the VHCF region) [14]. It is also extremely common that turbine blades experience more than N = 10⁷ stress cycles by vibration. Nevertheless, fatigue test data are normally derived for a number of cycles less than 10⁷, including those for subsea pipeline steels. For the scenario of free-spanning pipelines, the fatigue by VIVs is of great concern in engineering design [15]. The allowable span length of a suspended pipeline was investigated by considering VIV hysteresis effects [16]. The natural frequency of free spans was assessed by using a finite element model [17], and the random ocean current-induced fatigue life of free spans was further predicted on the basis of the Palmgren–Miner cumulative damage rule. Nevertheless, the correlation between the free span length and the corresponding service life has not been understood well, especially in the VHCF region.

In this study, the fatigue life of subsea-spanning pipeline undergoing VIVs is investigated, with a focus on VHCF. Based on the existing flume results for transverse VIV responses, a dimensionless vibration amplitude A/D–Vr curve with four branches is constructed in Section 2.1, and the maximum stress along the free span is then derived in Section 2.2. The bilinear stress range and number of cycles to failure (S-N) curves are recommended by considering the VHCF of high-strength steel pipelines with cathodic protection under seawater environments (see Section 3). Based on the constructed A/D–Vr curve and the recommended S-N curves, a flow chart is proposed in Section 4.1 to predict the fatigue life of a spanning pipeline undergoing VIVs. A parametric study is further performed in Section 4.2 to evaluate the VHCF life of the spanning pipeline. The correlations between the fatigue life and the flow velocity or the free span length are established. Finally, our concluding remarks are given in Section 5.

2. Vibration Amplitude and Stress Distribution along the Spanning Pipeline

2.1. Amplitude of VIVs

The vibration amplitude is correlated with the reduced velocity Vr, which is defined as

$$Vr={\displaystyle \frac{U}{f_{\mathrm{n}}D}}$$

(1)

where U is the characteristic velocity of the flow perpendicular to the cylinder’s axis; f_n is the structural natural frequency; and D is the outer diameter of the cylinder [4]. The natural frequency (f_n) of a free-spanning pipeline is a key factor in identifying the reduced velocity (see Equation (1)). The natural frequency or the first eigen frequency of a free-spanning pipeline is influenced by many factors (such as the span length, elastic modulus of the pipe steel, boundary conditions, effective mass, and moment of inertia of the pipe), which can be evaluated with the following equation [15]:

$$f_{\mathrm{n}}\approx {\displaystyle \frac{C_1}{L_{\mathrm{eff}}^2}}{\left({\displaystyle \frac{\left(1+F_{\mathrm{CS}}\right)EI}{m_{\mathrm{e}}}}\left[1+{\displaystyle \frac{S_{\mathrm{eff}}}{P_{\mathrm{cr}}}}+C_3{\left({\displaystyle \frac{\delta }{D}}\right)}^2\right]\right)}^{{\displaystyle \frac{1}{2}}}$$

(2)

where $L_{\mathrm{eff}}$ is the effective span length; $m_{\mathrm{e}}$ is the effective mass of the pipeline; E is the modulus of elasticity of the steel pipeline; I is the inertia moment of the steel pipeline; $S_{\mathrm{eff}}$ is the effective axial force (negative in compression); $P_{\mathrm{cr}}$ is the critical buckling load (positive sign); δ is the static deflection; $C_1$ and $C_3$ are the end boundary condition coefficients, which depend on the support conditions of the free-span boundary; $F_{\mathrm{CS}}$ denotes the stiffness of the concrete coating relative to the steel pipe stiffness; (1 + $F_{\mathrm{CS}}$) is the stress concentration factor due to the concrete coating and localized bending; $F_{\mathrm{CS}}=k_{\mathrm{c}}{\left({\displaystyle \frac{{\left(EI\right)}_{\mathrm{conc}}}{EI}}\right)}^{0.75}$; (EI)_conc denotes the stiffness of concrete coating; and the empirical constant k_c accounts for the deformation or slippage in the corrosion coating and the cracking of the concrete coating. With reference to DNV GL [15], the effective span length $L_{\mathrm{eff}}$ can be calculated as follows:

$${\displaystyle \frac{L_{\mathrm{eff}}}{L}}=\left\{\begin{array}{ll}{\displaystyle \frac{4.73}{-0.066{\beta }^2+1.02\beta +0.63}}\hfill & \beta \ge 2.7\hfill \\ {\displaystyle \frac{4.73}{0.036{\beta }^2+0.61\beta +1.0}}\hfill & \beta <2.7\hfill \end{array}\right.$$

(3)

In Equation (3), $\beta$ is the non-dimensional soil stiffness (i.e., $\beta ={\mathrm{l}\mathrm{o}\mathrm{g}}_{10}\left[{KL}^4/\left(\left(1+F_{\mathrm{C}\mathrm{S}}\right)EI\right)\right]$), where K is the soil stiffness (vertical or horizontal and static or dynamic). The values of $\beta$ are generally in the range of 2.0 < $\beta$ < 8.0. For the case of transverse VIV responses, the dynamic vertical stiffness (K_vd) can be used (i.e., K = K_vd), and the following expression can be applied for determination of the dynamic vertical stiffness; that is, K_vd = $0.88G/\left(1-\nu \right)$, where $G$ is the secant shear modulus of soil and $\nu$ is the Poisson ratio of the soil [18]. The basic rules for the ideal end conditions are as follows: (1) pinned-pinned, which is used for free spans where each end is allowed to rotate about the pipeline axis, and (2) fixed-fixed, which should be used for free spans which are fixed in place by some sort of anchor at both ends. The following values of C₁ in Equation (2) have been widely used for these typical end conditions: (1) C₁ = 1.57 for the pinned-pinned end condition and (2) C₁ = 3.56 for the fixed-fixed end condition (see DNV GL [15] and Guo et al. [19]). The effective mass per meter (m_e) is the sum of the mass of the steel pipe per meter (m_p), the mass of the content inside the pipe per meter (m_c), and the added mass per meter (m_a):

$$m_{\mathrm{e}}=m_{\mathrm{p}}+m_{\mathrm{c}}+m_{\mathrm{a}}$$

(4a)

$$m_{\mathrm{a}}=C_{\mathrm{A}}{\displaystyle \frac{\mathsf{\pi }{\rho }_{\mathrm{water}}{D}^2}{4}}$$

(4b)

where C_A is the added mass coefficient (for a circular pipeline, C_A = 1.0) and ${\rho }_{\mathrm{water}}$ is the density of the seawater. If the effective axial force and the static deflection are not taken into account, then Equation (2) can be simplified as follows:

$$f_{\mathrm{n}}\approx {\displaystyle \frac{C_1}{L_{\mathrm{eff}}^2}}{\left({\displaystyle \frac{\left(1+F_{\mathrm{CS}}\right)EI}{m_{\mathrm{e}}}}\right)}^{{\displaystyle \frac{1}{2}}}$$

(5)

In the past few decades, the VIVs of an elastically mounted cylinder have been investigated by quite a few researchers (see the comprehensive reviews of, for example, Williamson and Govardhan [5], Sarpkaya [20], Bearman [21] and Wu et al. [22]). In this study, the VIV amplitudes of wall-free cylinders in the region of the low mass damping parameter K_s (broadly K_s < 0.05; see Williamson and Govardhan [5]) were collected from some benchmark flume observations. The whole curve for the nonlinear variation of A/D with Vr was constructed as shown in Figure 2. Note that Wang et al. [23] experimentally studied the VIV of a neutrally buoyant circular cylinder in a recirculating open channel with the mass damping parameter K_s ≈ 3.46 × 10⁻². In the VIV experiment by de Oliveira Barbosa et al. [24], a freely vibrating cylinder was exposed to currents and placed near a plane boundary parallel to the cylinder axis, and K_s ≈ 1.45 × 10⁻². Daneshvar and Morton [25] studied the VIV of an elastically mounted circular cylinder in a water tunnel facility, where K_s ≈ 8.32 × 10⁻³. Liu and Gao [6] recently investigated the wall proximity effects on the triggering of transverse VIVs of a circular cylinder with an accessorial low structural damping VIV simulation device, where K_s ≈ 3.14 × 10⁻². As indicated in Figure 2, in such a low mass damping system, four distinct branches of VIV responses can be identified and marked with various colors, namely the initial excitation branch, the upper branch, the lower branch, and the desynchronization branch (see also Khalak and Williamson [26]). In the initial excitation branch (3.5 $\le$ Vr < 5.0; see Figure 2), the cylinder starts to vibrate with a relatively small amplitude (note: transverse VIVs do not occur at Vr < 3.5); in the upper branch (5.0 $\le$ Vr < 6.7), the cylinder vibrates with a large amplitude; in the lower branch (6.7 $\le$ Vr < 12), the cylinder vibrates with a moderate amplitude; and in the desynchronization branch (Vr ≥ 12), the vibration amplitude of a wall-free circular cylinder is further reduced. There exists a certain dispersion among the experimental data (e.g., at the upper branch; see Figure 2) which could be attributed to involving various values of K_s.

The above analyses indicate that once the pipeline parameters and the flow velocity (U) are given, the corresponding reduced velocity (Vr) can be calculated with Equation (1) and Equation (5), and the vibration amplitude can be obtained with reference to Figure 2.

2.2. Stress Distribution along the Free Span

It is supposed that the ocean current is perpendicular to the axis of the free spans, as illustrated in Figure 1. As stated above, the shoulders for a free span can be simplified as the pinned-pinned or fixed-fixed end condition. The maximum deflection in the middle of the span (w_max) under the action of a uniform load (q) can be calculated with Equation (6a) for the pinned-pinned end condition and with Equation (6b) for the fixed-fixed end condition [27]:

$$w_{\max }={\displaystyle \frac{{5q{L}_{\mathrm{eff}}}^4}{384EI}}$$

(6a)

$$w_{\max }={\displaystyle \frac{{q{L}_{\mathrm{eff}}}^4}{384EI}}$$

(6b)

where L_eff is the effective span length. But for the VIV responses, if the maximum deflection w_max is given (w_max = A), then the corresponding maximum bending moment (M_max) can be calculated with Equation (7a) for the pinned-pinned end condition and with Equation (7b) for the fixed-fixed end condition [27]:

$$M_{\max }={\displaystyle \frac{48EI}{{5L_{\mathrm{eff}}}^2}}{w}_{\max }$$

(7a)

$$M_{\max }=-{\displaystyle \frac{32EI}{{L_{\mathrm{eff}}}^2}}{w}_{\max }$$

(7b)

Figure 3b,c shows the distributions of the bending moment along the free span under these two end conditions. It is indicated that the bending moment reaches its peak value in the middle of the span under the pinned-pinned end condition. But under the fixed-fixed end condition, the two ends connected with the span shoulders undertake the maximum bending moment. Consequently, the maximum normal stresses (${\sigma }_{\max }$) correlated with the maximum deflections (w_max) can be expressed as

$${\sigma }_{\max }={\displaystyle \frac{24DE}{{5L_{\mathrm{eff}}}^2}}{w}_{\max }$$

(8a)

$${\sigma }_{\max }={\displaystyle \frac{16DE}{{L_{\mathrm{eff}}}^2}}{w}_{\max }$$

(8b)

for the pinned-pinned and fixed-fixed end conditions, respectively.

3. Recommended S-N Curves

The fatigue design is generally based on the use of S-N curves, which were obtained from fatigue tests. The number of cycles to failure (N) at a stress range ($\Delta \sigma$) is defined by an S-N curve. As mentioned earlier, significant fatigue damage to offshore structures usually occurs at $N\ge {10}^7$ cycles (VHCF region). For the VHCF mechanism of metallic materials, a few studies have already been carried out (e.g., Murakami et al. [9], Shiozawa et al. [10], Sakai et al. [28], and Hong et al. [29]). It has been generally recognized that cracks usually initiate from the surface of materials in the LCF and HCF regions, while the cracks preferably emerge in the interior of the materials with a fish eye morphology in the VHCF region. Cracks may initiate from several types of internal defects, which mainly include nonmetallic inclusions, process-related defects, second-phase particles, and microstructural inhomogeneities [30,31,32,33]. As illustrated in Figure 4a,b, an inclusion was found at the center of the fish eye zone. The rough area surrounding the inclusion within the fish eye zone, named the fine granular area (FGA), is the crack initiation region, which is critical to VHCF occurrence. The initiation and early growth of cracks could be attributed to the grain refinement caused by the dislocation interaction over a number of cyclic loads followed by the formation of micro-cracks, but the micro-cracks could also form irrespective of the grain refinement during cyclic loading [13].

Submarine pipelines may suffer from corrosion in engineering practices. A distinctive difference between corrosion fatigue and inert environment fatigue is that there is no safe stress range at which metallic materials have infinite fatigue life during engineering service in corrosive environments. The fatigue strength of metallic materials in corrosive environments decreases dramatically for the synergistic interactions between mechanical loading and the environmental effect. Cracks usually occur on the surfaces of metallic materials in very high cycle corrosion fatigue (VHC–CF) [34,35,36,37]. The surfaces of metallic materials may be degraded because of corrosion or erosion-corrosion, leading to the formation of pits, grooves, cracks, or rougher surfaces [38]. As illustrated in Figure 4c,d, surface corrosion pits are the origins of cracks for specimens tested in a seawater environment [36,37]. From the microscopic point of view, the corrosive media’s attack on the material’s surfaces can intensify the formation of microstructurally small cracks (MSCs), contributing to the acceleration of crack initiation and growth. The decrease in metal’s ductility or fracture toughness caused by atomic hydrogen is called hydrogen embrittlement (HE), which can make cracks spread quickly with minimal macroscopic signs. In brief, corrosion fatigue is a combination of mechanical and environmental phenomena.

Figure 4. (a) The fracture surface of high-strength steel in an air environment. (b) A close-up of the crack initiation region in (a) [39]. (c) Specimen tested under seawater flow; (d) A close-up of the crack initiation regions in (c) [36].

Figure 4. (a) The fracture surface of high-strength steel in an air environment. (b) A close-up of the crack initiation region in (a) [39]. (c) Specimen tested under seawater flow; (d) A close-up of the crack initiation regions in (c) [36].

For the sake of service safety, an unfavorable scenario was considered in this case study, such as the weld joint happening to be installed in the middle of the span for the pinned-pinned boundary condition and at one end of the span for the fixed-fixed boundary condition. The service environment was considered to be in the seawater and with cathodic protection (CP).

The fatigue test data were normally derived for a number of cycles less than 10⁷. As such, how to extrapolate the fatigue test data into the VHCF region was important for achieving a reliable assessment procedure [14]. Bilinear S-N curves (i.e., F3 and C1 (see Figure 5)) were recommended to define the number of cycles to failure at a stress range $\Delta \sigma$ (=2σ_max). Note that the F3 curve is for the circumferential butt weld made from one side without a backing bar, and the C1 curve is for the circumferential butt welds made from one side which were machined or ground flush to remove defects and weld overfill [14]. Meanwhile, the existing fatigue test data for the high-strength steels under various environments are also provided in Figure 5 for reference. For the pipe welds with different hot spot geometries, the structural stress concentration factor (SCF) should be accounted for. The SCF is defined as the ratio of local stress (hot spot stress) to nominal stress. It is recommended that SCF = 1.61 for the S-N curve F3 and SCF = 1.0 for the S-N curve C1 [14]. As shown in Figure 5, the S-N curves in the VHCF region became more horizontal as the number of cycles to failure N became larger than 10⁶. The recommended S-N curves can be expressed in the following form [14]:

$${\mathrm{Log}}_{10}\left(N\right)={\mathrm{Log}}_{10}\left(\overline{a}\right)-m{\mathrm{Log}}_{10}\left(\Delta \sigma {\left({\displaystyle \frac{t}{t_{\mathrm{ref}}}}\right)}^k\right)$$

(9)

where m is the negative inverse slope of the S-N curve; ${\mathrm{Log}}_{10}\left(\overline{a}\right)$ is the intercept of the axis; ${\mathrm{Log}}_{10}\left(N\right)$ is the reference wall thickness of the pipe for welded connections; k is the thickness exponent for the fatigue strength; t is the wall thickness of the pipe; and t_ref is the reference wall thickness of the pipe (0.025 m).

4. Fatigue Life Prediction and Discussions

4.1. A Flow Chart for Fatigue Life Prediction

A flow chart is proposed for the fatigue life prediction of a free-spanning pipeline undergoing VIVs, as illustrated in Figure 6. A brief description of the analysis procedure is given below:

(1)

For given structural parameters of the spanned pipeline, the natural frequency (f_n) of the free span can first be evaluated with Equations (2)–(4a). The reduced velocity (Vr) is calculated with Equation (1) for a certain flow velocity ($U$).

(2)

Once the value of Vr is obtained, the dimensionless vibration amplitude (A/D) can then be evaluated with reference to the recommended A/D–Vr curve (Figure 2). Subsequently, the maximum stress range $\Delta \sigma$ (= 2σ_max) can be calculated with Equation (8a) or Equation (8b), where the maximum deflection count (w_max) is substituted by the vibration amplitude A.

(3)

By referring to the recommended S-N curves (Figure 5), the fatigue life in terms of the vibration cycles (N) of the spanned pipeline can finally be obtained with Equation (9).

4.2. Parametric Study and Discussions

A subsea pipeline for gas transportation with an outer diameter of 20 inches (0.508 m) was considered for a parametric study. The parameters of the steel pipe, the seawater, the gas inside the pipe, and the soil stiffness are given in

Table 1. Input data for the parameters of the pipe spans.

Parameters	Values	Units
Outer diameter of the steel pipe (D)	0.508	m
Wall thickness of the steel pipe (t)	0.0379	m
Density of the steel (${\rho }_{\mathrm{steel}}$)	7.870 × 103	kg/m3
Density of the gas inside the pipe (${\rho }_{\mathrm{cont}}$)	0.200 × 103	kg/m3
Density of the seawater (${\rho }_{\mathrm{water}}$)	1.024 × 103	kg/m3
Kinematic viscosity of the seawater (${\upsilon }_{\mathrm{water}}$)	1.565 × 10−6	m2/s
Elastic modulus of the steel (E)	2.10 × 1011	Pa
Moment of inertia of the steel pipe (I)	1.56 × 10−3	m4
Non-dimensional soil stiffness ($\beta$)	4.0
Reference wall thickness of the pipe (tref)	0.025	m

, and the values of the key parameters for the S-N curves are listed in

Table 2. Parameters for the S-N curves.

S-N Curve	N ≤ 106		N > 106		k
	m1	${\mathbf{Log}}_{10}\left({\overline{\mathit{a}}}_1\right)$	m2	${\mathbf{Log}}_{10}\left({\overline{\mathit{a}}}_2\right)$
C1	3	12.049	5	16.081	0.10
F3	3	11.146	5	14.576	0.25

. For the extreme scenario of large free spans, the span length values (L) were examined in a range from 40 m to 130 m. The correlations between the fatigue life (N) and the flow velocity (U) or span length (L) could be obtained by following the flow chart shown in Figure 6.

Table 3. Calculation process for the fatigue life of a free-spanning pipeline with various span length values (U = 0.57 m/s).

L (m)	Leff (m)	C1		fn (Hz)		Vr		A/D		w (m)		σmax (MPa)		N (Predicted with S-N Curve C1)		N (Predicted with S-N Curve F3)
		P-P	F-F	P-P	F-F	P-P	F-F	P-P	F-F	P-P	F-F	P-P	F-F	P-P	F-F	P-P	F-F
40	51.78	1.57	3.56	0.407	0.916	2.76	1.23	0.019	0	0.010	0	1.82	0	1.52 × 1013		3.49 × 1011
45	58.25	1.57	3.56	0.322	0.724	3.49	1.55	0.025	0	0.013	0	1.95	0	1.07 × 1013		2.46 × 1011
50	64.72	1.57	3.56	0.261	0.586	4.31	1.91	0.410	0	0.208	0	25.44	0	2.87 × 107		7.78 × 105
55	71.19	1.57	3.56	0.215	0.484	5.21	2.32	0.922	0.017	0.468	0.008	47.30	2.84	1.29 × 106	1.67 × 1012	1.21 × 105	3.81 × 1010
60	77.66	1.57	3.56	0.181	0.407	6.20	2.76	0.926	0.019	0.471	0.010	39.95	2.70	3.01 × 106	2.13 × 1012	2.01 × 105	4.88 × 1010
65	84.14	1.57	3.56	0.154	0.347	7.28	3.24	0.788	0.023	0.400	0.012	28.95	2.77	1.50 × 107	1.86 × 1012	5.28 × 105	4.26 × 1010
70	90.61	1.57	3.56	0.133	0.299	8.44	3.75	0.658	0.103	0.334	0.053	20.86	10.92	7.74 × 107	1.97 × 109	1.77 × 106	4.51 × 107
75	97.08	1.57	3.56	0.116	0.261	9.69	4.31	0.571	0.413	0.290	0.210	15.75	37.97	3.16 × 108	3.88 × 106	7.22 × 106	2.34 × 105
80	103.55	1.57	3.56	0.102	0.229	11.02	4.90	0.438	0.871	0.223	0.443	10.63	70.46	2.26 × 109	3.53 × 105	5.16 × 107	3.66 × 104
85	110.02	1.57	3.56	0.090	0.203	12.44	5.53	0.157	0.928	0.080	0.472	3.37	66.49	7.01 × 1011	4.20 × 105	1.60 × 1010	4.36 × 104
90	116.50	1.57	3.56	0.080	0.181	13.95	6.20	0.124	0.926	0.063	0.471	2.37	59.17	4.06 × 1012	5.96 × 105	9.28 × 1010	6.18 × 104
95	122.97	1.57	3.56	0.072	0.162	15.54	6.91	0.120	0.857	0.061	0.435	2.06	49.12	8.24 × 1012	1.07 × 106	1.88 × 1011	1.08 × 105
100	129.44	1.57	3.56	0.065	0.147	17.22	7.66	0.111	0.727	0.057	0.369	1.73	37.61	1.98 × 1013	4.07 × 106	4.53 × 1011	2.41 × 105
105	135.91	1.57	3.56	0.059	0.133	18.99	8.44		0.658		0.334		30.89		1.09 × 107		4.34 × 105
110	142.38	1.57	3.56	0.054	0.121	20.84	9.27		0.592		0.301		25.34		2.93 × 107		7.87 × 105
115	148.86	1.57	3.56	0.049	0.111	22.78	10.13		0.548		0.278		21.44		6.75 × 107		1.54 × 106
120	155.33	1.57	3.56	0.045	0.102	24.80	11.03		0.437		0.222		15.72		3.19 × 108		7.30 × 106
125	161.80	1.57	3.56	0.042	0.094	26.91	11.96		0.225		0.114		7.45		1.34 × 1010		3.05 × 108
130	168.27	1.57	3.56	0.039	0.087	29.11	12.94		0.142		0.072		4.35		1.97 × 1011		4.52 × 109

Note: “P-P” is the abbreviation for the “pinned-pinned” end condition, and “F-F” is the abbreviation for the “fixed-fixed” end condition.

shows the calculation process for the free-spanning pipeline with various values for the span length under the pinned-pinned and the fixed-fixed end conditions. Note that a moderate flow velocity U (=0.57 m/s) was considered as an example.

Figure 7a,b illustrates the correlations of the flow velocity (U) and the reduced velocity (Vr) for four selected span length values (i.e., L = 50 m, 70 m, 90 m, and 110 m) under the pinned-pinned and fixed-fixed end conditions, respectively. Note that the end condition of the free span had much influence on the natural frequency (f_n) of the free-spanning pipeline, which would further alter the values of Vr (see Equation (1)). As mentioned earlier, the four VIV branches are marked with various colors, which are related to various ranges of Vr (see Figure 2). A shorter free span L would increase the natural frequency f_n (Equation (5)), and thus a higher flow velocity U was required to keep the reduced velocity Vr unchanged (Equation (1)). As indicated in Figure 7, for a given flow velocity U (e.g., U = 0.57 m/s; see the horizontal arrow), the corresponding reduced velocity Vr increased with the increase in L.

The variations in the predicted fatigue life (N) for two recommended S-N curves (see Figure 5) with a span length L at a given flow velocity U = 0.57 m/s are presented in Figure 8. Both the pinned-pinned and fixed-fixed end conditions were taken into account. It is indicated that the relationships between the fatigue life N and the span length L under a given flow velocity U were essentially nonlinear. As depicted in Figure 8, for 45 $\le$ L < 54 (m) under the pinned-pinned end condition or 68 $\le$ L < 81 (m) under the fixed-fixed end condition, the free span vibrated in the initial excitation branch (marked in blue; see Figure 8) such that the fatigue life N decreased with an increase in the span length L. For the span length 54 $\le$ L < 62 (m) (pinned-pinned) or 81 $\le$ L < 94 (m) (fixed-fixed), the free span vibrated in the upper branch (yellow) with larger amplitudes such that the corresponding fatigue life first became shorter to a minimum value and then increased with the increase in L. But for 62 $\le$ L < 83 (m) (pinned-pinned) or 94 $\le$ L < 125 (m) (fixed-fixed), the free span vibrated in the lower branch (green) (i.e., the vibration amplitudes were reduced). Therefore, the fatigue life became longer with an increasing L, and for L $\ge$ 83 (m) (pinned-pinned) or L $\ge$ 125 (m) (fixed-fixed), the free span vibrated in the desynchronization branch (gray) (i.e., the vibration amplitude continued to decrease with an increasing L) such that the fatigue life became much larger or even N > 10⁸. From Figure 8a,b, it can be recognized that the critical values of the span length to differentiate the four VIV branches under the pinned-pinned end condition were much smaller than those under the fixed-fixed end condition. This was due to the fact that for a specified value of L, the natural frequency f_n for the pinned-pinned end condition was much smaller than that for the fixed-fixed end condition. As such, to keep the natural frequency f_n unchanged (or the reduced velocity Vr unchanged for a certain flow velocity U; see Equation (1)), a shorter span length L under the pinned-pinned end condition was required than that under the fixed-fixed end condition.

The effect of the flow velocity (U) on the fatigue life (N) was further investigated. The variations in the fatigue life N with the flow velocity U for three span length values L = 70 m, 90 m, and 110 m are shown in Figure 9a–c, respectively. If the free span vibrated in the initial excitation branch (3.5 $\le$ Vr < 5.0) (i.e., 0.53 $\le U$ < 0.76 (m/s) for L = 70 m (Figure 9a), 0.32 $\le U$ < 0.46 (m/s) for L = 90 m (Figure 9b), and 0.22 $\le U$ < 0.31 (m/s) for L = 110 m (Figure 9c)), then the fatigue life N decreased with the increase in the flow velocity U. If the free span vibrated in the upper branch with large amplitudes (5.0 $\le$ Vr < 6.7) (i.e., 0.76 $\le U$ < 1.02 (m/s) for L = 70 m (Figure 9a), 0.46 $\le U$ < 0.62 (m/s) for L = 90 m (Figure 9b), and 0.31 $\le U$ < 0.41 (m/s) for L = 110 m (Figure 9c)), then the fatigue life N was first reduced slightly to a minimum value and then gradually increases with an increasing U. In the lower branch of VIVs (6.7 $\le$Vr < 12) (i.e., 1.02 $\le U$ < 1.82 (m/s) for L = 70 m (Figure 9a), 0.62 $\le U$ < 1.10 (m/s) for L = 90 m (Figure 9b), and 0.41 $\le U$ < 0.74 (m/s) for L = 110 m (Figure 9c)), the free span vibrated with moderate amplitudes, and the corresponding fatigue life N became longer with the increase in U. Similarly, in the desynchronization branch (Vr ≥ 12) (i.e., $U$ > 1.10 (m/s) for L = 90 m (Figure 9b) and $U$ > 0.74 (m/s) for L = 110 m (Figure 9c)), the free span vibrated with much smaller amplitudes, and the fatigue life N further became longer with the increase in U. It should be noted that for L = 70 m (Figure 9a), the desynchronization branch of VIVs was not involved in the examined range of U < 1.3 m/s.

The above parametric study shows that the fatigue life (N) of the free spans of subsea pipelines is quite sensitive to the span length (L) and the flow velocity (U). By following the flow chart for fatigue life prediction (Section 4.1), the variations in N with U and L could be obtained, as shown in Figure 10. Note that the S-N curve C1 (see Figure 5) was adopted to predict the fatigue life N of the extreme scenario of large free spans. As illustrated by the two-dimensional contour map (Figure 10), for the span length range 70 $\le$ L $\le$ 110 (m) and the flow velocity range 0.2 $\le$ U $\le$ 1.3 (m/s), the minimum value of N ($\approx$1.28$\times$10⁵) emerged at U $\approx$ 0.9 m/s for L = 70 m (see also Figure 9a). With a further decrease or increase in U for a specified span length value (L), the fatigue life N would be remarkedly enhanced, which could be within the regime of very high cycle fatigue (VHCF) (i.e., N > $~$10⁷; see Figure 10). As discussed above, such nonlinearity of the relationship between the fatigue life (N) and the flow velocity (U) and span length (L) is mainly attributed to the intrinsic characteristics of VIVs (see Figure 2) (i.e., the nonlinear variation in the dimensionless vibration amplitude with the reduced velocity).

5. Concluding Remarks

In subsea environments, free-spanning subsea pipelines may suffer from vortex-induced vibrations (VIVs), which could further result in fatigue failure after a substantial period of service. In this study, the fatigue failure of free spans caused by VIVs was investigated, with a focus on very high cycle fatigue (VHCF) in the extreme scenario of large free spans under flows with moderate velocities. The following conclusions can be drawn:

(1)

With reference to the benchmark flume observations of the VIV response amplitudes of a low mass damping system, the whole curve for the nonlinear variation in A/D with Vr (A/D–Vr curve) is recommended, in which four distinct branches of VIV responses can be identified (i.e., the initial excitation branch, the upper branch, the lower branch, and the desynchronization branch). Bilinear S-N curves were chosen to describe the relationship between the stress range and the fatigue life of the free spans in a complex seawater environment.

(2)

On the basis of the recommended A/D–Vr curve and the S-N curve for VHCF of high-strength steel in seawater and with cathodic protections, a prediction method was proposed for the fatigue life of free spans undergoing VIVs. If the pipeline and environmental parameters are given, then the fatigue life in terms of the number of cycles to failure N can be evaluated by following the flow chart.

(3)

The nonlinear relationships between the fatigue life N and the flow velocity U and span length L were obtained in a parametric study. For the examined range of span lengths and flow velocities, the minimum fatigue life emerged at a certain flow with a moderate velocity for a given span length L. With a further decrease or increase in U, the fatigue life N would be enhanced correspondingly, which could be within the regime of VHCF. Such nonlinearity of the relationship of N with U and L is mainly attributed to the intrinsic characteristics of VIVs (i.e., the nonlinear variation in the dimensionless vibration amplitude with a reduced velocity).

In the present investigation, a quantitative correlation was established between the VHCF life of a spanning pipeline undergoing VIVs and the key influential parameters (i.e., the span length and the flow velocity). The VHCF life of a spanning pipeline was examined under two extreme (F-F and P-P) end conditions by taking into account the effective span length. This study may provide an instructive reference or methodology for evaluating the service life of subsea-spanning pipelines. It should be noted that the above conclusions are predominantly limited to the engineering scenario of an isolated single span on a relatively flat seabed. Much uncertainty still exists for accurate prediction of the fatigue life of free spans (e.g., the complex pipe–soil interaction at the span shoulders), which needs to be further investigated.