An improved model for VIV fatigue life prediction of slender marine structures in time-varying flows

Abstract

The objective of this study is to propose an improved model to evaluate the fatigue life of slender marine structures with high efficiency and accuracy. We establish two Weibull functions to represent the probability density function (PDF) of time-varying flow velocities and fatigue stress. Next, a modified Latin hypercube sampling method is developed to improve the efficiency by reducing the sampling number based on the characteristics of vortex-induced vibrations. In addition, we employ the response surface method to predict the dominant mode and select the truncation order. This method could make the calculation time of each stress amplitude within 0.1–4 s, which meets the requirement for efficiency. To further improve the calculation accuracy, the Strouhal number used in the wake oscillator model is considered to change with the reduced velocity by introducing a correction factor. Besides, the influence of the tension fluctuation from axial vibration on fatigue damage is taken into consideration. A good agreement is achieved in the comparison between the numerical and experimental results. Moreover, the improvement of the accuracy and the efficiency of the proposed model is validated. In the end, a parametric study is conducted to investigate the effect of flow PDF parameters on the fatigue life.

Appendix

The solution of the y(z,t) in Eq. 14 can be written as [28]:

$$y(z,t)=\sum\limits_{{i=1}}^{\infty } {{{\tilde {\phi }}_i}} (z){\bar {y}_i}(t),$$

(23)

where ${\bar {\phi }_i}(z)$ is the mode function; ${\bar {y}_i}(t)$ is the weight function.

Next, the authors select the eigenvalue and propose eigenfunction expansion. The eigenvalue problem is chosen as:

$$\frac{{{{\text{d}}^4}{\phi _i}(z)}}{{d{z^4}}}=\alpha _{i}^{4}{\phi _i}(z),z \subset (0,1),$$

(24)

where ${\phi _i}(z)$ is the eigenfunction of Eq. 24, which satisfies the orthogonality property.

The selection of eigenfunction is computed based on the auxiliary eigenvalue problem with the boundary condition satisfied. The eigenfunction could be written as:

$${\phi _i}(z)=\sin ({\alpha _i}z),{\alpha _i}=i\pi ,\quad i=1,2,3 \ldots K,$$

(25)

where α_i is the eigenvalue.

The eigenfunction problem allows the definition of the following integral transform pair:

$${\bar {y}_i}(t)={\int_{0}^{1} {\mathop \phi \limits^{\sim } } _i}(z)y(z,t){\text{d}}z,\;{\text{transform}}$$

(26)

$$y(z,t)=\sum\limits_{{i=1}}^{\infty } {{{\tilde {\phi }}_i}} (z){\bar {y}_i}(t),{\text{inversion,}}$$

(27)

where ${\tilde {\phi }_i}(z)=\frac{{{\phi _i}(z)}}{{N_{i}^{{1/2}}}}=\sqrt 2 {\phi _i}(z)$, ${N_i}=\int_{0}^{1} {\phi _{i}^{2}} {\text{d}}z$.

After establishing the integral transform pair for y(z,t), the dimensionless form of Eq. 14 is multiplied by $\int_{0}^{1} {{{\tilde {\phi }}_i}} {\text{d}}z$, and then some mathematical manipulation is conducted. The modal analysis for Eq. 13 follows the same method by introducing another eigenfunction ${\mathop \Psi \limits^{\sim } _k}(z)$.

The new equation form after modal analysis is:

$$\begin{aligned} & \frac{{{{\text{d}}^2}{{\bar {y}}_i}(t)}}{{{\text{d}}{t^2}}}+\delta \frac{{{\text{d}}{{\bar {y}}_i}(t)}}{{{\text{d}}t}} - c\sum\limits_{{j=1}}^{\infty } {{P_{ij}}{{\bar {y}}_j}(t)} {\text{+}}b\alpha _{i}^{4}{{\bar {y}}_i}(t) \\ & \quad {\text{=}}\,\,M\sum\limits_{{k=1}}^{\infty } {{Q_{ik}}{{\bar {q}}_k}(t),} \\& \quad \quad i=1,2,3, \ldots K \\ \end{aligned}$$

(28)

$$\begin{aligned} & \frac{{{{\text{d}}^2}{{\bar {q}}_k}(t)}}{{{\text{d}}{t^2}}}+\varepsilon \sum\limits_{{l=1}}^{\infty } {\sum\limits_{{r=1}}^{\infty } {\sum\limits_{{s=1}}^{\infty } {{R_{klrs}}{{\bar {q}}_l}(t){{\bar {q}}_r}(t)} } } \frac{{{\text{d}}{{\bar {q}}_s}(t)}}{{{\text{d}}t}} - \varepsilon \frac{{{\text{d}}{{\bar {q}}_k}(t)}}{{{\text{d}}t}}+{{\bar {q}}_k}(t) \\ & \quad =A\sum\limits_{{i=1}}^{\infty } {{S_{ki}}\frac{{{{\text{d}}^2}\bar {y}(t)}}{{{\text{d}}{t^2}}}} , \\ & \quad \quad k=1,2,3, \ldots K \\ \end{aligned}$$

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where the K is the truncation order, the coefficients matrices are given by the following expressions:

$$\begin{aligned} {P_{ij}} & = & \int_{0}^{1} {\tilde {\phi }} (z)\frac{{{{\text{d}}^2}{{\tilde {\phi }}_j}(z)}}{{{\text{d}}{z^2}}}{\text{d}}z,\quad {Q_{ik}}=\int_{0}^{1} {\tilde {\phi }} (z){{\tilde {\Psi }}_k}(z){\text{d}}z \\ {R_{klrs}} & = & \int_{0}^{1} {{{\tilde {\Psi }}_k}} {{\tilde {\Psi }}_l}{{\tilde {\Psi }}_r}{{\tilde {\Psi }}_s}(z){\text{d}}z,\quad {S_{ki}}=\int_{0}^{1} {{{\tilde {\Psi }}_k}} (z)\tilde {\phi }(z){\text{d}}z. \\ \end{aligned}$$

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