Simulation-based time-dependent reliability analysis for composite hydrokinetic turbine blades

Abstract

The reliability of blades is vital to the system reliability of a hydrokinetic turbine. A time-dependent reliability analysis methodology is developed for river-based composite hydrokinetic turbine blades. Coupled with the blade element momentum theory, finite element analysis is used to establish the responses (limit-state functions) for the failure indicator of the Tsai–Hill failure criterion and blade deflections. The stochastic polynomial chaos expansion method is adopted to approximate the limit-state functions. The uncertainties considered include those in river flow velocity and composite material properties. The probabilities of failure for the two failure modes are calculated by means of time-dependent reliability analysis with joint upcrossing rates. A design example for the Missouri river is studied, and the probabilities of failure are obtained for a given period of operation time.

Appendices

Appendix A: MCS for time-dependent reliability analysis

The MCS for time-dependent reliability analysis involves both a stochastic process (river flow discharge) and random variables. To generate samples for the stochastic process, we discretize the time interval [t ₀, t _s ] into N points. Then the samples of the normalized and standardized river flow discharge process D _m is generated by

$$ \label{eq33} {{\bf D}}_{\bf m} ={\bf m}_{D_m } +{{\bf M}}\boldsymbol\varsigma $$

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where $\boldsymbol\varsigma =\left( {\varsigma_1 ,\varsigma_2 ,\cdots ,\varsigma_N } \right)^T$ is the vector of N independent standard normal random variables; ${\bf m}_{D_m } =( \mu_{D_m } ( {t_1 } )$, ${\mu}_{D_m } ( {t_2 } ),\cdots ,{\mu}_{D_m } ( {t_N } ) )^T$ is the vector of mean values of ${{\bf D}}_{{\bf m}} =\left( {D_m \left( {t_1 } \right),D_m \left( {t_2 } \right),\cdots ,D_m \left( {t_N } \right)} \right)^T$; and M is a lower triangular matrix obtained from the covariance matrix of D _m .

Let the covariance matrix of D _m at the N points be C _N×N , we have

$$ \label{eq34} \,\begin{aligned}[b] &{{\bf C}}_{N\times N} \\ &\quad =\left(\, {{\footnotesize\begin{array}{*{20}c} {\rho_{D_m } \left( {t_1 ,\;t_1 } \right)} \hfill & {\rho_{D_m } \left( {t_1 ,\;t_2 } \right)} \hfill & \cdots \hfill & {\rho_{D_m } \left( {t_1 ,\;t_N } \right)} \hfill \\ {\rho_{D_m } \left( {t_2 ,\;t_1 } \right)} \hfill & {\rho_{D_m } \left( {t_2 ,\;t_2 } \right)} \hfill & \cdots \hfill & {\rho_{D_m } \left( {t_2 ,\;t_N } \right)} \hfill \\ \vdots \hfill & \vdots \hfill & \ddots \hfill & \vdots \hfill \\ {\rho_{D_m } \left( {t_N ,\;t_1 } \right)} \hfill & {\rho_{D_m } \left( {t_N ,\;t_2 } \right)} \hfill & \cdots \hfill & {\rho_{D_m } \left( {t_N ,\;t_N } \right)} \hfill \\ \end{array} }} \,\right)_{\,\,N\times N} \end{aligned} $$

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Then M can be obtained by

$$ \label{eq35} {{\bf C}}_{N\times N} ={{\bf PDP}}^{-1}={{\bf MM}}^T $$

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in which D is a diagonal eigenvalue matrix of the covariance matrix C _N×N , and P is the N×N square matrix whose i-th column is the i-th eigenvector of C _N×N .

After samples of the stochastic process of river flow discharge are generated, they are plugged into the limit-state functions, and then the samples (trajectories) of the limit-state functions are obtained. A trajectory is traced from the initial time to the end of the time period. Once the trajectory upcrosses the limit state, then a failure occurs; and the remaining curve will not be checked anymore. The process is illustrated in Fig. 17.

Fig. 17

A trajectory of a limit-state function

Appendix B: Computation of v ⁺⁺ (t ₁, t ₂)

Madsen has derived the expression for $\emph{v}^{++}(t_{1}$, t ₂) as follows (Madsen and Krenk 1984)

$$ \label{eq36} \,\begin{aligned}[b] &\emph{v}^{++}\left( {t_1 ,t_2 } \right)\\ &\quad =\lambda_1 \lambda_2 f_{{\bf W}} \left( {\boldsymbol\upbeta} \right)\Psi \big( {{\big( {{\mathop{\beta}\limits^\cdot}_1 -\mu_1 } \big)} \mathord{\big/} {\lambda_1 }} \big)\Psi \big( {{\big( {{\mathop{\beta}\limits^\cdot}_2 -\mu_2 } \big)} \mathord{\big/} {\lambda_2 }} \big) \\[10pt] &\qquad +\lambda_1 \lambda_2 f_{{\bf W}} \left( {\boldsymbol\upbeta} \right)\kappa \Phi \big( {{\big( {\mu_1 -{\mathop{\beta}\limits^\cdot}_1 } \big)} \mathord{\big/} {\lambda_1 }} \big)\Phi \big( {{\big( {\mu_2 -{\mathop{\beta}\limits^\cdot}_2 } \big)} \mathord{\big/} {\lambda_2 }} \big) \\[10pt] &\qquad +\lambda_1^2 \lambda_2^2 f_{{\bf W}} \left( {\boldsymbol\upbeta} \right)\displaystyle\int_0^\kappa {\left( {\kappa -K} \right)f_{\ddot{\bf W} \big|{{\bf W}}} \big( { {\mathop{\boldsymbol\upbeta}\limits^\cdot } \big|\boldsymbol\upbeta ;K} \big)} dK \end{aligned} $$

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in which

$$ \label{eq37} \,\begin{aligned}[b] f_{{\bf W}} \left(\boldsymbol\upbeta\right)={}& \big\{ {\exp \big[ {{\big( {\beta _1^2 -2\rho \beta_1 \beta_2 +\beta_2^2 } \big)} \mathord{\big/} {\big( {2-2\rho^2} \big)}} \big]} \big\} \\ & \mathord{\big/} {\big( {2\pi \sqrt {1-\rho^2} } \big)} \end{aligned} $$

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$\boldsymbol\upbeta=[\beta_{1}, \beta_{2}]$ represents the time-invariant reliability index at time t ₁ and t ₂. μ ₁ and μ ₂, and λ ₁ and λ ₂, κ are the mean values, standard deviations, and correlation coefficient of $\left. {\dot{\bf L} \left( {t_1 } \right)} \right| \boldsymbol\upbeta$ and $\left. {\dot{\bf L}\left( {t_2 } \right)} \right| \boldsymbol\upbeta$, respectively. They are calculated by the following equations (Hu and Du 2011):

$$ \label{eq38} \,\begin{aligned}[b] \mu &=\left[\, {{\begin{array}{*{20}c} {\mu_1 } \hfill \\ {\mu_2 } \hfill \\ \end{array} }} \,\right]={{\bf c}}_{\dot{\bf L}\bf L} {{\bf c}}_{{{\bf LL}}}^{-1}\boldsymbol\upbeta\\ &={\left[\, {{\begin{array}{*{20}c} {\left( {\beta_2 -\rho \beta_1 } \right)\rho_1 } \hfill \\ {\left( {\beta_1 -\rho \beta_2 } \right)\rho_2 } \hfill \\ \end{array} }}\, \right]} \mathord{\big/} {\big( {1-\rho^2}\big)} \end{aligned} $$

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$$ \label{eq39} \,\begin{aligned}[b] \sum &= \,{{\bf c}}_{_{\left. {\dot{\bf L}} \right|{{\bf L}}} } ={{\bf c}}_{{\dot{\bf L}} {\dot{\bf L}}} -{{\bf c}}_{{\dot{\bf L}} {{\bf L}}} {{\bf c}}_{{{\bf LL}}}^{-1} {{\bf c}}_{{{\bf L}}{\dot{\bf L}}} \\ &=\left[\, {{\begin{array}{*{20}c} {\lambda_1^2 } \hfill & {\kappa \lambda_1 \lambda_2 } \hfill \\ {\kappa \lambda_1 \lambda_2 } \hfill & {\lambda_2^2 } \hfill \\ \end{array} }} \,\right] \end{aligned} $$

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where

$$ \label{eq40} \left[ {{\begin{array}{*{20}c} {{{\bf c}}_{{\dot{\bf L}} {\dot{\bf L}}}} \hfill & {{{\bf c}}_{{\dot{\bf L}} {{\bf L}}} } \hfill\\ {{{\bf c}}_{{{\bf L}}{\dot{\bf L}}}} \hfill & {{{\bf c}}_{{{\bf LL}}} } \hfill \\ \end{array} }} \right]=\left[ {{\begin{array}{*{20}c} {\omega^2\left( {t_1 } \right)} \hfill & {\rho_{12} } \hfill & 0 \hfill &{\rho_1 } \hfill \\ {\rho_{21} } \hfill & {\omega^2\left( {t_2 } \right)} \hfill & {\rho_2 }\hfill & 0 \hfill \\ 0 \hfill & {\rho_2 } \hfill & 1 \hfill & \rho \hfill \\ {\rho_1 } \hfill & 0 \hfill & \rho \hfill & 1 \hfill \\ \end{array} }} \right] $$

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$$ \label{eq41} \,\begin{aligned}[b] \rho_1 &=\mathop{\boldsymbol\upalpha}\limits^\cdot \left( {t_1 } \right){{\bf C}}\left( {t_1 ,t_2 } \right)\boldsymbol\upalpha^T\left( {t_2 } \right)\\ &\quad +\boldsymbol\upalpha \left( {t_1 } \right){\dot{\bf C}}_1 \left( {t_1 ,t_2 } \right)\boldsymbol\upalpha ^T\left( {t_2 } \right) \end{aligned} $$

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$$ \label{eq42} \,\begin{aligned}[b] \rho_2 &=\boldsymbol\upalpha \left( {t_1 } \right){{\bf C}}\left( {t_1 ,t_2 } \right){\mathop{\boldsymbol\upalpha}\limits^\cdot}^T\left( {t_2 } \right)\\ &\quad + \boldsymbol\upalpha\left( {t_1 } \right){\dot{\bf C}}_2 \left( {t_1 ,t_2 } \right)\boldsymbol\upalpha ^T\left( {t_2 } \right) \end{aligned} $$

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$$ \label{eq43} \,\begin{aligned}[b] \rho_{12} &=\mathop{\boldsymbol\upalpha}\limits^\cdot \left( {t_1 } \right){\dot{\bf C}}_2 \left( {t_1 ,t_2 } \right) {\boldsymbol\upalpha}^T\left( {t_2 } \right)+\mathop{{\boldsymbol\upalpha}}\limits^\cdot \left( {t_1 } \right){{\bf C}}\left( {t_1 ,t_2 } \right){\mathop{{\boldsymbol\upalpha}}\limits^\cdot}^T\left( {t_2 } \right) \\ &\quad + {\boldsymbol\upalpha} \left( {t_1 } \right){\ddot{\bf C}}_{12} \left( {t_1 ,\,t_2 } \right){\boldsymbol\upalpha}^T\left( {t_2 } \right)\\ &\quad + {\boldsymbol\upalpha} \left( {t_1 } \right){\dot{\bf C}}_1 \left( {t_1 ,\,t_2 } \right){\mathop{{\boldsymbol\upalpha}}\limits^\cdot}^T\left( {t_2 } \right) \end{aligned} $$

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$$ \label{eq44} \,\begin{aligned}[b] \rho_{21} &=\mathop{\boldsymbol\upalpha}\limits^\cdot \left( {t_1 } \right){{\bf C}}\left( {t_1 ,t_2 } \right){\mathop{\boldsymbol\upalpha}\limits^\cdot}^T\left( {t_2 } \right)+{\boldsymbol\upalpha} \left( {t_1 } \right){\dot{\bf C}}_1 \left( {t_1 ,t_2 } \right){\mathop{\boldsymbol\upalpha}\limits^\cdot}^T\left( {t_2 } \right) \\ &\quad + {\boldsymbol\upalpha} \left( {t_1 } \right){\ddot{\bf C}}_{21} \left( {t_1 ,t_2 } \right){\boldsymbol\upalpha}^T\left( {t_2 } \right)\\ &\quad +\mathop{\boldsymbol\upalpha}\limits^\cdot \left( {t_1 } \right){\dot{\bf C}}_2 \left( {t_1 ,t_2 } \right){\boldsymbol\upalpha}^T\left( {t_2 } \right) \end{aligned} $$

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$$ \label{eq45} \,\begin{aligned}[b] &{{\bf C}}\left( {t_1 ,t_2 } \right)\\ &\quad =\left[ {{\footnotesize\begin{array}{*{20}c} {{\bf 1}} \hfill & {{\bf 0}} \hfill & \cdots \hfill & {{\bf 0}} \hfill \\ {{\bf 0}} \hfill & {\rho^{Y_1 }\left( {t_1 ,t_2 } \right)} \hfill &\cdots \hfill & 0 \hfill \\ \vdots \hfill & \vdots \hfill & \ddots \hfill & \vdots \hfill \\ {{\bf 0}} \hfill & 0 \hfill & \cdots \hfill & {\rho^{Y_m }\left( {t_1 ,t_2 } \right)} \hfill \\ \end{array} }} \right]_{\left( {n+m} \right)\times \left( {n+m} \right)} \end{aligned} $$

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and

$$ \label{eq46} \,\begin{aligned}[b] &{\dot{\bf C}}_j \left( {t_1 ,t_2 } \right) \\ &\quad = {\partial {{\bf C}}\left( {t_1 ,t_2 } \right)} \mathord{\left/ {\vphantom {{\partial {{\bf C}}\left( {t_1 ,t_2 } \right)} {\partial t_j }}} \right. \kern-\nulldelimiterspace} {\partial t_j } \\ &\quad = \left[ {{\footnotesize\begin{array}{*{20}c} {{\bf 0}} \hfill & {{\bf 0}} \hfill & \cdots \hfill & {{\bf 0}} \hfill \\ {{\bf 0}} \hfill & {\dfrac{\partial \rho^{Y_1 }\left( {t_1 ,t_2 } \right)}{\partial t_j }} \hfill & \cdots \hfill & 0 \hfill \\ \vdots \hfill & \vdots \hfill & \ddots \hfill & \vdots \hfill \\ {{\bf 0}} \hfill & 0 \hfill & \cdots \hfill & {\dfrac{\partial \rho ^{Y_m }\left( {t_1 ,t_2 } \right)}{\partial t_j }} \hfill \\ \end{array} }} \right]_{\left( {n+m} \right)\times \left( {n+m} \right)} ,\\ &\qquad j=1,\;2 \end{aligned} $$

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