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


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.

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  1. Allen PM, Arnold JG, Byars BW (1994) Downstream channel geometry for use in planning-level models. Water Resour Bull 30:663–671

    Article  Google Scholar 

  2. Andrieu-Renaud C, Sudret B, Lemaire M (2004) The PHI2 method: a way to compute time-variant reliability. Reliab Eng Syst Saf 84:75–86

    Article  Google Scholar 

  3. Araújo AL, Mota Soares CM, Mota Soares CA, Herskovits J (2010) Optimal design and parameter estimation of frequency dependent viscoelastic laminated sandwich composite plates. Compos Struct 92:2321–2327

    Article  Google Scholar 

  4. Arora VK, Boer GJ (1999) A variable velocity flow routing algorithm for GCMs. J Geophys Res Atmos 104:30965–30979

    Article  Google Scholar 

  5. Beersma JJ, Buishand TA (2004) Joint probability of precipitation and discharge deficits in the Netherlands. Water Resour Res 40:1–11

    Article  Google Scholar 

  6. Blade Tidal (2011) Version 4.1, Demonstration version, GL Garrad Hassan. Accessed 5 Sep 2011

  7. Brooks DA (2011) The hydrokinetic power resource in a tidal estuary: the Kennebec River of the central Maine coast. Renewable Energ 36:1492–1501

    Article  Google Scholar 

  8. Eldred MS (2009) Recent advances in non-intrusive polynomial chaos and stochastic collocation methods for uncertainty analysis and design. In: 50th AIAA/ASME/ASCE/AHS/ASC Structures, structural dynamics and materials conference, art. no. 2009–2274

  9. Eldred MS, Burkardt J (2009) Comparison of non-intrusive polynomial chaos and stochastic collocation methods for uncertainty quantification. In: 47th AIAA aerospace sciences meeting including the new horizons forum and aerospace exposition, art. no. 2009–0976

  10. Chen W, Tsui K-L, Allen JK, Mistree F (1995) Integration of the response surface methodology with the compromise decision support problem in developing a general robust design procedure. American Society of Mechanical Engineers, Design Engineering Division (Publication) DE 82(1), pp 485–492

  11. Chiachio M, Chiachio J, Rus G (2012) Reliability in composites—a selective review and survey of current development. Compos B Eng 43:902–913

    Article  Google Scholar 

  12. Choi SK, Grandhi RV, Canfield RA (2007) Reliability-based structural design. Springer

  13. Du X, Sudjianto A (2005) Reliability-based design with the mixture of random and interval variables. Trans ASME J Mech Des 127:1068–1076

    Article  Google Scholar 

  14. Du X, Sudjianto A, Chen W (2004) An integrated framework for optimization under uncertainty using inverse reliability strategy. Trans ASME J Mech Des 126:562–570

    Article  Google Scholar 

  15. Eamon CD, Rais-Rohani M (2009) Integrated reliability and sizing optimization of a large composite structure. Mar Struct 22:315–334

    Article  Google Scholar 

  16. Engelund S, Rackwitz R, Lange C (1995) Approximations of first-passage times for differentiable processes based on higher-order threshold crossings. Probabilist Eng Mech 10:53–60

    Article  Google Scholar 

  17. Ginter VJ, Pieper JK (2011) Robust gain scheduled control of a hydrokinetic turbine. IEEE Trans Contr Syst Tech 19:805–817

    Article  Google Scholar 

  18. Grujicic M, Arakere G, Pandurangan B, Sellappan V, Vallejo A, Ozen M (2010) Multidisciplinary design optimization for glass-fiber epoxy-matrix composite 5 MW horizontal-axis wind-turbine blades. J Mater Eng Perform 19:1116–1127

    Article  Google Scholar 

  19. Guney MS (2011) Evaluation and measures to increase performance coefficient of hydrokinetic turbines. Renew Sustain Energ Rev 15:3669–3675

    Article  Google Scholar 

  20. Hantoro R, Utama IKAP, Erwandi, Sulisetyono A (2011) An experimental investigation of passive variable-pitch vertical-axis ocean current turbine. ITB J Eng Sci 43B:27–40

    Article  Google Scholar 

  21. Hu Z, Du X (2011) Time-dependent reliability analysis with joint upcrossing rates. Struct Multidisc Optim, revision under review

  22. Hu Z, Du X (2012) Reliability analysis for hydrokinetic turbine blades. Renewable Energ 48:251–262

    Article  Google Scholar 

  23. Kam TY, Su HM, Wang BW (2011) Development of glass-fabric composite wind turbine blade. Adv Mater Res:2482–2485

  24. Kriegesmann B, Rolfes R, Hühne C, Kling A (2011) Fast probabilistic design procedure for axially compressed composite cylinders. Compos Struct 93:3140–3149

    Google Scholar 

  25. Lago LI, Ponta FL, Chen L (2010) Advances and trends in hydrokinetic turbine systems. Energ Sustain Dev 14:287–296

    Article  Google Scholar 

  26. Lee CK (2008) Corrosion and wear-corrosion resistance properties of electroless Ni-P coatings on GFRP composite in wind turbine blades. Surf Coating Tech 202:4868–4874

    Article  Google Scholar 

  27. Madsen PH, Krenk S (1984) An integral equation method for the first passage problem in random vibration. J Appl Mech 51:674–679

    MathSciNet  MATH  Article  Google Scholar 

  28. Martin OLH (2008) Aerodynamics of wind turbines, 2nd edn. Earthscan, Sterling

  29. Mitosek HT (2000) On stochastic properties of daily river flow processes. J Hydrol 228:188–205

    Article  Google Scholar 

  30. Motley MR, Young YL (2010) Reliability-based global design of self-adaptive marine rotors. In: ASME 2010 3rd Joint US-European Fluids Engineering Summer Meeting, FEDSM 2010, Montreal, QC, pp 1113–1122

  31. Motley MR, Young YL (2011a) Performance-based design and analysis of flexible composite propulsors. J Fluid Struct 27:1310–1325

    Article  Google Scholar 

  32. Motley MR, Young YL (2011b) Influence of uncertainties on the response and reliability of self-adaptive composite rotors. Compos Struct 94:114–120

    Article  Google Scholar 

  33. Mühlberg K (2010) Corrosion protection of offshore wind turbines—a challenge for the steel builder and paint applicator. JPCL 27:20–32

    Google Scholar 

  34. Muste M, Yu K, Pratt T, Abraham D (2004) Practical aspects of ADCP data use for quantification of mean river flow characteristics; Part II: fixed-vessel measurements. Flow Meas Instrum 15:17–28

    Article  Google Scholar 

  35. Otache MY, Bakir M, Li Z (2008) Analysis of stochastic characteristics of the Benue River flow process. Chin J Oceanol Limnol 26:142–151

    Article  Google Scholar 

  36. Pimenta RJ, Diniz SMC, Queiroz G, Fakury RH, Galvão A, Rodrigues FC (2012) Reliability-based design recommendations for composite corrugated-web beams. Probab Eng Mech 28:185–193

    Article  Google Scholar 

  37. Preumont A (1985) On the peak factor of stationary Gaussian processes. J Sound Vib 100:15–34

    Article  Google Scholar 

  38. Rice SO (1944) Mathematical analysis of random noise. Bell Syst Tech J 23:282–332

    MathSciNet  MATH  Google Scholar 

  39. Rice SO (1945) Mathematical analysis of random noise. Bell Syst Tech J 24:146–156

    MathSciNet  Google Scholar 

  40. Ronold KO, Christensen CJ (2001) Optimization of a design code for wind-turbine rotor blades in fatigue. Eng Struct 23:993–1004

    Article  Google Scholar 

  41. Ronold KO, Larsen GC (2000) Reliability-based design of wind-turbine rotor blades against failure in ultimate loading. Eng Struct 22:565–574

    Article  Google Scholar 

  42. Saranyasoontorn K, Manuel L (2006) Design loads for wind turbines using the environmental contour method. J Sol Energ Eng Trans ASME 128:554–561

    Article  Google Scholar 

  43. Schall G, Faber MH, Rackwitz R (1991) The ergodicity assumption for sea states in the reliability estimation of offshore structures. J Offshore Mech Arctic Eng 113:241–246

    Article  Google Scholar 

  44. Schulze K, Hunger M, Döll P (2005) Simulating river flow velocity on global scale. Adv Geosci 5:133–136

    Article  Google Scholar 

  45. Singh A, Mourelatos Z, Li J (2010) Design for lifecycle cost using time-dependent reliability. Trans ASME J Mech Des 132:091008

    Article  Google Scholar 

  46. Sudret B (2008a) Analytical derivation of the outcrossing rate in time-variant reliability problems. Struct Infrastruct Eng 4:353–362

    Article  Google Scholar 

  47. Sudret B (2008b) Global sensitivity analysis using polynomial chaos expansions. Reliab Eng Syst Saf 93:964–979

    Article  Google Scholar 

  48. Toft HS, Sørensen JD (2011) Reliability-based design of wind turbine blades. Struct Saf 33:333–342

    Article  Google Scholar 

  49. Val DV, Chernin L (2011) Reliability of tidal stream turbine blades. In: 11th international conference on applications of statistics and probability in civil engineering, ICASP, Zurich, pp 1817–1822

  50. Vanmarcke EH (1975) On the distribution of the first-passage time for normal stationary random processes. J Appl Mech Trans ASME 42(Ser E):215–220

    MATH  Article  Google Scholar 

  51. Veldkamp D (2008) A probabilistic evaluation of wind turbine fatigue design rules. Wind Energ 11:655–672

    Article  Google Scholar 

  52. Wang W, Van Gelder PHAJM, Vrijling JK (2005) Long-memory in streamflow processes of the yellow river. In: IWA international conference on water economics, statistics, and finance rethymno, Greece, 8–10 July, pp 481–490

  53. Yang JN, Shinozuka M (1971) On the first excursion probability in stationary narrow- band random vibration. J Appl Mech Trans ASME 38(Ser E):1017–1022

    MATH  Article  Google Scholar 

  54. Young YL, Baker JW, Motley MR (2010) Reliability-based design and optimization of adaptive marine structures. Compos Struct 92:244–253

    Article  Google Scholar 

  55. Zhang J, Du X (2011) Time-dependent reliability analysis for function generator mechanisms. Trans ASME J Mech Des 133

  56. Zhang YX, Yang CH (2009) Recent developments in finite element analysis for laminated composite plates. Compos Struct 88:147–157

    Article  Google Scholar 

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The authors gratefully acknowledge the support from the Office of Naval Research through contract ONR N000141010923 (Program Manager - Dr. Michele Anderson) and the Intelligent Systems Center at the Missouri University of Science and Technology.

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Correspondence to Xiaoping Du.


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

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

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

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

Then M can be obtained by

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

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)

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

in which

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

\(\boldsymbol\upbeta=[\beta_{1}, \beta_{2}]\) 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):

$$ \label{eq38} \,\begin{aligned}[b] \mu &=\left[\, {{\begin{array}{*{20}c} {\mu_1 } \hfill \\ {\mu_2 } \hfill \\ \end{array} }} \,\right]={{\bf c}}_{\dot{\bf L}\bf L} {{\bf c}}_{{{\bf LL}}}^{-1}\boldsymbol\upbeta\\ &={\left[\, {{\begin{array}{*{20}c} {\left( {\beta_2 -\rho \beta_1 } \right)\rho_1 } \hfill \\ {\left( {\beta_1 -\rho \beta_2 } \right)\rho_2 } \hfill \\ \end{array} }}\, \right]} \mathord{\big/} {\big( {1-\rho^2}\big)} \end{aligned} $$
$$ \label{eq39} \,\begin{aligned}[b] \sum &= \,{{\bf c}}_{_{\left. {\dot{\bf L}} \right|{{\bf L}}} } ={{\bf c}}_{{\dot{\bf L}} {\dot{\bf L}}} -{{\bf c}}_{{\dot{\bf L}} {{\bf L}}} {{\bf c}}_{{{\bf LL}}}^{-1} {{\bf c}}_{{{\bf L}}{\dot{\bf L}}} \\ &=\left[\, {{\begin{array}{*{20}c} {\lambda_1^2 } \hfill & {\kappa \lambda_1 \lambda_2 } \hfill \\ {\kappa \lambda_1 \lambda_2 } \hfill & {\lambda_2^2 } \hfill \\ \end{array} }} \,\right] \end{aligned} $$


$$ \label{eq40} \left[ {{\begin{array}{*{20}c} {{{\bf c}}_{{\dot{\bf L}} {\dot{\bf L}}}} \hfill & {{{\bf c}}_{{\dot{\bf L}} {{\bf L}}} } \hfill\\ {{{\bf c}}_{{{\bf L}}{\dot{\bf L}}}} \hfill & {{{\bf c}}_{{{\bf LL}}} } \hfill \\ \end{array} }} \right]=\left[ {{\begin{array}{*{20}c} {\omega^2\left( {t_1 } \right)} \hfill & {\rho_{12} } \hfill & 0 \hfill &{\rho_1 } \hfill \\ {\rho_{21} } \hfill & {\omega^2\left( {t_2 } \right)} \hfill & {\rho_2 }\hfill & 0 \hfill \\ 0 \hfill & {\rho_2 } \hfill & 1 \hfill & \rho \hfill \\ {\rho_1 } \hfill & 0 \hfill & \rho \hfill & 1 \hfill \\ \end{array} }} \right] $$
$$ \label{eq41} \,\begin{aligned}[b] \rho_1 &=\mathop{\boldsymbol\upalpha}\limits^\cdot \left( {t_1 } \right){{\bf C}}\left( {t_1 ,t_2 } \right)\boldsymbol\upalpha^T\left( {t_2 } \right)\\ &\quad +\boldsymbol\upalpha \left( {t_1 } \right){\dot{\bf C}}_1 \left( {t_1 ,t_2 } \right)\boldsymbol\upalpha ^T\left( {t_2 } \right) \end{aligned} $$
$$ \label{eq42} \,\begin{aligned}[b] \rho_2 &=\boldsymbol\upalpha \left( {t_1 } \right){{\bf C}}\left( {t_1 ,t_2 } \right){\mathop{\boldsymbol\upalpha}\limits^\cdot}^T\left( {t_2 } \right)\\ &\quad + \boldsymbol\upalpha\left( {t_1 } \right){\dot{\bf C}}_2 \left( {t_1 ,t_2 } \right)\boldsymbol\upalpha ^T\left( {t_2 } \right) \end{aligned} $$
$$ \label{eq43} \,\begin{aligned}[b] \rho_{12} &=\mathop{\boldsymbol\upalpha}\limits^\cdot \left( {t_1 } \right){\dot{\bf C}}_2 \left( {t_1 ,t_2 } \right) {\boldsymbol\upalpha}^T\left( {t_2 } \right)+\mathop{{\boldsymbol\upalpha}}\limits^\cdot \left( {t_1 } \right){{\bf C}}\left( {t_1 ,t_2 } \right){\mathop{{\boldsymbol\upalpha}}\limits^\cdot}^T\left( {t_2 } \right) \\ &\quad + {\boldsymbol\upalpha} \left( {t_1 } \right){\ddot{\bf C}}_{12} \left( {t_1 ,\,t_2 } \right){\boldsymbol\upalpha}^T\left( {t_2 } \right)\\ &\quad + {\boldsymbol\upalpha} \left( {t_1 } \right){\dot{\bf C}}_1 \left( {t_1 ,\,t_2 } \right){\mathop{{\boldsymbol\upalpha}}\limits^\cdot}^T\left( {t_2 } \right) \end{aligned} $$
$$ \label{eq44} \,\begin{aligned}[b] \rho_{21} &=\mathop{\boldsymbol\upalpha}\limits^\cdot \left( {t_1 } \right){{\bf C}}\left( {t_1 ,t_2 } \right){\mathop{\boldsymbol\upalpha}\limits^\cdot}^T\left( {t_2 } \right)+{\boldsymbol\upalpha} \left( {t_1 } \right){\dot{\bf C}}_1 \left( {t_1 ,t_2 } \right){\mathop{\boldsymbol\upalpha}\limits^\cdot}^T\left( {t_2 } \right) \\ &\quad + {\boldsymbol\upalpha} \left( {t_1 } \right){\ddot{\bf C}}_{21} \left( {t_1 ,t_2 } \right){\boldsymbol\upalpha}^T\left( {t_2 } \right)\\ &\quad +\mathop{\boldsymbol\upalpha}\limits^\cdot \left( {t_1 } \right){\dot{\bf C}}_2 \left( {t_1 ,t_2 } \right){\boldsymbol\upalpha}^T\left( {t_2 } \right) \end{aligned} $$
$$ \label{eq45} \,\begin{aligned}[b] &{{\bf C}}\left( {t_1 ,t_2 } \right)\\ &\quad =\left[ {{\footnotesize\begin{array}{*{20}c} {{\bf 1}} \hfill & {{\bf 0}} \hfill & \cdots \hfill & {{\bf 0}} \hfill \\ {{\bf 0}} \hfill & {\rho^{Y_1 }\left( {t_1 ,t_2 } \right)} \hfill &\cdots \hfill & 0 \hfill \\ \vdots \hfill & \vdots \hfill & \ddots \hfill & \vdots \hfill \\ {{\bf 0}} \hfill & 0 \hfill & \cdots \hfill & {\rho^{Y_m }\left( {t_1 ,t_2 } \right)} \hfill \\ \end{array} }} \right]_{\left( {n+m} \right)\times \left( {n+m} \right)} \end{aligned} $$


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

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Hu, Z., Li, H., Du, X. et al. Simulation-based time-dependent reliability analysis for composite hydrokinetic turbine blades. Struct Multidisc Optim 47, 765–781 (2013).

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  • Reliability
  • Composite
  • Hydrokinetic turbine
  • Time-dependent