Structural reliability analysis of rotor blades in ultimate loading

Transcription

1 EWEA 2011 Brussels, Belgium: Europe s Premier Wind Energy Event Structural reliability analysis of rotor blades in ultimate loading K. C. Bacharoudis 1, D. J. Lekou 2, T. P. Philippidis 1 1. University of Patras, Dept. of Mechanical Engng & Aeronautics, Greece 2. Centre for Renewable Energy Sources, Wind Energy Division, Greece

2 Probabilistic Strength Analysis The objectives are: Reliability assessment of a given design (safety factors, Target reliability) New probabilistic design, sensitivity studies (Improve/optimize blade structural design)

3 Probabilistic Strength Analysis Sectional analysis based on thin wall multi cellular theory was performed by a numerical tool (THIN) Monte Carlo, Edgeworth expansion method and Response Surface method/monte Carlo were used Stress resultants, engineering elastic constants (E 1, E 2, v 12, G 12 ) and failure stresses (X T, X C, Y T, Y C, S) in the principal coordinate system of the UD ply were considered random variables (RV)

4 Probabilistic Strength Analysis The effort consists of three major tasks: Stochastic representation of material properties Stochastic modeling of the stress resultants acting on the blade section Implementation of fast and accurate reliability methods

5 Rotor Blade Model Blade model Sec 9.2m Blade section Thin model

6 Stochastic Material Properties Material properties at the ply level Sec 9.2m Material Property Mean Std. R d E x (GPa) Laminate E y (GPa) v xy G xy (GPa) X T (Mpa) Blade section X C (Mpa) Y T (Mpa) Y C (Mpa) S (Mpa) Material properties were considered normally distributed THIN model

7 Extreme Load Analysis (IEC ed.3) N z M z z N x N y M y x M x y Directions of the stress resultants of the blade under extreme loading

8 Extreme Load Analysis (IEC ed.3) 2,000 1,800 1,600 x N x M x N y M y N z M z z Flap moment [kn*m] 1,400 1,200 1, y Sample # Time series from aero elastic codes

9 Extreme Load Analysis (IEC ed.3) 2,000 1,800 1,600 Sample values Threshold Max x N x M x N y M y N z M z z Flap moment [kn*m] 1,400 1,200 1, y Sample # Pick local maxima according IEC Ed.3 Thrs.=mv+1.4*sd

10 Extreme Load Analysis (IEC ed.3) 14 N x N y M y N z M z z Ln( Ln(F max )) Empirical Lognormal Weibull 3pWeibull Gumbel Gumbel_1 x M x 2 y Flap moment [kn*m] Find best-fit probability distributions for every mean wind speed bin

11 Extreme Load Analysis (IEC ed.3) 1.00E E E 02 x N x M x N y N z M z M y z P(F ext >=F/T=10 min) 1.00E E E E E E E E 10 Lognormal Weibull 3pWeibull Gumbel Gumbel_1 P=3.8e 7 Maximum observation y 1.00E E 12 Flap moment [kn*m] Vout ext e ext Pr ob F F T P F, T Pr ob F F V, T p V dv V in

12 Extreme Load Analysis (IEC ed.3) 1.0 N x N y N z M z M y z P(F ext <F/T=20 yrs) Empirical Normal Lognormal Weibull 3pWeibull Gumbel x M x y Flap moment [kn*m] ext 20 1 ext 10min PF FT yr PF FT N

13 Extreme Load Analysis (IEC ed.3) Stress resultants are modeled as Gumbel distribution N x N y M y N z M z z Fx e e xb a x M x y Section 9.2m F d a b N x [kn] N y [kn] N z [kn] M x [knm] M z [knm] M y [knm]

14 Extreme Load Analysis (IEC ed.3) Tor. Flap M. Flap F Edge M. Edge F. Tor Flap M N z Flap F N x N y M z z Edge M Edge F M y x M x 1 Correlation matrix of stress resultants for section 9.2m from root y Correlation Edge M. Torsion Edge M. Flap M. Edge M. Flap F. Edge M. Edge M. Edge M. Edge F Sample #

15 Element failure probability i j i j N z Laminate x N x M x N y M y y M z z Assuming : Laminate is a series system of layers Each ply has one failure mode described by the specified failure criterion Positive correlated failure modes among the layers

16 Element failure probability i j i j N z Laminate x N x M x N y M y y M z z max,, P P P f f f node i layer 1 layer n max,, P P P f f f node j layer 1 layer n max, element node i node j P P P f f f

17 Failure criterion i j i j N z Laminate x N x M x N y M y y M z z max, element node i node j P P P f f f max,, P P P node i layer 1 layer n max,, node j layer 1 layer n f f f P P P f f f Limit state function for every ply formulated by Tsai-Hahn failure criterion R1 g X g 0 fail, g 0 safe

18 Reliability methods: Monte Carlo Basic Variables (RVs) THIN analysis Output Variable (RV) E 1, E 2, v 12, G 12 X T, X C, Y T, Y C, S N x, N y, N z, M x, M y, M z, ε x, ε y, ε s Limit state function gxr 1 Random number generation for the basic variables THIN analysis Evaluation of limit state function Layer failure probability P f layer n P g 0 fails total Estimation of element failure probability 2,000,000 simulations n

19 Reliability methods: EDW Basic Variables (RVs) Output Variable (RV) E 1, E 2, v 12, G 12 X T, X C, Y T, Y C, S N x, N y, N z, M x, M y, M z, Limit state function gxr F g g g g 3 g g 2 4! 2 6! 2 3! M 2 M M 2 1 g g g g 1 g 2 2, i, 2 2, i 2 3, i,... 2 i1xi i1xi i1xi Xi P P g 0 f layer 2

20 Reliability methods: RSM/MC Basic Variables (RVs) Regression models Output Variable (RV) E 1, E 2, v 12, G 12 X T, X C, Y T, Y C, S N x, N y, N z, M x, M y, M z, ε x, ε y, ε s (top and bottom layers) Limit state function gxr 1 Random number generation for the basic variables Stress-strain analysis through regression models Evaluation of limit state function Layer failure probability P f layer n P g 0 fails total Estimation of element failure probability 2,000,000 simulations n

21 Reliability methods: RSM/MC Input Variables (RVs) Regression models Output Variable (RV) E 1, E 2, v 12, G 12 N x, N y, N z, M x, M y, M z, ε x, ε y, ε s (top and bottom layers) Limit state function gxr 1 Building regression models Design of Experiment 10 input variables 5 levels to be tested for every input variable (circumscribed CCD) 6 output parameters (strains at the lower face of the bottom and the upper face of the top ply of the laminate). 149 THIN analyses

22 Failure Probability (IFF): Section 9.2m 6 Element # Failure probability MC RSM EDW Layer # [45] [45] [45] [45] [45] [-45] [45] [45] [45] [45] [-45] [45] [-45] [45] [-45] [-45] [-45] # element Very good agreement between MC and RSM/MC. EDW is less accurate. (Correlation was not considered)

23 Failure Probability (IFF): Section 9.2m 6 Element # Failure probability MC RSM EDW Layer # [45] [45] [45] [45] [45] [-45] [45] [45] [45] [45] [-45] [45] [-45] [45] [-45] [-45] [-45] # element Reliability analysis 3h-MC, 30 min-rsm/mc, 10 sec-edw

24 Conclusions Assessment of the reliability level of a rotor blade already designed according to IEC ed. 3 at the ply level was performed The stochastic modeling of sectional stress resultants of the blade under extreme loading was achieved The probabilistic analysis was performed by using MC, RSM/MC, EDW A numerical tool was developed that can be combined with aero elastic codes and can be used for reliability analysis and probabilistic design