Probabilistic Fatigue Methodology and Wind Turbine Reliability_ThesisPhd_lange

c. 4 CONTRACTOR REPORT SAND96-1246 Unlimited Release UC–1211

Probabilistic Fatigue Methodology and Wind Turbine Reliability

Clifford H. Lange Civil Engineering Department Stanford University Stanford, CA 94305

Prepared by Sandia National Laboratories Albuquerque, New Mexico 87185 and Livermore, California 94550 for the United States Department of Energy under Contract DE-AC04-94AL85000

Printed May 1996

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Distribution Category UC-121 Unlimited Release Printed March 1996 PROBABILISTIC AND WIND

FATIGUE TURBINE

METHODOLOGY RELIABILITY

Clifford H. Lange Civil Engineering Department Stanford University Stanford, CA 94305 Sandia

Contract:

AC-9051

Abstract Wind turbines gyroscopic

subjected

to highly irregular

effects are especially vulnerable

study is to develop and illustrate of fatigue-sensitive estimates

wind turbine

fatigue reliability

oped. A FORM/SORM

eral techniques

analysis is used to compute

of load distributions

of relatively

least two moments

resistance

a new, four-moment

fatigue damage, statistical

and impor-

includes uncertainty

by matching Weibull”

methodology

wind turbine

damage.

Improved

bility is found to be notably

fits

High b values require bet-

still higher moments. model is introduced.

at

For this Load and

for design against fatigue is proposed wind turbines.

blade loads have been represented

load distribution

Sev-

Common one-parameter

and fatigue

using data from two horizontal-axis

Weibull”

has been devel-

models are shown to produce dramat-

moments across a range of wind conditions.

new “quadratic

(CYCLES) that

failure probabilities

Weibull model.

“generalized

factor design (LRFD)

and demonstrated

and design

large stress ranges; this is effectively done by matching

(Weibull) and better

and

of this

response, and local fatigue properties.

may be achieved with the two-parameter

purpose,

components

are shown to better study fatigue loads data.

estimates

analysis

program

The limit state equation

models, such as the Rayleigh and exponential

ter modeling

A computer

and mechanical

loading, gross structural

ically different

The objective

for the probabilistic

components.

tance factors of the random variables. in environmental

to fatigue damage.

methods

of structural

loadings due to wind, gravity,

To estimate

by their first three

Based on the moments PI.. .p3,

models are introduced.

The fatigue relia-

affected by the choice of load distribution

model.

Foreword This report comprises Department

the Ph.D. thesis dissertation

of Civil Engineering

advisor of this research

of Stanford University

at Stanford

reliability

2 describes a computer of structural

general reliability

upper-bound speed).

in March 1996. The principal Other

CYCLES,useful for estimating

components.

fatigue reliability

has been tailored

to the

thesis

and Paul S. Veers at Sandia.

for various applications,

of the more broadly distributed The FAROWprogram

program,

and mechanical

program

submitted

has been Steven R. Winterstein.

readers have been C. Allin Cornell at Stanford, Chapter

of the author,

This program,

the fatigue

developed

served as a preliminary program

to wind turbine

for wind turbines,

applications

The FAROWcode also has a more user-friendly

the two codes are nearly identical and the description

interface.

version FAROW.

by including

cutoff wind speed and a variable cycle rate (e.g., as a function

an

of wind

The capabilities

given in Chapter

as a

of

2 is applicable

to both the CYCLES and FAROWprograms. This project, Laboratories Structures

developed at Stanford University,

Wind Energy Program

Technology

was supported

Department,

of the Civil Engineering

and the Reliability

Department

ii

—

by Sandia National

at Stanford

of Marine

University.

Contents

Foreword

ii

1

1

Introduction l.lOrganization

2

3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CYCLES Fatigue

Reliability

CYCLES @erview

2.2

General Fatigue Formulation:

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

. . . . . . . . . . . . . . .

8

. . . . . . . . . . . . . . . . . . . . . . . .

8

Assumptions

2.2.1

Load Environment

2.2.2

Gross Response

. . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.2,3

Failure Measure

. . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.3

General Fatigue Formulation:

2.4

Analytical

2.5

Solution

Algorithm

2.6

Program

CYCLES capabilities

2.7

Example

Application:

Load

5

Formulation

2.1

3

Fatigue Formulation

2.7.1

Definition

2.7.2

Results:

2.7.3

Lognormal

Models

. . . . . . . . . . . . . . . . . .

10

. . . . . . . . . . . . . . . . . . . . .

11

Results

for Failure Probability

. . . . . . . . . . . . . . . . .

17

. . . . . . .

21

. . . . . . . . . . . . . . . . . . . . . . . .

21

. . . . . . . . . . . . . . ...’...

The Sandia 34-m Test Bed VAWT

of Input

Base Case.....

. . . . . . . . . . . . . . . . . . .

versus Weibull Distribution

for Fatigue

for S–N Parameter:

C

27 27 30

Reliability

3.1

Fatigue Data and Damage Densities

3.2

One and Two Parameter

3.3

Generalized

Four-Parameter

15

. . . . . . . . . . . . . . . . . . .

Load Models

. . . . . . . . . . . . . . . . .

Load Models . ..

111

. . . . . . . . . . . . . . .

31 34 38

CONTENTS

iv

4

Model versus Statistical

3.3.2

Underlying

Methodology

3.3.3

Generalized

Weibull Model for Fatigue Loads

. . . . . . . . .

42

3.3.4

Generalized

Gumbel and Generalized

Load Models

43

3.4

Fatigue Damage Estimates

3.5

Uncertainty

3.6

Summary

LRFD

Dueto

Uncertainty

. . . . . . . . . . . . . .

39

. . . . . . . . . . . . . . . . . . . . .

40

Gaussian

.

. . . . . . . . . . . . . . . . . . . . . . . .50

Limited Data..

. . . . . . . . . . . . . . . . . .

51

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...54 55

for Fatigue

4.1

Scope and Organization

4.2

Background:

4.3

5

3.3.1

. . . . . . . . . . . . . . . . . . . . . . . . .55

Probabilistic

Design

. . . . . . . . . . . . . . . . . . . .

4.2.1

Probabilistic

Design against

. . . . . . . . . . . . .

57

4,2.2

Probabilistic

Design against Fatigue . . . . . . . . . . . . . . .

57

Fatigue Load Modeling

Overloads.

57

. . . . . . . . . . . . . . . . . . . . . . . ...60

4.3.1

Fatigue Loads for Given Wind Climate

4.3.2

Fatigue Loads Across Wind Climates

4.4

LRFD Assumptions

4.5

Variations

4.6

LRFD Calculations

and Computational

. . . . . . . . . . . . .

60

. . . . . . . . . . . . . .

63

, . . . . . . . . .

69

, . . . . . . . . . . . . . . . . . . .

72

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

with Load Distribution

Procedure

4.6.1

Turbine

l Results

. . . . . . . . . . . . . . . . . . . . . . ...75

4.6.2

Turbine

2 Results

. . . . . . . . . . . . . . . . . . . . . . ...77

4.7

Effects of Limited Data . . . . . . . . . . . . . . . . . . . . . . . ...81

4.8

Summary

Summary

and

.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...86 88

Recommendations

5.1

Overview of Important

5.2

General Recommendations

of Future Work

5.3

Specific Recommendations

to Extend

5.4

Challenges

References

for Future

Conclusions

Study...

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Current

88 90

Work . . . . . . . . . .

91

. . . . . . . . . . . . . . . . . . . .

92 96

v

CONTENTS

A Statistical

Moment

Estimation

100

.

#

List of Tables . .

26

. . . . . . .

28

2.1

Sandia 34-m Test Bed VAWT, CYCLES Base Case Input Summary

2.2

Sandia 34-m Test Bed VAWT, CYCLES Reliability

2.3

Effect of S-N Intercept

4.1

Random

variables in reliability

4.2

Turbine

1 reliability

results; effect of load distribution

4.3

Turbine

1 reliability

results;

Distribution

Results

.

29

. . . . . . . . . . . . . . . .

70

Type with Reduced

analyses.

Uncertainty

type

. . . . . .

all results with same normalized

fatigue

load LnO~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4

4.5

Turbine

2 reliability

results;

all results with same normalized

72

74

fatigue

load LnO~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

Most Damaging

84

Wind Speeds (m/s) from Reliability

vi

Analyses

. . . .

List of Figures 2.1

FORM and SORM approximations

2.2

Flow chart, general CYCLEScode execution.

2.3

Flow chart, initial processing

2.4

Wind speed distributions Rayleigh distribution Measured

2.6

Effective stress amplitude

19

variables.

from Amarillo, Bushland,

RMS Stresses at the Blade Upper Root

. . . . . .

. . . . . . . . . . . . . . . .............”..””

3.2

Damage Density; Flapwise Data.

3.3

Exponential

3.4

Weibull Model of Flapwise Data.

3.5

Weibull and Generalized

3.6

Histogram;

3.7

Damage Density; Edgewise Data.

3.8

Weibull and Generalized

and the theoretical

. . . . . . . . . . .

“..

Flapwise Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 33 33

. . . . . . . . . . . . . . . . . . . .

36

data . . . . . . . .

42

. . . . . . . . . . . . . . . . . . . . . . . .

44

Weibull models; Flapwise

edgewise data

. . . . . . . . . . . . . . . . . . . .

44

Weibull Models; Edgewise Data (Ranges Plot-

Gumbel Distribution

of Generalized

tions for Annual Extreme

23

36

45

for Annual Extreme Wave Height–19

Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Comparison

23

. . . . . . . . . . .

and Rayleigh Models; Flapwise Data.

Shifted Axis, S-S~a~) . . . . . . . . . . . . . . . . . . . . . . . .

Generalized

20

versus cycles to failure for 6063 aluminum

Histogram;

3.9

. . . . . . . . . . . . . . .

of CYCLES random

3.1

tedon

16

with ~ = 6.3 m/s . . . . . . . . . . . . . . . . . .

2.5

alloy

. . . . . . . . . . . .

to g(U) z] = exp{–[-&]”’};

The parameter

jx(fc) =

‘QaZ x

(2.5)

exp[–(~)a’] x

~. in this result is related to the mean ~ as follows:

(2.6)

Resulting

~, a. = mean, Weibull shape parameter

parameters:

mental parameter

of environ-

X (wave height, wind speed, etc.).

o The RMS of the (global) stress process is assumed to be of the form

C7g(z)= a,ef(—

x

(2.7)

Z..f)p

i.e., increasing

7

in power-law fashion with the load variable x. The local stress

at the fatigue-sensitive K. The resulting

detail is further

scaled by a stress concentration

factor

RMS O(Z) is then finally

(2.8)

o(z) = K . Og(z). Resulting

Xr.f, O,.f, p, K = reference level of load variable,

parameters:

erence level of RMS stress, power-law exponent, . Given load environment Weibull distribution.

and stress concentration

X, the stress amplitude

From random vibration

S is also assumed

theory, the mean-square

E[S2] = 20(z)2,

ref-

factor. to have value

(2.9)

. .

ANALYTICAL

2.4.

is assumed

FATIGUE FORMULATION

13

with o-(z) from Eq. 2.8. The resulting

form given in Eq. 2.5, with shape parameter

density

~slx (sIz) is of the

CZ,, and scale parameter

(2.10)

/%= 0(4[2/(2/%)!]1/? Resulting

as = Weibull shape parameter

parameter:

Typical range: between a$=l

(exponential

of stress S given X.

stress distribution)

and a~=2 (Rayleigh

stress distribution). ●

The S–N curve is taken here as a straight intercept

Co that includes the Goodman

Nf(s) =

s

C(l – KISm\/su

)-’ =

line on log-log scale, with an effective correction

Cos-’;

co=C(I

in which S~ and SU are the mean and ultimate Resulting

parameters:

for mean stress effects:

-

(’2.11)

Kpm[/s.)b

stress levels.

C’, b = S–N curve parameters;

S~, Su = mean, ulti-

mate stress levels. Substituting

Eqs. 2.5–2,11 into Eqs. 2.1 and 2.3, the following expression

for fa-

tigue life Tf is found:

(2.12) Note that parameters tration

and the stress concen-

factor K, are raised to the power b arising from the S–N curve. In contrast,

parameters

scaling the environmental

the composite If p >1,

directly scaling stress, such as o,.f

variable X, such as its mean ~,

power bp, reflecting the combined nonlinear

this suggests that the uncertainty

have significant

effect on fatigue life.

are raised to

effect of Eqs. 2.7 and 2.11.

in these environmental

parameters

may

CHAPTER 2. CYCLES FATIGUE RELIABILITY

14

Finally, under the foregoing assumptions

the damage

FORMULATION

density ~(~)

from Eq. 2.2

may also be found: exp[–(~)”r] D(z) cx xbp+”’-l z By finding the maximum

of this function,

(2.13)

the environmental

level x~.x that produces

the largest damage is found to be

x max =pz(@+-yaz_

~

—

(l/az)!

CYZ

For example,

if X has exponential

distribution x maz

Thus, the most damaging

=

az=l,

~bP+%–jl/..

(2.14)

CYz

so that

bpx

(2.15)

X level depends not only its average value, ~, but also

on the exponents

b and p of the S–N curve and RMS stress relation. Note that xm~z — –20~ if b=10 and p=2. This may be a rather may far exceed the mean X; e.g., x~aZ—

extreme case, however. For example in a wind turbine ~Z=2 (Rayleigh distribution)

and p=l

application,

common values of

(linear increase in stress with X) would result

in xmZ between 2 and 3 times ~ for b values within a realistic range (4 < b < 13).

15

SOLUTION ALGORITHM FOR FAILURE PROBABILITY

2.5.

f

2.5

y

For the reliability

Solution Algorithm

the computed

for Failure Probability

analysis the failure criterion

is taken to be the difference between

fatigue life (eqn. 2.12) and a specified target

lifetime, Tt.

G(X) = 7“ – Tt

(2.16)

The vector X contains the resulting parameters from the analytical lation of Section

2.4; X=[~,

az, ~ref, aref, p, K, as, C, b,Sm,S., A,

2.16, known as the failure state function,

G(X),

is positive

fatigue formu-

fo,

A].

Equation

when the component

is

safe and negative when it has failed. The solution described thorough

for the failure probability

that

has been

in Veers, (1990) and is reviewed here briefly for completeness.

A more

description

is a four step procedure

of reliability methods can be found in several references (Madsen

et al, 1986, Ang and Tang, 1990, Thoft-Christensen

and Baker, 1982, and, Melchers,

1987). The first step of the solution equation

procedure

of the limit state

given above as Eq. 2.16. In the second step, the “transformation”

that each random variable be associated distributed

is the “formulation”

random

the cumulative normal variate.

variable.

distribution

with a uncorrelated,

For independent

variables

if only the marginal

coefficients among the Zi are known, transformation methods,

dard normal variable a typically

this is achieved by equating

can be included by working with conditional

(Madsen et al, 1986). Alternatively

. With conventional

unit variance, normally

functions of the input variable and its associated

Correlation

somewhat

greater

extent

than

methods

correlation

due to non-normal

standard

distributions

and correlation

may proceed in two steps:

~ variables

variables xi. Analytical “distortion”

distributions

each xi can be transformed

Vi. The resulting

requires

marginally

to a stan-

will also be correlated,

the original

have been developed physical

physical

to

(non-normal)

to efficiently predict this variables

(Winterstein

et

al, 1989). . Correlation

among the Vi’s may be removed by standard

methods

(e.g., Cholesky

CHAPTER 2. CYCLES FATIGUE

16

RELIABILITY

FORMULATION

P g(u)=o u,

Figure 2.1: FORM and SORM approximations

decomposition

of the covariance matrix)

This is the approach

to g(U) .— g 0.05 n

THE SANDIA 34-M TEST BED VAWT

23

Rayleigh Distribution — Amarillo Airport ----Bushland Test Site ----””-”-

..... ...” ,-..\\ ... ;’ ,’ \, :1

0.01

0 5

20

Wi#Speed,

Figure 2.4: Wind speed distributions from Amarillo, Rayleigh distribution with ~ = 6.3 m/s F

I

1

25

{%s) Bushland,

i

and the theoretical

v

I

14 28 rpm ---34 rpm — 38 rpm -----

12 10

./”. ./ ./ .. / .=.. /’ ..../. .,.’

8 6 4

M.

,,.

2 +

0

o

5

Figure 2.5: Measured

20

25

Wir#Speed, !%s) RMS Stresses at the Blade Upper Root

CHAPTER 2. CYCLES FATIGUE RELIABILITY

24

FORMULATION

the tower. The stress states for the upper root were predicted Sullivan,

1984), a Sandia written

assumes steady winds.

response

(Lobitz and

finite element

code that

Strain gages were employed to measure stress states at

the upper root location. speeds of operation

frequency

using FFEVD

Figure 2.5 shows the measured

RMS stresses for fixed

at 28, 34, and 38 rpm. The trend is seen to be nearly linear

for all three modes of operation. For the power-law relationship

used by CYCLES (Eq. 2.7), the following param-

eters are used to characterize

the response:

Xr,i = 10 m/s,

and p = 1. Only the reference RMS level Or,f is treated p, and the characteristic

while the exponent, constants.

crr,f = 4.5 MF’a,

as a random

wind speed, x,~f, are defined as

Because there is a great deal of data available a relatively

ation, COY = 0.05, was chosen for oref which is assumed normally In chapter

4 procedures

useful for determining

variable

uncertainty

small varidistributed.

measures from data

will be shown. The COV used here for oref is an assumed value believed to be representative

of the existing uncertainty.

The remaining

variables

associated

chine are the stress concentration Weibull stress amplitude been predicted

with the gross stress response

factor, K, and the shape factor, as, of the

distribution.

or measured

of the ma-

The stress concentration

with accuracy.

As an approximation

factor has not for the heavily

bolted joint a mean K of 3.5 with 10% GOV is used.

Histograms

counted stress time histories show very good agreement

with a Rayleigh distri-

bution (Veers, 1982). In Chapter axis wind turbines

(HAWT)

there is an abundance

distribution

3 we will see that typical data from horizontal

is more exponential

in nature.

of stress data and the observed

Weibull shape parameter

of rainflow

is set to the constant

Again,

because

fits are very good, the

value of 2.0 making the stress

Rayleigh.

-t

. S—iV Curve The blades and joints are made of 6063-T6 aluminum tensive fatigue test data are available.

extrusions

for which ex-

The test data are shown in Figure 2.6,

2.7.

EXAMPLE APPLICATION:

40000 g

25000

f

20000

LO 3 % 15000 : .= v :

.. *.>

.

‘. ‘.

30000

: =*

THE SANDIA 34-M TEST BED VAWT

‘a

‘.

“:’”’”’”‘... ‘.

‘.

k

@ “~ ““ “ “ “ & ‘.. ❑

Teledyne Engr. Southern Univ Runout Specimen -2 Stn Dev Least Square Fit +2 Stn Dev

-.. -.. ,

8000

,

10000

1

I

,

100000

, I

, 1

,

a commonly

tigue lives are considerably assessments

observed

extended.

, I

,

1 e+08

1 e+09

alloy

as used by CYCLES (see Eq. 2.11). “fatigue limit” below which the fa-

This effect is not included

with occasionally

case here for wind turbines)

n

versus cycles to failure for 6063 aluminum

to an effective stress amplitude

This data displays

,

1 e+06 1 e+07 Cycles to Failure

Figure 2.6: Effective stress amplitude

lative damage

❑

o A ---— ----

‘.-

10000

normalized

25

applied

may effectively eliminate

since cumu-

larger stresses

(as is the

the fatigue limit (Dowl-

ing, 1988). The higher stress peaks alter the way most materials

respond

result in greater rates of fatigue damage than would be concluded

based solely

on constant N exponent, MPa).

amplitude

The least squares

b = 7.3, with intercept

The distribution

distribution

results.

C = 5.0E+21

and

fit to the data gives an S– (based on stress units of

of the data about the least squares line fits a Weibull

with COV = 0.613.

The mean stress and ultimate

strength

amplitudes

rule.

using Goodman’s

the joint but may vary substantially

are needed to define the effective stress

The mean stress has been measured along the blade span, resulting

near

in high

CHAPTER 2. CYCLES FATIGUE RELIABILITY FORMULATION

26

Definition Mean Wind Speed Wind Shape Factor Ref Stress RMS exponent Stress Cone Stress Shape Fact S-n Coeff S-n Exponent Mean Stress Ultimate Strs Cycle Rate Miners Damage Availability Target Life

Symbol x

Variable 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Zf ; as c b s. s. f. A A Tt

Distribution Normal Normal Normal Normal Normal Normal Weibull Normal Normal Normal Normal Normal Normal Normal

Mean 6.3 m/s 2.0 4.5 MPa 1.0 3.5 2.0 5.0E+21 7.3 7.0 MPa 285 MPa 2.0 Hz 1.0 1.0 20.

Cov 0.05 0.1 0.05 0.1 0.61 0.2 0.2

Table 2.1: Sandia 34-m Test Bed VAWT, CYCLESBase Case Input Summary

uncertainty

for the actual local mean. The mean stress is defined to be normally

distributed

with mean 7.0 MPa and 20% COV. The ultimate

extruded

aluminum

material has been measured

stress for the

and is set to the constant

value

285 MPa.

The three remaining

parameters

used to calculate the average fatigue damage rate

are the average response cycle rate, ~o, the actual value of Miner’s damage at failure, A, and fraction of time the turbine is available, A. The cycle rate frequency has been measured

and found to be relatively

sample to sample.

f. is assumed

COV. For simplicity, summarizes

independent

normally

of wind speed but does vary from

distributed

with mean 2.0 (Hz) and 20?Z0

both A and A are set to unity with no variation.

all the parameters

used in this wind turbine

fatigue reliability

Table 2.1 example.

EXAMPLE APPLICATION:

2.7.

2.7.2

Results:

The output

Base

THE SANDIA 34-M TEST BED VAWT

Case

from the CYCLESanalysis for this example is given in Table 2.2 under the

column heading approximately

identified

as “Weibull”.

The results show a probability

analysis identifies the likelihood or probability

will achieve some desired lifetime. of each random

variable

Of equal importance

about half of the total variability

K, and wind speed shape factor,

and 1470 contribution provides valuable

respectively.

Results

is the most important in this example. av,

are next,

that the turbine

is the relative

on the fatigue life of the turbine.

leading coefficient in the S-N relationship

factor,

of failure of

3% for a target lifetime of 20 years with a median excess lifetime of 294

years. The reliability

supplying

27

importance

show that

the

source of variability

The stress concentration

having approximately

The relative importance

of each random

insight to the designer who is attempting

21%

variable

to reduce the effects of

fatigue damage.

2.7.3

Lognormal eter:

Finally, reliability

versus

Weibull

Distribution

for

S–N

Param-

C

in order to investigate analysis,

the effect of distribution

type on the results

the example problem run here was repeated

type for the S-N coefficient changed

to lognormal.

of the

with the distribution

Table 2.2 summarizes

how the

results vary between the two cases. As might be expected

the failure probability

to the shift in probabilities lognormal

distribution

corresponding

of the resistance

variable C towards larger values (e.g. the

has both a narrower lower tail and a fatter upper tail than the

Weibull distribution),

it is applied to a resistance favorable (non-failures). to the stress concentration

This model here predicts

greater

reliability,

as

variable (fatigue life at given S) for which large values are

Note also the shifting of importance

from the S–N intercept

factor, K, and wind speed shape factor, av, as the analysis

shifts from a blade with marginal having potentially

decreased to 1.5%. This result is due

resistance

much higher resistance

The influence of distribution

(e.g., a Weibull distribution (e.g., the lognormal

type on reliability

for C) to one

distribution

for C’).

is clearly case dependent

as the

CHAPTER 2. CYCLES FATIGUE RELIABILITY

28

Weibull 294 yrs 2.56 3.01 Importance 4.9 14.4 4.9 21.2 52.2 0.9 1.4 -

Excess life over 20 years: FORM pi (in %) SORM pf (in %) Definition Symbol x Mean Wind Speed Wind Shape Factor ;,:j Ref Stress RMS exponent 2 Stress Cone as Stress Shape Fact c’ S-n Coeff b S-n Exponent Mean Stress Sm s. Ultimate Strs Cycle Rate f. A Miners Damage A Availability Tt Target Life

FORMULATION

Lognormal 277 yrs 1.50 1.55 Factors: (in?%) 7.9 27.6 7.9 32.6 20.3 1.6 2.2

Table 2.2: Sandia 34-m Test Bed VAWT, CYCLES Reliability following example demonstrates. testing was performed

Results

In this second example, it is assumed that sufficient

to reduce the uncertainty

in the previous example for the stress

factor, K, and wind speed shape factor, av. Both (20V’S are now taken

concentration

to be 5%. A new reliability

analysis shows that the S–N intercept,

distribution,

dominates

the relative

importance

importance.

Results for this hypothetical

of all random

G’, with a Weibull variables,

case and one with a lognormal

e.g.

79%

distribution

for C are shown in Table 2.3. Note that

when the overall uncertainty

failure probability of importance more dramatic.

has also been reduced.

away from the resistance This is expected,

C’ in the reliability The caution

in the problem Furthermore,

has been reduced,

the

with a lower pf the shifting

variable to those constituting

the load is much

due to the large role played by the random variable

calculations.

here is two-fold.

First, the choice of distribution

model for a critical

,.

2.7.

EXAMPLE APPLICATION:

THE SAiVDIA 34-M TEST BED VAWT

Excess life over 20 years: FORM pf (in %) SORM .,, w{ (in %), Definition Symbol Mean Wind Speed x Wind Shape Factor Ref Stress ;,;j RMS exponent Stress Cone i Stress Shape Fact as S-n Coeff c S-n Exponent b Mean Stress Sm Ultimate Strs s. Cycle Rate f. Miners Damage A Availability A Target Life Tt

Table 2.3: Effect of S-N Intercept

LoKnormal $77 yrs 0.17 0.16 Factors: (in%) 14.2 9.4 14.2

Weibull I 294 yrs 1.29 1.37 Importance 5.0 2.8 5.0 6.0

17.3

79.0

38.4

0.8

2.6

1.4

3.8

Distribution

Type with Reduced

random variable, when fit to the same mean and variance, bility estimates

by one or more orders of magnitude.

model has been widely used in these fatigue strength be considerably moment

unconservative.

information

a lognormal

suggests

plausible

range of reliability

can change failure proba-

Second, although

the lognormal

and S–N formulations,

it is perhaps

and Weibull models, and estimate

experience

Uncertainty

it may

At the least, if one knows nothing more than second-

(mean and variance),

Practical

29

that

prudent

the reliability

these two models provide

index, in view of distribution

to at least fit both

under each assumption. useful bounds

model uncertainty.

on the

Chapter

3

Load Models The fatigue reliability bility distribution, operating

life.

for Fatigue Reliability

of wind turbines

depends

on the relative frequency,

of various cyclic load levels to be encountered These are typically

required

or proba-

during the turbine’s

across a range of representative

wind

conditions. For time scales of the order of minutes, these loads may be measured short-term

experimental

Practical

ques-

tions then arise as to how these limited data should best be used (e.g., Jackson,

1992;

Sutherland,

1993; Sutherland

seeking to estimate histogram

by analytical

and Butterfield,

a representative

sufficiently

1994; Thresher

theoretical

If a smooth distribution

of fatigue life, what is the uncertainty seeks to address

et al, 1991). First,

probability

distribution

is to be fit, what functional

flexible and how should it be fit? Finally,

This chapter

methods.

in

fatigue life, is it sufficient to use the observed

of cyclic loads, or should a smooth

fit to the limited data?

estimate

studies, or predicted

by relatively

form is

beyond forming a single best

in this estimate

these concerns.

be

Subsequent

due to limited data? sections

address

the

following points in turn: 1.

Fatigue

Data

and

Damage

Densities.

proaches to study fatigue load data and modeling constructed, dition, standard

We first show several useful apneeds.

Damage

to suggest which stress ranges are most important

Weibull scale plots are used to show systematic Weibull models.

While not completely 30

deviations

density plots are to model.

In ad-

from a range of

new, these approaches

have yet to

FATIGUE DATA AND DAMAGE DENSITIES

3.1.

gain widespread 2. Model

use among wind turbine Uncertainty

to a particular predict.

Effects.

scatter

such as the Rayleigh

the resulting

scatter

is found among common

and exponential

range or RMS level. This motivates or more parameters

load modelers. We fit a number of conventional

data set, and determine

Considerable

31

load models

in fatigue damage they

one-parameter

load models,

models, when they are fit to the mean stress the need for more general load models, with two

fit to the data.

3. New Statistical

Loads

We introduce

Models.

model, which preserves the first four statistical these higher, more tail-sensitive

here a new generalized

moments

of the data.

load

By retaining

moments, it is more faithful to the observed frequency

of relatively

large load levels. At the same time, it is a rather mild perturbation

conventional

2-parameter

load model, e.g., Weibull, Gumbel, or Gaussian.

of a

Of primary

interest here is the generalized Weibull model which has found favor in various fatigue applications.

Other

applications

values, and a generalized

include

Gaussian

a generalized

Gumbel

model useful for analyzing

model for extreme nonlinear

vibration

problems. 4. Uncertainty of limited

data.

fatigue damage

due

to Limited

Techniques

are shown to estimate Acceptable

estimates.

Finally, we discuss the implications

Data.

the associated

levels of this uncertainty

uncertainty

are also discussed,

together with the data needs these imply. The result is strongly dependent properties;

in

on material

e.g., the slope of the S–N curve that governs fatigue behavior.

Our application axis wind turbine

here concerns both flapwise and edgewise loads on a horizontal (HAWT).

A companion

plies similar models to estimate

study

(Sutherland

and Veers, 1995) ap-

loads on the Sandia 34-m vertical axis wind turbine

(VAWT).

3.1

Fatigue Data and Damage Densities

In 1989 an extensive Altamont

data set was obtained

Pass, California

by Northern

upwind HAWT with a teetering

for a 100-kW wind turbine

Power Systems.

This turbine

operated

at

is a two-bladed,

hub design utilizing full-span hydraulic

passive pitch

CHAPTER 3. LOAD MODELS FOR FATIGUE RELIABILITY

32

control.

The fiberglass

rotor blades span 17.8 meters

bending moment in both the flapwise (out-of-plane) were measured

in various wind conditions

(Coleman

We consider first the flapwise moment. counted

ranges, taken from a 71-minute

of 10 m/s. implying

Various studies an exponential

a log-linear

Note, in particular,

directions

and McNiff, 1989). of rainflow

adequately

captures

fit on this semi-log scale,

We may question, the frequency

however,

whether

of rare, large stresses.

the single extreme stress range of around 32 [kN-m], which pairs in the history.

One may ask how much impact

the concept

of the damage density. Fatigue

rare loads have.

are typically

constant

and edgewise (in-plane)

a straight-line

model.

We address this issue here through tests

The root

history during an average inflow wind speed

probability

the largest peak and smallest trough such extreme,

diameter).

Figure 3.1 shows the histogram

have suggested

extrapolation

(rotor

used to estimate

stress amplitude

N(S),

the number

of cycles to failure

under

S. We adopt here a common power-law form of IV(S):

N(s) = + Here c and b are material properties

(3.1)

where c = I/C’, C used in the earlier definition

of the S–N Law, Eq. 2.11, from Chapter As shown in Figure 3.1, actual

‘

2.

load histories

produce

a number

of load cycles,

n(Sz), at various stress levels Sz. For each stress level, Miner’s rule assigns fatigue damage TL(S2) D(si) = jqzjj “– cSjn(Si) The latter estimated

as DtOt=~i

In fitting incurred

form uses the S–N relation D(Si),

loads models,

(3.2)

from Eq. 3.1. The total damage

is then

the sum across all stress levels S2. it is useful to focus on the relative fraction

at different levels. This is the damage density, herein denoted D(SZ) S)n(Si) d(s~) = ~i D(SZ) = xi $n(si)

of damage

d(Si):

(3.3)

..

3.1.

FATIGUE DATA AND DAMAGE DENSITIES

33

1000 . 100

10

1 5

0

I

10

Bending

15

20

Moment

Figure 3.1: Histogram;

25

30

35

Range [kN-m]

Flapwise Data.

nA U.*

0.35 * .— : n a

m

‘

~-----

‘

‘

Ib=7— Ib = 2 ““-” -----

0.3 0.25

0.2

?! 0.15 2 0.1

...: ., : ;.,..:: .. ;;:, :: ::!-: :: :::;.,

,..,...:: ;., ;;;;;::: :, :., ....;:::; ;; :::.. . . :: ::,.. ;~y:. ., ::.

0.05

m...!

0 05

10 15 20 25 30 Bending Moment Range [kN-m]

l?i~w-~3.2: Damage Density; Flapwise Data.

35

CHAPTER 3. LOAD MODELS FOR FATIGUE RELIABILITY

34

Thus, the relative damage is independent

of the intercept

c of the S-N

curve, but

on its slope b.

depends importantly

Figure 3.2 shows the damage densities of the flapwise loading for fatigue exponents b=2 and b=7. Fatigue exponents

exponents

below 5 are typical

of welded steel details,

of 6 or above have been found from coupon tests of aluminums

wind turbine

blades (VanDenAvyle

typical fiberglass blade materials

and Sutherland,

have suggested

1987).

while used in

Some fatigue studies of

even higher exponents;

e.g., b values

of 10 or above (Mandell et al, 1993). Figure 3.2 shows that the most damaging the fatigue exponent,

with

b. For welded steels, loads that lie within the body of the data In contrast,

(3-6 [kN-m]) are most damaging. largest cycle contributes

for aluminums

over 37% of the damage.

given limited data.

This effect is quantified

general, however, any loads model—whether

with b=7, the single

For composites

This leads to increasing

may give still larger contribution. life estimates

stress level changes dramatically

this single cycle

uncertainty

in subsequent

observed or fitted-should

on fatigue sections.

In

be used with

care if the largest observed load drives fatigue damage.

3.2

One and Two Parameter

As noted above, a common

probability

the exponential

model.

reflecting alternative

probability

the slope of the expected one-parameter

Load Models

model suggested

This model has a single parameter, histogram

on semi-log scale (Figure

model, based on random vibration

is the Rayleigh distribution.

Here we investigate

more general, two-parameter

Weibull distribution.

function

F(s)

essentially 3.1).

An

theory of linear systems,

the adequacy

For a general Weibull load model with parameters tribution

for HAWT blade loads is

of these through

the

Q and ~, the cumulative

dis-

is given by

F(s) = P[ Load This model includes the exponential

< s] = 1 – e-(’lp)a

(3.4)

and Rayleigh as special cases, corresponding

3.2.

ONE AND TWO PARAMETER LOAD MODELS

to a=l

and c2=2 respectively.

Note that

F(s)

35

is the cumulative

probability

loads less than specified s. It is also useful to ask the inverse question: SP has specified cumulative

probability

F’(s)=p.

of all

what fractile

Setting the left-side of Eq. 3.4 to p,

solving for s yields

SP = @[– In(l – p)]l/a To determine

the adequacy

the data on an appropriate constructed

graph between cumulative variable.

of a model such as Eq. 3.4, it is convenient

“probability

for any distribution

(3.5)

scale” plot.

by transforming

probabilities

In general,

to display

probability

scale is

one or both axes to obtain a linear

and the corresponding

values of the physical

For the specific case of Eq. 3.5, a linear result arises from taking logarithms:

ln(s,) = ln(~) + ~ ln[– ln(l – P)] Thus, the observed

(3.6)

load values are first sorted into ascending

order (.sI < Sz s

., .SN). We then plot ln(sz) versus ln[– In(l – pi)] on linear scale, with pi=i/(N Equivalently, chosen here. Rayleigh

+ 1).

we may plot Sz versus – ln(l –pi) on log-log scale; this is the alternative The result

and exponential

is a linear plot in any Weibull as important

special cases.

case, including

It also includes

both the

a still wider

range of models, with the slope of the line directly showing whether the model should be Rayleigh,

exponential,

or something

between (or outside).

Figure 3.3 shows the flapwise data from Figure 3.1 on this “Weibull scale.” This plot also shows corresponding the mean of the data.

Observe that

it is fairly easy to discriminate exponential

exponential

and Rayleigh distributions,

with this graphical

which preserve

“goodness-of-fit”

between the two distributions,

method

and confirm that the

model follows the data far better than the Rayleigh model.

Figure 3.4 repeats the flapwise data on Weibull scale, now plotted with the “best” Weibull distribution.

The two parameters

match both the mean and standard ently better

of this Weibull model have been chosen to

deviation

of the data.

fit, we will consider below whether

While providing

the difference between

model (Figure 3.3) and Weibull model (Figure 3.4) is statistically

an apparexponential

significant.

In the

CHAPTER 3. LOAD MODELS FOR FATIGUE RELIABILITY

.99999

d

I

I

I

I

..

Data Exponential -e--” Rayleigh

I

1

,.. ,..

0

J

/

/-

B’

,,,”

ji’ G .40 I

.5

10 2030 5 2 Bending Moment Range, [kN-m] 1

Figure 3.3: Exponential

1’

and Rayleigh Models; Flapwise Data.

I

I

I

I

I

I )

.99999 Weibull

& : .5 = ~

-A---

.95 .90 .80

.60

(5 .40 .5

1

2

5

10

20 30

Bending Moment Range, [kN-m] Figure 3.4: Weibull Model of Flapwise Data.

I

3.2.

>

next section, we will also consider still more general models. show the advantage

*

.

*

37

ONE AND TWO PARAMETER LOAD MODELS

discriminate

of using probability

between different models. ~

In general, these figures

scale instead of the traditional

histogram

to

CHAPTER 3. LOAD MODELS FOR FATIGUE RELIABILITY

38

3.3

Generalized Four-Parameter

By using data to fit two parameters—both

a as well as /3 in Eq. 3.4—it is not surpris-

ing that the Weibull fit seems visually superior Rayleigh models. ultimately

leading to seemingly

more parameters

uncertainty

perfect agreement

grows.

The practical

of its uncertainty,

Comparing

as the number

our uncertainty

or confidence interval)

more parameters

of parameters

in estimating

ap-

damage estimate

each

by the resulting

on our mean damage

to a probabilistic

model is no

does not vary significantly,

in view

from that given by a simpler model.

Figures 3.3 and 3.4, we may ask whether

and Weibull models is statistically

significant.

Figure 3.13) that at least for high exponents and exponential

and

still more parameters,

effect of this should be measured

adding

longer beneficial if the resulting

ponential

of data,

(e.g., coefficient of variation In general,

exponential

The tradeoff, of course, is that as one seeks to estimate

from a fixed amount

rate estimate.

to the l-parameter

In the same way, one may seek to introduce

proaches the number of data.

parameter

Load Models

models are statistically

ing a two-parameter

Weibull model.

significant.

These generalized

“generalized

4-parameter

ex-

We will show below (e.g., differences between

This supports

To test in turn whether

sufficient, we require a still more detailed provided by the four-moment,

b damage

the difference between

Weibull

the effort of seek-

the Weibull model is

model with which to compare

it. This is

Weibull” model defined below.

load models are perturbations

of 2-parameter

“par-

ent” models that are based on fitting not only to the mean p and and standard

devi-

ation a, but also the skewness a3 and kurtosis a4 of the data.

(For a general random

variable X, an is defined as the average value of ((X —p)/o)n. ) This section contains applications

for not only a generalized

and generalized application.

Gaussian

The emphasis

Weibull model.

Weibull model but also a generalized

Gumbel

model as well. Each model is shown useful for a specific however is on fatigue applications

using the generalized

GENERALIZED FOUR-PARAMETER

3.3.

3.3.1

Model

Generalized commonly

versus

Statistical

four-parameter

In particular,

cubic distortions

The problem

of the data.

estimates

are more difficult to estimate

one is led to try

as well. This will help to discriminate

between various

uncertainties.

to rare extreme

outcomes,

and hence

This is known as statistical

data set.

of our data set.

model reflects an attempt Practical

experience

at balance between

(e.g., Winterstein,

are often sufficient to define upper distribution

This experience

It is again supported

heights are insensitive

The benefit does not come without cost,

from a limited

Thus, our search for an “optimal”

here.

This scatter

which is difficult to quantify,

uncertainty, which reflects the limitations

the range of interest.

only one or two moments.

by model uncertainty. This is prevalent

are more sensitive

suggests that four moments

not two, three, five,

models, with very different tail behavior

models, and hence reduce model uncertainty.

model and statistical

of X—and

models,

To avoid this model uncertainty, to preserve higher moments

are sought

We may then ask why precisely four

is said to be produced

higher moments

distributions

can be fit to the same first two moments.

in low-order, one- or two-moment

however:

“parent”

distribution

is that a number of plausible

to modify standard,

match observed tail behavior.

models are of lower order, requiring

and hence fatigue reliability, in reliability

to better

of these standard

are used to fit the probability

ten, etc. Conventional

have been developed

distributions

to match the first four moments moments

Uncertainty

distributions

used two-parameter

39

LOAD MODELS

motivates

the generalized

1988)

tails over

models developed

by the results of Section 3.3.4, in which extreme

to the choice of parent

distribution,

wave

once four moments

have

been specified. This issue of statistical in Appendix of a data

uncertainty

A. Methodology set, especially

Of particular

interest

bias into estimated

with moment

useful for estimating

when the number

are cases with data

values of normalized

is discussed

the first four statistical

of data limitations

moments.

estimates

points that

moments

is limited,

is outlined.

introduce

considerable

The generalized

Gumbel example

(Section 3.3.4) is one such case and the effects of bias as well as corrective to compensate

for it are presented.

further

measures

CHAPTER 3. LOAD MODELS FOR FATIGUE RELIABILITY

40

3.3.2

Underlying

Development

Methodology

of a four-parameter

parameter

“parent”

Gumbel,

and Gaussian

distribution

model begins with a theoretical,

Implementation

distribution.

parent distributions.

has been achieved

Denoting

variable X is related to U through

physical random

for Weibull,

the parent variable as U, the

a cubic transformation:

x = co + c~u + CJJ2+ CJ73 An automated

optimization

developed

X.

Such a routine, at Stanford

(3.7)

routine then adjusts the coefficients c. to minimize the

difference between the measured variable

skewness and kurtosis

FITTING (Winterstein

University.

we require C3 > 0. The positive

and those of the generalized

et al, 1994) has been recently

Note also that

for Eq. 3.7 to remain

cubic term implies that X eventually

tails than U. When X has narrower tails than the parent distribution, automatically

fits the alternate

tribution

has broader the program

(3.8)

Cjxs

cubic coefficient c! plays the opposite

role, expanding

“

the dis-

of the physical variable X to recover the parent model.

This switching matically

monotone

relation:

u = c; + C;x + 4X2+ Here the positive

two-

between

two dual models,

within the FITTING program.

based on the size of Q4, occurs auto-

Adding such a dual model has been found to

greatly increase modeling flexibility for small kurtosis

cases. These have been found

to arise both in extreme and fatigue loading applications. Finally, in whichever

form the model is defined, the coefficients

c. are chosen to

minimize the error c, defined as

6 =

The speed of executing tion; i.e., by the amount small tolerance,

6tO~.

/(a,

-

a3~)’

+

(0!.

–

cl!,~)2

(3.9)

FITTING is governed largely by the speed of this optimizaof effort (trial Cn values) needed to achieve an acceptably

---

3.3.

GENERALIZED FOUR-PARAMETER

The FITTING report (Winterstein routine

documentation.

probabilistic

The basic goal of these generalized

etc.)—and

extreme values through

Numerical Another

Four-Moment

among

load models is to reflect

a cubic distortion

their higher statistical

distinction

program

Implementation

is numerical

sum of squared timization,

to better

reflect rare

Models.

four-moment

models

concerns

of Eqs. 3.7–3.8 as described

whether

their coeffi-

or from some numerabove for the FITTING

where the coefficients c. or c: are found by minimizing

errors in skewness and kurtosis.

requiring

model

moments.

cients (e.g., Cn in Eq. 3.7 or c: in Eq. 3.8) are found analytically ical algorithm.

details and sub-

the choice of basic two-parameter

then introduce

vs Analytical

41

et al, 1994) supplies additional

engineering judgment—through

(Weibull, Gumbel,

LOAD MODELS

62, the

This is done with constrained

Eq. 3.7 or 3.8 to remain monotone

op-

and often achieves perfect

moment fits; i.e., 62=0. Although four-moment

the numerical

approach

is not computationally

models have been pursued

in the case where U is standard

(Winterstein

Gaussian

burdensome,

analytical

and Lange, 1995), particularly

and a4x > a4u = 3. Here it is useful to

rewrite Eq. 3.7 in terms of Hermite polynomials: X = mx + KOXIU + c~(U2 – 1) + CA(U3– 3U)];

~ = (1 + 2c~ + 6c~)-112

Results for c~ and C4 have been found to make 62 vanish to first-order 1985) and second-order expressions

(Winterstein,

1.43& c’=

solutions

optimization:

c13x 1 – .o151a3xl + .3c& — 1 1 + o.2(c14x – 3) 6[ l–f).la~~

.

c40[1 - (Q!’x - 3)1

The above expressions

(Winterstein,

1988) in Cn. The most recent (and accurate)

have been fit to “exact” results from constrained c3.

C40 =

‘

were produced

for a range of requested

(3.10)

[1+

l.%(qx

(3.11)

-

3)]1/3 -1

10 using FITTING to generate

skewness

and kurtosis.

a matrix

The results

(3.12) of exact were then

CHAPTER 3. LOAD MODELS FOR FATIGUE RELIABILITY

42

I

1’ * ~

I

2030 10 5 2 Bending Moment Range, [kN-m] 1

curvefit to produce expressions

Generalized

Weibull

Perhaps

support

Somewhat

data

Fatigue

Loads

model of the NPS flapwise data, to the 2-parameter

fit to its Weibull

or visual fit of a cubic model on this scale, this 4-

Thus this single largest

match the cumulative

stress level; while visually

to affect even a four-moment

the general adequacy

for

most notable is its similarity

Unlike a least-squares

sufficient impact

Model

Weibull

moment fit does not bend to better level.

Weibull models; Flapwise

for C3 and C4 to be used in Eq. 3.10.

3.5 shows a generalized

first four moments. model.

1 1

.999

Figure 3.5: Weibull and Generalized

Figure

I

.99999

.5

3.3.3

I

I

probability

at the highest

striking,

does not have

fit to the data.

The net result is to

of the Weibull model for this flapwise case.

different findings arise for the edgewise bending

component,

however.

GENERALIZED FOUR-PARAMETER

3.3.

Figure

3.6 shows the histogram

minute duration. amplitude

LOAD MODELS

of the edgewise bending

This bimodal histogram

history

load frequency

oscillations

superposed.

distribution

model, Weibull or other, can describe this entire

the small-amplitude

load cycles need not be modeled for fatigue ap-

pattern.

Fortunately, plications.

over the same 71-

shows the presence of fairly regular, large-

stress cycles due to gravity, with small-amplitude

Clearly, no single-mode

43

This is seen in Figure 3.7, which shows that negligible fatigue damage is

caused by cyclic loads below about S~an=l 2 [kN-m]. Therefore,

we consider models

of load ranges above S~i. only. Above this value, Figure 3.8 shows that the generalized Weibull model gives quite a reasonable the data.

It captures

characteristic significant

3.3.4

the curvature

of the observed

trends

in

seen in the body of the data and the flattening

The generalized

of its upper tail.

improvement

extrapolation

over the ordinary

Weibull

model appears

to offer a

Weibull result in this case.

Generalized Gumbel and Generalized Gaussian Load Models

The primary Weibull

motivation

model for fatigue

ent distributions demonstrate certainty

for a four-parameter

load model

Generalized

applications.

in normalized

uses and reiterate

moment

estimates

load models with other

They are presented

are useful in other applications.

these alternative

here is the generalized

the important

par-

here to both

issue of statistical

and its significance

un-

to the four-moment

models.

Extreme

Values:

A Generalized

Based on probabilistic bution

is the natural

example considers

engineering

Gumbel

judgement

choice for modeling

annual maximum

Example the two-parameter

extreme

significant

value problems.

Gumbel Although

wave heights, this generalized

distrithis Gum-

bel model may also prove useful for extreme wind loads. This application Sea location,

concerns

the significant

for which 19 years of hindcast

wave height data

H. in a Southern

are available

(Danish

North

Hydraulic

44

CHAPTER 3. LOAD MODELS FOR FATIGUE RELIABILITY

I

I

I

1

I

1

4

i

,

1000

rhl

1 100

110

1

n-Ill 5

10

15

20

25

30

35

Bending Moment Range [kN-m] Figure 3.6: Histogram;

edgewise data

0.2 ,..,.-: ;; : ~,-. ::!; :;, ::;: ::.

0.15

:

.::.

b=7—

0.1

0.05

0 0

5 10 15 20 25 30 Bending Moment Range [kN-m]

Figure 3.7: Damage Density; Edgewise Data.

35

GENERALIZED FOUR-PARAMETER

3.3.

I .99999 .2 = Q j!j u

.999 .99 I .95

I

LOAD MODELS

I

I

I

45

I

/

Generalized Weibull ----Weibull ~ Data — /1

,. . .

--

.80 .60 .40 .20

.10 1 2 10 5 Bending Moment Range, [kN-m] Figure 3.8: Weibull and Generalized on Shifted Axis, S-S~zn).

Institute,

Weibull Models; Edgewise Data (Ranges Plotted

1989). For each of these 19 years, a single storm event has been identified

with maximum

significant

wave height H. (i.e. the annuaI maximum

value ranges from H. = 6.92m (1972/1973) The generalized

Gumbel distribution,

fairly well the systematic

curvature

As there are only 19 data

estimated

moments

and quantifies

is of concern.

This

plotted in Figure 3.9 along with the observed

of the data points,

It appears

on the Gumbel

the statistical

Appendix

A examines

to capture

probability

scale

uncertainty or bias in the the magnitude

of this bias

its significance.

It should be noted that an analytically

based generalized

viously been fit to this data set (Winterstein here are an improvement formation,

values).

to 9.66m (1981/1982).

data values, is of the inverse cubic form given by Eq. 3.8.

used.

20

Gumbel model has pre-

and Haver, 1991).

The results shown

in two senses: (1) FITTING includes an inverse cubic trans-

which is particularly

important

in reflecting

the narrower-than-Gumbel

CHAPTER 3. LOAD MODELS FOR FATIGUE RELIABILITY

46

.999 [ .998

GeneralizedGumbel Model of Annual SignificantWave Height I , I , I

.995

.30 .10

e

.001

11

10 9 7 8 Annual SignificantWave Heigh~ Hs [m]

6

Figure 3.9: Generalized Data.

Gumbel

Distribution

for Annual

Extreme

Wave Height-19

tails; and (2) FITTING permits greater accuracy to be achieved in matching

moments.

Because we deal here with annual extreme values, the Gumbel distribution natural

choice of parent

distribution.

We may ask, however, what effect is achieved

if a different choice of parent distribution The three distributions

of the underlying

upper tail and hence predicts the largest.

is selected.

are shown in Figure 3.10, The figure shows wave height

results up to the 1000-year fractile, follows that

is the

i.e. for which p= .999. The pattern

parent

distributions:

of variation

the Weibull has the narrowest

the lowest extreme values, while the Gumbel

Most notably, however, all three parent distributions

predicts

predict quite similar

wave heights over this domain of interest. This suggests behavior couraging,

that

of interest. particularly

knowledge

This apparent

of four moments robustness

is sufficient

to control

of the four-moment

in cases where the optimal parent distribution

description

the tail is en-

is not obvious.

3.3.

GENERALIZED FOUR-PARAMETER

.999 ~ .998

*

47

.. ;/ /1

GeneralizedGumbel Model of Annual SignificantWave Height I r

GeneralizedGumbel GeneralizedGaussian GeneralizedWeibull Wave Height Data

.995 .99[ .98 .95 .90

LOAD MODELS

1

;’

/

1

I

:: ;/ ;:~~ ;’ !#J

— ---------= ~

L

.80 .70 .50 .30 .10

(U-)1 .-y -

9 10 7 8 Annual SignificantWave Height, Hs [m]

6

Figure 3.10: Comparison of Generalized for Annual Extreme Wave Height.

Of course this conclusion

Vibration:

for the problem

A Generalized

cations, Adopting

loads.

Such non-Gaussian

due to nonlinear

relations

a simple lDOF

the user is encouraged

to vary

at hand.

Gaussian

As a final example, we consider the vibration non-Gaussian

Gumbel, and Weibull Distributions

may be problem-dependent;

the choice of parent distribution

Nonlinear

Gaussian,

11

Example

response of a linear structure

under

loading may arise, in wind and wave appli-

between wind/wave

velocities

and applied forces.

model of the response X,

x(t) + WJnx(t) + (d;x(t) in terms of the zero-mean

Gaussian

= Y(t)2

process Y(t), and the system natural

(3.13) frequency

CHAPTER 3. LOAD MODELS FOR FATIG UE RELIABILITY

48

Un and damping aratnam,

ratio ~. We consider

1987), for which Un=l.26

and Y(t+~)

is exp(–O.12]~1).

moment

their respective

relations.

response

whose moments

This yields a~X=2.7

As expected,

fractiles

ized Gaussian

and Ari-

and cr~x = 14.3, quite far from

ratio .— ~

#

.10

.*’ .01 ,+’ ‘, z .001 s“ /“ 6.00001 ,, i,/,,,,,,,:

Response Data Entropy Model Gaussian Model Cubic Gaussian

-4-202468101214 Standardized Figure 3.11: Oscillator

Response;

— -*--+------

Response 30% Damping.

>.99999 .— = .999 a .99 2

0

.90

t g

.50

.= ~

.10

z

,

.01 .001

Gaussian Model -+-Cubic Gaussian -----

= 0.00001

-4-202468101214 Standardized Figure 3.12: Oscillator

Response;

Response 10% Damping.

CHAPTER 3. LOAD MODELS FOR FATIGUE RELIABILITY

50

3.4

Fatigue Damage Estimates

Ultimately,

the impact

and adequacy

the context

of the application

of any probabilistic

at hand.

ranges Sa are to be used to estimate

model must be viewed in

In our case, probabilistic

models of stress

the total fatigue damage:

DtOt= c ~ S~n(Si) = cNiot~ z Thus, our interest focuses on estimating

(3.15)

the normalized

damage per cycle ~,

the

long-run average value of S6 over all stress cycles. Figure 3.13 shows estimates ted probabilistic

models.

of ~

found from the data, and from the various fit-

The generalized

The basic Weibull model also shows good

over the entire range of b values shown. agreement,

with mild departure

Weibull model follows the data quite well

for exponents

above around

pected from Figure 3.3, the Rayleigh model is extremely surprisingly, to potentially

the visually plausible underestimate

b=l O. As might be ex-

inaccurate.

exponent ial model (Figures

Somewhat

more

3.1 and 3.3) appears

damage, by about an order of magnitude

for b z 7 and

still more for higher b values. Note that we need not expect more damage when the observed is “filled in” with a continuous

probability

damage, D~~= ~, is essentially

a sample moment

an unbiased

estimate

long-run probability

model.

distribution

Indeed the observed

normalized

(of order b), and as such is always

of the true long-run value, Dtr.,=f

Sbj(s)ds. (Here ~(s) is the

density of stress S.) A caveat is in order here: in practice

may vary rat her asymmetrically

around

tail

DtTUe. Consider,

for example,

Dob~

an extreme

case in which damage is governed by a single stress level bin, whose mean recurrence rate is one per 10 histories.

(Multiple

If we collect many such histories, critical

bin—and

the remaining

occurrences

per history

are thus negligible.)

on average 90V0 will show no occurrences

hence will give mildly non-conservative

10?ZOshow one occurrence,

damage

and hence strongly

in this

estimates—while

overestimate

Thus, while the correct average is achieved, the chance of (mild) non-conservatism typically

higher than that of (stronger)

By fitting

a smooth,

continuous

.,

damage. is

conservatism.

probabilistic

model to all stress data,

we hope

3.5.

UNCERTAINTY DUE TO LIMITED DATA

10

,

I

I

I

I

I

i

I

I

Flapwise Data Weibull Generalized Weibuil Exponential Rayleigh

0.1

...,\ 111! 111 +..

‘\

51

I

I

I

I

— -*--*- -~ .A-----x-#’@

I

1

1

I

I

/

/{ > ,$3” x

1

I

I

1 2 3 4 5 6 7 8 9101112131415 Fatigue Exponent: b Figure 3.13: Normalized

to avoid this extreme sensitivity metrically)

of observed damage,

critical to well-represent

High b values require better

effectively done by matching moments

(generalized

Uncertainty

Finally,

we consider

history

of 71 minutes.

the most damaging

of relatively

at least two moments

stress lev-

large stress ranges;

(Weibull) and still better

Due to Limited

the impact

of uncertainty

this is by four

variance among segment damages,

Data

in damage

To do this we divide this history

segments each with IV/4 data—and

variance

modeling

In fitting this continuous

Weibull).

3.5

damage

Dob~,to the widely (and asym-

varying tails of the observed stress distribution.

model, however, it remains els.

Damage per cycle; flapwise data

due to our limited into subsets—e.g.,

produce damage estimates

n.eg=4

for each. The resulting

&.9, is then resealed to estimate

based on all nseg segments.

data

~2=&&/n,e~,

the

While we take n~~g=4 here, the final

CHAPTER 3. LOAD MODELS FOR FATIGUE RELIABILITY

52

I

I

I

i

I

1

I

[

1

I

I

I

i

&“ /

Flapwise Data Weibull

-A---

:/ / I

1

1

1 2

3

4

I

1

1

I

5 6

7

8

1

1

I

I

I

I

9101112131415

Fatigue Exponent: b Figure 3.14: Damage coefficient of variation;

variance 02 should be relatively As the exponent

b increases,

tainty due to their sensitivity

insensitive

to the number of segments

we may expect damage estimates

to rare, high stresses.

This figure shows V’&, the coefficient of variation the normalized

damage,

~.

flapwise data

used.

to grow in uncer-

This is confirmed in Figure 3.14. (ratio of o to mean damage)

Vm is shown to grow systematically

of

with exponent

b,

reaching 5070 for b of about 5 and nearing 1007o for b values above 10. Note also that these results

do not depend

significantly

on whether

the damage

estimates

use the

observed data or a Weibull model. This worst-case Consider

by a simple heuristic

argument.

a stress history whose highest ranges fall into a bin at level S~.X, in which

there are n(S~w) damage ~

value of VT= I can be supported

data among the total of NtOt cycles.

becomes increasingly

dominated

As b grows, the normalized

by cycles at Sm.. only:

UNCERTAINTY DUE TO LIMITED DATA

3.5.

bn(si)

~=~sax

~

n(Smaz

#

‘m

i The coefficient of variation

53

)

(3.16)

Ntot

can then be estimated

as

v~ =

(3.17) ‘“(S””’) = &

Ifn(S~.Z)=l

observation

coefficient of variation

is acceptable.

Figure 3.1, this suggests

100%

data needs we need to consider what level of damage uncer-

This can only be addressed

tainty sources in fatigue life estimation. at failure.

bin, ~in

in our damage estimate.

Finally, to determine tainty

inthislargest

Setting Dtot=A

by comparing

it with other uncer-

Here we define A as the actual damage level

in Eq. 3.15 and solving for the fatigue life Ntot=N1zf ~ (in

cycles):

A Nlife

The coefficient of variation

=

(3.18)

~

of Nlife is then roughly

‘Nlife

=

4I’q+v:+v$

Here the three coefficient of variations,

(3.19)

VC, VA, and V-,

respectively

reflect (1)

scatter in the S–N curve, (2) errors in Miner’s rule, and (3) load modeling uncertainty due to limited data (e.g., as shown in Figure 3.14 for our 71-min. history). that load uncertainty

has only moderate

impact,

VW< @ For most materials,

is about

we require

(3.20)

+ v~

typical values of ~~

only seek a Vw value of 0.5, or perhaps

To ensure

are 0.5 or higher.

Thus, we need

a bit less. For cases in Figure 3.14 where V~

1.0, this suggests that we obtain at least n&t=4 times our current duration

of 71 minutes.

This should reduce Vm to about

multiple observations

of the most damaging

1/=,

or 0.5. It should also yield

stress, unlike the current situation

shown

CHAPTER

54

3. LOAD MODELS

FOR FATIGUE

in Figures 3. 1–3.2. In other words, we seek at least n(S~u

) =4 observations

at the

stress level S~.Z; this should give V~ of about 0.5 (Eq. 3.17).

most damaging

3.6

RELIABILITY

Summary

Several techniques

have been shown to better study fatigue loads data.

sities show which stress ranges are most important

Damage den-

to model, reflecting fundamental

differences between flapwise and edgewise data (Figures 3.2 and 3.7). Common

one-parameter

models,

such as the Rayleigh

should be used with care. They may produce dramatically distributions

(Figure 3.3) and damage

(Figure 3.13).

and exponential different estimates

of load

While the exponential

model

seems visually plausible for the flapwise data, it can potentially by an order of magnitude

for S–N exponent

“Filling in” the distribution

models,

underestimate

damage

b >7 and still more for higher b values.

tail with a continuous

model need not result in more

damage (Figure 3.13). In fitting such a model, however, it is crucial to well-represent the most damaging

stress levels. High b values require better

large stress ranges; this is effectively done by matching and better

by matching

still higher moments.

modeling

of relatively

at least two moments

For this purpose,

(Weibull)

a new, four-moment

“generalized Weibull” model has been introduced. For edgewise data, damage

(Figure

3.7).

the lower mode of the observed Over the important

Weibull model gives quite a reasonable

histogram

gives negligible

range of larger stresses,

extrapolation

of the observed

data (Figure 3.8). It appears to offer a notable improvement

the generalized trends

in the

over the ordinary Weibull

result in this case. As the exponent their sensitivity and resulting

b increases,

damage estimates

show growing uncertainty

to rare, high stresses (Figure 3.14). This is quantified data needs are discussed.

due to

by Figure 3.14,

Chapter

4

for Fatigue

LRFD

This chapter considers particular, fatigue.

the design of wind turbine

blades to resist fatigue failures.

factors are developed for load and resistance

factor design (LRFD)

The use of separate load and resistance factors is consistent

of current,

probability-based

This chapter

combines

In

against

with a wide range

design codes (e.g., API, 1993). several novel approaches

to the blade fatigue

problem.

These include:

1,

the use of FORM/SORM

(first- and second-order

mate failure probabilities 2. new moment-based

and dominant

uncertainty

model of wind turbine

limited load data (Chapter

reliability

methods)

sources (Chapter

loads, designed

especially

to esti2); to reflect

3); and

3. a parallel LRFD study of a specific Danish wind turbine,

also based on FORM/SORM

(Ronold et al, 1994).

4.1

Scope and Organization

The dominant consider

fatigue

blade loading

the following three

which measured

is assumed

different

here to be flapwise bending.

horizontal-axis

load data are available. 55

wind turbines

(HAWTS),

We for

CHAPTER 4. LRFD FOR FATIGUE

56

Turbine

1:

Turbine diameter

1 is the AWT-26 machine,

teetered

rotor,

a downwind,

and power rating

two-bladed,

of 275kW (McCoy, 1995).

is used for our base case study, in view of the relatively available (197 10-minute

Turbine

large amount

This turbine of load data

segments).

2:

Turbine 2 is an upwind, two-bladed power rating of 100kW, operated 1989).

free-yaw, with 26-m

It has a teetering

It affords a contrast

HAWT with a rotor diameter

by Northern

1 both in mechanical

hydraulic

and McNiff,

passive pitch control.

design, and in the amount

of

with hub height of 35-m and power rating

of

available load data (only 20 10-minute

Turbine

Power Systems (Coleman

hub design with full-span

to Turbine

of 17.8-m and a

segments).

3:

Turbine

3 is a Danish

machine,

500kW. This has been the subject

of a similar LRFD study on wind turbine

blade

fatigue (Ronold et al, 1994). This study has been supported

as one of four sub-projects

within the 1994–1995 European

(EWTS)

Wind Turbine

We seek here both to demonstrate

Standards

basic methodology

for fatigue reliability,

to identify the general impact of different load models on reliability thus consider normalized probability hypothetical

distributions

bending

loads from these various turbines,

models of wind environment

and blade properties

We

and fit different

in each case.

to apply specifically to any of the machines

they should be seen as the result of applying

to the same (hypothetical)

calculations.

and

to each. To focus on load modeling only, we adopt the same,

our results are not intended Rather,

project.

wind turbine

blade.

various plausible

Thus,

in question. load models

BACKGROUND:

4.2.

4.2

PROBABILISTIC

Background:

DESIGN

Probabilistic

57

Design

4.2.1 Probabilistic Design against Overloads. Historically,

codified probabilistic

design has been most widely applied to “overload”

failures, caused when the worst load, L, in the service life of a component capacity

R. This capacity

may be associated

exceeds its

with first yield, excessive deformation,

buckling, or a similar criterion. If both the load L and resistance

R were known perfectly

at the time of design,

we would merely require that R z L. More generally, in load- and resistance-factor design (LRFD) resistance,

separate

factors, ~~ and @R, are used to scale the nominal

load and

Lnm and ~0~:

Of course, in any single situation

(4.1)

k ~LLnm

@R&~

Eq. 4.1 can be replaced

by a checking equation

involving a single design factor SFde~ on the net safety factor:

SF...

%.

= ~

nom

(4.2)

> SFd.. ; SFd,. = ~

With the two factors ~~ and ~R, however, Eq. 4.1 can more readily give uniform reliability across various cases—specifically, may dominate

include separate or the separate

or vice versa. Similarly,

over that of resistance,

be applied to separate

covering cases in which uncertainty

load contributions

different factors may

which show different variability.

factors for dead and live loads on offshore structures factors recently

suggested

for static,

loads on floating structures

(Banon et al, 1994).

4.2.2

Design

Probabilistic

Because fatigue is the cumulative load L and resistance

against

in load

wave-frequency,

Examples

(API, 1993), and slow-drift

Fatigue

result of many loads, the choice of an “equivalent”

R is somewhat

ambiguous.

Fatigue

predictions

are generally

CHAPTER 4. LRFD FOR FATIG UE

58

based on tests with constant

stress amplitude

ing number of cycles to fail, N(S),

is commonly

N(S) = N,efS;:m Here S,,j

S (and mean stress S~=O).

modeled with a power-law relation:

s ; Snorm= — Sref ‘

is a reference stress level, and Nr.f the number

level. In Chapter

2 the S-N

Law was written

Both N,ef and the power-law exponent, generally be considered

The result-

(4.3)

of cycles to fail at that

as N = CS’-b where c = N,~f@,f.

b, are material

properties,

both of which may

uncertain.

Miner’s rule then assigns damage D= I/N(S)

per cycle, and hence average damage

(4.4)

over the service life of the specimen. values.)

(Overbars

are used here to denote

More generally, this damage rate ~ can be adjusted

1. a stress concentration

to reflect;

factor K relating local to far-field stresses;

factor A, the fraction of time the wind turbine

2. an availability

average

component

oper-

ates; and 3. the effect of a non-zero

mean stress S~, w hich with the Goodman

the fatigue life by (1 – KISml)/SU (here Sw=ultimate

as was done in the CYCLES limit state

formulation

rule scales

stress);

(Chapter

2).

The result

is an

effective damage rate cJ~wm

‘eff

= ~

(1 –

i ‘eff

= ‘ref

Finally, we can identify load and resistance

K\sm]/sJb AKb

variables,

L and R, such that L z R

implies fatigue failure. If we seek the specimen to withstand life, Miner’s rule predicts criterion

to read

(4.5)

N~e, cycles in its service

failure if ~ef f IV8., 2 1. Here we generalize

this failure

BACKGROUND:

4.2.

PROBABILISTIC

DESIGN

59

~ef f N,e, 2 A, where randomness rule.

(4.6)

in A reflects possible errors (both bias and uncertainty)

This implies a failure criterion

of the form L z R, in terms

in Miner’s

of the following

“fatigue” load and resistance:

L= S~mm” N..T; Note that Alternative

in this formulation,

formulations

in terms of stresses Ronold et al, 1994.

fatigue

can instead

by taking

(4.7)

R= Neff” A load and resistance

have units of cycles.

assign load and resistance

factors ~~ and y~

the bt~ root of the above expressions

Numerical

values of these factors

(Eq. 4.7); e.g.,

may differ notably;

we may

expect ~~ = y~llb because damage is related here to the bth power of stress. Our current N.ff

that

formulation

seeks to reflect common

may be based on a lower-fractile

yL then serves to inflate a number

usage; e.g., a nominal

S–N curve.

The resulting

of load cycles to be withstood.

value of

load factor For example,

critical offshore facilities are often designed against -yL=10 times the service life (e.g., demonstration

of 200-year nominal life if the actual service life is 20 years).

In the following section we seek to; 1.

model load variability

given limited wind and load data;

2. study sensitivity y to various modeling 3. suggest convenient

load and resistance fatigue failure.

assumptions,

different machines,

choices of nominal fatigue load and resistance, factors

~~ and #~, to achieve desired

etc.; and

and associated

reliability

against

CHAPTER

60

4. LRFD FOR FATIGUE

Fatigue Load Modeling

4.3

Previously

(Chapter

veniently

3) the use of smooth,

fit to a limited number

to fatigue load modeling.

of statistical

at least two moments

the four-moment

“generalized

calculated

moments

(Weibull)

and sometimes

becomes

computationally

here. Unfortunately,

repeated

fits of the distribution.

fice for fatigue load ranges.

conditions

equations

burdensome

However, our subsequent

con-

with respect for high b

This is achieved

improved

to utilize the four-moment

performed

solution of nonlinear

is presented

was evaluated

large stress ranges is required.

by the FITTING routine is not easily integrated

multaneous

distributions

further

with

Weibull”.

For this reason it is desirable safety factor calculations

probability

When fitting such models it was shown that,

values, proper modeling of relatively by matching

analytical

experience

model in the partial

the generalized

Weibull as

into a FORM analysis.

Si-

to preserve the third and fourth moments

as the iterative

FORM

calculations

suggests that three-moment

models may suf-

Such a model, herein referred to as a “quadratic

below in the following subsection.

Its implementation

require

Weibull”,

for various wind

is also described.

4.3.1

Fatigue

Loads

for Given Wind Climate

We work here with the first three moments,

defined as follows:

pl=~

(4.8) (4.9)

(s -33

~,=

(4.10)

0;

Note that both M2, the coefficient of variation, normalized information estimate

to be unitless. about

Successively

and p3, the skewness coefficient, are

higher moments

provide increasingly

rare large loads, at the expense of being increasingly

from limited data.

Alternatively,

one may estimate

detailed

difficult to

the b-th moment

(and

4.3.

FATIGUE

LOAD MODELING

.99999 =.999 L.

:

61

Amplitude Data Quadratic Weibull Weibull [ Lognormal

. . ..-. ........... -----

I

.95 t /

...)% .*.

-.7

,-..

1

.40 ~’.’ .5

1

I

1

1 I

1 2 3 5 10 Normalized (to Mean) Load Amplitude

Figure 4.1: Distribution

of normalized

loads (Turbine

1: V = 11.5 m/see, I = .16).

hence S$O,~) directly from data, thus avoiding the need to fit any theoretical ity model. Our use of lower-moment associated

with estimating

models, however, seeks to reduce the variability

Sbnor7n>Particularly

10 or above) found for some composite Following common wind turbine segments.

Rainflow-counted

the results

binned

for the relatively

materials

practice,

stress ranges,

we divide load histories S, are identified

to the minimum

distribution

1 at a wind climate bin centered at V=ll.5 occurring

into 10-minute

for each segment, intensity

employed an eight by eight bin scheme where the minimum

Figure 4.1 shows a resulting

high ~ values (e.g. ~

(e.g., Mandell et al, 1993).

by mean wind speed V and turbulence

the bins (for V and 1) correspond

probabil-

I. This study

and maximum

and maximum

and

values of

values of the data.

of flapwise bending loads, found for Turbine m/s and 1=.16.

This is a fairly frequently

bin, and Figure 4.1 reflects a total of approximately

5 hours of data.

The

results are shown on “Weibull scale,” along which any Weibull model of the form

CHAPTER 4. LRF13 FOR FATIGUE

62

Prob [ load will appear

as a straight

line.

exponential

(Q=l ) and Rayleigh

(4.11)

> s] = exp[-(s/~)”]

Recall that special cases of the Weibull (a=2)

models.

include the

Both of these have been previously

applied to model HAWT and VAWT loads (Jackson,

1992; Kelley, 1995; Malcolm,

1990; Veers, 1982). The Weibull model in Figure 4.1, fit to the first two moments

PI and p2, appears

to match the data fairly well. It fails, however, to reflect the systematic data display on this scale. curvature

An alternate

in the other direction,

fractile levels. (A 4-moment Danish wind turbine

suggesting

variation

model, the Iognormal,

it notably

overestimates

on this lognormal

the

shows

loads at high-

model has been used in the

study of Ronold et al, 1994.)

We also show results from a “quadratic moments

two-moment

curvature

of the data.

Weibull”

model, based on the first three

It begins with the Weibull model SW.~bof Figure

4.1, fit to

pl and p2. If the skewness p3 of the data exceeds that of SW.i~, a quadratic added to SWe~bto broaden

its probability

term is

distribution:

+ &eib] S = Sm~n + ~[swezb

(4.12)

When the skewness p~ of the data is less than that of SW.zb,the roles of S and SW,~b in Eq. 4.12 are interchanged:

swez~ = Smzn + /$[s + 6s2] (This quadratic

equation

(4.13)

is readily inverted to yield an explicit result for S in terms

of SW.ab.) In either case, the fitting proceeds in 3 steps: 1. e is chosen to preserve the skewness, #3; 2.

K is

chosen to recover the correct variance,

3. the shift parameter

s ~in is finally introduced

a;; and to recover the correct mean, PI.

FATIGUE LOAD MODELING

4.3.

Figure 4.1 shows that the quadratic the data.

63

Weibull model indeed provides an improved fit to

Note here that the best Weibull model, SW.~b,overestimates

of large loads, and hence the load skewness. ensure that

S has narrower

for this turbine

distribution

Thus we select Eq. 4.13, with c > 0 to

tails than SWeab. Similar trends

in other wind conditions,

as shown in the next section.

Section 4.5 on how such differences, among the three load distributions result ing estimates

of fatigue reliability.

4.3.2

We focus in

shown, impact

tails than the commonly

used

model.

Fatigue

To implement estimates

are found

Note also that the data show p2=l. 1, so that

the best Weibull model SW.~bhas broader distribution exponential

the frequency

Across Wind Climates

Loads

3-moment

the preceding

E[pi] (“expected

values”)

load model for various wind conditions, of the three

moments

found for each V-I bin. The following power-law relation E[L2] = ati(v

pi (z=1,2,3)

best

have been

has then been fit:

v,e~)ali(fi)a2i

(4.14) Te

Figures 4.2-4.4 show resulting estimates loads have been normalized that

all results

in Figure

V= V7ef.) This approach conditions

of E[pi] for the three wind turbines. mean values at V= V..f=7.5m/s,

by their respective 4.2 predict

of modeling

(All

unit values of the normalized the load moments,

so

mean load at

pi, ascross different wind

closely parallels that of the Danish fatigue reliability

study (Ronold et al,

1994). This permits us to include results for that machine (Turbine 3), by substituting appropriate

expressions

V at a reference relatively

for E[pt]. Also, all results are shown versus mean wind speed

turbulence

moderate

variation

shown in the figures (although Most notable

intensity

l=l,e$=0.15.

with 1, (e.g., .1 s azi s it is kept in the subsequent

in these figures is the similarity

and 2 show similar estimates p2 (albeit apparently

Because

of all 3 moments,

lower p3 estimates).

all quantities

.3) this dependence

showed is not

analyses).

of the various turbines:

Turbines

and Turbine 3 gives consistent

A notable distinction

two studys is that Ronold et al, (1994) employed a polynomial

1

pl and

however between the expression

of the form

CHAPTER 4. LRFD FOR FATIG UE

64

3.0

i

I

Turbine Turbine

2.5

, ./ # . ,,

.

1.0

i--.........................

0.5

‘“”

,’

1 — 2 -----

2.0 1.5

I

. #,

,I;

~’”

,,

##.

.: ~“

,, ,# , ,

(Turbulence

Intensity

=.1 5)

0. o

10 15 20 5 10 Minute Mean Wind Speed (m/s)

Figure 4.2: Estimated

mean of normalized

25

loads.

2 1.75

‘(

1,5

Turbine 1 — Turbine2 ----Turbine

3 -----------

.....

1.25 1. 0.75 0.5

0.25

(Turbulence

Intensity

=.1 5)

0 0

5 10 15 20 10 Minute Mean Wind Speed (m/s)

Figure 4.3: Estimated

load coefficient of variation

(COV).

25

4.3. FATIGUE LOAD MODELING

2.5 -

I

65

I

I

1

‘2 UY m E 1.5 g %~ Turbine 1 — m Turbine 2 ----g 0.5 :...... Turbine 3 ----------1 c a 0 ...-..-..-” ......... -.-......... .........2 ............................................... -0.5

-

(Turbulence Intensity =.1 5) -1 0

1 5 10 15 20 10 Minute Mean Wind Speed (m/s)

Figure 4.4: Estimated

25

load skewness,

E[pz] = aji + a{iV + a&V2 + ajiI + aji12

(4,15)

which explains

the nonzero mean load (at V=O) for Turbine

also important

to note that Turbine 3 results are inferred from cited results (Ronold

et al, 1994) for the mean P!, standard Figures 4.2-4.7 have been constructed

deviation

3 in Figure

pi, and skewness p! of S’=ln S.

from Taylor series approximations,

gest the mean load PI N exp(p~), while the higher unitless moments and 3. These approximations

4.2. It is

may add to the discrepancy,

which sug-

pz = p; for i=2

however, for example in

Figure 4.4. In general, Figure 4.3 suggests that for all 3 turbines,

the coefficient of variation

uz generally exceeds 1.0, its value for an exponential model. Thus the “best” twomoment Weibull model is broader in its tail, or more damaging, than the commonly

CHAPTER 4. LRFD FOR FATIGUE

66

used exponential. sponding

However, the skewness p3 is generally

value for an exponential

while more damaging

variable.

less than

2.0, the corre-

This implies that the basic Weibull fit,

than a body-fit exponential,

in turn overestimates

damage due

to large load levels. In other words, the result shown in Figure 4.1 for Turbine symptomatic

of various turbines

in diverse wind climates:

the data show curvature

on Weibull scale, toward lower load levels at high fractiles predicts.

This effect will grow in importance

1 is

than the Weibull model

as the fatigue exponent

b increases; e.g.,

as we move from common metals to composites. Analogous standard

to Eq. 4.14, power-law fits have also been made of the corresponding

deviations,

D[pi], of the 3 moments

each V–I bin by a bootstrapping

technique,

pl. ..p3. These have been estimated

in which “equally likely” rainflow range

data (the same number as observed) are found by resampling (Efron and Tibshirani,

1986).

limited data, and approach cases with little data

These quantities

from the observed ranges

D[pz] directly

reflect the impact

(e.g., Turbine

2) to show relatively

higher uncertainty

1 and 3). Again Turbines

found to yield consistent

4.5-4.7.

clear whether moments

especially

of

zero as the amount of data grows. We should thus expect

D[pz], than those with more data (Turbines

more variable,

for

results

in Figures

Turbine

in Figures 4.5–4.6 and at extreme

this reflects the observed

of S from those reported

1 and 2 are

3 appears

wind speeds.

data for this turbine,

of in S, or the extrapolation

levels,

relatively It is not

our approximation of functional

of

forms

beyond the range of observed data. This difference, however, appears calculation.

In particular,

to propagate

the next section will show LRFD factors for both Turbine 1

(dense load data) and Turbine 2 (sparse load data). bine 3 appears

to the load- and resistance-factor

Because the uncertainty

closer in Figures 4.5–4.7 to that of Turbine

in the Danish study more closely parallel our sparse-data

in Tur-

2, LRFD factors reported case.

4.3.

FATIGUE LOAD MODELING

u 0.0125

8 -J ~ z

0.01

{ ().0()75 .5 E a 0.005 z z $0.0025 n c z 0“

I

67

I

1 .’ , ..I ,,, / ... .“ .. .’ .. .’ ,., .’ ... #,’ ,.,... .’ ..-. .“ -.. -’ ..............------.........-.’....... ,,.””’” . .’ ‘“” Turbine 1 — .’ Turbine 2 ----- ,,“ .’ Turbine 3 ..----,“ ,,,”’ +“Turbulence Intensity= .15 . .’ .’ 1 1 I 1 “’” 10 15 20 5 10 Minute Mean Wind Speed (m/s)

0

Figure 4.5: Standard

0.04

r

[

deviation,

estimated

I

[

,, #.-

25

mean load.

1

,.. .?. ... .... .,, ...

0.035 0.03

/ /. ... . . . -----

....

0.025

--------

--------

.-

Turbine 1 —

0.02

Turbine 3 -v..0.015

(Turbulence Intensity =.1 5)

0.01 0.005 1

0 0

J

1

J

10 20 10 M%ute Mean Win~5Speed (m/s)

Figure 4.6: Standard

deviation,

estimated

load COV.

J

25

68

CHAPTER 4. LRFD FOR FATIGUE

0.25

I

--------

I

I

I

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -------

. . .

0.2

Turbinel — Turbine2 ----Turbine3 ““----(Turbulence Intensity =.1 5)

0.15

0.1

.... .....-.----...........-.”..............................................------------

0.05

1

1

1

1

5

10

15

20

0 0

25

10 Minute Mean Wind Speed (m/s) Figure 4.7: Standard

deviation,

.,.

4.4. LRFD ASSUMPTIONS AND COMPUTATIONAL

LRFD

4.4

PROCEDURE

69

Assumptions and Computational

Proce-

dure

~

——

From Eq. 4.7, the fatigue weights the conditional joint probability

loading

involves the bt~ moment

S~Wm=Sb/S$ef.

This

Sb IV, 1, given various values of V and I, by their

moment

density fv,r(v, z):

H

jy=

Syv, I

“

jv,~(v, i) Ch (ii

(4.16)

all V I

We assume here that the mean wind speed V has Weibull distribution, with aver— age V and shape parameter CZv. The turbulence intensity 1 is assumed independent of V, and assigned

lognormal

distribution

tion COV1. Finally, we estimate lognormal,

and quadratic

with average ~ and coefficient of varia-

Sb IV,I from three moment-based

fits: the Weibull,

Weibull models as in Figure 4.1.

From the previous section, the necessary moments

pi are modeled as

(4.17)

Pi = E[Pi] + D[Li] . IX; i = 1,2,3 in terms of standard timated

among the Uaare included, es-

normal variables Ui. Correlations

from all the moment data irrespective

directly parallels that suggested

of their V and 1 values. This approach

in the Danish fatigue reliability

study (Ronold et al,

1994). Thus, the fatigue uncertain

load L=~”

lV~.r/S$ef is modeled

here as a function

of seven

quantities:

L = L(X) = in which X=[av,

=(X)

“

N.e,

Sb ref

(4.18)

~, COV1, ~, U1, U2, U3]. Note that ~/S#.f

is unitless,

so that this

s

load definition has units of cyczes. An alternative ?

terms of stress, will be described terize the Weibull distribution which are assumed

definition of load and resistance,

in

later (cf. Eq. 4.23). The first four of these charac-

of V and the lognormal

to be statistically

independent.

distribution The latter

of 1, respectively,

three are used, with

CHAPTER 4. LRF’D FOR FATIGUE

70

Variable

Mean 1.8 7.5 0.25 0.15 0.0 0.0 0.0 2.42E+18 4.1 OE+O9 8.0

Distribution Weibull Normal Weibull Normal Normal Normal Normal Weibull Constant Constant

Table 4.1: Random

variables in reliability

Eq. 4.17, to reflect load uncertainty

due to limited data.

Table 4.1 shows the distribution

types and parameters

as well as of the net resistance

Stn Dev 0.135 0.563 0.019 0.011 1,0 1.0 1.0 1.69E+18

analyses

for each of these variables,

variable R = ~~f ~” A in Eq. 4.7. Generally, this would

in turn require joint modeling of its various components;

e.g., IVref, K, S~, SU, and

A in Eq. 4.5; cf Veers et al, 1993. Here, however, for simplicity b=8 and the resistance

alternative enforced

we choose the value

variable R to have net coefficient of variation

Note that the computational tion somewhat

.“.

procedure

outlined

different from the CYCLESformulation

of 0.70.

will result in a limit state equadescribed in Chapter

version of CYCLES has been used here where the restrictive in CYCLES (see Section 2.4) have been relaxed to permit

(Incorporating of future

such enhancements

in future

work; see the recommendations

here is to study the impact

2. Thus an assumptions

more generality.

CYCLES versions may be a useful topic

that follow in Chapter

of various load models on reliability

5.) As our intent calculations

it was

necessary to include those model types into the program

input and modify the limit

state formulation

types.

complexity

to accommodate

generated

different distribution

With the additional

by the use of both wind speed, V, and turbulence

intensity,

1,

4.4.

●

LRE’D ASSUMPTIONS AND COMPUTATIONAL

as environmental ticable.

*

parameters,

Therefore

evaluated

closed form expressions

PROCEDURE

for fatigue life become imprac-

the ensuing FORM analyses employ a limit state equation

numerically

of numerical

of the FORM analyses.

integration

significantly

as well. types that

The distributions exists within

subroutine

has been added to compute

determine

the bt~ moment

quadratic

Weibull distribution

the three distributions

from the internal

CYCLES program.

the conditional

types library

A separate

Sb IV, I required

moment

This subroutine

as well as the Weibull and lognormal

Of course this added capability

this is not a big factor.

the original

types of the load

variables distribution

are selected

of the load in Eq. 4.16.

considered

requires longer solution

enhances the flexibility

For example not only can the distribution

be selected as an input option but also the environmental

of distribution

that is

using quadrature.

This implementation

can be changed

71

includes

the

distributions,

in this study.

does not come without cost. Numerical

integration

times; however, typical runs are on the order of minutes Run-time

to

so

costs are tied directly to the number of quadrature

points selected; this number was varied and several runs were made to ensure stable results.

Also, required

variable definitions, three moments

input to the program

sidered as making

FORM

program

the original

deviations,

CYCLES program

Weibull load distribution.

tively unimportant

on the contrary,

similar results for Turbine

with the assumptions

represents

for the academic

a useful compromise

existing knowledge of many structures

and mechanical

results

2 with an

intensity

in this case, and the loads data here show relatively

version of CYCLES was necessary

the CYCLES program

obsolete.

should not be con-

This is because the turbulence

(see Figures 4.2 and 4.3) in accordance enhanced

to the random

D[pi].

with its added generality

show that CYCLES would likely have produced assumed

In addition

the ati’s, ali’s, and azi’s of Eq. 4.14 must also be input for the

I?[pa] and their standard

This alternative

is increased.

constant

is relaCOV

of CYCLES. While this study

performed

here,

between level of detail and the components.

CHAPTER 4. LRFD FOR FATIGUE

72

Stress Distribution Type Log Weibull 4.4 x 10-1 2.5 X 10-7 Uncertainty Percentage 0.5 0.1 0.0 0.0 19.9 0.1 1.1 0.0 0.9 0.0 4.3 0.0

Pf ~ Variable

1

73.3

Table 4.2: Turbinel

4.5

reliability

99.8

results; effect ofload

distribution

type

Variations with Load Distribution 1, comparing

the effect of switching

between

the three load models as in Figure 4.1: lognormal,

Weibull, and quadratic

Weibull.

We first consider

As in Figure parameter

results

4.1, we pursue

models,

moment-based

and PI through

models yield identical quadratic

for Turbine

fits, matching

p3 for the quadratic

fatigue damage

results

PI and pz for the twoWeibull.

for fatigue exponents

Thus

all three

b=l and 2; the

Weibull would also agree with the observed damage when b=3.

With the exponent different estimates

b=8 chosen here, however, these models yield dramatically

of fatigue damage, and hence of the probability

pf of fatigue failure

within 20 years (iV~.r= 4.1 x 109 cycles at an average rate of 6.5 Hz). Typical pf values differ by more than 5 orders of magnitude: 4.2 shows the failure probabilities variables importance

for the lognormal

and relative

importance

and Weibull load distributions.

of the load variables

the importance

of the different Note that

shifts from 2770 for the lognormal

less than 1% for the Weibull distribution This highlights

from less than 10-6 to above 10-1. Table random

the relative

distribution

to

of loads.

of choosing an appropriate

load model. It reflects

._

LRFD CALCULATIONS

4.6.

.

the well-known

effect of tail-sensitivity

S have been preserved *

which is a relatively

LRFD

while the first two moments can vary greatly

of

among various

This effect grows with b; note that we choose here b=8,

high value for metals but relatively

low for many composites.

Calculations

The foregoing shows that calculated

in reliability,

here, the higher moment ~

models fit to the same data.

4.6

73

if we consider

fatigue reliability

an identically

designed

turbine

blade, the

is altered notably by the choice of load distribution.

Con-

versely, vastly different load factors would be needed to achieve the same reliability for different load distributions—i.

e., much higher load factors if the lognormal

were correct, much lower for the Weibull model, and so forth. later with the results for Turbine To reduce this sensitivity reflect the load distribution parameter

of load factors to distribution

choice, we can seek to

type in our nominal load. Assume we consider a design

We may then seek to adjust

our best estimates

This is demonstrated

1.

W, which relates the observed bending

i.e., S= Al/W.

model

of the distributions

moment

M to resulting

stress S;

W to preserve the mean damage,

given

of V, I and S IV, I—in other words, choose W

to preserve

L ..rn =

Siom

. N..,

=

L(X given

Xi=

~)

(4.19)

M truly follow a lognormal model we would require a —— higher W to preserve S~=M~/Wb—than if a Weibull model of M

Thus, if applied moments sturdier

blade—i.e.,

were correct.

Moreover, these differences between blade designs would increase with

b, to reflect increasing

sensitivity

to distribution

choice as b grows. Other design rules

are also possible; e.g., choose W to preserve a specific upper fractile of the long-run stress distribution.

An advantage

of preserving

reflects not only the choice of load distribution

Ln~ in Eq. 4.19, however, is that it but also the fatigue material

behavior

(i.e., choice of b). Table 4.3 shows the resulting advantage

of the nominal load definition in Eq. 4.19.

CHAPTER 4. LRFD FOR FATIGUE

74

Stress Distribution Type Log Weibull Quad Weib 4 x 10-4 I 4 x 10-4 1 5 x 10-4 Uncertainty Percentage 2.7 0.0 0.; 1.6 0.1 3.3 0.2 0.0 0.0 2.6 1.0 1.2 0.1 0.0 0.2 0.6 0.0 0.1 0.3 96.4 96.6 92.5

Table 4.3: Turbinel L nom.

reliability

results; allresults

with same normalized

By preserving the nominal load L nw in Eq. 4.19 for each distribution in Table 4.3 correspond

to three different blade designs.

fatigue load

type, the results

Resizing the blade in each

case scales the relative stress levels so that the l?[Sb] for each stress distribution (and therefore

L.m ) is the same for each blade.

mildly sensitive to the choice of load distribution. (lognormal (96-97%).

and Weibull)

The three-moment

to the additional contribution

uncertainty

In fact both two-moment

x 10-4, with almost

model (quadratic

are seen to be only

all uncertainty

models due to R

Weibull) gives slightly higher pf, due

in the higher-moment

statistic

p3.

(The uncertain

y

of R is thus reduced to 92.570 in this case. )

We therefore

suggest that load factors ~~ should be based on the nominal

defined in Eq. 4.19. materials,

give pf=4

The results

type

This should promote

load modeling

assumptions,

relatively

stable

and blade designs.

results

load

across different

.

LRFD CALCULATIONS

4.6.

75

Turbine 1 Results

4.6.1

Finally, we consider the inverse problem of probabilistic

design: what factors ~~and

~~ should be used in Eq. 4.1, with nominal fatigue load L.~ achieve a target failure probability

Am,

pf over the service life? FORM/SORM

are particularly

useful for these purposes.

and uncertainty

contributions,

In addition

to providing

to

methods of pf

estimates

values, L* and R*,

they provide load and resistance

most likely to cause failure at the design lifetime design load and resistance

and resistance

(see Section 2.5).

By setting

to these most likely values, we can estimate

our

the necessary

factors:

(4.20)

Regarding

nominal loads and resistances,

we again use Eq. 4.19 to define a “mean”

nominal load LnOn. Recall that Lno~ has units of “cycles” so that corresponding factors

are applied

following common

to an expected practice

number

the nominal

of service lifetime

fatigue resistance,

cycles.

load

In contrast,

RnO~, is set at its lower

2.3% fractile, i.e., the underlying

normal variable lies two standard

the mean.

units of “stress” ES ‘ given in the next section using

An example

results for turbine

utilizing

deviations

below

2.

Figures 4.8–4.9 show resulting load and resistance factors, respectively, 1. As may be expected,

the resistance

factor OR decreases

pf is lowered.

This reflects that while the nominal

conservatively

set (2.370 fractile),

steadily

resistance

still lower resistances

for Turbine as the target

~Om was somewhat

must be designed

against

if

we require still rarer failure events. In Figures the assumed quadratic relative

4.8–4.9 both the load and resistance 2 parameter

stress distribution

Weibull are somewhat levels of uncertainty

and Weibull distributed

distribution between

reported

are nearly identical

This can be explained

for these results

by observing

factors.

the

in Table 4.3. The lognormal importance

(= 96.5 %) and load (N 3.5 %) variables

load and resistance

for

cases while those for the 3 parameter

cases have nearly the same levels of relative

the resistance

ing in nearly identical

different.

factors

result-

Also from Figure 4.8, note that

CHAPTER 4. LRFD FOR FATIG UE

76

2 1.8

.$

.

1.6 1.4 8 ~

1.2

-

+.

Q-””=-=+”-”

21 v ~ 0.8 J

‘$? -----””-=

a+- ~...z-

Quadratic Weibull Weibull Lognormal

0.6

+---+---n...

0.4

1

0.2 0 .03

.1

.01

.003

.001

Failure Probability Figure 4.8: Load factors, Turbinel. in units of cycles (Eqs. 4.18–4.19).

Note that these factors apply toa

load defined

3.5 Quadratic Weibull Weibull Lognormal

3 -k\

2.5 a) c

+-- -+------ -

~,\,\ .\

2

CJ

(d

5 .—

1.5

cl)

r!?

1 0.5 0 .1

I

1

.03

.01

1 .003

.001

Failure Probability Figure 4.9: Resistance factors, Turbine 1. Note that these factors apply to a resistance defined in units of cycles (Eq. 4.7).

.

LRFD CALCULATIONS

4.6.

the required

load factors~~

values shown. uncertainty

77

are effectively constant

This is because load uncertainty

contributions

from all 7wind

throughout).

The actual load factor~~

(~~=1.2-l.8)

areneeded

over the several decades

is relatively

unimportant

and Ioadvariables

(i.e., the

remains less than 7.570

is not 1.0, however; somewhat

to cover not the uncertainty

ofpt

larger values

in load, but rather

the bias

“mean” load L ~~ and the actual load L* most likely to cause

between the nominal failure.

In Section 4.5 the choice of load model on fatigue critical as pf estimates designed blades.

To demonstrate,

for identically

that do not include load distribution

type

will produce

load factors that also differ by orders of magnitude.

an alternative

nominal load is defined where the design parameter,

W, is chosen to preserve the conditional m/see.

was shown to be

differed by more than 5 orders of magnitudes

Nominal load definitions

in their definition

reliability

mean stress given a mean wind speed V=50

Using Eq. 4.14 to evaluate ~ =i!l[pl] with V = 50 m/s and 1 = ~, we define

the nominal load by;

L~~ =

[~lV = 50 m/sec]b. N~.r

(4.21)

%?f The resulting alternative

load factors

load (L~~)

nominal

critical distinction

in Figure

4.10 differ by orders of magnitude

does not reflect the distribution

between ~b here and@

distribution

4.6.2

Figures Eq. 4.19.

load and resistance factors

of ~

for

of including

as Eq. 4.19 does.

4.11 and 4.12 show load and resistance Results

the advantage

Eq. 4.21

2 Results

vary markedly

uncertainty in Figures

factors for Turbine

from those for Turbine

sparse load data in this case, FORM

resistance

which is a poor predictor

b. These results clearly demonstrate

type in nominal load definitions

Turbine

Note the

in earlier nominal load definitions;

reflects only the mean stress ~ at V=50 m/see, high stress exponents

type.

as our

results show roughly

(see Table 4.4).

1. Due to the relatively equal contribution

As a result,

4. 11–4.12 vary similarly

2 based on

from

the implied load and

over the range of pf values

CHAPTER 4. LRFD FOR I?ATIG UE

78

100000 10000

5 z 2 u a !3

1000 100 10

,:

El- ---E! ---------El--------Q---------El--

l...+l Quadratic Weibull Weibull Lognormal

+ ----x-----+

-+-+-. - ❑ --

------ --

---x-+ --x--+

. L_————d .1

.03

.003

.01

.001

Failure Probability Figure 4.10: Load factors needed, for Turbine mean stress for wind speed V=50 m/s.

reported:

1, if nominal load is based only on the

~~ varies by about a factor of 3, and @R by about a factor of 7. Note also

that for relatively conservative; S–N curve.

high pf values, the 2.3% fractile nominal

However, to cover load uncertainty

Finally, note again that our formulation cycles: @R is applied to the number curve, and 7L to the number tively, we may define nominal

is too

we may need ~~ on the order of 5; greater than the service life.

applies all safety factors to a number

of cycles N to resist failure in a nominal

of cycles to be withstood loads and resistances

recall the failure criterion

> A, given by Eq. 4.6. Substituting

terms results in;

for ~,ff

of

S–N

in the service life. Alternain terms

of stresses,

set of factors ~~ and ~~. In order to define nominal

tances in terms of stresses ~.ffN$..

~~

OR factors above 1.0 show that we may design with a less conservative

i.e., ensure that our design life is an order of magnitude

an equivalent

resistance

and find

loads and resis-

defined in Section 4.2.2, e.g., from Eq. 4.5 and transposing

4.6.

LRFD CALCULATIONS

79

20

I

15 -

1

1

Quadratic Weibull +--Weibull --f---Lognormal -H--

5 -U ~ u a 3

10 “

5 +-

---I

0 .1

1

.01 .03 Failure Probability

I .003

.001

Figure 4.11: Load factors, Turbine 2. Note that these factors apply to a load defined in units of cycles (Eqs. 4.18–4.19).

4.5 r

I

3 2.5 -

I

Quadratic Weibull Weibull Lognormal

4 h

3.5 -

1

\.\* \*\\ ‘, . ‘Qzl. \\’. \ ‘.

I

+--l

-+--B---

2 1.5 1 0.5 01 .1

I

.03

1

.01

1

.003

1 .00 1

Failure Probability Figure 4.12: Resistance factors, Turbine 2. Note that these factors apply to a resistance defined in units of cycles (Eq. 4.7).

CHAPTER

80

Pf ~ Variable

Table 4.4: Turbine L nom.

4. LRFD FOR FATIGUE

Stress Distribution Type I Weibull I Quad Weib Log I 4 x 14–’ I 1(I x 10–’ 17 x lU–” Uncertainty Perce nt age I 2;6 28.3 7.3 7.3 7.1 0.0 0.0 0.0 1.4 0.4 0.4 3.5 0.3 0.3 5.2 2.1 8.9 7.0 57.2 61.7 52.5

2 reliability

results; all results with same normalized

fatigue load

A ~ Taking the @ root produces

k @.fN.ff

expressions

(4.22)

“~

for load and resistance

in units of stress;

1 .A~/b, L’=~ ; R’= s:ef Neff ~ [1 se~ 1 lib

I/b

(4.23)

[

The load term is simply the @ root of the expected the environmental the constant

variables

amplitude

V and 1. For the resistance

S–N relationship,

N(S’) = Neff The resulting

resistance

s

() ~

R’ reflects the material

term recall our definition

1

or S =

S~efiVeff “ ~ [

term then can be interpreted

properties

over of

Eqs. 4.3 and 4.5;

–b

with N~er as given by the S–N relationship

value of Sb integrated

I/b

1

(4.24)

as the stress level associated

of Eq. 4.3 (and scaled by Al/b). Therefore

(e.g. the resistance)

through

the required

service

EFFECTS OF LIMITED DATA

4.7.

81

cycles N~er. Figures

4.13 and 4.14 show load and resistance

factors

for Turbine

2 based on

Eq. 4.23 Because damage is related to the bth power of stresses, y~ = #b. this equality

consider the load definitions

given by Eqs. 4.18 and 4.23 and observe

that L1/b= L’ o [N~~~/S..f]. Since the load factors, 7L are computed

drops out of the calculation

and the equality

can be made for the resistance

term.

5 to 20. For b=8, corresponding

as the ratio of

load Lm~ (see Eq. 4.20) the constant

the design load L* to the nominal

To verify

~~ = ~~I’b holds.

N~#’/Sr.f

A similar argument

In Figure 4.11 typical values of ~~ range from

factors y~ on stresses range from 1.2 to 1.4. These

factors lie in a similar range as those of the Danish study (Ronold et al, 1994).

Effects of Limited Data

4.7

The use of Eqs. 4.14 and 4.17 to model fatigue loads across wind climates is intended to include uncertainty

in the loads due to limited data through the standard

(D[pi]’s) of the moments -4.7,

is expected

This approach

pl...p~.

to propagate

apparently

more data than Turbine Closer inspection case are more subtle.

The difference in D[pz] ‘s, observed

through

the load- and resistance-factor

2, has much lower load factors (Figure:

calculations.

Table 4.4 shows reliability

One would expect Turbine

types, by appropriate

10 times

4.8 and 4.11).

of the results shows that the effects of data limitations

the three load distribution

moderate

in Figures 4.5

works quite well as Turbine 1, with approximately

results and uncertainty

for Turbine 2. As for Turbine 1, the nominal load L.-

of importance

deviations

in this

contributions

has again been preserved across

choice of blade section modulus,

2, with higher D[pi]’s and load factors to show a shift

away from the resistance

R to the moments

shift to the Uz’s and a dramatic

Ui. Results show only a

shift to the wind speed shape factor, crv.

These results would suggest that there are sufficient blade load data for Turbine and the other sources of uncertainty that the dependence an important

are dictating

the reliability.

upon wind speeds and the uncertainty

factor to consider for Turbine

The dependence

W.

of Turbine

2

Table 4.4 shows

in those wind speeds is

2.

2 upon wind speeds, in contrast

to Turbine

1, can be

CHAPTER 4. LRFD FOR FATIGUE

1.6 --

1.4 1.2 8

G i?

u

(a

3

-------

-

1

Quadratic Weibull Weibull Lognormal

0.8 0.6

-+---+-----

0.4 0.2 0

.1

.03

.01

.003

.001

Failure Probability Figure 4.13: Load factors, Turbine2. in units of stress (Eq. 4.23).

Note that these factors apply to a load defined

1.4 1.2 1 0.8

Quadratic Weibull -+--Weibull -+-LOgnormal -•- -.

0.6 0.4 0.2 0 .1

.03

.01

.003

.001

Failure Probability Figure 4.14: Resistance factors, Turbine 2. Note that these factors apply to a resistance defined in units of stress (Eq. 4.23).

4.7.

EFFECTS OF LIMITED DATA

explained

by considering

deviations.

the trends in the mean blade loads rather than the standard

That is, it is the l?[pi]’s in Figures 4.2-4.4 and not the D[pi]’s (from Fig-

ures 4.5–4.7) that are responsible through

83

an additional

for the shift in uncertainty.

This has been confirmed

case study, in which the FORM analyses used to generate

results of Table 4.4 are repeated

with the D[pi]’s set to zero (to assume perfect data). (e.g., Ul,

In this case the results in Table 4.4 change very little since load uncertainty Uz, and Us) is small. Since identical distribution variables

(except

parameters

the loads) in the reliability

differences in mean blade damage, ~, It is also useful to examine turbines

and compare

Table 4.5 gives the most damaging

(I?[pl]) relationships

analyses,

are attributed

most damaging

them against

analyses at their converged,

the

were used for all random

we conclude

to different turbine

wind speeds obtained failure points.

from Figure 4.2 plotted

apparent

designs.

wind speeds (Eq. 2.14) for the two

the range of available

most-likely

that

data for each machine.

from the associated

FORM

Figure 4.15 shows mean load

over the range of data used to produce

the power-law fits. The most damaging wind speeds for Turbine 1 are within the range of measured

data.

This is desirable

as the results of the regression

analysis

used to

produce the curve-fits in Figure 4.15 are not valid outside the range of available data. Turbine

2 on the other hand is predicted

that far exceed the associated There are two important associated

to receive most damage from wind speeds

data seriously compromising

the results.

points to be made here. First, there are data limitations

with the reliability analysis of Turbine 2 but it is the lack of blade load data

at higher wind speeds rather than the paucity of the data at the more moderate speeds for which measurements are available. has been chosen arbitrarily

,

in which Turbine

2 is planned

need for prototype

machines,

to wind conditions

that span the range of envisioned

when using automated

to operate. ) In general these results reflect the clear on which load measurements

curve-fitting

range of wind conditions. outside

model

in this study, with no regard to the specific environment

A second point to be considered *

(Recall that the wind environment

wind

is the potential

the range of observations

operating

as Turbine

conditions.

danger that may be encountered

models to describe

When the critical

are made, to be subjected

blade loads across a broad

(most damaging) 2 is, results

environment

is far

are likely to be governed

CHAPTER 4. LRF13 FOR FATIGUE

84

m Distribution

Table 4.5: Most Damaging

method,

etc..

1

Turbine

2

Wind Speeds (m/s) from Reliability

not by the data but by the curve-fit. are possible depending

Turbine

Analyses

In such cases it seems likely a range of answers

upon the assumed functional

form of the relationship,

fitting

4.7.

85

EFFECTS OF LIMITED DATA

3 Turbine 1 +-Turbine 2 ----

2.5 2

B

a

lap

1.5 1 &

0.5

(Turbulence Intensity =.1 5)

0 o

Figure 4.15:

10 15 20 5 10 Minute Mean Wind Speed (m/s)

25

range of measured

data

CHAPTER

86

4. LRI?D FOR. FATIGUE

Summary

4.8

We have directly studied the fatigue reliability one with rather dense load data (Turbine

of two horizontal-axis

1) and another

wind turbines,

with relatively

sparse data

(Turbine 2). We have also tried to infer similar results for a Danish machine, reported in a parallel fatigue reliability

study (Ronold et al, 1994).

Our findings include the

following. To estimate

fatigue damage,

three statistical

flapwise loads have been represented

moments across a range of wind conditions.

and p2, show similar trends for all 3 machines different designs, Turbines This tends to support

at least for specific load components, Based on the moments models.

of observed

loads (e.g., Figure

than

our earlier

By preserving

particular

4-moment

model.

p~, they more faithfully

two-moment

(e.g., Winterstein

is found to be notably

the same 3 moments

models such as the simpler to implement

et al, 1994).

intensities

Weibull” load

reflect the upper fractile

At the same time, they are rather

When lognormal,

p~ in Figure 4.4.

new “quadratic

savings when many fits are required; e.g., to estimate

The fatigue reliability

Despite their rather

HAWT designs, etc.

4.1) than common

models

pl

a fairly general set of load distributions,

over a range of wind speeds V and turbulence

tribution

(Figures 4.2–4.3).

PI.. .p~, we have introduced

distribution

Weibull or the lognormal.

The first two moments,

1 and 2 also show similar third moment

the goal of establishing

by their first

This leads to

damage contributions

1.

affected by the choice of load dis-

Weibull, and quadratic

Weibull models are fit to

of loads data, typical failure probabilities

found to differ by more than 5 orders of magnitude:

for a 20-year life were

from less than

10-6 to above

10-1. This effect will grow with b; note that we choose here b=8, which is a relatively high value for metals but relatively load distribution

Once an appropriate

has been selected, this choice can be directly reflected in the fatigue

design by seeking to preserve and resistance

low for many composites.

the nominal

load Ln~

in Eq. 4.19.

Resulting

load-

factors are then only mildly sensitive to the choice of load distribution

(e.g., Figures 4.8-4.12). Because a broad range of load data is available for Turbine

1, its fatigue reliability

4.8.

SUMMARY

87

is governed by the uncertainty Miner’s rule, etc.). curve-is

in fatigue resist ante R (e.g., uncertainty

The required

resistance

factor @~—applied to the nominal

S–N

as the target pf is lowered.

This

shown in Figure 4.9 to decrease steadily

reflects that while the nominal resistance ~ ~ was somewhat fractile), still lower resistances events. In contrast,

conservatively

set (2.3Y0

must be designed against if we require still rarer failure

the load factor 7L in this case is relatively

bias between the nominal

in S–N curve,

flat, reflecting only the

“mean” load L ~~ and the actual load most likely to cause

failure. Because relatively uncertainties

sparse load data is available for Turbine

are found to be of comparable

data however is associated than with the quantity

importance

2, load and resistance

in this case.

(This lack of

more with the range over which the data was acquired

of data itself.)

Thus the implied load and resistance

in Figures 4. 11–4.12 vary similarly over the range of pf values reported: about a factor of 3, and OR by about a factor of 7.

factors

y~ varies by

Chapter

5

Summary

and Recommendations

Overview

5.1

CYCLES Fatigue A computer mechanical compute

Fteliabilit program

y Formulation

has been developed.

failure probabilities

and importance

includes uncertainty

fatigue reliability

A FORM/SORM

of structural analysis

factors of the random

in environmental

loading,

and

is used to

variables.

The

gross structural

and local fatigue properties.

The CYCLES fatigue reliability the controlling damage

Conclusions

(CYCLES) that estimates

components

limit state equation response,

of Important

quantities

can be derived.

the CYCL12Sformulation probabilistic

modeling

formulation

assumes

specific functional

forms for

of fatigue life so that a closed form expression

for fatigue

While these assumptions

generality,

represents

limit the program’s

a useful compromise

and the state of knowledge

between

level of detail

of many components

in

to which it

may be applied.

Load

Models

for Fatigue

Several techniques

.

Reliability

have been shown to better

study fatigue loads data.

densities show which stress ranges are most important tal differences between flapwise and edgewise data. 88

Damage

to model, reflecting fundamen-

89

OVERVIEW OF IMPORTANT CONCLUSIONS

5.1.

Common should

one-parameter

models,

be used with care.

load distributions parameter

such as the Rayleigh

They may produce

and fatigue damage.

and exponential

dramatically

Improved

different

models,

estimates

of

fits may be achieved with the two-

Weibull model.

“Filling in” the distribution

tail with a continuous

model need not result in more

damage.

In fitting

damaging

stress levels. High b values require better modeling of relative~y large stress

ranges;

such a model, however, it is crucial to well-represent

this is most effectively

and better

by matching

“generalized

done by matching

still higher moments.

at least two moments

For this purpose,

the most

(Weibull)

a new, four-moment

Weibull” model has been introduced.

As the exponent their sensitivity

b increases,

damage estimates

show growing uncertainty

This effect has been quantified

to rare, high stresses.

due to

and resulting

data needs have been discussed.

LRFD

for Fatigue

We have directly studied the fatigue reliability of two horizontal-axis (Turbines

1 and 2), and tried to infer similar results for a Danish machine,

in a parallel fatigue reliability To estimate

moments

flapwise loads have been represented

across a range of wind conditions.

PI and ~2, show similar trends for all 3 machines. third moment

p3. This tends to support

of load distributions,

models.

Turbines

pl...ps,

By preserving

we have introduced

1 and 2 also show similar a fairly general set

HAWT designs, etc.

new “quadratic

p~, they more faithfully

tile of observed loads than common two-moment

by their first

The first two moments,

the goal of establishing

at least for specific load components,

Based on the moments

reported

study (Ronold et al, 1994).

fatigue damage,

three statistical

distribution

wind turbines

Weibull” load

reflect the upper frac-

models such as the Weibull or the

lognormal. The fatigue tribution

model.

reliability

is found to be notably

When lognormal,

the same 3 moments

affected by the choice of load dis-

Weibull, and quadratic

Weibull models are fit to

of loads data, typical failure probabilities

found to differ by more than 5 orders of magnitude.

for a 20-year life were

This effect will grow with b; note

90

CHAPTER 5. SUMMARY AND RECOMMENDATIONS

that we choose here b=8. Because a broad range of load data is available for Turbine is governed by the uncertainty Miner’s rule, etc.). curv~decreases

The required

steadily

in this case is relatively load L .~

in fatigue resistance resistance

R (e.g., uncertainty

factor #~—applied

flat, reflecting

only the bias between

load data is available for Turbine 2, load and resistance

5.2

importance

The are a few recommendations

First it would be instructive

to repeat the reliability

4 with data from additional

turbines

considered

predictions quantity

the feasibility

errors, and resulting

loads, statistically,

of trends

for the two

a general set of load distributions data from different turbines

machines.

uncertainty

uncertainty,

variability

assumptions predicted different

in the offshore industry,

in the ocean climate,

inferred from measured

load

due to a limited

due to simplifying

practice

on

It may also be useful to

load factors, that arise when analytical

would parallel common

in load prediction

before

of this study, are presented.

The similarity

Such results could be used to improve design against ures.

will be mentioned

for different wind regimes (and possibly

which LRFD design reflects both natural uncertainty

are found to be

This modeling error could be assessed by comparing

Results

sparse

and load factors have been based here entirely

of observed data, for modeling

types).

Because relatively

analyses for the LRFD study in

are used. In this case one trades statistical

and measured

“mean”

of such distributions.

loads data from prototype

made in the analysis.

turbine

that

blade designs, etc. Additional

Note too that our predictions

consider modeling

turbines.

gives promise to establishing

(empirical)

the nominal

uncertainties

that are direct extensions

Chapter

observed

S–N

of Future Work

of a general nature

more specific recommendations,

would help establish

to the nominal

in this case.

General Recommendations

for specific components,

in S–N curve,

M the target Pf is lowered. In contra.% the 10ad factor ~~

and the actual load most likely to cause failure.

of comparable

1, its fatigue reliability

in

and model

loads on offshore structures. both overload and fatigue fail-

SPECIFIC RECOMMENDATIONS

5.3.

A final recommendation

TO EXTEND CURRENT WORK

of general interest

the set of wind characteristics

is to further

study, and characterize,

that best explain blade loads and hence fatigue damage.

There is nothing in CYCLESthat limits the analysis to considering turbulence

intensity;

other wind parameters

wind speed and/or

could in principle be propagated

the analysis in a similar way. The main challenge remains in identifying parameters

best “explain”

classical statistics

likely contribute

to address.

which wind that

have also been devel-

of structural

applied to predict

through

this is a question

and display, all sets of environmental

While commonly

such concepts

damage;

New methods

to overload failure independent

stein et al, 1993). structures,

blade loads and resulting

is often well-suited

oped to efficiently predict,

91

variables concept

that may

(e.g., Winter-

overload failures of offshore

may prove useful for the wind industry

as well—again

for

both overload and fatigue failure applications.

5.3

Specific Recommendations

to Extend

Current

Work Through

the course of this thesis, a number of basic developments

e.g., in modeling ficiently

loads from limited

into fatigue

here on suggesting ing/implementing challenging

reliability

and in propagating

calculations.

specific avenues of further our improved algorithms

Based

this uncertainty

on this experience,

ef-

we focus

work, both in the form of document-

and in further studying the areas we found

to model.

Directing attention work could profitably of an “enhanced”

first to the reliability algorithm

version of CYCLES. The capabilities

that it is conceptually

analysis

straightforward,

load models to include dependence

of Chapter

methods

4.

used here, further and documentation

of such a version could parOur research

and of little numerical

to calculate

We believe this generalization, the mean damage

efforts suggest

cost, to generalize our

on both wind speed and turbulence

vector set of wind attributes).

merical quadrature

(g–function)

be directed to include formal development

allel those used in the LRFD

another

data,

have been made;

required

intensity

(or

through

nu-

in our fatigue

CHAPTER 5. SUMMARY AND RECOMMENDATIONS

92

g–function,

should be made to significantly

ods. (This more general quadrature e.g., non-smooth

generality

load models.

has been offered through These distributions

tion routine. numerical

distinct

The simpler,

3-moment

into its standard

distribution robustness,

library

error, from a numerical

fits (e.g., quadratic

load distributions);

uncertainties

and (2) to model short-term

case (2), these generalized

distributions

when

optimiza-

Weibull model) avoid this

and are believed to be good candidates

(1) to model long-term

for dis-

particularly

to include in an au-

package such as CYCLES. Note also that these distributions uses:

of generalized,

have been used in research versions

are sought to be fit, with minimum

optimization,

tomated

the creation

A cause of concern had been its numerical

four moments

of these meth-

routine would also permit other generalizations;

of CYCLES, and deserve incorporation semination.

the applicability

S–N relations. )

An additional moment-based

broaden

may have two

(e.g., in parameters variations

of wind or

in loads given wind.

In

need to be included not in the CYCLESdistri-

bution library but rather in its g–function.

This has already been incorporated

in-house CYCLESversion; it is recommended

in our

that it be included in a newly distributed

version of CYCLES as well.

5.4

ChaHenges for Future

Study

Models. Finally, we seek to elaborate

Parametric

of our turbine-specific future study. Important development

studies,

to best model the practical The technique a power-law

(and their standard

with the two turbines

4 is based on a parametric

is used to define the variation

note here that this was not the first model adopted. parametric

model, in which it was assumed

systematically

as a function

during the required

in Chapter

4.

load model, in which

of predicted

across different wind climates.

aspects

avenues for

version of CYCLES for the LRFD analyses,

in Chapter

deviations)

challenging

loads modeling were encountered

cases encountered

reported

relationship

and how they may suggest

issues regarding

of the ‘ienhanced”

on some of the practical

load moments

It is perhaps

useful to

We first sought a still simpler

that only the mean load p~ (V, 1) varied

of mean wind speed V and turbulence

intensity

1. Data

93

CHALLENGES FOR FUTURE STUDY

5.4.

of the normalized,

(An analogous

over all wind conditions. data:

load Lnwn =L/p~(V, 1) were then created,

unit-mean

treatment

could be made of fatigue

observed cycles IVto fail at different stresses

predicted

and pooled

Scould

be normalized

mean, pN(S), and pooled to fit a single distribution

of normalized

life

by their resistance

N .w~=N/pN(S).) When this single (normalized) sidered here, uncertainty impact.

load model was implemented

due to limited data was typically

This is to be expected:

the assumption

tion model reduces data needs enormously, proper normalization,

into a common

for the turbines

found to have little or no

of a common

(parametric)

that the unit-mean

distribu-

as all data can be pooled/lumped,

“bin” for fitting purposes.

danger of this method is that it is only as good as its assumptions; any true variations

normalized

con-

after

The corresponding namely, it obscures

load, L.w~, may show with wind

conditions. Note that our final model was also of parametric law form to observed original

l-parameter

statistics.

form, fitting analytical,

It was, however, somewhat

normalized

load model considered

moments

pn (n=l ,2,3) were permitted

relations

were fit to both the mean and uncertainty

for two turbines

remained

nearly constant

This would tend to support

more general than the

above:

the first three load

to vary with wind conditions,

results showed some promise for simpler l-moment

and power-law

of each pn (V, 1). Our limited models; e.g., the observed COV

(near unity) for various wind conditions.

the assumption

of a common

(exponential)

distribution

of blade loads, in which only the mean load needs to be fit as a function conditions.

Preliminary

this conclusion

experience

with other turbine

may not be universal.

constant,

such as V and 1.

We believe, however, that sufficient loads data

An associated

case of Turbine

greater

importance

question

concerns

most appropriate

As a final caution regarding

I--e.g.,

are permissible

and which must be kept free to vary with wind parameters

power-law or polynomial—is

limited-data

of wind

blade loads data suggest that

exist to permit critical study of this question; i.e., which load statistics to be assumed

power-

parametric

2. We predict

which parametric

form—e.g.,

if one seeks to fit one such form.

models, we note our experiences rather

with the

different results than for Turbine

of wind vs fatigue resistance

uncertainty.

However, for

CHAPTER 5. SUMMARY AND RECOMMENDATIONS

94

Turbine 2 the greatest

damage contribution

have little or no data.

Thus our conclusions

than by our assumed curve-fits. easy to automate,

comes from wind conditions

for which we

in this case are driven less by our data

This is a common danger of parametric

models: while

they create a danger that users will apply them indiscriminately,

outside the range for which the data can validate them. Based on this experience, minimum

we propose that future CYCLESversions will also report the most damaging

wind conditions

(at the design point most likely to cause fatigue failure).

can then be cautioned which adequate

as to whether

Models.

Finally,

metric load models described loads models.

dimensions

and uncertainty

we note that

as an alternative

above, we have also considered

These assume no specific functional

involving bins of V and 1 values.

in their parameters—are

bin. The results are then propagated numerical

are based on wind conditions

for

to the para-

and implemented (“parametric”)

nonform of

with V or 1. Instead they simply sort data into discrete bins; here,

any load statistic in 2 discretized

the results

The user

data are available.

lVon-Parametric

parametric

at a

quadrature

through

routine to estimate

from each bin. (Several numerical

separately

from the data

the fatigue reliability

in each

analysis,

fatigue damage by summing

strategies

points may be chosen for the integration, may be interpolated

sought

Loads distributions—

using a

contributions

are possible here: 1. optimal quadrature and load statistics

from those observed at prescribed

may be kept as given, and the probabilities

lumped

at these optimal

bin locations;

optimally

points

or 2. the bins

into these prescribed

bins.) Our preliminary load representation, loads data.

experience

with these models was that

more faithful to observed trends (and uncertainties)

However, due to their discrete nature,

to achieve converged,

FORM-based

typical with “noisy,” non-smooth

analysis g–functions:

in g and its derivatives

Therefore,

fatigue

gradient-based difficulty,

with one or more uncertain

we believe that both nonparametric

a flexible

found in the

it was found difficult to use them

to predict

likely failure point are likely to have convergence variations

they provide

reliability.

searches for the most due to non-systematic variables.

as well as parametric

methods provide useful topics for future study. Regarding

This is

load modeling

the non-parametric

models,

5.4.

CHALLENGES FOR FUTURE STUDY

new inverse-#’0Rl14methods, and the constraint .

(Winterstein

95

which switch the objective function (failure probability)

(limit state function g), may be more stable for noisy g–functions

et al, 1993). Simulation

and other analysis methods

are also appropriate

for these cases. A practical

implementation

question in these non-parametric

binning

i.e., load statistics

are at least assumed

prescribed dimensions)

wind-condition

strategy.

Indeed,

choice of optimal

bin.

are accumulated

is effectively

parametric;

to be equal for all data that fall within the

Thus, the extent

reflects the modeler’s

are with varying wind conditions.

any binning

models concerns the

assumption

of the bin width

as to how uniform

(in one or more the turbine

loads

This bin width will in turn govern how much data

per bin, and hence the requisite data needs.

References (1993). Recommended practice for planning, designing and con-

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structing fixed oflshore platforms–-load and resistance factor design, 1st edition, American

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Wind Energy 1995, ASME, SED

Vol. 16, 107-114. Lobitz, D. W. and W.N. Sullivan (1984). Comparisons of Finite Element Predictions and Experimental Data jor the Forced Response oj the DOE 100 k W Vertical Axis Wind Turbine, Report

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Madsen, H.C)., S. Krenk, and N.C. Lind (1986). Methods of structural safety, PrenticeHall, Inc., New Jersey. Malcolm, turbine

D.J. (1990). Predictions under turbulent

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D.D. Samborsky,

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perfor-

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Proc., Wind Energy 1995, ASME, SED Vol. 16, 281-290. Melchers, R. (1987). Structural Reliability Analysis and Predictions, Limited, Halsted Press: a division of John Wiley & Sons, Inc. New York. Ronold, K. O., J. Wedel-Heinen,

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moment

phase relation-

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Appendix

A

Statistical

Moment

A brief background estimate

in statistical

the ordinary

The difficulties not ordinary

estimated

Xi/n.

arise when we instead

for the generalized

first moment

are required,

models developed

p before seeking to estimate we typically

because our p value is artificially Those exposed

to a standard

when estimating

the variance

bias, the sum of squared While unbiased

are available in the statistical

estimate

is

moment,

average ~~=1 X~/n. to estimate

e.g., k=4: Qz=p~12, aS=jLS/p~”5, in Section 3.3.

we must first estimate

the unknown

p~=13[(X – p)~]. And, if we use the same find to~low

estimates

of PZ, PS, PA, etc.

tuned to best match the mean of the observations.

statistics

course will best recognize this phenomenon

p2: to inflate the sample variance to account

deviations

estimates

X., a natural

seek, as in many applications,

here lies in its circular aspect:

data set for both purposes,

here. If we seek to

i.e., of the form p~=17[(X – p)~] for k=2, 3, 4 . . . .

Note that only the first four moments

The problem

is provided

Similarly, the k-th order “ordinary”

by the corresponding

but central moments;

and a4=p4/p~

estimation

mean value 13[X]=p from data Xl...

the simple average value ~=~~=1 13[X~], is naturally

moment

Estimation

for this

is divided by n – 1 rather than n.

of the higher moments literature

(Fisher,

~3, PA, . . . are less familiar, they

1928):

(Al)

100

101

n /42 =

(A.2)

>77-J2

n2 ‘3=

‘3P;=(n-

“

(nn2

I)(n - 2)(n-3)[(n+1)~4

in terms of the sample central conventional

(A.3)

I)(n - 2)m3

moment

(A.4)

- 3(n - l)~;] (Xi – ~)k/n.

rn~=~~=l

Eq. A.1 is the

result for the sample variance.

Remaining Bias. The routine optimization,

FIITING which produces

exmploys a companion

of a given data set. the quantities nonlinearly

The routine

generalized

load models using constrained

routine, (MLMOM,to compute statistical CALMOM uses Equations

A. 1 thru A.4 to estimate

CTZby M~”5,a3 by p3/(p~-5), and a4 by p4/(p~).

with pn, they may still contain

moments

some bias although

Because these vary the pn estimates

do

not. For example,

if we fit a Gumbel model to the 19 wave height data from Section

3.3.4, the true skewness and kurtosis 10000 data sets of size n=19

values are 1.14 and 5.40. However, simulating

and running

each through

the skewness 0.79 and kurtosis 3.89 (Winterstein

CALMOM, we find on average

and Haver, 1991).

To address this problem, the FITTING routine has an automatic ing bias through from the input this distribution.

simulation. data,

After FITTING constructs

a distribution

many similar data sets (of identical

If the moments

predicted

age from the input values, new theoretical This estimation-simulation

loop is continued

check for remainwith moments

size) are simulated

from CALMOM differ appreciably estimates

of the moments

iteratively

from

on aver-

are constructed.

until satisfactory

convergence

is found. Figure A. 1 shows the effect of enabling

this “unbiased”

option and disabling

it

APPENDIX A. STATISTICAL MOMENT ESTIMATION

102

.

Generalized Gumbel Model of Annual Significant Wave Height -%

.999 .998

Generalized Gumbel (unbiased) —

.995 .99 .98 .95

.90 .80

o

.70 .50 .30 .10

.001 6

7 8 9 10 Annual Significant Wave Height, Hs [m]

11

Figure A,l: Effect of Ignoring Bias: Wave Height Example.

(using “raw” moments duced for the example

from CALMOM directly) given in Section

for the generalized

3.3.4.

There

Gumbel model pro-

is relatively

little

difference

found in these cases. Larger effects may be found for cases of (1) fewer data and/or (2) distributions

with broader tails.

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