Risk for fatigue failure - sensitivity analysis

Igor Rychlik

Chalmers

Department of Mathematical Sciences

I Motivation, introduction to sensitivity analysis

II Example 1. moving vehicle on a rough road, linear responses toGaussian - LMA loads

III Example 2. Blade of a wind turbine, non-linear structure,square Gaussian load, if there is time.

Presentation is based on:

Aberg S., Podgorski K. and Rychlik I. (2009) Fatigue damage assessmentfor a spectral model of non-Gaussian random loads, Prob. Eng. Mech,Vol 24, pp. 608-617.

Bogsjo, K., and Rychlik, I. (2009) Vehicle fatigue damage caused by roadirregularities. Fatigue and Fracture of Engineering Materials andStructures, Vol. 32, pp. 391-402.

Sarkar, S. Gupta, S. and Rychlik, I. (2010) Wiener Chaos Expansions for

Estimating Rain-flow Fatigue Damage in Randomly Vibrating Structures

with Uncertain Parameters. Mathematical Sciences rep. 2010:07.

Failure; overloading, fatigue?

I - Estimation of risk for extreme response X

P( max0≤t≤T

X (t) > ucrt) ≤ E[NT (ucrt)],

E[NT (u)] - the expected number of upprossings of level u in time T .

II - Safety index for fatigue failure

IT =E[a]− E[ln(DT )]√σ2

a + R(DT )2,

a - strength of a component, ln(DT ) measures detoriation of the strength.

DT =∑N

i=1 hki , (hi rainflow cycles ranges) - nominal rainflow damage.

III - Narrow band bound Bendat JS (1964).; for symmetrical loads

E[DT ] ≤ k2k

∫ ∞0

uk−1E[NT (u)] du = T dnb.

Conclusion: Estimate of risk of extreme responses depends on tails ofE[NT (u)] while expected damage depends of its weighted average.

Expected damage E[ln(DT )] ≈ ln(E[DT ]):,

If stress variability is stationary and its correlation has short memory thenV(DT ) ≈ σ2 T and E[DT ] = d T hence

R(DT )2 ≈ σ2

d2

1

T,

and is often neglected for new structures with long service time T , whileE[ln(DT )] ≈ ln(d) T . We call d - damage growth rate.

At the design stage d is estimated from mathematical models for externalloads and structures response (systems of linear, non linear diff.equation). Often the parameters (mass, stiffness and damping) may notbe exactly known and hence the damage growth rate d becomesuncertain.

Sensitivity analysis - Taylor expansion:

Corrections to account for the parameter uncertainties are oftenestimated by means of Gauss error propagation formula (accuratefor small parameter variations).

For simplicity assume that damage growth rate d depends only onone normally distributed error, Z ∈ N(0, 1), say.

Then E[d ] and V(d) can be approximated by means of

d(Z ) ≈ d(0) +∂d

∂z(0) Z +

1

2

∂2d

∂z2(0) Z 2.

Fourier expansion using Hermite polynomials1 :

For damage rate d(Z ) such that E[d(Z )2] <∞,

d(Z ) =∞∑j=0

cjHj(Z ) ≈n∑

j=0

cjHj(z) = dn(Z ),

cj = E[d(Z )Hj(Z )]. The truncated polynomial dn converges in L2 to d .The rate of convergence is quite fast for a smooth functions.

1In the one-dimensional case, the normalized Hermite polynomials are

H0(z) = 1, H1(z) = z , H2(z) = (z2 − 1)/√

2, H3(z) = (z3 − 3z)/√

6,

and higher order polynomials can be generated recursively

√n + 1Hn+1(z) = zHn(z)−

√nHn−1(x).

Example I: vehicle on a rough road.

Symbol Value Unitms 3400 kgks 270 000 N/mcs 6000 Ns/mmu 350 kgkt 950000 N/mct 300 Ns/m

0 1000 2000 3000 4000−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

distans i meter

vägh

öjd

i met

er

0 2 4 6 8 10

10−8

10−6

10−4

10−2

100

Typical road spectrum SMIRA(ξ) = 10a0( ξξ0)−w , ξ > 0.01, and

a0 = −2.3, w = 3.3, ξ0 = 0.2m−1.

Transfer function

H(ω) =msω

2(kt + iωct)

kt − (ks+iωcs)ω2ms

msω2−ks−iωcs−muω2 + iωct

(1 +

msω2

ks −msω2 + iωcs

).

Response spectrum for a vehicle at speed v ,

Sσ(ω) =1

v|H(ω)|2SMIRA (ω/(2πv)) .

For Gaussian road d ≤ dnb = 2(3/2)kΓ(1+k/2)2π

√λ2 λ

(k−1)/20 where

λn =

∫ +∞

0ωnSσ(ω) dω.

−4 −2 0 2 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

v = 25 + 5 · Z m/s

velocities between 55 and 125 km/h

Hermite polynomial expansion:

d3(Z ) = 0.08 + 0.04H1(Z ) + 0.01H2(Z ) + 0.001H3(Z ).

Coefficient of variation is 0.52. Error propagation formula

dG (Z ) = 0.075 + 0.04 · Z + 0.007 · Z 2.

However, likely, the measured road profile is non-Gaussian (est.kurtosis is ca 3.6, skewness 0.1!)

Sensitivity for kurtosis - Transformed Gaussian loads

Winterstein S. et. al. (1994) - transformation: X (t) Gaussian process,Y (t) = G (X (t)) non-Gaussian road irregularity model, viz.

Y (t) = m H0 + K σ [H1(X (t)) + c2H2(X (t)) + c3 H3(X (t))]

Input: [m, σ2, sk, ekrt], mean, variance, skewness, excess kurtisis. Output[K , c2, c3]2 see e.g. WAFO

Choice of spectrum SX (ω)?

2K = 1/√

1 + 2 ∗ c22 + 6 ∗ c2

3 ,c2 = (sk/6) · (1− 0.015|sk|+ 0.3sk2)/(1 + 0.2ekrt),

c3 = 0.1c4((1 + 1.25ekrt)1/3 − 1), c4 = (1− 1.43sk2/ekrt)1−0.1(3+ekrt)0.8

.

Simulation of Gaussian processes:Spectral method: discretize spectrum and use of spectral representation,

X (t) ≈N∑

i=1

σiRi cos(ωi t + φi ),

Ri iid standard Gaussian and φi are iid uniform on [0, 2π].

Moving Averages (MA): smooth Gaussian white noise;

X (t) = σ

∫f (t − x) dB(x) ≈ σ

√∆x

N∑i=1

fi Xi ,

f (x) - kernel function∫

f (x)2 dx = 1; B(x) - Brownian motion; Xi iid std normal; ∆x = xi+1 − xi , fi = f (t − xi ).

The spectrum of X (t)

SX (ω) =σ2

2π|F f (ω)|2,

F is the Fourier transform.

Symmetrical Laplace noise - LMA

Replace Brownian motion by symmetrical Laplace motion, practically use

σ√ν√

∆iXi instead of σ√

∆xXi .

Here ∆i iid Gamma variables Γi ∈ Gamma(∆x/ν, 1). LMA process

X (t) =

∫f (t − x) dΛ(x) ≈ σ

√ν

N∑i=1

fi√

∆i Xi ,3

Useful property of LMA: conditionally Γi = γi , X (t) is zero meannon-stationary Gaussian process with V(X (t)) = σ2ν

∑f (t − xi )

2 γi .

3E[X (t)] = E[X (t)3] = 0, V(X (t)) = σ2, exc. kurt. κ = 3ν∫

f 4(x) dx .

Sensitivity for excess kurtosis ekrt > 0

0 0.5 1 1.5 20

20

40

60

80

100

120

140

160

180

200

Excess kurtosis κ

dam

age

grow

th r

ate

in %

In this example slight non-Gaussianity of

loads has moderate influence on damage

rate.

The damage increase is of similar magnitude as conservatism of thenarrow band bound. Hence it can be safer to not correct the nb. boundby using some specific properties of the spectrum (and Gaussian model).

Future work:

I Sensitivity of damage rates and extreme responses for nonGaussian loading.

II Investigation how to choose (estimate) kernel f (x) -symmetrical, asymmetrical.

III Work on algorithms to make LMA load modelling accessiblefor engineers.

Example II: Wind load on a blade of a wind turbine - stallflutter.

U

r

V

pitch

2b

elastic axis

α′′ + α/U2 = 2Cm/(πµr 2α) + F sin(kτ),

α(τ), non dimensional time τ = tV /b

U(τ) =1

bωα

√V 2

rot + (Vg + V (t))2,

Cm(τ) is defined by a system of non linearequations.

Some dynamical properties of the blade

0 5 10 15 20 25 300

0.05

0.1

0.15

0.2

0.25

0.3

0.35

U

Response

The bifurcation plot U no gusts.

0 5 10 15 20 25 300

1

2

3

4

5

6 x 10 3

U

dam

age

rate

Damage rate no gusts.

Turbine rotating with 25.4 RPM and average head wind Vg below 25m/s, then 16.1 < U < 17.9.

For rotor at rest and the mean gusts 15 < Vg < 45 m/s, 4.8 < U < 14.5.

It seems that the blade parameters are well chosen in respect to minimize

risk for fatigue damages

Damage rate - gusts includedGusts are stationary zero mean Gaussian processes having Devenportspectrum, e.g. SV (ω) depends on Vg .

10 20 30 40 50 60 70 8010 8

10 7

10 6

10 5

10 4

10 3

10 2

Vg

Rotor at rest.

10 20 30 40 50 60 70 8010 4

10 3

10 2

Vg

Rotor working with 25.4 RPM.

Dots: damage rates obtained when gusts fluctuations are neglected;

Crosses: estimated damage rates with variable gusts; Solid line: Hermite

polynomial approximations of 7th, 5th order, respectively.

Acknowledgements:

This presentation uses results obtained by our group working onrandom loads from Chalmers TH, Lund TH and IIT Madras.

The group consists of:

Anastassia Baxevani; Anders Bengtsson (LTH); Klas Bogsjo(Scania); Thomas Galtier (CTH); Sayan Gupta (IITM); WengangMao (CTH); Krzysztof Podgorski (LTH); Igor Rychlik (CTH);Sunetra Sarkar (IITM); Joerg Wegener (LTH); Jonas Wallin (LTH)and Sofia Aberg.

THANK YOU FOR ATTENTION!

Example III: Wind load on antenna mast.

m1 m2m10

c1 c2 c10c3

k1 k2 k10k3

F1(t) F2(t) F10(t)

123

54

76

98

10

Loads: Fi (t) = ci (Vg + V (t))2

Vg - average wind speed,

V (t) - zero-mean wind gusts

V (t) =∫

f (t − x) dΛ(x)

Response:X (t) = c0 + c1

∫h(t − x)V (x) dx

+c2

∫h(t − x)V 2(x) dx .

X (t) = c +

∫q(t − x) dΛ(x) +

∫ ∫Q(t − x , t − y) dΛ(x) dΛ(y),

where q = c1 h ∗ f , Q(x , y) = c2

∫h(t)f (t − x)f (t − y) dt.

Measured wind gusts 200 km away from cyclone, left plot.

Mean speed Vg = 6.5 m/s; St. Dev. 2.52; Skew. 0.28; Exc. kurt. -0.04.

Possible model: asymmetric LMA load

V (t) =

∫f (t − x) dΛ(x), Λ(x) = ζ x + µΓ(x) + σ B(Γ(x)).

(we used κ = 0.01 > 0.) Simulations of 45 minutes of LMA-gusts, right

plot.

0 10 20 30 40−10

−8

−6

−4

−2

0

2

4

6

8

10

win

d gu

sts

spee

d m

/s

45 minutes of measured wind gusts

0 10 20 30 40−10

−8

−6

−4

−2

0

2

4

6

8

10

m/s

45 minutes of sim. wind gusts

Estimated spectrum compared with ”Devenport type” spectrum

S(ω) = 0.05 · (7200/2/pi) · Vg/(2 + (75ω/Vg )2)1.4.

(Devenport spectrum CHa = 0.002,w1 = 5/6, c0 = 286), left plot.

The kernel f (x) in LMA-model - right plot.

0 5 10 15 20

10−4

10−3

10−2

10−1

100

101

ω

S(ω

)

Wind Spectrum

−50 0 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9kernel f(x)

(Asymmetrical loads then rainflow damage needs to be used!)

Galtier, T. Gupta, S. Rychlik, I. (2010) Crossings of Second-order Response Processes Subjected to LMA Loadings,

to appear in Journal of Probability and Statistics.