Download FATIGUE ANALYSIS FOR STRUCTURES UNDER STOCHASTIC LOADING

Proceedings of the 6th Annual ISC Research Symposium
ISCRS 2012
April 13, 2012, Rolla, Missouri
FATIGUE ANALYSIS FOR STRUCTURES UNDER STOCHASTIC LOADING
ABSTRACT
Predicting the peak distribution of stochastic stress responses is
critical for various fatigue analysis problems with stochastic
uncertainties. In this paper, a method, based on the first-order
reliability method (FORM), is proposed for estimating the peak
distribution of stochastic response. The method linearizes the
stochastic process at the associated most probable point. After
linearization, the non-Gaussian, nonlinear stochastic response is
transformed into an equivalent standard Gaussian process. The
peak distribution is then approximated using the rice’s formula
by considering the correlation of the stochastic process at time
instants. The method is capable of dealing with problems with
both random variables and stochastic processes. A fatigue
analysis example of the hydrokinetic turbine blades is adopted
to demonstrate the proposed method. The results show that the
solutions obtained from the proposed method are close to their
counterparts from Monte Carlo simulations (MCS).
1. INTRODUCTION
Stochastic responses, as the outputs of functions with random
variables and stochastic processes, are very common in
practical applications. Studying the statistical properties such as
the correlation, extreme, moments, of the response processes is
of great interest to many engineers and researchers [1, 2]. From
the aspect of reliability analysis, the extreme value of a
stochastic process is very important, because it is directly
related to the probability of failure of the stochastic response.
The extreme value of a stochastic process can be divided into
two categories, the global extreme value and the local extreme
value. The global extreme value is associated with the firstpassage failure of the process, while the local one corresponds
to the accumulated upcrossings of a process, such as
accumulated fatigue damage. This paper mainly focuses on the
local extreme value or peak of a process as it is the basis for
estimating the fatigue damage of structures [3].
In the past decades, many progresses have been made in
estimating the peak distribution of stochastic processes. For
instance, Winterstein [4] proposed Hermit moment models for
analyzing the extremes and fatigue of structures under
nonlinear random vibrations. In his method, the non-linear
narrowband stochastic process is represented as a monotonic
function in terms of Gaussian stochastic process. Due to the
monotonic characteristic, a peak of the Gaussian process is
associated with a peak of the non-Gaussian stochastic process.
Madsen [5] derived an expression for the peak distribution of
Gaussian stochastic process with different regularity factors.
Based on his derivation, the peak distribution of a Gaussian
process can be easily obtained. Later, in order to estimate the
extreme valued distribution of more general cases, Dunne [6]
developed an extreme value prediction method for the
nonlinear beam vibrations using the measured random response
histories. Similarly, Gupta [7] proposed a numerical algorithm
for approximating the peak distribution of Gaussian loads
combinations. In the numerical algorithm, the importance
sampling method is employed to estimate the multidimensional
integrals.
Even if the above developed analytical methods can
estimate the peak distribution of stochastic processes, they are
limited to some special cases like the Gaussian stochastic
process, combination of Gaussian stochastic processes,
monotonic function of Gaussian process, and so on. The
simulation method including the Monte Carlo simulation,
importance sampling method [8, 9], can offer good accuracy of
estimations. The efficiency of these simulation methods,
however, is not acceptable. Especially when the evaluated time
period is long, the computational effort of the simulation
method is very expensive.
The aim of this paper is to develop an analytical method to
predict the peak distribution of stochastic responses, which are
functions in terms of both random variables and stochastic
processes. Since the method would not make any assumption
about the shape of the stochastic response, the response
function does not have to be a monotonic function of stochastic
process. The proposed method is expected to be applied to
various applications with time-dependent uncertainties.
Representative examples include the fatigue analysis of
offshore structures under random wave and wind loading, the
reliability analysis of dynamic structures under random
vibrations and so on. It will play a vital role in evaluating the
performance of stochastic responses when there is not enough
experimental data available.
In section 2, we will briefly introduce the peak prediction
problem of stochastic response and its application in the fatigue
analysis of structures. In section 3, the idea of linearizing the
nonlinear stochastic process at the most probable point will be
discussed. We will then investigate the way of transforming the
non-Gaussian process into standard Gaussian one. The peak
distribution is finally approximated after the derivation of the
correlation function for the transformed stochastic process.
Following section 3, a design example of hydrokinetic turbine
blades is studied to validate the proposed methodology in
1
section 4. The results of the example are discussed in section 5
and the future work is also given in this section.
2. STATEMENT OF ROBLEM
2.1. Peak Distribution of Stochastic Response
For problems under uncertainties, the stochastic responses can
be represented as a general function as
(1)
Z (t )  g (X, Y(t ))
where X  {X1 , X 2 ,
X n } is the vector of random variables,
Y(t )  {Y1 (t ), Y2 (t ), Ym (t )} is the vector of stochastic
processes and Z (t ) is the stochastic response.
Since Z (t ) is also a stochastic process, as indicated in
Fig. 1, the trajectory of Z (t ) fluctuates over the time period
[0, t], and local maxima or peaks occur during the time period.
Sometimes we are interested in these peaks, especially when
we are estimating the time-dependent reliability and fatigue
damage of structures.
Z (t )
t
t
Fig. 1 Peaks of a stochastic process
For a peak larger than z , it has the following mathematical
characteristics
 Z (t )  z

(2)
 Z (t )  0

 Z (t )  0
Based on Eq. (2), the peak distribution of stochastic
response Z (t ) is given by [5]
0

f Z , Z , Z ( z, 0,  )d 
According to the level crossing counting method, the
number of counted cycles with amplitude greater than z is
given by
(5)
np  v pT [ f p ( z)  fv ( z )]
where v p is the number of upcrossings for the mean value
level over a unit time, T is the evaluated time period.
The expected fatigue damage E ( D) under continuous
variable loading is

E ( D)  v pT
 wz [ f
k
p
( z )  f v ( z )]dz
(6)
0
in which v p stands for the number of cycles per unit time, w
and k are SN curve parameters related to the material
property. These two parameters are obtained from the fatigue
tests of materials.
It is apparent from Eq. (6) that the most critical part for the
fatigue damage analysis is predicting the peak distribution of
stochastic response. In section 3, the first order reliability
method (FORM) will be employed to evaluate the peak
distribution.
Peaks
1
f p ( z) 
vm
2.2. Expected Fatigue Damage
Once the peak distribution of the stochastic response is
available, we can easily get the corresponding valley
distribution because the distribution of valleys is symmetrical
with that of peaks [3]:
(4)
f v ( z )  f p ( z )
(3)

where vm is the rate of local maxima, and f Z , Z , Z () is the
joint PDF of Z (t ) , Z (t ) and Z (t ) .
Even though researchers have proposed formulations for
Eq. (3) under some special cases [10-12], there is no close form
expression for it. We will discuss how to approximate Eq. (3)
using FORM in Section 3. In the following subsection, the
application of peak distribution in fatigue damage analysis will
be investigated.
3. PREDICTION OF PEAK DISTRIBUTION BASED ON
FORM
In this section, we first review FORM. After that, we discuss
the way of transforming the non-Gaussian stochastic process
into Gaussian one. We then derive the equations for predicting
the peak distribution.
3.1. Review of the First Order Reliability Method
FORM is the most widely used approach to evaluating the
reliability. For a limit-state function, Z (t )  g (X, Y(t )) , the
random variables X and Y(t ) are transformed into standard
normal variables U(t )  (UX , UY (t )) at each time instant t.
After the transformation, the limit-state function becomes a
function of standard normal variables
(7)
Z (t )  g (T (UX ), T [UY (t )])
in which T () denotes the operator of transforming standard
normal variables into the original ones.
Once the limit-state function is transformed into the U
space, we search for the Most Probable Point (MPP)
U*  (U*X , U*Y (t )) by solving the following optimization
problem [13, 14]

u*
min
u
(8)

*
*

subject to g (T (u X ), T [u Y (t )])  z
2
C Y1 (t1 , t2 )

0
CY (t1 , t2 )  


0

At the MPP point, the limit-state function has its highest
probability density and the instantaneous probability of failure
p f  Pr{Z (t )  z} is approximated as
p f  Pr{Z (t )  z}  ( )
(9)
and
  u*
(10)
in which () is the CDF of standard normal variable.
3.2. Linearization and Transformation
By employing the Taylor expansion method, the limit-state
function Z (t )  g (X, Y(t )) is linearized at the MPP. Based on
the linearization, we have the following equivalences
For z  Z
Pr{Z (t )  z}  ( )
(11)
for z  Z
Pr{Z (t )  z}  1  Pr{Z (t )  z}
 1  (  )
(12)
 1  (1  (  ))
 (  )
Therefore, for z  Z , the failure event Z (t )  z is
equivalent to
(13)
a (t )UT  
(18)

0
0 


0
0 






Ym
0
 
 0
The occurrence of a peak indicates a downcrossing by
W (t ) / 0 (t ) of zero level and thus the rate of peak is given by
[5]
 (t )
(19)
vm  m
2
where
 2  (t , t )
0 (t )2 
(20)
t1t2
and
 2  (t1 , t2 ) 
2 
1
m (t )2 
(21)


t1t2  0 (t1 )0 (t2 ) t1t2  t t t
1
2
Substitute Eq. (16) into Eq. (20) and Eq. (21), we have
0 (t )2  a (t )C2 (t , t )a (t )T  a (t )C(t , t )a (t )T
a (t )C12 (t , t )a (t )T  a (t )C1 (t , t )a (t )T
(22)
(14)
m (t ) 2
in the above equations,
a (t ) 
Since U  (UX , UY (t ))
T
T
u
*
u
*
(15)
is a vector of Gaussian random
a (t )  1 , the process W (t )  a (t )UT is a
standard Gaussian process. The computation of the probability
that the peak of stochastic process Z (t ) is less than z , is thus
transformed into solving the probability that the peak of
standard Gaussian process W (t ) is less than  or   .
In the following section, we will discuss the way of
approximating the probability Pr{Wp (t )   } , where Wp (t )
variables and

 2  (t1 , t2 )
 3  (t1 , t2 ) 
1
1

 2

0 (t1 )02 (t2 ) t1t2
02 (t1 )0 (t2 ) t1 2t2 



 3  (t1 , t2 )
 4  (t1 , t2 ) 
1
1



2
2
0 (t1 )0 (t2 )  2t1 2t2  t t
 0 (t1 )0 (t2 )  t1t2
1
(23)
C1 (t1 , t2 ) 
0

C(t1 , t2 ) 0


Y
t1
0 C 1 (t1 , t2 ) 
(24)
0

C(t1 , t2 ) 0


Y
0
C
(
t
,
t
)
t2

2 1 2 
0
0

C12 (t1 , t2 )  

Y
 0 C 12 (t1 , t2 ) 
C2 (t1 , t2 ) 
3.3. Approximation of Peak Distribution
Recall that we have W (t )  a (t )UT , the auto-correlation
function of the standard Gaussian process W (t ) is
 (t1 , t2 )  a (t1 )C(t1 , t2 )a (t2 )T
2 t
in which
stands for the peak of W (t ) .
(25)
(26)
C 1Yi (t1 , t2 )   Yi (t1 , t2 ) / t1 , i  1, 2,
,m
(27)
C (t1 , t2 )   (t1 , t2 ) / t2 , i  1, 2,
,m
(28)
Yi
2
(16)
Yi
and
in which
where I nn
0





C Ym (t1 , t2 ) 
0
0
Y1
for z  Z , the failure event Z (t )  z is equivalent to
a (t )UT  
0
0
I

C(t1 , t2 )   nn

Y
 0 C (t1 , t2 ) 
is an identity matrix and
C12Yi (t1 , t2 )   2  Yi (t1 , t2 ) / t1t2 , i  1, 2,
(17)
3
,m
(29)
Provide that Y(t ) are stationary processes and there is no
explicit
t
involved
in
the
limit-state
function
Z (t )  g (X, Y(t )) , we have
(30)
a (t )  0
After derivations and simplification, Eqs. (22) and (23)
turn out to be
(31)
0 (t )2  a (t )C12 (t , t )a (t )T
m (t )2 
a (t ) C1122 (t1 , t2 )
0 (t )
α  t  and  .
 Step 3: Calculate the regularity factor  and estimate the
CDF and PDF of the Gaussian points.
 Step 4: Evaluate the expected fatigue damage using Eq. (6),
and finally compute the estimated fatigue life of the structure.
Initialize parameters
T
a (t )
t1  t2  t
(32)
2
Gaussian points for
integration
zi , i  1, 2, , N
The regular factor of process W (t ) is given by

0 (t )
m (t )
a (t ) C1122 (t1 , t2 )
t1  t2  t
Limit-state function
Z  g (X, Y(t ))  z
(33)
a (t )C12 (t , t )a (t )T

a (t )T
The regular factor  is defined as the ratio between the
rate of zero-upcrossings and the rate of local maxima. For a
standard Gaussian process with regular factor  , the
cumulative density function (CDF) of its peaks can be
computed by

F ( )  1  (
)e

2
2
(

1 
1  2
Having derived the following equation,
Pr{Z p (t )  z}  Pr{Wp (t )   }
we obtain
Pr{Z p (t )  z}
2


2

)
α, 
Calculate the PDF and
CDF of peaks
(34)
Expected fatigue damage
(35)
Estimated fatigue life
(36)
)   e (
)
1  2
1  2
in which  is obtained from Eq. (9) and (10), and  is
estimated from Eq. (33).
We now have all the equations needed for estimating the
peak distribution of stochastic process Z (t ) . Based on these
equations, the stress peak distribution of structures under
stochastic loading can be approximated. The expected fatigue
damage per unit time then can be evaluated using Eq. (6) given
that we know the SN curve of the structure material.
 1  (
2
First Order Reliability
Method (FORM)
Fig. 2.
Flowchart of Fatigue Analysis for Structures under
Stochastic Loading
4. CASE STUDY
In this paper, a hydrokinetic turbine blade designed for the
Missouri River is adopted as a demonstration of the proposed
method. The blade is subjected to the stochastic river flow
loading. Fig. 3 shows the river flow loading on the turbine
blade. The stress response at the root of the blade is a stochastic
process due to the time-variant characteristics of river flow
velocity. The cross section at the root of the turbine blade is
presented in Fig. 4. The peak distribution of the stress needs to
be evaluated to analyze the fatigue life of the blade.
3.4. Numerical Procedure
The numerical produce for the fatigue analysis of structures
under stochastic loading is summarized as follows and depicted
in Fig. 2.
 Step 1: Initialize the random variables and stochastic
processes (i.e. transform non-Gaussian into Gaussian,
analyze the correlation characteristics of the stochastic
processes. Get the Gaussian points needed for the integration
in Eq. (6).
 Step 2: MPP search for the limit-state functions associated
with every Gaussian point. Calculate the corresponding
4
Table 1 Deterministic variables and parameters

Variable
110 kg / m
3
Value
3
R
c
CL
1m
0.23 m
1.5
vf
Table 2 Random variables
R
Variable
Mean
l1
0.18 m
M flap
l1
l2
4.1. Data
According to the Danish code [15, 16], the flapwise bending
moment at the root of the turbine blade is given by

R2
(37)
M flap  v 2f cCL
2
3
in which  is the density of water, v f is the river flow
velocity, R is the radius of turbine blade, c is the chord
length at 2R / 3 and CL is the lift coefficient at 2R / 3 .
Amongst these variables, the river flow velocity is
assumed to be a Gaussian process with known mean, standard
deviation and auto-correlation function. A trajectory of river
flow velocity over a certain time period is shown in Fig. 5.
4
Lognormal
Autocorrelation
N/A
Distribution
l2
0.05 m
110 m
Lognormal
N/A
vf
2.5 m/s
0.5 m/s
Gaussian
Eq. (38)
The auto-correlation function of river flow velocity is
 (t  t )2 
(38)
v (t1 , t2 )  exp   2 21 
0.5 

The stress of the turbine blade is computed by
 2
R2
v f cCL
M flap
l
S 3
 2  2 2 3
(39)
l1l2 /12 2
l1l2 / 6
Fig. 3. River flow loading on the turbine blade
Fig. 4. Cross section of hydrokinetic turbine blade at the root
Standard
deviation
1103 m
4.3. Results and Discussion
The peak distribution of the turbine blade is approximated
using the proposed new method. In order to evaluate the
accuracy of the proposed method, the obtained results are
compared with their counterparts from MCS. In the MCS, the
Expansion Optimal Linear Estimation method (EOLE) [17] is
employed to generate the trajectories of river flow velocity.
More details about this method can be found in [18]. Figs. 6
and 7 indicate the comparison of the PDFs and CDFs of stress
peaks computed from the proposed new method and the MCS
with 106 samples, respectively.
2.5
x 10
-7
New method
MCS
Velocity with correlation
2.1
2
2
1.5
PDF
velocity (m/s)
1.9
1.8
1
1.7
1.6
0.5
1.5
1.4
0
10
20
30
40
50
t (s)
60
70
80
90
0
100
Fig. 5. A trajectory of river flow velocity
Due to manufacturing process, the dimensions of turbine
blades are random variables. The random variables and
deterministic parameters of this example are given in Table 1
and 2, respectively.
0
0.5
1
1.5
2
Stress peak
2.5
3
3.5
x 10
Fig. 6. PDFs of peaks of turbine blade stress
5
7
1
New method
MCS
0.8
CDF
0.6
0.4
0.2
0
0
0.5
1
1.5
2
Stress peak
2.5
3
3.5
7
x 10
Fig. 7. CDFs of peaks of turbine blade stress
From the results shown in Figs. 6 and 7, it is apparent that
the results of the proposed new method are close to those of
MCS. It implys that the proposed approximation method can be
employed to substitute the MCS, to improve the efficiency.
There are some errors for the proposed new method when
comparing with the benchmark of MCS. The errors may arise
from the linearization in FORM.
5. CONCLUSIONS
We proposed an analytical method for the peak
distribution analysis of problems with both random variables
and stochastic processes. The method is based on the first order
reliability method. The nonlinear and non-Gaussian stochastic
process is linearized and transformed into an equivalent
Gaussian process. After analyzing the correlation of
transformed process at time instants, the peak distribution of
the stochastic process is approximated using the Rice’s formula.
Even if there are some negligible errors exist, the propose
method can be applied to many applciations with stochastic
loadings and uncertainties.
Our future work include improving the accuracy of the
proposed method and applying the proposed method to fatigue
analysis of hydrokinetic turbine blades.
6. ACKNOWLEDGMENTS
The authors gratefully acknowledge the support from the
Intelligent Systems Center at the Missouri University of
Science and Technology and the Office of Naval Research
through contract ONR N000141010923 (Program Manager –
Dr. Michele Anderson).
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