Probabilistic Assessment of Fracture Mechanics of Low Pressure Turbine
Disk Keyway
Lieh Chen (陳烈)1, Lih-jier Young(楊立杰)2 and Shih-teng Lin (林士騰)2
1
Department of Industrial Engineering and Management, Ta Hwa Institute of Technology,
No. 1, Dahua Rd., Qionglin Shiang, Hsinchu County, 307, Taiwan, R.O.C.
Tel: 03-592-7700-2755, Fax: 03-592-6848
Email: ietch@thit.edu.tw
2
Department of Applied Mathematics, Chung-Hua University,
No. 707, Sec. 2, Wufu Rd., Hsinchu, 30012, Taiwan, R.O.C.
Tel: 03-518-6392, Fax: 03-518-6435
Email: young@chu.edu.tw, steng.lin@msa.hinet.net
Abstract
This study presents the development of a fracture mechanics-based probability analysis which can be used
to estimate the remaining life of the low pressure turbine against unstable fracture in the rotor disk keyway.
Analysis focuses on the stability of crack growth of a semi-elliptical flaw in the rotor disk keyway. The crack
stability is judged on the basis of the linear elastic fracture mechanics. Then the Monte Carlo technique is
introduced to deal with several random variables. Based on the evaluation results, the probability of failure of
observed crack indications can be determined. The inspection interval could be justified.
Keywords: Monte Carlo simulation, probabilistic fracture mechanics, stress corrosion cracking, fracture
appearance transition temperature
1. Introduction
The electric power and the electric component
demand have been increasing
remaining life of low pressure turbine disks such as
fatigue crack growth rate, fracture toughness, stress
sharply. However,
intensity factor as a function of crack size, material
the new materials which are fully capable of
stress corrosion cracking rate as function of
withstanding stress corrosion cracking for large low
environment, and loading conditions. To examine the
pressure turbines and rotors have not yet been
integrity of the cracked low pressure turbine disk,
developed. As an observation, many low pressure
both
turbines are subjected to stress corrosion cracking in
mechanics are used. Deterministic fracture mechanics
wet bore, keyway, and dovetail area experienced
(DFM) method is used to ascertain allowable flaw
wetness in operation. Catastrophic failure of the disk
size, while probabilistic fracture mechanics (PFM) is
could occur if the cracks grow to critical size. It is,
an alternative for determining inspection interval.
therefore, one of the major concerns for the power
The Monte Carlo technique is one of numerical
industry to limit the probability that disk will burst
probabilistic analysis approaches that are amendable
and generate turbine missiles.
to statistical problems governed by a certain amount
Many variables are involved in estimating the
deterministic
of random variables.
and
probabilistic
fracture
There are many literatures focusing on choosing
Equation (1) indicates that the local stress near a flaw
random variables in Monte Carlo simulation.
depend on the product of the nominal stress and the
However, only a few studies adopted Monte Carlo
square root of the flaw size a. This product is called
method in analyzing the structural integrity of
“stress
cracked structures. Viswanathan (1984) estimated the
fundamental relationship.
intensity
factor”
to
emphasize
this
remaining life of the rotor with Monte Carlo
The crack growth of the disk due to fatigue or
simulation [1]. Sire and Kokorakis (1991) assessed
stress corrosion is predicted in accordance with the
the fatigue factor of container ship [2]. Zhu and Lin
theory of the conventional fracture mechanics. The
(1992) studied the aviation structure of fatigue crack
dependence of the fatigue crack growth rate [8, 9, 10]
growth under random loading [3]. Liao and Yang
on stress intensity factor can be conveniently
(1992) performed a probabilistic evaluation on
represented as follows:
fatigue
crack
growth
of
aluminum-alloy
[4].
Rosario and Roberts [5] conducted a remaining life
evaluation for a low pressure turbine disk in 1997.
Low pressure turbine disks are exposed to a
n1

 1-  Kth 
 K 

da


 C (K )m 
n2
dN


K

max 
1

  K 
c 
 







n3
(2)
damaging
where da / dN is the fatigue crack growth rate,
mechanisms include high cycle fatigue, low cycle,
K th is the threshold value of stress Intensity factor,
and stress corrosion cracking. These conditions can
K c is critical value, and n and C are constants
cause cracking of blade-fit area and keyway at bore
depending on the material and environmental
of disks. Therefore, our study tries to integrate
conditions. This typically results in a graph of the
deterministic fracture mechanics with probabilistic
form in Fig. 2. It is noted that the graph exhibits a
analysis in a consistent and convenient manner to
threshold stress intensity factor below which cracks
facilitate remaining life evaluation. Fig. 1 is a
will not propagate.
highly
wetted
environment,
whose
schematic drawing of a typical low pressure turbine
2.1 Finite element stress analysis
disk with a keyway and a rotor.
This section is not an integral part of fracture
2. Theoretical Investigations
analysis method, and will not be treated in detail.
The theory proposed by Griffith [6] in 1920
However, the determination of stress profile through
explained that it was the total energy release causing
the disk constituting one of the basic inputs to the
the crack growth. Then the stress intensity factor, KI,
fracture mechanics analysis. The general purpose
developed by Irwin [7], is a one parameter
finite element computer program ANSYS [11], was
representation of the stresses in the area of a crack tip.
utilized to perform finite element analysis.
It is a purely numerical quantity providing complete
knowledge of the stress field at the crack tip, which is
Two-dimensional axisymmetric finite element
defined as follows:
i j 
2.1.1 Loading and Boundary Conditions
K
model of the subject disks was constructed to
fi j    ..........
(1)
determine near bore disk stresses due to centrifugal
Where r is a distance from crack tip, and θ is the
effects using ANSYS computer program. It is noted
2 r
angle with respect to crack plane.
that an equivalent traction is applied on the disk OD
to simulate the blade centrifugal loading, which is
b 
N  Fcentrif
radial depth, R is outside diameter, t is the
thickness,
(3)
Acontact

is
the
angle,
is
Fi
the
boundary-correction factor, and G j is the influence
where N is the number of blades, Fcentrif is the
coefficients
centrifugal force per blade, and
distribution. The coefficients of the polynomial
Acontact is rim
contact area.
corresponding
to
the
jth
stress
expression are used in equation (5) to calculate the
The axial displacements at the center of the disk
stress intensity factor corresponding to the crack
are all fixed to simulate half-symmetry as the
depth, b, and to the stress profile represented by both
imposed boundary condition. The resulting disk/shaft
finite
finite element model is shown in Fig. 3.
concentration factor given in equation (4).
element
for  hoop
solution
and
stress
The
same procedure can then repeated at various crack
2.1.2 Stress Concentrating Effect of Keyway
aspect ratios. Fig. 5 shows stress intensity factor
In order to perform stress intensity factor
profiles obtained at various crack aspect ratios.
calculation for cracks emanating from the high stress
gradient area in the disk keyway/bore, a stress
2.3 Fracture Toughness
concentration factor ( K t ) of 2.2 [12] for GE axial
The material property pertinent to the present
keyway design was utilized. The stress concentration
fracture mechanics analysis method is the plane strain
factor profile can be obtained by varying radial
fracture toughness which is a function of temperature.
distance ( x ) from the crown of the keyway which is
The method predicts that crack instability occurs
expressed as:
when the stress intensity factor K I
2
4

 R 
 R  
Kt  x   1   Kt  1 0.25 

0
.
75

 Rx 
 Rx

 

(4)
equals the
material fracture toughness K IC .
The great majority of low pressure turbine
materials are built from ASTM A471 (3.5%
Ni-Cr-Mo-V) forgings for which a large number of
Where R is the keyway depth. Then the stress
toughness
data
have
been
generated.
Since
profile acting normally to the plane containing the
disk-specific fracture mechanics ( K IC ) data is not
crack can be immediately determined.
available, generic disk toughness for GE low turbines
were utilized to estimate K IC [14]. Fig. 6 is a
2.2 Stress Intensity Factor
schematic
Stress intensity factor expressions for surface flaw
in a cylinder shown in Fig. 4 have been determined
using finite element models [13], the stress intensity
K I   hoop
disk
toughness
versus
They can be fitted to a hyperbolic tangent cure such
that,
 T  T0 
K IC  A  B tanh 

 C 
(5)
where
of
temperature for typical low pressure turbine materials.
factor is expressed as:
b
b b t
 b 

Fi  , , ,  , G j   
Q
 t 
a t R
drawing
Where
T0
is
fracture
appearance
(6)
transition
temperature (FATT), T is disk temperature, A, B and
 a
Q  1  1. 464 b
1. 65
 hoop is the hoop stress, a is the surface length, b is
C are constants.
2.4 Crack Growth Rate
Crack growth due to stress corrosion cracking
uncertainty growth from each input variable to stress
(SCC) is the dominant mechanism in this analysis.
corrosion cracking life. The proposed method
The most widely accepted SCC crack growth
combines deterministic fracture mechanics with
equation used in the analysis of cracking in LP
probabilistic
turbines is given by Clark et al. of
relationships in matrix form between the various key
Westinghouse
[15] :
parameters.
 da 
 7302 
ln    C1 - 
  0.0278 y
dt
 
 T 
(7)
fracture
mechanics
to
develop
The essence of this approach is to
assign mean values and statistical distributions to all
of the key variables affecting the problem. Some key
where C1 is a material constant with a mean value of
parameters used in deterministic fracture mechanics
-4.968, and a standard deviation of 0.587, T is the
analyses of LP turbine rotor disks are known to vary
R ,  y is the
significantly, and can be assumed to behave in a
yield strength in ksi, and da/ dt is the growth rate
this paper are hoop stress, overspeed possibility,
in inch/hour.
fracture appearance transition temperature, operating
operating temperature of the disk in

random manner. These random variables adopted in
temperature, fracture toughness with and without
2.5 Critical crack size determination
prewarming, crack model, initial crack size, stress
Initiation crack depth can be determined during
the service time at the intersection between the stress
intensity factor K I and fracture toughness K IC .
From the determination of acr , it should be possible
to conclude that the low pressure turbine disk bore
integrity would be maintained in the case of severe
event such as the overspeed resulting from an
abnormal operation. However, the critical flaw crack
size is essentially dependent on stresses, flaw
geometry and disk material fracture toughness. These
parameters are dependent one another, and have
strong influence on critical crack depth and therefore
remaining life.
growth coefficient, and yield strength of the material.
The stress intensity factor and crack growth are
calculated deterministically for a few chosen standard
deviation of the above mentioned random variables.
These results were put into matrix form for use in the
probabilistic failure evaluation of low pressure
turbine. Performing each Monte Carlo iteration, a
random number is generated for each random
variable. The corresponding standard deviation is
calculated and is used to determine the stress
intensity factor at any time between initial crack
growth and final failure. Therefore, the time for the
crack to propagate from an initial flaw size to the
critical flaw size is obtained.
2.6 Estimating remaining life
Remaining life of LP turbine disks for keyway
3.1 Linear Congruential Generator
cracking is calculated by the initial crack size ( ai ),
the critical size ( acr ) and the crack growth rate
( da/ dt ), expressed by the following relationship,
trem
a  ai
 cr
da / dt
In this paper, the Random Number Generator is
made by Linear Congruential Generator, the equation
is
(8)
3. Monte Carlo Simulation
Monte Carlo simulation is used to analyze the
Yi 1   AYi  C 
where Y0 is the seed,
MODM
(9)
A is the constant multiplier,
C is the increment, and M is the modulus.
prewarming
3.2 Flow diagram for life assessment
By applying the input data to the analysis scheme
and
without
prewarming
respectively.
as shown in Fig. 6, the critical crack size, the
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[1]
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Fig. 10 and 11
for
Container
Ship
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inch, respectively.
Fracture Mechanics, Vol. 43, No. 1, Sep, 1992,
Fig. 12 and 13 show the
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pp.1-12.
prewarming and each with an initial crack of 0.34
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inch, respectively. By carrying out this investigation,
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three major findings are:
Mechanics,
1.
pp.651-655.
LP turbine with prewarming can significantly
increase the remaining life of LP turbine disk.
2.
[5]
Rosario,
No.
4,
Blaine
Nov,
W.
1992,
Roberts
failure of 10 , continued operation for another
Corrosion Cracking in Shrunk-On Disks of Low
9 and 8 year-period can be justified for the case
Pressure Turbines”, EPRI Steam Turbine Stress
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Corrosion Cracking Conference, March 19-20,
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case
of
0.25
inch
initial
withprewarming
and
without
A221, 1920, p.163.
[7] G.. R. Irwin, “Analysis of Stresses and Strains
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Near The End of a Crack Transversing a Plate,”
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crack
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43,
“Probabilistic Assessment of Failure by Stress
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4
3.
Darryl
Vol.
With limiting probability of catastrophic disk
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[8]
D.
W.
Heoppnre
and
W.
E.
Krupp,
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“Predication
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Application
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Component
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R2
y
ASME, Journal of Engineering Industrial, Series
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Rotor
x
[10] R. G. Forman and V. E. Kearney and R. M.
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da
dN
II
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K
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Fig. 2 Schematic representation of fatigue crack
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[14]
Stienstra,
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Auen,
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Fig. 3 Two-dimensional FEM model for Rotor-Disk
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[15] W. G. Clark, B. B. Seth, and D. M.
Shaffer,“
Procedures
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Stress Corrosion Cracking”, presented at Joint.
ASME/IEEE
October 1981.
Power
Generation
Conference,

2a
t
b
R1
Fig. 4 Crack model
K
IC
B
A
B
C
Temperature
Fig. 6 Hyperbolic tangent curve showing the
graphical interpretation of the parameters.

D o t/ Li n e s s h

5 0. 0 0


Fig. 5 Stress intensity factor profiles obtained at
4 0. 0 0


various crack aspect ratios (900 rpm).

NO_TIM
Input data

3 0. 0 0







2 0. 0 0




Times：1~10000
1 0. 0 0


Select var. min/max value
0 . 0 0             
0 .0 0


5 .0 0
Read random var.
1 0. 0 0
1 5. 0 0
2 0. 0 0
Y EA R
Fig. 8 Probabilistic frequency versus remaining
Calculate the physical property
life without prewarming (0.12 in)
Simulate the crack model

40.00


Calculate the crack size


30.00

SO_ TIM
Stress Corrosion Cracking




20.00



Estimating remaining life


10.00


Output data
0.00
             
0.00
Fig. 7 Flow diagram for a scheme of remaining life
evaluation
5.00



10.00
15.00
20.00
Y EA R
Fig. 9 Probabilistic frequency versus remaining
life with prewarming (0.12 in)
Dot/Lines show Me


Dot/Lines show Means
Dot/Lines show Me

60.00

40.00



 





30.00
SO_ TIM
NO _TIM

40.00




20.00


20.00















10.00



           
0.00
0.00


0.00
5.00
10.00
15.00
20.00
            
0.00
Fig. 10 Probabilistic frequency versus remaining
life without prewarming (0.25 in)




Dot/Lines show Means





SO_ TIM




20.00




10.00
0.00
0.00


5.00
10.00
15.00
20.00
Y EA R
Fig. 11 Probabilistic frequency versus remaining life
with prewarming (0.25 in)


Dot/Lines show Means
60.00







NO _TIM
40.00







20.00




0.00
          
0.00
5.00

15.00
20.00
Fig. 13 Probabilistic frequency versus remaining

30.00
            
10.00
life with prewarming (0.34 in)
40.00


Y EA R
Y EA R

5.00




10.00
15.00
20.00
Y EA R
Fig. 12 Probabilistic frequency versus remaining
life without prewarming (0.34 in)