FRACTURE MECHANICS EVALUATION OF THE XYZ UNIT 1 HP/IP ROTOR
SUMMARY
A bore condition assessment analysis has been performed on the XYZ Unit 1 HP/IP turbine
rotor. The results of the analysis indicate that the bore related failure probability is
essentially zero for thirty years from the date of the April, 2003 boresonic inspection. For
conservatism one-half of this time will be used to set the reinspection interval. Therefore the
rotor bore should be ultrasonically inspected again fifteen years from the date of this
inspection.
This recommendation is made on the assumption that the turbine rotor will be started ten
times per year in the future and that it will be started in accordance with the manufacturer’s
start-up procedures.
FRACTURE MECHANICS EVALUATION OF THE XYZ UNIT 1 HP/IP ROTOR
Fracture Mechanics Methodology
The stable growth rate of a crack in a component can be estimated knowing the depth of the
crack, the nominal stress surrounding the crack and the fracture toughness of the component
material. As the crack grows to a calculated critical depth, the growth becomes unstable and
the component is considered to have failed.
A typical fracture mechanics evaluation consists of finding the stress intensity factor for a
given crack depth and comparing that value to the material crack resistance known as the
fracture toughness. This is similar to comparing the stress applied to a component to the
yield strength of the component material.
The stresses which drive a crack in the bore of the turbine rotor are caused by the nonuniform distribution of temperatures particularly during a start up, the spinning of the
rotor and the pressure of the steam. These stresses are estimated using a general purpose
finite element program or one written specifically for turbine rotor bore analysis.
Required inputs for performing a thermal and mechanical stress analysis are the
following:
1. Geometry of the turbine rotor
2. Seal clearances of inlet, outlet and interstage seals
3. Total weight of the blades on each wheel
4. A profile of steam temperatures and pressures into and out of the turbine stages
versus time
5. A profile of steam temperatures and pressures into and out of the turbine seals versus
time.
6. A profile of turbine rotational speed with time
7. Thermal properties of the turbine rotor steel
Once the stress, material properties and the crack depth are known, a fracture mechanics
evaluation can be performed. This consists of calculating the growth of the crack with
time due to start-stop cycles and to creep during the steady state operation of the turbine.
At various points in time during the calculated growth of the crack, the stress intensity
factor is compared to the fracture toughness of the material.
From the above information a thermal transient heat transfer analysis is performed to
determine the distribution of temperatures in the turbine rotor at various times during a
start up and during steady state operation of the turbine. This is done using finite element
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analysis (FEA). FEA is a numerical procedure for analyzing structures that are too
complicated to be solved by classical analytical methods. A basic finite element concept
is discretization. The turbine rotor is discretized by modeling it as a series of finite
elements connected at each corner by nodes. Thermal material properties of the elements
such as thermal conductivity, specific heat, and density determine the rate of heat flux
between nodes. The result of a thermal transient analysis is the distribution of
temperatures at each node for given points in time.
As stated above, thermal properties will determine how heat flows from one node to the
other but the heat flux into the rotor surface from the surrounding steam is determined by
boundary conditions at the steam-rotor interface. To specify these boundary conditions in
the finite element computer code, the heat transfer coefficients between the steam and
rotor must be calculated. The six heat transfer surfaces found on a turbine rotor are listed
below.
1.
2.
3.
4.
5.
6.
labyrinth seal
rotating disk with a finite wall clearance
shrouded rotating disk with steam flow
metal to metal contact
rotating shaft in an infinite fluid
insulated surface
Once the temperature distributions are established at the nodes, the stresses within the
rotor body due to temperature are calculated for various points in time during the start up
and steady state operation. Next the stresses due to rotation are calculated. This requires
as input the profile of the rotor speed with time during a start up. The stresses due to
external pressure are calculated and lastly the stresses due to the weights of the blades.
All stresses are then combined to give a total stress calculated for various points in time
during the start up and steady state operation.
Information about the dimensions of existing cracks comes from a boresonic
examination. The various “hits” reported by the boresonic ultrasonic system are clustered
to some designated criteria. These clusters are then sized and the dimensions are
determined. These dimensions are inputs to the fracture mechanics evaluation.
Fracture mechanics is the engineering study of the load carrying capability of a structure
containing a crack. A parameter used in fracture mechanics is the stress intensity factor
which describes the intensification of an applied stress field near a crack tip. The stress
intensity factor is given by
KI   G  a
(1)
K I  stress intensity factor
  stress
G  geometry factor
a  crack depth
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Stress Analysis of the XYZ Unit 1 HP/IP Rotor
The thermal and mechanical stress analysis of the XYZ Unit 1 HP/IP rotor shown in
Figure 1 was performed using a general-purpose finite element program. The finite
element model of the rotor is shown in Figure 2.
Some details of the rotor periphery have been ignored in the finite element model. Any
errors introduced by this are negligible since the major concern is with the bore surface of
the rotor. The blades are not modeled and are accounted for by placing a lumped mass on
the outside surface of the wheels. It was assumed that the rotor is at 70  F before the
steam temperature profiles shown in Figure 3 were applied. This start was obtained from
Reference 2. The temperature profile in Figure 3 is for the HP portion of the rotor but was
used for the IP portion as well since thermocouple data was not available for the IP
section.
Some results of the finite element stress analysis are shown in Figure 4, Figure 5, Figure
7 and Figure 8.
Peak Stress
The maximum tangential stress occurs at 13.25 hours into the start and is located at 81.4
inches from the coupling. The stress at this time is about 84 ksi. This stress is due to the
thermal transient which begins at about 12 hours into the start and ends at about 13 hours
into the start.
Boresonic Examination of the XYZ Unit 1 HP/IP Rotor
The bore of the HP/IP rotor was inspected by ABC Boresonic Vendors and no significant
indications were found.
Fracture Mechanics of the XYZ Unit 1 HP/IP Rotor
Since the Fracture Appearance Transition Temperature (FATT) was not known for this
rotor, a value of 250 degrees F. was assumed. This value represents the worst-case
scenario for this vintage of rotors. Using information from publicly available literature, a
plot of fracture toughness versus temperature was constructed. This is shown in Figure 9.
Using this curve and the results of the thermal and stress analysis the critical crack was
calculated along the rotor bore. This is done by finding the temperature and stress of each
bore element during the start up and looking up the fracture toughness for each
temperature. This fracture toughness is then substituted for KI in Equation (1) and the
critical crack depth calculated. The result is shown in Figure 10. The stress intensity
factor model used was for a single edge crack. This method accounts for the fact that the
most susceptible area of a rotor bore may not be the point of highest stress but is the point
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at which the stress intensity factor of a possible crack is closest to the fracture toughness
of the rotor material. As the rotor heats up the fracture toughness increases.
Since the single edge crack formula was used to calculate these values, they should not be
taken as the depth at which an elliptical fatigue crack on or near the bore surface would
experience unstable growth. These values are used to locate the most susceptible areas of
the rotor. It can be seen that the lowest value of critical crack depth occurs below the
latter stages of the IP section at about 76 inches from the coupling. This occurs at 3 hours
into the start when the temperature of the bore is 129 degrees F. and the tangential stress
is 52,580 psi. The stress and temperature as a function of time at this location in the rotor
is shown in Figure 6. The fracture toughness at the time of the lowest critical crack is 53
KSI square root inch.
Since no significant indications were found, a hypothetical radial-axial crack 0.050
inches in depth was assumed to exist at this location. This is the smallest crack that could
be missed by the boresonic inspection. Crack growth calculations were then simulated
using a Monte Carlo technique.
The inputs to the Monte Carlo crack growth simulation are the following:
Input Description
Stress to grow crack
Stress at critical time
Fracture toughness
Initial Crack depth
Number of cycles
Number of simulations
Mean Value
72,881 psi
52,580 psi
53 KSI Square root inch
0.05 inch
400
10,000
Standard Deviation
7,281 psi
5,258 psi
5.3 KSI square root inch
0.01 inch
NA
NA
Probabilistic Estimate of Risk of Rotor Failure
Using crack growth equations from Reference 3 the hypothetical crack was grown
mathematically for 400 cycles representing forty years of starts at 10 starts per year.
There was no significant crack growth. Since the steady state temperature in these areas
is about 672  F only fatigue crack growth would be expected due to the start-stop cycles
of the rotor.
The failure probability curve as a function of time is shown in Figure 11. The experience
curve is from Reference 4.
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References
1. Email from Mr. Plant Manager to George Montgomery, Wednesday, April 9,
2003, 4:22 PM, Subject: Life Assessment Requirements.
2. Viswanathan, R., Damage Mechanisms and Life Assessment of HighTemperature Components, 1989, ASM International, Metals Park, Ohio
3. Hellan, Kare, Introduction to Fracture Mechanics, 1984, McGraw-Hill Book
Company.
4. Bush, S. H., A Reassessment of Turbine-Generator Failure Probability, Nuclear
Safety, Volume 19, No. 6, 1978.
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List of Figures
Figure 1 XYZ Unit 1 HP/IP Turbine Rotor ............................................. 8
Figure 2 Finite Element Model of HP/IP Rotor ........................................ 9
Figure 3 Start Temperatures for HP/IP Rotor ...................................... 10
Figure 4 Temperature Distribution at 13.25 Hours .............................. 11
Figure 5 Tangential Stress Distribution at 13.25 Hours ....................... 12
Figure 6 Temperature and Stress in Bore of IP..................................... 13
Figure 7 Temperature Distribution at Steady State .............................. 15
Figure 8 Tangential Stress Distribution at Steady State....................... 15
Figure 9 Fracture Toughness versus Temperature ............................... 16
Figure 10 Critical Crack Depth Along HP/IP Rotor ............................. 17
Figure 11 Failure Probability versus Time ............................................ 18
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Figure 1 XYZ Unit 1 HP/IP Turbine Rotor
8
Figure 2 Finite Element Model of HP/IP Rotor
9
Figure 3 Start Temperatures for HP/IP Rotor
10
Figure 4 Temperature Distribution at 13.25 Hours
11
Figure 5 Tangential Stress Distribution at 13.25 Hours
12
Figure 6 Temperature and Stress in Bore of IP (76” from coupling)
13
Figure 7 Temperature Distribution at Steady State
14
Figure 8 Tangential Stress Distribution at Steady State
15
Figure 9 Fracture Toughness versus Temperature
16
Figure 10 Critical Crack Depth Along HP/IP Rotor
17
Figure 11 Failure Probability versus Time
18