Performing a stress analysis and fracture mechanics evaluation of a turbine rotor bore can extend the life of the turbine rotor or increase the turbine rotor bore inspection period. Typical recommended inspection periods from the original equipment manufacturer may vary from seven to ten years and are based on the condition of a fleet of turbine rotors rather than an individual rotor. A stress analysis and fracture mechanics evaluation of a specific rotor can produce inspection periods of ten to fifteen years delaying the cost of an inspection or the replacement of a problem rotor.

In the early years of making forgings for turbine rotors a bore hole was drilled axially through the forging to remove any inclusions that may have been entrapped during the forging process. This bore increased the centrifugal stresses due to the rotation of the rotor but provided a means of examining the rotor for defects that may have been formed during manufacture or developed as a result of turbine rotor operation. In 1974 a 275 MW unit placed in service in 1957 experienced a rotor burst predominantly due to axial crack propagation from a near-bore flaw. This incident increased interest in the analysis and prevention of turbine rotor near-bore flaws.

Once in service turbine rotor bores are examined ultrasonically on a periodic basis to look for any indications of cracks or other defects. Based upon the results of this examination the rotor is allowed to continue in operation, is repaired if necessary or is replaced. Often this decision is made by comparing the condition of a rotor under examination to the condition of a fleet of similar rotors rather than performing a detailed fracture mechanics analysis. A detailed evaluation will examine an individual rotor. This means that its unique material properties, start up characteristics and detected near-bore flaws are used in the evaluation.

The advantages of a detailed stress analysis and fracture mechanics evaluation on an individual turbine rotor bore are that the rotor bore reinspection intervals are increased, rotor bore reinspection intervals are not a limiting factor in increasing the time between turbine overhauls, no rotors are replaced without a strong technical basis and problem rotors can have their life extended if operational procedures are changed. The evaluation process is not a simple one but is a worthwhile endeavor over the life of the turbine rotor.

A typical fracture mechanics evaluation consists of finding the stress intensity factor for a crack that may exist in the rotor bore 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 stress intensity factor describes the intensification of an applied stress field near a crack tip and is a function of the applied stress and crack depth. The crack depth is obtained from the results of the ultrasonic exam and the stress is obtained from a finite element stress analysis. The fracture toughness of the rotor material is a temperature-dependent material property and is obtained from Charpy impact and tensile tests. Once the stress intensity factor and the fracture toughness are known, a comparison of the two can be made to determine the structural integrity of the rotor containing a near-bore crack.

Due to the uncertainty in operating and material property parameters, a probabilistic approach is used which will provide a failure probability as a function of time. This means that instead of a single value for the stress, crack size, or fracture toughness the probability density functions (PDF's) of those values consisting of a mean and standard deviation will be used in a Monte Carlo simulation.

Ideally this stress analysis and fracture mechanics evaluation is performed before the turbine outage so that decisions made during the outage have a sound technical basis.

The stresses which drive a crack in the bore of the turbine rotor are caused by the non-uniform 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. 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

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 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 and specific heat 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. labyrinth seal

2. rotating disk
with a finite wall clearance

3. shrouded
rotating disk with steam flow

4. metal to metal contact

5.
rotating cylinder in an infinite fluid

6. insulated surface

The heat transfer coefficient must be calculated for each surface area for each time step used in the transient thermal analysis. For the rotor and start shown above this is twenty-five time steps on eighty-six heat transfer surfaces.

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. A plot of a typical temperature distribution is shown here. 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.

As mentioned earlier fracture toughness can be measured by Charpy impact tests in which material samples are broken at various temperatures. From the results of these Charpy impact and tensile tests a temperature transition curve is developed. This curve is a three parameter Weibull fit to the test data and provides an expression for the fracture toughness as a function of temperature. The advantage of the Weibull curve is that below the lower shelf and above the upper shelf the value of the fracture toughness is constant. The upper shelf value of fracture toughness is determined from a Rolfe-Novak correlation and the lower shelf from a Begley-Logsdon correlation. Another material property, the fracture appearance transition temperature (FATT), is the temperature half way between the upper and lower shelves.

If no material properties are available, they can be obtained from samples machined from a ring removed from the bore hole on the coupling end of the rotor.

Information about the dimensions of existing cracks comes from a boresonic vendor. The various “hits” reported by the boresonic ultrasonic system are clustered to some designated criteria. These clusters are then sized and the dimensions in the radial and axial direction are determined. These dimensions are inputs to the fracture mechanics evaluation. A radial-axial crack is considered more detrimental than a circumferential one.

Fracture mechanics is the engineering study of the load carrying capability of a structure containing a crack. In this case we are interested in the capability of a turbine rotor with a near-bore radial-axial crack to operate without a failure. As mentioned above this determination requires a knowledge of the stress in the rotor and the size of the existing crack. At this point the stress and temperature distribution in the rotor is known for a thermal transient start and steady state condition. The size of any existing near-bore crack is known from the ultrasonic examination.

The results of the finite element analysis can be employed to provide some insight into the operating conditions of the rotor. In order to understand the susceptibility of the turbine rotor to rapid crack growth, it is useful to calculate the critical crack depth along the length of the rotor. This is done by substituting the value of fracture toughness for the stress intensity factor in the equation relating stress intensity factor, stress and crack depth and solving for the crack depth. The fracture toughness is a function of temperature so its value will vary during the start-up period. This means that the critical crack depth must be calculated for each node or element along the bore surface of the rotor for each time step. For the rotor shown this would be 154 nodes and twenty-five time steps. Then the smallest critical crack depth is chosen from each time step. The result is shown here for the given rotor.

It can be seen that the smallest critical crack depth for the rotor is located in the IP section. From the finite element output it is also known that it occurs at a distance of 191 inches from the coupling face at a time 13.25 hours into the start and the stress and temperature at that time are 72 KSI and 476 ° F respectively. Even though 72 KSI is a high stress, it occurs at a time when the fracture toughness is on the upper shelf. This is due to the long warm-up period which suggests that the results of the finite element analysis can be used to determine the optimum warm-up time. The reason for the high stress is the steep temperature ramp between the 12th and 14th hour of the start.

At high temperatures and long exposure times turbine rotor metal undergoes a phenomenon called embrittlement which basically means that the curve of fracture toughness versus time shifts to the right lowering the fracture toughness for a given temperature. Using the steady state temperatures from the finite element thermal analysis, an evaluation can be made on the necessity of including a shift in the fracture toughness versus time curve for rotor embrittlement. The measured FATT for this rotor is 153 degrees F. and a shift of 100 degrees F. was obtained from the original equipment manufacturer. Therefore, a FATT of 253 degrees F. is used in subsequent calculations.

In the above stress contour plot it can be seen that the stress decreases away from the bore surface of the rotor. This means that as a crack grows, it will grow into a lower stress field. Advantage should be taken of this in any crack growth predictions. Using the constant stress field at the bore surface will produce overly conservative results. The results of the finite element analysis provide this stress distribution away from the bore surface.

Any suspected crack along the bore is evaluated using the stress and fracture toughness corresponding to the most critical time at the location of the suspected crack. If no crack is found during the boresonic examination, a crack equal to the size of the maximum missed flaw is assumed to exist at the rotor bore. The location of this assumed crack corresponds to the smallest critical crack size.

The structural integrity of the rotor bore in the presence of a crack
is determined by comparing the stress intensity factor K_{I}
with the fracture toughness K_{IC} from
the fracture toughness versus temperature curve. This comparison
is carried out many times during the mathematically simulated growth of
the crack using a Monte Carlo simulator on a personal computer. The simulated
growth uses distributions of the temperature, stress,
crack depth, etc. Finally the failure probability
versus time curve is constructed.

For further information, please contact :

*George Montgomery,PE*

gmontgomery@nc.rr.com

**Home Page of George A. Montgomery, PE**