Fracture Mechanics in Turbine Blade Analysis

I would like to take some time to start a discussion on fracture mechanics. The calculations for a basic fracture mechanics analysis are fairly simple, but they can play a very important role in failure analyses. This topic comes up quite frequently in turbine blade work.

So what is fracture mechanics? In short it’s a method of determining the time it takes a crack in a part to grow to failure under a specific loading condition. The crack growth stage of fatigue can make up a significant portion of a products life. This happens in products ranging from bicycles, to airplanes, to steam turbine blades.

At the heart of fracture mechanics is the stress intensity factor K defined as:

Where:

f(g) is a correction factor based on crack geometry. This value tends to be between 1 and 1.4.

a is the crack length

s is the remote stress

Fatigue crack growth is divided into 3 regions as shown in the figure below. In this figure, crack growth rate (da/dN) is plotted on the vertical axis in log scale and Stress Intensity Range (DK=K_max-K_min) is plotted on the horizontal axis in log scale. Region I is associated with crack threshold effects (the area where a crack first begins to grow), Region II is an area of linear growth (Paris region), and Region III exhibts extremely high/unstable crack growth.

For design purposes the focus is on Regions I and II. Crack growth is so fast in Region III that it does not have a significant effect on the total crack propagation life. Noted on the graph is DK_th, the threshold stress intensity which is determined through testing. This value marks the beginning of crack growth. K_c is the critical stress intensity and values higher than this predict fracture.

When performing a turbine blade analysis we often want to determine if a dynamic stress condition is severe enough to grow a crack (Refer to previous posts on blade analysis). To determine that, we run a fracture mechanics analysis for Region I of the above graph. If the stress condition and initial flaw size is not capable of growing a crack then we need not be concerned with removing the near resonant condition.

This analysis starts with calculating the stress intensity factor range:

In the case of an edge crack on a turbine blade airfoil the a correction factor f(g)=1.12 is typically used. Ds is the stress range, or dynamic stress for turbine blades (again refer to my last post on dynamic stress analysis). If DK is greater than DK_th, then a crack will propagate under the given loading condition.

One other thing to consider is the R ratio. Test results for DK_th values are very dependant on the conditions which they were tested at. A particular DK_th value will only apply to a loading condition that has the same R ratio as the test. The R ratio calculation is shown below:

Where s_m is the mean steady stress and s_d is the alternating or dynamic stress. If the R ratio for the DK_th test is different than calculated above, the DK_th will need to be adjusted to account for the difference. One common method for the compensation of DK_th is Walker’s Equation:

Where is the value at R=0, g is a material constant and is typically between 0.3 and 1. Steels are typically around 0.5. Using this relationship, and assuming that is a constant, you can calculate the DK_th value for any R ratio.

Thanks for reading,

Turbine Blade Dynamic Stress Analysis

Here is the second installment on our turbine blade analysis discussion. (Part one is here: Turbine Blade Modal Analysis)
This time I will focus on a dynamic stress analysis. Once you have created the interference diagram that was discussed in my last post, you will be able to identify conditions where resonance may occur. Typically I find any case where the resonant condition is less than 3% different from the forcing frequency (impulse line on interference diagram). I then run a dynamic stress analysis on each of those conditions using BLADE.

Resonant conditions were covered in my last post, but I think it is important enough to summarize it here. The dynamic amplitude and stress response of a structure depends on the following factors:

1. The natural frequencies of the system

2. The damping properties

3. The forcing amplitudes or stimulus ratio, defined as the ratio of the dynamic forces to the static steam loads on the blade

4. The phase angles, defined by the harmonic content (nodal diameter) of the modes.

The steam flow field is non-uniform due to nozzle asymmetry and irregular spacing geometry within the steam flow path. Other factors may include geometry variations of wakes, leakage flows and disturbances in the turbine structure such as joints and steam extractions. Since so many variables are involved and some of the fluid phenomena are still unknown, it is extremely difficult to estimate accurately the dynamic forces and consequently the stimulus ratio.

When calculating the alternating stresses using BLADE, I typically assume a 1% stimulus ratio so that results can be easily scaled. In practice the stimulus ratio varies for different machines and for different blade rows. When BLADE calculates these stresses it assumes that the system is at a resonant condition. Therefore, the resonant stresses that are output by BLADE need to be detuned (i.e. reduced) if the particular stimulus is not precisely at resonant frequency. For example let’s say the conditions that were selected to run a dynamic analysis on were within 1% of resonance and this occurs at 3500 Hz. This could lead to a significant detuning since the forcing frequency would be 35 Hz away from resonance.

The detuning of these resonant stresses is accomplished through the transmissibility function or sometimes referred to as the magnification factor. A derivation of the transmissibility function can be found in a mechanical vibrations text typically in the harmonic vibration chapter. For convenience here is the final result:

Where:

s_d= Dynamic Stress

s_r= Resonant Stress

h=Frequency Ratio (excitation frequency/natural frequency)

z =Critical damping ratio

Once the stress values are detuned you will now know the frequency and nodal diameter for each possible resonant condition and the dynamic stresses that occur there. With this you will be able to judge if this near resonant condition is significant and if a design needs to be modified to reduce the stresses or shift the frequencies to detune the resonant condition.

Thanks for reading and welcome your comments and suggestions.

Turbine Blade Modal Analysis

I have recently had the opportunity to work on a few steam turbine blade failure investigations and have found a whole new world of engineering simulation that I had not been exposed to. Analysis of turbine blades is its own animal and even though steam turbines have been in use in power generation for a very long time, the physics of power generation is so complex there are many areas that are still not well understood. Being new to turbine analysis I thought my experiences as I lean steam turbine analysis may be useful to others. The first topic I would like to describe has been fundamental to all of the turbine projects that I have been involved with: Modal analysis of a bladed disk row. In each investigation, modal analysis was used to calculate the natural frequencies and mode shapes for a particular stage of blades. This is a valuable tool in determining if the blades are operating near a resonant condition that could be responsible for a failure.

As you can find in any vibrations textbook, modal analysis is an eigenvalue procedure in which the eigenvalues of the equation of motion are the square of the natural frequencies and the eigenvectors are the mode shapes. For blade analysis, SimuTech Group uses an in-house developed code to run a modal Finite Element Analysis on turbine blades. This program is called BLADE.

A portion of the bladed disk is modeled in BLADE which usually consists of a 360°/N sector, where N is the number of blades in the row. The mass and stiffness matrices for the bladed disk sector are then reduced to a superelement containing selected fewer degrees of freedom. These selected degrees of freedom are called master degrees of freedom. They are selected in such a way as to be able to represent and predict the dynamic behavior of the bladed disk.

At operating speed, the rotating system stiffens because of the centrifugal effects. This stress stiffening causes the natural frequencies to be higher than their corresponding values calculated at zero RPM. The effect of the stress stiffening is evaluated and incorporated in the analysis.

If a rotating bladed disk is excited by a forcing which is fixed spatially, there are 3 conditions that need to be satisfied in order to produce a resonant condition:

1. The natural frequency of the blade row is equal to some per-rev forcing frequency

2. The number of nodal diameter of the natural mode equals the forcing harmonic number.

3. The excitation must be able to couple with the blade disk mode shape. For example, the forcing on the blade row must be in a direction that matches the mode shape deflection. If the forcing is along the axial direction of the turbine and the mode shape shows deflection only in the tangential direction, the mode can not be excited. Of course many modes contain components in both axial and tangential directions.

If all of the above conditions are met, resonance will occur. This natural frequency-forcing relationship is usually illustrated as an Interference diagram. In the Interference diagram, the natural frequency is plotted against the number of nodal diameters (harmonic content of a mode). An example of an interference diagram is shown above. The line through the origin is called the Impulse Line which corresponds to the operating speed. Whenever the Impulse Line intersects the natural frequency curves, a resonant condition may exist.

The Interference diagram is used to locate frequencies of interest for a more detailed stress analysis. Resonant stresses are calculated for these conditions and detuned as necessary to estimate the true dynamic stress in each blade. The dynamic stress will also depend on the damping present in the system as well as the stimulus ratio (the ratio of dynamic forcing to the steam bending force in the blade). Dynamic stress analysis is a discussion on its own and will likely be a topic of future entries.

Thanks for reading.