W. Loren Francis
6.5 Discussion of Results
The combustor liner has to satisfy all the structural reliability and safety requirements under the thermo- mechanical service loads. Initially, the liner was analyzed probabilistically to obtain the cumulative distribution functions (CDFs) of the probabilistic buckling loads for the first five modes (Figure 6.5).
The eigenvalues (critical load/applied load) for the five modes are given in Table 6.2. The sensitivity factors from Figure 6.6 show that the uncertainty in the liner thickness has the highest impact on the probabilistic distribution of the buckling load for the first mode followed by the pressure load and hoop modulus of the liner material. Figure 6.7 through Figure 6.11 show the buckled mode shapes of the liner for modes 1 through 5, respectively. Figure 6.7 indicates rigid body motion, whereas Figure 6.8 and Figure 6.9 depict the edge buckling phenomena, which correspond to a pair of equal eigenvalues (Table 6.2).
TABLE 6.1 Primitive Variables and Uncertainties for Probabilistic Structural Analysis of Combustor Liner (random input data).
Primitive
Variables Mean Value
Density 0.1002 lbs.sec2/in4
Coefficient of thermal expansion 2.43×106 in./in./°F
Thickness 0.8 in.
Pressure load 8.7 psi
Thermal load (inside) 1455–2400°F
Thermal load (outside) 1436–2386°F
Axial modulus 35.81 ksi
Hoop modulus 35.8 ksi
Poisson’s contribution 5.37 ksi
Shear modulus 12.8 ksi
Shear modulus 10.2 ksi
Shear modulus 10.2 ksi
Note: All variables were assumed normal with standard deviations equal to 5% of their mean values.
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FIGURE 6.5 Cumulative distribution functions of cylindrical liner segment buckling loads.
TABLE 6.2 Probabilistic Buckling Analysis [single cup liner segment (pressure load —1 psi)]
Mode Number
Mean Values of Eigenvalue Critical Loads Normalized
by the Applied Load
1 1.46
2 106.88
3 106.88
4 116.56
5 118.10
FIGURE 6.6 Probabilistic mode 1 buckling load sensitivities of a cylindrical liner segment.
0.000 0.20 0.40 0.60 0.80 1.00
120 240 360 480 600
Mode 4 Mode 3
Mode 2 Mode 5
Mode 1
Eigenvalue (critical load/applied load) Cumulative ProbabilitySensitivity
1.00
0.50
0.00
–0.50
–1.00 Thickness Thickness
0.001 0.999
Probability Level Modulus
(Hoop)
Modulus (Hoop)
Pressure Pressure
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FIGURE 6.7 First buckling mode, indicating rigid body motion.
FIGURE 6.8 Mode 2, indicating edge buckling.
FIGURE 6.9 Mode 3, indicating edge buckling.
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On the other hand, modes 4 and 5, as shown in Figure 6.10 and Figure 6.11, exhibit a combination of circumferential/radial deformations that are typical for cylindrical shell structures. It is important to note that these mode shapes are due to the types of boundary conditions used at the support locations as well as the combination of pressure and thermal loads.
Subsequently, the liner was probabilistically analyzed to obtain the CDFs of the probabilistic vibration frequencies for the first five modes. These CDFs are shown in Figure 6.12. The mean values of the first five natural frequencies are shown in Table 6.3. According to Figure 6.13, the first natural frequency was very sensitive to the scatter in the following properties of the liner material; density, thickness, hoop modulus, and axial modulus. However, for higher probabilities of failure, the impact of the variation in the density of the liner material on the vibration frequencies for the first mode dominates. The respective first five mode shapes are shown in Figure 6.14 through Figure 6.18. The first mode appears to be a rigid body type, and modes 2 through 5 involve circumferential/radial deformations.
In addition to satisfying the design requirements about the allowable vibration frequencies and buck- ling loads, the stress level in the combustor liner should be below some critical stress level. Therefore, the probabilistic longitudinal stress at a critical location on the inside of the liner was determined for the thermomechanical loads (see Figure 6.19). Furthermore, this stress distribution showed a wide scatter between the lowest and highest longitudinal stresses. According to Figure 6.20, the longitudinal stress distributions were very sensitive to the scatter in the thermal load profile, the variations in the liner material thermal expansion coefficient, and axial modulus. However, the CDFs of the longitudinal stresses FIGURE 6.10 Mode 4, indicating circumferential radial deformations.
FIGURE 6.11 Mode 5, indicating circumferential radial deformations.
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FIGURE 6.12 Cumulative distribution functions of cylindrical liner segment natural frequencies.
TABLE 6.3 Probabilistic Vibration Frequency Analysis [single cup liner segment (pressure load — 1 psi)]
Mode Number
Mean Values of Vibration Frequencies
(cps)
1 0.97
2 11.08
3 18.97
4 36.43
5 61.17
FIGURE 6.13 Probabilistic mode 1 vibration frequency sensitivities of a cylindrical liner segment.
Vibration Frequency (cps) 48 60 36
24 0 12
0.00 0.20 0.40 0.60 0.80
1.00 Mode 1
Mode 3
Mode 2
Mode 4 Mode 5
Cumulative Probability
0.999 0.102
Probability Level
Sensitivity
Thickness
Thickness Modulus
(Hoop) Modulus
(Axial)
Modulus (Axial) Density
Density 1.00
0.50
0.00
–0.50
–1.00
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FIGURE 6.14 First vibration frequency, indicating rigid body motion.
FIGURE 6.15 Second vibration frequency, indicating circumferential/radial deformations.
FIGURE 6.16 Third vibration frequency, indicating circumferential/radial deformations.
FIGURE 6.17 Fourth vibration frequency, indicating circumferential/radial deformations.
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FIGURE 6.18 Fifth vibration frequency, indicating circumferential/radial deformations.
FIGURE 6.19 Cumulative distribution function of Node 441 longitudinal stress.
FIGURE 6.20 Probabilistic longitudinal stress sensitivities for Node 441 at 0.001 probability.
1.00
0.75
0.50
0.25
0.00–60.00 –42.50 –25.00 Stress (ksi)
–7.50 10.00
Cumulative Probability
–1.00
Sensitivity
–0.50 0.00 0.50 1.00
Thermal
CTE
Modulus
(Axial) Thickness
Modulus (Hoop) 51326_C006.fm Page 11 Thursday, August 9, 2007 2:20 PM
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for the outer surface of the liner (see Figure 6.21) clearly indicated that the stresses were compressive at the lower probability of failure level, whereas they were tensile at the remaining probability of failure level. Lastly, the probabilistic hoop stress at a critical location indicated the reversal of the stresses (from compressive to tensile) from lower to higher probability of failure levels (see Figure 6.22). Once again, the variation in the thermal load profile has the highest impact on the CDFs of the probabilistic hoop stress (see Figure 6.23). The CDFs of the probabilistic hoop stress at the critical location of the outside of the liner showed a wide variation, and the stress is reversible from the lower probability of failure levels to higher probability of failures (see Figure 6.24).
FIGURE 6.21 Cumulative distribution function of Node 421 (outer surface of the liner) longitudinal stress.
FIGURE 6.22 Cumulative distribution function of Node 601 hoop stress.
36.00 24.80
13.60 2.40
Stress (ksi) –8.80
–20.00 0.00 0.20 0.40 0.60 0.80 1.00
Cumulative Probability
20 0
–20 –40
0.00–60 0.25 0.50
Cumulative Probability
0.75 1.00
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