Numerical simulation of fatigue cracks initiation and propagation for horizontal axial turbines shafts.

Bordeasu, Ilare ; Popoviciu, Mircea Octavian ; Marsavina, Liviu 等

1. INTRODUCTION

From previous numerical modeling of horizontal hydraulic turbines shafts it was concluded that in the fillet corner of the flange connecting the turbine shaft with the runner stress concentration can occur. This lead to cracks, under fatigue loading, which reaches such dimensions that the safe work of the shaft is endangered (Bordeasu, et al., 2009). The examination of these models reaches to the conclusion that this failure occurs through the inevitable fatigue stresses of the shaft running. Similar damages where reported for Palo Alto nuclear power plant (Ratiu, 1987). In the present work are presented the results obtained using Finite Element Method (FEM) of the cracks initiation in the fillet zone and a fatigue crack propagation study. The programs INVENTOR, ANSYS and AFGROW were applied on a shaft model with the same characteristics as those of the Romanian power plant Iron Gates II hydraulic turbines. The shafts material taken into consideration was AISI 1022 having the following mechanical characteristics: ultimate tensile strength [R.sub.m] = 440 MPa, yield strength [R.sub.p0,2] = 262 MPa, elastic modulus E = 207000 MPa, Poisson's ratio v = 0.3, fatigue ductility coefficient [[epsilon]'.sub.f] = 0.337, fatigue stress exponent b = -0.156, fatigue ductility exponent c = -0.485, fatigue strength [[sigma]'.sub.f] = 1384 MPa.

2. GEOMETRICAL MODELING

The principal dimensions of the shaft are: 7572 mm lengths and 2300 mm the maximum diameter. Between the flanges the shaft has a ring shaped cross section with the external diameter of 1200 mm and an internal one of 600 mm.

[FIGURE 1 OMITTED]

The 3D geometrical modeling of the shaft, at the scale 1:1, in 3D, was done in INVENTOR software. This geometry was imported in the FEM program ANSYS v.11. The 3D model and the computational mesh are presented in figure 1. The model contains 177344 tetrahedral elements connected with 290848 nodes. In the neighborhood of the flange the mesh was refined (the magnitude of the elements was taken of 20 mm) in order to obtain a better evidence of the stresses concentration.

3. NUMERICAL SIMULATION

3.1 Estimation of crack initiation due to fatigue loads

The strain based approach based on Coffin--Mason equation was used for fatigue calculations, (Shigley & Mischke, 1989). The fatigue computation was realized for the z stress component, produced by the superposition of bending (produced by the hidro-aggregate weight) and tensile (due to axial loads). The stress cycle characteristics were [[sigma].sub.max] = 88.86 MPa, [[sigma].sub.min] = -6.52 MPa and R = [[sigma].sub.min]/[[sigma].sub.max] = -0.07 according with the numerical results shown in figure 2.

From the computation performed with the FATIGUE TOOL of ANSYS v.11 software resulted the minimum duration, till crack initiation (a crack with the length of 4 mm and the depth of 1 mm) of [N.sub.i] = 3.0139 [10.sup.8] cycles which represents 80370 running hours (figure 3).

3.2 Estimation of the number of cycles for failure crack propagation with constant amplitude loading

For the failure propagation it was considered a ring cross section with a part through elliptical crack on the external circumference, figure 4. For this geometry the solution of the stress intensity factor was chosen according with (Raju & Newman, 1984) in the form:

[K.sub.I] = {[[sigma].sub.t] + H [[sigma].sub.b])[square root of [pi] a/Q] F(a, c, [D.sub.i], [D.sub.e], [phi]) (1)

where: [[sigma].sub.t] is the applied tensile stress,

[[sigma].sub.b] is the bending stress,

H and F are parameters for the crack geometry (the depth and length), the shaft wall thickness, and the frontal failure position,

a is the crack depth and c half of crack length.

Q = 1 + 1,464[(a/c).sup.1.65] for a/c [less than or equal to] 1 (2)

The numerical simulation was performed with the AFGROW code, Version 4.12.15, developed by Hartner, 2008. The input data were: the shaft and crack dimensions (the inner diameter [D.sub.i] = 0.6 m, the outside diameter [D.sub.o] = 1.2 m, the crack depth a = 0.001 m, the half of the crack length c = 0.002 m. On the basis of the preliminary stress computation (Bordeasu et al, 2009), for the crack propagation was taken into consideration a composed tensile and bending stress, with the condition that the static tensile load is superposed over the bending one, resulting a pulsating cycle with [[sigma].sub.max] = 88.9 MPa, [[sigma].sub.min] = -6.5 MPa, R = -0.07 with constant amplitude loading.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

The fracture mechanics parameters taken into considerations were those for the AISI 1022 steel: for Forman equation c = 1.447 x [10.sup.-12], n = 3.6, m = 1 (Forman et al., 1967); the plane stress fracture toughness [K.sub.C] = 110 MPa [m.sup.1/2], respectively the plane strain fracture toughness [K.sub.IC] = 77 MPa [m.sup.1/2], the threshold stress intensity factor range under which the crack do not propagates [DELTA]K = 1.5 MPa [m.sup.1/2].

Figure 5 presents the crack evolution for load with a bending symmetric alternating fatigue cycle. The initial crack values were reached after 80370 running hours in conformity with the studies regarding the crack initiation. The time necessary for the crack propagation was added to the initial time. So, a failure with the dimensions a = 16 mm, 2c = 64 mm is obtained after 153243 hours. After 159737 running hours the crack pierces the shaft wall. The difference from the previous situation is only of 6494 hours.

[FIGURE 5 OMITTED]

4. CONCLUSIONS

1. Failure compulsory occurs on the horizontal axial hydraulic turbines as a result of fatigue loads.

2. The present study was based on Fracture Mechanics, in the linear elastic domain, for the case of combined loading (bending and tensile) considering a constant amplitude fatigue cycle.

3. Working with the Fatigue Tool of ANSYS v.11 resulted a minimum period till crack initiation, in the filleted corner, of [N.sub.i] = 3.0136 [10.sup.8] cycles, which represents 80370 running hours.

4. The increase of the failure dimensions till a = 16 mm (depth) and 2c = 64 mm (length) occur after 153243 running hours and the piercing of the shaft wall occur after approximately 159737 hours.

5. The present results can be used for establishing the inspection and repair work periods for the power plant Iron Gates II bulb hydraulic turbines.

5. ACKNOWLEDGMENT

The present work has been supported from the National University Research Council Grant (CNCSIS) PNII, ID 34/77/2007 (Models Development for the Evaluation of Materials Behavior to Cavitation), and Nr. RU 177/10.10.2008, BC 146/13.10.2008 (Analyze regarding the reliability of the bulb turbines)

6. REFERENCES

Bordeasu, I., Popoviciu, M.O., Novac, D.M., Fatigue Studies upon Horizontal Hydraulic Turbines Shafts and estimation of Crack Initiation, Machine Design, Monograph, University of Novi Sad, 2009, pp. 191-196

Forman R.G., Hearney V.E., Engle R.M., Numerical Analysis of Crack Propagation in Cyclic-loaded Structures, Journal of Basic Engineering, Trans. ASME, Vol. 89, 1967

Hartner J. A., AFGROW Users Guide and Technical Manual, Wright-Patterson Air Force BASE, Ohio, 2008

Raju I. S., Newman J. C., Stress Intensity Factors Circumferential Surface Cracks in Pipes and Rods, Proc. of 17th National Symposium on Fracture Mechanics, Albany, 1984

Ratiu M., Analytical evaluation of potential shaft cracks on CE/KSB reactor coolant pump for the Palo-Verde nuclear power plant, Impell Report No. 01-1650-1630, California, 1987

Shigley J. E., Mischke C. R., Mechanical Engineering Design, Fifth Edition, McGraw-Hill, New Zourk, 1989