Introduction

The GE-F9 is an axial gas turbine, which consists of three blades and nozzle rows in the turbine part and seventeen stator and rotor blade rows in the compressor part. The maximum output power of the generator is 123.4 [MW], with an output voltage of 13.8 [KV], a nominal current of 6443 [A], and a generator speed of 3000 [rpm] [1,2,3]. The casing consists of two semi-casings, which are made of ductile cast iron and connected by flanges to each other [3].

Creep and fatigue are two of the most common causes of failure in gas turbines [4]. High stress under high-temperature conditions leads to the creep phenomenon, and extreme temperature changes contribute to thermal fatigue [3,4,5]. The existence of such factors causes a limitation in the lifetime of the turbine components, so having lifetime estimations for each component is essential for an optimized utilization. In the industrial gas turbines, the components go under cyclic loads because of the frequent startup and shutdown processes. Moreover, the components working under high temperatures and being exposed to the creep phenomenon have a time-dependent failure, so these components are designed based on the allowed strain or the rupture strain [1].

By studying the behavior of creep and fatigue crack growth, the lifetime of turbine component could be predictable [6, 7]. Torshizi and Jahangiri [8] analyzed the propagation steps of a creep–fatigue crack on the nozzle junction. Kruch et al. [9] performed an experiment and a numerical analysis based on FEM to study the crack propagation, under creep and fatigue loading. Raju and Newman [10] modeled a cylindrical conjunction using the exponential model for creep with the analytic method and compared the various estimation results with the results of FEM. Kanchanomai et al. [11] examined the effect of creep and crack growth in mode I on lead–tin eutectic alloys. In 1996, Poquillon et al. [12] studied the numerical simulation of the creep–fatigue crack initiation and growth on the 316L stainless steel. Salari et al. [13] studied fatigue crack growth of a reinforced cylindrical shell subjected to random loading. Stress analysis of the shell was carried out using ABAQUS. Fatigue crack growth was simulated in 3D using ZENCRACK, and the fatigue life was estimated. In 2004, Hyde et al. [14] studied the creep crack growth of CT tension specimens using the numerical method. In this study, the finite element analysis was done in steady state and the creep behavior was investigated by the Norton equation. Nikbin et al. [15] studied the creep information for mono-axial carbon–manganese steel with 9% of chrome in a temperature of 360 centigrade degrees. In this study, C* parameter was used for the assessment of the (C-Mn) steel creep crack behavior. Tan et al. [16] studied the creep crack growth information for some materials, including P22, 1CMV, 316LN, and P91, in order to estimate the lifetime of cracked components in high-performance temperature. They indicated that the results of the crack’s initiation as well as its growth are fully describable by the C* parameter. Wasmer et al. [17] studied the failure and the creep cracks growth in under-pressure pipes and in high-performance temperature piping systems in 2017. Zhao et al. [18] used FEM to predict the lifetime of a turbine rotor. They studied stress distribution of the rotor using ABAQUS and discussed the effect of a damage caused by the interaction of creep and fatigue.

In the present study, the location, direction, and size of a crack on the outer surface of a GE-Frame9 gas turbine casing were determined using nondestructive test. A finite element model was built in order to study the behavior of creep–fatigue growth of the detected crack. The temperature changes occurring over a normal cycle of the gas turbine were measured using thermography method, and measured data were used as input of FEM model. CTOD method was implemented in order to calculate the stress intensity factors. Moreover, the Paris and time-dependent Paris equations were used to estimate the fatigue and creep life, respectively. The interaction of creep and fatigue was calculated, and the growth rate of the crack was discussed.

Determination of Performance Conditions

A gas turbine is commonly a two- or three-stage axial turbine with an air-cooled system [19]. The cooling air is provided by two centrifuge fans in two sides of the turbine cabin [2]. The cooling process is shown schematically in Fig. 1.

Fig. 1
figure 1

Side section view of the gas turbine [2]

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In this project, a GE-F9 gas turbine, which is located in Shahid Rajaee combinational cycle power plant in Iran, was studied. The casing of this turbine has been cracked in both inner and outer sides. Each semi-casing has a semi-cylinder shape with thick wall, an inner diameter of 3 meters and an approximate weight of 6 tons [3]. The crack positions are almost visible with the naked eye (Fig. 2).

Fig. 2
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Crack in casing identified after penetrant testing

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The results of chemical analysis related to the sample obtained from the turbine’s outer casing are provided in Table 1 [3].

Table 1 Chemical analysis results of the casing
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The mentioned results resemble those of the A395 cast iron with an appropriate approximation. This cast iron belongs to the ductile cast iron group and can be regarded as ferrite ductile cast iron. The properties of A395 cast iron are shown in Table 2 [20].

Table 2 Mechanical properties of the ASTM A395 ductile cast iron
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The turbine goes through the three stages of a startup, baseload, and shutdown in each cycle. So, in each cycle, the turbine experiences the three stages of transient temperature increase, steady-state temperature, and transient temperature decrease. The temperature of the semi-casings’ outer surface was measured using two kinds of Fluke model thermo-vision cameras. One of the photographs of the thermography is shown in Fig. 3 where the unit was under an 80 MW load.

Fig. 3
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Temperature contours are taken by a thermograph camera (a). Schematic view of temperature distribution on the outer surface of the casing (b)

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Moreover, for measuring the temperature of the casing’s inner surface, 10 K-type thermocouples with a thermal range of 0 to 1100 °C were installed on the gas turbine casing. Figure 4 shows the thermocouple installation positions in which the sensors’ tip has been pitched into the bolts.

Fig. 4
figure 4

(a) The thermocouple installed on the position of the eccentric pin. (b) Position of installed thermocouples on the casing

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The diagram in Fig. 5 shows the changes in the temperature of outer surface of the casing measured by the thermocouples in terms of time during one cycle, including shutdown (a) and start up (b) for a specific point on the casing’s outer surface, which is approximately located in the place of the shroud segments of the first row.

Fig. 5
figure 5

Temperature variations measured by thermocouples as a function of time, (a) during an emergency shutdown and (b) during starting up

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Numerical Analysis

The ductile cast iron properties—which were provided to the ABAQUS software—are presented in Table 3.

Table 3 The ductile cast iron properties
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It should be noted that due to the minor changes in the properties within the temperature range used in this modeling, they were assumed to be the same during the simulation. The diagram of Fig. 6a shows the value of von Mises stress which was obtained for an element on the casing’s inner surface in terms of the number of elements used in meshing. As it can be seen in Fig. 6, for almost 350,000 elements, the results are independent of the number of elements. The C3d8RT elements were chosen for element types. Figure 6b shows the mesh of casing.

Fig. 6
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(a) The amount of von Mises stress for a single point on the casing’s inner surface in terms of the number of elements (b) the meshed model of casing

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The sub-modeling technique is used for analyzing a partial region of the model. In this approach, first, the model is analyzed by employing a relatively coarse meshing called global model; subsequently, the boundary conditions of the sub-model are applied from the outputs of global model (Fig. 7).

Fig. 7
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(a) The results of von Mises stress distribution around the crack location, (b) stress distribution around the crack location

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Figure 8a shows the stress for the outer surface in the stress concentration region for a typical cycle. During the startup, the outer surfaces have a lower temperature compared to the inner surfaces so the stress on the casing’s outer surfaces will be positive in this step. During working time because of the constant thermal gradient, the outer surface is under a constant tensional stress. Shutting down process is the reverse of starting up so during shutdown the stress on the casing’s outer surfaces will be negative.

Fig. 8
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Stress variation along a turbine working cycle, in a point on the position of the eccentric pin hole, (a) casing’s outer surface and (b) casing’s inner surface

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The whole duration of a turbine’s performance cycle was considered to be about 90,000 s, and it has been assumed that the turbine went under load in 10,000 s, worked for 25,000 s and unloaded in 55,000 s.

The melting temperature for ductile cast iron is 1477 K [21] and the temperature of the crack region is about 623 K, so the temperature of the crack region is about 0.421 times of the melting temperature. Considering the previous explanations, the chosen area has all of the required factors for the creep phenomenon including time, stress, and the temperature factors.

According to the time-based stress changes in Fig. 8a, during the time of turbine’s start up, the value of stress increases, meets the maximum value of tensional stress, and then decreases, which could lead to the fatigue crack propagation. During the working hours, because of the constant positive stress, the creep phenomenon could affect the growth of the crack. Since the compressive stress has no effect on the crack propagation, at the time of the turbine’s shutdown there is no crack growth. These explanations are shown in a schematic way in the diagrams of Fig. 9a and b.

Fig. 9
figure 9

(a) A schematic diagram of the effect of creep during the functioning time of turbine. (b) Thermal changes as a function of time for a node in stress concentration region

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We used the method introduced in [22] to assess the growth of the fatigue crack. In this paper, the time-dependent Paris equation was used for the creep crack growth as a function of Ct integral and linear superposition damage model was used in order to study the interaction of creep and fatigue.

The stress intensity factor can be calculated at each point in the direction of the crack front using the J-integral method, and at the LEFM state, crack tip opening displacement (CTOD) solution is applicable [23, 24].

$$ {\text{CTOD}} = \frac{4}{\pi}\frac{{K_{I}^{2}}}{{E\sigma_{\rm ys}}} $$
(1)

3D Crack Growth Simulation

In this study, for crack growth stimulation, large crack growth method and a crack block with (105_q672x8) code were used [25]. Then the fatigue and creep crack growth properties were introduced to the software [26]. In Fig. 10a the crack growth path and in Fig. 10b the opening of the crack’s front are shown.

Figure 11a, b and c shows some views of the crack front. According to Fig. 11a, the crack front lines get more closed at the end of the path which is indicative of a crack growth rate decreasing by time. First, the stress intensity factor decreases due to the longitudinal growth of the crack—which in and of itself leads to the reduction of stress value—and then, it decreases the growth rate of the crack. In Fig. 11b and c the crack growth path is shown from another view.

Fig. 10
figure 10

(a) Crack growth path and (b) crack’s front opening

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It can be seen in these pictures that the crack doesn’t follow a straight path which approves the crack growth behavior shown in Fig. 2.

In order to evaluate the effect of creep on crack growth, the performance of the casing was studied during 3000 cycles (20-year period) using the time-dependent Paris equation [23].

$$ \frac{{{\text{d}}a}}{{{\text{d}}t}} = C\left({C_{\rm t}} \right)^{n} $$
(2)

where n and C are material constants. After the transition time, the value of Ct approaches the C*. Further details are explained in [4, 27, 28].

In Fig. 11b, given the fact that turbine working hours is equal to 87,600 (within 20 years), in which creep crack growth is less than 0.77 mm, the creep crack growth is estimated during 3000 cycles; in Fig. 11a, the crack growth is calculated for 3000 cycles, in which the crack grows about 2.38 mm under fatigue conditions.

Fig. 11
figure 11

Crack front

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According to Fig. 12, the effect of fatigue on the crack growth is more significant than the creep. Equation 4 is used in order to investigate the interaction of creep and fatigue [29].

$$ \left[{\frac{{{\text{d}}a}}{{{\text{d}}N}}} \right]_{\text{total}} = \left({\frac{{{\text{d}}a}}{{{\text{d}}N}}} \right)_{\text{fatigue}} + \left({\frac{{{\text{d}}a}}{{{\text{d}}N}}} \right)_{\text{creep}} $$
(3)
$$ \Delta a = \frac{{{\text{d}}a}}{{{\text{d}}N}} = C\left({\Delta K} \right)^{n} + A\left[{\left({C_{\rm t}} \right)_{\text{avg}}} \right]^{m} t_{h} $$
(4)

where C, A, n, and m are material constants and th is the duration of the loading cycle.

Fig. 12
figure 12

The crack growth is calculated during 3000 cycles. (a) Fatigue crack, (b) creep crack

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In Fig. 12 under creep condition, the time required for crack to grow to 2.38, which is equal to the value of fatigue crack growth, has been calculated.

According to the diagram of Fig. 12, the crack under the creep conditions will grow 2.38 mm after 370,762 working hours. This means that the creep crack and fatigue crack grow 6.42 × 10−6 [mm/h] and 7.93 × 10−4 [mm/cycle] by the average value, respectively. So crack growth in each fatigue cycle equals to 123.5 working hours under creep conditions. Fatigue lifetime is calculated using the linear superposition model according to Eq 5.

$$ \frac{1}{{N_{\rm f}}} = \frac{1}{{N_{\rm f}^{\text{fatigue}}}} + \frac{1}{{N_{\rm f}^{\text{Creep}}}} $$
(5)

In this model, the damage which has been caused by three failure mechanisms, including creep, fatigue and oxidation, is superposed. In this project, it is assumed that the effect of oxidation is negligible. In Eq 5, Nf is the material’s fatigue lifetime, which is representative of the number of loading cycles before the failure time. For a crack growth of about 2.38 [mm] by having N fatiguef from the diagram in Fig. 13a and N creepf from Fig. 14 and by replacing in Eq 5, the value of Nf can be calculated, which equals to 2071 cycles. The effect of creep, fatigue, and the creep–fatigue interaction is shown in Table 4.

Fig. 13
figure 13

Creep crack growth as a function of time

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Fig. 14
figure 14

Stress intensity factor changes as a function of crack length

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Table 4 The time required for the propagation of crack by 2.38 [mm]
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In Fig. 14, stress intensity factor changes in terms of crack length are shown.

As shown in Fig. 14, given the crack direction and the fact that crack growth has occurred in mode 1, it can be concluded that the stress components perpendicular to the crack plane can be recognized as the effective factor on the crack growth. Stress intensity factor has the maximum value in point A of the diagram in Fig. 14 and after that stress intensity factor decreases while the crack grows until it reaches an amount less than Kth in point B, where the crack is arrested. The diagram of crack growth in terms of the number of cycles is shown in Fig. 15. It can be seen that the crack arrests after about 380,000 cycles.

In Fig. 16 the diagrams of \( \frac{{{\text{d}}a}}{{{\text{d}}t}} \) in terms of Ct and \( \frac{{{\text{d}}a}}{{{\text{d}}N}} \) in terms of stress intensity factor are shown. Crack arresting could be interpreted from Fig. 16a and b.

Fig. 15
figure 15

Crack growth diagram as a function of cycle number

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Fig. 16
figure 16

(a) Diagram da/dN as a function of stress intensity factor. (b) Diagram of da/dt as a function of Ct

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Conclusions

Stress analysis was conducted based on real boundary conditions, and the growth of the creep–fatigue crack was studied and it is concluded that

  • Regarding the results of stress analysis, the growth of the cracks located on the outer surface of the casing is due to creep and fatigue phenomenon. Since the casing’s outer surface is exposed to tensional stress at the time of performance and due to the high-performance temperature, all conditions will be suitable for the creep phenomenon to appear.

  • The effect of fatigue phenomenon on crack growth is more significant than the creep phenomenon. By average, the crack growth rate under the creep, fatigue and the interaction of fatigue–creep circumstances is 1.8 × 10−4, 7.9 × 10−4 and 9.766 × 10−6 [mm/cycle], respectively.

  • The crack growth is mainly under mode I, and it will be arrested after about 380,000 cycles.