Low-cycle fatigue behaviors of the elbow in a nuclear power plant piping system using the moment and deformation angle

Engineering Failure Analysis 96 (2019) 348–361

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Engineering Failure Analysis journal homepage: www.elsevier.com/locate/engfailanal

Low-cycle fatigue behaviors of the elbow in a nuclear power plant piping system using the moment and deformation angle

T

Sung-Wan Kima, Hyoung-Suk Choia, Bub-Gyu Jeona, , Dae-Gi Hahmb ⁎

a

Seismic Research and Test Center, Pusan National University, 49 Busandaehak-ro, Mulgeum Yangsan Kyungnam 50612, Republic of Korea Integrated Safety Assessment Division, Korea Atomic Energy Research Institute, 111 Daedeok-daero 989 beon-gil, Yusung-gu, Daejon 34057, Republic of Korea

b

ARTICLE INFO

ABSTRACT

Keywords: Nuclear power plant piping system 3-in steel pipe elbow Leakage line Low-cycle fatigue curve Moment-deformation angle

Maintaining the structural integrity of the major devices of a nuclear power plant is perceived as a particularly important issue in relation to the stability of the structure, and the structural integrity of the piping system is especially important for the safety of the nuclear power plant. In this study, the limit state of a steel pipe elbow, which is a fragile part of the nuclear power plant piping system, was defined as leakage, and an in-plane cyclic loading test was conducted. As it is difficult to measure the moment and deformation angle of the steel pipe elbow using conventional sensors, an image measurement system was used. A deformation angle measurement algorithm that uses an image measurement system applied an improved deformation angle measurement algorithm that uses image filter processing to facilitate noise reduction in images as well as the boundary line of the piping system. This study proposed a leakage line and low-cycle fatigue curves using the relationship between the moment and the deformation angle of a 3-in steel pipe elbow.

1. Introduction A nuclear power plant consists of numerous pipes. As these pipes are connected to major devices as pressure boundaries and transport high-energy fluids containing radiation, their structural integrity is extremely important for the performance of their own functions as well as for the stability of the nuclear power plant. Sufficient seismic performance must be secured for these pipes in order to prevent serious accidents caused by radiation leakage or functional loss against earthquakes assumed at the design stage [1,2]. Owing to the 2011 Fukushima nuclear power plant accident caused by the Great East Japan Earthquake, interest in research that considers beyond-design-basis events has been increasing. For such events, however, it is necessary to clearly define the failure mechanisms and failure modes of the nuclear power plant components [3]. A probabilistic seismic safety assessment is performed for nuclear power plants to prevent fatal failure and to reduce the damage caused by accidents. For such an assessment, the seismic fragility of the major components and systems related to safety is analyzed. For a reliable seismic fragility analysis, it is essential to define the failure modes and failure criteria than can represent actual serious accidents. To improve the reliability of a seismic fragility analysis, the failure modes need to be defined considering the performance or functional requirements of the target structure or device. The failure modes can be categorized as the cracking, yielding, and buckling of the structure or device. The seismic fragility is analyzed to predict how, under what circumstances, and to what degree of

⁎

Corresponding author. E-mail address: [email protected] (B.-G. Jeon).

https://doi.org/10.1016/j.engfailanal.2018.10.021 Received 31 July 2018; Received in revised form 9 October 2018; Accepted 26 October 2018 Available online 29 October 2018 1350-6307/ © 2018 Elsevier Ltd. All rights reserved.

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probability the components of the target device or structure will fail, and it is important to select the fragile components and clearly define the failure modes. If the failure modes are defined without clear grounds, inappropriate data capable of causing errors in seismic fragility assessment can be used [4]. A representative study on the seismic behavior of pipes is the Piping and Fitting Dynamic Reliability Program (PFDRP) performed by both the Electric Power Research Institute (EPRI) and the General Electric Company (GE) in the 1980s. The results of this study confirmed that the damage from repeated dynamic loads such as seismic loads is caused by fatigue [5]. The results of PFDRP revealed that the repeated application of a high load that may belong to a plastic zone develops a ratcheting effect, and it confirmed that the damage to pipe components under seismic loads is caused by low-cycle ratcheting fatigue. A vibration test of a piping system revealed that the damage to the piping system is not collapse but fatigue failure accompanied by ratcheting. In Japan, since 1997, studies have been conducted on the limit state of piping systems containing elbow components. Static and dynamic tests as well as analytical studies were conducted on various elbow components and piping systems [6]. In South Korea, tests with multiplatform shaking tables were conducted to study the seismic behavior of nuclear power plant pipes [7], and a study to assess the seismic safety of a seismic isolated nuclear power plant piping system was conducted [8]. In addition, studies for defining the quantitative failure criteria of the elbow were conducted until recently to assess the probabilistic seismic safety of nuclear power plant piping systems [9,10]. The results of these studies revealed that the loads applied to pipes owing to earthquakes have the characteristic of displacement control, and that the failure mode is low-cycle fatigue failure with ratcheting. Furthermore, it was revealed that the elbow is subject to concentrated nonlinear behavior and failure in piping systems [11–19]. In pipe design criteria such as those of the American Society of Mechanical Engineers (ASME) and the Japan Electric Association (JEAG), the limit state of pipes against earthquakes is assumed to be plastic collapse [20,21]. In the case of seismic loads, however, it is highly probable that pipes are subject to low-cycle fatigue failure accompanied by the ratcheting effect. Based on this background, studies are being conducted to identify the ratcheting effect that arises in pipes under excessive seismic loads, as well as the actual failure. Varelis et al. [22] conducted an in-plane cyclic loading test on a pipe elbow using a constant amplitude. They conducted research on a finite element model to simulate the ratcheting effect, and verified that the ratcheting effect of pipes could be analytically simulated. They predicted the fatigue life of pipes against earthquakes by simply creating a fatigue curve using the relationship between the applied load and the failure cycle. Urabe et al. [23,24] applied internal pressure to a pipe elbow, conducted an in-plane cyclic loading test, and verified that the ratcheting effect could be analytically simulated. They predicted the low-cycle fatigue life of the pipe bend considering the local wall thinning from the correlation between the strain range and the number of cycles. Gupta et al. [25] proposed a low-cycle fatigue curve for the ratcheting strain and the number of cycles. As previously mentioned, many studies have been conducted to identify the fatigue life of pipe components and to define their limit states against earthquakes considering the ratcheting strain in nuclear power plant piping systems. This resulted in an increasing number of visible achievements. Strain is local, however, and it is difficult to measure the limit state using conventional sensors. In addition, it may be difficult to clearly define certain phenomena such as stress concentration during analysis. Therefore, a component-based assessment may bring more reliable results in terms of defining the failure of pipes. The limit state of pipes is mostly assessed through the correlation between the applied load and the loading amplitude from a component-based failure test of the pipe elbow. It may be difficult, however, to directly simulate the direction of the applied force of the test condition for a pipe elbow included in the piping system. This may not be effective or economical in analyzing the seismic fragility of the piping system. As with the study that the Network for Earthquake Engineering Simulation (NEES) conducted to assess the limit states of pipe components and to analyze their seismic fragilities using an experimental method, studies capable of directly describing the deformation of the pipe components and considering the low-cycle fatigue failure effect are required [26]. Therefore, defining the failure criteria and limit states of pipes against earthquakes using the correlation between the moment and the deformation angle will present a fatigue life that is more reliable and easier to use in the field. In most component-based tests, however, it is difficult to directly measure the deformation angle using conventional sensors. The angle is usually inferred from the measured amplitude, and in most cases, there is no separate report on the rotation measurement method. As the deformation angle is induced by calculating the amplitude at a measurable position, it may be different from the deformation angle that occurs in the actual piping system. Therefore, it is necessary to directly measure the rotation of the pipes for key areas using noncontact sensors instead of calculating such rotation using deformation. It may be difficult for the existing deformation angle measurement method, which uses images, to detect the lines of a piping system because of the noise that occurs while acquiring optics, lighting, and images [10]. Therefore, this study proposes an improved line detection method to facilitate the deformation angle measurement of a piping system using images. The results of PFDRP revealed that the entire system was safer than the component test results because the loads were redistributed to other components owing to the local plastic deformation that occurred inside the components of the piping system. Therefore, in this study, a component-based test was conducted for a steel pipe elbow, and a low-cycle fatigue test was performed at a constant amplitude. 2. Moment and deformation angle measurement using images 2.1. Moment measurement using image processing method The measurement method that uses the image correlation method [27–29] analyzes the correlation using the gray levels of the object surface. In the case of the commonly used 8-bit image, the gray level of the object is expressed using steps between 0 and 255. 349

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Fig. 1. Schematic of relation of pixel point in reference and deformed windows

As the object surface has numerous points with the same gray levels, unit-pixel-based correlation analysis is not possible. Therefore, the deformation of an object is measured by separating small [(2M + 1)(2M + 1) in size] images called “windows” and analyzing their correlation. In other words, when a control point that the user wants to find in the initial or reference image is designated, the (2M + 1)(2M + 1) reference window with its center at the control point is separated. As shown in Fig. 1, to analyze the correlation between the images before and after deformation, the deformed window is separated from the image after deformation based on the coordinates of the reference window, and its correlation with the reference window is analyzed. In this study, the Normalized Sum of Squared Differences (NSSD) method of Eq. (1) was used as the method of analyzing the correlation between two images, and the coordinates with the minimum value represent the deformation of the object by an external force. In Eqs. (1) and (2), f(xi, yj) and g (xi′, yj′) represent the gray level values of the windows before and after deformation. M

M

CNSSD =

g x i , yj

f

g

i= M j= M

M

(

f (x i , yj )

M

(1) M

[f (xi , yj )]2 ,

f =

)

i= M j= M

M

(

)

[g xi , yj ]2

g = i= M j= M

(2)

To calculate the subpixel, the shape function [30,31] is used to predict the deformation of the surrounding points based on the measured displacement at each point. In the measurement method that uses the image correlation method, a zero- to second-order shape function is often used. The quadratic shape function of Eq. (3) was applied to consider the effect of the bending and nonlinear behavior of the object by an external force. ξ(xi, yj) is the shape function of the x coordinate, η(xi, yj) is the shape function of the y coordinate, u and v are the displacements of each point obtained by the image correlation method, and M is the size of the lattice point for which the displacement is to be measured from the image before deformation.

1 1 u xx x 2 + u yy y 2 + u xy y 2 + u xy x y, (i, j = 2 2 1 1 2 y + vxx x + vyy y 2 + vxy y 2 + vxy x y 2 2

(x i , yj ) = u + u x x + u y y + = v + vx x + vy

M : M ) (x i , yj ) (3)

2.2. Deformation angle measurement using image filter processing The purpose of image enhancement technology [32] is to process the original image and to convert the result so that it fits a special application goal. In this study, image filter processing was applied to measure the deformation angle of a pipe component. As image filter processing, an image enhancement algorithm in the spatial domain was applied. During the test, a Gaussian filter [33] was used to reduce the noise that occurred while acquiring optics, lighting, and images. The Gaussian filter reduces noise and connects the disconnected edges in the image in exchange for a lowered sharpness of the image. Owing to these characteristics, noise reduction using the Gaussian filter is performed in the noise-sensitive pretreatment process of edge detection. In noise reduction using a Gaussian filter, filtering is performed using a mask with a two-dimensional (2D) Gaussian distribution, which can be expressed as shown in Eq. (4). σ is the standard deviation.

Fig. 2. Gaussian mask. 350

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Fig. 3. Prewitt mask: (a) horizontal mask and (b) vertical mask.

Fig. 4. High-boost filter.

Fig. 5. Hough transform: (a) x − y plane and (b) θ − ρ plane.

G (x , y ) =

1 2

x 2 + y2

e

2 2

(4)

where (x, y) is the position of the pixel, and G(x, y) is the Gaussian function whose center position is (0, 0) and whose standard deviation is 1. As the Gaussian function is a continuous function with an infinite amount of information, it is expressed as a finite matrix through discrete approximation for actual implementation. Fig. 2 shows the 3 × 3 matrix obtained through the discrete approximation of a Gaussian function with a standard deviation of 0.4, and a weight was given to the image using the convolution mask. In this study, a Gaussian filter was applied to the image to facilitate the edge detection of the piping system, and edge lines were detected using a Prewitt mask [34]. In the image, edges generally occur between the object and the background as well as in the area inside the object where the pixel value abruptly changes. The Prewitt edge detection method applies horizontal and vertical slope masks to the spatial domain. To obtain edges using this method, the horizontal and vertical slopes of image I are first obtained, as shown in Eq. (5).

Gi = I (i

1, j

1) + I (i, j

= I (i

1, j

1) + I (i

1) + I (i + 1, j 1, j) + I (i

1)

1, j + 1)

{ I (i

1. j + 1) + I (i, j + 1) + I (i + 1, j + 1) } Gj

{I (i + 1. j

1) + I (i + 1, j ) + I (i + 1, j + 1) }

(5)

where i and j are the position indices of the pixel in the image. The final edge image for the Prewitt edge detection method is shown in Eq. (6), and the discretized form is shown in Fig. 3. 351

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Fig. 6. Lines detected using Hough transform.

O (i, j ) =

Gx2 + Gy2

(6)

To sharpen the edge lines [35] of the piping system, unsharp mask processing was used. Unsharp mask processing subtracts blurred images from an image and is expressed as Eq. (7).

gus (x , y ) = f (x , y )

(7)

f (x , y )

where f(x, y) is the original image, f (x , y ) is the blurred image of f(x, y), and gus(x, y) is the sharp image obtained through unsharp mask processing. The generalization of unsharp mask processing is called “high-boost filter processing” and can be expressed as Eq. (8):

ghb (x , y ) = Af (x , y )

(8)

f (x , y )

where A ≥ 1 holds. This equation can be expressed as Eq. (9):

ghb (x , y ) = (A

1) f (x , y ) + f (x , y )

(9)

f (x , y )

By substituting Eq. (7) into Eq. (9), the high-boost filter can be expressed as Eq. (10):

ghb (x , y ) = (A

(10)

1) f (x , y ) + gus (x , y )

Fig. 7. Displacement and deformation angle measurement algorithm. 352

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Fig. 8. Program for measuring deformation angle using MATLAB.

ELBOW PIPE - ASME B36. 10. SA-106 STEEL 3 in. SCH. 40, O.D. = 88.9mm, THK. = 5.49mm 57,33 60

28

270 3D (3*88.9 = 266.7)

11

4,

3

36 9,

84,38

24

102,33

40

Elbow connection jig

Coupling φ 15 tap

25

φ

382,33

φ 15

200

90

85

SECTION

5,49

45

8,9

77,92

0

φ8 30 30 30

φ3

30

30 30 30 30

10

? 30

Fig. 9. Manufactured steel pipe elbow and jig. 353

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The discrete approximated 3 × 3 matrix of Eq. (10) can be expressed as shown in Fig. 4. A power-law transformation [36] was used to increase the edge detection in the enhanced shape of the piping system. The method that uses the power-law transformation is shown in Eq. (11), which adjusts the gray level. c and γ are positive constants, s is the output, and γ is the gray level of the input pixel. This type of function can enlarge the narrow range of the bright input value depending on the value of γ, and vice versa. In the case of γ = 1, the input and output values are absolutely the same. In the case of γ < 1, the wide range becomes wider when the input value is small, and the narrow range becomes wider when the input value is large. (11)

s = cr

For the shape of the piping system enhanced through the previous pretreatment process, the line of the shape is detected using the Hough transform of Eq. (12). The Hough transform algorithm [37,38] is used to detect straight lines, curves, and other simple shapes from an image. In this study, lines were detected from the output shape through the Hough transform, and the deformation angle of the crossed lines was measured. The change value ρ is the distance of the line from the origin along the vector perpendicular to the line, and θ is the angle between the x-axis and this vector. The Hough transform expresses a straight line on the (x, y) plane as a curve in the Hough domain by creating the θ and ρ parameters through Eq. (12):

= x cos

(12)

+ y sin

The straight line expressed through Eq. (12) is represented on the (x,y) plane, as shown in Fig. 5(a), where x and y represent the horizontal and vertical axes in the image, respectively. Points A, B, and C located in the above straight line are expressed as a curve with a sinusoidal form in Fig. 5(b). Straight line l is expressed as a point where the curves are crossed in the Hough domain, and this point is called a “cumulative cell.” In the conventional method, which finds all straight lines in an image by computing all θ values, shape detection is difficult because straight lines are detected not only from the boundary of the shape but also from the noise in the image. For the lines detected through the Hough transform, not only the valid lines corresponding to the shape of the piping system but also many and extremely diverse lines are detected owing to the noise. The valid lines of the deformation angle tend to be closer to diagonal lines than to horizontal or straight lines. Therefore, the angles of the lines that can be judged as the shape of the elbow were determined to be between 30° and 60°, and the detected lines whose internal angles exceeded this range were removed using Eq. (13). In addition, lines with 300 or more pixels, which could be judged as the lines of the piping system, were stored and extended. These two extended lines crossed each other. By removing the lines with no intersection, the lines created by noise and other environments were eliminated. As four lines were detected in the piping system, as shown in Fig. 6, the average of θ1 and θ2, which were the internal angles of the intersecting lines, were obtained and used as the deformation angle.

Fig. 10. Sensor installation position: (a) installed steel pipe elbow and (b) steel pipe elbow acquired image measurement system. 354

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Fig. 11. Elbow with through leakage.

Table 1 Test results for each load case. Loading amplitude

Number of cycles to failure (N)

Moment range (kN‧m)

Relative deformation angle range (rad)

± 20

97.500 107.750 109.750 37.500 45.500 46.000 15.500 18.250 18.500 10.750 11.000 11.250 7.250 8.000 8.250 5.000 5.500 5.500 3.750 5 5.500 3.250 3.500 4.750 2.750 3.000 3.250

20.546 19.974 19.798 21.847 23.815 23.414 23.291 23.297 24.516 25.576 25.107 24.671 27.142 26.114 27.016 29.990 30.189 29.307 31.376 30.518 30.649 31.210 31.459 31.479 34.050 34.194 33.697

0.144 0.145 0.145 0.221 0.216 0.213 0.294 0.292 0.294 0.357 0.365 0.367 0.450 0.467 0.456 0.526 0.528 0.520 0.604 0.606 0.597 0.684 0.686 0.689 0.765 0.758 0.765

± 30 ± 40 ± 50 ± 60 ± 70 ± 80 ± 90 ± 100

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30° (0.524rad ) < Each Line Angle > 60° (1.047rad), Line Length > 300pixel, else = remove n

= arctan i=0

ly1n lx1n n

+ arctan i=0

ly3n lx3n

ly2n

ly3n

lx 2n

lx 3n

n

+ arctan i=0

ly2n lx2n

ly3n lx3n

Defomation Angle =

n

,

2

= arctan i=0

1

ly1n

ly3n

lx1n

lx 3n

| 1| + | 2 | 2

(13)

2.3. Algorithm summary Fig. 7 shows the algorithms for measuring the displacement of the piping system and the deformation angle of the shape. They are classified into five and seven steps, respectively. The displacement measurement algorithm arranges the acquired images in chronological order and designates control points to the points on the reference image where the displacement is to be known. To analyze the correlation between the reference and deformed windows in the area where the displacement of a pipe component is to be known, coordinates are represented by calculating NSSD. The coordinates are integers, and the subpixels are calculated using the shape function. For the coordinates expressed with integers, the subpixels are calculated using a quadratic shape function, and the displacement is analyzed. To measure the deformation angle of the piping system at the displacement point expressed with integers, a Region of Interest (ROI) window with 650 × 1000 pixels is extracted. The shape of the piping system is enhanced using a Gaussian filter to reduce the noise in the image and to improve the disconnected points. From the shape of the piping system with reduced noise, the edges are detected using a Prewitt mask. To sharpen the detected edges, a high-boost filter with an A = 2 factor is applied to the image. To facilitate line detection from the image, the gray level of the image is adjusted using the power-law transformation with the α = 1 and γ = 0.67 factors. The deformation angle is measured by detecting the lines generated through the Hough transform of the shape of the piping system. Based on the displacement measured at the elbow center of the image over time, the deformation angle of the elbow is measured in the ROI window with 650 × 1000 pixels. In this study, the algorithm described above was used after it was coded into an automated program using MATLAB. For the deformation angle measurement shown in Fig. 8, the reference point marked with the color blue represents the reference point designated in the reference image, and the color cyan represents the point where the reference point is optimally matched in the deformed image. 3. Low-cycle fatigue life of pipe elbow components 3.1. Pipe elbow component In this study, the elbow was determined to be the fragile position of the piping system in the case of seismic loads. A pipe component consisting of a 3-in SA106, Grade B, and SCH 40 ASME B36.10 pipe and an elbow was manufactured and used as shown in Fig. 9(a). The diameter of the pipe was 88.9 mm, and the thickness was 5.49 mm. To enable the plastic behavior of the elbow, the length of the straight pipe was made at more than three times the diameter (3D-270 mm), and the pipe was welded onto the elbow. Jigs were manufactured and welded onto both ends of the specimen, as shown in Fig. 9(b), to implement pin connection, and the specimen was installed in a Universal Testing Machine (UTM) for the test. In addition, to assess the low-cycle fatigue life of the piping system under seismic loads, it is essential to identify the relationships among the number of cycles, moment, and deformation angle until leakage occurs. This is accomplished through an in-plane cyclic loading test. Therefore, the piping system was manufactured by a company that produces and supplies pipes to actual nuclear power plants in order to minimize the effects of parameters other than the selected ones. Welding was performed by a welding company with certification for nuclear power plants because each part was connected through welding when the pipe component specimen was manufactured.

Fig. 12. Comparison of response measured from LVDT of UTM: (a) ± 30 mm, (b) ± 60 mm, and (c) ± 90 mm. 356

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3.2. Experiment setup The positions of the targets for measuring the loading amplitude of the UTM as well as the moment of the steel pipe elbow using the image measurement system are shown in Fig. 10. Fig. 10(a) shows the steel pipe elbow installed in the UTM and the air pump that was used to apply the 3-MPa internal pressure. Fig. 10(b), on the other hand, shows the steel pipe elbow that was measured using the image measurement system, and the positions for the measurement of the loading amplitude, moment, and deformation angle using the image measurement system. The image measurement system used a Complementary Metal Oxide Semiconductor (CMOS) camera (IMB-7050G, 2448 × 2048 pixels) and a laptop as the sensor for measurement, considering portability and ease of installation. The CMOS camera and laptop performed data transfer and control using LAN communication, and a lens (M5018-MP2) was used to measure the deformation angle of the pipe component positioned at a distance. The UTM performed the in-plane cyclic loading test through displacement control, and the loading amplitude of the UTM had to be measured in order to obtain the load applied by the UTM. As shown in Fig. 10(b), target1 was installed in the jig for UTM connection, and the displacement measured using the image measurement system was compared with that measured at the LVDT installed inside the UTM for synchronization. In addition, the

Fig. 13. Moment-deformation angle hysteresis loops: (a) ± 20 mm, (b) ± 30 mm, (c) ± 40 mm, (d) ± 50 mm, (e) ± 60 mm, (f) ± 70 mm, (g) ± 80 mm, (h) ± 90 mm, and (i) ± 100 mm. 357

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Fig. 14. Leaking line using moment-deformation angle relationship.

Fig. 15. Low-cycle fatigue curves: (a) moment-number of cycles and (b) deformation angle-number of cycles.

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moment was measured by multiplying distance d, which is the difference in horizontal distance between target1 and the center of the elbow, which is the position of control point, and the force applied by the UTM. Based on the control point to measure the moment, the ROI window was separated and the deformation angle was measured. The deformation angle was measured by enhancing the image of the piping system in the spatial domain using image filter processing and applying the Hough transform. By applying the Hough transform and obtaining the average of θ1 and θ2, which are the internal angles of the two intersecting lines, the deformation angle was measured. In the test, 2448 × 2048 pixel images were acquired at a rate of two frames per second using the image measurement system, while the data acquisition speed of the UTM was 10 Hz. The loading amplitude of the UTM and the displacement measured through image analysis had different sampling rates and measurement initiation timings. The displacement measured by the LVDT installed inside the UTM and the displacement measured at the target1 point using the image measurement system were resampled and synchronized through correlation analysis [39]. The cross-correlation function of Eq. (14) was used for the starting point of each response, and it was used at a point when Rxy(τ) was the highest. Eq. (14) represents the correlation between two signals in the time domain. This is the function used to determine the degree of similarity of two signals to each other.

Rxy ( ) =

1 N

n

x ( k ) y (k + )

(14)

k=1

where x(k) is the displacement measured by the LVDT of the UTM, y(k) is the response measured from target1, N is the number of data items to be used in the cross-correlation function, and τ is the time to be newly defined for the cross-correlation function. 3.3. In-plane cyclic loading test An in-plane cyclic loading test was conducted to define the low-cycle fatigue curves of the steel pipe elbow against seismic loads from the correlation between the moment and the deformation angle. It assumed that the relative amplitude applied to a pipe elbow by an earthquake was 20 mm, and conducted research related to the low-cycle fatigue failure of a nuclear power plant pipe elbow [14,23]. Therefore, in this study, it was assumed that the relative loading amplitude was ± 20 mm or more at the elbow of a pipe that penetrated both seismic-isolated and general structures owing to the seismic anchor motion. The loading amplitude ranged from ± 20 to ± 100 mm and increased by ± 10 mm, and the in-plane cyclic loading test was conducted with three specimens for each load case. In the study, the failure mode of the steel pipe elbow was defined as leakage, and the in-plane cyclic loading test was conducted until leakage occurred. Fig. 11 shows the representative specimens for the load cases with leakage. For the penetrating cracks, leakage occurred at or near the crown of the steel pipe elbow in the intrados direction, and it was confirmed that the cracks propagated in the axial direction. Table 1 shows the number of cycles, moment range, and deformation angle range when leakage of the steel pipe elbow occurred in the in-plane cyclic loading test, according to the loading amplitude. In the in-plane cyclic loading test, the number of cycles to failure ranged from 2.75 to 109.75 N, the moment range was from 19.798 to 34.194 kN‧m, and the relative deformation angle range was from 0.144 rad to 0.758 rad. Table 1. Test results for each load case. Fig. 12 is a graph showing the displacement response measured at target1 using the image measurement system, and the displacement response measured by the LVDT installed inside the UTM for the representative specimens. It was confirmed that the two responses are in good agreement with each other. Fig. 13 shows the hysteresis loops of the moment and deformation angle measured during the in-plane cyclic loading test with the representative specimens for a loading amplitude ranging from ± 20 to ± 100 mm. Fig. 14 shows the leakage line that uses the relationship between the moment and the deformation angle. The derived leakage line was based on the data when leakage occurred for a total of 27 specimens. The average regression curve equation calculated using the least-square method is shown in Eq. (15). Eq. (15) shows the relationship between the moment and the deformation angle when leakage occurs in the steel pipe elbow. The determination coefficient (R2) was 0.95 or higher, indicating that the moment and the deformation angle had a linear relationship.

Moment (kN m) = 21.156 Deformation Angle (rad ) + 17.793, R2 = 0.968

(15)

For a reliability assessment of the results of the in-plane cyclic loading test, the number of cycles to failure are expressed for the moment and the deformation angle in Fig. 15. The determination coefficient was 0.90 or higher, indicating remarkably high reliability. In both graphs, the middle black solid curve represents 50% failure probability, while the red solid and blue solid curves represent 90 and 10% failure probabilities, respectively. It can be seen that the results under all conditions are located between the 10 and 90% failure probabilities in the low-cycle fatigue curves. The relations of the low-cycle fatigue curves of the piping system shown in Fig. 15 can be expressed as Eqs. (16) and (17). It appears that the low-cycle fatigue curves for the moment and the deformation angle can be predicted through these relations.

Moment (kN m) = 37.856N

0.145,

Deformation Angle (rad ) = 1.279N

(16)

R2 = 0.912 0.506,

(17)

R2 = 0.971 359

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4. Conclusions In this study, an image-based noncontact measurement method was used to measure the deformation angle of a piping system while performing an in-plane cyclic loading test. In addition, the limit state and low-cycle fatigue life against seismic loads were proposed by presenting the leakage line and low-cycle fatigue curves for the moment and deformation angle of the steel pipe elbow of a seismic-isolated nuclear power plant. An improved deformation angle measurement algorithm was applied using image filter processing to facilitate noise reduction in images as well as the detection of boundary lines. It was confirmed that the deformation angle can be measured at a distance using this algorithm, without installing conventional sensors. To define the low-cycle fatigue curves for the moment and deformation angle of the elbow, which is the fragile component of the piping system, a 3-MPa internal pressure was applied, and an in-plane cyclic loading test was conducted for a 3-in piping elbow. The loading amplitude ranged from ± 20 to ± 100 mm and was increased by 10 mm until leakage occurred in the elbow of the piping system. As a result, leakage occurred at or near the crown of the piping system in the intrados direction, and it was confirmed that the cracks propagated in the axial direction. In addition, the leakage line and low-cycle fatigue curves were represented using the relationship between the moment and the deformation angle, which exhibited high reliability in the test. Therefore, it is expected that the leakage line and low-cycle fatigue curves proposed in this study can be used as data for analyzing the limit state and fatigue failure behavior of a nuclear power plant piping system against earthquakes. 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