Journal of Mechanical Science and Technology 25 (5) (2011) 1193~1199

www.springerlink.com/content/1738-494x DOI 10.1007/s12206-011-0222-5

Buckling limit evaluations for reactor vessel of ABTR†

Gyeong-Hoi Koo* and Suk-Hoon Kim Department of Liquid Metal Reactor Development, Korea Atomic Energy Research Institute, 1045 Daedeok-daero, Yuseong-gu, Daejeon 305-353, Korea

(Manuscript Received September 6, 2010; Revised December 19, 2010; Accepted January 9, 2011)

----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract We evaluated the buckling limit of a conceptually designed reactor vessel of the Advanced Burner Test Reactor (ABTR) for a horizon-

tal safe shutdown earthquake (SSE) seismic load. In this evaluation, both seismic isolation and non-isolation designs were considered for thin reactor vessels subjected to elevated temperature services. For calculating the buckling load, two kinds of methods, a numerical simulation method using finite element analysis and an evaluation formula driven from a theoretical basis, were used. To consider the material aging effect caused by a 60-year design lifetime and 510oC normal operating temperature, an isochronous stress-strain curve corresponding to these conditions was used for the nonlinear elastic-plastic buckling analysis method. From the evaluation results of the buckling load, it was found that plasticity behavior significantly affects the buckling strength, but that the initial geometrical imperfec-tions have little effect. Also, the non-seismic isolation design does not satisfy the buckling limit rules of the ASME BPV Section III, Subsection NH, but the seismic isolation design does satisfy it with sufficient margins.

Keywords: ABTR; Seismic buckling; Liquid metal reactor; Seismic isolation design; Elevated temperature design ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Introduction

Most high temperature reactor vessels cooled by a liquid metal are basically designed using a relatively thin thickness of less than 5 cm to prevent excessive thermal bending stress caused by elevated temperatures of over 500oC [1]. In general, liquid metal reactors are operated under a lower design pres-sure of less than 5bar, and therefore a thin design is available. However, this design feature inevitably creates an issue re-lated to the seismic design. To overcome this weakness of a thin design, most liquid metal faster reactors tend to adapt the seismic isolation design concept.

In this paper, seismic buckling evaluations are carried out to investigate the seismic isolation design performance of the advanced burner test reactor (ABTR) vessel design for a hori-zontal safe shutdown earthquake (SSE) seismic load. To do this, the buckling evaluations are performed on both seismic isolation and non-isolation design conditions using two meth-ods, a numerical simulation method using finite element analysis and an evaluation formula driven by a theoretical basis.

For a buckling design, many researchers have developed simple evaluation formulae [2-4]. Okada et al. proposed a

buckling strength equation for cylindrical vessels under static shear loads, which was developed on the basis of theoretical considerations for plasticity and geometrical imperfections [2]. Recently, Michel et al. studied the buckling of a cylindrical shell under static and dynamic shear loading to understand the effects of frequency excitation in seismic events [5]. To verify the evaluation formulae, various buckling tests have been also carried out using the scaled test models [6]. In this paper, sen-sitivity studies on the buckling strength for various reactor vessel geometric dimensions are performed and discussed in detail.

For the numerical simulations, three types of finite element analysis methods, an eigenvalue buckling analysis, nonlinear elastic buckling analysis, and nonlinear elastic-plastic buck-ling analysis, are used to ascertain the pertinent method for use in an actual design. Actually, one of the main issues in a high-temperature buckling analysis is how to consider the material degradation effects caused by service at elevated temperatures at the end of a design lifetime. In this paper, the isochronous stress-strain curve provided by ASME BPV Section III, Sub-section NH [7] is used in the nonlinear elastic-plastic buckling analysis. To confirm the finite element model, two types of finite element models are used, a simple shell element model without a support flange and bottom head parts, and a detailed solid element model. The results of the numerical simulations are compared with those of the evaluation formulae to confirm the buckling loads of the ABTR reactor vessel.

† This paper was recommended for publication in revised form by Associate Editor Chang-Wan Kim

*Corresponding author. Tel.: +82 42868 2950 E-mail address: ghkoo@kaeri.re.kr

© KSME & Springer 2011

1194 G.-H. Koo and S.-H. Kim / Journal of Mechanical Science and Technology 25 (5) (2011) 1193~1199

To assure the structural seismic integrity of the ABTR reac-tor vessel, especially in terms of buckling, the ASME-NH rules for load-controlled time-independent buckling limits are adapted in this paper.

2. Methods for buckling load calculations

2.1 Numerical simulation methods

Buckling analysis can be carried out using numerical finite element methods such as eigenvalue buckling analysis and nonlinear elastic-plastic buckling analysis. More detailed de-scriptions for these methods are as follows: 2.1.1 Eigenvalue buckling analysis

Eigenvalue buckling analysis predicts the theoretical buck-ling strength (the bifurcation point) of an ideal linear elastic structure. Bifurcation buckling refers to the unbounded growth of a new deformation pattern. The buckling problem is formu-lated as an eigenvalue problem as follows:

( [ ] [ ] ) { } {0}i iK Sλ ψ+ = (1) where matrices [K] and [S] indicate the system stiffness matrix and stress stiffness matrix, respectively, λi is the ith eigenvalue, and {ψ}i is the ith eigenvector of displacement. The eigenvec-tors are normalized so that the largest component is 1.0. Im-perfections and material nonlinearities cannot be included in this analysis. Thus, the buckling strength obtained from an eigenvalue buckling analysis may differ from that of a real structure, and often yields unconservative results. Therefore, care is required when using this method in an actual evalua-tion of buckling strength. 2.1.2 Nonlinear buckling analysis

Nonlinear buckling analysis including geometric and mate-rial nonlinearities is usually the more accurate analysis ap-proach and is therefore recommended for the design or evalua-tion of actual structures. There are two methods for obtaining buckling strength based on nonlinear buckling analysis. One basic approach is to constantly increment the applied loads until the solution begins to diverge, which can be obtained from a load-controlled buckling analysis. In this approach, a simple static analysis is conducted with large deflections ex-tended to the point where the structure reaches its limit load. Another approach is to constantly increase the displacement sufficiently to obtain a snap-through buckling curve, which can be obtained from a displacement-controlled buckling analysis. In a nonlinear buckling analysis, a sufficiently fine load or displacement increment should be used to obtain the expected buckling strength.

In this report, two approaches to nonlinear buckling analysis are carried out, with and without the plasticity effect. The results of the nonlinear buckling analysis without plasticity effect will be compared with the theoretical elastic bending

and shear buckling load from the evaluation formula.

2.2 Evaluation formula method

The buckling strength of a cylindrical shell structure under shear force can be obtained on the basis of theoretically calcu-lated elastic buckling loads. By including the effects of initial geometrical imperfections and plasticity, the critical shear buckling load, which causes buckling of a cylindrical shell, can be represented as follows:

, ,[ , ]b scr cr o cr oQ Min Q Qα= (2)

where α is the imperfection reduction factor. Qb

cr,o and Qscr,o in

Eq. (2) represent the critical shear loads that induce elastic-plastic bending buckling and shear buckling, respectively. These critical shear loads can be represented as follows:

, ,b bcr o b c cr eQ y Qη= , (3)

, ,s scr o s s cr eQ y Qη= . (4)

In Eqs. (3) and (4), Qb

cr,e and Qscr,e represent the theoretical

elastic bending and shear buckling load for a perfect cylindri-cal shell shape, ηc and ηs represent the plasticity reduction factors for axial compressive buckling and shear buckling, and yb and ys represent the axial and shear stress distribution fac-tors, respectively. These factors can be determined analyti-cally as functions of the following geometric and material parameters of a cylindrical shell: R (radius), t (thickness), L (length), E (Young’s modulus), σ0.2 (0.2% proof stress) and ν (Poisson’s ratio). For LMR component materials such as 304SS, 316SS, and Mod.9Cr-1Mo steel, these factors can be obtained approximately based on an assumption of a Ram-berg-Osgood type stress-strain relation as follows [2]:

0.7 ,[1.0, 1.04 tanh(0.98 / )]c

c E cr eMinη σ σ= (5)

0.7 , 0.7 ,[1.14 tanh( / ), tanh(1.6 / )]s ss E cr e E cr eMinη τ τ τ τ= (6)

0.7 ,1.0 0.21sech(3.5 / )cb E cr ey σ σ= + (7)

0.7E ,

0.7E ,

[1.0 0.22sech(1.7 / )

1.0 13.0sech(6.4 / ) ]

ss cr e

scr e

y Min τ τ

τ τ

= +

+ (8)

where

1/ 9 10 / 90.7 0.21.815E Eσ σ−= (9)

0.7 0.7 / 3E Eτ σ= . (10)

In Eqs. (5)-(8), ,ccr eσ and ,

scr eτ are theoretical elastic buck-

ling stresses for axial compression and shear, respectively, and can be expressed as [8, 9]:

2 1/ 2

, [3(1 )]ccr e

E tR

σ ν −= − (11)

G.-H. Koo and S.-H. Kim / Journal of Mechanical Science and Technology 25 (5) (2011) 1193~1199 1195

5 / 4 1/ 22

, 2 5 / 80.07708(1 )

scr e

E R Lt R

πτν

− −⎛ ⎞ ⎛ ⎞= ⎜ ⎟ ⎜ ⎟− ⎝ ⎠ ⎝ ⎠

. (12)

For the effects of initial geometrical imperfections on buck-

ling strength, an imperfection similar to the buckling mode is known as the most severe geometrical shape reducing the buckling strength [10]. In this paper, the imperfection reduc-tion factor, α in Eq. (2) developed by Okada etc., is used as follows:

20.66 0.9 1.0α γ γ= − + (13)

where

0.2 RE tσγ = . (14)

The applicable dimensional ranges of the cylindrical shell

for the evaluation formulae of buckling strength are 0.5 < L/R < 5.0 and 50.0 < R/t < 500.0.

3. Buckling limit evaluations

3.1 Descriptions of ABTR reactor vessel

The design material of the ABTR reactor vessel is 316 stainless steel, and the height and outer diameter of the side cylinder are 12m and 5.67m, respectively. The thickness of the ABTR reactor vessel is very thin at 0.05m, which is de-signed to protect excessive thermal bending stress during ser-vice at elevated temperatures.

Fig. 1 shows the conceptual shape and dimensions of the ABTR reactor vessel. The reactor vessel is supported on a skirt-type reactor support structure on a top flange, such as in

a cantilever structure. The slenderness ratio, L/R, is 4.27. In addition to thinness of the reactor vessel, large weights are concentrated on the bottom head of the reactor vessel. The entire core assemblies and reactor internal structures are sup-ported by a core support structure. The core support structure is welded to the bottom head of the reactor vessel, and the total inertia mass of the core and reactor internal structures is exerted on the bottom head of the reactor vessel through the core support structure. Therefore, it is assumed that the exter-nal load or displacement by a horizontal seismic load is mainly applied to the bottom head of the reactor vessel in this buckling evaluation, as shown in Fig. 2. A top hanging reactor vessel with a large weight effect on the bottom head is an un-profitable design feature for a lateral seismic load. Therefore, a seismic buckling evaluation of the ABTR reactor vessel is very important to assure its structural integrity.

3.2 Buckling load calculations

The normal operating temperature of the hot liquid primary coolant is 510oC, and the design pressure is very low, 0.5 MPa. The service design lifetime of the ABTR is 60 years. There-fore, due to the long holding time with elevated temperatures over 500oC it is expected that accumulated inelastic strain and creep-fatigue damage during a total service lifetime may be severe. The existence of geometric changes and structural damage at the end of its service lifetime may affect the buck-ling strength of the ABTR reactor vessel.

In this paper, the effects of maximum accumulated inelastic strain and creep-fatigue damage are not considered directly in the evaluation of the buckling strength. However, the isochro-nous stress-strain curves provided in ASME Subsection NH,

Fig. 1. Pre-conceptually designed ABTR reactor system.

Displacement

Fixed B.C.

Displacement

Fixed B.C.

Fig. 2. Applied boundary conditions for buckling analysis.

1196 G.-H. Koo and S.-H. Kim / Journal of Mechanical Science and Technology 25 (5) (2011) 1193~1199

which are intended to provide designers with information regarding the total strain caused by stress under elevated tem-perature conditions, assuming average material properties, are used in a nonlinear elastic-plastic buckling analysis. The deg-radation effect of a material at the end of its service lifetime is considered using an isochronous stress-strain curve corre-sponding to a service lifetime of 60 years and an assumed average metal temperature for normal operation of 510oC. Fig. 3 shows the used isochronous stress-strain curve extrapolated to 500,000 hours.

For numerical simulations, the commercial finite element computer program, ANSYS [11] is used. In this paper, three types of buckling analyses, an eigenvalue buckling analysis, nonlinear elastic buckling analysis, and nonlinear elastic-plastic buckling analysis, are carried out. For the modeling of

the reactor vessel, two types of finite element model, a simple shell model and a detailed 3-dimensional solid model includ-ing a top support flange, are used to investigate the buckling limit loads. Fig. 4 presents the finite element models used in the analysis. Fig. 4(a) shows a half symmetric shell model constructed using a 4-node elastic shell element (SHELL63) for the eigenvalue and nonlinear buckling analysis without plasticity effect, and a 4-node plastic large strain shell element (SHELL43) for nonlinear buckling analysis with plasticity effect. Fig. 4(b) shows a half symmetric solid model con-structed using a 3-D structural solid element (SOLID 45), including a top flange and bottom head. In the buckling analy-ses, a displacement-controlled analysis is used under the con-servative assumption that the total lateral seismic load induced by the inertia mass of the core assemblies and the reactor in-ternals is exerted on the bottom head of the reactor vessel. Therefore, the displacement load is applied to the bottom head horizontally for the buckling analyses.

Fig. 5 presents the results of the buckling modes for the shell element model. As shown, the eigenvalue method (Fig. 5(a)) and nonlinear elastic method (Fig. 5(b)) apparently give the mixed buckling mode shape representing the shear deflec-tion mode formed throughout the cylindrical body and the bending deflection at the top edge of cylinder due to the axial compression by bending. However, the buckling mode shapes by the nonlinear elastic-plastic buckling analyses show a pure bending mode as illustrated in Fig. 5(c). Fig. 6 presents the buckling mode shapes obtained using a detailed 3-D solid

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00

20

40

60

80

100

120

140

160

180

200

316 Stainless SteelTemp=510oCTime=5x105 hours

Stre

ss, M

Pa

Strain, %

Yield stress (117 MPa)

Fig. 3. Isochronous stress-strain curve used in buckling analyses.

(a) (b) Fig. 4. Used finite element analysis models: half-symmetry (a) Simple shell element model; (b) Detailed solid element model.

(a) (b) (c) Fig. 5. Buckling mode shapes for shell element model (a) Eigenvalue; (b) Nonlinear elastic; (c) Nonlinear elastic-plastic.

(a) (b) (c) Fig. 6. Buckling mode shapes for solid element model (a) Eigenvalue; (b) Nonlinear elastic; (c) Nonlinear elastic-plastic.

G.-H. Koo and S.-H. Kim / Journal of Mechanical Science and Technology 25 (5) (2011) 1193~1199 1197

analysis model. As shown in the results, the eigenvalue method gives a pure shear buckling mode slightly different from that in the shell element model; however, the nonlinear analysis methods present similar results to the shell element model.

The snap through curves of the applied displacement versus the reaction force obtained by the elastic-plastic buckling analyses are presented in Fig. 7. With an increase in displace-ment, the reaction forces reach critical buckling loads, 8.8 MN for a simple shell model, and 10.1 MN for a detailed solid model. As another method described in section 1.2, the buck-ling load is calculated by using the evaluation formula to con-firm the buckling loads. The input parameters for the calcula-tions are as follows:

- Aspect ratio: L/R = 4.72, R/t = 56.2, - Elastic modulus: E =160 GPa, - Yield Strength: σ0.2 = 117 MPa, - Poisson’s ratio: ν = 0.3. The results of the critical buckling load obtained from the

evaluation formula and the numerical simulations for the ABTR reactor vessel are as follows:

- Evaluation formula : Qcr = 11.7 MN (bending mode), - Detailed solid analysis model : Qcr = 10.1 MN (bending

mode),

- Simple shell analysis model : Qcr = 8.8 MN (bending mode).

As shown in the above results, the numerical finite element analyses give more conservative results than the evaluation formula. Table 1 presents a summary of the buckling loads. We can see that the linear eigenvalue buckling analysis sig-nificantly overestimates the critical buckling load compared with nonlinear buckling analyses. Also, the nonlinear elastic buckling analysis without plasticity effect reveals good agreement in terms of buckling mode shape with the elastic-plastic analysis method, but still shows a significantly overes-timated buckling load. Since the eigenvalue buckling analysis and nonlinear elastic buckling analysis methods may result in less conservative buckling load, especially at elevated tem-peratures, these methods are not suitable for use in the actual buckling design stage. Therefore, it is strongly recommended to use the nonlinear elastic-plastic buckling analysis method for a reactor vessel design with a long lifetime and elevated temperatures. In this case, considering the material degrada-tions at the end of a design lifetime, how to define and use the stress-strain curve for a nonlinear elastic-plastic buckling analysis is important.

To investigate the effects of geometric dimensions on the buckling strength, sensitivity analyses are carried out using the evaluation formula calculations. Fig. 8 shows the calculation results of buckling loads with or without the effects of initial geometrical imperfections and plasticity for various slender-ness ratios. From the figure, an initial geometrical imperfec-tion has little effect on the buckling strength, but plasticity has a great effect on an ABTR reactor vessel. From the results corresponding to the shape factor L/R=4.27, it is found that the bending buckling mode will be dominant in the ABTR reactor vessel due to its long cylinder type with large slenderness ratio.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.72

3

4

5

6

7

8

9

10

11

12

Simple Shell Model

Detailed Solid Model

Elastoplastic BucklingTemp = 510oCTime = 5x105 hours

Rea

ctio

n Fo

rce,

MN

Displacement, m Fig. 7. Snap through curves obtained by the nonlinear elastic-plastic analyses.

Shear mode Bending mode

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0-100

0

100

200

300

400

500

L/R = 4.27

Buck

ling

Load

s (M

N)

Slenderness Ratio (L/R)

Elastic Bending Elastic Shear Plastic w/o Imperfection Plastic with Imperfection

Fig. 8. Buckling loads for various slenderness ratios by evaluationformulae.

: Spring & Damper

: Horizontal Couple

: Nodal Point

CV

Rx Building

Seismic Isolator

Pump

N18

N19

N20

N21

N22

N17

RI

N7

N8

N9

N10

N11

IHX

N13

N14

N15

N16

N12

RV

Core

N24

N25

N26

N28

N31

N27

N1

N2

N3

N4

N5

N6

UIS

N23

N29

N30

N32

N33

N34

N35 N36

N37

N38

N39

N40

: Spring & Damper

: Horizontal Couple

: Nodal Point

: Spring & Damper

: Horizontal Couple

: Nodal Point

CV

Rx Building

Seismic Isolator

Pump

N18

N19

N20

N21

N22

N17

RI

N7

N8

N9

N10

N11

N7

N8

N9

N10

N11

IHX

N13

N14

N15

N16

N12

IHX

N13

N14

N15

N16

N12

RV

Core

N24

N25

N26

N28

N31

N27

N1

N2

N3

N4

N5

N6

UIS

N23

N29

N30

N32

N33

N34

N35 N36

N37

N38

N39

N40

Fig. 9. Seismic time history analysis model.

1198 G.-H. Koo and S.-H. Kim / Journal of Mechanical Science and Technology 25 (5) (2011) 1193~1199

3.3 Buckling limit evaluations

The ASME Boiler and Pressure Vessel Code, Section III, Division 1, Subsection NH provides rules for buckling and instability limits for an elevated temperature design. Accord-ing to this code, the load factor shall equal or exceed the val-ues given in the code rules for the specified design and service loadings to guard against time-independent (instantaneous) buckling as follows:

cr

d

QLoad FactorQ

= (15)

where Qcr indicates a load that will cause instant instability in the design or under actual service temperature, and Qd indi-cates the design or expected load. The load factor given in the ASME code is 3.0 for the design and service loadings Level A and B, 2.5 for Level C, 1.5 for Level D, and 2.25 for the test conditions.

In this paper, the design input load for Eq. (15) is obtained using the quasi-static assumption of the seismic load as fol-lows:

Qd = M x A (16)

where M is the total mass, 687 tons, including the core (200 tons), internal structures (230 tons), and reactor vessel with primary sodium coolant (257 tons); and A is the acceleration value in the design floor response spectrum corresponding to the fundamental frequency of the ABTR reactor vessel.

To determine the acceleration values for the ABTR reactor vessel for the horizontal seismic load of SSE, which is classi-fied as service loading level D, seismic time history analyses are performed. Fig. 9 shows the seismic time history analysis model used in this paper to calculate the floor response spectra for a subsystem seismic design. Using this model, the floor response spectra at the reactor support location are obtained from the detailed seismic time history analysis for both the non-seismic isolation design and seismic isolation design con-dition. Fig. 10 presents the calculated design floor response

spectra. From the results, the acceleration responses corre-sponding to the natural frequency of the reactor vessel (6.8Hz) are determined as 1.02g for the non-seismic isolation design and 0.17g for the seismic isolation design.

Using Eq. (16), the design seismic load is determined as Qd =6.87 MN for non-seismic isolation design and Qd =1.14 MN for the seismic isolation design. The critical buckling load of the ABTR reactor vessel, Qcr, is determined to be 10.1 MN by the nonlinear elastic-plastic buckling analysis for the detailed 3-D solid model. Finally, the results of the buckling limit evaluations for the ABTR reactor vessel are as follows:

- Non-seismic isolation design : Load Factor = 1.47 < 1.5 (Not Satisfied),

- Seismic isolation design : Load Factor = 8.86 > 1.5 (Sat-isfied).

From the above results, the calculated load factor is 1.47 for the case of the non-seismic isolation design, which is less than the value of the design code, 1.5, and does not satisfy the buck-ling limit rule of the ASME-NH. However, in the case of the seismic isolation design, the calculated load factor is 8.86, which is much larger than the buckling limit load factor of 1.5. It satisfies the design rule with a large margin of about 591%. Thus, the seismic isolation design is the most important design feature for a thin reactor vessel design of an ABTR to withstand horizontal seismic load against buckling. Table 2 presents a summary of buckling limit evaluations of ABTR reactor vessel. 4. Conclusions

Seismic buckling evaluations for an ABTR reactor vessel were carried out to resolve one of the structural integrity issues in a thin reactor vessel design based on the seismic isolation design. From the results of the buckling analysis, the plasticity effect is a dominant factor in the determination of buckling load, and a detailed nonlinear elastic-plastic analysis method is re-quired for the buckling analysis. The numerical simulation method using the finite element analysis shows good agreement with the evaluation formula method. This means that the isochronous stress-strain curve provided by the ASME BPV Section III, Subsection NH rules is adequate for use in the nonlinear elastic-plastic buckling analysis method as a material property. From the results of the buckling limit evaluations by the ASME-NH rules, the ABTR reactor vessel satisfies the buckling instability design rules with a sufficient margin when the seismic isolation design is adapted. However, the non-seismic isolation design does not satisfy the buckling limit rules. Therefore, it is concluded that the seismic isolation design is mandatory for a thin reactor vessel design subjected to elevated temperature services in the sodium-cooled fast reactors.

Acknowledgment

This project has been carried out under the Nuclear R & D Program by Ministry of Education, Science and Technology of South Korea.

0.1 1 10 1000.1

1

10

100

Seismic isolation (0.33 Hz)SSE = 0.3g (X-direction)Damping = 5%

Seismic Isolation Non-Seismic Isolation

Acce

lera

tion,

m/s

2

Frequency, Hz Fig. 10. Floor response spectra at reactor support for SSE.

G.-H. Koo and S.-H. Kim / Journal of Mechanical Science and Technology 25 (5) (2011) 1193~1199 1199

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Gyeong-Hoi Koo received a Ph.D. de-gree from KAIST in 1996. Dr. Koo is working for Korea Atomic Energy Re-search Institute since 1989 as a principal engineer in field of the nuclear mechani-cal engineering. He is currently acting as a member of ASME Nuclear Codes and Standards and a chairman of Korea

ASME Mirror Committ (KAMC). He is an adjunct professor at division of electronics and control engineering in Hanbat National University since 2007.