격자무늬를 갖는 에어포일의 단면 해석 및 워핑 모형 제작

(beam) ,

, .

(Bending), (Torsion),

(Tension), (Compression) .

,

. I

X

I

Y

,

J

. ,

. - (Euler-Bernoulli beam)

. (Saint-Venant)

. (thin-walled beam)

(warping) .

. KSec2D

. ,

. (warping displacement)

3D .

2.1 Fig 1

(reference axis) (tension center) (shear modulus weighted centroid),

(shear center) .

(centroid) .

1 (area

moment of inertia) .

2 2

, ,

Y Z YZ

A A A

I = òò z d A I = òò y d A I = òò yzd A ⁽¹⁾

� $ � $ �

2 2

, ,

Y Z YZ

A A A

I = òò z d A I = òò y d A I = òò yzd A

$ $ $ $ $

(2)

, Ī

Y

, Ī

Z

, Ī

YZ

,

Î

Y

, Î

Z

, Î

YZ

. Ā, Â

. ȳ, z̄

ŷ, ẑ

. (1), (2) .

. (Trefftz) .

(

xy xy xz xz

) ⁰

A A

t t dA dA

dy dz

w w

t + t = ^é ê ^æ ç ^{¶ ö} ÷ t + ^æ ç ^{¶ ö} ÷ t ^ù ú =

è ø

ê è ø ú

ë û

òò òò ⁽³⁾

xy

,

xz

t G z t G y

dy dz

w w

q ^{æ ¶ ö} q ^{æ ¶ ö}

= ç è - ÷ ø = ç è + ÷ ø (4)

t

xy

t

xz

(torsional shear

stress) t

xy

t

xz

(flexural shear stress), q ω

(out-of-plane warping function) . (Saint-Venant)

P .

( , ), ,

x y z

u = qw y z u = zx u q = yx q (5)

x .

(3) -

.

0,

^y

0, 0

x z

x y z

u u u

x y z

e = ^¶ = e = ^¶ = e = ^¶ =

¶ ¶ ¶ (6)

y x

xy

x xz z

z y yz

u u z

x y y

u

u y

x z z

u u

y z

g q w

g q w

g

¶ ¶ æ ¶ ö

= ¶ + ¶ = ç è - + ¶ ÷ ø

¶

¶ æ ¶ ö

= ¶ + ¶ = ç è - + ¶ ÷ ø

¶ ¶

= +

¶ ¶

(7)

(4) (5) (torque)

.

( )

� �

A

A

T G y z dA

G y y z z dA

z y

GJ GJ

q t t

w w

q

q f

= -

é æ ¶ ö æ ¶ ö ù

= ê ê ë ç è ¶ + ÷ ø - ç è ¶ - ÷ ø ú ú û

= =

òò

òò ⁽⁸⁾

Ĵ (torsional constant) .

�

A

d d

J y y z z dA

dz dy

w w

é æ ö æ ö ù

= ê ç + ÷ - ç - ÷ ú

è ø

ê è ø ú

ë û

òò ⁽⁹⁾

.

2

Y r Y r

A

EI = E I = E òò z d A (10)

2

Z r Z r

A

EI = E I = E òò y d A (11)

YZ r YZ r

A

EI = E I = E òò yzd A (12)

� $ �

ˆ ˆ

r r

ˆ

Y Z

A

GJ G J G I I z y d A

y z

w w

é æ ¶ ¶ ö ù

= = ê ê ë + - òò ç è ^$ ¶ - ¶ ÷ ø ú ú û ⁽¹³⁾

2.2.

(DAE31) . I

. ,

90

^o

, 45

^o

.

Fig 2 ,

. 2 mm 2 mm

12 mm t

.

. 1 m

,

.

Z

Y Z

Y R

R T.C

S.M.W.C y

z

ˆy ˆz

ysc S.C zsc

Y Z

Yˆ Zˆ

T.C : Tension center

S.M.W.C : Shear modulus weighted centroid S.C : Shear center (ysc, zsc)

R: Reference axes

Fig 1. KSec2D

CATIA Fig 2 .

MSC.PATRAN

(node) (element) .

Fig 3 .

Fig 3 MSC.PATRAN Fig 2

.

(linear

triangular element) .

EDISON KSec2D

.

.

KSec2D .

ParaView 3D

CATIA 3D

3D .

2.3 KSec2D

,

. y , z

(steel)

. .

Table 1.

E [GPa] u ρ [kg/m

³

]

210 0.3 7,765

MSC.PATRAN

KSec2D (input file) . KSec2D

(output file) out ,

warp ,

ParaView case .

. .

.

, y , z ,

.

Table 2.

Thickness [mm]

Area [mm

²

]

EI

Y

[N·mm

²

]

EI

Z

[N·mm

²

]

GJ [N·mm

²

] 2 4.05×10

³

3.43×10

¹²

2.71×10

¹⁴

1.64×10

¹²

4 7.97×10

³

6.73×10

¹²

5.19×10

¹⁴

3.19×10

¹²

6 1.18×10

⁴

9.91×10

¹²

7.47×10

¹⁴

4.64×10

¹²

8 1.55×10

⁴

1.30×10

¹³

9.59×10

¹⁴

5.97×10

¹²

10 1.91×10

⁴

1.59×10

¹³

1.16×10

¹⁵

7.20×10

¹²

12 2.26×10

⁴

1.88×10

¹³

1.34×10

¹⁵

8.32×10

¹²

Fig 2. : (a) (b) 90

^o

(c) 45

^o

(a)

(b)

(c)

Fig 3. : (a) (b) 90

^o

(c) 45

^o

Table 3. 90

^o

Thickness

[mm]

Area [mm

²

]

EI

Y

[N-mm

²

]

EI

Z

[N-mm

²

]

GJ [N-mm

²

] 2 8.92×10

³

7.08×10

¹²

4.84×10

¹⁴

2.46×10

¹²

4 1.71×10

⁴

1.37×10

¹³

8.96×10

¹⁴

4.69×10

¹²

6 2.44×10

⁴

1.97×10

¹³

1.24×10

¹⁵

6.64×10

¹²

8 3.12×10

⁴

2.53×10

¹³

1.55×10

¹⁵

8.39×10

¹²

10 3.76×10

⁴

3.05×10

¹³

1.84×10

¹⁵

9.94×10

¹²

12 4.32×10

⁴

3.47×10

¹³

2.12×10

¹⁵

1.12×10

¹³

Table 4. 45

^o

Thickness

[mm]

Area [mm

²

]

EI

Y

[N·mm

²

]

EI

Z

[N·mm

²

]

GJ [N·mm

²

] 2 7.75×10

³

6.39×10

¹²

4.35×10

¹⁴

2.23×10

¹²

4 1.49×10

⁴

1.22×10

¹³

8.17×10

¹⁴

4.24×10

¹²

6 2.14×10

⁴

1.75×10

¹³

1.15×10

¹⁵

6.04×10

¹²

8 2.73×10

⁴

2.23×10

¹³

1.43×10

¹⁵

7.64×10

¹²

10 3.28×10

⁴

2.66×10

¹³

1.69×10

¹⁵

9.05×10

¹²

12 3.80×10

⁴

3.08×10

¹³

1.94×10

¹⁵

1.04×10

¹³

Table 2, 3, 4

, , .

,

. 90

^o

, . 90

^o

.

Fig 4 .

90

^o

,

. z

, .

90

^o

z

21~23%, 42~47%, 45

^o

16~19%, 35~41% .

y

, 90

^o

0.9%,

Fig 4.

0.4% 0.1%, 0.9%

. 45

^o

0.4~0.5%

0.1% . 90

^o

45

^o

y

10 mm .

2.4

ParaView

.

KSec2D

CATIA .

KSec2D warp

x .

x, y, z CATIA

. 3D

3D . warp x

.

CAITA Fig 5 .

Fig 5 CATIA STL 3D

(a)

(b)

Fig 5. : (a) 90

^o

(b) 45

^o

(a)

(b)

Fig 6. 3D : (a) 90

^o

(b) 45

^o

3D Fig 6 .

Fig 2

Fig 6 .

, y , z ,

.

. z

.

45

^o

, 90

^o

.

z

90

^o

45

^o

.

(buckling) .

I ,

. I

, I

, .

. ,

, .

CATIA, 3D

.

2015 ( )

·

(No, 2014M3C1A6038344)

C. T. Sun, (2006) Mechanics of Aircraft Structure

, , , (1996)

, , pp.195~199

, , , , , (2011)

, , 39 , 3 pp.261~268

, (2012)

, ,

pp.578~582

, , , (2009)

, ,

37 , 5 , pp.442~449

, , (2001) H

, , 21 , 2 ,

pp.289~292

Park. I. J., M.. K. Dhadwal, Jung S. N., Kim D. H. (2011) Experimental

Validation of Cross-sectional Analysis for Composite Rotor Blades,

The 18

^th

International Conference on Composite Materials.