Metal Forming CAE Lab.Department of Mechanical EngineeringGyeongsang National University, Korea

5. ,

I

Metal Forming CAE Lab., Gyeongsang National University

(, Slender member)

(Truss member) (Rod)

(Circular shaft)

(Beam)

-,

,

I-beam, H-beam

P P

zTrJ

= , ,bx xy xzzz zz zz

M y VQ VQI bI tI

= = =

(Column)

x xxFA

= =

ColumnBuckled

t

b

x

z

xFlange

Web

: 7800 , : 2720

(Concentrated load)

(Distributed load)

(Line distribution) :

(Area distribution) :

(Volume distribution) : (body force, : (weight))

(Specific weight) :

3kg m

33 2 3

[ ] [ ] [ ], [ ] , N m[ ] [ ] [ ]M L Fg gL T L

= =

N m, kg m

2 2 2N m , kg m , 1Pa = 1N m

Saint Venant Principle

, ,

3kg m

. .

( )

( )L A

L A

xq x dx xdAX x

q x dx dA= = =

( )q x dx dA=

( )L

R q x dx ( )q x =

( )L

X R xq x dx=

loading diagram .

(Resultant force) ,

. ,

:

y

xBA( )q x

R

X

x

y

x

Loading diagram

( )q x dx

x dx

, ,

, ,

V V V V V V

V V V

V V V

xdW xdm y dW y dm z dW z dmx y z

W m W m W m

x dV y dV z dVx y z

dV dV dV

= = = = = =

= = =

VW x xdW

W mgVg

=

==

xx

dW

VW

x

y

' ' ', ,= = = = L L Lx dL y dL z dLx y z

L L L

, ,x y z', ', 'x y z

(centroid) : . , ,

:

:()

:

2

, , ,x y z

,

x yy z

: . .

' ' ', ,= = = = A A Ax dA y dA z dAx y z

A A A' ' '

, ,= = = = V V Vx dV y dV z dV

x y zV V V

'y

yx

y

'x

x

x

y

z

y

( ) ( )y x x yA CF F dxdy F dx F dyx y

= +

2 2A

V r dA rA = =

Green

(Divergence theorem)

( ) ( )b

a

df dx f b f adx

=

( ) ( )yx z x x y y z zV SFF F dv F n F n F n dS

x y z

+ + = + +

xV SV dv xn dS= =

:

:

:

1 2

1 2

1 1 2 2

1 2

n

n

A A AA

A A A A

i in n

n i

x dA xdA xdAxdAx

dA dA dA dA

A xx A x A x AA A A A

+ + += =

+ + +

+ + += =

+ + +

11 1

A

A

xdA xA

xdA x A

=

=

10 1200 6 1600 / 2 16800R= + =12000 5 4800 (4 2 /3 6) 41

1200 10 4800 7x + + = =

+

5.9 :

I

II

()

- I

-II

16800 0y A BF R R= + =41 16800 10 0

7

= + = A BM R9,840 N, 6,960 NB AR R = =

12000 4800 0y A BF R R= + =5 12000 8 4800 10 0A BM R= + =

4 10

0 4

4001200 (2 1) 03

= + + = y A BF R R dx x dx

4 10

0 4

40010 1200 (2 1) 03A B

M R xdx x xdx= + =

Rx

AR BR

AR BR

12000 4800

5 3 2

x dx x dx

1000dx400 (2 1)3 +x dx

5.93 : A B

0 ; 2000 0

0 ; 0.55 ( 2000) 0.8 0

625 N, 1375 N

y A B

A B

A B

F R R

M R

R R

= + =

= + =

= =

AR BR0.50.3

2000NR =

4000N/m

0 ; 6 2 0

0 ; 2 ( 6) 4.5 ( 2) 0

8 kN, 13 kN m

y A

A A

A A

F R

M M

R M

= =

= + + =

= =

5.96 : A ?

AR

1 4 3 6(kN)2

R = =

2 2.5

2kN4kN/m

AM

1 2

2

0 ; 1.5cos60 0

0 ; 1.5sin 60 0

0 ; (0.4 0.6)

3.6 1.5sin 60 4.8 0

0.75 kN, 1.22 kN, 3.08 kN

x A

y A B

A

B

A B A

F H

F V R R R

M R

R

H R V

= =

= + =

= +

+ =

= = =

5.98 : A B

AV BRAH

1R2R

2kN/m

1.5kN

11 1.8 2 1.82

R = =

21 1.2 2 1.22

R = =

:, :

:, :

xx

xz xy

xx

xz xy

M twisting momentM M bending moment

F axial forceF F shear force

( )

2

: :

x

y

z

Negative x-face

x

y

z

Positive x-face

1x

y

x

VF

bM

VF

bM

xxMxzM

xyM

y

xz

xxF

xyF

xzF, ,= = =xx xy b xzF F V F M M

3

, ,

,

(F ): +

(Mt ): +

(V ), (Mb ): + + +, +

+ . +

FF ( )+F ( )+tM ( )+V ( )+bM

-, -

(c) (d)

(a) (b)

VbM

bM

V

F

F

, .

() ,

.

, ,

.

+ .

.

-- .

0 .

0 .

1:1 .

.

38x L=

o 2L

L( )V x

L0

( )bM x

L0

8L0

38

L0

29128 L0

V+

bM +