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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 +