Motion and Stress Analysis by Vector Mechanics Edward C. Ting Professor Emeritus of Applied...
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Transcript of Motion and Stress Analysis by Vector Mechanics Edward C. Ting Professor Emeritus of Applied...
Motion and Stress Analysis by Vector Mechanics
Edward C. Ting
Professor Emeritus of Applied MechanicsPurdue University, West Lafayette, IN
National Central University, ChungLi, Taiwan
a computer framework
for the study of a multi-component structural system with
• component motion • component interactions: connection, contact, collision, penetration• geometrical changes: deformation, displacement, fragmentation, collapse
• stress distribution• behavior and material property changes
a physics approach of mechanics
motion analysis and VFIFE
* vector mechanics ---- particle mechanics * discrete description * intrinsic finite element ---- physical structural element
example: a rod in plane motion
Newton’s law
1. displacement is a motion
1,2,3
, 1,2,3
2. ,
3. 0,
j j j rj jr
rj jr r j
m
x F S
S S
m1
m2
m3
m1
m3
F3
S32F2
F1S21
S12
S23
y
x
x2
m2
analytical mechanics:
For motion analysis, assume
1. rigid body, 2. functional description
ˆc
c jj
jj
x
m
I M
x x e
x F
e
1
xxc
x
01 1
cm
I M
x x
x F
1 1
1 1
1 1
ˆ
( sin 0)
x
I M
I mgd
x x e
pendulum problem: hinged at end 1
motion analysis1. general formulation:
2. complete formulation:
e
1
x
xc
1xd1
x1
1
1 1
21 1
ˆ
ˆ( , )
ˆ ˆ( )
dx
x ds
component
ds f x dx
x dF dS
x x e dS e
e
1ˆ, ( )s sdu dxA AE
stress analysis
assume: 1. deformable body, 2. Hooke’s law
e
1
1ˆdx
1ˆdx
1ˆf dx s ds
sx
1x
X1
dF
1. An approximation
→ separate motion analysis and stress analysis
► continuous bodies: motion--rigid body; stress--deformable body
► variables: motion--displacement; stress--deformation
► governing equations: motion--translation and rotation; stress--equilibrium
2. Described by continuous functions
→ discretization
computation based on analytical mechanics
1,2,3,j j j rj j
r
m
x F S
12 21 1
23 32 2
f
f
S S e
S S e
1 2 1 1
2 3 2 2
u
u
x x
x x
1 11
2 22
( )
( )
f AE u
f AE u
01 1x x
1 2e e
1. Newton’s law
2. behavior model
3. kinematics
4. Hooke’s law
5. pendulum: constraint conditions hinged end: straight rod:
vector mechanics
1
1
2
3
f2e2
-f2e2
f1e1
-f1e1
2
3
e1
e2
l1
l 2
properties:
1. structure: a set of particles
2. always a dynamic process
3. always deformable
advantages:
1. suitable for computation
2. a general and systematic formulation
3. explicit constraint conditions
development needs:
1. describe structural geometry: intrinsic finite element
2. kinematics: fictitious reversed motion
3. continuity requirements
4. mechanics requirements
5. material model: standard tests
elements:
plane rod, plane frame, plane solid, space rod, space frame, 3d membrane, 3d solid, 3d plate shell
V-5 research group:
e. c. ting, c. y. wang, t. y. wu, r. z. wang, c. j. chuang
motion analysis procedure: a simple rod structure
A
B
B
C0t
t
0x
xu
0P
P
x
y
discrete model: mass particles and structural elements
A
B
B
C0t
t
0P
P( )tu
A
B
B
C0t
t( )tu
vector form equation of motion
Am
B
C1f
t
( )tu
P
1f2f
2fB
2
1 22( ) ( ) [ ( ) ( )]
dm t P t t t
dt
uf f
A
B
C
0t
( )tu
at
bt
ct
ft
B
a bt t t
path element
1. element geometry remains unchanged
2. small deformation
discrete path:
kinematics and force calculation
1 material frame: configuration at
2. variable: nodal deformation
3. fictitious reversed motion to define deformation
4. infinitesimal strain and engineering stress
5. nodal forces: use finite element
6. internal forces are in equilibrium
at
reversed motion for nodal deformation
A
A
vB
vB
aB
tB
a
( )
te
ae
u
( )r u
du
l
al
l
f
f( )
d r
a al l
u u u
e
A
A
vB
tB
te
ae
l
al laf
fal
ae
aB
af
f
f
f
a
a
l l
l
ˆˆa
a a
f ff
A A
aE
ˆˆ ( ) aa a a a a a a a
a
l lf f A E A
l
f e e e
ˆtff e
governing equations
21 2
21 2
x x x
y y y
P f fudm
P f fvdt
1 11 11 1 1
1 11
cos
sinx a
a a ay a
f l lE A
f l
2 22 22 2 2
2 22
cos
sinx a
a a ay a
f l lE A
f l
2 2 21 1 1
2 2 22 2 2
( ) ( )
( ) ( )
l b u h v
l b u h v
a bt t t
,a a
a a
u uu ud
v vv vdt
difference equation (symmetrical case)
2
22 , a b
d v h vm P f t t t
dt l
aa a a
a
l lf E A
l
2 2 2( )l b h v , ,a a a
dvt t v v v
dt
2
1 1
( )2 2n
n n n n nn
h vtv P f v v
m l
n a
n a a aa
l lf E A
l
2 2 2( )n nl b h v
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
0
5
10
15
20
25
P*
M otion Analysis
Analytica l so l.
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25
-0 .50
-0.25
0.00
0.25
0.50
0.75
P*
M otion Ana lysis
Analytica l so l.
0 .00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25
-0 .50
-0 .25
0.00
0.25
0.50
0.75
P*
M otion Ana lysis
Analytica l so l.
3* 1
3
PlP
AEh v
h
,
4 node plane solid element
x
y
1u
01
0t
t
02
3u
3au
3u
1au
1u
at
03 04
1a2a
3a
4a
1
2
3
4
estimate the rigid body motion
4
1a
2a
3a
4a
2
3
4
2
3
1
1
1ae
1e
4ae
3ae
2ae
4e
3e
2e1
4
1
1
4 jj
element translation
element rotation
u
fictitious reversed motion
1 ,1,1a
2a
3a
4a
2
3
4
2
3
4
nodal deformation
1( )( )r Ti i η R I x x
cos( ) sin( )
sin( ) cos( )
R
1 ,1,1a
2a
3a
4a
2
3
4
2
3
4
4η
4( )r η
4dη1
1 1
1 1
0
0
( ) , 2,3,4
dxdy
di ix T
di iy i
u u x xi
v v y y
R I
deformation coordinates to define independent variables
ˆˆ x Qx
ˆ ˆ , 1,2,3,4ˆ
di x
di y i
ui
v
Q
1 ,1a
2a
3a
4a
2 2ˆd u η
1e
2
3
4
x
y
2e
x
y
2dη
3v3u
4dη
4v
4u
1 1 2ˆ ˆ ˆ 0u v v
4 4
1 1
ˆ ˆ ˆ ˆ;i i i ii i
x N x y N y
1 2
3 4
1 1;
4 41 1
;4 4
N l s l t N l s l t
N l s l t N l s l t
4 4
1 1
ˆ ˆ ˆ ˆ;i i i ii i
u N u v N v
shape functions:
2 2 3 3 4 4 2 2 3 3 4 4ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ;x N x N x N x y N y N y N y
2 2 3 3 4 4 3 3 4 4ˆ ˆ ˆ ˆ ˆ ˆ ˆ;u N u N u N u v N v N v
*ˆ ˆ n ε Bu
2 3 4
1
JB B B B
*2 3 3 4 5ˆ ˆ ˆ ˆ ˆ ˆ( )Tn u u v u vu
2 2
2
2 2
ˆ ˆ, ,
0
ˆ ˆ, ,
t s s t
s t t s
x N y N
x N x N
B
ˆ ˆ, , 0
ˆ ˆ0 , , , 3,4
ˆ ˆ ˆ ˆ, , , ,
t i s s i t
i s i t t i s
s i t t i s t i s s i t
y N y N
x N x N i
x N x N y N y N
B
ˆ ˆ ˆ
ˆ ˆ ˆ
ˆ ˆ ˆ
x xa x
y ya y
xy xya xy
1 2U U
* * *ˆ ˆ ˆ( ) ( ) ( )a
T T Tn n a a aA
d dA u f u B σ σ
ˆ a σ E ε
*2 3 3 4 4
ˆ ˆ ˆ ˆ ˆˆx x y x yf f f f ff
* * *ˆ ˆ ˆa f f f
* *ˆ ˆˆ ˆ;a a
T Ta a a a a a aA A
d dA d dA f B σ f B σ
ˆ 2 2 2 3 3 3 3 4 4 4 42
ˆ 1 2 3 4
ˆ 1 2 3 4
1ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ0,ˆ
ˆ ˆ ˆ ˆ0,
ˆ ˆ ˆ ˆ0,
z y x y x y x
x x x x x
y y y y y
M f f y f x f y f x f yx
F f f f f
F f f f f
nodal forces
1 ,1a
2a
3a
4a
2
3
4
x
y
3ˆ
axf
3ˆ
ayf
4ˆ
axf
4ˆ
ayf
1ˆaxf
1ˆayf
2ˆ
axf2ˆ
ayf2ˆ
xf
2ˆ
yf
1ˆxf
1ˆ
yf
4ˆ
xf
4ˆ
yf3ˆ
xf
3ˆ
yf
1 ,1a
2a
3a
4a
2
3
4 4xf
4 yf
4ˆ
xf4
ˆyf
2
3
4
1
x
y
x
y
x
y
x
y
4xf
4 yf
4xf
4 yf1u
V
aV
V
ˆˆ
ˆix ixT
iy iy
f f
f f
RQ
stress
x
y
2
3
1
4
2
3
4
1
V
V
x
x
x
x
y
x
y
1u
xSxyS
yS
xS
xyS
yS
ˆ xˆ xy
ˆ y
xxy
y
ˆ ˆˆ ,
ˆ ˆx xy x xy
xy y xy y
S S
S S
σ S
ˆ ˆˆTS Q σQ
x xy
xy y
σ
Tσ RSR
Thank You