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Derivation of stiffness and flexibility for rods and beams by using dual integral equations
海洋大學河海工程學系報 告 者：謝正昌指導教授：陳正宗 特聘教授日期： 2006/04/01
中工論文競賽 ( 土木工程組 )
Outlines
Introduction Dual boundary integral formulation
for rod and beam problems Discussion of the rigid body mode
and spurious mode Conclusions
Outlines
Introduction Dual boundary integral formulation
for rod and beam problems Discussion of the rigid body mode
and spurious mode Conclusions
Introduction
flexibilitystiffness
For undergraduate students,it is well-known in mechanics of material.
For graduate students, they revisited it in the finite element course.
Dual boundary integral equations were employed to derive the stiffness and flexibility of the rod and beam .
Influence matrix[ ]K[ ]F
[ ]F
Influence matrix
singular
nonsingular
? [ ]K
Degenerate scale problem
Fredholm theorem and SVD updating technique
Rigid body modeSpurious mode
Outlines
Introduction Dual boundary integral formulation
for rod and beam problems Discussion of the rigid body mode
and spurious mode Conclusions
Rod and beam problems
(0)u
(0)t ( )t L
(0)m ( )m L(0)
Governing equation:4 ( ) 0 ,u x x D
(0)v ( )u L
( )L
(0)u ( )v L
Domain( )u L
2 ( ) 0 ,u x x D Governing equation:
Domain
Rod Beam
2
2
( , )( ) ,
U x sx s x
xd
¶= - - ¥ < <¥
¶
4
4
( , )( ) ,
U x sx s x
xd
¶= - - ¥ < <¥
¶
Fundamental solution Fundamental solution1
( , )2
U x s x s= -31
( , )12
U x s x s= -
Boundary integral equations
0
0
( ) ( , ) ( ) ( , ) ( )
( ) ( , ) ( ) ( , ) ( )
x L
x
x L
x
u s T x s u x U x s t x
t s M x s u x L x s t x
Rod
Beam[ ]
[ ]
[ ]
( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( )0
( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( )0
( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( )0
(
x Lu s U x s u x x s u x M x s u x V x s u x
x
x Lu s U x s u x x s u x M x s u x V x s u x
x
x Lu s U x s u x x s u x M x s u x V x s u xm m m m x
u s
Q
Qq q q q
Q
=¢¢¢ ¢¢ ¢= - + - +
==
¢ ¢¢¢ ¢¢ ¢= - + - +=
=¢¢ ¢¢¢ ¢¢ ¢= - + - +
=
¢¢¢ [ ]) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( )0
x LU x s u x x s u x M x s u x V x s u xv v v v x
Q=
¢¢¢ ¢¢ ¢= - + - +=
Degenerate kernels
( , ) ( , )
( , ) ( , )
U x s T x s
L x s M x s
®
¯ ¯
®
x
( , )U x s
( , )U x s
( , )x sQ ( , )M x s ( , )V x s
( , )x sqQ ( , )M x s ( , )V x s
( , )mU x s ( , )m x sQ ( , )mM x s ( , )mV x s
( , )vU x s ( , )v x sQ ( , )vM x s ( , )vV x s
x
x
x
s
s
s
Rod Beam
s
Degenerate kernels for rod problem
Kernels
Domain
x s>
x s<
( , )U x s ( , )T x s ( , )L x s ( , )M x s
1
2x s
1
2
1
2 0
1
2s x
1
2
1
20
Degenerate kernels for beam problem
Kernel
Domain( , )U x s ( , )U x s ( , )mU x s ( , )vU x s
x > s
x < s
3
12
x s
3
12
x s
2
4
x s
2
4
x s
2
x s
2
s x
1
2
1
2
Kernel
Domain
Kernel
Domain
Kernel
Domain
( , )x sQ ( , )x sqQ ( , )m x sQ ( , )v x sQ
x > s
x < s
2
4
x s
2
4
x s
2
s x
2
x s
1
2
1
2 0
0
( , )M x s ( , )M x s ( , )mM x s ( , )vM x s
x > s
x < s
2
x s
2
s x
1
2
1
2
0
0 0
0
( , )V x s ( , )V x s ( , )mV x s ( , )vV x s
x > s
x < s
1
2
1
2
0
0
0
0
0
0
Influence matrices
L-0+s
(0) (0)
( ) ( )
u tA Bu L t L
By approaching
to
and into the boundary integral equations
(0) (0)
( ) ( )B
t uK
t L u L
(0) (0)
(0) (0)
( ) ( )
( ) ( )
u u
u uA B
u L u L
u L u L
Rod Beam
(0) (0)
(0) (0)
( ) ( )
( ) ( )
B
u u
u uK
u L u L
u L u L
Translation matrix
0(0)u u
0(0)t p ( ) Lt L p
0(0)m m ( ) Lm L m0(0)
0(0)v v( ) Lu L u
( ) LL
0(0)u u( ) Lv L v
( ) Lu L uRod Bea
m
0 0(0) 1 0
( ) 0 1 ruL L
u uuT
u L u u
é ù é ùé ù é ùê ú ê úê ú ê ú= =ê ú ê úê ú ê úë û ë ûë û ë û
0 0(0) 1 01
( ) 0 1 rtL L
p ptT
t L p pEA
é ù é ùé ù é ù- ê ú ê úê ú ê ú= =ê ú ê úê ú ê úë û ë ûë û ë û
0 0
0 0
(0) 1 0 0 0
(0) 0 1 0 0
( ) 0 0 1 0
0 0 0 1( )
buL L
L L
u u u
uT
u L u u
u L
q q
q q
é ù é ù é ùé ùê ú ê ú ê úê úê ú ê ú ê úê ú¢ê ú ê ú ê úê ú= =ê ú ê ú ê úê úê ú ê ú ê úê úê ú ê ú ê úê ú¢ ê úê ú ê úê ú ë ûë û ë ûë û
0 0
0 0
(0) 1 0 0 0
(0) 0 1 0 01
0 0 1 0( )
0 0 0 1( )
btL L
L L
u v v
m muT
v vEIu L
m mu L
é ù é ù é ù¢¢¢ é ùê ú ê ú ê úê úê ú ê ú ê úê ú¢¢ -ê ú ê ú ê úê ú= =ê ú ê ú ê úê ú¢¢¢ -ê ú ê ú ê úê úê ú ê ú ê úê ú¢¢ ê úê ú ê úê ú ë ûë û ë ûë û
The stiffness matrix of rods
Stiffness matrix for rod problems using dual BEM
A BBK
FK
1 1
2 21 1
2 2
10
21
02
L
L
1 11
1 1L
1 1
1 1
EA
L
( ) 1Rank A ( ) 2Rank B ( ) 1B
Rank K ( ) 1F
Rank K
0 0
0 0
1 1
2 21 1
2 2
( ) 0Rank A ( ) 1Rank B
NA NA
Stiffness matrix for the beam by using the direct method
Eqs.
u
|
θ
u
|
m
A B BK
FK
1 1
02 2 21 1
02 2 2
1 10 0
2 21 1
0 02 2
L
L
( ) 2Rank A
3
2
2
1 10 0
12 41 1
0 012 4
1 10 0
4 21 1
0 04 2
L
LL
L L
L L
( ) 4Rank B
2 2
3
2 2
12 6 12 6
6 4 6 21
12 6 12 6
6 2 6 4
L L
L L L L
L LL
L L L L
( ) 2B
Rank K
2 2
3
2 2
12 6 12 6
6 4 6 2
12 6 12 6
6 2 6 4
L L
L L L LEI
L LL
L L L L
( ) 2F
Rank K
1 10
2 2 21 1
02 2 2
0 0 0 0
0 0 0 0
L
L
( ) 2Rank A
3
3 2 3
2 3 3
1 10 0
12 41 1
0 012 4
1 1 10
2 2 21 1 1
02 2 2
L
LL
L L L
L L L
( ) 4Rank B
2 2
3
2 2
12 6 12 6
6 4 6 21
12 6 12 6
6 2 6 4
L L
L L L L
L LL
L L L L
( ) 2B
Rank K
2 2
3
2 2
12 6 12 6
6 4 6 2
12 6 12 6
6 2 6 4
L L
L L L LEI
L LL
L L L L
( ) 2F
Rank K
Singular value decomposition
1A
A
11
1
[ ]r
T
i i ii
A v u
1
rT
i i ii
A u v
The
matrix can be expressed as
The
denoted by X
, , ,T T
A X A A X A X X A X AX X A XA
†X A
satisfies the four Penrose conditions.
The pseudo-inverse is identified as
The flexibility matrix of rods
Flexibility matrix for rod problems using dual BEM
A BBF
FF
1 1
2 21 1
2 2
10
21
02
L
L
( ) 1Rank A ( ) 2Rank B
0 0
0 0
1 1
2 21 1
2 2
( ) 0Rank A ( ) 1Rank B
NA NA
1 1
1 14
L
( ) 1B
Rank F
1 1
1 14
L
EA
( ) 1F
Rank F
Outlines
Introduction Dual boundary integral formulation
for rod and beam problems Discussion of the rigid body mode
and spurious mode Conclusions
Fundamental solution
( , ) ( , )rU x s U x s ax c= + +2
3
( , ) ( , )bU x s U x s ax bx
dx c
= + +
+ +
1 1(0)2 2
1 1 (1)
2 2
1(0)2
1 (1)
2
a a u
ua a
c a c u
uc a c
Rod Beam
1 1 16 2 6 2 6
2 2 2(0)1 1 1
6 2 6 2 6 (0)2 2 21 1 (1)
0 02 2 (1)1 1
0 02 2
1 12 3
12 41 1
2 312 4
1 10 0
4 21 1
0 04 2
d b d b d
ud b d b d u
u
u
c a c a b d a b d
c a c a b d a b d
(0)
(0)
(1)
(1)
u
u
u
u
( )u q- formulation
Influence matrix
0a= 1/ 4c=- 0a=
Rod Beam
0b= 1/ 48c=0d =
1
1 1
2 21 1
2 2
A
1
1 1 10
2 2 21 1 1
02 2 2
1 10 0
2 21 1
0 02 2
A
1
1 5 10
48 48 45 1 1
048 48 48
1 10 0
4 21 1
0 04 2
B
When and ,Whenand
,
1
1 1
4 41 1
4 4
B
Mathematical SVD structures of the influence matrices
1 1A Bff =spurious mode
1
T
A
-0.632 0 -0.774 0
-0.632 0 0.258 0.577
-0.316 -0.516 0.577
0.316 0.258 0.577
é ùé ùé ùê úê úê úê úê úê úê úê úê úé ù= ê úê úê úê úë û ê úê úê úê úê úê úê úê úê úë ûë ûë û
L L L L L L L L
L L L L L L L L
L L L L L L L L L
L L L L L L L L L
[ ]1 1,A B
1
TB
-0.632 0
-0.632
-0.316
0.316
y
é ùé ùê úê úê úê úê úê úé ù é ù= ê úê úê úê ú ë ûë û ê úê úê úê úê úê úë ûë û
L L L L L L
L L L L L L L
L L L L L L L
L L L L L L L
2
T
A
0 -0.774 0
0 0.258 0.577
-0.516 0.577
0.258 0.577
f
é ùé ùê úê úê úê úê úê úé ù é ù= ê úê úê úê ú ë ûë û ê úê úê úê úê úê úë ûë û
L L L L L
L L L L L
L L L L L L
L L L L L L
SVD updating document1
2
A
A
é ùê úê úë û
1 2A Ay y=
SVD updating term
rigid body mode
According to Fredholm alternative theorem [ ] 0T
T
A
Bf
é ùê ú =ê úê úë û
ì üï ïï ïï ïï ïï ïé ù í ýê úë û ï ïï ïï ïï ïï ïî þ
A
1
2ψ = 1
2
é ùê úê úé ù ê úê úë û ê úê úê úë û
-0.632-0.632φ = -0.3160.316
Spurious modes and the rigid body modes for a rod and a beam in BEM
Spurious mode(generalized
force)
Rigid body mode(generalized
displacement)
1
2
1
2
ì üï ïï ïï ïï ïï ïé ù í ýê úë û ï ïï ïï ïï ïï ïî þ
1
2φ = 1
2
0.316
0.316-
0.632-
0.632-
1(1)
2u
1(0)
2u
(0) 0.774u = -
'(0) 0.258u ='(1) 0.258u =
(1) 0.516u = -
(0) 0u = (1) 0.577u =
'(0) 0.577u = '(1) 0.577u =
é ùê úê ú
é ù ê úê úë û ê úê úê úê úë û
A
-0.774
0.258ψ =
-0.516
0.258
é ùê úê ú
é ù ê úê úë û ê úê úê úê úë û
A
0
0.577ψ =
0.577
0.577
Rod
Beam
Conclusion Dual boundary integral equations were employed to derive
the stiffness and flexibility of the rod and beam which match well with those of FEM.
Both direct and indirect methods were used. The displacement-slope and displacement-moment
formulations in the direct method can construct the stiffness matrix.
The single-double layer approach and single-triple layer approach work for the constructing of stiffness matrix in
the indirect method.
Conclusion The rigid body mode and spurious mode are imbedded in
the right and left unitary vectors of the influence matrices through SVD.
The end
Thank you for your kind attention!
