A parallel algorithm for the reduction of a nonsymmetric matrix to
1 Department of Civil and Environmental Engineering Sungkyunkwan University 비대칭 박벽보의...
Transcript of 1 Department of Civil and Environmental Engineering Sungkyunkwan University 비대칭 박벽보의...
![Page 1: 1 Department of Civil and Environmental Engineering Sungkyunkwan University 비대칭 박벽보의 개선된 해석이론 및 방법 An Improved Theory and Analysis Procedures of Nonsymmetric.](https://reader036.fdocuments.net/reader036/viewer/2022081518/551c1e75550346ad4f8b5a14/html5/thumbnails/1.jpg)
Contents
IntroductionIntroduction-- Historical review Historical review
-- Scope of this research Scope of this research
Linearized Principle of Virtual Work Linearized Principle of Virtual Work
Displacement Field of Nonsymmetric Thin-walled beamDisplacement Field of Nonsymmetric Thin-walled beam
Derivation of Exact Dynamics Stiffness MatrixDerivation of Exact Dynamics Stiffness Matrix - 14 displacement parameters- 14 displacement parameters
- Force-displacement relations- Force-displacement relations
- Exact dynamic stiffness matrix- Exact dynamic stiffness matrix
Straight Beam ElementStraight Beam Element
Numerical ExamplesNumerical Examples
ConclusionsConclusions
22
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IntroductionHistorical review (1)Historical review (1)
Argyris, J.H.Argyris, J.H.Dunne, P.C.Dunne, P.C.
Scharpf, D.W.Scharpf, D.W.
Kim, M.Y.Kim, M.Y. Chang, S.P.Chang, S.P.Kim, S.B.Kim, S.B.
““On large displacement-small strain analysis of structures On large displacement-small strain analysis of structures with rotational degrees of freedom”with rotational degrees of freedom”
19781978
19961996
19921992
33
Saleeb, A.F.Saleeb, A.F. Chang, T.Y.P.Chang, T.Y.P.
Gendy, A.S.Gendy, A.S.
““Spatial stability and free vibration of shear flexible thin-wSpatial stability and free vibration of shear flexible thin-walled elastic beams. I: Analytical approachalled elastic beams. I: Analytical approach
II: Numerical approach”II: Numerical approach”
19941994
““Effective modelling of spatial buckling of beamEffective modelling of spatial buckling of beam assemblages accounting for warping constraints andassemblages accounting for warping constraints and
ratation-dependency of moments”ratation-dependency of moments”
Kim, M.Y.Kim, M.Y. Chang, S.P.Chang, S.P.Kim, S.B.Kim, S.B.
““Spatial stability analysis of thin-walled space frame”Spatial stability analysis of thin-walled space frame”
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Historical review (2)Historical review (2)
Friberg, P.O.Friberg, P.O.
Banejee, J.R.Banejee, J.R.
Kim, S.B.Kim, S.B.Kim, M.Y. Kim, M.Y.
““New numerical scheme based on the quadratic eigenproblNew numerical scheme based on the quadratic eigenproblem of thin-walled beam with open cross section”em of thin-walled beam with open cross section”
““Generalized formulation which is believed to improve FriGeneralized formulation which is believed to improve Friberg’s method”berg’s method”
““Improved formulation for spatial stability and free Improved formulation for spatial stability and free vibration of thin-walled tapered beams and space frame”vibration of thin-walled tapered beams and space frame”
19851985
20002000
19941994
44
Leung, A.Y.TLeung, A.Y.TZeng, S.P. Zeng, S.P.
““Coupled bending-torsional dynamic stiffness considering fCoupled bending-torsional dynamic stiffness considering for Timoshenko beam”or Timoshenko beam”
19981998
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55
In order to demonstrate the accuracy of this study, the natural frequencies In order to demonstrate the accuracy of this study, the natural frequencies and buckling loads are evaluated and compared with analytic solutions and buckling loads are evaluated and compared with analytic solutions and F.E solutionsand F.E solutions
Most of previous finite element formulations of thin-walled beam Most of previous finite element formulations of thin-walled beam introduce approximate displacement fields by using the shape functionintroduce approximate displacement fields by using the shape function
In this research, a improved formulation for free vibration and spatial In this research, a improved formulation for free vibration and spatial stability of thin-walled beam is developedstability of thin-walled beam is developed
For the general case of loading and boundary conditions, it is very difficult For the general case of loading and boundary conditions, it is very difficult to obtain closed form solutions for natural frequencies and buckling loads to obtain closed form solutions for natural frequencies and buckling loads of thin-walled beamof thin-walled beam
A clearly consistent numerical procedure which generates an exact dynamic A clearly consistent numerical procedure which generates an exact dynamic stiffness matrix of thin-walled beam is presented.stiffness matrix of thin-walled beam is presented.
Scope of this researchScope of this research
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Linearized Principle of Virtual WorkLinearized Principle of Virtual Work
66
Equilibrium Equation for General ContinuumEquilibrium Equation for General Continuum
wherewhere
StrainStrain
Linearized Equilirium EquationLinearized Equilirium Equation
2t t t t t tij ij i i i iV V S
dV U U dV T U dS
*, , ,t t t o t oi i i ij ij ij ij ij i i iU U U T T T
* * * *
*
1{( ), ( ), ( ), ( ), }
2ij i i j j j i k k i k k j
ij ij ij
U U U U U U U U
e e
* * *, , , , , ,
1 1 1( ) , , ( )
2 2 2ij i j j i ij k i k j ij i j j ie U U U U e U U
* * 2( )o o oij ij ij ij ij ij i i i i i iV S V S
e e dV T U dS U U dV T U dS
wherewhere
vv
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Displacement Field of Nonsymmetric Thin-walled beamDisplacement Field of Nonsymmetric Thin-walled beam
77
x2
e2
e3
x3
x1
UxUy
Uz
1 2
3
f
O
C
O : Shear centerC : Centroid
x2
e2
e3
x3
x1
F1F2
F3
1 2
3
M
O
C
O : Shear centerC : Centroid
Notation for displacement parameters and stress resultantsNotation for displacement parameters and stress resultants
(a) Displacement parameters(a) Displacement parameters (b) Stress resultants(b) Stress resultants
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88
Displacement FieldDisplacement Field
Warping function:Warping function:
First-order termsFirst-order terms of displacement parameter of displacement parameter
Second-order termsSecond-order terms of displacement parameter of displacement parameter
2 3 3 2s e x e x
' ' '1 2 3x y zU U x U x U
2 3yU U x
3 2zU U x
* ' '1 2 3
1
2 z yU U x U x
2* 2 ' ' '2 2 3
1( )
2 y z yU U x U U x
* ' ' 2 '23 2 3
1
2 z y zU U U x U x
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99
Cross-section constantsCross-section constants
22 3 ,
AI x dA 2
3 2AI x dA
23 2 3AI x x dA 2
AI dA
2 3AI x dA 3 2A
I x dA
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1010
Stress ResultantsStress Resultants
1 11 2 12,A A
F dA F dA
3 13 ,A
F dA 1 13 2 12 3( )A
M x x dA
2 11 3 ,A
M x dA 3 11 2AM x dA
2 211 2 3( ) ,p A
M x x dA 11AM dA
12 132 3
R AM dA
x x
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1111
Potential energy of the thin-walled beamPotential energy of the thin-walled beam
E G M ext
2 2 211 12 13 1
14 4
2E L AEe Ge Ge dAdx
WhereWhere
* * *11 11 11 12 12 12 13 13 13 1( ) 2 ( ) 2 ( )o o o
G L Ae e e dAdx
2 2 2 21 2 3 1
1
2M L AU U U dAdx
1 1 2 2 3 3
1
2ext ST U T U T U dS
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1212
Potential energy of the thin-walled beamPotential energy of the thin-walled beam
'2 "2 "2 '2 "22 3
" " " "2 3 1
1[
2
2 2 ]
E x z yL
z y
EAU EI U EI U GJ EI
EI U EI U dx
'2 '2 ' ' ' '' '' '1 2 3 1
'' ' ' '' ' ' '22 3 1
1[ ( ) ( )
2
( ) ( ) ]
o o o oG y z z y z y z yL
o o oy y z z p
F U U F U F U M U U U U
M U U M U U M dx
2 2 2 2 '2 '2 2 '22 3
' ' ' '2 3 1
1[ ( )
2
2 2 ]
M x y z z y oL
z y
A U U U I U I U I I
I U I U dx
1
2ext Te eU F
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'' 2
'''' '''' 2 '' ''3 3 3 3
'' ''' ''1 1 2
'''' '''' 2 '' ''2 2 2 2
'' ''' ''1 1 3
'''' '''''''' " 2 ''2 3
0
( )
0
( )
0
(
x x
y y y
o o oy z
z z z
o o oz y
z y o
EAU AU
EI U EI AU I U I
FU M U M
EI U EI AU I U I
FU M U M
EI GJ EI U EI U I I
'' '' '' '' ''
2 3 2 3) 0o o oz y p y zI U I U M M U M U
1313
Governing equations of thin-walled beamGoverning equations of thin-walled beam
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'1
''' ''' 2 ' 2 ' ' '' '2 3 3 3 3 1 1 2
''' ''' 2 ' 2 ' ' '' '3 2 2 2 2 1 1 3
''' ' ''' ''' 2 '1 3 2
2 ' 2 ' '2 3 0
x
o o oy y y z
o o oz z z y
y z
oz y p
F EAU
F EI U EI I U I FU M U M
F EI U EI I U I FU M U M
M EI GJ EI U EI U I
I U I U M
' '2 3
'' '' '2 2 2 1 3
'' '' '3 3 3 1 2
'' '' ''3 2
.5 0.5
0.5 0.5
0.5 0.5
o oy z
o oz y
o oy z
y z
M U M U
M EI U EI M U M
M EI U EI M U M
M EI EI U EI U
1414
Force-displacement relationship Force-displacement relationship
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Derivation of Exact Dynamic Stiffness MatrixDerivation of Exact Dynamic Stiffness Matrix
14 displacement parameters 14 displacement parameters
1 2 14{ , , ..., }Td d dd
'1 2
' '' '''3 4 5 6
' '' '''7 8 9 10
' '' '''11 12 13 14
,
, , ,
, , ,
, , ,
x x
y y y y
z z z z
d U d U
d U d U d U d U
d U d U d U d U
d d d d
where,where,
1515
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' 22 1
' ' 2 2 23 6 3 14 3 3 5 3 13 1 5 1 10 2 13
' ' 2 2 22 10 2 14 7 2 9 2 13 1 9 1 6 3 13
' ' ' 2 2 214 3 6 2 10 13 11 13 2 9
23 5
o o o
o o o
o
EAd Ad
EI d EI d Ad I d I d Fd M d M d
EI d EI d Ad I d I d Fd M d M d
EI d EI d EI d GJd I d I d I d
I d
13 2 5 3 9o o o
pM d M d M d
14131312121110998
8765544321
',',',',',',',',','
dddddddddd
dddddddddd
1616
and,and,
Differential equation of the first order with constant coefficientDifferential equation of the first order with constant coefficient
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27
36
52433
21 0.1
EIaEIaEIaEIaEIaEAa
a
'A d = B d
a 1
a 1a 1
a 1a 4
a 1a 1
a 1a 3
a 1a 1
a 1a 2
a 5
a 6
a 6 a 7
a 7
a 6
1717
A
where,where,
In matrix formIn matrix form
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2 21 2 3 3 1 4 1
2 2 25 3 2 6 2 1 7 2 3
2 28 9
1.0, , ,
, ,
,
o o
o o o
oo p
b b A b I F b M
b I M b I F b I M
b I b GJ I M
b1
b5
b7
b1
b4b1
b2 b3b1
b1b1
b2
b1b1
b1
b1b2
b6
b6
b4 b8
1818
B
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14
1
ixi
i
a e
id Z
General solutionGeneral solution
Complex eigen analysis by using IMSL subroutine DGVCRGComplex eigen analysis by using IMSL subroutine DGVCRG
( ) 0i iA B Z
1919
14 eigenvalues14 eigenvalues 14 14 1414 eigenvectors eigenvectors
Eigen problem of nonsymmetric matrixEigen problem of nonsymmetric matrix
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d(x) = X(x) a
2020
where, where,
eu = E a
3 2 1
3 2 1
{ (0), (0), (0), (0), (0), (0), (0),
( ), ( ), ( ), ( ), ( ), ( ), ( )}
x y z
Tx y z
U U U f
U L U L L U L L L f L
eu
In matrix formIn matrix form
E is obtained fromis obtained from X(x)
Nodal displacement vectorNodal displacement vector
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-1ea = E u
-1ed(x) = X(x) E u
Compute complex inverse matrix by using IMSL subroutine DLINCGCompute complex inverse matrix by using IMSL subroutine DLINCG
2121
Displacement state vectorDisplacement state vector
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1 2 3 3 2 1{ , , , , , , }TF F M F M M Mf
2 2 21 2 3 1 3 3 4 1 5 3 2 6 3 7 2 1
2 2 28 2 9 2 3 10 2 11 3 2 12 2 3
213 15 14 15 1 16 3
, , , , , ,
, , , 0.5 , 0.5
, , , 0.5 , 0.5 ,
o o o o
o o o
o o op
S EA S I F S EI S M S I M S EI S I F
S EI S I M S EI S I M S I M
S GJ I M S EI S EI S M S M
17 20.5 oS M
f(x) = S d(x)
2222
where,where,
S
In matrix formIn matrix form
S1S3S2
-S4S12
S15S8
-S6-S14
S10
S5
S13
-S3-S6 -S10
S17
S9
S16 S10
S10S6
S14
S4S7 S8
S11 S6S15
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e d eF = K (ω) U
2323
Exact dynamic stiffness matrixExact dynamic stiffness matrix
where, where,
-1
-1
0
l
d
-S X( ) EK (ω)
S X( ) E
Nodal force vectorNodal force vector
T p qeF = F , F
1 2 3 1 2 3{ , , , , , , } , ,TF F F M M M M p q eF
where, where,
Nodal force vectorNodal force vector10 0p -
eF = -f( ) = -S X( ) E U-1q l l eF = f( ) = S X( ) E U
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Straight Beam ElementStraight Beam Element
2424
Shape functionsShape functions
xU : Linear Hermitian polynomial: Linear Hermitian polynomial
, ,y zU U : Cubic Hermitian polynomial: Cubic Hermitian polynomial
Equilibrium equationEquilibrium equation
2e g e e e eK + K U - M U = F
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Numerical ExamplesNumerical Examples
2 2 2 2
3 4 42 3
4 5 5 42 3
10000.0 / , 5000.0 / , 30.0 , 10.0
100.0 , 0.00785 / , 100.0 , 800.0
83750.0 , 600.0 , 8000.0 , 900.0o
E N cm G N cm A cm J cm
L cm kg cm I cm I cm
I cm I cm I cm I cm
2525
1. Free vibration of simply supported thin-walled beam1. Free vibration of simply supported thin-walled beam
Sectional propertySectional property
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Analysis resultsAnalysis results
2626
Tab.1 Natural frequencies of simply supported beamTab.1 Natural frequencies of simply supported beam 2( / sec)rad
Zero axial force F1 = -200 N at (x2,x3) = (-5,7) mode
Present study Analytic solution Present study Analytic solution
1.17287 1.17287 0.29079 0.29079
5.46156 5.46156 5.11979 5.11979
N=1
136.963
136.963
134.184 134.184
5.50341 5.50341 2.32697 2.32697
82.1628 82.1628 80.4626 80.4626
N=2
1621.25
1621.25 1612.80 1612.80
14.2024 14.2024 7.20439 7.20439
404.560 404.560 400.665 400.665
N=3
5835.63 5835.63 5822.04 5822.04
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2727
2. Free vibration of thin-walled cantilever and fixed beam2. Free vibration of thin-walled cantilever and fixed beam
L = 200 cm L = 200 cm
x2
x3
10 cm
4 cm
0.5 cm
2 cmCantilever beamCantilever beam Fixed beamFixed beam
Nonsymmetric channel sectionNonsymmetric channel section
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Tab.2 Natural frequencies of cantilever beamTab.2 Natural frequencies of cantilever beam 2( / sec)rad
mode Present study 5 beam element 10 beam element ABAQUS
1 0.027 0.027 0.027 0.028
2 0.336 0.337 0.336 0.331
3 0.707 0.709 0.707 0.696
4 1.074 1.076 1.075 1.074
5 4.859 4.880 4.860 4.766
6 7.186 7.232 7.189 7.083
7 18.22 18.45 18.24 17.95
8 20.15 20.26 20.16 19.36
9 24.39 24.73 24.42 23.58
10 47.34 48.77 47.54 46.52
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Tab.3 Natural frequencies of fixed beamTab.3 Natural frequencies of fixed beam 2( / sec)rad
mode Present study 5 beam element 10 beam element ABAQUS
1 0.910 0.912 0.910 0.914
2 3.115 3.129 3.116 3.046
3 5.803 5.850 5.806 5.780
4 17.66 17.82 17.68 17.24
5 18.60 19.03 18.63 18.31
6 20.61 20.66 20.61 19.20
7 44.83 46.77 45.02 43.99
8 59.01 60.61 59.12 56.81
9 91.21 115.0 92.07 87.77
10 150.1 152.9 150.6 127.6
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3. Buckling of thin-walled cantilever beam under axial load3. Buckling of thin-walled cantilever beam under axial load
mode Present study 8 Beam elements ABAQUS
1 13.800 13.800 14.001
2 112.55 112.56 113.10
3 191.84 191.84 190.08
4 258.54 258.77 256.67
5 414.76 416.05 408.53
6 526.71 526.73 509.74
7 571.33 571.33 546.49
Tab.4 Flexural-torsional buckling loads for cantilever beam [N]Tab.4 Flexural-torsional buckling loads for cantilever beam [N]
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ConclusionsConclusions
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A consistent numerical procedure which generates an exact dynamic sA consistent numerical procedure which generates an exact dynamic stiffness matrix of nonsymmetric thin-walled beam is presentedtiffness matrix of nonsymmetric thin-walled beam is presented
Numerical results by the present method are in a good agreement withNumerical results by the present method are in a good agreement withthose by thin-walled beam elements and ABAQUS’s shell elements.those by thin-walled beam elements and ABAQUS’s shell elements.
Present procedure is general and provides a systematic tool for the Present procedure is general and provides a systematic tool for the numerical evaluation of exact solution of ordinary differential equationnumerical evaluation of exact solution of ordinary differential equation
A improved formulation for spatial stability and free vibration of noA improved formulation for spatial stability and free vibration of nonsymmetric thin-walled beam is developednsymmetric thin-walled beam is developed