Dynamic Analysis of Structures bySuperposition of Modified Lanczos Vectors

Dynamic Analysis of Structures bySuperposition of Modified Lanczos Vectors

2003 한국전산구조공학회 봄 학술발표회 2003 년 4 월 12 일2003 년 4 월 12 일

Byoung-Wan Kim1), Hyung-Jo Jung2), Woon-Hak Kim3) and In-Won Lee4)

1) Ph.D. Candidate, Dept. of Civil and Environmental Eng., KAIST2) Assist. Professor, Dept. of Civil and Environmental Eng., Sejong Univ.3) Professor, Dept. of Civil Engineering, Hankyong National Univ.4) Professor, Dept. of Civil and Environmental Eng., KAIST

Byoung-Wan Kim1), Hyung-Jo Jung2), Woon-Hak Kim3) and In-Won Lee4)

1) Ph.D. Candidate, Dept. of Civil and Environmental Eng., KAIST2) Assist. Professor, Dept. of Civil and Environmental Eng., Sejong Univ.3) Professor, Dept. of Civil Engineering, Hankyong National Univ.4) Professor, Dept. of Civil and Environmental Eng., KAIST

2 2

Introduction Proposed method Numerical examples Conclusions

Contents

3 3

Introduction

Background

• The Lanczos vector superposition method is very efficient.• The Lanczos vector superposition method is very efficient.

• Dynamic analysis of structures- Direct integration method- Vector superposition method

• Dynamic analysis of structures- Direct integration method- Vector superposition method

• Vector superposition method- Eigenvector superposition method- Ritz vector superposition method- Lanczos vector superposition method

• Vector superposition method- Eigenvector superposition method- Ritz vector superposition method- Lanczos vector superposition method

Eigenvalue analysisEigenvalue analysis

No eigenvalue analysisNo eigenvalue analysis

4 4

Literature review

• Nour-Omid(1984) first proposed.• Nour-Omid(1995): Unsymmetric nonclassically damped system• Chen(1990): Symmetric nonclassically damped system• Ibrahimbegovic(1990), Mehai(1995): Dam-foundation system

• Nour-Omid(1984) first proposed.• Nour-Omid(1995): Unsymmetric nonclassically damped system• Chen(1990): Symmetric nonclassically damped system• Ibrahimbegovic(1990), Mehai(1995): Dam-foundation system

• Drawback: The method is costly when multi-input-loadedstructures are analyzed.

• Drawback: The method is costly when multi-input-loadedstructures are analyzed.

Objective

• Improvement of the Lanczos vector superposition methodto overcome the shortcoming.

• Improvement of the Lanczos vector superposition methodto overcome the shortcoming.

5 5

Proposed method

Conventional method

fKuuKMuM )(

• Dynamic equation of motion of structures• Dynamic equation of motion of structures

mass matrix(n n) mass matrix(n n) stiffness matrix(n n) stiffness matrix(n n)

MK

f ,

force vector(n 1) force vector(n 1)

Rayleigh damping coefficientsRayleigh damping coefficients

u displacements vector(n 1) displacements vector(n 1)

6 6

11~

iiiiii xxxx

ii xMKx 1

)( ijjTi Mxx

• Lanczos algorithm• Lanczos algorithm

iTii xMx

i

ii

xx

~1

2/1)~~( iTii xMx

ix ith Lanczos vectorith Lanczos vector

7 7

TMXMKX 1T

IMXX T

• Reduced tridiagonal equation of motion• Reduced tridiagonal equation of motion

][ 21 mxxxX

fKuuKMuM )( Xqu

fMKXqqITqT 1)( T

1MKXTpremultiplypremultiply

mm

mmm

1

112

221

11

T

8 8

• Single input loads• Single input loads

ppTT10

11)( eaMKXfMKXqqITqT

ap

spatial load distribution vector(n 1)spatial load distribution vector(n 1)

time variation function(scalar)time variation function(scalar)

• Multi-input loads• Multi-input loads

ApMKXfMKXqqITqT 11)( TT

Ap

spatial load distribution matrix(n k)spatial load distribution matrix(n k)time variation function vector(k 1)time variation function vector(k 1)the number of input loadsthe number of input loadsk

9 9

Proposed method

main idea main idea

• Modified Lanczos algorithm• Modified Lanczos algorithm

ijjTi Kyy

111~

iiiiii yyMyKy

ii MyKy 1

iTii yMy

i

ii

yy

~1

2/1)~~( iTii yKy

ijjTi Mxxconventional:conventional:

10 10

• Reduced tridiagonal equation of motion• Reduced tridiagonal equation of motion

][ 21 myyyY IKYY T

mm

mmm

1

112

221

11

S

fYrrISrS T )(

fKuuKMuM )( Yru

TYpremultiplypremultiply

SMYY T

• Single input load• Single input load

• Multi-input loads• Multi-input loads

ppp TTT10

1)( eaKKYaYfYrrISrS

ApYfYrrISrS TT )(

11 11

Summary

• Single input loads• Single input loads

• Multi-input loads• Multi-input loads

p10)( eqqITqT

p10)( errISrS

ApMKXqqITqT 1)( T ApYrrISrS T )(

- conventional- conventional

- proposed- proposed

- conventional- conventional

- proposed- proposed

12 12

Numerical examples

• Simple span beam(Pan and Li, 2002)• Multi-span continuous bridge(Park et al., 2002)• Simple span beam(Pan and Li, 2002)• Multi-span continuous bridge(Park et al., 2002)

Structures

Normalized RMS error

dtT

dtT

d

d

T

exactTexact

d

T

exactT

exactd

0

0

1

)()(1

uu

uuuu

exactu results by the direct integration methodresults by the direct integration method

dT time durationtime duration

13 13

Simple span beam

1 0 0 0 .1 1 6 8 m = 1 1 .6 8 m

Material properties System data

Young’s modulus (N/m2) Section inertia (m4) Section area (m2) Mass density (kg/m3)

1.72108 1.00 1.00 3105

Number of nodes Number of beam elements Number of degrees of freedom Half-bandwidth of system matrices

101 100 200 3

• Geometry and material properties• Geometry and material properties

14 14

• Loading configurations• Loading configurations

- Single input load(concentrated sinusoidal force)- Single input load(concentrated sinusoidal force)

- Multi-input load(moving load)- Multi-input load(moving load)

p = 8 7 3 0 9 s in 7 t (N )

p = 8 7 3 0 9 NV = 1 9 m /se c

15 15

0 4 8 1 2 1 6 2 0 2 4 2 8 3 2 3 6 4 0N u m b er o f u s ed v ecto rs

1 e -9

1 e -7

1 e -5

1 e -3

1 e -1

Err

or

R itz , L an czo s , P ro p o sed

0 4 8 1 2 1 6 2 0 2 4 2 8 3 2 3 6 4 0N u m b er o f u s ed v ecto rs

1 e -9

1 e -7

1 e -5

1 e -3

1 e -1

Err

or

R itz , L an czo s , P ro p o sed

• Error• Error

(a) Single input load(a) Single input load (b) Multi-input load(b) Multi-input load

P ro p o sed L an czo s v ec to r su p e rp o s itio n m e th o dC o n v en tio n a l L an czo s v ec to r su p e rp o s itio n m e th o d

E ig en v ec to r su p e rp o sitio n m e th o dM o d e acce le ra tio n m e th o dR itz v ec to r su p e rp o s itio n m e th o d

16 16

• Computing time• Computing time

(a) Single input load(a) Single input load (b) Multi-input load(b) Multi-input load

0 4 8 1 2 1 6 2 0 2 4 2 8 3 2 3 6 4 0N u m b er o f u s ed v ecto rs

0 .1

1

1 0

1 0 0

Com

putin

g tim

e (s

ec)

0 4 8 1 2 1 6 2 0 2 4 2 8 3 2 3 6 4 0N u m b er o f u s ed v ecto rs

0 .1

1

1 0

1 0 0

Com

putin

g tim

e (s

ec)

P ro p o sed L an czo s v ec to r su p e rp o s itio n m e th o dC o n v en tio n a l L an czo s v ec to r su p e rp o s itio n m e th o d

E ig en v ec to r su p e rp o sitio n m e th o dM o d e acce le ra tio n m e th o dR itz v ec to r su p e rp o s itio n m e th o d

17 17

Multi-span continuous bridge

• Geometry and material properties• Geometry and material properties

Dongjin bridge(PSC box girder type)Dongjin bridge(PSC box girder type)

3 7 .5 m 3 7 .5 m1 3 5 0 m = 6 5 0 m

7 2 5 m

h

P o t b e a rin gh = 8 .5 3 8 m ~ 1 0 .0 7 8 m

18 18

Material properties Decks Piers

Young’s modulus (GN/m2) Section inertia (Ixx) (m

4) Section inertia (Iyy) (m

4) Torsion constant (m4) Section area (m2) Mass density (kg/m3)

27.50 80.67, 88.71, 105.70 11.06, 11.54, 15.44 19.48, 20.87, 34.66 8.62, 9.95, 18.24 2300

23.24 92.17, 100.00, 84.42 14.29, 25.01, 23.60 11.87, 66.73, 52.94 14.00, 24.50, 17.00 2300

System data

Number of nodes Number of beam elements Number of connection elements Number of degrees of freedom Half-bandwidth of system matrices

177 160 16 870 35

19 19

• Loading configurations• Loading configurations

- Single input load(El Centro earthquake)- Single input load(El Centro earthquake)

- Multi-input load(moving load)- Multi-input load(moving load)

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0T im e (se c )

-4

-2

0

2

4

Acc

eler

atio

n(m

/sec

2 )

V = 3 3 .3 3 3 m /se c (1 2 0 k m /h r)p = 2 3 5 .4 4 k N (2 4 to n f)

20 20

• Error• Error

(a) Single input load(a) Single input load (b) Multi-input load(b) Multi-input load

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 1 8 0N u m b er o f u s ed v ecto rs

1 e -1 2

1 e -9

1 e -6

1 e -3

1 e 0

Err

or

R itz , L an czo s , P ro p o sed

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 1 8 0N u m b er o f u s ed v ecto rs

1 e -1 2

1 e -9

1 e -6

1 e -3

1 e 0

Err

or

R itz , L an czo s , P ro p o sed

P ro p o sed L an czo s v ec to r su p e rp o s itio n m e th o dC o n v en tio n a l L an czo s v ec to r su p e rp o s itio n m e th o d

E ig en v ec to r su p e rp o sitio n m e th o dM o d e acce le ra tio n m e th o dR itz v ec to r su p e rp o s itio n m e th o d

21 21

• Computing time• Computing time

(a) Single input load(a) Single input load (b) Multi-input load(b) Multi-input load

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 1 8 0N u m b er o f u s ed v ecto rs

1

1 0

1 0 0

1 0 0 0

Com

putin

g tim

e (s

ec)

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 1 8 0N u m b er o f u s ed v ecto rs

1

1 0

1 0 0

1 0 0 0

Com

putin

g tim

e (s

ec)

P ro p o sed L an czo s v ec to r su p e rp o s itio n m e th o dC o n v en tio n a l L an czo s v ec to r su p e rp o s itio n m e th o d

E ig en v ec to r su p e rp o sitio n m e th o dM o d e acce le ra tio n m e th o dR itz v ec to r su p e rp o s itio n m e th o d

22 22

Conclusions

• The Lanczos and Ritz vector superposition methods have almost the same accuracy.

• For the single input loading case, the Lanczos and Ritz vector superposition methods have better accuracy than the eigenvector superposition and mode acceleration methods.

• For the multi-input loading case, the eigenvector superposition and mode acceleration methods have better accuracy than the Lanczos and Ritz vector superposition methods.

• The Lanczos and Ritz vector superposition methods have almost the same accuracy.

• For the single input loading case, the Lanczos and Ritz vector superposition methods have better accuracy than the eigenvector superposition and mode acceleration methods.

• For the multi-input loading case, the eigenvector superposition and mode acceleration methods have better accuracy than the Lanczos and Ritz vector superposition methods.

23 23

• The Lanczos and Ritz vector superposition methods have better computing efficiency than the eigenvector superposition and mode acceleration methods.

• For the single input loading case, proposed and conventional Lanczos vector superposition methods have almost the same computing efficiency.

• For the multi-input loading case, proposed method has better computing efficiency than the conventional Lanczos vector superposition method.

• The Lanczos and Ritz vector superposition methods have better computing efficiency than the eigenvector superposition and mode acceleration methods.

• For the single input loading case, proposed and conventional Lanczos vector superposition methods have almost the same computing efficiency.

• For the multi-input loading case, proposed method has better computing efficiency than the conventional Lanczos vector superposition method.