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

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”

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

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

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

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

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

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

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

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

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

'' 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

'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

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

' 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

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

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

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

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

-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

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

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

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

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

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

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

2828

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

2929

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

3030

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]

ConclusionsConclusions

3131

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