990731_262423_380v3.i

NATIONAL RESEARCH LABORATORY

Jaehong LeeDept. of Architectural EngineeringSejong University

October 12, 2000

Energy-Based Approach for Buckling Problems in Steel Structures

990731_262423_380v3.i

OBJECTIVES

To Present Energy Method in Buckling Analysis

State-of-the Art Review of the Analysis of Thin-walled Structures

Structural Behavior of Cold-formed Channel Section Beams

990731_262423_380v3.i

CONTENTS

• Introduction

– Impact of cold-formed steel

– Structural Consideration of Channel section

• Lateral Buckling

• Flexural-Torsional Buckling

• Stress Analysis

• Local Buckling & Effective Width

• Analysis & Design of Cold-formed Channel

• Next Steps

990731_262423_380v3.i

Cold-formed steel represents over 45 percent of the steel construction market in U.S.

Sophisticated structures such as schools, churches and complex manufacturing facilities.

COLD-FORMED STEEL OFFERS VERSATILITY IN BUILDINGS

Ease of Prefabrication and Mass Production

Light Weight

Uniform Quality

Economy in Transformation and Handling

Quick and simple erection

990731_262423_380v3.i

ANALYSIS & DESIGN OF COLD-FORMED CHANNEL-SECTION BEAMS ARE NOT EASY

How to Take Care of These Complecated Behavior

Finite Element Analysis & AISI

Aisi Code

Effective Width

Linear Method & Iterative Method

Bending + Torsion

Things to Consider in Analysis and Design of Beams

Elastic Lateral Buckling

Inelastic Lateral Buckling

Local Buckling

Sectional Properties

Center of Gravity Shear Center

990731_262423_380v3.i

CONTENTS

• Introduction

– Impact of cold-formed steel

– Structural Consideration of Channel section

• Lateral Buckling

• Flexural-Torsional Buckling

• Stress Analysis

• Local Buckling & Effective Width

• Analysis & Design of Cold-formed Channel

• Next Steps

990731_262423_380v3.i

LATERAL BUCKLING MAY OCCUR WELL BELOW THE YIELD STRENGTH LEVEL

u

v

Original position

Final position for inplane bending

Fy

Elastic Lateral Buckling Strength

990731_262423_380v3.i

FINITE ELEMENT MODEL IS THE BEST

Kinematics

Constitutive Relations

Variational Formulation

Lateral Buckling Equations

Finite Element Model

• Build the appropriate displacement fields• Derive the strain tensor

• Strain energy• Potential of transverse load at shear center

• Stress resultants vs. strains

• Can be derived by integrating by parts• Coupled differential equations

• Setup the eigenvalue problem• Buckling loads and mode shapes

Kinematics Variational Formulation

Constitutive Relations

Lateral Buckling Equations

Finite Element Model

990731_262423_380v3.i

KINEMATICS OF THIN-WALLED SECTION

x

y

P

z

O

sn

qr

( , ) ( )sin ( ) ( )cos ( ) ( ) ( )

( , ) ( )cos ( ) ( )sin ( ) ( ) ( )

( , ) ( ) '( ) ( ) '( ) ( ) '( ) ( )

u s z U z s V z s z q s

v s z U z s V z s z r s

w s z W z U z x s V z y s z s

( , , ) ( , )

( , )( , , ) ( , )

( , )( , , ) ( , )

u s z n u s z

u s zv s z n v s z n

s

u s zw s z n w s z n

z

Contour Coordinate

Displacement Field Plate Action

Beam Action

•Kirchhoff-Love assumption•Shear strain at midsurface is zero.

Basic Assumptions

990731_262423_380v3.i

VARIATIONAL FORMULATION IS USED TO FORMULATE THE GOVERNING EQUATIONS

0

( ''

U V

[ ' '' '' '' '

]'' )

l

y t

b

z xN W M U

M U U ap

M V M M

dz

( )

( )bM f z

p g z

( )f z ( )g z

1 0

221

( )2 4

lz 1

1( )2 2

lz 0

WEAK FORM

CONSTITUTIVE MODEL

'

''

''

''

'4

z

y y

x x

t

N EAW

M EI U

M EI V

M EI

GJM

Load Type

s.c

a

990731_262423_380v3.i

GOVERNING LATERAL BUCKLING EQUATIONS CAN BE DERIVED BY INTEGRATION BY PARTS THE VARIED QUATITIES

'

''

''

'' '

0

( ) '' 0

0

2 '' 0

z

y b

x

t b

N

M M

M

M M M U ap

( ) '' 0

'' (

0

0

'' 0

''

)

iv

iv

x

y

iv

EI U f

EI GJ fU ag

EAW

EI V

Lateral Buckling Equations

990731_262423_380v3.i

FINITE ELEMENT MODEL IS DERIVED FROM THE WEAK FORM

([ ] [ ]){ } {0}K G

Finite Element Model (Standard Eigenvalue Problem)

: eigenvalue (buckling parameter): eigenfunction (buckling mode shape)

11

22

12

21 22

0[ ]

0

0[ ]

ij

ij

ij

ij ij

KK

K

GG

G G

990731_262423_380v3.i

CLOSED-FORM SOLUTION FOR ELASTIC LATERAL BUCKLING IS LIMITED

2 2

21cr y

n n EIM EI GJ

l GJl

M M

2

2 2

sin

sin

y

n zc

l

n zcMl

lUn EI

Simply-supported Beam Under Pure Bending

Buckling moment Buckling mode shape

2

22y

crxc

EhI

l S

(For H-section)

2 4 2

2 4

1 y y wcrcr

b b

EI GJ E I IM

S S l l

Buckling stress

990731_262423_380v3.i

UNEQUAL END MOMENTS AND VARIOUS BOUNDARY CONDITIONS SHOULD BE CONSIDERED

Bending coefficient (moment gradient factor) Cb

21 2 1 21.75 1.05( / ) 0.3( / ) 2.3bC M M M M

()

(+)

M2 >M1

M >M2 Cb=1

M1

M1

M2

M2

AISI Specification

1968-1980 editionSt. Venant torsion neglected

2

22b y

cr

C EhIM

l

1989 editionPekoz & Winter For singly-symmetric section:torsional-flexural buckling considered

cr b o y tM C r A

2

2

2

22

/

1

y

y y y

to t t

E

K l r

EIGJ

Ar K l

990731_262423_380v3.i

BEAM UNDER UNEQUAL END MOMENTS

21 1 2 1 21.75 1.05( / ) 0.3( / ) 2.3 bC M M M M

max

max

12.5

2.5 3 4 3

bA B c

MC

M M M M

0

2

4

6

8

10

12

- 1.0 - 0.5 0.0 0.5 1.0

β

Mcr

Cb1

Cb2

present

MM

990731_262423_380v3.i

BUCKLING MODES OF A BEAM UNDER UNEQUAL END MOMENTS

MM

-0.08

-0.04

0

0.04

0.08

0 0.2 0.4 0.6 0.8 1

Z/L

U

β=-1β=-1/2β=0β=1/2β=1

990731_262423_380v3.i

EFFECT OF LOADING POINT ON A CANTILEVER BEAM UNDER POINT LOAD AT FREE END

P

0

10

20

30

40

50

0.0 0.5 1.0 1.5 2.0 2.5

w2

topshear centerbottom

22

2

2

crcr

y

EIW

l GJ

P lP

EI GJ

990731_262423_380v3.i

EFFECT OF LOADING POINT ON A SIMPLY-SUPPORTED BEAM UNDER UNIFORMLY-DISTRIBUTED LOAD

w

0

50

100

150

200

0.0 0.5 1.0 1.5 2.0 2.5

W2

topshear centerbottom

22

2

3

crcr

y

EIW

l GJ

w lw

EI GJ

990731_262423_380v3.i

LATERALLY-SUPPORTED BEAM UNDER UNEQUAL END MOMENTS

MM

0.0

1.0

2.0

3.0

4.0

0.1 0.3 0.5 0.7 0.9

α/L

M/MO

β=-1β=-0.5β=0β=0.5β=1

x

990731_262423_380v3.i

LATERALLY-SUPPORTED BEAM UNDER UNEQUAL END MOMENTS

MMx

0.0

0.1

0.2

0.3

0.4

0.5

0.6

- 1 - 0.5 0 0.5 1β

(α/L)opt

990731_262423_380v3.i

INELASTIC LATERAL BUCKLING SHOULD BE CONSIDERED FOR REAL PROBLEMS

Fy

Elastic Lateral Buckling

Inelastic Lateral Buckling

When buckling stress exceeds the proportional limit

The beam behavior is governed by inelastic buckling

pr

For accurate solution, rigorous iterative method is required

990731_262423_380v3.i

AISI CODE PROVIDES CONSERVATIVE INELASTIC BUCKLING MOMENT

Mcr/My

My/Me

14

yc y

e

MM M

M

c eM M

0.5

1.0

21 3

990731_262423_380v3.i

CONTENTS

• Introduction

– Impact of cold-formed steel

– Structural Consideration of Channel section

• Lateral Buckling

• Flexural-Torsional Buckling

• Stress Analysis

• Local Buckling & Effective Width

• Analysis & Design of Cold-formed Channel

• Next Steps

990731_262423_380v3.i

SAME PROCEDURE EXCEPT THE WORK DONE BY FORCES

Kinematics

Constitutive Relations

Variational Formulation

Lateral Buckling Equations

Finite Element Model

• Build the appropriate displacement fields• Derive the strain tensor

• Strain energy• Potential of external forces

• Stress resultants vs. strains

• Can be derived by integrating by parts• Coupled differential equations

• Setup the eigenvalue problem• Buckling loads and mode shapes

Kinematics Variational Formulation

Constitutive Relations

Lateral Buckling Equations

Finite Element Model

990731_262423_380v3.i

VARIATIONAL FORMULATION IS USED TO FORMULATE THE GOVERNING EQUATIONS

0

2[ ' ' ' ' ' ' ( ' ' ' ') ( '

U V

' ' ')]}

[ ' '' '' '' 2 '

l

z y t

p p

x

z pN U U V V r x V V y U U

N W M U M V M M

dz

WEAK FORM

CONSTITUTIVE MODEL

'

''

''

''

'4

z

y y

x x

t

N EAW

M EI U

M EI V

M EI

GJM

s.c c.g

990731_262423_380v3.i

GOVERNING FLEXURAL-TORSIONAL BUCKLING EQUATIONS CAN BE DERIVED BY INTEGRATION BY PARTS THE VARIED QUATITIES

'

''

''

'' ' 2

0

( '' ") 0

( '' ") 0

2 ( " " ") 0

z

oy z p

ox z p

ot z p p p

N

M N U y

M N V x

M M N r y U x V

2

'' 0

'' 0

( ) '

' 0

'

'

0

iv o

y z

iv ox z

iv oz p

EI U N U

EI V N V

EI GJ N r

EAW

2

'' 0

( '' '') 0

( ) '' " 0

'' 0

iv o

y z

iv ox z p

iv o oz p z p

EI U N U

EI V N V x

EI GJ N r N x V

EAW

2

( '' '') 0

'' 0

( ) '

'

' 0

' 0

''

iv o

y z p

iv ox z

iv o oz p z p

EI U N U y

EI V N V

EI GJ N r N y U

EAW

990731_262423_380v3.i

FINITE ELEMENT MODEL IS DERIVED FROM THE WEAK FORM

([ ] [ ]){ } {0}K G

Finite Element Model (Standard Eigenvalue Problem)

: eigenvalue (buckling parameter): eigenfunction (buckling mode shape)

11

22

22

11 13

22 23

31 32 33

0 0

[ ] 0 0

0 0

0

[ ] 0

ij

ij

ij

ij ij

ij ij

ij ij ij

K

K K

K

G G

G G G

G G G

990731_262423_380v3.i

CONTENTS

• Introduction

– Impact of cold-formed steel

– Structural Consideration of Channel section

• Lateral Buckling

• Flexural-Torsional Buckling

• Stress Analysis

• Local Buckling & Effective Width

• Analysis & Design of Cold-formed Channel

• Next Steps

990731_262423_380v3.i

WHEN THE TRANSVERSE LOADS DO NOT PASS THROUGH THE SHEAR CENTER, THE MEMBER WILL BE SUBJECTED TO BOTH BENDING AND TORSION

v

c.g

s.cv

Bending

Bending + Torsion

Loads applied at shear center

Loads applied at center of gravity

Lateral Buckling

990731_262423_380v3.i

VARIOUS NORMAL AND SHEAR STRESSES CAN BE GENERATED

BENDING

TORSION

1. Longitudinal bending stress

2. Shear stress

1. Longitudinal bending stress

2. Shear stress

1. Warping longitudinal stress

2. Pure torsional shear stress

3. Warping shear stress

1. Warping longitudinal stress

2. Pure torsional shear stress

3. Warping shear stress

Myb I

VQv It

'''ES

t

tM tt J

''nE

990731_262423_380v3.i

WARPING CHARACTERISTICS OF CHANNEL SECTION

0y

x

Ix

I0y

x

Ix

I y

px

Ix

I

2

2 6x

d wdI b t

2 2

2 6 2y

d b twdI x b t

Shear Center Location

Normalized Unit Warping

n1

n2

n3

n4

1

1

2n

b d

2 2n

b d

3 2n

b d

4

1

2n

b d

x

y

px x b 1

2 / 3wd b t

b

d’c.gs.c

xp

t

2x AI y dA y A

I y dA

Definition :

Channel Section:

( )sr s ds

n stds Definition :

Channel Section:

990731_262423_380v3.i

WARPING CHARACTERISTICS OF CHANNEL SECTION

0y

x

Ix

I0y

x

Ix

I

2nA

I dA

Warping Moment of Inertia

b3

Warping Static Moment

2 2 2 21 1 2 2 12 12 2 2 3 3 23 23

2 23 3 4 4 34 34

1

3

n n n n n n n n

n n n

t b t bI

t b

2

3 2 1 31

6 2 6

d wI b d t

b t

nAS dA

2

2,3 1 24

b d tS

25 1 2

4 2

b d t d wS

b t

2

2

6 14

b d tS

1,4 0S

S1S2

S3 S4

S5

S6

Definition :

Channel Section:

Definition :

Channel Section:

990731_262423_380v3.i

STRESS ANALYSIS OF CHANNEL SECTION BEAM – EXAMPLE PROBLEM

Load applied at shear center Load applied at center of gravity

Shear stress

+

+

+

--

+ +

++

+

-

-

++

+

--

Normal stress

0.3k/ft

10’

1.5’’

7’’

0.135’’

b wb

v wv t

990731_262423_380v3.i

RESULTS OF STRESS ANALYSIS – EXAMPLE PROBLEM

point b w b+ w

1 20.10 -14.64 5.46

2 20.10 5.64 25.74

3 -20.10 -5.64 -25.74

4 -20.10 14.64 -5.46

point v w t v+w + t

6 0.68 0.369 9.24 9.551

2 0.941 0.314 9.24 9.867

5 2.070 0.158 9.24 11.468

12

3 4

5

6

A member exhibiting bending-torsion coupling shows significantly different stress distribution

990731_262423_380v3.i

CONTENTS

• Introduction

– Impact of cold-formed steel

– Structural Consideration of Channel section

• Lateral Buckling

• Stress Analysis

• Local Buckling & Effective Width

• Analysis & Design of Cold-formed Channel

• Next Steps

990731_262423_380v3.i

LOCAL BUCKLING CAN OCCUR BEFORE GLOBAL BUCKLING

Reduce the ultimate load-carrying capacity significantly

990731_262423_380v3.i

BEHAVIOR OF STIFFENED AND UNSTIFFENED COMPRESSION ELEMENTS ARE NOT IDENTICAL

A flat compression elements stiffened by other components (web, flange, lip, stiffener) along both longitudinal edges

Stiffened compression elements (s.c.e)

Unstiffened compression elements (u.c.e)

A flat compression element stiffened only along one of the two longitudinal edges

u.c.e

u.c.e

s.c.e s.c.e

990731_262423_380v3.i

PLATES DO NOT COLLAPSE WHEN BUCKLING OCCURS, BUT CAN STILL CARRY LOAD AFTER BUCKLING - POSTBUCKLING STRENGTH

p

pcr

4 4 4 2

4 2 2 4 22 0xw w w t w

x x x y D x

Plate Buckling Equation

2

212(1 )( / )cr

k E

w t

Plate Buckling Stress

•Rigorous solution of postbuckling is difficult (Nonlinear numerical Analysis needed)•Can define EFFECTIVE width

990731_262423_380v3.i

EFFECTIVE DESIGN WIDTH “b” CONCEPT IS WIDELY USED IN DESIGN PROCEDURE DUE TO THEIR SIMPLICITY

effective width, b, represents a width of the plate which just buckles when = y

w

1

1 cr

w

2

cr 2 y

w

3

3 = y

w

w

b/2

First introduced by von Karman (1932)

max0

wdx b

max

The initially uniform compressive stresses become redistributed

2

2 23(1 )( / )cr y

E

b t

cr

y

b

w

Relation of b and w

990731_262423_380v3.i

AISI SPECIFICATION FOR EFFECTIVE WIDTH HAS BEEN DEVELOPED

Winter (1946) presented the formula for effective width

AISI design provision (1946-1968)

Winter (1970) presented more realistic equation

AISI design provision (1970- )

max max

1.9 1 0.475E t E

b tw

max max

1 0.25cr crb

w

max max

1.9 1 0.415E t E

b tw

max max

1 0.22cr crb

w

990731_262423_380v3.i

AISI DESIGN PROVISION FOR EFFECTIVE WIDTH

0.673

0.673

b w

b w

0.673

1

Effective Design Width Equation

(1 0.22 / ) / 1

max1.052 w

t Ek

Individual plates subjected to different boundary conditions

Need to calculate k

990731_262423_380v3.i

BUCKLING STRESSES CAN BE DETERMINED VIA COEFFICIENT K

Boundary conditionTypes of

stressk

Comp. 4.0

Comp. 6.97

Comp. 0.425

Comp. 1.277

Comp. 5.42

s.s.

s.s.

s.s.

s.s.

s.s.

s.s.

s.s.

s.s.s.s.

s.s.

s.s.

s.s. s.s.

s.s.

free

fixed

fixed

fixed

fixed

free

Boundary conditionTypes of

stressk

Shear 5.34

Shear 8.98

Bending 23.9

Bending 41.8

s.s.

s.s.

s.s.

s.s.

fixed

fixedfixed fixed

s.s.

s.s.

s.s.

s.s.

fixed

fixedfixed fixed

2

212(1 )( / )cr

k E

w t

990731_262423_380v3.i

CALCULATION OF EFFECTIVE WIDTH OF COMPRESSION FLANGE FOR CHANNEL IS STRAIGHTFORWARD

k=0.425

max1.052 w

t Ek

lim

w w

t t

0.673 0.673

b wb w

Check if

Buckling coefficient for ss-ss-ss-free

Check the width-to-thickness ratio

Calculation of slenderness ratio

Determine the effective width

wb

(1 0.22 / ) / 1 Calculation of efffective width parameter

990731_262423_380v3.i

EFFECTIVE WIDTH OF WEB SHOULD BE CALCULATED BY ITERATION PROCESS (NOT SIMPLE)

Assume fully effective

lim

h h

t t

34 2(1 ) 2(1 )k

2 1/f f

max1.052 h

t Ek

0.673 0.673

eb w

eb w

wbf1

f2

b1

b2

1 /(3 )eb b

0.2360.236

2 / 2eb b

2 1eb b b

Check if

1 2 cb b h

hc

Check if

Recalculate the neutral axis

Web is fully effective!

1 2 1 2

1 2

( ) ( )

( )c p

c

b b b b

b b

b1 & b2 calculated

no

no

yes yes

(1 0.22 / ) / 1

n=1 n>1

990731_262423_380v3.i

CALCULATION OF EFFECTIVE WIDTH OF COMPRESSION FLANGE FOR LIPPED CHANNEL IS DEPENDENT TO THE RIGIDITY OF THE LIP

lim

w w

t t

0.673 0.673

b wb w

(1 0.22 / ) / 1

Check if

w

1.28 /S E

D

d

/ 3 /S w t S

/ / 3w t S/ 0.25D w

/w t S

/ / 3w t S / 0.25D wand

0aI

b w

0aI

b w

No edge stiffnener needed

3 4399 / / 0.33aI w t S t

1/ 24.82 5 / /s ak D w I I 0.8 ( / ) 0.25D w

for1/ 23.57( / ) 0.43 4s ak I I

for

( / ) 0.25D w

/s s s a sd d I I d for lip stiffener

ds

ds’ 4115 / / 5aI w t S t

1/34.82 5 / /s ak D w I I 0.8 ( / ) 0.25D w

for1/33.57( / ) 0.43 4s ak I I

for

( / ) 0.25D w

/s s s a sd d I I d for lip stiffener

max1.052 w

t Ek

For edge stiffener k=0.425

990731_262423_380v3.i

ANALYSIS AND DESIGN OF COLD-FORMED STEELS ARE INTEGRATED PROCEDURE

AISI code

Design

FEM

Analysis

Stresses

Ideas for Inelastic buckling

Ideas for local buckling

Ideas for stress analysis

Ideas for lateral buckling

990731_262423_380v3.i

CONTENTS

• Introduction

– Impact of cold-formed steel

– Structural Consideration of Channel section

• Lateral Buckling

• Stress Analysis

• Local Buckling & Effective Width

• Analysis & Design of Cold-formed Channel

• Next Steps

990731_262423_380v3.i

DESIGN STRENGTH CAN BE CALCULATED VIA COMPLICATED PROCEDURE

Sectional PropertiesCalculate the sectional properties (A, x, y, S, J, Ix ,Iy ,Iw) of full section by linear method

Elastic Lateral Buckling Moment

Inelastic Lateral Buckling Moment

0.5

0.5

e y

e y

M M

M M

Effective Width of Flange and Lip

Assume fully effective web and check the effectiveness by iteration

Effective Width of Web

Effective Sectional Modulus

cr b o y tM C r A

([ ] [ ]){ } {0}K G

1 / 4c y y eM M M M

c eM M

Recalculate the neutral axis until the effective web width is determined

cn e

f

MM S

SNominal and

Design Strengthn

af

MM

xe

cg

IS

Y

1 /(3 )eb b 0.2360.236

2 / 2eb b

2 1eb b b

0.673 0.673

b wb w

Determine buckling moment and mode using accurate finite element analysis or AISI code

Determine inelastic buckling moment using AISI code

Determine the effective width of compression flange and edge stiffener

The interaction of the local and overall lateral buckling results in a reduction of the lateral strength

990731_262423_380v3.i

DESIGN STRENGTH OF CHANNEL BEAM - EXAMPLE PROBLEM

2.5’ 2.5’ 2.5’ 2.5’

Sectional Properties

Elastic Critical Moment is calculated from AISI code or FEM

Inelastic Critical Moment is calculated from AISI code

Nominal Moment is based on the effective sectional modulus

2/8 /81.75 1.05( ) 0.3( ) 1.30

/ 4 / 4b

pl plC

pl pl

cr b o y tM C r A

2 2

2 2

29500104.74

2.5 12/ 0.569/y

y y y

Eksi

K l r

2

22

1125 .t

o t t

EIGJ k in

Ar K l

14

yc y

e

MM M

M

y f yM S

1.21748.62

1.532c

n ef

MM S

S

By linear method20.706A in 31.532fS in40.000848J in 62.66I in

0.5e yM M

P

(4% reduction)

(31% reduction)

329.75 .k in

50.56 .k in

38.62 .k in

48.62 .k in

990731_262423_380v3.i

RIGOROUSLY ANALYSE THE TECHNOLOGY TREE OF COLD-FORMED STEEL MEMBER

Shear Diaphragms

Beam Webs

Lateral Buckling

Bending + Torsion

Flexural Members

Cylindrical tubular members

Flexural BucklingTorsional Buckling

Effective Length

Compressive Members

BendingBending Strength

Stress Analysis

Deflection

COLD-FORMED STEEL MEMBER

Composite Design

Corrugated Sheets

Local Buckling

Warping

Pure Torsion

Purlins

Wall Studs

Compressive Strength Cold Work

Local Buckling

Effective Sectional Prop.

other cross-sections:

Distortional Buckling

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CONTENTS

• Introduction

– Impact of cold-formed steel

– Structural Consideration of Channel section

• Lateral Buckling

• Stress Analysis

• Local Buckling & Effective Width

• Analysis & Design of Cold-formed Channel

• How do we take care of the combined effects?

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VARIOUS TYPES OF BUCKLING CAN OCCUR

Global Buckling: profile of cross section does not change

Distortional Buckling:Lateral deflection of the unsupported flange

Local Buckling:Each plate element can buckle

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DIFFERENT AVAILABLE NUMERICAL METHODS

• Plate Finite Elements

• Finite Strip Method

• Beam Models

– Effective Width Concept

– Special Constitutive Law

– Enriched displacement field

– Plate FE with static condensation of d.o.f.’s

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PLATE FINITE ELEMENTS

– Can model local effects

– Requires a fine mesh

– Practical Difficulties

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FINITE STRIP METHOD

– D.O.F.’s can be reduced

– Limited to prismatic simply-supported members with constant forces

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BEAM MODELS

– Good for overall stability

– Nondeformability of the profile cross section

– Cannot account for local effects

Effective Width Concept:

limited to local buckling

Try to represent the effect rather than the phenomenon itself

Enriched displacement field:

Local deformation of the cross section is superimposed in a displacement field

Assumed that the shape of the local field is unchanged during the process

Plate FE with static condensation of d.o.f.’s:

Modeled as plate finite elements with restrained d.o.f.

Classical beam d.o.f. + magnitude of the local deformation

Timoshenko beam model

Transverse shear deformation

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CONCLUDING REMARKS

The geometric coupling depends on the shape of the cross section.

Needs fully geometrically nonlinear model to predict the structural behavior accurately. (tremendous efforts)

Beam model seems reasonable, but can be improved by considering local effects or shear deformation.

Consideration of material nonlinearity including inelastic buckling can be achieved by stress analysis or global assumption of plastic process.

Structural members with anisotropic materials (pultruded composites) awaits future attention.

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