Module 4 Member Design - Lecture 9 Local Buckling and Section Classification

8/19/2019 Module 4 Member Design - Lecture 9 Local Buckling and Section Classification

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Structural Steelwork Eurocodes

Development of

A Trans-national Approach

Course: Eurocode 3

Module 4 : Member design

Lecture 9 : Local Buckling and Section

Classification

Summary:

• Structural sections may be considered as an assembly of individual plate elements.• Plate elements may be internal (e.g. the webs of open beams or the flanges of boxes and others are

outstand (e.g. the flanges of open sections and the legs of angles.

• !hen loaded in compression these plates may buckle locally.

• "ocal buc#ling within the cross-section may limit the load carrying capacity of the section by

preventing the attainment of yield strength.

• Premature failure (arising from the effects of local buc#ling may be avoided by limiting the width to

thic#ness ratio - or slenderness - of individual elements within the cross section.

• This is the basis of the section classification approach.

• $%& defines four classes of cross-section. The class into which a particular cross-section falls depends

upon the slenderness of each element and the compressive stress distribution.

Pre-reuisites:

• A wor#ing #nowledge of $uler buc#ling theory.

• The idea of normalising stresses against yield stress

!otes for "utors:

This material comprises one &' minute lecture.The lecturer can brea# up the session with one formative exercise part way through (calculation of normalised plate slendernessThe lecturer may apply a summative assessment at the end of the session reuiring that students confirm the

$%& limits for %lass ) hot rolled sections


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Structural Steelwor# $urocodes *Development of a Trans-+ational ApproachDesign of ,embers

Section classification

#b$ecti%es:

• explain that sections may fail by compressive buc#ling of plates within the section.

• distinguish between internal and outstand elements.

• demonstrate that plate slenderness and edge restraints control the buc#ling behaviour.

• s#etch the relationship between normalised ultimate compressive stress and normalised plate slenderness.

• explain (in terms of the above s#etch the meaning of different section classifications.

• derive a result from $%& Table xxx for hot rolled sections.

• use the section classification method to choose appropriate sections.

• describe the effective width approach for %lass sections.

&eferences:

• $urocode & Design of steel structures Part /./ 0eneral rules and rules for buildings

• $SD$P "ectures 1.) and 1.&• The 2ehaviour and Design of Steel Structures3 %hapter - "ocal buc#ling of thin plate elements3

+ S Trahair and , A 2radford3 $ 4 5+ Spon 6evised Second $dition /77

Contents:

/. 8ntroduction

). %lassification

&. 2ehaviour of plate elements in compression

. $ffective width approach to design of %lass sections9. %oncluding summary

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'( )ntroduction

Structural sections3 be they rolled or welded3 may be considered as an assembly of individual

plate elements3 some of which are internal (e.g. the webs of open beams or the flanges of boxes and others are outstand (e.g. the flanges of open sections and the legs of angles - see

figure /. As the plate elements in structural sections are relatively thin compared with their width3 when loaded in compression (as a result of axial loads applied to the whole section

and:or from bending they may buc#le locally. The disposition of any plate element within the

cross section to buc#le may limit the axial load carrying capacity3 or the bending resistance of

the section3 by preventing the attainment of yield. Avoidance of premature failure arising fromthe effects of local buc#ling may be achieved by limiting the width-to-thic#ness ratio for

individual elements within the cross section. This is the basis of the section classification

approach.

;utstand

8nternal

!eb

5lange

!eb

8nternal

5lange

(a) Rolled I-section (b) Hollow section

5lange

(c) Welded box section

8nternal;utstand

8nternal!eb

*igure ' )nternal or outstand elements

+( Classification$%& defines four classes of cross section. The class into which a particular cross section falls

depends upon the slenderness of each element (defined by a width-to-thic#ness ratio and the

compressive stress distribution i.e. uniform or linear. The classes are defined in terms of performance reuirements for resistance of bending moments

5.3.2 (1)

Class 1 cross-sections are those which can form a plastic hinge with the reuired rotational

capacity for plastic analysis.

Class 2 cross-sections are those which3 although able to develop a plastic moment3 havelimited rotational capacity and are therefore unsuitable for structures designed by plastic

analysis.

Class 3 cross-sections are those in which the calculated stress in the extreme compression fibre

can reach yield but local buc#ling prevents the development of the plastic moment resistance.

Class 4 cross-sections are those in which local buc#ling limits the moment resistance (or compression resistance for axially loaded members. $xplicit allowance for the effects of local

buc#ling is necessary.

Table / summarises the classes in terms of behaviour3 moment capacity and rotational capacity.

/':'7:77 &


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f y

,oment

"ocal2uc#ling

f y

,oment

"ocal2uc#ling

f y

,oment

"ocal2uc#ling

,el

f y

,oment

"ocal2uc#ling

,el

odel o!

"e#a$iour

o%ent

ResistanceRotation Ca&acit' Class

/

/

/

/

/

/

/

/

Sufficient

"imited

+one

+one

,

, p l

,, p l

,, pl

,, pl

/

)

&

Plastic moment

on gross section

Plastic moment

on gross section

$lastic moment

on gross section

Plastic moment on

effective section

, pl

, pl

, pl

, pl

,el elastic moment resistance of cross-section

, pl plastic mom ent resistance of cross-section

, applied moment

φ rotation (curvature of sectionφ rotation (curvature of section reuired to generate fully plastic stress distribution across section pl

φ

φ

φ

φ

φφ pl

φφ

φφ

φ

φ

φφ

rot

pl

pl

pl

pl

"able ' Cross-section classifications in terms of moment resistance androtation ca,acity

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3( Bea%iour of ,late elements in com,ressionA thin flat rectangular plate subs coefficient 3.2.5 (1)

$ = ?oung>s modulus 3.2.5 (1)

The elastic critical buc#ling stress (σcr is thus inversely proportional to (b:t)

and analogous to

the slenderness ratio (":i for column buc#ling.;pen structural sections comprise a number of plates which are free along one longitudinal edge

(see figure )b and tend to be very long compared with their width. The buc#led shape for sucha plate is illustrated in figure )c. The relationship between aspect ratio and buc#ling parameter

for a long thin outstand element of this type is shown in figure )d3 from which it is clear that the buc#ling parameter tends towards a limiting value of '.)9 as the plate aspect ratio increases.

5or a section to be classified as class & or better the elastic critical buc#ling stress ( σcr must

exceed the yield stress f y . 5rom euation (/ (substituting ν = '.& and rearranging this will be

so if

3.2.5 (1)

( )'39$:f # '37)@ b:t σ ()This expression is general as the effect of stress gradient3 boundary conditions and aspect ratio

are all encompassed within the buc#ling parameter # σ. Table ) gives values for high aspect

ratios of internal and outstand elements under various elastic stress distributions.

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/) ) /

/ )

8 88 888



(d)

(c)

(b)(a)

Simply supported onall four edges

t

"

b

Simply supported

edge

5ree

edge

b

"

/

)

&

9

/ ) &'

9

Plate aspect ratio " : b

2uc#ling coefficient #

b

" 5ree$xact

# = '.)9 (b:")

'.)9

*igure + Bea%iour of ,late elements in com,ression

Trahair and2radford

σ = is maximum stress, compression

ψ = σ2 / σ1 +1 1 > ψ > 0 0 0 > ψ > −1 -1

Case IInternalelement

4,08,02

1,05 + ψ 7,81 7,81+6,29ψ+9,78ψ 2

Case IIutstan!element

0,4" 0,57-0,21ψ+0,07ψ 2 0,57 0,57-0,21ψ+0,07ψ 2 0,85

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Case IIIutstan!

element

0,4"0,578

ψ+0,341,70 1,7-5ψ+17,1ψ 2 2",8

"able + Buckling factors and stress distribution

The elastic-plastic behaviour of a perfect plate element sub


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*igure 3 .imensionless re,resentation of te elastic-,lasticbuckling stress

*ormati%e test ' / .eri%e te normalised euation for ,lateslenderness

• 0iven that( ) 9.': cr y p f σ λ =

• Plot the pvs p N λ

relationship

• Appreciating( ) 9.':)&9 y f =ε for strength normalisation3 use euation (/

( ))

)

)

//)

−=

b

t E k cr

ν

π σ σ

to derive euation (9Plates in sections are not perfectly flat nor is steel elastic-perfectly plastic (it strain hardens.

These factors3 coupled with the ability of plates to carry loads beyond the level causing elastic

buc#ling (postbuc#ling behaviour3 reuire λ p values to be reduced in order to delay the onsetof local buc#ling until the reuisite strain distribution through the section - yield at the extremefibre or fully plastic distribution - has been attained. $%& uses the following normalised plate

slenderness> as limits for classifications

5.2.1.4 (7)

Class 1λ p

@ '39

Class 2λ p@ '3B

$SD$P

Class 3λ p

@ '37 for elements under a stress gradient this is further reduced to '31 for

elements in compression throughout.

"ecture 1.)

2y substituting the appropriate values of # σ into euation (9 and noting theλ p

to be used for

each class3 limiting b:t ratios can be calculated. Table & presents limiting values for a rolledsection sub


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5lange c : tf = /' ε c : tf = // ε c : tf = /9 ε

!eb sub


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• Substituting the appropriate values of # σ into euation (93 useλ p

to determine limiting b:t

ratios for a flange

• 6evise in terms of d:tw for a web in compression

• %lassify the section in exercise xxxx (this refers to one of the overall design exercises

( Concluding summary

• Structural sections may be considered as an assembly of individual plate elements.

• Plate elements may be internal (e.g. the webs of open beams or the flanges of boxes and

others are outstand (e.g. the flanges of open sections and the legs of angles.

• !hen loaded in compression these plates may buckle locally.

• "ocal buc#ling within the cross-section may limit the load carrying capacity of the section

by preventing the attainment of yield strength.

• Premature failure (arising from the effects of local buc#ling may be avoided by limiting

the width to thic#ness ratio - or slenderness - of individual elements within the crosssection.

• This is the basis of the section classification approach.

• $%& defines four classes of cross-section. The class into which a particular cross-section

falls depends upon the slenderness of each element and the compressive stress distribution.

/':'7:77 /'


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a Webs (internal elements perpendicular to axis of bending

tw twd

tw d tw

tf

hdAxis of 2ending

d = h-&t (t = tf = tw

ClassWeb sub*ect to

bendin+

Web sub*ect to

co%&ression

Web sub*ect to bendin+

and co%&ression

Stressdistribution inelement

(compression positive

f y

f y -

d h

f y f y

f y - f y -

d h hdαd

when α > 0,5:

d:t w < 396ε/(13α − 1)

when α < 0,5:d:t w < 36ε/α

E

E

/ d:t w @ 72ε E d:t w @ && ε E

d:t w @ C& ε E d:t w @ &C ε E )

when α > 0,5:

d:t w < 456ε/(13α − 1) E

when α < 0,5:

d:t w < 41,5ε/α E

Stressdistribution inelement(compression

positive

f y

f y -

f y

f y

ψ f y -

d:)

d:)h

d h d h

& d:t w < 124 ε E d:t w @ ) ε E

when ψ > −1:d:t w < 42ε/(0,67 + 0,33ψ) E

when ψ < −1: E

d:tw

< 62ε/(1 − ψ) (− ψ

ε =f y )&9 )19 &99

E

"able 4 Ma0

/':'7:77 //


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b Internal !lan+e ele%ents (internal elements parallel to axis of bending

axis of bending

b tf t f btf

b

b

Class ,'&e Section in bendin+ Section in co%&

Stress distributionin element andacross section(compression positive

-

f y

-

-

f y

/6olled hollow section

;ther (b - &t f : tf b : tf


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c utstand !lan+es

t f

c c

t f t f c

t f

Rolled sections Welded sections

Class ,'&e o! section .lan+e sub*ect

to co%&ression Tip incompression

Tiptens

.lan+e sub*ect toco%&ression and bendin

Stress distributionin element(compression positive

-

c-

c

α c αc

/6olled

!elded

c:t f @ 10ε E

c:t f @ 9ε E

c:t f @

c:t f @ E

E 10εα7e

α

c:t f @ E

c:t f @ E

α

α

)6olled

!elded

c:t f @ 11ε E

c:t f @ 10ε E

c:t f @

c:t f @ E

E 11εα

10εα

c:t f @ E

c:tf @ E

α

α

Stress distributionin element(compression positive

-

c

-

c

&6olled

!elded

c:t f @ 15ε E

c:t f @ 14ε E

c:t f @ E

c:tf @ E

)&ε σ k

)&ε σ k

ε = )&9: f y

f y

ε

)&9 )19 &9

/ '37) '3C

.or k see !i+and table /

"able Ma0imum 1idt-to-tickness ratios for com,ression eleme/':'7:77 /&


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d 0n+les

6efer also to c.

G;utstand flangesG(Table B

t

t

h

b

(Does not apply toangles in continuous

contact with other components.

Class Section in co%&ression

Stress distributionacross section

(compression positive

t

f y-

f y

&

h

t

b h

t≤

+

≤/9 ) //9ε ε 3

e ,ubular sections

t d

Class Section in bendin+ and1or co%&ression

/ d t: ≤ 9')ε

) d t: ≤ 1')ε

& d t: ≤ 7')ε

ε =f y )&9 )19 &99

ε

"able Ma

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Stress distribution

(co%&ression &ositi$e)

E!!ecti$e widt# be!!

c

beff σ

/

σ )

/ '> ≥ψ #

beff $ cρ

σ /

σ )

bt %c

beff

ψ < ' #

b b ceff c= = −ρ ρ ψ : ( /

ψ σ σ =) /:

2uc#ling factor # σ

/

'3&

'

'391

-/

'3C9

/ /≥ ≥ −ψ

' 91 ' )/ '3 3 3− +ψ

c

beff

σ / σ

)

/ '> ≥ψ #

beff $ cρ

beff

σ /

σ )

bt bc

ψ < ' #

b b ceff c= = −ρ ρ ψ : ( /

ψ σ σ=) /:

/ / '> >ψ

'91C

0 ' /> > −ψ

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Stress distri bution

(co%&ression &ositi$e)E!!ecti$e widt# b

e!!

be/ be)

σ /

σ )

b

ψ = /

b = b - &t

beff = ρ b

be/ = '39 beff

be) = '39 beff

be/ be)

σ /

σ )

b

/ 'Hψ >

b = b - &t beff = ρ b

be/ =)

9

beff

- ψ

be) = beff - be/

be/ be)

σ /

σ )

b

bc bt ψ @ '

b = b - &t

b b beff c= = - ρ ρ ψ : ( / be/ = '3b eff

be) = '3Bb eff

ψ σ σ =) /:

2uc#lingfactor # σ

/

3'

/ 'H Hψ

C )

/ '9

3

3 ψ

'

13C/

' /H H -ψ

1 C/ B 7) 7 1C )3 3 3- ψ ψ

-/

)&37

- H/ ψ

9 7C /3 (

Alternatively3 for / /-> ψ > k σ ψ ψ ψ =

- /B

/ '//) / /) ) ' 9

I( 3 ( J ( 3

8llustrated as rhs.5or other sections b = d for webs

b= b for internal flange elements (except rhs

E

E E

"able 9 Effecti%e 1idts of com,ression elements/':'7:77 /B


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(b) Class 4 cross-sections - bendin+ %o%ent

(a) Class 4 cross-sections - axial !orce

0ross cross-section

0ross cross-section

%entroidal axis ofgross cross-section

%entroidal axis ofgross cross-section

%entroidal axis ofeffective cross-section

+on-effective Kones

+on-effective Kone

+on-effective Kone

%entroidal axis

%entroidal axis%entroidal axis oeffective section

e +

e,

e ,

%entroidal axis oeffective section

*igure 4 Effecti%e cross-sections for class 4 in com,ression and ben

/':'7:77 /1

Module 4 Member Design - Lecture 9 Local Buckling and Section Classification

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