Material course Ch7&8

8/13/2019 Material course Ch7&8

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Teaching Resource in Design of Steel Structures

IIT Madras, SERC Madras, Anna Univ., INSDAG 1

INTRODUCTION TO PLATE BUCKLINGLOCAL BUCKLINGAND SECTION

CLASSIFICATION


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Introduction

Critical Stress for Plate Buckling

Post-buckling Behaviour and Effective Width

Stability and Ultimate Strength of PlatesBuckling of Web Plates in Shear

Local Buckling

Basic Concepts of Plastic Theory

Section Classification based on width thickness ratio

Implications of Local buckling on design

CONTENTS


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Buckling of Plate under Uni-axial Compression

Nx

y

z

b

a

A

B

C D

RECTANGULAR PLATES ON FOUR SIDESx


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2

2

3

2

4

4

22

4

4

4 )1(122

x

wNEty

w

yx

w

x

wx

,...3,2,1 ,....3,2,1

sinsinm n

mnb

yn

a

xmww

2

22

3

2

4

44

22

422

4

44

)()1(12

2a

mN

Etb

n

ba

nm

a

mcrx

22

2

2

32

22

22222

2

32

)1(12/

//

)1(12

)(

mb

an

a

mEt

am

bnamEtN crx

2

22

321

)1(12

b

a

ma

bm

b

EtN

cr

22

2

)/)(1(12 tb

Ek

cr

CRITICAL BUCKLING STRESSEquilibrium Equation

w can be assumed as (satisfies end conditions)

Lowest value for n=1

(Ncr= crt)


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b

a = 2b

b

a = 3b

Buckling Modes for Long Plates

INFLUENCE OF ASPECT RATIO ON

PLATE BUCKLING


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m = 1 m = 2 m = 3k

4.0

1.0 2.0 3.0 a/b

k-values for a Simply Supported Plate

VARIATION OF PLATE BUCKLINGCOEFF. WITH ASPECT RATIO


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web

(a) (b) (c ) (d)

Plate Elements with Different Edge Conditions

(e)

Rotation of

free edge

Load Condi t ion Support Condi t ion Buc kl ing Coeff ic ient ,

kUniaxial Compressive Stress (

x) Hinged-hinged 4.00

Fixed-fixed 6.97Hinged-free 0.43

Fixed free 1.27

Shear Stress (xy

) Hinged-hinged 5.35

Fixed-fixed 8.99

Table 1 Values of k for Different Load and Support Conditions

PLATES WITH OTHER SUPPORT CONDITIONS


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21

2

2lim

)1(12

yf

Ek

t

b

DESIGN OF PLATE ELEMENTSLimiting width-thickness ratio to ensure yielding before platebuckling

Codes prescribe limiting b/t ratios as C/fy


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beff= Effective width

beff/2b

(a) Actual (b) Assumed

beff/2

Actual and Assumed Stress Distribution

in the Post-buckling Range

y

cr

y

creff

ffb

b 22.01

Plates with initial imperfections

POST-BUCKLING BEHAVIOUR & EFFECTIVE WIDTH

Winters formula

Von Karman

Ult Strength = fy beff


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Actual

paths

Secondary

path

Secondary

path

Funda

-mental

path

(a) Column

Actual paths

cor respond ing to

levels o f in i t ia l

imper fect ion

(b) Plate

P/PcrP/Pe

w0 w

1.0 1.0

w

Load versus Out-of-plane Displacement Curves

STABILITY OF PLATES


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2.0

1.0

Flat

plate

P/Pcr

Plates with

imperfections

1.0 2.0 3.0U/Ucr

Load versus Axial Deformation Diagram

STABILITY OF PLATES - 1

E/2

E


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Column and Plate Strength Curves

1.00

0.75

0.50

0.25

0 3.02.01.0 1.0

0.75

0.50

0.25

0 3.02.0

1.0

Test d ata

( E/fy)(cr/fy)

fy/ E fy/

cr

Test data

(b)

f/fy f/fy

(a)

STABILITY OF PLATES - 2


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Shear buckling of a plate

BUCKLING OF WEBPLATES IN SHEAR

cr


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POST-SHEAR BUCKLING BEHAVIOUR

Tension Field Action

0.3)/(andwhen

0.3)/(andwhen/1

/1

2

3

cr

y2

da

dada

ycr

crycrycr


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1

22

,

crcrb

b

cr

PLATE SUBJECTED TO COMBINED STRESSES

where = applied axial compressive stressb= applied bending compressive stress = applied shear stress

For a four-side supported plate


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LOCAL BUCKLINGAND SECTION CLASSIFICATION

OPEN AND CLOSED SECTIONS

Strength of compression members depends on slenderness ratio


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(b)(a)

Local buckling of Compression Members

LOCAL BUCKLING

Beamscompression flange buckles locally

Fabricated and cold-formed sections prone to local buckling

Local buckling gives distortion of c/s but need not lead to collapse


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L

Bending Moment Diagram

Plastic hinges

Mp

Collapse mechanism

Plastic hinges

Mp

Formation of a Collapse Mechanism in a Fixed Beam

w

Bending Moment Diagram

BASIC CONCEPTS OF PLASTIC THEORY

First yield moment MyPlastic moment Mp

Shape factor S = Mp/My

Rotation Capacity (a) at My(b) My< M


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SECTION CLASSIFICATION

Mp

Rotation

My

y u

Slender

Semi-compact

Compact

Plastic

Section Classification based on Moment-Rotation Characteristics


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Moment Capacities of Sections

My

Mp

1 2 3 =b/t

Semi-Compact SlenderPlastic Compact

SECTION CLASSIFICATION BASED ON

WIDTH -THICKNESS RATIO

For Compression members use compact or plastic sections


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Type of Element Type of

Section

Class of Section

Plastic ( 1) Compact( 2) Semi-compact (3)Outstand element of

compression flange

Welded b/t 7.9 b/t 8.9 b/t 13.6Rolled b/t 8.9 b/t 9.5 b/t 15.0

Internal element of

compression flange

Welded b/t 24.2 b/t 26.3 b/t 29.4Rolled b/t 27.3 b/t 33.6 b/t 41.0

Web with neutral

axis at mid depth

All d/t 83.0 d/t 102.9 d/t 126.0Web under uniform

compression

Welded d/t 29.4Rolled d/t 41.0

Single/double angle

T-stems

Rolled b/t 8.9d/t 8.9 b/t 10.0d/t 10.0 b/t 15.8d/t 15.8

Circular tube with

outer diameter DD/t 442 D/t 632 D/t 882

Table 2 Limits on Width to Thickness Ratio of Plate Elements

yf250


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IMPLICATIONS OF LOCAL BUCKLING ON DESIGN

Local Buckling can be prevented by limiting the b/t ratio

One can also use longitudinal stiffeners

Semi-compact and slender sections cannot be used in plastic design

Semi-compact sections can be used in elastic design knowing My

Slender sections also have stiffness problems and are not used in

hot-rolled structural steel work but are popular in cold-formed steels

Plate girders are usually designed taking advantage of tension field

Documents

Material course Ch7&8