1C8 Advanced design of steel structures prepared by Josef Machacek.

1C8 Advanced design

of steel structures

prepared by

Josef Machacek

2

List of lessons

1) Lateral-torsional instability of beams.2) Buckling of plates.3) Thin-walled steel members.4) Torsion of members. 5) Fatigue of steel structures.6) Composite steel and concrete structures. 7) Tall buildings. 8) Industrial halls.9) Large-span structures.10) Masts, towers, chimneys.11) Tanks and pipelines.12) Technological structures. 13) Reserve.

3

1. Lateral-torsional buckling

• Introduction (stability and strength).

• Critical moment.

• Resistance of the actual beam.

• Interaction of moment and axial force.

• Eurocode approach.

Objectives

Introduction

Critical moment

Resistance of actual beam

Interaction M+N

Assessment

Notes

4

IntroductionObjectives

Introduction

Critical moment


Interaction M+N

Assessment

Notes

Stability of ideal (straight) beam under bending

impulse

segment laterally supported in bending and torsion

Mcr

L

Strength of real beam (with imperfections 0, 0)

1M

yyLTRdb,

f

WM bifurcation under bending

strength

initial

yyLT fW

M

Mcr,1

0,0,

cr

yyLT

M

fW

reduction factor LT depends on:

5

Critical moment

Stability of ideal beam under bending (determination of Mcr)

"Basic beam" - with y-y axis of symmetry(simply supported in bending and torsion, loaded only by M)

Two equations of equilibrium (for lateral and torsional buckling)may be unified into one equation:

0d

d

d

d

z

2

2

2

t4

4

w EI

M

xGI

xEI

The first non-trivial solution gives M = Mcr:

L

GIEI

GIL

EI

L

GIEIM tz

crt

2w

2tz

cr 1

2wt

t2

w2

cr 11 GIL

EI

t

wwt GI

EI

L

where

Fz

S G

z

y

Fz

SG

z

y

Fz

S G

z

y

F z

hf

z

y

F z

S=G

z

y

Fz

S G

z

y

Fz

S G

z

y

F z

S G

z

y

F z

S G

z

yS=G

y y

z

z

Objectives

Introduction

Critical moment


Interaction M+N

Assessment

Notes

6

Critical moment

Generally (EN 1993-1-1) for beams with cross-sections according to picture:

zs

(C )

S

G

(T )

z

y

zs

S

G

(T )

z

y

hf

zg

F z

zg

F z

(C )

zaza

hf

sh

=

Fz

S G

z

y

Fz

SG

z

y

Fz

S G

z

y

F z

hf

z

y

F z

S=G

z

y

Fz

S G

z

y

Fz

S G

z

y

F z

S G

z

y

F z

S G

z

yS=G

Fzza zg

z

yG

S

zs

j3g2

2j3g2

2wt

z

1cr 1 CCCC

k

C

t

w

LTwwt GI

EI

Lk

t

z

LTz

gg GI

EI

Lk

z

t

z

LTz

jj GI

EI

Lk

z

C1 represents mainly shape of bending moment,

C2 comes in useful only if loading is not applied in shear centre,

C3 comes in useful only for cross-sections non-symmetrical about y-y.

symmetry about z-z symmetry about k y-y, loading through shear centre

Objectives

Introduction

Critical moment


Interaction M+N

Assessment

Notes

Procedure to determine Mcr:

1. Divide the beam into segments of lengths LLT according to lateral support:

7

Critical moment

2. Define shape of moment in the segment:

from table factor C1 1:

1 ~1,77 ~2,56

lateral support (bracing) in bending and torsion (lateral support "near" compression flange is sufficient)

segment 1segment 2

e.g. segment 1: (LLT,1)

e.g. segment 2: a) usually linear distribution

b) almost never

(because loading here creates continuous support)

~1,13 ~1, 35

segment 3

Objectives

Introduction

Critical moment


Interaction M+N

Assessment

Notes

Critical moment

Other cases of k :

angle of torsion is zero

stiffener rigidin torsion (½ tube)

stiffener non-rigidin torsion

kz = 1kw = 1

kz = 1kw = 0,5

possible lateral buckling possible lateral buckling

3. Determine support of segment ends: (actually ratio of "effective length")

kz = 1 (pins for lateral bending)kw = 1 (free warping of cross section)

Objectives

Introduction

Critical moment


Interaction M+N

Assessment

Notes

9

Critical moment

Cantilever: - only if free end is not laterally and torsionally supported (otherwise concerning Mcr this case is not a cantilever but normal beam segment),

- for cantilever with free end: kz = kw = 2.

konzervativní hodnotykz = 0,7kw = 1

kz = 1kw = 0,7 (conservatively 1)

conservatively(theor. kz = kw = 0,5)

structure torsionallyrigid torsionally

rigidtorsionallynon-rigid

possible lateral buckling

Objectives

Introduction

Critical moment


Interaction M+N

Assessment

Notes

10

Critical moment

4. Formula for Mcr depends also on position of loading with respect to shear centre (zg):

- lateral loading acting to shear centre S (zg > 0) is destabilizing: it increases the torsional moment

- lateral loading acting from shear centre S (zg < 0) is stabilizing: it decreases the torsional moment

Factor C2 formoment shape M:(valid for I cross-section)

Mel 0,46 0,55 1,56 1,63 0,88 1,15Mpl (plast. hinges) 0,98 1,63 0,70 1,08

Come in useful for lateral loading (loading byend moments is considered in shear centre).

S

S

F

F

Objectives

Introduction

Critical moment


Interaction M+N

Assessment

Notes

11

Critical moment

5. Cross-sections non-symmetrical about y-y

For I cross section with unequal flanges:

warping constant

parameter of asymmetry

2sz

2fw 21 )/h(I)(I

I+ I

I I ftfc

ftfcf

second mom. of area of compress. and tens. flange about z-z

ffA

h,dAz)zy(I

,zz 450

50 22

ysj

Factor C3 greatly depends on f and moment shape (below for kz = kw = 1):

f = -1 1,00 1,47 2,00 0,93

f = 0 1,00 1,00 0,00 0,53 f = 1 1,00 1,00 -2,00 0,38

Mcr =+1Mcr = 0 Mcr = -1

z s

(C )

S

G

(T )

z

y

z s

S

G

(T )

z

y

hf

zg

F z

zg

F z

(C )

zaza

hf

sh

=

Fzza zg

z

yG

S

zs hf

Objectives

Introduction

Critical moment


Interaction M+N

Assessment

Notes

12

Critical moment

Cross sections with imposed axis of rotation

V (imposed axis)

suck

oftenSMcr is affected by positionof imposed axis of rotation(Mcr is always greater, holdingis favourable)

For a simple beam with doubly symmetric cross section and generalimposed axis:

coefficients for shape of M: 1 2

2,00 0,00

0,93 0,81

0,60 0,811

t

2

LTw

2

zw

cr

2

2

h

IGLk

hIEIE

M

vg2v1

t

2

LTw

2vzw

cr zzz

IGLk

zIEIE

M

z

zg zv

G ≡ S axis y

For suck loading applied at tension flange:

Objectives

Introduction

Critical moment


Interaction M+N

Assessment

Notes

13

Critical moment

Approximate approach for lateral-torsional buckling

In buildings, the reduction factor forlateral buckling corresponding to"equivalent compression flange" (defined as flange with 1/3 ofcompression web)may be taken instead:

1

LT

zf,f

i

L

9931 ,f

E

y

Note: According to Eurocode the reduction factor is taken from curve c, but for cross sections with web slenderness h/tw ≤ 44 from curve d.The factor due to conservatism may be increased by 10%.

hw/6

impulse roughly

Objectives

Introduction

Critical moment


Interaction M+N

Assessment

Notes

14

Critical moment

Practical case of a continuous beam (or a rafter of a frame):The top beam flange is usually laterally supported bycladding (or decking). Instead of calculating stability toimposed axis a conservative approach may be used, considering loading at girder shear centre and neglecting destabilizing load location (C2 = 0):

It is desirable to secure laterally the dangerous zones ofbottom free compression flanges against instability:

• by bracing in the level of bottom flanges,• or by diagonals (sufficiently strong) between bottom flange and crossing beams (e.g. purlins).

Length LLT then corresponds to the distance of supports).

Note:For suck stabilizing effect should be applied (C2, zg < 0).

Mel C1 = 2,23Mpl C1 = 1,21

Mel C1 = 2,58Mpl C1 = 1,23

according to M distribution(for different M may be usedgraphs by A. Bureau)

Objectives

Introduction

Critical moment


Interaction M+N

Assessment

Notes

15

Critical moment

Beams that do not lose lateral stability:

1. Hollow cross sections Reason: high It high Mcr

2. Girders bent about their minor axis Reason: high It high Mcr

3. Short segment ( ) - all cross sections, e.g. Reason: LT 1

4. Full lateral restraint: "near" to the compression flange is sufficient ( approx. within h/4 )

40LT ,

compression flange loaded tension flange loaded

zg

zg

z v

z v

or higher

or anywhere higher

zv ≥ 0,47 zg zv ≤ 0,47 zg

Objectives

Introduction

Critical moment


Interaction M+N

Assessment

Notes

16

Resistance of the actual beam (Mb,Rd)

Similarly as for compression struts: actual strength Mb,Rd < Mcr

(due to imperfections)

e.g. DIN: n = 2,0 (rolled)

= 2,5 (welded)

1/n2nLTRdpl,Rdb, 1

MM

Eurocode EN 1993:

The procedure is the same as for columns: acc. to is determined

LT with respect to shape of the cross section (see next slide).

Note: For a direct 2. order analysis the imperfections e0d are available.

LT

10,1

but 1

2LT

LT

LT

2LT

2LTLT

LT

215,0 LTLT,0LTLTLT

1M

yyLTRdb,

f

WM ... Wy is section modulus acc. to cross section class

Objectives

Introduction

Critical moment


Interaction M+N

Assessment

Notes

17


Similarly as for compression struts: actual strength Mb,Rd < Mcr

(due to imperfections)

e.g. DIN: n = 2,0 (rolled)

= 2,5 (welded)

1/n2nLTRdpl,Rdb, 1

MM

Eurocode EN 1993:

The procedure is the same as for columns: acc. to is determined

LT with respect to shape of the cross section (see next slide).

Note: For a direct 2. order analysis the imperfections e0d are available.

LT

101

but 1

2LT

LT

LT

2LT

2LTLT

LT

,

2

150

LT

LT,0LTLTLT

,

1M

yyLTRdb,

f

WM ... Wy is section modulus acc. to cross section class

For common rolled and welded cross sections: = 0,75For non-constant M the factor may be reduced to LT,mod (see Eurocode).

400LT ,,

Objectives

Introduction

Critical moment


Interaction M+N

Assessment

Notes

Choice of buckling curve:

rolled I sections shallow h/b ≤ 2 (up to IPE300, HE600B) b

high h/b > 2 c

welded I sections h/b ≤ 2 c h/b > 2

d

18


Lm

greater residual stressesdue to welding

rigid cross section

For common rolled and welded cross sections: = 0,75For non-constant M the factor may be reduced to LT,mod (see Eurocode).

400LT ,,

Objectives

Introduction

Critical moment


Interaction M+N

Assessment

Notes

19


Lm

In plastic analysis (considering redistribution of moments and"rotated" plastic hinges) lateral torsional buckling in hinges must beprevented and designed for 2,5 % Nf,Ed:

0,025 Nf,Edgirder in location of Mpl

strut providingsupport force in compression

flange

Complicated structures (e.g. haunched girders) may be verified using "stable length" Lm

(in which LT = 1) - formulas see Eurocode.

hh

hsL h

L yLm

Objectives

Introduction

Critical moment


Interaction M+N

Assessment

Notes

20

Interaction M + N ("beam columns")

Lm

Always must be verified simple compression and bending in the moststressed cross section – see common non-linear relations.

In stability interaction two simultaneous formulas should be considered:

Usual case M + Ny:

1Rdy,LT

Edy,yy

Rdy

Ed

M

Mk

N

N

1Rdy,LT

Edy,zy

Rdz

Ed

M

Mk

N

N

1

11

M

Rkz,

Edz,Edz,yz

M

Rky,LT

Edy,Edy,yy

M1

Rky

Ed

M

MMk

M

MMk

NN

1

11

M

Rkz,

Edz,Edz,zz

M1

Rky,LT

Edy,Edy,zy

M

Rkz

Ed

M

MMk

M

MMk

NN

for class 4 only

factors kyy ≤ 1,8; kzy ≤ 1,4

(for relations see EN 1993-1-1, Annex B)

Note: historical unsuitable linear relationship (without 2nd order factors kyy, kzy)

1,b

Rdb,

Edy,Ed

M

M

N

N

Rd

MyMz

N

Objectives

Introduction

Critical moment


Interaction M+N

Assessment

Notes

21

Interaction M + N ("beam columns")

Lm

Complementary note:

Generally FEM may be used (complicated structures, non-uniform members etc.) to analyse lateral and lateral torsional buckling.First analyse the structure linearly, second critical loading. Then determine:

ult,k - minimum load amplifier of design loading to reach characteristic

resistance (without lateral and lateral-torsional buckling);

cr,op - minimum load amplifier of design loading to reach elastic critical

loading (for lateral or lateral torsional buckling).

opcr,

kult,op

op = min(, LT)

opM1Rky,

Edy,

1MRk

Ed

M

M

N

N

Resulting relationship:

Objectives

Introduction

Critical moment


Interaction M+N

Assessment

Notes

Assessment

• Ideal and actual beam – differences.

• Procedure for determining of critical moment.

• Destabilizing and stabilizing loading.

• Approximate approach for lateral torsional buckling.

• Resistance of actual beam.

• Interaction M+N according to Eurocode.

22

Objectives

Introduction

Critical moment


Interaction M+N

Assessment

Notes

Notes to users of the lecture

• This session requires about 90 minutes of lecturing.

• Within the lecturing, design of beams subjected to lateral torsional buckling is described. Calculation of critical moment under general loading and entry data is shown. Finally resistance of actual beam and design under interaction of moment and axial force in accordance with Eurocode 3 is presented.

• Further readings on the relevant documents from website of www.access-steel.com and relevant standards of national standard institutions are strongly recommended.

• Keywords for the lecture:

lateral torsional instability, critical moment, ideal beam, real beam, destabilizing loading, imposed axis, beam resistance, stability interaction.

Objectives

Introduction

Critical moment


Interaction M+N

Assessment

Notes

24

Notes for lecturers

• Subject: Lateral torsional buckling of beams.

• Lecture duration: 90 minutes.

• Keywords: lateral torsional instability, critical moment, ideal beam, real beam, destabilizing loading, imposed axis, beam resistance, stability interaction.

• Aspects to be discussed: Ideal beam and real beam with imperfections. Stability and strength. Critical moment and factors which influence its determination. Eurocode approach.

• After the lecturing, calculation of critical moments under various conditions or relevant software should be practised.

• Further reading: relevant documents www.access-steel.com and relevant standards of national standard institutions are strongly recommended.

• Preparation for tutorial exercise: see examples prepared for the course.

Objectives

Introduction

Critical moment


Interaction M+N

Assessment

Notes

1C8 Advanced design of steel structures prepared by Josef Machacek.

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Transcript of 1C8 Advanced design of steel structures prepared by Josef Machacek.