SN002

NCCI: Determination of non-dimensional slenderness of I and H sections SN002a-EN-EU

NCCI: Determination of non-dimensional slenderness of I and H sections

This NCCI presents a method for determining the non-dimensional slenderness without explicit determination of Mcr. The basic, conservative method can be refined to take account of section geometry and bending moment distribution.

Contents

1. Simplified method 2

2. Economy from more complexity 3

3. Allowance for the effect of destabilizing loads 6

NCCI: Determination of non-dimensional slenderness of I and H sectionsC

reat

ed o

n F

riday

, Sep

tem

ber

02, 2

011

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent


1. Simplified method For straight segments of hot-rolled doubly symmetric I and H sections with lateral restraints to the compression flange at both ends of the segment considered and with no destabilizing loads, the value of LTλ required by EN1993-1-1 §6.3.2.2 or §6.3.2.3 may be conservatively taken from Table 1.1.

S 235 S 275 S 355 S420 S 460

104z

LTiL

=λ 96

zLT

iL=λ

85z

LTiL

=λ 78z

LTiL

=λ 75

zLT

iL=λ

Table 1.1 LTλ for different grades of steel

where

L is the distance between points of restraint of the compression flange

iz is the radius of gyration of the section about the minor axis.

NOTES

Table 1.1 is derived from equation (1) taking C1 = 1,0, U = 0,9, V = 1,0 and wβ = 1,0.

Improved economy can be gained by increasing the complexity of the slenderness calculation. For beams designed as “simply supported”, there may be little gain, but for columns with large moments, the gain may be significant.

It is advisable to detail structures to avoid “destabilising” loading. This may be achieved by detailing so that the load and the beam flange are not free to move laterally. For example, where a floor acts as a horizontal diaphragm restraining the beam, the loading is not “destabilising”.

For further information, see also:

Economy from more complexity

Non-uniform bending moment distribution reduces LTλ by up to 40% where there is significant reversal of moment.

Section geometry reduces LTλ by up to 15%.

Lower yield strengths for thicker elements reduce LTλ by up to 5%.

Allowance for the effects of destabilising loads

Destabilising loads are rare but when they do exist the bending resistance is reduced. Destabilising loads need to be taken into account in design.

Background Theory Derivation of above simplified equations


reat

ed o

n F

riday

, Sep

tem

ber

02, 2

011

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent

http://www.access-steel.com/discovery/linklookup.aspx?id=EC012


2. Economy from more complexity A less conservative value of LTλ may be obtained by taking account of bending moment diagram, section geometry and lower yield strengths.

There is little economy to be gained for simply supported beams by use of 1

1C

, but in

columns with negative values of ψ (see Table 2.1) and large bending moments, the economy may be significant.

NOTE: For beams in “simple” construction (designed as Simple Supported beams), see EN1993-1-8 §5.1.1(2).

When the loading is not “destabilising”, LTλ is given by

wz1

w1

z

1LT

11 βλβλλλ UV

CUV

C==

(1)

where

C1 is a parameter dependent on the shape of the bending moment diagram. Values of

1

1C

for some bending moment diagrams are given in Table 2.1 and Table 2.2.

Values for other bending moment diagrams can be obtained from [SN003].

Conservatively, C1 = 1,0 (this value has been used in the simplified method above).

U is a parameter dependent on the section geometry and is given by:

w

zypl,

II

AgW

U =

In which g allows for the curvature of the beam if it has zero vertical deflection

before it is loaded and is given by ⎟⎟⎠

⎞⎜⎜⎝

⎛−=

y

z1IIg or, conservatively, g = 1,0

Conservatively, U = 0,9 (this value has been used in the simplified method above).

V is a parameter related to the slenderness. Where the loading is not “destabilising”, it may be taken as:

either, conservatively, = 1,0 for all sections symmetric about the major axis,

or as

42

f

z2011

1

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=

th

Vλ

for doubly symmetric hot rolled I and H sections


reat

ed o

n F

riday

, Sep

tem

ber

02, 2

011

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent

http://www.access-steel.com/discovery/linklookup.aspx?id=EC029

http://www.access-steel.com/discovery/linklookup.aspx?id=SN003


The exact definition of V, where the loading is not “destabilising”, is:

( )4

z

w

t

2

2z

2

w

1

II

IA

GEπk

kV

λ+⎟⎟

⎠

⎞⎜⎜⎝

⎛= If k = kw, then

( )4

z

w

t

2

2z1

1

II

IA

GEπ

Vλ

+

=

zz i

kL=λ ,in which

L is the distance between points of restraint to the compression flange

k is the effective length parameter and should be taken as 1,0 unless it can be demonstrated otherwise

ypl,

yw W

Wβ =

Wy is the modulus used to calculate Mb,Rd

For Class 1 and 2 sections Wy = Wpl,y

For Class 3 sections Wy = Wel,y

y1 f

Eπλ = in which fy is the yield strength appropriate to the thickness of the steel.


reat

ed o

n F

riday

, Sep

tem

ber

02, 2

011

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent


Table 2.1 Values of 1

1C

for end moment loading, to be used with k=1,0

1

1C

ψ

+1,00 1,00

+0,75 0,94

M ψ M

-1 ≤ ψ ≤ +1

+0,50 0,87

+0,25 0,81

0,00 0,75

-0,25 0,70

-0,50 0,66

-0,75 0,62

-1,00 0,63

Table 2.2 Values of 1

1C

for cases with transverse loading, to be used with k=1,0

Bending moment diagram 1

1C

Loading and support conditions

0,94

0,62

0.86

0,77


reat

ed o

n F

riday

, Sep

tem

ber

02, 2

011

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent


3. Allowance for the effect of destabilizing loads The effect of a ‘destabilising’ load may be taken into account by increasing the value of the non-dimensional slenderness.

3.1 Beams with destabilizing loads A beam with the load acting at a distance above the shear centre of the section is shown in Figure 3.1b. If both the load and the beam are free to move laterally, such a load is described as a “destabilising” load. The destabilising effect arises because when the beam buckles, deflecting laterally and twisting, the line of action of the load remains vertical but moves relative to the shear centre of the section. The load therefore applies an additional torque,

increasing the effect of lateral torsional buckling.

w

w

e

a) Load acting through b) Load acting at top flange shear centre (destabilising load)

Figure 3.1 An example of a destabilising load


reat

ed o

n F

riday

, Sep

tem

ber

02, 2

011

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent


3.2 Slenderness with destabilising loads Where the loading is “destabilising”, LTλ is given by

wz1

w1

z

1LT

11 βλUVDC

βλλUVD

Cλ ==

(2)

where

( ) ( )

25,0

w

z22

z

w

t

2

22

w

1

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

++⎟⎟⎠

⎞⎜⎜⎝

⎛

=

IIzC

II

IA

GEk

k

V

gz

πλ

For doubly symmetric hot rolled I and H sections, V may be taken conservatively as:

( )4w

z22

2

f2011

1

IIzC

th

V

gz +⎟⎟

⎠

⎞⎜⎜⎝

⎛+

=λ

C2 is a parameter dependent on the shape of the bending moment diagram. Values of C2 are given in SN003.

zg is the height of the “destabilising” load above the shear centre

5,0

w

zg2

21

1

⎟⎟⎠

⎞⎜⎜⎝

⎛−

=

II

zCV

D


reat

ed o

n F

riday

, Sep

tem

ber

02, 2

011

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent

http://www.access-steel.com/discovery/linklookup.aspx?id=SN003


Annex A Background Theory

The theoretical consistency between the simplified method and the explicit method using Mcr for calculating values of LTλ is demonstrated below.

The elastic critical buckling moment may be written:

( )( ) ( )

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−++⎟⎟

⎠

⎞⎜⎜⎝

⎛= g2

2g2

z2

t2

z

w2

w2

z2

1cr zCzCEIπ

GIkLII

kk

gkLEIπCM

where g is the correction factor for the increase in critical buckling moment caused by

increased curvature, which may be taken as ⎟⎟⎠

⎞⎜⎜⎝

⎛−=

y

z1IIg , or conservatively as g =1,0.

EN 1993-1-1 defines the “non-dimensional” slenderness ascr

yyLT

MfW

=λ

( )( ) ( )

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−++⎟⎟

⎠

⎞⎜⎜⎝

⎛=

g22

g2z

2t

2

z

w2

w2

2

1 zCzCEIπ

GIkLII

kk

gkLEIπC

fW

z

yy

( )

( ) ( )⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+⎟⎠⎞

⎜⎝⎛

+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛

=

g2w

z2g2

w

z2

z

22

wz

wy

2

2

1

111

zCIIzC

II

EAπGI

AI

kLkk

IIf

EπAI

kLA

gWC

t

z

y

( )( ) ( )

( )( )

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡++⎟⎟

⎠

⎞⎜⎜⎝

⎛=

w

zg2

2g2

w

zt22

z

22

wz

w

y

22

2

1

1111

IIzC

IIzC

AIII

EπG

ikL

kk

II

fEπi

kLA

gWC

w

zz

y

defining y

1 fEπ=λ and

zz i

kL=λ


reat

ed o

n F

riday

, Sep

tem

ber

02, 2

011

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent


( )( )

( ) ( )⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−++⎟⎟⎠

⎞⎜⎜⎝

⎛

=

w

zg2

w

z2g2

z

w

t

2

22

w

21

2z

y

1

111

IIzC

IIzC

II

IA

GEπλ

kk

λλ

II

AgW

C

z

w

z

defining ( ) ( )

w

z2g2

z

w

t

2

22

w

1

IIzC

II

IA

GEπλ

kk

V

z ++⎟⎟⎠

⎞⎜⎜⎝

⎛=

( )( )

⎟⎟⎠

⎞⎜⎜⎝

⎛−

=

w

zg22

21

2z

w

zypl,

pl.y

y

1LT

111

IIzC

Vλλ

II

AgW

WW

Cλ

defining ypl,

yw W

Wβ = and

w

zypl,

II

AgW

U =

( )( )

⎟⎟⎠

⎞⎜⎜⎝

⎛−

=

w

zg2

2

2

21

2z2

W1

LT

1

1

IIzCV

VλλUβ

Cλ

defining

⎟⎟⎠

⎞⎜⎜⎝

⎛−

=

w

zg2

21

1

IIzCV

D

( )( )

222

1

2z2

W1

LT1 DV

λλUβ

C=λ

W1

z

1LT

1 βλλUVD

Cλ =∴


reat

ed o

n F

riday

, Sep

tem

ber

02, 2

011

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent


V may be simplified as follows. Where k = kw and the load is applied through the shear centre of the section, V reduces to

( ) ( )4

z

w

t

2

2z

z

w

t

2

2z 1

1

1

1

II

IA

GEπλ

II

IA

GEπλ

V

+

=

+

=

For hot-rolled I-sections, 2

fz

w

t

220 ⎟⎟

⎠

⎞⎜⎜⎝

⎛≈

th

II

IA

GEπ

Therefore, for hot-rolled I-sections, and where the loads are not “destabilising”, V may be taken as:

42

f

z2011

1

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=

thλ

V


reat

ed o

n F

riday

, Sep

tem

ber

02, 2

011

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent


Quality Record RESOURCE TITLE NCCI: Determination of non-dimensional slenderness of I and H

sections

Reference(s)

ORIGINAL DOCUMENT

Name Company Date

Created by James Lim The Steel Construction Institute

Technical content checked by Charles King The Steel Construction Institute

Editorial content checked by D C Iles SCI 2/3/05

Technical content endorsed by the following STEEL Partners:

1. UK G W Owens SCI 1/3/05

2. France A Bureau CTICM 1/3/05

3. Sweden A Olsson SBI 1/3/05

4. Germany C Mueller RWTH 1/3/05

5. Spain J Chica Labein 1/3/05

Resource approved by Technical Coordinator

G W Owens SCI 21/04/06

TRANSLATED DOCUMENT

This Translation made and checked by:

Translated resource approved by:


reat

ed o

n F

riday

, Sep

tem

ber

02, 2

011

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent

SN002

Documents

Transcript of SN002