SN002a (NCCi - Determination of Non-dimensional Slenderness of I and H Sections)

8/18/2019 SN002a (NCCi - Determination of Non-dimensional Slenderness of I and H Sections)

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NCCI: Determination of non-dimensional slenderness of I and H sectionsSN002a-EN-EU

NCCI: Determination of non-dimensional slenderness of Iand H sections

This NCCI presents a method for determining the non-dimensional slenderness withoutexplicit 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

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1. Simplified methodFor straight segments of hot-rolled doubly symmetric I and H sections with lateral restraintsto the compression flange at both ends of the segment considered and with no destabilizingloads, the value of LTλ required by EN1993-1-1 §6.3.2.2 or §6.3.2.3 may be conservativelytaken from Table 1.1.

S 235 S 275 S 355 S420 S 460

104z

LTi L=λ

96z

LTi L=λ

85z

LTi L=λ

78z

LTi L=λ

75z

LTi L=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 C 1 = 1,0, U = 0,9, V = 1,0 and w β = 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 withlarge moments, the gain may be significant.

It is advisable to detail structures to avoid “destabilising” loading. This may be achieved bydetailing 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 morecomplexity

Non-uniform bending moment distribution reduces LTλ byup 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 byup to 5%.

Allowance for the effects ofdestabilising 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

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2. Economy from more complexity

A less conservative value of LTλ may be obtained by taking account of bending momentdiagram, section geometry and lower yield strengths.

There is little economy to be gained for simply supported beams by use of1

1

C , but in

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

NOTE: For beams in “simple” construction (designed as Simple Supported beams), seeEN1993-1-8 §5.1.1 (2).

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

wz1

w1

z

1LT

11 β λ β

λ λ

λ UV C

UV C

== (1)

where

C 1 is a parameter dependent on the shape of the bending moment diagram. Values of

1

1

C 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, C 1 = 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

zy pl,

I I

A

g W 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

z1 I I

g 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”, itmay be taken as:

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

or as

42

f

z201

1

1

⎟⎟

⎠ ⎞

⎜⎜

⎝ ⎛ +

=

t h

V λ

for doubly symmetric hot rolled I and H sections

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

( )4

z

w

t

2

2z

2

w

1

I I

I A

G E π k

k V

λ +⎟⎟ ⎠ ⎞

⎜⎜

⎝ ⎛

= If k = k w, then( )

4

z

w

t

2

2z1

1

I I

I A

G E π

V λ +

=

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 bedemonstrated otherwise

y pl,

yw W

W β =

W y is the modulus used to calculate M b,Rd

For Class 1 and 2 sections W y = W pl,y

For Class 3 sections W y = W el,y

y1

f

E π λ = in which f y is the yield strength appropriate to the thickness of the steel.

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Table 2.1

Values of 1

1

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

1

1

C

ψ

+1,00 1,00

+0,75 0,94

M M

-1 +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 of1

1

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

Bending moment diagram 1

1

C Loading and support

conditions

0,94

0,62

0.86

0,77

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3. Allowance for the effect of destabil izing loadsThe effect of a ‘destabilising’ load may be taken into account by increasing the value of thenon-dimensional slenderness.

3.1 Beams with destabilizing loadsA beam with the load acting at a distance above the shear centre of the section is shown inFigure 3.1b. If both the load and the beam are free to move laterally, such a load is describedas 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 movesrelative 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 flangeshear centre (destabilising load)

Figure 3.1 An example of a destabilising load

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3.2 Slenderness with destabilising loads

Where the loading is “destabilising”, LTλ is given by

wz1

w1

z

1LT

11 β λUVD

C β

λ λ

UVDC

λ == (2)

where

( )( )

25,0

w

z22

z

w

t

2

22

w

1

⎥⎥

⎥⎥

⎦

⎤

⎢⎢

⎢⎢

⎣

⎡

++⎟⎟ ⎠ ⎞

⎜⎜⎝ ⎛

=

I I

z C

I I

I A

G E k

k

V

g z

π

λ

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

( )4w

z22

2

f 201

1

1

I I

z C t h

V

g z +⎟⎟ ⎠

⎞⎜⎜

⎝ ⎛ +

=λ

C 2 is a parameter dependent on the shape of the bending moment diagram. Values of C 2 are given in SN003 .

z g is the height of the “destabilising” load above the shear centre

5,0

w

zg2

21

1

⎟⎟

⎠

⎞⎜⎜

⎝

⎛ −

=

I I

z C V

D

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Annex A Background Theory

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

The elastic critical buckling moment may be written:

( )( ) ( )

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ −++⎟⎟ ⎠

⎞⎜⎜

⎝ ⎛ = g2

2g2

z2

t2

z

w

2

w2

z2

1cr z C z C EI π GI kL

I I

k k

g kL

EI π C M

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

increased curvature, which may be taken as⎟⎟

⎠

⎞⎜⎜

⎝

⎛ −=

y

z1 I I

g , or conservatively as g =1,0.

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

yyLT

M

f W =λ

( )( ) ( )

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ −++⎟⎟ ⎠

⎞⎜⎜

⎝ ⎛

=

g22

g2z

2t

2

z

w2

w2

2

1 z C z C EI π

GI kL I I

k k

g kL

EI π C

f W

z

y y

( )

( ) ( )⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+⎟

⎠

⎞⎜

⎝

⎛ +⎟⎟ ⎠

⎞⎜⎜

⎝ ⎛

⎟ ⎠ ⎞

⎜⎝ ⎛

=

g2w

z2g2

w

z2

z

22

wz

wy

2

2

1

111

z C I I

z C I I

E Aπ

GI

A

I kL

k k

I I f

E π A I kL

A

g W

C

t

z

y

( )( ) ( )

( )( )

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ −⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

++⎟⎟ ⎠ ⎞

⎜⎜

⎝ ⎛

=

w

zg2

2g2

w

zt22

z

22

wz

w

y

22

2

1

1111

I I

z C I I

z C AI

I I

E π

G

i

kLk k

I I

f E π i

kL A

g W

C

w

z z

y

definingy

1 f E

π =λ andz

z ikL=λ

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( )( )

( ) ( )⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−++⎟⎟ ⎠ ⎞

⎜⎜

⎝ ⎛

=

w

zg2

w

z2g2

z

w

t

2

22

w

21

2z

y

1

111

I I

z C I I

z C

I I

I A

G E π

λk k

λ λ

I I

A

g W

C

z

w

z

defining

( ) ( )w

z2g2

z

w

t

22

2

w

1

I I z C

I I

I A

G E π

λk k

V

z ++⎟⎟ ⎠ ⎞

⎜⎜

⎝ ⎛

=

( )( )

⎟⎟

⎠

⎞⎜⎜

⎝

⎛ −

=

w

zg22

21

2z

w

zy pl,

pl.y

y

1LT

1

11

I I

z C V

λ

λ I I

A

g W

W

W

C λ

definingy pl,

yw W

W β = andw

zy pl, I I

A g W U =

( )( )

⎟⎟

⎠

⎞⎜⎜

⎝

⎛ −

=

w

zg2

2

2

21

2z2

W1

LT

1

1

I I

z C V

V

λ

λU β

C λ

defining

⎟⎟ ⎠

⎞⎜⎜⎝

⎛ −

=

wzg221

1

I I

z C V

D

( )( )

222

1

2z2

W1

LT1

DV λ

λU β

C =λ

W1

z

1LT

1 β

λ λ

UVDC

λ =∴

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

( ) ( )4

z

w

t

2

2z

z

w

t

2

2z 1

1

1

1

I I

I A

G E π

λ

I I

I A

G E π

λV

+=

+=

For hot-rolled I-sections,2

f z

w

t

220 ⎟⎟ ⎠

⎞⎜⎜

⎝ ⎛ ≈

t h

I I

I A

G E π

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

4

2

f

z201

1

1

⎟⎟ ⎠ ⎞

⎜⎜⎝ ⎛

+

=

t h λ

V

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Quality Record

RESOURCE TITLE NCCI: Determination of non-dimensional slenderness of I and Hsections

Reference(s)

ORIGINAL DOCUMENT

Name Company Date

Created by James Lim The Steel ConstructionInstitute

Technical cont ent checked by Charles King The Steel ConstructionInstitute

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

Technical content endorsed by thefollowing 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 TechnicalCoordinator

G W Owens SCI 21/04/06

TRANSLATED DOCUMENT

This Translation made and checked by:

Translated resource approved by:

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SN002a (NCCi - Determination of Non-dimensional Slenderness of I and H Sections)