SN001a (NCCI - Critical Axial Load for Torsional and Flexural Torsional Buckling Modes)

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NCCI: Critical axial load for torsional and flexural torsional buckling modes

SN001a-EN-EU

NCCI: Crit ical axial load for torsional and flexural torsionalbuckling modes

This NCCI gives the expressions for the critical axial load for the torsional buckling mode and the flexural-torsional buckling mode.

Contents

1. General 2

2. Torsional buckling 2

3. Flexural-torsional buckling 3

4. References 4

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1. General

For the following common cases, torsional and flexural torsional buckling will not give alower mode than flexural buckling :

Doubly symmetric I and H sections (provided that both flanges are restrained at positionsof lateral restraint)

Hollow sections

However, in some particular cases, the torsional buckling mode or the flexural-torsional

buckling mode of an axially loaded member may correspond to a critical load lower than the

one corresponding to the flexural buckling mode, especially for open sections. This document

gives rules to determine the critical load for such cases.

This document deals with uniform members only and for which the conditions of restraint ateach end of the member are at least :

restrained against lateral movement restrained against rotation about the longitudinal axis

2. Torsional buckling

The critical axial load N cr.T for torsional buckling mode may be calculated from :

⎟⎟ ⎠

⎞⎜⎜⎝

⎛ +=2

2

2

T

wt

o

Tcr,

1

l

EI GI

i N

π

(1)

with:

2

o

2

o

2

z

2

y

2

o z yiii +++= (2)

where:

E is the Young modulus (E = 210000 N/mm2)

G is the shear modulus (G = 80770 N/mm2)

I t is the torsion constant

I w is the warping constant

lT is the buckling length regarding the torsional buckling mode. In general, lT should be

taken as the system length, except when a special device prevents the warping at the

ends of the member.

yo and zo are the coordinates of the shear centre with respect to the centroid (see Figure 2.1).

For a doubly symmetric cross-section, the shear centre coincides with the centroid;

then yo = 0 and zo = 0

iy is the radius of gyration of the cross-section about the strong axis

iz is the radius of gyration of the cross-section about the weak axis

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G

S

z

y

yo zo

Figure 2.1 Coordinates of the shear centre S with respect to the centroid G

3. Flexural-torsional buckling

The flexural-torsional buckling mode should be considered only when the shear centre does

not coincide with the centroid.

The critical axial load N cr.TF for the flexural-torsional buckling mode is the smallest root of the

following cubic equation in N :

( )( )( ) ( ) ( ) 022222 =−−−−−−− ycr,ozcr,oTcr,zcr,ycr, N N z N N N y N N N N N N N io (3)

where

N cr,y and N cr,z are the critical axial loads for flexural buckling about yy and zz axes

respectively

N cr.T is the critical axial load for torsional buckling mode, see § 2.

The equation may also be written as follows :

( ) ( ) 22232

N N N N y N z N i

N i

ii⎥⎦

⎤⎢⎣

⎡++−++

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ +Tcr,zcr,ycr,ozcr,oycr,2

oo

2

z

2

y 1

0Tcr,zcr,ycr,ycr,Tcr,Tcr,zcr,zcr,ycr, =−+++ N N N N N N N N N N

(4)

When the cross-section is symmetric about the y-y-axis (see Figure 3.1), the critical axial load

may be obtained from :

( ) ( )

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ +−+−+

+=

2

o

2

z

2

y

Tcr,ycr,

2

Tcr,ycr,Tcr,ycr,2

z

2

y

2

oTFcr,

i

ii N N 4 N N N N

ii2

i N (5)

When the cross-section is symmetric about the z-z-axis, N cr.y should be replaced by N cr.z in theabove expression.

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yo

GS y y

z

z

Figure 3.1

Cross-section symmetric about y-y-axis

4. References

1 Timoshenko, S.P. and Gere, J.M.

Theory of elastic stability. 2nd

Edition. Mc Graw-Hill. 1961.

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Quality Record

RESOURCE TITLE NCCI: Critical axial load for torsional and flexural torsional bucklingmodes

Reference(s)

ORIGINAL DOCUMENT

Name Company Date

Created by A. BUREAU CTICM 02/02/05

Technical content checked by Y. GALEA CTICM 02/02/05

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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SN001a (NCCI - Critical Axial Load for Torsional and Flexural Torsional Buckling Modes)