sn001a-en-eu_ncci- critical axial load for torsional and flexural torsional

NCCI: Critical axial load for torsional and flexural torsional buckling modes SN001a-EN-EU

NCCI: Critical axial load for torsional and flexural torsional buckling 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

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

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1. General For the following common cases, torsional and flexural torsional buckling will not give a lower mode than flexural buckling :

Doubly symmetric I and H sections (provided that both flanges are restrained at positions of 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 at each 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 Ncr.T for torsional buckling mode may be calculated from :

⎟⎟⎠

⎞⎜⎜⎝

⎛+= 2

2

2T

wt

oTcr,

1lEIGI

iN π (1)

with:

2o

2o

2z

2y

2o zyiii +++= (2)

where:

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

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

It is the torsion constant

Iw 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 Ncr.TF for the flexural-torsional buckling mode is the smallest root of the following cubic equation in N :

( )( )( ) ( ) ( ) 022222 =−−−−−−− ycr,ozcr,oTcr,zcr,ycr, NNzNNNyNNNNNNNio (3)

where

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

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

The equation may also be written as follows :

( ) ( ) 22232 NNNNyNzN

iN

iii

⎥⎦

⎤⎢⎣

⎡++−++⎟

⎟⎠

⎞⎜⎜⎝

⎛ +Tcr,zcr,ycr,ozcr,oycr,2

oo

2z

2y 1

( ) 0Tcr,zcr,ycr,ycr,Tcr,Tcr,zcr,zcr,ycr, =−+++ NNNNNNNNNN

(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

2z

2y

Tcr,ycr,2

Tcr,ycr,Tcr,ycr,2z

2y

2o

TFcr, iii

NN4NNNNii2

iN (5)

When the cross-section is symmetric about the z-z-axis, Ncr.y should be replaced by Ncr.z in the above expression.


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yo

G S 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 buckling modes

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 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:


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sn001a-en-eu_ncci- critical axial load for torsional and flexural torsional

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