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NCCI: Elastic critical moment of cantilevers SN006a-EN-EU

NCCI: Elastic critical moment of cantilevers

This NCCI gives a simple method for the calculation of the critical moment of a cantilever beam.

Contents

1. Scope 2

2. Calculation of Mcr 2

3. Tables for C factor 3

4. Hot-rolled I profiles : design aids for Mcr,0 and wt 10 5. Application example 11

6. Information about LTBeam freeware for solving general LTB problems 12

7. References 12

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NCCI: Elastic critical moment of cantilevers SN006a-EN-EU

1. Scope This NCCI provides information for dealing with cantilever beams satisfying the following conditions:

The beam has a uniform and doubly symmetrical cross-section The beam is unrestrained except at support The beam is torsionally restrained at support Loads are applied normal to the beam to cause bending about the major axis. The line of

action passes through the shear centre (applied at, above or below the shear centre)

Favourable effect of the in-plane deflection not accounted for (beam assumed straight in the plane of loading at buckling)

Normal force negligible Note : Solutions for cases where torsion or lateral deflection is prevented at the end may be

found in specific literature (detailed information are available for instance in the Swedish Manual on Buckling |4|). Those cases, and numerous other cases outside the scope of this NCCI, may also be treated with specific software such as the one presented in Section 6.

2. Calculation of Mcr The critical value Mcr of the maximum moment M at support of the cantilever beam is given herein by :

0,crcr MCM = (1)where

Mcr,0 is the elastic critical moment for lateral torsional buckling of a simply supported beam (with fork support conditions at ends) under uniform moment, ignoring the warping stiffness of the beam.

C is a global factor that takes account of the influences of :

the moment distribution i.e. the shape of bending moment diagram, the warping rigidity of the beam, the introduction of loads above or below the shear centre, which coincides here with

the centre of gravity, and

the restraint conditions at support (warping)

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NCCI: Elastic critical moment of cantilevers SN006a-EN-EU

Mcr,0 is defined by:

tz0,cr IGIELM = (2)

in which

L is the length of the cantilever beam E is the Young modulus G is the shear modulus It is the torsion constant Iz is the second moment of area about the minor axis

3. Tables for C factor Values of C factor have been calculated for many cases using a specific software for lateral torsional buckling of beams named LTBeam (see Section 6), and tables given hereafter have been built for a direct use by the designer.

Two cases of restraint condition for warping have been considered at support :

warping totally free warping fully restrained In practice, warping at support is generally neither totally free nor fully restrained but rather in between depending on the detailing conception at support. Figure 3.1 and Figure 3.2 show cases for which warping may be considered as free or restrained. Where there is doubt about the restraint, it is recommended that warping be taken as free.

3

1 2

1 Column 2 Thin flanges 3 Cantilever beam (buckling)

Figure 3.1 Case of free warping conditions at support

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NCCI: Elastic critical moment of cantilevers SN006a-EN-EU

1

3 2

1

4

2

a) b)

1 Column 2 Cantilever beam (buckling) 3 Stiffening plate (on both sides) 4 Stiffeners (on both sides)

Figure 3.2 Case of restrained warping conditions at support

For a given loading case and a given support condition for warping, the C factor depends on two parameters :

one parameter wt for accounting for the warping rigidity of the beam

t

ww

1IGIE

Lt= (3)

where

Iw is the warping constant of the cross-section

one parameter for characterising the destabilizing or stabilizing effect of loads applied above or below the shear centre

2/s

a

hz= ( is given a sign, as explained below) (4)

where :

za is the distance of application of the load from the shear centre

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NCCI: Elastic critical moment of cantilevers SN006a-EN-EU

hs is defined by convention as:

z

ws 2 I

Ih = (5)

whatever the section shape is.

For doubly-symmetrical I cross-sections, hs may be taken as the distance between shear centres of the flanges :

fs thh =h is the total height of the section

tf is the flange thickness

Sign of The sign of is illustrated in Figure 3.3. For a load applied at the shear centre S of the cross-section, za = 0 and = 0. Otherwise, the sign of depends on the effect of the load when considering that a torsion angle due to buckling develops:

any load acting towards the shear centre S generates a twisting moment that amplifies the torsion angle : the load has a destabilising effect (case a), and is positive by convention,

on the contrary, any load acting outwards the shear centre S generates a twisting moment that acts against the torsion : the load has a stabilising effect (case c), and is negative by convention.

za

S hs

F

za=0 S hs

F

za

S hs

F

a) > 0 b) = 0 c) < 0 Destabilizing effect Stabilizing effect

Figure 3.3 Case of restrained warping conditions at support

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NCCI: Elastic critical moment of cantilevers SN006a-EN-EU

Values for the C factor are given for the following three simple loading cases:

uniformly distributed load q along the beam point load F applied at the free end of the beam external moment CM applied at the free end of the beam Values of parameters wt and are taken within the following ranges (most of practical cases):

0 wt 1 -2 3

C values for intermediate values of these parameters may be obtained by interpolation.

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NCCI: Elastic critical moment of cantilevers SN006a-EN-EU

3.2 C-factor for simple loading cases Table 3.1 C values for a uniformly distributed load q

q

stabilizing effect * destabilizing effect ** wt -2,00 -1,50 -1,00 -0,50 0,00 0,25 0,50 0,75 1,00 1,25 1,50 2,00 2,50 3,00

0 2,04 2,04 2,04 2,04 2,04 2,04 2,04 2,04 2,04 2,04 2,04 2,04 2,04 2,04

0,05 2,42 2,34 2,25 2,16 2,06 2,02 1,97 1,92 1,87 1,82 1,77 1,67 1,58 1,49

0,1 2,87 2,71 2,53 2,34 2,13 2,03 1,92 1,82 1,71 1,61 1,52 1,35 1,20 1,07