Table Chair

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Table 16.6

Types of beam not susceptible to lateral–torsional bucklingloading produces bending about the minor axisbeam provided with clos

Fig.16.4

Lateral–torsional buckling

The central feature in the above process is the determination of a measure of thebeam’s lateral–torsional buckling strength (p

b

) in terms of a parameter (

l

LT

) whichrepresents those factors which control this strength.Modications to the basicprocess permit the method to be used for une

p

b

and

l

LT

of BS 5950:Part 1 (and between

s

li

/

s

yc

and

l

LT

÷

(

s

yc

/

355

) in BS 5400:Part 3) assumes the beam between lateral restraints to besubject to uniform moment.Other patterns,such as a linear

n

,the value of which has been selected so as to ensure that theresulting value of

p

bcorrectly reects the enhanced strength due to the non-uniformmoment loading.An alternative approach consists of basing

l

LT

on the geometricaland support conditions alone but making allowance for the benecial effects of non-uniform moment by compari

M

b

with a suitably adjustedvalue of design moment

.

is taken as a factor





;then aseparate check that the capacity of the beam cross-section

M

c

is at least equal to

M

max

must also be made.In cases where is taken asM

max

,then the bucklingcheck will be more severe than (or in the ease of a stocky beam for which

M

b

=

M

c

,identical to) the cross-section capacity check.Allowance for non-uniform moment loading on cantilevers is normally treatedsomew

M M M M M

444

Beams

an end moment such as horizontal wind load acting on a façade,should be regardedas an ordinary beam since it does not have th

M

E

.Values of

M

E

may conveniently be obtainedfrom summaries of research data.

6

For example,BS 5950:Part 1 permits

l

LT

to becalculated from

-16.3

As an example of the use of this approach Fig.16.6 shows how signicantly higherload-carrying capacities may be obtained for a c

16.3.7Fully restrained beams

The design of beams is considerably simplied if lateral–torsional buckling effectsdo not have to be considered explicitly – a situati

Mb

may be taken as itsmoment capacity

M

c

and,in the absence of any reductions in

M

c

due to local buck-

l p



LTypE

#NAME?

( )

÷

( )

2

EpMM//

Basic design

445

Fig.16.5

Design modications using

m

-factor or

n

-factor methods

ling,high shear or torsion,it should be designed for its full in-plane bending strength.Certain of the conditions corresponding to the c

l

below which buckling will not affect

M

b

of Table 38 of BS 5950:Part 1,are sufciently high (

l

=

340,225 and 170 for

D/B

ratios of 2,3 and 4,and

p

y

=

275N/mm

2

) that only in very rare cases will lateral–torsional buckling be a design consideration.Situations in which the form of construction e

446

Beams

Fig.16.6Lateral–torsional buckling of a tip-loaded cantilever

that the restraints will effectively prevent movement at the braced cross-sections,thereby acting as if they were rigid supports.In pr

Basic design

447

Fig.16.7

Effect of type of cross-section on theoretical elastic critical moment



16.4Lateral bracing

For design to BS 5950:Part 1,unless the engineer is prepared to supplement the coderules with some degree of working from rst

7,8

Where properly designed restraint systems areused the limits on

l

LTfor

M

b

=

M

c

(or more correctly

p

b

=

p

y

)are given in Table 16.7.For beams in plastically-designed structures it is vital that premature failure dueto plastic lateral–torsional

L

/

r

y

to ensure satisfactory behaviour;it is not necessarilycompatible with the elastic design rules of section 4 of the code since accepta

M

p

.The expression of clause 5.3.3 of BS 5950:Part 1,

-16.4

makes no allowance for either of two potentially benecial effects:(1)moment gradient(2)restraint against lateral deection provided

Lr fpx

mycy

£+

( ) ( )

[ ]

3813027536

2212

///448

Beams

Table 16.7

Maximum values of

l

LT

forwhich

p

b



=

p

y

for rolledsections

p

y

(N/mm2

) Value of

l

LT

up to which

p

b

=

p

y

245 37265 35275 34325 32340 31365 30415 28430 27450 26

of Brown,

9

the basis of which is the original work on plastic instability of Horne.

10

This is covered explicitly in clause 5.3.3.A method of allowing for both effects whenthe beam segment being checked is either elas

L

m

with an enhanced value

L

s

obtained from clause 5.3.4 of BS 5950:Part 1.In both cases the presence of a change in cross-section,for example,as producedby

16.5Bracing action in bridges – U-frame design

The main longitudinal beams in several forms of bridge construction will,by virtueof the structural arrangement employed,receive a

U-frame

action.Figure 16.8 illustrates the original concept based on the half-through girderform of construction.(See Chapter 4 for a discuss

Bracing action in bridges



ly spaced or continuous lateral restraintclosed section

ual anged sections including tees,fabricated Is for which the section properties must be calculated,sections contain-ing slender pl

moment gradient reduc-ing from a maximum at one end or the parabolic distribution produced by a uniformload,are generally less

ng the resulting value of



,whether on the slenderness axis of the

t



at differently.For example,the set of effective length factors given in Table14 of Reference 1 includes allowances for the variation f

benet of non-uniform momentloading.For more complex arrangements that cannot reasonably be approximated by oneof the sta

ntilever with a tip load applied toits bottom ange,a case not specically covered by BS 5950:Part 1.

on which will occur if one or moreof the conditions of Table 16.6 are met.In these cases the beam’s buckling resistance moment



ase where a beam may be regardedas ‘fully restrained’ are virtually self-evident but others require either judgement orcalculation.L

ployed automatically providessome degree of lateral restraint or for which a bracing system is to be used toenhance a beam’s str

ctice,bracing will possess a nitestiffness.A more fundamental discussion of the topic,which explains the exactnature of bracing sti



rinciples,only restraints capable of acting as rigid supports are acceptable.Despite the absence of a specic stiffnessrequirement,

uckling does not impair the formation of the full plasticcollapse mechanism and the attainment of the plastic collapse load.Clause

lebehaviour can include the provision of adequate rotation capacity at momentsslightly below

by secondary structural membersattached to one ange as by the purlins on the top ange of a portal framerafter.The rst effect



ic or partially plastic is given inAppendix G of BS 5950:Part 1;alternatively the effect of intermittent tension angerestraint alone m

the type of haunch usually used in portal frame construction,may be allowedfor.When the restraint is such that lateral deection of t

signicant measure of restraintagainst lateral–torsional buckling by a device commonly referred to as

ion of different bridge types.) Ina simply-supported span,the top (compression) anges of the main girders,althoughlaterally unbrac



ate elements,members with properties that vary along their length,closed sections and ats.Various techniques for allowing for the

evere in terms of their effect on lateral stability;a givenbeam is likely to be able to withstand a larger peak moment before becomin





rom the arrangement usedas the basis for the s trength–slenderness relationship due to both the lateral supportconditions and the f

dard cases covered by correction factors,codes normally permit the directuse of the elastic critical moment



ateral–torsional buckling cannot occur in beams loaded in their weaker princi-pal plane;under the action of increasing load they will

ngth require careful consideration.The fundamental require-ment of any form of restraint if it is to be capable of increasing the stre

fness and bracing strength,may be found in References 7 and8.Noticeably absent from the code clauses is a quantitative denition



dherence to the strength requirement together with an awareness thatadequate stiffness is also necessary,avoiding obviously very

.3.3provides a basic limit on

ay be included in Equation (16.4) by adding the correction term



y be allowed for by replacing

he beam’s compressionange is prevented at intervals,then Equation (16.4) applies between the pointsof effective lateral restraint.

d in the sense that no bracing may be attached directly to them,cannot buckle freely in the manner of Fig.16.4 since their lower a



form of the appliedloading are also possible;some care is required in their use.The relationship between

lat-erally unstable.One means of allowing for this in design is to adjust the beam’s slen-derness by a factor





rm of the applied loading.When a cantilever is subdivided byone or more intermediate lateral restraints positioned between its root



collapse simply by plasticaction and excessive in-plane deformation.Much the same is true for rectangularbox sections even when

gth of themain member is that it limits the buckling type deformations.An appreciation of exactly how the main member would buc

of ‘adequatestiffness’,although it has subsequently been suggested that a bracing system that is25 times stiffer than the braced b



exible yet strongarrangements,should lead to satisfactory designs.Doubtful cases will merit exami-nation in a more fundamental



discussion of the application of this and otherapproaches for checking the stability of both rafters and columns in portal framesde

nges arerestrained by the deck.Buckling must therefore involve some distortion of thegirder web into the mode given in Fig.16.8 (a







and tip,thensegments other than the tip segment should be treated as ordinary beam segmentswhen assessing lateral–torsional b



bent about their strong axis.Figure 16.7,which is based onelastic critical load theory analogous to the Euler buckling of struts,shows

le if unbraced is a prerequisite for theprovision of an effective system.Since lateral–torsional buckling involves bothlateral deectio

am would meet this requirement.Examinationof Reference 7 shows that while such a check does cover the majority of cases,it iss t



ay.



signed according to the principles of either elastic or plastic theory is given insection 18.7.

suming that the end frames preventlateral movement of the top ange).An approximate way of dealing with this is to regard each lo







ckling strength.Similarly a cantilever subject to



thattypical RHS beams will be of the order of ten times more stable than UB or UCsections of the same area.The limits on

and twist,as shown in Fig.16.4,either or both deformations maybe addressed.Clauses 4.3.2 and 4.3.3 of BS 5950:Part 1 set out th

ill possible to provide arrangements in which even much stiffer bracing cannotsupply full restraint.





ngitudinal girder as a truss in which the tension chord is fully









e principles gov-erning the action of bracing designed to provide either lateral restraint or torsionalrestraint.In common with most ap













proaches to bracing design these clauses assume



Table 16.6


Fig.16.4



b


l

LT


p

b

and

l

LT


s

li

/

s

yc

and

l

LT

÷

(

s

yc

/

355


n


p


l

LT


M

b


.




m

times the maximum momentwithin the beam

M

max

;

m

=1.0 for uniform moment and

m

<

1.0 for non-uniformmoment.Provided that suitably chosen values of

m

and

n

are used,both methodscan be made to yield identical results;the difference arises simply in the way inwhich the correction is made

p

b

versus

l

LT

relationship for the

n

-factor method or on the strength axis for the

m

-factor method.Figure 16.5 illustrates both concepts,although for the purpose of the gure the

m

-factor method has been shown as an enhancement of

p

b

by 1/

m

rather than a reduc-tion in the requirement of checking

M

b

against

=

mM

max.BS 5950:Part 1 uses the

m

-factor method for all cases,while BS 5400:Part 3 includes only the

n

-factormethod.When the

m

-factor method is used the buckling check is conducted in terms of a moment less than the maximum moment in the beam segme

M

max




M

c


M

max


max


M

b

=

M

c


M M M M M

444

Beams


M

E

.Values of

M

E


6


l

LT


-16.3




Mb


M

c


M

c

due to local buck-

l p



LTypE

#NAME?

( )

÷

( )

2

EpMM//

Basic design

445

Fig.16.5


m

-factor or

n

-factor methods


l


M

b


l

=

340,225 and 170 for

D/B


p

y

=

275N/mm

2


446

Beams



Basic design

447

Fig.16.7




16.4Lateral bracing


7,8


l

LTfor

M

b

=

M

c

(or more correctly

p

b

=

p

y


L

/

r

y


M

p


-16.4


Lr fpx

mycy

£+

( ) ( )

[ ]

3813027536

2212

///448

Beams

Table 16.7

Maximum values of

l

LT

forwhich

p

b



=

p

y

for rolledsections

p

y

(N/mm2

) Value of

l

LT

up to which

p

b

=

p

y

245 37265 35275 34325 32340 31365 30415 28430 27450 26

of Brown,

9


10


L

m


L

s




U-frame












t

















































































ay.














































Table 16.6


Fig.16.4



b


l

LT


p

b

and

l

LT


s

li

/

s

yc

and

l

LT

÷

(

s

yc

/

355


n


p


l

LT


M

b


.




m


M

max

;

m


m

<


m

and

n


p

b

versus

l

LT


n


m


m


p

b

by 1/

m


M

b

against

=

mM


m


n


m


M

max




M

c


M

max


max


M

b

=

M

c


M M M M M

444

Beams


M

E

.Values of

M

E


6


l

LT


-16.3




Mb


M

c


M

c

due to local buck-

l p



LTypE

#NAME?

( )

÷

( )

2

EpMM//

Basic design

445

Fig.16.5


m

-factor or

n

-factor methods


l


M

b


l

=

340,225 and 170 for

D/B


p

y

=

275N/mm

2


446

Beams



Basic design

447

Fig.16.7




16.4Lateral bracing


7,8


l

LTfor

M

b

=

M

c

(or more correctly

p

b

=

p

y


L

/

r

y


M

p


-16.4


Lr fpx

mycy

£+

( ) ( )

[ ]

3813027536

2212

///448

Beams

Table 16.7

Maximum values of

l

LT

forwhich

p

b












t

















































































ay.














































Table 16.6


Fig.16.4



b


l

LT


p

b

and

l

LT


s

li

/

s

yc

and

l

LT

÷

(

s

yc

/

355


n


p


l

LT


M

b


.




m


M

max

;

m


m

<


m

and

n


p

b

versus

l

LT


n


m


m


p

b

by 1/

m


M

b

against

=

mM


m


n


m


M

max




M

c


M

max


max


M

b

=

M

c


M M M M M

444

Beams


M

E

.Values of

M

E


6


l

LT


-16.3




Mb


M

c


M

c

due to local buck-

l p



LTypE

#NAME?

( )

÷

( )

2

EpMM//

Basic design

445

Fig.16.5


m

-factor or

n

-factor methods


l


M

b


l

=

340,225 and 170 for

D/B


p

y

=

275N/mm

2


446

Beams



Basic design

447

Fig.16.7




16.4Lateral bracing


7,8


l

LTfor

M

b

=

M

c

(or more correctly

p

b

=

p

y


L

/

r

y


M

p


-16.4


Lr fpx

mycy

£+

( ) ( )

[ ]

3813027536

2212

///448

Beams

Table 16.7

Maximum values of

l

LT

forwhich

p

b



=

p

y

for rolledsections

p

y

(N/mm2

) Value of

l

LT

up to which

p

b

=

p

y

245 37265 35275 34325 32340 31365 30415 28430 27450 26

of Brown,

9


10


L

m


L

s




U-frame












t
















































































ay.













































Table 16.6


Fig.16.4



b


l

LT


p

b

and

l

LT


s

li

/

s

yc

and

l

LT

÷

(

s

yc

/

355


n


p


l

LT


M

b


.




m


M

max

;

m


m

<


m

and

n


p

b

versus

l

LT


n


m


m


p

b

by 1/

m


M

b

against

=

mM


m


n


m


M

max




M

c


M

max


max


M

b

=

M

c


M M M M M

444

Beams


M

E

.Values of

M

E


6


l

LT


-16.3




Mb


M

c


M

c

due to local buck-

l p



LTypE

#NAME?

( )

÷

( )

2

EpMM//

Basic design

445

Fig.16.5


m

-factor or

n

-factor methods


l


M

b


l

=

340,225 and 170 for

D/B


p

y

=

275N/mm

2


446

Beams



Basic design

447

Fig.16.7




16.4Lateral bracing


7,8


l

LTfor

M

b

=

M

c

(or more correctly

p

b

=

p

y


L

/

r

y


M

p


-16.4


Lr fpx

mycy

£+

( ) ( )

[ ]

3813027536

2212

///448

Beams

Table 16.7

Maximum values of

l

LT

forwhich

p

b



=

p

y

for rolledsections

p

y

(N/mm2

) Value of

l

LT

up to which

p

b

=

p

y

245 37265 35275 34325 32340 31365 30415 28430 27450 26

of Brown,

9


10


L

m


L

s




U-frame












t
















































































ay.













































Table 16.6


Fig.16.4



b


l

LT


p

b

and

l

LT


s

li

/

s

yc

and

l

LT

÷

(

s

yc

/

355


n


p


l

LT


M

b


.




m


M

max

;

m


m

<


m

and

n


p

b

versus

l

LT


n


m


m


p

b

by 1/

m


M

b

against

=

mM


m


n


m


M

max




M

c


M

max


max


M

b

=

M

c


M M M M M

444

Beams


M

E

.Values of

M

E


6


l

LT


-16.3




Mb


M

c


M

c

due to local buck-

l p



LTypE

#NAME?

( )

÷

( )

2

EpMM//

Basic design

445

Fig.16.5


m

-factor or

n

-factor methods


l


M

b


l

=

340,225 and 170 for

D/B


p

y

=

275N/mm

2


446

Beams



Basic design

447

Fig.16.7




16.4Lateral bracing


7,8


l

LTfor

M

b

=

M

c

(or more correctly

p

b

=

p

y


L

/

r

y


M

p


-16.4


Lr fpx

mycy

£+

( ) ( )

[ ]

3813027536

2212

///448

Beams

Table 16.7

Maximum values of

l

LT

forwhich

p

b



=

p

y

for rolledsections

p

y

(N/mm2

) Value of

l

LT

up to which

p

b

=

p

y

245 37265 35275 34325 32340 31365 30415 28430 27450 26

of Brown,

9


10


L

m


L

s




U-frame












t
















































































ay.














































Table 16.6


Fig.16.4



b


l

LT


p

b

and

l

LT


s

li

/

s

yc

and

l

LT

÷

(

s

yc

/

355


n


p


l

LT


M

b


.




m


M

max

;

m


m

<


m

and

n


p

b

versus

l

LT


n


m


m


p

b

by 1/

m


M

b

against

=

mM


m


n


m


M

max




M

c


M

max


max


M

b

=

M

c


M M M M M

444

Beams


M

E

.Values of

M

E


6


l

LT


-16.3




Mb


M

c


M

c

due to local buck-

l p



LTypE

#NAME?

( )

÷

( )

2

EpMM//

Basic design

445

Fig.16.5


m

-factor or

n

-factor methods


l


M

b


l

=

340,225 and 170 for

D/B


p

y

=

275N/mm

2


446

Beams



Basic design

447

Fig.16.7




16.4Lateral bracing


7,8


l

LTfor

M

b

=

M

c

(or more correctly

p

b

=

p

y


L

/

r

y


M

p


-16.4


Lr fpx

mycy

£+

( ) ( )

[ ]

3813027536

2212

///448

Beams

Table 16.7

Maximum values of

l

LT

forwhich

p

b



=

p

y

for rolledsections

p

y

(N/mm2

) Value of

l

LT

up to which

p

b

=

p

y

245 37265 35275 34325 32340 31365 30415 28430 27450 26

of Brown,

9


10


L

m


L

s




U-frame












t




















































































ay.








































Table Chair

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