UNRESTRAINED BEAMS

An unrestrained beam is susceptible to lateral torsional

buckling. Lateral torsional buckling (LTB) is the combined

lateral (sideways) deflection and twisting of an

unrestrained member subject to bending about its major

axis, as shown in Figure can either occur over the full

length of a member or between points of intermediate

lateral restraint.

When a steel beam is designed, it is usual to first consider

the need to provide adequate strength and stiffness against

vertical bending. This leads to a member in which the

stiffness in the vertical plane is much greater than that in

the horizontal plane. Sections normally used as beams have

the majority of their material concentrated in the flanges

that are made relatively narrow so as to prevent local

buckling. Open sections (i.e. I or H sections) are usually

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buckling. Open sections (i.e. I or H sections) are usually

used because of the need to connect beams to other members.

The combination of all these factors results in a section

whose torsional and lateral stiffnesses are relatively low,

which has a major affect on the buckling resistance of an

unrestrained member.

Local Buckling

Buckling

Local instability can occur in a cross-section when one or

more individual elements in a cross-section (e.g. the flange

or web of an I section), as shown in Figure , buckles without

any overall deflection.

An element within a cross-section which has a high width to

thickness ratio (i.e. slender) is susceptible to local buckling,

the effect of which is to reduce the load-carrying capacity of

the section.

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Failure in a flange occurs due to excessive compression and

in a web due to excessive shear or combined shear and

bending. In addition, a web may buckle as a result of

vertical compression due to the application of a

concentrated load.

In the case of hot-rolled steel sections the flange and

web proportions (i.e. flange outstand : thickness and

web thickness : depth ratios) are normally selected to

minimize the possibility of local buckling, although web

stiffening is sometimes required at points of

concentrated load such as reactions and column

positions on beams.

In the case of welded plate girders, additional web stiffening

is usually necessary to prevent shear buckling of the web.

In design there are two approaches generally considered appropriate to allow

for the possibility of local buckling. They are:

adopting a reduced design strength when calculating the member capacity, 1.

adopting ‘effective’ section properties in which an ‘actual’ plate width is

replaced by a narrower ‘effective’ plate width which is then used to calculate

modified section properties with which to determine the section capacity.

2.

A beam subject to bending is partly in tension and partly in

compression as shown in Figure .

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The tendency of an unrestrained compression flange in these

circumstances is to deform sideways and to twist about the

longitudinal axis as shown in Figure

This type of failure is called lateral torsional buckling and

will normally occur at a value of applied moment less than

the moment capacity (Mc) of the section, based on the yield

strength of the material. The reduced moment at failure is

known as the buckling resistance moment

Lateral Torsional Buckling of Beams

Lateral Restraint

The lateral restraint to the compression flange of a beam

prevents a sideways movement of the flange relative to the

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prevents a sideways movement of the flange relative to the

tension flange.

Full Lateral Restraint

It is always desirable where possible to provide full lateral

restraint to the compression flange of a beam. The existence

of either a cast-in-situ or precast concrete slab which is

supported directly on the top flange, as indicated in Figure

Intermittent Lateral Restraint

Most beams in buildings which do not have full lateral

restraint are provided with intermittent restraint in the

form of secondary beams, ties or bracing members as shown

in Figure

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Torsional Restraint

A beam is assumed to have torsional restraint about its

longitudinal axis at any location where both flanges are

held in their relative positions by external members during

bending, as illustrated in Figure

Beams without Torsional Restraint

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In situations where a beam is supported by a wall as in

Figure, no torsional restraint is provided to the flanges and

buckling is more likely to occur.

Factors influencing buckling resistance

The following factors all influence the buckling resistance

of an unrestrained beam:

• The length of the unrestrained span, i.e. the distance

between points at which lateral deflection is prevented.

The lateral bending stiffness of the section. •

The torsional stiffness of the section. •

The conditions of the restraint provided by the end

connections.

•

The position of application of the applied load and

whether or not it is free to move with the member as it

buckles.

•

All the factors above are brought together in a single

parameter λ LT, the ‘equivalent slenderness’ of the beam.

The shape of the bending moment diagram also has

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The shape of the bending moment diagram also has

an effect on the buckling resistance. Members that are

subject to non-uniform moments will have a varying

force in the compression flange and will therefore be

less likely to buckle than members that have a uniform

force in the compression flange. This is taken into

account by the parameter mLT

Behaviour of beams

The buckling resistance moment of an unrestrained

beam depends on its equivalent slenderness λ LT and

this relationship can be expressed as a ‘buckling

curve’, as shown by the solid line in Figure

• Short stocky members will attain the full plastic

moment MP.

• Long slender members will fail at moments

approximately equal to the elastic critical moment

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approximately equal to the elastic critical moment

Mcr. This is a theoretical value that takes no

account of imperfections and residual stress.

• Beams of intermediate slenderness fail through a

combination of elastic and plastic buckling.

Imperfections and residual stresses are most

significant in this region.

Design requirements

General

The Code states that an unrestrained beam must be

checked for local moment capacity of the section and

also for buckling resistance. However, lateral

torsional buckling need not be checked for the

following situations:

Circular or square hollow sections or solid bars. •

Section bending only about the minor axis. •

I, H or channel sections when the equivalent

slenderness λ LT is less than a limiting slenderness

value λ L0

•

Rectangular hollow sections when LE/ry is less than a

limiting value, as given in Table 15 of BS 5950-1:2000.

•

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Moment capacity

The section classification and moment capacity of

the section should be determined and checked in

the same way as for restrained beams i.e.

Mx ≤ Mcx

where:

Mx is the maximum major axis moment in the

segment under consideration Mcx is the major axis

moment capacity of the cross-section

Any reductions for high shear forces should be

included in this check.

Buckling resistance

The buckling resistance of the member between

either the ends of the member or any intermediate

restraints, a ‘segment’, should be checked as:

Mx ≤ Mb/mLT

where:

Mx is the maximum major axis moment in the

segment under consideration

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Mb is the buckling resistance moment

mLT is the equivalent uniform moment factor for

LTB

Buckling resistance moment

The buckling resistance moment Mb is dependent

on the section classification of the member and a

bending strength pb that depends on the

slenderness of the beam. Mb is calculated as

follows:

For Class 1 plastic

Mb = pb Sx

For Class 2 compact sections

Mb = pb Sx

For Class 3 semi-compact sections

Mb = pb Sx,eff or Mb = pb Zx (conservatively)

For Class 4 slender sections

Mb = pb Zx,eff

where:

pb is the bending strength

Sx is the section plastic modulus

Sx,eff is the section effective plastic modulus

Zx is the section elastic modulus

Zx,eff is the section effective elastic modulus

Bending strength

The value of the bending strength pb is obtained

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The value of the bending strength pb is obtained

from Tables 16 and 17 of BS 5950-1 and depends

on the value of the equivalent slenderness λ LT

and the design strength py

For I and H sections, the equivalent slenderness is given by:

λ LT = u v λ ( β w)0.5

where:

u is a buckling parameter obtained from section property

tables

v is a slenderness factor obtained from Table 19 of BS

5950-1 and depends on λ /x

x is the torsional index, obtained from section property

tables

λ is the slenderness, taken as LE/ry

LE is the effective length between points of restraint

ry is the radius of gyration about the minor axis

β w = 1.0 for Class 1 and Class 2 sections

β w = Sx,eff/Sx for Class 3 sections when Sx,eff

is used to calculate Mb

β w = Zx/Sx for Class 3 sections when Zx is

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β w = Zx/Sx for Class 3 sections when Zx is

used to calculate Mb

β w = Zx,eff/Sx for Class 4 sections

Effective length

Beams without intermediate lateral restraints

Values of effective length LE are given in BS

5950-1 Table 13 for beams and Table 14 for

cantilevers. Part of Table 13 is reproduced

here as Table 7.1.

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Destabilising loads

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Summary of design procedure

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