structural steelwork eurocodes

Structural Steelwork Eurocodes Development of

A Trans-national Approach Course: Eurocode 3

Module 4: Member design

Lecture 12: Unrestrained Beams

Summary: Beams bent about the major axis may fail by buckling in a more flexible plane This form of buckling involves both lateral deflection and twisting - lateral-torsional buckling The applied moment at which a beam buckles by deflecting laterally and twisting reached is the elastic

critical moment A design approach for beams prone to failure by lateral-torsional buckling must account for a large

number of factors - including section shape, the degree of lateral restraint, type of loading, residual stress pattern and initial imperfections

Stocky beams are unaffected by lateral torsional buckling and capacity is governed by the plastic resistance moment of the cross section

Slender beams have capacities close to the theoretical elastic critical moment Many practical beams are significantly adversely affected by inelasticity and geometrical

imperfections, hence elastic theory provides an upper band solution. A design expression linking the plastic capacity of stocky beams with the elastic behaviour of slender

beams is provided by a reduction factor for lateral torsional buckling, LT.

Pre-requisites: Bending theory Buckling of structural elements Restrained beam behaviour

Notes for Tutors: This material comprises one 30 minute lecture.

SSEDTA

Structural Steelwork Eurocodes Development of a Trans-National Approach Design of Members Unrestrained Beams

Objectives: The student should: be aware of the phenomenon of lateral torsional stability be able to identify the controlling paprameters understand the significance of the terms in the elastic torsional buckling equations be able to apply the EC3 rules to the design of a simply supported laterally unrestrained beam recognise practical applications where lateral torsional buckling is unlikely to present a problem

References: Narayanan, R., editor, Beams and Beam Columns: Stability and Strength, Applied Science

Publishers, 1983 Chen, W. F. and Atsuta, T., Thoery of Beam Columns Volume 2, Space Behaviour and Design,

McGraw Hill, 1977 Timoshenko, S.P. and Gere, J.M., Theory of Elastic Stability, Second Edition, McGraw Hill, 1962 Trahair, N.S. and Bradford, M.A., The Behaviour and Design of Steel Structures, E&F Spon, 1994 Kirby, P.A. and Nethercot, D.A., Design for Structural Stability Contents:

1.Introduction 2.Elastic buckling of a simply supported beam 3.Development of a design approach 4.Extension to other cases

4.1 Load pattern 4.2 Level of application of Load 4.3 End support conditions 4.4 Beams with intermediate lateral support 4.5 Continuous beams

5. Concluding summary

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1. Introduction

Whenever a slender structural element is loaded in its stiff plane there exist a tendency for it to fail by buckling in a more flexible plane. In the case of beam bent about its major axis, failure may occur by a form of buckling which involves both lateral deflection and twisting - lateral-torsional buckling. Figure 1 illustrates the phenomenon with a slender cantilever beam loaded by a vertical end load.

Dead w eightload appliedvert ically

Buckledposit ion

Unloadedposit ion

Clam p atroot

Figure 1 Lateral-torsional buckling of a slender cantilever beam If the cantilever was perfectly straight and the cross section initially stress free and perfectly elastic, the tip of the cantilever would deflect only in the vertical plane with no out of plane deflection until the applied moment reached a critical value at which the beam buckles by deflecting laterally and twisting. A design approach for beams prone to failure by lateral-torsional buckling must of necessity account for a large number of factors - including section shape, the degree of lateral restraint, type of loading, residual stress pattern and initial imperfections - and is therefore relatively complex. It is instructive to first consider a simple basic model which may then be developed to include more general cases.

2. Elastic buckling of a simply supported beam

Figure 2 shows a perfectly elastic, initially straight I beam loaded by equal and opposite end moments about its major axis (ie in the plane of the web). The beam is unrestrained along its length except at each end where the sections is prevented from twisting and lateral deflection but is free to rotate both in the plane of the web and on plan. The buckled shape and resultant deformations are also shown in the figure (note only half of the beam is shown, the deformations are at the midspan).

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Plan

M M

L

Elevation Section

u

y

z

x

Figure 2 Lateral torsional buckling of a simple I beam under uniform moment

The moment necessary to cause buckling may be determined by equating the disturbing effect of the applied end moments, acting through the buckling deformations, to the internal (bending and torsional) resistance of the section. The critical value of applied end moments, the elastic critical moment (Mcr), is found to be

Mcr = 5,0

2

2

2

2

+z

t

z

wz

EIGIL

II

LEI

(1) F.1

where

It is the torsion constant ; Iw is the warping constant Iz is the second moment of area about the minor axis; L is the unrestrained length of beam. The presence of the flexural stiffness (EIz) and torsional stiffness (GIt and EIw) in the equation is a direct consequence of the lateral and torsional components of the buckling deformations. The relative importance of these items will be a reflection of the type of cross section considered. Figure 3 illustrates this point by comparing the elastic critical moment of a box section (which has high flexural and torisional stiffness) with open sections of various shapes.

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0 10 20 30 40 50 60 700.001

0.01

0.1

1.0

Rat io of length to

Rat io of M toM for box

sect ion

cr

cr

Figure 3 Effect of cross section shape on theoretical elastic critical moment

Figure 4 compares values of the elastic critical moment (Mcr) for an I beam and a column section with similar in plane plastic moment capacities. Lateral-torsional buckling is a potentially more significant design consideration for a beam section which is much less stiff laterally and torsionally.

2 4 6 8 10 12 14 160

2

4

6

8

10

12

14

254x254 UC 89

457x152 UB 60L

254x254 UC 89457x152 UB 60

W (cm )pl

y

J (cm )

w

1284

25464

794

386700

1228

14307

4849

716400

H - Section

M M

18 20L (m)

MM

31,5 97,6

z

(cm )

(cm )

(cm )

- Section

4

4

4

4

3

crp

Figure 4 Comparison of elastic critical moments for I and H sections

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3. Development of a design approach

Real beams are not perfectly straight nor is the material elastic. Figure 5 shows the effects of residual stresses and strain hardening on the lateral buckling strength. Note that at high slenderness values the behaviour is well represented by elastic buckling theory but for stocky beams there is a complex interplay as inelastic behaviour causes a reduction in capacity, and for very stocky beams the capacity is limited by the plastic resistance of the section. Application of a theoretical treatment of the problem would be too complex for routine design so a combination of theory and test results is required to produce a reliable (safe) design approach.

Figure 5 Lateral buckling strengths of simply supported I beams Figure 6 compares a typical set of lateral torsional buckling test data with the theoretical elastic critical moments given by eqn 1. A non-dimensionalised form of plot has been used which permits results from different test series (which have different cross-sections and material strengths) to be compared directly via a non-dimensional slenderness, LT. For stocky beams ( LT 0.4) the capacity is unaffected by lateral torsional buckling and is governed by the plastic resistance moment of the cross section. Slender beams ( LT 1.2) have capacities close to the theoretical elastic critical moment, Mcr. However, beams of intermediate slenderness, which covers many practical beams, are significantly adversely affected by inelasticity and geometrical imperfections and therefore elastic theory provides an upper band solution. A design expression linking the plastic capacity of stocky beams with the elastic behaviour of slender beams is required. EC3 achieves this by use of a reduction factor for lateral torsional buckling, LT.

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0

S t oc k y In t erm ed iat e S lend er

c r

p l

Mp l

p l

c r

1 ,0

0 ,8

0 ,6

0 ,4

0 ,2

0 ,2 0 ,4 0 ,6 0 ,8 1 ,2 1 ,41 ,0MM

MM

M

=LT

Figure 6 Comparison of test data with theoretical elastic critical moments

The design buckling resistance moment (Mb.Rd) of a laterally unrestrained beam is thus taken as: Mb.Rd = LT w Wpl.y fy/m1 (2)

5.5.2 (1) (5.48)

which is effectively the plastic resistance of the section multiplied by the reduction factor ( LT). Figure 7 shows the relationship between LT and the non-dimensional slenderness, LT .

The curves shown are expressed by

LT = [ ] 5,0221 LTLTLT + (3) 5.5.2. (2) (5.49)

where

+++= 2))2.0(15,0 LTLTLTLTLT a (4)

in which a LT is an imperfection factor, taken as 0,21 for rolled sections and 0,49 for welded sections, with their more severe residual stresses.

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0

Welded beams

Rolled sections

Slenderness LT

LT

Red

uctio

n fa

ctor

0,5 1,0 1,5 2,0

0,2

0,4

0,6

1,0

1,0

Figure 7 Lateral-torsional buckling reduction factor LT , the non-dimensional slenderness, defined as ,/ crplRd MM may be calculated either by calculating the plastic resistance moment and elastic critical moment from first principles (see Appendix F.1) or more conveniently by the relationship:

LT = 5,01

wLT

(5) 5.5.2 (5)

where

5,0

1

=fyE (6)

and Lt may be calculated using apppropriate expressions for a variety of section shapes (see Appendix F.2.2.). For example, for any plain I or H section with equal flanges, and subject to uniform moment with simple end restraints,

F.2.2

LT= 25,02

//

2011

/

+

f

z

z

thiL

iL (7)

F.2.2 (5) (F.21)

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4. Extension to other cases

4.1 Load pattern

Uniform moment applied to an unrestrained beam is the most severe for consideration of lateral torsional buckling. An elastic analysis of alternative load cases results in higher values of elastic critical moments. For example, the elastic critical moment for uniform moment is (rearranging eqn (1))

Mcr = L

EI GIEI

L GIz r

w

t1

2

2+ (8)

but for a beam with a central point load the maximum moment at the centre on the point of buckling is

Mcr=4 24 1

2

2.L

EI GIEI

L GIz r

w

t+ (9)

which is 4.24/ higher than the base case. EC3 uses this ratio expressed as a factor, C1, to allow for the loading arrangement (shape of the bending moment diagram) for a variety of loading cases, as shown in Figure 8. C1 appears as a simple multiplier in expressions for Mcr (see EC3 eqn F.2) or as 1 1/ C in expressions for LT.

Beamand loads

Bendingmoment

M Cmax 1

M M

M

M -M

F

F

FF

M

M

M

FL4

FL8

1,00

1,879

2,752

1,365

1,132

21+ EIwL2 GJ

M = C L

EI GJcr 1

Note: The above values correspond to a effective length factor k of 1,0

Figure 8 Equivalent uniform moment factors C1

Table F.1.1 Table F.1.2

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4.2 Level of application of Load

Lateral stability of a beam is dependent not only on the arrangement of loads within the span but also on the level of application of the load relative to the centroid. Figure 9 illustrates the effect of placing the load above and below the centroid for a simple span with a central point load.

F

F

F

10 100 10001

a=d/2

a=0

a=d/2

F

L GIEI2

t

w

Equ

ival

ent u

nifo

rm m

omen

t fac

tor m

1,4

1,2

1,0

0,8

0,6

0,4

Figure 9 Effect of level of load application on beam stability Loads applied to the top flange add to the destabilising effect due to the additional twisting moment arising from the action of the load not passing through the section centroid. The influence of this behaviour becomes more significant as the depth of the section increases and/or the span reduces ie as L2GIt/EIw becomes smaller. Again EC3 accounts for this by introduction of a factor C2 into the general equation for the elastic critical moment (see EC3 eqn F.2) and expressions for LT (see EC3 eqns F.27 - F.31).

F.1.2 (F,2) F.2.2 (8) (F.27-F.31)

4.3 End support conditions

All of the foregoing has assumed end conditions which prevent lateral movement and twist but permit rotation on plan. End conditions which prevent rotation on plan enhance the elastic buckling resistance (in much the same way that column capacities are enhanced by rotational end restraints). A convenient way of including the effect of different support conditions is to redefine the unrestrained length as an effective length, or more precisely with two effective length factors, k and kw. The two factors reflect the two possible types of end fixity, lateral bending restraint and warping restraint. However it should be noted that it is recommended that kw be taken as 1,0 unless special provision for warping fixing is made. EC3 recommends k values of 0,5 for fully fixed ends, 0,7 for one free and one fixed end and of course 1,0 for two free ends. The choice of k is at the designers discretion.

F.1.2 (4) F.1.2 (2)

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4.4 Beams with intermediate lateral support

Where beams have lateral restraints at intervals along the span the segments of the beam between restraints may be treated in isolation, the design of the beam being based on the most critical segment. Lengths of beams between restraints should use an effective length factor k of 1,0 not 0,7, as in the buckled shape the adjacent unrestrained length will buckle in sympathy.

4.5 Continuous beams

Beams continuous over a number of spans may be treated as individual spans taking into account the shape of the bending moment diagram within each span as a result of continuity using the C1 factor.


Beams bent about the major axis may fail by buckling in a more flexible plane - lateral-torsional buckling

Moment at which buckling occurs is the elastic critical moment Design approach must account for a large number of factors - section shape, the degree of lateral restraint, type of loading, residual stress pattern and initial imperfections Stocky beams are unaffected by lateral torsional buckling Slender beams have capacities close to the theoretical elastic critical moment Practical beams are significantly adversely affected by inelasticity and geometrical

imperfections - elastic theory is an upper band solution. A design expression linking the plastic capacity of stocky beams with the elastic behaviour

of slender beams is provided by a reduction factor for lateral torsional buckling, LT

11/02/08 11

Course: Eurocode 3Module 4: Member design Lecture 12: Unrestrained BeamsSummary: Pre-requisites:Notes for Tutors: Objectives:References:

1. Introduction Figure 1 Lateral-torsional buckling of a slender cantilever beam

2. Elastic buckling of a simply supported beam Figure 2 Lateral torsional buckling of a simple I beam under uniform momentF.1

Figure 3 Effect of cross section shape on theoretical elastic critical momentFigure 4 Comparison of elastic critical moments for I and H sections

3. Development of a design approachFigure 5 Lateral buckling strengths of simply supported I beamsFigure 6 Comparison of test data with theoretical elastic critical momentsFigure 7 Lateral-torsional buckling reduction factor

4. Extension to other cases4.1 Load patternFigure 8 Equivalent uniform moment factors C14.2 Level of application of LoadFigure 9 Effect of level of load application on beam stability 4.3 End support conditions4.4 Beams with intermediate lateral support4.5 Continuous beams


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