NFATEC – L12 – Unrestrained beams (11/05/2004)library.tee.gr/digital/m2247/m2247_l12_eng.pdf · NFATEC – L12 – Unrestrained beams (11/05/2004) ... elastic critical moment

NFATEC – L12 – Unrestrained beams (11/05/2004)

{LASTEDIT}Roger 11/05/04{/LASTEDIT}

{LECTURE}

{LTITLE}Unrestrained Beams{/LTITLE}

{AUTHOR}Roger{/AUTHOR}

{EMAIL}[email protected]{/EMAIL}

{OVERVIEW}

• 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 (under ideal conditions and

assuming perfectly elastic behaviour) by deflecting laterally and twisting 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 bending strength 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, and hence elastic theory provides an upper bound 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 {EQN}xsiLT.gif{/EQN}.

• The reduction factor is related to the elastic critical moment, {EQN}Mcr.gif{/EQN}, but no explicit guidance is given in EC3 for calculating {EQN}Mcr.gif{/EQN}.

{/OVERVIEW}

{PREREQUISITES}

• Bending theory • Buckling of structural elements • Restrained beam behaviour

{/PREREQUISITES}

{OBJECTIVES}

On successful completion of this lecture you should:

• be aware of the phenomenon of lateral torsional stability • understand the significance of the terms in the elastic torsional buckling equations • be able to apply the EC3 rules to the design of a laterally unrestrained beam • recognise practical applications where lateral torsional buckling is unlikely to

present a problem

{/OBJECTIVES}

{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", Blackwell, 1979

{/REFERENCES}

{SECTION}

{STITLE}Introduction{/STITLE}

{SUMMARY}

{SUMTITLE}The nature of lateral-torsional buckling{/SUMTITLE}

A slender beam bent about its major axis may fail by lateral-torsional buckling which involves both lateral deflection and twisting. A large number of factors influence this behaviour, so the design approach is relatively complex.

{PPT}Lecture12Intro.pps{/PPT}

{DETAIL}

Whenever a slender structural element is loaded in its stiff plane there is 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. The figure below illustrates the phenomenon with a slender cantilever beam loaded by a vertical end load.

{IMAGE}LTB slender cantilever.gif{/IMAGE}

{FIGURE}Figure 1. Lateral-torsional buckling of a slender cantilever beam, clamped at the support A and loaded with a dead weight applied vertically at the free end B. The unloaded position is shown in red, and the buckled position in blue.{/FIGURE}

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.

{/DETAIL}

{/SUMMARY}

{/SECTION}

{SECTION}

{STITLE}Elastic buckling of a simply supported beam{/STITLE}

{SUMMARY}

{SUMTITLE}Factors influencing lateral-torsional buckling{/SUMTITLE}

Lateral torsional buckling is generally described in terms of the elastic critical moment {EQN}Mcr.gif{/EQN}. It is a theoretical concept rather than of direct use in code based design. For a beam under uniform moment the elastic critical moment depends on the flexural and torsional stiffnesses {EQN}EIz.gif{/EQN} {EQN}GIt.gif{/EQN}, {EQN}EIw.gif{/EQN}, and the unrestrained length {EQN}L.gif{/EQN}. Their relative importance depends on the type of cross section.

Lateral-torsional buckling is more significant for sections which are less stiff laterally and torsionally compared with their flexural stiffness.

{PPT}Lecture12ElasticMcr.pps{/PPT}

{DETAIL}

The figure below 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, and the deformations are at the midspan).

{IMAGE}LTB uniform BM.gif{/IMAGE}

{TIMAGE}Lateral-torsional buckling under uniform bending. (a) Elevation (b) Section (c) Plan{/TIMAGE}

{IMAGE}LTB deformed shape.gif{/IMAGE}

{TIMAGE}Lateral-torsional buckling under uniform bending - deformed shape{/TIMAGE}

{FIGURE}Figure 2. Lateral torsional buckling of a simple I beam under uniform moment{/FIGURE}

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 is referred to as the elastic critical moment {EQN}Mcr.gif{/EQN}.

The critical value of applied end moments, the elastic critical moment {EQN}Mcr.gif{/EQN}, is found to be

{EQN}Mcr.gif{/EQN} = {EQN}L12eqn2.gif{/EQN} (1)

where

{EQN}It.gif{/EQN} is the torsion constant;

{EQN}Iw.gif{/EQN} is the warping constant

{EQN}Iz.gif{/EQN} is the second moment of area about the minor axis;

{EQN}L.gif{/EQN} is the unrestrained length of beam.

The presence of the flexural stiffness {EQN}EIz.gif{/EQN} and torsional stiffness {EQN}GIt.gif{/EQN} and {EQN}EIw.gif{/EQN} 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. The figure below illustrates this point by comparing the elastic critical moment of a box section (which has high flexural and torsional stiffness) with open sections of various shapes.

{IMAGE}Effect of X sec on Mcr.gif{/IMAGE}

{FIGURE}Figure 3. Effect of cross section shape on theoretical elastic critical moment. {EQN}Mcrbox.gif{/EQN} is the ratio of {EQN}Mcr.gif{/EQN} for the section to {EQN}Mcr.gif{/EQN} for an equivalent box section.{/FIGURE}

The sections in the above figure all have the same cross-sectional area but have different section properties as shown.

The figure below compares values of the elastic critical moment {EQN}Mcr.gif{/EQN} 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.

{IMAGE}Mcr for I and H sections.gif{/IMAGE}

{FIGURE}Figure 4. Comparison of elastic critical moments for I and H sections{/FIGURE}

{/DETAIL}

{/SUMMARY}

{TEST}

{TTITLE}Factors influencing lateral-torsional buckling resistance{/TTITLE}

{QUESTION}

{QTITLE}Factors influencing lateral-torsional buckling resistance{/QTITLE}

{QTYPE}MC{/QTYPE}

{QTEXT}How is lateral torsional buckling resistance affected by increasing the span of the beam?{/QTEXT}

{ANSWER}

Increasing the span of the beam has no effect on lateral torsional buckling resistance

{CHECKMARK}0{/CHECKMARK}

{CHECK}No – lateral torsional buckling resistance decreases with increasing span length{/CHECK}

{UNCHECKMARK}1{/UNCHECKMARK}

{UNCHECK}Lateral torsional buckling resistance decreases with increasing span length.{/UNCHECK}

{/ANSWER}

{ANSWER}

Increasing the span of the beam increases lateral torsional buckling resistance


{CHECK}No – lateral torsional buckling resistance decreases with increasing span length{/CHECK}


{UNCHECK}Lateral torsional buckling resistance decreases with increasing span length {/UNCHECK}

{/ANSWER}

{ANSWER}

Increasing the span of the beam reduces lateral torsional buckling resistance


{CHECK}Yes –lateral torsional buckling resistance decreases with increasing span length{/CHECK}


{UNCHECK}Lateral torsional buckling resistance decreases with increasing span length{/UNCHECK}

{/ANSWER}

{/QUESTION}

{QUESTION}

{QTITLE}Lateral-torsional buckling resistance of beams{/QTITLE}

{QTYPE}MC{/QTYPE}

{QTEXT}How is lateral torsional buckling resistance affected by increasing the load on the beam?{/QTEXT}

{ANSWER}

Increasing the load on the beam has no effect on lateral torsional buckling resistance


{CHECK}Yes – lateral torsional buckling resistance is not affected by load level{/CHECK}


{UNCHECK}Lateral torsional buckling resistance is not affected by load level {/UNCHECK}

{/ANSWER}

{ANSWER}

Increasing the load on the beam increases lateral torsional buckling resistance


{CHECK}No – lateral torsional buckling resistance is not affected by load level{/CHECK}



{/ANSWER}

{ANSWER}

Increasing the load on the beam reduces lateral torsional buckling resistance


{CHECK}No – lateral torsional buckling resistance is not affected by load level{/CHECK}



{/ANSWER}

{/QUESTION}

{QUESTION}

{QTITLE}Factors influencing lateral-torsional buckling resistance of beams{/QTITLE}

{QTYPE}MC{/QTYPE}

{QTEXT}Which property of the cross-section most influences the elastic critical moment – the torsion constant, the second moment of area about the minor axis, or the second moment of area about the major axis?{/QTEXT}

{ANSWER}

torsion constant


{CHECK}No – increasing the torsion constant only affects the second term in the expression for lateral torsional buckling resistance{/CHECK}


{UNCHECK}Increasing the torsion constant only affects the second term in the expression for lateral torsional buckling resistance{/UNCHECK}

{/ANSWER}

{ANSWER}

the second moment of area about the minor axis


{CHECK}Yes – increasing the second moment of area about the minor axis affects the whole expression for lateral torsional buckling resistance {/CHECK}


{UNCHECK}Increasing the second moment of area about the minor axis affects the whole expression for lateral torsional buckling resistance {/UNCHECK}

{/ANSWER}

{ANSWER}

the second moment of area about the major axis


{CHECK}No - increasing the second moment of area about the major axis does not affect the lateral torsional buckling resistance at all.{/CHECK}


{UNCHECK}Increasing the second moment of area about the major axis does not affect the lateral torsional buckling resistance at all.{/UNCHECK}

{/ANSWER}

{/QUESTION}

{QUESTION}

{QTITLE}Influence of cross-section shape on lateral-torsional buckling resistance{/QTITLE}

{QTYPE}M{/QTYPE}

{QTEXT}Place the following types of cross-section in order of lateral torsional buckling resistance (1 = most resistant, 4 = least resistant){/QTEXT}

{ANSWER}

Square hollow section

{MARK}1{/MARK}

{MATCH}1{/MATCH}

{REASON}The second moment of area about the major and minor principal axes is the same so lateral torsional buckling cannot occur before yield.{/REASON}

{/ANSWER}

{ANSWER}

H- section

{MARK}1{/MARK}

{MATCH}2{/MATCH}

{REASON}The second moment of area about the minor principal axes is of a similar order of magnitude to that about the major principal axis, providing good resistance to lateral torsional buckling.{/REASON}

{/ANSWER}

{ANSWER}

I section

{MARK}1{/MARK}

{MATCH}3{/MATCH}

{REASON}The second moment of area about the minor principal axes is significantly less than that about the major principal axis, providing poor resistance to lateral torsional buckling.{/REASON}

{/ANSWER}

{ANSWER}

Flat plate loaded in the direction of the longer cross-sectional dimension

{MARK}1{/MARK}

{MATCH}4{/MATCH}

{REASON} The second moment of area about the minor principal axes is very much less than that about the major principal axis, providing very poor resistance to lateral torsional buckling.{/REASON}

{/ANSWER}

{/QUESTION}

{/TEST}

{/SECTION}

{SECTION}

{STITLE}Development of a design approach{/STITLE}

{SUMMARY}

{SUMTITLE}Design approaches{/SUMTITLE}

The lateral-torsional buckling behaviour of real beams at high slenderness is well represented by elastic buckling. For very stocky beams the capacity is limited by the bending strength (plastic, elastic or effective resistance moment, depending on section classification) while for beams with an intermediate slenderness residual stresses reduce the theoretical moment capacity. Routine design procedures are based on a combination of theory and test results. EC3 does this by applying a reduction factor for lateral torsional buckling. The reduction factor is a function of an imperfection factor (which depends on the type of section) and the non-dimensional slenderness, which in turn depends on the elastic critical moment.

{PPT}Lecture12Design.pps{/PPT}

{DETAIL}

Real beams are not perfectly straight nor is the material elastic. The figure below 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 bending strength (plastic, elastic or effective resistance moment, depending on section classification). 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.

{IMAGE}LTB of SS I beams.gif{/IMAGE}

{FIGURE}Figure 5. Lateral buckling strengths of simply supported I beams. A – Full plasticity; B – Elastic buckling; C – Strain hardening; D – Beams without residual stress; E – Hot rolled beams with residual stress; F – Welded beams with residual stress{/FIGURE}

The figure below compares a typical set of lateral torsional buckling test data with the theoretical elastic critical moments given by equation 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, {EQN}lambdabarLT.gif{/EQN}, defined as:

{EQN}L12eqn5.gif{/EQN} (2)

{ECLINK}6.3.2.2{/ECLINK}

{IMAGE}Mcr test and theory.gif{/IMAGE}

{FIGURE}Figure 6. Comparison of test data with theoretical elastic critical moments. A – Stocky beams; B – Intermediate beams; C – Slender beams.{/FIGURE}

For stocky beams (for rolled or equivalent welded sections {EQN}lambdabarLT.gif{/EQN} < 0,4) the capacity is unaffected by lateral torsional buckling and is governed by the bending strength of the cross section. Slender beams ({EQN}lambdabarLT.gif{/EQN} > 1,2) have capacities close to the theoretical elastic critical moment, {EQN}Mcr.gif{/EQN}. 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 bound 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, {EQN}xsiLT.gif{/EQN}, which, for the case of rolled or equivalent welded sections, is given by:


{ECLINK}6.3.2.3 (1) (6.57){/ECLINK}

where


in which {EQN}alphaLT.gif{/EQN} is an imperfection factor.

Values of {EQN}alphaLT.gif{/EQN} are given in the following table.

Cross-section Limits {EQN}alphaLT.gif{/EQN} {EQN}hb.gif{/EQN}<2 0,34 Rolled I-sections {EQN}hb.gif{/EQN}>2 0,49 {EQN}hb.gif{/EQN}<2 0,49 Welded I-sections {EQN}hb.gif{/EQN}>2 0,76

The parameters {EQN}beta.gif{/EQN} and {EQN}lambdabarLT0.gif{/EQN} and any limitation on the validity concerning beam depth or {EQN}hb.gif{/EQN} ratio may be given in the National Annex. However, the following values are recommended:

{EQN}lambdabarLT0.gif{/EQN} = 0,4

{EQN}beta.gif{/EQN} = 0,75

A similar approach applies to sections other than rolled sections and equivalent welded sections {ECLINK}EC3 clause 6.3.2.2{/ECLINK}.

The design buckling resistance moment {EQN}MbRd.gif{/EQN} of a laterally unrestrained beam is thus taken as:


{ECLINK}6.3.2.1 (3) (6.55){/ECLINK}

which is effectively the bending strength of the section multiplied by the reduction factor {EQN}xsiLT.gif{/EQN}.

{/DETAIL}

{/SUMMARY}

{TEST}

{TTITLE}Design methods for lateral-torsional buckling{/TTITLE}

{QUESTION}

{QTITLE}Failure modes{/QTITLE}

{QTYPE}M{/QTYPE}

{QTEXT}Match the following modes of failure with the beam slendernesses{/QTEXT}

{ANSWER}

Bending moment resistance (failure by material yielding)

{MARK}1{/MARK}

{MATCH}short{/MATCH}

{REASON}Beams with low slenderness are not prone to lateral torsional buckling and will fail by yielding.{/REASON}

{/ANSWER}

{ANSWER}

Mixed

{MARK}1{/MARK}

{MATCH}Intermediate slenderness{/MATCH}

{REASON} For beams with intermediate slenderness, the theoretical failure loads due to both lateral torsional buckling and yielding are of a similar magnitude and failure therefore generally involves a combination of the two.{/REASON}

{/ANSWER}

{ANSWER}

Slender

{MARK}1{/MARK}

{MATCH}Lateral torsional buckling{/MATCH}

{REASON} Beams with high slenderness are prone to lateral torsional buckling and will fail in this mode well before yielding. {/REASON}

{/ANSWER}

{/QUESTION}

{QUESTION}

{QTITLE}Imperfection factor{/QTITLE}

{QTYPE}MC{/QTYPE}

{QTEXT}Which of the following are accounted for by the imperfection factor?{/QTEXT}

{ANSWER}

Residual stresses


{CHECK}Yes – residual stress is an important imperfection{/CHECK}


{UNCHECK}Residual stress is an important imperfection{/UNCHECK}

{/ANSWER}

{ANSWER}

Initial out-of-straightness


{CHECK}Yes – Initial out-of-straightness is an important imperfection{/CHECK}


{UNCHECK}Initial out-of-straightness is an important imperfection{/UNCHECK}

{/ANSWER}

{ANSWER}

Shape of cross-section


{CHECK}No – the shape of the cross-section is accounted for by the flexural stiffness and the torsion and warping constants{/CHECK}


{UNCHECK}The shape of the cross-section is not accounted for by the imperfection factor (but is accounted for through the flexural stiffness and the torsion and warping constants).{/UNCHECK}

{/ANSWER}

{ANSWER}

Length of the beam


{CHECK}No – the length of the beam is accounted for in the general expression for buckling resistance moment{/CHECK}


{UNCHECK}The length of the beam is not accounted for by the imperfection factor (but is accounted for in the general expression for buckling resistance moment.{/UNCHECK}

{/ANSWER}

{/QUESTION}

{QUESTION}

{QTITLE}Design parameters{/QTITLE}

{QTYPE}MC{/QTYPE}

{QTEXT}What name is given to the parameter defined by {EQN}lambdabarLT.gif{/EQN}{/QTEXT}

{ANSWER}

Load ratio


{CHECK}No - {EQN}lambdabarLT.gif{/EQN}refers to the non-dimensional slenderness.{/CHECK}


{UNCHECK}{EQN}lambdabarLT.gif{/EQN}refers to the non-dimensional slenderness {/UNCHECK}

{/ANSWER}

{ANSWER}

Non-dimensional slenderness


{CHECK}Yes - {EQN}lambdabarLT.gif{/EQN}refers to the non-dimensional slenderness {/CHECK}


{UNCHECK}{EQN}lambdabarLT.gif{/EQN}does refer to the non-dimensional slenderness {/UNCHECK}

{/ANSWER}

{ANSWER}

Factor of safety


{CHECK}No - {EQN}lambdabarLT.gif{/EQN}refers to the non-dimensional slenderness {/CHECK}


{UNCHECK}{EQN}lambdabarLT.gif{/EQN}refers to the non-dimensional slenderness {/UNCHECK}

{/ANSWER}

{/QUESTION}

{QUESTION}

{QTITLE}Non-dimensional slenderness{/QTITLE}

{QTYPE}MC{/QTYPE}

{QTEXT}Does an increase in the length of a beam increase or decrease the non-dimensional slenderness{/QTEXT}

{ANSWER}

Increasing the length of a beam increases the non-dimensional slenderness


{CHECK}Yes{/CHECK}


{UNCHECK}Non-dimensional slenderness is proportional to length{/UNCHECK}

{/ANSWER}

{ANSWER}

Increasing the length of a beam decreases the non-dimensional slenderness


{CHECK}No - non-dimensional slenderness is proportional to length {/CHECK}


{UNCHECK}Non-dimensional slenderness is proportional to length{/UNCHECK}

{/ANSWER}

{/QUESTION}

{QUESTION}

{QTITLE}Second moment of area{/QTITLE}

{QTYPE}MC{/QTYPE}

{QTEXT}Does an increase in the second moment of area about the minor axis of a beam cross section {EQN}Iz.gif{/EQN} increase or decrease the non-dimensional slenderness{/QTEXT}

{ANSWER}

Increasing {EQN}Iz.gif{/EQN} increases the non-dimensional slenderness


{CHECK}No - increasing the second moment of area about the minor axis of a beam cross section {EQN}Iz.gif{/EQN} decreases the beam's slenderness and hence the non-dimensional slenderness.{/CHECK}


{UNCHECK}Increasing the second moment of area about the minor axis of a beam cross section {EQN}Iz.gif{/EQN} decreases the beam's slenderness and hence the non-dimensional slenderness.{/UNCHECK}

{/ANSWER}

{ANSWER}

Increasing {EQN}Iz.gif{/EQN} decreases the non-dimensional slenderness


{CHECK}Yes - increasing the second moment of area about the minor axis of a beam cross section {EQN}Iz.gif{/EQN} decreases the beam's slenderness and hence the non-dimensional slenderness.{/CHECK}


{UNCHECK}Increasing the second moment of area about the minor axis of a beam cross section {EQN}Iz.gif{/EQN} decreases the beam's slenderness and hence the non-dimensional slenderness.{/UNCHECK}

{/ANSWER}

{/QUESTION}

{/TEST}

{/SECTION}

{SECTION}

{STITLE}Influence of load pattern{/STITLE}

{SUMMARY}

{SUMTITLE}Influence of load type{/SUMTITLE}

The most severe loading case for lateral torsional buckling is a uniform moment. EC3 uses a factor, {EQN}f.gif{/EQN}, to modify the value of {EQN}xsiLT.gif{/EQN}, and hence allow for the beneficial effect of other loading cases.

{PPT}Lecture12Loadtype.pps{/PPT}

{DETAIL}

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 equation (1))


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


which is 4.24/{EQN}pi.gif{/EQN} higher than the base case. In order to allow for the loading arrangement (shape of the bending moment diagram) EC3 uses a factor, {EQN}f.gif{/EQN}, applied to the reduction factor {EQN}xsiLT.gif{/EQN}:


{ECLINK}6.3.2.3 (2) (6.58){/ECLINK}

for a variety of loading cases. The value of {EQN}f.gif{/EQN} may be defined by the National Annex, but the following should be used as a minimum:

{EQN}L12eqn24.gif{/EQN}

but {EQN}L12eqn25.gif{/EQN} (9)

where {EQN}kc.gif{/EQN} is a correction factor according to the following table.

Moment distribution {EQN}kc.gif{/EQN} {IMAGE}Tab66a.gif{/IMAGE}

{EQN}psi.gif{/EQN}=1

10

{IMAGE}Tab66b.gif{/IMAGE}

-1<{EQN}psi.gif{/EQN}<1

{EQN}L12eqn26.gif{/EQN}

{IMAGE}Tab66c.gif{/IMAGE} 0,94 {IMAGE}Tab66d.gif{/IMAGE} 0,92 {IMAGE}Tab66e.gif{/IMAGE} 0,91 {IMAGE}Tab66f.gif{/IMAGE} 0,86 {IMAGE}Tab66g.gif{/IMAGE} 0,77 {IMAGE}Tab66h.gif{/IMAGE} 0,82

{FIGURE}Table. Correction factors {EQN}kc.gif{/EQN}{/FIGURE}

{ECLINK}Table 6.6{/ECLINK}

{/DETAIL}

{/SUMMARY}

{/SECTION}

{SECTION}

{STITLE}Guidance on calculating the elastic critical moment{/STITLE}

{SUMMARY}

{SUMTITLE}Simplified procedures for elastic critical moment{/SUMTITLE}

The calculation of {EQN}Mcr.gif{/EQN} from equation (1) can be a tedious process. Simplified methods are not specifically referred to in EC3, but were included in earlier drafts, and may be acceptable.

{PPT}Lecture12Simple.pps{/PPT}

{DETAIL}

EC3 prescribes no method for determining {EQN}Mcr.gif{/EQN} and hence {EQN}lambdabarLT.gif{/EQN}. National Annexes may include some guidance, but if not the engineer may use any generally accepted method. Earlier drafts of EC3 included specific details of alternative calculation methods for {EQN}Mcr.gif{/EQN}, but there was no consensus on their inclusion in the final draft of the code. However, these methods may be acceptable in some member countries, and in the absence of any other detailed guidance one such approach is described here.

{EQN}lambdabarLT.gif{/EQN}, the non-dimensional slenderness, defined as {EQN}L12eqn14.gif{/EQN} may be calculated either by calculating the plastic resistance moment and elastic critical moment from first principles or more conveniently by the relationship:

{EQN}lambdabarLT.gif{/EQN} = {EQN}L12eqn16.gif{/EQN} (10)

where


and {EQN}lambdaLT.gif{/EQN}may be calculated using appropriate expressions for a variety of section shapes. For example, for any plain I or H section with equal flanges, and subject to uniform moment with simple end restraints,

{EQN}lambdaLT.gif{/EQN}= {EQN}L12eqn18.gif{/EQN} (12)

{EQN}betaw.gif{/EQN} = 1 for Class 1 or Class 2 cross-sections

{EQN}betaw.gif{/EQN} = {EQN}WelWpl.gif{/EQN} for Class 3 cross-sections

{EQN}betaw.gif{/EQN} = {EQN}WeffWpl.gif{/EQN} for Class 4 cross-sections

{/DETAIL}

{/SUMMARY}

{SUMMARY}

{SUMTITLE} Influence of level of application of load in relation to centroid{/SUMTITLE}

Loads applied to the top flange cause an additional twisting moment because the load does not pass through the section centroid. EC3 accounts for this by a factor in the expressions for the elastic critical moment

{PPT}Lecture12Loadlevel.pps{/PPT}

{DETAIL}

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. The following figure illustrates the effect of placing the load above and below the centroid for a simple span with a central point load.

{IMAGE}Effect of height of load on LTB.gif{/IMAGE}

{FIGURE}Figure 9. Effect of level of load application on beam stability, represented as the equivalent uniform moment factor, m.{/FIGURE}

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 {EQN}L12eqn27.gif{/EQN} becomes smaller. Again EC3 makes no specific recommendations for this. However earlier drafts used a factor applied to the general equation for the elastic critical moment and expressions for {EQN}lambdaLT.gif{/EQN}.

{/DETAIL}

{/SUMMARY}

{SUMMARY}

{SUMTITLE}Influence of end support conditions{/SUMTITLE}

EC3 gives no explicit reference to the effect of end conditions on the elastic critical moment. However, it is generally accepted that different support conditions can be accounted for through effective length factors. Earlier drafts of EC3 included factors

{EQN}k.gif{/EQN} and {EQN}kw.gif{/EQN}, relating to in plane bending restraint and warping restraint respectively. {EQN}kw.gif{/EQN} is usually taken as 1,0 whilst values of 0,5 - 1,0 are recommended for {EQN}k.gif{/EQN}.

{PPT}Lecture12Endconditions.pps{/PPT}

{DETAIL}

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, {EQN}k.gif{/EQN} and {EQN}kw.gif{/EQN}. The two factors reflect the two possible types of end fixity, in plane bending restraint and warping restraint. No specific guidance is given in EC3, but in earlier drafts it was recommended that {EQN}kw.gif{/EQN} be taken as 1,0 unless special provision for warping fixing is made. These earlier drafts of EC3 also recommended {EQN}k.gif{/EQN} 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 {EQN}k.gif{/EQN} is at the designer’s discretion.

{/DETAIL}

{/SUMMARY}

{SUMMARY}

{SUMTITLE}Beams with intermediate lateral support{/SUMTITLE}

Beams with intermediate lateral restraints are treated as a series of individual segments with an effective length factor {EQN}k.gif{/EQN} of 1,0.

{PPT}Lecture12Restraints.pps{/PPT}

{DETAIL}

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 {EQN}k.gif{/EQN} of 1,0 not 0,7, as in the buckled shape the adjacent unrestrained length will buckle in sympathy.

{/DETAIL}

{/SUMMARY}

{SUMMARY}

{SUMTITLE}Continuous beams{/SUMTITLE}

Continuous beams may be treated as individual spans with proper consideration of the bending moment distribution.

{PPT}Lecture12Continuousbeams.pps{/PPT}

{DETAIL}

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 {EQN}kc.gif{/EQN} factor.

{/DETAIL}

{/SUMMARY}

{/SECTION}

{SECTION}

{STITLE}Concluding summary{/STITLE}

{SUMMARY}

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

• Moment at which buckling occurs (under ideal conditions and assuming elastic behaviour) 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 bound 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,{EQN}L12eqn1.gif{/EQN}

{/SUMMARY}

{/SECTION}

{/LECTURE}

Documents

NFATEC – L12 – Unrestrained beams (11/05/2004)library.tee.gr/digital/m2247/m2247_l12_eng.pdf · NFATEC – L12 – Unrestrained beams (11/05/2004) ... elastic critical moment