en1993 practice paper-buckling analysis of steel bridges

C Hendy, S Denton, D MacKenzie, D Iles 1

EN1993 PRACTICE PAPER: BUCKLING ANALYSIS OF

STEEL BRIDGES C Hendy, Atkins, Epsom, UK S Denton, Parsons Brinckerhoff, Bristol, UK D MacKenzie, Flint and Neill, London, UK D Iles, SCI, Ascot, UK

Abstract Eurocode 3 presents definitions of slenderness in terms of critical forces or critical stresses to

facilitate determination of slenderness from a computer elastic critical buckling analysis. This

analysis will not always be required (there are often simpler provisions), but its availability

allows more accurate slenderness determination than might otherwise be obtained by simple

codified equations. Determination of slenderness can be avoided by carrying out second

order analysis allowing for imperfections; this is a more time-consuming approach but will

often given a more economic result, although not always depending on the imperfections

used. The exceptions are discussed in the paper.

Introduction Accounting for buckling is a key aspect of the design of steel structures. Eurocode 3 offers

considerable flexibility to designers in the way that this can be done, and it is therefore

important for designers to have an understanding of the fundamental concepts underpinning

buckling behaviour, such as the implications of imperfections and slenderness, the effects of

geometric and material non-linearity, and the possibility that buckling will occur at a global,

member and/or local level.

Eurocode 3 presents definitions of slenderness in terms of critical forces or critical stresses to

facilitate determination of slenderness from a computer elastic critical buckling analysis. This

analysis will not always be required (there are often simpler provisions), but its availability

allows more accurate slenderness determination than might otherwise be obtained by simple

codified equations. Determination of slenderness can be avoided by carrying out second

order analysis allowing for imperfections; this is a more time-consuming approach but will

often given a more economic result, although not always depending on the imperfections

used. The exceptions are discussed in the paper.

This paper provides some guidance on the use of both calculation methods including some

areas where caution is required. A brief overview of buckling behaviour and analysis is also

given. References to clauses in EN 1993 have been abbreviated below. For example, 3-1-

5/3.3(1) is a reference to clause 3.3(1) of EN 1993-1-5.

General Overview of Buckling Behaviour and Analysis In this section buckling behaviour and analysis requirements are considered in general terms;

the specific requirements of Eurocode 3 are discussed in the following sections of the paper.


It is easiest to first explore buckling behaviour in the context of a simple pin ended member

under axial load, and such an approach is taken here. The observations are however, more

generally relevant.

If a pin ended member with some initial (bow) imperfection is subjected to an increasing axial

load, the member will tend to bow outwards until a point is reached when, with increasing

lateral deflection, the load that can be sustained will reduce. The maximum axial load

(referred to here as the buckling load) will be dependent upon the slenderness of the member,

the initial bow and the material strength. The buckling load is usually reached when, or soon

after, yield first occurs at an extreme fibre of the cross-section; yielding leads to a reduction in

the (tangent) flexural modulus of the member and therefore the rate of change of lateral

deflection with load increases.

The axial buckling load that can be sustained may be considerably lower than the (theoretical)

maximum axial load that could be sustained by a perfectly straight member that remains in the

elastic state. This theoretical maximum axial load is the elastic critical force (in Eurocode

terminology) and for a pin ended member is the Euler buckling load, given by 2EI / Lcr

2,

where Lcr is the member length.

Of course, there are other factors that affect the buckling load, for example:

(i) Residual (self-equilibrating) stresses in the member due to the way it has been

manufactured can result in first yield, and therefore lateral instability, occurring at a

lower axial load.

(ii) Local buckling of the plates that make up the member might occur, and whilst this

local plate buckling might itself stabilise, it can result in a reduction in the effective

stiffness of the member and therefore a reduction in the buckling load. This effect is

usually very small.

(iii) If the member forms part of a larger structure, it is possible that some global

buckling instability will occur, before the member reaches its buckling load.

The effects of geometric imperfections and residual stresses can be accounted for by

incorporating appropriate geometric imperfections in the member buckling analysis as

discussed in the section on imperfections below. It should be noted that the use of equivalent

geometrical imperfections to represent residual stresses is usually more conservative than

modelling the pattern of residual stress directly in the analysis.

There are essentially two methods that can be used to calculate the buckling load accounting

for imperfections. The first is to use buckling curves that give a reduction factor that is

applied to the resistance of the cross section (squash load); the reduction factor depends on

the so-called „non-dimensional slenderness‟ that expresses the relationship between elastic

critical force and the squash load. In this way, the effect of buckling is taken into account

through a reduction in the member resistance. This is the approach that has generally been

used in past UK practice. The derivation of the buckling curves in EN 1993-1-1 is presented

in reference 3.


The second approach is to model the imperfections in the member in a numerical analysis

package that can take account of geometric non-linearity (i.e. the additional force effects

arising from the lateral deflection of the member under axial load) and material non-linearity

(i.e. yielding of the steel). There is generally no need to consider material non-linearity if the

analysis is stopped when yield is first reached – the further increase in load is small. Both

methods are discussed in this paper; if the imperfections are suitably chosen, both approaches

can give identical results.

When the first of these two methods is used there are several ways in which the elastic critical

force can be determined. In past UK practice, this was generally done through establishing an

effective buckling length (typically using tables and graphs) and the same method can still

effectively be used in designs to Eurocodes. However, with the increasing availability of

software that can perform elastic critical buckling analysis, it is expected that the elastic

critical force will increasingly be determined directly through numerical methods. There are a

number of pitfalls when using software to perform elastic critical buckling analysis, and these

are discussed later. There can also be considerable advantages.

It is, however, absolutely crucial that designers recognise that the results of elastic critical

buckling analyses do not give the buckling load of the structure directly – they give ideal

results (equivalent to the Euler buckling load) that must then be factored to account for

imperfections.

There is one further aspect that merits comment, although it does tend to be more relevant to

building than bridge design. This concerns the global response of the structure and the

influence that it may have on member buckling. If a structure is globally sensitive to second

order effects (i.e. if changes in its geometry under load give rise to increasing load effects),

then it will be important that such second order effects are taken into account in determining

the boundary conditions used for member buckling verifications, and in doing so, that account

is taken of the effects of global imperfections. Finally, of course, it will be understood that

consideration of imperfections is not only important in second order analysis, it is also

important in evaluating the loads in bracing members etc.

A general overview of the approaches that can be taken to account for member and global

buckling behaviour, the effects of imperfections and second order effects, is shown in Figure

A.1 in Appendix A of this paper. In this figure, the term „first order analysis‟ is used to refer

to an analysis in which the deformed geometry is not taken into account, the term „second

order analysis‟ is used to refer to an analysis in which the deformed geometry is taken into

account, and „non-linear analysis‟ is used to refer to an analysis in which both geometric (i.e.

second order) non-linearity and material non-linearity are modelled.


Elastic Critical Buckling Analysis

Use for buckling checks on members Eurocode 3 presents the expressions for non-dimensional slenderness expressions in terms of

critical forces, e.g. Ncr, Mcr, or, in the case of shear, in terms of critical stress,cr. Some

examples are shown below:

Buckling in compression:

Buckling in bending:

Buckling in shear:

It is therefore often beneficial to be able to evaluate these critical forces and stresses directly

to determine the most accurate slenderness. The resistance to the mode of buckling being

considered is then determined from equations for reduction factor for that particular mode,

which is usually theoretically based and adjusted for test observations. The reduction factor

curves for buckling of compression members in clause 6.3.1 of EN 1993-1-1 are a good

example, having been derived from the Perry-Robertson theory using values of imperfections

which provide good correlation with test results.

The critical stresses and forces can sometimes be obtained by hand calculation using

mathematical expressions. This is quite easy for members in compression where the concept

of effective length (Lcr) can be used to determine the critical buckling force Ncr as 2

cr

2

L

EI.

However, for bending the situation is different and it is difficult to determine an expression

for Mcr for real bridge situations and hence determining the value directly from an elastic

critical buckling analysis can be desirable and will often bring economic benefit.

Practical example of use A good example of use is the buckling of paired beams during construction of the concrete

deck slab. This may be a critical check as the girders will often be most susceptible to lateral

torsional buckling (LTB) failure when the deck slab is being poured. Beams are normally

braced in pairs with discrete torsional restraints, often in the form of X bracing or K bracing

(as shown in Figure 1), but for shallower girders single horizontal channels connecting the

beams at mid-height is an economic, but less rigid, alternative.

cr

y

N

Af

cr

y

M

WfLT

cr

y


Figure 1. Pairs of braced beams awaiting deck slab construction

Paired girders with torsional bracing as above generally fail by rotation of the braced pair over

a span length as shown in Figure 2. With widely spaced torsional bracing, buckling of the

compression flange between bracing points is also possible. There are no formulae for the

former situation given in EN 1993 so there are two possible approaches to determine a

slenderness and hence the reduction factor for buckling:

(i) utilise the hand calculation method of PD 6695-2[1]

to determine the slenderness

directly;

(ii) determine Mcr by computer analysis for use in slenderness calculation.

The second method will produce the most economic design.

Figure 2. Buckling of paired beams prior to concrete hardening

Point of

rotation


An example composite bridge case is considered below. It is a simple single span bridge with

two steel plate girders braced together by cross bracing. The dimensions are representative of

typical UK construction, being based on a recently constructed bridge. A uniformly

distributed vertical load was applied to both girders, representing ULS factored load from

concreting of the span and steel self weight, and an elastic critical buckling analysis carried

out. The lowest global mode of buckling, corresponding to the attainment of Mcr, is shown in

Figure 3; the girder pair is seen to rotate together over the whole span. The second lowest

global mode is shown in Figure 4 and corresponds to lateral buckling of the compression

flange between braces. Mcr is obtained from the computer analysis as the largest initial first

order bending moment multiplied by the load factor at buckling in the mode of interest.

Table 1 shows a comparison of the final bending resistances produced from method (i), (ii)

and a full non-linear analysis (method iii), the latter being a very close approximation to the

real bending resistance of the girders and is discussed more in below. The elastic critical

bucking analysis method has a clear economic advantage over the hand calculation method.

More detail on this particular example can be found in reference 2.

Figure 3. Lowest global mode of buckling for single span beams


Figure 4. Second lowest global mode of buckling for single span beams

Calculation method Bending resistance

(kNm)

(i) Hand calculation to PD 6695-2 5260

(ii) EN 1993-1-1 clause 6.3.2 with Mcr determined from elastic buckling analysis

7470

(iii) Non Linear FE (with strain hardening) 9591

Table 1. Comparison of resistances obtained by different methods for paired beams

Use for buckling checks on entire systems – BS EN 1993-1-1, 6.3.4 Clause 6.3.4 of EN 1993-1-1 is written as a general method for checking out of plane (lateral)

buckling of members and frames when the axial force and bending moment both give rise to

out of plane buckling of the element(s) i.e. the axial force or bending moment applied

separately would lead to lateral buckling of the element(s). An example of this is given in the

Designers‟ Guide to EN 1993-2[3]

, section 6.3.4.1, example 6.3-4. The slenderness for

buckling is given by:

opcr

kultop

,

,

where ult,k is the load factor to apply to the factored ULS loads to cause cross section failure

and cr,op is the load factor to apply to the same loads to give elastic critical buckling. In such

cases, it is logical that the cross section resistance used in the slenderness calculation be based


upon both the axial force and the bending moment together, because both cause lateral

buckling of the system i.e.

An important caveat to this approach is that if there are significant in-plane second order

effects (i.e. if the moment My,Ed is significantly amplified by the presence of the axial force

and in-plane deflections and imperfections) then these must be included in deriving My,Ed and

hence kult, . This is because since the moment My,Ed leads to lateral buckling, its full value

including second order effects must be used when checking lateral buckling.

The UK National Annex to EN 1993-1-1 limits the application of the rule to nominally

straight members. This restriction was not intended by the Eurocode drafters; moments from

initial curvature are included in the calculation of My,Ed perfectly satisfactorily. Indeed, the

example of application of the clause prepared by the Project Team[4]

features a curved

member.

The above format was not intended to be used to check other situations where the axial force

and moment do not both promote out of plane buckling. The checking of arches is one such

area, noting the limitations above in the UK NA regarding applicability only to straight

members! The format could, with care, however be applied to arches. The application of the

clause to the design of an arch is discussed in reference 5 where the method was shown to be

acceptable.

Pitfalls in elastic critical buckling analysis For those inexperienced in elastic critical buckling analysis, there are many pitfalls and some

examples are given below:

(i) Not using the correct mode in calculations The lowest global mode of buckling for the paired beams example above was shown in Figure

3. However, where shell elements have been used throughout, numerous local buckling

modes such as that shown in Figure 5 will usually be found at much lower load factors. These

typically correspond to buckling of the top of the web plate in compression or potentially to

torsional buckling of the top flange and may be ignored for the purposes of determining Mcr;

these buckling effects are considered in the effective section properties and flange outstand

shape limits in codified approaches. These modes may occur at much lower load factors than

the overall mode of buckling sought and their use in calculation would be very conservative.

It is important that this is understood. Simpler models can sometimes be used to avoid

determining modes that are of no interest e.g. the use of beam elements for flanges in plate

girders to eliminate flange torsional buckling modes.


Figure 5. Typical local elastic buckling modes for beams

(ii) Not appreciating the limitations of software Many, if not most, software programmes which can perform elastic critical buckling analysis

do so based on the initial un-deformed geometry. Where a structure (e.g. an arch) or element

flattens under load due to elastic shortening or abutment movements, the geometry changes

and the compressive forces can increase as illustrated in the simplified system in Figure 6.

Snap through buckling then becomes a possibility and this will be undetected by the software

unless it can include the effects of geometric non-linearity.

Figure 6. Flattening of arch (idealised as two pin-jointed members) due to abutment movement and elastic shortening

(iii) Not appreciating the limitations of code methods The slenderness of arches can be determined by first obtaining Ncr from an elastic critical

buckling analysis. This would be fine for the arch in Figure 7 with pin jointed hangers (not

shown). The slenderness is determined, then the reduction factor is obtained and the arch

strength is checked.

initial

shape


Figure 7. In-plane buckling of arch with pinned hangers

The same analysis can be used for the arch in Figure 8. However, this design has rigid

connections at the ends of the hangers. The buckling deformations induce moments in the

hangers but the analysis gives no information directly useful for checking the hangers.

Consequently, arch buckling may be checked as above but the additional effects on the

hangers are then missed if they are only designed for first order effects only. For this case, a

second order analysis should be used to determine the hanger moments.

buckled shape

Figure 8. In-plane buckling of arch with rigid hanger connections

(iv) Not understanding the software output The output of an eigenvalue buckling analysis is a series of buckling mode shapes and their

corresponding load factors. Often, software also displays moments and forces with each

mode. These are the internal effects associated with the mode shape when the peak

displacement in that mode has been set to unity in some set of units. It is not therefore

information that can be used directly in the design. Inexperienced engineers have however

been seen to try and design against the moments produced.

buckled shape


Non-Linear Analysis

Imperfections The modelling of imperfections is a key aspect to the non-linear analysis of structures.

Imperfections comprise geometric imperfections and residual stresses. The term “geometric

imperfection” is used to describe departures from the exact centreline setting out dimensions

found on drawings which occur during fabrication and erection. This is inevitable as all

construction work can only be executed within certain tolerances. Geometric imperfections

include lack of verticality, lack of straightness, lack of fit and minor joint eccentricities. The

behaviour of members under load is also affected by residual stresses within the members.

Residual stresses can lead to yielding of regions of members at lower applied external load

than predicted from stress analysis ignoring such effects, leading to a reduction in the member

stiffness. The effects of these residual stresses can be modelled by additional equivalent

geometric imperfections and these are given throughout EN 1993 for the overall design of

members (e.g 3-1-1/5.3.1(2)) and for local buckling of plates (e.g. 3-1-5/C.5). Member

imperfections can apply to overall structure geometries (global imperfection) or locally to

members (local imperfection).

Imperfections must be included in global analysis unless they are included by use of the

appropriate resistance formulae in clause 6.3 when checking the members. For example, the

flexural buckling curves provided in 3-1-1/Figure 6.4 include all imperfections for a given

member effective length of buckling. It should be noted that the equivalent geometric

imperfections given in EN 1993 are not slenderness dependent, being a function of length

only, whereas the imperfections in the resistance formulae are a function of the slenderness

with a cut-off level such that below a certain slenderness, no imperfection is applied in order

to replicate the results of tests for stocky elements. It should therefore be noted that if the

compression resistance of a simple pin-ended member of a given slenderness is obtained

using second order analysis with the imperfections given in Table 5.1 of EN 1993-1-1 for a

particular buckling curve, the resulting resistance will usually be slightly lower than that

obtained from the corresponding resistance curve in 3-1-1/Figure 6.4. For this reason, the UK

NA to EN 1993-1-1 requires the following: For elastic analysis of the cross-section, the initial imperfections for an individual section about a

particular axis should be back-calculated from the formula for the buckling curves given in BS EN

1993-1-1:2005, 6.3 using the elastic section modulus.

It may not be immediately apparent to designers how to do this but in fact 3-1-1/5.3.1(11)

itself provides guidance through an alternative method. To overcome this moderate

conservatism caused by the difference between imperfections recommended for global

analysis and those used in the resistance curves, EN 1993-1-1 provides an alternative method

whereby the imperfection for the whole structure (global and local imperfections) or an

element is based on the shape of the critical elastic buckling mode and with a magnitude

directly relating to that used in the resistance curves for the particular slenderness. This

unique imperfection is given by:


cr

maxcr,

Rk

2

1M

2

2init

1

120

''EI

M.

(D5.3-1)

cr represents the local ordinates of the mode shape and '' is the curvature produced by the

mode shape such that maxcr,''EI is the greatest bending moment due to cr at the critical cross

section. Other terms are as follows:

is the imperfection factor taken from 3-1-1/Tables 6.1 and 6.2 for the relevant mode of

buckling. For varying cross section, the greatest value can conservatively be taken.

cr

kult,

where kult, is the load amplifier to reach the characteristic squash load NRk of the

most axially stressed section and cr is the load amplifier for elastic critical buckling.

is the reduction factor for the above slenderness determined using the relevant buckling

curve appropriate to .

The derivation of this equation is given in reference 3.

This method and the proposed modification in the UK NA have the disadvantage that the

slenderness of the structure has to be determined first from an eigenvalue analysis which

tends to reduce the appeal of second order analysis as a practical design method. Second

order analysis of a pin-ended member with imperfections determined in this way will however

produce the same resistance as obtained from the resistance curves.

The above discussions relate in the main to flexural buckling. If lateral torsional buckling is

to be taken into account by second order analysis, the compression flange can be given a bow

imperfection about the beam minor axis. A value of 0.5 e0 is recommended in 3-1-1/5.3.4(3)

where e0 is again taken from 3-1-1/Table 5.1 (or back-calculated according to the UK NA,

which will improve the resistance) but the UK NA modifies this to the full value of 1.0 e0.

Example non-linear analysis for global buckling The same FE model of paired girders discussed in section 2.1 above was analysed under the

same loading considering non-linear material properties including strain hardening in

accordance with 3-1-5/Annex C (and in this case including the partial material factor for steel)

and non-linear geometry and including an initial deformation with shape corresponding to the

first elastic global buckling mode. This was used to determine the collapse load. The

magnitude of the largest bow deflection in this mode was taken as L/150 for curve d of Table

5.1 of EN 1993-1-1. The maximum moment reached and the moment at which first yield

occurred were noted. Failure occurred by rotation of the braced pair over a span in the same

shape as the elastic buckling mode of Figure 3; this equivalence in shape between eigenmode


and ultimate collapse mode will not generally occur in all buckling problems. Where there is

not equivalence, a refined (lower) prediction of the ultimate load will usually be obtained by

using the collapse geometry as a revised imperfection geometry for a new analysis.

Figure 9 shows the load-deflection curve up to failure for the bridge. The ultimate resistance

obtained by this method is given in Table 1 above. Non-linear analysis can be used to extract

greater resistance from beams for a number of reasons which include benefit from:

partial plastification of the tension zone in non-compact sections

strain hardening

moment redistribution in statically indeterminate structures (but not in the

above example).

Figure 9. Load-deflection curve for non-linear analysis of single span model

Local buckling Analysis of local buckling problems often requires a greater degree of experience and

understanding, particularly in the application of imperfections.

3-1-5/C.5 gives guidance on imperfections for the local modelling of plate elements. In

general, the distribution (or shape) of the imperfections to be used can be determined by one

of four methods:

Single Span - LTB - Non Linear

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 100 200 300 400 500 600

Vertical Displacement at most affected node (mm)

Lo

ad

Facto

r


1) Using the same distribution as the mode shapes found from elastic critical buckling analysis Elastic critical buckling analysis can be used to determine a unique imperfection distribution,

with the same form as the buckling mode shape, in the same manner as discussed in section

3.1 above for frames. It is often assumed that this method of applying imperfections will

maximise the reduction in resistance but this is not always true and there are difficulties in

implementation. The imperfection distribution will vary with each load case and it is difficult

to specify the imperfection magnitude for coupled modes involving both overall stiffened

panel buckling and local sub-panel buckling. The elastic buckling mode with the lowest load

factor may not also be the critical mode shape for reducing ultimate strength. Often, a slightly

lower resistance is produced using method 4).

2) Using assumed imperfection shapes based on buckling under direct stress The imperfection distribution can be based on the local and global plate buckling mode

shapes for compression acting alone in the longitudinal direction. This method will not

necessarily maximise the loss of resistance, but the resulting resistance will usually not be far

from the true resistance.

3) Applying transverse loading A variation on 2) above is to apply transverse loading so that the first order effects of such

loading replicate the first order effects of imperfections.

4) Application of the deformed shape at failure In this method, the deformed shape of the structure obtained at failure from a previous

analysis is used as the initial imperfection shape. This frequently gives the lowest resistance

(but rarely significantly lower than the other methods). It has the disadvantage that the

method is iterative, as an initial analysis to failure is required to produce the imperfection

shape.

A more detailed description of a typical application of non-linear analysis to a local buckling

problem (buckling of transversely stiffened webs in bending and shear) is given in reference

6.

Conclusions Eurocode 3 offers some real improvements in the codification for the design of steel plate

structures in that it provides a framework in which advanced methods may be used. This

paper has shown that there are some significant benefits that can be realised from this

approach but that there are also warnings that the more advanced methods require a greater

level of understanding of the fundamental physics behind the code. There are risks for the

unwary in unlocking the benefits of the code. Like all codes and standards, there is a

reasonable expectation that the user is familiar with the subject matter and competent in its

use.


References [1] PD 6695-2: Recommendations for the design of steel bridges. British Standards

Institution, London.

[2] Hendy C.R and Jones R.P (2009) Lateral buckling of plate girders with flexible restraints,

ICE Bridge Engineering, March 2009, Thomas Telford, London.

[3] Hendy C.R. and Murphy C.J. (2007) Designers’ Guide to EN1993-2, Eurocode 3: Design

of steel structures. Part 2: Steel bridges. Thomas Telford, London. ISBN 9780727731609

[4] CEN/TC250/SC3/N1639E, CEN background document

[5] Baird B, Hendy C.R, Wong P, Jones R.P, Sollis A.J, Nuttall H, Design of the Olympic

Park Bridges H01 and L01, to be published in Structural Engineering International

[6] Presta F., Hendy C.R. and Turco E. (2008) Numerical validation of simplified theories for

design rules of transversely stiffened plate girders, The Structural Engineer, Volume 86,

Number 21 pp 37 – 46


Appendix A – Overview of Analysis Approaches

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ed

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ly fr

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Glo

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imp

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ion

s in

corp

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ted

into

mo

de

l ge

om

etr

y

Me

mb

er i

mp

erf

ect

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m

ate

rial

no

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ear

ity

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rate

d in

to m

od

el

De

sign

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ctly

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Safe

ty d

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ed

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ty

form

at.

Inse

nsi

tive

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al

seco

nd

ord

er

eff

ect

s?

Y

N

Figure A.1. Overview of analysis options to account for global and member buckling, second order effects and imperfections

en1993 practice paper-buckling analysis of steel bridges

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