Beam Columns II

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STEEL CONSTRUCTION: ELEMENTS

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STEEL CONSTRUCTION:

ELEMENTS

Lecture 7.10.2: Beam Columns II

OBJECTIVE/SCOPE

To extend the introductory coverage of beam columns given in Lecture 7.10.1 to cover

the full three-dimensional case.

PREREQUISITES

Simple bending and torsion theory

Lecture 7.2: Cross-Section Classification

Lectures 7.5: Columns

Lectures 7.8: Restrained Beams

Lectures 7.9: Unrestrained Beams

Lecture 7.10.1: Beam columns I

RELATED LECTURES

Lecture 7.11: Frames

RELATED WORKED EXAMPLES

Worked Example 7.10: Beam Columns

SUMMARY

This lecture expands on the treatment of beam-columns given in Lectures 7.10.1 to

include the cases of out-of-plane buckling and biaxial bending. The basis for the

Eurocode 3 interaction formulae is discussed and related to physical behaviour [1].

1. INTRODUCTION

Lecture 7.10.1 introduced all the main aspects of beam-column behaviour and design

within the context of the uniaxial in-plane case. More general forms of response are,

however, possible. This lecture broadens the coverage to include all of the main cases.


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2. FORMS OF BEHAVIOUR

Three separate forms of beam-column behaviour are illustrated in Figure 1.

If the member is bent about its weaker principal axis, or is prevented from deflecting

laterally when bent about its stronger principal axis as shown in Figure 1a, then its

response will be confined to the plane of bending. This case has been covered in Lecture

7.10.1.

When a laterally unbraced beam-column of open cross-section is bent about its stronger

principal axis as shown in Figure 1b, then it may buckle prematurely out of the plane of

loading by deflecting laterally and twisting. Such behaviour is conceptually and

mathematically very similar to the lateral-torsional buckling of beams described in

Lectures 7.9.1and 7.9.2.


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The most general situation is illustrated in Figure 1c. When bending is applied about both

principal axes the member's response will be 3-dimensional in nature, involving biaxial

bending and twisting.

In Figure 1 the nature of the interaction in each case is listed in the caption. Clearly the

behaviour shown as Figure 1c is the most general, with that of Figures 1a and 1b being

simpler and more limited cases. For a full treatment of the in-plane case of Figure 1a refer

back to Lectures 7.10.1.


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3. FLEXURAL-TORSIONAL BUCKLING

When a laterally unrestrained I-section beam-column is bent about its major axis, it may

buckle by deflecting laterally and twisting at a load which is significantly less than the

maximum load predicted by an in-plane analysis. Assuming elastic behaviour and the

arrangement of applied loading and support conditions given in Figure 2, the critical

combinations of N and M may be obtained from the solution of:

= (1)


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in which io= [(Iy+ Iz)/A]1/2 - is the polar radius of gyration

Nz= 2EIz/L2 - is the minor axis critical load

No= (GJ/io2) (1 + 2EIw/GItL2) - is the torsional buckling load

Equation (1) reduces to the buckling of a beam when N 0 and to the buckling of acolumn in either flexure (Nz) or torsion (No) as M 0. In the first case the critical valueof M will be given by:

Mcr= (2)

in which EIz - is the minor axis flexural rigidity

GIt - is the torsional rigidity

EIw - is the warping rigidity


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In deriving Equation (1) no allowance was made for the amplification of the in-plane

moments M by the axial load acting through the in-plane deflections. As explained in

Lecture 7.10.1this may be approximated as M/(1-N/Ny). Equation (1) can, therefore, be

modified to:

(3)

Noting the relative magnitudes of Ny, Nz and No and re-arranging gives the following

approximation:

N/Nz+ {1/(1-N/Ny)}{M/i(NzNo)1/2} = 1 (4)

or

N/ Nz+ {1/(1-N/ Ny)}M/Mcr= 1 (5)

4. DESIGN

For design purposes it is necessary to make suitable allowances for effects such as initial

lack of straightness, partial yielding, residual stresses, etc., as has been fully discussed in

earlier lectures in the context of columns and beams. Thus some modification to Equation

(5) is necessary to make it suitable for design. In particular, the end points (corresponding

to the cases of M = 0 and N = 0) must conform to the established procedures for columns(Lectures 7.5.1and 7.5.2) and beams (Lectures 7.9.1and 7.9.2).

Eurocode 3 [1] uses the interaction equation:

1 (6)

In which kLT - is a coefficient whose value depends upon:

the level of axial load as measured by the ratio Nsd/ zA fy.

the member slenderness z. the pattern of primary moments.

and LT - is the reduction factor for lateral-torsional beam buckling.

For the most severe combination kLT adopts the value unity, corresponding to a linear

combination of the compressive and bending terms. This reflects the reduced scope for

amplification effects in this case, since the value of Nsdcannot exceed zA fy, which will,in turn, be significantly less than the elastic critical load for in-plane bucking N y.

It is, of course, also necessary to ensure against the possibility of in-plane failure by

excessive deflection in the plane of the web at a lower load than that given by Equation


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(6). This might occur, for example, in situations where different bracing and/or support

conditions are provided in the xy and xz planes as illustrated in Figure 3.

Such cases should be treated by checking, in addition to Equation (6), an in-plane

equation of the form:

1 (7)

in which mindepends on the in-plane conditions. Usually, however, Equation (6) willgovern.

5. BIAXIAL BENDING

Analysis for the full 3-dimensional case, even for the simple elastic version, is extremely

complex and closed-form solutions are not available. Rather than starting analytically it is

more convenient to approach the question of a suitable design approach from

considerations of behaviour and the use of the methods already derived for the simplercases of Figures 1a and 1b.


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Figure 4 presents a diagrammatic version of the design requirement. The N-Mzand N-My

axes correspond to the two uniaxial cases already examined. Interaction between the two

moments Mzand Mycorresponds to the horizontal plane. When all three load components

N, Mzand Myare present the resulting interaction plots somewhere in the 3-dimensionalspace represented by the diagram. Any point falling within the boundary corresponds to a

safe combination of loads.

Assuming proportional loading any load combination may be regarded as a straight line

starting at the origin, the orientation of which depends upon the relative sizes of the three

load components. Increasing the loads extends this line from the origin until it just

reaches and then exceeds the boundary. Non-proportional loading would correspond to a

series of lines.

In each case the axes have been taken as the ratio of the applied component to the

member's resistance under the load component alone, e.g. Nsd/ minAfyin the case of thecompressive loading. Thus Figure 4 actually represents the situation for one particular

example with particular values of cross-sectional properties, slenderness and load

arrangement. Changing some or all of these will alter the shape of the interaction surface

shown, but not the general principle involved.

6. DESIGN FOR BIAXIAL BENDING AND

COMPRESSION

For design purposes, it is necessary to have a convenient representation of the situation

described in Section 5 by using an interaction equation containing the three load


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components N, Mzand My. Parts of this equation, corresponding to the two 2-dimensional

cases represented by the N, Mzand N, Myplanes, have already been discussed. The full

equation must clearly reduce to these in the absence of the third load component.

Eurocode 3 [1] uses the pair of formulae:

1 (8)

1 (9)

Two checks are necessary because, under the action of compression plus major axismoment on an I-section with different support conditions in the zx and yx planes, it is not

known whether the in-plane or out-of-plane interaction will be the more critical; that is to

say whether, in the absence of Mz, failure would occur as shown in Figure 1a or Figure

1b. For the same conditions in both planes and z> y, minwill correspond to zandEquation (9) will govern since LT, the reduction factor for lateral-torsional buckling

under pure bending, will be less than or (if LTis small) equal to unity.

For cross-sections not susceptible to lateral-torsional buckling, e.g. tubes, only Equation

(8) is required since LT= 1.

7. TREATMENT OF OTHER THAN CLASS 1 OR 2

SECTIONS

The design formulae given as Equations (6) - (9) relate specifically to the case of Class 1

or 2 sections, i.e. those for which the proportions of the plate elements meet the

limitations necessary to ensure the development of full cross-sectional plasticity, as

explained in Lecture 7.2. When using either Class 3 or Class 4 sections some

modifications are necessary.

For Class 3 cross-sections the quantities Wpl.y and Wpl.z should be replaced by the

equivalent elastic quantities Wel.yand Wel.z.

When Class 4 sections are being employed the section properties A and W must relate to

the effective cross-section; the shift of the neutral axis of the effective cross-section from

its original position due to loss of effectiveness of some parts of the cross-section must

also be allowed for. Thus Equations (8) and (9) become:

1 (10)


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1 (11)

in which Aeff, Weff.yand Weff.zare the effective properties in the presence of only uniform

compression or moment about the y and z axes respectively and eN is the shift of the

neutral axis when the cross-section is subject to uniform compression.

An important point to note from the definition of Aeff and Weff above is that the

calculation of cross-sectional properties, and thus also cross-sectional classification,

should be undertaken on a separate basis for each of the three load components N, Myand

Mz. This does, of course, mean that the same member may be classified as (say) Class 1

for major-axis bending, Class 2 for minor-axis bending and Class 3 for compression. In

such cases the safe design approach is to conduct all beam-column checks using the

procedures for the least favourable class.

8. DETERMINATION OF k-FACTORS

The value of kLTfor use in Equation (6) is actually given by:

(but kLT1) (12)

in which LT= 0,15 zM.LT- 0,15 (but LT0,90) (13)

and M.LT - is the equivalent uniform moment factor for lateral-torsional bucklingdetermined from Table 2.

In Equations (7) - (11) the values of kyand kzshould be obtained from:

k = but k 1,15 (14)

= (2M- 4) + (Wpl- Wel)/Wel but 0,9 (15)

in which , , , M, Wpland Welall relate to the axis under consideration, i.e. y or z, andMis determined from Table 2.

For Class 3 or 4 cross-sections the second term in Equation (15) should be omitted.

9. CROSS-SECTION CHECKS

If allowance has been made when determining the k-factors (through the use of M) forthe less severe effect of patterns of moment other than uniform single curvature bending,it is necessary further to check that the cross-section is everywhere capable of locally


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resisting the combination of compression and primary moment(s) present at any point.

This location will usually be one or other of the ends as explained in Lecture 7.10.1.

Expressions for checking several types of cross-section under compression plus uniaxialbending were given in Lecture 7.10.1. For biaxial bending Eurocode 3 [1] uses:

1 (16)

in which the values of and depend upon the type of cross-section as indicated in Table1.

A simpler but conservative alternative is:

1 (17)

10. CONCLUDING SUMMARY

Depending on the form of the applied loading, 3 types of beam-column problemmay be identified.

The biaxial bending case is the most general and includes the 2 others as simplerand more restricted component cases.

Interaction equations are used for design purposes. These make use (as end points) of the design procedures for beams (N = 0) and

columns (M = 0).

The class of cross-section will affect some of the values used in the interactionequations.

11. REFERENCES

[1] Eurocode 3: "Design of Steel Structures": ENV 1993-1-1: Part 1.1: General rules and

rules for buildings, CEN, 1992.

12. ADDITIONAL READING

1. Chen, W. F. and Atsuta, T., "Theory of Beam-Columns" Vol. 2, McGraw-Hill,

1977.

Comprehensive treatment of the beam-column problem for the cases of flexural-

torsional bucking and biaxial bending.

2.

Trahair, N. S. and Bradford, M. A., "Behaviour and Design of Steel Structures",2nd edition, Chapman and Hall, 1988.


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Chapter 7 refers to beam-columns, including a comparison of the subject's

treatment in three design codes (not including Eurocode 3).

3.

Ballio, G. and Mazzolani, F. M., "Theory and Design of Steel Structures",Chapman and Hall, 1983.

Gives basis of original European approaches to the use of interaction formulae,

including derivations.

4. Galambos, T. V., "Guide to Stability Design Criteria for Metal Structures", 4th

edition, Wiley Interscience.

Chapter 8 presents a comprehensive review of theoretical, experimental and

design-oriented contributions to the topic of beam-column behaviour.

5. Dowling, P. J., Owens, G. W. and Knowles, P., "Structural Steel Design",

Butterworths, 1988.

Chapter 24 deals with beam-column behaviour and design, including explanations

of the physical significance of the concepts of interaction and slenderness.

6. Nethercot, D. A., "Limit State Design of Structural Steelwork", 2nd edition,

Chapman and Hall, 1991.

Chapter 6 deals with beam-column behaviour and design.

Table 1 Values of and for use in Equation (16)

Type of cross-section

I and H - sections

Circular tubes

Rectangular hollow sections

Solid rectangles and plates

2

2

but 6

1,73 + 1,8n3

5n but 1

2

but 6

1,73 + 1,8n3

n = Nsd/ Npl.Rd


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Table 2 Equivalent uniform moment factors M

Moment diagram Equivalent uniform moment factor M

End moments

M1 M

1

1 1

M,= 1,8 - 0,7

Moments due to in-planelateral loads

M,Q= 1,3

M,Q= 1,4

Moments due to in-planelateral loads plus end

moments

MQ

MQ

MQ

M1

M1

M1

M

M

M

M= m, + (M,Q- M, )

MQ= Max Mdue to lateral load only

M = Max Mfor moment diagramwithout change of sign

M = Max M+ Min Mwhere sign ofmoment diagram changes

Beam Columns II

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