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Member designFrom Steelconstruction.info

This article describes the verification of steel members subject to shear, bending moments and axial forces.The member must provide adequate compression, tension, bending and shear resistance. Where the memberis subjected to axial and lateral loading simultaneously, additional resistance requirements checks, takinginto account the combination of these loading effects will be required.

Member design follows the requirements given in BS EN 1993­1­1[1]. The overall process of memberdesign includes:

Classification of cross sectionsCross­section resistanceMember buckling (buckling under axial compression or lateral torsional buckling under bending)Combined axial loading and bending, where applicable.

SCI P362 forms the background to member design presented in this article and provides morecomprehensive guidance.

Contents

1 Partial factors for resistance2 Classification of cross sections3 Resistance of cross sections

3.1 General3.2 Material strength3.3 Section properties

3.3.1 Gross cross­section3.3.2 Net sections

3.4 Tension3.5 Compression3.6 Bending3.7 Torsion3.8 Shear3.9 Bending and shear3.10 Bending and axial force3.11 Bending, shear and axial force

4 Buckling resistance of members4.1 Uniform members in compression

4.1.1 Buckling resistance4.1.2 Flexural buckling (only)

4.2 Uniform members in bending4.2.1 Lateral torsional buckling resistance4.2.2 Reduction factor for lateral torsional buckling of rolled sections

4.3 Uniform members in bending and axial compression4.4 Columns in simple construction

5 References6 Further reading7 Resources8 See also9 External links10 CPD

Partial factors for resistance

The partial factors γM that are applied to the various characteristic values of resistance in member designare:

Resistance of cross­section: γM0 = 1.00Resistance of members to instability: γM1 = 1.00 (used in buckling resistance)Resistance of cross­sections in tension to fracture: γM2 = 1.10

The values are those given in the UK National Annex to BS­EN 1993­1­1[2].

Classification of cross sections

Four classes of cross­sections of are defined in BS EN 1993­1­1[1]:

Class 1 cross­sections are those which can form a plastic hinge with the rotation capacity required forplastic analysis without reduction of the resistance.Class 2 cross­sections are those which can develop their plastic moment resistance, but which havelimited rotation capacity because of local buckling.Class 3 cross­sections are those in which the stress in the extreme compression fibre of the steelmember assuming an elastic distribution of stresses can reach the yield strength, but local bucklingprevents development of the plastic moment resistance.Class 4 cross­sections are those in which local buckling will occur before the attainment of yieldstress in one or more parts of the cross­section.

The class of the cross section is determined from Table 5.2 of BS EN 1993­1­1[1] , where a cross section isclassified according to the highest (least favourable) class of its compression parts. See also SCI P362 .

Section classification is also given in resistance tables, such as SCI P363 (the 'Blue Book' ). SCI P363 givesaxial load ratios where (under increasing levels of axial load) a section becomes Class 3 and Class 4.

Class 4 cross sections are not considered in this article.

Resistance of cross sections

General

The design value of an action effect in each cross­section should not exceed the corresponding resistanceand if several action effects act simultaneously, the combined effect should not exceed the resistance forthat combination. As a conservative approximation for all cross­sections, a linear summation of theutilization ratios for each resistance may be used. For the combination of NEd, My,Ed and Mz,Ed this methodmay be applied using the following criteria:

NRd, My,Rd and Mz,Rd are the design values of the resistance depending on the cross­sectional classificationand including any reduction that might be caused by shear effects.

More generally, the Eurocodes provide specific clauses for common combined effects (for exampleBending and shear, Bending and axial force and Bending, shear and axial force) which should be used inpreference to this simplified approach.

Material strength

According to the UK National Annex to BS­EN 1993­1­1[2], yield strength fy and ultimate strength fu must

be taken from the product standard not Table 3.1 of the design standard. Moreover, if a range of ultimatestrengths is given in the product standard, the lowest value must be adopted. Yield strengths and ultimatestrengths for hot­rolled steelwork are given in BS EN 10025­2[3].

Section properties

Gross cross­section

The properties of the gross cross­section should be determined using the nominal dimensions. Holes forfasteners need not be deducted, but allowance should be made for larger openings. Splice materials shouldnot be included.

Net sections

The net area of a cross section should be taken as its gross area less appropriate deductions for all holes andother openings. For calculating net section properties, the deduction for a single fastener hole should be thegross cross­sectional area of the hole in the plane of its axis. For countersunk holes, appropriate allowanceshould be made for the countersunk portion.

Tension

The design value of the tension force NEd at each cross­section should satisfy:

For sections without holes, the design tension resistance Nt,Rd should be taken as the design plasticresistance of the gross cross­section:

where

A is the gross cross section

For sections with holes, the design tension resistance Nt,Rd should be taken as the smaller of:

The plastic resistance of the gross cross­section (given above); andThe design ultimate resistance of the net cross­section at holes for fasteners:

For angles connected through one leg, refer to BS EN 1993­1­8[4] Clause 3.10.3.

Similar consideration should also be given to other types of sections connected through outstands.

Compression

The design value of the compression force NEd at each cross­section should satisfy:

The design resistance of the cross­section for uniform compression Nc,Rd should be determined as follows:

for class 1, 2 or 3 cross­sections

Section classification is given in resistance tables, such as SCI P363 (the 'Blue Book').

For members of uniform cross­sections in axial compression the design buckling resistance, Nb,Rd almostalways governs.

Compression resistance design tool

Bending

The design value of the bending moment MEd at each cross­section should satisfy:

Where the design resistance for bending about one principal axis of a cross­section Mc,Rd is determined asfollows:

for Class 1 or 2 cross­sections

for Class 3 cross­sections

and

Wel,min corresponds to the fibre with the maximum elastic stress.

For bending about both axes, the following criterion may be used for I and H sections.

Bending resistance design tool

Torsion

Beams subject to loads which do not act through the point on the cross­section known as the shear centrenormally suffer some twisting. For doubly symmetrical sections such as UKB or UKC, the shear centrecoincides with the centroid, while for channels it is situated on the opposite side of the web from thecentroid.

If torsion cannot be avoided, a torsionally stiff section, such as a hollow section, should be used. The twistof an open section may be very significant and must be considered if this type of section is used.

More information on torsional resistance is given in SCI P385.

Shear

The design value of the shear force VEd at each cross­section should satisfy:

where:

Vc,Rd is the design plastic shear resistance Vpl,Rd.

In the absence of torsion, the design plastic shear resistance is given by:

where:

Av is the shear area.

For rolled I and H sections, with load parallel to the web, the shear area Av is given by:

Av = A ­ 2b tf + (tw + 2r)tf

The shear resistance may be limited by shear buckling. For such situations, reference is to be made toBS EN 1993­1­5[5]. Shear buckling is rarely a consideration with hot rolled sections.

Bending and shear

Where shear is present, allowance should be made for its effect on the bending resistance.

Where VEd < 0.5Vpl,Rd the effect of the shear force on the bending resistance may be neglected, exceptwhere shear buckling reduces the section resistance.

Where VEd ≥ 0.5Vpl,Rd the reduced moment resistance should be taken as the design resistance of the cross­section, calculated using a reduced yield strength:

(1 ­ ρ) fy for the shear area, where:

and Vpl,Rd is calculated as described here.

Bending and axial force

When an axial force is present, allowance should be made for its effect on the plastic moment resistance.For Class 1 and 2 cross­sections, the following criteria should be satisfied:

MEd ≤ MN,Rd

where:

MN,Rd is the design plastic moment resistance reduced due to the axial force NEd.

For doubly symmetric I­ and H­sections within certain limits, the effect of axial force may be neglected.This is covered in clause 6.2.9 of BS EN 1993­1­1[1]:

For Class 3 cross­sections the maximum longitudinal stress due to moment and axial force, taking accountof fastener holes, where relevant, should not exceed fy/γM0.

Combined axial compression and bending resistance design tool

Bending, shear and axial force

Where VEd ≤ 0.5Vpl,Rd, no reduction of the resistances defined for bending and axial force need be made.

Where VEd > 0.5Vpl,Rd, the design resistance of the cross­section to combinations of moment and axialforce should be calculated using a reduced yield strength, as given for bending and shear.

Buckling resistance of members

Uniform members in compression

BS EN 1993­1­1[1] covers three modes of buckling when subject to axial compression:

Flexural buckling (commonly known as strut buckling)Torsional buckling, which may be critical for cruciform sections subject to axial compressionTorsional­flexural buckling, which may be critical for asymmetric sections subject to axialcompression.

Buckling resistance

A compression member is verified against buckling by the relationship:

where:

NEd is the design value of the compression forceNb,Rd is the design buckling resistance of the compression member where:

for Class 1, 2 and 3 cross­sections, and

χ is the reduction factor for the relevant buckling mode

For axial compression in members, the value of χ for the appropriate non­dimensional slenderness isdetermined from the appropriate buckling curve according to:

but χ ≤ 1

where:

where α is an imperfection factor.

The non­dimensional slenderness is given by:

where Ncr is the elastic critical force for the relevant buckling mode.

For each mode of buckling, the value of Ncr is determined.

Open sections (UKB, UKC) (bi­symmetric sections) are not subject to torsional flexural buckling. Opensections do exhibit torsional buckling, but for any given length, minor axis flexural buckling is critical. SCIP363 (the Blue Book) provides flexural buckling resistances in both axis and the torsional bucklingresistance.

For angles, an effective slenderness should be calculated from Annex BB.1.2 of BS EN 1993­1­1[1]. Asimilar effective slenderness can be calculated for channels which are only connected through the web.

Selection of flexural buckling curve for a cross sectionPermission to reproduce extracts from British Standards is granted by the British Standards

Institution (BSI). No other use of this material is permitted. British Standards can be obtained in PDFor hard copy formats from the BSI online shop: http://shop.bsigroup.com or by contacting BSI

Customer Services for hard copies only: Tel: +44 (0)20 8996 9001, Email: cservices@bsigroup.com

See the Compression resistance design tool.

Flexural buckling (only)

For flexural, or strutbuckling, Ncr, the Euler

load, is equal to and the non­dimensionalslenderness is givenby:

forClass 1, 2 and 3 cross­sections, where:

Lcr is the bucklinglength in the axisconsideredi is the radius ofgyration about therelevant axis,determined usingthe properties ofthe gross cross­sectionλ1 = 86 for gradeS275 steelλ1 = 76 for gradeS355 steel

The imperfection factorα corresponding to theappropriate bucklingcurve is obtained fromthe table below. Thechoice of buckling curveis obtained from thetable to the right.

Imperfection factors for flexural buckling curvesBuckling curve a b c d

Imperfection factor α 0.21 0.34 0.49 0.76

Buckling curves for χ

The value of χ may be calculated, or may be obtained from a graph or a table. The graphical presentation isshown in the figure below, taken from SCI P362.

Uniform members in bending

Lateral torsional buckling resistance

A laterally unrestrained member subject to major axis bending is verified against lateral­torsional bucklingby the relationship:

where:

MEd is the design value of the momentMb,Rd is the design buckling resistance moment.

Beams with sufficient restraint to the compression flange are not susceptible to lateral­torsional buckling.

The design buckling resistance of a laterally unrestrained beam is given by:

where:

Wy is the appropriate section modulus as follows:Wy = Wpl,y for Class 1 and 2 cross­sectionsWy = Wel,y for Class 3 cross­sections

χLT is the reduction factor for lateral­torsional buckling.

See the Bending resistance design tool.

Reduction factor for lateral torsional buckling of rolled sections

For rolled sections of constant cross­section in bending, the value of χLT for the appropriate non­

dimensional slenderness LT is determined from:

but χLT ≤ 1

where:

For rolled sections:

λLT,0 = 0.4β = 0.75

The use of these values is endorsed by the UK National Annex[2].

where:

Wy is the appropriate section modulus for the section classificationMcr is the elastic critical moment for lateral­torsional buckling

An expression to evaluate Mcr is not given in BS EN 1993­1­1[1], however, methods which enable Mcr to bedetermined include:

Method 1

Values of C1 and for various moment conditions (load isnot destabilizing)

NCCI document SN003 providesappropriate expressions to calculate Mcr.For loads which are not destabilizing,and for doubly symmetric sections, i.e.UKB and UKC :

where:

E, G are material propertiesIz, It, Iw are section propertiesobtained from SCI P363 (the Bluebook)L is the buckling length of thememberC1 is a factor that depends on theshape of the bending momentdiagram ­ see figure on the right.

Method 2

Mcr may be determined using thesoftware LTBeam.

Alternatively, Mcr may be determinedusing the Elastic critical moment forlateral­torsional buckling (Mcr) calculation tool.

Other (simplified) approaches are described in SCI P362 Section 6.3.2.3.

The value of the imperfection parameter αLT corresponding to the appropriate buckling curve is given bythe table below and the choice of buckling curve given in the subsequent table.

Imperfection factors for lateral torsional bucklingcurves

Buckling curve a b c dImperfection factor αLT 0.21 0.34 0.49 0.76

Recommendations for the selection of lateral torsional curve

Cross­section Limits Buckling curve

Rolled doubly symmetric I and H sectionsand hot finished hollow sections

h/b ≤ 2 b2 < h/b ≤ 3.1 c

h/b > 3.1 d

Lateral torsional buckling curves for rolled sections

h/b > 3.1 dAngles (for moments in the major principal plane) dAll other hot­rolled sections d

Cold­formed hollow sectionsh/b ≤ 2 c

2 ≤ h/b < 3.1 d

The lateral torsional buckling curves for rolled sections are shown in the figure below, taken from SCIP362.

Having calculated λLT and selected the appropriate curve, the reduction factor χLT may be calculated ordetermined from look­up tables in SCI P362, or by using the above figure.

Uniform members in bending and axial compression

For members of structural systems, verification of buckling resistance of doubly symmetric cross­sectionsmay be carried out on the basis of the individual single span members regarded as cut out of the system.Second order effects of the sway system (P­Δ effects) should be taken into account, either by the endmoments of the member or by means of appropriate buckling lengths about each axis for the globalbuckling mode.

Members which are subjected to combined bending and axial compression should satisfy:

Where:

NEd, My,Ed and Mz,Ed are the design values of the compression force and the maximum momentsabout the y­y and z­z axes along the member, respectivelyNb,y,Rd and Nb,z,Rd are the design buckling resistances of the member about the major and minor axisrespectivelyMb,Rd is the design buckling resistance moment

Mcb,z,Rd for Class 1 and 2 sections

Mcb,z,Rd for Class 3 sectionskyy, kyz, kzy, kzz are interaction factors, which may be determined from Annex A or B ofBS EN 1993­1­1[1].

Annex B is recommended as the simpler approach for manual calculations. Use of either Annex ispermitted by the UK National Annex[2].

In some cases, a conservative value of the k factors may be sufficient for initial design. The following tablegives maximum values, based on Annex B of the Standard, and assuming the sections are susceptible totorsional deformations, i.e. not hollow sections.

k factors

Interaction factorMaximum values

Class 3 Class 1 and 2kyy Cmy × 1.6 Cmy × 1.8kyz kzz 0.6 × kzzkzy 1.0 1.0kzz Cmz × 1.6 Cmz × 2.4

The equations to calculate the interaction factors are given in SCI P362 Appendix D. A series of graphs areprovided in SCI P362 from which accurate values of the interaction factors may be determined as analternative to calculation.

See the Combined axial compression and bending resistance design tool.

Columns in simple construction

Design of columns in simple construction is based on NCCI document SN048 in which a column in simpleconstruction subject to nominal bending moments and axial compression may be verified using simplifiedinteraction criteria.

See the Columns in simple construction design tool.

References

1. ^ 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 BS EN 1993­1­1: 2005, Eurocode 3: Design of steel structures. Generalrules and rules for buildings, BSI

2. ^ 2.0 2.1 2.2 2.3 NA to BS EN 1993­1­1:2006. UK National Annex to Eurocode 3: Design of steelstructures General rules and rules for buildings, BSI

3. ^ BS EN 10025­2:2004 Hot rolled products of structural steels. Technical delivery conditions fornon­alloy structural steels, BSI.

4. ^ BS EN 1993­1­8:2005. Eurocode 3: Design of steel structures. Design of joints, BSI5. ^ BS EN 1993­1­5:2006. Eurocode 3: Design of steel structures Plated structural elements. BSI

Further reading

LTBeam (http://www.steelbizfrance.com/telechargement/desclog.aspx?idrub=1&lng=2) is a softwaretool which deals with the elastic 'Lateral Torsional Buckling of Beams under bending action abouttheir major axis.Steel Designers' Manual 7th Edition. (http://shop.steel­sci.com/products/231­steel­designers­manual­7th­edition.aspx) Editors B Davison & G W Owens. The Steel Construction Institute 2012, Chapters14, 15, 16, 17 and 19

ResourcesSCI P361 Steel Building Design: Introduction to the Eurocodes, 2009SCI P362 Steel Building Design: Concise Eurocodes, 2009SCI P363 Steel Building Design: Design Data, 2013.An interactive online version, or eBlue Book (http://tsbluebook.steel­sci.org/) , is also available.SCI P364 Steel Building Design: Worked Examples ­ Open Sections, 2009SCI P385 Design of steel beams in torsion, 2011.NCCI: SN003b­EN­EU Elastic critical moment for lateral torsional buckling.NCCI: SN048b­EN­GB Verification of columns in simple construction ­ a simplified interactioncriterion.

Member design tools:

Compression resistance design toolBending resistance design toolCombined axial compression and bending resistance design toolColumns in simple construction design toolElastic critical moment for lateral torsional buckling calculation toolConcrete filled structural hollow section design tool

See also

Steel construction productsDesign codes and standardsModelling and analysisDesign software and tools

External links

LTBeam (http://www.cticm.com/content/ltbeam­version­1011)

CPD

Introduction to Eurocode 3Worked examples to Eurocode 3

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