Principles of steelbeam design

The key to understanding the design of steel beams is to determine whether or not the beam is restrained against lateral torsional buckling. This subject is covered in the Level 1 (No. 16) Technical Guidance Note: Lateral torsional buckling. If a beam is restrained, all that needs to be checked is the bending moment and shear resistance of the beam as well as serviceability limits against the applied load/actions. All of which are based on the beam’s core properties.

If the beam is unrestrained along any part of its length however, then there is a risk it will fail due to lateral torsional buckling. To address this, Eurocode 3 establishes a reduction factor that is applied to the bending moment resistance of the beam. Calculating this factor is the cornerstone of unrestrained steel beam design within Eurocode 3.

Frequent references will be made on the section variables throughout this guide. You are advised therefore to examine Figure 1 for the defi nition and nomenclature of these variables.

Steel material propertiesThe density of steel is 7850 kg/m3 and the Young’s Modulus (E) is 210,000 N/mm2.

S275 and S355 are the two strength

Designing a steel beamIntroduction

This Technical Guidance Note is the fi rst of the Level 2 guides. Guides in this next level build on what has been described previously in the Level 1 series. The topics covered at Level 2 are of a more complex nature as they typically deal with the design of elements as opposed to core concepts such as loading and stability. As such, the amount of prior knowledge the reader is assumed to have is at the very least the contents of relevant Level 1 Technical Guidance Notes.

The subject of this guide is the design of non-composite steel beams to BS EN 1993-1-1 – Eurocode 3: Design of Steel Structures – Part 1-1: General Rules for Buildings. It covers both restrained and unrestrained rolled steel ‘I’ and ‘H’ beam sections. It does not encompass the design of ‘T’ sections, hollow sections, castellated beams, angles and welded sections.

W Applied practice

W Worked example

W Further reading

W Web resources

ICON LEGEND

grades of steel most commonly used in the construction industry within the UK. S275’s nominal yield strength is 275 N/mm2 and Grade S355’s nominal yield strength is 355 N/mm2. The actual yield strength is dependent on the maximum thickness of an element within a steel section, as the thicker the element the lower the yield

strength. Table 1 defi nes what the yield strength should be for the most common ranges of thicknesses found in open rolled steel sections. These fi gures are based on the values given in BS EN 10025 Hot Rolled

Products of Structural Steels, which is the product standard for rolled steel sections of various sub-grades.

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Technical Guidance Note January 2013

Note 1 Level 2

Description Variable

b Width of fl ange

h Depth of beam

z-z Minor axis

y-y Major axis

d Depth of web

tw Thickness of web

tf Thickness of fl ange

r Radius of root fi llet between web and fl ange

Wpl,y Plastic section modulus about the y-y axis

Wel,y Elastic section modulus about the y-y axis

iz Radius of gyration about the z-z axis

A Cross sectional area of the beam

Iyy Second moment of area about the y-y axis

N Figure 1Beam section notation used in Eurocode 3

W Principles of steel beam design

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Technical Guidance Note

Technical

January 2013

Note 1 Level 2

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of the elements must lie. These limits are further modifi ed based on the yield strength of the material; this is defi ned via coeffi cient ε thus:

Where:ε is the coeffi cient for section classifi cationfy is the yield strength of the steel, based on element thickness

Table 2 indicates the limiting values for various classes of section for both of the elements identifi ed in Figure 2. If any of the ratios go beyond those stated in Table 2 the section is considered to be in the Class 4 category.

Classifi cation of beam sectionsClause 5.5.2 in BS EN 1993-1-1 groups steel beams into four classifi cations. These classifi cations are based on a steel beam section’s resistance to suff ering from a local failure due to buckling:

Class 1/‘Plastic’ These sections can form a plastic hinge when a bending moment is applied to them without suff ering from local buckling failure.

Class 2/‘Compact’ These sections cannot properly develop a plastic hinge as their ability to rotate is limited before local buckling failure occurs.

Class 3/‘Semi-Compact’ These are sections that can withstand elastic stresses, specifi cally at the extreme fi bres of the section, but cannot form a plastic hinge. This has the eff ect of negating their plastic bending moment capacity.

Class 4/‘Slender’ Sections that will fail due to local buckling prior to the point of yield stress. Their plastic bending capacity therefore is non-existent.

When determining the classifi cation of a section, typically two parts of a rolled steel beam section are considered. For a simply supported beam, these are the edge of the top fl ange and the web, both of which are subjected to compression stress due to bending. Figure 2 indicates where these elements lie within a rolled steel beam section.

Table 5.2 in Clause 5.5 of BS EN 1993-1-1 defi nes the limits within which the geometry

Shear capacityTypically the component of the beam that takes the majority of the applied shear force is its web. There are instances where stiff eners are installed in order to support high shear loads, but this is very much the exception rather than the rule.

E Figure 2Elements

of a rolled steel beam that determine its classifi cation

Table 1: Yield strength fy vs. element thickness

GradeYield strength fy for element thickness < 16mm (N/mm2)

Yield strength fy for element thickness > 16mm

and < 40mm (N/mm2)

Yield strength fy for element thickness > 40mm

and < 63mm (N/mm2)

S275 275 265 255

S355 355 345 335

Table 2: Limiting values of geometries

for section classes 1-3

Class Web Flange

1 c/t ≤ 72ε c/t ≤ 9ε

2 c/t ≤ 83ε c/t ≤ 10ε

3 c/t ≤ 124ε c/t ≤ 14ε

/

f

N mm235

y

2

f =

"Guides in this next level build on what has been described previously in the Level 1 series"

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Bending moment resistance of steel beamsBS EN 1993-1-1 defi nes the bending moment resistance of restrained steel beams (Mc,Rd) in clause 6.2.5(2) as:

Lateral torsional bucklingClause 6.3.2.3 of BS EN 1993-1-1 describes how the value of χLT is related to the slenderness of the beam. This is related to the distance between restraints to the element of the beam that is subject to compression. For simply supported beams it is its upper-most fl ange. This is known as the non-dimensional slenderness ( ) and is defi ned thus:

Where:Wy and fy are as per previous defi nitionsMcr is the elastic critical moment for lateral

torsional buckling, which is based on the slenderness of the beam

Mcr is not defi ned within Eurocode 3, which off ers no guidance in calculating its value. There are however, many direct methods for calculating slenderness, the most simple of which is described in this guide.

SlendernessFor ‘H’ and ‘I’ sections it is possible to use simplifi ed methods to calculate the relative slenderness of the beam. The most conservative method is defi ned in Table 1.1 of NCCI: Determination of non-dimensional

slenderness of I and H sections SN002a-

EN-EU. It is based on applying the following equations that vary depending on the grade of steel being used:

For S275 Grade steel:

For S355 Grade steel:

Where:L is the distance between restraints to the

compression fl ange of the beamiz is the radius of gyration about the minor

axis of the beam

While valid, this method is very conservative as it ignores the bending moment diagram of the beam and can therefore result in oversized members. There is however, a more accurate yet complex method that does take into account the bending moment diagram and is described in NCCI: Determination of non-dimensional

slenderness of I and H sections SN002a-

EN-EU. It is strongly recommended that you examine this method in order to design more effi ciently sized steel beams.

( / )V

A f 3v y,pl Rd

M0c=

M Wf

,c Rd yM

y

0c=

M Wf

,b Rd LT yM

y

1|

c=

LTm

M

W f

cr

y yLTm =

Li

96LT

z

m =

: D

Li

85z

LTm =

: D

N Figure 3Approximate extent of web

resisting shear in a steel beam

All steel beams must satisfy the following expression:

Where:VEd is the applied shear forceVc,Rd is the design shear resistance

In the case of Class 1 and 2 rolled steel beams, the design shear resistance is designated as Vpl,RD and is defi ned in Clause 6.2.6, equation 6.18 of BS EN 1993-1-1 as:

Where:Av is the cross section area of the part of the beam that is resisting shear. For ‘I’ and ‘H’ sections this can conservatively be taken to be htw, which is the cross sectional area of the web and the thickness of the fl ange (Figure 3).

For all other classes of beam sections, you are referred to Clause 6.2.6 (4) of BS EN 1993-1-1 for determining their design shear resistance.

Clause 6.2.6 of BS EN 1993-1-1 has a more accurate equation that takes into account the radii of the root fi llet to the web-to-fl ange interface of the ‘H’ and ‘I’ sections. These can be used if you are fi nding it diffi cult to satisfy the shear resistance requirements.

V

V1

,c Rd

Ed# Where:

Wy is the major axis section modulus of the beam based on its classifi cation:

Wy = Wpl,y (Plastic section modulus) for Class 1 or 2

Wy = Wel,y (Elastic section modulus) for Class 3

Wy = Weff,y (Minimum eff ective section modulus) for Class 4

fy is the yield strength of the steel, based on element thicknessγM0 is the partial factor for the resistance of cross-sections, which in the UK is set at 1.0

The bending moment resistance should be reduced if the applied shear force is more than half of the plastic shear resistance of the beam. Where it exceeds this value, Clause 6.2.8 of BS EN 1993-1-1 applies. This places a modifi cation factor against the yield strength, thus:

Modifi ed yield strength = (1-ρ)fy

Where:

This modifi ed yield strength is then inserted into the calculation that determines bending moment resistance.

For Class 4 sections you are required to follow the guidance given in BS EN 1993-1-5 – Eurocode 3: Design of Steel Structures – Part 1-5: Plated Structural Elements. Class 4 sections are not found in rolled ‘I’ and ‘H’ elements and are therefore beyond the scope of this guide.

In the case of an unrestrained portion of a beam, a factor is applied to the bending moment resistance (Mb,Rd) that takes into account the risk of lateral torsional buckling. This is described Clause 6.3.2.1 (3) of BS EN 1993-1-1, in equation 6.55 as:

Where:Wy is the major axis section modulus of the

beam based on its classifi cation and is the same for restrained beams

fy is the yield strength of the steel, based on element thickness

γM1 is the partial factor for resistance of members subject to instability, which in the UK is set at 1.0

χLT is the reduction factor that takes into account lateral torsional buckling

V

V21

,pl Rd

Ed2

t = -` j

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Technical Guidance Note

Technical

Worked example

A simply supported, unrestrained steel beam spanning 8m is supporting another steel beam in the middle of its span. The ultimate load from this beam is 500 kN, while the serviceability load due to variable actions is 200 kN. The fl oor structure consists of one way spanning precast concrete planks with a screed and tiled fi nish. These planks span parallel to the steel beam and hence do not provide lateral restraint. Determine what size of beam is required to support this load, assuming the steel grade to be S355.

January 2013

Note 1 Level 2

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Table 3: Vertical defl ection to steel beams

Beam type/structure Defl ection limit

Cantilever Length/180

Beams supporting brittle fi nishes

Span/360

All other conditions Span/200

Purlins and cladding rails

To suit cladding system

In addition to defl ection, it is prudent to check the vibration of the beam i.e. its dynamic response. Technical Guidance Note No. 11, Level 1 explains how to do this.

There are further methods in determining Mcr described in the Non-Contradictory Complimentary Information website for structural steelwork.

Once the non-dimensional slenderness is established, the value of χLT is determined using equation 6.57 of Clause 6.3.2.3 of BS EN 1993-1-1 thus:

Where:The values of β and are defi ned as 0.4 and 0.75 respectively, as described in Clause NA.2.17 of NA to BS EN 1993-1-1.

αLT is the imperfection factor and is found in Table 6.3 of BS EN 1993-1-1, which reads against the steel beam’s buckling curve.

The buckling curves are labelled ‘a’ to ‘d’ and can be found in Clause NA.2.17 of NA to BS EN 1993-1-1. The buckling curve is dependent upon the h/b ratio of the beam section.

1LT

LT LT LT2 2|

bmU U=

+ -

. ( )0 5 1 ,LT LT LT LT LT02

m m ma bU = + - +6 @

ServiceabilityThe vertical defl ection limits for steel beams can be found in the Clause NA.2.23 of the UK National Annexe to BS EN 1993-1-1. Table 3 is based on these stated limits, which are for the defl ection due to unfactored imposed loads/variable actions only.

"Class 4 sections are not found in rolled ‘I’ and ‘H’ elements and are beyond the scope of this guide"

Eurocode 0.Applied practice

The applicable codes of practice for designing steel beams are as follows:

BS EN 1993-1-1 Eurocode 3: Design of Steel Structures – Part 1-1: General Rules for Buildings

BS EN 1993-1-1 UK National Annex to

Eurocode 3: Design of Steel Structures – Part 1-1: General Rules for Buildings

m ,LT 0

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Eurocode 0.Web

resources

Tata Steel Interactive ‘Blue Book’:www.tatasteelconstruction.com/en/design_

guidance/the_blue_book/

Non-Contradictory Complimentary Information website for structural steelwork:www.steel-ncci.co.uk/

Glossary and further reading

Reduction factor – A variable applied to the bending moment resistance of a beam due to the fact that it is unrestrained and hence subject to lateral torsional buckling.

Rolled steel section – A steel element that is cast to a pre-set size and not built up from separate plate elements.

Section classifi cation – A categorisation of steel elements that is based on the element’s ability to develop a plastic hinge when placed under load.

Further Reading The Institution of Structural Engineers (2010) Manual for the Design of Steelwork

Building Structures to Eurocode 3 London: The Institution of Structural Engineers

Steel Construction Institute (2009) Steel

Building Design. Worked Examples for

Students (P387) Ascot, Berkshire: SCI

The Institution of Structural Engineers (2012) ‘Principles of design’ The Structural

Engineer Vol. 90 (1) pp. 40-41

The Institution of Structural Engineers (2012) ‘Derivation of dead loads’ The

Structural Engineer Vol. 90 (1) pp. 43-45

The Institution of Structural Engineers (2012) ‘Derivation of imposed loads’ The

Structural Engineer Vol. 90 (2) pp. 46-48

The Institution of Structural Engineers (2012) ‘Derivation of wind load’ The

Structural Engineer Vol. 90 (2) pp. 49-52

The Institution of Structural Engineers (2012) ‘Derivation of snow load’ The

Structural Engineer Vol. 90 (3) pp. 22-24

The Institution of Structural Engineers (2012) ‘Lateral torsional buckling’ The

Structural Engineer Vol. 90 (16) pp. 28-30

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