Eurocode 2 - Beams

8/11/2019 Eurocode 2 - Beams

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4. Beams

IntroductionThe introduction of European standards to UKconstruction is a significant event.The ten designstandards, known as the Eurocodes, will affectall design and construction activities as currentBritish standards for design are due to bewithdrawn in 2010.

This publication is part of the series of guidesentitled How to design concrete structures usingEurocode 2. Their aim is to make the transition toEurocode 2: Design of concrete structuresas easyas possible by drawing together in one place keyinformation and commentary required for thedesign of typical concrete elements.

The cement and concrete industry recognised that

a substantial effort was required to ensure thatthe UK design profession would be able to useEurocode 2 quickly, effectively, efficiently andwith confidence. With support from government,consultants and relevant industry bodies, theConcrete Industry Eurocode 2 Group (CIEG) wasformed in 1999 and this Group has provided theguidance for a co-ordinated and collaborativeapproach to the introduction of Eurocode 2.Asa result, a range of resources is to be madeavailable through The Concrete Centre to helpdesigners during the transition period (see backcover for details).

How to design concrete structures using Eurocode 2

4. BeamsR M MossBSc, PhD, CEng, MICE, MIStructEO Brooker BEng, CEng, MICE, MIStructE

Designing to Eurocode 2This guide covers the analysis and design of concrete beams to Eurocode 21

which is essentially the same as with BS 81102 . However, the layout andcontent of Eurocode 2 may appear unusual to designers familiar with BS 8110.Eurocode 2 does not contain the derived formulae or specific guidance ondetermining moments and shear forces. This has arisen because it hasbeen European practice to give principles in the codes and for the detailedapplication to be presented in other sources such as textbooks.

The first guide in this series,How to design concrete structures using Eurocode 2:Introduction to Eurocodes3 , highlighted the key differences between Eurocode 2and BS 8110, including terminology.

It should be noted that values from the UK National Annex (NA) have beenused throughout this guide, including values that are embedded in derivedformulae (derivations can be found at www.eurocode2.info). A list of symbolsrelated to beam design is given at the end of this guide.

Design procedureA procedure for carrying out the detailed design of beams is shown in Table 1.This assumes that the beam dimensions have previously been determinedduring conceptual design. Concept designs prepared assuming detailed designwould be to BS 8110 may be continued through to detailed design usingEurocode 2. More detailed advice on determining design life, actions, materialproperties, methods of analysis, minimum concrete cover for durability andcontrol of crack widths can be found in the accompanying guideHow todesign concrete structures using Eurocode 2: Getting started 4 .

Fire resistanceEurocode 2, Part 12:Structural fire design6 , gives a choice of advanced,simplified or tabular methods for determining the fire resistance. Using tablesis the fastest method for determining the minimum dimensions and coverfor beams. There are, however, some restrictions and if these apply furtherguidance on the advanced and simplified methods can be obtained fromspecialist literature. Rather than giving a minimum cover, the tabular methodis based on nominal axis distance,a (see Figure 1).This is the distance from

the centre of the main reinforcing bar to the top or bottom surface of theContinues page 3


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Step Task Further guidance

How to guide Standard1 Determine design life Getting started NA to BS EN 1990 Table NA.2.1

2 Assess actions on the beam Getting started BS EN 1991 (10 parts) and National Annexes3 Determine which combinations of actions apply Introduction to Eurocodes NA to BS EN 1990 Tables NA.A1.1 and NA.A1.2 (B)4 Determine loading arrangements Getting started NA to BS EN 1992115 Assess durability requirements and determine concrete strengthGetting started BS 8500: 20026 Check cover requirements for appropriate fire resistance periodGetting started and Fire resistance Approved Document B. BS EN 199211: Section 5

section of this guide7 Calculate min. cover for durability,fire and bond requirements Getting started BS EN 199211 Cl 4.4.18 Analyse structure to obtain critical moments and shear forces Getting started and Table 3 of this guide BS EN 199211 section 59 Design flexural reinforcement See Flexure section of this guide BS EN 199211 section 6.110 Check shear capacity See Vertical shear section of this guide BS EN 199211 section 6.211 Check deflection See Deflection section of this guide BS EN 199211 section 7.412 Check spacing of bars Getting started BS EN 199211 section 7.3

NoteNA = National Annex

Table 1Beam design procedure

Table 2Minimum dimensions and axis distances for beams made with reinforced concrete for fire resistance

Figure 3Simplified rectangular stress block for concrete up to class C50/60 from Eurocode 2

d h

x

Section

As2

As

Neutral axis

b

d 2

Strain

c

s

sc

0.8 x F sc

f cd

Stress block and forces

z

F c

F st

Figure 1Section through structural member,showing nominal axis distances a and asd

b

a

asd

h > b

Standard fire resistance Minimum dimensions (mm)Possible combinations of a and bmin where a is the average axis distance and bmin is the width of the beamSimply supported beams Continuous beamsA B C D E F G H

R60 bmin = 120 160 300 300 120 200a = 40 35 30 25 25 12a

R90 bmin = 150 200 300 400 150 250a = 55 45 40 35 35 25

R120 bmin = 200 240 300 500 200 300 450 500a = 65 60 55 50 45 35 35 30

R240 bmin = 280 350 500 700 280 500 650 700a = 90 80 75 70 75 60 60 50

Notes1 This table is taken from BS EN 199212 Tables 5.5 and 5.6.2 The axis distance,asd, from the side of the beam to the corner bar should bea +10 mm except wherebmin is greater than the values in columns C and F.3 The table is valid only if the detailing requirements (see note 4) are observed and, in normal temperature design, redistribution of bending moments does not exceed 15%.4 For fire resistance of R90 and above, for a distance of 0.3leff from the centre line of each intermediate support, the area of top reinforcement should not be less than the following:

As,req(x) = As,req(0)(1 2.5(x/leff ) )where:

x is the distance of the section being considered from the centre line of the support. As,req(0 ) is the area of reinforcement required for normal temperature design. As,req(x) is the minimum area of reinforcement required at the section being considered but not less than that required for normal temperature design.leff is the greater of the effective lengths of the two adjacent spans.

5 For fire resistances R120 R240, the width of the beam at the first intermediate support should be at least that in column F, if both the following conditions exist:a there is no fixity at the end support; andb the acting shear at normal temperatureV sd > 0.67 V Rd,max.

Key a Normally the requirements of BS EN 199211 will determine the cover.


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4. Beams

Carry out analysis of beam to determinedesign moments (M) (see Table 3)

Obtain lever arm z from Table 5 or use

Calculate tension reinforcementrequired from

Check minimum reinforcementrequirements (see Table 6)

No compression reinforcement required

Check maximum reinforcement requirements As,max = 0.04 Acfor tension or compression reinforcement outside lap locations

Determine K from Table 4 orK = 0.60d 0.18d2 0.21

where d 1.0

Outside scope of this guide

Compression reinforcementrequired

Calculate lever arm z from

START

Concrete classC50/60?

Is K K ?

Yes

Yes

No

No

Determine K from K = Mbd 2 f ck

Calculate compressionreinforcement required from

where

Figure 2Procedure for determining flexural reinforcement

Table 3Bending moment and shear coefficients for beams

member. It is a nominal (not minimum) dimension, so the designershould ensure that:a cnom + f link+ f bar /2 and asd = a + 10 mmTable 2 gives the minimum dimensions for beams to meet the

standard fire periods.

FlexureThe design procedure for flexural design is given in Figure 2; this includesderived formulae based on the simplified rectangular stress block fromEurocode 2. Table 3 may be used to determine bending moments andshear forces for beams, provided the notes to the table are observed.

Table 4Values for K

As2 =(K K ) f ck bd 2

f sc(d d 2)

Calculate tensionreinforcement required from

f ck f ctm Minimum percentage (0.26 f ctm / f yka)25 2.6 0.13%28 2.8 0.14%30 2.9 0.15%32 3.0 0.16%35 3.2 0.17%40 3.5 0.18%45 3.8 0.20%

50 4.1 0.21%

1 + 1 3.53 K z =2d [ ]

1 + 1 3.53 K 0.95d z = 2d [ ]

As,min = where f ck 25 f yk

0.26 f ctm bt d

As = f yd z

M

As = + As2 f yd zK f ckbd 2

f yd

f sc

f sc = 700 f ydxx d2[ ]

Moment Shear

Outer support 25% of span moment 0.45 (G + Q)

Near middle of end span 0.900Gl+ 0.100 Ql

At first interior support 0.094 (G + Q)l 0.63 (G + Q)a

At middle of interior spans 0.066Gl+ 0.086 Ql

At interior supports 0.075 (G + Q)l 0.50 (G + Q)Keya 0.55 (G + Q) may be used adjacent to the interior span.Notes1 Redistribution of support moments by 15% has been included.2 Applicable to 3 or more spans only and whereQk Gk.3 Minimum span 0.85 longest span.4 l is the span,G is the total of the ULS permanent actions,Q is the total

of the ULS variable actions.

% redistribution d (redistribution ratio) K 0 1.00 0.2055 0.95 0.193

10 0.90 0.18015 0.85 0.16620 0.80 0.15125 0.75 0.136

K z/d

0.01 0.950a

0.02 0.950a

0.03 0.950a

0.04 0.950a

0.05 0.9540.06 0.9440.07 0.9340.08 0.9240.09 0.913

0.10 0.902

K z/d

0.11 0.8910.12 0.8800.13 0.8680.14 0.8560.15 0.8430.16 0.8300.17 0.8160.18 0.8020.19 0.787

0.20 0.771Keya Limiting z to 0.95d is not a requirement of Eurocode 2, but is considered to be good practice.

Keya Assuming f yk = 500 MPa

Table 5 z/d for singly reinforced rectangular sectionsTable 6Minimum percentage of required reinforcement


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Figure 5Procedure for determining vertical shear reinforcement

Yes (cot y = 2.5)

No No

START

Determine v Ed where:v Ed = shear stress at d from face of support [ v Ed = V Ed/(bw z) = V Ed/(0.9 bwd )]

Determine the concrete strut capacity v Rd,max cot y = 2.5from Table 7

Redesignsection

Determine y from:

Calculate area of shear reinforcement:

Check maximum spacing for vertical shear reinforcement: sl,max = 0.75 d

Table 7Minimum and maximum concrete strut capacity in terms of stress

f ck v Rd,max cot y = 2.5 v Rd,max cot y = 1.020 2.54 3.6825 3.10 4.5028 3.43 4.9730 3.64 5.2832 3.84 5.5835 4.15 6.0240 4.63 6.7245 5.08 7.3850 5.51 8.00

Figure 4Strut inclination method

Longitudinalreinforcement in tension

Vertical shearreinforcement

Concrete strut in compression

y

Eurocode 2 offers various methods for determining the stress-strainrelationship of concrete. For simplicity and familiarity the methodpresented here is the simplified rectangular stress block, which issimilar to that found in BS 8110 (see Figure 3).

Eurocode 2 gives recommendations for the design of concrete up toclass C90/105. However, for concrete greater than class C50/60, thestress block is modified. It is important to note that concrete strengthis based on the cylinder strength and not the cube strength (i.e. forclass C30/37 the cylinder strength ( f ck) is 30 MPa, whereas the cubestrength is 37 MPa).

Vertical shear Eurocode 2 introduces the strut inclination method for shear capacitychecks. In this method the shear is resisted by concrete struts acting incompression and shear reinforcement acting in tension.

The angle of the concrete strut varies, depending on the shear forceapplied (see Figure 4). The procedure for determining the shear capacityof a section is shown in Figure 5 (which includes UK NA values) and isin terms of shear stress in the vertical plane rather than a vertical forceas given in Eurocode 2. Where shear reinforcement is required, thenthe angle of the concrete strut should be calculated. For many typicalbeams the minimum angle of strut will apply (when coty = 2.5 or y =21.8) i.e. for class C30/37 concrete the strut angle exceeds 21.8 only

when the shear stress is greater than 3.27 N/mm2

(refer to Table 7).As with BS 8110, there is a maximum permitted shearcapacity,v Rd,max,(when cot y =1.0 ory = 45), but this is not restricted to 5 MPa as inBS 8110.

DeflectionEurocode 2 has two alternative methods for checking deflection,either a limiting span-to-depth ratio may be used or the theoreticaldeflection can be assessed using the expressions given in the Code.The latter is dealt with in detail in another guide in this series,How todesign concrete structures using Eurocode 2:Deflection6 .

The span-to-depth ratios should ensure that deflection is limited tospan/250 and this is the procedure presented in Figure 6.

Flanged beamsFlanged beams can be treated in much the same way as in BS 8110.The main differences compared with BS 8110 are that the assessmentof the flange width is more sophisticated (see Figures 9 and 10) and

that Eurocode 2 contains a check to confirm that the shear stress at

= s Asw

y = 0.5 sin-1 0.18 f ck (1 f ck /250)v Ed[ ]

0.9d f ywd cot y v Ed bw

Continues page 7

Is

v Ed< v Rd,max coty = 2.5?

Is

v Ed < v Rd,max coty = 1.0?(see Table 7)

Yes


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Selected symbols for slabs are shown overleaf.

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4. Beams

Figure 14Procedure for determining longitudinal shear capacity of flanged beams

Table 8Concrete strut capacity for longitudinal shear in flanged beams

f ck v Rd,max (when cot y = 2.5)Flange in tension Flange in compression

20 2.94 3.5925 3.60 4.3928 3.98 4.8530 4.22 5.1532 4.46 5.4435 4.82 5.8740 5.38 6.5545 5.90 7.2050 6.40 7.80

Yes

No No

No

Calculate the longitudinal shear stressfrom: v Ed = D F d/(hf D x )

(see Figure 13)

Determine the concrete strut capacityfrom Table 8 or from:

v Rd = 0.195 f ck (1 f ck /250)

Calculate area of transverse reinforcement from:

Yes (cot y f = 2.5) Yes (coty f = 2.0)

Is v RD > v Ed? Is v RD > v Ed?

Is length offlange under consideration

in tension?

Determine y f from:

Determine the concretestrut capacity from Table 8

or from:v Rd = 0.16 f ck (1 f ck /250)

= s Asf

y f =0.5 sin-1 0.2 f ck (1 f ck /250)v Ed[ ]

f yd cot y f

v Ed hf

the interface of the flange and web can be resisted by the transversereinforcement in the flange. The position of the neutral axis shouldbe determined, and then the area of reinforcement can be calculateddepending whether it lies in the flange or web (see Figure 11).

At supports the tension reinforcement to resist hogging momentsshould be distributed across the full width of the effective flange asshown in Figure 12.The span-to-depth deflection checks using ratio of tension reinforcement should be based on area of concrete abovecentre of tension steel.

Longitudinal shear The shear stress in the vertical plane between the flange and web shouldbe assessed according to section 6.2.4 and Figure 6.7 of the Eurocode(reproduced here as Figure 13). The change in force in the flange canbe assessed from the moment and lever arm at a particular location.The Eurocode states that the maximum length that can be consideredfor the change in force is half the distance between the maximummoment and the point where the moment is zero. Clearly, the maximumlongitudinal force will occur where the change in moment, andtherefore force, is the greatest; for a uniformly distributed load on acontinuous beam this will be the length of beam closest to the support.

Figure 14 shows a flow chart for assessing the longitudinal shearcapacity; in many cases the transverse reinforcement in the slab will besufficient to resist the shear force.This check is included to ensure thatwhere particularly thin flanges are used there is adequate reinforcement.

The longitudinal shear capacity is based on the variable strut inclinationmethod, which was described in the section on vertical shear.

Rules for spacing andquantity of reinforcementMinimum area of longitudinal reinforcementThe minimum area of reinforcement is As,min = 0.26 f c tm b t d / f y k butnot less than 0.0013b td , wherebt is the mean width of the tensionzone (see Table 6). For a T-beam with the flange in compression, onlythe width of the web is taken into account in calculating the value ofbt.

Maximum area of longitudinal reinforcementThe maximum area of tension or compression reinforcement, outsidelap locations should not exceed As,max = 0.04 Ac

Minimum spacing of reinforcementThe minimum clear distance between bars should be the greater of: Bar diameter Aggregate size plus 5 mm 20 mm


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4. Beams

References1 BRITISH STANDARDS INSTITUTION. BS EN 199211, Eurocode 2:Design of concrete structures Part 11 General rules and rules for

buildings. BSI, 2004.

2 BRITISH STANDARDS INSTITUTION. BS 81101:The structural use of concrete Part 1, Code of practice for design and construction.BSI, 1997.

3 NARAYANAN, R S & BROOKER, O.How to design concrete structures using Eurocode 2: Introduction to Eurocodes. The Concrete Centre, 2005.

4 BROOKER,O.How to design concrete structures using Eurocode 2: Getting started . The Concrete Centre, 2005.

5 BRITISH STANDARDS INSTITUTION. BS EN 199212, Eurocode 2:Design of concrete structures. General rules structural fire design. BSI, 2004

6 WEBSTER, R & BROOKER, O.How to design concrete structures using Eurocode 2: Deflection. The Concrete Centre, 2006.

Further guidance and advice Guides in this series cover:Introduction to Eurocodes, Getting started,Slabs, Beams,Columns, Foundations, Flat slabsand Deflection. For free

downloads, details of other publications and more information on Eurocode 2 visitwww.eurocode2.info This guide is taken from The Concrete Centres publication,How to design concrete structures using Eurocode 2(Ref. CCIP-006) For information on all the new Eurocodes visitwww.eurocodes.co.uk

AcknowledgementsThe content of this publication was produced as part of the project Eurocode 2: transition from UK to European concrete design standards. Thisproject was part funded by the DTI under the Partners in Innovation scheme.The lead partner was the British Cement Association. The work wascarried out under the guidance of the Concrete Industry Eurocode 2 Group, which consists of representatives from:Alan Baxter and Associates Arup British Cement Association British Precast Building Research Establishment Clark Smith Partnership Concrete Innovation and Design Construct Department for Trade and Industry Office of the Deputy Prime Minister The Concrete Centre The Concrete Society Quarry Products Association.

All advice or information from The Concrete Centre is intended for those who will evaluate the significance and limitations of itscontents and take responsibility for its use and application. No liability (including that for negligence) for any loss resulting from suchadvice or information is accepted by The Concrete Centre or its subcontractors, suppliers or advisors. Readers should note thatpublications from The Concrete Centre are subject to revision from time to time and they should therefore ensure that they are inpossession of the latest version. This publication has been produced following a contract placed by the Department for Trade andIndustry (DTI); the views expressed are not necessarily those of the DTI.

Ref: TCC/03/19ISBN 1-904818-29-3Published February 2006 The Concrete CentreTM

Published by The Concrete Centre

Riverside House, 4 Meadows Business Park,Station Approach, Blackwater,Camberley,Surrey GU17 9ABTel: +44 (0)1276 606800Fax: +44 (0)1276 606801www.concretecentre .com

lo Distance between points of zero momentl/d Span-to-depth ratioM Design moment at the ULS

x Depth to neutral axis (d - z)/0.4 x max Limiting value for depth to neutral axis (d 0.4)d whered 1.0 z Lever arma cc Coefficient taking account of long term 0.85 for flexure and

effects on compressive strength and of axial loadsunfavourable effects resulting from the 1.0 for other phenomenaway load is applied (From UK National Annex)

d Ratio of the redistributed moment to theelastic bending moment

g m Partial factor for material properties 1.15 for reinforcement (g s)1.5 for concrete (g c)

r 0 Reference reinforcement ratio R f ck/1000r Required tension reinforcement at mid-span As/bd

to resist the moment due to the designloads (or at support for cantilevers)

r Required compression reinforcement at As2/bd mid-span to resist the moment due to thedesign loads (or at support for cantilevers)

Ac Cross sectional area of concrete As Area of tension steel As2 Area of compression steel

As, prov Area of tension steel provided As, reqd Area of tension steel requiredbeff Effective flange widthbt Mean width of the tension zonebmin Width of beam or ribbw Width of section, or width of web on flanged beamsd Effective depthd 2 Effective depth to compression reinforcement f cd Design value of concrete compressive strength a cc f ck/g c for f ck C50/60 f ck Characteristic cylinder strength of concrete f ctm Mean value of axial tensile strength 0.30 f ck(2/3) for f ck C50/60

(from Table 3.1, Eurocode 2)hf Flange thickness

K Factor to take account of the different See table NA.4 instructural systems UK National Annexleff Effective span of member See Section 5.3.2.2 (1)

Symbol Definition Value Symbol Definition Value

Selected symbols

For more information on Eurocode 2 andother questions relating to the design, useand performance of concrete contact theNational Helpline on:0700 4 500 500 or 0700 4 CONCRETE

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