How_Beams_Rev2.pdf

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1 March 2005

How to use Eurocode 2: Beams

R Moss xxx xxxx xxxx , O Brooker BEng CEng MICE

Introduction

Common Intro. for all publications - TBC

Designing to Eurocode 2

The design of beam elements with Eurocode 2 1 is essentially the same as with BS 8110 2 , however,

the layout and content of Eurocode 2 will initially appear unusual to designers familiar with BS 8110.

British designers will also find it strange that Eurocode 2 does not contain the derived formulae, this

has arisen because it has been European practice to give principles in the codes and for the

detailed application to be presented in other sources such as text books.

The checks for flexural capacity and deflection can be carried out using similar procedures to those

in BS 8110. However, assessment of the shear capacity of a section uses the ‘strut inclination

method’. This guide will lead the designer through the design steps required for beam elements

and highlight key points that will enable Eurocode 2 to be used with confidence.

The first guide in this series How to use Eurocode 2: Introduction 3 highlighted the key differences

between Eurocode 2 and BS 8110, including terminology. Eurocode 2 terminology will be used

throughout this guide to aid familiarity, eg units of stress will be quoted in MPa, not N/mm 2 . It

should also be noted that values from the UK National Annex have been used throughout this

guide, this includes values that are embedded in derived formulae.

Flexural Capacity

Eurocode 2 offers alternative methods for determining the stress-strain relationship of concrete.

For simplicity and familiarity the method presented here is the simplified rectangular stress block

which is very similar to that found in BS 8110 (see figure 1).

Eurocode 2 gives recommendations for the design of concrete up to grade C90/105, however, the

factors, λ and η , modify the strength capacity for concretes above grade C50/60. For the purpose

of this guide only concrete up to grade C50/60 is considered. It is important to note that concrete

strength is based on the cylinder strength and not the cube strength (eg for grade C28/35 the

cylinder strength is 28 MPa, whereas the cube strength is 35 MPa).

As with BS 8110, K can be determined from the design ultimate moment, concrete strength and the

width and depth of the section. K ’ is similarly dependant of the amount of redistribution carried out.

Knowing K and K ’ and the lever arm, the area of reinforcement can be determined using formulae

derived from the stress block (see Figure 2). Finally, checks can be carried out to ensure that the

area of reinforcement is within the maximum and minimum areas permitted by the code. Note that

minimum area of reinforcement is greater than under BS 8110 where the cylinder strength is over

25 MPa. Selected symbols are given in Table 1

For the design of flanged beams refer to appropriate section below.

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Figure 1: Simplified Rectangular Stress Block from Eurocode 2

Table 1: Selected Symbols

Symbol

Definition

Value

Depth to neutral axis

( d - z )/0.4

x max

Limiting value for depth to neutral axis

( δ - 0.4) d where δ≤ 1.0

Effective depth

d 2

Effective depth to compression reinforcement

Ratio of the redistributed moment to the elastic

bending moment

A s

Area of tension steel

A s2

Area of compression steel

f cd

Design value of concrete compressive strength α cc f ck / γ c

f ck

Characteristic cylinder strength of concrete

α cc

Coefficient taking account of long term effects

on compressive strength and of unfavourable

effects resulting from the way load is applied

0.85 for the resistance of flexural

and axial forces. 1.0 for the

resistance of shear forces. (From UK

National Annex)

f yk

Characteristic yield strength of reinforcement

500 MPa for grade XXX

f yd

Design yield strength of reinforcement

f yk / γ m = 435 MPa for grade XXX

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)

γ m

Partial factor for material properties

1.15 for reinforcement ( γ y )

1.5 for concrete ( γ c )

A c

Cross sectional area of concrete

b.h

Factor to take account of the different structural

systems

See table 7.4N

ρ 0

Reference reinforcement ratio

√ f ck /1000

Required tension reinforcement at mid-span to

resist the moment due to the design loads (or at

support for cantilevers

A s / bd

ρ ’

Required compression reinforcement at mid-

span to resist the moment due to the design

loads (or at support for cantilevers

A s2 / bd

l eff

Effective length of member

See section 5.3.2.2 (1)

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Concrete grade

≤ C50/60?

Outside scope of this guide

Yes

Determine K and K ’ from:

K = M /( bd 2 f ck ) & K ’ = 0.6 δ – 0.18 δ – 0.21 where δ ≤ 1.0

(Refer to Table 2 for values of K ’)

Compression reinforcement

required

Is K ≤ K ’ ?

Yes

Calculate lever arm Z from

[ ]

No compression reinforcement required

−

Calculate lever arm Z from

[ ]

−

≤

Calculate compression reinforcement required from

(Refer to Table 3 for values of z/d )

(

−

)

where f sc = 700(( x - d 2 )/ x ) ≤ f yd

( )

−

Calculate tension reinforcement required from

A s = M / f yd . z

Calculate tension reinforcement required from

Check minimum reinforcement requirements:

A s,min = 0.26 f ctm b t d / f yk where f ck ≥ 25

(Refer to Table 4 for values of f ctm )

Check maximum reinforcement requirements A s,max = 0.4 A c

for tension of compression reinforcement outside lap locations

Figure 2: Procedure for Determining Flexural Reinforcement

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Table 2: Values for K ’

% Redistribution

K ’

1.00

0.2052

0.95

0.1931

0.90

0.1800

0.85

0.1661

0.80

0.1512

0.75

0.1355

0.70

0.1188

Table 3: z / d for singly

reinforced rectangular

sections

z / d

0.010

0.991

0.015

0.987

.020

0.982

0.025

0.977

0.030

0.973

0.035

0.968

0.040

0.963

0.045

0.959

0.050

0.954

0.055

0.949

0.060

0.944

0.065

0.939

0.070

0.934

0.075

0.929

0.080

0.924

0.085

0.918

0.090

0.913

0.095

0.908

0.100

0.902

0.105

0.897

0.110

0.891

0.115

0.885

0.120

0.880

0.125

0.874

0.130

0.868

0.135

0.862

0.140

0.856

0.145

0.849

0.150

0.843

0.155

0.836

0.160

0.830

0.165

0.823

0.170

0.816

0.175

0.809

0.180

0.802

0.185

0.795

0.190

0.787

0.195

0.779

0.200

0.771

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Table 4: Values of f ctm

f ck

f ctm

2.6

2.8

2.9

3.0

3.2

3.5

3.8

4.1

Deflection

In principle the assessment of deflection is similar in both Eurocode 2 and BS 8110. Eurocode 2

gives two methods for determining whether deflection is acceptable, either by limiting the span to

depth ratio or by calculation of actual deflection. The span to depth ratios should ensure that

deflection is limited to span over 250 under the quasi-permanent and this is the method presented

in this guide. The deflection by calculation method is described in How to use Eurocode 2:

Deflection 4 .

Table 7.4N from the code presents some basic span to depth ratios and introduces a factor, K ,

which takes account of the different structural systems and is reproduced here for convenience. A

more accurate method is to use the formulae on which Table 7.4N is based and these are given in

Figure 3 which presents a flow chart for checking the deflection of beam elements. Elected

symbols are given in Table 5.

Table 6: Basic ratios of span/effective depth for reinforced concrete members without axial compression

(Table 7.4N (BS) form UK National Annex)

Note 4: The values of K in the table may not be appropriate when the form-work is struck at an early age or

when construction loads exceed design load. In these cases the deflections may need to be calculated

using the advice in specialist literature.

Note 5 : The ratio of area of reinforcement provided to that required should be limited to 1.5 when the span

to depth ratio is adjusted.

Note 6 : Values are obtained using expressions (7.16) for grade C30/37 concrete and σ s = 310 MPa

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