TheStructuralEngineer38

Technical Guidance Note

Technical

April 2013

Note 4 Level 2

›

Principles of concrete beam design

The principles of reinforced concrete design are explained in Technical Guidance Note 3

(Level 2) which covers the design of concrete slabs. You are directed to that text prior to reading this guide in order to appreciate the concepts that are introduced within it. The key topics to note from that guide are:

• Rules governing how to determine the tension reinforcement required• Concrete cover for bond, corrosion and fi re protection • Material properties• Serviceability• Minimum area of steel reinforcement

A concrete beam is defi ned as an element whose width is less than 5 times its depth. In all other instances the element is a slab and therefore must be treated as such.

Designing a concrete beamIntroduction

The subject of this guide is the design of reinforced concrete beams to BS EN 1992-1-1 – Eurocode 2: Design of Concrete Structures – Part 1-1: General Rules for Buildings. It covers the design of multispan beams that have both ‘L’ and ‘T’ cross section profi les.

W

W Applied practice

W Worked example

W Further reading

W Web resources

ICON LEGEND

Principles of concrete beam design

Analysis of concrete beamsLike concrete slabs, there are a set of coeffi cients that can be used to determine the shear and bending moments in a beam. Provided the geometry of the beam spans are within 15% of each other and the dead and imposed loads are similar on all spans, the coeffi cients described in Table 1 can be used. In addition the imposed load must be less than or equal to the dead load in order for these coeffi cients to remain valid.Where the geometry of the beam falls outside of the parameters that allow for the use of coeffi cients described in Table 1, traditional analysis methods need to be employed in order to determine the forces in the beam.

It should be noted that bending moment redistribution is not covered by this guide. Bending moment redistribution allows for plastic deformation of reinforcement concrete elements, which results in the bending moments being redistributed

along the length of the element. Due to the complexity in analysing this phenomenon it will not be considered here.

Design of concrete beamsThere are three forms of reinforced concrete beam: rectangular, ‘T’ and ‘L’. These are defi ned in Clause 5.3.2.1 of BS EN 1992-1-1. The ‘T’ beam is the most common as it forms part of a downstand to a reinforced concrete slab. The width of the compression fl ange beff is derived using the following equation:

Where:bw is the width of the beambeff,i is defi ned as 0.2bi+0.1l0 ≤ 0.2l0

b is the distance between the midspan points of the slab the beam is supporting

The distance between beams is defi ned as 2bi. This is an important variable as it infl uences the width of compression fl anges to ‘T’ and ‘L’ beams due to the fact that they cannot overlap one another. The length over which the compression fl ange occurs lo is defi ned as the distance between the points of contrafl exure within the bending moment diagram of the beam structure. This is described in Figure 1, which is a replication of Figure 5.2 in BS EN 1992-1-1. Figure 2 is derived from Figure 5.3 of BS EN 1992-1-1 and defi nes the width of the compression fl anges that form part of ‘T’ and ‘L’ beams in relation to ones that are adjacent to them.

Table 1: Bending moment and shear coeffi cients for beams with uniform loading and spans

LocationOuter

supportEnd span

First interior

support

Typical

mid span

Interior

support

Bending moment 0 0.09Fl -0.11Fl 0.07Fl 0.1Fl

Shear 0.45F - 0.6F - 0.55F

Note: F is the total ultimate load and l is the span of the beam. When using these coeffi cients to determine bending moments, it is not permissible to redistribute them.

b b b b,eff w eff i #= + /

TSE16_38-41.indd 38TSE16_38-41.indd 38 22/03/2013 13:4422/03/2013 13:44

www.thestructuralengineer.org

39

Before it can be assumed that a compression fl ange exists within a concrete beam section, it must be determined where the neutral axis is located within it. This is found by using the stress block within the beam as load is applied to it. This distance s is defi ned as 2(d-z), with z being the lever arm of the beam cross section. If s ≤ h then the compression fl ange is not being eff ective and the beam needs to be treated as a rectangular one.

There are some instances where compression reinforcement is required in beams due to geometric and load application design constraints. In order to determine whether or not compression reinforcement is required in a reinforced concrete beam, the following equation can be employed:

If then compression reinforcement is required.

Where:M is the applied ultimate bending momentfck is the cylinder compressive strength of concreteb is the width of the beamd is the eff ective depth of the beam

The following equation is used to determine the area of steel As2 needed:

Where:fyk is the characteristic yield strength of reinforcement, which is typically taken to be 500 N/mm2

d2 is the eff ective depth to the compression reinforcement, which is from the top most surface of the beam for sagging moments and the beam’s soffi t for hogging moments

For ‘T’ and ‘L’ beams, in instances where the following expression is true, compression reinforcement is required:

Where:bf is the width of the compression fl angehf is the thickness of the compression fl ange, which can be no more than 0.36d

When no compression reinforcement is required, the minimum amount of steel reinforcement is placed within the compression zone of the beam.

Detailing requirementsWhen designing reinforced concrete beams it is vitally important that you take into account how they are to be constructed. If you specify reinforcement that is too diffi cult to install you risk signifi cant delays during construction; to the point where some re-design may need to be carried out in order to accommodate the constraints due to buildability related issues. Such delays can be avoided provided the designer considers how the reinforcement is to be installed in the beam.

Steel reinforcement bars come in certain serial sizes: 6,8,10,12,16,20,25,32,40 and 50mm, with the 6mm and 50mm being the least common. Primary reinforcement bars are typically sized between 20 and 40mm in diameter and are detailed to be no longer than 6m. The length of bar dictates the need to lap them, which impacts on the amount of space between bars that is available. Where bars become congested, it is advisable to place a spacer bar between the lapped bars to allow the passage of concrete between the reinforcement. Figure 3 shows how this impacts the layout of reinforcement and the design of the beam.

The minimum spacing between bars is one of the following:

• The maximum bar size in the element• The maximum aggregate size +5mm• 20mm

Whichever is the greater is the chosen spacing.

The maximum spacing is dependent upon the stress σs that is being applied to the reinforcement bar. This can be determined using the following equation:

γms is the material factor for steel reinforcementAs,req is the area of tension reinforcement requiredAs,prov is the area of tension reinforcement provided

E Figure 1Distance

between points of contrafl exure lo for continuous beams

E Figure 2‘T’ and ‘L’

beam defi nitions

.M f bd0 167 ck22

. ( ).

Af d d

M f bd0 87

0 167s

yk

ck2

2

2

=-

-

. ( . )M f b h d h0 567 0 5> ck f f f-

. .f

Q GQ G

AA

1 5 1 351

,

,s

ms

yk

k k

k k

s prov

s req2

c}

d= +

+v c c am m k' 1

E Figure 3Beam

with spacer bar

Spacer bar

TSE16_38-41.indd 39TSE16_38-41.indd 39 22/03/2013 13:4422/03/2013 13:44

TheStructuralEngineer40

Technical Guidance Note

Technical

April 2013

Note 4 Level 2

›

A span in a continuous beam is 8m long, placed at 6m spacing and has an ultimate bending moment of 675kNm at its supports and 550kNm at its mid span. The shear forces are 550kN at the points of support. Its size is 700mm deep by 500mm wide and it is monolithically cast with a fl oor slab, making it a ‘T’ beam. It has a fi re rating of 1 hour, the

Worked example

δ is the ratio of ultimate bending moment vs. the elastic bending moment redistribution. (As this note does not cover bending moment redistribution, assume this ratio value to be 1).

The second section of the expression concerns the factors for dead loads (permanent actions) and imposed loads (variable actions). The relevant coeffi cients for these loads need to be taken from BS EN 1990-1-1.

The stress is read against the bar size in Table 2, which describes the maximum distance between bars.

In beams that have an overall depth of more than 750mm, side lacer bars need to be installed in order to prevent cracking as the concrete sets. These bars need to be evenly distributed from the neutral axis to the bottom of the beam. The size of bars should be based on the minimum area of steel reinforcement.

Further information on detailing of reinforcement in concrete beams can be gleaned from the Institution’s Standard

Method of Detailing Structural Concrete (3rd ed.). You are strongly advised to review the relevant sections of this text before attempting to design a reinforced concrete beam.

Table 2: Maximum space between bars

vs. stress within reinforcement

Stress range

(N/mm2)

Maximum distance

between bars

(mm)

<160 300

160-180 275

180-200 250

200-220 225

220-240 200

240-260 175

260-280 150

280-300 125

300-320 100

320-340 75

340-360 50

Shear in concrete beamsShear is an important component in concrete beam design as the vast majority of beams require some form of shear reinforcement to be placed within them. Small diameter reinforcement bars (no greater than 16mm in diameter) are bent around the primary reinforcement bars to form a cage. These are called ‘shear links’ and provide the tension component to the concrete beam as it resists shear. BS EN 1992-1-1 uses the ‘variable strut inclination method’ to determine the amount of shear reinforcement required in a concrete beam. Figure 4 describes this concept, which is based on a truss being developed within the concrete beam as it resists shear.

Shear reinforcement for concrete beams VRd,s is defi ned as follows:

Where:VEd is the applied ultimate shearAsw is the area of shear reinforcements is the spacing between shear links in mmz is the lever arm to the primary reinforcement in mmfywd is the design yield strength of reinforcement, which is typically taken to be 500 N/mm2 over the material factor γs, which is 1.15α is the angle of the shear links to the horizontal axis of the beam. For vertically orientated links, the value of cot α is 0 and sin α is 1

θ is the angle of the variable strut that is formed when the beam resists shear, as shown in Figure 4. BS EN 1992-1-1 limits its value to between 22º and 45º. When a beam is supporting a UDL, θ is 22º, whereas beams that support point loads veer towards higher values. The magnitude of θ can be calculated using the following equation:

If the result of the above equation is a negative, then the strut has failed and the beam needs to be resized.If the value of cot θ is <1 then the beam needs to be resized. If it is >1 then a check needs to be carried

out for the shear force at a distance z from the end of the beam. If the value of cot θ is >2.5 then it is assumed to be 2.5.

The minimum amount of shear reinforcement required in a beam Asw,min is defi ned thus:

The maximum spacing for shear links s1,max is defi ned as 0.75d(1 + cot α)

The maximum distance between vertical legs of shear links sb,max is 0.75d and must be less than 600mm.

( )cot cot sinV sA zf V,Rd s

swywd Ed$i a a= +a k

.( / )

.sinb d f f

V0 51 2505 56

w ck ck

Ed1i =-

-; E

.sinsb

Af

f0 08,.

min

w

sw

yk

ck0 5

$a

E Figure 4Variable strut inclination method

TSE16_38-41.indd 40TSE16_38-41.indd 40 22/03/2013 13:4422/03/2013 13:44

www.thestructuralengineer.org

41

Glossary and further reading

Contrafl exure – Point at which there is no bending moment being applied to the concrete beam.

Shear link – A form of reinforcement that consists of a hoop or series of hoops that provides the tension resistance when a beam is subjected to shear forces.

Side lacers – Bars placed on the sides of deeper beams to prevent cracking.

Further Reading The Institution of Structural Engineers (2006) Manual for the design of concrete

building structures to Eurocode 2 London: The Institution of Structural Engineers

The Concrete Centre (2009) Worked

Examples to Eurocode 2: Volume 1 [Online] Available at: www.concretecentre.com/pdf/Worked_Example_Extract_Slabs.pdf (Accessed: February 2013)

Mosley W., Bungey J. and Hulse R. (2007) Reinforced Concrete Design to Eurocode

2 (6th ed.) Basingstoke, UK: Palgrave Macmillan

Reynolds C.E., Steedman J.C. and Threlfall A.J. (2007) Reynolds’s Reinforced Concrete

Designer’s Handbook (11th ed.) Oxford, UK: Taylor & Francis

Institution of Structural Engineers (2012/13) Technical Guidance Notes 1-5 and 17 (Level 1) and 3 (Level 2) The Structural Engineer 90 (1-3, 10) and 91 (3)

Eurocode 0.Web resources

The Concrete Centre: www.concretecentre.com/

The Institution of Structural Engineers library:www.istructe.org/resources-centre/library

Eurocode 0.Applied practice

BS EN 1992-1-1 Eurocode 2: Design of Concrete Structures – Part 1-1: General Rules for Buildings

BS EN 1992-1-1 UK National Annex to

Eurocode 2: Design of Concrete Structures – Part 1-1: General Rules for Buildings

grade of concrete is C30/37 and the beam is not directly exposed to water. Determine the tension and shear reinforcement required in the beam.

TSE16_38-41.indd 41TSE16_38-41.indd 41 22/03/2013 13:4522/03/2013 13:45