› Note 4 Level 2
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TheStructuralEngineer April 2013
Technical Technical Guidance Note
Designing a concrete beam Introduction
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 profiles.
ICON LEGEND
Principles of W concrete beam design W Applied practice
W Worked example
W Further reading
W Web resources
Analysis of concrete beams
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 fire protection • Material properties • Serviceability • Minimum area of steel reinforcement A concrete beam is defined 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.
Like concrete slabs, there are a set of coefficients 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 coefficients 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 coefficients to remain valid. Where the geometry of the beam falls outside of the parameters that allow for the use of coefficients 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
Table 1: Bending moment and shear coefficients for beams with uniform loading and spans Location
Outer support
End 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 coefficients to determine bending moments, it is not permissible to redistribute them.
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along the length of the element. Due to the complexity in analysing this phenomenon it will not be considered here.
Design of concrete beams There are three forms of reinforced concrete beam: rectangular, ‘T’ and ‘L’. These are defined 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 flange beff is derived using the following equation:
b eff = b w + / b eff,i # b
Where:
bw is the width of the beam beff,i is defined 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 defined as 2bi. This is an important variable as it influences the width of compression flanges to ‘T’ and ‘L’ beams due to the fact that they cannot overlap one another. The length over which the compression flange occurs lo is defined as the distance between the points of contraflexure 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 defines the width of the compression flanges that form part of ‘T’ and ‘L’ beams in relation to ones that are adjacent to them.
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E
Figure 1 Distance between points of contraflexure lo for continuous beams
Detailing requirements When 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 difficult to install you risk significant delays during construction; to the point where some redesign 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.
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Figure 2 ‘T’ and ‘L’ beam definitions
Before it can be assumed that a compression flange 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 defined as 2(d-z), with z being the lever arm of the beam cross section. If s ≤ h then the compression flange is not being effective 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: 2
If M 2 0.167fck bd then compression reinforcement is required. Where: M is the applied ultimate bending moment fck is the cylinder compressive strength of concrete b is the width of the beam d is the effective depth of 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.
Where:
fyk is the characteristic yield strength of reinforcement, which is typically taken to be 500 N/mm2 d2 is the effective depth to the compression reinforcement, which is from the top most surface of the beam for sagging moments and the beam’s soffit for hogging moments
The minimum spacing between bars is one of the following:
For ‘T’ and ‘L’ beams, in instances where the following expression is true, compression reinforcement is required:
Whichever is the greater is the chosen spacing.
M > 0.567fck b f h f (d - 0.5h f ) Where: bf is the width of the compression flange hf is the thickness of the compression flange, which can be no more than 0.36d
• The maximum bar size in the element • The maximum aggregate size +5mm • 20mm
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:
fyk A s,req m 1 }2 Qk + Gk a k vs = c c ms m' 1.5Q + 1.35G 1c A k k s, prov d γms is the material factor for steel reinforcement
When no compression reinforcement is required, the minimum amount of steel reinforcement is placed within the compression zone of the beam.
As,req is the area of tension reinforcement required
As,prov is the area of tension reinforcement provided
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Figure 3 Beam with spacer bar
The following equation is used to determine the area of steel As2 needed:
M - 0.167fck bd 2 A s2 = 0.87fyk (d - d 2)
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Spacer bar
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› Note 4 Level 2
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Technical Technical Guidance Note
TheStructuralEngineer April 2013
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).
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Figure 4 Variable strut inclination method
The second section of the expression concerns the factors for dead loads (permanent actions) and imposed loads (variable actions). The relevant coefficients 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.
Shear in concrete beams 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
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Shear 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.
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:
i = 0.5 sin -1 ;
5.56V Ed E b w d (1 - fck /250) fck
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. Shear reinforcement for concrete beams If it is >1 then a check needs to be carried out for the shear force at a VRd,s is defined as follows: distance z from the end of A sw k the beam. a V Rd,s = s zfywd (cot i + cot a) sin a $ V Ed If the value of cot θ is >2.5 then it is assumed to be 2.5. Where: VEd is the applied ultimate shear The minimum amount of shear reinforcement Asw is the area of shear reinforcement required in a beam Asw,min is defined thus: s is the spacing between shear links in mm z is the lever arm to the primary 0.08fck0.5 A sw,min reinforcement in mm $ fywd is the design yield strength of sb w sin a fyk reinforcement, which is typically taken to be 500 N/mm2 over the material factor γs, The maximum spacing for shear links s1,max is which is 1.15 defined as 0.75 d (1 + cot α) α is the angle of the shear links to the horizontal axis of the beam. For vertically The maximum distance between vertical orientated links, the value of cot α is 0 and legs of shear links sb,max is 0.75d and must be less than 600mm. sin α is 1
Worked example 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 floor slab, making it a ‘T’ beam. It has a fire rating of 1 hour, the
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Eurocode 0. 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.
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
Glossary and further reading Contraflexure – 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
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