Structural Design of
Concrete and Masonry:
Structural Design of Concrete and Masonry
Contents
About The Concrete Centre
4
How to calculate anchorage and lap lengths to Eurocode 2
12
Deflection – the span-to-effective-depth method and Eurocode 2
17
Fire Design of concrete columns and walls to Eurocode 2
23
Eurocode 6: Design of masonry structures for vertical loads
29
Eurocode 6: Design of masonry structures for lateral loads and other factors
34
Design of post-tensioned slabs
40
Guidance on the design of liquid-retaining structures
45
An introduction to strut-and-tie modelling
The Concrete Centre provides material, design and construction guidance with the aim
of enabling those involved in the design, use and performance of concrete and masonry
to realise the potential of the material.
Through funding from the cement, aggregates, ready-mixed and precast concrete
sectors, The Concrete Centre is able to invest in the development of services and
resources that support the design and construction of robust, sustainable, cost-effective
structures throughout the built environment.
Resources available for structural engineers are highlighted on the inside back cover
of this document (page 51) and there is a wealth of material available at
www.concretecentre.com I 3
Structural Design of Concrete and Masonry
Structural Design of Concrete and Masonry
This compendium includes articles that were first
published in the renowned journal ‘The Structural
Engineer’ following its invitation to The Concrete Centre
to write a series of technical papers on structural design
in concrete.
The series includes topics chosen represent topical issues
and respond to frequently asked questions that we
receive from designers, such as guidance on anchorage
and lap lengths, post-tensioning and column fire design.
The compendium also includes papers on deflections,
Eurocode 6, liquid retaining structures and strut-and-tie.
Acknowledgements
The Concrete Centre would like to thank the authors
and peer reviewers for their contribution to these
technical papers including:
John Roberts; RS Narayanan; Robert Vollum,
Imperial College.
Structural Design of Concrete and Masonry
How to calculate anchorage and lap
lengths to Eurocode 2
Introduction
EC2 provides information about reinforcement detailing in Sections 8 and 9 of Part 1-1 (BS EN 1992-1-1)1. Section 8 provides information on
the general aspects of detailing and this is where the rules for anchorage and lap lengths are given. Section 9 sets out the rules for detailing different types of elements, such as beams, slabs and columns. The calculation for anchorage and lap lengths is as described in EC2 and is fairly extensive. There are shortcuts to the process, the first being to use one of the tables produced by others2–4. These are based
on the bar being fully stressed and the cover being 25mm or ‘normal’. These assumptions are conservative, particularly the assumption that the bar is fully stressed, as bars are normally anchored or lapped away from the points of high stress. Engineering judgement should be used when applying any of the tables to ensure that the assumptions are reasonable and not overly conservative.
This article discusses how to calculate an anchorage and lap length for steel ribbed reinforcement subjected to predominantly static loading using the information in Section 8. Coated steel bars (e.g. coated with paint, epoxy or zinc) are not considered. The rules are applicable to normal buildings and bridges.
An anchorage length is the length of bar required to transfer the force in the bar into the concrete. A lap length is the length required to transfer the force in one bar to another bar. Anchorage and lap lengths are both calculated slightly differently depending on whether the bar is in compression or tension.
For bars in tension, the anchorage length is measured along the centreline of the bar. Figure 1 shows a tension anchorage for a bar in a pad base. The anchorage length for bars in tension can include bends and hooks (Figure 2), but bends and hooks do not contribute to compression anchorages. For a foundation, such as a pile cap or pad base, this can affect the depth of concrete that has to be provided. Most tables that have been produced in the UK for anchorage and lap lengths have been based on the assumption that the bar is fully stressed at the start of the anchorage or at the lap length. This is rarely the case, as good detailing principles put laps at locations of low stress and the area of steel provided tends to be greater than the area of steel required.
Ultimate bond stress
Both anchorage and lap lengths are determined by the ultimate bond stress fbd which depends on the concrete strength and whether the anchorage or lap length is in a ‘good’ or ‘poor’ bond condition. ƒbd = 2.25η1η2ƒctd (Expression 8.2 from BS EN 1992-1-1) where:
ƒctd is the design tensile strength of concrete, ƒctd = αctƒctk,0,05/γC
ƒctk,0,05 is the characteristic tensile strength of concrete,
ƒctk,0,05 = 0.7 × ƒctm
ƒctm is the mean tensile strength of concrete, ƒctm = 0.3 × ƒck(2/3)
ƒck is the characteristic cylinder strength of concrete γC is the partial safety factor for concrete
(γC = 1.5 in UK National Annex5)
αct is a coefficient taking account of long-term effects on the tensile strength and of unfavourable effects resulting from the way the load is applied (αct = 1.0 in UK National Annex) In EC2, anchorage and lap lengths are proportional to the stress in the bar at the start of the anchorage or lap. Therefore, if the bar is stressed to only half its ultimate capacity, the lap or anchorage length will be half what it would have needed to be if the bar were fully stressed.
This article provides guidance on how to calculate
anchorage and lap lengths to Eurocode 2.
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Structural Design of Concrete and Masonry
Table 1 gives the design tensile strengths for structural concretes up to C50/60.
η1 is the coefficient relating to the bond condition and η1 = 1 when the bond condition is ‘good’ and η1 = 0.7 when
the bond condition is ‘poor’
It has been found by experiment that the top section of a concrete pour provides less bond capacity than the rest of the concrete and therefore the coefficient reduces in the top of a section. Figure 8.2 in BS EN 1992-1-1 gives the locations where the bond condition can be considered ‘poor’ (Figure 3). Any reinforcement that is vertical or in the bottom of a section can be considered to be in ‘good’ bond condition. Any horizontal reinforcement in a slab 275mm thick or thinner can be considered to be in ‘good’ bond condition. Any horizontal reinforcement in the top of a thicker slab or beam should be considered as being in ‘poor’ bond condition.
η2 = 1.0 for bar diameters ø ≤ 32mm
η2 = (132–ø)/100 for ø > 32mm (η2 = 0.92 for 40mm bars) ø is the diameter of the bar
Confinement of concrete results in the characteristic compression strength being greater than ƒck and is known as ƒck.c. If the concrete surrounding a steel reinforcing bar is confined, the characteristic strength of the concrete is increased and so will be the ultimate bond stress between the bar and the concrete. Increasing the ultimate bond stress will reduce the anchorage length. Concrete can be confined by external pressure, internal stresses or reinforcement.
Anchorage lengths
Figure 4 gives the basic design procedure for calculating the anchorage length for a bar. There are various shortcuts, such as making all α coefficients = 1, that can be made to this procedure in order to ease the design process, although this will result in a more conservative answer. Both anchorage and lap lengths are determined from the ultimate bond strength ƒbd. The basic required anchorage length lb,rqd can be calculated
from:
lb,rqd = (ø/4) (σsd/ƒbd)
where σsd is the design stress in the bar at the position from where the anchorage is measured. If the design stress σsd is taken as the maximum
allowable design stress:
σsd = ƒyd = ƒyk/γs = 500/1.15 = 435MPa
This number is used for most of the published anchorage and lap length tables, but the design stress in the bar is seldom the maximum allowable design stress, as bars are normally anchored and lapped away from positions of maximum stress and the As,prov is normally greater than As,req.
The design anchorage length lbd is taken from the basic required
anchorage length lb,rqd multiplied by up to five coefficients, α1 to α5. lbd = α1 α2 α3 α4 α5 lb,rqd ≥ lb,min
where the coefficients α1 to α5 are influenced by: α1 – shape of the bar
90° ≤ a < 150° a
b) Hook
a) Bend or ‘L’ bar c) Loop or ‘U’ bar
≥5Ø
≥5Ø ≥150
90° ≤ a < 150° a
b) Hook
a) Bend or ‘L’ bar c) Loop or ‘U’ bar
≥5Ø
≥5Ø ≥150
a) 45º < a < 90º
Direction of concreting Direction of concreting
Direction of concreting Direction of concreting Key b) h < 250 mm d) h > 600 mm h h ≥ 300 c) h > 250 mm 250 a
‘Good’ bond conditions ‘Poor’ bond conditions Figure 3 ‘Good’ and ‘Poor’ bond conditions
Figure 2 Typical bends and hooks bent through 90o or more
Structural Design of Concrete and Masonry
α2 – concrete cover
α3 – confinement by transverse reinforcement α4 – confinement by welded transverse reinforcement α5 – confinement by transverse pressure
The minimum anchorage length lb,min is:
max {0.3lb,rqd; 10ø; 100mm} for a tension anchorage max {0.6lb,rqd; 10ø; 100mm} for a compression anchorage
The maximum value of all the five alpha coefficients is 1.0. The minimum is never less than 0.7. The value to use is given in Table 8.2 of BS EN 1992-1-1. In this table there are different values for α1 and α2 for straight bars and bars called other than straight. The other shapes are bars with a bend of 90° or more in the anchorage length. Any benefit in the α coefficients from the bent bars is often negated by the effects of cover. Note that the product of α2 α3 and α5 has to be ≥ 0.7.
To calculate the values of α1 and α2 the value of cd is needed. cd is obtained from Figure 8.3 in BS EN 1992-1-1 and shown here in Figure 5. cd is often the nominal cover to the bars. In any published anchorage tables, a conservative value for the nominal bar cover has to be assumed and 25mm is used in the Concrete Centre tables. If the cover is larger than 25mm, the anchorage length may be less than the value quoted in most published tables. For hooked or bent bars in wide elements, such as slabs or walls, cd is governed by the spacing between the bars. In Table 8.2 of BS EN 1992-1-1 anchorage length alpha coefficients are given for bars in tension and compression. The alpha values for a compression anchorage are all 1.0, the maximum value, except for α4 which is 0.7, the same as a tension anchorage. Hence, the anchorage length for a compression anchorage can always conservatively be used as the anchorage length for a bar in tension.
Start
Is the bar in ‘good’ position?
Is bar diameter
ø ≤ 32mm η2 = (132‐ø)/100
Determine fctd from Table 1
Yes Yes No No No No No No No No No Yes Take lbd = lb,rqd α4 = 1.0 Is the bar in compression? α1 = 1.0 α2 =1‐0.15(cd‐ø)/ø 0.7≤ α2 ≤1.0 α1= 0.7 if cd > 3ø α1=1.0 if cd ≤ 3ø α2 =1 ‐ 0.15 (cd‐3ø)/ø 0.7≤ α2 ≤1.0 α3 = 1 – Kλ 0.7≤ α3 ≤1.0 α3 = 1.0 Is the bar straight?
Does the bar have another bar between the
surface of the concrete and itself?
Is the bar confined by transverse pressure? α5=1– 0.04p 0.7≤ α5 ≤1.0 α5=1.0 Is α2∙α3∙α5 < 0.7 Take α2·α3·α5 = 0.7 lbd = α1∙α2·α3·α4∙α5·lb,rqd Check lbd > max{0.3lb,rqd;10ø;100mm} α1, α2, α3 and α5=1.0 α4 = 0.7 Yes Yes Yes Yes Yes Yes END η1 = 1.0 η2 = 1.0
Determine ultimate bond stress fbd = 2.25 η1 η2 fctd
Determine As,req and As,prov where the anchorage starts
Determine ultimate design stress in bar σsd = 435 As,req / As,prov
Determine basic anchorage length lb,rqd = (ø/4) (σsd/fbd)
(This can be conservatively used as the design anchorage length, lbd)
Determine the coefficients α1 to α5 (see Table 2)
Does the bar have transverse reinforcement
welded to it?
Can lb,rqd be used as the design
anchorage length lbd?
η1= 0.7
Figure 4: flow chart for anchorage lengths.
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Structural Design of Concrete and Masonry
Alpha values for tension anchorage
Alpha values for tension anchorage are provided in Table 8.2 of BS EN 1992-1-1.
α1 – shape of the bar Straight bar, α1 = 1.0
There is no benefit for straight bars; α1 is the maximum value of 1.0. Bars other than straight, α1 = 0.7 if cd > 3ø; otherwise α1 = 1.0
If we assume that the value of cd is 25mm, then the only benefit for bars other than straight is for bars that are 8mm in diameter or less. For bars larger than 8mm, α1 = 1.0. However, for hooked or bobbed bars in wide elements, where cd is based on the spacing of the bars, α1 will be 0.7 if the spacing of the bars is equal to or greater than 7ø.
α2 – concrete cover
Straight bar, α2 = 1 – 0.15(cd – ø)/ø ≥ 0.7 ≤ 1.0
There is no benefit in the value of α2 for straight bars unless (cd – ø) is positive, which it will be for small diameter bars. If cd is 25mm, then there will be some benefit for bars less than 25mm in diameter, i.e. for 20mm diameter bars and smaller, α2 will be less than 1.0. Bars other than straight, α2 = 1 – 0.15(cd – 3ø)/ø ≥ 0.7 ≤ 1.0
Start
Is the bar in ‘good’ position?
Is bar diameter
ø ≤ 32mm η2 = (132‐ø)/100
Determine fctd from Table 1
Yes Yes No No No No No No No No No Yes Take lbd = lb,rqd α4 = 1.0 Is the bar in compression? α1 = 1.0 α2 =1‐0.15(cd‐ø)/ø 0.7≤ α2 ≤1.0 α1= 0.7 if cd > 3ø α1=1.0 if cd ≤ 3ø α2 =1 ‐ 0.15 (cd‐3ø)/ø 0.7≤ α2 ≤1.0 α3 = 1 – Kλ 0.7≤ α3 ≤1.0 α3 = 1.0 Is the bar straight?
Does the bar have another bar between the
surface of the concrete and itself?
Is the bar confined by transverse pressure? α5=1– 0.04p 0.7≤ α5 ≤1.0 α5=1.0 Is α2∙α3∙α5 < 0.7 Take α2·α3·α5 = 0.7 lbd = α1∙α2·α3·α4∙α5·lb,rqd Check lbd > max{0.3lb,rqd;10ø;100mm} α1, α2, α3 and α5=1.0 α4 = 0.7 Yes Yes Yes Yes Yes Yes END η1 = 1.0 η2 = 1.0
Determine ultimate bond stress fbd = 2.25 η1 η2 fctd
Determine As,req and As,prov where the anchorage starts
Determine ultimate design stress in bar σsd = 435 As,req / As,prov
Determine basic anchorage length lb,rqd = (ø/4) (σsd/fbd)
(This can be conservatively used as the design anchorage length, lbd)
Determine the coefficients α1 to α5 (see Table 2)
Does the bar have transverse reinforcement
welded to it?
Can lb,rqd be used as the design
anchorage length lbd?
η1= 0.7
Figure 4: flow chart for anchorage lengths.
Table 1: Design tensile strength, ƒctd
C20/25 C25/30 C28/35 C30/37 C32/40 C35/45 C40/50 C50/60
ƒctm 2.21 2.56 2.77 2.90 3.02 3.21 3.51 4.07
ƒctk, 0.05 1.55 1.80 1.94 2.03 2.12 2.25 2.46 2.85
Structural Design of Concrete and Masonry
C
1C
1C
C
a
a
a) Straight barscd = min (a/2, c1, c) b) Bent or hooked bars cd = min (a/2, c1) c) Looped bars cd = c
K = 0.1
As Øt, Ast As Øt, Ast As Øt, Ast
K = 0.05 K = 0
Table 2: Anchorage and lap lengths for locations of maximum stress Bond
Condition
Reinforcement in tension, bar diameter, Ф (mm) Reinforcement
in compression 8 10 12 16 20 25 32 40 Anchorage length, lbd Straight bars only Good 230 320 410 600 780 1010 1300 1760 40Ф Poor 330 450 580 850 1120 1450 1850 2510 58Ф Other bars only Good 320 410 490 650 810 1010 1300 1760 40Ф Poor 460 580 700 930 1160 1450 1850 2510 58Ф Lap length, lo 50% lapped in one location (a6=1.4) Good 320 440 570 830 1090 1420 1810 2460 57Ф Poor 460 630 820 1190 1560 2020 2590 3520 81Ф 100% lapped in one location (a6=1.5) Good 340 470 610 890 1170 1520 1940 2640 61Ф Poor 490 680 870 1270 1670 2170 2770 3770 87Ф Concrete class C20/25 C28/35 C30/37 C32/40 C35/45 C40/50 C45/55 C50/60 Factor 1.16 0.93 0.89 0.85 0.80 0.73 0.68 0.63 Notes
1) Nominal cover to all sides and distance between bars ≥2mm (i.e. α2<1). At laps, clear distance between bars ≤50mm. 2) α1 = α3 = α4 = α5 = 1.0. For the beneficial effects of shape of bar, cover and confinement see Eurocode 2, Table 8.2.
3) Design stress has been taken as 435MPa. Where the design stress in the bar at the position from where the anchorage is measured, σsd, is less than 435MPa the figures in this table can be factored by σsd/435. The minimum lap length is given in cl. 8.7.3 of Eurocode 2. 4) The anchorage and lap lengths have been rounded up to the nearest 10mm.
5) Where 33% of bars are lapped in one location, decrease the lap lengths for ‘50% lapped in one location’ by a factor of 0.82. 6) The figures in this table have been prepared for concrete class C25/30.
Figure 5 Values of cd (c and c1 are taken to be cnom)
Figure 6 Values of K
Source: EC2-1-1, Figure 8.3.
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Structural Design of Concrete and Masonry
There is no benefit in the value of α2 for bars other than straight unless (cd – 3ø) is positive. If we assume that the value of cd is 25mm, then the only benefit for bars other than straight is for bars that are 8mm in diameter or less. For bars larger than 8mm α2 = 1.0. Again, for hooked or bobbed bars in wide elements, where cd is based on the spacing of the bars, α2 will be less than 1.0 if the spacing of the bars is equal to or greater than 7ø.
α3 – confinement by transverse reinforcement
All bar types, α3 = 1 – Kλ ≥ 0.7 ≤ 1.0 where:
K depends on the position of the confining reinforcement. The value of K is given in Figure 8.4 of BS EN 1991-1-1 and shown here in Figure 6. A corner bar in a beam has the highest value for K of 0.1. Bars which are in the outermost layer in a slab are not confined and the K value is zero λ is the amount of transverse reinforcement providing
confinement to a single anchored bar of area As = (ΣAst – ΣAst,min) / As
ΣAst is the cross-sectional area of the transverse reinforcement with diameter øt along the design anchorage length lbd
ΣAst,min is the cross-sectional area of the minimum transverse
reinforcement = 0.25 As for beams and zero for slabs
For example, if anchoring an H25 bar in a beam with H10 links at 300mm centres:
As = 491mm2 for a 25mm diameter bar
ΣAst,min = 0.25 × 491 = 123mm2
ΣAst = 4 × 78.5 = 314mm2, assuming links will provide at least four 10mm
diameter transverse bars in the anchorage length λ = (ΣAst – ΣAst,min)/ As = (314 – 123)/491 = 0.38 α3 = 1 – Kλ = 1 – 0.1 × 0.38 = 0.96
l
bdα4 – confinement by welded transverse reinforcement
α4 = 0.7 if the welded transverse reinforcement satisfies the requirements given in Figure 8.1e of BS EN 1992-1-1. Otherwise α4 = 1.0.
α5 – confinement by transverse pressure
All bar types, α5 = 1 – 0.04p ≥ 0.7 ≤ 1.0 where p is the transverse pressure (MPa) at the ultimate limit state along the design anchorage length, lbd. One place where the benefit of α5 can be used is when calculating the design anchorage length lbd of bottom bars at end supports. This benefit is given in BS EN 1992-1-1 cl. 9.2.1.4(3) and Figure 9.3, and is shown here in Figure 7. It applies to beams and slabs.
Lap lengths
A lap length is the length two bars need to overlap each other to transfer a force F from one bar to the other. If the bars are of different diameter, the lap length is based on the smaller bar. The bars are typically placed next to each other with no gap between them. There can be a gap, but if the gap is greater than 50mm or four times the bar diameter, the gap distance is added to the lap length.
Lapping bars, transferring a force from one bar to another via concrete, results in transverse tension and this is illustrated in Figure 8 which is a plan view of a slab. Cl.8.7.4.1 of BS EN 1992-1-1 gives guidance on the amount and position of the transverse reinforcement that should be provided. Following these rules can cause practical detailing issues if you have to lap bars where the stress in the bar is at its maximum. If possible, lapping bars where they are fully stressed should be avoided and, in
Figure 8 Plan view of slab illustrating transverse tension
Figure 7 Anchorage of bottom reinforcement at end supports in beams and slabs where directly supported by wall or column
Structural Design of Concrete and Masonry Start η1 = 0.7 η2 = (132-ø)/100 η2 = 1.0 η1 = 1.0 No No No No No Yes Yes Yes Yes Yes No α3 =1.0 α5 =1.0 Is α2·α3∙α5 < 0.7 Take α2·α3∙α5 = 0.7 l0 = α1∙α2∙α3∙α5∙α6∙lb,rqd Check l0 > max{0.3α6∙lb,rqd; 15ø; 200mm} α5 = 1 – 0.04p 0.7≤α5≤1.0
Is the bar confined by transverse pressure? α3 = 1 – Kλ 0.7≤ α3 ≤1.0 No No Yes Yes END Yes
Determine fctd from Table 1
Is the bar in ‘good’ position?
Is smaller bar diameter ø ≤ 32mm
Determine ultimate bond stress fbd = 2.25 η1 η2 fctd
Determine As,req and As,prov where the lap starts
Determine ultimate design stress in bar σsd = 435 As,req / As,prov
Determine basic anchorage length lb,rqd = (ø/4) (σsd/fbd)
Determine the coefficients α1, α2, α3 and α5
(see Table 2)
Take l0 = lb,rqd·α6
Determine α6
α6 = 1.4 for 50% lapped at a section
α6 = 1.5 for 100% lapped at a section
Is lb,rqd· α6 satisfactory as the lap length? Is the bar in compression? Is the bar straight? α1, α2, α3 and α5=1.0 α1 =1.0 α2 = 1-0.15(cd‐ø)/ø 0.7≤ α2 ≤1.0 α1 = 0.7 if cd > 3ø α1 = 1.0 if cd ≤ 3ø α2 = 1‐ 0.15(cd‐3ø)/ø 0.7≤ α2 ≤1.0
Does the bar have another bar between the
surface of the concrete and itself? Figure 9: flow chart for lap lengths.
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Structural Design of Concrete and Masonry
typical building structures, there is usually no need to lap bars where they are fully stressed, e.g lapping bars in the bottom of a beam or slab near mid-span. Examples where bars are fully stressed and laps are needed are in raft foundations and in long-span bridges.
The wording of this clause regarding guidance on the provision of transverse reinforcement is that it should be followed rather than it must be followed. This may allow the designer some scope to use engineering judgement when detailing the transverse reinforcement, e.g increasing the lap length may reduce the amount of transverse reinforcement.
All the bars in a section can be lapped at one location if the bars are in one layer. If more than one layer is required, then the laps should be staggered.
A design procedure to determine a lap length is given in Figure 9 and, as can be seen in the flow chart, the initial steps are the same as for the calculation of an anchorage length.
Design lap length, l0 = α1 α2 α3 α5 α6 lb,rqd ≥l0,min (Eq. 8.10 in BS EN 1992-1-1)
The coefficients α1, α2, and α5 are calculated in the same way as for anchorage lengths and, again, all the coefficients can be taken as = 1.0 as a simplification.
α3 is calculated slightly differently. When calculating α3 for a lap length ΣAst,min = As(σsd /fyd), with As = area of one lapped bar.
The design lap length can therefore be determined by multiplying the design anchorage length by one more alpha coefficient α6, provided α3 has been calculated for a lap rather than an anchorage.
Design lap length, l0 = α6 lbd ≥ l0,min
Minimum anchorage length, l0,min = max {0.3 α6 lb,rqd; 15ø; 200mm} α6 – coefficient based on the percentage of lapped bars in one lapped section, ρ1
α6 = (ρ1/25)0.5 ≥ 1.0 ≤ 1.5
where:
ρ1 is the percentage of reinforcement lapped within 0.65l0 from the centre of the lap length considered
In most cases either the laps will all occur at the same location, which is 100% lapped and where α6 = 1.5, or the laps will be staggered, which is 50% lapped and where α6 = 1.4.
For vertically cast columns, good bond conditions exist at laps.
“The largest possible savings in lap and anchorage length can be obtained
by considering the stress in the bar where it is lapped or anchored.”
Recommendations
The largest possible savings in lap and anchorage length can be obtained by considering the stress in the bar where it is lapped or anchored.
For most locations, the old rule of thumb of lap lengths being equal to 40ø should be sufficient. For this to be the case, the engineer should use their judgement and should satisfy themselves that the lap and anchorage locations are away from locations of high stress for the bars being lapped or anchored. Where it is not possible to lap or anchor away from those areas of high stress, the lengths will need to be up to the values given in Table 2.
This article presents the rules currently set out in EC2. However, there has been significant recent research which may find its way into the next revision of the Eurocode. For example, research into the effect of staggering on the strength of the lap (α6) was discussed by John Cairns in Structural Concrete (the fib journal) in 20146. In the review of the
Eurocodes, the detailing rules have been the subject of 208 comments (18% of the total for EC2) and it is acknowledged that the rules need to be simplified in the next revision.
References:
1) British Standards Institution (2004) BS EN 1992-1-1:2004 Design of concrete structures. General rules and rules for buildings, London, UK: BSI 2) Bond A. J., Brooker O., Harris A. J. et al. (2011) How to Design Concrete Structures using Eurocode 2, Camberley, UK: MPA The Concrete Centre 3) The Institution of Structural Engineers and the Concrete Society (2006) Standard method of detailing structural concrete: A manual for best practice. (3rd ed.), London, UK: The Institution of Structural Engineers
4) The Institution of Structural Engineers (2006) Manual for the design of concrete building structures to Eurocode 2, London, UK: The Institution of Structural Engineers
5) British Standards Institution (2005) NA to BS EN 1992-1-1:2004 UK National Annex to Eurocode 2. Design of concrete structures. General rules and rules for buildings, London, UK: BSI
6) Cairns J. (2014) ‘Staggered lap joints for tension reinforcement’, Structural Concrete, 15 (1), pp 45–54
Structural Design of Concrete and Masonry
Deflection – the span-to-effective-depth
method and EC2
Introduction
Essentially, the span-to-effective-depth method is a hand method based on experience, justified by various reports1,2. The L/d method also
serves as a very useful and valuable hand check on computer outputs. According to Section 7.4.2 of BS EN 1992-1-13 (Eurocode 2) and fib Model
Code 20104, its use “will be adequate for avoiding deflection problems
in normal circumstances”. The main attraction of the method is that it avoids the need to undertake laborious calculations.
While according to Eurocode 05, deflection limits should be agreed with
clients, generally the limits implicit in the L/d verification of deflection of concrete structures are L/250 overall and L/500 post partitions (i.e. for deflection affecting partitions, brittle finishes, etc.).
The current L/d method
In simple terms, the current BS EN 1992 L/d method means verifying that:
Allowable L/d = N x K x F1 x F2 x F3 ≥ actual L/d (1) where:
N = basic span-to-effective-depth ratio derived for K = 1.0 from the formulae:
if ρ ≤ ρ0
N = L/d = K[11 + 1.5ƒck0.5 ρ0/ρ + 3.2ƒck0.5 (ρ0 / ρ – 1)1.5] (2a)
or if ρ > ρ0
N = L/d = K[11 + 1.5ƒck0.5 ρ0/(ρ – ρ’) + ƒck0.5 (ρ’ / ρ0)0.5 /12] (2b)
for ρ’ = 0, N may be determined from Figure 1 where:
L = span
d = effective depth
ƒck = characteristic compressive cylinder strength of
concrete at 28 days ρ0 = fck0.5/1000
ρ = As,req/bd ρ’ = As2req/bd
K = factor to account for structural system (Table 1)
F1 = factor to account for flanged sections. When beff/bw = 1.0, factor
F1 = 1.0. When beff/bw > 3.0, factor F1 = 0.80. For values of beff/bw between 1.0 and 3.0, interpolation may be used
F2 = factor to account for brittle partitions in association with long spans. Generally F2 = 1.0 but if brittle partitions are liable to be damaged by excessive deflection, F2 should be determined as follows:
a) in flat slabs in which the longer span is greater than 8.5m, F2 = 8.5/leff
b) in beams and other slabs with spans in excess of 7.0m, F2 = 7.0/leff
F3 = factor to account for service stress in tensile reinforcement = 310/σs. It is considered conservative to assume that
310/σs = 500As,prov/(ƒykAs,req) where:
σs = tensile stress in reinforcement at mid-span (at support
for cantilevers) under design load at serviceability limit state (SLS) calculated using the characteristic value of serviceability load6
F3 is restricted to ≤1.56
Notes
Factors F1, F2 and F3 have been used here for convenience, they are not symbols used in BS EN 1992-1-1. According to the notes to Table NA.5 of the UK National Annex (NA)6 warnings are given that the values
of K may not be appropriate when formwork is struck at an early age. L/d may not exceed 40K
Basis and current issues
The L/d method is outlined in Eurocode 2 Commentary7. The method is
based on parametric studies by Corres et al.2, rather than theory. There
have been many comments relating to the soundness of the method, which is now acknowledged to have some limitations and deficiencies8,9:
• The expressions (7.16a) and (7.16b) in BS EN 1992-1-1 (Equations 2a and 2b) assume a certain ratio between total load and dead load, superimposed dead load (SDL) and imposed load (IL). It would be desirable to introduce different possibilities for these ratios in order to widen the application field of these formulae
The span-to-effective-depth (L/d) method is a very
popular way of verifying the limit state of deformation
(i.e. deflection) of concrete slabs and beams.
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Structural Design of Concrete and Masonry
• The expressions do not account for excess reinforcement in tension or compression. (UK practice allowed up to 100% additional reinforcement.) This parameter should be included • The expressions do not account for peak loading during
construction and the cracking induced during that process (Figure 2). This parameter should also be introduced
• The effects of ƒctm,ƒl (mean flexural tensile strength of concrete) were ignored in the background document, whereas the effects are very noticeable for sections with <0.6% reinforcement, i.e. they are very noticeable in slabs. The mean 28-day direct concrete tensile strength was used in deflection calculations
Figure 1 Basic span-to-effective-depth ratios, N, for K = 1, ρ’ = 0
Figure 3 L/d for simply supported slabs
Figure 2 Typical loading and deflection history for slab in multistorey building
Figure 4 L/d for simply supported slabs supporting imposed load of 2.5kN/m2
• The analysis of the section to determine whether the section was cracked or not looked at the “centre span of the beam only”, and conservatively used those properties throughout
• The assessment of Ec,eff (effective modulus) was questionable • The relative humidity (RH) was taken as 70%. In the UK, RH is often taken as being 50% internally and 80% or 85% externally • Results using this method do not give a good match with
span-to-depth ratios derived by calculating deflections rigorously under quasi-permanent loading (Figures 3 and 4)
• No allowance appears to have been made for the use of loading expressions (6.10a) and (6.10b) in BS EN 1990
• The method for adjustment when providing more reinforcement than required for flexure (based on steel service stress) is not conservative
The most substantiated comments came from Vollum10. The significant
reductions in slab thickness initially allowed by BS EN 1992-1-1, compared to those allowed by BS 8110, were met with some scepticism in the UK and modifications were made via the UK NA to EC26 (as
outlined earlier). Vollum showed that the EC2 span-to-depth rules do not account for cracking during construction; variations in effective depth over thickness (d/h), varying serviceability/ultimate loading ratio (w/wu) or the effect of restraint at the external supports.
Table 1: K factors to be applied to basic ratios of span to effective depth for different structural systems
Element K
Simply supported beams or slabs 1.0
End span of continuous beams or slabs 1.3
Interior spans of continuous beams or slabs 1.5
Flat slabs (based on longer span) 1.2
Cantilevers 0.4
Notes
1For two-way spanning slabs, the check should be carried out on the basis of the shorter span.
2This graph assumes simply supported span condition (K = 1.0).
K = 1.5 for interior span condition K = 1.3 for end span condition K = 0.4 for cantilevers
3Compression reinforcement, r’, has been taken as 0.
4Curves based on the following expressions: where r ≤ r0 and where r > r0. 11 + + 3.2 – 1 = K r fckr0 1.5 d l fck r r0
[
( ) ]
11 + + = K 1.5( r – r’)fckr0 d l 12 fck r0 r’[
]
1.5Structural Design of Concrete and Masonry
These issues have led some to suggest that the L/d method should be deleted from standards. However, to do so would be to deny designers a valuable tool and ‘feel’ for their designs, although there is clearly room for improvement.
Rigorous method of assessing deflection
Here, it is worth explaining the rigorous method according to BS EN 1992-1-1, Cl. 7.4.3. A section will crack if it experiences a serviceability moment exceeding its moment capacity at the time Mcr(t). If a section is cracked, then its inertia is much less than that of the uncracked section and so curvature is much greater in cracked sections. Cracked sections and greater degrees of cracking lead to larger curvatures and deflections (Figure 5).
Economically designed horizontal elements act somewhere between wholly uncracked and wholly cracked. Slabs tend to be less highly stressed and are cracked along only part of their spans. Beams tend to be more highly stressed and crack along much more of their spans. Actions are applied at different times and these actions may or may not cause cracking depending upon the flexural tensile strength of the concrete at the time. Once cracked, a section is assumed to stay cracked but some tensile stiffening occurs in the concrete between cracks. So the mean inertia of the segment is somewhere between those for wholly uncracked or wholly cracked sections. When considering curvatures, these different actions incur different creep coefficients, which affect the applicable effective modulus of the concrete used in assessing curvatures.
BS EN 1992-1-1 (and MC2010) state that an adequate prediction of behaviour and the mean curvature in a discreet element (Figure 6) is given by: 1/rm = ζ(ψ2 + ψ2cs) + (1 – ζ)(ψ1 + ψ1cs) where: rm = mean radius ζ = 1 – β(Mcr/M)2 where:
β = 1.0 for short-term and β = 0.5 for long-term loading. For construction loads, conservatively10 β = 0.70
Mcr = cracking moment
M = SLS moment
ψ1 = M/EceffI1 = curvature of uncracked section
ψ2 = M/EceffI2 = curvature of cracked section where:
Eceff = Ecm/(1 + φ)
where:
Ecm = modulus at 28 days φ = creep coefficient
I1, I2 = inertias of the uncracked and cracked sections ψ1cs, ψ2cs = shrinkage curvature
This ‘rigorous’ method is described in greater detail elsewhere11,12
and is supported by site-based research10,13,14.
Greater accuracy may be achieved by considering small increments of span and computing relevant curvatures and thus overall deflections. The method involves numerical integration, which is tedious by hand but can, of course, be undertaken by computer, notably by spreadsheet software.
Default assumptions for rigorous analysis
Deflections depend significantly on cracking, material properties and loading: all of which makes for difficulties and uncertainties at the design stage.
However, Vollum10 suggested that in the absence of better information,
the following assumptions should be made in deflection calculations of slabs in multistorey construction:
• The slab is struck at seven days; the superimposed dead load is applied at 60 days
• Creep and shrinkage strains are calculated with a relative humidity of 50% (internal environment assumed)
• Two levels of backprops are used • The floor above is cast after 10 days
• When slabs are supported by slabs below during construction, the peak construction load ωpeak is the peak UDL action for the SLS, which should be taken as 0.04h kN/m2 where h is the slab thickness
in mm
• The permanent load ωperm should be taken as the quasi permanent load combination and be applied at one year
• Peak deflections are calculated under the frequent load case; the increment in load ωfreq – ωperm should be treated as an instantaneous load in the calculation of ELT
where:
ωfreq = the frequent UDL action for the SLS and
ωperm = the permanent UDL action, including quasi-permanent variable actions, for the SLS
ELT = the equivalent long-term modulus of the concrete, dependant on loading and age at time of loading11,12
• It is difficult to assess the effective tensile strength of concrete in slabs due to its inherent variability, and there are uncertainties in the tensile stress induced by internal and external restraint and shrinkage. However, back-analysis of deflection data showed that the effective flexural strength of concrete in reinforced concrete slabs typically lies somewhere between the indirect and flexural strengths
Using these default values, rigorous methods of calculating deflection can be applied in order to judge the span-to-depth method. The differences between the L/d and the rigorous methods can then be compared.
Differences in values between methods
The data in Table 2 were derived for simply supported slabs by using: • spreadsheet TCC31R15 to determine outcome L/d ratios using the
rigorous method (Section 7.4.3 of BS EN 1992-1-1) and the default values described earlier, and
• spreadsheet TCC3115 to determine those using the L/d method in
Section 7.4.2 of BS EN 1992-1-1
For each span and imposed load the depth of the slab was iterated such that all design criteria were met and As,prov = As,req.
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Structural Design of Concrete and Masonry
As may been seen from Fig. 3, the agreement between the current L/d method and the current rigorous analysis method is not good at low spans or low imposed loads. As may be seen from Fig. 4, the current Cl. 7.4.2(2) method appears to underestimate the L/d required by as much as 15% for an imposed load of 2.5kN/m2 at about 8m.
Using this case as a worse case and using the L/d = 26 indicated by Cl. 7.4.2(2) would lead theoretically to long-term deflection of 42mm or L/190 in a 340mm thick simply supported slab (d = 308mm). Post-construction deflection would be L/487. For non-brittle finishes, L/190 compares with limits of L/200 for variable actions for steel beams and L/150 for timber. L/487 would appear acceptable. Actual deflections are often moderated by end restraints, stronger concrete, lower loads, etc. The data and graphs show an apparent anomaly. The L/d required for 2.5kN/m2 is smaller than for 5.0kN/m2 (Fig. 4). Close examination
revealed that, in line with Vollum, construction load was critical. If the slabs were the same thickness, cracking during construction would be the same, but the effect of lower cracked inertia of the 2.5kN/m2 slab
is greater than the additional creep in the more heavily reinforced and loaded slab.
With respect to continuity, rigorous analysis showed good correlation with the K factors in Table 1.
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33
construction defl ection would be L/487. For non-brittle fi nishes, L/190 compares with limits of L/200 for variable actions for steel beams and L/150 for timber. L/487 would appear acceptable. Actual defl ections are often moderated by end restraints, stronger concrete, lower loads, etc.
The data and graphs show an apparent anomaly. The L/d required for 2.5kN/m2 is smaller than for 5.0kN/m2(Fig. 4). Close examination
revealed that, in line with Vollum, construction load was critical. If the slabs were the same thickness, cracking during construction would be the same, but the eff ect of lower cracked inertia of the 2.5kN/m2 slab
is greater than the additional creep in the more heavily reinforced and loaded slab.
With respect to continuity, rigorous analysis showed good correlation with the K factors in Table 1.
Conclusion
Given the complexity and variability of concrete as a material, loading and the environment, it is perhaps unsurprising that the
current L/d method is inaccurate. Nonetheless, as discussed, it appears that the use of L/d methods “will be adequate for avoiding defl ection problems in normal circumstances”. Compliance with span-to-depth ratios means that defl ections in members may be considered not to exceed the implicit limits stated.
However, more rigorous methods are necessary in unusual circumstances or where defl ection limits other than those implicit in the simplifi ed methods are appropriate.
Work continues to provide a more accurate L/d method – particularly at low imposed loads. Part of that process is to consider the harmonisation of defl ection limits across all materials.
Acknowledgement
Parts of this paper were included in: Goodchild C., Vollum R. and Webster R. (2014) ‘Improving the L/d method’, fi b-Congress, Mumbai, India E=0.5 Tension stiffening Mu Fully cracked response Uncracked response Curvature \ \cs1 \cs2 Momen t
EC2 Moment curvature response
Table 2: ‘Basic’ L/d ratios
Span (m) 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0
L/d ratio using rigorous analysis:
Imposed load = 2.5kN/m2 37.8 30.7 27.1 24.3 22.2 20.7 19.6 18.4 17.8 17.2 16.7 Imposed load = 5kN/m2 37.8 31.2 27.6 25.0 22.9 21.4 20.1 18.9 18.2 17.6 17.2 Imposed load = 7.5kN/m2 30.3 27.5 25.4 23.6 22.0 20.5 19.4 18.4 17.7 17.2 16.8 Imposed load = 10kN/m2 24.6 23.2 22.5 21.5 20.4 19.5 18.6 17.8 17.2 16.7 16.3 L/d ratio using Cl 7.4.2(2): Imposed load = 2.5kN/m2 31.9 30.6 29.3 28.2 26.0 23.9 22.2 20.8 19.5 18.3 17.3 Imposed load = 5kN/m2 28.3 27.3 26.5 25.6 23.8 22.3 20.9 19.6 18.5 17.5 16.4 Imposed load = 7.5kN/m2 25.9 25.2 24.5 23.9 22.3 20.9 19.7 18.6 17.6 16.7 15.9 Imposed load = 10kN/m2 24.1 23.5 23.0 22.4 21.0 19.7 18.6 17.6 16.7 15.9 15.1 Notes: fck = 30MPa; fyk = 500MPa; As,prov = As,req; SDL = 1.5kN/m2 and long-term defl ection limit L/250, post-construction defl ection limit L/500
�
Figure 5Typical moment–curvature response
�
Figure 6Curvature in simply supported slabTSE44_30-34 CDG v1.indd 33 24/07/2015 09:10
Figure 6 Curvature in simply supported slab Figure 5 Typical moment–curvature response
Conclusion
Given the complexity and variability of concrete as a material, loading and the environment, it is perhaps unsurprising that the current L/d method is inaccurate. Nonetheless, as discussed, it appears that the use of L/d methods “will be adequate for avoiding deflection problems in normal circumstances”. Compliance with span-to-depth ratios means that deflections in members may be considered not to exceed the implicit limits stated.
However, more rigorous methods are necessary in unusual
circumstances or where deflection limits other than those implicit in the simplified methods are appropriate.
Work continues to provide a more accurate L/d method – particularly at low imposed loads. Part of that process is to consider the harmonisation of deflection limits across all materials.
Acknowledgement
Parts of this paper were included in: Goodchild C., Vollum R. and Webster R. (2014) ‘Improving the L/d method’, fib-Congress, Mumbai, India
Table 2: Basic’ L/d ratios
Span (m) 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0
L/d ratio using rigorous analysis:
Imposed load = 2.5kN/m2 37.8 30.7 27.1 24.3 22.2 20.7 19.6 18.4 17.8 17.2 16.7 Imposed load = 5kN/m2 37.8 31.2 27.6 25.0 22.9 21.4 20.1 18.9 18.2 17.6 17.2 Imposed load = 7.5kN/m2 30.3 27.5 25.4 23.6 22.0 20.5 19.4 18.4 17.7 17.2 16.8 Imposed load = 10kN/m2 24.6 23. 22.5 21.5 20.4 19.5 18.6 17.8 17.2 16.7 16.3 L/d ratio using Cl 7.4.2(2): Imposed load = 2.5kN/m2 31.9 30.6 29.3 28.2 26.0 23.9 22.2 20.8 19.5 18.3 17.3 Imposed load = 5kN/m2 28.3 27.3 26.5 25.6 23.8 22.3 20.9 19.6 18.5 17.5 16.4 Imposed load = 7.5kN/m2 25.9 25.2 24.5 23.9 22.3 20.9 19.7 18.6 17.6 16.7 15.9 Imposed load = 10kN/m2 24.1 23.5 23.0 22.4 21.0 19.7 18.6 17.6 16.7 15.9 15.1
Structural Design of Concrete and Masonry
References:
1) Beeby A. W. (1971) TR456: Modified Proposals for Controlling Deflections by Means of Ratios of Span to Effective Depth, Wexham Springs, UK: Cement and Concrete Association
2) Corres Peiretti H., Pérez Caldentey A., López Agüí J. C. and Edtbauer J. (2002) EC2 serviceability limit states: deflections. Supporting document: first draft, 15 June 2002, Madrid, Spain: Grupo de Hormigón Estructural – ETSICCP – UPM
3) British Standards Institution (2004) BS EN 1992-1-1:2004+A1:2014 Eurocode 2: Design of concrete structures. General rules and rules for buildings, London, UK: BSI
4) fib (2013) Model Code for Concrete Structures 2010, Berlin, Germany: Ernst & Sohn
5) British Standards Institution (2002) BS EN 1990:2002+A1:2005 Eurocode 0. Basis of structural design, London, UK: BSI
6) British Standards Institution (2004) BS EN 1992-1-1:2004+A1:2014 UK National Annex to Eurocode 2. Design of concrete structures. General rules and rules for buildings, London, UK: BSI
7) European Concrete Platform (2008) Eurocode 2 Commentary [Online] Available at: www.europeanconcrete.eu/publications/
eurocodes/114-commentarytoeurocode2 (Accessed: June 2015) 8) Beal A. N. (2009) ‘Eurocode 2: Span/depth ratios for RC slabs and beams’, The Structural Engineer, 87 (20), pp. 35–40
9) Goodchild C. and Webster R. (2012) BSI Committee B525/2 paper: Interpretation of BS EN 1992-1-1 with respect to span:depth (L/d) ratios (Unpublished)
10) Vollum R. L. (2009) ‘Comparison of deflection calculations and span-to-depth ratios in BS 8110 and Eurocode 2’, Magazine of Concrete Research, 61 (6), pp. 465–476
11) The Concrete Society (2005) TR58: Deflections in concrete slabs and beams, Camberley, UK: Concrete Society
12) Webster R. and Brooker O. (2006) How to design concrete structures using Eurocode 2: No. 8. Deflection calculations, Camberley, UK: The Concrete Centre
13) Vollum R. L., Moss R. M. and Hossain T. R. (2002) ‘Slab deflections in the Cardington in-situ concrete frame building’, Magazine of Concrete Research, 54 (1), pp. 23–34
14) Vollum R. L. (2003) ‘Investigation into backprop forces and deflections at St George Wharf’, Magazine of Concrete Research, 55 (5), pp. 449–460
15) Goodchild C. H. and Webster R. M. (2006) User Guide to RC Spreadsheets: v3, Camberley, UK: The Concrete Centre Reinforced concrete frame residential building. Courtesy of Coinford Construction.
www.concretecentre.com I 17
Structural Design of Concrete and Masonry
Fire design of concrete columns and
walls to Eurocode 2
Introduction
Concrete does not normally need any further protection against fire due to its thermal conductivity properties and the fact that it does not burn. The design of concrete slabs and beams is not generally affected by fire design requirements. However, fire design requirements can be a governing factor in the sizing of columns, particularly in multistorey buildings. This article therefore concentrates on the guidance given in Eurocode 2 on the sizing of concrete columns for different fire resistance periods.
Methods
Guidance on fire design to EC2 is given in part 1-2 (BS EN 1992-1-2)1
and is much more extensive than in the previous codes. For the design of columns and walls there are basically three design methods available to the engineer:
• tabular data
• simplified calculation methods • advanced calculation methods
This article covers some of the tabular methods and simplified calculation methods for columns and walls. Table 1 shows limitations on the different tabulated data for columns. Outside these limitations the simplified calculation methods can be used.
The tabulated data for columns are given in Chapter 5 of part 1-2, split into Method A and Method B. Both methods are based on tests and either can be used for the design of columns, but they have slightly different limitations on their use.
The two simplified methods given in Annex B are the 500°C isotherm method and the zone method. The zone method gives a more accurate analysis of the effect of the fire on the element than the 500°C isotherm method, but both can provide savings to the sizing of columns compared to the tabulated data in Methods A and B.
For all the different method types, the axial load on the element compared to the capacity of the column or wall is key to the design. A lightly loaded column will be able to resist a fire for a much longer period than the same column when fully loaded.
Most of the columns that have been tested have been square columns; therefore, the tabulated data for columns assume square or circular columns. Rectangular columns are not covered in Method B, but can be modelled, to a certain extent, in Method A.
Fin or blade columns are not covered by the tabulated data until they are greater than a width-to-thickness ratio of 4:1 (h:b). At this point, EC2 part 1-1 (BS EN 1992-1-1)2 states that they are walls, and the column
should be designed as a wall at both normal temperatures and in a fire. If a column needs to be designed to fit into a partition, the use of blade columns with a ratio of 4:1 or greater has been common for many years, as by definition these are walls.
The tabulated data are given for braced structures only. However, the background document for the UK, PD 6687-13, states that the tabulated
data can be used to size unbraced columns, at the discretion of the designer. In critical cases it recommends that Annex B, which details the simplified methods, be used. It justifies the use of tabulated data for both braced and unbraced columns on historical grounds.
In fire, concrete does not burn and performs well, both as
an engineered structure and as a material.
Figure 1 Reduction factor ηfi when Exp. 6.10 of EC2 has been used
fi= 0.9 0.7 0.5 0.2 0 0.5 1.0 1.5 2.0 2.5 3.0 Qk,1/Gk R eduction factor fi 0.8 0.7 0.6 0.5 0.4 0.3 0.2 G Q = 1.35 = 1.50 Jenny Burridge Concrete & Fire Version 1 Chap 7 Fig 7.1 08.09.08 Amendments
Structural Design of Concrete and Masonry
Previous codes did not distinguish between braced and
unbraced columns and the tabulated data from BS EN 1992-1-2 give larger column sizes than those from previous codes. There is also an argument that, in most unbraced structures, the fire will only affect a few of the columns at any one time. The columns in the fire can therefore be said to be braced by the columns unaffected by the fire. Loading
The load under fire conditions can be reduced from the loads taken for normal temperature design. Generally, the effect of the loads Ed,fi = ηfiEd where Ed is the design moment, axial load, shear force, etc. under normal temperature loads.
The factor can simply be taken as 0.7, or can be calculated: i) If Expression 6.10 of EC04 has been used in the normal
temperature design (Figure 1)
ii) If Expression 6.10b of EC0 has been used in the normal temperature design
where:
Qk,1 is the main variable action under consideration. Only one variable action need be considered in the fire design
ψ1,1 is the appropriate factor for the frequent value of the main variable action
Tabular methods Columns: Method A
Method A has the more stringent limitations of the two tabulated data methods:
• the effective length of column under fire conditions l0,fi≤ 3m. For a braced structure, the effective length can be taken as 0.5l, i.e. l ≤ 6m for intermediate floors and 0.5l ≤ l0,fi ≤ 0.7l, i.e. l ≤ 4.2m for top floors, where l is the actual length of the column
• the first order eccentricity M0Ed,fi / N0Ed,fi ≤ 0.15h or 0.15b, where
M0Ed,fi is the first order bending moment for the fire condition and
N0Ed,fi is the axial load under the fire condition • the amount of reinforcement As < 0.04Ac
The fire resistance period is based on the degree of utilisation μfi = NEd,fi / NRd and the table gives values for μfi = 0.2, 0.5 and 0.7. Table 5.2a in BS EN 1992-1-2 assumes that αcc = 1.0. In the UK a value of αcc =
0.85 for bending and compression has been chosen5. However, the
values in the table are conservative for the UK so can be used. Table 2 gives values for the UK.
Other values can be calculated from BS EN 1992-1-2 Expression 5.7, and this expression and method can be used for rectangular columns:
R = 120(
(
Rηfi + Ra+ Rl + Rb + Rn)
/ 120)1.8where R is the fire resistance period in minutes
Figure 2 Reduced concrete section for column exposed on four sidesc) Fire exposure on four sides (beam or column)
500 Co 500 Co 500 Co h h fi b fi b fi b fi b b b
a) Fire exposure on three sides with tension zone exposed
T - Tension C - Compression C T T C d d fi= d fi d
b) Fire exposure on three sides with the compression zone exposed
Jenny Burridge Concrete & Fire Version 1 Chap 7 Fig 7.7 10.09.08 Amendments 23.02.09, 25.06.09
Table 1: Summary of the tabulated data for columns in BS EN 1992-1-2
Slenderness ratio Effective length ≤ 3m λ ≤ 30 30 ≤ λ ≤ 80 30 ≤ λ ≤ 80 30 ≤ λ ≤ 80
Minimum dimensions 200 ≤ b ≤ 450 150 ≤ b ≤ 600 150 ≤ b ≤ 600 150 ≤ b ≤ 600 150 ≤ b ≤ 600 Eccentricity e ≤ 0.15b e ≤ 0.25b e ≤ 0.025b but e ≥ 10mm e ≤ 0.25b but e ≤ 100mm e ≤ 0.5b but e ≤ 200mm ω = 0.1
Table 5.2a* Table 5.2b
Table C1 Table C2 Table C3
ω = 0.5 Table C4 Table C5 Table C6
ω = 1.0 Table C7 Table C8 Table C9
Note
* ≤ 4% reinforcement All columns must be braced
b is the smallest dimension of a rectangular column, or the diameter of the column Mechanical reinforcement ratio
Slenderness ratio ω = Acfcd Asfyd λ = Loifi ηfi = 1.35Gk + 1.5 Qk,1 Gk + ψ1,1 Qk,1 ηfi = 1.25Gk + 1.5 Qk,1 Gk + ψ1,1 Qk,1
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Structural Design of Concrete and Masonry
This simplifies to Rη,fi =83(1–µfi) in the UK as αcc = 0.85. Ra = 1.6(a–30), where a is the axis distance Rl = 9.6(5–l0,fi) where 2.0m ≤ l0,fi ≤ 6.0m
Rn = 0 for columns with four longitudinal bars and Rn = 12 for columns with more than four bars
Rb = 0.09b’ where b’ is the modified column width
b’ = 2Ac/(b + h) for rectangular sections and the diameter for circular sections
h is limited to h ≤ 1.5b and 200mm ≤ b’ ≤ 450mm
However, as proved by tests, blade columns have a longer period of fire resistance than columns of the same thickness but less width. It therefore seems reasonable to use Exp. 5.7 for columns where h > 1.5b, but to limit b’ in this expression. If h > 1.5b then b’ should be limited to 2b × 1.5b/(b + 1.5b) = 1.2b. This will give a conservative answer for the fire resistance.
Example: Blade column design
Assume a 600 × 200 column, fully loaded in the normal temperature design condition, designed in the UK (αcc = 0.85)
with an effective length in fire of 2m (4m floor-to-floor height). μfi = 0.7 as the column is fully loaded ➝ Rηfi = 83(1–μfi) = 24.9 Axis distance a = 25mm cover + 10mm link + 8mm (H16 bar) = 43mm
➝ Ra = 1.6(a–30) = 20.8 l0,fi = 2m ➝ Rl = 9.6(5–l0,fi) = 28.8
b’ is kept to 1.2b = 240mm ➝ Rb = 0.09b’ = 21.6 Rn = 12 as there are more than four bars in the column R = 120 ((Rηfi + Ra + Rl + Rb + Rn)/120)1.8
R = 120((24.9 + 20.8 + 28.8 + 21.6 + 12)/120)1.8 = 99min
Columns: Method B
Method B provides a more comprehensive method for the design of columns in that the restrictions on eccentricity of the first order moments are less onerous. For most columns Table 5.2b will be adequate, but there are tables in Annex C of EC2 which give more options where the limitations of Table 5.2b are exceeded.
The restrictions on the use of Table 5.2b are that:
• the slenderness of the column under fire conditions should be λfi = l0,fi / i ≤ 30, where i is the minimum radius of inertia
• the first order eccentricity under fire conditions should satisfy the limit: e = M0Ed,fi / N0Ed,fi ≤ emax, where emax = 100mm, e/b ≤ 0.25 and b = minimum column dimension
The load level at normal temperature conditions, n, is used in the determination of the minimum values (Table 3).
n = N0Ed,fi / [0.7(Ac ƒcd + As ƒyd)]
Note that in the table the mechanical reinforcement ratio, ω, is one of the required parameters:
ω = ASƒyd / Ac ƒcd
In BS EN 1992-1-1, a conservative value in the determination of limiting slenderness for the column takes ω = 0.1. For a class C30/37 concrete, this represents approximately 0.4% reinforcement, whereas when ω = 1.0, the column would require approximately 4% reinforcement.
Walls
Tabulated data for load-bearing walls are given in Table 5.4 of BS EN 1992-1-2 (Table 4). The degree of utilisation μfi is the same as that for Method A for columns. Another restriction is that:
clear wall height wall thickness x z dfi z z bfi As=As1+As2 fcd,fi(20) x xb ffi cd,fi(20) A fs1 sd,fi( Fs=A fs scd,fi( Fs=A fs2 sd,fi( Mu=Mu1+Mu2 Jenny Burridge Concrete & Fire Version 1 Chap 7 Fig 7.8 Amendments 24.02.09, 26.02.09, 29.06.09, 13.11.09 m) m) m) As’ As
Figure 3 Stress distribution at ultimate limit state for rectangular concrete cross-section with compression reinforcement
≤ 40 Rη,fi = 83 1 – µfi
(0.85 / acc) + ω (1 + ω)
Table 2: Minimum column dimensions and axis distances for square and circular columns
Exposed condition Load level μfi Fire resistance period (minutes)
R 30 R 60 R 90 R 120 R 180 R 240
More than one side exposed 0.2 200/25 200/25 200/30 300/25 250/38 350/33 350/52 450/47 350/68 0.5 200/25 200/32 300/27 300/40 400/35 350/49 450/43 350/68 450/62 450/78 0.7 200/26 300/25 250/40 350/35 350/48 450/43 350/59 450/54 450/75 450/73 –
Only one side exposed
Structural Design of Concrete and Masonry
Simplified calculation methods 500°C isotherm method
In the isotherm method, concrete at a temperature above 500°C is neglected in the calculation of section resistance, while concrete at or below 500°C is assumed to retain its full, ambient temperature strength. In BS EN 1992-1-2 the method is illustrated with reference to rectangular sections. Thus, the calculation process is to first check that the section meets the minimum cross-sectional width requirements in Table 5.
If the minimum requirements are met, the area not damaged by heat, i.e. within the 500°C isotherm, is determined to give a reduced section size (bfi, dfi) where the concrete retains its original properties. All the reinforcement can be taken as acting with the section, including the reinforcement in the zone outside the 500°C isotherm, but the strength of the bars is reduced. The strength can be taken from Figure 4.2a of BS EN 1992-1-2.
While the temperature gradient through a section denoted by isotherms may be determined from testing, BS EN 1992-1-2 provides temperature profiles for a number of typical member types and cross-sections in Annex A.
The rounded corners of the residual section reflect the real profile of the isotherm and may be approximated to a rectangle (Figure 2); some interpretation may be required.
M1 kc( m1) kc(m1) az1 az1 az1 az1 az1 az1 az1 az1 W1 W1 W1 W1 W1 W1
Wall Wall end
Column
Jenny Burridge Concrete & Fire Version 1
Figure 4 Reduction of cross-section when using zone method
Table 2: Minimum column dimensions and axis distances for square and circular columns Load level at normal
temperature conditions (n)
Reinforcement ratio Fire resistance period (minutes)
R 30 R 60 R 90 R 120 R 180 R 240 0.15 0.1% 150/25 150/30 200/25 200/40 250/25 250/50 350/25 400/50 500/25 500/60 550/25 0.5% 150/25 150/25 150/35 200/25 200/45 300/25 300/45 450/25 450/45 500/25 1.0% 150/25 150/25 200/25 200/40 250/25 300/35 400/25 400/45 500/25 0.3 0.1% 150/25 200/40 300/25 300/40 400/25 400/50 550/25 500/60 550/25 550/40 600/25 0.5% 150/25 150/35 200/25 200/45 300/25 300/45 550/25 450/50 600/25 550/55 600/25 1.0% 150/25 150/30 200/25 200/40 300/25 250/50 400/25 450/50 550/25 500/40 600/30 0.5 0.1% 200/30 250/25 300/40 500/25 500/50 550/25 550/25 550/60 600/30 600/75 0.5% 150/25 250/35 350/25 300/45 550/25 450/50 600/25 500/60 600/50 600/70 1.0% 150/25 250/40 400/25 250/40 550/25 450/45 600/30 500/60 600/45 600/60 0.7 0.1% 300/30 350/25 500/25 550/40 600/25 550/60 600/45 >600 >600 0.5% 200/30 250/25 350/40 550/25 500/50 600/40 500/60 600/50 600/75 >600 1.0% 200/30 300/25 300/50 600/30 500/50 600/45 600/60 >600 >600
The section resistance may then be determined using conventional calculation methods (Figure 3) and compared against the design load in the fire situation in this figure, where: