approaches to the design of rc flat slabs

2001

Approaches to the designof reinforced concreteflat slabs

R M Moss, BSc, PhD, DIC, CEng, MICE, MIStructE

BRE Centre for Concrete Construction

BREGarston, Watford, WD25 9XX

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BR 422ISBN 1 86081 498 0

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CONTENTS

SUMMARY 1

Key messages 1

Best practice 1

1. USING FLAT SLAB CONSTRUCTION 3

Advantages 3

Disadvantages 3

2. CHOICE OF DESIGN METHOD 4

3. RATIONALISATION OF REINFORCEMENT 5

4. OPTIMISING SLAB THICKNESS 6

5. DESIGNING FOR THE ULTIMATE LIMIT STATE 7

5.1 Loadings 7

5.2 Designing for flexure 8

5.2.1 Alternative design methods 8

5.2.1.1 Elastic Analyses with Moment Redistribution 8

5.2.1.2 Yield line method 15

5.2.1.3 Finite Element methods 26

5.2.1.4 Other Methods 32

5.3 Designing for punching shear 33

6. DESIGNING FOR SERVICEABILITY 34

6.1 Deflections 34

6.1.1 Use of span/effective depth ratios 34

6.1.2 Deflection calculation methods 34

6.2 Cracking 40

7. CONCLUSIONS AND RECOMMENDATIONS 41

8. ACKNOWLEDGEMENTS 42

9. REFERENCES 43

Annex 1: List of currently available RCC spreadsheets 44

SUMMARY

This report considers issues surrounding the use of reinforced concrete flat slabconstruction. The report is not intended to be prescriptive and it is recognised thatthere are many alternative approaches to the design of reinforced concrete flat slabsthat are equally valid. The intention is that this fuller report will be summarised as aBest Practice guide as part of a series of guides emerging from the EuropeanConcrete Building Project (Reference 1).

Key messages

• Thin flat slab construction is the most cost effective form of in situ concrete floorconstruction for spans from 5 to 9m, especially where a square or near squaregrid is used. For spans in excess of 9m post-tensioning should be considered.

• In general terms in situ concrete flat slab construction without drops hasadvantages in simplifying falsework and formwork arrangements, therebypromoting rapid floor construction, and also giving maximum flexibility to the enduser.

• The additional construction process benefits associated with flat slabconstruction may well outweigh the benefits of constructing a more structurallyefficient, but more complicated, solution for the floors. Related to this are theoptions pursued for providing bracing elements within the structure (if any), whichtraditionally have slowed down the overall construction process, and the methodsfor forming the columns. Where bracing elements are used steel cross bracingmight offer one possible solution. Use of precast concrete columns is anotherpossibility to speed up the column construction and patented connection systemshave been developed to allow these to be used in conjunction with in situconcrete slabs.

• Experience from Cardington (Reference 1), where a low level of imposed loadwas chosen and in some cases the slabs were struck very early, has highlightedthe importance of the Permanent Works Designer considering the effects oftemporary construction loads on long-term serviceability performance1.

Best practice

• Investigate the benefits of using in situ concrete flat slab construction withoutdrops if possible. If column heads are considered essential consider the scopefor incorporating them as part of the column formwork to allow the advantages offlat soffits for the floors to be retained. Proprietary disposable systems areavailable for this purpose.

• Further guidance on the best structural solutions to adopt under differentcircumstances is given in Reference 2.

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1 Paper to be published in Magazine of Concrete Research

Approaches to the Design of Reinforced Concrete Flat Slabs

• If a flat slab solution is chosen, look at the construction process in its entirety,including the contractual arrangements, to decide what level of reinforcementrationalisation2 is most appropriate. This is further explained in Reference 3.

• Optimise the slab thickness wherever possible. In practice this will depend onmany factors such as the method of design, the presence or absence of holes,the importance placed on deflections and previous experience. Further guidanceis given in Reference 2.

• Consider whether a particular design method (e.g. yield line) is likely to lead tomore rationalised reinforcement layouts and the benefits that might entailparticularly on large projects.

• Improve the flow of information between the various parties in the reinforcementsupply chain3. Move towards a more integrated approach to design and detailingso that the contractor is better able to estimate weights of reinforcement whentendering.

• If possible consider at the outset how the structure will be built, what the criticalloading conditions will be and when they will occur including the possible effectson long-term deflection of early striking. Further guidance on the issuessurrounding early age loading is given in Reference 4.

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2 Reinforcement rationalisation in practical terms means reducing the number of bar marks used 3 The use of a common, agreed data exchange format is recommended. Background research leadingto this recommendation is described in the report Improving Rebar Information and Supply (IRIS) byA.Kalian, T.Thorpe and S.Austin, BRE report BR 401.


1. USING FLAT SLAB CONSTRUCTION Advantages The benefits of using flat slab construction are becoming increasingly recognised.Flat slab construction without drops can be built faster because formworkrequirements are simplified and minimised, and rapid turn around can be achievedusing a combination of early striking, discussed in Reference 4, and flying formsystems. Use of this form of construction also places no restrictions on the positioning ofhorizontal services and partitions and can minimise floor/floor heights when thereisn't a requirement for a deep false ceiling. This can potentially have knock-onbenefits in terms of reduced cladding costs, but this may in turn be offset by the lackof edge beams and the greater deflections that result. In situ concrete construction has inherently good fire resistance, resistance to soundtransmission and transmission of vibrations and high thermal mass, and thesebenefits are maximised with flat slab construction because of the uniformity of theslab thickness. Disadvantages The provision of large vertical service holes can pose a considerable potentialproblem with flat slab construction, particularly where there is a requirement tolocate these immediately adjacent to columns. Methods of dealing with the flexuraldesign to accommodate holes are covered in this report. Guidance on the use ofstructural steel shear heads to allow large openings to be formed adjacent tocolumns without compromising the punching shear resistance is intended to beincluded within the Report of a Concrete Society Working Party4. Another perceived disadvantage is the lack of flexibility in being able toaccommodate late design changes and modifications post-construction such asholes and increased loads. This is one reason why it may be worth not minimisingthe slab thickness (see section on optimising slab thickness below). Novelstrengthening systems such as use of carbon-fibre plate bonding have been usedfor example to allow a lift shaft opening to be formed in a slab post-construction. Very heavy point or line loads can also be difficult to accommodate economicallyand may dictate the limited use of upstand or downstand beams. For exampleupstand beams were found to be necessary at Cardington (Reference 1) toaccommodate the precast cladding loads in the two end bays with large stair and liftshaft openings. Because of the two-way spanning nature of the construction anticipated deflectionsmay also be greater than for other less economic one-way spanning beam and slabsystems.

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4 Concrete Society Technical Report: Shear Reinforcement Systems for Flat Plates (to be published).


2. CHOICE OF DESIGN METHOD The choice of design method is very much one of personal preference. It should bebased on what is appropriate for the structure to be designed, and on the designer'sown previous experience as well as what is likely to most benefit the client. For a small regular frame, the sub-frame method in accordance with BS 8110(Reference 5) is likely to be the most convenient though not necessarily the mosteconomic. In this case the RCC spreadsheets described below are ideal. For large buildings where the scale dictates the most efficient design, the yield linemethod will allow the optimum distribution of reinforcement. The yield line methodmay however require a separate elastic analysis of the serviceability limit states ofcracking and deflection, and separate consideration of the column and punchingshear design. Use of finite element analysis has particular advantages when the floor is supportedon an irregular grid of columns, the plan geometry of the slab itself is complicated,there are large openings or the slab carries heavy concentrated loads. These issuescan also be dealt with using the yield line method and, to a certain extent byjudicious choice of sub-frames, but may not be straightforward.

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3. RATIONALISATION OF REINFORCEMENT There is evidence to suggest that some design methods result in more rationalisedreinforcement layouts than others. As noted in Reference 3, there are processbenefits in terms of blanket provision of main longitudinal reinforcement which maywell outweigh savings in minimising reinforcement provision because of the requiredtailoring of loose bars. In practice even where loose bars are used the designer anddetailer will almost always rationalise to a certain extent and clearly there is scopefor sensible judgement as to what is appropriate. The benefits of rationalisation need to be clear to all those involved in the process,including the Quantity Surveyor, to overcome the misconception that the leastmaterial option necessarily results in cheapest price overall. Where rationalisation isemployed it needs to be done at an appropriate stage if the full benefits are to berealised in practice. Current typical contractual arrangements and in particular theformal appointment of the frame contractor after much of the detailing work hasbeen completed are seen as a potential barrier to this. There are considered to be definite process benefits to be gained by adopting wideruse of prefabricated mats. Prefabricated mats with tailored sizes and bar spacinghave been available for a considerable period, but to date have not been widelyspecified. One reason for this is the perception of the required minimum quantities tomake their supply and use commercially viable. There are also new developments, (e.g. the Bamtec system, Figure 1), whichinvolves one-way spanning reinforcement mats rolled out like a carpet into position.This is sold as a complete package involving Finite Element analysis5 for the designof the reinforcement and is claimed to result in 80% savings in fixing time. Thispotentially offers savings in transport costs, because of the greater density of matsthat can be transported. A more rationalised layout of reinforcement will also simplify the amount of detailingand the number of bending schedules required. Where possible the contractorshould be given the freedom to undertake the detailing as recommended in theConstruct report: A guide to contractor detailing of reinforcement in concrete, BCA,Crowthorne, 1997.

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5 A note of caution when using the Bamtec system in conjunction with BS8110 is the need to considerthe maximum design moment which can be transferred to an edge or corner column. It may benecessary to manually take account of Mtmax requirements (Part 1: Clause 3.7.4.2) adjusting end spanand penultimate support moments accordingly.


Figure 1: The Bamtec system

4. OPTIMISING SLAB THICKNESS Having chosen a flat slab solution, the next key issue is determining an appropriateslab thickness. Studies have been carried out looking to optimise the thicknessbased on designs to BS8110 (Reference 5) and other codes and the generalconclusion is that thinner slabs save money (References 6 and 7). This savingresults both from the direct savings in material costs of the concrete in the slabs,and the knock-on benefits in terms of reduced overall height of the structure.Cladding costs can as a result be reduced but note the earlier comment concerningslab deflections, which are likely to increase as the thickness is reduced. Thereduced self-weight of the slabs will mean lower column loads and permit smallerfoundations. There is of course a lower limit to the slab thickness and as the slab thicknessreduces the savings identified above become outweighed by the additional amountof reinforcement required and the increased difficulty in designing and fixing it. Thereis also a case for having some margin, particularly at outline scheme stage, toaccommodate late changes in architectural requirements and provision of holes inthe slab. Consideration could also be given to possible future alterations andchanges of use post-construction. Guidance on appropriate slab thickness is given in Reference 2 by way ofspan/depth charts. Slab depths for spans in the range of 5-9m range from aminimum of 200mm to approximately 380mm depending on the level of imposedloading. For the in situ concrete building at Cardington (Reference 1), which hadspans of 7.5m and a design imposed load of 2.5kN/m2, a slab thickness of 250mmwas chosen.

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5. DESIGNING FOR THE ULTIMATE LIMIT STATE 5.1 Loadings An example of the load history of a typical slab at Cardington is illustrated inFigure 2.

9

6.75

10.74

98.75

6

0

2

4

6

8

10

12

1 10 100 1000

Time (days)

Load

(kN

/m2 )

Figure 2: Typical load history for a floor at Cardington From Figure 2 distinct stages of loading can be clearly identified. The initial loadingis due to the self-weight when the slab is struck, together with an allowance forconstruction loads. The slab is then subject to a short peak load as a result of thecasting of the floor immediately above. A further smaller peak load is then applied asa result of loads from backprops when casting the second floor above. In the particular case of the Cardington slabs, no further load was applied untiladditional sandbags were placed on the building a considerable time afterconstruction. Nevertheless this type of load history, at least during construction, islikely to be typical of flat slabs in practice. It is an inherent feature of flat slabs, and indeed reinforced concrete construction ingeneral, that the dead/live load ratio is high. With moves towards lower designimposed loads, the "spare capacity" of a given slab over and above its self-weight isfurther reduced. This has implications for the way in which flat slab structuresperform and are built. Findings from Cardington also suggest that traditionalassumptions concerning the distribution of load through supporting slabs areincorrect, with the slab immediately beneath that being cast carrying a higherproportion of the load than usually assumed. Further guidance on this topic is givenin Reference 4. As a result of the above it is possible that the temporary loading conditions duringconstruction actually govern the design and the Permanent Works Designer (PWD)should be fully aware of the implications of this. In the interests of the client the PWD should not put unnecessary barriers in the wayof the contractor restricting the way he goes about constructing the frame, for

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example in relation to early striking. However he should consider the possible effectsof early striking upon issues such as long-term deflections. The Second Edition ofthe National Structural Concrete Specification6 sets out a balanced approachrecognising the interests of all the parties involved. When designing for the ultimate limit states of flexure and shear it will obviously berequired to apply appropriate load factors to determine the design loads to be used. 5.2 Designing for flexure 5.2.1 Alternative design methods 5.2.1.1 Elastic Analyses with Moment Redistribution 5.2.1.1.1 Introduction Any suitable form of elastic analysis in combination with an appropriate level ofmoment redistribution may be used to determine bending moment and shear forceenvelopes. In general the geometry of the structure and the different loadingarrangements on it will need to be considered, but under certain conditions simplecoefficients may suffice, (e.g. those given in Table 3.12 of BS8110). There is a range of user-friendly design tools available such as the RCC'sspreadsheets (Reference 8) which are used to illustrate many of the points madelater in the document. These automate the process and effectively mean that someform of frame analysis should be used in all but the simplest cases. 5.2.1.1.2 The equivalent frame method and use of sub-frame analysis The structure is divided up longitudinally and transversely into a series of two-dimensional frames, consisting of columns and strips of slab. Each frame may thenbe analysed in its entirety using the Hardy Cross or other suitable elastic method.Alternatively each strip of floor or roof may be analysed as a separate sub-frame. Sub-frame analysis is generally well understood and relatively simple to apply. It isalso very much tried and tested. On the other hand it is in some ways a very cruderepresentation of the true behaviour of the slabs. Having input the basic data on geometry and loads (Figure 3) and the appropriatelevel of moment redistribution, this sub-frame analysis is done automatically withinthe relevant RCC spreadsheet program RCC33.xls and the information presentedgraphically (Figure 4).

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6 Construct, BCA, BRE and RCC National Structural Concrete specification for building construction(NSCS), Crowthorne, BCA, 2000, BCA Publication Ref. 97.378.


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Figure 3: RCC Spreadsheet - Data entry


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Figure 4: RCC Spreadsheet - Automatic sub-frame analysis


5.2.1.1.3 Division of slab into column and middle strips Where simple coefficients or the equivalent frame method have been used todetermine bending moments, the three-dimensional nature of the construction willneed to be represented by assigning proportions of the moment to be carried withinimaginary column and middle strips of slab. Guidance on determining appropriatewidths of strip and these proportions is given in BS8110 and again is consideredautomatically within the RCC's spreadsheet program (Figure 5).

Figure 5: RCC Spreadsheet analysis - automatic division into columnand middle strips

5.2.1.1.4 Other design issues in relation to BS8110 When designing to BS8110 there are other requirements that must be met (e.g. inrelation to the placing of the determined hogging reinforcement within the columnstrip for internal panels and maximum design edge moments). Again these issuesare dealt with automatically within the RCC spreadsheet, which even goes as far asdetermining appropriate sizes and weights of reinforcement taking into accountsimple rules in relation to curtailment of reinforcement, and other detailingrequirements (Figure 6). This information can be extremely valuable when schemingthe structure.

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Figure 6: RCC spreadsheet determination of approximate bar weights
An area that cannot be dealt with automatically by the existing spreadsheetpackage, or indeed any other form of one dimensional sub-frame analysis, is theprovision of openings. From the point of view of flexure, the effect of small openingscan often be ignored or dealt with simply by proportioning additional transferreinforcement around the opening. To avoid stress concentrations this is oftenplaced diagonally at each corner of the opening. Major openings can sometimes bedealt with by judicious adjustment of the assumptions made in the sub-frameanalysis in each direction, or a more rigorous analysis may be justified.
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The RCC spreadsheet is intended to be used in combination with a companionsheet RCC13.xls dealing with punching shear where traditional shear links arespecified. This latter sheet can deal with the effects on punching shear of holes nearcolumns. 5.2.1.1.5 Experience from use at Cardington Sub-frame analysis was used successfully for the base design of the Cardingtonslabs carried out by Buro Happold using the ENV version of EC2 (Reference 9). Itwas also used for the variations on the base design. These involved sub-frameanalysis with rationalisation of reinforcement and the relaxation of deflectionrequirements on floor 3, the use of blanket cover loose bar for half of floor 4, andone-way mats on floor 5. The weight penalty introduced by rationalising the loose bar provision obtained wassmall and it is recommended that for smaller projects this method of analysis, inconjunction with an appropriate level of rationalisation of the loose bar, is adopted.The level appropriate must be a matter of engineering judgement, but should aim toreduce the number of bar marks used by about one-third compared with a minimummaterial solution. In the RCC spreadsheet the reinforcement suggested to be provided at a givenlocation is the minimum area consistent with the maximum preferred bar size, and itis for the user to determine a suitable level of rationalisation. In relation to the outputsuggested above this might be:

• two bar size/spacing combinations to cover all sagging reinforcement in end baysand internal bays

• two basic bar size/spacing combinations and lengths for the hoggingreinforcement in all column strips and middle strip regions. This spacing can bereduced for the outer regions of the column strips.

The hogging reinforcement in middle strip areas could be made continuouseverywhere providing the top reinforcement at centre span also, if needed.

5.2.1.1.6 Implications of EC2 EC2 is currently in the process of being converted into a full EN standard. As part ofthe Cardington project, and to assist in the conversion process, the use of the ENVversion of the code by Buro Happold, in conjunction with the National Applicationdocument, was reviewed. The findings have been reported to the relevant BSIcommittee. In principle sub frame analysis and the design approaches suggested in BS8110may still be used, but there are some additional requirements. In particular the ENVversion of EC2 requires different loading arrangements and combinations to beconsidered, and there are more possible permutations than in BS8110. There are plans to extend the RCC spreadsheet to deal with the design of flat slabsin accordance with the final EN version of EC2 when it is issued.

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5.2.1.1.7 Serviceability issues These are dealt with in more detail in the separate section on serviceability below. Inmost cases the simplified approach to controlling deflections using span/depth ratioswill be adequate. The results from an elastic analysis can however be used as thebasis for prediction of deflections.

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5.2.1.2 Yield line method 5.2.1.2.1 Introduction Yield line theory deals with two-dimensional slab or plate structures where plasticyielding is assumed to occur along a system of “yield lines”. When all the yield linesnecessary to initiate a mechanism have formed, the whole, or a portion, of the slabcan notionally collapse, which represents the ultimate condition. Provided the correct yield line pattern is chosen, use of the yield line method gives abetter representation of the influence of structure's geometry on the flexuralbehaviour of the slab at the ultimate condition than say sub-frame analysis, wherethe division of the slab into column and middle strips is somewhat arbitrary. It is essentially an analytical tool that is well suited to a simple hand method ofdesign and can deliver elegant and economic designs very quickly (Reference 10). Economy and speed come in both the design and the construction processes. Yield line design is a technique that has been around for many years but itscommercial exploitation appears to have been curtailed by a lack of understanding,the fear that it is an upper bound solution and the lack of computer support. It is notpossible within the scope of this guide to fully explain the method, but it is intendedto try and re-introduce practical designers to yield line techniques in the context offlat slabs supported on a rectangular grid of columns. Irregular grids of columns can be dealt with by the Yield Line method and someguidance on this is given in Reference 10. The technique challenges designers to use judgement, but once grasped it isexceedingly easy to use and put into practice. Since the yield line method only considers collapse mechanisms, it may benecessary to consider serviceability issues such as cracking and deflection in moredetail than with other forms of analysis that are based on elastic behaviour. 5.2.1.2.2 Use of the Method In essence failure of a slab can be considered analogous to that of a beam exceptthat the behaviour is now two-dimensional rather than one-dimensional. In thebeam plastic hinges will develop in the most highly stressed regions until amechanism is formed allowing the beam to collapse. In a similar way yield linesdevelop in a slab again eventually allowing a mechanism to be formed and failure tooccur. A simple case to analyse is that of a two-way spanning square slab simplysupported along all 4 sides, subject to a uniformly distributed load. As one wouldanticipate, as the load is gradually increased towards failure, yield lines developemanating from the centre of the slabs towards the supporting corners (Figure 7).

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When analysing a slab using yield line, the theoretical procedure is based on theprinciple that the work done in deforming the slab equates to the internal energydissipated forming the yield lines.

Figure 7: Simple yield line pattern

5.2.1.2.2.1 Application to Flat Slabs In the context of a flat slab supported on columns, the column connection to the slabis usually treated as pinned so that no moment transfer is taken into account. Itshould be emphasised that some form of elastic analysis, in combination with anappropriate level of moment redistribution, is still likely be required to derivemoments for the design of the columns themselves. However this does not in itselfundermine the value of the use of the yield line technique for the design of the slabs.Where a sub-frame analysis is unavailable or unwarranted, one solution might be touse the RCC spreadsheet RCC51.xls for this purpose. This is included within thesuite of spreadsheet packages, which have been widely disseminated by the RCC,and are available on request on a CD-ROM. For a rectangular grid of columns, depending on the size of the building there will bea certain number of bays in each direction and the spans may vary. Three possiblefailure modes may occur with such an arrangement as indicated in Figure 8. In the first (Figure 8a) the yield line pattern consists of parallel positive and negativemoments, with negative yield lines forming along the axis of rotation passing alongthe faces of a line of internal columns. A corresponding pattern could take place atright angles, or a combination of both these collapse modes could develop

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simultaneously (Figure 8b) in which case the positive and negative yield lines willform simultaneously in both directions. In the latter case the axes of rotation for the failure of any given panel will lie alongthe leading diagonals of adjacent panels. It can be shown theoretically that thecollapse load to generate this failure pattern cannot be less than that to generate theone-way spanning failure pattern, so that only the latter condition needs to beconsidered in practice. At slab edges simple line supports may be assumed. Since the edge and cornercolumns can rotate, it is usual to assume the axis of rotation for these simplesupports to be at the centreline of the columns. The spans are taken between theaxes of rotation when establishing the magnitude of the moments. The third possible mechanism is one in which conical failure surfaces develop overeach column (Figure 8c). The check on this possible failure mechanism will result inadditional U bars being provided at edge and corner column locations. 5.2.1.2.2.1.1 Design of bottom reinforcement It is common practice to assume the ratio of support to mid-span moment is equal to1, which is generally satisfactory for flat slabs unless there is a significant differencein the length of adjacent spans. Theoretically for any corner or edge bay in thedirection considered the ultimate mid-span moment will then be given by wl2/11.66where w is the design ultimate load, and l is the effective span (taken between axesof rotation). This will then allow determination of the required area of reinforcementin the normal way. Similarly for an internal bay in the direction considered it can be shown that theultimate mid-span moment may be conservatively taken as wl2/16. Reference 10gives further details of how these moments are derived. The reinforcement will be required to be distributed uniformly across the bay foreach direction, making the use of uniform one or two-way spanning mats veryattractive. As an example for comparison one could take the design using the equivalent framemethod presented above. Applying the same modification factor for compressionreinforcement of 1.05, the equivalent yield line design would give blanket saggingreinforcement in end bays of T20 at 300 centres, instead of the T20 at 350 in middlestrips and T20 at 250 in column strips.

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Figure 8a: Possible flat slab yield line pattern (yielding in one direction)

Figure 8b: Possible flat slab yield line pattern (yielding in two directions)


Figure 8c: Possible flat slab yield line pattern (yielding around columns)
5.2.1.2.2.1.2 Design of top reinforcement Top reinforcement is best concentrated around columns. This will result in anoptimal solution except where concern exists about incidental cracking, (e.g. a floorthat is to be power-floated), in which case a more even distribution would beappropriate (Reference 10). One common assumption is to concentrate it over an area of side equal to 0.5L x0.5L for an internal column, 0.5L x (0.2L+ edge distance E.D.) for an externalcolumn and (0.2L + E.D.) x (0.2L+ E.D.) for a corner column. L in this context is thecolumn centreline to centreline distance in the direction considered which differsslightly from the effective span. The concentrations of the top reinforcement areillustrated in Figure 9. Additional top distribution reinforcement can be providedbetween these concentrations if considered necessary.
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Figure 9: Concentrations of top reinforcement
The design of the top reinforcement then proceeds on the basis of considering thetotal moment along the yield lines at supports and then concentrating it in thecolumn head areas discussed above. This improves the design for punching shear,which can be further improved by further concentrating reinforcement directly overthe column, although there is no specific requirement to do this. With the assumption that the ratio of support to span moment is equal to 1, thesupport moments for end and internal bays will be the same as the mid-spansagging moments given above (i.e. wl2/11.66 and wl2/16 respectively). The topreinforcement passes over the gridlines and into the next bay. As a result in a 4 bayby 3 bay structure such as the Cardington building (Figure 10), the moment per unitlength along gridlines 2, 4, B and C will be the higher moment wl2/11.66 and onlythat along gridline 3 will be wl2/16. This moment multiplied by the total length of theyield lines is concentrated in the column head areas as defined above. The effective moment in these areas (Figure 10) along gridlines 2 and 4, and B andC is likely to be very similar. This means that the same top reinforcement can beprovided over these gridlines in each direction. As an example for comparison one could take the design using the equivalent framemethod presented above which is for an internal bay. The equivalent yield linedesign would give top reinforcement at the internal support in end spans of T16 at100 centres, instead of the T16 at 550 in middle strips and T16 at 125:250 in columnstrips. Since simple line supports are assumed along the edges of the slab, therewould be no requirement for top reinforcement (other than U bars) at the externalsupport using yield line. This compares with the T12 at 325 in middle strips and theT16 at 125 in the column strips using the equivalent frame method.
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Figure 10: Cardington 4 bay by 3 bay structure

Mech. duct

Lift shaft

PH duct

Stairs

Internal columns400mm x 400mm

External columns400mm x 250mm

Steelcross-bracing

7500m7500m

7500m

A B C D1

4

3

2

5

grid-lines

Total length of yield lines along B and C is 7.5 x 4 + 0.25=30.25m along 2, 3 and 4 is 7.5 x 3 + 0.25= 22.75m

Total width of column concentrations along B and C is (1.5 +0.125) x 2 + 3.75 x 3=14.5mTotal width of column concentrations along 2, 3 and 4 is (1.5 +0.125) x 2 + 3.75 x 2=10.75m

Ratio of length of yield lines to width of column concentrationsalong B and C is 2.09 and along 2, 3 and 4 is 2.12

Support moment for yield lines along B, C, 2 and 4 isw x 7.32/11.66Support moment for yield lines along 3 is w x 7.12/16

Effective support moment in column concentration = ratio xsupport moment established above (e.g. 2.09 x w x 7.32/11.66)

Shortened one-wayspanning yield lines

Extratopbars


It should also be recognised that the same density of reinforcement will be providedwhen designing using the yield line method for both internal and edge bays, butgreater densities will be required for an edge bay when using the equivalent framemethod. Yield line design will result in a highly rationalised layout of the main reinforcement,with only 4 different bar size/spacing combinations required. This compares with 8different arrangements for a single span direction involving only 3 spans using theexample of the equivalent frame method above. 5.2.1.2.2.1.3 Check on local failure mechanism This needs to be checked around all columns. It can be shown (Reference 10) thatthe formula for doing this in the case of internal columns is:

m+m' = S(1- (wA/S)0.33) /2� In the above formula: m is the average moment of resistance per unit length in the two directions providedby the bottom reinforcement in the bays adjacent to the column. m' is the average moment of resistance per unit length in the two directions providedby the top reinforcement over the column. w is the design ultimate load A is the area of column cross-section S is the ultimate load transferred from the slab to the column. Load may beassumed to be transferred equally between columns according to the area theysupport, except that in an end bay 55% of the load in a given direction may beassumed to be transferred to an internal column and the remaining 45% to theexternal column. In the case of corner columns the support moment m' per unit length to be providedin the form of U bars in each direction is calculated from: m' = S(1- (wA/S)0.33) /2 In the case of edge columns the support moment m' per unit length to be provided inthe form of U bars perpendicular to the slab edge is calculated from m' = S(1- (wA/S)0.33) /5.14 It should be checked that this moment does not exceed the resistance momentprovided by the top reinforcement over the column in the direction parallel to theslab edge. Nominal U bars should also be provided along the edges of the slabbetween the concentrations of bars at column supports. A full worked example isgiven in Reference 10.

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5.2.1.2.2.1.4 Dealing with perimeter loads So far only uniformly distributed loads on the slab have been considered. Perimeterloads and other load concentrations can usually be dealt with by considering them toact over a suitable width. However for very heavy perimeter loads or other forms ofconcentrated load a more rigorous analysis is required. 5.2.1.2.3 Experience from use at Cardington The yield line design carried out by Powell Tolner & Associates was found to givethe simplest bending reinforcement arrangements. Furthermore use of yield lineanalysis resulted in dramatically lower masses of steel for blanket cover, suggestingthat this is the most efficient design method where this approach to rationalisation isadopted. Although not used directly in combination with yield line at Cardington, blanket twoway mats were found to be about 40% quicker to install than loose bar. This speedadvantage is worthwhile for larger projects. Top reinforcement was concentrated over columns although some meshreinforcement was also provided to supplement the steel area required and to limitthe cracking. No cracking as a result of the concentration of top reinforcement hasbeen observed. Rigorous deflection calculations were carried out but it was concluded that thesewere unnecessary and that span/effective depth ratios were effective in controllingpredicted deflections. 5.2.1.2.4 Implications of EC2 As with BS8110, EC2 gives the designer scope to use a range of design methodsand yield line is one of the plastic methods of analysis recommended. When usingplastic methods of analysis the current ENV version of EC2 states the reinforcementshould be high ductility. 5.2.1.2.5 Definition of and methods of satisfying serviceability criteria Since the yield line method is by definition a plastic method of analysis, it cannotdeal with the Serviceability Limit State. Where explicit calculation of deflections isnot required the same simple approach using span/depth ratios for dealing withserviceability may in principle be adopted as for other design methods (see sectionbelow on serviceability design). However it should be recognised that the provisionof the bottom reinforcement has been determined on the basis of preventing acollapse mechanism forming along the entire length of the slab. A uniform momentand hence reinforcement requirement results whereas in elastic analysis (forexample sub-frame analysis) a greater concentration of sagging moment and hencereinforcement is assumed in certain areas (55% in column strips as opposed to 45%in middle strips). In deriving the steel stress and hence modifica��

- 23 -


�� b is normally taken as 1.1 for end spans and 1.2 for internalspans. 5.2.1.2.6 Dealing with provision of holes Potentially the yield line method can deal more satisfactorily with this than someother methods of design (e.g. sub-frame analysis) because they can be takenaccount of directly when determining a yield line pattern. For example for theCardington building (Figure 10), the stair and lift shaft openings at either end of thebuilding were dealt with as follows: In relation to the yield lines which could develop across the width of the building onlythat at mid-span in the end bays will be affected as a result of the yield line being ofreduced length. To reduce the complexity of the calculations the same blanket loadmay be conservatively taken to apply over the area of the hole. With this assumptiona standard solution may be used to determine a revised value for m. Assuming thatm=m' as previously this reduces to:

m=wl2/(11.66-8.24d/Ltot) In the above equation d is the missing length of yield line, in this case 3.6m or 5mand Ltot is the total length of the uninterrupted yield line, in this case 22.75m. Thecorresponding moments to be designed for are wl2/10.36 and wl2/9.85 instead ofwl2/11.66 as previously. In relation to the yield lines that could develop along the length of the building, onlythose at supports along gridlines B and C will be affected. This will require additionaltop reinforcement adjacent to the existing internal column concentrations next to theholes to replace that no longer effective in the edge columns. This is illustrated inblue in Figure 10. The size of the opening will have a bearing on the approach taken. For example ifthe depth of the openings had extended beyond half the span then an alternativeapproach would have been required. This would have involved deriving the localmoments in and around the corner panel in both directions from first principles.Further guidance is given in Reference 10. 5.2.1.2.7 Increasing the scope for using the yield line method Although the yield line method provides an upper bound solution, it can be usedreliably and with confidence. Further information and guidance can be found inReference 10. There is no reason in principle why the method of yield line design using thestandard solutions given in Reference 10 and summarised above could not beincluded within a new modified existing RCC spreadsheet, for example RCC33.xls.This would remove much of the hand computation normally associated with themethod.

- 24 -


Recent research at Nottingham Trent University has focused on the development ofa lower bound approach to yield line design that is capable of full automation(Reference 11). The advantage of such an approach is that it does not require thecritical collapse mechanism to be established, removing any lingering uncertaintyinherent in the yield line technique. There would be merit in developing software thatcould assist practical designers using this approach also.

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5.2.1.3 Finite Element methods 5.2.1.3.1 Introduction

The finite element method is a powerful computer method of analysis that can beused to obtain solutions to a wide range of structural problems. The principle of thefinite element method (Reference 12) is to model the structure as a series ofdiscrete elements, the pattern of displacement in each being represented as afunction of the element’s nodal displacements. Different types of elements can beused depending on the required level of complexity and the type of problem beingsolved. A set of equilibrium equations is formed in terms of the nodaldisplacements, using assumed material properties, and these equations are thensolved to provide the displacement, and subsequently stress distribution, throughoutthe slab.

Finite element techniques are particularly useful for analysing flat slabs that havecomplex geometries and openings or unusual loadings (such as large concentratedloads) and boundary conditions (such as irregular spans and column spacings).Computing power has increased greatly over the last decade and it is now possibleto model large areas of slab on standard desktop computers. The software has alsodeveloped significantly with graphical pre and post-processors and automatic meshgenerators now standard.

The commercially available finite element analysis (FEA) computer packages fallinto three categories in order of increasing sophistication.

• Elastic analysis (e.g., Lusas, Robot, Staad, etc).• Elastic analysis with cracked section capability (e.g. Skanska FEM design, FE

Designer)• Non-linear finite element analysis (e.g., DIANA, ABAQUS, etc)

The elastic analysis packages are probably the most common. They can be quickand easy to use since the reinforced concrete slab can be modelled as an isotropicmaterial, which greatly simplifies the model. The limitations of this assumption needto be understood by the engineer and are discussed below. At the other end of thescale the reinforcement and concrete can be analysed as a composite materialusing non-linear behaviour.

Experience from the use of the finite element method at Cardington showed that themass of reinforcement was substantially the same whether derived by simple elasticanalyses as described above or finite element analysis assuming elastic behaviour.This concurs with separate research carried out by Whitby Bird and Partners7 whichdemonstrated that the results from finite element methods for a rectangular buildingwere broadly in line with results obtained by using the guidance in BS8110.

- 26 -

7 Comparative Study of Flat Slab Design Whitby Bird & Partners 11 June 1999 (Internal Publication)


5.2.1.3.2 Setting up the finite element model

The choice of element type will affect the results obtained. It is generally acceptedthat for flat slabs a plate element should be used. This element type ignores themembrane effects of the material.

The term ‘mesh’ is used to describe the spacing of elements, the finer the mesh themore accurate the results. The engineer has to assess how fine a mesh to use: acoarse mesh will not give maximum forces, especially hogging moments at columnpositions. However, a very fine mesh will take an excessive time to compute, andincurs the law of diminishing returns. More sophisticated packages provide simplefacilities to refine the mesh close to discrete supports or openings.

It should be remembered that theoretically the peak hogging moment occurs at thecentre of the column so the moment at the face of the column is actually somewhatlower than the peak moment. The maximum sagging moment will not be as sensitiveto the mesh size because the rate of change of the moment is far smaller than forthe hogging moments.

The elements should be ‘well conditioned’, that is the ratio of maximum to minimumlength of the sides should not exceed 2 to 1. Element sizes in the range 100 to500mm would be expected for most situations.

Where a linear elastic analysis is used the edge columns will attract more moment inthe analysis than they would actually be able to transfer. This can lead to an over-estimate of column moments and an under-estimate of span moments. This can beroughly dealt with by using pinned columns with (or without) an applied momentequivalent to the maximum moment that can be transferred. Another option is theuse of rotational spring supports, although this can become cumbersome when arange of different load combinations needs to be considered.

More sophisticated methods of analysis enable the effects of cracking and yieldingto be taken account of directly, potentially allowing more accurate predictions ofdeflection.

All software will allow a number of load cases to be considered, and the engineermust assess how to treat pattern loading. It requires engineering judgement todetermine the ‘most unfavourable arrangement of design loads’ (clause 3.7.2.1 – BS8110: Part 1) for a floor plate with unusual geometry. Again the latest packagesprovide facilities to deal automatically with patch loading, which previously could bevery onerous to define.

5.2.1.3.3 Ultimate limit state design

The principal moments generated in the slab will not correspond to the direction ofthe reinforcement. However most software packages will give the local moments Mxand My. They will also give the local twist moment Mxy, which is often overlooked bythe inexperienced. Mxy must be considered in the reinforcement design, and DrWood proposed a suitable method8. This method together with some correctionsput forward by Mr G S T Armer has become known as Wood Armer moments. Othermethods can also be used.

- 27 -

8 The reinforcement of slabs in accordance with a pre-determined field of moments Dr R H Wood,Concrete, February 1968


Most software packages are either capable of calculating Wood Armer moments, orthe effects of torsional moments are seamlessly automated into the design toaccount of torsional effects in the design of the reinforcement.

The results from the analysis will generally be in the form of contour plots of stressesand forces, although more modern programmes can convert this directly intorequired reinforcement areas in mm2/m.

The engineer will need to rationalise the reinforcement in a similar manner to thedivision of the panel into column strips and middle strips, as would normally be donefor a simple elastic analysis.

Where a linear elastic analysis is used, peak stresses at columns can beexaggerated and sagging stresses underestimated as explained above, in whichcase, as with other forms of elastic analysis, some level of redistribution of thederived moments is likely to be appropriate. Edge transfer moments (BS8110: Part1: Clause 3.7.4.2) should be checked separately. Where cracked or non-linearanalysis is employed redistribution will effectively have been performedautomatically and there should be no need to check transfer moments separately.The columns can be modelled directly to improve the accuracy of the columnmoments and reactions predicted.

Punching shear can be dealt with in the normal way using the reactions from themodel, remembering to fully consider the effective shear forces (Clause. 3.7.6. - BS8110: Part 1). Although the model will give shear results, they will not be particularlyuseful if the columns are modelled as pins with no effective shear perimeter.However where the columns are modelled directly some programs can alsocalculate shear perimeters and punching requirements.

5.2.1.3.4 Serviceability limit state design

Elastic analysis can be used with adjustments made to the material propertiesassumed or non-linear analysis can be used.

Use of elastic analysis programs has traditionally required adjustments to allow forthe effects of cracking and other long-term effects such as creep and shrinkage tobe made manually, but there are now programs available which allow this process tobe automated.

5.2.1.3.4.1 Linear Elastic analysis with Manual Adjustment of MaterialProperties

For simple elastic analysis most finite element packages calculate section propertiesfrom the thickness of the elements. In this case it is recommended that the grossdepth of the concrete is used as this gives the correct torsional constant (C) but theinertia (I) will be too large as a result of the effects of cracking. One method ofreducing the value of the inertia (I) is to reduce the elastic modulus (E) to give thecorrect the bending stiffness (EI). Failure to do so will give misleading and non-conservative results.

In addition, the elastic modulus needs to take account of creep (a function of theduration of loading) and shrinkage. CIRIA report 110 (Reference 13) gives someuseful guidance on the values of elastic modulus and inertia to use when modelling

- 28 -


assuming elastic behaviour. There are two alternative methods, a simplified method,which is particularly useful for preliminary and scheme stage, and a more accuratemethod.

The simplified method suggests that long term elastic modulus is in the range of onethird to two thirds of short term elastic modulus, and that the cracked inertia is halfthe gross inertia. This would suggest that for grade C40 concrete (Mean short termE = 28 kN/mm2 – BS 8110: Part 2) the values to use for the finite element modelshould be in the range 5 kN/mm2 (storage loading) to 10 kN/mm2 (short termloading).

The more accurate method requires the engineer to undertake a cracked sectionanalysis in accordance with BS8110: Part 2. In this case, there could be a number ofdifferent values of effective elastic modulus throughout the slab in each direction asit is a function of stress in the reinforcement under service loads. Experiencesuggests that the effective E will not normally be below 5 kN/mm2 for grade C40concrete.

5.2.1.3.4.2 Linear Elastic Analysis with Cracked Section Capability

An iterative analysis can be used whereby the moment in each element is comparedwith the likely cracking moment. If the element is cracked then a cracked sectionmodulus is used and the analysis is repeated. This can be very time consuming tobe done by hand but programs are now available which permit this process to beautomated. This can lead to more appropriate moment distributions for the UltimateLimit State, as well as better deflection prediction at the Serviceability Limit State.

The user inputs the slab reinforcement, and then the long term deflections and crackwidths are calculated automatically. Examples of recent software that has becomeavailable include FEM Design and FE Designer.

5.2.1.3.4.3 Non-linear Analysis

Non-linear analysis programs consider the effects of yielding, in addition to crackingand redistribution. This approach should model flat slabs well, as they often havelittle bottom cracking, but have extensive cracking and partial yielding ofreinforcement in the top, over columns. However, these solutions require a greatdeal of computing time and the programs require specialist expertise to utilise themreliably. Whichever form of analysis is used to derive the predicted deflections, they shouldbe limited to meet the guidance in Clause 3.2 of BS 8110: Part 2. In most cases it isthe effect of deflection on finishes which is most relevant. Using the simple elasticapproach this can be estimated by subtracting the slab self-weight deflection with ashort-term modulus from the total service deflection. The three primary variables governing the deflection are the elastic short-termmodulus, the creep factor and the concrete flexural tensile strength. Accurateassessment of the influence of these properties is essential if accurate deflectionsare to be predicted.

- 29 -


5.2.1.3.5 Model validation and checking

As with any analysis it is necessary to validate the results in order to avoid errors inthe modelling and input of data. There are number of simple checks that must becarried out, which include an equilibrium check of loads and reactions and a checkthat the ‘free’ bending moment diagram is correct (e.g. for a uniformly distributedload the sum of the hogging and sagging moments in any given span should add upto wl2/8).

5.2.1.3.6 Implications of EC2

The current ENV version of EC2 permits both linear and non-linear methods ofnumerical analysis, and hence there is no particular barrier in terms of using finiteelements. It should be noted that EC2 requires different standard combinations ofloaded and unloaded bays to be considered than BS8110.

EC2 also gives guidance on determining cracked section properties. The methodgiven in the ENV version of EC2 is more sophisticated than that in BS 8110. Whencalculating crack widths and deflections EC2 seeks to determine the extent ofcracking based on the level to which the steel is stressed. It is reported that the latest draft of EC2 (which is due to be issued in EN form inFebruary 2003) is more sophisticated still, providing a full method for rigorousanalysis, including the prediction of creep factors and shrinkage.

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5.2.1.3.7 Summary

• The use of finite element analysis has no particular benefits in terms of designeconomy for a typical rectangular building, but can offer significant advantageswhen used for the design of unusual flat slabs. However in the hands of anexperienced operator with modern graphical input methods it can provide theadvantage of speed, particularly where the design of the reinforcement isintegrated into the supply as for example with the Bamtec system.

• It is important to understand the limitations of the assumptions made, the choiceof element type and the mesh size.

• Where not considered automatically, use Wood-Armer moments or other methodto calculate ultimate moments for reinforcement design.

• If using linear elastic analysis to predict deflections, use the gross depth of theslab and modify the long-term elastic modulus to account for cracked sectionproperties.

• Uncracked section properties can under-estimate the deflection, whilst crackedsection properties could over-estimate the deflection. This can be useful to setboundaries on the deflections that could occur.

• Always validate your results.

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5.2.1.4 Other Methods

5.2.1.4.1 Grillage analysis

There are standard computer programs available to carry out such analyses.Reference 13 gives detailed guidance on the application of the method to design offlat slabs. It is not as sophisticated as finite element modelling using two dimensionalplate elements, but this may offer advantages in terms of transparency of the resultsgenerated.

5.2.1.4.2 Hillerborg Strip method

The Hillerborg strip method is an alternative form of design based on a plasticapproach. It has advantages over the yield line method in that it provides a lowerbound solution. However, its application to flat slabs (Reference 14) is notparticularly straightforward and the method has not yet been found capable ofautomation.

It is reported that an important new group of Finite Element type programs isbeginning to appear based on "perfectly plastic" plate theory. These lower boundprograms may become common as they will be better able to model the partialyielding and redistribution of moment that occurs locally around supporting columnsin flat slabs, thus improving prediction of behaviour at both the Ultimate andServiceability Limit States.

5.2.1.4.3 Empirical methods

It is claimed by some that the most efficient way to design and construct flat slabs isto use the standard 'empirical' method which was widely used until CP110 wasreplaced by BS8110 in 1985. An updated version of it can be found in the IStructE'Gold Book' Recommendations for the Permissible Stress Design of ReinforcedConcrete Building Structures, clauses 3C.12-16. The 'empirical design' can be usedin either permissible stress design or limit state design.

It is claimed that it is not only quicker to design than the BS8110 'simplified method'but it also requires less bending steel, less shear steel (because more steel isprovided at columns, increasing shear resistance) and it generates standardrationalised reinforcement arrangements. For example, comparing with the designusing the equivalent frame method given above the empirical method predicts that14% less steel is required, yet offers higher shear resistance and reduced shearreinforcement requirements.

- 32 -


5.3 Designing for punching shear

Punching shear failure at column locations is an important design criterion in flatslabs. Where punching shear reinforcement has needed to be provided this hastraditionally taken the form of a large number of individual shear links arranged on aseries of perimeters from the edge of the column. However a range of proprietarysystems is now available which can greatly speed up the fixing process9.

Methods for the design of standard punching shear reinforcement are welldocumented in existing codes, and a companion spreadsheet RCC13.xls is availableto assist in the design process when using BS8110 (Reference 5). This includesdealing with holes when present in the slab. A list of the currently availablespreadsheets from the RCC is included as Annex 1

Where a sub-frame analysis or other form of elastic analysis has been used theeffective shear force Veff can be calculated directly from the redistributed momentsand shear forces. In some cases this is done automatically for example within theRCC spreadsheet RCC33.xls.

Where yield line design has been used these moments may not have beencalculated directly. In this case one approach is to use the load magnification factorsgiven in Clause. 3.7.6 of BS 8110: Part 1 or other codes. The basic shear force Vt towhich the magnification factor is applied may be calculated assuming load to betransferred equally between columns according to the area they support, except thatin an end bay 55% of the load in a given direction may be assumed to be transferredto an internal column and the remaining 45% to the external column.

In practice the designer has considerable freedom to exercise his own judgement inthis area.

- 33 -

9 Concrete Society Technical Report: Shear Reinforcement Systems for Flat Plates (to be published).


6. DESIGNING FOR SERVICEABILITY

For thin flat slabs, as advocated in this guide, serviceability criteria are very likely togovern the design. In practice this may limit the advantage gained from using non-linear or plastic analysis for the Ultimate Limit State. There is also some evidencefrom the work at Cardington that early striking of the slabs can result in greater long-term deflections, though the increase in total deflection is likely to be small. Thisincrease in deflection should be negligible when considering deflections subsequentto application of the finishes.

The design loads to be used when checking the serviceability limit states of crackingand deflection, are obviously those without any load factors applied.

6.1 Deflections

6.1.1 Use of span/effective depth ratios

The simplest method of controlling deflections is via span/effective depth (L/d) ratios.

Using the simple L/d approach for commercially efficient flat slabs will in many casesrequire additional tension reinforcement to be provided at mid-span, as was the caseat Cardington. The basis for the provision of this additional reinforcement isindependent of the design method used and is well documented in existing codes.However as has already been noted some caution should be exercised whenconsidering the reinforcement provision resulting from yield line design.

It is interesting to note that where the requirement for additional tensionreinforcement was relaxed for one of the floors at Cardington, this does not appearto have had any undue effect on the deflection performance, questioning the needfor the provision of this additional reinforcement. The critical parameter determiningthe level of deflection was the concrete tensile strength (related to the compressivestrength) at the times at which the peak loads were applied.

6.1.2 Deflection calculation methods

Explicit calculation of deflections may sometimes be required for example to meetclient requirements or those of other design disciplines (e.g. cladding, partitions orservices). In addition if deflections are governing the design more accurateprediction of deflections should permit design economies. The big disadvantage oftrying to predict deflections is the number of unknowns at the design stage and thevariability of the data and estimates. Caution should therefore be exercised in thereliance placed on the deflection predictions and a margin of up to 30% may beadvisable to allow for the uncertainties. An example of the predictions given bydifferent available methods is given in Reference 15.

To predict deflections one approach is to model the slab as one-way spanning. Thiswill by definition only give an approximation of the deflection as the two-wayspanning action of the slab is not considered explicitly. However there are ways ofcombining the effects from the two orthogonal directions (Reference 15).

- 34 -


Where a simple elastic analysis has been used, the moments determined for theUltimate Limit State design can be used, but with the load factors removed. Thedeflections will generally be greatest at the centre of each panel. However, giventhat partitions are usually along column lines, it may be necessary to calculatedeflections here also.

One approach to determining the deflection at the centre of the panel is to calculatethe deflections on two parallel column strips, using BS8110 Pt 2 or EC2, and addingthe average of these to the deflection of the middle strip spanning at right angles tothe two column strips. This is illustrated in Figure 11.

Figure 11: Calculation of the deflection at the centre of a flat slab panel

The deflection at the centre of the panel is given by:

2DCAB

EFmid

δδδδ ++=

or

2BCAD

HImid

δδδδ ++=

Assuming a parabolic variation in curvature along the length of a slab strip thedeflection at the centre of a slab strip is given by:

( )rightmidleft

L ψψψδ ++= 1096

2

- 35 -


where: E = elastic modulus (N/mm2)I = second moment of area at point being considered (mm4)L = length of span being considered (mm)M = moment at point being considered (Nm)δ = deflection (mm)ψ = curvature = M / EIψleft = curvature at the left supportψmid = curvature at mid-spanψright = curvature at the right support

This method requires the calculation of between six and nine curvatures dependingon the symmetry of the bay considered. However, this is not as tedious as it mayseem if a simple spreadsheet is used.

Even where such an elastic analysis has been used the assumptions made willeffect the ratio of support to mid-span moments and hence the deflectionsdetermined and this should be borne in mind. Linked to this is the fact that themoments may sometimes (e.g. when using simple coefficients) effectively havealready been redistributed.

Finite element analyses are particularly useful when there are irregular geometriesand holes. They can also deal directly with the two-way spanning nature of theconstruction.

Appropriate modelling of cracked section properties is important whichever methodis used to predict deflections. In practice this will mean making initial assumptionsabout determining what areas of slab are cracked and to what extent.

In the case of two-way spanning models there is scope for refining theseassumptions on an iterative basis. As described in the section on Finite ElementAnalysis, there are programs that allow this process to be automated.

Flat slab structures are in practice likely to receive their maximum loads duringconstruction. Two conditions can be identified:

1. When the slab is first struck and required to carry its self weight plus anallowance for construction load.

2. When the slab is required to (partially) carry the weight of fresh slabs being castabove.

These load conditions are illustrated in Figure 2.

More detailed analyses of deflection, where considered justified, should if possibletake account of these peaks in the load-time history. They should also considerwhen these are likely to occur in practice in relation to the strength gain of theconcrete.

Since the effective tensile strength of the concrete is the key parameter, a simplifiedapproach is to derive this parameter fctmodified :

- 36 -


fctmodified = Kminwp

Where wp = quasi permanent load (as defined in Reference 9) and Kmin = minimumof (Kstrike, Kpeak and Kperm), K = fct/w where w = load, and fct = tensile splittingstrength. Kstrike should be calculated with the self-weight of the slab plus the weightof the falsework. The construction load corresponding to Kpeak can be calculatedfollowing the recommendations given in Reference 4 and occurs when the slab issubjected to the loading relating to the casting of the slab immediately above. Kperm

relates to what is assumed to be the permanent condition.

This approach is based on analysis of data from Cardington and should provide areasonable upper bound to long-term slab deflections when this modified value of fct

is taken, and the approach to the calculation of deflections set out in Appendix 4 ofReference 9 is followed.

Since the reinforcement has relatively little influence on the flexural properties of anuncracked section, the tensile strength may be used to calculate the crackingmoment Mcr as follows:

Mcr = fctbh2/6.

Alternatively, more precisely, the method of transformed sections may be used. Inthe case of a rectangular section the following formulae apply:

Depth to neutral axis,

hh

d

h

d

xe

e

+−+

+−+

=)))(1(1(2

)1(21

'

''

ρρα

ρρα

Second moment of area,

( ) ( ) ( )[ ] bdxxddxh

hh

I e

−+−−+

−+= 22

23

''1212

ρρα

Cracking moment,xh

I

y

IM cr −

== fct fct

- 37 -


In the above equations:

�e is the modular ratio Es/Ec

��s/bd for the tension reinforcement��s'/bd for the compression reinforcementb is the width of the section (usually taken per m)h is the overall depth of the slabd' is the depth to the compression reinforcementd is the depth to the tension reinforcement

The method of transformed sections must be used to calculate the flexuralproperties of the cracked section. In this case a common assumption is to ignore theinfluence of the concrete in tension completely and assume a triangular stress blockfor the concrete in compression.

To avoid overestimating slab deflection, the peak construction load should not beoverestimated and realistic concrete strengths should be used.

Example: At floor 2 Cardington , Kpeak was critical (i.e. smallest value). Therefore

fctmodified = Kpeakwp = (fctpeak /wpeak) wp = (fctpeak/10.34) x 9

fctpeak was 3.43 MPa

Hence, fctmodified = 2.98 MPa

The mean measured 28 day tensile strength was 3.93 MPa. This emphasises theimportance of having accurate data on early age tensile strengths as well ascompressive strengths so that accurate long-term deflections may be predicted. Inthe absence of direct information on tensile strength it may be estimated fromfct =0.3 (0.8 x fc)

0.67 where fc is the estimated actual concrete cube strength that willbe achieved at each of the times to be considered. This equation is the standardequation relating tensile strength to cylinder strength given in the ENV version ofEC2 and with the cylinder strength assumed to be 80% of the equivalent cubestrength.

Predictions of the 300-day deflections for floor 3 of the Cardington building using anelastic analysis with cracked section capabilities is given in Figure 12. Thesepredictions were calculated using input values of fctm appropriate to the constructionoverload stage, with quasi-permanent loading (equivalent to experimental appliedloading). These results agree very well with the measured values.

- 38 -


- 39 -

Figure 12: Predictions of the 300 day deflections on floor 3 of the ECBPCardington


6.2 Cracking

Whichever method of design is used, adequate control of cracking will be achievedprovided good detailing practice is followed with regard to spacing of reinforcement.Guidance on this is given in existing codes.

Methods exist for predicting crack widths, which for example in relation to EC2 followsimilar principles to those for dealing with deflections (Reference 9). However theexplicit calculation of crack widths and spacing is not usually warranted.

- 40 -


7. CONCLUSIONS AND RECOMMENDATIONS

1. There are many alternative approaches to the design of reinforced concrete flatslabs that are equally valid.

2. The choice of design method is very much one of personal preference. It shouldbe based on what is appropriate for the structure to be designed, and on thedesigner's own previous experience as well as what is likely to most benefit theclient.

3. This guide has given some pointers as to how existing design guidance andmethods could be developed and made more user-friendly, particularly with theadvent of EC2 in the next few years. The guide also flags up issues for thePermanent Works Designer to consider as a result of the desire to strike slabsearlier and speed up the construction process.

4. There is evidence to suggest that existing design approaches, such as yield line,do not give a full representation of the behaviour of flat slab structures becauseof their material behaviour assumptions and also due to the influence of in-planeeffects, which result in membrane action. This potentially provides scope forfurther economies in the design of flat slab structures.

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8. ACKNOWLEDGEMENTS

The author would like to acknowledge the assistance provided by a large number ofindividuals and organisations in putting this guide together. In particular he wouldlike to acknowledge the contribution made by Mr Gerard Kennedy of Powell Tolner &Associates in relation to yield line design.

The author would also like to acknowledge the funding provided by the DETR whichled to the production of this report. The particular project was entitled "DerivingFurther Learning and Benefit from the in situ Concrete Building at Cardington".

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9. REFERENCES

1. The European Concrete Building Project, The Structural Engineer, Vol.78, No.218 January 2000.

2. Goodchild C H, Economic Concrete Frame Elements, BCA, 1997. 3. Goodchild C H, Rationalisation of Flat Slab Reinforcement, BCA, 2000 (Ref.

97.376). 4. Concrete Structures Group, Guide to Flat Slab Formwork and Falsework, The

Concrete Society, Crowthorne, to be published. 5. BSI. Structural Use of Concrete. Part 1: Code of Practice for Design and

Construction. London, BSI, 1997. BS 8110-1:1997. 6. Goodchild C.H, Cost Model Study, BCA, 1993 7. Webster M, Further Cost Model Studies: RCC's findings, Concrete, March/April

1995. 8. Goodchild C.H and Webster R M, Spreadsheets for Concrete Design to BS8110

and EC2, BCA, 2000. 9. BSI. Eurocode 2: Design of Concrete Structures. Part 1: General Rules and

Rules for Buildings. London, BSI, 1992. DD ENV 1992-1-1: 1992. 10. Kennedy G, Introduction to Yield Line Design, Proposed RCC publication, 2001. 11. Johnson D, Lower Bound Collapse Analysis of Concrete Slabs, BCA Higher

Education Conference, Cardiff, 1999. 12. Johnson D, Advanced Structural Mechanics, 2nd edition, Thomas Telford, 2000 13. Whittle R T, Design of Reinforced Concrete Flat Slabs to BS 8110, CIRIA report

110 (2nd edition, 1994). 14. Hillerborg, A. Strip Method Design Handbook, E&FN Spon, 1996.

15. Deflections in Concrete Slabs and Beams, proposed Concrete Societypublication.

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Annex 1: List of currently available RCC spreadsheets

Spreadsheets to BS 8110

Elements RCC11 Element Design.xlsRCC12 Bending and Axial Force.xlsRCC13 Punching Shear.xlsRCC14 Crack Width.xls

Analysis RCC21 Subframe Analysis.xlsSlabs RCC31 One-way Solid Slabs (A & D).xls #

RCC32 Ribbed Slabs (A & D).xlsRCC33 Flat Slabs (A & D).xls

Beams RCC41 Continuous Beams (A & D).xlsRCC42 Post-tensioned Slabs & Beams (A &

D).xlsColumns RCC51 Column Load Take-down &

Design.xlsRCC52 Column Chart generation.xlsRCC53 Column Design.xls

Walls RCC61 Basement Wall.xlsRCC62 Retaining Wall.xls

Stairs RCC71 Stair Flight & Landing - Single.xlsRCC72 Stairs & Landings - Multiple.xls

Foundations RCC81 Foundation Pads.xlsTabular versions RCC91 One-way Solid Slabs (Tables).xls

RCC92 Ribbed Slabs (Tables).xlsRCC93 Flat Slabs (Tables).xlsRCC94 Two-way Slabs (Tables).xlsRCC95 Continuous Beams (Tables).xls

Spreadsheets to EC2 (ENV 1992)

Elements RCCe11 Element design.xlsAnalysis RCCe21 Subframe analysis.xlsBeams RCCe41 Continuous beams (A & D).xls

Notesin the text of this publication, spreadsheets are often referred to by their initial referencenumber rather than the full names given above# A & D = Analysis and Design

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approaches to the design of rc flat slabs

Documents

flat slab construction

copyright bre

good building guides

alternative design methods

discount onother bre

wide range of building

finite element methods

deflection calculation