BRE Garston, Watford, WD25 9XX
2001
Ap Approaches to the design of reinforced concrete flat slabs
R M Moss Moss,, BSc, PhD, DIC, CEng, MICE, MIStructE
BRE Centre Centre for Concrete Construc Construction tion
BRE Garston, Watford, WD25 9XX
2001
Ap Approaches to the design of reinforced concrete flat slabs
R M Moss Moss,, BSc, PhD, DIC, CEng, MICE, MIStructE
BRE Centre Centre for Concrete Construc Construction tion
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BR 422 ISBN 1 86081 498 0
© Copyright BRE, 2001 First published 2001
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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 5.2.1.1 Elastic Analyses with Moment Redistribution
8 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 6. DESIGNING FOR SERVICEABILITY 6.1 Deflections
33 34 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 slab construction. The report is not intended to be prescriptive and it is recognised that there are many alternative approaches to the design of reinforced concrete flat slabs that are equally valid. The intention is that this fuller report will be summarised as a Best Practice guide as part of a series of guides emerging from the European Concrete Building Project (Reference 1). Key messages
•
Thin flat slab construction is the most cost effective form of in situ concrete floor construction for spans from 5 to 9m, especially where a square or near square grid is used. For spans in excess of 9m post-tensioning should be considered.
•
In general terms in situ concrete flat slab construction without drops has advantages in simplifying falsework and formwork arrangements, thereby promoting rapid floor construction, and also giving maximum flexibility to the end user.
•
The additional construction process benefits associated with flat slab construction may well outweigh the benefits of constructing a more structurally efficient, but more complicated, solution for the floors. Related to this are the options pursued for providing bracing elements within the structure (if any), which traditionally have slowed down the overall construction process, and the methods for forming the columns. Where bracing elements are used steel cross bracing might offer one possible solution. Use of precast concrete columns is another possibility to speed up the column construction and patented connection systems have been developed to allow these to be used in conjunction with in situ concrete slabs.
•
Experience from Cardington (Reference 1), where a low level of imposed load was chosen and in some cases the slabs were struck very early, has highlighted the importance of the Permanent Works Designer considering the effects of 1 temporary construction loads on long-term serviceability performance .
Best practice
•
Investigate the benefits of using in situ concrete flat slab construction without drops if possible. If column heads are considered essential consider the scope for incorporating them as part of the column formwork to allow the advantages of flat soffits for the floors to be retained. Proprietary disposable systems are available for this purpose.
•
Further guidance on the best structural solutions to adopt under different circumstances is given in Reference 2.
1
Paper to be published in Magazine of Concrete Research -1-
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 reinforcement 2 rationalisation is most appropriate. This is further explained in Reference 3.
•
Optimise the slab thickness wherever possible. In practice this will depend on many factors such as the method of design, the presence or absence of holes, the importance placed on deflections and previous experience. Further guidance is given in Reference 2.
•
Consider whether a particular design method (e.g. yield line) is likely to lead to more rationalised reinforcement layouts and the benefits that might entail particularly on large projects.
•
Improve the flow of information between the various parties in the reinforcement 3 supply chain . Move towards a more integrated approach to design and detailing so that the contractor is better able to estimate weights of reinforcement when tendering.
•
If possible consider at the outset how the structure will be built, what the critical loading conditions will be and when they will occur including the possible effects on long-term deflection of early striking. Further guidance on the issues surrounding early age loading is given in Reference 4.
2
Reinforcement rationalisation in practical terms means reducing the number of bar marks used The use of a common, agreed data exchange format is recommended. Background research leading to this recommendation is described in the report Improving Rebar Information and Supply (IRIS) by A.Kalian, T.Thorpe and S.Austin, BRE report BR 401. 3
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Approaches to the Design of Reinforced Concrete Flat Slabs
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 formwork requirements are simplified and minimised, and rapid turn around can be achieved using a combination of early striking, discussed in Reference 4, and flying form systems. Use of this form of construction also places no restrictions on the positioning of horizontal services and partitions and can minimise floor/floor heights when there isn't a requirement for a deep false ceiling. This can potentially have knock-on benefits in terms of reduced cladding costs, but this may in turn be offset by the lack of edge beams and the greater deflections that result. In situ concrete construction has inherently good fire resistance, resistance to sound transmission and transmission of vibrations and high thermal mass, and these benefits are maximised with flat slab construction because of the uniformity of the slab thickness. Disadvantages The provision of large vertical service holes can pose a considerable potential problem with flat slab construction, particularly where there is a requirement to locate these immediately adjacent to columns. Methods of dealing with the flexural design to accommodate holes are covered in this report. Guidance on the use of structural steel shear heads to allow large openings to be formed adjacent to columns without compromising the punching shear resistance is intended to be 4 included within the Report of a Concrete Society Working Party . Another perceived disadvantage is the lack of flexibility in being able to accommodate late design changes and modifications post-construction such as holes and increased loads. This is one reason why it may be worth not minimising the slab thickness (see section on optimising slab thickness below). Novel strengthening systems such as use of carbon-fibre plate bonding have been used for 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 economically and may dictate the limited use of upstand or downstand beams. For example upstand beams were found to be necessary at Cardington (Reference 1) to accommodate the precast cladding loads in the two end bays with large stair and lift shaft openings. Because of the two-way spanning nature of the construction anticipated deflections may also be greater than for other less economic one-way spanning beam and slab systems.
4
Concrete Society Technical Report: Shear Reinforcement Systems for Flat Plates (to be published). -3-
Approaches to the Design of Reinforced Concrete Flat Slabs
2. CHOICE OF DESIGN METHOD The choice of design method is very much one of personal preference. It should be based on what is appropriate for the structure to be designed, and on the designer's own 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 most economic. In this case the RCC spreadsheets described below are ideal. For large buildings where the scale dictates the most efficient design, the yield line method will allow the optimum distribution of reinforcement. The yield line method may however require a separate elastic analysis of the serviceability limit states of cracking and deflection, and separate consideration of the column and punching shear design. Use of finite element analysis has particular advantages when the floor is supported on 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 issues can also be dealt with using the yield line method and, to a certain extent by judicious choice of sub-frames, but may not be straightforward.
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Approaches to the Design of Reinforced Concrete Flat Slabs
3. RATIONALISATION OF REINFORCEMENT There is evidence to suggest that some design methods result in more rationalised reinforcement layouts than others. As noted in Reference 3, there are process benefits in terms of blanket provision of main longitudinal reinforcement which may well outweigh savings in minimising reinforcement provision because of the required tailoring of loose bars. In practice even where loose bars are used the designer and detailer will almost always rationalise to a certain extent and clearly there is scope for 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 least material option necessarily results in cheapest price overall. Where rationalisation is employed it needs to be done at an appropriate stage if the full benefits are to be realised in practice. Current typical contractual arrangements and in particular the formal appointment of the frame contractor after much of the detailing work has been completed are seen as a potential barrier to this. There are considered to be definite process benefits to be gained by adopting wider use of prefabricated mats. Prefabricated mats with tailored sizes and bar spacing have been available for a considerable period, but to date have not been widely specified. One reason for this is the perception of the required minimum quantities to make their supply and use commercially viable. There are also new developments, (e.g. the Bamtec system, Figure 1), which involves one-way spanning reinforcement mats rolled out like a carpet into position. 5 This is sold as a complete package involving Finite Element analysis for the design of the reinforcement and is claimed to result in 80% savings in fixing time. This potentially offers savings in transport costs, because of the greater density of mats that can be transported. A more rationalised layout of reinforcement will also simplify the amount of detailing and the number of bending schedules required. Where possible the contractor should be given the freedom to undertake the detailing as recommended in the Construct report: A guide to contractor detailing of reinforcement in concrete, BCA, Crowthorne, 1997.
5
A note of caution when using the Bamtec system in conjunction with BS8110 is the need to consider the maximum design moment which can be transferred to an edge or corner column. It may be necessary to manually take account of Mtmax requirements (Part 1: Clause 3.7.4.2) adjusting end span and penultimate support moments accordingly.
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Approaches to the Design of Reinforced Concrete Flat Slabs
Figure 1: The Bamtec system
4. OPTIMISING SLAB THICKNESS Having chosen a flat slab solution, the next key issue is determining an appropriate slab thickness. Studies have been carried out looking to optimise the thickness based on designs to BS8110 (Reference 5) and other codes and the general conclusion is that thinner slabs save money (References 6 and 7). This saving results 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 concerning slab deflections, which are likely to increase as the thickness is reduced. The reduced self-weight of the slabs will mean lower column loads and permit smaller foundations. There is of course a lower limit to the slab thickness and as the slab thickness reduces the savings identified above become outweighed by the additional amount of reinforcement required and the increased difficulty in designing and fixing it. There is also a case for having some margin, particularly at outline scheme stage, to accommodate late changes in architectural requirements and provision of holes in the slab. Consideration could also be given to possible future alterations and changes of use post-construction. Guidance on appropriate slab thickness is given in Reference 2 by way of span/depth charts. Slab depths for spans in the range of 5-9m range from a minimum of 200mm to approximately 380mm depending on the level of imposed loading. For the in situ concrete building at Cardington (Reference 1), which had 2 spans of 7.5m and a design imposed load of 2.5kN/m , a slab thickness of 250mm was chosen.
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Approaches to the Design of Reinforced Concrete Flat Slabs
5. DESIGNING FOR THE ULTIMATE LIMIT STATE 5.1 Loadings An example of the load history of a typical slab at Cardington is illustrated in Figure 2.
12
10.74
10 ) m / N k ( d a o L
2
8.75 8
6.75
9
9
6
6 4 2 0 1
10
100
1000
Time (days)
Figure 2: Typical load history for a floor at Cardington From Figure 2 distinct stages of loading can be clearly identified. The initial loading is due to the self-weight when the slab is struck, together with an allowance for construction loads. The slab is then subject to a short peak load as a result of the casting of the floor immediately above. A further smaller peak load is then applied as a result of loads from backprops when casting the second floor above. In the particular case of the Cardington slabs, no further load was applied until additional sandbags were placed on the building a considerable time after construction. Nevertheless this type of load history, at least during construction, is likely to be typical of flat slabs in practice. It is an inherent feature of flat slabs, and indeed reinforced concrete construction in general, that the dead/live load ratio is high. With moves towards lower design imposed loads, the "spare capacity" of a given slab over and above its self-weight is further reduced. This has implications for the way in which flat slab structures perform and are built. Findings from Cardington also suggest that traditional assumptions concerning the distribution of load through supporting slabs are incorrect, with the slab immediately beneath that being cast carrying a higher proportion of the load than usually assumed. Further guidance on this topic is given in Reference 4. As a result of the above it is possible that the temporary loading conditions during construction 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 way of the contractor restricting the way he goes about constructing the frame, for
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Approaches to the Design of Reinforced Concrete Flat Slabs
example in relation to early striking. However he should consider the possible effects of early striking upon issues such as long-term deflections. The Second Edition of 6 the National Structural Concrete Specification sets out a balanced approach recognising the interests of all the parties involved. When designing for the ultimate limit states of flexure and shear it will obviously be required 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 of moment redistribution may be used to determine bending moment and shear force envelopes. In general the geometry of the structure and the different loading arrangements on it will need to be considered, but under certain conditions simple coefficients 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's spreadsheets (Reference 8) which are used to illustrate many of the points made later in the document. These automate the process and effectively mean that some form 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 twodimensional frames, consisting of columns and strips of slab. Each frame may then be 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 is also very much tried and tested. On the other hand it is in some ways a very crude representation of the true behaviour of the slabs. Having input the basic data on geometry and loads (Figure 3) and the appropriate level of moment redistribution, this sub-frame analysis is done automatically within the relevant RCC spreadsheet program RCC33.xls and the information presented graphically (Figure 4).
6
Construct, BCA, BRE and RCC National Structural Concrete specification for building construction (NSCS), Crowthorne, BCA, 2000, BCA Publication Ref. 97.378. -8-
Approaches to the Design of Reinforced Concrete Flat Slabs
Figure 3: RCC Spreadsheet - Data entry
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Approaches to the Design of Reinforced Concrete Flat Slabs
Figure 4: RCC Spreadsheet - Automatic sub-frame analysis
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Approaches to the Design of Reinforced Concrete Flat Slabs
5.2.1.1.3 Division of slab into column and middle strips Where simple coefficients or the equivalent frame method have been used to determine bending moments, the three-dimensional nature of the construction will need to be represented by assigning proportions of the moment to be carried within imaginary column and middle strips of slab. Guidance on determining appropriate widths of strip and these proportions is given in BS8110 and again is considered automatically within the RCC's spreadsheet program (Figure 5).
Figure 5: RCC Spreadsheet analysis - automatic division into column and 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. in relation to the placing of the determined hogging reinforcement within the column strip for internal panels and maximum design edge moments). Again these issues are dealt with automatically within the RCC spreadsheet, which even goes as far as determining appropriate sizes and weights of reinforcement taking into account simple rules in relation to curtailment of reinforcement, and other detailing requirements (Figure 6). This information can be extremely valuable when scheming the structure.
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Approaches to the Design of Reinforced Concrete Flat Slabs
Figure 6: RCC spreadsheet determination of approximate bar weights An area that cannot be dealt with automatically by the existing spreadsheet package, or indeed any other form of one dimensional sub-frame analysis, is the provision of openings. From the point of view of flexure, the effect of small openings can often be ignored or dealt with simply by proportioning additional transfer reinforcement around the opening. To avoid stress concentrations this is often placed diagonally at each corner of the opening. Major openings can sometimes be dealt with by judicious adjustment of the assumptions made in the sub-frame analysis in each direction, or a more rigorous analysis may be justified.
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Approaches to the Design of Reinforced Concrete Flat Slabs
The RCC spreadsheet is intended to be used in combination with a companion sheet RCC13.xls dealing with punching shear where traditional shear links are specified. This latter sheet can deal with the effects on punching shear of holes near columns. 5.2.1.1.5 Experience from use at Cardington Sub-frame analysis was used successfully for the base design of the Cardington slabs carried out by Buro Happold using the ENV version of EC2 (Reference 9). It was also used for the variations on the base design. These involved sub-frame analysis with rationalisation of reinforcement and the relaxation of deflection requirements on floor 3, the use of blanket cover loose bar for half of floor 4, and one-way mats on floor 5. The weight penalty introduced by rationalising the loose bar provision obtained was small and it is recommended that for smaller projects this method of analysis, in conjunction 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 to reduce the number of bar marks used by about one-third compared with a minimum material solution. In the RCC spreadsheet the reinforcement suggested to be provided at a given location is the minimum area consistent with the maximum preferred bar size, and it is for the user to determine a suitable level of rationalisation. In relation to the output suggested above this might be:
• •
two bar size/spacing combinations to cover all sagging reinforcement in end bays and internal bays two basic bar size/spacing combinations and lengths for the hogging reinforcement in all column strips and middle strip regions. This spacing can be reduced for the outer regions of the column strips.
The hogging reinforcement in middle strip areas could be made continuous everywhere 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 of the Cardington project, and to assist in the conversion process, the use of the ENV version of the code by Buro Happold, in conjunction with the National Application document, was reviewed. The findings have been reported to the relevant BSI committee. In principle sub frame analysis and the design approaches suggested in BS8110 may still be used, but there are some additional requirements. In particular the ENV version of EC2 requires different loading arrangements and combinations to be considered, and there are more possible permutations than in BS8110. There are plans to extend the RCC spreadsheet to deal with the design of flat slabs in accordance with the final EN version of EC2 when it is issued.
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Approaches to the Design of Reinforced Concrete Flat Slabs
5.2.1.1.7 Serviceability issues These are dealt with in more detail in the separate section on serviceability below. In most cases the simplified approach to controlling deflections using span/depth ratios will be adequate. The results from an elastic analysis can however be used as the basis for prediction of deflections.
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Approaches to the Design of Reinforced Concrete Flat Slabs
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 plastic yielding is assumed to occur along a system of “yield lines”. When all the yield lines necessary to initiate a mechanism have formed, the whole, or a portion, of the slab can notionally collapse, which represents the ultimate condition. Provided the correct yield line pattern is chosen, use of the yield line method gives a better representation of the influence of structure's geometry on the flexural behaviour of the slab at the ultimate condition than say sub-frame analysis, where the 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 of design 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 its commercial 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 not possible within the scope of this guide to fully explain the method, but it is intended to try and re-introduce practical designers to yield line techniques in the context of flat slabs supported on a rectangular grid of columns. Irregular grids of columns can be dealt with by the Yield Line method and some guidance on this is given in Reference 10. The technique challenges designers to use judgement, but once grasped it is exceedingly easy to use and put into practice. Since the yield line method only considers collapse mechanisms, it may be necessary to consider serviceability issues such as cracking and deflection in more detail 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 except that the behaviour is now two-dimensional rather than one-dimensional. In the beam plastic hinges will develop in the most highly stressed regions until a mechanism is formed allowing the beam to collapse. In a similar way yield lines develop in a slab again eventually allowing a mechanism to be formed and failure to occur. A simple case to analyse is that of a two-way spanning square slab simply supported along all 4 sides, subject to a uniformly distributed load. As one would anticipate, as the load is gradually increased towards failure, yield lines develop emanating from the centre of the slabs towards the supporting corners (Figure 7).
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Approaches to the Design of Reinforced Concrete Flat Slabs
When analysing a slab using yield line, the theoretical procedure is based on the principle that the work done in deforming the slab equates to the internal energy dissipated 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 slab is usually treated as pinned so that no moment transfer is taken into account. It should be emphasised that some form of elastic analysis, in combination with an appropriate level of moment redistribution, is still likely be required to derive moments for the design of the columns themselves. However this does not in itself undermine 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 to use the RCC spreadsheet RCC51.xls for this purpose. This is included within the suite 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 be a certain number of bays in each direction and the spans may vary. Three possible failure 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 negative moments, with negative yield lines forming along the axis of rotation passing along the faces of a line of internal columns. A corresponding pattern could take place at right angles, or a combination of both these collapse modes could develop
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Approaches to the Design of Reinforced Concrete Flat Slabs
simultaneously (Figure 8b) in which case the positive and negative yield lines will form simultaneously in both directions. In the latter case the axes of rotation for the failure of any given panel will lie along the leading diagonals of adjacent panels. It can be shown theoretically that the collapse load to generate this failure pattern cannot be less than that to generate the one-way spanning failure pattern, so that only the latter condition needs to be considered in practice. At slab edges simple line supports may be assumed. Since the edge and corner columns can rotate, it is usual to assume the axis of rotation for these simple supports to be at the centreline of the columns. The spans are taken between the axes of rotation when establishing the magnitude of the moments. The third possible mechanism is one in which conical failure surfaces develop over each column (Figure 8c). The check on this possible failure mechanism will result in additional 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 to 1, which is generally satisfactory for flat slabs unless there is a significant difference in the length of adjacent spans. Theoretically for any corner or edge bay in the 2 direction considered the ultimate mid-span moment will then be given by wl /11.66 where w is the design ultimate load, and l is the effective span (taken between axes of rotation). This will then allow determination of the required area of reinforcement in the normal way. Similarly for an internal bay in the direction considered it can be shown that the 2 ultimate mid-span moment may be conservatively taken as wl /16. Reference 10 gives further details of how these moments are derived. The reinforcement will be required to be distributed uniformly across the bay for each direction, making the use of uniform one or two-way spanning mats very attractive. As an example for comparison one could take the design using the equivalent frame method presented above. Applying the same modification factor for compression reinforcement of 1.05, the equivalent yield line design would give blanket sagging reinforcement in end bays of T20 at 300 centres, instead of the T20 at 350 in middle strips and T20 at 250 in column strips.
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Approaches to the Design of Reinforced Concrete Flat Slabs
Figure 8a: Possible flat slab yield line pattern (yielding in one direction)
Figure 8b: Possible flat slab yield line pattern (yielding in two directions)
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Approaches to the Design of Reinforced Concrete Flat Slabs
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 an optimal solution except where concern exists about incidental cracking, (e.g. a floor that is to be power-floated), in which case a more even distribution would be appropriate (Reference 10). One common assumption is to concentrate it over an area of side equal to 0.5L x 0.5L for an internal column, 0.5L x (0.2L+ edge distance E.D.) for an external column and (0.2L + E.D.) x (0.2L+ E.D.) for a corner column. L in this context is the column centreline to centreline distance in the direction considered which differs slightly from the effective span. The concentrations of the top reinforcement are illustrated in Figure 9. Additional top distribution reinforcement can be provided between these concentrations if considered necessary.
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Approaches to the Design of Reinforced Concrete Flat Slabs
Figure 9: Concentrations of top reinforcement The design of the top reinforcement then proceeds on the basis of considering the total moment along the yield lines at supports and then concentrating it in the column head areas discussed above. This improves the design for punching shear, which can be further improved by further concentrating reinforcement directly over the 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, the support moments for end and internal bays will be the same as the mid-span 2 2 sagging moments given above (i.e. wl /11.66 and wl /16 respectively). The top reinforcement passes over the gridlines and into the next bay. As a result in a 4 bay by 3 bay structure such as the Cardington building (Figure 10), the moment per unit 2 length along gridlines 2, 4, B and C will be the higher moment wl /11.66 and only 2 that along gridline 3 will be wl /16. This moment multiplied by the total length of the yield 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 and C is likely to be very similar. This means that the same top reinforcement can be provided over these gridlines in each direction. As an example for comparison one could take the design using the equivalent frame method presented above which is for an internal bay. The equivalent yield line design would give top reinforcement at the internal support in end spans of T16 at 100 centres, instead of the T16 at 550 in middle strips and T16 at 125:250 in column strips. Since simple line supports are assumed along the edges of the slab, there would be no requirement for top reinforcement (other than U bars) at the external support using yield line. This compares with the T12 at 325 in middle strips and the T16 at 125 in the column strips using the equivalent frame method.
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Approaches to the Design of Reinforced Concrete Flat Slabs
grid-lines
B
A
C
D
1 Extra top bars
External columns 400mm x 250mm
7500m
Shortened one-way spanning yield lines 2 Internal columns 400mm x 400mm
7500m
Steel 3 crossbracing
PH duct
Mech. duct 4 Stairs
Lift shaft
5 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=14.5m Total 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 concentrations along 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 is 2 w x 7.3 /11.66 2 Support moment for yield lines along 3 is w x 7.1 /16 Effective support moment in column concentration = ratio x 2 support moment established above (e.g. 2.09 x w x 7.3 /11.66)
Figure 10: Cardington 4 bay by 3 bay structure
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Approaches to the Design of Reinforced Concrete Flat Slabs
It should also be recognised that the same density of reinforcement will be provided when designing using the yield line method for both internal and edge bays, but greater densities will be required for an edge bay when using the equivalent frame method. 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 8 different arrangements for a single span direction involving only 3 spans using the example 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) that the formula for doing this in the case of internal columns is: 0.33
m+m' = S(1- (wA/S)
) /2
In the above formula: m is the average moment of resistance per unit length in the two directions provided by the bottom reinforcement in the bays adjacent to the column. m' is the average moment of resistance per unit length in the two directions provided by 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 be assumed to be transferred equally between columns according to the area they support, except that in an end bay 55% of the load in a given direction may be assumed to be transferred to an internal column and the remaining 45% to the external column. In the case of corner columns the support moment m' per unit length to be provided in the form of U bars in each direction is calculated from: 0.33
m' = S(1- (wA/S)
) /2
In the case of edge columns the support moment m' per unit length to be provided in the form of U bars perpendicular to the slab edge is calculated from 0.33
m' = S(1- (wA/S)
) /5.14
It should be checked that this moment does not exceed the resistance moment provided by the top reinforcement over the column in the direction parallel to the slab edge. Nominal U bars should also be provided along the edges of the slab between the concentrations of bars at column supports. A full worked example is given in Reference 10.
- 22 -
Approaches to the Design of Reinforced Concrete Flat Slabs
5.2.1.2.2.1.4 Dealing with perimeter loads
So far only uniformly distributed loads on the slab have been considered. Perimeter loads and other load concentrations can usually be dealt with by considering them to act over a suitable width. However for very heavy perimeter loads or other forms of concentrated 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 give the simplest bending reinforcement arrangements. Furthermore use of yield line analysis resulted in dramatically lower masses of steel for blanket cover, suggesting that this is the most efficient design method where this approach to rationalisation is adopted. Although not used directly in combination with yield line at Cardington, blanket two way mats were found to be about 40% quicker to install than loose bar. This speed advantage is worthwhile for larger projects. Top reinforcement was concentrated over columns although some mesh reinforcement was also provided to supplement the steel area required and to limit the cracking. No cracking as a result of the concentration of top reinforcement has been observed. Rigorous deflection calculations were carried out but it was concluded that these were unnecessary and that span/effective depth ratios were effective in controlling predicted deflections. 5.2.1.2.4 Implications of EC2 As with BS8110, EC2 gives the designer scope to use a range of design methods and yield line is one of the plastic methods of analysis recommended. When using plastic methods of analysis the current ENV version of EC2 states the reinforcement should 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 cannot deal with the Serviceability Limit State. Where explicit calculation of deflections is not required the same simple approach using span/depth ratios for dealing with serviceability may in principle be adopted as for other design methods (see section below on serviceability design). However it should be recognised that the provision of the bottom reinforcement has been determined on the basis of preventing a collapse mechanism forming along the entire length of the slab. A uniform moment and hence reinforcement requirement results whereas in elastic analysis (for example sub-frame analysis) a greater concentration of sagging moment and hence reinforcement 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 -
Approaches to the Design of Reinforced Concrete Flat Slabs
b is normally taken as 1.1 for end spans and 1.2 for internal spans. 5.2.1.2.6 Dealing with provision of holes Potentially the yield line method can deal more satisfactorily with this than some other methods of design (e.g. sub-frame analysis) because they can be taken account of directly when determining a yield line pattern. For example for the Cardington building (Figure 10), the stair and lift shaft openings at either end of the building were dealt with as follows: In relation to the yield lines which could develop across the width of the building only that at mid-span in the end bays will be affected as a result of the yield line being of reduced length. To reduce the complexity of the calculations the same blanket load may be conservatively taken to apply over the area of the hole. With this assumption a standard solution may be used to determine a revised value for m. Assuming that m=m' as previously this reduces to: 2
m=wl /(11.66-8.24d/Ltot) In the above equation d is the missing length of yield line, in this case 3.6m or 5m and Ltot is the total length of the uninterrupted yield line, in this case 22.75m. The 2 2 corresponding moments to be designed for are wl /10.36 and wl /9.85 instead of 2 wl /11.66 as previously. In relation to the yield lines that could develop along the length of the building, only those at supports along gridlines B and C will be affected. This will require additional top reinforcement adjacent to the existing internal column concentrations next to the holes to replace that no longer effective in the edge columns. This is illustrated in blue in Figure 10. The size of the opening will have a bearing on the approach taken. For example if the depth of the openings had extended beyond half the span then an alternative approach would have been required. This would have involved deriving the local moments 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 used reliably and with confidence. Further information and guidance can be found in Reference 10. There is no reason in principle why the method of yield line design using the standard solutions given in Reference 10 and summarised above could not be included within a new modified existing RCC spreadsheet, for example RCC33.xls. This would remove much of the hand computation normally associated with the method.
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Approaches to the Design of Reinforced Concrete Flat Slabs
Recent research at Nottingham Trent University has focused on the development of a 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 the critical collapse mechanism to be established, removing any lingering uncertainty inherent in the yield line technique. There would be merit in developing software that could assist practical designers using this approach also.
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Approaches to the Design of Reinforced Concrete Flat Slabs
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 be used to obtain solutions to a wide range of structural problems. The principle of the finite element method (Reference 12) is to model the structure as a series of discrete elements, the pattern of displacement in each being represented as a function of the element’s nodal displacements. Different types of elements can be used depending on the required level of complexity and the type of problem being solved. A set of equilibrium equations is formed in terms of the nodal displacements, using assumed material properties, and these equations are then solved to provide the displacement, and subsequently stress distribution, throughout the slab. Finite element techniques are particularly useful for analysing flat slabs that have complex geometries and openings or unusual loadings (such as large concentrated loads) and boundary conditions (such as irregular spans and column spacings). Computing power has increased greatly over the last decade and it is now possible to model large areas of slab on standard desktop computers. The software has also developed significantly with graphical pre and post-processors and automatic mesh generators now standard. The commercially available finite element analysis (FEA) computer packages fall into 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 quick and easy to use since the reinforced concrete slab can be modelled as an isotropic material, which greatly simplifies the model. The limitations of this assumption need to be understood by the engineer and are discussed below. At the other end of the scale the reinforcement and concrete can be analysed as a composite material using non-linear behaviour. Experience from the use of the finite element method at Cardington showed that the mass of reinforcement was substantially the same whether derived by simple elastic analyses as described above or finite element analysis assuming elastic behaviour. 7 This concurs with separate research carried out by Whitby Bird and Partners which demonstrated that the results from finite element methods for a rectangular building were broadly in line with results obtained by using the guidance in BS8110.
7
Comparative Study of Flat Slab Design Whitby Bird & Partners 11 June 1999 (Internal Publication) - 26 -
Approaches to the Design of Reinforced Concrete Flat Slabs
5.2.1.3.2 Setting up the finite element model The choice of element type will affect the results obtained. It is generally accepted that for flat slabs a plate element should be used. This element type ignores the membrane effects of the material. The term ‘mesh’ is used to describe the spacing of elements, the finer the mesh the more accurate the results. The engineer has to assess how fine a mesh to use: a coarse mesh will not give maximum forces, especially hogging moments at column positions. However, a very fine mesh will take an excessive time to compute, and incurs the law of diminishing returns. More sophisticated packages provide simple facilities to refine the mesh close to discrete supports or openings. It should be remembered that theoretically the peak hogging moment occurs at the centre of the column so the moment at the face of the column is actually somewhat lower than the peak moment. The maximum sagging moment will not be as sensitive to the mesh size because the rate of change of the moment is far smaller than for the hogging moments. The elements should be ‘well conditioned’, that is the ratio of maximum to minimum length of the sides should not exceed 2 to 1. Element sizes in the range 100 to 500mm would be expected for most situations. Where a linear elastic analysis is used the edge columns will attract more moment in the analysis than they would actually be able to transfer. This can lead to an overestimate of column moments and an under-estimate of span moments. This can be roughly dealt with by using pinned columns with (or without ) an applied moment equivalent to the maximum moment that can be transferred. Another option is the use of rotational spring supports, although this can become cumbersome when a range of different load combinations needs to be considered. More sophisticated methods of analysis enable the effects of cracking and yielding to be taken account of directly, potentially allowing more accurate predictions of deflection. All software will allow a number of load cases to be considered, and the engineer must assess how to treat pattern loading. It requires engineering judgement to determine the ‘most unfavourable arrangement of design loads’ (clause 3.7.2.1 – BS 8110: Part 1) for a floor plate with unusual geometry. Again the latest packages provide facilities to deal automatically with patch loading, which previously could be very 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 of the reinforcement. However most software packages will give the local moments Mx and My. They will also give the local twist moment Mxy, which is often overlooked by the inexperienced. Mxy must be considered in the reinforcement design, and Dr 8 Wood proposed a suitable method . This method together with some corrections put forward by Mr G S T Armer has become known as Wood Armer moments. Other methods can also be used.
8
The reinforcement of slabs in accordance with a pre-determined field of moments Dr R H Wood, Concrete, February 1968 - 27 -
Approaches to the Design of Reinforced Concrete Flat Slabs
Most software packages are either capable of calculating Wood Armer moments, or the effects of torsional moments are seamlessly automated into the design to account of torsional effects in the design of the reinforcement. The results from the analysis will generally be in the form of contour plots of stresses and forces, although more modern programmes can convert this directly into 2 required reinforcement areas in mm /m. The engineer will need to rationalise the reinforcement in a similar manner to the division of the panel into column strips and middle strips, as would normally be done for a simple elastic analysis. Where a linear elastic analysis is used, peak stresses at columns can be exaggerated and sagging stresses underestimated as explained above, in which case, as with other forms of elastic analysis, some level of redistribution of the derived moments is likely to be appropriate. Edge transfer moments (BS8110: Part 1: Clause 3.7.4.2) should be checked separately. Where cracked or non-linear analysis is employed redistribution will effectively have been performed automatically and there should be no need to check transfer moments separately. The columns can be modelled directly to improve the accuracy of the column moments and reactions predicted. Punching shear can be dealt with in the normal way using the reactions from the model, remembering to fully consider the effective shear forces (Clause. 3.7.6. - BS 8110: Part 1). Although the model will give shear results, they will not be particularly useful if the columns are modelled as pins with no effective shear perimeter. However where the columns are modelled directly some programs can also calculate 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 properties assumed or non-linear analysis can be used. Use of elastic analysis programs has traditionally required adjustments to allow for the effects of cracking and other long-term effects such as creep and shrinkage to be made manually, but there are now programs available which allow this process to be automated.
5.2.1.3.4.1 Linear Elastic analysis with Manual Adjustment of Material Properties For simple elastic analysis most finite element packages calculate section properties from the thickness of the elements. In this case it is recommended that the gross depth of the concrete is used as this gives the correct torsional constant (C) but the inertia (I) will be too large as a result of the effects of cracking. One method of reducing the value of the inertia (I) is to reduce the elastic modulus (E) to give the correct the bending stiffness (EI). Failure to do so will give misleading and nonconservative results. In addition, the elastic modulus needs to take account of creep (a function of the duration of loading) and shrinkage. CIRIA report 110 (Reference 13) gives some useful guidance on the values of elastic modulus and inertia to use when modelling
- 28 -
Approaches to the Design of Reinforced Concrete Flat Slabs
assuming elastic behaviour. There are two alternative methods, a simplified method, which is particularly useful for preliminary and scheme stage, and a more accurate method. The simplified method suggests that long term elastic modulus is in the range of one third to two thirds of short term elastic modulus, and that the cracked inertia is half the gross inertia. This would suggest that for grade C40 concrete (Mean short term 2 E = 28 kN/mm – BS 8110: Part 2) the values to use for the finite element model 2 2 should be in the range 5 kN/mm (storage loading) to 10 kN/mm (short term loading). The more accurate method requires the engineer to undertake a cracked section analysis in accordance with BS8110: Part 2. In this case, there could be a number of different values of effective elastic modulus throughout the slab in each direction as it is a function of stress in the reinforcement under service loads. Experience 2 suggests that the effective E will not normally be below 5 kN/mm for grade C40 concrete.
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 compared with the likely cracking moment. If the element is cracked then a cracked section modulus is used and the analysis is repeated. This can be very time consuming to be done by hand but programs are now available which permit this process to be automated. This can lead to more appropriate moment distributions for the Ultimate Limit 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 crack widths are calculated automatically. Examples of recent software that has become available 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 cracking and redistribution. This approach should model flat slabs well, as they often have little bottom cracking, but have extensive cracking and partial yielding of reinforcement in the top, over columns. However, these solutions require a great deal of computing time and the programs require specialist expertise to utilise them reliably. Whichever form of analysis is used to derive the predicted deflections, they should be limited to meet the guidance in Clause 3.2 of BS 8110: Part 2. In most cases it is the effect of deflection on finishes which is most relevant. Using the simple elastic approach this can be estimated by subtracting the slab self-weight deflection with a short-term modulus from the total service deflection. The three primary variables governing the deflection are the elastic short-term modulus, the creep factor and the concrete flexural tensile strength. Accurate assessment of the influence of these properties is essential if accurate deflections are to be predicted.
- 29 -
Approaches to the Design of Reinforced Concrete Flat Slabs
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 in the modelling and input of data. There are number of simple checks that must be carried out, which include an equilibrium check of loads and reactions and a check that the ‘free’ bending moment diagram is correct (e.g. for a uniformly distributed load the sum of the hogging and sagging moments in any given span should add up 2 to wl /8). 5.2.1.3.6 Implications of EC2 The current ENV version of EC2 permits both linear and non-linear methods of numerical analysis, and hence there is no particular barrier in terms of using finite elements. It should be noted that EC2 requires different standard combinations of loaded and unloaded bays to be considered than BS8110. EC2 also gives guidance on determining cracked section properties. The method given in the ENV version of EC2 is more sophisticated than that in BS 8110. When calculating crack widths and deflections EC2 seeks to determine the extent of cracking 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 in February 2003) is more sophisticated still, providing a full method for rigorous analysis, including the prediction of creep factors and shrinkage.
- 30 -
Approaches to the Design of Reinforced Concrete Flat Slabs
5.2.1.3.7 Summary
•
The use of finite element analysis has no particular benefits in terms of design economy for a typical rectangular building, but can offer significant advantages when used for the design of unusual flat slabs. However in the hands of an experienced operator with modern graphical input methods it can provide the advantage of speed, particularly where the design of the reinforcement is integrated into the supply as for example with the Bamtec system.
•
It is important to understand the limitations of the assumptions made, the choice of element type and the mesh size.
•
Where not considered automatically, use Wood-Armer moments or other method to calculate ultimate moments for reinforcement design.
•
If using linear elastic analysis to predict deflections, use the gross depth of the slab and modify the long-term elastic modulus to account for cracked section properties.
•
Uncracked section properties can under-estimate the deflection, whilst cracked section properties could over-estimate the deflection. This can be useful to set boundaries on the deflections that could occur.
•
Always validate your results.
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Approaches to the Design of Reinforced Concrete Flat Slabs
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 of flat slabs. It is not as sophisticated as finite element modelling using two dimensional plate elements, but this may offer advantages in terms of transparency of the results generated. 5.2.1.4.2 Hillerborg Strip method The Hillerborg strip method is an alternative form of design based on a plastic approach. It has advantages over the yield line method in that it provides a lower bound solution. However, its application to flat slabs (Reference 14) is not particularly straightforward and the method has not yet been found capable of automation. It is reported that an important new group of Finite Element type programs is beginning to appear based on "perfectly plastic" plate theory. These lower bound programs may become common as they will be better able to model the partial yielding and redistribution of moment that occurs locally around supporting columns in flat slabs, thus improving prediction of behaviour at both the Ultimate and Serviceability 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 is to use the standard 'empirical' method which was widely used until CP110 was replaced by BS8110 in 1985. An updated version of it can be found in the IStructE 'Gold Book' Recommendations for the Permissible Stress Design of Reinforced Concrete Building Structures, clauses 3C.12-16. The 'empirical design' can be used in 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 is provided at columns, increasing shear resistance) and it generates standard rationalised reinforcement arrangements. For example, comparing with the design using the equivalent frame method given above the empirical method predicts that 14% less steel is required, yet offers higher shear resistance and reduced shear reinforcement requirements.
- 32 -
Approaches to the Design of Reinforced Concrete Flat Slabs
5.3 Designing for punching shear Punching shear failure at column locations is an important design criterion in flat slabs. Where punching shear reinforcement has needed to be provided this has traditionally taken the form of a large number of individual shear links arranged on a series of perimeters from the edge of the column. However a range of proprietary 9 systems is now available which can greatly speed up the fixing process . Methods for the design of standard punching shear reinforcement are well documented in existing codes, and a companion spreadsheet RCC13.xls is available to assist in the design process when using BS8110 (Reference 5). This includes dealing with holes when present in the slab. A list of the currently available spreadsheets from the RCC is included as Annex 1 Where a sub-frame analysis or other form of elastic analysis has been used the effective shear force Veff can be calculated directly from the redistributed moments and shear forces. In some cases this is done automatically for example within the RCC spreadsheet RCC33.xls. Where yield line design has been used these moments may not have been calculated directly. In this case one approach is to use the load magnification factors given in Clause. 3.7.6 of BS 8110: Part 1 or other codes. The basic shear force Vt to which the magnification factor is applied may be calculated assuming load to be transferred equally between columns according to the area they support, except that in an end bay 55% of the load in a given direction may be assumed to be transferred to an internal column and the remaining 45% to the external column. In practice the designer has considerable freedom to exercise his own judgement in this area.
9
Concrete Society Technical Report: Shear Reinforcement Systems for Flat Plates (to be published). - 33 -
Approaches to the Design of Reinforced Concrete Flat Slabs
6. DESIGNING FOR SERVICEABILITY For thin flat slabs, as advocated in this guide, serviceability criteria are very likely to govern the design. In practice this may limit the advantage gained from using nonlinear or plastic analysis for the Ultimate Limit State. There is also some evidence from the work at Cardington that early striking of the slabs can result in greater longterm deflections, though the increase in total deflection is likely to be small. This increase in deflection should be negligible when considering deflections subsequent to application of the finishes. The design loads to be used when checking the serviceability limit states of cracking and 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 cases require additional tension reinforcement to be provided at mid-span, as was the case at Cardington. The basis for the provision of this additional reinforcement is independent of the design method used and is well documented in existing codes. However as has already been noted some caution should be exercised when considering the reinforcement provision resulting from yield line design. It is interesting to note that where the requirement for additional tension reinforcement was relaxed for one of the floors at Cardington, this does not appear to have had any undue effect on the deflection performance, questioning the need for the provision of this additional reinforcement. The critical parameter determining the level of deflection was the concrete tensile strength (related to the compressive strength) 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 meet client requirements or those of other design disciplines (e.g. cladding, partitions or services). In addition if deflections are governing the design more accurate prediction of deflections should permit design economies. The big disadvantage of trying to predict deflections is the number of unknowns at the design stage and the variability of the data and estimates. Caution should therefore be exercised in the reliance placed on the deflection predictions and a margin of up to 30% may be advisable to allow for the uncertainties. An example of the predictions given by different available methods is given in Reference 15. To predict deflections one approach is to model the slab as one-way spanning. This will by definition only give an approximation of the deflection as the two-way spanning action of the slab is not considered explicitly. However there are ways of combining the effects from the two orthogonal directions (Reference 15).
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Approaches to the Design of Reinforced Concrete Flat Slabs
Where a simple elastic analysis has been used, the moments determined for the Ultimate Limit State design can be used, but with the load factors removed. The deflections will generally be greatest at the centre of each panel. However, given that partitions are usually along column lines, it may be necessary to calculate deflections here also. One approach to determining the deflection at the centre of the panel is to calculate the deflections on two parallel column strips, using BS8110 Pt 2 or EC2, and adding the average of these to the deflection of the middle strip spanning at right angles to the 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:
δ mid
= δ EF +
δ AB
δ mid
= δ HI +
δ AD
+ δ DC 2
or
+ δ BC 2
Assuming a parabolic variation in curvature along the length of a slab strip the deflection at the centre of a slab strip is given by:
δ =
L2 96
(ψ + 10ψ + ψ ) left
mid
right
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Approaches to the Design of Reinforced Concrete Flat Slabs
where:
E
I L M
δ ψ ψ left ψ mid ψ ri ht
= = = = = = = = =
2
elastic modulus (N/mm ) 4 second moment of area at point being considered (mm ) length of span being considered (mm) moment at point being considered (Nm) deflection (mm) curvature = M / EI curvature at the left support curvature at mid-span curvature at the right support
This method requires the calculation of between six and nine curvatures depending on the symmetry of the bay considered. However, this is not as tedious as it may seem if a simple spreadsheet is used. Even where such an elastic analysis has been used the assumptions made will effect the ratio of support to mid-span moments and hence the deflections determined and this should be borne in mind. Linked to this is the fact that the moments may sometimes (e.g. when using simple coefficients) effectively have already been redistributed. Finite element analyses are particularly useful when there are irregular geometries and holes. They can also deal directly with the two-way spanning nature of the construction. Appropriate modelling of cracked section properties is important whichever method is used to predict deflections. In practice this will mean making initial assumptions about determining what areas of slab are cracked and to what extent. In the case of two-way spanning models there is scope for refining these assumptions on an iterative basis. As described in the section on Finite Element Analysis, there are programs that allow this process to be automated. Flat slab structures are in practice likely to receive their maximum loads during construction. Two conditions can be identified: 1. When the slab is first struck and required to carry its self weight plus an allowance for construction load. 2. When the slab is required to (partially) carry the weight of fresh slabs being cast above. These load conditions are illustrated in Figure 2. More detailed analyses of deflection, where considered justified, should if possible take account of these peaks in the load-time history. They should also consider when these are likely to occur in practice in relation to the strength gain of the concrete. Since the effective tensile strength of the concrete is the key parameter, a simplified approach is to derive this parameter f ctmodified :
- 36 -
Approaches to the Design of Reinforced Concrete Flat Slabs
f ctmodified = K min w p Where wp = quasi permanent load (as defined in Reference 9) and Kmin = minimum of (Kstrike, Kpeak and Kperm), K = fct /w where w = load, and fct = tensile splitting strength. Kstrike should be calculated with the self-weight of the slab plus the weight of the falsework. The construction load corresponding to Kpeak can be calculated following the recommendations given in Reference 4 and occurs when the slab is subjected 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 a reasonable 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 of Reference 9 is followed. Since the reinforcement has relatively little influence on the flexural properties of an uncracked section, the tensile strength may be used to calculate the cracking moment Mcr as follows: 2 Mcr = fctbh /6. Alternatively, more precisely, the method of transformed sections may be used. In the case of a rectangular section the following formulae apply: Depth to neutral axis,
d ' d ' 1 + 2(α e − 1) ρ h + ρ h h x = ' 2(1 + (α e − 1)( ρ + ρ )) Second moment of area,
h 3 h 2 + h − x + (α e − 1)d [ρ (d − x )2 + ρ ' ( x − d ')2 ] b I = 12 2 Cracking moment , M cr
=
fct I
y
=
fct I
h − x
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Approaches to the Design of Reinforced Concrete Flat Slabs
In the above equations:
e is the modular ratio Es /Ec s /bd for the tension reinforcement s'/bd for the compression reinforcement b is the width of the section (usually taken per m) h is the overall depth of the slab d' is the depth to the compression reinforcement d is the depth to the tension reinforcement The method of transformed sections must be used to calculate the flexural properties of the cracked section. In this case a common assumption is to ignore the influence of the concrete in tension completely and assume a triangular stress block for the concrete in compression. To avoid overestimating slab deflection, the peak construction load should not be overestimated and realistic concrete strengths should be used. Example: At floor 2 Cardington , Kpeak was critical (i.e. smallest value). Therefore f ctmodified = K peak wp = (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 the importance of having accurate data on early age tensile strengths as well as compressive strengths so that accurate long-term deflections may be predicted. In the absence of direct information on tensile strength it may be estimated from 0.67 fct =0.3 (0.8 x fc) where fc is the estimated actual concrete cube strength that will be achieved at each of the times to be considered. This equation is the standard equation relating tensile strength to cylinder strength given in the ENV version of EC2 and with the cylinder strength assumed to be 80% of the equivalent cube strength. Predictions of the 300-day deflections for floor 3 of the Cardington building using an elastic analysis with cracked section capabilities is given in Figure 12. These predictions were calculated using input values of f ctm appropriate to the construction overload stage, with quasi-permanent loading (equivalent to experimental applied loading). These results agree very well with the measured values.
- 38 -
Approaches to the Design of Reinforced Concrete Flat Slabs
Figure 12: Predictions of the 300 day deflections on floor 3 of the ECBP Cardington
- 39 -
Approaches to the Design of Reinforced Concrete Flat Slabs
6.2 Cracking Whichever method of design is used, adequate control of cracking will be achieved provided 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 follow similar principles to those for dealing with deflections (Reference 9). However the explicit calculation of crack widths and spacing is not usually warranted.
- 40 -
Approaches to the Design of Reinforced Concrete Flat Slabs
7. CONCLUSIONS AND RECOMMENDATIONS 1. There are many alternative approaches to the design of reinforced concrete flat slabs that are equally valid. 2. The choice of design method is very much one of personal preference. It should be based on what is appropriate for the structure to be designed, and on the designer's own previous experience as well as what is likely to most benefit the client. 3. This guide has given some pointers as to how existing design guidance and methods could be developed and made more user-friendly, particularly with the advent of EC2 in the next few years. The guide also flags up issues for the Permanent Works Designer to consider as a result of the desire to strike slabs earlier 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 because of their material behaviour assumptions and also due to the influence of in-plane effects, which result in membrane action. This potentially provides scope for further economies in the design of flat slab structures.
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Approaches to the Design of Reinforced Concrete Flat Slabs
8. ACKNOWLEDGEMENTS The author would like to acknowledge the assistance provided by a large number of individuals and organisations in putting this guide together. In particular he would like 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 which led to the production of this report. The particular project was entitled "Deriving Further Learning and Benefit from the in situ Concrete Building at Cardington".
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Approaches to the Design of Reinforced Concrete Flat Slabs
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