Guide to the design and construction of reinforced concrete flat slabs (1)

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Licensedcopy:wspgroup,WSPManagementServices,31/10/2010,UncontrolledCopy,®TheConcreteSociety

Acknowledgements
The work of preparingthis Reportwas funded byThe Concrete Centre.
The ConcreteSociety is grateful for the assistance of RodWebster (Concrete
Innovation and Design) and of lan Feltham and Andrew Fraser (Arup).
The ConcreteSociety is grateful to the following for providing photographsfor inclusion
in the Report:
Arup (Figures IA, 1B andA5.1)
The Concrete Centre (Figure 45)
Published byThe ConcreteSociety
CCIP-022
PublishedApril 2007
ISBN 1-904482-33-3
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Printed by Cromwell Press,Trowbridge, UK.

Guide to the Design and Construction
of Reinforced Concrete Flat Slabs
Contents
Membersof the ProjectSteeringGroup
List of tables vii
iv
List of figures V
1. Introduction 1
2. Issuesaffecting design and construction 3
2.1 General 3
2.2 Influenceof procurement on design 3
2.3 Choice of type of flat slab 5
2.4 Construction method assumed in design 7
3. Typical behaviour of a flat slab 8
3.1 Typical bendingfailure mode 8
3.2 Typical deflected shape of an interior panel 8
3.3 Moment contours 9
3.4 Flexuralbehaviour of a flat slab as the vertical load is increased 10
3.5 Sway frames 11
3.6 Slab at edge columns 11
3.7 Core and shear walls 12
3.71 General 12
3.7.2 Moment transfer from slab 12
3.7.3 Localeffects 12
3.8 Effect of edge beams 13
3.9 Effect of early striking of formwork 13

4. Design 14
4.1 Introduction and scope 14
4.2 Designprocedure 15
4.3 Cover to reinforcement (Clause 4.4 of Eurocode2 and the UK NationalAnnex) 16
4.4 DeDth of slab 17
4.5 Loading 19
4.5.1 Ultimate limit state 19
4.5.2 Serviceabilitylimit state 22
4.6 Methods of analysis 22
4.6.1 General 22
4.6.2 Hoggingmoments over the supports 22
4.6.3 Coefficient method 23
4.6.4 Equivalentframe method 26
4.6.5 Finiteelement method 29
4.6.6 Grillage method 38
4.7 Specific considerations 39
4.7.1 Columns 39
4.7.2 Voided slabs 43
45
4.7.4 Holes in flat slabs 46
4.7.3 Slabs with drops panels and/or column heads
4.7.5 Solar gain 47
4.7.6 Hvbrid construction 48
4.8 Punchingshear 48
4.83 General 48
4.8.2 Effective applied shear stress 48
4.8.3 Punchingshear resistance 51
4.9 Cracking 54
4.10 Deflection 54
4.10.1 General 54
4.10.2 Acceptance criteria 54
4.10.3 Pre-camber 55
4.10.4 Accuracy of results 55
4.10.5 Use of eauivalent frame method 55

5. Detailing 56
5.1 General 56
5.2 Edge beams 59
5.3 Column/slabjoints 59
5.4 Distribution reinforcement 60
5.5 Punchingshear reinforcement 61
5.6 Holes in slabs 62
6. Construction 63
6.1 General 63
6.2 Construction process 63
6.2.1 General 63
6.2.2 Construction loads 64
6.2.3 Column size 64
6.2.4 Cement types and concrete strength in practice 64
6.2.5 Reinforcementdetailing 64
6.3 Strikingof formwork 65
6.4 Pre-cambers 67
6.5 Proprietarypunchingshear systems 67
6.6 Edge beams 68
6.7 Concrete in the column/slabjoint 68
6.8 Column head/droDDanels 68
6.9 Quality of construction 68
7. References 70
Appendices:
Al. Idealcalculationsequence usingnon-linear plate designsoftware 72
A2. Knowyour software -software features 73
74A3. Preferredmethodsof construction:Views of two contractors
A4. Interpretation of grillage analysisresults
A5. ExampleusingFinite ElementAnalysis
78
82
iii

Membersof the Project SteeringGroup
Full members JohnMason
Owen Brooker
JohnClarke
JohnColding
Charles Coodchild
Christer lsgren
TonyJones
Suqlain Mahmood
John Morrison
Nary Narayanan
RobertVollum
BjornWatson
RodWebster
RobinWhittle
Correspondingmembers Alan Cilbertson
Adrian Long
Alan Baxter &Associates (Chairman)
The ConcreteCentre
The ConcreteSociety (Secretary)
WSP Croup
The ConcreteCentre
Byrne Brothers
Arup
Sir Robert McAlpine Ltd
Buro Happold
Clark Smith Partnership
Imperial College
Antony Hunt Associates
Concrete Innovation and Design
Consultant (LeadAuthor)
ClRlA
Queen’s University, Belfast
iv

List of figures
Figure1
Figure2
Figure3
Figure4
Figure5
Figure6
Figure7
Figure 8
Figure9
Figure10
Figure11
Figure12
Figure13
Figure14
Figure15
Figure16
Figure17
Figure18
Figure19
Figure20
Figure21
Figure22
Figure23
Figure24
Figure25
Figure26
Figure27
Figure28
Figure29
Figure30
Figure31
Figure32
Figure33
Figure34
Figure35
Figure36
Figure37
Figure38
Figure39
Figure40
Figure41
HollidayWharf apartments, Birmingham.
(a) Under construction
(b) Completedbuilding '
Typicalforms of flat slabs.
Typical mechanismfor flexuralfailureof a flat slab.
Typicaldeflectedshape of an interiorflat slab panel.
Typicaldistribution of bendingstressfor a flat slab.
Typical load/deflectionbehaviourof flat slab.
Typicalyield-line pattern at edge column.
Warpingof flat slab alonga free edge.
Concentrationin slab stresses at core walls.
Flat slab thickness (solidwith flat soffit) for given imposed loads (IL).
Spadeffectivedepth ratiosfor a flat slab (K= 1.2).
Loadingon alternatestrips (two combinationsin each direction).
Effectivewidth of solid slabwith a concentratedload near an unsupported
edge.
Reduction in maximum hoggingmoment at columns.
Layoutof column and middle strips.
Lateraldistribution of momentsdependingon panelaspect ratio.
Effective width, be,for moment transfer at edge and corner columns.
Yield-line mechanism at edge column.
Behaviour at edge column under sway conditions.
Behaviourof two-bay slabs.
Plate or shell element moment output.
Typical arrangementsof elements.
Arrangements of elements from a meshgenerator.
Exampleof equivalentdepths to simulate stiffness.
Plate elementtypes.
Column head regions.
Modellingcolumn stiffness for fixed and pinnedsituations.
Section at column.
Approximatemodelsfor largecolumns.
Moment adjustmentfor large columns.
Confinementforces at column/slabjoint.
Typical cofferedslab arrangement.
Effectivedimensions of column head.
Possibleshear failure planes.
Effect of holes in flat slabs.
Relationshipbetweenpredictedtemperature differencedue to solargain and
slab thickness for differentsurfacingtypes for a specific location in the UK.
Typical basiccontrol perimetersaround a column.
Simplified methodfor determiningthe value of p.
Effect of applied momentson shear at internalcolumns.
Basic control perimeter,U,,for edge and corner columnsof flat slabs.
Reduced control perimeter,U,*,to take accountof the effectsof moment
transfer.
V

Figure42
Figure43
Figure44
Figure45
Figure46
Figure47
Figure48
Figure49
Figure50
Figure51
Figure52
FigureA4.1
FigureA4.2
FigureA4.3
FigureA4.4
FigureA4.5
FigureA4.6
FigureA4.7
I FigureA4.8
Outer limit for placingreinforcement.
Limitation on outer perimeterfor irregular layoutsof shear reinforcement.
Effectof pre-camber on deflections.
Flat slab at St JamesUniversityHospitalpriorto concreting.
Averagingof bending momentsover flat internalsupport.
Arrangement of 'U' bar reinforcementalongslab edge.
Required link shape for torsion.
Column/slab joint showingconfinement reinforcementin slab.
Arrangement of punchingshear reinforcement.
Reinforcementdetails at holes in slabs.
Indicationof final deflectionrelatedto time of striking backdrops.
Bending moments.
Compatibilityof torsionalstresses.
In-planestresses on section.
Averagingof moments.
Grillage data and results.
Applied momentsto member per unit width.
Average applied moment per unit width.
Equivalentapplied design moments per unit width.
FigureA51
FigureA5.2
FigureA5.3
FigureA5.4
FigureA5.5
FigureA5.6
FigureA5.7
FigureA5.8
FigureA5.9
FigureA510
FigureA511
FigureA512
FigureA513
FigureA514
FigureA515
FigureA516
FigureA517
FigureA518
FigureA5.19
FigureA5.20
FigureA5.21
FigureA5.22
FigureA5.23
FigureA5.24
FigureA5.25
FigureA5.26
Modelof flat slab project.
Planof 1stfloor.
Partof plan consideredin detail.
Required bottom cover.
Equivalentframe moment diagram alonggrid lineC for 300mm thick slab.
Equivalentframe moment diagram alonggrid line 2 for 300mm thick slab.
Mesh layoutfor selected area.
Moment contours.
ColumnC1 and C2 transfer moments.
Maximum hoggingmoment at face of column C2-1.
Maximum hogging moments inthe middlestrip 2C-D.
Maximumsaggingmoments in span CD12.
Required bottom reinforcementinthe y-direction for panelCD12.
Bendingmomentson lineC1-2.
Locationsof momentssummarised inTable A5.2.
Layout of designflexuralreinforcementfor 300mm deep slab.
Layoutof designflexuralreinforcementfor 250mm deep slab.
Layoutof punchingshear linksfor internalcolumn (10mmdia.).
Layout of punchingshear links for edge column (10mmdia.).
DeflectioncontoursfromType 1analysis.
Moment contours usingconcrete uncrackedsectionproperties.
Moment/Stiffnessdiagrams for typical saggingand hoggingregions.
Moment coefficientsto modify section propertiesfor first iterationof
analysis.
DeflectioncontoursfromType 2 analysiswith the quasi-permanentload.
Deflectioncontours usingType 2 analysisfor the frequent load.
Deflectioncontours usingType 2 analysisfor self-weight.
vi

List of tables
Table 1
Table 2
Table 3
Table 4
Table 5
Table 6
Table 7
Table A51
Table A5.2
Minimum slabthickness and axial distancesfor flat slabs.
Bendingmoment and shear force coefficientsfor flat slab panelsof three
or more approximatelyequal spans.
Distributionof design momentsfor solidflat slabs with flat soffits.
Valuesof k to determinetorsionalconstant.
Finiteelementdesignwatchpoints.
Valuesof J for usewith grillage analysis.
Recommendedpitchfor distribution bars (mm).
Initial punchingshear check summary.
Comparisonof design momentsfrom analyses.
vii

1. Introduction
The purposeof this Report is to provide informationand current best practiceon the
design and constructionof reinforcedconcrete flat slabs in accordancewith Eurocode 2
(BS EN 1992(’))and the National Structural ConcreteSpecification(2).The Report is intended
for use by designers (bothof the permanent andtemporaryworks) who already have a
basic understandingof structuralbehaviour (e.g.Reinforcedandprestressed concrete by
Kongand Evand3)).
Figurela
HdlidayWh.rf.p.rtments, Birmingham-
underconstruction.
Figure1b
HdlidayWharf apartments, Birmingham-
completedbuilding.
7
f

Flat slabs design is appropriatefor most floor situations and is also suitablefor irregular
column layouts,curvedfloor shapes, ramps, etc.The benefitsof choosingflat slabs
includethe following:
0 A minimum depth solution leadsto least claddingand has advantages in planning
0 Speed of construction
0 Flexibility in the plan layout, both in terms of the shape and column layout
CI Flat soffit, allowing clean finishes or freedom of layout of services
0 The use of economic large spans (see Section2.3)
0 Scope for cutting holes, alterations and repairs (see Section 4.7.4)
0 Scope for much repetition
a Reduction in the need for drop beams (and up-stand beams)
0 Scope and space for the use of flying forms
0 Good for sound attenuation.
height
The flexibility of flat slab construction can lead to high economy andyet allow the
architect great freedom of form.
The scope of this Report includes flat slabs with orthogonal and irregular layoutsof
columns, plain soffits,waffle slabs,voided slabs and slabs with edge beams. It should be
noted that solid flat slabs with a flat soffit are the most popular.This Report does not
cover the design of prestressedpost-tensionedslabs. For such slabs, reference should be
made to the ConcreteSocietyTechnical Report No.43(4).

2. Issues affectinrr desirrn and constructionU U
2.1 General There are a numberof factors that influencethe choice of design and construction of a
structure, including:
The type of structure
0 The client’s specific requirements
BuildingControl requirements
0 Localplanning rules
0 The ground and site conditions
The architect’sspecification
0 The structural engineer’s constraints
The contractor‘s preferred methodsof construction.
2.2 lnf~UenCeOf The chosen form of flat slab can depend on the form of contract, dependingon whether
the design is architect-led or contractor-led.A ‘traditional’ form of contract is where the
design follows the architect’s and engineer’s interpretation of the client’s brief and is taken
to tender documentswith reinforcement estimates.A ‘Design and build’ form of contract
is where the contractor is responsiblefor both the design and construction.There are many
different forms of contract (e.g.two-stage tender process) that lie somewhere between
these extremes, eachwith different amountsof influenceon the design and build approach.
procurement on design
The following points,which affect both the design and construction,should be considered
at an early stage in the procurement process:
0 The appropriatemethod of design should relyon the balance between the costs of
materialand labour,which are continuallychanging. In addition, the cost of programme
time is different on every project.Therefore, there is no universallycorrect approachto
design. Designsthat have heavily rationalisedreinforcement,incorporate proprietary
systems for shear design, and are designedto allow early removalof formwork are
likelyto be the most labour efficient and fastest to build.Those with full curtailment,
loose shear links and are designed assuming longer striking times are likelyto be the
most efficient in terms of materials(seealso Rationalisationofflat slabreinforcernent(5)).
Different construction approaches will be adopted dependingon the form of building,
e.g. high or low rise.
U The effects of designer-constructor communication to minimisethe formwork costs
of beams,walls, columns, etc.
The appropriateform of construction will depend on what it is intendedfor. If the
building is for low cost housingto a tight budget, the choice of construction method is
paramount. If it is a landmark building, part of the attraction of concreteflat slabs is
the ability to dealwith irregular bays,stiffening critical panelswith beams.
It should be noted that the cost of the structural frame is about 10%of the cost of a
multi-storey building and is likely to be small in comparisonwith the cost of the
cladding,which may affect the choice of frame.
3

0 Indecidingthe appropriatedesign approach, the designer needsto consider the overall
sustainabilityof energy for which the minimum use of materials is only one considera-
tion.The type of contract and the effect of the programmeon costs should be consi-
dered in addition to the balance betweenmaterial and labour costs. However, when
producinga design for tender it is recommendedthat the designer should generally err
on the side of minimum material cost with due regard to future changes.This gives the
firm tenderingthe maximumopportunity to developtheir own projectspecific approach;
it also protects the client in that additional material costs to improve construction
efficiencyare easily highlighted.When the designer is working on a project where the
concrete frame contractor is alreadyappointed, it is most efficient generally if the
designer can incorporatethe contractor’s preferredconstruction approach,although
certain design assumptionssuch as slab thicknessmay be fixed.This may, for example,
include the contractor’s preferredproprietaryshear reinforcementsystem. However,in
such a situation, it is essentialto have a clear understandingas to how additional
material costs are covered in the contract.The designer may be working directly for
the frame contractor, in which case the opportunity for tailoring the full design to suit
the construction method is greatest.
0 One part of the sustainabilityargumentwould also tend to pushthe form of construction
towards minimum material use, and may see the more frequent use of voided, coffered
or ribbedslabs to reduce material use.These forms of constructionwould also enhance
the useablethermal mass of the floor and reducethe running costs in use. However,
they would incur greater costs in construction as they are less easy to construct.
0 The use of post-tensionedflat slabs in the UK has become popular in recentyears.
These can be more economicfor longer spans. However, it should be noted that the
use of high strength concreteallows reinforcedconcreteflat slabs to be economic up
to 12m (seeConcreteSocietyTechnical Report 59(6)).
0 The specificationfor the claddingsystemcan often imposeserviceability restrictionson
the design of flat slabs,where the range of slab edge deflection must be limited.The
claddingcontractor may expect unreasonablyhigh levels of accuracy when predicting
deflections.Accurate estimation of deflections requires an accurate knowledgeof the
material properties,which may not be availableat the design stage. Stiff edge beams
or the addition of an extra edge column may be a suitable method of controlling
deflections.
0 Hybrid systems usinga combination of steel/precast concretecolumns, precast edges
or edge beams have been successfully developed.The use of precast slabs for the soffit
avoids the use of removableformwork.
0 Other subjects that should be consideredcarefully at the design stage are the acoustic
(see Concreteandso~ndinsu/ation(~)),vibration requirements (see the ConcreteSociety
Technical ReportTR43(4))and M&E services.
0 The design must result in a robust structure with suitable structural details that do not
compromisethe robustness.
4

2.3 Choice Of type Of flat
slab
Figure2 showstypical forms of flat slabs. Proprietary systems may include other accep-
table forms (e.g.voided slabs). Referenceshould be madeto Economic concreteframe
elements(*)for preliminarysizing.
Figure2
Typicalforms of flat slabs.
I' Solidflat slab Solidflat slab with drop panel Solid flat slab with column head
Coffered flat slab Coffered flat slab with
solid panels
Banded coffered flat slab
It should be notedthat, for flat slabs 2OOmmthick and over,the needto include punching
shear reinforcementis common. For slabs less than ZOOmm,thick shear reinforcementis
ineffective.
Flat slabwith flat soffit is economicalfor spansfrom 5mto 9m for office buildings,
hospitals, hotels, blocksof flats, etc usingC30/37 concrete. Longerspans may be
achieved using higher strengthconcrete (e.g.span of 10mto 12mwith C50/60 concrete)
(see ConcreteSocietyTechnical Report 59(6)).
Advantages:
0 Simple and fast construction
0 Absence of beams allows lowerstorey heights
0 Flexibilityof partition locationand horizontalservicedistribution
Architecturalfinish can be applied directly to the undersideof the slab.
Disadvantages:
0 Shear provisionaround columns may needto be resolved usinglarger columns,
Deflections,especially of edges supportingcladding, may cause problemsthat may be
column heads,drop panels,shear linksor proprietaryshear systems
resolved by includingmore edge columnsor edge beams.
5

Note: Itshould be notedthat all the following types of flat slab are likelyto be more
expensiveto buildthan a flat slab with a flat soffit as they addtime to construction.
Flat slab with dropsare economicalfor more heavily-loadedspans from 5m to 10m for
office buildings,hospitals,hotels, etc.
Advantages:
Reduction in the clear span leadingto a reduction in reinforcement required
U Increase in shear capacityand stiffnessof the slab
Absence of beams allows lower storey heights
Flexibilityof partition location and horizontalservice distribution.
Disadvantages:
Holes near column difficult to include
Drops cause disruption to formwork and extra cost
Flat slab with columnheadsare economicalfor more heavily-loadedspans from 6m
to 10m for office buildings, retaildevelopments, hospitals,hotels,etc.
Advantages:
Increase in shear capacity
U Absence of beams allows lower storey heights
U Flexibilityof partition location and horizontal servicedistribution.
Disadvantages:
Holes near column difficult to include
Column heads may cause extra cost to column formwork.
Flat slab with edge beamsare economicalfor spans up to 10m for office buildings,
retaildevelopments, hospitals, hotels, etc.
Advantages:
Provides stiff edge for support of cladding
U Absence of internal beams allows lower storey heights
Flexibilityof partition location and horizontal service distribution.
Disadvantage:
Edge beam may cause disruption to formwork (e.g.use of table forms)
Coffered (waffle) slabs are economicalfor spans up to 12m
Advantages:
Reductionin self-weight
Profile may be expressedarchitecturally.
Disadvantages:
Higher formwork costs
More difficult to fabricate reinforcement
More difficult to build partitions to fit unlessthey are positionedon the line of ribs
Provides lower sound insulation.
6

2.4 COnStrUCtiOn method The construction should be carried out in accordance with the National Structural
assumed in design Concrete Specification(2).
It should be noted that the performanceof a flat slab, particularly one designedfor rela-
tively low imposedloading, may be affected significantly (e.g.crackingand deflection) by
the construction method (seeChapter 6). In order to achieve consistencybetweendesign
and construction of structures, it is important for the designer to includeas part of the
project specificationa method statement indicating the assumptionsregardingconstruc-
tion.This will bringclarity to the project and set a benchmarkfor pricing.Of course,the
contractor is free to submit an alternative price based on any different assumptions(e.g.
based on early striking of formwork and proppingthe shuttering for the upperfloors off
the newly constructedfloors) to the original design. In this process,the performance
criteria agreed with the client should not be compromised.
Generally,the design assumptionsshould include the following information:
Sequenceof construction
0 Formworkstrikingtimes and back-proppingrequirements
U Pour sizes assumed
Cementtype in the concrete, 28-day strength and the strengthof concrete assumed
Breakdown of loading, including allowancefor construction loads
0 Loadinghistory assumed.
at striking
7

3JYP cal behaviour of a flat slab
3.1Typical bendingfailure
mode
A flat slab spans between columnsupports without the needfor beams.For a regular
layout of columns, failure can occur by the formation of hinge lines alongthe linesof
maximum hoggingand saggingmoments.This can be most easily presented usingthe
folded platetheory as shown in Figure3.A complementaryset of yield linescan form in
the orthogonaldirection.
0 0
4 - - : Saggingyield lines
0 0
0 0 0 0-I- - - Hoggingyield lines
m -
OI Y -
m -
-Column supports
ri 0 ri 0
One misconceptionof some engineersis to consider a reduced loadingwhen analysingin
a particulardirection.The momentsappliedineachorthogonaldirectionmusteachsustain
the totalloadingto maintainequilibrium.Thereisnosharingof the loadbypartialresistance
ineach orthogonaldirection.
3.2Typicaldeflected shape
of an interior panel
The deflectedshape of an interior panelof a flat slab on a regulargrid of columnsunder
typical in-serviceconditions is a function of the sum of the deflectionsin each orthogonal
directionas shown in Figure4. Similar deflectedshapeswill obtain from an irregular grid
of columns, but the interactionbetween adjacent bays may be more complex (see also
Section 4.10).
8

Figure4
Typicaldeflectedshapeof aninteriorflat slab
panel.
3.3 Momentcontours The useof finite element methodsshowsthat the distributionof bendingmoments per
unit width is characterisedby hoggingmomentsthat are sharply peaked inthe imme-
diatevicinityof the columns.The magnitude of the hoggingmoments locally to the
columnface can be severaltimes that of the saggingmoments inthe mid-spanzones.
These momentsdo occur in practiceandthe designshouldtake them into account.
Redistributionallows a more uniformspread of reinforcement but increasesthe likeli-
hood of cracking.
A typicaldistributionof bendingstresses for a uniformlydistributedloadon a flat slab
with a regular layoutof columns is illustratedin Figure 5.
Figure5
Typicaldistrlbutionof bendingrtrurfor aflat
dab.
9

3.4 Flexural behaviourof a
flat slab asthe vertical load
is increased
Figure6
Typical load/d.floction bahrviourofflat slab.
A typicalload/deflectioncurveof a flat slab is shown in Figure 6.
As thevertical loadonthe slab increases,the followingchangesoccur:
rn Moments at the support and mid-span increaseelastically untilthe first cracks occur.
These are likelyto appear first at thetop of the slab closeto the columnand mayoccur
duringconstruction if the removalof formwork takes placeearly.Otherwise,this limited
crackingmayoccur underthe quasi-permanent combinationof actions (see BS EN
1990(g)andthe UK NationalAnnex).
rn As the loadingis increasedbeyondthe characteristiccombinationof actions, cracking
may increaseto some way intothe span from the column, and cracks mayalso have
started to appear at mid-span.This is unlikely underthe frequent or quasi-permanent
combinationof actions (see BS EN 1990and the UK NationalAnnex) unless caused by
other effects (e.g. temperature or shrinkage).The crackingincreasesthe non-linear
behaviourof the slab, although it still behaveselasticallyas the load increasesbetween
the formationof new cracks, and can be modelled elastically takingaccount of the
tension stiffeningof the concrete.
rn As the loadingisfurther increased, the reinforcementfirst starts to yield inthe top bars
closeto the columnsandthejunction of the slab at edge columnsstarts to behave as
a plastic hinge.Apart from this, the slab still behaveselastically as the load increases
betweenthe formationof new cracks but with reducingtension stiffening.
rn Failurewill occur once a failure mechanismis reached(e.g.as shown in Figure 3)
A linearelastic modelgives satisfactory resultsfor the flexural behaviour (moments not
deflections) upto the stage where the number of cracks has reachedits maximum,
providedadjustment is madefor the plastic rotationof the slab at the edge columns and
yieldingof the bars localto the internalcolumns (seeChapter 4). It is reasonableto use
this modelto representthe ultimate limit state (ULS).
Maximumnumber
of cracks
Formationof
betweenformationof new cracks
(modelledbytensionstiffening)
cracking
Deflection -
10

Typical behaviour of a flat slab
3.5 Swayframes The designof a flat slab as a sway frame reliesonthe moment capacityof the column-
slabjoints for stability,for which specialcare is required both inthe designanddetailing.
Generally,connections betweencolumnsand aflat slabare unsuitedto resistinglarge
bendingmoments, especiallyat edge columns (seeSection 3.6). Even if the moment
capacity of internalcolumns is sufficient,the maximumpunchingshear capacity may be
exceeded becauseof the effectof momenttransfer. Hence,wherever possible, horizontal
loadingshould be resistedby shear or corewalls makingthe structure a bracedsystem.
However, manysway-frame flat slabs have performed satisfactorily in service.
Where a sway frame is beingdesigned, it is importantto considerthe combinationof
loadingfor equilibrium (EQU) (seeTableA.l.2 (A) of BS EN 1990(g)).
If it is not possibleto incorporatea bracingstructure,the followingpointsshouldbe noted:
Edgecolumns havevery limited momenttransfer capacity.
Internalcolumns/slab joints provide mostof the momenttransfer, which reducesthe
Holesinthe slab closeto a columncan reducethe momenttransfer capacity
shear capacity of the adjacent slab.
dramatically.
In additionto the normalstructural(STR) load cases those for equilibrium (EQU) should
be checked.The loadfactorsfor equilibriumaregiven inTable A.1.2 (A) of BS EN1990.
The contributionto momenttransfer by torsionof the slab at the sidesof the column is
normallysmall comparedwith direct momenttransfer.
3.6Slab at edge CO~UmnS Flexuralandtorsionalcrackingof the slabcloseto the faces of an edge columnreduce
the transfer moment capacity. Cracksform early on, sometimesbeforethe working load
is reached, and plastic rotationtakes placewith increaseof load. Figure 7 shows a typical
yield-line pattern.
Below Figure7
Typicalyield-line pattornat edgecolumn.
Right Figure8
Warpingofflat slabalongafmedge.
The actualbehaviour alonga free edge of a flat slab is complicatedby the warping of the
slabas shown in Figure 8.
Centre of rotation j
at mid-span
Profile of top of slab
at column line,
Slab at mid-span
shown in seaion
Dejlectionamting claddingis reduced
by thetorsionalrotationat mid-span.
Tovisualise.securelysupporta sheetofpaper
just withineachofitsfour corners.Any load
appliednearthecentrewillinducean upward
depeccionat theedger.
Deflection
affecting claddin8
Rdualondw
toadgatonion
11

?-
Typical behaviourofa fLaA slab
3.7 Core and shear walls
3.71 General Provided that the flat slab is sufficiently stiff to distribute in-planeforces to core walls or
shear walls,they should be designedto take the imposed lateralforces (e.g.from wind
loads).They should be arranged such as to avoid excessive twisting and warpingof the
structure.
3.7.2 Moment transfer from
slab
With multi-storey buildings, it is increasinglycommon to construct concretecores in
advance, usingslip-forming or jump-forming techniques.The connection to the slab is
achieved by means of bent-out bars cast into the corewalls. Proprietarysystems such as
‘continuity strips’ are used,which are typically limited toT16 bars at 150mm centres.
This can limit the amount of moment that can be transferred from the slabs to the core
walls. It is recommended,therefore, that these connectionsare initially modelled as pinned
with regard to the design of the slab.This also simplifiesthe design of the walls,which
often have to be designed (and sometimes constructed)beforethe slab design is complete.
In such a situation, the walls should be designedto take the maximum moment of the
slab sectionthat can be generatedwith the chosen set of bent-out bars. It should be
noted that this approach is likelyto lead to minor crackingof the slab as a result of
redistribution of the elastic forces.
Care should also be taken where deeper transfer slabs are supported by core walls.Where
fixity has been assumedfor the slab, it is important to check the capacityof the core wall.
In heavily-reinforcedslabs the detailing requirementto anchor 40% of the bottom steel
into the wall can exceedthe capacity of proprietary bent-out bars.Other solutions are
then required,such as leavingpocketswithin the walls for reinforcementto be fitted later.
3.7.3 Local effects Stress concentrationscan occur in a flat slab at core walls, particularly at the ends and in
the regions ‘A’ and ‘B’ shown in Figure9. It may be necessary to concentratethe reinforce-
ment in these areas to control cracking.The variation of shear around the walls is far from
uniform, and at point ‘B’ uplift mayoccur (i.e.reaction is reversed), leadingto increased
shear in adjacent partsof the slab alongthe wall. Although very unlikelyto lead to failure
of the slab, crackingis likelyto occur in these areas. Normally, this crackingis controlled
by placingsmall diameter bars at close centres.
To reduce/avoidthis crackingthe following actions may be taken:
0 Leave out the local area of concrete during the initialconstruction and complete this
0 Provide a minimum reinforcementof 0.25% of the concrete cross section, in each
0 Provide nominal shear links in the area of concentratedstressto increaseductility.
0 Design specificallyfor the high stresses (e.g.with the use of finite element programs).
at a later stage.
direction, top and bottom.
12

Figure9
Concentrationinslab stressesat corewalls.
P-
I
3.8 Effect Of edge beams The effect of an edge beamon the behaviourof a flat slab isto change the moment
transfer mechanismand results in a reductioninthe deflectionof the edge panel.The
momenttransfer mechanismmay change inthe followingways:
rn Ifthe edge beam has hightorsionaland lowflexuralstiffness (e.g.square in section), it
transmitsmomentto the columnthroughtorsion.
rn If the edge beam has high flexural and low torsionalstiffness (e.g.thin and deep), it
attracts load from the slab and transmitsit to the columnthrough directshear and
flexure with littletorsion. If such a beam is assumedto have zero torsionalstiffness at
the ultimate limit state, the design may assume that the load passes from the slab to
the edge beamand then to the column, i.e. the full shear is taken on the column
throughthe edge beam. Inthis situation,thevalue of the momenttransfer should be
based on the slab and columngeometries ignoringthe edge beam.Torsionalcracking
of such beams may occur underworking conditionsand should be considered.
In both cases, the momentdirectlytransferredfrom the slabto the column is reduced.
13
3.9 Effect Of early striking
of formwork
The time of removalof formworkand propscan affectthe finaldeflection.Theslab may
be subjectedto loadingat an early age, which causes crackingwhen the concrete has not
reachedits full strength.This can causethe deflectionunderconstructionloadsto be cri-
ticalto the design and subsequent behaviour.This is discussed in moredetailin Section6.3.

4. Design
4.1 introduction and scope The structural function of a flat slab is to support vertical loads with a suitablefactor of
safety and transfer these into the supporting columns and shear walls.Transverse loads
may be resisted either by their transfer through the slab to shear walls (or core structures),
or by frame action between the slab and columns (and/or shear walls).
The design approach,adopted in this Report is in accordance with Eurocode 2, unless
otherwisestated, and the appropriateclauses, figures, tables and expressionsfrom the code
are indicated. It gives guidancefor the design of flat slabs typically with spans up to 9m.
For such spans, a concrete strengthof C30/37 is commonly used but the Report is appli-
cable to other strengths. For longer spans, it may be beneficialto increasethe concrete
strength in order to reducethe slab depth (subjectto cost and supply conditions). Fire
rating of up to 2 hours is assumed.
In general,to ensure that the appearance and general utility is not impaired,the deflection
should be limited to span/250 when subjectedto quasi-permanent loads. Inorder to limit
damage to adjacent partsof the structure,the deflection after construction should be
limitedto span/500 when subjectedto quasi-permanentloads. Other limitations may be
requiredfor a specific purpose(e.g.where slab edge deflection is limited to ensure no
damage to cladding).
It is assumedthat the average crack width limitationfor serviceabilityconditions is
generally 0.3mm. Normally, this will be achieved by conforming to the recommendations
given in Standardmethodof detailingstructuralconcrete(lO).
Where de-icingsalts are likelyto be present (e.g.in car parks),the limitation to crack width
should be reduced to Olmm in addition to other methodsof protection (e.g.use of stain-
less steel reinforcement, additional protective layer,etc). See also Designrecommendations
for multi-storey andundergroundcarparks(ll).
Suitable limits to the spadeffective depth ratiosare providedin Clause 7.4.2 of Eurocode 2,
and explained in Section 4.4 of this Report.
It should be noted that the use of short cantilevers (e.g.LIS) at the edge of flat slabs can
providea very economic structurewhere the span/depth ratio based on that of an internal
bay may be usedfor an external bay.
There are a numberof different methodsfor the design of flat slabs.These include:
0 Simplified moment coefficients (based on tests, experience and yield-line methods).
This method is suitable for regular layoutsof columnswhere the spans are constant.
0 Equivalentframe.This method is suitable for regular layoutsof columns, but requires
engineeringjudgement for irregular layouts (see Section 4.6.4).
14

0 Finite element analysis.This method allows the design of irregular column layoutsand
can provide the design of reinforcementdetails.Where the appropriatesoftware is
available, it is possibleto obtain reasonableassessment of deflections(see also Concrete
SocietyTechnical Report 58(12)).Moreover,it is possibleto model in-plane effects such
as those caused by shrinkage and temperature.
0 Grillage analysis.This method has similar facilities to finite element models and can
also be usedfor irregular layoutsof columns.
0 Yield-line methods.These can providesuitabledesigns for ULS but do not give adequate
information for serviceabilitydesign. For further information, see Practicalyieldline
design(13).
4.2 Design procedure The following list outlines the points to be considered during the design process.
0 Consider buildability throughout the design process.
0 Consider the effects of solar radiation on flat slabs exposed to direct sunshine and the
Calculatethe required cover for bond of the reinforcement,durability and for fire
0 Assess requireddepth of slab from simplified spaddepth charts and spadeffective
Check for punchingshear and consider the effect on this, of a 2OOmm square hole
0 Consider effect of drops and heads on shear, bearingin mind the cost (see Section 6.8).
Calculatetypical top reinforcement required at internal column support, and check
possible congestionof the reinforcement.Waffle slabs require a special check for
lappingof top mesh (e.g.three layers).
0 Calculatetypical mid-span reinforcement,and check if hoggingcould occur at mid-
span.
Check moments and shears at a typical edge and corner column. Ensurethat the slab
is capable of transferringthe required moment.
0 If a waffle slab is considered:
0 Ensuresufficient solid section adjacent to columns. It should extendto at least 2.5
o Check sufficient depth of topping and thickness of ribs to ensure compliancewith
choice of surfacingto control temperature differentials.
resistance (see Section 4.3).
depth limitations (see Section 4.4).
close to column.
times the slab effective depth from each column face.
durability and fire resistance requirements.Depth of topping may be influencedby
detailing requirements.
Check effects of large columns (> 500mm) and rigid corners, such as core walls.
Check likely position and effect of holes. Make allowance for this in shear assessment.
0 Carry out detailed design (includingcalculations,drawings and reinforcementschedules).

4.3 Coverto reinforcement The required nominalcover should be specified by the designer.
(Clause 4.4 of Eurocode2
and the UK National
Coverto reinforcementis requiredto ensure:
0 The safe transmission of bond forces;the minimum cover should not be less than the
Annex) bar size
The protectionof the steel against corrosion (see Eurocode2 andTables NA.2 and NA.3
0 Adequate fire resistance (BS EN 1992-1-2(’)refersto ‘axis distance’ for cover,which is
in the UK NationalAnnex)
the distancefrom the centre of the reinforcingbar to the surfaceof concrete).
The importanceof achievingcover cannot be overstressedbecausethe durabilityof the
structure is often determined by this.
It should be notedthat specialcare is requiredto ensurethat adequatecover is specified
where drainage channelsare usedwith ‘falls’which run alongthe surface of the slab. In
addition,where the surfacefinish affects the cover this should be statedon the drawings
The followingruleswill normallyprovidea satisfactoryspecification usingC30/37 concrete:
Internal situations
The nominalcover to reinforcementfor internaluse (no risk of corrosion or attack, XO)
should not be less than (15mmor bar diameter)+Ac,,,,.
Externalsituations
The nominalcover to reinforcementfor externaluse (corrosioninduced by carbonation,
XC3) should not be less than 35mm +Acdev.
Car parks
The nominalcover to reinforcementfor car parks (corrosioninduced by chlorides,XD3)
should not normallybe less than 50mm +Acdev.This cover may be reduced if suitable
changesare madeto the concrete grade (see Eurocode2 andTables NA.2 and NA.3 in
the UK NationalAnnex.
Deviation, Dcdev
Normally,the allowance made in designfor deviation, Ac,,,,, should betaken as 10mm.This
may be reducedto 5mmwhere it is specifiedthat only a contractorwith a recognised
quality system for the inspectionof reinforcementshall do the work (e.g.a member of
5peCC, the Specialist ConcreteContractors Certflcation Scheme).
Fire resistance
Table 1gives the minimum dimensions and axis distances, a, of the reinforcementinthe
lower layer for flat slabs (takenfromTable 5.9 of Eurocode2, Part 1.2).
16

Design
Table 1 ire resistance(minutes) M-inimumdimensions [mm)
Minimumslabthicknessand axisdistancesfor
flat slabs. Slabthickness
3
EP(0t.
Nomrally,the Mver
requiredwubeEdntmlled
bydurabilityrqutnments II
-Fire resistancewith highstrengthconcrete
For concretestrengthsC55/67 and C60/75, an increaseinthe minimum slab depthof
O.lais required. For concretestrengths C70/85 and C80/95, an increaseinthe minimum
slabdepthof 0.30 is required.This is explained in moredetail in Section6 of Part 1.2 of
Eurocode2 andthe UK NationalAnnex.
Inorder to avoid ‘explosive‘ spallingfor concretegrades C55/67 to C 80/95, the content
of silica fume should not be greater than 6% byweight of cement.
Forconcretegrades greaterthat C80/95, referenceshouldbe madeto Section6of Part 1.2
of Eurocode2 andthe UK NationalAnnex.
Specialattentionto providingsufficientcoverfor columnsandwalls should be given
where high strength concrete is used.
4.4 Depth of slab The informationgiven in Figures 10aand 10bwill assistthe designerto make a preliminary
choice of depthfor aflat slab (solidslabwith flat soffit).
2 0 0 y
150 1 I I I 1 I I
I
4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0
Sp.n (4
so0
450
- 400
E
E- 350
5n
4 300
n
z 250
200
150 I I ! I
1 I I I
4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0
Span (m)
...........IL=2.5 kN/mZ ..- IL I5.0 kN/mZ
---I1=7.5 kN/mZ -IL=10.0kN/mZ
...........IL = 2.5 kN/m2 .--lL = 5.0 kN/m2
IL = 10.0kN/m2IL = 7.5 kN/m2 -
a) ForconcreteC30137 b) ForconcreteC50160
Figure10
Fbtslabthkknoss(solidwithflat roffit)for
givenimposedloads(11).
17

Figure11
Spanleffectivedepthratiosfor aflat slab
(K=1.2).
Inorderto establishthat punchingshear reinforcement will not be excessive, the initial
check may assume vRd,c= 0.7MPa for f,, = 30MPa (add 0.1 MPafor each increase of
10MPa in fck)
where:
vRd,c
fck = characteristic (cylinder) strength of the concrete.
= designvalue of the punchingshear resistanceof the slabwithout shear
reinforcement
The limitingspadeffectivedepth ratiosfor flat slabs with flat soffitsand a regular layout
of columns are given in Figure 11.Thisis basedon Expression7.16 of Eurocode 2 with a
value of K= 1.2 (appropriatefor flat slabs) and a practicallimit of 48 imposed.
Where the greater effective span, lefi(see Clause 5.3.2.2 of Eurocode 2) exceeds 8.5m and
supports partitions liable to bedamaged by excessive deflections,the values of Vd should
be multiplied by 8.5//,, (/,, in metres).
50
45
40
35
30
1
25
20
15
10
r;llimit
C30137 C50I60 C70185 C901105 ,
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
100A,,,,&! (meanvaluefor span)
It should be notedthat the strikingtime of the formwork and falsework, proppingand/or
back propping can affectthe final deflectionof the flat slab (see Section6.3).
Whereformwork is left in place,or propping is usedandthe props remain in place, until
the concrete attainsthe specifieddesignstrength, the Limitingspan/effectivedepth ratios
given in Figure 11should be satisfactory, i.e.the deflections are likelyto bewithin the
limits given in Eurocode2. (Props are defined as beingverticalload-bearingparts of the
falsework, such as dropheads,which are left in placewhile formworkand other falsework
are removed.The use of props reduces temporary spans.)
18

Where back propping is used (without props,perse) then referenceshould be madeto Early
striking andimprovedba~kpropping(’~).This gives a method of determiningthe characteristic
concrete strength requiredto be able to strike the formwork and falsework by relating
temporary construction loads to permanent loads.Thereby ensuringthat any crackingis
no worse than that intrinsic to the permanentworks design (back propping is defined as
proppingbelow a recentlycast slab installedas soon as reasonably possibleafter the form-
work and falseworkto that slab has been struck). It should be noted, however,that the
use of early striking and back proppingwill have an impact on early-agedeflectionsand
this will need to be considered in the specificationof finishes which are sensitiveto slab
tolerance, e.g.faqades. Furthermore,as such slabs are crackedfrom an early age, they
may move more undervarying live load than traditionally constructed slabs (seeJones’
contribution to the discussion on Criteriafor the loading of slabsduring construction(15)).
Slabs supported by, and supporting,formwork, proppingor back proppingshould also be
checkedsince a temporary load case may be critical, especially where the superimposed
permanentand variable actions are low.
Where:
0 neither props nor back proppingare used,and the formwork is struck beforethe
0 props are used but the structure is loaded beforethe concreteattains the specified
0 estimates of deflection, at any stage, are required,or
smaller deflectionsthan those inherent in usingspan-to-effective depth ratios are
specified strength is reached,or
design strength,or
required,
it will be necessaryto undertakedetailed calculationfor deflections(seeSection 410).These
calculationsshould be done in the light of the presumedor actual Method Statement
and, where possible, in consultationwith the constructor.Appropriate early-age properties
of concreteshould be used.The UK NationalAnnex to Eurocode 2 has limited the ratio of
reinforcement providedto that required by adjustingthe values from Expression716 or
Figure 11to 1.5.This is discussedfurther in Section6.3.
4.5 Loading
4.51 Ultimate limit State Particularloading patterns are applied to determine the design moment envelope. Any of
the load combinations permitted by the UK NationalAnnex to Eurocode2 may be used.
Unlessthere are specific abnormal loads present,to obtain the maximum span moments
for flat slabs it will be sufficient generallyto evaluatethe combinations of the full-
factored dead load over the whole slab together with the factored live loadingon
alternate full width strips of the slab, consideredseparately in each orthogonal direction
(not ‘chequer-board’) as shown in Figure 12.
19

Design
Figure12
Lordingondtornatostrips(two combinations
ineachdirection).
Note:
C, =characteristicvalue of permanent action
Q,=characteristicvalue of a singlevariable action
It maybe possibleto reducey ,the partialfactor for permanent actions,to 1.25 using
Expressions610a and 6.10bo?BS EN 1990.
‘Singleload’ case
The ‘single load’ case given in Clause 5.1.3 (l)P of the UK NationalAnnex to Eurocode 2
can be appliedto allforms of flat slab analysis.This is the case for maximumloading
(factored dead and live load) on all spans.
The UK NationalAnnex states that support momentsshould be reducedby 20% together
with a corresponding increasein span moments. However, it is recommendedthat for
flat slabsthe support momentsfor the inner columnstrips (see Figure 22), resultingfrom
elastic grillageor finite element analysisare not reduced, but the saggingmoments are
increasedbythe corresponding moment.The reasonfor this is that the rules given in
Section 4.6 allow for lateral redistributionof the peak momentsfor the inner column strip.
This results inthe design momentof resistancebeingless thanthe peak elastic design
moment. Itwill be conservativeand simple to fulfil this rule by increasingthe sagging
momentdiagram by 20% of the maximumsaggingmoment and to usethe rulesgiven in
Section 4.6 without reducingthe outer columnstrip or hoggingmiddlestrip moments.
Where the transfer momentto edge columns is limited (see Moment in slabat edge
columns in Section 4.6.3), which results in a redistributionof morethan 20% of the
elastic moment,thenthe correspondingspan moment should be increasedaccordingly.
The ‘single load’ case may only be applied when:
The area of each bay exceeds 30m2.Inthis context, a bay means a strip acrossthe full
rn The ratioof the characteristic imposedloadto the characteristicdead loaddoes not
The characteristic imposed load (excludingpartitions)does not exceed SkN/m2.
width of a structure boundedon the other two sides by linesof support.
exceed 1.25.
20

Figure13
Effectivewidth of solid slab with a
:oncentrated load nearan unsupportededge.
Concentratedloads
Where a slab carries one or more concentrated loads in a line inthe directionof the span,
it should bedesignedto resist the maximumbendingmoment caused bythe loading
system.Such a bendingmoment in a singleway systemmay be assumedto be resisted
by an effectivewidth of slab (measuredparallelto the supports) as follows (see Figure13):
rn For rectangularsolid slabs on simple supports,the effectivewidth may betaken as the
sum of the loadwidth and 2.4x(1-dowherex is the distancefrom the nearerlineof
support to the section under considerationand I is the span.
For other slabs, except where specially providedfor, the effective width will dependon
the ratio of the transverse and longitudinalflexural rigiditiesof the slab.When these
areapproximately equal, thevaluefor the effectivewidth as givenfor solidslabs may be
used, but as the ratiodecreases a smallervalue should betaken.The minimumvalue
which needsto betaken, however, isthe loadwidth plus 4dI(1- dometreswherex
and I are as defined above sothat, for a section at mid-span, the effectivewidth is equal
to I m plusthe loadwidth.
rn Wherethe concentrated load is near an unsupportededge of a slab the effective width
should not exceedthe value inthe first two point above as appropriate,nor half that
value plusthe distanceof the centre of the loadfrom the unsupportededge.
When consideringthe effects of concentrated loadson a flat slab this method is helpful
to determine howthe load is spread and howthe reinforcement should be arranged.
Unsupported 7
edge
l.Zx(1-4)
Effectivewidth
* r
, J i
-I I
21

4.5.2 Serviceability limit state
4.6 Methodsof analysis
4.6.1General
4.6.2 Hogging momentsover
the supports
Clause 7.4 of Eurocode2 recommendscheckingdeflections using ‘quasi-permanent’
loads. However,it should be noted that the cracked state of the slab will depend on the
maximum load that the slab has experienced.The worst design situation is when the full
serviceabilityload is applied and the calculation of I (see Expression7.18 in Eurocode2)
should be based on this.
Normally,crack width limits will be satisfied usingappropriatedetailing rules (seeStandard
methodofdetailing structuralcontrete(l0)).However,where these need to be checked
then this should be done usingthe ‘frequent’ load case (see Clause 7.3.4 of Eurocode2).
The analysis methodsfor flat slabs consideredin this Report include:
0 The simplified coefficient method (for slabs with a regular grid of columns)
The equivalentframe method (normally for slabs with a regular grid of columns)
0 Finiteelement methods (for slabs with a regular or irregulargrid of columns)
0 Grillage method (for slabs with a regularor irregulargrid of columns).
There is a particular problem concerningthe modelling of support moments over the
supports.The hogging moment at a support peaksvery sharply.Clause9.41 (2) of
Eurocode2, states that, for internal columns, half the total applieddesign moment must
be resisted within a width over the column of a quarter of the panelwidth. The width, so
calculated, should be based on the lesser dimensionof the panel (see Figurel5).
Hoggingmomentsgreaterthan those at a distance hc/3from the centrelineof the column
may be ignored providedthat the sum of the maximum positivedesign moment and the
average of the negativedesign moments, Msum,in any span of the slab for the whole
panelwidth is not less than given in Equation1 (see Figure 14).This requirement becomes
important for columnswith a large sectiondepth (see also Section4.71).
MSum1 n12(I, - 2hc/3)2/8 (Equation1)
where:
n
1,
12 = panelwidth, measuredfrom centres of columns
hc
= design ultimate load on the full width of panel betweenadjacent bay
centrelines
= panel length parallelto span, measuredfrom centres of columns
= effectivediameter of a column or column head (see Figure 14).

Figure14
Reductioninmaximumhoggingmomentat
columns.
Design
1' I
r
4.6.3 Coefficient method General
Simplified momentcoefficients can providean economic solutionfor simple orthogonal
flat slabs.Table 2 provides moment and shear force coefficients for flat slabs of three or
more approximately equal spans.This may be used providedthat:
H the spans do not differ by more 15%
m the ratioof the panelsize does not exceed 2 (i.e. 0.5$II, I 2/J.
I
Table 2
Bondingmomentandskar forcecoefflclents
for flat slab pmlsoft h mor mon
approximatelyqualspans.
E
Ie
Outer ..dar middle At first At middle At internal
support Iof endspan 1interior Iof interior supports
It should be notedthat the permanent loadof 1.35Gkmay be reducedto 1.25Gkwhere it
can be demonstratedto besafe bythe use of Expressions610a and 610b in BS EN1990i9).
Lateraldistributionof moments
The recommended methodof lateraldistributionof moments and reinforcement is
achieved by dividingeach panel into columnand middlestrips as shown in Figure 15.This
is an extendedversionof Figure 1.1 inAnnex I.of Eurocode 2, allowingfor panelaspect
ratios greaterthan 2.This Figure is also applicableto equivalent frame analysiswhich
may be usedwhere the ratioof the panelsides exceeds 2 (seealso Type 7 -Linearelastic
in Section4.6.5).The nominalstrip spans one way inthey-directionas a single-way slab.
23

Figure15
byoutof columnandmiddlestrips.
Table 3
Distributionof designmomentsfor solidflat
dabswithflat soffks.
Figure16
lateraldistributionof mamentsdependingon
pmlaspact ratio.
.....
.....
.-
i 144 I 3lY14 DI I
?-T I
I I
I I
...................... .-..+ .........1............................ 1..................I..................................... ....
144 1 [3: I I
.......................... -.......
I Middle
I strip
I I
I I
I I
I I
I I
-1- - - - f - - - - -- - - - - -
.....................................................................
I I Column
I I strip
I I
I - I - 1- =
Column Middle Nominalstrip
strip strip (oneway)
_ _ _ - _ _
- - - + - -
The design moments for the slab obtained fromTable 2 should be divided betweenthe
columnand middlestrips in the proportionsgiven inTable 3.This providessimilar but
more specific informationthan Figure 1.1 in Eurocode 2, where the values of k, and k, may
be taken from Figure 16.
Design moment IColumn striD for internal columns% IMiddle striD %
Longspan =k,
ShartrpWl-7:
Longspan=k,
jul0nSp&S
1.0 1.5 2.0
Aspect ratio(l,&)
Inadditionto the rulesgiven inTable 3, it is recommendedthat at internalcolumns half
of the design moment for the full width of panelshould be resistedwithin a width over
the columnof a quarter of the panelwidth.
The effectivewidth of the columnstrip is affectedby the presenceof a columndrop or
the width of the solid section for acoffered flat slab (see also Sections 4.7.2 and 4.7.3).
24

Design
Momentinslab at edge columns
Ingeneral,the momentthat can betransferred betweena slab and an edge or corner
column is considerablysmaller than that for an internalcolumn. Figures17and 18show a
typicalyield-line patternat an edge column.
For the analysisof slabswithout edge beams,the momenttransferredto an edge or
corner column, M,,,,, should normallybe limitedto 0.17bedzfk.The breadthof this strip,
be,for varioustypicalcases is shown in Figure 17.beshould not betaken as greater than
the columnstripwidth appropriate for an interior panel.Where the applied moment
transfer is greaterthan Mt,,,x moment redistributionmay be applied and the sagging
moment in the end span should be adjusted accordingly.Where Mt,,,,/0.4 FI (seeTable2)
exceeds the redistributionof moments limits accordingto the normalrules (Clause 5.5 of
Eurocode 2 andthe UK NationalAnnex) the designshould be altered (e.g. alterthe slab
thicknessor the columndimension). It should be notedthat the transfer moment for the
designfor the columnshould be determined in accordancewith Section 4.71.
Figure17 I
Slabedge
Slabedge
Figure 18shows the effectivewidth (c, +2r)within whichthe design reinforcement may
be placedto resist the transfer moment (see ClRlA Report 89(16)).The value of r should be
limited to the value of cy.It should be notedthat, togetherwith the limitation given for
momenttransfer (see above),this is consideredto be a reasonableextension to Eurocode
2, Clause9.4.2.
Figure18
Y k L d 4 h e m a d u n h m ~ ~ c d u m n .

Table 4
Valuesof k to determinetorsionalconstant.
4.6.4 Equivalent frame
method
Where an edge beam exists (see Section 3.6)which is adequatelydesigned for torsion,
the moment transfer from slab to column may be increased. However, flexural cracking
of the edge beam reduces its torsional stiffness, GJ,close to the column (where G is the
shear modulus and J is the torsional constant) and it is recommendedto take the
torsional constant of the beam equalto half the St. Venantvalue.The value of J for a
rectangularsection may be calculated using Equation2:
J = 0.5 ku3b (Equation2)
where:
U = the smaller dimension
b = the largerdimension
k = a constant as given Equation3 orTable 4.
k = 1/3- 3 . 3 6 ~{I - ( ~ / b ) ~ / 1 2 } / 1 6 b (Equation3)
1 1.25 1.5 1.75 2.0 2.25 2.5 3.0 4.0 5.0 10
0.141 0166 0.196 0.214 0.228 0.240 0.249 0.263 0.281 0.291 0.312 0.333
CeneraI
This method (see also Annex I of Eurocode 2) gives a reasonable representationof the
behaviour at the ultimate limit state by a systemof columns and beams analysedtwice;
once a5 a frame in the x-direction and once as a frame in the y-direction. The following
points should be noted:
As alreadystated in Section3.2, a flat slab supportedon columns, rather than peri-
meter beams, can fail as a one-way mechanismjust as a single-way slab, and it should
be reinforcedto resist the moment from the full load in each orthogonal direction.
0 The equivalentframe method does not provideany information concerningthe lateral
distribution of the total moments resultingfrom the analysis.The hogging moments
over a support from the equivalent frame analysiswill not represent the true situation,
and the actual moment per metrewidth will be much greater close to the support than
some lateral distance away.This is in contrast with the results from a finite element or
grillage analysis.Specific rules for the lateral distribution of the moments are required
to ensure a suitable arrangementof reinforcement.Two thirds of the total applied
design moment should be resisted within a width over the column of a quarter of the
panelwidth.
The equivalentframe overestimates the moment transfer at edge columnsas the model
assumes a line support of a wall ratherthan the point support of a column.Allowance
should be made for this inaccuracyin the modelling of the edge slab/columnjoint. A
reasonable approximation isto reducethe support moment by a factor equalto 0.7 of
the elastic moment found from the equivalent frame analysis.This should be treated
as a redistribution of the support moment and the moment in the span increased
appropriately.Further redistribution of moments is permissiblein accordingto the
normal rules (Clause 5.5 of Eurocode 2, and the UK NationalAnnex). If the moment
26

Design
Figure19
C d i t k n s .
~vkuratedgecolumnund.rnmy
transfer afterthese adjustments is greaterthan 0.17b,d2fc,(see Moment in slabat edge
column in Section 4.6.3), it is likelythat excessive crackingwill occur inthe slab around
the edge column and, in extreme situations, the shear capacity of the slab will be
reduced.Considerationshould be givento changingthe geometryof the slab and edge
columns.Wherethe ‘single load’ case is used (see Section4.5),the increase in the
edge-span moment from redistribution of the edge-columnmomentshouldcorrespond
to boththe effect of the above 0.7 reductionfactor and any further reductionrequired
(see Moment in slabat edgecolumn in Section 4.6.3).
Layout of structure
The structure should be divided longitudinallyandtransversely into frames consistingof
columnsand sectionsof slabs contained betweencentrelines of adjacent panels (area
bounded by four adjacent supports). For this reason, the slab stiffness should be reduced
(see below).
Sway deflections
These are likelyto be largerthan predicted usingan equivalent frame analysis.This is
because of the increasein rotationof a flat slab closeto the columncomparedwith a
continuously supported slab. Figure 19 showsthis effectfor an edge panel.
Slab stiffness
The choiceof slab stiffnessfor ultimate limit state analysisdepends on engineering
judgement. It is consideredreasonableto basethe designon the uncrackedconcrete
section properties (excludingreinforcement):
Forverticalloading, the stiffness may be basedon the full width of the panels.
For horizontalloading, it is more appropriateto take 40% of this width to take
account of the reducedstiffness at the slabkolumnjunction.
For panelswith an aspect ratiogreater than 2 (see Figure 16),the stiffnessof the slab is
basedon the columnand middlestrips only, but the loadingover the whole slab area
should be included.The nominalstrip should be designedas a single-way slab in they-
directionandwith nominalreinforcement in the x-direction.

Design
Figure20
Behaviourof two-bay slabs.
Lateral distribution of span hoggingmoments
The lateral distribution of moments where hoggingmoments exist at mid-spandoes not
conform to the above rules.The actualdistribution depends on the geometry and loading.
Provided the total hoggingmoment at mid-span is not greater than 20% of the hogging
moment at the support, it is reasonableto assumethat the moment is distributed evenly
across the slab.Where such hogging moment exceeds 20%,the distribution of moment
is concentrated more in the middle strip.
Analysis of athree-bay slab can show that hoggingmoments mayoccur in the centre
span, particularlyfor arrangementswhere the centre span is shorterthan the span on
either side.The lateral distribution of hoggingmoments and reinforcement across the
centre span may normally be assumedto be uniform across the full width of the panel.
Influenceof number of slab bays
Elasticanalysis shows that the centre columnsof a two-bay flat slab carry a load of more
than half the bay on either side as shown in Figure 20.
I I
SectionA-A
4 r Slab edge
i x
b j i -
...........p.............................. p................................ p............................... p.....................I 1
x j
...i....................... .t........i.................................. i................................. i....
Trl n n n
' 4Plan
The elastic value of k for two bayswith no moment restraint at edge columns is 1.25.This
reduces if moment is transferredto the edge columns.
28

When analysingsuch a system by the equivalent frame method in the longitudinal
direction (x-x), the section propertiesshould be basedon I as shown in Figure 20. However,
the loading,W,on this width is likelyto underestimatethe moments,particularlythose
at the internalsupports. Inorder to obtain a more accuratevalue of the loadingin this
direction, an analysisshould first be carried out in the transversedirectionto determine
the value of k.There is a consequent reductioninthe loadingalongthe lineof edge
columnsthat may be taken into account.
Slabs with morethan two bays across are only affected in this way at the first internal
column.
Edge beams
Inanalysingthe slab by an equivalentframe method perpendicular to an edge beam, it is
reasonable,normally,to assume that the lateraldistribution of the bending momentsand
reinforcementalongthe edge is as for the internalcolumn line.However,if there is doubt
about the moment capacityof the column,or about the amount of loadtransfer on to
the edge beam, a finite element or grillageanalysisprovidesa moreaccurate solution
(see Section 4.6.5or 4.6.6).
4.6.5 Finite element method General
The useof finite elementanalysisfor flat slabs producessimilar designs to other analytical
methodssuch as equivalentframe andyield-line analyses.Traditionally,its use has been
mainlyfor slabs with irregular geometry or with awkwardopenings, or where the estima-
tion of deflections(as opposedto keepingwithin span/depthlimits) has been required.
However,many contemporary packages usegraphicalmodellingmethods, the facility to
use CAD files, reinforcementdesignand other features that maketheir usequicker and
easier.This,togetherwith the ease of makingmodelchangesplusthe reducingcost of the
software,has ledto a moregeneral useof finite elements for flat slab design. Reference
should also be madeto the How to design concreteflat slabsusingfinite elernentanalysis(17).
Check list:
U Doyou really needto do a finite element model?
U What output do you want?Canyou get it?
U Knowyour software.
U Modelwith care (seeTable5).
U Use appropriateproperties.
0 Be aware of the pitfalls.
0 Carry out hand checks.
29

Design
Carry out hand checks
Be careful with edge beams
Designto M,* and My*.Not M, and My
(see Figure 21)
Check moments to perimeter columns
Model columns carefully
Do not over-reinforce at supports
Linear analysisoverestimates support
moments
Use meanvalues for E,,, f,, and E=
Use a realistic creep factor
Knowyour software
Adopt standard procedures
Remember GIGO
Table 5
Finite element design watchpoints.
Figure21
Plateor shellelement moment output.
Reactions,t M ;r w1/8etc
If modelledwith torsional stiffness,edge beams
must be designed for the inducedtorsions
Platedesign moments must be adjusted to include
the effects of torsion
Transfer moment must be s M,,,
The rotational stiffnessof columns should be
modelled
Do not reinforce for peak moments, but concentrate
as Figure 45
If using a linear package, consider reducing support
moments and increasingspan moments
Characteristic values for these parameters are not
appropriate for calculating deflections
The effective value of cp is a composite, based on
sequence and duration of loading
Not all packages do the same things or give the
same results. Knowyour packageand its limitations
Standard in-house procedurescan avoid many
common errors
Garbage in
Usersshould understand flat slab behaviour
Alternatively, set beamtorsional stiffness to zero
Most packages will do this automatically
If exceeded, both support and span moments will need to
be adjusted
Point or knife-edge supportswill produce very different
results
Over-congestion over supports may makeadequate
compaction of concrete very difficult
When reinforcement and crackingare modelled, support
moments are reducedand span moments are increased,
giving more manageablesteel arrangements
fa, should be for the age at which first cracking is
expected.SeeTR 58(”)
SeeTR 58
See Appendix A2 for some of the featuresone should know
Also saves ‘reinvention of the wheel‘ on every project
Garbageout
X
Treating reinforcedconcrete as an elastic isotropic materialcan leadto problems in inter-
pretingthe bendingmoment results.The output from a finite elementanalysisof plate
elements will give bendingmoments in the x- andy-directions,Mxand My.However,it
will also give the localtwisting moment Mxy.This moment is significantand must be con-
sidered in the reinforcementdesign.Mxydoes not act in the directionof the reinforcement
and a method is requiredto allow for Mxyin the design.A popularmethod inthe UK is
knownas Wood Armer moments,although it is not the only method used.Most software
will calculateWoodArmer momentsfor the user.They havefour components,top (hogging)
moments inthe x- andy-directions,Mx(T)and My,,,,and bottom (sagging)moments in
each direction,Mx(B)and My(B).The method is slightlyconservativeand these moments
form an envelope of the worst-casedesign moments.It is possibleto have both Mx(T)and
momentsat the same location in the slab (usually near the point of zero shear).
30

Alternatively,and more conservatively,Mxycan simply be addedto Mxand MYand the
design momentsare then Mx*= lMx(+ lMxyland My* = IMyl+ lMxyl.
Types of slab (‘plate’ and ‘shell’)design software
The types of analysiscan be described under five broad headings:
0 Type 1- Linear elastic: Simple assumptionsmadewith regardto the slab and column
stiffnessesfollowed by a linearelastic analysis.‘Plate’elements exclude in-plane
forces; ‘shell’elements includein-planeforces.
it has cracked.This leadsto a non-linearelastic analysisby iteration.
includingwhere the reinforcementmay haveyielded.
elements which can modelthe different interactionbetweenthe different layersof
concrete and reinforcement.
0 Type 2 - Non-linearelastic: Account is taken of the changes in the slab stiffness where
0 Type 3 - Non-linearelastic/plastic:Account is taken of the actualsteel stresses
0 Type 4 - 3D element:More accurate modellingis carried out using3D plate/shell
0 Type 5 -Whole structure packages.
Type 1- Linear elastic
Linear analysis is the mostwidely used methodof finite elementanalysisand can provide
reasonableresultsfor ultimate limit state (ULS)design. It is less sophisticatedthan non-
linearanalysis,which can provide a more realisticassessmentof deflections.Reinforced
concrete is treated as an elastic isotropic materialand a number of assumptionsare
madeto allow this methodto be used. Unlessthe designer is experienced and is able to
choose more realisticpropertiesfor the model,the following rules are recommendedfor
ULS analysis:
0 The stiffness of the slab and columnsmay be taken as:
Slab stiffness: O.S(€cm,s,ah/c,,,ah)/(l+cp), which allows for cracking in the slab
Columnstiffness: (Ec,,c,, /c,co,)/(l+cp)
where:
Ecm =
I, =
cp = long-term creep coefficientof concrete.
secant value of the modulusof elasticity of concrete at 28 days
2nd moment of area of the concrete sectiononly
The stiffness is normallyrepresented in the data by the actualgeometry and an
effective modulusof elasticity,Ee, (e.g.for the slab Eetr=0.5 €cm,slahl(l+cp)). Where an
accurate predictionof deflection (within Smm) is requiredthese simplifications are
probably not appropriate(see Type 2 - Non-linearelastic).
0 The choice of mesh arrangement and howthe column is modelledaffects the slab
moments at the face of the column.Thefiner the mesh,the more peakythe moments
at the support appearto become. Inorder to interpretthe results sensiblythe following
practical(but not exact) procedure is recommended.
Figure22 shows a reasonablearrangementof elements usingeight-noded plateelements
with a simple orthogonalrectangular mesh.An automaticallygenerated mesh is likely
to give a very different lookingmesh (see Figure23), but similar principles may be
applied which will give similar results.The maximum node moment should be taken
for each strip except for the inner column strip as shown.
31

Figure22
Typicalarrangementof elements.
W
Middle strip
Column strip, C ,
Inconsistent node moments
acrossface of columns
Meanof node moments
i Middle strip, M
Ml2 j MI2
v
Figure23
Arrangement of elementsfrom a mesh
generator.
The following recommendationsare normallyapplicable:
0 The panelwidth is divided into eight strips, four representingthe middleand four
representingthe column strip. In addition,two elements are placed adjacent to the
columnthus makingsix elements in the column width.
0 The slab design moment at the face of the column is calculatedfor the central half
of the columnstrip.This includesthe elements adjacent to the column plusthe
next element out.The meanof the node resultsfor all these elements alongthe
face of the column are used for the designof reinforcementin this strip.
nodesof the adjacent elements.
0 The column is represented by a centralnode together with rigid linksto all the
32

If the slab moment at the column face in the middle half of this edge column strip, as
calculatedabove, or the moment transferred to perimetercolumnsis greater than the
transfercapacityof the junction, 037b,d2fc, (seeClause 1.1.2 (5) of Eurocode 2 Annex I,), it
should be reduced,providedthat this reductionis not greaterthan 30%,and the moments
in adjacent spans are increasedby a correspondingamount. If the reduction is greater
than 30%,then it is likely that excessive crackingwill occur in the slab aroundthe column
and, in extremesituations,the shear capacityof the slab reduced.Consideration should
begiven to changingthe geometryof the slab and edge columns and/or settingup a more
realisticmodel.Where the 'single load' case is used (see Section 4.5) the increase inthe
edge-spanmomentfrom redistributionof the edge-column momentshould correspondto
the total redistributionof the edge moment if the reductionis requiredto be greater than
20% (see Moment in slab at edgecolumn in Section 4.6.3,and Clause 5.5 of Eurocode2
and the UK NationalAnnex).
For serviceabilitylimit state (SLS) analysis,it should be notedthat cracking, percentageof
reinforcementand creep are not considereddirectly by Type 7 analysis.Usually,their
effects on deflectionare includedby modifyingthe elastic modulusinthe same way as
for ULS (e.g.takinga value of half for the slab and full value for the column).
Type 2 - Non-linear elastic
Programsare described as non-linearwhen they solve problemsto which there is no
direct solution. ForthisType 2 modelling,it is assumedthat the reinforcementremains in
the elastic field and anyyielding is not modelled.Changes in the geometric dimensions
(e.g.solid sectionof waffle slabs) are reflectedin the element properties.
Once concrete cracks,sectionstiffness is a function of moment, and moment depends
upon sectionstiffness. Non-linearfinite element programsstart with un-crackedsection
properties, then after an initial runthey back-substituterecalculated propertiesand run
again.This process is iterated until assumedand actualelement properties match each
other within a predeterminedtolerance.
The 'elastic'versionsof these programsassume that materialshave a constant modulus
of elasticity regardlessof strain applied by moment.The (factor (see Clause 7.4.3of
Eurocode2) should then be used to derive elementstiffness by interpolatingbetweenthe
un-crackedand fully crackedvalues (see also Section 4.5). In the calculationof (to
Expression739 of Eurocode2, the factor p allowingfor long and short term tension
stiffeningshould normally be taken equalto 0.5 since the long-termvalue is appropriate
only after a few days (see ConcreteSocietyTechnical Report 59n). An exception might be
when examiningthe incrementaldeflectionafter installationof brittle partitions.The
variation in stiffness of the sectioncan be expressedinterms of equivalentdepth in the
softwaredata, see Figure24.
33

(2sigrl
-
Figure24
Exampleof equivalentdepthsto simulate
stiffness.
This method produces reasonably credible results (Ie calculateddeflections) in most
situations It does not, however, model the yielding of support reinforcementthat can
occur adjacent to most flat slab columns (even at SLS), as this method does not allow an
element’s stiffnessto be reducedbelow the fully cracked elastic value
Figure24
Exampleof equivalentdepthsto simulate
stiffness.
34
Moment (kNm)
This method produces reasonably credible results (i.e.calculateddeflections) in most
situations. It does not, however, model the yielding of support reinforcementthat can
occur adjacent to most flat slab columns (even at SLS), as this method does not allow an
element’s stiffnessto be reducedbelow the fully cracked elasticvalue.
The above approach should not disguisethe fact that there are manyvariables not included
in the analyses and that there could be significantvariation from the resultsfound. For
example, the E value of the concrete is influencedby the type of aggregate used (see
Clause 31.3 (2) of Eurocode2). Similarly,the development of tensile strength with time is
strongly influencedby curing and drying conditions as well as the thickness of the slab.
Between the 5% and 95% confidence limits (seeTable 31 of Eurocode2) it can vary by a
factor of 2. If a slab is cracked during construction by high temporary loading,the cracked
concretepropertiesshould be used even if the analysisshows that it is uncracked under
serviceabilityloads.This will not only affect the stiffness up to the crackingmoment but
will also affect the tension stiffening.The levelof refinement in the computer model
should reflectthese uncertainties;for example,if accuratedata for the E value is available
for the concrete, it may be worth carryingout additional refinement and includingthe
effects of shrinkage. If,as is normal, this information is not available, it is probablybetter
to bracketthe deflectionsfor the rangeof €values.
Type 3 - Non-linear elastic/plastic
These should producesimilar results toType 2 above as longas strain in the reinforcement
remainswithin the elastic range. However,once above the elastic limiting stress, Sk,the
stresshtrainrelationshipsindicatedin Figure 3.8 of Eurocode2 are usedto model reinforce-
ment yielding. Figure 3.2 of Eurocode 2 should be usedto determine concrete stresses and
strains, and tensionstiffeningshould be modelledas ConcreteSocietyTechnical Report 59(@.
These assumptionsintroduce a second order of non-linearity as the section stresses and
strains must then be solved by trial and error.The calculation of section curvaturealso
becomes more complex.
This type of program should providea good estimate of deflections in flat slabs of normal
thickness, as moment peaksover columns can shed laterally to more closely match the
reinforcement arrangement.

Initial assumptionsas to how the reinforcement is arranged must be madefor the first
run of a non-linear design program.Generally,this is done either by setting AS=As,rqdor
by setting As,provto a nominal levelof around 0.5%
where:
AS = area of tension reinforcement
As.rqd = area of tension reinforcement required
As.prov = area of tension reinforcement provided.
When a rationalisedreinforcement arrangement is decided,it must be applied to the
model (e.g.as a rough drawing).The program must then be rerunto take this chosen
arrangement into account.The updated results (usuallyshown graphically) should then
be checkedto ensure the adequacy of the chosen reinforcement arrangement.
Type 4 -3D element
These are suitablefor thickerflat slabs such as transfer floors,where effects such as internal
arching may needto be considered.WhereasTypes 1, 2 and 3 use two-dimensional plate
or shell elements with a thickness, this type of program has several layers of three-dimen-
sional concreteand reinforcement elements within the depth of the slab. Stresses, strains
and thence curvaturesare therefore deriveddirectly from the frame analysis.
Type 5 -Whole structure packages
Three-dimensionalframe analysis programsthat includefinite elements for floors and
walls can be very usefulfor modelling the global behaviour of structures and for easily
collecting column and foundation loads. However, becauseof the very large model size, it
is usually necessaryand practicalto use relativelycoarse finite element meshes. For this
reason, they may not be suitable for the final design of flat slabs. It should be noted that
these packagesdo not take account of construction sequence.
Modelling
Meshing
Most plate design packageswill include an automatic mesh generator together with tools
for refiningor altering the layout of individualelements.Some meshgeneratorswill be
better than others, but it is always important to check the suitability of the mesh used
and the assumptionsmade by the package.
The average size of element to be used will be a matter of engineeringjudgement, as
there has always to be a balance between requiredprecisionof results (fine mesh) and
speed of calculation (coarsemesh). Better resultswill be achieved by reducingmesh size
in the proximity of supports and applied point loads, and by ensuringthat each of these
has a node located at its centre.
It should be noted that elements may have 3/4 nodes or 6/8 nodes each, dependingon
software (see Figure 25). Elementswith additional side nodeswill not requireas fine a
mesh as those that haveonly a single node at each corner.
35

Figure25
Plate element types.
fnodes
Columns
It is important that columns (andwalls) above and below a plate floor are modelled, in
order that:
Slab moments are assessed correctly
0 Design column moments are derived
0 Punchingshear stresses can be realisticallyevaluated.
Most programswill represent columns as point supports linkedto rotational stiffnesses.
For internal columns,this form of modelling is reasonableonly where the spans are
approximately equal. Otherwisethe steps set out for edge columns below should be
considered.It should be noted that shear output from elements close to supports should
be checked before it is accepted.
For edge columns, modelling a column with a point support may lead to large inaccuracies
in calculatingthe moment transfer, span moments and displacements.These inaccuracies
will occur where the support node is placedat the edge of the slab model (unlessthe
column centre of action is reallyon the edge).These inaccuracies can be greatly reduced
by one of the following methods (see also Section 4.71):
0 Providevery stiff linkdbeams betweenthe support node and the nodeson the peri-
0 Insertdeep regions of slab in the plan areas occupied by the columns.
meter of the column.
Column heads
Although regular drop panelsare readily modelled,full column heads can be moredifficult.
If the software used is able to createtrapezoidalregions of varyingthickness, heads may
be modelled as the group of five plate regions as shown in Figure 26.This group can then
be copied to similar supports. Failingthis, beam memberscan be insertedto simulate the
increasedstiffnessof the head area.
Figure26
Column head regions.
36

Input data
For linear elastic programs (Type 1above),only the concretemodulus (€,J and an appro-
priate creep factor (cp) are normally required.Additionally, non-linear softwarewill require
that valuesfor concretetensilestrength (fctm)and free shrinkage strain (E,) are input before
defining slab regions, as well as bar diameters, covers and layeringof reinforcement.
Appropriatevalues for creepfactors,concretetensile strength and free shrinkage strain can
all depend on the loadinghistoryof the slab. When a frame contractor has been appointed,
a detailed construction programmetogether with knowledgeof the concrete and con-
struction techniquesto be used can enable the calculation of these values with relative
precision.Guidanceon how to derivecreep factors and tensile strength at the critical load
stage can be found in the ConcreteSocietyTechnical Report 58(”).
For designs at earlier stages or when the calculation of deflections is not considered
important, more conservativeassumptionscan be made.Suggestedvalues for these
situations are:
Creep factor, cp = 2.5 (a composite value allowing for striking at around 7 days).
Concretetensile strength, fctm= 0.78 x (valuefrom Table 3.1of Eurocode2) (first
Shrinkage strain, E,, = E , ~+E,, (fullvalues in accordance with Clause 3.1.4(6)of
crackingat 7 days).
Eurocode2; the software should default to this).
Load combinations and moment redistribution (see also Section 4.5)
Several load combinations will be includedin most analyses and it is important when
usingnon-linear programs,that SLS combinations are analysed separatelyto those at
ULS. Behaviourand the degree of crackingcan be very different at the two limit states,
necessitatingthe compilation of two discretestiffness matrices.Some finite element
packageshavethe facility to keepthese separatewithin a single run. Failingthis, separate
runs haveto be made.
Generally, it is not appropriateto redistributethe momentsfrom a finite element or grillage
analysis,although reducingthe stiffnessof certain elements may simulate it (see also
Section4.5). If redistribution is carriedout, then the more critical values for the column
design reactions,moments and shears should be taken from the elastic and redistributed
cases.
For serviceability, it is appropriateto consider permanent load everywhere and variable
load on the bay under consideration.This can be very time consumingand should only be
carried out where specifically required.
37

4.6.6 Grillage method General
The grillage method uses a modelwhere the slab is representedby a seriesof inter-
connectedbeams.Where the layoutof columns is not regularor there are significant holes,
or where actualdeflectioninformation is requiredthen the use of either grillage or finite
element is likelyto be the most suitable method.
Inorder to modelthe structurewith any accuracy, it is important to be able to apply the
loads in stageswith differingelement properties.This enables short-term and long-term
loadsto be applied to elementswith the appropriatestiffness.
Mesh generation
If the momentsfound from the grillage are goingto be useddirectly to calculate reinforce-
ment, the orientation of the grillagemembers should follow the directions inwhich it is
'
plannedto reinforce the slab.The spacingof elements should be as constant as possibly
to facilitate simple post-processing.It is recommendedthat the spacing between the
element passingthrough a column and the next element is approximatelyequalto the
size of the column. If a relativelyconstantgrid is being used it is acceptableto increase
the element spacing, but 1/8 of the span would seem a sensiblemaximum.Similarly,
there is little point in reducingthe spacing belowthe depth of the slab.The meshshould
includenodes at column lines and approximatelyat the mid-spanof each bay.
Section properties
An approximatemethod of calculatingthe slab-member sectionpropertiesis to hand
calculatethe moment and reinforcementrequiredfor hoggingand saggingin a typical
column strip.These momentsand reinforcementratioscan then be usedto find two
equivalentstiffnesses.An averagecan then be taken for the slab elements in the model.
Many of the available softwarepackagesonly allowthe stiffness, €1, to be varied by
alteringthe value of € or the depth of the section. If the stiffness is varied by usingan
equivalentslab depth care should be taken to ensurethat the self-weight is entered
separately by hand.
Untilthey crack, flat slabs have approximatelyequalpropertiesin all directions,so the
torsionalconstant,J, can be found from J = €//G.This may be consideredas the default
value.After cracking,this torsionalstiffness reducesto approximatelyhalf this value.
For serviceability analyses, this is best incorporatedwithin the analysisdata by multiplying
the defaultJvalues by the factors given inTable 6. hmaxis the width of the elementand
hminis the actualdepth of the slab. In general, it is acceptableto apply the crackedfactor.
As the factor is independent of the equivalent slab depth,the factor is not affected by
any refinementof member stiffness.These factors assume that the stiffness is corrected
by inputtingan equivalentslab depth as described above.
>51.5 2 3 5
1 0.87 0.77 0.69 0.61
Table 6
Values ofJ for usewith grillageanalysis.
1.43
L 1
710.71 0.5 0.43 0.38 0.34 0.30
38

I
If staged loading is used,care should be taken in assessing the short- and long-term
propertiesof the concrete.The appropriatecreep factor may be determined from Figure
3.1 of Eurocode2 (Clause 3.1.4 (2)).The expectedshrinkage strain should also be included
in the long-term properties(see Clause 3.1.4(6)).
Once the model has been run, it is possibleto refinethe member propertiesfurther by
usingthe actual momentsand proposed reinforcementlayoutto recalculate the stiffnesses
(see Clause 7.4.3 of Eurocode2) and hence equivalentdepths of groups of members (for
typical example, see Figure 24). Usually, it is sufficient to split the elements into five or six
groupsand create separate propertiesfor each group.
Calculationof ultimate design moments from a grillage incorporatingsignificanttorsional
stiffnesswould require substantialpost-processingto combine the torsional and flexural
moments (seeAppendix A.4). For the ultimate limit state, therefore,the torsional stiffness
can be reduced to about 5-10% of its default value.The peak torsion or torsion stress
should be compared to f,,, (seeClause 6.3.2 (5) of Eurocode2). If the torsion stress is above
this, the cause of the torsion should be investigatedand the consequences of cracking
considered before loweringthe torsional stiffnessfurther.
,
In order to assess the design slab moment at the column face, a similar procedureto that
described in Typesofslab (‘plate’and ‘shell’) design software in Section 4.6.5 may be used.
The moment transfer at edge columns should also be checked as described in the same
Section.
Loading(see also Section 4.5)
Providedthe grillage element spacing is similar in the two orthogonal directions,it is
appropriateto split the loadingequally between members.Whilst the ultimate hogging
moments can be found from a load case of ultimate load everywhere,the calculation of
sagging moments and column moments requires considerationof patch loads. Since to
envelope relevant patch-loading combinations could take a considerabletime, it will
normally be sufficient to look at the patch loads on typical (worsecase) bays and factor
up the sagging moments and column moments in the rest of the model accordingly.
4.7 Specific considerations
4.71 Columns General
The columns should be designed in each direction for the sum of the following:
The design moments from the slab analysis for gravity loads at each floor
The moment resultingfrom geometric imperfections (see Clause 5.2 of Eurocode2)
The design moments from a sway analysis (if columns are unbraced)
Any second order moments if the columns are slender.
= /,/400
39

In addition, if the structure is a sway frame,the load cases for equilibrium(EQU) should
be checked usingthe load factors given inTableA.1.2 (A) of BS EN 1990(g).
The momentstransferredto edge columns are limited by the maximum momentof resis-
tance of the slab as described in Sections3.6 and 4.6.3.However, it should be notedthat
the method provided in Section 4.6.3 gives a lower boundmoment for the column.The
design moment for the column may normallybe taken as the lesser of the moment taken
from the elastic analysisfor ULS or y,Z x 0.17be d2fck(see Section 4.6.3),where y, = 115.
Wherethere is an unsymmetricallayoutof columnsand/or shear walls, a three-dimensional
global analysis should be carried out to ensurethat allthe forces are taken into account.
Where the columnsection is subject to biaxial bendingit should bedesigned in accordance
with Clause 5.8.9of Eurocode 2.
Column stiffness for sub-frame analysis
The choice of stiffnessof the columns where an elastic analysis is carried out (e.g.sub-
frame usingequivalentframe, finite element or grillage)dependson the restraint at their
far ends (see Figure27). Normally, it is based on the far end beingfixed. If it is clear that
the far end is pinned(e.g.founded on a smallspreadfooting), its stiffness should be
reduced as shown in Figure27(b).
Figure27
Modellingcolumn stiffnessfor fixed and
pinned situations.
..........................
a) Far endfixed b) Farend pinned
Large columns
Where the effect of the size of the column is not specifically modelled,the following
approach should be adopted (e.g.in an equivalentframe or finite element analysis,where
the columnsare modelled as linear elements meetingat a point).For columnswith a depth
(dimensionin the directionof bending,h) greater than 500 or span/lO (see Figure28), ’
differentshear forces at the oppositefaces of the column cause a moment transfer in
additionto Mt,calculatedby modellingwith ‘stick’ elements.This extra transfer moment
should be includedinthe calculation of the momenttransfer (andits effect on the punching
shear calculation).
40

Figure28
Section at column.
Largecolumn if
This may be achieved by:
0 Modellingthe column realistically(i.e.not just assuming line elements to represent
Introducing a rigid arm (see Figure 29a),or
By the following approximate adjustment (see Equation4).This adjustment should be
made to the moment obtained from an analysiswhich has ignoredthe depth of the
column.
the column ignoringthe depth of the column)
Mt z M,’ +(V2’- V,’)h/3 (Equation4)
where Mt’,V,’ and V,’ (see Figure 29) are taken directly from the analysis.M,and M,’
are positivewhen the slab transfersa clockwise moment to the column, as shown in
Figure 29b.
The value of M, obtainedfrom any of the above methodsshould be used when calculating
the effective shear (see Section 4.6).
The adjustment in the latter method was obtained by consideringthe sub-framesshown
in Figure 29.The moment transfer was calculatedfor the two frames for a range of spans,
column widths, beam end fixities and column stiffnesses. For each case the value of the
function
was calculated.
41
The values were found to be fairly insensitiveto column widths but highly dependenton
the ratio of column stiffnessto total joint stiffness.This may be taken as:
where:
40, = full height of the column
Lslab = full span of the slab.
Figure 30 showsvalues of the abovefunction plotted againstthe column stiffness ratio. For
large columns,where this ratio approaches 1, the value of the function is between 3/10
and 1/3. It is this upper boundvalue which has been used in the approximateadjustment
formula given in Equation 4.

Figure29
Approximate modelsfor largecolumns.
Figure30
Momentadjustmentfor largecolumns.
Loadingappliedto
rigidarm where 7
Equivalentframe
or grillagemodel
Bendingmoments
Shearforces
a) Rigidarm modal
dEquivalentframe
or grillagemodel
Bendingmoments
Shearforces
b) Momentadjustmentmodel
0.4
- 0.2
I
--L
E
I
fd
-0.2
-0.4
olumndepth/shorter span=0.25
-1--- 1
-r ~ -
Longerspan/shorter span
1 (farendfixed)
1 (farend pinned)
---3 (farendfixed)
---3 (farend pinned)
1
‘t
0.2 0.4 0.6 0.8 10
Columnstifmcss/totaljoint stiffness

Guide to the design and construction of reinforced concrete flat slabs (1)

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