TECHNICAL MANUAL - Geoplast

English

GEOPLAST SLAB SOLUTIONS

TECHNICAL MANUALNEW NAUTILUS - NEW NAUTILUS EVOMODELING, CALCULATION, ON-SITE INSTALLATION

INDEXNEW NAUTILUS EVO TECHNICAL MANUAL1. Introduction ......................................................................................pag. 05 1.1 Slabs ...........................................................................................................................pag. 061.1.1 Strengths ....................................................................................................................................pag. 061.1.2 Weaknesses ...............................................................................................................................pag. 061.2 Lightened slabs ...........................................................................................................pag. 071.2.1 Advantages ................................................................................................................................pag. 071.3 Reticular slabs ............................................................................................................pag. 071.3.1 Advantages ................................................................................................................................pag. 071.3.2 Weak points ................................................................................................................................ pag. 081.3.3 Conclusions ................................................................................................................................pag. 081.4 Piastre alleggerite a corpi cavi ...................................................................................pag. 091.4.1 Advantages ................................................................................................................................pag. 091.4.2 Weak points ................................................................................................................................pag. 091.4.3 Conclusions ................................................................................................................................pag. 091.5 Lightened slabs with hollow articles .........................................................................pag. 101.5.1 General characteristics .............................................................................................................pag. 10

2. Technical data ..................................................................................pag. 11 Dimension tables ........................................................................................................................pag. 12

3. Calculation Manual ...........................................................................pag. 153.1 Slab theory ..................................................................................................................pag. 163.2 Predimensioning .........................................................................................................pag. 173.2.1 Determination of thickness ........................................................................................................pag. 173.2.2 Determination of the size of the minimum capital .....................................................................pag. 183.2.3 Determination of the size of the capital so as not to have to place shear reinforcement ........pag. 183.3 Stress calculation .......................................................................................................pag. 193.3.1 Stripe method.............................................................................................................................pag. 193.3.1.a Calculation of bending moments ...............................................................................................pag. 19 Plateonpillarssimplifiedcalculationprocedure .......................................................................pag. 213.3.1.b Calculation of shear stress .........................................................................................................pag. 223.3.1.c Calculation of punching .............................................................................................................pag. 223.3.2 Grashof’s method .......................................................................................................................pag. 223.3.3 F.E.M. modeling. .........................................................................................................................pag. 243.3.4 Practical example .......................................................................................................................pag. 273.4 Verification ..................................................................................................................pag. 293.4.1 Calculationofflexuralreinforcements .......................................................................................pag. 293.4.2 Calculation of the reinforcement stripe method ........................................................................pag. 303.4.2.a Calculation of bending moments ...............................................................................................pag. 303.4.3 Calculationandverificationofshearreinforcements ................................................................pag. 333.4.4 Types of transverse reinforcements ...........................................................................................pag. 34

3.4.5 Verificationoftheinterfacebetweenconcretelayerscastatdifferenttimes ...........................pag. 34 Regulatory requirements ............................................................................................................pag. 35 Haveanideaofthedistanceinwhichtheshearreinforcementisinserted .............................pag. 353.4.6 Calculationandverificationtopunching ...................................................................................pag. 35 Calculationmodel-verificationoftheresistanceonthecriticalperimeter ..............................pag. 36 Punching resistance ...................................................................................................................pag. 373.4.7 Deflectioncontrol .......................................................................................................................pag. 383.5.7 Construction details ...................................................................................................................pag. 39

4. On - site application .........................................................................pag. 414.1 Preparation of the formwork ......................................................................................pag. 424.2 Laying the lower layer of the reinforcement ..............................................................pag. 434.3 Additional reinforcement ............................................................................................pag. 434.4 Laying of the lightening ..............................................................................................pag. 444.5 Laying the upper layer of the reinforcement ..............................................................pag. 454.6 Spacers .......................................................................................................................pag. 454.7 Shear and punching reinforcement ............................................................................pag. 464.8 Perimeter reinforcement ............................................................................................pag. 464.9 Requirements for concrete casting ............................................................................pag. 464.10 Concrete mix design ...................................................................................................pag. 474.11 Opening of cavities before casting .............................................................................pag. 474.12 Opening of cavities after the construction of the slab ..............................................pag. 484.13 Use of dowels and anchoring .....................................................................................pag. 485. Special applications .........................................................................pag. 495.1 Prefabrication .............................................................................................................pag. 505.2 Post-tensioning ...........................................................................................................pag. 505.3 Thermal activation of the masses ..............................................................................pag. 506. Certifications ....................................................................................pag. 516.1 Behaviorduringfireloadandfireresistance .............................................................pag. 526.1.1 500 ° C Isotherm method ...........................................................................................................pag. 556.2 Acoustic behavior .......................................................................................................pag. 586.3 Thermal conductivity ..................................................................................................pag. 597. Packaging and logistics ...................................................................pag. 617.1 New Nautilus Evo - standard packaging conditions for land shipment .....................pag. 628. Appendix ..........................................................................................pag. 63 8.1 Kirchoff slab ...............................................................................................................pag. 648.2 Stress regime ..............................................................................................................pag. 648.3 Internal actions ...........................................................................................................pag. 658.4 Equilibrium equations .................................................................................................pag. 658.5 Boundary conditions ...................................................................................................pag. 668.6 Internal actions on a generic section .........................................................................pag. 66

1. INTRODUCTION

6 IIntroduction • chap 1

1.1 SLABSNowadays, reinforced concrete slabs are very commonstructures in the construction market. Although the tra-ditionalbeamslabsandhollowblockslabsarestillveryused,othertechnologieshavecertainlygainedtheupperhand.Ifwegetacloserlookatwhathappensaroundtheworld,beamslabsremaininuseinthecountrieswherethecostof materials are higher than labor costs.Ontheotherhand,therichestandthemostindustrializedcountries,mostlyusedeckplatesystems.Reinforced concrete slabs:

1.1.1 STRENGTHSThereasonsareeasilyidentifiable,asthereinforcedcon-crete plates:

1. they are very solid thanks to the side deformation prevention, which allows them to deform less and reduce thickness:

a thickness reduction allows the economization of materials;

b massing reduction,allows themaximizationof the groundsurfaceexploitation,whichrepresentsanim- portant cost;

2. there is no need of beams:

aonceagaintheyallowthereductionof the volumetric footprint of the deck;

b evitano tempi e costi di casseratura delle travi;

cfacilitano il passaggio degli impianti, riducendone notevolmente i tempi di posa;

3. avoid the scaffolding of the beams:

afacilitatethepassageoftheinstallations,significantly reducing installation times;

b meshes and straight bars are easier and faster to install;

c itispossibletousepre-fabricatedreinforcedsystems, inordertomaketheworkfaster(likeBAMTEClayers of reinforcements);

4. theyhaveexcellentfireandacousticbehavior,thanks to their mass.

Ifwereadthepointsabove,itwouldseemthattherearenoreasonswhyweshouldnotmakeourfloorswithslabs,however these structures also have someweak points,whichinfactlimittheiruse,inrespecttoothermoreeffi-cient methods:

1.1.2 WEAK POINTS1. They are massive structures:

a they consume high quantity of concrete;

b theyareveryheavy:therearelargespansbetweenthe pillars,but theself-weightprevailsand the result is very expensive.

2. Theyarenon-ductilestructurestowhichtheprinciples of the hierarchy of resistances are not applicable:

a cannotworkonaloom;

b theyneedbracingwithseptaorsimilar;

c theyneedlowstructures.

Figura 1 - lightened slab with New Nautilus Evo of Geoplast.

7Introduction • chap 1

1.2 LIGHTENED SLABS1.2.1 ADVANTAGESTo create a structure with all the characteristics andstrengthpointsofstandardslabs,butwithout theirself-weight would look like an ideal solution, but lightenedslabs:

1. limit volumes;

2. eliminatebeams/straightsoffit;

3. concrete is less expensive;

4. allowlargerspans;

5. allowoptimizationofverticalstructures;

6. reduce foundation load;

7. reduce excavation load.

1.3 RETICULAR SLABSThe slabs maintain their bidirectional structure and create an orthogonal grid through the installation of disposable blocks(inconcreteorterracotta)orreusable(inplasticorfiberglass).Leavingmassivecapitalsinthepillarsforthepunching.

1.3.1 ADVANTAGESTheadvantagesofferedbythistypeofsolution,aremultiple:

1. theseareslabswithoutbeams;

2. the quantity of concrete needed is reduced;

3. they are very light;

4. less steel is used;

5. the blocks to make the grids are very cheap.


1.3.2 WEAK POINTSThese structures also have some disadvantages:

1. they lose a lot of inertia compared to the massive slabs ofequal thickness,sotheymustcompensate for the greaterdeformabilitywithhigherthicknesses;

2. iftheydonotcomplywithcertaingeometricparame- ters, theydonothavesufficient torsionalstiffness to get a SLAB, therefore they have lower performance thantheequivalentFULLslab;

3. they must be reinforced grid by grid similar to the beams,withconsequentslowingofthelayingprocess;

4. they do not have good acoustic behavior;

5. theyhaveamediocrefirebehavior (nomorethanREI90’);

Figure 2 - example of reticular slab

6. due to the considerable deformability they have a limited range of use of lights and loads;

7. in the case of recoverable blocks, except for some particularapplications,theyneedafalseceiling.

1.3.3 CONCLUSIONSThis type of solution is ideal alternative to the slab and maintains its characteristics and advantages but in a well-definedfieldofapplication,apartfromtheseapplica-tions they become less competitive.

9Introduction • chap 1

1.4 LIGHTENED SLABS WITH HOLLOW ARTICLESHollowarticlesareembeddedinthepour,theyareusuallymadeofcubicshapedlow-densitypolystyreneorplastic.Blocksremainembeddedinthepourandcreateagridofribbings,whichareenclosedbetweentwomassiveupperandlowerslabs.

1.4.1 ADVANTAGESThissolutionismoreefficientthanmostreticularslabs:

1. presenceofthelowerslabmakes it perfect for all the purposes;

2. the same thicknessor even lower thicknessof a full slab can be maintained;

3. lightness and concrete savings are guaranteed;

4. theycanbereinforcedinthesamewayasmassiveslabs;

5. quantity of steel is reduced;

6. they have good acoustic behavior;

7. greatfirebehavior(uptoREI240’);

8. there is no need of a false ceiling.

1.4.2 WEAK POINTSThese structures also have some disadvantages:

1. in comparison to reticular slabs they consume more concreteandweightmore;

2. they consume more steel with the same concrete consumption and inertia compared to the reticular slabs,duetothelowerlever.

1.4.3 CONCLUSIONSThistypeofsolutionisidealfornarrowspansandreducedloads can be economically less interesting than the reticu-larslab,despitehavingclearlysuperiorperformances.Onthecontrary,itisabsolutelycompetitiveifcomparedtothefullslabsolution,especiallyintherangeofthicknessesfrom28to60cm,andspansbetween8and14m.


1.5 LIGHTENED SLABS WITH HOLLOW ARTICLES IN PLASTIC 1.5.1 GENERAL CHARACTERISTICSInthelast10-15yearsago,lightenedslabswerecreatedthroughtheuseofcubicblocksinlowdensitypolystyrene.

This construction technology had some disadvantages:

1. blockswerefragileandsufferfromweathering(water imbibition);

2. blocks were occupying a lot of space and did not facilitateworksitelogistics;

3. It was difficult to block them and keep them lifted by inferior reinforcements. Thismade it necessary tocast the lower slab, layand block the lightened slabs, complete the installation of the reinforcements and complete the casting. Thispracticemadeworktime-consuming;

4. incaseoffireithasbeenshownthatthepolystyrene sublimatescreatingoverpressureinsidethehollowthat cancauseexplosionsoftheslabs,furthermorethese gases are toxic.

Duringrecentyearsanewtechnicalmethodhasarrivedinthemarket.Thissolutionfinallyallowstheoverrunoftheselimitations.Thosearerecycledpolypropyleneformworks,52x52cmwithvariableheight.Theycanbe“single”,or“double”,byputtingtogethertwo“single”.

Figura 3 - layout of lightening New Nautilus Evo double

Figura 4 - plastic lightening “single” type

Figura 5 - plastic lightening “double” type

2. TECHNICAL DATA

12 Technical data • chap 2

HEIGHT Foot

p (mm)Spacerd (mm)

Actual size (mm)

Weight (kg)

± 10%Beam width

N (mm)Formwork

bearing (pz./m2)

Concreteconsumption CLS (m3/m2)

Formwork volume (m3/pz.)

H10 SINGLE

0-50-60-70-80-90-100 8 520 x 520 x H100 1.12

100120140160180200

2.602.442.302.162.041.93

0.0380.0410.0450.0480.0510.054

0.024

H13 SINGLE

0-50-60-70-80-90-100 8 520 x 520 x H130 1.18

100120140160180200

2.602.442.302.162.041.93

0.0570.0620.0660.0690.0730.076

0.028

H16 SINGLE

0-50-60-70-80-90-100 8 520 x 520 x H160 1.25

100120140160180200

2.602.442.302.162.041.93

0.0770.0820.0870.0910.0950.098

0.032

H20 SINGLE

0-50-60-70-80-90-100 8 520 x 520 x H200 1.35

100120140160180200

2.602.442.302.162.041.93

0.0990.1050.1100.1160.1200.125

0.039

H24 SINGLE

0-50-60-70-80-90-100 8 520 x 520 x H240 1.45

100120140160180200

2.602.442.302.162.041.93

0.1200.1280.1340.1410.1460.151

0.046

H28 SINGLE

0-50-60-70-80-90-100 8 520 x 520 x H280 1.55

100120140160180200

2.602.442.302.162.041.93

0.1420.1510.1580.1650.1720.178

0.053

DIMENSIONAL TABLES NEW NAUTILUS EVO SINGLE NEW NAUTILUS EVO DOUBLE

8mm

EC2

In the upper section of the form-workthereareuniformlydistributedspacers 8 mm thick. Theseelementsallowtheupperre-inforcement to be placed directly overtheformworkinordertoguar-antee a suitable concrete covering.

2x pz.

100-200mm

PPpolipropilene

Everyformworkisprovidedwithsidespacersthatallowthecorrectinstalla-tion of the elements according to the widthofthebeams,whichistobecal-culated during the design stage. The elements are marked from 100 to 200 mm and can be hooked to the side loops.

Thelowerspacerfeetareintegralele-mentsoftheformwork:theyaremold-edwiththerestoftheformworkandallow the creation of the lower slabwithathicknessevaluatedduringthedesign stage.The feet have a variable height from 50 to 100 mm.

R.E.I. 1804

x pz.

50-100mm

THE UPPER SPACER THE SIDE TAB THE LOWER FOOT

Geoplast Nuovo Nautilus Evo

52 cm

52 c

m

Geoplast Nuovo Nautilus Evo Double

Geoplast Nuovo Nautilus Evo DoubleGeoplast

i

NpH

d

Geoplast Nuovo Nautilus Evo Geoplast Nuovo Nautilus Evo

i

NpH

d

13Technical data • chap 2

HEIGHTFoot

p (mm)Spacerd (mm)

Actual size (mm)

Weight (kg)

± 10%Beam width

N (mm)Formwork

bearing (pz./m2)

Concreteconsumption CLS (m3/m2)

Formwork volume (m3/pz.)

H20 DOUBLE

0-50-60-70-80-90-100 8 520 x 520 x H100

+ H100 2.24

100120140160180200

2.602.442.302.162.041.93

0.0990.1050.1100.1160.1200.125

0.048

H23 DOUBLE

0-50-60-70-80-90-100 8 520 x 520 x H100

+ H130 2.30

100120140160180200

2.602.442.302.162.041.93

0.0950.1030.1110.1180.1240.130

0.052

H26 DOUBLE

0-50-60-70-80-90-100 8 520 x 520 x H130

+ H130 2.36

100120140160180200

2.602.442.302.162.041.93

0.1140.1230.1310.1390.1460.152

0.056

H29 DOUBLE

0-50-60-70-80-90-100 8 520 x 520

x H130 + H160 2.43

100120140160180200

2.602.442.302.162.041.93

0.1340.1440.1520.1600.1680.174

0.060

H30 DOUBLE

0-50-60-70-80-90-100 8 520 x 520

x H100 + H200 2.47

100120140160180200

2.602.442.302.162.041.93

0.1360.1460.1550.1640.1710.178

0.063

H32 DOUBLE

0-50-60-70-80-90-100 8 520 x 520

x H160 + H160 2.50

100120140160180200

2.602.442.302.162.041.93

0.1540.1640.1730.1820.1890.197

0.064

H33 DOUBLE

0-50-60-70-80-90-100 8 520 x 520

x H130 + H200 2.53

100120140160180200

2.602.442.302.162.041.93

0.1560.1660.1760.1850.1930.201

0.067

H34 DOUBLE

0-50-60-70-80-90-100 8 520 x 520

x H100 + H240 2.57

100120140160180200

2.602.442.302.162.041.93

0.1580.1690.1790.1890.1970.205

0.070

H36 DOUBLE

0-50-60-70-80-90-100 8 520 x 520

x H160 + H200 2.60

100120140160180200

2.602.442.302.162.041.93

0.1750.1870.1970.2060.2150.223

0.071

H37 DOUBLE

0-50-60-70-80-90-100 8 520 x 520

x H130 + H240 2.63

100120140160180200

2.602.442.302.162.041.93

0.1770.1890.2000.2100.2190.227

0.074

H38 DOUBLE

0-50-60-70-80-90-100 8 520 x 520

x H100 + H280 2.67

100120140160180200

2.602.442.302.162.041.93

0.1800.1920.2030.2130.2230.231

0.077

H40 DOUBLE

0-50-60-70-80-90-100 8 520 x 520

x H200 + H200 2.70

100120140160180200

2.602.442.302.162.041.93

0.1970.2100.2210.2310.2410.250

0.078

H41 DOUBLE

0-50-60-70-80-90-100 8 520 x 520

x H130 + H280 2.73

100120140160180200

2.602.442.302.162.041.93

0.1990.2120.2240.2350.2450.254

0.081

H44 DOUBLE

0-50-60-70-80-90-100 8 520 x 520

x H200 + H240 2.80

100120140160180200

2.602.442.302.162.041.93

0.2190.2320.2450.2560.2670.276

0.085

H48 DOUBLE

0-50-60-70-80-90-100 8 520 x 520

x H240 + H240 2.90

100120140160180200

2.602.442.302.162.041.93

0.2410.2550.2690.2810.2920.303

0.092

H52 DOUBLE

0-50-60-70-80-90-100 8 520 x 520

x H240 + H280 3.00

100120140160180200

2.602.442.302.162.041.93

0.2620.2760.2930.3060.3180.329

0.099

H56 DOUBLE

0-50-60-70-80-90-100 8 520 x 520

x H280 + H280 3.10

100120140160180200

2.602.442.302.162.041.93

0.2840.3010.3170.3310.3440.356

0.106

3. CALCULATION MANUAL

16 Calculation manual • chap 3

3.1 SLAB THEORYTheslabsareatypeoftwo-dimensionalstructuralelements which,preservingtheirthicknesses,combineexcellentstruc-turalperformance,fastinstallationaswellaseconomicandfunctional advantages.The structural behavior of the slab elements is characterized by the prevailing flexural behavior (flexion, shear, torsion),whichensuresatransferofloadsalongorthogonalpathsac-cordingtoasinglepreferentialdirectionortwoormorepref-erential directions, thereforedividing into “mono-directionalslabs”,inthefirstcase,or,“bidirectionalslabs”inthesecondcase.Theloadsthatweighontheslabcanbetransmittedinvariousways,bothpoint-wiseandcontinuously,thereforethefollow-ing categories of slabs can be identified:

� slab with constant thickness on columns with or withoutcapital(flatslab);

� slabwithvariable thickness,with local thickening in correspondenceofthecolumns(mushroomslab);

� slabsonedgebeams,placedontwoorfoursides;

� slabsonload-bearingwalls.

Asfarasthecalculationphaseisconcerned,thecodesaregenerallyrelatedtotheelasticanalysis,thereforeintheab-senceofcracking,and ignoring the reinforcement, accept-able hypotheses in the verification phase at the operating limit states.Onthecontrary,whentheultimatelimitstatesaretakenintoconsideration, thebehavioral non-linearity, due to the con-cretecrackingandtheplasticizingofthesteel,isnecessarilytaken into consideration.Certainly,thekeytounderstandingthemanyaspectsofthemechanical behavior of the slab is still provided by the the-ory of elasticity, in the hypothesis of small displacements.Theelasticmodel,basedonthehypothesisofdisplacementcontinuity, tensionsanddeformations,makes itpossible topreserve the concept of internal action as the local resultant force of the tensions that are having effect on a unitary sec-tion,whateverthenatureofthestress.A reference to the elastic method is obliged.

Asforthelightenedslabs,theshapeoftheboxconfiguresthe slab as a series of crossed ribs closed above and be-lowbytwoslabsofvariablethicknessbychoice.

Experimental results found in the appendix certify that this structure maintains the same nature and behavior as an orthotropic slab according to EC2.According toEurocode2, in fact, in the structural anal-ysis may not be necessary to decompose the ribbed or lightenedslabsintodiscreteelements,aslongasthewingor the upper structural part and the transverse ribs have adequatetorsionalstiffness.This assumption is valid if:

� thepitchoftheribsdoesnotexceed1500mm;

� theheightoftherib,belowthewing,isnotmorethan 4timesitswidth;

� the thickness of the wing has equal or greater valuethanthehighestvaluebetween1/10ofthenet spanbetweentheribsand50mm;

� thereare transverse ribsdistant fromeachotherno more than 10 times the total thickness of the slab.

A different pitch of the ribs leads to a different bending-re-sistantmechanism;infact,inamono-directionalslab,thezones of slab that are more distant from the rib itself do not appear to be fully collaborating.

Following the same logic, the distribution of the normalstresses in slab and counter-slab that are created due to the flexing of the deck are concentrated near the ribs and aredecreasingwhilemovingfurtherfromthem(aphenom-enon called shear moment).This kind of behavior is less important for the structures in question,sincetheribsontwosidespreventthebehaviorfrombeingpurelyone-way.

17Calculation manual • chapter 3

3.2 PREDIMENSIONING3.2.1 DETERMINATION OF THICKNESS The constructive configuration in which the technologyNew Nautilus EVO expresses its potential to the maxi-mum,thatofabidirectionalplate,inthesituationsuchastohavearelationbetweenthespansinthetwoorthogonaldirections expressed as:

Outside this range the behavior becomes purely mono-di-rectional.Afirstwaytoobtainan indicativesizeof theslabthick-nessiswithsimpleproportions,basedonstructuraltypesin use and spans to be covered:

� fullplateonpillars

� lightenedplateonpillars

� fullplateonbeams

� lightenedplateonbeams

Theminimumsizeof theupperand lowerslabs iscon-strained by minimum covering requirements to be secured to the bars. Inside the slab the minimum reinforcement covering required by the regulations must be ensured for the category of exposure relating to the calculation hy-pothesesmade,plusthetwodiametersofthebasicrein-forcementinthetwodirections.Thethicknessoftheslabisconsiderablyreduced,com-paredtoabasicreinforcementcomposedofelementswithelectro-weldedmeshes,ifaloosebarsolutionareused.

Figura 6 - arrangement of electrowelded meshes.

Asyoucanseefromthedrawing,thepositionofarmaturescontaining electro-welded meshes gives the necessaryoverlapping,atleastpartial,intheborderareasbetweentwoadjacentmeshes.Inthesmallsectionbelongingtoallfourofthedesignedmeshesitwillbenecessarytohavethespacetoplaceeightbarsintheslab,plusthespacefor necessary distancesbetween the lower bar and theupper sideof theslab,andbetween theupperbarandthe formwork, inaway thatallowsacorrect flowof theconcrete pour.

To thesegeometricestimationswemustadd fire safetyconsiderations;infact,greaterthicknessesfitwithbetterfire load resistance.

The size of the ribs is also related to the type of additional reinforcements required; in fact, if it is requiredand it isdecidedtoreinforcetheribs,itmustbedoneinawaythatthebarsinsidethemcanbeplaced,ensuringatthesametimeacorrectdistancefromtheformworks,aswellasre-inforcement gap.The minimum spacing must be such that:

Furthermore,moremassiveribs,asitwillbeshownlater,correspondtogreatershearresistance,thereforealargersize may be required to handle high stresses.


3.2.2 DETERMINATION OF THE SIZE OF THE MINIMUM CAPITALIt isadvisabletoleave,inthevicinityofthesupportare-asof theslab (columns, load-bearingwalls),areaswith-outlightenedslabs,inordertoallowthetransmissionofshearing actions by having the resistance of the full zone.The extension of the capitals over the pillars can be calcu-lated in the first instance by taking as the minimum exten-sion the one that contains the punching perimeter able to resistwithoutreinforcement,andinanycasetheonenotinferior than 2.75 fromtheedgeofthepillar,wheredisthe useful height of the section. Thecriticalperimeter, ,hasthefollowingexpression,asdefinedinEurocode2atpoint6.4.5(4),asitwasalsoshownpreviously:

Therefore, when the dimension of the circumference isobtained,itiseasytogobacktotheradius,andthereforeto the minimum size of the capital:

Figura 7 - non-lightened concrete area dimensions - capital

3.2.3 DETERMINATION OF THE SIZE OF THE CAPITAL SO AS NOT TO HAVE TO PLACE SHEAR REINFORCEMENTIfyoudonotwanttoimplementashearverification,andreinforcetheslabribs,youcanenlargethefullslabarea,sothatthestressinthelightenedareaisalwaysinferiororequal to the resistant shear of the sole concrete.Anideaofwhatthesizeofthecapitalshouldbeinordertoachievethisresultcanbeobtainedasfollows:Let’s consider a slab consisting of an infinite series offields,withspansgoinginthetwodirections, and, let’sanalyzethispillar;thispillarwillhaveacapitalwiththefollowingdimensions, subjecttoaload;weexcludethecontributionoftheloadonthecapitals,which,stronglyreinforcedfornegativebendingandpunching,canbehereconsidered as rigid constraints.The vertical action directed to the remaining portion of the deck is:

Figura 8 - area subject to a considered shearing action


This action is counterbalanced by the resistant shear of theribsattheinterfacebetweenthecapitalsandthere-mainingportionoftheslab,whichcanbeexpressedas:

Where is resistantcutofthesoleconcreteoftherib,isthe,numberofribsthatcontribute,thatcanbecal-

culated as the perimeter of the capitals in the considered field,dividedbyinteraxialspacingbetweentheribs .Pertanto l’azione resistente totale è la somma delle azioni resistenti delle nervature.Therefore,thetotalresistantactionisthesumoftheresist-ant actions of the ribs.Byequatingstressandresistance,Icanlookforthesizeof the capital ,whichsatisfiestherelation;incaseyouwanttoprovidesquarecapitals,theequationissimplifiedto:

This is a second-degree equation solvable in :

Alternatively, it ispossible to look for thecapitaldimen-sions,keepingthedimensionsdistinct and ,butafurther equation should be inserted to make the system resolvable; for example it can be set that the relation be-tween and ,isthesameasthatbetween and .

3.3 STRESS CALCULATION3.3.1 STRIP METHOD

3.3.1.a Calculation of bending momentsWithintheStaticMethod,theultimatelimitstateofaplateis characterized by the achievement of the limiting mo-ment in one or more sections.Therefore,therangeofmomentsmustobeytheequationoftheflexuralequilibriumoftheplate:

There are a priori infinite moment fields able to satisfy this equation,butthereisonlyoneexactsolution,correspond-ing to the formation of the kinematics.Withtheexceptionofa limitednumberofsimplecases,it is practically impossible to determine the exact solution soluwithouttheuseofautomaticcalculation.

However, it is possible to obtain satisfactory solutions,evenifapproximate,regardlessoftheformationofamech-anism,asproposedforexamplebyHillerborg,whoseStripMethod is easy to use.This method disregards the torsion, so the equation ofequilibrium becomes:

In the slotted plate there are at least three torsion-resistant mechanisms,andtherefore,inprinciple,thetorsioncan-notbeignoredHowever,whenthereinforcementisbeingsized(objectiveoftheHillerborgmethod),itiscautiousnotto take into account the torsion as the three mechanisms mentionedabove(torsioninthecompressedzone,aggre-gatemashingintheslottedplatesandwedgeactionofthereinforcements) too often depend on factors that are not partofthedesigner’scontrol,suchastheextensionofthecompressedzone,theopeningoftheslotsandthelocalflexural stiffness of the reinforcement.This hypothesis of neglected torsion corresponds to a hypothesis of a mechanical model including the plate madeofasetofstripsarrangedinthexandydirections,eachsubjecttobendingandshearing.Therefore, the stripsarrangedalong the linexabsorbapartoftheload, ,andthestripsintheydirectiontheremaining part ;thiscanbetranslatedwiththeequations:

Forpracticalreasons,variablevaluescannotbeassignedtothecoefficient withcontinuity,butconstantvaluesinzones. The plate must therefore be divided into zones,eachwithitsownvalue .The choice of this value must be made on the basis of the mosteffectiveflexioninthetransmissionoftheboundaryload,accordingtothefollowingcriteria:sincetwobentstripsintersectineachdxdyareolaoftheplate,oneparalleltothexaxisandtheotherparalleltotheyaxis,theloadactingonthatareolaistransmittedtotheboundarymainlybythestripthatismorerigidduringflex-ion(withlessspan,and/orwithmorerigidendconstraintsand/orconnectingtheareolawiththenearestconstraint);therefore, inthetransmissionoftheloadtothebounda-ry,prevailsthedirection or direction dependingonwhetherthestripalignedwithxorthestripalignedwithyislocallymorerigid.


Figura 9 - representation of the strip method according to Hillerborg.

In details: �plateuniformlyconstrainedtotheboundary:tothear-

easclosesttothesidesalignedwiththeyaxishighervalues of mustbeassigned (between0.5and1),whiletotheareasclosesttothesidesalignedwiththex,smallvaluesof mustbeassigned(between0and0.5) figure A.

For zones that are have the same average distance between twoequallyboundsides, it isalso reason-able to set and to adopt a distribution as in figureB.

Figura 10 - possible distribution (A) in a rectangular plate leaning on the boundary.

Figura 11 - possible distribution (B) in a rectangular plate leaning on the boundary.

�Platewithconstraintstomixedboundaries:theprox-imitytoa jointalignedwithyresults in largervaluesof whiletheproximitytoafreeedgealignedwithyresults in a null value of ,sinceitisnotpossible(dueto lack of constraint and structural continuity) any load transmissioninthexdirection(orthogonaltotheedgeitself).

Figura 12 - possible redistribution in a rectangular plate with con-straints to mixed boundaries.

The strip method also gives reason for the so-called “han-dles”,whichareboundaryareasfortheslots,andactasreal support beams for the strips that are orthogonal to them and that are interrupted by the presence of the slot.

Figura 13 - example of plate with slot and handles (assigned geometric dimensions).

In each strip the reinforcement must be sized on the basis ofthemaximummoment(thesizingisconservative):

or

Itshouldbenotedthatwehavealwaystalkedabout re-inforcement sizing, although in reality the strip methodmakes it possible to evaluate the resistant moments of the requested reinforcement.

;

In these formulas the thickness of the plate appears through the internal lever arm therefore if, asisusual,thethicknessoftheplate ,isassigned,there-inforcements are dimensioned; but if the reinforcement is assigned the thickness is dimensioned.


Plate on pillars simplifiedcalculationprocedureThe design of a plate field on pillars under the action of a vertical load canbesimplifiedas follows.Forobvi-ousbalancereasons, thesumof the resultantmoments

e ,and,alongthesectionsatthepillarsandinthespan,inthexdirection,mustbeequaltothetotalmomentcausedbyload, ; an analogous diagram can be taken into account for the moments in the y direction.Taking into consideration the dimensions of the sections ofthepillars,followsareductionofthemaximummomentcaused by loads compared to the case of a point support; it is possible to consider a value of span reduced by half thesizeofthepillarateachend.Considering,forexample,theydirection,withspan andpillarofsize, ,itispossi-ble to derive a total bending value in the direction equal to:

The sizing of the reinforcement is carried out by dividing thistotalmomentintoapositivemomentinthespan,andanegativeoneatthesupportsection,identifiedbythelineconnectingtwopillars.Thepercentageof carried by each section is determined by referring to the distribution of the bending moments of the elastic solution. According to the assumption of Parker and Gamble it is possible to dividethetotalmomentinthetwodirections:

or

attributing65%tothenegativemomentsinthesupport,and the remaining 35% to positive moments in the span. Fromthisobservationadesignprocesswasbornwhichdividestheplateintotwostrips,alateralone,whichcon-nectsthepillars,andacentralone,eachofwhichcarriespart of the total moment; the basis of this method is there-fore the use for the design of the average values of bend-ing moments. The percentages of the total moment that arecarriedbyeachstriparealwayschosenwithreferenceto the elastic solution: 70% of the negative moment is assignedtothelateralstripatthecolumns,and30%tothe central strip.

Ontheotherhand,thepositivemomentisdividedequally,50%to the lateralstrips,50%to thecentralones. Thelateralstripcanberepresentedasdividedintotwopartsofwhicheachoneoccupies20%ofthewidthofthefield(intotaltheyoccupy40%ofthewidth,thecentralone60%).

Figura 14 - diagram for the calculation of design moments in the y direction - width of the central and lateral strips.

Figura 15 - diagram for the calculation of the design moments - mo-ments of competence of each strip.

Since the previous divisions are made starting from the elasticsolution, it isadvisable toproceedwithout redis-tributingthemomentsbetweenthevaluesinthesupportsandthoseinthespan,inordertohaveagoodbehaviorofthe structure,with cracking and limited deflections. An-other reason for avoiding redistributions is the danger of platespunching in correspondencewith the support onthepillars,whichcanbe facilitatedby thecrackingdueto the flexion and yielding of the reinforcements in these areas.


3.3.1.b Calculation of shear stressLet’s consider a slab consisting of an infinite series offields,withspansinthetwodirections and ,andcapital withdimensions , ,subjecttoaload :assuming that the shear resulting from the reaction of the pillarsisabsorbedbythecapitals(verificationtopunchingofthepillar),thelightenedplatemustbeabletowithstandthe shear force resulting from the vertical load and acting ontheportionoftheslabshownonthefollowingfigure:

Figura 16 - area subject to a considered shearing action

This action must be absorbed by the ribs at the interface betweenthecapitalsandtheremainingportionoftheslab.Theshearforceforeachoftheribswillthereforebe:

3.3.1.c Calculation of punching If isthelastloadactingontheplate,thecapitalsmustbeabletowithstandthereactionofthepillarequalto:

3.3.2 GRASHOF’S METHODAnalternativesimplifiedmethodisbasedonananalysisthatcomparestheplatebehaviortoafictitiousbeam.Thisanalysisisbasedonthedeformationoftheplatefieldsdueto evenly distributed loads. It is imagined a division of the slabfieldintomanystrips,attheinfinitesimallimit,aandb,thatareorthogonaltoeachother.Thedistributedloadwillbe carried by the strip of the full slab according to a bidi-rectionalbehavior;thetwostripsinwhichthegenericplatefieldhasbrokendownwillcarryaloadratebasedonthestiffnessofthestrips,theserateswillbefoundthroughthe applicationofthefollowingapproximateformula(Grashof):

Once the competent load has been determined for strips a andb,thesimplificationofacontinuousbeammodelcanbe applied.


Figura 17 - constraint schemes


3.3.3 F.E.M. MODELINGÈpossibile,poi,modellareaglielementifinitilepiastreincalcestruzzo armato alleggerite con elementi Nuovo Nau-tilus EVO,elements,asfullplateswithreducedvaluesofstiffness andmass.Youcanadoptoneof the followingsolutions:

a. shapetheplatewithNew Nautilus EVO as a slab of the construction thickness, but withmultiplicative coefficients for mass and inertia reduction;

b. shape the plate withNew Nautilus EVO as a full slab of reduced thickness, in order to obtain equivalentstiffnessandweight;

c. shapetheplatewithNew Nautilus EVO as a slab of the construction thickness but introduce coefficients on the reinforced concrete material of the portion of theplateoccupiedby the lightening, inorder to reducetheYoung’smodulusandself-weight.

The solution aisobtainedasfollows:remembering that the flexural stiffness value is:

Where: � is Young’s Module;

� is the Moment of Inertia;

� is the Poisson’s Module.

Our goal is to shape an isotropic plate to the finite ele-ments,withthesamestiffnessasalightplate:

The coefficient obtained is the reductive coefficient to be applied to a full plate in the programwith finite ele-ments in order to obtain the equivalent flexural stiffness of theplatewithNew Nautilus EVO,and, if thematerial isthesame,itbecomes:

CALCULATE THE INERTIA OF THE LIGHTENED SLABELEMENT INERTIA [mm4] BARYCENTER [mm] AREA [mm2] VOLUME [m3]

H10 38577300 48.3 46361 0.024

H13 84834500 62.9 60350 0.028

H16 158272000 77.6 74339 0.032

H20 309335100 97.1 92909 0.039

H23 448121900 115.3 106712 0.052

H24 534784000 116.6 111643 0.046

H26 648180500 130.0 120701 0.056

H28 849526400 136.2 130295 0.053

H29 901349400 145.0 134690 0.060

H30 1002580000 151.2 139353 0.063

H32 1212130200 160.0 148600 0.064

H33 1332396800 165.8 153342 0.067

H34 1465266900 171.8 158004 0.070

H36 1729333800 180.5 167331 0.071

H37 1883915700 186.3 171994 0.074

H38 2052377400 192.2 176649 0.077

H40 2373891500 200.0 185982 0.078

H41 2570833000 206.8 190645 0.081

H44 3163889700 220.0 204635 0.085

H48 4109701700 240.0 223200 0.092

H52 5230090000 260.5 241938 0.099

H56 6534840000 280.0 280590 0.106


Therefore,theinertiaofthelightenedribcanbefound,us-ingHuygens-Steiner,as:

� : total thickness of the slab;

� :interaxialspacingbetweentheribs;

� : barycenter of the final lightened section;

� : barycenter of the empty zone;

� : moment of inertia in the x or y direction of the final lightened section;

� : moment of inertia in the x or y direction of the empty zone;

� : surface of the empty zone.

Dividing the obtained quantity by the length of the interaxial spacingbetween the formworks,we find the inertiavalueper linearmeter.While the inertia (per linearmeter)of thesolid section can be calculated as that of a rectangle of hei-ghtequaltotheheightoftheconcretesection,andofwidthequaltothewidthofanelementNew Nautilus EVO plus thesizeoftherib,theequivalentquantityforalightenedslab can be calculated as that of an ,profile,withlowerandupperwingsgivenbytheslabs,andacorewithdi-mensions of the rib.

The solution b is obtained by finding a thickness of a full plate,thathasthesameflexuralstiffensofthelightenedplate:

Resolvingit,givestheexpressionofthefictitiousthickness:

Inacompletely similarway to theonedone for flexuralstiffnesses, torsionalandshearstiffnessesalsomustbereduced in order to shape the behavior of the lightened plate correctly

Figura 18 - section type - calculation of the reduction coefficients

The torsional stiffness of the full slab is determined ac-cording to the formula:

Where the factor is a function of the relation :

1.5 2.0 3.0 4.0

0.196 0.229 0.263 0.281

6.0 8.0 10.0 ∞0.299 0.307 0.313 0.333

Similarly,itcanbeobtainedforthelightenedslab,usingtheBredt’sformula:


So,thereductionfactorturnsouttobe: The multiplicative factor that takes into account the reduc-tion of the shear strength is obtained by comparing the areas resistant to the shear stress of the full slab and the rib of the lightened slab:

Whileaportionofunitarywidthoffullslabreactsentirelytotheshear,theshearareaofthelightenedslabisgivenonlyby the full area of the section.

Regardingtheweightofthelightenedslabitself,thiscanbe calculated by subtracting the volume of the New Nau-tilus EVOformworkpersquaremeter,fromtheweightofthe corresponding full slab:

Thisvalueallowsyou tofind the reduction factorof theself-weight:


3.3.4 PRACTICAL EXAMPLELet’s imagine thatweneed toperformthepredimensio-ningofthefollowingslabfield:

� � � �Concrete:strengthclassC32/40

Fromtheestimatesofpredimensioningwecanassumethatthe lightened slab should have a thickness equal to:

Potentiallyrefinedlater,onceourF.E.M.calculationmodelwascreated.Let us assume that we perform reinforcement with thebasemeshofØ820x20,withaconcretecoverof30mm,the slab must be of a size such that can accommodate twodiametersofthebasemesh,theconcretecover,plusadistancediameterbetweenthebasemeshandthelight-ening element:

Roundedup, theminimumsize for the lowerandupperslabs is 60 mm. Therefore:

Sincethe17cmlighteningdoesnotexistinthecatalog,let’s suppose thatweareusing theNew Nautilus EVO H16lightening.Inordertohaveatotalthicknessof29cm,itshouldbeproceededwiththeacquiringof7cmoflowerslaband6cmofupperslab;infact,itispreferabletousethelargerofthetwothicknessesinthelowerslab,soastoensuregreatercoverageofthereinforcingbars,witharesulting improvement of fire performance.Again, as a first estimate, we can assume an interaxialspacingof14cm,and,ifthisisnotsufficient,thethicknesswill be increased. In this casewewill have amaximumtheoretical impact of lightening equal to:

Theweightoftheplatewillthereforebe:

Thisself-weightvaluewillnotbethecorrectoneasitdoesnotconsiderthefullareas,however,wecanuseitasafirstestimate.Atthispointwecancalculatetheminimumsizeoftheca-pital required:

Value obtained by multiplying the ultimate load by the area of influence of a central pillar

This means that at least one area circumscribed by a cir-cumferenceofdiametershouldbeleftaroundthepillarwi-thout any lightening.

Figura 19 - critical per meter and area without lightening


If,ontheotherhand,our intent istofindthesizeofthecapital such as to avoid having to insert shear reinforce-ment,thecalculationisthefollowing(asalreadyexplainedabove):

Where:

Resolvingit,givesthemaximumsizeofthecapital:

Wecanthenproceed,havingdefinedthegeometryofthedeck,withthecreationofthemodelforthefiniteelements.Taking into consideration that the minimum estimated capitalis3.10mwide,assumingtheeliminationabovethepillarofawholenumberoflightening,itshallbetakenaminimum capital equal to:

Atthispointwecancalculateaself-weightinamorepre-ciseway:

1. The total reference surface is equal to:

2. Theareawithoutlighteningwillbeequalto:

3. Thenetlightenedsurfacewillthereforebe:

4. Theweightofthemassiveslabwillthereforebe:

Theaverageweightoftheplatewillthereforebe:

It is possible to notice that compared to the initial esti-mate,theactualeffectiveweightdiffersbyapproximately6%,therefore,anacceptableinaccuracy.Inthisphase,however,theminimumcapitalrequirementsmust be combined, with the convenience of maximumlightening,respectingminimumfootingdistancesfromtheedgeoftheplate(intheorderof30-40cm).

Figura 20 - FEM model, 2D.

Figura 21 - FEM model, 3D.


3.4 VERIFICATION3.4.1 CALCULATION OF FLEXURAL REINFORCEMENTS The calculation is made in the assumption that in the pres-enceofthelastloadsthereinforcement,inthetwodirec-tionsxandy,developsitsownlastresistantmoments:

The assumption of simultaneous development of ultimate resistant moments implies infinite ductility of the reinforc-ingbars, therefore theelasticbehavior isnegligible; fur-thermore,thecongruencebetweentheextensionsofthetwobar familiesmaynotbeconsidered, since thecon-crete cracks under tension and is not considered as a con-straint for the reinforcement.Thefollowingisthemethodofthenormalmoment,usedfor the local dimensioning of the reinforcement failure: thereforeitisdifferentfromothermethods,socalledglob-almethods,inwhichthereinforcementissizedandbasedon thebehaviorof theentire structurecollapse (like themethod of fault lines).The last resistant moments developed by the reinforce-ment on a position having normal axis eand tangent axisassumethefollowingexpressions:

Itcanbeobservedthatasecondmemberhasnotorque,sincethebarsareimaginedastensionedthreads,togeth-erwiththecompressedconcrete,theonlyresistantmech-anism that the reinforcement develops is bending.Foranyposition,andforanyorientationof ,thecalcula-tionofthereinforcementmustcomplywiththefollowingconditions:

It shouldbenoted that, however, in the reinforcedcon-crete, evencracked, there are variousmechanisms thatallowresistancetotorsion:

�Dowelactiondevelopedbythereinforcementthat is stretched in the cracked area;

�continuityofthematerialinthecompressedzone;

�aggregateinterlockofcrackedsides.

Such mechanisms, although they are not being at thesametimeatthemaximumoftheiractions,contributetothesatisfactionofthesecondwritteninequality.Thenextstepistoidentifythemostcriticalposition,incorrespondenceofwhichhereistheleastdistancebe-tweenaresistantmomentandanactingmoment,whichcan be expressed as:

The optimal utilization of the resources of the section in questionsuggests imposingtheequalitybetweenstressmomentandresistantmoment,thatis:

The third equation is obtained by imposing that the previ-ousequationsaresatisfiedwiththeminimumamountofreinforcement:

However,rememberingthedefinitionoftheresistantmo-mentsgivenatthebeginningoftheparagraph,thereisadirectproportionalitybetweenthereinforcementsandthementioned moments:

Finally,itisnotedthat obtained:

Itcanbeshownthat,for, ,thatis,atthelower stretchededge, theexpressions for the reinforce-mentdesignoccurinthefollowingform:

And it entails the need to have a position of reinforcement thatdevelops,inbothdirections,apositiveresistantmo-ment.


Onthecontrary, todetermine the reinforcementson theupperedge(negativeresistantmoment),wehavethefol-lowingexpressions:

Themathematicalmodeljustillustratedcorrespondstoaprecisephysicalmodel,withtheformationofaframeworkofties(reinforcingbars)andstruts,onthestretchededge.The resolving expressions that are found must be applied jointly in every point of the structure: there are indeedpoints inwhichthere isaweakflexionandastrongtor-sion,andthatisuntilthefollowinginequalitiesareverified:

In this case the stretched reinforcements are required by thetwistingstress.

3.4.2 CALCULATION OF THE REINFORCEMENTS STRIP METHOD3.4.2.a Calculation of bending momentsWithintheStaticMethod,theultimatelimitstateofaplateis characterized by the achievement of the limit moment in one or more sections.Therefore,therangeofmomentsmustobeytheequationof the flexural equilibrium of the plate:

There are a priori infinite moment fields able to satisfy this equation,butthereisonlyoneexactsolution,correspond-ing to the formation of the kinematics.Withtheexceptionofa limitednumberofsimplecases,it is practically impossible to determine the exact solution without the use of automatic calculation. However, it ispossibletoobtainsatisfactorysolutions,evenifapproxi-mate,regardlessoftheformationofamechanism,aspro-posedforexamplebyHillerborg,whoseStripMethod iseasy to use.

This method disregards the torsion, so the equation ofequilibrium becomes:

In the slotted plate there are at least three torsion-resistant mechanisms,andtherefore,inprinciple,thetorsioncan-notbeignored.However,whenthereinforcementisbeingsized(objectiveoftheHillerborgmethod),itiscautiousnotto take into account the torsion as the three mechanisms mentionedabove(torsioninthecompressedzone,aggre-gatemashingintheslottedplatesandwedgeactionofthereinforcements) too often depend on factors that are not partofthedesigner’scontrol,suchastheextensionofthecompressedzone,theopeningoftheslotsandthelocalflexural stiffness of the reinforcement.This hypothesis of neglected torsion corresponds to a hy-pothesis of a mechanical model including the plate made ofasetofstripsarrangedinthexandydirections,eachsubjecttobendingandshearing.Therefore, the stripsarrangedalong the linexabsorbapartoftheload, ,andthestripsintheydirection-the remaining part ;thiscanbetranslatedwiththe equations:

For practical reasons, variable values cannot be as-signed to the coefficient with continuity, but constantvalues in zones. The plate must therefore be divided into zones,eachwith itsownvalueof . The choice of this valuemust bemadeon thebasis of themost effectiveflexioninthetransmissionoftheboundaryload,accord-ing to the following criteria: Since twobent strips inter-sectineachdxdyareolaoftheplate,oneparalleltothexaxisandtheotherparalleltotheyaxis,theloadactingonthat areola is transmitted to the boundary mainly by the stripthatismorerigidduringflexion(withlessspan,and/orwithmorerigidendconstraintsand/orconnectingtheareolawiththenearestconstraint);therefore,inthetrans-missionoftheloadtotheboundary,prevailsthedirection

or direction dependingonwhetherthestripalignedwithxorthestripalignedwithyislocallymore rigid.


Figura 22 - representation of the strip method according to Hillerborg

In details: �Plateuniformlyconstrainedtotheboundary:tothear-

easclosesttothesidesalignedwiththeyaxishighervalues of mustbeassigned (between0.5and1),whiletotheareasclosesttothesidesalignedwiththexaxsis,smallvaluesof mustbeassigned(between0 and 0.5) figure A.

For zones that are have the same average distance between twoequallyboundsides, it isalso reason-able to set and to adopt a distribution as in figureB.

Figura 23 - possible distribution (A) in a rectangular plate leaning on the boundary

Figura 24 - possible distribution (B) in a rectangular plate leaning on the boundary

� Platewithconstraintstomixedboundaries:theprox-imitytoa jointalignedwithyresults in largervaluesof whiletheproximitytoafreeedgealignedwithyresults in a null value of ,sinceitisnotpossible(dueto lack of constraint and structural continuity) any load transmissioninthexdirection(orthogonaltotheedgeitself).

Figura 25 - possible redistribution in a rectangular plate with con-straints to mixed

The strip method also gives reason for the so-called “han-dles”,whichareboundaryareasfortheslots,andactasreal support beams for the strips that are orthogonal to them and that are interrupted by the presence of the slot.

Figura 26 - example of plate with slot and handles (assigned geometric dimensions)

In each strip the reinforcement must be sized based on the maximummoment(thesizingisconservative):

or


ItOncethestressmoments,calculationsandverificationofreinforcementcanbedonewiththeusualfailuremethodonthedesignsectionqualitativelyillustratedinthefigurebellow.

Figura 23 - shows the seasonal use for calculating the failure method


3.4.3 CALCULATION AND VERIFICATION OF SHEAR REINFORCEMENTSUponextractionofthevalueofthestressedshearaction,fromabacusorfromanalysisoffiniteelements,thecalcu-lation of the shear reinforcement can be obtained through the same procedure applied to an I profile.The rib resistance in the absence of transverse reinforce-mentiscalculatedinthefollowingway:

Where:The terms that appear in this equation are:

� isthecharacteristicstrengthofconcrete,expressedin Mega Pascal;

�thepercentageofstretchedreinforcement,readytore-sist bending is:

as this value increases, the resistance to shearing in-creases,duetotheincreaseinthemeshingeffect;

�theterm

takes into account the dimensional effect. In relation to thephenomenaofbrittlefracture,theplatesofgreaterthickness have less resistance to punching per surface unit along the critical perimeter. This term cannot be greater than 2;

� istheshear-resistantarea,wherethefirsttermisthewidthofthejoist’score,whilethesecondistheuse-ful height.

TheEurocode(6.2.1(4))saysthat:“If,onthebasisofthesheardesigncalculations,noshearreinforcement is required, it is recommended to use atleast a minimum reinforcement according to point 9.2.2.This minimum reinforcement can be omitted in elements suchasslabs (full, ribbed,hollow)where the transversedistribution of loads can occur. […]”

Inareaswheretheshearexceedstheresistantshearoftheconcrete,itisnecessarytoinsertashearreinforcement.Thedesignofelementswithshearreinforcementisbasedonatrussmodel,and,forelementswithverticalshearre-inforcement,shearstrengthisdefinedas:

Where:

The terms that appear in this equation are:

� ,crosssectionareaoftheshearreinforcement;

� ,thetiepitch;

� ,theyieldstressoftheshearreinforcementdesign;

�itisrecommendedforthe ,angle,whichrepresentstheangle of inclination of the concrete strut compared the axisofthebeam,tobelimited:

It should be remembered that setting the maximum val-ue corresponds to finding the minimum amount of rein-forcement necessary to satisfy a given request for shear action.

� itistheinternalleverarm,foranelementofconstantheight,correspondingtothemaximumbendingmomentin the considered element.

While the maximum strength of the concrete is:

Where:� is a coefficient for reduction of the concrete strength crackedduetoshear,thevalueofwhichcanbefoundintheEurocodenationalappendix,whileitsrecommendedvalue is:

� is a coefficient that takes into account the interaction betweenthetensioninthecurrentstressandanyaxialcompression tension. Its value is unitary for non-pre-stressed structures.

Inamoregeneralsituation inwhichtheshearreinforce-ment is inserted at an angle ,theshearstrengthisequaltothelowestvaluebetween:


3.4.4 TYPES OF TRANSVERSE REINFORCEMENTSThere are various types of transverse reinforcement in use: shearstuds,shearhooks,closedties;theseconstructiondetailscanalsobefoundin“FibBulletin2:StructuralCon-crete”

Figura 28 - sheer-resistant hooks

Figura 29 - shear- resistant open ties

Figura 31 - shear-resistant prefabricated pins

3.4.5 VERIFICATION OF THE INTERFACE BETWEEN CONCRETE LAYERS CAST AT DIFFERENT TIMESThe value of the stressed slip force generated betweenlayers of concrete cast at different times can be estimated withreferencetoEC2par.6.2.5:

Where:

� is the ratiobetween the longitudinal force in the lastconcrete casting and the total longitudinal force in the com-pressedor stretchedarea,bothcalculated in theconsid-ered section;

� is the transverse shear force;

� it is the internal lever arm ;

� isthewidthoftheinterface(inourcaseequaltothe interaxialspacingbetweenthelightning);

The shear resistance of the interface design is given by:

Where:

� =0,40

� =0,7

� tensile strength of the concrete design

� tension produced by the minimum external force acting in the interface thatcanactsimultaneouslywith theshearforce,positiveiffromtensioning,butsuchthat is negative if from tension. If is from traction it is recommended to assume equal to 0;

� ratio between the reinforcement area that crosses the interface and the area of the interface;

� coefficient of reduction of the resistance referred to in paragraph6.2.2(6);

Figura 30 - sheer-resistant hooks


� is a factor, which considers the indented constructionjointsuchthat .

Check that it is less than .

REGULATORY REQUIREMENTSInfull,ribbed,orlightenedslab,itisnotnecessarytoplaceminimumshearreinforcementwherethetransversedistri-butionoftheloadispossible,asreportedintheEurocodeinpoint6.2.1(4):“If,basedonthesheardesigncalculations,noshearrein-forcementisrequired,itisrecommendedthatthereshouldstill be a minimum reinforcement according to point 9.2.2. This minimum reinforcement can be omitted in elements suchasslabs (full, ribbed,hollow)where the transversedistribution of loads can occur.” ThesameapproachcanbefoundintheBritishregulationBS81101997,Structuraluseofconcrete-Part1:Codeofpractice for design and construction at the point3.6Ribbedslabs(withsolidorhollowblocksorvoids).

HAVE AN IDEA OF THE DISTANCE IN WHICH THE SHEAR REINFORCEMENT IS INSERTEDInthepreviousparagraphitwascalculatedhowmuchrein-forcementisnecessarytowithstandthemaximumshearstress,however,itisnotnecessaryforthisquantitytobereproposedonallthelightenedspans,infacttheshear,farfromthesupports,decreasesinvalue,untilbeingpos-sible to besupported solely by the contribution of concrete. If you wanttohaveanindicativeideaofwhatisthedistancewi-thinwhichthereinforcementmustnecessarilybeplaced,thefollowingreasoningcanbefollowed:Let’sconsideraslabconsistingofaninfiniteseriesoffields, with spans in the two directions and , let’sanalyzeapillar; thatpillarwill haveacapitalwith thedi-mensions , .Imagining that theshearon theplatevaries linearlywiththeloadp,theactionwillhaveamaximumvalueincorre-spondencewiththepillar(withvalue ,giventhatwehavetwoslabfieldsthatdischargeonthepillar),anditwillcancelitselfincorrespondencewiththemidspansection.Iftheobjectiveistofindthedistancefromthepillarbeyondwhichtheshearreinforcementisnolongernecessary,withasimpletriangularproportionwehave:

Figura 32 - distance calculation x

Obviously,thisisonlyacontrolofthefirstapproximationand is not intended as a regulatory constraint.

3.4.6 CALCULATION AND VERIFICATION TO PUNCHINGShear forces are often dominant factors in the behavior of theplates,whichstronglyinfluencethedesign.Theshearforces are the biggest in correspondence to the pillars or toothersupportsorconcentratedloads,orwhenthein-troduction of forces concentrated in the transverse direc-tion to in- plane of a plate occurs.Ifweconsideraplaterestingonpillars,thereactiontrans-mitted by the pillar must be distributed in the concrete at the interfacebetween theplate and thepillar; thereareconsiderable shear stresses, aswell as very high nega-tivemoments,andconsequentlyaconsiderableconcen-tration of stresses. This can cause a sudden break due topenetrationthroughouttheplate,withtheformationofdiagonal cracks that cross the thickness of the concrete. This phenomenon is named punching failure.Initially it manifests itself with the opening of circularcracksaround theheadof thepillar; theseare followedby the cracks opening in the radial direction starting from the pillar; at about 2/3 of the breaking load the opening of truncatedcone-shapedcracksoccurs,fromthelowpartoftheinterfacebetweenthepillarandtheplatetowardsoutside; as the load increases, the crack is suddenlyreachedwithanabruptincreaseofthewidthofthetrun-catedcone-shapedcracks,withoutanywarningintermsof deformations.The presence of flexural reinforcement increases the punching resistance. Insomecases,a too lowbendingpercentage of flexural reinforcement above the columns can contribute to failure; it is therefore necessary to size the reinforcement in these areas also in regard to the dan-gerofpunchingfailure,aswellasofbendingstresses.Basedontheseobservationsitispossibletoschematizethepunchingphenomenonindifferentways:


CALCULATION MODEL - VERIFICATIONOF THE RESISTANCE ON THE CRITICAL PERIMETERGiven the complexity of the failure phenomenon due to punching, the standards acquire simplified verificationprocedures,definingaverificationsurfaceinthethicknessoftheplate,orthogonaltothemiddleplane,atwhichthecutting stresses must not exceed a predetermined mate-rial strength value.ThesurfacedefinedbytheEC2isatadistanceoftwicetheusefulheightoftheplate.Inthiswayareferencepe-rimeteruisevaluated,whichshouldbemultipliedbytheusefulheightdtoobtainthereferencesurface.Obviously,the verification surface changes when the columns areplacedattheedgeorinacorneroftheplate,orifthereareopeningsintheplatewhichreducetheavailablesurface,orwhentheeffectofhorizontalloadsissensitive.

Figura 33 - punching

Tosimplify thecalculation,Eurocode2proposesas thefirstsimplifiedapproachoftheamplificationcoefficientsoftheaveragestress,thatisobtainedbyuniformdistributionofthetransmittedactionfromthepillartotheverificationsurface.The stress can therefore be calculated as:

Dove:

� is the calculation value of the total force of the acting shear;

� is the perimeter of the critical section;

� is the coefficient that considers the non-uniform distri-bution of stress.

� considertheequilibriuminconditionsoffailurebetweenthe stress at the base of the cone and the tractive force in the upper reinforcements that are ready to absorb the corresponding bending moments;

� representing the resistantmechanism by a truss withcompressed concrete struts and rods stretched due to the resistant contribution of tractive concrete and of the reinforcements arranged to contrast the opening of the diagonal cracks.

It is necessary to carefully check that the safety conditions arerespected,duetothefragilenatureof thephenome-non,whichdoesnotensureanywarningonplatefailure.Punching in extreme conditions can cause a deck to fall ontheunderlyingone,triggeringitsfallwithaknock-oneffectandstartinganincrementalfailure.Inordertoavoidtheriskoftheaforementionedbrittlefailures,particularre-inforcementsareprovided,socalledpunching reinforce-ments,which pass through the breaking cone verticallyin order to absorb the tensile stress through the crack; fromthestaticpointofviewtheyplaytheroleofextendedconnecting rods in the truss scheme. In order to perform thisfunction,thetransversereinforcementsmustbeade-quately anchored.Ifclassictiesareused,theymustbeconnectedtotheflex-uralreinforcementwhichthereforealsoactsasatieholder.This type of solution is possible in plates and foundations ofaveragethickness,ifthethicknessissmall,itisneces-sarytousespecialdevices,forexamplesystemsconsist-ingofshorttransversebarsequippedwithanchorplatesto be placed in the pour and run parallel to the middle plain,orbycreatingreinforcingbasketsmadeofweavingbars that are transverse and parallel to the support surface.


The values of are proposed by Eurocode 2 in different ways,dependingonwhetherthestructureinthepresenceoflateralloadsreliesonsuitablebracingelements,oronlyon the stiffness of the plate-column assembly.In the first case the coefficient is:

� = 1.15 for internal pillar;

� = 1.4 for lateral pillar;

� = 1.5 corner pillar.

PUNCHING RESISTANCEThe choice of the reference surface is not very equivalent totheactualfailuremode.However,someelementsthatcorrespond to the physical reality are introduced into the verification,one linked to thestrengthof thematerial, acontribution from punching reinforcements, and finallyoneowedtothepresenceofreinforcementsprovidedforbending.

In case of given punching stress equal to the reaction ofthecolumn,andastrength ,thestressandstrengthvalues,dimensionallystresses, and ,arecalculatedby dividing these actions by the area of the reference sur-face,equalto .

The resistance of the material to the punching is obtained bytheshearstrengthoftheconcreteincreasedwiththemultiplicationbyacoefficientk to take intoaccount thebeneficialeffectsofthemeshingmechanismofthecon-crete along the diagonal crack.An important contribution to the punching resistance is gi-venbythestretchedflexuralreinforcementsetatthetopof the pillar,which, by passing through the cone shapecracking,actsasasupporttotheplate(withaneffectsi-milartothatofthereinforcementdowelintheshearfailureofthebeams).Theverificationformulatakesintoaccountthe , medium reinforcement percentage, between thepercentagesinthetwoorthogonaldirections.

The strength of concrete alone, in the absence of pun-chingreinforcement,isdefinedas:

Where:

� (with d in mm) coefficient that takes into account the meshing;

� medium longitudinal reinforcement above the column;

It is recommended to take , as average values for thepercentageofstressedreinforcement,adequatelyan-chored,abovethepillar,calculated inrelationtoawidthequal to the transverse dimension of the column increased by 3d on each side.

In case the punching reinforcements are used:

The punching resistance offered by the concrete must also becheckedimmediatelyjustbeforethepillar:

Where:

� is the calculation value of the total force of the acting shear;

� is the perimeter of the critical section close to thepillar(various,dependingonthepillarposition);

� is the coefficient that takes into account the stress distribution;

� takes into account the resistance of concrete cracked due to the shear.


3.4.7 DISPLACEMENT CONTROL The verification of the maximum displacements of a struc-ture is essential to ensure the preservation of appearance andfunctionalityrequirements,avoidingdamagetoparti-tions,finishingsandwindowanddoorframes.Insomecases,certainlimitsmayberequiredtoensurethecorrect operation of machinery or installations supported bythestructure,orwaterstagnationontheroof.Finally, sometimes it is also necessary to impose limitsfromthepointofviewofvibrations, inordertoavoidin-conveniencetotheinhabitants,orevenstructuraldamage.TheEuropeanlegislationallowsomittingofthecalculationandverificationofdisplacements,iftheplateelementsdonot exceed a limit value of the ratio between span andthickness. When the structure is slimmer than these limits it is necessary to perform the calculation.Fromageneralpointofview,thestrictcalculationofdisplacements occur through the integration of the cur-vatures of the element under the quasi-permanent load. For thispurpose, for the calculationof thedeflection, itis possible to apply substantially the same methods used forthebeams,appliedtoastripofplatealongwhichthecurvatures are evaluated.Consideringtheresponseofabentelement,thepresenceof intact concrete areas and cracks is observable; in the calculation it is necessary to consider both the behavior ofthefullyreactivetractionsection(stageI-elastic),andof the partialized section, without concrete that reactstotraction(stageII-cracked),assessinginthiswaythegreater deformability due to crack opening. It is therefore important to consider the percentage of reinforcement present in the sections and its tension commitment.It is also necessary to take into account the increase in displacements due to viscous deformations. In this re-gard,permanentandalmostpermanent loadsactingonthe structure must be considered.

Figure 34 - moment- curvature relation “average” for the calculation of displacements

The moment-curvature relation defines the response in terms of deformation of the section for a given stress value. First, the linear elastic response is represented

, corresponding toa fully reactivesection,withoutconsidering the cracking state. If the effects of viscosity are considered, always under the hypothesis of a fullyreactivesection, thestraight linewillbeobtained..StageII,ontheotherhand,considersthefullycrackedstate. The global response can be assessed by defining an “average” behavior betweenthese.The average behavior is defined in Eurocode 2 and in the Italian legislation as:

The distribution coefficient isbetween0and1andin-creaseswiththe increaseofstress inthesteel,which isevaluated in the calculation of the bent sections; for in-creasing values of the bending moment, the averagebehavior approaches the level . From the physical pointofviewthisreflectstheprogressivereductionofthestiffeningeffectgivenbytheconcretewiththeincreasingof the stresses and the opening of the cracks.The formula to derive isthefollowing:

39Special applications • chap 5

3.5.7 CONSTRUCTION DETAILSFor theplatesonpillars, thestructural analysiswithflatfiniteelementsleadstoevaluatebendingandtwistingmo-mentswithhighvaluesnearthesupportsonthepillarsoronwalls.Attheultimatelimitstate,theuseoftheelasticmethod provides for the possibility of redistributing the internal actions, inorder to take intoaccount, thegreatinternal hyperstaticityof theplate,whichhowever is re-duced by the progression of the concrete cracking and oftheplasticizationofthereinforcement.Eurocode2,forthe minimum and maximum amount of bending reinforce-ment, makes reference to the paragraph regarding therequirementsforbeams,butwiththeadditionofrequire-ments for transverse reinforcement; for unidirectional load bearingplates, in fact, itmust beplanned in a quantitynot less than 20% of the main reinforcement area. Further prescriptions concern the maximum recommended bar pitch:

� Formainreinforcement, ,being the total height of the plate;

� for secondary reinforcement.

Inareaswithconcentrated loadsorwithmaximummo-ment the previous values become respectively:

� Formainreinforcement, ,being the total height of the plate;

�Forsecondaryreinforcement .

At a free edge it is necessary to arrange reinforcement in a direction parallel and orthogonal to the same edge; for this purpose, thereinforcementscalculatedfor theplatecanbeused,preparedtoextenduptotheedgeandtocreatea C-shaped closing steel. This reinforcement effectively resiststorsion,whichisparticularlyhighnearthesupports,and requires sufficient anchoring.

Figure 35 -detail of free edge closing reinforcement

� = 1 forbarswithimprovedadherence,0.5forstraight bars;

� = 1 forshort-term load,0.5 forpermanentorcyclic loads;

� stressinthesteelinstageIIincorrespondencewith the cracking load;

� stressinthesteelinstageIIincorrespondencewith the acting load.

The quantity canbereplaced,asafirstapproxima-tion,by .

The plates resting on pillars are economical constructions due to the small thickness of the structure and the simplic-ityoftheconstruction.Forthisreason,thecontrolofthedeflections is particularly important.In conditions of load and symmetrical constraints, themaximum deflection in the center of the field should not be taken into consideration in relation to the span measured inparalleltothesidesoftheplate,butrelativelyinrelationto a span equal to the diagonal line of the plate.As an alternative it is possible to take into consideration therelativedisplacementbetweenthecenteroftheplateand its edges (whichwill be smaller than themaximumdisplacement),relatingtoaspanequaltothesideoftheplate. The maximum displacement on this strip corre-sponding to the median length is about 0.75 times the maximum deflection.Finally, it is possible to measure the ratio between themaximumloweringonthelinejoiningtwopillars,andthespan equal to the side of the plate. The maximum deflec-tion for the plate strip on one side of the field is about 0.71 times the maximum deflection.

For the verification it is therefore appropriate and concep-tually correct to use a ratio equal to 0.75 ,lessstrictthan ,sinceamongthecasespreviouslyexhibited,the biggest ratio deflection-span is precisely 0.75 .


Figure 36 - constructive detail of lightening with stiffening spandrel beam

Figure 37 - construction detail of lightening with thick beam

Thereinforcementincorrespondencewiththepillarsmustbesuitablythickened,giventhepeaksofsolicitationthatoccurintheseareas.EC2,inthechapter9.4 says:

“[..] it is recommended to place extrados reinforcements ofarea0.5Atwithinawidthequaltothesumof0.125timesthewidthsofthepanelstakenoneachsideofthepillars. A t represents the area of reinforcement necessary tosupporttheentirenegativemomentactingonawidth,equaltothesumoftwohalfpanelstakenoneachsideofthe pillar.Itisrecommendedthatatinternalpillarslowerreinforce-ments (≥ 2 bars) are placed in each direction and thatthese reinforcements cross the pillar.”

Incorrespondencewithedgeorcornerpillarssomege-ometric indications to be respected in order to transmit the bending moments from the plate to the pillar are made explicit,whichtranslatesintothedefinitionofaneffectivewidth be, where the reinforcement perpendicular to thefree board should be concentrated.LargereffectivewidthsarepresentedintheBritishstand-ardBS8110,againforunbalancedmomentsinadirectionperpendicular to the edge.

If theresultsareobtainedfromafiniteelementanalysis,thenegativemomentatthecolumnsorwallsmustbetak-ennotatthesupport itself,as it isaffectedbythepeakeffectduetothediscontinuity,butatadistancefromtheedgeofthesupportwhichisusuallyapproximabletotheone thickness of the slab.

4. ON-SITE APPLICATION

42 On-site application • chap 4

4.1 PREPARATION OF THE FORMWORK

The slab lightenedwith elements of the New Nautilus Evo isexecutedinasimilarwaytoasolidslab.Thecastingneedsacontinuoushorizontalformwork,pos-siblymadewiththeuseofprefabricatedreinforcedcon-creteslabs(predalles).Inthiscase,refertotheappropriatesection.Theformworkcanbeofthetraditionaltype,woodplanksor plywoodpanels laid on a suitablemeshofH-beamsmadeofwoodaswellormadeofsteeloraluminum.It is strongly recommended,where possible, the use ofspecific formworks for the construction of slabs in-si-tu,withmodularpanelsanddroppedheadersupportingbeams.These systems maximize on-site productivity.

Figure 38 - formwork of cast-in-situ slab with modular metal formworks

Figure 39 - formwork of cast-in-situ slab with traditional wooden formwork and plywood panels

Figure 40 - formwork of cast-in-situ slab with modular wooden formworks

43On-site application • chap 4

4.2 LAYING THE LOWER LAYER OF THE REINFORCEMENTOncethelowerformworkhasbeenprepared,thespacersof the reinforcing bars and the basic reinforcement mesh ofthelowerlayermustbeplaced.It is recommended to choose suitable spacers to let the concrete pass easily during the casting of the first layer.Usually there is a basic reinforcement and appropriate lo-calrefinementswherenecessary.The basic reinforcement mesh can be laid by single crossedbars,bymeansofelectro-weldedmeshorbycar-pet,crossingtwolayers:

� Reinforcementbysinglebars:diameter and pitch accordingtotheproject, restingonthe spacers suitable to guarantee the coverage foreseen byproject. Theminimumthicknessofthelowerslab(andconse- quentlyofthefootingoflightening)inthiscasewillbe:

� Reinforcement with electrowelded mesh: diameter and pitch and pitch according to

theproject,supportedonthespacerssuitabletoguar-antee the coverage accordingtotheproject.Theuse of electrowelded meshes makes the installationofreinforcementfaster,butnotethatthepanelsmustbe overlapped for at least ,thereforeinthecouplingpoints of 4 panels the overlapping of the rebars occu-pies a space equal to therefore, themini-mumthicknessofthelowerslab(andconsequentlyofthefootingofthelightening)willbe:

Toavoidthisinconvenience,itissuggestedtouseap-

propriate rationalized special type meshes.

� Carpet reinforcement: the carpet reinforcement is a special patent of theGerman companyBAMTEC©:these are rolls of rebars of variable pitch and diameter producedwithnumericalcontrolwithdirect interfaceto the structural calculation program F.E.M. They are producedinsuchawayastobeabletoprovidetheslabwiththeprojectstrengthoncepositioned.

Figure 41 - laying of the lower reinforcement mesh

4.3 ADDITIONAL REINFORCEMENT Any additional reinforcement to the basic mesh can be madeasfollows:

1. Byvaryingpitchanddiameterofthebasereinforcement locally:inthiscase,checkthatthethicknessofthelower slab(heightofthefooting)isadequatetoensureade- quate covering of the bars in the area below the lightening.

2. Adding bars in numbers and diameters suitable for obtaining the required moment of resistance inside the ribs that are created between one lightening and the other. The width of the rib must be such as to guarantee the space necessary to respect the reinforcement gap set by the regulations.

Figure 42 - example of additional reinforcement with loose bars and “single” lightening

Figure 43 - example of additional reinforcement with loose bars and “double” lightening


4.4 LAYING OF LIGHTENING Once the lower layer of reinforcement is positioned,the lightening can be applied.NOTE:inthecaseofDOUBLElightening,thetwohalf-shel-ls “UP” and “DOWN” must be assembled before being laid,deliveredontheconstructionsiteonseparatepallets.Thecouplingoccursinasimplewaythrougha“clip”joint.

Fortheinstallationofthelightening,followtheprocedurebelow:

1. IdentifyonthedrawingsuppliedbyGEOPLASTS.p.A. orintheprojectthestartingpoint(s)oftheinstallation, suitably listed.

2. Placethelighteningdirectlyonthelowerformwork.

3. Tie the elements together using the appropriate spacing straps,takingcarethatthepinengagesincorrespond-encewiththenumbercorrespondingtothewidthoftheribplannedintheprojectandshowninthesectionsofthedrawingssuppliedbyGEOPLASTS.p.A.or inthestructuralproject.

Figure 48 - spacer strap for connecting the lightening

3. By laying prefabricated cages with reinforcing bars welded on open (U) brackets, crossed. This system allows to install at the same time any reinforcement, wherenecessary,bycutting.Thewidthoftheribmust be such as to guarantee the space necessary to respect the reinforcement gap set by the regulations.

Figure 44 -example of additional prefabricated reinforcement and “single” type lightening

Figure 45 - example of additional prefabricated reinforcement and “double” type lightening

Figure 46 - layout of lightening

Figure 47 - layout of lightening


4.5 LAYING THE UPPER LAYER OF THE REINFORCEMENT The basic reinforcement mesh could be placed for sin-gle-crossedbars,bymeansofelectroweldedmeshorwithacarpet,crossingtwolayers:

� Reinforcement for single bars: in diameter and pitch accordingtotheproject,restingon the adequate spacers to ensure coverage providedbytheproject. The minimum thickness of the lower slab (and con-

sequently of the footing of the lightening) in this case willbe:

� Reinforcementwithelectroweldedmeshesindiameter

and pitch according to the project,resting on the adequate spacers to ensure coverage

providedbytheproject.Theuseofelectroweldedmeshesmakesthe installationofreinforcementfaster,

but note that the panels must be overlapped for at least , therefore in thecouplingpointsof4panelsthe overlapping of the rebars occupies a space equal to ,thereforetheminimumthicknessofthelowerslab(andconsequentlyofthefootingofthelight-ening)willbe:

Toavoidthisinconvenience,itissuggestedtouseap-propriate rationalized special type meshes.

� Carpet reinforcement: the carpet reinforcement is aspecialpatentoftheGermancompanyBAMTEC©:

these are rolls of rebars of variable pitch and diameter producedwithnumericalcontrolwithdirectinterfacetothe structural calculation program F.E.M.

Theyareproduced insuchaway toprovide theslabwiththeprojectresistanceoncetheyarepositioned.

Figure 49 - placing of the upper reinforcement bars

Figure 50 - laying of the upper reinforcement bars

4.6 SPACERS Theelectroweldedmeshup to8mm indiametercanbeplaceddirectlyabovethelightening,equippedwithaspe-cial radial spacer.Alternatively,specifictrellis(oneevery2-3ribs)ortheclas-sic “OMEGA” trestles must be used.

Figure 51 - plastic and fiber cement spacers

Figure 52 - plastic and fiber cement spacers


4.7 SHEAR AND PUNCHING REINFORCEMENT

Sincethesebarsmustanchortotheloweranduppermesh,they must be placed last.Normallyhooksorgrapplesareused,orspecialwidened-he-adspikesequippedwithspecialpositioningbrackets.

Figure 53 - shear reinforcement: longitudinal and transverse section

4.8 PERIMETER REINFORCEMENT Theperimeterstripsofthelightweightslabmustbeleft insolidconcreteatawidthatleastequaltothethicknessHofthe slab.Appropriate U-shaped bars designed to ensure the correct anchoringofthelowerandupperlayerbarsmustbeprovi-ded,asindicatedbythestandards.

4.9 REQUIREMENTS FOR CONCRETE CASTINGThelighteningmaterialstendtofloatinfreshconcrete,whichiswhyitisnecessarytotakesomeprecautionsduringthecasting,whichwillbecarriedoutintwophasesonthesameday.NOTE: under no circumstances should you puncture the lightening! The air leakage in fact eliminates the floata-tion but allows the concrete to penetrate the cavities, causing an increase in the weight of the structure not foreseen by the project. This could cause the structure to collapse.Pourtheconcretebydirectingthepumporbuckettowardstheribs,insuchawayastoexploitthepressureofthecon-creteandmakeitslidewellbelowthelightening.Fillconcreteup to a maximum of 4 cm beyond the height of the footing. Iftheformworkstendtorise,donotinsistandcontinuefur-ther.Vibratewithcare,visuallycheckingthat theconcreteflowswellbelowthelighteningandcheckingthatthecon-cretegoesupthroughthecone,uptothelevelofthecastingaroundtheformwork(siphonprinciple).Continueinthiswayuntil the entire surface of the floor is complete.Tocompletethecasting,waitthenecessarytimeuntil theconcrete,eventhoughitisstillfresh,hasstartedtosetandhas lost some of its fluidity.Anindicativetimeintervalcanbethefollowing:

1. From 90’ to 120’ for the temperature higher than 20°C.

2. From120’to240’forthetemperaturelowerthan20°C.

A practical indication of the correct state of maturation of the concrete consists in sticking a rebar into the casting and seeingif,onceextracted,leavesahole.

Figure 55 - cone correctly filled


4.10 CONCRETE MIXDESIGNFor thecorrectmixofconcretedesign, relyonthecon-crete supplier technologist. Grain size, fluidity, additivesmustbechosenbasedonthethicknessofthelowerslab,diameterandpitchofthebars,environmentalconditionsinwhichthecastingwilltakeplace.For the casting of the first layer of concrete the ideal is to use a consistency class S5 and it is advisable to use con-creteswithafluidityclasslowerthanS4.

4.11 OPENING OF CAVITIES BEFORE CASTINGThelightenedslabwithNewNautilusEvoelementsallowsyou to easily create holes and openings of any size. If the sizeoftheholecanbeinscribedwithinthelighteningplanmeasurementsanddoesnotinterrupttheribs,nospecif-icreinforcementwillberequired.Forlargerdimensionsitis necessary to check the need for appropriate reinforce-mentsduringthedesignphase.Inanycase,areinforce-mentofthetypeshownintheFigurebelowmustbepro-vided around these holes.

Figure 59 - preparation of cavities before casting

Figure 56 - cone not correctly filled, insist on vibration until reaching the situation of the previous figure

Figure 57 - the impression in the concrete indicates the right time to resume the casting

For casting surfaces over 500 m2 it is not necessary to fore-seeinterruptionsofthepour,asthetimerequiredtopreparethe first layer is normally greater than 120 ‘. For castings surfacesoflessthan500m2,itisadvisabletoorganizeinanticipationofapproximately60minutesofwaitingforthesecond layer. If the second layer is carried out more than 4 hourswithrespecttothefirst,itisrecommendedtoprovidesuitableconstructionbarsasincorrespondencewithacoldjunction.

Figure 58 - casting of the first layer


4.12 OPENING OF CAVITIES AFTER THE CONSTRUCTION OF THE SLAB

Itwillalwaysbepossibletoopennewholesincorrespond-encewiththelightening,ofmaximumsizethatcanbein-scribedinthelighteningplanmeasurements,aslongastheydonotcuttheribs.Largercavitiescanalsobemadeaftercastingtheslab,takingcaretoperformastaticcheckonthemigration of forces on the edges of the opening due to the interruption of the continuity of the slab and possible resto-ration of its strength through special reinforcement. The slab must be suitably propped up if necessary.Theprocedureisshowninthefiguresbelow.

Figure 62 - in dark purple the opening to be made, in light purple the hole to be made

Figure 63 - temporary hole

4.13 USE OF DOWELS AND ANCHORING

Figure 64 - final hole with reinforcement

It is possible to use dowels and anchors, both chemical(epoxyresins)andmechanical,tohangloadsonthelowerslab.Refertothetechnicaldatasheetsofthesuppliers(suchasHilti,Fisher,etc.)inordertocorrectlyuse(model,size,lengthetc.)thedowelaccordingtothethicknessofthelower/up-per slab and the loads carried by it.

5. SPECIAL APPLICATIONS

50 Special applications • chap 5

5.1 PREFABRICATIONIt is possible to create slabs cast-in-situ using semi-prefabri-catedslabs(predalles)asaformworkbottom.Themainad-vantagesofthissolutionarethelowerformworkcosts(es-peciallyintermsoftimeaftertheformworkhasbeenstruck).The use of this construction technique requires some addi-tional precautions:

1. For lifting, theslabsneed tohave laterally twosemi-ribs trellis. Since the New Nautilus Evo element has a widthof52cm,itisonlypossibletouse240cmwideslabsand80cmformworkinteraxialspacing.

2. In the transverse direction it is possible to reduce the interaxial spacing to reduce the consumption of con-crete,byshapingananisotropicslab.

3. The reinforcement steel must necessarily be placed exclusivelyintheribsor,atmost,thefirstlayeroftheprojectalready included in theplantcanbe foreseenwithintheprefabricatedslabs.

4. Onceplacedtogethertheslabsmustbejoinedtogetherwithappropriatelycalculatedseambarsalongthepe-rimeter.

Basedontheabove,themaindisadvantageofthissolu-tionwillbethegreaterconsumptionofconcreteandsteelcompared to the equivalent solution totally cast-in-situ.

5.2 POST TENSIONINGItispossibletomakeposttensionslabslightenedwithNew Nautilus Evoelementswithposttensioncablesplacedinoneortwodirections.Thistechnologyovercomestheintrin-siclimitsofspanandloadofthelightenedslabsystem(slabslightenedwithspansgreaterthan15-16mgointoshearandpunchingcrisis,unlesshigh-resistanceconcretesandveryhighreinforcementpercentagesareused),ortofurtherre-duce the thickness of the slab by pushing the advantages to the maximum due to the lightened structure. The cables can beplaceddirectlyintheribs,orinmassivebandssuitablypositioned and sized.

Figure 64 - post-tension cables on plate slab

5.3 THERMAL ACTIVATION OF THE MASSES The last frontier of energy saving is the thermal activation of the masses. This technology involves the embedding of pipes in the concrete structure inwhichwaterwill flowat a suitabletemperature.Inthiswaytheentirestructureofthebuildingcontributes to air-conditioning the rooms,with conside-rable energy savings.ThistypeoftechnologyiscompatiblewiththeNewNau-tilus Evo system and has been used successfully several times.Thestructuraldesignmustobviouslybeflankedfromthebeginningwiththatoftheplant.Ifyouareinterestedinthissolution,weadviseyoutocon-tactthemarketleaders,theFinnishcompanyUPONOR:https://www.uponor.com/

6. CERTIFICATIONS

52 Certifications• chap 6

Theslab lightenedwithelementsNewNautilusEvohasbeentestedontwooccasionsandinaccordancewithcur-rent European regulations.ThebehaviorunderloadfromfireoftheslablightenedwithNewNautilus Evo elements is attestedby the followingdocuments:

� Testreportn°CSI1890FRperformedaccordingtoUNIEN 1365-2: 2002 and UNI EN 1363-1: 2012

issuedbyCSIBOLLATE(MI);

� Testreportn°RS17-011GEOPLAST-MC-RE according to the regulation NF - EN 1363-1 : 2013 - 03 NF - EN 1365 - 2 : 2014-12 issuedbyCSTB(France);

� Sizing abacus n° 26067076_AL1 for fire resistance issuedbyCSTB(France);

Inparticular,thetestscarriedouthighlightthefollowing:

1. TheREI fire resistance,as for fullslabs, isa functionoftheconcretecover,thenumberanddiameterofthereinforcement, thetotal thicknessoftheslabandthewidthoftheribs;

2. It is possible to obtain any REI fire resistance class re-quiredbythestandards,playingontheaboveparame-ters;

3. Thepresenceofemptycavitiesdoesnotworsen theconduction of heat inside the structure and does not generate excess pressures that cause the explosion of the slab;

4. Thespallingoftheconcretecoversfollowsthebehav-ior of traditional reinforced concrete structures.

Copiesoftestreportsareavailableuponrequestbywrit-ing to: engineering@geoplast.it.Sizingabacusforfireresistanceisshowninthefollowingpages

6.1 BEHAVIOUR DURING FIRE LOAD AND FIRE RESISTANCE

53Certifications• chap 6

It is possible, as an alternative and in EU and non-EUcountrieswherethisprocedureisallowed,touseanalyti-calmethods,suchasfiniteelementthermalanalysiswiththe application of standardized temperature curves.

Figure 66 - modeling of the cross section of the slab (vacuum, concrete and steel)

Figure 67 - model F.E.M. of the section


The analytical calculation of the section can be per-formedbyourtechnicaldepartmentuponrequest,aftersending all the necessary data via email to: [email protected].

Figure 68 - calculation of thermal conduction over time for application of standardized temperature T (° C, t) curve

Figure 69 - calculation of Moment and Shear resistant under fire


6.1.1 500°C ISOTHERM METHOD

The abacus obtained from the laboratory test and sub-sequent numerical analysis allows to carry out the hotverificationofthesectionofthelightenedslabwithNuo-voNautilusEvoinaccordancewiththemethodreportedinAppendixB1ofUNIEN19921-2:2005,towhichwerefer for details.It is basically a matter of reducing the resistant section neglectingthecontributionoftheconcretewhichisesti-mated to have a temperature above 500 ° C and reducing the strength of the reinforcing steel as a function of the verificationtemperature,thenperformingaverificationofthesectionasifitwasmadecold.Thanks to the abacus it is possible to have the tempera-tureprofileinthesectionforagivenlowerslabthicknessandforagivenfireduration(30-60-90-120-180min.)withwhichitwillbeeasytoobtaintheappropriatereductionsto operate on the section

CONFIGURATION 8-180

Figure 70 - isothermal configuration for S1 = 80 mm at 180 min


Figure 71 - abacus for section reduction and estimated iron temperatures - S1 = 80 mm at 180 min


Figure 72 - reduction of the section subject to fire

Figure 73 - static scheme for hot verification

Fordetails,seeUNIEN19921-2:2005, inparticulartheparagraphs:

�4.1

�4.2

�4.2.4

�APPENDIX B.1


6.2 ACOUSTIC BEHAVIOURForthecalculationofthesound-insulatingpowerRwandofthetramplingnoiselevelLn,wofthebareslab(withoutthecontributionsgivenbythefinishes),itisnecessarytorefertothefullslabhavingthesameweight,determiningthe surface mass m’ in kN / m2.Toestimatethereferencevalues,formulasofprovenvali-ditypresentinCEN,IENGalileoFerraris,DINcanbeused.Also available on request are some test reports carried out insituonlightenedslabstothenudeandwithscreed.


6.3 THERMAL CONDUCTIVITYTo calculate the thermal conductivity of the slab lightned withNewNautilusEvo,itisnecessarytocalculateitsther-mal resistance. It appears to be the weighted averagebetween the thermal resistanceofa solidconcreteslabandthatofacavityclosedbetweentwoconcreteslabs.

Figure 74 - typical profile of a lightened plate with New Nautilus Evo

The value is defined as thermal transmittance:

In the case of a lightened slabwithNew Nautilus Evo elements,wewillhavetwothermalresistancesthatworkin parallel:

1.

2.

Where:

� istheinternalsurfaceresistanceequaltoabout0,13;

� is the external surface resistance equal to about 0.04;

� isthethermalresistanceoftheairspacewithinthe cavityleftbytheformwork,equaltoabout0.16infavorofsafety.

Figure 75 - overall surfaces of thermal resistances in parallel

The total thermal resistance is equal to the contribution ofeachofthetworesistances,whichisproportionaltothesurface that each of them covers:

Figure 76 - overall surfaces of thermal resistances in parallel

7. PACKAGINGAND LOGISTICS

62 Packaging and logistics • 7 chap

NEW NAUTILUS EVO

SINGLE DOUBLEElements per pallet pz 400 200

Dimensions per pallet cm 100 X 120 X H max 250Weight of pallet kg 560 max

STANDARD LOAD FACTORS FOR DIFFERENT MEANS OF TRANSPORT

MEANS OF TRANSPORT NUMBER OF PALLETS DIMENSION OF MEANSTruck 18 9,50x2,45xh2,70m

Truck + Trailer 14+14Truck7,70x2,45xh2,70mTrailer7,70x2,45xh2,70m

Semitrailer 24 13,60x2,45xh2,70mContainer 20 ft. 24 5,90x2,35xh2,39m(portah2,28m)Container 40 ft. 22 12,03x2,35xh2,39m(portah2,28m)

Container 40 ft. high cube 22 12,03x2,35xh2,70m(portah2,58m)Container 45 ft. 24 13,54x2,35xh2,70m(portah2,58m)

Standardconditioningconditionsforallmodelsforcompletepallets.Thedatashownhererelatestocompletepalletsofasingle model.

7.1 NEW NAUTILUS EVO STANDARD PACKAGING CONDITIONS FOR LAND SHIPMENT

8. APPENDIX

64 Appendix • chap 9

8.1 KIRCHOFF SLABThefollowingcalculationhypothesesareconsidered:

� smallthicknessslab(lessthan1/5oftheminimumef-fective span);

� smalldisplacementsanddeformations(lessthan1/5ofthe slab thickness);

� transverseloadstotheslab;

� decoupling between flexional and membranous pro-blem(thereforesymmetryoftheslabwithrespecttothemiddle plane);

� negligible shear deformations;

� predominantly flexural regime (so segments originallyorthogonal to the middle plane remain straight in the deformed configuration);

� linearelasticmaterial,homogeneousandisotropic;

� negligible normal efforts on the middle plane;

therefore, thedisplacementcomponentscanbewritten,withreferencetotheCartesianreferenceaxes:

In the previous equations is the displacement of the points of the middle plane in the direction and are the rotations of the tangent to the axis in the deformed position.Onthebasisofthehypothesisofamainlyflexuralregime,flexuralrotationscanbewrittenasderivativesofthetran-sverse displacement of the middle plane.

Therefore,thedeformationcomponentswillbe:

Thelightweightslabsrepresentafurtherevolutionoftheaforementionedstructuralsystem,withwhichnotonlyim-portantspansareobtainable,butitallowsitwithasavingintermsofconsumptionofconcreteandsteel,comparedto a full slab solution.Infact,intheanalysisofthebehaviorofareinforcedcon-creteslab, thepartof thesection that iscompressed isonly a small portionwith respect to the total thickness,theremainingpart isnotconsideredco-operating (giventhepoorperformanceoftheconcreteattraction),anditisnecessarytoconsideritonlyinresistingshearstress,anactionwhichishoweververylowawayfromthesupports(areawherenolighteningisforeseen).Therefore,withtheNew Nautilus EVO elementswe are going to reduce avirtuallyuselessweight,managingtoobtainaleanerslab,and greater possible spans.

8.2 STRESS REGIMEThe stress regime is immediately deducible from the de-formationstate,oncethelinearelasticbondisintroduced:

65Appendix • chap 9

8.3 INTERNAL ACTIONSAnalogously to what happens for a beam, also for theslabweoperatebyreferringtocomponentsoftheinternalaction.Thebendingmoment(perunitoflength,measuredalongthe middle plane) is defined as:

Dimensionallythebendingmomentwillbeaforce,sin-ce .

Thebendingmomenthasapositivesignwhenapositi-vesigntensionactswithapositivearmsopositive stretchesthelowerfibers.Thepreviousintegralgivesasa result:

Inasimilarway,thebendingmoment is defined as:

Inwhichisindicatedas flexural stiffness of the part of the slab of unit length. The shearing actions:

Theycannotbeexpresseddirectly,asthesheardeforma-tions have been considered negligible, thereforewewillhave to go through equilibriumequations.However, theinternal actions corresponding to the tangential tension

distributedlinearlyalongthethicknessandwithzero middle value remain to be defined.Therefore, their resultant force (membrane type) will bezero,butwillcreatetorque:

8.4 EQUILIBRIUM EQUATIONSInordertogetthesixunknowns itwill be necessary to add threemore equations to thethree that link the moments to .o write these equations, consider an infinitesimal elementsubjecttotheactionofatransverseload

Figure 77 - moments acting on infinitesimal elements

Figure 78 - shear agents on infinitesimal element

The equilibrium in rotation around the iswritten(byomit-tingtheinfinitesimalsofhigherorder,andsimplifyingtheterms of opposite sign):

Similarly, from theequilibriumto the rotationaround theaxis it results:

Finally the equilibrium to the translation in the direction :

66 Appendix • chap 9

Theseequationsallowwritinginfunctionof of the she-ars and to derive the resolvent equation:

Replacing in the equation of equilibrium to the translation in direction :

This equation to the partial derivatives in x and in y is in-tegrableonlywithapproximatemethods,suchasseriesdevelopments or discretization methods.

8.5 BOUNDARY CONDITIONSAdistinctioncanbemadebetweenkinematicboundaryconditions and static boundary conditions.Startingwiththekinematicboundaryconditions,onecanimagine being able to impose on a border ,forexam-pleattheedge,displacementsorrotations:

Whileitisassigned,onceyouset rotation in the plane :

The independent kinematic conditions on the boundary arethereforeonlytwo.Wethenpasstothestaticboun-dary conditions; considering for example the boundary

,thereseemtobethreequantitiesthatcanbefixed.However,Kirchoffwasabletodemonstrate

ho are connected,defininga sort of equivalentshear:

Which corresponds to the resultant of transversal agent forces(perunitoflength),whichcanbewritten,replacingthe previously developed formulas:

This equation can be used both to impose a certain action ontheboundary(asitcanbefreeedge),andforthecal-culation of a constraining reaction distributed on the edge.Inthecaseofdiscontinuities,suchasinthecorners,con-centrated forces emerge at the corners equal to:

8.6 INTERNAL ACTIONS ON THE GENERIC SECTION

Let’sassumeweconsideragenericsection,whosenor-mal forms a generic angle withaxis .

Figure 79 - generic section

Considering that e ,firstofall,wehaveabalanceinthetransversedirection :

The equilibrium to the rotation around the tangent gives:

Thebalanceinrotationaroundthenormal,instead,leadstowriting:

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