3) 4 Two-way Floor Slabs 3

TWO-WAY FLOOR SLABS

ETS CAMINS, CANALS I PORTS DE BARCELONABUILDING STRUCTURES

BUILDING STRUCTURES Two-way floor slabs

INDEX

1. Introduction

2. General typology

3. Waffle slab typology

Waffle slab with lost lightening blocks Waffle slab with framework system Waffle slab with special framework Sole plates (or sill plates)

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INDEX

4. Design of structural members

- Pillars

- Edge beams

- Drop panels

- Slab depth

5. Structural analysis

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1- INTRODUCTIONTwo-way floor slabs

Family of R.C. that work along two directions because of being directlysupported on pillars, or because of being supported bidirectional beamsystems. They are reinforced in two directions. They are usually lightened.

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2- GENERAL TYPOLOGYTypes of two-way floor slabs:

Solid two-way slab Two-way slab with drop panels

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Max. span length = 7-8m o 8-9m with drop panels

BUILDING STRUCTURES Two-way floor slabs6

Two-way slab with drop caps Two-way slab with drop caps and drop panels


Two-way slab with flat beams Two-way slab with deep beams

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Lightened two-way slab Waffle slab

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SUMMARY OF TYPOLOGY

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TWO-WAY FLOOR SLABS

SOLID SLABS

REINFORCED POST-TENSIONED

WAFFLE SLABS

LOST LIGHTENING BLOCKS

Ceramic

Concrete

RECOVERABLE LIGHTENING BLOCKS

SPECIAL BLOCKS

Polystyrene

Metallic

Fibre

Plastic


RibCompression layer

Lightening blocks Reinforcement

3- WAFFLE SLAB TYPOLOGY

Elements

Cast-in-place ribsCast-in-place compression layerLightening blocksReinforcement: top and bottom principal reinforcement

possible shear reinforcement ribsspecific reinforcement in edge beams and drop panelsdistribution reinforcement in the compression layer (mesh)

e=inter-axis distance


3.1. WAFFLE SLAB WITH LOST LIGHTENING BLOCKS

Ceramic flooring blocks. They are lighter but fragile. Not used in practice. Concrete blocks. Classic inter-axis = 80 x 80 cm = 70 x70 + 10 cm rib

(max. 1 m according to EHE; max. 1.5 m according to EC2)

Span length: up to 6-7mDepth: 23-35 cmCompression topping with minimum thickness of 5 cm according to EHE-08.

Scantlings:

3 pieces 4 pieces 6 pieces

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3.2. WAFFLE SLAB WITH RECOVERABLE LIGHTENNING BLOCKS

Span length from 7-8 m to 12 m; if > 12 m prestressing needed Classic inter-axis : 80 x 80 cm = 68 x 68 cm + 12 cm (rib minimum thickness)

(max. 1 m EHE - max. 1.5 m -EC2) )Fire protection regulation

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3.3. WAFFLE SLAB WITH SPECIAL BLOCKS:

POLYSTYRENE: UNE -53974/April-1998

- Low adhesion to plaster cracks- Not resistant- Lower shear resistance- Lower stiffness against vertical and horizontal actions- Lower fire resistance and production of smoke and gases- Lower corrosion protection

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3.3. WAFFLE SLAB WITH SPECIAL BLOCKS:

FIBRE

- RF-240- Thermal insulation- Acoustic insulation- More expensive- Non resistant

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3.4. SOLE PLATES

-IGLU and similar systems(pieces of recycled polypropylene) Lower volume of concrete Installations and conductions can be placed

inside

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4- DESIGN OF STRUCTURAL MEMBERS- Need to verify both resistance and deformability - Maximum economy: arrangement of cantilevers L = 1 a 2 m- Theoretical distribution:

EHE deviation in the position of pilars < 10% of their spans

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4.1. PILLARS

Steel:

connection to concrete slab by means of special elements need to study buckling location: water pipes normally 2UPN in a box section or HEB

Concrete:

Minimum depth: 25 x 25 cm to avoid buckling: 30 x 30 cm dimension: a < 30 cm if b > 100 cm squared pillar vs. pillars. In circular pillars, increase width by 30%

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Setbacks in order to cancel or diminish the supplementary bending stresses

?M ?M

?MM+M M+M M+M

M1M2

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4.2. EDGE BEAM

Code ACI 318-05 Fictitious rigidity or Moments redistributionWidth of the virtual frame lintel (EHE). Distinction between vertical and horizontal actions.

)

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Pillars Flexion

Torsion


Purpose of the edge beams

Link and tying of the the pillars Improve the connection of the pillars with the slab Support of closures Improving resistance against punching shear Improve the resistance in seismic zones Opening of any hole in the slab

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Sections:

- Plane or edge beams

= d

= 0.25m

0.25

(= 0.25m)

1. Armadura a torsin + flexin

2. Torsin + cortante

(s


b/2L

Luz de clculo a flexin y

torsin de la viga de borde

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Edge beam flexional and torsional calculation span


4.3. DROP PANEL

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3.4. SLAB DEPTH h

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Minimum required depth (EHE-08)

h L/32 in solid slabs of constant depth h L/28 in lightened slabs of constant depth

L is the maximum span in the two directions


According to EHE: art. 50, it will not be necessary to check the deflection if the slab has the following span/depth (L/d) ratio:

STRUCTURAL SYSTEML/d

Heavily reinforced elements: =1,5%

Weakly reinforced elements: =0,5%

Simply supported beam.One-way or two-way simply

supported slab.14 20

Continuous beam on one endContinuous unidirectional slab on

only one side.18 26

Continuous beam on both sides. One-way or two way continuous slab 20 30

Exterior panels and corner panels in slabs without beams supported

on isolated supports.16 23

Interior panels on slabs without beams supported isolated

supports.17 24

Cantilever 6 8

Table 50.2.2.1.a EHE. Relations L/d on concrete beams and reinforced concrete slabs subjected to simple bending.

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STRUCTURAL SYSTEML/d

Simply supported beam.One-way or two-way simply

supported slab.

Continuous beam on one endContinuous unidirectional slab on

only one side.

Continuous beam on both sides. One-way or two way continuous

slab

Exterior panels and corner panels in slabs without beams

supported on isolated supports.

Interior panels on slabs without beams supported

isolated supports.

Cantilever

Heavilyreinforced elements: =1,5%

Weakly reinforced elements: =0,5%


3.5. REINFORCEMENT

1.- Classic simple in design and execution2.- Implies a loss of mechanical arm.3.- According to the slab lines of stresses. It is not used due to its constructive difficulty.

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5- STRUCTURAL ANALYSIS

STRUCTURAL ANALYSIS METHODS

1. Virtual frame method. The two-way slab is modelled as system of 2D frames.

2. Grillage. The slab is modelled as an equivalent system of linear elements

3. Finite element method. The slab is modelled by means of plate elements supported on beam elements (pillars)


1. Linear analysis

2. Non-linear analysis

3. Linear analysis with limited redistribution

4. Plastic analysis

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CALCULATION METHODS REGARDING MATERIAL DESCRIPTION


VERIFICATION AND PROPORTIONING OF REINFORCEMENT

Ultimate limit states

1. Bending ultimate limit state and determination of bending reinforcement. It is necessary to consider the effect of the bending and torsion forces at each point of the slab.

2. Shear ultimate limit state. It is necessary to verify the ribs at their connection to the drop panes and the edge beams.

3. Torsion ultimate limit state in edge beams

4. Punching limit state in drop panels

Serviceability limit states

5. Limit states of cracking, deformation and vibrations


Effective width on T sections

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Compressed wingTensioned wing

Border rib Interior rib

b0 = distance between points of null moment


Calculation span

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Calculation span

Free span

Calculation span

Pillar Pillar

Pillar Wall


Linear analysis with limited redistribution:

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tension reinforcement

compression reinforcement

RedistributionIS admitted

RedistributionIs NOT admitted


Linear analysis with limited redistribution:

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REDISTRIBUTED LAW

INITIAL LAW


EQUIVALENT FRAME METHOD

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central strip

support strip

Central strip

exterior supports strip

Virtual frame


REQUESTS FOR THE APPLICATION OF THE EQUIVALENT FRAME METHOD

- The fundamental hypothesis is non interaction between equivalent frames.

- Interaction between frames may appear in the following conditions: Lack of symmetry in plan or elevation on geometry and /or stiffness Translational structures. Structures showing some translationality Presence of lateral stiffening elements such as cores and walls Non-gravitational actions on non-uniform structures Significant differences in loads or span lengths in the different spans

EQUIVALENT FRAME METHOD


Application to complex geometries

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Maximum deviation admitted by the code


A-Beam working in torsion

The torsional flexibility of the beam A is taken into account, indirectly, by defining an equivalent stiffness for the pillars

1 / Keq = 1 / Kc + 1 / Kt

Kt = [ (9 Ec C) / (l2 (1 - c2 / l2 )3 ) ]

Equivalent pillar stiffness


C = (1-0.63 x/y) x3 (y/3) when x < y

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Torsional inertia of virtual beam


Inertia of lintels

Vertical actions (gravity)The lintel stiffness is the actual stiffness of the slab corresponding to theequivalent frame width

Horizontal actions (gravity)The lintel stiffness is taken as 35% of the stiffness of the slabcorresponding to the equivalent frame width

The equivalent support stiffness already defined is used for both verticaland horizontal actions.


a) Moments due to vertical forces (gravitational action)

Moment distribution on the plate

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Negative moments In interior support In exterior support

Support strip 75% 100%

Central strip 25% 20%

Positive moments In interior support In exterior support

Support strip 60% 60%

Central strip 40% 40%

b) Moments due to horizontal forces: 100% absorbed by the supports strip


Moment transmission from plates to pillars

Transmission of bending from the support to the slab: of the moment at the end of the pillar Md , only a part k Md causes bending to the slab. Supplementary reinforcement is needed to resist this bending.

The rest is applied as shear stresses on the torsional element (1-k) Md.

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c1/c'2 0,5 1,0 2,0 3,0

K 0,55 0,40 0,30 0,20

c1 = dimension of the support in the direction parallel to the framec2 = dimension of the support in the direction perpendicular to the frame. In faade pillars,

c2 is taken as two times this dimension

The tension forces corresponding to k Md must be resisted with reinforcementplaced in a strip equal to the support width plus 1.5 times the depth of slab ordrop panel at each side

The part (1-k) Md must be resisted as torsion by edge and torsional beams. Itmust also be taken into account in the punching verification.


REQUESTS FOR THE APPLICATION OF THE DIRECT FRAME METHOD

- The mest defined by the supports has to be ortogonal with devations in support location limited to 1/10 of the span

- The ratio between the longest and shortest sides of the pannel must no te greater than 2

- The difference between the span lengths of consecutive spans must not be greater than 1/3 of the longest span

- Only for uniformly distributed loading. The live load must not be greater than 2 times the permanen load.

- There must be at least three spans in each direction

Simplified calculation of forces by the direct method


Simplified calculation of forces by the direct method

The method is only applicable to vertical loading

Calculation of moments in lintels:

The negative moments in supports and positive moments in spans are taken as a percentage (see table) of the isostatic moment Mo = (q a L2) / 8,Where q is the load, a the width and L the span length of the lintel.

Case A Case B Case CNegative moment in exterior support 30% 0% 65%

Positive moment in span 52% 63% 35%Negative moment in interior support 70% 75% 65%


Case A: slab on exterior supports

Case A: slab supported on wall

Case A: interior panel(slab on interior supports)


Calculation of moments in pillars

A Exterior suport

M pillar superior = (Ks / (Ks + Ki)) 0.3 MoM pillar inferior = (Ki / (Ks + Ki)) 0.3 MoK = 4 E Ipillar / Lpillar Lpillar = pilar length

B Interior supports:

Mo = 0.07 [ (gd + 0.5 qd) Lp1 L211 gd Lp2 L2l2]

gd=permanent load; qd=live load


Reinforcement detailing The separation between the principal reinforcing bars shall not be > 25 cm or > 2

times the depth h of the slab.

The diameter of the principal bars should must be lesser than h/10

The inferior and superior reinforcement in one direction must have a total section of at least 25% of the reinforcement in the other direction,

In the edges of the slabs additional reinforcement will be needed to resist possible concentrated loads

The reinforcement in the support strip must be continuous with the needed overlaps close to the supports. It must include at least two bars across the pillars, which must be anchored in the case of the exterior pillar.

Lower side armor should be continuous media ensolapada. 2 bars through the pillars and is anchored in the outer pillar.

In edge beans of waffle slabs the distance between hoops must be


Supports strip

% of minimum reinforcement

50

The rest

100

100

50

The rest

Support face

top

bottom

Central striptop

bottom

Exterior support axis Interior support axis

3) 4 Two-way Floor Slabs 3

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