2-Way Slab (Flat Slab)

Case Study #1

• Under what circumstances, the flat plate/slab or ribbed slab system are applicable ?

• What are the merits and demerits if they are adopted?

Primary StructureFloor system-material: composite structural steel/concrete- type: beams, composite metal deck/concrete floor- pattern: radial beams, edge beams and one-way floor slab- beam clear span 14 m- floor slab span 2.2 - 3.25m

Core- material reinforced concrete

Note:- If the floor plan is rectangular, ribbed slab system could have been adopted.

Design of Two-Way Floor Slab System

(including ribbed and waffle slab system)

Lecture GoalsLecture Goals

• One-way and two-way slab• Slab thickness, h • Simplified Coefficient Method

Comparison of OneComparison of One--way and Twoway and Two--way way slab behaviorslab behavior

One-way slabs carry load in one direction.

Two-way slabs carry load in two directions.


One-way and two-way slab action carry load in two directions.

One-way slabs: Generally, long side/short side > 1.5


Flat slab Two-way slab with beams

Comparison between a twoComparison between a two--way way slab verses a oneslab verses a one--way slabway slab

For flat plates and slabs the column connections can vary between:


Flat Plate Waffle slab


The two-way ribbed slab and waffled slab system: General thickness of the slab is 50 to 100 mm.

Comparison of OneComparison of One--way and Twoway and Two--way way slab behavior Economic Choices slab behavior Economic Choices

• Flat Plate suitable span 6 to 7.5m with LL= 3 ~ 5 kN/m2

Advantages– Low cost formwork– Exposed flat ceilings – Fast

Disadvantages– Low shear capacity– Low Stiffness (notable deflection)


• Flat Slab suitable span 6 to 9m with LL= 4 ~ 7.5 kN/m2

Advantages– Low cost formwork– Exposed flat ceilings – Fast

Disadvantages– Need more formwork for capital and panels


• Waffle Slab suitable span 9 to 15m with LL= 4 ~ 7.5 kN/m2

Advantages– Carries heavy loads– Attractive exposed ceilings – Fast

Disadvantages– Formwork with panels is expensive


• One-way Slab on beams suitable span 3 to 6m with LL= 3 ~ 5 kN/m2

– Can be used for larger spans with relatively higher cost and higher deflections

• One-way joist floor system is suitable span 6 to 9m with LL= 4 ~ 6 kN/m2

– Deep ribs, the concrete and steel quantities are relative low

– Expensive formwork expected.


ws =load taken by short direction

wl = load taken by long direction

δA = δB

Rule of Thumb: For B/A > 2, design as one-way slab

EI

Bw

EI

Aw

384

5

384

5 4l

4s =

ls4

4

l

s 162A BFor wwA

B

w

w=⇒==

TwoTwo--Way Slab DesignWay Slab DesignStatic Equilibrium of Two-Way Slabs

Analogy of two-way slab to plank and beam floor

Section A-A:

Moment per m width in planks

Total Moment

mper kNm 8

21wlM =⇒

( ) kNm 8

21

2flwlM =⇒


Analogy of two-way slab to plank and beam floor

Uniform load on each beam

Moment in one beam (Sec: B-B) kN 82

221

lblwlM ⎟

⎠⎞

⎜⎝⎛=⇒

kN/m 2

1wl⇒


Total Moment in both beams

Full load was transferred east-west by the planks and then was transferred north-south by the beams;

The same is true for a two-way slab or any other floor system.

( ) kNm 8

22

1lwlM =⇒

General Design ConceptsGeneral Design Concepts(1) Simplified Coefficient Method (SCM)

Limited to slab systems to uniformly distributed loads and supported on equally spaced columns. Method uses a set of coefficients to determine the design moment at critical sections. Two-way slab system that do not meet the limitations of the ACI Code 13.6.1 or BS8110 must be analyzed more accurate procedures.

General Design ConceptsGeneral Design Concepts(2) Equivalent Frame Method (EFM)

A three dimensional building is divided into a series of two-dimensional equivalent frames by cutting the building along lines midway between columns. The resulting frames are considered separately in the longitudinal and transverse directions of the building and treated floor by floor.

Equivalent Frame Method (EFM)Equivalent Frame Method (EFM)

Longitudinal equivalent frame

Transverse equivalent frame

Equivalent Frame Method (EFM)Equivalent Frame Method (EFM)

Elevation of the frame Perspective view

Method of AnalysisMethod of Analysis

(1) Elastic Analysis

Concrete slab may be treated as an elastic plate. Use Timoshenko’s method of analyzing the structure. Finite element analysis

Method of AnalysisMethod of Analysis(2) Plastic Analysis

The yield method used to determine the limit state of slab by considering the yield lines that occur in the slab as a collapse mechanism.

The strip method, where slab is divided into strips and the load on the slab is distributed in two orthogonal directions and the strips are analyzed as beams.

The optimal analysis presents methods for minimizing the reinforcement based on plastic analysis

Method of AnalysisMethod of Analysis(3) Nonlinear analysis

Simulates the true load-deformation characteristics of a reinforced concrete slab with finite-element method takes into consideration of nonlinearities of the stress-strain relationship of the individual members.

Simplified Coefficient MethodSimplified Coefficient Method

Basic Steps in TwoBasic Steps in Two--way Slab Designway Slab Design

Choose layout and type of slab.Choose slab thickness to control deflection. Also,

check if thickness is adequate for shear.Choose Design method

– Equivalent Frame Method- use elastic frame analysis to compute positive and negative moments

– Direct Design Method - uses coefficients to compute positive and negative slab moments

1.2.

3.

Basic Steps in TwoBasic Steps in Two--way Slab Designway Slab Design

Calculate positive and negative moments in the slab.Determine distribution of moments across the width of

the slab. - Based on geometry and beam stiffness.Assign a portion of moment to beams, if present.Design reinforcement for moments from steps 5 and 6.Check shear strengths at the columns

4.5.

6.7.8.

Minimum Thickness, hMinimum Thickness, h(only available in ACI code)(only available in ACI code)

Minimum Slab Thickness for twoMinimum Slab Thickness for two--way way constructionconstruction

The ACI Code 9.5.3 specifies a minimum slab thickness to control deflection. There are three empirical limitations for calculating the slab thickness (h), which are based on experimental research. If these limitations are not met, it will be necessary to compute deflection.


The definitions of the terms are:

h = Minimum slab thickness without interior beams

ln =

β =

αm=

Clear span in the long direction measured face to face of column

the ratio of the long to short clear span

The average value of α for all beams on the sides of the panel.

Definition of BeamDefinition of Beam--toto--Slab Stiffness Ratio, Slab Stiffness Ratio, αα

Accounts for stiffness effect of beams located along slab edge reduces deflections of panel

adjacent to beams.

slab of stiffness flexural

beam of stiffness flexural=α


With width bounded laterally by centerline of adjacent panels on each side of the beam.

scs

bcb

scs

bcb

E

E

/4E

/4E

I

I

lI

lI==α

slab uncracked of inertia ofMoment Ibeam uncracked of inertia ofMoment I

concrete slab of elasticity of Modulus Econcrete beam of elasticity of Modulus E

s

b

sb

cb

====

Beam and Slab Sections for calculation of Beam and Slab Sections for calculation of αα

Beam and Slab Sections for calculation of Beam and Slab Sections for calculation of αα Beam and Slab Sections for calculation of Beam and Slab Sections for calculation of αα

Definition of beam cross-section

Charts may be used to calculate α Fig. 13-21

Minimum Slab Thickness for twoMinimum Slab Thickness for two--way way constructionconstruction22.0 m ≤≤ α(a) For

( )2.05361400

8.0

m

yn

−+

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=αβ

fl

h

fy in MPa. But not less than 125 mm.


m2 α<(b) For

β9361400

8.0 yn

+

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=

fl

h

fy in MPa. But not less than 88 mm.


2.0m <α(c) For

Use the following table


Slabs without interior beams spanning between supports and ratio of long span to short span < 2

See section 9.5.3.3 For slabs with beams spanning between supports on all sides.

270 MPa

413 MPa

517 MPa


Slabs without drop panels meeting 13.3.7.1 and 13.3.7.2,

tmin = 125 mm

Slabs with drop panels meeting 13.3.7.1 and 13.3.7.2,

tmin = 100 mm

ExampleExample

A flat plate floor system with panels 7.2m by 6m is supported on 500mm square columns. Determine the minimum slab thickness required for the interior and corner panels. Use fcu = 30 MPa and fy = 460 MPa.

ExampleExample

The floor system consists of solid slabs and beams in two directions supported on 500mm square columns. Determine the minimum slab thickness required for an interior panel. Use fcu = 30 MPa and fy = 460 MPa.

ExampleExample

The cross-sections are:

ExampleExample

The resulting cross section:


Maximum Spacing of Reinforcement

At points of max. +/- M:

Max. and Min Reinforcement Requirements

( )( )

( )7.12.3 ACI mm. 450 and8110 BS 213.3.2 ACI 2

≤≤≤

sdsts

( ) ( ) ( )

( ) ( )balsmaxs

S&Tsmins

75.0

13.3.1 ACI 7.12 ACI from

AA

AA

=

=

Simplified Coefficient Method for TwoSimplified Coefficient Method for Two--way Slabway Slab

Minimum of 3 continuous spans in each direction. (3 x 3 panel)

Rectangular panels with long span/short span 2

Method of dividing total static moment Mo into positive and negative moments.

Limitations on use of Simplified Coefficient method

1.

2. ≤

Simplified Method for TwoSimplified Method for Two--way Slabway Slab

Successive span in each direction shall not differ by more than 1/3 the longer span.

Columns may be offset from the basic rectangular grid of the building by up to 0.1 times the span parallel to the offset.

3.

4.



All loads must be due to gravity only (N/A to unbraced laterally loaded frames, from mats or pre-stressed slabs)

Service (unfactored) live load 2 service dead load

5.

6. ≤



For panels with beams between supports on allsides, relative stiffness of the beams in the 2perpendicular directions.

Shall not be less than 0.2 nor greater than 5.0

Limitations on use of Simplified method

7.

212

221

l

l

α

α


Accounts for stiffness effect of beams located along slab edge reduces deflections of panel

adjacent to beams.

slab of stiffness flexural

beam of stiffness flexural=α


With width bounded laterally by centerline of adjacent panels on each side of the beam.

scs

bcb

scs

bcb

4E

4E

/4E

/4E

I

I

lI

lI==α

slab uncracked of inertia ofMoment Ibeam uncracked of inertia ofMoment I

concrete slab of elasticity of Modulus Econcrete beam of elasticity of Modulus E

s

b

sb

cb

====

Beam and Slab Sections for calculation of Beam and Slab Sections for calculation of αα Beam and Slab Sections for calculation of Beam and Slab Sections for calculation of αα

Beam and Slab Sections for calculation of Beam and Slab Sections for calculation of αα

Definition of beam cross-section

Charts may be used to calculate α Fig. 13-21

Column and Middle StripsColumn and Middle Strips

The slab is broken up into column and middle strips for analysis

Distribution of MomentsDistribution of Moments

Slab is considered to be a series of frames in two directions:


Slab is considered to be a series of frames in two directions:


Total static Moment, Mo

( )3-13 ACI 8

2n2u

0llwM =

( )cn

n

2

u

0.886d h using calc. columns, circular forcolumns between span clear

strip the of widthtransverse area unit per load design

==

=

=

lll

wwhere

Column Strips and Middle StripsColumn Strips and Middle Strips

Moments vary across width of slab panel

Design moments are averaged over the width of column strips over the columns & middle strips between column strips.

∴


Column strips Design w/width on either side of a column centerline equal to smaller of

⎩⎨⎧

1

2

25.0 25.0

ll

l1= length of span in direction moments are being determined.

l2= length of span transverse to l1


Middle strips: Design strip bounded by two column strips.

Positive and Negative Moments in PanelsPositive and Negative Moments in Panels

Moment coefficients are assigned to + M and -M Rules given in BS8110 Table 3.12

-0.086 FL

+0.063 FL

-0.063 FL

Shear in PanelsShear in Panels

Shear coefficients are given in BS8110 Table 3.12

0.6 F

F is the total design load on the strip of slab between adjacent columns.

0.5 F

Distribution of moment in 2Distribution of moment in 2--way way slab without beam (flat slab)slab without beam (flat slab)

45%55%+ M

25%75%- M

Middle strip

Column strip

2-Way Slab (Flat Slab)

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