93395851-2-Way-Slab-Flat-Slab.pdf

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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?

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Primary Structure

Floor 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 Goals

• One-way and two-way slab

• Slab thickness, h

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Comparison of One

-

way and Two

-

way

slab behavior

One-way slabs carry

load in one direction. Two-way slabs carry load in two directions.

Comparison of One

-

way and Two

-

way

slab behavior

One-way and two-way

slab action carry load in two directions.

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

Comparison of One

-

way and Two

-

way

slab behavior

Flat slab Two-way slab with beams

Comparison between a two

Comparison between a two-

-way

way

slab verses a one

slab verses a one-

-way slab

way slab

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

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Comparison of One

-

way and Two

-

way

slab behavior

Flat Plate Waffle slab

Comparison of One

-

way and Two

-

way

slab behavior

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

Comparison of One

-

way and Two

-

way

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)

Comparison of One

-

way and Two

-

way

slab behavior Economic Choices

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

Advantages

– Low cost formwork – Exposed flat ceilings – Fast

Disadvantages

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Comparison of One

-

way and Two

-

way

slab behavior Economic Choices

• 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

Comparison of One

-

way and Two

-

way

slab behavior Economic Choices

• 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.

Comparison of One

-

way and Two

-

way

slab behavior

w_s=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 B w EI A w 384 5 384 5 l 4 4 s = l s 4 4 l s 16 2A B For w w A B w w ₌ ₌ _⇒ ₌

Two

Two-

-Way Slab Design

Way Slab Design

Static 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 m per kNm 8 2 1 wl M = ⇒

( )

kNm 8 2 1 2 f l wl M = ⇒

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Two

Two-

-Way Slab Design

Way Slab Design

Analogy of two-way slab to plank and beam floor Uniform load on each beam

Moment in one beam (Sec: B-B) kN 8 2 2 2 1 lb l wl M ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ⇒ kN/m 2 1 wl ⇒

Two

Two-

-Way Slab Design

Way Slab Design

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 2 2 1 l wl M = ⇒

General 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 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.

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Equivalent Frame Method (EFM)

Longitudinal equivalent frame

Transverse equivalent frame

Equivalent Frame Method (EFM)

Elevation of the frame Perspective view

Method 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 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

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Method 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 Method

Basic Steps in Two

-

way 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 Two

-

way 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.

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Minimum Thickness, h

(only available in ACI code)

Minimum Slab Thickness for two

-

way

construction

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.

Minimum Slab Thickness for two

-

way

construction

The definitions of the terms are:

h = Minimum slab thickness without interior beams l_n =

β = α_m=

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 Beam

-

to

-

Slab Stiffness Ratio,

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

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Definition of Beam

-

to

-

Slab Stiffness Ratio,

α

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

s cs b cb s cs b cb

E

/

4E

/

4E

I

l

I

l

I

₌

=

α

slab uncracked of inertia of Moment I beam uncracked of inertia of Moment I concrete slab of elasticity of Modulus E concrete beam of elasticity of Modulus E s b sb cb = = = =

Beam and Slab Sections for calculation of

α

Beam and Slab Sections for calculation of

α

Beam and Slab Sections for calculation of

α

Definition of beam cross-section

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Minimum Slab Thickness for two

-

way

construction

2 2 . 0 ≤

α

_m ≤ (a) For

(

0.2

)

5 36 1400 8 . 0 m y n − + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + =

α

β

f l h

f_yin MPa. But not less than 125 mm.

Minimum Slab Thickness for two

-

way

construction

m 2<

α

(b) For

β

9 36 1400 8 . 0 y n + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + = f l h

f_yin MPa. But not less than 88 mm.

Minimum Slab Thickness for two

-

way

construction

2 . 0 m <

α

(c) For

Use the following table

Minimum Slab Thickness for two

-

way

construction

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

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Minimum Slab Thickness for two

-

way

construction

Slabs without drop panels meeting 13.3.7.1 and 13.3.7.2, t_min = 125 mm

Slabs with drop panels meeting 13.3.7.1 and 13.3.7.2, t_min = 100 mm

Example

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 f_cu= 30 MPa and f_y= 460 MPa.

Example

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 f_cu= 30 MPa and f_y= 460 MPa.

Example

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Example

The resulting cross section:

Minimum Slab Thickness for two

-

way

construction

Maximum Spacing of Reinforcement At points of max. +/- M:

Max. and Min Reinforcement Requirements

(

)

(

)

(

ACI7.12.3

)

mm. 450 and 8110 BS 2 13.3.2 ACI 2 ≤ ≤ ≤ s d s t s ( ) ( )

(

)

(max) s( )bal s S & T s min s

75 .

0

13.3.1 ACI

7.12 ACI

from

A

=

Simplified Coefficient Method for Two

-

-way Slab

way 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 M_ointo positive and negative moments.

Limitations on use of Simplified Coefficient method 1.

2.

≤

Simplified Method for Two

-

way 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.

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Simplified Method for Two

-

way Slab

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.

≤

Limitations on use of Simplified Coefficient method

Simplified Method for Two

-

way Slab

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

Shall not be less than 0.2 nor greater than 5.0 Limitations on use of Simplified method

7. 2 1 2 2 2 1

l

α

Definition of Beam

-

to

-

Slab Stiffness Ratio,

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

Definition of Beam

-

to

-

Slab Stiffness Ratio,

α

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

s cs b cb s cs b cb

4E

/

4E

/

4E

I

l

I

l

I

₌

=

α

slab uncracked of inertia of Moment I beam uncracked of inertia of Moment I concrete slab of elasticity of Modulus E concrete beam of elasticity of Modulus E s b sb cb = = = =

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Beam and Slab Sections for calculation of

α

Beam and Slab Sections for calculation of

α

Beam and Slab Sections for calculation of

α

Definition of beam cross-section

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

Column and Middle Strips

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

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Distribution of Moments

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

Distribution of Moments

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

Distribution of Moments

Total static Moment, M_o

(

ACI13-3

)

8 2 n 2 u 0 l l w M =

(

n c

)

n 2 u 0.886d h using calc. columns, circular for columns between span clear strip the of width transverse area unit per load design = = = = l l l w where

Column 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.

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Column Strips and Middle Strips

Column strips Design

w/width on either side of a column centerline equal to smaller of ⎩ ⎨ ⎧ 1 2 25 . 0 25 . 0 l l l1= length of span in

direction moments are being determined.

l₂= length of span transverse to l₁

Column Strips and Middle Strips

Middle strips: Design

strip bounded by two column strips.

Positive 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 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.

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Distribution of moment in 2

-

way

slab without beam (flat slab)

45% 55% + M 25% 75% - M Middle strip Column strip