4643two way slabs1

8/9/2019 4643Two Way Slabs1

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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 intwo directions.


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slab behavior

Flat slab Two-way slab with beams


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Comparison of One-way and Two-way slab

behavior

For flat plates and slabs the column connections can vary

between:


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slab behavior

Flat Plate Waffle slab


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slab behavior

The two-way ribbed slab and waffled slab system: eneral

thic!ness of the slab is " to #$cm.


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behavior %conomic Choices

Flat Plate &for relatively li'ht loads as in apartments or offices(

suitable span )."m to *.$m with ++, -"/0m1.

Advantages

+ow cost formwor!

%2posed flat ceilin's

Fast

Disadvantages +ow shear capacity

+ow Stiffness &notable deflection(


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Flat Slab &for heavy industrial loads( suitable span * to 3m with

++, "-4."/0m1.

Advantages

+ow cost formwor!

%2posed flat ceilin's

Fast Disadvantages

/eed more formwor! for capital and panels


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Waffle Slab &two-way 5oist system( suitable span 4."m to #1m

with ++, )-4."/0m1.

Advantages

Carries heavy loads

6ttractive e2posed ceilin's

Fast

Disadvantages Formwor! with panels is e2pensive


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One-way Slab on beams suitable span to *m with ++, -"/0m1.

Can be used for lar'er spans with relatively hi'her cost and hi'herdeflections

One-way 5oist system suitable span * to 3m with ++, )-*/0m1.

7eep ribs8 the concrete and steel 9uantities are relative low

%2pensive formwor! e2pected.


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?ehaviour of slabs loaded to

failure p*1)-*14


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behavior

ws ,load ta!en by short direction

wl , load ta!en by lon' direction

δ6 ,

δ?

Rule of Thumb: For ?06 @ 18

desi'n as one-way slab

EI

Bw

EI

Aw

A)

"

A)

" )l)

s =

ls)

)

l

s

#*16?For ww A

B

w

w

=⇒==


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Two-Way Slab 7esi'n

Static Equilibrium of Two-Way Slabs

6nalo'y of two-way slab to plan! and beam floor

Section 6-6:

Boment per m width in plan!s

Total Boment

1

# !/-m0mA

wl m⇒ =

( )

1

#

6-6 1 !/-mA

l

M wl ⇒ =


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Two-Way Slab 7esi'n


6nalo'y of two-way slab to plan! and beam floor

niform load on each beam

Boment in one beam &Sec: ?-?(1

# 1lb !/.m

1 A

wl l M

⇒ = ÷

# !/0m1

wl ⇒


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Two-Way Slab 7esi'n


Total Boment in both beams

Full load was transferred east-west by the plan!s and then wastransferred north-south by the beamsD

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

where:

( )1

1# !/.m

A B B

l M wl −⇒ =

( ) ( )

1 1

# 1

6-6 1 #

!/-m8 !/.mA A B B

l l M wl M wl

−= =


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7istribution of moments in slabs

Read p631-637


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eneral 7esi'n Concepts


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eneral 7esi'n Concepts

&1( %9uivalent Frame Bethod &%FB(

6 7 buildin' is divided into a series of 17 e9uivalent

frames by cuttin' the buildin' alon' lines midway

between columns. The resultin' frames are consideredseparately in the lon'itudinal and transverse directions of

the buildin' and treated floor by floor.

+imited to slab systems to uniformly distributed loads andsupported on e9ually spaced columns. Bethod uses a set

of coefficients to determine the desi'n moment at critical

sections.

( 7irect 7esi'n Bethod &77B(


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%9uivalent Frame Bethod &%FB(

%levation of the frame Perspective view


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Bethods of 6nalysis

&1( Plastic 6nalysis

The yield method used to determine the limit state of slab by

considerin' the yield lines that occur in the slab as a collapse

mechanism.

The strip method 8 where slab is divided into strips and the

load on the slab is distributed in two ortho'onal directions

and the strips are analy>ed as beams.


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Column and Biddle Strips

The slab is bro!en up

into column and

middle strips for

analysis


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?asic Steps in Two-way Slab 7esi'n

). Calculate positive and ne'ative moments in the slab.". 7etermine distribution of moments across the width of the

slab. ?ased on 'eometry and beam stiffness.

*. 6ssi'n a portion of moment to beams8 if present.

4. 7esi'n reinforcement for moments from steps " and *. Steps-4 need to be done for both principal directions.

A. Chec! shear stren'ths at the columns


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7efinition of ?eam-to-Slab Stiffness Eatio8 α

With width bounded laterally by centerline of ad5acent

panels on each side of the beam.

cb b cb b

cs s cs s

)% 0 %

)% 0 % f

I l I

I l I α = =

slabuncrac!edof inertiaof Boment;

beamuncrac!edof inertiaof Boment;

concreteslabof elasticityof Bodulus%concrete beamof elasticityof Bodulus%

s

b

sb

cb

=

===


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?eam and Slab Sections for calculation of α


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7efinition of beam cross-section

Charts may be used to calculate α


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%2ample #-#


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Binimum Slab Thic!ness for Two-way

Construction

The 6C; Code 3.". specifies a minimum slab thic!ness to

control deflection. There are three empirical limitations for

calculatin' the slab thic!ness &h(8 which are based on

e2perimental research. ;f these limitations are not met8 it will be

necessary to compute deflection.


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Construction

f $.1 1α ≤ ≤&a( For

( )

y

n

fm

$.A#)$$

* " $.1

f l

hβ α

+

÷ =+ −

f y in BPa. ?ut not less than #1."cm


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Construction

fm1 α


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Binimum Slab Thic!ness for 1-way Construction

fm $.1α


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Binimum Slab Thic!ness for two-way

construction

The definitions of the terms are:

h , Binimum slab thic!ness without interior beams

ln ,Clear span in the lon' direction measured face to face

of column

β = the ratio of the lon' to short clear span

αfm,the avera'e value of α for all beams on the sides ofthe panel.


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7irect 7esi'n Bethod for Two-way Slab

+imitations on use of 7irect 7esi'n method

. Successive span in each direction shall not differ by more

than #0 the lon'er span.

). Columns may be offset from the basic rectan'ular 'rid ofthe buildin' by up to $.# times the span parallel to the offset.

". 6ll loads must be due to 'ravity only &/06 to unbraced

laterally loaded frames8 foundation mats or pre-stressedslabs(

*. Service &unfactored( live load 1 service dead load≤


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7irect 7esi'n Bethod for Two-way Slab

4. For panels with beams between supports on all sides8 relative

stiffness of the beams in the 1 perpendicular directions.

Shall not be less than $.1 nor 'reater than ".$

+imitations 1 and 4 do not allow use of 77B for slab panels

that transmit loads as one way slabs.

+imitations on use of 7irect 7esi'n method

1# 1

1

1 #

f

f

l l

α α


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7istribution of Boments

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


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7istribution of Boments

Total static Boment8 M o

( )-#6C;

A

1

n1u$

l l w M =

( )cn

n

1

u

$.AA*dhusin'calc.columns8circularfor

columns betweenspanclear

striptheof widthetransvers

areaunit perloadfactored

==

=

=

l

l

l

wwhere


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%2ample #.1


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Column Strips and Biddle Strips

Boments vary across width of slab panel

7esi'n moments are avera'ed over the width

of column strips over the columns middle

strips between column strips.∴


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Column Strips and

Biddle Strips

Column strips 7esi'n

width on either side of a

column centerline e9ual to

smaller of

#

1

1".$

1".$

l

l

l#, len'th of span in

direction moments are bein' determined.

l1, len'th of span

transverse to l#


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Column Strips

and Biddle

StripsBiddle strips: 7esi'n

strip bounded by two

column strips.


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Boment 7istribution


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7istribution of B$


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Positive and /e'ative Boments in Panels

B$ is divided into G B and -B Eules 'iven in 6C; sec. #.*.

( ) A

1

n1u

$av'uu

l l w

M M M =≥−+⊕


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Transverse 7istribution of Boments

The lon'itudinal moment values mentioned are for the entire

width of the e9uivalent buildin' frame. The width of two half

column strips and two half-middle stripes of ad5acent panels.


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Transverse 7istribution of Boments

Transverse distribution of the lon'itudinal moments to middle andcolumn strips is a function of the ratio of len'th l10l#8αf#8 and bt.

torsional constant

cb b cb# t

cs s cs s

1

$.*

#

f E I E C

E I E I

x x y

C y

α β = =

= − ÷ ÷ ∑


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7istribution of B$

6C; Sec #.*..)

For spans framin' into a common support ne'ative moment

sections shall be desi'ned to resist the lar'er of the 1 interior B u=s

6C; Sec. #.*.."

%d'e beams or ed'es of slab shall be proportioned to resist in

torsion their share of e2terior ne'ative factored moments


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Factored Boment

in Column Strip

F t d B t i C l St i


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Factored Boment in Column Strip

Eatio of torsional stiffness of ed'e beam to fle2ural stiffness of

slab&width, to beam len'th(

βt

=


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Factored Boments

Factored Boments in

beams &6C; Sec.

#.*.(: resist a

percenta'e of columnstrip moment plus

moments due to loads

applied directly to

beams.


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Factored Boments

Factored Boments in Biddle strips &6C; Sec. #.*.(

The portion of the G Bu and - Bu not resisted by column

strips shall be proportionately assi'ned to correspondin' halfmiddle strips.

%ach middle strip shall be proportioned to resist the sum of

the moments assi'ned to its 1 half middle strips.


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%2ample #- and #-) pa'e *"$


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6C; Provisions for %ffects of Pattern +oads

T f f l


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Transfer of moments to columns

%2terior columns shall be desi'ned for $.Bo assumed

to be about the centroid of the shear perimeter8 and should be

divided between columns above and below the slab in proportion to

their stiffnesses.

;nterior Columns: refer to te2tboo! pa'e *"4


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Shear Stren'th of Slabs

;n two-way floor systems8 the slab must have ade9uate thic!nessto resist both bendin' moments and shear forces at critical section.

There are three cases to loo! at for shear.

#. Two-way Slabs supported on beams

1. Two-Way Slabs without beams

. Shear Eeinforcement in two-way slabs without beams.


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#. Two-way slabs supported on beams

The critical location is found at d distance from the column8 where

The supportin' beams are stiffand are capable of transmittin' floor loads to the columns.

( )c c 0 *V f bd φ φ =

Sh St th f Sl b


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There are two types of shear that need to be addressed

1. Two-Way Slabs without beams

#. One-way shear or beam shear at distance d from the column

1. Two-way or punch out shear which occurs alon' a truncated

cone.


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One-way shear considers critical section a distance d from the

column and the slab is considered as a wide beam spannin'

between supports.

( )ud c c 0 *V V f bd φ φ ≤ =


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Two-way shear fails alon' a a truncated cone or pyramid around the

column. The critical section is located d01 from the column face8

column capital8 or drop panel.


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Shear Stren'th of SlabsShear Stren'th of Slabs

;f shear reinforcement is not provided8 the shear stren'th ofconcrete is the smaller of:

#.

1.

bo , perimeter of the critical section

βc ,ratio of lon' side of column to short side

( )c c o c o

c

1$.#4 # $.V f b d f b d φ φ φ

β

= + ≤

÷

sc c o

o

$.$A 1d V f b d b

α φ φ = + ÷

αs is )$ for interior columns8 $ for ed'e columns8 and 1$ for

corner columns.

Sh St th f Sl b


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. Shear Eeinforcement in two-way slabs without beams.

For plates and flat slabs8 which do not meet the condition for

shear8 one can either

-;ncrease slab thic!ness

- 6dd reinforcement

Eeinforcement can be done by shearheads8 anchor bars8conventional stirrup ca'es and studded steel strips &see 6C;

##.##.).

Sh St th f Sl b


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Shearhead consists of steel ;-beams or channel welded into four

cross arms to be placed in slab above a column. 7oesnot apply to e2ternal columns due to lateral loads and

torsion.

Sh St th f Sl b


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6nchor bars consists of steel reinforcement rods or bent barreinforcement

Sh St th f Sl b


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Conventional stirrup ca'es

Sh St th f Sl b


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Studded steel strips


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Eeinforcement 7etails +oads

6fter all percenta'es of the static moments in the column and middle

strip are determined8 the steel reinforcement can be calculated for

ne'ative and positive moments in each strip.

Ba2imum Spacin' of Eeinforcement

6t points of ma2. G0- B:

Bin Eeinforcement Ee9uirements

( )

( )

1 6C; #..1

and )" 6C; 4.#1.

s t

s cm

≤

≤

( ) ( ) ( )s min s TFS from 6C; 4.#1 6C; #..# A A=

Binimum e2tension for reinforcement in


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slabs without beams&Fi'. #..A(

4643two way slabs1

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