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Design of flat slabs by IS: 456

The term flat slab means a reinforced concrete slab with or without drops, supported generally without

beams, by columns with or without flared column heads (see Fig. 12). A flat slab may be solid slab or

may have recesses formed on the soffit so that the sof fit comprises a series of ribs in two directions.The recesses may be formed by removable or permanent filler blocks.

Components of flat slab design:

a) Column strip :

Column strip means a design strip having a width of 0.25 I,, but not greater than 0.25 1, on each side

of the column centre-line, where I, is the span in the direction moments are being determined,

measured centre to centre of supports and 1, is the -span transverse to 1,, measured centre to centre of

supports.

b) Middle strip :

Middle strip means a design strip bounded on each of its opposite sides by the column strip.

c) Panel:

Panel means that part of a slab bounded on-each of its four sides by the centre -line of a

Column or centre-lines of adjacent-spans.

Division into column and middle strip along:

Longer span Shorter span

1L =6.6 m , 2L

=5.6 m

( i ) column strip

= 0.252L = 1.4 m

But not greater than 0.251

L = 1.65 m

(ii) Middle strip

= 5.6 (1.4+1.4) = 2.8 m

1L =5.6 m , 2L

=6.6 m

( i ) column strip

= 0.25 2L = 1.65 m

But not greater than 0.251

L = 1.4 m

(ii) Middle strip

= 6.6 (1.4+1.4) = 3.8 m

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d) Drops :

The drops when provided shall be rectangular in plan, and have a length in each direction not less than

one- third of the panel length in that direction. For exterior panels, the width of drops at right angles to

the non- continuous edge and measured from the centre -line of the columns shall be equal to one -half

the width of drop for interior panels.

Since the span is large it is desirable to provide drop.

Drop dimensions along:


1L =6.6 m , 2L =5.6 m

Not less than 1L /3 = 2.2 m

1L =5.6 m , 2L =6.6 m

Not less than1L /3 = 1.866 m

Hence provide a drop of size 2.2 x 2.2 m i.e. in column strip width.

e) column head :

Where column heads are provided, that portion of a column head which lies with in the largest right

circular cone or pyramid that has a vertex angle of 90and can be included entirely within the outlines

of the column and the column head, shall be considered for design purposes (see Fig. 2).

5.6 m

3.8 m

M.S

1.4 m

C.S

1.4 m

C.S

6.6 m

2.8 m

M.S

1.4 m

C.S

1.4 m

C.S

Fig 1:

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Fig 2:

Column head dimension along:


1L =6.6 m ,

2L =5.6 m

Not greater than1

L /4 = 1.65 m

1L =5.6 m ,

2L =6.6 m

Not greater than1

L /4 = 1.4 m

Adopting the diameter of column head = 1.30 m =1300 mm

f) Depth of flat slab:

The thickness of the flat slab up to spans of 10 m shall be generally controlled by considerations of span

(L ) to effective depth ( d) ratios given as below:

Cantilever 7; simply supported 20; Continuous 26

For slabs with drops, span to effective depth ratios given above shall be applied directly; otherwise the

span to effective depth ratios in accordance with above shall be multiplied by 0.9. For this purpose, the

longer span of the panel shall be considered. The minimum thickness of slab shall be 125 mm.

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Depth of flat slab:

Considering the flat slab as a continuous slab over a span not exceeding 10 m

L

d= 26

26

Ld

Depth considering along:


1L =6.6 m , 2L =5.6 m

6600

26 26

Ld =253.8 mm

Say 260 mm

1L =5.6 m , 2L =6.6 m

5600

26 26

Ld =215.3 mm

Say 220 mm

Taking effective depth of 25mm

Overall depth D = 260 +25 = 285 mm 125 mm (minimum slab thickness as per IS: 456)

It is safe to provide depth of 285 mm.

g) Estimation of load acting on the slab:

Dead load acting on the slab = 0.285 x 25 = 6.25 2/KN m = 1dw

Floor finishes etc. load on slab = 1.45 2/KN m = 2dw

Live load on slab = 7.75 2/KN m = lw

Total dead load = 1dw + 2dw =7.72/KN m = dw

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The design live load shall not exceed three times the design dead load.

Check:

7 . 7 51 .0 0 6 3

7 .7

l

d

wo k

w

Total design load =2

15.45 KN/md l

w w

h) Total Design Moment for a Span

The absolute sum of the positive and average and is given by negative bending moments in each

direction shall be taken as:

0

8

nW l

M

0M = total moment.

W= design load on an area 1 2l l

nl = clear span extending from face to face of columns, capitals, brackets or walls, but not less than

0.651

l

1l = length of span in the direction of

0M .

2l = length of span transverse to 1l .

Circular supports shall be treated as square supports havi ng the same area.Equivalent side of the column head having the same area:

2 2(1.3) 1.1524 4

a d m

Clear span along long span = nl =1 1

6.6 (1.152) (1.152) 5.448 4.292 2

m

(Should not be less than 0.651

l ) ok

Clear span along long span = nl =1 15.6 (1.152) (1.152) 4.44 3.642 2

m m

(Should not be less than 0.651

l ) ok

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Total design load along:


nl =5.448 m , 2l =5.6 m

2 nW w l l

15.45 5.6 5.448 471.36W K N

nl =4.44 m , 2l =6.6 m

2 nW w l l

1 5 . 4 5 6 . 6 4 . 4 4 4 5 2 . 7 4W K N

The absolute sum of ve and +ve moment in a panel along:


nl =5.448 m , 2l =5.6 m

0

4 7 1 . 3 6 5 . 4 4 8

8 8

nW l

M

03 2 0 . 9 9M K N m

nl =4.44 m , 2l =6.6 m

0

4 5 2 . 7 4 4 . 4 4

8 8

nW l

M

02 5 1 . 2M K N m

(i) Negative and Positive Design Moments:

The negative design moment shall be at the fac e of rectangular supports, circular supports being

treated as square supports having the same 31.4.5.1 Columns built integrally with the slab system

area. Shall be designed to-resist moments arising from loads .

In an interior span, the total design moment 0M shall be distributed in the following proportions:

Negative design moment 0.65

Positive design moment 0.35

In an end span, the total design moment0

M shall be distributed in the following proportions:

Interior negative design moment: 1

0.100.75

1c

Positive design moment: 1

0.280.63

1c

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Exterior negative design moment: 1

0.65

1c

c Is the ratio of flexural stiffness of the exterior columns to the flexural stiffness of the slab at a

joint taken in the direction moments are being determined and is given by:

c

c

s

K

K

cK =sum of the flexural stiffness of the columns meeting at the joint.

sK =flexural stiffness of the slab, expres sed as moment per unit rotation

It shall be permissible to modify these design moments by up to 10 percent, so lon g as the total

design moment0M for the panel in the direction considered is not less than that required by:

0

8

n

W lM

The negative moment section shall be designed to resist the larger of the two interior negative

design moments determined for the spans framing into a common support unless an analysis is

made to distribute the unbalanced moment in accordance with the stiffness of the adjoining parts.

Column strip :

Negative moment at an interior support: At an interior support, the column strip shall be

designed to resist 75 percent of the total negative moment in the panel at that support.

Negative moment at an exterior support:

a) At an exterior support, the column strip shall be designed to resist the t otal negative moment inthe panel at that support.

b) Where the exterior support consists of a column or a wall extending for a distance equal to or

greater than three-quarters of the value of 2l . The length of span transverse t o the direction

moments are being determined, the exterior negative moment shall be considered to be uniformly

distributed across the length 2l .

Positive moment for each span: For each span, the column strip shall be designed to r esist 60

percent of the total positive moment in the panel.

Moments in the middle strip:

a) That portion of-the design moment not resisted by the column strip shall be assigned to the

adjacent middle strips.

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b) Each middle strip shall be proportione d to resist the sum of the moments assigned to its two half

middle strips. cl The middle strip adjacent and parallel to an edge supported by a wall shall be

proportioned, to resist twice the moment assigned to half the middle strip corresponding to the fir st

row of interior columns.

Stiffness calculation:

let the height of the floor = 4.0 m

clear height of the column = height of floor depth of drop thickness of slab thickness of head.

= 4000 140 285 300 = 3275 mm

Effective height of column = 0.8 x 3275 = 2620 mm

(Assuming one end hinged and other end fixed)

stiffness coefficient

sum of f lexura l s t i f fness of co lum n a c t i ng a t the jo in t

f lexu ra l s t if fness o f the s labc

c

s

K

K

Longer span

34 2 4 520 104 4 4 4 502 2

12 327.5c

BOTTOM TOP

EEI EI EI EK

L L L L

34 660 28.5 2 4 1587.731.39

12 560 4 2273.5

cS C

s

E K EK

K E

From table 17 of IS: 456-2000

2 L

1 D

, m i n

, m i n

W0 . 8 4 8 & 1 . 0 0

W

0 . 7c

c c

L

L

Hence correction for pattern of loading in the direction of longer span is not required.

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Shorter span

42 (50)3975.8

12 262cK

3

560 28.5 1421.412 760

3975.82.79

1421.4

c

c

K

From table 17 of IS: 456-2000 for

2 L

1 D

,m in

,m in

W1 .17 & 1 .00

W

0 . 7 5

o k

c

c c

L

L

Hence the correction for pattern loading in the direction of short span is not required.

From table 17 of IS 456-2000

Imposed load/dead loadRatio 2

1

l

lValue of ,min

c

(1) (2) (3)

0.5

1.0

1.0

1.0

1.0

1.0

2.0

2.0

2.0

2.02.0

3.0

3.0

3.0

3.0

3.0

0.5 to 2.0

0.5

0.8

1.0

1.25

2.0

0.5

0.8

1.0

1.252.0

0.5

0:8

1.0

1.25

2.0

0

0.6

0.7

0.7

0.8

1.2

1.3

1.5

1.6

1.94.9

1.8

2.0

2.3

2.8

13.0

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Distrubution of bending moment across the panel width

It is an exterior panel.

Longer span

column strip

-ve B.M at exterior support = 00 . 6 5 0 .6 5 3 2 0 .9 91 .0 1 .0 1 2 1 .3 4 K N m1 1

1 11 . 3 9

c

M

+ve span BM =0

0 .2 8 0 .2 80 .6 3 0 .6 0 0 .6 3 3 2 0 .9 9 0 .6 0 9 0

1 11 1

1 . 3 9c

M K N m

-ve span BM at interior support =

0

0 .1 0 0 .1 00 .7 5 0 .7 5 0 .7 5 3 2 0 .9 9 0 .7 5 1 6 6 .5 0 K N m

1 11 1

c c

M

Middle strip

-ve BM at exterior support = 00 . 6 5 0 . 0 0 . 0 K N m1

1c

M

+ve span BM =0

0 .2 8 0 .2 80 .6 3 0 .4 0 0 .6 3 3 2 0 .9 9 0 .4 0 5 9 .9 6

1 11 1

1 . 3 9c

M K N m

-ve BM at interior support =

0

0 .1 0 0 .1 00 .7 5 0 .7 5 0 .7 5 3 2 0 .9 9 0 .2 5 5 5 .5 0 K N m

1 1

1 1 1 . 3 9c

M

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Short span

column strip

-ve moment at exterior support = 00 . 6 5 0 .6 5 2 5 1 .21 .0 1 .0 1 2 0 .1 9 K N m1 1

1 12 . 7 9

c

M

+ve moment0.28

0.63 251.2 0.60 63.881

12.79

KNm

-ve moment at exterior support =

0

0 .1 0 0 .1 00 .7 5 0 .7 5 0 .7 5 2 5 1 .2 0 .7 5 1 2 7 .4 3 K N m

1 11 1

2 . 7 9c

M

Middle strip

-ve moment at exterior support = 00 . 6 5 0 . 0 0 . 0 K N m1

1c

M

+ve mid-span moment =0

0 .2 8 0 .2 80 .6 3 0 .4 0 0 .6 3 2 5 1 .2 0 .4 0 4 2 .5 9

1 11 1

2 . 7 9c

M K N m

-ve moment at interior span =

0

0 .1 0 0 .1 00 .7 5 0 .7 5 0 .7 5 2 5 1 .2 0 .2 5 4 2 .4 4 K N m

1 11 1

2 . 7 9c

M

j) Effective depth of the slab

Thickness of the slab, from consideration of maximum positive moment any where in the slab.

Maximum +ve BM occurs in the column strip (long span) = 90.91 KNm

factored moment = 1.50 x 90.91 = 136.36 KNm

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2

0

6

0.138 ( 2800 mm)

136.36 10d= (M-20 grade concrete)

0.138 20 2800

d=132.83 mm 140 mm

ckM f bd b

Using 12 mm (diameter) main bars.

Overall thickness of slab =12

140 15 161 mm 170 mm2

Depth (along longitudinal direction) =12

170 15 150 mm2

Depth (along longitudinal direction) = 150 12 138 mm

k) Thickness of drop from maximum ve moment consideration

Thickness of drop from consideration of maximum ve moment any where in the panel.

Max ve BM occurs in the column strip = 166.6 KNm2

6 2

0.138

1.5 166.6 10 0.138 20 1400

254.3 mm

u ckM f bd

d

d

Say 260 mm. Use 12 mm bars

Over all thickness of flat slab:12

260 15 281 mm2

D

1300 mm

300 mm

300

mm

045 5.6 m

6.6 m

2340 mm

2200 mm

d/2

d/2

1.3 m

d/2

d/2

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l) Shear in Flat Slab

The critical section for shear shall be at a distance d/2 from the periphery of the

column/capital/ drop panel, perpendicular to the plane of the slab where d is the effective

depth of the section (Fig. 2). The shape in plan is geometrically similar to the support

immediately below the slab.

check for shear stress developed in slab

The critical section for shear for the slab will be at a distance d/2 from the face of drop.

Perimeter of critical section = 4 x 2340 = 9340 mm

Total factored shear force:

0 1 21.5 15.45 [ (2.34)(2.34)]

= 1.5 15.45 [6.6 5.6-(5.47)]

= 729.78 KN

V L L

Nominal shear stress =3

2729.78 10 0.55 N/mm9340 140

uv

V

bd

shear strength of concrete = 20.25 =0.25 20 =1.11 N/mmc ckf

Permissible shear stress = v s ck

2

v

(0.5 ), 0.848

(0.5 0.848)

1.348 1 1

1 1.11

1.11 N/mm

safe design ok

if 1.5 then the slab should be re-designed

s c c

s

s

v c

c

k

k

k

m) check for shear in drop

0 0

2

( ) (1.3 0.26) 4.89 m

V=1.5 15.45[5.6 6.6- (1.3 0.26) ]4

V 812.27 KN

b D d

Nominal shear stress :

32

2

812.27 100.683 N/mm

4890 260

0.25 1.11 N/mm

[safe in shear]

v

c ck

v c

f

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n) Reinforcement details

Longer span

-ve exterior reinforcement:

6

2

0.87 [ 0.42 ]

1.5 121.34 10 0.87 415 [150 0.42 0.48 150]4209 mm

u y st u

st

st

M f A d x

AA

Use 12 mm bars =4209

38 No.s113

1.4 1000c/c spacing is = 36 mm c/c

38

+ve steel:-6

2

1.5 90 10 43239.3

3122 mm

3122

Use 12 mm bars = 28 .113

3.8 1000/ spacing = 135 mm c/c

28

st

st

A

A

No s

c c

Reinforcement along shorter span:

Column strip:

6

2

0.87 [ 0.42 ]

1.5 127.5 10 0.87 415 [140 0.42 0.48 140]

3768.9 mm

u y st u

st

st

M f A d x

A

A

Use 12 mm bars =2

3768.933 No.s

(12)

4


33

Middle strip:

6

2

0.87 [ 0.42 ]

1.5 63.88 10 0.87 415 [281 0.42 0.48 281]1182 mm

u y st u

st

st

M f A d x

AA

Use 12 mm bars =2

118210 No.s

(12)

4


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