Download - Chapter 3 RC Flat Slab

Chapter 3 RC Flat Slab

GENERAL

In reinforced concrete flat slab buildings, floors are directly supported by columns as shown

in Fig. 3.1 without the use of intermediary beams. Flat slab systems are popular for use in

office and residential buildings, hospitals, schools and hotels. Omission of the beams yields

larger thickness of slabs.

Fig. 3.1 Simple flat slab

For covering a large area, three methods are usually adopted. These are

(1) Conventional Tee beam - slab construction.

(2) Flat slabs where the beams are omitted except the edge beams which may or may not be

provided.

(3) Grid slabs where deeper beams are used with closer spacing and columns are omitted.

3.1 ADVANTAGES OF FLAT SLAB

(1) The formwork is simpler than the tee beam-slab construction which gives economy and

simplicity in formwork.

(2) Construction of flat slab is simple and speedy.

(3) The architectural finish can be directly applied to the underside of the slab.

Design of Reinforced Concrete & Post-Tensioned flat slab using software 1

Supporting columns

Slab


(4) Absence of beams allows lower storey heights and, as a result, cost saving in vertical

cladding, partition walls, mechanical systems, plumbing and a large number of other

items of construction especially for medium and high rise buildings.

(5) They provide flexibility for partition location and allow passing and fixing services

easily.

(6) Windows can be extended up to the underside of the ceiling.

(7) The absence of sharp corners gives better fire resistance and less danger of concrete

sapling and exposing the reinforcement.

(8) Flat slab can result in more storey being accommodated within a restricted height of the

building.

3.2 TYPE OF FLAT SLAB CONSTRUCTION

(1) Simply the slab is supported on columns having no drop and no column head.

(2) The slab is strengthened by thickening the slab around the column, known as the drop.

(3) The slab is additionally strengthened by providing flared column head also known as

capital.

Fig. 3.2 Types of flat slab construction

Design of Reinforced Concrete & Post-Tensioned flat slab using software. 2


The flat slab is a two-way slab bending in both the directions and hence the reinforcement in

both the directions is necessary. The exact theoretical analysis is quite complex and can be

made by numerical techniques like finite element method or finite difference method. In fact

such exact analysis is not must all the times because of the moment redistribution

phenomenon. The code gives two methods for designing such slabs.

(1) Direct design method {D.D.M.}: In this method. Empirical coefficients are used to find

out the design moments at various points.

(2) Equivalent frame method {E.F.M.}: In this' method. The structure is divided into plane

continuous frames and the analysis is carried out.

3.3 COLUMN AND MIDDLE STRIPS

The flat slab panels are divided as set out in clause 31.1 IS:456-2000, into column strips and

middle strips as shown in fig. 3.3.



Fig. 3.3 Elements of flat slab

Column strip is a design strip with a width 0.25 lx or 0.25 ly whichever is smaller, on each

side of the column. When the drop is present, the width of the column strip shall be taken as

the width of the drop provided that the width of drop is not less than one-third of the panel

length in that direction.

Middle strip is the part of the slab bounded on each of its opposite sides by a column strip.

Panel is the part of a slab bounded on each four sides by the centre-line of a column or

centre-lines of adjacent spans.

3.4 PROPORTIONING OF FLAT SLAB ELEMENTS



The elements of flat slab are proportioned as set out in clause 30.2, IS: 456 and described as

follows:

(1) Thickness of flat slab: The thickness of slab shall be generally controlled by deflection

requirements same as that for solid slabs. The minimum thickness should be 125 mm. If the

flat slab contains drops and if the width of the drops in both directions is at least equal to

one-third of the respective spans, the deflection rules as applied to solid slabs are directly

applied to such slabs, otherwise the permissible span to effective depth ratios should be

multiplied by 0.9. For this purpose, the longer span should be considered (unlike the two-way

slab where the short span is considered), Also for finding out the modification factor for

tension reinforcement, the average percentage of steel across the whole width of panel at

mid-span should be used.

(2) Drops: The drops are provided to reduce the shear stresses around the column supports

and also to reduce the negative moment reinforcement. When the drops are provided in

flat slabs, they should satisfy the following requirements:

(a) They should be rectangular in plan and have a length in each direction not less than one-

third of the panel length in that direction.

(b) 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.

(c) The minimum projection of drop below the slab may be taken as one-fourth the thickness

of slab (not specified by code but referred in SP:24). The maximum thickness of drop for

the purpose of calculating negative moment reinforcement shall be the thickness of slab

plus one quarter the distance between the edge of drop and the edge of capital (column

head).

(3) Column head: Where column heads are provided, that portion of a column head which

lies within the largest right circular cone or pyramid that has a vertex angle of 90° and



can be included entirely within the outlines of the column and the column head shall be

considered for design purpose.

3.5 METHODS OF ANALYSIS AND DESIGN

It shall be permissible to design the slab system by one of the following methods:

a) The direct design method as specified in clause 31.4 of IS 456-2000 and

b) The equivalent frame method as specified in clause 31.5 of IS 456-2000.

3.6 LIMITATIONS OF DIRECT DESIGN METHOD

Slab system designed by the direct design method shall fulfill the following conditions:

a) There shall be minimum of three continuous spans in each direction,

b) The panels shall be rectangular, and the ratio of the longer span to the shorter span within

a panel shall not be greater than 2.0,

c) It shall be permissible to offset columns to a maximum of 10 percent of the span in the

direction of the offset not withstanding the provision in (b),

d) The successive span lengths in each direction shall not differ by more than one-third of the

longer span. The end spans may be shorter but not longer than the interior spans, and

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

3.7 DISTRIBUTION OF MOMENTS IN SLABS

This is dealt with in clause 31.4.2, IS: 456 and described below.

In the direct design method, the total design moment for a span shall be determined for a strip

bounded laterally by the centre-line of the panel on each side of the centre-line of the

supports. The total moment Mo to be resisted by the slab equals the sum of positive and



average negative bending moments in the span. It is the same as that for a simply supported

span. For a uniform load it is given by

Mo = (w ly ) lx2 / 8

where lx and ly are as defined in fig. 3.3. In the interior span, the total design moment Mo

shall be distributed in the following proportions:Negative design moment 0.65 Mo. Positive

design moment 0.35 Mo

Fig. 3.4 Distribution of total moments

Note that the negative design moment sha1l be located at the face of rectangular supports,

circular supports being treated as square supports having the same area. Refer to fig. 3.4. For

external span, the slab is not completely fixed at discontinuous edge. Fixity at the edge

depends on torsional restraint supplied at the discontinuous edge which is provided by

flexural stiffness of the exterior panel and exterior column. If the stiffness of edge support

were infinite, the slab at the edge can be considered as fixed, e.g. the discontinuous edge is



supported by a large beam having a large value of torsional stiffness or by an R.C.C. wall.

When the discontinuous edge is supported only at the column, fixity may be created only in

the region around the column. In such a case, rotation of the slab at the column will be zero

and maximum at the centre of discontinuous edge. Thus, distribution of total moment in the

exterior panel depends on relative stiffness of columns and slab meeting at a joint.

To take into account, the above facts, code defines the ratioac=∑ Kc

Ks.

Where αc = 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.

Kc = sum of the flexural stiffness of the columns meeting at the joint (upper and

lower columns)

Ks = flexural stiffness of the slab, expressed as moment per unit rotation.

The total design moment Mo shall be now distributed as follows in the end span. Refer to

fig.3.4.

Interior negative design moment:

= (0.75−0.10

1+1ac

)M 0 -------------------------------(1)

Positive design moment:

= (0.63−0.28

1+1ac

)M 0 -------------------------------(2)

Exterior negative design moment:

= ( 0.65

1+1ac

)M 0 -------------------------------(3)

It shall be permissible to modify these design moments by up to 10 per cent so long ,as the

total design moment Mo for the panel in the direction considered is not less than that



calculated by equation (1). This relaxation given by the code is due to the moment

redistribution phenomenon. The negative moment section shall be designed to resist the

larger of the two interior negative design moments determined for the spans framing in to a

common support unless an analysis is made to distribute the unbalanced moment in

accordance with the stiffness of the adjoining parts. The total moment distribution in to

negative and positive moments for internal and external spans is shown in fig.3.4. We have

now distributed the total moment into positive and negative moments. Let us consider, in

what proportion these moments are allotted to the column strip and the middle strip i.e.

distribution of moment across the panel width. The column strip takes more than 50% of the

longitudinal moment. The column strip takes a larger share of negative longitudinal moment

than the positive longitudinal moment. The variation of moment across the panel width is

dealt with in clause 31.5.5, IS: 456. Accordingly the following distribution shall be made in

design:

(a) Column strip: Negative moment at an interior support

At an interior support, the column strip shall be designed to resist 75 per cent of the total

negative moment in the panel at that support.

(b) Column strip: Negative moment at an exterior support

At an exterior support, the column strip shall be designed to resist the total negative moment

in the panel at that support. 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 1y' the length of

span transverse to the direction moments are being determined, the exterior negative moment

shall be considered to be uniformly distributed across the length 1y.

(c) Column strip: Positive moment for each span



For each span the column strip shall be designed to resist 60 per cent of the total positive

moment in the panel.

(d) Moments in the middle strip

The middle strip shall be designed on the following bases:

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

the adjacent middle strips.

(2) Each middle strip shall be proportioi1ed to resist the sum of the moments assigned to its

two half middle strips.

(3) 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

first row of interior columns.

3.8 EFFECT OF PATTERN LOADING

In direct design method, the moments are found out by considering all the spans loaded with

dead load and live load. When the pattern loading (the live load on certain spans but not on

others) is used, there can be substantial rise in the moments inducing overstressing in the

slabs. This overstressing is not much at supports but substantial at the mid-spans. A moderate

increase in the moments can be accommodated by the slab because of the moment

redistribution phenomenon. Code permits the overstressing up to 33 per cent. To limit the

overstressing up to 33 per cent, the code requires that the columns should have some

minimum stiffness as long as the live load does not exceed one-half that of the dead load.

The reasons for specifying these limits are as follows:

(1) If the columns are stiff, the joint of slab and columns approaches to fixed conditions. This

reduces the effects on the moments in the span when pattern loading is used.



(2) The smaller the ratio of live load to the dead load, the smaller the effect of live load

variations.

Should the ratio of live load to the dead load exceed 0.5, some modifications in design

moments are necessary. It can be shown by calculating a number of cases with all the spans

loaded with DL + LL and with pattern loading that the increase in negative moment due to

pattern loading is not much but the positive moment increases substantially. This means that

when the ratio of live load to dead load exceeds 0.5, the modifications in only positive

moment will be sufficient as an increase in negative moment can be redistributed to positive

moment. According to clause 31.4.6, IS: 456, in direct design method, when the ratio of live

load to dead load exceeds 0.5;

(a) The sum of flexural stiffness of the columns above and below the slab, ∑ Kc, shall be

such that αc is not less than the appropriate minimum value αc, min specified in table 11 of IS:

456 and reproduced in table 3.1.

(b) If the sum of the flexural stiffness of the columns, ∑ Kc does not satisfy (a), the positive

design moments for the panel shall be multiplied by the coefficient β s given by the following

equation:

βs = 1 + ( 2−wd

w1

4+wd

w1

)(1− α c

α c ,min) -------------------------------(4)

αc is the ratio of flexural stiffness of the columns above and below the slab to the flexural

stiffness of the slabs at a joint taken in the direction moments are being determined and is

given by

ac=∑ Kc

∑ Ks



Where Kc and Ks are flexural stiffness of column and slab respectively.

TABLE 3.1

MINIMUM PERMISSIBLE VALUES OF αc

Imposed load / Dead load Ratio ly / lx Value of αc,min

(1) (2) (3)0.5 0.5 to 2.0 01.0 0.5 0.61.0 0.8 0.71.0 1.0 0.71.0 1.25 0.81.0 2.0 1.22.0 0.5 1.32.0 0.8 1.52.0 1.0 1.62.0 1.25 1.92.0 2.0 4.93.0 0.5 1.83.0 0.8 2.03.0 1.0 2.33.0 1.25 2.83.0 2.0 13.0

3.9 TRANSFER OF FLOOR LOADS INTO COLUMNS

Two kinds of the loads are arising at the joint of columns and slab which must be safely

transmitted to the column, these are:

(1) Vertical load acting at the centre-line of the column and

(2) The unbalanced moment.

For the internal columns, there is a slab on both the sides, thus most of the moment is

balanced and unbalanced moment is comparatively small. This unbalanced moment may be

due to pattern loading, wind and earthquake loading or may other eccentric loading. For the

exterior columns, however, the unbalanced moment may have substantial value.



3.9.1 TRANSFER OF VERTICAL LOAD

The vertical load from the slab tends to displace the slab downward. This leads to produce

the shear stresses on the critical section around the perimeter of the column. The slab must be

strong enough to resist this punching shear stress. Although the failure occurs on the diagonal

plane, the code assumes the failure on a fictitious surface called critical section lying at a

distance d/2 from the periphery of the column/capital or drop panel perpendicular to the

plane of the slab. where d is the effective depth of the section.

3.9.2 TRANSFER OF MOMENT

If unbalanced moment exists, it is transmitted to the column by two ways. These are

(a)Flexural couple and (b) Shear couple. These are discussed below.

(a) Flexural couple



Fig. 3.5 Unbalanced moment and transfer of moment

A certain portion of unbalanced moment equal to αM is transferred through flexural couple

as shown in fig. 3.5, where M is unbalanced moment. To resist this flexural couple, a slab

width (effective strip) between lines that are one and one-half slab or drop panel thickness,

1.5 D on each side of the column or capital may be considered effective D being the depth of

slab or drop. To resist this flexural couple the additional reinforcement shall be designed in

the effective strip. The value of α is given by

α= 1

1+23 √ a1

a2

Where a1 = overall dimension of the critical section for shear in which moment acts

the direction in

a2 = overall dimension of the critical section for shear transverse to the direction in

which moment acts.

(b) Shear couple



The fraction of unbalanced moment (1- α) M shall be considered to be transferred by a shear

couple. It is stated by code that the shear stresses shall be taken as varying linearly about the

centroid of the critical section.

3.10 DESIGN FOR SHEAR

The shear stress is induced by two ways. i.e. nominal shear stress and shear due to transfer of

unbalanced moment. The total maximum shear stress will be the sum of the two.

3.10.1 CALCULATION OF SHEAR STRESS

The nominal shear stress due to vertical shear is given by

τ v 1=v

b0 d -------------------------------(5.1)

Where, V = design vertical shear

bo = perimeter of a critical section

d = effective depth of slab

The shear stress due to fraction of unbalanced moment is given by

τ v 2=(1−α ) M c

J c

-------------------------------(5.2)

Where, c= distance of extreme fiber of critical section from the centroidal axis of critical section

about which the unbalanced moment is acting

Jc = polar moment of inertia of the areas parallel to applied moment plus moment of inertia of

end area about centroidal axis of critical section For rectangular sections.

Jc = Iyy + Ixx

The total maximum and minimum shear stresses are given by

τ v=τ v1+τ v 2

The total maximum shear stress should not be more than the permissible shear stress. It may

be noted that as long as the ratio of live load to the dead load does not exceed 0.5, the shear



stress due to unbalanced moment for gravity load may not be considered, only nominal shear

stress may be considered.

3.10.2 PERMISSIBLE SHEAR STRESS

This is dealt with in clause 31.6.3 IS: 456-2000 and is described below:

When shear reinforcement is not provided, the calculated shear stress at the critical section

shall not exceed ks τc.

Where, ks = (0.5 + βc) but not greater than 1, βc being the ratio of short side to long side of the

column capital

τc = 0.25 √ f ck in limit state method of design. And 0.16√ f ck in working stress method

of design.

When the shear stress at the critical section exceeds the above permissible stress, but less

than 1.5 τc shear reinforcement shall be provided. If the shear exceeds 1.5 τc, the flat slab

shall be redesigned (either by increasing the depth of slab or by using the richer concrete

mix). To design the shear reinforcement, the shear stresses shall be investigated at successive

sections more distant from the support and shear reinforcement shall be provided up to a

section where the shear stress does not exceed 0.5 τc. While designing the shear

reinforcement, the shear stress carried by the concrete shall be assumed to be 0.5 τc and

reinforcement shall carry the remaining shear.

The area of shear reinforcement may be given by

A sv=vu−k s τ c b0 d

0.87 f y

xsv

d -------------------------------(6)

Where, Asv = the total cross-sectional area of all stirrup legs in the perimeter. The maximum

spacing of stirrups shall be 0.75 d and also the stirrups should be continued up to a distance d

beyond the section at which the shear stress is within allowable limit (ie. ks τ).



3.11 PROVISION OF REINFORCEMENT

The rules regarding provision of flat slab reinforcement are dealt with in clause 31.7.IS: 456-

2000 and are given below.



Fig. 3.6 Minimum bend joint locations and extensions for reinforcement in flat slabs


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