Flat Slab resting
directly on
columns
1. What is a flat slab?
31.1 General
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
Flat slab with
drop panel and
column head
A flat slab may be solid slab
or may have recesses
formed on the soffit so that
the soffit comprises a series
of ribs in two directions.
The recesses may be formed
by removable or permanent
filler blocks.
Flat Slab with
drop panels
Flat slab with
column head
What is mean by Flat Slab?
• A reinforced concrete slab supported directly by concrete
columns without the use of beams.
• Flat slabs are highly versatile elements widely used in
construction, providing minimum depth, fast construction and
allowing flexible column grids.
Construction:
The benefits of choosing flat slabs include a minimum depth
solution, speed of construction, flexibility in the plan layout
(both in terms of the shape and column layout).
HISTORICAL DEVELOPMENT
• Flat slabs were originally invented in the U.S.A in year 1906.
• This was the start of these type of construction.
• Many slabs were load-tested between 1910-20 in U.S.A.
• 1914 Nicholas proposed a method of analysis of these slab based on simple
statics, this method is know as direct design method.
BASIC DEFINITION OF FLAT SLAB
The term flat slab means a reinforces concrete slab with or without drop,
supported generally without beams, by column with or without flared column
heads.
APPLICATION OF FLAT SLAB
 In the case of high rise building thinner slabs are required so that additional floors can

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be added.
The distance that be spanned by post-tensioned slabs exceeds that of reinforced
constructions with the same thickness.
For increasing span lengths so as to increases the usable unencumbered floor space in
buildings.
For diminishing the number of joints in the structure.
For the speedy construction of the project.
The amount of steel required is much less than in normal RCC structure.
The moulds can be used No. of times as per the demand.
Due to reduce beam section the load transferred to foundation is less compared to that
of RCC structure.
The structure is crack free as the whole structure is in compression.
Large span of slab can obtain easily.
ADVANTAGES OF FLAT SLAB
 Floor to floor height reduction
 Faster construction
 Early formwork stripping
 Water resistant properties
 Saving in materials.
 Reduced foundation load
 Greater column free areas
 Architectural freedom
 Reduced construction costs
LIMITATION OF FLAT SLAB
 Careful handling of prefabricated components such as concrete panels or steel
and glass panels is required.
 Attention has to be paid to the strength and corrosion-resistance of the joining
of prefabricated sections to avoid failure of the joint.
 Similarly, leaks can form at joints in prefabricated components.
 Transportation costs may be higher for voluminous prefabricated sections than
for the materials of which they are made, which can often be packed more
compactly.
Types:
1. Typical flat slab
2. Slab without drop and column with column head.
3. Slab with drop and column without column head
4. Slab with drop and column with column head.
Uses of column head:
•Increase shear strength of slab .
•Reduce the moment in the slab by reducing the clear or effective span.
Uses of drop panels :
•Increase shear strength of slab
•Increase negative moment capacity of slab.
1. Typical flat slab:• This may be called as beam-slab construction. Hence in warehouses,
offices and public halls some times beams are avoided and slabs are
directly supported by columns.
•
This types of construction is aesthetically appealing also.
These slabs which are directly supported by columns are called
Flat Slabs.
2.Slab without drop and column with column head:
• The column head is some times widened so as to reduce the punching
shear in the slab.
• The widened portions are called column heads. The column heads may be
provided with any angle from the consideration of architecture but for the
design, concrete in the portion at 45º on either side of vertical only is
considered as effective for the design.
3.Slab with drop and column without column head:
• Moments in the slabs are more near the column. Hence the slab
is thickened near the columns by
providing the drops.
• Sometimes the drops are called as capital of the column.
4.Slab with drop and column with column head:
Flat slab system:
• 1.One way system:
• One way slab is supported on two opposite side only thus
structural action is only at one direction. Total load is carried in the
direction perpendicular to the supporting beam.
• If a slab is supported on all the four sides but the ratio of longer
span to shorten span
is greater than, then the slab
will be considered as one way slab.
2. Two way slab:
• Two way slabs are the slabs that are supported on four sides and
the ratio of longer span to shorter span is less than.
•
In two way slabs, load will be carried in both the directions. So,
main reinforcement is provided in both direction for two way
slabs.
Advantages:
• Simple formwork
• No beams—simplifying under-floor services outside the
drops
• Minimum structural depth.
Disadvantages:
• Drop panels may interfere with larger mechanical ducting.
• Vertical penetrations need to avoid area around columns.
INTRODUCTION
What is a flat slab?
• a reinforced concrete slab supported directly
concrete columns without the use of beams
by
INTRODUCTION
Flat slab
Flat slab with column head
Flat slab with drop panels
Flat slab with drop panel and column head
INTRODUCTION
Uses of column heads :
• increase shear strength of slab
• reduce the moment in the slab by reducing the
clear or effective span
Flat slab with column head
INTRODUCTION
Uses of drop panels :
• increase shear strength of slab
• increase negative moment capacity of slab
• stiffen the slab and hence reduce deflection
BENEFITS
BENEFITS
•
•
•
•
•
•
Flexibility in room layout
Saving in building height
Shorter construction time
Ease of installation of M&E services
Prefabricated welded mesh
Buildable score
Benefits . . .
FLEXIBILITY IN ROOM LAYOUT
• allows Architect to introduce partition walls anywhere
required
• allows owner to change the size of room layout
• allows choice of omitting false ceiling and finish soffit of
slab with skim coating
Benefits . . .
SAVING IN BUILDING HEIGHT
• Lower storey height will reduce building weight due to
lower partitions and cladding to façade
• approx. saves 10% in vertical members
• reduce foundation
load
Slab
Slab
Beam
3.4m
2.8 m
Conventional
2.8 m
3.2 m
Beam-Free
Benefits . . .
SHORTER CONSTRUCTION TIME
flat plate design will
facilitate the use of big
table formwork to
increase productivity
Benefits . . .
SINGLE SOFFIT LEVEL
Kitchen 30
26
0
Living
Room
30 Toilet
Sho wer
75
155
Balcony
30
Yard
30
Single Level
Ceiling
FlatPlate Slab
• Simplified the table formwork needed
Benefits . . .
EASE OF INSTALLATION
OF M&E SERVICES
• all M & E services can be mounted directly on the
underside of the slab instead of bending them to avoid
the beams
• avoids hacking through beams
Benefits . . .
PRE-FABRICATED WELDED MESH
• Prefabricated in
standard sizes
• Minimised
installation time
• Better quality
control
Benefits . . .
BUILDABLE SCORE
• allows standardized structural members and
prefabricated sections to be integrated into the
design for ease of construction
• this process will make the structure more buildable,
reduce the number of site workers and increase the
productivity at site
• more tendency to achieve a higher Buildable score
DESIGN
CONSIDERATIONS
Design Considerations. . . .
WALL AND COLUMN POSITION
• Locate position of wall to maximise the structural stiffness for lateral
loads
• Facilitates the rigidity to be located to the centre of building
Typical floor plan of Compass the Elizabeth
Design Considerations. . . .
OPTIMISATION OF
STRUCTURAL LAYOUT PLAN
• the sizes of vertical and structural structural members can
be optimised to keep the volume of concrete for the entire
superstructure inclusive of walls and lift cores to be in the
region of 0.4 to 0.5 m3 per square metre
• this figure is considered to be economical and
comparable to an optimum design in conventional of
beam and slab systems
Design Considerations. . . .
DEFLECTION CHECK
• necessary to include checking of the slab deflection for
all load cases both for short and long term basis
• In general, under full service load,  < L/250 or 40 mm
whichever is smaller
• Limit set to prevent unsightly occurrence of cracks on nonstructural walls and floor finishes
Design Considerations. . . .
CRACK CONTROL
• advisable to perform crack width calculations based on
spacing of reinforcement as detailed and the moment
envelope obtained from structural analysis
• good detailing of reinforcement will
– restrict the crack width to within acceptable
tolerances as specified in the codes and
– reduce future maintenance cost of the building
Design Considerations. . . .
FLOOR OPENINGS
• No opening should encroach upon a column head or drop
• Sufficient reinforcement must be provided to take care of stress
concentration
Design Considerations. . . .
PUNCHING SHEAR
• always a critical consideration in flat plate design
around the columns
• instead of using thicker section, shear reinforcement in
the form of shear heads, shear studs or stirrup cages
may be embedded in the slab to enhance shear capacity
at the edges of walls and columns
Design Considerations. . . .
PUNCHING SHEAR
Shear
Studs
Design Considerations. . . .
CONSTRUCTION LOADS
• critical for fast track project where removal of forms at early
strength is required
• possible to achieve 70% of specified concrete cube
strength within a day or two by using high strength
concrete
• alternatively use 2 sets of forms
Design Considerations. . . .
LATERAL STABILITY
• buildings with flat plate design is generally less rigid
• lateral stiffness depends largely on the configuration of lift
core position, layout of walls and columns
• frame action is normally insufficient to resist lateral loads
in high rise buildings, it needs to act in tendam with walls
and lift cores to achieve the required stiffness
Design Considerations. . . .
LATERAL STABILITY
MULTIPLE FUNCTION PERIMETER BEAMS
• adds lateral rigidity
• reduce slab deflection
DESIGN
METHODOLOGY
Design methodology .. .
METHODS OF DESIGN
• the finite element analysis
• the simplified method
• the equivalent frame method
Design methodology .. .
FINITE ELEMENT METHOD
• Based upon the division of complicated structures into smaller
simpler pieces (elements) whose behaviour can be formulated.
• E.g of software includes SAFE, ADAPT, etc
• results includes
– moment and shear envelopes
– contour of structural deformation
and
Design methodology .. .
SIMPLIFIED METHOD
Table 3.19 may be used provided
• Live load > 1.25 Dead load
• Live load (excluding partitions) > 5KN/m2
• there are at least 3 rows of panels of approximately
equal span in direction considered
• lateral stability is independent of slab column
connections
Design methodology .. ..
SIMPLIFIED METHOD
Table 3.19: BM and SF coefficients for flat slab or 3 or more equal spans
Outer Support
Near centre of
1st span
Column
Wall
Moment
-0.04Fl*
0.086Fl
0.083Fl*
Shear
0.45F
0.4F
-
0.04Fl
-
-
Total
column
moments
First interior
span
-0.063Fl
0.6F
0.022Fl
Centre of
interior
span
Interior
span
0.071Fl
-0.055Fl
-
0.5F
-
0.022Fl
* the design moments in the edge panel may have to be adjusted according to 3.7.4.3
F is the total design ultimate load on the strip of slab between adjacent columns considered (1.4gk +
1.6 qk)
l is the effective span
Design methodology .. .
EQUIVALENT FRAME METHOD
• most commonly used method
• the flat slab structure is divided longitudinally and
transversely into frames consisting of columns and
strips of slabs with :
– stiffness of members based on concrete alone
– for vertical loading, full width of the slab is used to
evaluate stiffness
– effect of drop panel may be neglected if dimension < lx/3
Design methodology .. .
EQUIVALENT FRAME METHOD
Plan of floor slab
Step 1 : define line of support in X & Y
directions
Design methodology .. .
EQUIVALENT FRAME METHOD
9
10
10
9.2
0.8
DESIGN STRIP IN PROTOTYPE
9
10
10.6
10.5 0.8
STRAIGHTENED DESIGN STRIP
Step 2 : define design strips in X & Y
directions
DESIGN STRIP IN ELEVATION
ANALYSIS OF
FLAT SLAB
Analysis of flat slab..
COLUMN HEAD
Effective dimension of a head , lh (mm) =lesser of lho or lh max
where lho = actual dimension, lh max = lc + 2(dh-40)
(i) lh = lh, max
lh max
lho
lc
(ii) lh = lho
dh
lh max
lho
lc
dh
Analysis of flat slab..
COLUMN HEAD
(iii) lh = lh, max
dh
lh max
lho
40
(iv) lh = lho
lh max
lc
For circular column or column head,
effective diameter ,
hc = 4 x area/ < 0.25lx
lho
lc
dh
Analysis of flat slab..
DIVISION OF PANELS
The panels are divided into ‘column strips’ and middle strips’ in
both direction.
(a)
Slab Without Drops
Column strip
middle strip (ly-lx/2)
Column
strip
lx/4
lx/4
middle
strip
lx/4
ly (longer span)
lx (shorter
span)
lx/4
Analysis of flat slab..
Slab With Drops
Drop
middle strip (ly-drop size)
lx/4
middle
strip
Column strip
= drop size
ly (longer span)
note : ignore drop if dimension is less than lx/3
l
x
Dro
p
(b)
Analysis of flat slab..
MOMENT DIVISION
Apportionment between column and middle
strip expressed as % of the total negative
design
moment
Column strip
Middle strip
Negative
75%
25%
Positive
55%
45%
• Note : For slab with drops where the width of the middle strip exceeds L/2,
the distribution of moment in the middle strip should be increased in
proportion to its increased width and the moment resisted by the column
strip should be adjusted accordingly.
Analysis of flat slab..
MOMENT DIVISION - EXAMPLE
6000 6000 6000
6000
6000
5000
7000
Layout of building
5000
A floor slab in a building where stability is provided by shear walls in one
direction (N-S). The slab is without drops and is supported internally and
on the external long sides by square columns . The imposed loading on the
floor is 5 KN/m2 and an allowance of 2.5KN/m2 for finishes, etc. fcu = 40
KN/m2, fy = 460KN/m2
Analysis of flat slab..
MOMENT DIVISION - EXAMPLE
6000
6000
6000
6000
1250
5000
2500
3500
2500
2750
7000
4000
3000 3000
1500
3500
2500
Division of panels into strips in x and y direction
Analysis of flat slab..
MOMENT DIVISION - EXAMPLE
6000
6000
200
200
35
3500
35
2500
200
Column strip
3000 3000
exterior support
centre of 1st span
200
369
= 0.75*35 on 2.5m strip = 10.5Knm
= 0.55*200 on 2.5 strip = 44KNm
= 0.75*200 on 3m strip = 50KNm
1st interior support
centre of interior span = 0.55 *369 on 3m strip = 67.7KNm
3500
2500
Middle strip
exterior support
centre of 1st span
= 0.25*35 on 2.5m strip = 3.5KNm
= 0.45*200 on 2.5 strip = 36KNm
1st interior support
= 0.25*200 on 3m strip = 16.7KNm
centre of interior span = 0.45 *369 on 3m strip = 55.4KNm
Analysis of flat slab..
DESIGN FOR BENDING
INTERNAL PANELS
• columns and middle strips should be designed to
withstand design moments from analysis
Analysis of flat slab..
DESIGN FOR BENDING
EDGE PANELS
• apportionment of moment exactly the same as internal
columns
• max. design moment transferable between slab and edge
column by a column strip of breadth be is
Mt, max = 0.15 be d2 fcu
< 0.5 design moment (EFM)
< 0.7 design moment (FEM)
Otherwise structural arrangements shall be changed.
Analysis of flat slab..
PUNCHING SHEAR
1. Calculate Veff =kVt at column
perimeter (approx. equal span)
Vt = SF transferred from slab
k = 1.15columns
for internal
1.25 where
corner
and column,
edge columns
M acts parallel to free edge and
1.4 for edge columns where M acts at
right angle to free edge
Column perimeter
Perimeter A
Perimeter B
3d 3d
4 2
Column perimeter
Perimeter A
Perimeter B
Perimeter C
3d 3d 3d
4 4 2
 lx/3
2. Determine vmax= Veff /uod where uo is the
Checkof
vma
< 0.8 perimeter
f cu or 5 N/mm2
length
column
3. Determine v=(Veff -V/ud) where u is the
length of perimeter A and V is the column
load and check v < vc
4. Repeat step 3 for perimeter B and C
Analysis of flat slab..
DEFLECTION
Span/depth ratio
Cantilever
7
Simply supported
20
Continuous
26
(i) use normal span/effective depth ratio if drop width >1/3 span
each way; otherwise
(ii) to apply
0.9 modification factor for flat slab, or
where drop panel width < L/3
1.0 otherwise
Analysis of flat slab..
OPENINGS
Holes in areas bounded by the column strips may be formed
providing :
• greatest dimension < 0.4 span length and
lx (shorter
span)
• total positive and negative moments are redistributed between the
remaining structure to meet the changed conditions
ly (longer span)
Analysis of flat slab..
OPENINGS
Holes in areas common to two column strips may be formed providing :
that their aggregate their length or width does not exceed one-tenth of the width of the column
strip;
•
that the reduced sections are capable of resisting with the moments; and
•
that the perimeter for calculating the design shear stress is reduced if appropriate
lx (shorter
span)
•
ly (longer span)
Analysis of flat slab..
OPENINGS
Holes in areas common to the column strip and the middle strip may be
formed providing :
that in aggregate their length or width does not exceed one-quarter of the width
of the column strip and
•
that the reduced sections are capable of resisting the design moments
lx (shorter
span)
•
ly (longer span)
Analysis of flat slab..
OPENINGS
For all other cases of openings, it should be framed on all
sides with beams to carry the loads to the columns.
DETAILING OF
FLAT SLAB
Detailing of flat slab .. .
TYPE OF REINFORCEMENT
F-mesh
- A mesh formed by main wire with cross wire at a
fixed spacing of 800 mm
Main wire - hard drawn ribbed wire with diameter and
spacing as per design
Cross wire - hard drawn smooth wire as holding wire
H8-800mm c/c for main wire diameter > 10mm
Detailing of flat slab .. .
TYPE OF REINFORCEMENT
F-Mesh 2
Main Wire
Holding Wire
Holding Wire
(800mm c/c)
Main Wire
F-Mesh 1
Holding Wire
TensionLap
= 45 dia.
Main Wire
Holding
Wire
Main
Wire
Plan View of Mesh Layout
Main
Wire
F - Mesh
Main Wire
Cross Wire
F - Mesh
Main Wire
Cross Wire
Detailing of flat slab .. .
REINFORCEMENT FOR
INTERNAL PANELS
• Reinforcement are arranged in 2 directions parallel to each
span; and
• 2/3 of the reinforcement required to resist negative
moment in the column strip must be placed in the
centre half of the strip
• for slab with drops, the top reinforcement should be
placed evenly across the column strip
STANDARD LAPPING OF MESH
(FOR FLAT SLAB)
TYPICAL DETAIL SHOWING RECESS AT SLAB
SOFFIT FOR SERVICES
TYPICAL SECTION AT STAIRCASE
DETAILS OF INSPECTION CHAMBER AT
APRON
DETAILS OF INSPECTION CHAMBER AT
APRON
DETAILS OF INSPECTION CHAMBER AT
APRON
DETAILS OF INSPECTION CHAMBER AT
APRON
DETAILS OF INSPECTION CHAMBER AT PLAY
AREA
1ST STOREY (DWELLING UNIT) SLAB
DETAILS OF HOUSEHOLD SHELTER
TYPICAL DETAILS OF 125X250 RC CHANNEL FOR
GAS PIPE ENTRY
INTRODUCTION
CLASSIFICATION OF FLAT SLAB
SYSTEM
1. Solid flat slab(or flat plate)
2. Solid flat slab with drop panels
3. Solid flat slab with column heads
4. Banded flat slab
1.Solid flat slab(or flat plate)
2.Solid flat slab with drop panels
3. Solid flat slab with column heads
3.Banded flat slab
METHOD OF CONSTRUCTION SYSTEM
The flat slabs can be cast-in-situ (cast-in-place). Else, the slabs can be precast at
ground level and lifted to the final height. The later type of slabs is called lift
slabs.
1. POST TENSIONING SYSTEM
2. PRE TENSIONING SYSTEM
1. POST TENSIONING SYSTEM
2. PRE TENSIONING SYSTEM
2.
PURPOSE TENSIONING SYSTEM
 To reduce the deflection.
 To reduce the punching shear.
 Reduced structure depth
 To Greater column free areas.
 Reduces the required number of columns and foundations.
 To Increase load bearing capacity.
 To bear seismic forces.
COMPONENT PARTS OF FLAT SLAB
Column strip
2. Middle strip
3. Panel.
1.
ANLYSIS OF FLAT SLAB
The steps of analysis slab is
1. Determine the factored negative (Mu–) and positive moment (Mu+) demands at the critical sections in
a slab-beam member from the analysis of an equivalent frame. The values of Mu– are calculated at the
faces of the columns. The values of Mu+ are calculated at the spans. The following sketch shows a
typical moment diagram in a level of an equivalent frame due to gravity loads.
ANLYSIS OF FLAT SLAB
2. Distribute Mu– to the CS and the MS. These components are represented as Mu,–CS and Mu,–
MS, respectively. Distribute Mu+ to the CS and the MS. These components are represented as
Mu, +CS and Mu, +MS, respectively.
ANLYSIS OF FLAT SLAB
3.If there is a beam in the column line in the spanning direction, distribute
Mu, + CS between the beam and rest of the CS.
4) Add the moments Mu,–MS and Mu, +MS for the two portions of the MS
frames).
5) Calculate the design moments per unit width of the CS and MS.
each of Mu,–CS and
(from adjacent equivalent
Code provision for flat slab
1.Thickness of flat slab
As per IS-456 : 2000
Code provision for flat slab
2. For drop
3. For column head
DESIGN OF FLAT SLAB
Direct design method
Limitations
Slab system designed by the direct design method shall fulfill the following
condition.
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 column to a maximum of 10 percent of span
in the direction of offset not with standing the provision in (b).
Direct design method
d) The successive span length 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 span , and
e) The design lived load shall not exceed three times the design dead load.
Design steps for flat slab
1.
2.
3.
4.
5.
6.
7.
8.
Check preliminary dimension
Check for applicability of DDm
Divide the slab with frame in X and Y directions and obtain dimension of X and Y
frames.
Analysis the interior and exterior panel.
a. Longitudinal distribution
b. Transverse distribution
Estimate the design moment in the external column
Estimate the design moment in the internal column
Design for shear
Detailing should be done as per code requirement.
Design for shear
Punching shear
Punching shear reinforcement
CONCLUSIONS
 As per Indian code we are using cube strength but in international standards





cylindered are used which gives higher strength than cube.
Drops are important criteria in increasing the shear strength of the slab.
Enhance resistance to punching failure at the junct ion of concrete slab & column.
By incorporating heads in slab, we are increasing rigidity of slab.
In the interior span, the total design moments (Mo).
The negative moment’s section shall be designed to resist the larger of the two interior
negative design moments for the span framing into common supports.
CONCLUSIONS
 According to Indian standard (IS 456) for RCC code has recommended
characteristic strength of concrete as 20, 25, and 30 and above 30 for high
strength concrete. For design purpose strength of concrete is taken as 2/3 of
actual strength this is to compensate the difference between cube strength and
actual strength of concrete in structure. After that we apply factor of safety of
1.5. So in practice Indian standard actually us es 46% of total concrete
characteristic strength. While in International practice is to take 85% of total
strength achieved by test and then apply factor of safety which is same as
Indian standard so in actual they use 57% of total strength.
 Pre fabricated sections to be integrated into the design for ease of construction.
References
 Indian standards 456,875.
 S.P 16.
 Advanced Reinforced Concrete Design-P.C Varghese.
 A.K. Jain - Limit state design of Reinforced concrete.
 Reinforced concrete design - S.unnikrishna Pillai, Devdas Menon
 S.Ramamrutham & R. Narayan - Design of Reinforced concrete Structures.
 V.N, Vazirani & S.P Chandola – Hand book of civil Engineering.
Introduction
Common practice of design and construction is to
support the slabs by beams and support the beams by
columns.
 This may be called as beam-slab construction.
 The beams reduce the available net clear ceiling height.

Hence in warehouses, offices and public halls some times
beams are avoided and slabs are directly supported by
columns.
 This types of construction is aesthetically appealing also.
 These slabs which are directly supported by columns are
called Flat Slabs.

Typical Flat slab (without drop & column head)
The column head is some times widened so as to reduce
the punching shear in the slab.
 The widened portions are called column heads.
 The column heads may be provided with any angle from
the consideration of architecture but for the design,
concrete in the portion at 45º on either side of vertical
only is considered as effective for the design

Slab without drop & column with column head
Moments in the slabs are more near the column.
 Hence the slab is thickened near the columns by
providing the drops
 Sometimes the drops are called as capital of the column.

Slab with drop & column without column head
Types of flat slabs
Slabs without drop and column head
 Slabs without drop and column with column head
 Slabs with drop and column without column head
 Slabs with drop and column head

Slab with drop & column with column head
The portion of flat slab that is bound on each of its four
sides by centre lines of adjacent columns is called a
panel.
 A panel may be divided into column strips and middle
strips.
 Column Strip means a design strip having a width of
0.25L1 or 0.25L2, whichever is less.
 The remaining middle portion which is bound by the
column strips is called middle strip.

Panels, column strip & middle strip in y-direction
Proportioning of Flat slab
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 in that direction.
 For exterior panels, the width of drops at right angles to
the non continuous edge and measured from the centreline of the columns shall be equal to one half of the
width of drop for interior panels.

Column Heads

Where column heads are provided, that portion of the
column head which lies within the largest right circular
cone or pyramid entirely within the outlines of the
column and the column head, shall be considered for
design purpose
Thickness of Flat Slab
From the consideration of deflection control IS 456-2000
specifies minimum thickness in terms of span to effective
depth ratio.
 For this purpose larger span is to be considered.
 If drop is provided, then the maximum value of ratio of
larger span to thickness shall be
 40, if mild steel is used
 32, if Fe 415 or Fe 500 steel is used

If drops are not provided or size of drops do not satisfy
the specification, then the ratio shall not exceed 0.9
times the value specified above i.e.,
 40 X 0.9 = 36, if mild steel is used.
 32 X 0.9 = 28.8, if HYSD bars are used
 It is also specified that in no case, the thickness of flat
slab shall be less than 125 mm.

Determination of BM & SF
For this IS 456-2000 permits use of any one of the
following two methods:
 The Direct Design Method
 The Equivalent Frame Method

Direct Design Method
This method has the limitation that it can be used only if
the following conditions are fulfilled:
 There shall be minimum of three continuous spans in
each directions.
 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.
 The successive span length in each direction shall not
differ by more than one-third of longer span.

The design live load shall not exceed three times the
design dead load.
 The end span must be shorter but not greater than the
interior span.
 It shall be permissible to offset columns a maximum of
10 percent of the span in the direction of the offset not
withstanding the provision in (b).

Total Design moment

The absolute sum of the positive and negative moment in
each direction is given by
M0 = Total moment
 W = Design load on the area L2 X Ln
 Ln = Clear span extending from face to face of columns,
capitals, brackets or walls but not less than 0.65 L1
 L1 = Length of span in the direction of M0; and
 L2 = Length of span transverse to L1

In taking the values of Ln, L1 and L2, the following clauses
are to be carefully noted:
 Circular supports shall be treated as square supports
having the same area i.e., squares of size 0.886D.
 When the transverse span of the panel on either side of
the centre line of support varies, L2 shall be taken as the
average of the transverse spans.
 When the span adjacent and parallel to an edge is being
considered, the distance from the edge to the centre-line
of the panel shall be substituted for L2.

Distribution of BM in to +ve & -ve moments
The total design moment M0 in a panel is to be
distributed into –ve moment and +ve moment as
specified below:
 In an interior span:
 Negative Design Moment 0.65 M0
 Positive Design Moment 0.35 M0

In an end span

Interior negative design moment

Positive design moment

Exterior negative design moment

where I is the ratio of flexural stiffness at 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
Kc = Sum of the flexural stiffness of the columns meeting at
the joint; and
 Ks = Flexural stiffness of the slab, expressed as moment per
unit rotation

Distribution of Moments Across the Panel Width
in a Column Strip
Moments in Columns

In this type of constructions column moments are to be
modified as suggested in IS 456–2000 [Clause No.
31.4.5].
Shear Force
The critical section for shear shall be at a distance d/2
from the periphery of the column/capital drop panel.
 Hence if drops are provided there are two critical
sections near columns.
 The shape of the critical section in plan is similar to the
support immediately below the slab as shown in Fig.


For columns sections with re-entrant angles, the critical
section shall be taken as indicated in Fig

In case of columns near the free edge of a slab, the
critical section shall be taken as shown in Fig.

The nominal shear stress may be calculated as
V – is shear force due to design
 bo – is the periphery of the critical section
 d – is the effective depth


The permissible shear stress in concrete may be
calculated as ksτc, where ks = 0.5 + but not greater than 1,
where is the ratio of short side to long side of the
column/capital; and

If shear stress
– no shear reinforcement are required. If
, shear reinforcement shall be provided. If shear stress
exceeds 1.5 flat slab shall be redesigned.
2. Types of flat slab
•
Flat Slab resting
directly on columns
• Flat slab with column head
•
Flat Slab with drop panels
• Flat slab with drop panel
and column head
Drop is a local thickening of the slab in
the region of column
Structural Advantages
• increase shear strength of slab
• increase negative moment capacity
of slab
• stiffen the slab and hence reduce
deflection
Column head is a local enlargement of
the column at the junction with the
slab
Structural Advantages
• increase shear strength of slab
(punching shear)
• reduce the moment in the slab by
reducing the clear or effective span
A flat slab may have recesses formed on the soffit so that the soffit comprises a series of ribs
in two directions ( waffle Slabs).
Flat slabs with capitals, drop panels, or both. These slabs are very satisfactory for
heavy loads and long spans.
Although the formwork is more expensive than for flat plates, flat slabs will require
less concrete and reinforcing than would be required for flat plates with the same
loads and spans.
They are particularly economical for warehouses, parking and industrial buildings,
and similar structures, where exposed drop panels or capitals are acceptable.
3. Benefits of flat slab
Flexibility in room layout
•
•
•
Introduce partition walls anywhere required
Change the size of room layout
Omit false ceiling
 Saving
in building height
• Lower storey height will reduce building weight
• approx. saves 10% in vertical members
• reduce foundation load
Shorter construction time
•
flat plate design will facilitate the use of
big table formwork to increase productivity
Ease of installation of M&E services
• all M & E services can be mounted directly on
the underside of the slab instead of bending
them to avoid the beams
• avoids hacking through beams
The main disadvantage is their lack of resistance to lateral loads due to
wind and earthquakes. Lateral load resisting systems such as shear walls
are oDen necessary
When the loads or spans or both become quite large, the slab thickness
and column sizes required for flat plates or flat slabs are of such
magnitude that it is more economical to use two-‐way slabs with beams,
despite the higher formwork costs.
4. Behaviour of Slab supported on Stiff , Flexible and no beams
Case Study:
• Panel Size = 4 m x 4m
• Slab Thickness = 125 mm
• Load = 5 kN/m2
• Stiff Supports ( Bearing wall)
• Flexible Supports (Beam) : 300 x 300 , 300 x 450 , 300 x 600 , 300 x 1000 mm
• Column supports at corners
A. Two way Slab on Rigid Supports (bearing Walls)
Mx = 3.616 kNm/m
My = 3.616 kNm/m
IS 456 Values (Table 27): 0.062 x 5 x 16 = 4.96
Slab Deflection = 1.4 mm
B. Two way Slab on Flexible Supports (Beams on all sides)
1. Beam Size : 300 x300 mm
Mx = 4.45 kNm/m
My = 4.45 kNm/m
IS 456 Values (Type 9): 0.056 x 5 x 16 = 4.48
Beam Moment = 12.2 kNm
Beam Deflection = 1.33 mm
Slab deflection= 2.9 mm
Mxy = 0.37 kNm/m
2. Beam Size : 300 x450 mm
Mx = 3 kNm/m
My = 3 kNm/m
IS 456 Values (Type 9): 0.056 x 5 x 16 = 4.48
Beam Moment = 15.6 kNm
Beam Deflection = 0.5 mm
Mxy = 0.73 kNm/m
Slab deflection= 1.5 mm
3. Beam Size : 300 x 600 mm
Mx = 2.43 kNm/m
My = 2.43 kNm/m
Mxy = 0.8 kNm/m
IS 456 Values (Type 9): 0.056 x 5 x 16 = 4.48
Beam Moment = 17 kNm
Beam Deflection = 0.24 mm
Slab Deflection = 0.98 mm
4. Beam Size : 300 x 1000 mm
Mx = 2 kNm/m
My = 2 kNm/m
IS 456 Values (Type 9): 0.056 x 5 x 16 = 4.48
Beam Moment = 18 kNm
Mxy = 0.8 kNm/m
5. Beam Size : 300 x 125 mm (Concealed Beams)
Mx = 9.8 kNm/m
My = 9.8 kNm/m
IS 456 Values (Type 9): 0.056 x 5 x 16 = 4.48
Beam Moment = 2.9 kNm
Slab Deflection = 7.0 mm
Mxy = 3 kNm/m
B. Two way Slab on Point Supports at corners (Flat Slab)
Mx = 9.075 kNm/m (Middle)
=12.4 kNm/m (Edge Strip)
My = 9.075 kNm/m (Middle)
=12.4 kNm/m (Edge Strip)
Slab Deflection = 8.67 mm
Mxy = 7.76 kNm/m
Results Summary
Type of
Support
Rigid
300 x125
Mx
My
Mxy
Beam
Moment
Deflection
Slab
Beam
3.616
3.616
2.6
-‐
1.4
-‐
(Concealed
Beams)
9.84
9.85
3.0
2.88
7.0
4.3
300 x300
4.45
4.45
0.37
12.2
2.9
1.33
300 x450
3
3
0.73
15.6
1.5
0.50
300 x600
2.43
2.43
0.8
17.0
0.98
0.24
300x1000
2
2
0.8
18.0
0.60
0.05
Flat Slab
9.0
9.0
7.76
-‐
8.676
-‐
Observations
• Two way Rectangular Slab supported on stiff beams, the shorter spans (stiffer portion
of the slab) carry larger load and subjected to larger moments. The longer spans carry
less load and subjected to less moment.
• Results indicate that decrease in supporting beams stiffness leads to an increase in
bending moments of slabs and decrease in bending moment of the beams (behavior that
is not captured using code recommendations).
•
If the slab is supported on bearing walls, slab moments are distributed in similar way.
• If the slab is supported only by the columns, the slab behaves like a two way slab with
an essential difference that all the load is carried in both directions to accumulate it
at the columns.
• With Concealed beams it is reveled that the behaviour is close to Flat slabs rather than any useful
beam action.
4. Structural Behaviour of Flat Slab
Column Strip
Column Strip
Middle Strip
Column Strip
Middle
Strip
Deflected Shape
5. Distribution of Total Panel Moment in different zones
-‐m8
A
-‐m4
-‐m2
-‐m4
B
m3
m7
C
-‐m8
-‐m6
m1
-‐m8
Column
Strip
m3
-‐m4
Column Strip
m7
m5
-‐m2
D
A
A
-‐m6
Middle Strip
Middle Strip
C
B
A -‐m4
Column Strip
-‐m8
Column
Strip
A Zone of –ve BM (Hogging) in both directions
B Zone of +ve BM(Sagging) and –ve BM
C Zone of -‐veBM and +ve BM
D Zone of +ve BM in both directions
m3
A
-‐m8
Column
Strip
-‐m2
-‐m4
C
B
A
-‐m6
Middle Strip
-‐m8
Column
Strip
-‐m2
m1
C
-‐m4
m3
A
-‐m4
A
m7
Middle Strip
A
D
B
A
-‐m8
-‐m2
D
C
C
-‐m2
-‐m4
B
m7
m5
Column Strip
-‐m8
m1
Column Strip
m5
C
-‐m4
B
m7
A
B
-‐m6
-‐m4
m3
D
A
-‐m8
-‐m6
-‐m6
-‐m8
A Zone of –ve BM (Hogging) in both directions
B Zone of +ve BM(Sagging) and –ve BM
C Zone of -‐veBM and +ve BM
D Zone of +ve BM in both directions
Moment Direction
6. Definitions
Moment Direction
COLUMN STRIP
0.25L1  0.25L2
L2
SPAN
Region
MIDDLE
STRIP
SPAN Region:
Bounded on all the four sides by middle strips
COLUMN STRIP
0.25L1  0.25L2
COLUMN STRIP
0.25L2  0.25L1
MIDDLE STRIP
L1
7. General Design Considerations
CL 31.2 Proportioning
31.2.1 Thickness of Flat Slab
• The thickness of the flat slab shall be
generally controlled by considerations of
span to effective depth ratios given in 23.2.
• For slabs with drops conforming to 31.2.2,
span to effective depth ratios given in 23.2
shall be applied directly; otherwise the span
to effective depth ratios obtained in
accordance with provisions in 23.2 shall be
multiplied by 0.9. For this purpose, the
longer span shall be considered.
• The minimum thickness of slab shall be 125
mm.
31.2.2 Drop
• The drops when provided shall be
rectangular in plan, and have a length
in each direction not less than onethird 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 centreline of the columns shall be equal to
one-half the width of drop for interior
panels.
• Minimum thickness of Drop
> ¼ of Slab thickness and
> 100 mm
31.2.3 Column Heads
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 900and can be included
entirely within the outlines of the column and the column head, shall be considered for
design purposes.
8. Determination of Bending Moment CL 31.3
31.3.1. 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 31.4, and
b) The equivalent frame method as specified in 31.5.
In each case the applicable limitations given in 31.4 and 31.5 shall be met.
9. Direct Design Method CL 31.4
A. Limitations : 31.4.1
Slab system designed by the direct design method shall fulfil 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 10percent of the span in the
direction of the offset notwithstanding 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.
Note:
Applicable to gravity loading condition alone (and not to the lateral loading condition)
Ly3
Ly2
3
For any Panel
Longer Span/Shorter Span 2
 0.1Lx1
Lx1  Lx2
Lx3  Lx2
Ly1  Ly2
Ly3  Ly2
2
 0.1Ly2
Lx1  2Lx2/3
Lx3  2Lx2/3
Ly1
1
Lx1
2
Lx2
3
Lx3
Ly1  2Ly2/3
Ly3  2Ly2/3
wuL/wuD  3
B. Total Design Moment for a Span: CL31.4.2
31.4.2.1 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.
1
1
CL of Panel 1
DESIGN
STRIP
M0x
M0y
CL of Panel 2
2
31.4.2.2 The absolute sum of the
positive and average negative
bending moments in each
direction shall be taken as:
lnx
2
wu kN/m
Ln
Note:
1. It is the same as the total moment that occurs in a
simply supported slab
Ln
L2
(L1)
L1
(L2)
2. The moment that actually occurs in such a slab has been shown by experience and tests
to be somewhat less than the value determined by the Mo expression. For this reason, l1
is replaced with ln
10. Distribution of Total Panel MomentM0
• It is next necessary to know what proportions of these total moments are positive and
what proportions are negative.
Interior Panel
• If a slab was completely fixed at the end of each
panel, the division would be as it is in a fixed-‐end
beam, two-‐thirds negative and one-‐third positive, as
shown in Figure.
• This division is reasonably accurate for interior
panels where the slab is continuous for several
spans in each direction with equal span lengths and
loads.
Exterior Panel
• The relative stiffnesses of the columns and slabs of exterior panels are of
far greater significance in their effect on the moments than is the case for
interior panels.
• The magnitudes of the moments are very sensitive to the amount of
torsional restraint supplied at the discontinuous edges.
• This restraint is provided both by the flexural stiffness of the slab and by
the flexural stiffness of the exterior column.
Code Recommendations
Distribution of Bending Moments across panel width Code Recommendations
11. Rebar Detailing - Code Recommendations
Bent bars are also used.
There seems to be a trend
among designers to use
straight bars more than
bent bars.
ELEVATION
Rebar Detailing - Code Recommendations
e
b
e
e
b
b
Ln greater of adjacent clear spans CL 31.7.3 (b)
e
e
b
b
Section through Middle Strip
12. Two way Shear in Flat Slab
• Flat plates present a possible problem in transferring the
shear at the perimeter of the columns.
• There is a danger that the columns may punch through
the slabs.
• As a result, it is frequently necessary to increase column
sizes or slab thicknesses or to use shear heads. Shear
heads consist of steel I or channel shapes placed in the
slab over the columns
Note:
Flat Slab with drop panel and capital, shear is required to be checked at two sections
1. at a distance d/2 from the face of column capital
2. at a distance d/2 from the face of drop panel
Design Example #1
Design by DDM flat plate supported on
columns 450 mm square, for a Live Load
= 3 kN/m2, Floor Finish = 1 kN/m2 use
M20 and Fe415. Assume clear cover = 20
mm. Effective Column Height = 3.35m.
Bay spacing in X and Y direction = 5m c/c
• Interior Panel P5
• Corner Panel P7
3 bays @ 5 m c/c
A. Interior Panel Design
Zone A – Corner Strip
Zone B – Middle Strip along X
Zone C – Middle Strip along Y
A
2.5
B m
Zone D – Interior Region
A
Step 1: Panel Division into Strips
C
2.5m
A
D
B
5m
31.1.1(a)
5m
C
Moment
direction
Along
L1
L2
X
5
5
1.25 and 1.25 m
Adopt 1.25 m
2.5m
Y
5
5
1.25 and 1.25 m
Adopt 1.25 m
2.5m
A
Width of Column Strip on
either side of Centre Line
= 0.25L2 and  0.25 L1
Middle
Strip
Step 2: Trial Depth CL 31.2.1
•
•
•
•
•
L/d = 26
Modification Factor = 1.33, Assuming pt = 0.4%, FIG 4 IS 456
d = 5000/(26 x 1.33) = 145 mm > 125
CL 31.2.1
DS= 145 + 20 + 18 = 183 mm ( assume #12 bars, and bars in two layers)
Provide Ds= 200 mm d = 200-‐20-‐18= 162 mm
Step 3 Design Loads / m width of Slab
• wuD = 1.5(25x 0.2 + 1) = 9kN/m
• wuL = 1.5 x 3 = 4.5kN/m
• wu = 13.5 kN/m
Step 4: Check for Applicability of DDM: CL 31.4.1
•
•
•
•
No. of Continuous Spans in each direction = 3 ; OK
Long Span/Short Span = 5/5 = 1 <2 ; OK
Successive spans in each direction = Equal; OK
wuL/wuD = 4.5/9 = 0.5 < 3 ; OK
31.4.1(a)
31.4.1(b)
31.4.1(d)
31.4.1(e)
Contributory Area
Step 5: Check for punching shear around Column
Assumed d = 162 mm
Section 1:
5m
• Critical Section at d/2 around the column
• Perimeter of Critical Section = 4 x 0.612= 2.448 m
• Design Shear at critical section Vu
• Vu = 13.5 ( 52 – 0.6122) = 333kN
• c = 0.25fck = 1.12 MPa
• ks = 0.5 +1 = 1.5 <=1 ; ks=1 ; ks c = 1.12
• Shear Resistance of Concrete = 1.12 x 2448 x 162 = 444kN > 333 kN
Critical Section
0.612m
0.612m
5m
OK
Step 6:Design Moments CL 31.4.2.2
wu = 13.5 kN /m
Parameters
L1 (Span in direction of Mo)
0.65L1
Ln (clear span extending from face to
face of columns, capitals)
Ln > 0.65L1
L2 (Span transverse to L1)
W = wu L2Ln
M0 = W Ln / 8
Along X
Along Y
5
3.25
5
3.25
m
m
4.55
4.55
m
4.55
5
307.2
174.72
4.55
5
307.2
174.72
(5-‐0.45) =
m
m
kN
kNm
Step 7 : Distribution of Bending Moment across panel width ;
CL: 31.4.3.2, 31.5.5
X
Y
113.6
113.6
kNm
31.4.3.2
85.2
85.2
kNm
31.5.5.1
2x1.25 =2.5
2x1.25 =2.5
• -‐m1 = M1/ Csw (Zone A)
34.1
34.1
kNm/m
• Middle Strip M2 = 0.25MN
28.4
28.4
kNm
Moment Direction along
Negative Design Moment
MN = -‐0.65*M0
• Column Strip M1 = 0.75MN
Width of Column Strip resisting M1 (Csw)
m
31.5.5.4(a)
Positive Design Moment
MP = 0.35*M0
61.2
61.2
kNm
31.4.3.2
• Column Strip M1 = 0.6MP
• +m1 = M1/ Csw
36.7
14.7
36.7
14.7
kNm
31.5.5.3
• Mid Span M2 = 0.4MP
• +m2 = M2/Msw (Zone D)
24.5
9.8
24.5
9.8
kNm
(Zone B &C)
-‐34.1
-‐34.1
A
14.7
14.7
C
9.8
-‐11.4
-‐34.1
-‐34.1
A
-‐11.4
A
B
-‐34.1
D
C
-‐11.4
-‐34.1
B
31.5.5.4(a)
kNm/m
-‐vesign : Hogging Moment (tension at top)
+ve sign : Sagging Moment (tension at bottom)
Step 8 : Check for adequacy of Depth
14.7
9.8
-‐11.4
14.7
-‐34.1
kNm/m
A -‐34.1
• Max Design Bending moment = 34.1 kNm/m
• Mu,lim = 72.41 kNm/m > 34.1,
• Depth is adequate G-‐1.1(c)
Step 9 :Rebar Details
26.5.2.1
• Ast,min = 0.12 x 200 x 1000 /100 = 240 mm2/m
• Minimum Effective Depth of Slab = 162 mm
G-‐1.1(b)
• 7.5 Ast2 – 58490Ast + Mu =0
Location
Moment
Ast
(kNm/m)
(mm2/m)
(-‐)34.1
14.7
(-‐)11.4
9.8
635
260
200
171
(-‐)34.1
(-‐)11.4
14.7
9.8
635
200
260
171
Bar
dia
Spacing
635
260
240
240
10
8
8
8
120 -‐T
190 -‐B
200 -‐T
200 -‐B
635
240
260
240
10
8
8
8
120 -‐T
200 -‐T
190 -‐B
200 -‐B
Ast
(prov)
mm
Along X
Zone A
Zone B
Zone C
Zone D
Along Y
Zone A
Zone B
Zone C
Zone D
#8@190
A
A
B
0.125Ln
#8@190
#8@200
#8@190
#8@190
C
#8@200
0.15Ln
A
Bottom Rebar Details in
Interior Panel
D
0.15Ln
B
C
0.125Ln
A
#10@120
A
B
#8@200
0.3Ln
0.2Ln
#10@120
A
D
C
C
#8@200
0.2Ln
0.3Ln
TOP Rebar Details in
Interior Panel
Note:
Distances for curtailment of rebars are
measured from column face
#8@200
B. Corner Panel Design
Step 5: Check for punching shear around Column
Assumed d = 162 mm
Section 1:
• Critical Section at d/2 around the column
• Perimeter of Critical Section = 2 x 0.531= 1.062 m
• Design Shear at critical section Vu
• Vu = 13.5 ( 2.52 – 0.5312) =81kN
• c = 0.25fck = 1.12 MPa
• ks = 0.5 +1 = 1.5 <=1 ; ks=1 ; ks c = 1.12
• Shear Resistance of Concrete = 1.12 x 1062 x 162 =
192kN > 81 kN
OK
162/2 = 81 mm
450
Step 6:Design Moments CL 31.4.2.2
M0 = 174.72 kNm
Step 7 : Distribution of Bending Moment across panel width ; CL 31.4.3.3 , 31.5.5
𝛼↓𝑐 =∑↑▒𝑘↓𝑐 /𝑘↓𝑠
A
C
1.25m
Assume Columns and Slab panels are with same
modulus of elasticity
A
B 1.25m
D
B
5m
A
5m
C
A
Parameters
Sum of c o l u m n
stiffness above and
below the slab
Along X
Along Y
(2 x 4 x Ec x 450 x 4503/12) /3350 = 8.16 Ec x 106
2 (4EcIc)/Lc
Slab stiffness
ks = 4EsIs/Ls
(4 Es x 5000 x 2003/12)/5000
= 2.67Es x
106
2.67Es x 106
c = kc /ks
3.06
3.06
 = 1+ (1/c)
1.33
1.33
-‐m1
A. Exterior negative design moment:
Exterior
-‐m2
-‐m1
Y
Exterior
-‐m2
X
• Column Strip M1 = MN
Width of Column Strip Csw resisting M1
-‐m1= M1/Csw
• Middle Strip M2=0
-‐m2 = 0
1.25
Interior
-‐m1
1.25
Moment Direction along
Negative Design Moment
MN = -‐0.65*M0/
-‐m1
Interior
X
Y
85.4
85.4
kNm
31.4.3.3
85.4
85.4
kNm
31.55.2(a)
2x1.25 = 2.5
2.5
34.2
0
0
34.2
0
0
m
kNm/m
kNm
kNm/m
31.5.5.4(a)
Exterior
B. Interior negative design moment:
Y
-‐m1
-‐m2
Interior
Exterior
X
-‐m1
-‐m1
Moment Direction along
Negative Design Moment
MN = -‐(0.75 – 0.1/)Mo
• Column Strip M1 = 0.75 MN
Width of Column Strip Csw resisting M1
• -‐m1= M1/ Csw
• Middle Strip M2 = 0.25 MN
Width of Middle Strip Msw resisting M2
• -‐m2 = M2/Msw
-‐m2
Interior
-‐m1
X
Y
118
118
kNm
31.4.3.3
88.5
2x1.25 =2.5
-‐35.4
22.12
2.5
-‐8.85
88.5
2.5
-‐35.4
22.12
2.5
-‐8.85
kNm
31.5.5.1
m
kNm/m
kNm
m
kNm/m
31.5.5.4(a)
Exterior
m1
C. Positive Moment in Mid Span:
m
Exteri
1
m2
Y
m2
o
m1
Interior
X
m1
r
m1
Interior
Moment Direction along
Design Moment
MP = (0.63 – 0.28/)Mo
• Column Strip M1 = 0.6 MP
Width of Column Strip Csw resisting M1
• m1 = M1/ Csw
• Middle Strip M2 = 0.4 MP
Width of Middle Strip Msw resisting M2
• m2 = M2/Msw
X
Y
73.29
73.29
kNm
31.4.3.3
43.98
2x1.25 =2.5
17.6
29.32
2.5
11.73
43.98
2.5
17.6
29.32
2.5
11.73
kNm
31.5.5.3
m
kNm/m
kNm
m
kNm/m
31.5.5.4(a)
Exterior
-‐34.2
-‐34.2
A
-‐34.2
-‐0
17.6
A
B
-‐35.4
-‐vesign : Hogging Moment (tension at top)
+ve sign : Sagging Moment (tension at bottom)
Exterior
0
17.6
C
11.73
17.6
11.73
D
C
-‐8.85
Interior
-‐35.4
-‐34.2
A
-‐35.4
-‐8.85
17.6
B
A
-‐35.4
Interior
Step 7 : Check for adequacy of Depth
• Max Design Bending moment = 35.4 kNm/m
• Mu,lim = 72.41 kNm/m > 35.4, Depth is adequate G-‐1.1(c)
Step 8 :Rebar Details
26.5.2.1
• Ast,min = 0.12 x 200 x 1000 /100 = 240 mm2/m
7.5 Ast2 – 58490Ast + Mu =0
Strip Location
Moment
Ast
(kNm/m)
(mm2/m)
(-‐)34.2
(-‐)35.4
17.6
(-‐)8.85
11.73
637
662
314
155
206
(-‐)34.2
(-‐)35.4
(-‐)8.85
17.6
11.73
637
662
155
314
206
Bar
dia
Spacing
637
662
314
240
240
10
120(T)
115(T)
160(B)
200(T)
200(B)
637
662
240
314
240
10
Ast
(prov)
mm
Along X
Zone A(Exterior)
Zone A(Interior)
Zone B
Zone C(Interior)
Zone D
10
8
8
8
Along Y
Zone A (Exterior)
Zone A(Interior)
Zone B (Interior)
Zone C
Zone D
10
8
8
8
120(T)
115(T)
200(T)
160(B)
200(B)
Exterior
A
17.6
17.6
Exterior
C
A
A
B
11.73
11.73
D
17.6
C
Interior
17.6
B
Interior
A
Exterior
-‐34.2
#8@200
#10@120
#10@115
#10@120
#8@200
#10@120
#8@200
-‐34.2
Exterior
0
A
C
-‐34.2
#10@115
#8@200
A
B
D
-‐35.4
C
-‐8.85
Interior
-‐35.4
#10@115
-‐34.2
-‐0
A
-‐35.4
-‐8.85
B
A -‐35.4
Interior
TOP Rebar details in Corner Panel
Design Example #2
Design by DDM flat plate supported on
columns 500 mm square, for a Live Load
= 4 kN/m2, Floor Finish = 1 kN/m2 use
M25 and Fe415. Floor slab is exposed to
moderate environment. Column Height =
3.5m (c/c). Bay spacing in X and Y
direction = 5.5m c/c. Assume that
building is not restrained against sway
• Interior Panel P5
• Corner Panel P7
3 bays @ 5.5 m c/c
A. Interior Panel Design
Zone A – Corner Strip
Zone B – Middle Strip along X
Zone C – Middle Strip along Y
A
2.75
B m
Zone D – Interior Region
A
Step 1: Panel Division into Strips
C
2.75m
A
D
B
5.5 m
5.5 m
C
31.1.1(a)
Moment
direction
Along
L1
L2
X
5
5
1.375 and 1.375 m
Adopt 1.375 m
2.75m
Y
5
5
1.375 and 1.375 m
Adopt 1.375 m
2.75m
Width of Column Strip on
either side of Centre Line
= 0.25L2 and  0.25 L1
Middle
Strip
A
Step 2: Trial Depth CL 31.2.1
•
•
•
•
•
L/d = 26
Modification Factor = 1.33, Assuming pt = 0.4%, FIG 4 IS 456
d = 5500/(26 x 1.33) = 160 mm > 125
CL 31.2.1
DS= 160 + 30 + 18 = 208 mm ( assume #12 bars, and bars in two layers)
Provide Ds= 225 mm d = 225-‐30-‐18= 177 mm
Step 3 Design Loads / m width of Slab
• wuD = 1.5(25x 0.225 + 1) = 9.94kN/m
• wuL = 1.5 x 4 = 6kN/m
• wu = 15.94  16 kN/m
Step 4: Check for Applicability of DDM: CL 31.4.1
•
•
•
•
No. of Continuous Spans in each direction = 3 ; OK
Long Span/Short Span = 5.5/5.5 = 1 <2 ; OK
Successive spans in each direction = Equal; OK
wuL/wuD = 6/9.94 = 0.6 < 3 ; OK
31.4.1(a)
31.4.1(b)
31.4.1(d)
31.4.1(e)
Contributory Area
Step 5: Check for punching shear around Column
Assumed d = 177 mm
Section 1:
5.5m
• Critical Section at d/2 around the column
• Perimeter of Critical Section = 4 x 0.677= 2.708 m
• Vu = 16 ( 5.52 – 0.6772) =477kN
• c = 0.25fck = 1.25 MPa
• ks = 0.5 +1 = 1.5 <=1 ; ks=1 ; ks c = 1.25
• Shear Resistance of Concrete = 1.25 x 2708 x 177 = 599kN > 477 kN
Critical Section
0.677m
0.677m
5.5m
OK
Step 6:Design Moments CL 31.4.2.2
wu = 16 kN /m
Parameters
L1 (Span in direction of Mo)
0.65L1
Ln (clear span extending from face to
face of columns, capitals)
Ln > 0.65L1
L2 (Span transverse to L1)
W = wu L2Ln
M0 = W Ln / 8
Along X
5.5
3.575
(5.5-‐0.5) =
Along Y
5.5
m
3.575 m
5
5
5
5.5
440
275
5
5.5
440
275
m
m
m
kN
kNm
Step 7 : Distribution of Bending Moment across panel width ;
CL: 31.4.3.2, 31.5.5
X
Y
Moment Direction along
Negative Design Moment
179
179
MN = -‐0.65*M0
134.3
134.3
• Column Strip M1 = 0.75MN
kNm
31.4.3.2
kNm
31.5.5.1
2.75
2.75
m
• -‐m1 = M1/ Csw (Zone A)
48.8
48.8
kNm/m
• Middle Strip M2 = 0.25MN
44.8
44.8
kNm
Width of Middle Strip resisting M2 (Msw)
2.75
2.75
m
16.3
16.3
kNm/m
Width of Column Strip resisting M1 (Csw)
• -‐m2 = M2/Msw
(Zone B & C)
31.5.5.4(a)
Positive Design Moment
MP = 0.35*M0
96.3
96.3
kNm
31.4.3.2
• Column Strip M1 = 0.6MP
• +m1 = M1/ Csw
57.8
57.8
kNm
31.5.5.3
21
21
• Mid Span M2 = 0.4MP
38.5
14
38.5
14
(Zone B &C)
• +m2 = M2/Msw (Zone D)
-‐48.8
-‐48.8
A
21
21
C
14
-‐16.3
-‐48.8
-‐48.8
A
-‐16.3
-‐48.8
C
B
31.5.5.4(a)
kNm/m
-‐vesign : Hogging Moment (tension at top)
+ve sign : Sagging Moment (tension at bottom)
Step 8 : Check for adequacy of Depth
-‐16.3
-‐48.8
-‐16.3
21
A
21
D
kNm
-‐48.8
B
14
kNm/m
A -‐48.8
• Max Design Bending moment = 48.8 kNm/m
• Mu,lim = 108 kNm/m > 48.8
• Depth is adequate G-‐1.1(c)
Step 9 :Rebar Details
26.5.2.1
• Ast,min = 0.12 x 225 x 1000 /100 = 270 mm2/m
• Minimum Effective Depth of Slab = 177 mm
G-‐1.1(b)
• 6 Ast2 – 63906Ast + Mu = 0
Location
Moment
Ast
(kNm/m)
(mm2/m)
(-‐)48.8
21
(-‐)16.3
14
828
340
262
224
(-‐)48.8
-‐16.3
21
14
828
262
340
224
Bar
dia
Spacing
828
340
270
270
10
8
8
8
90 -‐T
145 -‐B
180 -‐T
180 -‐B
828
270
340
270
10
8
8
8
90 -‐T
180 -‐T
145 -‐B
180 -‐B
Ast
(prov)
mm
Along X
Zone A
Zone B
Zone C
Zone D
Along Y
Zone A
Zone B
Zone C
Zone D
#8@145
A
A
B
0.125Ln
#8@145
#8@180
#8@145
#8@145
C
#8@180
0.15Ln
A
Bottom Rebar Details in
Interior Panel
D
0.15Ln
B
C
0.125Ln
A
#10@90
A
B
#8@180
0.3Ln
0.2Ln
#10@90
A
D
C
C
#8@180
0.2Ln
0.3Ln
TOP Rebar Details in
Interior Panel
Note:
Distances for curtailment of rebars are
measured from column face
#8@200
B. Corner Panel Design
Step 5: Check for punching shear around Column
Assumed d = 177 mm
Section 1:
• Critical Section at d/2 around the column
• Perimeter of Critical Section = 2 x 0.5885= 1.177 m
• Vu = 16 ( 2.752 – 0.58852) =115.5kN
• c = 0.25fck = 1.25 MPa
• ks = 0.5 +1 = 1.5 <=1 ; ks=1 ; ks c = 1.25
• Shear Resistance of Concrete = 1.25 x 1177 x 177 =
260kN > 115.5 kN
OK
177/2 = 88.5 mm
500
2.75m
2.75m
Step 6:Design Moments CL 31.4.2.2
M0 = 275 kNm
Step 7 : Distribution of Bending Moment across panel width ; CL 31.4.3.3 , 31.5.5
𝛼↓𝑐 =∑↑▒𝑘↓𝑐 /𝑘↓𝑠
A
C
1.25m
Assume Columns and Slab panels are with same
modulus of elasticity
A
B 1.25m
D
B
5m
A
5m
C
A
Parameters
Sum of c o l u m n
stiffness above and
below the slab
Along X
Along Y
Leff = 1.2 Lc (CL E1)
Lc = 3.5-‐0.225 = 3.275
(2 x 4 x Ec x 500 x 5003/12) /1.2*3275 = 10.6 Ec x 106
2 (4EcIc)/Lc
Slab stiffness
(4 Es x 5500 x 2253/12)/5500
ks = 4EsIs/Ls
= 2.67Es x 106
c = kc /ks
c min (Table 17)
 = 1+ (1/c)
2.8
3.8Es x 106
2.8
(0.7/0.5)*0.1 =0.14 < c. Adopt c = 2.8
1.36
1.36
-‐m1
A. Exterior negative design moment:
Exterior
-‐m2
-‐m1
Y
X
Interior
-‐m1
Interior
1.25
• Column Strip M1 = MN
Width of Column Strip Csw resisting M1
-‐m1= M1/Csw
• Middle Strip M2=0
-‐m2 = 0
1.25
Exterior
-‐m2
M0 = 275 kNm
Moment Direction along
Negative Design Moment
MN = -‐0.65*M0/
-‐m1
X
Y
131.4
131.4
kNm
31.4.3.3
131.4
131.4
kNm
31.55.2(a)
2.75
2.75
m
47.8
0
0
47.8
0
0
kNm/m
kNm
kNm/m
31.5.5.4(a)
Exterior
B. Interior negative design moment:
Y
-‐m1
-‐m2
Interior
Exterior
X
-‐m1
-‐m1
Moment Direction along
Negative Design Moment
MN = -‐(0.75 – 0.1/)Mo
• Column Strip M1 = 0.75 MN
Width of Column Strip Csw resisting M1
• -‐m1= M1/ Csw
• Middle Strip M2 = 0.25 MN
Width of Middle Strip Msw resisting M2
• -‐m2 = M2/Msw
-‐m2
Interior
-‐m1
X
Y
186
186
kNm
31.4.3.3
139.5
139.5
kNm
31.5.5.1
2.75
50.73
46.5
2.75
50.73
46.5
m
2.75
17
2.75
17
m
kNm/m
kNm
kNm/m
31.5.5.4(a)
Exterior
m1
C. Positive Moment in Mid Span:
m
Exteri
m2
1
Y
m2
m1
Interior
o
X
m1
r
m1
Interior
Moment Direction along
Design Moment
MP = (0.63 – 0.28/)Mo
• Column Strip M1 = 0.6 MP
Width of Column Strip Csw resisting M1
• m1 = M1/ Csw
• Middle Strip M2 = 0.4 MP
Width of Middle Strip Msw resisting M2
• m2 = M2/Msw
X
Y
116.6
116.6
kNm
31.4.3.3
70
2.75
25.5
46.7
2.75
17
70
2.75
25.5
46.7
2.75
17
kNm
31.5.5.3
m
kNm/m
kNm
m
kNm/m
31.5.5.4(a)
Exterior
-‐47.8
-‐47.8
A
-‐47.8
-‐0
25.5
A
B
-‐50.73
-‐vesign : Hogging Moment (tension at top)
+ve sign : Sagging Moment (tension at bottom)
Exterior
0
25.5
C
17
25.5
17
D
C
-‐17
Interior
-‐50.73
-‐47.8
A
-‐50.73
-‐17
25.5
B
A
-‐50.73
Interior
Step 7 : Check for adequacy of Depth
• Max Design Bending moment = 50.73 kNm/m
• Mu,lim = 108 kNm/m > 50.73, Depth is adequate G-‐1.1(c)
Step 8 :Rebar Details
26.5.2.1
• Ast,min = 0.12 x 225 x 1000 /100 = 270 mm2/m
6 Ast2 – 63906Ast + Mu = 0
Strip Location
Moment
Ast
(kNm/m)
(mm2/m)
(-‐)47.8
(-‐)50.73
25.5
(-‐)17
17
810
864
415
273
273
(-‐)47.8
(-‐)50.73
(-‐)17
25.5
17
810
864
273
415
273
Bar
dia
Spacing
810
864
415
273
273
10
10
10
8
8
90(T)
90(T)
180(B)
180(T)
180(B)
810
864
273
415
273
10
10
8
10
8
90(T)
90(T)
180(T)
180(B)
180(B)
Ast
(prov)
mm
Along X
Zone A(Exterior)
Zone A(Interior)
Zone B
Zone C(Interior)
Zone D
Along Y
Zone A (Exterior)
Zone A(Interior)
Zone B (Interior)
Zone C
Zone D
Design Example #3
Design by DDM flat plate supported on
columns of dia = 450 mm, Column head =
1.5 m dia, Drop panel size = 3.2 x 3.2 m,
for a Live Load = 4 kN/m2, Floor Finish =
1 kN/m2 use M20 and Fe415. Assume
clear cover = 20 mm. Column Height =
3.35m
6.4m
6.4m
• Interior Panel P5
• Exterior Panel P2/P4
• Corner Panel P1
6.4m
7.2m
7.2m
7.2m
1.6
A. Interior Panel Design
1.6
Step 1: Panel Division into Strips 31.1.1(a)
MSy
1.6
1.6
CSy
Moment
direction
Along
L1
L2
Width of Column
Strip on either side of
Centre Line
= 0.25L2 and
 0.25 L1
CSx
Middle
Strip
MSx
CSx
Lx = 7.2
Zone A – Corner Strip
Zone B – Middle Strip along X
Zone C – Middle Strip along Y
X
7.2
6.4 1.6 < 1.8 m; 1.6 m
4m
Y
6.4
7.2 1.8 > 1.6 m; 1.6 m
3.2m
Zone D – Interior Region
Ly = 6.4
CSy
Step 2: Trial Depth CL 31.2.1
•
•
•
•
•
L/d = 26
Modification Factor = 1.4, Assuming pt 0.4%, FIG 4 IS 456
d = 7200/(26 x 1.4) = 198 mm > 125
DS= 198+20+18= 236 mm ( assume #12 bars)
Provide Ds= 240 mm , d = 198mm
CL 31.2.1
Step 3: Design Loads / m width of Slab
• wuD = 1.5(25 x 0.24 + 1) = 10.5kN
• wuL = 1.5 x 4 = 6.0kN
• wu = 16.5 kN
Step 4: Check for Applicability of DDM: CL 31.4.1
•
•
•
•
No. of Continuous Spans in each direction = 3 ; OK
Long Span/Short Span = 7.2/6.4 = 1.125 <2 ; OK
Successive spans in each direction = Equal; OK
wuL/wuD = 6/10.5 = 0.571 < 3 ; OK
31.4.1(a)
31.4.1(b)
31.4.1(d)
31.4.1(e)
Step 5: Drop Panel Size : CL 31.2.2
•
•
•
•
•
Length along X ≥ Lx/3 = 2.4 m
Length along Y ≥ Ly/3 = 2.13 m
Generally Drop Panel Size is set equal to Width of Column Strip
Proposed size 3.2 x 3.2 meets all the requirements.
Minimum thickness = ¼ DS = 60 mm or 100 mm; Adopt 100 mm
Step 6:Column Head
• 1/4 to 1/5 of average span = 7.2/5 = 1.44 m
• Provided = 1.5 m ; Ok
• Equivalent Square Capital =0.89D = 1.335 m
• Minimum Effective Depth of Slab = 198 mm
• Effective Depth at Drop location = 298 mm
Section 1:
• Critical Section at d/2 around the column capital
• Perimeter of Critical Section =  ( 1.5 + 0.298) = 5.65 m
• Weight of Drop Projection below slab = 0.1x 25 x 1.5 = 3.75 kN/m2
• Design Shear at critical section around capital Vu
• Vu = 16.5 ( 7.2 x 6.4 -‐ x 1.7982/4) + 3.75(3.2 x 3.2 -‐ x 1.7982/4)
•
= 747 kN
• c = 0.25fck = 1.12 MPa
• ks = 0.5 +1 = 1.5 <=1 ; ks=1 ; ks c = 1.12
• Shear Resistance of Concrete = 1.12 x 5650 x 298 = 1885kN > 747 kN
Capital
1.5
3.2 m
Step 7 : Check for Shear around Column Capital
1.798
Critical Section
DROP
3.2 m
OK
Section 2 : Check for Shear around drop
Critical Section at d/2 around the drop
d = 198mm
Perimeter of Critical Section = 4 x 3.4 = 13.6m
Design Shear at critical section
Vu = 16.5 ( 7.2 x 6.4 – 3.42) = 569 kN
Shear Resistance of Concrete = 1.12 x 13600 x 198 =
3015kN > 569 kN
3.2 m
•
•
•
•
•
•
Capital
1.5
DROP
3.2 m
3.2 + 0.198 = 3.4
Critical Section
Step 8:Design Moments CL 31.4.2.2
wu = 16.5 kN /m
Parameters
L1 (Span in direction of Mo)
0.65L1
Ln (clear span extending from face to
face of columns, capitals)
Ln > 0.65L1
L2 (Span transverse to L1)
W = wu L2Ln
M0 = W Ln / 8
Along X
7.2
4.68
Along Y
6.4
m
4.16 m
(7.2-‐1.335)
=
(6.4-‐1.335)
=
m
5.865
5.065
5.865
6.4
619.34
454
5.065
7.2
601.72
381
m
m
kN
kNm
Step 9 : Distribution of Bending Moment across panel width ;
CL: 31.4.3.2, 31.5.5
X
Y
295.1
247.65
kNm
31.4.3.2
221.33
185.74
kNm
31.5.5.1
2x1.6 =3.2
69.17
2x1.6 =3.2
58.04
73.78
61.91
Width of Middle Strip Msw
3.2
4
• -‐m2 = M2/Msw
23.06
15.48
Moment Direction along
Negative Design Moment
MN = -‐0.65*M0
• Column Strip M1 = 0.75MN
Width of Column Strip Csw
• -‐m1= M1/ Csw
• Middle Strip M2 = 0.25MN
m
kNm/m
kNm
kNm/m
31.5.5.4(a)
Positive Design Moment
MP = 0.35*M0
• Column Strip M1 = 0.6MP
• +m1 = M1/ Csw
• Middle Strip M2 = 0.4MP
• +m2 = M2/Msw
-‐58.04
-‐69.17
A
29.79
-‐15.48
158.9
133.35
kNm
31.4.3.2
95.34
29.79
63.56
19.86
80.01
25
53.34
13.34
kNm
31.5.5.3
-‐58.04
A
B
-‐69.17
kNm/m
kNm
31.5.5.4(a)
kNm/m
-‐vesign : Hogging Moment (tension at top)
+ve sign : Sagging Moment (tension at bottom)
Step 10 : Check for adequacy of Depth
25
C
19.86
-‐23.06
-‐58.04
-‐69.17
A
25
13.34
D
-‐23.06
-‐58.04
-‐15.48
29.79
C
B
A -‐69.17
• Max Design Bending moment = 69.17 kNm/m
• Mu,lim = 126.36 kNm/m > 69.17, G-‐1.1(c)
• Depth is adequate
Moment
Direction
FE Results from
ETAB
MS
CS
Moment
Directio
n
CS
Step 11 :Rebar Details
26.5.2.1
• Ast,min = 0.12 x 240 x 1000 /100 = 288 mm2/m
• Minimum Effective Depth of Slab = 198 mm
G-‐1.1(b)
• 7.5 Ast2 – 71488Ast + Mu =0
Strip Location
Moment
Ast
(kNm/m)
(mm2/m)
(-‐)69.17
29.79
(-‐)23.06
19.86
1093
437
334
286
(-‐)58.04
(-‐)15.48
25
13.34
896
222
364
190
Bar
dia
Spacing
1093
437
334
288
10
8
8
8
70 -‐T
110 -‐B
150 -‐T
170 -‐B
896
288
364
288
10
8
8
8
85 -‐T
170-‐T
135 -‐B
170 -‐B
Ast
(prov)
mm
Along X
Zone A
Zone B
Zone C
Zone D
Along Y
Zone A
Zone B
Zone C
Zone D
#8@110
A
A
B
0.125Ln
#8@170
#8@110
C
0.15Ln
#8@170
#8@135
A
#8@135
LAP ZONE
D
C
0.15Ln
B
7.2 m
0.125Ln
A
Bottom Rebar Details in
Interior Panel
6.4 m
#10@ 70
0.2Ln
0.2Ln
#10@85
0.22Ln
0.22Ln
#8@150
0.33Ln
0.33Ln
A
0.33Ln
0.2Ln
0.33Ln
0.2Ln
0.22Ln
0.22Ln
B
A
#8@170
Top Rebar Details in
Interior Panel
Note:
Distances for curtailment of rebars are
measured from column face
#8@150
#8@135
#8@340
Section Through
Middle Strip -‐CDC
#8@170
7.2 m
#10@140
#10@85
#10@70
Section Through
Column Strip -‐ABA
#8@340
#8@170
#8@170
2. Corner Panel Design
Step 7 : Check for Shear around Column Capital
• Minimum Effective Depth of Slab = 198 mm
• Effective Depth at Drop location = 298 mm
Section 1:
• Critical Section at d/2 around the column capital
• Perimeter of Critical Section =  ( 1.5 + 0.298)/4 = 1.412 m
• Weight of Drop Projection below slab = 0.1x 25 x 1.5 = 3.75 kN/m2
• Design Shear at critical section around capital Vu
• Vu = 16.5 ( 3.6x 3.2 – ( x 1.7982/4)/4) + 3.75(1.6 x 1.6 – ( x1.7982/4)/4))
•
= 187 kN
• c = 0.25fck = 1.12 MPa
• ks = 0.5 +1 = 1.5 <=1 ; ks=1 ; ks c = 1.12
• Shear Resistance of Concrete = 1.12 x 1412 x 298 = 471kN > 187 kN
OK
Section 2 : Check for Shear around drop
free edge
•
•
•
•
•
•
Critical Section at d/2 around the drop
free edge
d = 198mm
Perimeter of Critical Section = 2 (1.7)=3.4m
Design Shear at critical section
=1.6 + 0.198/2
= 1.7 m
Vu = 16.5 ( 3.6 x 3.2 – 1.72) = 143 kN
Shear Resistance of Concrete = 1.12 x 3400 x 198 = 754kN > 143 kN
Step 8:Design Moments CL 31.4.2.2
M0 = W Ln / 8
Along X
454
Along Y
381
kNm
CRITICAL
SECTION
drop
Step 9 : Distribution of Bending Moment across panel width ;
CL 31.4.3.3 , 31.5.5
𝛼↓𝑐 =∑↑▒𝑘↓𝑐 /𝑘↓𝑠
Equivalent side of circular column = 0.89D = 0.89x 450 = 400 mm
Assume Ec = Es
Parameters
Sum of c o l u m n
stiffness above and
below the slab
Along X
Along Y
(2 x 4 x Ec x 400 x 4003/12) /3350 = 5.09 Ec x 106
2 (4EcIc)/Lc
Slab stiffness
(4 Es x 6400 x 2403/12)/7200
ks = 4EsIs/Ls
= 4.1Es x 106
c = kc /ks
c min (Table 17)
 = 1+ (1/c)
(4 Es x 7200 x 2403/12)/6400
= 5.184Es x 106
1.24
0.98
l2/l1 = 6.4/7.2 = 0.89,
WuL/WuD = 0.571
(0.7/0.5)*0.071 = 0.1 <c
7.2/6.4 = 1.125, WuL/WuD = 0.571
Adopt c
(0.8/0.5)*0.071 = 0.113 <c
Adopt c
1.8
2.02
-‐m1
A. Exterior negative design moment:
Exterior
-‐m2
-‐m1
Y
-‐m1
Exterior
-‐m2
X
Interior
-‐m1
Interior
1.6
Moment Direction along
Negative Design Moment
MN = -‐0.65*M0/
• Column Strip M1 = MN
Width of Column Strip Csw resisting M1
-‐m1= M1/ 3.2
• Middle Strip M2=0
Width of Middle Strip Msw resisting M2
-‐m2 = 0
1.6
X
Y
164
122.6
kNm
31.4.3.3
164
122.6
kNm
31.55.2(a)
2x1.6 =3.2
2x1.6 =3.2 m
-‐51.3
0
-‐38.3
0
3.2
4
0
0
kNm/m
kNm
kNm/m
31.5.5.4(a)
Exterior
B. Interior negative design moment:
Y
-‐m1
-‐m2
Interior
Exterior
-‐m1
-‐m1
Moment Direction along
Negative Design Moment
MN = -‐(0.75 – 0.1/)Mo
• Column Strip M1 = 0.75 MN
Width of Column Strip Csw resisting M1
• -‐m1= M1/ Csw
• Middle Strip M2 = 0.25 MN
Width of Middle Strip Msw resisting M2
• -‐m2 = M2/Msw
-‐m2
Interior
-‐m1
X
Y
315.3
266.9
kNm
31.4.3.3
236.5
2x1.6 =3.2
-‐73.9
78.83
3.2
-‐24.7
200.2
2x1.6 =3.2
-‐62.6
66.7
4
-‐16.7
kNm
31.5.5.1
m
kNm/m
kNm
m
kNm/m
31.5.5.4(a)
X
Exterior
m1
C. Positive Moment in Mid Span:
m
Exteri
m2
1
Y
m2
m1
Interior
o
m1
r
m1
Interior
Moment Direction along
Design Moment
MP = (0.63 – 0.28/)Mo
• Column Strip M1 = 0.6 MP
Width of Column Strip Csw resisting M1
• m1 = M1/ Csw
• Middle Strip M2 = 0.4 MP
Width of Middle Strip Msw resisting M2
• m2 = M2/Msw
X
Y
215.4
187.2
kNm
31.4.3.3
129.3
2x1.6 =3.2
40.4
86.2
3.2
26.94
112.3
2x1.6 =3.2
35.1
74.9
4
18.7
kNm
31.5.5.3
m
kNm/m
kNm
m
kNm/m
31.5.5.4(a)
X
Exterior
-‐38.3
-‐51.3
A
-‐38.3
-‐0
40.4
A
B
-‐73.9
-‐vesign : Hogging Moment (tension at top)
+ve sign : Sagging Moment (tension at bottom)
Exterior
0
35.1
C
35.1
18.7
26.94
D
C
-‐24.7
Interior
-‐62.6
-‐51.3
A
-‐62.6
-‐16.7
40.4
B
A
-‐73.9
Interior
Step 10 : Check for adequacy of Depth
• Max Design Bending moment = 73.9 kNm/m
• Mu,lim = 126.36 kNm/m > 73.9, Depth is adequate G-‐1.1(c)
Step 11 :Rebar Details
26.5.2.1
• Ast,min = 0.12 x 240 x 1000 /100 = 288 mm2/m
• Minimum Effective Depth of Slab = 198 mm
• 7.5 Ast2 – 71488Ast + Mu =0
Strip Location
Moment
Ast
(kNm/m)
(mm2/m)
(-‐)51.3
(-‐)73.9
40.4
(-‐)24.7
26.94
782
1180
604
359
393
(-‐)38.3
(-‐)62.6
(-‐)16.7
35.1
18.7
570
976
240
520
270
G-‐1.1(b)
Bar
dia
Spacing
782
1180
604
359
393
10
10
8
8
8
100 -‐T
65 -‐T
80 -‐B
140 -‐T
125-‐B
570
976
288
520
288
10
10
8
8
8
135
80
170
95
170
Ast
(prov)
mm
Along X
Zone A(Exterior)
Zone A(Interior)
Zone B
Zone C(Interior)
Zone D
Along Y
Zone A (Exterior)
Zone A(Interior)
Zone B (Interior)
Zone C
Zone D
#8@80
#8@125
#8@170
#8@80
#8@95
Spacing
(kNm/m)
Bar
dia
40.4
26.94
8
8
80 -‐B
125-‐B
35.1
18.7
8
8
95
170
Strip Location Moment
#8@95
mm
Along X
Zone B
Zone D
Along Y
Zone C
Zone D
#10@65
#10@100
#8@170(Min)*
-‐38.3
-‐51.3
#10@80
Exterior
0
A
35.1
C
40.4
-‐38.3
A-‐73.9
B
35.1
18.7
26.94
D
C
-‐24.7
Interior
#8@140
#8@170(Min)*
#10@135
Exterior
-‐0
-‐62.6
-‐51.3
A
-‐16.7
40.4
-‐62.6
A -‐73.9
B
Interior
#10@80
#10@135
#8@170
Spacing
(kNm/m)
Bar
dia
(-‐)51.3
(-‐)73.9
(-‐)24.7
10
10
8
100 -‐T
65 -‐T
140 -‐T
(-‐)38.3
(-‐)62.6
(-‐)16.7
10
10
8
135
80
170
Moment
mm
Along X
#10@65
#10@100
* Optional Top Rebars
Strip Location
Zone A(Exterior)
Zone A(Interior)
Zone C(Interior)
Along Y
Zone A (Exterior)
Zone A(Interior)
Zone B (Interior)
Transfer of Moments and Shears between Slabs and Columns
• The maximum load that a flat slab can support is dependent upon the strength of
the joint between the column and the slab.
• Load is transferred by shear from the slab to the column along an area around the
column
• In addition moments are also transferred.
• The moment situation is usually most critical at the exterior columns.
• Shear forces resulting from moment transfer must be considered in the design of
the lateral column reinforcement (i.e., ties and spirals).
EXAMPLE
Compute moment transferred to Interior and corner Column in example 2
Interior Column
• As spans are same in both directions
• M = 0.08 (0.5 w L L2 Ln 2 /(1+1/c ) = 0.08 x 0.5 x 6 x 5.5 x 52 / 1.36 = 24.3 kNm
• this moment is distributed to top and bottom column at junction in proportion to their
stiffness.
• M = 24.3/2 = 12.2 kNm
Corner Column
M = 131.4 kNm
Equivalent Frame Method (EFM)
CL 31.5
•
•
•
Edge Frame
Transverse Frame
More Comprehensive and Logical method
Used when limitations of DDM are not
complied with
Applicable when subjected to horizontal
loads
31.5.1 (a)
Idealizing the 3D slab –column system to 2D
frames along column Centre lines in both
longitudinal and transverse directions.
Longitudinal Frame
31.5.1(b)
For vertical loads, each floor, together
with the columns above and below, is
analyzed separately. For such an analysis,
the far ends of the columns are considered
fixed.
If there are large number of panels, the
moment at a particular joint in a slab
beam can be satisfactorily obtained by
assuming that the member is fixed two
panels away.
This simplification is permissible because vertical
loads in one panel only appreciably affect the
forces in that panel and in the one adjacent to it on
each side.
For lateral loads, it is necessary to consider an equivalent
frame that extends for the entire height of the building,
because the forces in a particular member are affected by the
lateral forces on all the stories above the floor being
considered.
Entire Frame Analysis
Gravity + Lateral Loads
31.5.1(C and d)
variation of the flexural
moment of inertia
I2 = moment of inertia at the face of the column / column capital
c2 = dimension of column capital in the transverse direction
l2 = width of equivalent frame.
• Variations of moment of inertia along the axis 0f the slab on account of provision of
drops shall be taken into account
• The stiffening effect of flared column heads may be ignored
31.5.2 Loading Paiern
wu LL > ¾ wu,DL
Critical Section
Interior Column Centre Line
Column /Capital face
< = C/2
Results in Significant reduction of design moments
Design Positive Moment (Span region)
M3 = M0 – (M1+M2)/2
C
Distribution of Moment
Similar to DDM
Example 3 : Compute moments in exterior/interior Panel along Longitudinal Span
Longitudinal Span = 7.2m, Transverse Span = 6.4 m, Interior Column = 450mm dia, Column Capital =
1500mm dia, Exterior Column = 400x400mm, Column Capital = 870mm(square), Floor to Floor = 3.35 m,
Slab Thickness = 240 mm, number of Panels = 4 in each direction
6.4m
6.4m
6.4m
7.2 m
7.2 m
7.2 m
7.2 m
Step 1: Stiffness Computations
Exterior Column (Kce) = 4E x (4004 /12) /3350 = 2.55E106 = 1
Interior Column (KcI) = 4E x (4504 /64) /3350 = 2.4E106 = 0.957
Slab(Ks) =4E x (6400 x 2403/12) /7200 = 4.1E106 = 1.608
Step 2: Simplified frame for analysis 31.5.1 (b)
Joint
A
3350
Relative
Stiffness
1-‐A
1
1-‐2
1.608
1-‐C
1
0.277
2-‐B
0.957
0.187
2-‐1
1.608
0.314
2-‐3
1.608
2-‐D
0.957
Sum
Distribution
Factors
B
1
2
3
1
3350
C
Member
7200
D
7200
2
Fixed End Moments = (16.5 x 6.4) x 7.22/12 = 456.2 kNm
0.277
3.608
5.13
0.446
0.314
0.187
Joint
Members
1
FIXED
1A+1C
2
1-‐2
2-‐1
FIXED
2B+2D
3
2-‐3
FIXED
3-‐2
Counter Clockwise
end moments are
positive
DF
FEM
Bal
CO
Bal
CO
Bal
CO
Bal
Final end
Moments
0.554
0.446
0.314
-‐252.74
-‐
-‐
-‐
-‐8.85
-‐
456.2
-‐203.46
-‐
-‐
15.97
-‐7.12
-‐
-‐456.2
-‐
-‐101.73
31.94
-‐
-‐
-‐3.56
1.12
261.6
-‐528.43528.4339.37
489.26 489.26
-‐261.6
261.6
1
0.374
0.314
-‐
-‐
38.04
-‐
-‐
-‐
1.33
456.2
-‐
-‐
31.94
-‐
-‐
-‐
1.12
2
-‐
-‐456.2
-‐
-‐
-‐
15.97
-‐
-‐
-‐440.23440.23
3
Step 3: Design Moments in Exterior Panel
A. Design Negative Moments at Critical Section
At Exterior Support : CL 31.5.3.2
Critical Section from Column Centre line = 435 mm
235
400
470
870
261.6
16.5 x 6.4 = 105.6 kN/m
528.43
0.435
105.6 x 7.2/2 -‐(528.43-‐261.6)/7.2
= 343 kN
Design Moment = 343 x 0.435 -‐261.6 -‐105.6x0.4352/2 = -‐122.4 kNm (Hogging)
At Interior Support : CL 31.5.3.1
Critical Section location is at capital face
261.6
Width of equivalent square
= 0.89D = 1335 mm
16.5 x 6.4 = 105.6 kN/m
343 kN
528.43
0.6675
417.32
667.5
mm
 0.175x7200 = 1260mm
Design Moment = 417.32 x 0.6675 -‐528.3 -‐105.6x0.66752/2 = -‐273.26 kNm (Hogging)
B. Design Positive Moment
M(+) = (16.5 x 6.4x7.2)7.2/8 – ( 528.43 + 261.6)/2 = 289.3 kNm
Moments
DDM
215.4
EFM
289.3
Negative Moment(Exterior Support)
164
122.4
Negative Moment (Interior Support)
315.3
273.3
Positive Moment (Span)
Step 4: Design Moments in Interior Panel
A. Design Negative Moments at Critical Section
At Interior Support : CL 31.5.3.1
489.26
440.23
A 16.5 x 6.4 = 105.6 kN/m
0.6675
387 kN
B
0.6675
373.32
Design Moment at A= 387 x 0.6675 -‐489.26 -‐105.6x0.66752/2 = -‐254.5 kNm (Hogging)
Design Moment at B = 373.32 x 0.6675 -‐440.23 -‐105.6x0.66752/2 = -‐214.6 kNm (Hogging)
B. Design Positive Moment
M(+) = (16.5 x 6.4x7.2)7.2/8 – ( 489.26 + 440.23)/2 = 219.5 kNm
Moments
Positive Moment (Span)
DDM
158.9
EFM
219.5
Negative Moment (Interior Support)
295.1
254.5/214.6
Need for Computer Analysis
The equivalent frame method is not satisfactory for hand calculations.
It is possible, however, to use computers and plane frame analysis
programs if the structure is modeled such that various nodal points in
the structure can account for the changing moments of inertia along the
member axis.
Column
SLAB
Drop Panel
Column Head
Column