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REINFORCED CONCRETE STRUCTURES DESIGN
AND DRAWING (ACE009)
B.Tech V Semester
Department of Civil Engineering
Prepared by
Gude Rama Krishna
Associate Professor
P. Vinay Kumar
Assistant Professor

 To introduce students to various concepts of Limit state
design.
 To impart knowledge regarding the design of structures of
buildings like slabs , beams and columns etc
COURSE GOAL
1

Course outline
UNIT TITLE CONTENTS
1 INTRODUCTION Concepts of RC Design –
Limit state method –
Material Stress–Strain
curves – Safety factors –
Characteristic values –
Stress block parameters –
IS-456:2000 – Working
stress method.
BEAMS: Limit state
analysis and design of
singly reinforced, doubly
reinforced, T, and L beam
sections
2

2 SHEAR TORSION AND BOND Limit state analysis and design of
section for shear and torsion –
concept of bond, anchorage and
development length, I.S. code
provisions. Design examples in
simply supported and continuous
beams, detailing Limit state design
for serviceability for deflection,
cracking and codal provision.
3 DESIGN OF SLABS Design of Two-way Slabs, one-way
slabs, Continuous slabs using I.S.
coefficients, Cantilever slab/ Canopy
slab.
3

4 SHORT AND LONG COLUMNS Axial loads, uni-axial and bi-
axial bending I.S. Code
provisions.
5 DESIGN OF FOOTINGS Isolated(square, rectangle)
and Combined Footings.
Design of Stair Case
4

Plain Cement Concrete
➢ Plain Cement Concrete is a hardened mass obtained from a
mixture of cement, sand, gravel and water in definite
proportions.
➢ These ingredients are mixed together to form a plastic mass
which is poured into desired shape moulds called as forms. This
plastic mass hardens on setting and we get plain cement
concrete.
7

➢ The hardening of this mixture is caused by a chemical
➢ reaction between cement and water.
➢ Plain cement concrete has good compressive strength but very
little tensile strength, thus limiting its use in construction.
➢ Plain concrete is used where good compressive strength and
weight are the main requirements and tensile stresses are very
low.
8

REINFORCED CEMENT CONCRETE
➢ Plain cement concrete has very low tensile strength.
➢ To improve the tensile strength of concrete, some sort of
reinforcement is needed which can take up the tensile stresses
developed in the structure.
➢ The most common type of reinforcement is in the form of steel bars
which are quite strong in tension.
➢ The reinforcing steel is placed in the forms and fresh concrete is
poured around it.
➢ This solidified composite mass is called as Reinforced cement
concrete and is abbreviated as R.C.C.
9

➢ Thus, Reinforced cement concrete is a composite material which
is made up of concrete and steel reinforcement.
➢ The steel reinforcement generally in the form of steel bars, are
placed in the tensile zone of the structure and take up the tensile
stresses R.C C. is a versatile construction material which is strong
in compression as well as tension.
➢ The use of reinforcement in concrete not only increases its
strength but also helps in preventing the temperature and
shrinkage stresses.
10

The composite action of steel and concrete in a reinforced concrete
section is dependent on the following important factors.
(i) The bond between steel and concrete.
(ii) Prevention of corrosion of steel bars embedded in the concrete.
(iii) Practically equal thermal expansion of both concrete and steel.
11

Uses of Reinforced Concrete
Reinforced cement concrete has innumerable uses in construction
some of which are list below:
1. Buildings
2 Flyovers
3. Water Tanks
4 Road and Rail Bridges
5 Chimneys and Towers
6 Retaining Walls
7 Bunkers and Silos
12

Advantages
Reinforced cement concrete has following advantages over other
construction materials.
1. Strength: R.C.C. has very good strength in tension as well as
compression.
2. Durability: R.C.C. structures are durable if designed and laid
properly. They can last up to 100 years.
3. Mouldability: R.C.C. sections can be given any shape easily by
properly designing the formwork. Thus, it is more suitable for
architectural requirements.
4. Ductility: The steel reinforcement imparts ductility to the R.C.C.
structures.
.
13

5. Economy: R.C.C. is cheaper as compared to steel and prestressed
concrete. There is an overall economy by using R.C.C. because its
maintenance cost is low.
6. Transportation: The raw materials which are required for R.C.C.
i.e. cement, sand, aggregate, water and steel are easily available
and can be transported easily. Nowadays Ready Mix Concrete (RMC
is used for faster and better construction. (RMC is the concrete
which is manufactured in the factory and transported to the site in
green or plastic state)
14

7. Fire Resistance: RCC structures are more fire resistant than
other commonly used construction materials like steel and wood
8 Permeability: RC.C is almost impermeable to moisture.
9. Seismic Resistance: Properly designed R.C C structures are
extremely resistant to earthquakes
15

Disadvantages:
Despite the above mentioned advantages, R.C.C. has following
disadvantages:
1. R.C.C. structures are heavier than structures of other materials like
steel, wood and
glass etc.
2. R.C.C. needs lot of formwork, centering and shuttering to be fixed,
thus require lot of site
space and skilled labour.
3. Concrete takes time to attain its full strength. Thus, R.C.C. structures
can't be used immediately after construction unlike steel structures.
16

MATERIALS USED IN R.C.C.
R.C.C. consists of concrete and reinforcing material. The strength of an
R.C.C. section depends on the kind of concrete and reinforcement used.
The properties of concrete depend upon the proportions and type of its
ingredients. A properly designed concrete mix is very durable. A good
concrete mix should satisfy the following requirements.
(i) The concrete should be mixed thoroughly to from a homogeneous
mix.
(ii) Concrete should be compacted properly to prevent it from being
porous.
(iii) Sufficient curing of concrete is required for developing full strength.
17

(iv) The water cement ratio should be appropriate, considering the
strength and workability criteria.
(v) The concrete mix should be designed properly and should have all
the ingredients in right proportions.
(vi) The water used for mixing should be free from all harmful organic
substances.
(vii) The aggregate should be hard, durable and properly graded. For
most R.C.C. works, 20 mm size of aggregate is suitable
(Viii) The cement used for R.C.C. work should be of good quality and
measured by weight only, and not by volume.
MATERIALS USED IN R.C.C
18

Concrete grades are expressed by letter M followed by a number.
The letter 'M' refers to the mix and the number represents the
characteristic compressive strength of concrete in N/mm2.
➢ The specified characteristic strength is determined for 150 mm size
cube at 28 days.
The characteristic compressive strength of concrete is defined as that
strength below which not more than 5 percent of the test results are
expected to fall.
19

The various grades of concrete as per their use are listed below:
1. For R.C.C. work - not lower than M20.
2. For post tensioning works - M35 and above.
3. For pretensioned prestressed concrete - M40 and above.
Concrete of grades lower than M20 may be used for plain concrete
works, lean concrete, simple foundations, masonry walls and other
simple constructions works.
20

Many traditional materials such as bamboo and natural fibres have been
tried as reinforcement in earlier times.
But steel is found to be the most appropriate form of reinforcement. It is
the most suitable reinforcing material in R.C.C. because of following
reasons:
1.Steel is very strong in tension, compression, shear and torsion.
2.Concrete develops very good bond with steel.
21
SUITABILITY OF STEEL AS REINFORCING MATERIAL

3. Steel is ductile in behavior. More ductility means more
elongation of steel before failure. This results in sufficient warning
time before failure.
4. The steel bars can be cut, bent, lifted and welded easily with
commonly available tools.
and machines.
5. Steel has longer life.
6 Steel is easily available.
22

Steel reinforcement has various advantages as listed above, which make
it a suitable reinforcing material. However, steel has a few disadvantages
which are listed below.
1. The biggest disadvantage of steel reinforcement is rusting. If concrete
is porous or if cover to the reinforcement is not sufficient, steel gets
rusted and loses strength.
2. Steel loses its strength at high temperatures.
23

Mild steel bars are also known as Fe 250 because the yield strength of
this steel is 250 N/mm2.
Although mild steel bars are very ductile are not preferred over high
yield strength deformed bars because of their less strength and weak
bond. The modulus of elasticity of mild steel is taken as equal to 2 x
105N/mm2. However, they are used as lateral ties in columns and also
places where nominal reinforcement is required. Mild steel plain bars
are represented by symbol
24
Mild Steel Reinforcement

The stress-strain curve for mild steel is given in Fig. 1.1. It shows a clear,
definite yield point.
25

METHODS OF R.C.C. DESIGN:
Design of any R.C.C. member comprises of the following:
(i) To decide the size (dimensions) of the member and the amount of
reinforcement required.
(ii) To check whether the adopted section will perform safely and
satisfactorily during the life time of the structure.
26

Various methods used for the design of R.C.C. structures are as follows:
(i) Working stress method.
(ii) Load factor or ultimate load method.
(iii) Limit state method.
27

Working Stress Method:
This method of design was the oldest one. It is based on the elastic
theory and assumes that both steel and concrete are elastic and obey
Hook's law. It means that the stress is directly
proportional to strain up to the point of collapse. Based on the elastic
theory, and assuming that the bond between steel and concrete is
perfect, permissible stresses of the materials are obtained.
28

• The basis of this method is that the permissible stresses are not
exceeded any where in the structure when it is subjected to worst
combination of working loads.
• In this method, the ultimate stresses of concrete and yield strength or
0.2% proof stress of steel are divided by factors of safety to obtain
permissible stresses.
• These factor of safety take into account the uncertainties in
manufacturing of these materials.
•Modular Ratio between steel and concrete remains constant.
29

The main drawbacks of the working stress method of design are as
follows:
(i) It assumes that concrete is elastic which is not true.
(ii) It uses factors of safety for stresses only which does not give true
margin of safety because we do not know the failure load.
(iii) It does not use any factor of safety with respect to loads. It means,
there is no provision for uncertainties associated with the load.
(iv) It does not account for shrinkage and creep which are time
dependent and plastic in nature.
30

(v) This method gives uneconomical sections.
(vi) It pays no attention to the conditions that arise at the time of
collapse.
 The working stress method is very simple and reliable but as per IS
456 : 2000 the working stress method is to be used only if it is not
possible to use limit state method of design. Working stress method
is the basic method and its knowledge is essential for understanding
the concepts of design.
31

Load Factor Method or Ultimate Load Method:
 In this method, ultimate load or collapse load is used as design load.
Ultimate load is obtained by multiplying working load with the load
factor. The load factor gives exact margin of safety in terms of load.
32

This method uses the real stress-strain curve and takes into
account the plastic behavior of materials. This method
gives very thin sections which result in excessive
deformation and cracking
thus, making the structure almost unserviceable. This
method is not at all used by designers.
33

 If the sections are designed based on ultimate strength design,
there is a danger that although the load factor is adequate. The
cracking and the deflections at the service loads may be excessive.
 Cracking may be excessive if the steel stresses are high or if the bars
are badly distributed.
Reasons

Limit State Method
This is the most rational method which takes into account the ultimate
strength of the structure and also the serviceability requirements. It is a
judicious combination of working stress and ultimate load method of
design. The acceptable limit of safety and serviceability requirements
before failure occurs is called a limit state. This method is based on the
concept of safety at ultimate loads (ultimate load method) and
serviceability at working loads (working stress method).
35

The two important limit states to be considered in design are:
(i) Limit state of collapse.
(ii) Limit state of serviceability.
36

What is Limit State?
 The acceptable limit of safety and serviceability requirements
before failure occurs is called a limit state.

DESIGN CONCEPTS
➢ Safety: implies that the likelihood of (partial or total) collapse of
structure is acceptably low not only under (normal loads) service
loads but also under overloads.
➢ Serviceability: satisfactory performance of structure under service
loads without discomfort to user due to excessive deflections,
cracking, vibration etc.
➢ Other considerations such as durability, impermeability, acoustic
and thermal insulation etc.
38

Limit State Design
Limit States
Purpose: to achieve acceptable probability that a structure will not
become unfit for its intended use i.e. that it will not reach a limit
state.
Thus, a structure ceases to be fit for use will constitute a limit state
and the design aims to avoid any such condition being reached
during the expected life of the structure.
Two principle types of limit state are:
I. Ultimate Limit State
II. Serviceability Limit State

Ultimate Limit State
This requires that the structure must be able to withstand, with an
adequate factor of safety against collapse, the loads for which it is
designed.
Limit state of Collapse: flexure, shear, compression, torsion, bearing,
etc.
Possibility of buckling or overturning must also be taken into
account, as must the possibility of accidental damage as caused, for
example, by an internal explosion.
Limit State Design

Most important serviceability limit states are
➢ Deflection: appearance or efficiency of any part of the structure
must not be adversely affected by deflections.
➢ Cracking: local damage due to cracking and spalling must not
affect the appearance, efficiency or durability of structure.
➢ Durability: this must be considered in terms of the proposed life of
the structure and its conditions of exposure.
Serviceability Limit States

Other limit states include
➢ Excessive vibration: which may cause discomfort or alarm as well
as damage.
➢ Fatigue: must be considered if cyclic loading is likely.
➢ Fire resistance: this must be considered in terms of resistance to
collapse, flame penetration and heat transfer.
➢ Special circumstances: any special requirements of the structure
which are not covered by any of the more common limit states,
such as earthquake resistance, must be taken into account.
Serviceability Limit States

What is Partial safety factor?
 In Limit State Design, the load actually used for each limit state is
called the “Design Load” for that limit state
 “Design Load” is the product of the characteristic load and the
relevant partial safety factor for loads
 Design load = γf x (characteristic load)
Partial safety factor

Why do we use partial safety factors?
➢ Partial safety factor is intended to cover those variations in loading
in design or in construction which are likely to occur after the
designer and the constructor have each exercised carefully their
skill and knowledge.
➢ Also takes into account nature of limit state in question.

Partial Factors of Safety
➢ Other possible variations such as constructional tolerances are
allowed for by partial factors of safety applied to the strength of
materials and to loadings.
➢ Lack of adequate data, however, makes this unrealistic and in
practice the values adopted are based on experience and
simplified calculations.

Partial factors of safety for loads (γf)
➢ Errors and inaccuracies may be due to a number of causes:
➢ Design assumptions and inaccuracy of calculation.
➢ Possible unusual load increases.
➢ Unforeseen stress redistributions.
➢ Constructional inaccuracies
➢ These are taken into account by applying a particular factor of
safety (γf) on the loadings, so that
➢ Design load = characteristic load x partial factor of safety (γf)
➢ This factor should also take into account the importance of the
limit state under consideration and reflect to some extent the
accuracy with which different type of loading can be predicted,
and the probability of particular load combinations occurring.

 The strength of material in an actual member will differ from that
measured in a carefully prepared test specimen and it is
particularly true for concrete where placing, compaction and
curing are so important to the strength. Steel, on the other hand,
is a relatively consistent material requiring a small partial factor of
safety.
 The severity of the limit state being considered. Thus, higher
values are taken for the ultimate limit state than for the
serviceability limit state.
Partial Factors of Safety for Materials(γm)

Partial Factor of Safety for loads
Load
Combination
Limit State of
Collapse
Limit State of
Serviceability
DL LL WL/EL DL LL WL/EL
DL + LL 1.5 1.5 1.0 1.0 1.0 --
DL + WL or EL
1.5 or
0.9*
- 1.5 1.0 --- 1.0
DL + LL + WL/EL 1.2 1.2 1.2 1.0 0.8 0.8

 In design calculations “Design Strength” for a given material and
limit state is obtained by dividing the characteristic strength by the
partial safety factor for strength, appropriate to that material and
that limit state.
 When assessing the strength of a structure or structural member
for the limit state of collapse, the partial safety factor should be
taken as 1.5 for concrete and 1.15 for steel.
What is Design Strength?

0.002 0.0035
Strain
fck
0.67 fck
0.67 fck/γm
Stress
Parabolic
Curve
Stress-Strain Curve for Concrete in Flexural Compression

DESIGN CONCEPTS
Safety: implies that the likelihood of (partial or total) collapse of
structure is acceptably low not only under (normal loads) service
loads but also under overloads.
Serviceability: satisfactory performance of structure under service
loads without discomfort to user due to excessive deflections,
cracking, vibration etc.
Other considerations such as durability, impermeability, acoustic and
thermal insulation etc.
51

1. At any cross-section, sections which are plane prior to bending
remain plane after bending. Or strain varies linearly with distance
from neutral axis i.e. plane sections remain plane in bending.
2. The maximum strain in concrete at the outermost fiber is 0.0035.
3. Stress-strain relationship in concrete could be either rectangular,
parabolic or combination of rectangular and parabolic curves
which should be agreeable with the experimental results.
Assumptions for Design in Flexure

4. The stresses in steel bars used for reinforcement are derived from
the representative stress-strain curve for the type of steel used.
5. Perfect bond between reinforced steel and adjoining concrete.
6. Tensile strength of concrete is neglected.
7. Minimum strain in steel reinforcement should not be less than
((0.87fy/Es) + 0.002).
Assumptions for Design in Flexure

Strain diagram and Stress blocks:
(a) Section; (b) Strain diagram; (c) Stress block
d
b
Ast
xu
0.0035
εcu
εs
0.446 fck
xu
0.87 fy
(a) (b)

0.42 xu
0.36 fck xu
xu
Stress Block Parameters

Characteristic Strength
 “Characteristic strength is defined as the strength of material
below which not more than 5 percent of the test results are
expected to fall”.
 Strength of concrete varies for the same concrete mix, which give
different compressive strength in laboratory tests.
 Variability in strength evidently depends on degree of quality
control.
 Variability in strength is measured in terms of either the “Standard
Deviation” or the Coefficient of Variation (COV), which is the ratio
of standard deviation to mean strength(fcm).
Characteristic strength

 It is well established that the probability distribution of concrete
strength (for a given mix ) is approximately “Normal”.
 Coefficient of variation is generally in the range of 0.01 to 0.02.
 Due to significant variability in strength, it is necessary to ensure
that the designer has a reasonable assurance of a certain
minimum strength of concrete.
 Characteristic strength provides minimum guaranteed strength.

 Accordingly, the mean strength of concrete, fcm (as obtained from
28 days tests) has to be significantly greater than the 5 percentile
characteristic strength, that is specified by the designer.
ck
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Normal Probability Curve

➢ Strength of materials upon which design is based on that
strength is assumed to be normal.
➢ Characteristic value is defined as that value below which it is
unlikely that more than 5% of the results will fall.
➢
➢
➢ = Characteristic Strength
➢ = Standard Deviation
➢ The relationship between and accounts for variations in
results of test specimens and with the method, and control of
manufacture, quality of construction and type of materials

64
.
1
−
= m
ck
f
f
ck
f
m
f
ck
f m
f

 Loads on structures can also be assessed stastically.
 Characteristic Load = Mean Load ± 1.64 (standard deviation).
 In most cases, it is the maximum loading on a structural member
that is critical and the upper, positive value given by the above
expression.
 But the lower, minimum value may apply when considering the
stability of the behaviour of continuous members.
Characteristic Loads

Doubly Reinforced Section
The R.C.C. beams in which the steel reinforcement is placed in the
tension as well as compression zone are called as doubly reinforced
beams. The moment of resistance of a balanced R.C.C. beam of
dimension b x d is Rbd2. Sometimes due to head room constraints or
architectural considerations the size of the beam is restricted and the
same beam (b x d) is required to resist a moment greater than Rbd2.
There are only two ways in which it can be done.
68

(i) By using an over reinforced section.
(ii) By using a doubly reinforced section.
The option (i) is not a good choice because over reinforced sections
are uneconomical and the failure of these beams is sudden without
warning.
Therefore, it is better to use doubly reinforced beam section in such
circumstances.
The extra steel provided in the tension and compression zone
constitutes, the additional moment of resistance (greater than Rbd2)
required.

NECESSITY OF DOUBLY REINFORCED SECTION
Doubly reinforced sections are used in the following conditions:
1. When the dimensions (b x d) of the beam are restricted due to any
constraints like availability of head room, architectural or space
considerations and the moment of resistance of singly reinforced
section is less than the external moment.
2. When the external loads may occur on either face of the member i.
e. , the loads are alternating or reversing and may cause tension on
both faces of the member.
3. When the loads are eccentric.

4. When the beam is subjected to accidental or sudden
lateral loads.
5. In the case of continuous beams or slab, the sections at
supports are generally designed as doubly reinforced
sections.

Section 1: Section 1 consists of a singly reinforced balanced section
having area of steel Ast1 and moment of resistance Mu lim.
Section 2: Section 2 consists of compression steel Asc and additional
tensile steel Ast2 corresponding to Asc. The moment of resistance of
this section is Mu2 such that
Mu=Mu lim + Mu2

Let us consider a doubly reinforced beam shown in Fig.

where
b= Width of beam
xu=Depth of neutral axis
d=Effective depth of beam
fsc= Stress in compression steel
d'= Effective cover to compression steel
fcc=Stress in concrete at the level of steel
Asc=Area of compression steel
Ast=Area of tension steel

Depth of Neutral Axis (xu)
The depth of neutral axis of a doubly reinforced beam section is
obtained by equating the total compression and total tension.
Total compression C1 + C2
Where
C1 is the force carried by the concrete area
C2 is the compressive force carried by compression steel Asc.
C1 = 0.36 fck. b. xu
C2 = fsc . Asc – fcc . Asc
The term fcc. Asc accounts for the loss of concrete area occupied by
compression steel.

DESIGN OF FLANGED BEAMS
K. Anand Goud
(IARE 10537)
77

 In reinforced concrete construction, slab is supported over beams.
 Simple concrete slabs of moderate depth and weight are limited to
spans of 3m to 5m
 If it is desired for long spans without excessive weight and material,
slab is built monolithically with RC beams and beams are considered
as flanged beams.
 At the interior portions of floor, slab with beam acts as a T-beam and
at an end the portion acts as an L-beam.
 Shear reinforcement of beams and bent bars extend into slab and
Complete construction is cast integrally. A part of slab acts with upper
part in bending compressive stresses.
Design of Flanged Beams

 Theoretically width of flange is supposed to act as top flange of
beam.
 Elements of flange midway between webs of two adjacent beams
are less highly stressed in longitudinal compression than those
elements directly over webs of beams.
 An effective width of flange, bf is used in the design of flanged
beam and is treated to be uniformly stressed at the maximum
value, which is smaller than actual width of flange.
 Effective width of flange primarily depends on span of the beam,
breadth of web, bw and thickness of flange, Df.
Effective Width of Flange

 IS: 456-2000 recommends for effective width of flanges of T- and
L-beams.
• For symmetrical T-beams
bf = [(l06) + bw + 6Df]
• For beams with slab on one side only
bf = [(l012) + bw + 3Df]
• For isolated T-beams
bf = [(l0((l0b)+4)) + bw]
• For Isolated L-beams
bf = [(0.5l0((l0b)+4)) + bw]

 Calculated effective flange width, bf shall be not greater than the
breadth of web plus half the sum of clear distances to the
adjacent beams on either side
• bf < 0.5 [l1 + l2] + bw
• bf < 0.5 [l2 + L3] + bw

 Depending upon proportions of cross-section, area of steel
reinforcement in tension, strength of materials
1. Neutral axis of a T-beam in one case may lie in the flange i.e.
depth of NA, xu is less than or equal to thickness of flange or depth
of slab, Df (Neutral axis lies within flange (xu < Df))
2. NA may lie in web i.e. depth of neutral axis, xu is more than
thickness of slab, Df.
 Stress diagram consists of a rectangular portion of depth 0.43xu
and a parabolic portion of depth 0.57xu.
Location of Neutral Axis

 When NA of T-section lies outside flange, it lies in web of T-beam.
However, there are two possibilities depending upon whether
depth of flange Df is less than or equal to 0.43xu or Df is more than
0.43 xu.
 Comparison of Df with 0.43xu (i.e. 3/7xu) is more rational as 0.43xu
is actual depth of rectangular portion of stress block.
 In IS:456-2000, if (Df/d) is less than 0.2, the flange of T-beam is
considered as small.
i. Df is less than 0.43xu
 Total area in compression consists of sum of compressive force
in concrete in web of width, bw, Cw, cu and compressive force in
concrete in the flange excluding web, Cf, cu.
(2) NEUTRAL AXIS LIES OUT SIDE FLANGE [i.e. xu > Df]

Df > 0.43 xu or (Df > 0.2d)
Depth of flange Df is more than 0.43xu, some portion is subjected to
uniform stress equal to 0.446fck (0.43xu) and remaining portion is
subjected to parabolic stress.
To obtain compressive force in portion of flange, concept of
modified thickness of flange equal to
yf = (0.15xu + 0.65Df)
is recommended by IS456-2000.
Average stress is assumed to be 0.446fck

Moment of Resistance
I. A singly reinforced slab 120mm thick is supported by T-beam spaced at 3.5m
c/c has an effective depth, d= 550mm, width, bw = 400mm. The beam is
provided with steel reinforcement consisting of 5 bars of 20mm diameter in
one layer, d’ = 50mm. le = 3.7m. Use M20 grade concrete and Fe415 steel.
Determine the depth of neutral axis and the moment of resistance of the
beam, MR?
II. Calculate the moment of resistance of a T-beam for M20 and Fe415, Df = 120
mm, bf = 750mm, bw = 250mm, d’ = 50mm, D = 500mm
III. T-beam floor, Df = 150mm, bw = 250mm spacing = 3.5m c/c, le = 8.0m. LL = 6.5
kN/m. Design an intermediate beam using M20 and Fe415 steel.
IV. T-beam d = 750mm, bf= 1400mm, Df = 100mm, bw = 300mm, Ast = ? M= 100kN-
m. Use M20 and Fe 415 HYSD bars.
Moment of Resistance

DESIGN OF
SINGLY REINFORCED BEAM
90

BEAM:-
A Beam is any structural member which resists load mainly by bending.
Therefore it is also called flexural member. Beam may be singly
reinforced or doubly reinforced. When steel is provided only in tensile
zone (i.e. below neutral axis) is called singly reinforced beam, but when
steel is provided in tension zone as well as compression zone is called
doubly reinforced beam.
91

The aim of design is:
➢To decide the size (dimensions) of the member and the amount of
reinforcement required.
➢To check whether the adopted section will perform safely and
satisfactorily during the life time of the structure.
92

OVER ALL DEPTH :-
THE NORMAL DISTANCE FROM THE TOP EDGE OF THE BEAM TO THE
BOTTOM EDGE OF THE BEAM IS CALLED OVER ALL DEPTH. IT IS
DENOTED BY ‘D’.
EFFECTIVE DEPTH:-
THE NORMAL DISTANCE FROM THE TOP EDGE OF BEAM TO THE
CENTRE OF TENSILE REINFORCEMENT IS CALLED EFFECTIVE DEPTH. IT
IS DENOTED BY ‘d’.
94

CLEAR COVER:-
THE DISTANCE BETWEEN THE BOTTOM OF THE BARS AND BOTTOM
MOST THE EDGE OF THE BEAM IS CALLED CLEAR COVER.
CLEAR COVER = 25mm OR DIA OF MAIN BAR, (WHICH EVER IS
GREATER).
EFFECTIVE COVER:-
THE DISTANCE BETWEEN CENTRE OF TENSILE REINFORCEMENT AND
THE BOTTOM EDGE OF THE BEAM IS CALLED EFFECTIVE COVER.
EFFECTIVE COVER = CLEAR COVER + ½ DIA OF BAR.
95

END COVER:-
END COVER = 2XDIA OF BAR OR 25mm (WHICH EVER IS GREATER)
NEUTRAL AXIS:- THE LAYER / LAMINA WHERE NO STRESS EXIST IS
KNOWN AS NEUTRAL AXIS. IT DIVIDES THE BEAM SECTION INTO TWO
ZONES, COMPRESION ZONE ABOVE THE NETURAL AXIS & TENSION
ZONE BELOW THE NEUTRAL AXIS.
96

DEPTH OF NETURAL AXIS:- THE NORMAL DISTANCE
BETWEEN THE TOP EDGE OF THE BEAM & NEUTRAL AXIS IS
CALLED DEPTH OF NETURAL AXIS. IT IS DENOTED BY ‘n’.
LEVER ARM:- THE DISTANCE BETWEEN THE RESULTANT
COMPRESSIVE FORCE (C) AND TENSILE FORCE (T) IS KNOWN
AS LEVER ARM. IT IS DENOTED BY ‘z’. THE TOTAL
COMPRESSIVE FORCE (C) IN CONCRETE ACT AT THE C.G. OF
COMPRESSIVE STRESS DIAGRAM i.e. n/3 FROM THE
COMPRESSION EDGE. THE TOTAL TENSILE FORCE (T) ACTS AT
C.G. OF THE REINFORCEMENT.
LEVER ARM = d-n/3
97

TENSILE REINFORCEMENT:-
THE REINFORCEMENT PROVIDED TENSILE ZONE IS CALLED
TENSILE REINFORCEMENT. IT IS DENOTED BY Ast.
COMPRESSION REINFORCEMENT :-
THE REINFORCEMENT PROVIDED COMPRESSION ZONEIS CALLED
COMPRESSION REINFORCEMENT. IT IS DENOTED BY Asc
98

TYPES OF BEAM SECTION:- THE BEAM SECTION CAN BE OF THE
FOLLOWING TYPES:
1.BALANCED SECTION
2.UNBALNCED SECTION
(a) UNDER- REINFORCED SECTION
(b) OVER-REINFORCED SECTION
1.BALANCED SECTION:- A SECTION IS KNOWN AS BALANCED SECTION
IN WHICH THE COMPRESSIVE STREE IN CONCRETE (IN COMPRESSIVE
ZONES) AND TENSILE STRESS IN STEEL WILL BOTH REACH THE
MAXIMUM PERMISSIBLE VALUES SIMULTANEOUSLY.
99

THE NEUTRAL AXIS OF BALANCED (OR CRITICAL) SECTION IS KNOWN AS
CRITICAL NEUTRAL AXIS (nc). THE AREA OF STEEL PROVIDED AS
ECONOMICAL AREA OF STEEL. REINFORCED CONCRETE SECTIONS ARE
DESIGNED AS BALANCED SECTIONS.
2. UNBALNCED SECTION:-THIS IS A SECTION IN WHICH THE QUANTITY
OF STEEL PROVIDED IS DIFFERENT FROM WHAT IS REQUIRED FOR THE
BALANCED SECTION.
UNBALANCED SECTIONS MAY BE OF THE FOLLOWING TWO TYPES:
(a) UNDER-REINFORCED SECTION
(b) OVER-REINFORCED SECTION
100

(a)UNDER-REINFORCED SECTION:- IF THE AREA OF STEEL PROVIDED IS
LESS THAN THAT REQUIRED FOR BALANCED SECTION, IT IS KNOWN AS
UNDER-REINFORCED SECTION. DUE TO LESS REINFORCEMENT THE
POSITION OF ACTUAL NEUTRAL AXIS (n) WILL SHIFT ABOVE THE
CRITICAL NEUTRAL AXIS (nc)i.e. n< nc. IN UNDER-REINFORCED SECTION
STEEL IS FULLY STRESSED AND CONCRETE IS UNDER STRESSED (i.e.
SOME CONCRETE REMAINS UN-UTILISED). STEEL BEING DUCTILE, TAKES
SOME TIME TO BREAK. THIS GIVES SUFFICIENT WARNING BEFORE THE
FINAL COLLAPSE OF THE STRUCTURE. FOR THIS REASON AND FROM
ECONOMY POINT OF VIEW THE UNDER-REINFORCED SECTIONS ARE
DESIGNED.
101

(b) OVER-REINFORCED SECTION:- IF THE AREA OF STEEL PROVIDED IS
MORE THAN THAT REQUIRED FOR A BALANCED SECTION, IT IS KNOWN
AS OVER-REINFORCED SECTION. AS THE AREA OF STEEL PROVIDED IS
MORE, THE POSITION OF N.A. WILL SHIFT TOWARDS STEEL,
THEREFORE ACTUAL AXIS (n) IS BELOW THE CRITICAL NEUTRAL AXIS
(nc)i.e. n > nc. IN THIS SECTION CONCRETE IS FULLY STRESSED AND
STEEL IS UNDER STRESSED. UNDER SUCH CONDITIONS, THE BEAM WILL
FAIL INITIALLY DUE TO OVER STRESS IN THE CONCRETE. CONCRETE
BEING BRITTLE, THIS HAPPENS SUDDENLY AND EXPLOSIVELY WITHOUT
ANY WARNING.
102

Basic rules for design of beam
1. Effective span:- In the case of simply supported beam the effective
length,
l = i. Distance between the centre of support
ii. Clear span + eff. Depth
eff. Span = least of i. & ii.
2. Effective depth:- The normal distance from the top edge of beam to
the centre of tensile reinforcement is called effective depth. It is
denoted by ‘d’.
d= D- effect. Cover
where D= over all depth
103

3. Bearing :- Bearings of beams on brick walls may be taken as follow:
Up to 3.5 m span, bearing = 200mm
Up to 5.5 m span, bearing =300mm
Up to 7.0 m span, bearing =400mm
4. Deflection control:- The vertical deflection limits assumed to be
satisfied if (a) For span up to 10m
Span / eff. Depth = 20
(For simply supported beam)
Span / eff. Depth = 7
(For cantilever beam)
104

(b) For span above 10m, the value in (a) should be multiplied
by 10/span (m), except for cantilever for which the deflection
calculations should be made.
(c) Depending upon the area and type of steel the value of
(a&b) modified as per modification factor.
5. Reinforcement :-
(a) Minimum reinforcement:- The minimum area of tensile
reinforcement shall not be less than that given by the
following:
Ast = 0.85 bd / fy
105

(b)Maximum reinforcement:- The maximum area of tensile
reinforcement shall not be more than 0.4bD
(c)Spacing of reinforcement bars:-
i. The horizontal distance between to parallel main bars shall not be less
than the greatest of the following:
➢ Diameter of the bar if the bars are of same diameter.
➢ Diameter of the larger bar if the diameter are unequal.
➢ 5mm more than the nominal maximum size of coarse
aggregate.
106

ii. When the bars are in vertical lines and the minimum vertical
distance between the bars shall be greater of the following:
➢ 15mm.
➢ 2/3rd of nominal maximum size of aggregate.
➢ Maximum diameter of the bar.
6. Nominal cover to reinforcement :- The Nominal cover is provided in
R.C.C. design:
➢ To protect the reinforcement against corrosion.
➢ To provide cover against fire.
➢ To develop the sufficient bond strength along the surface area of the
steel bar.
107

As per IS 456-2000, the value of nominal cover
to meet durability requirements as follow:-
Exposure conditions Nominal cover(mm)
Not less than
Mild
Moderate
Severe
Very severe
Extreme
20
30
45
50
75
108

Procedure for Design of Singly Reinforced Beam by
Working Stress Method
Given :
(i) Span of the beam (l)
(ii) Loads on the beam
(iii)Materials-Grade of Concrete and type of steel.
1. Calculate design constants for the given materials (k, j and R)
k = m σcbc / m σcbc + σst
where k is coefficient of depth of Neutral Axis
109

j = 1- k/3
where j is coefficient of lever arm.
R= 1/2 σcbc kj
where R is the resisting moment factor.
2. Assume dimension of beam:
d = Span/10 to Span/8
Effective cover = 40mm to 50mm
b = D/2 to 2/3D
3. Calculate the effective span (l) of the beam.
4. Calculate the self weight (dead load) of the beam.
Self weight = D x b x 25000 N/m
110

5. Calculate the total Load & maximum bending moment for
the beam.
Total load (w) = live load + dead load
Maximum bending moment, M = wl2 / 8 at the centre of beam
for simply supported beam.
M = wl2 / 2 at the support of beam for
cantilever beam.
6. Find the minimum effective depth
M = Mr
= Rbd2
dreqd. = √ M / R.b
111

7. Compare dreqd. With assumed depth value.
(i) If it is less than the assumed d, then assumption is correct.
(ii) If dreqd. is more than assumed d, then revise the depth value and
repeat steps 4, 5 & 6.
8. Calculate the area of steel required (Ast).
Ast = M / σst jd
Selecting the suitable diameter of bar calculate the number of bars
required
Area of one bar = π/4 x φ2 = Aφ
No. of bars required = Ast /Aφ
112

9. Calculate minimum area of steel (AS) required by the relation:
AS = 0.85 bd / fy
Calculate maximum area of steel by the area relation:
Maximum area of steel = 0.04bD
Check that the actual ASt provided is more than minimum and
less than maximum requirements.
113

10. Check for shear and design shear reinforcement.
11. Check for development length.
12. Check for depth of beam from deflection.
13. Write summary of design and draw a neat sketch.
114

 Doubly Reinforced beams
 T Beams and L Beams
Lecture Goals

Less concrete is needed to
resist the T and thereby
moving the neutral axis
(NA) up.
T
C
f
A
T
=
= y
s
Analysis of Doubly Reinforced Sections
116

Effect of Compression Reinforcement on the Strength
and Behavior
( )
1
2
2
s
c
1
c
and
2
;
C
Reinforced
Doubly
2
;
Reinforced
Singly
a
a
a
d
f
A
M
C
C
a
d
f
A
M
C
C
y
s
n
y
s
n







−
=

+
=







−
=
=

Analysis of Doubly Reinforced Sections
117

➢ Reduced sustained load deflections.
▪ Creep of concrete in compression zone
▪ transfer load to compression steel
▪ reduced stress in concrete
▪ less creep
▪ less sustained load deflection
Reasons for Providing Compression
Reinforcement

Reasons for Providing Compression Reinforcement
Effective of compression reinforcement on sustained
load deflections.
Reinforcement
119

 Increased Ductility
reduced stress
block depth
increase in steel strain
larger curvature are
obtained.
Reinforcement

Effect of compression reinforcement on strength and ductility
of under reinforced beams.
r < rb
Reinforcement
121

 Change failure mode from compression to tension. When
r > rbal addition of As strengthens.
 Eases in Fabrication - Use corner bars to hold &
anchor stirrups.
Effective reinforcement ratio = (r − r’)
Compression
zone
allows tension steel to
yield before crushing of
concrete.
Reinforcement

Compare the strain distribution in two beams with the
same As
Effect of Compression Reinforcement
123

Section 1: Section 2:
Addition of A’s strengthens compression zone so that less
concrete is needed to resist a given value of T. NA
goes up (c2 <c1) and es increases (es2 >es1).
1
c
s
s
1
1
1
c
c
1
c
s
s
85
.
0
85
.
0
85
.
0


b
f
f
A
c
c
b
f
ba
f
C
T
f
A
T

=

=

=
=
=
1
c
s
s
s
s
2
2
1
c
s
s
2
c
s
s
1
c
s
s
s
85
.
0
85
.
0
85
.
0


b
f
f
A
f
A
c
c
b
f
f
A
ba
f
f
A
C
C
T
f
A
T



−
=

+


=

+


=
+

=
=
124

Doubly Reinforced Beams
➢ Under reinforced Failure
➢ ( Case 1 ) Compression and tension steel yields
➢ ( Case 2 ) Only tension steel yields
➢ Over reinforced Failure
➢ ( Case 3 ) Only compression steel yields
➢ ( Case 4 ) No yielding Concrete crushes
Four Possible Modes of Failure

 Floor systems with slabs and beams are placed in monolithic
pour.
 Slab acts as a top flange to the beam; T-beams, and
Inverted L(Spandrel) Beams.
Analysis of Flanged Section

Analysis of Flanged Sections
Positive and Negative Moment Regions in a T-beam
127

If the neutral axis falls within
the slab depth analyze the
beam as a rectangular beam,
otherwise as a T-beam.
128

Effective Flange Width
Portions near the webs are more highly stressed than
areas away from the web.
129

Effective width (beff)
beff is width that is stressed uniformly to give the same
compression force actually developed in compression zone of
width b(actual)
130

eff
f w
actual
4
16
L
b
h b
b

 +

ACI Code Provisions for Estimating beff
131

From ACI 318, Section 8.10.3
Inverted L Shape Flange
( )
eff w
f w
actual w
12
6
0.5* clear distance to next web
L
b b
h b
b b
 +
 +
 = +
132

From ACI 318, Section 8.10
Isolated T-Beams
w
eff
w
f
4
2
b
b
b
h


133

Single Tee
Twin Tee
Box
Various Possible Geometries of T-Beams
134

Beam subjected to pure bending moment
Stresses developed in beam under pure bending moment
Bending Stresses in Beams
136

Deflection of Beam under Pure BM
137

Sectional View of Beam Neutral Surface/Axis
Deflected Shape of Beam
138

Bending and Shear Stresses
✓Equilibrium of Forces
• Vertical Shear Force
• Bending Moment
Shear Stress in Beams-General Loading
139

Horizontal Shear Stress
Shear stresses develop on horizontal planes

Beams with and without Planks
Deflection of Beam with Planks

P P
a a
V = +P
V = -
P
M = V
a
Basic Definition of Shear span, a
Definition of Shear Span

Diagonal
Crack
Diagonal Tension
V
V
V
V
V
V
V
V
Principal Stress Trajectories

Cracking in RC Beams
Direction of potential cracks in a simple beam.

Types of Cracks
• Type and formation of cracks depend on span-to-depth ratio of
beam and loading.
• For simply supported beam under uniformly distributed load, three
types of cracks are identified.
1.Flexural cracks: form at the bottom near mid span and propagate
upwards.
2.Web shear cracks: form near neutral axis close to support and
propagate inclined to the beam axis.
3.Flexure shear cracks: These cracks form at bottom due to flexure
and propagate due to both flexure and shear.

• Formation of cracks for a beam with large span-to-depth ratio and
uniformly distributed loading is shown.
a)Initiation of flexural cracks
b) Growth of flexural cracks and formation of flexure shear and web
shear cracks.

c) Cracks before failure
Fig. 7 Formation of cracks in a reinforced concrete beam

1. Components of shear resistance at a flexure shear crack are shown in the
following figure.
Fig. 8 Internal forces at a flexure shear crack
The notations in the previous figure are as follows.
Vcz = Shear carried by uncracked concrete
Va = Shear resistance due to aggregate interlock
Vd = Shear resistance due to dowel action
Vs = Shear carried by stirrups
• Magnitude and relative value of each component change with increasing load.
Components of Shear Resistance

• For beams with low span-to-depth ratio or inadequate shear
reinforcement, the failure may be due to shear.
• Failure due to shear is sudden as compared to failure due to flexure.
• Five modes of failure due to shear are identified.
1. Diagonal tension failure
2. Shear compression failure
3. Shear tension failure
4. Web crushing failure
5. Arch rib failure
• Mode of failure depends on span-to-depth ratio, loading, cross-section
of beam, amount and anchorage of reinforcement.
Modes of Failure

Diagonal tension failure: inclined crack propagates rapidly due to
inadequate shear reinforcement.
Fig. 9 Diagonal tension failure.
Shear compression failure: crushing of concrete near the
compression flange above the tip of the inclined crack.
Fig. 10 Shear compression failure

Shear tension failure: inadequate anchorage of longitudinal bars,
diagonal cracks propagate horizontally along the bars.
Fig. 11 Shear tension failure
Web crushing failure: concrete in the web crushes due to inadequate
web thickness.
Fig. 12 Web crushing failure

Arch rib failure: in deep beams, web may buckle and subsequently
crush. There can be anchorage failure or failure of the bearing.
Fig. 13 Arch rib failure
• Design for shear is to avoid shear failure; beam should fail in flexure
at its ultimate flexural strength.
• Design involves not only design of the stirrups, but also limiting
average shear stress in concrete, providing adequate thickness of
web and adequate development length of longitudinal bars.

 Behavior of RC members under Shear (including combined loads
with other loads) is very complex
1. Non-homogeneity of materials
2. Presence of Cracks and Reinforcement
3. Nonlinearity in Material Response
 Current design (Code) procedures,
I. Based on results of extensive tests on small size members with
simplifying assumptions
II. No unified and universally accepted method for prediction of
shear strength of beams without web reinforcement
Shear in Reinforced Concrete Members

vcz = Shear in compression zone (20-40%)
va = Aggregate Interlock forces (35-50%)
vd = Dowel action from longitudinal bars (15-25%)
Total Resistance = (vcz + vay +vd ) (For Beams without stirrups)
Shear Transfer Mechanisms
155

1. Strength of Concrete (f’c)
2. Percentage of Flexural (Tensile) Reinforcement (ρt)
3. Shear Span-to-Depth Ratio (a/d)
4. Depth of Member (d)
5. Size of Aggregate (da)
Factors Influencing Shear Strength

Qualitative variation of ultimate and Diagonal cracking strength with
Compressive Strength of Concreet
f'
c
Shear
Strength
Ultimate Strength
Diagonal Cracking Strength
Shear Strength with Compressive Strength of Concrete
157

% Tension Steel
%Tens. Steel
Shear
Strength
Ultimate Strength
Shear Strength with Tension Reinforcement

a/d ratio
a / d
Shear
Strength
Ultimate Strength
Shear Strength vs. Shear Span-to-Depth Ratio cal

Failure Mechanism of RC Elements at different a/d Ratio
160

Deep Beams (a/d < 1) Arch action
Tension tie (yielding/Fracture)
Compression Chord
Anchorage Failure
Bearing Failure
Arch Rib Failure
Failure Mechanisms

Short Beams (1 < a/d < 2.5)
Loss of Bond due to Crack
Shear Tension Failure
W
V
Crushing of Concrete
Shear Compression Failure
W
V

Slender Beams (2.5 < a/d < 6)
d
W
V
h g
a
b
c
j
e
a
C
Diagonal Tension Failure

➢ A phenomenon related to change in strength with member size,
decrease in strength with increase in member size
➢ Size effect in real structures:
i) material heterogeneity and
ii) discontinuities (flaws) to flow of stress either in the form of
micro cracking before application of any load load induced micro
cracking or macro cracking
Does beam depth effect the strength?

▪ Size effect in shear is more serious due to
• failure mode highly brittle - small deflections and lack of ductility
• shear strength related to tensile strength of concrete
▪ Evidence
▪ Sudden failure of Wilkins Air force depot warehouse in Shelby,
Ohio (1955)
▪ Catastrophic failure of structures leading to loss of human lives
and property due to Hyogo-Ken Nambu Earth Quake in 1995

Wilkins Air Force Depot in Shelby, Ohio (1955)

Collapse of Superstructure - Hyogo-Ken Nambu EQ (1995)

 Ultimate Strength: “load corresponding to the total and complete
failure due to shear and diagonal tension”
 Diagonal Cracking Strength: “load corresponding to formation of
first fully developed inclined crack”
a. An inclined crack is considered to be fully developed when it has
progressed sufficiently towards both the mid span and the
support while intersecting the tensile reinforcement
Diagonal Cracking and Ultimate Strength

Diagonal Cracking and Ultimate Strength
 How is it measured in the laboratory?
1. Normally, visual methods are deployed to measure the diagonal
cracking loads in the laboratory
2. load at the onset of formation of first diagonal crack
3. when the diagonal crack crosses the mid height of the beam

 The design provisions in most of the codes
1) are based on diagonal tension cracking and
2) by using a suitable multiplication factor, strength of short or
deep beams is obtained
3) Need to be re-examined
 Owing to complex nature of stress distribution, Code Provisions
for prediction of shear strength are Empirical in Nature for Beams
without Web Reinforcement

ACI - 2002
(a)
6
MPa
f
0.17
v c
'
c →
=
1999)
-
(ACI
f
0.5
(7)
MPa
M
d
V
17.2
f
0.16
d
V
M
2.5
-
3.5
v
c
'
u
u
c
'
u
u
c

→








+








= r
2.5
d
V
M
2.5
3.5 -
where
u
u









1.0
M
d
V
and
u
u

(6)
MPa
f
0.3
M
d
V
17.2
f
v c
'
u
u
c
c →









+
= r
'
16
.
0
Simplified
Prediction of Shear Strength
171

BS 8110 - 1997
(8)
MPa
f
d
d
b
A
0.79
v cu
v
s
m
c →




















=
3
/
1
4
/
1
3
/
1
25
400
100

2.0
d
a
for
d
a
2
(Eqn.8)
vc 










=
MPa
40.0
f
and
1.25
1.0,
d
400
3.0,
d
b
A
Where cu
m
v
s

=

 
100
For Short Beams

IS 456-2000
ck
ck
cr
f
0.62
(9)
MPa
6
)
1
5
1
(
(
0.8f
0.85
v

→
−
+
=


1.0
6.89
0.8f
nd
a ck

=
r







=
a/d
2
(Eqn.9)
vcr
Where 0.8 fck = Cylinder strength in terms of cube strength and
0.85 reduction factor = 1/γm ~ 1/1.2
For Short Beams

 Nominal Shear stress
Where
 vu = shear force due to design loads
 b = breadth of the member which for flanged sections shall be
taken as the breadth of web bw and
 d = effective depth
d
b
Vu
v =

Design of Beams for Shear

 Calculate the nominal shear stress and compare with the shear
strength of RC beams from Table 19 of IS 456-2000.
 If the nominal shear stress exceeds the shear strength of RC
beams without shear reinforcement, then the beam needs to be
designed for shear reinforcement.
 When is less than obtained from Table 19, minimum shear
reinforcement is provided which is given by
v
 c

v
 c

b
f
A
s
f
bs
A
y
sv
v
y
v
sv
4
.
0
4
.
0


Shear Strength of RC beams

IS 456-2000
Where 0.8 fck = Cylinder strength in terms of cube strength and
0.85 reduction factor = 1/γm ~ 1/1.2
For Short Beams

I. When the shear stress is greater than shear strength given in
Table 19 (IS 456), shear reinforcement shall be provided in any of
the following forms
a. Vertical stirrups
b. Bent-up bars along with stirrups and
c. Inclined stirrups
 Shear reinforcement shall be provided to carry a shear force equal to
 
bd
V c

−
Design of Shear Reinforcement

Forms of Shear Reinforcement
 For vertical stirrups
 For single bar or single group of parallel bars all bent up at the
same cross-section
 = angle between the inclined stirrup or bent up bar and the axis
of the member not less than 450
d
s
f
A
v
v
sv
sv
s
.
=

 sin
.
. sv
sv
s A
V =


1. Action of vertical and inclined (stirrups) shear reinforcement may
be described by the analogous truss action.
2. In a simple truss, the upper and lower chords are in compression
and tension respectively; the diagonal members, called web
members, are alternately in compression and tension.
3. Shear strength of RC beam may be increased by use of shear
reinforcement similar in action to tensile web members in a truss.
4. Shear reinforcement must be anchored in compression zone of
concrete and is usually hooped around longitudinal tension
reinforcement.
Truss Analogy

• Asv = total area of legs of shear links
• sv = spacing of links
• Number of links crossing 450 diagonal crack
• Total strength of vertical stirrups
• Spacing of stirrups required=
v
s
d
N =
v
SV
y
s
s
d
A
f
V 87
.
0
=
( )
( )
c
v
sv
y
v
c
v
v
sv
y
b
A
f
s
bd
s
d
A
f




−
=

−
=
87
.
0
87
.
0
Design of Stirrups

• Horizontal length over which the bar is effective can be taken as equal to d(cot β +
cot α), where β = direction of shear compression, α = angle at which the bars bent
• Let sv = spacing of bent bars. Then the number of effective bars in this region are
• The maximum shear carried by bent up bars =
( )( )
v
s
d
d
N
'
cot
cot −
+
=


( ) ( )( )
( )( )( ) ( )
d
d
d
s
d
f
A
s
d
d
f
A
V
v
y
sv
v
y
sv
s

−
=

+
=
−
+
=
)
'
(
;
45
;
sin
cos
87
.
0
cot
cot
sin
87
.
0
0
'






Design of Bent-up Bars

1. Restrains the growth of inclined cracking.
2. Ductility is increased and gives warning before failure.
3. In an unreinforced web, such reinforcement is of great value if a
member is subjected to an unexpected tensile force or an overload.
4. A minimum area of shear reinforcement is required whenever the
total factored shear force Vu is greater than one-half the shear
strength provided by concrete kVc.
5. Need to increase minimum shear reinforcement as concrete
strength increases to prevent sudden shear failure by inclined
cracking.
Minimum Shear Reinforcement

A slab is structural element whose thickness is small compared to its own
length and width. Slabs are usually used in floor and roof coverings.
Length of the longer span(l
One-way slabs:
When the ratio of the longer to the shorter side (L/ S) of the slab is at least
equal to 2.0, it is called one-way slab. Under the action of loads, it is
deflected in the short direction only, in a cylindrical form. Therefore, main
reinforcement is placed in the shorter direction, while the longer direction
is provided with shrinkage reinforcement to limit cracking. When the slab
is supported on two sides only, the load will be transferred to these sides
regardless of its longer span to shorter span ratio, and it will be classified
as one-way slab.
Introduction
187

One way slab; (a) classification; (b) reinforcement
Two-way Slabs:
When the ratio (L/ S) is less than 2.0, it is called two-way slab. Bending will
take place in the two directions in a dish-like form.
Accordingly, main reinforcement is required in the two directions.
189

In this section, two types will be discussed, one-way solid
slabs and one-way ribbed slabs.
One-way Solid Slabs
Minimum Thickness
To control deflection, ACI Code 9.5.2.1 specifies minimum
thickness values for one-way solid slabs, shown in Table.
Minimum thickness of one-way solid slabs
where l is the span length in the direction of bending.
One-way Slabs
191

One-way solid slabs are designed as a number of independent 1 m wide
strips which span in the short direction and supported on crossing beams.
Maximum Reinforcement Ratio
One-way solid slabs are designed as rectangular sections subjected
to shear and moment. Thus, the maximum reinforcement ratio ρmax
is not to exceed
sb
s
b A
A 75
.
0
and
75
.
0 max =
r
Shrinkage Reinforcement Ratio
According to ACI Code 7.12.2.1 and for steels yielding at
, the shrinkage reinforcement is taken not less than 0.0018 of
the gross concrete area, or
where, b = width of strip, and h = slab thickness.
2
/
4200 cm
kg
fy =
h
b
A shrinkage
s 0018
.
0
=
Design Concept
193

Minimum Reinforcement Ratio
According to ACI Code 10.5.4, the minimum flexural reinforcement is not
to be less than the shrinkage reinforcement, or
h
b
A m
s 0018
.
0
in =
Spacing Of Flexural Reinforcement Bars
Flexural reinforcement is to be spaced not farther than three times the
slab thickness, nor farther apart than 45 cm, center-to-center.
Spacing Of Shrinkage Reinforcement Bars
Shrinkage reinforcement is to be spaced not farther than five
times the slab thickness, nor farther apart than 45 cm, center-to-
center.
194

(1) Own weight of slab:
(2) Weight of slab covering materials:
- Sand fill with a thickness of about 5 cm,
-Cement mortar, 2.5 cm thick.
2
80
1
05
0 t/m
.
. 
2
10
2
025
0 t/m
.
. 
- Tiling
-A layer of plaster about 2 cm in thickness.
2
30
2
025
0 t/m
.
. 
2
10
2
02
0 t/m
.
. 
(3) Live Load:
Table shows typical values used by the Uniform Building Code (UBC).
Loads Assigned to Slabs
195

Minimum live Load
values on slabs
196

(4) Equivalent Partition Weight:
This load is usually taken as the weight of all walls carried by
the slab divided by the floor area and treated as a dead load
rather than a live load.
Loads Assigned to Beams
The beams are usually designed to carry the following loads:
- Their own weights.
- Weights of partitions applied directly on them.
- Floor loads.
The floor loads on beams supporting the slab in the shorter
direction may be assumed uniformly distributed throughout
their spans.
197

Approximate Structural Analysis
ACI Code 8.3.3 permits the use of the following approximate
moments and shears for design of continuous beams and
one-way slabs, provided:
1. Positive Moment:
a. End Spans:
When discontinuous end unrestrained,
When discontinuous end is integral with support,
where ln is the corresponding clear span length
b. Interior Spans:
11
2
/
l
w
M n
u
u =
14
2
/
l
w
M n
u
u =
16
2
/
l
w
M n
u
u =
2. Negative Moment:
a. Negative moment at exterior face of first interior support:
􀂃 Two spans,
9
/
2
n
u
u l
w
M =
198

More than two spans,
where ln is the average of adjacent clear span lengths.
b. Negative moment at other faces of interior supports:
c. Negative moment at interior face of exterior support:
Support is edge beam,
Support is a column,
3. Shear:
a. Shear in end members at face of first interior support:
b. Shear at face of all other supports:
where ln is the corresponding clear span length.
10
2
/
l
w
M n
u
u =
11
2
/
l
w
M n
u
u =
24
2
/
l
w
M n
u
u =
16
2
/
l
w
M n
u
u =
2
/
n
u
u l
w
V =
2
/
15
,
1 n
u
u l
w
V =
199

(a) Two spans, exterior edge unrestrained; (b) two spans,
support is spandrel beam; (c) more than two spans, exterior edge
unrestrained; (d) more than two spans, support is spandrel beam; (e)
two spans, shearing force diagram
200

Summary of One-way Solid Slab Design Procedure
Once design compressive strength of concrete and yield stress
of
reinforcement are specified, the next steps are followed:
1. Select representative 1 m wide design strip/strips to span in
the short direction.
2. Choose a slab thickness to satisfy deflection control
requirements.
When several numbers of slab panels exist, select the largest
calculated thickness.
3. Calculate the factored load Wu by magnifying service dead
and live loads according to this equation
.
l
d
u w
w
w 70
.
1
40
.
1 +
=
201

(a) Representative strip and reinforcement; (b) strip and loads
202

4. Draw the shear force and bending moment diagrams for each of the
strips.
5. Check adequacy of slab thickness in terms of resisting shear by
satisfying the following equation:
bd
f
V c
u
'
53
.
0 

203

6. Design flexural and shrinkage reinforcement:
Flexural reinforcement ratio is calculated from the following
equation:
Make sure that the reinforcement ratio is not larger than ¾ ρb
Compute the area of shrinkage reinforcement, where
Select appropriate bar numbers and diameters for both, main
and secondary reinforcement.
Check reinforcement spacing, modify your bar selection if
needed.
7. Draw a plan of the slab and representative cross sections
showing the dimensions and the selected reinforcement.
h
b
A m
s 0018
.
0
in =
204

Analysis by Coefficients
Restrained slabs are defined in codes as those slabs which are cast
integral with RC frames and which are not free to lift up at the
corners.
These slabs may be continuous or discontinuous at the edges.
Those which are discontinuous at edges are also referred to be
simply supported.
Coefficients specified in IS:456-2000, Table 26 in Annexure D can
be used for analysis of such slabs.
Moments in Two way Restrained Slabs with
Corners Held Down

Moments in Two way Restrained Slabs with Corners Held Down
Analysis by Coefficients
Conditions to be satisfied for use of these coefficients are:
1. Loading on adjacent spans should be the same
2. Span in each direction should be approximately equal.
The span moment per unit width and the negative
moments at continuous edges for the slabs are calculated
from the equations in terms of “lx”,
for short span
for longer span
2
x
x
x wl
M 
=
2
x
y
y wl
M 
=
Moments in Two way Restrained Slabs with
Corners Held Down

Restrained two way slab is divided into middle strip and
edge strips. Middle strip is forming three-fourth of slab
width in width directions.
Torsion steel must be provided at discontinuous edges as
specified in code.
Coefficients are given in Table 26 of IS: 456-2000. These
coefficients are derived from the yield line theory of
slabs.
Design of Two-way Slabs

 Table given for coefficients is applicable for slabs carrying uniformly
distributed loads not for concentrated loads, for which analysis
should be done, separately.
 Span moments/edge moments per unit width are calculated by
determining ratio of “ly” and “lx” and for different edge conditions.c

 Slabs restrained against corners lifted up
 nd = Number of discontinuous edges = 0,1,2,3 and 4.
 C = 1.0 for a discontinuous edge
= for continuous edge
 Subscripts “s” and “l” denote “short edge” and “long edge”
 Subscripts “1” and “2” represent two edges in short and longer
direction
 
1000
5
.
1
2
24
2
d
d
y
n
n +
+
=
+
( )
( ) 







+
+

−
=

+
+
2
2
1
2
1 1
18
3
9
2
l
l
s
s
y
x
C
C
r
C
C
3
7
Coefficients for Moments

1. Edge moments of continuous supports are 1.33 times the span
moments.
2. Long span moment coefficient”αy” is a constant for given end
conditions of slab, irrespective of the span ratios.
3. Short span coefficient varies sharply with variation of the ratio of
spans
Important Design Issues from Table 26

➢ While using Table 26 for a series of slabs, moments calculated at
an interior support will sometimes be different on two sides of
that support because of the differences in continuity condition of
slabs on opposite sides of support.

➢ While using design of two-way slabs with the help of coefficients,
restrained slabs are considered to be divided into middle and edge
strips.
➢ Moments given in Table 26 apply only to middle strips, and no
further redistribution is allowed for these moments.
➢ Edge strips have to be reinforced only with nominal minimum steel
for crack control.
Arrangement of Reinforcements

➢ Middle strip should have steel (+ve and –ve ) calculated for various
sections. In edge strips , steel is placed as positive steel at the
bottom of slab.
➢ Negative moments may be experienced at discontinuous edges
since, in practice, they are not supported on rollers but partially
restrained at their ends.
➢ The magnitude of this moment depends on the degree of fixity at
the edge of the slab and is intermediate.

Arrangement of Reinforcement
➢ Usual practice is to provide at these edges top reinforcement for
negative moment equal to
➢ As per IS:456.2000, 50 percent of the steel provided at mid span
should be extended along these edges, and the negative steel has
to extend into the span 0.1 times the span length from the face of
the beam.
24
2
wl

➢ Slab thickness should be calculated based on the greater value of
the negative B.M on the short span.
➢ Hence
➢ Total thickness = d (short) + 0.5ø + cover
➢ Total thickness = d (long) + 0.5ø + ø + cover
2
bd
Kf
M ck
u =

➢ The slab should satisfy span/effective depth ratio to control
deflection.
➢ Simply supported = 28
➢ Continuous = 32
➢ Depth of slab selected from deflection criterion will be generally
greater than the minimum required from strength criterion.
➢ Short span steel will be placed in the lower layer.

➢ Restrained moments are obtained for the middle strips only. The
reinforcement is distributed uniformly in the middle strips.
➢ Each direction is to be provided only with the minimum
reinforcement placed at the bottom of the slabs.

➢ In addition, corner steel reinforcement should be provided at the
discontinuous edges.
➢ Corner reinforcement consists of two mats, are placed on the top
and the other at the bottom of the slab, with each mat having
steel in both x and y directions.
➢ Where the slab is discontinuous on both sides of a corner, fall
torsion steel has to be provided.

 The area of the full torsion reinforcement per unit width in each of
the four layer should be as follows.
 (Area of full corner steel ) = [3/4][Area required for the maximum
span per unit width in each of four layers]
 These steels are to be provided for a distance of one-fifth the short
span .

The failure of members in compression are
due either to the load exceeding the ultimate
strength in compression (crushing) or due to
buckling under the load, because the applied
load is larger than the critical buckling load.
Long members are referred to as columns. Columns, normally fail in
buckling.
Compression Members
223

Compression Members
Buckling
failure
Crushing failure –
1985 Mexico
earthquake.
Compression Members
224

Pinned-Pinned, or
rounded-rounded
Euler’s
end
condition
Fixed-sliding Fixed-Free Fixed-pinned Fixed-Fixed
American Institute
of Steel
Construction
Pcr =
2
E I
(leff)2
Euler Column – End Conditions and
Effective Length
225

Biaxial bending in columns
227

Biaxial bending in columns
Design Procedure for biaxial bending:
A- Determine reinforcement based on the biaxial bending capacity:
1- Determine the dimensions based on a reasonable stress in the column.
2- Determine g in the week axis direction (any direction for square sections.
3- Calculate the biaxial bending moment in the week axis direction.
4- Use an interaction diagram to design the reinforcement for the section.
B- Use the Bresler equation to check the axial capacity of the section
1- Calculate Pnx from the interaction diagram assuming only Mnx is applied.
2- Calculate Pny from the interaction diagram assuming only Mny is applied.
3- Calculate P0.
4- Calculate Pn and check
u
n P
P 

228

Section near the supports can be assumed to have enhanced
shear strength.
Shear failure at sections of beams and cantilevers without shear
reinforcement normally takes place on a plane making an angle
300 with the horizontal.
Enhance shear capacity at sections near supports as would be
done in design of brackets, ribs, corbels etc.
Enhanced Shear Near Supports

Critical Sections for shear in beams which are supported on
members in compression, and tension
30
av
d
(a) Beams with compression at end region
(b) Face of support in compression
(c ) Face of support in tension

Beams of Varying Depth
 Beams with varying depth are encountered in RC.
 Beam depth is varied according to the variation of bending
moment and shear force.
1. Case (a): Bending moment increases numerically in the direction
in which effective depth increases.
2. Case (b). Bending moment decreases numerically in the
direction in which effective depth increases.

S.S. Beam
Cantilever
H
C
V
Case (a)
ß
M = H Jd
V
H
C
ß
Case (b)

Effective shear force for determining the shear stress
tan for case(a)
tan for case(b)
w
w
M
V V
d
M
V V
d


= −
= +

 When flexural reinforcement in beams is terminated in tension
region, at that section it should satisfy
a) shear at cut-off point does not exceed two-thirds of
combined strength of concrete and steel.
b) Additional stirrups should be provided along each terminated
direction over a distance from the cut-off point equal to
three-fourth effective depth, equal to
 Spacing of stirrups
y
su
f
s
b
A
4
.
0
'
=
 
c
s 

 −
 5
.
1

8
d

bars
of
area
Total
bars
off
cut
of
Area −
=

Design of Stirrups at Steel Cut-off Points
234

 Depth much greater than normal, in relation to their span, while
thickness is much smaller than either span or depth.
 Main loads and reactions act in plane of member to achieve a
state of plane stress in concrete
 Members with span-to-depth ratio of about 5 or less, or with
shear span, a, less than about twice depth are called deep
beams.
Deep Beams

l
h
a a
Fig. Placements of loads on deep beams.
(a). loads applied along the compression edge
(b). loads suspended along the tension edge
(c). loads distributed through depth

Deep Beams
➢ Examples of Deep Beams found in:
➢ Column offsets,
➢ Walls of rectangular tanks and bins,
➢ Floor diaphragms
➢ Shear walls,
➢ in folded plate roof structures
➢ Behavior of deep beams is significantly different from that of the
normal beams, requires special consideration in analysis, design,
and detailing of reinforcement.

Deep Beams (a/d < 1)
Arch action
Tension tie (yielding/Fracture)
Compression
Chord
Anchorage Failure
Bearing Failure
Arch Rib Failure

Lecture Goals
 Footing Classification
 Footing Design

Definition
Footings are structural members used to support columns and
walls and to transmit and distribute their loads to the soil in
such a way that the load bearing capacity of the soil is not
exceeded, excessive settlement, differential settlement,or
rotation are prevented and adequate safety against
overturning or sliding is maintained.
Footing
244

Types of Footing
Wall footings are used to
support structural walls that
carry loads for other floors or to
support nonstructural walls.
245

Isolated or single footings are
used to support single columns.
This is one of the most
economical types of footings and
is used when columns are spaced
at relatively long distances.
246

Types of Footing
Combined footings usually support
two columns, or three columns not
in a row. Combined footings are
used when two columns are so close
that single footings cannot be used
or when one column is located at or
near a property line.
247

Types of Footing
Cantilever or strap footings consist
of two single footings connected
with a beam or a strap and support
two single columns. This type
replaces a combined footing and is
more economical.
248

Types of Footing
Continuous footings support a
row of three or more columns.
They have limited width and
continue under all columns.
249

Types of Footing
Rafted or mat foundation consists of
one footing usually placed under the
entire building area. They are used,
when soil bearing capacity is low,
column loads are heavy single footings
cannot be used, piles are not used and
differential settlement must be
reduced.
250

Types of Footing
Pile caps are thick slabs used to
tie a group of piles together to
support and transmit column
loads to the piles.
251

When the column load P is applied on
the centroid of the footing, a uniform
pressure is assumed to develop on
the soil surface below the footing
area.
However the actual distribution of the soil is not uniform, but
depends on may factors especially the composition of the soil and
degree of flexibility of the footing.
Distribution of Soil Pressure
252

Soil pressure distribution in cohesionless
soil.
Soil pressure distribution in cohesive soil.
253

Footings must be designed to carry the column loads and
transmit them to the soil safely while satisfying code limitations.
The area of the footing based on the allowable bearing soil
capacity
Two-way shear or punch out shear.
One-way bearing
Bending moment and steel reinforcement required
1.
2.
3.
Design Considerations
254

Design Considerations
Footings must be designed to carry the column loads and
transmit them to the soil safely while satisfying code limitations.
Bearing capacity of columns at their base
Dowel requirements
Development length of bars
Differential settlement
1.
2.
3.
4.
255

The area of footing can be determined from the actual
external loads such that the allowable soil pressure is not
exceeded.
( )
pressure
soil
allowable
weight
-
self
including
load
Total
footing
of
Area =
footing
of
area
u
u
P
q =
Strength design requirements
Size of Footing
256

For two-way shear in slabs (& footings) Vc is smallest of
long side/short side of column concentrated
load or reaction area < 2
length of critical perimeter around the
column
where, c =
b0 =
ACI 11-35
d
b
f
V 0
c
c
c
4
2








+
=

When  > 2 the allowable Vc is reduced.
Two-Way Shear (Punching Shear)
257

Design of two-way shear
Assume d.
Determine b0.
b0 = 4(c+d)
b0 = 2(c1+d) +2(c2+d)
1.
2.
for square columns where
one side = c
for rectangular columns of
sides c1 and c2.
258

The shear force Vu acts at a section
that has a length b0 = 4(c+d) or
2(c1+d) +2(c2+d) and a depth d; the
section is subjected to a vertical
downward load Pu and vertical upward
pressure qu.
3.
259

Allowable
Let Vu=Vc
4. d
b
f
V 0
c
c 4
 =
0
c
u
4 b
f
V
d

=
If d is not close to the assumed d,
revise your assumptions
260

Design of one-way shear
For footings with bending action in
one direction the critical section is
located a distance d from face of
column
d
b
f
V 0
c
c 2
 =
261

The ultimate shearing force at section m-
m can be calculated








−
−
= d
c
L
b
q
V
2
2
u
u
If no shear reinforcement is to be
used, then d can be checked
262

b
f
V
d
2 c
u

=
If no shear reinforcement is to be
used, then d can be checked,
assuming Vu = fVc
263

Flexural Strength and Footing reinforcement
2
y
u
s








−
=
a
d
f
M
A

The bending moment in each direction
of the footing must be checked and the
appropriate reinforcement must be
provided.
264

b
f
A
f
a
85
.
0 c
s
y
=
Another approach is to calculated
Ru = Mu / bd2 and determine the
steel percentage required r .
Determine As then check if
assumed a is close to calculated a
265

The minimum steel percentage required in
flexural members is 200/fy with minimum area
and maximum spacing of steel bars in the
direction of bending shall be as required for
shrinkage temperature reinforcement.
266

The reinforcement in one-way footings and two-way
footings must be distributed across the entire width of the
footing.
1
2
direction
short
in
ent
reinforcem
Total
width
band
in
ent
Reinforcem
+
=

footing
of
side
short
footing
of
side
long
=

where
267

Bearing Capacity of Column at Base
The loads from the column act on the footing at the base of
the column, on an area equal to area of the column cross-
section. Compressive forces are transferred to the footing
directly by bearing on the concrete. Tensile forces must be
resisted by reinforcement, neglecting any contribution by
concrete.
268

Force acting on the concrete at the base of the column must not
exceed the bearing strength of the concrete
( )
1
c
1 85
.
0 A
f
N 
=
where f = 0.7 and
A1 =bearing area of column
269

The value of the bearing strength may be multiplied by a factor for bearing on footing
when the supporting surface is wider on all sides than the loaded area.
0
.
2
/ 1
2 
A
A
The modified bearing strength
( )
( )
1
c
2
1
2
1
c
2
85
.
0
2
/
85
.
0
A
f
N
A
A
A
f
N




270

Dowels in Footings
A minimum steel ratio r = 0.005 of the column section as
compared to r = 0.01 as minimum reinforcement for the column
itself. The number of dowel bars needed is four these may be
placed at the four corners of the column. The dowel bars are
usually extended into the footing, bent at the ends, and tied to
the main footing reinforcement. The dowel diameter shall not
exceed the diameter of the longitudinal bars in the column by
more than 0.15 in.
271

The development length for compression bars was given
but not less than Dowel bars must be checked for proper
development length.
c
b
y
d /
02
.
0 f
d
f
l =
in.
8
003
.
0 b
y 
d
f
Development length of the Reinforcing Bars
272

Footing usually support the following loads:
Dead loads from the substructure and superstructure
Live load resulting from material or occupancy
Weight of material used in back filling
Wind loads
1.
2.
3.
4.
Differential Settlement
273

General Requirements for Footing Design
A site investigation is required to determine the
chemical and physical properties of the soil.
Determine the magnitude and distribution of loads form
the superstructure.
Establish the criteria and the tolerance for the total and
differential settlements of the structure.
1.
2.
3.
274

Determine the most suitable and economic type of
foundation.
Determine the depth of the footings below the ground
level and the method of excavation.
Establish the allowable bearing pressure to be used in
design.
4.
5.
275

Determine the pressure distribution beneath the footing
based on its width
Perform a settlement analysis.
7.
8.
276

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