Module

Shear, Bond,

Anchorage,
Development Length
and

Torsion
Version 2 CE IIT, Kharagpur

Lesson

Limit
State
of

Instructional Objectives:
At the end of this lesson, the student should be able to:

name and explain the three different failure modes of reinforced concrete
beams under the combined effects of bending moment and shear force,
define nominal shear stress TVof rectangular and T-beams of uniform and
varying depths under the combined effects of bending moment and shear
force,

name the two parameters on which the design shear strength of concrete
depends,

-

o
-

nd out the maximum shear stress of concrete
reinforcement,
locate the critical sections for shear in beams,

-

Tcmaxwith

shear

explain when and why do we consider enhanced shear strength of concrete,
explain why the minimum shear reinforcement is provided in any beam,
determine the amount of minimum shear reinforcement to be provided in
any beam,
specify the three different ways of providing shear reinforcement in a beam,
design the shear reinforcement in a beam for each of the three methods
mentioned

0
0

beams

above,

design the shear reinforcement closed to the support of a beam,
specify the conditions to be satisfied for the curtailment of tension
reinforcement when designing shear reinforcement,
place the vertical stirrups in a beam.

6.13.1

Introduction

This lesson explains the three failure modes due to shear force in beams
and defines different shear stresses needed to design the beams for shear. The
critical sections for shear and the minimum shear reinforcement to be provided in

beams are mentioned as per IS 456. The design of shear reinforcement has
been illustrated in Lesson 14 through several numerical problems including the
curtailment

of tension

reinforcement

in exural

members.

Version 2 CE IIT, Kharagpur

6.13.2

Failure

Modes

due to Shear

iF*ig..it{e;fj.:Ft:ex.ul:e3
tI§T§§~lES3§E:T§
{concrete crushes in epmereeséenfj
Failtjre

rnees

Bending in reinforced concrete beams is usually accompanied by shear,
the exact analysis of which is very complex. However, experimental studies
conrmed the following three different modes of failure due to possible
combinations of shear force and bending moment at a given section (Figs.
6.13.1a to c):
(i) Web shear (Fig. 6.13.1a)
(ii) Flexural tension shear (Fig. 6.13.1b)
(iii) Flexural compression shear (Fig. 6.13.1c)
Web shear causes cracks which progress along the dotted line shown in

Fig. 6.13.1a. Steel yields in flexural tension shear as shown in Fig. 6.13.1b, while
concrete crushes in compression due to flexural compression shear as shown in
Fig. 6.13.1c. An in-depth presentation of the three types of failure modes is
beyond the scope here. Only the salient points needed for the routine design of
beams in shear are presented here.

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6.13.3

Shear

Stress

The distribution of shear stress in reinforced concrete rectangular, T and
L-beams of uniform and varying depths depends on the distribution of the normal
stress. However, for the sake of simplicity the nominal shear stress n, is

considered which is calculated as follows (IS 456, cls. 40.1 and 40.1.1):

fig
E
5

Hate:

Ey}.I&EEE§l
§E~':i:§fE'§lE.lil.tl1i¬.V1"E'§
{iii}tverage isrtrisbutian
{as}e.«r:ta:ngu;Iar beam

Maiatne:

ijgiia..u:;:§uaél
dE$§r§m.:tr.imm:
igiiAverage ietrihmazéen

Fig.

3.3: Eliiffgbulélf shear 5tr'@s«s
and
aviarage shears stress

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(i) In beams of uniform depth (Figs. 6.13.2a and b):
2'
=

Vu
bd

(6.1)
where V,, = shear force due to design loads,
b = breadth of rectangular beams and breadth of the web b.,., for flanged
beams, and

d

= effective depth.

(ii) In beams of varying depth:
M

V i d tan/2
TV =
bd

(6.2)

where TV,V, b or b.,.,and d are the same as in (i),
Mu = bending moment at the section, and

B

= angle between the top and the bottom edges.

The positive sign is applicable when the bending moment Mu decreases
numerically in the same direction as the effective depth increases, and the
negative sign is applicable when the bending moment Mu increases numerically
in the same direction as the effective depth increases.

6.13.4 Design Shear Strength of Reinforced Concrete
Recent laboratory experiments confirmed that reinforced concrete in
beams has shear strength even without any shear reinforcement. This shear
strength (Tc) depends on the grade of concrete and the percentage of tension
steel in beams. On the other hand, the shear strength of reinforced concrete with
the reinforcement is restricted to some maximum value rcmaxdepending on the
grade of concrete. These minimum and maximum shear strengths of reinforced

concrete (IS 456, cls. 40.2.1 and 40.2.3, respectively) are given below:
6.13.4.1

Design shear strength without shear reinforcement

(ES 456, cl.

40.2.1)

Version 2 CE IIT, Kharagpur

Table 19 of IS 456 stipulatesthe design shear strengthof concrete Tofor
different grades of concrete with a wide range of percentages of positive tensile
steel reinforcement. It is worth mentioning that the reinforced concrete beams
must be provided with the minimum shear reinforcement as per cl. 40.3 even
when 1.,is less than Tcgiven in Table 6.1.

Table6.1 Designshearstrength
of concrete,
1,,in N/mmz
Grade

(100 As/b

M 25

of concrete

M 30

M 35

d)
S 0.15

0.29

0.29

0.29

0.25

0.36

0.37

0.37

0.50

0.49

0.50

0.50

0.75

0.57

0.59

0.59

1.00

0.64

0.66

0.67

1.25

0.70

0.71

0.73

1.50

0.74

0.76

0.78

1.75

0.78

0.80

0.82

2.00

0.82

0.84

0.86

In Table 6.1, As is the area of longitudinal tension reinforcement which
continues at least one effective depth beyond the section considered except at
support where the full area of tension reinforcement may be used provided the

detailingis as per IS 456, cls. 26.2.2 and 26.2.3.
6.13.4.2

Maximum shear stress rcma,with shear reinforcement (cls. 40.2.3,
40.5.1 and 41.3.1)

Table 20 of IS 456 stipulates the maximum shear stress of reinforced

concrete in beams rcmaxas given below in Table 6.2. Under no circumstances,
the nominal shear stress in beams r., shall exceed rcmax given in Table 6.2 for
different grades of concrete.

Table6.2 Maximumshearstress,rcmax
in N/mmz
Grade

of

M 20

M 25

M 40 and

concrete

Tcmax,

above

2.8

3.1

3.5

3.7

4.0

N/mm2

Version 2 CE IIT, Kharagpur

6.13.5

Critical

Section

for Shear

37.

£51331

Fig.

itlpport conditions.for rilzrmastisnsg
fa.ot;ore;:
.she.arFUSE

Clauses 22.6.2 and 22.6.2.1 stipulate the critical section for shear and are
as follows:

For beams generally subjected to uniformly distributed loads or where the
principal load is located further than 2d from the face of the support, where d is
the effective depth of the beam, the critical sections depend on the conditions of
supports as shown in Figs. 6.13.3 a, b and c and are mentioned below.
(i) When the reaction in the direction of the applied shear introduces tension
(Fig. 6.13.3a) into the end region of the member, the shear force is to be
computed at the face of the support of the member at that section.
(ii)
When the reaction in the direction of the applied shear introduces
compression into the end region of the member (Figs. 6.13.3b and c), the shear
force computed at a distance d from the face of the support is to be used for the
design of sections located at a distance less than d from the face of the support.
The enhanced shear strength of sections close to supports, however, may be
considered as discussed in the following section.

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6.13.6
Enhanced Shear Strength of Sections Close to
Supports (cl. 40.5 of IS 456)

6.13.4: Sheer fsiiurvenear stuppert
Figure 6.13.4 shows the shear failure of simply supported and cantilever
beams without shear reinforcement. The failure plane is normally inclined at an

angle of 30° to the horizontal. However, in some situations the angle of failure is
more steep either due to the location of the failure section closed to a support or
for some other reasons. Under these situations, the shear force required to
produce failure is increased.

Such enhancement of shear strength near a support is taken into account
by increasing the design shear strength of concrete to (2drc/av) provided that the
design shear stress at the face of the support remains less than the value of

rcmaxgiven in Table 6.2 (Table 20 of IS 456). In the above expression of the
enhanced shear strength
d = effective depth of the beam,
To= design shear strength of concrete before the enhancement as given in

Table 6.1 (Table 19 of IS 456),
av = horizontal distance of the section from the face of the support (Fig.
6.13.4).

Similar enhancement of shear strength is also to be considered for
sections closed to point loads. It is evident from the expression (2drc /av) that
when av is equal to 2d, the enhanced shear strength does not come into picture.
Further, to increase the effectivity, the tension reinforcement is recommended to
be extended on each side of the point where it is intersected by a possible failure
plane for a distance at least equal to the effective depth, or to be provided with
an equivalent anchorage.
Version 2 CE IIT, Kharagpur

6.13.7 Minimum Shear Reinforcement (cls. 40.3, 26.5.1.5
and 26.5.1.6 of IS 56)
Minimum shear reinforcement

has to be provided even when n, is less

than Tcgiven in Table 6.1 as recommended in cl. 40.3 of IS 456. The amount of
minimum shear reinforcement, as given in cl. 26.5.1.6, is given below.
The minimum shear reinforcement in the form of stirrups shall be provided
such

that:

A, > 0.4
bs, 0.87fy
(6.3)
where A3,, = total cross-sectional area of stirrup legs effective in shear,
s., = stirrup spacing along the length of the member,
b = breadth of the beam or breadth of the web of the web of flanged
beam

b.,.,,and

fy = characteristic
strengthof the stirrupreinforcement
in N/mmzwhich
shallnot be takengreaterthan415 N/mm2.
The above provision is not applicable for members of minor structural
importance such as lintels where the maximum shear stress calculated is less
than half the permissible value.
The minimum shear reinforcement is provided for the following:
(i)

Any sudden failure of beams is prevented if concrete cover bursts
and the bond

(ii)

to the tension

steel

is lost.

Brittle shear failure is arrested which would have occurred without
shear

reinforcement.

(iii)

Tension failure is prevented which would have occurred due to
shrinkage, thermal stresses and internal cracking in beams.

(iv)

To hold the reinforcement in place when concrete is poured.

(v)

Section becomes effective with the tie effect of the compression
steel.

Version 2 CE IIT, Kharagpur

Further, cl. 26.5.1.5 of IS 456 stipulates that the maximum spacing of
shear reinforcement measured along the axis of the member shall not be more

than 0.75 d for vertical stirrups and d for inclined stirrups at 45°, where d is the
effective depth of the section. However, the spacing shall not exceed 300 mm in
any case.

6.13.8 Design of Shear Reinforcement (cl. 40.4 of IS 456)
When 1., is more than Tcgiven in Table 6.1, shear reinforcement shall be
provided in any of the three following forms:
(a) Vertical stirrups,
(b) Bent-up bars along with stirrups, and
(c) Inclined stirrups.
In the case of bent-up bars, it is to be seen that the contribution towards
shear resistance of bent-up bars should not be more than fifty per cent of that of
the total shear

reinforcement.

The amount of shear reinforcement to be provided is determined to carry a
shear force Vusequal to

(6.4)
where

b

is the breadth of rectangular beams or

b.,.,in the case of flanged

beams.

The strengths of shear reinforcement
reinforcement

Vus for the three types of shear

are as follows:

(a) Vertical stirrups:

V US

0.87 fy AWd
SV

(6.5)

(b)
For
inclined
stirrups
or
aseries
ofbars
bent-up
at
different
cross-sectio
0.87 f Awd _
V =ey(s1nor+cosor)

SV

(c) For single bar or single group of parallel bars, all bent-up at the same crosssection:

V

= 0.87 fy AWsv sina

(6.7)
where

A3,, =

within a distance

total cross-sectional

area of stirrup legs or bent-up bars

s.,,

s., = spacing of stirrups or bent-up bars along the length of the member,
1,, = nominal shear stress,

Tc = design shear strength of concrete,
b = breadth of the member which for the anged beams shall be taken as
the breadth of the web b.,.,,

fy = characteristic strength of the stirrup or bent-up reinforcement which

shallnot be takengreaterthan415 N/mm2,
o = angle between the inclined stirrup or bent-up bar and the axis of the

member, not less than 45°, and
d = effective depth.
The following two points are to be noted:
(i)

The total shear resistance shall be computed as the sum of the
resistance for the various types separately where more than one
type of shear reinforcement is used.

(ii)

The area of stirrups shall not be less than the minimum specified in
cl. 26.5.1.6.

6.13.9 Shear Reinforcement for Sections Close to Supports
As stipulated in cl. 40.5.2 of IS 456, the total area of the required shear
reinforcement

As is obtained

from:

As = av b (1., 2d Tc/av)/0.87 fy
and

2 0.4 av b/0.87 fy

(6.8)

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For flanged beams, b will be replaced by b.,.,,the breadth of the web of
flanged beams.
This reinforcement should be provided within the middle three quarters of
av, where

av is less than d, horizontal

shear reinforcement

will be effective

than

vertical.

Alternatively, one simplified method has been recommended in cl. 40.5.3

of IS 456 and the same is given below.
The following method is for beams carrying generally uniform load or
where the principal load is located further than 2d from the face of support. The
shear stress is calculated at a section a distance d from the face of support. The
value of Tc is calculated in accordance with Table 6.1 and appropriate shear
reinforcement is provided at sections closer to the support. No further check for
shear at such sections is required.

6.13.10

Curtailment

of

Tension

einforcement

in

Flexural Members (cl. 26.2.3.2 of IS 456)
Curtailment of tension reinforcement is done to provide the required
reduced area of steel with the reduction of the bending moment. However, shear
force increases with the reduction of bending moment. Therefore, it is necessary
to satisfy any one of
following three conditions while terminating the flexural
reinforcement

in tension

zone:

(i) The shear stress 1., at the cut-off point should not exceed two-thirds of
the permitted value which includes the shear strength of the web reinforcement.
Accordingly,

TV S (2/3) (Tc + Vus/b d)
Or
(6.9)

Vus 2

TV 'Tc)bd

(ii) For each of the terminated bars, additional stirrup area should be
provided over a distance of three-fourth of effective depth from the cut-off point.

The additional stirrup area shall not be less than 0.4 b s/fy, where b is the breadth
of rectangular beams and is replaced by b.,.,,the breadth of the web for flanged

beams, s = spacing of additional stirrups and fy is the characteristic strength of

stirrupreinforcement
in N/mm2.The valueof s shall not exceed d/(8 Bb),where
(3,,is the ratio of area of bars cut-off to the total area of bars at that section, and d
is the effective depth.

Version 2 CE IIT, Kharagpur

(iii) For bars of diameters 36 mm and smaller, the continuing bars provide
double the area required for flexure at the cut-off point. The shear stress should
not exceed three-fourths that permitted. Accordingly,

TV S (3/4) (Tc + Vus/b (1)
Or
Vus 2
(6.10)

TV '
Tc)b d

In the above expression b is the breadth of the rectangular beams which will be
b.,.,in the case of flanged beams.

6.13.11 Placement of Stirrups
E:

Plagcemst
ofvstéirrupa
The stirrups in beams shall be taken around the outer-most tension and
compression bars. In T and L-beams, the stirrups will pass around longitudinal
bars located close to the outer face of the flange. In the rectangular beams, two
holder bars of diameter 10 or 12 mm are provided if there is no particular need
for compression reinforcement (Fig. 6.13.5).

6.13.12
Q.1:

Practice

Questions

and Problems

with Answers

Name and explain the three different failure modes of reinforced concrete
beams under the combined effects of bending moment and shear force.
Version 2 CE IIT, Kharagpur

A.1:

Q.2:

Sec.

6.13.2

A.2:

Q.3:

Define nominal shear stress 1., of rectangular and T-beams of (i) uniform
depth and (ii) varying depth subjected to bending moment and shear
force.

A.3:
Sec.

6.13.3

Q.4:

What is meant by Design shear strength of concrete Tc"?
A.4:

Sec.

6.13.4

On what parameters

Tc of beams without shear reinforcement depends ?

How do you get Tc for different grades of concrete ?
Q.5:

Tc depends on (i) grade of concrete and (ii) percentage of tensile steel in
the beam.

Table 19 of cl. 40.2.1 of IS 456 gives the values of Tcand the same table is

A.5:

presented in Table 6.1 of sec. 6.13.4.1 of this lesson.
How do you know the maximum shear stress of concrete beams

with shear

Tcmax

reinforcement?

Q.6:
Sec.

6.13.4.2

How do you determine the critical sections for shear in a beam ?

A.6:

Sec.

Q.7:

When and why do we consider enhanced shear strength of concrete ?

A.7:

Sec.

Q.8:

6.13.5

6.13.6

How do we determine the minimum shear reinforcement

in rectangular

and T-beams ? Why do we provide the minimum shear reinforcement?
A.8:

Sec.

6.13.7

Q.9:

What are the three different ways to provide shear reinforcement? Explain
the method of design of each of them.

A.9:

Sec.

6.13.8

Q.10:
?
A.10:

Q.11:

How do we design the shear reinforcement close to the support of a beam

A.11:

Sec. 6.13.9
State

the

conditions

to

be

satisfied

for

the

curtailment

of

tension

reinforcement when designing the shear reinforcement.

Q.12:

Sec. 6.13.10

How do we place the vertical stirrups in a beam ?

A.12:

Sec. 6.13.11

6.13.13

References:

1. Reinforced
ConcreteLimitStateDesign,6 Edition,by AshokK. Jain,
Nem Chand & Bros, Roorkee, 2002.

2. LimitStateDesignof Reinforced
Concrete,
2" Edition,
by P.C.Varghese,
Prentice-Hall

of India Pvt. Ltd., New Delhi, 2002.

3. Advanced Reinforced Concrete Design, by P.C.Varghese, Prentice-Hall of
India Pvt. Ltd., New Delhi, 2001.

4. Reinforced
ConcreteDesign,2" Edition,by S.Unnikrishna
Pillaiand
Devdas Menon, Tata McGraw-Hill Publishing Company Limited, New
Delhi, 2003.

5. Limit State Design of ReinforcedConcrete Structures,by P.Dayaratnam,
Oxford& |.B.H. PublishingCompany Pvt. Ltd., New Delhi, 2004.

6. Reinforced
ConcreteDesign,13 RevisedEdition,by S.N.Sinha,Tata
McGraw-Hill Publishing Company. New Delhi, 1990.

7. Reinforced
Concrete,
6 Edition,
byS.K.Mallick
andA.P.Gupta,
Oxford&
IBH Publishing Co. Pvt. Ltd. New Delhi, 1996.

8. Behaviour,Analysis& Designof ReinforcedConcreteStructuralElements,
by |.C.Sya| and R.K.Ummat,A.H.Whee|er & Co. Ltd.,Allahabad, 1989.

9. Reinforced
ConcreteStructures,
3" Edition,by |.C.Syaland A.K.Goel,
A.H.Wheeler & Co. Ltd., Allahabad, 1992.

10.Textbook of R.C.C, by G.S.Birdie and J.S.Birdie, Wiley Eastern Limited,
New Delhi, 1993.

11.Designof ConcreteStructures,
13* Edition,by ArthurH. Nilson,David
Danivin and Charles W. Dolan, Tata McGraw-Hill Publishing Company
Limited, New Delhi, 2004.

12. Concrete Technology,
Longman, 1994.

by A.M.Nevi||e and J.J.Brooks, ELBS with

13.Properties
of Concrete,4" Edition,13 Indianreprint,by A.M.Nevi||e,
Longman, 2000.

14.Reinforced
Concrete
DesignersHandbook,
10" Edition,
byC.E.Reynolds
and J.C.Steedman, E & FN SPON, London, 1997.

15.lndianStandardPlain and ReinforcedConcrete Code of Practice(4
Revision), IS 456: 2000, BIS, New Delhi.
16. Design Aids for Reinforced Concrete to IS: 456
6.13.14

Test

Maximum

13 with

Marks = 50,

1978, BIS, New Delhi.

Solutions

Maximum

Time = 30 minutes

Answer all questions.

TQ.1:

Define nominal shear stress
uniform
shear

1., of rectangular and T-beams of (i)

depth and (ii) varying depth subjected to bending moment and

force.

(5
marks)
A.TQ.1:

Sec. 6.13.3

TQ.2:

How do you determine the critical sections for shear in a beam ?

(5

marks)
A.TQ.2:

Sec. 6.13.5

TQ.3:

When and why do we consider enhanced shear strength of concrete?

(5

marks)
A.TQ.3:

TQ.4:

Sec. 6.13.6

How do we determine the minimum shear reinforcement in rectangular
and T-beams? Why do we provide the minimum shear reinforcement ?
(5 marks)

A.TQ.4:

TQ.5:

Sec. 6.13.7

What are the three different ways to provide shear reinforcement ?
Explain the

A.TQ.5:

TQ.6:

method of design of each of them.

(5 marks)

Sec. 6.13.8

How do we design the shear reinforcement close to the support of a beam?
(5 marks)

A.TQ.6:
TQ.7:

Sec. 6.13.9
State

the

conditions

to

be satisfied

for

the

curtailment

reinforcement when designing the shear reinforcement.

of tension

(5 marks)

Version 2 CE IIT, Kharagpur

A.TQ.7:

Sec. 6.13.10

6.13.15 Summary of this Lesson
Learning the different failure modes, the shear stresses and the design
procedure of beams subjected to shear, this lesson explains the design
procedure with special reference to curtailment of tension reinforcement in

flexural members. Solutions of problems as illustrated in Lesson 14 and given in
the practice problems and test, students will be thoroughly conversant with the
design of rectangular and T-beams subjected to shear following the limit state of

collapse as recommended by IS 456..

Version 2 CE IIT, Kharagpur