Limit State of Collapse
Flexure (Theories and
Examples)
Version 2 CE IIT, Kharagpur

Computation

of

Parameters

of

Governing

Equations
Version 2 CE IIT, Kharagpur

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

o identify the primary load carrying mechanisms of reinforced concrete beams
and slabs,

o name three different types of reinforced concrete beam with their specific
applications,
-

identify the parameters influencing the effective widths of Tand Lbeams,

o differentiate between oneway and twoway slabs,
o state and explain the significance of six assumptions of the design,
o draw the stress-strain diagrams across the depth of a cross-section of
rectangular beam,
o write the three equations of equilibrium,
o write and derive the expressions of total compression and tension forces C
and T, respectively.

3.4.1

Introduction
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3.4.3: Stab l;E§ld%l"°
bending {st.ippp:rtedat mrrters)
Reinforced concrete beams and slabs carry loads primarily by bending
(Figs. 3.4.1 to 3). They are, therefore, designed on the basis of limit state of
collapse in flexure. The beams are also to be checked for other limit states of
shear and torsion. Slabs under normal design loadings (except in bridge decks
etc.) need not be provided with shear reinforcement. However, adequate
torsional reinforcement must be provided wherever needed.
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Version 2 CE IIT, Kharagpur

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Version 2 CE IIT, Kharagpur

This lesson explains the basic governing equations and the computation of
parameters required for the design of beams and oneway slabs employing limit
state of collapse in flexure. There are three types of reinforced concrete beams:
(i)
(ii)
(iii)

Singly or doubly reinforced rectangular beams (Figs. 3.4.4 to 7)
Singly or doubly reinforced Tbeams (Figs. 3.4.8 to 11)
Singly or doubly reinforced Lbeams (Figs. 3.4.12 to 15)

nncrete in compression
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Version 2 CE IIT, Kharagpur

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During construction of reinforced concrete structures, concrete slabs and beams
are cast monolithic making the beams a part of the floor deck system. While
bending under positive moments near midspan, bending compression stresses at
the top are taken by the rectangular section of the beams above the neutral axis
and the slabs, if present in T or L-beams (Figs. 3.4.4, 5, 8, 9, 12 and 13).
However, under the negative moment over the support or elsewhere, the bending
compression stresses are at the bottom and the rectangular sections of
rectangular, Tand L-beams below the neutral axis only resist that compression
(Figs. 3.4.6, 7, 10, 11, 14 and 15). Thus, in a slab-beam system the beam will be
Version 2 CE IIT, Kharagpur

considered as rectangular for the negative moment and T for the positive
moment. While for the intermediate spans of slabs the beam under positive
moment is considered as T, the end span edge beam is considered as Lbeam if
the slab is not projected on both the sides of the beam. It is worth mentioning that
the effective width of flange of these T or L-beams is to be determined which
depends on:
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Version 2 CE IIT, Kharagpur

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Version 2 CE IIT, Kharagpur

(a)
(b)
(c)
(d)

if it is an isolated or continuous beam
the distance between points of zero moments in the beam
the width of the web
the thickness of the flange

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3.4.2 Assumptions

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Version 2 CE IIT, Kharagpur

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The following are the assumptions of the design of flexural members (Figs.
3.4.18 to 20) employing limit state of collapse:
(i) Plane sections normal to the axis remain plane after bending.
This assumption ensures that the crosssection of the member does not
warp due to the loads applied. It further means that the strain at any point on the
crosssection is directly proportional to its distance from the neutral axis.
(ii)
The maximum strain in concrete at the outer most compression fibre is
taken as 0.0035 in bending (Figs. 3.4.19 and 20).
This is a clearly defined limiting strain of concrete in bending compression
beyond which the concrete will be taken as reaching the state of collapse. It is
very clear that the specified limiting strain of 0.0035 does not depend on the
strength of concrete.
(iii)
The acceptable stress-strain curve of concrete is assumed to be parabolic
as shown in Fig. 1.2.1 of Lesson 2.
The maximum compressive stress-strain curve in the structure is obtained
by reducing the values of the top parabolic curve (Figs. 21 of IS 45622000) in two
stages. First, dividing by 1.5 due to size effect and secondly, again dividing by
1.5 considering the partial safety factor of the material. The middle and bottom
curves (Fig. 21 of IS 456:2000) represent these stages. Thus, the maximum
compressive stress in bending is limited to the constant value of 0.446 fckfor the
strain ranging from 0.002 to 0.0035 (Figs. 3.4.19 and 20, Figs. 21 and 22 of IS
45622000).
Version 2 CE IIT, Kharagpur

(iv) The tensile strength of concrete is ignored.
Concrete has some tensile strength (very small but not zero). Yet, this
tensile strength is ignored and the steel reinforcement is assumed to resist the
tensile stress. However, the tensile strength of concrete is taken into account to
check the deflection and crack widths in the limit state of serviceability.
(v)
The design stresses of the reinforcement are derived from the
representative stress-strain curves as shown in Figs. 1.2.3 and 4 of Lesson 2 and
Figs. 23A and B of IS 456:2000, for the type of steel used using the partial safety
factor ymas 1.15.
In the reinforced concrete structures, two types of steel are used: one with
definite yield point (mild steel, Figs. 1.2.3 of Lesson 2 and Figs. 23B of IS
456:2000) and the other where the yield points are not definite (cold work
deformed bars). The representative stress-strain diagram (Fig. 1.2.4 of Lesson 2
and Fig. 23A of IS 45622000) defines the points between 0.8 fy and 1.0 fy in case
of cold work

deformed

bars where

the curve

is inelastic.

(vi)
The maximum strain in the tension reinforcement in the section at failure
shall not be less than fy/(1.15 Es) + 0.002, where fy is the characteristic strength
of steel and E5= modulus of elasticity of steel (Figs. 3.4.19 and 20).
This assumption ensures ductile failure in which the tensile reinforcement
undergoes a certain degree of inelastic deformation before concrete fails in
compression.

3.4.3 Singly Reinforce Rectangular Beams
Figure 3.4.18 shows the singly reinforced rectangular beam in flexure. The
following notations are used (Figs. 3.4.19 and 20):
As, = area of tension steel
width

of the beam

total compressive force of concrete
effective depth of the beam
centre to centre distance between supports
two constant loads acting at a distance of U3 from the two supports

*uI~o.c)c~
of the beam

~|

total

tensile

force

of steel

depth of neutral axis from the top compression fibre

II

II

"II

Xu

only shear force is there at the support and bending moment is zero, (ii) both
bending moment (increasing gradually) and shear force (constant = P) are there
between the support and the loading point and (iii) a constant moment (= PL/3) is
there

in the middle

third

zone

i.e. between

the two

loads

where

the shear

force

is

zero (Fig. 1.1.1 of Lesson 1). Since the beam is in static equilibrium, any crosssection of the beam is also in static equilibrium. Considering the cross-section in
the middle zone (Fig. 3.4.18) the three equations of equilibrium are the following
(Figs. 3.4.19 and 20):
(i) Equilibrium of horizontal forces: EH:
(3.1)

0 gives T= C

(ii) Equilibrium of vertical shear forces: 2 V= 0
(3.2)
This equation gives an identity 0 = 0 as there is no shear in the middle
third

zone

of the beam.

(iii) Equilibrium of moments: EM:
(3.3)

0,

This equation shows that the applied moment at the section is fully resisted by
moment of the resisting couple T a = C a , where a is the operating lever arm
between Tand C (Figs. 3.4.19 and 20).

3.4.5 Computations of Cand T

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.3

ga}Str%a.:ir2
diagram:

{btma

diagram

Fig. 3.-4..2l:Stress and .st;raint:li{a:g:r;atg*l.5
atbsaveneutral axle
Version 2 CE IIT, Kharagpur

Figures 3.4.21a and b present the enlarged view of the compressive part
of the strain and stress diagrams. The convex parabolic part of the stress block of
Fig. 3.4.21b is made rectangular by dotted lines to facilitate the calculations
adding another concave parabolic stress zone which is really non-existent as
marked by hatch in Fig. 3.4.21b.
The different compressive forces C, C1, C2 and C3 and distances x1 to x5
and x., as marked in Fig. 3.4.21b are explained in the following:
C = Total compressive force of concrete = C1 + C2
C1 = Compressive force of concrete due to the constant stress of 0.446
fckand up to a depth of x3 from the top fibre
C2 = Compressive force of concrete due to the convex parabolic stress
block of values ranging from zero at the neutral axis to 0.446 fckat a
distance of x3 from the top fibre
C3 = Compressive force of concrete due to the concave parabolic stress
block (actually non-existent) of values ranging from 0.446 fck at the
neutral axis to zero at a distance of x3 from the top fibre
x1 = Distance of the line of action of C1from the top compressive fibre
x2 =

Distance of the line of action of
compressive fibre

C (= C1 + C2) from the top

x3 = Distance of the fibre from the top compressive fibre, where the strain
= 0.002 and stress = 0.446 fck

x4 = Distance of the line of action of C2 from the top compressive fibre
x5 = Distance of the line of action of C3from the top compressive fibre
x1,= Distance of the neutral axis from the top compressive fibre.
From the strain triangle of Fig. 3.4.21a, we have
x" _ 3

xu

°&#39;°°2
1
0.0035
7

= 057

&#39;
g ivin9

X3 = 0.43 X

(3.4)
Since C1 is due to the constant stress acting from the top to a distance of x3, the
distance x1 of the line of action of C1 is:
Version 2 CE IIT, Kharagpur

X1 = 0.5 X3 = 0.215

X

(3.5)
From Fig. 3.4.21a:

or

X5

X3+ %(XuX3)= 0.43X+0.75(0.57
X)

X5

0.86 X

(3.6)
The compressive force 01 due to the rectangular stress block is:
C1 = bX3(0.446 fck) = 0.191 bxu fck
(3.7)
The compressive force C2 due to parabolic stressblock is:

C2= b(X.,X3)
§ (0.446
rck)= 0.17bxurck
(3.8)
Adding C1 and C2,we have
C = C1+ C2 = 0.361 bxu fck = 0.36 bxu fck (say)
(3.9)
The non-existent compressive force C3 due to parabolic (concave) stress block
is:

C3= b(X.,X3)§ (0.446
rck)= 0.085
bxurck
(3.10)
Now, we can get X4by taking moment of C2 and C3about the top fibre as follows:

C2(X4)
+ C3(X5)= (O2+C3)(X3+ x";x3)
which gives X4= 0.64 X
(3.11)
Similarly, X2 is obtained by taking moment of C1 and 02 about the top fibre as
follows:

C1(X1)+ C2(X4)= C(X2)

which
gives
X2
=0.4153
Xu
Version
2CE
IIT,
Kharagpur

Cl
(3.12) x2 = 0.42 xu

(say).

Thus, the required parameters of the stress block (Fig. 3.4.19) are
C = 0.36 b X fck

(3.9)

X2 = 0.42 X.)

(3.12)

and lever

(d - X2) = (d- 0.42 X)

arm

(3.13)
The tensile force T is obtained by multiplying the design stress of steel with the
area of steel. Thus,
fy
T: (E)
Ax,= 0.87
fyA_v,

(3.14)

3.4.6

Practice

Questions

and Problems

with Answers

Q.1:

How do the beams and slabs primarily carry the transverse loads ?

A.1:

The beams and slabs carry the transverse loads primarily by bending.

Q.2:

Name three different types of reinforced concrete beams and their specific
applications.

A.2:

They are:
(i)

Singly reinforced and doubly reinforced rectangular beams used in
resisting negative moments in intermediate spans of continuous
beams over the supports or elsewhere in slab-beam monolithic
constructions, and positive moments in midspan of isolated or
intermediate spans of beams with inverted slab (monolithic)
constructions

and lintels.

Singly reinforced and doubly reinforced Tbeams used in resisting
positive moments in isolated or intermediate spans (midspan) in
slab-beam monolithic constructions and negative moments over the
support for continuous spans with inverted slab (monolithic)
constructions.

Version 2 CE IIT, Kharagpur

(iii)

Singly reinforced and doubly reinforced Lbeams
Same as (ii)
above except that these are for end spans instead of intermediate
spans.

Q.3:

Name four parameters which determine the effective widths of T and L-

beams.
A.3:

The four parameters are:
(i) isolated or continuous beams,
(ii) the distance between points of zero moments in the beam,
(iii) the breadth of the web,
(iv) the thickness of the flange.

Q.4:

Differentiate between oneway and twoway slabs.

A.4:

One-way slab spans in one direction and twoway slab spans in both the
directions. Slabs whose ratio of longer span (ly) to shorter span (IX)is more
than 2 are called oneway. Slabs of this ratio up to 2 are called twoway
slabs.

Q.5:

State and explain the significance of the six assumptions of design of
flexural members employing limit state of collapse.

A.5:

Sec. 3.4.2 gives the full answer.

Q.6:

Draw a crosssection of singly reinforced rectangular beam and show the
strain and stress diagrams.

A.6:

Fig. 3.4.19.

Q.7:

Write the three equations of equilibrium needed to design the reinforced
concrete

beams.

A.7:

Vide sec. 3.4.4 and Eqs. 3.1 to 3.

Q.8:

Write the final expression of the total compressive force C and tensile
force T for a rectangular reinforced concrete beam in terms of the
designing parameters.

A.8:

Eq. 3.9 for Cand Eq. 3.14 for T.

Version 2 CE IIT, Kharagpur

3.4.7

References

1. ReinforcedConcreteLimitState Design,6" Edition,by Ashok K. Jain,
Nem Chand & Bros, Roorkee, 2002.

2. LimitStateDesignof ReinforcedConcrete,2 Edition,by P.C.Varghese,
PrenticeHa|I 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. ReinforcedConcreteDesign,2 Edition,by S.UnnikrishnaPillai and
Devdas Menon, Tata McGraw-Hill Publishing Company Limited, New
Delhi, 2003.

5. Limit State Design of Reinforced Concrete Structures, by P.Dayaratnam,
Oxford & I.B.H. Publishing Company Pvt. Ltd., New Delhi, 2004.

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

7. ReinforcedConcrete,6" Edition,by S.K.Ma|Iickand A.P.Gupta,Oxford &
IBH Publishing Co. Pvt. Ltd. New Delhi, 1996.
8. Behaviour, Analysis & Design of Reinforced Concrete Structural Elements,
by |.C.Sya| and R.K.Ummat, A.H.Whee|er & Co. Ltd., Allahabad, 1989.

9. ReinforcedConcreteStructures,3 Edition,by |.C.Sya|and A.K.Goe|,
A.H.Whee|er & 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 Arthur H. Nilson,David
Darwin and Charles W. Dolan, Tata McGraw-Hill Publishing Company
Limited, New Delhi, 2004.

12.Concrete Technology, by A.M.Neville and J.J.Brooks, ELBS with
Longman, 1994.

13.Propertiesof Concrete,4" Edition, 13 Indian reprint, by A.M.Neville,
Longman, 2000.

14.Reinforced
ConcreteDesignersHandbook,10"Edition,by C.E.Reyno|ds
and J.C.Steedman, E & FN SPON, London, 1997.

15.IndianStandardPlain and ReinforcedConcrete Codeof Practice(4"
Revision), IS 456: 2000, BIS, New Delhi.
16. Design Aids for Reinforced Concrete to IS: 456

3.4.8

Test

4 with

Maximum Marks = 50,

1978, BIS, New Delhi.

Solutions
Maximum Time = 30 minutes

Answer all questions.
TQ.1: Tick the correct answer:
marks)

(4 x 5

20

Version 2 CE IIT, Kharagpur

(i) Beams and slabs carry the transverse loads primarily by
(a)
(b)
(c)
(d)

truss action
balance of shear action
bending
slab-beam interaction

A.TQ.1: (i):

(c)

(ii) The ratio of longer span (ly) to shorter span (IX)of a two-way slab is
(a)
(b)
(c)
(d)

up to 2
more than 2
equal to 1
more than 1

A.TQ.1: (ii): (a)
(iii) An inverted 7&#39;beam
is considered as a rectangular beam for the design
(a) over the intermediate support of a continuous beam where the bending
moment is negative
(b) at the midspan of a continuous beam where the bending moment is
positive
(c) at the point of zero bending moment
(d) over the support of a simply supported beam
A.TQ.1: (iii): (b)
(iv)

The maximum strain in the tension reinforcement
shall

in the section at failure

be

(a) more than fy/(1.15 Es) + 0.002
(b) equal to 0.0035
(c) more than fy/Es + 0.002
(d) less than fy/(1.15 Es) + 0.002
A.TQ.1: (iv): (d)

Version 2 CE IIT, Kharagpur

TQ.2: Draw a crosssection of singly reinforced rectangular beam and show the
strain and stress diagrams.
(10)
A.TQ.2: Fig. 3.4.19
TQ.3: Name four parameters which determine the effective widths of Tand Lbeams. (6)
A.TQ.3: The four parameters are:
(i) isolated or continuous beams,
(ii) the distance between points of zero moments in the beam,
(iii) the breadth of the web,
(iv) the thickness of the flange.
TQ.4: Derive the final expressions of the total compressive force C and tensile
force T for a rectangular reinforced concrete beam in terms of the
designing parameters.
(10 +
4 = 14)
A.TQ.4:

Section 3.4.5 is the full answer.

3.4.9 Summary of this Lesson
Lesson 4 illustrates the primary load carrying principle in a slab-beam
structural system subjected to transverse loadings. It also mentions three
different types of singly and doubly reinforced beams normally used in
construction. Various assumptions made in the design of these beams employing
limit state of collapse are explained. The stress and strain diagrams of a singly
reinforced rectangular beam are explained to write down the three equations of
equilibrium. Finally, the computations of the total compressive and tensile forces
are illustrated.

Version 2 CE IIT, Kharagpur