Module
10
Compression Members
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Lesson
23
Short Compression
Members under
Axial Load with
Uniaxial Bending
Version 2 CE IIT, Kharagpur

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

draw the strain profiles for different locations of the depth of the neutral
axis,

•

explain the behaviour of such columns for any one of the strain profiles,

•

name and identify the three modes of failure of such columns,

•

explain the interaction diagram and divide into the three regions indicating
three modes of failure,

•

identify the three modes of failure from the depth of neutral axis,

•

identify the three modes of failure from the eccentricity of the axial load,

•

determine the area of compressive stress block, distance of the centroid of
the area of the compressive stress block from the highly compressed edge
when the neutral axis is within and outside the cross-section of the
column,

•

determine the compressive stress of concrete and tensile/compressive
stress of longitudinal steel for any location of the neutral axis within or
outside the cross-section of the column,

•

write the two equations of equilibrium,

•

explain the need to recast the equations in non-dimensional form for their
use in the design of such columns.

10.23.1 Introduction
Short reinforced concrete columns under axial load with uniaxial bending
behave in a different manner than when it is subjected to axial load, though
columns subjected to axial load can also carry some moment that may appear
during construction or otherwise. The behaviour of such columns and the three
modes of failure are illustrated in this lesson. It is explained that the moment M,
equivalent to the load P with eccentricity e (= M/P), will act in an interactive
manner. A particular column with specific amount of longitudinal steel, therefore,
can carry either a purely axial load Po (when M = 0), a purely moment Mo (when
P = 0) or several pairs of P and M in an interactive manner. Hence, the needed
interaction diagram of columns, which is a plot of P versus M, is explained
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discussing different positions of neutral axis, either outside or within the crosssection of the column.
Depending on the position of the neutral axis, the column may or may not
have tensile stress to be taken by longitudinal steel. In the compression region
however, longitudinal steel will carry the compression load along with the
concrete as in the case of axially loaded column.

10.23.2 Behaviour of Short Columns under Axial Load and
Uniaxial Moment
Normally, the side columns of a grid of beams and columns are subjected
to axial load P and uniaxial moment Mx causing bending about the major axis xx,
hereafter will be written as M. The moment M can be made equivalent to the
axial load P acting at an eccentricity of e (= M/P). Let us consider a symmetrically
reinforced short rectangular column subjected to axial load Pu at an eccentricity
of e to have Mu causing failure of the column.
Figure 10.21.11b of Lesson 21 presents two strain profiles IN and EF. For
the strain profile IN, the depth of the neutral axis kD is less than D, i.e., neutral
axis is within the section resulting the maximum compressive strain of 0.0035 on
the right edge and tensile strains on the left of the neutral axis forming cracks.
This column is in a state of collapse for the axial force Pu and moment Mu for
which IN is the strain profile. Reducing the eccentricity of the load Pu to zero, we
get the other strain profile EF resulting in the constant compressive strain of
0.002, which also is another collapse load. This axial load Pu is different from the
other one, i.e., a pair of Pu and Mu, for which IN is the profile. For the strain profile
EF, the neutral axis is at infinity (k = α ).
Figure 10.21.11c of Lesson 21 presents the strain profile EF with two
more strain profiles IH and JK intersecting at the fulcrum point V. The strain
profile IH has the neutral axis depth kD = D, while other strain profile JK has kD >
D. The load and its eccentricity for the strain profile IH are such that the
maximum compressive strain reaches 0.0035 at the right edge causing collapse
of the column, though the strains throughout the depth is compressive and zero
at the left edge. The strain profile JK has the maximum compressive strain at the
right edge between 0.002 and 0.0035 and the minimum compressive strain at the
left edge. This strain profile JK also causes collapse of the column since the
maximum compressive strain at the right edge is a limiting strain satisfying
assumption (ii) of sec. 10.21.10 of Lesson 21.
The four strain profiles, IN, EF, JK and IH of Figs.10.21.11b and c,
separately cause collapse of the same column when subjected to four different
pairs of Pu and Mu. This shows that the column may collapse either due to a
uniform constant strain throughout (= 0.002 by EF) or due to the maximum
compressive strain at the right edge satisfying assumption (ii) of sec.10.21.10 of
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Lesson 21 irrespective of the strain at the left edge (zero for IH and tensile for
IN). The positions of the neutral axis and the eccentricities of the load are widely
varying as follows:
(i) For the strain profile EF, kD is infinity and the eccentricity of the load is
zero.
(ii) For the strain profile JK, kD is outside the section (D < kD < α ), with
appropriate eccentricity having compressive strain in the section.
(iii) For the strain profile IH, kD is just at the left edge of the section (kD =
D), with appropriate eccentricity having zero and 0.0035 compressive
strains at the left and right edges, respectively.
(iv) For the strain profile IN, kD is within the section (kD < D), with
appropriate eccentricity having tensile strains on the left of the neutral
axis and 0.0035 compressive strain at the right edge.

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It is evident that gradual increase of the eccentricity of the load Pu from zero is
changing the strain profiles from EF to JK, IH and then to IN. Therefore, we can
accept that if we increase the eccentricity of the load to infinity, there will be only
Mu acting on the column. Designating by Po as the load that causes collapse of
the column when acting alone and Mo as the moment that also causes collapse
when acting alone, we mark them in Fig.10.23.1 in the vertical and horizontal
axes. These two points are the extreme points on the plot of Pu versus Mu, any
point on which is a pair of Pu and Mu (of different magnitudes) that will cause
collapse of the same column having the neutral axis either outside or within the
column.
The plot of Pu versus Mu of Fig.10.23.1 is designated as interaction
diagram since any point on the diagram gives a pair of values of Pu and Mu
causing collapse of the same column in an interactive manner. Following the
same logic, several alternative column sections with appropriate longitudinal
steel bars are also possible for a particular pair of Pu and Mu. Accordingly for the
purpose of designing the column, it is essential to understand the different modes
of failure of columns, as given in the next section.

10.23.3 Modes of Failure of Columns
The two distinct categories of the location of neutral axis, mentioned in the
last section, clearly indicate the two types of failure modes: (i) compression
failure, when the neutral axis is outside the section, causing compression
throughout the section, and (ii) tension failure, when the neutral axis is within the
section developing tensile strain on the left of the neutral axis. Before taking up
these two failure modes, let us discuss about the third mode of failure i.e., the
balanced failure.
(A) Balanced failure
Under this mode of failure, yielding of outer most row of longitudinal steel
near the left edge occurs simultaneously with the attainment of maximum
compressive strain of 0.0035 in concrete at the right edge of the column. As a
result, yielding of longitudinal steel at the outermost row near the left edge and
crushing of concrete at the right edge occur simultaneously. The different yielding
strains of steel are determined from the following:
(i) For mild steel (Fe 250): ε y = 0.87fy/Es
(10.12)
(ii) For cold worked deformed bars: ε y = 0.87fy/Es + 0.002
(10.13)

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The corresponding numerical values are 0.00109, 0.0038 and 0.00417 for Fe
250, Fe 415 and Fe 500, respectively. Such a strain profile is known a balanced
strain profile which is shown by the strain profile IQ in Fig.10.23.2b with a number
5. This number is shown in Fig.10.23.1 lying on the interaction diagram causing
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collapse of the column. The depth of the neutral axis is designated as kbD and
shown in Fig.10.23.2b. The balanced strain profile IQ in Fig.10.23.2b also shows
the strain ε y whose numerical value would change depending on the grade of
steel as mentioned earlier. It is also important to observe that this balanced
profile IQ does not pass trough the fulcrum point V in Fig.10.23.2b, while other
profiles 1, 2 and 3 i.e., EF, LM and IN pass through the fulcrum point V as none
of them produce tensile strain any where in the section of the column. The
neutral axis depth for the balanced strain profile IQ is less than D, while the same
for the other three are either equal to or more than D.
To have the balanced strain profile IQ causing balanced failure of the
column, the required load and moment are designated as Pb and Mb ,
respectively and shown in Fig.10.23.1 as the coordinates of point 5. The
corresponding eccentricity of the load Pb is defined by the notation eb (= Mb/Pb).
The four parameters of the balanced failure are, therefore, Pb, Mb, eb and kb (the
coefficient of the neutral axis depth kbD).
(B) Compression failure
Compression failure of the column occurs when the eccentricity of the load
Pu is less than that of balanced eccentricity (e < eb) and the depth of the neutral
axis is more than that of balanced failure. It is evident from Fig.10.23.2b that
these strain profiles may develop tensile strain on the left of the neutral axis till
kD = D. All these strain profiles having 1 > k > kb will not pass through the
fulcrum point V. Neither the tensile strain of the outermost row of steel on the left
of the neutral axis reaches ε y .
On the other hand, all strain profiles having kD greater than D pass
through the fulcrum point V and cause compression failure (Fig.10.23.2b). The
loads causing compression failure are higher than the balanced load Pb having
the respective eccentricities less than that of the load of balanced failure. The
extreme strain profile is EF marked by 1 in Fig.10.23.2b. Some of these points
causing compression failure are shown in Fig.10.23.1 as 1, 2, 3 and 4 having k >
kb, either within or outside the section.
Three such strain profiles are of interest and need further elaboration. One
of them is the strain profile IH (Fig.10.23.2b) marked by point 3 (Fig.10.23.1) for
which kD = D. This strain profile develops compressive strain in the section with
zero strain at the left edge and 0.0035 in the right edge as explained in sec.
10.23.2. Denoting the depth of the neutral axis by D and eccentricity of the load
for this profile by eD, we observe that the other strain profiles LM and EF
(Fig.10.23.2b), marked by 2 and 1 in Fig.10.23.1, have the respective kD > D and
e < eD.

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The second strain profile is EF (Fig.10.23.2b) marked by point 1 in
Fig.10.23.1 is for the maximum capacity of the column to carry the axial load Po
when eccentricity is zero and for which moment is zero and the neutral axis is at
infinity. This strain profile has also been discussed earlier in sec.10.23.2.
The third important strain profile LM, shown in Fig.10.23.2b and by point 2
in Figs.10.23.1 and 2, is also due to another pair of collapse Pm and Mm, having
the capacity to accommodate the minimum eccentricity of the load, which hardly
can be avoided in practical construction or for other reasons. The load Pm , as
seen from Fig.10.23.1, is less than Po and the column can carry Pm and Mm in an
interactive mode to cause collapse. Hence, a column having the capacity to carry
the truly concentric load Po (when M = 0) shall not be allowed in the design.
Instead, its maximum load shall be restricted up to Pm (< Po) along with Mm (due
to minimum eccentricity). Accordingly, the actual interaction diagram to be used
for the purpose of the design shall terminate with a horizontal line 22’ at point 2 of
Fig.10.23.1. Point 2 on the interaction diagram has the capacity of Pm with Mm
having eccentricity of em (= Mm/Pm) and the depth of the neutral axis is >> D
(Fig.10.23.2b).
It is thus seen that from points 1 to 5 (i.e., from compression failure to
balanced failure) of the interaction diagram of Fig.10.23.1, the loads are
gradually decreasing and the moments are correspondingly increasing. The
eccentricities of the successive loads are also increasing and the depths of
neutral axis are decreasing from infinity to finite but outside and then within the
section up to kbD at balanced failure (point 5). Moreover, this region of
compression failure can be subdivided into two zones: (i) zone from point 1 to
point 2, where the eccentricity of the load is less than the minimum eccentricity
that should be considered in the actual design as specified in IS 456, and (ii)
zone from point 2 to point 5, where the eccentricity of the load is equal to or more
than the minimum that is specified in IS 456. It has been mentioned also that the
first zone from point 1 to point 2 should be avoided in the design of column.

(C) Tension failure
Tension failure occurs when the eccentricity of the load is greater than the
balanced eccentricity eb. The depth of the neutral axis is less than that of the
balanced failure. The longitudinal steel in the outermost row on the left of the
neutral axis yields first. Gradually, with the increase of tensile strain, longitudinal
steel of inner rows, if provided, starts yielding till the compressive strain reaches
0.0035 at the right edge. The line IR of Fig.10.23.2b represents such a profile for
which some of the inner rows of steel bars have yielded and compressive strain
has reached 0.0035 at the right edge. The depth of the neutral axis is designated
by (kminD).

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It is interesting to note that in this region of the interaction diagram (from 5
to 6 in Fig.10.23.1), both the load and the moment are found to decrease till point
6 when the column fails due to Mo acting alone. This important behaviour is
explained below starting from the failure of the column due to Mo alone at point 6
of Fig.10.23.1.
At point 6, let us consider that the column is loaded in simple bending to
the point (when M = Mo) at which yielding of the tension steel begins. Addition of
some axial compressive load P at this stage will reduce the previous tensile
stress of steel to a value less than its yield strength. As a result, it can carry
additional moment. This increase of moment carrying capacity with the increase
of load shall continue till the combined stress in steel due to additional axial load
and increased moment reaches the yield strength.

10.23.4 Interaction Diagram
It is now understood that a reinforced concrete column with specified
amount of longitudinal steel has different carrying capacities of a pair of Pu and
Mu before its collapse depending on the eccentricity of the load. Figure 10.23.1
represents one such interaction diagram giving the carrying capacities ranging
from Po with zero eccentricity on the vertical axis to Mo (pure bending) on the
horizontal axis. The vertical axis corresponds to load with zero eccentricity while
the horizontal axis represents infinite value of eccentricity. A radial line joining the
origin O of Fig.10.23.1 to point 2 represents the load having the minimum
eccentricity. In fact, any radial line represents a particular eccentricity of the load.
Any point on the interaction diagram gives a unique pair of Pu and Mu that causes
the state of incipient failure. The interaction diagram has three distinct zones of
failure: (i) from point 1 to just before point 5 is the zone of compression failure, (ii)
point 5 is the balanced failure and (iii) from point 5 to point 6 is the zone of
tension failure. In the compression failure zone, small eccentricities produce
failure of concrete in compression, while large eccentricities cause failure
triggered by yielding of tension steel. In between, point 5 is the critical point at
which both the failures of concrete in compression and steel in yielding occur
simultaneously.
The interaction diagram further reveals that as the axial force Pu becomes
larger the section can carry smaller Mu before failing in the compression zone.
The reverse is the case in the tension zone, where the moment carrying capacity
Mu increases with the increase of axial load Pu. In the compression failure zone,
the failure occurs due to over straining of concrete. The large axial force
produces high compressive strain of concrete keeping smaller margin available
for additional compressive strain line to bending. On the other hand, in the
tension failure zone, yielding of steel initiates failure. This tensile yield stress
reduces with the additional compressive stress due to additional axial load. As a

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result, further moment can be applied till the combined stress of steel due to axial
force and increased moment reaches the yield strength.
Therefore, the design of a column with given Pu and Mu should be done
following the three steps, as given below:
(i) Selection of a trial section with assumed longitudinal steel,
(ii) Construction of the interaction diagram of the selected trial column
section by successive choices of the neutral axis depth from infinity
(pure axial load) to a very small value (to be found by trial to get P = 0
for pure bending),
(iii) Checking of the given Pu and Mu, if they are within the diagram.
We will discuss later whether the above procedure should be followed or
not. Let us first understand the corresponding compressive stress blocks of
concrete for the two distinct cases of the depth of the neutral axis: (i) outside the
cross-section and (ii) within the cross-section in the following sections.

10.23.5 Compressive Stress Block of Concrete when the
Neutral Axis Lies Outside the Section

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Figure 10.23.3c presents the stress block for a typical strain profile JK
having neutral axis depth kD outside the section (k > 1). The strain profile JK in
Fig.10.23.3b shows that up to a distance of 3D/7 from the right edge (point AO),
the compressive strain is ≥ 0.002 and, therefore, the compressive stress shall
remain constant at 0.446fck. The remaining part of the column section of length
4D/7, i.e., up to the left edge, has reducing compressive strains (but not zero).
The stress block is, therefore, parabolic from AO to H which becomes zero at U
(outside the section). The area of the compressive stress block shall be obtained
subtracting the parabolic area between AO to H from the rectangular area
between G and H. To establish the expression of this area, it is essential to know
the equation of the parabola between AO and U, whose origin is at AO. The
positive coordinates of X and Y are measured from the point AO upwards and to
the left, respectively. Let us assume that the general equation of the parabola as
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X = aY2 + bY + c
(10.14)
The values of a, b and c are obtained as follows:
(i)

At Y = 0, X = 0, at the origin: gives c = 0

(ii)

At Y = 0, dX/dY = 0, at the origin: gives b = 0

(iii)
At Y = (kD – 3D/7), i.e., at point U, X = 0.446fck: gives a = 0.446fck/D2(k2
3/7) .
Therefore, the equation of the parabola is:
X = {0.446fck/D2(k – 3/7)2}Y2
(10.15)
The value of X at the point H (left edge of the column), g is now
determined from Eq.10.15 when Y = 4D/7, which gives
g = 0.446 fck {4/(7k – 3)}2
(10.16)
Hence, the area of the compressive stress block = 0.446 fck D [1 – (4/21){4/(7k –
3)}2]
= C1 fck D
(10.17)
where C1 = 0.446[1 – (4/21){4/(7k – 3)}2]
(10.18)
Equation 10.17 is useful to determine the area of the stress block for any
value of k > 1 (neutral axis outside the section) by substituting the value of C1
from Eq.10.18. The symbol C1 is designated as the coefficient for the area of the
stress block.
The position of the centroid of the compressive stress block is obtained by
dividing the moment of the stress block about the right edge by the area of the
stress block. The moment of the stress block is obtained by subtracting the
moment of the parabolic part between AO and H about the right edge from the
moment of the rectangular stress block of full depth D about the right edge. The
expression of the moment of the stress block about the right edge is:
0.446 fck D(D/2) – (1/3)(4D/7) 0.446 fck {4/(7k – 3)}2 {3D/7 + (3/4)(4D/7)}

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= 0.446 fck D2 [(1/2) – (8/49){4/(7k – 3)}2]
(10.19)
Dividing Eq.10.19 by Eq.10.17, we get the distance of the centroid from
the right edge is:
D[(1/2) – (8/49){4/(7k – 3)}2]/[1 – (4/21){4/(7k – 3)}2]
(10.20)
= C2 D
(10.21)
where C2 is the coefficient for the distance of the centroid of the compressive
stress block of concrete measured from the right edge and is:
C2 = [(1/2) – (8/49){4/(7k – 3)}2]/[1 – (4/21){4/(7k – 3)}2]
(10.22)
Table 10.4 presents the values of C1 and C2 for different values of k greater than
1, as given in Table H of SP-16. For a specific depth of the neutral axis, k is
known. Using the corresponding values of C1 and C2 from Table 10.4, area of the
stress block of concrete and the distance of centroid from the right edge are
determined from Eqs.10.17 and 10.21, respectively.
Table 10.4 Stress block parameters C1 and C2 when the neutral axis is outside
the section

K
1.00
1.05
1.10
1.20
1.30
1.40
1.50
2.00
2.50
3.00
4.00

C1
0.361
0.374
0.384
0.399
0.409
0.417
0.422
0.435
0.440
0.442
0.444

C2
0.416
0.432
0.443
0.458
0.468
0.475
0.480
0.491
0.495
0.497
0.499

It is worth mentioning that the area of the stress block is 0.446fckD and the
distance of the centroid from the right edge is 0.5D, when k is infinite. Values of
C1 and C2 at k = 4 are very close to those when k = ∞ . In fact, for the practical

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interaction diagrams, it is generally adequate to consider values of k up to about
1.2.

10.23.6 Determination of Compressive Stress Anywhere in
the Section when the Neutral Axis Lies outside the
Section
The compressive stress of concrete at any point between G and AO of
Fig.10.23.3c is constant at 0.446fck as the strain in this zone is equal to or greater
than 0.002. So, we can write
fc = 0.446fck if 0.002 ≤ ε c ≤ 0.0035
(10.23)
However, compressive stress of concrete between AO and H is to be
determined using the equation of parabola. Let us determine the concrete stress
fc at a distance of Y from the origin AO. From Fig.10.23.3c, we have
fc = 0.446 fck - gc
(10.24)
where gc is as shown in Fig.10.23.3c and obtained from Eq.10.15. Thus, we get
fc = 0.446 fck – {0.446 fck/D2(k – 3/7)2}Y2
or

fc = 0.446 fck {1 – Y2/(kD – 3D/7)2}
(10.25)

Designating the strain of concrete at this point by ε c (Fig.10.23.3b), we
have from similar triangles

ε c /0.002 = 1 – Y/(kD – 3D/7)
which gives
Y = {1 – ( ε /0.002)}(kD – 3D/7)
(10.26)
Substituting the value of Y from Eq.10.26 in Eq.10.25, we have
fc = 0.446 fck [2( ε c /0.002) - ( ε c /0.002)2], if 0 ≤ ε c < 0.002
(10.27)

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10.23.7 Compressive Stress Block of Concrete when the
Neutral Axis is within the Section

Figure 10.23.4c presents the stress block for a typical strain profile IN
having neutral axis depth = kD within the section (k < 1). The strain profile IN in
Fig.10.23.4b shows that from a to AO, i.e., up to a distance of 3kD/7 from the
right edge, the compressive strain is ≥ 0.002 and, therefore, the compressive
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stress shall remain constant at 0.446fck. From AO to U, i.e., for a distance of
4kD/7, the strain is reducing from 0.002 to zero and the stress in this zone is
parabolic as shown in Fig.10.23.4c. The area of the stress block shall be
obtained subtracting the parabolic area between AO and U from the total
rectangular area between G and U. As in the case when the neutral axis is
outside the section (sec.10.23.5), we have to establish the equation of the
parabola with AO as the origin and the positive coordinates X and Y are
measured from the point AO upwards for X and from the point AO to the left for
Y, as shown in Fig.10.23.4c. Proceeding in the same manner as in sec.10.23.5
and assuming the same equation of the parabola as in Eq.10.14, the values of a,
b and c are obtained as:
(i) At Y = 0, X = 0, at the origin: gives c = 0
(ii) At Y = 0, dX/dY = 0, at the origin: gives b = 0
(iii) At U, (i.e., at Y = 4kD/7), X = 0.446 fck: gives a = 0.446 fck/(4kD/7)2.
Therefore, the equation of the parabola OR is:
X = {0.446 fck/(4kD/7)2}Y2
(10.28)
The area of the stress block = 0.446 fck kD – (1/3) 0.446 fck (4kD/7) = 0.36 fck
kD, the same as obtained earlier in Eq.3.9 of Lesson 4 for flexural members.
Similarly, the distance of the centroid can be obtained by dividing the moment of
area of stress block about the right edge by the area of the stress block. The
result is the same as in Eq.3.12 for the flexural members. Therefore, we have
Area of the stress block = 0.36 fck kD
(10.29)
The distance of the centroid of the stress block from the right edge = 0.42kD
(10.30)
Thus, the values of C1 and C2 of Eqs.10.17 and 10.21, respectively, are
0.36 and 0.42 when the neutral axis is within the section. It is to be noted that the
coefficients C1 and C2 are multiplied by Dfck and D, respectively when the neutral
axis is outside the section. However, they are to be multiplied here, when the
neutral axis is within the section, by kDfck and kD, respectively.
It is further to note that though the expressions of the area of stress block
and the distance of the centroid of the stress block from the right edge are the
same as those for the flexural members, the important restriction of the maximum
depth of the neutral axis xumax in the flexural members is not applicable in case of
column. By this restriction, the compression failure of the flexural members is
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avoided. In case of columns, compression failure is one of the three modes of
failure.

10.23.8 Determination of Compressive Stress Anywhere in
the Compressive Zone when the Neutral Axis is within
the Section
The compressive stress at any point between G and AO of Fig.10.23.4c is
constant at 0.446fck as the strain in this zone is equal to or greater than 0.002.
So, we can write
fc = 0.446 fck if 0.002 ≤ ε c ≤ 0.0035
….
(10.23)
However, the compressive stress between AO and U is to be determined from
the equation of the parabola. Let us determine the compressive stress fci at a
distance of Y from the origin AO. From Fig.10.23.4c, we have
fc = 0.446 fck - gc
(10.31)
where gc as shown in Fig.10.23.4c, is obtained from Eq.10.28. Thus, we get,
fc = {0.446 fck – 0.446 fck (4kD/7)2}Y2
(10.32)
Designating the strain of concrete at this point by ε c (Fig.10.23.4b), we
have from similar triangles

ε c /0.002 = 1 – Y/(4kD/7), which gives
Y = {1 - ε c /0.002}(4kD/7)
(10.33)
Substituting the value of Y from Eq.10.33 in Eq.10.32, we get the same equation,
Eq.10.27 of sec.10.23.6, when the neutral axis is outside the section. Therefore,
fc = 0.446 fck [2( ε c /0.002) – ( ε c /0.002)2] …. (10.26)
From the point U to the left edge H of the cross-section of the column, the
compressive stress is zero. Thus, we have
fc = 0 if ε c ≤ 0

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fc = 0.446 fck if ε c ≥ 0.002
fc = 0.446 fck{2( ε c /0.002) – ( ε c /0.002)2}, if 0 ≤ ε c < 0.002
(10.34)

10.23.9 Tensile and Compressive Stresses of Longitudinal
Steel
Stresses are compressive in all the six rows (A1 to A6 of Figs.10.23.3a
and c) of longitudinal steel provided in the column when the neutral axis depth
kD ≥ D. However, they are tensile on the left side of the neutral axis and
compressive on the right side of the neutral axis (Figs.10.23.4a and c) when kD <
D. These compressive or tensile stresses of longitudinal steel shall be calculated
from the strain ε si at that position of the steel which is obtained from the strain
profile considered for the purpose.
It should be remembered that the linear strain profiles are based on the
assumption that plane sections remain plane. Moreover, at the location of steel in
a particular row, the strain of steel ε si shall be the same as that in the adjacent
concrete ε ci . Thus, the strain of longitudinal steel can be calculated from the
particular strain profile if the neutral axis is within or outside the cross-section of
the column.
The corresponding stresses f si of longitudinal steel are determined from
the strain ε si (which is the same as that of ε ci in the adjacent concrete) from the
respective stress-strain diagrams of mild steel (Fig.1.2.3 of Lesson 2) and High
Yield Strength Deformed bars (Fig.1.2.4 of Lesson2). The values are
summarized in Table 10.5 below as presented in Table A of SP-16.
Table 10.5 Values of compressive or tensile f si from known values of ε si of
longitudinal steel (Fe 250, Fe 415 and Fe 500)
Fe 250

Fe 415

Fe 500

Strain

ε si

Stress
(N/mm2)
f si

Strain

ε si

Stress
(N/mm2)
f si

Strain

ε si

Stress
(N/mm2)
f si

< 0.00109

ε si (Es)

< 0.00144

ε si (Es)

< 0.00174

ε si (Es)

≥ 0.00109

217.5
(= 0.87 fy)

0.00144

288.7

0.00174

347.8

0.00163
0.00192
0.00241

306.7
324.8
342.8

0.00195
0.00226
0.00277

369.6
391.3
413.0

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0.00276
0.00380

351.8
360.9

0.00312
0.00417

423.9
434.8

Notes: 1. Linear interpolation shall be done for intermediate values.
2. Strain at initial yield = fy/Es
3. Strain at final yield = fy/Es + 0.002

10.23.10 Governing Equations
A column subjected to Pu and Mu (= Pu e) shall satisfy the two equations of
equilibrium, viz., ∑V = 0 and ∑M = 0, taking moment of vertical forces about the
centroidal axis of the column. The two governing equation are, therefore,
Pu = Cc + Cs
(10.35)
Mu = Cc (appropriate lever arm) + Cs (appropriate lever arm)
(10.36)
where Cc = Force due to concrete in compression
Cs = Force due to steel either in compression when kD ≥ D or in tension
and compression when kD < D
However, two points are to be remembered while expanding the equation
∑V = 0. The first is that while computing the force of steel in compression, the
force of concrete that is not available at the location of longitudinal steel has to
be subtracted. The second point is that the total force of steel shall consist of the
summation of forces in every row of steel having different stresses depending on
the respective distances from the centroidal axis. These two points are also to be
considered while expanding the other equation ∑M = 0. Moreover, negative sign
should be used for the tensile force of steel on the left of the neutral axis when
kD < D.
It is now possible to draw the interaction diagram of a trial section for the
given values of Pu and Mu following the three steps mentioned in sec.10.23.4.
However, such an attempt should be avoided for the reason explained below.
It has been mentioned in sec.10.23.2 that any point on the interaction
diagram gives a pair of values of Pu and Mu causing collapse. On the other hand,
it is also true that for the given Pu and Mu, several sections are possible. Drawing
of interaction diagrams for all the trial sections is time consuming. Therefore, it is
necessary to recast the interaction diagram selecting appropriate nondimensional parameters instead of Pu versus Mu as has been explained in this
lesson. Non-dimensional interaction diagram has the advantage of selecting
alternative sections quickly for a given pair of Pu and Mu. It is worth mentioning
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that all the aspects of the behaviour of column and the modes of failure shall
remain valid in constructing the more versatile non-dimensional interaction
diagram, which is taken up in Lesson 24.

10.23.11 Practice Questions and Problems with Answers
Q.1: Draw four typical strain profiles of a short, rectangular and symmetrically
reinforced concrete column causing collapse subjected to different pairs of
Pu and Mu when the depths of the neutral axis are (i) less than the depth of
column D, (ii) equal to the depth of column D, (iii) D < kD < ∞ and (iv) kD
= ∞ . Explain the behaviour of column for each of the four strain profiles.
A.1:

See sec. 10.23.2.

Q.2: Name and explain the three modes of failures of short, rectangular and
symmetrically reinforced concrete columns subjected to axial load Pu
uniaxial moment Mu.
A.2:

See sec.10.23.3.

Q.3:

Draw a typical interaction diagram, and explain the three zones
representing three modes of failure of a short, rectangular and
symmetrically reinforced concrete column subjected to axial load Pu and
uniaxial moment Mu.

A.3:

See sec.10.23.4.

Q.4: (a) Draw the compressive stress block of concrete of a short, rectangular
and symmetrically reinforced concrete column subjected to axial load Pu
and uniaxial moment Mu, when the neutral axis lies outside the section.
(b) Derive expressions of determining the area of the compressive stress
block of concrete and distance of the centroid of the compressive stress
block from the highly compressed right edge for a column of Q4(a).
A.4:

See sec.10.23.5.

Q.5: Derive expression of determining the stresses anywhere within the section
of a column of Q4.
A.5:

See sec.10.23.6.

Q.6: (a) Draw the compressive stress block of concrete of a short, rectangular
and symmetrically reinforced concrete column subjected to axial load
Pu and uniaxial moment Mu, when the neutral axis is within the section.

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(b) Derive expressions of determining the area of the compressive stress
block of concrete and distance of the centroid of the compressive
stress block from the compressed right edge for a column of Q6(a).
A.6:

See sec.10.23.7.

Q.7:

Derive expression of determining the compressive stress in the
compression zone of a column of Q6.

A.7:

See sec.10.23.8.

Q.8:

Explain the principle of determining the stresses (both tensile and
compressive) of longitudinal steel of a short, rectangular and
symmetrically reinforced concrete column subjected to axial load Pu and
uniaxial moment Mu.

A.8:

See sec.10.23.9.

Q.9: (a) Write the governing equations of equilibrium of a short, rectangular and
symmetrically reinforced concrete column subjected to axial load Pu
and uniaxial moment Mu.
(b) Would you use the equations of equilibrium for the design of a short,
rectangular and symmetrically reinforced concrete column for a given
pair of Pu and Mu? Justify your answer.
A.9:

See sec.10.23.10.

10.23.12 References
1. Reinforced Concrete Limit State Design, 6th Edition, by Ashok K. Jain,
Nem Chand & Bros, Roorkee, 2002.
2. Limit State Design of Reinforced Concrete, 2nd 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 Concrete Design, 2nd Edition, by S.Unnikrishna Pillai 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. Reinforced Concrete Design, 1st Revised Edition, by S.N.Sinha, Tata
McGraw-Hill Publishing Company. New Delhi, 1990.
7. Reinforced Concrete, 6th Edition, by S.K.Mallick and A.P.Gupta, Oxford &
IBH Publishing Co. Pvt. Ltd. New Delhi, 1996.

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8. Behaviour, Analysis & Design of Reinforced Concrete Structural Elements,
by I.C.Syal and R.K.Ummat, A.H.Wheeler & Co. Ltd., Allahabad, 1989.
9. Reinforced Concrete Structures, 3rd Edition, by I.C.Syal and 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. Design of Concrete Structures, 13th 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. Properties of Concrete, 4th Edition, 1st Indian reprint, by A.M.Neville,
Longman, 2000.
14. Reinforced Concrete Designer’s Handbook, 10th Edition, by C.E.Reynolds
and J.C.Steedman, E & FN SPON, London, 1997.
15. Indian Standard Plain and Reinforced Concrete – Code of Practice (4th
Revision), IS 456: 2000, BIS, New Delhi.
16. Design Aids for Reinforced Concrete to IS: 456 – 1978, BIS, New Delhi.

10.23.13 Test 23 with Solutions
Maximum Marks = 50,

Maximum Time = 30 minutes

Answer all questions.
TQ.1: Each of the following statements has four possible answers. Choose the
correct answer
(2
x 5 = 20 marks)
(a) The designed axial load of a short column has the theoretical carrying
capacity before it collapses
(i)
axis.

P = Po only as obtained from the interaction diagram on the vertical

(ii) P = Designed axial load with the code stipulated minimum eccentricity
only.
(iii) A pair of Pb and Mb only.
(iv) All of the above.
A.TQ.1a: (iv)
(b) A short column in compression failure due to an axial load Pu and
uniaxial moment Mu may have
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(i) kD = 0 and e = 0
(ii) kD = ∞ and e = 0
(iii) kD = 0 and e = ∞
(iv) kD = ∞ and e = ∞
A.TQ.1b: (ii)
(c) The fulcrum of the strain profile of a short column is a point through
which
(i) The strain profiles causing compression failure will pass.
(ii) The strain profile causing balanced failure will pass.
(iii) The strain profiles having no tension and causing compression failure
will pass.
(iv) The strain profiles causing tension failure will pass.
A.TQ.1c: (iii)
(d) The maximum compressive strain of concrete in balanced failure of a
short column subjected to Pb and Mb is
(i) 0.0035
(ii) 0.0035 minus 0.75 times the tensile strain of steel
(iii) 0.002
(iv) None of the above
A.TQ.1d: (i)
TQ.2: (a) Draw the compressive stress block of concrete of a short, rectangular
and symmetrically reinforced concrete column subjected to axial load
Pu and uniaxial moment Mu, when the neutral axis is within the section.
(b) Derive expressions of determining the area of the compressive stress
block of concrete and distance of the centroid of the compressive
stress block from the compressed right edge for a column of TQ.2 (a).
(10 + 20 = 30)
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A.TQ.2:

See sec.10.23.7.

10.23.14 Summary of this Lesson
Illustrating the behaviour of short, rectangular and symmetrically
reinforced rectangular columns under axial load Pu and uniaxial bending Mu, this
lesson explains the three modes of failure and the interaction diagram of such
columns. The different possible strain profiles, and the compressive stress blocks
are drawn and explained when the neutral axis is within and outside the crosssection of the column. Determination of compressive stresses of concrete and
tensile/compressive stresses of longitudinal steel are explained. The governing
equations of equilibrium are introduced to illustrate the need for recasting them in
non-dimensional form for the purpose of design of such columns.

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