Analysis and Design of Rectangular and L-Shaped Columns Subjected to Axial Load and Biaxial Bending

IJSEA (2016) 15–22 © JournalsPub 2016. All Rights Reserved Page 15
International Journal of Structural Engineering and Analysis
Vol. 2: Issue 1
www.journalspub.com
Analysis and Design of Rectangular and L-Shaped Columns
Subjected to Axial Load and Biaxial Bending
N. Dahiya*, V.K. Sehgal, B. Saini
Department of Civil Engineering, National Institute of Technology, Kurukshetra, Haryana, India
Abstract
Next to rectangular and circular columns, L-shaped columns may be the most frequently
encountered reinforced concrete columns, since they can be used as a corner column in
framed structures. The behaviour of irregular shaped reinforced concrete columns has been
a constant concern for a structural engineer, to design a safe and economic structure in
modern buildings and bridge piers. L-shaped reinforced concrete column subjected to biaxial
bending and axial compression is a common design problem. Axial load capacity and
Moment capacity of rectangular and L-shaped reinforced concrete columns have been done
in this work. A computer program has been developed to obtain the axial load capacity and
moment capacity of reinforced concrete columns of rectangular and L-shaped.
*Corresponding Author
E-mail: nitindahiya1@gmail.com
INTRODUCTION
Columns are important structural elements
which support floors and roofs. They are
the compression members and their failure
may endanger the whole structure.
Columns are members used primarily to
support axial compression and have a ratio
of height to the least lateral dimension of 3
or greater. In reinforced concrete
buildings, vertical member are subjected to
combine axial loads and bending
moments. These forces develop due to
external loads, such as dead load, live
load, and wind load.[1]
The strength of
column depends on the strength of the
material, shape, size of the cross section,
length and the degree of positional and
directional restraints at its ends. Column
may be defined as an element used to
support axial load and moments. Columns
are usually subjected to bending moments
about two perpendicular axes (X and Y) as
well as an axial force in the vertical (Z)
direction. A column located in the building
corner, encounters biaxial bending. In
recent past, the designers have started
using irregular shaped columns at the
corners of the buildings and at enclosure of
elevator shafts. The research work done by
many researchers has made it possible to
develop different design criteria for
biaxially loaded columns using working
stress and limit state design methods. The
present IS code (IS 456-2000) and design
aid (SP-16) follows the strength criteria as
a basis for designing reinforced concrete
columns in which the failure is defined in
terms of a limiting strain and stress in
concrete and the reinforcement.[2,3]
The methods available for design of
biaxially loaded columns are based on
(1) the equilibrium equations, which lead
to iterative method and
(2) Ultimate load capacity, which lead to
determining failure surfaces in
columns.
The concept of using failure surfaces has
been presented by Boris Bresler in 1960.
Bresler proposed two methods. The
reciprocal method (first method) uses

Analysis and Design of Rectangular and L-Shaped Columns Dahiya et al.
simple equations and gives satisfactory
results.[4]
1
𝑃𝑖
=
1
𝑃𝑥
+
1
𝑃𝑦
−
1
𝑃𝑜
where Pi = Ultimate axial load capacity
under biaxial eccentricities ex and ey
Px and Py=Ultimate axial load capacity
under uni-axial eccentricities ex and ey,
respectively
Po = Ultimate concentric axial load
capacity
The second method used the load contour
method and gives a general non-
dimensional equation, which has been
used by IS 456-2000 for design of
biaxially loaded columns.[5]
(
𝑀𝑥
𝑀𝑥𝑜
)
∝
+ (
𝑀 𝑦
𝑀 𝑦𝑜
)
𝛽
= 1.0
However, this equation has been modified
by Bureau of Indian standards.
Ramamurthy and Khan (1983) presented
two methods to represent the load contour
in L-shaped columns. First method is
based on the failure surfaces in the column
and second method proposes to be
replaced by the simple analysis of an
equivalent rectangular section.[6]
Thomas
Hsu in 1985 presented a computer
program which has been developed to
determine the ultimate axial load capacity
of L-section. In 1992, Mallikarjuna
presented a method based on limit state
analysis. A computer program has been
developed to determine the axial load
capacity of L-shaped section under biaxial
bending and axial compression. There are
less design aids for L-shaped reinforced
concrete column subjected to axial load
and biaxial bending, manual analysis of L-
shaped column was lengthy and
cumbersome process.[6,7]
CODAL PROVISIONS
IS 456-2000 recommends the following
assumptions
 Plane sections normal to the axis
remain plane after bending.
 The maximum strain in concrete at the
outermost compression fibre is taken
as 0.0035in bending.
 The relationship between the
compressive stress distribution in
concrete and the strain in concrete may
be assumed to be rectangle, trapezoid,
parabola or any other shape which
results in prediction of strength in
substantial agreement with the results
of test. For design purposes, the
compressive strength of concrete in the
structure shall be assumed to be 0.67
times the characteristic strength. The
partial safety factor 𝛾 𝑚(= 1.5) shall be
applied in addition tothis.[8–11]
 The tensile strength of the concrete is
ignored.
 The stresses in the reinforcement are
derived from representative stress-
strain curve for the type of steel used.
For design purposes the partial safety
factor 𝛾 𝑚(= 1.15) shall be applied.
 The maximum strain in the tension
reinforcement in the section at failure
shall not be less than:
(Fy/1.15 Es)+0.002
 The maximum compressive strain in
concrete in axial compression is taken
as 0.002.
 The maximum compressive strain at
the highly compressed extreme fibre in
concrete subjected to axial
compression and bending and when
there is no tension on the section shall
be 0.0035 minus 0.75 times the strain
at the least compressed extreme fibre.
 The maximum compressive strain at
the highly compressed extreme fibre in
concrete subjected to axial
compression and bending, when part of
the section is in tension, is taken as
0.0035. In the limiting case, when the
neutral axis lies along one edge of the
section, the strain varies from 0.0035
at the highly compressed edge to zero
at the opposite edge.[12–17]

Vol. 2: Issue 1
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METHOD OF ANALYSIS
If the columns have axial load and bending
moment about either the x-axis or y-axis
only they are classified as uniaxially
eccentrically loaded columns. The
behaviour of column subjected to axial
load and moment depends on the
magnitude of bending moment in relation
to axial load. If the bending moment is
large, a part of the column section may be
under tension with neutral axis lying inside
the section while for small magnitude of
bending moment the complete section will
be in compression. Thus, the stress
distribution across the section depends on
whether the neutral axis lies outside or
inside the section. The analysis process for
determining the ultimate strength of R.C.
columns subjected to axial compression
with bending is based on limiting the
maximum strain in the concrete to some
prescribed value. A program has been
developed to design a short column under
biaxial bending and axial compression.
Case 1
When neutral axis lies outside the section,
the axial load capacity and moment
capacity can be determined as follows:
Axial load capacity:
𝑃𝑢 = 𝑃𝑢𝑐 + ∑ 𝑃𝑢𝑠𝑡
𝑃𝑢𝑐 = (𝐶1 ∗ 𝐹𝑐𝑘 ∗ 𝐴 𝑔)
𝐶1 = 0.446(1 − 𝐶3/6) and 𝐶3 =
8/7(4/7𝑘 − 3)2
∑𝑃𝑢𝑠𝑡 = ∑𝐴 𝑠𝑖(𝑓𝑠𝑖 − 𝑓𝑐𝑖)
where Puc and Pust are axial force taken by
concrete and reinforcement, respectively.
Moment capacity:
𝑀 𝑢𝑥 = 𝑀 𝑢𝑐𝑥 + ∑𝑀 𝑢𝑠𝑡𝑥
𝑀 𝑢𝑐𝑥 = 𝑃𝑢𝑐 ∗ 𝐷((𝐶/𝐷) − 𝐶2)
𝐶2 = ((𝐶/𝐷) − 𝐶3/7)/(1 − 𝐶3/6)
∑𝑀 𝑢𝑠𝑡𝑥 = ∑𝑃𝑢𝑠𝑡 ∗ 𝑥𝑖
where Mucx and Mustx are the moment of
resistance due to concrete and
reinforcement, respectively
Similarly
𝑀 𝑢𝑦 = 𝑀 𝑢𝑐𝑦 + ∑𝑀 𝑢𝑠𝑡𝑦
where
𝑀 𝑢𝑐𝑦 = 𝑃𝑢𝑐 ∗ 𝐵((𝐶/𝐵) − 𝐶2)
𝐶2 = ((𝐶/𝐵) − 𝐶3/7)/(1 − 𝐶3/6)
∑𝑀 𝑢𝑠𝑡𝑦 = ∑𝑃𝑢𝑠𝑡 ∗ 𝑥𝑖
where Mucy and Musty are moment of
resistance due to concrete and
reinforcement respectively.
Case 2
When neutral axis lies inside the section,
the axial load capacity and moment
capacity can be computed as follows:
Axial load capacity
𝑃𝑢 = 𝑃𝑢𝑐 + ∑ 𝑃𝑢𝑠𝑡
𝑃𝑢𝑐 = (𝐶1 ∗ 𝐹𝑐𝑘 ∗ 𝐴 𝑔)
𝐶1 = 0.361 ∗ 𝑘
∑𝑃𝑢𝑠𝑡 = ∑𝐴 𝑠𝑖(𝑓𝑠𝑖 − 𝑓𝑐𝑖)
The tension taken by concrete is neglected.
Moment capacity:
𝑀 𝑢𝑥 = 𝑀 𝑢𝑐𝑥 + ∑𝑀 𝑢𝑠𝑡𝑥
where
𝑀 𝑢𝑐𝑥 = 𝑃𝑢𝑐 ∗ 𝐷((𝐶/𝐷) − 𝐶2)
𝐶2 = 0.416 ∗ 𝑘
∑𝑀 𝑢𝑠𝑡𝑥 = ∑𝑃𝑢𝑠𝑡 ∗ 𝑥𝑖
Similarly
𝑀 𝑢𝑦 = 𝑀 𝑢𝑐𝑦 + ∑𝑀 𝑢𝑠𝑡𝑦
where
𝑀 𝑢𝑐𝑦 = 𝑃𝑢𝑐 ∗ 𝐵((𝐶/𝐵) − 𝐶2)
𝐶2 = 0.416 ∗ 𝑘

𝑀 𝑢𝑠𝑡𝑥 = ∑𝑃𝑢𝑠𝑡 ∗ 𝑥𝑖
If the applied load is greater than the
computed axial load capacity, it means that
the compressive resistance of the section,
Pu needs to be increased by increasing the
depth of the neutral axis and the
compressive resistance of the concrete are
recalculated using given equations.[18]
Alternatively, if the applied load is less
than the computed axial load capacity, the
compressive resistance of concrete is
required to be reduced by decreasing the
depth of neutral axis and Pu is recalculated.
The process is repeated till the difference
is within the desired accuracy and final
value of the depth of neutral axis gets
fixed.
The next step is to calculate the ultimate
moment of resistance of the section, Mu
corresponding to the depth of neutral axis.
If the moment capacity is less than the
applied moment, then the area of steel is
increased and the whole process is
repeated till the requisite condition is
achieved.[19]
EXAMPLES AND VERIFICATION
Six examples are presented in detail and
show the validity of the proposed method.
Example 1
Design a short column under biaxial
bending with dimension 400mm × 600
mm nominal cover of 50 mm and
subjected to an axial load of 1600 kN and
a factored moments of 120 and 90 kNm
about x-axis and y-axis respectively.
Compressive strength of concrete (Fck) is
15 N/mm2
andYield strength of steel (Fy) is
415 N/mm2
.
Solution: The input data for the computer
program is as follows:
Fck=15 N/mm2
Fy= 415 N/mm2
Axial load =1600 kN
Moment about X-axis=120 kNm
Moment about Y-axis=90 kNm
A factored moment of 75 and 60 kNm
about the x- and y-axis, respectively.
15 N/mm2andYield strengths of steel (Fy)
is 415 N/mm2
Design parameters Proposed method IS 456-2000 Mallikarjuna (1989)
Moment capacity about x-axis (Nmm) 1.8938E8 1.94E8 2.0521E8
Moment capacity about y-axis (Nmm) 1.3067E8 1.19E8 1.2246E8
Area of longitudinal steel (mm2
) 3024 3053 3048
Value of alpha 1.7062 1.745 1.735
L.H.S of inequality equation 0.99 1.04 0.98

Vol. 2: Issue 1
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Example 2
bending with dimension 450 mm × 450
a factored moments of 75 and 60 kNm
about x- and y-axis, respectively.
15 N/mm2
and yield strength of steel (Fy)
is 415 N/mm2
Design parameters Proposed method A.K. Jain
Moment capacity about x-axis (Nmm) 1.14969E8 1.093E8
Moment capacity about y-axis (Nmm) 1.14969E8 1.093E8
) 1620 1620
Value of alpha 1.5579 1.55
L.H.S of inequality equation 0.88 0.94
Example 3
bending with dimension 400mm × 400
a factored moments of 190 kNm and 110
kNm about x-axis and y-axis respectively.
25 N/mm2
andYield strength of steel (Fy) is
415 N/mm2
.[20–23]
Solution
The moment capacities about x-axis and y-
axis are slightly different due to the fact,
the proposed method uses equilibrium
equations for analysis and, while Pillai and
Menon uses interaction curves given in
SP-16.
The value of alpha is well comparable in
both the cases.
The value of L.H.S of inequality equation
is 0.99, and the result is well comparable
with the result reported by Pillai and
Menon
Design parameters Proposed method Pillai and Menon
Moment capacity about x-axis (Nmm) 3.248E8 2.64E8
Moment capacity about y-axis (Nmm) 1.752E8 2.64E8
) 5568 5600
Value of alpha 1.393 1.273
L.H.S of inequality equation 0.99 0.986
Example 4
bending with dimension 1000mm ×
1000mm × 250mm nominal cover of 50
mm subjected to an axial load of 4000 kN
and a factored moments of 750 and 750
kNm about x- and y-axis, respectively.
25 N/mm2
is 415 N/mm2
.

Solution
The proposed method compute moment
capacities about x-axis and y-axis using
equilibrium equations while S.K. Sinha
uses interaction curves developed by him.
S.K. Sinha gave only area of longitudinal
steel. The value of L.H.S of inequality
equation is 0.99, and the result is well
comparable with the result reported by
S.K. Sinha.
Design parameters Proposed method S.K. Sinha
Moment capacity about x-axis (Nmm) 1.18963E9 –
Moment capacity about y-axis (Nmm) 1.18963E9 –
) 10,063 14,875
Value of alpha 1.5028 –
L.H.S of inequality equation 0.99 –
Example 5
mm × 75 mm nominal cover of 20 mm and
subjected to an axial load of 106.7 kN and
a factored moments of 13.56 and 4.14
25 N/mm2
is 415 N/mm2
.
Solution
Using the proposed method the moment
capacity, axial load capacity, area of
longitudinal reinforcement has been
calculated and results are checked with the
results of Hsu.
Hsu used value of alpha (α) as 1.5 for all
the cases, whereas value of alpha (α)
depends on the value of Pu/Puz. It can be
seen that value of interaction equation is
0.99 whereas value of interaction equation
reported by Hsu is 1.02.
The moment capacity and area of steel
were in good agreement with those
reported by Hsu
Design parameters Proposed method Hsu (1985) Interaction curve
Moment capacity about x-axis (N.mm) 2.735E7 2.11E7 2.21E7
Moment capacity about y-axis (N.mm) 7.856E6 6.554E6 14.53E6
) 1018 1098 1017
Value of alpha 1.04 1.5 1.01
L.H.S of inequality equation 0.99 1.02 0.91
Example 6
mm × 200 mm nominal cover of 50 mm
and subjected to an axial load of 890.8 kN
and a factored moments of 90.4 and 56.5
25 N/mm2
is 415 N/mm2
.

Vol. 2: Issue 1
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Solution
The moment capacities about x-axis and y-
axis are slightly different due to the fact,
the proposed method uses equilibrium
equations for analysis and design, while
Ramamurthy (method 1) uses failure
surface for developing actual shapes of
load contour and method 2 uses equivalent
Square or rectangular column for analysis
and design. Ramamurthy calculated
moments about the major axis only
neglecting the effect of minor axis. The
value of L.H.S of inequality equation is
0.99. The moment capacity and area of
steel were in good agreement with those
reported by Ramamurthy and Khan using
method 1 and method 2.[23–25]
Designers Proposed
method
Ramamurthy
(method 1)
Ramamurthy
(method 2)
Interaction
curve
Moment capacity about x-axis
(Nmm)
1.616E8 1.18E8 1.22E8 1.59E8
Moment capacity about y-axis
(Nmm)
0.919E8 – – 1.19E8
Ares of longitudinal steel
(mm2
)
3890 3920 3920 3890
Value of alpha 1.302 – – 1.31
L.H.S of inequality equation 0.99 – – 0.89
CONCLUSIONS
The analysis and design of L-shaped
column under biaxial bending and axial
compression are cumbersome and time
consuming. So, a computer program has
been developed for the analysis and design
of L-shaped column under biaxial bending
and axial compression. From the analytical
results following conclusions can be
deduced.
(1) The computer program accurately
gives the axial load and moment
capacity of the section.
(2) The algorithm allows the designer to
determine the area of longitudinal
reinforcement required by the selected
cross section.
(3) The results of this investigation can be
used to develop the interaction curves
for a given reinforcement layout.
(4) Designer can economise on the
reinforcement for a given section using
this computer program.
(5) This program can be used as an
important tool by practising engineer.
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1. Jain A.K. Reinforced Concrete Limit
State Design. Nemchand & Bros.
Publication.
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3. Horowitz B. Design of columns
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5. Dunar C., Sahin B. Arbitrary shaped
reinforced concrete members subjected
to biaxial bending and axial load,
Comput Struct. 1992; 44(4): 643–62p.
6. Yau C.Y., Chan S.L., So A.K.W.
Biaxial bending design of arbitrarily
shaped reinforced concrete column,
ACI Struct J. 1993; 90(3): 269–78p.
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and reinforced concrete, Bureau of
Indian Standards, New Delhi.

8. Weber D.C. Ultimate strength design
charts for columns with biaxial
bending, ACI J. 1966; 1205–29p.
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concrete to IS 456-1978, Bureau of
Indian Standards, New Delhi.
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11. Hsu C.-T.T. Analysis and design of
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21. Ramamurthy, Khan. L-shaped column
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