This article proposes a method to determine the storey-wise column size for displacement-based design of reinforced concrete frame buildings accounting for all components of frame deformation. The method considers deformation due to column flexure, shear, beam flexure, plastic rotation, and joint shear. It also accounts for dynamic amplification of drift and moments using empirical relationships developed from nonlinear time history analyses. The column size is determined through an iterative computer program-based algorithm to ensure columns remain elastic while other members are allowed to deform inelastically up to code-specified drift limits. The method was validated for buildings up to 12 stories with target drifts up to 3%.

1. Determination of column size for displacement-based design of
reinforced concrete frame buildings
S. S. Mayengbam*,†
and S. Choudhury
Civil Engineering Department, National Institute of Technology, Silchar, Assam, India
SUMMARY
This article reports a method to determine the storey-wise column size for displacement-based design
of reinforced concrete frame buildings with a wide range of storey drift and building plan. The method
uses a computer program based algorithm. The basic relation used in the algorithm is formulated by
considering the various possible deformation components involved in the overall frame deformation.
As a necessity to represent the deformation component due to plastic rotation of beam members, a
relation between the beam plastic rotation and the target-drift is adopted. To control the dynamic
amplification of interstorey drift, a target-drift dependant design-drift reduction factor is used. The
dynamic amplification of column moment is accounted with the help of an approximate conversion
of fundamental period of the building from the effective period of the equivalent SDOF system. To
avoid the formation of plastic hinge in column members, a design-drift dependant column–beam
moment capacity ratio is used. The method successfully determines the storey-wise column size for
buildings of four plans of different varieties, heights up to 12 storeys and target-drift up to 3%.
Copyright © 2013 John Wiley & Sons, Ltd.
Received 18 August 2012; Revised 19 October 2013; Accepted 20 October 2013
KEY WORDS: column size; storey drift; beam plastic rotation; displacement based design; frame
deformation components; dynamic amplification
1. INTRODUCTION
With any design method of reinforced concrete (RC) frame buildings, when the member sections have
to fulfil the required design constraints (mainly, the demand and the maximum reasonable percentage
reinforcement), the process essentially becomes an iterative one. Mayengbam and Choudhury [1]
introduced a method for calculating the column size using a computer program for displacement-
based design (DBD) of RC frame buildings with intermediate drift of 2% interstory drift ratio (IDR).
They assumed that the overall column drift is composed of only flexure and shear components. The
input parameters are beam size, design IDR, building geometry and those parameters generally
required for DBD.
The present study propose an upgrade to the method suggested by Mayengbam and
Choudhury [1] in such a way that the column dimensioning method can be used for
determining column size of RC frame buildings with a wide range of design IDR. The
proposed method attempts to consider all the possible components of frame displacement
including the dynamic amplification phenomena.
*Correspondence to: S. S. Mayengbam, Civil Engineering Department, National Institute of Technology, Silchar,
Assam, India.
†
E-mail: sunil_mayengbam@hotmail.com
Copyright © 2013 John Wiley & Sons, Ltd.
EARTHQUAKE ENGINEERING & STRUCTURAL DYNAMICS
Earthquake Engng Struct. Dyn. 2014; 43:1149–1172
Published online 21 November 2013 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/eqe.2391

2. 2. DEVELOPMENT OF COLUMN SIZE DETERMINATION PROCEDURE
2.1. Displacement components considered by Mayengbam and Choudhury [1]
Mayengbam and Choudhury [1] considered only two deformation components. The first one is the
column tip displacement (Δf) within elastic limit due to flexure, according to Park and Paulay [2]
given by Equation (1). Here, εy represents the yield strain at expected strength of rebar steel, lcc the
distance between the point of contra flexure and the point of maximum bending moment, and hc the
column depth in the direction of earthquake under consideration.
Δf ¼
0:7εyl2
cc
hc
: (1)
The second deformation component is the average column displacement because of shear (Δs)
according to Matamoros [3]. The relation according to Matamoros [3] is modified (given in
Equation (2)) by assuming the effective depth of the column section (the distance from extreme
compression fibre to centre of tension steel) as 0.9 times hc. This assumption is based on 525
possible combinations of different column depth (500 to 1500 mm with an increment of 50 mm),
rebar size (diameter: 20 to 40 mm with an increment of 5 mm) and clear cover (50 to 90 mm with an
increment of 10 mm). Considering the possible combinations, the average effective depth of the
column sections is found to be 0.905608 (which can be taken as 0.9) times the overall depth. For
this assumption, high eccentricity is not considered on the tension side of the column section, and
the possible effect of axial force on the effective depth is ignored. Such assumption is also made by
Mayengbam and Choudhury [1].
Δs ¼
24V 1 þ νð Þcr
0:9h2
cE
: (2)
Where V represents the maximum shear force applied to the member, ν the Poisson’s ratio, cr the
ratio of column depth to column width and E the modulus of elasticity of the material.
2.2. Additional displacement components considered in the proposed method
In the proposed method, it is assumed that the overall lateral deformation of RC frame is mainly a
function of deformation due to column flexure, column shear, elastic joint rotation comprising of
beam flexure and joint shear, beam plastic rotation and bond slip.
The yield drift (θby) due to beam flexure (ignoring strain hardening), according to Priestley, Calvi
and Kowalsky [4], for flanged beam section can be represented by Equation (3). Here, lb and hb
represent the length and depth of the adjoining beam, respectively. And the beam member rotation
due to joint shear (θjy) can be taken as 0.25 times θby.
θby ¼
0:283εylb
hb
: (3)
Plastic rotation of beam members has a significant contribution in the overall deformation of frames
with inelastic beam deformation. In order to consider the component of beam plastic rotation in the
proposed method, it is necessary to identify its correlation with interstorey drift. If the beam
members are designed by following the special ductile detailing rule given in design codes, their
plastic rotation capacities for different limit states are expected to follow a similar trend or
characteristic. The criteria for selecting beam member size are given in Section 3. RC frame
buildings of 4 storey, 8 storey and 12 storey with different plans are designed for various design-
drifts and nonlinear time history analyses (THA) are conducted. The design procedure, material
properties, modelling and spectrum-compatible ground motions (SCGMs) used are similar to those
mentioned in Section 3. The plastic rotations of beam members (θbp) are compared with the
1150 S. S. MAYENGBAM AND S. CHOUDHURY
Copyright © 2013 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2014; 43:1149–1172
DOI: 10.1002/eqe

3. corresponding storey drifts achieved (θta). The graphical plots for the comparison are shown in
Figure 1. When the plots are represented by a curve of degree five polynomial, it shows a
gradually increasing pattern of θbp with the increase in θta. After passing around 2.5% drift
mark, the curve bends up towards vertical, probably showing a significant decline in the beam
member’s capability of taking further more deflection or storey drift. It is practically difficult to
suggest a precise representation for this type of uncertain scattering plots, even with other
complex equations. To fulfil the necessity of considering the contribution of θbp in the overall
frame deformation, a linear representation (Equation (4)) of the plots is adopted in the present
study, as an approximate deformation component of the beam plastic rotation. A compromise
to the column size determination procedure for this uncertain scattering of the plots is
presented in Section 2.4.
Equation (4), along with the requirement of Section 2.4, agrees well with the non linear THA results
presented later in Sections 4 and 5. However, with special studies, a better correlation based on a higher
degree of precision, firm engineering background and experimental results, can always be adopted,
which may bring more light to the proposed method.
θbp ¼ 0:00616θta À 0:00402: (4)
Sezen [5] experimentally showed that within elastic range, the displacement due to bond slip (Δb) in
column varies approximately from about 0.3 to 0.4 times the displacement due to column flexure, and
is two times the displacement due to column shear. Taking this ratio into account, the contribution of
bond slip (fb) can be assumed as 0.4 times that of column flexure within elastic range. Experimental
results by Sezen [5] literally suggest that if the deformations resulting from longitudinal bar slip are
ignored in the analysis or member modelling, the predicted member deformations may be
significantly smaller or the predicted lateral member stiffness may be larger than the actual one.
Consideration for bond slip deformation has not been mentioned specifically in DBD procedures.
On the other hand, available ideal method for computer modelling of bond slip (Mitra and
Lowes [6]), though appropriate at microlevel, is too complex to use in RC frame modelling at
macro level. Also, modelling should be carried out according to how the RC frame is designed
in order to reflect the exact performance of the frame under consideration. Looking into this
matter, deformation component due to bond slip is omitted both in designing and modelling in
the present study. Such practice is generally followed in analysing frame buildings by using
DBD. However, provision for incorporating the bond slip behaviour is given later in Equations
(13) and (14), where fb can be taken as 0.4 for slip up to member yield point. This provision
may be used with proper designing and modelling by considering bond slip behaviour.
Figure 1. Achieved interstory drift ratio in percentage versus beam plastic rotation in radian.
COLUMN SIZE FOR DISPLACEMENT-BASED DESIGN OF RC FRAME BUILDINGS 1151
Copyright © 2013 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2014; 43:1149–1172
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4. As per the purpose of this study, it is required that the column members remain elastic, irrespective
of other influencing actions including plastic deformation of the adjoining beam members. Therefore,
the contribution of plastic deformation of column member is assumed nil. The assumed deformation
components of a typical frame, framing the column–beam joint up to the adjacent member point of
inflections are shown in Figure 2. Here, the points of inflection are assumed to be located at the mid
span of each frame members so that the distance between two consecutive points of inflection along
the member can represent the member length. Taking together the various components considered
with respect to the storey height, the overall interstorey lateral frame deformation (Δtotal) can be
assumed as given in Equation (5). In Equation (5), lc represents the length of the column within a
storey (storey height).
Δtotal ¼ θby þ θbp þ θjy
À Á
 lc þ Δf þ Δs þ Δb: (5)
In DBD, the design is carried out for a required displacement under a prescribed hazard level.
Because DBD evolves with the logic that displacement can be directly related to structural damage,
it is important to maintain the structural displacement within the specified target displacement to
avoid unexpected damage in the structural members. But, if the dynamic behaviour of the structure
is not considered during the design, it is not possible to maintain the IDR without significant
deviation from the target limit. The main reason for this phenomenon is the effect of dynamic
amplifications of storey drift because of higher modes. It can be incorporated in the design by
simply reducing the design-drift from the target-drift by some amount and design with the reduced
design-drift. Pettinga and Priestley [7] used a design-drift reduction factor of 0.85 to control the
target IDR of perimeter seismic frames of various heights designed with 2% target-drift and achieve
10% maximum exceedance of IDR from the target limit. To represent the achieved IDR, they
considered the average IDR from the nonlinear THA results under five artificial ground motions.
They further suggested a design-drift reduction factor of {1 À 0.01(H/2)}. Here, H represents the
building height. According to FEMA-356 [8], if less than seven THA are performed, the
maximum response of the parameter of interest shall be used for design. Therefore, the maximum
response should also be used as an ideal value for checking the performance of buildings. To
predict the value of reduced design-drift (θdr), in order to achieve a required target-drift (θta), a
simple empirical relation is introduced through regression analysis based on 180 nonlinear THA
results of buildings with various heights and design-drifts (up to 3%). The design procedure,
material properties, modelling and SCGMs are similar to those mentioned in Section 3. A
graphical plot of the regression is shown in Figure 3, where the design IDR (θdr) is represented
Figure 2. Assumed deformation components of a typical frame, framing the column–beam joint up to the
adjacent member point of inflections.
1152 S. S. MAYENGBAM AND S. CHOUDHURY
Copyright © 2013 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2014; 43:1149–1172
DOI: 10.1002/eqe

5. along the vertical axis, whereas the corresponding achieved IDR (θta) is represented along the
horizontal axis. Here, the achieved IDR corresponds to the maximum out of five nonlinear THA
results. For controlling the target IDR (θta), θdr is to be used as the reduced design IDR, in order
to achieve an approximate IDR value of around θta. The relation is also found to be valid for
buildings of different geometry, heights and design IDR, where the maximum exceedance of IDR
from the target IDR is found to be less than 2% of the target IDR.
The dynamic amplification of IDR (ad) because of higher modes can be calculated from Equation
(6).
ad ¼
θta
θdr
(6)
Where; θdr ¼ 0:77θta À 0:03: (7)
Equation (7) represents the straight line regression shown in Figure 3. Equation (7) is also required
for the direct application of Equation (4) because in the design stage, θdr will be used. Beside
interstorey drift, column shear and moment are also amplified because of higher mode effect. The
dynamic amplification of column moment (am) and shear (as) because of higher modes can be taken
as per Paulay and Priestley [9], for one-way frame as given in Equations (8), (9) and (10). Here, φo
represents the design over strength factor, ωm the dynamic moment magnification factor and T1 the
fundamental period of the building. In this study, seismic forces are assumed to act along one of the
two principal directions of the building at a time.
am ¼ φoωm (8)
as ¼ 1:3φo (9)
Where; 1:3 ≤ωm ¼ 0:6T1 þ 0:85 ≤ 1:8: (10)
am is to be used during the column size determination before designing. It is not suitable to
calculate the T1 without the exact modal analysis. Because direct DBD will be involved in
each iteration of the proposed column size determination procedure, the effective period (Te)
of the equivalent single degree of freedom system will be readily available. Therefore, an
approximate transformation of effective period to elastic period is introduced by analysing 30 buildings
of different heights (up to 12 storey) and design-drifts (up to 3%), with elastic modal analysis and
Figure 3. Design versus achieved interstory drift ratio in percentage.
COLUMN SIZE FOR DISPLACEMENT-BASED DESIGN OF RC FRAME BUILDINGS 1153
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DOI: 10.1002/eqe

6. direct DBD. The material properties and building geometry are similar to those mentioned in Section 3.
The graphical plots of the observed T1 values are shown in Figure 4. The scattered plots can be
represented by a straight line as given by Equation (11.1). The comparison of this straight line with the
trend line of a typical rough estimation of T1 (Equation (11.2)) is also shown in Figure 4. In Equation
(11.2), μ represents the displacement ductility.
T1 ¼ 0:495Te þ 0:273 (11:1)
T1 ¼ Te=
ffiffiffi
μ
p
: (11:2)
While applying Equations (8), (9) and (10), some necessary assumptions are made. As a measure to deal
with a wide range of IDR, no minimum and maximum bound of ωm value is used. Also, no over strength of
materials is considered (taking φo =1) at this stage of formulation (or column size determination), though
unavoidable minimal over strength is allowed later at the reinforcement layout design stage.
Considering the dynamic amplifications of IDR, column moment and column shear, Equation (5)
can be written as given in Equation (12).
θdrlc  ad ¼ θby þ θbp þ θjy
À Á
 lc þ Δf  am þ Δs  as þ Δb: (12)
Equation (12) can be written by using Equations (1), (2), (3) and (4) as given in Equation (13), then
as Equation (14).
θdrlcad ¼ 0:283
εylb
hb
1 þ 0:25ð Þ þ 0:00616θta À 0:00402ð Þ
& '
lc þ
0:7εyl2
cc am þ f bð Þ
hc
þ
24V 1 þ νð Þcras
0:9h2
cE
(13)
⇒ θdrad À
0:35375εylb
hb
À 0:00616θta À 0:00402ð Þ
& '
lch2
c À am þ f bð Þ0:7εyl2
cchc
À
24V 1 þ νð Þcras
0:9E
¼ 0:
(14)
2.3. Solution of Equation (14)
Equation (14) can be solved by following a stepwise algorithm as described in the succeeding
text. In the following steps, subscript i indicates the current storey number, k indicates the
Figure 4. Approximate conversion of fundamental period from effective period.
1154 S. S. MAYENGBAM AND S. CHOUDHURY
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DOI: 10.1002/eqe

7. count within the Newton–Raphson iteration loop, which gives the storey-wise column depth, and
j indicates the count of comparison between two consecutive sets of storey-wise column depths.
Here, the next set of column depths is derived from the previous set of column depths through
DBD via DBD parameters resulting from the previous set of column depths.
(1) Take a trial ground storey column depth, hc1;1;1
À Á
, initially set hcj;i;k
for other storeys same
as hc1;1;1
.
(2) Calculate the seismic weight with respect to the current storey-wise column depths. Calculate
the floor-wise design lateral force by using DBD and the design column shear (distributed
equally among the columns in the storey).
(3) Calculate the corresponding column depth hcj;i;k
by using Equation (14) by Newton–Raphson
method with k number of iterations. Accordingly, Equations (15.1) and (15.2) represent the
idealisation of Equation (14) as a function of hcj;i;k
and its first derivative with respect to hcj;i;k
,
respectively. And, Equation (15.3) derives the next iterative value of hcj;i;k
, that is, hcj;i; kþ1ð Þ
.
f hcj;i;k
À Á
¼ θdrad À
0:35375εylb
hb
À 0:00616θta À 0:00402ð Þ
& '
lch2
cj;i;k
À am þ f bð Þ0:7εyl2
cchcj;i;k
À
24V 1 þ νð Þcras
0:9E
¼ 0 (15:1)
f ’
hcj;i;k
À Á
¼ θdrad À
0:35375εylb
hb
À 0:00616θta À 0:00402ð Þ
& '
2lchcj;i;k
À am þ f bð Þ0:7εyl2
cc
¼ 0 (15:2)
hcj;i; kþ1ð Þ
¼ hcj;i;k
À
f hcj;i;k
À Á
f ′
hcj;i;k
À Á : (15:3)
(4) If hcj;i; kþ1ð Þ
À hcj;i;k
0.0001, record the hcj;i; kþ1ð Þ
value.
Else, take hcj;i;k
¼ hcj;i; kþ1ð Þ
and go to step (3).
(5) Repeat steps (3) and (4) for i = 1 to n (n = total number of storey in the building), to find the
values of hcj;1;k
, hcj;2;k
, hcj;3;k
, … up to hcj;n;k
.
(6) If hc jþ1ð Þ;i;k
À hcj;i;k
0.0001, the resulting column depth for each storey matches for two consec-
utive iterations. If so, the last value of column depth for each storey gives the required depth for
the respective storey.
Else, take hcj;i;k
= hc jþ1ð Þ;i;k
and go to step (2).
Based on the described algorithm, a computer program is written and used for the determination of
column size.
2.4. Additional requirement
The uncertain scattering of θbp (shown in Figure 1) should affect the column size derived from
Equation (14), because the amount of plastic rotation of beam members has a substantial effect on
the overall deformation of the frame as a whole. An analysis for column depth derived from Equation
(14) is carried out for buildings of 4 storeys, 8 storeys and 12 storeys. The buildings are designed for
1%, 2% and 3% IDR in accordance with Section 3. It is found that significant fluctuations in θbp
value other than those given by Equation (4) causes instability in Equation (14), leading to unrealistic
COLUMN SIZE FOR DISPLACEMENT-BASED DESIGN OF RC FRAME BUILDINGS 1155
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8. column depth. The expected reason for such instability is that most of the displacement components
considered in the present study (except Δf in Equation (1), which is second order to lcc ) are linear
representations of their respective functions. Any uncooperative or unrelated fluctuation among the
components causes imbalance in Equation (14).
It is also found from the column depth analysis that 1% increase in θbp value leads to an average
increase of about 2% in the derived column depth. The maximum scattered θbp value in Figure 1 is
found to be 0.0242 radian at θta = 2.7%, and the corresponding θbp value given by Equation (4) at
θta = 2.7% is found to be 0.0126 radian. Thus, the maximum scattered θbp is about 1.92 times the value
given by Equation (4) at θta = 2.7%. In other words, maximum scattered θbp is found to be equal to
0.92 times the θbp given by Equation (4), plus the θbp given by Equation (4). If this maximum θbp
value is considered in determining the column depth, the resulting column depth will be very large
Figure 6. Column-to-beam moment capacity ratio: (a) required for ground floor column; and (b) maximum
required among the columns of the upper half floors of the building.
Figure 5. Eurocode 8 spectra (0.45 g acceleration level for type B soil); (a) design displacement spectra; and
(b) comparison of design spectrum with response spectra of spectrum-compatible ground motions used for
nonlinear time history analyses.
Table I. Details of spectrum-compatible ground motions (PEER database).
Name Background earthquake Record no. Magnitude (MW) Duration (seconds)
GM1 Duzce 1999 Duzce, 270 (ERD) 7.2 25.9
GM2 El Centro 1940 N-S Component 6.9 31.8
GM3 Gazli 1976 Karakyr, 090 7.1 16.3
GM4 Kocaeli 1999 Sakarya, 090 (ERD) 7.4 30.0
GM5 N. Palm Spring 1986 0920, USGS station 5070 5.9 20.0
GM, ground motion.
1156 S. S. MAYENGBAM AND S. CHOUDHURY
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9. and unrealistic. In the present study, 15% of the exceeding θbp value (15% of 0.92 = 13.8%) above
those given by Equation (4) is considered for every θta value, in determining the column depth.
This way, the calculated column depth will increase by about 27.6% (Two times 13.8%, as 1%
increase in θbp value leads to an average increase of about 2% in the derived column depth).
Hence, a compromise for the scattered θbp is made by adopting an average column depth
amplification factor of 1.3. This factor is multiplied to the resulting column depth as the last step of
the column depth determination procedure.
Because the derived column depth is based on several mathematical expressions, it will not be
favourable to use the exact dimension in practice. It can be rounded to the nearest 50 mm as far as
the section can accommodate some practicable ratio of design reinforcement, which is generally
4% of the cross-sectional area. The column section should also fulfil the requirements of
Figure 8. Typical force–deformation behaviour (FEMA-356 [8]).
Figure 7. Building plans used; (a) plan-A; (b) plan-B; (c) plan-C; and (d) plan-D.
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10. Table IV. Floor masses for plan-A buildings.
% IDR
Storey
height
Floor masses (kilo Newton)
Target Design 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th 11th 12th
1.0 0.74 4 764 733 702 556 — — — — — — — —
8 825 780 769 747 733 726 717 582 — — — —
12 875 836 805 781 768 765 761 757 752 747 741 600
2.0 1.51 4 534 512 490 361 — — — — — — — —
8 561 540 519 505 497 495 494 375 — — — —
12 615 580 552 532 523 522 521 521 520 519 518 387
3.0 2.28 4 436 425 414 310 — — — — — — — —
8 462 450 436 427 422 422 421 317 — — — —
12 502 479 460 447 441 441 440 440 440 439 439 326
IDR, interstory drift ratio.
Table III. Determined column size for plan-A buildings.
% IDR
Storey
height
Floor wise column depths in millimetres (square column)
Target Design 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th 11th 12th
1.0 0.74 4 1082 1010 900 811 — — — — — — — —
8 1156 1108 1008 939 867 845 816 784 — — — —
12 1279 1187 1082 1017 946 937 926 913 898 881 862 841
2.0 1.51 4 933 870 776 711 — — — — — — — —
8 952 907 816 757 699 694 687 681 — — — —
12 1136 1043 933 870 801 798 796 793 790 787 783 779
3.0 2.28 4 663 618 551 507 — — — — — — — —
8 755 719 645 598 554 551 549 545 — — — —
12 922 844 753 701 645 644 642 641 640 638 636 634
IDR, interstory drift ratio.
Table II. Overall beam depth used for plan-A buildings.
Beam span
in metres
Beam depths in millimetres
0.74% design IDR 1.51% design IDR 2.28% design IDR
4-storey 8-storey 12-storey 4-storey 8-storey 12-storey 4-storey 8-storey 12-storey
5.0 1000 1050 1050 600 650 650 500 525 525
6.0 1200 1260 1260 720 780 780 600 630 630
IDR, interstory drift ratio.
Figure 9. The biaxial moment interaction of a typical second storey column of eight-storey building with
1.125% design interstory drift ratio. (M2 and M3 are the moments in the two principal directions).
1158 S. S. MAYENGBAM AND S. CHOUDHURY
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DOI: 10.1002/eqe

11. TableV.Directdisplacement-baseddesignparametersforplan-Abuildings.
TargetIDR
(percentage)
DesignIDR
(percentage)
Storey
height
Design
displacement
(metres)
Effective
mass
(kiloNewton)
Effective
height
(metres)Ductility
Equivalent
viscous
damping
(percentage)
Effective
period
(seconds)
Effectivestiffness
in(kiloNewton
permetre)
Stability
ratio
Design
baseshear
(kiloNewton)
1.00.7440.070422769.511.1417.440.5043536530.006624893
80.1091486117.671.0015.020.6554474130.006048827
120.1604742326.160.9934.870.9423300980.008452947
2.01.5140.142315669.421.39710.91.066543930.03007741
80.2222328617.611.2659.241.585516210.035511477
120.3268508926.101.2559.102.303378750.050512390
3.02.2840.216113129.4801.76014.391.793161100.08433515
80.3363277517.651.54312.442.626158910.09715395
120.4942428226.151.53012.323.618129150.12446486
IDR,interstorydriftratio.
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12. Equations (17a) and (17b). With such type of application, the building performance is not expected
to change much. The present study focuses more on the initial development stage of the proposed
dimensioning method. So, the exact dimension is used to verify and validate the correlations
involved. It will be a logical step to a final design approach if the solution is simplified with some
general assumptions and displayed in a general design chart. Such approach will require another
level of similar study by considering wide ranges of design possibilities, which can be considered
as a future extension of the present work.
3. DESIGN CONSIDERATIONS AND METHODOLOGY
The design procedure, material properties, modelling and SCGMs (Table I) used in this study are
mostly similar to those used by Mayengbam and Choudhury [1]. Some important information and
dissimilarities are briefly summarised here for convenience. Design spectrum considered is of 0.45 g
acceleration level for type B soil as per Eurocode 8 (CEN [10]). In the design displacement spectra
set shown in Figure 5a, a corner period extension of 5 s is applied to tackle larger displacement
demand for higher storey buildings and also to incorporate the significance of magnitude on the
corner period as per Faccioli, Paolucci and Rey [11]. Materials used in the design include concrete
of cube strength of 30 MPa and reinforcing rebar steel with yield strength of 415 MPa. For both
materials, expected strengths are used as per FEMA-356 [8].
The selection of beam depth is based on the yield drift equation for RC frame (Priestley, Calvi and
Kowalsky [4]) given by Equation (16a). Here, θy represents the frame yield rotation.
θy ¼ 0:5εylb=hb: (16a)
Figure 10. Interstory drift ratio (IDR) diagram for plan-A buildings designed with 1% target IDR in long and
short directions; (a and d) 4 storey, (b and e) 8 storey, (c and f) and 12 storey.
1160 S. S. MAYENGBAM AND S. CHOUDHURY
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DOI: 10.1002/eqe

13. By assuming that the frame follows beam-sway mechanism after yielding, the design-drift (θd) can
be represented by Equation (16b).
θd ¼ θy þ θbp: (16b)
Substituting the θy value from Equation (16a) in Equation (16b), Equation (16c) (Choudhury and
Mayengbam [12]) can be obtained.
hb ¼
0:5εylb
θd À θbp
: (16c)
Here, it is assumed that in beam members, Immediate Occupancy (IO) level hinges form
when IDR varies from 1 to 1.5%, Life Safety (LS) level hinges with IDR from 1.5 to 2.5%
IDR, and Collapse Prevention (CP) level hinges with IDR beyond 2.5%. The corresponding
discrete values of θbp are taken from FEMA 356 [8] as averages of plastic rotation allowed
for beams controlled by flexure, which are 0.0063 radian for IO level, 0.0113 radian for LS
level and 0.02 radian for CP level. It should be noted that such assumptions are not followed
in determining the column size (where Equation (4) is used). It is expected that discrete θbp
values may cause imbalance in Equation (14). For a particular θd value, an approximate beam
depth is chosen in such a way that θbp lies within the assumed ranges. The beam width is
taken between 1/2 and 2/3 hb.
The direct DBD procedure used in this study is similar to that of Pettinga and Priestley [7]. Here,
similar dynamic amplifications of column shear and column moment are not considered. Capacity
Figure 11. Interstory drift ratio (IDR) diagram for plan-A buildings designed with 2% target IDR in long and
short directions; (a and d) 4 storey, (b and e) 8 storey, (c and f) and 12 storey.
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14. design can be carried out by amplifying the column moment and shear capacity by am and as from
Equations (8) and (9), respectively. Still, it has to fulfil some capacity design criteria to avoid
column-sway mechanism. In Mayengbam and Choudhury [1], where the design is carried out for
intermediate drift of 2% IDR, the capacity design criteria of Eurocode 8 (CEN [10]) is followed.
The present study, which deals with buildings of a wide range of design IDR, follows capacity
design recommendation given later in Equations (17a) and (17b). Dynamic amplification of IDR is
considered by reducing the target IDR (θta) to a design value of IDR (θdr) according to Equation (7),
and designing with the reduced IDR. For this reduction of IDR, no changes are required in the
developed method. The method will only determine column sizes corresponding to the reduced
design IDR. P-Δ effect is considered according to Pettinga and Priestley [13]. Modelling of
buildings and nonlinear THA are performed with SAP2000 (Computers and Structures, Inc.,
Berkeley, CA, USA) [14]. Figure 5b shows the matching comparison between design
acceleration spectrum and the SCGM acceleration spectra, both for 0.45g acceleration level with
5% damping.
An ideal way of avoiding column-sway mechanism is to make the column members remain elastic
regardless of how much member rotation or moment develops on the adjoining beam members. So, it
is required that proper column–beam moment capacity ratio (C/B ratio, MCB) is maintained to avoid
column-sway mechanism. C/B ratio can be maintained by making the design column moment
capacity greater than those of the adjacent beams, so that energy dissipates through beam
deformation, where the beam members are suitably designed and detailed for energy dissipation
under severe deformations. Eurocode 8 (CEN [10]) suggests that the sum of the design moments of
resistance of the columns framing into the joint should be at least 1.3 times the corresponding sum
of the design moments of resistance of the beams, whereas ACI 318 [15] suggests a value of 1.2
instead of 1.3, at the joint face. Codal design-drift is generally limited to 2% IDR. As the proposed
Figure 12. Interstory drift ratio (IDR) diagram for plan-A buildings designed with 3% target IDR in long and
short directions; (a and d) 4-storey, (b and e) 8-storey, (c and f) and 12-storey.
1162 S. S. MAYENGBAM AND S. CHOUDHURY
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15. method attempts to deal with a wide range of design and target-drift, a design-drift dependant C/B ratio
(Equation 17) is used instead of a constant value of 1.3 or 1.2. This is based on the fact that for RC
frames, member rotation is proportional to the overall frame displacement or drift.
Keeping in view that the C/B ratio required to avoid column-sway mechanism should change with
the variation of achieved IDR (and hence, design IDR), an analytical study of the relation of C/B ratio
with design IDR is conducted. An iterative search procedure is adopted to determine the required C/B
ratio for frame buildings of 4 storey, 8 storey and 12 storey designed with direct DBD, for a wide range
of design IDR. The procedure includes nonlinear THA of the buildings under five SCGMs for every
cycle to check for possibilities of column-sway mechanism. The design procedure, material
properties, modelling and SCGMs are similar to those mentioned in Section 3. It is found that the
value of C/B ratio required for the upper floor columns is more than those of the lower floor
column, except for the ground floor columns. Graphical plots of the required C/B ratio for columns
of the ground floor and the upper half floors of the buildings for different design IDR are shown in
Figures 6a and 6b, respectively. The scattered plots can be represented approximately as straight
lines given by Equations (17a) and (17b), respectively. In the present study, Equation (17b) is
adopted for all columns above ground floor.
For ground floor columns; MCB ¼ 0:49θdr þ 1:227 (17a)
For other floor columns; MCB ¼ 0:336θdr þ 1:007: (17b)
The use of moment frames at every bay of the building is still commonly in practice in several
developing countries. Such building frames have smaller members compared with the buildings with
CP hinge
LS hinge
IO hinge
Figure 13. Hinges in typical frames of plan-A buildings of 1% target IDR in long and short directions for the
mentioned heights and spectrum-compatible ground motions; (a and f) 4-storey GM2, (b and e) 8-storey
GM5, (c and d) and 12-storey GM3.
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16. perimeter seismic frames. When these two types of buildings are designed precisely for the same drift
level, they should achieve similar IDR. The internal frames sustaining gravity loads are generally not
modelled in designing buildings with perimeter seismic frames. The performance given by the
perimeter seismic frames may not be realistic because the same ground motion will also act on the
internal frames that are not modelled. Also, the gravity load sustaining internal frame has some of
its own lateral load carrying capacity even though not designed for seismic action. To reflect a
more realistic performance, buildings with moment frames at every bay are considered in the
present study.
Various building plans considered in the study are shown in Figure 7. Building with plan-A has
frames of regular geometry, spanning 5 and 6 m horizontally in the long and short direction,
respectively. Plan-B has frames of regular geometry with the same span of 6 m in both directions.
Plan-C and plan-D have frames of irregular spans in short direction. Frames in short direction of
plan-C comprise two 6 m and one 4 m spans, and three 6 m spans in long direction. Whereas in
plan-D, the frames in short direction have two 5.94 m spans and two 2.44 m corridor spans with no
corridor beams, and eight spans of 5.94 m each in the long direction. A constant floor height of
3.3 m is taken in all the buildings considered.
Square section is adopted for all columns in spite of the differences in geometry of the building
plans. The difference in moment demands of columns in the two principal directions of the
buildings is handled by providing adequate reinforcement to fulfil the required demand in each
direction. Modelling is carried out by providing the effective member section properties
according to Priestley, Calvi and Kowalsky [4], where the yield moments are derived from
design reinforcement. For beam members, yield moment is obtained from the SAP2000 [14]
hinge result. But for column members, the yield moment is obtained for the imposed axial force,
CP hinge
LS hinge
IO hinge
Figure 14. Hinges in typical frames of plan-A buildings of 2% target interstory drift ratio in long and short
directions for the mentioned heights and spectrum-compatible ground motions; (a and f) 4-storey GM1, (b
and e) 8-storey GM3, (c and d) and 12-storey GM5.
1164 S. S. MAYENGBAM AND S. CHOUDHURY
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DOI: 10.1002/eqe

17. from the column bidirectional moment interaction diagram. A full bidirectional interaction of the
moment capacities in the columns is considered in the modelling approach. The SAP2000 [14]
generated column biaxial moment interaction behaviour of the model is adopted to represent the
real life behaviour. The biaxial moment interaction of a typical second storey column of 8-storey
building with 1.125% design IDR is shown in Figure 8. Because the seismic action is
considered along the two principal axes one at a time, the biaxial moment interaction coming
into picture will be either corresponding to 0 and 180°, or 90 and 270°. It is important to
include the column biaxial moment interaction in the modelling because it reflects the effect
of axial load on the directional column moment. The modelling approach can be considered
improper if column biaxial moment interaction is not included.
The nonlinear material property of the RC members is taken as Takeda (option available in
SAP2000 [14]). The post-elastic force–deformation behaviour for the members (Figure 9) is adopted
as per FEMA-356 [8]. Nonlinear direct integration (Newmark, gamma = 0.5 and beta = 0.25) THA
along with period specified damping, is used for performance evaluation.
4. TRIAL VALIDATION
To verify Equation (14) along with Equations (4), 7 and the other relations involved, the method
is applied in determining the column sizes for frame buildings of plan-A (Figure 7a) with storey
heights of 4, 8 and 12. The buildings are designed with 1, 2 and 3% target IDRs. The overall
beam depths used for plan-A buildings are given in Table II, and the corresponding determined
CP hinge
LS hinge
IO hinge
Figure 15. Hinges in typical frames of plan-A buildings of 3% target interstory drift ratio in long and short
directions for the mentioned heights and spectrum-compatible ground motions; (a and f) 4-storey GM1, (b
and e) 8-storey GM3, (c and d) and 12-storey GM5.
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18. TableVI.Performanceachievedforplan-Abuildings.
Performanceparameters
1%targetIDR2%targetIDR3%targetIDR
4-storey8-storey12-storey4-storey8-storey12-storey4-storey8-storey12-storey
%IDR0.981.021.041.851.982.052.772.852.75
Max.hingelevelIOIOIOLSLSLSCPCPLS
Max.θbp×10À2
inradian0.18860.23060.22810.64210.96650.94481.62841.62231.5009
θbp×10À2
(Equation(4))inradian0.20170.22630.23860.73760.80540.86081.30431.35361.2920
CriticalGMGM1GM5GM3GM1GM3GM5GM1GM5GM5
DirectionLongShortShortLongLongLongLongLongLong
IDR,interstorydriftratio;IO,immediateoccupancy;LS,lifesafety;CP,collapseprevention;GM,groundmotion.
1166 S. S. MAYENGBAM AND S. CHOUDHURY
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19. floor-wise column sizes are given in Table III. The calculated floor masses and direct DBD
parameters are given in Tables IV and V.
Interstory drift ratio diagrams for plan-A buildings with 1, 2 and 3% target IDR are shown in
Figures 10, 11 and 12, respectively. The corresponding hinges pictures in typical frames are shown
in Figures 13, 14 and 15, respectively. The target IDRs of the buildings is achieved with a fair
precision. The overall maximum exceedance of IDR is found to be 4% of the target IDR, which is
in case of 12-storey building with 1% target IDR under GM5 in short direction (Figure 10f). No
column hinges develop in any of the building frames.
The performance achieved by the plan-A buildings can be summarised as given in Table VI.
The maximum level of plastic hinges formed in the beam members of plan-A buildings with 1,
2 and 3% target IDR are found to be IO, LS and CP, respectively. The average deviation of
plastic hinge rotation in beam members from the value given by Equation (4) is found to be
12.76%. The average value of 12.76% lies within the 15% value considered in section 2.4.
Table IX. Floor masses for buildings of plan-B, plan-C and plan-D.
% IDR
Building
plan
Storey
height
Floor masses (kilo Newton)
Target Design 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th
1.5 1.125 B 10 1243 1199 1158 1128 1111 1107 1102 1097 1093 877
3.0 2.28 B 6 826 809 788 777 774 600 — — — —
1.5 1.125 C 10 570 547 525 510 501 499 497 494 492 380
3.0 2.28 C 6 374 365 355 349 346 257 — — — —
1.5 1.125 D 10 1802 1714 1632 1572 1538 1529 1519 1509 1498 1057
3.0 2.28 D 6 1137 1108 1075 1055 1046 696 — — — —
IDR, interstory drift ratio.
Table VIII. Determined column size for buildings of plan-B, plan-C and plan-D.
% IDR
Building
plan
Storey
height
Floor wise column depth in millimetre (square column)
Target Design 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th
1.5 1.125 B 10 1128 1062 959 896 826 819 809 794 784 770
3.0 2.28 B 6 718 686 616 554 536 529 — — — —
1.5 1.125 C 10 1118 1052 949 887 818 810 801 788 777 763
3.0 2.28 C 6 712 680 611 563 530 525 — — — —
1.5 1.125 D 10 1300 1227 1108 1035 954 944 931 915 901 883
3.0 2.28 D 6 771 736 662 608 575 568 — — — —
IDR, interstory drift ratio.
Table VII. Overall beam depth used for buildings of plan-B, plan-C and plan-D.
Beam span
in metres
Beam depths in millimetres
Plan-B Plan-C Plan-D
10-storey,
1.125%
design IDR
6-storey,
2.28%
design IDR
10-storey,
1.125%
design IDR
6-storey,
2.28%
design IDR
10-storey,
1.125%
design IDR
6-storey,
2.28%
design IDR
6.0 960 600 960 600 — —
5.94 — — — — 950 600
4.0 — — 640 400 — —
IDR, Interstory drift ratio.
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20. TableX.Directdisplacement-baseddesignparametersforplan-B,plan-Candplan-Dbuildings.
TargetIDR
(percentage)
DesignIDR
(percentage)
Building
plan
Storey
height
Design
displacement
(metres)
Effective
mass(kilo
Newton)
Effective
height
(metres)Ductility
Equivalent
viscous
damping
(percentage)
Effective
period
(seconds)
Effective
stiffness
in(kilo
Newton
permetre)
Stability
ratio
Design
baseshear
(kiloNewton)
1.51.125B100.2052906621.951.1547.631.3821875370.021638488
3.02.28B60.2586385513.471.48011.801.992383390.07329958
1.51.125C100.2046408421.861.1547.641.378849400.021617376
3.02.28C60.2574172413.401.48211.821.985172820.07314469
1.51.125D100.20311247421.671.1567.671.369262800.021553373
3.02.28D60.2552514713.251.62313.212.047485100.078612464
IDR,interstorydriftratio.
1168 S. S. MAYENGBAM AND S. CHOUDHURY
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21. 5. VALIDATION FOR OTHER BUILDINGS
In Section 4, the verification is carried out for buildings of plan-A. This section presents the
validity of the method for buildings with different types of plans, heights and target IDRs.
Buildings of plan-B, plan-C and plan-D (Figures 7b, 7c and 7d) with storey height of 6 and
10 are considered. These buildings are designed for 1.5 and 3% target IDR. The overall beam
depth used and the determined column size for the buildings are given in Tables VII and VIII,
respectively. The calculated floor masses and direct DBD parameters are given in Tables IX
and X, respectively.
The buildings are subjected to nonlinear THA, the results of which are shown in the form of IDR
diagrams in Figures 16 and 17, and member hinges pictures in Figures 18 and 19. The maximum
exceedance of IDR is found to be 6.67% above target IDR. This is in case of 10-storey building
having plan-C with 1.125% design IDR, under GM1 in short direction (Figure 17e). The columns
members of the building frames remain elastic. The maximum level of plastic hinges formed in the
beam members of the buildings designed with 1.5 and 3% design IDR are found to be LS and CP,
respectively.
The performance achieved by the buildings is summarised in Table XI. The average deviation
of plastic hinge rotation in beam members from the value given by Equation (4) is found to be
15.14%. The average value of 15.14% agrees with the 15% value considered in Section 2.4.
The empirical requirements of the proposed column dimensioning method are mostly based on
plan-A buildings, for which the verification is carried out in Section 4. With the results
Figure 16. Interstory drift ratio (IDR) diagram for 6-storey buildings of 3% target IDR in long and short di-
rections of (a and d) plan-B, (b and e) plan-C, and (c and f) plan-D.
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22. CP hinge
LS hinge
IO hinge
Figure 18. Hinges in typical six-storey building frames of 3% target interstory drift ratio in long and short
directions for the mentioned plans and spectrum-compatible ground motions; (a and d) plan-B GM5, (b
and e) plan-C GM5, and (c and f) plan-D GM5.
GM-1
GM-2
GM-3
GM-4
GM-5
Target IDR
Figure 17. Interstory drift ratio (IDR) diagram for 10-storey buildings of 1.5% target IDR in long and short
directions of (a and d) plan-B, (b and e) plan-C, and (c and f) plan-D.
1170 S. S. MAYENGBAM AND S. CHOUDHURY
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23. presented in this section, it can be concluded that the proposed method of determining column
dimension is effective in case of the different building geometries and plans, and different
design IDR considered in the present study.
CP hinge
LS hinge
IO hinge
Figure 19. Hinges in typical 10-storey building frames of 1.5% target interstory drift ratio in long and short
directions for the mentioned plans and spectrum-compatible ground motions; (a and d) plan-B GM1, (b and
e) plan-C GM1, and (c and f) plan-D GM3.
Table XI. Performance achieved for buildings of plan-B, plan-C and plan-D.
Performance parameters
6-storey, 3% target IDR 10-storey, 1.5% target IDR
Plan-B Plan-C Plan-D Plan-B Plan-C Plan-D
% IDR 3.05 2.95 2.85 1.52 1.6 1.57
Max. hinge level CP CP CP LS LS IO
Max. θbp × 10À 2
in radian 1.7824 1.6211 1.6762 0.6376 0.6532 0.5023
θbp × 10À 2
(Equation (4)) in radian 1.4768 1.4152 1.3536 0.5343 0.5836 0.5650
Critical GM GM5 GM5 GM5 GM1 GM1 GM3
Direction Short Short Short Long Short Short
IDR, interstory drift ratio; CP, collapse prevention; LS, life safety; IO, immediate occupancy; GM, ground motion.
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24. 6. SUMMARY
An approach to determine the column member’s size for DBD of RC frame buildings having a wide
range of target IDR and building plan geometry is presented in this report. A relation to obtain the
storey-wise column size is formulated by considering the various deformation components
involved in the overall frame deformation. The plastic rotation of beam member is related with
target IDR, to represent the deformation component due to beam plastic rotation. To control
the dynamic amplification of IDR, a new target-drift dependant design-drift reduction factor is
used. An approximate conversion of fundamental period from the effective period of the
building is introduced to facilitate the consideration of dynamic amplification of column
moment. And, a design-drift dependant C/B ratio is used to avoid plastic hinge formation in
column members. The proposed method is validated for columns of RC frame buildings with
four plans of different varieties, heights up to 12 storey and target IDRs up to 3%. The
determined column size can be effectively used with minor changes in dimension for designing
RC frame buildings with direct DBD, as long as the section can accommodate the design
reinforcement to some practicable ratio.
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DOI: 10.1002/eqe