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ISSN: 0253-3839 (Print) 2158-7299 (Online) Journal homepage: http://www.tandfonline.com/loi/tcie20
Use of modified hybrid PSOGSA for optimum
design of RC frame
Sonia Chutani & Jagbir Singh
To cite this article: Sonia Chutani & Jagbir Singh (2018): Use of modified hybrid PSOGSA
for optimum design of RC frame, Journal of the Chinese Institute of Engineers, DOI:
10.1080/02533839.2018.1473804
To link to this article: https://doi.org/10.1080/02533839.2018.1473804
Published online: 28 Jun 2018.
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3. search, ACO for local search and HS for tackling variables’ bound-
aries. Kaveh and Sabji (2011a; 2011b) compared HBB-BC and
Heuristic Particle Swarm Ant Colony algorithms by considering
the optimization problem related to reinforced cement concrete
frames. Esfandiary, Sheikholarefin, and Bondarabadi (2016) used
the basic concept of multi-criterion decision-making and com-
bined it with PSO to develop another algorithm (DMPSO) that
accelerated convergence toward the optimum solution of a
multi-objective structural optimization problem. Sung, Wang,
and Teo (2016) modeled optimum construction planning of
cable-stayed bridges using PSO and SA in the mutation opera-
tion of GA to improve escape ability from local minima.
Rashedi, Nezamabadi-Pour, and Saryazdi (2009) introduced
gravitational search algorithm (GSA), an algorithm based on
law of gravity and mass interactions. The algorithm is laden
with many features like memory-less adaption, self-learning
profile, and fast convergence. Mirjalili and Hashim (2010)
combined distinct capacities of PSO (social thinking capacity)
and GSA (local search capacity) to propose PSOGSA.
Meanwhile, the advantages of PSO like easy adaptability,
fewer parameters to be adjusted, and ability to go for global
optima were put to good use by the authors. The perfor-
mance of the PSOGSA algorithm was tested on several stan-
dard test cases involving distinct dimensions and
complexities, to check its usefulness. The test results indi-
cated improved exploration and exploitation abilities of the
hybrid algorithm. The present paper applies a hybrid of
democratic PSO (DPSO) and self-adaptive gravitational search
(SA-GSA) algorithms – herein called modified hybrid PSOGSA
(MPSOGSA) – for optimum design of RC frames. The objec-
tive of this study is to explore combined advantages of
different algorithms that have not been earlier considered
for hybridization, especially for the design of RC frames. This
approach – with involvement of the whole swarm – extends
the idea of adopting the algorithm’s constant parameters
stochastically to generate faster exploration, resulting in
speedy flow of information among agents and thereby redu-
cing computational time. Vetting of the current modified
algorithm (MPSOGSA) – using some nonlinear mathematical
functions – has been done to test its global optimization
capabilities. Methodology of the present study consists in
formulating the optimization problem on the basis of design
variables, choosing modified hybrid PSOGSA algorithm, and
comparing the results using examples considered in the
efarlier studies.
2. Formulation of the problem
2.1. Frame analysis and design
A computer-aided analysis and optimum design procedure for
plane RC frames subjected to gravity and lateral loads has
been attempted in the present investigation. Dead load due
to weight of every element within the structure and imposed
load acting on the structure when in service constitute the
gravity load. Lateral loads on the building frame are supposed
to act at floor levels and are considered to be caused due to
wind and seismic forces. The analysis of RC frames has been
performed using the stiffness matrix method. Two program
modules, one for analysis and the other for optimal design of a
given frame are codified in C++. The structural response
quantities such as axial force, shear force, and bending
moments in each element of the frame are obtained through
analysis. Due to complexities of a large number of design
variables associated with RC frame structures, it has been
discretized into beams and columns. In doing so, a number
of design variables and constraints got reduced. The design
process has been developed in such a way that it works as one
cohesive unit with unrestricted flow of parameters from one
section to another. The optimization procedure seeks the
optimum width, optimum depth, and optimum longitudinal
reinforcement of a member section while ensuring that stres-
ses and displacements, apart from other constraints, are within
defined limits.
Thus, a computer-based optimal method has been pre-
sented for optimal cost design of plane RC frame, wherein
beams are designed and optimized, followed by columns.
Elastic behavior of the structure has been considered, and
Limit State Method has been adopted for design of different
elements. Formulation of design problem includes the defini-
tion of objective function, design variables, and all code-
related constraints of IS 456:2000 (Plain and Reinforced
Concrete – Code of Practice, 2000). Some of the important
design considerations for all frame elements are:
● The lower and upper bound of cross sectional dimen-
sions are 300 mm and 1000 mm, respectively.
● At least four bars are used on the four cross sides.
● The minimum cover of concrete is taken as 40 mm.
● Minimum diameter of transverse steel is 10 mm.
2.1.1. Objective function
The cost of a RC structural element primarily includes the cost
of concrete and steel and has been calculated as
C ¼ CstVst þ CCVC (1)
C: total cost of structural element; Cst : cost of steel per unit
volume of steel; Vst : total volume of steel; CC : cost of concrete
per unit volume of concrete; VC : total volume of concrete.
Dividing Equation (1) by CC,
C
CC
¼
Cst
CC
Vst þ VC (2)
Substituting C
CC
¼ Z (Objective function), Cst
Cc
¼ α (Cost ratio),
and VC ¼ VG Vst in Equation (2), it becomes
Z ¼ α 1
ð ÞVst þ VG (3)
Since CC is a constant parameter for a given place, the objec-
tive function Z represents total cost of the frame that needs to
be minimized. Volume of steel (Vst) depends upon area of steel
and its provided length. Area of steel includes both longitu-
dinal as well as transverse steel. Similarly, gross volume of the
element (VG) depends upon its cross sectional area and length.
Since the RC frame is composed of several beams and
columns, the total cost of the RC frame has been considered
as Equation (4):
2 S. CHUTANI AND J. SINGH
4. ZTotal ¼
X
NB
n¼1
ZBeam þ
X
NC
n¼1
ZColumn (4)
2.1.2. Design variables and constraints for beam
optimization
In the present study, all input design parameters have been
considered fixed. These include span of beam, grade of rein-
forcement and concrete, intensity of dead and live loads,
effective cover of concrete, and cost ratio. The independent
design variables of the beam considered in the present model
are width (bB) and effective depth (dB) of the beam. The areas
of longitudinal reinforcement and shear reinforcement are
calculated as dependent design parameters. Design con-
straints considered in the present study not only consider
Indian code provisions for RC beam design (IS 456:2000;
Varghese 2013), but also a few practical aspects.
2.1.2.1. Moment capacity consideration. For a given beam,
the cross-sectional dimensions (depth and width) and area of
steel to be provided at the ends and at bottom shall be such
that the design moment of resistance is greater than the
actual moments to be borne by it at the respective sections.
0:87 fy Astend dB
fy Astend
fck bB
Mh and 0:87 fy Astmid dB
fy Astmid
fck bB
MS
(5)
2.1.2.2. Deflection consideration. For spans up to 10 m, the
vertical deflection of a continuous beam shall be considered
within limits if the ratio of its span (l) to its effective depth dB is
less than 26. For spans above 10 m, factor 26 is multiplied by 10
l .
2.1.2.3. Minimum width of beam. From practical considera-
tions, the beam shall be wide enough to accommodate at
least two bars of tensile steel of the given diameter.
bB bBmin
(6)
2.1.2.4. Slenderness limit of beam from lateral stability
consideration. As per IS 456:2000, a continuous beam shall
be so proportioned that the clear distance between lateral
restraints does not exceed 60bB or
250b2
B
dB
, whichever is less.
dB : effective depth of the beam; bB : width of the compres-
sion face midway between the lateral restraints.
2.1.2.5. Depth of neutral axis. To ensure that tensile steel
does not reach its yield stress before concrete fails in compres-
sion so as to avoid brittle failure, the maximum depth of
neutral axis has been restrained.
0:87 fy Astend
0:36 fck bB dB
xm
dB
and
0:87 fy Astmid
0:36 fck bB dB
xm
dB
(7)
xm
dB
varies with the grade of steel and is given below:
xm
dB
= 0.53, if fy = 250 N/mm2
; xm
dB
= 0.48, if fy = 415 N/mm2
;
xm
dB
= 0.46, if fy = 500 N/mm2
.
2.1.2.6. Minimum and maximum reinforcement steel. The
minimum and maximum area of tensile steel to be provided is
taken as
AstendðminÞ
0:85 bB dB
fy
; AstendðmaxÞ 0:04 bB DB; AstmidðminÞ
0:85 bB dB
fy
; AstmidðmaxÞ 0:04 bB DB
(8)
2.1.2.7. Maximum shear stress consideration. The nominal
shear stress in concrete shall not exceed the maximum per-
missible shear stress, i.e.
τc τc;max (9)
where τc;max ¼ 0:6375
p
fck
2.1.3. Design variables and constraints for column
optimization
Column optimization consists in determination of depth and
width of the columns, with ‘percentage area of longitudinal
reinforcement’ and ‘ratio of depth of neutral axis to depth of
column’ as design variables. Following constraints have been
considered:
2.1.3.1. Axial load capacity of column. The ‘axial load car-
rying capacity’ of the column shall be greater than the load to
be carried.
0:36fckbckDc þ
X
n
i¼1
ðfsi fciÞ
pi bcDC
100
P; (10)
bc and Dc : Width and depth of column; kDc : depth of NA
from extreme compression fiber; fsi and fci : stresses in the
reinforcement and concrete at the ith row of reinforcement;
n : number of rows of reinforcement; P : actual value of axial
load as applied on the column; pi : percentage area of steel in
the ithrow of reinforcement.
2.1.3.2. Moment capacity of column. The ‘moment carry-
ing capacity’ of the column shall be greater than the moment
to be carried.
0:36fckbckD2
C 0:5 0:416k
ð Þ þ
X
n
i¼1
ðfsi fciÞð pi
100fck
Þ
yi
DC
M;
(11)
yi : distance of the ithrow of reinforcement steel, measured
from the centroid of the section. It is positive toward the
highly compressed edge and negative toward the least com-
pressed edge.
M: actual value of bending moment as applied on the
column.
2.1.3.3. Longitudinal reinforcement in column. The cross-
sectional area of longitudinal reinforcement shall vary
between 0.8% and 4%of the gross cross-sectional area of the
column (the Indian code denotes a higher limit of 6%, but due
to practical difficulties in placing and compacting of concrete
JOURNAL OF THE CHINESE INSTITUTE OF ENGINEERS 3
5. at places where bars are to be lapped, a lower percentage has
been recommended).
p 0:8 and p 4:0 (12)
2.1.3.4. Minimum number of longitudinal reinforcement
bars. The number of longitudinal bars provided in a column
shall not be less than 4.
Total area of long: re inf
Area of one bar
4: (13)
2.1.3.5. Maximum peripheral distance between longitudi-
nal reinforcement bars. The spacing of longitudinal bars
measured along the periphery of column shall not be more
than 300 mm.
dp 300: (14)
2.1.3.6. Cross-section of the column. From a practical point
of view, the width of a column shall be equal to or greater
than the width of beams joining it and also its cross-sectional
dimensions shall be in sync with the size of the column lying
immediately beneath it.
3. Optimization algorithms
3.1. Democratic particle swarm optimization
PSO – developed by Kennedy and Eberhart (1995; 2001) – has
proved to be a powerful search technique through its application
in various fields over a wide variety of optimization problems.
The ‘interaction between particles’ to determine the best posi-
tion is the crux of this technique. All particles communicate
among themselves in search of their best position and adjust
their velocities accordingly. Though ‘simplicity’ and ‘fair search
potential’ are positive traits of the algorithm, ‘smaller exploration
capability’ and ‘chances to get trapped in local optima’ make it
susceptible to premature convergence. Kaveh and Zolghadr
(2014) introduced democratic PSO, an extended version of stan-
dard PSO, to improve these limitations. DPSO, in pursuit of
avoiding premature convergence, considers all good and bad
experiences of the particles and thereby tries to provide a better
tactic for exploring the solution domain. The improvement is
thus obtained by adding another term c3r3d0d
i t
ð Þ to the velocity
vector. Velocity vector is thereby expressed as
vd
i t þ 1
ð Þ ¼ χ½w vd
i t
ð Þ þ c1rðpd
i t
ð Þ xd
i t
ð ÞÞ
þ c2r pd
g t
ð Þ xd
i t
ð Þ
þ c3 r d0d
i t
ð Þ (15)
vd
i t
ð Þ and xd
i t
ð Þ describe velocity and position of particle i at
any time t. Parameters χ and w are the constriction factor and
inertia weight, respectively. c1 and c2 are the positive numbers
illustrating the weights of the acceleration terms that guide
each particle toward the individual best and swarm best posi-
tions, respectively. r is the uniformly distributed random num-
ber in the range 0–1. pd
i t
ð Þ and pd
g t
ð Þ are own best and the
global best solutions, respectively. c3 helps to control the
weight of democratic vector, and d
0
d
i is dth variable for the
ith particle of vector D. Vector D denotes the influence (demo-
cratic) of other particles of the swarm on movement of the ith
particle, and is considered as
Di ¼
X
n
k¼1
QikðXk XiÞ (16)
X is position vector of the particle, whereas Qikis weight of kth
particle in democratic movement of the ith particle, and is
calculated as
Qik ¼
Eik
fbest
f k
ð Þ
Pn
j¼1 Eij
fbest
f j
ð Þ
(17)
f is a cost function value, and f best represents it for the best
particle in current iteration. E is the eligibility parameter, which
for a minimization problem is defined as
Eik ¼ 1
f k
ð Þ f i
ð Þ
fworst fbest
rand [ f k
ð Þ f i
ð Þ
0 else
(18)
fworst is value of cost function for the worst particle in current
iteration. After calculating velocity by Equation (15), the parti-
cle’s new position in DPSO algorithm is defined similarly as in
the standard PSO, and is given as
xd
i t þ 1
ð Þ ¼ xd
i t
ð Þ þ vd
i t þ 1
ð Þ (19)
in which the time interval is equal to 1.0 and thus the velocity
vector can be added to the position vector. It is clear that the
information produced by all members of the swarm is utilized
by the PSO with the purpose of determining new position of
each particle, and thus the phrase democratic PSO.
3.2. Self-adaptive gravitational search algorithm
GSA, based on Newton’s gravitational law, is an algorithm
designed in a way where each agent (mass) attracts other
agents with a certain gravitational force. The algorithm, pro-
posed by Rashedi, Nezamabadi-Pour, and Saryazdi (2009), thus
uses ‘law of gravitation’ for notion of mass interactions in the
search space. The performance of all agents is measured by
their masses and the algorithm works as
(1) A system with N agents (search space) is considered,
whereby position of the ith agent is decided as
Xi ¼ x1
i ; :: x2
i . . . ; xn
i . . . . . . ::xd
i
; i ¼ 1; 2; 3 . . . : N; (20)
xd
i denotes position of ith agent in dth dimension.
Individual position of each agent ðXi Þ represents a probable
solution, which gets improved over the iterations.
(2) The mass calculation depends upon a function mi t
ð Þ; which
considers the best, worst, and current values of the objective
function.
4 S. CHUTANI AND J. SINGH
6. mi t
ð Þ ¼
fiti t
ð Þ worst t
ð Þ
best t
ð Þ worst t
ð Þ
(21)
For a minimization problem, lowest value of the objective
function is considered as best and highest value as worst. Thus,
best t
ð Þ ¼ minj2 1...:N
ð Þ fitj t
ð Þ; worst t
ð Þ ¼ maxj2 1...:N
ð Þ fitj t
ð Þ (22)
All agents accelerate, with certain acceleration, in the search
space to look for optima. The accelerations depend upon
attraction forces between the masses. The heavier masses
move slowly than the lighter ones. Total force exerted on an
agent – from a set of heavier masses – is computed on the
basis of Newton’s law as given:
Fd
ij t
ð Þ ¼
G t
ð Þ Mj t
ð Þ Mi t
ð Þ
Rij t
ð Þþ 2
xd
j t
ð Þ xd
i t
ð Þ
; (23)
Fd
i t
ð Þ ¼
X
j2Kbest;jÞi
randjFd
ij t
ð Þ (24)
Rij t
ð Þ represents Euclidian distance between agents i and j,
ε is a small constant, G t
ð Þ represents the gravitational con-
stant, and Kbest is a set of first K agents with best fitness value
(biggest mass).
(1) Acceleration of each agent is hereby calculated as
ad
i t
ð Þ ¼
Fd
i t
ð Þ
Mi t
ð Þ
(25)
(1) New velocities of the agents are calculated by adding a
fraction of current velocity to earlier acceleration as
vd
i t þ 1
ð Þ ¼ randivd
i t
ð Þ þ ad
i t
ð Þ (26)
(1) Agent’s new position is thereby calculated as
xd
i t þ 1
ð Þ ¼ xd
i t
ð Þ þ vd
i t þ 1
ð Þ: (27)
There are two tuning parameters in GSA, namely G t
ð Þ and
Kbest which greatly affects its performance. Researchers have
mostly used two ways of defining these parameters. Either
their values have been kept constant throughout the process
or varied linearly using certain concept (Rashedi and
Nezamabadi-pour, 2012). For example, G t
ð Þ initialized at the
start is made to vary linearly with time as in Equation (28):
G t
ð Þ ¼ GO t
ð Þ þ
t
tmax
β
(28)
t represents current iterations and tmax is maximum number
of iterations. Similarly, Kbest is reduced linearly starting from
total number of agents at the start to one at the end. There is
no clarity on what rule for linear variation of these parameters
shall be followed to get better results. The self-adaptive
approach tries to overcome this problem by defining a range
of these parameters and updating them stochastically at each
iteration (within the range), thereby bringing the self-adaptive
concept to GSA (Niknam et al. 2013).
3.3. Modified hybrid MPSOGSA
Determination of global optimal solution is the aim of imple-
menting any optimization algorithm, and hybridization of two
or more algorithms is performed to improve the performance.
Several heuristic algorithms have been combined to form
hybrid methods for optimization problems. The basic idea of
combining Standard PSO with GSA was suggested by Mirjalili
and Hashim (2010). They combined social thinking ability of
PSO and search capability of GSA.
Since democratic PSO (DPSO) and self-adaptive GSA
(SA-GSA) are improved versions of standard PSO and stan-
dard GSA, respectively, this paper has integrated the two
enhanced versions to evaluate their combined perfor-
mance through a new modified hybrid, i.e. MPSOGSA.
The hybrid is a stochastic algorithm with a feature to
randomly select important parameters that have an influ-
ence on the search procedure. The advantage of MPSOGSA
is that it avoids getting trapped in local optima, and also
improves upon premature convergence probability. It
thereby reaches a better optimal solution in a reasonable
time. The functionality of both the algorithms is combined
and they run parallel. The modified velocity equation is
given hereby:
vd
i t þ 1
ð Þ ¼ wvd
i t
ð Þ þ c1:r:ad
i t
ð Þ þ c2:r: pd
g t
ð Þ xd
i t
ð Þ
þ c3:r:d0d
i t
ð Þ (29)
vd
i t
ð Þ represents velocity of agent i at iteration t, c3 is the
weighing factor, w is the weighing function, r is random
number between 0 and 1, a i t
ð Þ is the acceleration of agent i
at iteration t, and p g is the best solution so far. d0
id t
ð Þ includes
democratic influence of other particles on ith particle in dth
dimension. Every iteration updates the position of particles as
in Equation 30, and the algorithm has been explained in terms
of flow chart in Figure 1.
xd
i t þ 1
ð Þ ¼ xd
i t
ð Þ þ vd
i t þ 1
ð Þ (30)
To test the performance of MPSOGSA, a set of nonlinear
benchmark functions were tested and the results were com-
pared with earlier studies (Table 1).
The results for bench mark functions indicate an improve-
ment over PSOGSA (Mirjalili and Hashim 2010) and DPSO
(Kaveh and Zolghadr 2014). The faster communication strat-
egy among the particles due to their democratic nature and
enhanced exploration caused by randomness of algorithm’s
parameters made the proposed technique more attractive to
get quality solutions.
4. Optimal design solution
For the application of current optimization technique, the
constrained optimization problem has first been converted
to an unconstrained one. The defined constraints are nor-
malized and exterior penalty function is incorporated for
any constraint violation, thereby constituting the uncon-
strained objective function (penalized objective function)
as follows:
JOURNAL OF THE CHINESE INSTITUTE OF ENGINEERS 5
7. Figure 1. General flow chart of RC member design using MPSOGSA.
Table 1. Minimization results of some benchmark functions.
Mathematical function Dim Range of function
PSOGSA
(Mirjalili and Hashim 2010)
DPSO
(Kaveh and Zolghadr 2014)
MPSOGSA
(Current study)
Fi x
ð Þ ¼
P
n
i¼1
½x2
i 10 cos 2πxi
ð Þ þ 10
(Rastrigin)
30 [−5.12,5.12] 19.1371 – 16.472
Fi x
ð Þ ¼
P
4
i¼1
ciexp
P
6
j¼1
aij xj pij
2
!
(Hartman)
6 [0,1] −2.6375 −3.322368 −3.3145
Fi x
ð Þ ¼ 4x2
1 2:1x4
1 þ 1
3 x6
1 þ x1x2 4x2
2 þ 4x4
2
2 [−5,5] −1.036 – −1.036
6 S. CHUTANI AND J. SINGH
8. Z0
¼ Z 1 þ C
ð Þδ
(31)
δ ¼ 2 (for structural design problems); C ¼ Sum of all con-
straint violations
The details of exterior penalty function method are not in
the scope of this paper. Further, the independent design
variables (given in Section 2) are searched from defined search
space to get the optimum results. Optimum sections are
chosen from all possible sections in the practical range unlike
a countable number of sections as available in the literature.
All the beams and columns of the given frame have been
designed as per Limit State Design philosophy, and optimum
solution has been obtained through MPSOGSA. The constant
parameters fine-tuned to get best and consistent results from
the two algorithms are given below:
c1 ¼ 0:5; c2 ¼ 1:5; c3 ¼ 4; 2¼ 10(MPSOGSA constants)
G t
ð ÞMin ¼ 1; G t
ð ÞMax ¼ 100; KbestMax
¼ max number of agents; KbestMin ¼ 1
The population size ‘N’ and maximum number of iterations
‘itrmax’ have been fixed at 20 and 1000, respectively, and upper
and lower bounds for the design variables have been defined
for random selection of the population. Stopping criterion has
been defined as ‘maximum number of iterations.’ Design pro-
cedure for different components of the frame has been devel-
oped in a generalized manner that accepts different
parametric values related to geometry of the frame, loads,
and properties of material. All ‘optimization runs’ have been
carried out on a standard PC with a Intel® Core™ i3 CPU M350
@2.27 GHz frequency and 3 GB RAM memory. The algorithm
has been coded in Turbo C++ installed in Windows 7 at 32 bit
operating system.
4.1. Optimal beam design
In order to evaluate the performance of the applied technique,
a beam (5 m span) which is part of a given frame has been
selected. The given set of loads for the beam, namely gravity
load ‘w’ (30 kN/m) and end moments ‘M1ʹ (50 kN-m) and
‘M2ʹ (100 kN-m) are shown in Figure 2. The configuration and
steel reinforcement are the design variables to be optimized
to satisfy the objective criteria. Grades of concrete and steel
(fck ¼ 30 N/mm2
and fy ¼ 415 N/mm2
, respectively) as well as
cost ratio (100) have been considered as input variables.
Effective cover to the reinforcement has been considered as
50 mm. The maximum depth to width ratio has been
restricted to 3, to avoid thin sections.
For the above-mentioned parameters, optimum algorithm
(MPSOGSA) suggested the optimum depth and optimum width
Figure 2. Loading conditions of beam.
Figure 3. Convergence curve for optimum design of RC beam using MPSOGSA.
Figure 4. Convergence trend for optimum design of RC column using MPSOGSA.
JOURNAL OF THE CHINESE INSTITUTE OF ENGINEERS 7
9. of the beam as 500 mm and 300 mm, respectively, and the
optimum percentage of steel as 1.53% of cross sectional area.
The design improvement by MPSOGSA is shown in Figure 3.
The design example of a simply supported beam with one
row of reinforcing steel (Camp, Pezeshk, and Hansson 2003) was
also tested and optimized by MPSOGSA technique. Although,
the results of present optimum design when compared with
previous study (RC-GA program) were found to be in good
agreement, the required computational time reduced consider-
ably. The present optimum design procedure required about ‘4 s’
of computing time for 20,000 evaluations as compared to ‘25 s’
for 100 generations quoted in the previous study.
4.2. Optimal column design
Columns of the RC frame have been considered as uniaxial ones,
and their designs are dependent on stresses in the reinforcing
steel. The computer-aided design program based on MPSOGSA
considers different combinations of design parameters (cross-
sectional dimensions and steel percentage), for calculating the
strength of column. Interaction between ‘axial force’ and ‘bend-
ing moment’ has been considered in the design and optimiza-
tion process of RC columns. The column design is considered
feasible only if the axial force and bending moment reside within
the load–moment interaction diagram. A column (part of a given
frame) is designed for a given axial load of 960 kN and uniaxial
moment of 250 kN-m. The minimum cross-sectional dimension
of the column is considered to be 300 mm. Similarly, the ‘cover
ratio’ and minimum ‘column depth to width ratio’ were set as 0.1
and 1.0, respectively. The grades of concrete and steel were
taken as fck = 30 N/mm2
and fy = 415 N/mm2
, respectively. The
unsupported length of column was considered to be 3 m. Also,
effective length ratio for the columns was kept as 1.2 and cost
ratio as 100. For these given set of input values, optimum depth
and width of the column are obtained as 730 mm and 300 mm,
respectively, whereas optimum percentage of longitudinal rein-
forcement is 0.8% of cross-sectional area. The MPSOGSA
algorithm showed convergence at 344 iterations and the
Figure 5. Geometry and loading of two bay-six story frame.
8 S. CHUTANI AND J. SINGH
11. convergence curve is shown in Figure 4. Time taken for getting
the optimum design was ‘4 s.’
4.3. Optimal design of RC frame
The efficacy and efficiency of the present design algorithm
(MPSOGSA) has been measured by considering design exam-
ples from reviewed literature. Necessary changes (in objective
function, constraints, cost ratios, etc.) required to bring in the
sense of compatibility – for comparing two designs – have
been done accordingly.
An example consists of a two bay-six story RC frame, with
given geometry and loads as shown in Figure 5. This example has
been considered in a few of the earlier studies as well [(Camp,
Pezeshk, and Hansson 2003) and (Rajeev and Krishnamoorthy
1998)], and a comparison between the present work and earlier
works is shown to be been done in Table 2. It also indicates cost
parameters, grade of materials, and grouping of members for the
frame under consideration. The RC-GA design (Camp, Pezeshk
and Hansson, ACI code) helped to reduce the cost of a given
frame by 4.2% as compared with GA design (Rajeev and
Krishnamoorthy, IS456: 1978), but a further cost saving of 4.8%
has been achieved using MPSOGSA design (present study).
Further significance lies in saving processing time for optimiza-
tion. The current computational time to carry out 1000 genera-
tions for a population size of 50 is about 6 min, which is much
less than 13 h required for 300 generations with a population size
of 300 in the RC-GA design.
5. Conclusion
The analysis of RC frame structure has been performed using
direct stiffness approach and the design procedure follows
Indian standard IS 456:2000 regulations. Optimum design results
are obtained with the use of new modified hybrid technique
(MPSOGSA). The proposed algorithm overcomes the limitations
of two individual algorithms (modified PSO and modified GSA) by
considering their hybrid, and thereby improves the overall perfor-
mance. Necessary changes have been incorporated to make the
study compatible with some earlier studies, and to help compare
the results. A comparison with some other algorithms reveals that
the time taken to carry out optimization in the present study – by
the use of MPSOGSA– has reduced significantly. Also, reduction in
total cost has been achieved in the design of RC frames using this
technique. Reduction in steel area plays a greater role in optimiza-
tion as compared to reduction in cross sectional area of frame
elements as verified in the design examples.
Nomenclature
Astend Area of steel at the beam end
Astend(max) Maximum area of steel at the beam end
Astend(min) Minimum area of steel at the beam end
Astmid Area of steel in the middle of the beam
Astmid(max) Maximum area of steel at the beam mid
Astmid(min) Minimum area of steel at the beam mid
bBmin
Minimum width of beam
DB Overall depth of beam
dp Maximum peripheral distance among longitudinal bars of the
column
fck Characteristic compressive strength of concrete in N/mm2
fy Characteristic strength of steel in N/mm2
Mh Hogging moment applied at the beam end
Ms Maximum sagging moment
p Percentage area of longitudinal reinforcement
xm Limiting depth of neutral axis
Disclosure statement
No potential conflict of interest was reported by the authors.
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